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+Project Gutenberg's Memorabilia Mathematica, by Robert Edouard Moritz
+
+This eBook is for the use of anyone anywhere at no cost and with
+almost no restrictions whatsoever. You may copy it, give it away or
+re-use it under the terms of the Project Gutenberg License included
+with this eBook or online at www.gutenberg.org
+
+
+Title: Memorabilia Mathematica
+ or the Philomath's Quotation-Book
+
+Author: Robert Edouard Moritz
+
+Release Date: January 22, 2014 [EBook #44730]
+
+Language: English
+
+Character set encoding: ASCII
+
+*** START OF THIS PROJECT GUTENBERG EBOOK MEMORABILIA MATHEMATICA ***
+
+
+
+
+Produced by Peter Vachuska, Richard Hulse and the Online
+Distributed Proofreading Team at http://www.pgdp.net
+
+
+
+
+
+
+
+
+
+
+ MEMORABILIA MATHEMATICA
+
+
+
+
+ THE MACMILLAN COMPANY
+
+ NEW YORK. BOSTON. CHICAGO. DALLAS
+ ATLANTA. SAN FRANCISCO
+
+ MACMILLAN & CO., LIMITED
+
+ LONDON. BOMBAY. CALCUTTA
+ MELBOURNE
+
+ THE MACMILLAN CO. OF CANADA, LTD.
+
+ TORONTO
+
+
+
+
+ MEMORABILIA MATHEMATICA
+
+ OR
+
+ THE PHILOMATH'S QUOTATION-BOOK
+
+
+
+
+ BY
+
+ ROBERT EDOUARD MORITZ, PH. D., PH. N. D.
+
+ PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF WASHINGTON
+
+ New York
+ THE MACMILLAN COMPANY
+ 1914
+
+ _All rights reserved_
+
+ COPYRIGHT, 1914, BY
+ ROBERT EDOUARD MORITZ
+
+
+
+
+ PREFACE
+
+
+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.
+
+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.
+
+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 grouped under
+twenty heads, and cross indexed under nearly seven hundred topics.
+
+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.
+
+The absence of similar English works has made the author's work largely
+that of the pioneer. Rebiere's "Mathematiques et Mathematiciens"
+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.
+
+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 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."
+
+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.
+
+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.
+
+ ROBERT EDOUARD MORITZ,
+ _September, 1913_.
+
+
+
+
+
+ CONTENTS
+
+ CHAPTER PAGE
+
+ I. DEFINITIONS AND OBJECT OF MATHEMATICS 1
+
+ II. THE NATURE OF MATHEMATICS 10
+
+ III. ESTIMATES OF MATHEMATICS 39
+
+ IV. THE VALUE OF MATHEMATICS 49
+
+ V. THE TEACHING OF MATHEMATICS 72
+
+ VI. STUDY AND RESEARCH IN MATHEMATICS 86
+
+ VII. MODERN MATHEMATICS 108
+
+ VIII. THE MATHEMATICIAN 121
+
+ IX. PERSONS AND ANECDOTES (A-M) 135
+
+ X. PERSONS AND ANECDOTES (N-Z) 166
+
+ XI. MATHEMATICS AS A FINE ART 181
+
+ XII. MATHEMATICS AS A LANGUAGE 194
+
+ XIII. MATHEMATICS AND LOGIC 201
+
+ XIV. MATHEMATICS AND PHILOSOPHY 209
+
+ XV. MATHEMATICS AND SCIENCE 224
+
+ XVI. ARITHMETIC 261
+
+ XVII. ALGEBRA 275
+
+ XVIII. GEOMETRY 292
+
+ XIX. THE CALCULUS AND ALLIED TOPICS 323
+
+ XX. THE FUNDAMENTAL CONCEPTS OF TIME AND SPACE 345
+
+ XXI. PARADOXES AND CURIOSITIES 364
+
+ INDEX 385
+
+
+
+
+ Alles Gescheite ist schon gedacht worden; man muss nur versuchen,
+ es noch einmal zu denken.--GOETHE.
+
+ _Sprueche in Prosa, Ethisches, I. 1._
+
+
+ A great man quotes bravely, and will not draw on his invention
+ when his memory serves him with a word as good.--EMERSON.
+
+ _Letters and Social Aims, Quotation and
+ Originality._
+
+
+
+
+ MEMORABILIA MATHEMATICA
+
+
+
+
+ MEMORABILIA MATHEMATICA
+
+ CHAPTER I
+
+ DEFINITIONS AND OBJECT OF MATHEMATICS
+
+
+=101.= 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.--SYLVESTER, J. J.
+
+ _Presidential Address to the British
+ Association, Exeter British Association
+ Report (1869); Collected Mathematical
+ Papers, Vol. 2, p. 659._
+
+
+=102.= ... 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, _mathematics_. 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; ....
+
+ --DESCARTES.
+
+ _Rules for the Direction of the Mind,
+ Philosophy of D. [Torrey] (New York,
+ 1892), p. 72._
+
+
+=103.= [Mathematics] has for its object the _indirect_
+measurement of magnitudes, and it _purposes to determine
+magnitudes by each other, according to the precise relations
+which exist between them_.--COMTE.
+
+ _Positive Philosophy [Martineau], Bk. 1,
+ chap. 1._
+
+
+=104.= 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.--COMTE.
+
+ _Positive Philosophy [Martineau], Bk. 1,
+ chap. 2._
+
+
+=105.= 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.--GRASSMANN, HERMANN.
+
+ _Stuecke aus dem Lehrbuche der
+ Arithmetik, Werke (Leipzig, 1904), Bd.
+ 2, p. 298._
+
+
+=106.= 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.
+
+ --BACON, FRANCIS.
+
+ _De Augmentis, Bk. 3; Advancement of
+ Learning, Bk. 2._
+
+
+=107.= 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 _Mixed Mathematics_, 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 _Pure Mathematics_.--WHEWELL, WILLIAM.
+
+ _The Philosophy of the Inductive
+ Sciences, Part 1, Bk. 2, chap. I, sect.
+ 4. (London, 1858)._
+
+
+=108.= 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.--MELLOR, J. W.
+
+ _Higher Mathematics for Students of
+ Chemistry and Physics (New York, 1902),
+ Prologue._
+
+
+=109.= Number, place, and combination ... the three intersecting
+but distinct spheres of thought to which all mathematical ideas
+admit of being referred.--SYLVESTER, J. J.
+
+ _Philosophical Magazine, Vol. 24 (1844),
+ p. 285; Collected Mathematical Papers,
+ Vol. 1, p. 91._
+
+
+=110.= 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.
+
+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.--SYLVESTER, J. J.
+
+ _A Probationary Lecture on Geometry,
+ York British Association Report (1844),
+ Part 2; Collected Mathematical Papers,
+ Vol. 2, p. 5._
+
+
+=111.= 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.
+
+ --PAPPERITZ, E.
+
+ _Ueber das System der rein mathematischen
+ Wissenschaften, Jahresbericht der
+ Deutschen Mathematiker-Vereinigung, Bd.
+ 1, p. 36._
+
+
+=112.= Pure mathematics is not concerned with magnitude. It is
+merely the doctrine of notation of relatively ordered thought
+operations which have become mechanical.--NOVALIS.
+
+ _Schriften (Berlin, 1901), Zweiter Teil,
+ p. 282._
+
+
+=113.= 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.--CHRYSTAL, GEORGE.
+
+ _Encyclopedia Britannica (9th edition),
+ Article "Mathematics."_
+
+
+=114.= 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.--HANKEL, HERMANN.
+
+ _Theorie der Complexen Zahlensysteme,
+ (Leipzig, 1867), p. 1._
+
+
+=115.= _Quantity is that which is operated with according to
+fixed mutually consistent laws._ 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 enables us to see how operators may be
+treated as quantities, and thus to understand the rationale of
+the so called symbolical methods.--CHRYSTAL, GEORGE.
+
+ _Encyclopedia Britannica (9th edition),
+ Article "Mathematics."_
+
+
+=116.= Mathematics--in a strict sense--is the abstract science
+which investigates deductively the conclusions implicit in the
+elementary conceptions of spatial and numerical relations.
+
+ --MURRAY, J. A. H.
+
+ _A New English Dictionary._
+
+
+=117.= Everything that the greatest minds of all times have
+accomplished toward the _comprehension of forms_ by means of
+concepts is gathered into one great science, _mathematics_.
+
+ --HERBART, J. F.
+
+ _Pestalozzi's Idee eines A B C der
+ Anschauung, Werke [Kehrbach],
+ (Langensalza, 1890), Bd. 1, p. 163._
+
+
+=118.= 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 _form_, 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.--HOBSON, E. W.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1910); Nature, Vol. 84, p.
+ 287._
+
+
+=119.= 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.
+
+ --WHITEHEAD, A. N.
+
+ _Universal Algebra (Cambridge, 1898),
+ Preface._
+
+
+=120.= Mathematics is the science which draws necessary
+conclusions.--PEIRCE, BENJAMIN.
+
+ _Linear Associative Algebra, American
+ Journal of Mathematics, Vol. 4 (1881),
+ p. 97._
+
+
+=121.= Mathematics is the universal art apodictic.--SMITH, W. B.
+
+ _Quoted by Keyser, C. J. in Lectures on
+ Science, Philosophy and Art (New York,
+ 1908), p. 13._
+
+
+=122.= Mathematics in its widest signification is the development
+of all types of formal, necessary, deductive reasoning.
+
+ --WHITEHEAD, A. N.
+
+ _Universal Algebra (Cambridge, 1898),
+ Preface, p. vi._
+
+
+=123.= Mathematics in general is fundamentally the science of
+self-evident things.--KLEIN, FELIX.
+
+ _Anwendung der Differential- und
+ Integralrechnung auf Geometrie (Leipzig,
+ 1902), p. 26._
+
+
+=124.= 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.--YOUNG, CHARLES WESLEY.
+
+ _Fundamental Concepts of Algebra and
+ Geometry (New York, 1911), p. 222._
+
+
+=125.= Pure mathematics is a collection of hypothetical,
+deductive theories, each consisting of a definite system of
+primitive, _undefined_, concepts or symbols and primitive,
+_unproved_, but self-consistent assumptions (commonly called
+axioms) together with their logically deducible consequences
+following by rigidly deductive processes without appeal to
+intuition.--FITCH, G. D.
+
+ _The Fourth Dimension simply Explained
+ (New York, 1910), p. 58._
+
+
+=126.= The whole of Mathematics consists in the organization of a
+series of aids to the imagination in the process of reasoning.
+
+ --WHITEHEAD, A. N.
+
+ _Universal Algebra (Cambridge, 1898), p.
+ 12._
+
+
+=127.= Pure mathematics consists entirely of such asseverations
+as that, if such and such a proposition is true of _anything_,
+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 _anything_
+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.--RUSSELL, BERTRAND.
+
+ _Recent Work on the Principles of
+ Mathematics, International Monthly, Vol.
+ 4 (1901), p. 84._
+
+
+=128.= Pure Mathematics is the class of all propositions of the form
+"_p_ implies _q_," where _p_ and _q_ are propositions containing one
+or more variables, the same in the two propositions, and neither _p_
+nor _q_ 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 _such that_, 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
+_uses_ a notion which is not a constituent of the propositions
+which it considers--namely, the notion of truth.--RUSSELL, BERTRAND.
+
+ _Principles of Mathematics (Cambridge,
+ 1903), p. 1._
+
+
+=129.= 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.--SYLVESTER, J. J.
+
+ _On a theorem, connected with Newton's
+ Rule, etc., Collected Mathematical
+ Papers, Vol. 3, p. 424._
+
+
+=130.= First of all, we ought to observe, that mathematical
+propositions, properly so called, are always judgments _a
+priori,_ and not empirical, because they carry along with them
+necessity, which can never be deduced from experience. If people
+should 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 _a
+priori_.--KANT, IMMANUEL.
+
+ _Critique of Pure Reason [Mueller], (New
+ York, 1900), p. 720._
+
+
+=131.= 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....
+
+ --WHITE, WILLIAM F.
+
+ _A Scrap-book of Elementary Mathematics,
+ (Chicago, 1908), p. 215._
+
+
+=132.= 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.
+
+ --KEYSER, C. J.
+
+ _Science, New Series, Vol. 35, p. 107._
+
+
+=133.= [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.--PEIRCE, C. S.
+
+ _Century Dictionary, Article
+ "Mathematics."_
+
+
+=134.= 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.]--HOWISON, G. H.
+
+ _The Departments of Mathematics, and
+ their Mutual Relations; Journal of
+ Speculative Philosophy, Vol. 5, p. 164._
+
+
+=135.= Mathematics is the science of the functional laws and
+transformations which enable us to convert figured extension and
+rated motion into number.--HOWISON, G. H.
+
+ _The Departments of Mathematics, and
+ their Mutual Relations; Journal of
+ Speculative Philosophy, Vol. 5, p. 170._
+
+
+
+
+ CHAPTER II
+
+ THE NATURE OF MATHEMATICS
+
+
+=201.= 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
+_revolution_, 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 _Thales_ 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 _a priori_, placed into
+that figure and represented in it, so that, in order to know
+anything with certainty _a priori_, 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.
+
+ --KANT, IMMANUEL.
+
+ _Critique of Pure Reason, Preface to the
+ Second Edition [Mueller], (New York,
+ 1900), p. 690._
+
+
+=202.= [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
+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.--SYLVESTER, J. J.
+
+ _A Plea for the Mathematician, Nature,
+ Vol. 1, p. 261._
+
+
+=203.= 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 _Lagrange_ or _Cauchy_, who with the
+keen discernment of a military commander marshalls a whole troop
+of completed operations in the execution of a new one.--MACH, E.
+
+ _Populaer-wissenschafliche Vorlesungen
+ (1908), pp. 224-225._
+
+
+=204.= 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.
+
+ --DILLMANN, E.
+
+ _Die Mathematik die Fackeltraegerin einer
+ neuen Zeit (Stuttgart, 1889), p. 94._
+
+
+=205.= The essence of mathematics lies in its freedom.
+
+ --CANTOR, GEORGE.
+
+ _Mathematische Annalen, Bd. 21, p. 564._
+
+
+=206.= 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.--HANKEL, HERMANN.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1884),
+ p. 16._
+
+
+=207.= 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.--CANTOR, GEORGE.
+
+ _Grundlagen einer allgemeinen
+ Manigfaltigkeitslehre (Leipzig, 1883),
+ Sect. 8._
+
+
+=208.= Mathematicians assume the right to choose, within the
+limits of logical contradiction, what path they please in
+reaching their results.--ADAMS, HENRY.
+
+ _A Letter to American Teachers of
+ History (Washington, 1910),
+ Introduction, p. v._
+
+
+=209.= 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.--HERBART, J. F.
+
+ _Werke [Kehrbach] (Langensalza, 1890),
+ Bd. 5, p. 105._
+
+
+=210.= 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.
+
+ --PEIRCE, BENJAMIN.
+
+ _Linear Associative Algebra, American
+ Journal of Mathematics, Vol. 4 (1881),
+ p. 97._
+
+
+=211.= Mathematics is a science continually expanding; and its
+growth, unlike some political and industrial events, is attended
+by universal acclamation.--WHITE, H. S.
+
+ _Congress of Arts and Sciences (Boston
+ and New York, 1905), Vol. 1, p. 455._
+
+
+=212.= 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.--HERBART, J. F.
+
+ _Werke [Kehrbach] (Langensalza, 1890),
+ Bd. 5, p. 101._
+
+
+=213.= 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.--BARROW, ISAAC.
+
+ _Mathematical Lectures (London, 1734),
+ p. 66._
+
+
+=214.= The dexterous management of terms and being able to _fend_
+and _prove_ 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.
+
+ --LOCKE, JOHN.
+
+ _Conduct of the Understanding, Sect.
+ 31._
+
+
+=215.= 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.
+
+ --REID, THOMAS.
+
+ _Essays on the Intellectual Powers of
+ Man, Essay 1, chap. 1._
+
+
+=216.= 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.--HANKEL, HERMANN.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1884),
+ p. 25._
+
+
+=217.= Mathematics, the priestess of definiteness and
+clearness.--HERBART, J. F.
+
+ _Werke [Kehrbach] (Langensalza, 1890),
+ Bd. 1, p. 171._
+
+
+=218.= ... 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.
+
+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.
+
+ --FOURIER, J.
+
+ _Theorie Analytique de la Chaleur,
+ Discours Preliminaire._
+
+
+=219.= 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....
+
+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 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.
+
+ --DESCARTES.
+
+ _Rules for the Direction of the Mind,
+ Philosophy of D. [Torrey] (New York,
+ 1892), pp. 64, 65._
+
+
+=220.= 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.--LAPLACE.
+
+ _A Philosophical Essay on Probabilities
+ [Truscott and Emory] (New York 1902), p.
+ 176._
+
+
+=221.= 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.
+
+ --MILL, J. S.
+
+ _System of Logic, Bk. 2, chap. 6, 2._
+
+
+=222.= 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 to this
+hypothetical character is to be ascribed the peculiar certainty,
+which is supposed to be inherent in demonstration.--MILL, J. S.
+
+ _System of Logic, Bk. 2, chap. 6, 1._
+
+
+=223.= 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.
+
+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.--WHEWELL, W.
+
+ _Thoughts on the Study of Mathematics.
+ Principles of English University
+ Education (London, 1838)._
+
+
+=224.= These sciences, Geometry, Theoretical Arithmetic and Algebra,
+have no principles besides definitions and axioms, and no process
+of proof but _deduction_; 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.--WHEWELL, W.
+
+ _The Philosophy of the Inductive
+ Sciences, Part 1, Bk. 2, chap. 1, sect.
+ 2 (London, 1858)._
+
+
+=225.= 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:
+_precision_, _sharpness_, _completeness_,[1] 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.--KEYSER, C. J.
+
+ _The Universe and Beyond; Hibbert
+ Journal, Vol. 3 (1904-1905), p. 309._
+
+ [1] i.e., in terms of the absolutely clear and
+ _in_definable.
+
+
+=226.= The reasoning of mathematicians is founded on certain 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.--ADAMS, JOHN.
+
+ _Diary, Works (Boston, 1850), Vol. 2, p.
+ 21._
+
+
+=227.= 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,
+_without passion, without apology_; knowing that their reasons,
+as Seneca testifies of them, are not brought to _persuade_, but
+to compel.--BARROW, ISAAC.
+
+ _Mathematical Lectures (London, 1734),
+ p. 64._
+
+
+=228.= What is exact about mathematics but exactness? And is not
+this a consequence of the inner sense of truth?--GOETHE.
+
+ _Sprueche in Prosa, Natur, 6, 948._
+
+
+=229.= ... 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 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.--SCHUBERT, H.
+
+ _Mathematical Essays and Recreations
+ (Chicago, 1898), p. 35._
+
+
+=230.= 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 _belief_ that
+nothing has been overlooked from the beginning. Every one knows
+how much this is the case even in arithmetic, the most elementary
+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 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.--HERBART, J. F.
+
+ _Werke [Kehrbach] (Langensalza, 1890),
+ Bd. 5, p. 105._
+
+
+=231.= [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.--HELMHOLTZ, H.
+
+ _Ueber das Verhaeltniss der
+ Naturwissenschaften zur Gesammtheit der
+ Wissenschaft, Vortraege und Reden, Bd. 1
+ (1896), p. 176._
+
+
+=232.= 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
+_self-evident_ [Selbstverstaendlichen]. Experience however, shows
+that for the majority of the cultured, even of scientists,
+mathematics remains the science of the _incomprehensible_
+[Unverstaendlichen].--PRINGSHEIM, ALFRED.
+
+ _Ueber Wert und angeblichen Unwert der
+ Mathematik, Jahresbericht der Deutschen
+ Mathematiker Vereinigung (1904), p.
+ 357._
+
+
+=233.= 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.--WHITEHEAD, A. N.
+
+ _Universal Algebra (Cambridge, 1898),
+ Preface, p. vi._
+
+
+=234.= 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.
+
+ --REID, THOMAS.
+
+ _Essays on the Intellectual Powers of
+ Man, Essay 1, chap. 3._
+
+
+=235.= ... 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.--LOCKE, JOHN.
+
+ _The Conduct of the Understanding, Sect.
+ 21._
+
+
+=236.= Those intervening ideas, which serve to show the agreement
+of any two others, are called _proofs_; and where the agreement or
+disagreement is by this means plainly and clearly perceived, it is
+called _demonstration_; it being _shown_ 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 _sagacity_.--LOCKE, JOHN.
+
+ _An Essay concerning Human
+ Understanding, Bk. 6, chaps. 2, 3._
+
+
+=237.= ... the speculative propositions of mathematics do not
+relate to _facts_; ... 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, _certain_ 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.--STEWART, DUGALD.
+
+ _Elements of the Philosophy of the Human
+ Mind, Part 3, chap. 1, sect. 3._
+
+
+=238.= No process of sound reasoning can establish a result not
+contained in the premises.--MELLOR, J. W.
+
+ _Higher Mathematics for Students of
+ Chemistry and Physics (New York, 1902),
+ p. 2._
+
+
+=239.= ... 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.--HOPKINSON, JOHN.
+
+ _James Forrest Lecture, 1894._
+
+
+=240.= 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
+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.
+
+ --HELMHOLTZ, H.
+
+ _Ueber das Verhaeltniss der
+ Naturwissenschaften zur Gesammtheit der
+ Wissenschaft, Vortraege und Reden, Bd. 1
+ (1896), p. 178._
+
+
+=241.= 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 _lemmas_, 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.--LEIBNITZ, G. W.
+
+ _New Essay on Human Understanding
+ [Langley], Bk. 4, chaps. 2, 8._
+
+
+=242.= 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.
+
+ --STEWART, DUGALD.
+
+ _The Elements of the Philosophy of the
+ Human Mind, Part 3, chap. 1, sect. 3._
+
+
+=243.= 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
+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.--SCHUBERT, H.
+
+ _Mathematical Essays and Recreations
+ (Chicago, 1898), p. 28._
+
+
+=244.= ... 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.
+
+ --KEYSER, CASSIUS J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 26._
+
+
+=245.= 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
+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.--MILL, JOHN STUART.
+
+ _An Examination of Sir William
+ Hamilton's Philosophy (London, 1878),
+ pp. 612, 613._
+
+
+=246.= 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."--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 31._
+
+
+=247.= ... 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
+in perceiving logical sequence, love of order, methodical
+arrangement and harmony, distinctness of conception.--PRICE, B.
+
+ _Treatise on Infinitesimal Calculus
+ (Oxford, 1868), Vol. 3, p. 6._
+
+
+=248.= Histories make men wise; poets, witty; the mathematics,
+subtile; natural philosophy, deep; moral, grave; logic and
+rhetoric, able to contend.--BACON, FRANCIS.
+
+ _Essays, Of Studies._
+
+
+=249.= 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.--HUXLEY, T. H.
+
+ _On the Educational Value of the Natural
+ History Sciences; Lay Sermons, Addresses
+ and Reviews; (New York, 1872), p. 87._
+
+
+=250.= [Mathematics] is that [subject] which knows nothing of
+observation, nothing of experiment, nothing of induction, nothing
+of causation.--HUXLEY, T. H.
+
+ _The Scientific Aspects of Positivism,
+ Fortnightly Review (1898); Lay Sermons,
+ Addresses and Reviews, (New York, 1872),
+ p. 169._
+
+
+=251.= 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 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.
+
+ --SYLVESTER, J. J.
+
+ _Presidential Address to British
+ Association, Exeter British Association
+ Report (1869), pp. 1-9.; Collected
+ Mathematical Papers, Vol. 2, p. 654._
+
+
+=252.= 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.
+
+ --HOBSON, E. W.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1910); Nature, Vol. 84, p.
+ 290._
+
+
+=253.= It has been asserted ... that the power of observation is not
+developed by mathematical studies; while the truth is, 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 _one_ and more than one, between _one_ and
+_two_, _two_ and _three_, 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 oning--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 _observations_. 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 _memory_ 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 _imagination_, the creative faculty of the 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
+_infinitesimal calculus_--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 _induction_,
+_analogy_, the _scrutinization_ of _premises_ or the _search_ for
+them, or the _balancing_ of _probabilities_, 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.--OLNEY, EDWARD.
+
+ _Kiddle and Schem's Encyclopedia of
+ Education, (New York, 1877), Article
+ "Mathematics."_
+
+
+=254.= 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 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.--HOBSON, E. W.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1910); Nature, Vol. 84, p.
+ 290._
+
+
+=255.= 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 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."--SYLVESTER, J. J.
+
+ _A Plea for the Mathematician, Nature,
+ Vol. 1, p. 238; Collected Mathematical
+ Papers, Vol. 2, pp. 655, 656._
+
+
+=256.= 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.--FISKE, JOHN.
+
+ _Darwinism and other Essays (Boston,
+ 1893), p. 296._
+
+
+=257.= The great science [mathematics] occupies itself at least
+just as much with the power of imagination as with the power of
+logical conclusion.--HERBART, F. J.
+
+ _Pestalozzi's Idee eines ABC der
+ Anschauung. Werke [Kehrbach]
+ (Langensaltza, 1890), Bd. 1, p. 174._
+
+
+=258.= The moving power of mathematical invention is not
+reasoning but imagination.--DE MORGAN, A.
+
+ _Quoted in Graves' Life of Sir W. R.
+ Hamilton, Vol. 3 (1889), p. 219._
+
+
+=259.= 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.--VOLTAIRE.
+
+ _A Philosophical Dictionary (Boston,
+ 1881), Vol. 3, p. 40. Article
+ "Imagination."_
+
+
+=260.= 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.
+
+ --SYLVESTER, J. J.
+
+ _A Plea for the Mathematician, Nature,
+ Vol. 1, p. 261; Collected Mathematical
+ Papers, Vol. 2 (Cambridge, 1908), p.
+ 717._
+
+
+=261.= 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.--DEE, JOHN.
+
+ _Euclid (1570), Preface._
+
+
+=262.= Mathematics stands forth as that which unites, mediates
+between Man and Nature, inner and outer world, thought and
+perception, as no other subject does.--FROEBEL.
+
+ _[Herford translation] (London, 1893),
+ Vol. 1, p. 84._
+
+
+=263.= 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.--SCHUBERT, H.
+
+ _Mathematical Essays and Recreations
+ (Chicago, 1898), p. 27._
+
+
+=264.= 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.
+
+ --KLEIN, F.
+
+ _Klein und Riecke: Ueber angewandte
+ Mathematik und Physik (1900), p. 228._
+
+
+=265.= 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
+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.
+
+ --CHAPMAN, C. H.
+
+ _Bulletin American Mathematical Society,
+ Vol. 2 (First series), p. 61._
+
+
+=266.= 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 _relationship_ of the _ideas_ 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.--HILBERT, D.
+
+ _Mathematical Problems, Bulletin
+ American Mathematical Society, Vol. 8,
+ p. 478._
+
+
+=267.= The mathematics have always been the implacable enemies of
+scientific romances.--ARAGO.
+
+ _Oeuvres (1855), t. 3, p. 498._
+
+
+=268.= 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.--COURNOT, AUGUSTIN.
+
+ _Mathematical Theory of the Principles
+ of Wealth [Bacon, N. T.], (New York,
+ 1897), p. 3._
+
+
+=269.= 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 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.--SPOTTISWOODE, W.
+
+ _Quoted in Sonnenschein's Encyclopedia
+ of Education (London, 1906), p. 208._
+
+
+=270.= 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
+reason is that, partly through the exigencies of academic
+instruction, but mainly through the martinet traditions of
+antiquity and the influence of mediaeval 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.--MCCORMACK, T. J.
+
+ _Preface to De Morgan's Elementary
+ Illustrations of the Differential and
+ Integral Calculus (Chicago, 1899)._
+
+
+=271.= 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.--LOCKE, JOHN.
+
+ _Second Reply to the Bishop of
+ Worcester._
+
+
+=272.= 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 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.--KEYSER, C. J.
+
+ _The Humanization of the Teaching of
+ Mathematics; Science, New Series, Vol.
+ 35, pp. 645-646._
+
+
+=273.= 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
+computes pennies or guineas, to rhetoric whether it defends truth
+or error.--GOETHE.
+
+ _Wilhelm Meisters Wanderjahre, Zweites
+ Buch._
+
+
+=274.= 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?
+
+ --HILL, THOMAS.
+
+ _The Imagination in Mathematics; North
+ American Review, Vol. 85, p. 226._
+
+
+=275.= ... 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.--HILL, THOMAS.
+
+ _The Uses of Mathesis; Bibliotheca
+ Sacra, Vol. 32, p. 523._
+
+
+=276.= It is in the inner world of pure thought, where all
+_entia_ 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.
+
+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 _solar_
+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 throughout are hues and tints transcending _sense_,
+painted there by radiant pencils of _psychic_ 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 aesthetic
+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 _Anders-Streben_, of all other knowledge of
+every kind.--KEYSER, C. J.
+
+ _The Universe and Beyond; Hibbert
+ Journal, Vol. 3 (1904-1905), pp.
+ 313-314._
+
+
+
+
+ CHAPTER III
+
+ ESTIMATES OF MATHEMATICS
+
+
+=301.= 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.--SYLVESTER, J. J.
+
+ _A Plea for the Mathematician, Nature,
+ 1, p. 262; Collected Mathematical Papers
+ (Cambridge, 1908), 2, p. 659._
+
+
+=302.= 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 _certainty_ of its results: for that is what, beyond all
+mental treasures, the human intellect craves for. Let us only be
+sure of _something_! More light, more light! [Greek: En de phaei
+kai olesson] "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.--DODGSON, C. L.
+
+ _A New Theory of Parallels (London,
+ 1895), Introduction._
+
+
+=303.= 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 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.--MERZ, J. T.
+
+ _A History of European Thought in the
+ Nineteenth Century (Edinburgh and
+ London, 1904), Vol. 1, p. 314._
+
+
+=304.= Mathematics ... the ideal and norm of all careful
+thinking.--HALL, G. STANLEY.
+
+ _Educational Problems (New York, 1911),
+ p. 393._
+
+
+=305.= Mathematics is the only true metaphysics.
+
+ --THOMSON, W. (LORD KELVIN).
+
+ _Thompson, S. P.: Life of Lord Kelvin
+ (London, 1910), p. 10._
+
+
+=306.= He who knows not mathematics and the results of recent
+scientific investigation dies without knowing _truth_.
+
+ --SCHELLBACH, C. H.
+
+ _Quoted in Young's Teaching of
+ Mathematics (London, 1907), p. 44._
+
+
+=307.= The reasoning of mathematics is a type of perfect
+reasoning.--BARNETT, P. A.
+
+ _Common Sense in Education and Teaching
+ (New York, 1905), p. 222._
+
+
+=308.= 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.--REID, THOMAS.
+
+ _Essays on the Intellectual Powers of
+ Man, 4th. Ed., p. 461._
+
+
+=309.= 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.--BUTLER, NICHOLAS MURRAY.
+
+ _The Meaning of Education and other
+ Essays and Addresses (New York, 1905),
+ p. 45._
+
+
+=310.= 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.--BACON, ROGER.
+
+ _Opus Majus, Part 4, Distinctia Prima,
+ cap. 1._
+
+
+=311.= 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.
+
+ --GOETHE.
+
+ _Sprueche in Prosa, Natur, III, 868._
+
+
+=312.= Do not imagine that mathematics is hard and crabbed, and
+repulsive to common sense. It is merely the etherealization of
+common sense.--THOMSON, W. (LORD KELVIN).
+
+ _Thompson, S. P.: Life of Lord Kelvin
+ (London, 1910), p. 1139._
+
+
+=313.= The advancement and perfection of mathematics are
+intimately connected with the prosperity of the State.--NAPOLEON I.
+
+ _Correspondance de Napoleon, t. 24
+ (1868), p. 112._
+
+
+=314.= 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.--LALANDE.
+
+ _Quoted in Bruhns' Alexander von
+ Humboldt (1872), Bd. 1, p. 232._
+
+
+=315.= 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.
+
+ --STEWART, DUGALD.
+
+ _Philosophy of the Human Mind, Part 3,
+ chap. 1, sect. 3; Collected Works
+ [Hamilton] (Edinburgh, 1854), Vol. 4._
+
+
+=316.= 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, 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.
+
+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.--MACAULAY.
+
+ _Lord Bacon: Edinburgh Review, July,
+ 1837. Critical and Miscellaneous Essays
+ (New York, 1879), Vol. 1, p. 397._
+
+
+=317.= _Ath._ 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.
+
+_Cle._ And what necessities of knowledge are there, Stranger,
+which are divine and not human?
+
+_Ath._ 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.
+
+ --PLATO.
+
+ _Republic, Bk. 7. Jowett's Dialogues of
+ Plato (New York, 1897), Vol. 4, p. 334._
+
+
+=318.= 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 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.--ARISTOTLE.
+
+ _Metaphysics [MacMahon] Bk. 12, chap.
+ 3._
+
+
+=319.= 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.--BILLINGSLEY, H.
+
+ _The Elements of Geometrie of the most
+ ancient Philosopher Euclide of Megara
+ (London, 1570), Note to the Reader._
+
+
+=320.= 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.--BRAHMAGUPTA.
+
+ _Quoted in Cajori's History of
+ Mathematics (New York, 1897), p. 92._
+
+
+=321.= 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.
+
+ --BACON, LORD.
+
+ _De Augmentis, Bk. 3; Advancement of
+ Learning, Bk. 2._
+
+
+=322.= 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.
+
+ --FRANKLIN, BENJAMIN.
+
+ _On the Usefulness of Mathematics, Works
+ (Boston, 1840), Vol. 2, p. 28._
+
+
+=323.= 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.
+
+ --FRANKLIN, BENJAMIN.
+
+ _On the Usefulness of Mathematics, Works
+ (Boston, 1840), Vol. 2, p. 69._
+
+
+=324.= What science can there be more noble, more excellent, more
+useful for men, more admirably high and demonstrative, than this
+of the mathematics?--FRANKLIN, BENJAMIN.
+
+ _On the Usefulness of Mathematics, Works
+ (Boston, 1840), Vol. 2, p. 69._
+
+
+=325.= 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."--EVERETT, EDWARD.
+
+ _Orations and Speeches (Boston, 1870),
+ Vol. 3, p. 514._
+
+
+=326.= There is no science which teaches the harmonies of nature
+more clearly than mathematics, ....--CARUS, PAUL.
+
+ _Andrews: Magic Squares and Cubes
+ (Chicago, 1908), Introduction._
+
+
+=327.= For it being the nature of the mind of man (to the extreme
+prejudice of knowledge) to delight in the spacious 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.--BACON, LORD.
+
+ _De Augmentis, Bk. 3; Advancement of
+ Learning, Bk. 2._
+
+
+=328.= 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.--CROMWELL, OLIVER.
+
+ _Letters and Speeches of Oliver Cromwell
+ (New York, 1899), Vol. 1, p. 371._
+
+
+=329.= 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.
+
+ --NOVALIS.
+
+ _Schriften (Berlin, 1901), Bd. 2, p.
+ 223._
+
+
+=330.= 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 unshaken
+Foundation of Sciences, and the plentiful Fountain of Advantage
+to human Affairs.--BARROW, ISAAC.
+
+ _Oration before the University of
+ Cambridge on being elected Lucasian
+ Professor of Mathematics, Mathematical
+ Lectures (London, 1734), p. 28._
+
+
+=331.= 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 _fiat_ 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.--WEBSTER, DANIEL.
+
+ _Works (Boston, 1872), Vol. 1, p. 180._
+
+
+=332.= 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.
+
+ --HILL, THOMAS.
+
+ _Imagination in Mathematics; North
+ American Review, Vol. 85, p. 228._
+
+
+=333.= 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.--HILL, THOMAS.
+
+ _Imagination in Mathematics; North
+ American Review, Vol. 85, p. 228._
+
+
+=334.= 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau],
+ Introduction, chap. 2._
+
+
+
+
+ CHAPTER IV
+
+ THE VALUE OF MATHEMATICS
+
+
+=401.= 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.--DILLMANN, E.
+
+ _Die Mathematik die Fackeltraegerin einer
+ neuen Zeit (Stuttgart, 1889), p. 40._
+
+
+=402.= 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.--BARROW, ISAAC.
+
+ _Prefatory Oration: Mathematical
+ Lectures (London, 1734), p. 31._
+
+
+=403.= 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.--MYERS, GEORGE.
+
+ _Arithmetic in Public School Education
+ (Chicago, 1911), p. 8._
+
+
+=404.= 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.--YOUNG, J. W. A.
+
+ _The Teaching of Mathematics (London,
+ 1907), p. 14._
+
+
+=405.= 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.--TODHUNTER, ISAAC.
+
+ _The Conflict of Studies and other
+ Essays (London, 1873), p. 12._
+
+
+=406.= Mathematics renders its best service through the immediate
+furthering of rigorous thought and the spirit of invention.
+
+ --HERBART J. F.
+
+ _Mathematischer Lehrplan fuer
+ Realschulen: Werke [Kehrbach]
+ (Langensalza, 1890), Bd. 5, p. 170._
+
+
+=407.= 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.--TODHUNTER, ISAAC.
+
+ _Conflict of Studies and other Essays
+ (London, 1873), pp. 6, 7._
+
+
+=408.= It is better to teach the child arithmetic and Latin
+grammar than rhetoric and moral philosophy, because they require
+exactitude of performance it is made certain that the lesson is
+mastered, and that power of performance is worth more than
+knowledge.--EMERSON, R. W.
+
+ _Lecture on Education._
+
+
+=409.= 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
+_a peu pres._ 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.--MILL, J. S.
+
+ _An Examination of Sir William
+ Hamilton's Philosophy (London, 1878), p.
+ 611._
+
+
+=410.= 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.--BACON, LORD.
+
+ _De Augmentis, Bk. 3; Advancement of
+ Learning, Bk. 2._
+
+
+=411.= 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.--BACON, LORD.
+
+ _Essays: On Studies._
+
+
+=412.= 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.--BACON, LORD.
+
+ _De Augmentis, Bk. 6; Advancement of
+ Learning, Bk. 2._
+
+
+=413.= 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.--JOWETT, B.
+
+ _Dialogues of Plato (New York, 1897),
+ Vol. 2, p. 78._
+
+
+=414.= 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.--TODHUNTER, ISAAC.
+
+ _Conflict of Studies (London, 1873), p.
+ 13._
+
+
+=415.= 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.--TODHUNTER, ISAAC.
+
+ _Conflict of Studies (London, 1873), p.
+ 2._
+
+
+=416.= 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.--POE, E. A.
+
+ _The Murders in Rue Morgue._
+
+
+=417.= 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.--COLTON, C. C.
+
+ _Lacon (New York, 1866)._
+
+
+=418.= 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.
+
+ _Congressional Committee on Military
+ Affairs, 1834; U. S. Bureau of
+ Education, Bulletin 1912, No. 2, p. 10._
+
+
+=419.= 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 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.--REIDT, F.
+
+ _Anleitung zum mathematischen Unterricht
+ in hoeheren Schulen (Berlin, 1906), p.
+ 17._
+
+
+=420.= 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.
+
+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....
+
+One would almost wish that our institutions of the science and
+art of public speaking would put over their doors the motto that
+Plato had over the entrance to his school of philosophy: "Let no
+one who is unacquainted with geometry enter here."--WHITE, W. F.
+
+ _A Scrap-book of Elementary Mathematics
+ (Chicago, 1908), p. 210._
+
+
+=421.= 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.--YOUNG, J. W. A.
+
+ _The Teaching of Mathematics (New York,
+ 1907), p. 42._
+
+
+=422.= 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.--TODHUNTER, ISAAC.
+
+ _Private Study of Mathematics: Conflict
+ of Studies and other Essays (London,
+ 1873), p. 82._
+
+
+=423.= 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.--LOCKE, JOHN.
+
+ _Conduct of the Understanding, Sect. 6._
+
+
+=424.= 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
+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.
+
+ --LOCKE, JOHN.
+
+ _Conduct of the Understanding, Sect. 7._
+
+
+=425.= 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.--LOCKE, JOHN.
+
+ _The Conduct of the Understanding, Sect. 7._
+
+
+=426.= 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 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; ....--LOCKE, JOHN.
+
+ _The Conduct of the Understanding, Sect.
+ 7._
+
+
+=427.= As an exercise of the reasoning faculty, pure mathematics
+is an admirable exercise, because it consists of _reasoning_
+alone, and does not encumber the student with an exercise of
+_judgment_: 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.--WHATELY, R.
+
+ _Annotations to Bacon's Essays (Boston,
+ 1873), Essay 1, p. 493._
+
+
+=428.= 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.--BERKELY, GEORGE.
+
+ _The Analyst, 2; Works (London, 1898),
+ Vol. 3, p. 10._
+
+
+=429.= 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 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.
+
+ --FITCH, J. C.
+
+ _Lectures on Teaching (New York, 1906),
+ pp. 291-292._
+
+
+=430.= 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 _learn to reason_ 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:
+
+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.
+
+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.
+
+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.
+
+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 gives
+confidence, and is absolutely necessary, if, as was said before,
+reason is not to be the instructor, but the pupil.
+
+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.--DE MORGAN, AUGUSTUS.
+
+ _On the Study and Difficulties of
+ Mathematics (Chicago, 1898), chap. 1._
+
+
+=431.= The instruction of children should aim gradually to
+combine knowing and doing [Wissen und Koennen]. Among all sciences
+mathematics seems to be the only one of a kind to satisfy this
+aim most completely.--KANT, IMMANUEL.
+
+ _Werke [Rosenkranz und Schubert], Bd. 9
+ (Leipzig, 1838), p. 409._
+
+
+=432.= Every discipline must be honored for reason other than its
+utility, otherwise it yields no enthusiasm for industry.
+
+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.--HERBART, J. F.
+
+ _Mathematischer Lehrplan fuer
+ Realgymnasien, Werke [Kehrbach],
+ (Langensalza, 1890), Bd. 5, p. 167._
+
+
+=433.= 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 develops emotion,
+understanding, and sympathy, so mathematics develops observation,
+imagination, and reason.--CHANCELLOR, W. E.
+
+ _A Theory of Motives, Ideals and Values
+ in Education (Boston and New York,
+ 1907), p. 406._
+
+
+=434.= 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.--HARRIS, W. T.
+
+ _Psychologic Foundations of Education
+ (New York, 1898), p. 325._
+
+
+=435.= 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.--YOUNG, J. W. A.
+
+ _The Teaching of Mathematics (New York,
+ 1907), p. 16._
+
+
+=436.= 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.--BACON, LORD.
+
+ _De Augmentis, Bk. 2; Advancement of
+ Learning, Bk. 3._
+
+
+=437.= 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.
+
+ --BOYLE, ROBERT.
+
+ _The Usefulness of Mathematicks to
+ Natural Philosophy; Works (London,
+ 1772), Vol. 3, p. 426._
+
+
+=438.= 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.
+
+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.
+
+ --BERTHELOT, M. P. E. M.
+
+ _Science as an Instrument of Education;
+ Popular Science Monthly (1897), p. 253._
+
+
+=439.= 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.--PRINGSHEIM, ALFRED.
+
+ _Ueber Wert und angeblichen Unwert der
+ Mathematik; Jahresbericht der Deutschen
+ Mathematiker Vereinigung, Bd. 13, p.
+ 381._
+
+
+=440.= 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.--STEWART, DUGALD.
+
+ _Philosophy of the Human Mind, Part 2,
+ chap. 2, sect. 3._
+
+
+=441.= 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 aesthetic 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.--HELMHOLTZ, H.
+
+ _On the Relation of Natural Science to
+ Science in general; Popular Lectures on
+ Scientific Subjects; Atkinson (New York,
+ 1900), pp. 25-26._
+
+
+=442.= 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, 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.--BAIN, ALEXANDER.
+
+ _Education as a Science (New York,
+ 1898), pp. 151-152._
+
+
+=443.= 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.
+
+ --EVERETT, EDWARD.
+
+ _Orations and Speeches (Boston, 1870),
+ Vol. 2, p. 510._
+
+
+=444.= 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 [Greek: ageometretos],
+as wanting in one of the most essential qualifications for the
+successful cultivation of the higher branches of philosophy.
+
+ --MILL, J. S.
+
+ _System of Logic, Bk. 3, chap. 24, sect.
+ 9._
+
+
+=445.= 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 _alphabet_ 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.--WHEWELL, WILLIAM.
+
+ _The Philosophy of the Inductive
+ Sciences, Part 1, Bk. 2, chap. 4, sect.
+ 8 (London, 1858)._
+
+
+=446.= 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.--DUTTON, S. T.
+
+ _Social Phases of Education in the
+ School and the Home (London, 1900), p.
+ 30._
+
+
+=447.= 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.--BURKE, EDMUND.
+
+ _On the Sublime and Beautiful, Part 3,
+ sect. 2._
+
+
+=448.= Out of the interaction of form and content in mathematics
+grows an acquaintance with methods which enable the 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.
+
+ --PRINGSHEIM, ALFRED.
+
+ _Ueber Wert und angeblichen Unwert der
+ Mathematik; Jahresbericht der Deutschen
+ Mathematiker Vereinigung (1904), p.
+ 374._
+
+
+=449.= 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
+"_abeunt studia in mores_."--SYLVESTER, J. J.
+
+ _A probationary Lecture on Geometry;
+ Collected Mathematical Papers
+ (Cambridge, 1908), Vol. 2, p. 9._
+
+
+=450.= Mathematical knowledge adds vigour to the mind, frees it
+from prejudice, credulity, and superstition.--ARBUTHNOT, JOHN.
+
+ _Usefulness of Mathematical Learning._
+
+
+=451.= 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, _potentia et actu_, has become clear to him;
+henceforth the philosopher can reveal him nothing new, as a
+geometrician he has discovered the basis of all thought.--GOETHE.
+
+ _Sprueche in Prosa, Ethisches, VI, 455._
+
+
+=452.= 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.--STEWART, DUGALD.
+
+ _Philosophy of the Human Mind, Part 3,
+ chap. 1, sect. 3._
+
+
+=453.= 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.--BAIN, ALEXANDER.
+
+ _Education as a Science (New York,
+ 1898), p. 153._
+
+
+=454.= 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
+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.
+
+ --MYERS, GEORGE.
+
+ _Arithmetic in Public Education
+ (Chicago), p. 13._
+
+
+=455.= A _general course_ 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.--ECHOLS, C. P.
+
+ _Mathematics at West Point and
+ Annapolis; U. S. Bureau of Education,
+ Bulletin 1912, No. 2, p. 11._
+
+
+=456.= 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.
+
+ --REIDT, F.
+
+ _Anleitung zum mathematischen Unterricht
+ in den hoeheren Schulen (Berlin, 1906),
+ p. 28._
+
+
+=457.= I do not maintain that the _chief value_ 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 _that is right_ as an answer to the
+questions with which one is _able_ 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 these ways and to arrive at the establishment of the
+true as opposed to the untrue, this relentless hewing _to_ the
+line and stopping _at_ 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.--MYERS, GEORGE.
+
+ _Arithmetic in Public Education
+ (Chicago), p. 18._
+
+
+=458.= In destroying the predisposition to anger, science of all
+kind is useful; but the mathematics possess this property in the
+most eminent degree.--DR. RUSH.
+
+ _Quoted in Day's Collacon (London, no
+ date)._
+
+
+=459.= 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.
+
+ --ARBUTHNOT, JOHN.
+
+ _Usefulness of Mathematical Learning._
+
+
+=460.= There is no prophet which preaches the superpersonal God
+more plainly than mathematics.--CARUS, PAUL.
+
+ _Reflections on Magic Squares; Monist
+ (1906), p. 147._
+
+
+=461.= 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.
+
+ --VOLTAIRE.
+
+ _Oeuvres Completes (Paris, 1880), t. 35,
+ p. 219._
+
+
+
+
+ CHAPTER V
+
+ THE TEACHING OF MATHEMATICS
+
+
+=501.= 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.
+
+ _International Commission on the
+ Teaching of Mathematics, American
+ Report; U. S. Bureau of Education,
+ Bulletin 1912, No. 4, p. 7._
+
+
+=502.= 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.
+
+ _International Commission on the
+ Teaching of Mathematics, American
+ Report; U. S. Bureau of Education,
+ Bulletin 1912, No. 4, p. 7._
+
+
+=503.= 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.
+
+ _Resolution adopted by the German
+ Association for the Advancement of
+ Scientific and Mathematical Instruction;
+ Jahresbericht der Deutschen Mathematiker
+ Vereinigung (1896), p. 41._
+
+
+=504.= 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
+the latter shows its traces in the comprehension of the development
+of civilization and the ability to participate in the further tasks
+of civilization.
+
+ _Unterrichtsblaetter fuer Mathematik und
+ Naturwissenschaft (1904), p. 128._
+
+
+=505.= 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.--MAGNUS, PHILIP.
+
+ _Perry's Teaching of Mathematics
+ (London, 1902), p. 84._
+
+
+=506.= The participation in the _general development of the
+mental powers_ without special reference to his future vocation
+must be recognized as the essential aim of mathematical
+instruction.--REIDT, F.
+
+ _Anleitung zum Mathematischen Unterricht
+ an hoeheren Schulen (Berlin, 1906), p.
+ 12._
+
+
+=507.= 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.--GOETTING, E.
+
+ _Ueber das Lehrziel im mathematischen
+ Unterricht der hoeheren Realanstalten;
+ Jahresbericht der Deutschen Mathematiker
+ Vereinigung, Bd. 2, p. 192._
+
+
+=508.= Mathematics will not be properly esteemed in wider circles
+until more than the _a b c_ 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 _formal_ training
+of the mind. The aim of mathematics is its _content_, 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.--HANKEL, H.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1884),
+ p. 6._
+
+
+=509.= 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.--HERBART, J. F.
+
+ _Umriss paedagogischer Vorlesungen; Werke
+ [Kehrbach] (Langensalza, 1902), Bd. 10,
+ p. 101._
+
+
+=510.= 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
+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.--PERRY, JOHN.
+
+ _The Teaching of Mathematics (London,
+ 1902), p. 14._
+
+
+=511.= 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.--PERRY, JOHN.
+
+ _The Teaching of Mathematics (London,
+ 1904), p. 12._
+
+
+=512.= 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 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.
+
+ --HUDSON, W. H. H.
+
+ _Perry's Teaching of Mathematics
+ (London, 1904), p. 33._
+
+
+=513.= "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."--LANGLEY, E. M.
+
+ _Perry's Teaching of Mathematics
+ (London, 1904), p. 43._
+
+
+=514.= 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.--SPEER, W. W.
+
+ _Primary Arithmetic (Boston, 1897), pp.
+ 26-27._
+
+
+=515.= 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.
+
+ --KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 29._
+
+
+=516.= 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.--HERBART, J. F.
+
+ _Letters and Lectures on Education
+ [Felkin] (London, 1908), p. 117._
+
+
+=517.= First, as concerns the _success_ 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.
+
+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 _correct thinking based on true
+premises secures mastery over the outer world_. To accomplish
+this the outer world must receive its share of attention from the
+very beginning.
+
+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.--KLEIN, FELIX.
+
+ _Ueber den Mathematischen Unterricht an
+ den hoeheren Schulen; Jahresbericht der
+ Deutschen Mathematiker Vereinigung, Bd.
+ 11, p. 131._
+
+
+=518.= 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.
+
+ --HOLZMUeLLER, GUSTAV.
+
+ _Methodisches Lehrbuch der
+ Elementar-Mathematik (Leipzig, 1904),
+ Teil 1, Vorwort, pp. 4-5._
+
+
+=519.= 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.--PERRY, JOHN.
+
+ _The Teaching of Mathematics (London,
+ 1902), pp. 19-20._
+
+
+=520.= Poor teaching leads to the inevitable idea that the
+subject [mathematics] is only adapted to peculiar minds, when it
+is 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.--SAFFORD, T. H.
+
+ _Mathematical Teaching (Boston, 1907),
+ p. 19._
+
+
+=521.= 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.--DE MORGAN, A.
+
+ _Study and Difficulties of Mathematics
+ (Chicago, 1902), Preface._
+
+
+=522.= 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.--BAIN, ALEXANDER.
+
+ _Education as a Science (New York,
+ 1898), p. 298._
+
+
+=523.= 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 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.
+
+ --TODHUNTER, ISAAC.
+
+ _Conflict of Studies (London, 1873), pp.
+ 10-11._
+
+
+=524.= 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.
+
+ _Report of the Committee of Ten on
+ Secondary School Subjects, (New York,
+ 1894), p. 109._
+
+
+=525.= 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.--DE MORGAN, A.
+
+ _Trigonometry and Double Algebra
+ (London, 1849), p. 92._
+
+
+=526.= 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 _Arithmetically_
+more simple which is determined by the more simple Equation, but
+that is _Geometrically_ more simple which is determined by the
+more simple drawing of Lines; and in Geometry, that ought to be
+reckoned best which is geometrically most simple.--NEWTON.
+
+ _On the Linear Construction of
+ Equations; Universal Arithmetic (London,
+ 1769), Vol. 2, p. 470._
+
+
+=527.= As long as algebra and geometry proceeded along separate
+paths, their advance was slow and their applications limited.
+
+But when these sciences joined company, they drew from each other
+fresh vitality and thenceforward marched on at a rapid pace
+toward perfection.--LAGRANGE.
+
+ _Lecons Elementaires sur les
+ Mathematiques, Lecon cinquieme.
+ [McCormack]._
+
+
+=528.= 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.
+
+ --WALKER, F. A.
+
+ _Discussions in Education (New York,
+ 1899), pp. 253-254._
+
+
+=529.= 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
+to him physics and mechanics as sciences it is useless to present
+to him any physical or mechanical principles.
+
+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,
+_the withered gourd_, fruitless endeavor, wasted strength.
+
+ --WALKER, F. A.
+
+ _Industrial Education; Discussions in
+ Education (New York, 1899), p. 132._
+
+
+=530.= 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.--NEWTON, ISAAC.
+
+ _Letter to Collins, 1670; Macclesfield,
+ Correspondence of Scientific Men
+ (Oxford, 1841), Vol. 2, p. 307._
+
+
+=531.= 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.
+
+ --CHRYSTAL, GEORGE.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, 1885; Nature, Vol. 32, pp.
+ 447-448._
+
+
+=532.= 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.
+
+ --SMITH, D. E.
+
+ _The Teaching of Elementary Mathematics
+ (New York, 1902), p. 219._
+
+
+=533.= 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.--CHRYSTAL, GEORGE.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, 1885; Nature, Vol. 32, p. 448._
+
+
+=534.= 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.--COLERIDGE, S. T.
+
+ _Lectures on Shakespere (Bohn Library),
+ p. 52._
+
+
+=535.= 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.
+
+ --PRINGSHEIM, ALFRED.
+
+ _Jahresbericht der Deutschen
+ Mathematiker Vereinigung (1898), p.
+ 143._
+
+
+=536.= 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.--D'ALEMBERT.
+
+ _Quoted by De Morgan: Trigonometry and
+ Double Algebra (London, 1849), Title
+ page._
+
+
+=537.= 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.--HILBERT, D.
+
+ _Mathematical Problems; Bulletin
+ American Mathematical Society, Vol. 8,
+ p. 441._
+
+
+=538.= 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.--GRASSMANN, HERMANN.
+
+ _Stuecke aus dem Lehrbuche der
+ Arithmetik; Werke, Bd. 2 (Leipsig,
+ 1904), p. 296._
+
+
+=539.= 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 _effective teaching_
+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
+_humour_ (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.
+
+ --MINCHIN, G. M.
+
+ _Perry's Teaching of Mathematics
+ (London, 1902), pp. 59-61._
+
+
+=540.= Remember this, the rule for giving an extempore lecture
+is--let the 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.
+
+ --DE MORGAN, A.
+
+ _Letter to Hamilton; Graves: Life of W.
+ R. Hamilton (New York, 1882-1889), Vol.
+ 3, p. 487._
+
+
+
+
+ CHAPTER VI
+
+ STUDY AND RESEARCH IN MATHEMATICS
+
+
+=601.= 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.--DE MORGAN, A.
+
+ _Study and Difficulties of Mathematics
+ (Chicago, 1902), chap. 12._
+
+
+=602.= 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.--DE MORGAN, A.
+
+ _Study and Difficulties of Mathematics
+ (Chicago, 1902), chap. 12._
+
+
+=603.= 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.--DE MORGAN, A.
+
+ _Study and Difficulties of Mathematics
+ (Chicago, 1902), chap. 3._
+
+
+=604.= 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.--LOCKE, JOHN.
+
+ _Conduct of the Understanding, sect.
+ 24._
+
+
+=605.= 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.--TODHUNTER, ISAAC.
+
+ _Private Study of Mathematics; Conflict
+ of Studies and other Essays (London,
+ 1873), p. 67._
+
+
+=606.= 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 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.--TODHUNTER, ISAAC.
+
+ _Private Study of Mathematics; Conflict
+ of Studies and other Essays (London,
+ 1873), p. 68._
+
+
+=607.= 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.
+
+ --CHRYSTAL, GEORGE.
+
+ _Algebra, Part 2 (Edinburgh, 1889),
+ Preface, p. 8._
+
+
+=608.= 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.--TODHUNTER, ISAAC.
+
+ _Private Study of Mathematics; Conflict
+ of Studies and other Essays (London,
+ 1873), p. 74._
+
+
+=609.= 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.--CREMONA, L.
+
+ _Projective Geometry [Leudesdorf]
+ (Oxford, 1885), Preface._
+
+
+=610.= 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.--VIOLA, J.
+
+ _Mathematische Sophismen (Wien, 1864),
+ Vorwort._
+
+
+=611.= 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.
+
+ --WHITWORTH, W. A.
+
+ _Modern Analytic Geometry (Cambridge,
+ 1866), p. 154._
+
+
+=612.= 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.--- FOURIER, J.
+
+ _Theorie Analytique de la Chaleur,
+ Discours Preliminaire._
+
+
+=613.= 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.
+
+ --DOLBEAR, A. E.
+
+ _Matter, Ether, Motion (Boston, 1894),
+ p. 89._
+
+
+=614.= 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.
+
+ --BELTRAMI.
+
+ _Giornale di matematiche, Vol. 11, p.
+ 153. [Young, J. W.]_
+
+
+=615.= 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.--CAJORI, F.
+
+ _History of Mathematics (New York,
+ 1897), pp. 1-2._
+
+
+=616.= 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.--CAJORI, F.
+
+ _History of Mathematics (New York,
+ 1897), p. 4._
+
+
+=617.= 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.--TODHUNTER, ISAAC.
+
+ _Private Study of Mathematics; Conflict
+ of Studies and other Essays (London,
+ 1873), p. 73._
+
+
+=618.= 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.--BURKHARDT, H.
+
+ _Mathematisches und wissenschaftliches
+ Denken; Jahresbericht der Deutschen
+ Mathematiker Vereinigung, Bd. 11, p.
+ 55._
+
+
+=619.= 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.--SEGRE, CORRADI.
+
+ _Some Recent Tendencies in Geometric
+ Investigation; Rivista di Matematica
+ (1891), p. 63. Bulletin American
+ Mathematical Society, 1904, p. 465.
+ [Young, J. W.]._
+
+
+=620.= 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.--SYLVESTER, J. J.
+
+ _Notes to the Exeter Association
+ Address; Collected Mathematical Papers
+ (Cambridge, 1908), Vol. 2, p. 715._
+
+
+=621.= 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.
+
+ --HANKEL, HERMANN.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1884),
+ p. 15._
+
+
+=622.= 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 _esthetic
+tact_ and which is the only thing in our science which cannot be
+taught or acquired, and is yet the indispensable endowment of
+every mathematician.--HANKEL, HERMANN.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1884),
+ p. 21._
+
+
+=623.= 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.--GLAISHER, J. W. L.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A, (1890); Nature, Vol.
+ 42, p. 467._
+
+
+=624.= 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.
+
+ --HILBERT, D.
+
+ _Mathematical Problems; Bulletin
+ American Mathematical Society, Vol. 8,
+ p. 438._
+
+
+=625.= In mathematics as in other fields, to find one self lost
+in wonder at some manifestation is frequently the half of a new
+discovery.--DIRICHLET, P. G. L.
+
+ _Werke, Bd. 2 (Berlin, 1897), p. 233._
+
+
+=626.= 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.
+
+ --MACH, ERNST.
+
+ _Popular Scientific Lectures (Chicago,
+ 1910), p. 196._
+
+
+=627.= 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 _ignorabimus_.--HILBERT, D.
+
+ _Mathematical Problems; Bulletin
+ American Mathematical Society, Vol. 8,
+ pp. 444-445._
+
+
+=628.= He who seeks for methods without having a definite problem
+in mind seeks for the most part in vain.--HILBERT, D.
+
+ _Mathematical Problems; Bulletin
+ American Mathematical Society, Vol. 8,
+ p. 444._
+
+
+=629.= A mathematical problem should be difficult in order to
+entice us, yet not completely inaccessible, lest it mock at our
+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.--HILBERT, D.
+
+ _Mathematical Problems; Bulletin
+ American Mathematical Society, Vol. 8,
+ p. 438._
+
+
+=630.= The great mathematicians have acted on the principle
+"_Divinez avant de demontrer_," and it is certainly true that
+almost all important discoveries are made in this fashion.
+
+ --KASNER, EDWARD.
+
+ _The Present Problems in Geometry;
+ Bulletin American Mathematical Society,
+ Vol. 11, p. 285._
+
+
+=631.= "Divide _et impera_" is as true in algebra as in
+statecraft; but no less true and even more fertile is the maxim
+"auge _et impera_." The more to do or to prove, the easier the
+doing or the proof.--SYLVESTER, J. J.
+
+ _Proof of the Fundamental Theorem of
+ Invariants; Philosophic Magazine (1878),
+ p. 186; Collected Mathematical Papers,
+ Vol. 3, p. 126._
+
+
+=632.= 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.--LAMPE, E.
+
+ _Die reine Mathematik in den Jahren
+ 1884-1899, p. 10._
+
+
+=633.= 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 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.--HILBERT, D.
+
+ _Mathematical Problems; Bulletin
+ American Mathematical Society, Vol. 8,
+ p. 479._
+
+
+=634.= 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.--GLAISHER, J. W. L.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A., (1890), Nature,
+ Vol. 42, p. 466._
+
+
+=635.= 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.--GLAISHER, J. W. L.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A (1890); Nature, Vol.
+ 42, p. 466._
+
+
+=636.= 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.--GORDAN, PAUL.
+
+ _Formensystem binaerer Formen (Leipzig,
+ 1875), p. 2._
+
+
+=637.= 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.--SMITH, H. J. S.
+
+ _Nature, Vol. 8 (1873), p. 452._
+
+
+=638.= 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.--HILBERT, D.
+
+ _Die Theorie der algebraischen
+ Zahlkorper, Vorwort; Jahresbericht der
+ Deutschen Mathematiker Vereinigung, Bd.
+ 4._
+
+
+=639.= 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.
+
+ --WHITEHEAD, A. N.
+
+ _Introduction to Mathematics (New York,
+ 1911), p. 157._
+
+
+=640.= 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 _ensemble_ 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, 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.--POINCARE, H.
+
+ _The Future of Mathematics; Monist, Vol.
+ 20, p. 80. [Halsted]._
+
+
+=641.= 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.
+
+ --SEGRE, CORRADI.
+
+ _Some Recent Tendencies in Geometric
+ Investigations. Rivista di Matematica,
+ Vol. 1, p. 44. Bulletin American
+ Mathematical Society, 1904, p. 444.
+ [Young, J. W.]._
+
+
+=642.= 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
+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.
+
+ --SEGRE, CORRADI.
+
+ _On some Recent Tendencies in Geometric
+ Investigations; Rivista di Matematica,
+ 1891, p. 43. Bulletin American
+ Mathematical Society, 1904, p. 443
+ [Young, J. W.]._
+
+
+=643.= 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.--SEGRE, CORRADI.
+
+ _On some recent Tendencies in
+ Geometrical Investigations; Rivista di
+ Matematica, 1891, p. 43. Bulletin
+ American Mathematical Society, 1904, p.
+ 443 [Young, J. W.]._
+
+
+=644.= 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
+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.--CLEBSCH, A.
+
+ _Zum Gedaechtniss an Julius Pluecker;
+ Goettinger Abhandlungen, 16, 1871,
+ Mathematische Classe, p. 6._
+
+
+=645.= During a conversation with the writer in the last weeks of
+his life, _Sylvester_ 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.
+
+ --MACMAHON, P. A.
+
+ _Proceedings London Royal Society, Vol.
+ 63, p. 17._
+
+
+=646.= 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.
+
+Other principles, similar in their nature, are added, and the
+original principle itself receives such modifications and
+extensions 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.--CRAIG, THOMAS.
+
+ _A Treatise on Projection, Preface. U.
+ S. Coast and Geodetic Survey, Treasury
+ Department Document, No. 61._
+
+
+=647.= The aim of research is the discovery of the equations
+which subsist between the elements of phenomena.--MACH, ERNST.
+
+ _Popular Scientific Lectures (Chicago,
+ 1910), p. 205._
+
+
+=648.= 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.
+
+ --SYLVESTER, J. J.
+
+ _Philosophic Magazine (1863), p. 460._
+
+
+=649.= 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
+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. "_Pauca sed matura_" was
+Gauss's motto.--GLAISHER, J. W. L.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A, (1890); Nature, Vol.
+ 42, p. 467._
+
+
+=650.= It is the man not the method that solves the problem.
+
+ --MASCHKE, H.
+
+ _Present Problems of Algebra and
+ Analysis; Congress of Arts and Sciences
+ (New York and Boston, 1905), Vol. 1, p.
+ 530._
+
+
+=651.= 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.--KLEIN, FELIX.
+
+ _Riemannsche Flaechen (Leipzig, 1906),
+ Bd. 1, p. 234._
+
+
+=652.= 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.--CLIFFORD, W. K.
+
+ _Conditions of Mental Development;
+ Lectures and Essays (London, 1901), Vol.
+ 1, p. 115._
+
+
+=653.= 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.--VOLTAIRE.
+
+ _A Philosophical Dictionary, Article
+ "Geometry"; (Boston, 1881), Vol. 1, p.
+ 374._
+
+
+=654.= 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.
+
+ --FORSYTH, A. R.
+
+ _Perry's Teaching of Mathematics
+ (London, 1902), p. 35._
+
+
+=655.= It is no paradox to say that in our most theoretical moods
+we may be nearest to our most practical applications.
+
+ --WHITEHEAD, A. N.
+
+ _Introduction to Mathematics (New York),
+ p. 100._
+
+
+=656.= 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.--EVERETT, EDWARD.
+
+ _Orations and Speeches, Vol. 3 (Boston,
+ 1870), p. 513._
+
+
+=657.= 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
+analysis.--EULER.
+
+ _Novi Comm. Petr., Vol. 4, Preface._
+
+
+=658.= 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.
+
+ --SYLVESTER, J. J.
+
+ _A Probationary Lecture on Geometry;
+ Collected Mathematical Papers, Vol. 2
+ (Cambridge, 1908), p. 7._
+
+
+=659.= 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 necessary
+key with which to attain the knowledge of the most important laws
+of nature.--WHITEHEAD, A. N.
+
+ _Introduction to Mathematics (New York,
+ York, 1911), pp. 136-137._
+
+
+=660.= 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.--WHEWELL, W.
+
+ _History of Scientific Ideas, Bk. 2,
+ chap. 14, sect. 3._
+
+
+=661.= The greatest mathematicians, as Archimedes, Newton, and
+Gauss, always united theory and applications in equal measure.
+
+ --KLEIN, FELIX.
+
+ _Elementarmathematik vom hoeheren
+ Standpunkte aus (Leipzig, 1909), Bd. 2,
+ p. 392._
+
+
+=662.= 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.--WHEWELL, W.
+
+ _History of Scientific Ideas, Bk. 2,
+ chap. 14, sect. 8._
+
+
+=663.= 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.--HOBSON, E. W.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A, (1910); Nature, Vol.
+ 84, p. 286._
+
+
+=664.= 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 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
+Poincare 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.--FORSYTH, A. R.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A, (1897); Nature, Vol.
+ 56, p. 377._
+
+
+
+
+ CHAPTER VII
+
+ MODERN MATHEMATICS
+
+
+=701.= Surely this is the golden age of mathematics.
+
+ --PIERPONT, JAMES.
+
+ _History of Mathematics in the
+ Nineteenth Century; Congress of Arts and
+ Sciences (Boston and New York, 1905),
+ Vol. 1, p. 493._
+
+
+=702.= 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 _Jahrbuch ueber die Fortschritte der
+Mathematik_, 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
+_Encyclopaedie der Mathematischen Wissenschaften_ 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.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 8._
+
+
+=703.= I have said that mathematics is the oldest of the sciences; a
+glance at its more recent history will show that it has the 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.--HOBSON, E. W.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A, (1910); Nature, Vol.
+ 84, p. 285._
+
+
+=704.= 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.--FORSYTH, A. R.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A, (1897); Nature, Vol.
+ 56, p. 378._
+
+
+=705.= 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.
+
+ --RUSSELL, BERTRAND.
+
+ _International Monthly, Vol. 4 (1901),
+ p. 83._
+
+
+=706.= One of the chiefest triumphs of modern mathematics
+consists in having discovered what mathematics really is.
+
+ --RUSSELL, BERTRAND.
+
+ _International Monthly, Vol. 4 (1901),
+ p. 84._
+
+
+=707.= 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.--BUTLER, N. M.
+
+ _The Meaning of Education and other
+ Essays and Addresses (New York, 1905),
+ p. 44._
+
+
+=708.= 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.--PIERPONT, J.
+
+ _The History of Mathematics in the
+ Nineteenth Century; Congress of Arts and
+ Sciences (Boston and New York, 1905),
+ Vol. 1, p. 474._
+
+
+=709.= The Modern Theory of Functions--that stateliest of all the
+pure creations of the human intellect.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 16._
+
+
+=710.= 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.
+
+ --KASNER, EDWARD.
+
+ _The Present Problems in Geometry;
+ Bulletin American Mathematical Society,
+ 1905, p. 285._
+
+
+=711.= 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 fundamental difference between the
+two methods. Euclid avoids it, in modern mathematics it is
+systematically introduced, for only thus is generality obtained.
+
+ --CAYLEY, ARTHUR.
+
+ _Encyclopedia Britannica (9th edition),
+ Article "Geometry."_
+
+
+=712.= 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 _De sectione determinata_ of Appolonius
+constitute only one problem which is resolved by a single
+equation.--CHASLES, M.
+
+ _Histoire de la Geometrie, chap. 1,
+ sect. 35._
+
+
+=713.= 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.--WHITEHEAD, A. N.
+
+ _Introduction to Mathematics (New York,
+ 1911), p. 119._
+
+
+=714.= 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; 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 _De sectione
+rationis_, 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.
+
+ --REYE, THEODORE.
+
+ _Die synthetische Geometrie im Altertum
+ und in der Neuzeit; Jahresbericht der
+ Deutschen Mathematiker Vereinigung, Bd.
+ 2, pp. 346-347._
+
+
+=715.= 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.
+
+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
+student by a point moving in accordance to this law, is the
+parabola.
+
+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.
+
+ --DILLMANN, E.
+
+ _Die Mathematik die Fackeltraegerin einer
+ neuen Zeit (Stuttgart, 1889), p. 37._
+
+
+=716.= The Excellence of _Modern Geometry_ 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 _Ancients_, 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.--HALLEY, E.
+
+ _An Instance of the Excellence of Modern
+ Algebra, etc.; Philosophical
+ Transactions, 1694, p. 960._
+
+
+=717.= One of the most conspicuous and distinctive features of
+mathematical thought in the nineteenth century is its critical
+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.--PIERPONT, J.
+
+ _History of Mathematics in the
+ Nineteenth Century; Congress of Arts and
+ Sciences (Boston and New York, 1905),
+ Vol. 1, p. 482._
+
+
+=718.= 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.
+
+ --HANKEL, HERMANN.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1884),
+ p. 9._
+
+
+=719.= 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 when men began to
+extend geometry into three dimensions.--BOOLE M. E.
+
+ _Logic of Arithmetic (Oxford, 1903),
+ Preface, pp. 18-19._
+
+
+=720.= 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 _variability_. 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.--HANKEL, HERMANN.
+
+ _Untersuchungen ueber die unendlich oft
+ oszillierenden und unstetigen
+ Functionen; Ostwald's Klassiker der
+ exacten Wissenschaften, No. 153, pp.
+ 44-45._
+
+
+=721.= 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.
+
+ --PIERPONT, J.
+
+ _History of Mathematics in the
+ Nineteenth Century; Congress of Arts and
+ Sciences (Boston and New York, 1905),
+ Vol. 1, p. 474._
+
+
+=722.= 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.--CAYLEY, ARTHUR.
+
+ _Presidential Address; Collected Works,
+ Vol. 11, p. 434._
+
+
+=723.= The solution of the difficulties which formerly surrounded
+the mathematical infinite is probably the greatest achievement of
+which our age has to boast.--RUSSELL, BERTRAND.
+
+ _The Study of Mathematics; Philosophical
+ Essays (London, 1910), p. 77._
+
+
+=724.= 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. "_Omne exit in infinitum_" is their favorite
+motto and accepted axiom.--SYLVESTER, J. J.
+
+ _A Plea for the Mathematician; Nature,
+ Vol. 1, p. 261._
+
+
+=725.= 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.--MERZ, J. T.
+
+ _History of European Thought in the
+ Nineteenth Century (Edinburgh and
+ London, 1903), p. 736._
+
+
+=726.= 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.--CLIFFORD, W. K.
+
+ _Philosophy of the Pure Sciences;
+ Lectures and Essays (London, 1901), Vol.
+ 1, p. 402._
+
+
+=727.= 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.--LIE, SOPHUS.
+
+ _Leipziger Berichte, No. 47 (1895), p.
+ 261._
+
+
+=728.= 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.
+
+ --HANKEL, HERMANN.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1884),
+ pp. 14-15._
+
+
+=729.= 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.--YOUNG, J. W.
+
+ _Fundamental Concepts of Algebra and
+ Geometry (New York, 1911), p. 225._
+
+
+=730.= Everybody praises the incomparable power of the
+mathematical method, but so is everybody aware of its
+incomparable unpopularity.--ROSANES, J.
+
+ _Jahresbericht der Deutschen
+ Mathematiker Vereinigung, Bd. 13, p.
+ 17._
+
+
+=731.= 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.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and
+ Arts (New York, 1908), p. 8._
+
+
+=732.= 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.
+
+ --PRINGSHEIM, ALFRED.
+
+ _Ueber Wert und angeblichen Unwert der
+ Mathematik; Jahresbericht der Deutschen
+ Mathematiker Vereinigung, (1904) p.
+ 357._
+
+
+=733.= [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.--LEFEVRE, ARTHUR.
+
+ _Number and its Algebra (Boston, 1903),
+ sect. 223._
+
+
+=734.= 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.
+
+ --SCHUBERT, H.
+
+ _Mathematical Essays and Recreations
+ (Chicago, 1898), p. 32._
+
+
+=735.= 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.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 6._
+
+
+=736.= 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.--FORSYTH, A. R.
+
+ _Report of the 67th meeting of the
+ British Association for the Advancement
+ of Science._
+
+
+=737.= 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.--WHITEHEAD, A. N.
+
+ _An Introduction to Mathematics (New
+ York, 1911), p. 252._
+
+
+=738.= 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.
+
+ _Provisional Report of the American
+ Subcommittee of the International
+ Commission on the Teaching of
+ Mathematics; Bulletin American Society
+ (1910), p. 97._
+
+
+=739.= Angling may be said to be so like the mathematics, that it
+can never be fully learnt.--WALTON, ISAAC.
+
+ _The Complete Angler, Preface._
+
+
+=740.= The flights of the imagination which occur to the pure
+mathematician are in general so much better described in his
+formulae 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, ....
+
+ --TAIT, P. G.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1871); Nature Vol. 4, p. 271._
+
+
+
+
+ CHAPTER VIII
+
+ THE MATHEMATICIAN
+
+
+=801.= The real mathematician is an enthusiast _per se_. Without
+enthusiasm no mathematics.--NOVALIS.
+
+ _Schriften (Berlin, 1901), Zweiter Teil,
+ p. 223._
+
+
+=802.= It is true that a mathematician, who is not somewhat of a
+poet, will never be a perfect mathematician.--WEIERSTRASS.
+
+ _Quoted by Mittag-Leffler; Compte rendu
+ du deuxieme congres international des
+ mathematiciens (Paris, 1902), p. 149._
+
+
+=803.= 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 _Lagrange_.--GOETHE.
+
+ _Wilhelm Meister's Wanderjahre, Zweites
+ Buch; Sprueche in Prosa; Natur, VI, 950._
+
+
+=804.= A thorough advocate in a just cause, a penetrating
+mathematician facing the starry heavens, both alike bear the
+semblance of divinity.--GOETHE.
+
+ _Wilhelm Meister's Wanderjahre, Zweites
+ Buch; Sprueche in Prosa; Natur, VI, 947._
+
+
+=805.= Mathematicians practice absolute freedom.--ADAMS, HENRY.
+
+ _A Letter to American Teachers of
+ History (Washington, 1910), p. 169._
+
+
+=806.= The mathematical method is the essence of mathematics. He
+who fully comprehends the method is a mathematician.--NOVALIS.
+
+ _Schriften (Berlin, 1901), Zweiter Teil,
+ p. 190._
+
+
+=807.= He who is unfamiliar with mathematics [literally, he who
+is a layman in mathematics] remains more or less a stranger to
+our time.--DILLMANN, E.
+
+ _Die Mathematik die Fackeltraegerin einer
+ neuen Zeit (Stuttgart, 1889), p. 39._
+
+
+=808.= 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.]--GILMAN, D. C.
+
+ _Report of the President of Johns
+ Hopkins University (1888), p. 29._
+
+
+=809.= 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.--MOEBIUS, P. J.
+
+ _Ueber die Anlage zur Mathematik
+ (Leipzig, 1900), p. 4._
+
+
+=810.= 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.--NOVALIS.
+
+ _Schriften, Zweiter Teil (Berlin, 1901),
+ p. 223._
+
+
+=811.= 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.--MILL, J. S.
+
+ _An Examination of Sir William
+ Hamilton's Philosophy (London, 1878), p.
+ 611._
+
+
+=812.= It is easier to square the circle than to get round a
+mathematician.--DE MORGAN, A.
+
+ _Budget of Paradoxes (London, 1872), p.
+ 90._
+
+
+=813.= Mathematicians are like Frenchmen: whatever you say to
+them they translate into their own language and forthwith it is
+something entirely different.--GOETHE.
+
+ _Maximen und Reflexionen, Sechste
+ Abtheilung._
+
+
+=814.= 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.
+
+ --SWIFT, JONATHAN.
+
+ _Gulliver's Travels, Part 3, chap. 2._
+
+
+=815.= 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.
+
+ --KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 22._
+
+
+=816.= 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 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.--STEWART, DUGALD.
+
+ _Elements of the Philosophy of the Human
+ Mind, Part 3, chap. 1, sect. 3._
+
+
+=817.= 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.--DESCARTES.
+
+ _Discourse upon Method, Part 2;
+ Philosophy of Descartes [Torrey] (New
+ York, 1892), p. 48._
+
+
+=818.= 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.--KEYSER, C. J.
+
+ _Lectures on Philosophy, Science and Art
+ (New York, 1908), p. 31._
+
+
+=819.= 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.--MILL, J. S.
+
+ _System of Logic, Bk. 2, chap. 4, sect.
+ 4._
+
+
+=820.= It may be true, that men, who are _mere_ 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 _mere_ philologists, _mere_ jurists,
+_mere_ soldiers, _mere_ merchants, etc. To such idle talk it
+might further be added: that whenever a certain exclusive
+occupation is _coupled_ with specific shortcomings, it is
+likewise almost certainly divorced from certain _other_
+shortcomings.--GAUSS.
+
+ _Gauss-Schumacher Briefwechsel, Bd. 4,
+ (Altona, 1862), p. 387._
+
+
+=821.= 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 _mere mathematician_ a very
+rare and scarcely possible occurrence.--STEWART, DUGALD.
+
+ _Elements of the Philosophy of the Human
+ Mind, part 3, chap. 1, sect. 3._
+
+
+=822.= 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:--
+
+ /+[infinity]
+ | -x squared
+ | e dx = [sq root][pi]
+ |
+ /-[infinity]
+
+Then putting his finger on what he had written, he turned to his
+class and said: "A mathematician is one to whom _that_ is as
+obvious as that twice two makes four is to you. Liouville was a
+mathematician.--THOMPSON, S. P.
+
+ _Life of Lord Kelvin (London, 1910), p.
+ 1139._
+
+
+=823.= It is not surprising, in view of the polydynamic constitution
+of the genuinely mathematical mind, that many of the major heroes
+of the science, men like Desargues and Pascal, Descartes and
+Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford,
+Riemann and Salmon and Pluecker and Poincare, 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, 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 admitting 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 _a priori_ 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 a la geometrie"; Lazare Carnot, author of the celebrated
+works, "Geometrie de position," and "Reflections sur la Metaphysique
+du Calcul infinitesimal"; Fourier, immortal creator of the
+"Theorie 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.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), pp. 32-33._
+
+
+=824.= If in Germany the goddess _Justitia_ had not the
+unfortunate habit of depositing the ministerial portfolios only
+in the cradles of her own progeny, who knows how many a German
+mathematician might not also have made an excellent minister.
+
+ --PRINGSHEIM, A.
+
+ _Jahresbericht der Deutschen
+ Mathematiker Vereinigung, Bd. 13 (1904),
+ p. 372._
+
+
+=825.= 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.--WHEWELL, W.
+
+ _History of the Inductive Sciences, Vol.
+ 1, Bk. 4, chap. 6, sect. 6._
+
+
+=826.= 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.--WHEWELL, W.
+
+ _Astronomy and General Physics (London,
+ 1833), Bk. 3, chap. 4, p. 327._
+
+
+=827.= 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.
+
+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....
+
+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....
+
+The country of Newton is still pre-eminent for its culture of
+mathematical physics, that of Gauss for the most abstract work in
+mathematics.--MERZ, J. T.
+
+ _History of European Thought in the
+ Nineteenth Century (Edinburgh and
+ London, 1903), p. 630._
+
+
+=828.= 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.--STEWART, DUGALD.
+
+ _Elements of the Philosophy of the Human
+ Mind, Part 3, chap. 1, sect. 3._
+
+
+=829.= 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.
+
+ --SYLVESTER, J. J.
+
+ _Baltimore Address; Mathematical Papers,
+ Vol. 3, pp. 72-73._
+
+
+=830.= An accomplished mathematician, i.e. a most wretched
+orator.--BARROW, ISAAC.
+
+ _Mathematical Lectures (London, 1734),
+ p. 32._
+
+
+=831.= _Nemo mathematicus genium indemnatus habebit._ [No
+mathematician[2] is esteemed a genius until condemned.]
+
+ _Juvenal, Liberii, Satura VI, 562._
+
+ [2] Used here in the sense of astrologer, or
+ soothsayer.
+
+
+=832.= 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.[3] 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.--WALLACE, A. R.
+
+ _Darwinism, chap. 15._
+
+ [3] 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.
+
+
+=833.= ... 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.--WALLACE, A. R.
+
+ _Darwinism, chap. 15._
+
+
+=834.= 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.
+
+ --PEARSON, KARL.
+
+ _Grammar of Science (London, 1911), Part
+ 1, p. 221._
+
+
+=835.= 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.--ELLIS, HAVELOCK.
+
+ _A Study of British Genius (London,
+ 1904), p. 142._
+
+
+=836.= 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.[4]--ELLIS, HAVELOCK.
+
+ _A Study of British Genius (London,
+ 1904), p. 69._
+
+ [4] The mathematical tendencies of Cambridge are
+ due to the fact that Cambridge drains the
+ ability of nearly the whole Anglo-Danish
+ district.
+
+
+=837.= 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.--PERRY, JOHN.
+
+ _The Teaching of Mathematics (London,
+ 1902), pp. 20-21._
+
+
+=838.= 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 naive 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 _Francis Galton_ in his researches on
+heredity, might be interesting.--KLEIN, FELIX.
+
+ _The Evanston Colloquium Lectures (New
+ York, 1894), p. 46._
+
+
+=839.= 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 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 Graecia, 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.--SYLVESTER, J. J.
+
+ _Presidential Address to the British
+ Association; Collected Mathematical
+ Papers, Vol. 2 (1908), p. 658._
+
+
+=840.= 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....
+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.--SHAW, J. B.
+
+ _What is Mathematics? Bulletin American
+ Mathematical Society Vol. 18 (1912), pp.
+ 386-387._
+
+
+=841.= 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?--HILL, THOMAS.
+
+ _Imagination in Mathematics; North
+ American Review, Vol. 85, p. 224._
+
+
+=842.= 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.--HILL, THOMAS.
+
+ _Imagination in Mathematics; North
+ American Review, Vol. 85, p. 227._
+
+
+=843.= 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 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.--HILL, THOMAS.
+
+ _Imagination in Mathematics; North
+ American Review, Vol. 85, p. 223._
+
+
+=844.= To think the thinkable--that is the mathematician's aim.
+
+ --KEYSER, C. J.
+
+ _The Universe and Beyond; Hibbert
+ Journal, Vol. 3 (1904-1905), p. 312._
+
+
+=845.= 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.
+
+ --ARISTIPPUS THE CYRENAIC.
+
+ _Quoted in Hicks, R. D., Stoic and
+ Epicurean, (New York, 1910) p. 210._
+
+
+
+
+ CHAPTER IX
+
+ PERSONS AND ANECDOTES
+
+ (A-M)
+
+
+=901.= Alexander is said to have asked Menaechmus to teach him
+geometry concisely, but Menaechmus replied: "O king, through the
+country there are royal roads and roads for common citizens, but
+in geometry there is one road for all."
+
+ _Stoboeus (Edition Wachsmuth, Berlin,
+ 1884), Ecl. 2, p. 30._
+
+
+=902.= 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.--SENECA.
+
+ _Epistle 91 [Thomas Lodge_].
+
+
+=903.= 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, 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.
+
+ --PLUTARCH.
+
+ _Life of Marcellus_ [_Dryden_].
+
+
+=904.= 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.--PLUTARCH.
+
+ _Life of Marcellus_ [_Dryden_].
+
+
+=905.= 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.--MACAULAY.
+
+ _Lord Bacon; Edinburgh Review, July
+ 1837; Critical and Miscellaneous Essays
+ (New York, 1879), Vol. 1, p. 380._
+
+
+=906.=
+
+ Call Archimedes from his buried tomb
+ Upon the plain of vanished Syracuse,
+ And feelingly the sage shall make report
+ How insecure, how baseless in itself,
+ Is the philosophy, whose sway depends
+ On mere material instruments--how weak
+ Those arts, and high inventions, if unpropped
+ By virtue.
+ --WORDSWORTH.
+
+ _The Excursion._
+
+
+=907.=
+
+ Zu Archimedes kam einst ein wissbegieriger
+ Juengling.
+ "Weihe mich," sprach er zu ihm, "ein in die
+ goettliche Kunst,
+ Die so herrliche Frucht dem Vaterlande
+ getragen,
+ Und die Mauern der Stadt vor der Sambuca
+ beschuetzt!"
+ "Goettlich nennst du die Kunst? Sie ists,"
+ versetzte der Weise;
+ "Aber das war sie, mein Sohn, eh sie dem
+ Staat noch gedient.
+ Willst du nur Fruechte von ihr, die kann auch
+ die Sterbliche zeugen;
+ Wer um die Goettin freit, suche in ihr nicht
+ das Weib."
+ --SCHILLER.
+
+ _Archimedes und der Schueler._
+
+ [To Archimedes once came a youth intent upon
+ knowledge.
+ Said he "Initiate me into the Science divine,
+ Which to our country has borne glorious fruits
+ in abundance,
+ And which the walls of the town 'gainst the
+ Sambuca protects."
+ "Callst thou the science divine? It is so,"
+ the wise man responded;
+ "But so it was, my son, ere the state by her
+ service was blest.
+ Would'st thou have fruit of her only? Mortals
+ with that can provide thee,
+ He who the goddess would woo, seek not the
+ woman in her."]
+
+
+=908.= 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.--PLUTARCH.
+
+ _Life of Marcellus_ [_Dryden_].
+
+
+=909.= 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
+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.--PLUTARCH.
+
+ _Life of Marcellus_ [_Dryden_].
+
+
+=910.= [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.--PLUTARCH.
+
+ _Life of Marcellus_ [_Dryden_].
+
+
+=911.= 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.
+
+ --WHITEHEAD, A. N.
+
+ _An Introduction to Mathematics (New
+ York, 1911), p. 38._
+
+
+=912.= 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 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.--PLUTARCH.
+
+ _Life of Marcellus [Dryden]._
+
+
+=913.= One feature which will probably most impress the
+mathematician accustomed to the rapidity and directness secured
+by the generality of modern methods is the _deliberation_ 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
+modern mathematicians found it easier to invent a new Analysis
+than to seek out the old.--HEATH, T. L.
+
+ _The Works of Archimedes (Cambridge,
+ 1897), Preface._
+
+
+=914.= 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.
+
+ --LOCKE, JOHN.
+
+ _Second Reply to the Bishop of
+ Worcester._
+
+
+=915.= 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 _De Augmentis_, 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,
+"_delicias et fastum mathematicorum_." 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.--MACAULAY.
+
+ _Lord Bacon: Edinburgh Review, July,
+ 1837; Critical and Miscellaneous Essays
+ (New York, 1879), Vol. 1, p. 380._
+
+
+=916.= 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.--MACAULAY.
+
+ _Lord Bacon, Edinburgh Review, July,
+ 1837; Critical and Miscellaneous Essays
+ (New York, 1879), Vol. 2, p. 381._
+
+
+=917.= 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 [Greek: eureka]
+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
+south pole below, as a reason why in our hemisphere the north
+winds predominate over the south.--SPEDDING, J.
+
+ _Works of Francis Bacon (Boston),
+ Preface to De Interpretatione Naturae
+ Prooemium._
+
+
+=918.= 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.
+
+ --DE MORGAN, A.
+
+ _Budget of Paradoxes (London, 1872), pp.
+ 53-54._
+
+
+=919.= 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.
+
+ --HUTTON, CHARLES.
+
+ _A Philosophical and Mathematical
+ Dictionary (London, 1815), Vol. 1, p.
+ 226._
+
+
+=920.= 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, "_Eadem mutata
+resurgo_," a happy allusion to the hope of the Christians, which
+is in a way symbolized by the properties of that curve.
+
+ --FONTENELLE.
+
+ _Eloge de M. Bernoulli; Oeuvres de
+ Fontenelle, t. 5 (1758), p. 112._
+
+
+=921.= 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 _intra semi-quadrantem horae!_
+the sum of the 10th powers of the first thousand integers, and
+found it to be
+
+ 91,409,924,241,424,243,424,241,924,242,500.
+ --CHRYSTAL, G.
+
+ _Algebra, Part 2 (Edinburgh, 1879), p.
+ 209._
+
+
+=922.= In the year 1692, James Bernoulli, discussing the logarithmic
+spiral [or equiangular spiral, [rho] = [alpha]^[theta]] ... 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, _simillima filia
+matri_. 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 [Greek:
+homoousios] with him, howsoever overshadowed. Or, if you prefer,
+since our _spira mirabilis_ 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 tombstone bear this spiral, with the motto,
+"Though changed, I arise again exactly the same, _Eadem numero
+mutata resurgo_."--HILL, THOMAS.
+
+ _The Uses of Mathesis; Bibliotheca
+ Sacra, Vol. 32, pp. 515-516._
+
+
+=923.= Babbage was one of the founders of the Cambridge
+Analytical Society whose purpose he stated was to advocate "the
+principles of pure _d_-ism as opposed to the _dot_-age of the
+university."--BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p. 451._
+
+
+=924.= 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.--HALSTED, G. B.
+
+ _Bolyai's Science Absolute of Space
+ (Austin, 1896), Introduction, p. 29._
+
+
+=925.= Bolyai [Janos] projected a universal language for speech
+as we have it for music and mathematics.--HALSTED, G. B.
+
+ _Bolyai's Science Absolute of Space
+ (Austin, 1896), Introduction, p. 29._
+
+
+=926.= [Bolyai's Science Absolute of Space]--the most
+extraordinary two dozen pages in the history of thought!
+
+ --HALSTED, G. B.
+
+ _Bolyai's Science Absolute of Space
+ (Austin, 1896), Introduction, p. 18._
+
+
+=927.= [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.--CAJORI, F.
+
+ _History of Elementary Mathematics (New
+ York, 1910), p. 273._
+
+
+=928.= Bernard Bolzano dispelled the clouds that throughout all
+the foregone centuries had enveloped the notion of Infinitude 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.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 42._
+
+
+=929.= Let me tell you how at one time the famous mathematician
+_Euclid_ 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 _Euclid's Elements_ and read for the
+first time his doctrine of _ratio_, 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.--BOLZANO, BERNARD.
+
+ _Selbstbiographie (Wien, 1875), p. 20._
+
+
+=930.= Mr. Cayley, of whom it may be so truly said, whether the
+matter he takes in hand be great or small, "_nihil tetigit quod
+non ornavit_,"....--SYLVESTER, J. J.
+
+ _Philosophic Transactions of the Royal
+ Society, Vol. 17 (1864), p. 605._
+
+
+=931.= It is not _Cayley's_ way to analyze concepts into their
+ultimate elements.... But he is master of the _empirical_
+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. _Cayley_ is thus the _natural philosopher_ among
+mathematicians.--NOETHER, M.
+
+ _Mathematische Annalen, Bd. 46 (1895),
+ p. 479._
+
+
+=932.= 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....
+
+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.--FORSYTH, A. R.
+
+ _Proceedings of London Royal Society,
+ Vol. 58 (1895), pp. 23-24._
+
+
+=933.= 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.
+
+ --FORSYTH, A. R.
+
+ _Proceedings London Royal Society, Vol.
+ 58 (1895), pp. 11-12._
+
+
+=934.= 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.--FORSYTH, A. R.
+
+ _Proceedings London Royal Society, Vol.
+ 58 (1895), p. 21._
+
+
+=935.= The mathematical talent of Cayley was characterized by
+clearness and extreme elegance of analytical form; it was
+re-enforced by an incomparable capacity for work which has
+caused the distinguished scholar to be compared with Cauchy.
+
+ --HERMITE, C.
+
+ _Comptes Rendus, t. 120 (1895), p. 234._
+
+
+=936.= 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
+_Euclid_. 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.
+
+ --CAJORI, F.
+
+ _History of Elementary Mathematics (New
+ York, 1910), p. 285._
+
+
+=937.= 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.
+
+ --THOMPSON, S. P.
+
+ _Life of Lord Kelvin (London, 1910), p.
+ 1137._
+
+
+=938.= 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 ideas, the conquering and
+masterful quality of the human mind which Goethe called in one
+word _das Daemonische_.--POLLOCK, F.
+
+ _Clifford's Lectures and Essays (New
+ York, 1901), Vol. 1, Introduction, pp.
+ 5-6._
+
+
+=939.= 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.--POLLOCK, F.
+
+ _Clifford's Lectures and Essays (New
+ York, 1901), Vol. 1, Introduction, p.
+ 10._
+
+
+=940.= 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;....--POLLOCK, F.
+
+ _Clifford's Lectures and Essays (New
+ York, 1901), Vol. 1, Introduction, pp.
+ 11-12._
+
+
+=941.= 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.
+
+ _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._
+
+
+=942.= [Comte] may truly be said to have created the philosophy
+of higher mathematics.--MILL, J. S.
+
+ _System of Logic (New York, 1846), p.
+ 369._
+
+
+=943.= 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
+_general_ 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 _views_ on any
+subject of pure or applied mathematics.--HAMILTON, W. R.
+
+ _Graves' Life of W. R. Hamilton (New
+ York, 1882-1889), Vol. 3, p. 475._
+
+
+=944.= 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.--BALL, W. W. R.
+
+ _History of Mathematics (London, 1911),
+ p. 394._
+
+
+=945.= 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 [pi], which, in answer to a question, he explained
+stood for the ratio of the circumference of a circle to its
+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?"
+
+ --BALL, W. W. R.
+
+ _Mathematical Recreations and Problems
+ (London, 1896), p. 180; See also De
+ Morgan's Budget of Paradoxes (London,
+ 1872), p. 172._
+
+
+=946.= 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
+_is_ 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;....--DE MORGAN, A.
+
+ _Budget of Paradoxes (London, 1872), p.
+ 172._
+
+
+=947.= [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:]
+
+Two of these I have joined in the title-page:
+
+ [Ut agendo surgamus arguendo gustamus.]
+
+A few of the others are personal remarks.
+
+ Great gun! do us a sum!
+
+is a sneer at my pursuit; but,
+
+ / n
+ | u
+ Go! great sum! | a du
+ /
+
+is more dignified....
+
+ Adsum, nugator, suge!
+
+is addressed to a student who continues talking after the lecture
+has commenced: ....
+
+ Graduatus sum! nego
+
+applies to one who declined to subscribe for an M. A. degree.
+
+ --DE MORGAN, AUGUSTUS.
+
+ _Budget of Paradoxes (London, 1872), p.
+ 82._
+
+
+=948.= 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.--MILL, J. S.
+
+ _An Examination of Sir W. Hamilton's
+ Philosophy (London, 1878), p. 626._
+
+
+=949.= To _Descartes_, the great philosopher of the 17th century,
+is due the undying credit of having removed the bann which until
+then rested upon geometry. The _analytical geometry_, 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.--HANKEL, H.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1884),
+ p. 10._
+
+
+=950.= [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.--MILL, J. S.
+
+ _An Examination of Sir W. Hamilton's
+ Philosophy (London, 1878), p. 617._
+
+
+=951.= ... [Greek: kai phasin hoti Ptolemaios ereto pote auton
+[Eukleiden], ei tis estin peri geometrian hodos syntomotera tes
+stoicheioseos; hode apekrinato me einai basiliken atrapon epi
+geometrian].
+
+[ ... 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."]--PROCLUS.
+
+ _(Edition Friedlein, 1873), Prol. II,
+ 39._
+
+
+=952.= Someone who had begun to read geometry with Euclid, when
+he had learned the first proposition, asked Euclid, "But 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."--STOBAEUS.
+
+ _(Edition Wachsmuth, 1884), Ecl. II._
+
+
+=953.= The sacred writings excepted, no Greek has been so much
+read and so variously translated as Euclid.[5]--DE MORGAN, A.
+
+ _Smith's Dictionary of Greek and Roman
+ Biology and Mythology (London, 1902),
+ Article, "Eucleides."_
+
+ [5] Riccardi's Bibliografia Euclidea (Bologna,
+ 1887), lists nearly two thousand editions.
+
+
+=954.= The thirteen books of Euclid must have been a tremendous
+advance, probably even greater than that contained in the
+"Principia" of Newton.--DE MORGAN, A.
+
+ _Smith's Dictionary of Greek and Roman
+ Biography and Mythology (London, 1902),
+ Article, "Eucleides."_
+
+
+=955.= 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.--REID, THOMAS.
+
+ _Essay on the Powers of the Human Mind
+ (Edinburgh, 1812), Vol. 2, p. 368._
+
+
+=956.= It is the invaluable merit of the great Basle mathematician
+Leonhard _Euler_, to have freed the analytical calculus from all
+geometrical bonds, and thus to have established _analysis_ as an
+independent science, which from his time on has maintained an
+unchallenged leadership in the field of mathematics.--HANKEL, H.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1884),
+ p. 12._
+
+
+=957.= 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.--RUDIO, F.
+
+ _Quoted by Ahrens W.: Scherz und Ernst
+ in der Mathematik (Leipzig, 1904), p.
+ 251._
+
+
+=958.= 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.--HUTTON, CHARLES.
+
+ _Philosophical and Mathematical
+ Dictionary (London, 1815), pp. 493-494._
+
+
+=959.= 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[6] gave him the first idea.
+
+ --BREWSTER, DAVID.
+
+ _Letters of Euler (New York, 1872), Vol.
+ 1, p. 24._
+
+ [6] The line referred to is:
+ "The anchor drops, the rushing keel is staid."
+
+
+=960.= 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.--CAJORI, F.
+
+ _History of Mathematics (New York,
+ 1897), pp. 253-254._
+
+
+=961.= Euler who could have been called almost without metaphor,
+and certainly without hyperbole, analysis incarnate.--ARAGO.
+
+ _Oeuvres, t. 2 (1854), p. 433._
+
+
+=962.= Euler calculated without any apparent effort, just as men
+breathe, as eagles sustain themselves in the air.--ARAGO.
+
+ _Oeuvres, t. 2 (1854), p. 133._
+
+
+=963.= 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.--BREWSTER, DAVID.
+
+ _Letters of Euler (New York, 1872), Vol.
+ 2, p. 22._
+
+
+=964.= 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.--CAJORI, F.
+
+ _History of Mathematics (New York,
+ 1897), p. 248._
+
+
+=965.= Euler's _Tentamen novae theorae musicae_ had no great
+success, as it contained too much geometry for musicians, and too
+much music for geometers.--FUSS, N.
+
+ _Quoted by Brewster: Letters of Euler
+ (New York, 1872), Vol. 1, p. 26._
+
+
+=966.= Euler was a believer in God, downright and
+straight-forward. The following story is told by Thiebault, in
+his _Souvenirs de vingt ans de sejour a Berlin_, .... 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:
+
+ a + b^n
+ _Monsieur_, ------- = x, _donc Dieu existe; repondez!_
+ n
+
+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.
+
+ --DE MORGAN, A.
+
+ _Budget of Paradoxes (London, 1872), p.
+ 251._
+
+
+=967.= Fermat died with the belief that he had found a
+long-sought-for law of prime numbers in the formula 2^2^n + 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
+2^2^5 + 1 = 4,294,967,297 = 6,700,417 times 641. The American
+lightning calculator _Zerah Colburn_, when a boy, readily found
+the factors but was unable to explain the method by which he made
+his marvellous mental computation.--CAJORI, F.
+
+ _History of Mathematics (New York,
+ 1897), p. 180._
+
+
+=968.= 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--
+
+ In Mathesi a sole fundes.[7]
+ --FLAMSTEED, J.
+
+ _Macclesfield: Correspondence of
+ Scientific Men (Oxford, 1841), Vol. 2,
+ p. 90._
+
+ [7] Johannes Flamsteedius.
+
+
+=969.= _To the Memory of Fourier_
+
+ Fourier! with solemn and profound delight,
+ Joy born of awe, but kindling momently
+ To an intense and thrilling ecstacy,
+ I gaze upon thy glory and grow bright:
+ As if irradiate with beholden light;
+ As if the immortal that remains of thee
+ Attuned me to thy spirit's harmony,
+ Breathing serene resolve and tranquil might.
+ Revealed appear thy silent thoughts of youth,
+ As if to consciousness, and all that view
+ Prophetic, of the heritage of truth
+ To thy majestic years of manhood due:
+ Darkness and error fleeing far away,
+ And the pure mind enthroned in perfect day.
+ --HAMILTON, W. R.
+
+ _Graves' Life of W. R. Hamilton, (New
+ York, 1882), Vol. 1, p. 596._
+
+
+=970.= Astronomy and Pure Mathematics are the magnetic poles
+toward which the compass of my mind ever turns.--GAUSS TO BOLYAI.
+
+ _Briefwechsel (Schmidt-Stakel), (1899),
+ p. 55._
+
+
+=971.= [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."--BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p. 458._
+
+
+=972.= The mathematical giant [Gauss], who from his lofty heights
+embraces in one view the stars and the abysses....--BOLYAI, W.
+
+ _Kurzer Grundriss eines Versuchs (Maros
+ Vasarhely, 1851), p. 44._
+
+
+=973.= Almost everything, which the mathematics of our century
+has brought forth in the way of original scientific ideas,
+attaches to the name of Gauss.--KRONECKER, L.
+
+ _Zahlentheorie, Teil 1 (Leipzig, 1901),
+ p. 43._
+
+
+=974.= 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.
+
+ --GAUSS TO BESSEL, 1810.
+
+ _Gauss-Bessel Briefwechsel (1880), p.
+ 107._
+
+
+=975.= 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."
+
+ --CANTOR, M.
+
+ _Allgemeine Deutsche Biographie, Bd. 8
+ (1878), p. 435._
+
+
+=976.= 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.--GOETHE.
+
+ _Letter to Naumann (1826); Vogel:
+ Goethe's Selbstzeugnisse (Leipzig,
+ 1903), p. 56._
+
+
+=977.= 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.
+
+ --KUMMER, E. E.
+
+ _Dirichlet: Werke, Bd. 2, p. 315._
+
+
+=978.= [The famous attack of Sir William Hamilton on the tendency
+of mathematical studies] affords the most express evidence of
+those fatal _lacunae_ 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.--MILL, J. S.
+
+ _Examination of Sir William Hamilton's
+ Philosophy (London, 1878), p. 607._
+
+
+=979.= 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.
+
+ --CLIFFORD, W. K.
+
+ _Aims and Instruments of Scientific
+ Thought; Lectures and Essays, Vol. 1
+ (London, 1901), p. 165._
+
+
+=980.= It is said of Jacobi, that he attracted the particular
+attention and friendship of Boeckh, 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.--SYLVESTER, J. J.
+
+ _Collected Mathematical Papers, Vol. 2
+ (Cambridge, 1908), p. 651._
+
+
+=981.= When Dr. Johnson felt, or fancied he felt, his fancy
+disordered, his constant recurrence was to the study of
+arithmetic.--BOSWELL, J.
+
+ _Life of Johnson (Harper's Edition,
+ 1871), Vol. 2, p. 264._
+
+
+=982.= 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."--ARAGO.
+
+ _Eulogy on Laplace: [Baden Powell]
+ Smithsonian Report, 1874, p. 132._
+
+
+=983.= The great masters of modern analysis are Lagrange,
+Laplace, and Gauss, who were contemporaries. It is interesting
+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.--BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p. 463._
+
+
+=984.= 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 _Il
+faut que j'y songe encore_, and put the paper in his pocket.
+
+ --DE MORGAN, A.
+
+ _Budget of Paradoxes (London, 1872), p.
+ 173._
+
+
+=985.= I never come across one of Laplace's "_Thus it plainly
+appears_" without feeling sure that I have hours of hard work
+before me to fill up the chasm and find out and show _how_ it
+plainly appears.--BOWDITCH, N.
+
+ _Quoted by Cajori: Teaching and History
+ of Mathematics in the U. S. (Washington,
+ 1896), p. 104._
+
+
+=986.= Biot, who assisted Laplace in revising it [The Mecanique
+Celeste] 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, "_Il est aise a voir._"
+
+ --BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p 427._
+
+
+=987.= It would be difficult to name a man more remarkable for
+the greatness and the universality of his intellectual powers
+than Leibnitz.--MILL, J. S.
+
+ _System of Logic, Bk. 2, chap. 5, sect.
+ 6._
+
+
+=988.= 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.--STEWART, DUGALD.
+
+ _Philosophy of the Human Mind, Part 3,
+ chap. 1, sect. 3._
+
+
+=989.= 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.
+
+ --MERZ, J. T.
+
+ _Leibnitz, Chap. 6._
+
+
+=990.= 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.[8] 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.
+
+ [8] This sentence has been reworded for the
+ purpose of this quotation.
+
+A mental attitude such as this is always highly favorable for
+mathematical as well as for philosophical investigations.
+Wherever 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.
+
+ --MERZ, J. T.
+
+ _Leibnitz (Philadelphia), pp. 44-45._
+
+
+=991.= 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.
+
+ --LAPLACE.
+
+ _Essai Philosophique sur les
+ Probabilites; Oeuvres (Paris, 1896), t.
+ 7, p. 119._
+
+
+=992.= Sophus Lie, great comparative anatomist of geometric
+theories.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 31._
+
+
+=993.= 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.--KLEIN, F.
+
+ _Lectures on Mathematics (New York,
+ 1911), p. 24._
+
+
+=994.= 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.--KLEIN, F.
+
+ _Lectures on Mathematics (New York,
+ 1911), p. 9._
+
+
+=995.= 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.--FIZI.
+
+ _Preface to the Lilawati. Quoted by A.
+ Hutton: A Philosophical and Mathematical
+ Dictionary, Article "Algebra" (London,
+ 1815)._
+
+
+=996.= Is there anyone whose name cannot be twisted into either
+praise or satire? I have had given to me,
+
+ _Thomas Babington Macaulay
+ Mouths big: a Cantab anomaly._
+ --DE MORGAN, A.
+
+ _Budget of Paradoxes (London, 1872), p.
+ 83._
+
+
+
+
+ CHAPTER X
+
+ PERSONS AND ANECDOTES
+
+ (N-Z)
+
+
+=1001.= 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.--ABBOTT, J. S. C.
+
+ _Napoleon Bonaparte (New York, 1904),
+ Vol. 1, chap. 10._
+
+
+=1002.= 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.--BREWSTER, D.
+
+ _Life of Sir Isaac Newton (London,
+ 1831), pp. 1, 2._
+
+
+=1003.= 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.--EMERSON.
+
+ _Essay on History._
+
+
+=1004.= 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.
+
+ --WHEWELL, W.
+
+ _History of the Inductive Sciences, Bk.
+ 7, chap. 2, sect. 5._
+
+
+=1005.= 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 point to which men's minds had been journeying
+for two thousand years.--WHEWELL, W.
+
+ _History of the Inductive Sciences, Bk.
+ 7, chap. 2, sect. 5._
+
+
+=1006.= The efforts of the great philosopher [Newton] were always
+superhuman; the questions which he did not solve were incapable
+of solution in his time.--ARAGO.
+
+ _Eulogy on Laplace, [Baden Powell]
+ Smithsonian Report, 1874, p. 133._
+
+
+=1007.=
+
+ Nature and Nature's laws lay hid in night:
+ God said, "Let Newton be!" and all was light.
+ --POPE, A.
+
+ _Epitaph intended for Sir Isaac Newton._
+
+
+=1008.=
+
+ There Priest of Nature! dost thou shine,
+ _Newton!_ a King among the Kings divine.
+ --SOUTHEY.
+
+ _Translation of a Greek Ode on
+ Astronomy._
+
+
+=1009.=
+
+ O'er Nature's laws God cast the veil of night,
+ Out-blaz'd a Newton's soul--and all was light.
+ --HILL, AARON.
+
+ _On Sir Isaac Newton._
+
+
+=1010.= Taking mathematics from the beginning of the world to the
+time when Newton lived, what he had done was much the better
+half.--LEIBNITZ.
+
+ _Quoted by F. R. Moulton: Introduction
+ to Astronomy (New York, 1906), p. 199._
+
+
+=1011.= 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.--LAGRANGE.
+
+ _Quoted by F. R. Moulton: Introduction
+ to Astronomy (New York, 1906), p. 199._
+
+
+=1012.= 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.--WILSON, JOHN.
+
+ _Noctes Ambrosianae._
+
+
+=1013.= Such men as Newton and Linnaeus are incidental, but
+august, teachers of religion.--WILSON, JOHN.
+
+ _Essays: Education of the People._
+
+
+=1014.= Sir Isaac Newton, the supreme representative of
+Anglo-Saxon genius.--ELLIS, HAVELOCK.
+
+ _Study of British Genius (London, 1904),
+ p. 49._
+
+
+=1015.= Throughout his life Newton must have devoted at least as
+much attention to chemistry and theology as to mathematics....
+
+ --BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p. 335._
+
+
+=1016.= 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.--REV. J. SPENCE.
+
+ _Anecdotes, Observations, and Characters
+ of Books and Men (London, 1868), p. 54._
+
+
+=1017.= 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.--PARTON, JAMES.
+
+ _Sir Isaac Newton._
+
+
+=1018.= 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.
+
+ --PARTON, JAMES.
+
+ _Sir Isaac Newton._
+
+
+=1019.= Newton lectured now and then to the few students who
+chose to hear him; and it is recorded that very frequently he
+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.--PARTON, JAMES.
+
+ _Sir Isaac Newton._
+
+
+=1020.= 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.--REV. J. SPENCE.
+
+ _Anecdotes, Observations, and Characters
+ of Books and Men (London, 1858), p.
+ 132._
+
+
+=1021.= 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, L1;" "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, L3;" "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, L2;" "Theatrum chemicum, L1 8s."--PARTON, JAMES.
+
+ _Sir Isaac Newton._
+
+
+=1022.= 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 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.--PARTON, JAMES.
+
+ _Sir Isaac Newton._
+
+
+=1023.= 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.--NEWTON, I.
+
+ _Quoted by Rev. J. Spence: Anecdotes,
+ Observations, and Characters of Books
+ and Men (London, 1858), p. 40._
+
+
+=1024.= If I have seen farther than Descartes, it is by standing
+on the shoulders of giants.--NEWTON, I.
+
+ _Quoted by James Parton: Sir Isaac
+ Newton._
+
+
+=1025.= 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."--WHEWELL, W.
+
+ _History of the Inductive Sciences, Bk.
+ 7, chap. 2, sect. 5._
+
+
+=1026.= Newton took no exercise, indulged in no amusements, and
+worked incessantly, often spending eighteen or nineteen hours out
+of the twenty-four in writing.--BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p. 358._
+
+
+=1027.= 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.--BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p. 356._
+
+
+=1028.= 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 _not
+new_; but Newton _went on_.--DE MORGAN, A.
+
+ _Budget of Paradoxes (London, 1872), p.
+ 82._
+
+
+=1029.= For other great mathematicians or philosophers, he
+[Gauss] used the epithets magnus, or clarus, or clarissimus; for
+Newton alone he kept the prefix summus.--BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p. 362._
+
+
+=1030.= 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.--HALSTED, G. B.
+
+ _F. Cajori's Teaching and History of
+ Mathematics in the U. S. (Washington,
+ 1890), p. 265._
+
+
+=1031.= 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.--CAJORI, F.
+
+ _Teaching and History of Mathematics in
+ the U. S. (Washington, 1890), p. 264._
+
+
+=1032.= 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
+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.--SYLVESTER, J. J.
+
+ _Johns Hopkins Commemoration Day
+ Address; Collected Mathematical Papers,
+ Vol. 3, p. 76._
+
+
+=1033.= 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....
+
+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 Goettingen? 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.--FRANKLIN, F.
+
+ _Johns Hopkins University Circulars 16
+ (1897), p. 54._
+
+
+=1034.= 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....
+
+The concept of _Function_ 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].--NOETHER, M.
+
+ _Mathematische Annalen, Bd. 50 (1898),
+ pp. 134-135._
+
+
+=1035.= Sylvester's _methods!_ 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
+_Substitutions-Theorie_, 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.--DAVIS, E. W.
+
+ _Cajori's Teaching and History of
+ Mathematics in the U.S. (Washington,
+ 1890), pp. 265-266._
+
+
+=1036.= 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 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.
+
+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.
+
+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 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.--HATHAWAY, A. S.
+
+ _F. Cajori's Teaching and History of
+ Mathematics in the U. S. (Washington,
+ 1890), pp. 266-267._
+
+
+=1037.= Professor Cayley has since informed me that the theorem
+about whose origin I was in doubt, will be found in Schlaefli'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 Schlaefli in the
+"Mathematical Journal," which bears my name as one of the editors
+upon the face.--SYLVESTER, J. J.
+
+ _Philosophical Transactions of the Royal
+ Society (1864), p. 642._
+
+
+=1038.= 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.
+
+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 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.--DURFEE, W. P.
+
+ _F. Cajori's Teaching and History of
+ Mathematics in the U. S. (Washington,
+ 1890), p. 268._
+
+
+=1039.= 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 [pi] and _e_, which have even found their way
+into Webster's Dictionary, made each young worker apply to himself
+the strictest tests.--HALSTED, G. B.
+
+ _F. Cajori's Teaching and History of
+ Mathematics in the U. S. (Washington,
+ 1890), p. 265._
+
+
+=1040.= 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.--MACMAHON. P. A.
+
+ _Nature, Vol. 55 (1897), p. 494._
+
+
+=1041.= The enthusiasm of Sylvester for his own work, which manifests
+itself here as always, indicates one of his characteristic qualities:
+a high degree of _subjectivity_ 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.
+
+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.
+
+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 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.
+
+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:
+
+"Admiratio generat quaestionem, quaestio investigationem
+investigatio inventionem."--NOETHER, M.
+
+ _Mathematische Annalen, Bd. 50 (1898),
+ pp. 155-160._
+
+
+=1042.= 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.--SYLVESTER, J. J.
+
+ _Nature, Vol. 37 (1887-1888), p. 162._
+
+
+=1043.= 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'.--MACFARLANE, A.
+
+ _Bibliotheca Mathematica, Bd. 3 (1903),
+ p. 189._
+
+
+=1044.= 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.
+
+ --MACFARLANE, A.
+
+ _Bibliotheca Mathematica, Bd. 3 (1903),
+ p. 189._
+
+
+=1045.= 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 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.--THOMPSON, S. P.
+
+ _Life of Lord Kelvin (London, 1910), p.
+ 1136._
+
+
+=1046.= The following is one of the many stories told of "old
+Donald McFarlane" the faithful assistant of Sir William Thomson.
+
+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!"
+
+ --THOMPSON, S. P.
+
+ _Life of Lord Kelvin (London, 1910), p.
+ 420._
+
+
+=1047.= The following story (here a little softened from the
+vernacular) was narrated by Lord Kelvin himself when dining at
+Trinity Hall:--
+
+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."--THOMPSON, S. P.
+
+ _Life of Lord Kelvin (London, 1910), p.
+ 1124._
+
+
+=1048.= Lord Kelvin, unable to meet his classes one day, posted
+the following notice on the door of his lecture room,--
+
+"Professor Thomson will not meet his classes today." The
+disappointed class decided to play a joke on the professor.
+Erasing the "c" they left the legend to read,--
+
+"Professor Thomson will not meet his lasses today." 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,--
+
+"Professor Thomson will not meet his asses today."[9]
+
+ --NORTHRUP, CYRUS.
+
+ _University of Washington Address,
+ November 2, 1908._
+
+ [9] 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.
+
+
+=1049.= 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.--LAMPE, E.
+
+ _Karl Weierstrass: Jahrbuch der
+ Deutschen Mathematiker Vereinigung, Bd.
+ 6 (1897), pp. 38-39._
+
+
+=1050.= 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.--LAMPE, E.
+
+ _Naturwissenschaftliche Rundschau, Bd.
+ 12 (1897), p. 361._
+
+
+
+
+ CHAPTER XI
+
+ MATHEMATICS AS A FINE ART
+
+
+=1101.= 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.--SYLVESTER, J. J.
+
+ _Presidential Address, British
+ Association Report (1869); Collected
+ Mathematical Papers, Vol. 2, p. 659._
+
+
+=1102.= 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.--POINCARE, HENRI.
+
+ _The Relation of Analysis and
+ Mathematical Physics; Bulletin American
+ Mathematical Society, Vol. 4 (1899), p.
+ 248._
+
+
+=1103.= 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.
+
+ --BOCHER, MAXIME.
+
+ _Fundamental Conceptions and Methods in
+ Mathematics; Bulletin American
+ Mathematical Society, Vol. 9 (1904), p.
+ 133._
+
+
+=1104.= 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 least, of our nobler
+impulses can escape from the dreary exile of the natural world.
+
+ --RUSSELL, BERTRAND.
+
+ _The Study of Mathematics: Philosophical
+ Essays (London, 1910), p. 73._
+
+
+=1105.= 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 Duerer, 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.--RUDIO, F.
+
+ _Virchow-Holtzendorf: Sammlung
+ gemeinverstaendliche wissenschaftliche
+ Vortraege, Heft 142, p. 19._
+
+
+=1106.= 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 _common_ plane to
+every three of these, _outside_ of which lies the fourth; and
+through every two may be drawn a common axis _opposite_ 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.
+
+ --SYLVESTER, J. J.
+
+ _Proof of the hitherto undemonstrated
+ Fundamental Theorem of Invariants:
+ Collected Mathematical Papers, Vol. 3,
+ p. 123._
+
+
+=1107.= 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.--MOEBIUS, P. J.
+
+ _Ueber die Anlage zur Mathematik
+ (Leipzig, 1900), p. 5._
+
+
+=1108.= 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.--PRINGSHEIM, A.
+
+ _Jahresbericht der Deutschen
+ Mathematiker Vereinigung, Bd. 32, p.
+ 381._
+
+
+=1109.= 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.--HOBSON, E. W.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1910) Nature, Vol. 84, p. 290._
+
+
+=1110.= 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, 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.--YOUNG, J. W. A.
+
+ _The Teaching of Mathematics (New York,
+ 1907), p. 44._
+
+
+=1111.= 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.
+
+ --KUMMER, E. E.
+
+ _Berliner Monatsberichte (1867), p.
+ 395._
+
+
+=1112.= 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.--ARBUTHNOT, JOHN.
+
+ _Usefulness of Mathematical Learning._
+
+
+=1113.= 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.--LAMPE, E.
+
+ _Die Entwickelung der Mathematik, etc.
+ (Berlin, 1893), p. 4._
+
+
+=1114.= 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 _beauty_," Schellbach
+responds with equal right, "Who does not know mathematics, and
+the results of recent scientific investigation, dies without
+knowing _truth_."--SIMON, MAX.
+
+ _Quoted in J. W. A. Young: Teaching of
+ Mathematics (New York, 1907), p. 44._
+
+
+=1115.= Buechsel 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,
+....--LAMPE, E.
+
+ _Die Entwickelung der Mathematik, etc.
+ (Berlin 1893), p. 22._
+
+
+=1116.= 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.... _Ultima se
+tangunt_. How expressive, how nicely characterizing withal is
+mathematics! As the musician recognizes Mozart, Beethoven,
+Schubert in the first chords, so the mathematician would
+distinguish 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....
+
+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.--BOLTZMANN, L.
+
+ _Gustav Robert Kirchhoff (Leipzig 1888),
+ pp. 28-30._
+
+
+=1117.=
+
+ On poetry and geometric truth,
+ And their high privilege of lasting life,
+ From all internal injury exempt,
+ I mused; upon these chiefly: and at length,
+ My senses yielding to the sultry air,
+ Sleep seized me, and I passed into a dream.
+ --WORDSWORTH.
+
+ _The Prelude, Bk. 5._
+
+
+=1118.= 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.--MILNER, FLORENCE.
+
+ _School Review, 1898, p. 114._
+
+
+=1119.= 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.
+
+ --WHITE, W. F.
+
+ _A Scrap-book of Elementary Mathematics
+ (Chicago, 1908), p. 208._
+
+
+=1120.= 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.--WORKMAN, W. P.
+
+ _F. Spencer: Aim and Practice of
+ Teaching (New York, 1897), p. 194._
+
+
+=1121.= 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.--POLLOCK, F.
+
+ _Clifford's Lectures and Essays (New
+ York, 1901), Vol. 1, Introduction, p.
+ 1._
+
+
+=1122.= 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.--HOFFMAN, F. S.
+
+ _Sphere of Science (London, 1898), p.
+ 107._
+
+
+=1123.= 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.--THOREAU, H. D.
+
+ _A Week on the Concord and Merrimac
+ Rivers (Boston, 1893), p. 477._
+
+
+=1124.= 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.--EMERSON, R. W.
+
+ _Society and Solitude, Chap. 7, Works
+ and Days._
+
+
+=1125.= 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.--HILL, THOMAS.
+
+ _North American Review, Vol. 85, p.
+ 230._
+
+
+=1126.= 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 [Greek: mathesis],
+learning, but [Greek: poiesis], a creation.--HILL, THOMAS.
+
+ _North American Review, Vol. 85, p.
+ 229._
+
+
+=1127.=
+
+ Music and poesy used to quicken you:
+ The mathematics, and the metaphysics,
+ Fall to them as you find your stomach serves you.
+ No profit grows, where is no pleasure ta'en:--
+ In brief, sir, study what you most affect.
+ --SHAKESPEARE.
+
+ _Taming of the Shrew, Act 1, Scene 1._
+
+
+=1128.= Music has much resemblance to algebra.--NOVALIS.
+
+ _Schriften, Teil 2 (Berlin, 1901), p.
+ 549._
+
+
+=1129.=
+
+ I do present you with a man of mine,
+ Cunning in music and in mathematics,
+ To instruct her fully in those sciences,
+ Whereof, I know, she is not ignorant.
+ --SHAKESPEARE.
+
+ _Taming of the Shrew, Act 2, Scene 1._
+
+
+=1130.= 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
+1/2, 2/3, 3/4, 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."--CAJORI, F.
+
+ _History of Mathematics (New York,
+ 1897), p. 67._
+
+
+=1131.= 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 _feels_ Mathematic, the mathematician _thinks_
+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!--SYLVESTER, J. J.
+
+ _On Newton's Rule for the Discovery of
+ Imaginary Roots; Collected Mathematical
+ Papers, Vol. 2, p. 419._
+
+
+=1132.= 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.--PRINGSHEIM, A.
+
+ _Jahresbericht der Deutschen
+ Mathematiker Vereinigung, Bd. 13, p.
+ 364._
+
+
+=1133.= 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.--HELMHOLTZ, H.
+
+ _Vortraege und Reden, Bd. 1
+ (Braunschweig, 1884), p. 82._
+
+
+=1134.= 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.
+
+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.--LAMPE, E.
+
+ _Die Entwickelung der Mathematik etc.
+ (Berlin, 1893), p. 4._
+
+
+=1135.= 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.
+
+ --WALLACE, A. R.
+
+ _Darwinism, Chap. 15._
+
+
+=1136.= 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.
+
+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 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.--WEISMANN, AUGUST.
+
+ _Essays upon Heredity [A. E. Shipley],
+ (Oxford, 1891), Vol. 1, p. 97._
+
+
+
+
+ CHAPTER XII
+
+ MATHEMATICS AS A LANGUAGE
+
+
+=1201.= 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.
+
+ --WELLS, H. G.
+
+ _Mankind in the Making (London, 1904),
+ pp. 191-192._
+
+
+=1202.= Mathematical language is not only the simplest and most
+easily understood of any, but the shortest also.--BROUGHAM, H. L.
+
+ _Works (Edinburgh, 1872), Vol. 7, p.
+ 317._
+
+
+=1203.= Mathematics is the science of definiteness, the necessary
+vocabulary of those who know.--WHITE, W. F.
+
+ _A Scrap-book of Elementary Mathematics
+ (Chicago, 1908), p. 7._
+
+
+=1204.= 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.
+
+ --DILLMANN, C.
+
+ _Die Mathematik die Fackeltraegerin einer
+ neuen Zeit (Stuttgart, 1889), p. 5._
+
+
+=1205.= 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.--SYLVESTER, J. J.
+
+ _Presidential Address, British
+ Association; Collected Mathematical
+ Papers, Vol. 2, p. 652._
+
+
+=1206.= 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.--SYLVESTER, J. J.
+
+ _A Probationary Lecture on Geometry;
+ Collected Mathematical Papers, Vol. 2,
+ p. 7._
+
+
+=1207.= 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.
+
+ --VENN, JOHN.
+
+ _Symbolic Logic (London and New York,
+ 1894), p. 74._
+
+
+=1208.= 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....
+
+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.--POINCARE, H.
+
+ _The Value of Science [Halsted] Popular
+ Science Monthly, 1906, pp. 195-196._
+
+
+=1209.= The most striking characteristic of the written language
+of algebra and of the higher forms of the calculus is the
+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.--HILL, THOMAS.
+
+ _Uses of Mathesis; Bibliotheca Sacra,
+ Vol. 32, p. 505._
+
+
+=1210.= 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.
+
+ --HILL, THOMAS.
+
+ _North American Review, Vol. 85, pp.
+ 224-225._
+
+
+=1211.= 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.--PRINGSHEIM, A.
+
+ _Jahresberichte der Deutschen
+ Mathematiker Vereinigung, Bd. 13, p.
+ 367._
+
+
+=1212.= The results of systematic symbolical reasoning must
+_always_ 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 _absolute
+universality of the interpretation of symbols_ is the fundamental
+principle of their use.--WHEWELL, WILLIAM.
+
+ _The Philosophy of the Inductive
+ Sciences, Part I, Bk. 2, chap. 12, sect.
+ 2 (London, 1858)._
+
+
+=1213.= Anyone who understands algebraic notation, reads at a
+glance in an equation results reached arithmetically only with
+great labour and pains.--COURNOT, A.
+
+ _Theory of Wealth [N. T. Bacon], (New
+ York, 1897), p. 4._
+
+
+=1214.= 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.--BERKELEY, GEORGE.
+
+ _Alciphron, or the Minute Philosopher,
+ Dialogue 7, sect. 12._
+
+
+=1215.= 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 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 Moebius'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.--GAUSS, C. J.
+
+ _Werke, Bd. 8, p. 298._
+
+
+=1216.= 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.
+
+ --DE MORGAN, A.
+
+ _Calculus of Functions; Encyclopedia
+ Metropolitana, Addition to Article 26._
+
+
+=1217.= 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
+with decimal fractions is the most miraculous result of a perfect
+notation.--WHITEHEAD, A. N.
+
+ _Introduction to Mathematics (New York,
+ 1911), p. 59._
+
+
+=1218.= 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.
+
+ --WHITEHEAD, A. N.
+
+ _Introduction to Mathematics (New York,
+ 1911), pp. 59-60._
+
+
+=1219.= 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.--RUSSELL, BERTRAND.
+
+ _International Monthly, 1901, p. 85._
+
+
+=1220.= 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.--COURNOT, A.
+
+ _Theory of Wealth [N. T. Bacon], (New
+ York, 1897), pp. 3-4._
+
+
+=1221.= An all-inclusive geometrical symbolism, such as Hamilton
+and Grassmann conceived of, is impossible.--BURKHARDT, H.
+
+ _Jahresbericht der Deutschen
+ Mathematiker Vereinigung, Bd. 5, p. 52._
+
+
+=1222.= 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.--LAPLACE.
+
+ _Oeuvres, t. 7 (Paris, 1896), p. xl._
+
+
+
+
+ CHAPTER XIII
+
+ MATHEMATICS AND LOGIC
+
+
+=1301.= 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.--PEIRCE, BENJAMIN.
+
+ _Linear Associative Algebra; American
+ Journal of Mathematics, Vol. 4 (1881),
+ p. 97._
+
+
+=1302.= 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.
+
+ --HELMHOLTZ, H.
+
+ _Vortraege und Reden, Bd. 1
+ (Braunschweig, 1896), p. 176._
+
+
+=1303.= 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,
+imagining those, who had not, men of imperfect judgments, and
+unfit to rule and govern.--FRANKLIN, BENJAMIN.
+
+ _Usefulness of Mathematics; Works
+ (Boston, 1840), Vol. 2, p. 68._
+
+
+=1304.= The mathematical conception is, from its very nature,
+abstract; indeed its abstractness is usually of a higher order
+than the abstractness of the logician.--CHRYSTAL, GEORGE.
+
+ _Encyclopedia Britannica (Ninth
+ Edition), Article "Mathematics."_
+
+
+=1305.= Mathematics, that giant pincers of scientific logic....
+
+ --HALSTED, G. B.
+
+ _Science (1905), p. 161._
+
+
+=1306.= 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.
+
+ --PASCAL.
+
+ _Quoted by A. Rebiere: Mathematiques et
+ Mathematiciens (Paris, 1898), pp.
+ 162-163._
+
+
+=1307.= 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.--GOETHE.
+
+ _Sprueche in Prosa, Natur IV, 946._
+
+
+=1308.= 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 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.--COMTE, A.
+
+ _Subjective Synthesis._
+
+
+=1309.= 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.--LEFEVRE, ARTHUR.
+
+ _Number and its Algebra (Boston, Sect.
+ 222.)_
+
+
+=1310.= 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.--BACON, FRANCIS.
+
+ _De Augmentis, Bk. 3._
+
+
+=1311.= 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.--D'ALEMBERT.
+
+ _Quoted in A. Rebiere: Mathematiques et
+ Mathematiciens (Paris, 1898), pp.
+ 151-152._
+
+
+=1312.= 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.--LEIBNITZ, G. W.
+
+ _New Essay on Human Understanding
+ [Langley], Bk. 4, chap. 2, sect. 12._
+
+
+=1313.= 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.--BOCHER, MAXIME.
+
+ _Bulletin of the American Mathematical
+ Society, Vol. 11, p. 120._
+
+
+=1314.= 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau],
+ (London, 1875), Vol. 1, pp. 321-322._
+
+
+=1315.= 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.
+
+ --DE MORGAN, A.
+
+ _Quoted in F. Cajori: History of
+ Mathematics (New York, 1897), p. 316._
+
+
+=1316.= 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.--LEIBNITZ, G. W.
+
+ _Philosophische Schriften [Gerhardt] Bd.
+ 8, p. 198._
+
+
+=1317.= 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 _calculations_, words
+of vague and uncertain meaning in favor of fixed symbols." Thus
+it will appear that "every paralogism is nothing but _an error of
+calculation_." "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 their counting-tables, and (having
+summoned a friend, if they like) say to one another: _Let us
+calculate_."--LATTA, ROBERT.
+
+ _Leibnitz, The Monadology, etc. (Oxford,
+ 1898), p. 85._
+
+
+=1318.= 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.
+
+ --RUSSELL, BERTRAND.
+
+ _International Monthly, 1901, p. 83._
+
+
+=1319.= Mathematics is but the higher development of Symbolic
+Logic.--WHETHAM, W. C. D.
+
+ _Recent Development of Physical Science
+ (Philadelphia, 1904), p. 34._
+
+
+=1320.= 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.
+
+ --WHITEHEAD, A. N.
+
+ _Universal Algebra (Cambridge, 1898),
+ Preface, p. 6._
+
+
+=1321.= ... 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.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 19._
+
+
+=1322.= 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.--KEYSER, C. J.
+
+ _Science, Vol. 35 (1912), p. 108._
+
+
+=1323.= I would express it as my personal view, which is probably
+not yet shared generally, that pure mathematics seems to me
+merely a _branch of general logic_; that branch which is based on
+the concept of _numbers_, 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.--SCHROeDER, E.
+
+ _Ueber Pasigraphie etc.; Verhandlungen
+ des 1. Internationalen
+ Mathematiker-Kongresses (Leipzig, 1898),
+ p. 149._
+
+
+=1324.= 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.--FISKE, JOHN.
+
+ _Darwinism and other Essays (Boston,
+ 1893), pp. 297-298._
+
+
+=1325.= 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 may in
+fact be regarded as an immense repository of logical resources,
+ready for use in scientific deduction and co-ordination.
+
+ --COMTE, A.
+
+ _Positive Philosophy [Martineau],
+ (London, 1875), Vol. 2, p. 439._
+
+
+=1326.= 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.--KEYSER, C. J.
+
+ _Science, Vol. 35 (1912), p. 110._
+
+
+
+
+ CHAPTER XIV
+
+ MATHEMATICS AND PHILOSOPHY
+
+
+=1401.= 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.--HERBART, J. F.
+
+ _Werke [Kehrbach], (Langensalza, 1890),
+ Bd. 5, p. 95._
+
+
+=1402.= It is the embarrassment of metaphysics that it is able to
+accomplish so little with the many things that mathematics offers
+her.--KANT, E.
+
+ _Metaphysische Anfangsgruende der
+ Naturwissenschaft, Vorrede._
+
+
+=1403.= 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.--WHITEHEAD, A. N.
+
+ _Introduction to Mathematics (New York,
+ 1911), p. 113._
+
+
+=1404.= 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 illustrious countryman
+discovered the infinitesimal calculus--and just these are the
+four greatest steps in the development of mathematics.
+
+ --HANKEL, HERMANN.
+
+ _Geschichte der Mathematik im Altertum
+ und im Mittelalter (Leipzig, 1874), pp.
+ 149-150._
+
+
+=1405.= 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.
+
+ --BORDAS-DEMOULINS.
+
+ _Quoted in A. Rebiere: Mathematiques et
+ Mathematiciens (Paris, 1898), p. 147._
+
+
+=1406.= In the end mathematics is but simple philosophy, and
+philosophy, higher mathematics in general.--NOVALIS.
+
+ _Schriften (Berlin, 1901), Teil 2, p.
+ 443._
+
+
+=1407.= It is a safe rule to apply that, when a mathematical or
+philosophical author writes with a misty profundity, he is
+talking nonsense.--WHITEHEAD, A. N.
+
+ _Introduction to Mathematics (New York,
+ 1911), p. 227._
+
+
+=1408.= The real finisher of our education is philosophy, but it
+is the office of mathematics to ward off the dangers of
+philosophy.--HERBART, J. F.
+
+ _Pestalozzi's Idee eines ABC der
+ Anschauung; Werke [Kehrbach],
+ (Langensalza, 1890), Bd. 1, p. 168._
+
+
+=1409.= 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.--CURTIUS, E.
+
+ _Berliner Monatsberichte (1873), p.
+ 517._
+
+
+=1410.= Geometry has been, throughout, of supreme importance in
+the history of knowledge.--RUSSELL, BERTRAND.
+
+ _Foundations of Geometry (Cambridge,
+ 1897), p. 54._
+
+
+=1411.= 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.--PLATO.
+
+ _Quoted by Sophie Germain: Memoire sur
+ les surfaces elastiques._
+
+
+=1412.= 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 zoologist when he gave names to the beasts of the field,
+nor were the Egyptian surveyors mathematicians.--HANKEL, H.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1884),
+ p. 7._
+
+
+=1413.= There are only two ways open to man for attaining a
+certain knowledge of truth: clear intuition and necessary
+deduction.--DESCARTES.
+
+ _Rules for the Direction of the Mind;
+ Torrey's The Philosophy of Descartes
+ (New York, 1892), p. 104._
+
+
+=1414.= 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.--REID, THOMAS.
+
+ _Essay on the Intellectual Powers of
+ Man, Essay 4, chap. 3._
+
+
+=1415.= 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 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.--HERBART, J. F.
+
+ _Werke Kehrbach (Langensalza, 1890), Bd.
+ 5, p. 105._
+
+
+=1416.= 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_{3}T_{1}, and I_{2}T_{1}, occur; but I_{1}T_{3} is more common,
+and I_{2}T_{15} and I_{1}T_{20} occur. In many cases metaphysics
+(M) occurs and I hold that I_{a}T_{b}M_{c} never occurs without b
++ c > 2a.
+
+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_{1}T_{3}M_{3} may easily be confounded with
+I_{1}T_{6}. As far as I dare say anything, those who have placed
+_Hegel, Fichte_, etc., in the rank of the extenders of _Kant_
+have imagined T and M to be identical.--DE MORGAN, A.
+
+ _Graves' Life of W. R. Hamilton (New
+ York, 1882-1889), Vol. 13, p. 446._
+
+
+=1417.= 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.--BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p. 458._
+
+
+=1418.=
+
+ But who shall parcel out
+ His intellect by geometric rules,
+ Split like a province into round and square?
+ --WORDSWORTH.
+
+ _The Prelude, Bk. 2._
+
+
+=1419.=
+
+ And Proposition, gentle maid,
+ Who soothly ask'd stern Demonstration's aid, ....
+ --COLERIDGE, S. T.
+
+ _A Mathematical Problem._
+
+
+=1420.= 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.--CAYLEY, ARTHUR.
+
+ _British Association Address (1888);
+ Collected Mathematical Papers, Vol. 11,
+ p. 430._
+
+
+=1421.= 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.--MILL, J. S.
+
+ _An Examination of Sir William
+ Hamilton's Philosophy (London, 1878), p.
+ 615._
+
+
+=1422.= 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 training an
+indispensable basis of real scientific education, and regarding,
+with Plato, one who is [Greek: ageometretos], as wanting in one of
+the most essential qualifications for the successful cultivation
+of the higher branches of philosophy.--MILL, J. S.
+
+ _System of Logic, Bk. 3, chap. 24, sect.
+ 9._
+
+
+=1423.= 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.
+
+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.--REID, THOMAS.
+
+ _Essays on the Powers of the Human Mind
+ (Edinburgh, 1812), Vol. 2, pp. 422-423._
+
+
+=1424.= 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.--STEWART, DUGALD.
+
+ _Philosophy of the Human Mind; Collected
+ Works (Edinburgh, 1854), Vol. 8, p. 5._
+
+
+=1425.= 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 suspected that they had
+knowledge of a mathematical science different from that of our
+time....
+
+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.--DESCARTES.
+
+ _Rules for the Direction of the Mind;
+ Philosophy of Descartes [Torrey], (New
+ York, 1892), pp. 70-71._
+
+
+=1426.= 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.
+
+ --DESCARTES.
+
+ _Rules for the Direction of the Mind;
+ Philosophy of Descartes [Torrey], (New
+ York, 1892), p. 62._
+
+
+=1427.= 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 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.
+
+ --MACAULAY.
+
+ _Lord Bacon; Edinburgh Review, July,
+ 1837._
+
+
+=1428.= 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
+_attention_, but not in the same manner, nor in the same degree.
+
+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 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.--STEWART, DUGALD.
+
+ _Philosophy of the Human Mind, Part 3,
+ chap. 1, sect. 3._
+
+
+=1429.= 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.
+
+ --WHEWELL, W.
+
+ _History of the Inductive Sciences, Vol.
+ 1, bk. 2, chap. 3._
+
+
+=1430.= For to pass by those Ancients, the wonderful _Pythagoras_,
+the sagacious _Democritus_, the divine _Plato_, the most subtle
+and very learned _Aristotle_, 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 _Aristotle's_
+Instrument but with a Mathematical Quill; or not be altogether
+deaf to the Lessons of natural _Philosophy_, while ignorant of
+_Geometry?_ Who void of (_Geometry_ shall I say, or) _Arithmetic_
+can comprehend _Plato's_ _Socrates_ lisping with Children
+concerning Square Numbers; or can conceive _Plato_ 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?--BARROW, ISAAC.
+
+ _Mathematical Lectures (London, 1734),
+ pp. 26-27._
+
+
+=1431.=
+
+ And Reason now through number, time, and space
+ Darts the keen lustre of her serious eye;
+ And learns from facts compar'd the laws to trace
+ Whose long procession leads to Deity
+ --BEATTIE, JAMES.
+
+ _The Minstrel, Bk. 2, stanza 47._
+
+
+=1432.= That Egyptian and Chaldean wisdom mathematical wherewith
+Moses and Daniel were furnished, ....--HOOKER, RICHARD.
+
+ _Ecclesiastical Polity, Bk. 3, sect. 8._
+
+
+=1433.= General and certain truths are only founded in the
+habitudes and relations of _abstract ideas_. 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, they would carry our thoughts
+further, and with greater evidence and clearness than possibly we
+are apt to imagine.--LOCKE, JOHN.
+
+ _An Essay concerning Human
+ Understanding, Bk. 4, chap. 12, sect.
+ 7._
+
+
+=1434.= 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.--DESCARTES.
+
+ _Discourse upon Method, part 2; The
+ Philosophy of Descartes [Torrey], (New
+ York, 1892), p. 47._
+
+
+=1435.= 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: _Quis leget haec_, and reply: _Vel duo vel nemo._
+
+ --LEIBNITZ.
+
+ _New Essay concerning Human
+ Understanding, Langley, Bk 2, chap. 29,
+ sect. 12._
+
+
+=1436.= It is commonly asserted that mathematics and philosophy
+differ from one another according to their _objects_, the former
+treating of _quantity_, the latter of _quality_. All this is
+false. The difference between these sciences cannot depend on
+their object; for philosophy applies to everything, hence also to
+_quanta_, and so does mathematics in part, inasmuch as everything
+has magnitude. It is only the _different kind of rational
+knowledge or application_ of reason in mathematics and philosophy
+which constitutes the specific difference between these two
+sciences. For philosophy is _rational knowledge from mere
+concepts_, mathematics, on the contrary, is _rational knowledge
+from the construction of concepts_.
+
+We construct concepts when we represent them in intuition _a
+priori_, 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 _in concreto_, however, the intuition is
+not empirical, but the object of contemplation is something _a
+priori_.
+
+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 _discursive_. But the reason why in mathematics we
+deal more with quantity lies in this, that magnitudes can be
+constructed in intuition _a priori_, while qualities, on the
+contrary, do not permit of being represented in intuition.--KANT, E.
+
+ _Logik; Werke [Hartenstein], (Leipzig,
+ 1868), Bd. 8, pp. 23-24._
+
+
+=1437.= 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.
+
+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.
+
+ --COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 1,
+ chap. 1._
+
+
+=1438.= 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.--HUME, DAVID.
+
+ _An Inquiry concerning Human
+ Understanding, sect. 7, part 1._
+
+
+=1439.= 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.--LOCKE, JOHN.
+
+ _An Essay concerning Human
+ Understanding, Bk. 4, chap. 3, sect.
+ 20._
+
+
+=1440.= 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,
+
+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.
+
+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 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.
+
+ --LOCKE, JOHN.
+
+ _An Essay concerning Human
+ Understanding, Bk. 4, chap. 3, sect.
+ 19._
+
+
+=1441.= 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.--LOCKE, JOHN.
+
+ _An Essay concerning Human
+ Understanding, Bk. 4, chap. 2, sect. 9._
+
+
+=1442.= 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.
+
+ --LEIBNITZ.
+
+ _New Essay concerning Human
+ Understanding, Bk. 4, chap. 2, sect. 9
+ [Langley]._
+
+
+
+
+ CHAPTER XV
+
+ MATHEMATICS AND SCIENCE
+
+
+=1501.= 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.--DESCARTES.
+
+ _Rules for the Direction of the Mind;
+ Philosophy of Descartes [Torrey], (New
+ York, 1892), p. 72._
+
+
+=1502.= 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.--HERBART, J. F.
+
+ _Werke (Kehrbach), (Langensalza, 1890),
+ Bd. 5, p. 106._
+
+
+=1503.= 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.--ARBUTHNOT.
+
+ _Quoted in Todhunter's History of the
+ Theory of Probability (Cambridge and
+ London, 1865), p. 51._
+
+
+=1504.= Mathematical Analysis is ... the true rational basis of
+the whole system of our positive knowledge.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 1,
+ chap. 1._
+
+
+=1505.= 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 1,
+ chap. 1._
+
+
+=1506.= 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau],
+ Introduction, chap. 2._
+
+
+=1507.= The concept of mathematics is the concept of science in
+general.--NOVALIS.
+
+ _Schriften (Berlin, 1901), Teil 2, p.
+ 222._
+
+
+=1508.= 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 _a
+priori_ knowledge, each natural science becomes real science only
+to the extent that it permits the application of mathematics.
+
+ --KANT, E.
+
+ _Metaphysische Anfangsgruende der
+ Naturwissenschaft, Vorrede._
+
+
+=1509.= 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.--SAFFORD, T. H.
+
+ _Mathematical Teaching etc. (Boston,
+ 1886), p. 9._
+
+
+=1510.= 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.--DAVIS, E. W.
+
+ _Proceedings Nebraska Academy of
+ Sciences for 1896 (Lincoln, 1897), p.
+ 282._
+
+
+=1511.= 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.--BACON, FRANCIS.
+
+ _Advancement of Learning, Bk. 2; De
+ Augmentis, Bk. 3._
+
+
+=1512.= 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.--GRASSMANN, H.
+
+ _Stuecke aus dem Lehrbuche der
+ Arithmetik; Werke (Leipzig, 1904), Bd.
+ 2, p. 298._
+
+
+=1513.= 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.--BOYLE, ROBERT.
+
+ _Works (London, 1772), Vol. 3, p. 429._
+
+
+=1514.= 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.--QUETELET.
+
+ _Quoted in E. Mailly's Eulogy on
+ Quetelet; Smithsonian Report, 1874, p.
+ 173._
+
+
+=1515.= 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.--MERZ, J. T.
+
+ _History of European Thought in the 19th
+ Century (Edinburgh and London, 1904),
+ Vol. 1, p. 333._
+
+
+=1516.= 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 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.--FOSTER, G. C.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A (1877); Nature, Vol.
+ 16, p. 312-313._
+
+
+=1517.= 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.--VENN, JOHN.
+
+ _Symbolic Logic (London and New York,
+ 1894), Introduction, p. xix._
+
+
+=1518.= 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 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.--HOBSON, E. W.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A (1910); Nature, Vol.
+ 84, pp. 285-286._
+
+
+=1519.= 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.
+
+ --MILL, J. S.
+
+ _System of Logic, Bk. 3, chap. 24, sect.
+ 9._
+
+
+=1520.= 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
+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.--KLEIN, F.
+
+ _Lectures on Mathematics (New York,
+ 1911), p. 47._
+
+
+=1521.= 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.--SMITH, H. J. S.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A (1873); Nature, Vol.
+ 8, p. 449._
+
+
+=1522.= 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 _disjecta membra_--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, 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.--FOSTER, G. C.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A (1877); Nature, Vol.
+ 16, p. 313._
+
+
+=1523.= In _Plato's_ 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
+_only_ 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 _only_ 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.--LINDEMAN, F.
+
+ _Lehren und Lernen in der Mathematik
+ (Muenchen, 1904), p. 14._
+
+
+=1524.= 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
+beyond the limits they were originally invented for,--thus
+opening out new and larger regions of thought.--MERZ, J. T.
+
+ _History of European Thought in the 19th
+ Century (Edinburgh and London, 1903),
+ Vol. 1, p. 698._
+
+
+=1525.= All the effects of nature are only mathematical results
+of a small number of immutable laws.--LAPLACE.
+
+ _A Philosophical Essay on Probabilities
+ [Truscott and Emory] (New York, 1902),
+ p. 177; Oeuvres, t. 7, p. 139._
+
+
+=1526.= What logarithms are to mathematics that mathematics are
+to the other sciences.--NOVALIS.
+
+ _Schriften (Berlin, 1901), Teil 2, p.
+ 222._
+
+
+=1527.= 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.
+
+ --MACAULAY.
+
+ _Milton; Critical and Miscellaneous
+ Essays (New York, 1879), Vol. 1, p. 13._
+
+
+=1528.= In questions of science the authority of a thousand is
+not worth the humble reasoning of a single individual.--GALILEO.
+
+ _Quoted in Arago's Eulogy on Laplace;
+ Smithsonian Report, 1874, p. 164._
+
+
+=1529.= Behind the artisan is the chemist, behind the chemist a
+physicist, behind the physicist a mathematician.--WHITE, W. F.
+
+ _Scrap-book of Elementary Mathematics
+ (Chicago, 1908), p. 217._
+
+
+=1530.= 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.--BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p. 503._
+
+
+=1531.= 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. 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.--FOSTER, G. C.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A (1877); Nature, Vol.
+ 16, p. 313._
+
+
+=1532.=
+
+ Give me to learn each secret cause;
+ Let Number's, Figure's, Motion's laws
+ Reveal'd before me stand;
+ These to great Nature's scenes apply,
+ And round the globe, and through the sky,
+ Disclose her working hand.
+ --AKENSIDE, M.
+
+ _Hymn to Science._
+
+
+=1533.= 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, and judge of them without easily
+acquiescing in anything but demonstration.--BOYLE, ROBERT.
+
+ _Works (London, 1772), Vol. 3, p. 426._
+
+
+=1534.= 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.--WHEWELL, W.
+
+ _History of the Inductive Sciences (New
+ York, 1894), Vol. 1, p. 416._
+
+
+=1535.= 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 use 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 3,
+ chap. 1._
+
+
+=1536.= 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 3,
+ chap. 1._
+
+
+=1537.= 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 _mathematical_. 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.--DE MORGAN, A.
+
+ _A Budget of Paradoxes (London, 1872),
+ p. 2._
+
+
+=1538.= 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.--DE MORGAN, A.
+
+ _Graves' Life of Sir William Rowan
+ Hamilton (New York, 1882-1889), Vol. 3,
+ p. 348._
+
+
+=1539.= 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
+forging of the weapon, he may find himself, sooner or later in
+the progress of his subject, without any weapon worth having.
+
+ --FORSYTH, A. R.
+
+ _Perry's Teaching of Mathematics
+ (London, 1902), p. 36._
+
+
+=1540.= 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.--FORSYTH, A. R.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A; Nature, Vol. 56
+ (1897), p. 377._
+
+
+=1541.= 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.--WHEWELL, W.
+
+ _History of the Inductive Science (New
+ York, 1894), Vol. 1, p. 311._
+
+
+=1542.= 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.
+
+ --SMITH, H. J. S.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A; Nature, Vol. 8
+ (1873), p. 450._
+
+
+=1543.= The silent work of the great Regiomontanus in his chamber
+at Nuremberg computed the Ephemerides which made possible the
+discovery of America by Columbus.--RUDIO, F.
+
+ _Quoted in Max Simon's Geschichte der
+ Mathematik im Altertum (Berlin, 1909),
+ Einleitung, p. xi._
+
+
+=1544.= 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.--WHATELY, R.
+
+ _Annotations to Bacon's Essays (Boston,
+ 1783), p. 492._
+
+
+=1545.= 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!...
+
+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.--HELMHOLTZ, H.
+
+ _Vortraege und Reden (Braunschweig,
+ 1884), Bd. 1, p. 142._
+
+
+=1546.= When the time comes that knowledge will not be sought for
+its own sake, and men will not press forward simply 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.--HUGHES, C. E.
+
+ _Quoted in D. E. Smith's Teaching of
+ Geometry (Boston, 1911), p. 9._
+
+
+=1547.= [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 (_intuitu
+intellectus_). "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."--WHEWELL, W.
+
+ _History of the Inductive Sciences (New
+ York, 1894), Vol. 1, p. 519. Bacon,
+ Roger: Opus Majus, Part 4, Distinctia
+ Prima, cap. 3._
+
+
+=1548.= 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 extremely
+heterogeneous, phenomena to a minimum of simple fundamental laws.
+
+ --PRINGSHEIM, A.
+
+ _Jahresbericht der Deutschen
+ Mathematiker Vereinigung, Bd. 13, p.
+ 366._
+
+
+=1549.= "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.]
+
+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.--WEBER, HEINRICH.
+
+ _Die partiellen Differentialgleichungen
+ der mathematischen Physik (Braunschweig,
+ 1882), Bd. 1, Vorrede._
+
+
+=1550.= 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.--PRICE, B.
+
+ _Treatise on Infinitesimal Calculus
+ (Oxford, 1858), Vol. 3, p. 5._
+
+
+=1551.= 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
+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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 3,
+ chap. 4._
+
+
+=1552.= 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 phenomena; and
+they connect forever with mathematical science one of the most
+important branches of natural philosophy.--FOURIER, J.
+
+ _Theory of Heat [Freeman], (Cambridge,
+ 1878), Chap. 3, p. 131._
+
+
+=1553.= 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.--FOURIER, J.
+
+ _Theory of Heat [Freeman], (Cambridge,
+ 1878), Chap. 1, p. 12._
+
+
+=1554.= 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
+_discovered_ a planet before unknown.--FARADAY.
+
+ _Some Thoughts on the Conservation of
+ Force._
+
+
+=1555.= 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....
+
+ --STEWART, DUGALD.
+
+ _Philosophy of the Human Mind, Part 3,
+ chap. 1, sect. 3._
+
+
+=1556.= 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 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.--SMITH, H. J. S.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A; Nature, Vol. 8
+ (1873), p. 450._
+
+
+=1557.= 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.--SMITH, H. J. S.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science, Section A; Nature, Vol. 8
+ (1873), p. 449._
+
+
+=1558.= 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 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.
+
+ --CLIFFORD, W. K.
+
+ _Lectures and Essays (New York, 1901),
+ Vol. 1, p. 144._
+
+
+=1559.= 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: "_Direct your telescope to a point on the ecliptic in
+the constellation of Aquarius, in longitude 326 deg., 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._" 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.
+
+ --YOUNG, C. A.
+
+ _General Astronomy (Boston, 1891), Art.
+ 653._
+
+
+=1560.= I am convinced that the future progress of chemistry as
+an exact science depends very much indeed upon the alliance with
+mathematics.--FRANKLAND, A.
+
+ _American Journal of Mathematics, Vol.
+ 1, p. 349._
+
+
+=1561.= It is almost impossible to follow the later developments
+of physical or general chemistry without a working knowledge of
+higher mathematics.--MELLOR, J. W.
+
+ _Higher Mathematics (New York, 1902),
+ Preface._
+
+
+=1562.=
+
+ ... Mount where science guides;
+ Go measure earth, weigh air, and state the tides;
+ Instruct the planets in what orb to run,
+ Correct old time, and regulate the sun.
+ --THOMSON, W.
+
+ _On the Figure of the Earth, Title
+ page._
+
+
+=1563.= Admission to its sanctuary [referring to astronomy] and
+to the privileges and feelings of a votary, is only to be gained
+by one means,--_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._
+
+ --HERSCHEL, J.
+
+ _Outlines of Astronomy, Introduction,
+ sect. 7._
+
+
+=1564.= 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, and to reduce it to the condition of
+an ordinary province of natural history.
+
+ _Edinburgh Review, Vol. 58 (1833-1834),
+ p. 168._
+
+
+=1565.= 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:....
+
+ _Edinburgh Review, Vol. 58 (1833-1834),
+ p. 169._
+
+
+=1566.= 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.
+
+ _Edinburgh Review, Vol. 58 (1833-1834),
+ p. 170._
+
+
+=1567.= 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 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.--HUMBOLDT, A.
+
+ _Cosmos [Otte], Vol. 2, part 2, sect.
+ 3._
+
+
+=1568.= Mighty are numbers, joined with art resistless.--EURIPIDES.
+
+ _Hecuba, Line 884._
+
+
+=1569.= 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.--PLATO.
+
+ _Laws [Jowett,] Bk. 5, p. 747._
+
+
+=1570.= 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 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.--SPENCER, HERBERT.
+
+ _Education, chap. 1._
+
+
+=1571.= [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.--EVERETT, EDWARD.
+
+ _Orations and Speeches (Boston, 1870),
+ Vol. 3, p. 47._
+
+
+=1572.= [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 _Mathematics_, 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 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 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.--BARROW, ISAAC.
+
+ _Mathematical Lectures (London, 1734),
+ pp. 27-30._
+
+
+=1573.= 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.--KARPINSKY, L.
+
+ _High School Education (New York, 1912),
+ chap. 6, p. 134._
+
+
+=1574.= 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.
+
+That is the fact he replied.
+
+But the guardian of ours happens to be both a military man and a
+philosopher.
+
+Unquestionably so.
+
+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, 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.--PLATO.
+
+ _Republic [Davis], Bk. 7, p. 525._
+
+
+=1575.= 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.--WILLIAMSON, JAMES.
+
+ _Elements of Euclid with Dissertations
+ (Oxford, 1781)._
+
+
+=1576.= _Where there is nothing to measure there is nothing to
+calculate_, 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 _tried_ eccentric circles, and Kepler _tried_ 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 _tried_ 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 _fundamental measures_ are in themselves
+exceedingly difficult, but fortunately, such investigations form
+a class of their own; our knowledge of _fundamental laws_ 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.
+
+ --HERBART, J. F.
+
+ _Werke [Kehrbach], (Langensalza, 1890),
+ Bd. 5, p. 97._
+
+
+=1577.= 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.--BOYLE, ROBERT.
+
+ _Works (London, 1772), Vol. 3, p. 426._
+
+
+=1578.= 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 5,
+ chap. 1._
+
+
+=1579.= In mathematics we find the primitive source of
+rationality; and to mathematics must the biologists resort for
+means to carry on their researches.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 5,
+ chap. 1._
+
+
+=1580.= 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.
+
+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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 5,
+ chap. 1._
+
+
+=1581.= 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 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 5,
+ chap. 1._
+
+
+=1582.= 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.--BOYLE, ROBERT.
+
+ _Works (London, 1772), Vol. 3, p. 430._
+
+
+=1583.= 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.--HERBART, J. F.
+
+ _Werke [Kehrbach], (Langensalza, 1890),
+ Bd. 5, p. 104._
+
+
+=1584.= All more definite knowledge must start with computation;
+and this is of most important consequences not only for the
+theory of memory, of imagination, of understanding, but as well
+for the doctrine of sensations, of desires, and affections.
+
+ --HERBART, J. F.
+
+ _Werke [Kehrbach], (Langensalza, 1890),
+ Bd. 5, p. 103._
+
+
+=1585.= 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.--DESSOIR, MAX.
+
+ _Westermann's Monatsberichte, Bd. 77, p.
+ 380; Ahrens: Scherz und Ernst in der
+ Mathematik (Leipzig, 1904), p. 395._
+
+
+=1586.= The social sciences mathematically developed are to be
+the controlling factors in civilization.--WHITE, W. F.
+
+ _A Scrap-book of Elementary Mathematics
+ (Chicago, 1908), p. 208._
+
+
+=1587.= 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 6,
+ chap. 4._
+
+
+=1588.= 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
+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.--WILSON, E. B.
+
+ _Bulletin American Mathematical Society,
+ Vol. 18 (1912), p. 463._
+
+
+=1589.= 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 role 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.--WOODWARD, R. S.
+
+ _Probability and Theory of Errors (New
+ York, 1906), Preface._
+
+
+=1590.= 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 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.--CROFTON, M. W.
+
+ _Encyclopedia Britannica, 9th Edition;
+ Article "Probability."_
+
+
+=1591.= 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.--ARAGO.
+
+ _Eulogy on Laplace [Baden-Powell],
+ Smithsonian Report, 1874, p. 164._
+
+
+=1592.= 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.--HERSCHEL, J.
+
+ _Quoted in Encyclopedia Britannica, 9th
+ Edition; Article "Probability."_
+
+
+=1593.= 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, ....--WILSON, E. B.
+
+ _Bulletin American Mathematical Society,
+ Vol. 18 (1912), p. 464._
+
+
+=1594.= The effort of the economist is to _see_, 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.--FISHER, IRVING.
+
+ _Transactions of Connecticut Academy,
+ Vol. 9 (1892), p. 119._
+
+
+=1595.= 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.
+
+ --MILL, J. S.
+
+ _An Examination of Sir William
+ Hamilton's Philosophy (London, 1878), p.
+ 622._
+
+
+=1596.= Let me pass on to say a word or two about the teaching of
+mathematics as an academic training for general professional
+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.--SHAW, W. H.
+
+ _Perry's Teaching of Mathematics
+ (London, 1902), p. 73._
+
+
+=1597.= 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.--JEFFERSON, THOMAS.
+
+ _Quoted in Cajori's Teaching and History
+ of Mathematics in the U. S. (Washington,
+ 1890), p. 35._
+
+
+=1598.= 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.--EVERETT, EDWARD.
+
+ _Orations and Speeches (Boston, 1870),
+ Vol. 2, p. 511._
+
+
+=1599.= All historic science tends to become mathematical.
+Mathematical power is classifying power.--NOVALIS.
+
+ _Schriften (Berlin, 1901), Teil 2, p.
+ 192._
+
+
+=1599a.= 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.--ADAM, HENRY.
+
+ _A Letter to American Teachers of
+ History (Washington, 1910), p. 115._
+
+
+=1599b.= Mathematics can be shown to sustain a certain relation
+to rhetoric and may aid in determining its laws.--SHERMAN L. A.
+
+ _University [of Nebraska] Studies, Vol.
+ 1, p. 130._
+
+
+
+
+ CHAPTER XVI
+
+ ARITHMETIC
+
+
+=1601.= 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.
+
+ --MACH, ERNST.
+
+ _Popular Scientific Lectures [McCormack]
+ (Chicago, 1898), p. 197._
+
+
+=1602.= 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 1,
+ chap. 1._
+
+
+=1603.= 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.--CARUS, PAUL.
+
+ _Reflections on Magic Squares; Monist,
+ Vol. 16 (1906), p. 139._
+
+
+=1604.= An ancient writer said that arithmetic and geometry are
+the _wings of mathematics_; 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.--LAGRANGE.
+
+ _Lecons Elementaires sur les
+ Mathematiques, Lecon seconde._
+
+
+=1605.= 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.--DILLMANN, E.
+
+ _Die Mathematik die Fackeltraegerin einer
+ neuen Zeit (Stuttgart, 1889), p. 12._
+
+
+=1606.=
+
+ Number, the inducer of philosophies,
+ The synthesis of letters, ....
+ --AESCHYLUS.
+
+ _Quoted in, Thomson, J. A., Introduction
+ to Science, chap. 1 (London)._
+
+
+=1607.= 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 _unity_, 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, _the most
+universal idea we have_.--LOCKE, JOHN.
+
+ _An Essay concerning Human
+ Understanding, Bk. 2, chap. 16, sect.
+ 1._
+
+
+=1608.= The _simple modes_ of _number_ 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.--LOCKE, JOHN.
+
+ _An Essay concerning Human
+ Understanding, Bk. 2, chap. 16, sect.
+ 3._
+
+
+=1609.= The number of a class is the class of all classes similar
+to the given class.--RUSSELL, BERTRAND.
+
+ _Principles of Mathematics (Cambridge,
+ 1903), p. 115._
+
+
+=1610.= Number is that property of a group of distinct things
+which remains unchanged during any change to which the group may
+be subjected which does not destroy the distinctness of the
+individual things.--FINE, H. B.
+
+ _Number-system of Algebra (Boston and
+ New York, 1890), p. 3._
+
+
+=1611.= The science of arithmetic may be called the science of
+exact limitation of matter and things in space, force, and time.
+
+ --PARKER, F. W.
+
+ _Talks on Pedagogics (New York, 1894),
+ p. 64._
+
+
+=1612.=
+
+ Arithmetic is the science of the Evaluation
+ of Functions,
+ Algebra is the science of the Transformation
+ of Functions.
+ --HOWISON, G. H.
+
+ _Journal of Speculative Philosophy, Vol.
+ 5, p. 175._
+
+
+=1613.= That _arithmetic_ rests on pure intuition of _time_ 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 _time_, it is intelligible only by means
+of this latter concept: hence counting is possible only by means
+of time.--This dependence of counting on _time_ is evidenced by
+the fact that in all languages multiplication is expressed by
+"times" [mal], that is, by a concept of time; sexies, [Greek:
+hexakis], six fois, six times.--SCHOPENHAUER, A.
+
+ _Die Welt als Vorstellung und Wille;
+ Werke (Frauenstaedt) (Leipzig, 1877),
+ Bd. 3, p. 39._
+
+
+=1614.= The miraculous powers of modern calculation are due to
+three inventions: the Arabic Notation, Decimal Fractions and
+Logarithms.--CAJORI, F.
+
+ _History of Mathematics (New York,
+ 1897), p. 161._
+
+
+=1615.= The grandest achievement of the Hindoos and the one
+which, of all mathematical investigations, has contributed most
+to the general progress of intelligence, is the invention of the
+principle of position in writing numbers.--CAJORI, F.
+
+ _History of Mathematics (New York,
+ 1897), p. 87._
+
+
+=1616.= 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 _Principia_ 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.
+
+ --GLAISHER, J. W. L.
+
+ _Encyclopedia Britannica, 9th Edition;
+ Article "Logarithms."_
+
+
+=1617.= 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.
+
+ --JOHNSON, SAMUEL.
+
+ _Boswell's Life of Johnson, Harper's
+ Edition (1871), Vol. 2, p. 33._
+
+
+=1618.= The method of arithmetical teaching is perhaps the best
+understood of any of the methods concerned with elementary
+studies.--BAIN, ALEXANDER.
+
+ _Education as a Science (New York,
+ 1898), p. 288._
+
+
+=1619.= 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
+_Arithmetike_. But if thei knewe how faree this seconde parte,
+doeeth excell the firste parte, thei would not accoumpte 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.
+
+ --RECORDE, ROBERT.
+
+ _Whetstone of Witte (London, 1557)._
+
+
+=1620.= 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.
+
+Aye, remarked he, it does this to a remarkable extent.
+
+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?
+
+Exactly so, he replied.
+
+And, moreover, I think you will not easily find that many things
+give the learner and student more trouble than this.
+
+Of course not.
+
+On all these accounts, then, we must not omit this branch of
+science, but those with the best of talents should be instructed
+therein.--PLATO.
+
+ _Republic [Davis], Bk. 7, chap. 8._
+
+
+=1621.= 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.--PLATO.
+
+ _Republic [Jowett], Bk. 7, p. 525._
+
+
+=1622.= 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.--MYERS, GEORGE.
+
+ _Monograph on Arithmetic in Public
+ Education (Chicago), p. 21._
+
+
+=1623.= 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.--MCMURRAY, C. A.
+
+ _Special Method in Arithmetic_ (_New
+ York, 1906_), _p. 18._
+
+
+=1624.= 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?
+
+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 _arts_, 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 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, _have_ 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.--FITCH, J. G.
+
+ _Lectures on Teaching (New York, 1906),
+ pp. 267-268._
+
+
+=1625.= 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 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.--FITCH, J. G.
+
+ _Lectures on Teaching (New York, 1906),
+ pp. 292-293._
+
+
+=1626.= Arithmetic and geometry, those wings on which the
+astronomer soars as high as heaven.--BOYLE, ROBERT.
+
+ _Usefulness of Mathematics to Natural
+ Philosophy; Works (London, 1772), Vol.
+ 3, p. 429._
+
+
+=1627.= Arithmetical symbols are written diagrams and geometrical
+figures are graphic formulas.--HILBERT, D.
+
+ _Mathematical Problems; Bulletin
+ American Mathematical Society, Vol. 8
+ (1902), p. 443._
+
+
+=1628.= 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.--DESCARTES.
+
+ _Rules for the Direction of the Mind;
+ Torrey's Philosophy of Descartes (New
+ York, 1892), p. 63._
+
+
+=1629.=
+
+ Why are _wise_ few, _fools_ numerous in the
+ excesse?
+ 'Cause, wanting _number_, they are
+ _numberlesse_.
+ --LOVELACE.
+
+ _Noah Bridges: Vulgar Arithmetike
+ (London, 1659), p. 127._
+
+
+=1630.= 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.--LOCKE, JOHN.
+
+ _An Essay concerning Human
+ Understanding, Bk. 2, chap. 16, sect.
+ 4._
+
+
+=1631.= Battalions of figures are like battalions of men, not
+always as strong as is supposed.--SAGE, M.
+
+ _Mrs. Piper and the Society for
+ Psychical Research [Robertson] (New
+ York, 1909), p. 151._
+
+
+=1632.= 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.--PARKER, F. W.
+
+ _Talks on Pedagogics (New York, 1894),
+ P. 64._
+
+
+=1633.= A rule to trick th' arithmetic.--KIPLING, R.
+
+ _To the True Romance._
+
+
+=1634.= God made integers, all else is the work of man.
+
+ --KRONECKER, L.
+
+ _Jahresberichte der Deutschen
+ Mathematiker Vereinigung, Bd. 2, p. 19._
+
+
+=1635.= Plato said "[Greek: aei ho theos geometre]." Jacobi
+changed this to "[Greek: aei ho theos arithmetizei]." Then came
+Kronecker and created the memorable expression "Die ganzen
+Zahlen hat Gott gemacht, alles andere ist Menschenwerk."--KLEIN, F.
+
+ _Jahresbericht der Deutschen
+ Mathematiker Vereinigung, Bd. 6, p.
+ 136._
+
+
+=1636.= Integral numbers are the fountainhead of all mathematics.
+
+ --MINKOWSKI, H.
+
+ _Diophantische Approximationen (Leipzig,
+ 1907), Vorrede._
+
+
+=1637.= The "Disquisitiones Arithmeticae" that great book with
+seven seals.--MERZ, J. T.
+
+ _A History of European Thought in the
+ Nineteenth Century (Edinburgh and
+ London, 1908), p. 721._
+
+
+=1638.= 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 _Disquisitiones Arithmeticae_; and
+inversely, every advance in the algebraic theory of forms is an
+acquisition to the arithmetical theory.--MATHEWS, G. B.
+
+ _Theory of Numbers (Cambridge, 1892),
+ Part 1, sect. 48._
+
+
+=1639.= Strictly speaking, the theory of numbers has nothing to
+do with negative, or fractional, or irrational quantities, _as
+such._ 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.
+
+ --MATHEWS, G. B.
+
+ _Theory of Numbers (Cambridge, 1892),
+ Part 1, sect. 1._
+
+
+=1640.= 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 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.--GLAISHER, J. W. L.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1890); Nature, Vol. 42, p.
+ 467._
+
+
+=1641.= 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.--KLEIN, F.
+
+ _Elementarmathematik vom hoeheren
+ Standpunkte aus. (Leipzig, 1908), Bd. 1,
+ p. 53._
+
+
+=1642.= 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.--GAUSS.
+
+ _Sartorius von Waltershausen: Gauss zum
+ Gedaechtniss. (Leipzig, 1866), p. 79._
+
+
+=1643.=
+
+ Zu Archimedes kam ein wissbegieriger Juengling,
+ Weihe mich, sprach er zu ihm, ein in die
+ goettliche Kunst,
+ Die so herrliche Dienste der Sternenkunde
+ geleistet,
+ Hinter dem Uranos noch einen Planeten entdeckt.
+ Goettlich nennst Du die Kunst, sie ist's,
+ versetzte der Weise,
+ Aber sie war es, bevor noch sie den Kosmos
+ erforscht,
+ Ehe sie herrliche Dienste der Sternenkunde
+ geleistet,
+ Hinter dem Uranos noch einen Planeten entdeckt.
+ Was Du im Kosmos erblickst, ist nur der
+ Goettlichen Abglanz,
+ In der Olympier Schaar thronet die ewige
+ Zahl.
+ --JACOBI, C. G. J.
+
+ _Journal fuer Mathematik, Bd. 101 (1887),
+ p. 338._
+
+ To Archimedes came a youth intent upon
+ knowledge,
+ Quoth he, "Initiate me into the science divine
+ Which to astronomy, lo! such excellent service
+ has rendered,
+ And beyond Uranus' orb a hidden planet
+ revealed."
+ "Call'st thou the science divine? So it is,"
+ the wise man responded,
+ "But so it was long before its light on the
+ Cosmos it shed,
+ Ere in astronomy's realm such excellent service
+ it rendered,
+ And beyond Uranus' orb a hidden planet
+ revealed.
+ Only reflection divine is that which Cosmos
+ discloses,
+ Number herself sits enthroned among Olympia's
+ hosts."
+
+
+=1644.= 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
+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.--GAUSS, C. F.
+
+ _Preface to Eisenstein's Mathematische
+ Abhandlungen (Berlin, 1847), [H. J. S.
+ Smith]._
+
+
+=1645.= 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.--SMITH, H. J. S.
+
+ _Report on the Theory of Numbers,
+ British Association, 1859; Collected
+ Mathematical Papers, Vol. 1, p. 38._
+
+
+=1646.= The invention of the symbol [congruent] 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.--MATHEWS, G. B.
+
+ _Theory of Numbers (Cambridge, 1892),
+ Part 1, sect. 29._
+
+
+=1647.= 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.--MATHEWS, G. B.
+
+ _Theory of Numbers (Cambridge, 1892),
+ Part 1, sect. 167._
+
+
+=1648.= 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, 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 _superficially_ as
+distinguished from our own limitation to _linearly_ extended
+time.--SYLVESTER, J. J.
+
+ _Collected Mathematical Papers, Vol. 4,
+ p. 600, footnote._
+
+
+
+
+ CHAPTER XVII
+
+ ALGEBRA
+
+
+=1701.= 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.
+
+ --DE MORGAN, A.
+
+ _Elements of Algebra (London, 1837),
+ Preface._
+
+
+=1702.= Algebra is generous, she often gives more than is asked
+of her.--D'ALEMBERT.
+
+ _Quoted in Bulletin American Mathematical
+ Society, Vol. 2 (1905), p. 285._
+
+
+=1703.= 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.--BOYLE, ROBERT.
+
+ _Works (London, 1772), Vol. 3, p. 426._
+
+
+=1704.= The human mind has never invented a labor-saving machine
+equal to algebra.--
+
+ _The Nation, Vol. 33, p. 237._
+
+
+=1705.= 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 _ideas of quantity_
+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.
+
+ --LOCKE, JOHN.
+
+ _An Essay concerning Human Understanding, Bk.
+ 4, chap. 3, sect. 18._
+
+
+=1706.= Algebra is but written geometry and geometry is but
+figured algebra.--GERMAIN, SOPHIE.
+
+ _Memoire sur la surfaces elastiques._
+
+
+=1707.= 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.
+
+ --LAGRANGE.
+
+ _Lecons elementaires sur les Mathematiques,
+ Lecon Cinquieme._
+
+
+=1708.= 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.--WHITEHEAD, A. N.
+
+ _Universal Algebra (Cambridge, 1898), p. 2._
+
+
+=1709.= 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.
+
+ --MATHEWS, G. B.
+
+ _F. Spencer: Chapters on Aims and Practice of
+ Teaching (London, 1899), p. 184._
+
+
+=1710.= 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.
+
+ --CHRYSTAL, GEORGE.
+
+ _Encyclopedia Britannica, 9th Edition;
+ Article "Algebra."_
+
+
+=1711.= 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.
+
+ --DE MORGAN, A.
+
+ _Elements of Algebra (London, 1837),
+ Preface._
+
+
+=1712.= We may always depend upon it that algebra, which cannot
+be translated into good English and sound common sense, is bad
+algebra.--CLIFFORD, W. K.
+
+ _Common Sense in the Exact Sciences (London,
+ 1885), chap. 1, sect. 7._
+
+
+=1713.= The best review of arithmetic consists in the study of
+algebra.--CAJORI, F.
+
+ _Teaching and History of Mathematics in U. S.
+ (Washington, 1896), p. 110._
+
+
+=1714.= [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 _Algebra is the Calculus
+of Functions_, and _Arithmetic the Calculus of Values_.--COMTE, A.
+
+ _Philosophy of Mathematics [Gillespie] (New
+ York, 1851), p. 55._
+
+
+=1715.= ... 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 _cannot_ be divested of,
+the thought of _possible_ Succession, or of pure, _ideal_
+Progression.--HAMILTON, W. R.
+
+ _Graves' Life of Hamilton (New York,
+ 1882-1889), Vol. 3, p. 633._
+
+
+=1716.= ... instead of seeking to attain consistency and
+uniformity of system, as some modern writers have attempted, by
+banishing this thought of time from the _higher_ Algebra, I seek
+to _attain_ the same object, by systematically introducing it
+into the _lower_ or earlier parts of the science.--HAMILTON, W. R.
+
+ _Graves' Life of Hamilton (New York,
+ 1882-1889), Vol. 3, p. 634._
+
+
+=1717.= 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."--CHRYSTAL, GEORGE.
+
+ _Encyclopedia Britannica, 9th Edition;
+ Article "Algebra."_
+
+
+=1718.= Time is said to have only _one dimension_, and space to
+have _three dimensions_.... The mathematical _quaternion_
+partakes of _both_ 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, _four
+dimensions_....
+
+ And how the One of Time, of Space the Three,
+ Might in the Chain of Symbols girdled be.
+ --HAMILTON, W. R.
+
+ _Graves' Life of Hamilton (New York,
+ 1882-1889), Vol. 3, p. 635._
+
+
+=1719.= 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.
+
+ --HILL, THOMAS.
+
+ _North American Review, Vol. 85, p.
+ 223._
+
+
+=1720.= 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.
+
+ --DE MORGAN, A.
+
+ _Graves' Life of Hamilton (New York,
+ 1882-1889), Vol. 3, p. 493._
+
+
+=1721.= 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.--THOMSON, WILLIAM.
+
+ _Thompson, S. P.: Life of Lord Kelvin
+ (London, 1910), p. 1138._
+
+
+=1722.= The whole affair [quaternions] has in respect to
+mathematics a value not inferior to that of "Volapuk" in respect
+to language.--THOMSON, WILLIAM.
+
+ _Thompson, S. P.: Life of Lord Kelvin
+ (London, 1910), p. 1138._
+
+
+=1723.= 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.--SEDGWICK.
+
+ _Graves' Life of Hamilton (New York,
+ 1882-1889), Vol. 3, p. 2._
+
+
+=1724.= 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 _all_ 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 _no choice_, 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.
+
+ --TAIT, P. G.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1871); Nature, Vol. 4, p. 270._
+
+
+=1725.= 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.--TAIT, P. G.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1871); Nature, Vol. 4, p. 271._
+
+
+=1726.= It is true that, in the eyes of the pure mathematician,
+Quaternions have one grand and fatal defect. They cannot be
+applied to space of _n_ 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 _Actual_ 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
+_Inconceivable_.--TAIT, P. G.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1871); Nature, Vol. 4, p. 271._
+
+
+=1727.= 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.--WHITEHEAD, A. N.
+
+ _An Introduction to Mathematics (New
+ York, 1911), p. 86._
+
+
+=1728.= 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.--BARROW, ISAAC.
+
+ _Mathematical Lectures (London, 1734),
+ p. 44._
+
+
+=1729.= 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.--HANKEL, HERMANN.
+
+ _Theorie der Complexen Zahlen (Leipzig,
+ 1867), p. 60._
+
+
+=1730.= 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, [sq root]-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.--GAUSS, C. F.
+
+ _Theoria residiorum biquadraticorum,
+ Commentatio secunda; Werke, Bd. 2
+ (Goettingen, 1863), p. 177._
+
+
+=1731.= ... the imaginary, this bosom-child of complex mysticism.
+
+ --DUeHRING, EUGEN.
+
+ _Kritische Geschichte der allgemeinen
+ Principien der Mechanik (Leipzig, 1877),
+ p. 517._
+
+
+=1732.= Judged by the only standards which are admissible in a
+pure doctrine of numbers _i_ 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).--FINE, H. B.
+
+ _The Number-System of Algebra (Boston,
+ 1890), p. 36._
+
+
+=1733.= This symbol [[sq root]-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.
+
+ --PEIRCE, BENJAMIN.
+
+ _On the Uses and Transformations of
+ Linear Algebras; American Journal of
+ Mathematics, Vol. 4 (1881), p. 216._
+
+
+=1734.= 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.--HILL, THOMAS.
+
+ _North American Review, Vol. 85, p.
+ 235._
+
+
+=1735.= 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 _memoria technica_ 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.--RUSSELL, BERTRAND.
+
+ _Foundations of Geometry (Cambridge,
+ 1897), p. 45._
+
+
+=1736.= 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.--HANKEL, HERMANN.
+
+ _Geschichte der Mathematik im Altertum
+ und Mittelalter (Leipzig, 1874), p.
+ 195._
+
+
+=1737.= 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.--CAJORI, F.
+
+ _History of Mathematics (New York,
+ 1897), p. 100._
+
+
+=1738.= 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.
+
+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.
+
+2. What square is that, which being increased or diminished by
+10, the sum and remainder are both square numbers?
+
+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.
+
+4. To divide a cube number into two cube numbers.
+
+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.
+
+6. There are three square numbers in continued geometric
+proportion, such, that the sum of the three is a square number.
+
+7. There is a square, such, that when it is increased and
+diminished by its root and 2, the sum and the difference are
+squares.--KHULASAT-AL-HISAB.
+
+ _Algebra; quoted in Hutton: A
+ Philosophical and Mathematical
+ Dictionary (London, 1815), Vol. 1, p.
+ 70._
+
+
+=1739.= The solution of such questions as these [referring to the
+solution of cubic equations] depends on correct judgment, aided
+by the assistance of God.--BIJA GANITA.
+
+ _Quoted in Hutton: A Philosophical and
+ Mathematical Dictionary (London, 1815),
+ Vol. 1, p. 65._
+
+
+=1740.= 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.--SYLVESTER, J. J.
+
+ _Philosophical Magazine, Vol. 1, (1851),
+ p. 300; Collected Mathematical Papers,
+ Vol. 1, p. 247._
+
+
+=1741.=
+
+ Fuchs. Fast moecht' ich nun _moderne Algebra_ studieren.
+
+ Meph. Ich wuenschte nicht euch irre zu fuehren.
+ Was diese Wissenschaft betrifft,
+ Es ist so schwer, die leere Form zu meiden,
+ Und wenn ihr es nicht recht begrifft,
+ Vermoegt die Indices ihr kaum zu unterscheiden.
+ Am Besten ist's, wenn ihr nur _Einem_ traut
+ Und auf des Meister's Formeln baut.
+ Im Ganzen--haltet euch an die _Symbole_.
+ Dann geht ihr zu der Forschung Wohle
+ Ins sichre Reich der Formeln ein.
+
+ Fuchs. Ein Resultat muss beim Symbole sein?
+
+ Meph. Schon gut! Nur muss man sich nicht alzu aengstlich
+ quaelen.
+ Denn eben, wo die Resultate fehlen,
+ Stellt ein Symbol zur rechten Zeit sich ein.
+ Symbolisch laesst sich alles schreiben,
+ Muesst nur im Allgemeinen bleiben.
+ Wenn man der Gleichung Loesung nicht erkannte,
+ Schreibt man sie als Determinante.
+ Schreib' was du willst, nur rechne _nie_ was aus.
+ Symbole lassen trefflich sich traktieren,
+ Mit einem Strich ist alles auszufuehren,
+ Und mit Symbolen kommt man immer aus.
+ --LASSWITZ, KURD.
+
+ _Der Faust-Tragoedie (-n)ter Teil;
+ Zeitschrift fuer mathematischen und
+ naturwissenschaftlichen Unterricht, Bd.
+ 14, p. 317._
+
+ Fuchs. To study _modern algebra_ I'm most persuaded.
+
+ Meph. 'Twas not my wish to lead thee astray.
+ But as concerns this science, truly
+ 'Tis difficult to avoid the empty form,
+ And should'st thou lack clear comprehension,
+ Scarcely the indices thou'll know apart.
+ 'Tis safest far to trust but _one_
+ And built upon your master's formulas.
+ On the whole--cling closely to your _symbols_.
+ Then, for the weal of research you may gain
+ An entrance to the formula's sure domain.
+
+ Fuchs. The symbol, it must lead to some result?
+
+ Meph. Granted. But never worry about results,
+ For, mind you, just where the results are wanting
+ A symbol at the nick of time appears.
+ To symbolic treatment all things yield,
+ Provided we stay in the general field.
+ Should a solution prove elusive,
+ Write the equation in determinant form.
+ Write what you please, but _never_ calculate.
+ Symbols are patient and long suffering,
+ A single stroke completes the whole affair.
+ Symbols for every purpose do suffice.
+
+
+=1742.= 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 at the Capitol of Modern Algebra over whose shining portal
+is inscribed "Theory of Invariants."--SYLVESTER, J. J.
+
+ _On Newton's Rule for the Discovery of
+ Imaginary Roots; Collected Mathematical
+ Papers, Vol. 2, p. 380._
+
+
+=1743.= 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.--TODHUNTER, I.
+
+ _Theory of Equations (London, 1904), p.
+ 250._
+
+
+=1744.= 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.
+
+ --CHRYSTAL, GEORGE.
+
+ _Algebra (London and Edinburgh, 1893),
+ Vol. 1, chap. 15, sect. 25._
+
+
+=1745.= _To a missing member of a family group of terms in an
+algebraical formula._
+
+ Lone and discarded one! divorced by fate,
+ Far from thy wished-for fellows--whither art
+ flown?
+ Where lingerest thou in thy bereaved estate,
+ Like some lost star, or buried meteor stone?
+ Thou mindst me much of that presumptuous one
+ Who loth, aught less than greatest, to be
+ great,
+ From Heaven's immensity fell headlong down
+ To live forlorn, self-centred, desolate:
+ Or who, like Heraclid, hard exile bore,
+ Now buoyed by hope, now stretched on rack of
+ fear,
+ Till throned Astaea, wafting to his ear
+ Words of dim portent through the Atlantic roar,
+ Bade him "the sanctuary of the Muse revere
+ And strew with flame the dust of Isis'
+ shore."
+ --SYLVESTER, J. J.
+
+ _Inaugural Lecture, Oxford, 1885;
+ Nature, Vol. 33, p. 228._
+
+
+=1746.= 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 _principle_ which is so
+valuable. It is the _idea_ of invariance that pervades today all
+branches of mathematics.--MACMAHON, P. A.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1901); Nature, Vol. 64, p.
+ 481._
+
+
+=1747.= [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.--FORSYTH, A. R.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1897); Nature, Vol. 56, p.
+ 378._
+
+
+=1748.= ... 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.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 28._
+
+
+=1749.= 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.
+
+ --KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 42._
+
+
+=1750.= 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
+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.--SYLVESTER, J. J.
+
+ _American Journal of Mathematics, Vol. 1
+ (1878), p. 126._
+
+
+=1751.= 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.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 12._
+
+
+=1752.= 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.--LIE, SOPHUS.
+
+ _Continuierliche Gruppen--Scheffers
+ (Leipzig, 1893), p. 665._
+
+
+=1753.= 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.--WHITEHEAD, A. N.
+
+ _Universal Algebra (Cambridge, 1898),
+ Preface, p. vi._
+
+
+=1754.= [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 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.--FORSYTH, A. R.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1897); Nature, Vol. 56, p.
+ 378._
+
+
+=1755.= 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.
+
+ --FORSYTH, A. R.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1897); Nature, Vol. 56, p.
+ 378._
+
+
+
+
+ CHAPTER XVIII
+
+ GEOMETRY
+
+
+=1801.= The science of figures is most glorious and beautiful.
+But how inaptly it has received the name geometry!--FRISCHLINUS, N.
+
+ _Dialog 1._
+
+
+=1802.= Plato said that God geometrizes continually.--PLUTARCH.
+
+ _Convivialium disputationum, liber 8,
+ 2._
+
+
+=1803.= [Greek: medeis ageometretos eisito mou ten stegen]. [Let
+no one ignorant of geometry enter my door.]--PLATO.
+
+ _Tzetzes, Chiliad, 8, 972._
+
+
+=1804.= 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.--BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p. 45._
+
+
+=1805.= Form and size constitute the foundation of all search for
+truth.--PARKER, F. W.
+
+ _Talks on Pedagogics (New York, 1894),
+ p. 72._
+
+
+=1806.= 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.
+
+Certainly, he said.
+
+Then must not a further admission be made?
+
+What admission?
+
+The admission that this knowledge at which geometry aims is of
+the eternal, and not of the perishing and transient.
+
+That may be easily allowed. Geometry, no doubt, is the knowledge
+of what eternally exists.
+
+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.--PLATO.
+
+ _Republic [Jowett-Davies], Bk. 7, p.
+ 527._
+
+
+=1807.= 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.
+
+ --CICERO.
+
+ _Tusculanae Disputationes, 1, 2, 5._
+
+
+=1808.=
+
+ Geometria,
+ Through which a man hath the sleight
+ Of length, and brede, of depth, of height.
+ --GOWER, JOHN.
+
+ _Confessio Amantis, Bk. 7._
+
+
+=1809.= 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.--D'ALEMBERT.
+
+ _Quoted in Rebiere: Mathematiques et
+ Mathematiciens (Paris, 1898), p. 10._
+
+
+=1810.= Geometry exhibits the most perfect example of logical
+stratagem.--BUCKLE, H. T.
+
+ _History of Civilization in England (New
+ York, 1891), Vol. 2, p. 342._
+
+
+=1811.= It is the glory of geometry that from so few principles,
+fetched from without, it is able to accomplish so much.--NEWTON.
+
+ _Philosophiae Naturalis Principia
+ Mathematica, Praefat._
+
+
+=1812.= 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.
+
+ --DE MORGAN, A.
+
+ _On the Study and Difficulties of
+ Mathematics (Chicago, 1902), p. 231._
+
+
+=1813.= 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 1,
+ chap. 3._
+
+
+=1814.= 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.--WHEWELL, WILLIAM.
+
+ _The Philosophy of the Inductive
+ Sciences, Part 1, Bk. 1, chap. 6, sect.
+ 1 (London, 1858)._
+
+
+=1815.= 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.--HALSTED, G. B.
+
+ _Proceedings of the American Association
+ for the Advancement of Science (1904),
+ p. 359._
+
+
+=1816.= A mathematical point is the most indivisible and unique
+thing which art can present.--DONNE, JOHN.
+
+ _Letters, 21._
+
+
+=1817.= 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.--KELLAND, P.
+
+ _Lectures on the Principles of
+ Demonstrative Mathematics (London,
+ 1843), p. 17._
+
+
+=1818.= 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.--TODHUNTER, I.
+
+ _Essay on Elementary Geometry; Conflict
+ of Studies and other Essays (London,
+ 1873), p. 167._
+
+
+=1819.= 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.
+
+ --DE MORGAN, A.
+
+ _Elements of Algebra (London, 1837),
+ Introduction._
+
+
+=1820.= This book [Euclid] has been for nearly twenty-two
+centuries the encouragement and guide of that scientific thought
+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."--CLIFFORD, W. K.
+
+ _Lectures and Essays (London, 1901),
+ Vol. 1, p. 354._
+
+
+=1821.= [Euclid] at once the inspiration and aspiration of
+scientific thought.--CLIFFORD, W. K.
+
+ _Lectures and Essays (London, 1901), Vol
+ 1, p. 355._
+
+
+=1822.= The "elements" of the Great Alexandrian remain for all
+time the first, and one may venture to assert, the _only_ 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.--HANKEL, H.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1884),
+ p. 7._
+
+
+=1823.= If we consider him [Euclid] as meaning to be what
+his commentators have taken him to be, a model of the most
+unscrupulous formal rigour, we can deny that he has altogether
+succeeded, though we admit that he made the nearest approach.
+
+ --DE MORGAN, A.
+
+ _Smith's Dictionary of Greek and Roman
+ Biography and Mythology (London, 1902);
+ Article "Eucleides."_
+
+
+=1824.= 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.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 8._
+
+
+=1825.= 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.--SYLVESTER, J. J.
+
+ _A Plea for the Mathematician; Nature,
+ Vol. 1, p. 261._
+
+
+=1826.= 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.--SYLVESTER, J. J.
+
+ _A Plea for the Mathematician; Nature,
+ Vol. 1, p. 262._
+
+
+=1827.= 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.--PARTON, JAMES.
+
+ _Sir Isaac Newton._
+
+
+=1828.= 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.--HEAVISIDE, OLIVER.
+
+ _Electro-Magnetic Theory (London, 1893),
+ Vol. 1, p. 148._
+
+
+=1829.= 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.
+
+ --SMITH, H. J. S.
+
+ _Nature, Vol. 8, p. 450._
+
+
+=1830.= 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.--WHEWELL, WILLIAM.
+
+ _The Philosophy of the Inductive
+ Sciences, (London, 1858), Part 1, Bk. 2,
+ chap. 1, sect. 4._
+
+
+=1831.= 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.--DE MORGAN, A.
+
+ _Smith's Dictionary of Greek and Roman
+ Biography and Mythology (London, 1902);
+ Article "Eucleides."_
+
+
+=1832.= 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.--GOW, JAMES.
+
+ _A Short History of Greek Mathematics
+ (Cambridge, 1884), p. 131._
+
+
+=1833.=
+
+ A figure and a step onward:
+ Not a figure and a florin.
+ --MOTTO OF THE PYTHAGOREAN BROTHERHOOD.
+
+ _W. B. Frankland: Story of Euclid
+ (London, 1902), p. 33._
+
+
+=1834.= 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.
+
+ --SMITH, H. J. S.
+
+ _Presidential Address British
+ Association for the Advancement of
+ Science (1873); Nature, Vol. 8, p. 451._
+
+
+=1835.= 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 _scare_-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 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.--BARROW, ISAAC.
+
+ _Mathematical Lectures (London, 1734),
+ p. 388._
+
+
+=1836.= 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.--HELMHOLTZ, H.
+
+ _The Origin and Significance of
+ Geometrical Axioms; Popular Scientific
+ Lectures [Atkinson], Second Series (New
+ York, 1881), p. 27._
+
+
+=1837.= 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.--BUCKLE, H. T.
+
+ _History of Civilization in England (New
+ York, 1891), Vol. 2, p. 343._
+
+
+=1838.= 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
+careless observer, and which compose the first principles and
+ultimate premises of the science.--MILL, J. S.
+
+ _System of Logic, Bk. 3, chap. 24, sect.
+ 7._
+
+
+=1839.= All such reasonings [natural philosophy, chemistry,
+agriculture, political economy, etc.] are, in comparison with
+mathematics, very complex; requiring so much _more_ 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.--WHATELY, R.
+
+ _Elements of Logic, Bk. 4, chap. 2,
+ sect. 5._
+
+
+=1840.=
+
+ [Geometry] that held acquaintance with the stars,
+ And wedded soul to soul in purest bond
+ Of reason, undisturbed by space or time.
+ --WORDSWORTH.
+
+ _The Prelude, Bk. 5._
+
+
+=1841.= 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
+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.
+
+ --CARSON, G. W. L.
+
+ _The Functions of Geometry as a Subject
+ of Education (Tonbridge, 1910), p. 3._
+
+
+=1842.= 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.--CLIFFORD, W. K.
+
+ _Quoted by Pollock in Clifford's
+ Lectures and Essays (London, 1901), Vol.
+ 1, Introduction, p. 43._
+
+
+=1843.=
+
+ On the lecture slate
+ The circle rounded under female hands
+ With flawless demonstration.
+ --TENNYSON.
+
+ _The Princess, II, l. 493._
+
+
+=1844.= 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 manoeuvres 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.--PLATO.
+
+ _Republic, Bk. 7, p. 526._
+
+
+=1845.= 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.
+
+Of what kind are they? he said.
+
+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.--PLATO.
+
+ _Republic [Jowett], Bk. 7, p. 527._
+
+
+=1846.= 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.--SMITH, D. E.
+
+ _The Teaching of Geometry (Boston,
+ 1911), p. 12._
+
+
+=1847.= 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.--HILL, THOMAS.
+
+ _The Uses of Mathesis; Bibliotheca
+ Sacra, Vol. 32, p. 504._
+
+
+=1848.=
+
+ Yet may we not entirely overlook
+ The pleasures gathered from the rudiments
+ Of geometric science. Though advanced
+ In these inquiries, with regret I speak,
+ No farther than the threshold, there I found
+ Both elevation and composed delight:
+ With Indian awe and wonder, ignorance pleased
+ With its own struggles, did I meditate
+ On the relations those abstractions bear
+ To Nature's laws.
+
+ * * * * *
+
+ More frequently from the same source I drew
+ A pleasure quiet and profound, a sense
+ Of permanent and universal sway,
+ And paramount belief; there, recognized
+ A type, for finite natures, of the one
+ Supreme Existence, the surpassing life
+ Which to the boundaries of space and time,
+ Of melancholy space and doleful time,
+ Superior and incapable of change,
+ Nor touched by welterings of passion--is,
+ And hath the name of God. Transcendent peace
+ And silence did wait upon these thoughts
+ That were a frequent comfort to my youth.
+
+ * * * * *
+
+ Mighty is the charm
+ Of those abstractions to a mind beset
+ With images and haunted by himself,
+ And specially delightful unto me
+ Was that clear synthesis built up aloft
+ So gracefully; even then when it appeared
+ Not more than a mere plaything, or a toy
+ To sense embodied: not the thing it is
+ In verity, an independent world,
+ Created out of pure intelligence.
+ --WORDSWORTH.
+
+ _The Prelude, Bk. 6._
+
+
+=1849.=
+
+ 'Tis told by one whom stormy waters threw,
+ With fellow-sufferers by the shipwreck spared,
+ Upon a desert coast, that having brought
+ To land a single volume, saved by chance,
+ A treatise of Geometry, he wont,
+ Although of food and clothing destitute,
+ And beyond common wretchedness depressed,
+ To part from company, and take this book
+ (Then first a self taught pupil in its truths)
+ To spots remote, and draw his diagrams
+ With a long staff upon the sand, and thus
+ Did oft beguile his sorrow, and almost
+ Forget his feeling:
+ --WORDSWORTH.
+
+ _The Prelude, Bk. 6._
+
+
+=1850.= 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....
+
+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, 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.--SMITH, D. E.
+
+ _The Teaching of Geometry (Boston,
+ 1911), p. 16._
+
+
+=1851.= 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.--D'ALEMBERT.
+
+ _Essai sur les Gens Lettres; Melages
+ (Amsterdam 1764), t. 1, p. 334._
+
+
+=1852.= If it were required to determine inclined planes of
+varying inclinations of such lengths that a free rolling body
+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.--KANT.
+
+ _Der einzig moegliche Beweisgrund zu
+ einer Demonstration des Daseins Gottes;
+ Werke (Hartenstein), Bd. 2, p. 137._
+
+
+=1853.= 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.--KANT.
+
+ _Der einzig moegliche Beweisgrund zu
+ einer Demonstration des Daseins Gottes;
+ Werke (Hartenstein), Bd. 2, p. 138._
+
+
+=1854.= 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 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 _slightly_ 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 _hecatomb_ of oxen! It would produce
+a quite inconvenient supply of beef.--DODGSON, C. L.
+
+ _A New Theory of Parallels (London,
+ 1895), Introduction, p. 16._
+
+
+=1855.= After Pythagoras discovered his fundamental theorem he
+sacrificed a hecatomb of oxen. Since that time all dunces[10]
+[Ochsen] tremble whenever a new truth is discovered.--BOERNE.
+
+ _Quoted in Moszkowski: Die unsterbliche
+ Kiste (Berlin, 1908), p. 18._
+
+ [10] In the German vernacular a dunce or blockhead
+ is called an ox.
+
+
+=1856.=
+
+ _Vom Pythagorieschen Lehrsatz._
+
+ Die Wahrheit, sie besteht in Ewigkeit,
+ Wenn erst die bloede Welt ihr Licht erkannt:
+ Der Lehrsatz, nach Pythagoras benannt,
+ Gilt heute, wie er galt in seiner Zeit.
+
+ Ein Opfer hat Pythagoras geweiht
+ Den Goettern, die den Lichtstrahl ihm gesandt;
+ Es thaten kund, geschlachtet und verbrannt,
+ Ein hundert Ochsen seine Dankbarkeit.
+
+ Die Ochsen seit den Tage, wenn sie wittern,
+ Dass eine neue Wahrheit sich enthuelle,
+ Erheben ein unmenschliches Gebruelle;
+
+ Pythagoras erfuellt sie mit Entsetzen;
+ Und machtlos, sich dem Licht zu wiedersetzen,
+ Verschiessen sie die Augen und erzittern.
+ --CHAMISSO, ADELBERT VON.
+
+ _Gedichte, 1835 (Haushenbusch), (Berlin,
+ 1889), p. 302._
+
+ Truth lasts throughout eternity,
+ When once the stupid world its light discerns:
+ The theorem, coupled with Pythagoras' name,
+ Holds true today, as't did in olden times.
+
+ A splendid sacrifice Pythagoras brought
+ The gods, who blessed him with this ray divine;
+ A great burnt offering of a hundred kine,
+ Proclaimed afar the sage's gratitude.
+
+ Now since that day, all cattle [blockheads] when they
+ scent
+ New truth about to see the light of day,
+ In frightful bellowings manifest their dismay;
+
+ Pythagoras fills them all with terror;
+ And powerless to shut out light by error,
+ In sheer despair they shut their eyes and tremble.
+
+
+=1857.= 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."--FRANKLAND, W. B.
+
+ _The Story of Euclid (London, 1902), p.
+ 70._
+
+
+=1858.=
+
+ As one
+ Who vers'd in geometric lore, would fain
+ Measure the circle; and, though pondering long
+ And deeply, that beginning, which he needs,
+ Finds not; e'en such was I, intent to scan
+ The novel wonder, and trace out the form,
+ How to the circle fitted, and therein
+ How plac'd: but the flight was not for my wing;
+ --DANTE.
+
+ _Paradise [Carey] Canto 33, lines
+ 122-129._
+
+
+=1859.= If geometry were as much opposed to our passions and
+present interests as is ethics, we should contest it and violate
+it 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.--LEIBNITZ.
+
+ _New Essays concerning Human
+ Understanding [Langley], Bk. 1, chap. 2,
+ sect. 12._
+
+
+=1860.= 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.--FREDERICK THE GREAT.
+
+ _Oeuvres (Decker), t. 7, p. 100._
+
+
+=1861.= 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.--MILL, J. S.
+
+ _System of Logic, Bk. 2, chap. 3, sect.
+ 3._
+
+
+=1862.= 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 their consequences a degree of exactness, of which
+these consequences are singly incapable.--HUME, D.
+
+ _A Treatise of Human Nature, Part 3,
+ sect. 1._
+
+
+=1863.= 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....
+
+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.--HUME, D.
+
+ _A Treatise of Human Nature, Part 3,
+ sect. 1._
+
+
+=1864.= 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.--RUSSELL, BERTRAND.
+
+ _Foundations of Geometry (Cambridge,
+ 1897), p. 120._
+
+
+=1865.= 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 moving of bodies, it comes to pass
+that Geometry is commonly referred to their magnitudes, and
+Mechanics to their motion.--NEWTON.
+
+ _Philosophiae Naturalis Principia
+ Mathematica, Praefat._
+
+
+=1866.= 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--BOCHER, MAXIME.
+
+ _Bulletin American Mathematical Society,
+ Vol. 2 (1904), p. 124._
+
+
+=1867.= 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.--POINCARE, H.
+
+ _On the Foundations of Geometry; Monist,
+ Vol. 9 (1898-1899), p. 41._
+
+
+=1868.= 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.--ARAGO.
+
+ _Oeuvres, t. 2 (1854), p. 694._
+
+
+=1869.= 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 division of knowledge into small elements
+renders it difficult of comprehension.--PROCLUS.
+
+ _Quoted in D. E. Smith: The Teaching of
+ Geometry (Boston, 1911), p. 71._
+
+
+=1870.= 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.--LAFAILLE, J. C.
+
+ _Theoremata de Centro Gravitatis
+ (Anvers, 1632), Praefat._
+
+
+=1871.= 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.--BUTLER, N. M.
+
+ _The Meaning of Education etc. (New
+ York, 1905), p. 171._
+
+
+=1872.= 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 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 cubed-b cubed is always divisible without a remainder by
+a-b.--DE MORGAN, A.
+
+ _On the Study and Difficulties of
+ Mathematics (Chicago, 1902), chap. 13._
+
+
+=1873.= The principal characteristics of the ancient geometry
+are:--
+
+(1) A wonderful clearness and definiteness of its concepts and an
+almost perfect logical rigour of its conclusions.
+
+(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.--CAJORI, F.
+
+ _History of Mathematics (New York,
+ 1897), p. 62._
+
+
+=1874.= It has been observed that the ancient geometers made use
+of a kind of analysis, which they employed in the solution of
+problems, although they begrudged to posterity the knowledge of
+it.--DESCARTES.
+
+ _Rules for the Direction of the Mind;
+ The Philosophy of Descartes [Torrey]
+ (New York, 1892), p. 68._
+
+
+=1875.= The ancients studied geometry with reference to the
+_bodies_ under notice, or specially: the moderns study it with
+reference to the _phenomena_ 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 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.--COMTE.
+
+ _Positive Philosophy [Martineau], Bk. 1,
+ chap. 3._
+
+
+=1876.= 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.
+
+ _Report of the Committee of Ten on
+ Secondary School Studies (Chicago,
+ 1894), p. 116._
+
+
+=1877.= 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.
+
+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.--STEINER, J.
+
+ _Werke, Bd. 1 (1881), p. 233._
+
+
+=1878.= 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" (Steiner), and
+has, as we may say without exaggeration, almost attained to the
+scientific ideal.--HANKEL, H.
+
+ _Die Entwickelung der Mathematik in den
+ letzten Jahrhunderten (Tuebingen, 1869)._
+
+
+=1879.= The two mathematically fundamental things in projective
+geometry are anharmonic ratio, and the quadrilateral construction.
+Everything else follows mathematically from these two.
+
+ --RUSSELL, BERTRAND.
+
+ _Foundations of Geometry (Cambridge,
+ 1897), p. 122._
+
+
+=1880.= ... 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.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and
+ Arts (New York, 1908), p. 2._
+
+
+=1881.= 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 1,
+ chap. 3._
+
+
+=1882.= 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 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."
+
+ --CLARKE, G. S.
+
+ _Quoted in W. S. Hall: Descriptive
+ Geometry (New York, 1902), chap. 1._
+
+
+=1883.= 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.--COMTE, A.
+
+ _Positive Philosophy [Martineau], Bk. 1,
+ chap. 3._
+
+
+=1884.= There is perhaps nothing which so occupies, as it were,
+the middle position of mathematics, as trigonometry.
+
+ --HERBART, J. F.
+
+ _Idee eines ABC der Anschauung; Werke
+ (Kehrbach) (Langensalza, 1890), Bd. 1,
+ p. 174._
+
+
+=1885.= Trigonometry contains the science of continually
+undulating magnitude: meaning magnitude which becomes alternately
+greater and less, without any termination to succession 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.--DE MORGAN, A.
+
+ _Trigonometry and Double Algebra
+ (London, 1849), Bk. 1, chap. 1._
+
+
+=1886.= Sin squared[phi] is odious to me, even though Laplace made use
+of it; should it be feared that sin[phi] squared might become ambiguous,
+which would perhaps never occur, or at most very rarely when
+speaking of sin ([phi] squared), well then, let us write (sin[phi]) squared,
+but not sin squared[phi], which by analogy should signify sin(sin[phi]).
+
+ --GAUSS.
+
+ _Gauss-Schumacher Briefwechsel, Bd. 3,
+ p. 292; Bd. 4, p. 63._
+
+
+=1887.= Perhaps to the student there is no part of elementary
+mathematics so repulsive as is spherical trigonometry.--TAIT, P. G.
+
+ _Encyclopedia Britannica, 9th Edition;
+ Article "Quaternions."_
+
+
+=1888.= "Napier's Rule of circular parts" is perhaps the happiest
+example of artificial memory that is known.--CAJORI, F.
+
+ _History of Mathematics (New York,
+ 1897), p. 165._
+
+
+=1889.= 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.--FOURIER.
+
+ _Theorie Analytique de la Chaleur,
+ Discours Preliminaire._
+
+
+=1890.= 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 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.--WHITEHEAD, A. N.
+
+ _An Introduction to Mathematics (New
+ York, 1911), p. 122._
+
+
+=1891.= 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.--PICARD, E.
+
+ _Bulletin American Mathematical Society,
+ Vol. 2 (1905), p. 409._
+
+
+=1892.= 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.--NEWTON.
+
+ _The Linear Constructions of Equations;
+ Universal Arithmetic (London, 1769),
+ Vol. 2, p. 468._
+
+
+=1893.= 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 depths of human cognition and so has an
+entirely different significance, than the most ingenious artifice
+which serves a special purpose.--BOIS-REYMOND, EMIL DU.
+
+ _Reden, Bd. 1 (Leipzig, 1885), p. 287._
+
+
+=1894.=
+
+ _Song of the Screw._
+
+ A moving form or rigid mass,
+ Under whate'er conditions
+ Along successive screws must pass
+ Between each two positions.
+ It turns around and slides along--
+ This is the burden of my song.
+
+ The pitch of screw, if multiplied
+ By angle of rotation,
+ Will give the distance it must glide
+ In motion of translation.
+ Infinite pitch means pure translation,
+ And zero pitch means pure rotation.
+
+ Two motions on two given screws,
+ With amplitudes at pleasure,
+ Into a third screw-motion fuse,
+ Whose amplitude we measure
+ By parallelogram construction
+ (A very obvious deduction).
+
+ Its axis cuts the nodal line
+ Which to both screws is normal,
+ And generates a form divine,
+ Whose name, in language formal,
+ Is "surface-ruled of third degree."
+ Cylindroid is the name for me.
+
+ Rotation round a given line
+ Is like a force along,
+ If to say couple you decline,
+ You're clearly in the wrong;--
+ 'Tis obvious, upon reflection,
+ A line is not a mere direction.
+
+ So couples with translations too
+ In all respects agree;
+ And thus there centres in the screw
+ A wondrous harmony
+ Of Kinematics and of Statics,--
+ The sweetest thing in mathematics.
+
+ The forces on one given screw,
+ With motion on a second,
+ In general some work will do,
+ Whose magnitude is reckoned
+ By angle, force, and what we call
+ The coefficient virtual.
+
+ Rotation now to force convert,
+ And force into rotation;
+ Unchanged the work, we can assert,
+ In spite of transformation.
+ And if two screws no work can claim,
+ Reciprocal will be their name.
+
+ Five numbers will a screw define,
+ A screwing motion, six;
+ For four will give the axial line,
+ One more the pitch will fix;
+ And hence we always can contrive
+ One screw reciprocal to five.
+
+ Screws--two, three, four or five, combined
+ (No question here of six),
+ Yield other screws which are confined
+ Within one screw complex.
+ Thus we obtain the clearest notion
+ Of freedom and constraint of motion.
+
+ In complex III, three several screws
+ At every point you find,
+ Or if you one direction choose,
+ One screw is to your mind;
+ And complexes of order III.
+ Their own reciprocals may be.
+
+ In IV, wherever you arrive,
+ You find of screws a cone,
+ On every line of complex V.
+ There is precisely one;
+ At each point of this complex rich,
+ A plane of screws have given pitch.
+
+ But time would fail me to discourse
+ Of Order and Degree;
+ Of Impulse, Energy and Force,
+ And Reciprocity.
+ All these and more, for motions small,
+ Have been discussed by Dr. Ball.
+ --ANONYMOUS.
+
+
+
+
+ CHAPTER XIX
+
+ THE CALCULUS AND ALLIED TOPICS
+
+
+=1901.= 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 _inverse_
+operations belongs to Newton and Leibnitz.--LIE, SOPHUS.
+
+ _Leipziger Berichte, 47 (1895),
+ Math.-phys. Classe, p. 53._
+
+
+=1902.= 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.
+
+ --LAPLACE.
+
+ _Les Integrales Definies, etc.; Oeuvres,
+ t. 12 (Paris, 1898), p. 359._
+
+
+=1903.= 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.
+
+ --MILL, J. S.
+
+ _System of Logic, Bk. 3, chap. 24, sect.
+ 6._
+
+
+=1904.= 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.--BALL, W. W. R.
+
+ _History of Mathematics (London, 1901),
+ p. 371._
+
+
+=1905.= 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.--WILLIAMSON, B.
+
+ _Encyclopedia Britannica, 9th Edition;
+ Article "Infinitesimal Calculus," sect.
+ 14._
+
+
+=1906.= 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.--LAGRANGE.
+
+ _Mechanique Analytique, Preface;
+ Oeuvres, t. 2 (Paris, 1888), p. 14._
+
+
+=1907.= 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....
+
+The objections which have been raised against the infinitesimal
+method are based on the false supposition that the errors due to
+neglecting infinitely small quantities during the actual
+calculation will continue to exist in the result of the
+calculation.--CARNOT, L.
+
+ _Reflections sur la Metaphysique du
+ Calcul Infinitesimal (Paris, 1813), p.
+ 215._
+
+
+=1908.= A limiting ratio is neither more nor less difficult to
+define than an infinitely small quantity.--CARNOT, L.
+
+ _Reflections sur la Metaphysique du
+ Calcul Infinitesimal (Paris, 1813), p.
+ 210._
+
+
+=1909.= 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.
+
+ --WHEWELL, W.
+
+ _The Philosophy of the Inductive
+ Sciences, Part 1, bk. 2, chap. 12, sect.
+ 1, (London, 1858)._
+
+
+=1910.= The differential calculus has all the exactitude of other
+algebraic operations.--LAPLACE.
+
+ _Theorie Analytique des Probabilites,
+ Introduction; Oeuvres, t. 7 (Paris,
+ 1886), p. 37._
+
+
+=1911.= 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.--HUTTON, CHARLES.
+
+ _A Philosophical and Mathematical
+ Dictionary (London, 1815), Vol. 1, p.
+ 525._
+
+
+=1912.= The states and conditions of matter, as they occur in
+nature, are in a state of perpetual flux, and these qualities
+may 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.--MELLOR, J. W.
+
+ _Higher Mathematics for Students of
+ Chemistry and Physics (London, 1902),
+ Prologue._
+
+
+=1913.= The calculus is the greatest aid we have to the
+appreciation of physical truth in the broadest sense of the word.
+
+ --OSGOOD, W. F.
+
+ _Bulletin American Mathematical Society,
+ Vol. 13 (1907), p. 467._
+
+
+=1914.= [Infinitesimal] analysis is the most powerful weapon of
+thought yet devised by the wit of man.--SMITH, W. B.
+
+ _Infinitesimal Analysis (New York,
+ 1898), Preface, p. vii._
+
+
+=1915.= 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.--BERKELEY, GEORGE.
+
+ _The Analyst, sect. 3._
+
+
+=1916.= 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.--DE MORGAN, A.
+
+ _Graves' Life of W. R. Hamilton (New
+ York, 1882-1889), Vol. 3, p. 479._
+
+
+=1917.= Pupils should be taught how to differentiate and how to
+integrate simple algebraic expressions before we attempt to
+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.
+
+ --THOMPSON, S. P.
+
+ _Perry's Teaching of Mathematics
+ (London, 1902), p. 49._
+
+
+=1918.= 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....
+
+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.--KLEIN, F.
+
+ _Jahresbericht der Deutschen
+ Mathematiker Vereinigung, Bd. 2 (1902),
+ p. 131._
+
+
+=1919.= Common integration is only the _memory of
+differentiation_ ... 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.--DE MORGAN, A.
+
+ _Transactions Cambridge Philosophical
+ Society, Vol. 8 (1844), p. 188._
+
+
+=1920.= 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.--LAPLACE.
+
+ _Theorie Analytique des Probabilites,
+ Introduction; Oeuvres, t. 7 (Paris,
+ 1886), p. 6._
+
+
+=1921.= 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?--BOIS-REYMOND, EMIL DU.
+
+ _Reden, Bd. 1 (Leipzig, 1885), p. 228._
+
+
+=1922.= 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.--SYLVESTER, J. J.
+
+ _Notes to the Meditation on Poncelet's
+ Theorem; Mathematical Papers, Vol. 2, p.
+ 214._
+
+
+=1923.= The experimental verification of a theory concerning
+any natural phenomenon generally rests on the result of an
+integration.--MELLOR, J. W.
+
+ _Higher Mathematics for Students of
+ Chemistry and Physics (New York, 1902),
+ p. 150._
+
+
+=1924.= 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....--LIE, SOPHUS.
+
+ _Leipziger Berichte, 47 (1895);
+ Math.-phys. Classe, p. 262._
+
+
+=1925.= 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.--MELLOR, J. W.
+
+ _Higher Mathematics for Students of
+ Chemistry and Physics (New York, 1902),
+ p. 157._
+
+
+=1926.= 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.--KLEIN, F.
+
+ _Lectures on Mathematics (New York,
+ 1911), p. 8._
+
+
+=1927.= 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.--KLEIN, F.
+
+ _Elementar Mathematik vom hoeheren
+ Standpunkte aus. (Leipzig. 1909) Vol. 2,
+ p. 354._
+
+
+=1928.= 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 crust, as
+subjects in their generality intractable without it, is to give
+but a feeble idea of its importance.--THOMSON AND TAIT.
+
+ _Elements of Natural Philosophy, chap.
+ 1._
+
+
+=1929.= 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.
+
+ --BYERLY, W. E.
+
+ _Integral Calculus (Boston, 1890), p.
+ 30._
+
+
+=1930.= 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....--WALCOTT, C. D.
+
+ _Smithsonian Mathematical Tables,
+ Hyperbolic Functions (Washington, 1909),
+ Advertisement._
+
+
+=1931.= 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.--SYLVESTER, J. J.
+
+ _Philosophic Magazine, Vol. 31 (1866),
+ p. 521._
+
+
+=1932.= 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.--MERZ, J. T.
+
+ _A History of European Thought in the
+ Nineteenth Century (Edinburgh and
+ London, 1903), p. 696._
+
+
+=1933.= That flower of modern mathematical thought--the notion of
+a function.--MCCORMACK, THOMAS J.
+
+ _On the Nature of Scientific Law and
+ Scientific Explanation, Monist, Vol. 10
+ (1899-1900), p. 555._
+
+
+=1934.=
+
+ Fuchs. Ich bin von alledem so consterniert,
+ Als wuerde mir ein Kreis im Kopfe quadriert.
+
+ Meph. Nachher vor alien andern Sachen
+ Muesst ihe euch an die Funktionen-Theorie machen.
+ Da seht, dass ihr tiefsinnig fasst,
+ Was sich zu integrieren nicht passt.
+ An Theoremen wird's euch nicht fehlen,
+ Muesst nur die Verschwindungspunkte zaehlen,
+ Umkehren, abbilden, auf der Eb'ne 'rumfahren
+ Und mit den Theta-Produkten nicht sparen.
+ --LASSWITZ, KURD.
+
+ _Der Faust-Tragoedie (-n)ter Tiel;
+ Zeitschrift fuer den math.-natur.
+ Unterricht, Bd. 14 (1883), p. 316._
+
+ Fuchs. Your words fill me with an awful dread,
+ Seems like a circle were squared in my head.
+
+ Meph. Next in order you certainly ought
+ On function-theory bestow your thought,
+ And penetrate with contemplation
+ What resists your attempts at integration.
+ You'll find no dearth of theorems there--
+ To vanishing-points give proper care--
+ Enumerate, reciprocate,
+ Nor forget to delineate,
+ Traverse the plane from end to end,
+ And theta-functions freely spend.
+
+
+=1935.= The student should avoid _founding results_ 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
+discovery, provided that their results be verified by other means
+before they are considered as established.--DE MORGAN, A.
+
+ _Trigonometry and Double Algebra
+ (London, 1849), p. 55._
+
+
+=1936.= 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.--DE MORGAN, A.
+
+ _Graves' Life of W. R. Hamilton (New
+ York, 1882-1889), Vol. 3, p. 249._
+
+
+=1937.= 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.--PIERPONT, J.
+
+ _Congress of Arts and Sciences (Boston
+ and New York, 1905), Vol. 1, p. 476._
+
+
+=1938.= 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.--RUSSELL, BERTRAND.
+
+ _International Monthly, Vol. 4 (1901),
+ p. 89._
+
+
+=1939.= 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.--HELMHOLTZ, H.
+
+ _Aim and Progress of Physical Science;
+ Popular Lectures [Flight] (New York,
+ 1900), p. 372._
+
+
+=1940.= 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.--PIERCE, C. F.
+
+ _The Law of Mind; Monist, Vol. 2
+ (1891-1892), pp. 543, 545._
+
+
+=1941.= The chief objection against all _abstract_ 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
+_dogmas_, 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 _in infinitum_; 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 _in infinitum_, this angle of
+contact becomes still less, even _in infinitum_, 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, _in infinitum_? 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 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.--HUME, DAVID.
+
+ _An Inquiry concerning Human
+ Understanding, Sect. 12, part 2._
+
+
+=1942.= He who can digest a second or third fluxion, a second or
+third difference, need not, methinks, be squeamish about any
+point in Divinity.--BERKELEY, G.
+
+ _The Analyst, sect. 7._
+
+
+=1943.= 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?--BERKELEY, G.
+
+ _The Analyst, sect. 35._
+
+
+=1944.= 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.,
+_ad infinitum_. 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 _status nascenti_, in their first origin or
+beginning to exist, before they become finite particles. And it
+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.--BERKELEY, G.
+
+ _The Analyst, sect, 4._
+
+
+=1945.= The _infinite_ divisibility of _finite_ 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.--BERKELEY, G.
+
+ _On the Principles of Human Knowledge,
+ Sect. 123._
+
+
+=1946.= 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....
+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 _variable indefinitely small quantities_. 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.
+
+ --WILLIAMSON, B.
+
+ _Encyclopedia Britannica, 9th Edition;
+ Article "Infinitesimal Calculus," Sect.
+ 12, 14._
+
+
+=1947.= 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.
+
+ --VOLTAIRE.
+
+ _A Philosophical Dictionary; Article
+ "Infinity." (Boston, 1881)._
+
+
+=1948.= 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!--CARUS, PAUL.
+
+ _Logical and Mathematical Thought;
+ Monist, Vol. 20 (1909-1910), p. 69._
+
+
+=1949.=
+
+ Great fleas have little fleas upon their backs
+ to bite 'em,
+ And little fleas have lesser fleas, and so _ad
+ infinitum._
+ And the great fleas themselves, in turn, have
+ greater fleas to go on;
+ While these again have greater still, and
+ greater still, and so on.
+ --DE MORGAN, A.
+
+ _Budget of Paradoxes (London, 1872), p.
+ 377._
+
+
+=1950.= We have adroitly defined the infinite in arithmetic by a
+loveknot, in this manner [infinity]; but we possess not therefore
+the clearer notion of it.--VOLTAIRE.
+
+ _A Philosophical Dictionary; Article
+ "Infinity." (Boston, 1881)._
+
+
+=1951.= I protest against the use of infinite magnitude as
+something completed, which in mathematics is never permissible.
+Infinity is merely a _facon de parler_, the real meaning being a
+limit which certain ratios approach indefinitely near, while
+others are permitted to increase without restriction.--GAUSS.
+
+ _Brief an Schumacher (1831); Werke, Bd.
+ 8 p. 216._
+
+
+=1952.= In spite of the essential difference between the
+conceptions of the _potential_ and the _actual_ infinite, the
+former signifying a _variable_ finite magnitude increasing beyond
+all finite limits, while the latter is a _fixed_, _constant_
+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 _illegitimate_ actual infinities and
+the influence of the modern epicuric-materialistic tendency, a
+certain _horror infiniti_ 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.--CANTOR, G.
+
+ _Zum Problem des actualen Unendlichen;
+ Natur und Offenbarung, Bd. 32 (1886), p.
+ 226._
+
+
+=1953.= 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 _withdrawing_ 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
+Infinite is a sign of that continuity of Existence which has been
+ideally divided into discrete parts in the affixing of limits.
+
+ --LEWES, G. H.
+
+ _Problems of Life and Mind (Boston,
+ 1875), Vol. 2, p. 384._
+
+
+=1954.= A great deal of misunderstanding is avoided if it be
+remembered that the terms _infinity_, _infinite_, _zero_,
+_infinitesimal_ must be interpreted in connexion with their
+context, and admit a variety of meanings according to the way in
+which they are defined.--MATHEWS, G. B.
+
+ _Theory of Numbers (Cambridge, 1892),
+ Part 1, sect. 104._
+
+
+=1955.= 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 _expansion_ and _duration_;
+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?
+
+ --LOCKE, JOHN.
+
+ _An Essay concerning Human
+ Understanding, Bk. 2, chap. 16, sect.
+ 8._
+
+
+=1956.= 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.--LOCKE, JOHN.
+
+ _An Essay concerning Human
+ Understanding, Bk. 2, chap. 17, sect.
+ 9._
+
+
+=1957.=
+
+ Willst du ins Unendliche schreiten?
+ Geh nur im Endlichen nach allen Seiten!
+ Willst du dich am Ganzen erquicken,
+ So musst du das Ganze im Kleinsten erblicken.
+ --GOETHE.
+
+ _Gott, Gemuet und Welt (1815)._
+
+ [Would'st thou the infinite essay?
+ The finite but traverse in every way.
+ Would'st in the whole delight thy heart?
+ Learn to discern the whole in its minutest part.]
+
+
+=1958.=
+
+ Ich haeufe ungeheure Zahlen,
+ Gebuerge Millionen auf,
+ Ich setze Zeit auf Zeit und Welt auf Welt zu Hauf,
+ Und wenn ich von der grausen Hoeh'
+ Mit Schwindeln wieder nach dir seh,'
+ Ist alle Macht der Zahl, vermehrt zu tausendmalen,
+ Noch nicht ein Theil von dir.
+ _Ich zieh' sie ab, und du liegst ganz vor mir._
+ --HALLER, ALBR. VON.
+
+ _Quoted in Hegel: Wissenschaft der
+ Logik, Buch 1, Abschnitt 2, Kap. 2, C,
+ b._
+
+ [Numbers upon numbers pile,
+ Mountains millions high,
+ Time on time and world on world amass,
+ Then, if from the dreadful hight, alas!
+ Dizzy-brained, I turn thee to behold,
+ All the power of number, increased thousandfold,
+ Not yet may match thy part.
+ _Subtract what I will, wholly whole thou art._]
+
+
+=1959.= 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.--RUSSELL, BERTRAND.
+
+ _International Monthly, Vol. 4 (1901),
+ p. 93._
+
+
+=1960.= 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 _equivalent_
+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.--KEYSER, C. J.
+
+ _The Axioms of Infinity; Hibbert
+ Journal, Vol. 2 (1903-1904), p. 539._
+
+
+=1961.= 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 the structure of Transfinite Being is open to exploration
+by the organon of Thought.--KEYSER, C. J.
+
+ _Lectures on Science, Philosophy and Art
+ (New York, 1908), p. 42._
+
+
+=1962.= 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.--CROFTON, M. W.
+
+ _Encyclopedia Britannica, 9th Edition;
+ Article, "Probability."_
+
+
+=1963.= 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.--LAPLACE.
+
+ _Theorie Analytique des Probabilities,
+ Introduction; Oeuvres, t. 7 (Paris,
+ 1886), p. 153._
+
+
+=1964.= 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.--DESCARTES.
+
+ _Discourse on Method, Part 3._
+
+
+=1965.= As _demonstration_ is the showing the agreement or
+disagreement of two ideas, by the intervention of one or more
+proofs, which have a constant, immutable, and visible connexion
+one with another; so _probability_ 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.--LOCKE, JOHN.
+
+ _An Essay concerning Human
+ Understanding, Bk. 4, chap. 15, sect.
+ 1._
+
+
+=1966.= 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.--LEIBNITZ.
+
+ _Philosophische Schriften [Gerhardt] Bd.
+ 7 (Berlin, 1890), p. 200._
+
+
+=1967.= 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.--CHRYSTAL, GEORGE.
+
+ _Algebra, Vol. 2 (Edinburgh, 1889),
+ chap. 36, sect. 1._
+
+
+=1968.= 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 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."--DAVIS, E. W.
+
+ _Bulletin American Mathematical Society,
+ Vol. 1 (1894-1895), p. 16._
+
+
+=1969.= 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.--LAPLACE.
+
+ _Theorie Analytique des Probabilities,
+ Introduction; Oeuvres, t. 7 (Paris,
+ 1886), p. 5._
+
+
+=1970.= 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.--CROFTON, M. W.
+
+ _Encyclopedia Britannica, 9th Edition;
+ Article "Probability."_
+
+
+=1971.= 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.--LAPLACE.
+
+ _Theorie Analytique des Probabilities,
+ Introduction; Oeuvres, t. 7 (Paris,
+ 1886), p. 152._
+
+
+=1972.= Not much has been added to the subject [of probability]
+since the close of Laplace's career. The history of science
+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.--CROFTON, M. W.
+
+ _Encyclopedia Britannica, 9th Edition;
+ Article "Probability."_
+
+
+=1973.= 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.--WOODWARD, R. S.
+
+ _Probability and Theory of Errors (New
+ York, 1906), p. 30._
+
+
+=1974.= 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 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.--WOODWARD, R. S.
+
+ _Probability and Theory of Errors (New
+ York, 1906), pp. 9-10._
+
+
+=1975.= 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.--REID, THOMAS.
+
+ _Essays on the Powers of the Human Mind
+ (Edinburgh, 1812), Vol. 2, p. 408._
+
+
+
+
+ CHAPTER XX
+
+ THE FUNDAMENTAL CONCEPTS, TIME AND SPACE
+
+
+=2001.= Kant's Doctrine of Time.
+
+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 _a
+priori_. Only when this representation _a priori_ is given,
+can we imagine that certain things happen at the same time
+(simultaneously) or at different times (successively).
+
+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.
+
+III. On this _a priori_ 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....
+
+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....
+
+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.--KANT, I.
+
+ _Critique of Pure Reason [Max Mueller]
+ (New York, 1900), pp. 24-25._
+
+
+=2002.= Kant's Doctrine of Space.
+
+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....
+
+II. Space is a necessary representation _a priori_, 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 _a priori_ which necessarily precedes all external
+phenomena.
+
+III. On this necessity of an _a priori_ representation of space
+rests the apodictic certainty of all geometrical principles, and
+the possibility of their construction _a priori_. For if the
+intuition of space were a concept gained _a posteriori_, 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.
+
+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 space, an intuition _a priori_, which is not
+empirical, must form the foundation of all conceptions of
+space....
+
+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 _intuition a priori_, and not a concept.--KANT, I.
+
+ _Critique of Pure Reason [Max Mueller]
+ (New York, 1900), pp. 18-20 and
+ Supplement 8._
+
+
+=2003.=
+
+ _Schopenhauer's Predicabilia a priori._[11]
+
+ OF TIME OF SPACE
+
+ 1. There is but _one time_, all 1. There is but _one space_,
+ different times are parts of all different spaces are
+ it. parts of it.
+
+ 2. Different times are not 2. Different spaces are not
+ simultaneous but successive. successive but
+ simultaneous.
+
+ 3. Everything in time may be 3. Everything in space may be
+ thought of as non-existent, thought of as non-existent,
+ but not time. but not space.
+
+ 4. Time has three divisions: 4. Space has three dimensions:
+ past, present and future, height, breadth, and
+ which form two directions length.
+ with a point of indifference.
+
+ 5. Time is infinitely 5. Space is infinitely
+ divisible. divisible.
+
+ 6. Time is homogeneous and a 6. Space is homogeneous and a
+ continuum: i.e. no part is continuum: i.e. no part
+ different from another, nor is different from another,
+ separated by something nor separated by something
+ which is not time. which is not space.
+
+ 7. Time has no beginning nor 7. Space has no limits
+ end, but all beginning and [Graenzen], but all limits
+ end is in time. are in space.
+
+ 8. Time makes counting 8. Space makes measurement
+ possible. possible.
+
+ 9. Rhythm exists only in time. 9. Symmetry exists only in
+ space.
+
+ 10. The laws of time are _a 10. The laws of space are _a
+ priori_ conceptions. priori_ conceptions.
+
+ 11. Time is perceptible _a 11. Space is immediately
+ priori_, but only by a perceptible _a priori_.
+ means of a line-image.
+
+ 12. Time has no permanence but 12. Space never passes but is
+ passes the moment it is permanent throughout
+ present. all time.
+
+ 13. Time never rests. 13. Space never moves.
+
+ 14. Everything in time has 14. Everything in space has
+ duration. position.
+
+ 15. Time has no duration, but 15. Space has no motion, but
+ all duration is in time; all motion is in space;
+ time is the persistence of space is the change in
+ what is permanent in position of that which
+ contrast with its restless moves in contrast to its
+ course. imperturbable rest.
+
+ 16. Motion is only possible in 16. Motion is only possible in
+ time. space.
+
+ 17. Velocity, the space being 17. Velocity, the time being
+ the same, is in the inverse the same, is in the direct
+ ratio of the time. ratio of the space.
+
+ 18. Time is not directly 18. Space is measurable directly
+ measurable by means of through itself and
+ itself but only by means of indirectly through motion
+ motion which takes place in which takes place in both
+ both space and time.... time and space....
+
+ 19. Time is omnipresent: each 19. Space is eternal: each
+ part of it is everywhere. part of it exists always.
+
+ 20. In time alone all things 20. In space alone all things
+ are successive. are simultaneous.
+
+ 21. Time makes possible the 21. Space makes possible the
+ change of accidents. endurance of substance.
+
+ 22. Each part of time contains 22. No part of space contains
+ all substance. the same substance as
+ another.
+
+ 23. Time is the _principium 23. Space is the _principium
+ individuationis_. individuationis_.
+
+ 24. The now is without 24. The point is without
+ duration. extension.
+
+ 25. Time of itself is empty and 25. Space is of itself empty
+ indeterminate. and indeterminate.
+
+ 26. Each moment is conditioned 26. The relation of each
+ by the one which precedes boundary in space to every
+ it, and only so far as this other is determined by its
+ one has ceased to exist. relation to any one.
+ (Principle of sufficient (Principle of sufficient
+ reason of being in time.) reason of being in space.)
+
+ 27. Time makes Arithmetic 27. Space makes Geometry
+ possible. possible.
+
+ 28. The simple element of 28. The element of Geometry
+ Arithmetic is unity. is the point.
+ --SCHOPENHAUER, A.
+
+ _Die Welt als Vorstellung und Wille;
+ Werke (Frauenstaedt) (Leipzig, 1877), Bd.
+ 2, p. 55._
+
+ [11] Schopenhauer's table contains a third column
+ headed "of matter" which has here been omitted.
+
+
+=2004.= 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.--WHEWELL, WILLIAM.
+
+ _The Philosophy of the Inductive
+ Sciences, Part 1, Bk. 2, chap. 4, sect.
+ 4 (London, 1858)._
+
+
+=2005.= Geometrical axioms are neither synthetic _a priori_
+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.
+
+That being so what ought one to think of this question: Is the
+Euclidean Geometry true?
+
+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.
+
+ --POINCARE, H.
+
+ _Non-Euclidean Geometry; Nature, Vol 45
+ (1891-1892), p. 407._
+
+
+=2006.= 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.--KLEIN, F.
+
+ _Elementarmathematik vom hoeheren
+ Standpunkte aus (Leipzig, 1909), Bd. 2,
+ p. 384._
+
+
+=2007.= Euclid's Postulate 5 [The Parallel Axiom].
+
+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.--EUCLID.
+
+ _The Thirteen Books of Euclid's Elements
+ [T. L. Heath] Vol. 1 (Cambridge, 1908),
+ p. 202._
+
+
+=2008.= 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....--HALL, H. S. and STEVENS, F. H.
+
+ _Euclid's Elements (London, 1892), p.
+ 55._
+
+
+=2009.= 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.--KLEIN, F.
+
+ _Elementarmathematik vom, hoeheren
+ Standpunkte aus (Leipzig, 1909), Bd. 2,
+ p. 382._
+
+
+=2010.= 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.--BALL, R. S.
+
+ _Encyclopedia Britannica, 9th Edition;
+ Article "Measurement."_
+
+
+=2011.= Mathematical and physiological researches have shown that
+the space of experience is simply an _actual_ case of many
+conceivable cases, about whose peculiar properties experience
+alone can instruct us.--MACH, ERNST.
+
+ _Popular Scientific Lectures (Chicago,
+ 1910), p. 205._
+
+
+=2012.= 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.--WILSON, E. B.
+
+ _Bulletin American Mathematical Society,
+ Vol. 2 (1904-1905), p. 81._
+
+
+=2013.= 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.--RUSSELL, BERTRAND.
+
+ _Recent Work on the Principles of
+ Mathematics; International Monthly, Vol.
+ 4 (1901), p. 86._
+
+
+=2014.= 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 is
+concerned, the famous problem of the parallel is settled for all
+time.--YOUNG, JOHN WESLEY.
+
+ _Fundamental Concepts of Algebra and
+ Geometry (New York, 1911), p. 32._
+
+
+=2015.= 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.--CLIFFORD, W. K.
+
+ _Lectures and Essays (New York, 1901),
+ Vol. 1, pp. 358-359._
+
+
+=2016.= 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.--CARUS, PAUL.
+
+ _Science, Vol. 18 (1903), p. 106._
+
+
+=2017.= 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, 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.
+
+ --KLEIN, F.
+
+ _Elementarmathematik vom hoeheren
+ Standpunkte aus (Leipzig, 1909), Bd. 2,
+ p. 377._
+
+
+=2018.= 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.
+
+ --RUSSELL, BERTRAND.
+
+ _Foundations of Geometry (Cambridge,
+ 1897), p. 6._
+
+
+=2019.= The most suggestive and notable achievement of the last
+century is the discovery of Non-Euclidean geometry.--HILBERT, D.
+
+ _Quoted by G. D. Fitch in Manning's "The
+ Fourth Dimension Simply Explained," (New
+ York, 1910), p. 58._
+
+
+=2020.= Non-Euclidean geometry--primate among the emancipators of
+the human intellect....--KEYSER, C. J.
+
+ _The Foundations of Mathematics; Science
+ History of the Universe, Vol. 8 (New
+ York, 1909), p. 192._
+
+
+=2021.= 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 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.
+
+ --KLEIN, F.
+
+ _Elementarmathematik vom hoeheren
+ Standpunkte_ aus _(Leipzig, 1909), Bd.
+ 2, p. 378._
+
+
+=2022.= 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.--CLIFFORD, W. K.
+
+ _Lectures and Essays (New York, 1901),
+ Vol. 1, pp. 356, 358._
+
+
+=2023.= 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 _nothing_, 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 always be below a
+certain limit. I have found several other such theorems, but none
+of them satisfies me.--GAUSS.
+
+ _Letter to Bolyai (1799); Werke, Bd. 8
+ (Goettingen, 1900), p. 159._
+
+
+=2024.= 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.--GAUSS.
+
+ _Letter to Gerling (1816); Werke, Bd. 8
+ (Goettingen, 1900), p. 169._
+
+
+=2025.= I am convinced more and more that the necessary truth of
+our geometry cannot be demonstrated, at least not _by_ the
+_human_ intellect _to_ 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.--GAUSS.
+
+ _Letter to Olbers (1817); Werke, Bd. 8
+ (Goettingen, 1900), p. 177._
+
+
+=2026.= There is no doubt that it can be rigorously established
+that the sum of the angles of a rectilinear triangle cannot
+exceed 180 deg. But it is otherwise with the statement that the sum
+of the angles cannot be less than 180 deg.; 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 deg. leads to a peculiar geometry, entirely
+different from the Euclidean, but throughout consistent with
+itself. I have developed this geometry to my own satisfaction 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 _definitely
+determinate_ (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 _absolutely impossible._ 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.
+
+ --GAUSS.
+
+ _Letter to Taurinus (1824); Werke, Bd. 8
+ (Goettingen, 1900), p. 187._
+
+
+=2027.= 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 _very extended_
+investigations concerning this matter in shape for publication,
+possibly not while I live, for I fear the cry of the Boeotians
+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 two points. This definition contains more than
+is necessary to the determination of the surface, and tacitly
+involves a theorem which demands proof.--GAUSS.
+
+ _Letter to Bessel (1829); Werke, Bd. 8
+ (Goettingen, 1900), p. 200._
+
+
+=2028.= I will add that I have recently received from Hungary a
+little paper on Non-Euclidean geometry, in which I rediscover all
+_my own ideas_ and _results_ 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.--GAUSS.
+
+ _Letter to Gerling (1832); Werke, Bd. 8
+ (Goettingen, 1900), p. 221._
+
+
+=2029.= 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. 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.--HELMHOLTZ, H.
+
+ _On the Origin and Significance of
+ Geometrical Axioms; Popular Scientific
+ Lectures, second series (New York,
+ 1881), pp. 57-59._
+
+
+=2030.= 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.--KLEIN, F.
+
+ _Mathematische Annalen, Bd. 43 (1893),
+ p. 95._
+
+
+=2031.= We are led naturally to extend the language of geometry
+to the case of any number of variables, still using the word
+_point_ to designate any system of values of n variables (the
+coordinates of the point), the word _space_ (of n dimensions) to
+designate the totality of all these points or systems of values,
+_curves_ or _surface_ to designate the spread composed of
+points whose coordinates are given functions (with the proper
+restrictions) of one or two parameters (the _straight line_ or
+_plane_, 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.
+
+ --SEGRE, CORRADI.
+
+ _Rivista di Matematica, Vol. I (1891),
+ p. 59. [J. W. Young.]_
+
+
+=2032.= 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 _impossible_ dimension, subject
+to all the laws of the three we find possible. And just as
+[sq root]-1 in common algebra, gives all its _significant_
+combinations _true_, so would it be with any number of dimensions
+of space which the speculator might choose to call into
+_impossible_ existence.--DE MORGAN, A.
+
+ _Trigonometry and Double Algebra
+ (London, 1849), Part 2, chap. 3._
+
+
+=2033.= 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.--SMITH, W. B.
+
+ _Introductory Modern Geometry (New York,
+ 1893), p. 274._
+
+
+=2034.= According to his frequently expressed view, Gauss considered
+the three dimensions of space as specific peculiarities of the
+human soul; people, which are unable to comprehend this, he
+designated in his humorous mood by the name Boeotians. 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.--SARTORIUS, W. V. WALTERSHAUSEN.
+
+ _Gauss zum Gedaechtniss (Leipzig, 1856),
+ p. 81._
+
+
+=2035.= _There is many a rational logos_, and the mathematician
+has high delight in the contemplation of _in_consistent _systems_
+of _consistent relationships_. 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.--KEYSER, C. J.
+
+ _The Universe and Beyond; Hibbert
+ Journal, Vol. 3 (1904-1905), p. 313._
+
+
+=2036.= 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.
+
+ --MERZ, J. T.
+
+ _History of European Thought in the
+ Nineteenth Century (Edinburgh and
+ London, 1903), p. 716._
+
+
+=2037.= 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 great
+concept designated ... hyperspace or multidimensional space.
+
+ --KEYSER, C. J.
+
+ _Mathematical Emancipations; Monist,
+ Vol. 16 (1906), p. 65._
+
+
+=2038.= 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.--KEYSER, C. J.
+
+ _Mathematical Emancipations; Monist,
+ Vol. 16 (1906), p. 83._
+
+
+=2039.= 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.--OSTWALD, W.
+
+ _Natural Philosophy [Seltzer], (New
+ York, 1910), p. 77._
+
+
+=2040.=
+
+ Fuchs. Was soll ich nun aber denn studieren?
+
+ Meph. Ihr koennt es mit _analytischer Geometrie_ probieren.
+ Da wird der Raum euch wohl dressiert,
+ In Coordinaten eingeschnuert,
+ Dass ihr nicht etwa auf gut Glueck
+ Von der Figur gewinnt ein Stueck.
+ Dann lehret man euch manchen Tag,
+ Dass, was ihr sonst auf einen Schlag
+ Construiertet im Raume frei,
+ Eine Gleichung dazu noetig sei.
+ Zwar war dem Menschen zu seiner Erbauung
+ Die dreidimensionale Raumanschauung,
+ Dass er sieht, was um ihn passiert,
+ Und die Figuren sich construiert--
+ Der Analytiker tritt herein
+ Und beweist, das koennte auch anders sein.
+ Gleichungen, die auf dem Papiere stehn,
+ Die muesst' man auch koennen im Raume sehn;
+ Und koennte man's nicht construieren,
+ Da muesste man's anders definieren.
+ Denn was man formt nach Zahlengesetzen
+ Muesst' uns auch geometrisch erletzen.
+ Drum in den unendlich fernen beiden
+ Imaginaeren Punkten muessen sich schneiden
+ Alle Kreise fein saeuberlich,
+ Auch Parallelen, die treffen sich,
+ Und im Raume kann man daneben
+ Allerlei Kruemmungsmasse erleben.
+ Die Formeln sind alle wahr und schoen,
+ Warum sollen sie nicht zu deuten gehn?
+ Da preisen's die Schueler aller Orten,
+ Dass das Gerade ist krumm geworden.
+ _Nicht-Euklidisch_ nennt's die Geometrie,
+ Spotted ihrer selbst, und weiss nicht wie.
+
+ Fuchs. Kann euch nicht eben ganz verstehn.
+
+ Meph. Das soll den Philosophen auch so gehn.
+ Doch wenn ihr lernt alles reducieren
+ Und gehoerig transformieren,
+ Bis die Formeln den Sinn verlieren,
+ Dann versteht ihr mathematish zu spekulieren.
+ --LASSWITZ, KURD.
+
+ _Der Faust-Tragoedie (-n)ter Teil;
+ Zeitschrift fuer den math-naturw.
+ Unterricht, Bd. 14 (1888), p. 316._
+
+ [Fuchs. To what study then should I myself apply?
+
+ Meph. Begin with _analytical geometry_.
+ There all space is properly trained,
+ By coordinates well restrained,
+ That no one by some lucky assay
+ Carry some part of the figure away.
+ Next thou'll be taught to realize,
+ Constructions won't help thee to geometrize,
+ And the result of a free construction
+ Requires an equation for proper deduction.
+ Three-dimensional space relation
+ Exists for human edification,
+ That he may see what about him transpires,
+ And construct such figures as he requires.
+ Enters the analyst. Forthwith you see
+ That all this might otherwise be.
+ Equations, written with pencil or pen,
+ Must be visible in space, and when
+ Difficulties in construction arise,
+ We need only define it otherwise.
+ For, what is formed after laws arithmetic
+ Must also yield some delight geometric.
+ Therefore we must not object
+ That all circles intersect
+ In the circular points at infinity.
+ And all parallels, they declare,
+ If produced must meet somewhere.
+ So in space, it can't be denied,
+ Any old curvature may abide.
+ The formulas are all fine and true,
+ Then why should they not have a meaning too?
+ Pupils everywhere praise their fate
+ That that now is crooked which once was straight.
+ Non-Euclidean, in fine derision,
+ Is what it's called by the geometrician.
+
+ Fuchs. I do not fully follow thee.
+
+ Meph. No better does philosophy.
+ To master mathematical speculation,
+ Carefully learn to reduce your equation
+ By an adequate transformation
+ Till the formulas are devoid of interpretation.]
+
+
+
+
+ CHAPTER XXI
+
+ PARADOXES AND CURIOSITIES
+
+
+=2101.= 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.
+
+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.--DE MORGAN, A.
+
+ _Budget of Paradoxes (London, 1872), p.
+ 473._
+
+
+=2102.= Proof requires a person who can give and a person who can
+receive....
+
+ A blind man said, As to the Sun,
+ I'll take my Bible oath there's none;
+ For if there had been one to show
+ They would have shown it long ago.
+ How came he such a goose to be?
+ Did he not know he couldn't see?
+ Not he.
+ --DE MORGAN, A.
+
+ _Budget of Paradoxes (London, 1872), p.
+ 262._
+
+
+=2103.= 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 _ignes fatui_--delusive phantoms, that
+float before us, and seem so fair, and are _all but_ in our
+grasp, so nearly that it never seems to need more than _one_ step
+further, and the prize shall be ours! Alas for him who has been
+turned 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!--DODGSON, C. L.
+
+ _A new Theory of Parallels (London,
+ 1895), Introduction._
+
+
+=2104.= 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.--SYLVESTER, J. J.
+
+ _On a Lady's Fan etc. Collected
+ Mathematical Papers, Vol. 3, p. 36._
+
+
+=2105.= 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.
+
+ --DE MORGAN, A.
+
+ _A Budget of Paradoxes (London, 1872),
+ p. 484._
+
+
+=2106.= Demonstrative reason never raises the cry of _Church in
+Danger!_ 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.--DE MORGAN, A.
+
+ _A Budget of Paradoxes (London, 1872),
+ p. 485._
+
+
+=2107.= 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 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!--DE MORGAN, A.
+
+ _A Budget of Paradoxes (London, 1872),
+ p. 71._
+
+
+=2108.= 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.--DE MORGAN, A.
+
+ _A Budget of Paradoxes (London, 1872),
+ p. 96._
+
+
+=2109.= 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.
+
+ --DE MORGAN, A.
+
+ _A Budget of Paradoxes (London, 1872),
+ p. 70._
+
+
+=2110.= 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.--BOLZANO, BERNARD.
+
+ _Selbstbiographie (Wien, 1875), p. 19._
+
+
+=2111.= The Theory of Parallels.
+
+It is known that to complete the theory it is only necessary to
+demonstrate the following proposition, which Euclid assumed as an
+axiom:
+
+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.
+
+[Illustration: A geometrical drawing of parallel lines and
+intersecting lines to accompany proof.]
+
+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.
+
+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.
+
+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.
+
+But if ACE is greater than ACBD, CE and BD must intersect, for
+otherwise ACE would be a part of ACBD.
+
+ _Journal fuer Mathematik, Bd. 2 (1834),
+ p. 198._
+
+
+=2112.= Are you sure that it is impossible to trisect the angle
+by _Euclid_? I have not to lament a single hour thrown away on
+the 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 _Gauss's_ inscription of the regular polygon of seventeen
+sides have seemed, a century ago, much less an impossible thing,
+by line and circle?--HAMILTON, W. R.
+
+ _Letter to De Morgan (1852)._
+
+
+=2113.= 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.
+
+ --NEWCOMB, SIMON.
+
+ _The Reminiscences of an Astronomer
+ (Boston and New York, 1903), p. 388._
+
+
+=2114.= 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 _screw_. 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.--DE MORGAN, A.
+
+ _Quoted in Graves' Life of Sir W. R.
+ Hamilton, Vol. 3 (New York, 1889), p.
+ 342._
+
+
+=2115.=
+
+ Mad Mathesis alone was unconfined,
+ Too mad for mere material chains to bind,
+ Now to pure space lifts her ecstatic stare,
+ Now, running round the circle, finds it square.
+ --POPE, ALEXANDER.
+
+ _The Dunciad, Bk. 4, lines 31-34._
+
+
+=2116.=
+
+ Or is't a tart idea, to procure
+ An edge, and keep the practic soul in ure,
+ Like that dear Chymic dust, or puzzling quadrature?
+ --QUARLES, PHILIP.
+
+ _Quoted by De Morgan: Budget of
+ Paradoxes (London, 1872), p. 436._
+
+
+=2117.=
+
+ Quale e'l geometra che tutto s' affige
+ Per misurar lo cerchio, e non ritruova,
+ Pensando qual principio ond' egli indige.
+ --DANTE.
+
+ _Paradise, canto 33, lines 122-125._
+
+ [As doth the expert geometer appear
+ Who seeks to square the circle, and whose skill
+ Finds not the law with which his course to steer.[12]]
+
+ _Quoted in Frankland's Story of Euclid
+ (London, 1902), p. 101._
+
+ [12] For another rendition of these same lines see
+ 1858.
+
+
+=2118.=
+
+ In _Mathematicks_ he was greater
+ Than _Tycho Brahe_, or _Erra Pater_:
+ For he, by _Geometrick_ scale,
+ Could take the size of _Pots of Ale_;
+ Resolve by Signs and Tangents streight,
+ If _Bread_ or _Butter_ wanted weight;
+ And wisely tell what hour o' th' day
+ The Clock doth strike, by _Algebra_.
+ --BUTLER, SAMUEL.
+
+ _Hudibras, Part 1, canto 1, lines_
+ 119-126.
+
+
+=2119.= 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.--COLERIDGE, SAMUEL.
+
+ _A Mathematical Problem._
+
+
+=2120.= At last we entered the palace, and proceeded into the
+chamber of presence where I saw the king seated on his throne,
+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.--SWIFT, JONATHAN.
+
+ _Gulliver's Travels; A Voyage to Laputa;
+ Chap. 2._
+
+
+=2121.= 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
+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.
+
+ --SWIFT, JONATHAN.
+
+ _Gulliver's Travels; A Voyage to Laputa,
+ Chap. 2._
+
+
+=2122.= 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.--SWIFT, JONATHAN.
+
+ _Gulliver's Travels; A Voyage to Laputa,
+ Chap. 2._
+
+
+=2123.= 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 _quantum_ 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.--SWIFT, JONATHAN.
+
+ _Gulliver's Travels; A Voyage to Laputa,
+ Chap. 5._
+
+
+=2124.= 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.
+
+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 _form of a
+cycloid_:" evidently taking the cycloid for a _figure_, instead
+of a _line_. This may help to explain the difficulty he is said
+to have had in obtaining his Degree.--WHATELY, R.
+
+ _Annotations to Bacon's Essays, Essay
+ L._
+
+
+=2125.= 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 _Principia_ 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.
+
+ --WHITEHEAD, A. N.
+
+ _An Introduction to Mathematics (New
+ York, 1911), p. 10._
+
+
+=2126.= [Illustration: A geometrical drawing including square and
+four triangles to demonstrate a graphical proof of the theorem of
+Pythagoras as described in the poem.]
+
+ Here I am as you may see
+ a squared + b squared - ab
+ When two Triangles on me stand
+ Square of hypothen^e is plann'd
+ But if I stand on them instead,
+ The squares of both the sides are read.
+ --AIRY, G. B.
+
+ _Quoted in Graves' Life of Sir W. R.
+ Hamilton, Vol. 3 (New York, 1889), p.
+ 502._
+
+
+=2127.= [pi] = 3.141 592 653 589 793 238 462 643 383 279 ...
+
+ 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.
+ --ORR, A. C.
+
+ _Literary Digest, Vol. 32 (1906), p.
+ 84._
+
+
+=2128.= I take from a biographical dictionary the first five
+names of poets, with their ages at death. They are
+
+ Aagard, died at 48.
+ Abeille, " " 76.
+ Abulola, " " 84.
+ Abunowas, " " 48.
+ Accords, " " 45.
+
+These five ages have the following characters in common:--
+
+1. The difference of the two digits composing the number divided
+by _three_, leaves a remainder of _one_.
+
+2. The first digit raised to the power indicated by the second,
+and then divided by _three_, leaves a remainder of _one_.
+
+3. The sum of the prime factors of each age, including _one_ as a
+prime factor, is divisible by _three_.--PEIRCE, C. S.
+
+ _A Theory of Probable Inference; Studies
+ in Logic (Boston, 1883), p. 163._
+
+
+=2129.= 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.
+
+ --MILLER, G. A.
+
+ _Some Thoughts on Modern Mathematical
+ Research; Science, Vol. 35 (1912), p.
+ 881._
+
+
+=2130.= _If the Indians hadn't spent the $24._ 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 x (1.07)^{280} = more than
+4,042,000,000.
+
+The latest tax assessment available at the time of writing gives
+the realty for the borough of Manhattan as $3,820,754.181. This
+is estimated to be 78% of the actual value, making the actual
+value a little more than $4,898,400,000.
+
+The amount of the Indians' money would therefore be more than the
+present assessed valuation but less than the actual valuation.
+
+ --WHITE, W. F.
+
+ _A Scrap-book of Elementary Mathematics
+ (Chicago, 1908), pp. 47-48._
+
+
+=2131.= See Mystery to Mathematics fly!--POPE, ALEXANDER.
+
+ _The Dunciad, Bk. 4, line 647._
+
+
+=2132.= 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.
+
+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.--REID, THOMAS.
+
+ _Essays on the Powers of the Human Mind
+ (Edinburgh, 1812), Vol. 2, p. 400._
+
+
+=2133.= In the Timaeus [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 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.
+
+ --WHEWELL, W.
+
+ _History of the Inductive Sciences, 3rd
+ Edition, Additions to Bk. 2._
+
+
+=2134.= 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.--KEPLER.
+
+ _Mysterium Cosmographicum [Whewell]._
+
+
+=2135.= 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
+_Republic_, 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.... 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
+astronomy.--WHEWELL, W.
+
+ _History of the Inductive Sciences, 3rd
+ Edition, Additions to Bk. 3._
+
+
+=2136.= 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, [Greek:
+Patroklos], [Greek: Hektor] and [Greek: Achilleus], the sum of
+the numbers in whose names are 861, 1225, and 1276 respectively.
+
+ --PEACOCK, GEORGE.
+
+ _Encyclopedia of Pure Mathematics
+ (London, 1847); Article "Arithmetic,"
+ sect. 38._
+
+
+=2137.= Round numbers are always false.--JOHNSON, SAMUEL.
+
+ _Johnsoniana; Apothegms, Sentiment,
+ etc._
+
+
+=2138.= Numero deus impare gaudet [God in number odd rejoices.]
+
+ --VIRGIL.
+
+ _Eclogue, 8, 77._
+
+
+=2139.= Why is it that we entertain the belief that for every
+purpose odd numbers are the most effectual?--PLINY.
+
+ _Natural History, Bk. 28, chap. 5._
+
+
+=2140.=
+
+ "Then here goes another," says he, "to make sure,
+ Fore there's luck in odd numbers," says Rory O'Moore.
+ --LOVER, S.
+
+ _Rory O'Moore._
+
+
+=2141.= 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.--SHAKESPEARE.
+
+ _The Merry Wives of Windsor, Act 5,
+ scene 1._
+
+
+=2142.= To add to golden numbers, golden numbers.--DECKER, THOMAS.
+
+ _Patient Grissell, Act 1, scene 1._
+
+
+=2143.=
+
+ I've read that things inanimate have moved,
+ And, as with living souls, have been inform'd,
+ By magic numbers and persuasive sound.
+ --CONGREVE, RICHARD.
+
+ _The Morning Bride, Act 1, scene 1._
+
+
+=2144.= ... the Yancos on the Amazon, whose name for three is
+
+ Poettarrarorincoaroac,
+
+of a length sufficiently formidable to justify the remark of La
+Condamine: Heureusement pour ceux qui ont a faire avec eux, leur
+Arithmetique ne va pas plus loin.--PEACOCK, GEORGE.
+
+ _Encyclopedia of Pure Mathematics
+ (London, 1847); Article "Arithmetic,"
+ sect. 32._
+
+
+=2145.= 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, _Pater_,
+_verbum_, and _spiritus sanctus_; 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, _sub-diaconati_, _diaconiti_, and
+_presbyterati_; there are three parts, not without mystery, of
+the most sacred body made by the priest in the mass; and three
+times he says _Agnus Dei_, and three times, _Sanctus_; 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,
+_Domine non sum dignus_; but who can say more of the ternary
+number in a shorter compass, than what the prophet says, _tu
+signaculum sanctae trinitatis_. There are three Furies in the
+infernal regions; three Fates, Atropos, Lachesis, and Clotho.
+There are three theological virtues: _Fides_, _spes_, and
+_charitas_. _Tria sunt pericula mundi: Equum currere; navigare,
+et sub tyranno vivere._ 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: _corde_, _ore_, _ope_. 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.
+
+ --PACIOLI, _Author of the first printed treatise on arithmetic._
+
+ _Quoted in Encyclopedia of Pure
+ Mathematics (London, 1847); Article
+ "Arithmetic," sect. 90._
+
+
+=2146.= Ah! why, ye Gods, should two and two make four?
+
+ --POPE, ALEXANDER.
+
+ _The Dunciad, Bk. 2, line 285._
+
+
+=2147.=
+
+ By him who stampt _The Four_ upon the mind,--
+ _The Four_, the fount of nature's endless stream.
+ --_Ascribed to_ PYTHAGORAS.
+
+ _Quoted in Whewell's History of the
+ Inductive Sciences, Bk. 4, chap. 3._
+
+
+=2148.=
+
+ Along the skiey arch the goddess trode,
+ And sought Harmonia's august abode;
+ The universal plan, the mystic Four,
+ Defines the figure of the palace floor.
+ Solid and square the ancient fabric stands,
+ Raised by the labors of unnumbered hands.
+ --NONNUS.
+
+ _Dionysiac, 41, 275-280. [Whewell]._
+
+
+=2149.= 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 are ten in number. The number eleven denotes sin,
+because it _transgresses_ 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....
+
+ --ST. AUGUSTINE.
+
+ _Sermon 41, art. 23._
+
+
+=2150.= Heliodorus says that the Nile is nothing else than the
+year, founding his opinion on the fact that the numbers expressed
+by the letters [Greek: Neilos], Nile, are in Greek arithmetic,
+[Nu] = 50; [Epsilon] = 5; [Iota] = 10; [Lambda] = 30; [Omicron] =
+70; [Sigma] = 200; and these figures make up together 365, the
+number of days in the year.
+
+ _Littell's Living Age, Vol. 117, p.
+ 380._
+
+
+=2151.= 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 1-9, from K to S 10-90, and
+from T to Z 100-500, we see--
+
+ M A R T I N L U T E R A
+ 30 1 80 100 9 40 20 200 100 5 80 1
+
+which gives 666. Again in Hebrew, _Lulter_ [Hebraized form of
+Luther] does the same:--
+
+ [resh] [tav] [lamed] [yod] [lamed]
+ 200 400 30 6 30
+ --DE MORGAN, A.
+
+ _Budget of Paradoxes (London, 1872), p. 37._
+
+
+=2152.= 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
+_Revelations_, it struck his mind that the _Beast_ must signify
+the Pope, Leo X.; He then proceeded in pious hope to make the
+calculation of the sum of the numeral letters in _Leo decimus_,
+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 bethought him
+again, that he has seen the name written Leo X., and that there
+were ten letters in _Leo decimus_, from either of which he could
+obtain the deficient number, and by interpreting the M to mean
+_mysterium_, 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.
+
+ --PEACOCK, GEORGE.
+
+ _Encyclopedia of Pure Mathematics
+ (London, 1847); Article "Arithmetic,"
+ sect. 89._
+
+
+=2153.= 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, _Honor est a Nilo_. Reading
+this, one is almost persuaded that the hit contained in it has a
+meaning provided by providence or fate.
+
+This is also amusingly illustrated in the case of the Frenchman
+Andre Pujom, who, using j as i, found in his name the anagram,
+Pendu a 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.
+
+ _New American Cyclopedia, Vol. 1;
+ Article "Anagram."_
+
+
+=2154.= 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,
+"Francois Marie Arouet," and which is now generally allowed to be
+an anagram of "Arouet, l. j.," that is, Arouet the younger.
+
+ _Encyclopedia Britannica, 11th Edition;
+ Article "Anagram."_
+
+
+=2155.= 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 impersonating
+youthful heroes. Each of them carried a shield on which appeared
+in gold one of the letters of the words _Domus Lescinia_. At the
+end of the first dance the children were so arranged that the
+letters on their shields spelled the words _Domus Lescinia_. At
+the end of the second dance they read: _ades incolumis_ (sound
+thou art here). After the third: _omnis es lucida_ (wholly
+brilliant art thou); after the fourth: _lucida sis omen_ (bright
+be the omen). Then: _mane sidus loci_ (remain our country's
+star); and again: _sis columna Dei_ (be a column of God); and
+finally: _I! scande solium_ (Proceed, ascend the throne). This
+last was the more beautiful since it proved a true prophecy.
+
+Even more artificial are the anagrams which transform one verse
+into another. Thus an Italian scholar beheld in a dream the line
+from Horace: _Grata superveniet, quae non sperabitur, hora_. This
+a friend changed to the anagram: _Est ventura Rhosina parataque
+nubere pigro._ This induced the scholar, though an old man, to
+marry an unknown lady by the name of Rosina.--HEIS, EDUARD.
+
+ _Algebraische Aufgaben (Koeln, 1898), p.
+ 331._
+
+
+=2156.= The following verses read the same whether read forward
+or backward:--
+
+ Aspice! nam raro mittit timor arma, nec ipsa
+ Si se mente reget, non tegeret Nemesis;[13]
+
+also,
+
+ Sator Arepo tenet opera rotas.
+ --HEIS, EDUARD.
+
+ _Algebraische Aufgaben (Koeln, 1898), p.
+ 328._
+
+ [13] The beginning of a poem which Johannes a Lasco
+ wrote on the count Karl von Suedermanland.
+
+
+=2157.= 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, 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.--HILL, THOMAS.
+
+ _Uses of Mathesis; Bibliotheca Sacra,
+ Vol. 32, p. 521._
+
+
+=2158.= 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.--HILL, THOMAS.
+
+ _Uses of Mathesis; Bibliotheca Sacra,
+ Vol. 32, p. 516._
+
+
+=2159.= Pure mathematics is the magician's real wand.--NOVALIS.
+
+ _Schriften, Zweiter Teil (Berlin, 1901),
+ p. 223._
+
+
+=2160.= 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.--NOVALIS.
+
+ _Schriften, Zweiter Teil (Berlin, 1911),
+ p. 222._
+
+
+
+
+ INDEX
+
+ =Black-faced numbers refer to authors=
+
+ Abbreviations:--m. = mathematics, math. = mathematical,
+ math'n. = mathematician.
+
+
+ Abbott, =1001=.
+
+ Abstract method, Development of, 729.
+
+ Abstract nature of m., Reason for, 638.
+
+ Abstract reasoning, Objection to, 1941.
+
+ Abstractness, math., Compared with logical, 1304.
+
+ Adams, Henry, M. and history, =1599=.
+ Math'ns practice freedom, =208=, =805=.
+
+ Adams, John, Method in m., =226=.
+
+ Aeneid, Euler's knowledge of, 859.
+
+ Aeschylus. On number, 1606.
+
+ Aim in teaching m., 501-508, 517, 844.
+
+ Airy, Pythagorean theorem, =2126=.
+
+ Akenside, =1532=.
+
+ Alexander, 901, 902.
+
+ Algebra, Chapter XVII.
+ Definitions of, 110, 1714, 1715.
+ Problems in, 320, 530, 1738.
+ Of use to grown men, 425.
+ And geometry, 525-527, 1610, 1707.
+ Advantages of, 1701, 1703, 1705.
+ Laws of, 1708-1710.
+ As an art, 1711.
+ Review of, 1713.
+ Designations of, 1717.
+ Origin of, 1736.
+ Burlesque on modern, 1741.
+ Hume on, 1863.
+
+ Algebraic notation, value of, 1213, 1214.
+
+ Algebraic treatises, How to read, 601.
+
+ Amusements in m., 904, 905.
+
+ Anagrams, On De Morgan, 947.
+ On Domus Lescinia, 2155.
+ On Flamsteed, 968.
+ On Macaulay, 996.
+ On Nelson, 2153.
+ On Newton, 1028.
+ On Voltaire, 2154.
+
+ Analysis, Invigorates the faculty of resolution, 416.
+ Relation of geometry to, 1931.
+
+ Analytical geometry, 1889, 1890, 1893.
+ Method of, 310.
+ Importance of, 949.
+ Burlesque on, 2040.
+
+ Ancient geometry,
+ Characteristics of, 712, 714.
+ Compared with modern, 1711-1716.
+ Method of, 1425, 1873-1875.
+
+ Ancients, M. among the, 321.
+
+ Anecdotes, Chapters, IX, X.
+
+ Anger, M. destroys predisposition to, 458.
+
+ Angling like m., 739.
+
+ Anglo-Danes, Aptitude for m., 836.
+
+ Anglo-Saxons, Aptitude for m., 837.
+ Newton as representative of, 1014.
+
+ Anonymous, Song of the screw, 1894.
+
+ Appolonius, 712, 714.
+
+ Approximate m., Why not sufficient, 1518.
+
+ Aptitude for m., 509, 510, 520, 836-838, 976, 1617.
+
+ Arabic notation, 1614.
+
+ Arago, M. the enemy of scientific romances, =267=.
+ Euler, "analysis incarnate," =961=.
+ Euler as a computer, =962=.
+ On Kepler's discovery, =982=.
+ Newton's efforts superhuman, =1006=.
+ On probabilities, =1591=.
+ Geometry as an instrument, 1868.
+
+ Arbuthnot, M. frees from prejudice, credulity and
+ superstition, =449=.
+ M. the friend of religion, =458=.
+ M. compared to music, =1112=.
+ On math, reasoning, =1503=.
+
+ Archimedes, His machines, 903, 904.
+ Estimate of math, appliances, 904-906, 908.
+ Wordsworth on, 906.
+ Schiller on, 907.
+ And engineering, 908.
+ Death of, 909.
+ His tomb, 910.
+ Compared with Newton, 911.
+ Character of his work, 912, 913.
+ Applied m., 1312.
+
+ Architecture and m., 276.
+
+ Archytas, 904.
+ And Plato, 1427.
+
+ Aristippus the Cyrenaic, 845.
+
+ Aristotle, 914.
+ On relation of m. to esthetics, =318=.
+
+ Arithmetic, Chapter XVI.
+ Definitions of, 106, 110, 1611, 1612, 1714.
+ Emerson on advantage of study of, 408.
+ Problems in, 528.
+ A master-key, 1571.
+ Based on concept of time, 1613.
+ Method of teaching, 1618.
+ Purpose of teaching, 454, 1624.
+ As logic, 1624, 1625.
+ The queen of m., 1642.
+ Higher, 1755.
+ Hume on, 1863.
+
+ Arithmetical theorems, 1639.
+
+ Art, M. as a fine, Chapter XI
+
+ Arts, M. and the, 1568-1570, 1573.
+
+ Astronomy and m., 1554, 1559, 1562-1567.
+
+ "Auge et impera.," 631.
+
+ Authority in science, 1528.
+
+ Axioms, 518, 2015.
+ In geometry, 1812, 2004, 2006.
+ Def. in disguise, 2005.
+ Euclid's, 2007-2010, 2014.
+ Nature of, 2012.
+ Proofs of, 2013.
+ And the idea of space, 2004.
+
+
+ Babbage, 923.
+
+ Bacon, Lord,
+ Classification of m., =106=.
+ M. makes men subtile, =248=.
+ View of m., 316, 915, 916.
+ M. held in high esteem by ancients, =321=.
+ On the generalizing power of m., =327=.
+ On the value of math, studies, =410=.
+ M. develops concentration of mind, =411=.
+ M. cures distraction of mind, =412=.
+ M. essential to study of nature, =436=.
+ His view of m., =915=, =916=.
+ His knowledge of m., 917, 918.
+ M. and logic, =1310=.
+ Growth of m., =1511=.
+
+ Bacon, Roger, Neglect of m. works injury to all science,
+ =310=.
+ On the value of m., =1547=.
+
+ Bain, Importance of m. in education, =442=.
+ On the charm of the study of m., =453=.
+ M. and science teaching, =522=.
+ Teaching of arithmetic, =1618=.
+
+ Ball, R. S., =2010=.
+
+ Ball, W. W. R., On Babbage, =923=.
+ On Demoivre's death, =944=.
+ De Morgan and the actuary, =945=.
+ Gauss as astronomer, =971=.
+ Laplace's "It is easy to see." =986=.
+ Lagrange, Laplace and Gauss contrasted, =993=.
+ Newton's interest in chemistry and theology, =1015=.
+ On Newton's method of work, =1026=.
+ On Newton's discovery of the calculus, =1027=.
+ Gauss's estimate of Newton, =1029=.
+ M. and philosophy, =1417=.
+ Advance in physics, =1530=.
+ Plato on geometry, 1804.
+ Notation of the calculus, =1904=.
+
+ Barnett, M. the type of perfect reasoning, =307=.
+
+ Barrow, On the method of m., =213=, =227=.
+ Eulogy of m., =330=.
+ M. as a discipline of the mind, =402=.
+ M. and eloquence, =830=.
+ Philosophy and m., =1430=.
+ Uses of m., =1572=.
+ On surd numbers, =1728=.
+ Euclid's definition of proportion, =1835=.
+
+ Beattie, =1431=.
+
+ Beauty of m., 453, 824, 1208.
+ Consists in simplicity, 242, 315.
+ Sylvester on, 1101.
+ Russell on, 1104.
+ Young on, 1110.
+ Kummer on, 1111.
+ White on, 1119.
+ And truth, 1114.
+ Boltzmann on, 1116.
+
+ Beltrami, On reading of the masters, =614=.
+
+ Berkeley, On geometry as logic, =428=.
+ On math. symbols, =1214=.
+ On fluxions, =1915=, =1942-1944=.
+ On infinite divisibility, =1945=.
+
+ Bernoulli, Daniel, 919.
+
+ Bernoulli, James,
+ Legend for his tomb, 920, 922.
+ Computation of sum of tenth powers of numbers, 921.
+ Discussion of logarithmic spiral, 922.
+
+ Berthelot, M. inspires respect for truth, =438=.
+
+ Bija Ganita, Solution of problems, =1739=.
+
+ Billingsley, M. beautifies the mind, =319=.
+
+ Binary arithmetic, 991.
+
+ Biology and m., 1579-1581.
+
+ Biot, Laplace's "It is easy to see," 986.
+
+ Bocher, M. likened to painting, =1103=.
+ Interrelation of m. and logic, =1313=.
+ Geometry as a natural science, =1866=.
+
+ Boerne, On Pythagoras, =1855=.
+
+ Bois-Reymond, On the analytic method, =1893=.
+ Natural selection and the calculus, =1921=.
+
+ Boltzmann, On beauty in m., =1116=.
+
+ Bolyai, Janos,
+ Duel with officers, 924.
+ Universal language, 925.
+ Science absolute of space, 926.
+
+ Bolyai, Wolfgang, 927.
+ On Gauss, =972=.
+
+ Bolzano, 928.
+ Cured by Euclid, =929=.
+ Parallel axiom, =2110=.
+
+ Book-keeping, Importance of the art of, 1571.
+
+ Boole, M. E. =719=.
+
+ Boole's Laws of Thought, 1318.
+
+ Borda-Demoulins, Philosophy and m., =1405=.
+
+ Boswell, =981=.
+
+ Bowditch, On Laplace's "Thus it plainly appears," =985=.
+
+ Boyle, Usefulness of m. to physics, =437=.
+ M. and science, =1513=, =1533=.
+ Ignorance of m., =1577=.
+ M. and physiology, =1582=.
+ Wings of m., =1626=.
+ Advantages of algebra, =1703=.
+
+ Brahmagupta, Estimate of m., =320=.
+
+ Brewster, On Euler's knowledge of the Aeneid, =959=.
+ On Euler as a computer, =963=.
+ On Newton's fame, =1002=.
+
+ Brougham, =1202=.
+
+ Buckle, On geometry, =1810=, =1837=.
+
+ Burke, On the value of m., =447=.
+
+ Burkhardt, On discovery in m., =618=.
+ On universal symbolism, =1221=.
+
+ Butler, N. M., M. demonstrates the supremacy of the human
+ reason, =309=.
+ M. the most astounding intellectual creation, =707=.
+ Geometry before algebra, =1871=.
+
+ Butler, Samuel, =2118=.
+
+ Byerly, On hyperbolic functions, =1929=.
+
+
+ Cajori, On the value of the history of m., =615=.
+ On Bolyai, =927=.
+ Cayley's view of Euclid, =936=.
+ On the extent of Euler's work, =960=.
+ On Euler's math. power, =964=.
+ On the Darmstaetter prize, =967=.
+ On Sylvester's first class at Johns Hopkins, =1031=.
+ On music and m. among the Pythagoreans, =1130=.
+ On the greatest achievement of the Hindoos, =1615=.
+ On modern calculation, =1614=.
+ On review in arithmetic, =1713=.
+ On Indian m., =1737=.
+ On the characteristics of ancient geometry, =1873=.
+ On Napier's rule, =1888=.
+
+ Calculating machines, =1641=.
+
+ Calculation, Importance of, 602.
+ Not the sole object of m., 268.
+
+ Calculus, Chapter XIX. Foundation of 253.
+ As a method, 309.
+ May be taught at an early age, 519, 1917, 1918.
+
+ Cambridge m., 836, 1210.
+
+ Cantor, On freedom in m., =205=, =207=.
+ On the character of Gauss's writing, =975=.
+ Zeno's problem, 1938.
+ On the infinite, =1952=.
+
+ Carlisle life tables, 946.
+
+ Carnot, On limiting ratios, =1908=.
+ On the infinitesimal method, =1907=.
+
+ Carson, Value of geometrical training, =1841=.
+
+ Cartesian method, 1889, 1890.
+
+ Carus, Estimate of m., =326=.
+ M. reveals supernatural God, =460=.
+ Number and nature, =1603=.
+ Zero and infinity, =1948=.
+ Non-euclidean geometry, =2016=.
+
+ Cathedral, "Petrified mathematics," 1110.
+
+ Causation in m., 251, 254.
+
+ Cayley, Advantage of modern geometry over ancient, =711=.
+ On the imaginary, =722=.
+ Sylvester on, 930.
+ Noether on, 931.
+ His style, 932.
+ Forsyth on, 932-934.
+ His method, 933.
+ Compared with Euler, 934.
+ Hermite on, 935.
+ His view of Euclid, 936.
+ His estimate of quaternions, 937.
+ M. and philosophy, =1420=.
+
+ Certainty of m., 222, 1440-1442, 1628, 1863.
+
+ Chamisso, Pythagorean theorem, =1856=.
+
+ Chancellor, M. develops observation, imagination and
+ reason, =433=.
+
+ Chapman, Different aspects of m., =265=.
+
+ Characteristics of m., 225, 229, 247, 263.
+
+ Characteristics of modern m., 720, 724-729.
+
+ Charm in m., 1115, 1640, 1848.
+
+ Chasles, Advantage of modern geometry over ancient, =712=.
+
+ Checks in m., 230.
+
+ Chemistry and m., 1520, 1560, 1561, 1750.
+
+ Chess, M. like, 840.
+
+ Chrystal, Definition of m., =113=.
+ Definition of quantity, =115=.
+ On problem solving, =531=.
+ On modern text-books, =533=.
+ How to read m., =607=.
+ His algebra, 635.
+ On Bernoulli's numbers, =921=.
+ On math. versus logical abstractness, =1304=.
+ Rules of algebra, =1710=.
+ On universal arithmetic, =1717=.
+ On Horner's method, =1744=.
+ On probabilities, =1967=.
+
+ Cicero, Decadence of geometry among Romans, =1807=.
+
+ Circle, Properties of, 1852, 1857.
+
+ Circle-squarers, 2108, 2109.
+
+ Clarke, Descriptive geometry, =1882=.
+
+ Classic problems, Hilbert on, 627.
+
+ Clebsch, On math. research, =644=.
+
+ Clifford, On direct usefulness of math. results, =652=.
+ Correspondence the central idea of modern m., =726=.
+ His vision, 938.
+ His method, 939.
+ His knowledge of languages, 940.
+ His physical strength, 941.
+ On Helmholtz, =979=.
+ On m. and mineralogy, =1558=.
+ On algebra and good English, =1712=.
+ Euclid the encouragement and guide of scientific thought, =1820=.
+ Euclid the inspiration and aspiration of scientific thought, =1821=.
+ On geometry for girls, =1842=.
+ On Euclid's axioms, =2015=.
+ On non-Euclidean geometry, =2022=
+
+ Colburn, 967.
+
+ Coleridge, On problems in m., =534=.
+ Proposition, gentle maid, =1419=.
+ M. the quintessence of truth, =2019=.
+
+ Colton, On the effect of math. training, =417=.
+
+ Commensurable numbers, 1966.
+
+ Commerce and m., 1571.
+
+ Committee of Ten, On figures in geometry, =524=.
+ On projective geometry, =1876=.
+
+ Common sense, M. the etherealization of, 312.
+
+ Computation, Not m., 515.
+ And m., 810.
+ Not concerned with significance of numbers, 1641.
+
+ Comte, On the object of m., =103=.
+ On the business of concrete m., =104=.
+ M. the indispensable basis of all education, =334=.
+ Mill on, 942.
+ Hamilton on, 943.
+ M. and logic, =1308=, =1314=, =1325=.
+ On Kant's view of m., =1437=.
+ Estimate of m., =1504=.
+ M. essential to scientific education, =1505=.
+ M. and natural philosophy, =1506=.
+ M. and physics, =1535=, =1551=.
+ M. and science, =1536=.
+ M. and biology, =1578=, =1580=, =1581=.
+ M. and social science, =1587=.
+ Every inquiry reducible to a question of number, =1602=.
+ Definition of algebra and arithmetic, =1714=.
+ Geometry a natural science, =1813=.
+ Ancient and modern methods, =1875=.
+ On the graphic method, =1881=.
+ On descriptive geometry, =1883=.
+ Mill's estimate of, 1903.
+
+ Congreve, =2143=.
+
+ Congruence, Symbol of, 1646.
+
+ Conic sections, 658, 660, 1541, 1542.
+
+ Conjecture, M. free from, 234.
+
+ Contingent truths, 1966.
+
+ Controversies in m., 215, 243, 1859.
+
+ Correlation in m., 525-527, 1707, 1710.
+
+ Correspondence, Concept of, 725, 726.
+
+ Coulomb, 1516.
+
+ Counting, Every problem can be solved by, 1601.
+
+ Cournot, On the object of m., =268=.
+ On algebraic notation, =1213=.
+ Advantage of math, notation, =1220=.
+
+ Craig, On the origin of a new science, =646=.
+
+ Credulity, M. frees mind from, 450.
+
+ Cremona, On English text-books, =609=.
+
+ Crofton,
+ On value of probabilities, =1590=.
+ On probabilities, =1952=, =1970=,=1972=.
+
+ Cromwell, On m. and public service, =328=.
+
+ Curiosities, Chapter XXI.
+
+ Curtius, M. and philosophy, =1409=.
+
+ Curve, Definition of, 1927.
+
+ Cyclometers, Notions of, 2108.
+
+ Cyclotomy depends on number theory, 1647.
+
+
+ D'Alembert, On rigor in m., =536=.
+ Geometry as logic, =1311=.
+ Algebra is generous, =1702=.
+ Geometrical versus physical truths, =1809=.
+ Standards in m., =1851=.
+
+ Dante, =1858=, =2117=.
+
+ Darmstaetter prize, 2129.
+
+ Davis, On Sylvester's method, =1035=.
+ M. and science, =1510=.
+ On probability, =1968=.
+
+ Decimal fractions, 1217, 1614.
+
+ Decker, =2142=.
+
+ Dedekind, Zeno's Problem, 1938.
+
+ Deduction, Why necessary, 219.
+ M. based on, 224.
+ And Intuition, 1413.
+
+ Dee, On the nature of m., =261=.
+
+ Definitions of m., Chapter I.
+ Also 2005.
+
+ Democritus, 321.
+
+ Demoivre, His death, 944.
+
+ Demonstrations, Locke on, 236.
+ Outside of m., 1312.
+ In m., 1423.
+
+ De Morgan, Imagination in m., =258=.
+ M. as an exercise in reasoning, =430=.
+ On difficulties in m., =521=.
+ On correlation in m., =525=.
+ On extempore lectures, =540=.
+ On reading algebraic works, =601=.
+ On numerical calculations, =602=.
+ On practice problems, =603=.
+ On the value of the history of m., =615=, =616=.
+ On math'ns., =812=.
+ On Bacon's knowledge of m., =918=.
+ And the actuary, 945.
+ On life tables, =946=.
+ Anagrams' on his name, =947=.
+ On translations of Euclid, =953=.
+ Euclid's elements compared with Newton's Principia, =954=.
+ Euler and Diderot, =966=.
+ Lagrange and the parallel axiom, =984=.
+ Anagram on Macaulay's name, =996=.
+ Anagrams on Newton's name, =1028=.
+ On math, notation, =1216=.
+ Antagonism of m. and logic, =1315=.
+ On German metaphysics, =1416=.
+ On m. and science, =1537=.
+ On m. and physics, =1538=.
+ On the advantages of algebra, =1701=.
+ On algebra as an art, =1711=.
+ On double algebra and quaternions, =1720=.
+ On assumptions in geometry, =1812=.
+ On Euclid in schools, =1819=.
+ Euclid not faultless, =1823=.
+ On Euclid's rigor, =1831=.
+ Geometry before algebra, =1872=.
+ On trigonometry, =1885=.
+ On the calculus in elementary instruction, =1916.=
+ On integration, =1919=.
+ On divergent series, =1935=, =1936=.
+ Ad infinitum, =1949=.
+ On the fourth dimension, =2032=.
+ Pseudomath and graphomath, =2101=.
+ On proof, =2102=.
+ On paradoxers, =2105=.
+ Budget of paradoxes, =2106=.
+ On D'Israeli's six follies of science, =2107=.
+ On notions of cyclometers, =2108=.
+ On St. Vincent, =2109=.
+ Where Euclid failed, =2114=.
+ On the number of the beast, =2151=.
+
+ Descartes, On the use of the term m., =102=.
+ On intuition and deduction, =219=, =1413=.
+ Math'ns alone arrive at proofs, =817=.
+ The most completely math. type of mind, 948.
+ Hankel on, 949.
+ Mill on, 950.
+ Hankel on, 1404.
+ On m. and philosophy, =1425=, =1434=.
+ Estimate of m., =1426=.
+ Unpopularity of, =1501=.
+ On the certainty of m., =1628=.
+ On the method of the ancients, =1874=.
+ On probable truth, =1964=.
+ Descriptive geometry, 1882, 1883.
+
+ Dessoir, M. and medicine, =1585=.
+
+ Determinants, 1740, 1741.
+
+ Diderot and Euler, 966.
+
+ Differential calculus, Chapter XIX.
+ And scientific physics, 1549.
+
+ Differential equations, 1549-1552, 1924, 1926.
+
+ Difficulties in m., 240, 521, 605-607, 634, 734, 735.
+
+ Dillmann, M. a royal science, =204=.
+ On m. as a high school subject, =401=.
+ Ancient and modern geometry compared, =715=.
+ On ignorance of, =807=.
+ On m. as a language, =1204=.
+ Number regulates all things, =1505=.
+
+ Dirichlet, On math, discovery, =625=.
+ As a student of Gauss, 977.
+
+ Discovery in m., 617-622, 625.
+
+ _D_-ism versus _dot_-age, 923.
+
+ Disquisitiones Arithmeticae, 975, 977, 1637, 1638.
+
+ D'Israeli, 2007.
+
+ Divergent series, 1935-1937.
+
+ "Divide et impera," 631.
+
+ Divine character of m., 325, 329.
+
+ "Divinez avant de demontrer," 630.
+
+ Division of labor in m., 631, 632.
+
+ Dodgson, On the charm of, =302=.
+ Pythagorean theorem, =1854=.
+ Ignes fatui in m., =2103=.
+
+ Dolbear, On experiment in math. research, =613=.
+
+ Domus Lescinia, Anagram on, 2155.
+
+ Donne, =1816=.
+
+ _Dot_-age versus _d_-ism, 923.
+
+ Durfee, On Sylvester's forgetfulness, =1038=.
+
+ Dutton, On the ethical value of m., =446=.
+
+
+ "Eadem mutata resurgo." 920, 922.
+
+ Echols, On the ethical value of m., =455=.
+
+ Economics and m., 1593, 1594.
+
+ Edinburgh Review, M. and astronomy, =1565=, =1566=.
+
+ Education, Place of m. in, 334, 408.
+ Study of arithmetic better than rhetoric, 408.
+ M. as an instrument in, 413, 414.
+ M. in primary, 431.
+ M. as a common school subject, 432.
+ Bain on m. in, 442.
+ Calculus in elementary, 1916, 1917.
+
+ Electricity, M. and the theory of, 1554.
+
+ Elegance in m., 640, 728.
+
+ Ellis, On precocity in m., =835=.
+ On aptitude of Anglo-Danes for m., =836=.
+ On Newton's genius, =1014=.
+
+ Emerson, On Newton and Laplace, =1003=.
+ On poetry and m., =1124=.
+
+ Endowment of math'ns, 818.
+
+ Enthusiasm, 801.
+
+ Equality, Grassmann's definition of, =105=.
+
+ Equations, 104, 526, 1891, 1892.
+
+ Errors, Theory of, 1973, 1974.
+
+ Esthetic element in m., 453-455, 640, 1102, 1105, 1852,
+ 1853.
+
+ Esthetic tact, 622.
+
+ Esthetic value of m., 1848, 1850.
+
+ Esthetics, Relation of m. to, 318, 319, 439.
+
+ Estimates of m., Chapter III.
+ See also 1317, 1324, 1325, 1427, 1504, 1508.
+
+ Ethical value of m., 402, 438, 446, 449, 455-457.
+
+ Euclid, Bolzano cured by, 929.
+ And Ptolemy, 951, 1878.
+ And the student, 952.
+
+ Euclid's Elements,
+ Translations of, 953.
+ Compared with the Principia, 954.
+ Greatness of, 955.
+ Greatest of human productions, 1817.
+ Performance in, 1818.
+ In English schools, 1819.
+ Encouragement and guide, 1820.
+ Inspiration and aspiration, 1821.
+ The only perfect model, 1822.
+ Not altogether faultless, 1823.
+ Only a small part of m., 1824.
+ Not fitted for boys, 1825.
+ Early study of, 1826.
+ Newton and, 1827.
+ Its place, 1828.
+ Unexceptional in rigor, 1829.
+ Origin of, 1831.
+ Doctrine of proportion, 1834.
+ Definition of proportion, 1835.
+ Steps in demonstration, 1839.
+ Parallel axiom, 2007.
+
+ Euclidean geometry, 711, 713, 715.
+
+ Eudoxus, 904.
+
+ Euler, the myriad-minded, 255.
+ Pencil outruns intelligence, 626.
+ On theoretical investigations, 657.
+ Merit of his work, 956.
+ The creator of modern math. thought, 957.
+ His general knowledge, 958.
+ His knowledge of the Aeneid, 959.
+ Extent of his work, 960.
+ "Analysis incarnate," 961.
+ As a computer, 962, 963.
+ His math. power, 964.
+ His _Tentamen novae theorae musicae_, 965.
+ And Diderot, 966.
+ Error in Fermat's law of prime numbers, 967.
+
+ Eureka, 911, 917.
+
+ Euripedes, 1568.
+
+ Everett, Estimate of m., =325=.
+ Value of math. training, =443=.
+ Theoretical investigations, =656=.
+ Arithmetic a master-key, =1571=.
+ On m. and law, =1598=.
+
+ Exactness, See precision.
+
+ Examinations, 407.
+
+ Examples, 422.
+
+ Experiment in m., 612, 613, 1530, 1531.
+
+ Extent of m., 737, 738.
+
+
+ Fairbairn, 528.
+
+ Fallacies, 610.
+
+ Faraday, M. and physics, 1554.
+
+ Fermat, 255, 967, 1902.
+
+ Fermat's theorem, 2129.
+
+ Figures, Committee of Ten on, 524.
+ Democritus view of, 321.
+ Battalions of, 1631.
+
+ Fine, Definition of number, =1610=.
+ On the imaginary, =1732=.
+
+ Fine Art, M. as a, Chapter XI.
+
+ Fisher, M. and economics, =1594=.
+
+ Fiske, Imagination in m., =256=.
+ Advantage of m. as logic, =1324=.
+
+ Fitch, Definition of m., =125=.
+ M. in education, =429=.
+ Purpose of teaching arithmetic, =1624=, =1625=.
+
+ Fizi, Origin of the Liliwati, =995=.
+
+ Flamsteed, Anagram on, =968=.
+
+ Fluxions, 1911, 1915, 1942-1944.
+
+ Fontenelle, Bernoulli's tomb, =920=.
+
+ Formulas, Compared to focus of a lens, 1515.
+
+ Forsyth, On direct usefulness of math. results, =654=.
+ On theoretical investigations, =664=.
+ Progress of m. =704=.
+ On Cayley, =932-934=.
+ On m. and physics, =1539=.
+ On m. and applications, =1540=.
+ On invariants, =1747=.
+ On function theory, =1754=, =1755=.
+
+ Foster, On m. and physics, =1516=, =1522=.
+ On experiment in m., =1531=.
+
+ Foundations of m., 717.
+
+ Four, The number, 2147, 2148.
+
+ Fourier, Math, analysis co-extensive with nature, =218=.
+ On math. research, =612=.
+ Hamilton on, 969.
+ On m. and physics, =1552=, =1553=.
+ On the advantage of the Cartesian method, =1889=.
+
+ Fourier's theorem, 1928.
+
+ Fourth dimension, 2032, 2039.
+
+ Frankland, A., M. and chemistry, =1560=.
+
+ Frankland, W. B., Motto of Pythagorean brotherhood, =1833=.
+ The most beautiful truth in geometry, =1857=.
+
+ Franklin, B., Estimate of m., =322=.
+ On the value of the study of m., =323=.
+ On the excellence of m., =324=.
+ On m. as a logical exercise, =1303=.
+
+ Franklin, F., On Sylvester's weakness, =1033=.
+
+ Frederick the Great, On geometry, =1860=.
+
+ Freedom in m., 205-208, 805.
+
+ French m., 1210.
+
+ Fresnel, 662.
+
+ Frischlinus, =1801=.
+
+ Froebel, M. a mediator between man and nature, =262=.
+
+ Function theory, 709, 1732, 1754, 1755.
+
+ Functional exponent, 1210.
+
+ Functionality, The central idea of modern m., 254.
+ Correlated to life, 272.
+
+ Functions, 1932, 1933.
+ Concept not used by Sylvester, 1034.
+
+ Fundamental concepts, Chapter XX.
+
+ Fuss, On Euler's _Tentamen novae theorae musicae_, =965=.
+
+
+ Galileo, On authority in science, =1528=.
+
+ Galton, 838.
+
+ Gauss, His motto, 649.
+ Mere math'ns, =820=.
+ And Newton compared, 827.
+ His power, 964.
+ His favorite pursuits, 970.
+ The first of theoretical astronomers, 971.
+ The greatest of arithmeticians, 971.
+ The math. giant, 972.
+ Greatness of, 973.
+ Lectures to three students, =974=.
+ His style and method, 983.
+ His estimate of Newton, 1029.
+ On the advantage of new calculi, =1215=.
+ M. and experiment, 1531.
+ His _Disquisitiones Arithmeticae_, 1639, 1640.
+ M. the queen of the sciences, =1642=.
+ On number theory, =1644=.
+ On imaginaries, =1730=.
+ On the notation sin squared[phi], =1886=.
+ On infinite magnitude, =1950=.
+ On non-euclidean geometry, =2023-2028=.
+ On the nature of space, 2034.
+
+ Generalization in m., 245, 246, 252, 253, 327, 728.
+
+ Genius, 819.
+
+ Geometrical investigations, 642, 643.
+
+ Geometrical training, Value of, 1841, 1842, 1844-1846.
+
+ Geometry, Chapter XVIII.
+ Bacon's definition of, 106.
+ Sylvester's definition of, 110.
+ Value to mankind, 332, 449.
+ And patriotism, 332.
+ An excellent logic, 428.
+ Plato's view of, 429.
+ The fountain of all thought, 451.
+ And algebra, 525-527.
+ Lack of concreteness, 710.
+ Advantage of modern over ancient, 711, 712.
+ And music, 965.
+ And arithmetic, 1604.
+ Is figured algebra, 1706.
+ Name inapt, 1801.
+ And experience, 1814.
+ Halsted's definition of, 1815.
+ And observation, 1830.
+ Controversy in, 1859.
+ A mechanical science, 1865.
+ A natural science, 1866.
+ Not an experimental science, 1867.
+ Should come before algebra, 1767, 1871, 1872.
+ And analysis, 1931.
+
+ Germain, Algebra is written geometry, =1706=.
+
+ Gilman, Enlist a great math'n, =808=.
+
+ Glaisher, On the importance of broad training, =623=.
+ On the importance of a well-chosen notation, =634=.
+ On the expansion of the field of m., =634=.
+ On the need of text-books on higher m., =635=.
+ On the perfection of math. productions, =649=.
+ On the invention of logarithms, =1616=.
+ On the theory of numbers, =1640=.
+
+ Goethe, On the exactness of m., =228=.
+ M. an organ of the higher sense, =273=.
+ Estimate of m., =311=.
+ M. opens the fountain of all thought, =451=.
+ Math'ns must perceive beauty of truth, =803=.
+ Math'ns bear semblance of divinity, =804=.
+ Math'ns like Frenchmen, =813=.
+ His aptitude for m., =976=.
+ M. like dialectics, =1307=.
+ On the infinite, 1957.
+
+ Golden age of m., 701, 702.
+ Of art and m. coincident, 1134.
+
+ Gordan, When a math. subject is complete, =636=.
+
+ Gow, Origin of Euclid, =1832=.
+
+ Gower, =1808=.
+
+ Grammar and m. compared, 441.
+
+ Grandeur of m., 325.
+
+ Grassmann, Definition of m., =105=.
+ Definition of magnitude, =105=.
+ Definition of equality, =105=.
+ On rigor in m., =538=.
+ On the value of m., =1512=.
+
+ Greek view of science, 1429.
+
+ Graphic method, 1881.
+
+ Graphomath, 2101.
+
+ Group, Notion of, 1751.
+
+ Growth of m., 209, 211, 703.
+
+
+ Hall, G. S., M. the ideal and norm of all careful
+ thinking, =304=.
+
+ Hall and Stevens, On the parallel axiom, =2008=.
+
+ Haller, On the infinite, =1958=.
+
+ Halley, On Cartesian geometry, 716.
+
+ Halsted, On Bolyai, =924-926=.
+ On Sylvester, =1030=, =1039=.
+ And Sylvester, =1031=, =1032=.
+ On m. as logic, =1305=.
+ Definition of geometry, =1815=.
+
+ Hamilton, Sir William, His ignorance of m., 978.
+
+ Hamilton, W. R., Importance of his quaternions, 333.
+ Estimate of Comte's ability, =943=.
+ To the memory of Fourier, =969=.
+ Discovery in light, 1558.
+ On algebra as the science of time, =1715=, =1716=.
+ On quaternions, =1718=.
+ On trisection of an angle, =2112=.
+
+ Hankel, Definition of m., =114=.
+ On freedom in m., =206=.
+ On the permanency of math. knowledge, =216=.
+ On aim in m., =508=.
+ On isolated theorems, =621=.
+ On tact in m., =622=.
+ On geometry, 714.
+ Ancient and modern m. compared, =718=, =720=.
+ Variability the central idea in modern m., =720=.
+ Characteristics of modern m., =728=.
+ On Descartes, =949=.
+ On Euler's work, =956=.
+ On philosophy and m., =1404=.
+ On the origin of m., =1412=.
+ On irrationals and imaginaries, =1729=.
+ On the origin of algebra, =1736=.
+ Euclid the only perfect model, =1822=.
+ Modern geometry a royal road, =1878=.
+
+ Harmony, 326, 1208.
+
+ Harris, M. gives command over nature, =434=.
+
+ Hathaway, On Sylvester, =1036=.
+
+ Heat, M. and the theory of, 1552, 1553.
+
+ Heath, Character of Archimedes' work, =913=.
+
+ Heaviside, The place of Euclid, =1828=.
+
+ Hebrew and Latin races, Aptitude for m., 838.
+
+ Hegel, =1417=.
+
+ Heiss, Famous anagrams, =2055=.
+ Reversible verses, =2056=.
+
+ Helmholtz, M. the purest form of logical activity, =231=.
+ M. requires perseverance and great caution, =240=.
+ M. should take more important place in education, =441=.
+ Clifford on, =979=.
+ M. the purest logic, =1302=.
+ M. and applications, =1445=.
+ On geometry, =1836=.
+ On the importance of the calculus, =1939=.
+ A non-euclidean world, =2029=.
+
+ Herbart, Definition of m., =117=.
+ M. the predominant science, =209=.
+ On the method of m., =212=, =1576=.
+ M. the priestess of definiteness and clearness, =217=.
+ On the importance of checks, =230=.
+ On imagination in m, =257=.
+ M. and invention, =406=.
+ M. the chief subject for common schools, =432=.
+ On aptitude for m., =509=.
+ On the teaching of m., =516=.
+ M. the greatest blessing, =1401=.
+ M. and philosophy, =1408=.
+ If philosophers understood m., =1415=.
+ M. indispensable to science, =1502=.
+ M. and psychology, =1583=, =1684=.
+ On trigonometry, =1884=.
+
+ Hermite, On Cayley, =935=.
+
+ Herschel, M. and astronomy, =1564=.
+ On probabilities, =1592=.
+
+ Hiero, 903, 904.
+
+ Higher m., Mellor's definition of, =108=.
+
+ Hilbert, On the nature of m., =266=.
+ On rigor in m., =537=.
+ On the importance of problems, =624=, =628=.
+ On the solvability of problems, =627=.
+ Problems should be difficult, =629=.
+ On the abstract character of m., =638=.
+ On arithmetical symbols, =1627=.
+ On non-euclidean geometry, =2019=.
+
+ Hill, Aaron, On Newton, =1009=.
+
+ Hill, Thomas, On the spirit of mathesis, =274=.
+ M. expresses thoughts of God, =275=.
+ Value of m., =332=.
+ Estimate of Newton's work, =333=.
+ Math'ns difficult to judge, =841=.
+ Math'ns indifferent to ordinary interests of life, =842=.
+ A geometer must be tried by his peers, =843=.
+ On Bernoulli's spiral, =922=.
+ On mathesis and poetry, =1125=.
+ On poesy and m., =1126=.
+ On m. as a language, =1209=.
+ Math, language untranslatable, =1210=.
+ On quaternions, =1719=.
+ On the imaginary, =1734=.
+ On geometry and literature, =1847=.
+ M. and miracles, =2157=, =2158=.
+
+ Hindoos, Grandest achievement of, 1615.
+
+ History and m., 1599.
+
+ History of m., 615, 616, 625, 635.
+
+ Hobson, Definition of m., =118=.
+ On the nature of m., =252=.
+ Functionality the central idea of m., =264=.
+ On theoretical investigations, =663=.
+ On the growth of m., =703=.
+ A great math'n a great artist, =1109=.
+ On m. and science, =1508=.
+ Hoffman, Science and poetry not antagonistic, =1122=.
+
+ Holzmueller, On the teaching of m., =518=.
+
+ Hooker, =1432=.
+
+ Hopkinson, M. a mill, =239=.
+
+ Horner's method, 1744.
+
+ Howison, Definition of m., =134=, =135=.
+ Definition of arithmetic, =1612=.
+
+ Hudson, On the teaching of m., =612=.
+
+ Hughes, On science for its own sake, =1546=.
+
+ Humboldt, M. and astronomy, =1567=.
+
+ Hume, On the advantage of math, science, =1438=.
+ On geometry, =1862=.
+ On certainty in m., =1863=.
+ Objection to abstract reasoning, =1941=.
+
+ Humor in m., 539.
+
+ Hutton, On Bernoulli, =919=.
+ On Euler's knowledge, =958=.
+ On the method of fluxions, =1911=.
+
+ Huxley, Negative qualities of m., =250=.
+
+ Hyper-space, 2030, 2031, 2033, 2036-2038.
+
+ Hyperbolic functions, 1929, 1930.
+
+
+ Ignes fatui in m., 2103.
+
+ Ignorabimus, None in m., 627.
+
+ Ignorance of m., 310, 331, 807, 1537, 1577.
+
+ Imaginaries, 722, 1729-1735.
+
+ Imagination in m., 246, 251, 253, 256-258, 433, 1883.
+
+ Improvement of elementary m., 617.
+
+ Incommensurable numbers, contingent truths like, 1966.
+
+ Indian m., 1736, 1737.
+
+ Induction in m., 220-223, 244.
+ And analogy, 724.
+
+ Infinite collection, Definition of, 1959, 1960.
+
+ Infinite divisibility, 1945.
+
+ Infinitesimal analysis, 1914.
+
+ Infinitesimals, 1905-1907, 1940, 1946, 1954.
+
+ Infinitum, Ad, 1949.
+
+ Infinity and infinite magnitude, 723, 928, 1947, 1948,
+ 1950-1958.
+
+ Integers, Kronecker on, 1634, 1635.
+
+ Integral numbers, Minkowsky on, 1636.
+
+ Integrals, Invention of, 1922.
+
+ Integration, 1919-1921, 1923, 1925.
+
+ International Commission on m., =501=, =502=, =938=.
+
+ Intuition and deduction, 1413.
+
+ Invariance, Correlated to life, 272.
+ MacMahon on, 1746.
+ Keyser on, 1749.
+
+ Invariants, Changeless in the midst of change, 276.
+ Importance of concept of, 727.
+ Sylvester on, 1742.
+ Forsyth on, 1747.
+ Keyser on, 1748.
+ Lie on, 1752.
+
+ Invention in m., 251, 260.
+
+ Inverse process, 1207.
+
+ Investigations, See research.
+
+ Irrationals, 1729.
+
+ Isolated theorems in m., 620, 621.
+
+ "It is easy to see," 985, 986, 1045.
+
+
+ Jacobi, His talent for philology, 980.
+ Aphorism, =1635=.
+ Die "Ewige Zahl," =1643=.
+
+ Jefferson, On m. and law, =1597=.
+
+ Johnson, His recourse to m., 981.
+ Aptitude for numbers, =1617=.
+ On round numbers, =2137=.
+
+ Journals and transactions, 635.
+
+ Jowett, M. as an instrument in education, =413=.
+
+ Judgment, M. requires, 823.
+
+ Jupiter's eclipses, 1544.
+
+ Justitia, The goddess, 824.
+
+ Juvenal, Nemo mathematicus etc., =831=.
+
+
+ Kant, On the a priori nature of m., =130=.
+ M. follows the safe way of science, =201=.
+ On the origin of scientific m., 201.
+ On m. in primary education, =431=.
+ M. the embarrassment of metaphysics, =1402=.
+ His view of m., =1436=, =1437=.
+ On the difference between m. and philosophy, =1436=.
+ On m. and science, =1508=.
+ Esthetic elements in m., =1852=, =1853=.
+ Doctrine of time, =2001=.
+ Doctrine of space, =2003=.
+
+ Karpinsky, M. and efficiency, =1673=.
+
+ Kasner, "Divinez avant de demontrer," =630=.
+ On modern geometry, =710=.
+
+ Kelland, On Euclid's elements, =1817=.
+
+ Kelvin, Lord, See William Thomson.
+
+ Kepler, His method, 982.
+ Planetary orbits and the regular solids, =2134=.
+
+ Keyser, Definition of m., =132=.
+ Three characteristics of m., =225=.
+ On the method of m., =244=.
+ On ratiocination, =246=.
+ M. not detached from life, =273=.
+ On the spirit of mathesis, =276=.
+ Computation not m., =515=.
+ Math, output of present day, =702=.
+ Modern theory of functions, =709=.
+ M. and journalism, =731=.
+ Difficulty of m., =735=.
+ M. appeals to whole mind, =815=.
+ Endowment of math'ns, =818=.
+ Math'ns in public service, =823=.
+ The aim of the math'n, =844=.
+ On Bolzano, =929=.
+ On Lie, =992=.
+ On symbolic logic, =1321=.
+ On the emancipation of logic, =1322=.
+ On the Principia Mathematica, =1326=.
+ On invariants, =1728=.
+ On invariance, =1729=.
+ On the notion of group, =1751=.
+ On the elements of Euclid, =1824=.
+ On protective geometry, =1880=.
+ Definition of infinite assemblage, =1960=.
+ On the infinite, =1961=.
+ On non-euclidean geometry, =2035=.
+ On hyper-space, =2037=, =2038=.
+
+ Khulasat-al-Hisab, Problems, =1738=.
+
+ Kipling, =1633=.
+
+ Kirchhoff, Artistic nature of his works, 1116.
+
+ Klein, Definition of m., =123=.
+ M. a versatile science, =264=.
+ Aim in teaching, =507=, =517=.
+ Analysts versus synthesists, =651=.
+ On theory and practice, =661=.
+ Math, aptitudes of various races, =838=.
+ Lie's final aim, =993=.
+ Lie's genius, =994=.
+ On m. and science, =1520=.
+ Famous aphorisms, =1635=.
+ Calculating machines, =1641=.
+ Calculus for high schools, =1918=.
+ On differential equations, =1926=.
+ Definition of a curve, =1927=.
+ On axioms of geometry, =2006=.
+ On the parallel axiom, =2009=.
+ On non-euclidean geometry, =2017=, =2021=.
+ On hyper-space, =2030=.
+
+ Kronecker, On the greatness of Gauss, =973=.
+ God made integers etc., =1634=.
+
+ Kummer, On Dirichlet, =977=.
+ On beauty in m., =1111=.
+
+
+ LaFaille, Mathesis few know, =1870=.
+
+ Lagrange, On correlation of algebra and geometry, =527=.
+ His style and method, 983.
+ And the parallel axiom, 984.
+ On Newton, =1011=.
+ Wings of m., =1604=.
+ Union of algebra and geometry, =1707=.
+ On the infinitesimal method, =1906=.
+
+ Lalande, M. in French army, =314=.
+
+ Langley, M. in Prussia, 513.
+
+ Lampe, On division of labor in m., =632=.
+ On Weierstrass, =1049=.
+ Weierstrass and Sylvester, =1050=.
+ Qualities common to math'ns and artists, =1113=.
+ Charm of m., =1115=.
+ Golden age of art and m. coincident, =1134=.
+
+ Language, Chapter XII. See also 311, 419, 443, 1523, 1804,
+ 1889.
+
+ Laplace, On instruction in m., =220=.
+ His style and method, =983=.
+ "Thus it plainly appears," 985, 986.
+ Emerson on, 1003.
+ On Leibnitz, =991=.
+ On the language of analysis, =1222=.
+ On m. and nature, =1525=.
+ On the origin of the calculus, =1902=.
+ On the exactitude of the differential calculus, =1910=.
+ The universe in a single formula, =1920=.
+ On probability, =1963=, =1969=, =1971=.
+
+ Laputa, Math'ns of, 2120-2122,
+ Math. school of, 2123.
+
+ Lasswitz, On modern algebra, =1741=.
+ On function theory, =1934=.
+ On non-euclidean geometry, =2040=.
+
+ Latin squares, 252.
+
+ Latta, On Leibnitz's logical calculus, =1317=.
+
+ Law and m., 1597, 1598.
+
+ Laws of thought, 719, 1318.
+
+ Leadership, M. as training for, 317.
+
+ Lecture, Preparation of, 540.
+
+ Lefevre, M. hateful to weak minds, =733=.
+ Logic and m., =1309=.
+
+ Leibnitz, On difficulties in m., =241=.
+ His greatness, 987.
+ His influence, 988.
+ The nature of his work, 989.
+ His math. tendencies, 990.
+ His binary arithmetic, 991.
+ On Newton, =1010=.
+ On demonstrations outside of m., =1312=.
+ Ars characteristica, =1316=.
+ His logical calculus, 1317.
+ Union of philosophical and m. productivity, 1404.
+ M. and philosophy, =1435=.
+ On the certainty of math. knowledge, =1442=.
+ On controversy in geometry, =1859=.
+ His differential calculus, 1902.
+ His notation of the calculus, 1904.
+ On necessary and contingent truth, =1966=.
+
+ Leverrier, Discovery of Neptune, 1559.
+
+ Lewes, On the infinite, =1953=.
+
+ Lie, On central conceptions in modern m., =727=.
+ Endowment of math'ns, =818=.
+ The comparative anatomist, 992.
+ Aim of his work, 993.
+ His genius, 994.
+ On groups, =1752=.
+ On the origin of the calculus, =1901=.
+ On differential equations, =1924=.
+
+ Liliwati, Origin of, 995.
+
+ Limitations of math. science, 1437.
+
+ Limits, Method of, 1905, 1908, 1909, 1940.
+
+ Lindeman, On m. and science, =1523=.
+
+ Liouville, 822.
+
+ Lobatchewsky, =2022=.
+
+ Locke, On the method of m., =214=, =235=.
+ On proofs and demonstrations, =236=.
+ On the unpopularity of m., =271=.
+ On m. as a logical exercise, =423=, =424=.
+ M. cures presumption, =425=.
+ Math, reasoning of universal application, =426=.
+ On reading of classic authors, =604=.
+ On Aristotle, =914=.
+ On m. and philosophy, =1433=.
+ On m. and moral science, =1439=, =1440=.
+ On the certainty of math. knowledge, =1440=, =1441=.
+ On unity, =1607=.
+ On number, =1608=.
+ On demonstrations in numbers, =1630=.
+ On the advantages of algebra, =1705=.
+ On infinity, =1955=, =1957=.
+ On probability, =1965=.
+
+ Logarithmic spiral, 922.
+
+ Logarithmic tables, 602.
+
+ Logarithms, 1526, 1614, 1616.
+
+ Logic and m., Chapter XIII.
+ See also 423-430, 442.
+
+ Logical calculus, 1316, 1317.
+
+ Longevity of math'ns, 839.
+
+ Lovelace, Why are wise few etc., =1629=.
+
+ Lover, =2140=.
+
+
+ Macaulay, Plato and Bacon, =316=.
+ On Archimedes, =905=.
+ Bacon's view of m., =915=, =916=.
+ Anagram on his name, =996=.
+ Plato and Archytas, =1427=.
+ On the power of m., =1527=.
+
+ Macfarlane, On Tait, Maxwell, Thomson, =1042=.
+ On Tait and Hamilton's quaternions, =1044=.
+
+ Mach, On thought-economy in m., =203=.
+ M. seems possessed of intelligence, =626=.
+ On aim of research, =647=.
+ On m. and counting, =1601=.
+ On the space of experience, =2011=.
+
+ MacMahon, Latin squares, 252.
+ On Sylvester's bend of mind, =645=.
+ On Sylvester's style, =1040=.
+ On the idea of invariance, =1746=.
+
+ Magnitude, Grassmann's definition, 105.
+
+ Magnus, On the aim in teaching m., =505=.
+
+ Manhattan Island, Cost of, 2130.
+
+ Marcellus, Estimate of Archimedes, =909=.
+
+ Maschke, Man above method, =650=.
+
+ Masters, On the reading of the, 614.
+
+ Mathematic, Sylvester on use of term, 101.
+ Bacon's use of term, 106.
+
+ Mathematical faculty, Frequency of, 832.
+
+ Mathematical mill, The, 239, 1891.
+
+ Mathematical productions, 648, 649.
+
+ Mathematical theory, When complete, 636, 637.
+
+ Mathematical training, 443, 444.
+ Maxims of math'ns, 630, 631, 649.
+ Not a computer, 1211.
+ Intellectual habits of math'ns, 1428.
+ The place of the, 1529.
+ Characteristics of the mind of a, 1534.
+
+ Mathematician, The, Chapter VIII.
+
+ Mathematics, Definitions of, Chapter I.
+ Objects of, Chapter I.
+ Nature of, Chapter II.
+ Estimates of, Chapter III.
+ Value of, Chapter IV.
+ Teaching of, Chapter V.
+ Study of, Chapter VI.
+ Research in, Chapter VI.
+ Modern, Chapter VII.
+ As a fine art, Chapter XI.
+ As a language, Chapter XII.
+ Also 445, 1814.
+ And logic, Chapter XIII.
+ And philosophy, Chapter XIV.
+ And science, Chapter XV.
+ And applications, Chapter XV.
+ Knowledge most in, 214.
+ Suppl. brevity of life, 218.
+ The range of, 269.
+ Compared to French language, 311.
+ The care of great men, 322.
+ And professional education, 429.
+ And science teaching, 522.
+ The queen of the sciences, 975.
+ Advantage over philosophy, 1436, 1438.
+ As an instrument, 1506.
+ For its own sake, 1540, 1541, 1545, 1546.
+ The wings of, 1604.
+
+ Mathesis, 274, 276, 1870, 2015.
+
+ Mathews, On Disqu. Arith. =1638=.
+ On number theory, =1639=.
+ The symbol [congruent], =1646=.
+ On Cyclotomy, =1647=.
+ Laws of algebra, =1709=.
+ On infinite, zero, infinitesimal, =1954=.
+
+ Maxims of great math'ns, 630, 631, 649.
+
+ Maxwell, 1043, 1116.
+
+ McCormack, On the unpopularity of m., =270=.
+ On function, =1933=.
+
+ Mechanique celeste, 985, 986.
+
+ Medicine, M. and the study of, 1585, 1918.
+
+ Mellor, Definition of higher m., =108=.
+ Conclusions involved in premises, =238=.
+ On m. and science, =1561=.
+ On the calculus, =1912=.
+ On integration, =1923=, =1925=.
+
+ Memory in m., 253.
+
+ Menaechmus, 901.
+
+ Mere math'ns, 820, 821.
+
+ Merz, On the transforming power of m., =303=.
+ On the dominant ideas in m., =725=.
+ On extreme views in m., =827=.
+ On Leibnitz's work, =989=.
+ On the math. tendency of Leibnitz, =990=.
+ On m. as a lens, =1515=.
+ M. extends knowledge, =1524=.
+ Disquisitiones Arithmeticae, =1637=.
+ On functions, =1932=.
+ On hyper-space, =2036=.
+
+ Metaphysics, M. the only true, 305.
+
+ Meteorology and m., 1557.
+
+ Method of m. 212-215, 226, 227, 230, 235, 244, 806, 1576.
+
+ Metric system, 1725.
+
+ Military training, M. in, 314, 418, 1574.
+
+ Mill, On induction in m., =221=, =222=.
+ On generalization in m., =245=.
+ On math. studies, =409=.
+ On m. in a scientific education, =444=.
+ Math'ns hard to convince, =811=.
+ Math'ns require genius, =819=.
+ On Comte, =942=.
+ On Descartes, =942=, =948=.
+ On Sir William Hamilton's ignorance of m., =978=.
+ On Leibnitz, =987=.
+ On m. and philosophy, =1421=.
+ On m. as training for philosophers, =1422=.
+ M. indispensable to science, =1519=.
+ M. and social science, =1595=.
+ On the nature of geometry, =1838=.
+ On geometrical method, =1861=.
+ On the calculus, =1903=.
+
+ Miller, On the Darmstaetter prize, =2129=.
+
+ Milner, Geometry and poetry, =1118=.
+
+ Minchin, On English text-books, =539=.
+
+ Mineralogy and m., 1558.
+
+ Minkowski, On integral numbers, =1636=.
+
+ Miracles and m., 2157, 2158, 2160.
+
+ Mixed m., Bacon's definition of, 106.
+ Whewell's definition of, 107.
+
+ Modern algebra, 1031, 1032, 1638, 1741.
+
+ Modern geometry, 1710-1713, 715, 716, 1878.
+
+ Modern m., Chapter VII.
+
+ Moebius, Math'ns constitute a favorite class, =809=.
+ M. a fine art, =1107=.
+
+ Moral science and m., 1438-1440.
+
+ Moral value of m., See ethical value.
+
+ Mottoes, Of math'ns, 630, 631, 649.
+ Of Pythagoreans, 1833.
+
+ Murray, Definition of m., =116=.
+
+ Music and m., 101, 276, 965, 1107, 1112, 1116, 1127, 1128,
+ 1130-1133, 1135, 1136.
+
+ Myers, On m. as a school subject, =403=.
+ On pleasure in m., =454=.
+ On the ethical value of m., =457=.
+ On the value of arithmetic, =1622=.
+
+ Mysticism and numbers, 2136-2141, 2143.
+
+
+ Napier's rule, 1888.
+
+ Napoleon, M. and the welfare of the state, =313=.
+ His interest in m., 314, 1001.
+
+ Natural science and m., Chapter XV.
+ Also 244, 444, 445, 501.
+
+ Natural selection, 1921.
+
+ Nature of m., Chapter II.
+ See also 815, 1215, 1308, 1426, 1525, 1628.
+
+ Nature, Study of, 433-436, 514, 516, 612.
+
+ Navigation and m., 1543, 1544.
+
+ Nelson, Anagram on, 2153.
+
+ Neptune, Discovery of, 1554, 1559.
+
+ Newcomb, On geometrical paradoxers, =2113=.
+
+ Newton,
+ Importance of his work, 333.
+ On correlation in m., =526=.
+ On problems in algebra, =530=.
+ And Gauss compared, 827.
+ His fame, 1002.
+ Emerson on, 1003.
+ Whewell on, 1004, 1005.
+ Arago on, 1006.
+ Pope on, 1007.
+ Southey on, 1008.
+ Hill on, 1009.
+ Leibnitz on, 1010.
+ Lagrange on, 1011.
+ No monument to, 1012.
+ Wilson on, 1012, 1013.
+ His genius, 1014.
+ His interest in chemistry and theology, 1015.
+ And alchemy, 1016, 1017.
+ His first experiment, 1018.
+ As a lecturer, 1019.
+ As an accountant, 1020.
+ His memorandum-book, 1021.
+ His absent-mindedness, 1022.
+ Estimate of himself, =1023-1025=.
+ His method of work, 1026.
+ Discovery of the calculus, 1027.
+ Anagrams on, 1028.
+ Gauss's estimate of, 1029.
+ On geometry, =1811=.
+ Compared with Euclid, 1827.
+ Geometry a mechanical science, =1865=.
+ Test of simplicity, =1892=.
+ Method of fluxions, 1902.
+
+ Newton's rule, 1743.
+
+ Nile, Origin of name, 2150.
+
+ Noether, On Cayley, =931=.
+ On Sylvester, =1034=, =1041=.
+
+ Non-euclidean geometry, 1322, 2016-2029, 2033, 2035, 2040.
+
+ Nonnus, On the mystic four, =2148=.
+
+ Northrup, On Lord Kelvin, =1048=.
+
+ Notation, Importance of, 634, 1222, 1646.
+ Value of algebraic, 1213, 1214.
+ Criterion of good, 1216.
+ On Arabic, 1217, 1614.
+ Advantage of math., 1220.
+ See also symbolism.
+
+ Notions, Cardinal of m., 110.
+ Indefinable, 1219.
+
+ Novalis, Definition of pure m., =112=.
+ M. the life supreme, =329=.
+ Without enthusiasm no m., =801=.
+ Method is the essence of m., =806=.
+ Math'ns not good computers, =810=.
+ Music and algebra, =1128=.
+ Philosophy and m., =1406=.
+ M. and science, =1507=, =1526=.
+ M. and historic science, =1599=.
+ M. and magic, =2159=.
+ M. and miracles, =2160=.
+
+ Number, Every inquiry reducible to a question of, 1602.
+ And nature, 1603.
+ Regulates all things, 1605.
+ Aeschylus on, 1606.
+ Definition of, 1609, 1610.
+ And superstition, 1632.
+ Distinctness of, 1707.
+ Of the beast, 2151, 2152.
+
+ Number-theory,
+ The queen of m., 975.
+ Nature of, 1639.
+ Gauss on, 1644.
+ Smith on, 1645.
+ Notation in, 1646.
+ Aid to geometry, 1647.
+ Mystery in, 1648.
+
+ Number-work, Purpose of, 1623.
+
+ Numbers, Pythagoras' view of, 321.
+ Mighty are, 1568.
+ Aptitude for, 1617.
+ Demonstrations in, 1630.
+ Prime, 1648.
+ Necessary truths like, 1966.
+ Round, 2137.
+ Odd, 2138-2141.
+ Golden, 2142.
+ Magic, 2143.
+
+
+ Obscurity in m. and philosophy, 1407.
+
+ Observation in m., 251-253, 255, 433, 1830.
+
+ Obviousness in m., 985, 986, 1045.
+
+ Olney, On the nature of m., =253=.
+
+ Oratory and m., 829, 830.
+
+ Order and arrangement, 725.
+
+ Origin of m., 1412.
+
+ Orr, Memory verse for [pi], =2127=.
+
+ Osgood, On the calculus, =1913=.
+
+ Ostwald, On four-dimensional space, 2039.
+
+
+ [pi]. In actuarial formula, 945.
+ Memory verse for, 2127.
+
+ Pacioli, On the number three, =2145=.
+
+ Painting and m., 1103, 1107.
+
+ Papperitz, On the object of pure m., 111.
+
+ Paradoxes, Chapter XXI.
+
+ Parallel axiom, Proof of, 984, 2110, 2111.
+ See also non-euclidean geometry.
+
+ Parker, Definition of arithmetic, =1611=.
+ Number born in superstition, =1632=.
+ On geometry, =1805=.
+
+ Parton, On Newton, =1917-1919=, =1021=, =1022=, =1827=.
+
+ Pascal, Logic and m., =1306=.
+
+ Peacock, On the mysticism of Greek philosophers, =2136=.
+ The Yankos word for three, =2144=.
+ The number of the beast, =2152=.
+
+ Pearson, M. and natural selection, =834=.
+
+ Peirce, Benjamin, Definition of m., =120=.
+ M. as an arbiter, =210=.
+ Logic dependent on m., =1301=.
+ On the symbol [sq root]-1, =1733=.
+
+ Peirce, C. S. Definition of m., =133=.
+ On accidental relations, =2128=.
+
+ Perry, On the teaching of m., =510=, =511=, =619=, =837=.
+
+ Persons and anecdotes, Chapters IX and X.
+
+ Philosophy and m., Chapter XIV.
+ Also 332, 401, 414, 444, 445, 452.
+
+ Physics and m., 129, 437, 1516, 1530, 1535, 1538, 1539,
+ 1548, 1549, 1550, 1555, 1556.
+
+ Physiology and m., 1578, 1581, 1582.
+
+ Picard, On the use of equations, =1891=.
+
+ Pierce, On infinitesimals, =1940=.
+
+ Pierpont, Golden age of m., =701=.
+ On the progress of m., =708=.
+ Characteristics of modern m., =717=.
+ On variability, =721=.
+ On divergent series, =1937=.
+
+ Plato, His view of m., 316, 429.
+ M. a study suitable for freemen, =317=.
+ His conic sections, 332.
+ And Archimedes, 904.
+ Union of math. and philosophical productivity, 1404.
+ Diagonal of square, =1411=.
+ And Archytas, 1427.
+ M. and the arts, =1567=.
+ On the value of m., =1574=.
+ On arithmetic, =1620=, =1621=.
+ God geometrizes, 1635, 1636. 1702.
+ On geometry, 429, 1803, 1804, =1806=, =1844=, =1845=.
+
+ Pleasure, Element of in m., 1622, 1629, 1848, 1850, 1851.
+
+ Pliny, =2039=.
+
+ Plus and minus signs, 1727.
+
+ Plutarch, On Archimedes, =903=, =904=, =908-910=, =912=.
+ God geometrizes, =1802=.
+
+ Poe, =417=.
+
+ Poetry and m.,
+ Weierstrass on, 802.
+ Pringsheim on, 1108.
+ Wordsworth on, 1117.
+ Milner on, 1118.
+ Workman on, 1120.
+ Pollock on, 1121.
+ Hoffman on, 1122.
+ Thoreau on, 1123.
+ Emerson on, 1124.
+ Hill on, 1125, 1126.
+ Shakespeare on, 1127.
+
+ Poincare, On elegance in m., =640=.
+ M. has a triple end, =1102=.
+ M. as a language, =1208=.
+ Geometry not an experimental science, =1867=.
+ On geometrical axioms, =2005=.
+
+ Point, 1816.
+
+ Political science, M. and, 1201, 1324.
+
+ Politics, Math'ns and, 814.
+
+ Pollock, On Clifford, =938-941=, =1121=.
+
+ Pope, =907=, =2015=, =2031=, =2046=.
+
+ Precision in m., 228, 639, 728.
+
+ Precocity in m., 835.
+
+ Predicabilia a priori, 2003.
+
+ Press, M. ignored by daily, 731, 732.
+
+ Price, Characteristics of m., =247=.
+ On m. and physics, =1550=.
+
+ Prime numbers, Sylvester on, 1648.
+
+ Principia Mathematica, 1326.
+
+ Pringsheim, M. the science of the self-evident, =232=.
+ M. should be studied for its own sake, =439=.
+ On the indirect value of m., =448=.
+ On rigor in m., =535=.
+ On m. and journalism, =732=.
+ On math'ns in public service, =824=.
+ Math'n somewhat of a poet, =1108=.
+ On music and m., =1132=.
+ On the language of m., =1211=.
+ On m. and physics, =1548=.
+
+ Probabilities, 442, 823, 1589, 1590-1592, 1962-1972, 1975.
+
+ Problem solving, 531, 532.
+
+ Problems, In m., 523, 534.
+ In arithmetic, 528.
+ In algebra, 530.
+ Should be simple, 603.
+ In Cambridge texts, 608.
+ On solution of, 611.
+ On importance of, 624, 628.
+ What constitutes good, 629.
+ Aid to research, 644.
+ Of modern m., 1926.
+
+ Proclus, Ptolemy and Euclid, =951=.
+ On characteristics of geometry, =1869=.
+
+ Progress in m., 209, 211, 212, 216, 218, 702-705, 708.
+
+ Projective geometry, 1876, 1877, 1879, 1880.
+
+ Proportion, Euclid's doctrine of, 1834.
+ Euclid's definition of, 1835.
+
+ Proposition, 1219, 1419.
+
+ Prussia, M. in, 513.
+
+ Pseudomath, Defined, 2101.
+
+ Psychology and m., 1576, 1583, 1584.
+
+ Ptolemy and Euclid, 951.
+
+ Public service, M. and, 823, 824, 1303, 1574.
+
+ Public speaking, M. and, 420, 829, 830.
+
+ Publications, Math. of present day, 702, 703.
+
+ Pure M., Bacon's definition of, 106.
+ Whewell's definition of, 107.
+ On the object of, 111, 129.
+ Novalis' conception of, 112.
+ Hobson's definition of, 118.
+ Russell's definition of, 127, 128.
+
+ Pursuit of m., 842.
+
+ Pythagoras,
+ Number the nature of things, 321.
+ Union of math, and philosophical productivity, 1404.
+ The number four, =2147=.
+
+ Pythagorean brotherhood, Motto of, 1833.
+
+ Pythagorean theorem, 1854-1856, 2026.
+
+ Pythagoreans, Music and M., 1130.
+
+
+ Quadrature, See Squaring of the circle.
+
+ Quantity, Chrystal's definition of, =115=.
+
+ Quarles, On quadrature, =2116=.
+
+ Quaternions, 333, 841, 937, 1044, 1210, 1718-1726.
+
+ Quetelet, Growth of m., =1514=.
+
+
+ Railway-making, 1570.
+
+ Reading of m., 601, 604-606.
+
+ Reason, M. most solid fabric of human, 308.
+ M. demonstrates supremacy of human, 309.
+
+ Reasoning, M. a type of perfect, 307.
+ M. as an exercise in, 423-427, 429, 430, 1503.
+
+ Recorde, Value of arithmetic, 1619.
+
+ Regiomontanus, 1543.
+
+ Regular solids, 2132-2135.
+
+ Reid, M. frees from sophistry, =215=.
+ Conjecture has no place in m., =234=.
+ M. the most solid fabric, =308=.
+ On Euclid's elements, =955=.
+ M. manifests what is impossible =1414=.
+ On m. and philosophy, =1423=.
+ Probability and Christianity, =1975=.
+ On Pythagoras and the regular solids, =2132=.
+
+ Reidt, M, as an exercise in language, =419=.
+ On the ethical value of m., =456=.
+ On aim in math. instruction, =506=.
+
+ Religion and m., 274-276, 459, 460, 1013.
+
+ Research in m., Chapter VI.
+
+ Reversible verses, 2156.
+
+ Reye, Advantages of modern over ancient geometry, =714=.
+
+ Rhetoric and m., 1599.
+
+ Riemann, On m. and physics, 1549.
+
+ Rigor in m., 535-538.
+
+ Rosanes, On the unpopularity of m., =730=.
+
+ Royal road, 201, 901, 951, 1774.
+
+ Royal science, M. a, 204.
+
+ Rudio, On Euler, =957=.
+ M. and great artists, =1105=.
+ On m. and navigation, =1543=.
+
+ Rush, M. cures predisposition to anger, =458=.
+
+ Russell, Definition of m., =127=, =128=.
+ On nineteenth century m., =705=.
+ Chief triumph of modern m., =706=.
+ On the infinite, =723=.
+ On beauty in m., =1104=.
+ On the value of symbols, =1219=.
+ On Boole's Laws of Thought, =1318=.
+ Principia Mathematica, 1326.
+ On geometry and philosophy, =1410=.
+ Definition of number, =1609=.
+ Fruitful uses of imaginaries, =1735=.
+ Geometrical reasoning circular, =1864=.
+ On projective geometry, =1879=.
+ Zeno's problems, =1938=.
+ Definition of infinite collection, =1959=.
+ On proofs of axioms, =2013=.
+ On non-euclidean geometry, =2018=.
+
+
+ Safford, On aptitude for m., =520=.
+ On m. and science, =1509=.
+
+ Sage, Battalions of figures, =1631=.
+
+ Sartorius, Gauss on the nature of space, =2034=.
+
+ Scepticism, 452, 811.
+
+ Schellbach, Estimate of m., =306=.
+ On truth, =1114=.
+
+ Schiller, Archimedes and the youth, =907=.
+
+ Schopenhauer, Arithmetic rests on the concept of time,
+ =1613=.
+ Predicabilia a priori, =2003=.
+
+ Schroeder, M. as a branch of logic, =1323=.
+
+ Schubert, Three characteristics of m., =229=.
+ On controversies in m., =243=.
+ Characteristics of m., =263=.
+ M. an exclusive science, =734=.
+
+ Science and m., Chapter XV.
+ M. an indispensible tool of, 309.
+ Neglect of m. works injury to, 310.
+ Craig on origin of new, 646.
+ Greek view of, 1429.
+ Six follies of, 2107.
+ See also 433, 436, 437, 461, 725.
+
+ Scientific education, Math. training indispensable basis
+ of, =444=.
+
+ Screw, The song of the, 1894.
+ As an instrument in geometry, 2114.
+
+ Sedgwick, Quaternion of maladies, =1723=.
+
+ Segre, On research in m., =619=.
+ What kind of investigations are important, =641=.
+ On the worthlessness of certain investigations, =642,
+ 643=.
+ On hyper-space, =2031=.
+
+ Seneca, Alexander and geometry, =902=.
+
+ Seventy-seven, The number, 2149.
+
+ Shakespeare, 1127, 1129, 2141.
+
+ Shaw, J. B., M. like game of chess, =840=.
+
+ Shaw, W. H., M. and professional life, =1596=.
+
+ Sherman, M. and rhetoric, =1599=.
+
+ Smith, Adam, 1324.
+
+ Smith, D. E., On problem solving, =532=.
+ Value of geometrical training, =1846=.
+ Reason for studying geometry, =1850=.
+
+ Smith, H. J. S., When a math. theory is completed, =637=.
+ On the growth of m., =1521=.
+ On m. and science, =1542=.
+ On m. and physics, =1556=.
+ On m. and meteorology, =1557=.
+ On number theory, =1645=.
+ Rigor in Euclid, =1829=.
+ On Euclid's doctrine of proportion, =1834=.
+
+ Smith, W. B., Definition of m., =121=.
+ On infinitesimal analysis, =1914=.
+ On non-euclidean and hyperspaces, =2033=.
+
+ Simon, On beauty and truth, =1114=.
+
+ Simplicity in m., 315, 526.
+
+ Sin squared[phi], On the notation of, 1886.
+
+ Six hundred sixty-six, The number, 2151, 2152.
+
+ Social science and m., 1201, 1586, 1587.
+
+ Social service, M. as an aid to, 313, 314, 328.
+
+ Social value of m., 456, 1588.
+
+ Solitude and m., 1849, 1851.
+
+ Sophistry, M. free from, 215.
+
+ Sound, M. and the theory of, 1551.
+
+ Southey, On Newton, =1008=.
+
+ Space, Of experience, 2011.
+ Kant's doctrine of, 2003.
+ Schopenhauer's predicabilia, 2004.
+ Whewell, On the idea of, 2004.
+ Non-euclidean, 2015, 2016, 2018.
+ Hyper-, 2030, 2031, 2033, 2036-2038.
+
+ Spedding, On Bacon's knowledge of m., =917=.
+
+ Speer, On m. and nature-study, =514=.
+
+ Spence, On Newton, =1016=, =1020=.
+
+ Spencer, On m. in the arts, =1570=.
+
+ Spherical trigonometry, 1887.
+
+ Spira mirabilis, 922.
+
+ Spottiswoode, On the kingdom of m., =269=.
+
+ Squaring the circle, 1537, 1858, 1934, 1948, 2115-2117.
+
+ St. Augustine, The number seventy seven, =2149=.
+
+ St. Vincent, As a circle-squarer, 2109.
+
+ Steiner, On projective geometry, =1877=.
+
+ Stewart, M. and facts, =237=.
+ On beauty in m., =242=.
+ What we most admire in m., =315=.
+ M. for its own sake, =440=.
+ M. the noblest instance of force of the human mind,
+ =452=.
+ Math'ns and applause, =816=.
+ Mere math'ns, =821=.
+ Shortcomings of math'ns, =828=.
+ On the influence of Leibnitz, =988=.
+ Reason supreme, =1424=.
+ M. and philosophy compared, =1428=.
+ M. and natural philosophy, =1555=.
+
+ Stifel, The number of the beast, =2152=.
+
+ Stobaeus, Alexander and Menaechmus, =901=.
+ Euclid and the student, =952=.
+
+ Study of m., Chapter VI.
+
+ Substitution, Concept of, 727.
+
+ Superstition, M. frees mind from, 450.
+ Number was born in, 1632.
+
+ Surd numbers, 1728.
+
+ Surprises, M. rich in, 202.
+
+ Swift, On m. and politics, =814=.
+ The math'ns of Laputa, =2120-2122=.
+ The math. school of Laputa, =2123=.
+ His ignorance of m., 2124, 2125.
+
+ Sylvester, On the use of the terms mathematic and
+ mathematics, =101=.
+ Order and arrangement the basic ideas of m., =109=,
+ =110=.
+ Definition of algebra, =110=.
+ Definition of arithmetic, =110=.
+ Definition of geometry, =110=.
+ On the object of pure m., =129=.
+ M. requires harmonious action of all the faculties,
+ =202=.
+ Answer to Huxley, =251=.
+ On the nature of m., =251=.
+ On observation in m., =255=.
+ Invention in m., =260=.
+ M. entitled to human regard, =301=.
+ On the ethical value of m., =449=.
+ On isolated theorems, =620=.
+ "Auge _et impera._" =631=.
+ His bent of mind, =645=.
+ Apology for imperfections, =648=.
+ On theoretical investigations, =658=.
+ Characteristics of modern m., =724=.
+ Invested m. with halo of glory, 740.
+ M. and eloquence, =829=.
+ On longevity of math'ns, =839=.
+ On Cayley, =930=.
+ His view of Euclid, 936.
+ Jacobi's talent for philology, =980=.
+ His eloquence, 1030.
+ Researches in quantics, =1032=.
+ His weakness, 1033, 1036, 1037.
+ One-sided character of his work, 1034.
+ His method, 1035, 1036, 1041.
+ His forgetfulness, 1037, 1038.
+ Relations with students, 1039.
+ His style, 1040, 1041.
+ His characteristics, 1041.
+ His enthusiasm, 1041.
+ The math. Adam, =1042=.
+ And Weierstrass, 1050.
+ On divine beauty and order in m., =1101=.
+ M. among the fine arts, =1106=.
+ On music and m., =1131=.
+ M. the quintessence of language, =1205=.
+ M. the language of the universe, =1206=.
+ On prime numbers, =1648=.
+ On determinants, =1740=.
+ On invariants, =1742=.
+ Contribution to theory of equations, 1743.
+ To a missing member etc., =1745=.
+ Invariants and isomerism, =1750=.
+ His dislike for Euclid, =1826=.
+ On the invention of integrals, =1922=.
+ On geometry and analysis, =1931=.
+ On paradoxes, =2104=.
+
+ Symbolic language, M. as a, 1207, 1212.
+ Use of, 1573.
+
+ Symbolic logic, 1316-1321.
+
+ Symbolism, On the nature of math., 1210.
+ Difficulty of math., 1218.
+ Universal impossible, 1221.
+ See also notation.
+
+ Symbols, Burlesque on, 1741.
+
+ Symbols, M. leads to mastery of, 421.
+ Value of math., 1209, 1212, 1219.
+ Essential to demonstration, 1316.
+ Arithmetical, 1627.
+
+
+ Tact in m., 622, 623.
+
+ Tait, On the unpopularity of m., =740=.
+ And Thomson, 1043.
+ And Hamilton, 1044.
+ On quaternions, =1724-1726=.
+ On spherical trigonometry, =1887=.
+
+ Talent, Math'ns men of, 825.
+
+ Teaching of m., Chapter V.
+
+ Tennyson, 1843.
+
+ Teutonic race, Aptitude for m., 838.
+
+ Text-books, Chrystal on, 533.
+ Minchin on, 539.
+ Cremona on English, 609.
+ Glaisher on need of, 635.
+
+ Thales, 201.
+
+ Theoretical investigations, 652-664.
+
+ Theory and practice, 661.
+
+ Thompson, Sylvanus, Lord Kelvin's definition of a math'n,
+ =822=.
+ Cayley's estimate of quaternions, =937=.
+ Thomson's "It is obvious that," =1045=.
+ Anecdote of Lord Kelvin, =1046=, =1047=.
+ On the calculus for beginners, =1917=.
+
+ Thomson, Sir William,
+ M. the only true metaphysics, =305=.
+ M. not repulsive to common sense, =312=.
+ What is a math'n? =822=.
+ And Tait, 1043.
+ "It is obvious that," 1045.
+ Anecdotes concerning, 1046, 1047, 1048.
+ On m. and astronomy, =1562=.
+ On quaternions, =1721, 1722=.
+
+ Thomson and Tait, 1043.
+ On Fourier's theorem, =1928=.
+
+ Thoreau, On poetry and m., =1123=.
+
+ Thought-economy in m., 203, 1209, 1704.
+
+ Three, The Yankos word for, 2144.
+ Pacioli on the number, 2145.
+
+ Time, Arithmetic rests on notion of 1613.
+ As a concept in algebra, 1715, 1716, 1717.
+ Kant's doctrine of, 2001.
+ Schopenhauer's predicabilia, 2003.
+
+ Todhunter, On m. as a university subject, =405=.
+ On m. as a test of performance, =408=.
+ On m. as an instrument in education, =414=.
+ M. requires voluntary exertion, =415=.
+ On exercises, =422=.
+ On problems, =523=, =608=.
+ How to read m., =605=, =606=.
+ On discovery in elementary m., =617=.
+ On Sylvester's theorem, =1743=.
+ On performance in Euclid, =1818=.
+
+ Transformation, Concept of, 727.
+
+ Trigonometry, 1881, 1884-1889.
+
+ Trilinear co-ordinates, 611.
+
+ Trisection of angle, 2112.
+
+ Truth, and m., 306.
+ Math'ns must perceive beauty of, 803.
+ And beauty, 1114.
+
+ Tzetzes, Plato on geom., =1803=.
+
+
+ Unity, Locke on the idea of, 1607.
+
+ Universal algebra, 1753.
+
+ Universal arithmetic, 1717.
+
+ Universal language, 925.
+
+ Unpopularity of m., 270, 271, 730-736, 738, 740, 1501,
+ 1628.
+
+ Usefulness, As a principle in research, 652-655, 659, 664.
+
+ Uses of m., See value of m.
+
+
+ Value of m., Chapter IV.
+ See also 330, 333, 1414, 1422, 1505, 1506, 1512, 1523,
+ 1526, 1527, 1533, 1541, 1542, 1543, 1547-1576,
+ 1619-1626, 1841, 1844-1851.
+
+ Variability, The central idea of modern m., 720, 721.
+
+ Venn, On m. as a symbolic language, =1207=.
+ M. the only gate, =1517=.
+
+ Viola, On the use of fallacies, =610=.
+
+ Virgil, =2138=.
+
+ Voltaire, Archimedes more imaginative than Homer, =259=.
+ M. the staff of the blind, =461=.
+ On direct usefulness of results, =653=.
+ On infinite magnitudes, =1947=.
+ On the symbol, =1950=.
+ Anagram on, 2154.
+
+
+ Walcott, On hyperbolic functions, =1930=.
+
+ Walker, On problems in arithmetic, =528=.
+ On the teaching of geometry, =529=.
+
+ Wallace, On the frequency of the math. faculty, =832=.
+ On m. and natural selection, =833, 834=.
+ Parallel growth of m. and music, =1135=.
+
+ Walton, Angling like m., =739=.
+
+ Weber, On m. and physics, =1549=.
+
+ Webster, Estimate of m., =331=.
+
+ Weierstrass, Math'ns are poets, =802=.
+ Anecdote concerning, 1049.
+ And Sylvester, 1050.
+ Problem of infinitesimals, 1938.
+
+ Weismann, On the origin of the math. faculty, =1136=.
+
+ Wells, On m. as a world language, =1201=.
+
+ Whately, On m. as an exercise, =427=.
+ On m. and navigation, =1544=.
+ On geometrical demonstrations, =1839=.
+ On Swift's ignorance of m., =2124=.
+
+ Whetham, On symbolic logic, =1319=.
+
+ Whewell, On mixed and pure math., =107=.
+ M. not an inductive science, =223=.
+ Nature of m., 224.
+ Value of geometry, 445.
+ On theoretical investigations, =660=, =662=.
+ Math'ns men of talent, =825=.
+ Fame of math'ns, =826=.
+ On Newton's greatness, =1004=.
+ On Newton's theory, =1005=.
+ On Newton's humility, =1025=.
+ On symbols, 1212.
+ On philosophy and m., =1429=.
+ On m. and science, =1534=.
+ Quotation from R. Bacon, =1547=.
+ On m. and applications, =1541=.
+ Geometry and experience, 1814.
+ Geometry not an inductive science, 1830.
+ On limits, 1909.
+ On the idea of space, 2004.
+ On Plato and the regular solids, =2133=,
+ =2135=.
+
+ White, H. S., On the growth of m., =211=.
+
+ White, W. F., Definition of m., =131=, =1203=.
+ M. as a prerequisite for public speaking, =420=.
+ On beauty in m., =1119=.
+ The place of the math'n, =1529=.
+ On m. and social science, =1586=.
+ The cost of Manhattan island, =2130=.
+
+ Whitehead, On the ideal of m., =119=.
+ Definition of m., =122=.
+ On the scope of m., =126=.
+ On the nature of m., =233=.
+ Precision necessary in m., =639=.
+ On practical applications, =655=.
+ On theoretical investigations, =659=.
+ Characteristics of ancient geometry, =713=.
+ On the extent of m., =737=.
+ Archimedes compared with Newton, =911=.
+ On the Arabic notation, =1217=.
+ Difficulty of math. notation, =1218=.
+ On symbolic logic, =1320=.
+ Principia Mathematica, 1326.
+ On philosophy and m., =1403=.
+ On obscurity in m. and philosophy, =1407=.
+ On the laws of algebra, =1708=.
+ On + and - signs, =1727=.
+ On universal algebra, =1753=.
+ On the Cartesian method, =1890=.
+ On Swift's ignorance of m., =2125=.
+
+ Whitworth, On the solution of problems, =611=.
+
+ Williamson, On the value of m., =1575=.
+ Infinitesimals and limits, =1905=.
+ On infinitesimals, =1946=.
+
+ Wilson, E. B., On the social value of m., =1588=.
+ On m. and economics, =1593=.
+ On the nature of axioms, =2012=.
+
+ Wilson, John, On Newton and Shakespeare, =1012=.
+ Newton and Linnaeus, =1013=.
+
+ Woodward, On probabilities, =1589=.
+ On the theory of errors, =1973=, =1974=.
+
+ Wordsworth, W., On Archimedes, =906=.
+ On poetry and geometric truth, =1117=.
+ On geometric rules, =1418=.
+ On geometry, =1840=, =1848=.
+ M. and solitude, =1859=.
+
+ Workman, On the poetic nature of m., =1120=.
+
+
+ Young, C. A., On the discovery of Neptune, =1559=.
+
+ Young, C. W., Definition of m., =124=.
+
+ Young, J. W. A., On m. as type a of thought, =404=.
+ M. as preparation for science study, =421=.
+ M. essential to comprehension of nature, =435=.
+ Development of abstract methods, =729=.
+ Beauty in m., =1110=.
+ On Euclid's axiom, =2014=.
+
+
+ Zeno, His problems, 1938.
+
+ Zero, 1948, 1954.
+
+
+ * * * * *
+
+
+
+
+ Transcriber's Notes
+
+ Punctuation has been standardised.
+
+ Characters in small caps have been replaced by all caps.
+
+ Italic text has been denoted by _underscores_ and bold text
+ by =equal signs=.
+
+ Em-dash added before all attribution names for consistency.
+
+ The two omitted illustrations have been identified by an
+ [Illustration:] tag with a short description.
+
+ Mis-alphabetized entries in the Index have been
+ corrected
+
+ Non-printable superscripts are represented by a
+ caret followed by the character , i.e. x^n.
+ If the superscript is more than one character,
+ they will be placed in {}, i.e. x^{23}.
+
+ Non-printable subscripts have been represented by an
+ underscore followed by the subscript in braces
+ i.e. _{15}.
+
+ Non-printable symbols have been presented in descriptive
+ brackets i.e. [infinity].
+
+ Book was written in a period when many words had not
+ become standarized in their spelling. Numerous words
+ have multiple spelling variations in the text. These
+ have been left unchanged unless noted below:
+
+ oe ligature --> oe
+
+ Sec.230 - "elmenetary" corrected to "elementary"
+ (the most elementary use of)
+
+ Sec.437 - "Mathematiks" corrected to "Mathematicks"
+ (The Usefulness of Mathematicks)
+ as in the quoted text.
+
+ Sec.511 - Block number shown as 517
+
+
+ Sec.517 - "hoheren" corrected to "hoeheren"
+ (hoeheren Schulen) for consistency
+
+ Sec.540 - duplicate word "the" removed
+ (let the mind)
+
+ Sec.657 - "anaylsis" corrected to "analysis"
+ (field of analysis.)
+
+ Sec.729 - "Geomtry" corrected to "Geometry"
+ (Algebra and Geometry)
+
+ Sec.822 - end of quote not identified;
+ placement unclear.
+
+ Sec.823 - "heros" corrected to "heroes"
+ (many of the major heroes)
+
+ Sec.986 - added missing end quote
+
+ Sec.1132 - "Vereiningung" corrected to "Vereinigung"
+ ( Deutschen Mathematiker Vereinigung)
+
+ Sec.1325 - "Philosphy" corrected to "Philosophy"
+ (Positive Philosophy)
+
+ Sec.1421 - "1427" corrected to block "1421"
+ (=1421.=)
+
+ Sec.1503 - "Todhunder's" corrected to "Todhunter's"
+ (Todhunter's History of)
+
+ Sec.1535 - "uses" corrected to "use"
+ (the use of analysis)
+
+ Sec.1803 - "tey" corrected to "ten"
+ (mou ten stegen)
+
+ Sec.1874 - "anaylsis" corrected to "analysis"
+ (a kind of analysis)
+
+ Sec.1930 - "Hyberbolic" corrected to "Hyperbolic"
+ (Mathematical Tables, Hyperbolic Functions)
+
+ Sec.2009 - "Stanfpunkte" corrected to "Standpunkte"
+ (hoeheren Standpunkte aus)
+
+ Sec.2126 - Omitted block number added
+
+ Sec.2135 - "astromomy" corrected to "astronomy"
+ (history of astronomy)
+
+ Sec.2151 - "10" corrected to "9"
+ (A to I represent 1-9)
+
+ p385 - Appolonius is also spelled Apollonius but not
+ referenced at 523 and 917
+
+ p387 - "Bocher" corrected to "Bocher"
+ as given in text
+
+ p395 - "Slyvester" corrected to "Sylvester"
+ (And Sylvester)
+
+ p397 - "Om" corrected to "On"
+ (On m. and law)
+
+ p403 - "philosphers" corrected to "philosophers"
+ (Greek philosophers)
+
+
+
+
+
+End of Project Gutenberg's Memorabilia Mathematica, by Robert Edouard Moritz
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