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--- 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

 

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-%                                                                         %

-%                                                                         %

-% Title: An Introduction to Mathematics                                   %

-%                                                                         %

-% Author: Alfred North Whitehead                                          %

-%                                                                         %

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

-%                                                                         %

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-%[** TN: Publisher's front matter]

-\noindent\footnotesize HOME UNIVERSITY LIBRARY \\

-OF MODERN KNOWLEDGE

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-\begin{center}

-\Large AN INTRODUCTION TO \\

-MATHEMATICS

-\medskip

-

-\normalsize

-\textsc{By A. N. WHITEHEAD, Sc.D., F.R.S.}

-\vfill

-

-\footnotesize

-\scshape London \\

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-\rule{0.5in}{0.5pt} \\[6pt]

-HENRY HOLT \& Co., New York \\

-Canada: WM. BRIGGS, Toronto \\

-India: R. \& T. WASHBOURNE, Ltd.

-\end{center}

-\normalsize

-\PageSep{ii}

-\iffalse

-HOME

-UNIVERSITY

-LIBRARY

-OF

-MODERN KNOWLEDGE

-

-Editors:

-

-HERBERT FISHER, M.A.. F.B.A.

-

-PROF. GILBERT MURRAY, D.LlTT.,

-LL.D., F.B.A.

-

-PROF. J. ARTHUR THOMSON, M.A.

-

-PROF. WILLIAM T. BREWSTER, M.A.

-

-\Add{(}COLUMBIA UNIVERSITY, U.S.A.)

-

-NEW YORK

-

-HENRY HOLT AND COMPANY

-\PageSep{iii}

-AN

-INTRODUCTION

-TO

-MATHEMATICS

-

-BY

-A. N. WHITEHEAD,

-Sc.D., F.R.S.,

-

-AUTHOR OF ``UNIVERSAL ALGEBRA,'' JOINT

-AUTHOR OF ``PRINCIPIA MATHEMATICA''

-

-NEW AND REVISED EDITION

-

-LONDON

-WILLIAMS AND NORGATE

-\PageSep{iv}

-PRINTED BY

-

-HALELL, WATSON AND VINEY, LD.,

-LONDON AND AYLESBURY.

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-%CHAP.                         PAGE

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-\ToCLine{I}{The Abstract Nature of Mathematics}{7}

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-\ToCLine{II}{Variables}{15}

-

-\ToCLine{III}{Methods of Application}{25}

-

-\ToCLine{IV}{Dynamics}{42}

-

-\ToCLine{V}{The Symbolism of Mathematics}{58}

-

-\ToCLine{VI}{Generalizations of Number}{71}

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-\ToCLine{VII}{Imaginary Numbers}{87}

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-\ToCLine{VIII}{Imaginary Numbers (Continued)}{101}

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-\ToCLine{IX}{Coordinate Geometry}{112}

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-%CHAP.                         PAGE

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-\ToCLine{XIV}{Series}{194}

-

-\ToCLine{XV}{The Differential Calculus}{217}

-

-\ToCLine{XVI}{Geometry}{236}

-

-\ToCLine{XVII}{Quantity}{245}

-

-\ToCLine{}{Notes}{250}

-

-\ToCLine{}{Bibliography}{251}

-

-\ToCLine{}{Index}{253}

-\PageSep{7}

-\MainMatter

-% [** TN: Text printed by \Chapter macro]

-% AN INTRODUCTION TO

-% MATHEMATICS

-

-\Chapter[Nature of Mathematics]{I}{The Abstract Nature of Mathematics}

-

-\First{The} study of mathematics is apt to commence

-in disappointment. The important

-applications of the science, the theoretical

-interest of its ideas, and the logical rigour of

-its methods, all generate the expectation of

-a speedy introduction to processes of interest.

-We are told that by its aid the stars are

-weighed and the billions of molecules in a

-drop of water are counted. Yet, like the

-ghost of Hamlet's father, this great science

-eludes the efforts of our mental weapons

-to grasp it---``\,'Tis here, 'tis there, 'tis

-gone''---and what we do see does not suggest

-the same excuse for illusiveness as sufficed

-for the ghost, that it is too noble for

-our gross methods. ``A show of violence,''

-if ever excusable, may surely be ``offered''

-to the trivial results which occupy the

-\PageSep{8}

-pages of some elementary mathematical

-treatises.

-

-The reason for this failure of the science to

-live up to its reputation is that its fundamental

-ideas are not explained to the student

-disentangled from the technical procedure

-which has been invented to facilitate their

-exact presentation in particular instances.

-Accordingly, the unfortunate learner finds

-himself struggling to acquire a knowledge of

-a mass of details which are not illuminated

-by any general conception. Without a doubt,

-technical facility is a first requisite for valuable

-mental activity: we shall fail to appreciate

-the rhythm of Milton, or the passion of

-Shelley, so long as we find it necessary to

-spell the words and are not quite certain of

-the forms of the individual letters. In this

-sense there is no royal road to learning. But

-it is equally an error to confine attention to

-technical processes, excluding consideration

-of general ideas. Here lies the road to

-pedantry.

-

-The object of the following Chapters is not

-to teach mathematics, but to enable students

-from the very beginning of their course to

-know what the science is about, and why it is

-necessarily the foundation of exact thought

-as applied to natural phenomena. All allusion

-in what follows to detailed deductions

-in any part of the science will be inserted

-\PageSep{9}

-merely for the purpose of example, and care

-will be taken to make the general argument

-comprehensible, even if here and there some

-technical process or symbol which the reader

-does not understand is cited for the purpose

-of illustration.

-

-The first acquaintance which most people

-\index{Abstractness (\emph{defined})}%

-have with mathematics is through arithmetic.

-That two and two make four is usually taken

-as the type of a simple mathematical proposition

-which everyone will have heard of.

-Arithmetic, therefore, will be a good subject

-to consider in order to discover, if possible,

-the most obvious characteristic of the science.

-Now, the first noticeable fact about arithmetic

-is that it applies to everything, to tastes and

-to sounds, to apples and to angels, to the

-ideas of the mind and to the bones of the

-body. The nature of the things is perfectly

-indifferent, of all things it is true that two

-and two make four. Thus we write down as

-the leading characteristic of mathematics

-that it deals with properties and ideas

-which are applicable to things just because

-they are things, and apart from any particular

-feelings, or emotions, or sensations, in any

-way connected with them. This is what

-is meant by calling mathematics an abstract

-science.

-

-The result which we have reached deserves

-attention. It is natural to think that an

-\PageSep{10}

-abstract science cannot be of much importance

-in the affairs of human life, because it

-has omitted from its consideration everything

-of real interest. It will be remembered

-that Swift, in his description of Gulliver's

-\index{Swift}%

-voyage to Laputa, is of two minds on this

-\index{Laputa}%

-point. He describes the mathematicians of

-that country as silly and useless dreamers,

-whose attention has to be awakened by

-flappers. Also, the mathematical tailor measures

-his height by a quadrant, and deduces

-his other dimensions by a rule and compasses,

-producing a suit of very ill-fitting clothes.

-On the other hand, the mathematicians of

-Laputa, by their marvellous invention of the

-magnetic island floating in the air, ruled the

-country and maintained their ascendency

-over their subjects. Swift, indeed, lived at

-a time peculiarly unsuited for gibes at contemporary

-mathematicians. Newton's \Title{Principia}

-\index{Newton}%

-had just been written, one of the great

-forces which have transformed the modern

-world. Swift might just as well have laughed

-at an earthquake.

-

-But a mere list of the achievements of

-mathematics is an unsatisfactory way of

-arriving at an idea of its importance. It is

-worth while to spend a little thought in

-getting at the root reason why mathematics,

-because of its very abstractness, must always

-remain one of the most important topics

-\PageSep{11}

-for thought. Let us try to make clear to

-ourselves why explanations of the order of

-events necessarily tend to become mathematical.

-

-Consider how all events are interconnected.

-When we see the lightning, we listen for the

-thunder; when we hear the wind, we look

-for the waves on the sea; in the chill autumn,

-the leaves fall. Everywhere order reigns, so

-that when some circumstances have been

-noted we can foresee that others will also be

-present. The progress of science consists in

-observing these interconnections and in showing

-with a patient ingenuity that the events

-of this evershifting world are but examples of

-a few general connections or relations called

-laws. To see what is general in what is particular

-and what is permanent in what is

-transitory is the aim of scientific thought. In

-the eye of science, the fall of an apple, the

-motion of a planet round a sun, and the clinging

-of the atmosphere to the earth are all

-seen as examples of the law of gravity. This

-possibility of disentangling the most complex

-evanescent circumstances into various examples

-of permanent laws is the controlling

-idea of modern thought.

-

-Now let us think of the sort of laws which

-we want in order completely to realize this

-scientific ideal. Our knowledge of the particular

-facts of the world around us is gained

-\PageSep{12}

-from our sensations. We see, and hear, and

-taste, and smell, and feel hot and cold, and

-push, and rub, and ache, and tingle. These

-are just our own personal sensations: my

-toothache cannot be your toothache, and my

-sight cannot be your sight. But we ascribe

-the origin of these sensations to relations between

-the things which form the external

-world. Thus the dentist extracts not the

-toothache but the tooth. And not only so,

-we also endeavour to imagine the world as

-one connected set of things which underlies

-all the perceptions of all people. There is not

-one world of things for my sensations and another

-for yours, but one world in which we

-both exist. It is the same tooth both for

-dentist and patient. Also we hear and we

-touch the same world as we see.

-

-It is easy, therefore, to understand that we

-want to describe the connections between

-these external things in some way which does

-not depend on any particular sensations, nor

-even on all the sensations of any particular

-person. The laws satisfied by the course of

-events in the world of external things are to

-be described, if possible, in a neutral universal

-fashion, the same for blind men as for

-deaf men, and the same for beings with

-faculties beyond our ken as for normal human

-beings.

-

-But when we have put aside our immediate

-\PageSep{13}

-\index{Abstractness (\emph{defined})}%

-\index{Dynamical Explanation}%

-sensations, the most serviceable part---from

-its clearness, definiteness, and universality---of

-what is left is composed of our general ideas

-of the abstract formal properties of things;

-in fact, the abstract mathematical ideas mentioned

-above. Thus it comes about that,

-step by step, and not realizing the full meaning

-of the process, mankind has been led to

-search for a mathematical description of the

-properties of the universe, because in this way

-only can a general idea of the course of events

-be formed, freed from reference to particular

-persons or to particular types of sensation.

-For example, it might be asked at dinner:

-``What was it which underlay my sensation

-of sight, yours of touch, and his of taste

-and smell?''\ the answer being ``an apple.''

-But in its final analysis, science seeks to

-describe an apple in terms of the positions

-and motions of molecules, a description which

-ignores me and you and him, and also ignores

-sight and touch and taste and smell.

-Thus mathematical ideas, because they

-are abstract, supply just what is wanted

-for a scientific description of the course of

-events.

-

-This point has usually been misunderstood,

-%[** TN: Entry listed on p. 18 in the original]

-\index{Pythagoras}%

-from being thought of in too narrow a way.

-Pythagoras had a glimpse of it when he proclaimed

-that number was the source of all

-things. In modern times the belief that the

-\PageSep{14}

-ultimate explanation of all things was to be

-found in Newtonian mechanics was an adumbration

-of the truth that all science as it

-grows towards perfection becomes mathematical

-\index{Dynamical Explanation}%

-in its ideas.

-\PageSep{15}

-

-

-\Chapter{II}{Variables}

-

-\First{Mathematics} as a science commenced when

-first someone, probably a Greek, proved propositions

-about \emph{any} things or about \emph{some}

-things, without specification of definite particular

-things. These propositions were first

-enunciated by the Greeks for geometry; and,

-accordingly, geometry was the great Greek

-mathematical science. After the rise of geometry

-centuries passed away before algebra

-made a really effective start, despite some

-faint anticipations by the later Greek mathematicians.

-

-The ideas of \emph{any} and of \emph{some} are introduced

-into algebra by the use of letters, instead

-of the definite numbers of arithmetic.

-Thus, instead of saying that $2 + 3 = 3 + 2$, in

-algebra we generalize and say that, if $x$ and~$y$

-stand for \emph{any} two numbers, then $x + y = y + x$.

-Again, in the place of saying that $3 > 2$, we

-generalize and say that if $x$~be \emph{any} number

-there exists \emph{some} number (or numbers)~$y$ such

-that $y > x$. We may remark in passing that

-this latter assumption---for when put in its

-strict ultimate form it is an assumption---is

-\PageSep{16}

-of vital importance, both to philosophy and

-to mathematics; for by it the notion of infinity

-is introduced. Perhaps it required the

-introduction of the arabic numerals, by which

-the use of letters as standing for definite

-numbers has been completely discarded in

-mathematics, in order to suggest to mathematicians

-the technical convenience of the

-use of letters for the ideas of \emph{any} number

-and \emph{some} number. The Romans would have

-stated the number of the year in which this

-is written in the form MDCCCCX., whereas

-we write it~1910, thus leaving the letters for

-the other usage. But this is merely a speculation.

-After the rise of algebra the differential

-calculus was invented by Newton and

-\index{Newton}%

-Leibniz, and then a pause in the progress

-\index{Leibniz}%

-of the philosophy of mathematical thought

-occurred so far as these notions are concerned;

-and it was not till within the last few years

-that it has been realized how fundamental

-\emph{any} and \emph{some} are to the very nature of mathematics,

-with the result of opening out still

-further subjects for mathematical exploration.

-

-Let us now make some simple algebraic

-statements, with the object of understanding

-exactly how these fundamental ideas occur.

-

-\Eq{(1)} For \emph{any} number~$x$, $x + 2 = 2 + x$;

-

-\Eq{(2)} For \emph{some} number~$x$, $x + 2 = 3$;

-

-\Eq{(3)} For \emph{some} number~$x$, $x + 2 > 3$.

-\PageSep{17}

-

-The first point to notice is the possibilities

-contained in the meaning of \emph{some}, as here

-used. Since $x + 2 = 2 + x$ for any number~$x$, it

-is true for \emph{some} number~$x$. Thus, as here used,

-\emph{any} implies \emph{some} and \emph{some} does not exclude

-\emph{any}. Again, in the second example, there is,

-in fact, only one number~$x$, such that $x + 2 = 3$,

-namely only the number~$1$. Thus the \emph{some}

-may be one number only. But in the third\Typo{,}{}

-example, any number~$x$ which is greater than~$1$

-gives $x + 2 > 3$. Hence there are an infinite

-number of numbers which answer to the \emph{some}

-number in this case. Thus \emph{some} may be anything

-between \emph{any} and \emph{one only}, including

-both these limiting cases.

-

-It is natural to supersede the statements

-\Eq{(2)} and \Eq{(3)} by the questions:

-

-\Eq{(2')} For what number~$x$ is $x + 2 = 3$;

-

-\Eq{(3')} For what numbers~$x$ is $x + 2 > 3$.

-

-%[** TN: No indent in the original]

-Considering~\Eq{(2')}, $x + 2 = 3$ is an equation, and

-\index{Unknown, The}%

-it is easy to see that its solution is $x = 3 - 2 = 1$.

-When we have asked the question implied in

-the statement of the equation $x + 2 = 3$, $x$~is

-called the unknown. The object of the solution

-of the equation is the determination of

-the unknown. Equations are of great importance

-in mathematics, and it seems as

-%[** TN: thoroughgoing hyphenated in the original; only instance]

-though \Eq{(2')}~exemplified a much more thoroughgoing

-and fundamental idea than the original

-statement~\Eq{(2)}. This, however, is a complete

-mistake. The idea of the undetermined

-\PageSep{18}

-``variable'' as occurring in the use of ``some''

-or ``any'' is the really important one in

-mathematics; that of the ``unknown'' in an

-equation, which is to be solved as quickly as

-possible, is only of subordinate use, though

-of course it is very important. One of the

-causes of the apparent triviality of much of

-elementary algebra is the preoccupation of

-the text-books with the solution of equations.

-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,

-\index{Relations between Variables|EtSeq}%

-\index{Variable, The}%

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

-involve more than one variable. For example,

-the consideration of the pairs of numbers

-$x$ and~$y$ (fractional or integral) which

-satisfy $x + y = 1$ involves the idea of two correlated

-variables, $x$~and~$y$. When two variables

-are present the same two main types of

-statement occur. For example, \Eq{(1)}~for

-\emph{any} pair of numbers, $x$~and~$y$, $x + y = y + x$,

-and \Eq{(2)}~for \emph{some} pairs of numbers, $x$~and~$y$,

-$x + y = 1$.

-

-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

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

-them formulæ of the second type can be

-\PageSep{19}

-thrown into an indefinite number of equivalent

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

-is equivalent to the relations

-\[

-y + x = 1,\quad

-(x - y) + 2y = 1,\quad

-6x + 6y = 6,

-\]

-and so on. Thus a skilful mathematician

-uses that equivalent form of the relation

-under consideration which is most convenient

-for his immediate purpose.

-

-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,

-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

-indefinite number of values remain open for

-the other.

-

-Again there is another important point to

-be noticed. If we restrict ourselves to positive

-numbers, integral or fractional, in considering

-the relation $x + y = 1$, then, if either

-$x$ or~$y$ be greater than~$1$, there is no positive

-number which the other can assume so as to

-satisfy the relation. Thus the ``field'' of

-the relation for~$x$ is restricted to numbers less

-than~$1$, and similarly for the ``field'' open

-to~$y$. Again, consider integral numbers only,

-positive or negative, and take the relation

-\PageSep{20}

-$y^{2} = x$, satisfied by pairs of such numbers.

-Then whatever integral value is given to~$y$,

-$x$~can assume one corresponding integral

-value. So the ``field'' for~$y$ is unrestricted

-among these positive or negative integers.

-But the ``field'' for~$x$ is restricted in two

-ways. In the first place $x$~must be positive,

-and in the second place, since $y$~is to be integral,

-$x$~must be a perfect square. Accordingly,

-the ``field'' of~$x$ is restricted to the set

-of integers $1^{2}$, $2^{2}$, $3^{2}$, $4^{2}$, and so on, \ie, to $1$,

-$4$, $9$, $16$, and so on.

-

-The study of the general properties of a

-relation between pairs of numbers is much

-facilitated by the use of a diagram constructed

-as follows:

-\Figure[3.5in]{1}

-

-Draw two lines $OX$ and $OY$ at right angles;

-let any number~$x$ be represented by $x$~units

-\PageSep{21}

-(in any scale) of length along~$OX$, any number~$y$

-by $y$~units (in any scale) of length along~$OY$.

-Thus if $OM$, along~$OX$, be $x$~units in

-length, and $ON$, along~$OY$, be $y$~units in length,

-by completing the parallelogram $OMPN$ we

-find a point~$P$ which corresponds to the pair

-of numbers $x$~and~$y$. To each point there

-corresponds one pair of numbers, and to each

-pair of numbers there corresponds one point.

-The pair of numbers are called the coordinates

-of the point. Then the points

-whose coordinates satisfy some fixed relation

-can be indicated in a convenient way,

-by drawing a line, if they all lie on a line,

-or by shading an area if they are all points

-in the area. If the relation can be represented

-by an equation such as $x + y = 1$, or

-$y^{2} = x$, then the points lie on a line, which is

-straight in the former case and curved in

-the latter. For example, considering only

-positive numbers, the points whose coordinates

-satisfy $x + y = 1$ lie on the straight

-line~$AB$ in \Fig{1}, where $0A = 1$ and $OB = 1$.

-Thus this segment of the straight line~$AB$

-gives a pictorial representation of the properties

-of the relation under the restriction to

-positive numbers.

-

-Another example of a relation between two

-variables is afforded by considering the variations

-in the pressure and volume of a given

-mass of some gaseous substance---such as air

-\PageSep{22}

-or coal-gas or steam---at a constant temperature.

-Let $v$~be the number of cubic feet in

-its volume and $p$~its pressure in lb.\ weight

-per square inch. Then the law, known as

-Boyle's law, expressing the relation between

-$p$ and~$v$ as both vary, is that the product~$pv$

-is constant, always supposing that the

-temperature does not alter. Let us suppose,

-for example, that the quantity of the gas

-and its other circumstances are such that

-we can put $pv = 1$ (the exact number on

-the right-hand side of the equation makes

-no essential difference).

-\Figure{2}

-

-Then in \Fig{2} we take two lines, $OV$ and~$OP$,

-at right angles and draw~$OM$ along~$OV$

-to represent $v$~units of volume, and $ON$ along~$OP$

-\PageSep{23}

-to represent $p$~units of pressure. Then

-the point~$Q$, which is found by completing the

-parallelogram $OMQN$, represents the state of

-the gas when its volume is $v$~cubic feet and its

-pressure is $p$~lb.\ weight per square inch. If

-the circumstances of the portion of gas considered

-are such that $pv = 1$, then all these

-points~$Q$ which correspond to any possible

-state of this portion of gas must lie on the

-curved line $ABC$, which includes all points

-for which $p$~and $v$ are positive, and $pv = 1$.

-Thus this curved line gives a pictorial representation

-of the relation holding between the

-volume and the pressure. When the pressure

-is very big the corresponding point~$Q$ must

-be near~$C$, or even beyond~$C$ on the undrawn

-part of the curve; then the volume will be

-very small. When the volume is big $Q$~will

-be near to~$A$, or beyond~$A$; and then the

-pressure will be small. Notice that an engineer

-or a physicist may want to know the

-particular pressure corresponding to some

-definitely assigned volume. Then we have

-the case of determining the \emph{unknown}~$p$ when

-\index{Unknown, The}%

-$v$~is a known number. But this is only in

-particular cases. In considering generally

-the properties of the gas and how it will behave,

-he has to have in his mind the general

-form of the whole curve $ABC$ and its general

-properties. In other words the really fundamental

-idea is that of the pair of \emph{variables}

-\PageSep{24}

-satisfying the relation $pv = 1$. This example

-illustrates how the idea of \emph{variables} is fundamental,

-\index{Variable, The}%

-both in the applications as well as in

-the theory of mathematics.

-\PageSep{25}

-

-

-\Chapter{III}{Methods of Application}

-

-\First{The} way in which the idea of variables

-satisfying a relation occurs in the applications

-of mathematics is worth thought, and by

-devoting some time to it we shall clear up

-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

-the complex circumstances which attend the

-building of a new house, amid all the various

-sensations and emotions of the owner, the

-architect, the builder, the workmen, and the

-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,

-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

-the house and the cost are not supposed to

-have been perceived or apprehended by any

-particular sensation or faculty, or by any

-\PageSep{26}

-particular man. They are stated in an abstract

-general way, with complete indifference

-to the owner's state of mind when he has

-to pay the bill.

-

-Now think a bit further as to what all this

-means. The building of a house is a complicated

-set of circumstances. It is impossible

-to begin to apply the law, or to test

-it, unless amid the general course of events

-it is possible to recognize a definite set of

-occurrences as forming a particular instance

-of the building of a house. In short, we must

-know a house when we see it, and must recognize

-the events which belong to its building.

-Then amidst these events, thus isolated in

-idea from the rest of nature, the two elements

-of the cost and cubic content must be determinable;

-and when they are both determined,

-if the law be true, they satisfy the general

-formula

-\[

-20y = x.

-\]

-But is the law true? Anyone who has had

-much to do with building will know that we

-have here put the cost rather high. It is

-only for an expensive type of house that it

-will work out at this price. This brings out

-another point which must be made clear.

-While we are making mathematical calculations

-connected with the formula $20y = x$, it

-is indifferent to us whether the law be true or

-\PageSep{27}

-false. In fact, the very meanings assigned

-to $x$~and~$y$, as being a number of cubic feet

-and a number of pounds sterling, are indifferent.

-During the mathematical investigation

-we are, in fact, merely considering the

-properties of this correlation between a pair

-of variable numbers $x$ and~$y$. Our results

-will apply equally well, if we interpret $y$ to

-mean a number of fishermen and $x$~the number

-of fish caught, so that the assumed law

-is that on the average each fisherman catches

-twenty fish. The mathematical certainty of

-the investigation only attaches to the results

-considered as giving properties of the correlation

-$20y = x$ between the variable pair of

-numbers $x$ and~$y$. There is no mathematical

-certainty whatever about the cost of the

-actual building of any house. The law is not

-quite true and the result it gives will not be

-quite accurate. In fact, it may well be hopelessly

-wrong.

-

-Now all this no doubt seems very obvious.

-But in truth with more complicated instances

-there is no more common error than to assume

-that, because prolonged and accurate mathematical

-calculations have been made, the

-application of the result to some fact of

-nature is absolutely certain. The conclusion

-of no argument can be more certain than the

-assumptions from which it starts. All mathematical

-calculations about the course of

-\PageSep{28}

-nature must start from some assumed law of

-nature, such, for instance, as the assumed

-law of the cost of building stated above.

-Accordingly, however accurately we have

-calculated that some event must occur, the

-doubt always remains---Is the law true? If

-the law states a precise result, almost certainly

-it is not precisely accurate; and thus

-even at the best the result, precisely as calculated,

-is not likely to occur. But then we

-have no faculty capable of observation with

-ideal precision, so, after all, our inaccurate

-laws may be good enough.

-

-We will now turn to an actual case, that

-of Newton and the Law of Gravity. This law

-states that any two bodies attract one another

-with a force proportional to the product

-of their masses, and inversely proportional to

-the square of the distance between them.

-Thus if $m$~and~$M$ are the masses of the two

-bodies, reckoned in lbs.\ say, and $d$~miles is

-the distance between them, the force on either

-body, due to the attraction of the other and

-directed towards it, is proportional to~$\dfrac{mM}{d^{2}}$;

-thus this force can be written as equal to

-$\dfrac{kmM}{d^{2}}$, where $k$~is a definite number depending

-on the absolute magnitude of this attraction

-and also on the scale by which we choose to

-measure forces. It is easy to see that, if we

-\PageSep{29}

-wish to reckon in terms of forces such as the

-weight of a mass of $1$~lb., the number which

-$k$~represents must be extremely small; for

-when $m$~and $M$ and~$d$ are each put equal to~$1$,

-$\dfrac{kmM}{d^{2}}$~becomes the gravitational attraction

-of two equal masses of $1$~lb.\ at the distance of

-one mile, and this is quite inappreciable.

-

-However, we have now got our formula for

-the force of attraction. If we call this force~$F$,

-it is $F = k\dfrac{mM}{d^{2}}$, giving the correlation between

-the variables $F$,~$m$,~$M$, and~$d$. We all

-know the story of how it was found out.

-Newton, it states, was sitting in an orchard

-and watched the fall of an apple, and then

-the law of universal gravitation burst upon

-\index{Gravitation}%

-his mind. It may be that the final formulation

-of the law occurred to him in an

-orchard, as well as elsewhere---and he must

-have been somewhere. But for our purposes

-it is more instructive to dwell upon the vast

-amount of preparatory thought, the product

-of many minds and many centuries, which

-was necessary before this exact law could be

-formulated. In the first place, the mathematical

-habit of mind and the mathematical

-procedure explained in the previous two

-chapters had to be generated; otherwise

-Newton could never have thought of a formula

-representing the force between \emph{any} two masses

-\PageSep{30}

-at \emph{any} distance. Again, what are the meanings

-\index{Distance}%

-of the terms employed, Force, Mass, Distance?

-\index{Force}%

-\index{Mass}%

-Take the easiest of these terms,

-Distance. It seems very obvious to us to

-conceive all material things as forming a definite

-geometrical whole, such that the distances

-of the various parts are measurable in

-terms of some unit length, such as a mile or

-a yard. This is almost the first aspect of a

-material structure which occurs to us. It is

-the gradual outcome of the study of geometry

-and of the theory of measurement. Even

-now, in certain cases, other modes of thought

-are convenient. In a mountainous country

-distances are often reckoned in hours. But

-leaving distance, the other terms, Force and

-Mass, are much more obscure. The exact

-comprehension of the ideas which Newton

-\index{Newton}%

-meant to convey by these words was of slow

-growth, and, indeed, Newton himself was the

-first man who had thoroughly mastered the

-true general principles of Dynamics.

-\index{Dynamics}%

-

-Throughout the middle ages, under the influence

-of Aristotle, the science was entirely

-\index{Aristotle}%

-misconceived. Newton had the advantage of

-coming after a series of great men, notably

-Galileo, in Italy, who in the previous two

-\index{Galileo}%

-centuries had reconstructed the science and

-had invented the right way of thinking about

-it. He completed their work. Then, finally,

-having the ideas of force, mass, and distance,

-\PageSep{31}

-clear and distinct in his mind, and realising

-their importance and their relevance to the

-fall of an apple and the motions of the planets,

-he hit upon the law of gravitation and proved

-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

-correct estimate of their relevance to the

-phenomena under observation. No less than

-ourselves, our remote ancestors were impressed

-with the importance of natural

-phenomena and with the desirability of taking

-energetic measures to regulate the sequence

-of events. Under the influence of irrelevant

-ideas they executed elaborate religious ceremonies

-to aid the birth of the new moon, and

-performed sacrifices to save the sun during

-the crisis of an eclipse. There is no reason to

-believe that they were more stupid than we

-are. But at that epoch there had not been

-opportunity for the slow accumulation of

-clear and relevant ideas.

-

-The sort of way in which physical sciences

-\index{Electromagnetism|EtSeq}%

-grow into a form capable of treatment by

-mathematical methods is illustrated by the

-history of the gradual growth of the science

-of electromagnetism. Thunderstorms are

-events on a grand scale, arousing terror in

-men and even animals. From the earliest

-times they must have been objects of wild

-\PageSep{32}

-\index{Electricity|EtSeq}%

-and fantastic hypotheses, though it may be

-doubted whether our modern scientific discoveries

-in connection with electricity are not

-more astonishing than any of the magical

-explanations of savages. The Greeks knew

-that amber (Greek, electron) when rubbed

-would attract light and dry bodies. In

-1600~\AD, Dr.~Gilbert, of Colchester, published

-\index{Gilbert, Dr.}%

-the first work on the subject in which any

-scientific method is followed. He made a

-list of substances possessing properties similar

-to those of amber; he must also have the

-credit of connecting, however vaguely, electric

-and magnetic phenomena. At the end of the

-seventeenth and throughout the eighteenth

-century knowledge advanced. Electrical

-machines were made, sparks were obtained

-from them; and the Leyden Jar was invented,

-by which these effects could be intensified.

-Some organised knowledge was

-being obtained; but still no relevant mathematical

-ideas had been found out. Franklin,

-\index{Franklin}%

-in the year 1752, sent a kite into the clouds

-and proved that thunderstorms were electrical.

-

-Meanwhile from the earliest epoch (2634~\BC)

-the Chinese had utilized the characteristic

-property of the compass needle, but do not

-seem to have connected it with any theoretical

-ideas. The really profound changes in human

-life all have their ultimate origin in knowledge

-\PageSep{33}

-pursued for its own sake. The use of the compass

-was not introduced into Europe till the end

-of the twelfth century~\AD, more than $3000$~years

-after its first use in China. The importance

-which the science of electromagnetism

-has since assumed in every department of

-human life is not due to the superior practical

-bias of Europeans, but to the fact that in the

-West electrical and magnetic phenomena

-were studied by men who were dominated by

-abstract theoretic interests.

-

-The discovery of the electric current is due

-\index{Electric Current}%

-to two Italians, Galvani in~1780, and Volta

-\index{Galvani}%

-\index{Volta}%

-in~1792. This great invention opened a new

-series of phenomena for investigation. The

-scientific world had now three separate,

-though allied, groups of occurrences on hand---the

-effects of ``statical'' electricity arising

-from frictional electrical machines, the magnetic

-phenomena, and the effects due to

-electric currents. From the end of the

-eighteenth century onwards, these three lines

-of investigation were quickly \Chg{inter-connected}{interconnected}

-and the modern science of electromagnetism

-was constructed, which now threatens to

-transform human life.

-

-Mathematical ideas now appear. During

-the decade 1780 to~1789, Coulomb, a Frenchman,

-\index{Coulomb}%

-proved that magnetic poles attract or

-repel each other, in proportion to the inverse

-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}%

-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}%

-also proved that two electric currents exerted

-forces on each other. ``The experimental investigation

-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

-experiment, seems as if it had leaped, full-grown

-and full armed, from the brain of

-the `Newton of Electricity.' It is perfect

-\index{Newton}%

-in form, and unassailable in accuracy, and it

-is summed up in a formula from which all

-the phenomena may be deduced, and which

-must always remain the cardinal formula of

-electro-dynamics.''\footnote

-  {\Title{Electricity and Magnetism}, Clerk Maxwell, Vol.~II.,

-\index{Clerk Maxwell}%

-  ch.~iii.}

-

-The momentous laws of induction between

-currents and between currents and magnets

-were discovered by Michael Faraday in 1831--82.

-\index{Faraday}%

-Faraday was asked: ``What is the use

-of this discovery?'' He answered: ``What is

-the use of a child---it grows to be a man.''

-Faraday's child has grown to be a man and

-is now the basis of all the modern applications

-\PageSep{35}

-of electricity. Faraday also reorganized the

-whole theoretical conception of the science.

-His ideas, which had not been fully understood

-by the scientific world, were extended

-and put into a directly mathematical form by

-Clerk Maxwell in~1873. As a result of his

-\index{Clerk Maxwell}%

-mathematical investigations, Maxwell recognized

-that, under certain conditions, electrical

-vibrations ought to be propagated. He at

-once suggested that the vibrations which

-form light are electrical. This suggestion has

-\index{Light}%

-since been verified, so that now the whole

-theory of light is nothing but a branch of the

-% [** TN: Herz [sic]]

-great science of electricity. Also Herz, a

-\index{Herz}%

-German, in~1888, following on Maxwell's

-ideas, succeeded in producing electric vibrations

-by direct electrical methods\Add{.} His

-experiments are the basis of our wireless

-telegraphy.

-

-In more recent years even more fundamental

-discoveries have been made, and the

-science continues to grow in theoretic importance

-and in practical interest. This rapid

-sketch of its progress illustrates how, by the

-gradual introduction of the relevant theoretic

-ideas, suggested by experiment and themselves

-suggesting fresh experiments, a whole

-mass of isolated and even trivial phenomena

-are welded together into one coherent science,

-in which the results of abstract mathematical

-deductions, starting from a few simple assumed

-\PageSep{36}

-laws, supply the explanation to the

-complex tangle of the course of events.

-

-Finally, passing beyond the particular

-sciences of electromagnetism and light, we

-can generalize our point of view still further,

-and direct our attention to the growth of

-mathematical physics considered as one great

-chapter of scientific thought. In the first

-place, what in the barest outlines is the story

-of its growth?

-

-It did not begin as one science, or as the

-product of one band of men. The Chaldean

-shepherds watched the skies, the agents of

-Government in Mesopotamia and Egypt

-measured the land, priests and philosophers

-brooded on the general nature of all things.

-The vast mass of the operations of nature

-appeared due to mysterious unfathomable

-forces. ``The wind bloweth where it listeth''

-expresses accurately the blank ignorance then

-existing of any stable rules followed in detail

-by the succession of phenomena. In broad outline,

-then as now, a regularity of events was

-patent. But no minute tracing of their interconnection

-was possible, and there was no

-knowledge how even to set about to construct

-such a science.

-

-Detached speculations, a few happy or unhappy

-shots at the nature of things, formed

-the utmost which could be produced.

-

-Meanwhile land-surveys had produced geometry,

-\index{Geometry}%

-\PageSep{37}

-and the observations of the heavens

-disclosed the exact regularity of the solar

-system. Some of the later Greeks, such as

-Archimedes, had just views on the elementary

-\index{Archimedes|EtSeq}%

-phenomena of hydrostatics and optics. Indeed,

-Archimedes, who combined a genius for

-mathematics with a physical insight, must

-rank with Newton, who lived nearly two

-\index{Newton}%

-thousand years later, as one of the founders

-of mathematical physics. He lived at Syracuse,

-the great Greek city of Sicily. When

-the Romans besieged the town (in 212~to

-210~\BC), he is said to have burned their ships

-by concentrating on them, by means of

-mirrors, the sun's rays. The story is highly

-improbable, but is good evidence of the reputation

-which he had gained among his contemporaries

-for his knowledge of optics. At

-the end of this siege he was killed. According

-to one account given by Plutarch, in his life of

-\index{Plutarch}%

-Marcellus, he was found by a Roman soldier

-\index{Marcellus}%

-absorbed in the study of a geometrical diagram

-which he had traced on the sandy floor of his

-room. He did not immediately obey the orders

-of his captor, and so was killed. For the credit

-of the Roman generals it must be said that

-the soldiers had orders to spare him. The

-internal evidence for the other famous story

-of him is very strong; for the discovery

-attributed to him is one eminently worthy of

-his genius for mathematical and physical research.

-\PageSep{38}

-Luckily, it is simple enough to be

-explained here in detail. It is one of the best

-easy examples of the method of application

-of mathematical ideas to physics.

-

-Hiero, King of Syracuse, had sent a quantity

-\index{Hiero}%

-of gold to some goldsmith to form the

-material of a crown. He suspected that the

-craftsmen had abstracted some of the gold

-and had supplied its place by alloying the

-remainder with some baser metal. Hiero

-sent the crown to Archimedes and asked him

-to test it. In these days an indefinite number

-of chemical tests would be available.

-But then Archimedes had to think out the

-matter afresh. The solution flashed upon

-him as he lay in his bath. He jumped

-up and ran through the streets to the

-palace, shouting \Foreign{Eureka! Eureka!} (I have

-found it, I have found it). This day, if we

-knew which it was, ought to be celebrated as

-the birthday of mathematical physics; the

-science came of age when Newton sat in his

-\index{Newton}%

-orchard. Archimedes had in truth made a

-great discovery. He saw that a body when

-immersed in water is pressed upwards by the

-surrounding water with a resultant force

-equal to the weight of the water it displaces.

-This law can be proved theoretically from the

-mathematical principles of hydrostatics and

-can also be verified experimentally. Hence,

-if $W$~lb.\ be the weight of the crown, as weighed

-\PageSep{39}

-in air, and $w$~lb.\ be the weight of the water

-which it displaces when completely immersed,

-$W - w$ would be the extra upward force

-necessary to sustain the crown as it hung in

-water.

-

-Now, this upward force can easily be ascertained

-by weighing the body as it hangs in

-water, as shown in the annexed figure. If

-\Figure{3}

-the weights in the right-hand scale come to

-$F$~lb., then the apparent weight of the crown

-in water is $F$~lb.; and we thus have

-\[

-F = W - w

-\]

-and thus

-\[

-w = W - F,

-\]

-and

-\[

-\frac{W}{w} = \frac{W}{W - F}

-\Tag{(A)}

-\]

-where $W$ and $F$ are determined by the easy,

-and fairly precise, operation of weighing.

-\PageSep{40}

-Hence, by equation~\Eq{(A)}, $\dfrac{W}{w}$~is known. But

-$\dfrac{W}{w}$~is the ratio of the weight of the crown to

-the weight of an equal volume of water.

-This ratio is the same for any lump of metal of

-the same material: it is now called the specific

-gravity of the material, and depends only on

-the intrinsic nature of the substance and not

-on its shape or quantity. Thus to test if the

-crown were of gold, Archimedes had only to

-take a lump of indisputably pure gold and

-find its specific gravity by the same process.

-If the two specific gravities agreed, the crown

-was pure; if they disagreed, it was debased.

-

-This argument has been given at length,

-because not only is it the first precise example

-of the application of mathematical ideas to

-physics, but also because it is a perfect and

-simple example of what must be the method

-and spirit of the science for all time.

-

-The death of Archimedes by the hands of a

-Roman soldier is symbolical of a world-change

-of the first magnitude: the theoretical Greeks,

-with their love of abstract science, were superseded

-in the leadership of the European world

-by the practical Romans. Lord Beaconsfield,

-\index{Beaconsfield, Lord}%

-in one of his novels, has defined a practical

-man as a man who practises the errors of

-his forefathers. The Romans were a great

-race, but they were cursed with the sterility

-\PageSep{41}

-\index{Specific Gravity}%

-which waits upon practicality. They did not

-improve upon the knowledge of their forefathers,

-and all their advances were confined

-to the minor technical details of engineering.

-They were not dreamers enough to arrive at

-new points of view, which could give a more

-fundamental control over the forces of nature.

-No Roman lost his life because he was absorbed

-in the contemplation of a mathematical

-diagram.

-\PageSep{42}

-

-

-\Chapter{IV}{Dynamics}

-

-\First{The} world had to wait for eighteen hundred

-years till the Greek mathematical physicists

-found successors. In the sixteenth and seventeenth

-centuries of our era great Italians, in

-particular Leonardo da~Vinci, the artist

-\index{Aristotle}%

-\index{Galileo|EtSeq}%

-\index{Leonardo da Vinci}%

-(born 1452, died 1519), and Galileo (born 1564,

-died 1642), rediscovered the secret, known to

-Archimedes, of relating abstract mathematical

-ideas with the experimental investigation of

-natural phenomena. Meanwhile the slow

-advance of mathematics and the accumulation

-of accurate astronomical knowledge had

-placed natural philosophers in a much more

-advantageous position for research. Also the

-very egoistic self-assertion of that age, its

-greediness for personal experience, led its

-thinkers to want to see for themselves what

-happened; and the secret of the relation of

-mathematical theory and experiment in inductive

-reasoning was practically discovered.

-It was an act eminently characteristic of the

-age that Galileo, a philosopher, should have

-\PageSep{43}

-dropped the weights from the leaning tower

-of Pisa. There are always men of thought

-and men of action; mathematical physics is

-the product of an age which combined in the

-same men impulses to thought with impulses

-to action.

-

-This matter of the dropping of weights from

-\index{Dynamics|EtSeq}%

-the tower marks picturesquely an essential

-step in knowledge, no less a step than the

-first attainment of correct ideas on the science

-of dynamics, the basal science of the whole

-subject. The particular point in dispute was

-as to whether bodies of different weights

-would fall from the same height in the same

-time. According to a dictum of Aristotle,

-universally followed up to that epoch, the

-heavier weight would fall the quicker. Galileo

-affirmed that they would fall in the same

-time, and proved his point by dropping

-weights from the top of the leaning tower.

-The apparent exceptions to the rule all arise

-when, for some reason, such as extreme lightness

-or great speed, the air resistance is important.

-But neglecting the air the law is

-exact.

-

-Galileo's successful experiment was not the

-\index{Motion, First Law of}%

-result of a mere lucky guess. It arose from

-his correct ideas in connection with inertia

-and mass. The first law of motion, as following

-Newton we now enunciate it, is---Every

-\index{Newton}%

-body continues in its state of rest or of uniform

-\PageSep{44}

-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

-over defeated heretics. The point at issue

-can be understood by deleting from the law

-the phrase ``or of uniform motion in a straight

-line.'' We there obtain what might be taken

-as the Aristotelian opposition formula:

-``Every body continues in its state of rest

-except so far as it is compelled by impressed

-force to change that state.''

-

-In this last false formula it is asserted that,

-apart from force, a body continues in a state

-of rest; and accordingly that, if a body is

-moving, a force is required to sustain the

-motion; so that when the force ceases, the

-motion ceases. The true Newtonian law

-takes diametrically the opposite point of view.

-The state of a body unacted on by force is

-that of uniform motion in a straight line, and

-no external force or influence is to be looked

-for as the cause, or, if you like to put it so, as

-the invariable accompaniment of this uniform

-rectilinear motion. Rest is merely a particular

-case of such motion, merely when the

-velocity is and remains zero. Thus, when a

-body is moving, we do not seek for any external

-influence except to explain changes in

-the rate of the velocity or changes in its direction.

-So long as the body is moving at the

-\PageSep{45}

-same rate and in the same direction there is

-no need to invoke the aid of any forces.

-

-The difference between the two points of

-view is well seen by reference to the theory of

-the motion of the planets. Copernicus, a

-\index{Copernicus}%

-Pole, born at Thorn in West Prussia (born

-1473, died 1543), showed how much simpler

-it was to conceive the planets, including the

-\Figure[2.25in]{4}

-earth as revolving round the sun in orbits

-which are nearly circular; and later, Kepler,

-\index{Kepler}%

-a German mathematician, in the year 1609

-proved that, in fact, the orbits are practically

-ellipses, that is, a special sort of oval curves

-\index{Ellipse}%

-which we will consider later in more detail.

-Immediately the question arose as to what

-are the forces which preserve the planets in

-this motion. According to the old false view,

-\PageSep{46}

-held by Kepler, the actual velocity itself required

-\index{Kepler}%

-preservation by force. Thus he looked

-for tangential forces as in the accompanying

-figure~(\FigNum{4}). But according to the Newtonian

-law, apart from some force the planet would

-move for ever with its existing velocity in a

-straight line, and thus depart entirely from

-the sun. Newton, therefore, had to search

-\index{Newton}%

-for a force which would bend the motion

-\Figure[2.25in]{5}

-round into its elliptical orbit. This he showed

-must be a force directed towards the sun as in

-the next figure~(\FigNum{5}). In fact, the force is the

-gravitational attraction of the sun acting

-according to the law of the inverse square of

-the distance, which has been stated above.

-

-The science of mechanics rose among the

-\index{Mechanics}%

-Greeks from a consideration of the theory of

-the mechanical advantage obtained by the use

-\PageSep{47}

-\index{Dynamical Explanation|EtSeq}%

-of a lever, and also from a consideration of

-various problems connected with the weights

-of bodies. It was finally put on its true basis

-at the end of the sixteenth and during the

-seventeenth centuries, as the preceding account

-shows, partly with the view of explaining

-the theory of falling bodies, but chiefly

-in order to give a scientific theory of planetary

-motions. But since those days dynamics has

-taken upon itself a more ambitious task, and

-now claims to be the ultimate science of which

-the others are but branches. The claim

-amounts to this: namely, that the various

-qualities of things perceptible to the senses

-are merely our peculiar mode of appreciating

-changes in position on the part of things

-existing in space. For example, suppose we

-look at Westminster Abbey. It has been

-standing there, grey and immovable, for centuries

-past. But, according to modern scientific

-theory, that greyness, which so heightens

-our sense of the immobility of the building, is

-itself nothing but our way of appreciating the

-rapid motions of the ultimate molecules, which

-form the outer surface of the building and

-communicate vibrations to a substance called

-the ether. Again we lay our hands on its

-stones and note their cool, even temperature,

-so symbolic of the quiet repose of the building.

-But this feeling of temperature simply marks

-our sense of the transfer of heat from the

-\PageSep{48}

-hand to the stone, or from the stone to the

-hand; and, according to modern science,

-heat is nothing but the agitation of the molecules

-of a body. Finally, the organ begins

-playing, and again sound is nothing but the

-result of motions of the air striking on the

-drum of the ear.

-

-Thus the endeavour to give a dynamical

-explanation of phenomena is the attempt to

-explain them by statements of the general

-form, that such and such a substance or body

-was in this place and is now in that place.

-Thus we arrive at the great basal idea of

-modern science, that all our sensations are

-the result of comparisons of the changed

-configurations of things in space at various

-times. It follows therefore, that the laws

-of motion, that is, the laws of the changes

-of configurations of things, are the ultimate

-laws of physical science.

-

-In the application of mathematics to the

-investigation of natural philosophy, science

-does systematically what ordinary thought

-does casually. When we talk of a chair, we

-usually mean something which we have been

-seeing or feeling in some way; though most

-of our language will presuppose that there

-is something which exists independently of

-our sight or feeling. Now in mathematical

-physics the opposite course is taken. The

-chair is conceived without any reference to

-\PageSep{49}

-\index{Variable, The}%

-anyone in particular, or to any special modes

-of perception. The result is that the chair

-becomes in thought a set of molecules in space,

-or a group of electrons, a portion of the ether

-in motion, or however the current scientific

-ideas describe it. But the point is that

-science reduces the chair to things moving in

-space and influencing each other's motions.

-Then the various elements or factors which

-enter into a set of circumstances, as thus

-conceived, are merely the things, like lengths

-of lines, sizes of angles, areas, and volumes, by

-which the positions of bodies in space can be

-settled. Of course, in addition to these geometrical

-elements the fact of motion and

-change necessitates the introduction of the

-rates of changes of such elements, that is to

-say, velocities, angular velocities, accelerations,

-and suchlike things. Accordingly, mathematical

-physics deals with correlations between

-variable numbers which are supposed

-to represent the correlations which exist in

-nature between the measures of these geometrical

-elements and of their rates of change.

-But always the mathematical laws deal with

-variables, and it is only in the occasional

-testing of the laws by reference to experiments,

-or in the use of the laws for special

-predictions that definite numbers are substituted.

-

-The interesting point about the world as

-\PageSep{50}

-thus conceived in this abstract way throughout

-the study of mathematical physics, where

-only the positions and shapes of things are

-considered together with their changes, is that

-the events of such an abstract world are sufficient

-to ``explain'' our sensations. When we

-hear a sound, the molecules of the air have

-been agitated in a certain way: given the

-agitation, or air-waves as they are called, all

-normal people hear sound; and if there are

-no air-waves, there is no sound. And, similarly,

-a physical cause or origin, or parallel

-event (according as different people might like

-to phrase it) underlies our other sensations.

-Our very thoughts appear to correspond to

-conformations and motions of the brain; injure

-the brain and you injure the thoughts.

-Meanwhile the events of this physical universe

-succeed each other according to the mathematical

-laws which ignore all special sensations

-and thoughts and emotions.

-

-Now, undoubtedly, this is the general aspect

-of the relation of the world of mathematical

-physics to our emotions, sensations, and

-thoughts; and a great deal of controversy

-has been occasioned by it and much ink

-spilled. We need only make one remark. The

-whole situation has arisen, as we have seen,

-from the endeavour to describe an external

-world ``explanatory'' of our various individual

-sensations and emotions, but a world

-\PageSep{51}

-also, not essentially dependent upon any

-particular sensations or upon any particular

-individual. Is such a world merely but

-one huge fairy tale? But fairy tales are

-fantastic and arbitrary: if in truth there

-be such a world, it ought to submit itself

-to an exact description, which determines

-accurately its various parts and their mutual

-relations. Now, to a large degree, this

-scientific world does submit itself to this

-test and allow its events to be explored

-and predicted by the apparatus of abstract

-mathematical ideas. It certainly seems that

-here we have an inductive verification of

-our initial assumption. It must be admitted

-that no inductive proof is conclusive; but

-if the whole idea of a world which has

-existence independently of our particular perceptions

-of it be erroneous, it requires careful

-explanation why the attempt to characterise

-it, in terms of that mathematical remnant

-of our ideas which would apply to it, should

-issue in such a remarkable success.

-

-It would take us too far afield to enter into

-\index{Parallelogram Law|EtSeq}%

-\index{Vectors|EtSeq}%

-a detailed explanation of the other laws of

-motion. The remainder of this chapter must

-be devoted to the explanation of remarkable

-ideas which are fundamental, both to mathematical

-physics and to pure mathematics:

-these are the ideas of vector quantities and

-the parallelogram law for vector addition. We

-\PageSep{52}

-have seen that the essence of motion is that

-a body was at~$A$ and is now at~$C$. This transference

-from $A$ to~$C$ requires two distinct

-elements to be settled before it is completely

-determined, namely its magnitude (\ie\ the

-length~$AC$) and its direction. Now anything,

-like this transference, which is completely

-given by the determination of a magnitude

-\Figure[2in]{6}

-and a direction is called a vector. For

-example, a velocity requires for its definition

-the assignment of a magnitude and of a

-direction. It must be of so many miles per

-hour in such and such a direction. The existence

-and the independence of these two

-elements in the determination of a velocity

-are well illustrated by the action of the captain

-of a ship, who communicates with different subordinates

-respecting them: he tells the chief

-engineer the number of knots at which he is

-to steam, and the helmsman the compass

-\PageSep{53}

-bearing of the course which he is to keep.

-Again the rate of change of velocity, that is

-velocity added per unit time, is also a vector

-quantity: it is called the acceleration. Similarly

-a force in the dynamical sense is another

-vector quantity. Indeed, the vector nature

-of forces follows at once according to dynamical

-principles from that of velocities and

-accelerations; but this is a point which we

-need not go into. It is sufficient here to say

-that a force acts on a body with a certain

-magnitude in a certain direction.

-

-Now all vectors can be graphically represented

-by straight lines. All that has to be

-done is to arrange: (i)~a scale according to

-which units of length correspond to units of

-magnitude of the vector---for example, one

-inch to a velocity of $10$~miles per~hour in the

-case of velocities, and one inch to a force of

-$10$~tons weight in the case of forces---and (ii)~a

-direction of the line on the diagram corresponding

-to the direction of the vector. Then

-a line drawn with the proper number of inches

-of length in the proper direction represents the

-required vector on the arbitrarily assigned scale

-of magnitude. This diagrammatic representation

-of vectors is of the first importance. By

-its aid we can enunciate the famous ``parallelogram

-law'' for the addition of vectors of the

-same kind but in different directions.

-

-Consider the vector~$AC$ in \Fig[figure]{6} as representative

-\PageSep{54}

-\index{Transportation, Vector of|EtSeq}%

-of the changed position of a body

-from $A$ to~$C$: we will call this the vector of

-transportation. It will be noted that, if the

-reduction of physical phenomena to mere

-changes in positions, as explained above, is

-correct, all other types of physical vectors are

-really reducible in some way or other to this

-single type. Now the final transportation

-from $A$ to~$C$ is equally well effected by a

-transportation from $A$ to~$B$ and a transportation

-from $B$ to~$C$, or, completing the parallelogram

-$ABCD$, by a transportation from $A$ to~$D$

-and a transportation from $D$ to~$C$. These

-transportations as thus successively applied

-are said to be added together. This is simply

-a definition of what we mean by the addition

-of transportations. Note further that, considering

-parallel lines as being lines drawn in

-the same direction, the transportations $B$~to~$C$

-and $A$~to~$D$ may be conceived as the same

-transportation applied to bodies in the two

-initial positions $B$ and~$A$. With this conception

-we may talk of the transportation

-$A$~to~$D$ as applied to a body in any position,

-for example at~$B$. Thus we may say that

-the transportation $A$~to~$C$ can be conceived

-as the sum of the two transportations $A$~to~$B$

-and $A$~to~$D$ applied in any order. Here

-we have the parallelogram law for the addition

-of transportations: namely, if the

-transportations are $A$~to~$B$ and $A$~to~$D$,

-\PageSep{55}

-complete the parallelogram $ABCD$, and then

-the sum of the two is the diagonal~$AC$.

-

-All this at first sight may seem to be

-very artificial. But it must be observed

-that nature itself presents us with the idea.

-For example, a steamer is moving in the

-direction~$AD$ (\Chg{cf.}{\cf}\ \Fig[fig.]{6}) and a man walks

-across its deck. If the steamer were still,

-in one minute he would arrive at~$B$; but

-during that minute his starting point~$A$ on

-the deck has moved to~$D$, and his path on

-the deck has moved from $AB$ to~$DC$. So

-that, in fact, his transportation has been from

-$A$ to~$C$ over the surface of the sea. It is,

-however, presented to us analysed into the

-sum of two transportations, namely, one from

-$A$ to~$B$ relatively to the steamer, and one

-from $A$ to~$D$ which is the transportation of

-the steamer.

-

-By taking into account the element of time,

-namely one minute, this diagram of the man's

-transportation~$AC$ represents his velocity.

-For if $AC$~represented so many feet of transportation,

-it now represents a transportation

-of so many feet per minute, that is to say, it

-represents the velocity of the man. Then

-$AB$ and $AD$ represent two velocities, namely,

-his velocity relatively to the steamer, and the

-velocity of the steamer, whose ``sum'' makes

-up his complete velocity. It is evident that

-diagrams and definitions concerning transportations

-\PageSep{56}

-are turned into diagrams and definitions

-concerning velocities by conceiving

-the diagrams as representing transportations

-per unit time. Again, diagrams and definitions

-concerning velocities are turned into

-diagrams and definitions concerning accelerations

-\Figure[3in]{7}

-by conceiving the diagrams as representing

-velocities added per unit time.

-

-Thus by the addition of vector velocities

-and of vector accelerations, we mean the

-addition according to the parallelogram law.

-

-Also, according to the laws of motion a

-force is fully represented by the vector

-acceleration it produces in a body of given

-mass. Accordingly, forces will be said to be

-added when their joint effect is to be reckoned

-according to the parallelogram law.

-

-Hence for the fundamental vectors of

-\PageSep{57}

-science, namely transportations, velocities,

-and forces, the addition of any two of the same

-kind is the production of a ``resultant''

-vector according to the rule of the parallelogram

-law.

-

-By far the simplest type of parallelogram

-is a rectangle, and in pure mathematics it is

-\index{Rectangle}%

-the relation of the single vector~$AC$ to the

-two component vectors, $AB$~and~$AD$, at right

-angles (\Chg{cf.}{\cf}\ \Fig[fig.]{7}), which is continually recurring.

-Let $x$,~$y$, and $r$~units represent the

-lengths of $AB$,~$AD$, and~$AC$, and let $m$~units

-of angle represent the magnitude of the angle

-$BAC$. Then the relations between $x$,~$y$,~$r$,

-and~$m$, in all their many aspects are the continually

-recurring topic of pure mathematics;

-and the results are of the type required for

-application to the fundamental vectors of

-mathematical physics. This diagram is the

-chief bridge over which the results of pure

-mathematics pass in order to obtain application

-to the facts of nature.

-\PageSep{58}

-

-

-\Chapter{V}{The Symbolism of Mathematics}

-

-\First{We} now return to pure mathematics, and

-consider more closely the apparatus of ideas

-out of which the science is built. Our first

-concern is with the symbolism of the science,

-and we start with the simplest and universally

-known symbols, namely those of arithmetic.

-

-Let us assume for the present that we have

-\index{Arabic Notation|EtSeq}%

-sufficiently clear ideas about the integral

-numbers, represented in the Arabic notation

-by $0$,~$1$, $2$,~\dots, $9$, $10$, $11$,~\dots\Add{,} $100$, $101$,~\dots\ and

-so on. This notation was introduced into

-Europe through the Arabs, but they apparently

-obtained it from Hindoo sources. The

-first known work\footnote

-  {For the detailed historical facts relating to pure

-  mathematics, I am chiefly indebted to \Title{A Short History

-\index{Ball, W. W. R.}%

-  of Mathematics}, by W.~W.~R. Ball.}

-% [** TN: http://www.gutenberg.org/ebooks/31246]

-in which it is systematically

-explained is a work by an Indian mathematician,

-Bhaskara (born 1114~\AD). But

-\index{Bhaskara}%

-the actual numerals can be traced back to the

-seventh century of our era, and perhaps were

-originally invented in Tibet. For our present

-\PageSep{59}

-purposes, however, the history of the notation

-is a detail. The interesting point to notice

-is the admirable illustration which this

-numeral system affords of the enormous importance

-of a good notation. By relieving

-the brain of all unnecessary work, a good

-notation sets it free to concentrate on more

-advanced problems, and in effect increases

-the mental power of the race. Before the

-introduction of the Arabic notation, multiplication

-was difficult, and the division even of

-integers called into play the highest mathematical

-faculties. Probably nothing in the

-modern world would have more astonished a

-Greek mathematician than to learn that, under

-the influence of compulsory education, a

-large proportion of the population of Western

-Europe could perform the operation of

-division for the largest numbers. This fact

-would have seemed to him a sheer impossibility.

-The consequential extension of

-the notation to decimal fractions was not

-accomplished till the seventeenth century.

-Our modern power of easy reckoning with

-decimal fractions is the almost miraculous

-result of the gradual discovery of a perfect

-notation.

-

-Mathematics is often considered a difficult

-and mysterious science, because of the

-numerous symbols which it employs. Of

-course, nothing is more incomprehensible than

-\PageSep{60}

-a symbolism which we do not understand.

-Also a symbolism, which we only partially

-understand and are unaccustomed to use, is

-difficult to follow. In exactly the same way

-the technical terms of any profession or trade

-are incomprehensible to those who have never

-been trained to use them. But this is not

-because they are difficult in themselves. On

-the contrary they have invariably been introduced

-to make things easy. So in mathematics,

-granted that we are giving any serious

-attention to mathematical ideas, the symbolism

-is invariably an immense simplification.

-It is not only of practical use, but is

-of great interest. For it represents an analysis

-of the ideas of the subject and an almost

-pictorial representation of their relations to

-each other. If anyone doubts the utility of

-symbols, let him write out in full, without any

-symbol whatever, the whole meaning of the

-following equations which represent some of

-\index{Algebra, Fundamental Laws of}%

-the fundamental laws of algebra\footnotemark:---

-\footnotetext{\Chg{Cf.}{\Cf}\ Note~A, \Pageref{noteA}.\Pagelabel{60}}

-%[** TN: left-aligned in the original]

-\begin{gather*}

-x + y = y + x\Add{,}

-\Tag{(1)} \\

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

-\Tag{(2)} \\

-x × y = y × x\Add{,}

-\Tag{(3)} \\

-(x × y) × z = x × (y × z)\Add{,}

-\Tag{(4)} \\

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

-\Tag{(5)}

-\end{gather*}

-

-Here \Eq{(1)}~and \Eq{(2)} are called the commutative

-and associative laws for addition, \Eq{(3)}~and \Eq{(4)}

-\PageSep{61}

-are the commutative and associative laws for

-multiplication, and \Eq{(5)}~is the distributive law

-relating addition and multiplication. For example,

-without symbols, \Eq{(1)}~becomes: If a

-second number be added to any given number

-the result is the same as if the first given

-number had been added to the second number.

-

-This example shows that, by the aid of symbolism,

-we can make transitions in reasoning

-almost mechanically by the eye, which otherwise

-would call into play the higher faculties

-of the brain.

-

-It is a profoundly erroneous truism, repeated

-by all copy-books and by eminent people when

-they are making speeches, that we should

-cultivate the habit of thinking of what we are

-doing. The precise opposite is the case.

-Civilization advances by extending the number

-of important operations which we can

-perform without thinking about them. Operations

-of thought are like cavalry charges in

-a battle---they are strictly limited in number,

-they require fresh horses, and must only

-be made at decisive moments.

-

-One very important property for symbolism

-to possess is that it should be concise, so as to

-be visible at one glance of the eye and to be

-rapidly written. Now we cannot place symbols

-more concisely together than by placing

-them in immediate juxtaposition. In a good

-symbolism therefore, the juxtaposition of important

-\PageSep{62}

-symbols should have an important

-meaning. This is one of the merits of the

-Arabic notation for numbers; by means of

-ten symbols, $0$,~$1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$,~$9$, and by

-simple juxtaposition it symbolizes any number

-whatever. Again in algebra, when we have

-two variable numbers $x$ and~$y$, we have to

-make a choice as to what shall be denoted by

-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

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

-written,

-\[

-xy = yx,\quad

-(xy)z = x(yz),\quad

-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 $30 × x$.

-

-It is evident that in substituting definite

-numbers for the variables some care must be

-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

-$20 + 3$.

-

-It is interesting to note how important for

-the development of science a modest-looking

-symbol may be. It may stand for the emphatic

-presentation of an idea, often a very

-\PageSep{63}

-subtle idea, and by its existence make it easy

-to exhibit the relation of this idea to all the

-complex trains of ideas in which it occurs.

-For example, take the most modest of all

-symbols, namely,~$0$, which stands for the \emph{number}

-\index{Zero|EtSeq}%

-zero. The Roman notation for numbers

-had no symbol for zero, and probably most

-mathematicians of the ancient world would

-have been horribly puzzled by the idea of the

-number zero. For, after all, it is a very

-subtle idea, not at all obvious. A great deal

-of discussion on the meaning of the zero of

-quantity will be found in philosophic works.

-Zero is not, in real truth, more difficult or

-subtle in idea than the other cardinal numbers.

-What do we mean by~$1$ or by~$2$, or by~$3$?

-But we are familiar with the use of these ideas,

-though we should most of us be puzzled to

-give a clear analysis of the simpler ideas

-which go to form them. The point about zero

-is that we do not need to use it in the operations

-of daily life. No one goes out to buy

-zero fish. It is in a way the most civilized

-of all the cardinals, and its use is only forced

-on us by the needs of cultivated modes of

-thought. Many important services are rendered

-by the symbol~$0$, which stands for the

-number zero.

-

-The symbol developed in connection with

-the Arabic notation for numbers of which it

-is an essential part. For in that notation the

-\PageSep{64}

-value of a digit depends on the position in

-which it occurs. Consider, for example, the

-digit~$5$, as occurring in the numbers $25$, $51$,

-$3512$, $5213$. In the first number~$5$ stands for

-five, in the second number $5$~stands for fifty,

-in the third number for five hundred, and in

-the fourth number for five thousand. Now,

-when we write the number fifty-one in the

-symbolic form~$51$, the digit~$1$ pushes the digit~$5$

-along to the second place (reckoning from

-right to left) and thus gives it the value fifty.

-But when we want to symbolize fifty by itself,

-we can have no digit~$1$ to perform this service;

-we want a digit in the units place to add

-nothing to the total and yet to push the~$5$

-along to the second place. This service is

-performed by~$0$, the symbol for zero. It is

-extremely probable that the men who introduced

-for this purpose had no definite conception

-in their minds of the number zero.

-They simply wanted a mark to symbolize the

-fact that nothing was contributed by the

-digit's place in which it occurs. The idea of

-zero probably took shape gradually from a

-desire to assimilate the meaning of this mark

-to that of the marks, $1$, $2$,~\dots\Add{,}~$9$, which do represent

-cardinal numbers. This would not

-represent the only case in which a subtle idea

-has been introduced into mathematics by a

-symbolism which in its origin was dictated by

-practical convenience.

-\PageSep{65}

-

-Thus the first use of~$0$ was to make the

-arable notation possible---no slight service.

-We can imagine that when it had been introduced

-for this purpose, practical men, of the

-sort who dislike fanciful ideas, deprecated the

-silly habit of identifying it with a number

-zero. But they were wrong, as such men

-always are when they desert their proper

-function of masticating food which others have

-prepared. For the next service performed by

-the symbol~$0$ essentially depends upon assigning

-to it the function of representing the

-number zero.

-

-This second symbolic use is at first sight

-so absurdly simple that it is difficult to make

-a beginner realize its importance. Let us

-start with a simple example. In \ChapRef{II}.\

-we mentioned the correlation between two

-variable numbers $x$ and $y$ represented by the

-equation $x + y = 1$. This can be represented

-in an indefinite number of ways; for example,

-$x = 1 - y$, $y = 1 - x$, $2x + 3y - 1 = x + 2y$, and so

-on. But the important way of stating it is

-\[

-x + y - 1 = 0.

-\]

-Similarly the important way of writing the

-equation $x = 1$ is $x - 1 = 0$, and of representing

-the equation $3x - 2 = 2x^{2}$ is $2x^{2} - 3x + 2 = 0$.

-The point is that all the symbols which represent

-variables, \eg\ $x$~and~$y$, and the symbols

-\PageSep{66}

-representing some definite number other than

-zero, such as $1$ or $2$ in the examples above,

-are written on the left-hand side, so that the

-whole left-hand side is equated to the number

-zero. The first man to do this is said to

-have been Thomas Harriot, born at Oxford

-\index{Harriot, Thomas}%

-in 1560 and died in~1621. But what is the

-importance of this simple symbolic procedure?

-It made possible the growth of the

-\index{Form, Algebraic|EtSeq}%

-modern conception of \emph{algebraic form}.

-

-This is an idea to which we shall have continually

-to recur; it is not going too far to

-say that no part of modern mathematics can

-be properly understood without constant recurrence

-to it. The conception of form is

-so general that it is difficult to characterize

-it in abstract terms. At this stage we shall

-do better merely to consider examples. Thus

-the equations $2x - 3 = 0$, $x - 1 = 0$, $5x - 6 = 0$,

-are all equations of the same form, namely,

-equations involving one unknown~$x$, which is

-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

-equations are called quadratic equations.

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

-yield another form, and so on. Among the

-three quadratic equations given above there

-is a minor difference between the last equation,

-\PageSep{67}

-$x^{2} - 4 = 0$, and the preceding two equations,

-due to the fact that~$x$ (as distinct

-from~$x^{2}$) does not appear in the last and

-does in the other two. This distinction is

-very unimportant in comparison with the

-great fact that they are all three quadratic

-equations.

-

-Then further there are the forms of equation

-stating correlations between two variables;

-for example, $x + y - 1 = 0$, $2x + 3y - 8 = 0$, and

-so on. These are examples of what is called

-the \emph{linear} form of equation. The reason for

-this name of ``linear'' is that the graphic

-method of representation, which is explained

-at the end of \ChapRef{II}\Add{.}, always represents

-such equations by a straight line. Then there

-are other forms for two variables---for example,

-the quadratic form, the cubic form, and so on.

-But the point which we here insist upon is

-that this study of form is facilitated, and,

-indeed, made possible, by the standard method

-of writing equations with the symbol~$0$ on

-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 $x + 0 = x$.

-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 =

-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

-the same general form as does $x^{2} - 3x + 2 = 0$.

-

-For these three reasons the symbol~$0$, representing

-the number zero, is essential to

-modern mathematics. It has rendered possible

-types of investigation which would have

-been impossible without it.

-

-The symbolism of mathematics is in truth

-the outcome of the general ideas which

-dominate the science. We have now two

-such general ideas before us, that of the variable

-and that of algebraic form. The junction

-of these concepts has imposed on mathematics

-another type of symbolism almost quaint in

-its character, but none the less effective. We

-have seen that an equation involving two

-variables, $x$~and~$y$, represents a particular

-correlation between the pair of variables.

-Thus $x + y - 1 = 0$ represents one definite correlation,

-and $3x + 2y - 5 = 0$ represents another

-definite correlation between the variables $x$

-and~$y$; and both correlations have the form

-of what we have called linear correlations.

-But now, how can we represent \emph{any} linear

-correlation between the variable numbers $x$

-and~$y$? Here we want to symbolize \emph{any}

-linear correlation; just as $x$~symbolizes \emph{any}

-\PageSep{69}

-number. This is done by turning the numbers

-which occur in the definite correlation $3x + 2y - 5 = 0$

-into letters. We obtain $ax + by -  c = 0$.

-Here $a$,~$b$,~$c$, stand for variable numbers just

-as do $x$ and~$y$: but there is a difference in the

-use of the two sets of variables. We study

-the general properties of the relationship between

-$x$ and $y$ while $a$,~$b$, and~$c$ have unchanged

-values. We do not determine what

-the values of $a$,~$b$, and~$c$ are; but whatever

-they are, they remain fixed while we study

-the relation between the variables $x$ and $y$

-for the whole group of possible values of $x$

-and~$y$. But when we have obtained the properties

-of this correlation, we note that, because

-$a$,~$b$, and~$c$ have not in fact been determined,

-we have proved properties which must

-belong to \emph{any} such relation. Thus, by now

-varying $a$,~$b$, and~$c$, we arrive at the idea that

-$ax + by - c = 0$ represents a variable linear

-correlation between $x$ and~$y$. In comparison

-with $x$ and~$y$, the three variables $a$,~$b$, and~$c$

-are called constants. Variables used in this

-\index{Constants}%

-way are sometimes also called parameters.

-\index{Parameters}%

-

-Now, mathematicians habitually save the

-trouble of explaining which of their variables

-are to be treated as ``constants,'' and which

-as variables, considered as correlated in their

-equations, by using letters at the end of the

-alphabet for the ``variable'' variables, and

-letters at the beginning of the alphabet for

-\PageSep{70}

-the ``constant'' variables, or parameters.

-The two systems meet naturally about the

-middle of the alphabet. Sometimes a word

-or two of explanation is necessary; but as a

-matter of fact custom and common sense are

-usually sufficient, and surprisingly little confusion

-is caused by a procedure which seems

-so lax.

-

-The result of this continual elimination of

-definite numbers by successive layers of parameters

-is that the amount of arithmetic performed

-by mathematicians is extremely small.

-Many mathematicians dislike all numerical

-computation and are not particularly expert

-at it. The territory of arithmetic ends where

-the two ideas of ``variables'' and of ``algebraic

-form'' commence their sway.

-\PageSep{71}

-

-

-\Chapter{VI}{Generalizations of Number}

-

-\First{One} great peculiarity of mathematics is the

-\index{Fractions|EtSeq}%

-set of allied ideas which have been invented

-in connection with the integral numbers from

-which we started. These ideas may be called

-extensions or generalizations of number. In

-the first place there is the idea of fractions.

-The earliest treatise on arithmetic which we

-possess was written by an Egyptian priest,

-named Ahmes, between 1700~\BC\ and 1100~\BC,

-\index{Ahmes}%

-and it is probably a copy of a much older

-work. It deals largely with the properties of

-fractions. It appears, therefore, that this

-concept was developed very early in the history

-of mathematics. Indeed the subject is

-a very obvious one. To divide a field into

-three equal parts, and to take two of the

-parts, must be a type of operation which had

-often occurred. Accordingly, we need not be

-surprised that the men of remote civilizations

-were familiar with the idea of two-thirds, and

-\PageSep{72}

-with allied notions. Thus as the first generalization

-of number we place the concept of

-fractions. The Greeks thought of this subject

-rather in the form of ratio, so that a

-Greek would naturally say that a line of

-two feet in length bears to a line of three

-feet in length the ratio of $2$~to~$3$. Under

-the influence of our algebraic notation we

-would more often say that one line was

-two-thirds of the other in length, and would

-think of two-thirds as a numerical multiplier.

-

-In connection with the theory of ratio, or

-\index{Incommensurable Ratios|EtSeq}%

-\index{Ratio|EtSeq}%

-fractions, the Greeks made a great discovery,

-which has been the occasion of a large amount

-of philosophical as well as mathematical

-thought. They found out the existence of

-``incommensurable'' ratios. They proved,

-in fact, during the course of their geometrical

-investigations that, starting with a line of any

-length, other lines must exist whose lengths

-do not bear to the original length the ratio

-of any pair of integers---or, in other words,

-that lengths exist which are not any exact

-fraction of the original length.

-

-For example, the diagonal of a square cannot

-be expressed as any fraction of the side of the

-same square; in our modern notation the

-length of the diagonal is $\sqrt{2}$~times the length

-of the side. But there is no fraction which

-exactly represents~$\sqrt{2}$. We can approximate

-\PageSep{73}

-to~$\sqrt{2}$ as closely as we like, but we never

-exactly reach its value. For example, $\dfrac{49}{25}$~is

-just less than~$2$, and $\dfrac{9}{4}$~is greater than~$2$, so

-that $\sqrt{2}$~lies between $\dfrac{7}{5}$ and~$\dfrac{3}{2}$. But the best

-systematic way of approximating to~$\sqrt{2}$ in

-obtaining a series of decimal fractions, each

-bigger than the last, is by the ordinary method

-of extracting the square root; thus the series

-is $1$, $\dfrac{14}{10}$, $\dfrac{141}{100}$, $\dfrac{1414}{1000}$, and so on.

-

-Ratios of this sort are called by the Greeks

-incommensurable. They have excited from

-the time of the Greeks onwards a great deal

-of philosophic discussion, and the difficulties

-connected with them have only recently been

-cleared up.

-

-We will put the incommensurable ratios

-\index{Real Numbers|EtSeq}%

-with the fractions, and consider the whole

-set of integral numbers, fractional numbers,

-and incommensurable numbers as forming

-one class of numbers which we will call ``real

-numbers.'' We always think of the real

-numbers as arranged in order of magnitude,

-starting from zero and going upwards, and

-becoming indefinitely larger and larger as we

-proceed. The real numbers are conveniently

-\PageSep{74}

-represented by points on a line. Let $OX$ be

-\Diagram{pg76}

-any line bounded at~$O$ and stretching away indefinitely

-in the direction~$OX$. Take any convenient

-point,~$A$, on it, so that $OA$~represents

-the unit length; and divide off lengths $AB$,

-$BC$, $CD$, and so on, each equal to~$OA$. Then

-the point~$O$ represents the number~$0$, $A$~the

-number~$1$, $B$~the number~$2$, and so on. In

-fact the number represented by any point is

-the measure of its distance from~$O$, in terms

-of the unit length~$OA$. The points between

-$O$ and~$A$ represent the proper fractions and

-the incommensurable numbers less than~$1$;

-the middle point of~$OA$ represents~$\dfrac{1}{2}$, that of~$AB$

-represents~$\dfrac{3}{2}$, that of~$BC$ represents~$\dfrac{5}{2}$, and

-so on. In this way every point on~$OX$ represents

-some one real number, and every real

-number is represented by some one point on~$OX$.

-

-The series (or row) of points along~$OX$,

-\index{Series|EtSeq}%

-starting from~$O$ and moving regularly in the

-direction from $O$ to~$X$, represents the real

-numbers as arranged in an ascending order

-\PageSep{75}

-of size, starting from zero and continually

-increasing as we go on.

-

-All this seems simple enough, but even at

-\index{Order, Type of|EtSeq}%

-this stage there are some interesting ideas to

-be got at by dwelling on these obvious facts.

-Consider the series of points which represent

-the integral numbers only, namely, the points,

-$O$,~$A$, $B$, $C$, $D$,~etc. Here there is a first point~$O$,

-a definite next point,~$A$, and each point,

-such as $A$ or~$B$, has one definite immediate

-predecessor and one definite immediate successor,

-with the exception of~$O$, which has no

-predecessor; also the series goes on indefinitely

-without end. This sort of order is

-called the type of order of the integers; its

-essence is the possession of next-door neighbours

-on either side with the exception of

-No.~1 in the row. Again consider the integers

-and fractions together, omitting the points

-which correspond to the incommensurable

-ratios. The sort of serial order which we now

-obtain is quite different. There is a first

-term~$O$; but no term has any immediate predecessor

-or immediate successor. This is

-easily seen to be the case, for between any

-two fractions we can always find another

-fraction intermediate in value. One very

-simple way of doing this is to add the fractions

-together and to halve the result. For example,

-%[** textstyle fractions start here]

-between $\frac{2}{3}$ and~$\frac{3}{4}$, the fraction $\frac{1}{2}(\frac{2}{3} + \frac{3}{4})$,

-that is~$\frac{17}{24}$, lies; and between $\frac{2}{3}$ and $\frac{17}{24}$ the

-\PageSep{76}

-\index{Compact Series}%

-fraction $\frac{1}{2}(\frac{2}{3} + \frac{17}{24})$, that is~$\frac{33}{48}$, lies; and so on

-indefinitely. Because of this property the

-series is said to be ``compact.'' There is no

-end point to the series, which increases indefinitely

-without limit as we go along the

-line~$OX$. It would seem at first sight as

-though the type of series got in this way from

-the fractions, always including the integers,

-would be the same as that got from all the

-real numbers, integers, fractions, and incommensurables

-taken together, that is, from all

-the points on the line~$OX$. All that we have

-hitherto said about the series of fractions

-applies equally well to the series of all real

-numbers. But there are important differences

-which we now proceed to develop. The

-absence of the incommensurables from the

-series of fractions leaves an absence of endpoints

-to certain classes. Thus, consider the

-incommensurable~$\sqrt{2}$. In the series of real

-numbers this stands between all the numbers

-whose squares are less than~$2$, and all the

-numbers whose squares are greater than~$2$.

-But keeping to the series of fractions alone

-and not thinking of the incommensurables, so

-that we cannot bring in~$\sqrt{2}$, there is no fraction

-which has the property of dividing off

-the series into two parts in this way, \ie\ so

-that all the members on one side have their

-squares less than~$2$, and on the other side

-greater than~$2$. Hence in the series of fractions

-\PageSep{77}

-there is a quasi-gap where $\sqrt{2}$~ought to

-come. This presence of quasi-gaps in the

-series of fractions may seem a small matter;

-but any mathematician, who happens to read

-this, knows that the possible absence of limits

-\index{Limits}%

-or maxima to a class of numbers, which yet

-does not spread over the whole series of numbers,

-is no small evil. It is to avoid this

-difficulty that recourse is had to the incommensurables,

-so as to obtain a complete series

-with no gaps.

-

-There is another even more fundamental

-difference between the two series. We can

-rearrange the fractions in a series like that of

-the integers, that is, with a first term, and

-such that each term has an immediate successor

-and (except the first term) an immediate

-predecessor. We can show how this can be

-done. Let every term in the series of fractions

-and integers be written in the fractional form

-by writing $\frac{1}{1}$ for~$1$, $\frac{2}{1}$ for~$2$, and so on for all the

-integers, excluding~$0$. Also for the moment

-we will reckon fractions which are equal in

-value but not reduced to their lowest terms

-as distinct; so that, for example, until further

-notice $\frac{2}{3}$, $\frac{4}{6}$, $\frac{6}{9}$, $\frac{8}{12}$, etc., are all reckoned as distinct.

-Now group the fractions into classes

-by adding together the numerator and denominator

-of each term. For the sake of

-brevity call this sum of the numerator and

-denominator of a fraction its index. Thus $7$~is

-\PageSep{78}

-the index of~$\frac{4}{3}$, and also of~$\frac{3}{4}$, and of~$\frac{2}{5}$. Let

-the fractions in each class be all fractions

-which have some specified index, which may

-therefore also be called the class index. Now

-arrange these classes in the order of magnitude

-of their indices. The first class has

-the index~$2$, and its only member is~$\frac{1}{1}$; the

-second class has the index~$3$, and its members

-are $\frac{1}{2}$ and~$\frac{2}{1}$; the third class has the index~$4$,

-and its members are $\frac{1}{3}$, $\frac{2}{2}$,~$\frac{3}{1}$; the fourth

-class has the index~$5$, and its members are

-$\frac{1}{4}$, $\frac{2}{3}$, $\frac{3}{2}$,~$\frac{4}{1}$; and so on. It is easy to see that

-the number of members (still including fractions

-not in their lowest terms) belonging to

-any class is one less than its index. Also the

-members of any one class can be arranged

-in order by taking the first member to be the

-fraction with numerator~$1$, the second member

-to have the numerator~$2$, and so on, up to~$(n - 1)$

-where $n$~is the index. Thus for the

-class of index~$n$, the members appear in the

-order\Typo{.}{}

-%[** TN: Reformatted slightly from the original]

-\[

-\frac{1}{n - 1},\quad

-\frac{2}{n - 2},\quad

-\frac{3}{n - 3},\

-\dots,\quad

-\frac{n - 1}{1}.

-\]

-The members

-of the first four classes have in fact been

-mentioned in this order. Thus the whole set

-of fractions have now been arranged in an

-order like that of the integers. It runs thus

-\begin{gather*}

-\frac{1}{1},\

-\frac{1}{2},\

-\frac{2}{1},\

-\frac{1}{3},\

-\left[\frac{2}{2}\right],\

-\frac{3}{1},\

-\frac{1}{4},\

-\frac{2}{3},\

-\frac{3}{2},\

-\frac{4}{1},\  \dots, \\

-%\PageSep{79}

-\frac{n - 1}{1},\

-\frac{1}{n - 1},\

-\frac{2}{n - 2},\

-\frac{3}{n - 3},\

-\dots,\

-\frac{n - 1}{1},\

-\frac{1}{n},

-\end{gather*}

-and so on.

-

-Now we can get rid of all repetitions of

-fractions of the same value by simply striking

-them out whenever they appear after their

-first occurrence. In the few initial terms

-written down above, $\frac{2}{2}$~which is enclosed above

-in square brackets is the only fraction not in

-its lowest terms. It has occurred before as~$\frac{1}{1}$.

-Thus this must be struck out. But the

-series is still left with the same properties,

-namely, (\textit{a})~there is a first term, (\textit{b})~each term

-has next-door neighbours, (\textit{c})~the series goes

-on without end.

-

-It can be proved that it is not possible to

-\index{Cantor, Georg}%

-arrange the whole series of real numbers in

-this way. This curious fact was discovered

-by Georg Cantor, a German mathematician

-still living; it is of the utmost importance

-in the philosophy of mathematical ideas. We

-are here in fact touching on the fringe of the

-great problems of the meaning of continuity

-and of infinity.

-

-Another extension of number comes from

-\index{Steps|EtSeq}%

-the introduction of the idea of what has been

-variously named an operation or a step,

-names which are respectively appropriate

-from slightly different points of view. We

-will start with a particular case. Consider

-\PageSep{80}

-the statement $2 + 3 = 5$. We add $3$ to~$2$ and

-obtain~$5$. Think of the operation of adding~$3$:

-let this be denoted by~$+3$. Again $4 - 3 = 1$.

-Think of the operation of subtracting~$3$:

-let this be denoted by~$-3$. Thus instead

-of considering the real numbers in themselves,

-we consider the \emph{operations} of adding or subtracting

-them: instead of~$\sqrt{2}$, we consider

-$+\sqrt{2}$ and~$-\sqrt{2}$, namely the operations of

-adding~$\sqrt{2}$ and of subtracting~$\sqrt{2}$. Then we

-can add these operations, of course in a

-different sense of addition to that in which we

-add numbers. The sum of two operations is

-the single operation which has the same effect

-as the two operations applied successively.

-In what order are the two operations to be

-applied? The answer is that it is indifferent,

-since for example

-\[

-2 + 3 + 1 = 2 + 1 + 3;

-\]

-so that the addition of the steps $+3$ and $+1$

-is commutative.

-

-Mathematicians have a habit, which is

-puzzling to those engaged in tracing out

-meanings, but is very convenient in practice,

-of using the same symbol in different though

-allied senses. The one essential requisite for

-a symbol in their eyes is that, whatever its

-possible varieties of meaning, the formal laws

-for its use shall always be the same. In

-\PageSep{81}

-accordance with this habit the addition of

-operations is denoted by~$+$ as well as the

-addition of numbers. Accordingly we can

-write

-\[

-(+3) + (+1) = +4;

-\]

-where the middle~$+$ on the left-hand side

-denotes the addition of the operations $+3$

-and~$+1$. But, furthermore, we need not be

-so very pedantic in our symbolism, except in

-the rare instances when we are directly tracing

-meanings; thus we always drop the first~$+$

-of a line and the brackets, and never write

-two $+$~signs running. So the above equation

-becomes

-\[

-3 + 1 = 4,

-\]

-which we interpret as simple numerical addition,

-or as the more elaborate addition of

-operations which is fully expressed in the

-previous way of writing the equation, or

-lastly as expressing the result of applying

-the operation~$+1$ to the number~$3$ and obtaining

-the number~$4$. Any interpretation

-which is possible is always correct. But the

-only interpretation which is always possible,

-under certain conditions, is that of operations.

-The other interpretations often give nonsensical

-results.

-

-This leads us at once to a question, which

-must have been rising insistently in the

-\PageSep{82}

-reader's mind: What is the use of all this

-elaboration? At this point our friend, the

-practical man, will surely step in and insist on

-sweeping away all these silly cobwebs of the

-brain. The answer is that what the mathematician

-is seeking is Generality. This is an

-\index{Generality in Mathematics}%

-idea worthy to be placed beside the notions

-of the Variable and of Form so far as concerns

-\index{Form, Algebraic}%

-\index{Variable, The}%

-its importance in governing mathematical

-procedure. Any limitation whatsoever upon

-the generality of theorems, or of proofs, or of

-interpretation is abhorrent to the mathematical

-instinct. These three notions, of the

-variable, of form, and of generality, compose

-a sort of mathematical trinity which preside

-over the whole subject. They all really

-spring from the same root, namely from the

-abstract nature of the science.

-

-Let us see how generality is gained by the

-introduction of this idea of operations. Take

-the equation $x + 1 = 3$; the solution is $x = 2$.

-Here we can interpret our symbols as mere

-numbers, and the recourse to ``operations''

-is entirely unnecessary. But, if $x$~is a mere

-number, the equation $x + 3 = 1$ is nonsense.

-For $x$~should be the number of things which

-remain when you have taken $3$~things away

-from $1$~thing; and no such procedure is

-possible. At this point our idea of algebraic

-form steps in, itself only generalization under

-another aspect. We consider, therefore, the

-\PageSep{83}

-\index{Positive and Negative Numbers|EtSeq}%

-general equation of the same form as $x + 1 = 3$.

-This equation is $x + a = b$, and its solution is

-$x = b - a$. Here our difficulties become acute;

-for this form can only be used for the numerical

-interpretation so long as $b$~is greater than~$a$,

-and we cannot say without qualification

-that $a$ and $b$ may be any constants. In other

-words we have introduced a limitation on

-the variability of the ``constants'' $a$~and~$b$,

-which we must drag like a chain throughout

-all our reasoning. Really prolonged mathematical

-investigations would be impossible

-under such conditions. Every equation

-would at last be buried under a pile of limitations.

-But if we now interpret our symbols

-as ``operations,'' all limitation vanishes like

-magic. The equation $x + 1 = 3$ gives $x = +2$,

-the equation $x + 3 = 1$ gives $x = -2$, the equation

-$x + a = b$ gives $x = b - a$ which is an operation

-of addition or subtraction as the case

-may be. We need never decide whether $b - a$

-represents the operation of addition or of

-subtraction, for the rules of procedure with

-the symbols are the same in either case.

-

-It does not fall within the plan of this work

-to write a detailed chapter of elementary

-algebra. Our object is merely to make plain

-the fundamental ideas which guide the formation

-of the science. Accordingly we do not

-further explain the detailed rules by which

-the ``positive and negative numbers'' are

-\PageSep{84}

-multiplied and otherwise combined. We have

-explained above that positive and negative

-numbers are operations. They have also

-been called ``steps.'' Thus $+3$~is the step

-by which we go from $2$ to~$5$, and $-3$~is the

-step backwards by which we go from $5$ to~$2$.

-Consider the line~$OX$ divided in the way explained

-in the earlier part of the chapter, so

-that its points represent numbers. Then~$+2$

-%[** TN: In original, negative numbers placed below axis, primed letters above]

-\Diagram{pg86}

-is the step from $O$ to~$B$, or from $A$ to~$C$, or

-(if the divisions are taken backwards along~$OX'$)

-from $C'$ to~$A'$, or from $D'$ to~$B'$, and so

-on. Similarly $-2$ is the step from $O$ to~$B'$,

-or from $B'$ to~$D'$, or from $B$ to~$O$, or from $C$

-to~$A$.

-

-We may consider the point which is reached

-by a step from~$O$, as representative of that

-step. Thus $A$~represents~$+1$, $B$~represents~$+2$,

-$A'$~represents~$-1$, $B'$~represents~$-2$, and

-so on. It will be noted that, whereas previously

-with the mere ``unsigned'' real numbers

-the points on one side of~$O$ only, namely along~$OX$,

-were representative of numbers, now

-with steps every point on the whole line

-stretching on both sides of~$O$ is representative

-of a step. This is a pictorial representation

-of the superior generality introduced by the

-positive and negative numbers, namely the

-\PageSep{85}

-operations or steps. These ``signed'' numbers

-are also particular cases of what have

-been called vectors (from the Latin \Foreign{veho}, I

-\index{Vectors}%

-draw or carry). For we may think of a

-particle as carried from $O$ to~$A$, or from $A$

-to~$B$.

-

-In suggesting a few pages ago that the

-practical man would object to the subtlety

-involved by the introduction of the positive

-and negative numbers, we were libelling that

-excellent individual. For in truth we are on

-the scene of one of his greatest triumphs. If

-the truth must be confessed, it was the practical

-man himself who first employed the actual

-symbols $+$ and~$-$. Their origin is not very

-certain, but it seems most probable that they

-arose from the marks chalked on chests of

-goods in German warehouses, to denote excess

-or defect from some standard weight. The

-earliest notice of them occurs in a book published

-at Leipzig, in \AD~1489. They seem

-first to have been employed in mathematics

-by a German mathematician, Stifel, in a book

-\index{Stifel}%

-published at Nuremburg in 1544~\AD. But

-then it is only recently that the Germans

-have come to be looked on as emphatically

-a practical nation. There is an old epigram

-which assigns the empire of the sea to the

-English, of the land to the French, and of the

-clouds to the Germans. Surely it was from

-the clouds that the Germans fetched $+$ and~$-$;

-\PageSep{86}

-the ideas which these symbols have

-generated are much too important for the

-welfare of humanity to have come from the

-sea or from the land.

-

-The possibilities of application of the positive

-and negative numbers are very obvious.

-If lengths in one direction are represented

-by positive numbers, those in the opposite

-direction are represented by negative numbers.

-If a velocity in one direction is positive, that

-in the opposite direction is negative. If a

-rotation round a dial in the opposite direction

-to the hands of a clock (anti-clockwise) is

-positive, that in the clockwise direction is

-negative. If a balance at the bank is positive,

-an overdraft is negative. If vitreous

-electrification is positive, resinous electrification

-is negative. Indeed, in this latter case,

-the terms positive electrification and negative

-electrification, considered as mere names,

-have practically driven out the other terms.

-An endless series of examples could be given.

-The idea of positive and negative numbers

-has been practically the most successful of

-mathematical subtleties.

-\PageSep{87}

-

-

-\Chapter{VII}{Imaginary Numbers}

-

-\First{If} the mathematical ideas dealt with in the

-\index{Imaginary Numbers|EtSeq}%

-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

-scandale}. Not only the practical man, but

-also men of letters and philosophers have expressed

-their bewilderment at the devotion

-of mathematicians to mysterious entities

-which by their very name are confessed to be

-imaginary. At this point it may be useful

-to observe that a certain type of intellect

-is always worrying itself and others by

-discussion as to the applicability of technical

-terms. Are the incommensurable numbers

-properly called numbers? Are the positive

-and negative numbers really numbers? Are

-the imaginary numbers imaginary, and are

-they numbers?---are types of such futile

-questions. Now, it cannot be too clearly

-understood that, in science, technical terms

-are names arbitrarily assigned, like Christian

-\PageSep{88}

-names to children. There can be no question

-of the names being right or wrong. They

-may be judicious or injudicious; for they can

-sometimes be so arranged as to be easy to

-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,

-``I pay them extra and make them mean

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

-whether imaginary numbers are imaginary,

-or as to whether they are numbers, but will

-take the phrase as the arbitrary name of a

-certain mathematical idea, which we will now

-endeavour to make plain.

-

-The origin of the conception is in every

-way similar to that of the positive and negative

-numbers. In exactly the same way it

-is due to the three great mathematical ideas

-of the variable, of algebraic form, and of

-generalization. The positive and negative

-numbers arose from the consideration of

-equations like $x + 1 = 3$, $x + 3 = 1$, and the

-general form $x + a = b$. Similarly the origin

-of imaginary numbers is due to equations like

-$x^{2} + 1 = 3$, $x^{2} + 3 = 1$, and $x^{2} + a = b$. Exactly

-the same process is gone through. The equation

-$x^{2} + 1 = 3$ 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

-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

-there is no positive or negative number which,

-when multiplied by itself, will give a negative

-square. Hence, if our symbols are to mean

-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

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

-cannot say unrestrictedly that the ``constants''

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

-the ``constants'' $a$~and~$b$ are not, as they

-ought to be, independent unrestricted ``variables'';

-and so again a host of limitations

-and restrictions will accumulate round our

-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

-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

-whether $a$~be positive or negative. Of

-course, the interpretation must be such that

-all the ordinary formal laws for addition, subtraction,

-multiplication, and division hold

-good; and also it must not interfere with the

-\PageSep{90}

-generality which we have attained by the use

-of the positive and negative numbers. In

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

-\end{align*}

-Hence, if we can so interpret our symbols that

-$\sqrt{(-1)}$~has a meaning, we have attained our

-object. Thus $\sqrt{(-1)}$~has come to be looked

-on as the head and forefront of all the

-imaginary quantities.

-

-This business of finding an interpretation

-for~$\sqrt{(-1)}$ is a much tougher job than the

-analogous one of interpreting~$-1$. In fact,

-while the easier problem was solved almost

-instinctively as soon as it arose, it at first

-hardly occurred, even to the greatest mathematicians,

-that here a problem existed which

-was perhaps capable of solution. Equations

-like $x^{2} = -3$, when they arose, were simply

-ruled aside as nonsense.

-

-However, it came to be gradually perceived

-during the eighteenth century, and even

-earlier, how very convenient it would be if

-an interpretation could be assigned to these

-nonsensical symbols. Formal reasoning with

-these symbols was gone through, merely

-assuming that they obeyed the ordinary

-\PageSep{91}

-algebraic laws of transformation; and it was

-seen that a whole world of interesting results

-could be attained, if only these symbols might

-legitimately be used. Many mathematicians

-were not then very clear as to the logic of

-their procedure, and an idea gained ground

-that, in some mysterious way, symbols which

-mean nothing can by appropriate manipulation

-yield valid proofs of propositions. Nothing

-can be more mistaken. A symbol

-which has not been properly defined is not a

-symbol at all. It is merely a blot of ink on

-paper which has an easily recognized shape.

-Nothing can be proved by a succession of

-blots, except the existence of a bad pen or a

-careless writer. It was during this epoch

-that the epithet ``imaginary'' came to be

-applied to~$\sqrt{(-1)}$. What these mathematicians

-had really succeeded in proving were

-a series of hypothetical propositions, of which

-this is the blank form: If interpretations

-exist for $\sqrt{(-1)}$ and for the addition, subtraction,

-multiplication, and division of~$\sqrt{(-1)}$

-which make the ordinary algebraic

-rules (\eg\ $x + y = y + x$, etc.)\ to be satisfied,

-then such and such results follows. It was

-natural that the mathematicians should not

-always appreciate the big ``If,'' which ought

-to have preceded the statements of their results.

-

-As may be expected the interpretation,

-\PageSep{92}

-when found, was a much more elaborate affair

-than that of the negative numbers and the

-reader's attention must be asked for some

-careful preliminary explanation. We have

-already come across the representation of a

-point by two numbers. By the aid of the

-\Figure{8}

-positive and negative numbers we can now

-represent the position of any point in a plane

-by a pair of such numbers. Thus we take

-the pair of straight lines $XOX'$ and $YOY'$, at

-right angles, as the ``axes'' from which we

-start all our measurements. Lengths measured

-along $OX$ and $OY$ are positive, and

-measured backwards along $OX'$ and $OY'$ are

-negative. Suppose that a pair of numbers,

-written in order, \eg~$(+3, +1)$, so that there

-\PageSep{93}

-\index{Ordered Couples|EtSeq}%

-is a first number ($+3$~in the above example),

-and a second number ($+1$~in the above example),

-represents measurements from~$O$

-along $XOX'$ for the first number, and along

-$YOY'$ for the second number. Thus (\Chg{cf.}{\cf}\ \Fig[fig.]{9}) in

-$(+3, +1)$ a length of $3$~units is to be measured

-along $XOX'$ in the positive direction, that

-is from~$O$ towards~$X$, and a length~$+1$

-measured along $YOY'$ in the positive direction,

-that is from~$O$ towards~$Y$. Similarly in

-$(-3, +1)$ the length of $3$~units is to be

-measured from~$O$ towards~$X'$, and of $1$~unit

-from towards~$Y$. Also in $(-3, -1)$ the

-two lengths are to be measured along $OX'$

-and $OY'$ respectively, and in $(+3, -1)$ along

-$OX$ and $OY'$ respectively. Let us for the

-moment call such a pair of numbers an

-``ordered couple.'' Then, from the two numbers

-$1$~and~$3$, eight ordered couples can be

-generated, namely

-\begin{gather*}

-(+1, +3),\ (-1, +3),\ (-1, -3),\ (+1, -3), \\

-(+3, +1),\ (-3, +1),\ (-3, -1),\ (+3, -1).

-\end{gather*}

-Each of these eight ``ordered couples'' directs

-a process of measurement along $XOX'$ and

-$YOY'$ which is different from that directed

-by any of the others.

-

-The processes of measurement represented

-by the last four ordered couples, mentioned

-above, are given pictorially in the figure.

-The lengths $OM$ and $ON$ together correspond

-\PageSep{94}

-to $(+3, +1)$, the lengths $OM'$ and $ON$

-together correspond to $(-3, +1)$, $OM'$~and

-$ON'$ together to $(-3, -1)$, and $OM$~and

-$ON'$ together to $(+3, -1)$. But by completing

-the various rectangles, it is easy to

-see that the point~$P$ completely determines

-and is determined by the ordered couple

-\Figure{9}

-$(+3, +1)$, the point~$P'$ by $(-3, +1)$, the

-point~$P''$ by $(-3, -1)$, and the point~$P'''$ by

-$(+3, -1)$. More generally in the previous

-figure~(\FigNum{8}), the point~$P$ corresponds to the

-ordered couple~$(x, y)$, where $x$~and~$y$ in the

-figure are both assumed to be positive, the

-point~$P'$ corresponds to $(x', y)$, where $x'$~in

-the figure is assumed to be negative, $P''$~to

-$(x' y')$, and $P'''$~to $(x, y')$. Thus an ordered

-\PageSep{95}

-couple $(x, y)$, where $x$~and~$y$ are any positive

-or negative numbers, and the corresponding

-point reciprocally determine each other. It

-is convenient to introduce some names at this

-juncture. In the ordered couple $(x, y)$ the

-first number~$x$ is called the ``abscissa'' of the

-\index{Abscissa}%

-corresponding point, and the second number~$y$

-is called the ``ordinate'' of the point, and

-\index{Ordinate}%

-the two numbers together are called the ``coordinates''

-\index{Coordinates}%

-of the point. The idea of determining

-the position of a point by its ``coordinates''

-was by no means new when the

-theory of ``imaginaries'' was being formed.

-It was due to Descartes, the great French

-\index{Descartes}%

-mathematician and philosopher, and appears

-in his \Title{Discours} published at Leyden in 1637~\AD.

-The idea of the ordered couple as a

-thing on its own account is of later growth

-and is the outcome of the efforts to interpret

-imaginaries in the most abstract way possible.

-

-It may be noticed as a further illustration

-of this idea of the ordered couple, that the

-point~$M$ in \Fig[fig.]{9} is the couple $(+3, 0)$, the

-point~$N$ is the couple $(0, +1)$, the point~$M'$

-the couple $(-3, 0)$, the point~$N'$ the couple

-$(0, -1)$, the point~$O$ the couple~$(0, 0)$.

-

-Another way of representing the ordered

-couple $(x, y)$ is to think of it as representing

-the dotted line~$OP$ (\Chg{cf.}{\cf}\ \Fig[fig.]{8}), rather than the

-point~$P$. Thus the ordered couple represents

-a line drawn from an ``origin,''~$O$, of a certain

-\index{Origin}%

-\PageSep{96}

-\index{Steps}%

-length and in a certain direction. The line~$OP$

-may be called the vector line from $O$ to~$P$,

-or the step from $O$ to~$P$. We see, therefore,

-that we have in this chapter only extended

-the interpretation which we gave formerly of

-the positive and negative numbers. This

-method of representation by vectors is very

-\index{Vectors}%

-useful when we consider the meaning to be

-assigned to the operations of the addition and

-multiplication of ordered couples.

-

-{\Loosen We will now go on to this question, and

-ask what meaning we shall find it convenient

-to assign to the addition of the two ordered

-couples $(x, y)$ and $(x', y')$. The interpretation

-must, (\textit{a})~make the result of addition

-to be another ordered couple, (\textit{b})~make the

-operation commutative so that $(x, y) + (x', y') = (x', y') + (x, y)$,

-(\textit{c})~make the operation

-associative so that}

-\[

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

-\]

-(\textit{d})~make the result of subtraction unique,

-so that when we seek to determine the

-unknown ordered couple $(x, y)$ so as to

-satisfy the equation

-\[

-(x, y) + (a, b) = (c, d),

-\]

-there is one and only one answer which we

-can represent by

-\[

-(x, y) = (c, d) - (a, b).

-\]

-\PageSep{97}

-All these requisites are satisfied by taking

-$(x, y) + (x', y')$ to mean the ordered couple

-$(x + x', y + y')$. Accordingly by definition we

-put

-\[

-(x, y) + (x', y') = (x + x', y + y').

-\]

-Notice that here we have adopted the mathematical

-habit of using the same symbol~$+$ in

-different senses. The $+$ on the left-hand side

-of the equation has the new meaning of~$+$

-which we are just defining; while the two~$+$'s

-on the right-hand side have the meaning

-of the addition of positive and negative numbers

-(operations) which was defined in the

-last chapter. No practical confusion arises

-from this double use.

-

-As examples of addition we have

-\begin{align*}

-(+3, +1) + (+2, +6) &= (+5, +7), \\

-(+3, -1) + (-2, -6) &= (+1, -7), \\

-(+3, +1) + (-3, -1) &= (0, 0).

-\end{align*}

-

-The meaning of subtraction is now settled

-for us. We find that

-\[

-(x, y) - (u, v) = (x - u, y - v).

-\]

-Thus

-\[

-(+3, +2) - (+1, +1) = (+2, +1),

-\]

-and

-\[

-(+1, -2) - (+2, -4) = (-1, +2),

-\]

-and

-\[

-(-1, -2) - (+2, +3) = (-3, -5).

-\]

-\PageSep{98}

-

-It is easy to see that

-\[

-(x, y) - (u, v) = (x, y) + (-u, -v).

-\]

-Also

-\[

-(x, y) - (x, y) = (0, 0).

-\]

-Hence $(0, 0)$~is to be looked on as the zero

-ordered couple. For example

-\[

-(x, y) + (0, 0) = (x, y).

-\]

-

-The pictorial representation of the addition

-of ordered couples is surprisingly easy.

-\Figure{10}

-

-{\Loosen Let $OP$ represent $(x, y)$ so that $OM = x$

-and $PM = y$; let $OQ$ represent $(x_{1}, y_{1})$ so that

-$OM_{1} = x_{1}$ and $QM_{1} = y_{1}$. Complete the parallelogram

-$OPRQ$ by the dotted lines $PR$ and~$QR$,

-then the diagonal~$OR$ is the ordered

-couple $(x + x_{1}, y + y_{1})$. For draw $PS$ parallel

-\PageSep{99}

-to~$OX$; then evidently the triangles $OQM_{1}$

-and $PRS$ are in all respects equal. Hence

-$MM' = PS = x_{1}$, and $RS = QM_{1}$ and therefore}

-\begin{gather*}

-OM' = OM + MM' = x + x_{1}, \\

-RM' = SM' + RS = y + y_{1}.

-\end{gather*}

-

-Thus $OR$~represents the ordered couple as

-required. This figure can also be drawn with

-$OP$ and $OQ$ in other quadrants.

-

-It is at once obvious that we have here

-come back to the parallelogram law, which

-\index{Parallelogram Law}%

-was mentioned in \ChapRef{VI}., on the laws of

-motion, as applying to velocities and forces.

-It will be remembered that, if $OP$ and $OQ$

-represent two velocities, a particle is said to

-be moving with a velocity equal to the two

-velocities added together if it be moving with

-the velocity~$OR$. In other words $OR$~is said

-to be the resultant of the two velocities $OP$

-and~$OQ$. Again forces acting at a point of a

-body can be represented by lines just as

-velocities can be; and the same parallelogram

-law holds, namely, that the resultant of the

-two forces $OP$ and $OQ$ is the force represented

-by the diagonal~$OR$. It follows that we can

-look on an ordered couple as representing a

-velocity or a force, and the rule which we

-have just given for the addition of ordered

-couples then represents the fundamental laws

-of mechanics for the addition of forces and

-\PageSep{100}

-velocities. One of the most fascinating

-characteristics of mathematics is the surprising

-way in which the ideas and results of

-different parts of the subject dovetail into

-each other. During the discussions of this

-and the previous chapter we have been guided

-merely by the most abstract of pure mathematical

-considerations; and yet at the end

-of them we have been led back to the most

-fundamental of all the laws of nature, laws

-which have to be in the mind of every engineer

-as he designs an engine, and of every naval

-architect as he calculates the stability of a

-ship. It is no paradox to say that in our

-most theoretical moods we may be nearest to

-our most practical applications.

-\PageSep{101}

-

-

-\Chapter[Imaginary Numbers]

-  {VIII}{Imaginary Numbers (\textit{C\MakeLowercase{ontinued}})}

-

-\First{The} definition of the multiplication of

-ordered couples is guided by exactly the same

-considerations as is that of their addition.

-The interpretation of multiplication must be

-such that

-

-\Eq{(\alpha)} the result is another ordered couple,

-

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

-\[

-(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)\},

-\]

-

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

-[with an exception for the case of the zero

-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),

-\]

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

-\]

-\PageSep{102}

-

-\Eq{(\epsilon)} Furthermore the law involving both

-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)\}.

-\end{multline*}

-

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

-be satisfied by an interpretation which,

-though it looks complicated at first, is capable

-of a simple geometrical interpretation.

-

-By definition we put

-\[

-(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

-ordered couples. It follows evidently from

-this definition that the result of multiplication

-is another ordered couple, and that the

-value of the right-hand side of equation~\Eq{(A)}

-is not altered by simultaneously interchanging

-$x$~with~$x'$, and $y$~with~$y'$. Hence conditions

-\Eq{(\alpha)} and \Eq{(\beta)} are evidently satisfied. The proof

-of the satisfaction of \Eq{(\gamma)}, \Eq{(\delta)}, \Eq{(\epsilon)} is equally

-easy when we have given the geometrical

-interpretation, which we will proceed to do

-in a moment. But before doing this it will

-be interesting to pause and see whether we

-have attained the object for which all this

-elaboration was initiated.

-

-We came across equations of the form

-$x^{2} = -3$, to which no solutions could be

-\PageSep{103}

-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 $\sqrt{(-1)} × \sqrt{(-1)} = -1$.

-

-Now let us consider the three special

-\index{Zero}%

-ordered couples\footnote

-  {For the future we follow the custom of omitting the

-  $+$~sign wherever possible, thus $(1, 0)$ stands for $(+1, 0)$

-  and $(0, 1)$ for $(0, +1)$.}

-$(0, 0)$, $(1, 0)$, and $(0, 1)$.

-

-We have already proved that

-\[

-(x, y) + (0, 0) = (x, y).

-\]

-

-Furthermore we now have

-\[

-(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

-$x × 0 = 0$.

-

-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

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

-\[

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

-\]

-

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

-\PageSep{104}

-

-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

-law of multiplication for ordered couples

-\[

-(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, $(0, -1) × (0, -1) = (-1, 0)$.

-

-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

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

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

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

-to~$+\sqrt{(-1)}$? The answer is that it is perfectly

-indifferent which symbolism we adopt.

-

-The ordered couples can be divided into

-three types, (i)~the ``complex imaginary''

-type~$(x, y)$, in which neither $x$ nor~$y$ is zero;

-(ii)~the ``real'' type~$(x, 0)$; (iii)~the ``pure

-imaginary'' type~$(0, y)$. Let us consider the

-relations of these types to each other. First

-multiply together the ``complex imaginary''

-\PageSep{105}

-couple $(x, y)$ and the ``real'' couple $(a, 0)$, we

-find

-\[

-(a, 0) × (x, y) = (ax, ay).

-\]

-

-Thus the effect is merely to multiply each

-term of the couple $(x, y)$ by the positive or

-negative real number~$a$.

-

-Secondly, multiply together the ``complex

-imaginary'' couple $(x, y)$ and the ``pure

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

-\[

-(0, b) × (x, y) = (-by, bx).

-\]

-

-Here the effect is more complicated, and is

-best comprehended in the geometrical interpretation

-to which we proceed after noting

-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).

-\]

-

-Fourthly, we multiply the two ``real''

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

-\[

-(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).

-\]

-

-We now turn to the geometrical interpretation,

-beginning first with some special cases.

-\PageSep{106}

-Take the couples $(1, 3)$ and $(2, 0)$ and consider

-the equation

-\[

-(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

-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 $(0, 2) × (1, 3)$,

-we have

-\[

-(0, 2) × (1, 3) = (-6, 2)\Add{.}

-\]

-

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

-the vector~$OR$ to~$(-6,2)$. Thus $OR$ which

-\PageSep{107}

-represents the new product is at right angles

-to~$OQ$ and of the same length. Notice that

-we have the same law regulating the length

-of~$OQ$ as in the previous case, namely, that

-its length is the product of the lengths of

-the two vectors which are multiplied together;

-but now that we have $ON_{1}$ along the

-``ordinate'' axis~$OY$, instead of $ON$ along

-the ``abscissa'' axis~$OX$, the direction of~$OP$

-has been turned through a right-angle.

-

-Hitherto in these examples of multiplication

-we have looked on the vector~$OP$ as modified

-by the vectors $ON$ and~$ON_{1}$. We shall get

-a clue to the general law for the direction by

-inverting the way of thought, and by thinking

-of the vectors $ON$ and~$ON_{1}$ as modified by

-the vector~$OP$. The law for the length remains

-unaffected; the resultant length is the

-length of the product of the two vectors.

-The new direction for the enlarged~$ON$ (\ie~$OQ$)

-is found by rotating it in the (anti-clockwise)

-direction of rotation from $OX$ towards~$OY$

-through an angle equal to the angle~$XOP$:

-it is an accident of this particular case that

-this rotation makes $OQ$ lie along the line~$OP$.

-Again consider the product of $ON_{1}$ and~$OP$;

-the new direction for the enlarged~$ON_{1}$ (\ie~$OR$)

-is found by rotating~$ON$ in the anti-clockwise

-direction of rotation through an

-angle equal to the angle~$XOP$, namely, the

-angle~$N_{1}OR$ is equal to the angle~$XOP$.

-\PageSep{108}

-

-The general rule for the geometrical representation

-of multiplication can now be enunciated

-thus:

-\Figure[3in]{12}

-

-The product of the two vectors $OP$ and~$OQ$

-is a vector~$OR$, whose length is the product

-of the lengths of $OP$ and~$OQ$ and whose

-direction~$OR$ is such that the angle~$XOR$ is

-equal to the sum of the angles $XOP$ and~$XOQ$.

-

-Hence we can conceive the vector~$OP$ as

-making the vector~$OQ$ rotate through an

-angle~$XOP$ (\ie\ $\text{the angle } QOR = \text{the angle } XOP$),

-or the vector~$OQ$ as making the vector~$OP$

-rotate through the angle~$XOQ$ (\ie $\text{the angle } POR = \text{the angle } XOQ$).

-

-We do not prove this general law, as we

-\PageSep{109}

-should thereby be led into more technical

-processes of mathematics than falls within the

-design of this book. But now we can immediately

-see that the associative law [numbered~\Eq{(\gamma)}

-above] for multiplication is satisfied.

-Consider first the length of the resultant

-vector; this is got by the ordinary process

-of multiplication for real numbers; and thus

-the associative law holds for it.

-

-Again, the direction of the resultant vector

-is got by the mere addition of angles, and the

-associative law holds for this process also.

-

-So much for multiplication. We have now

-rapidly indicated, by considering addition and

-multiplication, how an algebra or ``calculus''

-of vectors in one plane can be constructed,

-which is such that any two vectors in the

-plane can be added, or subtracted, and can

-be multiplied, or divided one by the other.

-

-We have not considered the technical details

-of all these processes because it would

-lead us too far into mathematical details;

-but we have shown the general mode of procedure.

-When we are interpreting our algebraic

-symbols in this way, we are said to be

-employing ``imaginary quantities'' or ``complex

-\index{Complex Quantities}%

-\index{Imaginary Quantities}%

-quantities.'' These terms are mere

-details, and we have far too much to think

-about to stop to enquire whether they are or

-are not very happily chosen.

-

-%[** TN: [sic] "nett", variant spelling]

-The nett result of our investigations is that

-\PageSep{110}

-any equations like $x + 3 = 2$ or $(x + 3)^{2} = -2$

-can now always be interpreted into terms of

-vectors, and solutions found for them. In

-seeking for such interpretations it is well to

-note that $3$~becomes $(3, 0)$, and $-2$~becomes

-$(-2, 0)$, and $x$~becomes the ``unknown''

-couple $(u, v)$: so the two equations become

-respectively $(u, v) + (3, 0) = (2, 0)$, and

-$\{(u, v) + (3, 0)\}^{2} = (-2, 0)$.

-

-We have now completely solved the initial

-difficulties which caught our eye as soon as

-we considered even the elements of algebra.

-The science as it emerges from the solution is

-much more complex in ideas than that with

-which we started. We have, in fact, created

-a new and entirely different science, which

-will serve all the purposes for which the old

-science was invented and many more in addition.

-But, before we can congratulate ourselves

-on this result to our labours, we must

-allay a suspicion which ought by this time to

-have arisen in the mind of the student. The

-question which the reader ought to be asking

-himself is: Where is all this invention of new

-interpretations going to end? It is true that

-we have succeeded in interpreting algebra so

-as always to be able to solve a quadratic

-equation like $x^{2} - 2x + 4 = 0$; but there are

-an endless number of other equations, for

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

-on without limit. Have we got to make a

-\PageSep{111}

-new science whenever a new equation appears?

-

-Now, if this were the case, the whole of our

-preceding investigations, though to some

-minds they might be amusing, would in truth

-be of very trifling importance. But the great

-fact, which has made modern analysis possible,

-is that, by the aid of this calculus of vectors,

-every formula which arises can receive its

-proper interpretation; and the ``unknown''

-quantity in every equation can be shown to

-indicate some vector. Thus the science is now

-complete in itself as far as its fundamental

-ideas are concerned. It was receiving its final

-form about the same time as when the steam

-engine was being perfected, and will remain

-a great and powerful weapon for the achievement

-of the victory of thought over things

-when curious specimens of that machine

-repose in museums in company with the

-helmets and breastplates of a slightly earlier

-epoch.

-\PageSep{112}

-

-

-\Chapter{IX}{Coordinate Geometry}

-

-\First{The} methods and ideas of coordinate geometry

-\index{Coordinate Geometry|EtSeq}%

-have already been employed in the

-previous chapters. It is now time for us to

-consider them more closely for their own

-sake; and in doing so we shall strengthen our

-hold on other ideas to which we have attained.

-In the present and succeeding chapters we

-will go back to the idea of the positive and

-negative real numbers and will ignore the

-imaginaries which were introduced in the last

-two chapters.

-

-We have been perpetually using the idea

-that, by taking two axes, $XOX'$ and~$YOY'$,

-in a plane, any point~$P$ in that plane can be

-determined in position by a pair of positive

-or negative numbers $x$ and~$y$, where (\Chg{cf.}{\cf}\

-\Fig[fig.]{13}) $x$~is the length~$OM$ and $y$~is the length~$PM$.

-This conception, simple as it looks, is

-the main idea of the great subject of coordinate

-geometry. Its discovery marks a

-momentous epoch in the history of mathematical

-thought. It is due (as has been

-\PageSep{113}

-already said) to the philosopher Descartes,

-\index{Descartes}%

-and occurred to him as an important mathematical

-method one morning as he lay in bed.

-Philosophers, when they have possessed a

-thorough knowledge of mathematics, have

-been among those who have enriched the

-\Figure{13}

-science with some of its best ideas. On the

-other hand it must be said that, with hardly

-an exception, all the remarks on mathematics

-made by those philosophers who have possessed

-but a slight or hasty and late-acquired

-knowledge of it are entirely worthless, being

-either trivial or wrong. The fact is a curious

-one; since the ultimate ideas of mathematics

-\PageSep{114}

-seem, after all, to be very simple, almost

-childishly so, and to lie well within the

-province of philosophical thought. Probably

-their very simplicity is the cause of error; we

-are not used to think about such simple

-abstract things, and a long training is necessary

-to secure even a partial immunity from

-error as soon as we diverge from the beaten

-track of thought.

-

-The discovery of coordinate geometry, and

-also that of projective geometry about the

-same time, illustrate another fact which is

-being continually verified in the history of

-knowledge, namely, that some of the greatest

-discoveries are to be made among the most

-well-known topics. By the time that the

-seventeenth century had arrived, geometry

-had already been studied for over two thousand

-years, even if we date its rise with the Greeks.

-Euclid, taught in the University of Alexandria,

-\index{Euclid}%

-being born about 330~\BC; and he only

-systematized and extended the work of a long

-series of predecessors, some of them men of

-genius. After him generation after generation

-of mathematicians laboured at the improvement

-of the subject. Nor did the

-subject suffer from that fatal bar to progress,

-namely, that its study was confined to a

-narrow group of men of similar origin and

-outlook---quite the contrary was the case;

-by the seventeenth century it had passed

-\PageSep{115}

-through the minds of Egyptians and Greeks,

-of Arabs and of Germans. And yet, after all

-this labour devoted to it through so many

-ages by such diverse minds its most important

-secrets were yet to be discovered.

-

-No one can have studied even the elements

-of elementary geometry without feeling the

-lack of some guiding method. Every proposition

-has to be proved by a fresh display of ingenuity;

-and a science for which this is true

-lacks the great requisite of scientific thought,

-namely, method. Now the especial point of

-coordinate geometry is that for the first

-time it introduced method. The remote

-deductions of a mathematical science are not

-of primary theoretical importance. The

-science has not been perfected, until it consists

-in essence of the exhibition of great allied

-methods by which information, on any desired

-topic which falls within its scope, can easily

-be obtained. The growth of a science is not

-primarily in bulk, but in ideas; and the more

-the ideas grow, the fewer are the deductions

-which it is worth while to write down. Unfortunately,

-mathematics is always encumbered

-by the repetition in text-books of

-numberless subsidiary propositions, whose importance

-has been lost by their absorption

-into the role of particular cases of more

-general truths---and, as we have already insisted,

-generality is the soul of mathematics.

-\PageSep{116}

-

-Again, coordinate geometry illustrates

-another feature of mathematics which has

-already been pointed out, namely, that mathematical

-sciences as they develop dovetail into

-each other, and share the same ideas in common.

-It is not too much to say that the

-various branches of mathematics undergo a

-perpetual process of generalization, and that

-as they become generalized, they coalesce.

-Here again the reason springs from the very

-nature of the science, its generality, that is

-to say, from the fact that the science deals

-with the general truths which apply to all

-things in virtue of their very existence as

-things. In this connection the interest of coordinate

-geometry lies in the fact that it

-relates together geometry, which started as

-the science of space, and algebra, which has

-its origin in the science of number.

-

-Let us now recall the main ideas of the two

-sciences, and then see how they are related

-by Descartes' method of coordinates. Take

-\index{Descartes}%

-algebra in the first place. We will not trouble

-ourselves about the imaginaries and will

-think merely of the real numbers with positive

-or negative signs. The fundamental idea

-is that of any number, the variable number,

-which is denoted by a letter and not by any

-definite numeral. We then proceed to the

-consideration of correlations between variables.

-For example, if $x$ and~$y$ are two variables,

-\PageSep{117}

-we may conceive them as correlated by

-the equations $x + y = 1$, or by $x - y = 1$, or in

-any one of an indefinite number of other ways.

-This at once leads to the application of the

-\index{Form, Algebraic}%

-idea of algebraic form. We think, in fact, of

-any correlation of some interesting type, thus

-rising from the initial conception of variable

-numbers to the secondary conception of

-variable correlations of numbers. Thus we

-generalize the correlation $x + y = 1$, into the

-correlation $ax + by = c$. Here $a$~and $b$ and~$c$,

-being letters, stand for any numbers and are

-in fact themselves variables. But they are

-the variables which determine the variable

-correlation; and the correlation, when determined,

-correlates the variable numbers $x$ and~$y$.

-Variables, like $a$,~$b$, and~$c$ above, which

-are used to determine the correlation are

-called ``constants,'' or parameters. The use

-\index{Constants}%

-\index{Parameters}%

-of the term ``constant'' in this connection

-for what is really a variable may seem at first

-sight to be odd; but it is really very natural.

-For the mathematical investigation is concerned

-with the relation between the variables

-$x$ and~$y$, after $a$,~$b$,~$c$ are supposed to have been

-determined. So in a sense, relatively to $x$

-and~$y$, the ``constants'' $a$,~$b$, and~$c$ are constants.

-Thus $ax + by = c$ stands for the general

-example of a certain algebraic form, that is,

-for a variable correlation belonging to a certain

-class.

-\PageSep{118}

-

-Again we generalize $x^{2} + y^{2} = 1$ into $ax^{2} + by^{2} = c$,

-or still further into $ax^{2} + 2hxy + by^{2} = c$,

-or, still further, into $ax^{2} + 2hxy + by^{2} + 2gx + 2fy = c$.

-

-Here again we are led to variable correlations

-which are indicated by their various algebraic

-forms.

-

-Now let us turn to geometry. The name

-of the science at once recalls to our minds

-the thought of figures and diagrams exhibiting

-triangles and rectangles and squares and

-circles, all in special relations to each other.

-The study of the simple properties of these

-figures is the subject matter of elementary

-geometry, as it is rightly presented to the

-beginner. Yet a moment's thought will show

-that this is not the true conception of the

-subject. It may be right for a child to commence

-his geometrical reasoning on shapes,

-like triangles and squares, which he has cut

-out with scissors. What, however, is a triangle?

-It is a figure marked out and bounded

-by three bits of three straight lines.

-

-Now the boundary of spaces by bits of

-lines is a very complicated idea, and not at

-all one which gives any hope of exhibiting

-the simple general conceptions which should

-form the bones of the subject. We want

-something more simple and more general. It

-is this obsession with the wrong initial ideas---very

-natural and good ideas for the creation

-\PageSep{119}

-of first thoughts on the subject---which was

-the cause of the comparative sterility of the

-study of the science during so many centuries.

-Coordinate geometry, and Descartes its inventor,

-must have the credit of disclosing the

-true simple objects for geometrical thought.

-

-In the place of a bit of a straight line, let

-us think of the whole of a straight line

-throughout its unending length in both directions.

-This is the sort of general idea from

-which to start our geometrical investigations.

-The Greeks never seem to have found any

-use for this conception which is now fundamental

-in all modern geometrical thought.

-Euclid always contemplates a straight line as

-drawn between two definite points, and is

-very careful to mention when it is to be produced

-beyond this segment. He never thinks

-of the line as an entity given once for all as a

-whole. This careful definition and limitation,

-so as to exclude an infinity not immediately

-apparent to the senses, was very characteristic

-of the Greeks in all their many

-activities. It is enshrined in the difference

-between Greek architecture and Gothic architecture,

-and between the Greek religion and

-the modern religion. The spire on a Gothic

-cathedral and the importance of the unbounded

-straight line in modern geometry

-are both emblematic of the transformation of

-the modern world.

-\PageSep{120}

-

-The straight line, considered as a whole,

-is accordingly the root idea from which

-modern geometry starts. But then other

-sorts of lines occur to us, and we arrive at the

-conception of the complete curve which at

-every point of it exhibits some uniform characteristic,

-just as the straight line exhibits

-at all points the characteristic of straightness.

-For example, there is the circle which

-\index{Circle}%

-at all points exhibits the characteristic of

-being at a given distance from its centre, and

-again there is the ellipse, which is an oval

-\index{Ellipse}%

-curve, such that the sum of the two distances

-of any point on it from two fixed points, called

-\index{Focus}%

-its \emph{foci}, is constant for all points on the curve.

-It is evident that a circle is merely a particular

-case of an ellipse when the two foci are

-superposed in the same point; for then the

-sum of the two distances is merely twice the

-radius of the circle. The ancients knew the

-properties of the ellipse and the circle and, of

-course, considered them as wholes. For example,

-Euclid never starts with mere segments

-(\ie,~bits) of circles, which are then prolonged.

-He always considers the whole circle

-as described. It is unfortunate that the

-circle is not the true fundamental line in

-geometry, so that his defective consideration

-of the straight line might have been of less

-consequence.

-

-This general idea of a curve which at any

-\PageSep{121}

-\index{Locus|EtSeq}%

-point of it exhibits some uniform property is

-expressed in geometry by the term ``locus.''

-A locus is the curve (or surface, if we do not

-confine ourselves to a plane) formed by points,

-all of which possess some given property.

-To every property in relation to each other

-which points can have, there corresponds

-some locus, which consists of all the points

-possessing the property. In investigating

-the properties of a locus considered as a whole,

-we consider \emph{any} point or points on the locus.

-Thus in geometry we again meet with the

-fundamental idea of the variable. Furthermore,

-in classifying loci under such headings

-as straight lines, circles, ellipses, etc., we again

-find the idea of form.

-

-Accordingly, as in algebra we are concerned

-with variable numbers, correlations between

-variable numbers, and the classification of

-correlations into types by the idea of algebraic

-form; so in geometry we are concerned with

-variable points, variable points satisfying

-some condition so as form to a locus, and the

-classification of \emph{loci} into types by the idea of

-conditions of the same form.

-

-Now, the essence of coordinate geometry

-is the identification of the algebraic correlation

-with the geometrical locus. The point

-on a plane is represented in algebra by its

-two coordinates, $x$~and~$y$, and the condition

-satisfied by any point on the locus is represented

-\PageSep{122}

-by the corresponding correlation

-between $x$~and~$y$. Finally to correlations

-expressible in some general algebraic form,

-such as $ax + by = c$, there correspond loci of

-some general type, whose geometrical conditions

-are all of the same form. We

-have thus arrived at a position where we

-can effect a complete interchange in ideas

-and results between the two sciences. Each

-science throws light on the other, and itself

-gains immeasurably in power. It is impossible

-not to feel stirred at the thought

-of the emotions of men at certain historic

-moments of adventure and discovery---Columbus

-\index{Columbus}%

-when he first saw the Western

-shore, Pizarro when he stared at the Pacific

-\index{Pizarro}%

-Ocean, Franklin when the electric spark came

-\index{Franklin}%

-from the string of his kite, Galileo when he

-\index{Galileo}%

-first turned his telescope to the heavens.

-Such moments are also granted to students

-in the abstract regions of thought, and high

-among them must be placed the morning when

-Descartes lay in bed and invented the method

-\index{Descartes}%

-of coordinate geometry.

-

-When one has once grasped the idea of coordinate

-geometry, the immediate question

-which starts to the mind is, What sort of

-loci correspond to the well-known algebraic

-forms? For example, the simplest among

-the general types of algebraic forms is $ax + by = c$.

-The sort of locus which corresponds

-\PageSep{123}

-to this is a straight line, and conversely to

-every straight line there corresponds an equation

-of this form. It is fortunate that the

-simplest among the geometrical loci should

-correspond to the simplest among the algebraic

-forms. Indeed, it is this general correspondence

-of geometrical and algebraic simplicity

-which gives to the whole subject its

-power. It springs from the fact that the

-connection between geometry and algebra is

-not casual and artificial, but deep-seated and

-essential. The equation which corresponds

-to a locus is called the equation ``of'' (or

-``to'') the locus. Some examples of equations

-of straight lines will illustrate the subject.

-\Figure[3.75in]{14}

-\PageSep{124}

-

-Consider $y - x = 0$; here the $a$,~$b$, and~$c$, of

-the general form have been replaced by $-1$,~$1$,

-and $0$ respectively. This line passes through

-the ``origin,''~$O$, in the diagram and bisects

-the angle~$XOY$. It is the line~$L'OL$ of the

-diagram. The fact that it passes through the

-origin,~$O$, is easily seen by observing that the

-equation is satisfied by putting $x = 0$ and

-$y = 0$ simultaneously, but $0$~and~$0$ are the coordinates

-of~$O$. In fact it is easy to generalize

-and to see by the same method that the

-equation of any line through the origin is of

-the form $ax + by = 0$. The locus of\Typo{}{ the} equation

-$y + x = 0$ also passes through the origin and

-bisects the angle~$X'OY$: it is the line~$L_{1}OL_{1}'$

-of the diagram.

-

-Consider $y - x = 1$: the corresponding locus

-does not pass through the origin. We therefore

-seek where it cuts the axes. It must cut

-the axis of~$x$ at some point of coordinates

-$x$~and~$0$. But putting $y = 0$ in the equation,

-we get $x = -1$; so the coordinates of this

-point~$(A)$ are $1$~and~$0$. Similarly the point~$(B)$

-where the line cuts the axis~$OY$ are $0$~and~$1$.

-The locus is the line~$AB$ in the figure and

-is parallel to~$LOL'$. Similarly $y + x = 1$ is the

-equation of line~$A_{1}B$ of the figure; and the

-locus is parallel to~$L_{1}OL_{1}'$. It is easy to prove

-the general theorem that two lines represented

-by equations of the forms $ax + by = 0$ and

-$ax + by = c$ are parallel.

-\PageSep{125}

-

-The group of loci which we next come upon

-are sufficiently important to deserve a chapter

-to themselves. But before going on to

-them we will dwell a little longer on the main

-ideas of the subject.

-

-The position of any point~$P$ is determined

-by arbitrarily choosing an origin,~$O$, two axes,

-\index{Axes}%

-$OX$~and~$OY$, at right-angles, and then by

-noting its coordinates $x$~and~$y$, \ie\ $OM$ and~$PM$

-(\cf\ \Fig[fig.]{13}). Also, as we have seen in the

-last chapter, $P$~can be determined by the

-``vector''~$OP$, where the idea of the vector

-includes a determinate direction as well as a

-determinate length. From an abstract

-mathematical point of view the idea of an

-arbitrary origin may appear artificial and

-clumsy, and similarly for the arbitrarily

-drawn axes, $OX$~and~$OY$. But in relation to

-the application of mathematics to the event

-of the Universe we are here symbolizing with

-direct simplicity the most fundamental fact

-respecting the outlook on the world afforded

-to us by our senses. We each of us refer

-our sensible perceptions of things to an origin

-which we call ``here'': our location in a

-particular part of space round which we

-group the whole Universe is the essential fact

-of our bodily existence. We can imagine

-beings who observe all phenomena in all space

-with an equal eye, unbiassed in favour of any

-part. With us it is otherwise, a cat at our

-\PageSep{126}

-feet claims more attention than an earthquake

-at Cape Horn, or than the destruction

-of a world in the Milky Way. It is true that

-in making a common stock of our knowledge

-with our fellowmen, we have to waive something

-of the strict egoism of our own individual

-``here.'' We substitute ``nearly

-here'' for ``here''; thus we measure miles

-from the town hall of the nearest town, or

-from the capital of the country. In measuring

-the earth, men of science will put the

-origin at the earth's centre; astronomers

-\index{Origin}%

-even rise to the extreme altruism of putting

-their origin inside the sun. But, far as this

-last origin may be, and even if we go further

-to some convenient point amid the nearer

-fixed stars, yet, compared to the immeasurable

-infinities of space, it remains true that

-our first procedure in exploring the Universe

-is to fix upon an origin ``nearly here.''

-

-Again the relation of the coordinates $OM$

-and~$MP$ (\ie\ $x$~and~$y$) to the vector~$OP$ is an

-instance of the famous parallelogram law, as

-\index{Parallelogram Law}%

-can easily be seen (\cf\ \Fig[fig.]{8}) by completing

-the parallelogram~$OMPN$. The idea of the

-``vector''~$OP$, that is, of a directed magnitude,

-is the root-idea of physical science.

-Any moving body has a certain magnitude

-of velocity in a certain direction, that is to

-say, its velocity is a directed magnitude, a

-vector. Again a force has a certain magnitude

-\PageSep{127}

-and has a definite direction. Thus,

-when in analytical geometry the ideas of the

-``origin,'' of ``coordinates,'' and of ``vectors''

-are introduced, we are studying the

-abstract conceptions which correspond to the

-fundamental facts of the physical world.

-\PageSep{128}

-

-

-\Chapter{X}{Conic Sections}

-

-\First{When} the Greek geometers had exhausted,

-\index{Conic Sections|EtSeq}%

-as they thought, the more obvious and interesting

-properties of figures made up of

-straight lines and circles, they turned to

-the study of other curves; and, with their

-almost infallible instinct for hitting upon

-things worth thinking about, they chiefly

-devoted themselves to conic sections, that

-is, to the curves in which planes would cut

-the surfaces of circular cones. The man

-who must have the credit of inventing the

-study is Menaechmus (born 375~\BC\ and

-\index{Menaechmus}%

-died 325~\BC); he was a pupil of Plato

-and one of the tutors of Alexander the

-Great. Alexander, by the by, is a conspicuous

-\index{Alexander the Great}%

-example of the advantages of good

-tuition, for another of his tutors was the

-philosopher Aristotle. We may suspect that

-\index{Aristotle}%

-Alexander found Menaechmus rather a dull

-teacher, for it is related that he asked for the

-\PageSep{129}

-proofs to be made shorter. It was to this

-request that Menaechmus replied: ``In the

-\index{Menaechmus}%

-country there are private and even royal

-roads, but in geometry there is only one road

-for all.'' This reply no doubt was true

-enough in the sense in which it would have

-been immediately understood by Alexander.

-But if Menaechmus thought that his proofs

-could not be shortened, he was grievously

-mistaken; and most modern mathematicians

-would be horribly bored, if they were compelled

-to study the Greek proofs of the properties

-of conic sections. Nothing illustrates

-better the gain in power which is obtained by

-the introduction of relevant ideas into a

-science than to observe the progressive

-shortening of proofs which accompanies the

-growth of richness in idea. There is a certain

-type of mathematician who is always

-rather impatient at delaying over the ideas

-of a subject: he is anxious at once to get on

-to the proofs of ``important'' problems. The

-history of the science is entirely against him.

-There are royal roads in science; but those

-who first tread them are men of genius and

-\index{Alexander the Great}%

-not kings.

-

-The way in which conic sections first presented

-themselves to mathematicians was as

-follows: think of a cone (\cf\ \Fig[fig.]{15}), whose

-vertex (or point) is~$V$, standing on a circular

-base~$STU$. For example, a conical shade to

-\PageSep{130}

-an electric light is often an example of such a

-surface. Now let the ``generating'' lines

-which pass through~$V$ and lie on the surface

-be all produced backwards; the result is a

-double cone, and $PQR$~is another circular cross

-section on the opposite side of~$V$ to the cross

-section~$STU$. The axis of the cone~$CVC'$

-passes through all the centres of these circles

-and is perpendicular to their planes, which

-are parallel to each other. In the diagram

-the parts of the curves which are supposed

-to lie behind the plane of the paper are dotted

-lines, and the parts on the plane or in front

-of it are continuous lines. Now suppose this

-double cone is cut by a plane not perpendicular

-to the axis~$CVC'$, or at least not

-necessarily perpendicular to it. Then three

-cases can arise:---

-

-(1) The plane may cut the cone in a closed

-\index{Ellipse|EtSeq}%

-oval curve, such as~$ABA'B'$ which lies entirely

-on one of the two half-cones. In this

-case the plane will not meet the other half-cone

-at all. Such a curve is called an ellipse; it is

-an oval curve. A particular case of such a

-section of the cone is when the plane is perpendicular

-to the axis~$CVC'$, then the section,

-such as $STU$ or $PQR$, is a circle. Hence a

-\index{Circle}%

-circle is a particular case of the ellipse.

-

-(2) The plane may be parallelled to a tangent

-plane touching the cone along one of its ``generating''

-lines as for example the plane of the

-\PageSep{131}

-\index{Parabola|EtSeq}%

-curve $D_{1}A_{1}D_{1}'$ in the diagram is parallel to

-the tangent plane touching the cone along the

-generating line~$VS$; the curve is still confined

-to one of the half-cones, but it is now not a

-closed oval curve, it goes on endlessly as long

-as the generating lines of the half-cone are

-produced away from the vertex. Such a

-conic section is called parabola.

-

-(3) The plane may cut both the half-cones,

-\index{Hyperbola|EtSeq}%

-so that the complete curve consists of two

-detached portions, or ``branches'' as they

-are called, this case is illustrated by the two

-branches $G_{2}A_{2}G_{2}'$ and $L_{2}A_{2}'L_{2}'$ which together

-make up the curve. Neither branch is closed,

-each of them spreading out endlessly as the

-two half-cones are prolonged away from the

-vertex. Such a conic section is called a

-hyperbola.

-

-There are accordingly three types of conic

-sections, namely, ellipses, parabolas, and

-hyperbolas. It is easy to see that, in a sense,

-parabolas are limiting cases lying between

-ellipses and hyperbolas. They form a more

-special sort and have to satisfy a more particular

-condition. These three names are

-apparently due to Apollonius of Perga (born

-\index{Apollonius of Perga}%

-about 260~\BC, and died about 200~\BC), who

-wrote a systematic treatise on conic sections

-which remained the standard work till the

-sixteenth century.

-%[** TN: Moved to top of paragraph]

-\Figure{15}

-

-It must at once be apparent how awkward

-\PageSep{132}

-and difficult the investigation of the properties

-of these curves must have been to the

-Greek geometers. The curves are plane

-curves, and yet their investigation involves

-the drawing in perspective of a solid figure.

-Thus in the diagram given above we have

-practically drawn no subsidiary lines and yet

-the figure is sufficiently complicated. The

-\PageSep{133}

-curves are plane curves, and it seems obvious

-that we should be able to define them without

-\Figure{16}

-going beyond the plane into a solid figure.

-At the same time, just as in the ``solid''

-\Figure[2.5in]{17}

-definition there is one uniform method of

-definition---namely, the section of a cone by

-\PageSep{134}

-a plane---which yields three cases, so in any

-``plane'' definition there also should be one

-uniform method of procedure which falls into

-three cases. Their shapes when drawn on

-their planes are those of the curved lines in

-the three figures \FigNum{16},~\FigNum{17}, and~\FigNum{18}. The

-points $A$~and~$A'$ in the figures are called

-%[** TN: Labels A, A', M, and line PM added to match the text]

-\Figure[4in]{18}

-the vertices and the line~$AA'$ the major axis.

-It will be noted that a parabola (\cf\ \Fig[fig.]{17})

-\index{Apollonius of Perga}%

-\index{Vertex}%

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

-remains constant both for the ellipse and the

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

-\PageSep{135}

-of $PM^{2}$ to~$AM$ is constant for the parabola

-of \Fig[fig.]{17}; and he bases most of his work

-on this fact. We are evidently advancing

-towards the desired uniform definition which

-does not go out of the plane; but have not

-yet quite attained to uniformity.

-

-In the diagrams \FigNum{16} and~\FigNum{18}, two points, $S$

-and~$S'$, will be seen marked, and in \Fig[diagram]{17}

-one point,~$S$. These are the \emph{foci} of the curves,

-and are points of the greatest importance.

-Apollonius knew that for an ellipse the sum

-of $SP$ and~$S'P$ (\ie\ $SP + S'P$) is constant as

-$P$~moves on the curve, and is equal to~$AA'$.

-Similarly for a hyperbola the difference $S'P - SP$

-is constant, and equal to~$AA'$ when $P$~is

-on one branch, and the difference $SP' - S'P'$

-is constant and equal to~$AA'$ when $P'$~is on

-the other branch. But no corresponding

-point seemed to exist for the parabola.

-

-Finally $500$~years later the last great Greek

-geometer, Pappus of Alexandria, discovered

-\index{Pappus}%

-the final secret which completed this line of

-thought. In the diagrams \FigNum{16} and~\FigNum{18} will be

-seen two lines, $XN$~and~$X'N'$, and in \Fig[diagram]{17}

-the single line,~$XN$. These are the directrices

-of the curves, two each for the ellipse

-and the hyperbola, and one for the parabola.

-Each directrix corresponds to its nearer focus.

-\index{Directrix}%

-\index{Focus}%

-The characteristic property of a focus,~$S$, and

-its corresponding directrix,~$XN$, for any one

-of the three types of curve, is that the ratio

-\PageSep{136}

-$SP$ to~$PN$ $\left(\ie\ \dfrac{SP}{PN}\right)$ is constant, where $PN$~is

-the perpendicular on the directrix from~$P$,

-and $P$~is any point on the curve. Here we

-have finally found the desired property of the

-curves which does not require us to leave

-the plane, and is stated uniformly for all

-three curves. For ellipses the ratio\footnote

-  {\Chg{Cf.}{\Cf}\ Note~B, \Pageref{noteB}.\Pagelabel{136}}

-is less

-than~$1$, for parabolas it is equal to~$1$, and for

-hyperbolas it is greater than~$1$.

-

-When Pappus had finished his investigations,

-\index{Pappus}%

-he must have felt that, apart from

-minor extensions, the subject was practically

-exhausted; and if he could have foreseen

-the history of science for more than a thousand

-years, it would have confirmed his belief.

-Yet in truth the really fruitful ideas in connection

-with this branch of mathematics had

-not yet been even touched on, and no one

-had guessed their supremely important applications

-in nature. No more impressive

-warning can be given to those who would

-confine knowledge and research to what is

-apparently useful, than the reflection that

-conic sections were studied for eighteen hundred

-years merely as an abstract science,

-without a thought of any utility other than

-to satisfy the craving for knowledge on the

-part of mathematicians, and that then at the

-end of this long period of abstract study, they

-\PageSep{137}

-were found to be the necessary key with

-which to attain the knowledge of one of the

-most important laws of nature.

-

-Meanwhile the entirely distinct study of

-astronomy had been going forward. The

-\index{Astronomy}%

-great Greek astronomer Ptolemy (died 168~\AD)

-\index{Ptolemy}%

-published his standard treatise on the

-subject in the University of Alexandria, explaining

-the apparent motions among the

-fixed stars of the sun and planets by the conception

-of the earth at rest and the sun and

-the planets circling round it. During the

-next thirteen hundred years the number and

-the accuracy of the astronomical observations

-increased, with the result that the description

-of the motions of the planets on

-Ptolemy's hypothesis had to be made more

-and more complicated. Copernicus (born

-\index{Copernicus}%

-1473~\AD\ and died 1543~\AD) pointed out

-that the motions of these heavenly bodies

-could be explained in a simpler manner if the

-sun were supposed to rest, and the earth and

-planets were conceived as moving round it.

-However, he still thought of these motions as

-essentially circular, though modified by a set

-of small corrections arbitrarily superimposed

-on the primary circular motions. So the

-matter stood when Kepler was born at Stuttgart

-\index{Kepler}%

-in Germany in 1571~\AD. There were

-two sciences, that of the geometry of conic

-sections and that of astronomy, both of which

-\PageSep{138}

-had been studied from a remote antiquity

-without a suspicion of any connection between

-the two. Kepler was an astronomer,

-\index{Kepler}%

-but he was also an able geometer, and on the

-subject of conic sections had arrived at ideas

-in advance of his time\Add{.} He is only one of

-many examples of the falsity of the idea that

-success in scientific research demands an exclusive

-absorption in one narrow line of study.

-Novel ideas are more apt to spring from

-an unusual assortment of knowledge---not

-necessarily from vast knowledge, but from a

-thorough conception of the methods and ideas

-of distinct lines of thought. It will be remembered

-that Charles Darwin was helped

-\index{Darwin}%

-to arrive at his conception of the law of

-evolution by reading Malthus' famous \Title{Essay

-\index{Malthus}%

-on Population}, a work dealing with a different

-subject---at least, as it was then

-thought.

-

-Kepler enunciated three laws of planetary

-\index{Kepler's Laws}%

-motion, the first two in~1609, and the third

-ten years later. They are as follows:

-

-(1) The orbits of the planets are ellipses,

-the sun being in the focus.

-

-(2) As a planet moves in its orbit, the

-radius vector from the sun to the planet

-sweeps out equal areas in equal times.

-

-(3) The squares of the periodic times of the

-several planets are proportional to the cubes

-of their major axes.

-\PageSep{139}

-

-These laws proved to be only a stage towards

-a more fundamental development of

-ideas. Newton (born 1642~\AD\ and died

-\index{Newton}%

-1727~\AD) conceived the idea of universal

-gravitation, namely, that any two pieces of

-\index{Gravitation}%

-matter attract each other with a force proportional

-to the product of their masses and

-inversely proportional to the square of their

-distance from each other. This sweeping

-general law, coupled with the three laws of

-motion which he put into their final general

-shape, proved adequate to explain all astronomical

-phenomena, including Kepler's laws,

-and has formed the basis of modern physics.

-Among other things he proved that comets

-might move in very elongated ellipses, or in

-parabolas, or in hyperbolas, which are nearly

-parabolas. The comets which return---such

-as Halley's comet---must, of course, move in

-\index{Halley}%

-ellipses. But the essential step in the proof of

-the law of gravitation, and even in the suggestion

-of its initial conception, was the verification

-of Kepler's laws connecting the

-motions of the planets with the theory of

-conic sections.

-

-From the seventeenth century onwards the

-abstract theory of the curves has shared in

-the double renaissance of geometry due to

-the introduction of coordinate geometry and

-of projective geometry. In projective geometry

-\index{Projective Geometry}%

-the fundamental ideas cluster round

-\PageSep{140}

-the consideration of sets (or pencils, as they

-\index{Pencils}%

-are called) of lines passing through a common

-point (the vertex of the ``pencil''). Now

-(\cf\ \Fig[fig.]{19}) if $A$,~$B$, $C$,~$D$, be any four fixed

-points on a conic section and $P$~be a variable

-point on the curve, the pencil of lines $PA$,

-\Figure[2.5in]{19}

-$PB$, $PC$, and~$PD$, has a special property,

-known as the constancy of its cross ratio. It

-\index{Cross Ratio}%

-will suffice here to say that cross ratio is a

-fundamental idea in projective geometry.

-For projective geometry this is really the definition

-of the curves, or some analogous property

-which is really equivalent to it. It

-\PageSep{141}

-will be seen how far in the course of ages of

-study we have drifted away from the old

-original idea of the sections of a circular cone.

-We know now that the Greeks had got hold

-of a minor property of comparatively slight

-importance; though by some divine good

-fortune the curves themselves deserved all

-the attention which was paid to them. This

-unimportance of the ``section'' idea is now

-marked in ordinary mathematical phraseology

-by dropping the word from their

-names. As often as not, they are now

-named merely ``conics'' instead of ``conic

-sections.''

-

-Finally, we come back to the point at

-\index{Locus}%

-which we left coordinate geometry in the last

-chapter. We had asked what was the type

-of \emph{loci} corresponding to the general algebraic

-form $ax + by = c$, and had found that it was

-the class of straight lines in the plane. We

-had seen that every straight line possesses an

-equation of this form, and that every equation

-of this form corresponds to a straight line.

-We now wish to go on to the next general

-type of algebraic forms. This is evidently

-to be obtained by introducing terms involving

-$x^{2}$~and $xy$ and~$y^{2}$. Thus the new general

-form must be written\Add{:}---

-\[

-ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0\Add{.}

-\]

-What does this represent? The answer is

-\PageSep{142}

-that (when it represents any locus) it always represents

-a conic section, and, furthermore,

-that the equation of every conic section can

-always be put into this shape. The discrimination

-of the particular sorts of conics as given

-by this form of equation is very easy. It entirely

-depends upon the consideration of $ab - h^{2}$,

-where $a$,~$b$, and~$h$, are the ``constants'' as

-written above. If $ab - h^{2}$ is a positive number,

-the curve is an ellipse; if $ab - h^{2} = 0$, the curve

-is a parabola: and if $ab - h^{2}$ is a negative

-number, the curve is a hyperbola.

-

-For example, put $a = b = 1$, $h = g = f = 0$,

-$c = -4$. 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

-positive. Hence the circle is a particular

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

-the equation of any circle can be

-put into the form $a(x^{2} + y^{2}) + 2gx + 2fy + c = 0$.

-Hence $ab - h^{2}$ becomes $a^{2} - 0$, that is,~$a^{2}$,

-which is necessarily positive. Accordingly

-all circles satisfy the condition for ellipses.

-The general form of the equation of a parabola

-is

-\[

-(dx + ey)^{2} + 2gx + 2fy + c = 0,

-\]

-so that the terms of the second degree, as

-\PageSep{143}

-they are called, can be written as a perfect

-square. For squaring out, we get

-\[

-d^{2} x^{2} + 2dexy + e^{2} y^{2} + 2gx + 2fy + c;

-\]

-so that by comparison $a = d^{2}$, $h = de$, $b = e^{2}$,

-and therefore $ab - h^{2} = d^{2} e^{2} - (de)^{2} = 0$. Hence

-the necessary condition is automatically satisfied.

-The equation $2xy - 4 = 0$, where $a = b = g = f = 0$,

-$h = 1$, $c = -4$, represents a hyperbola.

-For the condition $ab - h^{2}$ becomes

-$0 - 1^{2}$, that is,~$-1$, which is negative.

-

-{\Loosen The limitation, introduced by saying that,

-\index{Circular Cylinder}%

-\emph{when the general equation represents any locus},

-it represents a conic section, is necessary, because

-some particular cases of the general

-equation represent no real locus. For example

-$x^{2} + y^{2} + 1 = 0$ can be satisfied by no

-real values of $x$~and~$y$. It is usual to say that

-the locus is now one composed of imaginary

-points. But this idea of imaginary points in

-geometry is really one of great complexity,

-which we will not now enter into.}

-

-Some exceptional cases are included in the

-general form of the equation which may not

-be immediately recognized as conic sections.

-By properly choosing the constants the equation

-can be made to represent two straight

-lines. Now two intersecting straight lines

-may fairly be said to come under the Greek

-idea of a conic section. For, by referring to

-\PageSep{144}

-the picture of the double cone above, it will

-be seen that some planes through the vertex,~$V$,

-will cut the cone in a pair of straight lines

-intersecting at~$V$. The case of two parallel

-straight lines can be included by considering

-a circular cylinder as a particular case of a

-cone. Then a plane, which cuts it and is

-parallel to its axis, will cut it in two parallel

-straight lines. Anyhow, whether or no the

-%[** TN: [sic] "Greek", not "Greeks"]

-ancient Greek would have allowed these

-special cases to be called conic sections, they

-are certainly included among the curves represented

-by the general algebraic form of

-the second degree. This fact is worth noting;

-for it is characteristic of modern mathematics

-to include among general forms all sorts of

-particular cases which would formerly have

-received special treatment. This is due to

-its pursuit of generality.

-\PageSep{145}

-

-

-\Chapter{XI}{Functions}

-

-\First{The} mathematical use of the term function

-%[** TN: Index entry reads "p. 144" in the original]

-\index{Function|EtSeq}%

-has been adopted also in common life. For

-example, ``His temper is a function of his

-digestion,'' uses the term exactly in this

-mathematical sense. It means that a rule

-can be assigned which will tell you what his

-temper will be when you know how his

-digestion is working. Thus the idea of a

-``function'' is simple enough, we only have

-to see how it is applied in mathematics to

-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

-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

-($y$~years) is given by $y = x + 1$; and $y$~is a

-function of~$x$, namely, is the function~$x + 1$.

-

-In these examples $t$ and~$x$ are called the

-\PageSep{146}

-\index{Argument of a Function}%

-\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 $s$

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

-$20 × t$ and $x + 1$ respectively.

-

-Coming now to the general case, we can

-define a function in mathematics as a correlation

-between two variable numbers, called

-respectively the argument and the value of

-the function, such that whatever value be

-assigned to the ``argument of the function''

-the ``value of the function'' is definitely

-(\ie~uniquely) determined. The converse

-is not necessarily true, namely, that when

-the value of the function is determined

-the argument is also uniquely determined.

-Other functions of the argument~$x$ are $y = x^{2}$,

-%[** TN: log, sin italicized throughout in the original]

-$y = 2x^{2} + 3x + 1$, $y = x$, $y = \log x$, $y = \sin x$. The

-last two functions of this group will be

-readily recognizable by those who understand

-a little algebra and trigonometry. It is not

-worth while to delay now for their explanation,

-as they are merely quoted for the sake

-of example.

-

-Up to this point, though we have defined

-what we mean by a function in general, we

-have only mentioned a series of special functions.

-But mathematics, true to its general

-methods of procedure, symbolizes the general

-idea of any function. It does this by writing

-\PageSep{147}

-\index{Variable Function}%

-$F(x)$, $f(x)$, $g(x)$, $\phi(x)$,~etc., for any function of~$x$,

-where the argument~$x$ is placed in a bracket,

-and some letter like $F$,~$f$, $g$, $\phi$,~etc., is prefixed

-to the bracket to stand for the function.

-This notation has its defects. Thus it obviously

-clashes with the convention that the

-single letters are to represent variable numbers;

-since here $F$,~$f$, $g$, $\phi$,~etc., prefixed to a

-bracket stand for variable functions. It

-would be easy to give examples in which we

-can only trust to common sense and the context

-to see what is meant. One way of

-evading the confusion is by using Greek

-letters (\eg~$\phi$ as above) for functions; another

-way is to keep to $f$~and~$F$ (the initial

-letter of function) for the functional letter,

-and, if other variable functions have to be

-symbolized, to take an adjacent letter like~$g$.

-

-With these explanations and cautions, we

-write $y = f(x)$, to denote that $y$~is the value of

-some undetermined function of the argument~$x$;

-where $f(x)$ may stand for anything such

-as $x + 1$, $x^{2} - 2x + 1$, $\sin x$, $\log x$, or merely for

-$x$~itself. The essential point is that when $x$~is

-given, then $y$~is thereby definitely determined.

-It is important to be quite clear as

-to the generality of this idea. Thus in $y = f(x)$,

-we may determine, if we choose, $f(x)$~to

-mean that when $x$~is an integer, $f(x)$~is zero,

-and when $x$~has any other value, $f(x)$~is~$1$.

-Accordingly, putting $y = f(x)$, with this choice

-\PageSep{148}

-for the meaning of~$f$, $y$~is either $0$ or~$1$ according

-as the value of~$x$ is integral or otherwise.

-Thus $f(1) = 0$, $f(2) = 0$, $f(\frac{2}{3}) = 1$, $f(\sqrt{2}) = 1$, and

-so on. This choice for the meaning of~$f(x)$

-gives a perfectly good function of the argument~$x$

-according to the general definition of

-a function.

-

-A function, which after all is only a sort

-\index{Graphs|EtSeq}%

-of correlation between two variables, is represented

-like other correlations by a graph,

-that is in effect by the methods of coordinate

-geometry. For example, \Fig[fig.]{2} in \ChapRef{II}.\

-is the graph of the function~$\dfrac{1}{v}$ where $v$~is the

-argument and $p$~the value of the function.

-In this case the graph is only drawn for

-positive values of~$v$, which are the only values

-possessing any meaning for the physical application

-considered in that chapter. Again

-in \Fig[fig.]{14} of \ChapRef{IX}.\ the whole length of

-the line~$AB$, unlimited in both directions, is

-the graph of the function~$x + 1$, where $x$~is the

-argument and $y$~is the value of the function;

-and in the same figure the unlimited line~$A_{1}B$

-is the graph of the function~$1 - x$, and

-the line~$LOL'$ is the graph of the function~$x$,

-$x$~being the argument and $y$~the value of the

-function.

-

-These functions, which are expressed by

-simple algebraic formulæ, are adapted for representation

-by graphs. But for some functions

-\PageSep{149}

-this representation would be very

-misleading without a detailed explanation, or

-might even be impossible. Thus, consider the

-function mentioned above, which has the value~$1$

-for all values of its argument~$x$, except

-those which are integral, \eg\ except for $x = 0$,

-$x = 1$, $x = 2$, etc., when it has the value~$0$.

-Its appearance on a graph would be that of

-the straight line~$ABA'$ drawn parallel to the

-\Figure{20}

-axis~$XOX'$ at a distance from it of $1$~unit of

-length. But the points, $B$,~$C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$,~etc.,

-corresponding to the values $0$,~$1$, $2$, $3$, $4$,~etc., of

-the argument~$x$, are to be omitted, and instead

-of them the points $O$,~$B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$,~etc.,

-on the axis~$OX$, are to be taken. It is easy

-to find functions for which the graphical representation

-is not only inconvenient but

-impossible. Functions which do not lend

-themselves to graphs are important in the

-\PageSep{150}

-higher mathematics, but we need not concern

-ourselves further about them here.

-

-The most important division between functions

-\index{Continuous Functions|EtSeq}%

-\index{Discontinuous Functions|EtSeq}%

-is that between continuous and discontinuous

-functions. A function is continuous

-when its value only alters gradually for

-gradual alterations of the argument, and is

-discontinuous when it can alter its value by

-sudden jumps. Thus the two functions $x + 1$

-and $1 - x$, whose graphs are depicted as

-straight lines in \Fig[fig.]{14} of \ChapRef{IX}., are continuous

-functions, and so is the function~$\dfrac{1}{v}$,

-depicted in \ChapRef{II}., if we only think of

-positive values of~$v$. But the function depicted

-in \Fig[fig.]{20} of this chapter is discontinuous

-since at the values $x = 1$, $x = 2$, etc., of its

-argument, its value gives sudden jumps.

-

-Let us think of some examples of functions

-presented to us in nature, so as to get into

-our heads the real bearing of continuity and

-discontinuity. Consider a train in its journey

-along a railway line, say from Euston Station,

-the terminus in London of the London and

-North-Western Railway. Along the line in

-order lie the stations of Bletchley and Rugby.

-Let $t$~be the number of hours which the train

-has been on its journey from Euston, and $s$~be

-the number of miles passed over. Then $s$~is

-a function of~$t$, \ie~is the variable value

-corresponding to the variable argument~$t$.

-\PageSep{151}

-If we know the circumstances of the train's

-run, we know~$s$ as soon as any special value

-of~$t$ is given. Now, miracles apart, we may

-confidently assume that $s$~is a continuous

-function of~$t$. It is impossible to allow for

-the contingency that we can trace the train

-continuously from Euston to Bletchley, and

-that then, without any intervening time, however

-short, it should appear at Rugby. The

-idea is too fantastic to enter into our calculation:

-it contemplates possibilities not to be

-found outside the \Title{Arabian Nights}; and even

-in those tales sheer discontinuity of motion

-hardly enters into the imagination, they do

-not dare to tax our credulity with anything

-more than very unusual speed. But unusual

-speed is no contradiction to the great law of

-continuity of motion which appears to hold

-in nature. Thus light moves at the rate of

-about $190,000$ miles per~second and comes to

-us from the sun in seven or eight minutes;

-but, in spite of this speed, its distance travelled

-is always a continuous function of the time.

-

-It is not quite so obvious to us that the

-velocity of a body is invariably a continuous

-function of the time. Consider the train at

-any time~$t$: it is moving with some definite

-velocity, say $v$~miles per~hour, where $v$~is

-zero when the train is at rest in a station and

-is negative when the train is backing. Now

-we readily allow that $v$~cannot change its

-\PageSep{152}

-value suddenly for a big, heavy train. The

-train certainly cannot be running at forty

-miles per hour from 11.45~a.m.\ up to noon,

-and then suddenly, without any lapse of time,

-commence running at $50$~miles per~hour. We

-at once admit that the change of velocity

-will be a gradual process. But how about

-sudden blows of adequate magnitude? Suppose

-two trains collide; or, to take smaller

-objects, suppose a man kicks a football. It

-certainly appears to our sense as though the

-football began suddenly to move. Thus, in

-the case of velocity our senses do not revolt

-at the idea of its being a discontinuous function

-of the time, as they did at the idea of the

-train being instantaneously transported from

-Bletchley to Rugby. As a matter of fact,

-if the laws of motion, with their conception

-of mass, are true, there is no such thing as

-discontinuous velocity in nature. Anything

-that appears to our senses as discontinuous

-change of velocity must, according to them,

-be considered to be a case of gradual change

-which is too quick to be perceptible to us.

-It would be rash, however, to rush into the

-generalization that no discontinuous functions

-are presented to us in nature. A man who,

-trusting that the mean height of the land

-above sea-level between London and Paris

-was a continuous function of the distance

-from London, walked at night on Shakespeare's

-\PageSep{153}

-Cliff by Dover in contemplation of

-the Milky Way, would be dead before he had

-had time to rearrange his ideas as to the

-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,

-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

-\PageSep{154}

-now let $x$ have any value, positive or negative.

-The graph of the function is exhibited in \Fig[fig.]{21}.

-Suppose $x$ to change continuously from

-a large negative value through a numerically

-decreasing set of negative values up to~$0$, and

-thence through the series of increasing positive

-values. Accordingly, if a moving point,~$M$,

-represents~$x$ on~$XOX'$, $M$~starts at the

-extreme left of the axis~$XOX'$ and successively

-moves through $M_{1}$,~$M_{2}$, $M_{3}$, $M_{4}$,~etc.

-The corresponding points on the function are

-$P_{1}$,~$P_{2}$, $P_{3}$, $P_{4}$,~etc. It is easy to see that

-there is a point of discontinuity at $x = 0$, \ie~at

-the origin~$O$. For the value of the function

-on the negative (left) side of the origin becomes

-endlessly great, but negative, and the

-function reappears on the positive (right)

-side as endlessly great but positive. Hence,

-however small we take the length~$M_{2} M_{3}$,

-there is a finite jump between the values of

-the function at $M_{2}$ and~$M_{3}$. Indeed, this case

-has the peculiarity that the smaller we take the

-length between $M_{2}$ and~$M_{3}$, so long as they

-enclose the origin, the bigger is the jump in

-value of the function between them. This

-graph brings out, what is also apparent in

-\Fig[fig.]{20} of this chapter, that for many functions

-the discontinuities only occur at isolated

-points, so that by restricting the values of the

-argument we obtain a continuous function for

-these remaining values. Thus it is evident

-\PageSep{155}

-from \Fig[fig.]{21} that in $y = \dfrac{1}{x}$, if we keep to positive

-values only and exclude the origin, we obtain

-a continuous function. Similarly the same

-function, if we keep to negative values only,

-excluding the origin, is continuous. Again

-the function which is graphed in \Fig[fig.]{20} is continuous

-between $B$ and~$C_{1}$, and between $C_{1}$

-and~$C_{2}$, and between $C_{2}$ and $C_{3}$, and so on,

-always in each case excluding the end points.

-It is, however, easy to find functions such that

-their discontinuities occur at all points. For

-example, consider a function~$f(x)$, such that

-when $x$~is any fractional number $f(x) = 1$, and

-when $x$~is any incommensurable number

-$f(x) = 2$. This function is discontinuous at all

-points.

-

-Finally, we will look a little more closely

-at the definition of continuity given above.

-We have said that a function is continuous

-when its value only alters gradually for

-gradual alterations of the argument, and is

-discontinuous when it can alter its value by

-sudden jumps. This is exactly the sort of

-definition which satisfied our mathematical

-forefathers and no longer satisfies modern

-mathematicians. It is worth while to spend

-some time over it; for when we understand

-the modern objections to it, we shall have

-gone a long way towards the understanding

-of the spirit of modern mathematics. The

-\PageSep{156}

-whole difference between the older and the

-newer mathematics lies in the fact that vague

-half-metaphorical terms like ``gradually''

-are no longer tolerated in its exact statements.

-Modern mathematics will only admit statements

-and definitions and arguments which

-exclusively employ the few simple ideas about

-number and magnitude and variables on

-which the science is founded. Of two numbers

-one can be greater or less than the

-other; and one can be such and such a multiple

-of the other; but there is no relation of

-``graduality'' between two numbers, and

-hence the term is inadmissible. Now this

-may seem at first sight to be great pedantry.

-To this charge there are two answers. In

-the first place, during the first half of the

-nineteenth century it was found by some

-great mathematicians, especially Abel in

-\index{Abel}%

-Sweden, and Weierstrass in Germany, that

-\index{Weierstrass}%

-large parts of mathematics as enunciated in

-the old happy-go-lucky manner were simply

-wrong. Macaulay in his essay on Bacon

-\index{Bacon}%

-\index{Macaulay}%

-contrasts the certainty of mathematics with

-the uncertainty of philosophy; and by way

-of a rhetorical example he says, ``There has

-been no reaction against Taylor's theorem.''

-\index{Taylor's Theorem}%

-He could not have chosen a worse example.

-For, without having made an examination of

-English text-books on mathematics contemporary

-with the publication of this essay, the

-\PageSep{157}

-\index{Taylor's Theorem}%

-assumption is a fairly safe one that Taylor's

-theorem was enunciated and proved wrongly

-in every one of them. Accordingly, the

-anxious precision of modern mathematics is

-necessary for accuracy. In the second place

-it is necessary for research. It makes for

-clearness of thought, and thence for boldness

-of thought and for fertility in trying new

-combinations of ideas. When the initial

-statements are vague and slipshod, at every

-subsequent stage of thought common sense

-has to step in to limit applications and to

-explain meanings. Now in creative thought

-common sense is a bad master. Its sole

-criterion for judgment is that the new ideas

-shall look like the old ones. In other words

-it can only act by suppressing originality.

-

-In working our way towards the precise

-definition of continuity (as applied to functions)

-let us consider more closely the statement

-that there is no relation of ``graduality''

-between numbers. It may be asked, Cannot

-one number be only slightly greater than

-another number, or in other words, cannot

-the difference between the two numbers be

-small? The whole point is that in the abstract,

-apart from some arbitrarily assumed

-application, there is no such thing as a great

-or a small number. A million miles is a

-small number of miles for an astronomer

-investigating the fixed stars, but a million

-\PageSep{158}

-pounds is a large yearly income. Again, one-quarter

-is a large fraction of one's income to

-give away in charity, but is a small fraction

-of it to retain for private use. Examples can

-be accumulated indefinitely to show that

-great or small in any absolute sense have no

-abstract application to numbers. We can

-say of two numbers that one is greater or

-smaller than another, but not without specification

-of particular circumstances that any

-one number is great or small. Our task

-therefore is to define continuity without any

-mention of a ``small'' or ``gradual'' change

-in value of the function.

-

-In order to do this we will give names to

-some ideas, which will also be useful when

-we come to consider limits and the differential

-calculus.

-

-An ``interval'' of values of the argument~$x$

-\index{Interval|EtSeq}%

-of a function~$f(x)$ is all the values lying

-between some two values of the argument.

-For example, the interval between $x = 1$ and

-$x = 2$ consists of all the values which~$x$ can

-take lying between $1$ and~$2$, \ie\ it consists of

-all the real numbers between $1$ and~$2$. But

-the bounding numbers of an interval need

-not be integers. An interval of values of the

-argument \emph{contains} a number~$a$, when $a$~is a

-member of the interval. For example, the

-interval between $1$ and~$2$ contains $\frac{3}{2}$, $\frac{5}{3}$, $\frac{7}{4}$, and

-so on.

-\PageSep{159}

-

-A set of numbers approximates to a number~$a$

-\index{Standard of Approximation|EtSeq}%

-within a \emph{standard}~$k$, when the numerical

-difference between $a$ and every number of the

-set is less than~$k$. Here $k$~is the ``standard

-of approximation.'' Thus the set of numbers

-$3$,~$4$, $6$,~$8$, approximates to the number~$5$

-within the standard~$4$. In this case the

-standard~$4$ is not the smallest which could

-have been chosen, the set also approximates

-%[** TN: Original uses center dot for decimal point]

-to~$5$ within any of the standards $3.1$ or $3.01$

-or~$3.001$. Again, the numbers, $3.1$, $3.141$,

-$3.1415$, $3.14159$ approximate to $3.13102$ within

-the standard~$.032$, and also within the

-smaller standard~$.03103$.

-

-These two ideas of an interval and of

-\index{Neighbourhood|EtSeq}%

-approximation to a number within a standard

-are easy enough; their only difficulty is that

-they look rather trivial. But when combined

-with the next idea, that of the ``neighbourhood''

-of a number, they form the foundation

-of modern mathematical reasoning. What

-do we mean by saying that something is true

-for a function~$f(x)$ in the neighbourhood of

-the value~$a$ of the argument~$x$? It is this

-fundamental notion which we have now got to

-make precise.

-

-The values of a function~$f(x)$ are said to

-possess a characteristic in the ``neighbourhood

-of~$a$'' when some interval can be found,

-which (i)~contains the number~$a$ not as an

-end-point, and (ii)~is such that every value

-\PageSep{160}

-of the function for arguments, other than~$a$,

-lying within that interval possesses the characteristic.

-The value~$f(a)$ of the function for

-the argument~$a$ may or may not possess the

-characteristic. Nothing is decided on this

-point by statements about the \emph{neighbourhood}

-of~$a$.

-

-For example, suppose we take the particular

-function~$x^{2}$. Now \emph{in the neighbourhood of~$2$},

-the values of~$x^{2}$ are less than~$5$. For we can

-find an interval, \eg\ from $1$ to~$2.1$, which

-(i)~contains $2$ not as an end-point, and (ii)~is

-such that, for values of~$x$ lying within it, $x^{2}$~is

-less than~$5$.

-

-Now, combining the preceding ideas we

-know what is meant by saying that \emph{in the

-neighbourhood of~$a$} the function~$f(x)$ approximates

-to~$c$ within the \emph{standard}~$k$. It means

-that some interval can be found which (i)~includes

-$a$ not as an end-point, and (ii)~is such

-that all values of~$f(x)$, where $x$~lies in the interval

-and is not~$a$, differ from~$c$ by less than~$k$. For

-example, in the neighbourhood of~$2$, the function~$\sqrt{x}$

-approximates to~$1.41425$ within the

-standard~$.0001$. This is true because the

-square root of~$1.99996164$ is~$1.4142$ and the

-square root of~$2.00024449$ is~$1.4143$; hence

-for values of~$x$ lying in the interval

-$1.99996164$ to~$2.00024449$, which contains $2$

-not as an end-point, the values of the function~$\sqrt{x}$

-all lie between $1.4142$ and $1.4143$, and

-\PageSep{161}

-they therefore all differ from~$1.41425$ by less

-than~$.0001$. In this case we can, if we like,

-fix a smaller standard of approximation,

-namely $.000051$ or $.0000501$. Again, to take

-another example, in the neighbourhood of~$2$

-the function~$x^{2}$ approximates to~$4$ within the

-standard~$.5$. For $(1.9)^{2} = 3.61$ and $(2.1)^{2} = 4.41$,

-and thus the required interval $1.9$ to~$2.1$,

-containing $2$ not as an end-point, has

-been found. This example brings out the

-fact that statements about a function~$f(x)$ in

-the neighbourhood of a number~$a$ are distinct

-from statements about the value of~$f(x)$ when

-$x = a$. The production of an \emph{interval}, throughout

-which the statement is true, is required.

-Thus the mere fact that $2^{2} = 4$ does not by

-itself justify us in saying that in the \emph{neighbourhood}

-of~$2$ the function~$x^{2}$ is equal to~$4$.

-This statement would be untrue, because no

-interval can be produced with the required

-property. Also, the fact that $2^{2} = 4$ does not

-by itself justify us in saying that in the

-\emph{neighbourhood} of~$2$ the function~$x^{2}$ approximates

-to~$4$ within the standard~$.5$; although

-as a matter of fact, the statement has just

-been proved to be true.

-

-If we understand the preceding ideas, we

-understand the foundations of modern

-mathematics. We shall recur to analogous

-ideas in the chapter on Series, and again

-in the chapter on the Differential Calculus.

-\PageSep{162}

-\index{Continuous Functions@Continuous Functions (\emph{defined})}%

-Meanwhile, we are now prepared to define

-``continuous functions.'' A function~$f(x)$

-is ``continuous'' at a value~$a$ of its argument,

-when in the neighbourhood of~$a$

-its values approximate to~$f(a)$ (\ie~to its

-value at~$a$) within \emph{every} standard of approximation.

-

-This means that, whatever standard~$k$ be

-chosen, in the neighbourhood of~$a$ $f(x)$~approximates

-to~$f(a)$ within the standard~$k$.

-For example, $x^{2}$~is continuous at the value~$2$

-of its argument,~$x$, because however $k$~be

-chosen we can always find an interval, which

-(i)~contains $2$ not as an end-point, and (ii)~is

-such that the values of~$x^{2}$ for arguments lying

-within it approximate to~$4$ (\ie~$2^{2}$) within

-the standard~$k$. Thus, suppose we choose

-the standard~$.1$; now $(1.999)^{2} = 3.996001$,

-and $(2.01)^{2} = 4.0401$, and both these numbers

-differ from~$4$ by less than~$.1$. Hence, within

-the interval $1.999$ to $2.01$ the values of~$x^{2}$

-approximate to~$4$ within the standard~$.1$.

-Similarly an interval can be produced for any

-other standard which we like to try.

-

-Take the example of the railway train. Its

-velocity is continuous as it passes the signal

-box, if whatever velocity you like to assign

-(say one-millionth of a mile per hour) an interval

-of time can be found extending before

-and after the instant of passing, such that at

-all instants within it the train's velocity

-\PageSep{163}

-differs from that with which the train passed

-the box by less than one-millionth of a mile

-per hour; and the same is true whatever

-other velocity be mentioned in the place of

-one-millionth of a mile per hour.

-\PageSep{164}

-

-

-\Chapter{XII}{Periodicity in Nature}

-

-\First{The} whole life of Nature is dominated by

-\index{Periodicity|EtSeq}%

-the existence of periodic events, that is, by

-the existence of successive events so analogous

-to each other that, without any straining of

-language, they may be termed recurrences of

-the same event. The rotation of the earth

-produces the successive days. It is true that

-each day is different from the preceding days,

-however abstractly we define the meaning of

-a day, so as to exclude casual phenomena.

-But with a sufficiently abstract definition of

-a day, the distinction in properties between

-two days becomes faint and remote from

-practical interest; and each day may then

-be conceived as a recurrence of the phenomenon

-of one rotation of the earth. Again the

-path of the earth round the sun leads to the

-yearly recurrence of the seasons, and imposes

-another periodicity on all the operations of

-nature. Another less fundamental periodicity

-is provided by the phases of the moon.

-In modern civilized life, with its artificial light,

-these phases are of slight importance, but in

-\PageSep{165}

-ancient times, in climates where the days are

-burning and the skies clear, human life was

-apparently largely influenced by the existence of

-moonlight. Accordingly our divisions into

-weeks and months, with their religious associations,

-have spread over the European races from

-Syria and Mesopotamia, though independent

-observances following the moon's phases are

-found amongst most nations. It is, however,

-through the tides, and not through its phases

-of light and darkness, that the moon's periodicity

-has chiefly influenced the history of

-the earth.

-

-Our bodily life is essentially periodic.

-It is dominated by the beatings of the

-heart, and the recurrence of breathing.

-The presupposition of periodicity is indeed

-fundamental to our very conception of life.

-We cannot imagine a course of nature in

-which, as events progressed, we should be

-unable to say: ``This has happened before.''

-The whole conception of experience as a guide

-to conduct would be absent. Men would

-always find themselves in new situations

-possessing no substratum of identity with

-anything in past history. The very means of

-measuring time as a quantity would be absent.

-Events might still be recognized as occurring

-in a series, so that some were earlier and

-others later. But we now go beyond this

-bare recognition. We can not only say that

-\PageSep{166}

-\index{Time|EtSeq}%

-three events, $A$,~$B$,~$C$, occurred in this order,

-so that $A$~came before~$B$, and $B$~before~$C$;

-but also we can say that the length of time

-between the occurrences of $A$ and~$B$ was

-twice as long as that between $B$ and~$C$. Now,

-quantity of time is essentially dependent on

-observing the number of natural recurrences

-which have intervened. We may say

-that the length of time between $A$ and~$B$ was

-so many days, or so many months, or so

-many years, according to the type of recurrence

-to which we wish to appeal. Indeed,

-at the beginning of civilization, these three

-modes of measuring time were really distinct.

-It has been one of the first tasks of science

-among civilized or semi-civilized nations, to

-fuse them into one coherent measure. The

-full extent of this task must be grasped. It

-is necessary to determine, not merely what

-number of days (\eg~$365.25$\dots) go to some

-one year, but also previously to determine that

-the same number of days do go to the successive

-years. We can imagine a world in

-which periodicities exist, but such that no two

-are coherent. In some years there might be

-$200$~days and in others~$350$. The determination

-of the broad general consistency of the

-more important periodicities was the first step

-in natural science. This consistency arises

-from no abstract intuitive law of thought;

-it is merely an observed fact of nature

-\PageSep{167}

-guaranteed by experience. Indeed, so far is

-it from being a necessary law, that it is not

-even exactly true There are divergencies in

-every case. For some instances these divergencies

-are easily observed and are therefore

-immediately apparent. In other cases it requires

-the most refined observations and

-astronomical accuracy to make them apparent.

-Broadly speaking, all recurrences depending

-on living beings, such as the beatings

-of the heart, are subject in comparison with

-other recurrences to rapid variations. The

-great stable obvious recurrences---stable in

-the sense of mutually agreeing with great

-accuracy---are those depending on the motion

-of the earth as a whole, and on similar motions

-of the heavenly bodies.

-

-We therefore assume that these astronomical

-\index{Laws of Motion|EtSeq}%

-recurrences mark out equal intervals of

-time. But how are we to deal with their

-discrepancies which the refined observations

-of astronomy detect? Apparently we are

-reduced to the arbitrary assumption that one

-or other of these sets of phenomena marks out

-equal times---\eg\ that either all days are of

-equal length, or that all years are of equal

-length. This is not so: some assumptions

-must be made, but the assumption which

-underlies the whole procedure of the astronomers

-in determining the measure of time is

-that the laws of motion are exactly verified.

-\PageSep{168}

-Before explaining how this is done, it is interesting

-to observe that this relegation of

-the determination of the measure of time to

-the astronomers arises (as has been said) from

-the stable consistency of the recurrences with

-which they deal. If such a superior consistency

-had been noted among the recurrences

-characteristic of the human body, we

-should naturally have looked to the doctors

-of medicine for the regulation of our clocks.

-

-In considering how the laws of motion

-come into the matter, note that two inconsistent

-modes of measuring time will yield

-different variations of velocity to the same

-body. For example, suppose we define an

-hour as one twenty-fourth of a day, and take

-the case of a train running uniformly for two

-hours at the rate of twenty miles per hour.

-Now take a grossly inconsistent measure of

-time, and suppose that it makes the first hour

-to be twice as long as the second hour. Then,

-according to this other measure of duration,

-the time of the train's run is divided into

-two parts, during each of which it has traversed

-the same distance, namely, twenty

-miles; but the duration of the first part is

-twice as long as that of the second part.

-Hence the velocity of the train has not been

-uniform, and on the average the velocity

-during the second period is twice that during

-the first period. Thus the question as to

-\PageSep{169}

-whether the train has been running uniformly

-or not entirely depends on the standard of

-time which we adopt.

-

-Now, for all ordinary purposes of life on the

-earth, the various astronomical recurrences

-may be looked on as absolutely consistent;

-and, furthermore assuming their consistency,

-and thereby assuming the velocities and

-changes of velocities possessed by bodies, we

-find that the laws of motion, which have

-been considered above, are almost exactly

-verified. But only \emph{almost} exactly when we

-come to some of the astronomical phenomena.

-We find, however, that by assuming slightly

-different velocities for the rotations and

-motions of the planets and stars, the laws

-would be exactly verified. This assumption

-is then made; and we have, in fact thereby,

-adopted a measure of time, which is indeed

-defined by reference to the astronomical

-phenomena, but not so as to be consistent

-with the uniformity of any one of them. But

-the broad fact remains that the uniform flow

-of time on which so much is based, is itself

-dependent on the observation of periodic

-events.

-

-Even phenomena, which on the surface

-seem casual and exceptional, or, on the other

-hand, maintain themselves with a uniform

-persistency, may be due to the remote influence

-of periodicity. Take for example, the

-\PageSep{170}

-principle of resonance. Resonance arises

-\index{Resonance}%

-when two sets of connected circumstances

-have the same periodicities. It is a dynamical

-law that the small vibrations of all bodies

-when left to themselves take place in definite

-times characteristic of the body. Thus a

-pendulum with a small swing always vibrates

-in some definite time, characteristic of its shape

-and distribution of weight and length. A more

-complicated body may have many ways of

-vibrating; but each of its modes of vibration

-will have its own peculiar ``period.'' Those

-\index{Period}%

-periods of vibration of a body are called its

-``free'' periods. Thus a pendulum has but

-one period of vibration, while a suspension

-bridge will have many. We get a musical

-instrument, like a violin string, when the

-periods of vibration are all simple submultiples

-of the longest; \ie~if $t$~seconds be the longest

-period, the others are $\frac{1}{2}t$, $\frac{1}{3}t$, and so on, where

-any of these smaller periods may be absent.

-Now, suppose we excite the vibrations of a

-body by a cause which is itself periodic;

-then, if the period of the cause is very nearly

-that of one of the periods of the body, that

-mode of vibration of the body is very violently

-excited; even although the magnitude of the

-exciting cause is small. This phenomenon is

-called ``resonance.'' The general reason is

-easy to understand. Any one wanting to

-upset a rocking stone will push ``in tune''

-\PageSep{171}

-with the oscillations of the stone, so as always

-to secure a favourable moment for a push.

-If the pushes are out of tune, some increase

-the oscillations, but others check them. But

-when they are in tune, after a time all the

-pushes are favourable. The word ``resonance''

-\index{Resonance}%

-comes from considerations of sound:

-but the phenomenon extends far beyond the

-region of sound. The laws of absorption and

-emission of light depend on it, the ``tuning''

-of receivers for wireless telegraphy, the comparative

-importance of the influences of

-planets on each other's motion, the danger

-to a suspension bridge as troops march over

-it in step, and the excessive vibration of some

-ships under the rhythmical beat of their

-machinery at certain speeds. This coincidence

-of periodicities may produce steady

-phenomena when there is a constant association

-of the two periodic events, or it may

-produce violent and sudden outbursts when

-the association is fortuitous and temporary.

-

-Again, the characteristic and constant

-periods of vibration mentioned above are

-the underlying causes of what appear to

-us as steady excitements of our senses. We

-work for hours in a steady light, or we listen

-to a steady unvarying sound. But, if modern

-science be correct, this steadiness has no

-counterpart in nature. The steady light is

-due to the impact on the eye of a countless

-\PageSep{172}

-number of periodic waves in a vibrating ether,

-and the steady sound to similar waves in a

-vibrating air. It is not our purpose here to

-explain the theory of light or the theory of

-sound. We have said enough to make it

-evident that one of the first steps necessary

-to make mathematics a fit instrument for the

-investigation of Nature is that it should be

-able to express the essential periodicity of

-things. If we have grasped this, we can

-understand the importance of the mathematical

-conceptions which we have next to

-consider, namely, periodic functions.

-\PageSep{173}

-

-

-\Chapter{XIII}{Trigonometry}

-

-\First{Trigonometry} did not take its rise from

-\index{Trigonometry|EtSeq}%

-the general consideration of the periodicity of

-nature. In this respect its history is analogous

-to that of conic sections, which also had

-their origin in very particular ideas. Indeed,

-a comparison of the histories of the two

-sciences yields some very instructive analogies

-and contrasts. Trigonometry, like conic sections,

-had its origin among the Greeks. Its

-inventor was Hipparchus (born about 160~\BC),

-\index{Hipparchus}%

-a Greek astronomer, who made his

-observations at Rhodes. His services to

-astronomy were very great, and it left his

-\index{Astronomy}%

-hands a truly scientific subject with important

-results established, and the right method of

-progress indicated. Perhaps the invention

-of trigonometry was not the least of these

-services to the main science of his study. The

-next man who extended trigonometry was

-Ptolemy, the great Alexandrian astronomer,

-\index{Ptolemy}%

-whom we have already mentioned. We now

-\PageSep{174}

-see at once the great contrast between conic

-sections and trigonometry. The origin of

-trigonometry was practical; it was invented

-because it was necessary for astronomical research.

-The origin of conic sections was

-purely theoretical. The only reason for its

-initial study was the abstract interest of the

-ideas involved. Characteristically enough

-conic sections were invented about $150$~years

-earlier than trigonometry, during the very

-best period of Greek thought. But the importance

-of trigonometry, both to the theory

-and the application of mathematics, is only

-one of innumerable instances of the fruitful

-ideas which the general science has gained

-from its practical applications.

-

-We will try and make clear to ourselves

-what trigonometry is, and why it should be

-generated by the scientific study of astronomy.

-\index{Astronomy}%

-In the first place: What are the measurements

-which can be made by an astronomer?

-They are measurements of time and measurements

-of angles. The astronomer may adjust

-a telescope (for it is easier to discuss the

-familiar instrument of modern astronomers)

-so that it can only turn about a fixed axis

-pointing east and west; the result is that

-the telescope can only point to the south, with

-a greater or less elevation of direction, or, if

-turned round beyond the zenith, point to the

-north. This is the transit instrument, the

-\PageSep{175}

-great instrument for the exact measurement

-of the times at which stars are due south or

-due north. But indirectly this instrument

-measures angles. For when the time elapsed

-between the transits of two stars has been

-noted, by the assumption of the uniform

-rotation of the earth, we obtain the angle

-through which the earth has turned in that

-period of time. Again, by other instruments,

-the angle between two stars can be directly

-measured. For if $E$~is the eye of the astronomer,

-\Figure[2in]{22}

-and $EA$~and $EB$ are the directions in

-which the stars are seen, it is easy to devise

-instruments which shall measure the angle~$AEB$.

-Hence, when the astronomer is forming

-a survey of the heavens, he is, in fact,

-measuring angles so as to fix the relative

-directions of the stars and planets at any instant.

-Again, in the analogous problem of

-\PageSep{176}

-\index{Surveys|EtSeq}%

-\index{Triangle|EtSeq}%

-land-surveying, angles are the chief subject

-of measurements. The direct measurements

-of length are only rarely possible with any

-accuracy; rivers, houses, forests, mountains,

-and general irregularities of ground all get in

-the way. The survey of a whole country will

-depend only on one or two direct measurements

-of length, made with the greatest

-elaboration in selected places like Salisbury

-Plain. The main work of a survey is the

-measurement of angles. For example, $A$,~$B$,

-and~$C$ will be conspicuous points in the district

-\Figure[2in]{23}

-surveyed, say the tops of church towers.

-These points are visible each from the others.

-Then it is a very simple matter at~$A$ to

-measure the angle~$BAC$, and at~$B$ to measure

-the angle~$ABC$, and at~$C$ to measure the angle~$BCA$.

-Theoretically, it is only necessary to

-measure two of these angles; for, by a well-known

-proposition in geometry, the sum of

-the three angles of a triangle amounts to two

-\PageSep{177}

-right-angles, so that when two of the angles

-are known, the third can be deduced. It is

-better, however, in practice to measure all

-three, and then any small errors of observation

-can be checked. In the process of map-making

-a country is completely covered with

-triangles in this way. This process is called

-triangulation, and is the fundamental process

-\index{Triangulation}%

-in a survey.

-

-Now, when all the angles of a triangle are

-\index{Similarity|EtSeq}%

-known, the shape of the triangle is known---that

-is, the shape as distinguished from the

-size. We here come upon the great principle

-of geometrical similarity. The idea is very

-familiar to us in its practical applications.

-We are all familiar with the idea of a plan

-drawn to scale. Thus if the scale of a plan

-be an inch to a yard, a length of three inches

-in the plan means a length of three yards in

-the original. Also the shapes depicted in the

-plan are the shapes in the original, so that a

-right-angle in the original appears as a right-angle

-in the plan. Similarly in a map, which

-is only a plan of a country, the proportions

-of the lengths in the map are the proportions

-of the distances between the places indicated,

-and the directions in the map are the directions

-in the country. For example, if in the

-map one place is north-north-west of the

-other, so it is in reality; that is to say, in a

-map the angles are the same as in reality.

-\PageSep{178}

-\index{Scale of a Map}%

-Geometrical similarity may be defined thus:

-Two figures are similar (i)~if to any point

-in one figure a point in the other figure

-corresponds, so that to every line there is a

-corresponding line, and to every angle a

-corresponding angle, and (ii)~if the lengths

-of corresponding lines are in a fixed proportion,

-and the magnitudes of corresponding

-angles are the same. The fixed proportion

-of the lengths of corresponding lines in a map

-(or plan) and in the original is called the scale

-of the map. The scale should always be

-indicated on the margin of every map and

-plan. It has already been pointed out that

-two triangles whose angles are respectively

-equal are similar. Thus, if the two triangles

-\Figure{24}

-$ABC$ and~$DEF$ have the angles at $A$ and $D$

-equal, and those at $B$ and~$E$, and those at $C$

-and~$F$, then $DE$~is to~$AB$ in the same proportion

-\PageSep{179}

-as $EF$~is to~$BC$, and as $FD$~is to~$CA$.

-But it is not true of other figures that similarity

-is guaranteed by the mere equality of

-angles. Take for example, the familiar cases

-of a rectangle and a square. Let $ABCD$~be

-a square, and $ABEF$~be a rectangle. Then

-all the corresponding angles are equal. But

-\Figure[2.75in]{25}

-whereas the side~$AB$ of the square is equal to

-the side~$AB$ of the rectangle, the side~$BC$ of

-the square is about half the size of the side~$BE$

-of the rectangle. Hence it is not true

-that the square $ABCD$ is similar to the rectangle

-$ABEF$. This peculiar property of the

-triangle, which is not shared by other rectilinear

-figures, makes it the fundamental

-figure in the theory of similarity. Hence in

-surveys, triangulation is the fundamental

-process; and hence also arises the word ``trigonometry,''

-\PageSep{180}

-\index{Circle|EtSeq}%

-derived from the two Greek

-words \Foreign{trigonon} a triangle and \Foreign{metria} measurement.

-The fundamental question from which

-trigonometry arose is this: Given the magnitudes

-of the angles of a triangle, what can be

-stated as to the relative magnitudes of the

-sides. Note that we say ``\emph{relative} magnitudes

-of the sides,'' since by the theory of similarity

-it is only the proportions of the sides which

-are known. In order to answer this question,

-certain functions of the magnitudes of

-an angle, considered as the argument, are introduced.

-In their origin these functions

-were got at by considering a right-angled triangle,

-and the magnitude of the angle was

-defined by the length of the arc of a circle.

-In modern elementary books, the fundamental

-position of the arc of the circle as defining

-the magnitude of the angle has been

-pushed somewhat to the background, not to

-the advantage either of theory or clearness

-of explanation. It must first be noticed

-that, in relation to similarity, the circle holds

-the same fundamental position among curvilinear

-figures, as does the triangle among

-rectilinear figures. Any two circles are similar

-figures; they only differ in scale. The

-lengths of the circumferences of two circles,

-such as $APA'$ and $A_{1} P_{1} A_{1}'$ in the \Fig[fig.]{26} are

-in proportion to the lengths of their radii.

-Furthermore, if the two circles have the same

-\PageSep{181}

-centre~$O$, as do the two circles in \Fig[fig.]{26}, then

-the arcs $AP$ and $A_{1} P_{1}$ intercepted by the

-arms of any angle~$AOP$, are also in proportion

-to their radii. Hence the ratio of the

-\Figure{26}

-length of the arc~$AP$ to the length of the

-radius~$OP$, that is $\dfrac{\text{arc } AP}{\text{radius } OP}$ is a number which

-is quite independent of the length~$OP$, and is

-the same as the fraction $\dfrac{\text{arc } A_{1} P_{1}}{\text{radius } OP_{1}}$. This fraction

-of ``arc divided by radius'' is the proper

-theoretical way to measure the magnitude of

-\PageSep{182}

-\index{Cosine|EtSeq}%

-\index{Sine|EtSeq}%

-an angle; for it is dependent on no arbitrary

-unit of length, and on no arbitrary way of

-dividing up any arbitrarily assumed angle,

-such as a right-angle. Thus the fraction~$\dfrac{AP}{OA}$

-represents the magnitude of the angle~$AOP$.

-Now draw $PM$ perpendicularly to~$OA$. Then

-the Greek mathematicians called the line~$PM$

-the sine of the arc~$AP$, and the line~$OM$ the

-cosine of the arc~$AP$. They were well aware

-that the importance of the relations of these

-various lines to each other was dependent on

-the theory of similarity which we have just

-expounded. But they did not make their

-definitions express the properties which arise

-from this theory. Also they had not in their

-heads the modern general ideas respecting

-functions as correlating pairs of variable numbers,

-nor in fact were they aware of any

-modern conception of algebra and algebraic

-analysis. Accordingly, it was natural to

-them to think merely of the relations between

-certain lines in a diagram. For us the case

-is different: we wish to embody our more

-powerful ideas.

-

-Hence, in modern mathematics, instead

-of considering the arc~$AP$, we consider

-the fraction~$\dfrac{AP}{OP}$, which is a number the

-same for all lengths of~$OP$; and, instead of

-considering the lines $PM$ and~$OM$, we consider

-\PageSep{183}

-the fractions $\dfrac{PM}{OP}$ and~$\dfrac{OM}{OP}$, which again

-are numbers not dependent on the length of~$OP$,

-\ie~not dependent on the scale of our

-diagrams. Then we define the number $\dfrac{PM}{OP}$

-to be the \emph{sine} of the number $\dfrac{PA}{OP}$, and the

-number $\dfrac{OM}{OP}$ to be the \emph{cosine} of the number

-$\dfrac{PA}{OP}$. These fractional forms are clumsy to

-print; so let us put $u$ for the fraction~$\dfrac{AP}{OP}$,

-which represents the magnitude of the angle~$AOP$,

-and put $v$ for the fraction~$\dfrac{PM}{OM}$, and $w$~for

-the fraction~$\dfrac{OM}{OP}$. Then $u$,~$v$,~$w$, are numbers,

-and, since we are talking of \emph{any} angle~$AOP$,

-they are variable numbers. But a

-correlation exists between their magnitudes,

-so that when $u$ (\ie\ the angle~$AOP$) is given

-the magnitudes of $v$~and~$w$ are definitely determined.

-Hence $v$~and~$w$ are functions of the

-argument~$u$. We have called $v$ the \emph{sine} of~$u$,

-and $w$ the \emph{cosine} of~$u$. We wish to adapt

-the general functional notation $y = f(x)$ to

-these special cases: so in modern mathematics

-%[** TN: Function names italicized in the original]

-we write \Chg{$\sin$}{``$\sin$''} for~``$f$'' when we want to

-\PageSep{184}

-indicate the special function of ``sine,'' and

-``$\cos$'' for~``$f$'' when we want to indicate

-the special function of ``cosine.'' Thus, with

-the above meanings for $u$,~$v$,~$w$, we get

-\[

-v = \sin u,\quad\text{and}\quad

-w = \cos u,

-\]

-where the brackets surrounding the~$x$ in~$f(x)$

-are omitted for the special functions. The

-meaning of these functions $\sin$ and $\cos$ as

-correlating the pairs of numbers $u$~and~$v$, and

-$u$~and~$w$ is, that the functional relations are to

-be found by constructing (\cf\ \Fig[fig.]{26}) an angle~$AOP$,

-whose measure ``$AP$~divided by~$OP$''

-is equal to~$u$, and that then $v$~is the number

-given by ``$PM$~divided by~$OP$'' and $w$~is the

-number given by ``$OM$~divided by~$OP$.''

-

-It is evident that without some further definitions

-we shall get into difficulties when the

-number~$u$ is taken too large. For then the arc~$AP$

-may be greater than one-quarter of the

-circumference of the circle, and the point~$M$

-(\cf\ figs.\ \FigNum{26} and~\FigNum{27}) may fall between $O$ and~$A'$

-and not between $O$ and~$A$. Also $P$~may be

-below the line~$AOA'$ and not above it as in

-\Fig[fig.]{26}. In order to get over this difficulty

-we have recourse to the ideas and conventions

-of coordinate geometry in making our

-complete definitions of the sine and cosine.

-Let one arm~$OA$ of the angle be the axis~$OX$,

-and produce the axis backwards to

-obtain its negative part~$OX'$. Draw the

-\PageSep{185}

-other axis~$YOY'$ perpendicular to it. Let

-any point~$P$ at a distance~$r$ from~$O$ have

-coordinates $x$ and~$y$. These coordinates are

-both positive in the first ``quadrant'' of

-the plan, \eg\ the coordinates $x$ and~$y$ of~$P$

-\Figure{27}

-in \Fig[fig.]{27}. In the other quadrants, either

-one or both of the coordinates are negative,

-for example, $x'$~and~$y$ for~$P'$, and $x'$ and~$y'$

-for~$P''$, and $x$ and~$y'$ for~$P'''$ in \Fig[fig.]{27}, where

-$x'$ and~$y'$ are both negative numbers. The

-positive angle~$POA$ is the arc~$AP$ divided

-by~$r$, its sine is~$\dfrac{y}{r}$ and its cosine is~$\dfrac{x}{r}$; the positive

-\PageSep{186}

-angle~$AOP'$ is the arc~$ABP'$ divided by~$r$,

-its sine is~$\dfrac{y}{r}$ and cosine~$\dfrac{x'}{r}$; the positive angle~$AOP''$

-is the arc $ABA'P''$ divided by~$r$, its

-sine is~$\dfrac{y'}{r}$ and its cosine is~$\dfrac{x'}{r}$; the positive

-angle~$AOP'''$ is the arc $ABA'B'P'''$ divided

-by~$r$, its sine is~$\dfrac{y'}{r}$ and its cosine is~$\dfrac{x}{r}$.

-

-But even now we have not gone far enough.

-For suppose we choose~$u$ to be a number

-greater than the ratio of the whole circumference

-of the circle to its radius. Owing to

-the similarity of all circles this ratio is the

-same for all circles. It is always denoted in

-mathematics by the symbol~$2\pi$, where $\pi$~is

-the Greek form of the letter~\Foreign{p} and its

-name in the Greek alphabet is ``pi.'' It can

-be proved that $\pi$~is an incommensurable

-number, and that therefore its value cannot

-be expressed by any fraction, or by any

-terminating or recurring decimal. Its value

-to a few decimal places is~$3.14159$; for many

-purposes a sufficiently accurate approximate

-value is~$\dfrac{22}{7}$. Mathematicians can easily calculate~$\pi$

-to any degree of accuracy required,

-just as~$\sqrt{2}$ can be so calculated. Its value

-has been actually given to $707$~places of

-\PageSep{187}

-decimals. Such elaboration of calculation is

-merely a curiosity, and of no practical or

-theoretical interest. The accurate determination

-of~$\pi$ is one of the two parts of

-the famous problem of squaring the circle.

-\index{Squaring the Circle}%

-The other part of the problem is, by the

-theoretical methods of pure geometry to

-describe a straight line equal in length to the

-circumference. Both parts of the problem

-are now known to be impossible; and the

-insoluble problem has now lost all special

-practical or theoretical interest, having become

-absorbed in wider ideas.

-

-After this digression on the value of~$\pi$, we

-now return to the question of the general

-definition of the magnitude of an angle, so as

-to be able to produce an angle corresponding

-to any value~$u$. Suppose a moving point,~$Q$,

-to start from~$A$ on~$OX$ (\Chg{cf.}{\cf}\ \Fig[fig.]{27}), and to rotate

-in the positive direction (anti-clockwise, in

-the figure considered) round the circumference

-of the circle for any number of times, finally

-resting at any point, \eg~at $P$ or~$P'$ or~$P''$ or~$P'''$.

-Then the total length of the curvilinear

-circular path traversed, divided by the radius

-of the circle,~$r$, is the generalized definition of

-a positive angle of \emph{any} size. Let $x$,~$y$ be the

-coordinates of the point in which the point~$Q$

-rests, \ie~in one of the four alternative positions

-mentioned in \Fig[fig.]{27}; $x$~and~$y$ (as here used) will

-either \Typo{}{be} $x$~and~$y$, or $x'$~and~$y$, or $x'$~and~$y'$, or $x$~and~$y'$.

-\PageSep{188}

-Then the sign of this generalized

-angle is~$\dfrac{y}{r}$ and its cosine is~$\dfrac{x}{r}$. With these

-definitions the functional relations $v = \sin u$

-and $w = \cos u$, are at last defined for all positive

-real values of~$u$. For negative values of~$u$

-we simply take rotation of~$Q$ in the opposite

-(clockwise) direction; but it is not worth our

-while to elaborate further on this point, now

-that the general method of procedure has

-been explained.

-

-These functions of sine and cosine, as thus

-defined, enable us to deal with the problems

-concerning the triangle from which Trigonometry

-took its rise. But we are now in a

-position to relate Trigonometry to the wider

-idea of Periodicity of which the importance

-\index{Periodicity}%

-was explained in the last chapter. It is easy

-to see that the functions $\sin u$ and $\cos u$ are

-periodic functions of~$u$. For consider the

-position,~$P$ (in \Fig[fig.]{27}), of a moving point,~$Q$,

-which has started from~$A$ and revolved round

-the circle. This position,~$P$, marks the angles

-$\dfrac{\text{arc } AP}{r}$, and $2\pi + \dfrac{\text{arc } AP}{r}$, and $4\pi + \dfrac{\text{arc } AP}{r}$,

-and $6\pi + \dfrac{\text{arc } AP}{r}$, and so on indefinitely. Now,

-all these angles have the same sine and cosine,

-namely, $\dfrac{y}{r}$~and~$\dfrac{x}{r}$. Hence it is easy to see that,

-\PageSep{189}

-\index{Period|EtSeq}%

-if $u$ be chosen to have any value, the arguments

-$u$~and~$2\pi + u$, and $4\pi + u$, and $6\pi + u$,

-and $8\pi + u$ and so on indefinitely, have all the

-same values for the corresponding sines and

-cosines. In other words,

-\begin{alignat*}{4}

-\sin u &= \sin(2\pi + u) &&= \sin(4\pi + u) &&= \sin(6\pi + u) &&= \text{etc.}; \\

-\cos u &= \cos(2\pi + u) &&= \cos(4\pi + u) &&= \cos(6\pi + u) &&= \text{etc.}

-\end{alignat*}

-This fact is expressed by saying that $\sin u$ and

-$\cos u$ are periodic functions with their period

-equal to~$2\pi$.

-

-The graph of the function $y = \sin x$ (notice

-that we now abandon $v$~and~$u$ for the more

-familiar $y$~and~$x$) is shown in \Fig[fig.]{28}. We take

-on the axis of~$x$ any arbitrary length at pleasure

-to represent the number~$\pi$, and on the axis

-of~$y$ any arbitrary length at pleasure to represent

-the number~$1$. The numerical values of

-the sine and cosine can never exceed unity.

-The recurrence of the figure after periods of~$2\pi$

-will be noticed. This graph represents the

-simplest style of periodic function, out of

-which all others are constructed. The cosine

-gives nothing fundamentally different from the

-sine. For it is easy to prove that $\cos x = \sin(x + \dfrac{\pi}{2})$;

-hence it can be seen that the

-graph of $\cos x$ is simply \Fig[fig.]{28} modified by

-\PageSep{190}

-drawing the axis of~$OY$ through the point

-on~$OX$ marked~$\dfrac{\pi}{2}$, instead of drawing it in

-its actual position on the figure.

-

-It is easy to construct a `sine' function in

-\Figure{28}

-which the period has any assigned value~$a$.

-For we have only to write

-\[

-y = \sin \frac{2\pi x}{a},

-\]

-and then

-\[

-\sin \frac{2\pi (x + a)}{a}

-%[** TN: Changed curly braces to parentheses]

-  = \sin \left(\frac{2\pi x}{a} + 2\pi\right)

-  = \sin \frac{2\pi x}{a}.

-\]

-Thus the period of this new function is now~$a$.

-Let us now give a general definition of what

-\PageSep{191}

-we mean by a periodic function. The function~$f(x)$

-is periodic, with the period~$a$, if (i)~for \emph{any}

-value of~$x$ we have $f(x) = f(x + a)$, and (ii)~there

-is no number~$b$ smaller than~$a$ such that for

-\emph{any} value of~$x$, $f(x) = f(x + b)$.

-

-The second clause is put into the definition

-because when we have $\sin \dfrac{2\pi x}{a}$, it is not only

-periodic in the period~$a$, but also in the periods

-$2a$ and~$3a$, and so on; this arises since

-\[

-\sin \frac{2\pi (x + 3a)}{a}

-  = \sin \left(\frac{2\pi x}{a} + 6\pi\right)

-  = \sin \frac{2\pi x}{a}.

-\]

-So it is the smallest period which we want to

-get hold of and call \emph{the} period of the function.

-The greater part of the abstract theory of

-periodic functions and the whole of the applications

-of the theory to Physical Science are

-dominated by an important theorem called

-Fourier's Theorem; namely that, if $f(x)$ be a

-\index{Fourier's Theorem}%

-periodic function with the period~$a$ and if $f(x)$

-also satisfies certain conditions, which practically

-are always presupposed in functions suggested

-by natural phenomena, then $f(x)$ can

-be written as the sum of a set of terms in the

-form\Pagelabel{191}

-\begin{multline*}

-c_{0} + c_{1} \sin \left(\frac{2\pi x}{a} + e_{1}\right)

-  + c_{2} \sin \left(\frac{4\pi x}{a} + e_{2}\right) \\

-  + c_{3} \sin \left(\frac{6\pi x}{a} + e_{3}\right) + \text{etc.}

-\end{multline*}

-\PageSep{192}

-In this formula $c_{0}$,~$c_{1}$, $c_{2}$, $c_{3}$,~etc., and also

-$e_{1}$,~$e_{2}$, $e_{3}$,~etc., are constants, chosen so as to

-suit the particular function. Again we have

-to ask, How many terms have to be chosen?

-And here a new difficulty arises: for we can

-prove that, though in some particular cases a

-definite number will do, yet in general all we

-can do is to approximate as closely as we like

-to the value of the function by taking more

-and more terms. This process of gradual

-approximation brings us to the consideration

-of the theory of infinite series, an essential

-part of mathematical theory which we will

-consider in the next chapter.

-

-The above method of expressing a periodic

-\index{Harmonic Analysis}%

-function as a sum of sines is called the ``harmonic

-analysis'' of the function. For example,

-at any point on the sea coast the tides

-rise and fall periodically. Thus at a point

-near the Straits of Dover there will be two

-daily tides due to the rotation of the earth.

-The daily rise and fall of the tides are complicated

-by the fact that there are two tidal

-waves, one coming up the English Channel,

-and the other which has swept round the

-North of Scotland, and has then come southward

-down the North Sea. Again some high

-tides are higher than others: this is due to

-the fact that the Sun has also a tide-generating

-influence as well as the Moon. In this way

-monthly and other periods are introduced.

-\PageSep{193}

-We leave out of account the exceptional influence

-of winds which cannot be foreseen.

-The general problem of the harmonic analysis

-of the tides is to find sets of terms like those

-in the expression on \Pageref[page]{191} above, such that

-each set will give with approximate accuracy

-the contribution of the tide-generating influences

-of one ``period'' to the height of the

-tide at any instant. The argument~$x$ will

-therefore be the \emph{time} reckoned from any convenient

-commencement.

-

-Again, the motion of vibration of a violin

-string is submitted to a similar harmonic

-analysis, and so are the vibrations of the

-ether and the air, corresponding respectively

-to waves of light and waves of sound. We

-are here in the presence of one of the fundamental

-processes of mathematical physics---namely,

-nothing less than its general method

-of dealing with the great natural fact of

-Periodicity.

-\PageSep{194}

-

-

-\Chapter{XIV}{Series}

-

-\First{No} part of Mathematics suffers more from

-\index{Order|EtSeq}%

-\index{Series|EtSeq}%

-the triviality of its initial presentation to

-beginners than the great subject of series.

-Two minor examples of series, namely arithmetic

-and geometric series, are considered;

-these examples are important because they

-are the simplest examples of an important

-general theory. But the general ideas are

-never disclosed; and thus the examples, which

-exemplify nothing, are reduced to silly trivialities.

-

-The general mathematical idea of a series

-is that of a set of things ranged in order, that

-is, in sequence; This meaning is accurately

-represented in the common use of the term.

-Consider for example, the series of English

-Prime Ministers during the nineteenth century,

-arranged in the order of their first tenure of

-that office within the century. The series

-commences with William Pitt, and ends with

-\index{Pitt, William}%

-\index{Rosebery, Lord}%

-Lord Rosebery, who, appropriately enough,

-is the biographer of the first member. We

-\PageSep{195}

-might have considered other serial orders for

-the arrangement of these men; for example,

-according to their height or their weight.

-These other suggested orders strike us as

-trivial in connection with Prime Ministers,

-and would not naturally occur to the mind;

-but abstractly they are just as good orders

-as any other. When one order among terms

-is very much more important or more obvious

-than other orders, it is often spoken of as \emph{the}

-order of those terms. Thus \emph{the} order of the

-integers would always be taken to mean their

-order as arranged in order of magnitude. But

-of course there is an indefinite number of

-other ways of arranging them. When the

-number of things considered is finite, the

-number of ways of arranging them in order is

-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{,}

-\]

-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 $5! = 5 × 4 × 3 × 2 × 1 = 120$.

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

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

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

-\PageSep{196}

-

-It is easy to verify in the case of small

-values of~$n$ that $n!$ is the number of ways

-of arranging $n$~things in order. Thus consider

-two things $a$ and~$b$; these are capable

-of the two orders $ab$ and~$ba$, and $2! = 2$.

-

-Again, take three things $a$,~$b$, and~$c$; these

-are capable of the six orders, $abc$, $acb$, $bac$,

-$bca$, $cab$, $cba$, and $3! = 6$. Similarly for the

-twenty-four orders in which four things $a$,~$b$,~$c$,

-and~$d$, can be arranged.

-

-When we come to the infinite sets of things---like

-\index{Order, Type of}%

-the sets of all the integers, or all the

-fractions, or all the real numbers for instance---we

-come at once upon the complications of

-the theory of order-types. This subject was

-touched upon in \ChapRef{VI}. in considering

-the possible orders of the integers, and of the

-fractions, and of the real numbers. The

-whole question of order-types forms a comparatively

-new branch of mathematics of

-great importance. We shall not consider it

-any further. All the infinite series which we

-consider now are of the same order-type as

-the integers arranged in ascending order of

-magnitude, namely, with a first term, and

-such that each term has a couple of next-door

-neighbours, one on either side, with the

-exception of the first term which has, of

-course, only one next-door neighbour. Thus,

-if $m$~be any integer (not zero), there will be

-always an $m$th~term. A series with a finite

-\PageSep{197}

-number of terms (say $n$~terms) has the same

-characteristics as far as next-door neighbours

-are concerned as an infinite series; it only

-differs from infinite series in having a last

-term, namely, the~$n$th.

-

-The important thing to do with a series of

-numbers---using for the future ``series'' in

-the restricted sense which has just been mentioned---is

-to add its successive terms together.

-

-Thus if $u_{1}$,~$u_{2}$, $u_{3}$,~\dots\Add{,} $u_{n}$,~\dots\ are respectively

-the $1$st,~$2$nd, $3$rd, $4$th,~\dots\Add{,} $n$th,~\dots\

-terms of a series of numbers, we form successively

-the series $u_{1}$, $u_{1} + u_{2}$, $u_{1} + u_{2} + u_{3}$, $u_{1} + u_{2} + u_{3} + u_{4}$,

-and so on; thus the sum of the

-$1$st $n$~terms may be written\Typo{.}{}

-\[

-u_{1} + u_{2} + u_{3} + \dots + u_{n}.

-\]

-

-If the series has only a finite number of

-\index{Approximation|EtSeq}%

-terms, we come at last in this way to the

-sum of the whole series of terms. But, if

-the series has an infinite number of terms,

-this process of successively forming the sums

-of the terms never terminates; and in this

-sense there is no such thing as the sum of an

-infinite series.

-

-But why is it important successively to add

-the terms of a series in this way? The answer

-is that we are here symbolizing the fundamental

-mental process of approximation.

-This is a process which has significance far

-\PageSep{198}

-beyond the regions of mathematics. Our

-limited intellects cannot deal with complicated

-material all at once, and our method of

-arrangement is that of approximation. The

-statesman in framing his speech puts the

-dominating issues first and lets the details

-fall naturally into their subordinate places.

-There is, of course, the converse artistic

-method of preparing the imagination by the

-presentation of subordinate or special details,

-and then gradually rising to a crisis. In

-either way the process is one of gradual summation

-of effects; and this is exactly what

-is done by the successive summation of the

-terms of a series. Our ordinary method of

-stating numbers is such a process of gradual

-summation, at least, in the case of large

-numbers. Thus $568,213$ presents itself to

-the mind as\Add{:}---

-\[

-500,000 + 60,000 + 8,000 + 200 + 10 + 3\Add{.}

-\]

-

-In the case of decimal fractions this is so

-more avowedly. Thus $3.14159$ is\Add{:}---

-\[

-3 + \tfrac{1}{10} + \tfrac{4}{100} + \tfrac{1}{1000} + \tfrac{5}{10000} + \tfrac{9}{100000}\Add{.}

-\]

-Also, $3$ and~$3 + \frac{1}{10}$, and $3 + \tfrac{1}{10} + \tfrac{4}{100}$, and

-$3 + \tfrac{1}{10} + \tfrac{4}{100} + \tfrac{1}{1000}$,

-and $3 + \tfrac{1}{10} + \tfrac{4}{100} + \tfrac{1}{1000} + \tfrac{5}{10000}$ are

-successive approximations to the complete result

-$3.14159$. If we read $568,213$ backwards

-from right to left, starting with the $3$~units,

-\PageSep{199}

-we read it in the artistic way, gradually preparing

-the mind for the crisis of~$500,000$.

-

-The ordinary process of numerical multiplication

-proceeds by means of the summation

-of a series, Consider the computation

-\[

-\begin{array}{*{6}{@{}c@{}}}

- & & &3&4&2 \\

- & & &6&5&8 \\

-\cline{4-6}

-\Strut

- & &2&7&3&6 \\

- &1&7&1&0&  \\

-2&0&5&2& &  \\

-\cline{1-6}

-\Strut

-2&2&5&0&3&6

-\end{array}

-\]

-

-Hence the three lines to be added form a

-series of which the first term is the upper

-line. This series follows the artistic method

-of presenting the most important term last,

-not from any feeling for art, but because of

-the convenience gained by keeping a firm

-hold on the units' place, thus enabling us to

-omit some~$0$'s, formally necessary.

-

-But when we approximate by gradually

-\index{Limit of a Series|EtSeq}%

-adding the successive terms of an infinite

-series, what are we approximating to? The

-difficulty is that the series has no ``sum'' in

-the straightforward sense of the word, because

-the operation of adding together its terms

-can never be completed. The answer is that

-we are approximating to the \emph{limit} of the

-summation of the series, and we must now

-\PageSep{200}

-proceed to explain what the ``limit'' of a

-series is.

-

-The summation of a series approximates to

-a limit when the sum of any number of its

-terms, provided the number be large enough,

-is as nearly equal to the limit as you care to

-approach. But this description of the meaning

-of approximating to a limit evidently will

-not stand the vigorous scrutiny of modern

-mathematics. What is meant by \emph{large

-enough}, and by \emph{nearly equal}, and by \emph{care to

-approach}? All these vague phrases must be

-explained in terms of the simple abstract

-ideas which alone are admitted into pure

-mathematics.

-

-Let the successive terms of the series be

-$u_{1}$,~$u_{2}$, $u_{3}$, $u_{4}$,~\dots, $u_{n}$, etc., so that $u_{n}$~is the

-$n$th~term of the series. Also let $s_{n}$ be the

-sum of the $1$st $n$~terms, whatever $n$~may be.

-So that\Add{:}---

-\begin{gather*}

-s_{1} = u_{1},\quad

-s_{2} = u_{1} + u_{2},\quad

-s_{3} = u_{1} + u_{2} + u_{3},\quad\text{and} \\

-s_{n} = u_{1} + u_{2} + u_{3} + \dots + u_{n}.

-\end{gather*}

-

-Then the terms $s_{1}$,~$s_{2}$, $s_{3}$,~\dots\Add{,} $s_{n}$,~\dots\ form

-a new series, and the formation of this series

-is the process of summation of the original

-series. Then the ``approximation'' of the

-\emph{summation} of the original series to a ``limit''

-means the ``approximation of the \emph{terms} of

-this new series to a limit.'' And we have

-\PageSep{201}

-now to explain what we mean by the approximation

-to a limit of the terms of a series.

-

-Now, remembering the definition (given in

-\ChapRef[chapter]{XII}.)\ of a \emph{standard of approximation},

-\index{Standard of Approximation|EtSeq}%

-\index{Sum to Infinity|EtSeq}%

-the idea of a limit means this: $l$~is

-the limit of the terms of the series $s_{1}$,~$s_{2}$,

-$s_{3}$,~\dots\Add{,} $s_{n}$,~\dots, if, corresponding to each

-real number~$k$, taken as a standard of

-approximation, a term~$s_{n}$ of the series can

-be found so that all succeeding terms (\ie\

-$s_{n+1}$, $s_{n+2}$, etc.)\ approximate to~$l$ within

-that standard of approximation. If another

-smaller standard~$k^{1}$ be chosen, the term~$s_{n}$

-may be too early in the series, and a

-later term~$s_{m}$ with the above property will

-then be found.

-

-If this property holds, it is evident that as

-you go along to series $s_{1}$,~$s_{2}$, $s_{3}$,~\dots, $s_{n}$,~\dots\

-from left to right, after a time you come to

-terms \emph{all} of which are nearer to~$l$ than any

-number which you may like to assign. In

-other words you approximate to~$l$ as closely

-as you like. The close connection of this

-definition of the limit of a series with the

-definition of a continuous function given in

-\ChapRef[chapter]{XI}.\ will be immediately perceived.

-

-Then coming back to the original series $u_{1}$,~$u_{2}$,

-$u_{3}$,~\dots, $u_{n}$,~\dots, the limit of the terms of

-the series $s_{1}$,~$s_{2}$, $s_{3}$,~\dots, $s_{n}$,~\dots, is called

-the ``sum to infinity'' of the original series.

-But it is evident that this use of the word

-\PageSep{202}

-``sum'' is very artificial, and we must not

-assume the analogous properties to those of

-the ordinary sum of a finite number of terms

-without some special investigation.

-

-Let us look at an example of a ``sum to

-infinity.'' Consider the recurring decimal

-$.1111\dots$. This decimal is merely a way of

-symbolizing the ``sum to infinity'' of the series

-$.1$, $.01$, $.001$, $.0001$, etc. The corresponding

-series found by summation is $s_{1} = .1$,

-$s_{2} = .11$, $s_{3} = .111$, $s_{4} = .1111$, etc. The limit

-of the terms of this series is~$\frac{1}{9}$; this is easy to

-see by simple division, for

-\[

-\tfrac{1}{9}

-  = .1 + \tfrac{1}{90}

-  = .11 + \tfrac{1}{900}

-  = .111 + \tfrac{1}{9000} = \text{etc.}

-\]

-Hence, if $\frac{3}{17}$ is given (the $k$ of the definition),

-$.1$~and \emph{all} succeeding terms differ from~$\frac{1}{9}$ by

-less than~$\frac{3}{17}$; if $\frac{1}{1000}$ is given (another choice

-for the $k$ of the definition), $.111$ and all

-succeeding terms differ from~$\frac{1}{9}$ by less than~$\frac{1}{1000}$;

-and so on, whatever choice for~$k$ be

-made.

-

-It is evident that nothing that has been

-said gives the slightest idea as to how the

-``sum to infinity'' of a series is to be

-found. We have merely stated the conditions

-which such a number is to satisfy. Indeed,

-a general method for finding in all

-cases the sum to infinity of a series is intrinsically

-out of the question, for the simple reason

-that such a ``sum,'' as here defined, does not

-always exist. Series which possess a sum to

-\PageSep{203}

-\index{Convergent|EtSeq}%

-\index{Divergent|EtSeq}%

-infinity are called \emph{convergent}, and those which

-do not possess a sum to infinity are called

-\emph{divergent}.

-

-An obvious example of a divergent series

-is $1$,~$2$, $3$,~\dots, $n$~\dots\Add{,} \ie~the series of integers

-in their order of magnitude. For

-whatever number~$l$ you try to take as its

-sum to infinity, and whatever standard of

-approximation~$k$ you choose, by taking

-enough terms of the series you can always

-make their sum differ from~$l$ by more than~$k$.

-Again, another example of a divergent

-series is $1$,~$1$, $1$,~etc., \ie~the series of

-which each term is equal to~$1$. Then the

-sum of $n$~terms is~$n$, and this sum grows

-without limit as $n$~increases. Again, another

-example of a divergent series is $1$,~$-1$, $1$,~$-1$,

-$1$,~$-1$, etc., \ie~the series in which the terms

-are alternately $1$ and~$-1$. The sum of an

-odd number of terms is~$1$, and of an even

-number of terms is~$0$. Hence the terms of

-the series $s_{1}$,~$s_{2}$, $s_{3}$,~\dots\Add{,} $s_{n}$,~\dots\ do not approximate

-to a limit, although they do not

-increase without limit.

-

-It is tempting to suppose that the condition

-for $u_{1}$,~$u_{2}$,~\dots\Add{,} $u_{n}$,~\dots\ to have a sum

-to infinity is that $u_{n}$~should decrease indefinitely

-as $n$~increases. Mathematics would

-be a much easier science than it is, if this

-were the case. Unfortunately the supposition

-is not true.

-\PageSep{204}

-

-For example the series

-\[

-1,\quad

-\frac{1}{2},\quad

-\frac{1}{3},\quad

-\frac{1}{4},\ \dots,\quad

-\frac{1}{n},\ \dots

-\]

-is divergent. It is easy to see that this is

-the case; for consider the sum of $n$~terms

-%[** TN: "(n + 1)^{th} term" in the original

-beginning at the $(n + 1)$th term. These $n$~terms

-are $\dfrac{1}{n + 1}$, $\dfrac{1}{n + 2}$, $\dfrac{1}{n + 3}$,~\dots\Add{,} $\dfrac{1}{2n}$: there

-are $n$~of them and $\dfrac{1}{2n}$~is the least among them.

-Hence their sum is greater than $n$~times~$\dfrac{1}{2n}$,

-\ie~is greater than~$\dfrac{1}{2}$. Now, without

-altering the sum to infinity, if it exist, we

-can add together neighbouring terms, and

-obtain the series

-\[

-1,\quad

-\tfrac{1}{2},\quad

-\tfrac{1}{3} + \tfrac{1}{4},\quad

-\tfrac{1}{5} + \tfrac{1}{6} + \tfrac{1}{7} + \tfrac{1}{8},\quad

-\text{etc.},

-\]

-that is, by what has been said above, a series

-whose terms after the~$2$nd are greater than

-those of the series,

-\[

-1,\quad

-\tfrac{1}{2},\quad

-\tfrac{1}{2},\quad

-\tfrac{1}{2},\quad

-\text{etc.},

-\]

-where all the terms after the first are equal.

-But this series is divergent. Hence the

-original series is divergent.\footnote

-  {\Chg{Cf.}{\Cf}\ Note~C, \Pageref{noteC}.\Pagelabel{204}}

-

-This question of divergency shows how

-careful we must be in arguing from the properties

-\PageSep{205}

-of the sum of a finite number of terms

-to that of the sum of an infinite series. For

-the most elementary property of a finite

-number of terms is that of course they

-possess a sum: but even this fundamental

-property is not necessarily possessed by an

-infinite series. This caution merely states

-that we must not be misled by the suggestion

-of the technical term ``\emph{sum} of an infinite

-series.'' It is usual to indicate the sum of

-the infinite series

-\[

-u_{1},\quad

-u_{2},\quad

-u_{3},\ \dots\Add{,}\quad

-u_{n}\Add{,}\ \dots

-\]

-by

-\[

-u_{1} + u_{2} + u_{3} + \dots + u_{n} + \dots\Add{.}

-\]

-

-We now pass on to a generalization of the

-idea of a series, which mathematics, true to

-its method, makes by use of the variable.

-Hitherto, we have only contemplated series

-in which each definite term was a definite

-number. But equally well we can generalize,

-and make each term to be some mathematical

-expression containing a variable~$x$. Thus

-we may consider the series $1$,~$x$, $x^{2}$, $x^{3}$,~\dots,

-$x^{n}$,~\dots, and the series

-\[

-x,\quad

-\frac{x^{2}}{2},\quad

-\frac{x^{3}}{3},\ \dots,\quad

-\frac{x^{n}}{n},\ \dots\Add{.}

-\]

-

-In order to symbolize the general idea of

-any such function, conceive of a function of~$x$,

-$f_{n}(x)$~say, which involves in its formation

-a variable integer~$n$, then, by giving~$n$ the

-\PageSep{206}

-values $1$,~$2$, $3$,~etc., in succession, we get the

-series

-\[

-f_{1}(x),\quad

-f_{2}(x),\quad

-f_{3}(x),\ \dots,\quad

-f_{n}(x),\dots\Add{.}

-\]

-Such a series may be convergent for some

-values of~$x$ and divergent for others. It is,

-in fact, rather rare to find a series involving a

-variable~$x$ which is convergent for all values

-of~$x$,---at least in any particular instance it is

-very unsafe to assume that this is the case.

-For example, let us examine the simplest of

-all instances, namely, the ``geometrical''

-\index{Geometrical Series|EtSeq}%

-series

-\[

-1,\quad x,\quad x^{2},\quad x^{3},\ \dots,\quad x^{n},\ \dots\Add{.}

-\]

-The sum of $n$~terms is given by

-\[

-s_{n} = 1 + x + x^{2} + x^{3} + \dots + x^{n}.

-\]

-

-Now multiply both sides by~$x$ and we get

-\[

-xs_{n} = x + x^{2} + x^{3} + x^{4} + \dots + x^{n} + x^{n+1}\Add{.}

-\]

-Now subtract the last line from the upper

-line and we get

-\[

-s_{n}(1 - x) = s_{n} - xs_{n} = 1 - x^{n+1},

-\]

-and hence (if $x$~be not equal to~$1$)

-\[

-s_{n} = \frac{1 - x^{n+1}}{1 - x}

-  = \frac{1}{1 - x} - \frac{x^{n+1}}{1 - x}\Add{.}

-\]

-Now if $x$~be numerically less than~$1$, for sufficiently

-large values of~$n$, $\dfrac{x^{n+1}}{1 - x}$~is always numerically

-\PageSep{207}

-less than~$k$, however $k$~be chosen. Thus,

-if $x$~be numerically less than~$1$, the series $1$,~$x$,

-$x^{2}$,~\dots\Add{,} $x^{n}$,~\dots\ is convergent, and $\dfrac{1}{1 - x}$~is its

-limit. This statement is symbolized by

-\[

-\frac{1}{1 - x} = 1 + x + x^{2} + \dots + x^{n} + \dots,\quad

-(-1 < x < 1).

-\]

-But if $x$~is numerically greater than~$1$, or

-numerically equal to~$1$, the series is divergent.

-In other words, if $x$~lie between $-1$ and~$+1$,

-the series is convergent; but if $x$~be equal

-to~$-1$ or~$+1$, or if $x$~lie outside the interval

-$-1$~to~$+1$, then the series is divergent. Thus

-the series is convergent at all ``points''

-within the interval $-1$~to~$+1$, exclusive of

-the end points.

-

-At this stage of our enquiry another question

-arises. Suppose that the series

-\[

-f_{1}(x) + f_{2}(x) + f_{3}(x) + \dots + f_{n}(x) + \dots

-\]

-is convergent for all values of~$x$ lying within

-the interval $a$~to~$b$, \ie~the series is convergent

-for any value of~$x$ which is greater than~$a$ and

-less than~$b$. Also, suppose we want to be

-sure that in approximating to the limit we

-add together enough terms to come within

-some standard of approximation~$k$. Can we

-always state some number of terms, say~$n$,

-such that, if we take $n$~or more terms to

-form the sum, then \emph{whatever} value $x$~has

-\PageSep{208}

-within the interval we have satisfied the

-desired standard of approximation?

-

-Sometimes we can and sometimes we cannot

-\index{Non-Uniform Convergence|EtSeq}%

-\index{Uniform Convergence|EtSeq}%

-do this for each value of~$k$. When we

-can, the series is called uniformly convergent

-throughout the interval, and when we cannot

-do so, the series is called non-uniformly convergent

-throughout the interval. It makes

-a great difference to the properties of a series

-whether it is or is not uniformly convergent

-through an interval. Let us illustrate the

-matter by the simplest example and the

-simplest numbers.

-

-Consider the geometric series

-\[

-1 + x + x^{2} + x^{3} + \dots + x^{n} + \dots\Add{.}

-\]

-

-It is convergent throughout the interval

-$-1$~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

-$n$~terms, we have proved that the difference

-between $s_{n}(x)$ and the limit~$\dfrac{1}{1 - x}$ is~$\dfrac{x^{n+1}}{1 - x}$.

-Now suppose $n$~be any given number of terms,

-say~$20$, and let $k$~be any assigned standard

-of approximation, say~$.001$. Then, by taking

-$x$~near enough to~$+1$ or near enough to~$-1$,

-we can make the numerical value of~$\dfrac{x^{21}}{1 - x}$ to

-be greater than~$.001$. Thus $20$~terms will

-\PageSep{209}

-not do over the whole interval, though it is

-more than enough over some parts of it.

-

-The same reasoning can be applied whatever

-other number we take instead of~$20$,

-and whatever standard of approximation instead

-of~$.001$. Hence the geometric series

-$1 + x + x^{2} + x^{3} + \dots + x^{n} + \dots$ is non-uniformly

-convergent over its \emph{whole} interval of

-convergence $-1$~to~$+1$. But if we take any

-smaller interval lying at both ends within the

-interval $-1$~to~$+1$, the geometric series is

-uniformly convergent within it. For example,

-take the interval $0$~to~$+\frac{1}{10}$. Then any

-value for~$n$ which makes $\dfrac{x^{n+1}}{1 - x}$~numerically

-less than~$k$ \emph{at} these limits for~$x$ also serves

-for all values of~$x$ between these limits, since

-it so happens that $\dfrac{x^{n+1}}{1 - x}$ diminishes in numerical

-value as $x$~diminishes in numerical value.

-For example, take $k = .001$; then, putting

-$x = \frac{1}{10}$, we find:\Add{---}

-\begin{alignat*}{3}

-&\text{for $n = 1$,}\quad & \frac{x^{n+1}}{1 - x}

-  &= \frac{(\frac{1}{10})^{2}}{1 - \frac{1}{10}}

-  &&= \tfrac{1}{90} = .0111\dots, \\

-%

-&\text{for $n = 2$,}\quad & \frac{x^{n+1}}{1 - x}

-  &= \frac{(\frac{1}{10})^{3}}{1 - \frac{1}{10}}

-  &&= \tfrac{1}{900} = .00111\dots, \\

-%

-&\text{for $n = 3$,}\quad & \frac{x^{n+1}}{1 - x}

-  &= \frac{(\frac{1}{10})^{4}}{1 - \frac{1}{10}}

-  &&= \tfrac{1}{9000} = .000111\dots\Typo{,}{.}

-\end{alignat*}

-

-Thus three terms will do for the whole interval,

-\PageSep{210}

-though, of course, for some parts of

-the interval it is more than is necessary.

-Notice that, because $1 + x + x^{2} + \dots + x^{n} + \dots$

-is convergent (though not uniformly)

-throughout the interval $-1$~to~$+1$,

-for each value of~$x$ in the interval some number

-of terms~$n$ can be found which will satisfy

-a desired standard of approximation; but,

-as we take $x$ nearer and nearer to either end

-value $+1$ or~$-1$, larger and larger values of~$n$

-have to be employed.

-

-It is curious that this important distinction

-between uniform and non-uniform convergence

-was not published till 1847 by Stokes---afterwards,

-\index{Stokes, Sir George}%

-Sir~George Stokes---and later, independently

-in~1850 by Seidel, a German

-\index{Seidel}%

-mathematician.

-

-The critical points, where non-uniform convergence

-comes in, are not necessarily at the

-limits of the interval throughout which convergence

-holds. This is a speciality belonging

-to the geometric series.

-

-In the case of the geometric series $1 + x + x^{2} + \dots + x^{n} + \dots$,

-a simple algebraic

-expression~$\dfrac{1}{1 - x}$ can be given for its limit in

-its interval of convergence. But this is not

-always the case. Often we can prove a series

-to be convergent within a certain interval,

-though we know nothing more about its

-limit except that it is the limit of the series.

-\PageSep{211}

-But this is a very good way of defining a

-function; \viz.\ as the limit of an infinite convergent

-series, and is, in fact, the way in which

-most functions are, or ought to be, defined.

-

-Thus, the most important series in elementary

-\index{Exponential Series|EtSeq}%

-analysis is

-\[

-1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots + \frac{x^{n}}{n!} + \dots,

-\]

-where $n!$ has the meaning defined earlier in

-this chapter. This series can be proved to

-be absolutely convergent for \emph{all} values of~$x$,

-and to be uniformly convergent within any

-interval which we like to take. Hence it has

-all the comfortable mathematical properties

-which a series should have. It is called the

-exponential series. Denote its sum to infinity

-by~$\exp x$. Thus, by definition,

-\[

-\exp x = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \dots

-  + \frac{x^{n}}{n!} + \dots\Add{.}

-\]

-$\exp x$ 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).

-\Tag{(A)}

-\]

-In other words that

-\begin{multline*}

-(\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*}

-\PageSep{212}

-

-This property~\Eq{(A)} is an example of what

-is called an addition-theorem. When any

-\index{Addition-Theorem}%

-function [say~$f(x)$] has been defined, the first

-thing we do is to try to express $f(x + y)$ in terms

-of known functions of $x$~only, and known functions

-of $y$~only. If we can do so, the result

-is called an addition-theorem. Addition-theorems

-play a great part in mathematical

-analysis. Thus the addition-theorem for the

-sine is given by

-\[

-\sin(x + y) = \sin x \cos y + \cos x \sin y,

-\]

-and for the cosine by

-\[

-\cos(x + y) = \cos x \cos y - \sin x \sin y.

-\]

-

-As a matter of fact the best ways of defining

-$\sin x$ and $\cos x$ are not by the elaborate

-geometrical methods of the previous chapter,

-but as the limits respectively of the series

-\[

-x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \text{etc.} \dots,

-\]

-and

-\[

-1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \text{etc.} \dots,

-\]

-so that we put

-\begin{align*}

-\sin x &= x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \text{etc.} \dots, \\

-\cos x &= 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \text{etc.} \dots\Typo{,}{.}

-\end{align*}

-\PageSep{213}

-

-These definitions are equivalent to the geometrical

-definitions, and both series can be

-proved to be convergent for all values of~$x$,

-and uniformly convergent throughout any

-interval. These series for sine and cosine

-have a general likeness to the exponential

-series given above. They are, indeed, intimately

-connected with it by means of the

-theory of imaginary numbers explained in

-Chapters \ChapNum{VII}.\ and~\ChapNum{VIII}.

-\Figure{29}

-

-The graph of the exponential function is

-given in \Fig[fig.]{29}. It cuts the axis~$OY$ at the

-point $y = 1$, as evidently it ought to do, since

-when $x = 0$ every term of the series except

-the first is zero. The importance of the exponential

-function is that it represents any

-changing physical quantity whose rate of

-increase at any instant is a uniform percentage

-of its value at that instant. For

-\PageSep{214}

-example, the above graph represents the size

-at any time of a population with a uniform

-birth-rate, a uniform death-rate, and no emigration,

-where the $x$ corresponds to the time

-reckoned from any convenient day, and the

-$y$ represents the population to the proper

-scale. The scale must be such that $OA$~represents

-the population at the date which is

-taken as the origin. But we have here come

-upon the idea of ``rates of increase'' which

-is the topic for the next chapter.

-

-An important function nearly allied to the

-\index{Normal Error, Curve of}%

-exponential function is found by putting~$-x^{2}$

-for~$x$ as the argument in the exponential function.

-%[** TN: Omitted period following "exp"]

-We thus get $\exp (-x^{2})$. The graph

-$y = \exp(-x^{2})$ is given in \Fig[fig.]{30}.

-\Figure{30}

-

-The curve, which is something like a cocked

-hat, is called the curve of normal error. Its

-\PageSep{215}

-corresponding function is vitally important

-to the theory of statistics, and tells us in

-many cases the sort of deviations from the

-average results which we are to expect.

-

-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{.}

-\]

-\Figure{31}

-

-Its graph is given in \Fig[fig.]{31}. The points

-$A$,~$B$, $O$, $C$, $D$, $E$,~$F$, are placed at equal intervals~$\frac{1}{2}p$,

-and an unending series of them

-should be drawn forwards and backwards.

-This function represents the dying away of

-vibrations under the influence of friction or of

-``damping'' forces. Apart from the friction,

-the vibrations would be periodic, with a

-period~$p$; but the influence of the friction

-\PageSep{216}

-makes the extent of each vibration smaller

-than that of the preceding by a constant percentage

-of that extent. This combination

-of the idea of ``periodicity'' (which requires

-\index{Periodicity}%

-the sine or cosine for its symbolism) and of

-``constant percentage'' (which requires the

-exponential function for its symbolism) is the

-reason for the form of this function, namely,

-its form as a product of a sine-function into

-an exponential function.

-\PageSep{217}

-

-

-\Chapter{XV}{The Differential Calculus}

-

-\First{The} invention of the differential calculus

-\index{Differential Calculus|EtSeq}%

-marks a crisis in the history of mathematics.

-The progress of science is divided between

-periods characterized by a slow accumulation

-of ideas and periods, when, owing to the new

-material for thought thus patiently collected,

-some genius by the invention of a new method

-or a new point of view, suddenly transforms

-the whole subject on to a higher level. These

-contrasted periods in the progress of the

-history of thought are compared by Shelley

-to the formation of an avalanche.

-\begin{verse}

-\footnotesize

-\index{Shelley (quotation from)}%

-The sun-awakened avalanche! whose mass, \\

-Thrice sifted by the storm, had gathered there \\

-Flake after flake,---in heaven-defying minds \\

-As thought by thought is piled, till some great truth \\

-Is loosened, and the nations echo round, \\

-\dotfill

-\end{verse}

-

-The comparison will bear some pressing.

-The final burst of sunshine which awakens

-the avalanche is not necessarily beyond comparison

-in magnitude with the other powers

-of nature which have presided over its slow

-\PageSep{218}

-formation. The same is true in science. The

-genius who has the good fortune to produce

-the final idea which transforms a whole

-region of thought, does not necessarily excel

-all his predecessors who have worked at the

-preliminary formation of ideas. In considering

-the history of science, it is both silly and

-ungrateful to confine our admiration with a

-gaping wonder to those men who have made

-the final advances towards a new epoch\Add{.}

-

-In the particular instance before us, the

-\index{Leibniz|EtSeq}%

-\index{Newton|EtSeq}%

-subject had a long history before it assumed

-its final form at the hands of its

-two inventors. There are some traces of its

-methods even among the Greek mathematicians,

-and finally, just before the actual

-production of the subject, Fermat (born 1601~\AD,

-\index{Fermat}%

-and died 1665~\AD), a distinguished

-French mathematician, had so improved on

-previous ideas that the subject was all but

-created by him. Fermat, also, may lay

-claim to be the joint inventor of coordinate

-geometry in company with his contemporary

-and countryman, Descartes. It was, in fact,

-\index{Descartes}%

-Descartes from whom the world of science

-received the new ideas, but Fermat had certainly

-arrived at them independently.

-

-We need not, however, stint our admiration

-either for Newton or for Leibniz. Newton

-was a mathematician and a student of

-physical science, Leibniz was a mathematician

-\PageSep{219}

-and a philosopher, and each of them

-in his own department of thought was one of

-the greatest men of genius that the world

-has known. The joint invention was the

-occasion of an unfortunate and not very

-creditable dispute. Newton was using the

-methods of Fluxions, as he called the subject,

-\index{Fluxions}%

-in~1666, and employed it in the composition

-of his \Title{Principia}, although in the work as

-printed any special algebraic notation is

-avoided. But he did not print a direct statement

-of his method till~1693. Leibniz published

-his first statement in~1684. He was

-accused by Newton's friends of having got

-it from a MS. by Newton, which he had been

-shown privately. Leibniz also accused Newton

-of having plagiarized from him. There

-is now not very much doubt but that both

-should have the credit of being independent

-discoverers. The subject had arrived at a

-stage in which it was ripe for discovery, and

-there is nothing surprising in the fact that

-two such able men should have independently

-hit upon it.

-

-These joint discoveries are quite common

-in science. Discoveries are not in general

-made before they have been led up to

-by the previous trend of thought, and by

-that time many minds are in hot pursuit

-of the important idea. If we merely keep

-to discoveries in which Englishmen are

-\PageSep{220}

-concerned, the simultaneous enunciation of

-the law of natural selection by Darwin and

-\index{Darwin}%

-Wallace, and the simultaneous discovery of

-\index{Wallace}%

-Neptune by Adams and the French astronomer,

-\index{Adams}%

-Leverrier, at once occur to the mind.

-\index{Leverrier}%

-The disputes, as to whom the credit ought to

-be given, are often influenced by an unworthy

-spirit of nationalism. The really inspiring

-reflection suggested by the history of mathematics

-is the unity of thought and interest

-among men of so many epochs, so many nations,

-and so many races. Indians, Egyptians,

-Assyrians, Greeks, Arabs, Italians, Frenchmen,

-Germans, Englishmen, and Russians, have

-all made essential contributions to the progress

-of the science. Assuredly the jealous

-exaltation of the contribution of one particular

-nation is not to show the larger spirit.

-

-The importance of the differential calculus

-\index{Rate of Increase of Functions|EtSeq}%

-arises from the very nature of the subject,

-which is the systematic consideration of the

-rates of increase of functions. This idea is

-immediately presented to us by the study of

-nature; velocity is the rate of increase of the

-distance travelled, and acceleration is the

-rate of increase of velocity. Thus the fundamental

-idea of change, which is at the basis of

-our whole perception of phenomena, immediately

-suggests the enquiry as to the rate of

-change. The familiar terms of ``quickly''

-and ``slowly'' gain their meaning from a tacit

-\PageSep{221}

-reference to rates of change. Thus the differential

-calculus is concerned with the very

-key of the position from which mathematics

-can be successfully applied to the explanation

-of the course of nature.

-

-This idea of the rate of change was certainly

-in Newton's mind, and was embodied in the

-\Figure{32}

-language in which he explained the subject.

-It may be doubted, however, whether this

-point of view, derived from natural phenomena,

-was ever much in the minds of the preceding

-mathematicians who prepared the subject

-for its birth. They were concerned with the

-more abstract problems of drawing tangents

-\index{Tangents}%

-to curves, of finding the lengths of curves, and

-of finding the areas enclosed by curves. The

-\PageSep{222}

-last two problems, of the rectification of curves

-and the quadrature of curves as they are

-named, belong to the Integral Calculus, which

-\index{Integral Calculus}%

-is however involved in the same general subject

-as the Differential Calculus.

-

-The introduction of coordinate geometry

-\index{Tangents}%

-makes the two points of view coalesce. For

-(\Chg{cf.}{\cf}\ \Fig[fig.]{32}) let $AQP$ be any curved line and let

-$PT$ be the tangent at the point~$P$ on it. Let

-the axes of coordinates be $OX$ and~$OY$; and

-let $y = f(x)$ be the equation to the curve, so that

-$OM = x$, and $PM = y$. Now let $Q$ be any

-moving point on the curve, with coordinates

-$x_{1}$,~$y_{1}$; then $y_{1} = f(x_{1})$. And let $Q'$ be the point

-on the tangent with the same abscissa~$x_{1}$;

-suppose that the coordinates of~$Q'$ are $x_{1}$ and~$y'$.

-Now suppose that $N$~moves along the

-axis~$OX$ from left to right with a uniform

-velocity; then it is easy to see that the ordinate~$y'$

-of the point~$Q'$ on the tangent~$TP$ also

-increases uniformly as $Q'$~moves along the

-tangent in a corresponding way. In fact it is

-easy to see that the ratio of the rate of increase

-of~$Q'N$ to the rate of increase of~$ON$ is in the

-ratio of $Q'N$ to~$TN$, which is the same at all

-points of the straight line. But the rate of

-increase of~$QN$, which is the rate of increase

-of~$f(x_{1})$, varies from point to point of the curve

-so long as it is not straight. As $Q$~passes

-through the point~$P$, the rate of increase of~$f(x_{1})$

-(where $x_{1}$~coincides with~$x$ for the moment)

-\PageSep{223}

-is the same as the rate of increase of~$y'$ on the

-tangent at~$P$. Hence, if we have a general

-method of determining the rate of increase

-of a function~$f(x)$ of a variable~$x$, we can

-determine the slope of the tangent at any

-point $(x, y\Typo{,}{})$ on a curve, and thence can

-draw it. Thus the problems of drawing tangents

-to a curve, and of determining the

-rates of increase of a function are really

-identical.

-

-It will be noticed that, as in the cases of

-Conic Sections and Trigonometry, the more

-artificial of the two points of view is the one

-in which the subject took its rise. The really

-fundamental aspect of the science only rose

-into prominence comparatively late in the

-day. It is a well-founded historical generalization,

-that the last thing to be discovered

-in any science is what the science is really

-about. Men go on groping for centuries,

-guided merely by a dim instinct and a puzzled

-curiosity, till at last ``some great truth is

-loosened.''

-

-Let us take some special cases in order to

-familiarize ourselves with the sort of ideas

-which we want to make precise. A train is

-in motion---how shall we determine its velocity

-at some instant, let us say, at noon? We can

-take an interval of five minutes which includes

-noon, and measure how far the train has gone

-in that period. Suppose we find it to be five

-\PageSep{224}

-miles, we may then conclude that the train

-was running at the rate of $60$~miles per~hour.

-But five miles is a long distance, and we

-cannot be sure that just at noon the train

-was moving at this pace. At noon it may

-have been running $70$~miles per~hour, and

-afterwards the \Typo{break}{brake} may have been put on.

-It will be safer to work with a smaller interval,

-say one minute, which includes noon, and to

-measure the space traversed during that

-period. But for some purposes greater

-accuracy may be required, and one minute

-may be too long. In practice, the necessary

-inaccuracy of our measurements makes it

-useless to take too small a period for measurement.

-But in theory the smaller the period

-the better, and we are tempted to say that

-for ideal accuracy an infinitely small period

-is required. The older mathematicians, in

-particular Leibniz, were not only tempted,

-but yielded to the temptation, and did say

-it. Even now it is a useful fashion of speech,

-provided that we know how to interpret it

-into the language of common sense. It is

-curious that, in his exposition of the foundations

-of the calculus, Newton, the natural

-scientist, is much more philosophical than

-Leibniz, the philosopher, and on the other

-hand, Leibniz provided the admirable notation

-which has been so essential for the progress

-of the subject.

-\PageSep{225}

-

-Now take another example within the region

-of pure mathematics. Let us proceed to find

-the rate of increase of the function~$x^{2}$ for

-any value~$x$ of its argument. We have not

-yet really defined what we mean by rate of

-increase. We will try and grasp its meaning

-in relation to this particular case. When $x$~increases

-to $x + h$, the function~$x^{2}$ increases to

-$(x + h)^{2}$; so that the total increase has been

-$(x + h)^{2} - x^{2}$, due to an increase~$h$ in the argument.

-Hence throughout the interval $x$~to

-$(x + h)$ the average increase of the function per

-unit increase of the argument is $\dfrac{(x + h)^{2} - x^{2}}{h}$.

-But

-\[

-(x + h)^{2} = x^{2} + 2hx + h^{2},

-\]

-and therefore

-\[

-\frac{(x + h)^{2} - x^{2}}{h} = \frac{2hx + h^{2}}{h} = 2x + h.

-\]

-Thus $2x + h$ is the average increase of the

-function~$x^{2}$ per unit increase in the argument,

-the average being taken over by the interval

-$x$~to~$x + h$. But $2x + h$ depends on~$h$, the size

-of the interval. We shall evidently get what

-we want, namely the \emph{rate} of increase at the

-value~$x$ of the argument, by diminishing~$h$

-more and more. Hence \emph{in the limit} when $h$~has

-\PageSep{226}

-\index{Infinitely Small Quantities|EtSeq}%

-\emph{decreased indefinitely}, we say that $2x$~is the

-rate of increase of~$x^{2}$ at the value~$x$ of the

-argument.

-

-Here again we are apparently driven up

-against the idea of infinitely small quantities

-in the use of the words ``in the limit when $h$~has

-decreased indefinitely.'' Leibniz held that,

-mysterious as it may sound, there were actually

-existing such things as infinitely small

-quantities, and of course infinitely small numbers

-corresponding to them. Newton's language

-and ideas were more on the modern

-lines; but he did not succeed in explaining

-the matter with such explicitness so as to be

-evidently doing more than explain Leibniz's

-ideas in rather indirect language. The real

-explanation of the subject was first given by

-Weierstrass and the Berlin School of mathematicians

-\index{Weierstrass}%

-about the middle of the nineteenth

-century. But between Leibniz and Weierstrass

-a copious literature, both mathematical

-and philosophical, had grown up round these

-mysterious infinitely small quantities which

-mathematics had discovered and philosophy

-proceeded to explain. Some philosophers,

-\index{Berkeley, Bishop}%

-Bishop Berkeley, for instance, correctly denied

-the validity of the whole idea, though for

-reasons other than those indicated here. But

-the curious fact remained that, despite all

-criticisms of the foundations of the subject,

-there could be no doubt but that the mathematical

-\PageSep{227}

-procedure was substantially right. In

-fact, the subject was right, though the explanations

-were wrong. It is this possibility of

-being right, albeit with entirely wrong explanations

-as to what is being done, that so

-often makes external criticism---that is so far

-as it is meant to stop the pursuit of a method---singularly

-barren and futile in the progress of

-science. The instinct of trained observers,

-and their sense of curiosity, due to the fact

-that they are obviously getting at something,

-are far safer guides. Anyhow the general

-effect of the success of the Differential Calculus

-was to generate a large amount of bad philosophy,

-centring round the idea of the infinitely

-small. The relics of this verbiage

-may still be found in the explanations of

-many elementary mathematical text-books on

-the Differential Calculus. It is a safe rule to

-apply that, when a mathematical or philosophical

-author writes with a misty profundity,

-he is talking nonsense.

-\medskip

-

-Newton would have phrased the question

-\index{Limit of a Function|EtSeq}%

-by saying that, as $h$~approaches zero, in the

-limit $2x + h$ becomes~$2x$. It is our task so to

-explain this statement as to show that it does

-not in reality covertly assume the existence

-of Leibniz's infinitely small quantities. In

-reading over the Newtonian method of statement,

-it is tempting to seek simplicity by

-\PageSep{228}

-saying that $2x + h$ is~$2x$, when $h$~is zero. But

-this will not do; for it thereby abolishes the

-interval from $x$ to~$x + h$, over which the average

-increase was calculated. The problem is, how

-to keep an interval of length~$h$ over which to

-calculate the average increase, and at the same

-time to treat~$h$ as if it were zero. Newton did

-this by the conception of a limit, and we now

-\index{Weierstrass}%

-proceed to give Weierstrass's explanation of

-its real meaning.

-

-In the first place notice that, in discussing

-$2x + h$, we have been considering~$x$ as fixed in

-value and $h$~as varying. In other words $x$~has

-been treated as a ``constant'' variable,

-or parameter, as explained in \ChapRef{IX}.;

-and we have really been considering $2x + h$ as

-a function of the argument~$h$. Hence we can

-generalize the question on hand, and ask

-what we mean by saying that the function~$f(h)$

-tends to the limit~$l$, say, as its argument~$h$

-tends to the value zero. But again we shall

-see that the special value \emph{zero} for the argument

-does not belong to the essence of the subject;

-and again we generalize still further, and ask,

-what we mean by saying that the function~$f(h)$

-tends to the limit~$l$ as $h$~tends to the value~$a$.

-

-Now, according to the Weierstrassian explanation

-the whole idea of $h$~tending to the

-value~$a$, though it gives a sort of metaphorical

-picture of what we are driving at, is really off

-the point entirely. Indeed it is fairly obvious

-\PageSep{229}

-that, as long as we retain anything like ``$h$~tending

-to~$a$,'' as a fundamental idea, we are

-really in the clutches of the infinitely small;

-for we imply the notion of $h$~being infinitely

-near to~$a$. This is just what we want to get

-rid of.

-

-Accordingly, we shall yet again restate our

-phrase to be explained, and ask what we

-mean by saying that the limit of the function~$f(h)$

-at~$a$ is~$l$.

-

-The limit of~$f(h)$ at~$a$ is a property of the

-\index{Standard of Approximation|EtSeq}%

-neighbourhood of~$a$, where ``neighbourhood''

-is used in the sense defined in \ChapRef{XI}.\

-during the discussion of the continuity of

-functions. The value of the function~$f(h)$ at~$a$

-is~$f(a)$; but the limit is distinct in idea

-from the value, and may be different from

-it, and may exist when the value has not

-been defined. We shall also use the term

-``standard of approximation'' in the sense

-in which it is defined in \ChapRef{XI}. In

-fact, in the definition of ``continuity'' given

-towards the end of that chapter we have

-practically defined a limit. The definition of

-a limit is:---

-

-A function~$f(x)$ has the limit~$l$ at a value~$a$

-of its argument~$x$, when in the neighbourhood

-of~$a$ its values approximate to~$l$ within

-\emph{every} standard of approximation.

-

-Compare this definition with that already

-given for continuity, namely:---

-\PageSep{230}

-

-A function~$f(x)$ is continuous at a value~$a$

-of its argument, when in the neighbourhood

-of~$a$ its values approximate to its value at~$a$

-within \emph{every} standard of approximation.

-

-It is at once evident that a function is continuous

-at~$a$ when (i)~it possesses a limit at~$a$,

-and (ii)~that limit is equal to its value at~$a$.

-Thus the illustrations of continuity which

-have been given at the end of \ChapRef{XI}.\ are

-illustrations of the idea of a limit, namely,

-they were all directed to proving that $f(a)$~was

-the limit of~$f(x)$ at~$a$ for the functions

-considered and the value of~$a$ considered. It

-is really more instructive to consider the

-limit at a point where a function is not continuous.

-For example, consider the function

-of which the graph is given in \Fig[fig.]{20} of \ChapRef{XI}.

-This function~$f(x)$ is defined to have

-the value~$1$ for all values of the argument

-except the integers $0$,~$1$, $2$, $3$,~etc., and for these

-integral values it has the value~$0$. Now let

-us think of its limit when $x = 3$. We notice

-that in the definition of the limit the value

-of the function at~$a$ (in this case, $a = 3$) is excluded.

-But, excluding~$f(3)$, the values of~$f(x)$,

-when $x$~lies within any interval which

-(i)~contains $3$ not as an end-point, and (ii)~does

-not extend so far as $2$ and~$4$, are all

-equal to~$1$; and hence these values approximate

-to~$1$ within every standard of approximation.

-Hence $1$~is the limit of~$f(x)$ at the

-\PageSep{231}

-value~$3$ of the argument~$x$, but by definition

-$f(3) = 0$.

-

-This is an instance of a function which

-possesses both a value and a limit at the

-value~$3$ of the argument, but the value is not

-equal to the limit. At the end of \ChapRef{XI}.\

-the function~$x^{2}$ was considered at the

-value~$2$ of the argument. Its value at~$2$ is~$2^{2}$,

-\ie~$4$, and it was proved that its limit is also~$4$.

-Thus here we have a function with a

-value and a limit which are equal.

-

-Finally we come to the case which is essentially

-important for our purposes, namely, to

-a function which possesses a limit, but no

-defined value at a certain value of its argument.

-We need not go far to look for

-such a function, $\dfrac{2x}{x}$~will serve our purpose.

-Now in any mathematical book, we might

-find the equation, $\dfrac{2x}{x} = 2$, written without

-hesitation or comment. But there is a difficulty

-in this; for when $x$~is zero, $\dfrac{2x}{x} = \dfrac{0}{0}$; and

-$\dfrac{0}{0}$~has no defined meaning. Thus the value

-of the function~$\dfrac{2x}{x}$ at $x = 0$ has no defined

-\PageSep{232}

-meaning. But for every other value of~$x$,

-the value of the function~$\dfrac{2x}{x}$ is~$2$. Thus the

-limit of~$\dfrac{2x}{x}$ at $x = 0$ is~$2$, and it has no value

-at $x = 0$. Similarly the limit of~$\dfrac{x^{2}}{x}$ at $x = a$ is~$a$

-whatever $a$~may be, so that the limit of~$\dfrac{x^{2}}{x}$

-at $x = 0$ is~$0$. But the value of~$\dfrac{x^{2}}{x}$ at $x = 0$

-takes the form~$\dfrac{0}{0}$, which has no defined

-meaning. Thus the function~$\dfrac{x^{2}}{x}$ has a limit

-but no value at~$0$.

-

-We now come back to the problem from

-which we started this discussion on the nature

-of a limit. How are we going to define the

-rate of increase of the function~$x^{2}$ at any

-value~$x$ of its argument. Our answer is that

-this rate of increase is the limit of the function

-$\dfrac{(x + h)^{2} - x^{2}}{h}$ at the value zero for its

-argument~$h$. (Note that $x$~is here a ``constant.'')

-Let us see how this answer works

-\PageSep{233}

-in the light of our definition of a limit. We

-have

-\[

-\frac{(x + h)^{2} - x^{2}}{h}

-  = \frac{2hx + h^{2}}{h}

-  = \frac{h(2x + h)}{h}\Add{.}

-\]

-

-Now in finding the limit of~$\dfrac{h(2x + h)}{h}$ at the

-value~$0$ of the argument~$h$, the value (if any)

-of the function at $h = 0$ is excluded. But for

-all values of~$h$, except $h = 0$, we can divide

-through by~$h$. Thus the limit of~$\dfrac{h(2x + h)}{h}$ at

-$h = 0$ is the same as that of $2x + h$ at $h = 0$.

-Now, whatever standard of approximation~$k$

-we choose to take, by considering the interval

-from $-\frac{1}{2}k$ to~$+\frac{1}{2}k$ we see that, for values of~$h$

-which fall within it, $2x + h$~differs from~$2x$

-by less than~$\frac{1}{2}k$, that is by less than~$k$. This

-is true for \emph{any} standard~$k$. Hence in the neighbourhood

-of the value~$0$ for~$h$, $2x + h$~approximates

-to~$2x$ within \emph{every} standard of approximation,

-and therefore $2x$~is the limit of~$2x + h$

-at $h = 0$. Hence by what has been said above

-$2x$~is the limit of $\dfrac{(x + h)^{2} - x^{2}}{h}$ at the value~$0$

-for~$h$. It follows, therefore, that $2x$~is what

-we have called the rate of increase of~$x^{2}$ at

-the value~$x$ of the argument. Thus this

-method conducts us to the same rate of increase

-\PageSep{234}

-for~$x^{2}$ as did the Leibnizian way of

-making $h$~grow ``infinitely small.''

-

-The more abstract terms ``differential coefficient,''

-\index{Differential Coefficient}%

-or ``derived function,'' are generally

-\index{Derived Function}%

-used for what we have hitherto called the

-``rate of increase'' of a function. The

-general definition is as follows: the differential

-coefficient of the function~$f(x)$ is the

-limit, if it exist, of the function $\dfrac{f(x + h) - f(x)}{h}$

-of the argument~$h$ at the value~$0$ of its argument.

-

-How have we, by this definition and the

-subsidiary definition of a limit, really managed

-to avoid the notion of ``infinitely small numbers''

-which so worried our mathematical

-forefathers? For them the difficulty arose

-because on the one hand they had to use an

-interval $x$~to $x + h$ over which to calculate

-the average increase, and, on the other hand,

-they finally wanted to put $h = 0$. The result

-was they seemed to be landed into the notion

-of an existent interval of zero size. Now

-how do we avoid this difficulty? In this

-way---we use the notion that corresponding

-to \emph{any} standard of approximation, \emph{some} interval

-with such and such properties can be

-found. The difference is that we have

-\index{Variable, The}%

-grasped the importance of the notion of ``the

-variable,'' and they had not done so. Thus,

-\PageSep{235}

-at the end of our exposition of the essential

-notions of mathematical analysis, we are led

-back to the ideas with which in \ChapRef{II}.\

-we commenced our enquiry---that in mathematics

-the fundamentally important ideas

-are those of ``\emph{some} things'' and ``\emph{any}

-things.''

-\PageSep{236}

-

-

-\Chapter{XVI}{Geometry}

-

-\First{Geometry}, like the rest of mathematics, is

-\index{Geometry|EtSeq}%

-abstract. In it the properties of the shapes

-and relative positions of things are studied.

-But we do not need to consider who is observing

-the things, or whether he becomes acquainted

-with them by sight or touch or

-hearing. In short, we ignore all particular

-sensations. Furthermore, particular things

-such as the Houses of Parliament, or the

-terrestrial globe are ignored. Every proposition

-refers to any things with such and

-such geometrical properties. Of course it

-helps our imagination to look at particular

-examples of spheres and cones and triangles

-and squares. But the propositions do not

-merely apply to the actual figures printed in

-the book, but to any such figures.

-

-Thus geometry, like algebra, is dominated

-by the ideas of ``any'' and ``some'' things.

-Also, in the same way it studies the interrelations

-of sets of things. For example, consider

-any two triangles $ABC$ and~$DEF$.

-\PageSep{237}

-

-What relations must exist between some of

-\index{Triangle}%

-the parts of these triangles, in order that the

-triangles may be in all respects equal? This

-is one of the first investigations undertaken

-in all elementary geometries. It is a study

-\Figure{33}

-of a certain set of possible correlations between

-the two triangles. The answer is that

-the triangles are in all respects equal, if:---

-Either, (a)~Two sides of the one and the included

-angle are respectively equal to two

-sides of the other and the included angle:

-

-Or, (b)~Two angles of the one and the side

-joining them are respectively equal to two

-angles of the other and the side joining them:

-

-Or, (c)~Three sides of the one are respectively

-equal to three sides of the other.

-

-This answer at once suggests a further enquiry.

-What is the nature of the correlation

-between the triangles, when the three angles

-of the one are respectively equal to the three

-angles of the other? This further investigation

-leads us on to the whole theory of similarity

-\index{Similarity}%

-\PageSep{238}

-(\Chg{cf.}{\cf}\ \ChapRef{XIII}.), which is another

-type of correlation.

-

-Again, to take another example, consider

-the internal structure of the triangle~$ABC$.

-Its sides and angles are inter-related---the

-greater angle is opposite to the greater side,

-and the base angles of an isosceles triangle

-are equal. If we proceed to trigonometry

-this correlation receives a more exact determination

-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.

-

-Also there is the still simpler correlation

-between the angles of the triangle, namely,

-that their sum is equal to two right angles;

-and between the three sides, namely, that the

-sum of the lengths of any two is greater than

-the length of the third\Add{.}

-

-Thus the true method to study geometry is

-to think of interesting simple figures, such as

-the triangle, the parallelogram, and the circle,

-and to investigate the correlations between

-their various parts. The geometer has in his

-mind not a detached proposition, but a figure

-with its various parts mutually inter-dependent.

-Just as in algebra, he generalizes the

-triangle into the polygon, and the side into

-\PageSep{239}

-the conic section. Or, pursuing a converse

-route, he classifies triangles according as they

-are equilateral, isosceles, or scalene, and

-polygons according to their number of sides,

-and conic sections according as they are hyperbolas,

-ellipses, or parabolas.

-

-The preceding examples illustrate how the

-fundamental ideas of geometry are exactly

-the same as those of algebra; except that

-algebra deals with numbers and geometry

-with lines, angles, areas, and other geometrical

-entities. This fundamental identity

-is one of the reasons why so many geometrical

-truths can be put into an algebraic dress.

-Thus if $A$,~$B$, and~$C$ are the numbers of degrees

-respectively in the angles of the triangle~$ABC$,

-the correlation between the angles is represented

-by the equation

-\[

-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

-quoted above are other examples of the same

-\index{Variable, The}%

-fact. Thus the notion of the variable and

-the correlation of variables is just as essential

-in geometry as it is in algebra.

-

-But the parallelism between geometry and

-algebra can be pushed still further, owing to

-the fact that lengths, areas, volumes, and

-\PageSep{240}

-angles are all measurable; so that, for example,

-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

-above, are examples of this fact. But it receives

-its crowning application in analytical

-geometry. This great subject is often misnamed

-as Analytical Conic Sections, thereby

-\index{Analytical Conic Sections}%

-fixing attention on merely one of its subdivisions.

-It is as though the great science

-of Anthropology were named the Study of

-Noses, owing to the fact that noses are a

-prominent part of the human body.

-

-Though the mathematical procedures in

-geometry and algebra are in essence identical

-and intertwined in their development, there

-is necessarily a fundamental distinction between

-the properties of space and the properties

-of number---in fact all the essential difference

-between space and number. The ``spaciness''

-of space and the ``numerosity'' of

-number are essentially different things, and

-must be directly apprehended. None of the

-applications of algebra to geometry or of

-geometry to algebra go any step on the road

-to obliterate this vital distinction.

-

-One very marked difference between space

-and number is that the former seems to be so

-much less abstract and fundamental than the

-\PageSep{241}

-latter. The number of the archangels can be

-counted just because they are things. When

-we once know that their names are Raphael,

-Gabriel, and Michael, and that these distinct

-names represent distinct beings, we know without

-further question that there are three of

-them. All the subtleties in the world about

-the nature of angelic existences cannot alter

-this fact, granting the premisses.

-

-But we are still quite in the dark as to their

-relation to space. Do they exist in space at

-all? Perhaps it is equally nonsense to say

-that they are here, or there, or anywhere, or

-everywhere. Their existence may simply have

-no relation to localities in space. Accordingly,

-while numbers must apply to all things,

-space need not do so.

-

-The perception of the locality of things

-would appear to accompany, or be involved

-in many, or all, of our sensations. It is independent

-of any particular sensation in the

-sense that it accompanies many sensations.

-But it is a special peculiarity of the things

-which we apprehend by our sensations. The

-direct apprehension of what we mean by the

-positions of things in respect to each other

-is a thing \Foreign{sui generis}, just as are the apprehensions

-of sounds, colours, tastes, and smells.

-At first sight therefore it would appear that

-mathematics, in so far as it includes geometry

-in its scope, is not abstract in the sense in

-\PageSep{242}

-which abstractness is ascribed to it in

-\ChapRef{I}.

-

-This, however, is a mistake; the truth being

-\index{Abstract Nature of Geometry|EtSeq}%

-that the ``spaciness'' of space does not enter

-into our geometrical \emph{reasoning} at all. It

-enters into the geometrical intuitions of

-mathematicians in ways personal and peculiar

-to each individual. But what enter into the

-reasoning are merely certain properties of

-things in space, or of things forming space,

-which properties are completely abstract in

-the sense in which abstract was defined in

-\ChapRef{I}.; these properties do not involve

-any peculiar space-apprehension or space-intuition

-or space-sensation. They are on

-exactly the same basis as the mathematical

-properties of number. Thus the space-intuition

-which is so essential an aid to the study

-of geometry is logically irrelevant: it does

-not enter into the premisses when they are

-properly stated, nor into any step of the reasoning.

-It has the practical importance of an

-example, which is essential for the stimulation

-of our thoughts. Examples are equally necessary

-to stimulate our thoughts on number.

-When we think of ``two'' and ``three'' we

-see strokes in a row, or balls in a heap, or

-some other physical aggregation of particular

-things. The peculiarity of geometry is the

-fixity and overwhelming importance of the

-one particular example which occurs to our

-\PageSep{243}

-minds. The abstract logical form of the

-propositions when fully stated is, ``If any

-collections of things have such and such

-abstract properties, they also have such and

-such other abstract properties.'' But what

-appears before the mind's eye is a collection

-of points, lines, surfaces, and volumes in the

-space: this example inevitably appears, and

-is the sole example which lends to the proposition

-its interest. However, for all its overwhelming

-importance, it is but an example.

-

-Geometry, viewed as a mathematical science,

-is a division of the more general science of

-order. It may be called the science of dimensional

-order; the qualification ``dimensional''

-has been introduced because the limitations,

-which reduce it to only a part of the general

-science of order, are such as to produce the

-regular relations of straight lines to planes,

-and of planes to the whole of space.

-

-It is easy to understand the practical importance

-of space in the formation of the

-scientific conception of an external physical

-world. On the one hand our space-perceptions

-are intertwined in our various sensations

-and connect them together. We normally

-judge that we touch an object in the same

-place as we see it; and even in abnormal

-cases we touch it in the same space as we see

-it, and this is the real fundamental fact which

-ties together our various sensations. Accordingly,

-\PageSep{244}

-the space perceptions are in a sense the

-common part of our sensations. Again it

-happens that the abstract properties of space

-form a large part of whatever is of spatial

-interest. It is not too much to say that to

-every property of space there corresponds an

-abstract mathematical statement. To take

-the most unfavourable instance, a curve may

-have a special beauty of shape: but to this

-shape there will correspond some abstract

-mathematical properties which go with this

-shape and no others.

-

-Thus to sum up: (1)~the properties of space

-which are investigated in geometry, like those

-of number, are properties belonging to things

-as things, and without special reference to

-any particular mode of apprehension: (2)~Space-perception

-accompanies our sensations,

-perhaps all of them, certainly many; but it

-does not seem to be a necessary quality of

-things that they should all exist in one space

-or in any space.

-\PageSep{245}

-

-

-\Chapter{XVII}{Quantity}

-

-\First{In} the previous chapter we pointed out

-\index{Quantity|EtSeq}%

-that lengths are measurable in terms of some

-unit length, areas in term of a unit area, and

-volumes in terms of a unit volume.

-

-When we have a set of things such as

-lengths which are measurable in terms of any

-one of them, we say that they are quantities

-of the same kind. Thus lengths are quantities

-of the same kind, so are areas, and so are

-volumes. But an area is not a quantity of

-the same kind as a length, nor is it of the

-same kind as a volume. Let us think a little

-more on what is meant by being measurable,

-taking lengths as an example.

-

-Lengths are measured by the foot-rule. By

-transporting the foot-rule from place to place

-we judge of the equality of lengths. Again,

-three adjacent lengths, each of one foot, form

-one whole length of three feet. Thus to

-measure lengths we have to determine the

-equality of lengths and the addition of lengths.

-When some test has been applied, such as the

-transporting of a foot-rule, we say that the

-lengths are equal; and when some process

-\PageSep{246}

-has been applied, so as to secure lengths being

-contiguous and not overlapping, we say that

-the lengths have been added to form one

-whole length. But we cannot arbitrarily take

-any test as the test of equality and any

-process as the process of addition. The results

-of operations of addition and of judgments

-of equality must be in accordance with

-certain preconceived conditions. For example,

-the addition of two greater lengths must

-yield a length greater than that yielded by

-the addition of two smaller lengths. These

-preconceived conditions when accurately formulated

-may be called axioms of quantity.

-The only question as to their truth or falsehood

-\index{Axioms of Quantity|EtSeq}%

-which can arise is whether, when the axioms

-are satisfied, we necessarily get what ordinary

-people call quantities. If we do not, then

-the name ``axioms of quantity'' is ill-judged---that

-is all.

-

-These axioms of quantity are entirely abstract,

-just as are the mathematical properties

-of space. They are the same for all quantities,

-and they presuppose no special mode of perception.

-The ideas associated with the notion

-of quantity are the means by which a continuum

-like a line, an area, or a volume can

-be split up into definite parts. Then these

-parts are counted; so that numbers can be

-used to determine the exact properties of a

-continuous whole.

-\PageSep{247}

-

-Our perception of the flow of time and of

-\index{Time|EtSeq}%

-the succession of events is a chief example

-of the application of these ideas of quantity.

-We measure time (as has been said in considering

-periodicity) by the repetition of

-similar events---the burning of successive

-inches of a uniform candle, the rotation of

-the earth relatively to the fixed stars, the

-rotation of the hands of a clock are all examples

-of such repetitions. Events of these

-types take the place of the foot-rule in relation

-to lengths. It is not necessary to assume

-that events of any one of these types are

-exactly equal in duration at each recurrence.

-What is necessary is that a rule should be

-known which will enable us to express the

-relative durations of, say, two examples of

-some type. For example, we may if we like

-suppose that the rate of the earth's rotation

-is decreasing, so that each day is longer than

-the preceding by some minute fraction of a

-second. Such a rule enables us to compare

-the length of any day with that of any other

-day. But what is essential is that one series

-of repetitions, such as successive days, should

-be taken as the standard series; and, if the

-various events of that series are not taken as

-of equal duration, that a rule should be

-stated which regulates the duration to be

-assigned to each day in terms of the duration

-of any other day.

-\PageSep{248}

-

-What then are the requisites which such

-a rule ought to have? In the first place it

-should lead to the assignment of nearly equal

-durations to events which common sense

-judges to possess equal durations. A rule

-which made days of violently different lengths,

-and which made the speeds of apparently

-similar operations vary utterly out of proportion

-to the apparent minuteness of their

-differences, would never do. Hence the first

-requisite is general agreement with common

-sense. But this is not sufficient absolutely

-to determine the rule, for common sense is a

-rough observer and very easily satisfied. The

-next requisite is that minute adjustments of

-the rule should be so made as to allow of the

-simplest possible statements of the laws of

-nature. For example, astronomers tell us

-that the earth's rotation is slowing down, so

-that each day gains in length by some inconceivably

-minute fraction of a second. Their

-only reason for their assertion (as stated more

-fully in the discussion of periodicity) is that

-without it they would have to abandon the

-Newtonian laws of motion. In order to keep

-\index{Laws of Motion}%

-the laws of motion simple, they alter the

-measure of time. This is a perfectly legitimate

-procedure so long as it is thoroughly

-understood.

-

-What has been said above about the abstract

-nature of the mathematical properties

-\PageSep{249}

-of space applies with appropriate verbal

-changes to the mathematical properties of

-time. A sense of the flux of time accompanies

-all our sensations and perceptions, and practically

-all that interests us in regard to time

-can be paralleled by the abstract mathematical

-properties which we ascribe to it.

-Conversely what has been said about the two

-requisites for the rule by which we determine

-the length of the day, also applies to the rule

-for determining the length of a yard measure---namely,

-the yard measure appears to retain

-the same length as it moves about. Accordingly,

-any rule must bring out that, apart

-from minute changes, it does remain of invariable

-length; Again, the second requisite

-is this, a definite rule for minute changes

-shall be stated which allows of the simplest

-expression of the laws of nature. For example,

-in accordance with the second requisite

-the yard measures are supposed to

-expand and contract with changes of temperature

-according to the substances which

-they are made of.

-

-Apart from the facts that our sensations

-are accompanied with perceptions of locality

-and of duration, and that lines, areas, volumes,

-and durations, are each in their way quantities,

-the theory of numbers would be of very

-subordinate use in the exploration of the laws

-of the Universe, As it is, physical science

-\PageSep{250}

-reposes on the main ideas of number, quantity,

-space, and time. The mathematical

-sciences associated with them do not form

-the whole of mathematics, but they are the

-substratum of mathematical physics as at

-present existing.

-

-

-\BackMatter

-\Appendix{Notes}

-

-\Note{A} (\Pageref{60}).---In reading these equations it must be noted

-that a bracket is used in mathematical symbolism to

-mean that the operations within it are to be performed

-first. Thus $(1 + 3) + 2$ directs us first to add $3$ to~$1$, and

-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).

-\]

-We perform first the operations in brackets and obtain

-\[

-2 × 7 = 6 + 8

-\]

-which is obviously true.

-

-

-\Note{B} (\Pageref{136}).---This fundamental ratio~$\dfrac{SP}{PN}$ is called the

-eccentricity of the curve. The shape of the curve, as

-\index{Eccentricity}%

-distinct from its scale or size, depends upon the value of

-its eccentricity. Thus it is wrong to think of ellipses

-in general or of hyperbolas in general as having in either

-case one definite shape. Ellipses with different eccentricities

-have different shapes, and their sizes depend

-upon the lengths of their major axes. An ellipse with

-small eccentricity is very nearly a circle, and an ellipse

-of eccentricity only slightly less than unity is a long

-flat oval. All parabolas have the same eccentricity and

-are therefore of the same shape, though they can be

-drawn to different scales.

-\PageSep{251}

-

-\Note{C} (\Pageref{204}).---If a series with all its terms positive is

-\index{Absolute Convergence}%

-\index{Convergence, Absolute}%

-convergent, the modified series found by making some

-terms positive and some negative according to any

-definite rule is also convergent. Each one of the set of

-series thus found, including the original series, is called

-``absolutely convergent.'' But it is possible for a series

-with terms partly positive and partly negative to be

-convergent, although the corresponding series with all

-its terms positive is divergent. For example, the series

-\[

-1 - \tfrac{1}{2} + \tfrac{1}{3} - \tfrac{1}{4} + \text{etc.}

-\]

-is convergent though we have just proved that

-\[

-1 + \tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{4} + \text{etc.}

-\]

-is divergent. Such convergent series, which are not

-absolutely convergent, are much more difficult to deal

-with than absolutely convergent series.

-

-

-\Appendix[Note on the Study of Mathematics]{Bibliography}

-

-

-\First{The} difficulty that beginners find in the study of this

-science is due to the large amount of technical detail which

-has been allowed to accumulate in the elementary text-books,

-obscuring the important ideas.

-

-The first subjects of study, apart from a knowledge of

-arithmetic which is presupposed, must be elementary

-geometry and elementary algebra. The courses in both

-subjects should be short, giving only the necessary ideas;

-the algebra should be studied graphically, so that in

-practice the ideas of elementary coordinate geometry are

-also being assimilated. The next pair of subjects should

-be elementary trigonometry and the coordinate geometry

-of the straight line and circle. The latter subject is a

-short one; for it really merges into the algebra. The

-student is then prepared to enter upon conic sections, a

-very short course of geometrical conic sections and a longer

-one of analytical conics. But in all these courses great

-care should be taken not to overload the mind with more

-\PageSep{252}

-detail than is necessary for the exemplification of the

-fundamental ideas.

-

-The differential calculus and afterwards the integral

-calculus now remain to be attacked on the same system.

-A good teacher will already have illustrated them by the

-consideration of special cases in the course on algebra

-and coordinate geometry. Some short book on three-dimensional

-geometry must be also read.

-

-This elementary course of mathematics is sufficient for

-some types of professional career. It is also the necessary

-preliminary for any one wishing to study the subject for

-its intrinsic interest. He is now prepared to commence

-on a more extended course. He must not, however, hope

-to be able to master it as a whole. The science has grown

-to such vast proportions that probably no living mathematician

-can claim to have achieved this.

-

-Passing to the serious treatises on the subject to be read

-\emph{after} this preliminary course, the following may be mentioned:

-Cremona's \Title{Pure Geometry} (English Translation,

-Clarendon Press, Oxford), Hobson's \Title{Treatise on Trigonometry},

-Chrystal's \Title{Treatise on Algebra} (2~volumes), Salmon's

-\Title{Conic Sections}, Lamb's \Title{Differential Calculus}, and some

-book on \Title{Differential Equations}. The student will probably

-not desire to direct equal attention to all these subjects,

-but will study one or more of them, according as his interest

-dictates. He will then be prepared to select more advanced

-works for himself, and to plunge into the higher

-parts of the subject. If his interest lies in analysis, he

-should now master an elementary treatise on the theory

-of Functions of the Complex Variable; if he prefers to

-specialize in Geometry, he must now proceed to the

-standard treatises on the Analytical Geometry of three

-dimensions. But at this stage of his career in learning

-he will not require the advice of this note.

-

-I have deliberately refrained from mentioning any

-elementary works. They are very numerous, and of

-various merits, but none of such outstanding superiority

-as to require special mention by name to the exclusion

-of all the others.

-

-

-%[** TN: Index text]

-% ** Page 253

-\printindex

-\iffalse

-

-Abel 156

-

-Abscissa 95

-

-Absolute Convergence 251

-

-Abstract Nature of Geometry|EtSeq 242

-

-Abstractness (\emph{defined}) 9, 13

-

-Adams 220

-

-Addition-Theorem 212

-

-Ahmes 71

-

-Alexander the Great 128, 129

-

-Algebra, Fundamental Laws of 60

-

-Ampere@Ampère 34

-

-Analytical Conic Sections 240

-

-Apollonius of Perga 131, 134

-

-Approximation|EtSeq 197

-

-Arabic Notation|EtSeq 58

-

-Archimedes|EtSeq 37

-

-Argument of a Function 146

-

-Aristotle 30, 42, 128

-

-Astronomy 137, 173, 174

-

-Axes 125

-

-Axioms of Quantity|EtSeq 246

-

-Bacon 156

-

-Ball, W. W. R. 58

-

-Beaconsfield, Lord 41

-

-Berkeley, Bishop 226

-

-Bhaskara 58

-

-Cantor, Georg 79

-

-Circle 120, 130

-

-Circle@Circle|EtSeq 180

-

-Circular Cylinder 143

-

-Clerk Maxwell 34, 35

-

-Columbus 122

-

-Compact Series 76

-

-Complex Quantities 109

-

-Conic Sections|EtSeq 128

-

-Constants 69, 117

-

-Continuous Functions@Continuous Functions|EtSeq 150

-

-Continuous Functions@Continuous Functions (\emph{defined}) 162

-

-Convergence, Absolute 251

-

-Convergent|EtSeq 203

-

-Coordinate Geometry|EtSeq 112

-

-Coordinates 95

-

-Copernicus 45, 137

-

-%[** TN: Entry italicized in the original, "Sine" not italicized]

-Cosine|EtSeq 182

-

-Coulomb 33

-

-Cross Ratio 140

-

-Darwin 138, 220

-

-Derived Function 234

-

-Descartes 95, 113, 116, 122, 218

-

-Differential Calculus|EtSeq 217

-

-Differential Coefficient 234

-

-Directrix 135

-

-Discontinuous Functions|EtSeq 150

-

-Distance 30

-

-Divergent|EtSeq 203

-% ** Page 254

-

-Dynamical Explanation 13, 14

-

-Dynamical Explanation@Dynamical Explanation|EtSeq 47

-

-Dynamics 30

-

-Dynamics@Dynamics|EtSeq 43

-

-Eccentricity 250

-

-Electric Current 33

-

-Electricity|EtSeq 32

-

-Electromagnetism|EtSeq 31

-

-Ellipse 45, 120

-

-Ellipse@Ellipse|EtSeq 130

-

-Euclid 114

-

-Exponential Series|EtSeq 211

-

-Faraday 34

-

-Fermat 218

-

-Fluxions 219

-

-Focus 120, 135

-

-Force 30

-

-Form, Algebraic@Form, Algebraic|EtSeq 66

-

-Form, Algebraic 82, 117

-

-Fourier's Theorem 191

-

-Fractions|EtSeq 71

-

-Franklin 32, 122

-

-Function|EtSeq 144

-

-Galileo@Galileo|EtSeq 42

-

-Galileo 30, 122

-

-Galvani 33

-

-Generality in Mathematics 82

-

-Geometrical Series|EtSeq 206

-

-Geometry 36

-

-Geometry@Geometry|EtSeq 236

-

-Gilbert, Dr. 32

-

-Graphs|EtSeq 148

-

-Gravitation 29, 139

-

-Halley 139

-

-Harmonic Analysis 192

-

-Harriot, Thomas 66

-

-Herz 35

-

-Hiero 38

-

-Hipparchus 173

-

-Hyperbola|EtSeq 131

-

-Imaginary Numbers|EtSeq 87

-

-Imaginary Quantities 109

-

-Incommensurable Ratios|EtSeq 72

-

-Infinitely Small Quantities|EtSeq 226

-

-Integral Calculus 222

-

-Interval|EtSeq 158

-

-Kepler 45, 46, 137, 138

-

-Kepler's Laws 138

-

-Laputa 10

-

-Laws of Motion@Laws of Motion|EtSeq 167

-

-Laws of Motion 248

-

-Leibniz 16

-

-Leibniz@Leibniz|EtSeq 218

-

-Leonardo da Vinci 42

-

-Leverrier 220

-

-Light 35

-

-Limit of a Function|EtSeq 227

-

-Limit of a Series|EtSeq 199

-

-Limits 77

-

-Locus@Locus|EtSeq 121

-

-Locus 141

-

-Macaulay 156

-

-Malthus 138

-

-Marcellus 37

-

-Mass 30

-

-Mechanics 46

-

-Menaechmus 128, 129

-

-Motion, First Law of 43

-% ** Page 255

-

-Neighbourhood|EtSeq 159

-

-Newton 10, 16, 30, 34, 37, 38, 43, 46, 139

-

-Newton@Newton|EtSeq 218

-

-Non-Uniform Convergence|EtSeq 208

-

-Normal Error, Curve of 214

-

-Oersted@Öersted 34

-

-Order|EtSeq 194

-

-Order, Type of@Order, Type of|EtSeq 75

-

-Order, Type of 196

-

-Ordered Couples|EtSeq 93

-

-Ordinate 95

-

-Origin 95, 126

-

-Pappus 135, 136

-

-Parabola|EtSeq 131

-

-Parallelogram Law@Parallelogram Law|EtSeq 51

-

-Parallelogram Law 99, 126

-

-Parameters 69, 117

-

-Pencils 140

-

-Period 170

-

-Period@Period|EtSeq 189

-

-Periodicity@Periodicity|EtSeq 164

-

-Periodicity 188, 216

-

-Pitt, William 194

-

-Pizarro 122

-

-Plutarch 37

-

-Positive and Negative Numbers|EtSeq 83

-

-Projective Geometry 139

-

-Ptolemy 137, 173

-

-Pythagoras 18

-

-Quantity|EtSeq 245

-

-Rate of Increase of Functions|EtSeq 220

-

-Ratio|EtSeq 72

-

-Real Numbers|EtSeq 73

-

-Rectangle 57

-

-Relations between Variables|EtSeq 18

-

-Resonance 170, 171

-

-Rosebery, Lord 194

-

-Scale of a Map 178

-

-Seidel 210

-

-Series|EtSeq 74, 194

-

-Shelley (quotation from) 217

-

-Similarity@Similarity|EtSeq 177

-

-Similarity 237

-

-Sine|EtSeq 182

-

-Specific Gravity 41

-

-Squaring the Circle 187

-

-Standard of Approximation|EtSeq 159, 201, 229

-

-Steps@Steps|EtSeq 79

-

-Steps 96

-

-Stifel 85

-

-Stokes, Sir George 210

-

-Sum to Infinity|EtSeq 201

-

-Surveys|EtSeq 176

-

-Swift 10

-

-Tangents 221, 222

-

-Taylor's Theorem 156, 157

-

-Time|EtSeq 166, 247

-

-Transportation, Vector of|EtSeq 54

-

-Triangle@Triangle|EtSeq 176

-

-Triangle 237

-

-Triangulation 177

-

-Trigonometry|EtSeq 173

-

-Uniform Convergence|EtSeq 208

-

-Unknown, The 17, 23

-% ** Page 256

-

-Value of a Function 146

-

-Variable, The 18, 24, 49, 82, 234, 239

-

-Variable Function 147

-

-Vectors@Vectors|EtSeq 51

-

-Vectors 85, 96

-

-Vertex 134

-

-Volta 33

-

-Wallace 220

-

-Weierstrass 156, 226, 228

-

-Zero@Zero|EtSeq 63

-

-Zero 103

-%[** TN: End of index]

-\fi

-

-% Printed by Hazell, Walton \& Viney, Ld., London and Aylesbury.

-%[** TN: Raw OCR output of book catalog follows]

-\iffalse

-

-The

-

-Home University

-Library

-

-

-

-of Modern

-Knowledge

-

-

-

-jj Comprehensive Series of New

-and Specially Written (Books

-

-

-

-EDITORS:

-

-PROF. GILBERT MURRAY, D.Litt., LL.D., F.B.A.

-HERBERT FISHER, M.A., F.B.A.

-PKOF. J. ARTHUR THOMSON, M.A.

-PROF. WM. T. BREWSTER, M.A.

-

-The Home University Library

-

-" Is without the slightest doubt the pioneer in supplying serious literature

-for a large section of the public who are interested in the liberal educa-

-tion of the State." The Daily Mail.

-

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-University Library that its volumes do not merely attempt to feed

-ignorance with knowledge. The authors noticeably realise that the

-simple willing appetite of sharp-set ignorance is not specially common

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-

-"Each volume represents a three-hours' traffic with the talking-power

-of a good brain, operating with the ease and interesting freedom of a

-specialist dealing with his own subject. ... A series which promises to

-perform a real social service." The Times.

-

-"We can think of no series now being issued which better deserves

-support." The Observer.

-

-

-

-We think if they were given as prizes in place of the more costly

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-di:

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-id prol

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-

-

-

-ispensed on prize days, the pupils would

-rofit. If the publishers want a motto for the

-

-

-

-rubbish that is wont to be

-

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-

-series they might well take: ' Infinite riches in a little room.'" Irish

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-among earnest readers; but its wider and* sustained success, surely,

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-

-

-

-to give it

-

-

-

-]/- net

-in cloth

-

-

-256 Pages

-

-

-2/6 net

-in leather

-

-

-

-History and (geography

-

-

-

-3. THE FRENCH REVOLUTION

-

-By HILAIKE BELLOC, M.A. (With Maps.) "It is coloured with all the

-militancy of the author's temperament." Daily News.

-

-4. HISTORY OF WAR AND PEACE

-

-By G. H. FERRIS. The Rt. Hon. JAMES BRYCE writes: " I have read it with

-much interest and pleasure, admiring the skill with which you have managed

-to compress so many facts and views into so small a volume."

-

-8. POLAR EXPLORATION

-

-By Dr W. S. BRUCE, F.R.S.E., Leader of the "Scotia" Expedition. (With

-Maps.) "A very freshly written and interesting narrative." The Times.

-"A fascinating book." Portsmouth Times.

-

-12. THE OPENING-UP OF AFRICA

-

-By Sir H. H. JOHNSTON. G.C.M.G., K.C.B., D.Sc., F.Z.S. (With Maps.)

-" The Home University Library is much enriched by this excellent work."

-Daily Mail.

-

-13. MEDIAEVAL EUROPE

-

-By H. W. C. DAVIS, M.A. (With Maps.) "One more illustration of the

-fact that it takes a complete master of the subject to write briefly upon it."

-Manchester Guardian.

-

-14. THE PAPACY \&* MODERN TIMES (1303-1870)

-

-By WILLIAM BARRY, D.D. "Dr Barry has a wide range of knowledge

-and an artist's power of selection." Manchester Guardian.

-

-23. HISTORY OF OUR TIME, 1885-1911

-

-By G. P. GOOCH, M.A. " Mr Gooch contrives to breathe vitality into his story,

-and to give us the flesh as well as the bones of recent happenings." Observer.

-

-25. THE CIVILISATION OF CHINA

-

-By H. A. GILES, LL.D., Professor of Chinese in the University of Cambridge.

-"In all the mass of facts, Professor Giles never becomes dull. He is always

-ready with a ghost story or a street adventure for the reader's recreation."

-Spectator.

-

-29. THE DA WN OF HISTORY

-

-By J.L.MYRES, M. A., F.S. A., Wykeham Professor of Ancient History, Oxford.

-"There is not a page in it that is not suggestive." Manchester Guardian.

-

-33. THE HISTORY OF ENGLAND:

-A Study in Political Evolution.

-

-By Prof. A. F. POLLARD, M.A. With a Chronological Table. " It takes its

-place at once among the authoritative works on English history." Observer.

-

-34. CANADA

-

-By A. G. BRADLEY. " Who knows Canada, better than Mr A. G. Bradley? "

-Daily Chronicle. "The volume makes an immediate appeal to the man who

-wants to know something vivid and true about Canada." Canadian Gazette.

-

-

-

-37. PEOPLES 6* PROBLEMS OF INDIA

-

-By Sir T. W. HOLDERNESS, K.C.S.I., Secretary of the Revenue, Statistics,

-! and Commerce Department of the India Office. "Just the book which news-

-paper readers require to-day, and a marvel of comprehensiveness." Pall

-\ Mall Gazette.

-

-42. ROME

-

-By W. WARDE FOWLHR, M.A. " A masterly sketch of Roman character and

-of what it did for the world." The Spectator. "It has all the lucidity and

-charm of presentation we expect from this writer." Manchester Guardian.

-

-48. THE AMERICAN CIVIL WAR

-

-By F. L. PAXSON, Professor of American History, Wisconsin University.

-(With Maps.) "A stirring study." The Guardian.

-

-51. WARFARE IN BRITAIN

-

-By HILAIRE BELLOC, M.A. An account of how and where great battles of the

-past were fought on British soil, the roads and physical conditions determining

-the island's strategy, the castles, walled towns, etc.

-

-55. MASTER MARINERS

-

-By J. R. SPEARS. The romance of the sea, the great voyages of discovery,

-naval battles, the heroism of the sailor, and the development of the ship, from

-ancient times to to-day.

-

-IN PREPARATION

-

-ANCIENT GREECE. By Prof. GILBERT MURRAY, D.Litt., LL.D., F.B.A

-ANCIENT EGYPT. By F. LL. GRIFFITH, M.A.

-THE ANCIENT EAST. By D. G. HOGARTH, M.A., F.B.A.

-A SHORT h'ISTOR YOFEUROPE. By HERBERT FISHER, M. A., F.B.A.

-PREHISTORIC BRITAIN. By ROBERT MUNRO, M.A., M.D., LL.D.

-THE BYZANTINE EMPIRE. By NORMAN H. BAVNES.

-THE REFORM A TION. By Principal LINDSAY, LL.D.

-NAPOLEON. By HERBERT FISHER, M.A., F.B.A.

-A SHORT HISTORY OF RUSSIA. By Prof. MILYOUKOV.

-MODERN TURKEY. By D. G. HOGARTH, M.A.

-FRANCE OF TO-DAY. By ALBERT THOMAS.

-GERMANY OF TO-DA Y. By CHARLES TOWER.

-THE NAVY AND SEA POWER. By DAVID HANNAY.

-HISTORY OF SCOTLAND. By R. S. RAIT, M.A.

-SOUTH AMERICA. By Prof. W. R. SHEPHERD.

-LONDON. By Sir LAURENCE GOMME, F.S.A.

-

-HISTORY AND LITERATURE OF SPAIN. By J. FITZMAURICE-

-KELLY, F.B.A., Litt.D.

-

-

-

-Literature and

-

-

-

-2. SHAKESPEARE

-

-By JOHN MASEFIELD. " The book is a joy. We have had half-a-dozen more

-learned books on Shakespeare in the last few years, but not one so wise."

-Manchester Guardian.

-

-27. ENGLISH LITERATURE: MODERN

-

-By G. H. MAIR, M.A. " Altogether a fresh and individual book." Olstrver.

-

-35. LANDMARKS IN FRENCH LITERATURE

-

-By G. L. STRACHEY. " Mr Strachey is to be congratulated on his courage and

-success. It is difficult to imagine how a better account of French Literature

-could be given in 250 small pages than he has given here." The Times.

-

-

-

-39- ARCHITECTURE

-

-By Prof. W. R. LETHABY. (Over forty Illustrations.) " Popular guide-books

-to architecture are, as a rule, not worth ranch. This volume is a welcome excep-

-tion." Building News. " Delightfully bright reading." Christian World.

-

-43. ENGLISH LITERATURE: MEDIAEVAL

-

-By Prof. W. P. KER, M.A. "Prof. Ker has long proved his worth as one of

-the soundest scholars in English we have, and he is the very man to put an

-outline of English Mediaeval Literature before the uninstructed public. His

-knowledge and taste are unimpeachable, and his style is effective, simple, yet

-never dry." The Athemeum.

-

-45. THE ENGLISH LANGUAGE

-

-By L. PEARSALL SMI-TH, M.A. "A wholly fascinating study of the different

-streams that went to the making of the great river of the English speech."

-Daily News.

-

-52. GREAT WRITERS OF AMERICA

-

-By Prof. J. EKSKINE and Prof. W. P. TRENT. A popular sketch by two

-foremost authorities.

-

-IN PREPARATION

-

-ANCIENT ART AND RITUAL. By Miss JANE HARRISON, LL.D.,

-

-D.Litt.

-

-GREEK LITERA TURE. By Prof. GILBERT MURRAY, D.Litt.

-LA TIN LITER A TURE. By Prof. J. S. PHILLIMORE.

-CHA UCER AND HIS TIME. By Miss G. E. HADOW.

-THE RENAISSANCE. By Mrs R. A. TAYLOR.

-

-ITALIAN A RTOF THE RENAISSANCE. By ROGER E. FRY, M.A.

-THE ART OF PAINTING. By Sir FREUERICK WEDMORE.

-DR JOHNSON AND HIS CIRCLE. By JOHN BAILEY, M.A.

-THE VIC IORIAN AGE. By G. K- CHESTERTON.

-ENGLISH COMPOSITION. By Prof. WM. T. BREWSTER.

-GREA T WRITERS OF RUSSIA. By C. T. HAGBERG WRIGHT, LL.D.

-THE LITERATURE OF GERMANY. By Prof. J. G. ROBERTSON,

-

-M.A., Ph.D.

-SCANDINAVIAN HISTORY AND LITERATURE. By T. C.

-

-SNOW, M.A.

-

-

-

-Science

-

-

-

-7. MODERN GEOGRAPHY

-

-By Dr MARION NEWBIGIN. (Illustrated.) "Geography, again: what a dull,

-tedious study that was wont to be I . . . But Miss Marion Newbigin invests its

-dry bones with the flesh and blood of romantic interest, taking stock of

-geography as a fairy-book of science." Daily Telegraph.

-

-9. THE EVOLUTION OF PLANTS

-

-By Dr D. H. SCOTT, M.A., F.R.S., late Hon. Keeper of the Jodrell Laboratory,

-Kew. (Fully illustrated.) "The information which the book provides is as

-trustworthy as first-band knowledge can make it. ... Dr Scott's candid and

-familiar style makes the difficult subject both fascinating and easy."

-Gardeners' Chronicle.

-

-17. HEALTH AND DISEASE

-

-By W. LESLIE MACKKNZIE, M.D., Local Government Board, Edinburgh.

-"The science of public health administration has had no abler or more attractive

-exponent than Dr Mackenzie. He adds to a thorough grasp of the problems

-an illuminating style, and an arresting manner of treating a subject often

-dull and sometimes unsavoury." Economist.

-

-

-

-1 8. INTRODUCTION TO MATHEMATICS

-

-' By A. N. WHITEHEAD, Sc.D., F.R.S. (With Diagrams.) "MrWhitehead

-has discharged with conspicuous success the task he is so exceptionally qualified

-

-I to undertake. For he is one of our great authorities upon the foundations of the

-science, and has the breadth of view which is so requisite in presenting to the

-reader its aims. His exposition is clear and striking." Westminster Gazette.

-

-19. THE ANIMAL WORLD

-

-By Professor F. W. GAMBLE, D.Sc., F.R.S. With Introduction hy Sir Oliver

-Lodge. (Many Illustrations.) " A delightful and instructive epitome of animal

-(and vegetable) life. ... A most fascinating and suggestive survey." Morning

-Post.

-

-20. EVOLUTION

-

-By Professor J. ARTHUR THOMSON and Professor PATRICK GEDDES. "A

-many-coloured and romantic panorama, opening up, like no other book we know,

-a rational vision of world-development." Belfast News-Letter.

-

-22. CRIME AND INSANITY

-

-By Dr C. A. MERCIER, F.R.C.P., F.R.C.S., Author of "Text-Book of In-

-sanity," etc- " Furnishes much valuable information from one occupying the

-highest position among medico-legal psychologists." Asylum NCVJS.

-

-28. PSYCHICAL RESEARCH

-

-

-

-and thus what he has to say on thought-reading, hypnotism, telepathy, crystal-

-vision, spiritualism, divinings, and so on, will be read with avidity." Dundee

-

-

-

-

-31. ASTRONOMY

-

-By A. R. HINKS, M.A., Chief Assistant, Cambridge Observatory. "Original

-in thought, eclectic in substance, and critical in treatment. . . . No better

-little book is available." School World.

-

-32. INTRODUCTION TO SCIENCE

-

-By J. ARTHUR THOMSON, M.A., Regius Professor of Natural History, Aberdeen

-University. " Professor Thomson's delightful literary style is well known; and

-here he discourses freshly and easily on the methods of science and its relations

-with philosophy, art, religion, and practical life." Aberdeen Journal,

-

-36.

-

-

-

-By H. N. DICKSON, D.Sc. Oxon., M.A., F.R.S.E., President of the Royal

-Meteorological Society; Professor of Geography in University College, Reading.

-(With Diagrams.) "The author has succeeded in presenting in a very lucid

-and agreeable manner the causes of the movement of the atmosphere and of

-the more stable winds." Manchester Guardian.

-

-41. ANTHROPOLOGY

-

-By R R. MARETT, M.A., Reade

-"An absolutely perfect handboo

-fascinating and human that it bea

-

-44. THE PRINCIPLES OF PHYSIOLOGY

-

-By Prof. J. G. McKENDRiCK, M.D. " It is a delightful and wonderfully com-

-prehensive handling of a subject which, while of importance to all, does not

-readily lend itself to untechnical explanation. . . . The little book is more than

-a mere repository of knowledge; upon every page of it is stamped the impress

-of a creative imagination." Glasgow Herald.

-

-

-

-By R. R. MARETT, M.A., Reader in Social Anthropology in Oxford University.

-"An absolutely perfect handbook, so clear that a child could understand it, so

-fascinating and human that it beats fiction ' to a frazzle.' " Morning Leader.

-

-

-

-46. MATTER AND ENERGY

-

-By F. SODDY, M.A., F.R.S. "A most fascinating and instructive account or

-the great facts of physical science, concerning which our knowledge, of later

-years, has made such wonderful progress." The Bookseller.

-

-49. PSYCHOLOGY, THE STUDY OF BEHAVIOUR

-

-By Prof. W. McDouGALL, F.R.S., M.B. "A happy example of the non-

-technical handling of an unwieldy science, suggesting rather than dogmatising.

-It should whet appetites for deeper study." Christian World.

-

-53. THE MAKING OF THE EARTH

-

-ByProf.J.W. GREGORY, F.R.S. (With 38 Maps and Figures.) The Professor

-of Geology at Glasgow describes the origin of the earth, the formation and

-changes of its surface and structure, its geological history, the first appearance

-of life, and its influence upon the globe.

-

-57. THE HUMAN BODY

-

-By A. KEITH, M.D., LL,D., Conservator of Museum and Hunterian Pro-

-fessor, Royal College of Surgeons. (Illustrated.) The work of the dissecting-

-room is described, and among other subjects dealt with are: the development

-of the body; malformations and monstrosities; changes of youth and age; sex

-differences, are they increasing or decreasing? race characters; bodily features

-as indexes of mental character; degeneration and regeneration; and the

-genealogy and antiquity of man.

-

-58. ELECTRICITY

-

-By GisBERT KAPP, D.Eng., M.I.E.E., M.I.C.E., Professor of Electrical

-Engineering in the University of Birmingham. (Illustrated.) Deals with

-frictional and contact electricity; potential; electrification by mechanical

-means; the electric current; the dynamics of electric currents; alternating

-currents; the distribution of electricity, etc.

-

-IN PREPARATION

-

-CHEMISTRY. Py Prof. R. MELDOLA, F.R.S.

-

-THE MINERAL WORLD. By Sir T. H. HOLLAND, K.C.I. E., D.Sc.

-

-PLANT LII-'E. By Prof. J. B. FARMER, F.R.S.

-

-NERVES. By Prof. D. FRASER HARRIS, M.D., D.Sc.

-

-A STUDY OF SEX. By Prof. J. A. THOMSON and Prof. PATRICK GEDDES.

-

-THE GROWTH OF EUROPE. By Prof. GRKNVILLE COLE.

-

-

-

-Philosophy and "Religion

-

-

-

-ig's

-:tate

-

-

-

-15. MOHAMMEDANISM

-

-By Prof. D. S. MARGOLIOUTH, M.A., D.Litt. "This generous shilling':

-worth of wisdom. ... A delicate, humorous, and most responsible tractati

-by an illuminative professor." Daily Mail.

-

-40. THE PROBLEMS OF PHILOSOPHY

-

-By the Hon. BERTRAND RUSSELL, F.R.S.: 'A book that the ' man in the

-street ' will recognise at once to be a boon. . . . Consistently lucid and non-

-technical throughout." Christian World.

-

-47. BUDDHISM

-

-

-

-go. NONCONFORMITY: Its ORIGIN and PROGRESS

-

-I'.'- Principal W. B. SELBIE, M.A. "The historical part is brilliant in its

-:., clarity, and proportion, and in the later chapters on the present position

-.urns of Nonconformity Dr Selbie proves himself to be an ideal exponent

-of sound and moderate views." Christian World.

-

-54. ETHICS

-

-By G. E. MOORE, M.A., Lecturer in Moral Science in Cambridge University.

-Discusses Utilitarianism, the Objectivity of Moral Judgments, the Test of

-Right and Wrong, Free Will, and Intrinsic Value.

-

-56. THE MAKING OF THE NEW TESTAMENT

-

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-% End of the Project Gutenberg EBook of An Introduction to Mathematics, by 

-% Alfred North Whitehead                                                  %

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-% *** END OF THIS PROJECT GUTENBERG EBOOK AN INTRODUCTION TO MATHEMATICS ***

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