diff options
Diffstat (limited to '44730-h/44730-h.htm')
| -rw-r--r-- | 44730-h/44730-h.htm | 30138 |
1 files changed, 30138 insertions, 0 deletions
diff --git a/44730-h/44730-h.htm b/44730-h/44730-h.htm new file mode 100644 index 0000000..2bfb8fa --- /dev/null +++ b/44730-h/44730-h.htm @@ -0,0 +1,30138 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" + "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> +<html xmlns="http://www.w3.org/1999/xhtml" + xml:lang="en" + lang="en"> +<head> + <meta http-equiv="Content-Type" + content="text/html; charset=UTF-8" /> + <meta http-equiv="Content-Style-Type" + content="text/css" /> + <title> + Memorabilia Mathematica by Robert Edouard Moritz--A Project Gutenberg eBook.</title> + <link rel="coverpage" href="images/cover-page.jpg" /> + <style type="text/css"> + + + body { + text-align: justify; + margin-left: 10%; + margin-right: 10%; + } + + + /* Heading Styles */ + h1,h2,h3,h4,h5,h6 { + text-align: center; + clear: both; + } + + p { margin: 0% auto; } + + + /* Vertical alignment */ + .v0 { margin: 0% auto; } + .v1 { + margin-top: .5em; + text-indent: 2em; + } + .v2 { margin-top: 2em; } + .v4 { margin-top: 4em; } + .v6 { margin-top: 6em; } + .v8 { margin-top: 8em; } + + + /* Ruler Styles */ + hr { + margin: 3em auto 3em auto; + height: 0px; + border-width: 1px 0 0 0; + border-style: solid; + border-color: #708090; /* slategray */ + width: 33%; + clear: both; + } + + hr.tb { + margin: 1em auto; + width: 35%; + } + hr.chap {width: 50%;} + hr.full {width: 95%;} + hr.r10 { + margin: 1em auto; + width: 10%; + } + hr.blank { + margin: .5em auto; + visibility: hidden; + } + + + /* List Styles */ + ul.index { + list-style-type: none; + margin: auto 15%; + text-align: left; + } + li.ifrst { + margin-top: 1em; + text-indent: -4em; + } + li.indx { + margin-top: .5em; + text-indent: -4em; + } + li.isub1 {text-indent: -3em;} + li.isub2 {text-indent: -2em;} + + + /* table Styles */ + table { + margin: auto; + text-align: left; + vertical-align: top; + border-spacing: 1.5em .7em; + } + + td { + text-align: left; + vertical-align: top; + } + td.rt { text-align: right; } + td.hang { text-indent: -1.2em; } + + + /* numbering styles */ + .pagenum { + /* visibility: hidden; */ + position: absolute; + right: .5%; + font-size: xx-small; + font-weight: normal; + font-variant: normal; + font-style: normal; + letter-spacing: normal; + text-indent: 0em; + text-align: right; + color: #999999; } /* Grey */ + + + /* block Styles */ + blockquote { + margin-left: 5%; + margin-right: 5%; + } + .blockhang { + text-indent: -5%; + margin-left: 10%; + margin-right: 10%; + } + .block40 { + margin-left: 40%; + margin-right:10%; + text-align: left; + } + .blockright { + margin-left: 40%; + margin-right: 0%; + text-align: right; + } + .blockcite { + margin-left: 40%; + margin-right: 0%; + margin-top: 0em; + text-align: left; + font-style: italic; + } + + + /* Font Styles */ + .oldcentury { font-family: "England", "Gothic 32 Wide", serif; } + .Monofont { font-family: "Lucida Console", Monaco, monospace; } + + + /* text styling */ + sup { + line-height: .1em; + font-size: .7em; + } + sub { + line-height: .1em; + font-size: .7em; + } + .bold { font-weight: bold; } + .center { + text-align: center ; + text-indent: 0em; + vertical-align: middle; + } + .right { text-align: right; } + .xxs { font-size: xx-small; } + .xs { font-size: x-small; } + .small { font-size: small; } + .medium { font-size: medium; } + .large { font-size: large; } + .xl { font-size: x-large; } + .xxl { font-size: xx-large; } + .smcap { font-variant: small-caps; } + .u { text-decoration: underline; } + .italic { font-style: italic; } + + + /* Transcriber Notes */ + .msg { /* non-obtrusive */ + text-decoration: none; + border-bottom: 1px solid #A9A9A9; /* dark gray */ + } + .msg:link{color:inherit} + .msg:active{color:inherit} + .msg:visited{color:inherit} + .msg:hover{color:inherit} + + + /* Note Classes */ + .tn { /* transcribers notes */ + background-color: #E6E6FA; /* Lavender */ + border: solid 1px; + color: black; + margin: 5% 0%; + padding: 2%; + text-align: left; + } + + .footnote { /* Footnotes */ + background-color: #F0FFFF; /* Azure */ + border: dashed 1px; + color: #000; + margin: 5% 0%; + padding: 2%; + font-size: 1em; + text-align: left; + } + + + .fnanchor { + vertical-align: super; + line-height: 0.1em; + font-size: .8em; + text-decoration: none; + } + + .fnanch2 { /* closer to reference */ + margin-left: 0em; + vertical-align: super; + line-height: 0.1em; + font-size: .8em; + text-decoration: none; + position: relative; + left: -.4em; + margin-right: -.4em; + } + + + /* Poetry */ + .poem { + margin-left: 5%; + margin-right: 5%; + padding-left: 3em; + text-indent: -3em; + text-align: left; + } + .poem p { margin: 0em auto; } + .poem p.i0 { margin-left: 0em; } + .poem p.i2 { margin-left: 1em; } + .poem p.i4 { margin-left: 2em; } + .poem p.i6 { margin-left: 3em; } + .poem p.i7 { margin-left: 3.5em; } + .poem p.i8 { margin-left: 4em; } + .poem p.i12 { margin-left: 6em; } + + + /* Images */ + .figinline { + float: none; + clear: none; + margin: 0em; + vertical-align: middle; + text-align: left; + } + .figcenter { + margin: 1em auto; + text-align: center; + } + @media print, handheld + { + hr.tb { width: 35%; + margin-left: 32.5%; } + hr.chap { width: 50%; + margin-left: 25%; } + hr.full { width: 95%; + margin-left: 2.5%; } + hr.r10 { width: 10%; + margin-left: 45%; } + .msg { border-bottom: none} + .pagenum { visibility: hidden; } + } + + + </style> +</head> +<body> +<div>*** START OF THE PROJECT GUTENBERG EBOOK 44730 ***</div> + +<p> + <span class="pagenum"> + <a id="Page_i" /></span></p> + + <p class="xxl v6 bold center"> + MEMORABILIA MATHEMATICA</p> +<p> + <span class="pagenum"> + <a id="Page_ii" /></span></p> + + <div class="center v6 bold"> + <img src="images/img002.png" + width="183" + height="68" + alt="monogram" + id="img002" /> + <p class="small"> + THE MACMILLAN COMPANY</p> + <p class="xxs"> + NEW YORK · BOSTON. · CHICAGO · DALLAS</p> + <p class="xxs"> + ATLANTA · SAN FRANCISCO</p> + <p class="small smcap"> + MACMILLAN & CO., Limited</p> + <p class="xxs"> + LONDON · BOMBAY · CALCUTTA</p> + <p class="xxs"> + MELBOURNE</p> + <p class="small smcap"> + THE MACMILLAN CO. OF CANADA, Ltd.</p> + <p class="xxs"> + TORONTO</p> + </div> +<p> + <span class="pagenum"> + <a id="Page_iii" /></span></p> + + <div class="center"> + <hr class="r10" /> + <h1 class="v4"> + MEMORABILIA MATHEMATICA<br /> + <span class="small"> + OR</span><br /> + <span class="xl"> + THE PHILOMATH’S QUOTATION-BOOK</span></h1> + <p class="v8 small"> + BY</p> + <p class="large bold"> + ROBERT EDOUARD MORITZ, + <span class="smcap"> + Ph. D., Ph. N. D.</span></p> + <p class="xs"> + PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF WASHINGTON</p> + <p class="medium v8 oldcentury bold"> + New York</p> + <p class="xs"> + THE MACMILLAN COMPANY</p> + <p class="xs"> + 1914</p> + <p class="xxs italic v2"> + All rights reserved</p> + +<p> + <span class="pagenum"> + <a id="Page_iv" /></span></p> + + <hr class="r10" /> + <p class="v6 xs smcap"> + Copyright, 1914, by</p> + <p class="small"> + ROBERT EDOUARD MORITZ</p> + <img class="v6" + src="images/img004.png" + width="128" + height="146" + alt="emblem" + id="img004" /> + </div> + + <hr class="chap" /> + <h2 class="v2"> + <a name="PREFACE" + id="PREFACE">PREFACE</a></h2> + <p class="v1"> + +<span class="pagenum" + id="Page_v">v</span> + + Every one knows that the fine phrase “God geometrizes” is + attributed to Plato, but few know where this famous passage + is found, or the exact words in which it was first expressed. + Those who, like the author, have spent hours and even days in + the search of the exact statements, or the exact references, + of similar famous passages, will not question the timeliness + and usefulness of a book whose distinct purpose it is to + bring together into a single volume exact quotations, with + their exact references, bearing on one of the most + time-honored, and even today the most active and most + fruitful of all the sciences, the queen-mother of all the + sciences, that is, mathematics.</p> + <p class="v1"> + It is hoped that the present volume will prove indispensable + to every teacher of mathematics, to every writer on + mathematics, and that the student of mathematics and the + related sciences will find its perusal not only a source of + pleasure but of encouragement and inspiration as well. The + layman will find it a repository of useful information + covering a field of knowledge which, owing to the unfamiliar + and hence repellant character of the language employed by + mathematicians, is peculiarly inaccessible to the general + reader. No technical processes or technical facility is + required to understand and appreciate the wealth of ideas + here set forth in the words of the world’s great thinkers.</p> + <p class="v1"> + No labor has been spared to make the present volume worthy of + a place among collections of a like kind in other fields. Ten + years have been devoted to its preparation, years, which if + they could have been more profitably, could scarcely have + been more pleasurably employed. As a result there have been + brought together over one thousand more or less familiar + passages pertaining to mathematics, by poets, philosophers, + historians, statesmen, scientists, and mathematicians. These + have been gathered from over three hundred authors, and have + been + +<span class="pagenum"> + <a name="Page_vi" + id="Page_vi">vi</a></span> + + grouped under twenty heads, and + cross indexed under nearly seven hundred topics.</p> + <p class="v1"> + The author’s original plan was to give foreign quotations + both in the original and in translation, but with the growth + of material this plan was abandoned as infeasible. It was + thought to serve the best interest of the greater number of + English readers to give translations only, while preserving + the references to the original sources, so that the student + or critical reader may readily consult the original of any + given extract. In cases where the translation is borrowed the + translator’s name is inserted in brackets [] immediately + after the author’s name. Brackets are also used to indicate + inserted words or phrases made necessary to bring out the + context.</p> + <p class="v1"> + The absence of similar English works has made the author’s + work largely that of the pioneer. Rebière’s “Mathématiques et + Mathématiciens” and Ahrens’ “Scherz und Ernst in der + Mathematik” have indeed been frequently consulted but rather + with a view to avoid overlapping than to receive aid. Thus + certain topics as the correspondence of German and French + mathematicians, so excellently treated by Ahrens, have + purposely been omitted. The repetitions are limited to a + small number of famous utterances whose absence from a work + of this kind could scarcely be defended on any grounds.</p> + <p class="v1"> + No one can be more keenly aware of the shortcomings of a work + than its author, for none can have so intimate an + acquaintance with it. Among those of the present work is its + incompleteness, but it should be borne in mind that + incompleteness is a necessary concomitant of every collection + of whatever kind. Much less can completeness be expected in a + first collection, made by a single individual, in his leisure + hours, and in a field which is already boundless and is yet + expanding day by day. A collection of great thoughts, even if + complete today, would be incomplete tomorrow. Again, if some + authors are quoted more frequently than others of greater + fame and authority, the reason may be sought not only in the + fact that the writings of some authors peculiarly lent + themselves to quotation, a quality singularly absent in other + writers of the greatest merit and authority, but also in + this, that the greatest freedom has been exercised in the + choice of selections. The author has followed + +<span class="pagenum"> + <a name="Page_vii" + id="Page_vii">vii</a></span> + + the bent of his own fancy in + collecting whatever seemed to him sufficiently valuable + because of its content, its beauty, its originality, or + its terseness, to deserve a place in a “Memorabilia.”</p> + <p class="v1"> + Great pains has been taken to furnish exact readings and + references. In some cases where a passage could not be traced + to its first source, the secondary source has been given + rather than the reputed source. For the same reason many + references are to later editions rather than to inaccessible + first editions.</p> + <p class="v1"> + The author feels confident that this work will be of + assistance to his co-workers in the field of mathematics and + allied fields. If in addition it should aid in a better + appreciation of mathematicians and their work on the part of + laymen and students in other fields, the author’s foremost + aim in the preparation of this work will have been achieved.</p> + <p class="blockright"> + <span class="smcap"> + Robert Edouard Moritz,</span> + <br /> + <em>September, 1913.</em></p> + +<p> + <span class="pagenum"> + <a name="Page_viii" + id="Page_viii">viii</a> + <br /> + <a name="Page_ix" + id="Page_ix">ix</a></span></p> + + <hr class="tb" /> + <h2 title="Table of Contents"> + <a name="CONTENTS" + id="CONTENTS">CONTENTS</a></h2> + <table summary=""> + <tr class="smcap"> + <th> + Chapter</th> + <th> + </th> + <th> + Page</th></tr> + <tr> + <td class="rt"> + I.</td> + <td> + <span class="smcap"> + Definitions and Object of Mathematics</span></td> + <td class="rt"> + <a href="#Page_1">1</a></td></tr> + <tr> + <td class="rt"> + II.</td> + <td> + <span class="smcap"> + The Nature of Mathematics</span></td> + <td class="rt"> + <a href="#Page_10">10</a></td></tr> + <tr> + <td class="rt"> + III.</td> + <td> + <span class="smcap"> + Estimates of Mathematics</span></td> + <td class="rt"> + <a href="#Page_39">39</a></td></tr> + <tr> + <td class="rt"> + IV.</td> + <td> + <span class="smcap"> + The Value of Mathematics</span></td> + <td class="rt"> + <a href="#Page_49">49</a></td></tr> + <tr> + <td class="rt"> + V.</td> + <td> + <span class="smcap"> + The Teaching of Mathematics</span></td> + <td class="rt"> + <a href="#Page_72">72</a></td></tr> + <tr> + <td class="rt"> + VI.</td> + <td> + <span class="smcap"> + Study and Research in Mathematics</span></td> + <td class="rt"> + <a href="#Page_86">86</a></td></tr> + <tr> + <td class="rt"> + VII.</td> + <td> + <span class="smcap"> + Modern Mathematics</span></td> + <td class="rt"> + <a href="#Page_108">108</a></td></tr> + <tr> + <td class="rt"> + VIII.</td> + <td> + <span class="smcap"> + The Mathematician</span></td> + <td class="rt"> + <a href="#Page_121">121</a></td></tr> + <tr> + <td class="rt"> + IX.</td> + <td> + <span class="smcap"> + Persons and Anecdotes (A-M)</span></td> + <td class="rt"> + <a href="#Page_135">135</a></td></tr> + <tr> + <td class="rt"> + X.</td> + <td> + <span class="smcap"> + Persons and Anecdotes (N-Z)</span></td> + <td class="rt"> + <a href="#Page_166">166</a></td></tr> + <tr> + <td class="rt"> + XI.</td> + <td> + <span class="smcap"> + Mathematics as a Fine Art</span></td> + <td class="rt"> + <a href="#Page_181">181</a></td></tr> + <tr> + <td class="rt"> + XII.</td> + <td> + <span class="smcap"> + Mathematics as a Language</span></td> + <td class="rt"> + <a href="#Page_194">194</a></td></tr> + <tr> + <td class="rt"> + XIII.</td> + <td> + <span class="smcap"> + Mathematics and Logic</span></td> + <td class="rt"> + <a href="#Page_201">201</a></td></tr> + <tr> + <td class="rt"> + XIV.</td> + <td> + <span class="smcap"> + Mathematics and Philosophy</span></td> + <td class="rt"> + <a href="#Page_209">209</a></td></tr> + <tr> + <td class="rt"> + XV.</td> + <td> + <span class="smcap"> + Mathematics and Science</span></td> + <td class="rt"> + <a href="#Page_224">224</a></td></tr> + <tr> + <td class="rt"> + XVI.</td> + <td> + <span class="smcap"> + Arithmetic</span></td> + <td class="rt"> + <a href="#Page_261">261</a></td></tr> + <tr> + <td class="rt"> + XVII.</td> + <td> + <span class="smcap"> + Algebra</span></td> + <td class="rt"> + <a href="#Page_275">275</a></td></tr> + <tr> + <td class="rt"> + XVIII.</td> + <td> + <span class="smcap"> + Geometry</span></td> + <td class="rt"> + <a href="#Page_292">292</a></td></tr> + <tr> + <td class="rt"> + XIX.</td> + <td> + <span class="smcap"> + The Calculus and Allied Topics</span></td> + <td class="rt"> + <a href="#Page_323">323</a></td></tr> + <tr> + <td class="rt"> + XX.</td> + <td> + <span class="smcap"> + The Fundamental Concepts of Time and Space</span></td> + <td class="rt"> + <a href="#Page_345">345</a></td></tr> + <tr> + <td class="rt"> + XXI.</td> + <td> + <span class="smcap"> + Paradoxes and Curiosities</span></td> + <td class="rt"> + <a href="#Page_364">364</a></td></tr> + <tr> + <td class="rt"> + <span class="smcap"> + Index</span></td> + <td> + </td> + <td class="rt"> + <a href="#Page_385">385</a></td></tr></table> + +<p> + <span class="pagenum"> + <a name="Page_x" + id="Page_x">x</a> + <br /> + <a name="Page_xi" + id="Page_xi">xi</a></span></p> + + <hr class="tb" /> + <div class="v6"> + <p class="v2"> + Alles Gescheite ist schon gedacht worden; man muss nur + versuchen, es noch einmal zu denken.—<span class= + "smcap">Goethe.</span></p> + <p class="blockcite"> + Sprüche in Prosa, Ethisches, I. 1.</p> + <p class="v2"> + A great man quotes bravely, and will not draw on his + invention when his memory serves him with a word as + good.—<span class="smcap">Emerson.</span></p> + <p class="blockcite"> + Letters and Social Aims, Quotation and Originality.</p></div> + +<p> + <span class="pagenum"> + <a name="Page_xii" + id="Page_xii">xii</a> + <br /> + <a name="Page_xiii" + id="Page_xiii">xiii</a></span></p> + + <hr class="tb" /> + <p class="xl v6 center bold"> + MEMORABILIA MATHEMATICA</p> + +<p> + <span class="pagenum"> + <a name="Page_xiv" + id="Page_xiv">xiv</a> + <br /> + <a name="Page_1" + id="Page_1">1</a></span></p> + + <hr class="chap" /> + <p class="v4 xl center bold"> + MEMORABILIA MATHEMATICA</p> + <h2 class="v2" id="CHAPTER_I"> + CHAPTER I<br /> + <span class="large"> + DEFINITIONS AND OBJECT OF MATHEMATICS</span></h2> + + <p class="v2"> + <b><a name="Block_101" id="Block_101">101</a>.</b> + I think it would be desirable + that this form of word [mathematics] should be reserved + for the applications of the science, and that we should + use mathematic in the singular to denote the science + itself, in the same way as we speak of logic, rhetoric, or + (own sister to algebra) music.—<span + class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Presidential Address to the British Association, Exeter + British Association Report (1869); Collected Mathematical + Papers, Vol. 2, p. 659.</p> + + <p class="v2"> + <b><a name="Block_102" id="Block_102">102</a>.</b> + ... all the sciences which + have for their end investigations concerning order and + measure, are related to mathematics, it being of small + importance whether this measure be sought in numbers, + forms, stars, sounds, or any other object; that, + accordingly, there ought to exist a general science which + should explain all that can be known about order and + measure, considered independently of any application to a + particular subject, and that, indeed, this science has its + own proper name, consecrated by long usage, to wit, + <em>mathematics</em>. And a proof that it far surpasses in + facility and importance the sciences which depend upon it + is that it embraces at once all the objects to which these + are devoted and a great many others besides; + ....—<span class="smcap">Descartes.</span></p> + <p class="blockcite"> + Rules for the Direction of the Mind, Philosophy of D. + [Torrey] (New York, 1892), p. 72.</p> + + <p class="v2"> + <b><a name="Block_103" id="Block_103">103</a>.</b> + [Mathematics] has for its + object the <em>indirect</em> measurement of magnitudes, and + it <em>purposes to determine magnitudes by each other, + according to the precise relations which exist between + them</em>.—<span class="smcap">Comte.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 1, chap. 1.</p> + +<p> + <span class="pagenum"> + <a name="Page_2" + id="Page_2">2</a></span></p> + + <p class="v2"> + <b><a name="Block_104" id="Block_104">104</a>.</b> + The business of concrete + mathematics is to discover the equations which express the + mathematical laws of the phenomenon under consideration; + and these equations are the starting-point of the + calculus, which must obtain from them certain quantities + by means of others.—<span class="smcap">Comte.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 1, chap. 2.</p> + + <p class="v2"> + <b><a name="Block_105" id="Block_105">105</a>.</b> + Mathematics is the science of + the connection of magnitudes. Magnitude is anything that + can be put equal or unequal to another thing. Two things + are equal when in every assertion each may be replaced by + the other.—<span class="smcap">Grassmann, Hermann.</span></p> + <p class="blockcite"> + Stücke aus dem Lehrbuche der Arithmetik, Werke (Leipzig, + 1904), Bd. 2, p. 298.</p> + + <p class="v2"> + <b><a name="Block_106" id="Block_106">106</a>.</b> + Mathematic is either Pure or + Mixed: To Pure Mathematic belong those sciences which + handle Quantity entirely severed from matter and from + axioms of natural philosophy. These are two, Geometry and + Arithmetic; the one handling quantity continued, the other + dissevered.... Mixed Mathematic has for its subject some + axioms and parts of natural philosophy, and considers + quantity in so far as it assists to explain, demonstrate + and actuate these.—<span class="smcap">Bacon, + Francis.</span></p> + <p class="blockcite"> + De Augmentis, Bk. 3; Advancement of Learning, Bk. 2.</p> + + <p class="v2"> + <b><a name="Block_107" id="Block_107">107</a>.</b> + The ideas which these + sciences, Geometry, Theoretical Arithmetic and Algebra + involve extend to all objects and changes which we observe + in the external world; and hence the consideration of + mathematical relations forms a large portion of many of + the sciences which treat of the phenomena and laws of + external nature, as Astronomy, Optics, and Mechanics. Such + sciences are hence often termed <em>Mixed Mathematics</em>, + the relations of space and number being, in these branches + of knowledge, combined with principles collected from + special observation; while Geometry, Algebra, and the like + subjects, which involve no result of experience, are + called <em>Pure Mathematics</em>.—<span + class="smcap">Whewell, William.</span></p> + <p class="blockcite"> + The Philosophy of the Inductive Sciences, Part 1, Bk. 2, + chap. I, sect. 4. (London, 1858).</p> + +<p> + <span class="pagenum"> + <a name="Page_3" + id="Page_3">3</a></span></p> + + <p class="v2"> + <b><a name="Block_108" id="Block_108">108</a>.</b> + Higher Mathematics is the art + of reasoning about numerical relations between natural + phenomena; and the several sections of Higher Mathematics + are different modes of viewing these + relations.—<span class="smcap">Mellor, J. W.</span></p> + <p class="blockcite"> + Higher Mathematics for Students of Chemistry and Physics (New + York, 1902), Prologue</p> + + <p class="v2"> + <b><a name="Block_109" id="Block_109">109</a>.</b> + Number, place, and combination + ... the three intersecting but distinct spheres of thought + to which all mathematical ideas admit of being + referred.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Philosophical Magazine, Vol. 24 (1844), p. 285; Collected + Mathematical Papers, Vol. 1, p. 91.</p> + + <p class="v2"> + <b><a name="Block_110" id="Block_110">110</a>.</b> + There are three ruling ideas, + three so to say, spheres of thought, which pervade the + whole body of mathematical science, to some one or other + of which, or to two or all three of them combined, every + mathematical truth admits of being referred; these are the + three cardinal notions, of Number, Space and Order.</p> + <p class="v1"> + Arithmetic has for its object the properties of number in the + abstract. In algebra, viewed as a science of operations, + order is the predominating idea. The business of geometry is + with the evolution of the properties of space, or of bodies + viewed as existing in space.—<span + class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + A Probationary Lecture on Geometry, York British Association + Report (1844), Part 2; Collected Mathematical Papers, Vol. 2, + p. 5.</p> + + <p class="v2"> + <b><a name="Block_111" id="Block_111">111</a>.</b> + The object of pure mathematics + is those relations which may be conceptually established + among any conceived elements whatsoever by assuming them + contained in some ordered manifold; the law of order of + this manifold must be subject to our choice; the latter is + the case in both of the only conceivable kinds of + manifolds, in the discrete as well as in the + continuous.—<span class="smcap">Papperitz, E.</span></p> + <p class="blockcite"> + über das System der rein mathematischen Wissenschaften, + Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. 1, + p. 36.</p> + +<p> + <span class="pagenum"> + <a name="Page_4" + id="Page_4">4</a></span></p> + + <p class="v2"> + <b><a name="Block_112" id="Block_112">112</a>.</b> + Pure mathematics is not + concerned with magnitude. It is merely the doctrine of + notation of relatively ordered thought operations which + have become mechanical.—<span class="smcap">Novalis.</span></p> + <p class="blockcite"> + Schriften (Berlin, 1901), Zweiter Teil, p. 282.</p> + + <p class="v2"> + <b><a name="Block_113" id="Block_113">113</a>.</b> + Any conception which is + definitely and completely determined by means of a finite + number of specifications, say by assigning a finite number + of elements, is a mathematical conception. Mathematics has + for its function to develop the consequences involved in + the definition of a group of mathematical conceptions. + Interdependence and mutual logical consistency among the + members of the group are postulated, otherwise the group + would either have to be treated as several distinct + groups, or would lie beyond the sphere of + mathematics.—<span class="smcap">Chrystal, George.</span></p> + <p class="blockcite"> + Encyclopedia Britannica (9th edition), Article “Mathematics.”</p> + + <p class="v2"> + <b><a name="Block_114" id="Block_114">114</a>.</b> + The purely formal sciences, + logic and mathematics, deal with those relations which + are, or can be, independent of the particular content or + the substance of objects. To mathematics in particular + fall those relations between objects which involve the + concepts of magnitude, of measure and of + number.—<span class="smcap">Hankel, Hermann.</span></p> + <p class="blockcite"> + Theorie der Complexen Zahlensysteme, (Leipzig, 1867), p. 1.</p> + + <p class="v2"> + <b><a name="Block_115" id="Block_115">115</a>.</b> + <em>Quantity is that which is + operated with according to fixed mutually consistent + laws</em>. Both operator and operand must derive their + meaning from the laws of operation. In the case of + ordinary algebra these are the three laws already + indicated [the commutative, associative, and distributive + laws], in the algebra of quaternions the same save the law + of commutation for multiplication and division, and so on. + It may be questioned whether this definition is + sufficient, and it may be objected that it is vague; but + the reader will do well to reflect that any definition + must include the linear algebras of Peirce, the algebra of + logic, and others that may be easily imagined, although + they have not yet been developed. This general definition + of quantity + +<span class="pagenum"> + <a name="Page_5" + id="Page_5">5</a></span> + + enables us to see how operators + may be treated as quantities, and thus to understand the + rationale of the so called symbolical methods.—<span + class="smcap">Chrystal, George.</span></p> + <p class="blockcite"> + Encyclopedia Britannica (9th edition), Article “Mathematics.”</p> + + <p class="v2"> + <b><a name="Block_116" id="Block_116">116</a>.</b> + Mathematics—in a strict + sense—is the abstract science which investigates + deductively the conclusions implicit in the elementary + conceptions of spatial and numerical + relations.—<span class="smcap">Murray, J. A. H.</span></p> + <p class="blockcite"> + A New English Dictionary.</p> + + <p class="v2"> + <b><a name="Block_117" id="Block_117">117</a>.</b> + Everything that the greatest + minds of all times have accomplished toward the + <em>comprehension of forms</em> by means of concepts is + gathered into one great science, + <em>mathematics</em>.—<span class="smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Pestalozzi’s Idee eines A B C der Anschauung, Werke + [Kehrbach], (Langensalza, 1890), Bd. 1, p. 163.</p> + + <p class="v2"> + <b><a name="Block_118" id="Block_118">118</a>.</b> + Perhaps the least inadequate + description of the general scope of modern Pure + Mathematics—I will not call it a definition—would be to + say that it deals with <em>form</em>, in a very general + sense of the term; this would include algebraic form, + functional relationship, the relations of order in any + ordered set of entities such as numbers, and the analysis + of the peculiarities of form of groups of + operations.—<span class="smcap">Hobson, E. W.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science (1910); Nature, Vol. 84, p. 287.</p> + + <p class="v2"> + <b><a name="Block_119" id="Block_119">119</a>.</b> + The ideal of mathematics + should be to erect a calculus to facilitate reasoning in + connection with every province of thought, or of external + experience, in which the succession of thoughts, or of + events can be definitely ascertained and precisely stated. + So that all serious thought which is not philosophy, or + inductive reasoning, or imaginative literature, shall be + mathematics developed by means of a calculus.—<span class= + "smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Universal Algebra (Cambridge, 1898), Preface.</p> + +<p> + <span class="pagenum"> + <a name="Page_6" + id="Page_6">6</a></span></p> + + <p class="v2"> + <b><a name="Block_120" id="Block_120">120</a>.</b> + Mathematics is the science + which draws necessary conclusions.—<span class= + "smcap">Peirce, Benjamin.</span></p> + <p class="blockcite"> + Linear Associative Algebra, American Journal of Mathematics, + Vol. 4 (1881), p. 97.</p> + + <p class="v2"> + <b><a name="Block_121" id="Block_121">121</a>.</b> + Mathematics is the universal + art apodictic.—<span class="smcap">Smith, W. B.</span></p> + <p class="blockcite"> + Quoted by Keyser, C. J. in Lectures on Science, Philosophy + and Art (New York, 1908), p. 13.</p> + + <p class="v2"> + <b><a name="Block_122" id="Block_122">122</a>.</b> + Mathematics in its widest + signification is the development of all types of formal, + necessary, deductive reasoning.—<span class= + "smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Universal Algebra (Cambridge, 1898), Preface, p. vi.</p> + + <p class="v2"> + <b><a name="Block_123" id="Block_123">123</a>.</b> + Mathematics in general is + fundamentally the science of self-evident + things.—<span class="smcap">Klein, Felix.</span></p> + <p class="blockcite"> + Anwendung der Differential- und Integralrechnung auf Geometrie + (Leipzig, 1902), p. 26.</p> + + <p class="v2"> + <b><a name="Block_124" id="Block_124">124</a>.</b> + A mathematical science is any + body of propositions which is capable of an abstract + formulation and arrangement in such a way that every + proposition of the set after a certain one is a formal + logical consequence of some or all the preceding + propositions. Mathematics consists of all such + mathematical sciences.—<span class="smcap">Young, Charles + Wesley.</span></p> + <p class="blockcite"> + Fundamental Concepts of Algebra and Geometry (New York, + 1911), p. 222.</p> + + <p class="v2"> + <b><a name="Block_125" id="Block_125">125</a>.</b> + Pure mathematics is a + collection of hypothetical, deductive theories, each + consisting of a definite system of primitive, + <em>undefined</em>, concepts or symbols and primitive, + <em>unproved</em>, but self-consistent assumptions (commonly + called axioms) together with their logically deducible + consequences following by rigidly deductive processes + without appeal to intuition.—<span class="smcap">Fitch, G. + D.</span></p> + <p class="blockcite"> + The Fourth Dimension simply Explained (New York, 1910), p. + 58.</p> + + <p class="v2"> + <b><a name="Block_126" id="Block_126">126</a>.</b> + The whole of Mathematics + consists in the organization of a series of aids to the + imagination in the process of reasoning.—<span class= + "smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Universal Algebra (Cambridge, 1898), p. 12.</p> + +<p> + <span class="pagenum"> + <a name="Page_7" + id="Page_7">7</a></span></p> + + <p class="v2"> + <b><a name="Block_127" id="Block_127">127</a>.</b> + Pure mathematics consists + entirely of such asseverations as that, if such and such a + proposition is true of <em>anything</em>, then such and such + another proposition is true of that thing. It is essential + not to discuss whether the first proposition is really + true, and not to mention what the anything is of which it + is supposed to be true.... If our hypothesis is about + <em>anything</em> and not about some one or more particular + things, then our deductions constitute mathematics. Thus + mathematics may be defined as the subject in which we + never know what we are talking about, nor whether what we + are saying is true.—<span class="smcap">Russell, + Bertrand.</span></p> + <p class="blockcite"> + Recent Work on the Principles of Mathematics, International + Monthly, Vol. 4 (1901), p. 84.</p> + + <p class="v2"> + <b><a name="Block_128" id="Block_128">128</a>.</b> + Pure Mathematics is the class + of all propositions of the form “<em>p</em> implies + <em>q</em>,” where <em>p</em> and <em>q</em> are propositions + containing one or more variables, the same in the two + propositions, and neither <em>p</em> nor <em>q</em> contains + any constants except logical constants. And logical + constants are all notions definable in terms of the + following: Implication, the relation of a term to a class + of which it is a member, the notion of <em>such that</em>, + the notion of relation, and such further notions as may be + involved in the general notion of propositions of the + above form. In addition to these, Mathematics <em>uses</em> + a notion which is not a constituent of the propositions + which it considers—namely, the notion of + truth.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + Principles of Mathematics (Cambridge, 1903), p. 1.</p> + + <p class="v2"> + <b><a name="Block_129" id="Block_129">129</a>.</b> + The object of pure Physic is + the unfolding of the laws of the intelligible world; the + object of pure Mathematic that of unfolding the laws of + human intelligence.—<span class="smcap">Sylvester, J. + J.</span></p> + <p class="blockcite"> + On a theorem, connected with Newton’s Rule, etc., Collected + Mathematical Papers, Vol. 3, p. 424.</p> + + <p class="v2"> + <b><a name="Block_130" id="Block_130">130</a>.</b> + First of all, we ought to + observe, that mathematical propositions, properly so + called, are always judgments <i lang="la" xml:lang="la">a + priori,</i> and not + empirical, because they carry along with them necessity, + which can never be deduced from experience. If people should + +<span class="pagenum"> + <a name="Page_8" + id="Page_8">8</a></span> + + object to this, I am quite + willing to confine my statements to pure mathematics, the + very concept of which implies that it does not contain + empirical, but only pure knowledge <i lang="la" xml:lang="la">a + priori</i>.—<span class="smcap">Kant, Immanuel.</span></p> + <p class="blockcite"> + Critique of Pure Reason [Müller], (New York, 1900), p. 720.</p> + + <p class="v2"> + <b><a name="Block_131" id="Block_131">131</a>.</b> + Mathematics, the science of + the ideal, becomes the means of investigating, + understanding and making known the world of the real. The + complex is expressed in terms of the simple. From one + point of view mathematics may be defined as the science of + successive substitutions of simpler concepts for more + complex....—<span class="smcap">White, William F.</span></p> + <p class="blockcite"> + A Scrap-book of Elementary Mathematics, (Chicago, 1908), p. + 215.</p> + + <p class="v2"> + <b><a name="Block_132" id="Block_132">132</a>.</b> + The critical mathematician has + abandoned the search for truth. He no longer flatters + himself that his propositions are or can be known to him + or to any other human being to be true; and he contents + himself with aiming at the correct, or the consistent. The + distinction is not annulled nor even blurred by the + reflection that consistency contains immanently a kind of + truth. He is not absolutely certain, but he believes + profoundly that it is possible to find various sets of a + few propositions each such that the propositions of each + set are compatible, that the propositions of each such set + imply other propositions, and that the latter can be + deduced from the former with certainty. That is to say, he + believes that there are systems of coherent or consistent + propositions, and he regards it his business to discover + such systems. Any such system is a branch of + mathematics.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Science, New Series, Vol. 35, p. 107.</p> + + <p class="v2"> + <b><a name="Block_133" id="Block_133">133</a>.</b> + [Mathematics is] the study of + ideal constructions (often applicable to real problems), + and the discovery thereby of relations between the parts + of these constructions, before unknown.—<span class= + "smcap">Peirce, C. S.</span></p> + <p class="blockcite"> + Century Dictionary, Article “Mathematics.”</p> + +<p> + <span class="pagenum"> + <a name="Page_9" + id="Page_9">9</a></span></p> + + <p class="v2"> + <b><a name="Block_134" id="Block_134">134</a>.</b> + Mathematics is that form of + intelligence in which we bring the objects of the + phenomenal world under the control of the conception of + quantity. [Provisional definition.]—<span class= + "smcap">Howison, G. H.</span></p> + <p class="blockcite"> + The Departments of Mathematics, and their Mutual Relations; + Journal of Speculative Philosophy, Vol. 5, p. 164.</p> + + <p class="v2"> + <b><a name="Block_135" id="Block_135">135</a>.</b> + Mathematics is the science of + the functional laws and transformations which enable us to + convert figured extension and rated motion into + number.—<span class="smcap">Howison, G. H.</span></p> + <p class="blockcite"> + The Departments of Mathematics, and their Mutual Relations; + Journal of Speculative Philosophy, Vol. 5, p. 170.</p> + +<p> + <span class="pagenum"> + <a name="Page_10" + id="Page_10">10</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_II"> + CHAPTER II<br /> + <span class="large"> + THE NATURE OF MATHEMATICS</span></h2> + <p class="v2"> + <b><a name="Block_201" id="Block_201">201</a>.</b> + Mathematics, from the earliest + times to which the history of human reason can reach, has + followed, among that wonderful people of the Greeks, the + safe way of science. But it must not be supposed that it + was as easy for mathematics as for logic, in which reason + is concerned with itself alone, to find, or rather to make + for itself that royal road. I believe, on the contrary, + that there was a long period of tentative work (chiefly + still among the Egyptians), and that the change is to be + ascribed to a <em>revolution</em>, produced by the happy + thought of a single man, whose experiments pointed + unmistakably to the path that had to be followed, and + opened and traced out for the most distant times the safe + way of a science. The history of that intellectual + revolution, which was far more important than the passage + round the celebrated Cape of Good Hope, and the name of + its fortunate author, have not been preserved to us.... A + new light flashed on the first man who demonstrated the + properties of the isosceles triangle (whether his name was + <em>Thales</em> or any other name), for he found that he had + not to investigate what he saw in the figure, or the mere + concepts of that figure, and thus to learn its properties; + but that he had to produce (by construction) what he had + himself, according to concepts <i lang="la" xml:lang="la">a + priori</i>, placed + into that figure and represented in it, so that, in order + to know anything with certainty <i lang="la" xml:lang="la">a + priori</i>, he must + not attribute to that figure anything beyond what + necessarily follows from what he has himself placed into + it, in accordance with the concept.—<span class= + "smcap">Kant, Immanuel.</span></p> + <p class="blockcite"> + Critique of Pure Reason, Preface to the Second Edition + [Müller], (New York, 1900), p. 690.</p> + + <p class="v2"> + <b><a name="Block_202" id="Block_202">202</a>.</b> + [When followed in the proper + spirit], there is no study in the world which brings into + more harmonious action all the faculties of the mind than + the one [mathematics] of which I + +<span class="pagenum"> + <a name="Page_11" + id="Page_11">11</a></span> + + stand here as the humble + representative and advocate. There is none other which + prepares so many agreeable surprises for its followers, + more wonderful than the transformation scene of a + pantomime, or, like this, seems to raise them, by + successive steps of initiation to higher and higher states + of conscious intellectual being.—<span class= + "smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + A Plea for the Mathematician, Nature, Vol. 1, p. 261.</p> + + <p class="v2"> + <b><a name="Block_203" id="Block_203">203</a>.</b> + Thought-economy is most highly + developed in mathematics, that science which has reached + the highest formal development, and on which natural + science so frequently calls for assistance. Strange as it + may seem, the strength of mathematics lies in the + avoidance of all unnecessary thoughts, in the utmost + economy of thought-operations. The symbols of order, which + we call numbers, form already a system of wonderful + simplicity and economy. When in the multiplication of a + number with several digits we employ the multiplication + table and thus make use of previously accomplished results + rather than to repeat them each time, when by the use of + tables of logarithms we avoid new numerical calculations + by replacing them by others long since performed, when we + employ determinants instead of carrying through from the + beginning the solution of a system of equations, when we + decompose new integral expressions into others that are + familiar,—we see in all this but a faint reflection of the + intellectual activity of a <em>Lagrange</em> or + <em>Cauchy</em>, who with the keen discernment of a military + commander marshalls a whole troop of completed operations + in the execution of a new one.—<span class="smcap">Mach, + E.</span></p> + <p class="blockcite"> + Populär-wissenschafliche Vorlesungen (1908), pp. 224-225.</p> + + <p class="v2"> + <b><a name="Block_204" id="Block_204">204</a>.</b> + Pure mathematics proves itself + a royal science both through its content and form, which + contains within itself the cause of its being and its + methods of proof. For in complete independence mathematics + creates for itself the object of which it treats, its + magnitudes and laws, its formulas and symbols.—<span class= + "smcap">Dillmann, E.</span></p> + <p class="blockcite"> + Die Mathematik die Fackelträgerin einer neuen Zeit + (Stuttgart, 1889), p. 94.</p> + +<p> + <span class="pagenum"> + <a name="Page_12" + id="Page_12">12</a></span></p> + + <p class="v2"> + <b><a name="Block_205" id="Block_205">205</a>.</b> + The essence of mathematics + lies in its freedom.—<span class="smcap">Cantor, + George.</span></p> + <p class="blockcite"> + Mathematische Annalen, Bd. 21, p. 564.</p> + + <p class="v2"> + <b><a name="Block_206" id="Block_206">206</a>.</b> + Mathematics pursues its own + course unrestrained, not indeed with an unbridled licence + which submits to no laws, but rather with the freedom + which is determined by its own nature and in conformity + with its own being.—<span class="smcap">Hankel, + Hermann.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten Jahrhunderten + (Tübingen, 1884), p. 16.</p> + + <p class="v2"> + <b><a name="Block_207" id="Block_207">207</a>.</b> + Mathematics is perfectly free + in its development and is subject only to the obvious + consideration, that its concepts must be free from + contradictions in themselves, as well as definitely and + orderly related by means of definitions to the previously + existing and established concepts.—<span class= + "smcap">Cantor, George.</span></p> + <p class="blockcite"> + Grundlagen einer allgemeinen Manigfaltigkeitslehre (Leipzig, + 1883), Sect. 8.</p> + + <p class="v2"> + <b><a name="Block_208" id="Block_208">208</a>.</b> + Mathematicians assume the + right to choose, within the limits of logical + contradiction, what path they please in reaching their + results.—<span class="smcap">Adams, Henry.</span></p> + <p class="blockcite"> + A Letter to American Teachers of History (Washington, 1910), + Introduction, p. v.</p> + + <p class="v2"> + <b><a name="Block_209" id="Block_209">209</a>.</b> + Mathematics is the predominant + science of our time; its conquests grow daily, though + without noise; he who does not employ it for himself, will + some day find it employed against himself.—<span class= + "smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Werke [Kehrbach] (Langensalza, 1890), Bd. 5, p. 105.</p> + + <p class="v2"> + <b><a name="Block_210" id="Block_210">210</a>.</b> + Mathematics is not the + discoverer of laws, for it is not induction; neither is it + the framer of theories, for it is not hypothesis; but it + is the judge over both, and it is the arbiter to which + each must refer its claims; and neither law can rule nor + theory explain without the sanction of + mathematics.—<span class="smcap">Peirce, Benjamin.</span></p> + <p class="blockcite"> + Linear Associative Algebra, American Journal of Mathematics, + Vol. 4 (1881), p. 97.</p> + +<p> + <span class="pagenum"> + <a name="Page_13" + id="Page_13">13</a></span></p> + + <p class="v2"> + <b><a name="Block_211" id="Block_211">211</a>.</b> + Mathematics is a science + continually expanding; and its growth, unlike some + political and industrial events, is attended by universal + acclamation.—<span class="smcap">White, H. S.</span></p> + <p class="blockcite"> + Congress of Arts and Sciences (Boston and New York, 1905), + Vol. 1, p. 455.</p> + + <p class="v2"> + <b><a name="Block_212" id="Block_212">212</a>.</b> + Mathematics accomplishes + really nothing outside of the realm of magnitude; + marvellous, however, is the skill with which it masters + magnitude wherever it finds it. We recall at once the + network of lines which it has spun about heavens and + earth; the system of lines to which azimuth and altitude, + declination and right ascension, longitude and latitude + are referred; those abscissas and ordinates, tangents and + normals, circles of curvature and evolutes; those + trigonometric and logarithmic functions which have been + prepared in advance and await application. A look at this + apparatus is sufficient to show that mathematicians are + not magicians, but that everything is accomplished by + natural means; one is rather impressed by the multitude of + skilful machines, numerous witnesses of a manifold and + intensely active industry, admirably fitted for the + acquisition of true and lasting treasures.—<span class= + "smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Werke [Kehrbach] (Langensalza, 1890), Bd. 5, p. 101.</p> + + <p class="v2"> + <b><a name="Block_213" id="Block_213">213</a>.</b> + They [mathematicians] only + take those things into consideration, of which they have + clear and distinct ideas, designating them by proper, + adequate, and invariable names, and premising only a few + axioms which are most noted and certain to investigate + their affections and draw conclusions from them, and + agreeably laying down a very few hypotheses, such as are + in the highest degree consonant with reason and not to be + denied by anyone in his right mind. In like manner they + assign generations or causes easy to be understood and + readily admitted by all, they preserve a most accurate + order, every proposition immediately following from what + is supposed and proved before, and reject all things + howsoever specious and probable which can not be inferred + and deduced after the same manner.—<span class= + "smcap">Barrow, Isaac.</span></p> + <p class="blockcite"> + Mathematical Lectures (London, 1734), p. 66.</p> + +<p> + <span class="pagenum"> + <a name="Page_14" + id="Page_14">14</a></span></p> + + <p class="v2"> + <b><a name="Block_214" id="Block_214">214</a>.</b> + The dexterous management of + terms and being able to <em>fend</em> and <em>prove</em> with + them, I know has and does pass in the world for a great + part of learning; but it is learning distinct from + knowledge, for knowledge consists only in perceiving the + habitudes and relations of ideas one to another, which is + done without words; the intervention of sounds helps + nothing to it. And hence we see that there is least use of + distinction where there is most knowledge: I mean in + mathematics, where men have determined ideas with known + names to them; and so, there being no room for + equivocations, there is no need of + distinctions.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + Conduct of the Understanding, Sect. 31.</p> + + <p class="v2"> + <b><a name="Block_215" id="Block_215">215</a>.</b> + In mathematics it [sophistry] + had no place from the beginning: Mathematicians having had + the wisdom to define accurately the terms they use, and to + lay down, as axioms, the first principles on which their + reasoning is grounded. Accordingly we find no parties + among mathematicians, and hardly any disputes.—<span class= + "smcap">Reid, Thomas.</span></p> + <p class="blockcite"> + Essays on the Intellectual Powers of Man, Essay 1, chap. 1.</p> + + <p class="v2"> + <b><a name="Block_216" id="Block_216">216</a>.</b> + In most sciences one + generation tears down what another has built and what one + has established another undoes. In Mathematics alone each + generation builds a new story to the old + structure.—<span class="smcap">Hankel, Hermann.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten Jahrhunderten + (Tübingen, 1884), p. 25.</p> + + <p class="v2"> + <b><a name="Block_217" id="Block_217">217</a>.</b> + Mathematics, the priestess of definiteness and + clearness.—<span class="smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Werke [Kehrbach] (Langensalza, 1890), Bd. 1, p. 171.</p> + + <p class="v2"> + <b><a name="Block_218" id="Block_218">218</a>.</b> + ... mathematical analysis is + co-extensive with nature itself, it defines all + perceivable relations, measures times, spaces, forces, + temperatures; it is a difficult science which forms but + slowly, but preserves carefully every principle once + acquired; it increases and becomes stronger incessantly + amidst all the changes and errors of the human mind.</p> + +<p> + <span class="pagenum"> + <a name="Page_15" + id="Page_15">15</a></span></p> + + <p class="v1"> + Its chief attribute is clearness; it has no means for + expressing confused ideas. It compares the most diverse + phenomena and discovers the secret analogies which unite + them. If matter escapes us, as that of air and light because + of its extreme tenuity, if bodies are placed far from us in + the immensity of space, if man wishes to know the aspect of + the heavens at successive periods separated by many + centuries, if gravity and heat act in the interior of the + solid earth at depths which will forever be inaccessible, + mathematical analysis is still able to trace the laws of + these phenomena. It renders them present and measurable, and + appears to be the faculty of the human mind destined to + supplement the brevity of life and the imperfection of the + senses, and what is even more remarkable, it follows the same + course in the study of all phenomena; it explains them in the + same language, as if in witness to the unity and simplicity + of the plan of the universe, and to make more manifest the + unchangeable order which presides over all natural + causes.—<span class="smcap">Fourier, J.</span></p> + <p class="blockcite"> + Théorie Analytique de la Chaleur, Discours Préliminaire.</p> + + <p class="v2"> + <b><a name="Block_219" id="Block_219">219</a>.</b> + Let us now declare the means + whereby our understanding can rise to knowledge without + fear of error. There are two such means: intuition and + deduction. By intuition I mean not the varying testimony + of the senses, nor the deductive judgment of imagination + naturally extravagant, but the conception of an attentive + mind so distinct and so clear that no doubt remains to it + with regard to that which it comprehends; or, what amounts + to the same thing, the self-evidencing conception of a + sound and attentive mind, a conception which springs from + the light of reason alone, and is more certain, because + more simple, than deduction itself....</p> + <p class="v1"> + It may perhaps be asked why to intuition we add this other + mode of knowing, by deduction, that is to say, the process + which, from something of which we have certain knowledge, + draws consequences which necessarily follow therefrom. But we + are obliged to admit this second step; for there are a great + many things which, without being evident of themselves, + nevertheless bear the marks of certainty if only they are + deduced from + +<span class="pagenum"> + <a name="Page_16" + id="Page_16">16</a></span> + + true and incontestable + principles by a continuous and uninterrupted movement of + thought, with distinct intuition of each thing; just as we + know that the last link of a long chain holds to the + first, although we can not take in with one glance of the + eye the intermediate links, provided that, after having + run over them in succession, we can recall them all, each + as being joined to its fellows, from the first up to the + last. Thus we distinguish intuition from deduction, + inasmuch as in the latter case there is conceived a + certain progress or succession, while it is not so in the + former;... whence it follows that primary propositions, + derived immediately from principles, may be said to be + known, according to the way we view them, now by + intuition, now by deduction; although the principles + themselves can be known only by intuition, the remote + consequences only by deduction.—<span class= + "smcap">Descartes.</span></p> + <p class="blockcite"> + Rules for the Direction of the Mind, Philosophy of D. + [Torrey] (New York, 1892), pp. 64, 65.</p> + + <p class="v2"> + <b><a name="Block_220" id="Block_220">220</a>.</b> + Analysis and natural + philosophy owe their most important discoveries to this + fruitful means, which is called induction. Newton was + indebted to it for his theorem of the binomial and the + principle of universal gravity.—<span class= + "smcap">Laplace.</span></p> + <p class="blockcite"> + A Philosophical Essay on Probabilities [Truscott and Emory] + (New York 1902), p. 176.</p> + + <p class="v2"> + <b><a name="Block_221" id="Block_221">221</a>.</b> + There is in every step of an + arithmetical or algebraical calculation a real induction, + a real inference from facts to facts, and what disguises + the induction is simply its comprehensive nature, and the + consequent extreme generality of its language.—<span class= + "smcap">Mill, J. S.</span></p> + <p class="blockcite"> + System of Logic, Bk. 2, chap. 6, 2.</p> + + <p class="v2"> + <b><a name="Block_222" id="Block_222">222</a>.</b> + It would appear that Deductive + and Demonstrative Sciences are all, without exception, + Inductive Sciences: that their evidence is that of + experience, but that they are also, in virtue of the + peculiar character of one indispensable portion of the + general formulae according to which their inductions are + made, Hypothetical Sciences. Their conclusions are true + only upon certain suppositions, which are, or ought to be, + approximations to the truth, but are seldom, if ever, + exactly true; and + +<span class="pagenum"> + <a name="Page_17" + id="Page_17">17</a></span> + + to this hypothetical character + is to be ascribed the peculiar certainty, which is + supposed to be inherent in demonstration.—<span class= + "smcap">Mill, J. S.</span></p> + <p class="blockcite"> + System of Logic, Bk. 2, chap. 6, 1.</p> + + <p class="v2"> + <b><a name="Block_223" id="Block_223">223</a>.</b> + The peculiar character of + mathematical truth is, that it is necessarily and + inevitably true; and one of the most important lessons + which we learn from our mathematical studies is a + knowledge that there are such truths, and a familiarity + with their form and character.</p> + <p class="v1"> + This lesson is not only lost, but read backward, if the + student is taught that there is no such difference, and that + mathematical truths themselves are learned by + experience.—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + Thoughts on the Study of Mathematics. Principles of English + University Education (London, 1838).</p> + + <p class="v2"> + <b><a name="Block_224" id="Block_224">224</a>.</b> + These sciences, Geometry, + Theoretical Arithmetic and Algebra, have no principles + besides definitions and axioms, and no process of proof + but <em>deduction</em>; this process, however, assuming a most + remarkable character; and exhibiting a combination of simplicity + and complexity, of rigour and generality, quite unparalleled + in other subjects.—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + The Philosophy of the Inductive Sciences, Part 1, Bk. 2, + chap. 1, sect. 2 (London, 1858).</p> + + <p class="v2"> + <b><a name="Block_225" id="Block_225">225</a>.</b> + The apodictic quality of mathematical thought, the certainty + and correctness of its conclusions, are due, not to a special + mode of ratiocination, but to the character of the concepts + with which it deals. What is that distinctive characteristic? + I answer: <em>precision</em>, <em>sharpness</em>, + <em>completeness</em>,<a + href="#Footnote_1" + class="fnanch2" + title="i.e., in terms of the absolutely +clear and indefinable.">1</a> + + of definition. But how comes your mathematician by + such completeness? There is no mysterious trick involved; + some ideas admit of such precision, others do not; and the + mathematician is one who deals with those that + do.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + The Universe and Beyond; Hibbert Journal, Vol. 3 (1904-1905), + p. 309.</p> + + <p class="v2"> + <b><a name="Block_226" id="Block_226">226</a>.</b> + The reasoning of + mathematicians is founded on certain + +<span class="pagenum"> + <a name="Page_18" + id="Page_18">18</a></span> + + and infallible principles. + Every word they use conveys a determinate idea, and by + accurate definitions they excite the same ideas in the + mind of the reader that were in the mind of the writer. + When they have defined the terms they intend to make use + of, they premise a few axioms, or self-evident principles, + that every one must assent to as soon as proposed. They + then take for granted certain postulates, that no one can + deny them, such as, that a right line may be drawn from + any given point to another, and from these plain, simple + principles they have raised most astonishing speculations, + and proved the extent of the human mind to be more + spacious and capacious than any other science.—<span class= + "smcap">Adams, John.</span></p> + <p class="blockcite"> + Diary, Works (Boston, 1850), Vol. 2, p. 21.</p> + + <p class="v2"> + <b><a name="Block_227" id="Block_227">227</a>.</b> + It may be observed of + mathematicians that they only meddle with such things as + are certain, passing by those that are doubtful and + unknown. They profess not to know all things, neither do + they affect to speak of all things. What they know to be + true, and can make good by invincible arguments, that they + publish and insert among their theorems. Of other things + they are silent and pass no judgment at all, choosing + rather to acknowledge their ignorance, than affirm + anything rashly. They affirm nothing among their arguments + or assertions which is not most manifestly known and + examined with utmost rigour, rejecting all probable + conjectures and little witticisms. They submit nothing to + authority, indulge no affection, detest subterfuges of + words, and declare their sentiments, as in a court of + justice, <em>without passion, without apology</em>; knowing + that their reasons, as Seneca testifies of them, are not + brought to <em>persuade</em>, but to compel.—<span class= + "smcap">Barrow, Isaac.</span></p> + <p class="blockcite"> + Mathematical Lectures (London, 1734), p. 64.</p> + + <p class="v2"> + <b><a name="Block_228" id="Block_228">228</a>.</b> + What is exact about + mathematics but exactness? And is not this a consequence + of the inner sense of truth?—<span class= + "smcap">Goethe.</span></p> + <p class="blockcite"> + Sprüche in Prosa, Natur, 6, 948.</p> + + <p class="v2"> + <b><a name="Block_229" id="Block_229">229</a>.</b> + ... the three positive + characteristics that distinguish mathematical knowledge + from other knowledge ... may be briefly expressed as + follows: first, mathematical knowledge bears more + distinctly the imprint of truth on all its results than + any + +<span class="pagenum"> + <a name="Page_19" + id="Page_19">19</a></span> + + other kind of knowledge; + secondly, it is always a sure preliminary step to the + attainment of other correct knowledge; thirdly, it has no + need of other knowledge.—<span class="smcap">Schubert, + H.</span></p> + <p class="blockcite"> + Mathematical Essays and Recreations (Chicago, 1898), p. 35.</p> + + <p class="v2"> + <b><a name="Block_230" id="Block_230">230</a>.</b> + It is now necessary to + indicate more definitely the reason why mathematics not + only carries conviction in itself, but also transmits + conviction to the objects to which it is applied. The + reason is found, first of all, in the perfect precision + with which the elementary mathematical concepts are + determined; in this respect each science must look to its + own salvation.... But this is not all. As soon as human + thought attempts long chains of conclusions, or difficult + matters generally, there arises not only the danger of + error but also the suspicion of error, because since all + details cannot be surveyed with clearness at the same + instant one must in the end be satisfied with a + <em>belief</em> that nothing has been overlooked from the + beginning. Every one knows how much this is the case even + in arithmetic, the most + + <a id="TNanchor_1" + class="msg" + href="#TN_1" + title="originally spelled ‘elmenetary’">elementary</a> + + use of mathematics. No + one would imagine that the higher parts of mathematics + fare better in this respect; on the contrary, in more + complicated conclusions the uncertainty and suspicion of + hidden errors increases in rapid progression. How does + mathematics manage to rid itself of this inconvenience + which attaches to it in the highest degree? By making + proofs more rigorous? By giving new rules according to + which the old rules shall be applied? Not in the least. A + very great uncertainty continues to attach to the result + of each single computation. But there are checks. In the + realm of mathematics each point may be reached by a + hundred different ways; and if each of a hundred ways + leads to the same point, one may be sure that the right + point has been reached. A calculation without a check is + as good as none. Just so it is with every isolated proof + in any speculative science whatever; the proof may be ever + so ingenious, and ever so perfectly true and correct, it + will still fail to convince permanently. He will therefore + be much deceived, who, in metaphysics, or in psychology + which depends on metaphysics, hopes to see his greatest + care in the precise determination of the concepts and in + the logical conclusions rewarded by conviction, much less + by success in transmitting conviction to + +<span class="pagenum"> + <a name="Page_20" + id="Page_20">20</a></span> + + others. Not only must the + conclusions support each other, without coercion or + suspicion of subreption, but in all matters originating in + experience, or judging concerning experience, the results + of speculation must be verified by experience, not only + superficially, but in countless special cases.—<span class= + "smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Werke [Kehrbach] (Langensalza, 1890), Bd. 5, p. 105.</p> + + <p class="v2"> + <b><a name="Block_231" id="Block_231">231</a>.</b> + [In mathematics] we behold the + conscious logical activity of the human mind in its purest + and most perfect form. Here we learn to realize the + laborious nature of the process, the great care with which + it must proceed, the accuracy which is necessary to + determine the exact extent of the general propositions + arrived at, the difficulty of forming and comprehending + abstract concepts; but here we learn also to place + confidence in the certainty, scope and fruitfulness of + such intellectual activity.—<span class="smcap">Helmholtz, + H.</span></p> + <p class="blockcite"> + Ueber das Verhältniss der Naturwissenschaften zur Gesammtheit + der Wissenschaft, Vorträge und Reden, Bd. 1 (1896), p. 176.</p> + + <p class="v2"> + <b><a name="Block_232" id="Block_232">232</a>.</b> + It is true that mathematics, + owing to the fact that its whole content is built up by + means of purely logical deduction from a small number of + universally comprehended principles, has not unfittingly + been designated as the science of the <em>self-evident</em> + [Selbstverständlichen]. Experience however, shows that for + the majority of the cultured, even of scientists, + mathematics remains the science of the + <em>incomprehensible</em> [Unverständlichen].—<span class= + "smcap">Pringsheim, Alfred.</span></p> + <p class="blockcite"> + Ueber Wert und angeblichen Unwert der Mathematik, + Jahresbericht der Deutschen Mathematiker Vereinigung (1904), + p. 357.</p> + + <p class="v2"> + <b><a name="Block_233" id="Block_233">233</a>.</b> + Mathematical reasoning is + deductive in the sense that it is based upon definitions + which, as far as the validity of the reasoning is + concerned (apart from any existential import), needs only + the test of self-consistency. Thus no external + verification of definitions is required in mathematics, as + long as it is considered merely as + mathematics.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Universal Algebra (Cambridge, 1898), Preface, p. vi.</p> + +<p> + <span class="pagenum"> + <a name="Page_21" + id="Page_21">21</a></span></p> + + <p class="v2"> + <b><a name="Block_234" id="Block_234">234</a>.</b> + The mathematician pays not the + least regard either to testimony or conjecture, but + deduces everything by demonstrative reasoning, from his + definitions and axioms. Indeed, whatever is built upon + conjecture, is improperly called science; for conjecture + may beget opinion, but cannot produce + knowledge.—<span class="smcap">Reid, Thomas.</span></p> + <p class="blockcite"> + Essays on the Intellectual Powers of Man, Essay 1, chap. 3.</p> + + <p class="v2"> + <b><a name="Block_235" id="Block_235">235</a>.</b> + ... for the saving the long + progression of the thoughts to remote and first principles + in every case, the mind should provide itself several + stages; that is to say, intermediate principles, which it + might have recourse to in the examining those positions + that come in its way. These, though they are not + self-evident principles, yet, if they have been made out + from them by a wary and unquestionable deduction, may be + depended on as certain and infallible truths, and serve as + unquestionable truths to prove other points depending upon + them, by a nearer and shorter view than remote and general + maxims.... And thus mathematicians do, who do not in every + new problem run it back to the first axioms through all + the whole train of intermediate propositions. Certain + theorems that they have settled to themselves upon sure + demonstration, serve to resolve to them multitudes of + propositions which depend on them, and are as firmly made + out from thence as if the mind went afresh over every link + of the whole chain that tie them to first self-evident + principles.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + The Conduct of the Understanding, Sect. 21.</p> + + <p class="v2"> + <b><a name="Block_236" id="Block_236">236</a>.</b> + Those intervening ideas, which + serve to show the agreement of any two others, are called + <em>proofs</em>; and where the agreement or disagreement is + by this means plainly and clearly perceived, it is called + <em>demonstration</em>; it being <em>shown</em> to the + understanding, and the mind made to see that it is so. A + quickness in the mind to find out these intermediate + ideas, (that shall discover the agreement or disagreement + of any other) and to apply them right, is, I suppose, that + which is called <em>sagacity</em>.—<span class= + "smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 6, chaps. 2, 3.</p> + +<p> + <span class="pagenum"> + <a name="Page_22" + id="Page_22">22</a></span></p> + + <p class="v2"> + <b><a name="Block_237" id="Block_237">237</a>.</b> + ... the speculative + propositions of mathematics do not relate to <em>facts</em>; + ... all that we are convinced of by any demonstration in + the science, is of a necessary connection subsisting + between certain suppositions and certain conclusions. When + we find these suppositions actually take place in a + particular instance, the demonstration forces us to apply + the conclusion. Thus, if I could form a triangle, the + three sides of which were accurately mathematical lines, I + might affirm of this individual figure, that its three + angles are equal to two right angles; but, as the + imperfection of my senses puts it out of my power to be, + in any case, <em>certain</em> of the exact correspondence of + the diagram which I delineate, with the definitions given + in the elements of geometry, I never can apply with + confidence to a particular figure, a mathematical theorem. + On the other hand, it appears from the daily testimony of + our senses that the speculative truths of geometry may be + applied to material objects with a degree of accuracy + sufficient for the purposes of life; and from such + applications of them, advantages of the most important + kind have been gained to society.—<span class= + "smcap">Stewart, Dugald.</span></p> + <p class="blockcite"> + Elements of the Philosophy of the Human Mind, Part 3, chap. + 1, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_238" id="Block_238">238</a>.</b> + No process of sound reasoning + can establish a result not contained in the + premises.—<span class="smcap">Mellor, J. W.</span></p> + <p class="blockcite"> + Higher Mathematics for Students of Chemistry and Physics (New + York, 1902), p. 2.</p> + + <p class="v2"> + <b><a name="Block_239" id="Block_239">239</a>.</b> + ... we cannot get more out of + the mathematical mill than we put into it, though we may + get it in a form infinitely more useful for our + purpose.—<span class="smcap">Hopkinson, John.</span></p> + <p class="blockcite"> + James Forrest Lecture, 1894.</p> + + <p class="v2"> + <b><a name="Block_240" id="Block_240">240</a>.</b> + The iron labor of conscious + logical reasoning demands great perseverance and great + caution; it moves on but slowly, and is rarely illuminated + by brilliant flashes of genius. It knows little of that + facility with which the most varied instances come + thronging into the memory of the philologist or historian. + Rather is it an essential condition of the methodical + progress of + +<span class="pagenum"> + <a name="Page_23" + id="Page_23">23</a></span> + + mathematical reasoning that the + mind should remain concentrated on a single point, + undisturbed alike by collateral ideas on the one hand, and + by wishes and hopes on the other, and moving on steadily + in the direction it has deliberately chosen.—<span class= + "smcap">Helmholtz, H.</span></p> + <p class="blockcite"> + Ueber das Verhältniss der Naturwissenschaften zur Gesammtheit + der Wissenschaft, Vorträge und Reden, Bd. 1 (1896), p. 178.</p> + + <p class="v2"> + <b><a name="Block_241" id="Block_241">241</a>.</b> + If it were always necessary to + reduce everything to intuitive knowledge, demonstration + would often be insufferably prolix. This is why + mathematicians have had the cleverness to divide the + difficulties and to demonstrate separately the intervening + propositions. And there is art also in this; for as the + mediate truths (which are called <i lang="el" + xml:lang="el">lemmas</i>, since they + appear to be a digression) may be assigned in many ways, + it is well, in order to aid the understanding and memory, + to choose of them those which greatly shorten the process, + and appear memorable and worthy in themselves of being + demonstrated. But there is another obstacle, viz.: that it + is not easy to demonstrate all the axioms, and to reduce + demonstrations wholly to intuitive knowledge. And if we + had chosen to wait for that, perhaps we should not yet + have the science of geometry.—<span class= + "smcap">Leibnitz, G. W.</span></p> + <p class="blockcite"> + New Essay on Human Understanding [Langley], Bk. 4, chaps. 2, + 8.</p> + + <p class="v2"> + <b><a name="Block_242" id="Block_242">242</a>.</b> + In Pure Mathematics, where all + the various truths are necessarily connected with each + other, (being all necessarily connected with those + <em>hypotheses</em> which are the principles of the + science), an arrangement is beautiful in proportion as the + principles are few; and what we admire perhaps chiefly in + the science, is the astonishing variety of consequences + which may be demonstrably deduced from so small a number + of premises.—<span class="smcap">Stewart, Dugald.</span></p> + <p class="blockcite"> + The Elements of the Philosophy of the Human Mind, Part 3, + chap. 1, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_243" id="Block_243">243</a>.</b> + Whenever ... a controversy + arises in mathematics, the issue is not whether a thing is + true or not, but whether the proof might not be conducted + more simply in some other way, or whether the proposition + demonstrated is sufficiently important + +<span class="pagenum"> + <a name="Page_24" + id="Page_24">24</a></span> + + for the advancement of the + science as to deserve especial enunciation and emphasis, + or finally, whether the proposition is not a special case + of some other and more general truth which is as easily + discovered.—<span class="smcap">Schubert, H.</span></p> + <p class="blockcite"> + Mathematical Essays and Recreations (Chicago, 1898), p. 28.</p> + + <p class="v2"> + <b><a name="Block_244" id="Block_244">244</a>.</b> + ... just as the astronomer, + the physicist, the geologist, or other student of + objective science looks about in the world of sense, so, + not metaphorically speaking but literally, the mind of the + mathematician goes forth in the universe of logic in quest + of the things that are there; exploring the heights and + depths for facts—ideas, classes, relationships, + implications, and the rest; observing the minute and + elusive with the powerful microscope of his Infinitesimal + Analysis; observing the elusive and vast with the + limitless telescope of his Calculus of the Infinite; + making guesses regarding the order and internal harmony of + the data observed and collocated; testing the hypotheses, + not merely by the complete induction peculiar to + mathematics, but, like his colleagues of the outer world, + resorting also to experimental tests and incomplete + induction; frequently finding it necessary, in view of + unforeseen disclosures, to abandon one hopeful hypothesis + or to transform it by retrenchment or by + enlargement:—thus, in his own domain, matching, point for + point, the processes, methods and experience familiar to + the devotee of natural science.—<span class= + "smcap">Keyser, Cassius J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), p. + 26.</p> + + <p class="v2"> + <b><a name="Block_245" id="Block_245">245</a>.</b> + That mathematics “do not + cultivate the power of generalization,” ... will be + admitted by no person of competent knowledge, except in a + very qualified sense. The generalizations of mathematics, + are, no doubt, a different thing from the generalizations + of physical science; but in the difficulty of seizing + them, and the mental tension they require, they are no + contemptible preparation for the most arduous efforts of + the scientific mind. Even the fundamental notions of the + higher mathematics, from those of the differential + calculus upwards are products of a very high + abstraction.... To perceive the mathematical laws common + to the results of many mathematical + +<span class="pagenum"> + <a name="Page_25" + id="Page_25">25</a></span> + + operations, even in so simple a + case as that of the binomial theorem, involves a vigorous + exercise of the same faculty which gave us Kepler’s laws, + and rose through those laws to the theory of universal + gravitation. Every process of what has been called + Universal Geometry—the great creation of Descartes and his + successors, in which a single train of reasoning solves + whole classes of problems at once, and others common to + large groups of them—is a practical lesson in the + management of wide generalizations, and abstraction of the + points of agreement from those of difference among objects + of great and confusing diversity, to which the purely + inductive sciences cannot furnish many superior. Even so + elementary an operation as that of abstracting from the + particular configuration of the triangles or other + figures, and the relative situation of the particular + lines or points, in the diagram which aids the + apprehension of a common geometrical demonstration, is a + very useful, and far from being always an easy, exercise + of the faculty of generalization so strangely imagined to + have no place or part in the processes of + mathematics.—<span class="smcap">Mill, John Stuart.</span></p> + <p class="blockcite"> + An Examination of Sir William Hamilton’s Philosophy (London, + 1878), pp. 612, 613.</p> + + <p class="v2"> + <b><a name="Block_246" id="Block_246">246</a>.</b> + When the greatest of American + logicians, speaking of the powers that constitute the born + geometrician, had named Conception, Imagination, and + Generalization, he paused. Thereupon from one of the + audience there came the challenge, “What of reason?” The + instant response, not less just than brilliant, was: + “Ratiocination—that is but the smooth pavement on which + the chariot rolls.”—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), p. + 31.</p> + + <p class="v2"> + <b><a name="Block_247" id="Block_247">247</a>.</b> + ... the reasoning process + [employed in mathematics] is not different from that of + any other branch of knowledge, ... but there is required, + and in a great degree, that attention of mind which is in + some part necessary for the acquisition of all knowledge, + and in this branch is indispensably necessary. This must + be given in its fullest intensity; ... the other elements + especially characteristic of a mathematical mind are + quickness + +<span class="pagenum"> + <a name="Page_26" + id="Page_26">26</a></span> + + in perceiving logical sequence, + love of order, methodical arrangement and harmony, + distinctness of conception.—<span class="smcap">Price, + B.</span></p> + <p class="blockcite"> + Treatise on Infinitesimal Calculus (Oxford, 1868), Vol. 3, p. + 6.</p> + + <p class="v2"> + <b><a name="Block_248" id="Block_248">248</a>.</b> + Histories make men wise; + poets, witty; the mathematics, subtile; natural + philosophy, deep; moral, grave; logic and rhetoric, able + to contend.—<span class="smcap">Bacon, Francis.</span></p> + <p class="blockcite"> + Essays, Of Studies.</p> + + <p class="v2"> + <b><a name="Block_249" id="Block_249">249</a>.</b> + The Mathematician deals with + two properties of objects only, number and extension, and + all the inductions he wants have been formed and finished + ages ago. He is now occupied with nothing but deduction + and verification.—<span class="smcap">Huxley, T. H.</span></p> + <p class="blockcite"> + On the Educational Value of the Natural History Sciences; Lay + Sermons, Addresses and Reviews; (New York, 1872), p. 87.</p> + + <p class="v2"> + <b><a name="Block_250" id="Block_250">250</a>.</b> + [Mathematics] is that + [subject] which knows nothing of observation, nothing of + experiment, nothing of induction, nothing of + causation.—<span class="smcap">Huxley, T. H.</span></p> + <p class="blockcite"> + The Scientific Aspects of Positivism, Fortnightly Review + (1898); Lay Sermons, Addresses and Reviews, (New York, 1872), + p. 169.</p> + + <p class="v2"> + <b><a name="Block_251" id="Block_251">251</a>.</b> + We are told that “Mathematics + is that study which knows nothing of observation, nothing + of experiment, nothing of induction, nothing of + causation.” I think no statement could have been made more + opposite to the facts of the case; that mathematical + analysis is constantly invoking the aid of new principles, + new ideas, and new methods, not capable of being defined + by any form of words, but springing direct from the + inherent powers and activities of the human mind, and from + continually renewed introspection of that inner world of + thought of which the phenomena are as varied and require + as close attention to discern as those of the outer + physical world (to which the inner one in each individual + man may, I think, be conceived to stand somewhat in the + same relation of correspondence as a shadow to the object + from which it is projected, or as the hollow palm of one + hand to the closed fist which it grasps of the other), + that it is unceasingly calling forth the faculties of + observation + +<span class="pagenum"> + <a name="Page_27" + id="Page_27">27</a></span> + + and comparison, that one of its + principal weapons is induction, that it has frequent + recourse to experimental trial and verification, and that + it affords a boundless scope for the exercise of the + highest efforts of the imagination and + invention.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Presidential Address to British Association, Exeter British + Association Report (1869), pp. 1-9.; Collected Mathematical + Papers, Vol. 2, p. 654.</p> + + <p class="v2"> + <b><a name="Block_252" id="Block_252">252</a>.</b> + The actual evolution of + mathematical theories proceeds by a process of induction + strictly analogous to the method of induction employed in + building up the physical sciences; observation, + comparison, classification, trial, and generalisation are + essential in both cases. Not only are special results, + obtained independently of one another, frequently seen to + be really included in some generalisation, but branches of + the subject which have been developed quite independently + of one another are sometimes found to have connections + which enable them to be synthesised in one single body of + doctrine. The essential nature of mathematical thought + manifests itself in the discernment of fundamental + identity in the mathematical aspects of what are + superficially very different domains. A striking example + of this species of immanent identity of mathematical form + was exhibited by the discovery of that distinguished + mathematician ... Major MacMahon, that all possible Latin + squares are capable of enumeration by the consideration of + certain differential operators. Here we have a case in + which an enumeration, which appears to be not amenable to + direct treatment, can actually be carried out in a simple + manner when the underlying identity of the operation is + recognised with that involved in certain operations due to + differential operators, the calculus of which belongs + superficially to a wholly different region of thought from + that relating to Latin squares.—<span class= + "smcap">Hobson, E. W.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science (1910); Nature, Vol. 84, p. 290.</p> + + <p class="v2"> + <b><a name="Block_253" id="Block_253">253</a>.</b> + It has been asserted ... that + the power of observation is not developed by mathematical + studies; while the truth is, + +<span class="pagenum"> + <a name="Page_28" + id="Page_28">28</a></span> + + that; from the most elementary + mathematical notion that arises in the mind of a child to + the farthest verge to which mathematical investigation has + been pushed and applied, this power is in constant + exercise. By observation, as here used, can only be meant + the fixing of the attention upon objects (physical or + mental) so as to note distinctive peculiarities—to + recognize resemblances, differences, and other relations. + Now the first mental act of the child recognizing the + distinction between <em>one</em> and more than one, between + <em>one</em> and <em>two</em>, <em>two</em> and <em>three</em>, + etc., is exactly this. So, again, the first geometrical + notions are as pure an exercise of this power as can be + given. To know a straight line, to distinguish it from a + curve; to recognize a triangle and distinguish the several + forms—what are these, and all perception of form, but a + series of observations? Nor is it alone in securing these + fundamental conceptions of number and form that + observation plays so important a part. The very genius of + the common geometry as a method of reasoning—a system of + investigation—is, that it is but a series of observations. + The figure being before the eye in actual representation, + or before the mind in conception, is so closely + scrutinized, that all its distinctive features are + perceived; auxiliary lines are drawn (the imagination + leading in this), and a new series of inspections is made; + and thus, by means of direct, simple observations, the + investigation proceeds. So characteristic of common + geometry is this method of investigation, that Comte, + perhaps the ablest of all writers upon the philosophy of + mathematics, is disposed to class geometry, as to its + method, with the natural sciences, being based upon + observation. Moreover, when we consider applied + mathematics, we need only to notice that the exercise of + this faculty is so essential, that the basis of all such + reasoning, the very material with which we build, have + received the name <em>observations</em>. Thus we might + proceed to consider the whole range of the human + faculties, and find for the most of them ample scope for + exercise in mathematical studies. Certainly, the + <em>memory</em> will not be found to be neglected. The very + first steps in number—counting, the multiplication table, + etc., make heavy demands on this power; while the higher + branches require the memorizing of formulas which are + simply appalling to the uninitiated. So the + <em>imagination</em>, the creative faculty of the + +<span class="pagenum"> + <a name="Page_29" + id="Page_29">29</a></span> + + mind, has constant exercise in + all original mathematical investigations, from the + solution of the simplest problems to the discovery of the + most recondite principle; for it is not by sure, + consecutive steps, as many suppose, that we advance from + the known to the unknown. The imagination, not the logical + faculty, leads in this advance. In fact, practical + observation is often in advance of logical exposition. + Thus, in the discovery of truth, the imagination + habitually presents hypotheses, and observation supplies + facts, which it may require ages for the tardy reason to + connect logically with the known. Of this truth, + mathematics, as well as all other sciences, affords + abundant illustrations. So remarkably true is this, that + today it is seriously questioned by the majority of + thinkers, whether the sublimest branch of mathematics,—the + <em>infinitesimal calculus</em>—has anything more than an + empirical foundation, mathematicians themselves not being + agreed as to its logical basis. That the imagination, and + not the logical faculty, leads in all original + investigation, no one who has ever succeeded in producing + an original demonstration of one of the simpler + propositions of geometry, can have any doubt. Nor are + <em>induction</em>, <em>analogy</em>, the + <em>scrutinization</em> of <em>premises</em> or the + <em>search</em> for them, or the <em>balancing</em> of + <em>probabilities</em>, spheres of mental operations foreign + to mathematics. No one, indeed, can claim pre-eminence for + mathematical studies in all these departments of + intellectual culture, but it may, perhaps, be claimed that + scarcely any department of science affords discipline to + so great a number of faculties, and that none presents so + complete a gradation in the exercise of these faculties, + from the first principles of the science to the farthest + extent of its applications, as mathematics.—<span class= + "smcap">Olney, Edward.</span></p> + <p class="blockcite"> + Kiddle and Schem’s Encyclopedia of Education, (New York, + 1877), Article “Mathematics.”</p> + + <p class="v2"> + <b><a name="Block_254" id="Block_254">254</a>.</b> + The opinion appears to be + gaining ground that this very general conception of + functionality, born on mathematical ground, is destined to + supersede the narrower notion of causation, traditional in + connection with the natural sciences. As an abstract + formulation of the idea of determination in its most + general sense, the notion of + functionality includes and transcends + +<span class="pagenum"> + <a name="Page_30" + id="Page_30">30</a></span> + + the more special + notion of causation as a one-sided determination of future + phenomena by means of present conditions; it can be used + to express the fact of the subsumption under a general law + of past, present, and future alike, in a sequence of + phenomena. From this point of view the remark of Huxley + that Mathematics “knows nothing of causation” could only + be taken to express the whole truth, if by the term + “causation” is understood “efficient causation.” The + latter notion has, however, in recent times been to an + increasing extent regarded as just as irrelevant in the + natural sciences as it is in Mathematics; the idea of + thorough-going determinancy, in accordance with formal + law, being thought to be alone significant in either + domain.—<span class="smcap">Hobson, E. W.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science (1910); Nature, Vol. 84, p. 290.</p> + + <p class="v2"> + <b><a name="Block_255" id="Block_255">255</a>.</b> + Most, if not all, of the great + ideas of modern mathematics have had their origin in + observation. Take, for instance, the arithmetical theory + of forms, of which the foundation was laid in the + diophantine theorems of Fermat, left without proof by + their author, which resisted all efforts of the + myriad-minded Euler to reduce to demonstration, and only + yielded up their cause of being when turned over in the + blow-pipe flame of Gauss’s transcendent genius; or the + doctrine of double periodicity, which resulted from the + observation of Jacobi of a purely analytical fact of + transformation; or Legendre’s law of reciprocity; or + Sturm’s theorem about the roots of equations, which, as he + informed me with his own lips, stared him in the face in + the midst of some mechanical investigations connected (if + my memory serves me right) with the motion of compound + pendulums; or Huyghen’s method of continued fractions, + characterized by Lagrange as one of the principal + discoveries of that great mathematician, and to which he + appears to have been led by the construction of his + Planetary Automaton; or the new algebra, speaking of which + one of my predecessors (Mr. Spottiswoode) has said, not + without just reason and authority, from this chair, “that + it reaches out and indissolubly connects itself each year + with fresh branches of mathematics, that the theory of + equations has become almost new through it, algebraic + +<span class="pagenum"> + <a name="Page_31" + id="Page_31">31</a></span> + + geometry transfigured in + its light, that the calculus of variations, molecular + physics, and mechanics” (he might, if speaking at the + present moment, go on to add the theory of elasticity and + the development of the integral calculus) “have all felt + its influence.”—<span class="smcap">Sylvester, J. + J.</span></p> + <p class="blockcite"> + A Plea for the Mathematician, Nature, Vol. 1, p. 238; + Collected Mathematical Papers, Vol. 2, pp. 655, 656.</p> + + <p class="v2"> + <b><a name="Block_256" id="Block_256">256</a>.</b> + The ability to imagine + relations is one of the most indispensable conditions of + all precise thinking. No subject can be named, in the + investigation of which it is not imperatively needed; but + it can be nowhere else so thoroughly acquired as in the + study of mathematics.—<span class="smcap">Fiske, + John.</span></p> + <p class="blockcite"> + Darwinism and other Essays (Boston, 1893), p. 296.</p> + + <p class="v2"> + <b><a name="Block_257" id="Block_257">257</a>.</b> + The great science + [mathematics] occupies itself at least just as much with + the power of imagination as with the power of logical + conclusion.—<span class="smcap">Herbart, F. J.</span></p> + <p class="blockcite"> + Pestalozzi’s Idee eines ABC der Anschauung. Werke [Kehrbach] + (Langensaltza, 1890), Bd. 1, p. 174.</p> + + <p class="v2"> + <b><a name="Block_258" id="Block_258">258</a>.</b> + The moving power of + mathematical invention is not reasoning but + imagination.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Quoted in Graves’ Life of Sir W. R. Hamilton, Vol. 3 (1889), + p. 219.</p> + + <p class="v2"> + <b><a name="Block_259" id="Block_259">259</a>.</b> + There is an astonishing + imagination, even in the science of mathematics.... We + repeat, there was far more imagination in the head of + Archimedes than in that of Homer.—<span class= + "smcap">Voltaire.</span></p> + <p class="blockcite"> + A Philosophical Dictionary (Boston, 1881), Vol. 3, p. 40. + Article “Imagination.”</p> + + <p class="v2"> + <b><a name="Block_260" id="Block_260">260</a>.</b> + As the prerogative of Natural + Science is to cultivate a taste for observation, so that + of Mathematics is, almost from the starting point, to + stimulate the faculty of invention.—<span class= + "smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + A Plea for the Mathematician, Nature, Vol. 1, p. 261; + Collected Mathematical Papers, Vol. 2 (Cambridge, 1908), p. + 717.</p> + +<p> + <span class="pagenum"> + <a name="Page_32" + id="Page_32">32</a></span></p> + + <p class="v2"> + <b><a name="Block_261" id="Block_261">261</a>.</b> + A marveilous newtrality have + these things mathematicall, and also a strange + participation between things supernaturall, immortall, + intellectuall, simple and indivisible, and things + naturall, mortall, sensible, componded and + divisible.—<span class="smcap">Dee, John.</span></p> + <p class="blockcite"> + Euclid (1570), Preface.</p> + + <p class="v2"> + <b><a name="Block_262" id="Block_262">262</a>.</b> + Mathematics stands forth as + that which unites, mediates between Man and Nature, inner + and outer world, thought and perception, as no other + subject does.—<span class="smcap">Froebel.</span></p> + <p class="blockcite"> + [Herford translation] (London, 1893), Vol. 1, p. 84.</p> + + <p class="v2"> + <b><a name="Block_263" id="Block_263">263</a>.</b> + The intrinsic character of + mathematical research and knowledge is based essentially + on three properties: first, on its conservative attitude + towards the old truths and discoveries of mathematics; + secondly, on its progressive mode of development, due to + the incessant acquisition of new knowledge on the basis of + the old; and thirdly, on its self-sufficiency and its + consequent absolute independence.—<span class= + "smcap">Schubert, H.</span></p> + <p class="blockcite"> + Mathematical Essays and Recreations (Chicago, 1898), p. 27.</p> + + <p class="v2"> + <b><a name="Block_264" id="Block_264">264</a>.</b> + Our science, in contrast with + others, is not founded on a single period of human + history, but has accompanied the development of culture + through all its stages. Mathematics is as much interwoven + with Greek culture as with the most modern problems in + Engineering. She not only lends a hand to the progressive + natural sciences but participates at the same time in the + abstract investigations of logicians and + philosophers.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Klein und Riecke: Ueber angewandte Mathematik und Physik + (1900), p. 228.</p> + + <p class="v2"> + <b><a name="Block_265" id="Block_265">265</a>.</b> + There is probably no other + science which presents such different appearances to one + who cultivates it and to one who does not, as mathematics. + To this person it is ancient, venerable, and complete; a + body of dry, irrefutable, unambiguous reasoning. To the + mathematician, on the other hand, his science is yet in + the purple bloom of vigorous youth, everywhere + +<span class="pagenum"> + <a name="Page_33" + id="Page_33">33</a></span> + + stretching out after the + “attainable but unattained” and full of the excitement of + nascent thoughts; its logic is beset with ambiguities, and + its analytic processes, like Bunyan’s road, have a + quagmire on one side and a deep ditch on the other and + branch off into innumerable by-paths that end in a + wilderness.—<span class="smcap">Chapman, C. H.</span></p> + <p class="blockcite"> + Bulletin American Mathematical Society, Vol. 2 (First + series), p. 61.</p> + + <p class="v2"> + <b><a name="Block_266" id="Block_266">266</a>.</b> + Mathematical science is in my + opinion an indivisible whole, an organism whose vitality + is conditioned upon the connection of its parts. For with + all the variety of mathematical knowledge, we are still + clearly conscious of the similarity of the logical + devices, the <em>relationship</em> of the <em>ideas</em> in + mathematics as a whole and the numerous analogies in its + different departments. We also notice that, the farther a + mathematical theory is developed, the more harmoniously + and uniformly does its construction proceed, and + unsuspected relations are disclosed between hitherto + separated branches of the science. So it happens that, + with the extension of mathematics, its organic character + is not lost but manifests itself the more + clearly.—<span class="smcap">Hilbert, D.</span></p> + <p class="blockcite"> + Mathematical Problems, Bulletin American Mathematical + Society, Vol. 8, p. 478.</p> + + <p class="v2"> + <b><a name="Block_267" id="Block_267">267</a>.</b> + The mathematics have always + been the implacable enemies of scientific + romances.—<span class="smcap">Arago.</span></p> + <p class="blockcite"> + Oeuvres (1855), t. 3, p. 498.</p> + + <p class="v2"> + <b><a name="Block_268" id="Block_268">268</a>.</b> + Those skilled in + mathematical analysis know that its object is not simply + to calculate numbers, but that it is also employed to + find the relations between magnitudes which cannot be + expressed in numbers and between functions whose law is + not capable of algebraic expression.—<span class= + "smcap">Cournot, Augustin</span>.</p> + <p class="blockcite"> + Mathematical Theory of the Principles of Wealth + [Bacon, N. T.], (New York, 1897), p. 3.</p> + + <p class="v2"> + <b><a name="Block_269" id="Block_269">269</a>.</b> + Coterminous with space and + coeval with time is the Kingdom of Mathematics; within + this range her dominion is supreme; otherwise than + according to her order nothing can exist; in + contradiction to her laws nothing takes place. On her + +<span class="pagenum"> + <a name="Page_34" + id="Page_34">34</a></span> + + mysterious scroll is to be + found written for those who can read it that which has + been, that which is, and that which is to come. + Everything material which is the subject of knowledge + has number, order, or position; and these are her + first outlines for a sketch of the universe. If our + feeble hands cannot follow out the details, still her + part has been drawn with an unerring pen, and her work + cannot be gainsaid. So wide is the range of + mathematical sciences, so indefinitely may it extend + beyond our actual powers of manipulation that at some + moments we are inclined to fall down with even more + than reverence before her majestic presence. But so + strictly limited are her promises and powers, about so + much that we might wish to know does she offer no + information whatever, that at other moments we are + fain to call her results but a vain thing, and to + reject them as a stone where we had asked for bread. + If one aspect of the subject encourages our hopes, so + does the other tend to chasten our desires, and he is + perhaps the wisest, and in the long run the happiest, + among his fellows, who has learned not only this + science, but also the larger lesson which it directly + teaches, namely, to temper our aspirations to that + which is possible, to moderate our desires to that + which is attainable, to restrict our hopes to that of + which accomplishment, if not immediately practicable, + is at least distinctly within the range of + conception.—<span class="smcap">Spottiswoode, W.</span></p> + <p class="blockcite"> + Quoted in Sonnenschein’s Encyclopedia of Education + (London, 1906), p. 208.</p> + + <p class="v2"> + <b><a name="Block_270" id="Block_270">270</a>.</b> + But it is precisely mathematics, and the pure + science generally, from which the general educated public and + independent students have been debarred, and into which they + have only rarely attained more than a very meagre insight. The + reason of this is twofold. In the first place, the ascendant + and consecutive character of mathematical knowledge renders its + results absolutely insusceptible of presentation to persons who + are unacquainted with what has gone before, and so necessitates + on the part of its devotees a thorough and patient exploration + of the field from the very beginning, as distinguished from + those sciences which may, so to speak, be begun at the end, and + which are consequently cultivated with the greatest zeal. The + second + +<span class="pagenum"> + <a name="Page_35" id= + "Page_35">35</a></span> + + reason is that, partly through the + exigencies of academic instruction, but mainly through the + martinet traditions of antiquity and the influence of + mediæval logic-mongers, the great bulk of the + elementary text-books of mathematics have unconsciously assumed + a very repellant form,—something similar to + what is termed in the theory of protective mimicry in biology + “the terrifying form.” And it is + mainly to this formidableness and touch-me-not character of + exterior, concealing withal a harmless body, that the undue + neglect of typical mathematical studies is to be + attributed.—<span class="smcap">McCormack, T. J.</span></p> + <p class="blockcite"> + Preface to De Morgan’s Elementary Illustrations of the + Differential and Integral Calculus (Chicago, 1899).</p> + + <p class="v2"> + <b><a name="Block_271" id="Block_271">271</a>.</b> + Mathematics in gross, it is plain, are a grievance + in natural philosophy, and with reason: for mathematical + proofs, like diamonds, are hard as well as clear, and will be + touched with nothing but strict reasoning. Mathematical proofs + are out of the reach of topical arguments; and are not to be + attacked by the equivocal use of words or declaration, that + make so great a part of other + discourses,—nay, even of + controversies.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + Second Reply to the Bishop of Worcester.</p> + + <p class="v2"> + <b><a name="Block_272" id="Block_272">272</a>.</b> + The belief that mathematics, because it is + abstract, because it is static and cold and gray, is detached + from life, is a mistaken belief. Mathematics, even in its + purest and most abstract estate, is not detached from life. It + is just the ideal handling of the problems of life, as + sculpture may idealize a human figure or as poetry or painting + may idealize a figure or a scene. Mathematics is precisely the + ideal handling of the problems of life, and the central ideas + of the science, the great concepts about which its stately + doctrines have been built up, are precisely the chief ideas + with which life must always deal and which, as it tumbles and + rolls about them through time and space, give it its interests + and problems, and its order and rationality. That such is the + case a few indications will suffice to show. The mathematical + concepts of constant and variable are represented familiarly in + life by the notions of fixedness and change. The concept of + equation or that of an equational + +<span class="pagenum"> + <a name="Page_36" + id="Page_36">36</a></span> + + system, imposing + restriction upon variability, is matched in life by the concept + of natural and spiritual law, giving order to what were else + chaotic change and providing partial freedom in lieu of none at + all. What is known in mathematics under the name of limit is + everywhere present in life in the guise of some ideal, some + excellence high-dwelling among the rocks, an + “ever flying perfect” as Emerson + calls it, unto which we may approximate nearer and nearer, but + which we can never quite attain, save in aspiration. The + supreme concept of functionality finds its correlate in life in + the all-pervasive sense of interdependence and mutual + determination among the elements of the world. What is known in + mathematics as transformation—that is, + lawful transfer of attention, serving to match in orderly + fashion the things of one system with those of + another—is conceived in life as a process of + transmutation by which, in the flux of the world, the content + of the present has come out of the past and in its turn, in + ceasing to be, gives birth to its successor, as the boy is + father to the man and as things, in general, become what they + are not. The mathematical concept of invariance and that of + infinitude, especially the imposing doctrines that explain + their meanings and bear their names—What are + they but mathematicizations of that which has ever been the + chief of life’s hopes and dreams, of that + which has ever been the object of its deepest passion and of + its dominant enterprise, I mean the finding of the worth that + abides, the finding of permanence in the midst of change, and + the discovery of a presence, in what has seemed to be a finite + world, of being that is infinite? It is needless further to + multiply examples of a correlation that is so abounding and + complete as indeed to suggest a doubt whether it be juster to + view mathematics as the abstract idealization of life than to + regard life as the concrete realization of + mathematics.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + The Humanization of the Teaching of Mathematics; Science, + New Series, Vol. 35, pp. 645-646.</p> + + <p class="v2"> + <b><a name="Block_273" id="Block_273">273</a>.</b> + Mathematics, like dialectics, is an organ of the + inner higher sense; in its execution it is an art like + eloquence. Both alike care nothing for the content, to both + nothing is of value but the form. It is immaterial to + mathematics whether it + +<span class="pagenum"> + <a name="Page_37" + id="Page_37">37</a></span> + + computes pennies or guineas, to + rhetoric whether it defends truth or + error.—<span class="smcap">Goethe.</span></p> + <p class="blockcite"> + Wilhelm Meisters Wanderjahre, Zweites Buch.</p> + + <p class="v2"> + <b><a name="Block_274" id="Block_274">274</a>.</b> + The genuine spirit of Mathesis is devout. No + intellectual pursuit more truly leads to profound impressions + of the existence and attributes of a Creator, and to a deep + sense of our filial relations to him, than the study of these + abstract sciences. Who can understand so well how feeble are + our conceptions of Almighty Power, as he who has calculated the + attraction of the sun and the planets, and weighed in his + balance the irresistible force of the lightning? Who can so + well understand how confused is our estimate of the Eternal + Wisdom, as he who has traced out the secret laws which guide + the hosts of heaven, and combine the atoms on earth? Who can so + well understand that man is made in the image of his Creator, + as he who has sought to frame new laws and conditions to govern + imaginary worlds, and found his own thoughts similar to those + on which his Creator has acted?—<span class= + "smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + The Imagination in Mathematics; North American Review, + Vol. 85, p. 226.</p> + + <p class="v2"> + <b><a name="Block_275" id="Block_275">275</a>.</b> + ... what is physical is subject to the laws of + mathematics, and what is spiritual to the laws of God, and the + laws of mathematics are but the expression of the thoughts of + God.—<span class="smcap">Hill, + Thomas.</span></p> + <p class="blockcite"> + The Uses of Mathesis; Bibliotheca Sacra, Vol. 32, p. 523.</p> + + <p class="v2"> + <b><a name="Block_276" id="Block_276">276</a>.</b> + It is in the inner world of pure thought, where all + <em>entia</em> dwell, where is every type of order and manner of + correlation and variety of relationship, it is in this infinite + ensemble of eternal verities whence, if there be one cosmos or + many of them, each derives its character and mode of + being,—it is there that the spirit of + mathesis has its home and its life.</p> + <p class="v1"> + Is it a restricted home, a narrow life, static and cold and + grey with logic, without artistic interest, devoid of emotion + and mood and sentiment? That world, it is true, is not a world + of <em>solar</em> light, not clad in the colours that liven and + glorify the things of sense, but it is an illuminated + world, and over it all and everywhere + +<span class="pagenum"> + <a name="Page_38" id= + "Page_38">38</a></span> + + throughout are hues and + tints transcending <em>sense</em>, painted there by radiant + pencils of <em>psychic</em> light, the light in which it lies. It + is a silent world, and, nevertheless, in respect to the highest + principle of art—the interpenetration of + content and form, the perfect fusion of mode and + meaning—it even surpasses music. In a sense, + it is a static world, but so, too, are the worlds of the + sculptor and the architect. The figures, however, which reason + constructs and the mathematic vision beholds, transcend the + temple and the statue, alike in simplicity and in intricacy, in + delicacy and in grace, in symmetry and in poise. Not only are + this home and this life thus rich in æsthetic + interests, really controlled and sustained by motives of a + sublimed and supersensuous art, but the religious aspiration, + too, finds there, especially in the beautiful doctrine of + invariants, the most perfect symbols of what it + seeks—the changeless in the midst of change, + abiding things in a world of flux, configurations that remain + the same despite the swirl and stress of countless hosts of + curious transformations. The domain of mathematics is the sole + domain of certainty. There and there alone prevail the + standards by which every hypothesis respecting the external + universe and all observation and all experiment must be finally + judged. It is the realm to which all speculation and all + thought must repair for chastening and + sanitation—the court of last resort, I say + it reverently, for all intellection whatsoever, whether of + demon or man or deity. It is there that mind as mind attains + its highest estate, and the condition of knowledge there is the + ultimate object, the tantalising goal of the aspiration, the + <em>Anders-Streben</em>, of all other knowledge of every + kind.—<span class="smcap">Keyser, C. + J.</span></p> + <p class="blockcite"> + The Universe and Beyond; Hibbert Journal, Vol. 3 (1904-1905), + pp. 313-314.</p> + +<p> + <span class="pagenum"> + <a name="Page_39" + id="Page_39">39</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_III"> + CHAPTER III<br /> + <span class="large"> + ESTIMATES OF MATHEMATICS</span></h2> + + <p class="v2"> + <b><a name="Block_301" id="Block_301">301</a>.</b> + The world of ideas which it [mathematics] discloses + or illuminates, the contemplation of divine beauty and order + which it induces, the harmonious connection of its parts, the + infinite hierarchy and absolute evidence of the truths with + which mathematical science is concerned, these, and such like, + are the surest grounds of its title of human regard, and would + remain unimpaired were the plan of the universe unrolled like a + map at our feet, and the mind of man qualified to take in the + whole scheme of creation at a + glance.—<span class="smcap">Sylvester, J. + J.</span></p> + <p class="blockcite"> + A Plea for the Mathematician, Nature, 1, p. 262; Collected + Mathematical Papers (Cambridge, 1908), 2, p. 659.</p> + + <p class="v2"> + <b><a name="Block_302" id="Block_302">302</a>.</b> + It may well be doubted whether, in all the range of + Science, there is any field so fascinating to the + explorer—so rich in hidden + treasures—so fruitful in delightful + surprises—as that of Pure Mathematics. The + charm lies chiefly ... in the absolute <em>certainty</em> of its + results: for that is what, beyond all mental treasures, the + human intellect craves for. Let us only be sure of + <em>something</em>! More light, more light! + Ἐν δὲ φάει καὶ ὀλέεσσον + “And if our fate be death, give light and + let us die!” This is the cry that, through all the + ages, is going up from perplexed Humanity, and Science has + little else to offer, that will really meet the demands of its + votaries, than the conclusions of Pure + Mathematics.—<span class="smcap">Dodgson, C. L.</span></p> + <p class="blockcite"> + A New Theory of Parallels (London, 1895), + Introduction.</p> + + <p class="v2"> + <b><a name="Block_303" id="Block_303">303</a>.</b> + In every case the awakening touch has been the + mathematical spirit, the attempt to count, to measure, or to + calculate. What to the poet or the seer may appear to be the + very death of all his poetry and all his + visions—the cold touch of the calculating + +<span class="pagenum"> + <a name="Page_40" + id="Page_40">40</a></span> + + mind,—this has proved to be the spell by + which knowledge has been born, by which new sciences have been + created, and hundreds of definite problems put before the minds + and into the hands of diligent students. It is the geometrical + figure, the dry algebraical formula, which transforms the vague + reasoning of the philosopher into a tangible and manageable + conception; which represents, though it does not fully + describe, which corresponds to, though it does not explain, the + things and processes of nature: this clothes the fruitful, but + otherwise indefinite, ideas in such a form that the strict + logical methods of thought can be applied, that the human mind + can in its inner chamber evolve a train of reasoning the result + of which corresponds to the phenomena of the outer + world.—<span class="smcap">Merz, J. T.</span></p> + <p class="blockcite"> + A History of European Thought in the Nineteenth Century + (Edinburgh and London, 1904), Vol. 1, p. 314.</p> + + <p class="v2"> + <b><a name="Block_304" id="Block_304">304</a>.</b> + Mathematics ... the ideal and norm of all careful + thinking.—<span class="smcap">Hall, G. Stanley.</span></p> + <p class="blockcite"> + Educational Problems (New York, 1911), p. 393.</p> + + <p class="v2"> + <b><a name="Block_305" id="Block_305">305</a>.</b> + Mathematics is the only true + metaphysics.—<span class="smcap">Thomson, W. + (Lord Kelvin).</span></p> + <p class="blockcite"> + Thompson, S. P.: Life of Lord Kelvin (London, 1910), p. 10.</p> + + <p class="v2"> + <b><a name="Block_306" id="Block_306">306</a>.</b> + He who knows not mathematics and the results of + recent scientific investigation dies without knowing + <em>truth</em>.—<span class= + "smcap">Schellbach, C. H.</span></p> + <p class="blockcite"> + Quoted in Young’s Teaching of Mathematics (London, 1907), + p. 44.</p> + + <p class="v2"> + <b><a name="Block_307" id="Block_307">307</a>.</b> + The reasoning of mathematics is a type of perfect + reasoning.—<span class="smcap">Barnett, P. A.</span></p> + <p class="blockcite"> + Common Sense in Education and Teaching (New York, 1905), + p. 222.</p> + + <p class="v2"> + <b><a name="Block_308" id="Block_308">308</a>.</b> + Mathematics, once fairly established on the + foundation of a few axioms and definitions, as upon a rock, has + grown from age to age, so as to become the most solid fabric + that human reason can boast.—<span class= + "smcap">Reid, Thomas.</span></p> + <p class="blockcite"> + Essays on the Intellectual Powers of Man, 4th. Ed., p. 461.</p> + +<p><span class="pagenum"> + <a name="Page_41" + id="Page_41">41</a></span></p> + + <p class="v2"> + <b><a name="Block_309" id="Block_309">309</a>.</b> + The analytical geometry of Descartes and the + calculus of Newton and Leibniz have expanded into the marvelous + mathematical method—more daring than + anything that the history of philosophy + records—of Lobachevsky and Riemann, Gauss + and Sylvester. Indeed, mathematics, the indispensable tool of + the sciences, defying the senses to follow its splendid + flights, is demonstrating today, as it never has been + demonstrated before, the supremacy of the pure + reason.—<span class="smcap">Butler, Nicholas Murray.</span></p> + <p class="blockcite"> + The Meaning of Education and other Essays and Addresses + (New York, 1905), p. 45.</p> + + <p class="v2"> + <b><a name="Block_310" id="Block_310">310</a>.</b> + Mathematics is the gate and key of the sciences.... + Neglect of mathematics works injury to all knowledge, since he + who is ignorant of it cannot know the other sciences or the + things of this world. And what is worse, men who are thus + ignorant are unable to perceive their own ignorance and so do + not seek a remedy.—<span class="smcap">Bacon, Roger.</span></p> + <p class="blockcite"> + Opus Majus, Part 4, Distinctia Prima, cap. 1.</p> + + <p class="v2"> + <b><a name="Block_311" id="Block_311">311</a>.</b> + Just as it will never be successfully challenged + that the French language, progressively developing and growing + more perfect day by day, has the better claim to serve as a + developed court and world language, so no one will venture to + estimate lightly the debt which the world owes to + mathematicians, in that they treat in their own language + matters of the utmost importance, and govern, determine and + decide whatever is subject, using the word in the highest + sense, to number and + measurement.—<span class="smcap">Goethe.</span></p> + <p class="blockcite"> + Sprüche in Prosa, Natur, III, 868.</p> + + <p class="v2"> + <b><a name="Block_312" id="Block_312">312</a>.</b> + Do not imagine that mathematics is hard and + crabbed, and repulsive to common sense. It is merely the + etherealization of common + sense.—<span class="smcap">Thomson, W. (Lord Kelvin).</span></p> + <p class="blockcite"> + Thompson, S. P.: Life of Lord Kelvin (London, 1910), p. 1139.</p> + +<p><span class="pagenum"> + <a name="Page_42" + id="Page_42">42</a></span></p> + + <p class="v2"> + <b><a name="Block_313" id="Block_313">313</a>.</b> + The advancement and perfection of mathematics are + intimately connected with the prosperity of the + State.—<span class="smcap">Napoleon I.</span></p> + <p class="blockcite"> + Correspondance de Napoléon, t. 24 (1868), p. 112.</p> + + <p class="v2"> + <b><a name="Block_314" id="Block_314">314</a>.</b> + The love of mathematics is daily on the increase, + not only with us but in the army. The result of this was + unmistakably apparent in our last campaigns. Bonaparte himself + has a mathematical head, and though all who study this science + may not become geometricians like Laplace or Lagrange, or + heroes like Bonaparte, there is yet left an influence upon the + mind which enables them to accomplish more than they could + possibly have achieved without this + training.—<span class="smcap">Lalande.</span></p> + <p class="blockcite"> + Quoted in Bruhns’ Alexander von Humboldt (1872), Bd. 1, + p. 232.</p> + + <p class="v2"> + <b><a name="Block_315" id="Block_315">315</a>.</b> + In Pure Mathematics, where all the various truths + are necessarily connected with each other, (being all + necessarily connected with those hypotheses which are the + principles of the science), an arrangement is beautiful in + proportion as the principles are few; and what we admire + perhaps chiefly in the science, is the astonishing variety of + consequences which may be demonstrably deduced from so small a + number of + premises.—<span class="smcap">Stewart, Dugald.</span></p> + <p class="blockcite"> + Philosophy of the Human Mind, Part 3, chap. 1, sect. 3; + Collected Works [Hamilton] (Edinburgh, 1854), Vol. 4.</p> + + <p class="v2"> + <b><a name="Block_316" id="Block_316">316</a>.</b> + It is curious to observe how differently these + great men [Plato and Bacon] estimated the value of every kind + of knowledge. Take Arithmetic for example. Plato, after + speaking slightly of the convenience of being able to reckon + and compute in the ordinary transactions of life, passes to + what he considers as a far more important advantage. The study + of the properties of numbers, he tells us, habituates the mind + to the contemplation of pure truth, and raises us above the + material universe. He would have his disciples apply themselves + to this study, not that they may be able to buy or sell, not + that they may qualify themselves to be + shop-keepers or travelling merchants, + +<span class="pagenum"> + <a name="Page_43" + id="Page_43">43</a></span> + + but that they may learn + to withdraw their minds from the ever-shifting spectacle of + this visible and tangible world, and to fix them on the + immutable essences of things.</p> + <p class="v1"> + Bacon, on the other hand, valued this branch of knowledge only + on account of its uses with reference to that visible and + tangible world which Plato so much despised. He speaks with + scorn of the mystical arithmetic of the later Platonists, and + laments the propensity of mankind to employ, on mere matters of + curiosity, powers the whole exertion of which is required for + purposes of solid advantage. He advises arithmeticians to leave + these trifles, and employ themselves in framing convenient + expressions which may be of use in physical + researches.—<span class="smcap">Macaulay.</span></p> + <p class="blockcite"> + Lord Bacon: Edinburgh Review, July, 1837. Critical and + Miscellaneous Essays (New York, 1879), Vol. 1, p. 397.</p> + + <p class="v2"> + <b><a name="Block_317" id="Block_317">317</a>.</b> + <em>Ath.</em> There still remain three studies + suitable for freemen. Calculation in arithmetic is one of them; + the measurement of length, surface, and depth is the second; + and the third has to do with the revolutions of the stars in + reference to one another ... there is in them something that is + necessary and cannot be set aside, ... if I am not mistaken, + [something of] divine necessity; for as to the human + necessities of which men often speak when they talk in this + manner, nothing can be more ridiculous than such an application + of the words.</p> + <p class="v1"> + <em>Cle.</em> And what necessities of knowledge are there, + Stranger, which are divine and not human?</p> + <p class="v1"> + <em>Ath.</em> I conceive them to be those of which he who has no + use nor any knowledge at all cannot be a god, or demi-god, or + hero to mankind, or able to take any serious thought or charge + of them.—<span class="smcap">Plato.</span></p> + <p class="blockcite"> + Republic, Bk. 7. Jowett’s Dialogues of Plato (New York, 1897), + Vol. 4, p. 334.</p> + + <p class="v2"> + <b><a name="Block_318" id="Block_318">318</a>.</b> + Those who assert that the mathematical sciences + make no affirmation about what is fair or good make a false + assertion; for they do speak of these and frame demonstrations + of them in the most eminent sense of the word. For if they do + not actually employ these names, they do not exhibit even the + results and + +<span class="pagenum"> + <a name="Page_44" + id="Page_44">44</a></span> + + the reasons of these, and therefore can + be hardly said to make any assertion about them. Of what is + fair, however, the most important species are order and + symmetry, and that which is definite, which the mathematical + sciences make manifest in a most eminent degree. And since, at + least, these appear to be the causes of many + things—now, I mean, for example, order, and + that which is a definite thing, it is evident that they would + assert, also, the existence of a cause of this description, and + its subsistence after the same manner as that which is fair + subsists in.—<span class="smcap">Aristotle.</span></p> + <p class="blockcite"> + Metaphysics [MacMahon] Bk. 12, chap. 3.</p> + + <p class="v2"> + <b><a name="Block_319" id="Block_319">319</a>.</b> + Many arts there are which beautify the mind of man; + of all other none do more garnish and beautify it than those + arts which are called + mathematical.—<span class="smcap">Billingsley, H.</span></p> + <p class="blockcite"> + The Elements of Geometrie of the most ancient Philosopher + Euclide of Megara (London, 1570), Note to the Reader.</p> + + <p class="v2"> + <b><a name="Block_320" id="Block_320">320</a>.</b> + As the sun eclipses the stars by his brilliancy, so + the man of knowledge will eclipse the fame of others in + assemblies of the people if he proposes algebraic problems, and + still more if he solves them.—<span class= + "smcap">Brahmagupta.</span></p> + <p class="blockcite"> + Quoted in Cajori’s History of Mathematics (New York, 1897), + p. 92.</p> + + <p class="v2"> + <b><a name="Block_321" id="Block_321">321</a>.</b> + So highly did the ancients esteem the power of + figures and numbers, that Democritus ascribed to the figures of + atoms the first principles of the variety of things; and + Pythagoras asserted that the nature of things consisted of + numbers.—<span class="smcap">Bacon, Lord.</span></p> + <p class="blockcite"> + De Augmentis, Bk. 3; Advancement of Learning, Bk. 2.</p> + + <p class="v2"> + <b><a name="Block_322" id="Block_322">322</a>.</b> + There has not been any science so much esteemed and + honored as this of mathematics, nor with so much industry and + vigilance become the care of great men, and labored in by the + potentates of the world, viz. emperors, kings, princes, + etc.—<span class="smcap">Franklin, Benjamin.</span></p> + <p class="blockcite"> + On the Usefulness of Mathematics, Works (Boston, 1840), + Vol. 2, p. 28.</p> + +<p><span class="pagenum"> + <a name="Page_45" + id="Page_45">45</a></span></p> + + <p class="v2"> + <b><a name="Block_323" id="Block_323">323</a>.</b> + Whatever may have been imputed to some other + studies under the notion of insignificancy and loss of time, + yet these [mathematics], I believe, never caused repentance in + any, except it was for their remissness in the prosecution of + them.—<span class="smcap">Franklin, Benjamin.</span></p> + <p class="blockcite"> + On the Usefulness of Mathematics, Works (Boston, 1840), + Vol. 2, p. 69.</p> + + <p class="v2"> + <b><a name="Block_324" id="Block_324">324</a>.</b> + What science can there be more noble, more + excellent, more useful for men, more admirably high and + demonstrative, than this of the + mathematics?—<span class="smcap">Franklin, Benjamin.</span></p> + <p class="blockcite"> + On the Usefulness of Mathematics, Works (Boston, 1840), + Vol. 2, p. 69.</p> + + <p class="v2"> + <b><a name="Block_325" id="Block_325">325</a>.</b> + The great truths with which it [mathematics] deals, + are clothed with austere grandeur, far above all purposes of + immediate convenience or profit. It is in them that our limited + understandings approach nearest to the conception of that + absolute and infinite, towards which in most other things they + aspire in vain. In the pure mathematics we contemplate absolute + truths, which existed in the divine mind before the morning + stars sang together, and which will continue to exist there, + when the last of their radiant host shall have fallen from + heaven. They existed not merely in metaphysical possibility, + but in the actual contemplation of the supreme reason. The pen + of inspiration, ranging all nature and life for imagery to set + forth the Creator’s power and wisdom, finds + them best symbolized in the skill of the surveyor. + “He meted out heaven as with a + span;” and an ancient sage, neither falsely nor + irreverently, ventured to say, that “God is + a geometer.”—<span class="smcap">Everett, Edward.</span></p> + <p class="blockcite"> + Orations and Speeches (Boston, 1870), Vol. 3, p. 514.</p> + + <p class="v2"> + <b><a name="Block_326" id="Block_326">326</a>.</b> + There is no science which teaches the harmonies of + nature more clearly than + mathematics,....—<span class="smcap">Carus, Paul.</span></p> + <p class="blockcite"> + Andrews: Magic Squares and Cubes (Chicago, 1908), + Introduction.</p> + + <p class="v2"> + <b><a name="Block_327" id="Block_327">327</a>.</b> + For it being the nature of the mind of man (to the + extreme prejudice of knowledge) to delight in the + spacious + +<span class="pagenum"> + <a name="Page_46" + id="Page_46">46</a></span> + + liberty of generalities, as in a + champion region, and not in the enclosures of particularity; + the Mathematics were the goodliest fields to satisfy that + appetite.—<span class="smcap">Bacon, Lord.</span></p> + <p class="blockcite"> + De Augmentis, Bk. 3; Advancement of Learning, Bk. 2.</p> + + <p class="v2"> + <b><a name="Block_328" id="Block_328">328</a>.</b> + I would have my son mind and understand business, + read little history, study the mathematics and cosmography; + these are good, with subordination to the things of God.... + These fit for public services for which man is + born.—<span class="smcap">Cromwell, Oliver.</span></p> + <p class="blockcite"> + Letters and Speeches of Oliver Cromwell (New York, 1899), + Vol. 1, p. 371.</p> + + <p class="v2"> + <b><a name="Block_329" id="Block_329">329</a>.</b> + Mathematics is the life supreme. The life of the + gods is mathematics. All divine messengers are mathematicians. + Pure mathematics is religion. Its attainment requires a + theophany.—<span class="smcap">Novalis.</span></p> + <p class="blockcite"> + Schriften (Berlin, 1901), Bd. 2, p. 223.</p> + + <p class="v2"> + <b><a name="Block_330" id="Block_330">330</a>.</b> + The Mathematics which effectually exercises, not + vainly deludes or vexatiously torments studious Minds with + obscure Subtilties, perplexed Difficulties, or contentious + Disquisitions; which overcomes without Opposition, triumphs + without Pomp, compels without Force, and rules absolutely + without Loss of Liberty; which does not privately overreach a + weak Faith, but openly assaults an armed Reason, obtains a + total Victory, and puts on inevitable Chains; whose Words are + so many Oracles, and Works as many Miracles; which blabs out + nothing rashly, nor designs anything from the Purpose, but + plainly demonstrates and readily performs all Things within its + Verge; which obtrudes no false Shadow of Science, but the very + Science itself, the Mind firmly adheres to it, as soon as + possessed of it, and can never after desert it of its own + Accord, or be deprived of it by any Force of others: Lastly the + Mathematics, which depend upon Principles clear to the Mind, + and agreeable to Experience; which draws certain Conclusions, + instructs by profitable Rules, unfolds pleasant Questions; and + produces wonderful Effects; which is the fruitful Parent of, I + had almost said all, Arts, the + +<span class="pagenum"> + <a name="Page_47" + id="Page_47">47</a></span> + + unshaken Foundation of + Sciences, and the plentiful Fountain of Advantage to human + Affairs.—<span class="smcap">Barrow, Isaac.</span></p> + <p class="blockcite"> + Oration before the University of Cambridge on being + elected Lucasian Professor of Mathematics, Mathematical + Lectures (London, 1734), p. 28.</p> + + <p class="v2"> + <b><a name="Block_331" id="Block_331">331</a>.</b> + Doubtless the reasoning faculty, the mind, is the + leading and characteristic attribute of the human race. By the + exercise of this, man arrives at the properties of the natural + bodies. This is science, properly and emphatically so called. + It is the science of pure mathematics; and in the high branches + of this science lies the truly sublime of human acquisition. If + any attainment deserves that epithet, it is the knowledge, + which, from the mensuration of the minutest dust of the + balance, proceeds on the rising scale of material bodies, + everywhere weighing, everywhere measuring, everywhere detecting + and explaining the laws of force and motion, penetrating into + the secret principles which hold the universe of God together, + and balancing worlds against worlds, and system against system. + When we seek to accompany those who pursue studies at once so + high, so vast, and so exact; when we arrive at the discoveries + of Newton, which pour in day on the works of God, as if a + second <em>fiat</em> had gone forth from his own mouth; when, + further, we attempt to follow those who set out where Newton + paused, making his goal their starting-place, and, proceeding + with demonstration upon demonstration, and discovery upon + discovery, bring new worlds and new systems of worlds within + the limits of the known universe, failing to learn all only + because all is infinite; however we may say of man, in + admiration of his physical structure, that “in form and moving + he is express and admirable,” it is here, and here without + irreverence, we may exclaim, “In apprehension how like a god!” + The study of the pure mathematics will of course not be + extensively pursued in an institution, which, like this + [Boston Mechanics’ Institute], has a direct practical tendency + and aim. But it is still to be remembered, that pure + mathematics lie at the foundation of mechanical philosophy, + and that it is ignorance only which can speak or think of + that sublime science as useless research or barren + speculation.—<span class="smcap">Webster, Daniel.</span></p> + <p class="blockcite"> + Works (Boston, 1872), Vol. 1, p. 180.</p> + +<p><span class="pagenum"> + <a name="Page_48" + id="Page_48">48</a></span></p> + + <p class="v2"> + <b><a name="Block_332" id="Block_332">332</a>.</b> + The school of Plato has advanced the interests of + the race as much through geometry as through philosophy. The + modern engineer, the navigator, the astronomer, built on the + truths which those early Greeks discovered in their purely + speculative investigations. And if the poetry, statesmanship, + oratory, and philosophy of our day owe much to + Plato’s divine Dialogues, our commerce, our + manufactures, and our science are equally indebted to his Conic + Sections. Later instances may be abundantly quoted, to show + that the labors of the mathematician have outlasted those of + the statesman, and wrought mightier changes in the condition of + the world. Not that we would rank the geometer above the + patriot, but we claim that he is worthy of equal + honor.—<span class="smcap">Hill,Thomas.</span></p> + <p class="blockcite"> + Imagination in Mathematics; North American Review, Vol. 85, + p. 228.</p> + + <p class="v2"> + <b><a name="Block_333" id="Block_333">333</a>.</b> + The discoveries of Newton have done more for + England and for the race, than has been done by whole dynasties + of British monarchs; and we doubt not that in the great + mathematical birth of 1853, the Quaternions of Hamilton, there + is as much real promise of benefit to mankind as in any event of + Victoria’s reign.—<span class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + Imagination in Mathematics; North American Review, Vol. 85, + p. 228.</p> + + <p class="v2"> + <b><a name="Block_334" id="Block_334">334</a>.</b> + Geometrical and Mechanical phenomena are the most + general, the most simple, the most abstract of + all,—the most irreducible to others. It + follows that the study of them is an indispensable preliminary + to that of all others. Therefore must Mathematics hold the + first place in the hierarchy of the sciences, and be the point + of departure of all Education, whether general or + special.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Introduction, chap. 2.</p> + +<p> + <span class="pagenum"> + <a name="Page_49" + id="Page_49">49</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_IV"> + CHAPTER IV<br /> + <span class="large"> + THE VALUE OF MATHEMATICS</span></h2> + + <p class="v2"> + <b><a name="Block_401" id="Block_401">401</a>.</b> + Mathematics because of its nature and structure is + peculiarly fitted for high school instruction + [Gymnasiallehrfach]. Especially the higher mathematics, even if + presented only in its elements, combines within itself all + those qualities which are demanded of a secondary subject. It + engages, it fructifies, it quickens, compels attention, is as + circumspect as inventive, induces courage and self-confidence + as well as modesty and submission to truth. It yields the + essence and kernel of all things, is brief in form and + overflows with its wealth of content. It discloses the depth + and breadth of the law and spiritual element behind the surface + of phenomena; it impels from point to point and carries within + itself the incentive toward progress; it stimulates the + artistic perception, good taste in judgment and execution, as + well as the scientific comprehension of things. Mathematics, + therefore, above all other subjects, makes the student lust + after knowledge, fills him, as it were, with a longing to + fathom the cause of things and to employ his own powers + independently; it collects his mental forces and concentrates + them on a single point and thus awakens the spirit of + individual inquiry, self-confidence and the joy of doing; it + fascinates because of the view-points which it offers and + creates certainty and assurance, owing to the universal + validity of its methods. Thus, both what he receives and what + he himself contributes toward the proper conception and + solution of a problem, combine to mature the student and to + make him skillful, to lead him away from the surface of things + and to exercise him in the perception of their essence. A + student thus prepared thirsts after knowledge and is ready for + the university and its sciences. Thus it appears, that higher + mathematics is the best guide to philosophy and to the + philosophic conception of the world (considered as a + self-contained whole) and of one’s own + being.—<span class="smcap">Dillmann, E.</span></p> + <p class="blockcite"> + Die Mathematik die Fackelträgerin einer + neuen Zeit (Stuttgart, 1889), p. 40.</p> + +<p><span class="pagenum"> + <a name="Page_50" + id="Page_50">50</a></span></p> + + <p class="v2"> + <b><a name="Block_402" id="Block_402">402</a>.</b> + These Disciplines [mathematics] serve to inure and + corroborate the Mind to a constant Diligence in Study; to + undergo the Trouble of an attentive Meditation, and cheerfully + contend with such Difficulties as lie in the Way. They wholly + deliver us from a credulous Simplicity, most strongly fortify + us against the Vanity of Scepticism, effectually restrain from + a rash Presumption, most easily incline us to a due Assent, + perfectly subject us to the Government of right Reason, and + inspire us with Resolution to wrestle against the unjust + Tyranny of false Prejudices. If the Fancy be unstable and + fluctuating, it is to be poised by this Ballast, and steadied + by this Anchor, if the Wit be blunt it is sharpened upon this + Whetstone; if luxuriant it is pared by this Knife; if + headstrong it is restrained by this Bridle; and if dull it is + roused by this Spur. The Steps are guided by no Lamp more + clearly through the dark Mazes of Nature, by no Thread more + surely through the intricate Labyrinths of Philosophy, nor + lastly is the Bottom of Truth sounded more happily by any other + Line. I will not mention how plentiful a Stock of Knowledge the + Mind is furnished from these, with what wholesome Food it is + nourished, and what sincere Pleasure it enjoys. But if I speak + farther, I shall neither be the only Person, nor the first, who + affirms it; that while the Mind is abstracted and elevated from + sensible Matter, distinctly views pure Forms, conceives the + Beauty of Ideas, and investigates the Harmony of Proportions; + the Manners themselves are sensibly corrected and improved, the + Affections composed and rectified, the Fancy calmed and + settled, and the Understanding raised and excited to more + divine Contemplation. All which I might defend by Authority, + and confirm by the Suffrages of the greatest + Philosophers.—<span class="smcap">Barrow, Isaac.</span></p> + <p class="blockcite"> + Prefatory Oration: Mathematical Lectures (London, 1734), + p. 31.</p> + + <p class="v2"> + <b><a name="Block_403" id="Block_403">403</a>.</b> + No school subject so readily furnishes tasks whose + purpose can be made so clear, so immediate and so appealing to + the sober second-thought of the immature learner as the right + sort of elementary school + mathematics.—<span class="smcap">Myers, George.</span></p> + <p class="blockcite"> + Arithmetic in Public School Education (Chicago, 1911), p. 8.</p> + +<p><span class="pagenum"> + <a name="Page_51" + id="Page_51">51</a></span></p> + + <p class="v2"> + <b><a name="Block_404" id="Block_404">404</a>.</b> + Mathematics is a type of thought which seems + ingrained in the human mind, which manifests itself to some + extent with even the primitive races, and which is developed to + a high degree with the growth of civilization.... A type of + thought, a body of results, so essentially characteristic of + the human mind, so little influenced by environment, so + uniformly present in every civilization, is one of which no + well-informed mind today can be + ignorant.—<span class="smcap">Young, J. W. A.</span></p> + <p class="blockcite"> + The Teaching of Mathematics (London, 1907), p. 14.</p> + + <p class="v2"> + <b><a name="Block_405" id="Block_405">405</a>.</b> + Probably among all the pursuits of the University, + mathematics pre-eminently demand self-denial, patience, and + perseverance from youth, precisely at that period when they + have liberty to act for themselves, and when on account of + obvious temptations, habits of restraint and application are + peculiarly valuable.—<span class= + "smcap">Todhunter, Isaac.</span></p> + <p class="blockcite"> + The Conflict of Studies and other Essays (London, 1873), + p. 12.</p> + + <p class="v2"> + <b><a name="Block_406" id="Block_406">406</a>.</b> + Mathematics renders its best service through the + immediate furthering of rigorous thought and the spirit of + invention.—<span class="smcap">Herbart J. F.</span></p> + <p class="blockcite"> + Mathematischer Lehrplan für Realschulen: + Werke [Kehrbach] (Langensalza, 1890), Bd. 5, p. 170.</p> + + <p class="v2"> + <b><a name="Block_407" id="Block_407">407</a>.</b> + It seems to me that the older subjects, classics + and mathematics, are strongly to be recommended on the ground + of the accuracy with which we can compare the relative + performance of the students. In fact the definiteness of these + subjects is obvious, and is commonly admitted. There is however + another advantage, which I think belongs in general to these + subjects, that the examinations can be brought to bear on what + is really most valuable in these + subjects.—<span class="smcap">Todhunter, Isaac.</span></p> + <p class="blockcite"> + Conflict of Studies and other Essays (London, 1873), + pp. 6, 7.</p> + + <p class="v2"> + <b><a name="Block_408" id="Block_408">408</a>.</b> + It is better to teach the child arithmetic and Latin grammar + than rhetoric and moral philosophy, because they require + +<span class="pagenum"> + <a name="Page_52" + id="Page_52">52</a></span> + + exactitude of performance it + is made certain that the lesson is mastered, and that power of + performance is worth more than + knowledge.—<span class="smcap">Emerson, R. W.</span></p> + <p class="blockcite"> + Lecture on Education.</p> + + <p class="v2"> + <b><a name="Block_409" id="Block_409">409</a>.</b> + Besides accustoming the student to demand complete + proof, and to know when he has not obtained it, mathematical + studies are of immense benefit to his education by habituating + him to precision. It is one of the peculiar excellencies of + mathematical discipline, that the mathematician is never + satisfied with <i lang="fr" xml:lang="fr">à peu près.</i> He + requires the exact truth. Hardly any of the non-mathematical + sciences, except chemistry, has this advantage. One of the + commonest modes of loose thought, and sources of error both in + opinion and in practice, is to overlook the importance of + quantities. Mathematicians and chemists are taught by the whole + course of their studies, that the most fundamental difference + of quality depends on some very slight difference in + proportional quantity; and that from the qualities of the + influencing elements, without careful attention to their + quantities, false expectation would constantly be formed as to + the very nature and essential character of the result + produced.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + An Examination of Sir William Hamilton’s Philosophy + (London, 1878), p. 611.</p> + + <p class="v2"> + <b><a name="Block_410" id="Block_410">410</a>.</b> + In mathematics I can report no deficience, except + it be that men do not sufficiently understand the excellent use + of the Pure Mathematics, in that they do remedy and cure many + defects in the wit and faculties intellectual. For if the wit + be too dull, they sharpen it; if too wandering, they fix it; if + too inherent in the senses, they abstract it. So that as tennis + is a game of no use in itself, but of great use in respect it + maketh a quick eye and a body ready to put itself into all + positions; so in the Mathematics, that use which is collateral + and intervenient is no less worthy than that which is principal + and intended.—<span class="smcap">Bacon, Lord.</span></p> + <p class="blockcite"> + De Augmentis, Bk. 3; Advancement of Learning, Bk. 2.</p> + +<p><span class="pagenum"> + <a name="Page_53" + id="Page_53">53</a></span></p> + + <p class="v2"> + <b><a name="Block_411" id="Block_411">411</a>.</b> + If a man’s wit be wandering, let him study mathematics; for in + demonstrations, if his wit be called away never so little, he + must begin again.—<span class="smcap">Bacon, Lord.</span></p> + <p class="blockcite"> + Essays: On Studies.</p> + + <p class="v2"> + <b><a name="Block_412" id="Block_412">412</a>.</b> + If one be bird-witted, that is easily distracted + and unable to keep his attention as long as he should, + mathematics provides a remedy; for in them if the mind be + caught away but a moment, the demonstration has to be commenced + anew.—<span class="smcap">Bacon, Lord.</span></p> + <p class="blockcite"> + De Augmentis, Bk. 6; Advancement of Learning, Bk. 2.</p> + + <p class="v2"> + <b><a name="Block_413" id="Block_413">413</a>.</b> + The metaphysical philosopher from his point of view + recognizes mathematics as an instrument of education, which + strengthens the power of attention, develops the sense of order + and the faculty of construction, and enables the mind to grasp + under the simple formulae the quantitative differences of + physical phenomena.—<span class="smcap">Jowett, B.</span></p> + <p class="blockcite"> + Dialogues of Plato (New York, 1897), Vol. 2, p. 78.</p> + + <p class="v2"> + <b><a name="Block_414" id="Block_414">414</a>.</b> + Nor do I know any study which can compete with + mathematics in general in furnishing matter for severe and + continued thought. Metaphysical problems may be even more + difficult; but then they are far less definite, and, as they + rarely lead to any precise conclusion, we miss the power of + checking our own operations, and of discovering whether we are + thinking and reasoning or merely fancying and + dreaming.—<span class="smcap">Todhunter, Isaac.</span></p> + <p class="blockcite"> + Conflict of Studies (London, 1873), p. 13.</p> + + <p class="v2"> + <b><a name="Block_415" id="Block_415">415</a>.</b> + Another great and special excellence of mathematics + is that it demands earnest voluntary exertion. It is simply + impossible for a person to become a good mathematician by the + happy accident of having been sent to a good school; this may + give him a preparation and a start, but by his own individual + efforts alone can he reach an eminent + position.—<span class="smcap">Todhunter, Isaac.</span></p> + <p class="blockcite"> + Conflict of Studies (London, 1873), p. 2.</p> + +<p><span class="pagenum"> + <a name="Page_54" + id="Page_54">54</a></span></p> + + <p class="v2"> + <b><a name="Block_416" id="Block_416">416</a>.</b> + The faculty of resolution is possibly much + invigorated by mathematical study, and especially by that + highest branch of it which, unjustly, merely on account of its + retrograde operations, has been called, as if par excellence, + analysis.—<span class="smcap">Poe, E. A.</span></p> + <p class="blockcite"> + The Murders in Rue Morgue.</p> + + <p class="v2"> + <b><a name="Block_417" id="Block_417">417</a>.</b> + He who gives a portion of his time and talent to + the investigation of mathematical truth will come to all other + questions with a decided advantage over his opponents. He will + be in argument what the ancient Romans were in the field: to + them the day of battle was a day of comparative recreation, + because they were ever accustomed to exercise with arms much + heavier than they fought; and reviews differed from a real + battle in two respects: they encountered more fatigue, but the + victory was + bloodless.—<span class="smcap">Colton, C. C.</span></p> + <p class="blockcite"> + Lacon (New York, 1866).</p> + + <p class="v2"> + <b><a name="Block_418" id="Block_418">418</a>.</b> + Mathematics is the study which forms the foundation + of the course [West Point Military Academy]. This is necessary, + both to impart to the mind that combined strength and + versatility, the peculiar vigor and rapidity of comparison + necessary for military action, and to pave the way for progress + in the higher military sciences.</p> + <p class="blockcite"> + Congressional Committee on Military Affairs, 1834; U. S. + Bureau of Education, Bulletin 1912, No. 2, p. 10.</p> + + <p class="v2"> + <b><a name="Block_419" id="Block_419">419</a>.</b> + Mathematics, among all school subjects, is + especially adapted to further clearness, definite brevity and + precision in expression, although it offers no exercise in + flights of rhetoric. This is due in the first place to the + logical rigour with which it develops thought, avoiding every + departure from the shortest, most direct way, never allowing + empty phrases to enter. Other subjects excel in the development + of expression in other respects: translation from foreign + languages into the mother tongue gives exercise in finding the + proper word for the given foreign word and gives knowledge of + laws of syntax, the study of poetry and prose furnish fit + patterns for connected presentation and elegant form of + expression, composition is to exercise the pupil in a like + presentation of his own or borrowed thoughts + +<span class="pagenum"> + <a name="Page_55" + id="Page_55">55</a></span> + + and their development, the natural sciences teach description of + natural objects, apparatus and processes, as well as the + statement of laws on the grounds of immediate sense-perception. + But all these aids for exercise in the use of the mother + tongue, each in its way valuable and indispensable, do not + guarantee, in the same manner as mathematical training, the + exclusion of words whose concepts, if not entirely wanting, are + not sufficiently clear. They do not furnish in the same measure + that which the mathematician demands particularly as regards + precision of expression.—<span class="smcap">Reidt, F.</span></p> + <p class="blockcite"> + Anleitung zum mathematischen Unterricht in höheren Schulen + (Berlin, 1906), p. 17.</p> + + <p class="v2"> + <b><a name="Block_420" id="Block_420">420</a>.</b> + One rarely hears of the mathematical recitation as + a preparation for public speaking. Yet mathematics shares with + these studies [foreign languages, drawing and natural science] + their advantages, and has another in a higher degree than + either of them.</p> + <p class="v1"> + Most readers will agree that a prime requisite for healthful + experience in public speaking is that the attention of the + speaker and hearers alike be drawn wholly away from the speaker + and concentrated upon the thought. In perhaps no other + classroom is this so easy as in the mathematical, where the + close reasoning, the rigorous demonstration, the tracing of + necessary conclusions from given hypotheses, commands and + secures the entire mental power of the student who is + explaining, and of his classmates. In what other circumstances + do students feel so instinctively that manner counts for so + little and mind for so much? In what other circumstances, + therefore, is a simple, unaffected, easy, graceful manner so + naturally and so healthfully cultivated? Mannerisms that are + mere affectation or the result of bad literary habit recede to + the background and finally disappear, while those peculiarities + that are the expression of personality and are inseparable from + its activity continually develop, where the student frequently + presents, to an audience of his intellectual peers, a connected + train of reasoning....</p> + <p class="v1"> + One would almost wish that our institutions of the science and + art of public speaking would put over their doors the motto + +<span class="pagenum"> + <a name="Page_56" + id="Page_56">56</a></span> + + that Plato had over the entrance to his school of philosophy: + “Let no one who is unacquainted with geometry enter + here.”—<span class="smcap">White, W. F.</span></p> + <p class="blockcite"> + A Scrap-book of Elementary Mathematics (Chicago, 1908), + p. 210.</p> + + <p class="v2"> + <b><a name="Block_421" id="Block_421">421</a>.</b> + The training which mathematics gives in working + with symbols is an excellent preparation for other sciences; + ... the world’s work requires constant mastery of + symbols.—<span class="smcap">Young, J. W. A.</span></p> + <p class="blockcite"> + The Teaching of Mathematics (New York, 1907), p. 42.</p> + + <p class="v2"> + <b><a name="Block_422" id="Block_422">422</a>.</b> + One striking peculiarity of mathematics is its + unlimited power of evolving examples and problems. A student + may read a book of Euclid, or a few chapters of Algebra, and + within that limited range of knowledge it is possible to set + him exercises as real and as interesting as the propositions + themselves which he has studied; deductions which might have + pleased the Greek geometers, and algebraic propositions which + Pascal and Fermat would not have disdained to + investigate.—<span class="smcap">Todhunter, Isaac.</span></p> + <p class="blockcite"> + Private Study of Mathematics: Conflict of Studies and + other Essays (London, 1873), p. 82.</p> + + <p class="v2"> + <b><a name="Block_423" id="Block_423">423</a>.</b> + Would you have a man reason well, you must use him + to it betimes; exercise his mind in observing the connection + between ideas, and following them in train. Nothing does this + better than mathematics, which therefore, I think should be + taught to all who have the time and opportunity, not so much to + make them mathematicians, as to make them reasonable creatures; + for though we all call ourselves so, because we are born to it + if we please, yet we may truly say that nature gives us but the + seeds of it, and we are carried no farther than industry and + application have carried us.—<span class= + "smcap">Locke, John.</span></p> + <p class="blockcite"> + Conduct of the Understanding, Sect. 6.</p> + + <p class="v2"> + <b><a name="Block_424" id="Block_424">424</a>.</b> + Secondly, the study of mathematics would show them + the necessity there is in reasoning, to separate all the + distinct ideas, and to see the habitudes that all those + concerned in the present inquiry have to one another, and to + lay by those which relate not to the proposition in hand, and + wholly to leave them + +<span class="pagenum"> + <a name="Page_57" + id="Page_57">57</a></span> + + out of the reckoning. This is that + which, in other respects besides quantity is absolutely + requisite to just reasoning, though in them it is not so easily + observed and so carefully practised. In those parts of + knowledge where it is thought demonstration has nothing to do, + men reason as it were in a lump; and if upon a summary and + confused view, or upon a partial consideration, they can raise + the appearance of a probability, they usually rest content; + especially if it be in a dispute where every little straw is + laid hold on, and everything that can but be drawn in any way + to give color to the argument is advanced with ostentation. But + that mind is not in a posture to find truth that does not + distinctly take all the parts asunder, and, omitting what is + not at all to the point, draws a conclusion from the result of + all the particulars which in any way influence + it.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + Conduct of the Understanding, Sect. 7.</p> + + <p class="v2"> + <b><a name="Block_425" id="Block_425">425</a>.</b> + I have before mentioned mathematics, wherein + algebra gives new helps and views to the understanding. If I + propose these it is not to make every man a thorough + mathematician or deep algebraist; but yet I think the study of + them is of infinite use even to grown men; first by + experimentally convincing them, that to make anyone reason + well, it is not enough to have parts wherewith he is satisfied, + and that serve him well enough in his ordinary course. A man in + those studies will see, that however good he may think his + understanding, yet in many things, and those very visible, it + may fail him. This would take off that presumption that most + men have of themselves in this part; and they would not be so + apt to think their minds wanted no helps to enlarge them, that + there could be nothing added to the acuteness and penetration + of their + understanding.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + The Conduct of the Understanding, Sect. 7.</p> + + <p class="v2"> + <b><a name="Block_426" id="Block_426">426</a>.</b> + I have mentioned mathematics as a way to settle in + the mind a habit of reasoning closely and in train; not that I + think it necessary that all men should be deep mathematicians, + but that, having got the way of reasoning which that study + necessarily brings the mind to, they might be able to transfer + it to other parts of knowledge, as they shall have occasion. + For in + +<span class="pagenum"> + <a name="Page_58" + id="Page_58">58</a></span> + + all sorts of reasoning, every single argument should be managed + as a mathematical demonstration; the connection and dependence + of ideas should be followed till the mind is brought to the + source on which it bottoms, and observes the coherence all + along;....—<span class= "smcap">Locke, John.</span></p> + <p class="blockcite"> + The Conduct of the Understanding, Sect. 7.</p> + + <p class="v2"> + <b><a name="Block_427" id="Block_427">427</a>.</b> + As an exercise of the reasoning faculty, pure + mathematics is an admirable exercise, because it consists of + <em>reasoning</em> alone, and does not encumber the student with + an exercise of <em>judgment</em>: and it is well to begin with + learning one thing at a time, and to defer a combination of + mental exercises to a later + period.—<span class="smcap">Whately, R.</span></p> + <p class="blockcite"> + Annotations to Bacon’s Essays (Boston, 1873), Essay 1, + p. 493.</p> + + <p class="v2"> + <b><a name="Block_428" id="Block_428">428</a>.</b> + It hath been an old remark, that Geometry is an + excellent Logic. And it must be owned that when the definitions + are clear; when the postulata cannot be refused, nor the axioms + denied; when from the distinct contemplation and comparison of + figures, their properties are derived, by a perpetual + well-connected chain of consequences, the objects being still + kept in view, and the attention ever fixed upon them; there is + acquired a habit of reasoning, close and exact and methodical; + which habit strengthens and sharpens the mind, and being + transferred to other subjects is of general use in the inquiry + after truth.—<span class="smcap">Berkely, George.</span></p> + <p class="blockcite"> + The Analyst, 2; Works (London, 1898), Vol. 3, p. 10.</p> + + <p class="v2"> + <b><a name="Block_429" id="Block_429">429</a>.</b> + Suppose then I want to give myself a little + training in the art of reasoning; suppose I want to get out of + the region of conjecture and probability, free myself from the + difficult task of weighing evidence, and putting instances + together to arrive at general propositions, and simply desire + to know how to deal with my general propositions when I get + them, and how to deduce right inferences from them; it is clear + that I shall obtain this sort of discipline best in those + departments of thought in which the first principles are + unquestionably true. For in all + +<span class="pagenum"> + <a name="Page_59" + id="Page_59">59</a></span> + + our thinking, if we come + to erroneous conclusions, we come to them either by accepting + false premises to start with—in which case + our reasoning, however good, will not save us from error; or by + reasoning badly, in which case the data we start from may be + perfectly sound, and yet our conclusions may be false. But in + the mathematical or pure sciences,—geometry, + arithmetic, algebra, trigonometry, the calculus of variations + or of curves,—we know at least that there is + not, and cannot be, error in our first principles, and we may + therefore fasten our whole attention upon the processes. As + mere exercises in logic, therefore, these sciences, based as + they all are on primary truths relating to space and number, + have always been supposed to furnish the most exact discipline. + When Plato wrote over the portal of his school. + “Let no one ignorant of geometry enter + here,” he did not mean that questions relating to + lines and surfaces would be discussed by his disciples. On the + contrary, the topics to which he directed their attention were + some of the deepest problems,—social, + political, moral,—on which the mind could + exercise itself. Plato and his followers tried to think out + together conclusions respecting the being, the duty, and the + destiny of man, and the relation in which he stood to the gods + and to the unseen world. What had geometry to do with these + things? Simply this: That a man whose mind has not undergone a + rigorous training in systematic thinking, and in the art of + drawing legitimate inferences from premises, was unfitted to + enter on the discussion of these high topics; and that the sort + of logical discipline which he needed was most likely to be + obtained from geometry—the only mathematical + science which in Plato’s time had been + formulated and reduced to a system. And we in this country + [England] have long acted on the same principle. Our future + lawyers, clergy, and statesmen are expected at the University + to learn a good deal about curves, and angles, and numbers and + proportions; not because these subjects have the smallest + relation to the needs of their lives, but because in the very + act of learning them they are likely to acquire that habit of + steadfast and accurate thinking, which is indispensable to + success in all the pursuits of + life.—<span class="smcap">Fitch, J. C.</span></p> + <p class="blockcite"> + Lectures on Teaching (New York, 1906), pp. 291-292.</p> + +<p><span class="pagenum"> + <a name="Page_60" + id="Page_60">60</a></span></p> + + <p class="v2"> + <b><a name="Block_430" id="Block_430">430</a>.</b> + It is admitted by all that a finished or even a + competent reasoner is not the work of nature alone; the + experience of every day makes it evident that education + develops faculties which would otherwise never have manifested + their existence. It is, therefore, as necessary to <em>learn to + reason</em> before we can expect to be able to reason, as it is + to learn to swim or fence, in order to attain either of those + arts. Now, something must be reasoned upon, it matters not much + what it is, provided it can be reasoned upon with certainty. + The properties of mind or matter, or the study of languages, + mathematics, or natural history, may be chosen for this + purpose. Now of all these, it is desirable to choose the one + which admits of the reasoning being verified, that is, in which + we can find out by other means, such as measurement and ocular + demonstration of all sorts, whether the results are true or + not. When the guiding property of the loadstone was first + ascertained, and it was necessary to learn how to use this new + discovery, and to find out how far it might be relied on, it + would have been thought advisable to make many passages between + ports that were well known before attempting a voyage of + discovery. So it is with our reasoning faculties: it is + desirable that their powers should be exerted upon objects of + such a nature, that we can tell by other means whether the + results which we obtain are true or false, and this before it + is safe to trust entirely to reason. Now the mathematics are + peculiarly well adapted for this purpose, on the following + grounds:</p> + <p class="v1"> + 1. Every term is distinctly explained, and has but one meaning, + and it is rarely that two words are employed to mean the same + thing.</p> + <p class="v1"> + 2. The first principles are self-evident, and, though derived + from observation, do not require more of it than has been made + by children in general.</p> + <p class="v1"> + 3. The demonstration is strictly logical, taking nothing for + granted except self-evident first principles, resting nothing + upon probability, and entirely independent of authority and + opinion.</p> + <p class="v1"> + 4. When the conclusion is obtained by reasoning, its truth or + falsehood can be ascertained, in geometry by actual + measurement, in algebra by common arithmetical calculation. This + +<span class="pagenum"> + <a name="Page_61" + id="Page_61">61</a></span> + + gives confidence, and is absolutely necessary, if, as was said + before, reason is not to be the instructor, but the pupil.</p> + <p class="v1"> + 5. There are no words whose meanings are so much alike that the + ideas which they stand for may be confounded. Between the + meaning of terms there is no distinction, except a total + distinction, and all adjectives and adverbs expressing + difference of degrees are + avoided.—<span class="smcap">De Morgan, Augustus.</span></p> + <p class="blockcite"> + On the Study and Difficulties of Mathematics (Chicago, 1898), + chap. 1.</p> + + <p class="v2"> + <b><a name="Block_431" id="Block_431">431</a>.</b> + The instruction of children should aim gradually to + combine knowing and doing [Wissen und Können]. + Among all sciences mathematics seems to be the only one of a + kind to satisfy this aim most + completely.—<span class="smcap">Kant, Immanuel.</span></p> + <p class="blockcite"> + Werke [Rosenkranz und Schubert], Bd. 9 (Leipzig, 1838), + p. 409.</p> + + <p class="v2"> + <b><a name="Block_432" id="Block_432">432</a>.</b> + Every discipline must be honored for reason other than its + utility, otherwise it yields no enthusiasm for industry.</p> + <p class="v1"> + For both reasons, I consider mathematics the chief subject for + the common school. No more highly honored exercise for the mind + can be found; the buoyancy [Spannkraft] which it produces is + even greater than that produced by the ancient languages, while + its utility is + unquestioned.—<span class="smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Mathematischer Lehrplan für Realgymnasien, Werke [Kehrbach], + (Langensalza, 1890), Bd. 5, p. 167.</p> + + <p class="v2"> + <b><a name="Block_433" id="Block_433">433</a>.</b> + The motive for the study of mathematics is insight + into the nature of the universe. Stars and strata, heat and + electricity, the laws and processes of becoming and being, + incorporate mathematical truths. If language imitates the voice + of the Creator, revealing His heart, mathematics discloses His + intellect, repeating the story of how things came into being. + And the value of mathematics, appealing as it does to our + energy and to our honor, to our desire to know the truth and + thereby to live as of right in the household of God, is that it + establishes us in larger and larger certainties. As literature + +<span class="pagenum"> + <a name="Page_62" + id="Page_62">62</a></span> + + develops emotion, understanding, and + sympathy, so mathematics develops observation, imagination, and + reason.—<span class="smcap">Chancellor, W. E.</span></p> + <p class="blockcite"> + A Theory of Motives, Ideals and Values in Education + (Boston and New York, 1907), p. 406.</p> + + <p class="v2"> + <b><a name="Block_434" id="Block_434">434</a>.</b> + Mathematics in its pure form, as arithmetic, + algebra, geometry, and the applications of the analytic method, + as well as mathematics applied to matter and force, or statics + and dynamics, furnishes the peculiar study that gives to us, + whether as children or as men, the command of nature in this + its quantitative aspect; mathematics furnishes the instrument, + the tool of thought, which we wield in this + realm.—<span class="smcap">Harris, W. T.</span></p> + <p class="blockcite"> + Psychologic Foundations of Education (New York, 1898), + p. 325.</p> + + <p class="v2"> + <b><a name="Block_435" id="Block_435">435</a>.</b> + Little can be understood of even the simplest + phenomena of nature without some knowledge of mathematics, and + the attempt to penetrate deeper into the mysteries of nature + compels simultaneous development of the mathematical + processes.—<span class="smcap">Young, J. W. A.</span></p> + <p class="blockcite"> + The Teaching of Mathematics (New York, 1907), p. 16.</p> + + <p class="v2"> + <b><a name="Block_436" id="Block_436">436</a>.</b> + For many parts of nature can neither be invented + with sufficient subtility nor demonstrated with sufficient + perspicuity nor accommodated unto use with sufficient + dexterity, without the aid and intervening of + mathematics.—<span class="smcap">Bacon, Lord.</span></p> + <p class="blockcite"> + De Augmentis, Bk. 2; Advancement of Learning, Bk. 3.</p> + + <p class="v2"> + <b><a name="Block_437" id="Block_437">437</a>.</b> + I confess, that after I began ... to discern how useful + mathematicks may be made to physicks, I have often wished that + I had employed about the speculative part of geometry, and the + cultivation of the specious Algebra I had been taught very + young, a good part of that time and industry, that I had spent + about surveying and fortification (of which I remember I once + wrote an entire treatise) and other parts of practick + mathematicks.—<span class= "smcap">Boyle, Robert.</span></p> + <p class="blockcite"> + The Usefulness of + + <a id="TNanchor_2"></a> + <a class="msg" href="#TN_2" + title="originally spelled ‘Mathematiks’">Mathematicks</a> + + to Natural Philosophy; Works (London, 1772), Vol. 3, p. 426.</p> + +<p><span class="pagenum"> + <a name="Page_63" + id="Page_63">63</a></span></p> + + <p class="v2"> + <b><a name="Block_438" id="Block_438">438</a>.</b> + Mathematics gives the young man a clear idea of + demonstration and habituates him to form long trains of thought + and reasoning methodically connected and sustained by the final + certainty of the result; and it has the further advantage, from + a purely moral point of view, of inspiring an absolute and + fanatical respect for truth. In addition to all this, + mathematics, and chiefly algebra and infinitesimal calculus, + excite to a high degree the conception of the signs and + symbols—necessary instruments to extend the + power and reach of the human mind by summarizing an aggregate + of relations in a condensed form and in a kind of mechanical + way. These auxiliaries are of special value in mathematics + because they are there adequate to their definitions, a + characteristic which they do not possess to the same degree in + the physical and mathematical [natural?] sciences.</p> + <p class="v1"> + There are, in fact, a mass of mental and moral faculties that + can be put in full play only by instruction in mathematics; and + they would be made still more available if the teaching was + directed so as to leave free play to the personal work of the + student.—<span class="smcap">Berthelot, M. P. E. M.</span></p> + <p class="blockcite"> + Science as an Instrument of Education; Popular Science + Monthly (1897), p. 253.</p> + + <p class="v2"> + <b><a name="Block_439" id="Block_439">439</a>.</b> + Mathematical knowledge, therefore, appears to us of + value not only in so far as it serves as means to other ends, + but for its own sake as well, and we behold, both in its + systematic external and internal development, the most complete + and purest logical mind-activity, the embodiment of the highest + intellect-esthetics.—<span class="smcap">Pringsheim, + Alfred.</span></p> + <p class="blockcite"> + Ueber Wert und angeblichen Unwert der Mathematik; Jahresbericht + der Deutschen Mathematiker Vereinigung, Bd. 13, p. 381.</p> + + <p class="v2"> + <b><a name="Block_440" id="Block_440">440</a>.</b> + The advantages which mathematics derives from the peculiar + nature of those relations about which it is conversant, from + its simple and definite phraseology, and from the severe logic + so admirably displayed in the concatenation of its innumerable + theorems, are indeed immense, and well entitled to separate and + ample illustration.—<span class="smcap">Stewart, + Dugald.</span></p> + <p class="blockcite"> + Philosophy of the Human Mind, Part 2, chap. 2, sect. 3.</p> + +<p><span class="pagenum"> + <a name="Page_64" + id="Page_64">64</a></span></p> + + <p class="v2"> + <b><a name="Block_441" id="Block_441">441</a>.</b> + I do not intend to go deeply into the question how + far mathematical studies, as the representatives of conscious + logical reasoning, should take a more important place in school + education. But it is, in reality, one of the questions of the + day. In proportion as the range of science extends, its system + and organization must be improved, and it must inevitably come + about that individual students will find themselves compelled + to go through a stricter course of training than grammar is in + a position to supply. What strikes me in my own experience with + students who pass from our classical schools to scientific and + medical studies, is first, a certain laxity in the application + of strictly universal laws. The grammatical rules, in which + they have been exercised, are for the most part followed by + long lists of exceptions; accordingly they are not in the habit + of relying implicitly on the certainty of a legitimate + deduction from a strictly universal law. Secondly, I find them + for the most part too much inclined to trust to authority, even + in cases where they might form an independent judgment. In + fact, in philological studies, inasmuch as it is seldom + possible to take in the whole of the premises at a glance, and + inasmuch as the decision of disputed questions often depends on + an æsthetic feeling for beauty of expression, or + for the genius of the language, attainable only by long + training, it must often happen that the student is referred to + authorities even by the best teachers. Both faults are + traceable to certain indolence and vagueness of thought, the + sad effects of which are not confined to subsequent scientific + studies. But certainly the best remedy for both is to be found + in mathematics, where there is absolute certainty in the + reasoning, and no authority is recognized but that of one’s own + intelligence.—<span class="smcap">Helmholtz, H.</span></p> + <p class="blockcite"> + On the Relation of Natural Science to Science in general; + Popular Lectures on Scientific Subjects; Atkinson (New York, + 1900), pp. 25-26.</p> + + <p class="v2"> + <b><a name="Block_442" id="Block_442">442</a>.</b> + What renders a problem definite, and what leaves it + indefinite, may best be understood from mathematics. The very + important idea of solving a problem within limits of error is + an element of rational culture, coming from the same source. + The art of totalizing fluctuations by curves is capable of + being carried, in conception, far beyond the mathematical domain, + +<span class="pagenum"> + <a name="Page_65" + id="Page_65">65</a></span> + + where it is first learned. The + distinction between laws and coefficients applies in every + department of causation. The theory of Probable Evidence is the + mathematical contribution to Logic, and is of paramount + importance.—<span class="smcap">Bain, Alexander.</span></p> + <p class="blockcite"> + Education as a Science (New York, 1898), pp. 151-152.</p> + + <p class="v2"> + <b><a name="Block_443" id="Block_443">443</a>.</b> + We receive it as a fact, that some minds are so + constituted as absolutely to require for their nurture the + severe logic of the abstract sciences; that rigorous sequence + of ideas which leads from the premises to the conclusion, by a + path, arduous and narrow, it may be, and which the youthful + reason may find it hard to mount, but where it cannot stray; + and on which, if it move at all, it must move onward and + upward.... Even for intellects of a different character, whose + natural aptitude is for moral evidence and those relations of + ideas which are perceived and appreciated by taste, the study + of the exact sciences may be recommended as the best protection + against the errors into which they are most likely to fall. + Although the study of language is in many respects no mean + exercise in logic, yet it must be admitted that an eminently + practical mind is hardly to be formed without mathematical + training.—<span class="smcap">Everett, Edward.</span></p> + <p class="blockcite"> + Orations and Speeches (Boston, 1870), Vol. 2, p. 510.</p> + + <p class="v2"> + <b><a name="Block_444" id="Block_444">444</a>.</b> + The value of mathematical instruction as a + preparation for those more difficult investigations, consists + in the applicability not of its doctrines but of its methods. + Mathematics will ever remain the past perfect type of the + deductive method in general; and the applications of + mathematics to the simpler branches of physics furnish the only + school in which philosophers can effectually learn the most + difficult and important of their art, the employment of the + laws of simpler phenomena for explaining and predicting those + of the more complex. These grounds are quite sufficient for + deeming mathematical training an indispensable basis of real + scientific education, and regarding with Plato, one who is + ἀγεωμέτρητος, as wanting in one of the most essential + qualifications for the successful cultivation of the higher + branches of philosophy.—<span class="smcap">Mill, + J. S.</span></p> + <p class="blockcite"> + System of Logic, Bk. 3, chap. 24, sect. 9.</p> + +<p><span class="pagenum"> + <a name="Page_66" + id="Page_66">66</a></span></p> + + <p class="v2"> + <b><a name="Block_445" id="Block_445">445</a>.</b> + This science, Geometry, is one of indispensable use + and constant reference, for every student of the laws of + nature; for the relations of space and number are the + <em>alphabet</em> in which those laws are written. But besides + the interest and importance of this kind which geometry + possesses, it has a great and peculiar value for all who wish + to understand the foundations of human knowledge, and the + methods by which it is acquired. For the student of geometry + acquires, with a degree of insight and clearness which the + unmathematical reader can but feebly imagine, a conviction that + there are necessary truths, many of them of a very complex and + striking character; and that a few of the most simple and + self-evident truths which it is possible for the mind of man to + apprehend, may, by systematic deduction, lead to the most + remote and unexpected results.—<span class= + "smcap">Whewell, William.</span></p> + <p class="blockcite"> + The Philosophy of the Inductive Sciences, Part 1, Bk. 2, + chap. 4, sect. 8 (London, 1858).</p> + + <p class="v2"> + <b><a name="Block_446" id="Block_446">446</a>.</b> + Mathematics, while giving no quick remuneration, + like the art of stenography or the craft of bricklaying, does + furnish the power for deliberate thought and accurate + statement, and to speak the truth is one of the most social + qualities a person can possess. Gossip, flattery, slander, + deceit, all spring from a slovenly mind that has not been + trained in the power of truthful statement, which is one of the + highest utilities.—<span class="smcap">Dutton, S. T.</span></p> + <p class="blockcite"> + Social Phases of Education in the School and the Home + (London, 1900), p. 30.</p> + + <p class="v2"> + <b><a name="Block_447" id="Block_447">447</a>.</b> + It is from this absolute indifference and + tranquility of the mind, that mathematical speculations derive + some of their most considerable advantages; because there is + nothing to interest the imagination; because the judgment sits + free and unbiased to examine the point. All proportions, every + arrangement of quantity, is alike to the understanding, because + the same truths result to it from all; from greater from + lesser, from equality and + inequality.—<span class="smcap">Burke, Edmund.</span></p> + <p class="blockcite"> + On the Sublime and Beautiful, Part 3, sect. 2.</p> + + <p class="v2"> + <b><a name="Block_448" id="Block_448">448</a>.</b> + Out of the interaction of form and content in mathematics + grows an acquaintance with methods which enable the + +<span class="pagenum"> + <a name="Page_67" + id="Page_67">67</a></span> + + student to produce independently within + certain though moderate limits, and to extend his knowledge + through his own reflection. The deepening of the consciousness + of the intellectual powers connected with this kind of + activity, and the gradual awakening of the feeling of + intellectual self-reliance may well be considered as the most + beautiful and highest result of mathematical + training.—<span class="smcap">Pringsheim, Alfred.</span></p> + <p class="blockcite"> + Ueber Wert und angeblichen Unwert der Mathematik; Jahresbericht + der Deutschen Mathematiker Vereinigung (1904), p. 374.</p> + + <p class="v2"> + <b><a name="Block_449" id="Block_449">449</a>.</b> + He who would know what geometry is, must venture + boldly into its depths and learn to think and feel as a + geometer. I believe that it is impossible to do this, and to + study geometry as it admits of being studied and am conscious + it can be taught, without finding the reason invigorated, the + invention quickened, the sentiment of the orderly and beautiful + awakened and enhanced, and reverence for truth, the foundation + of all integrity of character, converted into a fixed principle + of the mental and moral constitution, according to the old and + expressive adage “<i lang="la" xml:lang="la">abeunt studia in + mores</i>.”—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + A probationary Lecture on Geometry; Collected Mathematical + Papers (Cambridge, 1908), Vol. 2, p. 9.</p> + + <p class="v2"> + <b><a name="Block_450" id="Block_450">450</a>.</b> + Mathematical knowledge adds vigour to the mind, frees it from + prejudice, credulity, and + superstition.—<span class="smcap">Arbuthnot, John.</span></p> + <p class="blockcite"> + Usefulness of Mathematical Learning.</p> + + <p class="v2"> + <b><a name="Block_451" id="Block_451">451</a>.</b> + When the boy begins to understand that the visible + point is preceded by an invisible point, that the shortest + distance between two points is conceived as a straight line + before it is ever drawn with the pencil on paper, he + experiences a feeling of pride, of satisfaction. And justly so, + for the fountain of all thought has been opened to him, the + difference between the ideal and the real, <i lang="la" + xml:lang="la">potentia et + actu</i>, has become clear to him; henceforth the philosopher + can reveal him nothing new, as a geometrician he has discovered + the basis of all + thought.—<span class="smcap">Goethe.</span></p> + <p class="blockcite"> + Sprüche in Prosa, Ethisches, VI, 455.</p> + +<p><span class="pagenum"> + <a name="Page_68" + id="Page_68">68</a></span></p> + + <p class="v2"> + <b><a name="Block_452" id="Block_452">452</a>.</b> + In mathematics, ... and in natural philosophy since + mathematics was applied to it, we see the noblest instance of + the force of the human mind, and of the sublime heights to + which it may rise by cultivation. An acquaintance with such + sciences naturally leads us to think well of our faculties, and + to indulge sanguine expectations concerning the improvement of + other parts of knowledge. To this I may add, that, as + mathematical and physical truths are perfectly uninteresting in + their consequences, the understanding readily yields its assent + to the evidence which is presented to it; and in this way may + be expected to acquire the habit of trusting to its own + conclusions, which will contribute to fortify it against the + weaknesses of scepticism, in the more interesting inquiries + after moral truth in which it may afterwards + engage.—<span class="smcap">Stewart, Dugald.</span></p> + <p class="blockcite"> + Philosophy of the Human Mind, Part 3, chap. 1, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_453" id="Block_453">453</a>.</b> + Those that can readily master the difficulties of + Mathematics find a considerable charm in the study, sometimes + amounting to fascination. This is far from universal; but the + subject contains elements of strong interest of a kind that + constitutes the pleasures of knowledge. The marvellous devices + for solving problems elate the mind with the feeling of + intellectual power; and the innumerable constructions of the + science leave us lost in wonder.—<span class="smcap">Bain, + Alexander.</span></p> + <p class="blockcite"> + Education as a Science (New York, 1898), p. 153.</p> + + <p class="v2"> + <b><a name="Block_454" id="Block_454">454</a>.</b> + Thinking is merely the comparing of ideas, + discerning relations of likeness and of difference between + ideas, and drawing inferences. It is seizing general truths on + the basis of clearly apprehended particulars. It is but + generalizing and particularizing. Who will deny that a child + can deal profitably with sequences of ideas like: How many + marbles are 2 marbles and 3 marbles? 2 pencils and 3 pencils? 2 + balls and 3 balls? 2 children and 3 children? 2 inches and 3 + inches? 2 feet and 3 feet? 2 and 3? Who has not seen the + countenance of some little learner light up at the end of such + a series of questions with the exclamation, “Why it’s always that + way. Isn’t it?” This is the glow of pleasure that the + generalizing step always affords + +<span class="pagenum"> + <a name="Page_69" + id="Page_69">69</a></span> + + him who takes the step himself. This is + the genuine life-giving joy which comes from feeling that one + can successfully take this step. The reality of such a + discovery is as great, and the lasting effect upon the mind of + him that makes it is as sure as was that by which the great + Newton hit upon the generalization of the law of gravitation. + It is through these thrills of discovery that love to learn and + intellectual pleasure are begotten and fostered. Good + arithmetic teaching abounds in such + opportunities.—<span class="smcap">Myers, George.</span></p> + <p class="blockcite"> + Arithmetic in Public Education (Chicago), p. 13.</p> + + <p class="v2"> + <b><a name="Block_455" id="Block_455">455</a>.</b> + A <em>general course</em> in mathematics should be + required of all officers for its practical value, but no less + for its educational value in training the mind to logical forms + of thought, in developing the sense of absolute truthfulness, + together with a confidence in the accomplishment of definite + results by definite + means.—<span class="smcap">Echols, C. P.</span></p> + <p class="blockcite"> + Mathematics at West Point and Annapolis; U. S. Bureau of + Education, Bulletin 1912, No. 2, p. 11.</p> + + <p class="v2"> + <b><a name="Block_456" id="Block_456">456</a>.</b> + Exercise in the most rigorous thinking that is + possible will of its own accord strengthen the sense of truth + and right, for each advance in the ability to distinguish + between correct and false thoughts, each habit making for + rigour in thought development will increase in the sound pupil + the ability and the wish to ascertain what is right in life and + to defend it.—<span class="smcap">Reidt, F.</span></p> + <p class="blockcite"> + Anleitung zum mathematischen Unterricht in den höheren Schulen + (Berlin, 1906), p. 28.</p> + + <p class="v2"> + <b><a name="Block_457" id="Block_457">457</a>.</b> + I do not maintain that the <em>chief value</em> of + the study of arithmetic consists in the lessons of morality + that arise from this study. I claim only that, to be impressed + from day to day, that there is something <em>that is right</em> + as an answer to the questions with which one is <em>able</em> to + grapple, and that there is a wrong answer—that there are ways + in which the right answer can be established as right, that + these ways automatically reject error and slovenliness, and that + the learner is able himself to manipulate + +<span class="pagenum"> + <a name="Page_70" + id="Page_70">70</a></span> + + these ways and to + arrive at the establishment of the true as opposed to the + untrue, this relentless hewing <em>to</em> the line and stopping + <em>at</em> the line, must color distinctly the thought life of + the pupil with more than a tinge of morality.... To be + neighborly with truth, to feel one’s self + somewhat facile in ways of recognizing and establishing what is + right, what is correct, to find the wrong persistently and + unfailingly rejected as of no value, to feel that one can apply + these ways for himself, that one can think and work + independently, have a real, a positive, and a purifying effect + upon moral character. They are the quiet, steady undertones of + the work that always appeal to the learner for the sanction of + his best judgment, and these are the really significant matters + in school work. It is not the noise and bluster, not even the + dramatics or the polemics from the teacher’s + desk, that abide longest and leave the deepest and stablest + imprint upon character. It is these still, small voices that + speak unmistakably for the right and against the wrong and the + erroneous that really form human character. When the school + subjects are arranged on the basis of the degree to which they + contribute to the moral upbuilding of human character good + arithmetic will be well up the + list.—<span class="smcap">Myers, George.</span></p> + <p class="blockcite"> + Arithmetic in Public Education (Chicago), p. 18.</p> + + <p class="v2"> + <b><a name="Block_458" id="Block_458">458</a>.</b> + In destroying the predisposition to anger, science + of all kind is useful; but the mathematics possess this + property in the most eminent + degree.—<span class="smcap">Dr. Rush.</span></p> + <p class="blockcite"> + Quoted in Day’s Collacon (London, no date).</p> + + <p class="v2"> + <b><a name="Block_459" id="Block_459">459</a>.</b> + The mathematics are the friends to religion, + inasmuch as they charm the passions, restrain the impetuosity + of the imagination, and purge the mind from error and + prejudice. Vice is error, confusion and false reasoning; and + all truth is more or less opposite to it. Besides, mathematical + truth may serve for a pleasant entertainment for those hours + which young men are apt to throw away upon their vices; the + delightfulness of them being such as to make solitude not only + easy but + desirable.—<span class="smcap">Arbuthnot, John.</span></p> + <p class="blockcite"> + Usefulness of Mathematical Learning.</p> + +<p><span class="pagenum"> + <a name="Page_71" + id="Page_71">71</a></span></p> + + <p class="v2"> + <b><a name="Block_460" id="Block_460">460</a>.</b> + There is no prophet which preaches the + superpersonal God more plainly than + mathematics.—<span class="smcap">Carus, Paul.</span></p> + <p class="blockcite"> + Reflections on Magic Squares; Monist (1906), p. 147.</p> + + <p class="v2"> + <b><a name="Block_461" id="Block_461">461</a>.</b> + Mathematics must subdue the flights of our reason; + they are the staff of the blind; no one can take a step without + them; and to them and experience is due all that is certain in + physics.—<span class="smcap">Voltaire.</span></p> + <p class="blockcite"> + Oeuvres Complètes (Paris, 1880), t. 35, p. 219.</p> + +<p><span class="pagenum"> + <a name="Page_72" + id="Page_72">72</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_V"> + CHAPTER V<br /> + <span class="large"> + THE TEACHING OF MATHEMATICS</span></h2> + + <p class="v2"> + <b><a name="Block_501" id="Block_501">501</a>.</b> + In mathematics two ends are constantly kept in + view: First, stimulation of the inventive faculty, exercise of + judgment, development of logical reasoning, and the habit of + concise statement; second, the association of the branches of + pure mathematics with each other and with applied science, that + the pupil may see clearly the true relations of principles and + things.</p> + <p class="blockcite"> + International Commission on the Teaching of Mathematics, + American Report; U. S. Bureau of Education, Bulletin 1912, + No. 4, p. 7.</p> + + <p class="v2"> + <b><a name="Block_502" id="Block_502">502</a>.</b> + The ends to be attained [in the teaching of + mathematics in the secondary schools] are the knowledge of a + body of geometrical truths, the power to draw correct + inferences from given premises, the power to use algebraic + processes as a means of finding results in practical problems, + and the awakening of interest in the science of mathematics.</p> + <p class="blockcite"> + International Commission on the Teaching of Mathematics, + American Report; U. S. Bureau of Education, Bulletin 1912, + No. 4, p. 7.</p> + + <p class="v2"> + <b><a name="Block_503" id="Block_503">503</a>.</b> + General preparatory instruction must continue to be + the aim in the instruction at the higher institutions of + learning. Exclusive selection and treatment of subject matter + with reference to specific avocations is disadvantageous.</p> + <p class="blockcite"> + Resolution adopted by the German Association for the + Advancement of Scientific and Mathematical Instruction; + Jahresbericht der Deutschen Mathematiker Vereinigung (1896), + p. 41.</p> + + <p class="v2"> + <b><a name="Block_504" id="Block_504">504</a>.</b> + In the secondary schools mathematics should be a + part of general culture and not contributory to technical + training of any kind; it should cultivate space intuition, + logical thinking, the power to rephrase in clear language + thoughts recognized as correct, and ethical and esthetic + effects; so treated, mathematics is a quite indispensable + factor of general education in so far as + +<span class="pagenum"> + <a name="Page_73" + id="Page_73">73</a></span> + + the latter shows its traces in the comprehension of the + development of civilization and the ability to participate in + the further tasks of civilization.</p> + <p class="blockcite"> + Unterrichtsblätter für Mathematik und Naturwissenschaft (1904), + p. 128.</p> + + <p class="v2"> + <b><a name="Block_505" id="Block_505">505</a>.</b> + Indeed, the aim of teaching [mathematics] should be + rather to strengthen his [the pupil’s] + faculties, and to supply a method of reasoning applicable to + other subjects, than to furnish him with an instrument for + solving practical + problems.—<span class="smcap">Magnus, Philip.</span></p> + <p class="blockcite"> + Perry’s Teaching of Mathematics (London, 1902), p. 84.</p> + + <p class="v2"> + <b><a name="Block_506" id="Block_506">506</a>.</b> + The participation in the <em>general development of + the mental powers</em> without special reference to his future + vocation must be recognized as the essential aim of mathematical + instruction.—<span class="smcap">Reidt, F.</span></p> + <p class="blockcite"> + Anleitung zum Mathematischen Unterricht an + höheren Schulen (Berlin, 1906), p. 12.</p> + + <p class="v2"> + <b><a name="Block_507" id="Block_507">507</a>.</b> + I am of the decided opinion, that mathematical + instruction must have for its first aim a deep penetration and + complete command of abstract mathematical theory together with + a clear insight into the structure of the system, and doubt not + that the instruction which accomplishes this is valuable and + interesting even if it neglects practical applications. If the + instruction sharpens the understanding, if it arouses the + scientific interest, whether mathematical or philosophical, if + finally it calls into life an esthetic feeling for the beauty + of a scientific edifice, the instruction will take on an + ethical value as well, provided that with the interest it + awakens also the impulse toward scientific activity. I contend, + therefore, that even without reference to its applications + mathematics in the high schools has a value equal to that of + the other subjects of + instruction.—<span class="smcap">Goetting, E.</span></p> + <p class="blockcite"> + Ueber das Lehrziel im mathematischen Unterricht der + höheren Realanstalten; Jahresbericht der + Deutschen Mathematiker Vereinigung, Bd. 2, p. 192.</p> + +<p><span class="pagenum"> + <a name="Page_74" + id="Page_74">74</a></span></p> + + <p class="v2"> + <b><a name="Block_508" id="Block_508">508</a>.</b> + Mathematics will not be properly esteemed in wider + circles until more than the <em>a b c</em> of it is taught in the + schools, and until the unfortunate impression is gotten rid of + that mathematics serves no other purpose in instruction than + the <em>formal</em> training of the mind. The aim of mathematics + is its <em>content</em>, its form is a secondary consideration + and need not necessarily be that historic form which is due to + the circumstance that mathematics took permanent shape under + the influence of Greek logic.—<span class= + "smcap">Hankel, H.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten + Jahrhunderten (Tübingen, 1884), p. 6.</p> + + <p class="v2"> + <b><a name="Block_509" id="Block_509">509</a>.</b> + The idea that aptitude for mathematics is rarer + than aptitude for other subjects is merely an illusion which is + caused by belated or neglected + beginners.—<span class="smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Umriss pädagogischer Vorlesungen; Werke + [Kehrbach] (Langensalza, 1902), Bd. 10, p. 101.</p> + + <p class="v2"> + <b><a name="Block_510" id="Block_510">510</a>.</b> + I believe that the useful methods of mathematics + are easily to be learned by quite young persons, just as + languages are easily learned in youth. What a wondrous + philosophy and history underlie the use of almost every word in + every language—yet the child learns to use + the word unconsciously. No doubt when such a word was first + invented it was studied over and lectured upon, just as one + might lecture now upon the idea of a rate, or the use of + Cartesian co-ordinates, and we may depend upon it that children + of the future will use the idea of the calculus, and use + squared paper as readily as they now cipher.... When Egyptian + and Chaldean philosophers spent years in difficult + calculations, which would now be thought easy by young + children, doubtless they had the same notions of the depth of + their knowledge that Sir William Thomson might now have of his. + How is it, then, that Thomson gained his immense knowledge in + the time taken by a Chaldean philosopher to acquire a simple + knowledge of arithmetic? The reason is plain. Thomson, when a + child, was taught in a few years more than all that was known + three thousand years ago of the properties of numbers. When it + is found essential to a boy’s future that + machinery should be given to his brain, it is given to him; he + is taught to use it, and his bright memory makes the use of it a + +<span class="pagenum"> + <a name="Page_75" + id="Page_75">75</a></span> + + second nature to him; but it is not + till after-life that he makes a close investigation of what + there actually is in his brain which has enabled him to do so + much. It is taken because the child has much faith. In after + years he will accept nothing without careful consideration. The + machinery given to the brain of children is getting more and + more complicated as time goes on; but there is really no reason + why it should not be taken in as early, and used as readily, as + were the axioms of childish education in ancient + Chaldea.—<span class="smcap">Perry, John.</span></p> + <p class="blockcite"> + The Teaching of Mathematics (London, 1902), p. 14.</p> + + <p class="v2"> + <b><a name="Block_511" + id="Block_511" + href="#TN_9" + class="msg" + title="originally shown as ‘517’">511</a>.</b> + The ancients devoted a lifetime to the study of + arithmetic; it required days to extract a square root or to + multiply two numbers together. Is there any harm in skipping + all that, in letting the school boy learn multiplication sums, + and in starting his more abstract reasoning at a more advanced + point? Where would be the harm in letting the boy assume the + truth of many propositions of the first four books of Euclid, + letting him assume their truth partly by faith, partly by + trial? Giving him the whole fifth book of Euclid by simple + algebra? Letting him assume the sixth as axiomatic? Letting + him, in fact, begin his severer studies where he is now in the + habit of leaving off? We do much less orthodox things. Every + here and there in one’s mathematical studies + one makes exceedingly large assumptions, because the methodical + study would be ridiculous even in the eyes of the most pedantic + of teachers. I can imagine a whole year devoted to the + philosophical study of many things that a student now takes in + his stride without trouble. The present method of training the + mind of a mathematical teacher causes it to strain at gnats and + to swallow camels. Such gnats are most of the propositions of + the sixth book of Euclid; propositions generally about + incommensurables; the use of arithmetic in geometry; the + parallelogram of forces, etc., + decimals.—<span class="smcap">Perry, John.</span></p> + <p class="blockcite"> + The Teaching of Mathematics (London, 1904), p. 12.</p> + + <p class="v2"> + <b><a name="Block_512" id="Block_512">512</a>.</b> + The teaching of elementary mathematics should be + conducted so that the way should be prepared for the building + upon them of the higher mathematics. The teacher should + always + +<span class="pagenum"> + <a name="Page_76" + id="Page_76">76</a></span> + + bear in mind and look forward to what + is to come after. The pupil should not be taught what may be + sufficient for the time, but will lead to difficulties in the + future.... I think the fault in teaching arithmetic is that of + not attending to general principles and teaching instead of + particular rules.... I am inclined to attack the teaching of + mathematics on the grounds that it does not dwell sufficiently + on a few general axiomatic + principles.—<span class="smcap">Hudson, W. H. H.</span></p> + <p class="blockcite"> + Perry’s Teaching of Mathematics (London, 1904), p. 33.</p> + + <p class="v2"> + <b><a name="Block_513" id="Block_513">513</a>.</b> + “Mathematics in Prussia! Ah, + sir, they teach mathematics in Prussia as you teach your boys + rowing in England: they are trained by men who have been + trained by men who have themselves been trained for generations + back.”—<span class="smcap">Langley, E. M.</span></p> + <p class="blockcite"> + Perry’s Teaching of Mathematics (London, 1904), p. 43.</p> + + <p class="v2"> + <b><a name="Block_514" id="Block_514">514</a>.</b> + A superficial knowledge of mathematics may lead to + the belief that this subject can be taught incidentally, and + that exercises akin to counting the petals of flowers or the + legs of a grasshopper are mathematical. Such work ignores the + fundamental idea out of which quantitative reasoning + grows—the equality of magnitudes. It leaves + the pupil unaware of that relativity which is the essence of + mathematical science. Numerical statements are frequently + required in the study of natural history, but to repeat these + as a drill upon numbers will scarcely lend charm to these + studies, and certainly will not result in mathematical + knowledge.—<span class="smcap">Speer, W. W.</span></p> + <p class="blockcite"> + Primary Arithmetic (Boston, 1897), pp. 26-27.</p> + + <p class="v2"> + <b><a name="Block_515" id="Block_515">515</a>.</b> + Mathematics is no more the art of reckoning and + computation than architecture is the art of making bricks or + hewing wood, no more than painting is the art of mixing colors + on a palette, no more than the science of geology is the art of + breaking rocks, or the science of anatomy the art of + butchering.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 29.</p> + +<p><span class="pagenum"> + <a name="Page_77" + id="Page_77">77</a></span></p> + + <p class="v2"> + <b><a name="Block_516" id="Block_516">516</a>.</b> + The study of mathematics—from + ordinary reckoning up to the higher + processes—must be connected with knowledge + of nature, and at the same time with experience, that it may + enter the pupil’s circle of + thought.—<span class="smcap">Herbart, J. + F.</span></p> + <p class="blockcite"> + Letters and Lectures on Education [Felkin] (London, 1908), + p. 117.</p> + + <p class="v2"> + <b><a name="Block_517" id="Block_517">517</a>.</b> + First, as concerns the <em>success</em> of teaching + mathematics. No instruction in the high schools is as difficult + as that of mathematics, since the large majority of students + are at first decidedly disinclined to be harnessed into the + rigid framework of logical conclusions. The interest of young + people is won much more easily, if sense-objects are made the + starting point and the transition to abstract formulation is + brought about gradually. For this reason it is psychologically + quite correct to follow this course.</p> + <p class="v1"> + Not less to be recommended is this course if we inquire into + the essential purpose of mathematical instruction. Formerly it + was too exclusively held that this purpose is to sharpen the + understanding. Surely another important end is to implant in + the student the conviction that <em>correct thinking based on + true premises secures mastery over the outer world</em>. To + accomplish this the outer world must receive its share of + attention from the very beginning.</p> + <p class="v1"> + Doubtless this is true but there is a danger which needs + pointing out. It is as in the case of language teaching where + the modern tendency is to secure in addition to grammar also an + understanding of the authors. The danger lies in grammar being + completely set aside leaving the subject without its + indispensable solid basis. Just so in the teaching of + mathematics it is possible to accumulate interesting + applications to such an extent as to stunt the essential + logical development. This should in no wise be permitted, for + thus the kernel of the whole matter is lost. Therefore: We do + want throughout a quickening of mathematical instruction by the + introduction of applications, but we do not want that the + pendulum, which in former decades may have inclined too much + toward the abstract side, should now swing to the other + extreme; we would rather pursue the proper middle + course.—<span class="smcap">Klein, Felix.</span></p> + <p class="blockcite"> + Ueber den Mathematischen Unterricht an den + + <a id="TNanchor_3"></a> + <a class="msg" href="#TN_3" + title="changed from ‘hoheren’ for consistency">höheren</a> + + Schulen; Jahresbericht der Deutschen Mathematiker Vereinigung, + Bd. 11, p. 131.</p> + +<p><span class="pagenum"> + <a name="Page_78" + id="Page_78">78</a></span></p> + + <p class="v2"> + <b><a name="Block_518" id="Block_518">518</a>.</b> + It is above all the duty of the methodical + text-book to adapt itself to the pupil’s + power of comprehension, only challenging his higher efforts + with the increasing development of his imagination, his logical + power and the ability of abstraction. This indeed constitutes a + test of the art of teaching, it is here where pedagogic tact + becomes manifest. In reference to the axioms, caution is + necessary. It should be pointed out comparatively early, in how + far the mathematical body differs from the material body. + Furthermore, since mathematical bodies are really portions of + space, this space is to be conceived as mathematical space and + to be clearly distinguished from real or physical space. + Gradually the student will become conscious that the portion of + the real space which lies beyond the visible stellar universe + is not cognizable through the senses, that we know nothing of + its properties and consequently have no basis for judgments + concerning it. Mathematical space, on the other hand, may be + subjected to conditions, for instance, we may condition its + properties at infinity, and these conditions constitute the + axioms, say the Euclidean axioms. But every student will + require years before the conviction of the truth of this last + statement will force itself upon + him.—<span class="smcap">Holzmüller, Gustav.</span></p> + <p class="blockcite"> + Methodisches Lehrbuch der Elementar-Mathematik (Leipzig, + 1904), Teil 1, Vorwort, pp. 4-5.</p> + + <p class="v2"> + <b><a name="Block_519" id="Block_519">519</a>.</b> + Like almost every subject of human interest, this + one [mathematics] is just as easy or as difficult as we choose + to make it. A lifetime may be spent by a philosopher in + discussing the truth of the simplest axiom. The simplest fact + as to our existence may fill us with such wonder that our minds + will remain overwhelmed with wonder all the time. A Scotch + ploughman makes a working religion out of a system which + appalls a mental philosopher. Some boys of ten years of age + study the methods of the differential calculus; other much + cleverer boys working at mathematics to the age of nineteen + have a difficulty in comprehending the fundamental ideas of the + calculus.—<span class="smcap">Perry, John.</span></p> + <p class="blockcite"> + The Teaching of Mathematics (London, 1902), pp. 19-20.</p> + + <p class="v2"> + <b><a name="Block_520" id="Block_520">520</a>.</b> + Poor teaching leads to the inevitable idea that the subject + [mathematics] is only adapted to peculiar minds, when it is + +<span class="pagenum"> + <a name="Page_79" + id="Page_79">79</a></span> + + the one universal science and the one + whose four ground-rules are taught us almost in infancy and + reappear in the motions of the + universe.—<span class="smcap">Safford, T. H.</span></p> + <p class="blockcite"> + Mathematical Teaching (Boston, 1907), p. 19.</p> + + <p class="v2"> + <b><a name="Block_521" id="Block_521">521</a>.</b> + The number of mathematical students ... would be + much augmented if those who hold the highest rank in science + would condescend to give more effective assistance in clearing + the elements of the difficulties which they + present.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Study and Difficulties of Mathematics (Chicago, 1902), + Preface.</p> + + <p class="v2"> + <b><a name="Block_522" id="Block_522">522</a>.</b> + He that could teach mathematics well, would not be + a bad teacher in any of the rest [physics, chemistry, biology, + psychology] unless by the accident of total inaptitude for + experimental illustration; while the mere experimentalist is + likely to fall into the error of missing the essential + condition of science as reasoned truth; not to speak of the + danger of making the instruction an affair of sensation, + glitter, or pyrotechnic + show.—<span class="smcap">Bain, Alexander.</span></p> + <p class="blockcite"> + Education as a Science (New York, 1898), p. 298.</p> + + <p class="v2"> + <b><a name="Block_523" id="Block_523">523</a>.</b> + I should like to draw attention to the + inexhaustible variety of the problems and exercises which it + [mathematics] furnishes; these may be graduated to precisely + the amount of attainment which may be possessed, while yet + retaining an interest and value. It seems to me that no other + branch of study at all compares with mathematics in this. When + we propose a deduction to a beginner we give him an exercise in + many cases that would have been admired in the vigorous days of + Greek geometry. Although grammatical exercises are well suited + to insure the great benefits connected with the study of + languages, yet these exercises seem to me stiff and artificial + in comparison with the problems of mathematics. It is not + absurd to maintain that Euclid and Apollonius would have + regarded with interest many of the elegant deductions which are + invented for the use of our students in geometry; but it seems + scarcely conceivable + +<span class="pagenum"> + <a name="Page_80" + id="Page_80">80</a></span> + + that the great masters in any other line of study could + condescend to give a moment’s attention to + the elementary books of the + beginner.—<span class="smcap">Todhunter, Isaac.</span></p> + <p class="blockcite"> + Conflict of Studies (London, 1873), pp. 10-11.</p> + + <p class="v2"> + <b><a name="Block_524" id="Block_524">524</a>.</b> + The visible figures by which principles are + illustrated should, so far as possible, have no accessories. + They should be magnitudes pure and simple, so that the thought + of the pupil may not be distracted, and that he may know what + features of the thing represented he is to pay attention to.</p> + <p class="blockcite"> + Report of the Committee of Ten on Secondary School + Subjects, (New York, 1894), p. 109.</p> + + <p class="v2"> + <b><a name="Block_525" id="Block_525">525</a>.</b> + Geometrical reasoning, and arithmetical process, + have each its own office: to mix the two in elementary + instruction, is injurious to the proper acquisition of + both.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Trigonometry and Double Algebra (London, 1849), p. 92.</p> + + <p class="v2"> + <b><a name="Block_526" id="Block_526">526</a>.</b> + Equations are Expressions of Arithmetical + Computation, and properly have no place in Geometry, except as + far as Quantities truly Geometrical (that is, Lines, Surfaces, + Solids, and Proportions) may be said to be some equal to + others. Multiplications, Divisions, and such sort of + Computations, are newly received into Geometry, and that + unwarily, and contrary to the first Design of this Science. For + whosoever considers the Construction of a Problem by a right + Line and a Circle, found out by the first Geometricians, will + easily perceive that Geometry was invented that we might + expeditiously avoid, by drawing Lines, the Tediousness of + Computation. Therefore these two Sciences ought not to be + confounded. The Ancients did so industriously distinguish them + from one another, that they never introduced Arithmetical Terms + into Geometry. And the Moderns, by confounding both, have lost + the Simplicity in which all the Elegance of Geometry consists. + Wherefore that is <em>Arithmetically</em> more simple which is + determined by the more simple Equation, but that is + <em>Geometrically</em> more simple which is determined by the + more simple drawing of Lines; and in Geometry, + +<span class="pagenum"> + <a name="Page_81" + id="Page_81">81</a></span> + + that ought to be reckoned best which is geometrically most + simple.—<span class="smcap">Newton.</span></p> + <p class="blockcite"> + On the Linear Construction of Equations; Universal + Arithmetic (London, 1769), Vol. 2, p. 470.</p> + + <p class="v2"> + <b><a name="Block_527" id="Block_527">527</a>.</b> + As long as algebra and geometry proceeded along + separate paths, their advance was slow and their applications + limited.</p> + <p class="v1"> + But when these sciences joined company, they drew from each + other fresh vitality and thenceforward marched on at a rapid + pace toward perfection.—<span class="smcap">Lagrange.</span></p> + <p class="blockcite"> + Leçons Élémentaires sur les Mathematiques, Leçon cinquiéme. + [McCormack].</p> + + <p class="v2"> + <b><a name="Block_528" id="Block_528">528</a>.</b> + The greatest enemy to true arithmetic work is found + in so-called practical or illustrative problems, which are + freely given to our pupils, of a degree of difficulty and + complexity altogether unsuited to their age and mental + development.... I am, myself, no bad mathematician, and all the + reasoning powers with which nature endowed me have long been as + fully developed as they are ever likely to be; but I have, not + infrequently, been puzzled, and at times foiled, by the subtle + logical difficulty running through one of these problems, given + to my own children. The head-master of one of our Boston high + schools confessed to me that he had sometimes been unable to + unravel one of these tangled skeins, in trying to help his own + daughter through her evening’s work. During + this summer, Dr. Fairbairn, the distinguished head of one of + the colleges of Oxford, England, told me that not only had he + himself encountered a similar difficulty, in the case of his + own children, but that, on one occasion, having as his guest + one of the first mathematicians of England, the two together + had been completely puzzled by one of these arithmetical + conundrums.—<span class="smcap">Walker, F. A.</span></p> + <p class="blockcite"> + Discussions in Education (New York, 1899), pp. 253-254.</p> + + <p class="v2"> + <b><a name="Block_529" id="Block_529">529</a>.</b> + It is often assumed that because the young child is not + competent to study geometry systematically he need be taught + nothing geometrical; that because it would be foolish to present + +<span class="pagenum"> + <a name="Page_82" + id="Page_82">82</a></span> + + to him physics and mechanics as sciences it is useless to + present to him any physical or mechanical principles.</p> + <p class="v1"> + An error of like origin, which has wrought incalculable + mischief, denies to the scholar the use of the symbols and + methods of algebra in connection with his early essays in + numbers because, forsooth, he is not as yet capable of + mastering quadratics!... The whole infant generation, wrestling + with arithmetic, seek for a sign and groan and travail together + in pain for the want of it; but no sign is given them save the + sign of the prophet Jonah, <em>the withered gourd</em>, fruitless + endeavor, wasted strength.—<span class= + "smcap">Walker, F. A.</span></p> + <p class="blockcite"> + Industrial Education; Discussions in Education (New York, + 1899), p. 132.</p> + + <p class="v2"> + <b><a name="Block_530" id="Block_530">530</a>.</b> + Particular and contingent inventions in the + solution of problems, which, though many times more concise + than a general method would allow, yet, in my judgment, are + less proper to instruct a learner, as acrostics, and such kind + of artificial poetry, though never so excellent, would be but + improper examples to instruct one that aims at Ovidean + poetry.—<span class="smcap">Newton, Isaac.</span></p> + <p class="blockcite"> + Letter to Collins, 1670; Macclesfield, Correspondence of + Scientific Men (Oxford, 1841), Vol. 2, p. 307.</p> + + <p class="v2"> + <b><a name="Block_531" id="Block_531">531</a>.</b> + The logic of the subject [algebra], which, both + educationally and scientifically speaking, is the most + important part of it, is wholly neglected. The whole training + consists in example grinding. What should have been merely the + help to attain the end has become the end itself. The result is + that algebra, as we teach it, is neither an art nor a science, + but an ill-digested farrago of rules, whose object is the + solution of examination problems.... The result, so far as + problems worked in examinations go, is, after all, very + miserable, as the reiterated complaints of examiners show; the + effect on the examinee is a well-known enervation of mind, an + almost incurable superficiality, which might be called + Problematic Paralysis—a disease which unfits a man to follow an + argument extending beyond the length of a printed octavo + page.—<span class="smcap">Chrystal, George.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science, 1885; Nature, Vol. 32, pp. 447-448.</p> + +<p><span class="pagenum"> + <a name="Page_83" + id="Page_83">83</a></span></p> + + <p class="v2"> + <b><a name="Block_532" id="Block_532">532</a>.</b> + It is a serious question whether America, following + England’s lead, has not gone into + problem-solving too extensively. Certain it is that we are + producing no text-books in which the theory is presented in the + delightful style which characterizes many of the French works + ..., or those of the recent Italian school, or, indeed, those + of the continental writers in + general.—<span class="smcap">Smith, D. E.</span></p> + <p class="blockcite"> + The Teaching of Elementary Mathematics (New York, 1902), + p. 219.</p> + + <p class="v2"> + <b><a name="Block_533" id="Block_533">533</a>.</b> + The problem for a writer of a text-book has come + now, in fact, to be this—to write a book so + neatly trimmed and compacted that no coach, on looking through + it, can mark a single passage which the candidate for a minimum + pass can safely omit. Some of these text-books I have seen, + where the scientific matter has been, like the + lady’s waist in the nursery song, compressed “so gent and + sma’,” that the thickness barely, if at all, surpasses what is + devoted to the publisher’s advertisements. We shall return, + I verily believe, to the Compendium of Martianus Capella. The + result of all this is that science, in the hands of + specialists, soars higher and higher into the light of day, + while educators and the educated are left more and more to + wander in primeval darkness.—<span class= + "smcap">Chrystal, George.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science, 1885; Nature, Vol. 32, p. 448.</p> + + <p class="v2"> + <b><a name="Block_534" id="Block_534">534</a>.</b> + Some persons have contended that mathematics ought + to be taught by making the illustrations obvious to the senses. + Nothing can be more absurd or injurious: it ought to be our + never-ceasing effort to make people think, not + feel.—<span class="smcap">Coleridge, S. T.</span></p> + <p class="blockcite"> + Lectures on Shakespere (Bohn Library), p. 52.</p> + + <p class="v2"> + <b><a name="Block_535" id="Block_535">535</a>.</b> + I have come to the conclusion that the exertion, + without which a knowledge of mathematics cannot be acquired, is + not materially increased by logical rigor in the method of + instruction.—<span class="smcap">Pringsheim, Alfred.</span></p> + <p class="blockcite"> + Jahresbericht der Deutschen Mathematiker Vereinigung + (1898), p. 143.</p> + +<p><span class="pagenum"> + <a name="Page_84" + id="Page_84">84</a></span></p> + + <p class="v2"> + <b><a name="Block_536" id="Block_536">536</a>.</b> + The only way in which to treat the elements of an + exact and rigorous science is to apply to them all the rigor + and exactness + possible.—<span class= "smcap">D’Alembert.</span></p> + <p class="blockcite"> + Quoted by De Morgan: Trigonometry and Double Algebra + (London, 1849), Title page.</p> + + <p class="v2"> + <b><a name="Block_537" id="Block_537">537</a>.</b> + It is an error to believe that rigor in proof is an + enemy of simplicity. On the contrary we find it confirmed by + numerous examples that the rigorous method is at the same time + the simpler and the more easily comprehended. The very effort + for rigor forces us to find out simpler methods of + proof.—<span class="smcap">Hilbert, D.</span></p> + <p class="blockcite"> + Mathematical Problems; Bulletin American Mathematical + Society, Vol. 8, p. 441.</p> + + <p class="v2"> + <b><a name="Block_538" id="Block_538">538</a>.</b> + Few will deny that even in the first scientific + instruction in mathematics the most rigorous method is to be + given preference over all others. Especially will every teacher + prefer a consistent proof to one which is based on fallacies or + proceeds in a vicious circle, indeed it will be morally + impossible for the teacher to present a proof of the latter + kind consciously and thus in a sense deceive his pupils. + Notwithstanding these objectionable so-called proofs, so far as + the foundation and the development of the system is concerned, + predominate in our textbooks to the present time. Perhaps it + will be answered, that rigorous proof is found too difficult + for the pupil’s power of comprehension. Should this be anywhere + the case,—which would only indicate some defect in the plan or + treatment of the whole,—the only remedy would be to merely + state the theorem in a historic way, and forego a proof with + the frank confession that no proof has been found which could + be comprehended by the pupil; a remedy which is ever doubtful + and should only be applied in the case of extreme necessity. + But this remedy is to be preferred to a proof which is no + proof, and is therefore either wholly unintelligible to the + pupil, or deceives him with an appearance of knowledge which + opens the door to all superficiality and lack of scientific + method.—<span class="smcap">Grassmann, Hermann.</span></p> + <p class="blockcite"> + Stücke aus dem Lehrbuche der Arithmetik; + Werke, Bd. 2 (Leipsig, 1904), p. 296.</p> + +<p><span class="pagenum"> + <a name="Page_85" + id="Page_85">85</a></span></p> + + <p class="v2"> + <b><a name="Block_539" id="Block_539">539</a>.</b> + The average English author [of mathematical texts] + leaves one under the impression that he has made a bargain with + his reader to put before him the truth, the greater part of the + truth, and nothing but the truth; and that if he has put the + facts of his subject into his book, however difficult it may be + to unearth them, he has fulfilled his contract with his reader. + This is a very much mistaken view, because <em>effective + teaching</em> requires a great deal more than a bare recitation + of facts, even if these are duly set forth in logical + order—as in English books they often are + not. The probable difficulties which will occur to the student, + the objections which the intelligent student will naturally and + necessarily raise to some statement of fact or + theory—these things our authors seldom or + never notice, and yet a recognition and anticipation of them by + the author would be often of priceless value to the student. + Again, a touch of <em>humour</em> (strange as the contention may + seem) in mathematical works is not only possible with perfect + propriety, but very helpful; and I could give instances of this + even from the pure mathematics of Salmon and the physics of + Clerk Maxwell.—<span class="smcap">Minchin, G. M.</span></p> + <p class="blockcite"> + Perry’s Teaching of Mathematics (London, 1902), pp. 59-61.</p> + + <p class="v2"> + <b><a name="Block_540" id="Block_540">540</a>.</b> + Remember this, the rule for giving an extempore + lecture is—let + + <a id="TNanchor_4"></a> + <a class="msg" href="#TN_4" + title="duplicate word ‘the’ removed">the</a> + + mind rest from the + subject entirely for an interval preceding the lecture, after + the notes are prepared; the thoughts will ferment without your + knowing it, and enter into new combinations; but if you keep + the mind active upon the subject up to the moment, the subject + will not ferment but stupefy.—<span class= + "smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Letter to Hamilton; Graves: Life of W. R. Hamilton (New + York, 1882-1889), Vol. 3, p. 487.</p> + +<p><span class="pagenum"> + <a name="Page_86" + id="Page_86">86</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_VI"> + CHAPTER VI<br /> + <span class="large"> + STUDY AND RESEARCH IN MATHEMATICS</span></h2> + + <p class="v2"> + <b><a name="Block_601" id="Block_601">601</a>.</b> + The first thing to be attended to in reading any + algebraic treatise is the gaining a perfect understanding of + the different processes there exhibited, and of their + connection with one another. This cannot be attained by the + mere reading of the book, however great the attention which may + be given. It is impossible in a mathematical work to fill up + every process in the manner in which it must be filled up in + the mind of the student before he can be said to have + completely mastered it. Many results must be given of which the + details are suppressed, such are the additions, + multiplications, extractions of square roots, etc., with which + the investigations abound. These must not be taken on trust by + the student, but must be worked out by his own pen, which must + never be out of his own hand while engaged in any mathematical + process.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Study and Difficulties of Mathematics (Chicago, 1902), + chap. 12.</p> + + <p class="v2"> + <b><a name="Block_602" id="Block_602">602</a>.</b> + The student should not lose any opportunity of + exercising himself in numerical calculation and particularly in + the use of logarithmic tables. His power of applying + mathematics to questions of practical utility is in direct + proportion to the facility which he possesses in + computation.—<span class="smcap">De Morgan,A.</span></p> + <p class="blockcite"> + Study and Difficulties of Mathematics (Chicago, 1902), + chap. 12.</p> + + <p class="v2"> + <b><a name="Block_603" id="Block_603">603</a>.</b> + The examples which a beginner should choose for + practice should be simple and should not contain very large + numbers. The powers of the mind cannot be directed to two + things at once; if the complexity of the numbers used requires + all the student’s attention, he cannot + observe the principle of the rule which he is + following.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Study and Difficulties of Mathematics (Chicago, 1902), + chap. 3.</p> + +<p><span class="pagenum"> + <a name="Page_87" + id="Page_87">87</a></span></p> + + <p class="v2"> + <b><a name="Block_604" id="Block_604">604</a>.</b> + Euclid and Archimedes are allowed to be knowing, + and to have demonstrated what they say: and yet whosoever shall + read over their writings without perceiving the connection of + their proofs, and seeing what they show, though he may + understand all their words, yet he is not the more knowing. He + may believe, indeed, but does not know what they say, and so is + not advanced one jot in mathematical knowledge by all his + reading of those approved + mathematicians.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + Conduct of the Understanding, sect. 24.</p> + + <p class="v2"> + <b><a name="Block_605" id="Block_605">605</a>.</b> + The student should read his author with the most + sustained attention, in order to discover the meaning of every + sentence. If the book is well written, it will endure and repay + his close attention: the text ought to be fairly intelligible, + even without illustrative examples. Often, far too often, a + reader hurries over the text without any sincere and vigorous + effort to understand it; and rushes to some example to clear up + what ought not to have been obscure, if it had been adequately + considered. The habit of scrupulously investigating the text + seems to me important on several grounds. The close scrutiny of + language is a very valuable exercise both for studious and + practical life. In the higher departments of mathematics the + habit is indispensable: in the long investigations which occur + there it would be impossible to interpose illustrative examples + at every stage, the student must therefore encounter and + master, sentence by sentence, an extensive and complicated + argument.—<span class="smcap">Todhunter, Isaac.</span></p> + <p class="blockcite"> + Private Study of Mathematics; Conflict of Studies and + other Essays (London, 1873), p. 67.</p> + + <p class="v2"> + <b><a name="Block_606" id="Block_606">606</a>.</b> + It must happen that in some cases the author is not + understood, or is very imperfectly understood; and the question + is what is to be done. After giving a reasonable amount of + attention to the passage, let the student pass on, reserving + the obscurity for future efforts.... The natural tendency of + solitary students, I believe, is not to hurry away prematurely + from a hard passage, but to hang far too long over it; the just + pride that does not like to acknowledge defeat, and the strong + will that cannot endure to be thwarted, both urge to a + continuance of effort even when success seems + hopeless. It is only by experience + +<span class="pagenum"> + <a name="Page_88" + id="Page_88">88</a></span> + + we gain the conviction that + when the mind is thoroughly fatigued it has neither the power + to continue with advantage its course in an assigned direction, + nor elasticity to strike out a new path; but that, on the other + hand, after being withdrawn for a time from the pursuit, it may + return and gain the desired + end.—<span class="smcap">Todhunter, Isaac.</span></p> + <p class="blockcite"> + Private Study of Mathematics; Conflict of Studies and + other Essays (London, 1873), p. 68.</p> + + <p class="v2"> + <b><a name="Block_607" id="Block_607">607</a>.</b> + Every mathematical book that is worth reading must + be read “backwards and + forwards,” if I may use the expression. I would + modify Lagrange’s advice a little and say, + “Go on, but often return to strengthen your + faith.” When you come on a hard or dreary passage, + pass it over; and come back to it after you have seen its + importance or found the need for it further + on.—<span class="smcap">Chrystal, George.</span></p> + <p class="blockcite"> + Algebra, Part 2 (Edinburgh, 1889), Preface, p. 8.</p> + + <p class="v2"> + <b><a name="Block_608" id="Block_608">608</a>.</b> + The large collection of problems which our modern + Cambridge books supply will be found to be almost an exclusive + peculiarity of these books; such collections scarcely exist in + foreign treatises on mathematics, nor even in English treatises + of an earlier date. This fact shows, I think, that a knowledge + of mathematics may be gained without the perpetual working of + examples.... Do not trouble yourselves with the examples, make + it your main business, I might almost say your exclusive + business, to understand the text of your + author.—<span class="smcap">Todhunter, Isaac.</span></p> + <p class="blockcite"> + Private Study of Mathematics; Conflict of Studies and + other Essays (London, 1873), p. 74.</p> + + <p class="v2"> + <b><a name="Block_609" id="Block_609">609</a>.</b> + In my opinion the English excel in the art of + writing text-books for mathematical teaching; as regards the + clear exposition of theories and the abundance of excellent + examples, carefully selected, very few books exist in other + countries which can compete with those of Salmon and many other + distinguished English authors that could be + named.—<span class="smcap">Cremona, L.</span></p> + <p class="blockcite"> + Projective Geometry [Leudesdorf] (Oxford, 1885), Preface.</p> + +<p><span class="pagenum"> + <a name="Page_89" + id="Page_89">89</a></span></p> + + <p class="v2"> + <b><a name="Block_610" id="Block_610">610</a>.</b> + The solution of fallacies, which give rise to + absurdities, should be to him who is not a first beginner in + mathematics an excellent means of testing for a proper + intelligible insight into mathematical truth, of sharpening the + wit, and of confining the judgment and reason within strictly + orderly limits.—<span class="smcap">Viola, + J.</span></p> + <p class="blockcite"> + Mathematische Sophismen (Wien, 1864), Vorwort.</p> + + <p class="v2"> + <b><a name="Block_611" id="Block_611">611</a>.</b> + Success in the solution of a problem generally + depends in a great measure on the selection of the most + appropriate method of approaching it; many properties of conic + sections (for instance) being demonstrable by a few steps of + pure geometry which would involve the most laborious operations + with trilinear co-ordinates, while other properties are almost + self-evident under the method of trilinear co-ordinates, which + it would perhaps be actually impossible to prove by the old + geometry.—<span class="smcap">Whitworth, W. A.</span></p> + <p class="blockcite"> + Modern Analytic Geometry (Cambridge, 1866), p. 154.</p> + + <p class="v2"> + <b><a name="Block_612" id="Block_612">612</a>.</b> + The deep study of nature is the most fruitful + source of mathematical discoveries. By offering to research a + definite end, this study has the advantage of excluding vague + questions and useless calculations; besides it is a sure means + of forming analysis itself and of discovering the elements + which it most concerns us to know, and which natural science + ought always to + conserve.—<span class="smcap">Fourier, J.</span></p> + <p class="blockcite"> + Théorie Analytique de la Chaleur, Discours Préliminaire.</p> + + <p class="v2"> + <b><a name="Block_613" id="Block_613">613</a>.</b> + It is certainly true that all physical phenomena + are subject to strictly mathematical conditions, and + mathematical processes are unassailable in themselves. The + trouble arises from the data employed. Most phenomena are so + highly complex that one can never be quite sure that he is + dealing with all the factors until the experiment proves it. So + that experiment is rather the criterion of mathematical + conclusions and must lead the + way.—<span class="smcap">Dolbear, A. E.</span></p> + <p class="blockcite"> + Matter, Ether, Motion (Boston, 1894), p. 89.</p> + +<p><span class="pagenum"> + <a name="Page_90" + id="Page_90">90</a></span></p> + + <p class="v2"> + <b><a name="Block_614" id="Block_614">614</a>.</b> + Students should learn to study at an early stage + the great works of the great masters instead of making their + minds sterile through the everlasting exercises of college, + which are of no use whatever, except to produce a new Arcadia + where indolence is veiled under the form of useless + activity.... Hard study on the great models has ever brought + out the strong; and of such must be our new scientific + generation if it is to be worthy of the era to which it is born + and of the struggles to which it is + destined.—<span class="smcap">Beltrami.</span></p> + <p class="blockcite"> + Giornale di matematiche, Vol. 11, p. 153. [Young, J. W.]</p> + + <p class="v2"> + <b><a name="Block_615" id="Block_615">615</a>.</b> + The history of mathematics may be instructive as + well as agreeable; it may not only remind us of what we have, + but may also teach us to increase our store. Says De Morgan, + “The early history of the mind of men with + regards to mathematics leads us to point out our own errors; + and in this respect it is well to pay attention to the history + of mathematics.” It warns us against hasty + conclusions; it points out the importance of a good notation + upon the progress of the science; it discourages excessive + specialization on the part of the investigator, by showing how + apparently distinct branches have been found to possess + unexpected connecting links; it saves the student from wasting + time and energy upon problems which were, perhaps, solved long + since; it discourages him from attacking an unsolved problem by + the same method which has led other mathematicians to failure; + it teaches that fortifications can be taken by other ways than + by direct attack, that when repulsed from a direct assault it + is well to reconnoitre and occupy the surrounding ground and to + discover the secret paths by which the apparently unconquerable + position can be taken.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Mathematics (New York, 1897), pp. 1-2.</p> + + <p class="v2"> + <b><a name="Block_616" id="Block_616">616</a>.</b> + The history of mathematics is important also as a + valuable contribution to the history of civilization. Human + progress is closely identified with scientific thought. + Mathematical and physical researches are a reliable record of + intellectual progress.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Mathematics (New York, 1897), p. 4.</p> + +<p><span class="pagenum"> + <a name="Page_91" + id="Page_91">91</a></span></p> + + <p class="v2"> + <b><a name="Block_617" id="Block_617">617</a>.</b> + It would be rash to say that nothing remains for + discovery or improvement even in elementary mathematics, but it + may be safely asserted that the ground has been so long and so + thoroughly explored as to hold out little hope of profitable + return for a casual adventurer.—<span class= + "smcap">Todhunter, Isaac.</span></p> + <p class="blockcite"> + Private Study of Mathematics; Conflict of Studies and + other Essays (London, 1873), p. 73.</p> + + <p class="v2"> + <b><a name="Block_618" id="Block_618">618</a>.</b> + We do not live in a time when knowledge can be + extended along a pathway smooth and free from obstacles, as at + the time of the discovery of the infinitesimal calculus, and in + a measure also when in the development of projective geometry + obstacles were suddenly removed which, having hemmed progress + for a long time, permitted a stream of investigators to pour in + upon virgin soil. There is no longer any browsing along the + beaten paths; and into the primeval forest only those may + venture who are equipped with the sharpest + tools.—<span class="smcap">Burkhardt, H.</span></p> + <p class="blockcite"> + Mathematisches und wissenschaftliches Denken; Jahresbericht + der Deutschen Mathematiker Vereinigung, Bd. 11, p. 55.</p> + + <p class="v2"> + <b><a name="Block_619" id="Block_619">619</a>.</b> + Though we must not without further consideration + condemn a body of reasoning merely because it is easy, + nevertheless we must not allow ourselves to be lured on merely + by easiness; and we should take care that every problem which + we choose for attack, whether it be easy or difficult, shall + have a useful purpose, that it shall contribute in some measure + to the up-building of the great + edifice.—<span class="smcap">Segre, Corradi.</span></p> + <p class="blockcite"> + Some Recent Tendencies in Geometric Investigation; Rivista + di Matematica (1891), p. 63. Bulletin American Mathematical + Society, 1904, p. 465. [Young, J. W.].</p> + + <p class="v2"> + <b><a name="Block_620" id="Block_620">620</a>.</b> + No mathematician now-a-days sets any store on the + discovery of isolated theorems, except as affording hints of an + unsuspected new sphere of thought, like meteorites detached + from some undiscovered planetary orb of + speculation.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Notes to the Exeter Association Address; Collected + Mathematical Papers (Cambridge, 1908), Vol. 2, p. 715.</p> + +<p><span class="pagenum"> + <a name="Page_92" + id="Page_92">92</a></span></p> + + <p class="v2"> + <b><a name="Block_621" id="Block_621">621</a>.</b> + Isolated, so-called “pretty theorems” have even less value in + the eyes of a modern mathematician than the discovery of a new + “pretty flower” has to the scientific botanist, though the + layman finds in these the chief charm of the respective + sciences.—<span class="smcap">Hankel, Hermann.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten + Jahrhunderten (Tübingen, 1884), p. 15.</p> + + <p class="v2"> + <b><a name="Block_622" id="Block_622">622</a>.</b> + It is, so to speak, a scientific tact, which must + guide mathematicians in their investigations, and guard them + from spending their forces on scientifically worthless problems + and abstruse realms, a tact which is closely related to + <em>esthetic tact</em> and which is the only thing in our science + which cannot be taught or acquired, and is yet the + indispensable endowment of every + mathematician.—<span class="smcap">Hankel, Hermann.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten + Jahrhunderten (Tübingen, 1884), p. 21.</p> + + <p class="v2"> + <b><a name="Block_623" id="Block_623">623</a>.</b> + The mathematician requires tact and good taste at + every step of his work, and he has to learn to trust to his own + instinct to distinguish between what is really worthy of his + efforts and what is not; he must take care not to be the slave + of his symbols, but always to have before his mind the + realities which they merely serve to express. For these and + other reasons it seems to me of the highest importance that a + mathematician should be trained in no narrow school; a wide + course of reading in the first few years of his mathematical + study cannot fail to influence for good the character of the + whole of his subsequent work.—<span class= + "smcap">Glaisher, J. W. L.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A, (1890); Nature, Vol. 42, p. 467.</p> + + <p class="v2"> + <b><a name="Block_624" id="Block_624">624</a>.</b> + As long as a branch of science offers an abundance + of problems, so long it is alive; a lack of problems + foreshadows extinction or the cessation of independent + development.—<span class="smcap">Hilbert, D.</span></p> + <p class="blockcite"> + Mathematical Problems; Bulletin American Mathematical + Society, Vol. 8, p. 438.</p> + +<p><span class="pagenum"> + <a name="Page_93" + id="Page_93">93</a></span></p> + + <p class="v2"> + <b><a name="Block_625" id="Block_625">625</a>.</b> + In mathematics as in other fields, to find one self lost in + wonder at some manifestation is frequently the half of a new + discovery.—<span class= "smcap">Dirichlet, P. G. L.</span></p> + <p class="blockcite"> + Werke, Bd. 2 (Berlin, 1897), p. 233.</p> + + <p class="v2"> + <b><a name="Block_626" id="Block_626">626</a>.</b> + The student of mathematics often finds it hard to + throw off the uncomfortable feeling that his science, in the + person of his pencil, surpasses him in + intelligence,—an impression which the great + Euler confessed he often could not get rid of. This feeling + finds a sort of justification when we reflect that the majority + of the ideas we deal with were conceived by others, often + centuries ago. In a great measure it is really the intelligence + of other people that confronts us in + science.—<span class="smcap">Mach, Ernst.</span></p> + <p class="blockcite"> + Popular Scientific Lectures (Chicago, 1910), p. 196.</p> + + <p class="v2"> + <b><a name="Block_627" id="Block_627">627</a>.</b> + It is probably this fact [referring to the + circumstance that the problems of the parallel axiom, the + squaring of the circle, the solution of the equation of the + fifth degree, have finally found fully satisfactory and + rigorous solutions] along with other philosophical reasons that + gives rise to the conviction (which every mathematician shares, + but which no one has yet supported by proof) that every + definite mathematical problem must necessarily be susceptible + of an exact settlement, either in the form of an actual answer + to the question asked, or by the proof of the impossibility of + its solution and therewith the necessary failure of all + attempts.... This conviction of the solvability of every + mathematical problem is a powerful incentive to the worker. We + hear within us the perpetual call: There is the problem. Seek + its solution. You can find it by pure reason, for in + mathematics there is no <i lang="la" + xml:lang="la">ignorabimus</i>.—<span + class="smcap">Hilbert, D.</span></p> + <p class="blockcite"> + Mathematical Problems; Bulletin American Mathematical + Society, Vol. 8, pp. 444-445.</p> + + <p class="v2"> + <b><a name="Block_628" id="Block_628">628</a>.</b> + He who seeks for methods without having a definite + problem in mind seeks for the most part in + vain.—<span class="smcap">Hilbert, D.</span></p> + <p class="blockcite"> + Mathematical Problems; Bulletin American Mathematical + Society, Vol. 8, p. 444.</p> + + <p class="v2"> + <b><a name="Block_629" id="Block_629">629</a>.</b> + A mathematical problem should be difficult in order to entice + us, yet not completely inaccessible, lest it mock at our + +<span class="pagenum"> + <a name="Page_94" + id="Page_94">94</a></span> + + efforts. It should be to us a guide + post on the mazy paths to hidden truths, and ultimately a + reminder of our pleasure in the successful + solution.—<span class="smcap">Hilbert, D.</span></p> + <p class="blockcite"> + Mathematical Problems; Bulletin American Mathematical + Society, Vol. 8, p. 438.</p> + + <p class="v2"> + <b><a name="Block_630" id="Block_630">630</a>.</b> + The great mathematicians have acted on the + principle “<i lang="fr" xml:lang="fr">Divinez avant de + demontrer</i>,” and it is certainly true that + almost all important discoveries are made in this + fashion.—<span class="smcap">Kasner, Edward.</span></p> + <p class="blockcite"> + The Present Problems in Geometry; Bulletin American + Mathematical Society, Vol. 11, p. 285.</p> + + <p class="v2"> + <b><a name="Block_631" id="Block_631">631</a>.</b> + “Divide <i lang="la" xml:lang="la">et + impera</i>” is as true in algebra as in statecraft; + but no less true and even more fertile is the maxim + “auge <i lang="la" xml:lang="la">et impera</i>.” The + more to do or to prove, the easier the doing or the + proof.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Proof of the Fundamental Theorem of Invariants; + Philosophic Magazine (1878), p. 186; Collected Mathematical + Papers, Vol. 3, p. 126.</p> + + <p class="v2"> + <b><a name="Block_632" id="Block_632">632</a>.</b> + As in the domains of practical life so likewise in + science there has come about a division of labor. The + individual can no longer control the whole field of + mathematics: it is only possible for him to master separate + parts of it in such a manner as to enable him to extend the + boundaries of knowledge by creative + research.—<span class="smcap">Lampe, E.</span></p> + <p class="blockcite"> + Die reine Mathematik in den Jahren 1884-1899, p. 10.</p> + + <p class="v2"> + <b><a name="Block_633" id="Block_633">633</a>.</b> + With the extension of mathematical knowledge will + it not finally become impossible for the single investigator to + embrace all departments of this knowledge? In answer let me + point out how thoroughly it is ingrained in mathematical + science that every real advance goes hand in hand with the + invention of sharper tools and simpler methods which at the + same time assist in understanding earlier theories and to cast + aside some more complicated developments. It is therefore + +<span class="pagenum"> + <a name="Page_95" + id="Page_95">95</a></span> + + possible for the individual + investigator, when he makes these sharper tools and simpler + methods his own, to find his way more easily in the various + branches of mathematics than is possible in any other + science.—<span class="smcap">Hilbert, D.</span></p> + <p class="blockcite"> + Mathematical Problems; Bulletin American Mathematical + Society, Vol. 8, p. 479.</p> + + <p class="v2"> + <b><a name="Block_634" id="Block_634">634</a>.</b> + It would seem at first sight as if the rapid + expansion of the region of mathematics must be a source of + danger to its future progress. Not only does the area widen but + the subjects of study increase rapidly in number, and the work + of the mathematician tends to become more and more specialized. + It is, of course, merely a brilliant exaggeration to say that + no mathematician is able to understand the work of any other + mathematician, but it is certainly true that it is daily + becoming more and more difficult for a mathematician to keep + himself acquainted, even in a general way, with the progress of + any of the branches of mathematics except those which form the + field of his own labours. I believe, however, that the + increasing extent of the territory of mathematics will always + be counteracted by increased facilities in the means of + communication. Additional knowledge opens to us new principles + and methods which may conduct us with the greatest ease to + results which previously were most difficult of access; and + improvements in notation may exercise the most powerful effects + both in the simplification and accessibility of a subject. It + rests with the worker in mathematics not only to explore new + truths, but to devise the language by which they may be + discovered and expressed; and the genius of a great + mathematician displays itself no less in the notation he + invents for deciphering his subject than in the results + attained.... I have great faith in the power of well-chosen + notation to simplify complicated theories and to bring remote + ones near and I think it is safe to predict that the increased + knowledge of principles and the resulting improvements in the + symbolic language of mathematics will always enable us to + grapple satisfactorily with the difficulties arising from the + mere extent of the subject.—<span class= + "smcap">Glaisher, J. W. L.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A., (1890), Nature, Vol. 42, p. 466.</p> + +<p><span class="pagenum"> + <a name="Page_96" + id="Page_96">96</a></span></p> + + <p class="v2"> + <b><a name="Block_635" id="Block_635">635</a>.</b> + Quite distinct from the theoretical question of the + manner in which mathematics will rescue itself from the perils + to which it is exposed by its own prolific nature is the + practical problem of finding means of rendering available for + the student the results which have been already accumulated, + and making it possible for the learner to obtain some idea of + the present state of the various departments of mathematics.... + The great mass of mathematical literature will be always + contained in Journals and Transactions, but there is no reason + why it should not be rendered far more useful and accessible + than at present by means of treatises or higher text-books. The + whole science suffers from want of avenues of approach, and + many beautiful branches of mathematics are regarded as + difficult and technical merely because they are not easily + accessible.... I feel very strongly that any introduction to a + new subject written by a competent person confers a real + benefit on the whole science. The number of excellent + text-books of an elementary kind that are published in this + country makes it all the more to be regretted that we have so + few that are intended for the advanced student. As an example + of the higher kind of text-book, the want of which is so badly + felt in many subjects, I may mention the second part of Prof. + Chrystal’s “Algebra” published last year, + which in a small compass gives a great mass of valuable and + fundamental knowledge that has hitherto been beyond the reach + of an ordinary student, though in reality lying so close at + hand. I may add that in any treatise or higher text-book it is + always desirable that references to the original memoirs should + be given, and, if possible, short historic notices also. I am + sure that no subject loses more than mathematics by any attempt + to dissociate it from its + history.—<span class="smcap">Glaisher, J. W. L.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A (1890); Nature, Vol. 42, p. 466.</p> + + <p class="v2"> + <b><a name="Block_636" id="Block_636">636</a>.</b> + The more a science advances, the more will it be + possible to understand immediately results which formerly could + be demonstrated only by means of lengthy intermediate + considerations: a mathematical subject cannot be considered as + finally completed until this end has been + attained.—<span class="smcap">Gordan, Paul.</span></p> + <p class="blockcite"> + Formensystem binärer Formen (Leipzig, 1875), p. 2.</p> + +<p><span class="pagenum"> + <a name="Page_97" + id="Page_97">97</a></span></p> + + <p class="v2"> + <b><a name="Block_637" id="Block_637">637</a>.</b> + An old French geometer used to say that a + mathematical theory was never to be considered complete till + you had made it so clear that you could explain it to the first + man you met in the street.—<span class= + "smcap">Smith, H. J. S.</span></p> + <p class="blockcite"> + Nature, Vol. 8 (1873), p. 452.</p> + + <p class="v2"> + <b><a name="Block_638" id="Block_638">638</a>.</b> + In order to comprehend and fully control + arithmetical concepts and methods of proof, a high degree of + abstraction is necessary, and this condition has at times been + charged against arithmetic as a fault. I am of the opinion that + all other fields of knowledge require at least an equally high + degree of abstraction as mathematics,—provided, that in these + fields the foundations are also everywhere examined with the + rigour and completeness which is actually + necessary.—<span class="smcap">Hilbert, D.</span></p> + <p class="blockcite"> + Die Theorie der algebraischen Zahlkorper, Vorwort; Jahresbericht + der Deutschen Mathematiker Vereinigung, Bd. 4.</p> + + <p class="v2"> + <b><a name="Block_639" id="Block_639">639</a>.</b> + The anxious precision of modern mathematics is + necessary for accuracy, ... it is necessary for research. It + makes for clearness of thought and for fertility in trying new + combinations of ideas. When the initial statements are vague + and slipshod, at every subsequent stage of thought, common + sense has to step in to limit applications and to explain + meanings. Now in creative thought common sense is a bad master. + Its sole criterion for judgment is that the new ideas shall + look like the old ones, in other words it can only act by + suppressing originality.—<span class= + "smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Introduction to Mathematics (New York, 1911), p. 157.</p> + + <p class="v2"> + <b><a name="Block_640" id="Block_640">640</a>.</b> + Mathematicians attach great importance to the + elegance of their methods and their results. This is not pure + dilettantism. What is it indeed that gives us the feeling of + elegance in a solution, in a demonstration? It is the harmony + of the diverse parts, their symmetry, their happy balance; in a + word it is all that introduces order, all that gives unity, + that permits us to see clearly and to comprehend at once both + the <em>ensemble</em> and the details. But this is exactly what + yields great results, in fact the more we see this aggregate + clearly and at a single glance, the better we perceive its + analogies with other neighboring objects, + +<span class="pagenum"> + <a name="Page_98" + id="Page_98">98</a></span> + + consequently the more chances we have of divining the possible + generalizations. Elegance may produce the feeling of the + unforeseen by the unexpected meeting of objects we are not + accustomed to bring together; there again it is fruitful, since + it thus unveils for us kinships before unrecognized. It is + fruitful even when it results only from the contrast between + the simplicity of the means and the complexity of the problem + set; it makes us then think of the reason for this contrast and + very often makes us see that chance is not the reason; that it + is to be found in some unexpected law. In a word, the feeling + of mathematical elegance is only the satisfaction due to any + adaptation of the solution to the needs of our mind, and it is + because of this very adaptation that this solution can be for + us an instrument. Consequently this esthetic satisfaction is + bound up with the economy of thought.—<span class= + "smcap">Poincaré, H.</span></p> + <p class="blockcite"> + The Future of Mathematics; Monist, Vol. 20, p. 80. [Halsted].</p> + + <p class="v2"> + <b><a name="Block_641" id="Block_641">641</a>.</b> + The importance of a result is largely relative, is + judged differently by different men, and changes with the times + and circumstances. It has often happened that great importance + has been attached to a problem merely on account of the + difficulties which it presented; and indeed if for its solution + it has been necessary to invent new methods, noteworthy + artifices, etc., the science has gained more perhaps through + these than through the final result. In general we may call + important all investigations relating to things which in + themselves are important; all those which have a large degree + of generality, or which unite under a single point of view + subjects apparently distinct, simplifying and elucidating them; + all those which lead to results that promise to be the source + of numerous consequences; etc.—<span class= + "smcap">Segre, Corradi.</span></p> + <p class="blockcite"> + Some Recent Tendencies in Geometric Investigations. + Rivista di Matematica, Vol. 1, p. 44. Bulletin American + Mathematical Society, 1904, p. 444. [Young, J. W.].</p> + + <p class="v2"> + <b><a name="Block_642" id="Block_642">642</a>.</b> + Geometric writings are not rare in which one would + seek in vain for an idea at all novel, for a result which + sooner or later might be of service, for anything in fact which + might be + +<span class="pagenum"> + <a name="Page_99" + id="Page_99">99</a></span> + + destined to survive in the science; and + one finds instead treatises on trivial problems or + investigations on special forms which have absolutely no use, + no importance, which have their origin not in the science + itself but in the caprice of the author; or one finds + applications of known methods which have already been made + thousands of times; or generalizations from known results which + are so easily made that the knowledge of the latter suffices to + give at once the former. Now such work is not merely useless; + it is actually harmful because it produces a real incumbrance + in the science and an embarrassment for the more serious + investigators; and because often it crowds out certain lines of + thought which might well have deserved to be + studied.—<span class="smcap">Segre, Corradi.</span></p> + <p class="blockcite"> + On some Recent Tendencies in Geometric Investigations; + Rivista di Matematica, 1891, p. 43. Bulletin American + Mathematical Society, 1904, p. 443 [Young, J. W.].</p> + + <p class="v2"> + <b><a name="Block_643" id="Block_643">643</a>.</b> + A student who wishes now-a-days to study geometry + by dividing it sharply from analysis, without taking account of + the progress which the latter has made and is making, that + student no matter how great his genius, will never be a whole + geometer. He will not possess those powerful instruments of + research which modern analysis puts into the hands of modern + geometry. He will remain ignorant of many geometrical results + which are to be found, perhaps implicitly, in the writings of + the analyst. And not only will he be unable to use them in his + own researches, but he will probably toil to discover them + himself, and, as happens very often, he will publish them as + new, when really he has only rediscovered + them.—<span class="smcap">Segre, Corradi.</span></p> + <p class="blockcite"> + On some recent Tendencies in Geometrical Investigations; + Rivista di Matematica, 1891, p. 43. Bulletin American + Mathematical Society, 1904, p. 443 [Young, J. W.].</p> + + <p class="v2"> + <b><a name="Block_644" id="Block_644">644</a>.</b> + Research may start from definite problems whose + importance it recognizes and whose solution is sought more or + less directly by all forces. But equally legitimate is the + other method of research which only selects the field of its + activity and, contrary to the first method, freely reconnoitres + in the search for problems which are capable of solution. + Different individuals + +<span class="pagenum"> + <a name="Page_100" + id="Page_100">100</a></span> + + will hold different views as to + the relative value of these two methods. If the first method + leads to greater penetration it is also easily exposed to the + danger of unproductivity. To the second method we owe the + acquisition of large and new fields, in which the details of + many things remain to be determined and explored by the first + method.—<span class="smcap">Clebsch, A.</span></p> + <p class="blockcite"> + Zum Gedächtniss an Julius Plücker; Göttinger Abhandlungen, + 16, 1871, Mathematische Classe, p. 6.</p> + + <p class="v2"> + <b><a name="Block_645" id="Block_645">645</a>.</b> + During a conversation with the writer in the last + weeks of his life, <em>Sylvester</em> remarked as curious that + notwithstanding he had always considered the bent of his mind + to be rather analytical than geometrical, he found in nearly + every case that the solution of an analytical problem turned + upon some quite simple geometrical notion, and that he was + never satisfied until he could present the argument in + geometrical language.—<span class= + "smcap">MacMahon, P. A.</span></p> + <p class="blockcite"> + Proceedings London Royal Society, Vol. 63, p. 17.</p> + + <p class="v2"> + <b><a name="Block_646" id="Block_646">646</a>.</b> + The origin of a science is usually to be sought for + not in any systematic treatise, but in the investigation and + solution of some particular problem. This is especially the + case in the ordinary history of the great improvements in any + department of mathematical science. Some problem, mathematical + or physical, is proposed, which is found to be insoluble by + known methods. This condition of insolubility may arise from + one of two causes: Either there exists no machinery powerful + enough to effect the required reduction, or the workmen are not + sufficiently expert to employ their tools in the performance of + an entirely new piece of work. The problem proposed is, + however, finally solved, and in its solution some new + principle, or new application of old principles, is necessarily + introduced. If a principle is brought to light it is soon found + that in its application it is not necessarily limited to the + particular question which occasioned its discovery, and it is + then stated in an abstract form and applied to problems of + gradually increasing generality.</p> + <p class="v1"> + Other principles, similar in their nature, are added, and the + original principle itself receives such + modifications and extensions + +<span class="pagenum"> + <a name="Page_101" + id="Page_101">101</a></span> + + as are from time to time + deemed necessary. The same is true of new applications of old + principles; the application is first thought to be merely + confined to a particular problem, but it is soon recognized + that this problem is but one, and generally a very simple one, + out of a large class, to which the same process of + investigation and solution are applicable. The result in both + of these cases is the same. A time comes when these several + problems, solutions, and principles are grouped together and + found to produce an entirely new and consistent method; a + nomenclature and uniform system of notation is adopted, and the + principles of the new method become entitled to rank as a + distinct science.—<span class="smcap">Craig, Thomas.</span></p> + <p class="blockcite"> + A Treatise on Projection, Preface. U. S. Coast and + Geodetic Survey, Treasury Department Document, No. 61.</p> + + <p class="v2"> + <b><a name="Block_647" id="Block_647">647</a>.</b> + The aim of research is the discovery of the + equations which subsist between the elements of + phenomena.—<span class="smcap">Mach, Ernst.</span></p> + <p class="blockcite"> + Popular Scientific Lectures (Chicago, 1910), p. 205.</p> + + <p class="v2"> + <b><a name="Block_648" id="Block_648">648</a>.</b> + Let him [the author] be permitted also in all + humility to add ... that in consequence of the large arrears of + algebraical and arithmetical speculations waiting in his mind + their turn to be called into outward existence, he is driven to + the alternative of leaving the fruits of his meditations to + perish (as has been the fate of too many foregone theories, the + still-born progeny of his brain, now forever resolved back + again into the primordial matter of thought), or venturing to + produce from time to time such imperfect sketches as the + present, calculated to evoke the mental co-operation of his + readers, in whom the algebraical instinct has been to some + extent developed, rather than to satisfy the strict demands of + rigorously systematic + exposition.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Philosophic Magazine (1863), p. 460.</p> + + <p class="v2"> + <b><a name="Block_649" id="Block_649">649</a>.</b> + In other branches of science, where quick + publication seems to be so much desired, there may possibly be + some excuse for giving to the world slovenly or ill-digested + work, but there is no such excuse in mathematics. The form + ought to be as + +<span class="pagenum"> + <a name="Page_102" + id="Page_102">102</a></span> + + perfect as the substance, and the + demonstrations as rigorous as those of Euclid. The + mathematician has to deal with the most exact facts of Nature, + and he should spare no effort to render his interpretation + worthy of his subject, and to give to his work its highest + degree of perfection. “<i lang="la" xml:lang="la">Pauca + sed matura</i>” was Gauss’s + motto.—<span class="smcap">Glaisher, J. W. L.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A, (1890); Nature, Vol. 42, p. 467.</p> + + <p class="v2"> + <b><a name="Block_650" id="Block_650">650</a>.</b> + It is the man not the method that solves the + problem.—<span class="smcap">Maschke, H.</span></p> + <p class="blockcite"> + Present Problems of Algebra and Analysis; Congress of Arts + and Sciences (New York and Boston, 1905), Vol. 1, p. 530.</p> + + <p class="v2"> + <b><a name="Block_651" id="Block_651">651</a>.</b> + Today it is no longer questioned that the + principles of the analysts are the more far-reaching. Indeed, + the synthesists lack two things in order to engage in a general + theory of algebraic configurations: these are on the one hand a + definition of imaginary elements, on the other an + interpretation of general algebraic concepts. Both of these + have subsequently been developed in synthetic form, but to do + this the essential principle of synthetic geometry had to be + set aside. This principle which manifests itself so brilliantly + in the theory of linear forms and the forms of the second + degree, is the possibility of immediate proof by means of + visualized constructions.—<span class= + "smcap">Klein, Felix.</span></p> + <p class="blockcite"> + Riemannsche Flächen (Leipzig, 1906), Bd. 1, p. 234.</p> + + <p class="v2"> + <b><a name="Block_652" id="Block_652">652</a>.</b> + Abstruse mathematical researches ... are ... often + abused for having no obvious physical application. The fact is + that the most useful parts of science have been investigated + for the sake of truth, and not for their usefulness. A new + branch of mathematics, which has sprung up in the last twenty + years, was denounced by the Astronomer Royal before the + University of Cambridge as doomed to be forgotten, on account + of its uselessness. Now it turns out that the reason why we + cannot go further in our investigations of molecular action is + that we do not know enough of this branch of + mathematics.—<span class="smcap">Clifford, W. K.</span></p> + <p class="blockcite"> + Conditions of Mental Development; Lectures and Essays + (London, 1901), Vol. 1, p. 115.</p> + +<p><span class="pagenum"> + <a name="Page_103" + id="Page_103">103</a></span></p> + + <p class="v2"> + <b><a name="Block_653" id="Block_653">653</a>.</b> + In geometry, as in most sciences, it is very rare + that an isolated proposition is of immediate utility. But the + theories most powerful in practice are formed of propositions + which curiosity alone brought to light, and which long remained + useless without its being able to divine in what way they + should one day cease to be so. In this sense it may be said, + that in real science, no theory, no research, is in effect + useless.—<span class="smcap">Voltaire.</span></p> + <p class="blockcite"> + A Philosophical Dictionary, Article “Geometry”; (Boston, 1881), + Vol. 1, p. 374.</p> + + <p class="v2"> + <b><a name="Block_654" id="Block_654">654</a>.</b> + Scientific subjects do not progress necessarily on + the lines of direct usefulness. Very many applications of the + theories of pure mathematics have come many years, sometimes + centuries, after the actual discoveries themselves. The weapons + were at hand, but the men were not able to use + them.—<span class="smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Perry’s Teaching of Mathematics (London, 1902), p. 35.</p> + + <p class="v2"> + <b><a name="Block_655" id="Block_655">655</a>.</b> + It is no paradox to say that in our most + theoretical moods we may be nearest to our most practical + applications.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Introduction to Mathematics (New York), p. 100.</p> + + <p class="v2"> + <b><a name="Block_656" id="Block_656">656</a>.</b> + Although with the majority of those who study and + practice in these capacities [engineers, builders, surveyors, + geographers, navigators, hydrographers, astronomers], + second-hand acquirements, trite formulas, and appropriate + tables are sufficient for ordinary purposes, yet these trite + formulas and familiar rules were originally or gradually + deduced from the profound investigations of the most gifted + minds, from the dawn of science to the present day.... The + further developments of the science, with its possible + applications to larger purposes of human utility and grander + theoretical generalizations, is an achievement reserved for a + few of the choicest spirits, touched from time to time by + Heaven to these highest issues. The intellectual world is + filled with latent and undiscovered truth as the material world + is filled with latent electricity.—<span class="smcap">Everett, + Edward.</span></p> + <p class="blockcite"> + Orations and Speeches, Vol. 3 (Boston, 1870), p. 513.</p> + +<p><span class="pagenum"> + <a name="Page_104" + id="Page_104">104</a></span></p> + + <p class="v2"> + <b><a name="Block_657" id="Block_657">657</a>.</b> + If we view mathematical speculations with reference + to their use, it appears that they should be divided into two + classes. To the first belong those which furnish some marked + advantage either to common life or to some art, and the value + of such is usually determined by the magnitude of this + advantage. The other class embraces those speculations which, + though offering no direct advantage, are nevertheless valuable + in that they extend the boundaries of analysis and increase our + resources and skill. Now since many investigations, from which + great advantage may be expected, must be abandoned solely + because of the imperfection of analysis, no small value should + be assigned to those speculations which promise to enlarge the + field of + + <a id="TNanchor_5"></a> + <a class="msg" href="#TN_5" + title="orignally spelled ‘anaylsis’">analysis</a>.—<span + + class="smcap">Euler.</span></p> + <p class="blockcite"> + Novi Comm. Petr., Vol. 4, Preface.</p> + + <p class="v2"> + <b><a name="Block_658" id="Block_658">658</a>.</b> + The discovery of the conic sections, attributed to + Plato, first threw open the higher species of form to the + contemplation of geometers. But for this discovery, which was + probably regarded in Plato’s time and long + after him, as the unprofitable amusement of a speculative + brain, the whole course of practical philosophy of the present + day, of the science of astronomy, of the theory of projectiles, + of the art of navigation, might have run in a different + channel; and the greatest discovery that has ever been made in + the history of the world, the law of universal gravitation, + with its innumerable direct and indirect consequences and + applications to every department of human research and + industry, might never to this hour have been + elicited.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + A Probationary Lecture on Geometry; Collected Mathematical + Papers, Vol. 2 (Cambridge, 1908), p. 7.</p> + + <p class="v2"> + <b><a name="Block_659" id="Block_659">659</a>.</b> + No more impressive warning can be given to those + who would confine knowledge and research to what is apparently + useful, than the reflection that conic sections were studied + for eighteen hundred years merely as an abstract science, + without regard to any utility other than to satisfy the craving + for knowledge on the part of mathematicians, and that then at + the end of this long period of abstract study, they were found + to be the + +<span class="pagenum"> + <a name="Page_105" + id="Page_105">105</a></span> + + necessary key with which to attain + the knowledge of the most important laws of + nature.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Introduction to Mathematics (New York, York, 1911), + pp. 136-137.</p> + + <p class="v2"> + <b><a name="Block_660" id="Block_660">660</a>.</b> + The Greeks in the first vigour of their pursuit of + mathematical truth, at the time of Plato and soon after, had by + no means confined themselves to those propositions which had a + visible bearing on the phenomena of nature; but had followed + out many beautiful trains of research concerning various kinds + of figures, for the sake of their beauty alone; as for instance + in their doctrine of Conic Sections, of which curves they had + discovered all the principal properties. But it is curious to + remark, that these investigations, thus pursued at first as + mere matters of curiosity and intellectual gratification, were + destined, two thousand years later, to play a very important + part in establishing that system of celestial motions which + succeeded the Platonic scheme of cycles and epicycles. If the + properties of conic sections had not been demonstrated by the + Greeks and thus rendered familiar to the mathematicians of + succeeding ages, Kepler would probably not have been able to + discover those laws respecting the orbits and motions of + planets which were the occasion of the greatest revolution that + ever happened in the history of + science.—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + History of Scientific Ideas, Bk. 2, chap. 14, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_661" id="Block_661">661</a>.</b> + The greatest mathematicians, as Archimedes, Newton, + and Gauss, always united theory and applications in equal + measure.—<span class="smcap">Klein, Felix.</span></p> + <p class="blockcite"> + Elementarmathematik vom höheren Standpunkte aus + (Leipzig, 1909), Bd. 2, p. 392.</p> + + <p class="v2"> + <b><a name="Block_662" id="Block_662">662</a>.</b> + We may see how unexpectedly recondite parts of pure + mathematics may bear upon physical science, by calling to mind + the circumstance that Fresnel obtained one of the most curious + confirmations of the theory (the laws of Circular Polarization + by reflection) through an interpretation of an algebraical + expression, which, according to the original conventional + meaning of the symbols, involved an impossible + quantity.—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + History of Scientific Ideas, Bk. 2, chap. 14, sect. 8.</p> + +<p><span class="pagenum"> + <a name="Page_106" + id="Page_106">106</a></span></p> + + <p class="v2"> + <b><a name="Block_663" id="Block_663">663</a>.</b> + A great department of thought must have its own + inner life, however transcendent may be the importance of its + relations to the outside. No department of science, least of + all one requiring so high a degree of mental concentration as + Mathematics, can be developed entirely, or even mainly, with a + view to applications outside its own range. The increased + complexity and specialisation of all branches of knowledge + makes it true in the present, however it may have been in + former times, that important advances in such a department as + Mathematics can be expected only from men who are interested in + the subject for its own sake, and who, whilst keeping an open + mind for suggestions from outside, allow their thought to range + freely in those lines of advance which are indicated by the + present state of their subject, untrammelled by any + preoccupation as to applications to other departments of + science. Even with a view to applications, if Mathematics is to + be adequately equipped for the purpose of coping with the + intricate problems which will be presented to it in the future + by Physics, Chemistry and other branches of physical science, + many of these problems probably of a character which we cannot + at present forecast, it is essential that Mathematics should be + allowed to develop freely on its own + lines.—<span class="smcap">Hobson, E. W.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A, (1910); Nature, Vol. 84, p. 286.</p> + + <p class="v2"> + <b><a name="Block_664" id="Block_664">664</a>.</b> + To emphasize this opinion that mathematicians would + be unwise to accept practical issues as the sole guide or the + chief guide in the current of their investigations, ... let me + take one more instance, by choosing a subject in which the + purely mathematical interest is deemed supreme, the theory of + functions of a complex variable. That at least is a theory in + pure mathematics, initiated in that region, and developed in + that region; it is built up in scores of papers, and its plan + certainly has not been, and is not now, dominated or guided by + considerations of applicability to natural phenomena. Yet what + has turned out to be its relation to practical issues? The + investigations of Lagrange and others upon the construction of + maps appear as a portion of the general property of conformal + representation; which is merely the general geometrical method of + +<span class="pagenum"> + <a name="Page_107" + id="Page_107">107</a></span> + + regarding functional relations in + that theory. Again, the interesting and important + investigations upon discontinuous two-dimensional fluid motion + in hydrodynamics, made in the last twenty years, can all be, + and now are all, I believe, deduced from similar considerations + by interpreting functional relations between complex variables. + In the dynamics of a rotating heavy body, the only substantial + extension of our knowledge since the time of Lagrange has + accrued from associating the general properties of functions + with the discussion of the equations of motion. Further, under + the title of conjugate functions, the theory has been applied + to various questions in electrostatics, particularly in + connection with condensors and electrometers. And, lastly, in + the domain of physical astronomy, some of the most conspicuous + advances made in the last few years have been achieved by + introducing into the discussion the ideas, the principles, the + methods, and the results of the theory of functions ... the + refined and extremely difficult work of Poincaré + and others in physical astronomy has been possible only by the + use of the most elaborate developments of some purely + mathematical subjects, developments which were made without a + thought of such applications.—<span class= + "smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A, (1897); Nature, Vol. 56, p. 377.</p> + +<p><span class="pagenum"> + <a name="Page_108" + id="Page_108">108</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_VII"> + CHAPTER VII<br /> + <span class="large"> + MODERN MATHEMATICS</span></h2> + + <p class="v2"> + <b><a name="Block_701" id="Block_701">701</a>.</b> + Surely this is the golden age of + mathematics.—<span class="smcap">Pierpont, James.</span></p> + <p class="blockcite"> + History of Mathematics in the Nineteenth Century; Congress + of Arts and Sciences (Boston and New York, 1905), Vol. 1, p. + 493.</p> + + <p class="v2"> + <b><a name="Block_702" id="Block_702">702</a>.</b> + The golden age of + mathematics—that was not the age of Euclid, + it is ours. Ours is the age when no less than six international + congresses have been held in the course of nine years. It is in + our day that more than a dozen mathematical societies contain a + growing membership of more than two thousand men representing + the centers of scientific light throughout the great culture + nations of the world. It is in our time that over five hundred + scientific journals are each devoted in part, while more than + two score others are devoted exclusively, to the publication of + mathematics. It is in our time that the <i lang="de" + xml:lang="de">Jahrbuch + über die Fortschritte der Mathematik</i>, though + admitting only condensed abstracts with titles, and not + reporting on all the journals, has, nevertheless, grown to + nearly forty huge volumes in as many years. It is in our time + that as many as two thousand books and memoirs drop from the + mathematical press of the world in a single year, the estimated + number mounting up to fifty thousand in the last generation. + Finally, to adduce yet another evidence of a similar kind, it + requires not less than seven ponderous tomes of the forthcoming + <i lang="de" xml:lang="de">Encyclopaedie der Mathematischen + Wissenschaften</i> to + contain, not expositions, not demonstrations, but merely + compact reports and bibliographic notices sketching + developments that have taken place since the beginning of the + nineteenth century.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 8.</p> + + <p class="v2"> + <b><a name="Block_703" id="Block_703">703</a>.</b> + I have said that mathematics is the oldest of the + sciences; a glance at its more recent history will show that it + has the + +<span class="pagenum"> + <a name="Page_109" + id="Page_109">109</a></span> + + energy of perpetual youth. The output + of contributions to the advance of the science during the last + century and more has been so enormous that it is difficult to + say whether pride in the greatness of achievement in this + subject, or despair at his inability to cope with the + multiplicity of its detailed developments, should be the + dominant feeling of the mathematician. Few people outside of + the small circle of mathematical specialists have any idea of + the vast growth of mathematical literature. The Royal Society + Catalogue contains a list of nearly thirty-nine thousand papers + on subjects of Pure Mathematics alone, which have appeared in + seven hundred serials during the nineteenth century. This + represents only a portion of the total output, the very large + number of treatises, dissertations, and monographs published + during the century being + omitted.—<span class="smcap">Hobson, E. W.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A, (1910); Nature, Vol. 84, p. 285.</p> + + <p class="v2"> + <b><a name="Block_704" id="Block_704">704</a>.</b> + Mathematics is one of the oldest of the sciences; + it is also one of the most active, for its strength is the + vigour of perpetual youth.—<span class= + "smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A, (1897); Nature, Vol. 56, p. 378.</p> + + <p class="v2"> + <b><a name="Block_705" id="Block_705">705</a>.</b> + The nineteenth century which prides itself upon the + invention of steam and evolution, might have derived a more + legitimate title to fame from the discovery of pure + mathematics.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + International Monthly, Vol. 4 (1901), p. 83.</p> + + <p class="v2"> + <b><a name="Block_706" id="Block_706">706</a>.</b> + One of the chiefest triumphs of modern mathematics + consists in having discovered what mathematics really + is.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + International Monthly, Vol. 4 (1901), p. 84.</p> + + <p class="v2"> + <b><a name="Block_707" id="Block_707">707</a>.</b> + Modern mathematics, that most astounding of + intellectual creations, has projected the + mind’s eye through infinite time and the + mind’s hand into boundless + space.—<span class="smcap">Butler, N. M.</span></p> + <p class="blockcite"> + The Meaning of Education and other Essays and Addresses + (New York, 1905), p. 44.</p> + +<p><span class="pagenum"> + <a name="Page_110" + id="Page_110">110</a></span></p> + + <p class="v2"> + <b><a name="Block_708" id="Block_708">708</a>.</b> + The extraordinary development of mathematics in the + last century is quite unparalleled in the long history of this + most ancient of sciences. Not only have those branches of + mathematics which were taken over from the eighteenth century + steadily grown, but entirely new ones have sprung up in almost + bewildering profusion, and many of them have promptly assumed + proportions of vast + extent.—<span class= "smcap">Pierpont, J.</span></p> + <p class="blockcite"> + The History of Mathematics in the Nineteenth Century; + Congress of Arts and Sciences (Boston and New York, 1905), + Vol. 1, p. 474.</p> + + <p class="v2"> + <b><a name="Block_709" id="Block_709">709</a>.</b> + The Modern Theory of Functions—that stateliest of all the pure + creations of the human + intellect.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 16.</p> + + <p class="v2"> + <b><a name="Block_710" id="Block_710">710</a>.</b> + If a mathematician of the past, an Archimedes or + even a Descartes, could view the field of geometry in its + present condition, the first feature to impress him would be + its lack of concreteness. There are whole classes of geometric + theories which proceed not only without models and diagrams, + but without the slightest (apparent) use of spatial intuition. + In the main this is due, to the power of the analytic + instruments of investigations as compared with the purely + geometric.—<span class="smcap">Kasner, Edward</span>.</p> + <p class="blockcite"> + The Present Problems in Geometry; Bulletin American + Mathematical Society, 1905, p. 285.</p> + + <p class="v2"> + <b><a name="Block_711" id="Block_711">711</a>.</b> + In Euclid each proposition stands by itself; its + connection with others is never indicated; the leading ideas + contained in its proof are not stated; general principles do + not exist. In modern methods, on the other hand, the greatest + importance is attached to the leading thoughts which pervade + the whole; and general principles, which bring whole groups of + theorems under one aspect, are given rather than separate + propositions. The whole tendency is toward generalization. A + straight line is considered as given in its entirety, extending + both ways to infinity, while Euclid is very careful never to + admit anything but finite quantities. The treatment of the + infinite is in fact another + +<span class="pagenum"> + <a name="Page_111" + id="Page_111">111</a></span> + + fundamental difference between the two methods. Euclid avoids + it, in modern mathematics it is systematically introduced, for + only thus is generality + obtained.—<span class= "smcap">Cayley, Arthur</span>.</p> + <p class="blockcite"> + Encyclopedia Britannica (9th edition), Article“Geometry.”</p> + + <p class="v2"> + <b><a name="Block_712" id="Block_712">712</a>.</b> + This is one of the greatest advantages of modern + geometry over the ancient, to be able, through the + consideration of positive and negative quantities, to include + in a single enunciation the several cases which the same + theorem may present by a change in the relative position of the + different parts of a figure. Thus in our day the nine principal + problems and the numerous particular cases, which form the + object of eighty-three theorems in the two books <cite>De + sectione determinata</cite> of Appolonius constitute only + one problem which is resolved by a single + equation.—<span class="smcap">Chasles, M.</span></p> + <p class="blockcite"> + Histoire de la Géométrie, chap. 1, sect. 35.</p> + + <p class="v2"> + <b><a name="Block_713" id="Block_713">713</a>.</b> + Euclid always contemplates a straight line as drawn + between two definite points, and is very careful to mention + when it is to be produced beyond this segment. He never thinks + of the line as an entity given once for all as a whole. This + careful definition and limitation, so as to exclude an infinity + not immediately apparent to the senses, was very characteristic + of the Greeks in all their many activities. It is enshrined in + the difference between Greek architecture and Gothic + architecture, and between Greek religion and modern religion. + The spire of a Gothic cathedral and the importance of the + unbounded straight line in modern Geometry are both emblematic + of the transformation of the modern + world.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Introduction to Mathematics (New York, 1911), p. 119.</p> + + <p class="v2"> + <b><a name="Block_714" id="Block_714">714</a>.</b> + The geometrical problems and theorems of the Greeks always + refer to definite, oftentimes to rather complicated figures. + Now frequently the points and lines of such a figure may assume + very many different relative positions; each of these possible + cases is then considered separately. On the contrary, present + day mathematicians generate their figures one from another, + and are accustomed to consider them subject to variation; + +<span class="pagenum"> + <a name="Page_112" + id="Page_112">112</a></span> + + in this manner they unite + the various cases and combine them as much as possible by + employing negative and imaginary magnitudes. For example, the + problems which Appolonius treats in his two books <cite>De + sectione rationis</cite>, are solved today by means of a single, + universally applicable construction; Apollonius, on the + contrary, separates it into more than eighty different cases + varying only in position. Thus, as Hermann Hankel has fittingly + remarked, the ancient geometry sacrifices to a seeming + simplicity the true simplicity which consists in the unity of + principles; it attained a trivial sensual presentability at the + cost of the recognition of the relations of geometric forms in + all their changes and in all the variations of their sensually + presentable + positions.—<span class="smcap">Reye, Theodore.</span></p> + <p class="blockcite"> + Die synthetische Geometrie im Altertum und in der Neuzeit; + Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. 2, + pp. 346-347.</p> + + <p class="v2"> + <b><a name="Block_715" id="Block_715">715</a>.</b> + It is known that the mathematics prescribed for the + high school [Gymnasien] is essentially Euclidean, while it is + modern mathematics, the theory of functions and the + infinitesimal calculus, which has secured for us an insight + into the mechanism and laws of nature. Euclidean mathematics is + indeed, a prerequisite for the theory of functions, but just as + one, though he has learned the inflections of Latin nouns and + verbs, will not thereby be enabled to read a Latin author much + less to appreciate the beauties of a Horace, so Euclidean + mathematics, that is the mathematics of the high school, is + unable to unlock nature and her laws. Euclidean mathematics + assumes the completeness and invariability of mathematical + forms; these forms it describes with appropriate accuracy and + enumerates their inherent and related properties with perfect + clearness, order, and completeness, that is, Euclidean + mathematics operates on forms after the manner that anatomy + operates on the dead body and its members.</p> + <p class="v1"> + On the other hand, the mathematics of variable + magnitudes—function theory or + analysis—considers mathematical forms in + their genesis. By writing the equation of the parabola, we + express its law of generation, the law according to which the + variable point moves. The path, produced before the eyes of the + +<span class="pagenum"> + <a name="Page_113" + id="Page_113">113</a></span> + + student by a point moving in accordance to this law, is + the parabola.</p> + <p class="v1"> + If, then, Euclidean mathematics treats space and number forms + after the manner in which anatomy treats the dead body, modern + mathematics deals, as it were, with the living body, with + growing and changing forms, and thus furnishes an insight, not + only into nature as she is and appears, but also into nature as + she generates and creates,—reveals her transition steps and in + so doing creates a mind for and understanding of the laws of + becoming. Thus modern mathematics + bears the same relation to Euclidean mathematics that + physiology or biology ... bears to anatomy. But it is exactly + in this respect that our view of nature is so far above that of + the ancients; that we no longer look on nature as a quiescent + complete whole, which compels admiration by its sublimity and + wealth of forms, but that we conceive of her as a vigorous + growing organism, unfolding according to definite, as delicate + as far-reaching, laws; that we are able to lay hold of the + permanent amidst the transitory, of law amidst fleeting + phenomena, and to be able to give these their simplest and + truest expression through the mathematical + formulas.—<span class="smcap">Dillmann, E.</span></p> + <p class="blockcite"> + Die Mathematik die Fackelträgerin einer neuen Zeit + (Stuttgart, 1889), p. 37.</p> + + <p class="v2"> + <b><a name="Block_716" id="Block_716">716</a>.</b> + The Excellence of <em>Modern Geometry</em> is in + nothing more evident, than in those full and adequate Solutions + it gives to Problems; representing all possible Cases in one + view, and in one general Theorem many times comprehending whole + Sciences; which deduced at length into Propositions, and + demonstrated after the manner of the <em>Ancients</em>, might + well become the subjects of large Treatises: For whatsoever + Theorem solves the most complicated Problem of the kind, does + with a due Reduction reach all the subordinate + Cases.—<span class="smcap">Halley, E.</span></p> + <p class="blockcite"> + An Instance of the Excellence of Modern Algebra, etc.; + Philosophical Transactions, 1694, p. 960.</p> + + <p class="v2"> + <b><a name="Block_717" id="Block_717">717</a>.</b> + One of the most conspicuous and distinctive features of + thought in the nineteenth century is its critical + +<span class="pagenum"> + <a name="Page_114" + id="Page_114">114</a></span> + + spirit. Beginning with the calculus, + it soon permeates all analysis, and toward the close of the + century it overhauls and recasts the foundations of geometry + and aspires to further conquests in mechanics and in the + immense domains of mathematical physics.... A searching + examination of the foundations of arithmetic and the calculus + has brought to light the insufficiency of much of the reasoning + formerly considered as + conclusive.—<span class="smcap">Pierpont, J.</span></p> + <p class="blockcite"> + History of Mathematics in the Nineteenth Century; Congress of + Arts and Sciences (Boston and New York, 1905), Vol. 1, + p. 482.</p> + + <p class="v2"> + <b><a name="Block_718" id="Block_718">718</a>.</b> + If we compare a mathematical problem with an + immense rock, whose interior we wish to penetrate, then the + work of the Greek mathematicians appears to us like that of a + robust stonecutter, who, with indefatigable perseverance, + attempts to demolish the rock gradually from the outside by + means of hammer and chisel; but the modern mathematician + resembles an expert miner, who first constructs a few passages + through the rock and then explodes it with a single blast, + bringing to light its inner + treasures.—<span class="smcap">Hankel, Hermann.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten + Jahrhunderten (Tübingen, 1884), p. 9.</p> + + <p class="v2"> + <b><a name="Block_719" id="Block_719">719</a>.</b> + All the modern higher mathematics is based on a + calculus of operations, on laws of thought. All mathematics, + from the first, was so in reality; but the evolvers of the + modern higher calculus have known that it is so. Therefore + elementary teachers who, at the present day, persist in + thinking about algebra and arithmetic as dealing with laws of + number, and about geometry as dealing with laws of surface and + solid content, are doing the best that in them lies to put + their pupils on the wrong track for reaching in the future any + true understanding of the higher algebras. Algebras deal not + with laws of number, but with such laws of the human thinking + machinery as have been discovered in the course of + investigations on numbers. Plane geometry deals with such laws + of thought as were discovered by men intent on finding out how + to measure surface; and solid geometry with such additional + laws of thought as were discovered + +<span class="pagenum"> + <a name="Page_115" + id="Page_115">115</a></span> + + when men began to extend geometry into three + dimensions.—<span class="smcap">Boole M. E.</span></p> + <p class="blockcite"> + Logic of Arithmetic (Oxford, 1903), Preface, pp. 18-19.</p> + + <p class="v2"> + <b><a name="Block_720" id="Block_720">720</a>.</b> + It is not only a decided preference for synthesis + and a complete denial of general methods which characterizes + the ancient mathematics as against our newer science [modern + mathematics]: besides this external formal difference there is + another real, more deeply seated, contrast, which arises from + the different attitudes which the two assumed relative to the + use of the concept of <em>variability</em>. For while the + ancients, on account of considerations which had been + transmitted to them from the philosophic school of the + Eleatics, never employed the concept of motion, the spatial + expression for variability, in their rigorous system, and made + incidental use of it only in the treatment of phonoromically + generated curves, modern geometry dates from the instant that + Descartes left the purely algebraic treatment of equations and + proceeded to investigate the variations which an algebraic + expression undergoes when one of its variables assumes a + continuous succession of values.—<span class="smcap">Hankel, + Hermann.</span></p> + <p class="blockcite"> + Untersuchungen über die unendlich oft oszillierenden und + unstetigen Functionen; Ostwald’s Klassiker der exacten + Wissenschaften, No. 153, pp. 44-45.</p> + + <p class="v2"> + <b><a name="Block_721" id="Block_721">721</a>.</b> + Without doubt one of the most characteristic + features of mathematics in the last century is the systematic + and universal use of the complex variable. Most of its great + theories received invaluable aid from it, and many owe their + very existence to + it.—<span class= "smcap">Pierpont, J.</span></p> + <p class="blockcite"> + History of Mathematics in the Nineteenth Century; Congress + of Arts and Sciences (Boston and New York, 1905), Vol. 1, p. + 474.</p> + + <p class="v2"> + <b><a name="Block_722" id="Block_722">722</a>.</b> + The notion, which is really the fundamental one + (and I cannot too strongly emphasise the assertion), underlying + and pervading the whole of modern analysis and geometry, is + that of imaginary magnitude in analysis and of imaginary space + in geometry.—<span class="smcap">Cayley, Arthur.</span></p> + <p class="blockcite"> + Presidential Address; Collected Works, Vol. 11, p. 434.</p> + +<p><span class="pagenum"> + <a name="Page_116" + id="Page_116">116</a></span></p> + + <p class="v2"> + <b><a name="Block_723" id="Block_723">723</a>.</b> + The solution of the difficulties which formerly + surrounded the mathematical infinite is probably the greatest + achievement of which our age has to + boast.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + The Study of Mathematics; Philosophical Essays (London, 1910), + p. 77.</p> + + <p class="v2"> + <b><a name="Block_724" id="Block_724">724</a>.</b> + Induction and analogy are the special characteristics of modern + mathematics, in which theorems have given place to theories + and no truth is regarded otherwise than as a link in an + infinite chain. “<i lang="la" xml:lang="la">Omne exit in + infinitum</i>” is their + favorite motto and accepted axiom.—<span class= + "smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + A Plea for the Mathematician; Nature, Vol. 1, p. 261.</p> + + <p class="v2"> + <b><a name="Block_725" id="Block_725">725</a>.</b> + The conception of correspondence plays a great part + in modern mathematics. It is the fundamental notion in the + science of order as distinguished from the science of + magnitude. If the older mathematics were mostly dominated by + the needs of mensuration, modern mathematics are dominated by + the conception of order and arrangement. It may be that this + tendency of thought or direction of reasoning goes hand in hand + with the modern discovery in physics, that the changes in + nature depend not only or not so much on the quantity of mass + and energy as on their distribution or + arrangement.—<span class="smcap">Merz, J. T.</span></p> + <p class="blockcite"> + History of European Thought in the Nineteenth Century + (Edinburgh and London, 1903), p. 736.</p> + + <p class="v2"> + <b><a name="Block_726" id="Block_726">726</a>.</b> + Now this establishment of correspondence between + two aggregates and investigation of the propositions that are + carried over by the correspondence may be called the central + idea of modern mathematics.—<span class= + "smcap">Clifford, W. K.</span></p> + <p class="blockcite"> + Philosophy of the Pure Sciences; Lectures and Essays + (London, 1901), Vol. 1, p. 402.</p> + + <p class="v2"> + <b><a name="Block_727" id="Block_727">727</a>.</b> + In our century the conceptions substitution and + substitution group, transformation and transformation group, + operation and operation group, invariant, differential + invariant and differential parameter, appear more and more + clearly as the most important conceptions of + mathematics.—<span class="smcap">Lie, Sophus.</span></p> + <p class="blockcite"> + Leipziger Berichte, No. 47 (1895), p. 261.</p> + +<p><span class="pagenum"> + <a name="Page_117" + id="Page_117">117</a></span></p> + + <p class="v2"> + <b><a name="Block_728" id="Block_728">728</a>.</b> + Generality of points of view and of methods, + precision and elegance in presentation, have become, since + Lagrange, the common property of all who would lay claim to the + rank of scientific mathematicians. And, even if this generality + leads at times to abstruseness at the expense of intuition and + applicability, so that general theorems are formulated which + fail to apply to a single special case, if furthermore + precision at times degenerates into a studied brevity which + makes it more difficult to read an article than it was to write + it; if, finally, elegance of form has well-nigh become in our + day the criterion of the worth or worthlessness of a + proposition,—yet are these conditions of the + highest importance to a wholesome development, in that they + keep the scientific material within the limits which are + necessary both intrinsically and extrinsically if mathematics + is not to spend itself in trivialities or smother in + profusion.—<span class="smcap">Hankel, Hermann.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten + Jahrhunderten (Tübingen, 1884), pp. 14-15.</p> + + <p class="v2"> + <b><a name="Block_729" id="Block_729">729</a>.</b> + The development of abstract methods during the past + few years has given mathematics a new and vital principle which + furnishes the most powerful instrument for exhibiting the + essential unity of all its + branches.—<span class="smcap">Young, J. W.</span></p> + <p class="blockcite"> + Fundamental Concepts of Algebra and + + <a id="TNanchor_6"></a> + <a class="msg" href="#TN_6" + title="originally spelled ‘Geomtry’">Geometry</a> + + (New York, 1911), p. 225.</p> + + <p class="v2"> + <b><a name="Block_730" id="Block_730">730</a>.</b> + Everybody praises the incomparable power of the mathematical + method, but so is everybody aware of its incomparable + unpopularity.—<span class= "smcap">Rosanes, J.</span></p> + <p class="blockcite"> + Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. + 13, p. 17.</p> + + <p class="v2"> + <b><a name="Block_731" id="Block_731">731</a>.</b> + Indeed the modern developments of mathematics + constitute not only one of the most impressive, but one of the + most characteristic, phenomena of our age. It is a phenomenon, + however, of which the boasted intelligence of a “universalized” + daily press seems strangely unaware; and there is no other + great human interest, whether of science or of art, regarding + which the mind of the educated public is permitted to hold so + many fallacious opinions and inferior + estimates.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Arts (New York, 1908), + p. 8.</p> + +<p><span class="pagenum"> + <a name="Page_118" + id="Page_118">118</a></span></p> + + <p class="v2"> + <b><a name="Block_732" id="Block_732">732</a>.</b> + It may be asserted without exaggeration that the + domain of mathematical knowledge is the only one of which our + otherwise omniscient journalism has not yet possessed + itself.—<span class="smcap">Pringsheim, Alfred.</span></p> + <p class="blockcite"> + Ueber Wert und angeblichen Unwert der Mathematik; Jahresbericht + der Deutschen Mathematiker Vereinigung, (1904) p. 357.</p> + + <p class="v2"> + <b><a name="Block_733" id="Block_733">733</a>.</b> + [The] inaccessibility of special fields of mathematics, except + by the regular way of logically antecedent acquirements, + renders the study discouraging or hateful to weak or indolent + minds.—<span class= "smcap">Lefevre, Arthur.</span></p> + <p class="blockcite"> + Number and its Algebra (Boston, 1903), sect. 223.</p> + + <p class="v2"> + <b><a name="Block_734" id="Block_734">734</a>.</b> + The majority of mathematical truths now possessed + by us presuppose the intellectual toil of many centuries. A + mathematician, therefore, who wishes today to acquire a + thorough understanding of modern research in this department, + must think over again in quickened tempo the mathematical + labors of several centuries. This constant dependence of new + truths on old ones stamps mathematics as a science of uncommon + exclusiveness and renders it generally impossible to lay open + to uninitiated readers a speedy path to the apprehension of the + higher mathematical truths. For this reason, too, the theories + and results of mathematics are rarely adapted for popular + presentation.... This same inaccessibility of mathematics, + although it secures for it a lofty and aristocratic place among + the sciences, also renders it odious to those who have never + learned it, and who dread the great labor involved in acquiring + an understanding of the questions of modern mathematics. + Neither in the languages nor in the natural sciences are the + investigations and results so closely interdependent as to make + it impossible to acquaint the uninitiated student with single + branches or with particular results of these sciences, without + causing him to go through a long course of preliminary + study.—<span class="smcap">Schubert, H.</span></p> + <p class="blockcite"> + Mathematical Essays and Recreations (Chicago, 1898), p. 32.</p> + +<p><span class="pagenum"> + <a name="Page_119" + id="Page_119">119</a></span></p> + + <p class="v2"> + <b><a name="Block_735" id="Block_735">735</a>.</b> + Such is the character of mathematics in its + profounder depths and in its higher and remoter zones that it + is well nigh impossible to convey to one who has not devoted + years to its exploration a just impression of the scope and + magnitude of the existing body of the science. An imagination + formed by other disciplines and accustomed to the interests of + another field may scarcely receive suddenly an apocalyptic + vision of that infinite interior world. But how amazing and how + edifying were such a revelation, if it only could be + made.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 6.</p> + + <p class="v2"> + <b><a name="Block_736" id="Block_736">736</a>.</b> + It is not so long since, during one of the meetings + of the Association, one of the leading English newspapers + briefly described a sitting of this Section in the words, + “Saturday morning was devoted to pure mathematics, and so there + was nothing of any general interest:” still, such toleration + is better than undisguised and ill-informed + hostility.—<span class="smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Report of the 67th meeting of the British Association for + the Advancement of Science.</p> + + <p class="v2"> + <b><a name="Block_737" id="Block_737">737</a>.</b> + The science [of mathematics] has grown to such vast + proportion that probably no living mathematician can claim to + have achieved its mastery as a + whole.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + An Introduction to Mathematics (New York, 1911), p. 252.</p> + + <p class="v2"> + <b><a name="Block_738" id="Block_738">738</a>.</b> + There is perhaps no science of which the development has been + carried so far, which requires greater concentration and will + power, and which by the abstract height of the qualities + required tends more to separate one from daily life.</p> + <p class="blockcite"> + Provisional Report of the American Subcommittee of the + International Commission on the Teaching of Mathematics; + Bulletin American Society (1910), p. 97.</p> + + <p class="v2"> + <b><a name="Block_739" id="Block_739">739</a>.</b> + Angling may be said to be so like the mathematics, that it can + never be fully + learnt.—<span class="smcap">Walton, Isaac.</span></p> + <p class="blockcite"> + The Complete Angler, Preface.</p> + +<p><span class="pagenum"> + <a name="Page_120" + id="Page_120">120</a></span></p> + + <p class="v2"> + <b><a name="Block_740" id="Block_740">740</a>.</b> + The flights of the imagination which occur to the + pure mathematician are in general so much better described in + his formulæ than in words, that it is not + remarkable to find the subject treated by outsiders as + something essentially cold and + uninteresting—... the only successful + attempt to invest mathematical reasoning with a halo of + glory—that made in this section by Prof. + Sylvester—is known to a comparative + few,....—<span class="smcap">Tait, P. G.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science (1871); Nature Vol. 4, p. 271.</p> + +<p><span class="pagenum"> + <a name="Page_121" + id="Page_121">121</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_VIII"> + CHAPTER VIII<br /> + <span class="large"> + THE MATHEMATICIAN</span></h2> + + <p class="v2"> + <b><a name="Block_801" id="Block_801">801</a>.</b> + The real mathematician is an enthusiast <em>per se</em>. Without + enthusiasm no + mathematics.—<span class= "smcap">Novalis</span>.</p> + <p class="blockcite"> + Schriften (Berlin, 1901), Zweiter Teil, p. 223.</p> + + <p class="v2"> + <b><a name="Block_802" id="Block_802">802</a>.</b> + It is true that a mathematician, who is not somewhat of a poet, + will never be a perfect mathematician.—<span class= + "smcap">Weierstrass.</span></p> + <p class="blockcite"> + Quoted by Mittag-Leffler; Compte rendu du deuxième congrês + international des mathématiciens (Paris, 1902), p. 149.</p> + + <p class="v2"> + <b><a name="Block_803" id="Block_803">803</a>.</b> + The mathematician is perfect only in so far as he + is a perfect being, in so far as he perceives the beauty of + truth; only then will his work be thorough, transparent, + comprehensive, pure, clear, attractive and even elegant. All + this is necessary to resemble + <em>Lagrange</em>.—<span class="smcap">Goethe.</span></p> + <p class="blockcite"> + Wilhelm Meister’s Wanderjahre, Zweites + Buch; Sprüche in Prosa; Natur, VI, 950.</p> + + <p class="v2"> + <b><a name="Block_804" id="Block_804">804</a>.</b> + A thorough advocate in a just cause, a penetrating + mathematician facing the starry heavens, both alike bear the + semblance of divinity.—<span class="smcap">Goethe.</span></p> + <p class="blockcite"> + Wilhelm Meister’s Wanderjahre, Zweites + Buch; Sprüche in Prosa; Natur, VI, 947.</p> + + <p class="v2"> + <b><a name="Block_805" id="Block_805">805</a>.</b> + Mathematicians practice absolute + freedom.—<span class="smcap">Adams, Henry.</span></p> + <p class="blockcite"> + A Letter to American Teachers of History (Washington, + 1910), p. 169.</p> + + <p class="v2"> + <b><a name="Block_806" id="Block_806">806</a>.</b> + The mathematical method is the essence of + mathematics. He who fully comprehends the method is a + mathematician.—<span class="smcap">Novalis.</span></p> + <p class="blockcite"> + Schriften (Berlin, 1901), Zweiter Teil, p. 190.</p> + +<p><span class="pagenum"> + <a name="Page_122" + id="Page_122">122</a></span></p> + + <p class="v2"> + <b><a name="Block_807" id="Block_807">807</a>.</b> + He who is unfamiliar with mathematics [literally, + he who is a layman in mathematics] remains more or less a + stranger to our time.—<span class="smcap">Dillmann, E.</span></p> + <p class="blockcite"> + Die Mathematik die Fackelträgerin einer + neuen Zeit (Stuttgart, 1889), p. 39.</p> + + <p class="v2"> + <b><a name="Block_808" id="Block_808">808</a>.</b> + Enlist a great mathematician and a distinguished + Grecian; your problem will be solved. Such men can teach in a + dwelling-house as well as in a palace. Part of the apparatus + they will bring; part we will furnish. [Advice given to the + Trustees of Johns Hopkins University on the choice of a + professorial staff.]—<span class="smcap">Gilman, D. C.</span></p> + <p class="blockcite"> + Report of the President of Johns Hopkins University + (1888), p. 29.</p> + + <p class="v2"> + <b><a name="Block_809" id="Block_809">809</a>.</b> + Persons, who have a decided mathematical talent, + constitute, as it were, a favored class. They bear the same + relation to the rest of mankind that those who are academically + trained bear to those who are + not.—<span class="smcap">Moebius, P. J.</span></p> + <p class="blockcite"> + Ueber die Anlage zur Mathematik (Leipzig, 1900), p. 4.</p> + + <p class="v2"> + <b><a name="Block_810" id="Block_810">810</a>.</b> + One may be a mathematician of the first rank + without being able to compute. It is possible to be a great + computer without having the slightest idea of + mathematics.—<span class="smcap">Novalis.</span></p> + <p class="blockcite"> + Schriften, Zweiter Teil (Berlin, 1901), p. 223.</p> + + <p class="v2"> + <b><a name="Block_811" id="Block_811">811</a>.</b> + It has long been a complaint against mathematicians + that they are hard to convince: but it is a far greater + disqualification both for philosophy, and for the affairs of + life, to be too easily convinced; to have too low a standard of + proof. The only sound intellects are those which, in the first + instance, set their standards of proof high. Practice in + concrete affairs soon teaches them to make the necessary + abatement: but they retain the consciousness, without which + there is no sound practical reasoning, that in accepting + inferior evidence because there is no better to be had, they do + not by that acceptance raise it to + completeness.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + An Examination of Sir William Hamilton’s Philosophy + (London, 1878), p. 611.</p> + +<p><span class="pagenum"> + <a name="Page_123" + id="Page_123">123</a></span></p> + + <p class="v2"> + <b><a name="Block_812" id="Block_812">812</a>.</b> + It is easier to square the circle than to get round + a mathematician.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), p. 90.</p> + + <p class="v2"> + <b><a name="Block_813" id="Block_813">813</a>.</b> + Mathematicians are like Frenchmen: whatever you say + to them they translate into their own language and forthwith it + is something entirely + different.—<span class="smcap">Goethe.</span></p> + <p class="blockcite"> + Maximen und Reflexionen, Sechste Abtheilung.</p> + + <p class="v2"> + <b><a name="Block_814" id="Block_814">814</a>.</b> + What I chiefly admired, and thought altogether + unaccountable, was the strong disposition I observed in them + [the mathematicians of Laputa] towards news and politics; + perpetually inquiring into public affairs; giving their + judgments in matters of state; and passionately disputing every + inch of party opinion. I have indeed observed the same + disposition among most of the mathematicians I have known in + Europe, although I could never discover the least analogy + between the two sciences.—<span class= + "smcap">Swift, Jonathan.</span></p> + <p class="blockcite"> + Gulliver’s Travels, Part 3, chap. 2.</p> + + <p class="v2"> + <b><a name="Block_815" id="Block_815">815</a>.</b> + The great mathematician, like the great poet or + naturalist or great administrator, is born. My contention shall + be that where the mathematic endowment is found, there will + usually be found associated with it, as essential implications + in it, other endowments in generous measure, and that the + appeal of the science is to the whole mind, direct no doubt to + the central powers of thought, but indirectly through sympathy + of all, rousing, enlarging, developing, emancipating all, so + that the faculties of will, of intellect and feeling learn to + respond, each in its appropriate order and degree, like the + parts of an orchestra to the “urge and ardor” of its leader and + lord.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 22.</p> + + <p class="v2"> + <b><a name="Block_816" id="Block_816">816</a>.</b> + Whoever limits his exertions to the gratification + of others, whether by personal exhibition, as in the case of + the actor and of the mimic, or by those kinds of literary + composition which are calculated for no end but to please or to + entertain, renders himself, in some measure, dependent on their + caprices and humours. The diversity among men, in their judgments + +<span class="pagenum"> + <a name="Page_124" + id="Page_124">124</a></span> + + concerning the objects of taste, is + incomparably greater than in their speculative conclusions; and + accordingly, a mathematician will publish to the world a + geometrical demonstration, or a philosopher, a process of + abstract reasoning, with a confidence very different from what + a poet would feel, in communicating one of his productions even + to a friend.—<span class="smcap">Stewart, Dugald.</span></p> + <p class="blockcite"> + Elements of the Philosophy of the Human Mind, Part 3, + chap. 1, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_817" id="Block_817">817</a>.</b> + Considering that, among all those who up to this time made + discoveries in the sciences, it was the mathematicians alone + who had been able to arrive at demonstrations—that is to say, + at proofs certain and evident—I did not doubt that I should + begin with the same truths that they have investigated, + although I had looked for no other advantage from them than + to accustom my mind to nourish itself upon truths and not to + be satisfied with false + reasons.—<span class= "smcap">Descartes.</span></p> + <p class="blockcite"> + Discourse upon Method, Part 2; Philosophy of Descartes + [Torrey] (New York, 1892), p. 48.</p> + + <p class="v2"> + <b><a name="Block_818" id="Block_818">818</a>.</b> + When the late Sophus Lie ... was asked to name the + characteristic endowment of the mathematician, his answer was + the following quaternion: Phantasie, Energie, Selbstvertrauen, + Selbstkritik.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Philosophy, Science and Art (New York, 1908), + p. 31.</p> + + <p class="v2"> + <b><a name="Block_819" id="Block_819">819</a>.</b> + The existence of an extensive Science of + Mathematics, requiring the highest scientific genius in those + who contributed to its creation, and calling for the most + continued and vigorous exertion of intellect in order to + appreciate it when created, + etc.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + System of Logic, Bk. 2, chap. 4, sect. 4.</p> + + <p class="v2"> + <b><a name="Block_820" id="Block_820">820</a>.</b> + It may be true, that men, who are <em>mere</em> + mathematicians, have certain specific shortcomings, but that is + not the fault of mathematics, for it is equally true of every + other exclusive occupation. So there are <em>mere</em> + philologists, <em>mere</em> jurists, <em>mere</em> soldiers, + <em>mere</em> merchants, etc. To such idle talk it might further + be added: that whenever a certain exclusive occupation is + +<span class="pagenum"> + <a name="Page_125" + id="Page_125">125</a></span> + + <em>coupled</em> with specific shortcomings, it is likewise + almost certainly divorced from certain <em>other</em> + shortcomings.—<span class= "smcap">Gauss.</span></p> + <p class="blockcite"> + Gauss-Schumacher Briefwechsel, Bd. 4, (Altona, 1862), p. 387.</p> + + <p class="v2"> + <b><a name="Block_821" id="Block_821">821</a>.</b> + Mathematical studies ... when combined, as they now + generally are, with a taste for physical science, enlarge + infinitely our views of the wisdom and power displayed in the + universe. The very intimate connexion indeed, which, since the + date of the Newtonian philosophy, has existed between the + different branches of mathematical and physical knowledge, + renders such a character as that of a <em>mere mathematician</em> + a very rare and scarcely possible + occurrence.—<span class="smcap">Stewart, Dugald.</span></p> + <p class="blockcite"> + Elements of the Philosophy of the Human Mind, part 3, + chap. 1, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_822" id="Block_822">822</a>.</b> + Once when lecturing to a class he [Lord Kelvin] used the word + “mathematician,” and then interrupting himself asked his class: + “Do you know what a mathematician is?” Stepping to the + blackboard he wrote upon it:—</p> + <div class="figcenter"> + <img src="images/img822.png" + width="128" + height="48" + alt="integral from minus to plus infinity of e to the power + minus x squared dx equals root pi" + id="img822" /></div> + <p class="v1"> + Then putting his finger on what he had written, he turned to + his class and said: “A mathematician is one + to whom <em>that</em> is as obvious as that twice two makes four + is to you. Liouville was a + + <a id="TNanchor_7"></a> + <a class="msg" href="#TN_7" + title="end of quote not identified; +placement unclear">mathematician</a>.—<span + + class="smcap">Thompson, S. P.</span></p> + <p class="blockcite"> + Life of Lord Kelvin (London, 1910), p. 1139.</p> + + <p class="v2"> + <b><a name="Block_823" id="Block_823">823</a>.</b> + It is not surprising, in view of the polydynamic + constitution of the genuinely mathematical mind, that many of + the major + + <a id="TNanchor_8"></a> + <a class="msg" href="#TN_8" + title="originally spelled ‘heros’">heroes</a> + + of the science, men like Desargues and Pascal, + Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz + and Clifford, Riemann and Salmon and Plücker and + Poincaré, have attained to high distinction in + other fields not only of science but of philosophy and letters + too. And when we reflect that the very greatest mathematical + achievements have been due, not alone to the peering, + microscopic, histologic vision of men like Weierstrass, + illuminating the hidden recesses, + +<span class="pagenum"> + <a name="Page_126" + id="Page_126">126</a></span> + + the minute and intimate + structure of logical reality, but to the larger vision also of + men like Klein who survey the kingdoms of geometry and analysis + for the endless variety of things that nourish there, as the + eye of Darwin ranged over the flora and fauna of the world, or + as a commercial monarch contemplates its industry, or as a + statesman beholds an empire; when we reflect not only that the + Calculus of Probability is a creation of mathematics but that + the master mathematician is constantly required to exercise + judgment—judgment, that is, in matters not of + certainty—balancing probabilities not yet reduced nor even + reducible perhaps to calculation; when we reflect that he is + called upon to exercise a function analogous to that of the + comparative anatomist like Cuvier, comparing theories and + doctrines of every degree of similarity and dissimilarity of + structure; when, finally, we reflect that he seldom deals with + a single idea at a time, but is for the most part engaged + in wielding organized hosts of them, as a general wields at + once the division of an army or as + a great civil administrator directs from his central office + diverse and scattered but related groups of interests and + operations; then, I say, the current opinion that devotion to + mathematics unfits the devotee for practical affairs should be + known for false on <i lang="la" xml:lang="la">a priori</i> + grounds. And one should be + thus prepared to find that as a fact Gaspard Monge, creator of + descriptive geometry, author of the classic “Applications de + l’analyse à la géométrie;” Lazare Carnot, author of the + celebrated works, “Géométrie de position,” and “Réflections + sur la Métaphysique du Calcul infinitesimal;” Fourier, immortal + creator of the “Théorie analytique de la chaleur;” Arago, + rightful inheritor of Monge’s chair of geometry; Poncelet, + creator of pure projective geometry; one should not be surprised, + I say, to find that these and other mathematicians in a land + sagacious enough to invoke their aid, rendered, alike in peace + and in war, eminent public + service.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + pp. 32-33.</p> + + <p class="v2"> + <b><a name="Block_824" id="Block_824">824</a>.</b> + If in Germany the goddess <em>Justitia</em> had not + the unfortunate habit of depositing the ministerial portfolios + only in the + +<span class="pagenum"> + <a name="Page_127" + id="Page_127">127</a></span> + + cradles of her own progeny, who knows how many a German + mathematician might not also have made an excellent + minister.—<span class= "smcap">Pringsheim, A.</span></p> + <p class="blockcite"> + Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. + 13 (1904), p. 372.</p> + + <p class="v2"> + <b><a name="Block_825" id="Block_825">825</a>.</b> + We pass with admiration along the great series of + mathematicians, by whom the science of theoretical mechanics + has been cultivated, from the time of Newton to our own. There + is no group of men of science whose fame is higher or brighter. + The great discoveries of Copernicus, Galileo, Newton, had fixed + all eyes on those portions of human knowledge on which their + successors employed their labors. The certainty belonging to + this line of speculation seemed to elevate mathematicians above + the students of other subjects; and the beauty of mathematical + relations and the subtlety of intellect which may be shown in + dealing with them, were fitted to win unbounded applause. The + successors of Newton and the Bernoullis, as Euler, Clairaut, + D’Alembert, Lagrange, Laplace, not to introduce living names, + have been some of the most remarkable men of talent which the + world has seen.—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + History of the Inductive Sciences, Vol. 1, Bk. 4, chap. 6, + sect. 6.</p> + + <p class="v2"> + <b><a name="Block_826" id="Block_826">826</a>.</b> + The persons who have been employed on these + problems of applying the properties of matter and the laws of + motion to the explanation of the phenomena of the world, and + who have brought to them the high and admirable qualities which + such an office requires, have justly excited in a very eminent + degree the admiration which mankind feels for great + intellectual powers. Their names occupy a distinguished place + in literary history; and probably there are no scientific + reputations of the last century higher, and none more merited, + than those earned by great mathematicians who have laboured + with such wonderful success in unfolding the mechanism of the + heavens; such for instance as D’Alembert, Clairaut, Euler, + Lagrange, Laplace.—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + Astronomy and General Physics (London, 1833), Bk. 3, chap. + 4, p. 327.</p> + +<p><span class="pagenum"> + <a name="Page_128" + id="Page_128">128</a></span></p> + + <p class="v2"> + <b><a name="Block_827" id="Block_827">827</a>.</b> + Two extreme views have always been held as to the + use of mathematics. To some, mathematics is only measuring and + calculating instruments, and their interest ceases as soon as + discussions arise which cannot benefit those who use the + instruments for the purposes of application in mechanics, + astronomy, physics, statistics, and other sciences. At the + other extreme we have those who are animated exclusively by the + love of pure science. To them pure mathematics, with the theory + of numbers at the head, is the only real and genuine science, + and the applications have only an interest in so far as they + contain or suggest problems in pure mathematics.</p> + <p class="v1"> + Of the two greatest mathematicians of modern times, Newton and + Gauss, the former can be considered as a representative of the + first, the latter of the second class; neither of them was + exclusively so, and Newton’s inventions in + the science of pure mathematics were probably equal to + Gauss’s work in applied mathematics. + Newton’s reluctance to publish the method of + fluxions invented and used by him may perhaps be attributed to + the fact that he was not satisfied with the logical foundations + of the Calculus; and Gauss is known to have abandoned his + electro-dynamic speculations, as he could not find a satisfying + physical basis....</p> + <p class="v1"> + Newton’s greatest work, the “Principia”, laid the foundation + of mathematical physics; Gauss’s greatest work, the + “Disquisitiones Arithmeticae”, that of higher arithmetic as + distinguished from algebra. Both works, written in the + synthetic style of the ancients, are difficult, if not + deterrent, in their form, neither of them leading the reader + by easy steps to the results. It took twenty or more years + before either of these works received due recognition; neither + found favour at once before that great tribunal of mathematical + thought, the Paris Academy of Sciences....</p> + <p class="v1"> + The country of Newton is still pre-eminent for its culture of + mathematical physics, that of Gauss for the most abstract work + in mathematics.—<span class="smcap">Merz, J. T.</span></p> + <p class="blockcite"> + History of European Thought in the Nineteenth Century + (Edinburgh and London, 1903), p. 630.</p> + +<p><span class="pagenum"> + <a name="Page_129" + id="Page_129">129</a></span></p> + + <p class="v2"> + <b><a name="Block_828" id="Block_828">828</a>.</b> + As there is no study which may be so advantageously + entered upon with a less stock of preparatory knowledge than + mathematics, so there is none in which a greater number of + uneducated men have raised themselves, by their own exertions, + to distinction and eminence.... Many of the intellectual + defects which, in such cases, are commonly placed to the + account of mathematical studies, ought to be ascribed to the + want of a liberal education in early + youth.—<span class="smcap">Stewart, Dugald.</span></p> + <p class="blockcite"> + Elements of the Philosophy of the Human Mind, Part 3, + chap. 1, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_829" id="Block_829">829</a>.</b> + I know, indeed, and can conceive of no pursuit so + antagonistic to the cultivation of the oratorical faculty ... + as the study of Mathematics. An eloquent mathematician must, + from the nature of things, ever remain as rare a phenomenon as + a talking fish, and it is certain that the more anyone gives + himself up to the study of oratorical effect the less will he + find himself in a fit state to mathematicize. It is the + constant aim of the mathematician to reduce all his expressions + to their lowest terms, to retrench every superfluous word and + phrase, and to condense the Maximum of meaning into the Minimum + of language. He has to turn his eye ever inwards, to see + everything in its dryest light, to train and inure himself to a + habit of internal and impersonal reflection and elaboration of + abstract thought, which makes it most difficult for him to + touch or enlarge upon any of those themes which appeal to the + emotional nature of his fellow-men. When called upon to speak + in public he feels as a man might do who has passed all his + life in peering through a microscope, and is suddenly called + upon to take charge of a astronomical observatory. He has to + get out of himself, as it were, and change the habitual focus + of his vision.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Baltimore Address; Mathematical Papers, Vol. 3, pp. 72-73.</p> + + <p class="v2"> + <b><a name="Block_830" id="Block_830">830</a>.</b> + An accomplished mathematician, i.e. a most wretched + orator.—<span class="smcap">Barrow, Isaac.</span></p> + <p class="blockcite"> + Mathematical Lectures (London, 1734), p. 32.</p> + +<p><span class="pagenum"> + <a name="Page_130" + id="Page_130">130</a></span></p> + + <p class="v2"> + <b><a name="Block_831" id="Block_831">831</a>.</b> + <i lang="la" xml:lang="la">Nemo mathematicus genium indemnatus + habebit.</i> [No mathematician<a + href="#Footnote_2" + class="fnanchor" + title="Used here in the sense of astrologer, +or soothsayer.">2</a> + is esteemed a genius until condemned.]</p> + <p class="blockcite"> + Juvenal, Liberii, Satura VI, 562.</p> + + <p class="v2"> + <b><a name="Block_832" id="Block_832">832</a>.</b> + Taking ... the mathematical faculty, probably fewer + than one in a hundred really possess it, the great bulk of the + population having no natural ability for the study, or feeling + the slightest interest in it.<a + href="#Footnote_3" + class="fnanchor">3</a> + And if we attempt to measure the amount of variation in the + faculty itself between a first-class mathematician and the + ordinary run of people who find any kind of calculation + confusing and altogether devoid of interest, it is probable + that the former could not be estimated at less than a hundred + times the latter, and perhaps a thousand times would more + nearly measure the difference between + them.—<span class="smcap">Wallace, A. R.</span></p> + <p class="blockcite"> + Darwinism, chap. 15.</p> + + <p class="v2"> + <b><a name="Block_833" id="Block_833">833</a>.</b> + ... the present gigantic development of the + mathematical faculty is wholly unexplained by the theory of + natural selection, and must be due to some altogether distinct + cause.—<span class="smcap">Wallace, A. R.</span></p> + <p class="blockcite"> + Darwinism, chap. 15.</p> + + <p class="v2"> + <b><a name="Block_834" id="Block_834">834</a>.</b> + Dr. Wallace, in his “Darwinism”, declares that he + can find no ground for the existence of pure scientists, + especially mathematicians, on the hypothesis of natural + selection. If we put aside the fact that great power in + theoretical science is correlated with other developments of + increasing brain-activity, we may, I think, still account for + the existence of pure scientists as Dr. Wallace would himself + account for that of worker-bees. Their function may not fit + them individually to survive in the struggle for existence, but + they are a source of strength and efficiency to the society + which produces + them.—<span class= "smcap">Pearson, Karl.</span></p> + <p class="blockcite"> + Grammar of Science (London, 1911), Part 1, p. 221.</p> + +<p><span class="pagenum"> + <a name="Page_131" + id="Page_131">131</a></span></p> + + <p class="v2"> + <b><a name="Block_835" id="Block_835">835</a>.</b> + It is only in mathematics, and to some extent in + poetry, that originality may be attained at an early age, but + even then it is very rare (Newton and Keats are examples), and + it is not notable until adolescence is + completed.—<span class="smcap">Ellis, Havelock.</span></p> + <p class="blockcite"> + A Study of British Genius (London, 1904), p. 142.</p> + + <p class="v2"> + <b><a name="Block_836" id="Block_836">836</a>.</b> + The Anglo-Dane appears to possess an aptitude for + mathematics which is not shared by the native of any other + English district as a whole, and it is in the exact sciences + that the Anglo-Dane triumphs.<a + href="#Footnote_4" + class="fnanchor">4</a>—<span class= + "smcap">Ellis, Havelock.</span></p> + <p class="blockcite"> + A Study of British Genius (London, 1904), p. 69.</p> + + <p class="v2"> + <b><a name="Block_837" id="Block_837">837</a>.</b> + In the whole history of the world there was never a + race with less liking for abstract reasoning than the + Anglo-Saxon.... Common-sense and compromise are believed in, + logical deductions from philosophical principles are looked + upon with suspicion, not only by legislators, but by all our + most learned professional + men.—<span class="smcap">Perry, John.</span></p> + <p class="blockcite"> + The Teaching of Mathematics (London, 1902), pp. 20-21.</p> + + <p class="v2"> + <b><a name="Block_838" id="Block_838">838</a>.</b> + The degree of exactness of the intuition of space + may be different in different individuals, perhaps even in + different races. It would seem as if a strong + naïve space-intuition were an attribute + pre-eminently of the Teutonic race, while the critical, purely + logical sense is more fully developed in the Latin and Hebrew + races. A full investigation of this subject, somewhat on the + lines suggested by <em>Francis Galton</em> in his researches on + heredity, might be interesting.—<span class= + "smcap">Klein, Felix.</span></p> + <p class="blockcite"> + The Evanston Colloquium Lectures (New York, 1894), p. 46.</p> + + <p class="v2"> + <b><a name="Block_839" id="Block_839">839</a>.</b> + This [the fact that the pursuit of mathematics + brings into harmonious action all the faculties of the human + mind] accounts for the extraordinary longevity of all the + greatest masters of the Analytic art, the Dii Majores of the + mathematical + +<span class="pagenum"> + <a name="Page_132" + id="Page_132">132</a></span> + + Pantheon. Leibnitz lived to the age + of 70; Euler to 76; Lagrange to 77; Laplace to 78; Gauss to 78; + Plato, the supposed inventor of the conic sections, who made + mathematics his study and delight, who called them the handles + or aids to philosophy, the medicine of the soul, and is said + never to have let a day go by without inventing some new + theorems, lived to 82; Newton, the crown and glory of his race, + to 85; Archimedes, the nearest akin, probably, to Newton in + genius, was 75, and might have lived on to be 100, for aught we + can guess to the contrary, when he was slain by the impatient + and ill-mannered sergeant, sent to bring him before the Roman + general, in the full vigour of his faculties, and in the very + act of working out a problem; Pythagoras, in whose school, I + believe, the word mathematician (used, however, in a somewhat + wider than its present sense) originated, the second founder of + geometry, the inventor of the matchless theorem which goes by + his name, the pre-cognizer of the undoubtedly mis-called + Copernican theory, the discoverer of the regular solids and the + musical canon who stands at the very apex of this pyramid of + fame, (if we may credit the tradition) after spending 22 years + studying in Egypt, and 12 in Babylon, opened school when 56 or + 57 years old in Magna Græcia, married a young + wife when past 60, and died, carrying on his work with energy + unspent to the last, at the age of 99. The mathematician lives + long and lives young; the wings of his soul do not early drop + off, nor do its pores become clogged with the earthy particles + blown from the dusty highways of vulgar + life.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Presidential Address to the British Association; Collected + Mathematical Papers, Vol. 2 (1908), p. 658.</p> + + <p class="v2"> + <b><a name="Block_840" id="Block_840">840</a>.</b> + The game of chess has always fascinated + mathematicians, and there is reason to suppose that the + possession of great powers of playing that game is in many + features very much like the possession of great mathematical + ability. There are the different pieces to learn, the pawns, + the knights, the bishops, the castles, and the queen and king. + The board possesses certain possible combinations of squares, + as in rows, diagonals, etc. The pieces are subject to certain + rules by which their motions are governed, and there are other + rules governing the players.... + +<span class="pagenum"> + <a name="Page_133" + id="Page_133">133</a></span> + + One has only to + increase the number of pieces, to enlarge the field of the + board, and to produce new rules which are to govern either the + pieces or the player, to have a pretty good idea of what + mathematics consists.—<span class="smcap">Shaw, J. B.</span></p> + <p class="blockcite"> + What is Mathematics? Bulletin American Mathematical + Society Vol. 18 (1912), pp. 386-387.</p> + + <p class="v2"> + <b><a name="Block_841" id="Block_841">841</a>.</b> + Every man is ready to join in the approval or condemnation of + a philosopher or a statesman, a poet or an orator, an artist + or an architect. But who can judge of a mathematician? Who will + write a review of Hamilton’s Quaternions, and show us wherein + it is superior to Newton’s + Fluxions?—<span class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + Imagination in Mathematics; North American Review, Vol. + 85, p. 224.</p> + + <p class="v2"> + <b><a name="Block_842" id="Block_842">842</a>.</b> + The pursuit of mathematical science makes its + votary appear singularly indifferent to the ordinary interests + and cares of men. Seeking eternal truths, and finding his + pleasures in the realities of form and number, he has little + interest in the disputes and contentions of the passing hour. + His views on social and political questions partake of the + grandeur of his favorite contemplations, and, while careful to + throw his mite of influence on the side of right and truth, he + is content to abide the workings of those general laws by which + he doubts not that the fluctuations of human history are as + unerringly guided as are the perturbations of the planetary + hosts.—<span class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + Imagination in Mathematics; North American Review, Vol. + 85, p. 227.</p> + + <p class="v2"> + <b><a name="Block_843" id="Block_843">843</a>.</b> + There is something sublime in the secrecy in which + the really great deeds of the mathematician are done. No + popular applause follows the act; neither contemporary nor + succeeding generations of the people understand it. The + geometer must be tried by his peers, and those who truly + deserve the title of geometer or analyst have usually been + unable to find so many as twelve living peers to form a jury. + Archimedes so far outstripped his competitors in the race, that + more than a thousand years elapsed before any man appeared, + able to sit in judgment on his work, and to say how far he had + really gone. And in judging of those men whose + names are worthy of being mentioned + +<span class="pagenum"> + <a name="Page_134" + id="Page_134">134</a></span> + + in connection with his,—Galileo, Descartes, Leibnitz, Newton, + and the mathematicians created by Leibnitz and Newton’s + calculus,—we are forced to depend upon their testimony of one + another. They are too far above our reach for us to judge of + them.—<span class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + Imagination in Mathematics; North American Review, + Vol. 85, p. 223.</p> + + <p class="v2"> + <b><a name="Block_844" id="Block_844">844</a>.</b> + To think the thinkable—that is the mathematician’s + aim.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + The Universe and Beyond; Hibbert Journal, Vol. 3 + (1904-1905), p. 312.</p> + + <p class="v2"> + <b><a name="Block_845" id="Block_845">845</a>.</b> + Every common mechanic has something to say in his + craft about good and evil, useful and useless, but these + practical considerations never enter into the purview of the + mathematician.—<span + class="smcap">Aristippus the Cyrenaic.</span></p> + <p class="blockcite"> + Quoted in Hicks, R. D., Stoic and Epicurean, (New York, + 1910) p. 210.</p> + +<p><span class="pagenum"> + <a name="Page_135" + id="Page_135">135</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_IX"> + CHAPTER IX<br /> + <span class="large"> + PERSONS AND ANECDOTES<br /> + (A-M)</span></h2> + + <p class="v2"> + <b><a name="Block_901" id="Block_901">901</a>.</b> + Alexander is said to have asked Menæchmus to teach him geometry + concisely, but Menæchmus replied: “O king, + through the country there are royal roads and roads for common + citizens, but in geometry there is one road for all.”</p> + <p class="blockcite"> + Stobœus (Edition Wachsmuth, Berlin, 1884), Ecl. 2, p. 30</p> + + <p class="v2"> + <b><a name="Block_902" id="Block_902">902</a>.</b> + Alexander the king of the Macedonians, began like a + wretch to learn geometry, that he might know how little the + earth was, whereof he had possessed very little. Thus, I say, + like a wretch for this, because he was to understand that he + did bear a false surname. For who can be great in so small a + thing? Those things that were delivered were subtile, and to be + learned by diligent attention: not which that mad man could + perceive, who sent his thoughts beyond the ocean sea. Teach me, + saith he, easy things. To whom his master said: These things be + the same, and alike difficult unto all. Think thou that the + nature of things saith this. These things whereof thou + complainest, they are the same unto all: more easy things can + be given unto none; but whosoever will, shall make those things + more easy unto himself. How? With uprightness of + mind.—<span class="smcap">Seneca</span>.</p> + <p class="blockcite"> + Epistle 91 [Thomas Lodge].</p> + + <p class="v2"> + <b><a name="Block_903" id="Block_903">903</a>.</b> + Archimedes ... had stated that given the force, any + given weight might be moved, and even boasted, we are told, + relying on the strength of demonstration, that if there were + another earth, by going into it he could remove this. Hiero + being struck with amazement at this, and entreating him to make + good this problem by actual experiment, and show some great + weight moved by a small engine, he fixed accordingly upon a + ship of burden out of the king’s arsenal, + which could not be drawn out of the dock without great labor + and many men; and, + +<span class="pagenum"> + <a name="Page_136" + id="Page_136">136</a></span> + + loading her with many passengers and + a full freight, sitting himself the while far off with no great + endeavor, but only holding the head of the pulley in his hand + and drawing the cords by degrees, he drew the ship in a + straight line, as smoothly and evenly, as if she had been in + the sea. The king, astonished at this, and convinced of the + power of the art, prevailed upon Archimedes to make him engines + accommodated to all the purposes, offensive and defensive, of a + siege ... the apparatus was, in most opportune time, ready at + hand for the Syracusans, and with it also the engineer + himself.—<span class="smcap">Plutarch.</span></p> + <p class="blockcite"> + Life of Marcellus [Dryden].</p> + + <p class="v2"> + <b><a name="Block_904" id="Block_904">904</a>.</b> + These machines [used in the defense of the + Syracusans against the Romans under Marcellus] he [Archimedes] + had designed and contrived, not as matters of any importance, + but as mere amusements in geometry; in compliance with king + Hiero’s desire and request, some time + before, that he should reduce to practice some part of his + admirable speculation in science, and by accommodating the + theoretic truth to sensation and ordinary use, bring it more + within the appreciation of people in general. Eudoxus and + Archytas had been the first originators of this far-famed and + highly-prized art of mechanics, which they employed as an + elegant illustration of geometrical truths, and as means of + sustaining experimentally, to the satisfaction of the senses + conclusions too intricate for proof by words and diagrams. As, + for example, to solve the problem, so often required in + constructing geometrical figures, given the two extremes, to + find the two mean lines of a proportion, both these + mathematicians had recourse to the aid of instruments, adapting + to their purpose certain curves and sections of lines. But what + with Plato’s indignation at it, and his + invectives against it as the mere corruption and annihilation + of the one good of geometry,—which was thus + shamefully turning its back upon the unembodied objects of pure + intelligence to recur to sensation, and to ask help (not to be + obtained without base supervisions and depravation) from + matter; so it was that mechanics came to be separated from + geometry, and, repudiated and neglected by philosophers, took + its place as a military + art.—<span class="smcap">Plutarch.</span></p> + <p class="blockcite"> + Life of Marcellus [Dryden].</p> + +<p><span class="pagenum"> + <a name="Page_137" + id="Page_137">137</a></span></p> + + <p class="v2"> + <b><a name="Block_905" id="Block_905">905</a>.</b> + Archimedes was not free from the prevailing notion + that geometry was degraded by being employed to produce + anything useful. It was with difficulty that he was induced to + stoop from speculation to practice. He was half ashamed of + those inventions which were the wonder of hostile nations, and + always spoke of them slightingly as mere amusements, as trifles + in which a mathematician might be suffered to relax his mind + after intense application to the higher parts of his + science.—<span class="smcap">Macaulay.</span></p> + <p class="blockcite"> + Lord Bacon; Edinburgh Review, July 1837; Critical and + Miscellaneous Essays (New York, 1879), Vol. 1, p. 380.</p> + + <p class="v2"> + <b><a name="Block_906" id="Block_906">906</a>.</b></p> + <div class="poem"> + <p class="i0"> + Call Archimedes from his buried tomb</p> + <p class="i0"> + Upon the plain of vanished Syracuse,</p> + <p class="i0"> + And feelingly the sage shall make report</p> + <p class="i0"> + How insecure, how baseless in itself,</p> + <p class="i0"> + Is the philosophy, whose sway depends</p> + <p class="i0"> + On mere material instruments—how weak</p> + <p class="i0"> + Those arts, and high inventions, if unpropped</p> + <p class="i0"> + By virtue.</p> + </div> + <p class="block40"> + —<span class="smcap">Wordsworth.</span></p> + <p class="blockcite"> + The Excursion.</p> + + <p class="v2"> + <b><a name="Block_907" id="Block_907">907</a>.</b></p> + <div class="poem"> + <p class="i0"> + Zu Archimedes kam einst ein wissbegieriger Jüngling.</p> + <p class="i0"> + “Weihe mich,” sprach er zu ihm, “ein in die göttliche + Kunst,</p> + <p class="i0"> + Die so herrliche Frucht dem Vaterlande getragen,</p> + <p class="i0"> + Und die Mauern der Stadt vor der Sambuca beschützt!”</p> + <p class="i0"> + “Göttlich nennst du die Kunst? Sie ists,” versetzte der + Weise;</p> + <p class="i0"> + “Aber das war sie, mein Sohn, eh sie dem Staat noch + gedient.</p> + <p class="i0"> + Willst du nur Früchte von ihr, die kann auch die Sterbliche + zeugen;</p> + <p class="i0"> + Wer um die Göttin freit, suche in ihr nicht das + Weib.”</p> + </div> + <p class="block40"> + —<span class="smcap">Schiller.</span></p> + <p class="blockcite"> + Archimedes und der Schüler.</p> + +<p><span class="pagenum"> + <a name="Page_138" + id="Page_138">138</a></span></p> + + <div class="poem"> + <br /> + <p class="i0"> + [To Archimedes once came a youth intent upon knowledge.</p> + <p class="i0"> + Said he “Initiate me into the Science divine,</p> + <p class="i0"> + Which to our country has borne glorious fruits in + abundance,</p> + <p class="i0"> + And which the walls of the town ’gainst the Sambuca + protects.”</p> + <p class="i0"> + “Callst thou the science divine? It is so,” the wise + man responded;</p> + <p class="i0"> + “But so it was, my son, ere the state by her service + was blest.</p> + <p class="i0"> + Would’st thou have fruit of her only? Mortals with + that can provide thee,</p> + <p class="i0"> + He who the goddess would woo, seek not the woman in + her.”]</p> + </div> + + <p class="v2"> + <b><a name="Block_908" id="Block_908">908</a>.</b> + Archimedes possessed so high a spirit, so profound + a soul, and such treasures of highly scientific knowledge, that + though these inventions [used to defend Syracuse against the + Romans] had now obtained him the renown of more than human + sagacity, he yet would not deign to leave behind him any + commentary or writing on such subjects; but, repudiating as + sordid and ignoble the whole trade of engineering, and every + sort of art that lends itself to mere use and profit, he placed + his whole affection and ambition in those purer speculations + where there can be no reference to the vulgar needs of life; + studies, the superiority of which to all others is + unquestioned, and in which the only doubt can be whether the + beauty and grandeur of the subjects examined, or the precision + and cogency of the methods and means of proof, most deserve our + admiration.—<span class="smcap">Plutarch.</span></p> + <p class="blockcite"> + Life of Marcellus [Dryden].</p> + + <p class="v2"> + <b><a name="Block_909" id="Block_909">909</a>.</b> + Nothing afflicted Marcellus so much as the death of Archimedes, + who was then, as fate would have it, intent upon working out + some problem by a diagram, and having fixed his mind alike and + his eyes upon the subject of his speculation, he never noticed + the incursion of the Romans, nor that the city was taken. In + this transport of study and contemplation, a soldier, + unexpectedly coming up to him, commanded him to follow to + +<span class="pagenum"> + <a name="Page_139" + id="Page_139">139</a></span> + + Marcellus, which he declined to do + before he had worked out his problem to a demonstration; the + soldier, enraged, drew his sword and ran him through. Others + write, that a Roman soldier, running upon him with a drawn + sword, offered to kill him; and that Archimedes, looking back, + earnestly besought him to hold his hand a little while, that he + might not leave what he was at work upon inconclusive and + imperfect; but the soldier, nothing moved by his entreaty, + instantly killed him. Others again relate, that as Archimedes + was carrying to Marcellus mathematical instruments, dials, + spheres, and angles, by which the magnitude of the sun might be + measured to the sight, some soldiers seeing him, and thinking + that he carried gold in a vessel, slew him. Certain it is, that + his death was very afflicting to Marcellus; and that Marcellus + ever after regarded him that killed him as a murderer; and that + he sought for his kindred and honoured them with signal + favours.—<span class="smcap">Plutarch.</span></p> + <p class="blockcite"> + Life of Marcellus [Dryden].</p> + + <p class="v2"> + <b><a name="Block_910" id="Block_910">910</a>.</b> + [Archimedes] is said to have requested his friends + and relations that when he was dead, they would place over his + tomb a sphere containing a cylinder, inscribing it with the + ratio which the containing solid bears to the + contained.—<span class="smcap">Plutarch.</span></p> + <p class="blockcite"> + Life of Marcellus [Dryden].</p> + + <p class="v2"> + <b><a name="Block_911" id="Block_911">911</a>.</b> + Archimedes, who combined a genius for mathematics + with a physical insight, must rank with Newton, who lived + nearly two thousand years later, as one of the founders of + mathematical physics.... The day (when having discovered his + famous principle of hydrostatics he ran through the streets + shouting Eureka! Eureka!) ought to be celebrated as the + birthday of mathematical physics; the science came of age when + Newton sat in his orchard.—<span class= + "smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + An Introduction to Mathematics (New York, 1911), p. 38.</p> + + <p class="v2"> + <b><a name="Block_912" id="Block_912">912</a>.</b> + It is not possible to find in all geometry more + difficult and more intricate questions or more simple and lucid + explanations [than those given by Archimedes]. Some ascribe + this to his natural genius; while others think that incredible + effort and toil produced these, to all appearance, easy and + unlaboured + +<span class="pagenum"> + <a name="Page_140" + id="Page_140">140</a></span> + + results. No amount of investigation + of yours would succeed in attaining the proof, and yet, once + seen, you immediately believe you would have discovered it; by + so smooth and so rapid a path he leads you to the conclusion + required.—<span class="smcap">Plutarch.</span></p> + <p class="blockcite"> + Life of Marcellus [Dryden].</p> + + <p class="v2"> + <b><a name="Block_913" id="Block_913">913</a>.</b> + One feature which will probably most impress the + mathematician accustomed to the rapidity and directness secured + by the generality of modern methods is the <em>deliberation</em> + with which Archimedes approaches the solution of any one of his + main problems. Yet this very characteristic, with its + incidental effects, is calculated to excite the more admiration + because the method suggests the tactics of some great + strategist who foresees everything, eliminates everything not + immediately conducive to the execution of his plan, masters + every position in its order, and then suddenly (when the very + elaboration of the scheme has almost obscured, in the mind of + the spectator, its ultimate object) strikes the final blow. + Thus we read in Archimedes proposition after proposition the + bearing of which is not immediately obvious but which we find + infallibly used later on; and we are led by such easy stages + that the difficulties of the original problem, as presented at + the outset, are scarcely appreciated. As Plutarch says: + “It is not possible to find in geometry more + difficult and troublesome questions, or more simple and lucid + explanations.” But it is decidedly a rhetorical + exaggeration when Plutarch goes on to say that we are deceived + by the easiness of the successive steps into the belief that + anyone could have discovered them for himself. On the contrary, + the studied simplicity and the perfect finish of the treatises + involve at the same time an element of mystery. Though each + step depends on the preceding ones, we are left in the dark as + to how they were suggested to Archimedes. There is, in fact, + much truth in a remark by Wallis to the effect that he seems + “as it were of set purpose to have covered + up the traces of his investigation as if he had grudged + posterity the secret of his method of inquiry while he wished + to extort from them assent to his results.” Wallis + adds with equal reason that not only Archimedes but nearly all + the ancients so hid away from posterity their method of + Analysis (though it is certain that they had one) that more + +<span class="pagenum"> + <a name="Page_141" + id="Page_141">141</a></span> + + modern mathematicians found it easier + to invent a new Analysis than to seek out the + old.—<span class="smcap">Heath, T. L.</span></p> + <p class="blockcite"> + The Works of Archimedes (Cambridge, 1897), Preface.</p> + + <p class="v2"> + <b><a name="Block_914" id="Block_914">914</a>.</b> + It is a great pity Aristotle had not understood + mathematics as well as Mr. Newton, and made use of it in his + natural philosophy with good success: his example had then + authorized the accommodating of it to material + things.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + Second Reply to the Bishop of Worcester.</p> + + <p class="v2"> + <b><a name="Block_915" id="Block_915">915</a>.</b> + The opinion of Bacon on this subject [geometry] was + diametrically opposed to that of the ancient philosophers. He + valued geometry chiefly, if not solely, on account of those + uses, which to Plato appeared so base. And it is remarkable + that the longer Bacon lived the stronger this feeling became. + When in 1605 he wrote the two books on the Advancement of + Learning, he dwelt on the advantages which mankind derived from + mixed mathematics; but he at the same time admitted that the + beneficial effect produced by mathematical study on the + intellect, though a collateral advantage, was + “no less worthy than that which was + principal and intended.” But it is evident that his + views underwent a change. When near twenty years later, he + published the <cite>De Augmentis</cite>, which is the Treatise on the + Advancement of Learning, greatly expanded and carefully + corrected, he made important alterations in the part which + related to mathematics. He condemned with severity the + pretensions of the mathematicians, “<i lang="la" + xml:lang="la">delicias et fastum + mathematicorum</i>.” Assuming the well-being of the + human race to be the end of knowledge, he pronounced that + mathematical science could claim no higher rank than that of an + appendage or an auxiliary to other sciences. Mathematical + science, he says, is the handmaid of natural philosophy; she + ought to demean herself as such; and he declares that he cannot + conceive by what ill chance it has happened that she presumes + to claim precedence over her + mistress.—<span class= "smcap">Macaulay.</span></p> + <p class="blockcite"> + Lord Bacon: Edinburgh Review, July, 1837; Critical and + Miscellaneous Essays (New York, 1879), Vol. 1, p. 380.</p> + +<p><span class="pagenum"> + <a name="Page_142" + id="Page_142">142</a></span></p> + + <p class="v2"> + <b><a name="Block_916" id="Block_916">916</a>.</b> + If Bacon erred here [in valuing mathematics only + for its uses], we must acknowledge that we greatly prefer his + error to the opposite error of Plato. We have no patience with + a philosophy which, like those Roman matrons who swallowed + abortives in order to preserve their shapes, takes pains to be + barren for fear of being + homely.—<span class="smcap">Macaulay.</span></p> + <p class="blockcite"> + Lord Bacon, Edinburgh Review, July, 1837; Critical and + Miscellaneous Essays (New York, 1879), Vol. 2, p. 381.</p> + + <p class="v2"> + <b><a name="Block_917" id="Block_917">917</a>.</b> + He [Lord Bacon] appears to have been utterly + ignorant of the discoveries which had just been made by + Kepler’s calculations ... he does not say a + word about Napier’s Logarithms, which had + been published only nine years before and reprinted more than + once in the interval. He complained that no considerable + advance had been made in Geometry beyond Euclid, without taking + any notice of what had been done by Archimedes and Apollonius. + He saw the importance of determining accurately the specific + gravities of different substances, and himself attempted to + form a table of them by a rude process of his own, without + knowing of the more scientific though still imperfect methods + previously employed by Archimedes, Ghetaldus and Porta. He + speaks of the + + <a class="msg" href="#TN_26" + title="originally shown as ‘εὓυρηκα’">εὔυρηκα</a> + + of Archimedes in a manner which implies that he did not clearly + appreciate either the problem to be solved or the principles + upon which the solution depended. In reviewing the progress of + Mechanics, he makes no mention either of Archimedes, or + Stevinus, Galileo, Guldinus, or Ghetaldus. He makes no allusion + to the theory of Equilibrium. He observes that a ball of one + pound weight will fall nearly as fast through the air as a ball + of two, without alluding to the theory of acceleration of + falling bodies, which had been made known by Galileo more than + thirty years before. He proposed an inquiry with regard to the + lever,—namely, whether in a balance with + arms of different length but equal weight the distance from the + fulcrum has any effect upon the + inclination—though the theory of the lever + was as well understood in his own time as it is now.... He + speaks of the poles of the earth as fixed, in a manner which + seems to imply that he was not acquainted with the precession + of the equinoxes; and in another place, of the north pole being + above and the + +<span class="pagenum"> + <a name="Page_143" + id="Page_143">143</a></span> + + south pole below, as a reason why in + our hemisphere the north winds predominate over the + south.—<span class="smcap">Spedding, J.</span></p> + <p class="blockcite"> + Works of Francis Bacon (Boston), Preface to De + Interpretatione Naturae Prooemium.</p> + + <p class="v2"> + <b><a name="Block_918" id="Block_918">918</a>.</b> + Bacon himself was very ignorant of all that had + been done by mathematics; and, strange to say, he especially + objected to astronomy being handed over to the mathematicians. + Leverrier and Adams, calculating an unknown planet into a + visible existence by enormous heaps of algebra, furnish the + last comment of note on this specimen of the goodness of + Bacon’s view.... Mathematics was beginning + to be the great instrument of exact inquiry: Bacon threw the + science aside, from ignorance, just at the time when his + enormous sagacity, applied to knowledge, would have made him + see the part it was to play. If Newton had taken Bacon for his + master, not he, but somebody else, would have been + Newton.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), pp. 53-54.</p> + + <p class="v2"> + <b><a name="Block_919" id="Block_919">919</a>.</b> + Daniel Bernoulli used to tell two little + adventures, which he said had given him more pleasure than all + the other honours he had received. Travelling with a learned + stranger, who, being pleased with his conversation, asked his + name; “I am Daniel Bernoulli,” answered he with great modesty; + “and I,” said the stranger (who thought he meant to + laugh at him) “am Isaac Newton.” + Another time, having to dine with the celebrated Koenig, the + mathematician, who boasted, with some degree of + self-complacency, of a difficult problem he had solved with + much trouble, Bernoulli went on doing the honours of his table, + and when they went to drink coffee he presented Koenig with a + solution of the problem more elegant than his + own.—<span class="smcap">Hutton, Charles.</span></p> + <p class="blockcite"> + A Philosophical and Mathematical Dictionary (London, + 1815), Vol. 1, p. 226.</p> + + <p class="v2"> + <b><a name="Block_920" id="Block_920">920</a>.</b> + Following the example of Archimedes who wished his + tomb decorated with his most beautiful discovery in geometry + and ordered it inscribed with a cylinder circumscribed by a + sphere, James Bernoulli requested that his tomb be inscribed + with his logarithmic spiral together with the words, + “<em>Eadem</em> + +<span class="pagenum"> + <a name="Page_144" + id="Page_144">144</a></span> + + <em>mutata resurgo</em>,” a happy allusion to the hope of the + Christians, which is in a way symbolized by the properties of + that curve.—<span class= "smcap">Fontenelle.</span></p> + <p class="blockcite"> + Eloge de M. Bernoulli; Oeuvres de Fontenelle, t. 5 (1758), + p. 112.</p> + + <p class="v2"> + <b><a name="Block_921" id="Block_921">921</a>.</b> + This formula [for computing + Bernoulli’s numbers] was first given by + James Bernoulli. He gave no general demonstration; but was + quite aware of the importance of his theorem, for he boasts + that by means of it he calculated <i lang="la" + xml:lang="la">intra semi-quadrantem + horae!</i> the sum of the 10th powers of the first thousand + integers, and found it to be</p> + <p class="center">91,409,924,241,424,243,424,241,924,242,500.</p> + <p class="right">—<span class="smcap">Chrystal, G.</span></p> + <p class="blockcite"> + Algebra, Part 2 (Edinburgh, 1879), p. 209.</p> + + <p class="v2"> + <b><a name="Block_922" id="Block_922">922</a>.</b> + In the year 1692, James Bernoulli, discussing the logarithmic + spiral [or equiangular spiral, ρ = α<sup>θ</sup>] ... shows + that it reproduces itself in its evolute, its involute, and its + caustics of both reflection and refraction, and then adds: "But + since this marvellous spiral, by such a singular and wonderful + peculiarity, pleases me so much that I can scarce be satisfied + with thinking about it, I have thought that it might not be + inelegantly used for a symbolic representation of various + matters. For since it always produces a spiral similar to + itself, indeed precisely the same spiral, however it may be + involved or evolved, or reflected or refracted, it may be taken + as an emblem of a progeny always in all things like the parent, + <i lang="la" xml:lang="la">simillima filia matri</i>. Or, + if it is not forbidden to + compare a theorem of eternal truth to the mysteries of our + faith, it may be taken as an emblem of the eternal generation + of the Son, who as an image of the Father, emanating from him, + as light from light, remains ὁμοούσιος + with him, howsoever overshadowed. Or, if you prefer, since our + <i lang="la" xml:lang="la">spira mirabilis</i> remains, + amid all changes, most + persistently itself, and exactly the same as ever, it may be + used as a symbol, either of fortitude and constancy in + adversity, or, of the human body, which after all its changes, + even after death, will be restored to its exact and perfect + self, so that, indeed, if the fashion of Archimedes were + allowed in these days, I should gladly have my + +<span class="pagenum"> + <a name="Page_145" + id="Page_145">145</a></span> + + tombstone bear this spiral, with the motto, “Though changed, + I arise again exactly the same, <i lang="la" + xml:lang="la">Eadem numero mutata + resurgo</i>.”—<span class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + The Uses of Mathesis; Bibliotheca Sacra, Vol. 32, + pp. 515-516.</p> + + <p class="v2"> + <b><a name="Block_923" id="Block_923">923</a>.</b> + Babbage was one of the founders of the Cambridge Analytical + Society whose purpose he stated was to advocate "the principles + of pure <em>d</em>-ism as opposed to the <em>dot</em>-age of the + university.”—<span class= "smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p. 451.</p> + + <p class="v2"> + <b><a name="Block_924" id="Block_924">924</a>.</b> + Bolyai [Janos] when in garrison with cavalry + officers, was provoked by thirteen of them and accepted all + their challenges on condition that he be permitted after each + duel to play a bit on his violin. He came out victor from his + thirteen duels, leaving his thirteen adversaries on the + square.—<span class="smcap">Halsted, G. B.</span></p> + <p class="blockcite"> + Bolyai’s Science Absolute of Space (Austin, 1896), + Introduction, p. 29.</p> + + <p class="v2"> + <b><a name="Block_925" id="Block_925">925</a>.</b> + Bolyai [Janos] projected a universal language for speech as we + have it for music and + mathematics.—<span class="smcap">Halsted, G. B.</span></p> + <p class="blockcite"> + Bolyai’s Science Absolute of Space (Austin, 1896), + Introduction, p. 29.</p> + + <p class="v2"> + <b><a name="Block_926" id="Block_926">926</a>.</b> + [Bolyai’s Science Absolute of + Space]—the most extraordinary two dozen + pages in the history of + thought!—<span class="smcap">Halsted, G. B.</span></p> + <p class="blockcite"> + Bolyai’s Science Absolute of Space + (Austin, 1896), Introduction, p. 18.</p> + + <p class="v2"> + <b><a name="Block_927" id="Block_927">927</a>.</b> + [Wolfgang Bolyai] was extremely modest. No + monument, said he, should stand over his grave, only an + apple-tree, in memory of the three apples: the two of Eve and + Paris, which made hell out of earth, and that of Newton, which + elevated the earth again into the circle of the heavenly + bodies.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Elementary Mathematics (New York, 1910), p. 273.</p> + + <p class="v2"> + <b><a name="Block_928" id="Block_928">928</a>.</b> + Bernard Bolzano dispelled the clouds that + throughout all the foregone centuries had enveloped the notion + of Infinitude + +<span class="pagenum"> + <a name="Page_146" + id="Page_146">146</a></span> + + in darkness, completely sheared the + great term of its vagueness without shearing it of its + strength, and thus rendered it forever available for the + purposes of logical discourse.—<span class= + "smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 42.</p> + + <p class="v2"> + <b><a name="Block_929" id="Block_929">929</a>.</b> + Let me tell you how at one time the famous + mathematician <em>Euclid</em> became a physician. It was during a + vacation, which I spent in Prague as I most always did, when I + was attacked by an illness never before experienced, which + manifested itself in chilliness and painful weariness of the + whole body. In order to ease my condition I took up + <cite>Euclid’s Elements</cite> and read for the + first time his doctrine of <em>ratio</em>, which I found treated + there in a manner entirely new to me. The ingenuity displayed + in Euclid’s presentation filled me with such + vivid pleasure, that forthwith I felt as well as + ever.—<span class="smcap">Bolzano, Bernard.</span></p> + <p class="blockcite"> + Selbstbiographie (Wien, 1875), p. 20.</p> + + <p class="v2"> + <b><a name="Block_930" id="Block_930">930</a>.</b> + Mr. Cayley, of whom it may be so truly said, + whether the matter he takes in hand be great or small, + “<i lang="la" xml:lang="la">nihil tetigit quod non + ornavit</i>,”....—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Philosophic Transactions of the Royal Society, Vol. 17 + (1864), p. 605.</p> + + <p class="v2"> + <b><a name="Block_931" id="Block_931">931</a>.</b> + It is not <em>Cayley’s</em> way to + analyze concepts into their ultimate elements.... But he is + master of the <em>empirical</em> utilization of the material: in + the way he combines it to form a single abstract concept which + he generalizes and then subjects to computative tests, in the + way the newly acquired data are made to yield at a single + stroke the general comprehensive idea to the subsequent + numerical verification of which years of labor are devoted. + <em>Cayley</em> is thus the <em>natural philosopher</em> among + mathematicians.—<span class="smcap">Noether, M.</span></p> + <p class="blockcite"> + Mathematische Annalen, Bd. 46 (1895), p. 479.</p> + + <p class="v2"> + <b><a name="Block_932" id="Block_932">932</a>.</b> + When Cayley had reached his most advanced + generalizations he proceeded to establish them directly by some + method or other, though he seldom gave the clue by which they + had first been obtained: a proceeding which does not tend to + make his papers easy reading....</p> + +<p><span class="pagenum"> + <a name="Page_147" + id="Page_147">147</a></span></p> + + <p class="v1"> + His literary style is direct, simple and clear. His legal + training had an influence, not merely upon his mode of + arrangement but also upon his expression; the result is that + his papers are severe and present a curious contrast to the + luxuriant enthusiasm which pervades so many of + Sylvester’s papers. He used to prepare his + work for publication as soon as he carried his investigations + in any subject far enough for his immediate purpose.... A paper + once written out was promptly sent for publication; this + practice he maintained throughout life.... The consequence is + that he has left few arrears of unfinished or unpublished + papers; his work has been given by himself to the + world.—<span class="smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Proceedings of London Royal Society, Vol. 58 (1895), + pp. 23-24.</p> + + <p class="v2"> + <b><a name="Block_933" id="Block_933">933</a>.</b> + Cayley was singularly learned in the work of other + men, and catholic in his range of knowledge. Yet he did not + read a memoir completely through: his custom was to read only + so much as would enable him to grasp the meaning of the symbols + and understand its scope. The main result would then become to + him a subject of investigation: he would establish it (or test + it) by algebraic analysis and, not infrequently, develop it so + to obtain other results. This faculty of grasping and testing + rapidly the work of others, together with his great knowledge, + made him an invaluable referee; his services in this capacity + were used through a long series of years by a number of + societies to which he was almost in the position of standing + mathematical advisor.—<span + class="smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Proceedings London Royal Society, Vol. 58 (1895), pp. + 11-12.</p> + + <p class="v2"> + <b><a name="Block_934" id="Block_934">934</a>.</b> + Bertrand, Darboux, and Glaisher have compared + Cayley to Euler, alike for his range, his analytical power, + and, not least, for his prolific production of new views and + fertile theories. There is hardly a subject in the whole of + pure mathematics at which he has not + worked.—<span class="smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Proceedings London Royal Society, Vol. 58 (1895), p. 21.</p> + + <p class="v2"> + <b><a name="Block_935" id="Block_935">935</a>.</b> + The mathematical talent of Cayley was characterized + by clearness and extreme elegance of + analytical form; it was re-enforced + +<span class="pagenum"> + <a name="Page_148" + id="Page_148">148</a></span> + + by an incomparable capacity + for work which has caused the distinguished scholar to be + compared with Cauchy.—<span class= "smcap">Hermite, C.</span></p> + <p class="blockcite"> + Comptes Rendus, t. 120 (1895), p. 234.</p> + + <p class="v2"> + <b><a name="Block_936" id="Block_936">936</a>.</b> + J. J. Sylvester was an enthusiastic supporter of + reform [in the teaching of geometry]. The difference in + attitude on this question between the two foremost British + mathematicians, J. J. Sylvester, the algebraist, and Arthur + Cayley, the algebraist and geometer, was grotesque. Sylvester + wished to bury Euclid “deeper than e’er plummet sounded” out + of the schoolboy’s reach; Cayley, an ardent admirer + of Euclid, desired the retention of Simson’s + <em>Euclid</em>. When reminded that this treatise was a mixture + of Euclid and Simson, Cayley suggested striking out + Simson’s additions and keeping strictly to + the original treatise.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Elementary Mathematics (New York, 1910), p. 285.</p> + + <p class="v2"> + <b><a name="Block_937" id="Block_937">937</a>.</b> + Tait once urged the advantage of Quaternions on Cayley (who + never used them), saying: “You know Quaternions are just like + a pocket-map.” “That may be,” replied Cayley, “but you’ve got + to take it out of your pocket, and unfold it, before it’s of + any use.” And he dismissed the subject with a + smile.—<span class="smcap">Thompson, S. P.</span></p> + <p class="blockcite"> + Life of Lord Kelvin (London, 1910), p. 1137.</p> + + <p class="v2"> + <b><a name="Block_938" id="Block_938">938</a>.</b> + As he [Clifford] spoke he appeared not to be + working out a question, but simply telling what he saw. Without + any diagram or symbolic aid he described the geometrical + conditions on which the solution depended, and they seemed to + stand out visibly in space. There were no longer consequences + to be deduced, but real and evident facts which only required + to be seen.... So whole and complete was his vision that for + the time the only strange thing was that anybody should fail to + see it in the same way. When one endeavored to call it up + again, and not till then, it became clear that the magic of + genius had been at work, and that the common sight had been + raised to that higher perception by the power that makes and + transforms + +<span class="pagenum"> + <a name="Page_149" + id="Page_149">149</a></span> + + ideas, the conquering and masterful quality of the human mind + which Goethe called in one word <i lang="de" xml:lang="de">das + Dämonische</i>.—<span class="smcap">Pollock, F.</span></p> + <p class="blockcite"> + Clifford’s Lectures and Essays (New York, 1901), Vol. 1, + Introduction, pp. 5-6.</p> + + <p class="v2"> + <b><a name="Block_939" id="Block_939">939</a>.</b> + Much of his [Clifford’s] best + work was actually spoken before it was written. He gave most of + his public lectures with no visible preparation beyond very + short notes, and the outline seemed to be filled in without + effort or hesitation. Afterwards he would revise the lecture + from a shorthand writer’s report, or + sometimes write down from memory almost exactly what he had + said. It fell out now and then, however, that neither of these + things was done; in such cases there is now no record of the + lecture at all.—<span class="smcap">Pollock, F.</span></p> + <p class="blockcite"> + Clifford’s Lectures and Essays (New + York, 1901), Vol. 1, Introduction, p. 10.</p> + + <p class="v2"> + <b><a name="Block_940" id="Block_940">940</a>.</b> + I cannot find anything showing early aptitude for + acquiring languages; but that he [Clifford] had it and was fond + of exercising it in later life is certain. One practical reason + for it was the desire of being able to read mathematical papers + in foreign journals; but this would not account for his taking + up Spanish, of which he acquired a competent knowledge in the + course of a tour to the Pyrenees. When he was at Algiers in + 1876 he began Arabic, and made progress enough to follow in a + general way a course of lessons given in that language. He read + modern Greek fluently, and at one time he was furious about + Sanskrit. He even spent some time on hieroglyphics. A new + language is a riddle before it is conquered, a power in the + hand afterwards: to Clifford every riddle was a challenge, and + every chance of new power a divine opportunity to be seized. + Hence he was likewise interested in the various modes of + conveying and expressing language invented for special + purposes, such as the Morse alphabet and shorthand.... I have + forgotten to mention his command of French and German, the + former of which he knew very well, and the latter quite + sufficiently;....—<span class="smcap">Pollock, F.</span></p> + <p class="blockcite"> + Clifford’s Lectures and Essays (New + York, 1901), Vol. 1, Introduction, pp. 11-12.</p> + +<p><span class="pagenum"> + <a name="Page_150" + id="Page_150">150</a></span></p> + + <p class="v2"> + <b><a name="Block_941" id="Block_941">941</a>.</b> + The most remarkable thing was his + [Clifford’s] great strength as compared with + his weight, as shown in some exercises. At one time he could + pull up on the bar with either hand, which is well known to be + one of the greatest feats of strength. His nerve at dangerous + heights was extraordinary. I am appalled now to think that he + climbed up and sat on the cross bars of the weathercock on a + church tower, and when by way of doing something worse I went + up and hung by my toes to the bars he did the same.</p> + <p class="blockcite"> + Quoted from a letter by one of Clifford’s friends to Pollock, F.: + Clifford’s Lectures and Essays (New York, + 1901), Vol. 1, Introduction, p. 8.</p> + + <p class="v2"> + <b><a name="Block_942" id="Block_942">942</a>.</b> + [Comte] may truly be said to have created the philosophy of + higher mathematics.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + System of Logic (New York, 1846), p. 369.</p> + + <p class="v2"> + <b><a name="Block_943" id="Block_943">943</a>.</b> + These specimens, which I could easily multiply, may + suffice to justify a profound distrust of Auguste Comte, + wherever he may venture to speak as a mathematician. But his + vast <em>general</em> ability, and that personal intimacy with + the great Fourier, which I most willingly take his own word for + having enjoyed, must always give an interest to his + <em>views</em> on any subject of pure or applied + mathematics.—<span class="smcap">Hamilton, W. R.</span></p> + <p class="blockcite"> + Graves’ Life of W. R. Hamilton (New + York, 1882-1889), Vol. 3, p. 475.</p> + + <p class="v2"> + <b><a name="Block_944" id="Block_944">944</a>.</b> + The manner of Demoivre’s death + has a certain interest for psychologists. Shortly before it, he + declared that it was necessary for him to sleep some ten + minutes or a quarter of an hour longer each day than the + preceding one: the day after he had thus reached a total of + something over twenty-three hours he slept up to the limit of + twenty-four hours, and then died in his + sleep.—<span class="smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1911), p. 394.</p> + + <p class="v2"> + <b><a name="Block_945" id="Block_945">945</a>.</b> + De Morgan was explaining to an actuary what was the + chance that a certain proportion of some group of people would + at the end of a given time be alive; and quoted the actuarial + formula, involving π, which, in answer to a + question, he explained stood for the ratio of the circumference + of a circle to its + +<span class="pagenum"> + <a name="Page_151" + id= "Page_151">151</a></span> + + diameter. His acquaintance, who had + so far listened to the explanation with interest, interrupted + him and exclaimed, “My dear friend, that + must be a delusion, what can a circle have to do with the + number of people alive at a given + time?”—<span class="smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + Mathematical Recreations and Problems (London, 1896), p. + 180; See also De Morgan’s Budget of + Paradoxes (London, 1872), p. 172.</p> + + <p class="v2"> + <b><a name="Block_946" id="Block_946">946</a>.</b> + A few days afterwards, I went to him [the same + actuary referred to in 945] and very gravely told him that I + had discovered the law of human mortality in the Carlisle + Table, of which he thought very highly. I told him that the law + was involved in this circumstance. Take the table of the + expectation of life, choose any age, take its expectation and + make the nearest integer a new age, do the same with that, and + so on; begin at what age you like, you are sure to end at the + place where the age past is equal, or most nearly equal, to the + expectation to come. “You don’t mean that this always + happens?”—“Try it.” He did try, again and again; and found it + as I said. “This is, indeed, a curious thing; + this <em>is</em> a discovery!” I might have sent him + about trumpeting the law of life: but I contented myself with + informing him that the same thing would happen with any table + whatsoever in which the first column goes up and the second + goes down;....—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), p. 172.</p> + + <p class="v2"> + <b><a name="Block_947" id="Block_947">947</a>.</b> + [De Morgan relates that some person had made up 800 + anagrams on his name, of which he had seen about 650. + Commenting on these he says:]</p> + <p class="v1"> + Two of these I have joined in the title-page:</p> + <p class="center"> + [Ut agendo surgamus arguendo gustamus.]</p> + <p class="v1"> + A few of the others are personal remarks.</p> + <p class="center"> + Great gun! do us a sum!</p> + <p class="v0"> + is a sneer at my pursuit; but,</p> + <p class="center"> + Go! great sum! + <img src="images/img947.png" + width="75" height="48" + alt="integral of a to the power of u to the power of n du" + class="figinline" + id="img947" /></p> + <p class="v0"> + is more dignified....<br /></p> + <p class="center"> + Adsum, nugator, suge!</p> + <p class="v0"> + is addressed to a student who continues talking after the + lecture has commenced: ... + +<span class="pagenum"> + <a name="Page_152" + id="Page_152">152</a></span></p> + + <p class="center"> + Graduatus sum! nego</p> + <p class="v0"> + applies to one who declined to subscribe for an M. A. + degree.—<span class="smcap">De Morgan, Augustus.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), p. 82.</p> + + <p class="v2"> + <b><a name="Block_948" id="Block_948">948</a>.</b> + Descartes is the completest type which history + presents of the purely mathematical type of + mind—that in which the tendencies produced + by mathematical cultivation reign unbalanced and + supreme.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + An Examination of Sir W. Hamilton’s + Philosophy (London, 1878), p. 626.</p> + + <p class="v2"> + <b><a name="Block_949" id="Block_949">949</a>.</b> + To <em>Descartes</em>, the great philosopher of the + 17th century, is due the undying credit of having removed the + bann which until then rested upon geometry. The <em>analytical + geometry</em>, as Descartes’ method was + called, soon led to an abundance of new theorems and + principles, which far transcended everything that ever could + have been reached upon the path pursued by the + ancients.—<span class="smcap">Hankel, H.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten + Jahrhunderten (Tübingen, 1884), p. 10.</p> + + <p class="v2"> + <b><a name="Block_950" id="Block_950">950</a>.</b> + [The application of algebra has] far more than any + of his metaphysical speculations, immortalized the name of + Descartes, and constitutes the greatest single step ever made + in the progress of the exact + sciences.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + An Examination of Sir W. Hamilton’s + Philosophy (London, 1878), p. 617.</p> + + <p class="v2"> + <b><a name="Block_951" id="Block_951">951</a>.</b> + ... καί φασιν ὅτι Πτολεμαῖος ἤρετό ποτε αύτόν [Εὐκλειδην], + εἴ τίς ἐστιν περὶ γεωμετρίαν ὁδὸς συντομωτέρα τῆς + στοιχειώσεως· ὁδὲ ἀπεκρὶνατο μὴ εἶναι βασιλικὴν ἀτραπὸν + ἐπὶ γεωμετρίαν.</p> + <p class="v1"> + [ ... they say that Ptolemy once asked him (Euclid) whether + there was in geometry no shorter way than that of the elements, + and he replied, “There is no royal road to + geometry.”]—<span class="smcap">Proclus.</span></p> + <p class="blockcite"> + (Edition Friedlein, 1873), Prol. II, 39.</p> + + <p class="v2"> + <b><a name="Block_952" id="Block_952">952</a>.</b> + Someone who had begun to read geometry with Euclid, + when he had learned the first proposition, asked Euclid, “But + +<span class="pagenum"> + <a name="Page_153" + id="Page_153">153</a></span> + + what shall I get by learning these + things?” whereupon Euclid called his slave and + said, “Give him three-pence, since he must + make gain out of what he + learns.”—<span class= "smcap">Stobæus.</span></p> + <p class="blockcite"> + (Edition Wachsmuth, 1884), Ecl. II.</p> + + <p class="v2"> + <b><a name="Block_953" id="Block_953">953</a>.</b> + The sacred writings excepted, no Greek has been so + much read and so variously translated as Euclid.<a + href="#Footnote_5" + title="Riccardi’s Bibliografia Euclidea (Bologna, 1887), +lists nearly two thousand editions." + class="fnanchor">5</a>—<span + class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Smith’s Dictionary of Greek and Roman Biology and Mythology + (London, 1902), Article, “Eucleides.”</p> + + <p class="v2"> + <b><a name="Block_954" id="Block_954">954</a>.</b> + The thirteen books of Euclid must have been a + tremendous advance, probably even greater than that contained + in the “Principia” of + Newton.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Smith’s Dictionary of Greek and Roman Biography and Mythology + (London, 1902), Article, “Eucleides.”</p> + + <p class="v2"> + <b><a name="Block_955" id="Block_955">955</a>.</b> + To suppose that so perfect a system as that of + Euclid’s Elements was produced by one man, + without any preceding model or materials, would be to suppose + that Euclid was more than man. We ascribe to him as much as the + weakness of human understanding will permit, if we suppose that + the inventions in geometry, which had been made in a tract of + preceding ages, were by him not only carried much further, but + digested into so admirable a system, that his work obscured all + that went before it, and made them be forgot and + lost.—<span class="smcap">Reid, Thomas.</span></p> + <p class="blockcite"> + Essay on the Powers of the Human Mind (Edinburgh, 1812), + Vol. 2, p. 368.</p> + + <p class="v2"> + <b><a name="Block_956" id="Block_956">956</a>.</b> + It is the invaluable merit of the great Basle + mathematician Leonhard <em>Euler</em>, to have freed the + analytical calculus from all geometrical bonds, and thus to + have established <em>analysis</em> as an independent science, + which from his time on has maintained an unchallenged + leadership in the field of + mathematics.—<span class="smcap">Hankel, H.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten + Jahrhunderten (Tübingen, 1884), p. 12.</p> + +<p><span class="pagenum"> + <a name="Page_154" + id="Page_154">154</a></span></p> + + <p class="v2"> + <b><a name="Block_957" id="Block_957">957</a>.</b> + We may safely say, that the whole form of modern + mathematical thinking was created by Euler. It is only with the + greatest difficulty that one is able to follow the writings of + any author immediately preceding Euler, because it was not yet + known how to let the formulas speak for themselves. This art + Euler was the first one to + teach.—<span class="smcap">Rudio, F.</span></p> + <p class="blockcite"> + Quoted by Ahrens W.: Scherz und Ernst in der Mathematik + (Leipzig, 1904), p. 251.</p> + + <p class="v2"> + <b><a name="Block_958" id="Block_958">958</a>.</b> + The general knowledge of our author [Leonhard + Euler] was more extensive than could well be expected, in one + who had pursued, with such unremitting ardor, mathematics and + astronomy as his favorite studies. He had made a very + considerable progress in medical, botanical, and chemical + science. What was still more extraordinary, he was an excellent + scholar, and possessed in a high degree what is generally + called erudition. He had attentively read the most eminent + writers of ancient Rome; the civil and literary history of all + ages and all nations was familiar to him; and foreigners, who + were only acquainted with his works, were astonished to find in + the conversation of a man, whose long life seemed solely + occupied in mathematical and physical researches and + discoveries, such an extensive acquaintance with the most + interesting branches of literature. In this respect, no doubt, + he was much indebted to an uncommon memory, which seemed to + retain every idea that was conveyed to it, either from reading + or from meditation.—<span + class="smcap">Hutton, Charles.</span></p> + <p class="blockcite"> + Philosophical and Mathematical Dictionary (London, 1815), + pp. 493-494.</p> + + <p class="v2"> + <b><a name="Block_959" id="Block_959">959</a>.</b> + Euler could repeat the Aeneid from the beginning to + the end, and he could even tell the first and last lines in + every page of the edition which he used. In one of his works + there is a learned memoir on a question in mechanics, of which, + as he himself informs us, a verse of Aeneid<a + href="#Footnote_6" + title="“The anchor drops, the rushing keel is staid.”" + class="fnanchor">6</a> + gave him the first + idea.—<span class="smcap">Brewster, David.</span></p> + <p class="blockcite"> + Letters of Euler (New York, 1872), Vol. 1, p. 24.</p> + +<p> + <span class="pagenum"> + <a name="Page_155" + id="Page_155">155</a></span></p> + + <p class="v2"> + <b><a name="Block_960" id="Block_960">960</a>.</b> + Most of his [Euler’s] memoirs + are contained in the transactions of the Academy of Sciences at + St. Petersburg, and in those of the Academy at Berlin. From + 1728 to 1783 a large portion of the Petropolitan transactions + were filled by his writings. He had engaged to furnish the + Petersburg Academy with memoirs in sufficient number to enrich + its acts for twenty years—a promise more + than fulfilled, for down to 1818 [Euler died in 1793] the + volumes usually contained one or more papers of his. It has + been said that an edition of Euler’s + complete works would fill 16,000 quarto + pages.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Mathematics (New York, 1897), pp. 253-254.</p> + + <p class="v2"> + <b><a name="Block_961" id="Block_961">961</a>.</b> + Euler who could have been called almost without + metaphor, and certainly without hyperbole, analysis + incarnate.—<span class="smcap">Arago.</span></p> + <p class="blockcite"> + Oeuvres, t. 2 (1854), p. 433.</p> + + <p class="v2"> + <b><a name="Block_962" id="Block_962">962</a>.</b> + Euler calculated without any apparent effort, just + as men breathe, as eagles sustain themselves in the + air.—<span class="smcap">Arago.</span></p> + <p class="blockcite"> + Oeuvres, t. 2 (1854), p. 133.</p> + + <p class="v2"> + <b><a name="Block_963" id="Block_963">963</a>.</b> + Two of his [Euler’s] pupils + having computed to the 17th term, a complicated converging + series, their results differed one unit in the fiftieth cipher; + and an appeal being made to Euler, he went over the calculation + in his mind, and his decision was found + correct.—<span class="smcap">Brewster, David.</span></p> + <p class="blockcite"> + Letters of Euler (New York, 1872), Vol. 2, p. 22.</p> + + <p class="v2"> + <b><a name="Block_964" id="Block_964">964</a>.</b> + In 1735 the solving of an astronomical problem, + proposed by the Academy, for which several eminent + mathematicians had demanded several months’ + time, was achieved in three days by Euler with aid of improved + methods of his own.... With still superior methods this same + problem was solved by the illustrious Gauss in one + hour.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Mathematics (New York, 1897), p. 248.</p> + +<p><span class="pagenum"> + <a name="Page_156" + id="Page_156">156</a></span></p> + + <p class="v2"> + <b><a name="Block_965" id="Block_965">965</a>.</b> + Euler’s <i lang="de" xml:lang="de">Tentamen novae + theorae musicae</i> had no great success, as it contained too + much geometry for musicians, and too much music for + geometers.—<span class="smcap">Fuss, N.</span></p> + <p class="blockcite"> + Quoted by Brewster: Letters of Euler (New York, 1872), + Vol. 1, p. 26.</p> + + <p class="v2"> + <b><a name="Block_966" id="Block_966">966</a>.</b> + Euler was a believer in God, downright and + straight-forward. The following story is told by Thiebault, in + his <i lang="fr" xml:lang="fr">Souvenirs de vingt ans de + séjour à + Berlin</i>,.... Thiebault says that he has no personal + knowledge of the truth of the story, but that it was believed + throughout the whole of the north of Europe. Diderot paid a + visit to the Russian Court at the invitation of the Empress. He + conversed very freely, and gave the younger members of the + Court circle a good deal of lively atheism. The Empress was + much amused, but some of her counsellors suggested that it + might be desirable to check these expositions of doctrine. The + Empress did not like to put a direct muzzle on her + guest’s tongue, so the following plot was + contrived. Diderot was informed that a learned mathematician + was in possession of an algebraical demonstration of the + existence of God, and would give it him before all the Court, + if he desired to hear it. Diderot gladly consented: though the + name of the mathematician is not given, it was Euler. He + advanced toward Diderot, and said gravely, and in a tone of + perfect conviction:</p> + <p class="center"> + <em>Monsieur</em>, + <img src="images/img966.png" + width="118" + height="48" + alt="a plus b to the power n, all over n, equals x" + class="figinline" id="img966" /> + <i lang="fr" xml:lang="fr">donc Dieu existe; repondez!</i></p> + <p class="v1"> + Diderot, to whom algebra was Hebrew, was embarrassed and + disconcerted; while peals of laughter rose on all sides. He + asked permission to return to France at once, which was + granted.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), p. 251.</p> + + <p class="v2"> + <b><a name="Block_967" id="Block_967">967</a>.</b> + Fermat died with the belief that he had found along-sought-for + law of prime numbers in the formula 2<sup>2<sup>n</sup></sup> + + 1 = a prime, but he admitted that + he was unable to prove it rigorously. The law is not true, as + was pointed out by Euler in the example 2<sup>2<sup>5</sup></sup> + + 1 = 4,294,967,297 = 6,700,417 times + 641. The American lightning calculator <em>Zerah Colburn</em>, + when a boy, + +<span class="pagenum"> + <a name="Page_157" + id="Page_157">157</a></span> + + readily found the factors but was + unable to explain the method by which he made his marvellous + mental computation.—<span class= "smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Mathematics (New York, 1897), p. 180.</p> + + <p class="v2"> + <b><a name="Block_968" id="Block_968">968</a>.</b> + I crave the liberty to conceal my name, not to + suppress it. I have composed the letters of it written in Latin + in this sentence—</p> + <blockquote> + <p class="v1"> + In Mathesi a sole fundes.<a + href="#Footnote_7" + title="Johannes Flamsteedius" + class="fnanchor">7</a>—<span + class="smcap">Flamsteed, J.</span></p></blockquote> + <p class="blockcite"> + Macclesfield: Correspondence of Scientific Men (Oxford, + 1841), Vol. 2, p. 90.</p> + + <p class="v2"> + <b><a name="Block_969" id="Block_969">969</a>.</b> + <em>To the Memory of Fourier</em></p> + <div class="poem"> + <p class="i0"> + Fourier! with solemn and profound delight,</p> + <p class="i0"> + Joy born of awe, but kindling momently</p> + <p class="i0"> + To an intense and thrilling ecstacy,</p> + <p class="i0"> + I gaze upon thy glory and grow bright:</p> + <p class="i0"> + As if irradiate with beholden light;</p> + <p class="i0"> + As if the immortal that remains of thee</p> + <p class="i0"> + Attuned me to thy spirit’s harmony,</p> + <p class="i0"> + Breathing serene resolve and tranquil might.</p> + <p class="i0"> + Revealed appear thy silent thoughts of youth,</p> + <p class="i0"> + As if to consciousness, and all that view</p> + <p class="i0"> + Prophetic, of the heritage of truth</p> + <p class="i0"> + To thy majestic years of manhood due:</p> + <p class="i0"> + Darkness and error fleeing far away,</p> + <p class="i0"> + And the pure mind enthroned in perfect day.</p> + </div> + <p class="block40"> + —<span class="smcap">Hamilton, W. R.</span></p> + <p class="blockcite"> + Graves’ Life of W. R. Hamilton, (New + York, 1882), Vol. 1, p. 596.</p> + + <p class="v2"> + <b><a name="Block_970" id="Block_970">970</a>.</b> + Astronomy and Pure Mathematics are the magnetic + poles toward which the compass of my mind ever + turns.—<span class="smcap">Gauss to Bolyai.</span></p> + <p class="blockcite"> + Briefwechsel (Schmidt-Stakel), (1899), p. 55.</p> + + <p class="v2"> + <b><a name="Block_971" id="Block_971">971</a>.</b> + [Gauss calculated the elements of the planet Ceres] + and his analysis proved him to be the first of theoretical + astronomers no less than the greatest of + “arithmeticians.”—<span class="smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p. 458.</p> + +<p><span class="pagenum"> + <a name="Page_158" + id="Page_158">158</a></span></p> + + <p class="v2"> + <b><a name="Block_972" id="Block_972">972</a>.</b> + The mathematical giant [Gauss], who from his lofty + heights embraces in one view the stars and the + abysses....—<span class="smcap">Bolyai, W.</span></p> + <p class="blockcite"> + Kurzer Grundriss eines Versuchs (Maros Vasarhely, 1851), + p. 44.</p> + + <p class="v2"> + <b><a name="Block_973" id="Block_973">973</a>.</b> + Almost everything, which the mathematics of our + century has brought forth in the way of original scientific + ideas, attaches to the name of + Gauss.—<span class="smcap">Kronecker, L.</span></p> + <p class="blockcite"> + Zahlentheorie, Teil 1 (Leipzig, 1901), p. 43.</p> + + <p class="v2"> + <b><a name="Block_974" id="Block_974">974</a>.</b> + I am giving this winter two courses of lectures to + three students, of which one is only moderately prepared, the + other less than moderately, and the third lacks both + preparation and ability. Such are the onera of a mathematical + profession.—<span class="smcap">Gauss to Bessel, 1810.</span></p> + <p class="blockcite"> + Gauss-Bessel Briefwechsel (1880), p. 107.</p> + + <p class="v2"> + <b><a name="Block_975" id="Block_975">975</a>.</b> + Gauss once said “Mathematics is + the queen of the sciences and number-theory the queen of + mathematics.” If this be true we may add that the + Disquisitiones is the Magna Charta of number-theory. The + advantage which science gained by Gauss’ + long-lingering method of publication is this: What he put into + print is as true and important today as when first published; + his publications are statutes, superior to other human statutes + in this, that nowhere and never has a single error been + detected in them. This justifies and makes intelligible the + pride with which Gauss said in the evening of his life of the + first larger work of his youth: “The Disquisitiones arithmeticae + belong to history.”—<span class= "smcap">Cantor, M.</span></p> + <p class="blockcite"> + Allgemeine Deutsche Biographie, Bd. 8 (1878), p. 435.</p> + + <p class="v2"> + <b><a name="Block_976" id="Block_976">976</a>.</b> + Here I am at the limit which God and nature has + assigned to my individuality. I am compelled to depend upon + word, language and image in the most precise sense, and am + wholly unable to operate in any manner whatever with symbols + and numbers which are easily intelligible to the most highly + gifted minds.—<span class="smcap">Goethe.</span></p> + <p class="blockcite"> + Letter to Naumann (1826); Vogel: Goethe’s Selbstzeugnisse + (Leipzig, 1903), p. 56.</p> + +<p><span class="pagenum"> + <a name="Page_159" + id="Page_159">159</a></span></p> + + <p class="v2"> + <b><a name="Block_977" id="Block_977">977</a>.</b> + Dirichlet was not satisfied to study + Gauss’ “Disquisitiones + arithmeticae” once or several times, but continued + throughout life to keep in close touch with the wealth of deep + mathematical thoughts which it contains by perusing it again + and again. For this reason the book was never placed on the + shelf but had an abiding place on the table at which he + worked.... Dirichlet was the first one, who not only fully + understood this work, but made it also accessible to + others.—<span class="smcap">Kummer, E. E.</span></p> + <p class="blockcite"> + Dirichlet: Werke, Bd. 2, p. 315.</p> + + <p class="v2"> + <b><a name="Block_978" id="Block_978">978</a>.</b> + [The famous attack of Sir William Hamilton on the + tendency of mathematical studies] affords the most express + evidence of those fatal <i lang="fr" xml:lang="fr">lacunae</i> + in the circle of his + knowledge, which unfitted him for taking a comprehensive or + even an accurate view of the processes of the human mind in the + establishment of truth. If there is any pre-requisite which all + must see to be indispensable in one who attempts to give laws + to the human intellect, it is a thorough acquaintance with the + modes by which human intellect has proceeded, in the case + where, by universal acknowledgment, grounded on subsequent + direct verification, it has succeeded in ascertaining the + greatest number of important and recondite truths. This + requisite Sir W. Hamilton had not, in any tolerable degree, + fulfilled. Even of pure mathematics he apparently knew little + but the rudiments. Of mathematics as applied to investigating + the laws of physical nature; of the mode in which the + properties of number, extension, and figure, are made + instrumental to the ascertainment of truths other than + arithmetical or geometrical—it is too much + to say that he had even a superficial knowledge: there is not a + line in his works which shows him to have had any knowledge at + all.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + Examination of Sir William Hamilton’s + Philosophy (London, 1878), p. 607.</p> + + <p class="v2"> + <b><a name="Block_979" id="Block_979">979</a>.</b> + Helmholtz—the physiologist who + learned physics for the sake of his physiology, and mathematics + for the sake of his physics, and is now in the first rank of + all three.—<span class="smcap">Clifford, W. K.</span></p> + <p class="blockcite"> + Aims and Instruments of Scientific Thought; Lectures and + Essays, Vol. 1 (London, 1901), p. 165.</p> + +<p><span class="pagenum"> + <a name="Page_160" + id="Page_160">160</a></span></p> + + <p class="v2"> + <b><a name="Block_980" id="Block_980">980</a>.</b> + It is said of Jacobi, that he attracted the + particular attention and friendship of Böckh, the + director of the philological seminary at Berlin, by the great + talent he displayed for philology, and only at the end of two + years’ study at the University, and after a + severe mental struggle, was able to make his final choice in + favor of mathematics.—<span class= + "smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Collected Mathematical Papers, Vol. 2 (Cambridge, 1908), + p. 651.</p> + + <p class="v2"> + <b><a name="Block_981" id="Block_981">981</a>.</b> + When Dr. Johnson felt, or fancied he felt, his + fancy disordered, his constant recurrence was to the study of + arithmetic.—<span class="smcap">Boswell, J.</span></p> + <p class="blockcite"> + Life of Johnson (Harper’s Edition, 1871), Vol. 2, p. 264.</p> + + <p class="v2"> + <b><a name="Block_982" id="Block_982">982</a>.</b> + Endowed with two qualities, which seemed + incompatible with each other, a volcanic imagination and a + pertinacity of intellect which the most tedious numerical + calculations could not daunt, Kepler conjectured that the + movements of the celestial bodies must be connected together by + simple laws, or, to use his own expression, by harmonic laws. + These laws he undertook to discover. A thousand fruitless + attempts, errors of calculation inseparable from a colossal + undertaking, did not prevent him a single instant from + advancing resolutely toward the goal of which he imagined he + had obtained a glimpse. Twenty-two years were employed by him + in this investigation, and still he was not weary of it! What, + in reality, are twenty-two years of labor to him who is about + to become the legislator of worlds; who shall inscribe his name + in ineffaceable characters upon the frontispiece of an immortal + code; who shall be able to exclaim in dithyrambic language, and + without incurring the reproach of anyone, + “The die is cast; I have written my book; it + will be read either in the present age or by posterity, it + matters not which; it may well await a reader, since God has + waited six thousand years for an interpreter of his + words.”—<span class="smcap">Arago.</span></p> + <p class="blockcite"> + Eulogy on Laplace: [Baden Powell] Smithsonian Report, + 1874, p. 132.</p> + + <p class="v2"> + <b><a name="Block_983" id="Block_983">983</a>.</b> + The great masters of modern analysis are Lagrange, + Laplace, and Gauss, who were contemporaries. It is interesting + +<span class="pagenum"> + <a name="Page_161" + id="Page_161">161</a></span> + + to note the marked contrast in their + styles. Lagrange is perfect both in form and matter, he is + careful to explain his procedure, and though his arguments are + general they are easy to follow. Laplace on the other hand + explains nothing, is indifferent to style, and, if satisfied + that his results are correct, is content to leave them either + with no proof or with a faulty one. Gauss is as exact and + elegant as Lagrange, but even more difficult to follow than + Laplace, for he removes every trace of the analysis by which he + reached his results, and studies to give a proof which while + rigorous shall be as concise and synthetical as + possible.—<span class="smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p. 463.</p> + + <p class="v2"> + <b><a name="Block_984" id="Block_984">984</a>.</b> + Lagrange, in one of the later years of his life, + imagined that he had overcome the difficulty [of the parallel + axiom]. He went so far as to write a paper, which he took with + him to the Institute, and began to read it. But in the first + paragraph something struck him which he had not observed: he + muttered <i lang="fr" xml:lang="fr">Il faut que j’y songe + encore</i>, and put the paper + in his pocket.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), p. 173.</p> + + <p class="v2"> + <b><a name="Block_985" id="Block_985">985</a>.</b> + I never come across one of Laplace’s “<em>Thus it + plainly appears”</em> without feeling sure that I + have hours of hard work before me to fill up the chasm and find + out and show <em>how</em> it plainly + appears.—<span class="smcap">Bowditch, N.</span></p> + <p class="blockcite"> + Quoted by Cajori: Teaching and History of Mathematics in + the U. S. (Washington, 1896), p. 104.</p> + + <p class="v2"> + <b><a name="Block_986" id="Block_986">986</a>.</b> + Biot, who assisted Laplace in revising it [The + Mécanique Céleste] for the press, + says that Laplace himself was frequently unable to recover the + details in the chain of reasoning, and if satisfied that the + conclusions were correct, he was content to insert the + constantly recurring formula, “<i lang="fr" xml:lang="fr">Il + est àisé a + voir.</i>”—<span class="smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p 427.</p> + + <p class="v2"> + <b><a name="Block_987" id="Block_987">987</a>.</b> + It would be difficult to name a man more remarkable + for the greatness and the universality of his intellectual + powers than Leibnitz.—<span class= "smcap">Mill, J. S.</span></p> + <p class="blockcite"> + System of Logic, Bk. 2, chap. 5, sect. 6.</p> + +<p><span class="pagenum"> + <a name="Page_162" + id="Page_162">162</a></span></p> + + <p class="v2"> + <b><a name="Block_988" id="Block_988">988</a>.</b> + The influence of his + [Leibnitz’s] genius in forming that peculiar + taste both in pure and in mixed mathematics which has prevailed + in France, as well as in Germany, for a century past, will be + found, upon examination, to have been incomparably greater than + that of any other individual.—<span class= + "smcap">Stewart, Dugald.</span></p> + <p class="blockcite"> + Philosophy of the Human Mind, Part 3, chap. 1, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_989" id="Block_989">989</a>.</b> + Leibnitz’s discoveries lay in + the direction in which all modern progress in science lies, in + establishing order, symmetry, and harmony, i.e., + comprehensiveness and perspicuity,—rather + than in dealing with single problems, in the solution of which + followers soon attained greater dexterity than + himself.—<span class="smcap">Merz, J. T.</span></p> + <p class="blockcite"> + Leibnitz, Chap. 6.</p> + + <p class="v2"> + <b><a name="Block_990" id="Block_990">990</a>.</b> + It was his [Leibnitz’s] love of + method and order, and the conviction that such order and + harmony existed in the real world, and that our success in + understanding it depended upon the degree and order which we + could attain in our own thoughts, that originally was probably + nothing more than a habit which by degrees grew into a formal + rule.<a + href="#Footnote_8" + title="This sentence has been reworded for the +purpose of this quotation." + class="fnanchor">8</a> + This habit was acquired + by early occupation with legal and mathematical questions. We + have seen how the theory of combinations and arrangements of + elements had a special interest for him. We also saw how + mathematical calculations served him as a type and model of + clear and orderly reasoning, and how he tried to introduce + method and system into logical discussions, by reducing to a + small number of terms the multitude of compound notions he had + to deal with. This tendency increased in strength, and even in + those early years he elaborated the idea of a general + arithmetic, with a universal language of symbols, or a + characteristic which would be applicable to all reasoning + processes, and reduce philosophical investigations to that + simplicity and certainty which the use of algebraic symbols had + introduced into mathematics.</p> + + <p class="v1"> + A mental attitude such as this is always highly favorable for + mathematical as well as for philosophical investigations. + Wherever + +<span class="pagenum"> + <a name="Page_163" + id="Page_163">163</a></span> + + progress depends upon + precision and clearness of thought, and wherever such can be + gained by reducing a variety of investigations to a general + method, by bringing a multitude of notions under a common term + or symbol, it proves inestimable. It necessarily imports the + special qualities of number—viz., their continuity, infinity + and infinite divisibility—like mathematical + quantities—and destroys the notion that + irreconcilable contrasts exist in nature, or gaps which cannot + be bridged over. Thus, in his letter to Arnaud, Leibnitz + expresses it as his opinion that geometry, or the philosophy of + space, forms a step to the philosophy of + motion—i.e., of corporeal things—and the philosophy of motion + a step to the philosophy of mind.—<span class= + "smcap">Merz, J. T.</span></p> + <p class="blockcite"> + Leibnitz (Philadelphia), pp. 44-45.</p> + + <p class="v2"> + <b><a name="Block_991" id="Block_991">991</a>.</b> + Leibnitz believed he saw the image of creation in + his binary arithmetic in which he employed only two characters, + unity and zero. Since God may be represented by unity, and + nothing by zero, he imagined that the Supreme Being might have + drawn all things from nothing, just as in the binary arithmetic + all numbers are expressed by unity with zero. This idea was so + pleasing to Leibnitz, that he communicated it to the Jesuit + Grimaldi, President of the Mathematical Board of China, with + the hope that this emblem of the creation might convert to + Christianity the reigning emperor who was particularly attached + to the sciences.—<span class= "smcap">Laplace.</span></p> + <p class="blockcite"> + Essai Philosophique sur les Probabilités; + Oeuvres (Paris, 1896), t. 7, p. 119.</p> + + <p class="v2"> + <b><a name="Block_992" id="Block_992">992</a>.</b> + Sophus Lie, great comparative anatomist of + geometric theories.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 31.</p> + + <p class="v2"> + <b><a name="Block_993" id="Block_993">993</a>.</b> + It has been the final aim of Lie from the beginning + to make progress in the theory of differential equations; as + subsidiary to this may be regarded both his geometrical + developments and the theory of continuous + groups.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Lectures on Mathematics (New York, 1911), p. 24.</p> + +<p><span class="pagenum"> + <a name="Page_164" + id="Page_164">164</a></span></p> + + <p class="v2"> + <b><a name="Block_994" id="Block_994">994</a>.</b> + To fully understand the mathematical genius of + Sophus Lie, one must not turn to books recently published by + him in collaboration with Dr. Engel, but to his earlier + memoirs, written during the first years of his scientific + career. There Lie shows himself the true geometer that he is, + while in his later publications, finding that he was but + imperfectly understood by the mathematicians accustomed to the + analytic point of view, he adopted a very general analytic form + of treatment that is not always easy to + follow.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Lectures on Mathematics (New York, 1911), p. 9.</p> + + <p class="v2"> + <b><a name="Block_995" id="Block_995">995</a>.</b> + It is said that the composing of the Lilawati was + occasioned by the following circumstance. Lilawati was the name + of the author’s [Bhascara] daughter, + concerning whom it appeared, from the qualities of the + ascendant at her birth, that she was destined to pass her life + unmarried, and to remain without children. The father + ascertained a lucky hour for contracting her in marriage, that + she might be firmly connected and have children. It is said + that when that hour approached, he brought his daughter and his + intended son near him. He left the hour cup on the vessel of + water and kept in attendance a time-knowing astrologer, in + order that when the cup should subside in the water, those two + precious jewels should be united. But, as the intended + arrangement was not according to destiny, it happened that the + girl, from a curiosity natural to children, looked into the + cup, to observe the water coming in at the hole, when by chance + a pearl separated from her bridal dress, fell into the cup, + and, rolling down to the hole, stopped the influx of water. So + the astrologer waited in expectation of the promised hour. When + the operation of the cup had thus been delayed beyond all + moderate time, the father was in consternation, and examining, + he found that a small pearl had stopped the course of the + water, and that the long-expected hour was passed. In short, + the father, thus disappointed, said to his unfortunate + daughter, I will write a book of your name, which shall remain + to the latest times—for a good name is a + second life, and the ground-work of eternal + existence.—<span class="smcap">Fizi.</span></p> + <p class="blockcite"> + Preface to the Lilawati. Quoted by A. Hutton: A Philosophical + and Mathematical Dictionary, Article “Algebra” (London, + 1815).</p> + +<p><span class="pagenum"> + <a name="Page_165" + id="Page_165">165</a></span></p> + + <p class="v2"> + <b><a name="Block_996" id="Block_996">996</a>.</b> + Is there anyone whose name cannot be twisted into + either praise or satire? I have had given to me,</p> + <div class="poem"> + <p class="i4"> + <em>Thomas Babington Macaulay</em></p> + <p class="i4"> + <em>Mouths big: a Cantab anomaly.</em></p> + </div> + <p class="block40"> + —<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), p. 83.</p> + +<p> + <span class="pagenum"> + <a name="Page_166" + id="Page_166">166</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_X"> + CHAPTER X<br /> + <span class="large"> + PERSONS AND ANECDOTES<br /> + (N-Z)</span></h2> + + <p class="v2"> + <b><a name="Block_1001" id="Block_1001">1001</a>.</b> + When he had a few moments for diversion, he + [Napoleon] not unfrequently employed them over a book of + logarithms, in which he always found + recreation.—<span class="smcap">Abbott, J. S. C.</span></p> + <p class="blockcite"> + Napoleon Bonaparte (New York, 1904), Vol. 1, chap. 10.</p> + + <p class="v2"> + <b><a name="Block_1002" id="Block_1002">1002</a>.</b> + The name of Sir Isaac Newton has by general + consent been placed at the head of those great men who have + been the ornaments of their species.... The philosopher + [Laplace], indeed, to whom posterity will probably assign a + place next to Newton, has characterized the Principia as + pre-eminent above all the productions of human + intellect.—<span class="smcap">Brewster, D.</span></p> + <p class="blockcite"> + Life of Sir Isaac Newton (London, 1831), pp. 1, 2.</p> + + <p class="v2"> + <b><a name="Block_1003" id="Block_1003">1003</a>.</b> + Newton and Laplace need myriads of ages and + thick-strewn celestial areas. One may say a gravitating solar + system is already prophesied in the nature of + Newton’s mind.—<span class="smcap">Emerson.</span></p> + <p class="blockcite"> + Essay on History.</p> + + <p class="v2"> + <b><a name="Block_1004" id="Block_1004">1004</a>.</b> + The law of gravitation is indisputably and + incomparably the greatest scientific discovery ever made, + whether we look at the advance which it involved, the extent of + truth disclosed, or the fundamental and satisfactory nature of + this truth.—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + History of the Inductive Sciences, Bk. 7, chap. 2, sect. 5.</p> + + <p class="v2"> + <b><a name="Block_1005" id="Block_1005">1005</a>.</b> + Newton’s theory is the circle + of generalization which includes all the others [as + Kepler’s laws, Ptolemy’s + theory, etc.];—the highest point of the + inductive ascent;—the catastrophe of the philosophic drama + to which Plato had prologized;—the + +<span class="pagenum"> + <a name="Page_167" + id="Page_167">167</a></span> + + point to which men’s minds had been journeying for two + thousand years.—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + History of the Inductive Sciences, Bk. 7, chap. 2, sect. 5.</p> + + <p class="v2"> + <b><a name="Block_1006" id="Block_1006">1006</a>.</b> + The efforts of the great philosopher [Newton] were + always superhuman; the questions which he did not solve were + incapable of solution in his + time.—<span class="smcap">Arago.</span></p> + <p class="blockcite"> + Eulogy on Laplace, [Baden Powell] Smithsonian Report, + 1874, p. 133.</p> + + <p class="v2"> + <b><a name="Block_1007" id="Block_1007">1007</a>.</b></p> + <div class="poem"> + <p class="i0"> + Nature and Nature’s laws lay hid in night:</p> + <p class="i0"> + God said, “Let Newton be!” and all was light.</p> + </div> + <p class="block40"> + —<span class="smcap">Pope, A.</span></p> + <p class="blockcite"> + Epitaph intended for Sir Isaac Newton.</p> + + <p class="v2"> + <b><a name="Block_1008" id="Block_1008">1008</a>.</b></p> + <div class="poem"> + <p class="i0"> + There Priest of Nature! dost thou shine,</p> + <p class="i0"> + <em>Newton!</em> a King among the Kings divine.</p> + </div> + <p class="block40"> + —<span class="smcap">Southey.</span></p> + <p class="blockcite"> + Translation of a Greek Ode on Astronomy.</p> + + <p class="v2"> + <b><a name="Block_1009" id="Block_1009">1009</a>.</b></p> + <div class="poem"> + <p class="i0"> + O’er Nature’s laws God cast the veil of night,</p> + <p class="i0"> + Out-blaz’d a Newton’s soul—and all was light.</p> + </div> + <p class="block40"> + —<span class="smcap">Hill, Aaron.</span></p> + <p class="blockcite"> + On Sir Isaac Newton.</p> + + <p class="v2"> + <b><a name="Block_1010" id="Block_1010">1010</a>.</b> + Taking mathematics from the beginning of the world + to the time when Newton lived, what he had done was much the + better half.—<span class="smcap">Leibnitz.</span></p> + <p class="blockcite"> + Quoted by F. R. Moulton: Introduction to Astronomy (New + York, 1906), p. 199.</p> + + <p class="v2"> + <b><a name="Block_1011" id="Block_1011">1011</a>.</b> + Newton was the greatest genius that ever existed, + and the most fortunate, for we cannot find more than once a + system of the world to establish.—<span class= + "smcap">Lagrange.</span></p> + <p class="blockcite"> + Quoted by F. R. Moulton: Introduction to Astronomy (New + York, 1906), p. 199.</p> + + <p class="v2"> + <b><a name="Block_1012" id="Block_1012">1012</a>.</b> + A monument to Newton! a monument to Shakespeare! + Look up to Heaven—look into the Human Heart. + Till the planets and the passions—the affections and the + fixed stars are extinguished—their names cannot + die.—<span class="smcap">Wilson, John.</span></p> + <p class="blockcite"> + Noctes Ambrosianae.</p> + +<p><span class="pagenum"> + <a name="Page_168" + id="Page_168">168</a></span></p> + + <p class="v2"> + <b><a name="Block_1013" id="Block_1013">1013</a>.</b> + Such men as Newton and Linnaeus are incidental, + but august, teachers of + religion.—<span class="smcap">Wilson, John.</span></p> + <p class="blockcite"> + Essays: Education of the People.</p> + + <p class="v2"> + <b><a name="Block_1014" id="Block_1014">1014</a>.</b> + Sir Isaac Newton, the supreme representative of + Anglo-Saxon genius.—<span class= + "smcap">Ellis, Havelock.</span></p> + <p class="blockcite"> + Study of British Genius (London, 1904), p. 49.</p> + + <p class="v2"> + <b><a name="Block_1015" id="Block_1015">1015</a>.</b> + Throughout his life Newton must have devoted at + least as much attention to chemistry and theology as to + mathematics....—<span class="smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p. 335.</p> + + <p class="v2"> + <b><a name="Block_1016" id="Block_1016">1016</a>.</b> + There was a time when he [Newton] was possessed + with the old fooleries of astrology; and another when he was so + far gone in those of chemistry, as to be upon the hunt after + the philosopher’s stone.—<span + class="smcap">Rev. J. Spence.</span></p> + <p class="blockcite"> + Anecdotes, Observations, and Characters of Books and Men + (London, 1868), p. 54.</p> + + <p class="v2"> + <b><a name="Block_1017" id="Block_1017">1017</a>.</b> + For several years this great man [Newton] was + intensely occupied in endeavoring to discover a way of changing + the base metals into gold.... There were periods when his + furnace fires were not allowed to go out for six weeks; he and + his secretary sitting up alternate nights to replenish + them.—<span class="smcap">Parton, James.</span></p> + <p class="blockcite"> + Sir Isaac Newton.</p> + + <p class="v2"> + <b><a name="Block_1018" id="Block_1018">1018</a>.</b> + On the day of Cromwell’s death, + when Newton was sixteen, a great storm raged all over England. + He used to say, in his old age, that on that day he made his + first purely scientific experiment. To ascertain the force of + the wind, he first jumped with the wind and then against it; + and, by comparing these distances with the extent of his own + jump on a calm day, he was enabled to compute the force of the + storm. When the wind blew thereafter, he used to say it was so + many feet strong.—<span class="smcap">Parton, James.</span></p> + <p class="blockcite"> + Sir Isaac Newton.</p> + + <p class="v2"> + <b><a name="Block_1019" id="Block_1019">1019</a>.</b> + Newton lectured now and then to the few students who chose + to hear him; and it is recorded that very frequently he + +<span class="pagenum"> + <a name="Page_169" + id="Page_169">169</a></span> + + came to the lecture-room and found it + empty. On such occasions he would remain fifteen minutes, and + then, if no one came, return to his + apartments.—<span class="smcap">Parton, James.</span></p> + <p class="blockcite"> + Sir Isaac Newton.</p> + + <p class="v2"> + <b><a name="Block_1020" id="Block_1020">1020</a>.</b> + Sir Isaac Newton, though so deep in algebra and + fluxions, could not readily make up a common account: and, when + he was Master of the Mint, used to get somebody else to make up + his accounts for him.—<span class= + "smcap">Rev. J. Spence.</span></p> + <p class="blockcite"> + Anecdotes, Observations, and Characters of Books and Men + (London, 1858), p. 132.</p> + + <p class="v2"> + <b><a name="Block_1021" id="Block_1021">1021</a>.</b> + We have one of his [Newton’s] college memorandum-books, + which is highly interesting. The following are some of the + entries: “Drills, gravers, a hone, a hammer, and a mandril, + 5s.;” “a magnet, 16s.;” “compasses, 2s.;” “glass bubbles, 4s.;” + “at the tavern several other times, £1;” “spent on my + cousin, 12s.;” “on other acquaintances, 10s.;” “Philosophical + Intelligences, 9s. 6d.;” “lost at cards twice, 15s.;” + “at the tavern twice, 3s. 6d.;” “to three prisms, £3;” + “four ounces of putty, 1s. 4d.;” “Bacon’s Miscellanies, 1s. 6d.;” + “a bible binding, 3s.;” “for oranges to my sister, 4s. 2d.;” + “for aquafortis, sublimate, oyle pink, fine silver, antimony, + vinegar, spirit of wine, white lead, salt of tartar, £2;“ + “Theatrum chemicum, £1 8s”—<span + class="smcap">Parton, James.</span></p> + <p class="blockcite"> + Sir Isaac Newton.</p> + + <p class="v2"> + <b><a name="Block_1022" id="Block_1022">1022</a>.</b> + On one occasion, when he was giving a dinner to + some friends at the university, he left the table to get them a + bottle of wine; but, on his way to the cellar, he fell into + reflection, forgot his errand and his company, went to his + chamber, put on his surplice, and proceeded to the chapel. + Sometimes he would go into the street half dressed, and on + discovering his condition, run back in great haste, much + abashed. Often, while strolling in his garden, he would + suddenly stop, and then run rapidly to his room, and begin to + write, standing, on the first piece of paper that presented + itself. Intending to dine in the public hall, he would go out + in a brown study, take the wrong turn, walk a while, and then + return to his room, having totally forgotten the dinner. Once + having dismounted from his horse to lead him + +<span class="pagenum"> + <a name="Page_170" + id="Page_170">170</a></span> + + up a + hill, the horse slipped his head out of the bridle; but Newton, + oblivious, never discovered it till, on reaching a tollgate at + the top of the hill, he turned to remount and perceived that + the bridle which he held in his hand had no horse attached to + it. His secretary records that his forgetfulness of his dinner + was an excellent thing for his old housekeeper, who + “sometimes found both dinner and supper + scarcely tasted of, which the old woman has very pleasantly and + mumpingly gone away with.” On getting out of bed in + the morning, he has been discovered to sit on his bedside for + hours without dressing himself, utterly absorbed in + thought.—<span class="smcap">Parton, James.</span></p> + <p class="blockcite"> + Sir Isaac Newton.</p> + + <p class="v2"> + <b><a name="Block_1023" id="Block_1023">1023</a>.</b> + I don’t know what I may seem to + the world, but, as to myself, I seem to have been only as a boy + playing on the seashore, and diverting myself in now and then + finding a smoother pebble or a prettier shell than ordinary, + whilst the great ocean of truth lay all undiscovered before + me.—<span class="smcap">Newton, I.</span></p> + <p class="blockcite"> + Quoted by Rev. J. Spence: Anecdotes, Observations, and + Characters of Books and Men (London, 1858), p. 40.</p> + + <p class="v2"> + <b><a name="Block_1024" id="Block_1024">1024</a>.</b> + If I have seen farther than Descartes, it is by + standing on the shoulders of + giants.—<span class="smcap">Newton, I.</span></p> + <p class="blockcite"> + Quoted by James Parton: Sir Isaac Newton.</p> + + <p class="v2"> + <b><a name="Block_1025" id="Block_1025">1025</a>.</b> + Newton could not admit that there was any + difference between him and other men, except in the possession + of such habits as ... perseverance and vigilance. When he was + asked how he made his discoveries, he answered, + “by always thinking about them;“ + and at another time he declared that if he had done anything, + it was due to nothing but industry and patient thought: + “I keep the subject of my inquiry constantly + before me, and wait till the first dawning opens gradually, by + little and little, into a full and clear + light”—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + History of the Inductive Sciences, Bk. 7, chap. 2, sect. 5.</p> + + <p class="v2"> + <b><a name="Block_1026" id="Block_1026">1026</a>.</b> + Newton took no exercise, indulged in no + amusements, and worked incessantly, often spending eighteen or + nineteen hours out of the twenty-four in + writing.—<span class="smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p. 358.</p> + +<p><span class="pagenum"> + <a name="Page_171" + id="Page_171">171</a></span></p> + + <p class="v2"> + <b><a name="Block_1027" id="Block_1027">1027</a>.</b> + Foreshadowings of the principles and even of the + language of [the infinitesimal] calculus can be found in the + writings of Napier, Kepler, Cavalieri, Pascal, Fermat, Wallis, + and Barrow. It was Newton’s good luck to + come at a time when everything was ripe for the discovery, and + his ability enabled him to construct almost at once a complete + calculus.—<span class="smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p. 356.</p> + + <p class="v2"> + <b><a name="Block_1028" id="Block_1028">1028</a>.</b> + Kepler’s suggestion of + gravitation with the inverse distance, and + Bouillaud’s proposed substitution of the + inverse square of the distance, are things which Newton knew + better than his modern readers. I have discovered two anagrams + on his name, which are quite conclusive: the notion of + gravitation was <em>not new</em>; but Newton <em>went + on</em>.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), p. 82.</p> + + <p class="v2"> + <b><a name="Block_1029" id="Block_1029">1029</a>.</b> + For other great mathematicians or philosophers, he + [Gauss] used the epithets magnus, or clarus, or clarissimus; + for Newton alone he kept the prefix + summus.—<span class="smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p. 362.</p> + + <p class="v2"> + <b><a name="Block_1030" id="Block_1030">1030</a>.</b> + To know him [Sylvester] was to know one of the + historic figures of all time, one of the immortals; and when he + was really moved to speak, his eloquence equalled his + genius.—<span class="smcap">Halsted, G. B.</span></p> + <p class="blockcite"> + F. Cajori’s Teaching and History of Mathematics in the U. S. + (Washington, 1890), p. 265.</p> + + <p class="v2"> + <b><a name="Block_1031" id="Block_1031">1031</a>.</b> + Professor Sylvester’s first + high class at the new university Johns Hopkins consisted of + only one student, G. B. Halsted, who had persisted in urging + Sylvester to lecture on the modern algebra. The attempt to + lecture on this subject led him into new investigations in + quantics.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + Teaching and History of Mathematics in the U. S. + (Washington, 1890), p. 264.</p> + + <p class="v2"> + <b><a name="Block_1032" id="Block_1032">1032</a>.</b> + But for the persistence of a student of this + university in urging upon me his desire to study with me the + modern algebra I should never have been led into this + investigation; and the + +<span class="pagenum"> + <a name="Page_172" + id="Page_172">172</a></span> + + new facts and principles which I + have discovered in regard to it (important facts, I believe), + would, so far as I am concerned, have remained still hidden in + the womb of time. In vain I represented to this inquisitive + student that he would do better to take up some other subject + lying less off the beaten track of study, such as the higher + parts of the calculus or elliptic functions, or the theory of + substitutions, or I wot not what besides. He stuck with perfect + respectfulness, but with invincible pertinacity, to his point. + He would have the new algebra (Heaven knows where he had heard + about it, for it is almost unknown in this continent), that or + nothing. I was obliged to yield, and what was the consequence? + In trying to throw light upon an obscure explanation in our + text-book, my brain took fire, I plunged with re-quickened zeal + into a subject which I had for years abandoned, and found food + for thoughts which have engaged my attention for a considerable + time past, and will probably occupy all my powers of + contemplation advantageously for several months to + come.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Johns Hopkins Commemoration Day Address; Collected + Mathematical Papers, Vol. 3, p. 76.</p> + + <p class="v2"> + <b><a name="Block_1033" id="Block_1033">1033</a>.</b> + Sylvester was incapable of reading mathematics in + a purely receptive way. Apparently a subject either fired in + his brain a train of active and restless thought, or it would + not retain his attention at all. To a man of such a + temperament, it would have been peculiarly helpful to live in + an atmosphere in which his human associations would have + supplied the stimulus which he could not find in mere reading. + The great modern work in the theory of functions and in allied + disciplines, he never became acquainted with....</p> + <p class="v1"> + What would have been the effect if, in the prime of his powers, + he had been surrounded by the influences which prevail in + Berlin or in Göttingen? It may be confidently + taken for granted that he would have done splendid work in + those domains of analysis, which have furnished the laurels of + the great mathematicians of Germany and France in the second + half of the present century.—<span class= + "smcap">Franklin, F.</span></p> + <p class="blockcite"> + Johns Hopkins University Circulars 16 (1897), p. 54.</p> + +<p><span class="pagenum"> + <a name="Page_173" + id="Page_173">173</a></span></p> + + <p class="v2"> + <b><a name="Block_1034" id="Block_1034">1034</a>.</b> + If we survey the mathematical works of Sylvester, + we recognize indeed a considerable abundance, but in + contradistinction to Cayley—not a + versatility toward separate fields, but, with few + exceptions—a confinement to + arithmetic-algebraic branches....</p> + <p class="v1"> + The concept of <em>Function</em> of a continuous variable, the + fundamental concept of modern mathematics, plays no role, is + indeed scarcely mentioned in the entire work of + Sylvester—Sylvester was combinatorist + [combinatoriker].—<span class="smcap">Noether, M.</span></p> + <p class="blockcite"> + Mathematische Annalen, Bd. 50 (1898), pp. 134-135.</p> + + <p class="v2"> + <b><a name="Block_1035" id="Block_1035">1035</a>.</b> + Sylvester’s <em>methods!</em> He + had none. “Three lectures will be delivered + on a New Universal Algebra,” he would say; then, + “The course must be extended to twelve.” It did last all the + rest of that year. The following year the course was to be + <em>Substitutions-Theorie</em>, by Netto. We all got the text. He + lectured about three times, following the text closely and + stopping sharp at the end of the hour. Then he began to think + about matrices again. “I must give one + lecture a week on those,” he said. He could not + confine himself to the hour, nor to the one lecture a week. Two + weeks were passed, and Netto was forgotten entirely and never + mentioned again. Statements like the following were not + unfrequent in his lectures: “I + haven’t proved this, but I am as sure as I + can be of anything that it must be so. From this it will + follow, etc.” At the next lecture it turned out + that what he was so sure of was false. Never mind, he kept on + forever guessing and trying, and presently a wonderful + discovery followed, then another and another. Afterward he + would go back and work it all over again, and surprise us with + all sorts of side lights. He then made another leap in the + dark, more treasures were discovered, and so on + forever.—<span class="smcap">Davis, E. W.</span></p> + <p class="blockcite"> + Cajori’s Teaching and History of + Mathematics in the U.S. (Washington, 1890), pp. 265-266.</p> + + <p class="v2"> + <b><a name="Block_1036" id="Block_1036">1036</a>.</b> + I can see him [Sylvester] now, with his white + beard and few locks of gray hair, his forehead wrinkled + o’er with thoughts, writing rapidly his + figures and formulae on the board, sometimes explaining as he + wrote, while we, his listeners, caught + +<span class="pagenum"> + <a name="Page_174" + id="Page_174">174</a></span> + + the reflected sounds from the board. But stop, something is not + right, he pauses, his hand goes to his forehead to help his + thought, he goes over the work again, emphasizes the leading + points, and finally discovers his difficulty. Perhaps it is + some error in his figures, perhaps an oversight in the + reasoning. Sometimes, however, the difficulty is not + elucidated, and then there is not much to the rest of the + lecture. But at the next lecture we would hear of some new + discovery that was the outcome of that difficulty, and of some + article for the Journal, which he had begun. If a text-book had + been taken up at the beginning, with the intention of following + it, that text-book was most likely doomed to oblivion for the + rest of the term, or until the class had been made listeners to + every new thought and principle that had sprung from the + laboratory of his mind, in consequence of that first + difficulty. Other difficulties would soon appear, so that no + text-book could last more than half of the term. In this way + his class listened to almost all of the work that subsequently + appeared in the Journal. It seemed to be the quality of his + mind that he must adhere to one subject. He would think about + it, talk about it to his class, and finally write about it for + the Journal. The merest accident might start him, but once + started, every moment, every thought was given to it, and, as + much as possible, he read what others had done in the same + direction; but this last seemed to be his real point; he could + not read without finding difficulties in the way of + understanding the author. Thus, often his own work reproduced + what had been done by others, and he did not find it out until + too late.</p> + <p class="v1"> + A notable example of this is in his theory of cyclotomic + functions, which he had reproduced in several foreign journals, + only to find that he had been greatly anticipated by foreign + authors. It was manifest, one of the critics said, that the + learned professor had not read Kummer’s + elementary results in the theory of ideal primes. Yet Professor + Smith’s report on the theory of numbers, which contained a full + synopsis of Kummer’s theory, was Professor Sylvester’s constant + companion.</p> + <p class="v1"> + This weakness of Professor Sylvester, in not being able to read + what others had done, is perhaps a concomitant of his peculiar + genius. Other minds could pass over little difficulties and + not be troubled by them, and so go on to a final understanding + +<span class="pagenum"> + <a name="Page_175" + id="Page_175">175</a></span> + + of the results of the author. But + not so with him. A difficulty, however small, worried him, and + he was sure to have difficulties until the subject had been + worked over in his own way, to correspond with his own mode of + thought. To read the work of others, meant therefore to him an + almost independent development of it. Like the man whose + pleasure in life is to pioneer the way for society into the + forests, his rugged mind could derive satisfaction only in + hewing out its own paths; and only when his efforts brought him + into the uncleared fields of mathematics did he find his place + in the Universe.—<span class= "smcap">Hathaway, A. S.</span></p> + <p class="blockcite"> + F. Cajori’s Teaching and History of + Mathematics in the U. S. (Washington, 1890), pp. 266-267.</p> + + <p class="v2"> + <b><a name="Block_1037" id="Block_1037">1037</a>.</b> + Professor Cayley has since informed me that the + theorem about whose origin I was in doubt, will be found in + Schläfli’s “De Eliminatione.” This is not + the first unconscious plagiarism I have been guilty of towards + this eminent man whose friendship I am proud to claim. A more + glaring case occurs in a note by me in the + “Comptes Rendus,” on the + twenty-seven straight lines of cubic surfaces, where I believe + I have followed (like one walking in his sleep), down to the + very nomenclature and notation, the substance of a portion of a + paper inserted by Schläfli in the “Mathematical Journal,” which + bears my name as one of the editors upon the + face.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Philosophical Transactions of the Royal Society (1864), + p. 642.</p> + + <p class="v2"> + <b><a name="Block_1038" id="Block_1038">1038</a>.</b> + He [Sylvester] had one remarkable peculiarity. He + seldom remembered theorems, propositions, etc., but had always + to deduce them when he wished to use them. In this he was the + very antithesis of Cayley, who was thoroughly conversant with + everything that had been done in every branch of mathematics.</p> + <p class="v1"> + I remember once submitting to Sylvester some investigations + that I had been engaged on, and he immediately denied my first + statement, saying that such a proposition had never been heard + of, let alone proved. To his astonishment, I showed him a + +<span class="pagenum"> + <a name="Page_176" + id="Page_176">176</a></span> + + paper of his own in which he had + proved the proposition; in fact, I believe the object of his + paper had been the very proof which was so strange to + him.—<span class="smcap">Durfee, W. P.</span></p> + <p class="blockcite"> + F. Cajori’s Teaching and History of + Mathematics in the U. S. (Washington, 1890), p. 268.</p> + + <p class="v2"> + <b><a name="Block_1039" id="Block_1039">1039</a>.</b> + A short, broad man of tremendous vitality, the + physical type of Hereward, the last of the English, and his + brother-in-arms, Winter, Sylvester’s + capacious head was ever lost in the highest cloud-lands of pure + mathematics. Often in the dead of night he would get his + favorite pupil, that he might communicate the very last product + of his creative thought. Everything he saw suggested to him + something new in the higher algebra. This transmutation of + everything into new mathematics was a revelation to those who + knew him intimately. They began to do it themselves. His ease + and fertility of invention proved a constant encouragement, + while his contempt for provincial stupidities, such as the + American hieroglyphics for π and <em>e</em>, which + have even found their way into Webster’s + Dictionary, made each young worker apply to himself the + strictest tests.—<span class="smcap">Halsted, G. B.</span></p> + <p class="blockcite"> + F. Cajori’s Teaching and History of + Mathematics in the U. S. (Washington, 1890), p. 265.</p> + + <p class="v2"> + <b><a name="Block_1040" id="Block_1040">1040</a>.</b> + Sylvester’s writings are + flowery and eloquent. He was able to make the dullest subject + bright, fresh and interesting. His enthusiasm is evident in + every line. He would get quite close up to his subject, so that + everything else looked small in comparison, and for the time + would think and make others think that the world contained no + finer matter for contemplation. His handwriting was bad, and a + trouble to his printers. His papers were finished with + difficulty. No sooner was the manuscript in the + editor’s hands than alterations, + corrections, ameliorations and generalizations would suggest + themselves to his mind, and every post would carry further + directions to the editors and + printers.—<span class="smcap">MacMahon. P. A.</span></p> + <p class="blockcite"> + Nature, Vol. 55 (1897), p. 494.</p> + + <p class="v2"> + <b><a name="Block_1041" id="Block_1041">1041</a>.</b> + The enthusiasm of Sylvester for his own work, which manifests + itself here as always, indicates one of his characteristic + +<span class="pagenum"> + <a name="Page_177" + id="Page_177">177</a></span> + + qualities: a high degree of + <em>subjectivity</em> in his productions and publications. + Sylvester was so fully possessed by the matter which for the + time being engaged his attention, that it appeared to him and + was designated by him as the summit of all that is important, + remarkable and full of future promise. It would excite his + phantasy and power of imagination in even a greater measure + than his power of reflection, so much so that he could never + marshal the ability to master his subject-matter, much less to + present it in an orderly manner.</p> + <p class="v1"> + Considering that he was also somewhat of a poet, it will be + easier to overlook the poetic flights which pervade his + writing, often bombastic, sometimes furnishing apt + illustrations; more damaging is the complete lack of form and + orderliness of his publications and their sketchlike + character,.... which must be accredited at least as much to + lack of objectivity as to a superfluity of ideas. Again, the + text is permeated with associated emotional expressions, + bizarre utterances and paradoxes and is everywhere accompanied + by notes, which constitute an essential part of + Sylvester’s method of presentation, + embodying relations, whether proximate or remote, which + momentarily suggested themselves. These notes, full of + inspiration and occasional flashes of genius, are the more + stimulating owing to their incompleteness. But none of his + works manifest a desire to penetrate the subject from all sides + and to allow it to mature; each mere surmise, conceptions which + arose during publication, immature thoughts and even errors + were ushered into publicity at the moment of their inception, + with utmost carelessness, and always with complete + unfamiliarity of the literature of the subject. Nowhere is + there the least trace of self-criticism. No one can be expected + to read the treatises entire, for in the form in which they are + available they fail to give a clear view of the matter under + contemplation.</p> + <p class="v1"> + Sylvester’s was not a harmoniously gifted or + well-balanced mind, but rather an instinctively active and + creative mind, free from egotism. His reasoning moved in + generalizations, was frequently influenced by analysis and at + times was guided even by mystical numerical relations. His + reasoning consists less frequently of pure intelligible + conclusions than of inductions, or rather conjectures incited + by individual observations and + +<span class="pagenum"> + <a name="Page_178" + id="Page_178">178</a></span> + + verifications. In this + he was guided by an algebraic sense, developed through long + occupation with processes of forms, and this led him luckily to + general fundamental truths which in some instances remain + veiled. His lack of system is here offset by the advantage of + freedom from purely mechanical logical activity.</p> + <p class="v1"> + The exponents of his essential characteristics are an intuitive + talent and a faculty of invention to which we owe a series of + ideas of lasting value and bearing the germs of fruitful + methods. To no one more fittingly than to Sylvester can be + applied one of the mottos of the Philosophic Magazine:</p> + <p class="v1"> + “Admiratio generat quaestionem, quaestio investigationem + investigatio inventionem”—<span class= + "smcap">Noether, M.</span></p> + <p class="blockcite"> + Mathematische Annalen, Bd. 50 (1898), pp. 155-160.</p> + + <p class="v2"> + <b><a name="Block_1042" id="Block_1042">1042</a>.</b> + Perhaps I may without immodesty lay claim to the + appellation of Mathematical Adam, as I believe that I have + given more names (passed into general circulation) of the + creatures of the mathematical reason than all the other + mathematicians of the age + combined.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Nature, Vol. 37 (1887-1888), p. 162.</p> + + <p class="v2"> + <b><a name="Block_1043" id="Block_1043">1043</a>.</b> + Tait dubbed Maxwell dp/dt, for according to + thermodynamics dp/dt = JCM (where C denotes + Carnot’s function) the initials of (J. C.) + Maxwell’s name. On the other hand Maxwell + denoted Thomson by T and Tait by T´; so that + it became customary to quote Thomson and + Tait’s Treatise on Natural Philosophy as T + and T´.—<span class="smcap">Macfarlane, A.</span></p> + <p class="blockcite"> + Bibliotheca Mathematica, Bd. 3 (1903), p. 189.</p> + + <p class="v2"> + <b><a name="Block_1044" id="Block_1044">1044</a>.</b> + In future times Tait will be best known for his + work in the quaternion analysis. Had it not been for his + expositions, developments and applications, + Hamilton’s invention would be today, in all + probability, a mathematical curiosity.—<span + class="smcap">Macfarlane, A.</span></p> + <p class="blockcite"> + Bibliotheca Mathematica, Bd. 3 (1903), p. 189.</p> + + <p class="v2"> + <b><a name="Block_1045" id="Block_1045">1045</a>.</b> + Not seldom did he [Sir William Thomson], in his + writings, set down some mathematical statement with the + prefacing remark “it is obvious that” to the + perplexity of mathematical + +<span class="pagenum"> + <a name="Page_179" + id="Page_179">179</a></span> + + readers, to whom the statement was + anything but obvious from such mathematics as preceded it on + the page. To him it was obvious for physical reasons that might + not suggest themselves at all to the mathematician, however + competent.—<span class="smcap">Thompson, S. P.</span></p> + <p class="blockcite"> + Life of Lord Kelvin (London, 1910), p. 1136.</p> + + <p class="v2"> + <b><a name="Block_1046" id="Block_1046">1046</a>.</b> + The following is one of the many stories told of “old Donald + McFarlane” the faithful assistant of Sir William Thomson.</p> + <p class="v1"> + The father of a new student when bringing him to the + University, after calling to see the Professor [Thomson] drew + his assistant to one side and besought him to tell him what his + son must do that he might stand well with the Professor. + “You want your son to stand weel with the + Profeessorr?” asked McFarlane. “Yes.” “Weel, + then, he must just have a guid bellyful o’ + mathematics!“—<span class="smcap">Thompson, S. P.</span></p> + <p class="blockcite"> + Life of Lord Kelvin (London, 1910), p. 420.</p> + + <p class="v2"> + <b><a name="Block_1047" id="Block_1047">1047</a>.</b> + The following story (here a little softened from + the vernacular) was narrated by Lord Kelvin himself when dining + at Trinity Hall:—</p> + <p class="v1"> + A certain rough Highland lad at the university had done + exceedingly well, and at the close of the session gained prizes + both in mathematics and in metaphysics. His old father came up + from the farm to see his son receive the prizes, and visited + the College. Thomson was deputed to show him round the place. + “Weel, Mr. Thomson,” asked the old man, “and what may these + mathematics be, for which my son has getten a prize?” “I told + him,” replied Thomson, “that mathematics meant reckoning with + figures, and calculating.” “Oo ay,” said the old man, “he’ll + ha’ getten that fra’ me: I were ever a braw hand at the + countin’.” After a pause he resumed: “And what, Mr. Thomson, + might these metapheesics be?” “I endeavoured,” replied Thomson, + “to explain how metaphysics was the attempt to express in + language the indefinite.” The old Highlander stood still and + scratched his head. “Oo ay: may be he’ll ha’ getten that fra’ + his mither. She were aye a bletherin’ + body”—<span class="smcap">Thompson, S. P.</span></p> + <p class="blockcite"> + Life of Lord Kelvin (London, 1910), p. 1124.</p> + +<p><span class="pagenum"> + <a name="Page_180" + id="Page_180">180</a></span></p> + + <p class="v2"> + <b><a name="Block_1048" id="Block_1048">1048</a>.</b> + Lord Kelvin, unable to meet his classes one day, + posted the following notice on the door of his lecture + room,—</p> + <p class="center"> + “Professor Thomson will not meet his classes today.”</p> + <p class="v0"> + The disappointed class decided to play a joke on the professor. + Erasing the “c” they left the legend to read,—</p> + <p class="center"> + “Professor Thomson will not meet his lasses today.”</p> + <p class="v0"> + When the class assembled the next day in + anticipation of the effect of their joke, they were astonished + and chagrined to find that the professor had outwitted them. + The legend of yesterday was now found to read,—</p> + <p class="center"> + “Professor Thomson will not meet his asses today.” + <a href="#Footnote_9" + class="fnanchor">9</a></p> + <p class="block40"> + —<span class="smcap">Northrup, Cyrus.</span></p> + <p class="blockcite"> + University of Washington Address, November 2, 1908.</p> + + <p class="v2"> + <b><a name="Block_1049" id="Block_1049">1049</a>.</b> + One morning a great noise proceeded from one of + the classrooms [of the Braunsberger gymnasium] and on + investigation it was found that Weierstrass, who was to give + the recitation, had not appeared. The director went in person + to Weierstrass’ dwelling and on knocking was + told to come in. There sat Weierstrass by a glimmering lamp in + a darkened room though it was daylight outside. He had worked + the night through and had not noticed the approach of daylight. + When the director reminded him of the noisy throng of students + who were waiting for him, his only reply was that he could + impossibly interrupt his work; that he was about to make an + important discovery which would attract attention in scientific + circles.—<span class="smcap">Lampe, E.</span></p> + <p class="blockcite"> + Karl Weierstrass: Jahrbuch der Deutschen Mathematiker + Vereinigung, Bd. 6 (1897), pp. 38-39.</p> + + <p class="v2"> + <b><a name="Block_1050" id="Block_1050">1050</a>.</b> + Weierstrass related ... that he followed + Sylvester’s papers on the theory of + algebraic forms very attentively until Sylvester began to + employ Hebrew characters. That was more than he could stand and + after that he quit him.—<span class= "smcap">Lampe, E.</span></p> + <p class="blockcite"> + Naturwissenschaftliche Rundschau, Bd. 12 (1897), p. 361.</p> + +<p><span class="pagenum"> + <a name="Page_181" + id="Page_181">181</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_XI"> + CHAPTER XI<br /> + <span class="large"> + MATHEMATICS AS A FINE ART</span></h2> + + <p class="v2"> + <b><a name="Block_1101" id="Block_1101">1101</a>.</b> + The world of idea which it discloses or + illuminates, the contemplation of divine beauty and order which + it induces, the harmonious connexion of its parts, the infinite + hierarchy and absolute evidence of the truths with which it is + concerned, these, and such like, are the surest grounds of the + title of mathematics to human regard, and would remain + unimpeached and unimpaired were the plan of the universe + unrolled like a map at our feet, and the mind of man qualified + to take in the whole scheme of creation at a + glance.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Presidential Address, British Association Report (1869); + Collected Mathematical Papers, Vol. 2, p. 659.</p> + + <p class="v2"> + <b><a name="Block_1102" id="Block_1102">1102</a>.</b> + Mathematics has a triple end. It should furnish an + instrument for the study of nature. Furthermore it has a + philosophic end, and, I venture to say, an end esthetic. It + ought to incite the philosopher to search into the notions of + number, space, and time; and, above all, adepts find in + mathematics delights analogous to those that painting and music + give. They admire the delicate harmony of number and of forms; + they are amazed when a new discovery discloses for them an + unlooked for perspective; and the joy they thus experience, has + it not the esthetic character although the senses take no part + in it? Only the privileged few are called to enjoy it fully, it + is true; but is it not the same with all the noblest arts? + Hence I do not hesitate to say that mathematics deserves to be + cultivated for its own sake, and that the theories not + admitting of application to physics deserve to be studied as + well as others. <span class="smcap">Poincaré, Henri.</span></p> + <p class="blockcite"> + The Relation of Analysis and Mathematical Physics; Bulletin + American Mathematical Society, Vol. 4 (1899), p. 248.</p> + +<p><span class="pagenum"> + <a name="Page_182" + id="Page_182">182</a></span></p> + + <p class="v2"> + <b><a name="Block_1103" id="Block_1103">1103</a>.</b> + I like to look at mathematics almost more as an art than + as a science; for the activity of the mathematician, constantly + creating as he is, guided though not controlled by the external + world of the senses, bears a resemblance, not fanciful I + believe but real, to the activity of an artist, of a painter + let us say. Rigorous deductive reasoning on the part of the + mathematician may be likened here to technical skill in drawing + on the part of the painter. Just as no one can become a good + painter without a certain amount of skill, so no one can become + a mathematician without the power to reason accurately up to a + certain point. Yet these qualities, fundamental though they + are, do not make a painter or mathematician worthy of the name, + nor indeed are they the most important factors in the case. + Other qualities of a far more subtle sort, chief among which in + both cases is imagination, go to the making of a good artist or + good mathematician.—<span class="smcap">Bôcher, + Maxime.</span></p> + <p class="blockcite"> + Fundamental Conceptions and Methods in Mathematics; Bulletin + American Mathematical Society, Vol. 9 (1904), p. 133.</p> + + <p class="v2"> + <b><a name="Block_1104" id="Block_1104">1104</a>.</b> + Mathematics, rightly viewed, possesses not only + truth, but supreme beauty—a beauty cold and + austere, like that of sculpture, without appeal to any part of + our weaker nature, without the gorgeous trappings of painting + or music, yet sublimely pure, and capable of a stern perfection + such as only the greatest art can show. The true spirit of + delight, the exaltation, the sense of being more than man, + which is the touchstone of the highest excellence, is to be + found in mathematics as surely as in poetry. What is best in + mathematics deserves not merely to be learned as a task, but to + be assimilated as a part of daily thought, and brought again + and again before the mind with ever-renewed encouragement. Real + life is, to most men, a long second-best, a perpetual + compromise between the real and the possible; but the world of + pure reason knows no compromise, no practical limitations, no + barrier to the creative activity embodying in splendid edifices + the passionate aspiration after the perfect from which all + great work springs. Remote from human passions, remote even + from the pitiful facts of nature, the generations have + gradually created an ordered cosmos, where pure thought can + dwell as in its natural home, and where one, at + +<span class="pagenum"> + <a name="Page_183" + id="Page_183">183</a></span> + + least, of our nobler impulses can escape from the dreary exile + of the natural world.—<span class= + "smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + The Study of Mathematics: Philosophical Essays (London, + 1910), p. 73.</p> + + <p class="v2"> + <b><a name="Block_1105" id="Block_1105">1105</a>.</b> + It was not alone the striving for universal + culture which attracted the great masters of the Renaissance, + such as Brunellesco, Leonardo de Vinci, Raphael, Michael Angelo + and especially Albrecht Dürer, with irresistible + power to the mathematical sciences. They were conscious that, + with all the freedom of the individual phantasy, art is subject + to necessary laws, and conversely, with all its rigor of + logical structure, mathematics follows esthetic + laws.—<span class="smcap">Rudio, F.</span></p> + <p class="blockcite"> + Virchow-Holtzendorf: Sammlung gemeinverständliche + wissenschaftliche Vorträge, Heft 142, p. 19.</p> + + <p class="v2"> + <b><a name="Block_1106" id="Block_1106">1106</a>.</b> + Surely the claim of mathematics to take a place + among the liberal arts must now be admitted as fully made good. + Whether we look at the advances made in modern geometry, in + modern integral calculus, or in modern algebra, in each of + these three a free handling of the material employed is now + possible, and an almost unlimited scope is left to the + regulated play of fancy. It seems to me that the whole of + aesthetic (so far as at present revealed) may be regarded as a + scheme having four centres, which may be treated as the four + apices of a tetrahedron, namely Epic, Music, Plastic, and + Mathematic. There will be found a <em>common</em> plane to every + three of these, <em>outside</em> of which lies the fourth; and + through every two may be drawn a common axis <em>opposite</em> to + the axis passing through the other two. So far is certain and + demonstrable. I think it also possible that there is a centre + of gravity to each set of three, and that the line joining each + such centre with the outside apex will intersect in a common + point—the centre of gravity of the whole + body of aesthetic; but what that centre is or must be I have + not had time to think out.—<span class= + "smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Proof of the hitherto undemonstrated Fundamental Theorem + of Invariants: Collected Mathematical Papers, Vol. 3, p. 123.</p> + +<p><span class="pagenum"> + <a name="Page_184" + id="Page_184">184</a></span></p> + + <p class="v2"> + <b><a name="Block_1107" id="Block_1107">1107</a>.</b> + It is with mathematics not otherwise than it is + with music, painting or poetry. Anyone can become a lawyer, + doctor or chemist, and as such may succeed well, provided he is + clever and industrious, but not every one can become a painter, + or a musician, or a mathematician: general cleverness and + industry alone count here for + nothing.—<span class="smcap">Moebius, P. J.</span></p> + <p class="blockcite"> + Ueber die Anlage zur Mathematik (Leipzig, 1900), p. 5.</p> + + <p class="v2"> + <b><a name="Block_1108" id="Block_1108">1108</a>.</b> + The true mathematician is always a good deal of an + artist, an architect, yes, of a poet. Beyond the real world, + though perceptibly connected with it, mathematicians have + intellectually created an ideal world, which they attempt to + develop into the most perfect of all worlds, and which is being + explored in every direction. None has the faintest conception + of this world, except he who knows + it.—<span class="smcap">Pringsheim, A.</span></p> + <p class="blockcite"> + Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. + 32, p. 381.</p> + + <p class="v2"> + <b><a name="Block_1109" id="Block_1109">1109</a>.</b> + Who has studied the works of such men as Euler, + Lagrange, Cauchy, Riemann, Sophus Lie, and Weierstrass, can + doubt that a great mathematician is a great artist? The + faculties possessed by such men, varying greatly in kind and + degree with the individual, are analogous with those requisite + for constructive art. Not every mathematician possesses in a + specially high degree that critical faculty which finds its + employment in the perfection of form, in conformity with the + ideal of logical completeness; but every great mathematician + possesses the rarer faculty of constructive + imagination.—<span class="smcap">Hobson, E. W.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science (1910) Nature, Vol. 84, p. 290.</p> + + <p class="v2"> + <b><a name="Block_1110" id="Block_1110">1110</a>.</b> + Mathematics has beauties of its + own—a symmetry and proportion in its + results, a lack of superfluity, an exact adaptation of means to + ends, which is exceedingly remarkable and to be found elsewhere + only in the works of the greatest beauty. It was a felicitous + expression of Goethe’s to call a noble cathedral “frozen music,” + but it might even better be called “petrified mathematics.” + The beauties of mathematics—of simplicity, + +<span class="pagenum"> + <a name="Page_185" + id="Page_185">185</a></span> + + of symmetry, of completeness—can and should be + exemplified even to young children. When this subject is + properly and concretely presented, the mental emotion should be + that of enjoyment of beauty, not that of repulsion from the + ugly and the unpleasant.—<span class= + "smcap">Young, J. W. A.</span></p> + <p class="blockcite"> + The Teaching of Mathematics (New York, 1907), p. 44.</p> + + <p class="v2"> + <b><a name="Block_1111" id="Block_1111">1111</a>.</b> + A peculiar beauty reigns in the realm of + mathematics, a beauty which resembles not so much the beauty of + art as the beauty of nature and which affects the reflective + mind, which has acquired an appreciation of it, very much like + the latter.—<span class="smcap">Kummer, E. E.</span></p> + <p class="blockcite"> + Berliner Monatsberichte (1867), p. 395.</p> + + <p class="v2"> + <b><a name="Block_1112" id="Block_1112">1112</a>.</b> + Mathematics make the mind attentive to the objects + which it considers. This they do by entertaining it with a + great variety of truths, which are delightful and evident, but + not obvious. Truth is the same thing to the understanding as + music to the ear and beauty to the eye. The pursuit of it does + really as much gratify a natural faculty implanted in us by our + wise Creator as the pleasing of our senses: only in the former + case, as the object and faculty are more spiritual, the delight + is more pure, free from regret, turpitude, lassitude, and + intemperance that commonly attend sensual + pleasures.—<span class="smcap">Arbuthnot, John.</span></p> + <p class="blockcite"> + Usefulness of Mathematical Learning.</p> + + <p class="v2"> + <b><a name="Block_1113" id="Block_1113">1113</a>.</b> + However far the calculating reason of the + mathematician may seem separated from the bold flight of the + artist’s phantasy, it must be remembered + that these expressions are but momentary images snatched + arbitrarily from among the activities of both. In the + projection of new theories the mathematician needs as bold and + creative a phantasy as the productive artist, and in the + execution of the details of a composition the artist too must + calculate dispassionately the means which are necessary for the + successful consummation of the parts. Common to both is the + creation, the generation, of forms out of + mind.—<span class="smcap">Lampe, E.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik, etc. (Berlin, 1893), p. 4.</p> + +<p><span class="pagenum"> + <a name="Page_186" + id="Page_186">186</a></span></p> + + <p class="v2"> + <b><a name="Block_1114" id="Block_1114">1114</a>.</b> + As pure truth is the polar star of our science + [mathematics], so it is the great advantage of our science over + others that it awakens more easily the love of truth in our + pupils.... If Hegel justly said, “Whoever + does not know the works of the ancients, has lived without + knowing <em>beauty</em>,” Schellbach responds with + equal right, “Who does not know mathematics, + and the results of recent scientific investigation, dies + without knowing <em>truth</em>”—<span class= + "smcap">Simon, Max.</span></p> + <p class="blockcite"> + Quoted in J. W. A. Young: Teaching of Mathematics (New + York, 1907), p. 44.</p> + + <p class="v2"> + <b><a name="Block_1115" id="Block_1115">1115</a>.</b> + Büchsel in his reminiscences from + the life of a country parson relates that he sought his + recreation in Lacroix’s Differential + Calculus and thus found intellectual refreshment for his + calling. Instances like this make manifest the great advantage + which occupation with mathematics affords to one who lives + remote from the city and is compelled to forego the pleasures + of art. The entrancing charm of mathematics, which captivates + every one who devotes himself to it, and which is comparable to + the fine frenzy under whose ban the poet completes his work, + has ever been incomprehensible to the spectator and has often + caused the enthusiastic mathematician to be held in derision. A + classic illustration is the example of + Archimedes,....—<span class="smcap">Lampe, E.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik, etc. (Berlin 1893), p. 22.</p> + + <p class="v2"> + <b><a name="Block_1116" id="Block_1116">1116</a>.</b> + Among the memoirs of Kirchhoff are some of + uncommon beauty. Beauty, I hear you ask, do not the Graces flee + where integrals stretch forth their necks? Can anything be + beautiful, where the author has no time for the slightest + external embellishment?... Yet it is this very simplicity, the + indispensableness of each word, each letter, each little dash, + that among all artists raises the mathematician nearest to the + World-creator; it establishes a sublimity which is equalled in + no other art,—something like it exists at + most in symphonic music. The Pythagoreans recognized already + the similarity between the most subjective and the most + objective of the arts.... <i lang="la" xml:lang="la">Ultima + se tangunt</i>. How + expressive, how nicely characterizing withal is mathematics! As + the musician recognizes Mozart, Beethoven, Schubert in the + first chords, so the mathematician would distinguish + +<span class="pagenum"> + <a name="Page_187" + id="Page_187">187</a></span> + + his Cauchy, Gauss, Jacobi, + Helmholtz in a few pages. Extreme external elegance, sometimes + a somewhat weak skeleton of conclusions characterizes the + French; the English, above all Maxwell, are distinguished by + the greatest dramatic bulk. Who does not know + Maxwell’s dynamic theory of gases? At first + there is the majestic development of the variations of + velocities, then enter from one side the equations of condition + and from the other the equations of central motions,—higher + and higher surges the chaos of formulas,—suddenly four words + burst forth: “Put n = 5.” The evil + demon V disappears like the sudden ceasing of the basso parts + in music, which hitherto wildly permeated the piece; what + before seemed beyond control is now ordered as by magic. There + is no time to state why this or that substitution was made, he + who cannot feel the reason may as well lay the book aside; + Maxwell is no program-musician who explains the notes of his + composition. Forthwith the formulas yield obediently result + after result, until the temperature-equilibrium of a heavy gas + is reached as a surprising final climax and the curtain + drops....</p> + <p class="v1"> + Kirchhoff’s whole tendency, and its true + counterpart, the form of his presentation, was different.... He + is characterized by the extreme precision of his hypotheses, + minute execution, a quiet rather than epic development with + utmost rigor, never concealing a difficulty, always dispelling + the faintest obscurity. To return once more to my allegory, he + resembled Beethoven, the thinker in tones.—He who doubts that + mathematical compositions can be beautiful, let him read his + memoir on Absorption and Emission (Gesammelte Abhandlungen, + Leipzig, 1882, p. 571-598) or the chapter of his mechanics + devoted to Hydrodynamics.—<span class= + "smcap">Boltzmann, L.</span></p> + <p class="blockcite"> + Gustav Robert Kirchhoff (Leipzig 1888), pp. 28-30.</p> + + <p class="v2"> + <b><a name="Block_1117" id="Block_1117">1117</a>.</b></p> + <div class="poem"> + <p class="i0"> + On poetry and geometric truth,</p> + <p class="i0"> + And their high privilege of lasting life,</p> + <p class="i0"> + From all internal injury exempt,</p> + <p class="i0"> + I mused; upon these chiefly: and at length,</p> + <p class="i0"> + My senses yielding to the sultry air,</p> + <p class="i0"> + Sleep seized me, and I passed into a dream.</p> + </div> + <p class="block40"> + —<span class= "smcap">Wordsworth.</span></p> + <p class="blockcite"> + The Prelude, Bk. 5.</p> + +<p><span class="pagenum"> + <a name="Page_188" + id="Page_188">188</a></span></p> + + <p class="v2"> + <b><a name="Block_1118" id="Block_1118">1118</a>.</b> + Geometry seems to stand for all that is practical, + poetry for all that is visionary, but in the kingdom of the + imagination you will find them close akin, and they should go + together as a precious heritage to every + youth.—<span class="smcap">Milner, Florence.</span></p> + <p class="blockcite"> + School Review, 1898, p. 114.</p> + + <p class="v2"> + <b><a name="Block_1119" id="Block_1119">1119</a>.</b> + The beautiful has its place in mathematics as + elsewhere. The prose of ordinary intercourse and of business + correspondence might be held to be the most practical use to + which language is put, but we should be poor indeed without the + literature of imagination. Mathematics too has its triumphs of + the creative imagination, its beautiful theorems, its proofs + and processes whose perfection of form has made them classic. + He must be a “practical” man who can see no poetry in + mathematics.—<span class="smcap">White, W. F.</span></p> + <p class="blockcite"> + A Scrap-book of Elementary Mathematics (Chicago, 1908), + p. 208.</p> + + <p class="v2"> + <b><a name="Block_1120" id="Block_1120">1120</a>.</b> + I venture to assert that the feelings one has when + the beautiful symbolism of the infinitesimal calculus first + gets a meaning, or when the delicate analysis of Fourier has + been mastered, or while one follows Clerk Maxwell or Thomson + into the strange world of electricity, now growing so rapidly + in form and being, or can almost feel with Stokes the + pulsations of light that gives nature to our eyes, or track + with Clausius the courses of molecules we can measure, even if + we know with certainty that we can never see + them—I venture to assert that these feelings + are altogether comparable to those aroused in us by an + exquisite poem or a lofty + thought.—<span class="smcap">Workman, W. P.</span></p> + <p class="blockcite"> + F. Spencer: Aim and Practice of Teaching (New York, 1897), + p. 194.</p> + + <p class="v2"> + <b><a name="Block_1121" id="Block_1121">1121</a>.</b> + It is an open secret to the few who know it, but a + mystery and stumbling block to the many, that Science and + Poetry are own sisters; insomuch that in those branches of + scientific inquiry which are most abstract, most formal, and + most remote from the grasp of the ordinary sensible + imagination, a higher power of imagination akin to the creative + insight of the poet is most needed and most fruitful of lasting + work.—<span class="smcap">Pollock, F.</span></p> + <p class="blockcite"> + Clifford’s Lectures and Essays (New + York, 1901), Vol. 1, Introduction, p. 1.</p> + +<p><span class="pagenum"> + <a name="Page_189" + id="Page_189">189</a></span></p> + + <p class="v2"> + <b><a name="Block_1122" id="Block_1122">1122</a>.</b> + It is as great a mistake to maintain that a high + development of the imagination is not essential to progress in + mathematical studies as to hold with Ruskin and others that + science and poetry are antagonistic + pursuits.—<span class="smcap">Hoffman, F. S.</span></p> + <p class="blockcite"> + Sphere of Science (London, 1898), p. 107.</p> + + <p class="v2"> + <b><a name="Block_1123" id="Block_1123">1123</a>.</b> + We have heard much about the poetry of + mathematics, but very little of it has as yet been sung. The + ancients had a juster notion of their poetic value than we. The + most distinct and beautiful statements of any truth must take + at last the mathematical form. We might so simplify the rules + of moral philosophy, as well as of arithmetic, that one formula + would express them both.—<span class= + "smcap">Thoreau, H. D.</span></p> + <p class="blockcite"> + A Week on the Concord and Merrimac Rivers (Boston, 1893), + p. 477.</p> + + <p class="v2"> + <b><a name="Block_1124" id="Block_1124">1124</a>.</b> + We do not listen with the best regard to the + verses of a man who is only a poet, nor to his problems if he + is only an algebraist; but if a man is at once acquainted with + the geometric foundation of things and with their festal + splendor, his poetry is exact and his arithmetic + musical.—<span class="smcap">Emerson, R. W.</span></p> + <p class="blockcite"> + Society and Solitude, Chap. 7, Works and Days.</p> + + <p class="v2"> + <b><a name="Block_1125" id="Block_1125">1125</a>.</b> + Mathesis and Poetry are ... the utterance of the + same power of imagination, only that in the one case it is + addressed to the head, and in the other, to the + heart.—<span class="smcap">Hill, + Thomas.</span></p> + <p class="blockcite"> + North American Review, Vol. 85, p. 230.</p> + + <p class="v2"> + <b><a name="Block_1126" id="Block_1126">1126</a>.</b> + The Mathematics are usually considered as being + the very antipodes of Poesy. Yet Mathesis and Poesy are of the + closest kindred, for they are both works of the imagination. + Poesy is a creation, a making, a fiction; and the Mathematics + have been called, by an admirer of them, the sublimest and most + stupendous of fictions. It is true, they are not only μάθησις, + learning, but ποίησις, a + creation.—<span class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + North American Review, Vol. 85, p. 229.</p> + +<p><span class="pagenum"> + <a name="Page_190" + id="Page_190">190</a></span></p> + + <p class="v2"> + <b><a name="Block_1127" id="Block_1127">1127</a>.</b></p> + <div class="poem"> + <p class="i0"> + Music and poesy used to quicken you:</p> + <p class="i0"> + The mathematics, and the metaphysics,</p> + <p class="i0"> + Fall to them as you find your stomach serves you.</p> + <p class="i0"> + No profit grows, where is no pleasure ta’en:—</p> + <p class="i0"> + In brief, sir, study what you most affect.</p> + </div> + <p class="block40"> + —<span class="smcap">Shakespeare.<br /></span></p> + <p class="blockcite"> + Taming of the Shrew, Act 1, Scene 1.</p> + + <p class="v2"> + <b><a name="Block_1128" id="Block_1128">1128</a>.</b> + Music has much resemblance to + algebra.—<span class="smcap">Novalis.</span></p> + <p class="blockcite"> + Schriften, Teil 2 (Berlin, 1901), p. 549.</p> + + <p class="v2"> + <b><a name="Block_1129" id="Block_1129">1129</a>.</b></p> + <div class="poem"> + <p class="i0"> + I do present you with a man of mine,</p> + <p class="i0"> + Cunning in music and in mathematics,</p> + <p class="i0"> + To instruct her fully in those sciences,</p> + <p class="i0"> + Whereof, I know, she is not ignorant.</p> + </div> + <p class="block40"> + —<span class="smcap">Shakespeare.</span></p> + <p class="blockcite"> + Taming of the Shrew, Act 2, Scene 1.</p> + + <p class="v2"> + <b><a name="Block_1130" id="Block_1130">1130</a>.</b> + Saturated with that speculative spirit then + pervading the Greek mind, he [Pythagoras] endeavoured to + discover some principle of homogeneity in the universe. Before + him, the philosophers of the Ionic school had sought it in the + matter of things; Pythagoras looked for it in the structure of + things. He observed the various numerical relations or + analogies between numbers and the phenomena of the universe. + Being convinced that it was in numbers and their relations that + he was to find the foundation to true philosophy, he proceeded + to trace the origin of all things to numbers. Thus he observed + that musical strings of equal lengths stretched by weights + having the proportion of ½, ⅔, ¾, produced intervals + which were an octave, a fifth and a fourth. Harmony, therefore, + depends on musical proportion; it is nothing but a mysterious + numerical relation. Where harmony is, there are numbers. Hence + the order and beauty of the universe have their origin in + numbers. There are seven intervals in the musical scale, and + also seven planets crossing the heavens. The same numerical + relations which underlie the former must underlie the latter. + But where number is, there is harmony. Hence his spiritual ear + discerned in the planetary motions a wonderful “Harmony of + spheres”—<span class= "smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Mathematics (New York, 1897), p. 67.</p> + +<p><span class="pagenum"> + <a name="Page_191" + id="Page_191">191</a></span></p> + + <p class="v2"> + <b><a name="Block_1131" id="Block_1131">1131</a>.</b> + May not Music be described as the Mathematic of + sense, Mathematic as Music of the reason? the soul of each the + same! Thus the musician <em>feels</em> Mathematic, the + mathematician <em>thinks</em> Music,—Music the + dream, Mathematic the working life—each to + receive its consummation from the other when the human + intelligence, elevated to its perfect type, shall shine forth + glorified in some future Mozart-Dirichlet or + Beethoven-Gauss—a union already not + indistinctly foreshadowed in the genius and labours of a + Helmholtz!—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + On Newton’s Rule for the Discovery of Imaginary Roots; + Collected Mathematical Papers, Vol. 2, p. 419.</p> + + <p class="v2"> + <b><a name="Block_1132" id="Block_1132">1132</a>.</b> + Just as the musician is able to form an acoustic + image of a composition which he has never heard played by + merely looking at its score, so the equation of a curve, which + he has never seen, furnishes the mathematician with a complete + picture of its course. Yea, even more: as the score frequently + reveals to the musician niceties which would escape his ear + because of the complication and rapid change of the auditory + impressions, so the insight which the mathematician gains from + the equation of a curve is much deeper than that which is + brought about by a mere inspection of the + curve.—<span class="smcap">Pringsheim, A.</span></p> + <p class="blockcite"> + Jahresbericht der Deutschen Mathematiker + + <a id="TNanchor_10"></a> + <a class="msg" href="#TN_10" + title="originally spelled ‘Vereiningung’">Vereinigung</a>. + + Bd. 13, p. 364.</p> + + <p class="v2"> + <b><a name="Block_1133" id="Block_1133">1133</a>.</b> + Mathematics and music, the most sharply contrasted + fields of scientific activity which can be found, and yet + related, supporting each other, as if to show forth the secret + connection which ties together all the activities of our mind, + and which leads us to surmise that the manifestations of the + artist’s genius are but the unconscious expressions of a + mysteriously acting + rationality.—<span class="smcap">Helmholtz, H.</span></p> + <p class="blockcite"> + Vorträge und Reden, Bd. 1 (Braunschweig, 1884), p. 82.</p> + + <p class="v2"> + <b><a name="Block_1134" id="Block_1134">1134</a>.</b> + Among all highly civilized peoples the golden age + of art has always been closely coincident with the golden age + of the pure sciences, particularly with mathematics, the most + ancient among them.</p> + +<p><span class="pagenum"> + <a name="Page_192" + id="Page_192">192</a></span></p> + + <p class="v1"> + This coincidence must not be looked upon as accidental, but as + natural, due to an inner necessity. Just as art can thrive only + when the artist, relieved of the anxieties of existence, can + listen to the inspirations of his spirit and follow in their + lead, so mathematics, the most ideal of the sciences, will + yield its choicest blossoms only when life’s + dismal phantom dissolves and fades away, when the striving + after naked truth alone predominates, conditions which prevail + only in nations while in the prime of their + development.—<span class="smcap">Lampe, E.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik etc. (Berlin, 1893), p. 4.</p> + + <p class="v2"> + <b><a name="Block_1135" id="Block_1135">1135</a>.</b> + Till the fifteenth century little progress appears + to have been made in the science or practice of music; but + since that era it has advanced with marvelous rapidity, its + progress being curiously parallel with that of mathematics, + inasmuch as great musical geniuses appeared suddenly among + different nations, equal in their possession of this special + faculty to any that have since arisen. As with the mathematical + so with the musical faculty—it is impossible + to trace any connection between its possession and survival in + the struggle for existence.—<span class= + "smcap">Wallace, A. R.</span></p> + <p class="blockcite"> + Darwinism, Chap. 15.</p> + + <p class="v2"> + <b><a name="Block_1136" id="Block_1136">1136</a>.</b> + In my opinion, there is absolutely no trustworthy + proof that talents have been improved by their exercise through + the course of a long series of generations. The Bach family + shows that musical talent, and the Bernoulli family that + mathematical power, can be transmitted from generation to + generation, but this teaches us nothing as to the origin of + such talents. In both families the high-watermark of talent + lies, not at the end of the series of generations, as it should + do if the results of practice are transmitted, but in the + middle. Again, talents frequently appear in some member of a + family which has not been previously distinguished.</p> + <p class="v1"> + Gauss was not the son of a mathematician; + Handel’s father was a surgeon, of whose + musical powers nothing is known; Titian was the son and also + the nephew of a lawyer, while he and his brother, Francesco + Vecellio, were the first painters in a + +<span class="pagenum"> + <a name="Page_193" + id="Page_193">193</a></span> + + family which produced a succession of seven other artists with + diminishing talents. These facts do not, however, prove that + the condition of the nerve-tracts and centres of the brain, + which determine the specific talent, appeared for the first + time in these men: the appropriate condition surely existed + previously in their parents, although it did not achieve + expression. They prove, as it seems to me, that a high degree + of endowment in a special direction, which we call talent, + cannot have arisen from the experience of previous generations, + that is, by the exercise of the brain in the same specific + direction.—<span class="smcap">Weismann, August.</span></p> + <p class="blockcite"> + Essays upon Heredity [A. E. Shipley], (Oxford, 1891), Vol. + 1, p. 97.</p> + +<p><span class="pagenum"> + <a name="Page_194" + id="Page_194">194</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_XII"> + CHAPTER XII<br /> + <span class="large"> + MATHEMATICS AS A LANGUAGE</span></h2> + + <p class="v2"> + <b><a name="Block_1201" id="Block_1201">1201</a>.</b> + The new mathematics is a sort of supplement to + language, affording a means of thought about form and quantity + and a means of expression, more exact, compact, and ready than + ordinary language. The great body of physical science, a great + deal of the essential facts of financial science, and endless + social and political problems are only accessible and only + thinkable to those who have had a sound training in + mathematical analysis, and the time may not be very remote when + it will be understood that for complete initiation as an + efficient citizen of one of the new great complex world wide + states that are now developing, it is as necessary to be able + to compute, to think in averages and maxima and minima, as it + is now to be able to read and to + write.—<span class="smcap">Wells, H. G.</span></p> + <p class="blockcite"> + Mankind in the Making (London, 1904), pp. 191-192.</p> + + <p class="v2"> + <b><a name="Block_1202" id="Block_1202">1202</a>.</b> + Mathematical language is not only the simplest and + most easily understood of any, but the shortest + also.—<span class="smcap">Brougham, H. L.</span></p> + <p class="blockcite"> + Works (Edinburgh, 1872), Vol. 7, p. 317.</p> + + <p class="v2"> + <b><a name="Block_1203" id="Block_1203">1203</a>.</b> + Mathematics is the science of definiteness, the + necessary vocabulary of those who + know.—<span class="smcap">White, W. F.</span></p> + <p class="blockcite"> + A Scrap-book of Elementary Mathematics (Chicago, 1908), p. 7.</p> + + <p class="v2"> + <b><a name="Block_1204" id="Block_1204">1204</a>.</b> + Mathematics, too, is a language, and as concerns + its structure and content it is the most perfect language which + exists, superior to any vernacular; indeed, since it is + understood by every people, mathematics may be called the + language of languages. Through it, as it were, nature herself + speaks; through it the Creator of the world has spoken, and + through it the Preserver of the world continues to + speak.—<span class="smcap">Dillmann, C.</span></p> + <p class="blockcite"> + Die Mathematik die Fackelträgerin einer + neuen Zeit (Stuttgart, 1889), p. 5.</p> + +<p><span class="pagenum"> + <a name="Page_195" + id="Page_195">195</a></span></p> + + <p class="v2"> + <b><a name="Block_1205" id="Block_1205">1205</a>.</b> + Would it sound too presumptuous to speak of + perception as a quintessence of sensation, language (that is, + communicable thought) of perception, mathematics of language? + We should then have four terms differentiating from inorganic + matter and from each other the Vegetable, Animal, Rational, and + Super-sensual modes of + existence.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Presidential Address, British Association; Collected + Mathematical Papers, Vol. 2, p. 652.</p> + + <p class="v2"> + <b><a name="Block_1206" id="Block_1206">1206</a>.</b> + Little could Plato have imagined, when, indulging + his instinctive love of the true and beautiful for their own + sakes, he entered upon these refined speculations and revelled + in a world of his own creation, that he was writing the grammar + of the language in which it would be demonstrated in after ages + that the pages of the universe are + written.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + A Probationary Lecture on Geometry; Collected Mathematical + Papers, Vol. 2, p. 7.</p> + + <p class="v2"> + <b><a name="Block_1207" id="Block_1207">1207</a>.</b> + It is the symbolic language of mathematics only + which has yet proved sufficiently accurate and comprehensive to + demand familiarity with this conception of an inverse + process.—<span class="smcap">Venn, John.</span></p> + <p class="blockcite"> + Symbolic Logic (London and New York, 1894), p. 74.</p> + + <p class="v2"> + <b><a name="Block_1208" id="Block_1208">1208</a>.</b> + Without this language [mathematics] most of the + intimate analogies of things would have remained forever + unknown to us; and we should forever have been ignorant of the + internal harmony of the world, which is the only true objective + reality....</p> + <p class="v1"> + This harmony ... is the sole objective reality, the only truth + we can attain; and when I add that the universal harmony of the + world is the source of all beauty, it will be understood what + price we should attach to the slow and difficult progress which + little by little enables us to know it + better.—<span class="smcap">Poincaré, H.</span></p> + <p class="blockcite"> + The Value of Science [Halsted] Popular Science Monthly, + 1906, pp. 195-196.</p> + + <p class="v2"> + <b><a name="Block_1209" id="Block_1209">1209</a>.</b> + The most striking characteristic of the written language + of algebra and of the higher forms of the calculus is the + +<span class="pagenum"> + <a name="Page_196" + id="Page_196">196</a></span> + + sharpness of definition, by which we + are enabled to reason upon the symbols by the mere laws of + verbal logic, discharging our minds entirely of the meaning of + the symbols, until we have reached a stage of the process where + we desire to interpret our results. The ability to attend to + the symbols, and to perform the verbal, visible changes in the + position of them permitted by the logical rules of the science, + without allowing the mind to be perplexed with the meaning of + the symbols until the result is reached which you wish to + interpret, is a fundamental part of what is called analytical + power. Many students find themselves perplexed by a perpetual + attempt to interpret not only the result, but each step of the + process. They thus lose much of the benefit of the labor-saving + machinery of the calculus and are, indeed, frequently + incapacitated for using it.—<span class= + "smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + Uses of Mathesis; Bibliotheca Sacra, Vol. 32, p. 505.</p> + + <p class="v2"> + <b><a name="Block_1210" id="Block_1210">1210</a>.</b> + The prominent reason why a mathematician can be + judged by none but mathematicians, is that he uses a peculiar + language. The language of mathesis is special and + untranslatable. In its simplest forms it can be translated, as, + for instance, we say a right angle to mean a square corner. But + you go a little higher in the science of mathematics, and it is + impossible to dispense with a peculiar language. It would defy + all the power of Mercury himself to explain to a person + ignorant of the science what is meant by the single phrase + “functional exponent.” How much + more impossible, if we may say so, would it be to explain a + whole treatise like Hamilton’s Quaternions, + in such a wise as to make it possible to judge of its value! + But to one who has learned this language, it is the most + precise and clear of all modes of expression. It discloses the + thought exactly as conceived by the writer, with more or less + beauty of form, but never with obscurity. It may be prolix, as + it often is among French writers; may delight in mere verbal + metamorphoses, as in the Cambridge University of England; or + adopt the briefest and clearest forms, as under the pens of the + geometers of our Cambridge; but it always reveals to us + precisely the writer’s + thought.—<span class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + North American Review, Vol. 85, pp. 224-225.</p> + +<p><span class="pagenum"> + <a name="Page_197" + id="Page_197">197</a></span></p> + + <p class="v2"> + <b><a name="Block_1211" id="Block_1211">1211</a>.</b> + The domain, over which the language of analysis + extends its sway, is, indeed, relatively limited, but within + this domain it so infinitely excels ordinary language that its + attempt to follow the former must be given up after a few + steps. The mathematician, who knows how to think in this + marvelously condensed language, is as different from the + mechanical computer as heaven from + earth.—<span class="smcap">Pringsheim, A.</span></p> + <p class="blockcite"> + Jahresberichte der Deutschen Mathematiker Vereinigung, Bd. + 13, p. 367.</p> + + <p class="v2"> + <b><a name="Block_1212" id="Block_1212">1212</a>.</b> + The results of systematic symbolical reasoning + must <em>always</em> express general truths, by their nature; and + do not, for their justification, require each of the steps of + the process to represent some definite operation upon quantity. + The <em>absolute universality of the interpretation of + symbols</em> is the fundamental principle of their + use.—<span class="smcap">Whewell, William.</span></p> + <p class="blockcite"> + The Philosophy of the Inductive Sciences, Part I, Bk. 2, + chap. 12, sect. 2 (London, 1858).</p> + + <p class="v2"> + <b><a name="Block_1213" id="Block_1213">1213</a>.</b> + Anyone who understands algebraic notation, reads + at a glance in an equation results reached arithmetically only + with great labour and pains.—<span class= + "smcap">Cournot, A.</span></p> + <p class="blockcite"> + Theory of Wealth [N. T. Bacon], (New York, 1897), p. 4.</p> + + <p class="v2"> + <b><a name="Block_1214" id="Block_1214">1214</a>.</b> + As arithmetic and algebra are sciences of great + clearness, certainty, and extent, which are immediately + conversant about signs, upon the skilful use whereof they + entirely depend, so a little attention to them may possibly + help us to judge of the progress of the mind in other sciences, + which, though differing in nature, design, and object, may yet + agree in the general methods of proof and + inquiry.—<span class="smcap">Berkeley, George.</span></p> + <p class="blockcite"> + Alciphron, or the Minute Philosopher, Dialogue 7, sect. 12.</p> + + <p class="v2"> + <b><a name="Block_1215" id="Block_1215">1215</a>.</b> + In general the position as regards all such new + calculi is this—That one cannot accomplish + by them anything that could not be accomplished without them. + However, the advantage is, that, provided such a calculus + corresponds to the + +<span class="pagenum"> + <a name="Page_198" + id="Page_198">198</a></span> + + inmost nature of frequent needs, + anyone who masters it thoroughly is + able—without the unconscious inspiration of + genius which no one can command—to solve the + respective problems, yea, to solve them mechanically in + complicated cases in which, without such aid, even genius + becomes powerless. Such is the case with the invention of + general algebra, with the differential calculus, and in a more + limited region with Lagrange’s calculus of variations, with my + calculus of congruences, and with Möbius’s calculus. Such + conceptions unite, as it were, into an organic whole countless + problems which otherwise would remain isolated and require for + their separate solution more or less application of inventive + genius.—<span class="smcap">Gauss, C. J.</span></p> + <p class="blockcite"> + Werke, Bd. 8, p. 298.</p> + + <p class="v2"> + <b><a name="Block_1216" id="Block_1216">1216</a>.</b> + The invention of what we may call primary or + fundamental notation has been but little indebted to analogy, + evidently owing to the small extent of ideas in which + comparison can be made useful. But at the same time analogy + should be attended to, even if for no other reason than that, + by making the invention of notation an art, the exertion of + individual caprice ceases to be allowable. Nothing is more easy + than the invention of notation, and nothing of worse example + and consequence than the confusion of mathematical expressions + by unknown symbols. If new notation be advisable, permanently + or temporarily, it should carry with it some mark of + distinction from that which is already in use, unless it be a + demonstrable extension of the + latter.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Calculus of Functions; Encyclopedia Metropolitana, + Addition to Article 26.</p> + + <p class="v2"> + <b><a name="Block_1217" id="Block_1217">1217</a>.</b> + Before the introduction of the Arabic notation, + multiplication was difficult, and the division even of integers + called into play the highest mathematical faculties. Probably + nothing in the modern world could have more astonished a Greek + mathematician than to learn that, under the influence of + compulsory education, the whole population of Western Europe, + from the highest to the lowest, could perform the operation of + division for the largest numbers. This fact would have seemed + to him a sheer impossibility.... Our modern power of easy + reckoning + +<span class="pagenum"> + <a name="Page_199" + id="Page_199">199</a></span> + + with decimal fractions is the most miraculous result of a + perfect notation.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Introduction to Mathematics (New York, 1911), p. 59.</p> + + <p class="v2"> + <b><a name="Block_1218" id="Block_1218">1218</a>.</b> + Mathematics is often considered a difficult and + mysterious science, because of the numerous symbols which it + employs. Of course, nothing is more incomprehensible than a + symbolism which we do not understand. Also a symbolism, which + we only partially understand and are unaccustomed to use, is + difficult to follow. In exactly the same way the technical + terms of any profession or trade are incomprehensible to those + who have never been trained to use them. But this is not + because they are difficult in themselves. On the contrary they + have invariably been introduced to make things easy. So in + mathematics, granted that we are giving any serious attention + to mathematical ideas, the symbolism is invariably an immense + simplification.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Introduction to Mathematics (New York, 1911), pp. 59-60.</p> + + <p class="v2"> + <b><a name="Block_1219" id="Block_1219">1219</a>.</b> + Symbolism is useful because it makes things + difficult. Now in the beginning everything is self-evident, and + it is hard to see whether one self-evident proposition follows + from another or not. Obviousness is always the enemy to + correctness. Hence we must invent a new and difficult symbolism + in which nothing is obvious.... Thus the whole of Arithmetic + and Algebra has been shown to require three indefinable notions + and five indemonstrable + propositions.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + International Monthly, 1901, p. 85.</p> + + <p class="v2"> + <b><a name="Block_1220" id="Block_1220">1220</a>.</b> + The employment of mathematical symbols is + perfectly natural when the relations between magnitudes are + under discussion; and even if they are not rigorously + necessary, it would hardly be reasonable to reject them, + because they are not equally familiar to all readers and + because they have sometimes been wrongly used, if they are able + to facilitate the exposition of problems, to render it more + concise, to open the way to more extended developments, and to + avoid the digressions of vague + argumentation.—<span class="smcap">Cournot, A.</span></p> + <p class="blockcite"> + Theory of Wealth [N. T. Bacon], (New York, 1897), pp. 3-4.</p> + +<p><span class="pagenum"> + <a name="Page_200" + id="Page_200">200</a></span></p> + + <p class="v2"> + <b><a name="Block_1221" id="Block_1221">1221</a>.</b> + An all-inclusive geometrical symbolism, such as Hamilton and + Grassmann conceived of, is + impossible.—<span class="smcap">Burkhardt, H.</span></p> + <p class="blockcite"> + Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. + 5, p. 52.</p> + + <p class="v2"> + <b><a name="Block_1222" id="Block_1222">1222</a>.</b> + The language of analysis, most perfect of all, + being in itself a powerful instrument of discoveries, its + notations, especially when they are necessary and happily + conceived, are so many germs of new + calculi.—<span class="smcap">Laplace.</span></p> + <p class="blockcite"> + Oeuvres, t. 7 (Paris, 1896), p. xl.</p> + +<p><span class="pagenum"> + <a name="Page_201" id="Page_201">201</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_XIII"> + CHAPTER XIII<br /> + <span class="large"> + MATHEMATICS AND LOGIC</span></h2> + + <p class="v2"> + <b><a name="Block_1301" id="Block_1301">1301</a>.</b> + Mathematics belongs to every inquiry, moral as + well as physical. Even the rules of logic, by which it is + rigidly bound, could not be deduced without its aid. The laws + of argument admit of simple statement, but they must be + curiously transposed before they can be applied to the living + speech and verified by observation. In its pure and simple form + the syllogism cannot be directly compared with all experience, + or it would not have required an Aristotle to discover it. It + must be transmuted into all the possible shapes in which + reasoning loves to clothe itself. The transmutation is the + mathematical process in the establishment of the + law.—<span class="smcap">Peirce, Benjamin.</span></p> + <p class="blockcite"> + Linear Associative Algebra; American Journal of + Mathematics, Vol. 4 (1881), p. 97.</p> + + <p class="v2"> + <b><a name="Block_1302" id="Block_1302">1302</a>.</b> + In mathematics we see the conscious logical + activity of our mind in its purest and most perfect form; here + is made manifest to us all the labor and the great care with + which it progresses, the precision which is necessary to + determine exactly the source of the established general + theorems, and the difficulty with which we form and comprehend + abstract conceptions; but we also learn here to have confidence + in the certainty, breadth, and fruitfulness of such + intellectual labor.—<span class="smcap">Helmholtz, H.</span></p> + <p class="blockcite"> + Vorträge und Reden, Bd. 1 (Braunschweig, 1896), p. 176.</p> + + <p class="v2"> + <b><a name="Block_1303" id="Block_1303">1303</a>.</b> + Mathematical demonstrations are a logic of as much + or more use, than that commonly learned at schools, serving to + a just formation of the mind, enlarging its capacity, and + strengthening it so as to render the same capable of exact + reasoning, and discerning truth from falsehood in all + occurrences, even in subjects not mathematical. For which + reason it is said, the Egyptians, Persians, and Lacedaemonians + seldom elected any new kings, but such as had some + knowledge in the mathematics, + +<span class="pagenum"> + <a name="Page_202" + id="Page_202">202</a></span> + + imagining those, who had not, men + of imperfect judgments, and unfit to rule and + govern.—<span class="smcap">Franklin, Benjamin.</span></p> + <p class="blockcite"> + Usefulness of Mathematics; Works (Boston, 1840), Vol. 2, + p. 68.</p> + + <p class="v2"> + <b><a name="Block_1304" id="Block_1304">1304</a>.</b> + The mathematical conception is, from its very + nature, abstract; indeed its abstractness is usually of a + higher order than the abstractness of the + logician.—<span class="smcap">Chrystal, George.</span></p> + <p class="blockcite"> + Encyclopedia Britannica (Ninth Edition), Article + “Mathematics”</p> + + <p class="v2"> + <b><a name="Block_1305" id="Block_1305">1305</a>.</b> + Mathematics, that giant pincers of scientific + logic....—<span class="smcap">Halsted, G. B.</span></p> + <p class="blockcite"> + Science (1905), p. 161.</p> + + <p class="v2"> + <b><a name="Block_1306" id="Block_1306">1306</a>.</b> + Logic has borrowed the rules of geometry without + understanding its power.... I am far from placing logicians by + the side of geometers who teach the true way to guide the + reason.... The method of avoiding error is sought by every one. + The logicians profess to lead the way, the geometers alone + reach it, and aside from their science there is no true + demonstration.—<span class="smcap">Pascal.</span></p> + <p class="blockcite"> + Quoted by A. Rebière: Mathématiques et Mathématiciens + (Paris, 1898), pp. 162-163.</p> + + <p class="v2"> + <b><a name="Block_1307" id="Block_1307">1307</a>.</b> + Mathematics, like dialectics, is an organ of the + higher sense, in its execution it is an art like eloquence. To + both nothing but the form is of value; neither cares anything + for content. Whether mathematics considers pennies or guineas, + whether rhetoric defends truth or error, is perfectly + immaterial to either.—<span class= "smcap">Goethe.</span></p> + <p class="blockcite"> + Sprüche in Prosa, Natur IV, 946.</p> + + <p class="v2"> + <b><a name="Block_1308" id="Block_1308">1308</a>.</b> + Confined to its true domain, mathematical + reasoning is admirably adapted to perform the universal office + of sound logic: to induce in order to deduce, in order to + construct.... It contents itself to furnish, in the most + favorable domain, a model of clearness, of precision, and + consistency, the close contemplation of which is alone able to + prepare the mind to render other conceptions also as perfect as + their nature permits. Its general reaction, more negative than + positive, must + +<span class="pagenum"> + <a name="Page_203" + id="Page_203">203</a></span> + + consist, above all, in inspiring us + everywhere with an invincible aversion for vagueness, + inconsistency, and obscurity, which may always be really + avoided in any reasoning whatsoever, if we make sufficient + effort.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Subjective Synthesis.</p> + + <p class="v2"> + <b><a name="Block_1309" id="Block_1309">1309</a>.</b> + Formal thought, consciously recognized as such, is + the means of all exact knowledge; and a correct understanding + of the main formal sciences, Logic and Mathematics, is the + proper and only safe foundation for a scientific + education.—<span class="smcap">Lefevre, Arthur.</span></p> + <p class="blockcite"> + Number and its Algebra (Boston, Sect. 222.)</p> + + <p class="v2"> + <b><a name="Block_1310" id="Block_1310">1310</a>.</b> + It has come to pass, I know not how, that + Mathematics and Logic, which ought to be but the handmaids of + Physic, nevertheless presume on the strength of the certainty + which they possess to exercise dominion over + it.—<span class="smcap">Bacon, Francis.</span></p> + <p class="blockcite"> + De Augmentis, Bk. 3.</p> + + <p class="v2"> + <b><a name="Block_1311" id="Block_1311">1311</a>.</b> + We may regard geometry as a practical logic, for + the truths which it considers, being the most simple and most + sensible of all, are, for this reason, the most susceptible to + easy and ready application of the rules of + reasoning.—<span class="smcap">D’Alembert.</span></p> + <p class="blockcite"> + Quoted in A. Rebière: Mathématiques et Mathématiciens + (Paris, 1898), pp. 151-152.</p> + + <p class="v2"> + <b><a name="Block_1312" id="Block_1312">1312</a>.</b> + There are notable examples enough of demonstration + outside of mathematics, and it may be said that Aristotle has + already given some in his “Prior + Analytics.” In fact logic is as susceptible of + demonstration as geometry,.... Archimedes is the first, whose + works we have, who has practised the art of demonstration upon + an occasion where he is treating of physics, as he has done in + his book on Equilibrium. Furthermore, jurists may be said to + have many good demonstrations; especially the ancient Roman + jurists, whose fragments have been preserved to us in the + Pandects.—<span class="smcap">Leibnitz, G. W.</span></p> + <p class="blockcite"> + New Essay on Human Understanding [Langley], Bk. 4, chap. + 2, sect. 12.</p> + +<p><span class="pagenum"> + <a name="Page_204" + id="Page_204">204</a></span></p> + + <p class="v2"> + <b><a name="Block_1313" id="Block_1313">1313</a>.</b> + It is commonly considered that mathematics owes + its certainty to its reliance on the immutable principles of + formal logic. This ... is only half the truth imperfectly + expressed. The other half would be that the principles of + formal logic owe such a degree of permanence as they have + largely to the fact that they have been tempered by long and + varied use by mathematicians. “A vicious + circle!” you will perhaps say. I should rather + describe it as an example of the process known by + mathematicians as the method of successive + approximation.—<span class="smcap">Bôcher, Maxime.</span></p> + <p class="blockcite"> + Bulletin of the American Mathematical Society, Vol. 11, + p. 120.</p> + + <p class="v2"> + <b><a name="Block_1314" id="Block_1314">1314</a>.</b> + Whatever advantage can be attributed to logic in + directing and strengthening the action of the understanding is + found in a higher degree in mathematical study, with the + immense added advantage of a determinate subject, distinctly + circumscribed, admitting of the utmost precision, and free from + the danger which is inherent in all abstract + logic,—of leading to useless and puerile + rules, or to vain ontological speculations. The positive + method, being everywhere identical, is as much at home in the + art of reasoning as anywhere else: and this is why no science, + whether biology or any other, can offer any kind of reasoning, + of which mathematics does not supply a simpler and purer + counterpart. Thus, we are enabled to eliminate the only + remaining portion of the old philosophy which could even appear + to offer any real utility; the logical part, the value of which + is irrevocably absorbed by mathematical + science.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], (London, 1875), Vol. 1, + pp. 321-322.</p> + + <p class="v2"> + <b><a name="Block_1315" id="Block_1315">1315</a>.</b> + We know that mathematicians care no more for logic + than logicians for mathematics. The two eyes of exact science + are mathematics and logic: the mathematical sect puts out the + logical eye, the logical sect puts out the mathematical eye; + each believing that it can see better with one eye than with + two.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Quoted in F. Cajori: History of Mathematics (New York, + 1897), p. 316.</p> + +<p><span class="pagenum"> + <a name="Page_205" + id="Page_205">205</a></span></p> + + <p class="v2"> + <b><a name="Block_1316" id="Block_1316">1316</a>.</b> + The progress of the art of rational discovery + depends in a great part upon the art of characteristic (ars + characteristica). The reason why people usually seek + demonstrations only in numbers and lines and things represented + by these is none other than that there are not, outside of + numbers, convenient characters corresponding to the + notions.—<span class="smcap">Leibnitz, G. W.</span></p> + <p class="blockcite"> + Philosophische Schriften [Gerhardt] Bd. 8, p. 198.</p> + + <p class="v2"> + <b><a name="Block_1317" id="Block_1317">1317</a>.</b> + The influence of the mathematics of Leibnitz upon + his philosophy appears chiefly in connection with his law of + continuity and his prolonged efforts to establish a Logical + Calculus.... To find a Logical Calculus (implying a universal + philosophical language or system of signs) is an attempt to + apply in theological and philosophical investigations an + analytic method analogous to that which had proved so + successful in Geometry and Physics. It seemed to Leibnitz that + if all the complex and apparently disconnected ideas which make + up our knowledge could be analysed into their simple elements, + and if these elements could each be represented by a definite + sign, we should have a kind of “alphabet of + human thoughts.” By the combination of these signs + (letters of the alphabet of thought) a system of true knowledge + would be built up, in which reality would be more and more + adequately represented or symbolized.... In many cases the + analysis may result in an infinite series of elements; but the + principles of the Infinitesimal Calculus in mathematics have + shown that this does not necessarily render calculation + impossible or inaccurate. Thus it seemed to Leibnitz that a + synthetic calculus, based upon a thorough analysis, would be + the most effective instrument of knowledge that could be + devised. “I feel,” he says, “that controversies can never be + finished, nor silence imposed upon the Sects, unless we give up + complicated reasonings in favor of simple <em>calculations</em>, + words of vague and uncertain meaning in favor of fixed + symbols.” Thus it will appear that “every paralogism is nothing + but <em>an error of calculation</em>.” “When + controversies arise, there will be no more necessity of + disputation between two philosophers than between two + accountants. Nothing will be needed but that they should take + pen in hand, sit down with + +<span class="pagenum"> + <a name="Page_206" + id="Page_206">206</a></span> + + their counting-tables, and (having summoned a friend, if they + like) say to one another: <em>Let us + calculate</em>”—<span class= "smcap">Latta, Robert.</span></p> + <p class="blockcite"> + Leibnitz, The Monadology, etc. (Oxford, 1898), p. 85.</p> + + <p class="v2"> + <b><a name="Block_1318" id="Block_1318">1318</a>.</b> + Pure mathematics was discovered by Boole in a work which he + called “The Laws of Thought“.... His work was concerned with + formal logic, and this is the same thing as + mathematics.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + International Monthly, 1901, p. 83.</p> + + <p class="v2"> + <b><a name="Block_1319" id="Block_1319">1319</a>.</b> + Mathematics is but the higher development of + Symbolic Logic.—<span class="smcap">Whetham, W. C. D.</span></p> + <p class="blockcite"> + Recent Development of Physical Science (Philadelphia, + 1904), p. 34.</p> + + <p class="v2"> + <b><a name="Block_1320" id="Block_1320">1320</a>.</b> + Symbolic Logic has been disowned by many logicians + on the plea that its interest is mathematical, and by many + mathematicians on the plea that its interest is + logical.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Universal Algebra (Cambridge, 1898), Preface, p. 6.</p> + + <p class="v2"> + <b><a name="Block_1321" id="Block_1321">1321</a>.</b> + ... the two great components of the critical + movement, though distinct in origin and following separate + paths, are found to converge at last in the thesis: Symbolic + Logic is Mathematics, Mathematics is Symbolic Logic, the twain + are one.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 19.</p> + + <p class="v2"> + <b><a name="Block_1322" id="Block_1322">1322</a>.</b> + The emancipation of logic from the yoke of + Aristotle very much resembles the emancipation of geometry from + the bondage of Euclid; and, by its subsequent growth and + diversification, logic, less abundantly perhaps but not less + certainly than geometry, has illustrated the blessings of + freedom.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Science, Vol. 35 (1912), p. 108.</p> + + <p class="v2"> + <b><a name="Block_1323" id="Block_1323">1323</a>.</b> + I would express it as my personal view, which is probably + not yet shared generally, that pure mathematics seems to + +<span class="pagenum"> + <a name="Page_207" + id="Page_207">207</a></span> + + me merely a <em>branch of general + logic</em>; that branch which is based on the concept of + <em>numbers</em>, to whose economic advantages is to be + attributed the tremendous development which this particular + branch has undergone as compared with the remaining branches of + logic, which until the most recent times have remained almost + stationary.—<span class="smcap">Schröder, E.</span></p> + <p class="blockcite"> + Ueber Pasigraphie etc.; Verhandlungen des 1. Internationalen + Mathematiker-Kongresses (Leipzig, 1898), p. 149.</p> + + <p class="v2"> + <b><a name="Block_1324" id="Block_1324">1324</a>.</b> + If logical training is to consist, not in + repeating barbarous scholastic formulas or mechanically tacking + together empty majors and minors, but in acquiring dexterity in + the use of trustworthy methods of advancing from the known to + the unknown, then mathematical investigation must ever remain + one of its most indispensable instruments. Once inured to the + habit of accurately imagining abstract relations, recognizing + the true value of symbolic conceptions, and familiarized with a + fixed standard of proof, the mind is equipped for the + consideration of quite other objects than lines and angles. The + twin treatises of Adam Smith on social science, wherein, by + deducing all human phenomena first from the unchecked action of + selfishness and then from the unchecked action of sympathy, he + arrives at mutually-limiting conclusions of transcendent + practical importance, furnish for all time a brilliant + illustration of the value of mathematical methods and + mathematical discipline.—<span class= + "smcap">Fiske, John.</span></p> + <p class="blockcite"> + Darwinism and other Essays (Boston, 1893), pp. 297-298.</p> + + <p class="v2"> + <b><a name="Block_1325" id="Block_1325">1325</a>.</b> + No irrational exaggeration of the claims of + Mathematics can ever deprive that part of philosophy of the + property of being the natural basis of all logical education, + through its simplicity, abstractness, generality, and freedom + from disturbance by human passion. There, and there alone, we + find in full development the art of reasoning, all the + resources of which, from the most spontaneous to the most + sublime, are continually applied with far more variety and + fruitfulness than elsewhere;.... The more abstract + portion of mathematics + +<span class="pagenum"> + <a name="Page_208" + id="Page_208">208</a></span> + + may in fact be regarded as an immense + repository of logical resources, ready for use in scientific + deduction and co-ordination.—<span class= + "smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive + + <a id="TNanchor_11"></a> + <a class="msg" href="#TN_11" + title="originally spelled ‘Philosphy’">Philosophy</a> + + [Martineau], (London, 1875), Vol. 2, p. 439.</p> + + <p class="v2"> + <b><a name="Block_1326" id="Block_1326">1326</a>.</b> + Logic it is called [referring to Whitehead and + Russell’s Principia Mathematica] and logic + it is, the logic of propositions and functions and classes and + relations, by far the greatest (not merely the biggest) logic + that our planet has produced, so much that is new in matter and + in manner; but it is also mathematics, a prolegomenon to the + science, yet itself mathematics in its most genuine sense, + differing from other parts of the science only in the respects + that it surpasses these in fundamentality, generality and + precision, and lacks traditionality. Few will read it, but all + will feel its effect, for behind it is the urgence and push of + a magnificent past: two thousand five hundred years of record + and yet longer tradition of human endeavor to think + aright.—<span class="smcap">Keyser, C. + J.</span></p> + <p class="blockcite"> + Science, Vol. 35 (1912), p. 110.</p> + +<p><span class="pagenum"> + <a name="Page_209" + id="Page_209">209</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_XIV"> + CHAPTER XIV<br /> + <span class="large"> + MATHEMATICS AND PHILOSOPHY</span></h2> + + <p class="v2"> + <b><a name="Block_1401" id="Block_1401">1401</a>.</b> + Socrates is praised by all the centuries for + having called philosophy from heaven to men on earth; but if, + knowing the condition of our science, he should come again and + should look once more to heaven for a means of curing men, he + would there find that to mathematics, rather than to the + philosophy of today, had been given the crown because of its + industry and its most happy and brilliant + successes.—<span class="smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Werke [Kehrbach], (Langensalza, 1890), Bd. 5, p. 95.</p> + + <p class="v2"> + <b><a name="Block_1402" id="Block_1402">1402</a>.</b> + It is the embarrassment of metaphysics that it is + able to accomplish so little with the many things that + mathematics offers her.—<span class= "smcap">Kant, E.</span></p> + <p class="blockcite"> + Metaphysische Anfangsgründe der Naturwissenschaft, Vorrede.</p> + + <p class="v2"> + <b><a name="Block_1403" id="Block_1403">1403</a>.</b> + Philosophers, when they have possessed a thorough + knowledge of mathematics, have been among those who have + enriched the science with some of its best ideas. On the other + hand it must be said that, with hardly an exception, all the + remarks on mathematics made by those philosophers who have + possessed but a slight or hasty or late-acquired knowledge of + it are entirely worthless, being either trivial or + wrong.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Introduction to Mathematics (New York, 1911), p. 113.</p> + + <p class="v2"> + <b><a name="Block_1404" id="Block_1404">1404</a>.</b> + The union of philosophical and mathematical + productivity, which besides in Plato we find only in + Pythagoras, Descartes and Leibnitz, has always yielded the + choicest fruits to mathematics: To the first we owe scientific + mathematics in general, Plato discovered the analytic method, + by means of which mathematics was elevated above the view-point + of the elements, Descartes created the analytical geometry, our + own + +<span class="pagenum"> + <a name="Page_210" + id="Page_210">210</a></span> + + illustrious countryman discovered the + infinitesimal calculus—and just these are + the four greatest steps in the development of + mathematics.—<span class="smcap">Hankel, Hermann.</span></p> + <p class="blockcite"> + Geschichte der Mathematik im Altertum und im Mittelalter + (Leipzig, 1874), pp. 149-150.</p> + + <p class="v2"> + <b><a name="Block_1405" id="Block_1405">1405</a>.</b> + Without mathematics one cannot fathom the depths + of philosophy; without philosophy one cannot fathom the depths + of mathematics; without the two one cannot fathom + anything.—<span class="smcap">Bordas-Demoulins.</span></p> + <p class="blockcite"> + Quoted in A. Rebière: Mathématiques et Mathématiciens + (Paris, 1898), p. 147.</p> + + <p class="v2"> + <b><a name="Block_1406" id="Block_1406">1406</a>.</b> + In the end mathematics is but simple philosophy, + and philosophy, higher mathematics in + general.—<span class="smcap">Novalis.</span></p> + <p class="blockcite"> + Schriften (Berlin, 1901), Teil 2, p. 443.</p> + + <p class="v2"> + <b><a name="Block_1407" id="Block_1407">1407</a>.</b> + It is a safe rule to apply that, when a + mathematical or philosophical author writes with a misty + profundity, he is talking + nonsense.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Introduction to Mathematics (New York, 1911), p. 227.</p> + + <p class="v2"> + <b><a name="Block_1408" id="Block_1408">1408</a>.</b> + The real finisher of our education is philosophy, + but it is the office of mathematics to ward off the dangers of + philosophy.—<span class="smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Pestalozzi’s Idee eines ABC der Anschauung; Werke [Kehrbach], + (Langensalza, 1890), Bd. 1, p. 168.</p> + + <p class="v2"> + <b><a name="Block_1409" id="Block_1409">1409</a>.</b> + Since antiquity mathematics has been regarded as + the most indispensable school for philosophic thought and in + its highest spheres the research of the mathematician is indeed + most closely related to pure speculation. Mathematics is the + most perfect union between exact knowledge and theoretical + thought.—<span class="smcap">Curtius, E.</span></p> + <p class="blockcite"> + Berliner Monatsberichte (1873), p. 517.</p> + + <p class="v2"> + <b><a name="Block_1410" id="Block_1410">1410</a>.</b> + Geometry has been, throughout, of supreme + importance in the history of + knowledge.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + Foundations of Geometry (Cambridge, 1897), p. 54.</p> + +<p><span class="pagenum"> + <a name="Page_211" + id="Page_211">211</a></span></p> + + <p class="v2"> + <b><a name="Block_1411" id="Block_1411">1411</a>.</b> + He is unworthy of the name of man who is ignorant + of the fact that the diagonal of a square is incommensurable + with its side.—<span class="smcap">Plato.</span></p> + <p class="blockcite"> + Quoted by Sophie Germain: Mémoire sur les surfaces + élastiques.</p> + + <p class="v2"> + <b><a name="Block_1412" id="Block_1412">1412</a>.</b> + Mathematics, considered as a science, owes its + origin to the idealistic needs of the Greek philosophers, and + not as fable has it, to the practical demands of Egyptian + economics.... Adam was no zoölogist when he gave names to the + beasts of the field, nor were the Egyptian surveyors + mathematicians.—<span class= "smcap">Hankel, H.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten + Jahrhunderten (Tübingen, 1884), p. 7.</p> + + <p class="v2"> + <b><a name="Block_1413" id="Block_1413">1413</a>.</b> + There are only two ways open to man for attaining + a certain knowledge of truth: clear intuition and necessary + deduction.—<span class="smcap">Descartes.</span></p> + <p class="blockcite"> + Rules for the Direction of the Mind; Torrey’s The Philosophy + of Descartes (New York, 1892), p. 104.</p> + + <p class="v2"> + <b><a name="Block_1414" id="Block_1414">1414</a>.</b> + Mathematicians have, in many cases, proved some + things to be possible and others to be impossible, which, + without demonstration, would not have been believed.... + Mathematics afford many instances of impossibilities in the + nature of things, which no man would have believed, if they had + not been strictly demonstrated. Perhaps, if we were able to + reason demonstratively in other subjects, to as great extent as + in mathematics, we might find many things to be impossible, + which we conclude, without hesitation, to be + possible.—<span class="smcap">Reid, Thomas.</span></p> + <p class="blockcite"> + Essay on the Intellectual Powers of Man, Essay 4, chap. 3.</p> + + <p class="v2"> + <b><a name="Block_1415" id="Block_1415">1415</a>.</b> + If philosophers understood mathematics, they would + know that indefinite speech, which permits each one to think + what he pleases and produces a constantly increasing difference + of opinion, is utterly unable, in spite of all fine words and + even in spite of the magnitude of the objects which are under + contemplation, to maintain a balance against a + science which instructs + +<span class="pagenum"> + <a name="Page_212" + id="Page_212">212</a></span> + + and advances through every word which + it utters and which at the same time wins for itself endless + astonishment, not through its survey of immense spaces, but + through the exhibition of the most prodigious human ingenuity + which surpasses all power of + description.—<span class="smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Werke Kehrbach (Langensalza, 1890), Bd. 5, p. 105.</p> + + <p class="v2"> + <b><a name="Block_1416" id="Block_1416">1416</a>.</b> + German intellect is an excellent thing, but when a + German product is presented it must be analysed. Most probably + it is a combination of intellect (I) and tobacco-smoke (T). + Certainly I<sub>3</sub>T<sub>1</sub>, and + I<sub>2</sub>T<sub>1</sub>, occur; + but I<sub>1</sub>T<sub>3</sub> is more common, + and I<sub>2</sub>T<sub>15</sub> and I<sub>1</sub>T<sub>20</sub> + occur. In many cases metaphysics (M) occurs and I hold that + I<sub>a</sub>T<sub>b</sub>M<sub>c</sub> never occurs + without b + c > 2a.</p> + <p class="v1"> + N. B.—Be careful, in analysing the compounds + of the three, not to confound T and M, which are strongly + suspected to be isomorphic. Thus, + I<sub>1</sub>T<sub>3</sub>M<sub>3</sub> may easily be confounded + with I<sub>1</sub>T<sub>6</sub>. As far as I dare say + anything, those who have placed <em>Hegel, Fichte</em>, etc., in + the rank of the extenders of <em>Kant</em> have imagined T and M + to be identical.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Graves’ Life of W. R. Hamilton (New York, 1882-1889), Vol. 13, + p. 446.</p> + + <p class="v2"> + <b><a name="Block_1417" id="Block_1417">1417</a>.</b> + The discovery [of Ceres] was made by G. Piazzi of + Palermo; and it was the more interesting as its announcement + occurred simultaneously with a publication by Hegel in which he + severely criticized astronomers for not paying more attention + to philosophy, a science, said he, which would at once have + shown them that there could not possibly be more than seven + planets, and a study of which would therefore have prevented an + absurd waste of time in looking for what in the nature of + things could never be found.—<span class= + "smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p. 458.</p> + + <p class="v2"> + <b><a name="Block_1418" id="Block_1418">1418</a>.</b></p> + <div class="poem"> + <p class="i0"> + But who shall parcel out</p> + <p class="i0"> + His intellect by geometric rules,</p> + <p class="i0"> + Split like a province into round and square?</p> + </div> + <p class="block40"> + —<span class="smcap">Wordsworth.</span></p> + <p class="blockcite"> + The Prelude, Bk. 2.</p> + +<p><span class="pagenum"> + <a name="Page_213" + id="Page_213">213</a></span></p> + + <p class="v2"> + <b><a name="Block_1419" id="Block_1419">1419</a>.</b></p> + <div class="poem"> + <p class="i0"> + And Proposition, gentle maid,</p> + <p class="i0"> + Who soothly ask’d stern Demonstration’s aid,....</p> + </div> + <p class="block40"> + —<span class="smcap">Coleridge, S. T.</span></p> + <p class="blockcite"> + A Mathematical Problem.</p> + + <p class="v2"> + <b><a name="Block_1420" id="Block_1420">1420</a>.</b> + Mathematics connect themselves on the one side + with common life and physical science; on the other side with + philosophy in regard to our notions of space and time, and in + the questions which have arisen as to the universality and + necessity of the truths of mathematics and the foundation of + our knowledge of them.—<span class= + "smcap">Cayley, Arthur.</span></p> + <p class="blockcite"> + British Association Address (1888); Collected Mathematical + Papers, Vol. 11, p. 430.</p> + + <p class="v2"> + <b><a name="Block_1421" + id="Block_1421" + href="#TN_12" + class="msg" + title="originally shown as ‘1427’">1421</a>.</b> + + Mathematical teaching ... trains the mind to + capacities, which ... are of the closest kin to those of the + greatest metaphysician and philosopher. There is some color of + truth for the opposite doctrine in the case of elementary + algebra. The resolution of a common equation can be reduced to + almost as mechanical a process as the working of a sum in + arithmetic. The reduction of the question to an equation, + however, is no mechanical operation, but one which, according + to the degree of its difficulty, requires nearly every possible + grade of ingenuity: not to speak of the new, and in the present + state of the science insoluble, equations, which start up at + every fresh step attempted in the application of mathematics to + other branches of knowledge.—<span class= + "smcap">Mill, J. S.</span></p> + <p class="blockcite"> + An Examination of Sir William + Hamilton’s Philosophy (London, 1878), p. 615.</p> + + <p class="v2"> + <b><a name="Block_1422" id="Block_1422">1422</a>.</b> + The value of mathematical instruction as a + preparation for those more difficult investigations, consists + in the applicability not of its doctrines, but of its methods. + Mathematics will ever remain the most perfect type of the + Deductive Method in general; and the applications of + mathematics to the simpler branches of physics, furnish the + only school in which philosophers can effectually learn the + most difficult and important portion of their art, the + employment of the laws of the simpler phenomena for explaining + and predicting those of the more complex. These grounds + are quite sufficient for deeming mathematical + +<span class="pagenum"> + <a name="Page_214" + id="Page_214">214</a></span> + + training an + indispensable basis of real scientific education, and + regarding, with Plato, one who is ἀγεωμέτρητος, + as wanting in one of the most essential qualifications for the + successful cultivation of the higher branches of + philosophy.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + System of Logic, Bk. 3, chap. 24, sect. 9.</p> + + <p class="v2"> + <b><a name="Block_1423" id="Block_1423">1423</a>.</b> + In metaphysical reasoning, the process is always + short. The conclusion is but a step or two, seldom more, from + the first principles or axioms on which it is grounded, and the + different conclusions depend not one upon another.</p> + <p class="v1"> + It is otherwise in mathematical reasoning. Here the field has + no limits. One proposition leads on to another, that to a + third, and so on without end. If it should be asked, why + demonstrative reasoning has so wide a field in mathematics, + while, in other abstract subjects, it is confined within very + narrow limits, I conceive this is chiefly owing to the nature + of quantity, ... mathematical quantities being made up of parts + without number, can touch in innumerable points, and be + compared in innumerable different + ways.—<span class="smcap">Reid, Thomas.</span></p> + <p class="blockcite"> + Essays on the Powers of the Human Mind (Edinburgh, 1812), + Vol. 2, pp. 422-423.</p> + + <p class="v2"> + <b><a name="Block_1424" id="Block_1424">1424</a>.</b> + The power of Reason ... is unquestionably the most + important by far of those which are comprehended under the + general title of Intellectual. It is on the right use of this + power that our success in the pursuit of both knowledge and of + happiness depends; and it is by the exclusive possession of it + that man is distinguished, in the most essential respects, from + the lower animals. It is, indeed, from their subserviency to + its operations, that the other faculties ... derive their chief + value.—<span class="smcap">Stewart, + Dugald.</span></p> + <p class="blockcite"> + Philosophy of the Human Mind; Collected Works (Edinburgh, + 1854), Vol. 8, p. 5.</p> + + <p class="v2"> + <b><a name="Block_1425" id="Block_1425">1425</a>.</b> + When ... I asked myself why was it then that the + earliest philosophers would admit to the study of wisdom only + those who had studied mathematics, as if this science was the + easiest of all and the one most necessary for preparing and + disciplining the mind to comprehend the more advanced, I + +<span class="pagenum"> + <a name="Page_215" + id="Page_215">215</a></span> + + suspected that they had knowledge of + a mathematical science different from that of our time....</p> + <p class="v1"> + I believe I find some traces of these true mathematics in + Pappus and Diophantus, who, although they were not of extreme + antiquity, lived nevertheless in times long preceding ours. But + I willingly believe that these writers themselves, by a + culpable ruse, suppressed the knowledge of them; like some + artisans who conceal their secret, they feared, perhaps, that + the ease and simplicity of their method, if become popular, + would diminish its importance, and they preferred to make + themselves admired by leaving to us, as the product of their + art, certain barren truths deduced with subtlety, rather than + to teach us that art itself, the knowledge of which would end + our admiration.—<span class="smcap">Descartes.</span></p> + <p class="blockcite"> + Rules for the Direction of the Mind; Philosophy of + Descartes [Torrey], (New York, 1892), pp. 70-71.</p> + + <p class="v2"> + <b><a name="Block_1426" id="Block_1426">1426</a>.</b> + If we rightly adhere to our rule [that is, that we + should occupy ourselves only with those subjects in reference + to which the mind is capable of acquiring certain and + indubitable knowledge] there will remain but few things to the + study of which we can devote ourselves. There exists in the + sciences hardly a single question upon which men of + intellectual ability have not held different opinions. But + whenever two men pass contrary judgment on the same thing, it + is certain that one of the two is wrong. More than that, + neither of them has the truth; for if one of them had a clear + and precise insight into it, he could so exhibit it to his + opponent as to end the discussion by compelling his + conviction.... It follows from this, if we reckon rightly, that + among existing sciences there remain only geometry and + arithmetic, to which the observance of our rule would bring + us.—<span class="smcap">Descartes.</span></p> + <p class="blockcite"> + Rules for the Direction of the Mind; Philosophy of + Descartes [Torrey], (New York, 1892), p. 62.</p> + + <p class="v2"> + <b><a name="Block_1427" id="Block_1427">1427</a>.</b> + The same reason which led Plato to recommend the + study of arithmetic led him to recommend also the study of + geometry. The vulgar crowd of geometricians, he says, will not + +<span class="pagenum"> + <a name="Page_216" id="Page_216">216</a></span> + + understand him. They have practice + always in view. They do not know that the real use of the + science is to lead men to the knowledge of abstract, essential, + eternal truth. (Plato’s Republic, Book 7). + Indeed if we are to believe Plutarch, Plato carried his feeling + so far that he considered geometry as degraded by being applied + to any purpose of vulgar utility. Archytas, it seems, had + framed machines of extraordinary power on mathematical + principles. (Plutarch, Sympos., VIII., and Life of Marcellus. + The machines of Archytas are also mentioned by Aulus Gellius + and Diogenes Laertius). Plato remonstrated with his friend, and + declared that this was to degrade a noble intellectual exercise + into a low craft, fit only for carpenters and wheelwrights. The + office of geometry, he said, was to discipline the mind, not to + minister to the base wants of the body. His interference was + successful; and from that time according to Plutarch, the + science of mechanics was considered unworthy of the attention + of a philosopher.—<span class="smcap">Macaulay.</span></p> + <p class="blockcite"> + Lord Bacon; Edinburgh Review, July, 1837.</p> + + <p class="v2"> + <b><a name="Block_1428" id="Block_1428">1428</a>.</b> + The intellectual habits of the Mathematicians are, + in some respects, the same with those [of the Metaphysicians] + we have been now considering; but, in other respects, they + differ widely. Both are favourable to the improvement of the + power of <em>attention</em>, but not in the same manner, nor in + the same degree.</p> + <p class="v1"> + Those of the metaphysician give capacity of fixing the + attention on the subjects of our consciousness, without being + distracted by things external; but they afford little or no + exercise to that species of attention which enables us to + follow long processes of reasoning, and to keep in view all the + various steps of an investigation till we arrive at the + conclusion. In mathematics, such processes are much longer than + in any other science; and hence the study of it is peculiarly + calculated to strengthen the power of steady and concatenated + thinking,—a power which, in all the pursuits + of life, whether speculative or active, is one of the most + valuable endowments we can possess. This command of attention, + however, it may be proper to add, is to be acquired, not by the + practice of modern methods, but by the study of Greek geometry, + more particularly, by accustoming ourselves to pursue long + trains of demonstration, without + +<span class="pagenum"> + <a name="Page_217" + id="Page_217">217</a></span> + + availing ourselves of + the aid of any sensible diagrams; the thoughts being directed + solely by those ideal delineations which the powers of + conception and of memory enable us to + form.—<span class="smcap">Stewart,Dugald.</span></p> + <p class="blockcite"> + Philosophy of the Human Mind, Part 3, chap. 1, sect.3.</p> + + <p class="v2"> + <b><a name="Block_1429" id="Block_1429">1429</a>.</b> + They [the Greeks] speculated and theorized under a + lively persuasion that a Science of every part of nature was + possible, and was a fit object for the exercise of a + man’s best faculties; and they were speedily + led to the conviction that such a science must clothe its + conclusions in the language of mathematics. This conviction is + eminently conspicuous in the writings of Plato.... Probably no + succeeding step in the discovery of the Laws of Nature was of + so much importance as the full adoption of this pervading + conviction, that there must be Mathematical Laws of Nature, and + that it is the business of Philosophy to discover these Laws. + This conviction continues, through all the succeeding ages of + the history of the science, to be the animating and supporting + principle of scientific investigation and + discovery.—<span class="smcap">Whewell,W.</span></p> + <p class="blockcite"> + History of the Inductive Sciences, Vol. 1, bk. 2, chap.3.</p> + + <p class="v2"> + <b><a name="Block_1430" id="Block_1430">1430</a>.</b> + For to pass by those Ancients, the wonderful + <em>Pythagoras</em>, the sagacious <em>Democritus</em>, the divine + <em>Plato</em>, the most subtle and very learned + <em>Aristotle</em>, Men whom every Age has hitherto acknowledged + as deservedly honored, as the greatest Philosophers, the + Ring-leaders of Arts; in whose Judgments how much these Studies + [mathematics] were esteemed, is abundantly proclaimed in + History and confirmed by their famous Monuments, which are + everywhere interspersed and bespangled with Mathematical + Reasonings and Examples, as with so many Stars; and + consequently anyone not in some Degree conversant in these + Studies will in vain expect to understand, or unlock their + hidden Meanings, without the Help of a Mathematical Key: For + who can play well on <em>Aristotle’s</em> + Instrument but with a Mathematical Quill; or not be altogether + deaf to the Lessons of natural <em>Philosophy</em>, while + ignorant of <em>Geometry?</em> Who void of (<em>Geometry</em> shall + I say, or) <em>Arithmetic</em> can comprehend + <em>Plato’s</em> + +<span class="pagenum"> + <a name="Page_218" + id="Page_218">218</a></span> + + <em>Socrates</em> lisping + with Children concerning Square Numbers; or can conceive + <em>Plato</em> himself treating not only of the Universe, but the + Polity of Commonwealths regulated by the Laws of Geometry, and + formed according to a Mathematical + Plan?—<span class="smcap">Barrow,Isaac.</span></p> + <p class="blockcite"> + Mathematical Lectures (London, 1734), pp. 26-27.</p> + + <p class="v2"> + <b><a name="Block_1431" id="Block_1431">1431</a>.</b></p> + <div class="poem"> + <p class="i0"> + And Reason now through number, time, and space</p> + <p class="i0"> + Darts the keen lustre of her serious eye;</p> + <p class="i0"> + And learns from facts compar’d the laws to trace</p> + <p class="i0"> + Whose long procession leads to Deity</p> + </div> + <p class="block40"> + —<span class="smcap">Beattie, James.</span></p> + <p class="blockcite"> + The Minstrel, Bk. 2, stanza 47.</p> + + <p class="v2"> + <b><a name="Block_1432" id="Block_1432">1432</a>.</b> + That Egyptian and Chaldean wisdom mathematical + wherewith Moses and Daniel were + furnished,....—<span class="smcap">Hooker, Richard.</span></p> + <p class="blockcite"> + Ecclesiastical Polity, Bk. 3, sect. 8.</p> + + <p class="v2"> + <b><a name="Block_1433" id="Block_1433">1433</a>.</b> + General and certain truths are only founded in the + habitudes and relations of <em>abstract ideas</em>. A sagacious + and methodical application of our thoughts, for the finding out + of these relations, is the only way to discover all that can be + put with truth and certainty concerning them into general + propositions. By what steps we are to proceed in these, is to + be learned in the schools of mathematicians, who, from very + plain and easy beginnings, by gentle degrees, and a continued + chain of reasonings, proceed to the discovery and demonstration + of truths that appear at first sight beyond human capacity. The + art of finding proofs, and the admirable method they have + invented for the singling out and laying in order those + intermediate ideas that demonstratively show the equality or + inequality of unapplicable quantities, is that which has + carried them so far and produced such wonderful and unexpected + discoveries; but whether something like this, in respect of + other ideas, as well as those of magnitude, may not in time be + found out, I will not determine. This, I think, I may say, that + if other ideas that are the real as well as the nominal + essences of their species, were pursued in the way + familiar to mathematicians, + +<span class="pagenum"> + <a name="Page_219" + id="Page_219">219</a></span> + + they would carry our thoughts + further, and with greater evidence and clearness than possibly + we are apt to imagine.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 4, chap. 12, + sect. 7.</p> + + <p class="v2"> + <b><a name="Block_1434" id="Block_1434">1434</a>.</b> + Those long chains of reasoning, quite simple and + easy, which geometers are wont to employ in the accomplishment + of their most difficult demonstrations, led me to think that + everything which might fall under the cognizance of the human + mind might be connected together in a similar manner, and that, + provided only that one should take care not to receive anything + as true which was not so, and if one were always careful to + preserve the order necessary for deducing one truth from + another, there would be none so remote at which he might not at + last arrive, nor so concealed which he might not + discover.—<span class="smcap">Descartes.</span></p> + <p class="blockcite"> + Discourse upon Method, part 2; The Philosophy of Descartes + [Torrey], (New York, 1892), p. 47.</p> + + <p class="v2"> + <b><a name="Block_1435" id="Block_1435">1435</a>.</b> + If anyone wished to write in mathematical fashion + in metaphysics or ethics, nothing would prevent him from so + doing with vigor. Some have professed to do this, and we have a + promise of mathematical demonstrations outside of mathematics; + but it is very rare that they have been successful. This is, I + believe, because they are disgusted with the trouble it is + necessary to take for a small number of readers where they + would ask as in Persius: <i lang="la" xml:lang="la">Quis + leget haec</i>, and reply: + <i lang="la" xml:lang="la">Vel duo vel + nemo</i>.—<span class="smcap">Leibnitz.</span></p> + <p class="blockcite"> + New Essay concerning Human Understanding, Langley, Bk 2, + chap. 29, sect. 12.</p> + + <p class="v2"> + <b><a name="Block_1436" id="Block_1436">1436</a>.</b> + It is commonly asserted that mathematics and + philosophy differ from one another according to their + <em>objects</em>, the former treating of <em>quantity</em>, the + latter of <em>quality</em>. All this is false. The difference + between these sciences cannot depend on their object; for + philosophy applies to everything, hence also to <em>quanta</em>, + and so does mathematics in part, inasmuch as everything has + magnitude. It is only the <em>different kind of rational + knowledge or application</em> of reason in mathematics and + philosophy which constitutes the specific difference between + these two + + <span class="pagenum"> + <a name="Page_220" + id="Page_220">220</a></span> + + sciences. For philosophy is + <em>rational knowledge from mere concepts</em>, mathematics, on + the contrary, is <em>rational knowledge from the construction of + concepts</em>.</p> + <p class="v1"> + We construct concepts when we represent them in intuition <i + lang="la" xml:lang="la">a + priori</i>, without experience, or when we represent in + intuition the object which corresponds to our concept of + it.—The mathematician can never apply his + reason to mere concepts, nor the philosopher to the + construction of concepts.—In mathematics the + reason is employed <i lang="la" xml:lang="la">in concreto</i>, + however, the intuition + is not empirical, but the object of contemplation is something + <i lang="la" xml:lang="la">a priori</i>.</p> + <p class="v1"> + In this, as we see, mathematics has an advantage over + philosophy, the knowledge in the former being intuitive, in the + latter, on the contrary, only <em>discursive</em>. But the reason + why in mathematics we deal more with quantity lies in this, + that magnitudes can be constructed in intuition <i lang="la" + xml:lang="la">a + priori</i>, while qualities, on the contrary, do not permit of + being represented in intuition.—<span class= + "smcap">Kant, E.</span></p> + <p class="blockcite"> + Logik; Werke [Hartenstein], (Leipzig, 1868), Bd. 8, pp.23-24.</p> + + <p class="v2"> + <b><a name="Block_1437" id="Block_1437">1437</a>.</b> + Kant has divided human ideas into the two + categories of quantity and quality, which, if true, would + destroy the universality of Mathematics; but + Descartes’ fundamental conception of the + relation of the concrete to the abstract in Mathematics + abolishes this division, and proves that all ideas of quality + are reducible to ideas of quantity. He had in view geometrical + phenomena only; but his successors have included in this + generalization, first, mechanical phenomena, and, more + recently, those of heat. There are now no geometers who do not + consider it of universal application, and admit that every + phenomenon may be as logically capable of being represented by + an equation as a curve or a motion, if only we were always + capable (which we are very far from being) of first + discovering, and then resolving it.</p> + <p class="v1"> + The limitations of Mathematical science are not, then, in its + nature. The limitations are in our intelligence: and by these we + find the domain of the science remarkably restricted, in + proportion as phenomena, in becoming special, become + complex.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 1, chap. 1.</p> + +<p><span class="pagenum"> + <a name="Page_221" + id="Page_221">221</a></span></p> + + <p class="v2"> + <b><a name="Block_1438" id="Block_1438">1438</a>.</b> + The great advantage of the mathematical sciences + over the moral consists in this, that the ideas of the former, + being sensible, are always clear and determinate, the smallest + distinction between them being immediately perceptible, and the + same terms are still expressive of the same ideas, without + ambiguity or variation. An oval is never mistaken for a circle, + nor an hyperbola for an ellipsis. The isosceles and scalenum + are distinguished by boundaries more exact than vice and + virtue, right or wrong. If any term be defined in geometry, the + mind readily, of itself, substitutes on all occasions, the + definition for the thing defined: Or even when no definition is + employed, the object itself may be represented to the senses, + and by that means be steadily and clearly apprehended. But the + finer sentiments of the mind, the operations of the + understanding, the various agitations of the passions, though + really in themselves distinct, easily escape us, when surveyed + by reflection; nor is it in our power to recall the original + object, so often as we have occasion to contemplate it. + Ambiguity, by this means, is gradually introduced into our + reasonings: Similar objects are readily taken to be the same: + And the conclusion becomes at last very wide off the + premises.—<span class="smcap">Hume, David.</span></p> + <p class="blockcite"> + An Inquiry concerning Human Understanding, sect. 7, part 1.</p> + + <p class="v2"> + <b><a name="Block_1439" id="Block_1439">1439</a>.</b> + One part of these disadvantages in moral ideas + which has made them be thought not capable of demonstration, + may in a good measure be remedied by definitions, setting down + that collection of simple ideas, which every term shall stand + for; and then using the terms steadily and constantly for that + precise collection. And what methods algebra, or something of + that kind, may hereafter suggest, to remove the other + difficulties, it is not easy to foretell. Confident, I am, that + if men would in the same method, and with the same + indifferency, search after moral as they do mathematical + truths, they would find them have a stronger connexion one with + another, and a more necessary consequence from our clear and + distinct ideas, and to come nearer perfect demonstration than + is commonly imagined.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 4, chap. 3, + sect. 20.</p> + +<p><span class="pagenum"> + <a name="Page_222" + id="Page_222">222</a></span></p> + + <p class="v2"> + <b><a name="Block_1440" id="Block_1440">1440</a>.</b> + That which in this respect has given the advantage + to the ideas of quantity, and made them thought more capable of + certainty and demonstration [than moral ideas], is,</p> + <p class="v1"> + First, That they can be set down and represented by sensible + marks, which have a greater and nearer correspondence with them + than any words or sounds whatsoever. Diagrams drawn on paper + are copies of the ideas in the mind, and not liable to the + uncertainty that words carry in their signification. An angle, + circle, or square, drawn in lines, lies open to the view, and + cannot be mistaken: it remains unchangeable, and may at leisure + be considered and examined, and the demonstration be revised, + and all the parts of it may be gone over more than once, + without any danger of the least change in the ideas. This + cannot be done in moral ideas: we have no sensible marks that + resemble them, whereby we can set them down; we have nothing + but words to express them by; which, though when written they + remain the same, yet the ideas they stand for may change in the + same man; and it is seldom that they are not different in + different persons.</p> + <p class="v1"> + Secondly, Another thing that makes the greater difficulty in + ethics is, That moral ideas are commonly more complex than + those of the figures ordinarily considered in mathematics. From + whence these two inconveniences + follow:—First, that their names are of more + uncertain signification, the precise collection of simple ideas + they stand for not being so easily agreed on; and so the sign + that is used for them in communication always, and in thinking + often, does not steadily carry with it the same idea. Upon + which the same disorder, confusion, and error follow, as would + if a man, going to demonstrate something of an heptagon, + should, in the diagram he took to do it, leave out one of the + angles, or by oversight make the figure with an angle more than + the name ordinarily imported, or he intended it should when at + first he thought of his demonstration. This often happens, and + is hardly avoidable in very complex moral ideas, where the same + name being retained, an angle, i.e. one simple idea is left + out, or put in the complex one (still called by the same name) + more at one time than another. Secondly, From the complexedness + of these moral ideas there follows another inconvenience, viz., + that the mind cannot easily retain + +<span class="pagenum"> + <a name="Page_223" + id="Page_223">223</a></span> + + those precise combinations so exactly and perfectly as is + necessary in the examination of the habitudes and + correspondences, agreements or disagreements, of several of + them one with another; especially where it is to be judged of + by long deductions and the intervention of several other + complex ideas to show the agreement or disagreement of two + remote ones.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 4, chap. 3, + sect. 19.</p> + + <p class="v2"> + <b><a name="Block_1441" id="Block_1441">1441</a>.</b> + It has been generally taken for granted, that + mathematics alone are capable of demonstrative certainty: but + to have such an agreement or disagreement as may be intuitively + perceived, being, as I imagine, not the privileges of the ideas + of number, extension, and figure alone, it may possibly be the + want of due method and application in us, and not of sufficient + evidence in things, that demonstration has been thought to have + so little to do in other parts of knowledge, and been scarce so + much as aimed at by any but mathematicians. For whatever ideas + we have wherein the mind can perceive the immediate agreement + or disagreement that is between them, there the mind is capable + of intuitive knowledge, and where it can perceive the agreement + or disagreement of any two ideas, by an intuitive perception of + the agreement or disagreement they have with any intermediate + ideas, there the mind is capable of demonstration: which is not + limited to the idea of extension, figure, number, and their + modes.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 4, chap. 2, + sect. 9.</p> + + <p class="v2"> + <b><a name="Block_1442" id="Block_1442">1442</a>.</b> + Now I shall remark again what I have already + touched upon more than once, that it is a common opinion that + only mathematical sciences are capable of a demonstrative + certainty; but as the agreement and disagreement which may be + known intuitively is not a privilege belonging only to the + ideas of numbers and figures, it is perhaps for want of + application on our part that mathematics alone have attained to + demonstrations.—<span class="smcap">Leibnitz.</span></p> + <p class="blockcite"> + New Essay concerning Human Understanding, Bk. 4, chap. 2, + sect. 9 [Langley].</p> + +<p><span class="pagenum"> + <a name="Page_224" + id="Page_224">224</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_XV"> + CHAPTER XV<br /> + <span class="large"> + MATHEMATICS AND SCIENCE</span></h2> + + <p class="v2"> + <b><a name="Block_1501" id="Block_1501">1501</a>.</b> + How comes it about that the knowledge of other + sciences, which depend upon this [mathematics], is painfully + sought, and that no one puts himself to the trouble of studying + this science itself? I should certainly be surprised, if I did + not know that everybody regarded it as being very easy, and if + I had not long ago observed that the human mind, neglecting + what it believes to be easy, is always in haste to run after + what is novel and advanced.—<span class= + "smcap">Descartes.</span></p> + <p class="blockcite"> + Rules for the Direction of the Mind; Philosophy of + Descartes [Torrey], (New York, 1892), p. 72.</p> + + <p class="v2"> + <b><a name="Block_1502" id="Block_1502">1502</a>.</b> + All quantitative determinations are in the hands + of mathematics, and it at once follows from this that all + speculation which is heedless of mathematics, which does not + enter into partnership with it, which does not seek its aid in + distinguishing between the manifold modifications that must of + necessity arise by a change of quantitative determinations, is + either an empty play of thoughts, or at most a fruitless + effort. In the field of speculation many things grow which do + not start from mathematics nor give it any care, and I am far + from asserting that all that thus grow are useless weeds, among + them may be many noble plants, but without mathematics none + will develop to complete + maturity.—<span class="smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Werke (Kehrbach), (Langensalza, 1890), Bd. 5, p. 106.</p> + + <p class="v2"> + <b><a name="Block_1503" id="Block_1503">1503</a>.</b> + There are few things which we know, which are not capable of + being reduc’d to a Mathematical Reasoning, and when they + cannot, it’s a sign our knowledge of them is very small and + confus’d; and where a mathematical reasoning can be had, it’s + as great folly to make use of any other, as to grope for a + thing in the dark, when you have a candle standing by + you.—<span class="smcap">Arbuthnot.</span></p> + <p class="blockcite"> + Quoted in + + <a id="TNanchor_13"></a> + <a class="msg" href="#TN_13" + title="originally spelled Todhunder’s">Todhunter’s</a> + + History of the Theory of Probability + (Cambridge and London, 1865), p. 51.</p> + +<p><span class="pagenum"> + <a name="Page_225" + id="Page_225">225</a></span></p> + + <p class="v2"> + <b><a name="Block_1504" id="Block_1504">1504</a>.</b> + Mathematical Analysis is ... the true rational + basis of the whole system of our positive + knowledge.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 1, chap. 1.</p> + + <p class="v2"> + <b><a name="Block_1505" id="Block_1505">1505</a>.</b> + It is only through Mathematics that we can + thoroughly understand what true science is. Here alone we can + find in the highest degree simplicity and severity of + scientific law, and such abstraction as the human mind can + attain. Any scientific education setting forth from any other + point, is faulty in its basis.—<span class= + "smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 1, chap. 1.</p> + + <p class="v2"> + <b><a name="Block_1506" id="Block_1506">1506</a>.</b> + In the present state of our knowledge we must + regard Mathematics less as a constituent part of natural + philosophy than as having been, since the time of Descartes and + Newton, the true basis of the whole of natural philosophy; + though it is, exactly speaking, both the one and the other. To + us it is of less use for the knowledge of which it consists, + substantial and valuable as that knowledge is, than as being + the most powerful instrument that the human mind can employ in + the investigation of the laws of natural + phenomena.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Introduction, chap. 2.</p> + + <p class="v2"> + <b><a name="Block_1507" id="Block_1507">1507</a>.</b> + The concept of mathematics is the concept of science in + general.—<span class="smcap">Novalis.</span></p> + <p class="blockcite"> + Schriften (Berlin, 1901), Teil 2, p. 222.</p> + + <p class="v2"> + <b><a name="Block_1508" id="Block_1508">1508</a>.</b> + I contend, that each natural science is real + science only in so far as it is mathematical.... It may be that + a pure philosophy of nature in general (that is, a philosophy + which concerns itself only with the general concepts of nature) + is possible without mathematics, but a pure science of nature + dealing with definite objects (physics or psychology), is + possible only by means of mathematics, and since each natural + science contains only as much real science as it contains <i + lang="la" xml:lang="la">a + priori</i> knowledge, each natural science becomes real science + only to the extent that it permits the application of + mathematics.—<span class="smcap">Kant, E.</span></p> + <p class="blockcite"> + Metaphysische Anfangsgründe der Naturwissenschaft, Vorrede.</p> + +<p><span class="pagenum"> + <a name="Page_226" + id="Page_226">226</a></span></p> + + <p class="v2"> + <b><a name="Block_1509" id="Block_1509">1509</a>.</b> + The theory most prevalent among teachers is that + mathematics affords the best training for the reasoning + powers;... The modern, and to my mind true, theory is that + mathematics is the abstract form of the natural sciences; and + that it is valuable as a training of the reasoning powers, not + because it is abstract, but because it is a representation of + actual things.—<span class="smcap">Safford, T. H.</span></p> + <p class="blockcite"> + Mathematical Teaching etc. (Boston, 1886), p. 9.</p> + + <p class="v2"> + <b><a name="Block_1510" id="Block_1510">1510</a>.</b> + It seems to me that no one science can so well + serve to co-ordinate and, as it were, bind together all of the + sciences as the queen of them all, + mathematics.—<span class="smcap">Davis, E. W.</span></p> + <p class="blockcite"> + Proceedings Nebraska Academy of Sciences for 1896 + (Lincoln, 1897), p. 282.</p> + + <p class="v2"> + <b><a name="Block_1511" id="Block_1511">1511</a>.</b> + And as for Mixed Mathematics, I may only make this + prediction, that there cannot fail to be more kinds of them, as + nature grows further disclosed.—<span class= + "smcap">Bacon, Francis.</span></p> + <p class="blockcite"> + Advancement of Learning, Bk. 2; De Augmentis, Bk. 3.</p> + + <p class="v2"> + <b><a name="Block_1512" id="Block_1512">1512</a>.</b> + Besides the exercise in keen comprehension and the + certain discovery of truth, mathematics has another formative + function, that of equipping the mind for the survey of a + scientific system.—<span class="smcap">Grassmann, H.</span></p> + <p class="blockcite"> + Stücke aus dem Lehrbuche der Arithmetik; Werke (Leipzig, 1904), + Bd. 2, p. 298.</p> + + <p class="v2"> + <b><a name="Block_1513" id="Block_1513">1513</a>.</b> + Mathematicks may help the naturalists, both to + frame hypotheses, and to judge of those that are proposed to + them, especially such as relate to mathematical subjects in + conjunction with others.—<span class= + "smcap">Boyle, Robert.</span></p> + <p class="blockcite"> + Works (London, 1772), Vol. 3, p. 429.</p> + + <p class="v2"> + <b><a name="Block_1514" id="Block_1514">1514</a>.</b> + The more progress physical sciences make, the more + they tend to enter the domain of mathematics, which is a kind + of centre to which they all converge. We may even judge of the + degree of perfection to which a science has arrived by the + facility with which it may be submitted to + calculation.—<span class="smcap">Quetelet.</span></p> + <p class="blockcite"> + Quoted in E. Mailly’s Eulogy on Quetelet; Smithsonian Report, + 1874, p. 173.</p> + +<p><span class="pagenum"> + <a name="Page_227" + id="Page_227">227</a></span></p> + + <p class="v2"> + <b><a name="Block_1515" id="Block_1515">1515</a>.</b> + The mathematical formula is the point through + which all the light gained by science passes in order to be of + use to practice; it is also the point in which all knowledge + gained by practice, experiment, and observation must be + concentrated before it can be scientifically grasped. The more + distant and marked the point, the more concentrated will be the + light coming from it, the more unmistakable the insight + conveyed. All scientific thought, from the simple gravitation + formula of Newton, through the more complicated formulae of + physics and chemistry, the vaguer so called laws of organic and + animated nature, down to the uncertain statements of psychology + and the data of our social and historical knowledge, alike + partakes of this characteristic, that it is an attempt to + gather up the scattered rays of light, the different parts of + knowledge, in a focus, from whence it can be again spread out + and analyzed, according to the abstract processes of the + thinking mind. But only when this can be done with a + mathematical precision and accuracy is the image sharp and + well-defined, and the deductions clear and unmistakable. As we + descend from the mechanical, through the physical, chemical, + and biological, to the mental, moral, and social sciences, the + process of focalization becomes less and less + perfect,—the sharp point, the focus, is + replaced by a larger or smaller circle, the contours of the + image become less and less distinct, and with the possible + light which we gain there is mingled much darkness, the sources + of many mistakes and errors. But the tendency of all scientific + thought is toward clearer and clearer definition; it lies in + the direction of a more and more extended use of mathematical + measurements, of mathematical + formulae.—<span class="smcap">Merz, J. T.</span></p> + <p class="blockcite"> + History of European Thought in the 19th Century (Edinburgh + and London, 1904), Vol. 1, p. 333.</p> + + <p class="v2"> + <b><a name="Block_1516" id="Block_1516">1516</a>.</b> + From the very outset of his investigations the + physicist has to rely constantly on the aid of the + mathematician, for even in the simplest cases, the direct + results of his measuring operations are entirely without + meaning until they have been submitted to more or less of + mathematical discussion. And when in this way some + interpretation of the experimental results has been arrived at, + and it has been proved that two or + +<span class="pagenum"> + <a name="Page_228" + id="Page_228">228</a></span> + + more physical quantities stand in a definite relation to each + other, the mathematician is very often able to infer, from the + existence of this relation, that the quantities in question + also fulfill some other relation, that was previously + unsuspected. Thus when Coulomb, combining the functions of + experimentalist and mathematician, had discovered the law of + the force exerted between two particles of electricity, it + became a purely mathematical problem, not requiring any further + experiment, to ascertain how electricity is distributed upon a + charged conductor and this problem has been solved by + mathematicians in several + cases.—<span class="smcap">Foster, G. C.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A (1877); Nature, Vol. 16, p. 312-313.</p> + + <p class="v2"> + <b><a name="Block_1517" id="Block_1517">1517</a>.</b> + Without consummate mathematical skill, on the part + of some investigators at any rate, all the higher physical + problems would be sealed to us; and without competent skill on + the part of the ordinary student no idea can be formed of the + nature and cogency of the evidence on which the solutions rest. + Mathematics are not merely a gate through which we may approach + if we please, but they are the only mode of approach to large + and important districts of + thought.—<span class="smcap">Venn, John.</span></p> + <p class="blockcite"> + Symbolic Logic (London and New York, 1894), Introduction, + p. xix.</p> + + <p class="v2"> + <b><a name="Block_1518" id="Block_1518">1518</a>.</b> + Much of the skill of the true mathematical + physicist and of the mathematical astronomer consists in the + power of adapting methods and results carried out on an exact + mathematical basis to obtain approximations sufficient for the + purposes of physical measurements. It might perhaps be thought + that a scheme of Mathematics on a frankly approximative basis + would be sufficient for all the practical purposes of + application in Physics, Engineering Science, and Astronomy, and + no doubt it would be possible to develop, to some extent at + least, a species of Mathematics on these lines. Such a system + would, however, involve an intolerable awkwardness and + prolixity in the statements of results, especially in view of + the fact that the degree of approximation necessary for various + purposes is very different, and thus that unassigned grades of + approximation + +<span class="pagenum"> + <a name="Page_229" + id="Page_229">229</a></span> + + would have to be provided for. + Moreover, the mathematician working on these lines would be cut + off from the chief sources of inspiration, the ideals of + exactitude and logical rigour, as well as from one of his most + indispensable guides to discovery, symmetry, and permanence of + mathematical form. The history of the actual movements of + mathematical thought through the centuries shows that these + ideals are the very life-blood of the science, and warrants the + conclusion that a constant striving toward their attainment is + an absolutely essential condition of vigorous growth. These + ideals have their roots in irresistible impulses and + deep-seated needs of the human mind, manifested in its efforts + to introduce intelligibility in certain great domains of the + world of thought.—<span class="smcap">Hobson, E. W.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A (1910); Nature, Vol. 84, pp. 285-286.</p> + + <p class="v2"> + <b><a name="Block_1519" id="Block_1519">1519</a>.</b> + The immense part which those laws [laws of number + and extension] take in giving a deductive character to the + other departments of physical science, is well known; and is + not surprising, when we consider that all causes operate + according to mathematical laws. The effect is always dependent + upon, or in mathematical language, is a function of, the + quantity of the agent; and generally of its position also. We + cannot, therefore, reason respecting causation, without + introducing considerations of quantity and extension at every + step; and if the nature of the phenomena admits of our + obtaining numerical data of sufficient accuracy, the laws of + quantity become the grand instruments for calculating forward + to an effect, or backward to a + cause.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + System of Logic, Bk. 3, chap. 24, sect. 9.</p> + + <p class="v2"> + <b><a name="Block_1520" id="Block_1520">1520</a>.</b> + The ordinary mathematical treatment of any applied + science substitutes exact axioms for the approximate results of + experience, and deduces from these axioms the rigid + mathematical conclusions. In applying this method it must not + be forgotten that the mathematical developments transcending + the limits of exactness of the science are of no practical + value. It follows that a large portion of abstract mathematics + remains without finding any practical + application, the amount of mathematics + +<span class="pagenum"> + <a name="Page_230" + id="Page_230">230</a></span> + + that can be usefully + employed in any science being in proportion to the degree of + accuracy attained in the science. Thus, while the astronomer + can put to use a wide range of mathematical theory, the chemist + is only just beginning to apply the first derivative, i.e. the + rate of change at which certain processes are going on; for + second derivatives he does not seem to have found any use as + yet.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Lectures on Mathematics (New York, 1911), p. 47.</p> + + <p class="v2"> + <b><a name="Block_1521" id="Block_1521">1521</a>.</b> + The bond of union among the physical sciences is + the mathematical spirit and the mathematical method which + pervades them.... Our knowledge of nature, as it advances, + continuously resolves differences of quality into differences + of quantity. All exact reasoning—indeed all + reasoning—about quantity is mathematical + reasoning; and thus as our knowledge increases, that portion of + it which becomes mathematical increases at a still more rapid + rate.—<span class="smcap">Smith, H. J. S.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A (1873); Nature, Vol. 8, p. 449.</p> + + <p class="v2"> + <b><a name="Block_1522" id="Block_1522">1522</a>.</b> + Another way of convincing ourselves how largely + this process [of assimilation of mathematics by physics] has + gone on would be to try to conceive the effect of some + intellectual catastrophe, supposing such a thing possible, + whereby all knowledge of mathematics should be swept away from + men’s minds. Would it not be that the + departure of mathematics would be the destruction of physics? + Objective physical phenomena would, indeed, remain as they are + now, but physical science would cease to exist. We should no + doubt see the same colours on looking into a spectroscope or + polariscope, vibrating strings would produce the same sounds, + electrical machines would give sparks, and galvanometer needles + would be deflected; but all these things would have lost their + meaning; they would be but as the dry bones—the <i lang="la" + xml:lang="la">disjecta + membra</i>—of what is now a living and growing science. To + follow this conception further, and to try to image to ourselves + in some detail what would be the kind of knowledge of physics + which would remain possible, supposing all mathematical ideas + to be blotted out, + +<span class="pagenum"> + <a name="Page_231" + id="Page_231">231</a></span> + + would be extremely interesting, but it would lead us directly + into a dim and entangled region where the subjective seems to be + always passing itself off for the objective, and where I at + least could not attempt to lead the way, gladly as I would + follow any one who could show where a firm footing is to be + found. But without venturing to do more than to look from a + safe distance over this puzzling ground, we may see clearly + enough that mathematics is the connective tissue of physics, + binding what would else be merely a list of detached + observations into an organized body of + science.—<span class="smcap">Foster, G. C.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A (1877); Nature, Vol. 16, p. 313.</p> + + <p class="v2"> + <b><a name="Block_1523" id="Block_1523">1523</a>.</b> + In <em>Plato’s</em> time + mathematics was purely a play of the free intellect; the + mathematic-mystical reveries of a Pythagoras foreshadowed a + far-reaching significance, but such a significance (except in + the case of music) was as yet entirely a matter of fancy; yet + even in that time mathematics was the prerequisite to all other + studies! But today, when mathematics furnishes the <em>only</em> + language by means of which we may formulate the most + comprehensive laws of nature, laws which the ancients scarcely + dreamed of, when moreover mathematics is the <em>only</em> means + by which these laws may be understood,—how + few learn today anything of the real essence of our + mathematics!... In the schools of today mathematics serves only + as a disciplinary study, a mental gymnastic; that it includes + the highest ideal value for the comprehension of the universe, + one dares scarcely to think of in view of our present day + instruction.—<span class="smcap">Lindeman, F.</span></p> + <p class="blockcite"> + Lehren und Lernen in der Mathematik (München, 1904), p. 14.</p> + + <p class="v2"> + <b><a name="Block_1524" id="Block_1524">1524</a>.</b> + All applications of mathematics consist in + extending the empirical knowledge which we possess of a limited + number or region of accessible phenomena into the region of the + unknown and inaccessible; and much of the progress of pure + analysis consists in inventing definite conceptions, marked by + symbols, of complicated operations; in ascertaining their + properties as independent objects of research; and in extending + their meaning + +<span class="pagenum"> + <a name="Page_232" + id="Page_232">232</a></span> + + beyond the limits they were originally invented for,—thus + opening out new and larger regions of + thought.—<span class="smcap">Merz, J. T.</span></p> + <p class="blockcite"> + History of European Thought in the 19th Century (Edinburgh + and London, 1903), Vol. 1, p. 698.</p> + + <p class="v2"> + <b><a name="Block_1525" id="Block_1525">1525</a>.</b> + All the effects of nature are only mathematical + results of a small number of immutable + laws.—<span class="smcap">Laplace.</span></p> + <p class="blockcite"> + A Philosophical Essay on Probabilities [Truscott and + Emory] (New York, 1902), p. 177; Oeuvres, t. 7, p. 139.</p> + + <p class="v2"> + <b><a name="Block_1526" id="Block_1526">1526</a>.</b> + What logarithms are to mathematics that + mathematics are to the other + sciences.—<span class="smcap">Novalis.</span></p> + <p class="blockcite"> + Schriften (Berlin, 1901), Teil 2, p. 222.</p> + + <p class="v2"> + <b><a name="Block_1527" id="Block_1527">1527</a>.</b> + Any intelligent man may now, by resolutely + applying himself for a few years to mathematics, learn more + than the great Newton knew after half a century of study and + meditation.—<span class="smcap">Macaulay.</span></p> + <p class="blockcite"> + Milton; Critical and Miscellaneous Essays (New York, + 1879), Vol. 1, p. 13.</p> + + <p class="v2"> + <b><a name="Block_1528" id="Block_1528">1528</a>.</b> + In questions of science the authority of a + thousand is not worth the humble reasoning of a single + individual.—<span class="smcap">Galileo.</span></p> + <p class="blockcite"> + Quoted in Arago’s Eulogy on Laplace; + Smithsonian Report, 1874, p. 164.</p> + + <p class="v2"> + <b><a name="Block_1529" id="Block_1529">1529</a>.</b> + Behind the artisan is the chemist, behind the + chemist a physicist, behind the physicist a + mathematician.—<span class="smcap">White, W. F.</span></p> + <p class="blockcite"> + Scrap-book of Elementary Mathematics (Chicago, 1908), p. 217.</p> + + <p class="v2"> + <b><a name="Block_1530" id="Block_1530">1530</a>.</b> + The advance in our knowledge of physics is largely + due to the application to it of mathematics, and every year it + becomes more difficult for an experimenter to make any mark in + the subject unless he is also a + mathematician.—<span class="smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p. 503.</p> + + <p class="v2"> + <b><a name="Block_1531" id="Block_1531">1531</a>.</b> + In very many cases the most obvious and direct + experimental method of investigating a given problem is + extremely difficult, or for some reason or other + untrustworthy. + +<span class="pagenum"> + <a name="Page_233" id="Page_233">233</a></span> + + In such cases the mathematician can + often point out some other problem more accessible to + experimental treatment, the solution of which involves the + solution of the former one. For example, if we try to deduce + from direct experiments the law according to which one pole of + a magnet attracts or repels a pole of another magnet, the + observed action is so much complicated with the effects of the + mutual induction of the magnets and of the forces due to the + second pole of each magnet, that it is next to impossible to + obtain results of any great accuracy. Gauss, however, showed + how the law which applied in the case mentioned can be deduced + from the deflections undergone by a small suspended magnetic + needle when it is acted upon by a small fixed magnet placed + successively in two determinate positions relatively to the + needle; and being an experimentalist as well as a + mathematician, he showed likewise how these deflections can be + measured very easily and with great + precision.—<span class="smcap">Foster, G. C.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A (1877); Nature, Vol. 16, p. 313.</p> + + <p class="v2"> + <b><a name="Block_1532" id="Block_1532">1532</a>.</b></p> + <div class="poem"> + <p class="i0"> + Give me to learn each secret cause;</p> + <p class="i0"> + Let Number’s, Figure’s, Motion’s laws</p> + <p class="i0"> + Reveal’d before me stand;</p> + <p class="i0"> + These to great Nature’s scenes apply,</p> + <p class="i0"> + And round the globe, and through the sky,</p> + <p class="i0"> + Disclose her working hand.</p> + </div> + <p class="block40"> + —<span class="smcap">Akenside, M.</span></p> + <p class="blockcite"> + Hymn to Science.</p> + + <p class="v2"> + <b><a name="Block_1533" id="Block_1533">1533</a>.</b> + Now there are several scores, upon which skill in + mathematicks may be useful to the experimental philosopher. For + there are some general advantages, which mathematicks may bring + to the minds of men, to whatever study they apply themselves, + and consequently to the student of natural philosophy; namely, + that these disciplines are wont to make men accurate, and very + attentive to the employment that they are about, keeping their + thoughts from wandering, and inuring them to patience in going + through with tedious and intricate demonstrations; besides, + that they much improve reason, by accustoming the mind to + deduce successive consequences, + +<span class="pagenum"> + <a name="Page_234" + id="Page_234">234</a></span> + + and judge of them without easily acquiescing in anything but + demonstration.—<span class="smcap">Boyle, Robert.</span></p> + <p class="blockcite"> + Works (London, 1772), Vol. 3, p. 426.</p> + + <p class="v2"> + <b><a name="Block_1534" id="Block_1534">1534</a>.</b> + It is not easy to anatomize the constitution and + the operations of a mind [like Newton’s] + which makes such an advance in knowledge. Yet we may observe + that there must exist in it, in an eminent degree, the elements + which compose the mathematical talent. It must possess + distinctness of intuition, tenacity and facility in tracing + logical connection, fertility of invention, and a strong + tendency to generalization.—<span + class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + History of the Inductive Sciences (New York, 1894), Vol. + 1, p. 416.</p> + + <p class="v2"> + <b><a name="Block_1535" id="Block_1535">1535</a>.</b> + The domain of physics is no proper field for + mathematical pastimes. The best security would be in giving a + geometrical training to physicists, who need not then have + recourse to mathematicians, whose tendency is to despise + experimental science. By this method will that union between + the abstract and the concrete be effected which will perfect + the uses of mathematical, while extending the positive value of + physical science. Meantime, the + + <a id="TNanchor_14"></a> + <a class="msg" href="#TN_14" + title="originally read ‘uses’">use</a> + + of analysis in physics is + clear enough. Without it we should have no precision, and no + co-ordination; and what account could we give of our study of + heat, weight, light, etc.? We should have merely series of + unconnected facts, in which we could foresee nothing but by + constant recourse to experiment; whereas, they now have a + character of rationality which fits them for purposes of + prevision.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 3, chap. 1.</p> + + <p class="v2"> + <b><a name="Block_1536" id="Block_1536">1536</a>.</b> + It must ever be remembered that the true positive + spirit first came forth from the pure sources of mathematical + science; and it is only the mind that has imbibed it there, and + which has been face to face with the lucid truths of geometry + and mechanics, that can bring into full action its natural + positivity, and apply it in bringing the most complex studies + into the reality of demonstration. No other discipline can + fitly prepare the intellectual + organ.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 3, chap. 1.</p> + +<p><span class="pagenum"> + <a name="Page_235" + id="Page_235">235</a></span></p> + + <p class="v2"> + <b><a name="Block_1537" id="Block_1537">1537</a>.</b> + During the last two centuries and a half, physical + knowledge has been gradually made to rest upon a basis which it + had not before. It has become <em>mathematical</em>. The question + now is, not whether this or that hypothesis is better or worse + to the pure thought, but whether it accords with observed + phenomena in those consequences which can be shown necessarily + to follow from it, if it be true. Even in those sciences which + are not yet under the dominion of mathematics, and perhaps + never will be, a working copy of the mathematical process has + been made. This is not known to the followers of those sciences + who are not themselves mathematicians, and who very often exalt + their horns against the mathematics in consequence. They might + as well be squaring the circle, for any sense they show in this + particular.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + A Budget of Paradoxes (London, 1872), p. 2.</p> + + <p class="v2"> + <b><a name="Block_1538" id="Block_1538">1538</a>.</b> + Among the mere talkers so far as mathematics are + concerned, are to be ranked three out of four of those who + apply mathematics to physics, who, wanting a tool only, are + very impatient of everything which is not of direct aid to the + actual methods which are in their + hands.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Graves’ Life of Sir William Rowan + Hamilton (New York, 1882-1889), Vol. 3, p. 348.</p> + + <p class="v2"> + <b><a name="Block_1539" id="Block_1539">1539</a>.</b> + Something has been said about the use of + mathematics in physical science, the mathematics being regarded + as a weapon forged by others, and the study of the weapon being + completely set aside. I can only say that there is danger of + obtaining untrustworthy results in physical science, if only + the results of mathematics are used; for the person so using + the weapon can remain unacquainted with the conditions under + which it can be rightly applied.... The results are often + correct, sometimes are incorrect; the consequence of the latter + class of cases is to throw doubt upon all the applications of + such a worker until a result has been otherwise tested. + Moreover, such a practice in the use of mathematics leads a + worker to a mere repetition in the use of familiar weapons; he + is unable to adapt them with any confidence when some new set + of conditions arise with a demand for a new method: for want of + adequate instruction in the + +<span class="pagenum"> + <a name="Page_236" + id="Page_236">236</a></span> + + forging of the weapon, he may find himself, sooner or later + in the progress of his subject, without any weapon worth + having.—<span class="smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Perry’s Teaching of Mathematics (London, 1902), p. 36.</p> + + <p class="v2"> + <b><a name="Block_1540" id="Block_1540">1540</a>.</b> + If in the range of human endeavor after sound + knowledge there is one subject that needs to be practical, it + surely is Medicine. Yet in the field of Medicine it has been + found that branches such as biology and pathology must be + studied for themselves and be developed by themselves with the + single aim of increasing knowledge; and it is then that they + can be best applied to the conduct of living processes. So also + in the pursuit of mathematics, the path of practical utility is + too narrow and irregular, not always leading far. The witness + of history shows that, in the field of natural philosophy, + mathematics will furnish the more effective assistance if, in + its systematic development, its course can freely pass beyond + the ever-shifting domain of use and + application.—<span class="smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A; Nature, Vol. 56 (1897), p. 377.</p> + + <p class="v2"> + <b><a name="Block_1541" id="Block_1541">1541</a>.</b> + If the Greeks had not cultivated Conic Sections, + Kepler could not have superseded Ptolemy; if the Greeks had + cultivated Dynamics, Kepler might have anticipated + Newton.—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + History of the Inductive Science (New York, 1894), Vol. 1, + p. 311.</p> + + <p class="v2"> + <b><a name="Block_1542" id="Block_1542">1542</a>.</b> + If we may use the great names of Kepler and Newton + to signify stages in the progress of human discovery, it is not + too much to say that without the treatises of the Greek + geometers on the conic sections there could have been no + Kepler, without Kepler no Newton, and without Newton no science + in the modern sense of the term, or at least no such conception + of nature as now lies at the basis of all our science, of + nature as subject in the smallest as well as in its greatest + phenomena, to exact quantitative relations, and to definite + numerical laws.—<span class="smcap">Smith, H. J. S.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A; Nature, Vol. 8 (1873), p. 450.</p> + +<p><span class="pagenum"> + <a name="Page_237" + id="Page_237">237</a></span></p> + + <p class="v2"> + <b><a name="Block_1543" id="Block_1543">1543</a>.</b> + The silent work of the great Regiomontanus in his + chamber at Nuremberg computed the Ephemerides which made + possible the discovery of America by + Columbus.—<span class="smcap">Rudio, F.</span></p> + <p class="blockcite"> + Quoted in Max Simon’s Geschichte der + Mathematik im Altertum (Berlin, 1909), Einleitung, p. xi.</p> + + <p class="v2"> + <b><a name="Block_1544" id="Block_1544">1544</a>.</b> + The calculation of the eclipses of + Jupiter’s satellites, many a man might have + been disposed, originally, to regard as a most unprofitable + study. But the utility of it to navigation (in the + determination of longitudes) is now well + known.—<span class="smcap">Whately, R.</span></p> + <p class="blockcite"> + Annotations to Bacon’s Essays (Boston, 1783), p. 492.</p> + + <p class="v2"> + <b><a name="Block_1545" id="Block_1545">1545</a>.</b> + Who could have imagined, when Galvani observed the + twitching of the frog muscles as he brought various metals in + contact with them, that eighty years later Europe would be + overspun with wires which transmit messages from Madrid to St. + Petersburg with the rapidity of lightning, by means of the same + principle whose first manifestations this anatomist then + observed!...</p> + <p class="v1"> + He who seeks for immediate practical use in the pursuit of + science, may be reasonably sure, that he will seek in vain. + Complete knowledge and complete understanding of the action of + forces of nature and of the mind, is the only thing that + science can aim at. The individual investigator must find his + reward in the joy of new discoveries, as new victories of + thought over resisting matter, in the esthetic beauty which a + well-ordered domain of knowledge affords, where all parts are + intellectually related, where one thing evolves from another, + and all show the marks of the mind’s + supremacy; he must find his reward in the consciousness of + having contributed to the growing capital of knowledge on which + depends the supremacy of man over the forces hostile to the + spirit.—<span class="smcap">Helmholtz, H.</span></p> + <p class="blockcite"> + Vorträge und Reden (Braunschweig, 1884), Bd. 1, p. 142.</p> + + <p class="v2"> + <b><a name="Block_1546" id="Block_1546">1546</a>.</b> + When the time comes that knowledge will not be sought for its + own sake, and men will not press forward simply + +<span class="pagenum"> + <a name="Page_238" + id="Page_238">238</a></span> + + in a desire of achievement, without + hope of gain, to extend the limits of human knowledge and + information, then, indeed, will the race enter upon its + decadence.—<span class="smcap">Hughes, C. E.</span></p> + <p class="blockcite"> + Quoted in D. E. Smith’s Teaching of Geometry + (Boston, 1911), p. 9.</p> + + <p class="v2"> + <b><a name="Block_1547" id="Block_1547">1547</a>.</b> + [In the Opus Majus of Roger Bacon] there is a + chapter, in which it is proved by reason, that all sciences + require mathematics. And the arguments which are used to + establish this doctrine, show a most just appreciation of the + office of mathematics in science. They are such as follows: + That other sciences use examples taken from mathematics as the + most evident:—That mathematical knowledge is, as it were, + innate to us, on which point he refers to the well-known + dialogue of Plato, as quoted by Cicero:—That this science, + being the easiest, offers the best introduction to the more + difficult:—That in mathematics, things as known to us are + identical with things as known to nature:—That we can here + entirely avoid doubt and error, and obtain certainty and + truth:—That mathematics is prior to other sciences in nature, + because it takes cognizance of quantity, which is apprehended + by intuition (<i lang="la" xml:lang="la">intuitu + intellectus</i>). “Moreover,” he adds, + “there have been found famous men, as + Robert, bishop of Lincoln, and Brother Adam Marshman (de + Marisco), and many others, who by the power of mathematics have + been able to explain the causes of things; as may be seen in + the writings of these men, for instance, concerning the Rainbow + and Comets, and the generation of heat, and climates, and the + celestial bodies”—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + History of the Inductive Sciences (New York, 1894), Vol. + 1, p. 519. Bacon, Roger: Opus Majus, Part 4, Distinctia + Prima, cap. 3.</p> + + <p class="v2"> + <b><a name="Block_1548" id="Block_1548">1548</a>.</b> + The analysis which is based upon the conception of + function discloses to the astronomer and physicist not merely + the formulae for the computation of whatever desired distances, + times, velocities, physical constants; it moreover gives him + insight into the laws of the processes of motion, teaches him + to predict future occurrences from past experiences and + supplies him with means to a scientific knowledge of nature, + i.e. it enables him to trace back whole groups of various, + sometimes + +<span class="pagenum"> + <a name="Page_239" + id="Page_239">239</a></span> + + extremely heterogeneous, phenomena to a minimum of simple + fundamental laws.—<span class="smcap">Pringsheim, A.</span></p> + <p class="blockcite"> + Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. + 13, p. 366.</p> + + <p class="v2"> + <b><a name="Block_1549" id="Block_1549">1549</a>.</b> + “As is known, scientific + physics dates its existence from the discovery of the + differential calculus. Only when it was learned how to follow + continuously the course of natural events, attempts, to + construct by means of abstract conceptions the connection + between phenomena, met with success. To do this two things are + necessary: First, simple fundamental concepts with which to + construct; second, some method by which to deduce, from the + simple fundamental laws of the construction which relate to + instants of time and points in space, laws for finite intervals + and distances, which alone are accessible to observation (can + be compared with experience).” [Riemann.]</p> + <p class="v1"> + The first of the two problems here indicated by Riemann + consists in setting up the differential equation, based upon + physical facts and hypotheses. The second is the integration of + this differential equation and its application to each separate + concrete case, this is the task of + mathematics.—<span class="smcap">Weber, Heinrich.</span></p> + <p class="blockcite"> + Die partiellen Differentialgleichungen der mathematischen + Physik (Braunschweig, 1882), Bd. 1, Vorrede.</p> + + <p class="v2"> + <b><a name="Block_1550" id="Block_1550">1550</a>.</b> + Mathematics is the most powerful instrument which + we possess for this purpose [to trace into their farthest + results those general laws which an inductive philosophy has + supplied]: in many sciences a profound knowledge of mathematics + is indispensable for a successful investigation. In the most + delicate researches into the theories of light, heat, and sound + it is the only instrument; they have properties which no other + language can express; and their argumentative processes are + beyond the reach of other + symbols.—<span class="smcap">Price, B.</span></p> + <p class="blockcite"> + Treatise on Infinitesimal Calculus (Oxford, 1858), Vol. 3, + p. 5.</p> + + <p class="v2"> + <b><a name="Block_1551" id="Block_1551">1551</a>.</b> + Notwithstanding the eminent difficulties of the + mathematical theory of sonorous vibrations, we owe to it such + progress as has yet been made in acoustics. The formation of the + +<span class="pagenum"> + <a name="Page_240" + id="Page_240">240</a></span> + + differential equations proper to the + phenomena is, independent of their integration, a very + important acquisition, on account of the approximations which + mathematical analysis allows between questions, otherwise + heterogeneous, which lead to similar equations. This + fundamental property, whose value we have so often to + recognize, applies remarkably in the present case; and + especially since the creation of mathematical thermology, whose + principal equations are strongly analogous to those of + vibratory motion.—This means of + investigation is all the more valuable on account of the + difficulties in the way of direct inquiry into the phenomena of + sound. We may decide the necessity of the atmospheric medium + for the transmission of sonorous vibrations; and we may + conceive of the possibility of determining by experiment the + duration of the propagation, in the air, and then through other + media; but the general laws of the vibrations of sonorous + bodies escape immediate observation. We should know almost + nothing of the whole case if the mathematical theory did not + come in to connect the different phenomena of sound, enabling + us to substitute for direct observation an equivalent + examination of more favorable cases subjected to the same law. + For instance, when the analysis of the problem of vibrating + chords has shown us that, other things being equal, the number + of oscillations is in inverse proportion to the length of the + chord, we see that the most rapid vibrations of a very short + chord may be counted, since the law enables us to direct our + attention to very slow vibrations. The same substitution is at + our command in many cases in which it is less + direct.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 3, chap. 4.</p> + + <p class="v2"> + <b><a name="Block_1552" id="Block_1552">1552</a>.</b> + Problems relative to the uniform propagation, or + to the varied movements of heat in the interior of solids, are + reduced ... to problems of pure analysis, and the progress of + this part of physics will depend in consequence upon the + advance which may be made in the art of analysis. The + differential equations ... contain the chief results of the + theory; they express, in the most general and concise manner, + the necessary relations of numerical analysis to a very + extensive class of + +<span class="pagenum"> + <a name="Page_241" + id="Page_241">241</a></span> + + phenomena; and they connect forever with mathematical science + one of the most important branches of natural + philosophy.—<span class="smcap">Fourier, J.</span></p> + <p class="blockcite"> + Theory of Heat [Freeman], (Cambridge, 1878), Chap. 3, p. 131.</p> + + <p class="v2"> + <b><a name="Block_1553" id="Block_1553">1553</a>.</b> + The effects of heat are subject to constant laws + which cannot be discovered without the aid of mathematical + analysis. The object of the theory is to demonstrate these + laws; it reduces all physical researches on the propagation of + heat, to problems of the integral calculus, whose elements are + given by experiment. No subject has more extensive relations + with the progress of industry and the natural sciences; for the + action of heat is always present, it influences the processes + of the arts, and occurs in all the phenomena of the + universe.—<span class="smcap">Fourier, J.</span></p> + <p class="blockcite"> + Theory of Heat [Freeman], (Cambridge, 1878), Chap. 1, p. 12.</p> + + <p class="v2"> + <b><a name="Block_1554" id="Block_1554">1554</a>.</b> + Dealing with any and every amount of static + electricity, the mathematical mind has balanced and adjusted + them with wonderful advantage, and has foretold results which + the experimentalist can do no more than verify.... So in + respect of the force of gravitation, it has calculated the + results of the power in such a wonderful manner as to trace the + known planets through their courses and perturbations, and in + so doing has <em>discovered</em> a planet before + unknown.—<span class="smcap">Faraday.</span></p> + <p class="blockcite"> + Some Thoughts on the Conservation of Force.</p> + + <p class="v2"> + <b><a name="Block_1555" id="Block_1555">1555</a>.</b> + Certain branches of natural philosophy (such as + physical astronomy and optics), ... are, in a great measure, + inaccessible to those who have not received a regular + mathematical education....—<span + class="smcap">Stewart, Dugald.</span></p> + <p class="blockcite"> + Philosophy of the Human Mind, Part 3, chap. 1, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_1556" id="Block_1556">1556</a>.</b> + So intimate is the union between mathematics and + physics that probably by far the larger part of the accessions + to our mathematical knowledge have been obtained by the efforts + of mathematicians to solve the problems set to them by + experiment, and to create “for each successive class of + phenomena, a new calculus or a new geometry, as the case + might be, which + +<span class="pagenum"> + <a name="Page_242" + id="Page_242">242</a></span> + + might prove not wholly inadequate to the subtlety of + nature.” Sometimes, indeed, the mathematician has + been before the physicist, and it has happened that when some + great and new question has occurred to the experimentalist or + the observer, he has found in the armoury of the mathematician + the weapons which he has needed ready made to his hand. But, + much oftener, the questions proposed by the physicist have + transcended the utmost powers of the mathematics of the time, + and a fresh mathematical creation has been needed to supply the + logical instrument requisite to interpret the new + enigma.—<span class="smcap">Smith, H. J. S.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A; Nature, Vol. 8 (1873), p. 450.</p> + + <p class="v2"> + <b><a name="Block_1557" id="Block_1557">1557</a>.</b> + Of all the great subjects which belong to the + province of his section, take that which at first sight is the + least within the domain of mathematics—I + mean meteorology. Yet the part which mathematics plays in + meteorology increases every year, and seems destined to + increase. Not only is the theory of the simplest instruments + essentially mathematical, but the discussions of the + observations—upon which, be it remembered, + depend the hopes which are already entertained with increasing + confidence, of reducing the most variable and complex of all + known phenomena to exact laws—is a problem + which not only belongs wholly to mathematics, but which taxes + to the utmost the resources of the mathematics which we now + possess.—<span class="smcap">Smith, H. J. S.</span></p> + <p class="blockcite"> + Presidential Address British Association for the Advancement + of Science, Section A; Nature, Vol. 8 (1873), p. 449.</p> + + <p class="v2"> + <b><a name="Block_1558" id="Block_1558">1558</a>.</b> + You know that if you make a dot on a piece of + paper, and then hold a piece of Iceland spar over it, you will + see not one dot but two. A mineralogist, by measuring the + angles of a crystal, can tell you whether or no it possesses + this property without looking through it. He requires no + scientific thought to do that. But Sir William Roman Hamilton + ... knowing these facts and also the explanation of them which + Fresnel had + +<span class="pagenum"> + <a name="Page_243" + id="Page_243">243</a></span> + + given, thought about the subject, and + he predicted that by looking through certain crystals in a + particular direction we should see not two dots but a + continuous circle. Mr. Lloyd made the experiment, and saw the + circle, a result which had never been even suspected. This has + always been considered one of the most signal instances of + scientific thought in the domain of + physics.—<span class="smcap">Clifford, W. K.</span></p> + <p class="blockcite"> + Lectures and Essays (New York, 1901), Vol. 1, p. 144.</p> + + <p class="v2"> + <b><a name="Block_1559" id="Block_1559">1559</a>.</b> + The discovery of this planet [Neptune] is justly + reckoned as the greatest triumph of mathematical astronomy. + Uranus failed to move precisely in the path which the computers + predicted for it, and was misguided by some unknown influence + to an extent which a keen eye might almost see without + telescopic aid.... These minute discrepancies constituted the + data which were found sufficient for calculating the position + of a hitherto unknown planet, and bringing it to light. + Leverrier wrote to Galle, in substance: + “<em>Direct your telescope to a point on the + ecliptic in the constellation of Aquarius, in longitude + 326°, and you will find within a degree of that + place a new planet, looking like a star of about the ninth + magnitude, and having a perceptible disc.</em>” The + planet was found at Berlin on the night of Sept. 26, 1846, in + exact accordance with this prediction, within half an hour + after the astronomers began looking for it, and only about + 52′ distant from the precise point that + Leverrier had indicated.—<span class= + "smcap">Young, C. A.</span></p> + <p class="blockcite"> + General Astronomy (Boston, 1891), Art. 653.</p> + + <p class="v2"> + <b><a name="Block_1560" id="Block_1560">1560</a>.</b> + I am convinced that the future progress of chemistry as an + exact science depends very much indeed upon the alliance with + mathematics.—<span class="smcap">Frankland, A.</span></p> + <p class="blockcite"> + American Journal of Mathematics, Vol. 1, p. 349.</p> + + <p class="v2"> + <b><a name="Block_1561" id="Block_1561">1561</a>.</b> + It is almost impossible to follow the later + developments of physical or general chemistry without a working + knowledge of higher + mathematics.—<span class="smcap">Mellor, J. W.</span></p> + <p class="blockcite"> + Higher Mathematics (New York, 1902), Preface.</p> + +<p><span class="pagenum"> + <a name="Page_244" + id="Page_244">244</a></span></p> + + <p class="v2"> + <b><a name="Block_1562" id="Block_1562">1562</a>.</b></p> + <div class="poem"> + <p class="i0"> + ... Mount where science guides;</p> + <p class="i0"> + Go measure earth, weigh air, and state the tides;</p> + <p class="i0"> + Instruct the planets in what orb to run,</p> + <p class="i0"> + Correct old time, and regulate the sun.</p> + </div> + <p class="block40"> + —<span class="smcap">Thomson, W.</span></p> + <p class="blockcite"> + On the Figure of the Earth, Title page.</p> + + <p class="v2"> + <b><a name="Block_1563" id="Block_1563">1563</a>.</b> + Admission to its sanctuary [referring to + astronomy] and to the privileges and feelings of a votary, is + only to be gained by one means,—<em>sound and + sufficient knowledge of mathematics, the great instrument of + all exact inquiry, without which no man can ever make such + advances in this or any other of the higher departments of + science as can entitle him to form an independent opinion on + any subject of discussion within their + range.</em>—<span class="smcap">Herschel, J.</span></p> + <p class="blockcite"> + Outlines of Astronomy, Introduction, sect. 7.</p> + + <p class="v2"> + <b><a name="Block_1564" id="Block_1564">1564</a>.</b> + The long series of connected truths which compose + the science of astronomy, have been evolved from the + appearances and observations by calculation, and a process of + reasoning entirely geometrical. It was not without reason that + Plato called geometry and arithmetic the wings of astronomy; + for it is only by means of these two sciences that we can give + a rational account of any of the appearances, or connect any + fact with theory, or even render a single observation available + to the most common astronomical purpose. It is by geometry that + we are enabled to reason our way up through the apparent + motions to the real orbits of the planets, and to assign their + positions, magnitudes and eccentricities. And it is by + application of geometry—a sublime geometry, + indeed, invented for the purpose—to the + general laws of mechanics, that we demonstrate the law of + gravitation, trace it through its remotest effects on the + different planets, and, comparing these effects with what we + observe, determine the densities and weights of the minutest + bodies belonging to the system. The whole science of astronomy + is in fact a tissue of geometrical reasoning, applied to the + data of observation; and it is from this circumstance that it + derives its peculiar character of precision and certainty. To + disconnect it from geometry, therefore, and to substitute + familiar illustrations and vague description for close and + logical reasoning, is to deprive it of its principal + advantages, + +<span class="pagenum"> + <a name="Page_245" + id="Page_245">245</a></span> + + and to reduce it to the condition of + an ordinary province of natural history.</p> + <p class="blockcite"> + Edinburgh Review, Vol. 58 (1833-1834), p. 168.</p> + + <p class="v2"> + <b><a name="Block_1565" id="Block_1565">1565</a>.</b> + But geometry is not only the instrument of + astronomical investigation, and the bond by which the truths + are enchained together,—it is also the + instrument of explanation, affording, by the peculiar brevity + and perspicuity of its technical processes, not only aid to the + learner, but also such facilities to the teacher as he will + find it very difficult to supply, if he voluntarily undertakes + to forego its assistance. Few undertakings, indeed, are + attended with greater difficulty than that of attempting to + exhibit the connecting links of a chain of mathematical + reasoning, when we lay aside the technical symbols and notation + which relieve the memory, and speak at once to the eyes and the + understanding:....</p> + <p class="blockcite"> + Edinburgh Review, Vol. 58 (1833-1834), p. 169.</p> + + <p class="v2"> + <b><a name="Block_1566" id="Block_1566">1566</a>.</b> + With an ordinary acquaintance of trigonometry, and + the simplest elements of algebra, one may take up any + well-written treatise on plane astronomy, and work his way + through it, from beginning to end, with perfect ease; and he + will acquire, in the course of his progress, from the mere + examples put before him, an infinitely more correct and precise + idea of astronomical methods and theories, than he could obtain + in a lifetime from the most eloquent general descriptions that + ever were written. At the same time he will be strengthening + himself for farther advances, and accustoming his mind to + habits of close comparison and rigid demonstration, which are + of infinitely more importance than the acquisition of stores of + undigested facts.</p> + <p class="blockcite"> + Edinburgh Review, Vol. 58 (1833-1834), p. 170.</p> + + <p class="v2"> + <b><a name="Block_1567" id="Block_1567">1567</a>.</b> + While the telescope serves as a means of + penetrating space, and of bringing its remotest regions nearer + us, mathematics, by inductive reasoning, have led us onwards to + the remotest regions of heaven, and brought a portion of them + within the range of our possibilities; nay, in our own + times—so propitious to the extension of + knowledge—the application of + +<span class="pagenum"> + <a name="Page_246" + id="Page_246">246</a></span> + + all the elements yielded by the present conditions of astronomy + has even revealed to the intellectual eyes a heavenly body, and + assigned to it its place, orbit, mass, before a single + telescope has been directed towards + it.—<span class="smcap">Humboldt, A.</span></p> + <p class="blockcite"> + Cosmos [Otte], Vol. 2, part 2, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_1568" id="Block_1568">1568</a>.</b> + Mighty are numbers, joined with art + resistless.—<span class="smcap">Euripides.</span></p> + <p class="blockcite"> + Hecuba, Line 884.</p> + + <p class="v2"> + <b><a name="Block_1569" id="Block_1569">1569</a>.</b> + No single instrument of youthful education has + such mighty power, both as regards domestic economy and + politics, and in the arts, as the study of arithmetic. Above + all, arithmetic stirs up him who is by nature sleepy and dull, + and makes him quick to learn, retentive, shrewd, and aided by + art divine he makes progress quite beyond his natural + powers.—<span class="smcap">Plato.</span></p> + <p class="blockcite"> + Laws [Jowett,] Bk. 5, p. 747.</p> + + <p class="v2"> + <b><a name="Block_1570" id="Block_1570">1570</a>.</b> + For all the higher arts of construction some + acquaintance with mathematics is indispensable. The village + carpenter, who, lacking rational instruction, lays out his work + by empirical rules learned in his apprenticeship, equally with + the builder of a Britannia Bridge, makes hourly reference to + the laws of quantitative relations. The surveyor on whose + survey the land is purchased; the architect in designing a + mansion to be built on it; the builder in preparing his + estimates; his foreman in laying out the foundations; the + masons in cutting the stones; and the various artisans who put + up the fittings; are all guided by geometrical truths. + Railway-making is regulated from beginning to end by + mathematics: alike in the preparation of plans and sections; in + staking out the lines; in the mensuration of cuttings and + embankments; in the designing, estimating, and building of + bridges, culverts, viaducts, tunnels, stations. And similarly + with the harbors, docks, piers, and various engineering and + architectural works that fringe the coasts and overspread the + face of the country, as well as the mines that run underneath + it. Out of geometry, too, as applied to astronomy, the art of + navigation has grown; and so, by this science, has been made + possible that enormous foreign commerce which supports a large + part of our population, and supplies us with many + +<span class="pagenum"> + <a name="Page_247" + id="Page_247">247</a></span> + + necessaries and most of our luxuries. And nowadays even the + farmer, for the correct laying out of his drains, has recourse + to the level—that is, to geometrical + principles.—<span class="smcap">Spencer, Herbert.</span></p> + <p class="blockcite"> + Education, chap. 1.</p> + + <p class="v2"> + <b><a name="Block_1571" id="Block_1571">1571</a>.</b> + [Arithmetic] is another of the great master-keys + of life. With it the astronomer opens the depths of the + heavens; the engineer, the gates of the mountains; the + navigator, the pathways of the deep. The skillful arrangement, + the rapid handling of figures, is a perfect + magician’s wand. The mighty commerce of the + United States, foreign and domestic, passes through the books + kept by some thousands of diligent and faithful clerks. Eight + hundred bookkeepers, in the Bank of England, strike the + monetary balance of half the civilized world. Their skill and + accuracy in applying the common rules of arithmetic are as + important as the enterprise and capital of the merchant, or the + industry and courage of the navigator. I look upon a well-kept + ledger with something of the pleasure with which I gaze on a + picture or a statue. It is a beautiful work of + art.—<span class="smcap">Everett, Edward.</span></p> + <p class="blockcite"> + Orations and Speeches (Boston, 1870), Vol. 3, p. 47.</p> + + <p class="v2"> + <b><a name="Block_1572" id="Block_1572">1572</a>.</b> + [Mathematics] is the fruitful Parent of, I had + almost said all, Arts, the unshaken Foundation of Sciences, and + the plentiful Fountain of Advantage to Human Affairs. In which + last Respect, we may be said to receive from the + <em>Mathematics</em>, the principal Delights of Life, Securities + of Health, Increase of Fortune, and Conveniences of Labour: + That we dwell elegantly and commodiously, build decent Houses + for ourselves, erect stately Temples to God, and leave + wonderful Monuments to Posterity: That we are protected by + those Rampires from the Incursions of the Enemy; rightly use + Arms, skillfully range an Army, and manage War by Art, and not + by the Madness of wild Beasts: That we have safe Traffick + through the deceitful Billows, pass in a direct Road through + the tractless Ways of the Sea, and come to the designed Ports + by the uncertain Impulse of the Winds: That we rightly cast up + our Accounts, do Business expeditiously, dispose, + tabulate, and calculate scattered + +<span class="pagenum"> + <a name="Page_248" + id="Page_248">248</a></span> + + Ranks of Numbers, and easily + compute them, though expressive of huge Heaps of Sand, nay + immense Hills of Atoms: That we make pacifick Separations of + the Bounds of Lands, examine the Moments of Weights in an equal + Balance, and distribute every one his own by a just Measure: + That with a light Touch we thrust forward vast Bodies which way + we will, and stop a huge Resistance with a very small Force: + That we accurately delineate the Face of this Earthly Orb, and + subject the Oeconomy of the Universe to our Sight: That we + aptly digest the flowing Series of Time, distinguish what is + acted by due Intervals, rightly account and discern the various + Returns of the Seasons, the stated Periods of Years and Months, + the alternate Increments of Days and Nights, the doubtful + Limits of Light and Shadow, and the exact Differences of Hours + and Minutes: That we derive the subtle Virtue of the Solar Rays + to our Uses, infinitely extend the Sphere of Sight, enlarge the + near Appearances of Things, bring to Hand Things remote, + discover Things hidden, search Nature out of her Concealments, + and unfold her dark Mysteries: That we delight our Eyes with + beautiful Images, cunningly imitate the Devices and portray the + Works of Nature; imitate did I say? nay excel, while we form to + ourselves Things not in being, exhibit Things absent, and + represent Things past: That we recreate our Minds and delight + our Ears with melodious Sounds, attemperate the inconstant + Undulations of the Air to musical Tunes, add a pleasant Voice + to a sapless Log and draw a sweet Eloquence from a rigid Metal; + celebrate our Maker with an harmonious Praise, and not unaptly + imitate the blessed Choirs of Heaven: That we approach and + examine the inaccessible Seats of the Clouds, the distant + Tracts of Land, unfrequented Paths of the Sea; lofty Tops of + the Mountains, low Bottoms of the Valleys, and deep Gulphs of + the Ocean: That in Heart we advance to the Saints themselves + above, yea draw them to us, scale the etherial Towers, freely + range through the celestial Fields, measure the Magnitudes, and + determine the Interstices of the Stars, prescribe inviolable + Laws to the Heavens themselves, and confine the wandering + Circuits of the Stars within fixed Bounds: Lastly, that we + comprehend the vast Fabrick of the Universe, admire and + contemplate the wonderful Beauty of the Divine + +<span class="pagenum"> + <a name="Page_249" + id="Page_249">249</a></span> + + Workmanship, and to learn the incredible Force and Sagacity of + our own Minds, by certain Experiments, and to acknowledge the + Blessings of Heaven with pious + Affection.—<span class="smcap">Barrow, Isaac.</span></p> + <p class="blockcite"> + Mathematical Lectures (London, 1734), pp. 27-30.</p> + + <p class="v2"> + <b><a name="Block_1573" id="Block_1573">1573</a>.</b> + Analytical and graphical treatment of statistics + is employed by the economist, the philanthropist, the business + expert, the actuary, and even the physician, with the most + surprisingly valuable results; while symbolic language + involving mathematical methods has become a part of wellnigh + every large business. The handling of pig-iron does not seem to + offer any opportunity for mathematical application. Yet + graphical and analytical treatment of the data from + long-continued experiments with this material at Bethlehem, + Pennsylvania, resulted in the discovery of the law that fatigue + varied in proportion to a certain relation between the load and + the periods of rest. Practical application of this law + increased the amount handled by each man from twelve and a half + to forty-seven tons per day. Such study would have been + impossible without preliminary acquaintance with the simple + invariable elements of + mathematics.—<span class="smcap">Karpinsky, L.</span></p> + <p class="blockcite"> + High School Education (New York, 1912), chap. 6, p. 134.</p> + + <p class="v2"> + <b><a name="Block_1574" id="Block_1574">1574</a>.</b> + They [computation and arithmetic] belong then, it + seems, to the branches of learning which we are now + investigating;—for a military man must + necessarily learn them with a view to the marshalling of his + troops, and so must a philosopher with the view of + understanding real being, after having emerged from the + unstable condition of becoming, or else he can never become an + apt reasoner.</p> + <p class="v1"> + That is the fact he replied.</p> + <p class="v1"> + But the guardian of ours happens to be both a military man and + a philosopher.</p> + <p class="v1"> + Unquestionably so.</p> + <p class="v1"> + It would be proper then, Glaucon, to lay down laws for this + branch of science and persuade those about to engage in the + most important state-matters to apply themselves to + computation, + +<span class="pagenum"> + <a name="Page_250" + id="Page_250">250</a></span> + + and study it, not in the common + vulgar fashion, but with the view of arriving at the + contemplation of the nature of numbers by the intellect + itself,—not for the sake of buying and + selling as anxious merchants and retailers, but for war also, + and that the soul may acquire a facility in turning itself from + what is in the course of generation to truth and real + being.—<span class="smcap">Plato.</span></p> + <p class="blockcite"> + Republic [Davis], Bk. 7, p. 525.</p> + + <p class="v2"> + <b><a name="Block_1575" id="Block_1575">1575</a>.</b> + The scientific part of Arithmetic and Geometry + would be of more use for regulating the thoughts and opinions + of men than all the great advantage which Society receives from + the general application of them: and this use cannot be spread + through the Society by the practice; for the Practitioners, + however dextrous, have no more knowledge of the Science than + the very instruments with which they work. They have taken up + the Rules as they found them delivered down to them by + scientific men, without the least inquiry after the Principles + from which they are derived: and the more accurate the Rules, + the less occasion there is for inquiring after the Principles, + and consequently, the more difficult it is to make them turn + their attention to the First Principles; and, therefore, a + Nation ought to have both Scientific and Practical + Mathematicians.—<span + class="smcap">Williamson, James.</span></p> + <p class="blockcite"> + Elements of Euclid with Dissertations (Oxford, 1781).</p> + + <p class="v2"> + <b><a name="Block_1576" id="Block_1576">1576</a>.</b> + <em>Where there is nothing to measure there is + nothing to calculate</em>, hence it is impossible to employ + mathematics in psychological investigations. Thus runs the + syllogism compounded of an adherence to usage and an apparent + truth. As to the latter, it is wholly untrue that we may + calculate only where we have measured. Exactly the opposite is + true. Every hypothetically assumed law of quantitative + combination, even such as is recognized as invalid, is subject + to calculation; and in case of deeply hidden but important + matters it is imperative to try on hypotheses and to subject + the consequences which flow from them to precise computation + until it is found which one of the various hypotheses coincides + with experience. Thus the ancient astronomers <em>tried</em> + eccentric circles, and Kepler + +<span class="pagenum"> + <a name="Page_251" + id="Page_251">251</a></span> + + <em>tried</em> the + ellipse to account for the motion of the planets, the latter + also compared the squares of the times of revolution with the + cubes of the mean distances before he discovered their + agreement. In like manner Newton <em>tried</em> whether a + gravitation, varying inversely as the square of the distance, + sufficed to keep the moon in its orbit about the earth; if this + supposition had failed him, he would have tried some other + power of the distance, as the fourth or fifth, and deduced the + corresponding consequences to compare them with the + observations. Just this is the greatest benefit of mathematics, + that it enables us to survey the possibilities whose range + includes the actual, long before we have adequate definite + experience; this makes it possible to employ very incomplete + indications of experience to avoid at least the crudest errors. + Long before the transit of Venus was employed in the + determination of the sun’s parallax, it was + attempted to determine the instant at which the sun illumines + exactly one-half of the moon’s disk, in + order to compute the sun’s distance from the + known distance of the moon from the earth. This was not + possible, for, owing to psychological reasons, our method of + measuring time is too crude to give us the desired instant with + sufficient accuracy; yet the attempt gave us the knowledge that + the sun’s distance from us is at least + several hundred times as great as that of the moon. This + illustration shows clearly that even a very imperfect estimate + of a magnitude in a case where no precise observation is + possible, may become very instructive, if we know how to + exploit it. Was it necessary to know the scale of our solar + system in order to learn of its order in general? Or, taking an + illustration from another field, was it impossible to + investigate the laws of motion until it was known exactly how + far a body falls in a second at some definite place? Not at + all. Such determinations of <em>fundamental measures</em> are in + themselves exceedingly difficult, but fortunately, such + investigations form a class of their own; our knowledge of + <em>fundamental laws</em> does not need to wait on these. To be + sure, computation invites measurement, and every easily + observed regularity of certain magnitudes is an incentive to + mathematical investigation.—<span class= + "smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Werke [Kehrbach], (Langensalza, 1890), Bd. 5, p. 97.</p> + +<p><span class="pagenum"> + <a name="Page_252" + id="Page_252">252</a></span></p> + + <p class="v2"> + <b><a name="Block_1577" id="Block_1577">1577</a>.</b> + Those who pass for naturalists, have, for the most + part, been very little, or not at all, versed in mathematicks, + if not also jealous of them.—<span class= + "smcap">Boyle, Robert.</span></p> + <p class="blockcite"> + Works (London, 1772), Vol. 3, p. 426.</p> + + <p class="v2"> + <b><a name="Block_1578" id="Block_1578">1578</a>.</b> + However hurtful may have been the incursions of + the geometers, direct and indirect, into a domain which it is + not for them to cultivate, the physiologists are not the less + wrong in turning away from mathematics altogether. It is not + only that without mathematics they could not receive their due + preliminary training in the intervening sciences: it is further + necessary for them to have geometrical and mechanical + knowledge, to understand the structure and the play of the + complex apparatus of the living, and especially the animal + organism. Animal mechanics, statical and dynamical, must be + unintelligible to those who are ignorant of the general laws of + rational mechanics. The laws of equilibrium and motion are ... + absolutely universal in their action, depending wholly on the + energy, and not at all on the nature of the forces considered: + and the only difficulty is in their numerical application in + cases of complexity. Thus, discarding all idea of a numerical + application in biology, we perceive that the general theorems + of statics and dynamics must be steadily verified in the + mechanism of living bodies, on the rational study of which they + cast an indispensable light. The highest orders of animals act + in repose and motion, like any other mechanical apparatus of a + similar complexity, with the one difference of the mover, which + has no power to alter the laws of motion and equilibrium. The + participation of rational mechanics in positive biology is thus + evident. Mechanics cannot dispense with geometry; and beside, + we see how anatomical and physiological speculations involve + considerations of form and position, and require a familiar + knowledge of the principal geometrical laws which may cast + light upon these complex + relations.—<span class="smcap">Comte,A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 5, chap. 1.</p> + + <p class="v2"> + <b><a name="Block_1579" id="Block_1579">1579</a>.</b> + In mathematics we find the primitive source of + rationality; and to mathematics must the biologists resort for + means to carry on their + researches.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 5, chap. 1.</p> + +<p><span class="pagenum"> + <a name="Page_253" id= + "Page_253">253</a></span></p> + + <p class="v2"> + <b><a name="Block_1580" id="Block_1580">1580</a>.</b> + In this school [of mathematics] must they + [biologists] learn familiarly the real characters and + conditions of scientific evidence, in order to transfer it + afterwards to the province of their own theories. The study of + it here, in the most simple and perfect cases, is the only + sound preparation for its recognition in the most complex.</p> + <p class="v1"> + The study is equally necessary for the formation of + intellectual habits; for obtaining an aptitude in forming and + sustaining positive abstractions, without which the comparative + method cannot be used in either anatomy or physiology. The + abstraction which is to be the standard of comparison must be + first clearly formed, and then steadily maintained in its + integrity, or the analysis becomes abortive: and this is so + completely in the spirit of mathematical combinations, that + practice in them is the best preparation for it. A student who + cannot accomplish the process in the more simple case may be + assured that he is not qualified for the higher order of + biological researches, and must be satisfied with the humbler + office of collecting materials for the use of minds of another + order. Hence arises another use of mathematical + training;—that of testing and classifying + minds, as well as preparing and guiding them. Probably as much + good would be done by excluding the students who only encumber + the science by aimless and desultory inquiries, as by fitly + instituting those who can better fulfill its + conditions.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 5, chap. 1.</p> + + <p class="v2"> + <b><a name="Block_1581" id="Block_1581">1581</a>.</b> + There seems no sufficient reason why the use of + scientific fictions, so common in the hands of geometers, + should not be introduced into biology, if systematically + employed, and adopted with sufficient sobriety. In mathematical + studies, great advantages have arisen from imagining a series + of hypothetical cases, the consideration of which, though + artificial, may aid the clearing up of the real subject, or its + fundamental elaboration. This art is usually confounded with + that of hypotheses; but it is entirely different; inasmuch as + in the latter case the solution alone is imaginary; whereas in + the former, the problem itself is radically ideal. Its use can + never be in biology comparable to what it is in mathematics: + but it seems to me that + +<span class="pagenum"> + <a name="Page_254" + id="Page_254">254</a></span> + + the abstract character of the + higher conceptions of comparative biology renders them + susceptible of such treatment. The process will be to + intercalate, among different known organisms, certain purely + fictitious organisms, so imagined as to facilitate their + comparison, by rendering the biological series more homogeneous + and continuous: and it might be that several might hereafter + meet with more or less of a realization among organisms + hitherto unexplored. It may be possible, in the present state + of our knowledge of living bodies, to conceive of a new + organism capable of fulfilling certain given conditions of + existence. However that may be, the collocation of real cases + with well-imagined ones, after the manner of geometers, will + doubtless be practised hereafter, to complete the general laws + of comparative anatomy and physiology, and possibly to + anticipate occasionally the direct exploration. Even now, the + rational use of such an artifice might greatly simplify and + clear up the ordinary system of biological instruction. But it + is only the highest order of investigators who can be trusted + with it. Whenever it is adopted, it will constitute another + ground of relation between biology and + mathematics.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 5, chap. 1.</p> + + <p class="v2"> + <b><a name="Block_1582" id="Block_1582">1582</a>.</b> + I think it may safely enough be affirmed, that he, + that is not so much as indifferently skilled in mathematicks, + can hardly be more than indifferently skilled in the + fundamental principles of + physiology.—<span class="smcap">Boyle, Robert.</span></p> + <p class="blockcite"> + Works (London, 1772), Vol. 3, p. 430.</p> + + <p class="v2"> + <b><a name="Block_1583" id="Block_1583">1583</a>.</b> + It is not only possible but necessary that + mathematics be applied to psychology; the reason for this + necessity lies briefly in this: that by no other means can be + reached that which is the ultimate aim of all speculation, + namely conviction.—<span class="smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Werke [Kehrbach], (Langensalza, 1890), Bd. 5, p. 104.</p> + + <p class="v2"> + <b><a name="Block_1584" id="Block_1584">1584</a>.</b> + All more definite knowledge must start with + computation; and this is of most important consequences not + only for + +<span class="pagenum"> + <a name="Page_255" + id="Page_255">255</a></span> + + the theory of memory, of imagination, + of understanding, but as well for the doctrine of sensations, + of desires, and affections.—<span class= + "smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Werke [Kehrbach], (Langensalza, 1890), Bd. 5, p. 103.</p> + + <p class="v2"> + <b><a name="Block_1585" id="Block_1585">1585</a>.</b> + In the near future mathematics will play an + important part in medicine: already there are increasing + indications that physiology, descriptive anatomy, pathology and + therapeutics cannot escape mathematical + legitimation.—<span class="smcap">Dessoir, Max.</span></p> + <p class="blockcite"> + Westermann’s Monatsberichte, Bd. 77, p. 380; Ahrens: Scherz + und Ernst in der Mathematik (Leipzig, 1904), p. 395.</p> + + <p class="v2"> + <b><a name="Block_1586" id="Block_1586">1586</a>.</b> + The social sciences mathematically developed are + to be the controlling factors in + civilization.—<span class="smcap">White, W. F.</span></p> + <p class="blockcite"> + A Scrap-book of Elementary Mathematics (Chicago, 1908), p. + 208.</p> + + <p class="v2"> + <b><a name="Block_1587" id="Block_1587">1587</a>.</b> + It is clear that this education [referring to + education preparatory to the science of sociology] must rest on + a basis of mathematical philosophy, even apart from the + necessity of mathematics to the study of inorganic philosophy. + It is only in the region of mathematics that sociologists, or + anybody else, can obtain a true sense of scientific evidence, + and form the habit of rational and decisive argumentation; can, + in short, learn to fulfill the logical conditions of all + positive speculation, by studying universal positivism at its + source. This training, obtained and employed with the more care + on account of the eminent difficulty of social science, is what + sociologists have to seek in + mathematics.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 6, chap. 4.</p> + + <p class="v2"> + <b><a name="Block_1588" id="Block_1588">1588</a>.</b> + It is clear that the individual as a social unit + and the state as a social aggregate require a certain modicum + of mathematics, some arithmetic and algebra, to conduct their + affairs. Under this head would fall the theory of interest, + simple and compound, matters of discount and amortization, and, + if lotteries hold a prominent place in raising moneys, as in + some states, questions of probability must be added. As the + state + +<span class="pagenum"> + <a name="Page_256" + id="Page_256">256</a></span> + + becomes more highly organized and + more interested in the scientific analysis of its life, there + appears an urgent necessity for various statistical + information, and this can be properly obtained, reduced, + correlated, and interpreted only when the guiding spirit in the + work have the necessary mathematical training in the theory of + statistics. (Figures may not lie, but statistics compiled + unscientifically and analyzed incompetently are almost sure to + be misleading, and when this condition is unnecessarily chronic + the so-called statisticians may well be called liars.) The + dependence of insurance of various kinds on statistical + information and the very great place which insurance occupies + in the modern state, albeit often controlled by private + corporations instead of by the government, makes the theories + of paramount importance to our social + life.—<span class="smcap">Wilson, E. B.</span></p> + <p class="blockcite"> + Bulletin American Mathematical Society, Vol. 18 (1912), p. + 463.</p> + + <p class="v2"> + <b><a name="Block_1589" id="Block_1589">1589</a>.</b> + The theory of probabilities and the theory of + errors now constitute a formidable body of knowledge of great + mathematical interest and of great practical importance. Though + developed largely through the applications to the more precise + sciences of astronomy, geodesy, and physics, their range of + applicability extends to all the sciences; and they are plainly + destined to play an increasingly important rôle in the + development and in the applications of the sciences of the + future. Hence their study is not only a commendable element + in a liberal education, but some knowledge of them is essential + to a correct understanding of daily + events.—<span class="smcap">Woodward, R. S.</span></p> + <p class="blockcite"> + Probability and Theory of Errors (New York, 1906), + Preface.</p> + + <p class="v2"> + <b><a name="Block_1590" id="Block_1590">1590</a>.</b> + It was not to be anticipated that a new science + [the science of probabilities] which took its rise in games of + chance, and which had long to encounter an obloquy, hardly yet + extinct, due to the prevailing idea that its only end was to + facilitate and encourage the calculations of gamblers, could + ever have attained its present status—that + its aid should be called for in every department of natural + science, both to assist in discovery, which it has repeatedly + done (even in pure mathematics), to minimize the unavoidable + errors of observation, and to detect the presence + +<span class="pagenum"> + <a name="Page_257" + id="Page_257">257</a></span> + + of causes as revealed by observed events. Nor are commercial and + other practical interests of life less indebted to it: wherever + the future has to be forecasted, risk to be provided against, + or the true lessons to be deduced from statistics, it corrects + for us the rough conjectures of common sense, and decides which + course is really, according to the lights of which we are in + possession, the wisest for us to + pursue.—<span class="smcap">Crofton, M.W.</span></p> + <p class="blockcite"> + Encyclopedia Britannica, 9th Edition; Article “Probability”</p> + + <p class="v2"> + <b><a name="Block_1591" id="Block_1591">1591</a>.</b> + The calculus of probabilities, when confined + within just limits, ought to interest, in an equal degree, the + mathematician, the experimentalist, and the statesman. From the + time when Pascal and Fermat established its first principles, + it has rendered, and continues daily to render, services of the + most eminent kind. It is the calculus of probabilities, which, + after having suggested the best arrangements of the tables of + population and mortality, teaches us to deduce from those + numbers, in general so erroneously interpreted, conclusions of + a precise and useful character; it is the calculus of + probabilities which alone can regulate justly the premiums to + be paid for assurances; the reserve funds for the disbursements + of pensions, annuities, discounts, etc. It is under its + influence that lotteries and other shameful snares cunningly + laid for avarice and ignorance have definitely + disappeared.—<span class="smcap">Arago.</span></p> + <p class="blockcite"> + Eulogy on Laplace [Baden-Powell], Smithsonian Report, + 1874, p. 164.</p> + + <p class="v2"> + <b><a name="Block_1592" id="Block_1592">1592</a>.</b> + Men were surprised to hear that not only births, + deaths, and marriages, but the decisions of tribunals, the + results of popular elections, the influence of punishments in + checking crime, the comparative values of medical remedies, the + probable limits of error in numerical results in every + department of physical inquiry, the detection of causes, + physical, social, and moral, nay, even the weight of evidence + and the validity of logical argument, might come to be surveyed + with the lynx-eyed scrutiny of a dispassionate + analysis.—<span class="smcap">Herschel, J.</span></p> + <p class="blockcite"> + Quoted in Encyclopedia Britannica, 9th Edition; Article + “Probability”</p> + +<p><span class="pagenum"> + <a name="Page_258" + id="Page_258">258</a></span></p> + + <p class="v2"> + <b><a name="Block_1593" id="Block_1593">1593</a>.</b> + If economists expect of the application of the + mathematical method any extensive concrete numerical results, + and it is to be feared that like other non-mathematicians all + too many of them think of mathematics as merely an arithmetical + science, they are bound to be disappointed and to find a + paucity of results in the works of the few of their colleagues + who use that method. But they should rather learn, as the + mathematicians among them know full well, that mathematics is + much broader, that it has an abstract quantitative (or even + qualitative) side, that it deals with relations as well as + numbers,....—<span class="smcap">Wilson, E. B.</span></p> + <p class="blockcite"> + Bulletin American Mathematical Society, Vol. 18 (1912), + p. 464.</p> + + <p class="v2"> + <b><a name="Block_1594" id="Block_1594">1594</a>.</b> + The effort of the economist is to <em>see</em>, to + picture the inter-play of economic elements. The more clearly + cut these elements appear in his vision, the better; the more + elements he can grasp and hold in his mind at once, the better. + The economic world is a misty region. The first explorers used + unaided vision. Mathematics is the lantern by which what before + was dimly visible now looms up in firm, bold outlines. The old + phantasmagoria disappear. We see better. We also see + further.—<span class="smcap">Fisher, Irving.</span></p> + <p class="blockcite"> + Transactions of Connecticut Academy, Vol. 9 (1892), p. 119.</p> + + <p class="v2"> + <b><a name="Block_1595" id="Block_1595">1595</a>.</b> + In the great inquiries of the moral and social + sciences ... mathematics (I always mean Applied Mathematics) + affords the only sufficient type of deductive art. Up to this + time, I may venture to say that no one ever knew what deduction + is, as a means of investigating the laws of nature, who had not + learned it from mathematics, nor can any one hope to understand + it thoroughly, who has not, at some time in his life, known + enough of mathematics to be familiar with the instrument at + work.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + An Examination of Sir William Hamilton’s Philosophy + (London, 1878), p. 622.</p> + + <p class="v2"> + <b><a name="Block_1596" id="Block_1596">1596</a>.</b> + Let me pass on to say a word or two about the + teaching of mathematics as an academic training for general + professional + +<span class="pagenum"> + <a name="Page_259" + id="Page_259">259</a></span> + + life. It has immense capabilities in + that respect. If you consider how much of the effectiveness of + an administrator depends upon the capacity for co-ordinating + appropriately a number of different ideas, precise accuracy of + definition, rigidity of proof, and sustained reasoning, strict + in every step, and when you consider what substitutes for these + things nine men out of every ten without special training have + to put up with, it is clear that a man with a mathematical + training has incalculable + advantages.—<span class="smcap">Shaw, W. H.</span></p> + <p class="blockcite"> + Perry’s Teaching of Mathematics (London, 1902), p. 73.</p> + + <p class="v2"> + <b><a name="Block_1597" id="Block_1597">1597</a>.</b> + Before you enter on the study of law a sufficient + ground work must be laid.... Mathematics and natural philosophy + are so useful in the most familiar occurrences of life and are + so peculiarly engaging and delightful as would induce everyone + to wish an acquaintance with them. Besides this, the faculties + of the mind, like the members of a body, are strengthened and + improved by exercise. Mathematical reasoning and deductions + are, therefore, a fine preparation for investigating the + abstruse speculations of the + law.—<span class="smcap">Jefferson, Thomas.</span></p> + <p class="blockcite"> + Quoted in Cajori’s Teaching and History + of Mathematics in the U. S. (Washington, 1890), p. 35.</p> + + <p class="v2"> + <b><a name="Block_1598" id="Block_1598">1598</a>.</b> + It has been observed in England of the study of + law,—though the acquisition of the most + difficult parts of its learning, the interpretation of laws, + the comparison of authorities, and the construction of + instruments, would seem to require philological and critical + training; though the weighing of evidence and the investigation + of probable truth belong to the province of the moral sciences, + and the peculiar duties of the advocate require rhetorical + skill,—yet that a large proportion of the + most distinguished members of the profession has proceeded from + the university (that of Cambridge) most celebrated for the + cultivation of mathematical + studies.—<span class="smcap">Everett, Edward.</span></p> + <p class="blockcite"> + Orations and Speeches (Boston, 1870), Vol. 2, p. 511.</p> + +<p><span class="pagenum"> + <a name="Page_260" + id="Page_260">260</a></span></p> + + <p class="v2"> + <b><a name="Block_1599" id="Block_1599">1599</a>.</b> + All historic science tends to become mathematical. + Mathematical power is classifying + power.—<span class="smcap">Novalis.</span></p> + <p class="blockcite"> + Schriften (Berlin, 1901), Teil 2, p. 192.</p> + + <p class="v2"> + <b><a name="Block_1599a" id="Block_1599a">1599a</a>.</b> + History has never regarded itself as a science of + statistics. It was the Science of Vital Energy in relation with + time; and of late this radiating centre of its life has been + steadily tending,—together with every form of physical and + mechanical energy,—toward mathematical + expression.—<span class= "smcap">Adam, Henry.</span></p> + <p class="blockcite"> + A Letter to American Teachers of History (Washington, + 1910), p. 115.</p> + + <p class="v2"> + <b><a name="Block_1599b" id="Block_1599b">1599b</a>.</b> + Mathematics can be shown to sustain a certain + relation to rhetoric and may aid in determining its + laws.—<span class="smcap">Sherman L. A.</span></p> + <p class="blockcite"> + University [of Nebraska] Studies, Vol. 1, p. 130.</p> + +<p><span class="pagenum"> + <a name="Page_261" + id="Page_261">261</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_XVI"> + CHAPTER XVI<br /> + <span class="large"> + ARITHMETIC</span></h2> + + <p class="v2"> + <b><a name="Block_1601" id="Block_1601">1601</a>.</b> + There is no problem in all mathematics that cannot + be solved by direct counting. But with the present implements + of mathematics many operations can be performed in a few + minutes which without mathematical methods would take a + lifetime.—<span class="smcap">Mach, Ernst.</span></p> + <p class="blockcite"> + Popular Scientific Lectures [McCormack] (Chicago, 1898), + p. 197.</p> + + <p class="v2"> + <b><a name="Block_1602" id="Block_1602">1602</a>.</b> + There is no inquiry which is not finally reducible + to a question of Numbers; for there is none which may not be + conceived of as consisting in the determination of quantities + by each other, according to certain + relations.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 1, chap. 1.</p> + + <p class="v2"> + <b><a name="Block_1603" id="Block_1603">1603</a>.</b> + Pythagoras says that number is the origin of all + things, and certainly the law of number is the key that unlocks + the secrets of the universe. But the law of number possesses an + immanent order, which is at first sight mystifying, but on a + more intimate acquaintance we easily understand it to be + intrinsically necessary; and this law of number explains the + wondrous consistency of the laws of + nature.—<span class="smcap">Carus, Paul.</span></p> + <p class="blockcite"> + Reflections on Magic Squares; Monist, Vol. 16 (1906), p. 139.</p> + + <p class="v2"> + <b><a name="Block_1604" id="Block_1604">1604</a>.</b> + An ancient writer said that arithmetic and + geometry are the <em>wings of mathematics</em>; I believe one can + say without speaking metaphorically that these two sciences are + the foundation and essence of all the sciences which deal with + quantity. Not only are they the foundation, they are also, as + it were, the capstones; for, whenever a result has been arrived + at, in order to use that result, it is necessary to translate + it into numbers or into lines; to translate it into numbers + requires the aid of arithmetic, to translate it into lines + necessitates the use of + geometry.—<span class="smcap">Lagrange.</span></p> + <p class="blockcite"> + Leçons Elémentaires sur les Mathématiques, Leçon seconde.</p> + +<p><span class="pagenum"> + <a name="Page_262" + id="Page_262">262</a></span></p> + + <p class="v2"> + <b><a name="Block_1605" id="Block_1605">1605</a>.</b> + It is number which regulates everything and it is + measure which establishes universal order.... A quiet peace, an + inviolable order, an inflexible security amidst all change and + turmoil characterize the world which mathematics discloses and + whose depths it unlocks.—<span class= + "smcap">Dillmann, E.</span></p> + <p class="blockcite"> + Die Mathematik die Fackelträgerin einer + neuen Zeit (Stuttgart, 1889), p. 12.</p> + + <p class="v2"> + <b><a name="Block_1606" id="Block_1606">1606</a>.</b></p> + <div class="poem"> + <p class="i0"> + Number, the inducer of philosophies,</p> + <p class="i0"> + The synthesis of letters,....</p> + </div> + <p class="block40"> + —<span class="smcap">Aeschylus.</span></p> + <p class="blockcite"> + Quoted in, Thomson, J. A., Introduction to Science, chap. + 1 (London).</p> + + <p class="v2"> + <b><a name="Block_1607" id="Block_1607">1607</a>.</b> + Amongst all the ideas we have, as there is none + suggested to the mind by more ways, so there is none more + simple, than that of <em>unity</em>, or one: it has no shadow of + variety or composition in it; every object our senses are + employed about; every idea in our understanding; every thought + of our minds, brings this idea along with it. And therefore it + is the most intimate to our thoughts, as well as it is, in its + agreement to all other things, <em>the most universal idea we + have</em>.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 2, chap. 16, + sect. 1.</p> + + <p class="v2"> + <b><a name="Block_1608" id="Block_1608">1608</a>.</b> + The <em>simple modes</em> of <em>number</em> are of + all other the most distinct; every the least variation, which + is an unit, making each combination as clearly different from + that which approacheth nearest to it, as the most remote; two + being as distinct from one, as two hundred; and the idea of two + as distinct from the idea of three, as the magnitude of the + whole earth is from that of a + mite.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 2, chap. 16, + sect. 3.</p> + + <p class="v2"> + <b><a name="Block_1609" id="Block_1609">1609</a>.</b> + The number of a class is the class of all classes + similar to the given class.—<span class= + "smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + Principles of Mathematics (Cambridge, 1903), p. 115.</p> + + <p class="v2"> + <b><a name="Block_1610" id="Block_1610">1610</a>.</b> + Number is that property of a group of distinct things which + remains unchanged during any change to which the + +<span class="pagenum"> + <a name="Page_263" + id="Page_263">263</a></span> + + group may be subjected which does not + destroy the distinctness of the individual + things.—<span class="smcap">Fine, H. B.</span></p> + <p class="blockcite"> + Number-system of Algebra (Boston and New York, 1890), p. 3.</p> + + <p class="v2"> + <b><a name="Block_1611" id="Block_1611">1611</a>.</b> + The science of arithmetic may be called the + science of exact limitation of matter and things in space, + force, and time.—<span class="smcap">Parker, F. W.</span></p> + <p class="blockcite"> + Talks on Pedagogics (New York, 1894), p. 64.</p> + + <p class="v2"> + <b><a name="Block_1612" id="Block_1612">1612</a>.</b></p> + <div class="poem"> + <p class="i0"> + Arithmetic is the science of the Evaluation of Functions,</p> + <p class="i0"> + Algebra is the science of the Transformation of Functions.</p> + </div> + <p class="block40"> + —<span class="smcap">Howison, G. H.</span></p> + <p class="blockcite"> + Journal of Speculative Philosophy, Vol. 5, p. 175.</p> + + <p class="v2"> + <b><a name="Block_1613" id="Block_1613">1613</a>.</b> + That <em>arithmetic</em> rests on pure intuition of + <em>time</em> is not so obvious as that geometry is based on pure + intuition of space, but it may be readily proved as follows. + All counting consists in the repeated positing of unity; only + in order to know how often it has been posited, we mark it each + time with a different word: these are the numerals. Now + repetition is possible only through succession: but succession + rests on the immediate intuition of <em>time</em>, it is + intelligible only by means of this latter concept: hence + counting is possible only by means of time.—This dependence + of counting on <em>time</em> is evidenced by the fact that in + all languages multiplication is expressed by “times” [mal], + that is, by a concept of time; sexies, ἑξακις, six fois, + six times.—<span class= "smcap">Schopenhauer, A.</span></p> + <p class="blockcite"> + Die Welt als Vorstellung und Wille; Werke (Frauenstaedt) + (Leipzig, 1877), Bd. 3, p. 39.</p> + + <p class="v2"> + <b><a name="Block_1614" id="Block_1614">1614</a>.</b> + The miraculous powers of modern calculation are + due to three inventions: the Arabic Notation, Decimal Fractions + and Logarithms.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Mathematics (New York, 1897), p. 161.</p> + + <p class="v2"> + <b><a name="Block_1615" id="Block_1615">1615</a>.</b> + The grandest achievement of the Hindoos and the one which, + of all mathematical investigations, has contributed most + +<span class="pagenum"> + <a name="Page_264" + id="Page_264">264</a></span> + + to the general progress of + intelligence, is the invention of the principle of position in + writing numbers.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Mathematics (New York, 1897), p. 87.</p> + + <p class="v2"> + <b><a name="Block_1616" id="Block_1616">1616</a>.</b> + The invention of logarithms and the calculation of + the earlier tables form a very striking episode in the history + of exact science, and, with the exception of the + <cite>Principia</cite> of Newton, there is no mathematical work + published in the country which has produced such important + consequences, or to which so much interest attaches as to + Napier’s + Descriptio.—<span class="smcap">Glaisher, J. W. L.</span></p> + <p class="blockcite"> + Encyclopedia Britannica, 9th Edition; Article + “Logarithms.”</p> + + <p class="v2"> + <b><a name="Block_1617" id="Block_1617">1617</a>.</b> + All minds are equally capable of attaining the + science of numbers: yet we find a prodigious difference in the + powers of different men, in that respect, after they have grown + up, because their minds have been more or less exercised in + it.—<span class="smcap">Johnson, Samuel.</span></p> + <p class="blockcite"> + Boswell’s Life of Johnson, Harper’s Edition (1871), Vol. 2, p. + 33.</p> + + <p class="v2"> + <b><a name="Block_1618" id="Block_1618">1618</a>.</b> + The method of arithmetical teaching is perhaps the + best understood of any of the methods concerned with + elementary studies.—<span + class="smcap">Bain, Alexander.</span></p> + <p class="blockcite"> + Education as a Science (New York, 1898), p. 288.</p> + + <p class="v2"> + <b><a name="Block_1619" id="Block_1619">1619</a>.</b> + What a benefite that onely thyng is, to haue the + witte whetted and sharpened, I neade not trauell to declare, + sith all men confesse it to be as greate as maie be. Excepte + any witlesse persone thinke he maie bee to wise. But he that + most feareth that, is leaste in daunger of it. Wherefore to + conclude, I see moare menne to acknowledge the benefite of + nomber, than I can espie willying to studie, to attaine the + benefites of it. Many praise it, but fewe dooe greatly practise + it: onlesse it bee for the vulgare practice, concernying + Merchaundes trade. Wherein the desire and hope of gain, maketh + many willying to sustaine some trauell. For aide of whom, I did + sette forth the first parte of <cite>Arithmetike</cite>. But + if thei + knewe how faree this seconde parte, doeeth excell the firste + parte, thei would not accoumpte + +<span class="pagenum"> + <a name="Page_265" + id="Page_265">265</a></span> + + any tyme loste, that + were emploied in it. Yea thei would not thinke any tyme well + bestowed till thei had gotten soche habilitie by it, that it + might be their aide in al other + studies.—<span class="smcap">Recorde, Robert.</span></p> + <p class="blockcite"> + Whetstone of Witte (London, 1557).</p> + + <p class="v2"> + <b><a name="Block_1620" id="Block_1620">1620</a>.</b> + You see then, my friend, I observed, that our real + need of this branch of science [arithmetic] is probably because + it seems to compel the soul to use our intelligence in the + search after pure truth.</p> + <p class="v1"> + Aye, remarked he, it does this to a remarkable extent.</p> + <p class="v1"> + Have you ever noticed that those who have a turn for arithmetic + are, with scarcely an exception, naturally quick in all + sciences; and that men of slow intellect, if they be trained + and exercised in this study ... become invariably quicker than + they were before?</p> + <p class="v1"> + Exactly so, he replied.</p> + <p class="v1"> + And, moreover, I think you will not easily find that many + things give the learner and student more trouble than this.</p> + <p class="v1"> + Of course not.</p> + <p class="v1"> + On all these accounts, then, we must not omit this branch of + science, but those with the best of talents should be + instructed therein.—<span class="smcap">Plato.</span></p> + <p class="blockcite"> + Republic [Davis], Bk. 7, chap. 8.</p> + + <p class="v2"> + <b><a name="Block_1621" id="Block_1621">1621</a>.</b> + Arithmetic has a very great and elevating effect, + compelling the soul to reason about abstract number, and if + visible or tangible objects are obtruding upon the argument, + refusing to be satisfied.—<span class="smcap">Plato.</span></p> + <p class="blockcite"> + Republic [Jowett], Bk. 7, p. 525.</p> + + <p class="v2"> + <b><a name="Block_1622" id="Block_1622">1622</a>.</b> + Good arithmetic contributes powerfully to + purposive effort, to concentration, to tenacity of purpose, to + generalship, to faith in right, and to the joy of achievement, + which are the elements that make up efficient citizenship.... + Good arithmetic exalts thinking, furnishes intellectual + pleasure, adds appreciably to love of right, and subordinates + pure memory.—<span class="smcap">Myers, George.</span></p> + <p class="blockcite"> + Monograph on Arithmetic in Public Education (Chicago), p. 21.</p> + +<p><span class="pagenum"> + <a name="Page_266" + id="Page_266">266</a></span></p> + + <p class="v2"> + <b><a name="Block_1623" id="Block_1623">1623</a>.</b> + On the one side we may say that the purpose of + number work is to put a child in possession of the machinery of + calculation; on the other side it is to give him a better + mastery of the world through a clear (mathematical) insight + into the varied physical objects and activities. The whole + world, from one point of view, can be definitely interpreted + and appreciated by mathematical measurements and estimates. + Arithmetic in the common school should give a child this point + of view, the ability to see and estimate things with a + mathematical eye.—<span + class="smcap">McMurray, C. A.</span></p> + <p class="blockcite"> + Special Method in Arithmetic (New York, 1906), p. 18.</p> + + <p class="v2"> + <b><a name="Block_1624" id="Block_1624">1624</a>.</b> + We are so accustomed to hear arithmetic spoken of + as one of the three fundamental ingredients in all schemes of + instruction, that it seems like inquiring too curiously to ask + why this should be. Reading, Writing, and + Arithmetic—these three are assumed to be of + co-ordinate rank. Are they indeed co-ordinate, and if so on + what grounds?</p> + <p class="v1"> + In this modern “trivium” the art + of reading is put first. Well, there is no doubt as to its + right to the foremost place. For reading is the instrument of + all our acquisition. It is indispensable. There is not an hour + in our lives in which it does not make a great difference to us + whether we can read or not. And the art of Writing, too; that + is the instrument of all communication, and it becomes, in one + form or other, useful to us every day. But Counting—doing + sums,—how + often in life does this accomplishment come into exercise? + Beyond the simplest additions, and the power to check the items + of a bill, the arithmetical knowledge required of any + well-informed person in private life is very limited. For all + practical purposes, whatever I may have learned at school of + fractions, or proportion, or decimals, is, unless I happen to + be in business, far less available to me in life than a + knowledge, say, of history of my own country, or the elementary + truths of physics. The truth is, that regarded as practical + <em>arts</em>, reading, writing, and arithmetic have no right to + be classed together as co-ordinate elements of education; for + the last of these is considerably less useful to the average + man or woman not only than the other two, but than + +<span class="pagenum"> + <a name="Page_267" + id="Page_267">267</a></span> + + many others that might be named. But reading, writing, and such + mathematical or logical exercise as may be gained in connection + with the manifestation of numbers, <em>have</em> a right to + constitute the primary elements of instruction. And I believe + that arithmetic, if it deserves the high place that it + conventionally holds in our educational system, deserves it + mainly on the ground that it is to be treated as a logical + exercise. It is the only branch of mathematics which has found + its way into primary and early education; other departments of + pure science being reserved for what is called higher or + university instruction. But all the arguments in favor of + teaching algebra and trigonometry to advanced students, apply + equally to the teaching of the principles or theory of + arithmetic to schoolboys. It is calculated to do for them + exactly the same kind of service, to educate one side of their + minds, to bring into play one set of faculties which cannot be + so severely or properly exercised in any other department of + learning. In short, relatively to the needs of a beginner, + Arithmetic, as a science, is just as valuable—it is certainly + quite as intelligible—as the higher mathematics to a university + student.—<span class="smcap">Fitch, J. G.</span></p> + <p class="blockcite"> + Lectures on Teaching (New York, 1906), pp. 267-268.</p> + + <p class="v2"> + <b><a name="Block_1625" id="Block_1625">1625</a>.</b> + What mathematics, therefore are expected to do for + the advanced student at the university, Arithmetic, if taught + demonstratively, is capable of doing for the children even of + the humblest school. It furnishes training in reasoning, and + particularly in deductive reasoning. It is a discipline in + closeness and continuity of thought. It reveals the nature of + fallacies, and refuses to avail itself of unverified + assumptions. It is the one department of school-study in which + the sceptical and inquisitive spirit has the most legitimate + scope; in which authority goes for nothing. In other + departments of instruction you have a right to ask for the + scholar’s confidence, and to expect many + things to be received on your testimony with the understanding + that they will be explained and verified afterwards. But here + you are justified in saying to your pupil + “Believe nothing which you cannot + understand. Take nothing for granted.” In short, + the proper office of arithmetic is to serve as elementary + +<span class="pagenum"> + <a name="Page_268" + id="Page_268">268</a></span> + + training in logic. All through your + work as teachers you will bear in mind the fundamental + difference between knowing and thinking; and will feel how much + more important relatively to the health of the intellectual + life the habit of thinking is than the power of knowing, or + even facility of achieving visible results. But here this + principle has special significance. It is by Arithmetic more + than by any other subject in the school course that the art of + thinking—consecutively, closely, logically—can be effectually + taught.—<span class="smcap">Fitch, J. G.</span></p> + <p class="blockcite"> + Lectures on Teaching (New York, 1906), pp. 292-293.</p> + + <p class="v2"> + <b><a name="Block_1626" id="Block_1626">1626</a>.</b> + Arithmetic and geometry, those wings on which the + astronomer soars as high as + heaven.—<span class="smcap">Boyle, Robert.</span></p> + <p class="blockcite"> + Usefulness of Mathematics to Natural Philosophy; Works + (London, 1772), Vol. 3, p. 429.</p> + + <p class="v2"> + <b><a name="Block_1627" id="Block_1627">1627</a>.</b> + Arithmetical symbols are written diagrams and + geometrical figures are graphic + formulas.—<span class="smcap">Hilbert, D.</span></p> + <p class="blockcite"> + Mathematical Problems; Bulletin American Mathematical + Society, Vol. 8 (1902), p. 443.</p> + + <p class="v2"> + <b><a name="Block_1628" id="Block_1628">1628</a>.</b> + Arithmetic and geometry are much more certain than + the other sciences, because the objects of them are in + themselves so simple and so clear that they need not suppose + anything which experience can call in question, and both + proceed by a chain of consequences which reason deduces one + from another. They are also the easiest and clearest of all the + sciences, and their object is such as we desire; for, except + for want of attention, it is hardly supposable that a man + should go astray in them. We must not be surprised, however, + that many minds apply themselves by preference to other + studies, or to philosophy. Indeed everyone allows himself more + freely the right to make his guess if the matter be dark than + if it be clear, and it is much easier to have on any question + some vague ideas than to arrive at the truth itself on the + simplest of all.—<span class="smcap">Descartes.</span></p> + <p class="blockcite"> + Rules for the Direction of the Mind; Torrey’s Philosophy of + Descartes (New York, 1892), p. 63.</p> + +<p><span class="pagenum"> + <a name="Page_269" + id="Page_269">269</a></span></p> + + <p class="v2"> + <b><a name="Block_1629" id="Block_1629">1629</a>.</b></p> + <div class="poem"> + <p class="i0"> + Why are <em>wise</em> few, <em>fools</em> numerous in the + excesse?</p> + <p class="i0"> + ’Cause, wanting <em>number</em>, they are + <em>numberlesse</em>.</p> + </div> + <p class="block40"> + —<span class="smcap">Lovelace.</span></p> + <p class="blockcite"> + Noah Bridges: Vulgar Arithmetike (London, 1659), p. 127.</p> + + <p class="v2"> + <b><a name="Block_1630" id="Block_1630">1630</a>.</b> + The clearness and distinctness of each mode of + number from all others, even those that approach nearest, makes + me apt to think that demonstrations in numbers, if they are not + more evident and exact than in extension, yet they are more + general in their use, and more determinate in their + application. Because the ideas of numbers are more precise and + distinguishable than in extension; where every equality and + excess are not so easy to be observed or measured; because our + thoughts cannot in space arrive at any determined smallness + beyond which it cannot go, as an unit; and therefore the + quantity or proportion of any the least excess cannot be + discovered.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 2, chap. 16, + sect. 4.</p> + + <p class="v2"> + <b><a name="Block_1631" id="Block_1631">1631</a>.</b> + Battalions of figures are like battalions of men, + not always as strong as is + supposed.—<span class="smcap">Sage, M.</span></p> + <p class="blockcite"> + Mrs. Piper and the Society for Psychical Research + [Robertson] (New York, 1909), p. 151.</p> + + <p class="v2"> + <b><a name="Block_1632" id="Block_1632">1632</a>.</b> + Number was born in superstition and reared in + mystery,... numbers were once made the foundation of religion + and philosophy, and the tricks of figures have had a marvellous + effect on a credulous people.—<span class= + "smcap">Parker, F. W.</span></p> + <p class="blockcite"> + Talks on Pedagogics (New York, 1894), P. 64.</p> + + <p class="v2"> + <b><a name="Block_1633" id="Block_1633">1633</a>.</b> + A rule to trick th’ arithmetic.—<span + class="smcap">Kipling, R.</span></p> + <p class="blockcite"> + To the True Romance.</p> + + <p class="v2"> + <b><a name="Block_1634" id="Block_1634">1634</a>.</b> + God made integers, all else is the work of + man.—<span class="smcap">Kronecker, L.</span></p> + <p class="blockcite"> + Jahresberichte der Deutschen Mathematiker Vereinigung, Bd. + 2, p. 19.</p> + + <p class="v2"> + <b><a name="Block_1635" id="Block_1635">1635</a>.</b> + Plato said “ἀεὶ ὁ θεὸς γεωμέτρε.” Jacobi changed this to + “ἀεὶ ὁ θεὸς ἀριθμητίζει.” Then came Kronecker + +<span class="pagenum"> + <a name="Page_270" + id="Page_270">270</a></span> + + and created the memorable expression “Die ganzen Zahlen hat + Gott gemacht, alles andere ist + Menschenwerk”—<span class= "smcap">Klein, F.</span></p> + <p class="blockcite"> + Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. + 6, p. 136.</p> + + <p class="v2"> + <b><a name="Block_1636" id="Block_1636">1636</a>.</b> + Integral numbers are the fountainhead of all + mathematics.—<span class="smcap">Minkowski, H.</span></p> + <p class="blockcite"> + Diophantische Approximationen (Leipzig, 1907), Vorrede.</p> + + <p class="v2"> + <b><a name="Block_1637" id="Block_1637">1637</a>.</b> + The “Disquisitiones Arithmeticae” that great book with seven + seals.—<span class="smcap">Merz, J. T.</span></p> + <p class="blockcite"> + A History of European Thought in the Nineteenth Century + (Edinburgh and London, 1908), p. 721.</p> + + <p class="v2"> + <b><a name="Block_1638" id="Block_1638">1638</a>.</b> + It may fairly be said that the germs of the modern + algebra of linear substitutions and concomitants are to be + found in the fifth section of the <cite>Disquisitiones + Arithmeticae</cite>; and inversely, every advance in the algebraic + theory of forms is an acquisition to the arithmetical + theory.—<span class="smcap">Mathews, G. B.</span></p> + <p class="blockcite"> + Theory of Numbers (Cambridge, 1892), Part 1, sect. 48.</p> + + <p class="v2"> + <b><a name="Block_1639" id="Block_1639">1639</a>.</b> + Strictly speaking, the theory of numbers has + nothing to do with negative, or fractional, or irrational + quantities, <em>as such.</em> No theorem which cannot be + expressed without reference to these notions is purely + arithmetical: and no proof of an arithmetical theorem, can be + considered finally satisfactory if it intrinsically depends + upon extraneous analytical + theories.—<span class="smcap">Mathews, G. B.</span></p> + <p class="blockcite"> + Theory of Numbers (Cambridge, 1892), Part 1, sect. 1.</p> + + <p class="v2"> + <b><a name="Block_1640" id="Block_1640">1640</a>.</b> + Many of the greatest masters of the mathematical + sciences were first attracted to mathematical inquiry by + problems relating to numbers, and no one can glance at the + periodicals of the present day which contain questions for + solution without noticing how singular a charm such problems + still continue to exert. The interest in numbers seems + implanted in the human mind, and it is a pity that it should + not have freer scope in this country. The + methods of the theory of numbers + +<span class="pagenum"> + <a name="Page_271" + id="Page_271">271</a></span> + + are peculiar to itself, and + are not readily acquired by a student whose mind has for years + been familiarized with the very different treatment which is + appropriate to the theory of continuous magnitude; it is + therefore extremely desirable that some portion of the theory + should be included in the ordinary course of mathematical + instruction at our University. From the moment that Gauss, in + his wonderful treatise of 1801, laid down the true lines of the + theory, it entered upon a new day, and no one is likely to be + able to do useful work in any part of the subject who is + unacquainted with the principles and conceptions with which he + endowed it.—<span class="smcap">Glaisher, J. W. L.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science (1890); Nature, Vol. 42, p. 467.</p> + + <p class="v2"> + <b><a name="Block_1641" id="Block_1641">1641</a>.</b> + Let us look for a moment at the general + significance of the fact that calculating machines actually + exist, which relieve mathematicians of the purely mechanical + part of numerical computations, and which accomplish the work + more quickly and with a greater degree of accuracy; for the + machine is not subject to the slips of the human calculator. + The existence of such a machine proves that computation is not + concerned with the significance of numbers, but that it is + concerned essentially only with the formal laws of operation; + for it is only these that the machine can obey—having been thus + constructed—an intuitive perception of the significance of + numbers being out of the + question.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Elementarmathematik vom höheren Standpunkte + aus. (Leipzig, 1908), Bd. 1, p. 53.</p> + + <p class="v2"> + <b><a name="Block_1642" id="Block_1642">1642</a>.</b> + Mathematics is the queen of the sciences and + arithmetic the queen of mathematics. She often condescends to + render service to astronomy and other natural sciences, but in + all relations she is entitled to the first + rank.—<span class="smcap">Gauss.</span></p> + <p class="blockcite"> + Sartorius von Waltershausen: Gauss zum + Gedächtniss. (Leipzig, 1866), p. 79.</p> + + <p class="v2"> + <b><a name="Block_1643" id="Block_1643">1643</a>.</b></p> + <div class="poem"> + <p class="i0"> + Zu Archimedes kam ein wissbegieriger Jüngling,</p> + <p class="i0"> + Weihe + mich, sprach er zu ihm, ein in die göttliche + Kunst, + +<span class="pagenum"> + <a name="Page_272" + id="Page_272">272</a></span></p> + + <p class="i0"> + Die so herrliche Dienste der Sternenkunde geleistet,</p> + <p class="i0"> + Hinter dem Uranos noch einen Planeten entdeckt.</p> + <p class="i0"> + Göttlich nennst Du die Kunst, sie ist’s, versetzte + der Weise,</p> + <p class="i0"> + Aber sie war es, bevor noch sie den Kosmos erforscht,</p> + <p class="i0"> + Ehe sie herrliche Dienste der Sternenkunde geleistet,</p> + <p class="i0"> + Hinter dem Uranos noch einen Planeten entdeckt.</p> + <p class="i0"> + Was Du im Kosmos erblickst, ist nur der Göttlichen Abglanz,</p> + <p class="i0"> + In der Olympier Schaar thronet die ewige Zahl.</p> + </div> + <p class="block40"> + —<span class="smcap">Jacobi, C. G. J.</span></p> + <p class="blockcite"> + Journal für Mathematik, Bd. 101 (1887), p. 338.</p> + + <div class="poem"> + <br /> + <p class="i0"> + To Archimedes came a youth intent upon + knowledge,</p> + <p class="i0"> + Quoth he, + “Initiate me into the science + divine</p> + <p class="i0"> + Which to astronomy, lo! + such excellent service has rendered,</p> + <p class="i0"> + And beyond Uranus’ orb a + hidden planet revealed.”</p> + <p class="i0"> + “Call’st + thou the science divine? So it is,” the wise man + responded,</p> + <p class="i0"> + “But so it was long before its light + on the Cosmos it shed,</p> + <p class="i0"> + Ere in + astronomy’s realm such excellent service + it rendered,</p> + <p class="i0"> + And beyond + Uranus’ orb a hidden planet + revealed.</p> + <p class="i0"> + Only reflection + divine is that which Cosmos discloses,</p> + <p class="i0"> + Number herself sits enthroned among + Olympia’s hosts.”</p> + </div> + + <p class="v2"> + <b><a name="Block_1644" id="Block_1644">1644</a>.</b> + The higher arithmetic presents us with an + inexhaustible store of interesting + truths,—of truths too, which are not + isolated, but stand in a close internal connexion, and between + which, as our knowledge increases, we are continually + discovering new and sometimes wholly unexpected ties. A great + part of its theories derives an additional charm from the + peculiarity that important propositions, with the impress of + simplicity upon them, are often easily discoverable by + induction, and yet are of so profound a character that we + cannot find their demonstration + +<span class="pagenum"> + <a name= "Page_273" + id="Page_273">273</a></span> + + till after many vain + attempts; and even then, when we do succeed, it is often by + some tedious and artificial process, while the simpler methods + may long remain concealed.—<span class= + "smcap">Gauss, C. F.</span></p> + <p class="blockcite"> + Preface to Eisenstein’s Mathematische + Abhandlungen (Berlin, 1847), [H. J. S. Smith].</p> + + <p class="v2"> + <b><a name="Block_1645" id="Block_1645">1645</a>.</b> + The Theory of Numbers has acquired a great and + increasing claim to the attention of mathematicians. It is + equally remarkable for the number and importance of its + results, for the precision and rigorousness of its + demonstrations, for the variety of its methods, for the + intimate relations between truths apparently isolated which it + sometimes discloses, and for the numerous applications of which + it is susceptible in other parts of + analysis.—<span class="smcap">Smith, H. J. S.</span></p> + <p class="blockcite"> + Report on the Theory of Numbers, British Association, + 1859; Collected Mathematical Papers, Vol. 1, p. 38.</p> + + <p class="v2"> + <b><a name="Block_1646" id="Block_1646">1646</a>.</b> + The invention of the symbol ≡ by Gauss affords a striking + example of + the advantage which may be derived from an appropriate + notation, and marks an epoch in the development of the science + of arithmetic.—<span class="smcap">Mathews, G. B.</span></p> + <p class="blockcite"> + Theory of Numbers (Cambridge, 1892), Part 1, sect. 29.</p> + + <p class="v2"> + <b><a name="Block_1647" id="Block_1647">1647</a>.</b> + As Gauss first pointed out, the problem of + cyclotomy, or division of the circle into a number of equal + parts, depends in a very remarkable way upon arithmetical + considerations. We have here the earliest and simplest example + of those relations of the theory of numbers to transcendental + analysis, and even to pure geometry, which so often + unexpectedly present themselves, and which, at first sight, are + so mysterious.—<span class="smcap">Mathews, G. B.</span></p> + <p class="blockcite"> + Theory of Numbers (Cambridge, 1892), Part 1, sect. 167.</p> + + <p class="v2"> + <b><a name="Block_1648" id="Block_1648">1648</a>.</b> + I have sometimes thought that the profound mystery + which envelops our conceptions relative to prime numbers + depends upon the limitations of our faculties in regard to + time, + +<span class="pagenum"> + <a name="Page_274" + id="Page_274">274</a></span> + + which like space may be in its + essence poly-dimensional, and that this and such sort of truths + would become self-evident to a being whose mode of perception + is according to <em>superficially</em> as distinguished from our + own limitation to <em>linearly</em> extended + time.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Collected Mathematical Papers, Vol. 4, p. 600, footnote.</p> + +<p><span class="pagenum"> + <a name="Page_275" + id= "Page_275">275</a></span> </p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_XVII"> + CHAPTER XVII<br /> + <span class="large"> + ALGEBRA</span></h2> + + <p class="v2"> + <b><a name="Block_1701" id="Block_1701">1701</a>.</b> + The science of algebra, independently of any of + its uses, has all the advantages which belong to mathematics in + general as an object of study, and which it is not necessary to + enumerate. Viewed either as a science of quantity, or as a + language of symbols, it may be made of the greatest service to + those who are sufficiently acquainted with arithmetic, and who + have sufficient power of comprehension to enter fairly upon its + difficulties.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Elements of Algebra (London, 1837), Preface.</p> + + <p class="v2"> + <b><a name="Block_1702" id="Block_1702">1702</a>.</b> + Algebra is generous, she often gives more than is + asked of her.—<span class="smcap">D’Alembert.</span></p> + <p class="blockcite"> + Quoted in Bulletin American Mathematical Society, Vol. 2 + (1905), p. 285.</p> + + <p class="v2"> + <b><a name="Block_1703" id="Block_1703">1703</a>.</b> + The operations of symbolic arithmetick seem to me + to afford men one of the clearest exercises of reason that I + ever yet met with, nothing being there to be performed without + strict and watchful ratiocination, and the whole method and + progress of that appearing at once upon the paper, when the + operation is finished, and affording the analyst a lasting, + and, as it were, visible + ratiocination.—<span class="smcap">Boyle, Robert.</span></p> + <p class="blockcite"> + Works (London, 1772), Vol. 3, p. 426.</p> + + <p class="v2"> + <b><a name="Block_1704" id="Block_1704">1704</a>.</b> + The human mind has never invented a labor-saving + machine equal to algebra.—</p> + <p class="blockcite"> + The Nation, Vol. 33, p. 237.</p> + + <p class="v2"> + <b><a name="Block_1705" id="Block_1705">1705</a>.</b> + They that are ignorant of Algebra cannot imagine + the wonders in this kind are to be done by it: and what further + improvements and helps advantageous to other parts of knowledge + the sagacious mind of man may yet find out, it is not easy to + determine. This at least I believe, that the <em>ideas of + quantity</em> + +<span class="pagenum"> + <a name="Page_276" + id="Page_276">276</a></span> + + are not those alone that are capable + of demonstration and knowledge; and that other, and perhaps + more useful, parts of contemplation, would afford us certainty, + if vices, passions, and domineering interest did not oppose and + menace such endeavours.—<span + class="smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 4, chap. 3, + sect. 18.</p> + + <p class="v2"> + <b><a name="Block_1706" id="Block_1706">1706</a>.</b> + Algebra is but written geometry and geometry is but figured + algebra.—<span class="smcap">Germain, Sophie.</span></p> + <p class="blockcite"> + Mémoire sur la surfaces élastiques.</p> + + <p class="v2"> + <b><a name="Block_1707" id="Block_1707">1707</a>.</b> + So long as algebra and geometry proceeded + separately their progress was slow and their application + limited, but when these two sciences were united, they mutually + strengthened each other, and marched together at a rapid pace + toward perfection.—<span class="smcap">Lagrange.</span></p> + <p class="blockcite"> + Leçons élémentaires sur les Mathématiques, Leçon Cinquième.</p> + + <p class="v2"> + <b><a name="Block_1708" id="Block_1708">1708</a>.</b> + The laws of algebra, though suggested by + arithmetic, do not depend on it. They depend entirely on the + conventions by which it is stated that certain modes of + grouping the symbols are to be considered as identical. This + assigns certain properties to the marks which form the symbols + of algebra. The laws regulating the manipulation of algebraic + symbols are identical with those of arithmetic. It follows that + no algebraic theorem can ever contradict any result which could + be arrived at by arithmetic; for the reasoning in both cases + merely applies the same general laws to different classes of + things. If an algebraic theorem can be interpreted in + arithmetic, the corresponding arithmetical theorem is therefore + true.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Universal Algebra (Cambridge, 1898), p. 2.</p> + + <p class="v2"> + <b><a name="Block_1709" id="Block_1709">1709</a>.</b> + That a formal science like algebra, the creation + of our abstract thought, should thus, in a sense, dictate the + laws of its own being, is very remarkable. It has required the + experience of centuries for us to realize the full force of + this appeal.—<span class="smcap">Mathews, G. B.</span></p> + <p class="blockcite"> + F. Spencer: Chapters on Aims and Practice of Teaching + (London, 1899), p. 184.</p> + +<p><span class="pagenum"> + <a name="Page_277" + id="Page_277">277</a></span></p> + + <p class="v2"> + <b><a name="Block_1710" id="Block_1710">1710</a>.</b> + The rules of algebra may be investigated by its + own principles, without any aid from geometry; and although in + many cases the two sciences may serve to illustrate each other, + there is not now the least necessity in the more elementary + parts to call in the aid of the latter in expounding the + former.—<span class="smcap">Chrystal, George.</span></p> + <p class="blockcite"> + Encyclopedia Britannica, 9th Edition; Article “Algebra”</p> + + <p class="v2"> + <b><a name="Block_1711" id="Block_1711">1711</a>.</b> + Algebra, as an art, can be of no use to any one in + the business of life; certainly not as taught in the schools. I + appeal to every man who has been through the school routine + whether this be not the case. Taught as an art it is of little + use in the higher mathematics, as those are made to feel who + attempt to study the differential calculus without knowing more + of the principles than is contained in books of + rules.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Elements of Algebra (London, 1837), Preface.</p> + + <p class="v2"> + <b><a name="Block_1712" id="Block_1712">1712</a>.</b> + We may always depend upon it that algebra, which + cannot be translated into good English and sound common sense, + is bad algebra.—<span class="smcap">Clifford, W. K.</span></p> + <p class="blockcite"> + Common Sense in the Exact Sciences (London, 1885), chap. + 1, sect. 7.</p> + + <p class="v2"> + <b><a name="Block_1713" id="Block_1713">1713</a>.</b> + The best review of arithmetic consists in the + study of algebra.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + Teaching and History of Mathematics in U. S. (Washington, + 1896), p. 110.</p> + + <p class="v2"> + <b><a name="Block_1714" id="Block_1714">1714</a>.</b> + [Algebra] has for its object the resolution of + equations; taking this expression in its full logical meaning, + which signifies the transformation of implicit functions into + equivalent explicit ones. In the same way arithmetic may be + defined as destined to the determination of the values of + functions.... We will briefly say that <em>Algebra is the + Calculus of Functions</em>, and <em>Arithmetic the Calculus of + Values</em>.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Philosophy of Mathematics [Gillespie] (New York, 1851), p. 55.</p> + +<p><span class="pagenum"> + <a name="Page_278" + id="Page_278">278</a></span></p> + + <p class="v2"> + <b><a name="Block_1715" id="Block_1715">1715</a>.</b> + ... the subject matter of algebraic science is the + abstract notion of time; divested of, or not yet clothed with, + any actual knowledge which we may possess of the real Events of + History, or any conception which we may frame of Cause and + Effect in Nature; but involving, what indeed it <em>cannot</em> + be divested of, the thought of <em>possible</em> Succession, or + of pure, <em>ideal</em> + Progression.—<span class="smcap">Hamilton, W. R.</span></p> + <p class="blockcite"> + Graves’ Life of Hamilton (New York, 1882-1889), Vol. 3, + p. 633.</p> + + <p class="v2"> + <b><a name="Block_1716" id="Block_1716">1716</a>.</b> + ... instead of seeking to attain consistency and + uniformity of system, as some modern writers have attempted, by + banishing this thought of time from the <em>higher</em> Algebra, + I seek to <em>attain</em> the same object, by systematically + introducing it into the <em>lower</em> or earlier parts of the + science.—<span class="smcap">Hamilton, W. R.</span></p> + <p class="blockcite"> + Graves’ Life of Hamilton (New York, + 1882-1889), Vol. 3, p. 634.</p> + + <p class="v2"> + <b><a name="Block_1717" id="Block_1717">1717</a>.</b> + The circumstances that algebra has its origin in + arithmetic, however widely it may in the end differ from that + science, led Sir Isaac Newton to designate it + “Universal Arithmetic,” a + designation which, vague as it is, indicates its character + better than any other by which it has been attempted to express + its functions—better certainly, to ordinary + minds, than the designation which has been applied to it by Sir + William Rowan Hamilton, one of the greatest mathematicians the + world has seen since the days of Newton—“the Science of Pure + Time;” or even than the title by which De Morgan would + paraphrase Hamilton’s words—“the Calculus of + Succession”—<span class= "smcap">Chrystal, George.</span></p> + <p class="blockcite"> + Encyclopedia Britannica, 9th Edition; Article “Algebra”</p> + + <p class="v2"> + <b><a name="Block_1718" id="Block_1718">1718</a>.</b> + Time is said to have only <em>one dimension</em>, + and space to have <em>three dimensions</em>.... The mathematical + <em>quaternion</em> partakes of <em>both</em> these elements; in + technical language it may be said to be “time plus space,” or + “space plus time:” and in this + sense it has, or at least involves a reference to, <em>four + dimensions</em>....</p> + +<p><span class="pagenum"> + <a name="Page_279" + id="Page_279">279</a></span></p> + + <div class="poem"> + <p class="i0"> + And how the One of Time, of Space the Three,</p> + <p class="i0"> + Might in the Chain of Symbols girdled be.</p> + </div> + <p class="block40"> + —<span class="smcap">Hamilton, W. R.</span></p> + <p class="blockcite"> + Graves’ Life of Hamilton (New York, + 1882-1889), Vol. 3, p. 635.</p> + + <p class="v2"> + <b><a name="Block_1719" id="Block_1719">1719</a>.</b> + It is confidently predicted, by those best + qualified to judge, that in the coming centuries + Hamilton’s Quaternions will stand out as the + great discovery of our nineteenth century. Yet how silently has + the book taken its place upon the shelves of the + mathematician’s library! Perhaps not fifty + men on this side of the Atlantic have seen it, certainly not + five have read it.—<span class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + North American Review, Vol. 85, p. 223.</p> + + <p class="v2"> + <b><a name="Block_1720" id="Block_1720">1720</a>.</b> + I think the time may come when double algebra will + be the beginner’s tool; and quaternions will + be where double algebra is now. The Lord only knows what will + come above the quaternions.—<span class= + "smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Graves’ Life of Hamilton (New York, + 1882-1889), Vol. 3, p. 493.</p> + + <p class="v2"> + <b><a name="Block_1721" id="Block_1721">1721</a>.</b> + Quaternions came from Hamilton after his really + good work had been done; and though beautifully ingenious, have + been an unmixed evil to those who have touched them in any way, + including Clerk Maxwell.—<span class= + "smcap">Thomson, William.</span></p> + <p class="blockcite"> + Thompson, S. P.: Life of Lord Kelvin (London, 1910), p. 1138.</p> + + <p class="v2"> + <b><a name="Block_1722" id="Block_1722">1722</a>.</b> + The whole affair [quaternions] has in respect to + mathematics a value not inferior to that of + “Volapuk” in respect to + language.—<span class="smcap">Thomson, William.</span></p> + <p class="blockcite"> + Thompson, S. P.: Life of Lord Kelvin (London, 1910), p. 1138.</p> + + <p class="v2"> + <b><a name="Block_1723" id="Block_1723">1723</a>.</b> + A quaternion of maladies! Do send me some formula + by help of which I may so doctor them that they may all become + imaginary or positively equal to + nothing.—<span class="smcap">Sedgwick.</span></p> + <p class="blockcite"> + Graves’ Life of Hamilton (New York, + 1882-1889), Vol. 3, p. 2.</p> + +<p><span class="pagenum"> + <a name="Page_280" + id="Page_280">280</a></span></p> + + <p class="v2"> + <b><a name="Block_1724" id="Block_1724">1724</a>.</b> + If nothing more could be said of Quaternions than + that they enable us to exhibit in a singularly compact and + elegant form, whose meaning is obvious at a glance on account + of the utter inartificiality of the method, results which in + the ordinary Cartesian co-ordinates are of the utmost + complexity, a very powerful argument for their use would be + furnished. But it would be unjust to Quaternions to be content + with such a statement; for we are fully entitled to say that in + <em>all</em> cases, even in those to which the Cartesian methods + seem specially adapted, they give as simple an expression as + any other method; while in the great majority of cases they + give a vastly simpler one. In the common methods a judicious + choice of co-ordinates is often of immense importance in + simplifying an investigation; in Quaternions there is usually + <em>no choice</em>, for (except when they degrade to mere + scalars) they are in general utterly independent of any + particular directions in space, and select of themselves the + most natural reference lines for each particular + problem.—<span class="smcap">Tait, P. G.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science (1871); Nature, Vol. 4, p. 270.</p> + + <p class="v2"> + <b><a name="Block_1725" id="Block_1725">1725</a>.</b> + Comparing a Quaternion investigation, no matter in + what department, with the equivalent Cartesian one, even when + the latter has availed itself to the utmost of the improvements + suggested by Higher Algebra, one can hardly help making the + remark that they contrast even more strongly than the decimal + notation with the binary scale, or with the old Greek + arithmetic—or than the well-ordered + subdivisions of the metrical system with the preposterous + no-systems of Great Britain, a mere fragment of which (in the + form of Table of Weights and Measures) form, perhaps the most + effective, if not the most ingenious, of the many instruments + of torture employed in our elementary + teaching.—<span class="smcap">Tait, P. G.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science (1871); Nature, Vol. 4, p. 271.</p> + + <p class="v2"> + <b><a name="Block_1726" id="Block_1726">1726</a>.</b> + It is true that, in the eyes of the pure + mathematician, Quaternions have one grand and fatal defect. + They cannot be + +<span class="pagenum"> + <a name="Page_281" + id="Page_281">281</a></span> + + applied to space of <em>n</em> + dimensions, they are contented to deal with those poor three + dimensions in which mere mortals are doomed to dwell, but which + cannot bound the limitless aspirations of a Cayley or a + Sylvester. From the physical point of view this, instead of a + defect, is to be regarded as the greatest possible + recommendation. It shows, in fact, Quaternions to be the + special instrument so constructed for application to the + <em>Actual</em> as to have thrown overboard everything which is + not absolutely necessary, without the slightest consideration + whether or no it was thereby being rendered useless for + application to the <em>Inconceivable</em>.—<span + class="smcap">Tait, P. G.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science (1871); Nature, Vol. 4, p. 271.</p> + + <p class="v2"> + <b><a name="Block_1727" id="Block_1727">1727</a>.</b> + There is an old epigram which assigns the empire + of the sea to the English, of the land to the French, and of + the clouds to the Germans. Surely it was from the clouds that + the Germans fetched + and −; the ideas which these symbols have + generated are much too important to the welfare of humanity to + have come from the sea or from the + land.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + An Introduction to Mathematics (New York, 1911), p. 86.</p> + + <p class="v2"> + <b><a name="Block_1728" id="Block_1728">1728</a>.</b> + Now as to what pertains to these Surd numbers + (which, as it were by way of reproach and calumny, having no + merit of their own are also styled Irrational, Irregular, and + Inexplicable) they are by many denied to be numbers properly + speaking, and are wont to be banished from arithmetic to + another Science, (which yet is no science) viz. + algebra.—<span class="smcap">Barrow, Isaac.</span></p> + <p class="blockcite"> + Mathematical Lectures (London, 1734), p. 44.</p> + + <p class="v2"> + <b><a name="Block_1729" id="Block_1729">1729</a>.</b> + If it is true as Whewell says, that the essence of + the triumphs of science and its progress consists in that it + enables us to consider evident and necessary, views which our + ancestors held to be unintelligible and were unable to + comprehend, then the extension of the number concept to include + the irrational, and we will at once add, the imaginary, is the + greatest forward step which pure mathematics has ever + taken.—<span class="smcap">Hankel, Hermann.</span></p> + <p class="blockcite"> + Theorie der Complexen Zahlen (Leipzig, 1867), p. 60.</p> + +<p><span class="pagenum"> + <a name="Page_282" + id="Page_282">282</a></span></p> + + <p class="v2"> + <b><a name="Block_1730" id="Block_1730">1730</a>.</b> + That this subject [of imaginary magnitudes] has + hitherto been considered from the wrong point of view and + surrounded by a mysterious obscurity, is to be attributed + largely to an ill-adapted notation. If for instance, +1,−1, + √−1 had been called direct, inverse, and + lateral units, instead of positive, negative, and imaginary (or + even impossible) such an obscurity would have been out of + question.—<span class="smcap">Gauss, C. F.</span></p> + <p class="blockcite"> + Theoria residiorum biquadraticorum, Commentatio secunda; + Werke, Bd. 2 (Goettingen, 1863), p. 177.</p> + + <p class="v2"> + <b><a name="Block_1731" id="Block_1731">1731</a>.</b> + ... the imaginary, this bosom-child of complex + mysticism.—<span class="smcap">Dühring, Eugen.</span></p> + <p class="blockcite"> + Kritische Geschichte der allgemeinen Principien der + Mechanik (Leipzig, 1877), p. 517.</p> + + <p class="v2"> + <b><a name="Block_1732" id="Block_1732">1732</a>.</b> + Judged by the only standards which are admissible + in a pure doctrine of numbers <em>i</em> is imaginary in the same + sense as the negative, the fraction, and the irrational, but in + no other sense; all are alike mere symbols devised for the sake + of representing the results of operations even when these + results are not numbers (positive + integers).—<span class="smcap">Fine, H. B.</span></p> + <p class="blockcite"> + The Number-System of Algebra (Boston, 1890), p. 36.</p> + + <p class="v2"> + <b><a name="Block_1733" id="Block_1733">1733</a>.</b> + This symbol [√−1] is + restricted to a precise signification as the representative of + perpendicularity in quaternions, and this wonderful algebra of + space is intimately dependent upon the special use of the + symbol for its symmetry, elegance, and power. The immortal + author of quaternions has shown that there are other + significations which may attach to the symbol in other cases. + But the strongest use of the symbol is to be found in its + magical power of doubling the actual universe, and placing by + its side an ideal universe, its exact counterpart, with which + it can be compared and contrasted, and, by means of curiously + connecting fibres, form with it an organic whole, from which + modern analysis has developed her surpassing + geometry.—<span class="smcap">Peirce, Benjamin.</span></p> + <p class="blockcite"> + On the Uses and Transformations of Linear Algebras; + American Journal of Mathematics, Vol. 4 (1881), p. 216.</p> + +<p><span class="pagenum"> + <a name="Page_283" + id="Page_283">283</a></span></p> + + <p class="v2"> + <b><a name="Block_1734" id="Block_1734">1734</a>.</b> + The conception of the inconceivable [imaginary], + this measurement of what not only does not, but cannot exist, + is one of the finest achievements of the human intellect. No + one can deny that such imaginings are indeed imaginary. But + they lead to results grander than any which flow from the + imagination of the poet. The imaginary calculus is one of the + masterkeys to physical science. These realms of the + inconceivable afford in many places our only mode of passage to + the domains of positive knowledge. Light itself lay in darkness + until this imaginary calculus threw light upon light. And in + all modern researches into electricity, magnetism, and heat, + and other subtile physical inquiries, these are the most + powerful instruments.—<span + class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + North American Review, Vol. 85, p. 235.</p> + + <p class="v2"> + <b><a name="Block_1735" id="Block_1735">1735</a>.</b> + All the fruitful uses of imaginaries, in Geometry, + are those which begin and end with real quantities, and use + imaginaries only for the intermediate steps. Now in all such + cases, we have a real spatial interpretation at the beginning + and end of our argument, where alone the spatial interpretation + is important; in the intermediate links, we are dealing in + purely algebraic manner with purely algebraic quantities, and + may perform any operations which are algebraically permissible. + If the quantities with which we end are capable of spatial + interpretation, then, and only then, our results may be + regarded as geometrical. To use geometrical language, in any + other case, is only a convenient help to the imagination. To + speak, for example, of projective properties which refer to the + circular points, is a mere <i lang="la" xml:lang="la">memoria + technica</i> for purely + algebraical properties; the circular points are not to be found + in space, but only in the auxiliary quantities by which + geometrical equations are transformed. That no contradictions + arise from the geometrical interpretation of imaginaries is not + wonderful; for they are interpreted solely by the rules of + Algebra, which we may admit as valid in their interpretation to + imaginaries. The perception of space being wholly absent, + Algebra rules supreme, and no inconsistency can + arise.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + Foundations of Geometry (Cambridge, 1897), p. 45.</p> + +<p><span class="pagenum"> + <a name="Page_284" + id="Page_284">284</a></span></p> + + <p class="v2"> + <b><a name="Block_1736" id="Block_1736">1736</a>.</b> + Indeed, if one understands by algebra the + application of arithmetic operations to composite magnitudes of + all kinds, whether they be rational or irrational number or + space magnitudes, then the learned Brahmins of Hindostan are + the true inventors of algebra.—<span class= + "smcap">Hankel, Hermann.</span></p> + <p class="blockcite"> + Geschichte der Mathematik im Altertum und Mittelalter + (Leipzig, 1874), p. 195.</p> + + <p class="v2"> + <b><a name="Block_1737" id="Block_1737">1737</a>.</b> + It is remarkable to what extent Indian mathematics + enters into the science of our time. Both the form and the + spirit of the arithmetic and algebra of modern times are + essentially Indian and not + Grecian.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Mathematics (New York, 1897), p. 100.</p> + + <p class="v2"> + <b><a name="Block_1738" id="Block_1738">1738</a>.</b> + There are many questions in this science [algebra] + which learned men have to this time in vain attempted to solve; + and they have stated some of these questions in their writings, + to prove that this science contains difficulties, to silence + those who pretend they find nothing in it above their ability, + to warn mathematicians against undertaking to answer every + question that may be proposed, and to excite men of genius to + attempt their solution. Of these I have selected seven.</p> + <p class="v1"> + 1. To divide 10 into two parts, such, that when each part is + added to its square-root and the sums multiplied together, the + product is equal to the supposed number.</p> + <p class="v1"> + 2. What square is that, which being increased or diminished by + 10, the sum and remainder are both square numbers?</p> + <p class="v1"> + 3. A person said he owed to Zaid 10 all but the square-root of + what he owed to Amir, and that he owed Amir 5 all but the + square-root of what he owed Zaid.</p> + <p class="v1"> + 4. To divide a cube number into two cube numbers.</p> + <p class="v1"> + 5. To divide 10 into two parts such, that if each is divided by + the other, and the two quotients are added together, the sum is + equal to one of the parts.</p> + <p class="v1"> + 6. There are three square numbers in continued geometric + proportion, such, that the sum of the three is a square + number.</p> + <p class="v1"> + 7. There is a square, such, that when it is increased + and + +<span class="pagenum"> + <a name="Page_285" + id="Page_285">285</a></span> + + diminished by its root and 2, the sum + and the difference are squares.—<span class= + "smcap">Khulasat-al-Hisab.</span></p> + <p class="blockcite"> + Algebra; quoted in Hutton: A Philosophical and + Mathematical Dictionary (London, 1815), Vol. 1, p. 70.</p> + + <p class="v2"> + <b><a name="Block_1739" id="Block_1739">1739</a>.</b> + The solution of such questions as these [referring + to the solution of cubic equations] depends on correct + judgment, aided by the assistance of + God.—<span class="smcap">Bija Ganita.</span></p> + <p class="blockcite"> + Quoted in Hutton: A Philosophical and Mathematical + Dictionary (London, 1815), Vol. 1, p. 65.</p> + + <p class="v2"> + <b><a name="Block_1740" id="Block_1740">1740</a>.</b> + For what is the theory of determinants? It is an + algebra upon algebra; a calculus which enables us to combine + and foretell the results of algebraical operations, in the same + way as algebra itself enables us to dispense with the + performance of the special operations of arithmetic. All + analysis must ultimately clothe itself under this + form.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Philosophical Magazine, Vol. 1, (1851), p. 300; Collected + Mathematical Papers, Vol. 1, p. 247.</p> + + <p class="v2"> + <b><a name="Block_1741" id="Block_1741">1741</a>.</b></p> + <div class="poem"> + <p class="i0"> + Fuchs.</p> + <p class="i7"> + Fast möcht’ ich nun <em>moderne Algebra</em> studieren.</p> + <p class="i0"> + Meph.</p> + <p class="i7"> + Ich wünschte nicht euch irre zu führen.</p> + <p class="i7"> + Was diese Wissenschaft betrifft,</p> + <p class="i7"> + Es ist so schwer, die leere Form zu meiden,</p> + <p class="i7"> + Und wenn ihr es nicht recht begrifft,</p> + <p class="i7"> + Vermögt die Indices ihr kaum zu unterscheiden.</p> + <p class="i7"> + Am Besten ist’s, wenn ihr nur <em>Einem</em> traut</p> + <p class="i7"> + Und auf des Meister’s Formeln baut.</p> + <p class="i7"> + Im Ganzen—haltet euch an die <em>Symbole</em>.</p> + <p class="i7"> + Dann geht ihr zu der Forschung Wohle</p> + <p class="i7"> + Ins sichre Reich der Formeln ein.</p> + <p class="i0"> + Fuchs.</p> + <p class="i7"> + Ein Resultat muss beim Symbole sein?</p> + <p class="i0"> + Meph.</p> + <p class="i7"> + Schon gut! Nur muss man sich nicht alzu ängstlich quälen.</p> + <p class="i7"> + Denn eben, wo die Resultate fehlen,</p> + <p class="i7"> + Stellt ein Symbol zur rechten Zeit sich ein.</p> + <p class="i7"> + Symbolisch lässt sich alles schreiben,</p> + <p class="i7"> + Müsst nur im Allgemeinen bleiben.</p> + +<span class="pagenum"> + <a name="Page_286" + id="Page_286">286</a></span> + + <p class="i7"> + Wenn man der Gleichung Lösung nicht erkannte,</p> + <p class="i7"> + Schreibt man sie als Determinante.</p> + <p class="i7"> + Schreib’ was du willst, nur rechne <em>nie</em> was aus.</p> + <p class="i7"> + Symbole lassen trefflich sich traktieren,</p> + <p class="i7"> + Mit einem Strich ist alles auszuführen,</p> + <p class="i7"> + Und mit Symbolen kommt man immer aus.</p> + </div> + <p class="block40"> + —<span class="smcap">Lasswitz, Kurd.</span></p> + <p class="blockcite"> + Der Faust-Tragödie (-n)ter Teil; Zeitschrift für mathematischen + und naturwissenschaftlichen Unterricht, Bd. 14, p. 317.</p> + + <div class="poem"> + <p class="i0"> + Fuchs.</p> + <p class="i7"> + To study <em>modern algebra</em> I’m most persuaded.</p> + <p class="i0"> + Meph.</p> + <p class="i7"> + ’Twas not my wish to lead thee astray.</p> + <p class="i7"> + But as concerns this science, truly</p> + <p class="i7"> + ’Tis difficult to avoid the empty form,</p> + <p class="i7"> + And should’st thou lack clear comprehension,</p> + <p class="i7"> + Scarcely the indices thou’ll know apart.</p> + <p class="i7"> + ’Tis safest far to trust but <em>one</em></p> + <p class="i7"> + And built upon your master’s formulas.</p> + <p class="i7"> + On the whole—cling closely to your <em>symbols</em>.</p> + <p class="i7"> + Then, for the weal of research you may gain</p> + <p class="i7"> + An entrance to the formula’s sure domain.</p> + <p class="i0"> + Fuchs.</p> + <p class="i7"> + The symbol, it must lead to some result?</p> + <p class="i0"> + Meph.</p> + <p class="i7"> + Granted. But never worry about results,</p> + <p class="i7"> + For, mind you, just where the results are wanting</p> + <p class="i7"> + A symbol at the nick of time appears.</p> + <p class="i7"> + To symbolic treatment all things yield,</p> + <p class="i7"> + Provided we stay in the general field.</p> + <p class="i7"> + Should a solution prove elusive,</p> + <p class="i7"> + Write the equation in determinant form.</p> + <p class="i7"> + Write what you please, but <em>never</em> calculate.</p> + <p class="i7"> + Symbols are patient and long suffering,</p> + <p class="i7"> + A single stroke completes the whole affair.</p> + <p class="i7"> + Symbols for every purpose do suffice.</p> + </div> + + <p class="v2"> + <b><a name="Block_1742" id="Block_1742">1742</a>.</b> + As all roads are said to lead to Rome, so I find, + in my own case at least, that all algebraic inquiries sooner or + later end + +<span class="pagenum"> + <a name="Page_287" + id="Page_287">287</a></span> + + at the Capitol of Modern Algebra over + whose shining portal is inscribed “Theory of + Invariants”—<span class= "smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + On Newton’s Rule for the Discovery of Imaginary Roots; + Collected Mathematical Papers, Vol. 2, p. 380.</p> + + <p class="v2"> + <b><a name="Block_1743" id="Block_1743">1743</a>.</b> + If we consider the beauty of the theorem + [Sylvester’s Theorem on + Newton’s Rule for the Discovery of Imaginary + Roots] which has now been expounded, the interest which belongs + to the rule associated with the great name of Newton, and the + long lapse of years during which the reason and extent of that + rule remained undiscovered by mathematicians, among whom + Maclaurin, Waring and Euler are explicitly included, we must + regard Professor Sylvester’s investigations + made to the Theory of Equations in modern times, justly to be + ranked with those of Fourier, Sturm and + Cauchy.—<span class="smcap">Todhunter, I.</span></p> + <p class="blockcite"> + Theory of Equations (London, 1904), p. 250.</p> + + <p class="v2"> + <b><a name="Block_1744" id="Block_1744">1744</a>.</b> + Considering the remarkable elegance, generality, + and simplicity of the method [Homer’s Method + of finding the numerical values of the roots of an equation], + it is not a little surprising that it has not taken a more + prominent place in current mathematical textbooks.... As a + matter of fact, its spirit is purely arithmetical; and its + beauty, which can only be appreciated after one has used it in + particular cases, is of that indescribably simple kind, which + distinguishes the use of position in the decimal notation and + the arrangement of the simple rules of arithmetic. It is, in + short, one of those things whose invention was the creation of + a commonplace.—<span class="smcap">Chrystal, George.</span></p> + <p class="blockcite"> + Algebra (London and Edinburgh, 1893), Vol. 1, chap. 15, + sect. 25.</p> + + <p class="v2"> + <b><a name="Block_1745" id="Block_1745">1745</a>.</b> + <em>To a missing member of a family group of terms + in an algebraical formula.</em></p> + <div class="poem"> + <p class="i4"> + Lone and discarded one! divorced by fate,</p> + <p class="i0"> + Far from thy wished-for fellows—whither art flown?</p> + <p class="i0"> + Where lingerest thou in thy bereaved estate,</p> + <p class="i0"> + Like some lost star, or buried meteor stone?</p> + +<span class="pagenum"> + <a name="Page_288" + id="Page_288">288</a></span> + + <p class="i0"> + Thou mindst me much of that presumptuous one</p> + <p class="i0"> + Who loth, aught less than greatest, to be great,</p> + <p class="i0"> + From Heaven’s immensity fell headlong down</p> + <p class="i0"> + To live forlorn, self-centred, desolate:</p> + <p class="i0"> + Or who, like Heraclid, hard exile bore,</p> + <p class="i0"> + Now buoyed by hope, now stretched on rack of fear,</p> + <p class="i0"> + Till throned Astæa, wafting to his ear</p> + <p class="i0"> + Words of dim portent through the Atlantic roar,</p> + <p class="i0"> + Bade him “the sanctuary of the Muse revere</p> + <p class="i0"> + And strew with flame the dust of Isis’ shore.”</p> + </div> + <p class="block40"> + —<span class= "smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Inaugural Lecture, Oxford, 1885; Nature, Vol. 33, p. 228.</p> + + <p class="v2"> + <b><a name="Block_1746" id="Block_1746">1746</a>.</b> + In every subject of inquiry there are certain + entities, the mutual relations of which, under various + conditions, it is desirable to ascertain. A certain combination + of these entities are submitted to certain processes or are + made the subjects of certain operations. The theory of + invariants in its widest scientific meaning determines these + combinations, elucidates their properties, and expresses + results when possible in terms of them. Many of the general + principles of political science and economics can be + represented by means of invariantive relations connecting the + factors which enter as entities into the special problems. The + great principle of chemical science which asserts that when + elementary or compound bodies combine with one another the + total weight of the materials is unchanged, is another case in + point. Again, in physics, a given mass of gas under the + operation of varying pressure and temperature has the + well-known invariant, pressure multiplied by volume and divided + by absolute temperature.... In mathematics the entities under + examination may be arithmetical, algebraical, or geometrical; + the processes to which they are subjected may be any of those + which are met with in mathematical work.... It is the + <em>principle</em> which is so valuable. It is the <em>idea</em> of + invariance that pervades today all branches of + mathematics.—<span class="smcap">MacMahon, P. A.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science (1901); Nature, Vol. 64, p. 481.</p> + +<p><span class="pagenum"> + <a name="Page_289" + id="Page_289">289</a></span></p> + + <p class="v2"> + <b><a name="Block_1747" id="Block_1747">1747</a>.</b> + [The theory of invariants] has invaded the domain + of geometry, and has almost re-created the analytical theory; + but it has done more than this for the investigations of Cayley + have required a full reconsideration of the very foundations of + geometry. It has exercised a profound influence upon the theory + of algebraic equations; it has made its way into the theory of + differential equations; and the generalisation of its ideas is + opening out new regions of most advanced and profound + functional analysis. And so far from its course being + completed, its questions fully answered, or its interest + extinct, there is no reason to suppose that a term can be + assigned to its growth and its + influence.—<span class="smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science (1897); Nature, Vol. 56, p. 378.</p> + + <p class="v2"> + <b><a name="Block_1748" id="Block_1748">1748</a>.</b> + ... the doctrine of Invariants, a theory filling + the heavens like a light-bearing ether, penetrating all the + branches of geometry and analysis, revealing everywhere abiding + configurations in the midst of change, everywhere disclosing + the eternal reign of the law of + form.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 28.</p> + + <p class="v2"> + <b><a name="Block_1749" id="Block_1749">1749</a>.</b> + It is in the mathematical doctrine of Invariance, + the realm wherein are sought and found configurations and types + of being that, amidst the swirl and stress of countless hosts + of transformations remain immutable, and the spirit dwells in + contemplation of the serene and eternal reign of the subtile + laws of Form, it is there that Theology may find, if she will, + the clearest conceptions, the noblest symbols, the most + inspiring intimations, the most illuminating illustrations, and + the surest guarantees of the object of her teaching and her + quest, an Eternal Being, unchanging in the midst of the + universal flux.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 42.</p> + + <p class="v2"> + <b><a name="Block_1750" id="Block_1750">1750</a>.</b> + I think that young chemists desirous of raising + their science to its proper rank would act wisely in making + themselves master betimes of the theory of + algebraic forms. What mechanics + +<span class="pagenum"> + <a name="Page_290" + id="Page_290">290</a></span> + + is to physics, that I think is + algebraic morphology, founded at option on the theory of + partitions or ideal elements, or both, is destined to be to the + chemistry of the future ... invariants and isomerism are sister + theories.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + American Journal of Mathematics, Vol. 1 (1878), p. 126.</p> + + <p class="v2"> + <b><a name="Block_1751" id="Block_1751">1751</a>.</b> + The great notion of Group, ... though it had + barely merged into consciousness a hundred years ago, has + meanwhile become a concept of fundamental importance and + prodigious fertility, not only affording the basis of an + imposing doctrine—the Theory of + Groups—but therewith serving also as a bond + of union, a kind of connective tissue, or rather as an immense + cerebro-spinal system, uniting together a large number of + widely dissimilar doctrines as organs of a single + body.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 12.</p> + + <p class="v2"> + <b><a name="Block_1752" id="Block_1752">1752</a>.</b> + In recent times the view becomes more and more + prevalent that many branches of mathematics are nothing but the + theory of invariants of special + groups.—<span class="smcap">Lie, Sophus.</span></p> + <p class="blockcite"> + Continuierliche Gruppen—Scheffers (Leipzig, 1893), p. 665.</p> + + <p class="v2"> + <b><a name="Block_1753" id="Block_1753">1753</a>.</b> + Universal Algebra has been looked on with some + suspicion by many mathematicians, as being without intrinsic + mathematical interest and as being comparatively useless as an + engine of investigation.... But it may be shown that Universal + Algebra has the same claim to be a serious subject of + mathematical study as any other branch of + mathematics.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + Universal Algebra (Cambridge, 1898), Preface, p. vi.</p> + + <p class="v2"> + <b><a name="Block_1754" id="Block_1754">1754</a>.</b> + [Function] theory was, in effect, founded by + Cauchy; but, outside his own investigations, it at first made + slow and hesitating progress. At the present day, its + fundamental ideas may be said almost to govern most departments + of the analysis of continuous quantity. On many of them, it has + shed a completely new light; it has educed relations between + them before unknown. It may be doubted whether any subject is + at the + +<span class="pagenum"> + <a name="Page_291" + id="Page_291">291</a></span> + + present day so richly endowed with + variety of method and fertility of resource; its activity is + prodigious, and no less remarkable than its activity is its + freshness.—<span class="smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science (1897); Nature, Vol. 56, p. 378.</p> + + <p class="v2"> + <b><a name="Block_1755" id="Block_1755">1755</a>.</b> + Let me mention one other contribution which this + theory [Theory of functions of a complex variable] has made to + knowledge lying somewhat outside our track. During the rigorous + revision to which the foundations of the theory have been + subjected in its re-establishment by Weierstrass, new ideas as + regards number and continuity have been introduced. With him + and with others influenced by him, there has thence sprung a + new theory of higher arithmetic; and with its growth, much has + concurrently been effected in the elucidation of the general + notions of number and quantity.... It thus appears to be the + fact that, as with Plato, or Descartes, or Leibnitz, or Kant, + the activity of pure mathematics is again lending some + assistance to the better comprehension of those notions of + time, space, number, quantity, which underlie a philosophical + conception of the universe.—<span class= + "smcap">Forsyth, A. R.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science (1897); Nature, Vol. 56, p. 378.</p> + +<p><span class="pagenum"> + <a name="Page_292" + id="Page_292">292</a></span> </p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_XVIII"> + CHAPTER XVIII<br /> + <span class="large"> + GEOMETRY</span></h2> + + <p class="v2"> + <b><a name="Block_1801" id="Block_1801">1801</a>.</b> + The science of figures is most glorious and + beautiful. But how inaptly it has received the name + geometry!—<span class="smcap">Frischlinus, N.</span></p> + <p class="blockcite"> + Dialog 1.</p> + + <p class="v2"> + <b><a name="Block_1802" id="Block_1802">1802</a>.</b> + Plato said that God geometrizes + continually.—<span class="smcap">Plutarch.</span></p> + <p class="blockcite"> + Convivialium disputationum, liber 8, 2.</p> + + <p class="v2"> + <b><a name="Block_1803" id="Block_1803">1803</a>.</b> + μηδεὶς ἐγεωμέτρητος εἰσίτω μοῦ + + <a id="TNanchor_15"></a> + <a class="msg" href="#TN_15" + title="originally read ‘τὴυ’">τὴν</a> + + στέγην. + [Let no one ignorant of geometry enter my + door.]—<span class="smcap">Plato.</span></p> + <p class="blockcite"> + Tzetzes, Chiliad, 8, 972.</p> + + <p class="v2"> + <b><a name="Block_1804" id="Block_1804">1804</a>.</b> + All the authorities agree that he [Plato] made a + study of geometry or some exact science an indispensable + preliminary to that of philosophy. The inscription over the + entrance to his school ran “Let none ignorant of geometry enter + my door,” and on one occasion an applicant who knew no geometry + is said to have been refused admission as a + student.—<span class="smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p. 45.</p> + + <p class="v2"> + <b><a name="Block_1805" id="Block_1805">1805</a>.</b> + Form and size constitute the foundation of all + search for truth.—<span class="smcap">Parker, F. W.</span></p> + <p class="blockcite"> + Talks on Pedagogics (New York, 1894), p. 72.</p> + + <p class="v2"> + <b><a name="Block_1806" id="Block_1806">1806</a>.</b> + At present the science [of geometry] is in flat + contradiction to the language which geometricians use, as will + hardly be denied by those who have any acquaintance with the + study: for they speak of finding the side of a square, and + applying and adding, and so on, as if they were engaged in some + business, and as if all their propositions had a practical end + in view: whereas in reality the science is pursued wholly for + the sake of knowledge.</p> + +<p> + <span class="pagenum"> + <a name="Page_293" + id="Page_293">293</a></span></p> + + <p class="v1"> + Certainly, he said.</p> + <p class="v1"> + Then must not a further admission be made?</p> + <p class="v1"> + What admission?</p> + <p class="v1"> + The admission that this knowledge at which geometry aims is of + the eternal, and not of the perishing and transient.</p> + <p class="v1"> + That may be easily allowed. Geometry, no doubt, is the + knowledge of what eternally exists.</p> + <p class="v1"> + Then, my noble friend, geometry will draw the soul towards + truth, and create the mind of philosophy, and raise up that + which is now unhappily allowed to fall + down.—<span class="smcap">Plato.</span></p> + <p class="blockcite"> + Republic [Jowett-Davies], Bk. 7, p. 527.</p> + + <p class="v2"> + <b><a name="Block_1807" id="Block_1807">1807</a>.</b> + Among them [the Greeks] geometry was held in + highest honor: nothing was more glorious than mathematics. But + we have limited the usefulness of this art to measuring and + calculating.—<span class="smcap">Cicero.</span></p> + <p class="blockcite"> + Tusculanae Disputationes, 1, 2, 5.</p> + + <p class="v2"> + <b><a name="Block_1808" id="Block_1808">1808</a>.</b></p> + <div class="poem"> + <p class="i12"> + Geometria,</p> + <p class="i0"> + Through which a man hath the sleight</p> + <p class="i0"> + Of length, and brede, of depth, of + height.</p> + </div> + <p class="block40"> + —<span class="smcap">Gower, John.</span></p> + <p class="blockcite"> + Confessio Amantis, Bk. 7.</p> + + <p class="v2"> + <b><a name="Block_1809" id="Block_1809">1809</a>.</b> + Geometrical truths are in a way asymptotes to + physical truths, that is to say, the latter approach the former + indefinitely near without ever reaching them + exactly.—<span class="smcap">D’Alembert.</span></p> + <p class="blockcite"> + Quoted in Rebière: Mathématiques + et Mathématiciens (Paris, 1898), p. 10.</p> + + <p class="v2"> + <b><a name="Block_1810" id="Block_1810">1810</a>.</b> + Geometry exhibits the most perfect example of + logical stratagem.—<span class="smcap">Buckle, H. T.</span></p> + <p class="blockcite"> + History of Civilization in England (New York, 1891), Vol. + 2, p. 342.</p> + + <p class="v2"> + <b><a name="Block_1811" id="Block_1811">1811</a>.</b> + It is the glory of geometry that from so few + principles, fetched from without, it is able to accomplish so + much.—<span class="smcap">Newton.</span></p> + <p class="blockcite"> + Philosophiae Naturalis Principia Mathematica, Praefat.</p> + +<p><span class="pagenum"> + <a name="Page_294" + id="Page_294">294</a></span></p> + + <p class="v2"> + <b><a name="Block_1812" id="Block_1812">1812</a>.</b> + Geometry is the application of strict logic to + those properties of space and figure which are self-evident, + and which therefore cannot be disputed. But the rigor of this + science is carried one step further; for no property, however + evident it may be, is allowed to pass without demonstration, if + that can be given. The question is therefore to demonstrate all + geometrical truths with the smallest possible number of + assumptions.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + On the Study and Difficulties of Mathematics (Chicago, + 1902), p. 231.</p> + + <p class="v2"> + <b><a name="Block_1813" id="Block_1813">1813</a>.</b> + Geometry is a true natural + science:—only more simple, and therefore + more perfect than any other. We must not suppose that, because + it admits the application of mathematical analysis, it is + therefore a purely logical science, independent of observation. + Every body studied by geometers presents some primitive + phenomena which, not being discoverable by reasoning, must be + due to observation alone.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 1, chap. 3.</p> + + <p class="v2"> + <b><a name="Block_1814" id="Block_1814">1814</a>.</b> + Geometry in every proposition speaks a language + which experience never dares to utter; and indeed of which she + but half comprehends the meaning. Experience sees that the + assertions are true, but she sees not how profound and absolute + is their truth. She unhesitatingly assents to the laws which + geometry delivers, but she does not pretend to see the origin + of their obligation. She is always ready to acknowledge the + sway of pure scientific principles as a matter of fact, but she + does not dream of offering her opinion on their authority as a + matter of right; still less can she justly claim to herself the + source of that authority.—<span + class="smcap">Whewell, William.</span></p> + <p class="blockcite"> + The Philosophy of the Inductive Sciences, Part 1, Bk. 1, + chap. 6, sect. 1 (London, 1858).</p> + + <p class="v2"> + <b><a name="Block_1815" id="Block_1815">1815</a>.</b> + Geometry is the science created to give + understanding and mastery of the external relations of things; + to make easy the explanation and description of such relations + and the transmission of this + mastery.—<span class="smcap">Halsted, G. B.</span></p> + <p class="blockcite"> + Proceedings of the American Association for the + Advancement of Science (1904), p. 359.</p> + +<p><span class="pagenum"> + <a name="Page_295" + id="Page_295">295</a></span></p> + + <p class="v2"> + <b><a name="Block_1816" id="Block_1816">1816</a>.</b> + A mathematical point is the most indivisible and + unique thing which art can + present.—<span class="smcap">Donne, John.</span></p> + <p class="blockcite"> + Letters, 21.</p> + + <p class="v2"> + <b><a name="Block_1817" id="Block_1817">1817</a>.</b> + It is certain that from its completeness, + uniformity and faultlessness, from its arrangement and + progressive character, and from the universal adoption of the + completest and best line of argument, Euclid’s + “Elements” stand pre-eminently at + the head of all human productions. In no science, in no + department of knowledge, has anything appeared like this work: + for upward of 2000 years it has commanded the admiration of + mankind, and that period has suggested little toward its + improvement.—<span class="smcap">Kelland, P.</span></p> + <p class="blockcite"> + Lectures on the Principles of Demonstrative Mathematics + (London, 1843), p. 17.</p> + + <p class="v2"> + <b><a name="Block_1818" id="Block_1818">1818</a>.</b> + In comparing the performance in Euclid with that + in Arithmetic and Algebra there could be no doubt that Euclid + had made the deepest and most beneficial impression: in fact it + might be asserted that this constituted by far the most + valuable part of the whole training to which such persons + [students, the majority of which were not distinguished for + mathematical taste and power] were + subjected.—<span class="smcap">Todhunter, I.</span></p> + <p class="blockcite"> + Essay on Elementary Geometry; Conflict of Studies and + other Essays (London, 1873), p. 167.</p> + + <p class="v2"> + <b><a name="Block_1819" id="Block_1819">1819</a>.</b> + In England the geometry studied is that of Euclid, + and I hope it never will be any other; for this reason, that so + much has been written on Euclid, and all the difficulties of + geometry have so uniformly been considered with reference to + the form in which they appear in Euclid, that the study of that + author is a better key to a great quantity of useful reading + than any other.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Elements of Algebra (London, 1837), Introduction.</p> + + <p class="v2"> + <b><a name="Block_1820" id="Block_1820">1820</a>.</b> + This book [Euclid] has been for nearly twenty-two centuries the + encouragement and guide of that scientific thought + +<span class="pagenum"> + <a name="Page_296" + id="Page_296">296</a></span> + + which is one thing with the progress + of man from a worse to a better state. The encouragement; for + it contained a body of knowledge that was really known and + could be relied on, and that moreover was growing in extent and + application. For even at the time this book was written—shortly + after the foundation of the Alexandrian Museum—Mathematics was + no longer the merely ideal science of the Platonic school, but + had started on her career of conquest over the whole world of + Phenomena. The guide; for the aim of every scientific student + of every subject was to bring his knowledge of that subject + into a form as perfect as that which geometry had attained. Far + up on the great mountain of Truth, which all the sciences hope + to scale, the foremost of that sacred sisterhood was seen, + beckoning for the rest to follow her. And hence she was called, + in the dialect of the Phythagoreans, “the + purifier of the reasonable + soul”—<span class="smcap">Clifford, W. K.</span></p> + <p class="blockcite"> + Lectures and Essays (London, 1901), Vol. 1, p. 354.</p> + + <p class="v2"> + <b><a name="Block_1821" id="Block_1821">1821</a>.</b> + [Euclid] at once the inspiration and aspiration of + scientific thought.—<span class="smcap">Clifford, W. K.</span></p> + <p class="blockcite"> + Lectures and Essays (London, 1901), Vol 1, p. 355.</p> + + <p class="v2"> + <b><a name="Block_1822" id="Block_1822">1822</a>.</b> + The “elements” of + the Great Alexandrian remain for all time the first, and one + may venture to assert, the <em>only</em> perfect model of logical + exactness of principles, and of rigorous development of + theorems. If one would see how a science can be constructed and + developed to its minutest details from a very small number of + intuitively perceived axioms, postulates, and plain + definitions, by means of rigorous, one would almost say chaste, + syllogism, which nowhere makes use of surreptitious or foreign + aids, if one would see how a science may thus be constructed + one must turn to the elements of + Euclid.—<span class="smcap">Hankel, H.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten + Jahrhunderten (Tübingen, 1884), p. 7.</p> + + <p class="v2"> + <b><a name="Block_1823" id="Block_1823">1823</a>.</b> + If we consider him [Euclid] as meaning to be what his + commentators have taken him to be, a + model of the most unscrupulous + +<span class="pagenum"> + <a name="Page_297" + id="Page_297">297</a></span> + + formal rigour, we can deny that + he has altogether succeeded, though we admit that he made the + nearest approach.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Smith’s Dictionary of Greek and Roman Biography and Mythology + (London, 1902); Article “Eucleides”</p> + + <p class="v2"> + <b><a name="Block_1824" id="Block_1824">1824</a>.</b> + The Elements of Euclid is as small a part of + mathematics as the Iliad is of literature; or as the sculpture + of Phidias is of the world’s total + art.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 8.</p> + + <p class="v2"> + <b><a name="Block_1825" id="Block_1825">1825</a>.</b> + I should rejoice to see ... Euclid honourably shelved or buried + “deeper than did ever plummet sound” out of the + schoolboys’ reach; morphology introduced + into the elements of algebra; projection, correlation, and + motion accepted as aids to geometry; the mind of the student + quickened and elevated and his faith awakened by early + initiation into the ruling ideas of polarity, continuity, + infinity, and familiarization with the doctrines of the + imaginary and inconceivable.—<span class= + "smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + A Plea for the Mathematician; Nature, Vol. 1, p. 261.</p> + + <p class="v2"> + <b><a name="Block_1826" id="Block_1826">1826</a>.</b> + The early study of Euclid made me a hater of geometry, ... and + yet, in spite of this repugnance, which had become a second + nature in me, whenever I went far enough into any mathematical + question, I found I touched, at last, a geometrical + bottom.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + A Plea for the Mathematician; Nature, Vol. 1, p. 262.</p> + + <p class="v2"> + <b><a name="Block_1827" id="Block_1827">1827</a>.</b> + Newton had so remarkable a talent for mathematics + that Euclid’s Geometry seemed to him “a trifling book,” and he + wondered that any man should have taken the trouble to + demonstrate propositions, the truth of which was so obvious to + him at the first glance. But, on attempting to read the more + abstruse geometry of Descartes, without having mastered the + elements of the science, he was baffled, and was glad to come + back again to his Euclid.—<span + class="smcap">Parton, James.</span></p> + <p class="blockcite"> + Sir Isaac Newton.</p> + +<p><span class="pagenum"> + <a name="Page_298" + id="Page_298">298</a></span></p> + + <p class="v2"> + <b><a name="Block_1828" id="Block_1828">1828</a>.</b> + As to the need of improvement there can be no + question whilst the reign of Euclid continues. My own idea of a + useful course is to begin with arithmetic, and then not Euclid + but algebra. Next, not Euclid, but practical geometry, solid as + well as plane; not demonstration, but to make acquaintance. + Then not Euclid, but elementary vectors, conjoined with + algebra, and applied to geometry. Addition first; then the + scalar product. Elementary calculus should go on + simultaneously, and come into the vector algebraic geometry + after a bit. Euclid might be an extra course for learned men, + like Homer. But Euclid for children is + barbarous.—<span class="smcap">Heaviside, Oliver.</span></p> + <p class="blockcite"> + Electro-Magnetic Theory (London, 1893), Vol. 1, p. 148.</p> + + <p class="v2"> + <b><a name="Block_1829" id="Block_1829">1829</a>.</b> + Geometry is nothing if it be not rigorous, and the + whole educational value of the study is lost, if strictness of + demonstration be trifled with. The methods of Euclid are, by + almost universal consent, unexceptionable in point of + rigour.—<span class="smcap">Smith, H. J. S.</span></p> + <p class="blockcite"> + Nature, Vol. 8, p. 450.</p> + + <p class="v2"> + <b><a name="Block_1830" id="Block_1830">1830</a>.</b> + To seek for proof of geometrical propositions by + an appeal to observation proves nothing in reality, except that + the person who has recourse to such grounds has no due + apprehension of the nature of geometrical demonstration. We + have heard of persons who convince themselves by measurement + that the geometrical rule respecting the squares on the sides + of a right-angles triangle was true: but these were persons + whose minds had been engrossed by practical habits, and in whom + speculative development of the idea of space had been stifled + by other employments.—<span + class="smcap">Whewell, William.</span></p> + <p class="blockcite"> + The Philosophy of the Inductive Sciences, (London, 1858), + Part 1, Bk. 2, chap. 1, sect. 4.</p> + + <p class="v2"> + <b><a name="Block_1831" id="Block_1831">1831</a>.</b> + No one has ever given so easy and natural a chain + of geometrical consequences [as Euclid]. There is a + never-erring truth in the + results.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Smith’s Dictionary of Greek and Roman Biography and Mythology + (London, 1902); Article “Eucleides”</p> + +<p><span class="pagenum"> + <a name="Page_299" + id="Page_299">299</a></span></p> + + <p class="v2"> + <b><a name="Block_1832" id="Block_1832">1832</a>.</b> + Beyond question, Egyptian geometry, such as it + was, was eagerly studied by the early Greek philosophers, and + was the germ from which in their hands grew that magnificent + science to which every Englishman is indebted for his first + lessons in right seeing and + thinking.—<span class="smcap">Gow, James.</span></p> + <p class="blockcite"> + A Short History of Greek Mathematics (Cambridge, 1884), + p. 131.</p> + + <p class="v2"> + <b><a name="Block_1833" id="Block_1833">1833</a>.</b></p> + <div class="poem"> + <p class="i0"> + A figure and a step onward:</p> + <p class="i0"> + Not a figure and a florin.</p> + </div> + <p class="block40"> + —<span class="smcap">Motto of the Pythagorean + Brotherhood.</span></p> + <p class="blockcite"> + W. B. Frankland: Story of Euclid (London, 1902), p. 33.</p> + + <p class="v2"> + <b><a name="Block_1834" id="Block_1834">1834</a>.</b> + The doctrine of proportion, as laid down in the + fifth book of Euclid, is, probably, still unsurpassed as a + masterpiece of exact reasoning; although the cumbrousness of + the forms of expression which were adopted in the old geometry + has led to the total exclusion of this part of the elements + from the ordinary course of geometrical education. A zealous + defender of Euclid might add with truth that the gap thus + created in the elementary teaching of mathematics has never + been adequately supplied.—<span class= + "smcap">Smith, H. J. S.</span></p> + <p class="blockcite"> + Presidential Address British Association for the + Advancement of Science (1873); Nature, Vol. 8, p. 451.</p> + + <p class="v2"> + <b><a name="Block_1835" id="Block_1835">1835</a>.</b> + The Definition in the Elements, according to + Clavius, is this: Magnitudes are said to be in the same Reason + [ratio], a first to a second, and a third to a fourth, when the + Equimultiples of the first and third according to any + Multiplication whatsoever are both together either short of, + equal to, or exceed the Equimultiples of the second and fourth, + if those be taken, which answer one another.... Such is + Euclid’s Definition of Proportions; that + <em>scare</em>-Crow at which the over modest or slothful + Dispositions of Men are generally affrighted: they are modest, + who distrust their own Ability, as soon as a Difficulty + appears, but they are slothful that will not give some + Attention for the learning of Sciences; as if while we are + involved in Obscurity we could clear ourselves without Labour. + Both of + +<span class="pagenum"> + <a name="Page_300" + id="Page_300">300</a></span> + + which Sorts of Persons are to be + admonished, that the former be not discouraged, nor the latter + refuse a little Care and Diligence when a Thing requires some + Study.—<span class="smcap">Barrow, Isaac.</span></p> + <p class="blockcite"> + Mathematical Lectures (London, 1734), p. 388.</p> + + <p class="v2"> + <b><a name="Block_1836" id="Block_1836">1836</a>.</b> + Of all branches of human knowledge, there is none + which, like it [geometry] has sprung a completely armed Minerva + from the head of Jupiter; none before whose death-dealing Aegis + doubt and inconsistency have so little dared to raise their + eyes. It escapes the tedious and troublesome task of collecting + experimental facts, which is the province of the natural + sciences in the strict sense of the word: the sole form of its + scientific method is deduction. Conclusion is deduced from + conclusion, and yet no one of common sense doubts but that + these geometrical principles must find their practical + application in the real world about us. Land surveying, as well + as architecture, the construction of machinery no less than + mathematical physics, are continually calculating relations of + space of the most varied kinds by geometrical principles; they + expect that the success of their constructions and experiments + shall agree with their calculations; and no case is known in + which this expectation has been falsified, provided the + calculations were made correctly and with sufficient + data.—<span class="smcap">Helmholtz, H.</span></p> + <p class="blockcite"> + The Origin and Significance of Geometrical Axioms; Popular + Scientific Lectures [Atkinson], Second Series (New York, + 1881), p. 27.</p> + + <p class="v2"> + <b><a name="Block_1837" id="Block_1837">1837</a>.</b> + The amazing triumphs of this branch of mathematics [geometry] + show how powerful a weapon that form of deduction is which + proceeds by an artificial reparation of facts, in themselves + inseparable.—<span class= "smcap">Buckle, H. T.</span></p> + <p class="blockcite"> + History of Civilization in England (New York, 1891), Vol. + 2, p. 343.</p> + + <p class="v2"> + <b><a name="Block_1838" id="Block_1838">1838</a>.</b> + Every theorem in geometry is a law of external + nature, and might have been ascertained by generalizing from + observation and experiment, which in this case resolve + themselves into comparisons and measurements. But it was found + practicable, and being practicable was desirable, to deduce + these truths by ratiocination from a small number of general + laws of nature, the certainty and universality of which was + obvious to the most + +<span class="pagenum"> + <a name="Page_301" + id="Page_301">301</a></span> + + careless observer, and which compose + the first principles and ultimate premises of the + science.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + System of Logic, Bk. 3, chap. 24, sect. 7.</p> + + <p class="v2"> + <b><a name="Block_1839" id="Block_1839">1839</a>.</b> + All such reasonings [natural philosophy, + chemistry, agriculture, political economy, etc.] are, in + comparison with mathematics, very complex; requiring so much + <em>more</em> than that does, beyond the process of merely + deducing the conclusion logically from the premises: so that it + is no wonder that the longest mathematical demonstration should + be much more easily constructed and understood, than a much + shorter train of just reasoning concerning real facts. The + former has been aptly compared to a long and steep, but even + and regular, flight of steps, which tries the breath, and the + strength, and the perseverance only; while the latter resembles + a short, but rugged and uneven, ascent up a precipice, which + requires a quick eye, agile limbs, and a firm step; and in + which we have to tread now on this side, now on + that—ever considering as we proceed, whether + this or that projection will afford room for our foot, or + whether some loose stone may not slide from under us. There are + probably as many steps of pure reasoning in one of the longer + of Euclid’s demonstrations, as in the whole + of an argumentative treatise on some other subject, occupying + perhaps a considerable volume.—<span class= + "smcap">Whately, R.</span></p> + <p class="blockcite"> + Elements of Logic, Bk. 4, chap. 2, sect. 5.</p> + + <p class="v2"> + <b><a name="Block_1840" id="Block_1840">1840</a>.</b></p> + <div class="poem"> + <p class="i0"> + [Geometry] that held acquaintance with the stars,</p> + <p class="i0"> + And wedded soul to soul in purest bond</p> + <p class="i0"> + Of reason, undisturbed by space or time.</p> + </div> + <p class="block40"> + —<span class="smcap">Wordsworth.</span></p> + <p class="blockcite"> + The Prelude, Bk. 5.</p> + + <p class="v2"> + <b><a name="Block_1841" id="Block_1841">1841</a>.</b> + The statement that a given individual has received + a sound geometrical training implies that he has segregated + from the whole of his sense impressions a certain set of these + impressions, that he has eliminated from their consideration + all irrelevant impressions (in other words, acquired a + subjective command of these impressions), that he has developed + on the basis of these impressions an ordered and continuous + system of logical deduction, and finally that he is capable of + expressing the + +<span class="pagenum"> + <a name="Page_302" + id="Page_302">302</a></span> + + nature of these impressions and his + deductions therefrom in terms simple and free from ambiguity. + Now the slightest consideration will convince any one not + already conversant with the idea, that the same sequence of + mental processes underlies the whole career of any individual + in any walk of life if only he is not concerned entirely with + manual labor; consequently a full training in the performance + of such sequences must be regarded as forming an essential part + of any education worthy of the name. Moreover the full + appreciation of such processes has a higher value than is + contained in the mental training involved, great though this + be, for it induces an appreciation of intellectual unity and + beauty which plays for the mind that part which the + appreciation of schemes of shape and color plays for the + artistic faculties; or, again, that part which the appreciation + of a body of religious doctrine plays for the ethical + aspirations. Now geometry is not the sole possible basis for + inculcating this appreciation. Logic is an alternative for + adults, provided that the individual is possessed of sufficient + wide, though rough, experience on which to base his reasoning. + Geometry is, however, highly desirable in that the objective + bases are so simple and precise that they can be grasped at an + early age, that the amount of training for the imagination is + very large, that the deductive processes are not beyond the + scope of ordinary boys, and finally that it affords a better + basis for exercise in the art of simple and exact expression + than any other possible subject of a school + course.—<span class="smcap">Carson, G. W. L.</span></p> + <p class="blockcite"> + The Functions of Geometry as a Subject of Education + (Tonbridge, 1910), p. 3.</p> + + <p class="v2"> + <b><a name="Block_1842" id="Block_1842">1842</a>.</b> + It seems to me that the thing that is wanting in + the education of women is not the acquaintance with any facts, + but accurate and scientific habits of thought, and the courage + to think that true which appears unlikely. And for supplying + this want there is a special advantage in geometry, namely that + it does not require study of a physically laborious kind, but + rather that rapid intuition which women certainly possess; so + that it is fit to become a scientific pursuit for + them.—<span class="smcap">Clifford, W. K.</span></p> + <p class="blockcite"> + Quoted by Pollock in Clifford’s Lectures and Essays + (London, 1901), Vol. 1, Introduction, p. 43.</p> + +<p><span class="pagenum"> + <a name="Page_303" + id="Page_303">303</a></span></p> + + <p class="v2"> + <b><a name="Block_1843" id="Block_1843">1843</a>.</b></p> + <div class="poem"> + <p class="i6"> + On the lecture slate</p> + <p class="i0"> + The circle rounded under female + hands</p> + <p class="i0"> + With flawless + demonstration.</p> + </div> + <p class="block40"> + —<span class="smcap">Tennyson.</span></p> + <p class="blockcite"> + The Princess, II, l. 493.</p> + + <p class="v2"> + <b><a name="Block_1844" id="Block_1844">1844</a>.</b> + It is plain that that part of geometry which bears + upon strategy does concern us. For in pitching camps, or in + occupying positions, or in closing or extending the lines of an + army, and in all the other manœuvres of an army + whether in battle or on the march, it will make a great + difference to a general, whether he is a geometrician or + not.—<span class="smcap">Plato.</span></p> + <p class="blockcite"> + Republic, Bk. 7, p. 526.</p> + + <p class="v2"> + <b><a name="Block_1845" id="Block_1845">1845</a>.</b> + Then nothing should be more effectually enacted, + than that the inhabitants of your fair city should learn + geometry. Moreover the science has indirect effects, which are + not small.</p> + <p class="v1"> + Of what kind are they? he said.</p> + <p class="v1"> + There are the military advantages of which you spoke, I said; + and in all departments of study, as experience proves, any one + who has studied geometry is infinitely quicker of + apprehension.—<span class="smcap">Plato.</span></p> + <p class="blockcite"> + Republic [Jowett], Bk. 7, p. 527.</p> + + <p class="v2"> + <b><a name="Block_1846" id="Block_1846">1846</a>.</b> + It is doubtful if we have any other subject that + does so much to bring to the front the danger of carelessness, + of slovenly reasoning, of inaccuracy, and of forgetfulness as + this science of geometry, which has been so polished and + perfected as the centuries have gone + on.—<span class="smcap">Smith, D. E.</span></p> + <p class="blockcite"> + The Teaching of Geometry (Boston, 1911), p. 12.</p> + + <p class="v2"> + <b><a name="Block_1847" id="Block_1847">1847</a>.</b> + The culture of the geometric imagination, tending + to produce precision in remembrance and invention of visible + forms will, therefore, tend directly to increase the + appreciation of works of + belles-letters.—<span class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + The Uses of Mathesis; Bibliotheca Sacra, Vol. 32, p. 504.</p> + + <p class="v2"> + <b><a name="Block_1848" id="Block_1848">1848</a>.</b></p> + <div class="poem"> + <p class="i0"> + Yet may we not entirely overlook</p> + <p class="i0"> + The pleasures gathered from the rudiments</p> + <p class="i0"> + Of geometric science. Though advanced</p> + <p class="i0"> + In these inquiries, with regret I speak, + +<span class="pagenum"> + <a name="Page_304" + id="Page_304">304</a></span></p> + + <p class="i0"> + No farther than the threshold, there I found</p> + <p class="i0"> + Both elevation and composed delight:</p> + <p class="i0"> + With Indian awe and wonder, ignorance pleased</p> + <p class="i0"> + With its own struggles, did I meditate</p> + <p class="i0"> + On the relations those abstractions bear</p> + <p class="i0"> + To Nature’s laws.</p> + + <hr class="tb" /> + + <p class="i0"> + More frequently from the same source I drew</p> + <p class="i0"> + A pleasure quiet and profound, a sense</p> + <p class="i0"> + Of permanent and universal sway,</p> + <p class="i0"> + And paramount belief; there, recognized</p> + <p class="i0"> + A type, for finite natures, of the one</p> + <p class="i0"> + Supreme Existence, the surpassing life</p> + <p class="i0"> + Which to the boundaries of space and time,</p> + <p class="i0"> + Of melancholy space and doleful time,</p> + <p class="i0"> + Superior and incapable of change,</p> + <p class="i0"> + Nor touched by welterings of passion—is,</p> + <p class="i0"> + And hath the name of God. Transcendent peace</p> + <p class="i0"> + And silence did wait upon these thoughts</p> + + <hr class="tb" /> + + <p class="i12"> + Mighty is the charm</p> + <p class="i0"> + Of those abstractions to a mind beset</p> + <p class="i0"> + With images and haunted by himself,</p> + <p class="i0"> + And specially delightful unto me</p> + <p class="i0"> + Was that clear synthesis built up aloft</p> + <p class="i0"> + So gracefully; even then when it appeared</p> + <p class="i0"> + Not more than a mere plaything, or a toy</p> + <p class="i0"> + To sense embodied: not the thing it is</p> + <p class="i0"> + In verity, an independent world,</p> + <p class="i0"> + Created out of pure intelligence.</p> + </div> + <p class="block40"> + —<span class="smcap">Wordsworth.</span></p> + <p class="blockcite"> + The Prelude, Bk. 6.</p> + + <p class="v2"> + <b><a name="Block_1849" id="Block_1849">1849</a>.</b></p> + <div class="poem"> + <p class="i0"> + ’Tis told by one whom stormy waters threw,</p> + <p class="i0"> + With fellow-sufferers by the shipwreck spared,</p> + <p class="i0"> + Upon a desert coast, that having brought</p> + <p class="i0"> + To land a single volume, saved by chance,</p> + <p class="i0"> + A treatise of Geometry, he wont, + +<span class="pagenum"> + <a name="Page_305" + id="Page_305">305</a></span></p> + + <p class="i0"> + Although of food and clothing destitute,</p> + <p class="i0"> + And beyond common wretchedness depressed,</p> + <p class="i0"> + To part from company, and take this + book</p> + <p class="i0"> + (Then first a self taught pupil in its truths)</p> + <p class="i0"> + To spots remote, and draw his diagrams</p> + <p class="i0"> + With a long staff upon the sand, and thus</p> + <p class="i0"> + Did oft beguile his sorrow, and almost</p> + <p class="i0"> + Forget his feeling:</p> + </div> + <p class="block40"> + —<span class="smcap">Wordsworth.</span></p> + <p class="blockcite"> + The Prelude, Bk. 6.</p> + + <p class="v2"> + <b><a name="Block_1850" id="Block_1850">1850</a>.</b> + We study art because we receive pleasure from the + great works of the masters, and probably we appreciate them the + more because we have dabbled a little in pigments or in clay. + We do not expect to be composers, or poets, or sculptors, but + we wish to appreciate music and letters and the fine arts, and + to derive pleasure from them and be uplifted by them....</p> + <p class="v1"> + So it is with geometry. We study it because we derive pleasure + from contact with a great and ancient body of learning that has + occupied the attention of master minds during the thousands of + years in which it has been perfected, and we are uplifted by + it. To deny that our pupils derive this pleasure from the study + is to confess ourselves poor teachers, for most pupils do have + positive enjoyment in the pursuit of geometry, in spite of the + tradition that leads them to proclaim a general dislike for all + study. This enjoyment is partly that of the + game,—the playing of a game that can always + be won, but that cannot be won too easily. It is partly that of + the aesthetic, the pleasure of symmetry of form, the delight of + fitting things together. But probably it lies chiefly in the + mental uplift that geometry brings, the contact with absolute + truth, and the approach that one makes to the Infinite. We are + not quite sure of any one thing in biology; our knowledge of + geology is relatively very slight, and the economic laws of + society are uncertain to every one except some individual who + attempts to set them forth; but before the world was fashioned + the square on the hypotenuse was equal to the sum of the + squares on the other two sides of a right triangle, and it will + be so after this world is dead; and the inhabitant of Mars, if + he exists, probably knows its truth as we know it. The uplift + of this contact with absolute truth, with truth + eternal, + +<span class="pagenum"> + <a name="Page_306" + id="Page_306">306</a></span> + + gives pleasure to humanity to a + greater or less degree, depending upon the mental equipment of + the particular individual; but it probably gives an appreciable + amount of pleasure to every student of geometry who has a + teacher worthy of the name.—<span + class="smcap">Smith, D. E.</span></p> + <p class="blockcite"> + The Teaching of Geometry (Boston, 1911), p. 16.</p> + + <p class="v2"> + <b><a name="Block_1851" id="Block_1851">1851</a>.</b> + No other person can judge better of either [the + merits of a writer and the merits of his works] than himself; + for none have had access to a closer or more deliberate + examination of them. It is for this reason, that in proportion + that the value of a work is intrinsic, and independent of + opinion, the less eagerness will the author feel to conciliate + the suffrages of the public. Hence that inward satisfaction, so + pure and so complete, which the study of geometry yields. The + progress which an individual makes in this science, the degree + of eminence which he attains in it, all this may be measured + with the same rigorous accuracy as the methods about which his + thoughts are employed. It is only when we entertain some doubts + about the justness of our own standard, that we become anxious + to relieve ourselves from our uncertainty, by comparing it with + the standard of another. Now, in all matters which fall under + the cognizance of taste, this standard is necessarily somewhat + variable; depending on a sort of gross estimate, always a + little arbitrary, either in whole or in part; and liable to + continual alteration in its dimensions, from negligence, + temper, or caprice. In consequence of these circumstances I + have no doubt, that if men lived separate from each other, and + could in such a situation occupy themselves about anything but + self-preservation, they would prefer the study of the exact + sciences to the cultivation of the agreeable arts. It is + chiefly on account of others, that a man aims at excellence in + the latter, it is on his own account that he devotes himself to + the former. In a desert island, accordingly, I should think + that a poet could scarcely be vain; whereas a geometrician + might still enjoy the pride of + discovery.—<span class="smcap">D’Alembert.</span></p> + <p class="blockcite"> + Essai sur les Gens Lettres; Melages (Amsterdam 1764), t. + 1, p. 334.</p> + + <p class="v2"> + <b><a name="Block_1852" id="Block_1852">1852</a>.</b> + If it were required to determine inclined planes of varying + inclinations of such lengths that a free rolling body + +<span class="pagenum"> + <a name="Page_307" + id="Page_307">307</a></span> + + would descend on them in equal times, + any one who understands the mechanical laws involved would + admit that this would necessitate sundry preparations. But in + the circle the proper arrangement takes place of its own accord + for an infinite variety of positions yet with the greatest + accuracy in each individual case. For all chords which meet the + vertical diameter whether at its highest or lowest point, and + whatever their inclinations, have this in common: that the free + descent along them takes place in equal times. I remember, one + bright pupil, who, after I had stated and demonstrated this + theorem to him, and he had caught the full import of it, was + moved as by a miracle. And, indeed, there is just cause for + astonishment and wonder when one beholds such a strange union + of manifold things in accordance with such fruitful rules in so + plain and simple an object as the circle. Moreover, there is no + miracle in nature, which because of its pervading beauty or + order, gives greater cause for astonishment, unless it be, for + the reason that its causes are not so clearly comprehended, + marvel being a daughter of + ignorance.—<span class="smcap">Kant.</span></p> + <p class="blockcite"> + Der einzig mögliche Beweisgrund zu einer Demonstration des + Daseins Gottes; Werke (Hartenstein), Bd. 2, p. 137.</p> + + <p class="v2"> + <b><a name="Block_1853" id="Block_1853">1853</a>.</b> + These examples [taken from the geometry of the + circle] indicate what a countless number of other such harmonic + relations obtain in the properties of space, many of which are + manifested in the relations of the various classes of curves in + higher geometry, all of which, besides exercising the + understanding through intellectual insight, affect the emotion + in a similar or even greater degree than the occasional + beauties of nature.—<span class="smcap">Kant.</span></p> + <p class="blockcite"> + Der einzig mögliche Beweisgrund zu einer Demonstration des + Daseins Gottes; Werke (Hartenstein), Bd. 2, p. 138.</p> + + <p class="v2"> + <b><a name="Block_1854" id="Block_1854">1854</a>.</b> + But neither thirty years, nor thirty centuries, + affect the clearness, or the charm, of Geometrical truths. Such + a theorem as “the square of the hypotenuse + of a right-angled triangle is equal to the sum of the squares + of the sides” is as dazzlingly beautiful now as it was in the + day when Pythagoras first discovered + +<span class="pagenum"> + <a name="Page_308" + id="Page_308">308</a></span> + + it, and celebrated its advent, + it is said, by sacrificing a hecatomb of + oxen—a method of doing honor to Science that + has always seemed to me <em>slightly</em> exaggerated and + uncalled-for. One can imagine oneself, even in these degenerate + days, marking the epoch of some brilliant scientific discovery + by inviting a convivial friend or two, to join one in a + beefsteak and a bottle of wine. But a <em>hecatomb</em> of oxen! + It would produce a quite inconvenient supply of + beef.—<span class="smcap">Dodgson, C. L.</span></p> + <p class="blockcite"> + A New Theory of Parallels (London, 1895), Introduction, + p. 16.</p> + + <p class="v2"> + <b><a name="Block_1855" id="Block_1855">1855</a>.</b> + After Pythagoras discovered his fundamental + theorem he sacrificed a hecatomb of oxen. Since that time all + dunces<a + href="#Footnote_10" + title="In the German vernacular a dunce or +blockhead is called an ox." + class="fnanchor">10</a> + [Ochsen] tremble whenever a new truth is + discovered.—<span class="smcap">Boerne.</span></p> + <p class="blockcite"> + Quoted in Moszkowski: Die unsterbliche Kiste (Berlin, + 1908), p. 18.</p> + + <p class="v2"> + <b><a name="Block_1856" id="Block_1856">1856</a>.</b></p> + <div class="poem"> + <p class="i8"> + <em>Vom Pythagorieschen Lehrsatz.</em></p> + + <hr class="blank" /> + + <p class="i2"> + Die Wahrheit, sie besteht in Ewigkeit,</p> + <p class="i0"> + Wenn erst die blöde Welt ihr Licht erkannt:</p> + <p class="i0"> + Der Lehrsatz, nach Pythagoras benannt,</p> + <p class="i0"> + Gilt heute, wie er galt in seiner Zeit.</p> + + <hr class="blank" /> + + <p class="i2"> + Ein Opfer hat Pythagoras geweiht</p> + <p class="i0"> + Den Göttern, die den Lichtstrahl ihm gesandt;</p> + <p class="i0"> + Es thaten kund, geschlachtet und verbrannt,</p> + <p class="i0"> + Ein hundert Ochsen seine Dankbarkeit.</p> + + <hr class="blank" /> + + <p class="i2"> + Die Ochsen seit den Tage, wenn sie wittern,</p> + <p class="i0"> + Dass eine neue Wahrheit sich enthülle,</p> + <p class="i0"> + Erheben ein unmenschliches Gebrülle;<br /></p> + + <hr class="blank" /> + + <p class="i2"> + Pythagoras erfüllt sie mit Entsetzen;</p> + <p class="i0"> + Und machtlos, sich dem Licht zu wiedersetzen,</p> + <p class="i0"> + Verschiessen sie die Augen und erzittern.</p> + </div> + <p class="block40"> + —<span class="smcap">Chamisso, Adelbert von.</span></p> + <p class="blockcite"> + Gedichte, 1835 (Haushenbusch), (Berlin, 1889), p. 302.</p> + +<p><span class="pagenum"> + <a name="Page_309" + id="Page_309">309</a></span></p> + + <div class="poem"> + + <hr class="blank" /> + + <p class="i2"> + Truth lasts throughout eternity,</p> + <p class="i0"> + When once the stupid world its light discerns:</p> + <p class="i0"> + The theorem, coupled with Pythagoras’ name,</p> + <p class="i0"> + Holds true today, as’t did in olden times.</p> + + <hr class="blank" /> + + <p class="i2"> + A splendid sacrifice Pythagoras brought</p> + <p class="i0"> + The gods, who blessed him with this ray divine;</p> + <p class="i0"> + A great burnt offering of a hundred kine,</p> + <p class="i0"> + Proclaimed afar the sage’s gratitude.</p> + + <hr class="blank" /> + + <p class="i2"> + Now since that day, all cattle [blockheads] when they scent</p> + <p class="i0"> + New truth about to see the light of day,</p> + <p class="i0"> + In frightful bellowings manifest their dismay;</p> + + <hr class="blank" /> + + <p class="i2"> + Pythagoras fills them all with terror;</p> + <p class="i0"> + And powerless to shut out light by error,</p> + <p class="i0"> + In sheer despair they shut their eyes and tremble.</p> + </div> + + <p class="v2"> + <b><a name="Block_1857" id="Block_1857">1857</a>.</b> + To the question “Which is the + signally most beautiful of geometrical truths?“ + Frankland replies: “One star excels another + in brightness, but the very sun will be, by common consent, a + property of the circle [Euclid, Book 3, Proposition 31] + selected for particular mention by Dante, that greatest of all + exponents of the + beautiful.”—<span class="smcap">Frankland, W. B.</span></p> + <p class="blockcite"> + The Story of Euclid (London, 1902), p. 70.</p> + + <p class="v2"> + <b><a name="Block_1858" id="Block_1858">1858</a>.</b></p> + <div class="poem"> + <p class="i12"> + As one</p> + <p class="i0"> + Who vers’d in geometric lore, would fain</p> + <p class="i0"> + Measure the circle; and, though pondering long</p> + <p class="i0"> + And deeply, that beginning, which he needs,</p> + <p class="i0"> + Finds not; e’en such was I, intent to scan</p> + <p class="i0"> + The novel wonder, and trace out the form,</p> + <p class="i0"> + How to the circle fitted, and therein</p> + <p class="i0"> + How plac’d: but the flight was not for my wing;</p> + </div> + <p class="block40"> + —<span class="smcap">Dante.</span></p> + <p class="blockcite"> + Paradise [Carey] Canto 33, lines 122-129.</p> + + <p class="v2"> + <b><a name="Block_1859" id="Block_1859">1859</a>.</b> + If geometry were as much opposed to our passions + and present interests as is ethics, we should contest it and + violate it + +<span class="pagenum"> + <a name="Page_310" + id="Page_310">310</a></span> + + but little less, notwithstanding all + the demonstrations of Euclid and of Archimedes, which you would + call dreams and believe full of paralogisms; and Joseph + Scaliger, Hobbes, and others, who have written against Euclid + and Archimedes, would not find themselves in such a small + company as at present.—<span class="smcap">Leibnitz.</span></p> + <p class="blockcite"> + New Essays concerning Human Understanding [Langley], Bk. + 1, chap. 2, sect. 12.</p> + + <p class="v2"> + <b><a name="Block_1860" id="Block_1860">1860</a>.</b> + I have no fault to find with those who teach + geometry. That science is the only one which has not produced + sects; it is founded on analysis and on synthesis and on the + calculus; it does not occupy itself with probable truth; + moreover it has the same method in every + country.—<span class="smcap">Frederick the Great.</span></p> + <p class="blockcite"> + Oeuvres (Decker), t. 7, p. 100.</p> + + <p class="v2"> + <b><a name="Block_1861" id="Block_1861">1861</a>.</b> + There are, undoubtedly, the most ample reasons for + stating both the principles and theorems [of geometry] in their + general form,.... But, that an unpractised learner, even in + making use of one theorem to demonstrate another, reasons + rather from particular to particular than from the general + proposition, is manifest from the difficulty he finds in + applying a theorem to a case in which the configuration of the + diagram is extremely unlike that of the diagram by which the + original theorem was demonstrated. A difficulty which, except + in cases of unusual mental powers, long practice can alone + remove, and removes chiefly by rendering us familiar with all + the configurations consistent with the general conditions of + the theorem.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + System of Logic, Bk. 2, chap. 3, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_1862" id="Block_1862">1862</a>.</b> + The reason why I impute any defect to geometry, + is, because its original and fundamental principles are + deriv’d merely from appearances; and it may + perhaps be imagin’d, that this defect must + always attend it, and keep it from ever reaching a greater + exactness in the comparison of objects or ideas, than what our + eye or imagination alone is able to attain. I own that this + defect so far attends it, as to keep it from ever aspiring to a + full certainty. But since these fundamental principles depend + on the easiest and least deceitful appearances, they bestow on + +<span class="pagenum"> + <a name="Page_311" id="Page_311">311</a></span> + + their consequences a degree of + exactness, of which these consequences are singly + incapable.—<span class="smcap">Hume, D.</span></p> + <p class="blockcite"> + A Treatise of Human Nature, Part 3, sect. 1.</p> + + <p class="v2"> + <b><a name="Block_1863" id="Block_1863">1863</a>.</b> + I have already observed, that geometry, or the + art, by which we fix the proportions of figures, + tho’ it much excels both in universality and + exactness, the loose judgments of the senses and imagination; + yet never attains a perfect precision and exactness. Its first + principles are still drawn from the general appearance of the + objects; and that appearance can never afford us any security, + when we examine the prodigious minuteness of which nature is + susceptible....</p> + <p class="v1"> + There remain, therefore, algebra and arithmetic as the only + sciences, in which we can carry on a chain of reasoning to any + degree of intricacy, and yet preserve a perfect exactness and + certainty.—<span class="smcap">Hume, D.</span></p> + <p class="blockcite"> + A Treatise of Human Nature, Part 3, sect. 1.</p> + + <p class="v2"> + <b><a name="Block_1864" id="Block_1864">1864</a>.</b> + All geometrical reasoning is, in the last resort, + circular: if we start by assuming points, they can only be + defined by the lines or planes which relate them; and if we + start by assuming lines or planes, they can only be defined by + the points through which they + pass.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + Foundations of Geometry (Cambridge, 1897), p. 120.</p> + + <p class="v2"> + <b><a name="Block_1865" id="Block_1865">1865</a>.</b> + The description of right lines and circles, upon + which Geometry is founded, belongs to Mechanics. Geometry does + not teach us to draw these lines, but requires them to be + drawn.... it requires that the learner should first be taught + to describe these accurately, before he enters upon Geometry; + then it shows how by these operations problems may be solved. + To describe right lines and circles are problems, but not + geometrical problems. The solution of these problems is + required from Mechanics; by Geometry the use of them, when + solved, is shown.... Therefore Geometry is founded in mechanical + practice, and is nothing but that part of universal Mechanics + which accurately proposes and demonstrates the art of measuring. + But since the manual arts are chiefly conversant in the + +<span class="pagenum"> + <a name="Page_312" + id="Page_312">312</a></span> + + moving of bodies, it comes to pass + that Geometry is commonly referred to their magnitudes, and + Mechanics to their motion.—<span class="smcap">Newton.</span></p> + <p class="blockcite"> + Philosophiae Naturalis Principia Mathematica, Praefat.</p> + + <p class="v2"> + <b><a name="Block_1866" id="Block_1866">1866</a>.</b> + We must, then, admit ... that there is an + independent science of geometry just as there is an independent + science of physics, and that either of these may be treated by + mathematical methods. Thus geometry becomes the simplest of the + natural sciences, and its axioms are of the nature of physical + laws, to be tested by experience and to be regarded as true + only within the limits of error of + observation—<span class="smcap">Bôcher, Maxime.</span></p> + <p class="blockcite"> + Bulletin American Mathematical Society, Vol. 2 (1904), + p. 124.</p> + + <p class="v2"> + <b><a name="Block_1867" id="Block_1867">1867</a>.</b> + Geometry is not an experimental science; + experience forms merely the occasion for our reflecting upon + the geometrical ideas which pre-exist in us. But the occasion + is necessary, if it did not exist we should not reflect, and if + our experiences were different, doubtless our reflections would + also be different. Space is not a form of sensibility; it is an + instrument which serves us not to represent things to + ourselves, but to reason upon + things.—<span class="smcap">Poincaré, H.</span></p> + <p class="blockcite"> + On the Foundations of Geometry; Monist, Vol. 9 (1898-1899), + p. 41.</p> + + <p class="v2"> + <b><a name="Block_1868" id="Block_1868">1868</a>.</b> + It has been said that geometry is an instrument. + The comparison may be admitted, provided it is granted at the + same time that this instrument, like Proteus in the fable, + ought constantly to change its + form.—<span class="smcap">Arago.</span></p> + <p class="blockcite"> + Oeuvres, t. 2 (1854), p. 694.</p> + + <p class="v2"> + <b><a name="Block_1869" id="Block_1869">1869</a>.</b> + It is essential that the treatment [of geometry] + should be rid of everything superfluous, for the superfluous is + an obstacle to the acquisition of knowledge; it should select + everything that embraces the subject and brings it to a focus, + for this is of the highest service to science; it must have + great regard both to clearness and to conciseness, for their + opposites trouble our understanding; it must aim to generalize + its theorems, for the + +<span class="pagenum"> + <a name="Page_313" id="Page_313">313</a></span> + + division of knowledge into small elements renders it difficult + of comprehension.—<span class="smcap">Proclus.</span></p> + <p class="blockcite"> + Quoted in D. E. Smith: The Teaching of Geometry (Boston, + 1911), p. 71.</p> + + <p class="v2"> + <b><a name="Block_1870" id="Block_1870">1870</a>.</b> + Many are acquainted with mathematics, but mathesis + few know. For it is one thing to know a number of propositions + and to make some obvious deductions from them, by accident + rather than by any sure method of procedure, another thing to + know clearly the nature and character of the science itself, to + penetrate into its inmost recesses, and to be instructed by its + universal principles, by which facility in working out + countless problems and their proofs is secured. For as the + majority of artists, by copying the same model again and again, + gain certain technical skill in painting, but no other + knowledge of the art of painting than what their eyes suggest, + so many, having read the books of Euclid and other + geometricians, are wont to devise, in imitation of them and to + prove some propositions, but the most profound method of solving + more difficult demonstrations and problems they are utterly + ignorant of.—<span class= "smcap">LaFaille, J. C.</span></p> + <p class="blockcite"> + Theoremata de Centro Gravitatis (Anvers, 1632), Praefat.</p> + + <p class="v2"> + <b><a name="Block_1871" id="Block_1871">1871</a>.</b> + The elements of plane geometry should precede + algebra for every reason known to sound educational theory. It + is more fundamental, more concrete, and it deals with things + and their relations rather than with + symbols.—<span class="smcap">Butler, N. M.</span></p> + <p class="blockcite"> + The Meaning of Education etc. (New York, 1905), p. 171.</p> + + <p class="v2"> + <b><a name="Block_1872" id="Block_1872">1872</a>.</b> + The reason why geometry is not so difficult as + algebra, is to be found in the less general nature of the + symbols employed. In algebra a general proposition respecting + numbers is to be proved. Letters are taken which may represent + any of the numbers in question, and the course of the + demonstration, far from making use of a particular case, does + not even allow that any reasoning, however general in its + nature, is conclusive, unless the symbols are as general as the + arguments.... In geometry on the contrary, at least in the + elementary parts, any proposition may be safely demonstrated on + reasonings on any one particular example.... It also affords + some facility that + +<span class="pagenum"> + <a name="Page_314" + id="Page_314">314</a></span> + + the results of elementary geometry + are in many cases sufficiently evident of themselves to the + eye; for instance, that two sides of a triangle are greater + than the third, whereas in algebra many rudimentary + propositions derive no evidence from the senses; for example, + that a<sup>3</sup>−b<sup>3</sup> is always divisible without + a remainder by a−b.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + On the Study and Difficulties of Mathematics (Chicago, + 1902), chap. 13.</p> + + <p class="v2"> + <b><a name="Block_1873" id="Block_1873">1873</a>.</b> + The principal characteristics of the ancient + geometry are:—</p> + <p class="v1"> + (1) A wonderful clearness and definiteness of its concepts and + an almost perfect logical rigour of its conclusions.</p> + <p class="v1"> + (2) A complete want of general principles and methods.... In + the demonstration of a theorem, there were, for the ancient + geometers, as many different cases requiring separate proof as + there were different positions of the lines. The greatest + geometers considered it necessary to treat all possible cases + independently of each other, and to prove each with equal + fulness. To devise methods by which all the various cases could + all be disposed of with one stroke, was beyond the power of the + ancients.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Mathematics (New York, 1897), p. 62.</p> + + <p class="v2"> + <b><a name="Block_1874" id="Block_1874">1874</a>.</b> + It has been observed that the ancient geometers + made use of a kind of + + <a id="TNanchor_16"></a> + <a class="msg" href="#TN_16" + title="originally read ‘anaylsis’">analysis</a>, + + which they employed in the + solution of problems, although they begrudged to posterity the + knowledge of it.—<span class="smcap">Descartes.</span></p> + <p class="blockcite"> + Rules for the Direction of the Mind; The Philosophy of + Descartes [Torrey] (New York, 1892), p. 68.</p> + + <p class="v2"> + <b><a name="Block_1875" id="Block_1875">1875</a>.</b> + The ancients studied geometry with reference to + the <em>bodies</em> under notice, or specially: the moderns study + it with reference to the <em>phenomena</em> to be considered, or + generally. The ancients extracted all they could out of one + line or surface, before passing to another; and each inquiry + gave little or no assistance in the next. The moderns, since + Descartes, employ themselves on questions which relate to any + figure whatever. They abstract, to treat by itself, every + question relating to the same + +<span class="pagenum"> + <a name="Page_315" + id="Page_315">315</a></span> + + geometrical phenomenon, + in whatever bodies it may be considered. Geometers can thus + rise to the study of new geometrical conceptions, which, + applied to the curves investigated by the ancients, have + brought out new properties never suspected by + them.—<span class="smcap">Comte.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 1, chap. 3.</p> + + <p class="v2"> + <b><a name="Block_1876" id="Block_1876">1876</a>.</b> + It is astonishing that this subject [projective + geometry] should be so generally ignored, for mathematics + offers nothing more attractive. It possesses the concreteness + of the ancient geometry without the tedious particularity, and + the power of the analytical geometry without the reckoning, and + by the beauty of its ideas and methods illustrates the esthetic + generality which is the charm of higher mathematics, but which + the elementary mathematics generally lacks.</p> + <p class="blockcite"> + Report of the Committee of Ten on Secondary School Studies + (Chicago, 1894), p. 116.</p> + + <p class="v2"> + <b><a name="Block_1877" id="Block_1877">1877</a>.</b> + There exist a small number of very simple + fundamental relations which contain the scheme, according to + which the remaining mass of theorems [in projective geometry] + permit of orderly and easy development.</p> + <p class="v1"> + By a proper appropriation of a few fundamental relations one + becomes master of the whole subject; order takes the place of + chaos, one beholds how all parts fit naturally into each other, + and arrange themselves serially in the most beautiful order, + and how related parts combine into well-defined groups. In this + manner one arrives, as it were, at the elements, which nature + herself employs in order to endow figures with numberless + properties with the utmost economy and + simplicity.—<span class="smcap">Steiner, J.</span></p> + <p class="blockcite"> + Werke, Bd. 1 (1881), p. 233.</p> + + <p class="v2"> + <b><a name="Block_1878" id="Block_1878">1878</a>.</b> + Euclid once said to his king Ptolemy, who, as is + easily understood, found the painstaking study of the + “Elements” repellant, “There exists no royal road to + mathematics.” But we may add: Modern geometry is a + royal road. It has disclosed “the organism, + by means of which the most heterogeneous phenomena in the world + of space are united one with another + +<span class="pagenum"> + <a name="Page_316" + id="Page_316">316</a></span>” + + (Steiner), and has, as + we may say without exaggeration, almost attained to the + scientific ideal.—<span class="smcap">Hankel, H.</span></p> + <p class="blockcite"> + Die Entwickelung der Mathematik in den letzten + Jahrhunderten (Tübingen, 1869).</p> + + <p class="v2"> + <b><a name="Block_1879" id="Block_1879">1879</a>.</b> + The two mathematically fundamental things in + projective geometry are anharmonic ratio, and the quadrilateral + construction. Everything else follows mathematically from these + two.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + Foundations of Geometry (Cambridge, 1897), p. 122.</p> + + <p class="v2"> + <b><a name="Block_1880" id="Block_1880">1880</a>.</b> + ... Projective Geometry: a boundless domain of + countless fields where reals and imaginaries, finites and + infinites, enter on equal terms, where the spirit delights in + the artistic balance and symmetric interplay of a kind of + conceptual and logical counterpoint,—an + enchanted realm where thought is double and flows throughout in + parallel streams.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Arts (New York, 1908), + p. 2.</p> + + <p class="v2"> + <b><a name="Block_1881" id="Block_1881">1881</a>.</b> + The ancients, in the early days of the science, + made great use of the graphic method, even in the form of + construction; as when Aristarchus of Samos estimated the + distance of the sun and moon from the earth on a triangle + constructed as nearly as possible in resemblance to the + right-angled triangle formed by the three bodies at the instant + when the moon is in quadrature, and when therefore an + observation of the angle at the earth would define the + triangle. Archimedes himself, though he was the first to + introduce calculated determinations into geometry, frequently + used the same means. The introduction of trigonometry lessened + the practice; but did not abolish it. The Greeks and Arabians + employed it still for a great number of investigations for + which we now consider the use of the Calculus + indispensable.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 1, chap. 3.</p> + + <p class="v2"> + <b><a name="Block_1882" id="Block_1882">1882</a>.</b> + A mathematical problem may usually be attacked by + what is termed in military parlance the method of + “systematic approach;” that is + to say, its solution may be gradually felt + +<span class="pagenum"> + <a name="Page_317" + id="Page_317">317</a></span> + + for, even though the successive steps leading to that solution + cannot be clearly foreseen. But a Descriptive Geometry problem + must be seen through and through before it can be attempted. + The entire scope of its conditions, as well as each step toward + its solution, must be grasped by the imagination. It must be + “taken by assault”—<span class="smcap">Clarke, G. S.</span></p> + <p class="blockcite"> + Quoted in W. S. Hall: Descriptive Geometry (New York, + 1902), chap. 1.</p> + + <p class="v2"> + <b><a name="Block_1883" id="Block_1883">1883</a>.</b> + The grand use [of Descriptive Geometry] is in its + application to the industrial arts;—its few + abstract problems, capable of invariable solution, relating + essentially to the contacts and intersections of surfaces; so + that all the geometrical questions which may arise in any of + the various arts of construction,—as + stone-cutting, carpentry, perspective, dialing, fortification, + etc.,—can always be treated as simple + individual cases of a single theory, the solution being + certainly obtainable through the particular circumstances of + each case. This creation must be very important in the eyes of + philosophers who think that all human achievement, thus far, is + only a first step toward a philosophical renovation of the + labours of mankind; towards that precision and logical + character which can alone ensure the future progression of all + arts.... Of Descriptive Geometry, it may further be said that + it usefully exercises the student’s faculty + of Imagination,—of conceiving of complicated + geometrical combinations in space; and that, while it belongs + to the geometry of the ancients by the character of its + solutions, it approaches to the geometry of the moderns by the + nature of the questions which compose + it.—<span class="smcap">Comte, A.</span></p> + <p class="blockcite"> + Positive Philosophy [Martineau], Bk. 1, chap. 3.</p> + + <p class="v2"> + <b><a name="Block_1884" id="Block_1884">1884</a>.</b> + There is perhaps nothing which so occupies, as it + were, the middle position of mathematics, as + trigonometry.—<span class="smcap">Herbart, J. F.</span></p> + <p class="blockcite"> + Idee eines ABC der Anschauung; Werke (Kehrbach) + (Langensalza, 1890), Bd. 1, p. 174.</p> + + <p class="v2"> + <b><a name="Block_1885" id="Block_1885">1885</a>.</b> + Trigonometry contains the science of continually + undulating magnitude: meaning magnitude which becomes + alternately greater and less, without + any termination to succession + +<span class="pagenum"> + <a name="Page_318" + id="Page_318">318</a></span> + + of increase and decrease.... All + trigonometric functions are not undulating: but it may be + stated that in common algebra nothing but infinite series + undulate: in trigonometry nothing but infinite series do not + undulate.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Trigonometry and Double Algebra (London, 1849), Bk. 1, + chap. 1.</p> + + <p class="v2"> + <b><a name="Block_1886" id="Block_1886">1886</a>.</b> + Sin<sup>2</sup>φ is odious to me, even though Laplace made use + of it; should it be feared that sinφ<sup>2</sup> might become + ambiguous, which would perhaps never occur, or at most very + rarely when speaking of sin (φ<sup>2</sup>), well then, let us + write (sinφ)<sup>2</sup>, but not sin<sup>2</sup>φ, which by + analogy should signify + sin(sinφ).—<span class= "smcap">Gauss.</span></p> + <p class="blockcite"> + Gauss-Schumacher Briefwechsel, Bd. 3, p. 292; Bd. 4, p. 63.</p> + + <p class="v2"> + <b><a name="Block_1887" id="Block_1887">1887</a>.</b> + Perhaps to the student there is no part of + elementary mathematics so repulsive as is spherical + trigonometry.—<span class="smcap">Tait, P. G.</span></p> + <p class="blockcite"> + Encyclopedia Britannica, 9th Edition; Article “Quaternions”</p> + + <p class="v2"> + <b><a name="Block_1888" id="Block_1888">1888</a>.</b> + “Napier’s Rule of circular parts” is perhaps the happiest + example of artificial memory that is + known.—<span class="smcap">Cajori, F.</span></p> + <p class="blockcite"> + History of Mathematics (New York, 1897), p. 165.</p> + + <p class="v2"> + <b><a name="Block_1889" id="Block_1889">1889</a>.</b> + The analytical equations, unknown to the ancients, + which Descartes first introduced into the study of curves and + surfaces, are not restricted to the properties of figures, and + to those properties which are the object of rational mechanics; + they apply to all phenomena in general. There cannot be a + language more universal and more simple, more free from errors + and obscurities, that is to say, better adapted to express the + invariable relations of nature.—<span class= + "smcap">Fourier.</span></p> + <p class="blockcite"> + Théorie Analytique de la Chaleur, Discours Préliminaire.</p> + + <p class="v2"> + <b><a name="Block_1890" id="Block_1890">1890</a>.</b> + It is impossible not to feel stirred at the + thought of the emotions of men at certain historic moments of + adventure and discovery—Columbus when he + first saw the Western shore, Pizarro when he stared at the + Pacific Ocean, Franklin when the + +<span class="pagenum"> + <a name="Page_319" + id="Page_319">319</a></span> + + electric spark came + from the string of his kite, Galileo when he first turned his + telescope to the heavens. Such moments are also granted to + students in the abstract regions of thought, and high among + them must be placed the morning when Descartes lay in bed and + invented the method of co-ordinate + geometry.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + An Introduction to Mathematics (New York, 1911), p. 122.</p> + + <p class="v2"> + <b><a name="Block_1891" id="Block_1891">1891</a>.</b> + It is often said that an equation contains only + what has been put into it. It is easy to reply that the new + form under which things are found often constitutes by itself + an important discovery. But there is something more: analysis, + by the simple play of its symbols, may suggest generalizations + far beyond the original limits.—<span class= + "smcap">Picard, E.</span></p> + <p class="blockcite"> + Bulletin American Mathematical Society, Vol. 2 (1905), p. + 409.</p> + + <p class="v2"> + <b><a name="Block_1892" id="Block_1892">1892</a>.</b> + It is not the Simplicity of the Equation, but the + Easiness of the Description, which is to determine the Choice + of our Lines for the Constructions of Problems. For the + Equation that expresses a Parabola is more simple than that + that expresses the Circle, and yet the Circle, by its more + simple Construction, is admitted before + it.—<span class="smcap">Newton.</span></p> + <p class="blockcite"> + The Linear Constructions of Equations; Universal + Arithmetic (London, 1769), Vol. 2, p. 468.</p> + + <p class="v2"> + <b><a name="Block_1893" id="Block_1893">1893</a>.</b> + The pursuit of mathematics unfolds its formative + power completely only with the transition from the elementary + subjects to analytical geometry. Unquestionably the simplest + geometry and algebra already accustom the mind to sharp + quantitative thinking, as also to assume as true only axioms + and what has been proven. But the representation of functions + by curves or surfaces reveals a new world of concepts and + teaches the use of one of the most fruitful methods, which the + human mind ever employed to increase its own effectiveness. + What the discovery of this method by Vieta and Descartes + brought to humanity, that it brings today to every one who is + in any measure endowed for such things: a life-epoch-making + beam of light [Lichtblick]. This method has its roots in the + farthest + +<span class="pagenum"> + <a name="Page_320" + id="Page_320">320</a></span> + + depths of human cognition and so has + an entirely different significance, than the most ingenious + artifice which serves a special + purpose.—<span class="smcap">Bois-Reymond, Emil du.</span></p> + <p class="blockcite"> + Reden, Bd. 1 (Leipzig, 1885), p. 287.</p> + + <p class="v2"> + <b><a name="Block_1894" id="Block_1894">1894</a>.</b></p> + <div class="poem"> + <p class="i8"> + <em>Song of the Screw.</em></p> + + <hr class="blank" /> + + <p class="i0"> + A moving form or rigid mass,</p> + <p class="i2"> + Under whate’er conditions</p> + <p class="i0"> + Along successive screws must pass</p> + <p class="i2"> + Between each two positions.</p> + <p class="i0"> + It turns around and slides along—</p> + <p class="i0"> + This is the burden of my song.</p> + + <hr class="blank" /> + + <p class="i0"> + The pitch of screw, if multiplied</p> + <p class="i2"> + By angle of rotation,</p> + <p class="i0"> + Will give the distance it must glide</p> + <p class="i2"> + In motion of translation.</p> + <p class="i0"> + Infinite pitch means pure translation,</p> + <p class="i0"> + And zero pitch means pure rotation.</p> + + <hr class="blank" /> + + <p class="i0"> + Two motions on two given screws,</p> + <p class="i2"> + With amplitudes at pleasure,</p> + <p class="i0"> + Into a third screw-motion fuse,</p> + <p class="i2"> + Whose amplitude we measure</p> + <p class="i0"> + By parallelogram construction</p> + <p class="i0"> + (A very obvious deduction).</p> + + <hr class="blank" /> + + <p class="i0"> + Its axis cuts the nodal line</p> + <p class="i2"> + Which to both screws is normal,</p> + <p class="i0"> + And generates a form divine,</p> + <p class="i2"> + Whose name, in language formal,</p> + <p class="i0"> + Is “surface-ruled of third degree.”</p> + <p class="i0"> + Cylindroid is the name for me.</p> + + <hr class="blank" /> + + + <p class="i0"> + Rotation round a given line</p> + <p class="i2"> + Is like a force along,</p> + <p class="i0"> + If to say couple you decline,</p> + <p class="i2"> + You’re clearly in the wrong;—</p> + <p class="i0"> + ’Tis obvious, upon reflection,</p> + <p class="i0"> + A line is not a mere direction.</p> + +<span class="pagenum"> + <a name="Page_321" + id="Page_321">321</a></span> + + <hr class="blank" /> + + <p class="i0"> + So couples with translations too</p> + <p class="i2"> + In all respects agree;</p> + <p class="i0"> + And thus there centres in the screw</p> + <p class="i2"> + A wondrous harmony</p> + <p class="i0"> + Of Kinematics and of Statics,—</p> + <p class="i0"> + The sweetest thing in mathematics.</p> + + <hr class="blank" /> + + <p class="i0"> + The forces on one given screw,</p> + <p class="i2"> + With motion on a second,</p> + <p class="i0"> + In general some work will do,</p> + <p class="i2"> + Whose magnitude is reckoned</p> + <p class="i0"> + By angle, force, and what we call</p> + <p class="i0"> + The coefficient virtual.</p> + + <hr class="blank" /> + + <p class="i0"> + Rotation now to force convert,</p> + <p class="i2"> + And force into rotation;</p> + <p class="i0"> + Unchanged the work, we can assert,</p> + <p class="i2"> + In spite of transformation.</p> + <p class="i0"> + And if two + screws no work can claim,</p> + <p class="i0"> + Reciprocal will be their name.</p> + + <hr class="blank" /> + + <p class="i0"> + Five numbers will a screw define,</p> + <p class="i2"> + A screwing motion, six;</p> + <p class="i0"> + For four will give the axial line,</p> + <p class="i2"> + One more the pitch will fix;</p> + <p class="i0"> + And hence we always can contrive</p> + <p class="i0"> + One screw reciprocal to five.</p> + + <hr class="blank" /> + + <p class="i0"> + Screws—two, three, four or five, combined</p> + <p class="i2"> + (No question here of six),</p> + <p class="i0"> + Yield other screws which are confined</p> + <p class="i2"> + Within one screw complex.</p> + <p class="i0"> + Thus we obtain the clearest notion</p> + <p class="i0"> + Of freedom and constraint of motion.</p> + + <hr class="blank" /> + + <p class="i0"> + In complex III, three several screws</p> + <p class="i2"> + At every point you find,</p> + <p class="i0"> + Or if you one direction choose,</p> + <p class="i2"> + One screw is to your mind; + +<span class="pagenum"> + <a name="Page_322" + id="Page_322">322</a></span></p> + + <p class="i0"> + And complexes of order III.</p> + <p class="i0"> + Their own reciprocals may be.</p> + + <hr class="blank" /> + + <p class="i0"> + In IV, wherever you arrive,</p> + <p class="i2"> + You find of screws a cone,</p> + <p class="i0"> + On every line of complex V.</p> + <p class="i2"> + There is precisely one;</p> + <p class="i0"> + At each point of this complex + rich,</p> + <p class="i0"> + A plane of screws have given pitch.</p> + + <hr class="blank" /> + + <p class="i0"> + But time would fail me to discourse</p> + <p class="i2"> + Of Order and Degree;</p> + <p class="i0"> + Of Impulse, Energy and Force,</p> + <p class="i2"> + And Reciprocity.</p> + <p class="i0"> + All these and more, for motions small,</p> + <p class="i0"> + Have been discussed by Dr. Ball.</p> + </div> + <p class="block40"> + —<span class="smcap">Anonymous.</span></p> + +<p><span class="pagenum"> + <a name="Page_323" + id="Page_323">323</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_XIX"> + CHAPTER XIX<br /> + <span class="large"> + THE CALCULUS AND ALLIED TOPICS</span></h2> + + <p class="v2"> + <b><a name="Block_1901" id="Block_1901">1901</a>.</b> + It may be said that the conceptions of + differential quotient and integral, which in their origin + certainly go back to Archimedes, were introduced into science + by the investigations of Kepler, Descartes, Cavalieri, Fermat + and Wallis.... The capital discovery that differentiation and + integration are <em>inverse</em> operations belongs to Newton and + Leibnitz.—<span class="smcap">Lie, Sophus.</span></p> + <p class="blockcite"> + Leipziger Berichte, 47 (1895), Math.-phys. Classe, p. 53.</p> + + <p class="v2"> + <b><a name="Block_1902" id="Block_1902">1902</a>.</b> + It appears that Fermat, the true inventor of the + differential calculus, considered that calculus as derived from + the calculus of finite differences by neglecting infinitesimals + of higher orders as compared with those of a lower order.... + Newton, through his method of fluxions, has since rendered the + calculus more analytical, he also simplified and generalized + the method by the invention of his binomial theorem. Leibnitz + has enriched the differential calculus by a very happy + notation.—<span class="smcap">Laplace.</span></p> + <p class="blockcite"> + Lés Intégrales + Définies, etc.; Oeuvres, t. 12 (Paris, 1898), p. 359.</p> + + <p class="v2"> + <b><a name="Block_1903" id="Block_1903">1903</a>.</b> + Professor Peacock’s Algebra, + and Mr. Whewell’s Doctrine of Limits should + be studied by every one who desires to comprehend the evidence + of mathematical truths, and the meaning of the obscure + processes of the calculus; while, even after mastering these + treatises, the student will have much to learn on the subject + from M. Comte, of whose admirable work one of the most + admirable portions is that in which he may truly be said to + have created the philosophy of the higher + mathematics.—<span class="smcap">Mill, J. S.</span></p> + <p class="blockcite"> + System of Logic, Bk. 3, chap. 24, sect. 6.</p> + +<p><span class="pagenum"> + <a name="Page_324" + id="Page_324">324</a></span></p> + + <p class="v2"> + <b><a name="Block_1904" id="Block_1904">1904</a>.</b> + If we must confine ourselves to one system of + notation then there can be no doubt that that which was + invented by Leibnitz is better fitted for most of the purposes + to which the infinitesimal calculus is applied than that of + fluxions, and for some (such as the calculus of variations) it + is indeed almost essential.—<span class= + "smcap">Ball, W. W. R.</span></p> + <p class="blockcite"> + History of Mathematics (London, 1901), p. 371.</p> + + <p class="v2"> + <b><a name="Block_1905" id="Block_1905">1905</a>.</b> + The difference between the method of + infinitesimals and that of limits (when exclusively adopted) + is, that in the latter it is usual to retain evanescent + quantities of higher orders until the end of the calculation + and then neglect them. On the other hand, such quantities are + neglected from the commencement in the infinitesimal method, + from the conviction that they cannot affect the final result, + as they must disappear when we proceed to the + limit.—<span class="smcap">Williamson, B.</span></p> + <p class="blockcite"> + Encyclopedia Britannica, 9th Edition; Article + “Infinitesimal Calculus,” sect. 14.</p> + + <p class="v2"> + <b><a name="Block_1906" id="Block_1906">1906</a>.</b> + When we have grasped the spirit of the + infinitesimal method, and have verified the exactness of its + results either by the geometrical method of prime and ultimate + ratios, or by the analytical method of derived functions, we + may employ infinitely small quantities as a sure and valuable + means of shortening and simplifying our + proofs.—<span class="smcap">Lagrange.</span></p> + <p class="blockcite"> + Méchanique Analytique, Preface; Oeuvres, t. 2 + (Paris, 1888), p. 14.</p> + + <p class="v2"> + <b><a name="Block_1907" id="Block_1907">1907</a>.</b> + The essential merit, the sublimity, of the + infinitesimal method lies in the fact that it is as easily + performed as the simplest method of approximation, and that it + is as accurate as the results of an ordinary calculation. This + advantage would be lost, or at least greatly impaired, if, + under the pretense of securing greater accuracy throughout the + whole process, we were to substitute for the simpler method + given by Leibnitz, one less convenient and less in harmony with + the probable course of natural events....</p> + <p class="v1"> + The objections which have been raised against the infinitesimal + method are based on the false supposition that the + errors + +<span class="pagenum"> + <a name="Page_325" + id="Page_325">325</a></span> + + due to neglecting infinitely small quantities during the actual + calculation will continue to exist in the result of the + calculation.—<span class="smcap">Carnot, L.</span></p> + <p class="blockcite"> + Réflections sur la Métaphysique + du Calcul Infinitésimal (Paris, 1813), p. 215.</p> + + <p class="v2"> + <b><a name="Block_1908" id="Block_1908">1908</a>.</b> + A limiting ratio is neither more nor less + difficult to define than an infinitely small + quantity.—<span class="smcap">Carnot, L.</span></p> + <p class="blockcite"> + Réflections sur la Métaphysique + du Calcul Infinitésimal (Paris, 1813), p. 210.</p> + + <p class="v2"> + <b><a name="Block_1909" id="Block_1909">1909</a>.</b> + A limit is a peculiar and fundamental conception, + the use of which in proving the propositions of Higher Geometry + cannot be superseded by any combination of other hypotheses and + definitions. The axiom just noted that what is true up to the + limit is true at the limit, is involved in the very conception + of a limit: and this principle, with its consequences, leads to + all the results which form the subject of the higher + mathematics, whether proved by the consideration of evanescent + triangles, by the processes of the Differential Calculus, or in + any other way.—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + The Philosophy of the Inductive Sciences, Part 1, bk. 2, + chap. 12, sect. 1, (London, 1858).</p> + + <p class="v2"> + <b><a name="Block_1910" id="Block_1910">1910</a>.</b> + The differential calculus has all the exactitude + of other algebraic operations.—<span class= + "smcap">Laplace.</span></p> + <p class="blockcite"> + Théorie Analytique des Probabilités, Introduction; + Oeuvres, t. 7 (Paris, 1886), p. 37.</p> + + <p class="v2"> + <b><a name="Block_1911" id="Block_1911">1911</a>.</b> + The method of fluxions is probably one of the + greatest, most subtle, and sublime discoveries of any age: it + opens a new world to our view, and extends our knowledge, as it + were, to infinity; carrying us beyond the bounds that seemed to + have been prescribed to the human mind, at least infinitely + beyond those to which the ancient geometry was + confined.—<span class="smcap">Hutton, Charles.</span></p> + <p class="blockcite"> + A Philosophical and Mathematical Dictionary (London, + 1815), Vol. 1, p. 525.</p> + + <p class="v2"> + <b><a name="Block_1912" id="Block_1912">1912</a>.</b> + The states and conditions of matter, as they occur + in nature, are in a state of perpetual flux, and these + qualities may + +<span class="pagenum"> + <a name="Page_326" + id="Page_326">326</a></span> + + be effectively studied by the + Newtonian method (Methodus fluxionem) whenever they can be + referred to number or subjected to measurement (real or + imaginary). By the aid of Newton’s calculus + the mode of action of natural changes from moment to moment can + be portrayed as faithfully as these words represent the + thoughts at present in my mind. From this, the law which + controls the whole process can be determined with unmistakable + certainty by pure calculation.—<span class= + "smcap">Mellor, J. W.</span></p> + <p class="blockcite"> + Higher Mathematics for Students of Chemistry and Physics + (London, 1902), Prologue.</p> + + <p class="v2"> + <b><a name="Block_1913" id="Block_1913">1913</a>.</b> + The calculus is the greatest aid we have to the + appreciation of physical truth in the broadest sense of the + word.—<span class="smcap">Osgood, W. F.</span></p> + <p class="blockcite"> + Bulletin American Mathematical Society, Vol. 13 (1907), p. + 467.</p> + + <p class="v2"> + <b><a name="Block_1914" id="Block_1914">1914</a>.</b> + [Infinitesimal] analysis is the most powerful + weapon of thought yet devised by the wit of + man.—<span class="smcap">Smith, W. B.</span></p> + <p class="blockcite"> + Infinitesimal Analysis (New York, 1898), Preface, p. vii.</p> + + <p class="v2"> + <b><a name="Block_1915" id="Block_1915">1915</a>.</b> + The method of Fluxions is the general key by help + whereof the modern mathematicians unlock the secrets of + Geometry, and consequently of Nature. And, as it is that which + hath enabled them so remarkably to outgo the ancients in + discovering theorems and solving problems, the exercise and + application thereof is become the main if not sole employment + of all those who in this age pass for profound + geometers.—<span class="smcap">Berkeley, George.</span></p> + <p class="blockcite"> + The Analyst, sect. 3.</p> + + <p class="v2"> + <b><a name="Block_1916" id="Block_1916">1916</a>.</b> + I have at last become fully satisfied that the + language and idea of infinitesimals should be used in the most + elementary instruction—under all safeguards + of course.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Graves’ Life of W. R. Hamilton (New + York, 1882-1889), Vol. 3, p. 479.</p> + + <p class="v2"> + <b><a name="Block_1917" id="Block_1917">1917</a>.</b> + Pupils should be taught how to differentiate and how to + integrate simple algebraic expressions before we attempt to + +<span class="pagenum"> + <a name="Page_327" + id="Page_327">327</a></span> + + teach them geometry and these other + complicated things. The dreadful fear of the symbols is + entirely broken down in those cases where at the beginning the + teaching of the calculus is adopted. Then after the pupil has + mastered those symbols you may begin geometry or anything you + please. I would also abolish out of the school that thing + called geometrical conics. There is a great deal of + superstition about conic sections. The student should be taught + the symbols of the calculus and the simplest use of these + symbols at the earliest age, instead of these being left over + until he has gone to the College or + University.—<span class="smcap">Thompson, S. P.</span></p> + <p class="blockcite"> + Perry’s Teaching of Mathematics (London, 1902), p. 49.</p> + + <p class="v2"> + <b><a name="Block_1918" id="Block_1918">1918</a>.</b> + Every one versed in the matter will agree that + even the elements of a scientific study of nature can be + understood only by those who have a knowledge of at least the + elements of the differential and integral calculus, as well as + of analytical geometry—i.e. the so-called + lower part of the higher mathematics.... We should raise the + question, whether sufficient time could not be reserved in the + curricula of at least the science high schools [Realanstalten] + to make room for these subjects....</p> + <p class="v1"> + The first consideration would be to entirely relieve from the + mathematical requirements of the university [Hochschule] + certain classes of students who can get along without extended + mathematical knowledge, or to make the necessary mathematical + knowledge accessible to them in a manner which, for various + reasons, has not yet been adopted by the university. Among such + students I would count architects, also the chemists and in + general the students of the so-called descriptive natural + sciences. I am moreover of the opinion—and + this has been for long a favorite idea of + mine—, that it would be very useful to + medical students to acquire such mathematical knowledge as is + indicated by the above described modest limits; for it seems + impossible to understand far-reaching physiological + investigations, if one is terrified as soon as a differential + or integration symbol appears.—<span class= + "smcap">Klein, F.</span></p> + <p class="blockcite"> + Jahresbericht der Deutschen Mathematiker Vereinigung, Bd. + 2 (1902), p. 131.</p> + +<p><span class="pagenum"> + <a name="Page_328" + id="Page_328">328</a></span></p> + + <p class="v2"> + <b><a name="Block_1919" id="Block_1919">1919</a>.</b> + Common integration is only the <em>memory of + differentiation</em> ... the different artifices by which + integration is effected, are changes, not from the known to the + unknown, but from forms in which memory will not serve us to + those in which it will.—<span + class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Transactions Cambridge Philosophical Society, Vol. 8 + (1844), p. 188.</p> + + <p class="v2"> + <b><a name="Block_1920" id="Block_1920">1920</a>.</b> + Given for one instant an intelligence which could + comprehend all the forces by which nature is animated and the + respective positions of the beings which compose it, if + moreover this intelligence were vast enough to submit these + data to analysis, it would embrace in the same formula both the + movements of the largest bodies in the universe and those of + the lightest atom: to it nothing would be uncertain, and the + future as the past would be present to its eyes. The human mind + offers a feeble outline of that intelligence, in the perfection + which it has given to astronomy. Its discoveries in mechanics + and in geometry, joined to that of universal gravity, have + enabled it to comprehend in the same analytical expressions the + past and future states of the world + system.—<span class="smcap">Laplace.</span></p> + <p class="blockcite"> + Théorie Analytique des Probabilités, Introduction; + Oeuvres, t. 7 (Paris, 1886), p. 6.</p> + + <p class="v2"> + <b><a name="Block_1921" id="Block_1921">1921</a>.</b> + There is perhaps the same relation between the + action of natural selection during one generation and the + accumulated result of a hundred thousand generations, that + there exists between differential and integral. How seldom are + we able to follow completely this latter relation although we + subject it to calculation. Do we on that account doubt the + correctness of our + integrations?—<span class="smcap">Bois-Reymond, Emil du.</span></p> + <p class="blockcite"> + Reden, Bd. 1 (Leipzig, 1885), p. 228.</p> + + <p class="v2"> + <b><a name="Block_1922" id="Block_1922">1922</a>.</b> + It seems to be expected of every pilgrim up the + slopes of the mathematical Parnassus, that he will at some + point or other of his journey sit down and invent a definite + integral or two towards the increase of the common + stock.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Notes to the Meditation on Poncelet’s + Theorem; Mathematical Papers, Vol. 2, p. 214.</p> + +<p><span class="pagenum"> + <a name="Page_329" + id="Page_329">329</a></span></p> + + <p class="v2"> + <b><a name="Block_1923" id="Block_1923">1923</a>.</b> + The experimental verification of a theory + concerning any natural phenomenon generally rests on the result + of an integration.—<span class="smcap">Mellor, J. W.</span></p> + <p class="blockcite"> + Higher Mathematics for Students of Chemistry and Physics + (New York, 1902), p. 150.</p> + + <p class="v2"> + <b><a name="Block_1924" id="Block_1924">1924</a>.</b> + Among all the mathematical disciplines the theory + of differential equations is the most important.... It + furnishes the explanation of all those elementary + manifestations of nature which involve + time....—<span class="smcap">Lie, Sophus.</span></p> + <p class="blockcite"> + Leipziger Berichte, 47 (1895); Math.-phys. Classe, p. 262.</p> + + <p class="v2"> + <b><a name="Block_1925" id="Block_1925">1925</a>.</b> + If the mathematical expression of our ideas leads + to equations which cannot be integrated, the working hypothesis + will either have to be verified some other way, or else + relegated to the great repository of unverified + speculations.—<span class="smcap">Mellor, J. W.</span></p> + <p class="blockcite"> + Higher Mathematics for Students of Chemistry and Physics + (New York, 1902), p. 157.</p> + + <p class="v2"> + <b><a name="Block_1926" id="Block_1926">1926</a>.</b> + It is well known that the central problem of the + whole of modern mathematics is the study of the transcendental + functions defined by differential + equations.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Lectures on Mathematics (New York, 1911), p. 8.</p> + + <p class="v2"> + <b><a name="Block_1927" id="Block_1927">1927</a>.</b> + Every one knows what a curve is, until he has + studied enough mathematics to become confused through the + countless number of possible exceptions.... A curve is the + totality of points, whose co-ordinates are functions of a + parameter which may be differentiated as often as may be + required.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Elementar Mathematik vom höheren Standpunkte + aus. (Leipzig. 1909) Vol. 2, p. 354.</p> + + <p class="v2"> + <b><a name="Block_1928" id="Block_1928">1928</a>.</b> + Fourier’s theorem is not only + one of the most beautiful results of modern analysis, but it + may be said to furnish an indispensable instrument in the + treatment of nearly every recondite question in modern physics. + To mention only sonorous vibrations, the propagation of + electric signals along telegraph wires, and the conduction of + heat by the earth’s + +<span class="pagenum"> + <a name="Page_330" + id="Page_330">330</a></span> + + crust, as subjects in their generality intractable without it, + is to give but a feeble idea of its + importance.—<span class="smcap">Thomson and Tait.</span></p> + <p class="blockcite"> + Elements of Natural Philosophy, chap. 1.</p> + + <p class="v2"> + <b><a name="Block_1929" id="Block_1929">1929</a>.</b> + The principal advantage arising from the use of + hyperbolic functions is that they bring to light some curious + analogies between the integrals of certain irrational + functions.—<span class="smcap">Byerly, W. E.</span></p> + <p class="blockcite"> + Integral Calculus (Boston, 1890), p. 30.</p> + + <p class="v2"> + <b><a name="Block_1930" id="Block_1930">1930</a>.</b> + Hyperbolic functions are extremely useful in every + branch of pure physics and in the applications of physics + whether to observational and experimental sciences or to + technology. Thus whenever an entity (such as light, velocity, + electricity, or radio-activity) is subject to gradual + absorption or extinction, the decay is represented by some form + of hyperbolic functions. Mercator’s + projection is likewise computed by hyperbolic functions. + Whenever mechanical strains are regarded great enough to be + measured they are most simply expressed in terms of hyperbolic + functions. Hence geological deformations invariably lead to + such expressions....—<span class="smcap">Walcott, C. D.</span></p> + <p class="blockcite"> + Smithsonian Mathematical Tables, + + <a id="TNanchor_17"></a> + <a class="msg" href="#TN_17" + title="originally read ‘Hyberbolic’">Hyperbolic</a> + + Functions + (Washington, 1909), Advertisement.</p> + + <p class="v2"> + <b><a name="Block_1931" id="Block_1931">1931</a>.</b> + Geometry may sometimes appear to take the lead + over analysis, but in fact precedes it only as a servant goes + before his master to clear the path and light him on the way. + The interval between the two is as wide as between empiricism + and science, as between the understanding and the reason, or as + between the finite and the + infinite.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + Philosophic Magazine, Vol. 31 (1866), p. 521.</p> + + <p class="v2"> + <b><a name="Block_1932" id="Block_1932">1932</a>.</b> + Nature herself exhibits to us measurable and + observable quantities in definite mathematical dependence; the + conception of a function is suggested by all the processes of + nature where we observe natural phenomena varying according to + distance or to time. Nearly all the + “known” functions have presented + themselves in the attempt to solve geometrical, mechanical, or + physical problems.—<span class="smcap">Merz, J. T.</span></p> + <p class="blockcite"> + A History of European Thought in the Nineteenth Century + (Edinburgh and London, 1903), p. 696.</p> + +<p><span class="pagenum"> + <a name="Page_331" + id="Page_331">331</a></span></p> + + <p class="v2"> + <b><a name="Block_1933" id="Block_1933">1933</a>.</b> + That flower of modern mathematical + thought—the notion of a + function.—<span class="smcap">McCormack, Thomas J.</span></p> + <p class="blockcite"> + On the Nature of Scientific Law and Scientific + Explanation, Monist, Vol. 10 (1899-1900), p. 555.</p> + + <p class="v2"> + <b><a name="Block_1934" id="Block_1934">1934</a>.</b></p> + <div class="poem"> + <p class="i0"> + Fuchs. Ich bin von alledem so consterniert,</p> + <p class="i7"> + Als würde mir ein Kreis im Kopfe quadriert.</p> + <p class="i0"> + Meph.</p> + <p class="i7"> + Nachher vor alien andern Sachen</p> + <p class="i7"> + Müsst ihe euch an die Funktionen-Theorie machen.</p> + <p class="i7"> + Da seht, dass ihr tiefsinnig fasst,</p> + <p class="i7"> + Was sich zu integrieren nicht passt.</p> + <p class="i7"> + An Theoremen wird’s euch nicht fehlen,</p> + <p class="i7"> + Müsst nur die Verschwindungspunkte zählen,</p> + <p class="i7"> + Umkehren, abbilden, auf der Eb’ne ’rumfahren</p> + <p class="i7"> + Und mit den Theta-Produkten nicht sparen.</p> + </div> + <p class="block40"> + —<span class="smcap">Lasswitz, Kurd.</span></p> + <p class="blockcite"> + Der Faust-Tragödie (-n)ter Tiel; Zeitschrift + für den math.-natur. Unterricht, Bd. 14 (1883), p. 316.</p> + + <div class="poem"> + <hr class="blank" /> + <p class="i0"> + Fuchs.</p> + <p class="i7"> + Your words fill me with an awful dread,</p> + <p class="i7"> + Seems like a circle were squared in my head.</p> + <p class="i0"> + Meph.</p> + <p class="i7"> + Next in order you certainly ought</p> + <p class="i7"> + On function-theory bestow your thought,</p> + <p class="i7"> + And penetrate with contemplation</p> + <p class="i7"> + What resists your attempts at integration.</p> + <p class="i7"> + You’ll find no dearth of theorems there—</p> + <p class="i7"> + To vanishing-points give proper care—</p> + <p class="i7"> + Enumerate, reciprocate,</p> + <p class="i7"> + Nor forget to delineate,</p> + <p class="i7"> + Traverse the plane from end to end,</p> + <p class="i7"> + And theta-functions freely spend.</p> + </div> + + <p class="v2"> + <b><a name="Block_1935" id="Block_1935">1935</a>.</b> + The student should avoid <em>founding results</em> + upon divergent series, as the question of their legitimacy is + disputed upon grounds to which no answer commanding anything + like general assent has yet been given. But they may be used as + means of + +<span class="pagenum"> + <a name="Page_332" + id="Page_332">332</a></span> + + discovery, provided that their + results be verified by other means before they are considered + as established.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Trigonometry and Double Algebra (London, 1849), p. 55.</p> + + <p class="v2"> + <b><a name="Block_1936" id="Block_1936">1936</a>.</b> + There is nothing now which ever gives me any + thought or care in algebra except divergent series, which I + cannot follow the French in + rejecting.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Graves’ Life of W. R. Hamilton (New + York, 1882-1889), Vol. 3, p. 249.</p> + + <p class="v2"> + <b><a name="Block_1937" id="Block_1937">1937</a>.</b> + It is a strange vicissitude of our science that + these [divergent] series which early in the century were + supposed to be banished once and for all from rigorous + mathematics should at its close be knocking at the door for + readmission.—<span class="smcap">Pierpont, J.</span></p> + <p class="blockcite"> + Congress of Arts and Sciences (Boston and New York, 1905), + Vol. 1, p. 476.</p> + + <p class="v2"> + <b><a name="Block_1938" id="Block_1938">1938</a>.</b> + Zeno was concerned with three problems.... These + are the problem of the infinitesimal, the infinite, and + continuity.... From him to our own day, the finest intellects + of each generation in turn attacked these problems, but + achieved broadly speaking nothing.... Weierstrass, Dedekind, + and Cantor, ... have completely solved them. Their solutions + ... are so clear as to leave no longer the slightest doubt of + difficulty. This achievement is probably the greatest of which + the age can boast.... The problem of the infinitesimal was + solved by Weierstrass, the solution of the other two was begun + by Dedekind and definitely accomplished by + Cantor.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + International Monthly, Vol. 4 (1901), p. 89.</p> + + <p class="v2"> + <b><a name="Block_1939" id="Block_1939">1939</a>.</b> + It was not till Leibnitz and Newton, by the + discovery of the differential calculus, had dispelled the + ancient darkness which enveloped the conception of the + infinite, and had clearly established the conception of the + continuous and continuous change, that a full and productive + application of the newly-found mechanical conceptions made any + progress.—<span class="smcap">Helmholtz, H.</span></p> + <p class="blockcite"> + Aim and Progress of Physical Science; Popular Lectures + [Flight] (New York, 1900), p. 372.</p> + +<p><span class="pagenum"> + <a name="Page_333" + id="Page_333">333</a></span></p> + + <p class="v2"> + <b><a name="Block_1940" id="Block_1940">1940</a>.</b> + The idea of an infinitesimal involves no + contradiction.... As a mathematician, I prefer the method of + infinitesimals to that of limits, as far easier and less + infested with snares.—<span class="smcap">Pierce, C. F.</span></p> + <p class="blockcite"> + The Law of Mind; Monist, Vol. 2 (1891-1892), pp. 543, 545.</p> + + <p class="v2"> + <b><a name="Block_1941" id="Block_1941">1941</a>.</b> + The chief objection against all <em>abstract</em> + reasonings is derived from the ideas of space and time; ideas, + which, in common life and to a careless view, are very clear + and intelligible, but when they pass through the scrutiny of + the profound sciences (and they are the chief object of these + sciences) afford principles, which seem full of obscurity and + contradiction. No priestly <em>dogmas</em>, invented on purpose + to tame and subdue the rebellious reason of mankind, ever + shocked common sense more than the doctrine of the infinite + divisibility of extension, with its consequences; as they are + pompously displayed by all geometricians and metaphysicians, + with a kind of triumph and exultation. A real quantity, + infinitely less than any finite quantity, containing quantities + infinitely less than itself, and so on <em>in infinitum</em>; + this is an edifice so bold and prodigious, that it is too + weighty for any pretended demonstration to support, because it + shocks the clearest and most natural principles of human + reason. But what renders the matter more extraordinary, is, + that these seemingly absurd opinions are supported by a chain + of reasoning, the clearest and most natural; nor is it possible + for us to allow the premises without admitting the + consequences. Nothing can be more convincing and satisfactory + than all the conclusions concerning the properties of circles + and triangles; and yet, when these are once received, how can + we deny, that the angle of contact between a circle and its + tangent is infinitely less than any rectilineal angle, that as + you may increase the diameter of the circle <em>in + infinitum</em>, this angle of contact becomes still less, even + <em>in infinitum</em>, and that the angle of contact between + other curves and their tangents may be infinitely less than + those between any circle and its tangent, and so on, <em>in + infinitum</em>? The demonstration of these principles seems as + unexceptionable as that which proves the three angles of a + triangle to be equal to two right ones, though the latter + +<span class="pagenum"> + <a name="Page_334" id= "Page_334">334</a></span> + + opinion be natural and easy, and the + former big with contradiction and absurdity. Reason here seems + to be thrown into a kind of amazement and suspense, which, + without the suggestion of any sceptic, gives her a diffidence + of herself, and of the ground on which she treads. She sees a + full light, which illuminates certain places; but that light + borders upon the most profound darkness. And between these she + is so dazzled and confounded, that she scarcely can pronounce + with certainty and assurance concerning any one + object.—<span class="smcap">Hume, David.</span></p> + <p class="blockcite"> + An Inquiry concerning Human Understanding, Sect. 12, part 2.</p> + + <p class="v2"> + <b><a name="Block_1942" id="Block_1942">1942</a>.</b> + He who can digest a second or third fluxion, a + second or third difference, need not, methinks, be squeamish + about any point in Divinity.—<span class= + "smcap">Berkeley, G.</span></p> + <p class="blockcite"> + The Analyst, sect. 7.</p> + + <p class="v2"> + <b><a name="Block_1943" id="Block_1943">1943</a>.</b> + And what are these fluxions? The velocities of + evanescent increments. And what are these same evanescent + increments? They are neither finite quantities, nor quantities + infinitely small, nor yet nothing. May we not call them ghosts + of departed quantities?—- <span class= + "smcap">Berkeley, G.</span></p> + <p class="blockcite"> + The Analyst, sect. 35.</p> + + <p class="v2"> + <b><a name="Block_1944" id="Block_1944">1944</a>.</b> + It is said that the minutest errors are not to be + neglected in mathematics; that the fluxions are celerities, not + proportional to the finite increments, though ever so small; + but only to the moments or nascent increments, whereof the + proportion alone, and not the magnitude, is considered. And of + the aforesaid fluxions there be other fluxions, which fluxions + of fluxions are called second fluxions. And the fluxions of + these second fluxions are called third fluxions: and so on, + fourth, fifth, sixth, etc., <em>ad infinitum</em>. Now, as our + Sense is strained and puzzled with the perception of objects + extremely minute, even so the Imagination, which faculty + derives from sense, is very much strained and puzzled to frame + clear ideas of the least particle of time, or the least + increment generated therein: and much more to comprehend the + moments, or those increments of the flowing quantities in + <i lang="la" xml:lang="la">status nascenti</i>, in their + first origin or beginning to + exist, before they become finite particles. And it + +<span class="pagenum"> + <a name="Page_335" + id="Page_335">335</a></span> + + seems + still more difficult to conceive the abstracted velocities of + such nascent imperfect entities. But the velocities of the + velocities, the second, third, fourth, and fifth velocities, + etc., exceed, if I mistake not, all human understanding. The + further the mind analyseth and pursueth these fugitive ideas + the more it is lost and bewildered; the objects, at first + fleeting and minute, soon vanishing out of sight. Certainly, in + any sense, a second or third fluxion seems an obscure Mystery. + The incipient celerity of an incipient celerity, the nascent + augment of a nascent augment, i.e. of a thing which hath no + magnitude; take it in what light you please, the clear + conception of it will, if I mistake not, be found impossible; + whether it be so or no I appeal to the trial of every thinking + reader. And if a second fluxion be inconceivable, what are we + to think of third, fourth, fifth fluxions, and so on without + end.—<span class="smcap">Berkeley, G.</span></p> + <p class="blockcite"> + The Analyst, sect, 4.</p> + + <p class="v2"> + <b><a name="Block_1945" id="Block_1945">1945</a>.</b> + The <em>infinite</em> divisibility of <em>finite</em> + extension, though it is not expressly laid down either as an + axiom or theorem in the elements of that science, yet it is + throughout the same everywhere supposed and thought to have so + inseparable and essential a connection with the principles and + demonstrations in Geometry, that mathematicians never admit it + into doubt, or make the least question of it. And, as this + notion is the source whence do spring all those amusing + geometrical paradoxes which have such a direct repugnancy to + the plain common sense of mankind, and are admitted with so + much reluctance into a mind not yet debauched by learning; so + it is the principal occasion of all that nice and extreme + subtility which renders the study of Mathematics so difficult + and tedious.—<span class="smcap">Berkeley, G.</span></p> + <p class="blockcite"> + On the Principles of Human Knowledge, Sect. 123.</p> + + <p class="v2"> + <b><a name="Block_1946" id="Block_1946">1946</a>.</b> + To avoid misconception, it should be borne in mind + that infinitesimals are not regarded as being actual quantities + in the ordinary acceptation of the words, or as capable of + exact representation. They are introduced for the purpose of + abridgment and simplification of our reasonings, and are an + ultimate phase of magnitude when it is conceived by the mind as + capable of diminution below any assigned quantity, however + small.... + +<span class="pagenum"> + <a name="Page_336" + id="Page_336">336</a></span> + + Moreover such quantities are + neglected, not, as Leibnitz stated, because they are infinitely + small in comparison with those that are retained, which would + produce an infinitely small error, but because they must be + neglected to obtain a rigorous result; since such result must + be definite and determinate, and consequently independent of + these <em>variable indefinitely small quantities</em>. It may be + added that the precise principles of the infinitesimal + calculus, like those of any other science, cannot be thoroughly + apprehended except by those who have already studied the + science, and made some progress in the application of its + principles.—<span class="smcap">Williamson, B.</span></p> + <p class="blockcite"> + Encyclopedia Britannica, 9th Edition; Article + “Infinitesimal Calculus,” Sect. 12, 14.</p> + + <p class="v2"> + <b><a name="Block_1947" id="Block_1947">1947</a>.</b> + We admit, in geometry, not only infinite + magnitudes, that is to say, magnitudes greater than any + assignable magnitude, but infinite magnitudes infinitely + greater, the one than the other. This astonishes our dimension + of brains, which is only about six inches long, five broad, and + six in depth, in the largest + heads.—<span class="smcap">Voltaire.</span></p> + <p class="blockcite"> + A Philosophical Dictionary; Article “Infinity.” (Boston, + 1881).</p> + + <p class="v2"> + <b><a name="Block_1948" id="Block_1948">1948</a>.</b> + Infinity is the land of mathematical hocus pocus. + There Zero the magician is king. When Zero divides any number + he changes it without regard to its magnitude into the + infinitely small [great?], and inversely, when divided by any + number he begets the infinitely great [small?]. In this domain + the circumference of the circle becomes a straight line, and + then the circle can be squared. Here all ranks are abolished, + for Zero reduces everything to the same level one way or + another. Happy is the kingdom where Zero + rules!—<span class="smcap">Carus, Paul.</span></p> + <p class="blockcite"> + Logical and Mathematical Thought; Monist, Vol. 20 + (1909-1910), p. 69.</p> + + <p class="v2"> + <b><a name="Block_1949" id="Block_1949">1949</a>.</b></p> + <div class="poem"> + <p class="i0"> + Great fleas have little fleas upon their backs to bite ’em,</p> + <p class="i0"> + And little fleas have lesser fleas, + and so <em>ad infinitum.</em></p> + <p class="i0"> + And the great fleas themselves, in turn, + have greater fleas to go on;</p> + <p class="i0"> + While these again have greater still, + and greater still, and so on.</p> + </div> + <p class="block40"> + —<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), p. 377.</p> + +<p><span class="pagenum"> + <a name="Page_337" + id="Page_337">337</a></span></p> + + <p class="v2"> + <b><a name="Block_1950" id="Block_1950">1950</a>.</b> + We have adroitly defined the infinite in + arithmetic by a loveknot, in this manner ∞; + but we possess not therefore the clearer notion of + it.—<span class="smcap">Voltaire.</span></p> + <p class="blockcite"> + A Philosophical Dictionary; Article “Infinity.” (Boston, + 1881).</p> + + <p class="v2"> + <b><a name="Block_1951" id="Block_1951">1951</a>.</b> + I protest against the use of infinite magnitude as + something completed, which in mathematics is never permissible. + Infinity is merely a <i lang="fr" xml:lang="fr">facon de + parler</i>, the real meaning + being a limit which certain ratios approach indefinitely near, + while others are permitted to increase without + restriction.—<span class="smcap">Gauss.</span></p> + <p class="blockcite"> + Brief an Schumacher (1831); Werke, Bd. 8 p. 216.</p> + + <p class="v2"> + <b><a name="Block_1952" id="Block_1952">1952</a>.</b> + In spite of the essential difference between the + conceptions of the <em>potential</em> and the <em>actual</em> + infinite, the former signifying a <em>variable</em> finite + magnitude increasing beyond all finite limits, while the latter + is a <em>fixed</em>, <em>constant</em> quantity lying beyond all + finite magnitudes, it happens only too often that the one is + mistaken for the other.... Owing to a justifiable aversion to + such <em>illegitimate</em> actual infinities and the influence of + the modern epicuric-materialistic tendency, a certain <em>horror + infiniti</em> has grown up in extended scientific circles, which + finds its classic expression and support in the letter of Gauss + [see 1951], yet it seems to me that the consequent uncritical + rejection of the legitimate actual infinite is no lesser + violation of the nature of things, which must be taken as they + are.—<span class="smcap">Cantor, G.</span></p> + <p class="blockcite"> + Zum Problem des actualen Unendlichen; Natur und + Offenbarung, Bd. 32 (1886), p. 226.</p> + + <p class="v2"> + <b><a name="Block_1953" id="Block_1953">1953</a>.</b> + The Infinite is often confounded with the + Indefinite, but the two conceptions are diametrically opposed. + Instead of being a quantity with unassigned yet assignable + limits, the Infinite is not a quantity at all, since it neither + admits of augmentation nor diminution, having no assignable + limits; it is the operation of continuously <em>withdrawing</em> + any limits that may have been assigned: the endless addition of + new quantities to the old: the flux of continuity. The Infinite + is no more a quantity than Zero is a quantity. If Zero is the + sign of a vanished quantity, the + +<span class="pagenum"><a + name="Page_338" + id="Page_338">338</a></span> + + Infinite is a sign of + that continuity of Existence which has been ideally divided + into discrete parts in the affixing of + limits.—<span class="smcap">Lewes, G. H.</span></p> + <p class="blockcite"> + Problems of Life and Mind (Boston, 1875), Vol. 2, p. 384.</p> + + <p class="v2"> + <b><a name="Block_1954" id="Block_1954">1954</a>.</b> + A great deal of misunderstanding is avoided if it + be remembered that the terms <em>infinity</em>, <em>infinite</em>, + <em>zero</em>, <em>infinitesimal</em> must be interpreted in + connexion with their context, and admit a variety of meanings + according to the way in which they are + defined.—<span class="smcap">Mathews, G. B.</span></p> + <p class="blockcite"> + Theory of Numbers (Cambridge, 1892), Part 1, sect. 104.</p> + + <p class="v2"> + <b><a name="Block_1955" id="Block_1955">1955</a>.</b> + This further is observable in number, that it is + that which the mind makes use of in measuring all things that + by us are measurable, which principally are <em>expansion</em> + and <em>duration</em>; and our idea of infinity, even when + applied to those, seems to be nothing but the infinity of + number. For what else are our ideas of Eternity and Immensity, + but the repeated additions of certain ideas of imagined parts + of duration and expansion, with the infinity of number; in + which we can come to no end of + addition?—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 2, chap. 16, + sect. 8.</p> + + <p class="v2"> + <b><a name="Block_1956" id="Block_1956">1956</a>.</b> + But of all other ideas, it is number, which I + think furnishes us with the clearest and most distinct idea of + infinity we are capable of.—<span class= + "smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 2, chap. 17, + sect. 9.</p> + + <p class="v2"> + <b><a name="Block_1957" id="Block_1957">1957</a>.</b></p> + <div class="poem"> + <p class="i0"> + Willst du ins Unendliche schreiten?</p> + <p class="i0"> + Geh nur im Endlichen nach allen Seiten!</p> + <p class="i0"> + Willst du dich am Ganzen erquicken,</p> + <p class="i0"> + So musst du das Ganze im Kleinsten erblicken.</p> + </div> + <p class="block40"> + —<span class="smcap">Goethe.</span></p> + <p class="blockcite"> + Gott, Gemüt und Welt (1815).</p> + + <div class="poem"> + <hr class="blank" /> + <p class="i0"> + [Would’st thou the infinite essay?</p> + <p class="i0"> + The finite but traverse in every way.</p> + <p class="i0"> + Would’st in the whole delight thy heart?</p> + <p class="i0"> + Learn to discern the whole in its minutest part.]</p> + </div> + +<p> + <span class="pagenum"> + <a name="Page_339" + id="Page_339">339</a></span></p> + + <p class="v2"> + <b><a name="Block_1958" id="Block_1958">1958</a>.</b></p> + <div class="poem"> + <p class="i0"> + Ich häufe ungeheure Zahlen,</p> + <p class="i0"> + Gebürge Millionen auf,</p> + <p class="i0"> + Ich setze Zeit auf Zeit und Welt auf Welt zu Hauf,</p> + <p class="i0"> + Und wenn ich von der grausen Höh’</p> + <p class="i0"> + Mit Schwindeln wieder nach dir seh,’</p> + <p class="i0"> + Ist alle Macht der Zahl, vermehrt zu tausendmalen,</p> + <p class="i0"> + Noch nicht ein Theil von dir.</p> + <p class="i0"> + <i>Ich zieh’ sie ab, und du liegst ganz vor mir</i>.</p> + </div> + <p class="block40"> + —<span class="smcap">Haller, Albr. Von.</span></p> + <p class="blockcite"> + Quoted in Hegel: Wissenschaft der Logik, Buch 1, Abschnitt + 2, Kap. 2, C, b.</p> + + <div class="poem"> + <hr class="blank" /> + <p class="i0"> + [Numbers upon numbers pile,</p> + <p class="i0"> + Mountains millions high,</p> + <p class="i0"> + Time on time and world on world + amass,</p> + <p class="i0"> + Then, if from the + dreadful hight, alas!</p> + <p class="i0"> + Dizzy-brained, I turn thee to behold,</p> + <p class="i0"> + All the power of number, increased + thousandfold,</p> + <p class="i0"> + Not yet may match + thy part.</p> + <p class="i0"> + <em>Subtract what I + will, wholly whole thou art</em>.]</p> + </div> + + <p class="v2"> + <b><a name="Block_1959" id="Block_1959">1959</a>.</b> + A collection of terms is infinite when it contains + as parts other collections which have just as many terms in it + as it has. If you can take away some of the terms of a + collection, without diminishing the number of terms, then there + is an infinite number of terms in the + collection.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + International Monthly, Vol. 4 (1901), p. 93.</p> + + <p class="v2"> + <b><a name="Block_1960" id="Block_1960">1960</a>.</b> + An assemblage (ensemble, collection, group, + manifold) of elements (things, no matter what) is infinite or + finite according as it has or has not a part to which the whole + is just <em>equivalent</em> in the sense that between the + elements composing that part and those composing the whole + there subsists a unique and reciprocal (one-to-one) + correspondence.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + The Axioms of Infinity; Hibbert Journal, Vol. 2 + (1903-1904), p. 539.</p> + + <p class="v2"> + <b><a name="Block_1961" id="Block_1961">1961</a>.</b> + Whereas in former times the Infinite betrayed its + presence not indeed to the faculties of Logic but only to the + spiritual Imagination and Sensibility, + mathematics has shown ... that + +<span class="pagenum"> + <a name="Page_340" + id="Page_340">340</a></span> + + the structure of Transfinite + Being is open to exploration by the organon of + Thought.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Lectures on Science, Philosophy and Art (New York, 1908), + p. 42.</p> + + <p class="v2"> + <b><a name="Block_1962" id="Block_1962">1962</a>.</b> + The mathematical theory of probability is a + science which aims at reducing to calculation, where possible, + the amount of credence due to propositions or statements, or to + the occurrence of events, future or past, more especially as + contingent or dependent upon other propositions or events the + probability of which is known.—<span class= + "smcap">Crofton, M. W.</span></p> + <p class="blockcite"> + Encyclopedia Britannica, 9th Edition; Article, “Probability”</p> + + <p class="v2"> + <b><a name="Block_1963" id="Block_1963">1963</a>.</b> + The theory of probabilities is at bottom nothing + but common sense reduced to calculus; it enables us to + appreciate with exactness that which accurate minds feel with a + sort of instinct for which ofttimes they are unable to account. + If we consider the analytical methods to which this theory has + given birth, the truth of the principles on which it is based, + the fine and delicate logic which their employment in the + solution of problems requires, the public utilities whose + establishment rests upon it, the extension which it has + received and which it may still receive through its application + to the most important problems of natural philosophy and the + moral sciences; if again we observe that, even in matters which + cannot be submitted to the calculus, it gives us the surest + suggestions for the guidance of our judgments, and that it + teaches us to avoid the illusions which often mislead us, then + we shall see that there is no science more worthy of our + contemplations nor a more useful one for admission to our + system of public education.—<span class="smcap">Laplace.</span></p> + <p class="blockcite"> + Théorie Analytique des Probabilitiés, Introduction; + Oeuvres, t. 7 (Paris, 1886), p. 153.</p> + + <p class="v2"> + <b><a name="Block_1964" id="Block_1964">1964</a>.</b> + It is a truth very certain that, when it is not in + our power to determine what is true, we ought to follow what is + most probable.—<span class="smcap">Descartes.</span></p> + <p class="blockcite"> + Discourse on Method, Part 3.</p> + + <p class="v2"> + <b><a name="Block_1965" id="Block_1965">1965</a>.</b> + As <em>demonstration</em> is the showing the + agreement or disagreement of two ideas, by the intervention of + one or more + +<span class="pagenum"> + <a name="Page_341" + id="Page_341">341</a></span> + + proofs, which have a constant, + immutable, and visible connexion one with another; so + <em>probability</em> is nothing but the appearance of such an + agreement or disagreement, by the intervention of proofs, whose + connexion is not constant and immutable, or at least is not + perceived to be so, and it is enough to induce the mind to + judge the proposition to be true or false, rather than + contrary.—<span class="smcap">Locke, John.</span></p> + <p class="blockcite"> + An Essay concerning Human Understanding, Bk. 4, chap. 15, + sect. 1.</p> + + <p class="v2"> + <b><a name="Block_1966" id="Block_1966">1966</a>.</b> + The difference between necessary and contingent + truths is indeed the same as that between commensurable and + incommensurable numbers. For the reduction of commensurable + numbers to a common measure is analogous to the demonstration + of necessary truths, or their reduction to such as are + identical. But as, in the case of surd ratios, the reduction + involves an infinite process, and yet approaches a common + measure, so that a definite but unending series is obtained, so + also contingent truths require an infinite analysis, which God + alone can accomplish.—<span class="smcap">Leibnitz.</span></p> + <p class="blockcite"> + Philosophische Schriften [Gerhardt] Bd. 7 (Berlin, 1890), + p. 200.</p> + + <p class="v2"> + <b><a name="Block_1967" id="Block_1967">1967</a>.</b> + The theory in question [theory of probability] + affords an excellent illustration of the application of the + theory of permutation and combinations which is the fundamental + part of the algebra of discrete quantity; it forms in the + elementary parts an excellent logical exercise in the accurate + use of terms and in the nice discrimination of shades of + meaning; and, above all, it enters into the regulation of some + of the most important practical concerns of modern + life.—<span class="smcap">Chrystal, George.</span></p> + <p class="blockcite"> + Algebra, Vol. 2 (Edinburgh, 1889), chap. 36, sect. 1.</p> + + <p class="v2"> + <b><a name="Block_1968" id="Block_1968">1968</a>.</b> + There is possibly no branch of mathematics at once + so interesting, so bewildering, and of so great practical + importance as the theory of probabilities. Its history reveals + both the wonders that can be accomplished and the bounds that + cannot be transcended by mathematical science. It is the link + between rigid deduction and the vast field of inductive + science. A complete theory of probabilities would be the + complete theory of + +<span class="pagenum"> + <a name="Page_342" + id="Page_342">342</a></span> + + the formation of belief. It is + certainly a pity then, that, to quote M. Bertrand, + “one cannot well understand the calculus of + probabilities without having read Laplace’s + work,” and that “one cannot read + Laplace’s work without having prepared + oneself for it by the most profound mathematical + studies”—<span class="smcap">Davis, E. W.</span></p> + <p class="blockcite"> + Bulletin American Mathematical Society, Vol. 1 (1894-1895), + p. 16.</p> + + <p class="v2"> + <b><a name="Block_1969" id="Block_1969">1969</a>.</b> + The most important questions of life are, for the + most part, really only problems of probability. Strictly + speaking one may even say that nearly all our knowledge is + problematical; and in the small number of things which we are + able to know with certainty, even in the mathematical sciences + themselves, induction and analogy, the principal means for + discovering truth, are based on probabilities, so that the + entire system of human knowledge is connected with this + theory.—<span class="smcap">Laplace.</span></p> + <p class="blockcite"> + Théorie Analytique des Probabilitiés, Introduction; + Oeuvres, t. 7 (Paris, 1886), p. 5.</p> + + <p class="v2"> + <b><a name="Block_1970" id="Block_1970">1970</a>.</b> + There is no more remarkable feature in the + mathematical theory of probability than the manner in which it + has been found to harmonize with, and justify, the conclusions + to which mankind have been led, not by reasoning, but by + instinct and experience, both of the individual and of the + race. At the same time it has corrected, extended, and invested + them with a definiteness and precision of which these crude, + though sound, appreciations of common sense were till then + devoid.—<span class="smcap">Crofton, M. W.</span></p> + <p class="blockcite"> + Encyclopedia Britannica, 9th Edition; Article “Probability”</p> + + <p class="v2"> + <b><a name="Block_1971" id="Block_1971">1971</a>.</b> + It is remarkable that a science [probabilities] + which began with the consideration of games of chance, should + have become the most important object of human + knowledge.—<span class="smcap">Laplace.</span></p> + <p class="blockcite"> + Théorie Analytique des Probabilitiés, Introduction; + Oeuvres, t. 7 (Paris, 1886), p. 152.</p> + + <p class="v2"> + <b><a name="Block_1972" id="Block_1972">1972</a>.</b> + Not much has been added to the subject [of + probability] since the close of Laplace’s + career. The history of science + +<span class="pagenum"> + <a name="Page_343" + id="Page_343">343</a></span> + + records more than one + parallel to this abatement of activity. When such a genius has + departed, the field of his labours seems exhausted for the + time, and little left to be gleaned by his successors. It is to + be regretted that so little remains to us of the inner workings + of such gifted minds, and of the clue by which each of their + discoveries was reached. The didactic and synthetic form in + which these are presented to the world retains but faint traces + of the skilful inductions, the keen and delicate perception of + fitness and analogy, and the power of imagination ... which + have doubtless guided such a master as Laplace or Newton in + shaping out such great designs—only the + minor details of which have remained over, to be supplied by + the less cunning hand of commentator and + disciple.—<span class="smcap">Crofton, M. W.</span></p> + <p class="blockcite"> + Encyclopedia Britannica, 9th Edition; Article “Probability”</p> + + <p class="v2"> + <b><a name="Block_1973" id="Block_1973">1973</a>.</b> + The theory of errors may be defined as that branch + of mathematics which is concerned, first, with the expression + of the resultant effect of one or more sources of error to + which computed and observed quantities are subject; and, + secondly, with the determination of the relation between the + magnitude of an error and the probability of its + occurrence.—<span class="smcap">Woodward, R. S.</span></p> + <p class="blockcite"> + Probability and Theory of Errors (New York, 1906), p. 30.</p> + + <p class="v2"> + <b><a name="Block_1974" id="Block_1974">1974</a>.</b> + Of all the applications of the doctrine of + probability none is of greater utility than the theory of + errors. In astronomy, geodesy, physics, and chemistry, as in + every science which attains precision in measuring, weighing, + and computing, a knowledge of the theory of errors is + indispensable. By the aid of this theory the exact sciences + have made great progress during the nineteenth century, not + only in the actual determinations of the constants of nature, + but also in the fixation of clear ideas as to the possibilities + of future conquests in the same direction. Nothing, for + example, is more satisfactory and instructive in the history of + science than the success with which the unique method of least + squares has been applied to the problems presented by the earth + and the other members of the solar system. So great, in fact, + are the practical value and theoretical importance + +<span class="pagenum"> + <a name="Page_344" + id="Page_344">344</a></span> + + of least squares, that it is frequently mistaken for + the whole theory of errors, and is sometimes regarded as + embodying the major part of the doctrine of probability + itself.—<span class="smcap">Woodward, R. S.</span></p> + <p class="blockcite"> + Probability and Theory of Errors (New York, 1906), pp. 9-10.</p> + + <p class="v2"> + <b><a name="Block_1975" id="Block_1975">1975</a>.</b> + Direct and inverse ratios have been applied by an + ingenious author to measure human affections, and the moral + worth of actions. An eminent Mathematician attempted to + ascertain by calculation, the ratio in which the evidence of + facts must decrease in the course of time, and fixed the period + when the evidence of the facts on which Christianity is founded + shall become evanescent, and when in consequence no faith shall + be found on the earth.—<span class="smcap">Reid, Thomas.</span></p> + <p class="blockcite"> + Essays on the Powers of the Human Mind (Edinburgh, 1812), + Vol. 2, p. 408.</p> + +<p><span class="pagenum"> + <a name="Page_345" + id="Page_345">345</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_XX"> + CHAPTER XX<br /> + <span class="large"> + THE FUNDAMENTAL CONCEPTS, TIME AND SPACE</span></h2> + + <p class="v2"> + <b><a name="Block_2001" id="Block_2001">2001</a>.</b> + Kant’s Doctrine of Time.</p> + <p class="v1"> + I. Time is not an empirical concept deduced from any + experience, for neither co-existence nor succession would enter + into our perception, if the representation of time were not + given <em>a priori</em>. Only when this representation <em>a + priori</em> is given, can we imagine that certain things happen + at the same time (simultaneously) or at different times + (successively).</p> + <p class="v1"> + II. Time is a necessary representation on which all intuitions + depend. We cannot take away time from phenomena in general, + though we can well take away phenomena out of time. In time + alone is reality of phenomena possible. All phenomena may + vanish, but time itself (as the general condition of their + possibility) cannot be done away with.</p> + <p class="v1"> + III. On this <em>a priori</em> necessity depends also the + possibility of apodictic principles of the relations of time, + or of axioms of time in general. Time has one dimension only; + different times are not simultaneous, but successive, while + different spaces are never successive, but simultaneous. Such + principles cannot be derived from experience, because + experience could not impart to them absolute universality nor + apodictic certainty....</p> + <p class="v1"> + IV. Time is not a discursive, or what is called a general + concept, but a pure form of sensuous intuition. Different times + are parts only of one and the same time....</p> + <p class="v1"> + V. To say that time is infinite means no more than that every + definite quantity of time is possible only by limitations of + one time which forms the foundation of all times. The original + representation of time must therefore be given as unlimited. + But when the parts themselves and every quantity of an object + can be represented as determined by limitation only, the whole + representation cannot be given by concepts (for in that case + the partial representation comes first), but must be founded on + immediate intuition.—<span class="smcap">Kant, I.</span></p> + <p class="blockcite"> + Critique of Pure Reason [Max Müller] (New + York, 1900), pp. 24-25.</p> + +<p><span class="pagenum"> + <a name="Page_346" + id="Page_346">346</a></span></p> + + <p class="v2"> + <b><a name="Block_2002" id="Block_2002">2002</a>.</b> + Kant’s Doctrine of Space.</p> + <p class="v1"> + I. Space is not an empirical concept which has been derived + from external experience. For in order that certain sensations + should be referred to something outside myself, i.e. to + something in a different part of space from that where I am; + again, in order that I may be able to represent them as side by + side, that is, not only as different, but as in different + places, the representation of space must already be there....</p> + <p class="v1"> + II. Space is a necessary representation <em>a priori</em>, + forming the very foundation of all external intuitions. It is + impossible to imagine that there should be no space, though one + might very well imagine that there should be space without + objects to fill it. Space is therefore regarded as a condition + of the possibility of phenomena, not as a determination + produced by them; it is a representation <em>a priori</em> which + necessarily precedes all external phenomena.</p> + <p class="v1"> + III. On this necessity of an <em>a priori</em> representation of + space rests the apodictic certainty of all geometrical + principles, and the possibility of their construction <em>a + priori</em>. For if the intuition of space were a concept gained + <em>a posteriori</em>, borrowed from general external experience, + the first principles of mathematical definition would be + nothing but perceptions. They would be exposed to all the + accidents of perception, and there being but one straight line + between two points would not be a necessity, but only something + taught in each case by experience. Whatever is derived from + experience possesses a relative generality only, based on + induction. We should therefore not be able to say more than + that, so far as hitherto observed, no space has yet been found + having more than three dimensions.</p> + <p class="v1"> + IV. Space is not a discursive or so-called general concept of + the relations of things in general, but a pure intuition. For, + first of all, we can imagine one space only, and if we speak of + many spaces, we mean parts only of one and the same space. Nor + can these parts be considered as antecedent to the one and + all-embracing space and, as it were, its component parts out of + which an aggregate is formed, but they can be thought of as + existing within it only. Space is essentially one; its + multiplicity, and therefore the general concept of spaces in + general, arises entirely from limitations. Hence it follows + that, with respect to + +<span class="pagenum"> + <a name="Page_347" + id="Page_347">347</a></span> + + space, an intuition <em>a + priori</em>, which is not empirical, must form the foundation of + all conceptions of space....</p> + <p class="v1"> + V. Space is represented as an infinite given quantity. Now it + is quite true that every concept is to be thought as a + representation, which is contained in an infinite number of + different possible representations (as their common + characteristic), and therefore comprehends them: but no + concept, as such, can be thought as if it contained in itself + an infinite number of representations. Nevertheless, space is + so thought (for all parts of infinite space exist + simultaneously). Consequently, the original representation of + space is an <em>intuition a priori</em>, and not a + concept.—<span class="smcap">Kant, I.</span></p> + <p class="blockcite"> + Critique of Pure Reason [Max Müller] (New + York, 1900), pp. 18-20 and Supplement 8.</p> + + <p class="v2"> + <b><a name="Block_2003" id="Block_2003">2003</a>.</b> + <em>Schopenhauer’s Predicabilia a priori</em>.<a + href="#Footnote_11" + title="Third column headed “of matter” has here been omitted." + class="fnanchor">11</a></p> + + <table summary=""> + <tr class="center"> + <th> </th> + <th> + <span class="smcap">OF TIME</span></th> + <th> + <span class="smcap">OF SPACE</span></th></tr> + <tr> + <td class="rt"> + 1.</td> + <td> + There is but <em>one time</em>, all different times + are parts of it.</td> + <td> + There is but <em>one space</em>, all different spaces + are parts of it.</td></tr> + <tr> + <td class="rt"> + 2.</td> + <td> + Different times are not simultaneous but + successive.</td> + <td> + Different spaces are not successive but + simultaneous.</td></tr> + <tr> + <td class="rt"> + 3.</td> + <td> + Everything in time may be thought of as non-existent, + but not time.</td> + <td> + Everything in space may be thought of as non-existent, + but not space.</td></tr> + <tr> + <td class="rt"> + 4.</td> + <td> + Time has three divisions: past, present and future, + which form two directions with a point of indifference.</td> + <td> + Space has three dimensions: height, breadth, + and length.</td></tr> + <tr> + <td class="rt"> + 5.</td> + <td> + Time is infinitely divisible.</td> + <td> + Space is infinitely divisible.</td></tr> + <tr> + <td class="rt"> + 6.</td> + <td> + Time is homogeneous and a continuum: i.e. no part is + different from another, nor separated by something + which is not time.</td> + <td> + Space is homogeneous and a continuum: i.e. no part is + different from another, nor separated by something + which is not space. + +<span class="pagenum"> + <a name="Page_348" + id="Page_348">348</a></span></td></tr> + + <tr> + <td class="rt"> + 7.</td> + <td> + Time has no beginning nor end, but all beginning and end + is in time.</td> + <td> + Space has no limits [Gränzen], but all limits are in + space.</td></tr> + <tr> + <td class="rt"> + 8.</td> + <td> + Time makes counting possible.</td> + <td> + Space makes measurement possible.</td></tr> + <tr> + <td class="rt"> + 9.</td> + <td> + Rhythm exists only in time.</td> + <td> + Symmetry exists only in space.</td></tr> + <tr> + <td class="rt"> + 10.</td> + <td> + The laws of time are <em>a priori</em> conceptions.</td> + <td> + The laws of space are <em>a priori</em> conceptions.</td></tr> + <tr> + <td class="rt"> + 11.</td> + <td> + Time is perceptible <em>a priori</em>, but only by means + of a line-image.</td> + <td> + Space is immediately perceptible <em>a priori</em>.</td></tr> + <tr> + <td class="rt"> + 12.</td> + <td> + Time has no permanence but passes the moment it is + present.</td> + <td> + Space never passes but is permanent throughout all + time.</td></tr> + <tr> + <td class="rt"> + 13.</td> + <td> + Time never rests.</td> + <td> + Space never moves.</td></tr> + <tr> + <td class="rt"> + 14.</td> + <td> + Everything in time has duration.</td> + <td> + Everything in space has position.</td></tr> + <tr> + <td class="rt"> + 15.</td> + <td> + Time has no duration, but all duration is in time; time + is the persistence of what is permanent in contrast with + its restless course.</td> + <td> + Space has no motion, but all motion is in space; space is + the change in position of that which moves in contrast to + its imperturbable rest.</td></tr> + <tr> + <td class="rt"> + 16.</td> + <td> + Motion is only possible in time.</td> + <td> + Motion is only possible in space.</td></tr> + <tr> + <td class="rt"> + 17.</td> + <td> + Velocity, the space being the same, is in the inverse + ratio of the time.</td> + <td> + Velocity, the time being the same, is in the direct + ratio of the space.</td></tr> + <tr> + <td class="rt"> + 18.</td> + <td> + Time is not directly measurable by means of itself but + only by means of motion which takes place in both space + and time....</td> + <td> + Space is measurable directly through itself and indirectly + through motion which takes place in both time and + space....</td></tr> + <tr> + <td class="rt"> + 19.</td> + <td> + Time is omnipresent: each part of it is everywhere.</td> + <td> + Space is eternal: each part of it exists always.</td></tr> + <tr> + <td class="rt"> + 20.</td> + <td> + In time alone all things are successive.</td> + <td> + In space alone all things are simultaneous. + +<span class="pagenum"> + <a name="Page_349" + id="Page_349">349</a></span></td></tr> + + <tr> + <td class="rt"> + 21.</td> + <td> + Time makes possible the change of accidents.</td> + <td> + Space makes possible the endurance of substance.</td></tr> + <tr> + <td class="rt"> + 22.</td> + <td> + Each part of time contains all substance.</td> + <td> + No part of space contains the same substance as + another.</td></tr> + <tr> + <td class="rt"> + 23.</td> + <td> + Time is the <em>principium individuationis</em>.</td> + <td> + Space is the <em>principium individuationis</em>.</td></tr> + <tr> + <td class="rt"> + 24.</td> + <td> + The now is without duration. </td> + <td> + The point is without extension.</td></tr> + <tr> + <td class="rt"> + 25.</td> + <td> + Time of itself is empty and indeterminate.</td> + <td> + Space is of itself empty and indeterminate.</td></tr> + <tr> + <td class="rt"> + 26.</td> + <td> + Each moment is conditioned by the one which precedes it, + and only so far as this one has ceased to exist. (Principle + of sufficient reason of being in time.)</td> + <td> + The relation of each boundary in space to every other is + determined by its relation to any one. (Principle of + sufficient reason of being in space.)</td></tr> + <tr> + <td class="rt"> + 27.</td> + <td> + Time makes Arithmetic possible.</td> + <td> + Space makes Geometry possible.</td></tr> + <tr> + <td class="rt"> + 28.</td> + <td> + The simple element of Arithmetic is unity.</td> + <td> + The element of Geometry is the point.</td></tr> + </table> + <p class="block40"> + —<span class="smcap">Schopenhauer, A.</span></p> + <p class="blockcite"> + Die Welt als Vorstellung und Wille; Werke + (Frauenstädt) (Leipzig, 1877), Bd. 2, p. 55.</p> + + <p class="v2"> + <b><a name="Block_2004" id="Block_2004">2004</a>.</b> + The clear possession of the Idea of Space is the + first requisite for all geometrical reasoning; and this + clearness of idea may be tested by examining whether the axioms + offer themselves to the mind as + evident.—<span class="smcap">Whewell, William.</span></p> + <p class="blockcite"> + The Philosophy of the Inductive Sciences, Part 1, Bk. 2, + chap. 4, sect. 4 (London, 1858).</p> + + <p class="v2"> + <b><a name="Block_2005" id="Block_2005">2005</a>.</b> + Geometrical axioms are neither synthetic <em>a + priori</em> conclusions nor experimental facts. They are + conventions: our choice, amongst all possible conventions, is + guided by experimental facts; but it remains free, and is only + limited by the necessity of avoiding all contradiction.... In + other words, axioms of geometry are only definitions in + disguise.</p> + +<p><span class="pagenum"> + <a name="Page_350" + id="Page_350">350</a></span></p> + + <p class="v1"> + That being so what ought one to think of this question: Is the + Euclidean Geometry true?</p> + <p class="v1"> + The question is nonsense. One might as well ask whether the + metric system is true and the old measures false; whether + Cartesian co-ordinates are true and polar co-ordinates + false.—<span class="smcap">Poincaré, H.</span></p> + <p class="blockcite"> + Non-Euclidean Geometry; Nature, Vol 45 (1891-1892), p. 407.</p> + + <p class="v2"> + <b><a name="Block_2006" id="Block_2006">2006</a>.</b> + I do in no wise share this view [that the axioms + are arbitrary propositions which we assume wholly at will, and + that in like manner the fundamental conceptions are in the end + only arbitrary symbols with which we operate] but consider it + the death of all science: in my judgment the axioms of geometry + are not arbitrary, but reasonable propositions which generally + have the origin in space intuition and whose separate content + and sequence is controlled by reasons of + expediency.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Elementarmathematik vom höheren Standpunkte + aus (Leipzig, 1909), Bd. 2, p. 384.</p> + + <p class="v2"> + <b><a name="Block_2007" id="Block_2007">2007</a>.</b> + Euclid’s Postulate 5 [The Parallel Axiom].</p> + <p class="v1"> + That, if a straight line falling on two straight lines make the + interior angles on the same side less than two right angles, + the two straight lines, if produced indefinitely, meet on that + side on which are the angles less than the two right + angles.—<span class="smcap">Euclid.</span></p> + <p class="blockcite"> + The Thirteen Books of Euclid’s Elements + [T. L. Heath] Vol. 1 (Cambridge, 1908), p. 202.</p> + + <p class="v2"> + <b><a name="Block_2008" id="Block_2008">2008</a>.</b> + It must be admitted that + Euclid’s [Parallel] Axiom is unsatisfactory + as the basis of a theory of parallel straight lines. It cannot + be regarded as either simple or self-evident, and it therefore + falls short of the essential characteristics of an + axiom....—<span class="smcap">Hall, H. S.</span> and + <span class="smcap">Stevens, F. H.</span></p> + <p class="blockcite"> + Euclid’s Elements (London, 1892), p. 55.</p> + + <p class="v2"> + <b><a name="Block_2009" id="Block_2009">2009</a>.</b> + We may still well declare the parallel axiom the + simplest assumption which permits us to represent spatial + relations, and so it will be true generally, that concepts and + axioms are not immediate facts of intuition, but rather the + idealizations of these facts chosen for reasons of + expediency.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Elementarmathematik vom, höheren + + <a id="TNanchor_18"></a> + <a class="msg" href="#TN_18" + title="originally read ‘Stanfpunkte’">Standpunkte</a> + + aus (Leipzig, 1909), Bd. 2, p. 382.</p> + +<p><span class="pagenum"> + <a name="Page_351" + id="Page_351">351</a></span></p> + + <p class="v2"> + <b><a name="Block_2010" id="Block_2010">2010</a>.</b> + The characteristic features of our space are not + necessities of thought, and the truth of + Euclid’s axioms, in so far as they specially + differentiate our space from other conceivable spaces, must be + established by experience and by experience + only.—<span class="smcap">Ball, R. S.</span></p> + <p class="blockcite"> + Encyclopedia Britannica, 9th Edition; Article “Measurement”</p> + + <p class="v2"> + <b><a name="Block_2011" id="Block_2011">2011</a>.</b> + Mathematical and physiological researches have + shown that the space of experience is simply an <em>actual</em> + case of many conceivable cases, about whose peculiar properties + experience alone can instruct + us.—<span class="smcap">Mach, Ernst.</span></p> + <p class="blockcite"> + Popular Scientific Lectures (Chicago, 1910), p. 205.</p> + + <p class="v2"> + <b><a name="Block_2012" id="Block_2012">2012</a>.</b> + The familiar definition: An axiom is a + self-evident truth, means if it means anything, that the + proposition which we call an axiom has been approved by us in + the light of our experience and intuition. In this sense + mathematics has no axioms, for mathematics is a formal subject + over which formal and not material implication + reigns.—<span class="smcap">Wilson, E. B.</span></p> + <p class="blockcite"> + Bulletin American Mathematical Society, Vol. 2 + (1904-1905), p. 81.</p> + + <p class="v2"> + <b><a name="Block_2013" id="Block_2013">2013</a>.</b> + The proof of self-evident propositions may seem, + to the uninitiated, a somewhat frivolous occupation. To this we + might reply that it is often by no means self-evident that one + obvious proposition follows from another obvious proposition; + so that we are really discovering new truths when we prove what + is evident by a method which is not evident. But a more + interesting retort is, that since people have tried to prove + obvious propositions, they have found that many of them are + false. Self-evidence is often a mere will-o’-the-wisp, which + is sure to lead us astray if we take it as our + guide.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + Recent Work on the Principles of Mathematics; + International Monthly, Vol. 4 (1901), p. 86.</p> + + <p class="v2"> + <b><a name="Block_2014" id="Block_2014">2014</a>.</b> + The problem [of Euclid’s + Parallel Axiom] is now at a par with the squaring of the circle + and the trisection of an angle by means of ruler and compass. + So far as the mathematical public + +<span class="pagenum"> + <a name="Page_352" + id="Page_352">352</a></span> + + is concerned, the famous problem of the parallel is settled for + all time.—<span class="smcap">Young, John Wesley.</span></p> + <p class="blockcite"> + Fundamental Concepts of Algebra and Geometry (New York, + 1911), p. 32.</p> + + <p class="v2"> + <b><a name="Block_2015" id="Block_2015">2015</a>.</b> + If the Euclidean assumptions are true, the + constitution of those parts of space which are at an infinite + distance from us, “geometry upon the plane + at infinity,” is just as well known as the geometry + of any portion of this room. In this infinite and thoroughly + well-known space the Universe is situated during at least some + portion of an infinite and thoroughly well-known time. So that + here we have real knowledge of something at least that concerns + the Cosmos; something that is true throughout the Immensities + and the Eternities. That something Lobatchewsky and his + successors have taken away. The geometer of to-day knows + nothing about the nature of the actually existing space at an + infinite distance; he knows nothing about the properties of + this present space in a past or future eternity. He knows, + indeed, that the laws assumed by Euclid are true with an + accuracy that no direct experiment can approach, not only in + this place where we are, but in places at a distance from us + that no astronomer has conceived; but he knows this as of Here + and Now; beyond this range is a There and Then of which he + knows nothing at present, but may ultimately come to know + more.—<span class="smcap">Clifford, W. K.</span></p> + <p class="blockcite"> + Lectures and Essays (New York, 1901), Vol. 1, pp. 358-359.</p> + + <p class="v2"> + <b><a name="Block_2016" id="Block_2016">2016</a>.</b> + The truth is that other systems of geometry are + possible, yet after all, these other systems are not spaces but + other methods of space measurements. There is one space only, + though we may conceive of many different manifolds, which are + contrivances or ideal constructions invented for the purpose of + determining space.—<span class="smcap">Carus, Paul.</span></p> + <p class="blockcite"> + Science, Vol. 18 (1903), p. 106.</p> + + <p class="v2"> + <b><a name="Block_2017" id="Block_2017">2017</a>.</b> + As I have formerly stated that from the + philosophic side Non-Euclidean Geometry has as yet not + frequently met with full understanding, so I must now emphasize + that it is universally recognized in the science of + mathematics; indeed, + +<span class="pagenum"> + <a name="Page_353" + id="Page_353">353</a></span> + + for many purposes, as for instance + in the modern theory of functions, it is used as an extremely + convenient means for the visual representation of highly + complicated arithmetical + relations.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Elementarmathematik vom höheren Standpunkte + aus (Leipzig, 1909), Bd. 2, p. 377.</p> + + <p class="v2"> + <b><a name="Block_2018" id="Block_2018">2018</a>.</b> + Everything in physical science, from the law of + gravitation to the building of bridges, from the spectroscope + to the art of navigation, would be profoundly modified by any + considerable inaccuracy in the hypothesis that our actual space + is Euclidean. The observed truth of physical science, + therefore, constitutes overwhelming empirical evidence that + this hypothesis is very approximately correct, even if not + rigidly true.—<span class="smcap">Russell, Bertrand.</span></p> + <p class="blockcite"> + Foundations of Geometry (Cambridge, 1897), p. 6.</p> + + <p class="v2"> + <b><a name="Block_2019" id="Block_2019">2019</a>.</b> + The most suggestive and notable achievement of the + last century is the discovery of Non-Euclidean + geometry.—<span class="smcap">Hilbert, D.</span></p> + <p class="blockcite"> + Quoted by G. D. Fitch in Manning’s “The Fourth Dimension Simply + Explained,” (New York, 1910), p. 58.</p> + + <p class="v2"> + <b><a name="Block_2020" id="Block_2020">2020</a>.</b> + Non-Euclidean geometry—primate + among the emancipators of the human + intellect....—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + The Foundations of Mathematics; Science History of the + Universe, Vol. 8 (New York, 1909), p. 192.</p> + + <p class="v2"> + <b><a name="Block_2021" id="Block_2021">2021</a>.</b> + Every high school teacher [Gymnasial-lehrer] must + of necessity know something about non-euclidean geometry, + because it is one of the few branches of mathematics which, by + means of certain catch-phrases, has become known in wider + circles, and concerning which any teacher is consequently + liable to be asked at any time. In physics there are many such + matters—almost every new discovery is of + this kind—which, through certain catch-words + have become topics of common conversation, and about which + therefore every teacher must of course be informed. Think of a + teacher of physics who knows + +<span class="pagenum"> + <a name="Page_354" + id="Page_354">354</a></span> + + nothing of Roentgen + rays or of radium; no better impression would be made by a + mathematician who is unable to give information concerning + non-euclidean geometry.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Elementarmathematik vom höheren + Standpunkte aus (Leipzig, 1909), Bd. 2, p. 378.</p> + + <p class="v2"> + <b><a name="Block_2022" id="Block_2022">2022</a>.</b> + What Vesalius was to Galen, what Copernicus was to + Ptolemy, that was Lobatchewsky to Euclid. There is, indeed, a + somewhat instructive parallel between the last two cases. + Copernicus and Lobatchewsky were both of Slavic origin. Each of + them has brought about a revolution in scientific ideas so + great that it can only be compared with that wrought by the + other. And the reason of the transcendent importance of these + two changes is that they are changes in the conception of the + Cosmos.... And in virtue of these two revolutions the idea of + the Universe, the Macrocosm, the All, as subject of human + knowledge, and therefore of human interest, has fallen to + pieces.—<span class="smcap">Clifford, W. K.</span></p> + <p class="blockcite"> + Lectures and Essays (New York, 1901), Vol. 1, pp. 356, 358.</p> + + <p class="v2"> + <b><a name="Block_2023" id="Block_2023">2023</a>.</b> + I am exceedingly sorry that I have failed to avail + myself of our former greater proximity to learn more of your + work on the foundations of geometry; it surely would have saved + me much useless effort and given me more peace, than one of my + disposition can enjoy so long as so much is left to consider in + a matter of this kind. I have myself made much progress in this + matter (though my other heterogeneous occupations have left me + but little time for this purpose); though the course which I + have pursued does not lead as much to the desired end, which + you assure me you have reached, as to the questioning of the + truth of geometry. It is true that I have found much which many + would accept as proof, but which in my estimation proves + <em>nothing</em>, for instance, if it could be shown that a + rectilinear triangle is possible, whose area is greater than + that of any given surface, then I could rigorously establish + the whole of geometry. Now most people, no doubt, would grant + this as an axiom, but not I; it is conceivable that, however + distant apart the vertices of the triangle might be chosen, its + area might yet + +<span class="pagenum"> + <a name="Page_355" + id="Page_355">355</a></span> + + always be below a certain limit. I + have found several other such theorems, but none of them + satisfies me.—<span class="smcap">Gauss.</span></p> + <p class="blockcite"> + Letter to Bolyai (1799); Werke, Bd. 8 (Göttingen, 1900), + p. 159.</p> + + <p class="v2"> + <b><a name="Block_2024" id="Block_2024">2024</a>.</b> + On the supposition that Euclidean geometry is not valid, it is + easy to show that similar figures do not exist; in that case + the angles of an equilateral triangle vary with the + side in which I see no absurdity at all. The angle is a + function of the side and the sides are functions of the angle, + a function which, of course, at the same time involves a + constant length. It seems somewhat of a paradox to say that a + constant length could be given a priori as it were, but in this + again I see nothing inconsistent. Indeed, it would be desirable + that Euclidean geometry were not valid, for then we should + possess a general a priori standard of + measure.—<span class="smcap">Gauss.</span></p> + <p class="blockcite"> + Letter to Gerling (1816); Werke, Bd. 8 + (Göttingen, 1900), p. 169.</p> + + <p class="v2"> + <b><a name="Block_2025" id="Block_2025">2025</a>.</b> + I am convinced more and more that the necessary + truth of our geometry cannot be demonstrated, at least not + <em>by</em> the <em>human</em> intellect <em>to</em> the human + understanding. Perhaps in another world we may gain other + insights into the nature of space which at present are + unattainable to us. Until then we must consider geometry as of + equal rank not with arithmetic, which is purely a priori, but + with mechanics.—<span class="smcap">Gauss.</span></p> + <p class="blockcite"> + Letter to Olbers (1817); Werke, Bd. 8 + (Göttingen, 1900), p. 177.</p> + + <p class="v2"> + <b><a name="Block_2026" id="Block_2026">2026</a>.</b> + There is no doubt that it can be rigorously + established that the sum of the angles of a rectilinear + triangle cannot exceed 180°. But it is otherwise + with the statement that the sum of the angles cannot be less + than 180°; this is the real Gordian knot, the rocks + which cause the wreck of all.... I have been occupied with the + problem over thirty years and I doubt if anyone has given it + more serious attention, though I have never published anything + concerning it. The assumption that the angle sum is less than + 180° leads to a peculiar geometry, entirely + different from the Euclidean, but throughout consistent + with itself. I have developed this + geometry to my own satisfaction + +<span class="pagenum"> + <a name="Page_356" + id="Page_356">356</a></span> + + so that I can solve every + problem that arises in it with the exception of the + determination of a certain constant which cannot be determined + a priori. The larger one assumes this constant the more nearly + one approaches the Euclidean geometry, an infinitely large + value makes the two coincide. The theorems of this geometry + seem in part paradoxical, and to the unpracticed absurd; but on + a closer and calm reflection it is found that in themselves + they contain nothing impossible.... All my efforts to discover + some contradiction, some inconsistency in this Non-Euclidean + geometry have been fruitless, the one thing in it that seems + contrary to reason is that space would have to contain a + <em>definitely determinate</em> (though to us unknown) linear + magnitude. However, it seems to me that notwithstanding the + meaningless word-wisdom of the metaphysicians we know really + too little, or nothing, concerning the true nature of space to + confound what appears unnatural with the <em>absolutely + impossible.</em> Should Non-Euclidean geometry be true, and this + constant bear some relation to magnitudes which come within the + domain of terrestrial or celestial measurement, it could be + determined a posteriori.—<span class="smcap">Gauss.</span></p> + <p class="blockcite"> + Letter to Taurinus (1824); Werke, Bd. 8 + (Göttingen, 1900), p. 187.</p> + + <p class="v2"> + <b><a name="Block_2027" id="Block_2027">2027</a>.</b> + There is also another subject, which with me is + nearly forty years old, to which I have again given some + thought during leisure hours, I mean the foundations of + geometry.... Here, too, I have consolidated many things, and my + conviction has, if possible become more firm that geometry + cannot be completely established on a priori grounds. In the + mean time I shall probably not for a long time yet put my + <em>very extended</em> investigations concerning this matter in + shape for publication, possibly not while I live, for I fear + the cry of the Bœotians which would arise should I + express my whole view on this matter.—It is + curious too, that besides the known gap in + Euclid’s geometry, to fill which all efforts + till now have been in vain, and which will never be filled, + there exists another defect, which to my knowledge no one thus + far has criticised and which (though possible) it is by no + means easy to remove. This is the definition of a plane as a + surface which wholly contains the line joining any + +<span class="pagenum"> + <a name="Page_357" + id="Page_357">357</a></span> + + two + points. This definition contains more than is necessary to the + determination of the surface, and tacitly involves a theorem + which demands proof.—<span class="smcap">Gauss.</span></p> + <p class="blockcite"> + Letter to Bessel (1829); Werke, Bd. 8 + (Göttingen, 1900), p. 200.</p> + + <p class="v2"> + <b><a name="Block_2028" id="Block_2028">2028</a>.</b> + I will add that I have recently received from + Hungary a little paper on Non-Euclidean geometry, in which I + rediscover all <em>my own ideas</em> and <em>results</em> worked + out with great elegance,.... The writer is a very young + Austrian officer, the son of one of my early friends, with whom + I often discussed the subject in 1798, although my ideas were + at that time far removed from the development and maturity + which they have received through the original reflections of + this young man. I consider the young geometer v. Bolyai a + genius of the first rank.—<span class="smcap">Gauss.</span></p> + <p class="blockcite"> + Letter to Gerling (1832); Werke, Bd. 8 + (Göttingen, 1900), p. 221.</p> + + <p class="v2"> + <b><a name="Block_2029" id="Block_2029">2029</a>.</b> + Think of the image of the world in a convex + mirror.... A well-made convex mirror of moderate aperture + represents the objects in front of it as apparently solid and + in fixed positions behind its surface. But the images of the + distant horizon and of the sun in the sky lie behind the mirror + at a limited distance, equal to its focal length. Between these + and the surface of the mirror are found the images of all the + other objects before it, but the images are diminished and + flattened in proportion to the distance of their objects from + the mirror.... Yet every straight line or plane in the outer + world is represented by a straight [?] line or plane in the + image. The image of a man measuring with a rule a straight line + from the mirror, would contract more and more the farther he + went, but with his shrunken rule the man in the image would + count out exactly the same number of centimeters as the real + man. And, in general, all geometrical measurements of lines and + angles made with regularly varying images of real instruments + would yield exactly the same results as in the outer world, all + lines of sight in the mirror would be represented by straight + lines of sight in the mirror. In short, I do not see how men in + the mirror are to discover that their bodies are not rigid + solids and their experiences good examples of the correctness + of Euclidean axioms. + +<span class="pagenum"> + <a name="Page_358" + id="Page_358">358</a></span> + + But if they could look out upon + our world as we look into theirs without overstepping the + boundary, they must declare it to be a picture in a spherical + mirror, and would speak of us just as we speak of them; and if + two inhabitants of the different worlds could communicate with + one another, neither, as far as I can see, would be able to + convince the other that he had the true, the other the + distorted, relation. Indeed I cannot see that such a question + would have any meaning at all, so long as mechanical + considerations are not mixed up with + it.—<span class="smcap">Helmholtz, H.</span></p> + <p class="blockcite"> + On the Origin and Significance of Geometrical Axioms; + Popular Scientific Lectures, second series (New York, 1881), + pp. 57-59.</p> + + <p class="v2"> + <b><a name="Block_2030" id="Block_2030">2030</a>.</b> + That space conceived of as a locus of points has + but three dimensions needs no argument from the mathematical + point of view; but just as little can we from this point of + view prevent the assertion that space has really four or an + infinite number of dimensions though we perceive only three. + The theory of multiply-extended manifolds, which enters more + and more into the foreground of mathematical research, is from + its very nature perfectly independent of such an assertion. But + the form of expression, which this theory employs, has indeed + grown out of this conception. Instead of referring to the + individuals of a manifold, we speak of the points of a higher + space, etc. In itself this form of expression has many + advantages, in that it facilitates comprehension by calling up + geometrical intuition. But it has this disadvantage, that in + extended circles, investigations concerning manifolds of any + number of dimensions are considered singular alongside the + above-mentioned conception of space. This view is without the + least foundation. The investigations in question would indeed + find immediate geometric applications if the conception were + valid but its value and purpose, being independent of this + conception, rests upon its essential mathematical + content.—<span class="smcap">Klein, F.</span></p> + <p class="blockcite"> + Mathematische Annalen, Bd. 43 (1893), p. 95.</p> + + <p class="v2"> + <b><a name="Block_2031" id="Block_2031">2031</a>.</b> + We are led naturally to extend the language of + geometry to the case of any number of variables, still using + the word <em>point</em> to designate any system of values of n + variables (the + +<span class="pagenum"> + <a name="Page_359" + id="Page_359">359</a></span> + + coördinates of the + point), the word <em>space</em> (of n dimensions) to designate + the totality of all these points or systems of values, + <em>curves</em> or <em>surface</em> to designate the spread + composed of points whose coördinates are given + functions (with the proper restrictions) of one or two + parameters (the <em>straight line</em> or <em>plane</em>, when they + are linear fractional functions with the same denominator), + etc. Such an extension has come to be a necessity in a large + number of investigations, in order as well to give them the + greatest generality as to preserve in them the intuitive + character of geometry. But it has been noted that in such use + of geometric language we are no longer constructing truly a + geometry, for the forms that we have been considering are + essentially analytic, and that, for example, the general + projective geometry constructed in this way is in substance + nothing more than the algebra of linear + transformations.—<span class="smcap">Segre, Corradi.</span></p> + <p class="blockcite"> + Rivista di Matematica, Vol. I (1891), p. 59. [J. W. Young.]</p> + + <p class="v2"> + <b><a name="Block_2032" id="Block_2032">2032</a>.</b> + Those who can, in common algebra, find a square + root of −1, will be at no loss to find a fourth dimension in + space in which ABC may become ABCD: or, if they cannot find it, + they have but to imagine it, and call it an <em>impossible</em> + dimension, subject to all the laws of the three we find + possible. And just as √−1 in common + algebra, gives all its <em>significant</em> combinations + <em>true</em>, so would it be with any number of dimensions of + space which the speculator might choose to call into + <em>impossible</em> existence—<span class= + "smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Trigonometry and Double Algebra (London, 1849), Part 2, + chap. 3.</p> + + <p class="v2"> + <b><a name="Block_2033" id="Block_2033">2033</a>.</b> + The doctrine of non-Euclidean spaces and of + hyperspaces in general possesses the highest intellectual + interest, and it requires a far-sighted man to foretell that it + can never have any practical + importance.—<span class="smcap">Smith, W. B.</span></p> + <p class="blockcite"> + Introductory Modern Geometry (New York, 1893), p. 274.</p> + + <p class="v2"> + <b><a name="Block_2034" id="Block_2034">2034</a>.</b> + According to his frequently expressed view, Gauss + considered the three dimensions of space as specific + peculiarities + +<span class="pagenum"> + <a name="Page_360" + id="Page_360">360</a></span> + + of the human soul; people, which are + unable to comprehend this, he designated in his humorous mood + by the name Bœotians. We could imagine ourselves, + he said, as beings which are conscious of but two dimensions; + higher beings might look at us in a like manner, and continuing + jokingly, he said that he had laid aside certain problems + which, when in a higher state of being, he hoped to investigate + geometrically.—<span class= + "smcap">Sartorius, W. v. Waltershausen.</span></p> + <p class="blockcite"> + Gauss zum Gedächtniss (Leipzig, 1856), p. 81.</p> + + <p class="v2"> + <b><a name="Block_2035" id="Block_2035">2035</a>.</b> + <em>There is many a rational logos</em>, and the + mathematician has high delight in the contemplation of + <em>in</em>consistent <em>systems</em> of <em>consistent + relationships</em>. There are, for example, a Euclidean geometry + and more than one species of non-Euclidean. As theories of a + given space, these are not compatible. If our universe be, as + Plato thought, and nature-science takes for granted, a + space-conditioned, geometrised affair, one of these geometries + may be, none of them may be, not all of them can be, valid in + it. But in the vaster world of thought, all of them are valid, + there they co-exist, and interlace among themselves and others, + as differing component strains of a higher, strictly + supernatural, hypercosmic, + harmony.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + The Universe and Beyond; Hibbert Journal, Vol. 3 + (1904-1905), p. 313.</p> + + <p class="v2"> + <b><a name="Block_2036" id="Block_2036">2036</a>.</b> + The introduction into geometrical work of + conceptions such as the infinite, the imaginary, and the + relations of hyperspace, none of which can be directly + imagined, has a psychological significance well worthy of + examination. It gives a deep insight into the resources and + working of the human mind. We arrive at the borderland of + mathematics and psychology.—<span + class="smcap">Merz, J. T.</span></p> + <p class="blockcite"> + History of European Thought in the Nineteenth Century + (Edinburgh and London, 1903), p. 716.</p> + + <p class="v2"> + <b><a name="Block_2037" id="Block_2037">2037</a>.</b> + Among the splendid generalizations effected by + modern mathematics, there is none more brilliant or more + inspiring or more fruitful, and none more commensurate with the + limitless immensity of being itself, than that which produced + the + +<span class="pagenum"> + <a name="Page_361" + id="Page_361">361</a></span> + + great concept designated ... hyperspace or multidimensional + space.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Mathematical Emancipations; Monist, Vol. 16 (1906), p. 65.</p> + + <p class="v2"> + <b><a name="Block_2038" id="Block_2038">2038</a>.</b> + The great generalization [of hyperspace] has made + it possible to enrich, quicken and beautify analysis with the + terse, sensuous, artistic, stimulating language of geometry. On + the other hand, the hyperspaces are in themselves immeasurably + interesting and inexhaustibly rich fields of research. Not only + does the geometrician find light in them for the illumination + of otherwise dark and undiscovered properties of ordinary + spaces of intuition, but he also discovers there wondrous + structures quite unknown to ordinary space.... It is by + creation of hyperspaces that the rational spirit secures + release from limitation. In them it lives ever joyously, + sustained by an unfailing sense of infinite + freedom.—<span class="smcap">Keyser, C. J.</span></p> + <p class="blockcite"> + Mathematical Emancipations; Monist, Vol. 16 (1906), p. 83.</p> + + <p class="v2"> + <b><a name="Block_2039" id="Block_2039">2039</a>.</b> + Mathematicians who busy themselves a great deal + with the formal theory of four-dimensional space, seem to + acquire a capacity for imagining this form as easily as the + three-dimensional form with which we are all + familiar.—<span class="smcap">Ostwald, W.</span></p> + <p class="blockcite"> + Natural Philosophy [Seltzer], (New York, 1910), p. 77.</p> + + <p class="v2"> + <b><a name="Block_2040" id="Block_2040">2040</a>.</b></p> + <div class="poem"> + <p class="i0"> + Fuchs.</p> + <p class="i7"> + Was soll ich nun aber denn studieren?</p> + <p class="i0"> + Meph.</p> + <p class="i7"> + Ihr könnt es mit <em>analytischer Geometrie</em> probieren.</p> + <p class="i7"> + Da wird der Raum euch wohl dressiert,</p> + <p class="i7"> + In Coordinaten eingeschnürt,</p> + <p class="i7"> + Dass ihr nicht etwa auf gut Glück</p> + <p class="i7"> + Von der Figur gewinnt ein Stück.</p> + <p class="i7"> + Dann lehret man euch manchen Tag,</p> + <p class="i7"> + Dass, was ihr sonst auf einen Schlag</p> + <p class="i7"> + Construiertet im Raume frei,</p> + <p class="i7"> + Eine Gleichung dazu nötig sei.</p> + <p class="i7"> + Zwar war dem Menschen zu seiner Erbauung</p> + <p class="i7"> + Die dreidimensionale Raumanschauung,</p> + +<span class="pagenum"> + <a name="Page_362" + id="Page_362">362</a></span> + + <p class="i7"> + Dass er sieht, was um ihn passiert,</p> + <p class="i7"> + Und die Figuren sich construiert—</p> + <p class="i7"> + Der Analytiker tritt herein</p> + <p class="i7"> + Und beweist, das könnte auch anders sein.</p> + <p class="i7"> + Gleichungen, die auf dem Papiere stehn,</p> + <p class="i7"> + Die müsst’ man auch können im Raume sehn;</p> + <p class="i7"> + Und könnte man’s nicht construieren,</p> + <p class="i7"> + Da müsste man’s anders definieren.</p> + <p class="i7"> + Denn was man formt nach Zahlengesetzen</p> + <p class="i7"> + Müsst’ uns auch geometrisch erletzen.</p> + <p class="i7"> + Drum in den unendlich fernen beiden</p> + <p class="i7"> + Imaginären Punkten müssen sich schneiden</p> + <p class="i7"> + Alle Kreise fein säuberlich,</p> + <p class="i7"> + Auch Parallelen, die treffen sich,</p> + <p class="i7"> + Und im Raume kann man daneben</p> + <p class="i7"> + Allerlei Krümmungsmasse erleben.</p> + <p class="i7"> + Die Formeln sind alle wahr und schön,</p> + <p class="i7"> + Warum sollen sie nicht zu deuten gehn?</p> + <p class="i7"> + Da preisen’s die Schüler aller Orten,</p> + <p class="i7"> + Dass das Gerade ist krumm geworden.</p> + <p class="i7"> + <em>Nicht-Euklidisch</em> nennt’s die Geometrie,</p> + <p class="i7"> + Spotted ihrer selbst, und weiss nicht wie.</p> + <p class="i0"> + Fuchs.</p> + <p class="i7"> + Kann euch nicht eben ganz verstehn.</p> + <p class="i0"> + Meph.</p> + <p class="i7"> + Das soll den Philosophen auch so gehn.</p> + <p class="i7"> + Doch wenn ihr lernt alles reducieren</p> + <p class="i7"> + Und gehörig transformieren,</p> + <p class="i7"> + Bis die Formeln den Sinn verlieren,</p> + <p class="i7"> + Dann versteht ihr mathematish zu spekulieren.</p> + </div> + <p class="block40"> + —<span class="smcap">Lasswitz, Kurd.</span></p> + <p class="blockcite"> + Der Faust-Tragödie (-n)ter Teil; Zeitschrift für den + math-naturw. Unterricht, Bd. 14 (1888), p. 316.</p> + + <div class="poem"> + <hr class="blank" /> + <p class="i0"> + [Fuchs.</p> + <p class="i7"> + To what study then should I myself apply?</p> + <p class="i0"> + Meph.</p> + <p class="i7"> + Begin with <em>analytical geometry</em>.</p> + <p class="i7"> + There all space is properly trained,</p> + <p class="i7"> + By coördinates well restrained,</p> + <p class="i7"> + That no one by some lucky assay</p> + <p class="i7"> + Carry some part of the figure away.</p> + +<span class="pagenum"> + <a name="Page_363" + id="Page_363">363</a></span> + + <p class="i7"> + Next thou’ll be taught to realize,</p> + <p class="i7"> + Constructions won’t help thee to geometrize,</p> + <p class="i7"> + And the result of a free construction</p> + <p class="i7"> + Requires an equation for proper deduction.</p> + <p class="i7"> + Three-dimensional space relation</p> + <p class="i7"> + Exists for human edification,</p> + <p class="i7"> + That he may see what about him transpires,</p> + <p class="i7"> + And construct such figures as he requires.</p> + <p class="i7"> + Enters the analyst. Forthwith you see</p> + <p class="i7"> + That all this might otherwise be.</p> + <p class="i7"> + Equations, written with pencil or pen,</p> + <p class="i7"> + Must be visible in space, and when</p> + <p class="i7"> + Difficulties in construction arise,</p> + <p class="i7"> + We need only define it otherwise.</p> + <p class="i7"> + For, what is formed after laws arithmetic</p> + <p class="i7"> + Must also yield some delight geometric.</p> + <p class="i7"> + Therefore we must not object</p> + <p class="i7"> + That all circles intersect</p> + <p class="i7"> + In the circular points at infinity.</p> + <p class="i7"> + And all parallels, they declare,</p> + <p class="i7"> + If produced must meet somewhere.</p> + <p class="i7"> + So in space, it can’t be denied,</p> + <p class="i7"> + Any old curvature may abide.</p> + <p class="i7"> + The formulas are all fine and true,</p> + <p class="i7"> + Then why should they not have a meaning too?</p> + <p class="i7"> + Pupils everywhere praise their fate</p> + <p class="i7"> + That that now is crooked which once was straight.</p> + <p class="i7"> + Non-Euclidean, in fine derision,</p> + <p class="i7"> + Is what it’s called by the geometrician.</p> + <p class="i0"> + Fuchs.</p> + <p class="i7"> + I do not fully follow thee.</p> + <p class="i0"> + Meph.</p> + <p class="i7"> + No better does philosophy.</p> + <p class="i7"> + To master mathematical speculation,</p> + <p class="i7"> + Carefully learn to reduce your equation</p> + <p class="i7"> + By an adequate transformation</p> + <p class="i7"> + Till the formulas are devoid of interpretation.]</p> + </div> + +<p><span class="pagenum"> + <a name="Page_364" + id="Page_364">364</a></span> </p> + + <hr class="chap" /> + <h2 class="v2" id="CHAPTER_XXI"> + CHAPTER XXI<br /> + <span class="large"> + PARADOXES AND CURIOSITIES</span></h2> + + <p class="v2"> + <b><a name="Block_2101" id="Block_2101">2101</a>.</b> + The pseudomath is a person who handles mathematics + as a monkey handles the razor. The creature tried to shave + himself as he had seen his master do; but, not having any + notion of the angle at which the razor was to be held, he cut + his own throat. He never tried it a second time, poor animal! + but the pseudomath keeps on in his work, proclaims himself + clean shaved, and all the rest of the world hairy.</p> + <p class="v1"> + The graphomath is a person who, having no mathematics, attempts + to describe a mathematician. Novelists perform in this way: + even Walter Scott now and then burns his fingers. His dreaming + calculator, Davy Ramsay, swears “by the + bones of the immortal Napier.” Scott thought that + the philomaths worshipped relics: so they do in one + sense.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), p. 473.</p> + + <p class="v2"> + <b><a name="Block_2102" id="Block_2102">2102</a>.</b> + Proof requires a person who can give and a person + who can receive....</p> + <div class="poem"> + <p class="i0"> + A blind man said, As to the Sun,</p> + <p class="i0"> + I’ll take my Bible oath there’s none;</p> + <p class="i0"> + For if there had been one to show</p> + <p class="i0"> + They would have shown it long ago.</p> + <p class="i0"> + How came he such a goose to be?</p> + <p class="i0"> + Did he not know he couldn’t see?</p> + <p class="i12"> + Not he.</p> + </div> + <p class="block40"> + —<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), p. 262.</p> + + <p class="v2"> + <b><a name="Block_2103" id="Block_2103">2103</a>.</b> + Mathematical research, with all its wealth of + hidden treasure, is all too apt to yield nothing to our + research: for it is haunted by certain <em>ignes + fatui</em>—delusive phantoms, that float + before us, and seem so fair, and are <em>all but</em> in our + grasp, so nearly that it never seems to need more than + <em>one</em> step further, and the prize shall be ours! Alas for + him who has been turned + +<span class="pagenum"> + <a name="Page_365" + id="Page_365">365</a></span> + + aside from real research by one of + these spectres—who has found a music in its + mocking laughter—and who wastes his life and + energy in the desperate chase!—<span class= + "smcap">Dodgson, C. L.</span></p> + <p class="blockcite"> + A new Theory of Parallels (London, 1895), Introduction.</p> + + <p class="v2"> + <b><a name="Block_2104" id="Block_2104">2104</a>.</b> + As lightning clears the air of impalpable vapours, + so an incisive paradox frees the human intelligence from the + lethargic influence of latent and unsuspected assumptions. + Paradox is the slayer of + Prejudice.—<span class="smcap">Sylvester, J. J.</span></p> + <p class="blockcite"> + On a Lady’s Fan etc. Collected + Mathematical Papers, Vol. 3, p. 36.</p> + + <p class="v2"> + <b><a name="Block_2105" id="Block_2105">2105</a>.</b> + When a paradoxer parades capital letters and + diagrams which are as good as Newton’s to + all who know nothing about it, some persons wonder why science + does not rise and triturate the whole thing. This is why: all + who are fit to read the refutation are satisfied already, and + can, if they please, detect the paradoxer for themselves. Those + who are not fit to do this would not know the difference + between the true answer and the new capitals and diagrams on + which the delighted paradoxer would declare that he had + crumbled the philosophers, and not they + him.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + A Budget of Paradoxes (London, 1872), p. 484.</p> + + <p class="v2"> + <b><a name="Block_2106" id="Block_2106">2106</a>.</b> + Demonstrative reason never raises the cry of + <em>Church in Danger!</em> and it cannot have any Dictionary of + heresies except a Budget of Paradoxes. Mistaken claimants are + left to Time and his extinguisher, with the approbation of all + non-claimants: there is no need of a succession of exposures. + Time gets through the job in his own workmanlike + manner.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + A Budget of Paradoxes (London, 1872), p. 485.</p> + + <p class="v2"> + <b><a name="Block_2107" id="Block_2107">2107</a>.</b> + D’Israeli speaks of the “six follies of + science,” —the quadrature, the + duplication, the perpetual motion, the + philosopher’s stone, magic, and astrology. + He might as well have added the trisection, to make the mystic + number seven; but had he done so, he would still have been very + lenient; only seven follies in all science, from mathematics to + chemistry! Science might have said to such a + judge—as convicts used to + +<span class="pagenum"> + <a name="Page_366" + id="Page_366">366</a></span> + + say who got seven years, expecting it for life, + “Thank you, my Lord, and may you sit there + until they are over,” —may the + Curiosities of Literature outlive the Follies of + Science!—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + A Budget of Paradoxes (London, 1872), p. 71.</p> + + <p class="v2"> + <b><a name="Block_2108" id="Block_2108">2108</a>.</b> + Montucla says, speaking of France, that he finds + three notions prevalent among cyclometers: 1. That there is a + large reward offered for success; 2. That the longitude problem + depends on that success; 3. That the solution is the great end + and object of geometry. The same three notions are equally + prevalent among the same class in England. No reward has ever + been offered by the government of either country. The longitude + problem in no way depends upon perfect solution; existing + approximations are sufficient to a point of accuracy far beyond + what can be wanted. And geometry, content with what exists, has + long passed on to other matters. Sometimes a cyclometer + persuades a skipper who has made land in the wrong place that + the astronomers are at fault, for using a wrong measure of the + circle; and the skipper thinks it a very comfortable solution! + And this is the utmost that the problem has to do with + longitude.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + A Budget of Paradoxes (London, 1872), p. 96.</p> + + <p class="v2"> + <b><a name="Block_2109" id="Block_2109">2109</a>.</b> + Gregory St. Vincent is the greatest of + circle-squarers, and his investigations led him into many + truths: he found the property of the arc of the hyperbola which + led to Napier’s logarithms being called + hyperbolic. Montucla says of him, with sly truth, that no one + ever squared the circle with so much genius, or, excepting his + principal object, with so much + success.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + A Budget of Paradoxes (London, 1872), p. 70.</p> + + <p class="v2"> + <b><a name="Block_2110" id="Block_2110">2110</a>.</b> + When I reached geometry, and became acquainted + with the proposition the proof of which has been sought for + centuries, I felt irresistibly impelled to try my powers at its + discovery. You will consider me foolish if I confess that I am + still earnestly of the opinion to have succeeded in my + attempt.—<span class="smcap">Bolzano, Bernard.</span></p> + <p class="blockcite"> + Selbstbiographie (Wien, 1875), p. 19.</p> + +<p><span class="pagenum"> + <a name="Page_367" + id="Page_367">367</a></span></p> + + <p class="v2"> + <b><a name="Block_2111" id="Block_2111">2111</a>.</b> + The Theory of Parallels.</p> + <p class="v1"> + It is known that to complete the theory it is only necessary to + demonstrate the following proposition, which Euclid assumed as + an axiom:</p> + <p class="v1"> + Prop. If the sum of the interior angles ECF and DBC which two + straight lines EC and DB make with a third line CP is less than + two right angles, the lines, if sufficiently produced, will + intersect.</p> + <div class="figcenter"> + <img id="img2111" + src="images/img2111.png" + width="600" + height="271" + alt="geometrical drawing of parallel lines and intersecting + lines to accompany proof"/> + </div> + <p class="v1"> + Proof. Construct PCA equal to the supplement PBD of CBD, and + ECF, FCG, etc. each equal to ACE, so that ACF = 2.ACE, ACG = + 3.ACE, etc. Then however small the angle ACE may be, there + exists some number n such that n.ACE = ACH will be equal to or + greater than ACP.</p> + <p class="v1"> + Again, take BI, IL, etc. each equal to CB, and draw IK, LM, + etc. parallel to BD, then the figures ACBD, DBIK, KILM, etc. + are congruent, and ACIK = 2.ABCD, ACLM = 3.ACBD, etc.</p> + <p class="v1"> + Take ACNO = n.ACBD, n having the same value as in the + expression ACH = n.ACE, then ACNO is certainly less than ACP, + since ACNO must be increased by ONP to be equal to ACP. It + follows that ACNO is also less than ACH, and by taking the nth + part of each of these, that ACBD is less than ACE.</p> + <p class="v1"> + But if ACE is greater than ACBD, CE and BD must intersect, for + otherwise ACE would be a part of ACBD.</p> + <p class="blockcite"> + Journal für Mathematik, Bd. 2 (1834), p. 198.</p> + + <p class="v2"> + <b><a name="Block_2112" id="Block_2112">2112</a>.</b> + Are you sure that it is impossible to trisect the + angle by <em>Euclid</em>? I have not to lament a single hour + thrown away on the + +<span class="pagenum"> + <a name="Page_368" + id="Page_368">368</a></span> + + attempt, but fancy that it is rather + a tact, a feeling, than a proof, which makes us think that the + thing cannot be done. But would + <em>Gauss’s</em> inscription of the regular + polygon of seventeen sides have seemed, a century ago, much + less an impossible thing, by line and + circle?—<span class="smcap">Hamilton, W. R.</span></p> + <p class="blockcite"> + Letter to De Morgan (1852).</p> + + <p class="v2"> + <b><a name="Block_2113" id="Block_2113">2113</a>.</b> + One of the most curious of these cases + [geometrical paradoxers] was that of a student, I am not sure + but a graduate, of the University of Virginia, who claimed that + geometers were in error in assuming that a line had no + thickness. He published a school geometry based on his views, + which received the endorsement of a well-known New York school + official and, on the basis of this, was actually endorsed, or + came very near being endorsed, as a text-book in the public + schools of New York.—<span class="smcap">Newcomb, Simon.</span></p> + <p class="blockcite"> + The Reminiscences of an Astronomer (Boston and New York, + 1903), p. 388.</p> + + <p class="v2"> + <b><a name="Block_2114" id="Block_2114">2114</a>.</b> + What distinguishes the straight line and circle + more than anything else, and properly separates them for the + purpose of elementary geometry? Their self-similarity. Every + inch of a straight line coincides with every other inch, and + off a circle with every other off the same circle. Where, then, + did Euclid fail? In not introducing the third curve, which has + the same property—the <em>screw</em>. The + right line, the circle, the screw—the + representations of translation, rotation, and the two + combined—ought to have been the instruments + of geometry. With a screw we should never have heard of the + impossibility of trisecting an angle, squaring the circle, + etc.—<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Quoted in Graves’ Life of Sir W. R. + Hamilton, Vol. 3 (New York, 1889), p. 342.</p> + + <p class="v2"> + <b><a name="Block_2115" id="Block_2115">2115</a>.</b></p> + <div class="poem"> + <p class="i0"> + Mad Mathesis alone was unconfined,</p> + <p class="i0"> + Too mad for mere material chains to bind,</p> + <p class="i0"> + Now to pure space lifts her ecstatic stare,</p> + <p class="i0"> + Now, running round the circle, finds it square.</p> + </div> + <p class="block40"> + —<span class="smcap">Pope, Alexander.</span></p> + <p class="blockcite"> + The Dunciad, Bk. 4, lines 31-34.</p> + +<p><span class="pagenum"> + <a name="Page_369" + id="Page_369">369</a></span></p> + + <p class="v2"> + <b><a name="Block_2116" id="Block_2116">2116</a>.</b></p> + <div class="poem"> + <p class="i0"> + Or is’t a tart idea, to procure</p> + <p class="i0"> + An edge, and keep the practic soul in ure,</p> + <p class="i0"> + Like that dear Chymic dust, or puzzling quadrature?</p> + </div> + <p class="block40"> + —<span class="smcap">Quarles, Philip.</span></p> + <p class="blockcite"> + Quoted by De Morgan: Budget of Paradoxes (London, 1872), + p. 436.</p> + + <p class="v2"> + <b><a name="Block_2117" id="Block_2117">2117</a>.</b></p> + <div class="poem"> + <p class="i0"> + Quale è’l geometra che tutto s’ affige</p> + <p class="i0"> + Per misurar lo cerchio, e non ritruova,</p> + <p class="i0"> + Pensando qual principio ond’ egli indige.</p> + </div> + <p class="block40"> + —<span class="smcap">Dante.</span></p> + <p class="blockcite"> + Paradise, canto 33, lines 122-125.</p> + <div class="poem"> + <hr class="blank" /> + <p class="i0"> + [As doth the expert geometer appear</p> + <p class="i0"> + Who seeks to square the circle, and whose skill</p> + <p class="i0"> + Finds not the law with which his course to steer.<a + href="#Footnote_12" + title="For another rendition of these same lines see 1858." + class="fnanchor">12</a>]</p> + </div> + <p class="blockcite"> + Quoted in Frankland’s Story of Euclid + (London, 1902), p. 101.</p> + + <p class="v2"> + <b><a name="Block_2118" id="Block_2118">2118</a>.</b></p> + <div class="poem"> + <p class="i0"> + In <em>Mathematicks</em> he was greater</p> + <p class="i0"> + Than <em>Tycho Brahe</em>, or <em>Erra Pater</em></p> + <p class="i0"> + For he, by <em>Geometrick</em> scale,</p> + <p class="i0"> + Could take the size of <em>Pots of Ale</em>;</p> + <p class="i0"> + Resolve by Signs and Tangents streight,</p> + <p class="i0"> + If <em>Bread</em> or <em>Butter</em> wanted weight;</p> + <p class="i0"> + And wisely tell what hour o’ th’ day</p> + <p class="i0"> + The Clock doth strike, by <em>Algebra</em>.</p> + </div> + <p class="block40"> + —<span class="smcap">Butler, Samuel.</span></p> + <p class="blockcite"> + Hudibras, Part 1, canto 1, lines 119-126.</p> + + <p class="v2"> + <b><a name="Block_2119" id="Block_2119">2119</a>.</b> + I have often been surprised that Mathematics, the + quintessence of truth, should have found admirers so few and so + languid. Frequent considerations and minute scrutiny have at + length unravelled the cause; viz. that though Reason is + feasted, Imagination is starved; whilst Reason is luxuriating + in its proper Paradise, Imagination is wearily travelling on a + dreary desert.—<span class= "smcap">Coleridge, Samuel.</span></p> + <p class="blockcite"> + A Mathematical Problem.</p> + + <p class="v2"> + <b><a name="Block_2120" id="Block_2120">2120</a>.</b> + At last we entered the palace, and proceeded into + the chamber of presence where I saw the king seated on his + throne, + +<span class="pagenum"> + <a name="Page_370" + id="Page_370">370</a></span> + + attended on each side by persons of + prime quality. Before the throne, was a large table filled with + globes and spheres, and mathematical instruments of all kinds. + His majesty took not the least notice of us, although our + entrance was not without sufficient noise, by the concourse of + all persons belonging to the court. But he was then deep in a + problem, and we attended an hour, before he could solve it. + There stood by him, on each side, a young page with flaps in + their hands, and when they saw he was at leisure, one of them + gently struck his mouth, and the other his right ear; at which + he started like one awaked on the sudden, and looking toward me + and the company I was in, recollected the occasion of our + coming, whereof he had been informed before. He spake some + words, whereupon immediately a young man with a flap came to my + side, and flapt me gently on the right ear, but I made signs, + as well as I could, that I had no occasion for such an + instrument; which, as I afterwards found, gave his majesty, and + the whole court, a very mean opinion of my understanding. The + king, as far as I could conjecture, asked me several questions, + and I addressed myself to him in all the languages I had. When + it was found, that I could neither understand nor be + understood, I was conducted by his order to an apartment in his + palace, (this prince being distinguished above all his + predecessors, for his hospitality to strangers) where two + servants were appointed to attend me. My dinner was brought, + and four persons of quality, did me the honour to dine with me. + We had two courses of three dishes each. In the first course, + there was a shoulder of mutton cut into an equilateral + triangle, a piece of beef into a rhomboides, and a pudding into + a cycloid. The second course, was, two ducks trussed up in the + form of fiddles; sausages and puddings, resembling flutes and + haut-boys, and a breast of veal in the shape of a harp. The + servants cut our bread into cones, cylinders, parallelograms, + and several other mathematical + figures.—<span class="smcap">Swift, Jonathan.</span></p> + <p class="blockcite"> + Gulliver’s Travels; A Voyage to Laputa; Chap. 2.</p> + + <p class="v2"> + <b><a name="Block_2121" id="Block_2121">2121</a>.</b> + Those to whom the king had entrusted me, observing + how ill I was clad, ordered a taylor to come next morning, and + take measure for a suit of cloaths. This operator did his + office + +<span class="pagenum"> + <a name="Page_371" + id="Page_371">371</a></span> + + after a different manner, from those + of his trade in Europe. He first took my altitude by a + quadrant, and then, with rule and compasses, described the + dimensions and outlines of my whole body, all which he entered + upon paper; and in six days, brought my cloaths very ill made, + and quite out of shape, by happening to mistake a figure in the + calculation. But my comfort was, that I observed such accidents + very frequent, and little + regarded.—<span class="smcap">Swift, Jonathan.</span></p> + <p class="blockcite"> + Gulliver’s Travels; A Voyage to Laputa, Chap. 2.</p> + + <p class="v2"> + <b><a name="Block_2122" id="Block_2122">2122</a>.</b> + The knowledge I had in mathematics, gave me great + assistance in acquiring their phraseology, which depended much + upon that science, and music; and in the latter I was not + unskilled. Their ideas are perpetually conversant in lines and + figures. If they would, for example, praise the beauty of a + woman, or any other animal, they describe it by rhombs, + circles, parallelograms, ellipses, and other geometrical terms, + or by words of art drawn from music, needless here to repeat. I + observed in the king’s kitchen all sorts of + mathematical and musical instruments, after the figures of + which, they cut up the joints that were served to his + majesty’s table.—<span class="smcap">Swift, Jonathan.</span></p> + <p class="blockcite"> + Gulliver’s Travels; A Voyage to Laputa, Chap. 2.</p> + + <p class="v2"> + <b><a name="Block_2123" id="Block_2123">2123</a>.</b> + I was at the mathematical school, where the master + taught his pupils, after a method, scarce imaginable to us in + Europe. The propositions, and demonstrations, were fairly + written on a thin wafer, with ink composed of a cephalic + tincture. This, the student was to swallow upon a fasting + stomach, and for three days following, eat nothing but bread + and water. As the wafer digested, the tincture mounted to his + brain, bearing the proposition along with it. But the success + has not hitherto been answerable, partly by some error in the + <em>quantum</em> or composition, and partly by the perverseness + of lads; to whom this bolus is so nauseous, that they generally + steal aside, and discharge it upwards, before it can operate; + neither have they been yet persuaded to use so long an + abstinence as the prescription + requires.—<span class="smcap">Swift, Jonathan.</span></p> + <p class="blockcite"> + Gulliver’s Travels; A Voyage to Laputa, Chap. 5.</p> + +<p><span class="pagenum"> + <a name="Page_372" + id="Page_372">372</a></span></p> + + <p class="v2"> + <b><a name="Block_2124" id="Block_2124">2124</a>.</b> + It is worth observing that some of those who + disparage some branch of study in which they are deficient, + will often affect more contempt for it than they really feel. + And not unfrequently they will take pains to have it thought + that they are themselves well versed in it, or that they easily + might be, if they thought it worth while;—in + short, that it is not from hanging too high that the grapes are + called sour.</p> + <p class="v1"> + Thus, Swift, in the person of Gulliver, represents himself, + while deriding the extravagant passion for Mathematics among + the Laputians, as being a good mathematician. Yet he betrays + his utter ignorance, by speaking “of a + pudding in the <em>form of a cycloid</em>:” evidently + taking the cycloid for a <em>figure</em>, instead of a + <em>line</em>. This may help to explain the difficulty he is said + to have had in obtaining his + Degree.—<span class="smcap">Whately, R.</span></p> + <p class="blockcite"> + Annotations to Bacon’s Essays, Essay L.</p> + + <p class="v2"> + <b><a name="Block_2125" id="Block_2125">2125</a>.</b> + It is natural to think that an abstract science + cannot be of much importance in the affairs of human life, + because it has omitted from its consideration everything of + real interest. It will be remembered that Swift, in his + description of Gulliver’s voyage to Laputa, + is of two minds on this point. He describes the mathematicians + of that country as silly and useless dreamers, whose attention + has to be awakened by flappers. Also, the mathematical tailor + measures his height by a quadrant, and deduces his other + dimensions by a rule and compasses, producing a suit of very + ill-fitting clothes. On the other hand, the mathematicians of + Laputa, by their marvellous invention of the magnetic island + floating in the air, ruled the country and maintained their + ascendency over their subjects. Swift, indeed, lived at a time + peculiarly unsuited for gibes at contemporary mathematicians. + Newton’s <cite>Principia</cite> had just been + written, one of the great forces which have transformed the + modern world. Swift might just as well have laughed at an + earthquake.—<span class="smcap">Whitehead, A. N.</span></p> + <p class="blockcite"> + An Introduction to Mathematics (New York, 1911), p. 10.</p> + +<p><span class="pagenum"> + <a name="Page_373" + id="Page_373">373</a></span></p> + + <p class="v2"> + <b><a id="Block_2126" + href="#TN_19" + class="msg" + title="Block number added">2126</a>.</b></p> + + + <div class="figcenter"> + <img id="img2126" + src="images/img2126.png" + width="590" + height="600" + alt="A geometrical drawing including square and four + triangles to demonstrate a graphical proof of the theorem + of Pythagoras as described in the poem."/> + </div> + <div class="poem"> + <p class="i0"> + Here I am as you may see</p> + <p class="i8"> + a<sup>2</sup> + b<sup>2</sup> − ab</p> + <p class="i0"> + When two Triangles on me stand</p> + <p class="i0"> + Square of hypothen<sup>e</sup> is plann’d</p> + <p class="i0"> + But if I stand on them instead,</p> + <p class="i0"> + The squares of both the sides are read.</p> + </div> + <p class="block40"> + —<span class="smcap">Airy, G. B.</span></p> + <p class="blockcite"> + Quoted in Graves’ Life of Sir W. R. + Hamilton, Vol. 3 (New York, 1889), p. 502.</p> + + <p class="v2"> + <b><a name="Block_2127" id="Block_2127">2127</a>.</b> + π = 3.141 592 653 589 793 238 462 643 383 279 ...</p> + <pre class="Monofont"> + 3 1 4 1 5 9 + Now I, even I, would celebrate + 2 6 5 3 5 + In rhymes inapt, the great + 8 9 7 9 + Immortal Syracusan, rivaled nevermore, + 3 2 3 8 4 + Who in his wondrous lore, + 6 2 6 + Passed on before, + 4 3 3 8 3 2 7 9 + Left men his guidance how to circles mensurate.</pre> + <p class="block40"> + —<span class="smcap">Orr, A. C.</span></p> + <p class="blockcite"> + Literary Digest, Vol. 32 (1906), p. 84.</p> + + <p class="v2"> + <b><a name="Block_2128" id="Block_2128">2128</a>.</b> + I take from a biographical dictionary the first + five names of poets, with their ages at death. They are</p> + <pre class="Monofont"> + Aagard, died at 48. + Abeille, “ “ 76. + Abulola, “ “ 84. + Abunowas, “ “ 48. + Accords, “ “ 45.</pre> + <p class="v0"> + These five ages have the following characters in + common:— + +<span class="pagenum"> + <a name="Page_374" + id="Page_374">374</a></span></p> + + <p class="v1"> + 1. The difference + of the two digits composing the number divided by <em>three</em>, + leaves a remainder of <em>one</em>.</p> + <p class="v1"> + 2. The first digit raised to the power indicated by the second, + and then divided by <em>three</em>, leaves a remainder of + <em>one</em>.</p> + <p class="v1"> + 3. The sum of the prime factors of each age, including + <em>one</em> as a prime factor, is divisible by + <em>three</em>.—<span class="smcap">Peirce, C. S.</span></p> + <p class="blockcite"> + A Theory of Probable Inference; Studies in Logic (Boston, + 1883), p. 163.</p> + + <p class="v2"> + <b><a name="Block_2129" id="Block_2129">2129</a>.</b> + In view of the fact that the offered prize [for + the solution of the problem of Fermat’s + Greater Theorem] is about $25,000 and that lack of marginal + space in his copy of Diophantus was the reason given by Fermat + for not communicating his proof, one might be tempted to wish + that one could send credit for a dime back through the ages to + Fermat and thus secure this coveted prize, if it actually + existed. This might, however, result more seriously than one + would at first suppose; for if Fermat had bought on credit a + dime’s worth of paper even during the year + of his death, 1665, and if this bill had been drawing compound + interest at the rate of six per cent, since that time, the bill + would now amount to more than seven times as much as the + prize.—<span class="smcap">Miller, G. A.</span></p> + <p class="blockcite"> + Some Thoughts on Modern Mathematical Research; Science, + Vol. 35 (1912), p. 881.</p> + + <p class="v2"> + <b><a name="Block_2130" id="Block_2130">2130</a>.</b> + <em>If the Indians hadn’t spent + the $24</em>. In 1626 Peter Minuit, first governor of New + Netherland, purchased Manhattan Island from the Indians for + about $24. The rate of interest on money is higher in new + countries, and gradually decreases as wealth accumulates. + Within the present generation the legal rate in the state has + fallen from 7% to 6%. Assume for simplicity a uniform rate of + 7% from 1626 to the present, and suppose that the Indians had + put their $24 at interest at that rate (banking facilities in + New York being always taken for granted!) and had added the + interest to the principal yearly. What would be the amount now, + after 280 years? 24 × (1.07)<sup>280</sup> = more than + 4,042,000,000.</p> + <p class="v1"> + The latest tax assessment available at the time of writing + gives the realty for the borough of Manhattan as + $3,820,754.181. + +<span class="pagenum"> + <a name="Page_375" + id="Page_375">375</a></span> + + This is estimated to be 78% of the + actual value, making the actual value a little more than + $4,898,400,000.</p> + <p class="v1"> + The amount of the Indians’ money would + therefore be more than the present assessed valuation but less + than the actual valuation.—<span + class="smcap">White, W. F.</span></p> + <p class="blockcite"> + A Scrap-book of Elementary Mathematics (Chicago, 1908), + pp. 47-48.</p> + + <p class="v2"> + <b><a name="Block_2131" id="Block_2131">2131</a>.</b> + See Mystery to Mathematics + fly!—<span class="smcap">Pope, Alexander.</span></p> + <p class="blockcite"> + The Dunciad, Bk. 4, line 647.</p> + + <p class="v2"> + <b><a name="Block_2132" id="Block_2132">2132</a>.</b> + The Pythagoreans and Platonists were carried + further by this love of simplicity. Pythagoras, by his skill in + mathematics, discovered that there can be no more than five + regular solid figures, terminated by plane surfaces which are + all similar and equal; to wit, the tetrahedron, the cube, the + octahedron, the dodecahedron, and the eicosihedron. As nature + works in the most simple and regular way, he thought that all + elementary bodies must have one or other of those regular + figures; and that the discovery of the properties and relations + of the regular solids must be a key to open the mysteries of + nature.</p> + <p class="v1"> + This notion of the Pythagoreans and Platonists has undoubtedly + great beauty and simplicity. Accordingly it prevailed, at least + to the time of Euclid. He was a Platonic philosopher, and is + said to have wrote all the books of his Elements, in order to + discover the properties and relations of the five regular + solids. The ancient tradition of the intention of Euclid in + writing his elements, is countenanced by the work itself. For + the last book of the elements treats of the regular solids, and + all the preceding are subservient to the + last.—<span class="smcap">Reid, Thomas.</span></p> + <p class="blockcite"> + Essays on the Powers of the Human Mind (Edinburgh, 1812), + Vol. 2, p. 400.</p> + + <p class="v2"> + <b><a name="Block_2133" id="Block_2133">2133</a>.</b> + In the Timæus [of Plato] it is + asserted that the particles of the various elements have the + forms of these [the regular] solids. Fire has the Pyramid; + Earth has the Cube; Water the Octahedron; Air the Icosahedron; + and the Dodecahedron is the plan of the Universe itself. It was + natural that when Plato had learnt that other mathematical + properties had a bearing upon the constitution of the Universe, + he should suppose that the + +<span class="pagenum"> + <a name="Page_376" + id="Page_376">376</a></span> + + singular property of + space, which the existence of this limited and varied class of + solids implied, should have some corresponding property in the + Universe, which exists in + space.—<span class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + History of the Inductive Sciences, 3rd Edition, Additions + to Bk. 2.</p> + + <p class="v2"> + <b><a name="Block_2134" id="Block_2134">2134</a>.</b> + The orbit of the earth is a circle: round the + sphere to which this circle belongs, describe a dodecahedron; + the sphere including this will give the orbit of Mars. Round + Mars describe a tetrahedron; the circle including this will be + the orbit of Jupiter. Describe a cube round + Jupiter’s orbit; the circle including this + will be the orbit of Saturn. Now inscribe in the + earth’s orbit an icosahedron; the circle + inscribed in it will be the orbit of Venus. Inscribe an + octahedron in the orbit of Venus; the circle inscribed in it + will be Mercury’s orbit. This is the reason + of the number of the planets.—<span class= + "smcap">Kepler.</span></p> + <p class="blockcite"> + Mysterium Cosmographicum [Whewell].</p> + + <p class="v2"> + <b><a name="Block_2135" id="Block_2135">2135</a>.</b> + It will not be thought surprising that Plato + expected that Astronomy, when further advanced, would be able + to render an account of many things for which she has not + accounted even to this day. Thus, in the passage in the seventh + Book of the <em>Republic</em>, he says that the philosopher + requires a reason for the proportion of the day to the month, + and the month to the year, deeper and more substantial than + mere observation can give. Yet Astronomy has not yet shown us + any reason why the proportion of the times of the + earth’s rotation on its axis, the + moon’s revolution round the earth, and the + earth’s revolution round the sun, might not + have been made by the Creator quite different from what they + are. But in asking Mathematical Astronomy for reasons which she + cannot give, Plato was only doing what a great astronomical + discoverer, Kepler, did at a later period. One of the questions + which Kepler especially wished to have answered was, why there + are five planets, and why at such particular distances from the + sun? And it is still more curious that he thought he had found + the reason of these things, in the relation of those five + regular solids which Plato was desirous of introducing into the + philosophy of the universe.... + +<span class="pagenum"> + <a name="Page_377" + id="Page_377">377</a></span> + + Kepler regards the law + which thus determines the number and magnitude of the planetary + orbits by means of the five regular solids as a discovery no + less remarkable and certain than the Three Laws which give his + name its imperishable place in the history of + + <a id="TNanchor_20"></a> + <a class="msg" href="#TN_20" + title="originally read ‘astromomy’">astronomy</a>.—<span + + class="smcap">Whewell, W.</span></p> + <p class="blockcite"> + History of the Inductive Sciences, 3rd Edition, Additions + to Bk. 3.</p> + + <p class="v2"> + <b><a name="Block_2136" id="Block_2136">2136</a>.</b> + Pythagorean philosophers ... maintained that of + two combatants, he would conquer, the sum of the numbers + expressed by the characters of whose names exceeded the sum of + those expressed by the other. It was upon this principle that + they explained the relative prowess and fate of the heroes in + Homer, Πατροκλος, Ἑκτορ and Αχιλλευς, + the sum of the numbers in whose names are 861, 1225, and 1276 + respectively.—<span class="smcap">Peacock, George.</span></p> + <p class="blockcite"> + Encyclopedia of Pure Mathematics (London, 1847); Article + “Arithmetic,” sect. 38.</p> + + <p class="v2"> + <b><a name="Block_2137" id="Block_2137">2137</a>.</b> + Round numbers are always + false.—<span class="smcap">Johnson, Samuel.</span></p> + <p class="blockcite"> + Johnsoniana; Apothegms, Sentiment, etc.</p> + + <p class="v2"> + <b><a name="Block_2138" id="Block_2138">2138</a>.</b> + Numero deus impare gaudet [God in number odd + rejoices.]—<span class="smcap">Virgil.</span></p> + <p class="blockcite"> + Eclogue, 8, 77.</p> + + <p class="v2"> + <b><a name="Block_2139" id="Block_2139">2139</a>.</b> + Why is it that we entertain the belief that for + every purpose odd numbers are the most + effectual?—<span class="smcap">Pliny.</span></p> + <p class="blockcite"> + Natural History, Bk. 28, chap. 5.</p> + + <p class="v2"> + <b><a name="Block_2140" id="Block_2140">2140</a>.</b></p> + <div class="poem"> + <p class="i0"> + “Then here goes another,” says he, “to make sure,</p> + <p class="i0"> + Fore there’s luck in odd numbers,” says Rory O’Moore.</p> + </div> + <p class="block40"> + —<span class="smcap">Lover, S.</span></p> + <p class="blockcite"> + Rory O’Moore.</p> + + <p class="v2"> + <b><a name="Block_2141" id="Block_2141">2141</a>.</b> + This is the third time; I hope, good luck lies in + odd numbers.... They say, there is divinity in odd numbers, + either in nativity, chance, or + death.—<span class="smcap">Shakespeare.</span></p> + <p class="blockcite"> + The Merry Wives of Windsor, Act 5, scene 1.</p> + +<p><span class="pagenum"> + <a name="Page_378" + id="Page_378">378</a></span></p> + + <p class="v2"> + <b><a name="Block_2142" id="Block_2142">2142</a>.</b> + To add to golden numbers, golden + numbers.—<span class="smcap">Decker, Thomas.</span></p> + <p class="blockcite"> + Patient Grissell, Act 1, scene 1.</p> + + <p class="v2"> + <b><a name="Block_2143" id="Block_2143">2143</a>.</b></p> + <div class="poem"> + <p class="i0"> + I’ve read that things inanimate have moved,</p> + <p class="i0"> + And, as with living souls, have been inform’d,</p> + <p class="i0"> + By magic numbers and persuasive sound.</p> + </div> + <p class="block40"> + —<span class="smcap">Congreve, Richard.</span></p> + <p class="blockcite"> + The Morning Bride, Act 1, scene 1.</p> + + <p class="v2"> + <b><a name="Block_2144" id="Block_2144">2144</a>.</b> + ... the Yancos on the Amazon, whose name for three is</p> + <p class="center"> + Poettarrarorincoaroac,</p> + <p class="v0"> + of a length sufficiently formidable to justify the remark of La + Condamine: Heureusement pour ceux qui ont à faire avec + eux, leur Arithmetique ne va pas plus + loin.—<span class="smcap">Peacock, George.</span></p> + <p class="blockcite"> + Encyclopedia of Pure Mathematics (London, 1847); Article + “Arithmetic,” sect. 32.</p> + + <p class="v2"> + <b><a name="Block_2145" id="Block_2145">2145</a>.</b> + There are three principal sins, avarice, luxury, + and pride; three sorts of satisfaction for sin, fasting, + almsgiving, and prayer; three persons offended by sin, God, the + sinner himself, and his neighbour; three witnesses in heaven, + <i lang="la" xml:lang="la">Pater</i>, <i lang="la" + xml:lang="la">verbum</i>, and <i lang="la" + xml:lang="la">spiritus sanctus</i>; three + degrees of penitence, contrition, confession, and satisfaction, + which Dante has represented as the three steps of the ladder + that lead to purgatory, the first marble, the second black and + rugged stone, and the third red porphyry. There are three + sacred orders in the church militant, <i lang="la" + xml:lang="la">sub-diaconati</i>, + <i lang="la" xml:lang="la">diaconiti</i>, and <i lang="la" + xml:lang="la">presbyterati</i>; there are three + parts, not without mystery, of the most sacred body made by the + priest in the mass; and three times he says <i lang="la" + xml:lang="la">Agnus Dei</i>, + and three times, <i lang="la" xml:lang="la">Sanctus</i>; + and if we well consider all + the devout acts of Christian worship, they are found in a + ternary combination; if we wish rightly to partake of the holy + communion, we must three times express our contrition, + <i lang="la" xml:lang="la">Domine non sum dignus</i>; + but who can say more of the + ternary number in a shorter compass, than what the prophet + says, <i lang="la" xml:lang="la">tu signaculum sanctae + trinitatis</i>. There are three + Furies in the infernal regions; three Fates, Atropos, Lachesis, + and Clotho. There are three theological virtues: + <i lang="la" xml:lang="la">Fides</i>, + +<span class="pagenum"> + <a name="Page_379" + id="Page_379">379</a></span> + + <i lang="la" xml:lang="la">spes</i>, and <i lang="la" + xml:lang="la">charitas</i>. + <i lang="la" xml:lang="la">Tria sunt pericula mundi: Equum + currere; navigare, et sub + tyranno vivere.</i> There are three enemies of the soul: the + Devil, the world, and the flesh. There are three things which + are of no esteem: the strength of a porter, the advice of a + poor man, and the beauty of a beautiful woman. There are three + vows of the Minorite Friars: poverty, obedience, and chastity. + There are three terms in a continued proportion. There are + three ways in which we may commit sin: <i lang="la" + xml:lang="la">corde</i>, + <i lang="la" xml:lang="la">ore</i>, <i lang="la" + xml:lang="la">ope</i>. Three principal things in Paradise: + glory, riches, and justice. There are three things which are + especially displeasing to God: an avaricious rich man, a proud + poor man, and a luxurious old man. And all things, in short, + are founded in three; that is, in number, in weight, and in + measure.—<span class="smcap">Pacioli</span>, <cite>Author of + the first printed treatise on arithmetic.</cite></p> + <p class="blockcite"> + Quoted in Encyclopedia of Pure Mathematics (London, 1847); + Article “Arithmetic,” sect. 90.</p> + + <p class="v2"> + <b><a name="Block_2146" id="Block_2146">2146</a>.</b></p> + <div class="poem"> + <p class="i0"> + Ah! why, ye Gods, should two and two make four?</p> + </div> + <p class="block40"> + —<span class="smcap">Pope, Alexander.</span></p> + <p class="blockcite"> + The Dunciad, Bk. 2, line 285.</p> + + <p class="v2"> + <b><a name="Block_2147" id="Block_2147">2147</a>.</b></p> + <div class="poem"> + <p class="i0"> + By him who stampt <em>The Four</em> upon the mind,—</p> + <p class="i0"> + <em>The Four</em>, the fount of nature’s endless stream.</p> + </div> + <p class="block40"> + —<em>Ascribed to</em> <span class= + "smcap">Pythagoras.</span></p> + <p class="blockcite"> + Quoted in Whewell’s History of the + Inductive Sciences, Bk. 4, chap. 3.</p> + + <p class="v2"> + <b><a name="Block_2148" id="Block_2148">2148</a>.</b></p> + <div class="poem"> + <p class="i0"> + Along the skiey arch the goddess trode,</p> + <p class="i0"> + And sought Harmonia’s august abode;</p> + <p class="i0"> + The universal plan, the mystic Four,</p> + <p class="i0"> + Defines the figure of the palace floor.</p> + <p class="i0"> + Solid and square the ancient fabric stands,</p> + <p class="i0"> + Raised by the labors of unnumbered hands.</p> + </div> + <p class="block40"> + —<span class="smcap">Nonnus.</span></p> + <p class="blockcite"> + Dionysiac, 41, 275-280. [Whewell].</p> + + <p class="v2"> + <b><a name="Block_2149" id="Block_2149">2149</a>.</b> + The number seventy-seven figures the abolition of + all sins by baptism.... The number ten signifies justice and + beatitude, resulting from the creature, which makes seven with + the Trinity, which is three: therefore + it is that God’s commandments + +<span class="pagenum"> + <a name="Page_380" + id="Page_380">380</a></span> + + are ten in + number. The number eleven denotes sin, because it + <em>transgresses</em> ten.... This number seventy-seven is the + product of eleven, figuring sin, multiplied by seven, and not + by ten, for seven is the number of the creature. Three + represents the soul, which is in some sort an image of + Divinity; and four represents the body, on account of its four + qualities....—<span class="smcap">St. Augustine.</span></p> + <p class="blockcite"> + Sermon 41, art. 23.</p> + + <p class="v2"> + <b><a name="Block_2150" id="Block_2150">2150</a>.</b> + Heliodorus says that the Nile is nothing else than + the year, founding his opinion on the fact that the numbers + expressed by the letters Νειλος, Nile, are in Greek arithmetic, + Ν = 50; Ε = 5; I = 10; Λ = 30; Ο = 70; + Σ = 200; and these figures make up together 365, + the number of days in the year.</p> + <p class="blockcite"> + Littell’s Living Age, Vol. 117, p. 380.</p> + + <p class="v2"> + <b><a name="Block_2151" id="Block_2151">2151</a>.</b> + In treating 666, Bungus [Petri Bungi Bergomatis + Numerorum mysteria, Bergamo, 1591] a good Catholic, could not + compliment the Pope with it, but he fixes it on Martin Luther + with a little forcing. If from A to I represent 1-<a + + class="msg" + href="#TN_27" + title="originally read ‘10’">9</a>, + + from K to + S 10-90, and from T to Z 100-500, we see—</p> + <pre class="Monofont"> + M A R T I N L U T E R A + 30 1 80 100 9 40 20 200 100 5 80 1</pre> + <p class="v0"> + which gives 666. Again in Hebrew, <em>Lulter</em> [Hebraized form + of Luther] does the same:—</p> + <pre class="Monofont"> + ל י ל ת ר + 200 400 30 6 30</pre> + <p class="block40"> + —<span class="smcap">De Morgan, A.</span></p> + <p class="blockcite"> + Budget of Paradoxes (London, 1872), p. 37.</p> + + <p class="v2"> + <b><a name="Block_2152" id="Block_2152">2152</a>.</b> + Stifel, the most acute and original of the early + mathematicians of Germany, ... relates ... that whilst a monk + at Esslingen in 1520, and when infected by the writings of + Luther, he was reading in the library of his convent the 13th + Chapter of <em>Revelations</em>, it struck his mind that the + <em>Beast</em> must signify the Pope, Leo X.; He then proceeded + in pious hope to make the calculation of the sum of the numeral + letters in <i lang="la" xml:lang="la">Leo decimus</i>, + which he found to be M, D, C, L, + V, I; the sum which these formed was too great by M, and too + little by X; but he + +<span class="pagenum"> + <a name="Page_381" + id="Page_381">381</a></span> + + bethought him again, that he has seen + the name written Leo X., and that there were ten letters in + <i lang="la" xml:lang="la">Leo decimus</i>, from either + of which he could obtain the + deficient number, and by interpreting the M to mean + <i lang="la" xml:lang="la">mysterium</i>, he found the + number required, a discovery + which gave him such unspeakable comfort, that he believed that + his interpretation must have been an immediate inspiration of + God.—<span class="smcap">Peacock, George.</span></p> + <p class="blockcite"> + Encyclopedia of Pure Mathematics (London, 1847); Article + “Arithmetic,” sect. 89.</p> + + <p class="v2"> + <b><a name="Block_2153" id="Block_2153">2153</a>.</b> + Perhaps the best anagram ever made is that by Dr. + Burney on Horatio Nelson, so happily transformed into the Latin + sentence so truthful of the great admiral, <i lang="la" + xml:lang="la">Honor est a + Nilo</i>. Reading this, one is almost persuaded that the hit + contained in it has a meaning provided by providence or fate.</p> + <p class="v1"> + This is also amusingly illustrated in the case of the Frenchman + André Pujom, who, using j as i, found in his name + the anagram, Pendu à Riom. Riom being the seat of + justice for the province of Auvergne, the poor fellow, impelled + by a sort of infatuation, actually committed a capital offence + in that province, and was hanged at Riom, that the anagram + might be fulfilled.</p> + <p class="blockcite"> + New American Cyclopedia, Vol. 1; Article “Anagram”</p> + + <p class="v2"> + <b><a name="Block_2154" id="Block_2154">2154</a>.</b> + The most remarkable pseudonym [of transposed names + adopted by authors] is the name of “Voltaire,” which the + celebrated philosopher assumed instead of his family name, + “François Marie Arouet,” and which is now generally allowed to + be an anagram of “Arouet, l. j.,” that is, Arouet the younger.</p> + <p class="blockcite"> + Encyclopedia Britannica, 11th Edition; Article “Anagram”</p> + + <p class="v2"> + <b><a name="Block_2155" id="Block_2155">2155</a>.</b> + Perhaps the most beautiful anagram that has ever + been composed is by Jablonsky, a former rector of the school at + Lissa. The occasion was the following: When while a young man + king Stanislaus of Poland returned from a journey, the whole + house of Lescinsky assembled to welcome the family heir. On + this occasion Jablonsky arranged for a school program, the + closing number of which consisted of a ballet by thirteen + pupils + +<span class="pagenum"> + <a name="Page_382" + id="Page_382">382</a></span> + + impersonating youthful heroes. Each + of them carried a shield on which appeared in gold one of the + letters of the words <i lang="la" xml:lang="la">Domus + Lescinia</i>. At the end of the + first dance the children were so arranged that the letters on + their shields spelled the words <i lang="la" + xml:lang="la">Domus Lescinia</i>. At the + end of the second dance they read: <i lang="la" + xml:lang="la">ades incolumis</i> (sound + thou art here). After the third: <i lang="la" + xml:lang="la">omnis es lucida</i> (wholly + brilliant art thou); after the fourth: <i lang="la" + xml:lang="la">lucida sis omen</i> + (bright be the omen). Then: <i lang="la" xml:lang="la">mane + sidus loci</i> (remain our + country’s star); and again: <i lang="la" xml:lang="la">sis + columna + Dei</i> (be a column of God); and finally: <i lang="la" + xml:lang="la">I! scande + solium</i> (Proceed, ascend the throne). This last was the more + beautiful since it proved a true prophecy.</p> + <p class="v1"> + Even more artificial are the anagrams which transform one verse + into another. Thus an Italian scholar beheld in a dream the + line from Horace: <i lang="la" xml:lang="la">Grata + superveniet, quae non sperabitur, + hora</i>. This a friend changed to the anagram: <i lang="la" + xml:lang="la">Est ventura + Rhosina parataque nubere pigro</i>. This induced the scholar, + though an old man, to marry an unknown lady by the name of + Rosina.—<span class="smcap">Heis, Eduard.</span></p> + <p class="blockcite"> + Algebraische Aufgaben (Köln, 1898), p. 331.</p> + + <p class="v2"> + <b><a name="Block_2156" id="Block_2156">2156</a>.</b> + The following verses read the same whether read + forward or backward:—</p> + <div class="poem"> + <p class="center"> + Aspice! nam raro mittit timor arma, nec ipsa</p> + <p class="center"> + Si se mente reget, non tegeret Nemesis;<a + href="#Footnote_13" + title="Beginning of a poem which Johannes a Lasco +wrote on the count Karl von Südermanland." + class="fnanchor">13</a></p> + </div> + <p class="v0"> + also,</p> + <div class="poem"> + <p class="center"> + Sator Arepo tenet opera rotas.</p> + </div> + <p class="block40"> + —<span class="smcap">Heis, Eduard.</span></p> + <p class="blockcite"> + Algebraische Aufgaben (Köln, 1898), p. 328.</p> + + <p class="v2"> + <b><a name="Block_2157" id="Block_2157">2157</a>.</b> + There is a certain spiral of a peculiar form on + which a point may have been approaching for centuries the + center, and have nearly reached it, before we discover that its + rate of approach is accelerated. The first thought of the + observer, on seeing the acceleration, would be to say that it + would reach the center sooner than he had before supposed. But + as the point comes near the center it suddenly, although still + moving under the same simple law as from the beginning, makes a + very short turn upon its path and flies off rapidly almost in a + straight line, + +<span class="pagenum"> + <a name="Page_383" + id="Page_383">383</a></span> + + out to an infinite distance. This + illustrates that apparent breach of continuity which we + sometimes find in a natural law; that apparently sudden change + of character which we sometimes see in + man.—<span class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + Uses of Mathesis; Bibliotheca Sacra, Vol. 32, p. 521.</p> + + <p class="v2"> + <b><a name="Block_2158" id="Block_2158">2158</a>.</b> + One of the most remarkable of + Babbage’s illustrations of miracles has + never had the consideration in the popular mind which it + deserves; the illustration drawn from the existence of isolated + points fulfilling the equation of a curve.... There are + definitions of curves which describe not only the positions of + every point in a certain curve, but also of one or more + perfectly isolated points; and if we should attempt to get by + induction the definition, from the observation of the points on + the curve, we might fail altogether to include these isolated + points; which, nevertheless, although standing alone, as + miracles to the observer of the course of the points in the + curve, are nevertheless rigorously included in the law of the + curve.—<span class="smcap">Hill, Thomas.</span></p> + <p class="blockcite"> + Uses of Mathesis; Bibliotheca Sacra, Vol. 32, p. 516.</p> + + <p class="v2"> + <b><a name="Block_2159" id="Block_2159">2159</a>.</b> + Pure mathematics is the magician’s real + wand.—<span class="smcap">Novalis.</span></p> + <p class="blockcite"> + Schriften, Zweiter Teil (Berlin, 1901), p. 223.</p> + + <p class="v2"> + <b><a name="Block_2160" id="Block_2160">2160</a>.</b> + Miracles, considered as antinatural facts, are + amathematical, but there are no miracles in this sense, and + those so called may be comprehended by means of mathematics, + for to mathematics nothing is + miraculous.—<span class="smcap">Novalis.</span></p> + <p class="blockcite"> + Schriften, Zweiter Teil (Berlin, 1911), p. 222.</p> + +<p> + <span class="pagenum"> + <a name="Page_384" + id="Page_384">384</a><br /> + <a name="Page_385" + id="Page_385">385</a></span></p> + + <hr class="chap" /> + <h2 class="v2" id="INDEX"> + INDEX</h2> + + <p class="v2"> + <b>Bold-faced numbers refer to authors</b></p> + <p class="v2"> + Abbreviations:— m. = mathematics, + math. = mathematical, + math’n. = mathematician.</p> + + <ul class="index"> + <li class="ifrst"> + Abbott, <b><a + href="#Block_1001">1001</a></b>.</li> + <li class="indx"> + Abstract method, Development of, <a + href="#Block_729">729</a>.</li> + <li class="indx"> + Abstract nature of m., Reason for, <a + href="#Block_638">638</a>.</li> + <li class="indx"> + Abstractness, math., Compared with logical, <a + href="#Block_1304">1304</a>.</li> + <li class="indx"> + Abstract reasoning, Objection to, <a + href="#Block_1941">1941</a>.</li> + <li class="indx"> + Adams, Henry,</li> + <li class="isub1"> + M. and history, <b><em><a + href="#Block_1599a">1599</a></em></b>.</li> + <li class="isub1"> + Math’ns practice freedom, <b><a + href="#Block_208">208</a></b>, <b><a + href="#Block_805">805</a></b>.</li> + <li class="indx"> + Adams, John, Method in m., <b><a + href="#Block_226">226</a></b>.</li> + <li class="indx"> + Aeneid, Euler’s knowledge of, <em><a + href="#Block_959">859</a></em>.</li> + <li class="indx"> + Aeschylus. On number, <a + href="#Block_1606">1606</a>.</li> + <li class="indx"> + Aim in teaching m., <a + href="#Block_501">501-508</a>, <a + href="#Block_517">517</a>, <a + href="#Block_844">844</a>.</li> + <li class="indx"> + Airy, Pythagorean theorem, <b><a + href="#Block_2126">2126</a></b>.</li> + <li class="indx"> + Akenside, <b><a + href="#Block_1532">1532</a></b>.</li> + <li class="indx"> + Alexander, <a + href="#Block_901">901</a>, <a + href="#Block_902">902</a>.</li> + <li class="indx"> + Algebra,</li> + <li class="isub1"> + Chapter <a + href="#CHAPTER_XVII">XVII</a>.</li> + <li class="isub1"> + Definitions of, <a + href="#Block_110">110</a>, <a + href="#Block_1714">1714</a>, <a + href="#Block_1715">1715</a>.</li> + <li class="isub1"> + Problems in, <a + href="#Block_320">320</a>, <a + href="#Block_530">530</a>, <a + href="#Block_1738">1738</a>.</li> + <li class="isub1"> + Of use to grown men, <a + href="#Block_425">425</a>.</li> + <li class="isub1"> + And geometry, <a + href="#Block_525">525-527</a>, <a + href="#Block_1610">1610</a>, <a + href="#Block_1707">1707</a>.</li> + <li class="isub1"> + Advantages of, <a + href="#Block_1701">1701</a>, <a + href="#Block_1703">1703</a>, <a + href="#Block_1705">1705</a>.</li> + <li class="isub1"> + Laws of, <a + href="#Block_1708">1708-1710</a>.</li> + <li class="isub1"> + As an art, <a + href="#Block_1711">1711</a>.</li> + <li class="isub1"> + Review of, <a + href="#Block_1713">1713</a>.</li> + <li class="isub1"> + Designations of, <a + href="#Block_1717">1717</a>.</li> + <li class="isub1"> + Origin of, <a + href="#Block_1736">1736</a>.</li> + <li class="isub1"> + Burlesque on modern, <a + href="#Block_1741">1741</a>.</li> + <li class="isub1"> + Hume on, <a + href="#Block_1863">1863</a>.</li> + <li class="indx"> + Algebraic notation, value of, <a + href="#Block_1213">1213</a>, <a + href="#Block_1214">1214</a>.</li> + <li class="indx"> + Algebraic treatises, How to read, <a + href="#Block_601">601</a>.</li> + <li class="indx"> + Amusements in m., <a + href="#Block_904">904</a>, <a + href="#Block_905">905</a>.</li> + <li class="indx"> + Anagrams,</li> + <li class="isub1"> + On De Morgan, <a + href="#Block_947">947</a>.</li> + <li class="isub1"> + On Domus Lescinia, <a + href="#Block_2155">2155</a>.</li> + <li class="isub1"> + On Flamsteed, <a + href="#Block_968">968</a>.</li> + <li class="isub1"> + On Macaulay, <a + href="#Block_996">996</a>.</li> + <li class="isub1"> + On Nelson, <a + href="#Block_2153">2153</a>.</li> + <li class="isub1"> + On Newton, <a + href="#Block_1028">1028</a>.</li> + <li class="isub1"> + On Voltaire, <a + href="#Block_2154">2154</a>.</li> + <li class="indx"> + Analysis,</li> + <li class="isub1"> + Invigorates the faculty of resolution, <a + href="#Block_416">416</a>.</li> + <li class="isub1"> + Relation of geometry to, <a + href="#Block_1931">1931</a>.</li> + <li class="indx"> + Analytical geometry, <a + href="#Block_1889">1889</a>, <a + href="#Block_1890">1890</a>, <a + href="#Block_1893">1893</a>.</li> + <li class="isub1"> + Method of, <a + href="#Block_310">310</a>.</li> + <li class="isub1"> + Importance of, <a + href="#Block_949">949</a>.</li> + <li class="isub1"> + Burlesque on, <a + href="#Block_2040">2040</a>.</li> + <li class="indx"> + Ancient geometry,</li> + <li class="isub1"> + Characteristics of, <a + href="#Block_712">712</a>, <a + href="#Block_714">714</a>.</li> + <li class="isub1"> + Compared with modern, <a + href="#Block_1711">1711-1716</a>.</li> + <li class="isub1"> + Method of, <a + href="#Block_1425">1425</a>, <a + href="#Block_1873">1873-1875</a>.</li> + <li class="indx"> + Ancients, M. among the, <a + href="#Block_321">321</a>.</li> + <li class="indx"> + Anecdotes, Chapters, <a + href="#CHAPTER_IX">IX</a>, <a + href="#CHAPTER_X">X</a>.</li> + <li class="indx"> + Anger, M. destroys predisposition to, <a + href="#Block_458">458</a>.</li> + <li class="indx"> + Angling like m., <a + href="#Block_739">739</a>.</li> + <li class="indx"> + Anglo-Danes, Aptitude for m., <a + href="#Block_836">836</a>.</li> + <li class="indx"> + Anglo-Saxons,</li> + <li class="isub1"> + Aptitude for m., <a + href="#Block_837">837</a>.</li> + <li class="isub1"> + Newton as representative of, <a + href="#Block_1014">1014</a>.</li> + <li class="indx"> + Anonymous, Song of the screw, <a + href="#Block_1894">1894</a>.</li> + <li class="indx"> + + <a id="TNanchor_21" + class="msg" href="#TN_21" + title="also spelled Apollonius in +blocks 523 & 917">Appolonius</a>, <a + href="#Block_712">712</a>, <a + href="#Block_714">714</a>.</li> + + <li class="indx"> + Approximate m., Why not sufficient, <a + href="#Block_1518">1518</a>. + +<span class="pagenum"> + <a name="Page_386" + id="Page_386">386</a></span></li> + + <li class="indx"> + Aptitude for m., <a + href="#Block_509">509</a>, <a + href="#Block_510">510</a>, <a + href="#Block_520">520</a>, <a + href="#Block_836">836-838</a>, <a + href="#Block_976">976</a>, <a + href="#Block_1617">1617</a>.</li> + <li class="indx"> + Arabic notation, <a + href="#Block_1614">1614</a>.</li> + <li class="indx"> + Arago,</li> + <li class="isub1"> + M. the enemy of scientific romances, <b><a + href="#Block_267">267</a></b>.</li> + <li class="isub1"> + Euler, “analysis incarnate,” <b><a + href="#Block_961">961</a></b>.</li> + <li class="isub1"> + Euler as a computer, <b><a + href="#Block_962">962</a></b>.</li> + <li class="isub1"> + On Kepler’s discovery, <b><a + href="#Block_982">982</a></b>.</li> + <li class="isub1"> + Newton’s efforts superhuman, <b><a + href="#Block_1006">1006</a></b>.</li> + <li class="isub1"> + On probabilities, <b><a + href="#Block_1591">1591</a></b>.</li> + <li class="isub1"> + Geometry as an instrument, <a + href="#Block_1868">1868</a>.</li> + <li class="indx"> + Arbuthnot,</li> + <li class="isub1"> + M. frees from prejudice, credulity and superstition, <b><em><a + href="#Block_450">449</a></em></b>.</li> + <li class="isub1"> + M. the friend of religion, <b><em><a + href="#Block_459">458</a></em></b>.</li> + <li class="isub1"> + M. compared to music, <b><a + href="#Block_1112">1112</a></b>.</li> + <li class="isub1"> + On math, reasoning, <b><a + href="#Block_1503">1503</a></b>.</li> + <li class="indx"> + Archimedes,</li> + <li class="isub1"> + His machines, <a + href="#Block_903">903</a>, <a + href="#Block_904">904</a>.</li> + <li class="isub1"> + Estimate of math, appliances, <a + href="#Block_904">904-906</a>, <a + href="#Block_908">908</a>.</li> + <li class="isub1"> + Wordsworth on, <a + href="#Block_906">906</a>.</li> + <li class="isub1"> + Schiller on, <a + href="#Block_907">907</a>.</li> + <li class="isub1"> + And engineering, <a + href="#Block_908">908</a>.</li> + <li class="isub1"> + Death of, <a + href="#Block_909">909</a>.</li> + <li class="isub1"> + His tomb, <a + href="#Block_910">910</a>.</li> + <li class="isub1"> + Compared with Newton, <a + href="#Block_911">911</a>.</li> + <li class="isub1"> + Character of his work, <a + href="#Block_912">912</a>, <a + href="#Block_913">913</a>.</li> + <li class="isub1"> + Applied m., <a + href="#Block_1312">1312</a>.</li> + <li class="indx"> + Architecture and m., <a + href="#Block_276">276</a>.</li> + <li class="indx"> + Archytas, <a + href="#Block_904">904</a>.</li> + <li class="isub1"> + And Plato, <a + href="#Block_1427">1427</a>.</li> + <li class="indx"> + Aristippus the Cyrenaic, <a + href="#Block_845">845</a>.</li> + <li class="indx"> + Aristotle, <a + href="#Block_914">914</a>.</li> + <li class="isub1"> + On relation of m. to esthetics, <b><a + href="#Block_318">318</a></b>.</li> + <li class="indx"> + Arithmetic,</li> + <li class="isub1"> + Chapter <a + href="#CHAPTER_XVI">XVI</a>.</li> + <li class="isub1"> + Definitions of, <a + href="#Block_106">106</a>, <a + href="#Block_110">110</a>, <a + href="#Block_1611">1611</a>, <a + href="#Block_1612">1612</a>, <a + href="#Block_1714">1714</a>.</li> + <li class="isub1"> + Emerson on advantage of study of, <a + href="#Block_408">408</a>.</li> + <li class="isub1"> + Problems in, <a + href="#Block_528">528</a>.</li> + <li class="isub1"> + A master-key, <a + href="#Block_1571">1571</a>.</li> + <li class="isub1"> + Based on concept of time, <a + href="#Block_1613">1613</a>.</li> + <li class="isub1"> + Method of teaching, <a + href="#Block_1618">1618</a>.</li> + <li class="isub1"> + Purpose of teaching, <a + href="#Block_454">454</a>, <a + href="#Block_1624">1624</a>.</li> + <li class="isub1"> + As logic, <a + href="#Block_1624">1624</a>, <a + href="#Block_1625">1625</a>.</li> + <li class="isub1"> + The queen of m.,<a + href="#Block_1642">1642</a>.</li> + <li class="isub1"> + Higher, <a + href="#Block_1755">1755</a>.</li> + <li class="isub1"> + Hume on, <a + href="#Block_1863">1863</a>.</li> + <li class="indx"> + Arithmetical theorems, <a + href="#Block_1639">1639</a>.</li> + <li class="indx"> + Art, M. as a fine, Chapter <a + href="#CHAPTER_XI">XI</a>.</li> + <li class="indx"> + Arts, M. and the, <a + href="#Block_1568">1568-1570</a>, <a + href="#Block_1573">1573</a>.</li> + <li class="indx"> + Astronomy and m., <a + href="#Block_1554">1554</a>, <a + href="#Block_1559">1559</a>, <a + href="#Block_1562">1562-1567</a>.</li> + <li class="indx"> + “Auge et impera.,” <a + href="#Block_631">631</a>.</li> + <li class="indx"> + Authority in science, <a + href="#Block_1528">1528</a>.</li> + <li class="indx"> + Axioms, <a + href="#Block_518">518</a>, <a + href="#Block_2015">2015</a>.</li> + <li class="isub1"> + In geometry, <a + href="#Block_1812">1812</a>, <a + href="#Block_2004">2004</a>, <a + href="#Block_2006">2006</a>.</li> + <li class="isub1"> + Def. in disguise, <a + href="#Block_2005">2005</a>.</li> + <li class="isub1"> + Euclid’s, <a + href="#Block_2007">2007-2010</a>, <a + href="#Block_2014">2014</a>.</li> + <li class="isub1"> + Nature of, <a + href="#Block_2012">2012</a>.</li> + <li class="isub1"> + Proofs of, <a + href="#Block_2013">2013</a>.</li> + <li class="isub1"> + And the idea of space, <a + href="#Block_2004">2004</a>.</li> + + <li class="ifrst"> + Babbage, <a + href="#Block_923">923</a>.</li> + <li class="indx"> + Bacon, Lord,</li> + <li class="isub1"> + Classification of m., <b><a + href="#Block_106">106</a></b>.</li> + <li class="isub1"> + M. makes men subtile, <b><a + href="#Block_248">248</a></b>.</li> + <li class="isub1"> + View of m., <a + href="#Block_316">316</a>, <a + href="#Block_915">915</a>, <a + href="#Block_916">916</a>.</li> + <li class="isub1"> + M. held in high esteem by ancients, <b><a + href="#Block_321">321</a></b>.</li> + <li class="isub1"> + On the generalizing power of m., <b><a + href="#Block_327">327</a></b>.</li> + <li class="isub1"> + On the value of math, studies, <b><a + href="#Block_410">410</a></b>.</li> + <li class="isub1"> + M. develops concentration of mind, <b><a + href="#Block_411">411</a></b>.</li> + <li class="isub1"> + M. cures distraction of mind, <b><a + href="#Block_412">412</a></b>.</li> + <li class="isub1"> + M. essential to study of nature, <b><a + href="#Block_436">436</a></b>.</li> + <li class="isub1"> + His view of m., <b><a + href="#Block_915">915</a></b>, <b><a + href="#Block_916">916</a></b>.</li> + <li class="isub1"> + His knowledge of m., <a + href="#Block_917">917</a>, <a + href="#Block_918">918</a>.</li> + <li class="isub1"> + M. and logic, <b><a + href="#Block_1310">1310</a></b>.</li> + <li class="isub1"> + Growth of m., <b><a + href="#Block_1511">1511</a></b>.</li> + <li class="indx"> + Bacon, Roger,</li> + <li class="isub1"> + Neglect of m. works injury to all science, <b><a + href="#Block_310">310</a></b>.</li> + <li class="isub1"> + On the value of m., <b><a + href="#Block_1547">1547</a></b>.</li> + <li class="indx"> + Bain,</li> + <li class="isub1"> + Importance of m. in education, <b><a + href="#Block_442">442</a></b>.</li> + <li class="isub1"> + On the charm of the study of m., <b><a + href="#Block_453">453</a></b>.</li> + <li class="isub1"> + M. and science teaching, <b><a + href="#Block_522">522</a></b>.</li> + <li class="isub1"> + Teaching of arithmetic, <b><a + href="#Block_1618">1618</a></b>.</li> + <li class="indx"> + Ball, R. S., <b><a + href="#Block_2010">2010</a></b>.</li> + <li class="indx"> + Ball, W. W. R.,</li> + <li class="isub1"> + On Babbage, <b><a + href="#Block_923">923</a></b>.</li> + <li class="isub1"> + On Demoivre’s death, <b><a + href="#Block_944">944</a></b>. + +<span class="pagenum"> + <a name="Page_387" + id="Page_387">387</a></span></li> + + <li class="isub1"> + De Morgan and the actuary, <b><a + href="#Block_945">945</a></b>.</li> + <li class="isub1"> + Gauss as astronomer, <b><a + href="#Block_971">971</a></b>.</li> + <li class="isub1"> + Laplace’s “It is easy to see” <b><a + href="#Block_986">986</a></b>.</li> + <li class="isub1"> + Lagrange, Laplace and Gauss contrasted, <b><em><a + href="#Block_983">993</a></em></b>.</li> + <li class="isub1"> + Newton’s interest in chemistry and theology, <b><a + href="#Block_1015">1015</a></b>.</li> + <li class="isub1"> + On Newton’s method of work, <b><a + href="#Block_1026">1026</a></b>.</li> + <li class="isub1"> + On Newton’s discovery of the calculus, <b><a + href="#Block_1027">1027</a></b>.</li> + <li class="isub1"> + Gauss’s estimate of Newton, <b><a + href="#Block_1029">1029</a></b>.</li> + <li class="isub1"> + M. and philosophy, <b><a + href="#Block_1417">1417</a></b>.</li> + <li class="isub1"> + Advance in physics, <b><a + href="#Block_1530">1530</a></b>.</li> + <li class="isub1"> + Plato on geometry, <a + href="#Block_1804">1804</a>.</li> + <li class="isub1"> + Notation of the calculus, <b><a + href="#Block_1904">1904</a></b>.</li> + <li class="indx"> + Barnett, M. the type of perfect reasoning, <b><a + href="#Block_307">307</a></b>.</li> + <li class="indx"> + Barrow,</li> + <li class="isub1"> + On the method of m., <b><a + href="#Block_213">213</a></b>, <b><a + href="#Block_227">227</a></b>.</li> + <li class="isub1"> + Eulogy of m., <b><a + href="#Block_330">330</a></b>.</li> + <li class="isub1"> + M. as a discipline of the mind, <b><a + href="#Block_402">402</a></b>.</li> + <li class="isub1"> + M. and eloquence, <b><a + href="#Block_830">830</a></b>.</li> + <li class="isub1"> + Philosophy and m., <b><a + href="#Block_1430">1430</a></b>.</li> + <li class="isub1"> + Uses of m., <b><a + href="#Block_1572">1572</a></b>.</li> + <li class="isub1"> + On surd numbers, <b><a + href="#Block_1728">1728</a></b>.</li> + <li class="isub1"> + Euclid’s definition of proportion, <b><a + href="#Block_1835">1835</a></b>.</li> + <li class="indx"> + Beattie, <b><a + href="#Block_1431">1431</a></b>.</li> + <li class="indx"> + Beauty of m., <a + href="#Block_453">453</a>, <a + href="#Block_824">824</a>, <a + href="#Block_1208">1208</a>.</li> + <li class="isub1"> + Consists in simplicity, <a + href="#Block_242">242</a>, <a + href="#Block_315">315</a>.</li> + <li class="isub1"> + Sylvester on, <a + href="#Block_1101">1101</a>.</li> + <li class="isub1"> + Russell on, <a + href="#Block_1104">1104</a>.</li> + <li class="isub1"> + Young on, <a + href="#Block_1110">1110</a>.</li> + <li class="isub1"> + Kummer on, <a + href="#Block_1111">1111</a>.</li> + <li class="isub1"> + White on, <a + href="#Block_1119">1119</a>.</li> + <li class="isub1"> + And truth, <a + href="#Block_1114">1114</a>.</li> + <li class="isub1"> + Boltzmann on, <a + href="#Block_1116">1116</a>.</li> + <li class="indx"> + Beltrami, On reading of the masters, <b><a + href="#Block_614">614</a></b>.</li> + <li class="indx"> + Berkeley,</li> + <li class="isub1"> + On geometry as logic, <b><a + href="#Block_428">428</a></b>.</li> + <li class="isub1"> + On math. symbols, <b><a + href="#Block_1214">1214</a></b>.</li> + <li class="isub1"> + On fluxions, <b><a + href="#Block_1915">1915</a></b>, <b><a + href="#Block_1942">1942-1944</a></b>.</li> + <li class="isub1"> + On infinite divisibility, + <b><a href="#Block_1945">1945</a></b>.</li> + <li class="indx"> + Bernoulli, Daniel, <a + href="#Block_919">919</a>.</li> + <li class="indx"> + Bernoulli, James,</li> + <li class="isub1"> + Legend for his tomb, <a + href="#Block_920">920</a>, <a + href="#Block_922">922</a>.</li> + <li class="isub1"> + Computation of sum of tenth powers of numbers, <a + href="#Block_921">921</a>.</li> + <li class="isub1"> + Discussion of logarithmic spiral, <a + href="#Block_922">922</a>.</li> + <li class="indx"> + Berthelot, M. inspires respect for truth, <b><a + href="#Block_438">438</a></b>.</li> + <li class="indx"> + Bija Ganita, Solution of problems, <b><a + href="#Block_1739">1739</a></b>.</li> + <li class="indx"> + Billingsley, M. beautifies the mind, <b><a + href="#Block_319">319</a></b>.</li> + <li class="indx"> + Binary arithmetic, <a + href="#Block_991">991</a>.</li> + <li class="indx"> + Biology and m., <a + href="#Block_1579">1579-1581</a>.</li> + <li class="indx"> + Biot, Laplace’s “It is easy to see,” <a + href="#Block_986">986</a>.</li> + <li class="indx"> + + <a id="TNanchor_22"></a> + <a class="msg" href="#TN_22" + title="originally read ‘Bocher’">Bôcher</a>,</li> + + <li class="isub1"> + M. likened to painting, <b><a + href="#Block_1103">1103</a></b>.</li> + <li class="isub1"> + Interrelation of m. and logic, <b><a + href="#Block_1313">1313</a></b>.</li> + <li class="isub1"> + Geometry as a natural science, <b><a + href="#Block_1866">1866</a></b>.</li> + <li class="indx"> + Boerne, On Pythagoras, <b><a + href="#Block_1855">1855</a></b>.</li> + <li class="indx"> + Bois-Reymond,</li> + <li class="isub1"> + On the analytic method, <b><a + href="#Block_1893">1893</a></b>.</li> + <li class="isub1"> + Natural selection and the calculus, <b><a + href="#Block_1921">1921</a></b>.</li> + <li class="indx"> + Boltzmann, On beauty in m., <b><a + href="#Block_1116">1116</a></b>.</li> + <li class="indx"> + Bolyai, Janos,</li> + <li class="isub1"> + Duel with officers, <a + href="#Block_924">924</a>.</li> + <li class="isub1"> + Universal language, <a + href="#Block_925">925</a>.</li> + <li class="isub1"> + Science absolute of space, <a + href="#Block_926">926</a>.</li> + <li class="indx"> + Bolyai, Wolfgang, <a + href="#Block_927">927</a>.</li> + <li class="isub1"> + On Gauss, <b><a + href="#Block_972">972</a></b>.</li> + <li class="indx"> + Bolzano, <a + href="#Block_928">928</a>.</li> + <li class="isub1"> + Cured by Euclid, <b><a + href="#Block_929">929</a></b>.</li> + <li class="isub1"> + Parallel axiom, <b><a + href="#Block_2110">2110</a></b>.</li> + <li class="indx"> + Book-keeping, Importance of the art of, <a + href="#Block_1571">1571</a>.</li> + <li class="indx"> + Boole, M. E. <b><a + href="#Block_719">719</a></b>.</li> + <li class="indx"> + Boole’s Laws of Thought, <a + href="#Block_1318">1318</a>.</li> + <li class="indx"> + Borda-Demoulins, Philosophy and m., <b><a + href="#Block_1405">1405</a></b>.</li> + <li class="indx"> + Boswell, <b><a href="#Block_981">981</a></b>.</li> + <li class="indx"> + Bowditch, On Laplace’s “Thus it plainly appears,” <b><a + href="#Block_985">985</a></b>.</li> + <li class="indx"> + Boyle,</li> + <li class="isub1"> + Usefulness of m. to physics, <b><a + href="#Block_437">437</a></b>.</li> + <li class="isub1"> + M. and science, <b><a + href="#Block_1513">1513</a></b>, <b><a + href="#Block_1533">1533</a></b>.</li> + <li class="isub1"> + Ignorance of m., <b><a + href="#Block_1577">1577</a></b>. + +<span class="pagenum"> + <a name="Page_388" + id="Page_388">388</a></span></li> + + <li class="isub1"> + M. and physiology, <b><a + href="#Block_1582">1582</a></b>.</li> + <li class="isub1"> + Wings of m., <b><a + href="#Block_1626">1626</a></b>.</li> + <li class="isub1"> + Advantages of algebra, <b><a + href="#Block_1703">1703</a></b>.</li> + <li class="indx"> + Brahmagupta, Estimate of m., <b><a + href="#Block_320">320</a></b>.</li> + <li class="indx"> + Brewster,</li> + <li class="isub1"> + On Euler’s knowledge of the Aeneid, <b><a + href="#Block_959">959</a></b>.</li> + <li class="isub1"> + On Euler as a computer, <b><a + href="#Block_963">963</a></b>.</li> + <li class="isub1"> + On Newton’s fame, <b><a + href="#Block_1002">1002</a></b>.</li> + <li class="indx"> + Brougham, <b><a + href="#Block_1202">1202</a></b>.</li> + <li class="indx"> + Buckle, On geometry, <b><a + href="#Block_1810">1810</a></b>, <b><a + href="#Block_1837">1837</a></b>.</li> + <li class="indx"> + Burke, On the value of m., <b><a + href="#Block_447">447</a></b>.</li> + <li class="indx"> + Burkhardt,</li> + <li class="isub1"> + On discovery in m., <b><a + href="#Block_618">618</a></b>.</li> + <li class="isub1"> + On universal symbolism, <b><a + href="#Block_1221">1221</a></b>.</li> + <li class="indx"> + Butler, N. M.,</li> + <li class="isub1"> + M. demonstrates the supremacy of the human reason, <b><a + href="#Block_309">309</a></b>.</li> + <li class="isub1"> + M. the most astounding intellectual creation, <b><a + href="#Block_707">707</a></b>.</li> + <li class="isub1"> + Geometry before algebra, <b><a + href="#Block_1871">1871</a></b>.</li> + <li class="indx"> + Butler, Samuel, <b><a + href="#Block_2118">2118</a></b>.</li> + <li class="indx"> + Byerly, On hyperbolic functions, <b><a + href="#Block_1929">1929</a></b>.</li> + <li class="ifrst"> + Cajori,</li> + <li class="isub1"> + On the value of the history of m., <b><a + href="#Block_615">615</a></b>.</li> + <li class="isub1"> + On Bolyai, <b><a + href="#Block_927">927</a></b>.</li> + <li class="isub1"> + Cayley’s view of Euclid, <b><a + href="#Block_936">936</a></b>.</li> + <li class="isub1"> + On the extent of Euler’s work, <b><a + href="#Block_960">960</a></b>.</li> + <li class="isub1"> + On Euler’s math. power, <b><a + href="#Block_964">964</a></b>.</li> + <li class="isub1"> + On the Darmstaetter prize, <b><a + href="#Block_967">967</a></b>.</li> + <li class="isub1"> + On Sylvester’s first class at Johns Hopkins, <b><a + href="#Block_1031">1031</a></b>.</li> + <li class="isub1"> + On music and m. among the Pythagoreans, <b><a + href="#Block_1130">1130</a></b>.</li> + <li class="isub1"> + On the greatest achievement of the Hindoos, <b><a + href="#Block_1615">1615</a></b>.</li> + <li class="isub1"> + On modern calculation, <b><a + href="#Block_1614">1614</a></b>.</li> + <li class="isub1"> + On review in arithmetic, <b><a + href="#Block_1713">1713</a></b>.</li> + <li class="isub1"> + On Indian m., <b><a + href="#Block_1737">1737</a></b>.</li> + <li class="isub1"> + On the characteristics of ancient geometry, <b><a + href="#Block_1873">1873</a></b>.</li> + <li class="isub1"> + On Napier’s rule, <b><a + href="#Block_1888">1888</a></b>.</li> + <li class="indx"> + Calculating machines, <b><a + href="#Block_1641">1641</a></b>.</li> + <li class="indx"> + Calculation,</li> + <li class="isub1"> + Importance of, <a + href="#Block_602">602</a>.</li> + <li class="isub1"> + Not the sole object of m., <a + href="#Block_268">268</a>.</li> + <li class="indx"> + Calculus,</li> + <li class="isub1"> + Chapter <a + href="#CHAPTER_XIX">XIX</a>.</li> + <li class="isub1"> + Foundation of <a + href="#Block_253">253</a>.</li> + <li class="isub1"> + As a method, <a + href="#Block_309">309</a>.</li> + <li class="isub1"> + May be taught at an early age,<a + href="#Block_519">519</a>, <a + href="#Block_1917">1917</a>, <a + href="#Block_1918">1918</a>.</li> + <li class="indx"> + Cambridge m., <a + href="#Block_836">836</a>, <a + href="#Block_1210">1210</a>.</li> + <li class="indx"> + Cantor,</li> + <li class="isub1"> + On freedom in m., <b><a + href="#Block_205">205</a></b>, <b><a + href="#Block_207">207</a></b>.</li> + <li class="isub1"> + On the character of Gauss’s writing, <b><a + href="#Block_975">975</a></b>.</li> + <li class="isub1"> + Zeno’s problem, <a + href="#Block_1938">1938</a>.</li> + <li class="isub1"> + On the infinite, <b><a + href="#Block_1952">1952</a></b>.</li> + <li class="indx"> + Carlisle life tables, <a + href="#Block_946">946</a>.</li> + <li class="indx"> + Carnot,</li> + <li class="isub1"> + On limiting ratios, <b><a + href="#Block_1908">1908</a></b>.</li> + <li class="isub1"> + On the infinitesimal method, <b><a + href="#Block_1907">1907</a></b>.</li> + <li class="indx"> + Carson, Value of geometrical training, <b><a + href="#Block_1841">1841</a></b>.</li> + <li class="indx"> + Cartesian method, <a + href="#Block_1889">1889</a>, <a + href="#Block_1890">1890</a>.</li> + <li class="indx"> + Carus,</li> + <li class="isub1"> + Estimate of m., <b><a + href="#Block_326">326</a></b>.</li> + <li class="isub1"> + M. reveals supernatural God, <b><a + href="#Block_460">460</a></b>.</li> + <li class="isub1"> + Number and nature, <b><a + href="#Block_1603">1603</a></b>.</li> + <li class="isub1"> + Zero and infinity, <b><a + href="#Block_1948">1948</a></b>.</li> + <li class="isub1"> + Non-euclidean geometry, <b><a + href="#Block_2016">2016</a></b>.</li> + <li class="indx"> + Cathedral, “Petrified mathematics,” <a + href="#Block_1110">1110</a>.</li> + <li class="indx"> + Causation in m., <a + href="#Block_251">251</a>, <a + href="#Block_254">254</a>.</li> + <li class="indx"> + Cayley,</li> + <li class="isub1"> + Advantage of modern geometry over ancient, <b><a + href="#Block_711">711</a></b>.</li> + <li class="isub1"> + On the imaginary, <b><a + href="#Block_722">722</a></b>.</li> + <li class="isub1"> + Sylvester on, <a + href="#Block_930">930</a>.</li> + <li class="isub1"> + Noether on, <a + href="#Block_931">931</a>.</li> + <li class="isub1"> + His style, <a + href="#Block_932">932</a>.</li> + <li class="isub1"> + Forsyth on, <a + href="#Block_932">932-934</a>.</li> + <li class="isub1"> + His method, <a + href="#Block_933">933</a>.</li> + <li class="isub1"> + Compared with Euler, <a + href="#Block_934">934</a>.</li> + <li class="isub1"> + Hermite on, <a + href="#Block_935">935</a>.</li> + <li class="isub1"> + His view of Euclid, <a + href="#Block_936">936</a>.</li> + <li class="isub1"> + His estimate of quaternions, <a + href="#Block_937">937</a>.</li> + <li class="isub1"> + M. and philosophy, <b><a + href="#Block_1420">1420</a></b>.</li> + <li class="indx"> + Certainty of m., <a + href="#Block_222">222</a>, <a + href="#Block_1440">1440-1442</a>, <a + href="#Block_1628">1628</a>, <a + href="#Block_1863">1863</a>.</li> + <li class="indx"> + Chamisso, Pythagorean theorem, <b><a + href="#Block_1856">1856</a></b>.</li> + <li class="indx"> + Chancellor, M. develops observation, imagination and + reason, <b><a + href="#Block_433">433</a></b>.</li> + <li class="indx"> + Chapman, Different aspects of m., <b><a + href="#Block_265">265</a></b>.</li> + <li class="indx"> + Characteristics of m., <a + href="#Block_225">225</a>, <a + href="#Block_229">229</a>, <a + href="#Block_247">247</a>, <a + href="#Block_263">263</a>. + +<span class="pagenum"> + <a name="Page_389" + id="Page_389">389</a></span></li> + + <li class="indx"> + Characteristics of modern m., <a + href="#Block_720">720</a>, <a + href="#Block_724">724-729</a>.</li> + <li class="indx"> + Charm in m., <a + href="#Block_1115">1115</a>, <a + href="#Block_1640">1640</a>, <a + href="#Block_1848">1848</a>.</li> + <li class="indx"> + Chasles, Advantage of modern geometry over ancient, <b><a + href="#Block_712">712</a></b>.</li> + <li class="indx"> + Checks in m., <a + href="#Block_230">230</a>.</li> + <li class="indx"> + Chemistry and m., <a + href="#Block_1520">1520</a>, <a + href="#Block_1560">1560</a>, <a + href="#Block_1561">1561</a>, <a + href="#Block_1750">1750</a>.</li> + <li class="indx"> + Chess, M. like, <a + href="#Block_840">840</a>.</li> + <li class="indx"> + Chrystal,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_113">113</a></b>.</li> + <li class="isub1"> + Definition of quantity, <b><a + href="#Block_115">115</a></b>.</li> + <li class="isub1"> + On problem solving, <b><a + href="#Block_531">531</a></b>.</li> + <li class="isub1"> + On modern text-books, <b><a + href="#Block_533">533</a></b>.</li> + <li class="isub1"> + How to read m., <b><a + href="#Block_607">607</a></b>.</li> + <li class="isub1"> + His algebra, <a + href="#Block_635">635</a>.</li> + <li class="isub1"> + On Bernoulli’s numbers, <b><a + href="#Block_921">921</a></b>.</li> + <li class="isub1"> + On math. versus logical abstractness, <b><a + href="#Block_1304">1304</a></b>.</li> + <li class="isub1"> + Rules of algebra, <b><a + href="#Block_1710">1710</a></b>.</li> + <li class="isub1"> + On universal arithmetic, <b><a + href="#Block_1717">1717</a></b>.</li> + <li class="isub1"> + On Horner’s method, <b><a + href="#Block_1744">1744</a></b>.</li> + <li class="isub1"> + On probabilities, <b><a + href="#Block_1967">1967</a></b>.</li> + <li class="indx"> + Cicero, Decadence of geometry among Romans, <b><a + href="#Block_1807">1807</a></b>.</li> + <li class="indx"> + Circle, Properties of, <a + href="#Block_1852">1852</a>, <a + href="#Block_1857">1857</a>.</li> + <li class="indx"> + Circle-squarers, <a + href="#Block_2108">2108</a>, <a + href="#Block_2109">2109</a>.</li> + <li class="indx"> + Clarke, Descriptive geometry, <b><a + href="#Block_1882">1882</a></b>.</li> + <li class="indx"> + Classic problems, Hilbert on, <a + href="#Block_627">627</a>.</li> + <li class="indx"> + Clebsch, On math. research, <b><a + href="#Block_644">644</a></b>.</li> + <li class="indx"> + Clifford,</li> + <li class="isub1"> + On direct usefulness of math. results, <b><a + href="#Block_652">652</a></b>.</li> + <li class="isub1"> + Correspondence the central idea of modern m., <b><a + href="#Block_726">726</a></b>.</li> + <li class="isub1"> + His vision, <a + href="#Block_938">938</a>.</li> + <li class="isub1"> + His method, <a + href="#Block_939">939</a>.</li> + <li class="isub1"> + His knowledge of languages, <a + href="#Block_940">940</a>.</li> + <li class="isub1"> + His physical strength, <a + href="#Block_941">941</a>.</li> + <li class="isub1"> + On Helmholtz, <b><a + href="#Block_979">979</a></b>.</li> + <li class="isub1"> + On m. and mineralogy, <b><a + href="#Block_1558">1558</a></b>.</li> + <li class="isub1"> + On algebra and good English, <b><a + href="#Block_1712">1712</a></b>.</li> + <li class="isub1"> + Euclid the encouragement and guide of scientific + thought, <b><a + href="#Block_1820">1820</a></b>.</li> + <li class="isub1"> + Euclid the inspiration and aspiration of scientific + thought, <b><a + href="#Block_1821">1821</a></b>.</li> + <li class="isub1"> + On geometry for girls, <b><a + href="#Block_1842">1842</a></b>.</li> + <li class="isub1"> + On Euclid’s axioms, <b><a + href="#Block_2015">2015</a></b>.</li> + <li class="isub1"> + On non-Euclidean geometry, <b><a + href="#Block_2022">2022</a></b></li> + <li class="indx"> + Colburn, <a + href="#Block_967">967</a>.</li> + <li class="indx"> + Coleridge,</li> + <li class="isub1"> + On problems in m., <b><a + href="#Block_534">534</a></b>.</li> + <li class="isub1"> + Proposition, gentle maid, <b><a + href="#Block_1419">1419</a></b>.</li> + <li class="isub1"> + M. the quintessence of truth, <b><em><a + href="#Block_2119">2019</a></em></b>.</li> + <li class="indx"> + Colton, On the effect of math. training, <b><a + href="#Block_417">417</a></b>.</li> + <li class="indx"> + Commensurable numbers, <a + href="#Block_1966">1966</a>.</li> + <li class="indx"> + Commerce and m., <a + href="#Block_1571">1571</a>.</li> + <li class="indx"> + Committee of Ten,</li> + <li class="isub1"> + On figures in geometry, <b><a + href="#Block_524">524</a></b>.</li> + <li class="isub1"> + On projective geometry, <b><a + href="#Block_1876">1876</a></b>.</li> + <li class="indx"> + Common sense, M. the etherealization of, <a + href="#Block_312">312</a>.</li> + <li class="indx"> + Computation,</li> + <li class="isub1"> + Not m., <a + href="#Block_515">515</a>.</li> + <li class="isub1"> + And m., <a + href="#Block_810">810</a>.</li> + <li class="isub1"> + Not concerned with significance of numbers, <a + href="#Block_1641">1641</a>.</li> + <li class="indx"> + Comte,</li> + <li class="isub1"> + On the object of m., <b><a + href="#Block_103">103</a></b>.</li> + <li class="isub1"> + On the business of concrete m., <b><a + href="#Block_104">104</a></b>.</li> + <li class="isub1"> + M. the indispensable basis of all education, <b><a + href="#Block_334">334</a></b>.</li> + <li class="isub1"> + Mill on, <a + href="#Block_942">942</a>.</li> + <li class="isub1"> + Hamilton on, <a + href="#Block_943">943</a>.</li> + <li class="isub1"> + M. and logic, <b><a + href="#Block_1308">1308</a></b>, <b><a + href="#Block_1314">1314</a></b>, <b><a + href="#Block_1325">1325</a></b>.</li> + <li class="isub1"> + On Kant’s view of m., <b><a + href="#Block_1437">1437</a></b>.</li> + <li class="isub1"> + Estimate of m., <b><a + href="#Block_1504">1504</a></b>.</li> + <li class="isub1"> + M. essential to scientific education, <b><a + href="#Block_1505">1505</a></b>.</li> + <li class="isub1"> + M. and natural philosophy, <b><a + href="#Block_1506">1506</a></b>.</li> + <li class="isub1"> + M. and physics, <b><a + href="#Block_1535">1535</a></b>, <b><a + href="#Block_1551">1551</a></b>.</li> + <li class="isub1"> + M. and science, <b><a + href="#Block_1536">1536</a></b>.</li> + <li class="isub1"> + M. and biology, <b><a + href="#Block_1578">1578</a></b>, <b><a + href="#Block_1580">1580</a></b>, <b><a + href="#Block_1581">1581</a></b>.</li> + <li class="isub1"> + M. and social science, <b><a + href="#Block_1587">1587</a></b>.</li> + <li class="isub1"> + Every inquiry reducible to a question of number, <b><a + href="#Block_1602">1602</a></b>.</li> + <li class="isub1"> + Definition of algebra and arithmetic, <b><a + href="#Block_1714">1714</a></b>.</li> + <li class="isub1"> + Geometry a natural science, <b><a + href="#Block_1813">1813</a></b>.</li> + <li class="isub1"> + Ancient and modern methods, <b><a + href="#Block_1875">1875</a></b>.</li> + <li class="isub1"> + On the graphic method, <b><a + href="#Block_1881">1881</a></b>.</li> + <li class="isub1"> + On descriptive geometry, <b><a + href="#Block_1883">1883</a></b>.</li> + <li class="isub1"> + Mill’s estimate of, <a + href="#Block_1903">1903</a>. + +<span class="pagenum"> + <a name="Page_390" + id="Page_390">390</a></span></li> + + <li class="indx"> + Congreve, <b><a + href="#Block_2143">2143</a></b>.</li> + <li class="indx"> + Congruence, Symbol of, <a + href="#Block_1646">1646</a>.</li> + <li class="indx"> + Conic sections, <a + href="#Block_658">658</a>, <a + href="#Block_660">660</a>, <a + href="#Block_1541">1541</a>, <a + href="#Block_1542">1542</a>.</li> + <li class="indx"> + Conjecture, M. free from, <a + href="#Block_234">234</a>.</li> + <li class="indx"> + Contingent truths, <a + href="#Block_1966">1966</a>.</li> + <li class="indx"> + Controversies in m., <a + href="#Block_215">215</a>, <a + href="#Block_243">243</a>, <a + href="#Block_1859">1859</a>.</li> + <li class="indx"> + Correlation in m., <a + href="#Block_525">525-527</a>, <a + href="#Block_1707">1707</a>, <a + href="#Block_1710">1710</a>.</li> + <li class="indx"> + Correspondence, Concept of, <a + href="#Block_725">725</a>, <a + href="#Block_726">726</a>.</li> + <li class="indx"> + Coulomb, <a + href="#Block_1516">1516</a>.</li> + <li class="indx"> + Counting, Every problem can be solved by, <a + href="#Block_1601">1601</a>.</li> + <li class="indx"> + Cournot,</li> + <li class="isub1"> + On the object of m., <b><a + href="#Block_268">268</a></b>.</li> + <li class="isub1"> + On algebraic notation, <b><a + href="#Block_1213">1213</a></b>.</li> + <li class="isub1"> + Advantage of math, notation, <b><a + href="#Block_1220">1220</a></b>.</li> + <li class="indx"> + Craig, On the origin of a new science, <b><a + href="#Block_646">646</a></b>.</li> + <li class="indx"> + Credulity, M. frees mind from, <a + href="#Block_450">450</a>.</li> + <li class="indx"> + Cremona, On English text-books, <b><a + href="#Block_609">609</a></b>.</li> + <li class="indx"> + Crofton,</li> + <li class="isub1"> + On value of probabilities, <b><a + href="#Block_1590">1590</a></b>.</li> + <li class="isub1"> + On probabilities, <b><a + href="#Block_1952">1952</a></b>, <b><a + href="#Block_1970">1970</a></b>, <b><a + href="#Block_1972">1972</a></b>.</li> + <li class="indx"> + Cromwell, On m. and public service, <b><a + href="#Block_328">328</a></b>.</li> + <li class="indx"> + Curiosities, Chapter <a + href="#CHAPTER_XXI">XXI</a>.</li> + <li class="indx"> + Curtius, M. and philosophy, <b><a + href="#Block_1409">1409</a></b>.</li> + <li class="indx"> + Curve, Definition of, <a + href="#Block_1927">1927</a>.</li> + <li class="indx"> + Cyclometers, Notions of, <a + href="#Block_2108">2108</a>.</li> + <li class="indx"> + Cyclotomy depends on number theory, <a + href="#Block_1647">1647</a>.</li> + + <li class="ifrst"> + D’Alembert,</li> + <li class="isub1"> + On rigor in m., <b><a + href="#Block_536">536</a></b>.</li> + <li class="isub1"> + Geometry as logic, <b><a + href="#Block_1311">1311</a></b>.</li> + <li class="isub1"> + Algebra is generous, <b><a + href="#Block_1702">1702</a></b>.</li> + <li class="isub1"> + Geometrical versus physical truths, <b><a + href="#Block_1809">1809</a></b>.</li> + <li class="isub1"> + Standards in m., <b><a + href="#Block_1851">1851</a></b>.</li> + <li class="indx"> + Dante, <b><a + href="#Block_1858">1858</a></b>, <b><a + href="#Block_2117">2117</a></b>.</li> + <li class="indx"> + Darmstaetter prize, <a + href="#Block_2129">2129</a>.</li> + <li class="indx"> + Davis,</li> + <li class="isub1"> + On Sylvester’s method, <b><a + href="#Block_1035">1035</a></b>.</li> + <li class="isub1"> + M. and science, <b><a + href="#Block_1510">1510</a></b>.</li> + <li class="isub1"> + On probability, <b><a + href="#Block_1968">1968</a></b>.</li> + <li class="indx"> + Decimal fractions, <a + href="#Block_1217">1217</a>, <a + href="#Block_1614">1614</a>.</li> + <li class="indx"> + Decker, <b><a + href="#Block_2142">2142</a></b>.</li> + <li class="indx"> + Dedekind, Zeno’s Problem, <a + href="#Block_1938">1938</a>.</li> + <li class="indx"> + Deduction,</li> + <li class="isub1"> + Why necessary, <a + href="#Block_219">219</a>.</li> + <li class="isub1"> + M. based on, <a + href="#Block_224">224</a>.</li> + <li class="isub1"> + And Intuition, <a + href="#Block_1413">1413</a>.</li> + <li class="indx"> + Dee, On the nature of m., <b><a + href="#Block_261">261</a></b>.</li> + <li class="indx"> + Definitions of m., Chapter <a + href="#CHAPTER_I">I</a>.</li> + <li class="isub1"> + Also <a + href="#Block_2005">2005</a>.</li> + <li class="indx"> + Democritus, <a + href="#Block_321">321</a>.</li> + <li class="indx"> + Demoivre, His death, <a + href="#Block_944">944</a>.</li> + <li class="indx"> + Demonstrations,</li> + <li class="isub1"> + Locke on, <a + href="#Block_236">236</a>.</li> + <li class="isub1"> + Outside of m., <a + href="#Block_1312">1312</a>.</li> + <li class="isub1"> + In m., <a + href="#Block_1423">1423</a>.</li> + <li class="indx"> + De Morgan,</li> + <li class="isub1"> + Imagination in m., <b><a + href="#Block_258">258</a></b>.</li> + <li class="isub1"> + M. as an exercise in reasoning, <b><a + href="#Block_430">430</a></b>.</li> + <li class="isub1"> + On difficulties in m., <b><a + href="#Block_521">521</a></b>.</li> + <li class="isub1"> + On correlation in m., <b><a + href="#Block_525">525</a></b>.</li> + <li class="isub1"> + On extempore lectures, <b><a + href="#Block_540">540</a></b>.</li> + <li class="isub1"> + On reading algebraic works, <b><a + href="#Block_601">601</a></b>.</li> + <li class="isub1"> + On numerical calculations, <b><a + href="#Block_602">602</a></b>.</li> + <li class="isub1"> + On practice problems, <b><a + href="#Block_603">603</a></b>.</li> + <li class="isub1"> + On the value of the history of m., <b><a + href="#Block_615">615</a></b>, <b><a + href="#Block_616">616</a></b>.</li> + <li class="isub1"> + On math’ns., <b><a + href="#Block_812">812</a></b>.</li> + <li class="isub1"> + On Bacon’s knowledge of m., <b><a + href="#Block_918">918</a></b>.</li> + <li class="isub1"> + And the actuary, <a + href="#Block_945">945</a>.</li> + <li class="isub1"> + On life tables, <b><a + href="#Block_946">946</a></b>.</li> + <li class="isub1"> + Anagrams’ on his name, <b><a + href="#Block_947">947</a></b>.</li> + <li class="isub1"> + On translations of Euclid, <b><a + href="#Block_953">953</a></b>.</li> + <li class="isub1"> + Euclid’s elements compared with Newton’s Principia, <b><a + href="#Block_954">954</a></b>.</li> + <li class="isub1"> + Euler and Diderot, <b><a + href="#Block_966">966</a></b>.</li> + <li class="isub1"> + Lagrange and the parallel axiom, <b><a + href="#Block_984">984</a></b>.</li> + <li class="isub1"> + Anagram on Macaulay’s name, <b><a + href="#Block_996">996</a></b>.</li> + <li class="isub1"> + Anagrams on Newton’s name, <b><a + href="#Block_1028">1028</a></b>.</li> + <li class="isub1"> + On math, notation, <b><a + href="#Block_1216">1216</a></b>.</li> + <li class="isub1"> + Antagonism of m. and logic, <b><a + href="#Block_1315">1315</a></b>.</li> + <li class="isub1"> + On German metaphysics, <b><a + href="#Block_1416">1416</a></b>.</li> + <li class="isub1"> + On m. and science, <b><a + href="#Block_1537">1537</a></b>.</li> + <li class="isub1"> + On m. and physics, <b><a + href="#Block_1538">1538</a></b>.</li> + <li class="isub1"> + On the advantages of algebra, <b><a + href="#Block_1701">1701</a></b>.</li> + <li class="isub1"> + On algebra as an art, <b><a + href="#Block_1711">1711</a></b>. + +<span class="pagenum"> + <a name="Page_391" + id="Page_391">391</a></span></li> + + <li class="isub1"> + On double algebra and quaternions, <b><a + href="#Block_1720">1720</a></b>.</li> + <li class="isub1"> + On assumptions in geometry, <b><a + href="#Block_1812">1812</a></b>.</li> + <li class="isub1"> + On Euclid in schools, <b><a + href="#Block_1819">1819</a></b>.</li> + <li class="isub1"> + Euclid not faultless, <b><a + href="#Block_1823">1823</a></b>.</li> + <li class="isub1"> + On Euclid’s rigor, <b><a + href="#Block_1831">1831</a></b>.</li> + <li class="isub1"> + Geometry before algebra, <b><a + href="#Block_1872">1872</a></b>.</li> + <li class="isub1"> + On trigonometry, <b><a + href="#Block_1885">1885</a></b>.</li> + <li class="isub1"> + On the calculus in elementary instruction, <b><a + href="#Block_1916">1916</a>.</b></li> + <li class="isub1"> + On integration, <b><a + href="#Block_1919">1919</a></b>.</li> + <li class="isub1"> + On divergent series, <b><a + href="#Block_1935">1935</a></b>, <b><a + href="#Block_1936">1936</a></b>.</li> + <li class="isub1"> + Ad infinitum, <b><a + href="#Block_1949">1949</a></b>.</li> + <li class="isub1"> + On the fourth dimension, <b><a + href="#Block_2032">2032</a></b>.</li> + <li class="isub1"> + Pseudomath and graphomath, <b><a + href="#Block_2101">2101</a></b>.</li> + <li class="isub1"> + On proof, <b><a + href="#Block_2102">2102</a></b>.</li> + <li class="isub1"> + On paradoxers, <b><a + href="#Block_2105">2105</a></b>.</li> + <li class="isub1"> + Budget of paradoxes, <b><a + href="#Block_2106">2106</a></b>.</li> + <li class="isub1"> + On D’Israeli’s six follies of science, <b><a + href="#Block_2107">2107</a></b>.</li> + <li class="isub1"> + On notions of cyclometers, <b><a + href="#Block_2108">2108</a></b>.</li> + <li class="isub1"> + On St. Vincent, <b><a + href="#Block_2109">2109</a></b>.</li> + <li class="isub1"> + Where Euclid failed, <b><a + href="#Block_2114">2114</a></b>.</li> + <li class="isub1"> + On the number of the beast, <b><a + href="#Block_2151">2151</a></b>.</li> + <li class="indx"> + Descartes,</li> + <li class="isub1"> + On the use of the term m., <b><a + href="#Block_102">102</a></b>.</li> + <li class="isub1"> + On intuition and deduction, <b><a + href="#Block_219">219</a></b>, <b><a + href="#Block_1413">1413</a></b>.</li> + <li class="isub1"> + Math’ns alone arrive at proofs, <b><a + href="#Block_817">817</a></b>.</li> + <li class="isub1"> + The most completely math. type of mind, <a + href="#Block_948">948</a>.</li> + <li class="isub1"> + Hankel on, <a + href="#Block_949">949</a>.</li> + <li class="isub1"> + Mill on, <a + href="#Block_950">950</a>.</li> + <li class="isub1"> + Hankel on, <a + href="#Block_1404">1404</a>.</li> + <li class="isub1"> + On m. and philosophy, + <b><a href="#Block_1425">1425</a></b>, <b><a + href="#Block_1434">1434</a></b>.</li> + <li class="isub1"> + Estimate of m., <b><a + href="#Block_1426">1426</a></b>.</li> + <li class="isub1"> + Unpopularity of, <b><a + href="#Block_1501">1501</a></b>.</li> + <li class="isub1"> + On the certainty of m., <b><a + href="#Block_1628">1628</a></b>.</li> + <li class="isub1"> + On the method of the ancients, <b><a + href="#Block_1874">1874</a></b>.</li> + <li class="isub1"> + On probable truth, <b><a + href="#Block_1964">1964</a></b>.</li> + <li class="isub1"> + Descriptive geometry, <a + href="#Block_1882">1882</a>, <a + href="#Block_1883">1883</a>.</li> + <li class="indx"> + Dessoir, M. and medicine, <b><a + href="#Block_1585">1585</a></b>.</li> + <li class="indx"> + Determinants, <a + href="#Block_1740">1740</a>, <a + href="#Block_1741">1741</a>.</li> + <li class="indx"> + Diderot and Euler, <a + href="#Block_966">966</a>.</li> + <li class="indx"> + Differential calculus,</li> + <li class="isub1"> + Chapter <a + href="#CHAPTER_XIX">XIX</a>.</li> + <li class="isub1"> + And scientific physics, <a + href="#Block_1549">1549</a>.</li> + <li class="indx"> + Differential equations, <a + href="#Block_1549">1549-1552</a>, <a + href="#Block_1924">1924</a>, <a + href="#Block_1926">1926</a>.</li> + <li class="indx"> + Difficulties in m., <a + href="#Block_240">240</a>, <a + href="#Block_521">521</a>, <a + href="#Block_605">605-607</a>, <a + href="#Block_634">634</a>, <a + href="#Block_734">734</a>, <a + href="#Block_735">735</a>.</li> + <li class="indx"> + Dillmann,</li> + <li class="isub1"> + M. a royal science, <b><a + href="#Block_204">204</a></b>.</li> + <li class="isub1"> + On m. as a high school subject, <b><a + href="#Block_401">401</a></b>.</li> + <li class="isub1"> + Ancient and modern geometry compared, <b><a + href="#Block_715">715</a></b>.</li> + <li class="isub1"> + On ignorance of, <b><a + href="#Block_807">807</a></b>.</li> + <li class="isub1"> + On m. as a language, <b><a + href="#Block_1204">1204</a></b>.</li> + <li class="isub1"> + Number regulates all things, <b><em><a + href="#Block_1605">1505</a></em></b>.</li> + <li class="indx"> + Dirichlet,</li> + <li class="isub1"> + On math, discovery, <b><a + href="#Block_625">625</a></b>.</li> + <li class="isub1"> + As a student of Gauss, <a + href="#Block_977">977</a>.</li> + <li class="indx"> + Discovery in m., <a + href="#Block_617">617-622</a>, <a + href="#Block_625">625</a>.</li> + <li class="indx"> + <em>D</em>-ism versus <em>dot</em>-age, <a + href="#Block_923">923</a>.</li> + <li class="indx"> + Disquisitiones Arithmeticae, <a + href="#Block_975">975</a>, <a + href="#Block_977">977</a>, <a + href="#Block_1637">1637</a>, <a + href="#Block_1638">1638</a>.</li> + <li class="indx"> + D’Israeli, <a + href="#Block_2007">2007</a>.</li> + <li class="indx"> + Divergent series, <a + href="#Block_1935">1935-1937</a>.</li> + <li class="indx"> + “Divide et impera,” <a + href="#Block_631">631</a>.</li> + <li class="indx"> + Divine character of m., <a + href="#Block_325">325</a>, <a + href="#Block_329">329</a>.</li> + <li class="indx"> + “Divinez avant de demontrer,” <a + href="#Block_630">630</a>.</li> + <li class="indx"> + Division of labor in m., <a + href="#Block_631">631</a>, <a + href="#Block_632">632</a>.</li> + <li class="indx"> + Dodgson,</li> + <li class="isub1"> + On the charm of, <b><a + href="#Block_302">302</a></b>.</li> + <li class="isub1"> + Pythagorean theorem, <b><a + href="#Block_1854">1854</a></b>.</li> + <li class="isub1"> + Ignes fatui in m., <b><a + href="#Block_2103">2103</a></b>.</li> + <li class="indx"> + Dolbear, On experiment in math. research, <b><a + href="#Block_613">613</a></b>.</li> + <li class="indx"> + Domus Lescinia, Anagram on, <a + href="#Block_2155">2155</a>.</li> + <li class="indx"> + Donne, <b><a + href="#Block_1816">1816</a></b>.</li> + <li class="indx"> + <em>Dot</em>-age versus <em>d</em>-ism, <a + href="#Block_923">923</a>.</li> + <li class="indx"> + Durfee, On Sylvester’s forgetfulness, <b><a + href="#Block_1038">1038</a></b>.</li> + <li class="indx"> + Dutton, On the ethical value of m., <b><a + href="#Block_446">446</a></b>.</li> + <li class="ifrst"> + “Eadem mutata resurgo.” <a + href="#Block_920">920</a>, <a + href="#Block_922">922</a>.</li> + <li class="indx"> + Echols, On the ethical value of m., <b><a + href="#Block_455">455</a></b>.</li> + <li class="indx"> + Economics and m., <a + href="#Block_1593">1593</a>, <a + href="#Block_1594">1594</a>. + +<span class="pagenum"> + <a name="Page_392" + id="Page_392">392</a></span></li> + + <li class="indx"> + Edinburgh Review, M. and astronomy, <b><a + href="#Block_1565">1565</a></b>, <b><a + href="#Block_1566">1566</a></b>.</li> + <li class="indx"> + Education,</li> + <li class="isub1"> + Place of m. in, <a + href="#Block_334">334</a>, <a + href="#Block_408">408</a>.</li> + <li class="isub1"> + Study of arithmetic better than rhetoric, <a + href="#Block_408">408</a>.</li> + <li class="isub1"> + M. as an instrument in, <a + href="#Block_413">413</a>, <a + href="#Block_414">414</a>.</li> + <li class="isub1"> + M. in primary, <a + href="#Block_431">431</a>.</li> + <li class="isub1"> + M. as a common school subject, <a + href="#Block_432">432</a>.</li> + <li class="isub1"> + Bain on m. in, <a + href="#Block_442">442</a>.</li> + <li class="isub1"> + Calculus in elementary, <a + href="#Block_1916">1916</a>, <a + href="#Block_1917">1917</a>.</li> + <li class="indx"> + Electricity, M. and the theory of, <a + href="#Block_1554">1554</a>.</li> + <li class="indx"> + Elegance in m., <a + href="#Block_640">640</a>, <a + href="#Block_728">728</a>.</li> + <li class="indx"> + Ellis,</li> + <li class="isub1"> + On precocity in m., <b><a + href="#Block_835">835</a></b>.</li> + <li class="isub1"> + On aptitude of Anglo-Danes for m., <b><a + href="#Block_836">836</a></b>.</li> + <li class="isub1"> + On Newton’s genius, <b><a + href="#Block_1014">1014</a></b>.</li> + <li class="indx"> + Emerson,</li> + <li class="isub1"> + On Newton and Laplace, <b><a + href="#Block_1003">1003</a></b>.</li> + <li class="isub1"> + On poetry and m., <b><a + href="#Block_1124">1124</a></b>.</li> + <li class="indx"> + Endowment of math’ns, <a + href="#Block_818">818</a>.</li> + <li class="indx"> + Enthusiasm, <a + href="#Block_801">801</a>.</li> + <li class="indx"> + Equality, Grassmann’s definition of, <b><a + href="#Block_105">105</a></b>.</li> + <li class="indx"> + Equations, <a + href="#Block_104">104</a>, <a + href="#Block_526">526</a>, <a + href="#Block_1891">1891</a>, <a + href="#Block_1892">1892</a>.</li> + <li class="indx"> + Errors, Theory of, <a + href="#Block_1973">1973</a>, <a + href="#Block_1974">1974</a>.</li> + <li class="indx"> + Esthetic element in m., <a + href="#Block_453">453-455</a>, <a + href="#Block_640">640</a>, <a + href="#Block_1102">1102</a>, <a + href="#Block_1105">1105</a>, <a + href="#Block_1852">1852</a>, + <a + href="#Block_1853">1853</a>.</li> + <li class="indx"> + Esthetic tact, <a + href="#Block_622">622</a>.</li> + <li class="indx"> + Esthetic value of m., <a + href="#Block_1848">1848</a>, <a + href="#Block_1850">1850</a>.</li> + <li class="indx"> + Esthetics, Relation of m. to, <a + href="#Block_318">318</a>, <a + href="#Block_319">319</a>, <a + href="#Block_439">439</a>.</li> + <li class="indx"> + Estimates of m., Chapter <a + href="#CHAPTER_III">III</a>.</li> + <li class="isub1"> + See also <a + href="#Block_1317">1317</a>, <a + href="#Block_1324">1324</a>, <a + href="#Block_1325">1325</a>, <a + href="#Block_1427">1427</a>, <a + href="#Block_1504">1504</a>, <a + href="#Block_1508">1508</a>.</li> + <li class="indx"> + Ethical value of m., <a + href="#Block_402">402</a>, <a + href="#Block_438">438</a>, <a + href="#Block_446">446</a>, <a + href="#Block_449">449</a>, <a + href="#Block_455">455-457</a>.</li> + <li class="indx"> + Euclid,</li> + <li class="isub1"> + Bolzano cured by, <a + href="#Block_929">929</a>.</li> + <li class="isub1"> + And Ptolemy, <a + href="#Block_951">951</a>, <a + href="#Block_1878">1878</a>.</li> + <li class="isub1"> + And the student, <a + href="#Block_952">952</a>.</li> + <li class="indx"> + Euclid’s Elements,</li> + <li class="isub1"> + Translations of, <a + href="#Block_953">953</a>.</li> + <li class="isub1"> + Compared with the Principia, <a + href="#Block_954">954</a>.</li> + <li class="isub1"> + Greatness of, <a + href="#Block_955">955</a>.</li> + <li class="isub1"> + Greatest of human productions, <a + href="#Block_1817">1817</a>.</li> + <li class="isub1"> + Performance in, <a + href="#Block_1818">1818</a>.</li> + <li class="isub1"> + In English schools, <a + href="#Block_1819">1819</a>.</li> + <li class="isub1"> + Encouragement and guide, <a + href="#Block_1820">1820</a>.</li> + <li class="isub1"> + Inspiration and aspiration, <a + href="#Block_1821">1821</a>.</li> + <li class="isub1"> + The only perfect model, <a + href="#Block_1822">1822</a>.</li> + <li class="isub1"> + Not altogether faultless, <a + href="#Block_1823">1823</a>.</li> + <li class="isub1"> + Only a small part of m., <a + href="#Block_1824">1824</a>.</li> + <li class="isub1"> + Not fitted for boys, <a + href="#Block_1825">1825</a>.</li> + <li class="isub1"> + Early study of, <a + href="#Block_1826">1826</a>.</li> + <li class="isub1"> + Newton and, <a + href="#Block_1827">1827</a>.</li> + <li class="isub1"> + Its place, <a + href="#Block_1828">1828</a>.</li> + <li class="isub1"> + Unexceptional in rigor, <a + href="#Block_1829">1829</a>.</li> + <li class="isub1"> + Origin of, <a + href="#Block_1831">1831</a>.</li> + <li class="isub1"> + Doctrine of proportion, <a + href="#Block_1834">1834</a>.</li> + <li class="isub1"> + Definition of proportion, <a + href="#Block_1835">1835</a>.</li> + <li class="isub1"> + Steps in demonstration, <a + href="#Block_1839">1839</a>.</li> + <li class="isub1"> + Parallel axiom, <a + href="#Block_2007">2007</a>.</li> + <li class="indx"> + Euclidean geometry, <a + href="#Block_711">711</a>, <a + href="#Block_713">713</a>, <a + href="#Block_715">715</a>.</li> + <li class="indx"> + Eudoxus, <a + href="#Block_904">904</a>.</li> + <li class="indx"> + Euler,</li> + <li class="isub1"> + the myriad-minded, <a + href="#Block_255">255</a>.</li> + <li class="isub1"> + Pencil outruns intelligence, <a + href="#Block_626">626</a>.</li> + <li class="isub1"> + On theoretical investigations, <a + href="#Block_657">657</a>.</li> + <li class="isub1"> + Merit of his work, <a + href="#Block_956">956</a>.</li> + <li class="isub1"> + The creator of modern math. thought, <a + href="#Block_957">957</a>.</li> + <li class="isub1"> + His general knowledge, <a + href="#Block_958">958</a>.</li> + <li class="isub1"> + His knowledge of the Aeneid, <a + href="#Block_959">959</a>.</li> + <li class="isub1"> + Extent of his work, <a + href="#Block_960">960</a>.</li> + <li class="isub1"> + “Analysis incarnate,” <a + href="#Block_961">961</a>.</li> + <li class="isub1"> + As a computer, <a + href="#Block_962">962</a>, <a + href="#Block_963">963</a>.</li> + <li class="isub1"> + His math. power, <a + href="#Block_964">964</a>.</li> + <li class="isub1"> + His <i lang="la" xml:lang="la">Tentamen novae theorae + musicae</i>, <a + href="#Block_965">965</a>.</li> + <li class="isub1"> + And Diderot, <a + href="#Block_966">966</a>.</li> + <li class="isub1"> + Error in Fermat’s law of prime numbers, <a + href="#Block_967">967</a>.</li> + <li class="indx"> + Eureka, <a + href="#Block_911">911</a>, <a + href="#Block_917">917</a>.</li> + <li class="indx"> + Euripedes, <a + href="#Block_1568">1568</a>.</li> + <li class="indx"> + Everett,</li> + <li class="isub1"> + Estimate of m., <b><a + href="#Block_325">325</a></b>.</li> + <li class="isub1"> + Value of math. training, <b><a + href="#Block_443">443</a></b>.</li> + <li class="isub1"> + Theoretical investigations, <b><a + href="#Block_656">656</a></b>.</li> + <li class="isub1"> + Arithmetic a master-key, <b><a + href="#Block_1571">1571</a></b>.</li> + <li class="isub1"> + On m. and law, <b><a + href="#Block_1598">1598</a></b>.</li> + <li class="indx"> + Exactness, See precision.</li> + <li class="indx"> + Examinations, <a + href="#Block_407">407</a>.</li> + <li class="indx"> + Examples, <a + href="#Block_422">422</a>. + +<span class="pagenum"> + <a name="Page_393" + id="Page_393">393</a></span></li> + + <li class="indx"> + Experiment in m., <a + href="#Block_612">612</a>, <a + href="#Block_613">613</a>, <a + href="#Block_1530">1530</a>, <a + href="#Block_1531">1531</a>.</li> + <li class="indx"> + Extent of m., <a + href="#Block_737">737</a>, <a + href="#Block_738">738</a>.</li> + <li class="ifrst"> + Fairbairn, <a + href="#Block_528">528</a>.</li> + <li class="indx"> + Fallacies, <a + href="#Block_610">610</a>.</li> + <li class="indx"> + Faraday, M. and physics, <a + href="#Block_1554">1554</a>.</li> + <li class="indx"> + Fermat, <a + href="#Block_255">255</a>, <a + href="#Block_967">967</a>, <a + href="#Block_1902">1902</a>.</li> + <li class="indx"> + Fermat’s theorem, <a + href="#Block_2129">2129</a>.</li> + <li class="indx"> + Figures,</li> + <li class="isub1"> + Committee of Ten on, <a + href="#Block_524">524</a>.</li> + <li class="isub1"> + Democritus view of, <a + href="#Block_321">321</a>.</li> + <li class="isub1"> + Battalions of, <a + href="#Block_1631">1631</a>.</li> + <li class="indx"> + Fine,</li> + <li class="isub1"> + Definition of number, <b><a + href="#Block_1610">1610</a></b>.</li> + <li class="isub1"> + On the imaginary, <b><a + href="#Block_1732">1732</a></b>.</li> + <li class="indx"> + Fine Art, M. as a, Chapter <a + href="#CHAPTER_XI">XI</a>.</li> + <li class="indx"> + Fisher, M. and economics, <b><a + href="#Block_1594">1594</a></b>.</li> + <li class="indx"> + Fiske,</li> + <li class="isub1"> + Imagination in m., <b><a + href="#Block_256">256</a></b>.</li> + <li class="isub1"> + Advantage of m. as logic, <b><a + href="#Block_1324">1324</a></b>.</li> + <li class="indx"> + Fitch,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_125">125</a></b>.</li> + <li class="isub1"> + M. in education, <b><a + href="#Block_429">429</a></b>.</li> + <li class="isub1"> + Purpose of teaching arithmetic, <b><a + href="#Block_1624">1624</a></b>, <b><a + href="#Block_1625">1625</a></b>.</li> + <li class="indx"> + Fizi, Origin of the Liliwati, <b><a + href="#Block_995">995</a></b>.</li> + <li class="indx"> + Flamsteed, Anagram on, <b><a + href="#Block_968">968</a></b>.</li> + <li class="indx"> + Fluxions, <a + href="#Block_1911">1911</a>, <a + href="#Block_1915">1915</a>, <a + href="#Block_1942">1942-1944</a>.</li> + <li class="indx"> + Fontenelle, Bernoulli’s tomb, <b><a + href="#Block_920">920</a></b>.</li> + <li class="indx"> + Formulas, Compared to focus of a lens, <a + href="#Block_1515">1515</a>.</li> + <li class="indx"> + Forsyth,</li> + <li class="isub1"> + On direct usefulness of math. results, <b><a + href="#Block_654">654</a></b>.</li> + <li class="isub1"> + On theoretical investigations, <b><a + href="#Block_664">664</a></b>.</li> + <li class="isub1"> + Progress of m. <b><a + href="#Block_704">704</a></b>.</li> + <li class="isub1"> + On Cayley, <b><a + href="#Block_932">932-934</a></b>.</li> + <li class="isub1"> + On m. and physics, <b><a + href="#Block_1539">1539</a></b>.</li> + <li class="isub1"> + On m. and applications, <b><a + href="#Block_1540">1540</a></b>.</li> + <li class="isub1"> + On invariants, <b><a + href="#Block_1747">1747</a></b>.</li> + <li class="isub1"> + On function theory, <b><a + href="#Block_1754">1754</a></b>, <b><a + href="#Block_1755">1755</a></b>.</li> + <li class="indx"> + Foster,</li> + <li class="isub1"> + On m. and physics, <b><a + href="#Block_1516">1516</a></b>, <b><a + href="#Block_1522">1522</a></b>.</li> + <li class="isub1"> + On experiment in m., <b><a + href="#Block_1531">1531</a></b>.</li> + <li class="indx"> + Foundations of m., <a + href="#Block_717">717</a>.</li> + <li class="indx"> + Four, The number, <a + href="#Block_2147">2147</a>, <a + href="#Block_2148">2148</a>.</li> + <li class="indx"> + Fourier, Math, analysis co-extensive with nature, <b><a + href="#Block_218">218</a></b>.</li> + <li class="isub1"> + On math. research, <b><a + href="#Block_612">612</a></b>.</li> + <li class="isub1"> + Hamilton on, <a + href="#Block_969">969</a>.</li> + <li class="isub1"> + On m. and physics, <b><a + href="#Block_1552">1552</a></b>, <b><a + href="#Block_1553">1553</a></b>.</li> + <li class="isub1"> + On the advantage of the Cartesian method, <b><a + href="#Block_1889">1889</a></b>.</li> + <li class="indx"> + Fourier’s theorem, <a + href="#Block_1928">1928</a>.</li> + <li class="indx"> + Fourth dimension, <a + href="#Block_2032">2032</a>, <a + href="#Block_2039">2039</a>.</li> + <li class="indx"> + Frankland, A., M. and chemistry, <b><a + href="#Block_1560">1560</a></b>.</li> + <li class="indx"> + Frankland, W. B., Motto of Pythagorean brotherhood, <b><a + href="#Block_1833">1833</a></b>.</li> + <li class="isub1"> + The most beautiful truth in geometry, <b><a + href="#Block_1857">1857</a></b>.</li> + <li class="indx"> + Franklin, B.,</li> + <li class="isub1"> + Estimate of m., <b><a + href="#Block_322">322</a></b>.</li> + <li class="isub1"> + On the value of the study of m., <b><a + href="#Block_323">323</a></b>.</li> + <li class="isub1"> + On the excellence of m., <b><a + href="#Block_324">324</a></b>.</li> + <li class="isub1"> + On m. as a logical exercise, <b><a + href="#Block_1303">1303</a></b>.</li> + <li class="indx"> + Franklin, F., On Sylvester’s weakness, <b><a + href="#Block_1033">1033</a></b>.</li> + <li class="indx"> + Frederick the Great, On geometry, <b><a + href="#Block_1860">1860</a></b>.</li> + <li class="indx"> + Freedom in m., <a + href="#Block_205">205-208</a>, <a + href="#Block_805">805</a>.</li> + <li class="indx"> + French m., <a + href="#Block_1210">1210</a>.</li> + <li class="indx"> + Fresnel, <a + href="#Block_662">662</a>.</li> + <li class="indx"> + Frischlinus, <b><a + href="#Block_1801">1801</a></b>.</li> + <li class="indx"> + Froebel, M. a mediator between man and nature, <b><a + href="#Block_262">262</a></b>.</li> + <li class="indx"> + Function theory, <a + href="#Block_709">709</a>, <a + href="#Block_1732">1732</a>, <a + href="#Block_1754">1754</a>, <a + href="#Block_1755">1755</a>.</li> + <li class="indx"> + Functional exponent, <a + href="#Block_1210">1210</a>.</li> + <li class="indx"> + Functionality,</li> + <li class="isub1"> + The central idea of modern m., <a + href="#Block_254">254</a>.</li> + <li class="isub1"> + Correlated to life, <a + href="#Block_272">272</a>.</li> + <li class="indx"> + Functions, <a + href="#Block_1932">1932</a>, <a + href="#Block_1933">1933</a>.</li> + <li class="isub1"> + Concept not used by Sylvester, <a + href="#Block_1034">1034</a>.</li> + <li class="indx"> + Fundamental concepts, Chapter <a + href="#CHAPTER_XX">XX</a>.</li> + <li class="indx"> + Fuss, On Euler’s <i lang="la" xml:lang="la">Tentamen + novae theorae musicae</i>, <b><a + href="#Block_965">965</a></b>.</li> + + <li class="ifrst"> + Galileo, On authority in science, <b><a + href="#Block_1528">1528</a></b>.</li> + <li class="indx"> + Galton, <a + href="#Block_838">838</a>.</li> + <li class="indx"> + Gauss,</li> + <li class="isub1"> + His motto, <a + href="#Block_649">649</a>.</li> + <li class="isub1"> + Mere math’ns, <b><a + href="#Block_820">820</a></b>.</li> + <li class="isub1"> + And Newton compared, <a + href="#Block_827">827</a>.</li> + <li class="isub1"> + His power, <a + href="#Block_964">964</a>.</li> + <li class="isub1"> + His favorite pursuits, <a + href="#Block_970">970</a>.</li> + <li class="isub1"> + The first of theoretical astronomers, <a + href="#Block_971">971</a>.</li> + <li class="isub1"> + The greatest of arithmeticians, <a + href="#Block_971">971</a>. + +<span class="pagenum"> + <a name="Page_394" + id="Page_394">394</a></span></li> + + <li class="isub1"> + The math. giant, <a + href="#Block_972">972</a>.</li> + <li class="isub1"> + Greatness of, <a + href="#Block_973">973</a>.</li> + <li class="isub1"> + Lectures to three students, <b><a + href="#Block_974">974</a></b>.</li> + <li class="isub1"> + His style and method, <a + href="#Block_983">983</a>.</li> + <li class="isub1"> + His estimate of Newton, <a + href="#Block_1029">1029</a>.</li> + <li class="isub1"> + On the advantage of new calculi, <b><a + href="#Block_1215">1215</a></b>.</li> + <li class="isub1"> + M. and experiment, <a + href="#Block_1531">1531</a>.</li> + <li class="isub1"> + His <cite>Disquisitiones Arithmeticae</cite>, <a + href="#Block_1639">1639</a>, <a + href="#Block_1640">1640</a>.</li> + <li class="isub1"> + M. the queen of the sciences, <b><a + href="#Block_1642">1642</a></b>.</li> + <li class="isub1"> + On number theory, <b><a + href="#Block_1644">1644</a></b>.</li> + <li class="isub1"> + On imaginaries, <b><a + href="#Block_1730">1730</a></b>.</li> + <li class="isub1"> + On the notation sin<sub>2</sub>φ, <b><a + href="#Block_1886">1886</a></b>.</li> + <li class="isub1"> + On infinite magnitude, <b><em><a + href="#Block_1951">1950</a></em></b>.</li> + <li class="isub1"> + On non-euclidean geometry, <b><a + href="#Block_2023">2023-2028</a></b>.</li> + <li class="isub1"> + On the nature of space, <a + href="#Block_2034">2034</a>.</li> + <li class="indx"> + Generalization in m., <a + href="#Block_245">245</a>, <a + href="#Block_246">246</a>, <a + href="#Block_252">252</a>, <a + href="#Block_253">253</a>, <a + href="#Block_327">327</a>, <a + href="#Block_728">728</a>.</li> + <li class="indx"> + Genius, <a + href="#Block_819">819</a>.</li> + <li class="indx"> + Geometrical investigations, <a + href="#Block_642">642</a>, <a + href="#Block_643">643</a>.</li> + <li class="indx"> + Geometrical training, Value of, <a + href="#Block_1841">1841</a>, <a + href="#Block_1842">1842</a>, <a + href="#Block_1844">1844-1846</a>.</li> + <li class="indx"> + Geometry,</li> + <li class="isub1"> + Chapter <a + href="#CHAPTER_XVIII">XVIII</a>.</li> + <li class="isub1"> + Bacon’s definition of, <a + href="#Block_106">106</a>.</li> + <li class="isub1"> + Sylvester’s definition of, <a + href="#Block_110">110</a>.</li> + <li class="isub1"> + Value to mankind, <a + href="#Block_332">332</a>, <a + href="#Block_449">449</a>.</li> + <li class="isub1"> + And patriotism, <a + href="#Block_332">332</a>.</li> + <li class="isub1"> + An excellent logic, <a + href="#Block_428">428</a>.</li> + <li class="isub1"> + Plato’s view of, <a + href="#Block_429">429</a>.</li> + <li class="isub1"> + The fountain of all thought, <a + href="#Block_451">451</a>.</li> + <li class="isub1"> + And algebra, <a + href="#Block_525">525-527</a>.</li> + <li class="isub1"> + Lack of concreteness, <a + href="#Block_710">710</a>.</li> + <li class="isub1"> + Advantage of modern over ancient, <a + href="#Block_711">711</a>, <a + href="#Block_712">712</a>.</li> + <li class="isub1"> + And music, <a + href="#Block_965">965</a>.</li> + <li class="isub1"> + And arithmetic, <a + href="#Block_1604">1604</a>.</li> + <li class="isub1"> + Is figured algebra, <a + href="#Block_1706">1706</a>.</li> + <li class="isub1"> + Name inapt, <a + href="#Block_1801">1801</a>.</li> + <li class="isub1"> + And experience, <a + href="#Block_1814">1814</a>.</li> + <li class="isub1"> + Halsted’s definition of, <a + href="#Block_1815">1815</a>.</li> + <li class="isub1"> + And observation, <a + href="#Block_1830">1830</a>.</li> + <li class="isub1"> + Controversy in, <a + href="#Block_1859">1859</a>.</li> + <li class="isub1"> + A mechanical science, <a + href="#Block_1865">1865</a>.</li> + <li class="isub1"> + A natural science, <a + href="#Block_1866">1866</a>.</li> + <li class="isub1"> + Not an experimental science, <a + href="#Block_1867">1867</a>.</li> + <li class="isub1"> + Should come before algebra, <em>1767</em>, <a + href="#Block_1871">1871</a>, <a + href="#Block_1872">1872</a>.</li> + <li class="isub1"> + And analysis, <a + href="#Block_1931">1931</a>.</li> + <li class="indx"> + Germain, Algebra is written geometry, <b><a + href="#Block_1706">1706</a></b>.</li> + <li class="indx"> + Gilman, Enlist a great math’n, <b><a + href="#Block_808">808</a></b>.</li> + <li class="indx"> + Glaisher,</li> + <li class="isub1"> + On the importance of broad training, <b><a + href="#Block_623">623</a></b>.</li> + <li class="isub1"> + On the importance of a well-chosen notation, <b><a + href="#Block_634">634</a></b>.</li> + <li class="isub1"> + On the expansion of the field of m., <b><a + href="#Block_634">634</a></b>.</li> + <li class="isub1"> + On the need of text-books on higher m., <b><a + href="#Block_635">635</a></b>.</li> + <li class="isub1"> + On the perfection of math. productions, <b><a + href="#Block_649">649</a></b>.</li> + <li class="isub1"> + On the invention of logarithms, <b><a + href="#Block_1616">1616</a></b>.</li> + <li class="isub1"> + On the theory of numbers, <b><a + href="#Block_1640">1640</a></b>.</li> + <li class="indx"> + Goethe,</li> + <li class="isub1"> + On the exactness of m., <b><a + href="#Block_228">228</a></b>.</li> + <li class="isub1"> + M. an organ of the higher sense, <b><a + href="#Block_273">273</a></b>.</li> + <li class="isub1"> + Estimate of m., <b><a + href="#Block_311">311</a></b>.</li> + <li class="isub1"> + M. opens the fountain of all thought, <b><a + href="#Block_451">451</a></b>.</li> + <li class="isub1"> + Math’ns must perceive beauty of truth, <b><a + href="#Block_803">803</a></b>.</li> + <li class="isub1"> + Math’ns bear semblance of divinity, <b><a + href="#Block_804">804</a></b>.</li> + <li class="isub1"> + Math’ns like Frenchmen, <b><a + href="#Block_813">813</a></b>.</li> + <li class="isub1"> + His aptitude for m., <b><a + href="#Block_976">976</a></b>.</li> + <li class="isub1"> + M. like dialectics, <b><a + href="#Block_1307">1307</a></b>.</li> + <li class="isub1"> + On the infinite, <a + href="#Block_1957">1957</a>.</li> + <li class="indx"> + Golden age of m., <a + href="#Block_701">701</a>, <a + href="#Block_702">702</a>.</li> + <li class="isub1"> + Of art and m. coincident, <a + href="#Block_1134">1134</a>.</li> + <li class="indx"> + Gordan, When a math. subject is complete, <b><a + href="#Block_636">636</a></b>.</li> + <li class="indx"> + Gow, Origin of Euclid, <b><a + href="#Block_1832">1832</a></b>.</li> + <li class="indx"> + Gower, <b><a href="#Block_1808">1808</a></b>.</li> + <li class="indx"> + Grammar and m. compared, <a + href="#Block_441">441</a>.</li> + <li class="indx"> + Grandeur of m., <a + href="#Block_325">325</a>.</li> + <li class="indx"> + Grassmann,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_105">105</a></b>.</li> + <li class="isub1"> + Definition of magnitude, <b><a + href="#Block_105">105</a></b>.</li> + <li class="isub1"> + Definition of equality, <b><a + href="#Block_105">105</a></b>.</li> + <li class="isub1"> + On rigor in m., <b><a + href="#Block_538">538</a></b>.</li> + <li class="isub1"> + On the value of m., <b><a + href="#Block_1512">1512</a></b>.</li> + <li class="indx"> + Greek view of science, <a + href="#Block_1429">1429</a>.</li> + <li class="indx"> + Graphic method, <a + href="#Block_1881">1881</a>.</li> + <li class="indx"> + Graphomath, <a + href="#Block_2101">2101</a>. + +<span class="pagenum"> + <a name="Page_395" + id="Page_395">395</a></span></li> + + <li class="indx"> + Group, Notion of, <a + href="#Block_1751">1751</a>.</li> + <li class="indx"> + Growth of m., <a + href="#Block_209">209</a>, <a + href="#Block_211">211</a>, <a + href="#Block_703">703</a>.</li> + <li class="ifrst"> + Hall, G. S., M. the ideal and norm of all careful + thinking, <b><a + href="#Block_304">304</a></b>.</li> + <li class="indx"> + Hall and Stevens, On the parallel axiom, <b><a + href="#Block_2008">2008</a></b>.</li> + <li class="indx"> + Haller, On the infinite, <b><a + href="#Block_1958">1958</a></b>.</li> + <li class="indx"> + Halley, On Cartesian geometry, <a + href="#Block_716">716</a>.</li> + <li class="indx"> + <a id="TNanchor_23">Halsted</a>,</li> + <li class="isub1"> + On Bolyai, <b><a + href="#Block_924">924-926</a></b>.</li> + <li class="isub1"> + On Sylvester, <b><a + href="#Block_1030">1030</a></b>, <b><a + href="#Block_1039">1039</a></b>.</li> + <li class="isub1"> + And + <a class="msg" href="#TN_23" + title="originally spelled ‘Slyvester’">Sylvester</a>, <b><a + + href="#Block_1031">1031</a></b>, <b><a + href="#Block_1032">1032</a></b>.</li> + <li class="isub1"> + On m. as logic, <b><a + href="#Block_1305">1305</a></b>.</li> + <li class="isub1"> + Definition of geometry, <b><a + href="#Block_1815">1815</a></b>.</li> + <li class="indx"> + Hamilton, Sir William, His ignorance of m., <a + href="#Block_978">978</a>.</li> + <li class="indx"> + Hamilton, W. R.,</li> + <li class="isub1"> + Importance of his quaternions, <a + href="#Block_333">333</a>.</li> + <li class="isub1"> + Estimate of Comte’s ability, <b><a + href="#Block_943">943</a></b>.</li> + <li class="isub1"> + To the memory of Fourier, <b><a + href="#Block_969">969</a></b>.</li> + <li class="isub1"> + Discovery in light, <a + href="#Block_1558">1558</a>.</li> + <li class="isub1"> + On algebra as the science of time, <b><a + href="#Block_1715">1715</a></b>, <b><a + href="#Block_1716">1716</a></b>.</li> + <li class="isub1"> + On quaternions, <b><a + href="#Block_1718">1718</a></b>.</li> + <li class="isub1"> + On trisection of an angle, <b><a + href="#Block_2112">2112</a></b>.</li> + <li class="indx"> + Hankel,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_114">114</a></b>.</li> + <li class="isub1"> + On freedom in m., <b><a + href="#Block_206">206</a></b>.</li> + <li class="isub1"> + On the permanency of math. knowledge, <b><a + href="#Block_216">216</a></b>.</li> + <li class="isub1"> + On aim in m., <b><a + href="#Block_508">508</a></b>.</li> + <li class="isub1"> + On isolated theorems, <b><a + href="#Block_621">621</a></b>.</li> + <li class="isub1"> + On tact in m., <b><a + href="#Block_622">622</a></b>.</li> + <li class="isub1"> + On geometry, <a + href="#Block_714">714</a>.</li> + <li class="isub1"> + Ancient and modern m. compared, <b><a + href="#Block_718">718</a></b>, <b><a + href="#Block_720">720</a></b>.</li> + <li class="isub1"> + Variability the central idea in modern m., <b><a + href="#Block_720">720</a></b>.</li> + <li class="isub1"> + Characteristics of modern m., <b><a + href="#Block_728">728</a></b>.</li> + <li class="isub1"> + On Descartes, <b><a + href="#Block_949">949</a></b>.</li> + <li class="isub1"> + On Euler’s work, <b><a + href="#Block_956">956</a></b>.</li> + <li class="isub1"> + On philosophy and m., <b><a + href="#Block_1404">1404</a></b>.</li> + <li class="isub1"> + On the origin of m., <b><a + href="#Block_1412">1412</a></b>.</li> + <li class="isub1"> + On irrationals and imaginaries, <b><a + href="#Block_1729">1729</a></b>.</li> + <li class="isub1"> + On the origin of algebra, <b><a + href="#Block_1736">1736</a></b>.</li> + <li class="isub1"> + Euclid the only perfect model, <b><a + href="#Block_1822">1822</a></b>.</li> + <li class="isub1"> + Modern geometry a royal road, <b><a + href="#Block_1878">1878</a></b>.</li> + <li class="indx"> + Harmony, <a + href="#Block_326">326</a>, <a + href="#Block_1208">1208</a>.</li> + <li class="indx"> + Harris, M. gives command over nature, <b><a + href="#Block_434">434</a></b>.</li> + <li class="indx"> + Hathaway, On Sylvester, <b><a + href="#Block_1036">1036</a></b>.</li> + <li class="indx"> + Heat, M. and the theory of, <a + href="#Block_1552">1552</a>, <a + href="#Block_1553">1553</a>.</li> + <li class="indx"> + Heath, Character of Archimedes’ work, <b><a + href="#Block_913">913</a></b>.</li> + <li class="indx"> + Heaviside, The place of Euclid, <b><a + href="#Block_1828">1828</a></b>.</li> + <li class="indx"> + Hebrew and Latin races, Aptitude for m., <a + href="#Block_838">838</a>.</li> + <li class="indx"> + Hegel, <b><a + href="#Block_1417">1417</a></b>.</li> + <li class="indx"> + Heiss,</li> + <li class="isub1"> + Famous anagrams, <b><em><a + href="#Block_2155">2055</a></em></b>.</li> + <li class="isub1"> + Reversible verses, <b><em><a + href="#Block_2156">2056</a></em></b>.</li> + <li class="indx"> + Helmholtz,</li> + <li class="isub1"> + M. the purest form of logical activity, <b><a + href="#Block_231">231</a></b>.</li> + <li class="isub1"> + M. requires perseverance and great caution, <b><a + href="#Block_240">240</a></b>.</li> + <li class="isub1"> + M. should take more important place in education, <b><a + href="#Block_441">441</a></b>.</li> + <li class="isub1"> + Clifford on, <b><a + href="#Block_979">979</a></b>.</li> + <li class="isub1"> + M. the purest logic, <b><a + href="#Block_1302">1302</a></b>.</li> + <li class="isub1"> + M. and applications, <b><em><a + href="#Block_1545">1445</a></em></b>.</li> + <li class="isub1"> + On geometry, <b><a + href="#Block_1836">1836</a></b>.</li> + <li class="isub1"> + On the importance of the calculus, <b><a + href="#Block_1939">1939</a></b>.</li> + <li class="isub1"> + A non-euclidean world, <b><a + href="#Block_2029">2029</a></b>.</li> + <li class="indx"> + Herbart,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_117">117</a></b>.</li> + <li class="isub1"> + M. the predominant science, <b><a + href="#Block_209">209</a></b>.</li> + <li class="isub1"> + On the method of m., <b><a + href="#Block_212">212</a></b>, <b><a + href="#Block_1576">1576</a></b>.</li> + <li class="isub1"> + M. the priestess of definiteness and clearness, <b><a + href="#Block_217">217</a></b>.</li> + <li class="isub1"> + On the importance of checks, <b><a + href="#Block_230">230</a></b>.</li> + <li class="isub1"> + On imagination in m, <b><a + href="#Block_257">257</a></b>.</li> + <li class="isub1"> + M. and invention, <b><a + href="#Block_406">406</a></b>.</li> + <li class="isub1"> + M. the chief subject for common schools, <b><a + href="#Block_432">432</a></b>.</li> + <li class="isub1"> + On aptitude for m., <b><a + href="#Block_509">509</a></b>.</li> + <li class="isub1"> + On the teaching of m., <b><a + href="#Block_516">516</a></b>.</li> + <li class="isub1"> + M. the greatest blessing, <b><a + href="#Block_1401">1401</a></b>.</li> + <li class="isub1"> + M. and philosophy, <b><a + href="#Block_1408">1408</a></b>.</li> + <li class="isub1"> + If philosophers understood m., <b><a + href="#Block_1415">1415</a></b>.</li> + <li class="isub1"> + M. indispensable to science, <b><a + href="#Block_1502">1502</a></b>. + +<span class="pagenum"> + <a name="Page_396" + id="Page_396">396</a></span></li> + + <li class="isub1"> + M. and psychology, <b><a + href="#Block_1583">1583</a></b>, <b><a + href="#Block_1584">1584</a></b>.</li> + <li class="isub1"> + On trigonometry, <b><a + href="#Block_1884">1884</a></b>.</li> + <li class="indx"> + Hermite, On Cayley, <b><a + href="#Block_935">935</a></b>.</li> + <li class="indx"> + Herschel,</li> + <li class="isub1"> + M. and astronomy, <b><em><a + href="#Block_1563">1564</a></em></b>.</li> + <li class="isub1"> + On probabilities, <b><a + href="#Block_1592">1592</a></b>.</li> + <li class="indx"> + Hiero, <a + href="#Block_903">903</a>, <a + href="#Block_904">904</a>.</li> + <li class="indx"> + Higher m., Mellor’s definition of, <b><a + href="#Block_108">108</a></b>.</li> + <li class="indx"> + Hilbert,</li> + <li class="isub1"> + On the nature of m., <b><a + href="#Block_266">266</a></b>.</li> + <li class="isub1"> + On rigor in m., <b><a + href="#Block_537">537</a></b>.</li> + <li class="isub1"> + On the importance of problems, <b><a + href="#Block_624">624</a></b>, <b><a + href="#Block_628">628</a></b>.</li> + <li class="isub1"> + On the solvability of problems, <b><a + href="#Block_627">627</a></b>.</li> + <li class="isub1"> + Problems should be difficult, <b><a + href="#Block_629">629</a></b>.</li> + <li class="isub1"> + On the abstract character of m., <b><a + href="#Block_638">638</a></b>.</li> + <li class="isub1"> + On arithmetical symbols, <b><a + href="#Block_1627">1627</a></b>.</li> + <li class="isub1"> + On non-euclidean geometry, <b><a + href="#Block_2019">2019</a></b>.</li> + <li class="indx"> + Hill, Aaron, On Newton, <b><a + href="#Block_1009">1009</a></b>.</li> + <li class="indx"> + Hill, Thomas,</li> + <li class="isub1"> + On the spirit of mathesis, <b><a + href="#Block_274">274</a></b>.</li> + <li class="isub1"> + M. expresses thoughts of God, <b><a + href="#Block_275">275</a></b>.</li> + <li class="isub1"> + Value of m., <b><a + href="#Block_332">332</a></b>.</li> + <li class="isub1"> + Estimate of Newton’s work, <b><a + href="#Block_333">333</a></b>.</li> + <li class="isub1"> + Math’ns difficult to judge, <b><a + href="#Block_841">841</a></b>.</li> + <li class="isub1"> + Math’ns indifferent to ordinary interests of life, <b><a + href="#Block_842">842</a></b>.</li> + <li class="isub1"> + A geometer must be tried by his peers, <b><a + href="#Block_843">843</a></b>.</li> + <li class="isub1"> + On Bernoulli’s spiral, <b><a + href="#Block_922">922</a></b>.</li> + <li class="isub1"> + On mathesis and poetry, <b><a + href="#Block_1125">1125</a></b>.</li> + <li class="isub1"> + On poesy and m., <b><a + href="#Block_1126">1126</a></b>.</li> + <li class="isub1"> + On m. as a language, <b><a + href="#Block_1209">1209</a></b>.</li> + <li class="isub1"> + Math, language untranslatable, <b><a + href="#Block_1210">1210</a></b>.</li> + <li class="isub1"> + On quaternions, <b><a + href="#Block_1719">1719</a></b>.</li> + <li class="isub1"> + On the imaginary, <b><a + href="#Block_1734">1734</a></b>.</li> + <li class="isub1"> + On geometry and literature, <b><a + href="#Block_1847">1847</a></b>.</li> + <li class="isub1"> + M. and miracles, <b><a + href="#Block_2157">2157</a></b>, <b><a + href="#Block_2158">2158</a></b>.</li> + <li class="indx"> + Hindoos, Grandest achievement of, <a + href="#Block_1615">1615</a>.</li> + <li class="indx"> + History and m., <a + href="#Block_1599">1599</a>.</li> + <li class="indx"> + History of m., <a + href="#Block_615">615</a>, <a + href="#Block_616">616</a>, <a + href="#Block_625">625</a>, <a + href="#Block_635">635</a>.</li> + <li class="indx"> + Hobson,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_118">118</a></b>.</li> + <li class="isub1"> + On the nature of m., <b><a + href="#Block_252">252</a></b>.</li> + <li class="isub1"> + Functionality the central idea of m., <b><a + href="#Block_264">264</a></b>.</li> + <li class="isub1"> + On theoretical investigations, <b><a + href="#Block_663">663</a></b>.</li> + <li class="isub1"> + On the growth of m., <b><a + href="#Block_703">703</a></b>.</li> + <li class="isub1"> + A great math’n a great artist, <b><a + href="#Block_1109">1109</a></b>.</li> + <li class="isub1"> + On m. and science, <b><em><a + href="#Block_1518">1508</a></em></b>.</li> + <li class="indx"> + Hoffman, Science and poetry not antagonistic, <b><a + href="#Block_1122">1122</a></b>.</li> + <li class="indx"> + Holzmüller, On the teaching of m., <b><a + href="#Block_518">518</a></b>.</li> + <li class="indx"> + Hooker, <b><a + href="#Block_1432">1432</a></b>.</li> + <li class="indx"> + Hopkinson, M. a mill, <b><a + href="#Block_239">239</a></b>.</li> + <li class="indx"> + Horner’s method, <a + href="#Block_1744">1744</a>.</li> + <li class="indx"> + Howison,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_134">134</a></b>, <b><a + href="#Block_135">135</a></b>.</li> + <li class="isub1"> + Definition of arithmetic, <b><a + href="#Block_1612">1612</a></b>.</li> + <li class="indx"> + Hudson, On the teaching of m., <b><a + href="#Block_512">512</a></b>.</li> + <li class="indx"> + Hughes, On science for its own sake, <b><a + href="#Block_1546">1546</a></b>.</li> + <li class="indx"> + Humboldt, M. and astronomy, <b><a + href="#Block_1567">1567</a></b>.</li> + <li class="indx"> + Hume,</li> + <li class="isub1"> + On the advantage of math, science, <b><a + href="#Block_1438">1438</a></b>.</li> + <li class="isub1"> + On geometry, <b><a + href="#Block_1862">1862</a></b>.</li> + <li class="isub1"> + On certainty in m., <b><a + href="#Block_1863">1863</a></b>.</li> + <li class="isub1"> + Objection to abstract reasoning, <b><a + href="#Block_1941">1941</a></b>.</li> + <li class="indx"> + Humor in m., <a + href="#Block_539">539</a>.</li> + <li class="indx"> + Hutton,</li> + <li class="isub1"> + On Bernoulli, <b><a + href="#Block_919">919</a></b>.</li> + <li class="isub1"> + On Euler’s knowledge, <b><a + href="#Block_958">958</a></b>.</li> + <li class="isub1"> + On the method of fluxions, <b><a + href="#Block_1911">1911</a></b>.</li> + <li class="indx"> + Huxley, Negative qualities of m., <b><a + href="#Block_250">250</a></b>.</li> + <li class="indx"> + Hyper-space, <a + href="#Block_2030">2030</a>, <a + href="#Block_2031">2031</a>, <a + href="#Block_2033">2033</a>, <a + href="#Block_2036">2036-2038</a>.</li> + <li class="indx"> + Hyperbolic functions, <a + href="#Block_1929">1929</a>, <a + href="#Block_1930">1930</a>.</li> + + <li class="ifrst"> + Ignes fatui in m., <a + href="#Block_2103">2103</a>.</li> + <li class="indx"> + Ignorabimus, None in m., <a + href="#Block_627">627</a>.</li> + <li class="indx"> + Ignorance of m., <a + href="#Block_310">310</a>, <a + href="#Block_331">331</a>, <a + href="#Block_807">807</a>, <a + href="#Block_1537">1537</a>, <a + href="#Block_1577">1577</a>.</li> + <li class="indx"> + Imaginaries, <a + href="#Block_722">722</a>, <a + href="#Block_1729">1729-1735</a>.</li> + <li class="indx"> + Imagination in m., <a + href="#Block_246">246</a>, <a + href="#Block_251">251</a>, <a + href="#Block_253">253</a>, <a + href="#Block_256">256-258</a>, <a + href="#Block_433">433</a>, <a + href="#Block_1883">1883</a>.</li> + <li class="indx"> + Improvement of elementary m., <a + href="#Block_617">617</a>. + +<span class="pagenum"> + <a name="Page_397" + id="Page_397">397</a></span></li> + + <li class="indx"> + Incommensurable numbers, contingent truths like, <a + href="#Block_1966">1966</a>.</li> + <li class="indx"> + Indian m., <a + href="#Block_1736">1736</a>, <a + href="#Block_1737">1737</a>.</li> + <li class="indx"> + Induction in m., <a + href="#Block_220">220-223</a>, <a + href="#Block_244">244</a>.</li> + <li class="isub1"> + And analogy, <a + href="#Block_724">724</a>.</li> + <li class="indx"> + Infinite collection, Definition of, <a + href="#Block_1959">1959</a>, <a + href="#Block_1960">1960</a>.</li> + <li class="indx"> + Infinite divisibility, <a + href="#Block_1945">1945</a>.</li> + <li class="indx"> + Infinitesimal analysis, <a + href="#Block_1914">1914</a>.</li> + <li class="indx"> + Infinitesimals, <a + href="#Block_1905">1905-1907</a>, <a + href="#Block_1940">1940</a>, <a + href="#Block_1946">1946</a>, <a + href="#Block_1954">1954</a>.</li> + <li class="indx"> + Infinitum, Ad, <a + href="#Block_1949">1949</a>.</li> + <li class="indx"> + Infinity and infinite magnitude, <a + href="#Block_723">723</a>, <a + href="#Block_928">928</a>, <a + href="#Block_1947">1947</a>, <a + href="#Block_1948">1948</a>, <a + href="#Block_1950">1950-1958</a>.</li> + <li class="indx"> + Integers, Kronecker on, <a + href="#Block_1634">1634</a>, <a + href="#Block_1635">1635</a>.</li> + <li class="indx"> + Integral numbers, Minkowsky on, <a + href="#Block_1636">1636</a>.</li> + <li class="indx"> + Integrals, Invention of, <a + href="#Block_1922">1922</a>.</li> + <li class="indx"> + Integration, <a + href="#Block_1919">1919-1921</a>, <a + href="#Block_1923">1923</a>, <a + href="#Block_1925">1925</a>.</li> + <li class="indx"> + International Commission on m., <b><a + href="#Block_501">501</a></b>, <b><a + href="#Block_502">502</a></b>, <b><em><a + href="#Block_738">938</a></em></b>.</li> + <li class="indx"> + Intuition and deduction, <a + href="#Block_1413">1413</a>.</li> + <li class="indx"> + Invariance,</li> + <li class="isub1"> + Correlated to life, <a + href="#Block_272">272</a>.</li> + <li class="isub1"> + MacMahon on, <a + href="#Block_1746">1746</a>.</li> + <li class="isub1"> + Keyser on, <a + href="#Block_1749">1749</a>.</li> + <li class="indx"> + Invariants,</li> + <li class="isub1"> + Changeless in the midst of change, <a + href="#Block_276">276</a>.</li> + <li class="isub1"> + Importance of concept of, <a + href="#Block_727">727</a>.</li> + <li class="isub1"> + Sylvester on, <a + href="#Block_1742">1742</a>.</li> + <li class="isub1"> + Forsyth on, <a + href="#Block_1747">1747</a>.</li> + <li class="isub1"> + Keyser on, <a + href="#Block_1748">1748</a>.</li> + <li class="isub1"> + Lie on, <a + href="#Block_1752">1752</a>.</li> + <li class="indx"> + Invention in m., <a + href="#Block_251">251</a>, <a + href="#Block_260">260</a>.</li> + <li class="indx"> + Inverse process, <a + href="#Block_1207">1207</a>.</li> + <li class="indx"> + Investigations, See research.</li> + <li class="indx"> + Irrationals, <a + href="#Block_1729">1729</a>.</li> + <li class="indx"> + Isolated theorems in m., <a + href="#Block_620">620</a>, <a + href="#Block_621">621</a>.</li> + <li class="indx"> + “It is easy to see,” <a + href="#Block_985">985</a>, <a + href="#Block_986">986</a>, <a + href="#Block_1045">1045</a>.</li> + + <li class="ifrst"> + Jacobi,</li> + <li class="isub1"> + His talent for philology, <a + href="#Block_980">980</a>.</li> + <li class="isub1"> + Aphorism, <b><a + href="#Block_1635">1635</a></b>.</li> + <li class="isub1"> + Die “Ewige Zahl,” <b><a + href="#Block_1643">1643</a></b>.</li> + <li class="indx"> + <a id="TNanchor_24">Jefferson</a>, + <a class="msg" + href="#TN_24" + title="originally spelled ‘Om’">On</a> + + m. and law, <b><a + href="#Block_1597">1597</a></b>.</li> + <li class="indx"> + Johnson,</li> + <li class="isub1"> + His recourse to m., <a + href="#Block_981">981</a>.</li> + <li class="isub1"> + Aptitude for numbers, <b><a + href="#Block_1617">1617</a></b>.</li> + <li class="isub1"> + On round numbers, <b><a + href="#Block_2137">2137</a></b>.</li> + <li class="indx"> + Journals and transactions, <a + href="#Block_635">635</a>.</li> + <li class="indx"> + Jowett, M. as an instrument in education, <b><a + href="#Block_413">413</a></b>.</li> + <li class="indx"> + Judgment, M. requires, <a + href="#Block_823">823</a>.</li> + <li class="indx"> + Jupiter’s eclipses, <a + href="#Block_1544">1544</a>.</li> + <li class="indx"> + Justitia, The goddess, <a + href="#Block_824">824</a>.</li> + <li class="indx"> + Juvenal, Nemo mathematicus etc., <b><a + href="#Block_831">831</a></b>.</li> + + <li class="ifrst"> + Kant,</li> + <li class="isub1"> + On the a priori nature of m., <b><a + href="#Block_130">130</a></b>.</li> + <li class="isub1"> + M. follows the safe way of science, <b><a + href="#Block_201">201</a></b>.</li> + <li class="isub1"> + On the origin of scientific m., <a + href="#Block_201">201</a>.</li> + <li class="isub1"> + On m. in primary education, <b><a + href="#Block_431">431</a></b>.</li> + <li class="isub1"> + M. the embarrassment of metaphysics, <b><a + href="#Block_1402">1402</a></b>.</li> + <li class="isub1"> + His view of m., <b><a + href="#Block_1436">1436</a></b>, <b><a + href="#Block_1437">1437</a></b>.</li> + <li class="isub1"> + On the difference between m. and philosophy, <b><a + href="#Block_1436">1436</a></b>.</li> + <li class="isub1"> + On m. and science, <b><a + href="#Block_1508">1508</a></b>.</li> + <li class="isub1"> + Esthetic elements in m., <b><a + href="#Block_1852">1852</a></b>, <b><a + href="#Block_1853">1853</a></b>.</li> + <li class="isub1"> + Doctrine of time, <b><a + href="#Block_2001">2001</a></b>.</li> + <li class="isub1"> + Doctrine of space, <b><em><a + href="#Block_2002">2003</a></em></b>.</li> + <li class="indx"> + Karpinsky, M. and efficiency, <b><a + href="#Block_1573">1573</a></b>.</li> + <li class="indx"> + Kasner,</li> + <li class="isub1"> + “Divinez avant de demontrer," <b><a + href="#Block_630">630</a></b>.</li> + <li class="isub1"> + On modern geometry, <b><a + href="#Block_710">710</a></b>.</li> + <li class="indx"> + Kelland, On Euclid’s elements, <b><a + href="#Block_1817">1817</a></b>.</li> + <li class="indx"> + Kelvin, Lord, See William Thomson.</li> + <li class="indx"> + Kepler,</li> + <li class="isub1"> + His method, <a + href="#Block_982">982</a>.</li> + <li class="isub1"> + Planetary orbits and the regular solids, <b><a + href="#Block_2134">2134</a></b>.</li> + <li class="indx"> + Keyser,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_132">132</a></b>.</li> + <li class="isub1"> + Three characteristics of m., <b><a + href="#Block_225">225</a></b>.</li> + <li class="isub1"> + On the method of m., <b><a + href="#Block_244">244</a></b>.</li> + <li class="isub1"> + On ratiocination, <b><a + href="#Block_246">246</a></b>.</li> + <li class="isub1"> + M. not detached from life, <b><em><a + href="#Block_272">273</a></em></b>.</li> + <li class="isub1"> + On the spirit of mathesis, <b><a + href="#Block_276">276</a></b>.</li> + <li class="isub1"> + Computation not m., <b><a + href="#Block_515">515</a></b>.</li> + <li class="isub1"> + Math, output of present day, <b><a + href="#Block_702">702</a></b>. + +<span class="pagenum"> + <a name="Page_398" + id="Page_398">398</a></span></li> + + <li class="isub1"> + Modern theory of functions, <b><a + href="#Block_709">709</a></b>.</li> + <li class="isub1"> + M. and journalism, <b><a + href="#Block_731">731</a></b>.</li> + <li class="isub1"> + Difficulty of m., <b><a + href="#Block_735">735</a></b>.</li> + <li class="isub1"> + M. appeals to whole mind, <b><a + href="#Block_815">815</a></b>.</li> + <li class="isub1"> + Endowment of math’ns, <b><a + href="#Block_818">818</a></b>.</li> + <li class="isub1"> + Math’ns in public service, <b><a + href="#Block_823">823</a></b>.</li> + <li class="isub1"> + The aim of the math’n, <b><a + href="#Block_844">844</a></b>.</li> + <li class="isub1"> + On Bolzano, <b><em><a + href="#Block_928">929</a></em></b>.</li> + <li class="isub1"> + On Lie, <b><a + href="#Block_992">992</a></b>.</li> + <li class="isub1"> + On symbolic logic, <b><a + href="#Block_1321">1321</a></b>.</li> + <li class="isub1"> + On the emancipation of logic, <b><a + href="#Block_1322">1322</a></b>.</li> + <li class="isub1"> + On the Principia Mathematica, <b><a + href="#Block_1326">1326</a></b>.</li> + <li class="isub1"> + On invariants, <b><em><a + href="#Block_1748">1728</a></em></b>.</li> + <li class="isub1"> + On invariance, <b><em><a + href="#Block_1749">1729</a></em></b>.</li> + <li class="isub1"> + On the notion of group, <b><a + href="#Block_1751">1751</a></b>.</li> + <li class="isub1"> + On the elements of Euclid, <b><a + href="#Block_1824">1824</a></b>.</li> + <li class="isub1"> + On protective geometry, <b><a + href="#Block_1880">1880</a></b>.</li> + <li class="isub1"> + Definition of infinite assemblage, <b><a + href="#Block_1960">1960</a></b>.</li> + <li class="isub1"> + On the infinite, <b><a + href="#Block_1961">1961</a></b>.</li> + <li class="isub1"> + On non-euclidean geometry, <b><a + href="#Block_2035">2035</a></b>.</li> + <li class="isub1"> + On hyper-space, <b><a + href="#Block_2037">2037</a></b>, <b><a + href="#Block_2038">2038</a></b>.</li> + <li class="indx"> + Khulasat-al-Hisab, Problems, <b><a + href="#Block_1738">1738</a></b>.</li> + <li class="indx"> + Kipling, <b><a + href="#Block_1633">1633</a></b>.</li> + <li class="indx"> + Kirchhoff, Artistic nature of his works, <a + href="#Block_1116">1116</a>.</li> + <li class="indx"> + Klein,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_123">123</a></b>.</li> + <li class="isub1"> + M. a versatile science, <b><a + href="#Block_264">264</a></b>.</li> + <li class="isub1"> + Aim in teaching, <b><a + href="#Block_507">507</a></b>, <b><a + href="#Block_517">517</a></b>.</li> + <li class="isub1"> + Analysts versus synthesists, <b><a + href="#Block_651">651</a></b>.</li> + <li class="isub1"> + On theory and practice, <b><a + href="#Block_661">661</a></b>.</li> + <li class="isub1"> + Math, aptitudes of various races, <b><a + href="#Block_838">838</a></b>.</li> + <li class="isub1"> + Lie’s final aim, <b><a + href="#Block_993">993</a></b>.</li> + <li class="isub1"> + Lie’s genius, <b><a + href="#Block_994">994</a></b>.</li> + <li class="isub1"> + On m. and science, <b><a + href="#Block_1520">1520</a></b>.</li> + <li class="isub1"> + Famous aphorisms, <b><a + href="#Block_1635">1635</a></b>.</li> + <li class="isub1"> + Calculating machines, <b><a + href="#Block_1641">1641</a></b>.</li> + <li class="isub1"> + Calculus for high schools, <b><a + href="#Block_1918">1918</a></b>.</li> + <li class="isub1"> + On differential equations, <b><a + href="#Block_1926">1926</a></b>.</li> + <li class="isub1"> + Definition of a curve, <b><a + href="#Block_1927">1927</a></b>.</li> + <li class="isub1"> + On axioms of geometry, <b><a + href="#Block_2006">2006</a></b>.</li> + <li class="isub1"> + On the parallel axiom, <b><a + href="#Block_2009">2009</a></b>.</li> + <li class="isub1"> + On non-euclidean geometry, <b><a + href="#Block_2017">2017</a></b>, <b><a + href="#Block_2021">2021</a></b>.</li> + <li class="isub1"> + On hyper-space, <b><a + href="#Block_2030">2030</a></b>.</li> + <li class="indx"> + Kronecker,</li> + <li class="isub1"> + On the greatness of Gauss, <b><a + href="#Block_973">973</a></b>.</li> + <li class="isub1"> + God made integers etc., <b><a + href="#Block_1634">1634</a></b>.</li> + <li class="indx"> + Kummer,</li> + <li class="isub1"> + On Dirichlet, <b><a + href="#Block_977">977</a></b>.</li> + <li class="isub1"> + On beauty in m., <b><a + href="#Block_1111">1111</a></b>.</li> + + <li class="ifrst"> + LaFaille, Mathesis few know, <b><a + href="#Block_1870">1870</a></b>.</li> + <li class="indx"> + Lagrange, On correlation of algebra and geometry, <b><a + href="#Block_527">527</a></b>.</li> + <li class="isub1"> + His style and method, <a + href="#Block_983">983</a>.</li> + <li class="isub1"> + And the parallel axiom, <a + href="#Block_984">984</a>.</li> + <li class="isub1"> + On Newton, <b><a + href="#Block_1011">1011</a></b>.</li> + <li class="isub1"> + Wings of m., <b><a + href="#Block_1604">1604</a></b>.</li> + <li class="isub1"> + Union of algebra and geometry, <b><a + href="#Block_1707">1707</a></b>.</li> + <li class="isub1"> + On the infinitesimal method, <b><a + href="#Block_1906">1906</a></b>.</li> + <li class="indx"> + Lalande, M. in French army, <b><a + href="#Block_314">314</a></b>.</li> + <li class="indx"> + Langley, M. in Prussia, <a + href="#Block_513">513</a>.</li> + <li class="indx"> + Lampe,</li> + <li class="isub1"> + On division of labor in m., <b><a + href="#Block_632">632</a></b>.</li> + <li class="isub1"> + On Weierstrass, <b><a + href="#Block_1049">1049</a></b>.</li> + <li class="isub1"> + Weierstrass and Sylvester, <b><a + href="#Block_1050">1050</a></b>.</li> + <li class="isub1"> + Qualities common to math’ns and artists, <b><a + href="#Block_1113">1113</a></b>.</li> + <li class="isub1"> + Charm of m., <b><a + href="#Block_1115">1115</a></b>.</li> + <li class="isub1"> + Golden age of art and m. coincident, <b><a + href="#Block_1134">1134</a></b>.</li> + <li class="indx"> + Language, </li> + <li class="isub1"> + Chapter <a + href="#CHAPTER_XII">XII</a>.</li> + <li class="isub1"> + See also <a + href="#Block_311">311</a>, <a + href="#Block_419">419</a>, <a + href="#Block_443">443</a>, <a + href="#Block_1523">1523</a>, <a + href="#Block_1804">1804</a>, <a + href="#Block_1889">1889</a>.</li> + <li class="indx"> + Laplace,</li> + <li class="isub1"> + On instruction in m., <b><a + href="#Block_220">220</a></b>.</li> + <li class="isub1"> + His style and method, <b><a + href="#Block_983">983</a></b>.</li> + <li class="isub1"> + “Thus it plainly appears,” <a + href="#Block_985">985</a>, <a + href="#Block_986">986</a>.</li> + <li class="isub1"> + Emerson on, <a + href="#Block_1003">1003</a>.</li> + <li class="isub1"> + On Leibnitz, <b><a + href="#Block_991">991</a></b>.</li> + <li class="isub1"> + On the language of analysis, <b><a + href="#Block_1222">1222</a></b>.</li> + <li class="isub1"> + On m. and nature, <b><a + href="#Block_1525">1525</a></b>.</li> + <li class="isub1"> + On the origin of the calculus, <b><a + href="#Block_1902">1902</a></b>.</li> + <li class="isub1"> + On the exactitude of the differential calculus, <b><a + href="#Block_1910">1910</a></b>. + +<span class="pagenum"> + <a name="Page_399" + id="Page_399">399</a></span></li> + + <li class="isub1"> + The universe in a single formula, <b><a + href="#Block_1920">1920</a></b>.</li> + <li class="isub1"> + On probability, <b><a + href="#Block_1963">1963</a></b>, <b><a + href="#Block_1969">1969</a></b>, <b><a + href="#Block_1971">1971</a></b>.</li> + <li class="indx"> + Laputa,</li> + <li class="isub1"> + Math’ns of, <a + href="#Block_2120">2120-2122</a>,</li> + <li class="isub1"> + Math. school of, <a + href="#Block_2123">2123</a>.</li> + <li class="indx"> + Lasswitz,</li> + <li class="isub1"> + On modern algebra, <b><a + href="#Block_1741">1741</a></b>.</li> + <li class="isub1"> + On function theory, <b><a + href="#Block_1934">1934</a></b>.</li> + <li class="isub1"> + On non-euclidean geometry, <b><a + href="#Block_2040">2040</a></b>.</li> + <li class="indx"> + Latin squares, <a + href="#Block_252">252</a>.</li> + <li class="indx"> + Latta, On Leibnitz’s logical calculus, <b><a + href="#Block_1317">1317</a></b>.</li> + <li class="indx"> + Law and m., <a + href="#Block_1597">1597</a>, <a + href="#Block_1598">1598</a>.</li> + <li class="indx"> + Laws of thought, <a + href="#Block_719">719</a>, <a + href="#Block_1318">1318</a>.</li> + <li class="indx"> + Leadership, M. as training for, <a + href="#Block_317">317</a>.</li> + <li class="indx"> + Lecture, Preparation of, <a + href="#Block_540">540</a>.</li> + <li class="indx"> + Lefevre,</li> + <li class="isub1"> + M. hateful to weak minds, <b><a + href="#Block_733">733</a></b>.</li> + <li class="isub1"> + Logic and m., <b><a + href="#Block_1309">1309</a></b>.</li> + <li class="indx"> + Leibnitz,</li> + <li class="isub1"> + On difficulties in m., <b><a + href="#Block_241">241</a></b>.</li> + <li class="isub1"> + His greatness, <a + href="#Block_987">987</a>.</li> + <li class="isub1"> + His influence, <a + href="#Block_988">988</a>.</li> + <li class="isub1"> + The nature of his work, <a + href="#Block_989">989</a>.</li> + <li class="isub1"> + His math. tendencies, <a + href="#Block_990">990</a>.</li> + <li class="isub1"> + His binary arithmetic, <a + href="#Block_991">991</a>.</li> + <li class="isub1"> + On Newton, <b><a + href="#Block_1010">1010</a></b>.</li> + <li class="isub1"> + On demonstrations outside of m., <b><a + href="#Block_1312">1312</a></b>.</li> + <li class="isub1"> + Ars characteristica, <b><a + href="#Block_1316">1316</a></b>.</li> + <li class="isub1"> + His logical calculus, <a + href="#Block_1317">1317</a>.</li> + <li class="isub1"> + Union of philosophical and m. productivity, <a + href="#Block_1404">1404</a>.</li> + <li class="isub1"> + M. and philosophy, <b><a + href="#Block_1435">1435</a></b>.</li> + <li class="isub1"> + On the certainty of math. knowledge, <b><a + href="#Block_1442">1442</a></b>.</li> + <li class="isub1"> + On controversy in geometry, <b><a + href="#Block_1859">1859</a></b>.</li> + <li class="isub1"> + His differential calculus, <a + href="#Block_1902">1902</a>.</li> + <li class="isub1"> + His notation of the calculus, <a + href="#Block_1904">1904</a>.</li> + <li class="isub1"> + On necessary and contingent truth, <b><a + href="#Block_1966">1966</a></b>.</li> + <li class="indx"> + Leverrier, Discovery of Neptune, <a + href="#Block_1559">1559</a>.</li> + <li class="indx"> + Lewes, On the infinite, <b><a + href="#Block_1953">1953</a></b>.</li> + <li class="indx"> + Lie,</li> + <li class="isub1"> + On central conceptions in modern m., <b><a + href="#Block_727">727</a></b>.</li> + <li class="isub1"> + Endowment of math’ns, <b><a + href="#Block_818">818</a></b>.</li> + <li class="isub1"> + The comparative anatomist, <a + href="#Block_992">992</a>.</li> + <li class="isub1"> + Aim of his work, <a + href="#Block_993">993</a>.</li> + <li class="isub1"> + His genius, <a + href="#Block_994">994</a>.</li> + <li class="isub1"> + On groups, <b><a + href="#Block_1752">1752</a></b>.</li> + <li class="isub1"> + On the origin of the calculus, <b><a + href="#Block_1901">1901</a></b>.</li> + <li class="isub1"> + On differential equations, <b><a + href="#Block_1924">1924</a></b>.</li> + <li class="indx"> + Liliwati, Origin of, <a + href="#Block_995">995</a>.</li> + <li class="indx"> + Limitations of math. science, <a + href="#Block_1437">1437</a>.</li> + <li class="indx"> + Limits, Method of, <a + href="#Block_1905">1905</a>, <a + href="#Block_1908">1908</a>, <a + href="#Block_1909">1909</a>, <a + href="#Block_1940">1940</a>.</li> + <li class="indx"> + Lindeman, On m. and science, <b><a + href="#Block_1523">1523</a></b>.</li> + <li class="indx"> + Liouville, <a + href="#Block_822">822</a>.</li> + <li class="indx"> + Lobatchewsky, <b><a + href="#Block_2022">2022</a></b>.</li> + <li class="indx"> + Locke,</li> + <li class="isub1"> + On the method of m., <b><a + href="#Block_214">214</a></b>, <b><a + href="#Block_235">235</a></b>.</li> + <li class="isub1"> + On proofs and demonstrations, <b><a + href="#Block_236">236</a></b>.</li> + <li class="isub1"> + On the unpopularity of m., <b><a + href="#Block_271">271</a></b>.</li> + <li class="isub1"> + On m. as a logical exercise, <b><a + href="#Block_423">423</a></b>, <b><a + href="#Block_424">424</a></b>.</li> + <li class="isub1"> + M. cures presumption, <b><a + href="#Block_425">425</a></b>.</li> + <li class="isub1"> + Math, reasoning of universal application, <b><a + href="#Block_426">426</a></b>.</li> + <li class="isub1"> + On reading of classic authors, <b><a + href="#Block_604">604</a></b>.</li> + <li class="isub1"> + On Aristotle, <b><a + href="#Block_914">914</a></b>.</li> + <li class="isub1"> + On m. and philosophy, <b><a + href="#Block_1433">1433</a></b>.</li> + <li class="isub1"> + On m. and moral science, <b><a + href="#Block_1439">1439</a></b>, <b><a + href="#Block_1440">1440</a></b>.</li> + <li class="isub1"> + On the certainty of math. knowledge, <b><a + href="#Block_1440">1440</a></b>, <b><a + href="#Block_1441">1441</a></b>.</li> + <li class="isub1"> + On unity, <b><a + href="#Block_1607">1607</a></b>.</li> + <li class="isub1"> + On number, <b><a + href="#Block_1608">1608</a></b>.</li> + <li class="isub1"> + On demonstrations in numbers, <b><a + href="#Block_1630">1630</a></b>.</li> + <li class="isub1"> + On the advantages of algebra, <b><a + href="#Block_1705">1705</a></b>.</li> + <li class="isub1"> + On infinity, <b><a + href="#Block_1955">1955</a></b>, <b><em><a + href="#Block_1956">1957</a></em></b>.</li> + <li class="isub1"> + On probability, <b><a + href="#Block_1965">1965</a></b>.</li> + <li class="indx"> + Logarithmic spiral, <a + href="#Block_922">922</a>.</li> + <li class="indx"> + Logarithmic tables, <a + href="#Block_602">602</a>.</li> + <li class="indx"> + Logarithms, <a + href="#Block_1526">1526</a>, <a + href="#Block_1614">1614</a>, <a + href="#Block_1616">1616</a>.</li> + <li class="indx"> + Logic and m., Chapter <a + href="#CHAPTER_XIII">XIII</a>.</li> + <li class="isub1"> + See also <a + href="#Block_423">423-430</a>, <a + href="#Block_442">442</a>.</li> + <li class="indx"> + Logical calculus, <a + href="#Block_1316">1316</a>, <a + href="#Block_1317">1317</a>.</li> + <li class="indx"> + Longevity of math’ns, <a + href="#Block_839">839</a>. + +<span class="pagenum"> + <a name="Page_400" + id="Page_400">400</a></span></li> + + <li class="indx"> + Lovelace, Why are wise few etc., <b><a + href="#Block_1629">1629</a></b>.</li> + <li class="indx"> + Lover, <b><a + href="#Block_2140">2140</a></b>.</li> + + <li class="ifrst"> + Macaulay,</li> + <li class="isub1"> + Plato and Bacon, <b><a + href="#Block_316">316</a></b>.</li> + <li class="isub1"> + On Archimedes, <b><a + href="#Block_905">905</a></b>.</li> + <li class="isub1"> + Bacon’s view of m., <b><a + href="#Block_915">915</a></b>, <b><a + href="#Block_916">916</a></b>.</li> + <li class="isub1"> + Anagram on his name, <b><a + href="#Block_996">996</a></b>.</li> + <li class="isub1"> + Plato and Archytas, <b><a + href="#Block_1427">1427</a></b>.</li> + <li class="isub1"> + On the power of m., <b><a + href="#Block_1527">1527</a></b>.</li> + <li class="indx"> + Macfarlane,</li> + <li class="isub1"> + On Tait, Maxwell, Thomson, <b><em><a + href="#Block_1043">1042</a></em></b>.</li> + <li class="isub1"> + On Tait and Hamilton’s quaternions, <b><a + href="#Block_1044">1044</a></b>.</li> + <li class="indx"> + Mach,</li> + <li class="isub1"> + On thought-economy in m., <b><a + href="#Block_203">203</a></b>.</li> + <li class="isub1"> + M. seems possessed of intelligence, <b><a + href="#Block_626">626</a></b>.</li> + <li class="isub1"> + On aim of research, <b><a + href="#Block_647">647</a></b>.</li> + <li class="isub1"> + On m. and counting, <b><a + href="#Block_1601">1601</a></b>.</li> + <li class="isub1"> + On the space of experience, <b><a + href="#Block_2011">2011</a></b>.</li> + <li class="indx"> + MacMahon,</li> + <li class="isub1"> + Latin squares, <a + href="#Block_252">252</a>.</li> + <li class="isub1"> + On Sylvester’s bend of mind, <b><a + href="#Block_645">645</a></b>.</li> + <li class="isub1"> + On Sylvester’s style, <b><a + href="#Block_1040">1040</a></b>.</li> + <li class="isub1"> + On the idea of invariance, <b><a + href="#Block_1746">1746</a></b>.</li> + <li class="indx"> + Magnitude, Grassmann’s definition, <a + href="#Block_105">105</a>.</li> + <li class="indx"> + Magnus, On the aim in teaching m., <b><a + href="#Block_505">505</a></b>.</li> + <li class="indx"> + Manhattan Island, Cost of, <a + href="#Block_2130">2130</a>.</li> + <li class="indx"> + Marcellus, Estimate of Archimedes, <b><a + href="#Block_909">909</a></b>.</li> + <li class="indx"> + Maschke, Man above method, <b><a + href="#Block_650">650</a></b>.</li> + <li class="indx"> + Masters, On the reading of the, <a + href="#Block_614">614</a>.</li> + <li class="indx"> + Mathematic,</li> + <li class="isub1"> + Sylvester on use of term, <a + href="#Block_101">101</a>.</li> + <li class="isub1"> + Bacon’s use of term, <a + href="#Block_106">106</a>.</li> + <li class="indx"> + Mathematical faculty, Frequency of, <a + href="#Block_832">832</a>.</li> + <li class="indx"> + Mathematical mill, The, <a + href="#Block_239">239</a>, <a + href="#Block_1891">1891</a>.</li> + <li class="indx"> + Mathematical productions, <a + href="#Block_648">648</a>, <a + href="#Block_649">649</a>.</li> + <li class="indx"> + Mathematical theory, When complete, <a + href="#Block_636">636</a>, <a + href="#Block_637">637</a>.</li> + <li class="indx"> + Mathematical training, <a + href="#Block_443">443</a>, <a + href="#Block_444">444</a>.</li> + <li class="isub1"> + Maxims of math’ns, <a + href="#Block_630">630</a>, <a + href="#Block_631">631</a>, <a + href="#Block_649">649</a>.</li> + <li class="isub1"> + Not a computer, <a + href="#Block_1211">1211</a>.</li> + <li class="isub1"> + Intellectual habits of math’ns, <a + href="#Block_1428">1428</a>.</li> + <li class="isub1"> + The place of the, <a + href="#Block_1529">1529</a>.</li> + <li class="isub1"> + Characteristics of the mind of a, <a + href="#Block_1534">1534</a>.</li> + <li class="indx"> + Mathematician, The, Chapter <a + href="#CHAPTER_VIII">VIII</a>.</li> + <li class="indx"> + Mathematics,</li> + <li class="isub1"> + Definitions of, Chapter <a + href="#CHAPTER_I">I</a>.</li> + <li class="isub1"> + Objects of, Chapter <a + href="#CHAPTER_I">I</a>.</li> + <li class="isub1"> + Nature of, Chapter <a + href="#CHAPTER_II">II</a>.</li> + <li class="isub1"> + Estimates of, Chapter <a + href="#CHAPTER_III">III</a>.</li> + <li class="isub1"> + Value of, Chapter <a + href="#CHAPTER_IV">IV</a>.</li> + <li class="isub1"> + Teaching of, Chapter <a + href="#CHAPTER_V">V</a>.</li> + <li class="isub1"> + Study of, Chapter <a + href="#CHAPTER_VI">VI</a>.</li> + <li class="isub1"> + Research in, Chapter <a + href="#CHAPTER_VI">VI</a>.</li> + <li class="isub1"> + Modern, Chapter <a + href="#CHAPTER_VII">VII</a>.</li> + <li class="isub1"> + As a fine art, Chapter <a + href="#CHAPTER_XI">XI</a>.</li> + <li class="isub1"> + As a language, Chapter <a + href="#CHAPTER_XII">XII</a>.</li> + <li class="isub2"> + Also <a + href="#Block_445">445</a>, <a + href="#Block_1814">1814</a>.</li> + <li class="isub1"> + And logic, Chapter <a + href="#CHAPTER_XIII">XIII</a>.</li> + <li class="isub1"> + And philosophy, Chapter <a + href="#CHAPTER_XIV">XIV</a>.</li> + <li class="isub1"> + And science, Chapter <a + href="#CHAPTER_XV">XV</a>.</li> + <li class="isub1"> + And applications, Chapter <a + href="#CHAPTER_XV">XV</a>.</li> + <li class="isub1"> + Knowledge most in, <a + href="#Block_214">214</a>.</li> + <li class="isub1"> + Suppl. brevity of life, <a + href="#Block_218">218</a>.</li> + <li class="isub1"> + The range of, <a + href="#Block_269">269</a>.</li> + <li class="isub1"> + Compared to French language, <a + href="#Block_311">311</a>.</li> + <li class="isub1"> + The care of great men, <a + href="#Block_322">322</a>.</li> + <li class="isub1"> + And professional education, <a + href="#Block_429">429</a>.</li> + <li class="isub1"> + And science teaching, <a + href="#Block_522">522</a>.</li> + <li class="isub1"> + The queen of the sciences, <a + href="#Block_975">975</a>.</li> + <li class="isub1"> + Advantage over philosophy, <a + href="#Block_1436">1436</a>, <a + href="#Block_1438">1438</a>.</li> + <li class="isub1"> + As an instrument, <a + href="#Block_1506">1506</a>.</li> + <li class="isub1"> + For its own sake, <a + href="#Block_1540">1540</a>, <a + href="#Block_1541">1541</a>, <a + href="#Block_1545">1545</a>, <a + href="#Block_1546">1546</a>.</li> + <li class="isub1"> + The wings of, <a + href="#Block_1604">1604</a>.</li> + <li class="indx"> + Mathesis, <a + href="#Block_274">274</a>, <a + href="#Block_276">276</a>, <a + href="#Block_1870">1870</a>, <a + href="#Block_2015">2015</a>.</li> + <li class="indx"> + Mathews,</li> + <li class="isub1"> + On Disqu. Arith. <b><a + href="#Block_1638">1638</a></b>.</li> + <li class="isub1"> + On number theory, <b><a + href="#Block_1639">1639</a></b>.</li> + <li class="isub1"> + The symbol ≡, <b><a + href="#Block_1646">1646</a></b>.</li> + <li class="isub1"> + On Cyclotomy, <b><a + href="#Block_1647">1647</a></b>.</li> + <li class="isub1"> + Laws of algebra, <b><a + href="#Block_1709">1709</a></b>.</li> + <li class="isub1"> + On infinite, zero, infinitesimal, <b><a + href="#Block_1954">1954</a></b>.</li> + <li class="indx"> + Maxims of great math’ns, <a + href="#Block_630">630</a>, <a + href="#Block_631">631</a>, <a + href="#Block_649">649</a>.</li> + <li class="indx"> + Maxwell, <a + href="#Block_1043">1043</a>, <a + href="#Block_1116">1116</a>. + +<span class="pagenum"> + <a name="Page_401" + id="Page_401">401</a></span></li> + + <li class="indx"> + McCormack,</li> + <li class="isub1"> + On the unpopularity of m., <b><a + href="#Block_270">270</a></b>.</li> + <li class="isub1"> + On function, <b><a + href="#Block_1933">1933</a></b>.</li> + <li class="indx"> + Méchanique céleste, <a + href="#Block_985">985</a>, <a + href="#Block_986">986</a>.</li> + <li class="indx"> + Medicine, M. and the study of, <a + href="#Block_1585">1585</a>, <a + href="#Block_1918">1918</a>.</li> + <li class="indx"> + Mellor,</li> + <li class="isub1"> + Definition of higher m., <b><a + href="#Block_108">108</a></b>.</li> + <li class="isub1"> + Conclusions involved in premises, <b><a + href="#Block_238">238</a></b>.</li> + <li class="isub1"> + On m. and science, <b><a + href="#Block_1561">1561</a></b>.</li> + <li class="isub1"> + On the calculus, <b><a + href="#Block_1912">1912</a></b>.</li> + <li class="isub1"> + On integration, <b><a + href="#Block_1923">1923</a></b>, <b><a + href="#Block_1925">1925</a></b>.</li> + <li class="indx"> + Memory in m., <a + href="#Block_253">253</a>.</li> + <li class="indx"> + Menæchmus, <a + href="#Block_901">901</a>.</li> + <li class="indx"> + Mere math’ns, <a + href="#Block_820">820</a>, <a + href="#Block_821">821</a>.</li> + <li class="indx"> + Merz,</li> + <li class="isub1"> + On the transforming power of m., <b><a + href="#Block_303">303</a></b>.</li> + <li class="isub1"> + On the dominant ideas in m., <b><a + href="#Block_725">725</a></b>.</li> + <li class="isub1"> + On extreme views in m., <b><a + href="#Block_827">827</a></b>.</li> + <li class="isub1"> + On Leibnitz’s work, <b><a + href="#Block_989">989</a></b>.</li> + <li class="isub1"> + On the math. tendency of Leibnitz, <b><a + href="#Block_990">990</a></b>.</li> + <li class="isub1"> + On m. as a lens, <b><a + href="#Block_1515">1515</a></b>.</li> + <li class="isub1"> + M. extends knowledge, <b><a + href="#Block_1524">1524</a></b>.</li> + <li class="isub1"> + Disquisitiones Arithmeticae, <b><a + href="#Block_1637">1637</a></b>.</li> + <li class="isub1"> + On functions, <b><a + href="#Block_1932">1932</a></b>.</li> + <li class="isub1"> + On hyper-space, <b><a + href="#Block_2036">2036</a></b>.</li> + <li class="indx"> + Metaphysics, M. the only true, <a + href="#Block_305">305</a>.</li> + <li class="indx"> + Meteorology and m., <a + href="#Block_1557">1557</a>.</li> + <li class="indx"> + Method of m. <a + href="#Block_212">212-215</a>, <a + href="#Block_226">226</a>, <a + href="#Block_227">227</a>, <a + href="#Block_230">230</a>, <a + href="#Block_235">235</a>, <a + href="#Block_244">244</a>, <a + href="#Block_806">806</a>, <a + href="#Block_1576">1576</a>.</li> + <li class="indx"> + Metric system, <a + href="#Block_1725">1725</a>.</li> + <li class="indx"> + Military training, M. in, <a + href="#Block_314">314</a>, <a + href="#Block_418">418</a>, <a + href="#Block_1574">1574</a>.</li> + <li class="indx"> + Mill,</li> + <li class="isub1"> + On induction in m., <b><a + href="#Block_221">221</a></b>, <b><a + href="#Block_222">222</a></b>.</li> + <li class="isub1"> + On generalization in m., <b><a + href="#Block_245">245</a></b>.</li> + <li class="isub1"> + On math. studies, <b><a + href="#Block_409">409</a></b>.</li> + <li class="isub1"> + On m. in a scientific education, <b><a + href="#Block_444">444</a></b>.</li> + <li class="isub1"> + Math’ns hard to convince, <b><a + href="#Block_811">811</a></b>.</li> + <li class="isub1"> + Math’ns require genius, <b><a + href="#Block_819">819</a></b>.</li> + <li class="isub1"> + On Comte, <b><a + href="#Block_942">942</a></b>.</li> + <li class="isub1"> + On Descartes, <b><a + href="#Block_942">942</a></b>, <b><a + href="#Block_948">948</a></b>.</li> + <li class="isub1"> + On Sir William Hamilton’s ignorance of m., <b><a + href="#Block_978">978</a></b>.</li> + <li class="isub1"> + On Leibnitz, <b><a + href="#Block_987">987</a></b>.</li> + <li class="isub1"> + On m. and philosophy, <b><a + href="#Block_1421">1421</a></b>.</li> + <li class="isub1"> + On m. as training for philosophers, <b><a + href="#Block_1422">1422</a></b>.</li> + <li class="isub1"> + M. indispensable to science, <b><a + href="#Block_1519">1519</a></b>.</li> + <li class="isub1"> + M. and social science, <b><a + href="#Block_1595">1595</a></b>.</li> + <li class="isub1"> + On the nature of geometry, <b><a + href="#Block_1838">1838</a></b>.</li> + <li class="isub1"> + On geometrical method, <b><a + href="#Block_1861">1861</a></b>.</li> + <li class="isub1"> + On the calculus, <b><a + href="#Block_1903">1903</a></b>.</li> + <li class="indx"> + Miller, On the Darmstaetter prize, <b><a + href="#Block_2129">2129</a></b>.</li> + <li class="indx"> + Milner, Geometry and poetry, <b><a + href="#Block_1118">1118</a></b>.</li> + <li class="indx"> + Minchin, On English text-books, <b><a + href="#Block_539">539</a></b>.</li> + <li class="indx"> + Mineralogy and m., <a + href="#Block_1558">1558</a>.</li> + <li class="indx"> + Minkowski, On integral numbers, <b><a + href="#Block_1636">1636</a></b>.</li> + <li class="indx"> + Miracles and m., <a + href="#Block_2157">2157</a>, <a + href="#Block_2158">2158</a>, <a + href="#Block_2160">2160</a>.</li> + <li class="indx"> + Mixed m.,</li> + <li class="isub1"> + Bacon’s definition of, <a + href="#Block_106">106</a>.</li> + <li class="isub1"> + Whewell’s definition of, <a + href="#Block_107">107</a>.</li> + <li class="indx"> + Modern algebra, <a + href="#Block_1031">1031</a>, <a + href="#Block_1032">1032</a>, <a + href="#Block_1638">1638</a>, <a + href="#Block_1741">1741</a>.</li> + <li class="indx"> + Modern geometry, <a + href="#Block_1710">1710-1713</a>, <a + href="#Block_715">715</a>, <a + href="#Block_716">716</a>, <a + href="#Block_1878">1878</a>.</li> + <li class="indx"> + Modern m., Chapter <a + href="#CHAPTER_VII">VII</a>.</li> + <li class="indx"> + Moebius,</li> + <li class="isub1"> + Math’ns constitute a favorite class, <b><a + href="#Block_809">809</a></b>.</li> + <li class="isub1"> + M. a fine art, <b><a + href="#Block_1107">1107</a></b>.</li> + <li class="indx"> + Moral science and m., <a + href="#Block_1438">1438-1440</a>.</li> + <li class="indx"> + Moral value of m., See ethical value.</li> + <li class="indx"> + Mottoes,</li> + <li class="isub1"> + Of math’ns, <a + href="#Block_630">630</a>, <a + href="#Block_631">631</a>, <a + href="#Block_649">649</a>.</li> + <li class="isub1"> + Of Pythagoreans, <a + href="#Block_1833">1833</a>.</li> + <li class="indx"> + Murray, Definition of m., <b><a + href="#Block_116">116</a></b>.</li> + <li class="indx"> + Music and m., <a + href="#Block_101">101</a>, <a + href="#Block_276">276</a>, <a + href="#Block_965">965</a>, <a + href="#Block_1107">1107</a>, <a + href="#Block_1112">1112</a>, <a + href="#Block_1116">1116</a>, <a + href="#Block_1127">1127</a>, <a + href="#Block_1128">1128</a>, <a + href="#Block_1130">1130-1133</a>, <a + href="#Block_1135">1135</a>, <a + href="#Block_1136">1136</a>.</li> + <li class="indx"> + Myers,</li> + <li class="isub1"> + On m. as a school subject, <b><a + href="#Block_403">403</a></b>.</li> + <li class="isub1"> + On pleasure in m., <b><a + href="#Block_454">454</a></b>.</li> + <li class="isub1"> + On the ethical value of m., <b><a + href="#Block_457">457</a></b>.</li> + <li class="isub1"> + On the value of arithmetic, <b><a + href="#Block_1622">1622</a></b>.</li> + <li class="indx"> + Mysticism and numbers, <a + href="#Block_2136">2136-2141</a>, <a + href="#Block_2143">2143</a>. + +<span class="pagenum"> + <a name="Page_402" + id="Page_402">402</a></span></li> + + <li class="ifrst"> + Napier’s rule, <a + href="#Block_1888">1888</a>.</li> + <li class="indx"> + Napoleon,</li> + <li class="isub1"> + M. and the welfare of the state, <b><a + href="#Block_313">313</a></b>.</li> + <li class="isub1"> + His interest in m., <a + href="#Block_314">314</a>, <a + href="#Block_1001">1001</a>.</li> + <li class="indx"> + Natural science and m.,</li> + <li class="isub1"> + Chapter <a + href="#CHAPTER_XV">XV</a>.</li> + <li class="isub1"> + Also <a + href="#Block_244">244</a>, <a + href="#Block_444">444</a>, <a + href="#Block_445">445</a>, <a + href="#Block_501">501</a>.</li> + <li class="indx"> + Natural selection, <a + href="#Block_1921">1921</a>.</li> + <li class="indx"> + Nature of m.,</li> + <li class="isub1"> + Chapter <a + href="#CHAPTER_II">II</a>.</li> + <li class="isub1"> + See also <a + href="#Block_815">815</a>, <a + href="#Block_1215">1215</a>, <a + href="#Block_1308">1308</a>, <a + href="#Block_1426">1426</a>,<a + href="#Block_1525">1525</a>, <a + href="#Block_1628">1628</a>.</li> + <li class="indx"> + Nature, Study of, <a + href="#Block_433">433-436</a>, <a + href="#Block_514">514</a>, <a + href="#Block_516">516</a>, <a + href="#Block_612">612</a>.</li> + <li class="indx"> + Navigation and m., <a + href="#Block_1543">1543</a>, <a + href="#Block_1544">1544</a>.</li> + <li class="indx"> + Nelson, Anagram on, <a + href="#Block_2153">2153</a>.</li> + <li class="indx"> + Neptune, Discovery of, <a + href="#Block_1554">1554</a>, <a + href="#Block_1559">1559</a>.</li> + <li class="indx"> + Newcomb, On geometrical paradoxers, <b><a + href="#Block_2113">2113</a></b>.</li> + <li class="indx"> + Newton,</li> + <li class="isub1"> + Importance of his work,<a + href="#Block_333">333</a>.</li> + <li class="isub1"> + On correlation in m., <b><a + href="#Block_526">526</a></b>.</li> + <li class="isub1"> + On problems in algebra, <b><a + href="#Block_530">530</a></b>.</li> + <li class="isub1"> + And Gauss compared, <a + href="#Block_827">827</a>.</li> + <li class="isub1"> + His fame, <a + href="#Block_1002">1002</a>.</li> + <li class="isub1"> + Emerson on, <a + href="#Block_1003">1003</a>.</li> + <li class="isub1"> + Whewell on, <a + href="#Block_1004">1004</a>,<a + href="#Block_1005">1005</a>.</li> + <li class="isub1"> + Arago on, <a + href="#Block_1006">1006</a>.</li> + <li class="isub1"> + Pope on, <a + href="#Block_1007">1007</a>.</li> + <li class="isub1"> + Southey on, <a + href="#Block_1008">1008</a>.</li> + <li class="isub1"> + Hill on, <a + href="#Block_1009">1009</a>.</li> + <li class="isub1"> + Leibnitz on, <a + href="#Block_1010">1010</a>.</li> + <li class="isub1"> + Lagrange on, <a + href="#Block_1011">1011</a>.</li> + <li class="isub1"> + No monument to, <a + href="#Block_1012">1012</a>.</li> + <li class="isub1"> + Wilson on, <a + href="#Block_1012">1012</a>, <a + href="#Block_1013">1013</a>.</li> + <li class="isub1"> + His genius, <a + href="#Block_1014">1014</a>.</li> + <li class="isub1"> + His interest in chemistry and theology, <a + href="#Block_1015">1015</a>.</li> + <li class="isub1"> + And alchemy, <a + href="#Block_1016">1016</a>, <a + href="#Block_1017">1017</a>.</li> + <li class="isub1"> + His first experiment, <a + href="#Block_1018">1018</a>.</li> + <li class="isub1"> + As a lecturer, <a + href="#Block_1019">1019</a>.</li> + <li class="isub1"> + As an accountant, <a + href="#Block_1020">1020</a>.</li> + <li class="isub1"> + His memorandum-book, <a + href="#Block_1021">1021</a>.</li> + <li class="isub1"> + His absent-mindedness, <a + href="#Block_1022">1022</a>.</li> + <li class="isub1"> + Estimate of himself, <b><a + href="#Block_1023">1023-1025</a></b>.</li> + <li class="isub1"> + His method of work, <a + href="#Block_1026">1026</a>.</li> + <li class="isub1"> + Discovery of the calculus, <a + href="#Block_1027">1027</a>.</li> + <li class="isub1"> + Anagrams on, <a + href="#Block_1028">1028</a>.</li> + <li class="isub1"> + Gauss’s estimate of, <a + href="#Block_1029">1029</a>.</li> + <li class="isub1"> + On geometry, <b><a + href="#Block_1811">1811</a></b>.</li> + <li class="isub1"> + Compared with Euclid, <a + href="#Block_1827">1827</a>.</li> + <li class="isub1"> + Geometry a mechanical science, <b><a + href="#Block_1865">1865</a></b>.</li> + <li class="isub1"> + Test of simplicity, <b><a + href="#Block_1892">1892</a></b>.</li> + <li class="isub1"> + Method of fluxions, <a + href="#Block_1902">1902</a>.</li> + <li class="indx"> + Newton’s rule, <a + href="#Block_1743">1743</a>.</li> + <li class="indx"> + Nile, Origin of name, <a + href="#Block_2150">2150</a>.</li> + <li class="indx"> + Noether,</li> + <li class="isub1"> + On Cayley, <b><a + href="#Block_931">931</a></b>.</li> + <li class="isub1"> + On Sylvester, <b><a + href="#Block_1034">1034</a></b>, <b><a + href="#Block_1041">1041</a></b>.</li> + <li class="indx"> + Non-euclidean geometry, <a + href="#Block_1322">1322</a>, <a + href="#Block_2016">2016-2029</a>, <a + href="#Block_2033">2033</a>, <a + href="#Block_2035">2035</a>, <a + href="#Block_2040">2040</a>.</li> + <li class="indx"> + Nonnus, On the mystic four, <b><a + href="#Block_2148">2148</a></b>.</li> + <li class="indx"> + Northrup, On Lord Kelvin, <b><a + href="#Block_1048">1048</a></b>.</li> + <li class="indx"> + Notation,</li> + <li class="isub1"> + Importance of, <a + href="#Block_634">634</a>, <a + href="#Block_1222">1222</a>, <a + href="#Block_1646">1646</a>.</li> + <li class="isub1"> + Value of algebraic, <a + href="#Block_1213">1213</a>, <a + href="#Block_1214">1214</a>.</li> + <li class="isub1"> + Criterion of good, <a + href="#Block_1216">1216</a>.</li> + <li class="isub1"> + On Arabic, <a + href="#Block_1217">1217</a>, <a + href="#Block_1614">1614</a>.</li> + <li class="isub1"> + Advantage of math., <a + href="#Block_1220">1220</a>.</li> + <li class="isub1"> + See also symbolism.</li> + <li class="indx"> + Notions,</li> + <li class="isub1"> + Cardinal of m., <a + href="#Block_110">110</a>.</li> + <li class="isub1"> + Indefinable, <a + href="#Block_1219">1219</a>.</li> + <li class="indx"> + Novalis, Definition of pure m., <b><a + href="#Block_112">112</a></b>.</li> + <li class="isub1"> + M. the life supreme, <b><a + href="#Block_329">329</a></b>.</li> + <li class="isub1"> + Without enthusiasm no m., <b><a + href="#Block_801">801</a></b>.</li> + <li class="isub1"> + Method is the essence of m., <b><a + href="#Block_806">806</a></b>.</li> + <li class="isub1"> + Math’ns not good computers, <b><a + href="#Block_810">810</a></b>.</li> + <li class="isub1"> + Music and algebra, <b><a + href="#Block_1128">1128</a></b>.</li> + <li class="isub1"> + Philosophy and m., <b><a + href="#Block_1406">1406</a></b>.</li> + <li class="isub1"> + M. and science, <b><a + href="#Block_1507">1507</a></b>, <b><a + href="#Block_1526">1526</a></b>.</li> + <li class="isub1"> + M. and historic science, <b><a + href="#Block_1599">1599</a></b>.</li> + <li class="isub1"> + M. and magic, <b><a + href="#Block_2159">2159</a></b>.</li> + <li class="isub1"> + M. and miracles, <b><a + href="#Block_2160">2160</a></b>.</li> + <li class="indx"> + Number,</li> + <li class="isub1"> + Every inquiry reducible to a question of, <a + href="#Block_1602">1602</a>.</li> + <li class="isub1"> + And nature, <a + href="#Block_1603">1603</a>.</li> + <li class="isub1"> + Regulates all things, <a + href="#Block_1605">1605</a>.</li> + <li class="isub1"> + Aeschylus on, <a + href="#Block_1606">1606</a>.</li> + <li class="isub1"> + Definition of, <a + href="#Block_1609">1609</a>, <a + href="#Block_1610">1610</a>.</li> + <li class="isub1"> + And superstition, <a + href="#Block_1632">1632</a>.</li> + <li class="isub1"> + Distinctness of, <a + href="#Block_1707">1707</a>.</li> + <li class="isub1"> + Of the beast, <a + href="#Block_2151">2151</a>, <a + href="#Block_2152">2152</a>.</li> + <li class="indx"> + Number-theory,</li> + <li class="isub1"> + The queen of m., <a + href="#Block_975">975</a>.</li> + <li class="isub1"> + Nature of, <a + href="#Block_1639">1639</a>.</li> + <li class="isub1"> + Gauss on, <a + href="#Block_1644">1644</a>. + +<span class="pagenum"> + <a name="Page_403" + id="Page_403">403</a></span></li> + + <li class="isub1"> + Smith on, <a + href="#Block_1645">1645</a>.</li> + <li class="isub1"> + Notation in, <a + href="#Block_1646">1646</a>.</li> + <li class="isub1"> + Aid to geometry, <a + href="#Block_1647">1647</a>.</li> + <li class="isub1"> + Mystery in, <a + href="#Block_1648">1648</a>.</li> + <li class="indx"> + Number-work, Purpose of, <a + href="#Block_1623">1623</a>.</li> + <li class="indx"> + Numbers,</li> + <li class="isub1"> + Pythagoras’ view of, <a + href="#Block_321">321</a>.</li> + <li class="isub1"> + Mighty are, <a + href="#Block_1568">1568</a>.</li> + <li class="isub1"> + Aptitude for, <a + href="#Block_1617">1617</a>.</li> + <li class="isub1"> + Demonstrations in, <a + href="#Block_1630">1630</a>.</li> + <li class="isub1"> + Prime, <a + href="#Block_1648">1648</a>.</li> + <li class="isub1"> + Necessary truths like, <a + href="#Block_1966">1966</a>.</li> + <li class="isub1"> + Round, <a + href="#Block_2137">2137</a>.</li> + <li class="isub1"> + Odd, <a + href="#Block_2138">2138-2141</a>.</li> + <li class="isub1"> + Golden, <a + href="#Block_2142">2142</a>.</li> + <li class="isub1"> + Magic, <a + href="#Block_2143">2143</a>.</li> + + <li class="ifrst"> + Obscurity in m. and philosophy, <a + href="#Block_1407">1407</a>.</li> + <li class="indx"> + Observation in m., <a + href="#Block_251">251-253</a>, <a + href="#Block_255">255</a>, <a + href="#Block_433">433</a>, <a + href="#Block_1830">1830</a>.</li> + <li class="indx"> + Obviousness in m., <a + href="#Block_985">985</a>, <a + href="#Block_986">986</a>, <a + href="#Block_1045">1045</a>.</li> + <li class="indx"> + Olney, On the nature of m., <b><a + href="#Block_253">253</a></b>.</li> + <li class="indx"> + Oratory and m., <a + href="#Block_829">829</a>, <a + href="#Block_830">830</a>.</li> + <li class="indx"> + Order and arrangement, <a + href="#Block_725">725</a>.</li> + <li class="indx"> + Origin of m., <a + href="#Block_1412">1412</a>.</li> + <li class="indx"> + Orr, Memory verse for π, <b><a + href="#Block_2127">2127</a></b>.</li> + <li class="indx"> + Osgood, On the calculus, <b><a + href="#Block_1913">1913</a></b>.</li> + <li class="indx"> + Ostwald, On four-dimensional space, <a + href="#Block_2039">2039</a>.</li> + + <li class="ifrst"> + π.</li> + <li class="isub1"> + In actuarial formula, <a + href="#Block_945">945</a>.</li> + <li class="isub1"> + Memory verse for, <a + href="#Block_2127">2127</a>.</li> + <li class="indx"> + Pacioli, On the number three, <b><a + href="#Block_2145">2145</a></b>.</li> + <li class="indx"> + Painting and m., <a + href="#Block_1103">1103</a>, <a + href="#Block_1107">1107</a>.</li> + <li class="indx"> + Papperitz, On the object of pure m., <a + href="#Block_111">111</a>.</li> + <li class="indx"> + Paradoxes, Chapter <a + href="#CHAPTER_XXI">XXI</a>.</li> + <li class="indx"> + Parallel axiom,</li> + <li class="isub1"> + Proof of, <a + href="#Block_984">984</a>, <a + href="#Block_2110">2110</a>, <a + href="#Block_2111">2111</a>.</li> + <li class="isub1"> + See also non-euclidean geometry.</li> + <li class="indx"> + Parker,</li> + <li class="isub1"> + Definition of arithmetic, <b><a + href="#Block_1611">1611</a></b>.</li> + <li class="isub1"> + Number born in superstition, <b><a + href="#Block_1632">1632</a></b>.</li> + <li class="isub1"> + On geometry, <b><a + href="#Block_1805">1805</a></b>.</li> + <li class="indx"> + Parton, On Newton, <b><em><a + href="#Block_1017">1917-1919</a></em></b>, <b><a + href="#Block_1021">1021</a></b>, <b><a + href="#Block_1022">1022</a></b>, <b><a + href="#Block_1827">1827</a></b>.</li> + <li class="indx"> + Pascal, Logic and m., <b><a + href="#Block_1306">1306</a></b>.</li> + <li class="indx"> + <a id="TNanchor_25">Peacock</a>,</li> + <li class="isub1"> + On the mysticism of Greek + + <a class="msg" + href="#TN_25" + title="originally spelled ‘Philosphers’">philosophers</a>, + <b><a + href="#Block_2136">2136</a></b>.</li> + + <li class="isub1"> + The Yankos word for three, <b><a + href="#Block_2144">2144</a></b>.</li> + <li class="isub1"> + The number of the beast, <b><a + href="#Block_2152">2152</a></b>.</li> + <li class="indx"> + Pearson, M. and natural selection, <b><a + href="#Block_834">834</a></b>.</li> + <li class="indx"> + Peirce, Benjamin,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_120">120</a></b>.</li> + <li class="isub1"> + M. as an arbiter, <b><a + href="#Block_210">210</a></b>.</li> + <li class="isub1"> + Logic dependent on m., <b><a + href="#Block_1301">1301</a></b>.</li> + <li class="isub1"> + On the symbol √-1, <b><a + href="#Block_1733">1733</a></b>.</li> + <li class="indx"> + Peirce, C. S.</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_133">133</a></b>.</li> + <li class="isub1"> + On accidental relations, <b><a + href="#Block_2128">2128</a></b>.</li> + <li class="indx"> + Perry, On the teaching of m., <b><a + href="#Block_510">510</a></b>, <b><a + href="#Block_511">511</a></b>, <b><a + href="#Block_519">519</a></b>, <b><a + href="#Block_837">837</a></b>.</li> + <li class="indx"> + Persons and anecdotes, Chapters <a + href="#CHAPTER_IX">IX</a>. and <a + href="#CHAPTER_X">X</a>.</li> + <li class="indx"> + Philosophy and m.,</li> + <li class="isub1"> + Chapter <a + href="#CHAPTER_XIV">XIV</a>.</li> + <li class="isub1"> + Also <a + href="#Block_332">332</a>, <a + href="#Block_401">401</a>, <a + href="#Block_414">414</a>, <a + href="#Block_444">444</a>, <a + href="#Block_445">445</a>, <a + href="#Block_452">452</a>.</li> + <li class="indx"> + Physics and m., <a + href="#Block_129">129</a>, <a + href="#Block_437">437</a>, <a + href="#Block_1516">1516</a>, <a + href="#Block_1530">1530</a>, <a + href="#Block_1535">1535</a>, <a + href="#Block_1538">1538</a>, <a + href="#Block_1539">1539</a>, <a + href="#Block_1548">1548</a>, <a + href="#Block_1549">1549</a>, <a + href="#Block_1550">1550</a>, <a + href="#Block_1555">1555</a>, <a + href="#Block_1556">1556</a>.</li> + <li class="indx"> + Physiology and m., <a + href="#Block_1578">1578</a>, <a + href="#Block_1581">1581</a>, <a + href="#Block_1582">1582</a>.</li> + <li class="indx"> + Picard, On the use of equations, <b><a + href="#Block_1891">1891</a></b>.</li> + <li class="indx"> + Pierce, On infinitesimals, <b><a + href="#Block_1940">1940</a></b>.</li> + <li class="indx"> + Pierpont,</li> + <li class="isub1"> + Golden age of m., <b><a + href="#Block_701">701</a></b>.</li> + <li class="isub1"> + On the progress of m., <b><a + href="#Block_708">708</a></b>.</li> + <li class="isub1"> + Characteristics of modern m., <b><a + href="#Block_717">717</a></b>.</li> + <li class="isub1"> + On variability, <b><a + href="#Block_721">721</a></b>.</li> + <li class="isub1"> + On divergent series, <b><a + href="#Block_1937">1937</a></b>.</li> + <li class="indx"> + Plato,</li> + <li class="isub1"> + His view of m., <a + href="#Block_316">316</a>, <a + href="#Block_429">429</a>.</li> + <li class="isub1"> + M. a study suitable for freemen, <b><a + href="#Block_317">317</a></b>.</li> + <li class="isub1"> + His conic sections, <a + href="#Block_332">332</a>.</li> + <li class="isub1"> + And Archimedes, <a + href="#Block_904">904</a>.</li> + <li class="isub1"> + Union of math. and philosophical productivity, <a + href="#Block_1404">1404</a>.</li> + <li class="isub1"> + Diagonal of square, <b><a + href="#Block_1411">1411</a></b>.</li> + <li class="isub1"> + And Archytas, <a + href="#Block_1427">1427</a>.</li> + <li class="isub1"> + M. and the arts, <b><a + href="#Block_1567">1567</a></b>.</li> + <li class="isub1"> + On the value of m., <b><a + href="#Block_1574">1574</a></b>.</li> + <li class="isub1"> + On arithmetic, <b><a + href="#Block_1620">1620</a></b>, <b><a + href="#Block_1621">1621</a></b>. + +<span class="pagenum"> + <a name="Page_404" + id="Page_404">404</a></span></li> + + <li class="isub1"> + God + geometrizes, <a + href="#Block_1635">1635</a>, <a + href="#Block_1636">1636</a>. <a + href="#Block_1702">1702</a>.</li> + <li class="isub1"> + On geometry, <a + href="#Block_429">429</a>, <a + href="#Block_1803">1803</a>, <a + href="#Block_1804">1804</a>, <b><a + href="#Block_1806">1806</a></b>, <b><a + href="#Block_1844">1844</a></b>, <b><a + href="#Block_1845">1845</a></b>.</li> + <li class="indx"> + Pleasure, Element of in m., <a + href="#Block_1622">1622</a>, <a + href="#Block_1629">1629</a>, <a + href="#Block_1848">1848</a>, <a + href="#Block_1850">1850</a>, <a + href="#Block_1851">1851</a>.</li> + <li class="indx"> + Pliny, <b><em><a + href="#Block_2139">2039</a></em></b>.</li> + <li class="indx"> + Plus and minus signs, <a + href="#Block_1727">1727</a>.</li> + <li class="indx"> + Plutarch,</li> + <li class="isub1"> + On Archimedes, <b><a + href="#Block_903">903</a></b>, <b><a + href="#Block_904">904</a></b>, <b><a + href="#Block_908">908-910</a></b>, <b><a + href="#Block_912">912</a></b>.</li> + <li class="isub1"> + God geometrizes, <b><a + href="#Block_1802">1802</a></b>.</li> + <li class="indx"> + Poe, <b><em><a + href="#Block_416">417</a></em></b>.</li> + <li class="indx"> + Poetry and m.,</li> + <li class="isub1"> + Weierstrass on, <a + href="#Block_802">802</a>.</li> + <li class="isub1"> + Pringsheim on, <a + href="#Block_1108">1108</a>.</li> + <li class="isub1"> + Wordsworth on, <a + href="#Block_1117">1117</a>.</li> + <li class="isub1"> + Milner on, <a + href="#Block_1118">1118</a>.</li> + <li class="isub1"> + Workman on, <a + href="#Block_1120">1120</a>.</li> + <li class="isub1"> + Pollock on, <a + href="#Block_1121">1121</a>.</li> + <li class="isub1"> + Hoffman on, <a + href="#Block_1122">1122</a>.</li> + <li class="isub1"> + Thoreau on, <a + href="#Block_1123">1123</a>.</li> + <li class="isub1"> + Emerson on, <a + href="#Block_1124">1124</a>.</li> + <li class="isub1"> + Hill on, <a + href="#Block_1125">1125</a>, <a + href="#Block_1126">1126</a>.</li> + <li class="isub1"> + Shakespeare on, <a + href="#Block_1127">1127</a>.</li> + <li class="indx"> + Poincaré,</li> + <li class="isub1"> + On elegance in m., <b><a + href="#Block_640">640</a></b>.</li> + <li class="isub1"> + M. has a triple end, <b><a + href="#Block_1102">1102</a></b>.</li> + <li class="isub1"> + M. as a language, <b><a + href="#Block_1208">1208</a></b>.</li> + <li class="isub1"> + Geometry not an experimental science, <b><a + href="#Block_1867">1867</a></b>.</li> + <li class="isub1"> + On geometrical axioms, <b><a + href="#Block_2005">2005</a></b>.</li> + <li class="indx"> + Point, <a + href="#Block_1816">1816</a>.</li> + <li class="indx"> + Political science, M. and, <a + href="#Block_1201">1201</a>, <a + href="#Block_1324">1324</a>.</li> + <li class="indx"> + Politics, Math’ns and, <a + href="#Block_814">814</a>.</li> + <li class="indx"> + Pollock, On Clifford, <b><a + href="#Block_938">938-941</a></b>, <b><a + href="#Block_1121">1121</a></b>.</li> + <li class="indx"> + Pope, <i><b><a + href="#Block_1007">907</a></b>, <b><a + href="#Block_2115">2015</a></b>, <b><a + href="#Block_2131">2031</a></b>, <b><a + href="#Block_2146">2046</a></b></i>.</li> + <li class="indx"> + Precision in m., <a + href="#Block_228">228</a>, <a + href="#Block_639">639</a>, <a + href="#Block_728">728</a>.</li> + <li class="indx"> + Precocity in m., <a + href="#Block_835">835</a>.</li> + <li class="indx"> + Predicabilia a priori, <a + href="#Block_2003">2003</a>.</li> + <li class="indx"> + Press, M. ignored by daily, <a + href="#Block_731">731</a>, <a + href="#Block_732">732</a>.</li> + <li class="indx"> + Price,</li> + <li class="isub1"> + Characteristics of m., <b><a + href="#Block_247">247</a></b>.</li> + <li class="isub1"> + On m. and physics, <b><a + href="#Block_1550">1550</a></b>.</li> + <li class="indx"> + Prime numbers, Sylvester on, <a + href="#Block_1648">1648</a>.</li> + <li class="indx"> + Principia Mathematica, <a + href="#Block_1326">1326</a>.</li> + <li class="indx"> + Pringsheim,</li> + <li class="isub1"> + M. the science of the self-evident, <b><a + href="#Block_232">232</a></b>.</li> + <li class="isub1"> + M. should be studied for its own sake, <b><a + href="#Block_439">439</a></b>.</li> + <li class="isub1"> + On the indirect value of m., <b><a + href="#Block_448">448</a></b>.</li> + <li class="isub1"> + On rigor in m., <b><a + href="#Block_535">535</a></b>.</li> + <li class="isub1"> + On m. and journalism, <b><a + href="#Block_732">732</a></b>.</li> + <li class="isub1"> + On math’ns in public service, <b><a + href="#Block_824">824</a></b>.</li> + <li class="isub1"> + Math’n somewhat of a poet, <b><a + href="#Block_1108">1108</a></b>.</li> + <li class="isub1"> + On music and m., <b><a + href="#Block_1132">1132</a></b>.</li> + <li class="isub1"> + On the language of m., <b><a + href="#Block_1211">1211</a></b>.</li> + <li class="isub1"> + On m. and physics, <b><a + href="#Block_1548">1548</a></b>.</li> + <li class="indx"> + Probabilities, <a + href="#Block_442">442</a>, <a + href="#Block_823">823</a>, <a + href="#Block_1589">1589</a>, <a + href="#Block_1590">1590-1592</a>, <a + href="#Block_1962">1962-1972</a>, <a + href="#Block_1975">1975</a>.</li> + <li class="indx"> + Problem solving, <a + href="#Block_531">531</a>, <a + href="#Block_532">532</a>.</li> + <li class="indx"> + Problems,</li> + <li class="isub1"> + In m., <a + href="#Block_523">523</a>, <a + href="#Block_534">534</a>.</li> + <li class="isub1"> + In arithmetic, <a + href="#Block_528">528</a>.</li> + <li class="isub1"> + In algebra, <a + href="#Block_530">530</a>.</li> + <li class="isub1"> + Should be simple, <a + href="#Block_603">603</a>.</li> + <li class="isub1"> + In Cambridge texts, <a + href="#Block_608">608</a>.</li> + <li class="isub1"> + On solution of, <a + href="#Block_611">611</a>.</li> + <li class="isub1"> + On importance of, <a + href="#Block_624">624</a>, <a + href="#Block_628">628</a>.</li> + <li class="isub1"> + What constitutes good, <a + href="#Block_629">629</a>.</li> + <li class="isub1"> + Aid to research, <a + href="#Block_644">644</a>.</li> + <li class="isub1"> + Of modern m., <a + href="#Block_1926">1926</a>.</li> + <li class="indx"> + Proclus,</li> + <li class="isub1"> + Ptolemy and Euclid, <b><a + href="#Block_951">951</a></b>.</li> + <li class="isub1"> + On characteristics of geometry, <b><a + href="#Block_1869">1869</a></b>.</li> + <li class="indx"> + Progress in m., <a + href="#Block_209">209</a>, <a + href="#Block_211">211</a>, <a + href="#Block_212">212</a>, <a + href="#Block_216">216</a>, <a + href="#Block_218">218</a>, <a + href="#Block_702">702-705</a>, <a + href="#Block_708">708</a>.</li> + <li class="indx"> + Projective geometry, <a + href="#Block_1876">1876</a>, <a + href="#Block_1877">1877</a>, <a + href="#Block_1879">1879</a>, <a + href="#Block_1880">1880</a>.</li> + <li class="indx"> + Proportion,</li> + <li class="isub1"> + Euclid’s doctrine of, <a + href="#Block_1834">1834</a>.</li> + <li class="isub1"> + Euclid’s definition of, <a + href="#Block_1835">1835</a>.</li> + <li class="indx"> + Proposition, <a + href="#Block_1219">1219</a>, <a + href="#Block_1419">1419</a>.</li> + <li class="indx"> + Prussia, M. in, <a + href="#Block_513">513</a>.</li> + <li class="indx"> + Pseudomath, Defined, <a + href="#Block_2101">2101</a>.</li> + <li class="indx"> + Psychology and m., <a + href="#Block_1576">1576</a>, <a + href="#Block_1583">1583</a>, <a + href="#Block_1584">1584</a>.</li> + <li class="indx"> + Ptolemy and Euclid, <a + href="#Block_951">951</a>.</li> + <li class="indx"> + Public service, M. and, <a + href="#Block_823">823</a>, <a + href="#Block_824">824</a>, <a + href="#Block_1303">1303</a>, <a + href="#Block_1574">1574</a>.</li> + <li class="indx"> + Public speaking, M. and, <a + href="#Block_420">420</a>, <a + href="#Block_829">829</a>, <a + href="#Block_830">830</a>.</li> + <li class="indx"> + Publications, Math. of present day, <a + href="#Block_702">702</a>, <a + href="#Block_703">703</a>.</li> + <li class="indx"> + Pure M.,</li> + <li class="isub1"> + Bacon’s definition of, <a + href="#Block_106">106</a>.</li> + <li class="isub1"> + Whewell’s definition of, <a + href="#Block_107">107</a>.</li> + <li class="isub1"> + On the object of, <a + href="#Block_111">111</a>, <a + href="#Block_129">129</a>.</li> + <li class="isub1"> + Novalis’ conception of, <a + href="#Block_112">112</a>.</li> + <li class="isub1"> + Hobson’s definition of, <a + href="#Block_118">118</a>. + +<span class="pagenum"> + <a name="Page_405" + id="Page_405">405</a></span></li> + + <li class="isub1"> + Russell’s definition of, <a + href="#Block_127">127</a>, <a + href="#Block_128">128</a>.</li> + <li class="indx"> + Pursuit of m., <a + href="#Block_842">842</a>.</li> + <li class="indx"> + Pythagoras,</li> + <li class="isub1"> + Number the nature of things, <a + href="#Block_321">321</a>.</li> + <li class="isub1"> + Union of math, and philosophical productivity, <a + href="#Block_1404">1404</a>.</li> + <li class="isub1"> + The number four, <b><a + href="#Block_2147">2147</a></b>.</li> + <li class="indx"> + Pythagorean brotherhood, Motto of, <a + href="#Block_1833">1833</a>.</li> + <li class="indx"> + Pythagorean theorem, <a + href="#Block_1854">1854-1856</a>, <a + href="#Block_2026">2026</a>.</li> + <li class="indx"> + Pythagoreans, Music and M., <a + href="#Block_1130">1130</a>.</li> + + <li class="ifrst"> + Quadrature, See Squaring of the circle.</li> + <li class="indx"> + Quantity, Chrystal’s definition of, <b><a + href="#Block_115">115</a></b>.</li> + <li class="indx"> + Quarles, On quadrature, <b><a + href="#Block_2116">2116</a></b>.</li> + <li class="indx"> + Quaternions, <a + href="#Block_333">333</a>, <a + href="#Block_841">841</a>, <a + href="#Block_937">937</a>, <a + href="#Block_1044">1044</a>, <a + href="#Block_1210">1210</a>, <a + href="#Block_1718">1718-1726</a>.</li> + <li class="indx"> + Quetelet, Growth of m., <b><a + href="#Block_1514">1514</a></b>.</li> + + <li class="ifrst"> + Railway-making, <a + href="#Block_1570">1570</a>.</li> + <li class="indx"> + Reading of m., <a + href="#Block_601">601</a>, <a + href="#Block_604">604-606</a>.</li> + <li class="indx"> + Reason,</li> + <li class="isub1"> + M. most solid fabric of human, <a + href="#Block_308">308</a>.</li> + <li class="isub1"> + M. demonstrates supremacy of human, <a + href="#Block_309">309</a>.</li> + <li class="indx"> + Reasoning,</li> + <li class="isub1"> + M. a type of perfect, <a + href="#Block_307">307</a>.</li> + <li class="isub1"> + M. as an exercise in, <a + href="#Block_423">423-427</a>, <a + href="#Block_429">429</a>, <a + href="#Block_430">430</a>, <a + href="#Block_1503">1503</a>.</li> + <li class="indx"> + Recorde, Value of arithmetic, <a + href="#Block_1619">1619</a>.</li> + <li class="indx"> + Regiomontanus, <a + href="#Block_1543">1543</a>.</li> + <li class="indx"> + Regular solids, <a + href="#Block_2132">2132-2135</a>.</li> + <li class="indx"> + Reid,</li> + <li class="isub1"> + M. frees from sophistry, <b><a + href="#Block_215">215</a></b>.</li> + <li class="isub1"> + Conjecture has no place in m., <b><a + href="#Block_234">234</a></b>.</li> + <li class="isub1"> + M. the most solid fabric, <b><a + href="#Block_308">308</a></b>.</li> + <li class="isub1"> + On Euclid’s elements, <b><a + href="#Block_955">955</a></b>.</li> + <li class="isub1"> + M. manifests what is impossible <b><a + href="#Block_1414">1414</a></b>.</li> + <li class="isub1"> + On m. and philosophy, <b><a + href="#Block_1423">1423</a></b>.</li> + <li class="isub1"> + Probability and Christianity, <b><a + href="#Block_1975">1975</a></b>.</li> + <li class="isub1"> + On Pythagoras and the regular solids, <b><a + href="#Block_2132">2132</a></b>.</li> + <li class="indx"> + Reidt,</li> + <li class="isub1"> + M, as an exercise in language, <b><a + href="#Block_419">419</a></b>.</li> + <li class="isub1"> + On the ethical value of m., <b><a + href="#Block_456">456</a></b>.</li> + <li class="isub1"> + On aim in math. instruction, <b><a + href="#Block_506">506</a></b>.</li> + <li class="indx"> + Religion and m., <a + href="#Block_274">274-276</a>, <a + href="#Block_459">459</a>, <a + href="#Block_460">460</a>, <a + href="#Block_1013">1013</a>.</li> + <li class="indx"> + Research in m., Chapter <a + href="#CHAPTER_VI">VI</a>.</li> + <li class="indx"> + Reversible verses, <a + href="#Block_2156">2156</a>.</li> + <li class="indx"> + Reye, Advantages of modern over ancient geometry, <b><a + href="#Block_714">714</a></b>.</li> + <li class="indx"> + Rhetoric and m., <a + href="#Block_1599">1599</a>.</li> + <li class="indx"> + Riemann, On m. and physics, <a + href="#Block_1549">1549</a>.</li> + <li class="indx"> + Rigor in m., <a + href="#Block_535">535-538</a>.</li> + <li class="indx"> + Rosanes, On the unpopularity of m., <b><a + href="#Block_730">730</a></b>.</li> + <li class="indx"> + Royal road, <a + href="#Block_201">201</a>, <a + href="#Block_901">901</a>, <a + href="#Block_951">951</a>, <em><a + href="#Block_1878">1774</a></em>.</li> + <li class="indx"> + Royal science, M. a, <a + href="#Block_204">204</a>.</li> + <li class="indx"> + Rudio,</li> + <li class="isub1"> + On Euler, <b><a + href="#Block_957">957</a></b>.</li> + <li class="isub1"> + M. and great artists, <b><a + href="#Block_1105">1105</a></b>.</li> + <li class="isub1"> + On m. and navigation, <b><a + href="#Block_1543">1543</a></b>.</li> + <li class="indx"> + Rush, M. cures predisposition to anger, <b><a + href="#Block_458">458</a></b>.</li> + <li class="indx"> + Russell,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_127">127</a></b>, <b><a + href="#Block_128">128</a></b>.</li> + <li class="isub1"> + On nineteenth century m., <b><a + href="#Block_705">705</a></b>.</li> + <li class="isub1"> + Chief triumph of modern m., <b><a + href="#Block_706">706</a></b>.</li> + <li class="isub1"> + On the infinite, <b><a + href="#Block_723">723</a></b>.</li> + <li class="isub1"> + On beauty in m., <b><a + href="#Block_1104">1104</a></b>.</li> + <li class="isub1"> + On the value of symbols, <b><a + href="#Block_1219">1219</a></b>.</li> + <li class="isub1"> + On Boole’s Laws of Thought, <b><a + href="#Block_1318">1318</a></b>.</li> + <li class="isub1"> + Principia Mathematica, <a + href="#Block_1326">1326</a>.</li> + <li class="isub1"> + On geometry and philosophy, <b><a + href="#Block_1410">1410</a></b>.</li> + <li class="isub1"> + Definition of number, <b><a + href="#Block_1609">1609</a></b>.</li> + <li class="isub1"> + Fruitful uses of imaginaries, <b><a + href="#Block_1735">1735</a></b>.</li> + <li class="isub1"> + Geometrical reasoning circular, <b><a + href="#Block_1864">1864</a></b>.</li> + <li class="isub1"> + On projective geometry, <b><a + href="#Block_1879">1879</a></b>.</li> + <li class="isub1"> + Zeno’s problems, <b><a + href="#Block_1938">1938</a></b>.</li> + <li class="isub1"> + Definition of infinite collection, <b><a + href="#Block_1959">1959</a></b>.</li> + <li class="isub1"> + On proofs of axioms, <b><a + href="#Block_2013">2013</a></b>.</li> + <li class="isub1"> + On non-euclidean geometry, <b><a + href="#Block_2018">2018</a></b>. + +<span class="pagenum"> + <a name="Page_406" + id="Page_406">406</a></span></li> + + <li class="ifrst"> + Safford,</li> + <li class="isub1"> + On aptitude for m., <b><a + href="#Block_520">520</a></b>.</li> + <li class="isub1"> + On m. and science, <b><a + href="#Block_1509">1509</a></b>.</li> + <li class="indx"> + Sage, Battalions of figures, <b><a + href="#Block_1631">1631</a></b>.</li> + <li class="indx"> + Sartorius, Gauss on the nature of space, <b><a + href="#Block_2034">2034</a></b>.</li> + <li class="indx"> + Scepticism, <a + href="#Block_452">452</a>, <a + href="#Block_811">811</a>.</li> + <li class="indx"> + Schellbach,</li> + <li class="isub1"> + Estimate of m., <b><a + href="#Block_306">306</a></b>.</li> + <li class="isub1"> + On truth, <b><a + href="#Block_1114">1114</a></b>.</li> + <li class="indx"> + Schiller, Archimedes and the youth, <b><a + href="#Block_907">907</a></b>.</li> + <li class="indx"> + Schopenhauer,</li> + <li class="isub1"> + Arithmetic rests on the concept of time, <b><a + href="#Block_1613">1613</a></b>.</li> + <li class="isub1"> + Predicabilia a priori, <b><a + href="#Block_2003">2003</a></b>.</li> + <li class="indx"> + Schröder, M. as a branch of logic, <b><a + href="#Block_1323">1323</a></b>.</li> + <li class="indx"> + Schubert,</li> + <li class="isub1"> + Three characteristics of m., <b><a + href="#Block_229">229</a></b>.</li> + <li class="isub1"> + On controversies in m., <b><a + href="#Block_243">243</a></b>.</li> + <li class="isub1"> + Characteristics of m., <b><a + href="#Block_263">263</a></b>.</li> + <li class="isub1"> + M. an exclusive science, <b><a + href="#Block_734">734</a></b>.</li> + <li class="indx"> + Science and m.,</li> + <li class="isub1"> + Chapter <a + href="#CHAPTER_XV">XV</a>.</li> + <li class="isub1"> + M. an indispensible tool of, <a + href="#Block_309">309</a>.</li> + <li class="isub1"> + Neglect of m. works injury to, <a + href="#Block_310">310</a>.</li> + <li class="isub1"> + Craig on origin of new, <a + href="#Block_646">646</a>.</li> + <li class="isub1"> + Greek view of, <a + href="#Block_1429">1429</a>.</li> + <li class="isub1"> + Six follies of, <a + href="#Block_2107">2107</a>.</li> + <li class="isub1"> + See also <a + href="#Block_433">433</a>, <a + href="#Block_436">436</a>, <a + href="#Block_437">437</a>, <a + href="#Block_461">461</a>, <a + href="#Block_725">725</a>.</li> + <li class="indx"> + Scientific education, Math. training indispensable + basis of, <a + href="#Block_444">444</a>.</li> + <li class="indx"> + Screw,</li> + <li class="isub1"> + The song of the, <a + href="#Block_1894">1894</a>.</li> + <li class="isub1"> + As an instrument in geometry, <a + href="#Block_2114">2114</a>.</li> + <li class="indx"> + Sedgwick, Quaternion of maladies, <b><a + href="#Block_1723">1723</a></b>.</li> + <li class="indx"> + Segre,</li> + <li class="isub1"> + On research in m., <b><a + href="#Block_619">619</a></b>.</li> + <li class="isub1"> + What kind of investigations are important, <b><a + href="#Block_641">641</a></b>.</li> + <li class="isub1"> + On the worthlessness of certain investigations, <b><a + href="#Block_642">642</a></b>, <b><a + href="#Block_643">643</a></b>.</li> + <li class="isub1"> + On hyper-space, <b><a + href="#Block_2031">2031</a></b>.</li> + <li class="indx"> + Seneca, Alexander and geometry, <b><a + href="#Block_902">902</a></b>.</li> + <li class="indx"> + Seventy-seven, The number, <a + href="#Block_2149">2149</a>.</li> + <li class="indx"> + Shakespeare, <a + href="#Block_1127">1127</a>, <a + href="#Block_1129">1129</a>, <a + href="#Block_2141">2141</a>.</li> + <li class="indx"> + Shaw, J. B., M. like game of chess, <b><a + href="#Block_840">840</a></b>.</li> + <li class="indx"> + Shaw, W. H., M. and professional life, <b><a + href="#Block_1596">1596</a></b>.</li> + <li class="indx"> + Sherman, M. and rhetoric, <b><a + href="#Block_1599b">1599</a></b>.</li> + <li class="indx"> + Smith, Adam, <a + href="#Block_1324">1324</a>.</li> + <li class="indx"> + Smith, D. E.,</li> + <li class="isub1"> + On problem solving, <b><a + href="#Block_532">532</a></b>.</li> + <li class="isub1"> + Value of geometrical training, <b><a + href="#Block_1846">1846</a></b>.</li> + <li class="isub1"> + Reason for studying geometry, <b><a + href="#Block_1850">1850</a></b>.</li> + <li class="indx"> + Smith, H. J. S.,</li> + <li class="isub1"> + When a math. theory is completed, <b><a + href="#Block_637">637</a></b>.</li> + <li class="isub1"> + On the growth of m., <b><a + href="#Block_1521">1521</a></b>.</li> + <li class="isub1"> + On m. and science, <b><a + href="#Block_1542">1542</a></b>.</li> + <li class="isub1"> + On m. and physics, <b><a + href="#Block_1556">1556</a></b>.</li> + <li class="isub1"> + On m. and meteorology, <b><a + href="#Block_1557">1557</a></b>.</li> + <li class="isub1"> + On number theory, <b><a + href="#Block_1645">1645</a></b>.</li> + <li class="isub1"> + Rigor in Euclid, <b><a + href="#Block_1829">1829</a></b>.</li> + <li class="isub1"> + On Euclid’s doctrine of proportion, <b><a + href="#Block_1834">1834</a></b>.</li> + <li class="indx"> + Smith, W. B.,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_121">121</a></b>.</li> + <li class="isub1"> + On infinitesimal analysis, <b><a + href="#Block_1914">1914</a></b>.</li> + <li class="isub1"> + On non-euclidean and hyperspaces, <b><a + href="#Block_2033">2033</a></b>.</li> + <li class="indx"> + Simon, On beauty and truth, <b><a + href="#Block_1114">1114</a></b>.</li> + <li class="indx"> + Simplicity in m., <a + href="#Block_315">315</a>, <a + href="#Block_526">526</a>.</li> + <li class="indx"> + Sin<sub>2</sub>φ, On the notation of, <a + href="#Block_1886">1886</a>.</li> + <li class="indx"> + Six hundred sixty-six, The number, <a + href="#Block_2151">2151</a>, <a + href="#Block_2152">2152</a>.</li> + <li class="indx"> + Social science and m., <a + href="#Block_1201">1201</a>, <a + href="#Block_1586">1586</a>, <a + href="#Block_1587">1587</a>.</li> + <li class="indx"> + Social service, M. as an aid to, <a + href="#Block_313">313</a>, <a + href="#Block_314">314</a>, <a + href="#Block_328">328</a>.</li> + <li class="indx"> + Social value of m., <a + href="#Block_456">456</a>, <a + href="#Block_1588">1588</a>.</li> + <li class="indx"> + Solitude and m., <a + href="#Block_1849">1849</a>, <a + href="#Block_1851">1851</a>.</li> + <li class="indx"> + Sophistry, M. free from, <a + href="#Block_215">215</a>.</li> + <li class="indx"> + Sound, M. and the theory of, <a + href="#Block_1551">1551</a>.</li> + <li class="indx"> + Southey, On Newton, <b><a + href="#Block_1008">1008</a></b>.</li> + <li class="indx"> + Space,</li> + <li class="isub1"> + Of experience, <a + href="#Block_2011">2011</a>.</li> + <li class="isub1"> + Kant’s doctrine of, <a + href="#Block_2003">2003</a>.</li> + <li class="isub1"> + Schopenhauer’s predicabilia, <a + href="#Block_2004">2004</a>.</li> + <li class="isub1"> + Whewell, On the idea of, <a + href="#Block_2004">2004</a>.</li> + <li class="isub1"> + Non-euclidean, <a + href="#Block_2015">2015</a>, <a + href="#Block_2016">2016</a>, <a + href="#Block_2018">2018</a>.</li> + <li class="isub1"> + Hyper-, <a + href="#Block_2030">2030</a>, <a + href="#Block_2031">2031</a>, <a + href="#Block_2033">2033</a>, <a + href="#Block_2036">2036-2038</a>.</li> + <li class="indx"> + Spedding, On Bacon’s knowledge of m., <b><a + href="#Block_917">917</a></b>.</li> + <li class="indx"> + Speer, On m. and nature-study, <b><a + href="#Block_514">514</a></b>.</li> + <li class="indx"> + Spence, On Newton, <b><a href="#Block_1016">1016</a></b>, <b><a + href="#Block_1020">1020</a></b>. + +<span class="pagenum"> + <a name="Page_407" + id="Page_407">407</a></span></li> + + <li class="indx"> + Spencer, On m. in the arts, <b><a + href="#Block_1570">1570</a></b>.</li> + <li class="indx"> + Spherical trigonometry, <a + href="#Block_1887">1887</a>.</li> + <li class="indx"> + Spira mirabilis, <a + href="#Block_922">922</a>.</li> + <li class="indx"> + Spottiswoode, On the kingdom of m., <b><a + href="#Block_269">269</a></b>.</li> + <li class="indx"> + Squaring the circle, <a + href="#Block_1537">1537</a>, <a + href="#Block_1858">1858</a>, <a + href="#Block_1934">1934</a>, <a + href="#Block_1948">1948</a>, <a + href="#Block_2115">2115-2117</a>.</li> + <li class="indx"> + St. Augustine, The number seventy seven, <b><a + href="#Block_2149">2149</a></b>.</li> + <li class="indx"> + St. Vincent, As a circle-squarer, <a + href="#Block_2109">2109</a>.</li> + <li class="indx"> + Steiner, On projective geometry, <b><a + href="#Block_1877">1877</a></b>.</li> + <li class="indx"> + Stewart,</li> + <li class="isub1"> + M. and facts, <b><a + href="#Block_237">237</a></b>.</li> + <li class="isub1"> + On beauty in m., <b><a + href="#Block_242">242</a></b>.</li> + <li class="isub1"> + What we most admire in m., <b><a + href="#Block_315">315</a></b>.</li> + <li class="isub1"> + M. for its own sake, <b><a + href="#Block_440">440</a></b>.</li> + <li class="isub1"> + M. the noblest instance of force of the human mind, <b><a + href="#Block_452">452</a></b>.</li> + <li class="isub1"> + Math’ns and applause, <b><a + href="#Block_816">816</a></b>.</li> + <li class="isub1"> + Mere math’ns, <b><a + href="#Block_821">821</a></b>.</li> + <li class="isub1"> + Shortcomings of math’ns, <b><a + href="#Block_828">828</a></b>.</li> + <li class="isub1"> + On the influence of Leibnitz, <b><a + href="#Block_988">988</a></b>.</li> + <li class="isub1"> + Reason supreme, <b><a + href="#Block_1424">1424</a></b>.</li> + <li class="isub1"> + M. and philosophy compared, <b><a + href="#Block_1428">1428</a></b>.</li> + <li class="isub1"> + M. and natural philosophy, <b><a + href="#Block_1555">1555</a></b>.</li> + <li class="indx"> + Stifel, The number of the beast, <b><a + href="#Block_2152">2152</a></b>.</li> + <li class="indx"> + Stobæus,</li> + <li class="isub1"> + Alexander and Menæchmus, <b><a + href="#Block_901">901</a></b>.</li> + <li class="isub1"> + Euclid and the student, <b><a + href="#Block_952">952</a></b>.</li> + <li class="indx"> + Study of m., Chapter <a + href="#CHAPTER_VI">VI</a>.</li> + <li class="indx"> + Substitution, Concept of, <a + href="#Block_727">727</a>.</li> + <li class="indx"> + Superstition,</li> + <li class="isub1"> + M. frees mind from, <a + href="#Block_450">450</a>.</li> + <li class="isub1"> + Number was born in, <a + href="#Block_1632">1632</a>.</li> + <li class="indx"> + Surd numbers, <a + href="#Block_1728">1728</a>.</li> + <li class="indx"> + Surprises, M. rich in, <a + href="#Block_202">202</a>.</li> + <li class="indx"> + Swift,</li> + <li class="isub1"> + On m. and politics, <b><a + href="#Block_814">814</a></b>.</li> + <li class="isub1"> + The math’ns of Laputa, <b><a + href="#Block_2120">2120-2122</a></b>.</li> + <li class="isub1"> + The math. school of Laputa, <b><a + href="#Block_2123">2123</a></b>.</li> + <li class="isub1"> + His ignorance of m., <a + href="#Block_2124">2124</a>, <a + href="#Block_2125">2125</a>.</li> + <li class="indx"> + Sylvester,</li> + <li class="isub1"> + On the use of the terms mathematic and mathematics, <b><a + href="#Block_101">101</a></b>.</li> + <li class="isub1"> + Order and arrangement the basic ideas of m., <b><a + href="#Block_109">109</a></b>, <b><a + href="#Block_110">110</a></b>.</li> + <li class="isub1"> + Definition of algebra, <b><a + href="#Block_110">110</a></b>.</li> + <li class="isub1"> + Definition of arithmetic, <b><a + href="#Block_110">110</a></b>.</li> + <li class="isub1"> + Definition of geometry, <b><a + href="#Block_110">110</a></b>.</li> + <li class="isub1"> + On the object of pure m., <b><a + href="#Block_129">129</a></b>.</li> + <li class="isub1"> + M. requires harmonious action of all the faculties, <b><a + href="#Block_202">202</a></b>.</li> + <li class="isub1"> + Answer to Huxley, <b><a + href="#Block_251">251</a></b>.</li> + <li class="isub1"> + On the nature of m., <b><a + href="#Block_251">251</a></b>.</li> + <li class="isub1"> + On observation in m., <b><a + href="#Block_255">255</a></b>.</li> + <li class="isub1"> + Invention in m., <b><a + href="#Block_260">260</a></b>.</li> + <li class="isub1"> + M. entitled to human regard, <b><a + href="#Block_301">301</a></b>.</li> + <li class="isub1"> + On the ethical value of m., <b><a + href="#Block_449">449</a></b>.</li> + <li class="isub1"> + On isolated theorems, <b><a + href="#Block_620">620</a></b>.</li> + <li class="isub1"> + “Auge <em>et impera</em>.” <b><a + href="#Block_631">631</a></b>.</li> + <li class="isub1"> + His bent of mind, <b><a + href="#Block_645">645</a></b>.</li> + <li class="isub1"> + Apology for imperfections, <b><a + href="#Block_648">648</a></b>.</li> + <li class="isub1"> + On theoretical investigations, <b><a + href="#Block_658">658</a></b>.</li> + <li class="isub1"> + Characteristics of modern m., <b><a + href="#Block_724">724</a></b>.</li> + <li class="isub1"> + Invested m. with halo of glory, <a + href="#Block_740">740</a>.</li> + <li class="isub1"> + M. and eloquence, <b><a + href="#Block_829">829</a></b>.</li> + <li class="isub1"> + On longevity of math’ns, <b><a + href="#Block_839">839</a></b>.</li> + <li class="isub1"> + On Cayley, <b><a + href="#Block_930">930</a></b>.</li> + <li class="isub1"> + His view of Euclid, <a + href="#Block_936">936</a>.</li> + <li class="isub1"> + Jacobi’s talent for philology, <b><a + href="#Block_980">980</a></b>.</li> + <li class="isub1"> + His eloquence, <a + href="#Block_1030">1030</a>.</li> + <li class="isub1"> + Researches in quantics, <b><a + href="#Block_1032">1032</a></b>.</li> + <li class="isub1"> + His weakness, <a + href="#Block_1033">1033</a>, <a + href="#Block_1036">1036</a>, <a + href="#Block_1037">1037</a>.</li> + <li class="isub1"> + One-sided character of his work, <a + href="#Block_1034">1034</a>.</li> + <li class="isub1"> + His method, <a + href="#Block_1035">1035</a>, <a + href="#Block_1036">1036</a>, <a + href="#Block_1041">1041</a>.</li> + <li class="isub1"> + His forgetfulness, <a + href="#Block_1037">1037</a>, <a + href="#Block_1038">1038</a>.</li> + <li class="isub1"> + Relations with students, <a + href="#Block_1039">1039</a>.</li> + <li class="isub1"> + His style, <a + href="#Block_1040">1040</a>, <a + href="#Block_1041">1041</a>.</li> + <li class="isub1"> + His characteristics, <a + href="#Block_1041">1041</a>.</li> + <li class="isub1"> + His enthusiasm, <a + href="#Block_1041">1041</a>.</li> + <li class="isub1"> + The math. Adam, <b><a + href="#Block_1042">1042</a></b>.</li> + <li class="isub1"> + And Weierstrass, <a + href="#Block_1050">1050</a>.</li> + <li class="isub1"> + On divine beauty and order in m., <b><a + href="#Block_1101">1101</a></b>.</li> + <li class="isub1"> + M. among the fine arts, + <b><a href="#Block_1106">1106</a></b>.</li> + <li class="isub1"> + On music and m., <b><a + href="#Block_1131">1131</a></b>. + +<span class="pagenum"> + <a name="Page_408" + id="Page_408">408</a></span></li> + + <li class="isub1"> + M. the quintessence of language, <b><a + href="#Block_1205">1205</a></b>.</li> + <li class="isub1"> + M. the language of the universe, <b><a + href="#Block_1206">1206</a></b>.</li> + <li class="isub1"> + On prime numbers, <b><a + href="#Block_1648">1648</a></b>.</li> + <li class="isub1"> + On determinants, <b><a + href="#Block_1740">1740</a></b>.</li> + <li class="isub1"> + On invariants, <b><a + href="#Block_1742">1742</a></b>.</li> + <li class="isub1"> + Contribution to theory of equations, <a + href="#Block_1743">1743</a>.</li> + <li class="isub1"> + To a missing member etc., <b><a + href="#Block_1745">1745</a></b>.</li> + <li class="isub1"> + Invariants and isomerism, <b><a + href="#Block_1750">1750</a></b>.</li> + <li class="isub1"> + His dislike for Euclid, <b><a + href="#Block_1826">1826</a></b>.</li> + <li class="isub1"> + On the invention of integrals, <b><a + href="#Block_1922">1922</a></b>.</li> + <li class="isub1"> + On geometry and analysis, <b><a + href="#Block_1931">1931</a></b>.</li> + <li class="isub1"> + On paradoxes, <b><a + href="#Block_2104">2104</a></b>.</li> + <li class="indx"> + Symbolic language,</li> + <li class="isub1"> + M. as a, <a + href="#Block_1207">1207</a>, <a + href="#Block_1212">1212</a>.</li> + <li class="isub1"> + Use of, <a + href="#Block_1573">1573</a>.</li> + <li class="indx"> + Symbolic logic, <a + href="#Block_1316">1316-1321</a>.</li> + <li class="indx"> + Symbolism,</li> + <li class="isub1"> + On the nature of math., <a + href="#Block_1210">1210</a>.</li> + <li class="isub1"> + Difficulty of math., <a + href="#Block_1218">1218</a>.</li> + <li class="isub1"> + Universal impossible, <a + href="#Block_1221">1221</a>.</li> + <li class="isub1"> + See also notation.</li> + <li class="indx"> + Symbols, Burlesque on, <a + href="#Block_1741">1741</a>.</li> + <li class="indx"> + Symbols,</li> + <li class="isub1"> + M. leads to mastery of, <a + href="#Block_421">421</a>.</li> + <li class="isub1"> + Value of math., <a + href="#Block_1209">1209</a>, <a + href="#Block_1212">1212</a>,<a + href="#Block_1219">1219</a>.</li> + <li class="isub1"> + Essential to demonstration, <a + href="#Block_1316">1316</a>.</li> + <li class="isub1"> + Arithmetical, <a + href="#Block_1627">1627</a>.</li> + + <li class="ifrst"> + Tact in m., <a + href="#Block_622">622</a>, <a + href="#Block_623">623</a>.</li> + <li class="indx"> + Tait,</li> + <li class="isub1"> + On the unpopularity of m., <b><a + href="#Block_740">740</a></b>.</li> + <li class="isub1"> + And Thomson, <a + href="#Block_1043">1043</a>.</li> + <li class="isub1"> + And Hamilton, <a + href="#Block_1044">1044</a>.</li> + <li class="isub1"> + On quaternions, <b><a + href="#Block_1724">1724-1726</a></b>.</li> + <li class="isub1"> + On spherical trigonometry, <b><a + href="#Block_1887">1887</a></b>.</li> + <li class="indx"> + Talent, Math’ns men of, <a + href="#Block_825">825</a>.</li> + <li class="indx"> + Teaching of m., Chapter <a + href="#CHAPTER_V">V</a>.</li> + <li class="indx"> + Tennyson, <a + href="#Block_1843">1843</a>.</li> + <li class="indx"> + Teutonic race, Aptitude for m., <a + href="#Block_838">838</a>.</li> + <li class="indx"> + Text-books,</li> + <li class="isub1"> + Chrystal on, <a + href="#Block_533">533</a>.</li> + <li class="isub1"> + Minchin on, <a + href="#Block_539">539</a>.</li> + <li class="isub1"> + Cremona on English, <a + href="#Block_609">609</a>.</li> + <li class="isub1"> + Glaisher on need of, <a + href="#Block_635">635</a>.</li> + <li class="indx"> + Thales, <a + href="#Block_201">201</a>.</li> + <li class="indx"> + Theoretical investigations, <a + href="#Block_652">652-664</a>.</li> + <li class="indx"> + Theory and practice, <a + href="#Block_661">661</a>.</li> + <li class="indx"> + Thompson, Sylvanus,</li> + <li class="isub1"> + Lord Kelvin’s definition of a math’n, <b><a + href="#Block_822">822</a></b>.</li> + <li class="isub1"> + Cayley’s estimate of quaternions, <b><a + href="#Block_937">937</a></b>.</li> + <li class="isub1"> + Thomson’s “It is obvious that,” <b><a + href="#Block_1045">1045</a></b>.</li> + <li class="isub1"> + Anecdote of Lord Kelvin, <b><a + href="#Block_1046">1046</a></b>, <b><a + href="#Block_1047">1047</a></b>.</li> + <li class="isub1"> + On the calculus for beginners, <b><a + href="#Block_1917">1917</a></b>.</li> + <li class="indx"> + Thomson, Sir William,</li> + <li class="isub1"> + M. the only true metaphysics, <b><a + href="#Block_305">305</a></b>.</li> + <li class="isub1"> + M. not repulsive to common sense, <b><a + href="#Block_312">312</a></b>.</li> + <li class="isub1"> + What is a math’n? <b><a + href="#Block_822">822</a></b>.</li> + <li class="isub1"> + And Tait, <a + href="#Block_1043">1043</a>.</li> + <li class="isub1"> + “It is obvious that,” <a + href="#Block_1045">1045</a>.</li> + <li class="isub1"> + Anecdotes concerning, <a + href="#Block_1046">1046</a>, <a + href="#Block_1047">1047</a>, <a + href="#Block_1048">1048</a>.</li> + <li class="isub1"> + On m. and astronomy, <b><a + href="#Block_1562">1562</a></b>.</li> + <li class="isub1"> + On quaternions, <b><a + href="#Block_1721">1721</a></b>, <b><a + href="#Block_1722">1722</a></b>.</li> + <li class="indx"> + Thomson and Tait, <a + href="#Block_1043">1043</a>.</li> + <li class="isub1"> + On Fourier’s theorem, <b><a + href="#Block_1928">1928</a></b>.</li> + <li class="indx"> + Thoreau, On poetry and m., <b><a + href="#Block_1123">1123</a></b>.</li> + <li class="indx"> + Thought-economy in m., <a + href="#Block_203">203</a>, <a + href="#Block_1209">1209</a>, <a + href="#Block_1704">1704</a>.</li> + <li class="indx"> + Three,</li> + <li class="isub1"> + The Yankos word for, <a + href="#Block_2144">2144</a>.</li> + <li class="isub1"> + Pacioli on the number, <a + href="#Block_2145">2145</a>.</li> + <li class="indx"> + Time,</li> + <li class="isub1"> + Arithmetic rests on notion of <a + href="#Block_1613">1613</a>.</li> + <li class="isub1"> + As a concept in algebra, <a + href="#Block_1715">1715</a>, <a + href="#Block_1716">1716</a>, <a + href="#Block_1717">1717</a>.</li> + <li class="isub1"> + Kant’s doctrine of, <a + href="#Block_2001">2001</a>.</li> + <li class="isub1"> + Schopenhauer’s predicabilia, <a + href="#Block_2003">2003</a>.</li> + <li class="indx"> + Todhunter,</li> + <li class="isub1"> + On m. as a university subject, <b><a + href="#Block_405">405</a></b>.</li> + <li class="isub1"> + On m. as a test of performance, <b><em><a + href="#Block_407">408</a></em></b>.</li> + <li class="isub1"> + On m. as an instrument in education, <b><a + href="#Block_414">414</a></b>.</li> + <li class="isub1"> + M. requires voluntary exertion, <b><a + href="#Block_415">415</a></b>. + +<span class="pagenum"> + <a name="Page_409" + id="Page_409">409</a></span></li> + + <li class="isub1"> + On exercises, <b><a + href="#Block_422">422</a></b>.</li> + <li class="isub1"> + On problems, <b><a + href="#Block_523">523</a></b>, <b><a + href="#Block_608">608</a></b>.</li> + <li class="isub1"> + How to read m., <b><a + href="#Block_605">605</a></b>, <b><a + href="#Block_606">606</a></b>.</li> + <li class="isub1"> + On discovery in elementary m., <b><a + href="#Block_617">617</a></b>.</li> + <li class="isub1"> + On Sylvester’s theorem, <b><a + href="#Block_1743">1743</a></b>.</li> + <li class="isub1"> + On performance in Euclid, <b><a + href="#Block_1818">1818</a></b>.</li> + <li class="indx"> + Transformation, Concept of, <a + href="#Block_727">727</a>.</li> + <li class="indx"> + Trigonometry, <a + href="#Block_1881">1881</a>, <a + href="#Block_1884">1884-1889</a>.</li> + <li class="indx"> + Trilinear co-ordinates, <a + href="#Block_611">611</a>.</li> + <li class="indx"> + Trisection of angle, <a + href="#Block_2112">2112</a>.</li> + <li class="indx"> + Truth,</li> + <li class="isub1"> + and m., <a + href="#Block_306">306</a>.</li> + <li class="isub1"> + Math’ns must perceive beauty of, <a + href="#Block_803">803</a>.</li> + <li class="isub1"> + And beauty, <a + href="#Block_1114">1114</a>.</li> + <li class="indx"> + Tzetzes, Plato on geom., <b><a + href="#Block_1803">1803</a></b>.</li> + + <li class="ifrst"> + Unity, Locke on the idea of, <a + href="#Block_1607">1607</a>.</li> + <li class="indx"> + Universal algebra, <a + href="#Block_1753">1753</a>.</li> + <li class="indx"> + Universal arithmetic, <a + href="#Block_1717">1717</a>.</li> + <li class="indx"> + Universal language, <a + href="#Block_925">925</a>.</li> + <li class="indx"> + Unpopularity of m., <a + href="#Block_270">270</a>, <a + href="#Block_271">271</a>, <a + href="#Block_730">730-736</a>, <a + href="#Block_738">738</a>, <a + href="#Block_740">740</a>, <a + href="#Block_1501">1501</a>, <a + href="#Block_1628">1628</a>.</li> + <li class="indx"> + Usefulness, As a principle in research, <a + href="#Block_652">652-655</a>, <a + href="#Block_659">659</a>, <a + href="#Block_664">664</a>.</li> + <li class="indx"> + Uses of m., See value of m.</li> + + <li class="ifrst"> + Value of m.,</li> + <li class="isub1"> + Chapter <a + href="#CHAPTER_IV">IV</a>.</li> + <li class="isub1"> + See also <a + href="#Block_330">330</a>, <a + href="#Block_333">333</a>, <a + href="#Block_1414">1414</a>, <a + href="#Block_1422">1422</a>, <a + href="#Block_1505">1505</a>, <a + href="#Block_1506">1506</a>, <a + href="#Block_1512">1512</a>, <a + href="#Block_1523">1523</a>, <a + href="#Block_1526">1526</a>, <a + href="#Block_1527">1527</a>, <a + href="#Block_1533">1533</a>, <a + href="#Block_1541">1541</a>, <a + href="#Block_1542">1542</a>, <a + href="#Block_1543">1543</a>, <a + href="#Block_1547">1547-1576</a>, <a + href="#Block_1619">1619-1626</a>, <a + href="#Block_1841">1841</a>, <a + href="#Block_1844">1844-1851</a>.</li> + <li class="indx"> + Variability, The central idea of modern m., <a + href="#Block_720">720</a>, <a + href="#Block_721">721</a>.</li> + <li class="indx"> + Venn,</li> + <li class="isub1"> + On m. as a symbolic language, <b><a + href="#Block_1207">1207</a></b>.</li> + <li class="isub1"> + M. the only gate, <b><a + href="#Block_1517">1517</a></b>.</li> + <li class="indx"> + Viola, On the use of fallacies, <b><a + href="#Block_610">610</a></b>.</li> + <li class="indx"> + Virgil, <b><a + href="#Block_2138">2138</a></b>.</li> + <li class="indx"> + Voltaire,</li> + <li class="isub1"> + Archimedes more imaginative than Homer, <b><a + href="#Block_259">259</a></b>.</li> + <li class="isub1"> + M. the staff of the blind, <b><a + href="#Block_461">461</a></b>.</li> + <li class="isub1"> + On direct usefulness of results, <b><a + href="#Block_653">653</a></b>.</li> + <li class="isub1"> + On infinite magnitudes, <b><a + href="#Block_1947">1947</a></b>.</li> + <li class="isub1"> + On the symbol, <b><a + href="#Block_1950">1950</a></b>.</li> + <li class="isub1"> + Anagram on, <a + href="#Block_2154">2154</a>.</li> + + <li class="ifrst"> + Walcott, On hyperbolic functions, <b><a + href="#Block_1930">1930</a></b>.</li> + <li class="indx"> + Walker,</li> + <li class="isub1"> + On problems in arithmetic, <b><a + href="#Block_528">528</a></b>.</li> + <li class="isub1"> + On the teaching of geometry, <b><a + href="#Block_529">529</a></b>.</li> + <li class="indx"> + Wallace, On the frequency of the math. faculty, <b><a + href="#Block_832">832</a></b>.</li> + <li class="isub1"> + On m. and natural selection, <b><a + href="#Block_833">833</a></b>, <b><a + href="#Block_834">834</a></b>.</li> + <li class="isub1"> + Parallel growth of m. and music, <b><a + href="#Block_1135">1135</a></b>.</li> + <li class="indx"> + Walton, Angling like m., <b><a + href="#Block_739">739</a></b>.</li> + <li class="indx"> + Weber, On m. and physics, <b><a + href="#Block_1549">1549</a></b>.</li> + <li class="indx"> + Webster, Estimate of m., <b><a + href="#Block_331">331</a></b>.</li> + <li class="indx"> + Weierstrass,</li> + <li class="isub1"> + Math’ns are poets, <b><a + href="#Block_802">802</a></b>.</li> + <li class="isub1"> + Anecdote concerning, <a + href="#Block_1049">1049</a>.</li> + <li class="isub1"> + And Sylvester, <a + href="#Block_1050">1050</a>.</li> + <li class="isub1"> + Problem of infinitesimals, <a + href="#Block_1938">1938</a>.</li> + <li class="indx"> + Weismann, On the origin of the math. faculty, <b><a + href="#Block_1136">1136</a></b>.</li> + <li class="indx"> + Wells, On m. as a world language, <b><a + href="#Block_1201">1201</a></b>.</li> + <li class="indx"> + Whately,</li> + <li class="isub1"> + On m. as an exercise, <b><a + href="#Block_427">427</a></b>.</li> + <li class="isub1"> + On m. and navigation, <b><a + href="#Block_1544">1544</a></b>.</li> + <li class="isub1"> + On geometrical demonstrations, <b><a + href="#Block_1839">1839</a></b>.</li> + <li class="isub1"> + On Swift’s ignorance of m., <b><a + href="#Block_2124">2124</a></b>.</li> + <li class="indx"> + Whetham, On symbolic logic, <b><a + href="#Block_1319">1319</a></b>.</li> + <li class="indx"> + Whewell,</li> + <li class="isub1"> + On mixed and pure math., <b><a + href="#Block_107">107</a></b>.</li> + <li class="isub1"> + M. not an inductive science, <b><a + href="#Block_223">223</a></b>.</li> + <li class="isub1"> + Nature of m., <a + href="#Block_224">224</a>.</li> + <li class="isub1"> + Value of geometry, <a + href="#Block_445">445</a>.</li> + <li class="isub1"> + On theoretical investigations, <b><a + href="#Block_660">660</a></b>, <b><a + href="#Block_662">662</a></b>.</li> + <li class="isub1"> + Math’ns men of talent, <b><a + href="#Block_825">825</a></b>.</li> + <li class="isub1"> + Fame of math’ns, <b><a + href="#Block_826">826</a></b>.</li> + <li class="isub1"> + On Newton’s greatness, <b><a + href="#Block_1004">1004</a></b>.</li> + <li class="isub1"> + On Newton’s theory, <b><a + href="#Block_1005">1005</a></b>.</li> + <li class="isub1"> + On Newton’s humility, <b><a + href="#Block_1025">1025</a></b>.</li> + <li class="isub1"> + On symbols, <a + href="#Block_1212">1212</a>.</li> + <li class="isub1"> + On philosophy and m., <b><a + href="#Block_1429">1429</a></b>.</li> + <li class="isub1"> + On m. and science, <b><a + href="#Block_1534">1534</a></b>.</li> + <li class="isub1"> + Quotation from R. Bacon, <b><a + href="#Block_1547">1547</a></b>.</li> + <li class="isub1"> + On m. and applications, <b><a + href="#Block_1541">1541</a></b>.</li> + <li class="isub1"> + Geometry and experience, <a + href="#Block_1814">1814</a>. + +<span class="pagenum"> + <a name="Page_410" + id="Page_410">410</a></span></li> + + <li class="isub1"> + Geometry not an inductive science, <a + href="#Block_1830">1830</a>.</li> + <li class="isub1"> + On limits, <a + href="#Block_1909">1909</a>.</li> + <li class="isub1"> + On the idea of space, <a + href="#Block_2004">2004</a>.</li> + <li class="isub1"> + On Plato and the regular solids, <b><a + href="#Block_2133">2133</a></b>, <b><a + href="#Block_2135">2135</a></b>.</li> + <li class="indx"> + White, H. S., On the growth of m., <b><a + href="#Block_211">211</a></b>.</li> + <li class="indx"> + White, W. F.,</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_131">131</a></b>, <b><a + href="#Block_1203">1203</a></b>.</li> + <li class="isub1"> + M. as a prerequisite for public speaking, <b><a + href="#Block_420">420</a></b>.</li> + <li class="isub1"> + On beauty in m., <b><a + href="#Block_1119">1119</a></b>.</li> + <li class="isub1"> + The place of the math’n, <b><a + href="#Block_1529">1529</a></b>.</li> + <li class="isub1"> + On m. and social science, <b><a + href="#Block_1586">1586</a></b>.</li> + <li class="isub1"> + The cost of Manhattan island, <b><a + href="#Block_2130">2130</a></b>.</li> + <li class="indx"> + Whitehead, On the ideal of m., <b><a + href="#Block_119">119</a></b>.</li> + <li class="isub1"> + Definition of m., <b><a + href="#Block_122">122</a></b>.</li> + <li class="isub1"> + On the scope of m., <b><a + href="#Block_126">126</a></b>.</li> + <li class="isub1"> + On the nature of m., <b><a + href="#Block_233">233</a></b>.</li> + <li class="isub1"> + Precision necessary in m., <b><a + href="#Block_639">639</a></b>.</li> + <li class="isub1"> + On practical applications, <b><a + href="#Block_655">655</a></b>.</li> + <li class="isub1"> + On theoretical investigations, <b><a + href="#Block_659">659</a></b>.</li> + <li class="isub1"> + Characteristics of ancient geometry, <b><a + href="#Block_713">713</a></b>.</li> + <li class="isub1"> + On the extent of m., <b><a + href="#Block_737">737</a></b>.</li> + <li class="isub1"> + Archimedes compared with Newton, <b><a + href="#Block_911">911</a></b>.</li> + <li class="isub1"> + On the Arabic notation, <b><a + href="#Block_1217">1217</a></b>.</li> + <li class="isub1"> + Difficulty of math. notation, <b><a + href="#Block_1218">1218</a></b>.</li> + <li class="isub1"> + On symbolic logic, <b><a + href="#Block_1320">1320</a></b>.</li> + <li class="isub1"> + Principia Mathematica, <a + href="#Block_1326">1326</a>.</li> + <li class="isub1"> + On philosophy and m., <b><a + href="#Block_1403">1403</a></b>.</li> + <li class="isub1"> + On obscurity in m. and philosophy, <b><a + href="#Block_1407">1407</a></b>.</li> + <li class="isub1"> + On the laws of algebra, <b><a + href="#Block_1708">1708</a></b>.</li> + <li class="isub1"> + On + and − signs, <b><a + href="#Block_1727">1727</a></b>.</li> + <li class="isub1"> + On universal algebra, <b><a + href="#Block_1753">1753</a></b>.</li> + <li class="isub1"> + On the Cartesian method, <b><a + href="#Block_1890">1890</a></b>.</li> + <li class="isub1"> + On Swift’s ignorance of m., <b><a + href="#Block_2125">2125</a></b>.</li> + <li class="indx"> + Whitworth, On the solution of problems, <b><a + href="#Block_611">611</a></b>.</li> + <li class="indx"> + Williamson,</li> + <li class="isub1"> + On the value of m., <b><a + href="#Block_1575">1575</a></b>.</li> + <li class="isub1"> + Infinitesimals and limits, <b><a + href="#Block_1905">1905</a></b>.</li> + <li class="isub1"> + On infinitesimals, <b><a + href="#Block_1946">1946</a></b>.</li> + <li class="indx"> + Wilson, E. B.,</li> + <li class="isub1"> + On the social value of m., <b><a + href="#Block_1588">1588</a></b>.</li> + <li class="isub1"> + On m. and economics, <b><a + href="#Block_1593">1593</a></b>.</li> + <li class="isub1"> + On the nature of axioms, <b><a + href="#Block_2012">2012</a></b>.</li> + <li class="indx"> + Wilson, John,</li> + <li class="isub1"> + On Newton and Shakespeare, <b><a + href="#Block_1012">1012</a></b>.</li> + <li class="isub1"> + Newton and Linnæus, <b><a + href="#Block_1013">1013</a></b>.</li> + <li class="indx"> + Woodward,</li> + <li class="isub1"> + On probabilities, <b><a + href="#Block_1589">1589</a></b>.</li> + <li class="isub1"> + On the theory of errors, <b><a + href="#Block_1973">1973</a></b>, <b><a + href="#Block_1974">1974</a></b>.</li> + <li class="indx"> + Wordsworth, W.,</li> + <li class="isub1"> + On Archimedes, <b><a + href="#Block_906">906</a></b>.</li> + <li class="isub1"> + On poetry and geometric truth, <b><a + href="#Block_1117">1117</a></b>.</li> + <li class="isub1"> + On geometric rules, <b><a + href="#Block_1418">1418</a></b>.</li> + <li class="isub1"> + On geometry, <b><a + href="#Block_1840">1840</a></b>, <b><a + href="#Block_1848">1848</a></b>.</li> + <li class="isub1"> + M. and solitude, <b><em><a + href="#Block_1849">1859</a></em></b>.</li> + <li class="indx"> + Workman, On the poetic nature of m., <b><a + href="#Block_1120">1120</a></b>.</li> + + <li class="ifrst"> + Young, C. A., On the discovery of Neptune, <b><a + href="#Block_1559">1559</a></b>.</li> + <li class="indx"> + Young, C. W., Definition of m., <b><a + href="#Block_124">124</a></b>.</li> + <li class="indx"> + Young, J. W. A.,</li> + <li class="isub1"> + On m. as type a of thought, <b><a + href="#Block_404">404</a></b>.</li> + <li class="isub1"> + M. as preparation for science study, <b><a + href="#Block_421">421</a></b>.</li> + <li class="isub1"> + M. essential to comprehension of nature, <b><a + href="#Block_435">435</a></b>.</li> + <li class="isub1"> + Development of abstract methods, <b><a + href="#Block_729">729</a></b>.</li> + <li class="isub1"> + Beauty in m., <b><a + href="#Block_1110">1110</a></b>.</li> + <li class="isub1"> + On Euclid’s axiom, <b><a + href="#Block_2014">2014</a></b>.</li> + + <li class="ifrst"> + Zeno, His problems, <a + href="#Block_1938">1938</a>.</li> + <li class="indx"> + Zero, <a + href="#Block_1948">1948</a>, <a + href="#Block_1954">1954</a>.</li> + </ul> + + <hr class="full" /> + + <div class="footnote"> + <h2 id="Footnotes"> + Footnotes</h2> + <table summary=""> + <tr> + <td class="rt"> + <a id="Footnote_1" + href="#Block_225">1</a></td> + <td> + i.e., in terms of the absolutely clear and + <em>in</em>definable.</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_2" + href="#Block_831">2</a></td> + <td> + Used here in the sense of astrologer, or + soothsayer.</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_3" + href="#Block_832">3</a></td> + <td> + This is the estimate furnished me by two mathematical + masters in one of our great public schools of the + proportion of boys who have any special taste or + capacity for mathematical studies. Many more, of course, + can be drilled into a fair knowledge of elementary + mathematics, but only this small proportion possess the + natural faculty which renders it possible for them ever + to rank high as mathematicians, to take any pleasure in + it, or to do any original mathematical work.</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_4" + href="#Block_836">4</a></td> + <td> + The mathematical tendencies of Cambridge are due to + the fact that Cambridge drains the ability of nearly + the whole Anglo-Danish district.</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_5" + href="#Block_953">5</a></td> + <td> + Riccardi’s Bibliografia Euclidea (Bologna, 1887), + lists nearly two thousand editions.</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_6" + href="#Block_959">6</a></td> + <td> + The line referred to is: <br /> + “The anchor drops, the rushing keel is staid.”</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_7" + href="#Block_968">7</a></td> + <td> + Johannes Flamsteedius.</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_8" + href="#Block_990">8</a></td> + <td> + This sentence has been reworded for the purpose + of this quotation.</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_9" + href="#Block_1048">9</a></td> + <td> + Author’s note. My colleague, Dr. E. T. Bell, informs + me that this same anecdote is associated with the name + of J. S. Blackie, Professor of Greek at Aberdeen and + Edinburgh.</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_10" + href="#Block_1855">10</a></td> + <td> + In the German vernacular a dunce or blockhead is + called an ox.</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_11" + href="#Block_2003">11</a></td> + <td> + Schopenhauer’s table contains a third column headed + “of matter” which has here been omitted.</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_12" + href="#Block_2117">12</a></td> + <td> + For another rendition of these same lines see + 1858.</td></tr> + <tr> + <td class="rt"> + <a id="Footnote_13" + href="#Block_2156">13</a></td> + <td> + The beginning of a poem which Johannes a Lasco wrote + on the count Karl von Südermanland.</td></tr> + </table> + </div> + + <hr class="chap" /> + <div class="tn"> + <h2 id="Tnotes"> + Transcriber’s Notes:</h2> + <blockquote class="blockhang"> + <p> + Punctuation has been standardised.</p></blockquote> + + <blockquote class="blockhang"> + <p> + Em-dash added before all attribution names for + consistency.</p></blockquote> + + <blockquote class="blockhang"> + <p> + Mis-alphabetized entries in the Index have been + corrected</p></blockquote> + + <blockquote class="blockhang"> + <p> + Several references in the Index refer to wrong quote blocks. + The reference number has been left unchanged, but italicized. + If the correct block could be easily identified, the link has + been updated to the correct block. If the reference refers + to a non-existing block, it is not linked. Only bold author + links were checked, all others were left as presented. + </p></blockquote> + + <blockquote> + <p> + Book was written in a period when many words had not + become standardized in their spelling. Numerous words have + multiple spelling variations in the text. These have been + left unchanged unless noted below:</p></blockquote> + + <table summary=""> + <tr> + <td class="rt"> + <a id="TN_1" + href="#Block_230">§230</a></td> + <td class="hang"> + — “elmenetary” corrected to “elementary” (the most elementary + use of)</td></tr> + <tr> + <td class="rt"> + <a id="TN_2" + href="#Block_437">§437</a></td> + <td class="hang"> + — “Mathematiks” corrected to “Mathematicks” (The Usefulness + of Mathematicks) as shown in the quoted text.</td></tr> + <tr> + <td class="rt"> + <a id="TN_9" + href="#Block_511">§511</a></td> + <td class="hang"> + — “517” corrected to block “511”</td></tr> + <tr> + <td class="rt"> + <a id="TN_3" + href="#Block_517">§517</a></td> + <td class="hang"> + — “hoheren” corrected to “höheren” (höheren Schulen) + for consistency</td></tr> + <tr> + <td class="rt"> + <a id="TN_4" + href="#Block_540">§540</a></td> + <td class="hang"> + — duplicate word “the” removed (let the mind)</td></tr> + <tr> + <td class="rt"> + <a id="TN_5" + href="#Block_657">§657</a></td> + <td class="hang"> + — “anaylsis” corrected to “analysis” + (field of analysis.)</td></tr> + <tr> + <td class="rt"> + <a id="TN_6" + href="#Block_729">§729</a></td> + <td class="hang"> + — “Geomtry” corrected to “Geometry” + (Algebra and Geometry)</td></tr> + <tr> + <td class="rt"> + <a id="TN_7" + href="#Block_822">§822</a></td> + <td class="hang"> + — end of quote not identified + - placement unclear.</td></tr> + <tr> + <td class="rt"> + <a id="TN_8" + href="#Block_823">§823</a></td> + <td class="hang"> + — “heros” corrected to “heroes” + (many of the major heroes)</td></tr> + <tr> + <td class="rt"> + <a id="TN_26" + href="#Block_917">§917</a></td> + <td class="hang"> + — “εὓυρηκα” corrected to “εὔυρηκα” + (speaks of the εὔυρηκα)</td></tr> + <tr> + <td class="rt"> + <a id="TN_10" + href="#Block_1132">§1132</a></td> + <td class="hang"> + — “Vereiningung” corrected to “Vereinigung” + (Deutschen Mathematiker Vereinigung )</td></tr> + <tr> + <td class="rt"> + <a id="TN_11" + href="#Block_1325">§1325</a></td> + <td class="hang"> + — “Philosphy” corrected to “Philosophy” + (Positive Philosophy)</td></tr> + <tr> + <td class="rt"> + <a id="TN_12" + href="#Block_1421">§1421</a></td> + <td class="hang"> + — “1427” corrected to block “1421”</td></tr> + <tr> + <td class="rt"> + <a id="TN_13" + href="#Block_1503">§1503</a></td> + <td class="hang"> + — “Todhunder’s” corrected to “Todhunter’s” + (Todhunter’s History of)</td></tr> + <tr> + <td class="rt"> + <a id="TN_14" + href="#Block_1535">§1535</a></td> + <td class="hang"> + — “uses” corrected to “use” + (the use of analysis)</td></tr> + <tr> + <td class="rt"> + <a id="TN_15" + href="#Block_1803">§1803</a></td> + <td class="hang"> + — “τὴυ” corrected to “τὴν” + (μοῦ τὴν στέγην)</td></tr> + <tr> + <td class="rt"> + <a id="TN_16" + href="#Block_1874">§1874</a></td> + <td class="hang"> + — “anaylsis” corrected to “analysis” + (a kind of analysis)</td></tr> + <tr> + <td class="rt"> + <a id="TN_17" + href="#Block_1930">§1930</a></td> + <td class="hang"> + — “Hyberbolic” corrected to “Hyperbolic” + (Mathematical Tables, Hyperbolic Functions)</td></tr> + <tr> + <td class="rt"> + <a id="TN_18" + href="#Block_2009">§2009</a></td> + <td class="hang"> + — “Stanfpunkte” corrected to “Standpunkte” + (höheren Standpunkte aus)</td></tr> + <tr> + <td class="rt"> + <a id="TN_19" + href="#Block_2126">§2126</a></td> + <td class="hang"> + — Block number 2126 added</td></tr> + <tr> + <td class="rt"> + <a id="TN_20" + href="#Block_2135">§2135</a></td> + <td class="hang"> + — “astromomy” corrected to “astronomy” + (history of astronomy)</td></tr> + <tr> + <td class="rt"> + <a id="TN_27" + href="#Block_2151">§2151</a></td> + <td class="hang"> + — “10” corrected to “9” + (A to I represent 1-9)</td></tr> + <tr> + <td class="rt"> + <a id="TN_21" + href="#TNanchor_21">Appolonius</a></td> + <td class="hang"> + — Also spelled “Apollonius” but not + referenced at blocks 523 and 917</td></tr> + <tr> + <td class="rt"> + <a id="TN_22" + href="#TNanchor_22">Bôcher</a></td> + <td class="hang"> + — “Bocher” corrected to “Bôcher” as given + in text</td></tr> + <tr> + <td class="rt"> + <a id="TN_23" + href="#TNanchor_23">Halsted</a></td> + <td class="hang"> + — “Slyvester” corrected to “Sylvester”</td></tr> + <tr> + <td class="rt"> + <a id="TN_24" + href="#TNanchor_24">Jefferson</a></td> + <td class="hang"> + — “Om” corrected to “On” + (On m. and law)</td></tr> + <tr> + <td class="rt"> + <a id="TN_25" + href="#TNanchor_25">Peacock</a></td> + <td class="hang"> + — “Philosphers” corrected to “philosophers” + (Greek philosophers)</td></tr> + </table> + </div> + +<div>*** END OF THE PROJECT GUTENBERG EBOOK 44730 ***</div> +</body> +</html> |