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-rw-r--r--40624-t/40624-t.tex436
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diff --git a/40624-t/40624-t.tex b/40624-t/40624-t.tex
index b63be49..efb0558 100644
--- a/40624-t/40624-t.tex
+++ b/40624-t/40624-t.tex
@@ -15,10 +15,11 @@
% Author: William F. White %
% %
% Release Date: August 30, 2012 [EBook #40624] %
+% Most recently updated: June 11, 2021 %
% %
% Language: English %
% %
-% Character set encoding: ISO-8859-1 %
+% Character set encoding: UTF-8 %
% %
% *** START OF THIS PROJECT GUTENBERG EBOOK A SCRAP-BOOK *** %
% %
@@ -115,7 +116,7 @@
\listfiles
\documentclass[12pt]{book}[2005/09/16]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\usepackage[latin1]{inputenc}[2006/05/05]
+\usepackage[utf8]{inputenc}[2006/05/05]
\usepackage[T1]{fontenc}
\usepackage{ifthen}[2001/05/26] %% Logical conditionals
@@ -665,8 +666,12 @@
\newcommand{\Brk}{\displaybreak[0] \\}
% Handle degree symbols and centered dots as Latin-1 characters
-\DeclareInputText{176}{\ifmmode{{}^\circ}\else\textdegree\fi}
-\DeclareInputText{183}{\ifmmode\cdot\else\textperiodcentered\fi}
+\DeclareUnicodeCharacter{00A3}{\pounds}
+\DeclareUnicodeCharacter{00B0}{{}^\circ}
+\DeclareUnicodeCharacter{00B1}{\pm}
+\DeclareUnicodeCharacter{00B7}{\cdot}
+\DeclareUnicodeCharacter{00D7}{\times}
+\DeclareUnicodeCharacter{00F7}{\div}
%% Upright capital letters in math mode
\DeclareMathSymbol{A}{\mathalpha}{operators}{`A}
@@ -719,10 +724,11 @@ Title: A Scrap-Book of Elementary Mathematics
Author: William F. White
Release Date: August 30, 2012 [EBook #40624]
+Most recently updated: June 11, 2021
Language: English
-Character set encoding: ISO-8859-1
+Character set encoding: UTF-8
*** START OF THIS PROJECT GUTENBERG EBOOK A SCRAP-BOOK ***
\end{PGtext}
@@ -827,7 +833,7 @@ The Open Court Publishing Company
\smallskip
\scriptsize London Agents \\
-Kegan Paul, Trench, Trübner \&~Co., Ltd. \\
+Kegan Paul, Trench, Trübner \&~Co., Ltd. \\
1908
\normalsize
\end{center}
@@ -1174,35 +1180,35 @@ 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.
+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.
+ 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~$714285$.)
\[
-356 × 142857 = 50857092
+356 × 142857 = 50857092
\]
(adding the~$50$ to the~$092$, $857142$).
\DPPageSep{018}{12}
-The one exception given above $(7 × 142857 = 999999)$
+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
@@ -1210,9 +1216,9 @@ 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 $\nicefrac{1}{29}$,
+$2 × 0588\dots = 1176470588235294$ (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
@@ -1241,12 +1247,12 @@ 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{.}''}
+ 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.}%
@@ -1307,27 +1313,27 @@ 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 $493 × \text{the number}$
+$493$ 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
+(\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
+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
+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}~$857$.
+$6 × 7 = 42$. 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
@@ -1340,7 +1346,7 @@ 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}~$53$. $7 × \text{the repetend}$
+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~$53$, $999946$.
@@ -1350,7 +1356,7 @@ to repeat. (Few such performances will bear repetition.)
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 $13 × 142857$
+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
@@ -1372,59 +1378,59 @@ 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 \\
+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 \\
+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},
+ {\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
+ 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
+ 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
+ 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
+12345679 × 8 &= 98765432 \\
+12345679 × 9 &= 111111111 \Brk
\intertext{to which may, of course, be added}
-12345679 × 18 &= 222222222 \\
-12345679 × 27 &= 333333333 \\
-12345679 × 36 &= 444444444 \\
+12345679 × 18 &= 222222222 \\
+12345679 × 27 &= 333333333 \\
+12345679 × 36 &= 444444444 \\
\text{etc.} &
\end{align*}
\DPPageSep{025}{19}
@@ -1445,8 +1451,8 @@ in a note entitled
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
+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.}
@@ -1460,11 +1466,11 @@ the lowest that are divisible by cubes other than~$1$:
\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~$851$;
+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~$41$: $17589 = 41 × 429$; $75891 = 41 × 1851$;
-$58917 = 41 × 1437$; $89175 = 41 × 2175$; $91758 = 41 × 2238$.
+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
@@ -1510,7 +1516,7 @@ of the same digits:
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},
+ in~1883, and completed in \Title{L'Intermédiaire des Mathématiciens},
1897~(4:168).}
\begin{align*}
11826^{2} &= 139854276 & 20316^{2} &= 412739856 \\
@@ -1533,7 +1539,7 @@ of the same digits:
\DPPageSep{027}{21}
2. Containing the ten digits:\footnote
- {\Title{L'Intermédiaire des Mathématiciens}, 1907~(14:135).}
+ {\Title{L'Intermédiaire des Mathématiciens}, 1907~(14:135).}
\begin{align*}
32043^{2} &= 1026753849 & 45624^{2} &= 2081549376 \\
32286^{2} &= 1042385796 & 55446^{2} &= 3074258916 \Brk
@@ -1620,7 +1626,7 @@ digits:
\begin{align*}
\frac{62}{31} - \frac{970}{485} &= 0 &
\frac{13485}{02697} &= 5 \\
-\frac{62}{31} × \frac{485}{970} &= 1 &
+\frac{62}{31} × \frac{485}{970} &= 1 &
\frac{34182}{05697} &= 6 \\
\frac{97062}{48531} &= 2 &
\frac{41832}{05976} &= 7\\
@@ -1630,7 +1636,7 @@ digits:
\frac{57429}{06381} &= 9 = \frac{95742}{10638}\Add{.}
\end{align*}
Lucas\footnote
- {\Title{Théorie des Nombres}, p.~40.}
+ {\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.
@@ -1667,7 +1673,7 @@ 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
+\Eg, $43268 ÷ 29$. The ordinary
\begin{table}[hbt!]
\[
\begin{array}[t]{r@{}r}
@@ -1697,19 +1703,19 @@ of the operation by the quotient figure for that step.
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~$1$
+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.
+$14 ÷ 3$,~$4$; 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
+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, $14 ÷ 3$,~$4$).
+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.
@@ -1855,7 +1861,7 @@ Let $n$ represent any number selected. The first
member of the following equality readily reduces to~$n$,
and the identity explains the trick.
\[
-\bigl\{[(2n + 4)5 + 12]10 - 320\bigr\} ÷ 100 = n.
+\bigl\{[(2n + 4)5 + 12]10 - 320\bigr\} ÷ 100 = n.
\]
2. \textit{Three dice being thrown on a table, to tell the
@@ -2010,14 +2016,14 @@ by~$7$.
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~$1001$.
+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, $4,728,350,169 ÷ 1001$.
+method may be made clear by an example, $4,728,350,169 ÷ 1001$.
\begin{gather*}
\begin{array}{l}
4728350169 \\
@@ -2146,7 +2152,7 @@ proposition was enunciated by Wilson,\footnote
He entered the law, became a judge, and attained a high reputation.}
first published
\DPPageSep{042}{36}
-by Waring in his \Title{Meditationes Algebraicæ}, and first
+by Waring in his \Title{Meditationes Algebraicæ}, and first
proved by Lagrange in~1771.
\index{Lagrange}%
@@ -2185,7 +2191,7 @@ comment:
(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
+\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
@@ -2203,7 +2209,7 @@ 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 =
+only one way, ceases to hold. \Eg, $26 = 2 × 13 =
(5 + \sqrt{-1})(5 - \sqrt{-1})$.
\Par{Asymptotic laws.} This happily chosen name describes
@@ -2325,7 +2331,7 @@ 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{Encyclopædia Britannica}
+Glaisher's article ``Tables'' in the \Title{Encyclopædia Britannica}
\index{Glaisher}%
is instructive.
@@ -2410,7 +2416,7 @@ 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
+$\text{which} = 641 × 6,700,417$. ``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
@@ -2420,7 +2426,7 @@ 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 $2^{75} · 5 + 1$.
+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
@@ -2641,7 +2647,7 @@ 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
+amount now, after $280$~years? $24 × 1.07^{280} = \text{more
than } 4,042,000,000$.
The latest tax assessment available at the time
@@ -2727,7 +2733,7 @@ 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
+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
@@ -3644,8 +3650,8 @@ 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{Encyclopædia
+far as $9 × 9$.''\footnote
+ {Dr.~Glaisher in his article ``Napier'' in the \Title{Encyclopædia
\index{Glaisher|FN}%
Britannica}.}
Though published
@@ -4229,12 +4235,12 @@ to sign, $+$~is understood before~$\surd$. Considering, then,
only the positive even root or the real odd root, it is
true that ``like roots of equals are equal,'' and
\[
-\sqrt[n]{ab} = \sqrt[n]{a\vphantom{b}} · \sqrt[n]{b}\Add{.}
+\sqrt[n]{ab} = \sqrt[n]{a\vphantom{b}} · \sqrt[n]{b}\Add{.}
\]
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)(-1)} &= \sqrt{-1} · \sqrt{-1} \\
\sqrt{1} &= (\sqrt{-1})^{2} \\
1 &= -1\Add{.}
\end{align*}
@@ -4294,7 +4300,7 @@ 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{Athenæum} of forty years ago.}
+ \Title{Athenæum} of forty years ago.}
Let
\[
@@ -4557,7 +4563,7 @@ 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.
+ {Albert Girard, \Title{Invention Nouvelle en l'Algèbre}, Amsterdam.
Perhaps also the first to distinctly recognize imaginary
roots of an equation.}
@@ -4576,12 +4582,12 @@ 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
+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
+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.}%
+interpretation of the imaginary was made by H.~Kühn
+\index{Kühn, H.}%
in~1750, in the \Title{Transactions of the St.~Petersburg
Academy.}
@@ -4597,9 +4603,9 @@ 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,
+These geometric interpretations by Kühn and Argand,
\index{Argand, J.~R.}%
-\index{Kühn, H.}%
+\index{Kühn, H.}%
and especially one made by Wessel,\footnote
{To the Copenhagen Academy of Sciences, 1797.}
\index{Wessel}%
@@ -4619,7 +4625,7 @@ 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.
+interpretations made by Kühn and Argand.
The contrasting terms ``real'' and ``imaginary'' as
applied to the roots of an equation were first used by
@@ -4660,7 +4666,7 @@ 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 $±a$ and~$±ai$.
+roots: $x^{2} - a^{2} = 0$ and $x^{2} + a^{2} = 0$ satisfied by $±a$ and~$±ai$.
The complete quadratic
\[
ax^{2} + bx + c =0
@@ -4767,8 +4773,8 @@ unit to obtain the other number (the multiplier.)\footnote
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.
+ $1 + 1^{2}$, 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
@@ -4776,7 +4782,7 @@ unit to obtain the other number (the multiplier.)\footnote
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
+ 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.
@@ -4964,7 +4970,7 @@ letters numbered~$19$ to be all sent.
\Chapter[Two negative conclusions.]{Two negative conclusions reached
in the nineteenth century.}
\index{Abel, N.~H.}%
-\index{Bocher@Bôcher, M.|FN}%
+\index{Bocher@Bôcher, M.|FN}%
\index{Higher equations}%
\index{Nineteenth century, negative conclusions reached}%
\index{Parallel postulates|(}%
@@ -4983,7 +4989,7 @@ 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.\
+ 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
@@ -5147,7 +5153,7 @@ quoted above.
\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
+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
@@ -5247,7 +5253,7 @@ 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},
+Schlömilch's \Title{Zeitschrift für Mathematik und Physik},
Vol.~13, p.~162.''
In an article in \Title{The Open Court}, August 1907, (from
@@ -5288,7 +5294,7 @@ $B$~and $C$ unity.
Solving the equation
\[
-x^{2} - xy - y^{2}= ±1
+x^{2} - xy - y^{2}= ±1
\]
we find by the Theory of Numbers that the $y$~and~$x$
may be taken as any two consecutive numbers in the
@@ -5343,10 +5349,10 @@ 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
+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{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.
@@ -5370,9 +5376,9 @@ 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}
+puzzlist.'' \Fig{A} is a square $8 × 8$, area~$64$. \Fig{B}
shows the pieces rearranged in a rectangle apparently
-$7 × 9$, area~$63$.
+$7 × 9$, area~$63$.
%[** TN: Moved figure up to associated text]
\Figures{0.9}{A}{0.9}{B}
\DPPageSep{123}{117}
@@ -5680,7 +5686,7 @@ Professor De~Morgan, in his \Title{\DPtypo{Buaget}{Budget} of Paradoxes}
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{Athenæum}, where
+first appeared in instalments in the \Title{Athenæum}, 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
@@ -6062,7 +6068,7 @@ 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{.}
+FP · FC &= FP' · FC'\Add{.}
\Tag{(1)}
\end{align*}
@@ -6077,11 +6083,11 @@ PB^{2} &= MP^{2} + MB^{2} \Brk
\therefore
FB^{2} - PB^{2} &= FM^{2} - MP^{2} \\
&= (FM + MP)(FM - MP) \\
- &= FP · FC
+ &= FP · FC
\Tag{(2)}
\end{align*}
-From \Eq{(1)} and \Eq{(2)}, $FP' · FC' = FB^{2} - PB^{2}$.
+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,
@@ -6144,8 +6150,8 @@ 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.}}%
+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
@@ -6337,9 +6343,9 @@ The Greeks used the entire chord of double the arc.
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
+\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
+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
@@ -6507,7 +6513,7 @@ 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 $\text{any
-constant} × AP \doteq \text{that constant} × AB$.
+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''~$AB$.
@@ -6603,8 +6609,8 @@ 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} ± 0.2\mu \\
-K \text{ no.}~20 &= 1\, \text{kg} - 0.039\, \text{mg} ± 0.002\, \text{mg}\Add{.}
+M \text{ no.}~27 &= 1\, \text{m} - 1.6\mu + 8.657\mu T + 0.00100\mu T^{2} ± 0.2\mu \\
+K \text{ no.}~20 &= 1\, \text{kg} - 0.039\, \text{mg} ± 0.002\, \text{mg}\Add{.}
\end{align*}
(U.S. coast and geodetic survey.)
@@ -6708,7 +6714,7 @@ 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} \\
+ &= \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
@@ -6835,7 +6841,7 @@ 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
+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
@@ -7124,14 +7130,14 @@ 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
+ 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|(}%
+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
@@ -7141,10 +7147,10 @@ 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
+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|)}%
+\index{Königsberg|)}%
Conceive the isles to shrink to points, and the problem
may be stated more conveniently with reference
@@ -7235,8 +7241,8 @@ 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}%
+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
@@ -7372,13 +7378,13 @@ 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@{Dædalus}}%
+\index{Daedalus@{Dædalus}}%
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{Encyclopædia Britannica} (article
+described in the \Title{Encyclopædia Britannica} (article
``Labyrinth'') as follows:
\Figure[1.0]{45}
@@ -7420,15 +7426,15 @@ apocalyptic literature, especially to such as have a
Pythagorean regard for hidden meaning in numbers.
There were centuries in which no satisfactory interpretation
-\index{Irenæus|(}%
+\index{Irenæus|(}%
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 (Cæsar)
-\index{Caesar@{Cæsar Neron}}%
+connected the number~$666$ with ``Emperor (Cæsar)
+\index{Caesar@{Cæsar 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
@@ -7442,7 +7448,7 @@ 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 Irenæus.
+number with Nero was apparently unknown to Irenæus.
\DPPageSep{187}{181}
He made several conjectures of words to fit
the number.
@@ -7497,7 +7503,7 @@ $V$\Add{,} and~$I$ have numerical values.
This and a similar derivation from Luther's name are
\index{Luther}%
perhaps the most famous of these performances.
-\index{Irenæus|)}%
+\index{Irenæus|)}%
De~Morgan cites a book by Rev.\ David Thom,\footnote
{\Title{The Number and Names of the Apocalyptic Beasts}, part~1,
@@ -7529,8 +7535,8 @@ are ``six of one and half a dozen of the other.''
\Chapter{Magic squares.}
-\index{Dela@{De la Loubère}}%
-\index{Loubère, de la}%
+\index{Dela@{De la Loubère}}%
+\index{Loubère, de la}%
\index{Magic!squares}%
\index{Squares!magic}%
@@ -7540,7 +7546,7 @@ 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{Encyclopædia Britannica.})
+(\Title{Encyclopædia Britannica.})
The term is often extended to include an assemblage
of numbers not consecutive but meeting all other requirements
@@ -7553,7 +7559,7 @@ each $q$ times those in the original
square.
\Figure[0.4]{46}
-One way (De~la Loubère's)
+One way (De~la Loubère's)
of constructing any odd-number
square is as follows:
@@ -7685,9 +7691,9 @@ 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{Encyclopædia Britannica}, together with some account
+\Title{Encyclopædia Britannica}, together with some account
of their history. See also Ball's \Title{Recreations}; Lucas's
-\Title{Récréations}, vol.~4, Cinquième Récréation, ``Les Carrés
+\Title{Récréations}, vol.~4, Cinquième Récréation, ``Les Carrés
magiques de Fermat''; and the comprehensive
\index{Fermat}%
article, ``A Mathematical Study of Magic Squares,''
@@ -8043,7 +8049,7 @@ 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
+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,
@@ -8523,8 +8529,8 @@ no one who is unacquainted with geometry enter here.''
\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}%
+ 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.})}
@@ -9351,7 +9357,7 @@ now.
\Figure[1.0]{59}
\begin{Remark}
-The dotted line~$QPQ$, if revolved~$90°$ about $XX'$ as axis, remaining
+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.
@@ -9492,11 +9498,11 @@ in the seventeenth and eighteenth centuries.\footnote
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
+ 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.}
+ 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
@@ -9507,7 +9513,7 @@ 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
+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.
@@ -9611,7 +9617,7 @@ Hints on Methods of Teaching. Macmillan, 1905 (1896).
\Bibitem Cajori, Florian. History of Mathematics. Macmillan, 1894.
\PgNo{37}, \PgNo{148}, \PgNo{193}.
-\Bibitem Cantor, Moritz. Vorlesungen über die Geschichte der Mathematik.
+\Bibitem Cantor, Moritz. Vorlesungen über die Geschichte der Mathematik.
$3$~vol. Teubner, Leipzig, 1880--92. \PgNo{49}, \PgNo{67}, \PgNo{148},
\PgNo{234}.
@@ -9631,7 +9637,7 @@ Julius Springer, Berlin, 1903. \PgNo{40}.
\Bibitem Dodgson, C.~L\@. Through the Looking-glass and What Alice
Found There. 1872. *\PgNo{218}.
-\Bibitem Encyclopædia Britannica. Ed.~9. \PgNo{39}, *\PgNo{71}, *\PgNo{176}, *\PgNo{183}, \PgNo{186}.
+\Bibitem Encyclopædia 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}.
@@ -9644,12 +9650,12 @@ Appleton, 1898. \PgNo{119}.
\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 Fourier. Analyse des Equations Determinées. \PgNo{23}.
-\Bibitem Fourrey, E\@. Curiositées Géométriques. Vuibert et Nony,
+\Bibitem Fourrey, E\@. Curiositées Géométriques. Vuibert et Nony,
Paris, 1907. \PgNo{235}.
-\Bibitem Girard, Albert. Invention Nouvelle en l'Algèbre. Amsterdam,
+\Bibitem Girard, Albert. Invention Nouvelle en l'Algèbre. Amsterdam,
1629. \PgNo{92}.
\Bibitem Gray, Peter. Tables for the Formation of Logarithms and
@@ -9691,19 +9697,19 @@ 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 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}.
+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,
+\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 Lucas, Edouard. Théorie des Nombres. \PgNo{17}, \PgNo{22}.
\Bibitem McLellan and Dewey. Psychology of Number. Appleton,
1895. \PgNo{154}.
@@ -9728,7 +9734,7 @@ Paris, 1891--6. *\PgNo{17}, *\PgNo{70}, \PgNo{141}, *\PgNo{171}, \PgNo{186}, *\P
\DPPageSep{245}{239}
% [** TN: http://www.gutenberg.org/ebooks/28233]
-\Bibitem Newton, Isaac. Philosophiæ Naturalis Principia Mathematica.
+\Bibitem Newton, Isaac. Philosophiæ Naturalis Principia Mathematica.
1687. *\PgNo{149}.
\Bibitem New York Education (now American Education). *\PgNo{210}, \PgNo{235}.
@@ -9749,7 +9755,7 @@ Teachers. \PgNo{206}.
\Bibitem Public School Journal. *\PgNo{206}.
-\Bibitem Rebiere. Mathématique et Mathématiciens. \PgNo{196}.
+\Bibitem Rebiere. Mathématique et Mathématiciens. \PgNo{196}.
\Bibitem Recorde, Robert. Grounde of Artes. 1540. \PgNo{68}\Add{.}
@@ -9762,7 +9768,7 @@ 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 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}.
@@ -9804,7 +9810,7 @@ Social Measurements. Science Press, New York, 1904.
\Bibitem Treviso Arithmetic. 1478. \PgNo{59}, \PgNo{67}.
-\Bibitem Waring, Edward. Meditationes Algebraicæ. \PgNo{36}.
+\Bibitem Waring, Edward. Meditationes Algebraicæ. \PgNo{36}.
\Bibitem Widmann, John. Arithmetic. Leipsic, 1489. \PgNo{162}.
@@ -9941,7 +9947,7 @@ Billion 9
Binomial theorem and statistics. 159
-Bocher@Bôcher, M.|FN 103, 212
+Bocher@Bôcher, M.|FN 103, 212
Bolingbroke, Lord 51
@@ -9966,7 +9972,7 @@ Buffon 126
Building Committee, advice to 201
-Caesar@{Cæsar Neron} 180
+Caesar@{Cæsar Neron} 180
Cajori, Florian 59, 124
@@ -10059,7 +10065,7 @@ Criterion for prime numbers 36
Curiosities, numerical 19
-Daedalus@{Dædalus} 178
+Daedalus@{Dædalus} 178
Days-work problem 213
@@ -10082,7 +10088,7 @@ Definition
Degree of accuracy of measurements 43-44
-Dela@{De la Loubère} 183
+Dela@{De la Loubère} 183
Delian problem|EtSeq 122
@@ -10381,7 +10387,7 @@ Interest, compound and simple 47
Involution not commutative 154
-Irenæus 180-181
+Irenæus 180-181
Isles and bridges 170
@@ -10407,9 +10413,9 @@ Knilling 57
Knowlton 197
-Königsberg 170-171, 174
+Königsberg 170-171, 174
-Kühn, H. 93, 94
+Kühn, H. 93, 94
Kulik 40
@@ -10458,7 +10464,7 @@ Logarithms 45, 47, 52, 69, 87, 102, 165
London and Wise 176
-Loubère, de la 183
+Loubère, de la 183
Lowest common multiples, two 89
@@ -10532,7 +10538,7 @@ Minotaur 178
Miscellaneous notes on number 34
-Mobius@{Möbius, A. F.} 140
+Mobius@{Möbius, A. F.} 140
Mohammed 175-176
@@ -11223,9 +11229,9 @@ ESSAYS. \Name{Paul Carus}. Cloth, 50c net (2s.\ 6d.\ net.)
Cloth, 1.00 net. (5s.)
217. KARMA, A STORY OF EARLY BUDDHISM. \Name{Paul Carus}.
-Illustrated. Crêpe paper, tied in silk. 75c (3s.\ 6d.)
+Illustrated. Crêpe paper, tied in silk. 75c (3s.\ 6d.)
-217G. KARMA, Eine buddhistische Erzählung. \Name{Paul Carus}. Illustrated.
+217G. KARMA, Eine buddhistische Erzählung. \Name{Paul Carus}. Illustrated.
35c.
291. NIRVANA, A STORY OF BUDDHIST PSYCHOLOGY, \Name{Paul
@@ -11351,15 +11357,15 @@ AND INTEGRAL CALCULUS. \Name{Augustus DeMorgan}. Cloth,
\$1.00 net (4s.\ 6d.\ net.)
-DESCARTES, RENE. %[** TN: [sic], not RENÉ]
+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
+\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,
+of \Name{René Descartes}. Transl.\ by \Name{John Veitch}. Cloth,
75c net. (3s.\ 6d.\ net.)
@@ -11389,7 +11395,7 @@ 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}.
+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.
@@ -11542,7 +11548,7 @@ Montgomery. Cloth}, 75c net. (3s.\ 6d.\ net.)
LEVY-BRUHL, LUCIEN.
273. HISTORY OF MODERN PHILOSOPHY IN FRANCE. \Name{Lucien
-Lévy-Bruhl}. With portraits. \$3.00 net. (12s.\ net.)
+Lévy-Bruhl}. With portraits. \$3.00 net. (12s.\ net.)
LOCKE, JOHN.
@@ -11588,26 +11594,26 @@ Avesta. \Name{Lawrence H. Mills}. 1906. Cloth, gilt top, \$4.00 net.
MUELLER, F. MAX.
231. THREE INTRODUCTORY LECTURES ON THE SCIENCE
-OF THOUGHT. \Name{F. Max Müller}. With a correspondence on
-\textsc{Thought without words} between F. Max Müller and Francis
+OF THOUGHT. \Name{F. Max Müller}. With a correspondence on
+\textsc{Thought without words} between F. Max Müller and Francis
Galton, the Duke of Argyll, G. J. Romanes and Others. Cloth,
75c. (3s.\ 6d.)
232. THREE LECTURES ON THE SCIENCE OF LANGUAGE.
-With a supplement, \textsc{My Predecessors.} \Name{F. Max Müller}. Cloth,
+With a supplement, \textsc{My Predecessors.} \Name{F. Max Müller}. Cloth,
75c. (3s.\ 6d.)
NAEGELI, CARL VON.
300. A MECHANICO-PHYSIOLOGICAL THEORY OF ORGANIC
-EVOLUTION. \Name{Carl von Nägeli. Cloth}, 50c net. (2s.\ 6d.\ net)
+EVOLUTION. \Name{Carl von Nägeli. Cloth}, 50c net. (2s.\ 6d.\ net)
NOIRE, LUDWIG.
397. ON THE ORIGIN OF LANGUAGE, and THE LOGOS THEORY.
-\Name{Ludwig Noiré}. Cloth, 50c net. (2s.\ 6d.\ net.)
+\Name{Ludwig Noiré}. Cloth, 50c net. (2s.\ 6d.\ net.)
OLDENBERG, PROF. H.
@@ -11846,7 +11852,7 @@ set. Single portraits, Japanese paper, 50c (2s.\ 6d.); single
portraits, on plate paper, 25c (1s.\ 6d.)
332a. FRAMING PORTRAIT OF HUGO DE VRIES. Platino finish.
-$10×12''$, unmounted. Postpaid, \$1.00. (4s.\ 6d.\ net.)
+$10×12''$, unmounted. Postpaid, \$1.00. (4s.\ 6d.\ net.)
SMITH, PROF. DAVID EUGENE.
@@ -11869,14 +11875,14 @@ THE RELIGION OF SCIENCE LIBRARY
(1s.\ 6d.)
2. THREE INTRODUCTORY LECTURES ON THE SCIENCE
-OF THOUGHT. \Name{F. Max Müller.} With a correspondence on
-``Thought Without Words'' between \Name{F. Max Müller} and \Name{Francis
+OF THOUGHT. \Name{F. Max Müller.} With a correspondence on
+``Thought Without Words'' between \Name{F. Max Müller} and \Name{Francis
Galton}, the \Name{Duke of Argyll}, \Name{George J. Romanes} and others.
25c, mailed 29c. (1s.\ 6d.)
\DPPageSep{266}{260}
3. THREE LECTURES ON THE SCIENCE OF LANGUAGE.
-With My Predecessors. \Name{F. Max Müller.} 25c, mailed 29c.
+With My Predecessors. \Name{F. Max Müller.} 25c, mailed 29c.
(1s.\ 6d.)
4. THE DISEASES OF PERSONALITY. \Name{Prof.\ Th.\ Ribot.} 25c,
@@ -11903,7 +11909,7 @@ mailed 60c. (2s.\ 6d.)
Snell.} 25c, mailed 29c. (1s.\ 6d.)
11. ON THE ORIGIN OF LANGUAGE and the Logos Theory. \Name{L.
-Noiré.} 15c, mailed 18c. (1s.\ 6d.)
+Noiré.} 15c, mailed 18c. (1s.\ 6d.)
12. THE FREE TRADE STRUGGLE IN ENGLAND. \Name{M. M. Trumbull.}
25c, mailed 31c. (1s.\ 6d.)
@@ -11974,7 +11980,7 @@ mailed 30c. (1s.\ 6d.)
volume. 60c, mailed 80c. (3s.)
32. A MECHANICO-PHYSIOLOGICAL THEORY OF ORGANIC
-EVOLUTION. \Name{Carl von Nägeli.} 15c, mailed 18c. (9d.)
+EVOLUTION. \Name{Carl von Nägeli.} 15c, mailed 18c. (9d.)
33. CHINESE FICTION. Rev. \Name{G. T. Candlin.} Illustrated. 15c,
mailed 18c. (9d.)
@@ -11993,7 +11999,7 @@ M. Stanley.} 20c, mailed 23c. (1s.) Out of print.
38. DISCOURSE ON THE METHOD OF RIGHTLY CONDUCTING
THE REASON, AND SEEKING TRUTH IN THE
-SCIENCES. \Name{René Descartes.} Transl.\ by \Name{Prof.\ John Veitch.}
+SCIENCES. \Name{René Descartes.} Transl.\ by \Name{Prof.\ John Veitch.}
25c, mailed 29c. (1s.\ 6d.)
39. THE DAWN OF A NEW RELIGIOUS ERA and other Essays.
@@ -12036,7 +12042,7 @@ KNOWLEDGE. \Name{George Berkeley.} 25c, mailed 31c. (1s.\ 6d.)
RELIGION. \Name{John P. Hylan.} 25c, mailed 29c. (1s.\ 6d.)
51. THE MEDITATIONS AND SELECTIONS FROM THE PRINCIPLES
-of \Name{René Descartes.} Transl.\ by \Name{Prof.\ John Veitch.}
+of \Name{René Descartes.} Transl.\ by \Name{Prof.\ John Veitch.}
35c, mailed 42c. (2s.)
52. LEIBNIZ: DISCOURSE ON METAPHYSICS, CORRESPONDENCE
@@ -12087,7 +12093,7 @@ THE OPEN COURT PUBLISHING COMPANY
1322 Wabash Avenue, Chicago
-London: Kegan Paul, Trench, Trübner \& Co., Ltd.
+London: Kegan Paul, Trench, Trübner \& Co., Ltd.
\DPPageSep{269}{263}
10 Cents Per Copy THE OPEN COURT \$1.00 Per Year
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-%%%%%%%%%%%%%%%%%%%%%%%% 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 Algebraicæ}, 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{Encyclopædia 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{Encyclopædia
-\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{Athenæum} 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 $±a$ and~$±ai$.
-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
-\[
-x^{2} - xy - y^{2}= ±1
-\]
-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{Athenæum}, 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} ± 0.2\mu \\
-K \text{ no.}~20 &= 1\, \text{kg} - 0.039\, \text{mg} ± 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@{Dædalus}}%
-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{Encyclopædia 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{Irenæus|(}%
-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 (Cæsar)
-\index{Caesar@{Cæsar 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 Irenæus.
-\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{Irenæus|)}%
-
-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{Encyclopædia 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{Encyclopædia 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 Encyclopædia 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. Philosophiæ 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 Algebraicæ. \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@{Cæsar 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@{Dædalus} 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
-
-Irenæus 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]
-
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generated by cgit v1.2.3 (git 2.25.1) at 2025-12-16 15:30:31 +0000