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diff --git a/40624-t/old/40624-t.tex b/40624-t/old/40624-t.tex deleted file mode 100644 index b63be49..0000000 --- a/40624-t/old/40624-t.tex +++ /dev/null @@ -1,13642 +0,0 @@ -% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
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-% The Project Gutenberg EBook of A Scrap-Book of Elementary Mathematics, by
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-% %
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-% %
-% %
-% Title: A Scrap-Book of Elementary Mathematics %
-% Notes, Recreations, Essays %
-% %
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-% %
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-\DeclareMathSymbol{U}{\mathalpha}{operators}{`U}
-\DeclareMathSymbol{V}{\mathalpha}{operators}{`V}
-\DeclareMathSymbol{W}{\mathalpha}{operators}{`W}
-\DeclareMathSymbol{X}{\mathalpha}{operators}{`X}
-\DeclareMathSymbol{Y}{\mathalpha}{operators}{`Y}
-\DeclareMathSymbol{Z}{\mathalpha}{operators}{`Z}
-
-%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%
-\begin{document}
-%%%% PG BOILERPLATE %%%%
-\PGBoilerPlate
-\begin{center}
-\begin{minipage}{\textwidth}
-\small
-\begin{PGtext}
-The Project Gutenberg EBook of A Scrap-Book of Elementary Mathematics, by
-William F. White
-
-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: A Scrap-Book of Elementary Mathematics
- Notes, Recreations, Essays
-
-Author: William F. White
-
-Release Date: August 30, 2012 [EBook #40624]
-
-Language: English
-
-Character set encoding: ISO-8859-1
-
-*** START OF THIS PROJECT GUTENBERG EBOOK A SCRAP-BOOK ***
-\end{PGtext}
-\end{minipage}
-\end{center}
-\clearpage
-
-%%%% Credits and transcriber's note %%%%
-\begin{center}
-\begin{minipage}{\textwidth}
-\begin{PGtext}
-Produced by Andrew D. Hwang, Joshua Hutchinson, and the
-Online Distributed Proofreading Team at http://www.pgdp.net
-(This file was produced from images from the Cornell
-University Library: Historical Mathematics Monographs
-collection.)
-\end{PGtext}
-\end{minipage}
-\vfill
-\TranscribersNote{\TransNoteText}
-\end{center}
-%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
-\FrontMatter
-\iffalse
-%\DPpngSep{001}%
-Production Note
-
-Cornell University Library produced
-this volume to replace the
-irreparably deteriorated original.
-It was scanned using Xerox software
-and equipment at 600 dots
-per inch resolution and compressed
-prior to storage using
-CCITT Group 4 compression. The
-digital data were used to create
-Cornell's replacement volume on
-paper that meets the ANSI Standard
-Z39.48-1984. The production
-of this volume was supported in
-part by the Commission on Preservation
-and Access and the Xerox
-Corporation. Digital file copyright
-by Cornell University
-Library 1991.
-\fi
-%\DPpngSep{002}%
-%[Blank Page]
-%\DPpngSep{003}%
-\iffalse
-% [** TN: Library stamp]
-Cornell University Library
-
-BOUGHT WITH THE INCOME
-FROM THE
-SAGE ENDOWMENT FUND
-THE GIFT OF
-Henry W. Sage
-1891
-
-A.220013. 10/2/08.
-MATHEMATICS LIBRARY
-\fi
-%\DPpngSep{004}%
-%[Blank Page]
-%\DPpngSep{005}%
-%[Blank Page]
-%\DPpngSep{006}%
-\cleardoublepage
-\null\vfill
-\begin{center}
-\Graphic[jpg]{0.8\textwidth}{frontispiece} \\
-NUMERALS OR COUNTERS? \\
-\scriptsize
-From the \Title{Margarita Philosophica}. (See page~\PgNo{67}.)
-\end{center}
-\vfill
-\DPPageSep{007}{1}
-\cleardoublepage
-\begin{center}
-\Huge\bfseries
-A Scrap-Book \\
-{\footnotesize of} \\
-Elementary Mathematics
-\vfill
-
-\normalfont
-
-\footnotesize Notes, Recreations, Essays
-\vfill\vfill
-
-By
-\smallskip
-
-{\Large William F. White, Ph.D.} \\
-State Normal School, New Paltz, New York
-\normalsize
-\vfill\vfill\vfill\vfill
-
-Chicago \\
-The Open Court Publishing Company
-
-\smallskip
-\scriptsize London Agents \\
-Kegan Paul, Trench, Trübner \&~Co., Ltd. \\
-1908
-\normalsize
-\end{center}
-\DPPageSep{008}{2}
-\clearpage
-\null\vfill
-\begin{center}
-\footnotesize\scshape Copyright by \\
-The Open Court Publishing Co. \\
-1908.
-\end{center}
-\vfill
-\DPPageSep{009}{3}
-
-
-\TableofContents
-\iffalse
-CONTENTS.
-
- PAGE
-
-Preface. 7
-
-The two systems of numeration of large numbers. 9
-
-Repeating products. 11
-
-Multiplication at sight: a new trick with an old principle. 15
-
-A repeating table. 17
-
-A few numerical curiosities. 19
-
-Nine. 25
-
-Familiar tricks based on literal arithmetic. 27
-
-General test of divisibility. 30
-Test of divisibility by 7. 31
-Test of divisibility by 7, 11, and 13. 32
-
-Miscellaneous notes on number. 34
-The theory of numbers. 34
-Fermat's last theorem. 35
-Wilson's theorem. 35
-Formulas for prime numbers. 36
-A Chinese criterion for prime numbers. 36
-Are there more than one set of prime factors of a number? 37
-Asymptotic laws. 37
-Growth of the concept of number. 37
-Some results of permutation problems. 37
-Tables. 39
-Some long numbers. 40
-How may a particular number arise? 41
-
-Numbers arising from measurement. 43
-Decimals as indexes of degree of accuracy of measure. 44
-Some applications. 45
-
-Compound interest. 47
-If the Indians hadn't spent the \$24. 47
-\fi
-\DPPageSep{010}{4}
-\iffalse
-Decimal separatrixes. 49
-
-Present trends in arithmetic. 51
-
-Multiplication and division of decimals. 59
-
-Arithmetic in the Renaissance. 66
-
-Napier's rods and other mechanical aids to calculation. 69
-
-Axioms in elementary algebra. 73
-
-Do the axioms apply to equations? 76
-
-Checking the solution of an equation. 81
-
-Algebraic fallacies. 83
-
-Two highest common factors. 89
-
-Positive and negative numbers. 90
-
-Visual representation of complex numbers. 92
-
-Illustration of the law of signs in algebraic multiplication. 97
-A geometric illustration. 97
-From a definition of multiplication. 98
-A more general form of the law of signs. 99
-Multiplication as a proportion. 100
-Gradual generalization of multiplication. 100
-
-Exponents. 101
-
-An exponential equation. 102
-
-Two negative conclusions reached in the 19th century. 103
-
-The three parallel postulates illustrated. 105
-
-Geometric puzzles. 109
-Paradromic rings. 117
-
-Division of plane into regular polygons. 118
-
-A homemade leveling device. 120
-
-``Rope stretchers.'' 121
-
-The three famous problems of antiquity. 122
-
-The circle squarer's paradox. 126
-
-The instruments that are postulated. 130
-
-The triangle and its circles. 133
-
-Linkages and straight-line motion. 136
-
-The four-colors theorem. 140
-
-Parallelogram of forces. 142
-
-A question of fourth dimension by analogy. 143
-\fi
-\DPPageSep{011}{5}
-\iffalse
-Symmetry illustrated by paper folding. 144
-
-Apparatus to illustrate line values of trigonometric functions. 146
-
-Sine. 148
-
-Growth of the philosophy of the calculus. 149
-
-Some illustrations of limits. 152
-
-Law of commutation. 154
-
-Equations of U. S. standards of length and mass. 155
-
-The mathematical treatment of statistics. 156
-
-Mathematical symbols. 162
-
-Beginnings of mathematics on the Nile. 164
-
-A few surprising facts in the history of mathematics. 165
-
-Quotations on mathematics. 166
-
-Autographs of mathematicians. 168
-
-Bridges and isles, figure tracing, unicursal signatures, labyrinths.
- 170
-
-The number of the beast. 180
-
-Magic squares. 183
-Domino magic squares. 187
-Magic hexagons. 187
-
-The square of Gotham. 189
-
-A mathematical game-puzzle. 191
-
-Puzzle of the camels. 193
-
-A few more old-timers. 194
-
-A few catch questions. 196
-
-Seven-counters game. 197
-
-To determine direction by a watch. 199
-
-Mathematical advice to a building committee. 201
-
-The golden age of mathematics. 203
-
-The movement to make mathematics teaching more concrete. 205
-
-The mathematical recitation as an exercise in public speaking. 210
-
-The nature of mathematical reasoning. 212
-
-Alice in the wonderland of mathematics. 218
-\fi
-\DPPageSep{012}{6}
-\iffalse
-Bibliographic notes. 234
-Mathematical recreations. 234
-Publication of foregoing sections in periodicals. 235
-
-Bibliographic Index. 236
-
-General index. 241
-\fi
-\DPPageSep{013}{7}
-%[** TN: Index cross-references]
-\index{Logarithms|seealso{$e$}}%
-\index{n@$n$ dimensions|seealso{Fourth dimension}}%
-\index{Napier, John.|see{\DPtypo{Logariths}{Logarithms}}}%
-\index{Orthotomic|seealso{Imaginary}}%
-\index{Scalar|seealso{Real numbers}}%
-
-\Preface
-\index{Escott, E.~B.|(}%
-\index{Taylor, J.~M.}%
-
-Mathematics is the language of definiteness, the necessary
-vocabulary of those who know. Hence the intimate
-connection between mathematics and science.
-
-The tendency to select the problems and illustrations
-of mathematics mostly from the scientific, commercial
-and industrial activities of to-day, is one with
-which the writer is in accord. It may seem that in the
-following pages puzzles have too largely taken the
-place of problems. But this is not a text-book. Moreover,
-amusement is one of the fields of applied mathematics.
-
-The author desires to express obligation to Prof.\
-James~M. Taylor, \DPchg{LL.~D.}{LL.D.}, of Colgate University
-(whose pupil the author was for four years and afterward
-his assistant for two years) for early inspiration
-and guidance in mathematical study; to many mathematicians
-who have favored the author with words of
-encouragement or suggestion while some of the sections
-of the book have been appearing in periodical
-form; and to the authors and publishers of books that
-have been used in preparation. Footnotes give, in
-most cases, only sufficient reference to identify the
-book cited. For full bibliographic data see pages
-\PgNo{236}--\PgNo{240}. Special thanks are due to E.~B. Escott,
-M.S., of the mathematics department of the University
-of Michigan, who read the manuscript. His comments
-were of especial value in the theory of numbers. Extracts
-\DPPageSep{014}{8}
-from his notes on that subject (many of them
-hitherto unpublished) were generously placed at the
-disposal of the present writer. Where used, mention
-of the name will generally be found at the place.
-%[** TN: [sic] "acknowledgement", archaic spelling]
-Grateful acknowledgement is made of the kindness
-and the critical acumen of Mr.~Escott.
-
-The arrangement in more or less distinct sections
-accounts for occasional repetitions. The author asks
-the favor of notification of any errors that may be
-found.
-
-The aim has been to present some of the most interesting
-and suggestive phases of the subject. To this
-aim, all others have yielded, except that accuracy has
-never intentionally been sacrificed. It is hoped that
-this little book may be found to possess all the unity,
-completeness and originality that its title claims.
-
-\Signature{The Author.}
-{\textsc{New Paltz}, N. Y., August, 1907.}
-\index{Escott, E.~B.|)}%
-\DPPageSep{015}{9}
-
-
-\MainMatter
-
-\Chapter[Numeration of large numbers.]{The two systems of numeration of
-large numbers.}
-\index{Two systems of numeration}%
-\index{Arithmetic|(}%
-\index{French!numeration}%
-\index{German!numeration}%
-\index{Italian!numeration}%
-\index{Numeration, two systems}%
-
-What does a billion mean?
-\index{Billion}%
-
-In Great Britain and usually in the northern countries
-of Europe the numeration of numbers is by
-groups of six figures ($10^{6} = \text{million}$, $10^{12} = \text{billion}$, $10^{18}
-= \text{trillion}$, etc.)\ while in south European countries and
-in America it is by groups of three figures ($10^{6} = \text{million}$,
-$10^{9} = \text{billion}$, $10^{12} = \text{trillion}$, etc.). Our names
-are derived from the English usage: \emph{billion}, the \emph{second}
-\index{English!numeration}%
-power of a million; \emph{trillion}, the \emph{third} power of a million;
-etc.
-
-As the difference appears only in such large numbers,
-which are best written and read by exponents,
-it is not a matter of practical importance---indeed the
-difference in usage is rarely noticed---except in the
-case of \emph{billion}. This word is often heard; and it
-means a thousand million when spoken by one half
-of the world, and a million million in the mouths of
-the other half.
-
-\Par{Billion.} ``A billion does not strike the average
-mind as a very great number in this day of billion
-dollar trusts, yet a scientist has computed that at 10:40~\am,
-April~29, 1902, only a billion minutes had elapsed
-since the birth of Christ.'' One wonders where he
-obtained the data for such accuracy, but the general
-correctness of his result is easily verified. ``Billion''
-\DPPageSep{016}{10}
-is here used in the French and American sense (thousand
-million).
-
-An English professor has computed that if Adam
-was created in 4004~\BC\ (Ussher's chronology), and
-if he had been able to work $24$~hours a day continuously
-till now at counting at the rate of three a
-second, he would have but little more than half completed
-the task of counting a billion in the English
-sense (million million).
-\DPPageSep{017}{11}
-
-
-\Chapter{Repeating products.}
-\index{Circulating decimals|(}%
-\index{Products, repeating|(}%
-\index{Recurring decimals|(}%
-\index{Repeating!decimals|(}%
-\index{Repeating!products|(}%
-
-If $142857$ be multiplied by successive numbers, the
-figures repeat in the same cyclic order;
-\begin{figure}[hbt!]
-\[
-\begin{array}{ccc}
- & 1 \\
-7 & & 4 \\
-5 & & 2 \\
- & 8
-\end{array}
-\]
-\end{figure}
-that is, they read around the circle in the
-margin in the same order, but beginning at
-a different figure each time.
-\begin{align*}
-2 × 142857 &= \Z285714 \\
-3 × \Ditto{142857} &= \Z428571 \\
-4 × \Ditto{142857} &= \Z571428 \Brk
-5 × \Ditto{142857} &= \Z714285 \Brk
-6 × \Ditto{142857} &= \Z857142 \Brk
-7 × \Ditto{142857} &= \Z999999 \\
-8 × \Ditto{142857} &= 1142856.
-\end{align*}
-(When we attempt to put this seven-place number
-in our six-place circle, the first and last figures
-occupy the same place. Add them, and we still have
-the circular order~$142857$.)
-\begin{alignat*}{2}
- 9 × 142857 &= \Z1285713\quad & (285714) \\
-10 × \Ditto{142857} &= \Z1428570 & (428571) \Brk
-11 × \Ditto{142857} &= \Z1571427 & (571428) \Brk
-23 × \Ditto{142857} &= \Z3285711 & (285714) \\
-89 × \Ditto{142857} &= 12714273.
-\end{alignat*}
-(Again placing in the six-place circular order and
-adding figures that would occupy the same place, or
-taking the~$12$ and adding it to the~$73$, we have~$714285$.)
-\[
-356 × 142857 = 50857092
-\]
-(adding the~$50$ to the~$092$, $857142$).
-\DPPageSep{018}{12}
-
-The one exception given above $(7 × 142857 = 999999)$
-to the circular order furnishes the clew to the identity
-of this ``peculiar'' number: $142857$~is the repetend of
-the fraction~$\nicefrac{1}{7}$ expressed decimally. Similar properties
-belong to any ``perfect repetend'' (repetend the
-number of whose figures is just one less than the denominator
-of the common fraction to which the circulate
-is equal). Thus $\nicefrac{1}{17} = .\dot{0}58823529411764\dot{7}$;
-$2 × 0588\dots = 1176470588235294$ (same circular order);
-$7 × 0588\dots = 4117647058823529$; while
-$17 × 0588 \dots = 9999999999999999$. So also with the repetend of $\nicefrac{1}{29}$,
-which is~$0344827586206896551724137931$.
-
-It is easy to see why, in reducing~$\SlantFrac{1}{p}$ ($p$~being a
-prime) to a decimal, the figures must begin to repeat
-in less than $p$~decimal places; for at every step
-in the process of division the remainder must be less
-than the divisor. There are therefore only $p - 1$ different
-numbers that can be remainder. After that
-the process repeats.
-\begin{gather*}
-\frac{1}{7}
- = .1 \frac{3}{7}
- = .14 \frac{2}{7}
- = .142 \frac{6}{7}
- = .1428 \frac{4}{7}
- = .14285 \frac{5}{7} \\
- = .142857 \frac{1}{7} = \dots\Add{.}
-\end{gather*}
-
-Hence if we multiply $142857$ by $3$,~$2$, $6$, $4$,~$5$, we get
-the repetend beginning after the $1$st,~$2$d, $3$d, $4$th,~$5$th
-figures respectively.
-
-``If a repetend contains $\dfrac{p - 1}{2}$ digits, all the multiples
-up to~$p - 1$ will give one of two numbers each consisting
-of $\dfrac{p - 1}{2}$ digits. Example: $\dfrac{1}{13} = .\dot{0}7692\dot{3}$\Add{.}
-\DPPageSep{019}{13}
-\begin{alignat*}{2}
- 1 × 76923 &= \Z76923 & 2 × 76923 &= 153846 \\
- 3 × \Ditto{76923} &= 230769 & 5 × \Ditto{76923} &= 384615 \Brk
- 4 × \Ditto{76923} &= 307692 & 6 × \Ditto{76923} &= 461538 \Brk
- 9 × \Ditto{76923} &= 692307 & 7 × \Ditto{76923} &= 538461 \Brk
-10 × \Ditto{76923} &= 769230 & 8 × \Ditto{76923} &= 615384 \Brk
-12 × \Ditto{76923} &= 923076\quad & 11 × \Ditto{76923} &= 846153\rlap{\Add{.}''}
-\end{alignat*}
-\Attrib{\DPchg{(Escott).}{(Escott.)}}
-\index{Escott, E.~B.}%
-
-``In the repetend for $\nicefrac{1}{7}$, if we divide the number
-into halves, their sum is composed of~$9$'s, viz., $142 + 857 = 999$.
-A similar property is true of the repetend
-for~$\nicefrac{1}{17}$\Add{,} etc. This property is true also of the two
-numbers obtained from~$\nicefrac{1}{13}$. However, when we find
-the repetends of fractions~$\SlantFrac{1}{p}$ where the repetend contains
-only $\dfrac{p - 1}{2}$~digits, but which is of the form~$4n + 3$,
-it is not the halves of the numbers which are complementary,
-but the two numbers themselves. Example:
-\[
-\begin{array}{r@{\,}c@{\,}l>{\quad}r@{\,}c@{\,}l}
-\dfrac{1}{31} &=& .\dot{0}3225806451612\dot{9} &
-\dfrac{3}{31} &=& .\dot{0}9677419354838\dot{7} \\
-\Strut[24pt]
-\dfrac{30}{31} &=& .\dot{9}6774193548387\dot{0} \\
-\cline{3-3}
-\Strut\text{Sum} &=& .\dot{9}9999999999999\dot{9}\rlap{''}
-\end{array}
-\]
-\Attrib{(Escott\Add{.})}
-
-``A useful application may be made of this property
-of repeating, in reducing a fraction~$\SlantFrac{1}{p}$ to a decimal.
-After a number of figures have been found, as many
-more may be found by multiplying those already found
-by the remainder. It is, of course, advantageous to
-carry on the work until a comparatively small remainder
-has been found. Example: In reducing~$\nicefrac{1}{97}$
-to a decimal, after we have obtained the digits
-$.01030927835$ we get a remainder~$5$. Therefore, from
-this point on the digits are the same as those of~$\nicefrac{1}{97}$
-multiplied by~$5$. Multiplying by~$5$ (or dividing by~$2$)
-\DPPageSep{020}{14}
-we get $11$~more digits at once. The lengths of the
-periods of the reciprocals of primes have been determined
-at least as far as $p = 100,000$.'' \Attrib{(Escott.)}
-\index{Escott, E.~B.}%
-\DPPageSep{021}{15}
-
-
-%[** TN: Running head does not appear in the original]
-\Chapter[Multiplication at sight.]{Multiplication at sight: a new
-trick with an old principle.}
-\index{Multiplication!at sight}%
-\index{New trick with an old principle}%
-\index{Trick, new with an old principle}%
-
-This property of repeating the figures, possessed
-by these numbers, enables one to perform certain operations
-that seem marvelous till the observer understands
-the process. For example, one says: ``I will
-write the multiplicand, you may write below it any
-multiplier you choose of---say---two or three figures,
-and I will immediately set down the complete product,
-writing from left to right.'' He writes for the multiplicand
-$142857$. Suppose the observers then write
-$493$ as the multiplier. He thinks of $493 × \text{the number}$
-as $\SlantFrac{493}{7} = 70 \nicefrac{3}{7}$; so he \emph{writes}~$70$ as the first figures
-of the product (writing from left to right). Now $\SlantFrac{3}{7}$
-(\ie, $3 × \nicefrac{1}{7}$) is thought of as $3 × \text{the repetend}$, and it
-is necessary to determine first where to begin in writing
-the figures in the circular order. This may be
-determined by thinking that, since $3 × 7 \text{(the units
-figure of the multiplicand)} = 21$, the last figure is~$1$;
-therefore the first figure is the next after~$1$ in the
-circular order, namely~$4$. (Or one may obtain the~$4$
-by dividing $3$ by~$7$.) So he \emph{writes} in the product
-(after the~$70$) $4285$. From the $71$~remaining, the $70$
-first written must be subtracted (compare the explanation
-of $89 × 142857$ given above). This leaves the
-last two figures~$01$, and the product stands~$70428501$.
-When the spectators have satisfied themselves by actual
-multiplication that this is the correct product, let
-\DPPageSep{022}{16}
-us suppose that they test the ``lightning calculator''
-with $825$~as a multiplier. $\SlantFrac{825}{7} = 117 \nicefrac{6}{7}$. \emph{Write}~$117$.
-$6 × 7 = 42$. Next figure after~$2$ in repetend is~$8$. \emph{Write}~$857$.
-From the remaining~$142$ subtract the~$117$. \emph{Write}~$025$.
-
-Note that after the figures obtained by division ($117$~in
-the last example) have been written, there remain
-just six figures to write, and that the number first
-written is to be subtracted from the six-place number
-found from the circular order ($117$~subtracted from
-$857142$ in the last example). After a little practice,
-products may be written in this way without hesitation.
-
-If the multiplier is a multiple of~$7$, the process is
-even simpler. Take $378$ for multiplier. $\SlantFrac{378}{7} = 54$.
-Think of it as $53 \nicefrac{7}{7}$. \emph{Write}~$53$. $7 × \text{the repetend}$
-gives six nines. Mentally subtracting~$53$ from~$999999$,
-\emph{write}, after the~$53$, $999946$.
-
-This trick may be varied in many ways, so as not
-to repeat. (Few such performances will bear repetition.)
-\Eg, the operator may say, ``I will give a
-multiplicand, you may write the multiplier, divide
-your product by~$13$, and I will write the quotient as
-soon as you have written the multiplier.'' He then
-writes as multiplicand $1857141$, which is $13 × 142857$
-and is written instantly by the rule above. Now, as
-the $13$ cancels, the quotient may be written as the
-product was written in the foregoing illustrations. Of
-course another number could have been used instead
-of~$13$.
-\index{Circulating decimals|)}%
-\index{Products, repeating|)}%
-\index{Recurring decimals|)}%
-\index{Repeating!decimals|)}%
-\index{Repeating!products|)}%
-\DPPageSep{023}{17}
-
-
-\Chapter{A repeating table.}
-\index{Repeating!table}%
-\index{Tables!repeating}%
-
-Some peculiarities depending on the decimal notation
-of number. The first is the sum of the digits in
-the $9$'s~table.
-\begin{alignat*}{2}
-9 × 1 &= \Z9 \\
-9 × 2 &= \Z18;\quad & 1 + 8 &= 9 \\
-9 × 3 &= \Z27; & 2 + 7 &= 9 \\
-9 × 4 &= \Z36; & 3 + 6 &= 9 \\
-\multispan{4}{\dotfill} \\
-9 × 9 &= \Z81; & 8 + 1 &= 9 \\
-9 × 10 &= \Z90; & 9 + 0 &= 9 \\
-9 × 11 &= \Z99; & 9 + 9 &= 18;\quad \rlap{$1 + 8 = 9$} \\
-9 × 12 &= 108; & 1 + 0 & + 8 = 9 \\
-9 × 13 &= 117; & 1 + 1 & + 7 = 9 \\
- & \text{etc.}
-\end{alignat*}
-
-The following are given by Lucas\footnote
- {\Title{Récréations Mathématiques}, IV, 232--3; \Title{Théorie des Nombres},
- I,~8.}
-in a note entitled
-\Title{Multiplications curieuses}:
-\begin{align*}
- 1 × 9 + 2 &= 11 \\
- 12 × 9 + 3 &= 111 \\
- 123 × 9 + 4 &= 1111 \Brk
- 1234 × 9 + 5 &= 11111 \Brk
- 12345 × 9 + 6 &= 111111 \Brk
- 123456 × 9 + 7 &= 1111111 \\
- 1234567 × 9 + 8 &= 11111111 \\
-12345678 × 9 + 9 &= 111111111\Add{.} \Brk
-\DPPageSep{024}{18}
- 9 × 9 + 7 &= 88 \\
- 98 × 9 + 6 &= 888 \\
- 987 × 9 + 5 &= 8888 \Brk
- 9876 × 9 + 4 &= 88888 \Brk
- 98765 × 9 + 3 &= 888888 \Brk
- 987654 × 9 + 2 &= 8888888 \\
- 9876543 × 9 + 1 &= 88888888 \\
-98765432 × 9 + 0 &= 888888888\Add{.} \Brk
-%
- 1 × 8 + 1 &= 9 \\
- 12 × 8 + 2 &= 98 \\
- 123 × 8 + 3 &= 987 \Brk
- 1234 × 8 + 4 &= 9876 \Brk
- 12345 × 8 + 5 &= 98765 \Brk
- 123456 × 8 + 6 &= 987654 \Brk
- 1234567 × 8 + 7 &= 9876543 \\
- 12345678 × 8 + 8 &= 98765432 \\
-123456789 × 8 + 9 &= 987654321\Add{.} \Brk
-%
-12345679 × 8 &= 98765432 \\
-12345679 × 9 &= 111111111 \Brk
-\intertext{to which may, of course, be added}
-12345679 × 18 &= 222222222 \\
-12345679 × 27 &= 333333333 \\
-12345679 × 36 &= 444444444 \\
-\text{etc.} &
-\end{align*}
-\DPPageSep{025}{19}
-
-
-\FNChapter[A few numerical curiosities.]{A few numerical curiosities.}
-
-\footnotetext{Nearly all of the numerical curiosities in this section were
- given to the writer by Mr.~Escott.}
-\index{Curiosities, numerical}%
-\index{Escott, E.~B.|FN}%
-\index{Forty-one, curious property of}%
-\index{Numbers arising from measurement!differing from their log.\ only in position of decimal point}%
-\index{Numerical curiosity}%
-\index{Thirty-seven, curious property of}%
-
-\begin{gather*}
-11^{2} = 121;\quad 111^{2} = 12321;\quad 1111^{2} = 1234321;\quad \text{etc.} \\
-1 + 2 + 1 = 2^{2};\quad 1 + 2 + 3 + 2 + 1 = 3^{2}; \\
-1 + 2 + 3 + 4 + 3 + 2 + 1 = 4^{2};\quad \text{ etc.} \\
-121 = \frac{22 × 22}{1 + 2 + 1};\quad
-12321 = \frac{333 × 333}{1 + 2 + 3 + 2 +1};\quad
-\text{etc.}\footnotemark
-\end{gather*}
-\footnotetext{\Title{The Monist}, 1906; XVI, 625.}
-
-The following three consecutive numbers are probably
-the lowest that are divisible by cubes other than~$1$:
-\[
-1375,\quad 1376,\quad 1377
-\]
-(divisible by the cubes of $5$,~$2$ and $3$ respectively).
-
-\Par{A curious property of $37$ and~$41$.} Certain multiples
-of~$37$ are still multiples of~$37$ when their figures
-are permuted cyclically: $259 = 7 × 37$; $592 = 16 × 37$;
-$925 = 25 × 37$. The same is true of $185$, $518$\Add{,} and~$851$;
-$296$, $629$\Add{,} and~$962$. A similar property is true of multiples
-of~$41$: $17589 = 41 × 429$; $75891 = 41 × 1851$;
-$58917 = 41 × 1437$; $89175 = 41 × 2175$; $91758 = 41 × 2238$.
-
-\Par{Numbers differing from their logarithms only in
-the position of the decimal point.} The determination
-of such numbers has been discussed by Euler and by
-\index{Euler}%
-Professor Tait. Following are three examples of a list
-\index{Tait}%
-that could be extended indefinitely.
-\DPPageSep{026}{20}
-\begin{align*}
-\log 1.3712885742 &= .13712885742 \\
-\log 237.5812087593 &= 2.375812087593 \\
-\log 3550.2601815865 &= 3.5502601815865\Add{.}
-\end{align*}
-
-
-\Section{Powers having same digits\Add{.}}
-\index{Digits!in powers}%
-\index{Digits!in square numbers}%
-\index{Powers having same digits}%
-
-Consecutive numbers whose squares have the same
-digits:
-\begin{align*}
-13^{2} &= 169 & 157^{2} &= 24649 & 913^{2} &= 833569 \\
-14^{2} &= 196 & 158^{2} &= 24964 & 914^{2} &= 835396\Add{.}
-\end{align*}
-Cubes containing the same digits:
-\begin{align*}
-345^{3} &= 41063625 & 331^{3} &= 36264691 \\
-384^{3} &= 56623104 & 406^{3} &= 66923416 \\
-405^{3} &= 66430125\Add{.}
-\end{align*}
-A pair of numbers two of whose powers are composed
-of the same digits:
-\begin{align*}
-32^{2} &= 1024 & 32^{4} &= 1048576\\
-49^{2} &= 2401 & 49^{4} &= 5764801
-\end{align*}
-
-\Section{Square numbers containing the digits not repeated\Add{.}}
-\index{Square numbers containing the digits not repeated}%
-
-1. Containing the nine digits:\footnote
- {Published in the \Title{Mathematical Magazine}, Washington, D.C.,
- in~1883, and completed in \Title{L'Intermédiaire des Mathématiciens},
- 1897~(4:168).}
-\begin{align*}
-11826^{2} &= 139854276 & 20316^{2} &= 412739856 \\
-12363^{2} &= 152843769 & 22887^{2} &= 523814769 \Brk
-12543^{2} &= 157326849 & 23019^{2} &= 529874361 \Brk
-14676^{2} &= 215384976 & 23178^{2} &= 537219684 \Brk
-15681^{2} &= 245893761 & 23439^{2} &= 549386721 \Brk
-15963^{2} &= 254817369 & 24237^{2} &= 587432169 \Brk
-18072^{2} &= 326597184 & 24276^{2} &= 589324176 \Brk
-19023^{2} &= 361874529 & 24441^{2} &= 597362481 \Brk
-19377^{2} &= 375468129 & 24807^{2} &= 615387249 \Brk
-19569^{2} &= 382945761 & 25059^{2} &= 627953481 \Brk
-19629^{2} &= 385297641 & 25572^{2} &= 653927184 \Brk
-% [** Original page break; macro moved out of math mode]
-25941^{2} &= 672935481 & 27273^{2} &= 743816529 \Brk
-26409^{2} &= 697435281 & 29034^{2} &= 842973156 \Brk
-26733^{2} &= 714653289 & 29106^{2} &= 847159236 \\
-27129^{2} &= 735982641 & 30384^{2} &= 923187456
-\end{align*}
-\DPPageSep{027}{21}
-
-2. Containing the ten digits:\footnote
- {\Title{L'Intermédiaire des Mathématiciens}, 1907~(14:135).}
-\begin{align*}
-32043^{2} &= 1026753849 & 45624^{2} &= 2081549376 \\
-32286^{2} &= 1042385796 & 55446^{2} &= 3074258916 \Brk
-33144^{2} &= 1098524736 & 68763^{2} &= 4728350169 \Brk
-35172^{2} &= 1237069584 & 83919^{2} &= 7042398561 \\
-39147^{2} &= 1532487609 & 99066^{2} &= 9814072356
-\end{align*}
-
-\Section{Arrangements of the digits\Add{.}}
-\index{Arrangements of the digits}%
-\index{Digits!arrangements of}%
-
-If the number $123456789$ be multiplied by all the integers
-less than~$9$ and prime to~$9$, namely $2$,~$4$, $5$, $7$,~$8$,
-each product contains the nine digits and uses each
-digit but once.
-
-Each term in the following subtraction contains
-each of the nine digits once.
-\[
-\begin{array}{c}
-987654321 \\
-123456789 \\
-\hline
-864197532
-\end{array}
-\]
-
-To arrange the nine digits additively so as to make~$100$:
-\[
-\begin{array}[t]{r}
-15 \\
-36 \\
-47 \\
-\hline
-98 \\
- 2 \\
-\hline
-100 \\
-\end{array}
-\Qquad
-\begin{array}[t]{r}
- 56 \\
- 8 \\
- 4 \\
- 3 \\
-\hline
- 71 \\
- 29 \\
-\hline
-100
-\end{array}
-\Qquad
-\begin{array}[t]{r@{}l}
- 95&\frac{1}{2} \\
- 4&\frac{38}{76}\Strut \\
-\hline
-100&
-\end{array}
-\]
-Many other solutions. See Fourrey and Lucas.
-\DPPageSep{028}{22}
-
-To arrange the ten digits additively so as to make~$100$:
-\[
-\begin{array}{r@{}l}
-50&\frac{1}{2} \\
-49&\frac{38}{76}\Strut \\
-\hline
-100
-\end{array}
-\Qquad
-\begin{array}{r@{}l}
-80&\frac{27}{54} \\
-19&\frac{3}{6}\Strut \\
-\hline
-100
-\end{array}
-\]
-Many ways of doing this also.
-
-To place the ten digits so as to produce each of the
-digits:
-\begin{align*}
-\frac{62}{31} - \frac{970}{485} &= 0 &
-\frac{13485}{02697} &= 5 \\
-\frac{62}{31} × \frac{485}{970} &= 1 &
-\frac{34182}{05697} &= 6 \\
-\frac{97062}{48531} &= 2 &
-\frac{41832}{05976} &= 7\\
-\frac{107469}{35823} &= 3 &
-\frac{25496}{03187} &= 8\\
-\frac{23184}{05796} &= 4 &
-\frac{57429}{06381} &= 9 = \frac{95742}{10638}\Add{.}
-\end{align*}
-Lucas\footnote
- {\Title{Théorie des Nombres}, p.~40.}
-also gives examples where the ten digits are
-used, the zero not occupying the first place in a number,
-for all of the ten numbers above except~$6$, which is impossible.
-It will be noticed that, in the example given
-above for~$3$, the digit~$3$ occurs twice.
-
-The nine digits arranged to form a perfect cube:
-\begin{gather*}
-% [** TN: Rebroken; equations set "three and one" in the original]
-\frac{8}{32461759} = \left(\frac{2}{319}\right)^{3} \qquad
-\frac{8}{24137569} = \left(\frac{2}{289}\right)^{3} \\
-\frac{125}{438976} = \left(\frac{5}{76}\right)^{3} \qquad
-\frac{512}{438976} = \left(\frac{8}{76}\right)^{3}\Add{.}
-\end{gather*}
-
-The ten digits arranged to form a perfect cube:
-\[
-\frac{9261}{804357} = \left(\frac{21}{93}\right)^{3}\Add{.}
-\]
-\DPPageSep{029}{23}
-
-The ten digits placed to give an approximate value
-\index{p@{$\pi$}!expressed with the ten digits}%
-of~$\pi$:
-\[
-\pi = \frac{67389}{21450} = 3.141678+\Add{.}
-\]
-
-\Par[.]{Fourier's method of division}\footnote
- {Fourier, p.~187.}
-\index{Division!Fourier's method}%
-\index{Fourier's method of division}%
-by a number of two
-digits of which the units digit is~$9$. Increase the divisor
-by~$1$, and increase the dividend used at each step
-of the operation by the quotient figure for that step.
-\Eg, $43268 ÷ 29$. The ordinary
-\begin{table}[hbt!]
-\[
-\begin{array}[t]{r@{}r}
- & 1492 \\
-\cline{2-2}
-29 &)43268 \\
- &29\Z\Z\Z \\
-\cline{2-2}
- &142\Z\Z \\
- &116\Z\Z \\
-\cline{2-2}
- &266\Z \\
- &261\Z \\
-\cline{2-2}
- & 58 \\
- & 58 \\
-\cline{2-2}
-\end{array}
-\Qquad
-\begin{array}[t]{r@{}r}
-29 &)43268 \\
-\cline{2-2}
- & 1492
-\end{array}
-\]
-\end{table}
-arrangement is \hyperref[page:23]{shown at the left} for comparison. The
-form at the right is all that need be written in Fourier's
-method. To perform the operation, one thinks
-of the divisor as~$30$; $4 ÷ 3$, ($43 ÷ 30$,)~$1$; write the~$1$
-in the quotient and add it to the~$43$; $44 - 30 = 14$;
-$14 ÷ 3$,~$4$; etc. The reason underlying it is easily seen.
-\Eg, at the second step we have, by the common
-method, $142 - 4 × 29$. By Fourier's method we have
-$142 + 4 - 4 × 30$. The addition of the same number
-(the quotient figure) to both minuend and subtrahend
-does not affect the remainder.
-
-In the customary method for the foregoing example
-one practically uses~$30$ as divisor in determining the
-\DPPageSep{030}{24}
-quotient figure (thinking at the second step, $14 ÷ 3$,~$4$).
-In Fourier's method this is extended to the whole
-operation and the work is reduced to mere short division.
-
-So also in dividing by $19$, $39$, $49$,~etc. The method
-is, of course, not limited to divisors of two places, nor
-to those ending in~$9$. It may be used in dividing by a
-number ending in $8$,~$7$\Add{,}~etc.\ by increasing the divisor by
-$2$,~$3$\Add{,} etc\Add{.}, and also the dividend used at each step by $2$,~$3$\Add{,}
-etc.\ times the quotient figure for that step. But the
-advantage of the method lies chiefly in the case first
-stated.
-
-``The method is rediscovered every little while by
-some one and hailed as a great discovery.''
-\DPPageSep{031}{25}
-
-
-\Chapter{Nine.}
-\index{Nine, curious properties of}%
-
-Curious properties of the number nine, and numerical
-tricks with it, are given and explained by many
-writers; among them Dr.\ Edward Brooks, in his \Title{Philosophy
-of Arithmetic}. Of all such properties, perhaps
-the most practical application is the check on division
-and multiplication by casting out nines, the Hindu check
-\index{Hindu!check on division and multiplication}%
-as it is called. Next might come the bookkeeper's
-\index{Book-keeper's clue to inverted numbers}%
-clue to inverted numbers. In double-entry book-keeping,
-if there has been inversion (\eg, \$$4.83$ written
-in the debit side of one account, and \$$4.38$ in the
-credit side of another) and no other mistake, the trial
-balance will be ``off'' by a multiple of nine. It can
-also be seen in what columns the transposition was
-made.
-
-Recently suggested, and of no practical interest, is
-another property of the ``magic number,'' easily explained,
-\index{Magic!number}%
-like the rest, but at first glance curious: invert
-the figures of any three-place number; divide the
-difference between the original number and the inverted
-number by nine; and you may read
-the quotient forward or backward. Moreover
-the figure that occurs in the quotient
-is the difference between the first and last
-figures of the number taken.
-% [** TN: Inset figure in the original]
-\[
-\begin{array}{r@{}r}
- & 845 \\
- & 548 \\
-\cline{2-2}
-9 &)297 \\
-\cline{2-2}
- &\Z33
-\end{array}
-\]
-Explanation:
-Let $a$, $b$, $c$ be the hundreds, tens, units figures
-respectively of any three-place number. Then
-the number is $100a + 10b + c$, and the number inverted
-is $100c + 10b + a$.
-\DPPageSep{032}{26}
-\begin{align*}
-\frac{(100a + 10b + c) - (100c + 10b + a)}{9}
- &= \frac{99(a - c)}{9} \\
- &= 11(a - c)\Add{.}
-\end{align*}
-The product of~$11$ and any one-place number will have
-both figures alike, and may be read either way.
-
-Better known are the following three---all old and
-all depending on the principle, that the remainder,
-after dividing any number by~$9$, is the same as the
-remainder after dividing the sum of its digits by~$9$.
-
-1. Find the difference \DPtypo{betweeen}{between} a number of two
-figures and the number made by inverting the figures,
-conceal the numbers from me, but tell me one figure
-of the difference. I will tell you whether there is another
-figure in the difference, and, if so, what it is.
-(This can scarcely be repeated without every spectator
-noticing that one merely subtracts the given
-figure of the difference from~$9$.)
-
-2. Write a number of three or more places, divide
-by~$9$, and tell me the remainder; erase one figure, not
-zero, divide the resulting number by~$9$, and tell me
-the remainder. I will tell you the figure erased
-(which is, of course, the first remainder minus the
-second, or if the first is not greater than the second,
-then the first $+9 - \text{the second}$).
-
-3. Write a number with a missing figure, and I
-will immediately fill in the figure necessary to make
-the number exactly divisible by~$9$. (Suppose $728\ 57$
-to be written. Write $7$ in the space; for the excess
-from the given number after casting out~$9$'s is~$2$, and
-$9 - 2 = 7$.) This may be varied by undertaking to fill
-the space with a figure that shall make the number
-divisible by nine and leaving a specified remainder.\footnote
- {Adapted from Hooper, I, 22\Add{.}}
-\DPPageSep{033}{27}
-
-
-\Chapter[Familiar tricks.]{Familiar tricks based on literal
-arithmetic.}
-\index{Familiar tricks based on literal arithmetic}%
-\index{Tricks based on literal arithmetic}%
-
-Besides the tricks with the number~$9$, there are many
-other well-known arithmetical diversions, most, but not
-all, of them, depending on the Arabic notation of numbers
-used. Those illustrated in this section are specially
-numerous, can be ``made while you wait'' by any
-one with a little ingenuity and an elementary knowledge
-of algebra (or, more properly, of literal arithmetic)
-and, when set forth, are transparent the moment
-they are expressed in literal notation. They are
-amusing to children, and it is no wonder that the
-supply of them is perennial. The following three may
-be given as fairly good types. The first two are taken
-from Dr.\ Hooper's book, which was published in~1774.
-Verbatim quotation of them is made in order to preserve
-the flavor of quaintness. Only the explanation
-in terms of literal arithmetic is by the present writer.
-
-1. \textit{A person privately fixing on any number, to tell
-him that number.}
-
-After the person has fixed on a number, bid him
-double it and add~$4$ to that sum, then multiply the
-whole by~$5$; to the product let him add~$12$, and multiply
-the amount by~$10$. From the sum of the whole
-let him deduct~$320$, and tell you the remainder, from
-which, if you cut off the two last figures, the number
-that remains will be that he fixed on.
-\DPPageSep{034}{28}
-
-Let $n$ represent any number selected. The first
-member of the following equality readily reduces to~$n$,
-and the identity explains the trick.
-\[
-\bigl\{[(2n + 4)5 + 12]10 - 320\bigr\} ÷ 100 = n.
-\]
-
-2. \textit{Three dice being thrown on a table, to tell the
-number of each of them, and the order in which they
-stand.}
-
-Let the person who has thrown the dice double the
-number of that next his left hand, and add~$5$ to that
-sum; then multiply the amount by~$5$, and to the
-product add the number of the middle die; then let
-the whole be multiplied by~$10$, and to that product
-add the number of the third die. From the total let
-there be subtracted~$250$, and the figures of the number
-that remains will answer to the points of the three
-dice as they stand on the table.
-
-Let $x$, $y$, $z$ represent the numbers of points shown
-on the three dice in order. Then the instructions,
-expressed in symbols, give
-\[
-[(2x + 5)5 + y]10 + z - 250.
-\]
-Removing signs of grouping, we have
-\[
-100x + 10y + z,
-\]
-the number represented by the three digits $x$,~$y$,~$z$ in
-order.
-
-%[** TN: Quoted, not italicized, in the original]
-3. ``Take the number of the month in which you
-were born ($1$~for January, $2$~for February, etc.),
-double it; add~$5$; multiply by~$50$; add your age in
-years; subtract~$365$; add~$115$. The resulting number
-indicates your age---month and years.'' \Eg, a person
-$19$~years old and born in August ($8$th~month)
-would have, at the successive stages of the operation,
-\DPPageSep{035}{29}
-$8$,~$16$, $21$, $1050$, $1069$, $704$; and for the final number,
-$819$ ($8$~for August, $19$~for the years).
-
-If we let $m$ represent the number of the month,
-and $y$~the number of years, we can express the rule
-as a formula:
-\[
-(2m + 5)50 + y - 365 + 115,
-\]
-which simplifies to
-\[
-100m + y,
-\]
-the number of hundreds being the number of the
-month, and the number expressed by the last two
-digits being the number of years.
-\DPPageSep{036}{30}
-
-
-\FNChapter[General test of divisibility.]{General test of divisibility.}
-
-\footnotetext{Divisible \emph{without remainder} is of course the meaning of
- ``divisible'' in such a connection.}
-\index{Divisibility, tests of}%
-\index{General test of divisibility}%
-\index{Tests of divisibility}%
-
-Let $M$~represent any integer containing no prime
-factor that is not a factor of~$10$ (that is, no primes
-but $5$~and~$2$). Then $\SlantFrac{1}{M}$ expressed decimally is
-terminate. Call the number of places in the decimal~$m$.
-Let $N$~represent any prime except $5$,~$2$,~$1$. Then
-the reciprocal of~$N$ expressed decimally is a circulate.
-Call the number of places in the repetend~$n$.
-
-1. The remainder obtained by dividing any integer,~$I$,
-by~$M$ is the same as that obtained by dividing the
-number represented by the last (right-hand) $m$~digits
-of~$I$ by~$M$. If the number represented by those $m$~digits
-is divisible by~$M$, $I$~is divisible by~$M$, and not
-otherwise.
-
-2. The remainder obtained by dividing~$I$ by~$N$ is
-the same as that obtained by dividing the sum of the
-numbers expressed by the successive periods of $n$~digits
-of~$I$ by~$N$. If that sum is divisible by~$N$, $I$~is
-divisible by~$N$, and not otherwise. This depends
-on Fermat's theorem, that $P^{p-1} - 1$ is divisible by~$p$
-when $p$~and~$P$ are prime to each other.
-
-3. If a number is composite and contains a prime
-factor other than $5$~and~$2$, the divisibility of~$I$ by it
-may be tested by testing with the factors separately
-by (1)~and~(2).
-\DPPageSep{037}{31}
-
-Thus it is possible to test the divisibility of any integer
-by any other integer. This is usually of only
-theoretic interest, as actual division is preferable. But
-in the case of $2$,~$3$, $4$, $5$, $6$, $8$, $9$\Add{,} and~$10$ the test is easy
-and practical. A simple statement of it for each of
-these particular cases is found in almost any arithmetic.
-
-For $11$ a test slightly easier than the special application
-\index{Eleven, tests of divisibility by|(}%
-of the general test is usually given. That is,
-subtract the sum of the even-numbered digits from the
-sum of the odd-numbered digits (counting from the
-right) and add~$11$ to the minuend if smaller than the
-subtrahend. The result gives the same remainder
-when divided by~$11$ as the original number gives.
-The original number is divisible by~$11$ if that result is,
-and not otherwise. These remainders may be used
-in the same manner as the remainders used in casting
-out the nines, but are not so conveniently obtained.
-
-\Par{Test of divisibility by~$7$.} No known form of the general
-\index{Seven, tests of divisibility by|(}%
-test in this case is as easy as actually dividing by~$7$.
-From the point of view of theory it may be worth
-noticing that, as $7$'s~reciprocal gives a complementary
-repetend, the general test admits of variety of form.\footnote
- {A chapter of Brooks's \Title{Philosophy of Arithmetic} is devoted
- to divisibility by~$7$.}
-Let us consider, however, the direct application.
-
-Since the repetend has $6$~places, the test for divisibility
-by~$7$ is as follows: A number is divisible by~$7$
-if the sum of the numbers represented by the successive
-periods of $6$~figures each is divisible by~$7$, and
-not otherwise;~\eg,
-
-%[** TN: Punctuation not fully consistent throughout]
-Given the number $26,436,080,216,581$\Add{.}
-\DPPageSep{038}{32}
-\[
-\begin{array}{r@{}r}
- & 216581 \\
- & 436080 \\
- & 26 \\
-\cline{2-2}
-7 &)652687 \\
-\cline{2-2}
- & 93241\rlap{\Add{.}}
-\end{array}
-\]
-
-No remainder; therefore the given number is divisible
-by~$7$.
-
-{\Loosen \Par[.]{Test of divisibility by $7$,~$11$\Add{,} and~$13$} at the same time.\footnote
- {This was given to the author by Mr.~Escott, who writes:
- ``I have never seen it published, but it is so simple that it would
- be surprising if it had not been.''}
-\index{Escott, E.~B.|FN}%
-\index{Thirteen, test of divisibility by}%
-Since $7 × 11 × 13 = 1001$, divide the given number by~$1001$.
-If the remainder is divisible by $7$,~$11$, or~$13$,
-the given number is also, and not otherwise.}
-
-To divide by~$1001$, subtract each digit from the
-third digit following. An inspection of a division by~$1001$
-will show why this simple rule holds. The
-method may be made clear by an example, $4,728,350,169 ÷ 1001$.
-\begin{gather*}
-\begin{array}{l}
-4728350169 \\
-472\rlap{$/$}4626543 \\
-\Z\Z\Z3
-\end{array} \\
-\text{Quotient, $4723626$; remainder, $543$\Add{.}}
-\end{gather*}
-The third digit before the~$4$ being~$0$ (understood),
-write the difference,~$4$, beneath the~$4$. Similarly for
-$7$~and~$2$. $8 - 4 = 4$ (which for illustration is here
-written beneath the~$8$). We should next have $3 - 7$.
-This changes the~$4$ just found to~$3$, and puts $6$ under
-the original~$3$ (that is, $83 - 47 = 36$). $5 - 2$, $0 - 3$
-(always subtracting from a digit of the \emph{original} number
-the third digit to the left in the \emph{difference}, or lower,
-number), $1 - 6$,~etc. Making the corrections mentally
-we have the number as written. The number represented
-by the last three digits,~$543$, is the \emph{remainder}
-\DPPageSep{039}{33}
-after dividing the given number by~$1001$, and the
-number represented by the other digits, $4723626$, is
-the \emph{quotient}. With a little practice, this method can
-be applied rapidly and without making erasures. The
-remainder,~$543$, which alone is needed for the test,
-may also be obtained by subtracting the sum of the
-even-numbered periods of three figures each in the
-original number from the sum of the odd-numbered
-periods. A rapid method of obtaining the remainder
-thus is easily acquired; but the way illustrated above
-is more convenient.
-
-However obtained, the remainder is divisible or not
-by $7$,~$11$\Add{,} or~$13$, according as the given number is
-divisible or not. (Here $543$~is not divisible by $7$,~$11$\Add{,}
-or~$13$; therefore $4728350169$~is not divisible by either
-of them.) The original number is thus replaced, for
-the purpose of investigation, by a number of three
-places at most. As this tests for three common primes
-at once, it is convenient for one factoring large numbers
-without a factor table.
-\index{Eleven, tests of divisibility by|)}%
-\index{Seven, tests of divisibility by|)}%
-\DPPageSep{040}{34}
-
-
-\Chapter{Miscellaneous notes on number.}
-\index{Miscellaneous notes on number}%
-\index{Number!miscellaneous notes on|(}%
-\index{Numbers arising from measurement!theory of|EtSeq}%
-
-\Par[.]{The theory of numbers} has been called a ``neglected
-\index{Theory!of numbers|EtSeq}%
-but singularly fascinating subject.''\footnote
- {Ball, \Title{Hist.}, p.~416.}
-``Magic charm''
-is the quality ascribed to it by the foremost mathematician
-of the nineteenth century.\footnote
- {``The most beautiful theorems of higher arithmetic have
- this peculiarity, that they are easily discovered by induction,
- while on the other hand their demonstrations lie in exceeding
- obscurity and can be ferreted out only by very searching investigations.
- It is precisely this which gives to higher arithmetic
- that magic charm which has made it the favorite science
- of leading mathematicians, not to mention its inexhaustible
- richness, wherein it so far excels all other parts of pure
- mathematics.'' (Gauss; quoted by Young, p.~155.)}
-\index{Gauss}%
-Gauss said also:
-``Mathematics the queen of the sciences, and arithmetic
-[\ie, theory of numbers] the crown of mathematics.''
-And he was master of the sciences of his time. ``While
-it requires some facility in abstract reasoning, it may
-be taken up with practically no technical mathematics, is
-easily amenable to numerical exemplifications, and
-leads readily to the frontier. It is perhaps the only
-branch of mathematics where there is any possibility
-that new and valuable discoveries might be made without
-an extensive acquaintance with technical mathematics.''\footnote
- {Young, p.~155.}
-
-An interesting exercise in higher arithmetic is to
-investigate theorems and the established properties of
-particular numbers to determine which have their
-\DPPageSep{041}{35}
-origin in the nature of number itself and which are
-due to the decimal scale in which the numbers are
-expressed.
-
-\Par{Fermat's last theorem.} Of the many theorems in
-\index{Fermat's theorem!last theorem}%
-numbers discovered by Fermat, nearly all have since
-been proved. A well-known exception is sometimes
-called his ``last theorem.'' It ``is to the effect that no
-integral values of $x$,~$y$,~$z$ can be found to satisfy the
-equation $x^{n} + y^{n} = z^{n}$, if $n$~is an integer greater than~$2$.
-This proposition has acquired extraordinary celebrity
-from the fact that no general demonstration of it has
-been given, but there is no reason to doubt that it is
-true.''\footnote
- {Ball, \Title{Recreations}, p.~37.}
-It has been proved for special cases, and proved
-generally if certain assumptions be granted. Fermat
-asserted that he had a valid proof. That may yet be
-\DPchg{re-discovered}{rediscovered}; or, more likely, a new proof will be
-found by some new method of attack. ``Interest in
-problems connected with the theory of numbers seems
-recently to have flagged, and possibly it may be found
-hereafter that the subject is approached better on
-other lines.''\footnote
- {Ball, \Title{Hist.}, p.~469.}
-
-\Par[.]{Wilson's theorem} may be stated as follows: If $p$~is
-\index{Wilson's theorem}%
-a prime, $1 + \Fac{p - 1}$ is a multiple of~$p$. This well-known
-proposition was enunciated by Wilson,\footnote
- {As he was not a professional mathematician, but little
-\index{Wilson, John, biographic note|FN}%
- mention of him is made in histories of the subject. The following
- items may be of interest. They are from De~Morgan's
- \Title{Budget of Paradoxes}, p.~132--3. John Wilson (1741--1793) was
- educated at Cambridge. While an undergraduate he ``was
- considered stronger in algebra than any one in the University,
- except Professor Waring, one of the most powerful algebraists
- of the century.'' Wilson was the senior wrangler of~1761.
- He entered the law, became a judge, and attained a high reputation.}
-first published
-\DPPageSep{042}{36}
-by Waring in his \Title{Meditationes Algebraicĉ}, and first
-proved by Lagrange in~1771.
-\index{Lagrange}%
-
-\Par{Formulas for prime numbers.} ``It is easily demonstrated
-\index{Escott, E.~B.}%
-\index{Formulas for prime numbers}%
-\index{Primes!formulas for}%
-that no rational algebraic formula can always,
-give primes. Several remarkable expressions have
-been found, however, which give a large number of
-primes for consecutive values of~$x$. Legendre gave
-\index{Legendre}%
-$2x^{2} + 29$, which gives primes for $x = 0$ to~$28$, or for
-$29$~values of~$x$. Euler found $x^{2} + x + 41$, which gives
-\index{Euler}%
-primes for $x = 0$ to~$39$, \ie, $40$~values of~$x$. I have
-found $6x^{2} + 6x + 31$, giving primes for $29$~values of~$x$; and
-$3x^{2} + 3x + 23$, giving primes for $22$~values of~$x$. These
-expressions give different primes. We can transform
-them so that they will give primes for more values of~$x$,
-but not different primes. For instance, in Euler's
-formula if we replace $x$ by~$x - 40$, we get $x^{2} - 79x + 1601$,
-which gives primes for $80$~consecutive values
-of~$x$.'' \Attrib{(Escott.)}
-
-\Par{A Chinese criterion for prime numbers.} With reference
-\index{Chinese criterion for prime numbers}%
-\index{Criterion for prime numbers}%
-\index{Primes!Chinese, criterion for}%
-to the so-called criterion, that a number~$p$ is
-prime when the condition, that $2^{p-1} - 1$ be divisible by~$p$,
-is satisfied, Mr.~Escott makes the following interesting
-comment:
-
-``This is a well-known property of prime numbers
-(Fermat's Theorem) but it is not sufficient. My
-\index{Fermat's theorem}%
-attention was drawn to the problem by a question in
-\Title{L'Intermédiaire des Mathématiciens}, which led to a
-little article by me in the \Title{Messenger of Mathematics}.
-As the smallest number which satisfies the condition
-and which is not prime is~$341$, and to verify it by
-ordinary arithmetic (not having the resources of the
-Theory of Numbers) would require the division of
-$2^{340} - 1$ by~$341$, it is probable that the Chinese obtained
-the test by a mere induction.''
-\DPPageSep{043}{37}
-
-\Par{Are there more than one set of prime factors of a
-number?} Most text-books answer no; and this answer
-\index{Factors!more than one set of prime}%
-\index{Prime factors of a number, more than one set}%
-is correct if only arithmetic numbers are considered.
-But when the conception of number is extended
-to include complex numbers, the proposition,
-that a number can be factored into prime factors in
-only one way, ceases to hold. \Eg, $26 = 2 × 13 =
-(5 + \sqrt{-1})(5 - \sqrt{-1})$.
-
-\Par{Asymptotic laws.} This happily chosen name describes
-\index{Asymptotic laws}%
-``laws which approximate more closely to accuracy
-as the numbers concerned become larger.''\footnote
- {Ball, \Title{Hist.}, p.~464.}
-Legendre is among the best-known names here. One
-\index{Legendre}%
-of the most celebrated of the original researches of
-Dirichlet, in the middle of the last century, was on
-\index{Dirichlet}%
-this branch of the theory of numbers.
-
-\Par[.]{Growth of the concept of number}, from the arithmetic
-\index{Growth of concept of number}%
-\index{Number!growth of concept of}%
-integers of the Greeks, through the rational
-\index{Greeks}%
-fractions of Diophantus, ratios and irrationals recognized
-\index{Diophantus}%
-as numbers in the sixteenth century, negative
-\Foreign{versus} positive numbers fully grasped by Girard and
-\index{Girard, Albert}%
-Descartes, imaginary and complex by Argand, Wessel,
-\index{Argand, J.~R.}%
-\index{Descartes}%
-\index{Wessel}%
-Euler and Gauss,\footnote
- {See p.~\PgNo{94}.}
-\index{Euler}%
-\index{Gauss}%
-has proceeded in recent times to
-new theories of irrationals and the establishing of the
-continuity of numbers without borrowing it from
-space.\footnote
- {See Cajori's admirable summary, \Title{Hist.\ of Math.}, p.~372.}
-
-\Par{Some results of permutation problems.} The formulas
-\index{Combinations and permutations}%
-\index{Permutations}%
-for the number of permutations, and the number
-of combinations, of $n$~dissimilar things taken $r$~at a
-time are given in every higher algebra. The most
-important may be condensed into one equality:
-\[
-{}^{n} P_{r} = n(n - 1)(n - 2) \dots (n - r + 1)
- = \frac{\Fac[]{n}}{\Fac{n - r}}
- = {}^{n} C_{r} \Fac[]{r}\Add{.}
-\]
-\DPPageSep{044}{38}
-
-There are $3,979,614,965,760$ ways of arranging a
-\index{Dominoes!number of ways of arranging}%
-set of $28$~dominoes (\ie, a set from double zero to
-double six) in a line, with like numbers in contact.\footnote
- {Ball, \Title{Recreations}, p.~30, citing Reiss, \Title{Annali di matematica},
-\index{Reiss}%
-Milan, Nov.~1871.}
-
-``Suppose the letters of the alphabet to be wrote so
-small that no one of them shall take up more space
-than the hundredth part of a square inch: to find
-how many square yards it would require to write all
-the permutations of the $24$~letters in that size.''\footnote
- {Hooper, I,~59.}
-Dr.~Hooper
-computes that ``it would require a surface
-$18620$~times as large as that of the earth to write all
-the permutations of the $24$~letters in the size above
-mentioned.''
-
-Fear has been expressed that if the epidemic of
-\index{Societies' initials}%
-organizing societies should persist, the combinations
-and permutations of initial letters might become exhausted.
-We have F.A.A.M., I.O.O.F., K.M.B., K.P.,
-I.O.G.T., W.C.T.U., Y.M.C.A., Y.W.C.A., A.B.A.,
-A.B.S., A.C.M.S., etc.,~etc. An almanac names more
-than a hundred as ``prominent in New York City,''
-and its list is exclusive of fraternal organizations, of
-which the number is known to be vast. Already there
-are cases of two societies having names with the
-same initial letters. But by judicious choice this can
-long be avoided. Hooper's calculation supposed the
-entire alphabet to be employed in every combination.
-Societies usually employ only $2$,~$3$ or $4$~letters. And
-a letter may repeat, as the~\emph{A} in the title of the A.L.A.
-or of the A.A.A\@. The present problem is therefore
-different from that above. The number of permutations
-of $26$~letters taken two at a time, the two being
-not necessarily dissimilar, is~$26^{2}$; three at a time, $26^{3}$;~etc.
-\DPPageSep{045}{39}
-As there is occasionally a society known by one
-letter and occasionally one known by five, we have
-\[
-26^{1} + 26^{2} + 26^{3} + 26^{4} + 26^{5} = 12,356,630.
-\]
-This total of possible permutations is easily beyond
-immediate needs. By lengthening the names of societies
-(as seems to have already begun) the total can
-be made much larger\DPtypo{,}{.} Since the time when Hooper's
-calculations were made, two letters have been
-added to the alphabet. When the number of societies
-reaches about the twelve million mark, it would be
-well to agitate for a further extension of the alphabet.
-With these possibilities one may be assured, on
-the authority of exact science, that there is no cause
-for immediate alarm. The author hastens to allay
-the apprehensions of prospective organizers.
-
-\Par{Tables.} Many computations would not be possible
-\index{Tables}%
-without the aid of tables. Some of them are monuments
-to the patient application of their makers. Once
-made, they are a permanent possession. The time
-saved to the computer who uses the table is the one
-item taken into account in judging of the value of a
-table. It is difficult to appreciate the variety and
-extent of the work that has been done in constructing
-tables. For this purpose an examination of Professor
-Glaisher's article ``Tables'' in the \Title{Encyclopĉdia Britannica}
-\index{Glaisher}%
-is instructive.
-
-Anything that facilitates the use of a book of tables
-is important. Spacing, marginal tabs (in-cuts), projecting
-tabs---all such devices economize a little time
-at each handling of the book; and in the aggregate
-this economy is no trifle. Among American collections
-of tables for use in elementary mathematics the best
-example of convenience of arrangement for ready
-\DPPageSep{046}{40}
-reference is doubtless Taylor's \Title{Five-place Logarithmic
-and Trigonometric Tables} (1905). Dietrichkeit's
-\Title{Siebenstellige Logarithmen und Antilogarithmen}
-(1903) is a model of convenience.
-
-When logarithms to many places are needed, they
-can be readily calculated by means of tables made
-for the purpose, such as Gray's for carrying them to
-$24$~places (London,~1876).
-
-Factor tables have been printed which enable one
-to resolve into prime factors any composite number
-as far as the $10$th~million. They were computed by
-different calculators. ``Prof.\ D.~N. Lehmer, of the
-\index{Lehmer, D.~N.}%
-\index{Escott, E.~B.}%
-\index{Primes!tables of}%
-University of California, is now at work on factor
-tables which will extend to the $12$th~million. When
-completed they will be published by the Carnegie Institution,
-Washington,~D.C\@. According to Petzval,
-\index{Petzval}%
-tables giving the smallest prime factors of numbers
-as far as $100,000,000$ have been calculated by Kulik,
-\index{Kulik}%
-but have remained in manuscript in the possession of
-the Vienna Academy\dots Lebesgue's \Title{Table des Diviseurs
-\index{Vienna academy}%
-des Nombres} goes as far as~$115500$ and is very compact,
-occupying only $20$~pages.'' \Attrib{(Escott.)}
-
-\Par{Some long numbers.} The computation of the value
-\index{p@{$\pi$}}%
-\index{Shanks, William}%
-of~$\pi$ to $707$~decimal places by Shanks\footnote
- {See page~\PgNo{124}.}
-and of $e$ to
-\index{e@{$e$}}%
-$346$~places by Boorman,\footnote
- {Mathematical Magazine, 1:204.}
-\index{Boorman}%
-are famous feats of calculation.
-
-``Paradoxes of calculation sometimes appear as illustrations
-\index{Circulating decimals}%
-\index{Recurring decimals}%
-\index{Repeating!decimals}%
-of the value of a new method. In 1863,
-Mr.\ G.~Suffield, M.A., and Mr.\ J.~R. Lunn, M.A., of
-\index{Lunn, J. R.}%
-\index{Shuffield, G.}%
-Clare College and of St.~John's College, Cambridge,
-published the whole quotient of $10000\dots$ divided by~$7699$,
-throughout the whole of one of the recurring
-\DPPageSep{047}{41}
-periods, having $7698$~digits. This was done in illustration
-\index{Escott, E.~B.|FN}%
-\index{Shuffield, G.}%
-of Mr.~Suffield's method of synthetic division.''\footnote
- {De~Morgan, p.~292. ``Suffield's `new' method was discovered
- by Fourier in the early part of the century and has been
-\index{Fourier's method of division|FN}%
- rediscovered many times since. It was published, apparently
- as a new discovery, a few years ago in the \Title{Mathematical
- Gazette}.'' \Attrib{(Escott.)}}
-
-Exceptions have been found to Fermat's theorem
-\index{Fermat's theorem!on binary powers}%
-on binary powers (which he was careful to say he
-had not proved). The theorem is, that all numbers
-of the form $2^{2^{n}} + 1$ are prime. Euler showed, in
-\index{Euler}%
-1732, that if $n = 5$, the formula gives $4,294,967,297$,
-$\text{which} = 641 × 6,700,417$. ``During the last thirty years
-it has been shown that the resulting numbers are composite
-when $n = 6$, $9$, $11$, $12$, $18$, $23$, $36$, and~$38$; the
-two last numbers contain many thousands of millions
-of digits.''\footnote
- {Ball, \Title{Recreations}, p.~37.}
-To these values of~$n$ for which $2^{2^{n}} + 1$~is
-composite, must now be added the value $n = 73$.
-``Dr.\ J.~C. Morehead has proved this year [1907] that
-\index{Morehead, J. C.}%
-this number is divisible by the prime number $2^{75} · 5 + 1$.
-\index{Number!How may a particular number arise?}%
-This last number contains $24$~digits and is probably
-the largest prime number discovered up to the present.''\footnote
- {Mr.~Escott.}
-If the number $2^{2^{73}} + 1$ itself were written in
-the ordinary notation without exponents, and if it
-were desired to print the number in figures the size
-of those on this page, how many volumes like this
-would be required? They would make a library many
-millions of times as large as the Library of Congress.
-
-\Par{How may a particular number arise?} (1)~From
-purely mathematical analysis---in the investigation of
-the properties of numbers, as in the illustrations
-just given, in the investigation of the properties of
-\DPPageSep{048}{42}
-some ideally constructed magnitude, as the ratio of
-the diagonal to the side of a square, or in any investigation
-involving only mathematical elements; (2)~from
-measurement of actual magnitude, time etc.:
-(3)~by arbitrary invention, as when a text-book writer
-or a teacher makes examples; or (4)~by combinations
-of these.
-
-Those of class~(3) are generally used to develop
-skill in the manipulation of numbers from classes (1)
-and~(2).
-
-Numbers from source~(2), measurement, are the
-subject of the next section.
-\index{Number!miscellaneous notes on|)}%
-\DPPageSep{049}{43}
-
-
-\Chapter{Numbers arising from measurement.}
-\index{Accuracy of measures|EtSeq}%
-\index{Degree of accuracy of measurements|(}%
-\index{Metric system}%
-\index{Time-pieces, accuracy of}%
-\index{Measurement!numbers arising from}%
-\index{Measurement!degree of accuracy of|(}%
-\index{Numbers arising from measurement}%
-
-There is no such thing as an exact measurement of
-distance, capacity, mass, time, or any such quantity.
-It is only a question of \emph{degree} of accuracy.
-
-``The best time-pieces can be trusted to measure a
-week within one part in~$756,000$.''\footnote
- {Prof.\ William Harkness, ``Art of Weighing and Measuring,''
- \Title{Smithsonian Report} for 1888, p.~616.}
-The equations of
-standards on page~\PgNo{155} show the degree of accuracy
-attained in two instances by the International Bureau
-of Weights and Measures. In the measure of length
-\index{Weights and measures}%
-(the distance between two lines on a bar of platinum-iridium)
-the range of error is shown to be $0.2$~in a
-million, or one in five million. In the measure of mass
-it is one in five hundred million. But these are measurements
-famous for their precision, made in cases
-in which accuracy is of prime importance, and the
-comparisons effected under the most favorable conditions.
-No such accuracy is attained in most work.
-In a certain technical school, two-tenths of a per cent
-is held to be fair tolerance of error for ``exact work''
-in chemical analysis. The accuracy in measurement
-attained by ordinary artisans in their work is of a
-somewhat lower degree.
-
-Now in a number expressing measurement the number
-of significant figures indicates the degree of accuracy.
-Hence the number of significant digits is
-limited. If any one were to assert that the distance
-\DPPageSep{050}{44}
-\index{Neptune, distance from sun}%
-of Neptune from the sun is $2,788,820,653$ miles, the
-statement would be immediately rejected. A distance
-of billions of miles can not, by any means now known,
-be measured to the mile. We should be sure that the
-last four or five figures must be unknown and that
-this number is not to be taken seriously. What astronomers
-\index{Astronomers}%
-do state is that the distance is $2,788,800,000$
-miles.
-
-The metrology of the future will doubtless be able
-to extend gradually the limits of precision, and therefore
-to expand the significant parts of numbers. But
-the principle will always hold.
-
-The numbers arising from the measurements of
-daily life have but few significant figures.
-
-The following paragraph is another illustration of
-the principle.
-
-\Par{Decimals as indexes of degree of accuracy of measure.}
-\index{Decimals as indexes of degree of accuracy}%
-The child is taught that $.42 = .420 = .4200$. True;
-but the scientist who reports that a certain distance is
-$.42$~cm, and the scientist who reports it as $.420$~cm,
-wish to convey, and do convey, to their readers different
-impressions. From the first we understand that
-the distance is $.42$~cm correct to the nearest hundredth
-of a~cm; that is, it is more than $.415$~cm and
-less than $.425$~cm. From the second we learn that it
-is $.420$~cm to the nearest thousandth; that is, more
-than $.4195$ and less than~$.4205$. Compare the decimals,
-including $0.00100$, in the equation of the U.S.~standard
-meter, p.~\PgNo{155}.
-\index{Degree of accuracy of measurements|)}%
-\index{Measurement!degree of accuracy of|)}%
-
-Exact measurement is an ideal. It is the \emph{limit}
-which an ever improving metrology is approaching
-forever nearer. The question always is of \emph{degree} of
-accuracy of measure. And this question is answered
-\DPPageSep{051}{45}
-by the number of decimal places in which the result is
-expressed.
-
-\Par{Some applications.} The foregoing principle explains
-why for very large and very small numbers the
-index notation is sufficient; in which it is said, for
-example, that a certain star is $5 x 10^{13}$~miles from the
-earth. This is easier to write than $5$~followed by $13$~ciphers,
-and there is no need to enumerate and read
-such a number. Similarly $10$~with a negative exponent
-serves to write such a decimal fraction as is used to
-express the length of a wave of light or any of the
-minute measurements of microscopy.
-
-The principle explains also why a table of logarithms
-\index{Logarithms}%
-for ordinary use need not tabulate numbers beyond
-four or five places (four or five places in the
-``arguments,'' to use the technical term of table makers;
-only the logarithms of numbers to~$10,000$, or $100,000$,
-to use the common phraseology). Interpolation extends
-them to one more place with fair accuracy, and
-for ordinary computation one rarely needs the logarithm
-of a number of more than five significant digits.
-
-It explains also why a method of approximation in
-multiplication is so desirable. If any of the data are
-\index{Multiplication!approximate}%
-furnished by measurement, the result can be only
-approximate at best. Example~VII on page~\PgNo{64}, explained
-on page~\PgNo{62}, is a case in point. To compute
-that product to six decimal places would waste time.
-Worse than that; to show such a result would \emph{pretend}
-to an accuracy \emph{not attained}, by conveying the impression
-that the circumference is known to six decimal
-places when in fact it is known to but two decimal
-places.\footnote
- {Even the second decimal place is in doubt, as may be
- seen by taking for multiplicand first $74.276$, then~$74.284$.}
-\DPPageSep{052}{46}
-
-In a certain village the tax rate, found by dividing
-\index{Tax rate}%
-the total appropriation for the year by the total assessed
-valuation, was $.01981$ for the year~1906. As
-always (unless the divisor contains no prime factor
-but $2$~and~$5$) the quotient is an interminate decimal.
-To how many places should the decimal be carried?
-Theoretically it should be carried far enough to give
-a product ``correct to cents'' when used to compute
-the tax of the highest taxpayer. In this case the
-decimal is accurate enough for all assessments not
-exceeding \$$1000$. As a matter of fact, there were
-several in excess of this amount.
-
-For an understanding of the common applications
-\index{Error, theory of}%
-\index{Theory!of error}%
-of arithmetic it is important that the learner appreciate
-the elementary considerations of the theory of
-error; at least that he habitually ask himself, ``To how
-many places may my result be regarded as accurate?''
-\DPPageSep{053}{47}
-
-
-\Chapter{Compound interest.}
-\index{Compound interest}%
-\index{Interest, compound and simple}%
-
-The enormous results obtained by computing compound
-interest---as well as the wide divergence between
-these or any results obtained from a geometric
-progression of many terms and the results found in
-actual life---may be seen from the following ``examples'':
-
-At $3$\% (the prevailing rate at present in savings
-banks) \$$1$~put at interest at the beginning of the
-Christian era to be compounded annually would now
-amount to \$$(1.03)^{1906}$, which by the use of logarithms
-\index{Logarithms}%
-is found to be, in \emph{round} numbers, \$$3,000,000,000,000,000,000,000,000$.
-The amount of~\$$1$ for the same
-time and rate but at \emph{simple} interest would be only~\$$58.18$.
-
-\Par{If the Indians hadn't spent the~\$$24$.} In 1626 Peter
-\index{If the Indians hadn't spent the \$$24$}%
-\index{Indians spent the \$$24$}%
-Minuit, first governor of New Netherland, purchased
-Manhattan Island from the Indians for about~\$$24$.
-\index{Manhattan, value of reality in 1626 and now|(}%
-\index{New York, value of realty in 1626 and now|(}%
-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
-\DPPageSep{054}{48}
-interest to the principal yearly. What would be the
-amount now, after $280$~years? $24 × 1.07^{280} = \text{more
-than } 4,042,000,000$.
-
-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. The Indians could have
-bought back most of the property now, with improvements;
-from which one might point the moral of
-saving money and putting it at interest! The rise
-in the value of the real estate of Manhattan, phenomenal
-as it is, has but little more than kept pace with
-the growth of money at~$7$\% compound interest. But
-New York realty values are now growing more rapidly:
-the Indians would better purchase soon!
-\index{Manhattan, value of reality in 1626 and now|)}%
-\index{New York, value of realty in 1626 and now|)}%
-\DPPageSep{055}{49}
-
-
-\Chapter{Decimal separatrixes.}
-\index{Decimal separatrixes}%
-\index{Separatrixes, decimal}%
-
-The term \emph{separatrix} in the sense of a mark between
-the integral and fractional parts of a number written
-decimally, was used by Oughtred in~1631. He used
-\index{Oughtred}%
-the mark~$\llcorner$ for the purpose. Stevin had used a
-figure in a circle over or under each decimal place to
-indicate the order of that decimal place. Of the various
-other separatrixes that have been used, four are
-in common use to-day, if (2)~and~(3) below may be
-counted separately:
-
-1. \emph{A vertical line}: \eg, that separating cents from
-\index{Witt, Richard}%
-dollars in ledgers, bills\Add{,}~etc. As a temporary separatrix
-the line appears in a work by Richard Witt,
-1613. Napier used the line in his \Title{Rabdologia}, 1617.
-This is a very common separatrix in every civilized
-country to-day.
-
-2. The \emph{period}. Fink, citing Cantor, says that the
-decimal point is found in the trigonometric tables of
-Pitiscus (in Germany) 1612. Napier, in the \Title{Rabdologia},
-\index{Pitiscus}%
-speaks of using the period or comma. His
-usage, however, is mostly of a notation now obsolete
-(but he uses the comma at least once). The period
-has always been the prevailing form of the decimal
-point in America.
-
-3. The \emph{Greek colon} (dot above the line). Newton
-\index{Newton}%
-advocated placing the point in this position ``to prevent
-it from being confounded with the period used
-\DPPageSep{056}{50}
-as a mark of punctuation'' (Brooks). It is commonly
-so written in England now.
-
-4. The \emph{comma}. The first known instance of its use
-as decimal separatrix is said to be in the Italian trigonometry
-\index{Italian!decimal separatrix}%
-of Pitiscus, 1608. Perhaps next by Kepler,
-\index{Kepler}%
-\index{Pitiscus}%
-1616, from which may be dated the German use of it.
-\index{German!decimal separatrix}%
-Briggs used it in his table of logarithms in~1624, and
-\index{Briggs}%
-early English writers generally employed the comma.
-\index{English!decimal separatrix}%
-English usage changed to the Greek colon; but the
-comma is the customary form of the decimal point on
-the continent of Europe.
-
-The usage as to decimal point is not absolutely
-\index{French!decimal separatrix}%
-uniform in any of the countries named; but, in general,
-one expects to see $1 \nicefrac{25}{100}$ written decimally in
-the form of~$1.25$ in America, $1 · 25$ in England, and
-$1,25$~in Germany, France or Italy.
-
-A mere space to indicate the separation may also
-be mentioned as common in print.
-
-The vertical line (for a column of decimals) and the
-\index{Miller, G. A.}%
-space should doubtless persist, and \emph{one} form of the
-``point.'' Prof.\ G.~A. Miller, of the University of
-Illinois, who argues for the comma as being the symbol
-used by much the largest number of mathematicians,
-remarks:\footnote
- {``On Some of the Symbols of Elementary Mathematics,''
- \Title{School Science and Mathematics}, May, 1907.
-
- Where the decimal point is a comma the separation of long
- numbers into periods of three (or six) figures for convenience
- of reading is effected by spacing.}
-``As mathematics is pre-eminently cosmopolitan
-and eternal it is very important that its symbols
-should be world symbols. All national distinctions
-along this line should be obliterated as rapidly as
-possible.''
-\DPPageSep{057}{51}
-
-
-\Chapter{Present trends in arithmetic.}
-\index{Arithmetic!present trends in}%
-\index{Present trends in arithmetic}%
-\index{Trends in arithmetic}%
-
-``History is past politics, and politics is present history.''
-Such is the apothegm of the famous historian
-Freeman. In the case of a science and an art, like
-\index{Freeman, E.~A.}%
-arithmetic or the teaching of arithmetic, history is
-past method, and method is present history. The fact
-that our generation is helping to make the history of
-arithmetic and of the teaching of arithmetic---as it is
-also making history in other matters that attract more
-public attention---is the reason for considering now
-some of the present trends in arithmetic. A present
-trend is a pointer pointing from what has been to
-what is to be, since the science is a continuum. Lord
-Bolingbroke said that we study history to know how
-\index{Bolingbroke, Lord}%
-to act in the future, to make the most of the future.
-That is why we study history in the making, or present
-trends, in so far as it is possible for us, living in
-the midst, to see those trends.
-%<tb>
-
-Very noticeable among them is the gradual decimalization
-\index{Decimalization of arithmetic|EtSeq}%
-of arithmetic. Counting by~$10$ is prehistoric
-in nearly all parts of the world, $10$~fingers being the
-evident explanation. If we had been present at the
-beginning of arithmetical history, we might have given
-the primitive race valuable advice as to the choice of
-a radix of notation! It would then have been opportune
-to call attention to the advantage of~$12$ over~$10$
-arising from the greater factorability of~$12$. Or if
-\DPPageSep{058}{52}
-\index{Gath giant}%
-\index{Giant with twelve fingers}%
-the pioneers of arithmetic had been like the Gath
-giant of 2~Sam.\ 21:20, with six fingers on each hand,
-they would doubtless have used $12$ as a radix. Lacking
-such counsel, and being equipped by nature with
-only $10$~fingers to use as counters, they started arithmetic
-on a decimal basis. History since has been a
-steady progress in the direction thus chosen (except
-in details like the table of time, where the incommensurable
-ratio between the units fixed by nature defied
-even the French Revolution).
-
-The Arabic notation\DPtypo{,}{} ``was brought to perfection in
-\index{Arabic!notation}%
-\index{Hindu!numerals (Arabic)}%
-the fifth or sixth century,''\footnote
- {Cajori, \Title{Hist.\ of Elem.\ Math.}, p.~154.}
-but did not become common
-in Europe till the sixteenth century. It is not
-quite universal yet, the Roman numerals being still
-used on the dials of \DPchg{timepieces}{time-pieces}, in the titles of sovereigns,
-the numbers of book chapters and subdivisions,
-and, in general, where an archaic effect is
-sought. But the Arabic numerals are so much more
-convenient that they are superseding the Roman in
-these places. The change has been noticeable even
-in the last ten or fifteen years.
-
-The extension of the Arabic system to include fractions
-was made in the latter part of the sixteenth
-century. But notwithstanding the superior convenience
-of decimal fractions, they spread but slowly;
-and it is only in comparatively recent times that they
-may be said to be more common than ``common fractions.''
-
-The next step was logarithms---a step taken in~1614.
-\index{Logarithms}%
-Within the next ten years they were \DPtypo{accomodated}{accommodated} to
-what \emph{we} should call ``the base''~$10$.
-
-The dawn of the nineteenth century found decimal
-\index{Coinage, decimal}%
-coinage well started in the United States, and a general
-\DPPageSep{059}{53}
-movement toward decimalization under way in
-France contemporaneous with the political revolution.
-The subsequent spread of the metric system over most
-\index{Metric system}%
-of the continent of Europe and over many other parts
-of the world has been the means of teaching decimal
-fractions.
-
-The movement is still on. The value and importance
-of decimals are now recognized more every year.
-And much remains to be decimalized. In stock quotations,
-fractions are not yet expressed decimally.
-Three great nations have still to adopt decimal weights
-and measures in popular use, and England has still to
-adopt decimal coinage. The history of arithmetic has
-been, in large part, a slow but well-marked growth of
-the decimal idea.
-
-Those who are working for world-wide decimal
-coinage, weights and measures---as a time-saver in
-\index{Weights and measures}%
-school-room, counting house and work-shop---as a
-boon that we owe to posterity as well as to ourselves---may
-learn from such a historical survey both caution
-and courage. Caution not to expect a sudden
-change. Multitudes move slowly in matters requiring
-a mental readjustment. The present reform movements---for
-decimal weights and measures in the
-United States, and decimal weights, measures and coins
-in Great Britain---are making more rapid progress than
-the Arabic numerals or decimal fractions made: and
-the opponents of the present reform are not so numerous
-or so prejudiced as were their prototypes who
-opposed the Arabic notation in the Middle Ages and
-later. Caution also against impatience with a conservatism
-whose arguments are drawn from the temporary
-inconvenience of making any change. Courage
-to work and wait---in line with progress.
-\DPPageSep{060}{54}
-
-In using fractions, the Egyptians and Greeks kept
-\index{Arithmetic!teaching|(}%
-\index{Babylonia}%
-\index{Egypt}%
-\index{Methods in arithmetic|(}%
-\index{Fractions}%
-\index{Greeks}%
-the numerators constant and operated with the denominators.
-The Romans and Babylonians preferred a
-constant denominator, and performed operations on
-the numerator. The Romans reduced their fractions
-to the common denominator~$12$, the Babylonians to~$60$ths.
-We also reduce our fractions to a common
-denominator; but we choose~$100$. One of the most
-characteristic trends of modern arithmetic is the rapid
-growth in the use of percentage---another development
-of the decimal idea. The broker and the biologist, the
-statistician and the salesman, the manufacturer and the
-mathematician alike express results in per~cents.
-%<tb>
-
-These and other changes in the methods of computers
-have brought about, though tardily, corresponding
-changes in the subject matter of arithmetic as
-taught in the schools. Scholastic puzzles are giving
-place to problems drawn from the life of to-day.
-
-Perhaps one may venture the opinion that, in order
-to merit a place in the arithmetic curriculum, a topic
-must be useful either (1)~in commerce or (2)~in industry
-or (3)~in science. Under~(3) may be included,
-conceivably, a topic whose sole, or chief, use is in later
-mathematical work. At least two other reasons have
-been given for retaining a subject: (4)~It is required
-for examination. But it will be found that subjects
-not clearly justified on one of the grounds above mentioned
-are rarely required by examining bodies of this
-generation; and such subjects, if pointed out, would
-doubtless be withdrawn from any syllabus. (5)~It
-gives superior mental training. But on closer scrutiny
-this argument becomes somewhat evanescent. A survey
-of results in that branch of educational psychology
-\index{Psychology}%
-\DPPageSep{061}{55}
-which treats of the coefficient of correlation between
-a pupil's attainments in various activities, weakens
-one's faith in our ability to give a certain amount of
-general discipline by a certain amount of special training.
-Moreover, that discipline can be as well acquired
-by the study of subjects that serve a direct, useful
-purpose. We may, then, limit our criteria to these:
-utility for business or industrial pursuits, and utility
-for work in science.
-
-Applying these tests to the topics contained in the
-schoolbooks of a generation ago, we see that many
-of them are not worthy of a place in the crowded curriculum
-of our generation. Turning to the schools,
-we find that many of these topics have, in fact, been
-dropped. Others are receiving less attention each
-year. Among such may be mentioned: ``true'' discount,
-partnership involving time, and equation of
-payments (all three giving, besides, a false idea of
-business), and Troy and apothecaries weight, cube
-root (except for certain purposes with advanced pupils)
-and compound proportion.
-
-At the same time, other topics in the arithmetic
-course are of increasing importance; notably those
-involving percentage and other decimal operations,
-and those relating to stock companies and other developments
-of modern economic activity.
-%<tb>
-
-School life is adjusting itself to present social conditions,
-not only in the topics taught, but in the problems
-used and the way in which the topics are treated.
-Good books no longer set problems in stocks involving
-the purchase of a fractional number of shares!
-
-As Agesilaus, king of Sparta, said, ``Let boys study
-\index{Agesilaus}%
-\index{Sparta}%
-what will be useful to men.''
-\DPPageSep{062}{56}
-
-The Greeks studied \textgreek{>arijmhtik'h}, or theory of numbers,
-\index{Greeks}%
-and \textgreek{logistik'h}, or practical calculation. Hence the modern
-definition of arithmetic, ``the science of numbers
-and the art of computation.'' As Prof.\ David Eugene
-Smith points out (in his \Title{Teaching of Elementary
-\index{Smith, D.~E.}%
-Mathematics}) ``the modern arithmetic of the schools
-includes much besides this.'' It includes the introduction
-of the pupil to the commercial, industrial and
-% [** TN: [sic] "quantitive"]
-scientific life of to-day on the quantitive side.
-%<tb>
-
-Characteristic of our time is the extensive use of
-arithmetical machines (such as adding machines and
-instruments from which per cents may be read) and
-of tables (of square roots for certain scientific work,
-interest tables for banks,~etc.). The initial invention
-of such appliances is not recent; it is their variety,
-adaptability and rapidly extending usefulness that may
-be classed as a present phenomenon.
-
-They have not, however, eliminated the necessity
-for training good reckoners. They may have narrowed
-the field somewhat, but in that remaining part
-which is both practical and necessary they have set
-the standard of attainment higher. Indeed, an important
-feature of the present situation is the insistent
-demand of business men that the schools turn out better
-computers. There must soon come, in school, a
-stronger emphasis on accuracy and rapidity in the
-four fundamental operations.
-
-Emphasis on accuracy and rapidity in calculation
-leads to the use of ``examples'' involving abstract
-numbers. Emphasis on the business applications alone
-leads to the almost exclusive use of ``problems'' in
-which the computative is but an incidental feature.
-Both are necessary. It has been well said that examples
-\DPPageSep{063}{57}
-are to the arithmetic pupil what exercises are to
-the learner on the piano, while problems are to the
-former what tunes are to the latter. Without exercises,
-no skill; with exercises alone, no accomplishment.
-The exercises are for the technique of the art.
-The teacher can not afford to neglect either.
-
-The last century or more has been the age of special
-methods in teaching. One has succeeded another in
-popular favor. Each has taught us an important
-lesson---something that will be a permanent acquisition
-to the pedagogy of the science. Few things are more
-interesting to the student of the history of arithmetic
-methods than to trace each school-room practice of
-to-day to its origin in some worthy contributor to the
-science (\eg, in the primary grades, the use of blocks
-to Trapp, 1780; the ``number pictures'' to Von~Busse;
-\index{Trapp}%
-\index{Von Busse}%
-counting by $2$'s, $3$'s\Add{,}~\dots\ as preparation for the multiplication
-tables to Knilling and Tanck;~etc.). More
-\index{Knilling}%
-\index{Tanck}%
-recently several famous methods have appeared which
-are still advocated. But the present trend is toward
-a choosing of the best from each---an eclectic method.
-%<tb>
-
-Most questions of method have never been adequately
-tested. It is, for instance, asserted by some
-and denied by others that pupils would know as much
-arithmetic at the end of the $8$th~school year if they
-were to begin arithmetic with the~$5$th or even later.
-History may well lead us to doubt the proposition;
-but who can tell? The greatest desideratum in all
-arithmetic teaching to-day is the thorough study of
-the subject by the scientific methods employed in
-educational psychology. Some one with facilities for
-\index{Psychology}%
-doing this service for arithmetic could be a benefactor
-\DPPageSep{064}{58}
-indeed. Questions that are matters for accurate test
-and measurement should not always remain questions.
-Meantime, empiricism is unavoidable.
-%<tb>
-
-To summarize the tendencies noted: the decimalization
-of arithmetic, growth of percentage, elimination
-of many topics from the school curriculum in
-arithmetic with increased emphasis on others, modernizing
-the treatment of remaining topics, demand
-for more accuracy and rapidity in computation, inclination
-toward an eclectic method in teaching arithmetic,
-present empiricism pending scientific investigation.
-This list is, of course, far from exhaustive,
-but it is believed to be true and significant.
-
-Lacking such exact information as that just asked
-for as the desideratum of to-day, we may make the best
-of mere observation of the trends of our time. And
-as to the great movements in the history of the art of
-arithmetic itself, the conclusions are definite and decisive.
-By orienting ourselves, by studying the past
-and noting the currents, we may acquaint ourselves
-with the direction of present forces and may take part
-in shaping our course. Our to-days are conditioned
-by our yesterdays, to-morrow by to-day.
-\index{Arithmetic!teaching|)}%
-\index{Methods in arithmetic|)}%
-\DPPageSep{065}{59}
-
-
-\Chapter{Multiplication and division of decimals.}
-\index{Multiplication!of decimals}%
-
-For the multiplication of whole numbers the Italians
-invented many methods.\footnote
- {For the historical facts in this section the author is indebted
- mainly to Professor Cajori and Prof.\ David Eugene
- Smith, the two leading American authorities on the history of
- mathematics.}
-\index{Cajori, Florian}%
-\index{Smith, D.~E.}%
-Pacioli (1494) gives eight.
-\index{Pacioli}%
-Of these, only one was in common use, and it alone
-has survived in commerce and the schools. Shown
-in~I on p.~\PgNo{64}. It was called \Foreign{bericuocolo} (honey cake or
-ginger bread) by the Florentines, and \Foreign{scacchiera} (chess
-or checker board) by the Venetians. The little squares
-in the partial products fell into disuse (and with them
-the names which they made appropriate) leaving the
-familiar form~II on p.~\PgNo{64}. The Treviso arithmetic
-(1478), the first arithmetic printed, contains a long
-example in multiplication, which appears about as it
-would appear on the blackboard of an American school
-to-day.
-
-In 1585 appeared Simon Stevin's immortal \Title{La
-Disme}, only seven pages, but the first publication to
-\index{Stevin, Simon|EtSeq}%
-expound decimal fractions, though the same author
-had used them in an interest table published the year
-before. III~on p.~\PgNo{64} is from \Title{La Disme}, and shows
-Stevin's notation (the numbers in circles, or parentheses,
-indicating the order of decimals, tenths the
-first order~etc.) IV~is the same example with the
-decimals expressed by the notation now prevalent in
-\DPPageSep{066}{60}
-America. Let us call this arrangement of work Stevin's
-method.
-
-An arrangement in which all decimal points are in
-\index{Romain, Adrian|EtSeq}%
-a vertical column (see V below) is said to have been
-used by Adrian Romain a quarter of a century later.
-He may not have been the inventor of this arrangement;
-but, for the sake of a name, call it Romain's
-method.
-
-Romain's method is advocated in a few of the best
-recent advanced arithmetics, but Stevin's is still vastly
-the more common; and these two are the only methods
-in use. Romain's has four slight advantages: (1)~A
-person setting down an example from dictation can
-begin to write the multiplier as soon as the place of its
-decimal point is seen, while in Stevin's method he
-waits to hear the entire multiplier before he writes
-any of it, in order to have its last (right-hand) figure
-stand beneath the last figure of the multiplicand
-(though this \DPtypo{positon}{position} may be regarded as a non-essential
-feature in Stevin's arrangement). (2)~Romain's
-method fits more naturally with the ``Austrian''
-method of division (decimal point of quotient over
-that of dividend). (3)~After the partial products
-are added, it is not necessary to count and point off
-in the product as many decimal places as there are in
-the multiplicand and multiplier together, since the
-decimal point in the product (as well as in the partial
-products) is directly beneath that in the multiplicand.
-(4)~Romain's method is more readily adapted to
-abridged multiplication where only approximate results
-are required. On the other hand, Stevin's method
-has one very decided advantage: the first figure written
-in each partial product is directly beneath its
-figure in the multiplier, so that it is not necessary
-\DPPageSep{067}{61}
-(as it is in Romain's) to determine the place of the
-decimal point in a partial product. So important is
-this, that Stevin's alone has been generally taught to
-children, notwithstanding the numerous points in favor
-of Romain's.
-
-It occurred to the writer recently to try to combine
-in one method the advantages of both of the Flemish
-methods, and he hit upon the following simple rule:\footnote
- {Since writing this the author has come upon the same
- method of multiplication in Lagrange's \Title{Lectures}, delivered in
- 1795 (p.~29--30 of the Open Court Publishing Co.'s edition).
- One who invents anything in elementary mathematics is likely
- to find that ``the ancients have stolen his ideas.''}
-Write the units figure of the multiplier under the last
-(right-hand) figure of the multiplicand, begin each
-partial product (as in the familiar method of Stevin)
-under the figure by which you are multiplying, and all
-decimal points in products will then be directly beneath
-that in the multiplicand. Decimal points in
-partial products may be written or not, as desired.
-The reason underlying the rule is apparent. VI~shows
-the arrangement of work.
-
-In this arrangement the placing of the partial products
-is automatic, as in Stevin's method, and the
-pointing off in the product is automatic, as in Romain's.
-It is available for use by the child in his first
-multiplication of decimals and by the skilled computer
-in his abridged work.
-
-To assist in keeping like decimal orders in the same
-column it is recommended that the vertical line shown
-in VII and VIII be drawn before the partial products
-are written. One of the earliest uses of the line
-as decimal separatrix is in an example in Napier's
-\Title{Rabdologia} (1617). He draws it through the partial
-and complete products. It is said to be the first
-\DPPageSep{068}{62}
-example of abridged multiplication. A circumference
-is computed whose diameter is~$635$.
-
-VII~illustrates the application of the method here
-\index{Multiplication!approximate}%
-advocated to multiplication in which only an approximation
-is sought. The diameter of a circle is found
-by measurement to be $74.28$~cm. This is correct to
-$0.01$~cm. No computation can give the circumference
-to any higher degree of accuracy. Partial products
-are kept to three places in order to determine the correct
-figure for the second place in the complete product.
-The arrangement of work shows what figures
-to omit.
-
-It should be remarked that all three methods of
-multiplication of decimals are alike---and like the multiplication
-of whole numbers---in that one may multiply
-first by the digit of lowest order in the multiplier
-or by the digit of highest order first. The method of
-multiplying by the highest order first was described
-by the Italian arithmeticians as \Foreign{a~dietro}. Though it
-may seem to be working \emph{backwards}, it is not so in
-fact; for it puts the more important before the less,
-and has practical advantage in abridged multiplication,
-like that shown in~VII\@. But that question is
-distinct from the one under consideration.
-
-Stevin writes the last figure of the multiplier under
-the last figure of the multiplicand; Romain writes
-units under units; the method here proposed writes
-units under last. In \emph{whole} numbers, units figure \emph{is}
-the last.
-
-Applied to the ordinary multiplication of decimals,
-as in VI or~VIII, the method here proposed seems
-to be well adapted to schoolroom use, possessing all
-the simplicity of Stevin's. Methods classes in this
-normal school to whom the method was presented,
-\DPPageSep{069}{63}
-immediately preferred it, and a grade in the training
-school used it readily. Of course this proves nothing,
-for every method is a success in the hands of its advocates.
-The changes here set forth are, however,
-not advocated; they are merely proposed as a possibility.
-
-The analogous method for division of decimals
-\index{Division!of decimals}%
-possesses analogous advantages. It avoids the necessity
-of multiplying the divisor and dividend by such
-a power of~$10$ as will make the divisor integral (as in
-the method now perhaps most in favor) and the necessity
-of counting to point off in the quotient a number
-of decimal places equal to the number in the
-dividend minus that in the divisor (as in the older
-method still common). Like the latter, it begins the
-division at once; and like the former, its pointing off
-is automatic. IX~shows the arrangement. The figure
-under the last figure of the divisor is \emph{units} figure of the
-quotient. This determines the place of the decimal
-point. That part of the quotient which projects beyond
-the divisor, is fractional.
-
-If the order of multiplication used has been \Foreign{a~dietro},
-as in~VIII, the division in~IX is readily seen to be
-the inverse operation. The partial products appear in
-the same order as partial dividends.
-
-Like each of the methods in use, it may be abbreviated
-by writing only the remainders below the dividend.
-Shown in~X\@.
-
-If the ``little castle'' method of multiplication of
-whole numbers, with multiplier above multiplicand,
-had prevailed, instead of the ``chess board,'' in the
-fifteenth century, the arrangement now proposed for
-the multiplication and division of decimals would have
-afforded slightly greater advantage.
-\DPPageSep{070}{64}
-
-\vfil
-\begin{Arithm}
-I
-\[
-\begin{array}{*{8}{c}}
- & & & & 3 & 2 & 5 & 7 \\
- & & & & 8 & 9 & 4 & 6 \\
-\cline{4-8}
- & & & \VC{1} & \VC{9} & \VC{5} & \VC{4} & \VCV{2} \\
-\cline{3-8}
- & & \VC{1} & \VC{3} & \VC{0} & \VC{2} & \VC{8} & \VC{} \\
-\cline{2-7}
- & \VC{2} & \VC{9} & \VC{3} & \VC{1} & \VC{3}& \VC{} & \\
-\cline{1-6}
-\VC{2} & \VC{6} & \VC{0} & \VC{5} & \VC{6}& \VC{} & & \\
-\hline
-2 & 9 & 1 & 3 & 7 & 1 & 2 & 2
-\end{array}
-\]
-\end{Arithm}
-\begin{Arithm}
-II
-\[
-\begin{array}{*{8}{c}}
- & & & & 3 & 2 & 5 & 7 \\
- & & & & 8 & 9 & 4 & 6 \\
-\cline{4-8}
- & & & 1 & 9 & 5 & 4 & 2 \\
- & & 1 & 3 & 0 & 2 & 8 \\
- & 2 & 9 & 3 & 1 & 3 \\
-2 & 6 & 0 & 5 & 6 \\
-\hline
-2 & 9 & 1 & 3 & 7 & 1 & 2 & 2
-\end{array}
-\]
-\end{Arithm}
-\index{Multiplication!approximate}%
-\vfil
-
-\begin{Arithm}
-III
-\[
-\begin{array}{*{8}{c}}
- & & & & & \Dgt{(0)} & \Dgt{(1)} & \Dgt{(2)} \\
- & & & & 3 & 2 & 5 & 7 \\
- & & & & 8 & 9 & 4 & 6 \\
-\cline{4-8}
- & & & 1 & 9 & 5 & 4 & 2 \\
- & & 1 & 3 & 0 & 2 & 8 \\
- & 2 & 9 & 3 & 1 & 3 \\
-2 & 6 & 0 & 5 & 6 \\
-\hline
-2 & 9 & 1 & 3 & 7 & 1 & 2 & 2 \\
- & & & \Dgt{(0)} & \Dgt{(1)} & \Dgt{(2)} & \Dgt{(3)} & \Dgt{(4)}
-\end{array}
-\]
-\end{Arithm}
-\begin{Arithm}
-IV
-\[
-\begin{array}{*{8}{c}}
-\\ %[** TN: Vertical alignment hack]
- & & & & 3 & 2\Dec & 5 & 7 \\
- & & & & 8 & 9\Dec & 4 & 6 \\
-\cline{4-8}
- & & & 1 & 9 & 5 & 4 & 2 \\
- & & 1 & 3 & 0 & 2 & 8 \\
- & 2 & 9 & 3 & 1 & 3 \\
-2 & 6 & 0 & 5 & 6 \\
-\hline
-2 & 9 & 1 & 3\Dec & 7 & 1 & 2 & 2
-\end{array}
-\]
-\end{Arithm}
-\vfil
-
-\begin{Arithm}
-V
-\[
-\begin{array}{*{8}{c}}
- & & 3 & 2\Dec & 5 & 7 \\
- & & 8 & 9\Dec & 4 & 6 \\
-\cline{3-8}
- & & & 1\Dec & 9 & 5 & 4 & 2 \\
- & & 1 & 3\Dec & 0 & 2 & 8 \\
- & 2 & 9 & 3\Dec & 1 & 3 \\
-2 & 6 & 0 & 5\Dec & 6\\
-\hline
-2 & 9 & 1 & 3\Dec & 7 & 1 & 2 & 2
-\end{array}
-\]
-\end{Arithm}
-\begin{Arithm}
-VI
-\[
-\begin{array}{*{8}{c}}
- & & 3 & 2\Dec & 5 & 7 \\
- & & & & 8 & 9\Dec & 4 & 6 \\
-\cline{4-8}
- & & & 1\Dec & 9 & 5 & 4 & 2 \\
- & & 1 & 3\Dec & 0 & 2 & 8 \\
- & 2 & 9 & 3\Dec & 1 & 3 \\
-2 & 6 & 0 & 5\Dec & 6 \\
-\hline
-2 & 9 & 1 & 3\Dec & 7 & 1 & 2 & 2
-\end{array}
-\]
-\end{Arithm}
-\vfil
-
-\begin{Arithm}
-VII
-\[
-\begin{array}{*{9}{c}}
- & 7 & 4\Dec & 2 & 8 & & & \\
- & & & & 3\Dec & 1 & 4 & 1 & 6 \\
-\hline
-2 & 2 & 2 & \VC{8} & 4 \\
- & & 7 & \VC{4} & 2 & 8 \\
- & & 2 & \VC{9} & 7 & 1 \\
- & & & \VC{} & 7 & 4 \\
- & & & \VC{} & 4 & 5 \\
-\cline{1-5}
-2 & 3 & 3 & 3 & 6
-\end{array}
-\]
-\end{Arithm}
-\begin{Arithm}
-VIII
-\[
-\begin{array}{*{8}{c}}
- & & 3 & 2\Dec & 5 & 7 \\
- & & & & 8 & 9\Dec & 4 & 6 \\
-\hline
-2 & 6 & 0 & 5 & \VC{6} & & & \\
- & 2 & 9 & 3 & \VC{1} & 3 & \\
- & & 1 & 3 & \VC{0} & 2 & 8\\
- & & & 1 & \VC{9} & 5 & 4 & 2\\
-\hline
-2 & 9 & 1 & 3 & \VC{7} & 1 & 2 & 2
-\end{array}
-\]
-\end{Arithm}
-\vfil
-\DPPageSep{071}{65}
-\index{Division!of decimals}%
-
-\begin{Arithm}
-IX
-\[
-\begin{array}{*{8}{c}}
- & & 3 & 2\Dec & 5 & 7 \\
- & & & & 8 & 9\Dec & 4 & 6 \\
-\hline
-2 & 9 & 1 & 3\Dec & 7 & 1 & 2 & 2 \\
-2 & 6 & 0 & 5 & 6 \\
-\cline{1-6}
- & 3 & 0 & 8 & 1 & 1 \\
- & 2 & 9 & 3 & 1 & 3 \\
-\cline{2-7}
- & & 1 & 4 & 9 & 8 & 2 \\
- & & 1 & 3 & 0 & 2 & 8 \\
-\cline{3-8}
- & & & 1 & 9 & 5 & 4 & 2 \\
- & & & 1 & 9 & 5 & 4 & 2 \\
-\cline{4-8}
-\end{array}
-\]
-\end{Arithm}
-\begin{Arithm}
-X
-\[
-\begin{array}{*{8}{c}}
- & & 3 & 2\Dec & 5 & 7 \\
- & & & & 8 & 9\Dec & 4 & 6 \\
-\hline
-2 & 9 & 1 & 3\Dec & 7 & 1 & 2 & 2 \\
- & 3 & 0 & 8 & 1 & 1 \\
- & & 1 & 4 & 9 & 8 & 2 \\
- & & & 1 & 9 & 5 & 4 & 2
-\end{array}
-\]
-\end{Arithm}
-\DPPageSep{072}{66}
-
-
-\Chapter{Arithmetic in the Renaissance.}
-\index{Arabic!notation|(}%
-\index{Hindu!numerals (Arabic)|(}%
-\index{Arithmetics of the Renaissance|(}%
-\index{Renaissance, arithmetic in}%
-\index{Arithmetic!in the Renaissance}%
-
-The invention of printing was important for arithmetic,
-not only because it made books more accessible,
-but also because it spread the use of the Hindu (``Arabic'')
-numerals with their decimal notation.
-
-The oldest text-book on arithmetic to use these numerals
-is said to be that of Avicenna, an Arabian
-\index{Avicenna}%
-physician of Bokhara, about 1000~\AD\ (Brooks).
-According to Cardan (sixteenth century) it was Leonardo
-\index{Cardan}%
-\index{Leonardo of Pisa}%
-of Pisa who introduced the numerals into Europe
-(by his \Title{Liber Abaci}, 1202). In England, though
-there is one instance of their use in a manuscript of~1282,
-and another in~1325, their use is somewhat exceptional
-even in the fifteenth century. Then came
-printed books and a more general acceptance of the
-decimal notation.
-
-The importance of this step can hardly be over-estimated.
-Even the Greeks, with all their mathematical
-\index{Greeks}%
-acumen, had contented themselves with mystic and
-philosophic properties of numbers and had made comparatively
-little progress in the art of computation.
-They lacked a suitable notation. When such a notation
-was adopted at the close of the Middle Ages, the
-art advanced rapidly. That advance was one feature
-of the Renaissance, a detail in the great intellectual
-awakening of that marvelous half century from 1450
-to~1500, ``the age of progress''.
-
-The choice between the old and the new in arithmetical
-\DPPageSep{073}{67}
-notation is well pictured by the illustration\footnote
- {See frontispiece.}
-of
-arithmetic in the first printed cyclopedia, the \Title{Margarita
-Philosophica}~(1503). Two accountants are at
-their tables. The old man is using the abacus; the
-young man, the Hindu numerals so familiar to us.
-The aged reckoner looks askance at his youthful rival,
-in whose face is hope and confidence; while on a dais
-behind both stands the goddess to decide which shall
-have the ascendency. Her eyes are fixed on the
-younger candidate, at her right, and there can be no
-doubt that to the new numerals is to be the victory.
-The background of the picture is characteristically
-medieval. It is an apt illustration of the passing of
-the old arithmetic. To us of four centuries after, it
-whispers (as one has said of the towers of old Oxford)
-``the last enchantment of the Middle Age.''
-
-The anonymous book known as the Treviso arithmetic,
-from its place of publication, is the first arithmetic
-ever printed. It appeared in~1478. In this
-Italian work of long ago the multiplication looks modern.
-\index{Multiplication!in first printed arithmetic}%
-But long division was by the galley (or ``scratch'')
-\index{Division!in first printed arithmetic}%
-method then prevalent.
-
-Pacioli's \Title{Summa di Arithmetica} was published in
-1494 (some say ten years earlier). It also uses the
-Hindu numerals.\footnote
- {In Pacioli's work, the words ``zero'' (\Foreign{cero}) and ``million''
-\index{Zero!first use of word in print}%
-\index{Million, first use of term in print}%
- (\Foreign{millione}) are found for the first time in print. Cantor, II,
- 284.}
-
-Tonstall's arithmetic (1522) was ``the first important
-arithmetical work of English authorship.''\footnote
- {Cajori, \Title{Hist.\ of Elem.\ Math.}, p.~180.}
-De~Morgan
-calls the book ``decidedly the most classical
-which ever was written on the subject in Latin, both
-in purity of style and goodness of matter.''
-\DPPageSep{074}{68}
-
-Recorde's celebrated \Title{Grounde of Artes} (1540) was
-written in English. It uses the Hindu numerals, but
-\index{Hindu!numerals (Arabic)|)}%
-teaches reckoning by counters. The exposition is in
-dialogue form.
-
-The first English work on double entry book-keeping,
-\index{Book-keeping, first English book on}%
-\index{Mellis, John}%
-by John Mellis (London, 1588), has an appendix
-on arithmetic.
-
-\Title{The Pathway to Knowledge}, anonymous, translated
-from Dutch into English by W.~P., was published in
-London in~1596. It contains two lines which are immortal.
-The translator has been said to be the author
-of the lines. In modernized form they are known to
-every schoolboy. Of all the arithmetical doggerel of
-that age, this is pre-eminently the classic:
-\begin{verse}
-``Thirtie daies hath September, Aprill, June, and November, \\
-\index{Thirtie daies hath September}%
-\PadTxt{``}{}Februarie, eight and twentie alone; all the rest thirtie
-and one.''
-\end{verse}
-
-On the subject of early arithmetics De~Morgan's
-\Title{Arithmetical Books} is the standard work. An interesting
-contribution to the subject is Prof.\ David Eugene
-Smith's illustrated article, ``The Old and the New
-Arithmetic,'' published by Ginn \&~Co.\ in their Textbook
-Bulletin, February,~1905.
-\index{Arabic!notation|)}%
-\index{Arithmetics of the Renaissance|)}%
-\DPPageSep{075}{69}
-
-
-\Chapter[Napier's rods.]{Napier's rods and other mechanical
-aids to calculation.}
-\index{Calculation, mechanical aids}%
-\index{Rods, Napier's}%
-\index{Napier's rods}%
-
-No mathematical invention to facilitate computation
-has been made for three centuries that is comparable
-to logarithms. Napier's rods, or ``Napier's bones,''
-\index{Logarithms}%
-once famous, owe their interest now largely to the
-fact that they are the invention of the man who gave
-logarithms to the world, John Napier, baron of Merchiston.
-The inventor's description of the rods is contained
-in his \Title{Rabdologia}, published in~1617, the year
-of his death.
-
-The rods consist of $10$~strips of wood or other material,
-with square ends. A rod has on each of its
-four lateral faces the multiples of one of the digits.
-One of the rods has, on the four faces respectively,
-the multiples of $0$,~$1$, $9$,~$8$; another, of $0$,~$2$, $9$,~$7$; etc.
-Each square gives the product of two digits, the two
-figures of the product being separated by the diagonal
-of the square. \Eg, in \Fig{2} the lowest right
-hand square contains the digits $7$~and~$2$, $72$~being the
-product of~$9$ (at the left of the same row) and~$8$ (at
-the top of the rod).
-
-\Fig{2} represents the faces of the rods giving the
-multiples of $4$,~$3$\Add{,} and~$8$, placed together and against a
-rod containing the nine digits to be used as multiplier,
-all in position to multiply 438 by any number---say~$26$.
-\DPPageSep{076}{70}
-The products are written off, from the rods. But
-the tens digit in each case is to be added to the next
-units digit; that is, the two figures in a rhomboid are
-to be added. The operation of multiplying $438$ by~$26$,
-\begin{figure}[htb!]
-\centering
-\Graphic{0.8\textwidth}{fig1}
-\caption{Fig.~1.\protect\footnotemark}
-\end{figure}
-\footnotetext{From Lucas, III, 76.}%
-after arranging the rods as in \Fig[Figure]{2}, would be
-somewhat as follows: beginning at the right hand and
-multiplying first by~$6$, we have~$8$, $4 + 8$, (carrying the~$1$)
-\DPPageSep{077}{71}
-$1 + 1 + 4$, $2$, giving the number (from left to right)
-$2628$, the first partial product. Similarly $876$~is read
-from the row of squares at the
-right of the multiplier~$2$. It is
-shifted one place to the left in
-writing it under the former partial
-product. Then these two
-numbers are added.
-\Figure[0.4]{2}
-
-Somewhat analogous is the
-use of the rods for division.
-
-``It is evident that they would
-be of little use to any one who
-knew the multiplication table as
-far as $9 × 9$.''\footnote
- {Dr.~Glaisher in his article ``Napier'' in the \Title{Encyclopĉdia
-\index{Glaisher|FN}%
- Britannica}.}
-Though published
-(and invented) later than
-logarithms, which we so much
-admire, the rods were welcomed
-more cordially by contemporaries.
-Several editions of the
-\Title{Rabdologia} were brought out
-on the Continent within a decade.
-``Nothing shows more clearly the rude state of
-arithmetical knowledge at the beginning of the seventeenth
-century than the universal satisfaction with
-which Napier's invention was welcomed by all classes
-%[** TN: Same footnote mark]
-\addtocounter{footnote}{-1}
-and regarded as a real aid to calculation.''\footnotemark{} It is from
-this point of view that the study of the rods is interesting
-and instructive to us.
-
-The \Title{Rabdologia} contains other matter besides the
-description of rods for multiplication and division.
-But such mechanical aids to calculation are soon superseded.
-\DPPageSep{078}{72}
-
-It is worthy of note in this connection, however, that
-in the absence of so facile an instrument for calculation
-as our Arabic notation, simple mechanical devices
-might be found so serviceable as to persist for centuries.
-The abacus, which is familiar to almost every
-one, but only as a historical relic, a piece of illustrative
-apparatus, or a toy, was a highly important aid to
-computation among the Greeks and Romans. Similar
-\index{Greeks}%
-to the abacus is the Chinese \emph{swan pan}. It is said that
-\index{Swan pan}%
-Oriental accountants are able, by its use, to make computations
-rivaling in accuracy and speed those performed
-by Occidentals with numerals on paper.
-
-Modern adding machines, per~cent devices, and the
-more complicated and costly calculating instruments
-have led up to such mechanical marvels as ``electrical
-calculating machines'' and the machines of Babbage
-\index{Babbage}%
-and Scheutz, which latter prepare tables of logarithms
-\index{Scheutz}%
-and of logarithmic functions without error arithmetical
-or typographical, computing, stereotyping and delivering
-them ready for the press.
-
-If Napier's rods be regarded as exemplars of such
-products of the nineteenth century, they are primitive
-members of a long line of honorable succession.
-\index{Arithmetic|)}%
-\DPPageSep{079}{73}
-
-
-\Chapter{Axioms in elementary algebra.}
-\index{Algebra|(}%
-\index{Axioms!in elementary algebra}%
-
-Many text-books on the subject introduce equations
-with a list of axioms such as the following:
-
-1. Things equal to the same thing or equal things
-are equal to each other.
-
-2. If equals be added to equals, the sums are equal.
-
-3. If equals be subtracted from equals, the remainders
-are equal.
-
-4. If equals be multiplied by equals, the products
-are equal.
-
-5. If equals be divided by equals, the quotients are
-equal.
-
-6. The whole is greater than any of its parts.
-
-7. Like powers, or like roots, of equals are equal.
-\index{Roots!of equal numbers}%
-
-These time-honored ``common notions'' are the foundation
-of logical arithmetic. On them is based also
-the reasoning of algebra. But it is most desirable
-that, when we extend their meaning to the comparison
-of algebraic numbers, we should notice the limitations
-of the axioms. Generalization is a characteristic of
-mathematics. When we generalize, we remove limitations
-that have been stated or implied. A proposition
-true with those limitations may or may not be
-true without them. For illustration: When we proceed
-from geometry of two dimensions to geometry
-of three dimensions, the limitation, always understood
-in plane geometry, that all figures considered are (except
-while employing the motion postulate for superposition)
-\DPPageSep{080}{74}
-in the plane of the paper or blackboard, is
-removed. Some of the propositions true in plane
-geometry hold also in solid, and some do not. Compare
-in this respect the two theorems, ``Through a
-given \emph{external} point only one perpendicular can be
-drawn to a given line,'' and, ``Through a given \emph{internal}
-point only one perpendicular can be drawn to a given
-line.''\footnote
- {Using the term \emph{perpendicular} in the sense customary in
- elementary geometry.}
-For another illustration see the paragraph (p.~\PgNo{37}),
-``Are there more than one set of prime factors
-of a number?'' \emph{No} when factor means arithmetic
-number; \emph{yes} when the meaning of the word is extended
-to include complex numbers. See also instances of
-the ``fallacy of accident,'' p.~\PgNo{85}~f.
-
-We might expect that some of the axioms of arithmetic
-would need qualification when we attempt to extend
-them so as to apply to algebraic numbers. And
-that is what we find. But we do not find that all
-authors have notified their readers of the limitations
-or have observed them in their own use of the axioms.
-Surely it is not too much to expect that the axioms of
-a science shall be true and applicable \emph{in the sense in
-which the terms are used in that science}.
-
-The fifth, or ``division axiom,'' should receive the
-important qualification given it by the best of the
-books, ``divided by equals, \emph{except zero}.'' Without such
-limitation the statement is far from axiomatic.
-
-A writer of the sixth ``axiom'' may also have
-on another page something like this: ``$+3$~is the
-whole, or \emph{sum}.'' Seeing that one of its parts is~$+7$,
-one wonders how the author, in a text-book
-on algebra, could ever have written the ``axiom,''
-``The whole is greater than any of its parts.''
-%[** TN: Inset in the original]
-\[
-\begin{array}{r}
-+7 \\
--5 \\
-+2 \\
--1 \\
-\hline
-+3
-\end{array}
-\]
-\DPPageSep{081}{75}
-
-In the seventh axiom, like roots of equals are equal
-\index{Complex numbers}%
-\index{Roots!of equal numbers}%
-\emph{arithmetically}. Otherwise worded: Like real roots
-of equals are equal, like signs being taken.\footnote
- {The defense often heard for the unqualified axiom, Like
- roots of equals are equal, in algebra---that \emph{like} here means
- \emph{equal}---would reduce the axiom to a platitude, Roots\DPnote{** [sic] capitalized} are
- equal if they are equal. Besides being insipid, this is insufficient.
- To be of any use, the axiom must mean, that if $C$~and
- $D$ are known to represent each a square root, or each a
- cube root, of $A$~and $B$ respectively, and if $A$~and $B$ are known
- to be equal, then $C$~and $D$ are as certainly known to be two
- expressions for the same number. Now in the case of square
- roots this inference is justified only when like signs are taken.
- For cube roots, if $A = B = 1$, then $-\dfrac{1}{2} + \dfrac{1}{2} \sqrt{-3}$ is a cube root
- of~$A$, and $-\dfrac{1}{2} - \dfrac{1}{2} \sqrt{-3}$ is a cube root of~$B$; but $-\dfrac{1}{2} + \dfrac{1}{2} \sqrt{-3}$
- and $-\dfrac{1}{2} - \dfrac{1}{2} \sqrt{-3}$ are not expressions for the same number.
- If their modulus (page~\PgNo{94}) be taken as their absolute value,
- they are equal to each other and to the real cube root~$1$ in
- absolute value. If our axiom be made to read, Like odd \emph{real}
- roots are equal, it is applicable to such roots without trouble.
- $A$~has but one cube root that is real, and $B$~has but one, and
- they are equal.
-
- It is interesting to notice in passing that the two numbers
- just used, $-\dfrac{1}{2} + \dfrac{1}{2} \sqrt{-3}$ and $-\dfrac{1}{2} - \dfrac{1}{2} \sqrt{-3}$, are a pair of
- unequal numbers each of which is the square of the other.}
-
-When we use the word ``equal'' in the axioms, do
-we mean anything else than ``same''---If two numbers
-are the same as a third number, they are the same as
-each other,~etc.?
-\DPPageSep{082}{76}
-
-
-\Chapter{Do the axioms apply to equations?}
-\index{Axioms!apply to equations?}%
-\index{Do the axioms apply to equations?}%
-\index{Equations!axioms apply to?}%
-
-Most text-books in elementary algebra use them as
-if they applied. Most of the algebras have, somewhere
-in the first fifty or sixty pages, something like this:
-\[
-3x + 4 = 19\Add{.}
-\]
-Subtracting~$4$ from each member,
-\[
-3x = 15\Add{.}
-\Ax{3}
-\]
-Dividing by~$3$,
-\[
-x = 5\Add{.}
-\Ax{5}
-\]
-
-This shows how common some very loose thinking
-on this subject is. For although no mistake has been
-made in the algebraic operation, the citation of axioms
-as authority for these steps opens the way for a pupil
-to divide both members of an equation by an unknown,
-in which case he drops a solution,\footnote
- {Every teacher of elementary algebra is aware of the
- tendency of pupils (unless carefully guided) to ``divide through
- by~$x$'' when possible, and to fail to note that they have lost out
- the solution $x = 0$.}
-or to apply one of
-the other axioms and introduce a solution.
-
-As a matter of fact, the axioms do not apply directly
-to equations: for (A)~one can follow the axioms,
-make no mistake, and arrive at a result which is incorrect:
-(B)~he can violate the axioms and come out
-right: (C)~the axioms, from their very nature, can
-not apply directly to equations.
-\DPPageSep{083}{77}
-
-
-\Section[\quad]{\Inum{(A)} To follow axioms and come out wrong:}
-\[
-x - 1 = 2\Add{.}
-\Tag{(1)}
-\]
-Multiplying each member by $x - 5$,
-\[
-x^{2} - 6x + 5 = 2x - 10\Add{.}
-\Ax{4}
-\]
-Subtracting $x - 7$ from each member,
-\[
-x^{2} - 7x + 12 = x - 3\Add{.}
-\Ax{3}
-\]
-Dividing each member by $x - 3$,
-\[
-x - 4 = 1\Add{.}
-\Ax{5}
-\]
-Adding~$4$ to each member,
-\[
-x = 5\Add{.}
-\Ax{2}
-\]
-But $x = 5$ does not satisfy~\Eq{(1)}. The only value of~$x$
-that satisfies~\Eq{(1)} is~$3$.
-
-Misunderstanding at this point is so common that
-\index{Equations!equivalency|(}%
-it is deemed best to be explicit at the risk of being
-tedious. The multiplication by~$x - 5$ introduces the
-solution $x = 5$, and the division by~$x - 3$ loses the solution
-$x = 3$. Now it may be argued, that the axioms
-of the preceding section when properly qualified exclude
-division by zero, and that $x - 3$~is here a form
-of zero since $3$~is the value of~$x$ for which equation~\Eq{(1)}
-is true. Exactly; but this only shows that in
-operating with equations the question for what value
-of~$x$ they are true is bound to be raised. The attempt
-to qualify the axioms and adjust them to this necessity
-will, if thoroughgoing, lead to principles of equivalency
-of equations.\footnote
- {Such, for example, as the following:
-
- To add or subtract the same expression (known or unknown)
- to both members of an equation, does not affect the value of~$x$
- (the resulting equation is equivalent to the original).
-
- To multiply or divide both members by a known number not
- zero, does not affect the value of~$x$.
-
- To multiply or divide both members by an integral function
- of~$x$, introduces or loses, respectively, solutions (namely, the
- solution of the equation formed by putting the multiplier
- equal to zero) it being understood that the equations are in
- the standard form.}
-Any objector is requested to study
-\DPPageSep{084}{78}
-carefully the principles of equivalency as set forth in
-one of the best algebras and notice their relation to the
-axioms on the one hand and to operations with equations
-on the other, and see whether he is not then prepared
-to say that the axioms do not apply \emph{directly} to
-equations.
-
-It should be noted that the foregoing is not an
-attack on the integrity of the axioms, but only on the
-application of them where they are not applicable.
-
-If it be objected that in~(A) the axioms are not really
-followed, the reply is, that they are here followed as
-they are naturally followed by pupils taught to apply
-them directly to equations, and as they are occasionally
-followed by the authors of some elementary algebras,
-only the errors are here made more glaring and the
-process reduced \Foreign{ad~absurdum}.
-
-
-\Section[\quad]{\Inum{(B)} To violate the axioms and come out right:}
-
-In order to avoid the objection that the errors made
-by violating two axioms may just balance each other,
-only \emph{one} axiom will be violated.
-\[
-x - 1 = 2\Add{.}
-\Tag{(1)}
-\]
-Add $10$ to one member \emph{and not to the other}. This will
-doubtless be deemed a sufficiently flagrant transgression
-of the ``addition axiom'':
-\[
-x + 9 = 2\Add{.}
-\Tag{(2)}
-\]
-Multiplying each member by~$x - 3$,
-\[
-x^{2} + 6x - 27 = 2x - 6\Add{.}
-\TagAx{(3)}{4}
-\]
-Subtracting $2x - 6$ from each member,
-\[
-x^{2} + 4x - 21 = 0\Add{.}
-\TagAx{(4)}{3}
-\]
-Dividing each member by $x + 7$,
-\[
-x - 3 = 0\Add{.}
-\TagAx{(5)}{5}
-\]
-Adding $3$ to each member,
-\[
-x = 3\Add{.}
-\Ax{2}
-\]
-\DPPageSep{085}{79}
-Inasmuch as $3$~is \emph{the correct root} of equation~\Eq{(1)}, the
-error in the first step must have been balanced by another,
-or by several. It was done in obtaining \Eq{(3)}~and~\Eq{(5)},
-though at both steps the axioms were applied.
-
-
-\Section[\quad]{\Inum{(C)} The axioms, from their very nature, can not
-have any direct application to equations.}
-
-The axioms say that---if equals be added to equals
-etc.---the results are equal. But the question in solving
-equations is, For what value of~$x$ are they equal?
-Of course they are equal for \emph{some} value of~$x$. So
-when something was added to one member and not
-to the other, the results were equal \emph{for some value of~$x$}.
-Arithmetic, dealing with numbers, needs to know
-that certain resulting numbers are equal to certain
-others; but algebra, dealing with the equation, the
-conditional equality of expressions, needs to know on
-\emph{what condition} the expressions represent the same
-number---in other words, for what values of the unknown
-the equation is true. In (B) above, the objection
-to equation~\Eq{(2)} is not that its two members
-are not equal (they are ``equal'' as much as are the
-two members of the first equation) but that they are
-not equal \emph{for the same value of~$x$} as in the first equation;
-that is \Eq{(2)}~is not \emph{equivalent} to~\Eq{(1)}.
-%<tb>
-
-The principles of equivalency of equations as given
-in a few of the best of the texts are not too difficult
-for the beginner. The \emph{proof} of them may well be deferred
-till later. Even if never proved, they would
-be, for the present purpose, vastly superior to axioms
-that do not apply. To give \emph{no} reasons would be preferable
-to the practice of quoting axioms that do not
-apply.
-\index{Equations!equivalency|)}%
-\DPPageSep{086}{80}
-
-The axioms have their place in connection with
-equations; namely, in the proof of the principles of
-equivalency. To apply the axioms directly in the solution
-of equations is an error.
-
-Pupils can hardly be expected to think clearly about
-the nature of the equation when they are so misled.
-How the authors of the great majority of the elementary
-texts can have made so palpable a mistake in so
-elementary a matter, is one of the seven wonders of
-algebra.
-\DPPageSep{087}{81}
-
-
-\Chapter{Checking the solution of an equation.}
-\index{Checking solution of equation}%
-\index{Equations!checking solution of}%
-
-The habit which many high-school pupils have of
-checking their solution of an equation by first substituting
-for~$x$ in both members of the given equation,
-performing like operations upon both members
-until a numerical identity is obtained, and then declaring
-their work ``proved,'' may be illustrated by the
-following ``proof,'' in which the absurdity is apparent:
-\[
-1 + \sqrt{x + 2} = 1 - \sqrt{12 - x}\Add{.}
-\Tag{(1)}
-\]
-Solution\Add{:}
-\begin{align*}
-\sqrt{x + 2} &= -\sqrt{12 - x}
-\Tag{(2)} \\
-x + 2 &= 12 - x
-\Tag{(3)} \\
-2x &= 10 \\
-x &= 5\Add{.}
-\end{align*}
-``Proof''\Add{:}
-\begin{align*}
-1 + \sqrt{5 + 2} &= 1 - \sqrt{12 - 5} \\
-\sqrt{5 + 2} &= -\sqrt{12 - 5} \\
-5 + 2 &= 12 - 5 \\
-7 &= 7\Add{.}
-\end{align*}
-
-Checking in the legitimate manner---by substituting
-in one member of the given equation and reducing the
-resulting number to its simplest form, then substituting
-in the other member and reducing to simplest form---we
-have $1 + \sqrt{7}$ for the first member, and $1 - \sqrt{7}$~for
-the second. As these are not equal numbers, $5$~is not
-a root of the equation. There is no root.
-\DPPageSep{088}{82}
-
-The $5$ was introduced in squaring. That is, $x = 5$
-satisfies equation~\Eq{(3)} but not \Eq{(2)} or~\Eq{(1)}. By the
-change of a sign in either \Eq{(1)} or \Eq{(2)} we obtain an
-equation that is true for $x = 5$:
-\[
-1 + \sqrt{x + 2} = 1 - \sqrt{12 - x}\Add{.}
-\]
-When rational equations are derived from irrational
-by involution, there are always other irrational equations,
-differing from these in the sign of a term, from
-which the same rational equations would be derived.
-
-In a popular algebra may be found the equation
-\[
-x + 5 - \sqrt{x + 5} = 6
-\]
-and in the answer list printed in the book, ``$4$, or~$-1$''
-is given for this equation. $4$~is a solution, but $-1$~is
-not. Unfortunately this instance is not unique.
-
-As the fallacy in the erroneous method shown above
-is in assuming that all operations are reversible, that
-method may be caricatured by the old absurdity,
-
-%[** TN: Reformatted from the original]
-To prove that
-\begin{align*}
-5 &= 1\Add{.}
-\intertext{Subtracting $3$ from each,}
-2 &= -2\Add{.}
-\intertext{Squaring}
-4 &= 4\Add{.} \\
-\therefore\ 5 &= 1!
-\end{align*}
-\DPPageSep{089}{83}
-
-
-\Chapter{Algebraic fallacies.}
-\index{Algebraic!fallacies}%
-\index{Fallacies!algebraic}%
-\index{Converse, fallacy of|EtSeq}%
-\index{Undistributed middle|EtSeq}%
-
-A humorist maintained that in all literature there
-are really only a few jokes with many variations, and
-proceeded to give a classification into which all jests
-could be placed---a limited list of type jokes. A fellow
-humorist proceeded to reduce this number (to three,
-if the writer's memory is correct). Whereupon a
-third representative of the profession took the remaining
-step and declared that there are none. Whether
-these gentlemen succeeded in eliminating jokes altogether
-or in adding another to an already enormous
-number, depends perhaps on the point of view.
-
-The writer purposes to classify and illustrate some
-of the commoner algebraic fallacies, in the hope, not
-of adding a striking original specimen, but rather of
-standardizing certain types, at the risk of blighting
-them. Fallacies, like ghosts, are not fond of light.
-Analysis is perilous to all species of the genus.
-
-Of the classes, or subclasses, into which Aristotle
-\index{Aristotle}%
-divided the fallacies of logic, only a few merit special
-notice here. Prominent among these is that variety
-of paralogism known as undistributed middle. In
-mathematics it masks as the fallacy of converse, or
-employing a process that is not uniquely reversible
-as if it were. For example, the following:\footnote
- {Taken, with several of the other illustrations, from the
- fallacies compiled by W.~W.~R. Ball. See his \Title{Mathematical
- Recreations and Essays}.}
-\DPPageSep{090}{84}
-
-Let $c$~be the arithmetic mean between two \emph{unequal}
-numbers $a$~and~$b$; that is, let
-\[
-a + b = 2c\Add{.}
-\Tag{(1)}
-\]
-Then
-\begin{align*}
-(a + b)(a - b) &= 2c(a - b) \\
-a^{2} - b^{2} &= 2ac - 2bc\Add{.}
-\intertext{Transposing,}
-a^{2} - 2ac &= b^{2} - 2bc\Add{.}
-\Tag{(2)} \\
-\intertext{Adding $c^{2}$ to each,}
-a^{2} - 2ac + c^{2} &= b^{2} - 2bc + c^{2}
-\Tag{(3)} \\
-\therefore\ a - c &= b - c \\
-\intertext{and}
-a &= b
-\end{align*}
-But $a$~and~$b$ were taken unequal.
-
-Of course the two members of~\Eq{(3)} are arithmetically
-equal but of opposite quality; their squares, the
-two members of~\Eq{(2)}, are equal. The fallacy here is
-so apparent that it would seem superfluous to expose
-it, were it not so common in one form or another.
-
-For another example take the absurdity used in the
-preceding section to caricature an erroneous method of
-checking a solution of an equation. Let us resort to
-a parallel column arrangement:
-\begin{gather*}
-\begin{tabular}{ll}
-\Stmnt{A bird is an animal;} &
-\Stmnt{Two equal numbers have
-equal squares;} \\
-\Stmnt{A horse is an animal;} &
-\Stmnt{These two numbers have
-equal squares;} \\
-\Stmnt{$\therefore$ A horse is a bird.} &
-\Stmnt{$\therefore$ These two numbers are equal.}
-\end{tabular} \\
-\begin{tabular}{ll}
-\Stmnt{The untutored man pooh-poohs
-at this, because the
-\emph{conclusion} is absurd, but fails
-to notice a like fallacy on the
-lips of the political speaker
-of his own party.} &
-\Stmnt{The first-year high-school
-pupil derides this whenever
-the \emph{conclusion} is absurd, but
-would allow to pass unchallenged
-the fallacious method
-of checking shown in the preceding
-section.}
-\end{tabular}
-\end{gather*}
-
-In case of indicated square roots the fallacy may be
-much less apparent. By the common convention as
-to sign, $+$~is understood before~$\surd$. Considering, then,
-\DPPageSep{091}{85}
-only the positive even root or the real odd root, it is
-true that ``like roots of equals are equal,'' and
-\[
-\sqrt[n]{ab} = \sqrt[n]{a\vphantom{b}} · \sqrt[n]{b}\Add{.}
-\]
-But if $a$~and~$b$ are negative, and $n$~even, the identity
-no longer holds, and by assuming it we have the absurdity
-\begin{align*}
-\sqrt{(-1)(-1)} &= \sqrt{-1} · \sqrt{-1} \\
-\sqrt{1} &= (\sqrt{-1})^{2} \\
-1 &= -1\Add{.}
-\end{align*}
-
-Or take for granted that $\sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}}$ for all values of
-the letters. The following is an identity, since each
-member $= \sqrt{-1}$:
-\[
-\sqrt{\frac{1}{-1}} = \sqrt{\frac{-1}{1}}\Add{.}
-\]
-Hence!
-\[
-\frac{\sqrt{1}}{\sqrt{-1}} = \frac{\sqrt{-1}}{\sqrt{1}}\Add{.}
-\]
-Clearing of fractions,
-\[
-(\sqrt{1})^{2} = (\sqrt{-1})^{2}\Add{.}
-\]
-Or
-\[
-1 = -1\Add{.}
-\]
-
-The ``fallacy of accident,'' by which one argues
-from a general rule to a special case where some circumstance
-renders the rule inapplicable, and its converse
-fallacy, and De~Morgan's suggested third variety
-\index{Demorgan@{De Morgan}}%
-of the fallacy, from one special case to another, all
-find exemplification in pseudo-algebra. As a general
-rule, if equals be divided by equals, the quotients are
-equal; but not if the equal divisors are any form of
-zero. The application of the general rule to this special
-case is the method underlying the largest number of
-the common algebraic fallacies.
-\DPPageSep{092}{86}
-\[
-x^{2} - x^{2} = x^{2} - x^{2}\Add{.}
-\]
-Factoring the first member as the difference of squares,
-and the second by taking out a common factor,
-\[
-(x + x)(x - x) = x(x - x)\Add{.}
-\Tag{(1)}
-\]
-Canceling $x - x$,
-\begin{align*}
-x + x &= x \\
-\Tag{(2)} \\
-2x &= x \\
-2 &= 1\Add{.}
-\Tag{(3)}
-\end{align*}
-Dividing by~$0$ changes identity~\Eq{(1)} into equation~\Eq{(2)},
-which is true for only one value of~$x$, namely~$0$. Dividing~\Eq{(2)}
-by~$x$ leaves the absurdity~\Eq{(3)}.
-
-Take another old illustration:\footnote
- {Referred to by De~Morgan as ``old'' in a number of the
- \Title{Athenĉum} of forty years ago.}
-
-Let
-\[
-x = 1\Add{.}
-\]
-Then
-\begin{align*}
-x^{2} &= x\Add{.}
-\intertext{And}
-x^{2} - 1 &= x - 1\Add{.} \\
-\intertext{Dividing both by $x - 1$,}
-x + 1 &= 1\Add{.} \\
-\intertext{But}
-x &= 1\Add{.} \\
-\intertext{Whence, by substituting,}
-2 &= 1\Add{.}
-\end{align*}
-
-The use of a divergent series furnishes another
-type of fallacy, in which one assumes something to
-be true of all series which in fact is true only of the
-convergent. For this purpose the harmonic series is
-perhaps oftenest employed.
-\[
-1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots\Add{.}
-\]
-Group the terms thus:
-\begin{gather*}
-1 + \frac{1}{2} + \left(\frac{1}{3} + \frac{1}{4}\right)
- + \left(\frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8}\right) + \\
- \left(\frac{1}{9} + \dots \text{ to $8$~terms}\right) +
- \left(\frac{1}{17} + \dots \text{ to $16$~terms}\right) + \dots\Add{.}
-\end{gather*}
-Every term (after the second) in the series as now
-written $> \nicefrac{1}{2}$. Therefore the sum of the first $n$~terms
-\DPPageSep{093}{87}
-increases without limit as $n$~increases indefinitely.\footnote
- {The sum of the first $2^{n}$~terms $> 1 + \nicefrac{1}{2}\, n$.}
-The series has no finite sum; it is divergent. But if
-the signs in this series are alternately $+$~and~$-$, the
-series
-\[
-1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \dots
-\]
-is convergent. With this in mind, the following fallacy
-is transparent enough:
-\begin{align*}
-\log 2 &= 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots \\
- &= \left(1 + \frac{1}{3} + \frac{1}{5} + \dots \right)
- - \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \dots \right) \\
- &= \left[\left(1 + \frac{1}{3} + \frac{1}{5} + \dots \right)
- + \left(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \dots \right)\right] \\
- &\qquad -2\left(\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \dots \right) \\
- &= \left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots \right)
- - \left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots \right) \\
- &= 0\Add{.}
-\intertext{But}
-\log 1 &= 0\Add{.}
-\end{align*}
-Suppose $\infty$ written in place of each parenthesis.
-
-$\infty$ and~$0$ are both convenient ``quantities'' for the
-\index{Infinite}%
-\index{Zero!in fallacies}%
-fallacy maker.
-
-By tacitly assuming that all real numbers have logarithms
-\index{Logarithms}%
-and that they are amenable to the same laws
-as the logarithms of arithmetic numbers, another type
-of fallacy emerges:
-\[
-(-1)^{2} = 1\Add{.}
-\]
-Since the logarithms of equals are equal,
-\begin{align*}
-2 \log (-1) &= \log 1 = 0 \\
-\therefore \log (-1) &= 0 \\
-\therefore \log (-1) &= \log 1 \\
-\text{and } -1 &=1\Add{.}
-\end{align*}
-\DPPageSep{094}{88}
-
-The idea of this type is credited to John Bernoulli.
-\index{Bernoulli}%
-\index{Geometric!multiplication}%
-\index{Multiplication!geometric}%
-Some great minds have turned out conceits like these
-as by-products, and many amateurs have found delight
-in the same occupation. To those who enjoy
-weaving a mathematical tangle for their friends to
-unravel, the diversion may be recommended as harmless.
-And the following may be suggested as promising
-points around which to weave a snarl: the tangent
-of an angle becoming a discontinuous function
-for those particular values of the angle which are represented
-by $(n + \frac{1}{2}) \pi$; discontinuous algebraic functions;
-the fact that when $h$,~$j$\Add{,} and~$k$ are rectangular
-unit vectors the commutative law does not hold, but
-\index{Commutative law}%
-\index{Quaternions}%
-\index{Vectors}%
-$hjk = -kjh$; the well-known theorems of plane geometry
-that are not true in solid geometry without qualification;~etc.
-
-Let us use one of these to make a fallacy to order.
-In the fraction~$\SlantFrac{1}{x}$, if the denominator be diminished,
-the fraction is increased.
-
-% [** TN: Reformatted from the original]
-When $x = 5$, $3$, $1$, $-1$, $-3$, $-5$, a decreasing series,
-then $\SlantFrac{1}{x} = \SlantFrac{1}{5}$, $\SlantFrac{1}{3}$, $1$, $-1$, $-\SlantFrac{1}{3}$, $-\SlantFrac{1}{5}$, an increasing series,
-as, by rule, each term of the second series is greater
-than the term before it: $\SlantFrac{1}{3} > \SlantFrac{1}{5}$, $1 > \SlantFrac{1}{3}$, $-\SlantFrac{1}{5} > -\SlantFrac{1}{3}$.
-Then the fourth term is greater than the third; that is
-$-1 > + 1$.
-
-Neither the fallacies of formal logic nor those of
-algebra invalidate sound reasoning. From the counterfeit
-coin one does not infer that the genuine is valueless.
-Scrutiny of the counterfeit may enable us to
-avoid being deceived later by some particularly clever
-specimen. Counterfeit coins also, if so stamped, make
-good playthings.
-\DPPageSep{095}{89}
-
-
-\Chapter{Two highest common factors.}
-\index{Factors!two highest common}%
-\index{Highest common factors, two}%
-\index{Lowest common multiples, two}%
-\index{Two \HCF}%
-
-If asked for the \HCF\ of $a^{2} - x^{2}$ and $x^{3} - a^{3}$, one
-pupil will give $a - x$, and another $x - a$. Which is
-right? Both. It is only in such a case that pupils
-raise the question; but the example is not peculiar in
-having two H.C.F\@. If the given expressions had been
-$x^{2} - a^{2}$ and $x^{3} - a^{3}$, $x - a$~would naturally be obtained,
-and would probably be the only H.C.F. offered; but
-$a - x$ is as much a common factor and is of as high
-a degree. Perhaps it is taken as a matter of course---certainly
-it is but rarely stated---that every set of algebraic
-expressions has two highest common factors,
-arithmetically equal but of opposite quality.
-
-As the term ``highest'' is used in a technical way,
-the purist will perhaps pardon the solecism ``two
-highest.''
-
-Similarly, of course, there are two L.C.M. of every
-set of algebraic expressions. By going through the
-answer list for exercises in L.C.M. in an algebra and
-changing the signs, one obtains another list of answers.
-\DPPageSep{096}{90}
-
-
-\Chapter{Positive and negative numbers.}
-\index{Negative and positive numbers}%
-\index{Real numbers}%
-\index{Positive and negative numbers}%
-
-To speak of arithmetical numbers as positive, is
-still so common an error as to need correction at every
-opportunity. The numbers of arithmetic are not positive.
-They are numbers \emph{without quality}. Negatives
-are not later than positives, either in the individual's
-conception or in that of the race. How can the idea
-of one of two \emph{opposites} be earlier than the other, or
-clearer? The terms ``positive'' and ``negative'' being
-correlative, neither can have meaning without the
-other.\footnote
- {A good exercise to develop clear thinking as to the relation
- between positive, negative and arithmetic numbers is, to
- consider the correspondence of the positive and negative solutions
- of an equation to the arithmetic solutions of the problem
- that gave rise to the equation, and the question to what
- primary assumptions this correspondence is due.}
-
-An ``algebraic balance'' has been patented and put
-\index{Algebraic!balance}%
-\index{Balance, algebraic}%
-on the market,\footnote
- {By F.~C. Donecker, Chicago. Described in \Title{School Science
-\index{Donecker, F.~C.|FN}%
- and Mathematics}. See also ``Another Algebraic Balance,'' by
- N.~J. Lennes, \Foreign{id.}, Nov.~1905; and ``Content-Problems for High
-\index{Lennes, N.~J.|FN}%
- School Algebra,'' by G.~W. Meyers, \Foreign{id.}, Jan.~1907, reprinted
-\index{Myers, G. W.|FN}%
- from \Title{School Review.}}
-designed to illustrate positive and
-negative numbers, also transposition and the other
-operations on an equation. It is composed of a system
-of levers and scale pans with weights. The value
-of this excellent apparatus in illustrating positive and
-negative numbers is in showing them to be opposites
-of each other. \Eg,~a weight in the positive scale
-pan neutralizes the pull on the beam exerted by a
-\DPPageSep{097}{91}
-weight of equal mass in the negative scale pan. The
-two weights are of equal mass, as the two numbers
-are of equal arithmetical value. When the weight is
-put into \emph{either} scale pan, it represents a ``real,'' or
-quality, number; it becomes either $+$~or~$-$.
-
-The unfortunate expression ``less than nothing''(due
-\index{Hindu!illustration of real numbers}%
-to Stifel), the attempt to consider negative numbers
-\index{Stifel}%
-apart from positive and to teach negative after positive,
-and the name ``fictitious'' for negative numbers,
-all seem absurd enough now; but they became so only
-when the real significance of positive and negative as
-opposites was clearly seen. The value of the illustration
-from debts and credits (due to the Hindus) and
-from the thermometer, lies in the aptness for bringing
-out the oppositeness of positive and negative.
-
-For the illustration from directed lines, see \Fig{3}
-on the following page.
-
-It is appropriate that the advertisements of the algebraic
-balance use the quotation from Cajori's \Title{History
-of Elementary Mathematics}: ``Negative numbers
-appeared `absurd' or `fictitious' so long as mathematicians
-had not hit upon a \emph{visual or graphical representation
-of them\dots}\Add{.} Omit all illustrations by lines,
-or by the thermometer, and negative numbers will be
-as absurd to modern students as they were to the early
-algebraists.''
-\DPPageSep{098}{92}
-
-
-\Chapter{Visual representation of complex
-numbers.}
-\index{Complex numbers}%
-\index{Geometric illustration!of complex numbers}%
-\index{Representation of complex numbers}%
-\index{Visual representation of complex numbers}%
-
-If the sect~$OR$, one unit long and extending to the
-right of~$O$, be taken to represent~$+1$, then $-1$~will be
-represented by~$OL$, extending one to the left of~$O$.
-$+a$~would be pictured
-by a line $a$~units long
-and to the right; $-a$,
-$a$~units long and to the
-left. This simplest and
-best-known use of directed
-lines gives us a
-geometric representation
-of real numbers.
-The Hindus early
-\index{Hindu!illustration of real numbers}%
-gave this interpretation
-to numbers of
-opposite quality; but
-it does not appear
-to have been given by a European until 1629, by
-\index{Girard, Albert}%
-Girard.\footnote
- {Albert Girard, \Title{Invention Nouvelle en l'Algèbre}, Amsterdam.
- Perhaps also the first to distinctly recognize imaginary
- roots of an equation.}
-
-\Figure[0.5]{3}
-
-Conceiving the line of unit length to be revolved
-in what is assumed as the positive direction (counter-clockwise)
-$-1$~may be called the factor that revolves
-from $OR$~($+1$) to $OL$~($-1$). Then $\sqrt{-1}$ is the factor
-which, being used \emph{twice}, produces that result; using
-\DPPageSep{099}{93}
-\index{Wallis}%
-it \emph{once} as a factor revolving the line through one
-of the two right angles. Then $OU$~pictures the
-number~$+\sqrt{-1}$. Similarly, since multiplication of~$-1$
-by $-\sqrt{-1}$ twice produces~$+1$, $-\sqrt{-1}$~may be considered
-as the factor which revolves from $OL$ through one
-right angle to~$OD$. If distances to the right are called~$+$,
-then distances to the left are~$-$, and $+\sqrt{-1} · b$ denotes
-\Figure{4}
-a line $b$~units long and extending up, and $-\sqrt{-1} · b$~a
-line $b$~units long extending down. The geometric
-interpretation of the imaginary was made by H.~Kühn
-\index{Kühn, H.}%
-in~1750, in the \Title{Transactions of the St.~Petersburg
-Academy.}
-
-To represent graphically the number $a + b\sqrt{-1}$ (see
-\Fig{4}), we lay off~$OA$ in the $+$~direction and $a$~units
-long; $AB$, $b$~units long and in the direction indicated
-by~$\sqrt{-1}$; and draw~$OB$. The directed line~$OB$ represents
-\DPPageSep{100}{94}
-the complex number $a + b\sqrt{-1}$. And the length
-of~$OB$, $\sqrt{a^{2} + b^{2}}$, is the \emph{modulus} of $a + b\sqrt{-1}$. The
-geometric interpretation of such a number was made
-by Jean Robert Argand, of Geneva, in his \Title{Essai}, 1806.
-The term ``modulus'' in this connection was first used
-by him, in~1814.
-
-These geometric interpretations by Kühn and Argand,
-\index{Argand, J.~R.}%
-\index{Kühn, H.}%
-and especially one made by Wessel,\footnote
- {To the Copenhagen Academy of Sciences, 1797.}
-\index{Wessel}%
-who extended
-the method to a representation in space of
-three dimensions, may be regarded as precursors of the
-beautiful methods of vector analysis given to the world
-by Sir William Rowan Hamilton in 1852 and 1866
-\index{Hamilton, W.~R.}%
-\index{Vectors}%
-under the name ``quaternions.''
-\index{Quaternions}%
-
-The letter~$i$ as symbol for the unit of imaginary
-\index{Imaginary}%
-numbers,~$\sqrt{-1}$, was suggested by Euler. It remained
-\index{Euler}%
-for Gauss to popularize the sign~$i$ and the geometric
-\index{Gauss}%
-interpretations made by Kühn and Argand.
-
-The contrasting terms ``real'' and ``imaginary'' as
-applied to the roots of an equation were first used by
-Descartes. The name ``imaginary'' was so well started
-\index{Descartes}%
-that it still persists, and seems likely to do so, although
-it has long been seen to be a misnomer.\footnote
- {It is interesting to notice the prestige of Descartes's usage
- in fixing the language of algebra: the first letters of the alphabet
- for knowns, the last letters for unknowns, the present
- form of exponents, the dot between factors for multiplication.}
-A few
-writers use the terms \emph{scalar} and \emph{orthotomic} in place
-\index{Orthotomic}%
-\index{Scalar}%
-of \emph{real} and \emph{imaginary}.
-
-The historical development of this subject furnishes
-an illustration of the general rule, that, as we advance,
-each new generalization includes as special cases what
-we have previously known on the subject. The general
-form of the complex number, $a + bi$, includes as
-special cases the real number and the imaginary. If
-\DPPageSep{101}{95}
-$b = 0$, $a + bi$~is real. If $a = 0$, $a + bi$~is imaginary.
-The common form of a complex number is the sum
-of a real number and an imaginary.\footnote
- {Professor Schubert (p.~\PgNo{24}) adds that ``we have found the
- most general numerical form to which the laws of arithmetic
- can lead.''}
-
-In 1799 Gauss published the first of his three proofs
-\index{Gauss}%
-that every algebraic equation has a root of the form
-$a + bi$.
-
-The linear equation forces us to the consideration
-of numbers of opposite quality: $x - a = 0$ and $x + a = 0$,
-satisfied by the values $+a$ and $-a$ respectively. The
-pure quadratic gives imaginary in contrast with real
-roots: $x^{2} - a^{2} = 0$ and $x^{2} + a^{2} = 0$ satisfied by $ħa$ and~$ħai$.
-The complete quadratic
-\[
-ax^{2} + bx + c =0
-\]
-has for its roots a pair of conjugate complex numbers
-when the discriminant, $b^{2} - 4ac$, is negative and $b$~is
-\index{Discriminant}%
-not~$= 0$.
-
-But though the recognition of imaginary and complex
-numbers is a necessary consequence of simple
-algebraic analysis, no complete understanding or appreciation
-of them is possible until there is some tangible
-or visible representation of them. History's
-lesson to us in this respect is plain: positive and negative,
-imaginary, and complex numbers must be graphically
-represented in teaching algebra.
-
-The algebraic balance mentioned on page~\PgNo{90} might
-\index{Algebraic!balance}%
-\index{Balance, algebraic}%
-be further developed by the addition of an appliance
-whereby imaginary numbers should be illustrated, a
-weight put into a certain pan having the effect of
-pulling the main beam to one side, and arrangements
-for pulling the beam in several other directions to illustrate
-complex numbers.
-\DPPageSep{102}{96}
-
-If in a football game we denote the forces exerted
-in the direction~$OR$ (in \Fig{3}) by positive real numbers,
-then the opponents' energy exerted in exactly
-the opposite direction,~$OL$, will be denoted by negative
-numbers. Forces in the line of $OU$ or $OD$ will be
-denoted by imaginary numbers; and all other forces
-in the game, acting in any other direction on the
-field, will be denoted by complex numbers of the general
-type.\footnote
- {Illustration from Taylor's \Title{Elements of Algebra}, where the
- visual representation of imaginary and complex numbers is
- made in full.}
-
-Each force represented by a general complex number
-is resolvable into two forces, one represented by
-a real number and the other by an imaginary, as $OB$
-(in \Fig{4}) is the resultant of $OA$ and~$AB$.
-
-A trigonometric representation of an imaginary number
-\index{Exponent, imaginary}%
-\index{Imaginary!exponent}%
-\index{Trigonometry}%
-as exponent is furnished by the formula
-\[
-e^{i} = \cos 1 + i \sin 1.
-\]
-\DPPageSep{103}{97}
-
-
-\Chapter[The law of signs.]{Illustrations of the law of signs in
-algebraic multiplication\DPchg{,}{.}}
-\index{Illustrations!of the law of signs}%
-\index{Law of signs}%
-\index{Multiplication!law of signs illustrated}%
-\index{Signs, illustrations of law of}%
-
-\Subsection{A geometric illustration.}{A Geometric Illustration.}
-\index{Geometric illustration!of law of signs in multiplication}%
-
-If distances to the right of~$O$ be called~$+$, then distances
-to the left will be~$-$. Call distances up from~$O$
-$+$, and those down~$-$. Rectangle~$OR$ has $ab$~units
-of area. \emph{Assume} that the product~$ab$ is~$+$.
-\Figure{5}
-
-Suppose $SR$ to move to the left until it is $a$~units
-to the left of~$O$, in the position~$S'R'$. The base diminished,
-became zero, and passed through that value,
-and therefore is now negative; so also the rectangle.
-The product of $-a$ and $+b$ is~$-ab$.
-\DPPageSep{104}{98}
-
-Suppose $TR'$ to move downward until it is $b$~units
-below~$O$. The rectangle, previously~$-$, has passed
-through zero, and must now be~$+$. The product of
-$-a$ and $-b$ is~$+ab$.
-
-Similarly $(+a)(-b) = -ab$.
-
-
-\Section{From a Definition of Multiplication.}
-\index{Definition!of multiplication}%
-\index{Multiplication!definition}%
-
-Multiplication is the process of performing upon
-one of two given numbers (the multiplicand) the
-same operation which is performed upon the primary
-unit to obtain the other number (the multiplier.)\footnote
- {In this definition, ``the same operation which is performed
- upon the primary unit to obtain the multiplier'' is to be understood
- to mean the most fundamental operation by which the
- multiplier may be obtained from unity, or that operation which
- is primarily signified by the multiplier. \Eg, If the multiplier
- is~$2$, this number primarily means unity taken twice, or
- the unit added to itself; multiplying $4$ by~$2$ therefore means
- adding $4$ to itself, giving the result~$8$. Dr.~Young, in his new
- book, \Title{The Teaching of Mathematics}, p.~227, says that as $2$~is
- $1 + 1^{2}$, therefore $2 × 4$ would by this definition be $4 + 4^{2}$, or~$20$;
- or, as $2$~is $1 + \SlantFrac{1}{1}$, therefore $2 × 4$ would be $4 + \SlantFrac{4}{4}$, or~$5$; etc.
- But while it is true that $1 + 1^{2}$ and $1 + \SlantFrac{1}{1}$ are each equal to~$2$,
- neither of them is the primary signification of~$2$, or represents~$2$
- in the sense of the definition. Neither of them is a proper
- statement of the multiplier ``within the meaning of the law.''
-
- It is not maintained that this definition has no difficulties,
- or that it directly helps a learner in comprehending the meaning
- of such a multiplication as $\sqrt{2} × \sqrt{3}$, but only that it is a
- generalization that is helpful for the purpose for which it is
- used, and that it is in line with the fundamental idea of multiplication
- so far as that idea is understood.
-
- The definition is only tentative, and this treatment does not
- pretend to be a proof.}
-
-When the multiplier is an arithmetical integer, the
-primary unit is that of arithmetic,~$1$, and we have the
-special case that is correctly defined in the primary
-school as, ``taking one number as many times as there
-are units in another.''
-
-Suppose we are to multiply $+4$ by~$+3$. \emph{Assuming}
-$+1$~as the primary unit, the multiplier is produced by
-\DPPageSep{105}{99}
-taking that unit additively ``three times,'' $(+1) + (+1)
-+ (+1)$. That is what the number~$+3$ means; and to
-multiply $+4$ by it, means to do that to~$+4$. $(+4) + (+4)
-+ (+4) = +12$. Similarly, the product of $-4 \text{ by } +3 =
-(-4) + (-4) + (-4) = -12$.
-
-To multiply $+4$ by~$-3$: The multiplier is the result
-obtained by taking three times additively the primary
-unit \emph{with its quality changed}. The product of~$+4$ by~$-3$
-is therefore the result obtained by taking three
-times additively~$+4$ \emph{with its quality changed}. $(-4) +
-(-4) + (-4) = -12$. Similarly, to multiply $-4$ by~$-3$
-is to take three times additively~$-4$ with its quality
-changed: $(+4) + (+4) + (+4) = +12$.
-
-Summarizing the four cases, we have ``the law of
-\index{General form of law of signs}%
-signs'': the product is~$+$ when the factors are of like
-quality, $-$~when they are of unlike quality.
-
-
-\Subsection{A more general form of the law of signs.}{A more General Form of the Law of Signs.}
-
-In deriving the law from the definition of multiplication,
-the primary unit was assumed as~$+1$. Assume
-$-1$~as the primary unit, and multiply $+4$ by~$+3$. The
-multiplier,~$+3$, is obtained from the primary unit,~$-1$,
-by taking three times additively the unit with its sign
-changed. Performing the same operation on the multiplicand,~$+4$,
-we have $(-4) + (-4) + (-4) = -12$. Similarly,
-the product of $-4 \text{ by } +3 = (+4) + (+4) + (+4) =
-+12$. To multiply $+4$ by~$-3$: The multiplier is the result
-obtained by taking three times additively the unit,~$-1$,
-without change of sign; therefore the product of
-$+4 \text{ by } -3 = (+4) + (+4) + (+4) = +12$. So also $-4$~multiplied
-by~$-3$ gives~$-12$. Summarizing \emph{these} four
-cases, we have the law of signs when $-1$~is taken as
-the primary unit: the product is~$-$ when the factors
-are of like quality, $+$~when they are of unlike quality.
-\DPPageSep{106}{100}
-
-In the geometric illustration above, we first assumed
-the rectangle $+a$ by $+b$ to be~$+$. Assuming the contrary,
-the sign of each subsequent product is reversed, and
-we have a geometric illustration of the law of signs
-when $-1$~is taken as the primary unit.
-
-The law of signs taking $+1$ as the primary unit, and
-that taking $-1$ as the primary unit, may be combined
-into one law thus: If the two factors are alike in quality,
-the product is like the primary unit in quality; if
-the two factors are opposite in quality, the quality of
-the product is opposite to that of the primary unit.
-Or: \emph{Like} signs give \emph{like} (like the primary unit); \emph{unlike}
-signs give \emph{unlike} (the unit).
-
-The assumption of still other numbers as primary
-unit leads to other laws---other ``algebras.''
-
-
-\Subsection{Multiplication as a proportion.}{Multiplication as a Proportion.}
-\index{Multiplication!as a proportion}%
-\index{Multiplication!gradual generalization of}%
-\index{Proportion, multiplication as}%
-
-Since by definition a product bears the same relation
-to the multiplicand that the multiplier bears to the
-primary unit, this equality of relation may be stated
-in the form of a proportion:
-\[
-\text{product} : \text{multiplicand} :: \text{multiplier} : \text{primary unit}
-\]
-or,
-\[
-\text{primary unit} : \text{multiplier} :: \text{multiplicand} : \text{product}.
-\]
-
-
-\Subsection{Gradual generalization of multiplication.}{Gradual Generalization of Multiplication.}
-
-From the time when Pacioli found it necessary (and
-\index{Pacioli}%
-difficult) to explain how, in the case of proper fractions
-in arithmetic, the product is less than the multiplicand,
-to the present with its use of the term \emph{multiplication}
-in higher mathematics, is a long evolution.
-It is one of the best illustrations of the generalization
-of a term that was etymologically restricted at the
-beginning.
-\DPPageSep{107}{101}
-
-
-\Chapter{Exponents.}
-\index{Exponents}%
-
-The definition of \emph{exponent} found in the elementary
-\index{Definition!of exponents}%
-algebras is sufficient for the case to which it is applied---the
-case in which the exponents are arithmetic integers.
-Our assumption of a primary unit for algebra
-being what it is, the distinction between arithmetic
-numbers as exponents and positive numbers as exponents
-is usually neglected. Or we may simply define
-positive exponent. The meaning of negative and fractional
-exponents is easily deduced. In fact those who
-first used exponents and invented an exponential notation
-(Oresme in the fourteenth century and Stevin independently
-\index{Oresme}%
-\index{Stevin, Simon}%
-in the sixteenth) had fractions as well as
-whole numbers as exponents. And negative exponents
-had been invented before Wallis studied them in the
-\index{Wallis}%
-seventeenth century. Each of these can be defined
-separately. And modern mathematics has used other
-forms of exponents. They have been made to follow
-the laws of exponents first proved for ordinary integral
-exponents, and their significance has been assigned
-in conformity thereto. Each separate species
-of exponent is defined. A unifying conception of them
-all might express itself in a definition covering all
-known forms as special cases. The general treatment
-of exponents is yet to come.
-
-\textsc{Wanted: a definition of exponent} that shall be
-general for elementary mathematics.
-\DPPageSep{108}{102}
-
-
-\Chapter{An exponential equation.}
-\index{Equation!exponential}%
-\index{Exponential equation}%
-\index{Logarithms}%
-
-The chain-letters, once so numerous, are now---it
-\index{Chain-letters}%
-is to be hoped---obsolete. In the form that was probably
-most common, the first writer sends three letters,
-each numbered~$1$. Each recipient is to copy and send
-three, numbered~$2$, and so on until number~$50$ is
-reached.
-
-Query: If every one were to do as requested, and it
-were possible to avoid sending to any person twice,
-what number of letter would be reached when every
-man, woman and child in the world should have received
-a letter?
-
-Let $n$ represent the number. Take the population
-of the earth to be fifteen hundred million. Then this
-large number is the sum of the series
-\begin{gather*}
-3,\quad 3^{2},\quad 3^{3}\Add{,}\ \dots\quad 3^{n} \\
-\begin{aligned}
-S = \frac{a(r^{n} - 1)}{r - 1} &= \frac{3(3^{n} - 1)}{2} \\
-\frac{3}{2}(3^{n} - 1) &= 1,500,000,000 \\
-3^{n} - 1 &= 1,000,000,000 \\
-n \log 3 &= \log(10^{9}) \\
-n &= \frac{9}{\log 3} = 18.86\Add{.}
-\end{aligned}
-\end{gather*}
-
-There are not enough people in the world for the
-letters numbered~$19$ to be all sent.
-\DPPageSep{109}{103}
-
-
-%[** TN: Original ToC entry reads, "... 19th century"]
-\Chapter[Two negative conclusions.]{Two negative conclusions reached
-in the nineteenth century.}
-\index{Abel, N.~H.}%
-\index{Bocher@Bôcher, M.|FN}%
-\index{Higher equations}%
-\index{Nineteenth century, negative conclusions reached}%
-\index{Parallel postulates|(}%
-\index{Roots!of higher equations}%
-\index{Negative conclusions in 19th century}%
-\index{Two negative conclusions reached in the 19th century}%
-
-1. That general equations above the fourth degree
-are insoluble by pure algebra.
-
-The solution of equations of the third and fourth
-degree had been known since 1545. Two centuries
-and a half later, young Gauss, in his thesis for the
-\index{Gauss}%
-doctorate, proved that every algebraic equation has
-\index{Equation!insolvability of general higher}%
-a root, real or imaginary.\footnote
- {Of this proof, published when Gauss was twenty-two years
- old, Professor Maxime Bôcher remarks (\Title{Bulletin of Amer.\
- Mathematical Society}, Dec.~1904, p.~118, \DPchg{note}{noted}): ``Gauss's first
- proof (1799) that every algebraic equation has a root gives a
- striking example of the use of intuition in what was intended
- as an absolutely rigorous proof by one of the greatest and at
- the same time most critical mathematical minds the world has
- ever seen.'' It should be added that Gauss afterward gave
- two other proofs of the theorem.}
-He made the conjecture
-in~1801 that it might be impossible to solve by radicals
-any general equation of higher degree than the fourth.
-This was proved by Abel, a Norwegian, whose proof
-was printed in~1824, when he was about twenty-two
-years old. Two years later the proof was published in
-an expanded form, with more detail.
-
-Thus inventive effort was turned in other directions.
-\index{Algebra|)}%
-
-2. That the ``parallel postulate'' of Euclid can never
-\index{Euclid|(}%
-\index{Euclid's postulate|(}%
-\index{Geometry|(}%
-\index{Ptolemy}%
-be proved from the other postulates and axioms.
-
-Ever since Ptolemy, in the second century, the attempt
-had been made to prove this postulate, or
-``axiom,'' and thus place it among theorems. In~1826,
-\DPPageSep{110}{104}
-Lobachevsky, professor and rector at the University
-\index{Euclidean and non-Euclidean geometry|(}%
-\index{Lobachevsky|(}%
-\index{Non-Euclidean geometry|(}%
-of Kasan, Russia, proved the futility of the attempt,
-and published his proof in~1829. He constructed a
-self-consistent geometry in which the other postulates
-and axioms are assumed and the contrary of this, thus
-showing that this is independent of them and therefore
-can not be proved from them. No notice of his researches
-appeared in Germany till~1840. In~1891
-Lobachevsky's work was made easily available to English
-readers through a translation by Prof.\ George
-Bruce Halsted.\footnote
- {Austin, Texas, 1892. It contains a most interesting introduction
- by the translator. Dr.~Halsted translated also Bolyai's
-\index{Bolyai|FN}%
-\index{Halsted, G.~B.}%
- work (1891), compiled a \Title{Bibliography of Hyperspace and
- Non-Euclidean Geometry} (1878) of 174~titles by 81~authors,
- and has himself written extensively on the subject, being
- probably the foremost writer in America on non-Euclidean
- geometry and allied topics.}
-
-The effort previously expended in attempting the
-\index{n@$n$ dimensions}%
-impossible was henceforth to be turned to the development
-of non-Euclidean geometry, to investigating the
-consequences of assuming the contrary of certain axioms,
-to $n$-dimensional geometry. ``As is usual in
-every marked intellectual advance, every existing difficulty
-removed has opened up new fields of research,
-new tendencies of thought and methods of investigation,
-and consequently new and more difficult problems
-calling for solution.''\footnote
- {Withers, p.~63--4.}
-
-High-school geometry must simply \emph{assume} (choose)
-Euclid's postulate of parallels, perhaps preferably in
-Playfair's form of it: Two intersecting lines can not
-both be parallel to the same line.
-\DPPageSep{111}{105}
-
-
-\Chapter{The three parallel postulates
-illustrated.}
-\index{Three parallel postulates illustrated}%
-
-In contrast to Euclid's postulate (just quoted) Lobachevsky's
-is, that through a given point an indefinite
-number of lines can be drawn in a plane, none of which
-cut a given line in the plane, while Riemann's postulate
-\index{Riemann's postulate|(}%
-is, that through the point no line can be drawn
-in the plane that will not cut the given line. Thus we
-have three elementary plane geometries.
-
-An excellent simple illustration of the contrast has
-been devised: Let $AB$ and $PC$ be two straight lines in
-\Figure{6}
-the same plane, both unlimited in both directions; $AB$~fixed
-in position; and $PC$~rotating about the point~$P$,
-say in the positive (counter-clockwise) direction, intersecting
-first toward the right as shown in \Fig[Figure]{6}.
-
-``Three different results are logically possible. When
-the rotating line ceases to intersect the fixed line in
-one direction [toward the right] it will immediately
-\DPPageSep{112}{106}
-intersect in the opposite direction [toward the left],
-or it will continue to rotate for a time before intersection
-takes place, or else there will be a period of
-time during which the two lines intersect in both directions.
-The first of these possibilities gives Euclid's,
-the second Lobachevsky's, and the third Riemann's
-geometry.
-
-``The mind's attitude toward these three possibilities
-taken successively illustrates in a curious way the
-essentially empirical nature of the straight line as we
-conceive it. Logically one of these \DPtypo{possibilties}{possibilities} is just
-as acceptable as the other. From this point of view
-strictly taken there is certainly no reason for preferring
-one of them to another. Psychologically, however,
-Riemann's hypothesis seems absolutely contradictory,
-and even Euclid's is not quite so acceptable
-as that of Lobachevsky.''
-
-As a slight test of the relative acceptability of these
-hypotheses to the unsophisticated mind, the present
-writer drew on the blackboard a figure like that above,
-mentioned in simple language the three possibilities,
-and asked pupils to express opinion on slips of paper.
-Forty-six out of~54 voted that the second is the true
-one. Two said they ``guessed'' it is, twenty-one
-``thought'' so, thirteen ``felt sure,'' and ten ``knew.''
-Six ``thought'' that the first supposition is correct, and
-two ``felt sure'' of it. No one voted for the third, and
-the writer has never heard but one person express
-opinion in favor of the third supposition. Some of
-the pupils had had a few weeks of plane geometry.
-Of these, most who voted in the majority wanted to
-change as soon as it was pointed out that the second
-supposition implies that two intersecting lines can
-both be parallel to the same line. Undoubtedly some
-\DPPageSep{113}{107}
-of the more immature were unable to grasp the idea
-that the lines are of unlimited length, and possibly it
-may be somewhat general that those who favor the
-second supposition do not fully grasp that idea. Such
-a test merely illustrates that Euclid's postulate is not
-in all its forms apodictic.
-
-The whole question of parallel postulates is admirably
-\index{Withers, J. W.}%
-treated by Dr.~Withers,\footnote
- {John William Withers, \Title{Euclid's Parallel Postulate: Its
- Nature, Validity, and Place in Geometrical Systems}, his thesis
- for the doctorate at Yale, published by The Open Court Publishing
- Co., 1905. It includes a bibliography of about 140~titles
- on this and more or less closely related subjects, mentioning
- Halsted's bibliography of 174~titles and Roberto
- Bonola's of 353~titles. To these lists might be added Manning's
-\index{Bonola, Roberto|FN}%
- \Title{Non-Euclidean Geometry} (1901) which is brief, elementary
- and interesting.}
-to whose book (p.~117)
-the writer is indebted for the two paragraphs
-quoted above.
-
-\Par{In trigonometry.} The familiar figure in trigonometry
-\index{Trigonometry}%
-representing the line values of the tangent of an
-angle at the center of a unit circle as the angle increases
-and passes through~$90°$ is another form of
-this figure. And the assumption that intersection of
-the final (revolving) side with the line of tangents
-begins at an infinite distance below at the instant it
-ceases above, places our trigonometry on a Euclidean
-basis.
-
-\Par{Parallels meet at infinity.} Kepler's definition would
-\index{Kepler}%
-\index{Parallels meet at infinity}%
-seem paradoxical if offered in elementary geometry,
-but is valuable in more advanced work, and is intelligible
-enough when made in the language of limits.
-Let $PP'$ be perpendicular to~$SQ$; let $Q$~move farther
-and farther to the right while $P$~remains fixed; and
-let $P'PR$~be the limit toward which angle~$P'PQ$ approaches
-as the distance of~$Q$ from~$P'$ increases without
-\DPPageSep{114}{108}
-limit.\footnote
- {In \Fig{6} the moving line rotated until after it ceased to
- intersect the fixed line toward the right. In the present illustration
- (\Fig{7}) $PQ$~rotates only as~$Q$, the point of intersection,
- recedes along the line~$SP'Q$.}
-Then $PR$~is parallel to~$SQ$. That is, parallelism
-is attributed to the limiting position of intersecting
-lines as the point of intersection recedes without
-limit; which, for the sake of brevity, we may express
-by the familiar sentence, ``Parallels meet at infinity.''
-
-\Par{The three postulates again.} Now suppose $PS$ to
-move, $P$~remaining fixed and $S$~moving to the left,
-$TPP'$~being the limit of angle~$SPP'$ as $P'S$~increases
-without \DPtypo{limt}{limit}. Then $PT$~is parallel to~$SQ$. According
-\Figure[1.0]{7}
-to Euclid's postulate $PT$~and $PR$ are one straight line;
-according to Lobachevsky's they are not; while according
-to Riemann's $Q$~and~$S$ can not recede to an
-infinite distance (but $Q$~comes around, so to speak,
-through~$S$, to $P'$ again) and there is no limiting position
-(in the terminology of the theory of limits) and
-no parallel in the Euclidean sense of the term.
-\index{Euclid|)}%
-\index{Euclid's postulate|)}%
-\index{Euclidean and non-Euclidean geometry|)}%
-\index{Lobachevsky|)}%
-\index{Non-Euclidean geometry|)}%
-\index{Parallel postulates|)}%
-\index{Riemann's postulate|)}%
-\DPPageSep{115}{109}
-
-
-\Chapter{Geometric puzzles.}
-\index{Geometric!puzzles}%
-\index{Puzzles, geometric}%
-
-``A rectangular hole $13$~inches long and $5$~inches
-wide was discovered in the bottom of a ship. The
-ship's carpenter had only one piece of board with
-which to make repairs, and that was but $8$~inches
-square ($64$~square inches) while the hole contained
-\Figure[0.66]{8}
-$65$~square inches. But he knew how to cut the board
-so as to make it fill the hole''! Or in more prosaic
-form:
-
-\Fig{8} is a square $8$~units on a side, area~$64$; cut it
-\DPPageSep{116}{110}
-through the heavy lines and rearrange the pieces as
-indicated by the letters in \Fig{9}, and you have a rectangle
-$5$~by~$13$, area~$65$. Explain.
-\Figures{0.875}{9}{0.9}{10}
-
-\Fig{10} explains. $EH$~is a straight line, and $HG$~is
-a straight line, but they are not parts of the same
-straight line. Proof:
-
-Let $X$~be the point at which $EH$~produced meets~$GJ$;
-\DPPageSep{117}{111}
-then from the similarity of triangles $EHK$ and~$EXJ$
-\begin{align*}
-XJ : HK &= EJ : EK \\
-XJ : 3 &= 13 : 8 \\
-XJ &= 4.875\Add{.}
-\intertext{But}
-GJ &= 5.
-\end{align*}
-
-Similarly, $EFG$~is a broken line.
-
-The area of the rectangle is, indeed,~$65$, but the area
-of the rhomboid~$EFGH$ is~$1$.
-
-Professor Ball\footnote
- {\Title{Recreations}, p.~49.}
-uses this to illustrate that proofs by
-dissection and superposition are to be regarded with
-suspicion until supplemented by mathematical reasoning.
-
-``This geometrical paradox~\dots\ seems to have been
-well known in~1868, as it was published that year in
-Schlömilch's \Title{Zeitschrift für Mathematik und Physik},
-Vol.~13, p.~162.''
-
-In an article in \Title{The Open Court}, August 1907, (from
-\index{Escott, E.~B.}%
-which the preceding four lines are quoted), Mr.~Escott
-generalizes this puzzle. The puzzle is so famous that
-his analysis can not but be of interest. With his permission
-it is here reproduced:
-
-In \Fig{11}, it is shown how we can arrange the same
-pieces so as to form the three figures, $A$,~$B$, and~$C$.
-If we take $x = 5$, $y = 3$, we shall have $A = 63$, $B = 64$,
-$C = 65$.
-
-Let us investigate the three figures by algebra.
-\begin{align*}
-A &= 2xy + 2xy + y(2y - x) = 3xy + 2y^{2} \\
-B &= (x + y)^{2} = x^{2} + 2xy + y^{2} \\
-C &= x(2x + y) = 2x^{2} + xy \\
-C - B &= x^{2} - xy - y^{2} \\
-B - A &= x^{2} - xy - y^{2}.
-\end{align*}
-\DPPageSep{118}{112}
-\Figure[0.7]{11}
-\DPPageSep{119}{113}
-
-These three figures would be equal if $x^{2} - xy - y^{2} = 0$,
-\ie, if
-\[
-\frac{x}{y} = \frac{1 + \sqrt{5}}{2}
-\]
-so the three figures cannot be made equal if $x$~and~$y$
-are expressed in rational numbers.
-
-We will try to find rational values of $x$~and $y$ which
-will make the difference between $A$~and $B$ or between
-$B$~and $C$ unity.
-
-Solving the equation
-\[
-x^{2} - xy - y^{2}= ħ1
-\]
-we find by the Theory of Numbers that the $y$~and~$x$
-may be taken as any two consecutive numbers in the
-series
-\[
-%[** TN: Six-dot ellipsis in the original]
-1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ 55,\ \dots
-\]
-where each number is the sum of the two preceding
-numbers.
-
-The values $y = 3$ and $x = 5$ are the ones commonly
-given. For these we have, as stated above, $A < B < C$.
-
-The next pair, $x = 8$, $y = 5$ give $A > B > C$, \ie,
-$A = 170$, $B = 169$, $C = 168$.
-
-\Fig{12} shows an interesting modification of the
-puzzle.
-\begin{align*}
-A &= 4xy + (y + x)(2y - x) = 2y^{2} + 2yz + 3xy - xz \\
-B &= (x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2yz + 2zx + 2xy \\
-C &= (x + 2z)(2x + y + z) = 2x^{2} + 2z^{2} + 2yz + 5zx + xy\Add{.}
-\end{align*}
-
-When $x = 6$, $y = 5$, $z = 1$ we have $A = B = C = 144$.
-
-When $x = 10$, $y = 10$, $z = 3$ we have $A > B > C$, viz.,
-\[
-A = 530,\ B = 529,\ C = 528.
-\]
-
-\Par[.]{Another puzzle} is made by constructing a cardboard
-rectangle $13$~by~$11$, cutting it through one of the
-diagonals (\Fig{13}) and sliding one triangle against
-the other along their common hypotenuse to the
-\DPPageSep{120}{114}
-\Figure[0.66]{12}
-\DPPageSep{121}{115}
-position shown in \Fig{14}. Query: How can \Fig{14}
-be made up of square~$VRXS$, $12$~units on a side, area~$144$,
-$+$~triangle $PQR$, area~$0.5$, $+$~triangle $STU$, area~$0.5$,
-$=$~total area~$145$; when the area of \Fig{13} is only~$143$?
-
-Inspection of the figures, especially if aided by the
-cross lines, will show that $VRXS$ is not a square. $VS$~is
-$12$~long; but $SX < 12$. $TX = 11$ (the shorter side
-in \Fig{13}) but $ST < 1$ (see~$ST$ in \Fig{13}).
-\Figure{13}
-\begin{align*}
-ST : VP &= SU : VU \\
-ST : 11 &= 1 : 13 \\
-ST &= \nicefrac{11}{13}\Add{.} \Brk
-\text{Rectangle }
-VRXS &= 12 × 11 \nicefrac{11}{13} = 142 \nicefrac{2}{13}\Add{.} \Brk
-\text{Triangle }
-PQR &= \text{triangle } STU
- = \nicefrac{1}{2} · \nicefrac{11}{13} · 1
- = \nicefrac{11}{26}\Add{.} \\
-\text{Fig.~14} &= \text{rectangle} + 2 \text{ triangles} \\
- &= 142 \nicefrac{2}{13} + \nicefrac{11}{13} = 143.
-\end{align*}
-\DPPageSep{122}{116}
-
-By sliding the triangles one place (to the first cross
-line) in the other direction we appear to have a rectangle
-$14$~by~$10$ and two small triangles with an area
-of $\nicefrac{1}{2}$ each, total area~$141$---as much smaller than \Fig{13}
-as \Fig{14} is larger. Slide the triangles one more
-place in the direction last used, and the apparent area
-\Figure{14}
-is~$139$. The explanation is of course similar to that
-given for \Fig{14}.
-
-This paradox also might be treated by an analysis
-\index{Escott, E.~B.}%
-resembling that by which Mr.~Escott has treated the
-preceding.
-
-Very similar is a puzzle due to S.~Loyd, ``the
-\index{Loyd, S.}%
-puzzlist.'' \Fig{A} is a square $8 × 8$, area~$64$. \Fig{B}
-shows the pieces rearranged in a rectangle apparently
-$7 × 9$, area~$63$.
-%[** TN: Moved figure up to associated text]
-\Figures{0.9}{A}{0.9}{B}
-\DPPageSep{123}{117}
-
-\Par{Paradromic rings.}\footnote
- {The theory of these rings is due to Listing, \Title{Topologie},
- part~10. See Ball's \Title{Recreations}, p.~75--6.}
-\index{Paradromic rings}%
-\index{Rings, paradromic}%
-\index{Surface with one face}%
-A puzzle of a very different
-sort is made as follows. Take a strip of paper, say
-half as wide and twice as long as this page; give one
-end a half turn and paste it to the other end. The ring
-thus formed is used in theory of functions to illustrate
-a surface that has only one face: a line can be drawn
-on the paper from any point of it to any other point
-of it, whether the two points were on the same side
-or on opposite sides of the strip from which the ring
-was made. The ring is to be slit---cut lengthwise all
-the way around, making the strip of half the present
-width. State in advance what will result. Try and
-see. Now predict the effect of a second and a third
-slitting.
-\DPPageSep{124}{118}
-
-
-\Chapter{Division of plane into regular polygons.}
-\index{Division of plane into regular polygons}%
-\index{Plane, division into regular polygons}%
-\index{Regular polygons, division of plane into}%
-
-The theorem seems to have been pleasing to the
-ancients, as it is to high-school pupils to-day, that a
-plane surface can be divided into equilateral triangles,
-squares, or regular hexagons, and that these are the
-\index{Hexagons!division of plane into}%
-only regular polygons into which the surface is divisible.
-As a regular hexagon is divided by its radii into
-\Figures{0.9}{15}{0.9}{16}
-six equilateral triangles, the division of the surface
-into triangles and hexagons gives the same arrangement
-(\Fig{16}).
-
-The hexagonal form of the bee's cell has long attracted
-\index{Bee's cell|(}%
-attention and admiration. The little worker
-could not have chosen a better form if he had had the
-advantage of a full course in Euclid! The hexagon is
-\index{Euclid}%
-best adapted to the purpose. It was discussed from a
-\DPPageSep{125}{119}
-mathematical point of view by Maclaurin in one of the
-\index{Maclaurin}%
-last papers he wrote.\footnote
- {In \Title{Philosophical Transactions} for~1743.}
-It has been pointed out\footnote
- {See for example E.~P. Evans's \Title{Evolutional Ethics and
- Animal Psychology}, p.~205.}
-that the hexagonal structure need not be attributed to mechanical
-instinct, but may be due solely to external pressure.
-(The cells of the human body, originally round,
-become hexagonal under pressure from morbid
-growth.)
-
-Agricultural journals are advising the planting of
-\index{Planting in hexagonal forms}%
-trees (as also corn~etc.)\ on the plan of the equilateral
-triangle instead of the square. Each tree is as far
-from its nearest neighbors in \Fig{16} as in \Fig{15}.
-The circles indicated in the corner of each figure
-represent the soil etc.\ on which each tree may be supposed
-to draw. The circles in \Fig{16} are as large
-as in \Fig{15} but there is not so much space lost between
-them. As the distance from row to row in
-\Fig{16} $=$ the altitude of one of the equilateral triangles
-$= \frac{1}{2}\sqrt{3} = 0.866$ of the distance between trees, it requires
-(beyond the first row) only $87$\%~as much
-ground to set out a given number of trees on this
-plan as is required to set them out on the plan of \Fig{15}.
-It may be predicted that, as land becomes scarce,
-\emph{pressure} will force the orchards, gardens and fields
-into a uniformly hexagonal arrangement.
-\index{Bee's cell|)}%
-\DPPageSep{126}{120}
-
-
-\Chapter{A homemade leveling device.}
-\index{Home-made leveling device}%
-\index{Leveling device}%
-
-The newspapers have been printing instructions for
-making a simple instrument useful in laying out the
-grades for ditches on a farm, or in \DPtypo{simlar}{similar} work in
-which a high degree of accuracy is not needed.
-
-Strips of thin board are nailed together, as shown
-in \Fig{17}, to form a triangle with equal vertical sides.
-The mid-point of the base is marked, and a plumb
-line is let fall from the opposite vertex. When the
-instrument is placed so that the line crosses the mark,
-\Figure[0.6]{17}
-the bar at the base is horizontal, being perpendicular to
-the plumb line. \emph{The median to the base of an isosceles
-triangle is perpendicular to the base.} From the lengths
-of the sides of the triangle it may be computed---or it
-may be found by trial---how far from the middle of the
-crossbar a mark must be placed so that when the plumb
-line crosses it the bar shall indicate a grade of $1$~in~$200$,
-$1$~in~$100$, etc\Add{.}
-\DPPageSep{127}{121}
-
-
-\Chapter{``Rope stretchers.''}
-\index{Pythagorean proposition}%
-\index{Rope stretchers}%
-
-If a rope $12$~units long be marked off into three segments
-of $3$,~$4$, and $5$~units, the end points brought together,
-and the rope stretched, the triangle thus formed
-is right-angled (\Fig{18}). This was used by the builders
-\Figure[0.6]{18}
-of the pyramids. The Egyptian word for surveyor
-means, literally, ``rope stretcher.'' Surveyors
-to this day use the same principle, counting off some
-multiple of these numbers in links of their chain.
-\DPPageSep{128}{122}
-
-
-\Chapter{The three famous problems of antiquity.}
-\index{Antiquity, three famous problems of}%
-\index{Circle-squaring|(}%
-\index{Delian problem|EtSeq}%
-\index{Duplication of cube\EtSeq}%
-\index{Quadrature of the circle|(}%
-\index{Squaring the circle|(}%
-\index{Trisection of angle|EtSeq}%
-\index{Problems!of antiquity}%
-\index{Three famous problems of antiquity}%
-
-1. To trisect an angle or arc.
-
-2. To ``duplicate the cube.''
-
-3. To ``square the circle.''
-
-The trisection of an angle is an ancient problem;
-``but tradition has not enshrined its origin in romance.''\footnote
- {Ball, \Title{Recreations}, p.~245.}
-The squaring of the circle is said to have
-been first attempted by Anaxagoras. The problem
-\index{Anaxagoras}%
-to duplicate the cube ``was known in ancient times as
-the Delian problem, in consequence of a legend that
-the Delians had consulted Plato on the subject. In
-\index{Plato}%
-one form of the story, which is related by Philoponus,
-\index{Philoponus}%
-it is asserted that the Athenians in 430~\BC, when
-suffering from the plague of eruptive typhoid fever,
-consulted the oracle at Delos as to how they could
-stop it. Apollo replied that they must double the size
-\index{Apollo}%
-of his altar which was in the form of a cube. To the unlearned
-suppliants nothing seemed more easy, and a new
-altar was constructed either having each of its edges
-double that of the old one (from which it followed
-that the volume was increased eightfold) or by placing
-a similar cubic altar next to the old one. Whereupon,
-according to the legend, the indignant god made
-the pestilence worse than before, and informed a
-fresh deputation that it was useless to trifle with him,
-as his new altar must be a cube and have a volume
-\DPPageSep{129}{123}
-exactly double that of his old one. Suspecting a mystery
-the Athenians applied to Plato, who referred them
-\index{Plato}%
-to the geometricians, and especially to Euclid, who had
-\index{Euclid}%
-made a special study of the problem.''\footnote
- {Ball, \Title{Hist.}, p.~43--4; nearly the same in his \Title{Recreations}, p.~239--240.}
-It is a hard-hearted
-historical criticism that would cast a doubt on
-a story inherently so credible as this on account of so
-trifling a circumstance as that Plato was not born till
-429~\BC\ and Euclid much later.
-
-Hippias of Elis invented the quadratrix for the trisection
-\index{Hippias of Elis}%
-\index{Quadratrix}%
-of an angle, and it was later used for the
-quadrature of the circle. Other Greeks devised other
-\index{Greeks}%
-curves to effect the construction required in (1)~and~(2).
-Eratosthenes and Nicomedes invented mechanical
-\index{Eratosthenes}%
-\index{Nicomedes}%
-instruments to draw such curves. But none of
-these curves can be constructed with ruler and compass
-alone. And this was the limitation imposed on
-the solution of the problems.
-
-Antiquity bequeathed to modern times all three problems
-unsolved. Modern mathematics, with its more
-efficient methods, has proved them all impossible of
-construction with ruler and compass alone---a result
-which the shrewdest investigator in antiquity could
-have only conjectured---has shown new ways of solving
-them if the limitation of ruler and compass be removed,
-and has devised and applied methods of approximation.
-It has \emph{dissolved} the problems, if that
-term may be permitted.
-
-It was not until 1882 that the transcendental nature
-\index{p@{$\pi$}|(}%
-of the number~$\pi$ was established (by Lindemann).
-\index{Lindemann}%
-The final results in all three of the problems, with
-mathematical demonstrations, are given in Klein's
-\Title{Famous Problems of Elementary Geometry}. A more
-\DPPageSep{130}{124}
-popular and elementary discussion is Rupert's \Title{Famous
-Geometrical Theorems and Problems}.
-
-It should be noted that the number~$\pi$, which the
-\index{Calculus of probability}%
-\index{Probability}%
-student first meets as the ratio of the circumference
-to the diameter of a circle, is a number that appears
-often in analysis in connections remote from elementary
-geometry; \eg,~in formulas in the calculus
-of probability.
-
-The value of~$\pi$ was computed to $707$~places of decimals
-by William Shanks. His result (communicated
-\index{Shanks, William}%
-in~1873) with a discussion of the formula he used
-(Machin's) may be found in the \Title{Proceedings of the
-Royal Society of London}, Vol.~21. No other problem
-of the sort has been worked out to such a degree of
-accuracy---``an accuracy exceeding the ratio of microscopic
-to telescopic distances.'' An illustration calculated
-to give some conception of the degree of accuracy
-attained may be found in Professor Schubert's
-\Title{Mathematical Essays and Recreations}, p.~140.
-
-Shanks was a computer. He stands in contrast to the
-circle-squarers, who expect to find a ``solution.'' Most
-of his computation serves, apparently, no useful purpose.
-But it should be a deterrent to those who---immune
-to the demonstration of Lindemann and others---still
-\index{Lindemann}%
-hope to find an exact ratio.
-
-The quadrature of the circle has been the most fascinating
-of mathematical problems. The ``army of
-circle-squarers'' has been recruited in each generation.
-``Their efforts remained as futile as though they had
-attempted to jump into a rainbow'' (Cajori); yet they
-\index{Cajori, Florian}%
-were undismayed. In some minds, the proof that no
-solution can be found seems only to have lent zest to
-the search.
-\DPPageSep{131}{125}
-
-That these problems are of perennial interest, is attested
-by the fact that contributions to them still appear.
-In 1905 a little book was published in Los
-Angeles entitled \Title{The Secret of the Circle and the
-Square}, in which also the division of ``any angle into
-any number of equal angles'' is considered. The
-author, J.~C. Willmon, gives original methods of approximation.
-\Title{School Science and Mathematics} for
-May~1906 contains a ``solution'' of the trisection problem
-by a high-school boy in Missouri, printed, apparently,
-to show that the problem still has fascination
-for the youthful mind. In a later number of that
-magazine the problem is discussed by another from
-the vantage ground of higher mathematics.
-
-While the three problems have all been proved to
-be insolvable under the condition imposed, still the
-attempts made through many centuries to find a solution
-have led to much more valuable results, not only
-by quickening interest in mathematical questions, but
-especially by the many and important discoveries that
-have been made in the effort. The voyagers were unable
-to find the northwest passage, and one can easily
-see now that the search was \emph{necessarily} futile; but in the
-attempt they discovered continents whose resources,
-when developed, make the wealth of the Indies seem
-poor indeed.
-\DPPageSep{132}{126}
-
-
-\Chapter{The circle-squarer's paradox.}
-\index{Circle-squarer's paradox}%
-\index{Calculus of probability|(}%
-\index{Demorgan@{De Morgan}|(}%
-\index{Paradox, circle-squarer's}%
-\index{Probability|(}%
-
-Professor De~Morgan, in his \Title{\DPtypo{Buaget}{Budget} of Paradoxes}
-(London, 1872) gave circle-squarers the honor of
-more extended individual notice and more complete
-refutation than is often accorded them. The Budget
-%[ ** TN: [sic] "instalments" with one l}
-first appeared in instalments in the \Title{Athenĉum}, where
-it attracted the correspondence and would-be contributions
-of all the circle-squarers, and the like amateurs,
-of the day. His facetious treatment of them
-drew forth their severest criticisms, which in turn
-gave most interesting material for the Budget. He
-says he means that the coming New Zealander shall
-know how the present generation regards circle-squarers.
-Theirs is one of the most amusing of the
-many paradoxes of which he wrote. The book is out
-of print, and so rare that the following quotations from
-it may be acceptable:
-
-``Mere pitch-and-toss has given a more accurate
-approach to the quadrature of the circle than has been
-reached by some of my paradoxers\dots\Add{.} The method is
-as follows: Suppose a planked floor of the usual kind,
-with thin visible seams between the planks. Let there
-be a thin straight rod, or wire, not so long as the
-breadth of the plank. This rod, being tossed up at
-hazard, will either fall quite clear of the seams, or
-will lay across one seam. Now Buffon, and after him
-\index{Buffon}%
-Laplace, proved the following: That in the long run
-\index{Laplace}%
-the fraction of the whole number of trials in which a
-\DPPageSep{133}{127}
-seam is intersected will be the fraction which twice
-the length of the rod is of the circumference of the
-circle having the breadth of a plank for its diameter.
-In 1855 Mr.\ Ambrose Smith, of Aberdeen, made
-\index{Smith, Ambrose}%
-$3,204$~trials with a rod three-fifths of the distance
-between the planks: there were $1,213$~clear intersections,
-and $11$~contacts on which it was difficult to
-decide. Divide these contacts equally~\dots\ this gives
-$\pi = 3.1553$. A pupil of mine made $600$~trials with a
-rod of the length between the seams, and got $\pi = 3.137$.''
-(P.~170--1.)\footnote
- {Ball, in his \Title{Mathematical Recreations and Essays} (p.~261,
- citing the \Title{Messenger of Mathematics}, Cambridge, 1873,~2:
- 113--4) adds that ``in~1864 Captain Fox made $1120$~trials with
-\index{Fox, Captain|FN}%
- some additional precautions, and obtained as the mean value
- $\pi = 3.1419$.''}
-
-``The celebrated interminable fraction $3.14159\dots$,
-which the mathematician calls~$\pi$, is the ratio of the
-circumference to the diameter. But it is thousands of
-things besides. It is constantly turning up in mathematics:
-and if arithmetic and algebra had been studied
-without geometry, $\pi$~must have come in somehow,
-though at what stage or under what name must have
-depended upon the casualties of algebraical invention.
-This will readily be seen when it is stated that $\pi$~is
-nothing but four times the series
-\[
-1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \dots
-\]
-\Foreign{ad infinitum}. It would be wonderful if so simple a
-series had but one kind of occurrence. As it is, our
-trigonometry being founded on the circle, $\pi$~first appears
-as the ratio stated. If, for instance, a deep
-study of probable fluctuation from the average had
-preceded geometry, $\pi$~might have emerged as a number
-perfectly indispensable in such problems as---What is
-\DPPageSep{134}{128}
-the chance of the number of aces lying between a
-$\text{million} + x$ and a $\text{million} - x$, when six million of
-throws are made with a die?'' (P.~171.)
-
-``More than thirty years ago I had a friend~\dots\ who
-was~\dots\ thoroughly up in all that relates to mortality,
-life assurance,~etc. One day, explaining to him how it
-should be ascertained what the chance is of the survivors
-of a large number of persons now alive lying
-between given limits of number at the end of a certain
-time, I came, of course, upon the introduction of~$\pi$,
-which I could only describe as the ratio of the circumference
-of a circle to its diameter. `Oh, my dear
-friend! that must be a delusion; what can the circle
-have to do with the numbers alive at the end of a given
-time?'---`I cannot demonstrate it to you; but it is
-demonstrated.'\,'' (P.~172.)
-\index{Calculus of probability|)}%
-\index{Probability|)}%
-
-``The feeling which tempts persons to this problem
-[exact quadrature] is that which, in romance, made it
-impossible for a knight to pass a castle which belonged
-to a giant or an enchanter. I once gave a lecture on
-the subject: a gentleman who was introduced to it by
-what I said remarked, loud enough to be heard all
-around, `Only prove to me that it is impossible, and I
-will set about it this very evening.'
-
-``This rinderpest of geometry cannot be cured, when
-once it has seated itself in the system: all that can be
-done is to apply what the learned call prophylactics to
-those who are yet sound.'' (P.~390.)
-
-``The finding of two mean proportionals is the preliminary
-to the famous old problem of the duplication
-of the cube, proposed by Apollo (not Apollonius)
-\index{Apollo}%
-himself. D'Israeli speaks of the `six follies of science,'---the
-\index{Disraeli@{D'Israeli}}%
-quadrature, the duplication, the perpetual motion,
-the philosopher's stone, magic, and astrology.
-\DPPageSep{135}{129}
-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 till they are over,'---may
-the Curiosities of Literature outlive the Follies
-of Science!'' (P.~71.)
-\index{Circle-squaring|)}%
-\index{Demorgan@{De Morgan}|)}%
-\index{p@{$\pi$}|)}%
-\index{Quadrature of the circle|)}%
-\index{Squaring the circle|)}%
-\DPPageSep{136}{130}
-
-
-\Chapter{The instruments that are postulated.}
-\index{Instruments that are postulated}%
-\index{Ruler unlimited and ungraduated|(}%
-\index{Trisection of angle|(}%
-
-The use of two instruments is allowed in theoretic
-elementary geometry, the ruler and the compass---a
-limitation said to be due to Plato.
-\index{Plato}%
-
-It is understood that the compass is to be of unlimited
-opening. For if the compass would not open
-as far as we please, it could not be used to effect the
-construction demanded in Euclid's third postulate, the
-\index{Euclid}%
-drawing of a circle with any center and \emph{any} radius.
-Similarly, it is understood that the ruler is of unlimited
-length for the use of the second postulate.
-
-Also that the ruler is \emph{ungraduated}. If there were
-even \emph{two} marks on the straight-edge and we were
-\index{Straight-edge|(}%
-allowed to use these and move the ruler \emph{so as to fit} a
-figure, the problem to trisect an angle (impossible
-to elementary geometry) could be readily solved, as
-follows:
-
-Let $ABC$ be the angle, and $P$,~$Q$ the two points
-on the straight-edge. (\Fig{19}.)
-
-On one arm of angle~$B$ lay off $BA = PQ$. Bisect~$BA$,
-at~$M$.
-
-Draw $MK \parallel BC$, and $ML \perp BC$.
-
-Adjust the straight-edge to fit the figure so that
-$P$~lies on~$MK$, $Q$~on~$ML$, and at the same time the
-straight-edge passes through~$B$. Then $BP$~trisects
-the angle.
-\DPPageSep{137}{131}
-
-\textit{Proof.} $\angle PBC = \text{its alternate } \angle BPM$.
-
-Mark $N$ the mid-point of~$PQ$, and draw~$NM$. Then
-$N$, the mid-point of the hypotenuse of the rt.~$\triangle\ PQM$,
-is equidistant from the vertexes of the triangle.
-\begin{align*}
-\therefore \angle BPM &= \angle PMN\Add{.} \\
-\text{Exterior } \angle BNM
- &= \angle BPM + \angle PMN \\
- &= 2\angle BPM\Add{.}
-\end{align*}
-\Figure[0.9]{19}
-\begin{align*}
-\because NM &= \tfrac{1}{2}PQ = BM \\
-\therefore \angle MBN &= \angle BNM \\
-\angle PBC = \angle BPM &= \tfrac{1}{2}\angle BNM
- = \tfrac{1}{2}\angle ABN = \tfrac{1}{3}\angle ABC.
-\end{align*}
-\DPPageSep{138}{132}
-
-A.~B. Kempe,\footnote
- {\Title{How to Draw a Straight Line}, note~(2).}
-\index{Kempe, A.~B.}%
-from whom this form of the well-known
-solution is adapted, raises the question whether
-Euclid does not use a graduated ruler and the fitting
-process when, in book~1, proposition~4, he fits side~$AB$
-of triangle~$AB\Gamma$ to side $\Delta E$ of triangle $\Delta EZ$---the
-first proof by superposition, with which every high-school
-pupil is familiar. It may be replied that Euclid
-does not determine a point (as $P$~is found in the
-angle above) by fitting and measuring. He superposes
-only in his reasoning, in his proof.
-
-Our straight-edge must be ungraduated, or it grants
-us too much; it must be unlimited or it grants us too
-little.
-\index{Ruler unlimited and ungraduated|)}%
-\index{Straight-edge|)}%
-\index{Trisection of angle|)}%
-\DPPageSep{139}{133}
-
-
-\Chapter{The triangle and its circles.}
-\index{Centers of triangle}%
-\index{Circles of triangle}%
-\index{Collinearity of centers of triangle}%
-\index{Triangle and its circles}%
-
-The following statement of notation and familiar
-definitions may be permitted:
-
-$O$, \emph{orthocenter} of the triangle~$ABD$, the point of
-concurrence of the three altitudes of the triangle.
-
-$G$, center of \emph{gravity}, center of mass, or centroid, of
-the triangle, the point of concurrence of the three
-medians.
-
-$C$, \emph{circumcenter} of the triangle, center of the circumscribed
-circle, point of concurrence of the perpendicular
-bisectors of the sides of the triangle.
-
-$I$, \emph{in-center} of the triangle, center of the inscribed
-circle, point of concurrence of the bisectors of the
-three interior angles of the triangle.
-
-$E$, $E$, $E$, \emph{ex-centers}, centers of the escribed circles,
-each~$E$ the point of concurrence of the bisectors of
-two exterior angles of the triangle and one interior
-angle.
-
-An obtuse angled triangle is used in the figure so
-that the centers may be farther apart and the figure
-less crowded.
-
-\Par{Collinearity of centers.} $O$,~$G$, and~$C$ are collinear,
-and $OG = \text{twice } GC$.
-
-Corollary: The distance from $O$ to a vertex of the
-triangle is twice the distance from $C$ to the side opposite
-that vertex.\footnote
- {Or this corollary may easily be proved independently and
- the proposition that $O$,~$G$\Add{,} and~$C$ are in a straight line of which
- $G$~is a trisection point be derived from it, as the writer once did
- when unacquainted with the results that had been achieved
- in this field.}
-\DPPageSep{140}{134}
-
-\Par{The nine-point circle.} Let $L$,~$M$,~$N$ be the mid-points
-\index{Nine-point circle|(}%
-of the sides; $A'$,~$B'$,~$D'$, the projections of the
-\Figure{20}
-vertexes on the opposite sides; $H$,~$J$,~$K$, the mid-points
-of $OA$,~$OB$,~$OD$, respectively. Then these nine points
-\DPPageSep{141}{135}
-are concyclic; and the circle through them is called
-the nine-point circle of the triangle (\Fig{20}).
-
-The center of the nine-point circle is the mid-point
-of~$OC$, and its radius is half the radius of the circumscribed
-circle.
-
-The discovery of the nine-point circle has been erroneously
-attributed to Euler. Several investigators
-\index{Euler}%
-discovered it independently in the early part of the
-nineteenth century. The name \emph{nine-point circle} is said
-to be due to Terquem (1842) editor of \Title{Nouvelles Annales}.
-\index{Terquem}%
-Karl Wilhelm Feuerbach proved, in a pamphlet
-of 1822, what is now known as ``Feuerbach's theorem'':
-\index{Feuerbach's theorem}%
-The nine-point circle of a triangle is tangent to the
-\index{Nine-point circle|)}%
-inscribed circle and each of the escribed circles of the
-triangle.
-
-So many beautiful theorems about the triangle have
-been proved that Crelle---himself one of the foremost
-\index{Crelle}%
-investigators of it---exclaimed: ``It is indeed wonderful
-that so simple a figure as the triangle is so inexhaustible
-in properties. How many as yet unknown properties
-of other figures may there not be!''
-
-The reader is referred to Cajori's \Title{History of Elementary
-Mathematics} and the treatises on this subject
-mentioned in his note, p.~259, and to the delightful
-monograph, \Title{Some Noteworthy Properties of the Triangle
-and Its Circles}, by W.~H. Bruce, president of
-the North Texas State Normal School, Denton. Many
-of Dr.~Bruce's proofs and some of his theorems are
-original.
-\DPPageSep{142}{136}
-
-
-\Chapter{Linkages and straight-line motion.}
-\index{Kempe, A.~B.}%
-\index{Straight-edge}%
-\index{Linkages and straight-line motion}%
-\index{Straight-line motion}%
-
-Under the title \Title{How to Draw a Straight Line},
-A.~B. Kempe wrote a little book which is full of theoretic
-interest to the geometer, as it touches one of the foundation
-postulates of the science.
-
-We occasionally run a pencil around a coin to draw
-a circumference, thus using one circle to produce another.
-But this is only a makeshift: we have an instrument,
-not itself circular, with which to draw a
-circle---the compass. Now, when we come to draw
-a straight line we say that that postulate grants us the
-\Figures{0.7}{21}{1.0}{22}
-use of a ruler. But this is demanding a straight edge
-for drawing a straight line---given a straight line to
-copy. Is it possible to construct an instrument, not
-itself straight, which shall draw a straight line? Such
-an instrument was first invented by Peaucellier, a
-\index{Peaucellier|(}%
-French army officer in the engineer corps. It is a
-``linkage.'' Since that time (1864) other linkages
-have been invented to effect rectilinear motion, some
-\index{Rectilinear motion|(}%
-of them simpler than Peaucellier's. But as his is
-earliest, it may be taken as the type.
-
-Preliminary to its construction, however, let us consider
-\DPPageSep{143}{137}
-a single link (\Fig{21}) pivoted at one end and
-carrying a pencil at the other. The pencil describes
-a circumference. If two links (\Fig{22}) be hinged
-at~$H$, and point~$F$ fastened to the plane, point~$P$ is
-\Figures{0.7}{23}{0.9}{24}
-\Figures{0.9}{25}{0.9}{26}
-free to move in any direction; its path is indeterminate.
-The number of links must be odd to give determinate
-motion. If a system of three links be
-\DPPageSep{144}{138}
-fastened at both ends, a point in the middle link describes
-a definite curve---say a loop. Five links can
-give the requisite straight-line motion; but Peaucellier's
-was a seven-link apparatus.
-
-Such a linkage can be made by any teacher. The
-writer once made a small one of links cut out of cardboard
-and fastened together by shoemaker's eyelets;
-also a larger one (about $30$~times the size of \Fig{23})
-of thin boards joined with bolts. $F$~and~$O$ (\Fig{23})
-were made to fasten in mouldings above the blackboard,
-and $P$~carried a piece of crayon. This proved
-very interesting to a geometry class for a lecture. It
-is needless to say that no one would think of any such
-appliance for daily class-room use. The ruler is the
-practical instrument.
-
-\Fig{24} is a diagram of the apparatus shown in
-\Fig{23}. $FA = FB$. In all positions $APBC$~is a rhombus.
-$F$~and~$O$ are fastened at points whose distance
-apart is equal to~$OC$. Then $C$~moves in an arc of a
-circle whose center is~$O$; $A$~and~$B$ move in an arc
-with center at~$F$. It is to be shown that $P$~moves in a
-straight line.
-
-Draw $PP' \perp FO$ produced.\footnote
- {\emph{Imagine} these lines drawn, if one objects to drawing a
- straight line as one step in the process of showing that a
- straight line can be drawn!}
-$FCC'$,~being inscribed
-in a semicircle, is a right angle. Hence $\triangle$s $FP'P$ and
-$FC'C$, having $\angle F$~in common, are similar, and
-\begin{align*}
-FP : FP' &= FC' : FC \\
-FP · FC &= FP' · FC'\Add{.}
-\Tag{(1)}
-\end{align*}
-
-$F$,~$C$\Add{,} and~$P$, being each equidistant from $A$ and~$B$,
-lie in the same straight line; and the diagonals of the
-rhombus~$APBC$ are perpendicular bisectors of each
-other. Hence
-\DPPageSep{145}{139}
-\begin{align*}
-FB^{2} &= FM^{2} + MB^{2} \\
-PB^{2} &= MP^{2} + MB^{2} \Brk
-\therefore
-FB^{2} - PB^{2} &= FM^{2} - MP^{2} \\
- &= (FM + MP)(FM - MP) \\
- &= FP · FC
-\Tag{(2)}
-\end{align*}
-
-From \Eq{(1)} and \Eq{(2)}, $FP' · FC' = FB^{2} - PB^{2}$.
-
-But as the linkage moves, $FC'$,~$FB$\Add{,} and~$PB$ all remain
-constant; therefore $FP'$~is constant. That is,
-$P'$,~the projection of~$P$ on~$FO$, is always the same
-point; or in other words, $P$~moves in a \emph{straight line}
-(perpendicular to~$FO$).
-
-If the distance between the two fixed points, $F$~and~$O$,
-be made less than the length of the link~$OC$, $P$~moves
-in an arc of a circle with concave toward~$O$
-(\Fig{25}). As $OC - OF$ approaches zero as a limit,
-the radius of the arc traced by~$P$ increases without
-limit.
-
-Then as would be expected, if $OF$~be made greater
-than~$OC$, $P$~traces an arc that is convex toward~$O$
-(\Fig{26}). The smaller $OF - OC$, the longer the
-radius of the arc traced by~$P$. It is curious that so
-small an instrument may be used to describe an arc
-of a circle with enormous radius and with center on
-the opposite side of the arc from the instrument.
-
-The straight line---the ``simplest curve'' of mathe\-ma\-ti\-cians---lies
-between these two arcs, and is the
-limiting form of each.
-
-Linkages possess many interesting properties. The
-subject was first presented to English-speaking students
-by the late Professor Sylvester. Mr.~Kempe
-\index{Kempe, A.~B.}%
-\index{Sylvester, J. J.}%
-showed ``that a link-motion can be found to describe
-any given algebraic curve.''
-\index{Peaucellier|)}%
-\index{Rectilinear motion|)}%
-\DPPageSep{146}{140}
-
-
-\Chapter{The four-colors theorem.}
-\index{Four-colors theorem}%
-\index{Colors in map drawing}%
-\index{Map makers' proposition}%
-
-This theorem, known also as the map makers'
-proposition, has become celebrated. It is, that four
-colors are sufficient for any map, no two districts having
-a common boundary line to be colored the same;
-and this no matter how numerous the districts, how
-irregular their boundaries or how complicated their
-arrangement.
-
-That four colors may be necessary can be seen from
-\Fig{27}. A few trials will convince
-\Figure[0.4]{27}
-most persons that it is probably
-impossible to draw a map requiring
-more than four. To give
-a mathematical proof of it, is quite
-another matter.
-
-The proposition is said to have
-been long known to map makers. It was mentioned
-as a mathematical proposition by A.~F. Möbius, in~1840,
-\index{Mobius@{Möbius, A. F.}}%
-and later popularized by De~Morgan. All that
-\index{Demorgan@{De Morgan}}%
-is needed to give a proposition celebrity is to proclaim
-it one of the unsolved problems of the science. Cayley's
-\index{Cayley}%
-remark, in~1878, that this one had remained
-unproved was followed by at least two published demonstrations
-within two or three years. But each had a
-flaw. The chance is still open for some one to invent
-a new method of attack.
-
-If the proposition were not true, it could be disproved
-by a single special case, by producing a ``map''
-\DPPageSep{147}{141}
-with five districts of which each bounds every other.
-Many have tried to do this.
-
-It has been shown that there are surfaces on which
-the proposition would not hold true. The theorem
-refers to a plane or the surface of a globe.
-
-For historical presentation and bibliographic notes,
-see Ball's \Title{Recreations}, pp.~51--3; or for a more extended
-discussion, Lucas, IV,~168 \Foreign{et~seq}.
-\DPPageSep{148}{142}
-
-
-\Chapter{Parallelogram of forces.}
-\index{Forces, parallelogram of}%
-\index{Parallelogram of forces}%
-
-One of the best-known principles of physics is, that
-if a ball,~$B$, is struck a blow which if acting alone
-would drive the ball to~$A$, and a blow which alone
-would drive it to~$C$, and both blows are delivered at
-once, the ball takes the direction~$BD$, the diagonal of
-\Figure[0.5]{28}
-the parallelogram of $BA$
-and~$BC$, and the force is
-just sufficient to drive the
-ball to~$D$. $BD$~is the \emph{resultant}
-of the two forces.
-
-If a third force, represented
-by some line~$BE$,
-operates simultaneously with those represented by $BA$
-and~$BC$, then the diagonal of the parallelogram of $BD$
-and~$BE$ is the resultant of the three forces. And
-so on.
-
-Hence the resultant of forces is always less than
-the sum of the forces unless the forces act in the same
-direction. The more nearly their lines of action approach
-each other, the more nearly does their resultant
-approach their sum.
-
-One is tempted to draw the moral, that social forces
-have a resultant and obey an analogous law, the result
-of all the educational or other social energy expended
-on a child, or in a community, being less than the
-sum, unless all forces act in the same line.
-\DPPageSep{149}{143}
-
-
-\Chapter{A question of fourth dimension by
-analogy.}
-\index{Dimension!fourth}%
-\index{Fourth dimension}%
-\index{Question of fourth dimension by analogy}%
-
-After class one day a normal-school pupil asked the
-writer the following question, and received the following
-reply:
-
-\textit{Q\@.} If the path of a moving point (no dimension)
-is a line (one dimension), and the path of a moving
-line is a surface (two dimensions), and the path of
-a moving surface is a solid (three dimensions), why
-isn't the path of a moving solid a four-dimensional
-magnitude?
-
-\textit{A\@.} If your hypotheses were correct, your conclusion
-should follow by analogy. The path of a moving point
-is, indeed, always a line. The path of a moving line
-is a surface \emph{except} when the line moves in its own
-dimension, ``slides in its trace.'' The path of a moving
-surface is a solid only when the motion is in a third
-dimension. The generation of a four-dimensional
-magnitude by the motion of a solid presupposes that
-the solid is to be moved in a fourth dimension.
-\DPPageSep{150}{144}
-
-
-\Chapter{Symmetry illustrated by paper
-folding.}
-\index{Symmetry illustrated by paper folding}%
-
-The following simple device has been found by the
-writer to give pupils an idea of symmetry with a
-certainty and directness which no verbal explanation
-unaided can approach. Require each pupil to take
-a piece of calendered or sized paper, fold and crease
-it once, straighten it out again, draw rapidly with ink
-any figure on one half of the paper, and fold together
-while the ink is still damp. The original drawing and
-the trace on the other half of the paper are symmetric
-with respect to the crease as an axis. Again: Fold
-a paper in two perpendicular creases. In one quadrant
-draw a figure whose two end points lie one in each
-crease. Quickly fold so as to make a trace in each
-of the other quadrants. A closed figure is formed
-which is symmetric with respect to the intersection
-of the creases as center.
-
-T.~Sundara Row, in his \Title{Geometric Exercises in
-Paper Folding} (edited and revised by Beman and
-\index{Paper folding}%
-Smith),\footnote
- {Chicago, The Open Court Publishing Co.}
-has shown how to make many of the constructions
-of plane geometry by paper folding, including
-beautiful illustrations of some of the regular
-\index{Illustrations!of symmetry}%
-polygons and the locating of points on some of the
-higher plane curves.
-\DPPageSep{151}{145}
-
-Illustrations of symmetry by the use of the mirror
-are well brought out in a brief article recently published
-in \Title{American Education}.\footnote
- {Number for March 1907, p.~464--5, article ``Symmetrical
-\index{Lathrop, H.~J.|FN}%
- Plane Figures,'' by Henry~J. Lathrop.}
-\index{Geometry|)}%
-\DPPageSep{152}{146}
-
-
-\Chapter[Line values of trigonometric functions.]
-{Apparatus to illustrate line values
-of trigonometric functions.}
-\index{Line values of trigonometric functions}%
-\index{Apparatus to illustrate line values of trigonometric functions}%
-\index{Illustrations!of trigonometric functions}%
-\index{Trigonometry|(}%
-
-A piece of apparatus to illustrate trigonometric
-lines representing the trigonometric ratios may be
-constructed somewhat as follows (\Fig{29}):
-
-To the center~$O$ of a disc is attached a rod~$OR$,
-which may be revolved. A tangent rod is screwed
-\Figure[0.6]{29}
-to the disc at~$A$. Along this a little block bearing the
-letter~$T$ is made to slide easily. The block is also
-connected to the rod~$OR$, so that $T$~marks the intersection
-of the two lines. Similarly a block~$R$ is moved
-along the tangent rod~$BR$. At~$P$, a unit's distance
-from~$O$ on the rod~$OR$, another rod~($PM$) is pivoted.
-\DPPageSep{153}{147}
-A weight at the lower end keeps the rod in a vertical
-position. It passes through a block which is made to
-slide freely along~$OA$ and which bears the letter~$M$.
-
-As the rod~$OR$ is revolved in the positive direction,
-increasing the angle~$O$, $MP$~represents the increasing
-sine, $OM$~the decreasing cosine, $AT$~the increasing
-tangent, $BR$~the decreasing cotangent, $OT$~the increasing
-secant, $OR$~the decreasing cosecant.
-\DPPageSep{154}{148}
-
-
-\Chapter{``Sine.''}
-\index{Al Battani}%
-\index{Hindu!word for sine}%
-\index{Sine, history of the word}%
-
-Students in trigonometry sometimes say: ``From
-\index{Trigonometry|)}%
-the line value, or geometric representation, of the
-trigonometric ratios it is easy to see why the tangent
-and secant were so named. And the co-functions are
-the functions of the complementary angles. But what
-is the origin of the name \emph{sine}?'' It is a good question.
-The following answer is that of Cantor, Fink,
-and Cajori; but Cantor deems it doubtful.
-
-The Greeks used the entire chord of double the arc.
-\index{Greeks}%
-The Hindus, though employing half the chord of
-double the arc (what we call \emph{sine} in a unit circle),
-used for it their former name for the entire chord,
-\Foreign{jîva}, which meant literally ``bow-string,'' a natural
-designation for chord. Their work came to us through
-the Arabs, who transliterated the Sanskrit \Foreign{jîva} into
-Arabic \Foreign{dschiba}. Arabic being usually written in ``unpointed
-\index{Arabic!word for sine}%
-text'' (without vowels) like a modern stenographer's
-notes, \Foreign{dschiba} having no meaning in Arabic,
-and the Arabic word \Foreign{dschaib} having the same
-consonants, it was easy for the latter to take the place
-of the former. But \Foreign{dschaib} means ``bosom.'' Al~Battani,
-the foremost astronomer of the ninth century,
-wrote a book on the motion of the heavenly bodies.
-In the twelfth century this was translated into Latin
-by Plato Tiburtinus, who rendered the Arabic word
-\index{Plato Tiburtinus}%
-by the Latin \Foreign{sinus} (bosom). And \Foreign{sinus}, Anglicized, is
-``sine.''
-\DPPageSep{155}{149}
-
-
-\Chapter{Growth of the philosophy of the
-calculus.}
-\index{Calculus|(}%
-\index{Growth of philosophy of the calculus}%
-\index{Philosophy of the calculus}%
-
-The latter half of the seventeenth century produced
-\index{Leibnitz|(}%
-\index{Newton|(}%
-that powerful instrument of mathematical research,
-the differential calculus.\footnote
- {Newton and Leibnitz invented it in the sense that they
- brought it to comparative perfection as an instrument of research.
- Like most epoch-making discoveries it had been foreshadowed.
- Cavalieri, Kepler, Fermat and many others had
-\index{Cavalieri|FN}%
-\index{Fermat|FN}%
-\index{Kepler|FN}%
- been working toward it. One must go a long way back into
- the history of mathematics to find a time when there was no
- suggestion of it. As this note is penned the newspapers bring
- a report that Mr.~Hiberg, a Danish scientist, says he has recently
-\index{Hiberg|FN}%
- discovered in a palimpsest in Constantinople, a hitherto
- unknown work on mathematics by Archimedes. ``The manuscript,
-\index{Archimedes|FN}%
-\index{Eratosthenes|FN}%
-\index{Greeks|FN}%
- which is entitled `On Method,' is dedicated to Eratosthenes,
- and relates to the applying of mechanics to the solution
- of certain problems in geometry. There is in this ancient
- Greek manuscript a method that bears a strong resemblance to
- the integral calculus of modern days, and is capable of being
- used for the solution of problems reserved for the genius of
- Leibnitz and Newton eighteen centuries later.'' (N.~Y. Tribune.)}
-The master minds that invented
-it, Newton and Leibnitz, failed to clear the
-subject of philosophical difficulties.
-
-Newton's reasoning is based on this initial theorem
-in the \Title{Principia}: ``Quantities, and the ratios of quantities,
-that during any finite time constantly approach
-each other, and before the end of that time approach
-nearer than any given difference, are ultimately
-equal.'' It is not surprising that neither this statement
-nor its demonstration gave universal satisfaction.
-The ``zeros'' whose ratio was considered in the
-\DPPageSep{156}{150}
-\index{Berkeley, George}%
-method of fluxions were characterized by the astute
-Bishop Berkeley as ``ghosts of departed quantities.''
-
-Leibnitz based his calculus on the principle that
-one may substitute for any magnitude another which
-differs from it only by a quantity infinitely small.
-This is assumed as ``a sort of axiom.'' Pressed for an
-explanation, he said that, in comparison with finite
-quantities, he treated infinitely small quantities as
-\emph{incomparables}, negligible ``like grains of sand in comparison
-with the sea.'' This, if consistently held,
-should have made the calculus a mere method of approximation.
-
-According to the explanations of both, strictly applied,
-the calculus should have produced results that
-were close approximations. But instead, its results
-were absolutely accurate. Berkeley first, and afterward
-L.~N.~M. Carnot, pointed out that this was due
-to compensation of errors. This phase of the subject
-is perhaps nowhere treated in a more piquant style
-than in Bledsoe's \Title{Philosophy of Mathematics}.
-
-The method of limits permits a rigor of demonstration
-not possible to the pure infinitesimalists. Logically
-the methods of the latter are to be regarded as
-abridgments. As treated by the best writers the calculus
-is to-day on a sound philosophical basis. It is
-admirable for its logic as well as for its marvelous
-efficiency.
-
-But many writers are so dominated by the thinking
-\index{Zero!meaning of symbol|(}%
-of the past that they still use the symbol~$0$ to mean
-sometimes ``an infinitely small quantity'' and sometimes
-absolute zero. Clearer thinking impels to the
-use of~$\iota$ (iota) or~$i$ or some other symbol to mean an
-infinitesimal, denoting by~$0$ only zero.
-\index{Leibnitz|)}%
-\index{Newton|)}%
-
-This distinction implies that between their reciprocals.
-\DPPageSep{157}{151}
-The symbol~$\infty$, first used for an infinite by
-\index{Infinite!symbols for}%
-\index{Symbols!for infinite}%
-Wallis in the seventeenth century, has long been used
-\index{Wallis}%
-both for a variable increasing without limit and for
-absolute infinity. The revised edition of Taylor's
-%[** TN: Hack to approximate one-off symbol]
-Calculus (Ginn 1898) introduced a new symbol~$o\kern-2pt\varphi$,
-a contraction of~$\SlantFrac{a}{0}$, for absolute infinity, using $\infty$
-only for an infinite (the reciprocal of an infinitesimal).
-It is to be hoped that this usage will become universal.
-
-In the book just referred to is perhaps the clearest
-and most concise statement to be found anywhere of
-the inverse problems of the differential calculus and
-the integral calculus, as well as of the three methods
-used in the calculus.
-\index{Zero!meaning of symbol|)}%
-\DPPageSep{158}{152}
-
-
-\Chapter{Some illustrations of limits.}
-\index{Constants and variables illustrated|(}%
-\index{Limits illustrated}%
-\index{Variables illustrated|(}%
-\index{Illustrations!of limits}%
-
-Physical illustrations of variables are numerous. But
-to find a similar case of a constant, is not easy. The
-long history of the determination of standards (yard,
-meter etc.)\ is the history of a search for physical constants.
-Constants are the result of abstraction or are
-limited by definition. Non-physical constants are numerous,
-and enter into most problems.
-
-If one person is just a year older than another, the
-ratio of the age of the younger to that of the older, at
-successive birthdays, is $\dfrac{0}{1}$, $\dfrac{1}{2}$, $\dfrac{2}{3}$, $\dfrac{3}{4}$, $\dfrac{4}{5}$~\dots\ $\dfrac{49}{50}$, $\dfrac{50}{51}$~\dots.
-In general: the ratio of the ages of any two persons
-is a variable approaching unity as limit. The sum of
-of their ages is a variable increasing without limit.
-The difference between their ages is a constant.
-\Figure[1.0]{30}
-
-When pupils have the idea of the time-honored
-point~$P$ which moves half way from $A$ to~$B$ the first
-second, half the remaining distance the next second,
-etc., but have trouble with the product of a constant
-and a variable, they have sometimes been helped by the
-following ``optical illustration'': Imagine yourself looking
-at \Fig{30} through a glass that makes everything
-look twice as large as it appears to the naked eye.
-\DPPageSep{159}{153}
-$AP$~still seems to approach~$AB$ as limit; that is, twice
-the ``real''~$AP$ is approaching twice the ``real''~$AB$ as
-limit. Now suppose your glass magnifies $3$~times,
-$n$~times. $AP$~still approaches $AB$ magnified the same
-number of times. That is, if $AP \doteq AB$, then $\text{any
-constant} × AP \doteq \text{that constant} × AB$.
-
-Reverse the glass, making $AP$ look one-$n$th part as
-large as at first. It approaches one-$n$th of the ``real''~$AB$.
-Putting this in symbols, with $x$~representing
-the variable, and $c$~the constant,
-\[
-%[** TN: Italicized "lt" in theoriginal]
-\lim \left(\frac{x}{c}\right) = \frac{\lim x}{c}\Add{.}
-\]
-Or in words: The limit of the ratio of a variable to a
-constant is the ratio of
-the limit of the variable
-to the constant.
-\Figure[0.5]{31}
-
-Let $x$ represent the
-broken line from $A$ to~$C$
-(\Fig{31}), composed
-first of $4$~parts, then of~$8$,
-then of~$16$ (the last
-division shown in the
-figure) then of~$32$, etc.
-The polygon bounded
-by $x$,~$AB$ and $BC \doteq
-\triangle ABC$. What of the
-length of~$x$? Most persons
-to whom this old figure is new answer off-hand,
-``$x \doteq AC$.'' But a minute's reflection shows that $x$~is
-constant and $= AB + BC$.
-\index{Calculus|)}%
-\index{Constants and variables illustrated|)}%
-\index{Variables illustrated|)}%
-\DPPageSep{160}{154}
-
-
-\Chapter{Law of commutation.}
-\index{Commutative law}%
-\index{Involution not commutative}%
-\index{Law of signs!of \DPtypo{commuation}{commutation}}%
-
-This law, emphasized for arithmetic in McLellan
-and Dewey's \Title{Psychology of Number}, and explicitly
-employed in all algebras that give attention to the
-logical side of the subject, is one whose importance
-is often overlooked. So long as it is used implicitly
-and regarded as of universal application, its import
-is neglected. An antidote: to remember that there are
-regions in which this law does not apply. \Eg:
-
-In the ``geometric multiplication'' of rectangular vectors
-\index{Geometric!multiplication}%
-\index{Multiplication!geometric}%
-\index{Vectors}%
-used in quaternions, the commutative property
-\index{Quaternions}%
-of factors does not hold, but a change in the cyclic
-order of factors reverses the sign of the product.
-
-Even in elementary algebra or arithmetic, the commutative
-principle is not valid in the operation of involution.
-Professor Schubert, in his \Title{Mathematical
-Essays and Recreations}, has called attention to the
-fact that this limitation---the impossibility of interchanging
-base and exponent---renders useless any high
-operation of continued involution.
-\DPPageSep{161}{155}
-
-
-\Chapter{Equations of U.S. standards of
-length and mass.}
-\index{Equations of U.S. standards of length and mass}%
-\index{Length, standard of}%
-\index{Mass, standard of}%
-\index{Measures, standard}%
-\index{Metric system}%
-\index{Standards of length and mass}%
-\index{United States standards of length and mass}%
-\index{Weights and measures}%
-
-By order approved by the secretary of the treasury
-April~5, 1893, the international prototype meter and
-kilogram are regarded as fundamental standards, the
-\index{Kilogram}%
-yard, pound etc.\ being defined in terms of them.
-
-All of the nations taking part in the convention
-have very accurate copies of the international standards.
-The degree of accuracy of the comparisons
-may be seen from the equations expressing the relation
-of meter no.~27 and kilogram no.~20, of the
-United States, to the international prototypes. $T$~represents
-the number of degrees of the centigrade scale
-of the hydrogen thermometer. The last term in each
-equation shows the range of error.
-\begin{align*}
-M \text{ no.}~27 &= 1\, \text{m} - 1.6\mu + 8.657\mu T + 0.00100\mu T^{2} ħ 0.2\mu \\
-K \text{ no.}~20 &= 1\, \text{kg} - 0.039\, \text{mg} ħ 0.002\, \text{mg}\Add{.}
-\end{align*}
-
-(U.S. coast and geodetic survey.)
-\DPPageSep{162}{156}
-
-
-\Chapter{The mathematical treatment of
-statistics.}
-\index{Analytic geometry|(}%
-\index{Calculus of probability}%
-\index{Combinations and permutations}%
-\index{Distribution curve for measures|(}%
-\index{Graph of equation|(}%
-\index{Mathematical treatment of statistics}%
-\index{Measurements treated statistically|(}%
-\index{Social sciences treated mathematically}%
-\index{Statistics, mathematical treatment of}%
-\index{Surface of \DPtypo{frequencey}{frequency}|(}%
-
-This is one of the most important and interesting
-applications of mathematics to the needs of modern
-civilization. Just as data gathered by an incompetent
-observer are worthless---or by a biased observer, unless
-the bias can be measured and eliminated from the result---so
-also conclusions obtained from even the best
-data by one unacquainted with the principles of statistics
-must be of doubtful value.
-
-The laws of statistics are applications of mathematical
-formulas, especially of permutations, combinations
-\index{Permutations}%
-and probability. Take for illustration two simple laws
-\index{Probability}%
-(the mathematical derivation of them would not be
-so simple):
-%<tb>
-
-1. Suppose a number of measurements have been
-made. If the measures be laid off as abscissas, and
-the number of times each measure occurs be represented
-graphically as the corresponding ordinate, the
-line drawn through the points thus plotted is called
-the \emph{distribution curve} for these measures. The area
-between this line and the axis of~$x$ is the \emph{surface of
-frequency}.
-
-If a quantity one is measuring is due to chance combinations
-of an infinite number of causes, equal in
-amount and independent, and all equally likely to
-occur, the surface of frequency is of the form shown
-\DPPageSep{163}{157}
-in \Fig{32}, the equation of the curve being $y = e^{-x^{2}}$.
-
-Most effects that are measured are not due to such
-combinations of causes, and their distribution curves
-are more or less irregular; but under favorable conditions
-they frequently approximate this, which may
-be called the normal, ``the normal probability integral.''
-\index{Normal probability integral}%
-In these cases the tables that have been computed for
-this surface are of great assistance.
-\index{Analytic geometry|)}%
-\index{Graph of equation|)}%
-%<tb>
-
-2. Every one knows that, other things being equal, the
-greater the number of measurements made, the greater
-the probability of their average (or other mean) being
-the true one. It is shown mathematically that the
-\Figure{32}
-probability \emph{varies as the square root} of the number
-of measures. \Eg,~\DPtypo{If}{if} in one investigation $64$~cases
-were measured, and in another $25$~cases, the returns
-from the first investigation will be more trustworthy
-than those from the second in the ratio of $8$~to~$5$.
-
-It is also apparent that, if the average deviation (or
-other measure of variability) of the measures from
-their average in one set is greater than in another,
-the average is less trustworthy in that set in which the
-variability is the greater. Expressed mathematically,
-the trustworthiness \emph{varies inversely} as the variability.
-\Eg,~in one investigation the average deviation of
-\DPPageSep{164}{158}
-the measures from their average is~$2$ ($2$~cm, $2$~grams,
-or whatever the unit may be) while in another investigation
-(involving the same number of measures
-etc.)\ the average deviation is~$2.5$. Then the probable
-approach to accuracy of the average obtained in the
-first investigation is to that of the average obtained in
-the second as $\SlantFrac{1}{2}$ is to~$\SlantFrac{1}{2.5}$, or as $5$~to~$4$.
-
-If the two investigations differed both in the number
-of measures and in the deviation from the average,
-both would enter as factors in determining the relative
-confidence to be reposed in the two results. \Eg,
-combine the examples in the two preceding paragraphs:
-An average was obtained from $64$~measures
-whose variability was~$2$, and another from $25$~measures
-whose variability was~$2.5$. Then
-\begin{align*}
-\TextBox{trustworthiness}{trustworthiness \\ of first average} :
-\TextBox{of second average}{trustworthiness \\ of second average}
- &= \sqrt{64} × \frac{1}{2} : \sqrt{25} × \frac{1}{2.5} \\
- &= 2 : 1\Add{.}
-\end{align*}
-The trustworthiness of the mean of a number of measures
-varies directly as the square root of the number
-of measures and inversely as their variability.
-%<tb>
-
-The foregoing principles---the A\;B\;C~of statistical
-science---show some of its method and its value and
-the direction in which it is working. Perhaps the
-most readable treatise on the subject is Professor
-Edward~L. Thorndike's \Title{Introduction to the Theory
-of Mental and Social Measurements}. It presupposes
-only an elementary knowledge of mathematics and
-contains references to more technical works on the
-subject.
-%<tb>
-
-Professor W.~S\Add{.} Hall, in ``Evaluation of Anthropometric
-\index{Hall, W.~S.}%
-Data'' (\Title{Jour.\ Am.\ Med.\ Assn.}, Chicago, 1901)
-\DPPageSep{165}{159}
-showed that the curve of distribution of biologic data
-is the curve of the coefficients in the expansion of an
-algebraic binomial. In a most interesting article, ``A
-Guide to the Equitable Grading of Students,'' in
-\index{Grading of students}%
-\Title{School Science and Mathematics} for June,~1906, he
-applies this principle to the distribution of student
-records in a class.
-
-In the expansion of $(a + b)^{5}$ there are $6$~terms, and
-\index{Binomial theorem and statistics.}%
-\index{Marking students}%
-\index{Student records}%
-the coefficients are $1$,~$5$, $10$, $10$, $5$,~$1$. Their sum is~$32$.
-If $320$~students do their work and are tested and
-graded under normal (though perhaps unusual) conditions,
-and $6$~different marks are used---say A,~B, C,
-D, E,~F---the number of pupils attaining each of these
-standings should approximate $10$, $50$, $100$, $100$, $50$, $10$,
-respectively. If $3200$~students were rated in $6$~groups
-under similar conditions, the numbers in the groups
-would be ten times as great---$100$, $500$, $1000$~etc., and
-the approximation would be relatively closer than
-when only $320$ were tested. The study of the conditions
-that cause deviation from this normal distribution
-of standings is instructive both statistically and
-pedagogically.
-\index{Distribution curve for measures|)}%
-\index{Surface of \DPtypo{frequencey}{frequency}|)}%
-%<tb>
-
-A rough-and-ready statistical method, available in
-certain cases, may be illustrated as follows: Suppose
-we are engaged in ascertaining the number of words
-in the vocabulary of normal-school juniors. (Such
-an investigation is now in progress under the direction
-of Dr.\ Margaret~K. Smith, of the normal faculty at
-\index{Smith, M.~K.}%
-New Paltz.) Let us select at random a page of the
-dictionary---say the $13$th---and by appropriate tests
-ascertain the number of words on this page that the
-pupil knows, divide this number by the number of
-words on the page, and thus obtain a convenient expression
-\DPPageSep{166}{160}
-for the \emph{part} of the words known. Suppose
-the quotient to be~$.3016$. Turn to page~113 and make
-similar tests, and divide the number known on \emph{both}
-pages by the number of words on both pages, giving---say---$.2391$.
-After trying page~213 the result is
-found for three pages. In each case the decimal
-represents the total result reached thus far in the
-experiment. Suppose the successive decimals to be
-\[
-\begin{array}{l}
-.3016 \\
-.2391 \\
-.2742 \\
-.2688 \\
-.2562 \\
-.2610 \\
-.2628 \\
-.2631 \\
-.2642 \\
-.2638
-\end{array}
-\]
-A few decimals thus obtained may convince the experimenter
-that the first figure has ``become constant.''
-Many more may be necessary to determine the second
-figure unless the ``series converges'' rapidly as above.
-If the first two figures be found to be~$26$, this student
-knows $26$\%~of the words in the dictionary. Multiplying
-the ``dictionary total'' by this coefficient, gives
-the extent of the student's vocabulary, correct to $1$\%~of
-the dictionary total. If a higher degree of accuracy
-had been required, a three-place coefficient would have
-been determined.
-
-This method has the practical advantage that the
-coefficient found at each step furnishes, by comparison
-with the coefficients previously obtained, an indication
-of the degree of accuracy that will be attained by
-\DPPageSep{167}{161}
-its use. The labor of division may be diminished by
-using, on each page, only the first $20$~words (or other
-multiple of~$10$). Similarly with each student to be
-examined. The method here described is applicable
-to certain classes of measures.
-\index{Measurements treated statistically|)}%
-\DPPageSep{168}{162}
-
-
-\Chapter{Mathematical symbols.}
-\index{Mathematical symbols}%
-\index{Symbols!mathematical}%
-
-The origin of most of the symbols in common use
-may be learned from any history of mathematics. The
-noteworthy thing is their recentness. Of our symbols
-of operation the oldest are $+$~and~$-$, which appear in
-Widmann's arithmetic (Leipsic,~1489).
-
-Consider the situation in respect to symbols at the
-middle of the sixteenth century. The radical sign
-had been used by Rudolff, $(\;)$,~$×$, $÷$,~$>$\Add{,} and~$<$ were
-\index{Rudolff}%
-still many years in the future, $=$~had not yet appeared
-(though another symbol for the same had been used
-slightly) and $+$~and~$-$ were not in general use. Almost
-everything was expressed by words or by mere abbreviations.
-Yet at that time both cubic and biquadratic
-equations had been solved and the methods published.
-It is astonishing that men with the intellectual
-acumen necessary to invent a solution of equations of
-the third or fourth degree should not have hit upon
-a device so simple as symbols of operation for the
-abridgment of their work.
-
-The inconvenience of the lack of symbols may be
-easily tested by writing---say---a quadratic equation
-and solving it without any of the ordinary symbols
-of algebra.
-
-Even after the introduction of symbols began, the
-process was slow. But recently it has moved with
-accelerating velocity, until now not only do we have
-a symbol for each operation---sometimes a choice of
-\DPPageSep{169}{163}
-symbols---but most of the letters of the alphabet are
-engaged for special mathematical uses. \Eg:
-\begin{itemize}
-\item[$a$] finite quantity, known number, side of triangle
-opposite~$A$, intercept on axis of~$x$, altitude~\dots
-
-\item[$b$] known number, side of triangle opposite~$B$, base,
-intercept on axis of~$y$~\dots
-
-\item[$c$] constant~\dots
-
-\item[$d$] differential, distance~\dots
-
-\item[$e$] base of Napierian logarithms.
-\end{itemize}
-
-A considerable inroad has been made on the Greek
-alphabet, \eg:
-\begin{itemize}
-\item[$\gamma$] inclination to axis of~$x$.
-
-\item[$\pi$] $3.14159$\dots
-
-\item[$\Sigma$] sum of terms similarly obtained.
-
-\item[$\sigma$] standard deviation (in theory of measurements).
-\end{itemize}
-But the supply of alphabets is by no means exhausted.
-There is no cause for alarm.
-\DPPageSep{170}{164}
-
-
-\Chapter{Beginnings of mathematics on the
-Nile.}
-\index{Beginnings of mathematics on the Nile}%
-\index{Nile, beginnings of mathematics on}%
-
-Whatever the excavations in Babylonia and Assyria
-\index{Assyria}%
-\index{Babylonia}%
-may ultimately reveal as to the state of mathematical
-learning in those early civilizations, it is established
-that in Egypt the knowledge of certain mathematical
-\index{Egypt}%
-facts and processes was so ancient as to have left no
-record of its origin.
-
-The truth of the Pythagorean theorem for the special
-case of the isosceles right triangle may have been
-widely known among people using tile floors (see Beman
-and Smith's \Title{New Plane Geometry}, p.~103). That
-$3$,~$4$, and~$5$ are the sides of a right triangle was known
-and used by the builders of the pyramids and temples.
-The Ahmes papyrus (1700~\BC\ and based on a work
-\index{Ahmes papyrus}%
-of perhaps 3000~\BC\ or earlier) contains many arithmetical
-problems, a table of unit-fractions, etc., and
-the solution of simple equations, in which \Foreign{hau} (heap)
-\index{Equations!solved in ancient Egypt}%
-represents the unknown. Though one may feel sure
-that arithmetic must be the oldest member of the
-mathematical family, still the beginnings of arithmetic,
-algebra and geometry are all prehistoric. When the
-curtain raises on the drama of human history, we see
-men computing, solving linear equations, and using
-a simple case of the Pythagorean proposition.
-\index{Pythagorean proposition}%
-\DPPageSep{171}{165}
-
-
-\Chapter{A few surprising facts in the history
-of mathematics.}
-\index{Astronomers}%
-\index{Decimals invented late}%
-\index{History of mathematics!surprising facts}%
-\index{Mathematical symbols}%
-\index{Napier, Mark}%
-\index{Surprising facts in the history of mathematics}%
-
-That spherical trigonometry was developed earlier
-\index{Trigonometry}%
-than plane trigonometry (explained by the fact that the
-former was used in astronomy).
-
-That the solution of equations of the third and
-fourth degree preceded the use of most of the symbols
-\index{Symbols!mathematical}%
-of operation, even of~$=$.
-
-That decimals---so simple and convenient---should
-not have been invented till after so much ``had been
-attempted in physical research and numbers had been
-so deeply pondered'' (Mark Napier).
-
-That logarithms were invented before exponents
-\index{Exponents}%
-\index{Logarithms}%
-were used; the derivation of logarithms from exponents---now
-always used in teaching logarithms---being
-first pointed out by Euler more than a century
-\index{Euler}%
-later.
-
-That the earliest systems of logarithms (Napier's,
-Speidell's), constructed for the sole object of facilitating
-\index{Speidell}%
-computation, should have missed that mark
-(leaving it for Briggs, Gellibrand, Vlacq, Gunter and
-\index{Briggs}%
-\index{Gellibrand}%
-\index{Gunter}%
-\index{Vlacq}%
-others) but should have attained theoretical importance,
-lending themselves to the purposes of modern
-analytical methods (Cajori).
-\DPPageSep{172}{166}
-
-
-\Chapter{Quotations on mathematics.}
-\index{Quotations on mathematics}%
-
-Following are some of the quotations that have been
-used at different times in the decoration of a frieze
-above the blackboard in the writer's recitation room:
-
-Let no one who is unacquainted with geometry
-\index{Plato}%
-\textsc{leave} here. (This near the door and on the inside---an
-adaptation of the motto that Plato is said to have
-had over the outside of the entrance to his school of
-philosophy, the Academy: ``Let no one who is unacquainted
-with geometry \emph{enter} here.'')
-
-God geometrizes continually. \Name{Plato.}
-
-There is no royal road to geometry. \Name{Euclid.}
-\index{Euclid}%
-
-Mathematics, the queen of the sciences. \Name{Gauss.}
-\index{Gauss}%
-
-Mathematics is the glory of the human mind. \Name{Leibnitz.}
-\index{Leibnitz}%
-
-Mathematics is the most marvelous instrument created
-by the genius of man for the discovery of truth.
-\Name{Laisant.}
-\index{Laisant}%
-
-Mathematics is the indispensable instrument of all
-physical research. \Name{Berthelot.}
-\index{Berthelot}%
-
-All my physics is nothing else than geometry. \Name{Descartes.}
-\index{Descartes}%
-
-There is nothing so prolific in utilities as abstractions.
-\Name{Faraday.}
-\index{Faraday}%
-
-The two eyes of exact science are mathematics and
-logic. \Name{De~Morgan.}
-\index{Demorgan@{De Morgan}}%
-
-All scientific education which does not commence
-\DPPageSep{173}{167}
-with mathematics is, of necessity, defective at its foundation.
-\Name{Compte.}
-\index{Compte}%
-
-It is in mathematics we ought to learn the general
-method always followed by the human mind in its
-positive researches. \Name{Compte.}
-
-A natural science is a science only in so far as it is
-mathematical. \Name{Kant.}
-\index{Kant}%
-
-The progress, the improvement of mathematics are
-linked to the prosperity of the state. \Name{Napoleon.}
-\index{Napoleon}%
-
-If the Greeks had not cultivated conic sections, Kepler
-\index{Greeks}%
-\index{Kepler}%
-could not have superseded Ptolemy. \Name{Whewell.}
-\index{Ptolemy}%
-\index{Whewell}%
-
-No subject loses more than mathematics by any attempt
-to dissociate it from its history. \Name{Glaisher.}
-\index{Glaisher}%
-\index{History of mathematics}%
-\DPPageSep{174}{168}
-
-
-\Chapter{Autographs of mathematicians.}
-\index{Autographs of mathematicians}%
-\index{Chirography of mathematicians}%
-\index{Handwriting of mathematicians}%
-\index{Signatures!of mathematicians|(}%
-
-For the photograph from which this cut (\Fig{33})
-was made the writer is indebted to Prof.\ David Eugene
-Smith. As an explorer in the bypaths of mathematical
-\index{Smith, D.~E.}%
-history and a collector of interesting specimens
-therefrom, Dr.~Smith is, perhaps, without a peer.\footnote
- {Several handsome sets of portraits of mathematicians,
- edited by Dr.~Smith, are published by The Open Court Publishing
- Company.}
-
-The reader will be interested to see a facsimile of
-the handwriting of Euler and Johann Bernoulli, Lagrange
-\index{Bernoulli}%
-\index{Euler}%
-\index{Lagrange}%
-and Laplace and Legendre, Clifford and Dodgson,
-\index{Clifford}%
-\index{Dodgson, C.~L.}%
-\index{Laplace}%
-\index{Legendre}%
-and William Rowan Hamilton, and others of the
-\index{Hamilton, W.~R.}%
-immortals, grouped together on one page. In the
-upper right corner is the autograph of Moritz Cantor,
-\index{Cantor, Moritz}%
-the historian of mathematics. On the sheet overlapping
-that, the name over the verses is faint; it is that
-of J.~J. Sylvester, late professor in Johns Hopkins
-\index{Sylvester, J. J.}%
-University.
-
-One who tries to decipher some of these documents
-may feel that he is indeed ``In the Mazes of Mathematics.''\footnote
- {This section first printed in a series bearing that title, in
- \Title{The Open Court}, March--July, 1907.}
-Mathematicians are not as a class noted
-for the elegance or the legibility of their chirography,
-and these examples are not submitted as models of
-penmanship. But each bears the sign manual of one
-of the builders of the proud structure of modern
-mathematics.
-\DPPageSep{175}{169}
-\jpgFigure[1.0]{33}
-\index{Signatures!of mathematicians|)}%
-\DPPageSep{176}{170}
-
-
-\Chapter[Bridges and isles, labyrinths etc.]{Bridges and isles, figure tracing,
-unicursal signatures,
-labyrinths.}
-\index{Bridges and isles}%
-\index{Figure tracing}%
-\index{Labyrinths}%
-\index{Signatures!unicursal}%
-\index{Isles and bridges}%
-\index{Unicursal signatures and figures}%
-
-This section presents a few of the more elementary
-results of the application of mathematical methods to
-these interesting puzzle questions.\footnote
- {For more extended discussion, and for proofs of the theorems
- here stated, see Euler's \Title{Solutio Problematis ad Geometriam
- Situs Pertinentis}, Listing's \Title{Vorstudien sur Topologie},
- Ball's \Title{Mathematical Recreations and Essays}, Lucas's \Title{Récréations
- Mathématiques}, and the references given in notes by the
- last two writers named. To these two the present writer is
- especially indebted.}
-\Figure{34}
-
-The city of Königsberg is near the mouth of the
-\index{Königsberg|(}%
-Pregel river, which has at that point an island called
-Kneiphof. The situation of the seven bridges is shown
-in \Fig{34}. A discussion arose as to whether it is
-\DPPageSep{177}{171}
-possible to cross all the bridges in a single promenade
-without crossing any bridge a second time. Euler's
-\index{Euler}%
-famous memoir was presented to the Academy of
-Sciences of St.~Petersburg in~1736 in answer to this
-question. Rather, the Königsberg problem furnished
-him the occasion to solve the general problem of any
-number and combination of isles and bridges.
-\index{Königsberg|)}%
-
-Conceive the isles to shrink to points, and the problem
-may be stated more conveniently with reference
-to a diagram as the problem of tracing a given figure
-\Figure[0.6]{35}
-without removing the pencil from the paper and without
-retracing any part; or, if not possible to do so
-with one stroke, to determine \emph{how many} such strokes
-are necessary. \Fig{35} is a diagrammatic representation
-of \Fig{34}, the isle Kneiphof being at the
-point~$K$.
-
-The number of lines proceeding from any point of
-a figure may be called the \emph{order} of that point. Every
-\DPPageSep{178}{172}
-point will therefore be of either an even order or an
-odd order. \Eg, as there are $3$~lines from point~$A$
-of \Fig{36}, the order of the point is odd; the order of
-point~$E$ is even. The well-known conclusions reached
-by Euler may now be stated as follows:
-\Figure[0.4]{36}
-\Figures{0.8}{37}{0.8}{38}
-
-\begin{Thm}
-In a closed figure \emph{(one with no free point or ``loose
-end'')} the number of points of odd order is even\end{Thm},
-whether the figure is unicursal or not. \Eg, \Fig{36},
-a multicursal closed figure, has \emph{four} points of odd
-order.
-
-\begin{Thm}
-A figure of which every point is of even order can
-\index{Magic!pentagon|(}%
-\index{Hexagons!magic|(}%
-\index{Magic!hexagons|(}%
-\index{Pentagon, magic|(}%
-be traced by one stroke starting from any point of the
-\DPPageSep{179}{173}
-\index{Carus, Paul}%
-figure.\end{Thm} \Eg, \Fig{37}, the magic pentagon, symbol of
-the Pythagorean school, and \Fig{38}, a ``magic hexagram
-commonly called the shield of David and frequently
-used on synagogues'' (Carus), have no points
-of odd order; each is therefore unicursal.
-
-\begin{Thm}
-A figure with only two points of odd order can be
-traced by one stroke by starting at one of those points.\end{Thm}
-\Eg, \Fig{39} (taken originally from Listing's \Title{Topologie})
-has but two points of odd order, $A$~and~$Z$; it
-may therefore be traced by one stroke beginning at
-either of these two points and ending at the other.
-\Figure{39}
-One may make a game of it by drawing a figure, as
-Lucas suggests, like \Fig{39}, but in larger scale on
-cardboard, placing a small counter on the middle of
-each line that joins two neighboring points, and setting
-the problem to determine the course to follow in removing
-all the counters successively (simply tracing
-continuously and removing each counter as it is passed,
-an objective method of recording which lines have
-been traced).
-
-\begin{Thm}
-A figure with more than two points of odd order
-is multicursal.\end{Thm} \Eg, \Fig{40} has more than two points
-of odd order and requires more than one course, or
-stroke, to traverse it.
-\index{Hexagons!magic|)}%
-\index{Pentagon, magic|)}%
-\index{Magic!pentagon|)}%
-\index{Magic!hexagons|)}%
-\DPPageSep{180}{174}
-
-The last two theorems just stated are special cases
-of Listing's:
-
-\begin{Thm}
-Let $2n$ represent the number of points of odd order;
-then $n$~strokes are necessary
-and sufficient to
-trace the figure.\end{Thm} \Eg,
-\Fig{39} with $2$~points of
-\Figure[0.5]{40}
-odd order, requires one
-stroke; \Fig{40}, representing
-a fragment of
-masonry, has $8$~points of odd order and requires four
-strokes.
-
-Return now to the Königsberg problem of \Fig{34}.
-\index{Königsberg}%
-By reference to the diagram in \Fig{35} it is seen that
-there are four points of odd order. Hence it is not
-possible to cross every bridge once and but once without
-taking two strolls.
-
-An interesting application of these theorems is the
-\index{Diagonals of a polygon|(}%
-consideration of the number of strokes necessary to
-describe an $n$-gon and its diagonals. As the points
-of intersection of the diagonals are all of even order,
-we need to consider only the vertexes. Since from
-each vertex there is a line to every
-other vertex, the number of lines
-from each vertex is~$n - 1$. Hence,
-if $n$~is odd, every point is of even
-order, and the entire figure can be
-\Figure[0.3]{41}
-traced unicursally beginning at any
-point; \eg, \Fig{41}, a pentagon
-with its diagonals. If $n$~is even,
-$n - 1$~is odd, every vertex is of odd
-order, the number of points of odd
-order is~$n$, and the figure can not be described in less
-\DPPageSep{181}{175}
-than $\SlantFrac{n}{2}$~courses; e.g., \Fig{36}, quadrilateral, requires
-two strokes.
-\index{Diagonals of a polygon|)}%
-
-\Par{Unicursal signatures.} A signature (or other writing)
-\index{Signatures!unicursal|(}%
-is of course subject to the same laws as are other
-figures with respect to the number of times the pen
-must be put to the paper. Since the terminal point
-could have been connected with the point of starting
-without lifting the pen, the signature may be counted
-as a closed figure if it has no free end but these two.
-The number of points of odd order will be found to
-be even. The dot over an~\textit{i}, the cross of a~\textit{t}, or any
-\Figures{0.8}{42}{0.3}{43}
-other mark leaving a free point, makes the signature
-multicursal. There are so many names not requiring
-separate strokes that one would expect more unicursal
-signatures than are actually found. De~Morgan's (as
-\index{Demorgan@{De Morgan}}%
-shown in the cut in the preceding section) is one;
-but most of the signatures there shown were made
-with several strokes each. Of the signatures to the
-Declaration of Independence there is not one that is
-\index{Declaration of Independence}%
-strictly unicursal; though that of \textit{Th~Jefferson} looks as
-\index{Jefferson, Thomas}%
-if the end of the~\textit{h} and the beginning of the~\textit{J} might
-often have been completely joined, and in that case
-\DPPageSep{182}{176}
-his signature would have been written in a single
-course of the pen.
-\index{Signatures!unicursal|)}%
-
-\Fig{42}, formed of two crescents, is ``the so-called
-\index{Crescents of Mohammed|(}%
-\index{Mohammed|(}%
-sign manual of Mohammed, said to have been originally
-traced in the sand by the point of his scimetar
-without taking the scimetar off the ground or retracing
-any part of the figure,'' which can easily be
-done beginning at any point of the figure, as it contains
-no point of odd order. The mother of the
-writer suggests that, if the horns of Mohammed's
-\Figure[0.9]{44}
-crescents be omitted, a figure (\Fig{43}) is left which
-can not be traced unicursally. There are then four
-points of odd order; hence two strokes are requisite
-to describe the figure.
-\index{Crescents of Mohammed|)}%
-\index{Mohammed|)}%
-
-\Par[.]{Labyrinths} such as the very simple one shown in
-\index{Labyrinths|(}%
-\index{Mazes|(}%
-\Fig{44} (published in~1706 by London and Wise)
-\index{London and Wise}%
-are familiar, as drawings, to every one. In some of
-the more complicated mazes it is not so easy to thread
-one's way, even in the drawing, where the entire
-\DPPageSep{183}{177}
-maze is in sight, while in the actual labyrinth, where
-walls or hedges conceal everything but the path one
-is taking at the moment, the difficulty is greatly increased
-and one needs a rule of procedure.
-
-The mathematical principles involved are the same
-as for tracing other figures; but in their application
-several differences are to be noticed in the conditions
-of the two problems. A labyrinth, as it stands, is not
-a closed figure; for the entrance and the center are
-free ends, as are also the ends of any blind alleys that
-the maze may contain. These are therefore points of
-odd order. There are usually other points of odd
-order. Hence in a single trip the maze can not be
-completely traversed. But it is not required to do so.
-The problem here is, to go from the entrance to the
-center, the shorter the route found the better. Moreover,
-the rules of the game do not forbid retracing
-one's course.
-
-It is readily seen (as first suggested by Euler) that
-\index{Euler}%
-by going over each line twice the maze becomes a
-closed figure, terminating where it begins, at the entrance,
-including the center as one point in the course,
-and containing only points of even order. Hence
-every labyrinth can be completely traversed by going
-over every path twice---once in each direction. It is
-only necessary to have some means of marking the
-routes already taken (and their direction) to avoid
-the possibility of losing one's way. This duplication
-of the entire course permits no failure and is so general
-a method that one does not need to know anything
-about the particular labyrinth in order to traverse it
-successfully and confidently. But if a plan of the
-labyrinth can be had, a course may be found that is
-shorter.
-\DPPageSep{184}{178}
-
-Theseus, as he \emph{threaded} the Cretan labyrinth in quest
-\index{Cretan labyrinth}%
-\index{Theseus}%
-of the Minotaur, would have regarded Euler's mathematical
-\index{Euler}%
-\index{Minotaur}%
-theory of mazes as much less romantic than
-the silken cord with Ariadne at the outer end; but
-\index{Ariadne}%
-there are occasions where a modern finds it necessary
-to ``go by the book.'' Doubtless the labyrinth of
-\index{Daedalus@{Dĉdalus}}%
-Daedalus was ``a mighty maze, but not without a plan.''
-
-\Fig{45} presents one of the most famous labyrinths,
-\index{Hampton Court labyrinth}%
-though by no means among the most puzzling. It is
-described in the \Title{Encyclopĉdia Britannica} (article
-``Labyrinth'') as follows:
-\Figure[1.0]{45}
-
-``The maze in the gardens at Hampton Court Palace
-is considered to be one of the finest examples in
-England. It was planted in the early part of the reign
-of William~III, though it has been supposed that a
-maze had existed there since the time of Henry~VIII\@.
-It is constructed on the hedge and alley system, and
-was, we believe, originally planted with hornbeam,
-but many of the plants have died out, and been replaced
-by hollies, yews, etc., so that the vegetation is
-mixed. The walks are about half a mile in length,
-and the extent of ground occupied is a little over a
-quarter of an acre. The center contains two large
-\DPPageSep{185}{179}
-trees, with a seat beneath each. The key to reach this
-resting place is to keep the right hand continuously in
-contact with the hedge from first to last, going round
-all the stops.''
-\index{Labyrinths|)}%
-\index{Mazes|)}%
-\DPPageSep{186}{180}
-
-
-\Chapter{The number of the beast.}
-\index{Beast, number of}%
-\index{Number!of the beast}%
-
-``Here is wisdom. He that hath understanding, let
-him count the number of the beast; for it is the number
-of a man: and his number is Six hundred and sixty
-and six.'' (Margin, ``Some ancient authorities read
-\emph{Six hundred and sixteen}.'') Revelation~13:18.
-
-No wonder that these words have been a powerful
-incentive to a class of interpreters who delight in
-apocalyptic literature, especially to such as have a
-Pythagorean regard for hidden meaning in numbers.
-
-There were centuries in which no satisfactory interpretation
-\index{Irenĉus|(}%
-was generally known. At about the same
-time, in~1835, Benary, Fritzsche, Hitzig and Reuss
-\index{Benary}%
-\index{Fritzsche}%
-\index{Hitzig}%
-\index{Reuss}%
-connected the number~$666$ with ``Emperor (Cĉsar)
-\index{Caesar@{Cĉsar Neron}}%
-Neron,'' \texthebrew{qsr nrwn}. In the number notation of the Hebrews
-the letter $\texthebrew{q} = 100$, $\texthebrew{s} = 60$, $\texthebrew{r} = 200$, $\texthebrew{n|} = 50$, $\texthebrew{r} = 200$,
-$\texthebrew{w} = 6$, $\texthebrew{n} = 50$. These numbers added give~$666$. Omitting
-the final letter from the name (making it ``Emperor
-Nero'') the number represented is~$616$, the marginal
-\index{Nero}%
-reading. The present writer's casual opinion
-is that the foregoing is the meaning intended in the
-passage; and that after the fear of Nero passed, the
-knowledge of the meaning of the number gradually
-faded, and had to be rediscovered long afterward.
-It is, however, strange, that only about a century after
-the writing of the Apocalypse the connection of the
-number with Nero was apparently unknown to Irenĉus.
-\DPPageSep{187}{181}
-He made several conjectures of words to fit
-the number.
-
-In the later Middle Ages and afterward, the number
-was made to fit heresies and individual heretics.
-Protestants in turn found that a little ingenuity could
-discover a similar correspondence between the number
-and symbols for the papacy or names of popes.
-So the exchange of these expressions of regard continued.
-When the name is taken in Greek, the number
-is expressed in Greek numerals, where every letter is
-a numeral; but when Latin is used, only $M$,~$D$, $C$, $L$, $X$,
-$V$\Add{,} and~$I$ have numerical values.
-\[
-\setlength{\arraycolsep}{1pt}
-\begin{array}{*{16}{r}l}
-\SSS{V} &
-\SSS{I} &
-\SSS{C} &
-\SSS{A} &
-\SSS{R} &
-\SSS{I} &
-\SSS{V} &
-\SSS[c]{S} &
-\SSS{F} &
-\SSS{I} &
-\SSS{L} &
-\SSS{I} &
-\SSS{I} &
-\SSS{D} &
-\SSS{E} &
-\SSS{I} \\
-\SSS{5} &
-\SSS{+1} &
-\SSS[c]{+100} &
-&
-&
-\SSS{+1} &
-\SSS{+5} &
-&
-&
-\SSS{+1} &
-\SSS[c]{+50} &
-\SSS{+1} &
-\SSS{+1} &
-\SSS[c]{+500} &
-&
-\SSS{+1} & = 666\Add{.}
-\end{array}
-\]
-This and a similar derivation from Luther's name are
-\index{Luther}%
-perhaps the most famous of these performances.
-\index{Irenĉus|)}%
-
-De~Morgan cites a book by Rev.\ David Thom,\footnote
- {\Title{The Number and Names of the Apocalyptic Beasts}, part~1,
- 8vo, 1848. See De~Morgan's \Title{Budget of Paradoxes}, p.~402--3\Add{.}}
-\index{Demorgan@{De Morgan}}%
-from which he quotes names, significant mottoes~etc.\
-that have been shown to spell out the number~$666$.
-He gives $18$~such from the Latin and $38$~from the
-Greek and omits those from the Hebrew. Some of
-these were made in jest, but many in grim earnest.
-He also gives a few from other sources than the book
-mentioned.
-
-The number of such interpretations is so great as
-to destroy the claim of any. ``We can not infer much
-from the fact that the key fits the lock, if it is a lock
-in which almost any key will turn.'' A certain interest
-still attaches to all such cabalistic hermeneutics, and
-they are not without their lesson to us, but it is not the
-lesson intended by the interpreter. When it comes to
-\DPPageSep{188}{182}
-the use of such interpretations by one branch of the
-Church against another, one would prefer as less irreverent
-the suggestion of De~Morgan, that the true
-\index{Demorgan@{De Morgan}}%
-explanation of the three sixes is that the interpreters
-are ``six of one and half a dozen of the other.''
-\DPPageSep{189}{183}
-
-
-\Chapter{Magic squares.}
-\index{Dela@{De la Loubère}}%
-\index{Loubère, de la}%
-\index{Magic!squares}%
-\index{Squares!magic}%
-
-``A magic square is one divided into any number
-of equal squares, like a chess-board, in each of which
-is placed one of a series of consecutive numbers from~$1$
-up to the square of the number of cells in a side,
-in such a manner that the sum of those in the same row
-or column and in each of the two diagonals is constant.''
-(\Title{Encyclopĉdia Britannica.})
-
-The term is often extended to include an assemblage
-of numbers not consecutive but meeting all other requirements
-of this definition. If every number in a
-magic square be multiplied by any number,~$q$, integral
-or fractional, arithmetical, real or imaginary, such
-an assemblage is formed, and by the distributive law
-of multiplication its sums are
-each $q$ times those in the original
-square.
-\Figure[0.4]{46}
-
-One way (De~la Loubère's)
-of constructing any odd-number
-square is as follows:
-
-1. In assigning the consecutive
-numbers, proceed in an
-oblique direction up and to
-the right (see $4$,~$5$,~$6$ in \Fig{46}).
-
-2. When this would carry a number out of the
-square, write that number in the cell at the opposite
-end of the column or row, as shown in case of the
-canceled figures in the margin of \Fig{46}.
-\DPPageSep{190}{184}
-
-3. When the application of rule~1 would place a
-number in a cell already occupied, write the new number,
-instead, in the cell beneath the one last filled.
-(The cell above and to the right of~$3$ being occupied,
-$4$~is written beneath~$3$.)
-
-4. Treat the marginal square marked~x as an occupied
-cell, and apply rule~3.
-
-5. Begin by putting~$1$ in the top cell of the middle
-column.
-
-This rule will fill any square having an odd number
-of cells in each row and column.
-
-The investigation of some of the properties of the
-simple squares just described is an interesting diversion.
-For example, after the $5$-square and $7$-square
-have been constructed and one is familiar with the rule,
-he may set himself the problem to find a formula for
-the sum of the numbers in each row, column or diagonal
-of any square. Noticing that the diagonal from lower
-left corner to upper right is composed of consecutive
-numbers, it will be easy to write the formula for the sum
-of that series (the required sum) if we can find the formula
-for the number in the lower left corner. Since the
-number of cells in each row or column of the squares
-we are considering is odd, we represent that number
-by the general formula for an odd number, $2n + 1$.
-Our square, then, is a $(2n + 1)$-square. If $n$~be taken
-$= 1$, we have a $3$-square; if $n = 2$, a $5$-square; etc.
-Now it is seen by inspection that the number in the
-lower left cell is $n(2n + 1) + 1$, the succeeding numbers
-in the diagonal being $n(2n + 1) + 2$, $n(2n + 1) + 3$, etc.
-Summing this series to $2n + 1$ terms, we have the required
-formula, $(2n + 1)(2n^{2} + 2n + 1)$. This might be
-tabulated as follows (including~$1$ as the limiting case
-of a magic square):
-\DPPageSep{191}{185}
-\[
-\begin{array}{@{}c*{3}{|c}@{}}
-\hline
-\hline
-\ColHead{ARBITRARY}{ARBITRARY
-VALUES OF
-$n$~(SUCCESSIVE
-INTEGERS)}
-&
-\ColHead{NO. OF CELLS}{NO. OF CELLS
-IN EACH
-ROW OR COLUMN
-(SUCCESSIVE
-ODD NUMBERS)}
-&
-\ColHead{THE NUMBER IN}{THE NUMBER IN
-THE LOWER
-LEFT CORNER}
-&
-\ColHead{SUM OF THE NUMBERS IN}{SUM OF THE NUMBERS IN
-ANY ROW, COLUMN OR DIAGONAL} \\
-\hline
-%
-n & 2n + 1 & n(2n + 1) + 1 & (2n + 1)(2n^{2} + 2n + 1) \\
-0 & 1 & \Z1 & \Z\Z1 \\
-1 & 3 & \Z4 & \Z15 \\
-2 & 5 & 11 & \Z65 \\
-3 & 7 & 22 & 175 \\
-4 & 9 & 37 & 369 \\
-5 & \PadTo[r]{1}{11} & 56 & 671 \\
-\text{etc.} & \text{etc.} & & \\
-\hline
-\end{array}
-\]
-
-%[** TN: Floating to improve page breaks]
-\begin{table}[ht!]
-Following is the $11$-square; sum,~$671$:
-\[
-\begin{array}{*{11}{r}}
- 68 & 81 & 94 & 107 & 120 & 1 & 14 & 27 & 40 & 53 & 66 \\
- 80 & 93 & 106 & 119 & 11 & 13 & 26 & 39 & 52 & 65 & 67 \\
- 92 & 105 & 118 & 10 & 12 & 25 & 38 & 51 & 64 & 77 & 79 \\
-104 & 117 & 9 & 22 & 24 & 37 & 50 & 63 & 76 & 78 & 91 \\
-116 & 8 & 21 & 23 & 36 & 49 & 62 & 75 & 88 & 90 & 103 \\
- 7 & 20 & 33 & 35 & 48 & 61 & 74 & 87 & 89 & 102 & 115 \\
- 19 & 32 & 34 & 47 & 60 & 73 & 86 & 99 & 101 & 114 & 6 \\
- 31 & 44 & 46 & 59 & 72 & 85 & 98 & 100 & 113 & 5 & 18 \\
- 43 & 45 & 58 & 71 & 84 & 97 & 110 & 112 & 4 & 17 & 30 \\
- 55 & 57 & 70 & 83 & 96 & 109 & 111 & 3 & 16 & 29 & 42 \\
- 56 & 69 & 82 & 95 & 108 & 121 & 2 & 15 & 28 & 41 & 54
-\end{array}
-\]
-\end{table}
-\DPPageSep{192}{186}
-
-There are also ``geometrical magic squares,'' in
-\index{Squares!geometrical magic}%
-which the \emph{product} of the numbers in every row, column
-and diagonal is the same. If a number be selected
-as base and the numbers in an ordinary magic square
-be used as exponents by which to affect it, the resulting
-powers form a geometric square (by the first law
-\index{Geometric!magic squares}%
-of exponents). \Eg, Take $2$ as base and the numbers
-in the square (\Fig{46}) as
-exponents. The resulting geometrical
-magic square (\Fig{47})
-has $2^{15}$ for the product of the
-numbers in each line.
-\Figure[0.4]{47}
-
-The theory of magic squares
-in general, including even-number
-squares, squares with additional
-properties, etc., and including
-the extension of the idea to
-cubes, is given in the article ``Magic Squares'' in the
-\Title{Encyclopĉdia Britannica}, together with some account
-of their history. See also Ball's \Title{Recreations}; Lucas's
-\Title{Récréations}, vol.~4, Cinquième Récréation, ``Les Carrés
-magiques de Fermat''; and the comprehensive
-\index{Fermat}%
-article, ``A Mathematical Study of Magic Squares,''
-\index{Frierson, L.~S.}%
-by L.~S. Frierson, in \Title{The Monist} for April,~1907, p.~272--293.
-
-The oldest manuscript on magic squares still preserved
-\index{Greeks}%
-dates from the fourth or fifth century. It is
-by a Greek named Moscopulus. Magic squares engraved
-\index{Moscopulus}%
-on metal or stone are said to be worn as talismans
-in some parts of India to this day. (\Title{Britannica.})
-
-Among the most prominent of the modern philosophers
-who have amused themselves by perfecting
-the theory of magic squares is Franklin, ``the model
-\index{Franklin, Benjamin}%
-of practical wisdom.''
-\DPPageSep{193}{187}
-
-\Par{Domino magic squares.} A pleasing diversion is the
-\index{Dominoes!in magic squares}%
-\index{Magic!hexagons|(}%
-\index{Squares!coin}%
-\index{Squares!domino}%
-forming of magic squares with dominoes. This phase
-of the subject has been set forth by several writers;
-among them Ball,\footnote
- {\Title{Recreations}, p.~165--6.}
-who also mentions \emph{coin magic
-squares}. The following are by Mr.~Escott, who remarks:
-\index{Escott, E.~B.}%
-\Figures{0.9}{48}{0.9}{49}
-``I do not know how many solutions there are.
-I give five [of which two are reproduced here], which
-I found after a few trials. In each of these magic
-squares the sum is the
-greatest possible,~$19$. If
-we subtract every number
-from~$6$, we get magic
-squares where the sum is
-the least possible,~$5$.''
-
-\Par{Magic hexagons.}\footnote
- {From Mr.~Escott, who says: ``The first appeared in \Title{Knowledge},
-\index{Loyd, S.|FN}%
- in~1895, and the second is due to Mr.\ S.~Lloyd.''}
-\index{Hexagons!magic|(}%
-Sum of any side of triangle
-$=$ sum of vertexes
-of either triangle $=$ sum of
-vertexes of convex hexagon
-$=$ sum of vertexes of
-any parallelogram $= 26$. ``There are only six solutions,
-of which this is one.'' (\Fig{50}.)
-\Figure[0.5]{50}
-\DPPageSep{194}{188}
-
-Place the numbers $1$ to $19$ on the sides of the equilateral
-triangles so that the sum on every side is the
-same.
-\Figures{0.9}{51}{0.9}{52}
-
-The sum on the sides of the triangles in \Fig{51}
-is~$22$. In \Fig{52} it is~$23$. If we subtract each of the
-above numbers from~$20$, we have solutions where the
-sums are respectively $38$ and~$37$.
-\index{Hexagons!magic|)}%
-\index{Magic!hexagons|)}%
-\DPPageSep{195}{189}
-
-
-\Chapter{The square of Gotham.}
-\index{Gotham, square of}%
-\index{Square of Gotham}%
-
-\begin{center}
-(From \Title{Teachers' Note Book}, by permission.)
-\end{center}
-
-The wise men of Gotham, famous for their eccentric
-blunders, once undertook the management of a school;
-they arranged their establishment in the form of a
-square divided into $9$~rooms. The playground occupied
-the center, and $24$~scholars the rooms around it,
-$3$~being in each. In spite of the strictness of discipline,
-it was suspected that the boys were in the habit of
-\Figures{0.8}{53}{0.8}{54}
-playing truant, and it was determined to set a strict
-watch. To assure themselves that all the boys were
-on the premises, they visited the rooms, and found $3$~in
-each, or $9$~in a row. Four boys then went out, and
-the wise men soon after visited the rooms, and finding
-$9$~in each row, thought all was right. The four boys
-then came back, accompanied by four strangers; and
-the Gothamites, on their third round, finding still $9$~in
-each row, entertained no suspicion of what had
-taken place. Then $4$~more ``chums'' were admitted,
-\DPPageSep{196}{190}
-but the wise men, on examining the establishment a
-fourth time, still found $9$~in each row, and so came
-to an opinion that their previous suspicions had been
-unfounded.
-
-Figures 53--56 show how all this was possible, as they
-\Figures{0.8}{55}{0.8}{56}
-represent the contents of each room at the four different
-visits; \Fig{53}, at the commencement of the
-watch; \Fig{54}, when four had gone out; \Fig{55},
-when the four, accompanied by another four had returned;
-and \Fig{56}, when four more had joined them.
-\DPPageSep{197}{191}
-
-
-\Chapter{A mathematical game-puzzle.}
-\index{Counters, games}%
-\index{Games with counters}%
-\index{Game-puzzle}%
-\index{Mathematical game-puzzle}%
-\index{Puzzle!game}%
-
-``Place $15$~checkers on the table. You are to draw
-(take away either $1$,~$2$\Add{,} or~$3$); then your opponent is to
-draw (take $1$,~$2$\Add{,} or~$3$ at his option); then you draw
-again; then your opponent. You are to force him to
-take the last one.''
-
-Solution: When your opponent makes his last draw,
-there must be just one checker left for him to take.
-Since at every draw you are limited to removing either
-$1$,~$2$\Add{,} or~$3$, you can, by your last draw, leave just~$1$ if,
-and only if, you find on the board before that draw
-either $2$,~$3$\Add{,} or~$4$. You must, therefore, after your next
-to the last draw, leave the board so that he can not
-but leave, after his next to the last draw, either $2$,~$3$\Add{,} or~$4$.
-$5$~is clearly the number that you must leave at that
-time; since if he takes~$1$, he leaves~$4$; if~$2$,~$3$; if~$3$,~$2$.
-Similarly, after your next preceding draw you must
-leave~$9$; after your \emph{next} preceding,~$13$; that is, \begin{Thm}you
-must first draw~$2$. Then after each draw that he makes,
-you draw the difference between $4$ and the number
-that he has just drawn\end{Thm}, (if he takes~$1$, you follow by
-taking~$3$; if he takes~$2$, you take~$2$; if he takes~$3$, you
-take~$1$). Four being the sum of the smallest number
-and the largest that may be drawn, you can always
-make the sum of two consecutive draws (your opponent's
-and yours)~$4$, and you can not always make it
-any other number.
-
-Following would be a more general problem: Let
-\DPPageSep{198}{192}
-your opponent place on the board \emph{any} number of
-checkers leaving you to choose who shall first draw
-($1$,~$2$\Add{,} or~$3$ as before). Required to leave the last
-checker for him. Solution: If the number he places
-on the board is a number of the form~$4n + 1$, choose
-that he shall draw first. Then keep the number left
-on the board in that form by making $\text{his draw} + \text{yours}
-= 4$, until $n = 0$; that is, until there is but one left. If
-the number that he places on the board is not of that
-form; draw first and reduce it to a number that is of
-that form, and proceed as before.
-
-The problem might be further generalized by varying
-the number that may be taken at a draw.
-\DPPageSep{199}{193}
-
-
-\Chapter{Puzzle of the camels.}
-\index{Arabic camel puzzle}%
-\index{Camels, puzzle of}%
-\index{Puzzle!of the camels}%
-
-There was once an Arab who had three sons. In
-his will he bequeathed his property, consisting of
-camels, to his sons, the eldest son to have one-half of
-them, the second son one-third, and the youngest one-ninth.
-The Arab died leaving $17$~camels, a number
-not divisible by either $2$,~$3$\Add{,} or~$9$. As the camels could
-not be divided, a neighboring sheik was called in consultation.
-
-He loaned them a camel, so that they had $18$~to divide.
-\[
-\begin{tabular}{p{0.66\textwidth}@{}r@{}}
-The first son took~$\nicefrac{1}{2}$ \dotfill & $9$ \\
-The second took~$\nicefrac{1}{3}$ \dotfill & $6$ \\
-The third took~$\nicefrac{1}{9}$ \dotfill & $2$ \\
-\cline{2-2}
-Total \dotfill & $17$
-\end{tabular}
-\]
-
-They had divided equitably, and were able to return
-the camel that had been loaned to them.
-
-It should be noted that $\nicefrac{1}{2} + \nicefrac{1}{3} + \nicefrac{1}{9} = \nicefrac{17}{18}$, not
-unity. The numbers $9$,~$6$,~$2$ are in the same ratio as
-$\nicefrac{1}{2}$,~$\nicefrac{1}{3}$,~$\nicefrac{1}{9}$.
-
-This is probably an imitation of the old Roman
-\index{Inheritance, Roman problem}%
-\index{Roman inheritance problem}%
-inheritance problem which may be found in Cajori's
-\Title{History of Mathematics}, p.~79--80, or in his \Title{History
-of Elementary Mathematics}, p.~41.
-\DPPageSep{200}{194}
-
-
-\Chapter{A few more old-timers.}
-\index{Old-timers}%
-
-A man had eight gallons of wine in a keg. He
-\index{Kegs-of-wine puzzle}%
-wanted to divide it so as to get one-half. He had no
-measures but a three gallon keg, a five gallon keg and
-a seven gallon keg. How did he divide it? (The
-five gallon keg is unnecessary.)
-
-\Prob
-\Par{Only one dimension on Wall street.} Broker (determined
-\index{Dimension!only one in Wall street}%
-\index{Wall street}%
-to see the bright side): ``Every time I
-bought stocks for a rise, they went down; and when
-sold them, they went up. Luckily they can't go
-sidewise.''
-
-\Prob
-\Par{The apple women.} Two apple women had $30$~apples
-\index{Apple women}%
-each for sale. If the first had sold hers at the rate of
-$2$~for $1$~cent, she would have received $15$~cents. If the
-other had sold hers at $3$~for $1$~cent, she would have
-received $10$~cents. Both would have had $25$~cents.
-But they put them all together and sold the $60$~apples
-at $5$~for $2$~cents, thus getting $24$~cents. What became
-of the other cent?
-
-\Prob
-\Par{\GD\ with same remainder.} Given three (or more)
-\index{Divisor, greatest, with remainder}%
-\index{Greatest divisor with remainder}%
-integers, as $27$,~$48$,~$90$; required to find their greatest
-integral divisor that will leave the same remainder.
-
-Solution: Subtract the smallest number from each
-of the others. The \GCD\ of the differences is the
-required divisor. $48 - 27 = 21$; $90 - 27 = 63$; \GCD~of
-\DPPageSep{201}{195}
-$21$~and~$63$ is~$21$. If the given numbers be divided
-by~$21$, there is a remainder of~$6$ in each case.
-
-\Prob
-``$15$~Christians and $15$~Turks, being at sea in one and
-\index{Christians and Turks at sea}%
-\index{Turks and Christians at sea}%
-the same ship in a terrible storm, and the pilot declaring
-a necessity of casting the one half of those
-persons into the sea, that the rest might be saved;
-they all agreed, that the persons to be cast away should
-be set out by lot after this manner, viz., the $30$~persons
-should be placed in a round form like a ring, and
-then beginning to count at one of the passengers,
-and proceeding circularly, every ninth person should
-be cast into the sea, until of the $30$~persons there remained
-only~$15$. The question is, how those $30$~persons
-should be placed, that the lot might infallibly fall
-upon the $15$~Turks and not upon any of the $15$~Christians.''
-
-The early history of this problem is given by Professor
-Cajori in his \Title{History of Elementary Mathematics},
-p.~221--2, who also quotes mnemonic verses
-giving the solution: $4$~Christians, then $5$~Turks, then
-$2$~Christians,~etc.
-
-The solution is really found by arranging $30$~numbers
-or counters in a ring, or in a row to be read
-in circular order. Count according to the conditions
-of the problem, marking every ninth one~``T'' until
-$15$~are marked, then mark the remaining $15$~``C\@.''
-
-The same problem has appeared in other forms.
-Sometimes other classes of persons take the places of
-the Christians and Turks, sometimes every tenth one
-is lost instead of every ninth.
-\DPPageSep{202}{196}
-
-
-\Chapter{A few catch questions.}
-\index{Catch questions}%
-\index{Fallacies!catch questions}%
-\index{Questions, catch}%
-
-What number can be divided by every other number
-without a remainder?
-
-\Prob
-``Four-fourths exceeds three-fourths by what fractional
-part?'' This question will usually divide a company.
-
-\Prob
-Can a fraction whose numerator is less than its denominator
-be equal to a fraction whose numerator
-is greater than its denominator? If not, how can
-\[
-\frac{-3}{+6} = \frac{+5}{-10}?
-\]
-
-\Prob
-In the proportion
-\[
-+6 : -3 :: -10 : +5
-\]
-is not either extreme greater than either mean? What
-has become of the old rule, ``greater is to less as greater
-is to less''?
-
-\Prob
-Where is the fallacy in the following?
-\begin{align*}
-\text{$1$ mile square} &= \text{$1$ square mile} \\
-\therefore \text{$2$ miles square} &= \text{$2$ square miles}\Add{.}
-\end{align*}
-(Axiom: If
-equals be multiplied by equals,~etc.)
-
-Or in this (which is from Rebiere):
-\begin{align*}
-\text{A glass $\nicefrac{1}{2}$ full of water}
- &= \text{a glass $\nicefrac{1}{2}$ empty} \\
-\therefore \text{A glass full} &= \text{a glass empty}\Add{.}
-\end{align*}
-(Axiom:
-If equals be multiplied.)
-\DPPageSep{203}{197}
-
-
-\Chapter{Seven-counters game.}
-\index{Counters, games}%
-\index{Games with counters}%
-\index{Seven-counters game}%
-
-Required to place seven counters on seven of the
-eight spots in conformity to the following rule: To
-place a counter, one must set out from a spot that is
-unoccupied and move along a straight line to the spot
-where the counter is to be placed.
-\Figure[0.6]{57}
-
-The writer remembers seeing this as a child when
-the game was probably new. The solution is so easy
-that it offered no difficulty then. A puzzle whose solution
-is seen by almost any one in a minute or two
-is hardly worth a name, and one wonders to see it in
-Lucas's \Title{Récréations mathématiques} and dignified by
-the title ``The American Game of Seven and Eight.''
-\index{American game of seven and eight}%
-Lucas explains that the game, invented by Knowlton,
-\index{Knowlton}%
-\DPPageSep{204}{198}
-of Buffalo, N.~Y., was published, in~1883, by an American
-journal offering at first a prize to the person who
-should send, within a fixed time, the solution expressed
-in the fewest words.
-
-Lucas's statement of the solution is, \emph{Take always
-for point of destination the preceding point of departure.}
-Starting, for example, from the point~$4$, following
-the line $4$--$1$, and placing a counter at~$1$; the
-spot~$4$ must be the second spot of arrival. As one can
-reach~$4$ only by the line $7$--$4$, the spot~$7$ will necessarily
-be the second point of departure; etc., the seven moves
-being
-\[
-\text{$4$--$1$, $7$--$4$, $2$--$7$, $5$--$2$, $8$--$5$, $3$--$8$, $6$--$3$.}
-\]
-
-Lucas\footnote
- {Vol.~3, sixth recreation, from which the figure and description
- in the text are taken.}
-generalizes the game somewhat and adds
-other amusements with counters, less trivial than ``the
-American game.''
-\DPPageSep{205}{199}
-
-
-\Chapter{To determine direction by a watch.}
-\index{Compass, watch as}%
-\index{Direction determined by a watch}%
-\index{Watch as compass}%
-
-Those who are familiar with this very elementary
-operation usually take it for granted that every one
-knows it. Inquiry made recently in a class of normal
-school students revealed the fact that but few had
-heard of it and not one could explain or state the
-method. The writer has not infrequently known well
-informed persons to express surprise and pleasure at
-hearing it.
-
-With the face of the watch up, point the hour hand
-to the sun. Then the point midway between the present
-hour mark and~XII is toward the south. \Eg, at
-$4$~o'clock, when the hour hand is held toward the sun,
-II~is toward the south.
-
-Or the rule may be stated thus: With the point that
-is midway between the present hour and~XII held
-toward the sun, XII~is toward the south. \Eg, at
-$4$~o'clock hold~II toward the sun; then the line from
-the center of the face to the mark~XII is the south
-line.
-
-The reason is apparent. At $12$~o'clock the sun, the
-hour hand and the XII~mark are all toward the south.
-The sun and the hour hand revolve in the same direction,
-but the hour hand makes the complete revolution
-in $12$~hours, the sun in~$24$. Hence the rule.
-
-The errors due to holding the watch horizontal instead
-of in the plane of the ecliptic, and to the difference
-\DPPageSep{206}{200}
-between standard time and solar time, are negligible
-for the purpose to which this rule is usually put.
-
-Ball\footnote
- {\Title{Recreations}, p.~355.}
-mentions that the ``rule is given by W.~H. Richards,
-\Title{Military Topography}, London, 1883.'' Being
-so simple and convenient, it was probably known
-earlier.
-
-Professor Ball also gives (p.~356) the rule for the
-southern hemisphere: ``If the watch is held so that
-the figure~XII points to the sun, then the direction
-which bisects the angle between the hour of the day
-and the figure~XII will point due north.''
-\DPPageSep{207}{201}
-
-
-\Chapter[Advice to a building committee.]{Mathematical advice to a building
-committee.}
-\index{Advice to a building committee}%
-\index{Building Committee, advice to}%
-\index{Carroll, Lewis}%
-\index{Dodgson, C.~L.}%
-\index{Mathematical advice to a building committee}%
-
-It will be remembered that the man who, under the
-pseudonym Lewis Carroll, wrote \Title{Alice's Adventures
-in Wonderland} was really Rev.\ Charles Lutwidge
-Dodgson, lecturer in mathematics at Oxford. To a
-building committee about to erect a new school building
-he gave some advice that added gaiety to the deliberations.
-Children who have laughed at the Mock
-Turtle's description of his school life in the sea, as
-given to Alice, will recognize the same humor in these
-suggestions to the building committee:
-
-``It is often impossible for students to carry on
-accurate mathematical calculations in close contiguity
-to one another, owing to their mutual interference and
-a tendency to general conversation. Consequently
-these processes require different rooms in which irrepressible
-% [** TN: [sic] "conversationists", archaic spelling}
-conversationists, who are found to occur
-in every branch of society, might be carefully and permanently
-fixed.
-
-``It may be sufficient for the present to enumerate
-the following requisites; others might be added as the
-funds permitted:
-
-``A\@. A very large room for calculating greatest
-common measure. To this a small one might be attached
-for least common multiple; this, however, might
-be dispensed with.
-
-``B\@. A piece of open ground for keeping roots and
-\DPPageSep{208}{202}
-practising their extraction; it would be advisable to
-keep square roots by themselves, as their corners are
-apt to damage others.
-
-``C\@. A room for reducing fractions to their lowest
-\index{Fractions}%
-terms. This should be provided with a cellar for keeping
-the lowest terms when found.
-
-``D\@. A large room, which might be darkened and
-fitted up with a magic lantern for the purpose of exhibiting
-circulating decimals in the act of circulation.
-\index{Circulating decimals}%
-
-``E\@. A narrow strip of ground, railed off and carefully
-leveled, for testing practically whether parallel
-lines meet or not; for this purpose it should reach, to
-use the expressive language of Euclid, `ever so far.'\,''
-\index{Euclid}%
-\DPPageSep{209}{203}
-
-
-\Chapter{The golden age of mathematics.}
-\index{Golden age of mathematics}%
-\index{Literature of mathematics}%
-
-``The eighteenth century was philosophic, the nineteenth
-scientific.'' Mathematics---itself ``the queen of
-the sciences,'' as Gauss phrased it---is the necessary
-\index{Gauss}%
-method of all exact investigation. Kepler exclaimed:
-\index{Kepler}%
-``The laws of nature are but the mathematical thoughts
-of God.'' No wonder, therefore, that the nineteenth
-century surpassed its predecessors in extent and variety
-of mathematical invention and application.
-
-One reads now of ``the recent renaissance of mathematics.''
-\index{Renaissance of mathematics}%
-Strictly, there is no new birth or awakening
-of mathematics, for its productivity has long been
-continuous. Being the index of scientific progress, it
-must rise with the rise of civilization. That rise has
-been so rapid of late that, speaking comparatively,
-one may be justified in characterizing the present great
-mathematical activity as a renaissance.
-
-``The committee appointed by the Royal Society to
-\index{Royal Society's catalog}%
-report on a catalogue of periodical literature estimated,
-in~1900, that more than $1500$~memoirs on pure
-mathematics were now issued annually.''\footnote
- {Ball, \Title{Hist.}, p.~455.}
-
-Poets put the golden age of the race in the past.
-Prophets have seen that it is in the future. The recent
-marvelous growth of mathematics has been said to
-place \emph{its} golden age in the present or the immediate
-\index{Pierpont, James}%
-future. Professor James Pierpont,\footnote
- {Address before the department of mathematics of the International
- Congress of Arts and Science, St.~Louis, Sept.~20,
- 1904, on ``The History of Mathematics in the Nineteenth Century,''
- \Title{Bull.\ Am.\ Mathem.\ Society}, 11:3:159.}
-after summing up
-\DPPageSep{210}{204}
-the mathematical achievements of the nineteenth century,
-exclaimed: ``We who stand on the threshold of
-a new century can look back on an era of unparalleled
-progress. Looking into the future an equally bright
-prospect greets our eyes; on all sides fruitful fields of
-research invite our labor and promise easy and rich
-returns. Surely this is the golden age of mathematics!''
-
-And this golden age must last as long as men interrogate
-nature or value precision or seek truth. ``Mathematics
-is \DPchg{preeminently}{pre-eminently} cosmopolitan and eternal.''
-\DPPageSep{211}{205}
-
-
-\Chapter[To make teaching more concrete.]{The movement to make mathematics
-teaching more concrete.}
-\index{Algebra!teaching of|EtSeq}%
-\index{Arithmetic!teaching|EtSeq}%
-\index{Concrete, mathematics teaching}%
-\index{Geometry!teaching|EtSeq}%
-\index{Movement to make teaching more concrete}%
-\index{Teaching made concrete|EtSeq}%
-\index{Mathematics!teaching more concrete}%
-
-With the increased mathematical production has
-come a movement for improved teaching. The impetus
-is felt in many lands. ``The world-wide movement
-in the teaching of mathematics, in the midst of
-which we stand,'' are the recent words of a leader in
-this department.\footnote
- {Dr.\ J.~W.~A. Young, assistant professor of the pedagogy
-\index{Young, J. W. A.}%
- of mathematics in the University of Chicago, in an address
- before the mathematical section of the Central Association of
- Science and Mathematics Teachers, Nov.~30, 1906.}
-
-The movement is, in large part, for more concrete
-teaching---for a closer correlation between the mathematical
-subjects themselves and between the mathematics
-and the natural sciences, for extensive use of
-graphical representation, the introduction of more problems
-pertaining to pupils' interests and experiences, a
-larger use of induction and appeal to intuition at the
-expense of rigorous proof in the earlier years, the
-postponement of the more abstract topics, and the
-constant aim to show the useful applications.
-
-Some of the more conservative things that are urged
-are what every good teacher has been doing for years.
-On the other hand, some of the more radical suggestions
-will doubtless prove impractical and be abandoned.
-Still the movement as a whole is healthful
-and full of promise.
-
-Among American publications that are taking part
-\DPPageSep{212}{206}
-in it may be mentioned the magazine \Title{School Science
-and Mathematics}, which is doing much for the correlation
-of elementary pure and applied mathematics,
-the \Title{Reports} of the various committees, the \Title{Proceedings}
-of the Central Association of Science and Mathematics
-Teachers and similar organizations, and Dr.\
-Young's new book, \Title{The Teaching of Mathematics}.
-
-The \Title{Public School Journal} says, ``The position of
-mathematics as a mental tonic would be strengthened,''
-and quotes Fourier, ``The deeper study of nature is the
-\index{Fourier}%
-most fruitful source of mathematical study.''
-
-The movement to teach the calculus through engineering
-\index{Calculus}%
-problems and the like has attracted wide
-attention.
-
-Some of the applications of the rudiments of descriptive
-geometry to drawing (mechanical, perspective
-\index{Descriptive geometry}%
-\index{Geometry!descriptive}%
-etc.)\ are not far to seek in works on drawing.
-
-The applications of geometry to elementary science
-have been given in outline. It would be well if there
-were available lists of the common applications in the
-trades. \Eg, (in the carpenter's trade):
-
-The chalk line to mark a \emph{straight} (etymologically,
-\emph{stretched}) line. Illustrating the old statement, ``The
-straight line is the shortest distance between two
-points.''
-
-Putting the spirit level in two non-parallel positions
-on a plane surface to see whether the surface is horizontal.
-``A plane is determined by two intersecting
-straight lines.''
-
-Etc.
-
-Perhaps most teachers of geometry have made, or
-induced pupils to make, some such list; but the writer
-is not aware that any extensive compilation is in print.
-
-Fairly complete lists of the applications of algebra
-\DPPageSep{213}{207}
-to the natural sciences may be found in the publications
-named above.
-
-The new industrial arithmetic is one of the educational
-\index{De@{DeKalb normal school}}%
-\index{Measurements treated statistically|(}%
-\index{Statistics, mathematical treatment of|(}%
-features of our time. There should be an arithmetic
-with problems drawn largely from agricultural
-life. The 1905 catalog of the Northern Illinois State
-Normal School, De Kalb, contains a valuable classified
-outline of child activities involving and illustrating
-number. Dr.\ Charles~A. McMurry, in his \Title{Special
-Method in Arithmetic}, mentions the need of ``much
-more abundant statistical data than the arithmetics
-contain.''
-
-If we could have these things as teaching \emph{material},
-without the affliction of a fad for teaching mathematics
-\emph{entirely} through its practical applications, it
-would be a boon indeed.
-
-While rejoicing in the movement for correlation of
-mathematics and the other sciences, these two points
-should not be overlooked:
-
-1. The sciences commonly called natural are not
-\index{Social sciences treated mathematically|(}%
-the only observational sciences. The field of applied
-mathematics is as broad as the field of definite knowledge
-or investigation. Some parts of this field are
-specially worthy of note in this connection. The statistical
-sciences, the social sciences treated mathematically,
-the application of the methods of exact science
-to social measurements such as those obtained in educational
-psychology and the study of population, public
-health, economic problems etc.---these are sciences
-aiming at accuracy. They seek to achieve expression
-in natural law. They offer some of the best opportunities
-of applied mathematics. The recent growth
-in the sciences of this group has been, if possible, more
-marked than that of the physical sciences. Nor are
-\DPPageSep{214}{208}
-they less characteristic of the spirit of our time. Indeed
-it has been said that the quotation beginning the
-preceding section should be extended so as to say,
-``The eighteenth century was philosophic, the nineteenth
-scientific, and the twentieth is to be sociologic.''
-
-The statistical sciences call for a broad acquaintance
-with mathematical lore which is sometimes regarded
-as abstract and impractical by certain critics of current
-mathematical curricula.\footnote
- {It is true that the statistical sciences are exposed to caricature,
- as in the story of the German statistician who announced
- that he had tabulated returns from the marriage records
- of the entire country for the year and had discovered
- that the number of men married that year was exactly equal
- to the number of women married in the same period of time!
- It is true that statisticians have (rarely) computed results
- that might have been deduced \Foreign{a~priori}. It is true also that
- some of the results of statistical science have not proved to
- be practical or yielded material returns. But these things
- might be said also of the natural sciences, whose inestimable
- value is everywhere recognized. The social sciences mathematically
- developed are to be one of the controlling factors in
- civilization.}
-The social sciences are not
-studied by those who are pursuing elementary mathematical
-courses. It is not proposed that elementary
-mathematics should be correlated with them instead
-of with the physical sciences, or in addition thereto.
-But it should be remembered that use in the physical
-sciences is by no means the only ultimate aim which
-makes mathematics practical.
-\index{Measurements treated statistically|)}%
-\index{Social sciences treated mathematically|)}%
-\index{Statistics, mathematical treatment of|)}%
-
-2. The beautiful has its place in mathematics as
-\index{Beauty in mathematics}%
-\index{Literature of mathematics|(}%
-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.
-\DPPageSep{215}{209}
-He must be a ``practical'' man who can see no poetry
-in mathematics!
-
-Let mathematics be correlated with physical science;
-let it be concrete; but let the movement be understood
-and the subject taught in the light of the broadest
-educational philosophy.
-\index{Literature of mathematics|)}%
-\DPPageSep{216}{210}
-
-
-\FNChapter[An exercise in public speaking.]{The mathematical recitation as an
-exercise in public speaking.}
-
-\footnotetext{Article by the writer in \Title{New York Education}, now \Title{American
- Education}, for January, 1899\Add{.}}
-\index{Exercise in public speaking}%
-\index{Mathematical recitation as an exercise in public speaking}%
-\index{Oratory, mathematical recitation as exercise}%
-\index{Recitation as an exercise in public speaking}%
-\index{Speaking, recitation as an exercise in}%
-
-The value of translating from a foreign language,
-in broadening the vocabulary by compelling the mind
-to move in unfrequented paths of thought; of drawing,
-in quickening the appreciation of exact relations,
-proportion and perspective; of the natural sciences, in
-developing independence of thought---this is all familiar
-to the student of oratory; often has he been told
-the value of pursuing these studies for one entering
-his profession. But one rarely hears of the mathematical
-recitation as a preparation for public speaking.
-Yet mathematics shares with these studies 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 speaker and hearers alike be drawn wholly
-away from the speaker and concentrated upon his
-thought. In perhaps no other class-room 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
-\DPPageSep{217}{211}
-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 habits 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.
-
-How interesting is a recitation from this point of
-view! I do not recall more than two pupils reciting
-mathematics in an affected manner. In both cases this
-passed away. One of these, a lady who was previously
-acquainted with the work done during the early part
-of the term, lost her mannerisms when the class took
-up a subject that was advance work to her, and that
-called out her higher powers.
-
-The continual use of diagrams to make the meaning
-clear stimulates the student's power of illustration.
-
-The effect of mathematical study on the orator in his
-ways of thinking is apparent---the cultivation of clear
-and vigorous deduction from known facts.
-
-One could almost wish that our institutions for the
-teaching of the science and the art of public speaking
-would put over their doors the motto that Plato had
-\index{Plato}%
-over the entrance to his school of philosophy: ``Let
-no one who is unacquainted with geometry enter here.''
-\DPPageSep{218}{212}
-
-
-\FNChapter[Nature of mathematical reasoning.]{The nature of mathematical reasoning.}
-\index{Definition!of mathematics|(}%
-\index{Mathematical reasoning, nature of}%
-\index{Mathematics!nature of|(}%
-\index{Nature of mathematical reasoning}%
-\index{Reasoning, mathematical}%
-
-\footnotetext{vanced section of teachers institutes. For a treatment of old
- and new definitions of mathematics, the reader is referred to
-\index{Mathematics!definitions|FN}%
- Prof.\ Maxime Bôcher's ``The Fundamental Conceptions and
-\index{Bocher@Bôcher, M.|FN}%
- Methods of Mathematics,'' \Title{Bull.\ Am.\ Math.\ Soc.}, II:3:115--135. \\
- (Footnote text is truncated in the original.---\textit{Transcriber.})}
-
-Why is mathematics ``the exact science''? Because
-\index{Exact science}%
-of its self-imposed limitations. Mathematics concerns
-itself, not with any problem of the nature of things
-in themselves, but with the simpler problems of the
-relations between things. Starting from certain definite
-assumptions, the mathematician seeks only to
-arrive by legitimate processes at conclusions that are
-surely right if the data are right; as in geometry.
-So the arithmetician is concerned only that the result
-of his computation shall be correct assuming the data
-to be correct; though if he is also a teacher, he is in
-that capacity concerned that the data of the problems
-set for his pupils shall correspond to actual commercial,
-industrial or scientific conditions of the present day.
-
-Mathematics is usually occupied with the consideration
-\index{Cistern problem|(}%
-of only one or a few of the phases of a situation.
-Of the many conditions involved, only a few of the
-most important and the most available are considered.
-All other variables are treated as constants. Take for
-illustration the ``cistern problem,'' which as it occurs
-in the writings of Heron of Alexandria (c.\ 2d~cent.\ \BC)
-\index{Heron of Alexandria}%
-must be deemed very respectable on the score
-of age: given the time in which each pipe can fill a
-\DPPageSep{219}{213}
-cistern separately, required the time in which they
-will fill it together. This assumes the flow to be constant.
-Other statements of the problem, in which
-one pipe fills while another empties, presuppose the
-outflow also to be constant whether the cistern is full
-or nearly empty; or at least the rate of outflow is
-taken as an average rate and treated as a constant.
-Or the ``days-work problem'' (which is only the cistern
-\index{Days-work problem}%
-problem disguised): given the time in which each man
-can do a piece of work separately, required the time
-in which they will do it together. This assumes that
-the men work at the same rate whether alone or together.
-Some persons who have employed labor know
-how violent an assumption this is, and are prepared to
-defend the position of the thoughtless school boy who
-says, ``If A~can do a piece of work in $5$~days which
-B~can do in $3$~days, it will take them $8$~days working
-together,'' as against the answer $1 \nicefrac{7}{8}$~days, which is
-deemed orthodox among arithmeticians. Or, to move
-up to the differential calculus for an illustration: ``The
-\index{Calculus}%
-differentials of variables which change non-uniformly
-are what \emph{would be} their corresponding increments if
-at the corresponding values considered the change of
-each became and continued uniform with respect to the
-same variable.''\footnote
- {Taylor's \Title{Calculus}, p.~8.}
-
-Mathematics resembles fine art in that each abstracts
-\index{Art and mathematics}%
-\index{Fine art and mathematics}%
-some one pertinent thing, or some few things, from the
-mass of things and concentrates attention on the element
-selected. The landscape painter gives us, not
-every blade of grass, but only those elements that
-serve to bring out the meaning of the scene. With
-mathematics also as with fine art, this may result in
-a more valuable product then any that could be obtained
-by taking into account every element. The
-\index{Cistern problem|)}%
-\DPPageSep{220}{214}
-portrait painted by the artist does not exactly reproduce
-the subject as he was at any one moment of his
-life, yet it may be a truer representation of the man
-than one or all of his photographs. So it is with one
-of Shakespeare's historical dramas and the annals
-which were its ``source.'' ``The truest things are things
-that never happened.''
-
-Mathematics is a science of the ideal. The magnitudes
-\index{Ideal, mathematics science of}%
-of geometry exist only as mental creations, a
-chalk mark being but a physical aid to the mind in
-holding the conception of a geometric line.
-
-The concrete is of necessity complex; only the abstract
-\index{Concrete necessarily complex}%
-can be simple. This is why mathematics is the
-simplest of all studies---simplest in proportion to the
-mastery attained. The same standard of mastery being
-applied, physics is much simpler than biology: it is
-more mathematical. As we rise in the scale mathematically,
-relations become simple, until in astronomy we
-find the nearest approach to conformity by physical
-nature to a \emph{single} mathematical law, and we see a
-meaning in Plato's dictum, ``God geometrizes continually.''
-\index{Plato}%
-
-Mathematics is thinking God's thought after him.
-\index{Orbits of planets}%
-\index{Planetary orbits}%
-When anything is \emph{understood}, it is found to be susceptible
-of mathematical statement. The vocabulary of
-mathematics ``is the ultimate vocabulary of the material
-universe.'' The planets had for many centuries been
-recognized as ``wanderers'' among the heavenly bodies;
-much had come to be known about their movements;
-Tycho Brahe had made a series of careful
-\index{Tycho Brahe}%
-\index{Brahe, Tycho}%
-observations of Mars; Kepler stated the law: Every
-\index{Kepler}%
-planet moves in an elliptical orbit with the sun at one
-focus, and the radius vector generates equal areas in
-equal times. When the motion was understood, it was
-\DPPageSep{221}{215}
-expressed in the language of mathematics. Gravitation
-\index{Gravitation}%
-waited long for a Newton to state its law. When the
-\index{Newton}%
-statement came, it was in terms of ``the ultimate vocabulary'':
-Every particle of matter in the universe attracts
-every other particle with a force varying directly as the
-masses, and inversely as the square of the distances.
-When any other science---say psychology---becomes as
-\index{Psychology}%
-definite in its results, those results will be stated in as
-mathematical language. After many experiments to
-determine the measure of the increase of successive
-sensations of the same kind when the stimulus increases,
-and after tireless effort in the application of the
-``just perceptible increment'' as a unit. Prof.\ G.~T.
-Fechner of Leipsic announced in~1860, in his \Title{Psychophysik},
-\index{Fechner, G.~T.}%
-that the sensation varies as the logarithm of
-the stimulus. Fechner's law has not been established
-by subsequent investigations; but it was the expression
-of definiteness in thinking, whether that thinking was
-correct or not, and it illustrates mathematics as the
-language of precision.
-
-Mathematics, the science of the ideal, becomes the
-\index{Ideal, mathematics science of}%
-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---a
-problem in arithmetic or algebra shown to depend
-on previous problems and to require only the fundamental
-operations, the theorems of geometry shown
-to depend on the definitions and axioms, the unknown
-parts of a triangle computed from the known, the
-simplifications and far-reaching generalizations of the
-calculus, etc. It is true that we often have successive
-substitutions of simpler concepts in other sciences
-\DPPageSep{222}{216}
-(\eg, the reduction of the forms of logical reasoning
-to type forms; the simplifications culminating in the
-formulas of chemistry; etc.)\ but we naturally apply
-the adjective \emph{mathematical} to those phases of any science
-in which this method predominates. In this view
-also it is seen why mathematical rigor of demonstration
-is itself an advancing standard. ``Archimedean proof''
-\index{Archimedean proof}%
-was to the Greeks a synonym for unquestionable demonstration.
-\index{Greeks}%
-
-If a relation between variables is stated in mathematical
-\index{Formula, principle and rule}%
-\index{Principle, rule and formula}%
-\index{Rule, principle and formula}%
-symbols, the statement is a formula. A formula
-translated into words becomes a principle if the indicative
-mode is used, a rule if the imperative mode.
-
-Mathematics is ``ultimate'' in the generality of its
-\index{Diagonals of a polygon}%
-reasoning. By the aid of symbols it transcends experience
-and the imaging power of the mind. It determines,
-for example, the number of diagonals of a
-polygon of $1000$~sides to be $498500$ by substitution in
-the easily deduced formula $\SlantFrac{n(n - 3)}{2}$, although one
-never has occasion to draw a representation of a
-$1000$-gon and could not make a distinct mental picture
-of its $498500$ diagonals.
-
-If there are other inhabited planets, doubtless ``these
-\index{e@{$e$}}%
-\index{p@{$\pi$}}%
-\index{Inhabited planets}%
-\index{Planets, inhabited}%
-all differ from one another in language, customs and
-laws.'' But one can not imagine a world in which $\pi$~is
-not equal to~$3.14159+$, or $e$~not equal to~$2.71828+$,
-though all the \emph{symbols} for number might easily be
-very different.
-
-In recent years a few ``astronomers,'' with an enterprise
-\index{Astronomers|(}%
-\index{Mars, signaling|(}%
-that would reflect credit on an advertising bureau,
-have discussed in the newspapers plans for communicating
-with the inhabitants of Mars. What symbols
-could be used for such communication? Obviously
-those which must be common to rational beings everywhere.
-\DPPageSep{223}{217}
-%[** TN: Next two lines transposed in the original]
-Accordingly it was proposed to lay out an
-equilateral triangle many kilometers on a side and
-illuminate it with powerful arc lights. If our Martian
-neighbors should reply with a triangle, we could then
-test them on other polygons. Apparently the courtesies
-exchanged would for some time have to be confined
-to the amenities of geometry.
-
-Civilization is humanity's response to the first---not
-\index{Civilization and mathematics}%
-\index{Concrete, mathematics teaching}%
-\index{Mathematics!teaching more concrete}%
-\index{Teaching made concrete}%
-the last, or by any means the greatest---command of
-its Maker, ``Subdue the earth and have dominion over
-it.'' And the aim of applied mathematics is ``the
-mastery of the world quantitatively.'' ``Science is only
-quantitative knowledge.'' Hence mathematics is an
-index of the advance of civilization.
-
-The applications of mathematics have furnished the
-chief incentive to the investigation of pure mathematics
-and the best illustrations in the teaching of it; yet
-the mathematician must keep the abstract science in
-advance of the need for its application, and must even
-push his inquiry in directions that offer no prospect
-of any practical application, both from the point of
-view of truth for truth's sake and from a truly far-sighted
-utilitarian viewpoint as well. Whewell said,
-\index{Whewell}%
-``If the Greeks had not cultivated conic sections, Kepler
-\index{Greeks}%
-\index{Kepler}%
-could not have superseded Ptolemy.'' Behind the
-\index{Ptolemy}%
-artisan is a chemist, ``behind the chemist a physicist,
-behind the physicist a mathematician.'' It was Michael
-Faraday who said, ``There is nothing so prolific in
-\index{Faraday}%
-utilities as abstractions.''
-\index{Astronomers|)}%
-\index{Definition!of mathematics|)}%
-\index{Mars, signaling|)}%
-\index{Mathematics!nature of|)}%
-\DPPageSep{224}{218}
-
-
-\Chapter{Alice in the wonderland of mathematics.}
-\index{Alice in the wonderland of mathematics}%
-\index{Carroll, Lewis}%
-\index{Dodgson, C.~L.}%
-\index{Mathematics!Alice in the wonderland of}%
-\index{Wonderland of mathematics}%
-
-Years after Alice had her ``Adventures in Wonderland''
-and ``Through the Looking-glass,'' described
-by ``Lewis Carroll,'' she went to college. She was a
-young woman of strong religious convictions. As she
-studied science and philosophy, she was often perplexed
-to reduce her conclusions in different lines to
-a system, or at least to find some analogy which would
-make the coexistence of the fundamental conceptions
-of faith and of science more thinkable. These questions
-have puzzled many a more learned mind than hers,
-but never one more earnest.
-
-Alice developed a fondness for mathematics and
-elected courses in it. The professor in that department
-had lectured on $n$-dimensional space, and Alice
-had read E.~A. Abbott's charming little book, \Title{Flatland;
-a Romance of Many Dimensions, by a Square}, which
-had been recommended to her by an instructor.
-
-The big daisy-chain which was to be a feature of the
-approaching class-day exercises was a frequent topic
-of conversation among the students. It was uppermost
-in her mind one warm day as she went to her room
-after a hearty luncheon and settled down in an easy
-chair to rest and think.
-
-``Why!'' she said, half aloud, ``I was about to make
-a daisy-chain that hot day when I fell asleep on the
-bank of the brook and went to Wonderland---so long
-\DPPageSep{225}{219}
-ago. That was when I was a little girl. Wouldn't it
-be fun to have such a dream now? If I were a child
-again, I'd curl up in this big chair and go to sleep
-this minute. `Let's pretend.'\,''
-
-So saying, and with the magic of this favorite phrase
-upon her, she fell into a pleasant revery. Present surroundings
-faded out of consciousness, and Alice was
-in Wonderland.
-
-``What a long daisy-chain this is!'' thought Alice.
-``I wonder if I'll ever come to the end of it. Maybe
-it hasn't any end. Circles haven't ends, you know.
-Perhaps it's like finding the end of a rainbow. Maybe
-I'm going off along one of the infinite branches of a
-curve.''
-
-Just then she saw an arbor-covered path leading off
-to one side. She turned into it; and it led her into
-a room---a throne-room, for there a fairy or goddess
-sat in state. Alice thought this being must be one
-of the divinities of classical mythology, but did not
-know which one. Approaching the throne she bowed
-very low and simply said, ``Goddess''; whereat that
-personage turned graciously and said, ``Welcome,
-Alice.'' It did not seem strange to Alice that such a
-being should know her name.
-
-``Would you like to go through Wonderland?''
-
-``Oh! yes,'' answered Alice eagerly.
-
-``You should go with an attendant. I will send the
-court jester, who will act as guide,'' said the fairy,
-at the same time waving a wand.
-
-Immediately there appeared---Alice could not tell
-how---a courtier dressed in the fashion of the courts of
-the old English kings. He dropped on one knee before
-the fairy; then, rising quickly, bowed to Alice, addressing
-her as, ``Your Majesty.''
-\DPPageSep{226}{220}
-
-It seemed pleasant to be treated with such deference,
-but she promptly answered, ``You mistake; I am only
-Miss~---''
-
-Here the fairy interrupted: ``Call her `Alice'. The
-name means `princess.'\,''
-
-``And you may call me `Phool.'\,'' said the courtier;
-``only you will please spell it with a~\textit{ph}.''
-
-``How can I spell it when I am only speaking it?''
-she asked.
-
-``\emph{Think} the~\textit{ph}.''
-
-``Very well,'' answered Alice rather doubtfully, ``but
-who ever heard of spelling `fool' with~\textit{ph}?''
-
-Then he smiled broadly as he replied: ``I am an
-anti-spelling-reformer. I desire to preserve the~\textit{ph}
-in words in place of~\textit{f} so that one may recognize their
-foreign origin and derivation.''
-
-``Y-e-s,'' said Alice, ``but what does \emph{phool} come
-from?''
-
-Again the fairy interrupted. Though always gracious,
-she seemed to prefer brevity and directness.
-``You will need the magic wand.''
-
-So saying, she handed it to the jester. The moment
-he had the wand, the fairy vanished. And the girl and
-the courtier were alone in the wonderful world, and
-they were not strangers. They were calling each other
-``Alice'' and ``Phool.'' And he held the magic wand.
-
-One flourish of that wand, and they seemed to be
-in a wholly different country. There were many
-beings, having length, but no breadth or thickness;
-or, rather, they were very thin in these two dimensions,
-and uniformly so. They were moving only in
-one line.
-
-``Oh! I know!'' exclaimed Alice, ``This is Lineland.
-I read about it.''
-\DPPageSep{227}{221}
-
-``Yes,'' said Phool; ``if you hadn't read about it or
-thought about it, I couldn't have shown it to you.''
-
-Alice looked questioningly at the wand in his hand.
-
-``It has marvelous power, indeed,'' he said. ``To
-show you in this way what you have thought about,
-that is magic; to show you what you had never thought
-of, would be---''
-
-Alice could not catch the last word. A little twitch
-of the wand set them down at a different point in the
-line, where they could get a better view of lineland.
-Alice thrust her hand across the line in front of one of
-the inhabitants. He stopped short. She withdrew it.
-He was amazed at the apparition: a body (or point)
-had suddenly appeared in his world and as suddenly
-vanished. Alice was interested to see how a linelander
-could be imprisoned between two points.
-
-``He never thinks to go around one of the obstacles,''
-she said.
-
-``The line is his world,'' said Phool. ``One never
-thinks of going out of the world to get around an
-obstacle.''
-
-``If I could communicate with him, could I teach
-him about a second dimension?''
-
-``He has no apperceiving mass,'' said Phool laconically.
-
-``Very good,'' said Alice, laughing; ``surely he has
-no mass. Then he can get out of his narrow world
-only by accident?''
-
-``Accident!'' repeated Phool, affecting surprise, ``I
-thought you were a philosopher.''
-
-``No,'' replied Alice, ``I am only a college girl.''
-
-``But,'' said Phool, ``you are a lover of wisdom.
-Isn't that what `philosopher' means? You see I'm a
-stickler for etymologies.''
-\DPPageSep{228}{222}
-
-``All right,'' said Alice, ``I am a philosopher then.
-But tell me how that being can ever appreciate space
-outside of his world.''
-
-``He might evolve a few dimensions.''
-
-Alice stood puzzled for a minute, though she knew
-that Phool was jesting. Then a serious look came
-into his face, and he continued:
-
-``One-dimensional beings can learn of another dimension
-only by the act of some being from without their
-world. But let us see something of a broader world.''
-
-So saying, he waved the wand, and they were in a
-country where the inhabitants had length and breadth,
-but no appreciable thickness.
-
-Alice was delighted. ``This is Flatland,'' she cried
-out. Then after a minute she said, ``I thought the
-Flatlanders were regular geometric figures.''
-
-Phool laughed at this with so much enjoyment that
-Alice laughed too, though she saw nothing very funny
-about it.
-
-Phool explained: ``You are thinking of the Flatland
-where all lawyers are square, and where acuteness
-is a characteristic of the lower classes while obtuseness
-is a mark of nobility. That would, indeed, be very
-flat; but we spell that with a capital~\textit{F}. This is flatland
-with a small~\textit{f}.''
-
-Alice fell to studying the life of the two-dimension
-people and thinking how the world must seem to them.
-She reasoned that polygons, circles and all other plane
-figures are always seen by them as line-segments; that
-they can not see an angle, but can infer it; that they
-may be imprisoned within a quadrilateral or any other
-plane figure if it has a closed perimeter which they
-may not cross; and that if a three-dimensional being
-were to cross their world (surface) they could appreciate
-\DPPageSep{229}{223}
-only the section of him made by that surface,
-so that he would appear to them to be two-dimensional
-but possessing miraculous powers of motion.
-
-Alice was pleased, but curious to see more. ``Let's
-see other dimensional worlds,'' she said.
-
-``Well, the three-dimensional world, you're in all
-\index{Dimension!fourth|(}%
-\index{Fourth dimension|(}%
-the time,'' said Phool, at the same time moving the
-wand a little and changing the scene, ``and now if you
-will show me how to wave this wand around through
-a fourth dimension, we'll be in that world straightway.''
-
-``Oh! I can't,'' said Alice.
-
-``Neither can I,'' said he.
-
-``Can anybody?''
-
-``They say that in four-dimensional space one can
-see the inside of a closed box by looking into it from
-a fourth dimension just as you could see the inside of
-a rectangle in flatland by looking down into it from
-above; that a knot can not be tied in that space; and
-that a being coming to our world from such a world
-would seem to us three-dimensional, as all we could
-see of him would be a section made by our space, and
-that section would be what we call a solid. He would
-appear to us---let us say---as human. And he would
-be not less human than we, nor less real, but more so;
-if `real' has degrees of comparison. The flatlander
-who crosses the linelander's world (line) appears to
-the native to be like the one-dimensional beings, but
-possessed of miraculous powers. So also the solid in
-flatland: the cross-section of him is all that a flatlander
-is, and that is only a section, only a phase of
-his real self. The ability of a being of more than
-three dimensions to appear and disappear, as to enter
-or leave a room when all doors were shut, might
-\DPPageSep{230}{224}
-make him seem to us like a ghost, but he would be
-more real and substantial than we are.''
-
-He paused, and Alice took occasion to remark:
-
-``That is all obtained by reason; I want to see a
-four-dimensional world.''
-
-Then, fearing that it might not seem courteous to
-her guide to appear disappointed, she added:
-
-``But I ought to have known that the wand couldn't
-show us anything we might wish to see; for then there
-would be no limit to our intelligence.''
-
-``Would unlimited intelligence mean the same thing
-as absolutely infinite intelligence?'' Phool asked.
-
-``That sounds to me like a conundrum,'' said Alice.
-``Is it a play on words?''
-\index{Dimension!fourth|)}%
-\index{Fourth dimension|)}%
-
-``There goes Calculus,'' said Phool. ``I'll ask him.---Hello!
-Cal.''
-\index{Calculus|EtSeq}%
-\index{Infinite|EtSeq}%
-
-Alice looked and saw a dignified old gentleman
-with flowing white beard. He turned when his name
-was called.
-
-While Calculus was approaching them, Phool said
-in a low tone to Alice: ``He'll enjoy having an eager
-pupil like you. This will be a carnival for Calculus.''
-
-When that worthy joined them and was made acquainted
-with the topic of conversation, he turned to
-Alice and began instruction so vigorously that Phool
-said, by way of caution:
-
-``Lass! Handle with care.''
-
-Alice did not like the implication that a girl could
-not stand as much mathematics as any one. But then
-she thought, ``That is only a joke,'' and she seemed
-vaguely to remember having heard it somewhere before.
-
-``If you mean,'' said Calculus, ``to ask whether a
-variable that increases without limit is the same thing
-\DPPageSep{231}{225}
-as absolute infinity, the answer is clearly No. A
-variable increasing without limit is always nearer to
-\emph{zero} than to absolute \emph{infinity}. For simplicity of illustration,
-compare it with the variable of uniform change,
-time, and suppose the variable we are considering
-doubles every second. Then, no matter how long it
-may have been increasing at this rate, it is still nearer
-zero than infinity.''
-
-``Please explain,'' said Alice.
-
-``Well,'' continued Calculus, ``consider its value at
-any moment. It is only half what it will be one second
-hence, and only quarter what it will be two seconds
-hence, when it will still be increasing. Therefore it
-is \emph{now} much nearer zero than infinity. But what is
-true of its value at the moment under consideration is
-true of any, and therefore of every, moment. An infinite
-is always nearer to zero than to infinity.''
-
-``Is that the reason,'' asked Alice, ``why one must
-say `increases without limit' instead of `approaches
-infinity as a limit'?''
-
-``Certainly,'' said Calculus; ``a variable can not approach
-infinity as a limit. Students often have to be
-reminded of this.''
-
-Alice had an uncomfortable feeling that the conversation
-was growing too personal, and gladly turned
-it into more speculative channels by remarking:
-
-``I see that one could increase in wisdom forever,
-though that seems miraculous.''
-
-``What do you mean by miraculous?'' asked Phool.
-
-``Why---'' began Alice, and hesitated.
-
-``People who begin an answer with `Why' are rarely
-able to give an answer,'' said Phool.
-
-``I fear I shall not be able,'' said Alice. ``An etymologist''
-(this with a sly look at Phool) ``might say
-\DPPageSep{232}{226}
-it means `wonderful'; and that is what I meant when
-speaking about infinites. But usually one would call
-that miraculous which is an exception to natural law.''
-
-``We must take the young lady over to see the curve
-\index{Analytic geometry|(}%
-tracing,'' said Calculus to Phool.
-
-``Yes, indeed!'' he replied. Then, turning to Alice,
-``Do you enjoy fireworks?''
-
-``Yes, thank you,'' said Alice, ``but I can't stay till
-dark.''
-
-``No?'' said Phool, with an interrogation. ``Well,
-we'll have them very soon.''
-
-``Fireworks in daytime?'' she asked.
-
-But at that moment Phool made a flourish with the
-wand, and it was night---a clear night with no moon
-or star. It seemed so natural for the magic wand to
-accomplish things that Alice was not \emph{very} much surprised
-at even this transformation. She asked:
-
-``Did you say you were to show me curve tracing?''
-
-``Yes,'' said Phool. ``Perhaps you don't attend the
-races, but you may enjoy seeing the \emph{traces}.''
-
-During this conversation the three had been walking,
-and they now came to a place where there was
-what appeared to be an enormous electric switchboard.
-A beautiful young woman was in charge.
-
-As they approached, Calculus said to Alice, ``That
-is Ana Lytic. You are acquainted with her, I presume.''
-
-``The name sounds familiar,'' said Alice, ``but I don't
-remember to have ever seen her. I should like to
-meet her.''
-
-On being presented, Alice greeted her new acquaintance
-as `Miss Lytic'; but that person said, in a very
-gracious manner:
-
-``Nobody ever addresses me in that way. I am always
-\DPPageSep{233}{227}
-called `Ana Lytic,' except by college students.
-They usually call me `Ana Lyt.' I presume they
-shorten my name thus because they know me so well.''
-
-In spite of the speaker's winning manner, the last
-clause made Alice somewhat self-conscious. Her
-cheeks felt very warm. She was relieved when, at that
-moment, Calculus said:
-
-\index{Graph of equation|(}%
-``This young lady would like to see some of your
-work.''
-
-``Some pyrotechnic curve tracing,'' interrupted the
-talkative Phool.
-
-Calculus continued: ``Please let us have an algebraic
-curve with a conjugate point.''
-
-Ana Lytic touched a button, and across the world
-of darkness (as it seemed to Alice) there flashed a
-sheet of light, dividing space by a luminous plane.
-It quickly faded, but left two rays of light perpendicular
-to each other, faint but apparently permanent.
-
-``These are the axes of coordinates,'' explained Ana
-Lytic.
-
-Then she pressed another button, and Alice saw
-what looked like a meteor. She watched it come from
-a great distance, cross the ray of light that had been
-called one of the axes, and go off on the other side
-as rapidly as it had come, always moving in the plane
-indicated by the vanished sheet of light. She thought
-of a comet; but instead of having merely a luminous
-tail, it left in its wake a permanent path of light. Ana
-Lytic had come close to Alice, and the two girls stood
-looking at the brilliant curve that stretched away
-across the darkness as far as the eye could reach.
-
-``Isn't it beautiful!'' exclaimed Alice.
-
-Any attempt to represent on paper what she saw
-\DPPageSep{234}{228}
-must be poor and inadequate. \Fig[Figure]{58} is such an
-attempt.
-
-Suddenly she exclaimed: ``What is that \emph{point} of
-light?'' indicating by gesture a bright point situated
-as shown in the figure by~$P$.
-\Figure[1.0]{58}
-
-``That is a point of the curve,'' said Ana Lytic.
-
-``But it is away from all the rest of it,'' objected
-Alice.
-
-Going over to her apparatus and taking something---Alice
-could not see what---Ana Lytic began to write
-on what, in the darkness, might surely be called a
-\DPPageSep{235}{229}
-blackboard. The characters were of the usual size
-of writing on school boards, but they were characters
-of light and could be plainly read in the night. This
-is what she wrote:
-\[
-y^{2} = (x - 2)^{2} (x - 3).
-\]
-
-Stepping back, she said: ``That is the equation of
-the curve.''
-
-Alice expressed her admiration at seeing the equation
-before her and its graph stretching across the
-world in a line of light.
-
-``I never imagined coordinate geometry could be so
-beautiful,'' she said.
-
-``This is throwing light on the subject for you,''
-said Phool.
-
-``The point about which you asked,'' said Ana Lytic
-to Alice, ``is the point~$(2, 0)$. You see that it satisfies
-the equation. It is a point of the graph.''
-
-Alice now noticed that units of length were marked
-off on the dimly seen axes by slightly more brilliant
-points of light. Thus she easily read the coordinates
-of the point.
-
-``Yes,'' she said, ``I see that; but it seems strange
-that it should be off away from the rest.''
-
-``Yes,'' said Calculus, who had been listening all the
-time. ``One expects the curve to be continuous. Continuity
-is the message of modern scientific thought.
-This point seems to break that law---to be `miraculous,'
-as you defined the term a few minutes ago. If all
-observed instances but one have some visible connection,
-we are inclined to call that one miraculous
-and the rest natural. As only that seems wonderful
-which is unusual, the miraculous in mathematics would
-be only an isolated case.''
-
-``I thank you,'' said Alice warmly. ``That is the
-\DPPageSep{236}{230}
-way I should like to have been able to say it. An
-isolated case is perplexing to me. I like to think that
-there is a universal reign of law.''
-
-``\emph{Evidently},'' said Phool, ``here is an exception. It
-is \emph{obvious} that there are several alternatives, such as,
-for example, that the point is not on the graph, that
-the graph has an isolated point, \emph{and so forth}.''
-
-Calculus, Ana Lytic and Phool all laughed at this.
-To Alice's inquiry, Phool explained:
-
-``We often say `evidently' or `obviously' when we
-can't give a reason, and we conclude a list with `and
-so forth' when we can't think of another item.''
-
-Alice felt the remark might have been aimed at her.
-Still she had not used either of these expressions in this
-conversation, and Phool had made the remark in a
-general way as if he were satirizing the foibles of the
-entire human race. Moreover, if she felt inclined to
-resent it as an impertinent criticism from a self-constituted
-teacher, she remembered that it was only the
-jest of a jester and treated it merely as an interruption.
-
-``Tell me about the isolated point,'' she said to
-Calculus.
-
-He proceeded in a teacher-like way, which seemed
-appropriate in him.
-
-\index{Imaginary!branch of graph|(}%
-\Name{Calculus.} For $x = 2$ in this equation, $y = 0$. For any
-other value of~$x$ less than~$3$, what would $y$~be?
-
-\Name{Alice.} An imaginary.
-
-\Name{Calculus.} And what is the geometric representation
-of an imaginary number?
-
-\Name{Alice.} A line whose length is given by the absolute,
-or arithmetic, value of the imaginary and whose direction
-is perpendicular to that which represents positives
-and negatives.
-
-\Name{Calculus.} Good. Then---
-\DPPageSep{237}{231}
-\index{Complex numbers!branch of graph|(}%
-\index{Representation of complex numbers|(}%
-
-\Name{Alice} (bounding with delight at the discovery). Oh!
-I see! I see! There must be points of the graph outside
-of the plane.
-
-\Name{Calculus.} Yes, there are imaginary branches, and
-perhaps Ana Lytic will be good enough to show you
-now.
-\Figure[1.0]{59}
-
-\begin{Remark}
-The dotted line~$QPQ$, if revolved~$90°$ about $XX'$ as axis, remaining
-in that position in plane perpendicular to paper, would
-be the ``imaginary part'' of the graph.
-
-The dot-and-dash line $SRPRS$ represents the projection
-on the plane of the paper of the two ``complex parts.'' At~$P$
-each branch is in the plane of the paper, at each point~$R$ one
-branch is about $0.7$~from the plane each side of the paper, at~$S$
-each branch is $1.5$~from the plane,~etc.
-\end{Remark}
-\DPPageSep{238}{232}
-
-That young lady touched something on her magic
-switchboard, and another brilliant curve stretched
-across the heavens. The plane determined by it was
-perpendicular to the plane previously shown. (The
-dotted line in \Fig{59} represents in a prosaic way
-what Alice saw.)
-
-``O, I see!'' exclaimed Alice. ``That point is not
-isolated. It is the point in which this `imaginary'
-branch, which is as \emph{real} as any, pierces the plane of
-the two axes.''
-
-``Now,'' said Calculus, ``if instead of substituting
-real values for~$x$ and solving the equation for~$y$, you
-were to substitute real numbers for~$y$ and solve for~$x$,
-\index{Real numbers}%
-you would, in general, obtain for each value of~$y$ one
-real and two complex numbers as the values of~$x$.
-The curve through all the points with complex abscissas
-is neither in the plane of the axes nor in a
-plane perpendicular to it. But you shall see.''
-
-(The dot-and-dash line in \Fig{59} represents these
-branches.)
-
-When Ana Lytic made the proper connection at the
-switchboard, these branches of the curve also stood
-out in lines of light.
-
-Alice was more deeply moved than ever. There was
-a note of deep satisfaction in her voice as she said:
-
-``The point that troubled me because of its isolation
-is a point common to several branches of the curve.''
-
-``The supernatural is more natural than anything
-else,'' said Phool.
-
-``The miraculous,'' thought Alice, ``is only a special
-case of a higher law. We fail to understand things
-because they are connected with that which is out of
-our plane.''
-\index{Complex numbers!branch of graph|)}%
-\index{Imaginary!branch of graph|)}%
-\index{Representation of complex numbers|)}%
-\DPPageSep{239}{233}
-
-She added aloud: ``This I should call the \emph{miracle
-curve}.''
-
-``Yet there is nothing exceptional about this curve,''
-said Calculus. ``Any algebraic curve with a conjugate
-point has similar properties.''
-
-Then Calculus said something to Ana Lytic---Alice
-could not hear what---and Ana Lytic was just touching
-something on the switchboard when there was a crash
-of thunder. Alice gave a start and awoke to find herself
-in her own room at midday, and to realize that
-the slamming of a door in the corridor had been the
-thunder that terminated her dream.
-
-She sat up in the big chair and, with the motion
-that had been characteristic of her as a little girl, gave
-``that queer little toss of her head, to keep back the
-wandering hair that \emph{would} always get into her eyes,''
-and said to herself:
-
-``There aren't any curves of light across the sky
-at all! And worlds of one or two dimensions exist
-only in the mind. They are abstractions. But at least
-they are thinkable. I'm glad I had the dream. Imagination
-\emph{is} a magic wand.---The future life will be a
-\emph{real} wonderland, and---''
-\index{Analytic geometry|)}%
-\index{Graph of equation|)}%
-
-Then the ringing of a bell reminded her that it was
-time to start for an afternoon lecture, and she heard
-some of her classmates in the corridor calling to her,
-``Come, Alice.''
-\DPPageSep{240}{234}
-
-
-\BackMatter
-\Appendix{Bibliographic notes.}
-\index{Bibliographic!notes}%
-
-%[** TN: Smaller type in the original]
-\Par{Mathematical recreations.} The Ahmes papyrus, oldest mathematical
-\index{Ahmes papyrus}%
-\index{Mathematical recreations}%
-\index{Recreations, mathematical}%
-work in existence, has a problem which Cantor interprets
-as one proposed for amusement. At which Cajori remarks:\footnote
- {\Title{Hist.\ of Elem.\ Math.}, p.~24.}
-``If the above interpretations are correct, it looks as
-if `mathematical recreations' were indulged in by scholars
-forty centuries ago.''
-
-The collection of ``Problems for Quickening the Mind'' Cantor
-\index{Problems!for quickening the mind}%
-thinks was by Alcuin (735--804). Cajori's interesting
-\index{Alcuin}%
-comment\footnote
- {\Title{Id.}, p.~113--4.}
-is: ``It has been remarked that the proneness to
-propound jocular questions is truly Anglo-Saxon, and that
-Alcuin was particularly noted in this respect. Of interest is
-the title which the collection bears: `Problems for Quickening
-the Mind.' Do not these words bear testimony to the fact
-that even in the darkness of the Middle Ages the mind-developing
-power of mathematics was recognized?''
-
-Later many collections of mathematical recreations were
-published, and many arithmetics contained some of the recreations.
-Their popularity is noticeable in England and Germany
-in the seventeenth and eighteenth centuries.\footnote
- {A book entitled \Title{Rara Arithmetica} by Prof.\ David Eugene Smith,
-\index{Smith, D.~E.|FN}%
- is to be published by Ginn \&~Co.\ the coming summer or fall (1907).
- It will contain six or seven hundred pages and have three hundred illustrations,
- presenting graphically the most interesting facts in the history
- of arithmetic. Its author's reputation in this field insures the book
- an immediate place among the classics of mathematical history.}
-
-A good bibliography of mathematical recreations is given
-by Lucas.\footnote
- {I:237--248. Extensive as his list is, it is professedly restricted in
- scope. He says\DPtypo{.}{,} Nous donnons ci-après, suivant l'ordre chronologique,
- l'indication des principaux livres, mémoires, extraits de correspondance,
- qui ont été publiés sur l'Arithmétique de position et sur la Géométrie de
- situation. Nous avons surtout choisi les documents qui se rapportent
- aux sujets que nous avons traités ou que nous traiterons ultérieurement.}
-There are $16$~titles from the sixteenth century, $33$~from
-the seventeenth, $38$~from the eighteenth, and $100$~from
-the nineteenth century, the latest date being~1890. Young
-\DPPageSep{241}{235}
-(p.~173--4) gives a list of $20$~titles, mostly recent, in no case
-duplicating those of Lucas's list (except where mentioning a
-later edition). This gives a total of over two hundred titles.
-Now turn to two other collections, and we find the list greatly
-extended. Ahrens' \Title{Mathematische Unterhaltungen} (1900) has
-a bibliography of $330$~titles, including nearly all those given
-by Lucas. Fourrey's \Title{Curiositées Géométriques} (1907) has the
-most recent bibliography. It is extensive in itself and mostly
-supplementary to the lists by Lucas and Ahrens.
-
-In all the vast number of published mathematical recreations,
-the present writer does not know of a book covering
-the subject in general which was written and published in
-America. We seem to have taken our mathematics very seriously
-on this side of the Atlantic.
-
-\Par{Publications of foregoing sections in periodicals.} The sections
-\index{Periodicals, publication of foregoing sections in}%
-\index{Publication of foregoing sections in periodicals}%
-of this book which have been printed in magazines are
-as follows. The month and year in each case are those of the
-magazine, and the page is the page of this book at which the
-section begins.
-
-\Title{The Open Court}, January 1907, p.~\PgNo{218}; February, p.~\PgNo{212};
-March, p.~\PgNo{73},~\PgNo{76}; April, p.~\PgNo{109}; May, p.~\PgNo{143}, \PgNo{154}, \PgNo{196},~\PgNo{122};
-June, p.~\PgNo{81},~\PgNo{83}; July, p.~\PgNo{168},~\PgNo{170}.
-
-\Title{The Monist}, January 1907, p.~\PgNo{11},~\PgNo{15}.
-
-\Title{New York Education} (now \Title{American Education}), January
-1899, p.~\PgNo{210}.
-
-\Title{American Education}, September 1906, p.~\PgNo{59}; March 1907,
-p.~\PgNo{51}.
-
-Some of the articles have been altered slightly since their
-publication in periodical form.
-\DPPageSep{242}{236}
-
-
-%[** TN: "Index" capitalized in the original table of contents]
-\Appendix{Bibliographic index.}
-\index{Bibliographic!index}%
-
-{\small
-List of the publications mentioned in this book, with the pages where
-mentioned. The pages of this book are given after the imprint in each
-entry. These references are not included in the general index.
-
-A date in (\;) is the date of copyright.
-
-In many cases a work is barely mentioned. *~indicates either more
-extended use made of the book in this case, or direct (though brief)
-quotation, or a figure taken from the book.}
-
-% [** TN: http://www.gutenberg.org/ebooks/201]
-\Bibitem [Abbott, E.~A\@.] Flatland; a Romance of Many Dimensions,
-by a Square. [London, 1884] Boston, 1899. *\PgNo{218}.
-
-\Bibitem Ahrens. Mathematische Unterhaltungen und Spiele. Leipzig,
-1900. \PgNo{235}.
-
-\Bibitem American Education (monthly). Albany, N.~Y. \PgNo{145}, *\PgNo{210},
-\PgNo{235}.
-
-\Bibitem Annali di Matematica. Milan. \PgNo{38}.
-
-\Bibitem Argand, J.~R\@. Essai. Geneva, 1806. \PgNo{94}\Add{.}
-
-% [** TN: http://www.gutenberg.org/ebooks/26839]
-\Bibitem Ball, W.~W.~R\@. Mathematical Recreations and Essays. Ed.~4.
-Macmillan, London, 1905. (A book both fascinating and
-scholarly, attractive to every one with any taste for
-mathematical studies.) *\PgNo{35}, *\PgNo{38}, *\PgNo{41}, *\PgNo{83}, \PgNo{111}, \PgNo{117}, *\PgNo{122},
-\PgNo{123}, *\PgNo{127}, *\PgNo{141}, *\PgNo{171}, \PgNo{186}, \PgNo{187}, *\PgNo{200}\Add{.}
-
-% [** TN: http://www.gutenberg.org/ebooks/31246]
-\Bibitem Ball, W.~W.~R\@. Short Account of the History of Mathematics.
-Ed.~3. Macmillan, London, 1901. *\PgNo{34}, *\PgNo{35}, *\PgNo{37}, *\PgNo{123},
-*\PgNo{203}\Add{.}
-
-\Bibitem Beman and Smith. New Plane Geometry. Ginn (1895, '99).
-\PgNo{164}\Add{.}
-
-\Bibitem Bledsoe, A.~T\@. Philosophy of Mathematics. Lippincott, 1891
-(1867). \PgNo{150}.
-
-\Bibitem Brooks, Edward. Philosophy of Arithmetic~\dots\ Sower, Philadelphia
-(1876.) (An admirable popular presentation of
-some of the elementary theory of numbers; also historical
-notes). \PgNo{25}, \PgNo{31}, *\PgNo{50}, \PgNo{66}.
-
-\Bibitem Bruce, W.~H\@. Some Noteworthy Properties of the Triangle
-and its Circles. Heath, 1903. (One of the series of
-Heath's Mathematical Monographs, $10$~cents each). \PgNo{135}\Add{.}
-\DPPageSep{243}{237}
-
-\Bibitem Bulletin of the American Mathematical Society (monthly).
-Lancaster, Pa., and New York City. *\PgNo{103}, *\PgNo{204}, \PgNo{212}.
-
-\Bibitem Cajori, Florian. History of Elementary Mathematics, with
-Hints on Methods of Teaching. Macmillan, 1905 (1896).
-(This suggestive book should be read by every teacher.)
-*\PgNo{52}, *\PgNo{67}, *\PgNo{91}, \PgNo{135}, \PgNo{148}, *\PgNo{165}, \PgNo{193}, \PgNo{195}, *\PgNo{234}.
-
-% [** TN: http://www.gutenberg.org/ebooks/31061]
-\Bibitem Cajori, Florian. History of Mathematics. Macmillan, 1894.
-\PgNo{37}, \PgNo{148}, \PgNo{193}.
-
-\Bibitem Cantor, Moritz. Vorlesungen über die Geschichte der Mathematik.
-$3$~vol. Teubner, Leipzig, 1880--92. \PgNo{49}, \PgNo{67}, \PgNo{148},
-\PgNo{234}.
-
-\Bibitem De~Morgan, Augustus. Arithmetical Books. \PgNo{68}.
-
-\Bibitem De~Morgan, Augustus. Budget of Paradoxes. Longmans,
-London, 1872. *\PgNo{35}, *\PgNo{41}, \PgNo{86}, *\PgNo{126}, *\PgNo{181}.
-
-\Bibitem Dietrichkeit, O\@. Siebenstellige Logarithmen und Antilogarithmen.
-Julius Springer, Berlin, 1903. \PgNo{40}.
-
-% [** TN: http://www.gutenberg.org/ebooks/28885]
-\Bibitem Dodgson, C.~L\@. Alice's Adventures in Wonderland. 1865.
-\PgNo{201}, *\PgNo{218}.
-
-% [** TN: http://www.gutenberg.org/ebooks/12]
-\Bibitem Dodgson, C.~L\@. Through the Looking-glass and What Alice
-Found There. 1872. *\PgNo{218}.
-
-\Bibitem Encyclopĉdia Britannica. Ed.~9. \PgNo{39}, *\PgNo{71}, *\PgNo{176}, *\PgNo{183}, \PgNo{186}.
-
-\Bibitem Euler, Leonhard. Solutio Problematis ad Geometriam Situs
-Pertinentis. St.~Petersburg, 1736. \PgNo{170}.
-
-\Bibitem Evans, E.~P\@. Evolutional Ethics and Animal Psychology.
-Appleton, 1898. \PgNo{119}.
-
-\Bibitem Fechner, G.~T\@. Psychophysik. 1860. \PgNo{215}.
-
-\Bibitem Fink, Karl. Brief History of Mathematics, tr.\ by Beman and
-Smith. Open Court Publishing Co., 1900. \PgNo{49}, \PgNo{93}, \PgNo{148}.
-
-\Bibitem Fourier. Analyse des Equations Determinées. \PgNo{23}.
-
-\Bibitem Fourrey, E\@. Curiositées Géométriques. Vuibert et Nony,
-Paris, 1907. \PgNo{235}.
-
-\Bibitem Girard, Albert. Invention Nouvelle en l'Algèbre. Amsterdam,
-1629. \PgNo{92}.
-
-\Bibitem Gray, Peter. Tables for the Formation of Logarithms and
-Antilogarithms to $24$ or any less Number of Places.
-C.~Layton, London, 1876. \PgNo{40}.
-
-\Bibitem Halsted, G.~B\@. Bibliography of Hyperspace and Non-Euclidean
-Geometry. 1878. \PgNo{104}, \PgNo{107}.
-
-\Bibitem Harkness, William. Art of Weighing and Measuring. Smithsonian
-Report for 1888. *\PgNo{43}.
-\DPPageSep{244}{238}
-
-\Bibitem Hooper, W\@. Rational Recreations, in which the Principles of
-Numbers and Natural Philosophy Are Clearly and
-Copiously Elucidated~\dots\ $4$~vol. London, 1774. (Only
-the first $166$~pages of vol.~1 treat of numbers.) \PgNo{26}, *\PgNo{27},
-*\PgNo{38}.
-
-\Bibitem Journal of the American Medical Association. Chicago. \PgNo{158}.
-
-% [** TN: http://www.gutenberg.org/ebooks/25155]
-\Bibitem Kempe, A.~B\@. How to Draw a Straight Line; a Lecture on
-Linkages. Macmillan, London, 1877. *\PgNo{132}, *\PgNo{136}, *\PgNo{139}.
-
-\Bibitem Klein, F\@. Famous Problems of Elementary Geometry; tr.\ by
-Beman and Smith. Ginn, 1897. \PgNo{123}.
-
-\Bibitem Knowledge. \PgNo{187}.
-
-% [** TN: http://www.gutenberg.org/ebooks/36640]
-\Bibitem Lagrange, J.~L\@. Lectures on Elementary Mathematics; tr.\
-by T.~J.~McCormack. Ed.~2. Open Court Publishing
-Co., 1901 (1898). \PgNo{61}.
-
-\Bibitem Lebesgue, V.~A\@. Table des Diviseurs des Nombres. Gauthier-Villars,
-Paris. \PgNo{40}.
-
-\Bibitem Leonardo Fibonacci. Algebra et Almuchabala (Liber Abaci).
-1202. \PgNo{66}.
-
-\Bibitem L'Intermédiaire des Mathématiciens. *\PgNo{20}, *\PgNo{21}, \PgNo{36}.
-
-\Bibitem Listing, J.~B\@. Vorstudien zur Topologie (Abgedruckt aus den
-Göttinger Studien). Göttingen, 1848. \PgNo{117}, \PgNo{170}, \PgNo{173}.
-
-\Bibitem Lobatschewsky, Nicholaus. Geometrical Researches on the
-Theory of Parallels; tr.\ by G.~B. Halsted. Austin,
-Texas, 1892 (date of dedication). \PgNo{104}.
-
-\Bibitem Lucas Edouard. Récréations Mathématiques. $4$~vol. Gauthier-Villars,
-Paris, 1891--6. *\PgNo{17}, *\PgNo{70}, \PgNo{141}, *\PgNo{171}, \PgNo{186}, *\PgNo{197}, *\PgNo{234}.
-
-\Bibitem Lucas, Edouard. Théorie des Nombres. \PgNo{17}, \PgNo{22}.
-
-\Bibitem McLellan and Dewey. Psychology of Number. Appleton,
-1895. \PgNo{154}.
-
-\Bibitem McMurry, C.~A\@. Special Method in Arithmetic. Macmillan,
-1905. *\PgNo{207}.
-
-\Bibitem Manning, H.~P\@. Non-Euclidean Geometry. Ginn, 1901. \PgNo{107}.
-
-\Bibitem Margarita Philosophica. 1503. \PgNo{67} and frontispiece.
-
-\Bibitem Mathematical Gazette. London. \PgNo{41}.
-
-\Bibitem Mathematical Magazine. Washington. *\PgNo{20}, \PgNo{40}.
-
-\Bibitem Messenger of Mathematics. Cambridge, \PgNo{36}, \PgNo{127}.
-
-\Bibitem Monist (quarterly). Open Court Publishing Co. *\PgNo{19}, \PgNo{186},
-\PgNo{235}.
-
-\Bibitem Napier, John. Rabdologia. 1617. \PgNo{49}, \PgNo{61}, \PgNo{69}, \PgNo{71}.
-\DPPageSep{245}{239}
-
-% [** TN: http://www.gutenberg.org/ebooks/28233]
-\Bibitem Newton, Isaac. Philosophiĉ Naturalis Principia Mathematica.
-1687. *\PgNo{149}.
-
-\Bibitem New York Education (now American Education). *\PgNo{210}, \PgNo{235}.
-
-\Bibitem Open Court (monthly). Open Court Publishing Co. \PgNo{111},
-\PgNo{168}, \PgNo{235}.
-
-\Bibitem Pacioli, Lucas. Summa di Arithmetica~\dots\ Venice, 1494. \PgNo{59}, \PgNo{67}.
-
-\Bibitem Pathway to Knowledge. London, 1596. *\PgNo{68}.
-
-\Bibitem Philosophical Transactions, 1743. \PgNo{119}.
-
-\Bibitem Proceedings of the Central Association of Science and Mathematics
-Teachers. \PgNo{206}.
-
-\Bibitem Proceedings of the Royal Society of London, vol.~21. \PgNo{124}.
-
-\Bibitem Public School Journal. *\PgNo{206}.
-
-\Bibitem Rebiere. Mathématique et Mathématiciens. \PgNo{196}.
-
-\Bibitem Recorde, Robert. Grounde of Artes. 1540. \PgNo{68}\Add{.}
-
-\Bibitem Richards, W.~H\@. Military Topography. London, 1883. \PgNo{200}.
-
-\Bibitem Row, T.~S\@. Geometric Exercises in Paper Folding. Ed.~1,
-Madras, 1893; ed.~2 (edited by Beman and Smith).
-Open Court Publishing Co., 1901. \PgNo{144}.
-
-\Bibitem Rupert, W.~W. Famous Geometrical Theorems and Problems,
-with their History. Heath, 1901. \PgNo{124}.
-
-\Bibitem Schlömilch. Zeitschrift für Mathematik und Physik. \PgNo{111}.
-
-\Bibitem School Science and Mathematics (monthly) Chicago. *\PgNo{50},
-\PgNo{90}, \PgNo{125}, \PgNo{159}, \PgNo{206}.
-
-% [** TN: http://www.gutenberg.org/ebooks/25387]
-\Bibitem Schubert, Hermann. Mathematical Essays and Recreations,
-tr.\ by T.~J. McCormack. Open Court Publishing Co.,
-1903 (1899). \PgNo{95}, \PgNo{124}, \PgNo{154}.
-
-\Bibitem Smith, D.~E\@. Rara Arithmetica. Ginn, 1907. \PgNo{234}.
-
-\Bibitem Smith, D.~E\@. Teaching of Elementary Mathematics. Macmillan,
-1905 (1900). *\PgNo{56}.
-
-\Bibitem Smith, D.~E\@. The Old and the New Arithmetic. Reprinted
-from Text-Book Bulletin for Feb.\ 1905. Ginn. *\PgNo{68}.
-
-\Bibitem Stevin, Simon. La Disme (part of a larger work). 1585. \PgNo{59}\Add{.}
-
-\Bibitem Taylor, J.~M\@. Elements of Algebra. Allyn (1900). *\PgNo{96}.
-
-\Bibitem Taylor, J.~M\@. Elements of the Differential and Integral Calculus.
-Rev.\ ed. Ginn, 1898. \PgNo{151}, *\PgNo{213}.
-
-\Bibitem Taylor, J.~M\@. Five-place Logarithmic and Trigonometric
-Tables. Ginn (1905). \PgNo{40}.
-
-\Bibitem Teachers' Note Book (an occasional publication). *\PgNo{189}.
-
-\Bibitem Thom, David. The Number and Names of the Apocalyptic
-Beasts. 1848. \PgNo{181}.
-\DPPageSep{246}{240}
-
-\Bibitem Thorndike, E.~L\@. Introduction to the Theory of Mental and
-Social Measurements. Science Press, New York, 1904.
-*\PgNo{156}--\PgNo{158}.
-
-\Bibitem Tonstall, Cuthbert. Arithmetic. 1522. \PgNo{67}.
-
-\Bibitem Treviso Arithmetic. 1478. \PgNo{59}, \PgNo{67}.
-
-\Bibitem Waring, Edward. Meditationes Algebraicĉ. \PgNo{36}.
-
-\Bibitem Widmann, John. Arithmetic. Leipsic, 1489. \PgNo{162}.
-
-\Bibitem Willmon, J.~C\@. Secret of the Circle and the Square. Author's
-edition. Los Angeles, 1905. \PgNo{125}.
-
-\Bibitem Withers, J.~W\@. Euclid's Parallel Postulate: Its Nature,
-Validity, and Place in Geometrical Systems. Open
-Court Publishing Co., 1905. *\PgNo{104}, *\PgNo{105}--\PgNo{106}, \PgNo{107}.
-
-\Bibitem Young. J.~W.~A\@. Teaching of Mathematics in the Elementary
-and Secondary School. Longmans, 1907. *\PgNo{34}, \PgNo{98}, \PgNo{206},
-\PgNo{235}.
-\DPPageSep{247}{241}
-
-\printindex
-\iffalse
-% Start of index text
-
-General index.
-
-%[** TN: Text printed by the \printindex macro]
-\textsc{Note}: 43f means page 43 and the page or pages immediately following.
-149n means note at bottom of page 149. References given in the
-Bibliographic Index (preceding pages) are not (except in rare instances)
-repeated here.
-
-Abel, N.~H. 103
-
-Accuracy of measures|EtSeq 43
-
-Advice to a building committee 201
-
-Agesilaus 55
-
-Ahmes papyrus 164, 234
-
-Al Battani 148
-
-Alcuin 234
-
-Algebra 73-103
- teaching of|EtSeq 205
-
-Algebraic
- balance 90, 95
- fallacies 83
-
-Alice in the wonderland of mathematics 218
-
-American game of seven and eight 197
-
-Analytic geometry 156-157, 226-233
-
-Anaxagoras 122
-
-Antiquity, three famous problems of 122
-
-Apollo 122, 128
-
-Apparatus to illustrate line values of trigonometric functions 146
-
-Apple women 194
-
-Arabic camel puzzle 193
-
-Arabic
- notation 52, 66-68
- word for sine 148
-
-Archimedean proof 216
-
-Archimedes|FN 149
-
-Argand, J.~R. 37, 94
-
-Ariadne 178
-
-Aristotle 83
-
-Arithmetic 9-72
- in the Renaissance 66
- present trends in 51
- teaching 54-58
- teaching|EtSeq 205
-
-Arithmetics of the Renaissance 66-68
-
-Arrangements of the digits 21
-
-Art and mathematics 213
-
-Assyria 164
-
-Astronomers 44, 165, 216-217
-
-Asymptotic laws 37
-
-Autographs of mathematicians 168
-
-Avicenna 66
-
-Axioms
- in elementary algebra 73
- apply to equations? 76
-
-Babbage 72
-
-Babylonia 54, 164
-
-Balance, algebraic 90, 95
-
-Beast, number of 180
-
-Beauty in mathematics 208
-
-Bee's cell 118-119
-
-Beginnings of mathematics on the Nile 164
-
-Benary 180
-
-Berkeley, George 150
-
-Bernoulli 88, 168
-
-Berthelot 166
-
-Bibliographic
- notes 234
- index 236
-
-Billion 9
-
-Binomial theorem and statistics. 159
-
-Bocher@Bôcher, M.|FN 103, 212
-
-Bolingbroke, Lord 51
-
-Bolyai|FN 104
-
-Bonola, Roberto|FN 107
-\DPPageSep{248}{242}
-
-Book-keeper's clue to inverted numbers 25
-
-Book-keeping, first English book on 68
-
-Boorman 40
-
-Brahe, Tycho 214
-
-Bridges and isles 170
-
-Briggs 50, 165
-
-Buffon 126
-
-Building Committee, advice to 201
-
-Caesar@{Cĉsar Neron} 180
-
-Cajori, Florian 59, 124
-
-Calculation, mechanical aids 69
-
-Calculus 149-153, 206, 213
-
-Calculus|EtSeq 224
-
-Calculus of probability 124, 126-128, 156
-
-Camels, puzzle of 193
-
-Cantor, Moritz 168
-
-Cardan 66
-
-Carroll, Lewis 201, 218
-
-Carus, Paul 173
-
-Catch questions 196
-
-Cavalieri|FN 149
-
-Cayley 140
-
-Centers of triangle 133
-
-Chain-letters 102
-
-Checking solution of equation 81
-
-Chinese criterion for prime numbers 36
-
-Chirography of mathematicians 168
-
-Christians and Turks at sea 195
-
-Circle-squarer's paradox 126
-
-Circle-squaring 122-129
-
-Circles of triangle 133
-
-Circulating decimals 11-16, 40, 202
-
-Cistern problem 212-213
-
-Civilization and mathematics 217
-
-Clifford 168
-
-Coinage, decimal 52
-
-Collinearity of centers of triangle 133
-
-Colors in map drawing 140
-
-Combinations and permutations 37, 156
-
-Commutative law 88, 154
-
-Compass, watch as 199
-
-Complex numbers 75, 92
- branch of graph 231-232
-
-Compound interest 47
-
-Compte 167
-
-Concrete, mathematics teaching 205, 217
-
-Concrete necessarily complex 214
-
-Constants and variables illustrated 152-153
-
-Converse, fallacy of|EtSeq 83
-
-Counters, games 191, 197
-
-Crelle 135
-
-Crescents of Mohammed 175-176
-
-Cretan labyrinth 178
-
-Criterion for prime numbers 36
-
-Curiosities, numerical 19
-
-Daedalus@{Dĉdalus} 178
-
-Days-work problem 213
-
-De@{DeKalb normal school} 207
-
-Decimal separatrixes 49
-
-Decimalization of arithmetic|EtSeq 51
-
-Decimals as indexes of degree of accuracy 44
-
-Decimals invented late 165
-
-Declaration of Independence 175
-
-Definition
- of multiplication 98
- of exponents 101
- of mathematics 212-217
-
-Degree of accuracy of measurements 43-44
-
-Dela@{De la Loubère} 183
-
-Delian problem|EtSeq 122
-
-Demorgan@{De Morgan} 85, 126-129, 140, 166, 175, 181, 182
-
-Descartes 37, 94, 166
-
-Descriptive geometry 206
-
-Diagonals of a polygon 174-175, 216
-
-Digits
- in powers 20
- in square numbers 20
- arrangements of 21
-
-Dimension
- fourth 143, 223-224
- only one in Wall street 194
-
-Diophantus 37
-
-Direction determined by a watch 199
-
-Dirichlet 37
-
-Discriminant 95
-
-Disraeli@{D'Israeli} 128
-\DPPageSep{249}{243}
-
-Distribution curve for measures 156-159
-
-Divisibility, tests of 30
-
-Division
- Fourier's method 23
- in first printed arithmetic 67
- of decimals 63, 65
-
-Division of plane into regular polygons 118
-
-Divisor, greatest, with remainder 194
-
-Do the axioms apply to equations? 76
-
-Dodgson, C.~L. 168, 201, 218
-
-Dominoes
- number of ways of arranging 38
- in magic squares 187
-
-Donecker, F.~C.|FN 90
-
-Duplication of cube\EtSeq 122
-
-e@{$e$} 40, 216
-
-Egypt 54, 164
-
-Eleven, tests of divisibility by 31-33
-
-English
- numeration 9
- decimal separatrix 50
-
-Equation
- exponential 102
- insolvability of general higher 103
-
-Equations
- axioms apply to? 76
- equivalency 77-79
- checking solution of 81
- solved in ancient Egypt 164
-
-Equations of U.S. standards of length and mass 155
-
-Eratosthenes 123
-
-Eratosthenes|FN 149
-
-Error, theory of 46
-
-Escott, E.~B. 7-8, 13, 14, 36, 40, 111, 116, 187
-
-Escott, E.~B.|FN 19, 32, 41
-
-Euclid 103-108, 118, 123, 130, 166, 202
-
-Euclidean and non-Euclidean geometry 104-108
-
-Euclid's postulate 103-108
-
-Euler 19, 36, 37, 41, 94, 135, 165, 168, 171, 177, 178
-
-Exact science 212
-
-Exercise in public speaking 210
-
-Exponent, imaginary 96
-
-Exponential equation 102
-
-Exponents 101, 165
-
-Factors
- more than one set of prime 37
- two highest common 89
-
-Fallacies
- algebraic 83
- catch questions 196
-
-Familiar tricks based on literal arithmetic 27
-
-Faraday 166, 217
-
-Fechner, G.~T. 215
-
-Fermat 186
-
-Fermat|FN 149
-
-Fermat's theorem 36
- last theorem 35
- on binary powers 41
-
-Feuerbach's theorem 135
-
-Figure tracing 170
-
-Fine art and mathematics 213
-
-Forces, parallelogram of 142
-
-Formula, principle and rule 216
-
-Formulas for prime numbers 36
-
-Forty-one, curious property of 19
-
-Four-colors theorem 140
-
-Fourier 206
-
-Fourier's method of division 23
-
-Fourier's method of division|FN 41
-
-Fourth dimension 143, 223-224
-
-Fox, Captain|FN 127
-
-Fractions 54, 202
-
-Franklin, Benjamin 186
-
-Freeman, E.~A. 51
-
-French
- numeration 9
- decimal separatrix 50
-
-Frierson, L.~S. 186
-
-Fritzsche 180
-
-Game-puzzle 191
-
-Games with counters 191, 197
-
-Gath giant 52
-
-Gauss 34, 37, 94, 95, 103, 166, 203
-
-Gellibrand 165
-
-General form of law of signs 99
-
-General test of divisibility 30
-
-Geometric illustration
- of complex numbers 92
- of law of signs in multiplication 97
-
-Geometric
- magic squares 186
- multiplication 88, 154
- puzzles 109
-
-Geometry 103-145
- teaching|EtSeq 205
- descriptive 206
-
-German
- numeration 9
- decimal separatrix 50
-
-Giant with twelve fingers 52
-\DPPageSep{250}{244}
-
-Girard, Albert 37, 92
-
-Glaisher 39, 167
-
-Glaisher|FN 71
-
-Golden age of mathematics 203
-
-Gotham, square of 189
-
-Grading of students 159
-
-Graph of equation 156-157, 227-233
-
-Gravitation 215
-
-Greatest divisor with remainder 194
-
-Greeks 37, 54, 56, 66, 72, 123, 148, 167, 186, 216, 217
-
-Greeks|FN 149
-
-Growth of concept of number 37
-
-Growth of philosophy of the calculus 149
-
-Gunter 165
-
-Hall, W.~S. 158
-
-Halsted, G.~B. 104
-
-Hamilton, W.~R. 94, 168
-
-Hampton Court labyrinth 178
-
-Handwriting of mathematicians 168
-
-Heron of Alexandria 212
-
-Hexagons
- division of plane into 118
- magic 172-173, 187-188
-
-Hiberg|FN 149
-
-Higher equations 103
-
-Highest common factors, two 89
-
-Hindu
- check on division and multiplication 25
- illustration of real numbers 91, 92
- numerals (Arabic) 52, 66-68
- word for sine 148
-
-Hippias of Elis 123
-
-History of mathematics 167
- surprising facts 165
-
-Hitzig 180
-
-Home-made leveling device 120
-
-Ideal, mathematics science of 214, 215
-
-If the Indians hadn't spent the \$$24$#Indians 47
-
-Illustrations
- of the law of signs 97
- of symmetry 144
- of trigonometric functions 146
- of limits 152
-
-Imaginary 94
- exponent 96
- branch of graph 230-232
-
-Indians spent the \$$24$#Indians 47
-
-Infinite 87
- symbols for 151
-
-Infinite|EtSeq 224
-
-Inhabited planets 216
-
-Inheritance, Roman problem 193
-
-Instruments that are postulated 130
-
-Interest, compound and simple 47
-
-Involution not commutative 154
-
-Irenĉus 180-181
-
-Isles and bridges 170
-
-Italian
- numeration 9
- decimal separatrix 50
-
-Jefferson, Thomas 175
-
-Kant 167
-
-Kegs-of-wine puzzle 194
-
-Kempe, A.~B. 132, 136, 139
-
-Kepler 50, 107, 167, 203, 214, 217
-
-Kepler|FN 149
-
-Kilogram 155
-
-Knilling 57
-
-Knowlton 197
-
-Königsberg 170-171, 174
-
-Kühn, H. 93, 94
-
-Kulik 40
-
-Labyrinths 170, 176-179
-
-Lagrange 36, 168
-
-Laisant 166
-
-Laplace 126, 168
-
-Lathrop, H.~J.|FN 145
-
-Law of signs 97
- of \DPtypo{commuation}{commutation} 154
-
-Legendre 36, 37, 168
-
-Lehmer, D.~N. 40
-
-Leibnitz 149-150, 166
-
-Length, standard of 155
-
-Lennes, N.~J.|FN 90
-
-Leonardo of Pisa 66
-
-Leveling device 120
-
-Limits illustrated 152
-
-Lindemann 123, 124
-
-Line values of trigonometric functions 146
-
-Linkages and straight-line motion 136
-
-Literature of mathematics 203, 208-209
-
-Lobachevsky 104-108
-\DPPageSep{251}{245}
-
-Logarithms 45, 47, 52, 69, 87, 102, 165
- |seealso $e$.
-
-London and Wise 176
-
-Loubère, de la 183
-
-Lowest common multiples, two 89
-
-Loyd, S. 116
-
-Loyd, S.|FN 187
-
-Lunn, J. R. 40
-
-Luther 181
-
-Maclaurin 119
-
-Magic
- number 25
- pentagon 172-173
- hexagons 172-173, 187-188
- squares 183
-
-Manhattan, value of reality in 1626 and now#Manhattan 47-48
-
-Map makers' proposition 140
-
-Marking students 159
-
-Mars, signaling 216-217
-
-Mass, standard of 155
-
-Mathematical advice to a building committee 201
-
-Mathematical game-puzzle 191
-
-Mathematical reasoning, nature of 212
-
-Mathematical recitation as an exercise in public speaking 210
-
-Mathematical recreations 234
-
-Mathematical symbols 162, 165
-
-Mathematical treatment of statistics 156
-
-Mathematics
- definitions|FN 212
- nature of 212-217
- teaching more concrete 205, 217
- Alice in the wonderland of 218
-
-Mazes 176-179
-
-Measurement
- numbers arising from 43
- degree of accuracy of 43-44
-
-Measurements treated statistically 156-161, 207-208
-
-Measures, standard 155
-
-Mellis, John 68
-
-Methods in arithmetic 54-58
-
-Metric system 43, 53, 155
-
-Miller, G. A. 50
-
-Million, first use of term in print 67
-
-Minotaur 178
-
-Miscellaneous notes on number 34
-
-Mobius@{Möbius, A. F.} 140
-
-Mohammed 175-176
-
-Morehead, J. C.#Morehead 41
-
-Moscopulus 186
-
-Movement to make teaching more concrete 205
-
-Multiplication
- at sight 15
- approximate 45, 62, 64
- of decimals 59
- in first printed arithmetic 67
- law of signs illustrated 97
- definition 98
- as a proportion 100
- gradual generalization of 100
- geometric 88, 154
-
-Myers, G. W.|FN 90
-
-n@{$n$ dimensions} 104
- |seealso Fourth dimension.
-
-Napier, John. |see \DPtypo{Logariths}{Logarithms}.
-
-Napier, Mark 165
-
-Napier's rods 69
-
-Napoleon 167
-
-Nature of mathematical reasoning 212
-
-Negative and positive numbers 90
-
-Negative conclusions in 19th century#negative conclusions 103
-
-Neptune, distance from sun 44
-
-Nero 180
-
-New trick with an old principle 15
-
-New York, value of realty in 1626 and now#Manhattan 47-48
-
-Newton 49, 149-150, 215
-
-Nicomedes 123
-
-Nile, beginnings of mathematics on 164
-
-Nine, curious properties of 25
-
-Nine-point circle 134-135
-
-Nineteenth century, negative conclusions reached 103
-
-Non-Euclidean geometry 104-108
-
-Normal probability integral 157
-
-Number
- miscellaneous notes on 34-42
- growth of concept of 37
- How may a particular number arise? 41
- of the beast 180
-
-Numbers arising from measurement 43
- differing from their log.\ only in position of decimal point 19
- theory of|EtSeq 34
-\DPPageSep{252}{246}
-
-Numeration, two systems 9
-
-Numerical curiosity 19
-
-Old-timers 194
-
-Oratory, mathematical recitation as exercise 210
-
-Orbits of planets 214
-
-Oresme 101
-
-Orthotomic 94
- |seealso Imaginary.
-
-Oughtred 49
-
-p@{$\pi$} 40, 123-129, 216
- expressed with the ten digits 23
-
-Pacioli 59, 100
-
-Paper folding 144
-
-Paradox, circle-squarer's 126
-
-Paradromic rings 117
-
-Parallel postulates 103-108
-
-Parallelogram of forces 142
-
-Parallels meet at infinity 107
-
-Peaucellier 136-139
-
-Pentagon, magic 172-173
-
-Periodicals, publication of foregoing sections in 235
-
-Permutations 37, 156
-
-Petzval 40
-
-Philoponus 122
-
-Philosophy of the calculus 149
-
-Pierpont, James 203
-
-Pitiscus 49, 50
-
-Plane, division into regular polygons 118
-
-Planetary orbits 214
-
-Planets, inhabited 216
-
-Planting in hexagonal forms 119
-
-Plato 122, 123, 130, 166, 211, 214
-
-Plato Tiburtinus 148
-
-Positive and negative numbers 90
-
-Powers having same digits 20
-
-Present trends in arithmetic 51
-
-Prime factors of a number, more than one set 37
-
-Primes
- formulas for 36
- Chinese, criterion for 36
- tables of 40
-
-Principle, rule and formula 216
-
-Probability 124, 126-128, 156
-
-Problems
- of antiquity 122
- for quickening the mind 234
-
-Products, repeating 11-16
-
-Proportion, multiplication as 100
-
-Psychology 54, 57, 215
-
-Ptolemy 103, 167, 217
-
-Publication of foregoing sections in periodicals 235
-
-Puzzle
- game 191
- of the camels 193
-
-Puzzles, geometric 109
-
-Pythagorean proposition 121, 164
-
-Quadratrix 123
-
-Quadrature of the circle 122-129
-
-Quaternions 88, 94, 154
-
-Question of fourth dimension by analogy 143
-
-Questions, catch 196
-
-Quotations on mathematics 166
-
-Real numbers 90, 232
-
-Reasoning, mathematical 212
-
-Recitation as an exercise in public speaking 210
-
-Recreations, mathematical 234
-
-Rectilinear motion 136-139
-
-Recurring decimals 11-16, 40
-
-Regular polygons, division of plane into 118
-
-Reiss 38
-
-Renaissance, arithmetic in 66
-
-Renaissance of mathematics 203
-
-Repeating
- decimals 11-16, 40
- products 11-16
- table 17
-
-Representation of complex numbers 92, 231-232
-
-Reuss 180
-
-Riemann's postulate 105-108
-
-Rings, paradromic 117
-
-Rods, Napier's 69
-
-Romain, Adrian|EtSeq 60
-
-Roman inheritance problem 193
-
-Roots
- of equal numbers 73, 75
- of higher equations 103
-
-Rope stretchers 121
-
-Royal Society's catalog 203
-
-Rudolff 162
-
-Rule, principle and formula 216
-
-Ruler unlimited and ungraduated 130-132
-
-Scalar 94
- |seealso Real numbers.
-
-Scheutz 72
-\DPPageSep{253}{247}
-
-Separatrixes, decimal 49
-
-Seven-counters game 197
-
-Seven, tests of divisibility by 31-33
-
-Shanks, William 40, 124
-
-Signatures
- of mathematicians 168-169
- unicursal 170, 175-176
-
-Signs, illustrations of law of 97
-
-Sine, history of the word 148
-
-Smith, Ambrose 127
-
-Smith, D.~E. 56, 59, 168
-
-Smith, D.~E.|FN 234
-
-Smith, M.~K. 159
-
-Social sciences treated mathematically 156, 207-208
-
-Societies' initials 38
-
-Sparta 55
-
-Speaking, recitation as an exercise in 210
-
-Speidell 165
-
-Square numbers containing the digits not repeated 20
-
-Square of Gotham 189
-
-Squares
- magic 183
- geometrical magic 186
- coin 187
- domino 187
-
-Squaring the circle 122-129
-
-Standards of length and mass 155
-
-Statistics, mathematical treatment of 156, 207-208
-
-Stevin, Simon 101
-
-Stevin, Simon|EtSeq 59
-
-Stifel 91
-
-Straight-edge 130-132, 136
-
-Straight-line motion 136
-
-Student records 159
-
-Shuffield, G. 40, 41
-
-Surface of \DPtypo{frequencey}{frequency} 156-159
-
-Surface with one face 117
-
-Surprising facts in the history of mathematics 165
-
-Swan pan 72
-
-Sylvester, J. J. 139, 168
-
-Symbols
- mathematical 162, 165
- for infinite 151
-
-Symmetry illustrated by paper folding 144
-
-Tables 39
- repeating 17
-
-Tait 19
-
-Tanck 57
-
-Tax rate 46
-
-Taylor, J.~M. 7
-
-Teaching made concrete 217
-
-Teaching made concrete|EtSeq 205
-
-Terquem 135
-
-Tests of divisibility 30
-
-Theory
- of error 46
- of numbers|EtSeq 34
-
-Theseus 178
-
-Thirteen, test of divisibility by 32
-
-Thirtie daies hath September 68
-
-Thirty-seven, curious property of 19
-
-Three famous problems of antiquity 122
-
-Three parallel postulates illustrated 105
-
-Time-pieces, accuracy of 43
-
-Trapp 57
-
-Trends in arithmetic 51
-
-Triangle and its circles 133
-
-Trick, new with an old principle 15
-
-Tricks based on literal arithmetic 27
-
-Trigonometry 96, 107, 146-148, 165
-
-Trisection of angle 130-132
-
-Trisection of angle|EtSeq 122
-
-Turks and Christians at sea 195
-
-Two H. C. F. 89
-
-Two negative conclusions reached in the 19th century#negative conclusions 103
-
-Two systems of numeration 9
-
-Tycho Brahe 214
-
-Undistributed middle|EtSeq 83
-
-Unicursal signatures and figures 170
-
-United States standards of length and mass 155
-
-Variables illustrated 152-153
-
-Vectors 88, 94, 154
-
-Vienna academy 40
-
-Visual representation of complex numbers 92
-
-Vlacq 165
-
-Von Busse 57
-
-Wall street 194
-
-Wallis 93, 101, 151
-
-Watch as compass 199
-\DPPageSep{254}{248}
-
-Weights and measures 43, 53, 155
-
-Wessel 37, 94
-
-Whewell 167, 217
-
-Wilson, John, biographic note|FN 35
-
-Wilson's theorem 35
-
-Withers, J. W. 107
-
-Witt, Richard 49
-
-Wonderland of mathematics 218
-
-Young, J. W. A. 205
-
-Zero
- in fallacies 87
- meaning of symbol 150-151
- first use of word in print 67
-% End of index text
-\fi
-\DPPageSep{255}{249}
-
-\iffalse
-% start of the catalog
-% [** TN: Proofread and formatted text retained in source]
-
-COMPLETE LIST OF BOOKS AND PAMPHLETS,
-PUBLISHED EXCLUSIVELY BY THE OPEN
-COURT PUBLISHING COMPANY, 1322 WABASH
-AVE., CHICAGO, ILL.; ARRANGED BY AUTHORS;
-INCLUDING ALSO A FEW IMPORTATIONS.
-NOVEMBER, 1907.
-
-Note:---The following numbers, 345, 356, 325, 226, 317,
-344, 363, 318, 339, and 214 are imported publications, to
-which the ``importation'' discount will apply. ``Trade'' rates
-on request.
-
-ABBOTT, DAVID P.
-
-377. BEHIND THE SCENES WITH THE MEDIUMS. \Name{David P.
-Abbott.} 328~pp. Cloth, \$1.50 net, postpaid.
-
-ANDREWS, W. S.
-
-337. MAGIC SQUARES AND CUBES. \Name{W.~S. Andrews.} With chapters
-by Paul Carus, L.~S. Frierson and C~A. Browne, Jr., and
-introduction by Paul Carus. Price, \$1.00 net. (5s.\ net.)
-
-
-ANESAKI, M.
-
-345. BUDDHIST AND CHRISTIAN GOSPELS, Being Gospel Parallels
-from Pali Texts. Now first compared from the originals
-by Albert J. Edmunds. Edited with parallels and notes from
-the Chinese Buddhist Triptaka by \Name{M. Anesaki}\Add{.}
-\$1.50 net.
-
-
-ASHCROFT, EDGAR A.
-
-356. THE WORLD'S DESIRES or The Results of Monism, an elementary
-treatise on a realistic religion and philosophy of human
-life, by \Name{Edgar A. Ashcroft.} 1905. Cloth, gilt top, \$1.00 net.
-
-
-BARCK, CARL.
-
-375. THE HISTORY OF SPECTACLES. \Name{Carl Barck, A.M., M.D\@.}
-Profusely illustrated. Price, 15 cents net.
-
-
-BAYNE, JULIA TAFT.
-
-323. HADLEY BALLADS. \Name{Julia Taft Bayne.} 75c net.
-
-
-BERKELEY, GEORGE.
-
-307. A TREATISE CONCERNING THE PRINCIPLES OF HUMAN
-KNOWLEDGE. \Name{George Berkeley.} Cloth, 60c net. (3s.\ net)
-
-308. THREE DIALOGUES BETWEEN HYLAS AND PHILONOUS.
-\Name{George Berkeley.} Cloth, 60c net. (3s.\ net.)
-
-
-BINET, ALFRED.
-
-201. THE PSYCHIC LIFE OF MICRO-ORGANISMS. \Name{Alfred Binet.}
-75c (3s.\ 6d.)
-
-270. THE PSYCHOLOGY OF REASONING. \Name{Alfred Binet.} Transl\Add{.}\
-by \Name{Adam Gowans Whyte.} 75c net. (3s.\ 6d.)
-
-296. ON DOUBLE CONSCIOUSNESS. \Name{Alfred Binet.} Cloth, 50c net.
-(2s.\ 6d.\ net.)
-
-
-BLOOMFIELD. MAURICE\DPtypo{,}{.}
-
-334. CERBERUS, THE DOG OF HADES. The History of an Idea.
-\Name{Prof.\ M. Bloomfield.} Boards, 50c net. (2s.\ 6d.\ net.)
-\DPPageSep{256}{250}
-
-
-BONNEY, HONORABLE CHARLES CARROLL.
-
-304. WORLD'S CONGRESS ADDRESSES, Delivered by the President,
-the \Name{Hon.\ C.~C. Bonney}. Cloth, 50c net. (2s.\ 6d.\ net.)
-
-
-BONNEY, FLORENCE PEORIA.
-
-286. MEDITATIONS (Poems). \Name{Florence Peoria Bonney}. Cloth, \$1.00
-net.\DPtypo{)}{}%[**no opening paren]
-
-
-BUDGE, E. A. WALLIS.%[** TN: 'E .A.' in the original]
-
-325. THE GODS OF THE EGYPTIANS OR STUDIES IN EGYPTIAN
-MYTHOLOGY. \Name{E.~A. Wallis Budge}. With plates and
-illustrations, 2 vols. Cloth, \$20.00 net.
-
-226. THE BOOK OF THE DEAD, a translation of the Chapters,
-Hymns, etc., of the Theban Recension. \Name{E.~A. Wallis Budge.}
-Illustrated. 3 vols. \$3.75 per set net. Vols.~VI, VII, VIII
-in the series of Books on Egypt and Chaldea.
-
-317. A HISTORY OF EGYPT, From the End of the Neolithic Period
-to the Death of Cleopatra VII, B.~C.\ 30. \Name{E.~A. Wallis Budge.}
-Richly illustrated. 8 vols. Cloth, \$10.00 net.
-
-I. Egypt in the Neolithic and Archaic Period.
-
-II. Egypt Under the Great Pyramid Builders.
-
-III. Egypt Under the Amenembats and Hyksos.
-
-IV. Egypt and her Asiatic Empire.
-
-V. Egypt Under Rameses the Great.
-
-VI. Egypt Under the Priest Kings and Tanites and Nubians.
-
-VII. Egypt Under the Saites, Persians and Ptolemies.
-
-VIII. Egypt Under the Ptolemies and Cleopatra VII.
-
-344. THE DECREES OF MEMPHIS AND CANOPUS, in three volumes.
-\textsc{The Rosetta Stone}, Vols.~I and II\@. \textsc{The Decree of
-Canopus}, Vol.~III, by \Name{E.~A. Wallis Budge}, With plates. 1904.
-\$1.25 per volume. Three volumes \$3.75 net.
-
-363. THE EGYPTIAN HEAVEN AND HELL. \Name{E.~A. Wallis Budge.}
-Vol.~I, \textsc{The Book of Am Tuat}; Vol.~II, \textsc{The Book of Gates};
-Vol.~III, \textsc{The Egyptian Heaven and Hell}. 1906. Cloth,
-Illustrated. \$5.00 per set
-
-
-CALKINS, MARY WHITON.
-
-150. THE METAPHYSICAL SYSTEM OF HOBBES, as contained in
-twelve chapters from his ``Elements of Philosophy Concerning
-Body'' and in briefer extracts from his ``Human Nature'' and
-``Leviathan,'' selected by \Name{Mary Whiton Calkins}. Cloth, 75 cents
-net. (3s.)
-
-351. LOCKE'S ESSAY CONCERNING HUMAN UNDERSTANDING.
-Books II and IV (with omissions). Selected by \Name{Mary
-Whiton Calkins}, 2d ed. Cloth, 75 cents net. (3s.)
-
-
-CARUS, DR. PAUL,
-
-204. FUNDAMENTAL PROBLEMS, the Method of Philosophy as a
-Systematic Arrangement of Knowledge. \Name{Paul Carus}. Cloth,
-\$1.50. (7s.\ 6d.)
-
-207. THE SOUL OF MAN, an Investigation of the Facts of Physiological
-and Experimental Psychology. \Name{Paul Carus}. Illustrated,
-Cloth, \$1.50 net. (6s.\ net.)
-\DPPageSep{257}{251}
-
-208. PRIMER OF PHILOSOPHY. \Name{Paul Carus}. Cloth, \$1.00. (5s.)
-
-210. MONISM AND MELIORISM, A Philosophical Essay on Causality
-and Ethics. \Name{Paul Carus}. Paper, 50c (2s.\ 6d.)
-
-213. (a) THE PHILOSOPHY OF THE TOOL. 10c. (6d.) (b) OUR
-NEED OF PHILOSOPHY. 5c (3d.) (c) SCIENCE A
-RELIGIOUS REVELATION. 5c.
-(3d.) \Name{Paul Carus}.
-
-290. THE SURD OF METAPHYSICS, An Inquiry into the Question
-\textsc{Are there Things-in-themselves?} \Name{Paul Carus}. Cloth, \$1.25
-net. (5s.\ 6d.\ net.)
-
-303. KANT AND SPENCER, A Study of the Fallacies of Agnosticism.
-\Name{Paul Carus}. Cloth, 50c net. (2s.\ 6d.\ net.)
-
-312. KANT'S PROLEGOMENA TO ANY FUTURE METAPHYSICS.
-Edited by \Name{Paul Carus}. Cloth, 75c net. (3s.\ 6d.\ net.)
-
-215. THE GOSPEL OF BUDDHA, According to Old Records, told by
-\Name{Paul Carus}. Cloth, \$1.00. (5s.)
-
-254. BUDDHISM AND ITS CHRISTIAN CRITICS. \Name{Paul Carus}.
-\$1.25. (6s.\ 6d.)
-
-261. GODWARD, A Record of Religious Progress. \Name{Paul Carus}. 50c.
-(2s.\ 6d.)
-
-278. THE HISTORY OF THE DEVIL AND THE IDEA OF EVIL,
-From the Earliest Times to the Present day. \Name{Paul Carus}. Illustrated.
-\$6.00. (30s.)
-
-280. HISTORY OF THE CROSS. \Name{Paul Carus}. (In preparation.)
-
-321. THE AGE OF CHRIST. A Brief Review of the Conditions
-under which Christianity originated. \Name{Paul Carus}. Paper, 15c
-net. (10d.)
-
-341. THE DHARMA, or the Religion of Enlightenment, An Exposition
-of Buddhism. \Name{Paul Carus}. 25c (1s.\ 6d.)
-
-216. DAS EVANGELIUM BUDDHAS. A German translation of The
-Gospel of Buddha. Cloth, \$1.25. (5 marks.)
-
-255. LAO-TZE'S TAO TEH KING. Chinese-English. With Introduction,
-Transliteration and Notes by \Name{Paul Carus}. \$3.00 (15s.)
-
-357. T'AI-SHANG KAN-YING P'IEN, Treatise of the Exalted One
-on Response and Retribution. \Name{Paul Carus} and \Name{Teitaro Suzuki}.
-Boards, 75c net (3s.\ 6d.)
-
-358. YIN CHIH WEN, The Tract of the Quiet Way. With extracts
-from the Chinese Commentary. \Name{Paul Carus} and \Name{Teitaro Suzuki}.
-Boards, 25c net (1s.\ 6d.)
-
-275. THE WORLD'S PARLIAMENT OF RELIGIONS AND THE
-RELIGIOUS PARLIAMENT EXTENSION, a Memorial Published
-by the Religious Parliament Extension Committee. Popular
-edition. C. C. Bonney and \Name{Paul Carus}.
-
-205. HOMILIES OF SCIENCE. \Name{Paul Carus}. Cloth, gilt top, \$1.50.
-(7s.\ 6d.)
-
-206. THE IDEA OF GOD. \Name{Paul Carus}. Paper, 15c. (9d.)
-
-211. THE RELIGION OF SCIENCE. \Name{Paul Carus}. Cloth, 50c net
-(2s.\ 6d.)
-
-212. KARMA, A STORY OF BUDDHIST ETHICS. \Name{Paul Carus}.
-Illustrated by Kwason Suzuki. American edition, 15c. (10d.)
-
-268. THE ETHICAL PROBLEM. Three Lectures on Ethics as a
-Science. \Name{Paul Carus}. Cloth. \$1.25. (6s.\ 6d.)
-\DPPageSep{258}{252}
-
-285. WHENCE AND WHITHER. An Inquiry into the Nature of
-the Soul, Its Origin and Its Destiny. \Name{Paul Carus}. Cloth, 75c
-net. (3s.\ 6d net.)
-
-302. THE DAWN OF A NEW RELIGIOUS ERA, AND OTHER
-ESSAYS. \Name{Paul Carus}. Cloth, 50c net (2s.\ 6d.\ net.)
-
-209. TRUTH IN FICTION, Twelve Tales with a Moral. \Name{Paul Carus}.
-Cloth, 1.00 net. (5s.)
-
-217. KARMA, A STORY OF EARLY BUDDHISM. \Name{Paul Carus}.
-Illustrated. Crêpe paper, tied in silk. 75c (3s.\ 6d.)
-
-217G. KARMA, Eine buddhistische Erzählung. \Name{Paul Carus}. Illustrated.
-35c.
-
-291. NIRVANA, A STORY OF BUDDHIST PSYCHOLOGY, \Name{Paul
-Carus}. Illustrated by \Name{Kwason Suzuki}. Cloth, 60c net. (3s.\ net.)
-
-313. AMITABHA, A Story of Buddhist Theology. \Name{Paul Carus}.
-Boards, 50c net. (2s.\ 6d.)
-
-246. THE CROWN OF THORNS, a story of the Time of Christ,
-\Name{Paul Carus}. Illustrated. Cloth 75c net. (3s.\ 6d.\ net.)
-
-247. THE CHIEF'S DAUGHTER, a Legend of Niagara. \Name{Paul Carus}.
-Illustrated. Cloth, \$1.00 net. (4s.\ 6d.)
-
-267. SACRED TUNES FOR THE CONSECRATION OF LIFE.
-Hymns of the Religion of Science. \Name{Paul Carus}. 50c.
-
-281. GREEK MYTHOLOGY. \Name{Paul Carus}. In preparation.
-
-282. EROS AND PSYCHE, A Fairy-Tale of Ancient Greece, Retold
-after Apuleius, by \Name{Paul Carus}. Illustrated. \$1.50 net. (6s.\ net.)
-
-295. THE NATURE OF THE STATE. \Name{Paul Carus}. Cloth 50c net.
-(2s.\ 6d.\ net\Add{.})
-
-224. GOETHE AND SCHILLER'S XENIONS. Selected and translated
-by \Name{Paul Carus}. Paper, 50c (2s.\ 6d.)
-
-343. FRIEDRICH SCHILLER, A Sketch of his Life and an Appreciation
-of his Poetry. \Name{Paul Carus}. Profusely illustrated.
-Boards, 75c net. (3s.\ 6d.)
-
-353. THE RISE OF MAN. A Sketch of the Origin of the Human
-Race. \Name{Paul Carus}. Illustrated. 1906. Boards, cloth back,
-75c net. (3s\Add{.} 6d.\ net.)
-
-365. OUR CHILDREN. Hints from Practical Experience for Parents
-and Teachers. \Name{Paul Carus}. \$1.00 net. (4s.\ 6d.\ net.)
-
-371. THE STORY OF SAMSON and Its Place in the Religious Development
-of Mankind. \Name{Paul Carus}. Illustrated. Boards,
-cloth back, \$1.00 net. (4s.\ 6d.\ net.)
-
-372. CHINESE THOUGHT. An Exposition of the Main Characteristic
-features of the Chinese World-Conception.
-\Name{Paul Carus}. Being a continuation of the author's
-essay Chinese Philosophy. Illustrated. Index. Pp.
-195. \$1.00 net. (4s.\ 6d.)
-
-373. CHINESE LIFE AND CUSTOMS. \Name{Paul Carus}. With illustrations
-by Chinese artists. Pp.~114. 75c.\ net (3s.\
-6d.\ net.)
-
-
-CLEMENT, ERNEST W.
-
-331. THE JAPANESE FLORAL CALENDAR. \Name{E. W. Clement}. Illustrated.
-Boards, 50c net. (2s.\ 6d.\ net.)
-
-
-CONWAY, MONCURE DANIEL.
-
-277. SOLOMON AND SOLOMONIC LITERATURE. \Name{M. D. Conway}.
-Cloth, \$1.50 net. (6s.)
-
-
-COPE, E. D.
-
-219. THE PRIMARY FACTORS OF ORGANIC EVOLUTION. \Name{E.
-D. Cope, Ph.\ D}. 2d ed. Illustrated. Cloth, \$2.00 net (10s.)
-\DPPageSep{259}{253}
-
-CORNILL, CARL HEINRICH.
-
-220. THE PROPHETS OF ISRAEL, Popular Sketches from Old
-Testament History. \Name{C. H. Cornill}. Transl.\ by S. F. Corkran.
-\$1.00 net. (5s.)
-
-259. THE HISTORY OF THE PEOPLE OF ISRAEL, From the
-Earliest Times to the Destruction of Jerusalem by the Romans.
-\Name{C. H. Cornill}. Transl.\ by \Name{W. H. Carruth}. Cloth, \$1.50 (7s.\ 6d.)
-
-262. GESCHICHTE DES VOLKES ISRAEL. \Name{C. H. Cornill}. Gebunden
-\$2.00. (8 Mark.)
-
-251. THE RISE OF THE PEOPLE OF ISRAEL. \Name{C. H. Cornill}, in
-Epitomes of Three Sciences: Comparative Philology, Psychology
-and Old Testament History. \Name{H. H. Oldenberg}, \Name{J.
-Jastrow}, \Name{C. H. Cornill}. \DPtypo{Colth}{Cloth}, 50c net. (2s.\ 6d.)
-
-
-CUMONT, FRANZ.
-
-319. THE MYSTERIES OF MITHRA. \Name{Prof.\ Franz Cumont}. Transl.\
-by \Name{T. J. McCormack}. Illus. Cloth, \$1.50 net. (6s.\ 6d.\ net.)
-
-
-DEDEKIND, RICHARD.
-
-287. ESSAYS ON THE THEORY OF NUMBERS. I. Continuity
-and Irrational Numbers. II. The Nature and Meaning of
-Numbers. \Name{R. Dedekind}. Transl.\ by \Name{W. W. Beman}. Cloth,
-75c net. (3s.\ 6d.\ net.)
-
-
-DELITZSCH, DR. FRIEDRICH.
-
-293b. BABEL AND BIBLE. Three Lectures on the Significance of
-Assyriological Research for Religion, Embodying the most important
-Criticisms and the Authors Replies. \Name{F. Delitzsch}.
-Translated from the German. Illustrated. 1906. \$1.00 net.
-
-
-DE MORGAN, AUGUSTUS.
-
-%[** TN: http://www.gutenberg.org/ebooks/39088]
-264. ON THE STUDY AND DIFFICULTIES OF MATHEMATICS.
-\Name{Augustus DeMorgan}. Cloth, \$1.25 net. (4s.\ 6d.\ net)
-
-%[** TN: http://www.gutenberg.org/ebooks/39041]
-271. ELEMENTARY ILLUSTRATIONS OF THE DIFFERENTIAL
-AND INTEGRAL CALCULUS. \Name{Augustus DeMorgan}. Cloth,
-\$1.00 net (4s.\ 6d.\ net.)
-
-
-DESCARTES, RENE. %[** TN: [sic], not RENÉ]
-
-301. DISCOURSE ON THE METHOD OF RIGHTLY CONDUCTING
-THE REASON AND SEEKING TRUTH IN THE SCIENCES.
-\Name{René Descartes}. Transl.\ by \Name{John Veitch}. Cloth, 60c
-net. (3s.\ net.)
-
-310. THE MEDITATIONS AND SELECTIONS FROM THE PRINCIPLES
-of \Name{René Descartes}. Transl.\ by \Name{John Veitch}. Cloth,
-75c net. (3s.\ 6d.\ net.)
-
-
-DE VRIES, HUGO.
-
-332. SPECIES AND VARIETIES, THEIR ORIGIN BY MUTATION.
-\Name{Prof.\ Hugo de Vries}. Edited by \Name{D. T. MacDougal}.
-\$5.00 net (21s.\ net.)
-
-369. PLANT-BREEDING. Comments on the Experiments of Nilsson
-and Burbank. \Name{Hugo de Vries}. Illustrated. Cloth. Gilt. \$1.50
-net. \$1.70 postpaid.
-\DPPageSep{260}{254}
-
-EDMUNDS, ALBERT J.
-
-218. HYMNS OF THE FAITH (DHAMMAPADA), being an Ancient
-Anthology Preserved in the Sacred Scriptures of the Buddhists.
-Transl.\ by \Name{Albert~J.~Edmunds}. Cloth, \$1.00 net. (4s.\ 6d.\ net.)
-
-345. BUDDHIST AND CHRISTIAN GOSPELS, Being Gospel Parallels
-from Pali Texts. Now first compared from the originals
-by \Name{Albert~J.~Edmunds}. Edited with parallels and notes from
-the Chinese Buddhist Triptaka by \Name{M.~Anesaki} \$1.50 net.
-
-
-EVANS, HENRY RIDGELY.
-
-330. THE NAPOLEON MYTH. \Name{H.~R.~Evans}. With ``The Grand
-Erratum,'' by \Name{J.~B.~Pérès}, and Introduction by \Name{Paul Carus}.
-Illustrated. Boards, 75c net. (3s.\ 6d.\ net.)
-
-347. THE OLD AND THE NEW MAGIC. \Name{Henry~R.~Evans}. Illustr.
-Cloth, gilt top. \$1.50 net, mailed \$1.70.
-
-
-FECHNER, GUSTAV THEODOR.
-
-349. ON LIFE AFTER DEATH. \Name{Gustav Theodor Fechner}. Tr.\ from
-the German by \Name{Hugo Wernekke}. Bds. 75c.
-
-
-FICHTE, JOHANN GOTTLIEB.
-
-361. THE VOCATION OF MAN. \Name{Johann Gottlieb Fichte}. Tr.\ by
-\Name{William Smith}, with biographical introduction by \Name{E. Ritchie},
-1906. Cloth, 75c net. (3s.\ 6d.)
-
-
-FINK, DR. CARL.
-
-272. A BRIEF HISTORY OF MATHEMATICS. \Name{Dr.~Karl~Fink}.
-Transl.\ from the German by \Name{W.~W.~Beman} and \Name{D.~E.~Smith}.
-Cloth, \$1.50 net. (5s.\ 6d.\ net.)
-
-
-FREYTAG, GUSTAV.
-
-248. MARTIN LUTHER. \Name{Gustav Freytag}. Transl.\ by \Name{H.~E.~O.~Heinemann}.
-Illustrated. Cloth, \$1.00 net. (5s.)
-
-221. THE LOST MANUSCRIPT. A Novel. \Name{Gustav Freytag}. Two
-vols. Cloth, \$4.00\DPtypo{,}{.} (21s.)
-
-221a. THE SAME. One vol. \$1.00. (5s.)
-
-
-GARBE, RICHARD.
-
-223. THE PHILOSOPHY OF ANCIENT INDIA. \Name{Prof.~R.~Garbe}.
-Cloth, 50c net. (2s.\ 6d.\ net.)
-
-222. THE REDEMPTION OF THE BRAHMAN. A novel. \Name{Richard~Garbe}. Cloth, 75c. (3s.\ 6d.)
-
-
-GUNKEL, HERMANN.
-
-227. THE LEGENDS OF GENESIS. \Name{Prof.~H.~Gunkel}. Transl.\ by
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