1 files changed, 86 insertions, 84 deletions

diff --git a/41568-t/41568-t.tex b/41568-t/41568-t.tex
index 9d9f532..dc4294e 100644
--- a/41568-t/41568-t.tex
+++ b/41568-t/41568-t.tex
@@ -13,10 +13,11 @@
 % Author: Alfred North Whitehead                                          %

 %                                                                         %

 % Release Date: December 6, 2012 [EBook #41568]                           %

+% 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 AN INTRODUCTION TO MATHEMATICS ***

 %                                                                         %

@@ -110,7 +111,7 @@
 \documentclass[12pt,leqno]{book}[2005/09/16]

 

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-\usepackage[latin1]{inputenc}[2006/05/05]

+\usepackage[utf8]{inputenc}[2006/05/05]

 

 \usepackage{ifthen}[2001/05/26]  %% Logical conditionals

 

@@ -511,10 +512,11 @@ Title: An Introduction to Mathematics
 Author: Alfred North Whitehead

 

 Release Date: December 6, 2012 [EBook #41568]

+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 AN INTRODUCTION TO MATHEMATICS ***

 \end{PGtext}

@@ -1023,7 +1025,7 @@ The same remark applies to the solution of
 the inequality~\Eq{(3')} as compared to the original

 statement~\Eq{(3)}.

 

-But the majority of interesting formulæ,

+But the majority of interesting formulæ,

 \index{Relations between Variables|EtSeq}%

 \index{Variable, The}%

 especially when the idea of \emph{some} is present,

@@ -1042,9 +1044,9 @@ The second type of statement invites consideration
 of the aggregate of pairs of numbers

 which are bound together by some fixed

 relation---in the case given, by the relation

-$x + y = 1$. One use of formulæ of the first

+$x + y = 1$. One use of formulæ of the first

 type, true for \emph{any} pair of numbers, is that by

-them formulæ of the second type can be

+them formulæ of the second type can be

 \PageSep{19}

 thrown into an indefinite number of equivalent

 forms. For example, the relation $x + y = 1$

@@ -1063,7 +1065,7 @@ It is not in general true that, when a pair
 of terms satisfy some fixed relation, if one of

 the terms is given the other is also definitely

 determined. For example, when $x$ and~$y$

-satisfy $y^{2} = x$, if $x = 4$, $y$~can be~$±2$, thus,

+satisfy $y^{2} = x$, if $x = 4$, $y$~can be~$±2$, thus,

 for any positive value of~$x$ there are alternative

 values for~$y$. Also in the relation

 $x + y > 1$, when either $x$ or~$y$ is given, an

@@ -1207,7 +1209,7 @@ our thoughts on the whole subject.
 

 Let us start with the simplest of examples:---Suppose

 that building costs $1$\textit{s.}\ per cubic

-foot and that $20$\textit{s.}\ make~£$1$. Then in all

+foot and that $20$\textit{s.}\ make~£$1$. Then in all

 the complex circumstances which attend the

 building of a new house, amid all the various

 sensations and emotions of the owner, the

@@ -1216,7 +1218,7 @@ onlookers as the house has grown to completion,
 this fixed correlation is by the law

 assumed to hold between the cubic content

 and the cost to the owner, namely that if $x$~be

-the number of cubic feet, and £$y$~the cost,

+the number of cubic feet, and £$y$~the cost,

 then $20y = x$. This correlation of $x$~and $y$ is

 assumed to be true for the building of any

 house by any owner. Also, the volume of

@@ -1402,7 +1404,7 @@ it to be the formula always satisfied in these
 various motions.

 

 The vital point in the application of mathematical

-formulæ is to have clear ideas and a

+formulæ is to have clear ideas and a

 correct estimate of their relevance to the

 phenomena under observation. No less than

 ourselves, our remote ancestors were impressed

@@ -1504,16 +1506,16 @@ square of their distances, and also that the
 \PageSep{34}

 same law holds for electric charges---laws

 curiously analogous to that of gravitation.

-In~1820, Öersted, a Dane, discovered that

-\index{Oersted@Öersted}%

+In~1820, Öersted, a Dane, discovered that

+\index{Oersted@Öersted}%

 electric currents exert a force on magnets,

 and almost immediately afterwards the

 mathematical law of the force was correctly

-formulated by Ampère, a Frenchman, who

-\index{Ampere@Ampère}%

+formulated by Ampère, a Frenchman, who

+\index{Ampere@Ampère}%

 also proved that two electric currents exerted

 forces on each other. ``The experimental investigation

-by which Ampère established the

+by which Ampère established the

 law of the mechanical action between electric

 currents is one of the most brilliant achievements

 in science. The whole, theory and

@@ -1828,7 +1830,7 @@ body continues in its state of rest or of uniform
 motion in a straight line, except so far

 as it is compelled by impressed force to

 change that state. This law is more than a

-dry formula: it is also a pæan of triumph

+dry formula: it is also a pæan of triumph

 over defeated heretics. The point at issue

 can be understood by deleting from the law

 the phrase ``or of uniform motion in a straight

@@ -2340,11 +2342,11 @@ x + y = y + x\Add{,}
 \Tag{(1)} \\

 (x + y) + z = x + (y + z)\Add{,}

 \Tag{(2)} \\

-x × y = y × x\Add{,}

+x × y = y × x\Add{,}

 \Tag{(3)} \\

-(x × y) × z = x × (y × z)\Add{,}

+(x × y) × z = x × (y × z)\Add{,}

 \Tag{(4)} \\

-x × (y + z) = (x × y) + (x × z)\Add{.}

+x × (y + z) = (x × y) + (x × z)\Add{.}

 \Tag{(5)}

 \end{gather*}

 

@@ -2398,7 +2400,7 @@ their juxtaposition~$xy$. Now the two most
 important ideas on hand are those of addition

 and multiplication. Mathematicians have

 chosen to make their symbolism more concise

-by defining $xy$ to stand for $x × y$. Thus the

+by defining $xy$ to stand for $x × y$. Thus the

 laws \Eq{(3)},~\Eq{(4)}, and~\Eq{(5)} above are in general

 written,

 \[

@@ -2409,14 +2411,14 @@ x(y + z) = xy + xz,
 thus securing a great gain in conciseness.

 The same rule of symbolism is applied to the

 juxtaposition of a definite number and a variable:

-we write~$3x$ for $3 × x$, and $30x$ for $30 × x$.

+we write~$3x$ for $3 × x$, and $30x$ for $30 × x$.

 

 It is evident that in substituting definite

 numbers for the variables some care must be

-taken to restore the~$×$, so as not to conflict

+taken to restore the~$×$, so as not to conflict

 with the Arabic notation. Thus when we

 substitute $2$~for~$x$ and $3$~for~$y$ in~$xy$, we must

-write $2 × 3$ for~$xy$, and not~$23$ which means

+write $2 × 3$ for~$xy$, and not~$23$ which means

 $20 + 3$.

 

 It is interesting to note how important for

@@ -2552,7 +2554,7 @@ not multiplied by itself, so that $x^{2}$, $x^{3}$,~etc., do
 not appear. Again $3x^{2} - 2x + 1 = 0$, $x^{2} - 3x + 2 = 0$,

 $x^{2} - 4 = 0$, are all equations of the same

 form, namely, equations involving one unknown~$x$

-in which $x × x$, that is~$x^{2}$, appears. These

+in which $x × x$, that is~$x^{2}$, appears. These

 equations are called quadratic equations.

 Similarly cubic equations, in which $x^{3}$~appears,

 yield another form, and so on. Among the

@@ -2586,15 +2588,15 @@ the right-hand side.
 

 There is yet another function performed by~$0$

 in relation to the study of form. Whatever

-number $x$ may be, $0 × x = 0$, and $x + 0 = x$.

+number $x$ may be, $0 × x = 0$, and $x + 0 = x$.

 By means of these properties minor differences

 of form can be assimilated. Thus the

 difference mentioned above between the quadratic

 equations $x^{2} - 3x + 2 = 0$, and $x^{2} - 4 = 0$,

 can be obliterated by writing the latter

 \PageSep{68}

-equation in the form $x^{2} + (0 × x) - 4 = 0$. For,

-by the laws stated above, $x^{2} + (0 × x) - 4 =

+equation in the form $x^{2} + (0 × x) - 4 = 0$. For,

+by the laws stated above, $x^{2} + (0 × x) - 4 =

 x^{2} + 0 - 4 = x^{2} - 4$. Hence the equation $x^{2} - 4 = 0$\Typo{,}{}

 is merely representative of a particular

 class of quadratic equations and belongs to

@@ -3231,7 +3233,7 @@ last chapter have been a popular success,
 those of the present chapter have excited

 almost as much general attention. But their

 success has been of a different character, it

-has been what the French term a \Foreign{succès de

+has been what the French term a \Foreign{succès de

 scandale}. Not only the practical man, but

 also men of letters and philosophers have expressed

 their bewilderment at the devotion

@@ -3258,7 +3260,7 @@ remember, or so as to suggest relevant and
 important ideas. But the essential principle

 involved was quite clearly enunciated in

 Wonderland to Alice by Humpty Dumpty,

-when he told her, à~propos of his use of words,

+when he told her, à~propos of his use of words,

 ``I pay them extra and make them mean

 what I like.'' So we will not bother as to

 whether imaginary numbers are imaginary,

@@ -3283,7 +3285,7 @@ $x^{2} + 1 = 3$ becomes $x^{2} = 2$, and this has two
 solutions, either $x = +\sqrt{2}$, or $x = -\sqrt{2}$. The

 statement that there are these alternative

 \PageSep{89}

-solutions is usually written $x = ±\sqrt{2}$. So far

+solutions is usually written $x = ±\sqrt{2}$. So far

 all is plain sailing, as it was in the previous

 case. But now an analogous difficulty arises.

 For the equation $x^{2} + 3 = 1$ gives $x^{2} = -2$ and

@@ -3294,7 +3296,7 @@ the ordinary positive or negative numbers,
 there is no solution to $x^{2} = -2$, and the equation

 is in fact nonsense. Thus, finally taking

 the general form $x^{2} + a = b$, we find the pair

-of solutions $x = ±\sqrt{(b - a)}$, when, and only

+of solutions $x = ±\sqrt{(b - a)}$, when, and only

 when, $b$~is not less than~$a$. Accordingly we

 cannot say unrestrictedly that the ``constants''

 $a$~and~$b$ may be any numbers, that is,

@@ -3306,7 +3308,7 @@ work as we proceed.
 

 The same task as before therefore awaits

 us: we must give a new interpretation to our

-symbols, so that the solutions $±\sqrt{(b -  a)}$ for

+symbols, so that the solutions $±\sqrt{(b -  a)}$ for

 the equation $x^{2} + a = b$ always have meaning.

 In other words, we require an interpretation

 of the symbols so that $\sqrt{a}$~always has meaning

@@ -3322,7 +3324,7 @@ fact, it must in a sense include them as
 special cases. When $a$~is negative we may

 write $-c^{2}$ for it, so that $c^{2}$~is positive. Then

 \begin{align*}

-\sqrt{a} &= \sqrt{(-c^{2})} = \sqrt{\{(-1) × c^{2}\}} \\

+\sqrt{a} &= \sqrt{(-c^{2})} = \sqrt{\{(-1) × c^{2}\}} \\

   &= \sqrt{(-1)} \sqrt{c^{2}} = c\sqrt{(-1)}.

 \end{align*}

 Hence, if we can so interpret our symbols that

@@ -3666,12 +3668,12 @@ such that
 

 \Eq{(\beta)} the operation is commutative, so that

 \[

-(x, y) × (x', y') = (x', y') × (x, y),

+(x, y) × (x', y') = (x', y') × (x, y),

 \]

 

 \Eq{(\gamma)} the operation is associative, so that

 \[

-\{(x, y) × (x', y')\} × (u, v) = (x, y) × \{(x', y') × (u, v)\},

+\{(x, y) × (x', y')\} × (u, v) = (x, y) × \{(x', y') × (u, v)\},

 \]

 

 \Eq{(\delta)} must make the result of division unique

@@ -3680,12 +3682,12 @@ couple $(0, 0)$], so that when we seek to determine
 the unknown couple $(x, y)$ so as to

 satisfy the equation

 \[

-(x, y) × (a, b) = (c, d),

+(x, y) × (a, b) = (c, d),

 \]

 there is one and only one answer, which we

 can represent by

 \[

-(x, y) = (c, d) ÷ (a, b),\quad\text{or by}\quad

+(x, y) = (c, d) ÷ (a, b),\quad\text{or by}\quad

 (x, y) = \frac{(c, d)}{(a, b)}\Add{.}

 \]

 \PageSep{102}

@@ -3694,8 +3696,8 @@ can represent by
 addition and multiplication, called the distributive

 law, must be satisfied, namely

 \begin{multline*}

-(x,y) × \{(a, b) + (c, d)\} \\

-= \{(x, y) × (a, b)\} + \{(x, y) × (c, d)\}.

+(x,y) × \{(a, b) + (c, d)\} \\

+= \{(x, y) × (a, b)\} + \{(x, y) × (c, d)\}.

 \end{multline*}

 

 All these conditions \Eq{(\alpha)}, \Eq{(\beta)}, \Eq{(\gamma)}, \Eq{(\delta)}, \Eq{(\epsilon)} can

@@ -3705,12 +3707,12 @@ of a simple geometrical interpretation.
 

 By definition we put

 \[

-(x, y) × (x', y') = \{(xx' - yy'), (xy' + x'y)\}\Add{.}

+(x, y) × (x', y') = \{(xx' - yy'), (xy' + x'y)\}\Add{.}

 \Tag{(A)}

 \]

 

 This is the definition of the meaning of the

-symbol~$×$ when it is written between two

+symbol~$×$ when it is written between two

 ordered couples. It follows evidently from

 this definition that the result of multiplication

 is another ordered couple, and that the

@@ -3733,7 +3735,7 @@ assigned in terms of positive and negative real
 numbers. We then found that all our difficulties

 would vanish if we could interpret the

 equation $x^{2} = -1$, \ie, if we could so define

-$\sqrt{(-1)}$ that $\sqrt{(-1)} × \sqrt{(-1)} = -1$.

+$\sqrt{(-1)}$ that $\sqrt{(-1)} × \sqrt{(-1)} = -1$.

 

 Now let us consider the three special

 \index{Zero}%

@@ -3750,22 +3752,22 @@ We have already proved that
 

 Furthermore we now have

 \[

-(x, y) × (0, 0) = (0, 0).

+(x, y) × (0, 0) = (0, 0).

 \]

 

 Hence both for addition and for multiplication

 the couple $(0, 0)$ plays the part of zero in

 elementary arithmetic and algebra; compare

 the above equations with $x + 0 = x$, and

-$x × 0 = 0$.

+$x × 0 = 0$.

 

 Again consider $(1, 0)$: this plays the part

 of~$1$ in elementary arithmetic and algebra.

 In these elementary sciences the special

-characteristic of~$1$ is that $x × 1 = x$, for all

+characteristic of~$1$ is that $x × 1 = x$, for all

 values of~$x$. Now by our law of multiplication

 \[

-(x, y) × (1, 0) = \{(x - 0), (y + 0)\} = (x, y).

+(x, y) × (1, 0) = \{(x - 0), (y + 0)\} = (x, y).

 \]

 

 Thus $(1, 0)$ is the unit couple.

@@ -3774,23 +3776,23 @@ Thus $(1, 0)$ is the unit couple.
 Finally consider $(0, 1)$: this will interpret

 for us the symbol~$\sqrt{(-1)}$. The symbol must

 therefore possess the characteristic property

-that $\sqrt{(-1)} × \sqrt{(-1)} = -1$. Now by the

+that $\sqrt{(-1)} × \sqrt{(-1)} = -1$. Now by the

 law of multiplication for ordered couples

 \[

-(0, 1) × (0, 1) = \{(0 - 1), (0 + 0)\} = (-1, 0).

+(0, 1) × (0, 1) = \{(0 - 1), (0 + 0)\} = (-1, 0).

 \]

 

 But $(1, 0)$ is the unit couple, and $(-1, 0)$

 is the negative unit couple; so that $(0, 1)$ has

 the desired property. There are, however,

 two roots of~$-1$ to be provided for, namely

-$±\sqrt{(-1)}$. Consider $(0, -1)$; here again remembering

-that $(-1)^{2} = 1$, we find, $(0, -1) × (0, -1) = (-1, 0)$.

+$±\sqrt{(-1)}$. Consider $(0, -1)$; here again remembering

+that $(-1)^{2} = 1$, we find, $(0, -1) × (0, -1) = (-1, 0)$.

 

 Thus $(0, -1)$ is the other square root of~$\Typo{\sqrt{(-1)}}{-1}$.

 Accordingly the ordered couples

 $(0, 1)$ and $(0, -1)$ are the interpretations of

-$±\sqrt{(-1)}$ in terms of ordered couples. But

+$±\sqrt{(-1)}$ in terms of ordered couples. But

 which corresponds to which? Does $(0, 1)$

 correspond to $+\sqrt{(-1)}$ and $(0, -1)$ to~$-\sqrt{(-1)}$,

 or $(0, 1)$ to~$-\sqrt{(-1)}$, and $(0, -1)$

@@ -3808,7 +3810,7 @@ multiply together the ``complex imaginary''
 couple $(x, y)$ and the ``real'' couple $(a, 0)$, we

 find

 \[

-(a, 0) × (x, y) = (ax, ay).

+(a, 0) × (x, y) = (ax, ay).

 \]

 

 Thus the effect is merely to multiply each

@@ -3819,7 +3821,7 @@ Secondly, multiply together the ``complex
 imaginary'' couple $(x, y)$ and the ``pure

 imaginary'' couple $(0, b)$, we find

 \[

-(0, b) × (x, y) = (-by, bx).

+(0, b) × (x, y) = (-by, bx).

 \]

 

 Here the effect is more complicated, and is

@@ -3830,19 +3832,19 @@ three yet more special cases.
 Thirdly, we multiply the ``real'' couple

 $(a, 0)$ by the imaginary $(0, b)$ and obtain

 \[

-(a, 0) × (0, b) =(0, ab).

+(a, 0) × (0, b) =(0, ab).

 \]

 

 Fourthly, we multiply the two ``real''

 couples $(a, 0)$ and $(a', 0)$ and obtain

 \[

-(a, 0) × (a', 0) =( aa', 0).

+(a, 0) × (a', 0) =( aa', 0).

 \]

 

 Fifthly, we multiply the two ``imaginary

 couples'' $(0, b)$ and $(0, \Typo{b}{b'})$ and obtain

 \[

-(0, b) × (0, b') = (-bb', 0).

+(0, b) × (0, b') = (-bb', 0).

 \]

 

 We now turn to the geometrical interpretation,

@@ -3851,22 +3853,22 @@ beginning first with some special cases.
 Take the couples $(1, 3)$ and $(2, 0)$ and consider

 the equation

 \[

-(2, 0) × (1, 3) = (2, 6)\Add{.}

+(2, 0) × (1, 3) = (2, 6)\Add{.}

 \]

 \Figure{11}

 

 In the diagram (\Fig[fig.]{11}) the vector~$OP$ represents~$(1, 3)$,

 and the vector~$ON$ represents~$(2, 0)$,

 and the vector~$OQ$ represents~$(2, 6)$.

-Thus the product $(2, 0) × (1, 3)$ is found geometrically

+Thus the product $(2, 0) × (1, 3)$ is found geometrically

 by taking the length of the vector~$OQ$

 to be the product of the lengths of the

 vectors $OP$ and~$ON$, and (in this case) by

 producing $OP$ to~$Q$ to be of the required

-length. Again, consider the product $(0, 2) × (1, 3)$,

+length. Again, consider the product $(0, 2) × (1, 3)$,

 we have

 \[

-(0, 2) × (1, 3) = (-6, 2)\Add{.}

+(0, 2) × (1, 3) = (-6, 2)\Add{.}

 \]

 

 The vector~$ON_{1}$, corresponds to~$(0, 2)$ and

@@ -4684,7 +4686,7 @@ has only one vertex. Apollonius proved\footnote
   {\Chg{Cf.}{\Cf}\ Ball, \Foreign{loc.\ cit.}, for this account of Apollonius and

   Pappus.}

 that

-the ratio of $PM^{2}$ to $AM·MA'$ $\left(\ie\ \dfrac{PM^{2}}{AM·\Typo{MA}{MA'}}\right)$

+the ratio of $PM^{2}$ to $AM·MA'$ $\left(\ie\ \dfrac{PM^{2}}{AM·\Typo{MA}{MA'}}\right)$

 remains constant both for the ellipse and the

 hyperbola (figs.\ \FigNum{16} and \FigNum{18}), and that the ratio

 \PageSep{135}

@@ -4944,7 +4946,7 @@ $c = -4$. We then get the equation $x^{2} + y^{2} - 4 = 0$.
 It is easy to prove that this is the equation

 of a circle, whose centre is at the origin,

 and radius is $2$~units of length. Now $ab - h^{2}$

-becomes $1 × 1 - 0^{2}$, that is,~$1$, and is therefore

+becomes $1 × 1 - 0^{2}$, that is,~$1$, and is therefore

 positive. Hence the circle is a particular

 case of an ellipse, as it ought to be. Generalising,

 the equation of any circle can be

@@ -5035,8 +5037,8 @@ variable numbers. Let us think first of some
 concrete examples: If a train has been travelling

 at the rate of twenty miles per hour, the

 distance ($s$~miles) gone after any number of

-hours, say~$t$, is given by $s = 20 × t$; and $s$~is

-called a function of~$t$. Also $20 × t$ is the function

+hours, say~$t$, is given by $s = 20 × t$; and $s$~is

+called a function of~$t$. Also $20 × t$ is the function

 of~$t$ with which $s$~is identical. If John

 is one year older than Thomas, then, when

 Thomas is at any age of $x$~years, John's age

@@ -5049,10 +5051,10 @@ In these examples $t$ and~$x$ are called the
 \index{Value of a Function}%

 ``arguments'' of the functions in which they

 appear. Thus $t$~is the argument of the function

-$20 × t$, and $x$~is the argument of the function

-$x + 1$. If $s = 20 × t$, and $y = x + 1$, then $s$

+$20 × t$, and $x$~is the argument of the function

+$x + 1$. If $s = 20 × t$, and $y = x + 1$, then $s$

 and~$y$ are called the ``values'' of the functions

-$20 × t$ and $x + 1$ respectively.

+$20 × t$ and $x + 1$ respectively.

 

 Coming now to the general case, we can

 define a function in mathematics as a correlation

@@ -5147,7 +5149,7 @@ $x$~being the argument and $y$~the value of the
 function.

 

 These functions, which are expressed by

-simple algebraic formulæ, are adapted for representation

+simple algebraic formulæ, are adapted for representation

 by graphs. But for some functions

 \PageSep{149}

 this representation would be very

@@ -5279,7 +5281,7 @@ necessity of caution in scientific conclusions.
 It is very easy to find a discontinuous

 function, even if we confine ourselves to the

 \Figure{21}

-simplest of the algebraic formulæ. For example,

+simplest of the algebraic formulæ. For example,

 take the function $y = \dfrac{1}{x}$, which we

 have already considered in the form $p = \dfrac{1}{v}$,

 where $v$~was confined to positive values. But

@@ -6501,14 +6503,14 @@ called the number of their permutations. The
 number of permutations of a set of $n$~things,

 where $n$~is some finite integer, is

 \[

-n × (n - 1) × (n - 2) × (n - 3) × \dots × 4 × 3 × 2 × 1\Add{,}

+n × (n - 1) × (n - 2) × (n - 3) × \dots × 4 × 3 × 2 × 1\Add{,}

 \]

 that is to say, it is the product of the first $n$

 integers; this product is so important in

 mathematics that a special symbolism, is used

 for it, and it is always written~`$n!$\Add{.}' Thus,

-$2! = 2 × 1 = 2$, and $3! = 3 × 2 × 1 = 6$, and $4! = 4 × 3 × 2 × 1 = 24$,

-and $5! = 5 × 4 × 3 × 2 × 1 = 120$.

+$2! = 2 × 1 = 2$, and $3! = 3 × 2 × 1 = 6$, and $4! = 4 × 3 × 2 × 1 = 24$,

+and $5! = 5 × 4 × 3 × 2 × 1 = 120$.

 As $n$~increases, the value of~$n!$ increases very

 quickly; thus $100!$~is a hundred times as

 large as~$99!$\Add{.}

@@ -7000,7 +7002,7 @@ Consider the geometric series
 \]

 

 It is convergent throughout the interval

-$-1$~to~$+1$, excluding the end values $x = ±1$.

+$-1$~to~$+1$, excluding the end values $x = ±1$.

 

 But it is not uniformly convergent throughout

 this interval. For if $s_{n}(x)$~be the sum of

@@ -7115,12 +7117,12 @@ $\exp x$ is called the exponential function.
 It is fairly easy to prove, with a little

 knowledge of elementary mathematics, that

 \[

-(\exp x) × (\exp y) = \exp(x + y).

+(\exp x) × (\exp y) = \exp(x + y).

 \Tag{(A)}

 \]

 In other words that

 \begin{multline*}

-(\exp x) × (\exp y) \\

+(\exp x) × (\exp y) \\

   = 1 + (x + y) + \frac{(x + y)^{2}}{2!} + \frac{(x + y)^{3}}{3!} + \dots

   + \frac{(x + y)^{n}}{n!} + \dots\Add{.}

 \end{multline*}

@@ -7218,7 +7220,7 @@ Another important function is found by
 combining the exponential function with the

 sine, in this way:\Add{---}

 \[

-y = \exp(-cx) × \sin \frac{2\pi x}{p}\Add{.}

+y = \exp(-cx) × \sin \frac{2\pi x}{p}\Add{.}

 \]

 \Figure{31}

 

@@ -7908,7 +7910,7 @@ in the familiar shape
 \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c},

 \]

 $a^{2} = b^{2} + c^{2} - 2bc \cos A$, with two similar

-formulæ.

+formulæ.

 

 Also there is the still simpler correlation

 between the angles of the triangle, namely,

@@ -7947,12 +7949,12 @@ respectively in the angles of the triangle~$ABC$,
 the correlation between the angles is represented

 by the equation

 \[

-A + B + C = 180°;

+A + B + C = 180°;

 \]

 and if $a$,~$b$,~$c$ are the number of feet respectively

 in the three sides, the correlation between the

 sides is represented by $a < b + c$, $b < c + a$,

-$c < a + b$. Also the trigonometrical formulæ

+$c < a + b$. Also the trigonometrical formulæ

 quoted above are other examples of the same

 \index{Variable, The}%

 fact. Thus the notion of the variable and

@@ -7968,7 +7970,7 @@ the size of any length can be determined
 by the number (not necessarily integral) of

 times which it contains some arbitrarily known

 unit, and similarly for areas, volumes, and

-angles. The trigonometrical formulæ, given

+angles. The trigonometrical formulæ, given

 above, are examples of this fact. But it receives

 its crowning application in analytical

 geometry. This great subject is often misnamed

@@ -8311,11 +8313,11 @@ then to add~$2$ to the result; and $1 + (3 + 2)$ directs us
 first to add $2$ to~$3$, and then to add the result to~$1$. Again

 a numerical example of equation~\Eq{(5)} is

 \[

-2 × (3 + 4) = (2 × 3) + (2 × 4).

+2 × (3 + 4) = (2 × 3) + (2 × 4).

 \]

 We perform first the operations in brackets and obtain

 \[

-2 × 7 = 6 + 8

+2 × 7 = 6 + 8

 \]

 which is obviously true.

 

@@ -8452,7 +8454,7 @@ Alexander the Great 128, 129
 

 Algebra, Fundamental Laws of 60

 

-Ampere@Ampère 34

+Ampere@Ampère 34

 

 Analytical Conic Sections 240

 

@@ -8691,7 +8693,7 @@ Non-Uniform Convergence|EtSeq 208
 

 Normal Error, Curve of 214

 

-Oersted@Öersted 34

+Oersted@Öersted 34

 

 Order|EtSeq 194