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-%                                                                         %

-% Project Gutenberg's An Introduction to Mathematics, by Alfred North Whitehead

-%                                                                         %

-% This eBook is for the use of anyone anywhere at no cost and with        %

-% almost no restrictions whatsoever.  You may copy it, give it away or    %

-% re-use it under the terms of the Project Gutenberg License included     %

-% with this eBook or online at www.gutenberg.org                          %

-%                                                                         %

-%                                                                         %

-% Title: An Introduction to Mathematics                                   %

-%                                                                         %

-% Author: Alfred North Whitehead                                          %

-%                                                                         %

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

-%                                                                         %

-% Language: English                                                       %

-%                                                                         %

-% Character set encoding: ISO-8859-1                                      %

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-Project Gutenberg's An Introduction to Mathematics, by Alfred North Whitehead

-

-This eBook is for the use of anyone anywhere at no cost and with

-almost no restrictions whatsoever.  You may copy it, give it away or

-re-use it under the terms of the Project Gutenberg License included

-with this eBook or online at www.gutenberg.org

-

-

-Title: An Introduction to Mathematics

-

-Author: Alfred North Whitehead

-

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

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-*** START OF THIS PROJECT GUTENBERG EBOOK AN INTRODUCTION TO MATHEMATICS ***

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-%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%

-\PageSep{i}

-\FrontMatter

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

-\noindent\footnotesize HOME UNIVERSITY LIBRARY \\

-OF MODERN KNOWLEDGE

-\vfill

-

-\begin{center}

-\Large AN INTRODUCTION TO \\

-MATHEMATICS

-\medskip

-

-\normalsize

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

-\vfill

-

-\footnotesize

-\scshape London \\

-{\normalsize WILLIAMS \& NORGATE} \\[6pt]

-\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.

-\fi

-\PageSep{v}

-\TableofContents

-

-%CHAP.                         PAGE

-

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

-

-\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}

-

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

-

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

-

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

-

-\ToCLine{X}{Conic Sections}{128}

-

-\ToCLine{XI}{Functions}{145}

-

-\ToCLine{XII}{Periodicity in Nature}{164}

-\PageSep{vi}

-

-%CHAP.                         PAGE

-\ToCLine{XIII}{Trigonometry}{173}

-

-\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.

-

-" It is a thing very favourable to the real success of The Home

-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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-indigestion. The food supplied is therefore frequently medicinal as

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-Manchester Guardian.

-

-"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

-

-di:

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-id prol

-series they might well take:

-

-

-

-ispensed on prize days, the pupils would

-rofit. If the publishers want a motto for the

-

-

-

-rubbish that is wont to be

-

-find more pleasure and

-

-series they might well take: ' Infinite riches in a little room.'" Irish

-

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-

-" The scheme was successful at the start because it met a want

-among earnest readers; but its wider and* sustained success, surely,

-comes from the fact that it has to a large extent created and certainly

-refined the taste by which it is appreciated." Daily Chronicle.

-

-" Here is the world's learning in little, and none too poor t&lt;

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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

-

-By Prof. B. W. BACON, LL. LX, D.D. An authoritative summary of the results

-of modern critical research with regard to the origins of the New Testament, in

-" the formative period when conscious inspiration was still in its full glow rather

-than the period of collection into an official canon," showing the mingling of the

-two great currents of Christian thought " Pauline and 'Apostolic,' the Greek-

-Christian gospel about Jesus, and the Jewish-Christian gospel of Jesus, the

-gospel of the Spirit and the gospel of au thority."

-

-jo. MISSIONS: THEIR RISE and DEVELOPMENT

-

-By Mrs CREIGHTON. The beginning of modern missions after the Reforma-

-tion and their growth are traced, and an account is given of their present

-work, its extent and character.

-

-IN PREPARATION

-

-THE OLD TESTAMENT. By Prof. GEORGE MOORE, D.D., LL.D.

-BETWEEN THE OLD AND NEW TESTAMENTS. By R. H.

-

-CHARLES, D.D.

-

-COMPARATIVE RELIGION. By Prof. J. ESTLIN CARPENTER, D.Litt.

-A HISTOR Y of FREEDOM of THOUGHT. By Prof. J. B. BURY, LL.D.

-A HISTORY OF PHILOSOPHY. By CLEMKNT WKBB, M.A.

-

-

-

-Social Science

-

-

-

-. PARLIAMENT

-

-Its History, Constitution, and Practice. By Sir COURTENAY P. ILBERT.

-K.C.B., K.C.S.I., Clerk of the House of Commons. "The best book on the

-history and practice of the House of Commons since Bagehot's 'Constitution.'"

-Yorkshire Post.

-

-. THE STOCK EXCHANGE

-

-By F. W. HIRST, Editor of " The Economist." " To an unfinancial mind must

-be a revelation. . . . The book is as clear, vigorous, and sane as Bagehot's ' Lom-

-bard Street,' than which there is no higher compliment." Morning Leader

-

-. IRISH NATIONALITY

-

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-% Alfred North Whitehead                                                  %

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-% *** END OF THIS PROJECT GUTENBERG EBOOK AN INTRODUCTION TO MATHEMATICS ***

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