diff options
Diffstat (limited to '40624-t/40624-t.tex')
| -rw-r--r-- | 40624-t/40624-t.tex | 436 |
1 files changed, 221 insertions, 215 deletions
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
|