zotero-db/storage/9FUVLHWR/.zotero-ft-cache

Project Gutenberg’s An Introduction to Mathematics, by Alfred North Whitehe 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 *** START OF THIS PROJECT GUTENBERG EBOOK AN INTRODUCTION TO MATHEMATICS **

Produced by Andrew D. Hwang. (This ebook was produced using OCR text generously provided by the University of California, Santa Barbara, through the Internet Archive.)
transcriber’s note
The camera-quality ﬁles for this public-domain ebook may be downloaded gratis at
www.gutenberg.org/ebooks/41568. This ebook was produced using scanned images and OCR text generously provided by the University of California, Santa Barbara, through the Internet Archive. Minor typographical corrections and presentational changes have been made without comment. This PDF ﬁle is optimized for screen viewing, but may be recompiled for printing. Please consult the preamble of the LATEX source ﬁle for instructions and other particulars.

HOME UNIVERSITY LIBRARY OF MODERN KNOWLEDGE
AN INTRODUCTION TO MATHEMATICS
By A. N. WHITEHEAD, Sc.D., F.R.S.
London
WILLIAMS & NORGATE
HENRY HOLT & Co., New York Canada: WM. BRIGGS, Toronto India: R. & T. WASHBOURNE, Ltd.

CONTENTS

CHAP.

PAGE

I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI XVII

THE ABSTRACT NATURE OF MATHEMATICS 1 VARIABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 METHODS OF APPLICATION . . . . . . . . . . . . . . . . . . . 15 DYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 THE SYMBOLISM OF MATHEMATICS . . . . . . . . . . 43 GENERALIZATIONS OF NUMBER . . . . . . . . . . . . . . 54 IMAGINARY NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . 67 IMAGINARY NUMBERS (CONTINUED) . . . . . . . . . 80 COORDINATE GEOMETRY . . . . . . . . . . . . . . . . . . . . . 90 CONIC SECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 PERIODICITY IN NATURE . . . . . . . . . . . . . . . . . . . . . . 134 TRIGONOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 THE DIFFERENTIAL CALCULUS . . . . . . . . . . . . . . . 179 GEOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 QUANTITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 NOTES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

AN INTRODUCTION TO
MATHEMATICS
CHAPTER I
THE ABSTRACT NATURE OF MATHEMATICS
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 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 concep-

NATURE OF MATHEMATICS

2

tion. 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 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 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,

INTRODUCTION TO MATHEMATICS

3

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 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 voyage to Laputa, is of two minds on this 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 Principia 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.

NATURE OF MATHEMATICS

4

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 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 from our sensations. We see, and hear, and taste, and smell, and feel hot and cold, and push, and rub, and ache, and tingle.

INTRODUCTION TO MATHEMATICS

5

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 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,

NATURE OF MATHEMATICS

6

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, 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 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 in its ideas.

CHAPTER II
VARIABLES
Mathematics as a science commenced when ﬁrst someone, probably a Greek, proved propositions about any things or about 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 any and of 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 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 any number there exists 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 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 let-

VARIABLES

8

ters for the ideas of any number and 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 Leibniz, and then a pause in the progress 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 any and 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.
(1) For any number x, x + 2 = 2 + x; (2) For some number x, x + 2 = 3; (3) For some number x, x + 2 > 3. The ﬁrst point to notice is the possibilities contained in the meaning of some, as here used. Since x+2 = 2+x for any number x, it is true for some number x. Thus, as here used, any implies some and some does not exclude 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 some may be one number only. But in the third 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 some number in this case. Thus some may be anything between

INTRODUCTION TO MATHEMATICS

9

any and one only, including both these limiting cases. It is natural to supersede the statements (2) and (3) by
the questions: (2 ) For what number x is x + 2 = 3; (3 ) For what numbers x is x + 2 > 3. Considering (2 ), x + 2 = 3 is an equation, and 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 though (2 ) exempliﬁed a much more thoroughgoing and fundamental idea than the original statement (2). This, however, is a complete mistake. The idea of the undetermined “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 (3 ) as compared to the original statement (3).
But the majority of interesting formulæ, especially when the idea of 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

10

variables are present the same two main types of statement occur. For example, (1) for any pair of numbers, x and y, x + y = y + x, and (2) for 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 any pair of numbers, is that by them formulæ of the second type can be thrown into an indeﬁnite number of equivalent forms. For example, the relation x + y = 1 is equivalent to the relations
y + x = 1, (x − y) + 2y = 1, 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 y2 = 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

INTRODUCTION TO MATHEMATICS

11

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 y2 = 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 12, 22, 32, 42, and so on, i.e., 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:
Y

B
xP 1N
yy

O xM 1 A

X

Fig. 1.

VARIABLES

12

Draw two lines OX and OY at right angles; let any number x be represented by x units (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 OM P N 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 y2 = 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 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

INTRODUCTION TO MATHEMATICS

13

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 righthand side of the equation makes no essential diﬀerence).

P

C

B

Q N
p

O

v

M

Fig. 2.

A V

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 to represent p units of pressure. Then the point Q, which is found by completing the parallelogram OM QN , 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

VARIABLES

14

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 unknown p when 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 variables satisfying the relation pv = 1. This example illustrates how the idea of variables is fundamental, both in the applications as well as in the theory of mathematics.

CHAPTER III
METHODS OF APPLICATION
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 1s. per cubic foot and that 20s. 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 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

METHODS OF APPLICATION

16

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 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

INTRODUCTION TO MATHEMATICS

17

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 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

METHODS OF APPLICATION

18

attraction of the other and directed towards it, is propor-

tional

to

mM d2

;

thus

this

force

can

be

written

as

equal

to

kmM d2

,

where

k

is

a

deﬁnite

number

depending

on

the

ab-

solute magnitude of this attraction and also on the scale by

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

we wish to reckon in terms of forces such as the weight of

a mass of 1 lb., the number which k represents must be ex-

tremely small; for when m and M and d are each put equal

to 1,

kmM d2

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

mM d2

,

giv-

ing 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

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 formu-

lated. In the ﬁrst place, the mathematical habit of mind and

the mathematical procedure explained in the previous two

INTRODUCTION TO MATHEMATICS

19

chapters had to be generated; otherwise Newton could never have thought of a formula representing the force between any two masses at any distance. Again, what are the meanings of the terms employed, Force, Mass, Distance? 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 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.
Throughout the middle ages, under the inﬂuence of Aristotle, the science was entirely misconceived. Newton had the advantage of coming after a series of great men, notably Galileo, in Italy, who in the previous two 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, clear and distinct in his mind, and realising their importance and their relevance to the fall of an apple and the motions of the plan-

METHODS OF APPLICATION

20

ets, 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 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 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 a.d., Dr. Gilbert, of Colchester, published the ﬁrst work on the subject in which any scientiﬁc method is followed.

INTRODUCTION TO MATHEMATICS

21

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, in the year 1752, sent a kite into the clouds and proved that thunderstorms were electrical.
Meanwhile from the earliest epoch (2634 b.c.) 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 pursued for its own sake. The use of the compass was not introduced into Europe till the end of the twelfth century a.d., 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 to two Italians, Galvani in 1780, and Volta in 1792. This great invention opened a new series of phenomena for investigation. The scientiﬁc world had now three separate, though allied,

METHODS OF APPLICATION

22

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 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, proved that magnetic poles attract or repel each other, in proportion to the inverse square of their distances, and also that the same law holds for electric charges—laws curiously analogous to that of gravitation. In 1820, O¨ ersted, a Dane, discovered that electric currents exert a force on magnets, and almost immediately afterwards the mathematical law of the force was correctly formulated by Amp`ere, a Frenchman, who also proved that two electric currents exerted forces on each other. “The experimental investigation by which Amp`ere 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 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.”∗
∗Electricity and Magnetism, Clerk Maxwell, Vol. II., ch. iii.

INTRODUCTION TO MATHEMATICS

23

The momentous laws of induction between currents and between currents and magnets were discovered by Michael Faraday in 1831–82. 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 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 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 since been veriﬁed, so that now the whole theory of light is nothing but a branch of the great science of electricity. Also Herz, a German, in 1888, following on Maxwell’s ideas, succeeded in producing electric vibrations by direct electrical methods. 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 mathe-

METHODS OF APPLICATION

24

matical deductions, starting from a few simple assumed 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, 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 phenomena of

INTRODUCTION TO MATHEMATICS

25

hydrostatics and optics. Indeed, Archimedes, who combined a genius for mathematics with a physical insight, must rank with Newton, who lived nearly two 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 b.c.), 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 Marcellus, he was found by a Roman soldier 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. 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 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

METHODS OF APPLICATION

26

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 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 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 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 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,

INTRODUCTION TO MATHEMATICS

27

Weights

The crown

Water

Fig. 3.

and

(A)

W w

=

W W −F

where W and F are determined by the easy, and fairly pre-

cise,

operation of

weighing.

Hence,

by

equation

(A),

W w

is

known.

But

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

METHODS OF APPLICATION

28

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, 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 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.

CHAPTER IV
DYNAMICS
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 (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 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 the tower marks picturesquely an essential step in knowledge, no less a step than the ﬁrst attainment of correct ideas on the science

DYNAMICS

30

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 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 body continues in its state of rest or of uniform motion in a straight line, except so far as it is compelled by impressed force to change that state. This law is more than a dry formula: it is also a pæan of triumph 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.

INTRODUCTION TO MATHEMATICS

31

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 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 Pole, born at Thorn in West Prussia (born 1473, died 1543), showed how much simpler it was to conceive the planets, including the earth as revolving round the sun in orbits which are nearly circular; and later, 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 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, held by Kepler, the actual velocity itself required preservation by force. Thus he looked for tangential forces as in the accompanying ﬁgure (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

DYNAMICS

32

Force (on false hypothesis) Planet

Sun

Fig. 4.
thus depart entirely from the sun. Newton, therefore, had to search for a force which would bend the motion round into
Planet
Force Sun

Fig. 5.
its elliptical orbit. This he showed must be a force directed

INTRODUCTION TO MATHEMATICS

33

towards the sun as in the next ﬁgure (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 Greeks from a consideration of the theory of the mechanical advantage obtained by the use 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

DYNAMICS

34

feeling of temperature simply marks our sense of the transfer of heat from the 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 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

INTRODUCTION TO MATHEMATICS

35

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 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 airwaves 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

DYNAMICS

36

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 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

INTRODUCTION TO MATHEMATICS

37

our ideas which would apply to it, should issue in such a remarkable success.
It would take us too far aﬁeld to enter into 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 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 (i.e. the length AC) and its direction. Now anything, like this transference, which is completely given by the determination of a magnitude and

D

C

A

B

Fig. 6.

a direction is called a vector. For example, a velocity requires for its deﬁnition the assignment of a magnitude and

DYNAMICS

38

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 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 “par-

INTRODUCTION TO MATHEMATICS

39

allelogram law” for the addition of vectors of the same kind but in diﬀerent directions.
Consider the vector AC in ﬁgure 6 as representative 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, 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

DYNAMICS

40

it must be observed that nature itself presents us with the idea. For example, a steamer is moving in the direction AD (cf. 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 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 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 paral-

INTRODUCTION TO MATHEMATICS

41

D

C

y

r

m A

x B

Fig. 7.

lelogram law. Also, according to the laws of motion a force is fully rep-
resented 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 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 the relation of the single vector AC to the two component vectors, AB and AD, at right angles (cf. ﬁg. 7), which is continually recurring. Let

DYNAMICS

42

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.

CHAPTER V
THE SYMBOLISM OF MATHEMATICS
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 suﬃciently clear ideas about the integral numbers, represented in the Arabic notation by 0, 1, 2, . . . , 9, 10, 11, . . . , 100, 101, . . . and so on. This notation was introduced into Europe through the Arabs, but they apparently obtained it from Hindoo sources. The ﬁrst known work∗ in which it is systematically explained is a work by an Indian mathematician, Bhaskara (born 1114 a.d.). But the actual numerals can be traced back to the seventh century of our era, and perhaps were originally invented in Tibet. For our present 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
∗For the detailed historical facts relating to pure mathematics, I am chieﬂy indebted to A Short History of Mathematics, by W. W. R. Ball.

THE SYMBOLISM OF MATHEMATICS

44

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 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

INTRODUCTION TO MATHEMATICS

45

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 the fundamental laws of algebra∗:—

(1)

x + y = y + x,

(2)

(x + y) + z = x + (y + z),

(3)

x × y = y × x,

(4)

(x × y) × z = x × (y × z),

(5)

x × (y + z) = (x × y) + (x × z).

Here (1) and (2) are called the commutative and associative laws for addition, (3) and (4) are the commutative and associative laws for multiplication, and (5) is the distributive law relating addition and multiplication. For example, without symbols, (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
∗Cf. Note A, p. 207.

THE SYMBOLISM OF MATHEMATICS

46

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 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 (3), (4), and (5) above are in general written,
xy = yx, (xy)z = x(yz), 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.

INTRODUCTION TO MATHEMATICS

47

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 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 number 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 SYMBOLISM OF MATHEMATICS

48

The symbol developed in connection with the Arabic notation for numbers of which it is an essential part. For in that notation the 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, . . . , 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.
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

INTRODUCTION TO MATHEMATICS

49

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 Chapter 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 = 2x2 is 2x2 − 3x + 2 = 0. The point is that all the symbols which represent variables, e.g. x and y, and the symbols 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 in 1560 and died in 1621. But what is the importance of this simple symbolic procedure? It made possible the growth of the modern conception of algebraic form.

THE SYMBOLISM OF MATHEMATICS

50

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 x2, x3, etc., do not appear. Again 3x2 − 2x + 1 = 0, x2 − 3x + 2 = 0, x2 − 4 = 0, are all equations of the same form, namely, equations involving one unknown x in which x×x, that is x2, appears. These equations are called quadratic equations. Similarly cubic equations, in which x3 appears, yield another form, and so on. Among the three quadratic equations given above there is a minor diﬀerence between the last equation, x2 − 4 = 0, and the preceding two equations, due to the fact that x (as distinct from x2) 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 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 Chapter II., always represents such equations by a straight line. Then there are

INTRODUCTION TO MATHEMATICS

51

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 x2 − 3x + 2 = 0, and x2 − 4 = 0, can be obliterated by writing the latter equation in the form x2 + (0 × x) − 4 = 0. For, by the laws stated above, x2 + (0 × x) − 4 = x2 + 0 − 4 = x2 − 4. Hence the equation x2 − 4 = 0 is merely representative of a particular class of quadratic equations and belongs to the same general form as does x2 − 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 vari-

THE SYMBOLISM OF MATHEMATICS

52

ables. 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 any linear correlation between the variable numbers x and y? Here we want to symbolize any linear correlation; just as x symbolizes any 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 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 way are sometimes also called 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

INTRODUCTION TO MATHEMATICS

53

their equations, by using letters at the end of the alphabet for the “variable” variables, and letters at the beginning of the alphabet for 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.

CHAPTER VI
GENERALIZATIONS OF NUMBER
One great peculiarity of mathematics is the 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 b.c. and 1100 b.c., 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 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 fractions, the

INTRODUCTION TO MATHEMATICS

55

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 “incommensu-

rable” 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 sq√uare; in our modern notation the length of the diagonal is 2 times the length

of the√side. But there is no fracti√on which exactly repre-

sents 2. We can approximate to 2 as closely as we like,

but less

we never exactly

than

2,

and

9 4

is

reach its value. greater than 2,

For

example, √

49 25

is

just

so that 2 lies between

7 5

and √

3 2

.

But the best systematic way of approximating

to 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,

14 10

,

141 100

,

1414 1000

,

and

so

on.

Ratios of this sort are called by the Greeks incommen-

surable. They have excited from the time of the Greeks

onwards a great deal of philosophic discussion, and the diﬃ-

culties connected with them have only recently been cleared

up.

GENERALIZATIONS OF NUMBER

56

We will put the incommensurable ratios 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 represented by points on a line. Let OX be any line bounded at O and stretching away indeﬁ-

0

1 2

1

3 2

2

5 2

3

7 2

4

OMAN B P C QD

X

nitely 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

1 2

,

that

of

AB

rep-

resents

3 2

,

that

of

BC

represents

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, starting from O

and moving regularly in the direction from O to X, repre-

INTRODUCTION TO MATHEMATICS

57

sents the real numbers as arranged in an ascending order of

size, starting from zero and continually increasing as we go

on.

All this seems simple enough, but even at 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 frac-

tions 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 suc-

cessor. This is easily seen to be the case, for between any

two fractions we can always ﬁnd another fraction interme-

diate in value. One very simple way of doing this is to add

the fractions together and to halve the result. For exam-

ple, and

between between

2 23 3

and and

1437,
24

the the

fraction fraction

2112((2323

+ +

213447)),,

that that

is is

17 2343 48

, ,

lies; 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

GENERALIZATIONS OF NUMBER

58

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 endpo√ints to certain classes. Thus, consider the incommensurable 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 thi√nking of the incommensurables, so that we cannot bring in 2, there is no fraction which has the property of dividing oﬀ the series into two parts in this way, i.e. so that all the members on one side have their squares less than 2, and on the other side greater than 2. He√nce in the series of fractions there is a quasi-gap where 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 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 be-

INTRODUCTION TO MATHEMATICS

59

tween 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

1 1

for

1,

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

2 3

,

4 6

,

6 9

,

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 the index

of

4 3

,

and

also

of

3 4

,

and

of

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 class class class

has has has has

the the the the

iiiinnnnddddeeeexxxx4352,,,,aaaannnnddddiiitttisstssmmmoeeenmmmlybbbeemerrrssesmaaarrbreeee13r1214,i,as22n32,d11,31;32;12t,;thh14tehe; seafeontcuhdorinrtsdhdo

on. It is easy to see that the number of members (still in-

cluding 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

GENERALIZATIONS OF NUMBER

60

index. Thus for the class of index n, the members appear in the order

n

1 −

1

,

n

2 −

2

,

n

3 −

3

,

...,

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

1 1

,

1 2

,

2 1

,

1 3

,

2 2

,

3 1

,

1 4

,

2 3

,

3 2

,

4 1

,

...,

n

− 1

1

,

n

1 −

1

,

n

2 −

2

,

n

3 −

3

,

...,

n

− 1

1

,

1 n

,

and so on.

Now we can get rid of all repetitions of fractions of the

same value by simply striking them out whenever they ap-

pear after their ﬁrst occurrence. In the few initial terms writ-

ten

down

above,

2 2

which

is

enclosed

above

in

square

brackets

is the only fraction not in its lowest terms. It has occurred

before

as

1 1

.

Thus

this

must

be

struck

out.

But

the

series

is

still left with the same properties, namely, (a) there is a ﬁrst

term, (b) each term has next-door neighbours, (c) the series

goes on without end.

It can be proved that it is not possible to 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

INTRODUCTION TO MATHEMATICS

61

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 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 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 consid√er the operations √of adding√or subtracting them: instead of 2,√we consider + 2 and −√ 2, namely the operations of adding 2 and of subtracting 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

GENERALIZATIONS OF NUMBER

62

is that, whatever its possible varieties of meaning, the formal laws for its use shall always be the same. In 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 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

INTRODUCTION TO MATHEMATICS

63

cobwebs of the brain. The answer is that what the mathematician is seeking is Generality. This is an idea worthy to be placed beside the notions of the Variable and of Form so far as concerns 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 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

GENERALIZATIONS OF NUMBER

64

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 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 is the step from O

−3 −2 −1

+1 +2 +3

X′

D′ C′ B′ A′ O A B C D E

X

to B, or from A to C, or (if the divisions are taken back-

INTRODUCTION TO MATHEMATICS

65

wards 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 operations or steps. These “signed” numbers are also particular cases of what have been called vectors (from the Latin veho, I 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,

GENERALIZATIONS OF NUMBER

66

in a.d. 1489. They seem ﬁrst to have been employed in mathematics by a German mathematician, Stifel, in a book published at Nuremburg in 1544 a.d. 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 −; 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.

CHAPTER VII
IMAGINARY NUMBERS
If the mathematical ideas dealt with in the 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 succ`es 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 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, a` propos of his use of words, “I pay them extra and make

IMAGINARY NUMBERS

68

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 x2 + 1 = 3, x2 + 3 = 1, and x2 + a = b. Exactly the same process is gone through. The equation x2 + 1 = 3 √becomes x2 =√ 2, and this has two solutions, either x = + 2, or x = − 2. The statement that th√ere are these alternative solutions is usually written x = ± 2. So far all is plain sailing, as it was in the previous case. But now an analogous diﬃculty arises. For the equation x2 + 3 = 1 gives x2 = −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 x2 = −2, and the equation is in fact nonsense. Thus, ﬁnally taking the general form x2 +a = b, we ﬁnd the pair of solutions x = ± (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

INTRODUCTION TO MATHEMATICS

69

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 ± (b − a) for the equation x2 + a = b always have

meaning. In othe√r words, we require an interpretation of the symbols so that a always has meaning whether a be posi-

tive 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 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 −c2 for it, so that c2 is positive. Then

√ a= =

(−c2) = {(−1) × c2} √
(−1) c2 = c (−1).

Hence, if we can so interpret our symbols that (−1) has
a meaning, we have attained our object. Thus (−1) has come to be looked on as the head and forefront of all the imaginary quantities.
This business of ﬁnding an interpretation for (−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

IMAGINARY NUMBERS

70

existed which was perhaps capable of solution. Equations like x2 = −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 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 (−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 (−1) and for the addition, sub-
traction, multiplication, and division of (−1) which make the ordinary algebraic rules (e.g. x + y = y + x, etc.) to be satisﬁed, then such and such results follows. It was natural

INTRODUCTION TO MATHEMATICS

71

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, 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 positive and negative numbers we can now represent the

Y

P′

N

P

y

y

y

M′

X′

x′

O

x

y′

y′

M

X

y′

N′

P ′′

P ′′′

Y′

Fig. 8.

position of any point in a plane by a pair of such numbers. Thus we take the pair of straight lines XOX and Y OY ,

IMAGINARY NUMBERS

72

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, e.g. (+3, +1), so that there 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 Y OY for the second number. Thus (cf. 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 Y OY 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
(+1, +3), (−1, +3), (−1, −3), (+1, −3),
(+3, +1), (−3, +1), (−3, −1), (+3, −1).
Each of these eight “ordered couples” directs a process of measurement along XOX and Y OY 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 pictori-

INTRODUCTION TO MATHEMATICS

73

ally in the ﬁgure. The lengths OM and ON together correspond 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 (+3, +1), the point P by (−3, +1), the point P by
Y

P′ X′ M′
P ′′

N +1

−3

+3

O

−1 N′

P MX
P ′′′

Y′
Fig. 9.
(−3, −1), and the point P by (+3, −1). More generally in the previous ﬁgure (8), the point P corresponds to the ordered couple (x, y), where x and y in the ﬁgure are both

IMAGINARY NUMBERS

74

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 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 corresponding point, and the second number y is called the “ordinate” of the point, and the two numbers together are called the “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 mathematician and philosopher, and appears in his Discours published at Leyden in 1637 a.d. 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. 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 (cf. ﬁg. 8), rather than the point P . Thus the ordered couple represents a line drawn from an “origin,” O, of a certain length and in a certain direction. The line OP may be called the vector line

INTRODUCTION TO MATHEMATICS

75

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 useful when we consider the meaning to be assigned to the operations of the addition and multiplication of ordered couples.
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, (a) make the result of addition to be another ordered couple, (b) make the operation commutative so that (x, y) + (x , y ) = (x , y ) + (x, y), (c) make the operation associative so that
{(x, y) + (x , y )} + (u, v) = (x, y) + {(x , y ) + (u, v)},
(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).
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 ).

IMAGINARY NUMBERS

76

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 righthand 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

(+3, +1) + (+2, +6) = (+5, +7), (+3, −1) + (−2, −6) = (+1, −7), (+3, +1) + (−3, −1) = (0, 0).

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).

It is easy to see that

(x, y) − (u, v) = (x, y) + (−u, −v).

Also (x, y) − (x, y) = (0, 0).

INTRODUCTION TO MATHEMATICS

77

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.

Y

R

Q

P

S

X′

O

M1 M

M′

X

Y′
Fig. 10.
Let OP represent (x, y) so that OM = x and P M = y; let OQ represent (x1, y1) so that OM1 = x1 and QM1 = y1. Complete the parallelogram OP RQ by the dotted lines P R and QR, then the diagonal OR is the ordered couple (x + x1, y + y1). For draw P S parallel to OX; then evidently

IMAGINARY NUMBERS

78

the triangles OQM1 and P RS are in all respects equal. Hence M M = P S = x1, and RS = QM1 and therefore
OM = OM + M M = x + x1,
RM = SM + RS = y + y1.
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 was mentioned in Chapter 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 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

INTRODUCTION TO MATHEMATICS

79

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.

CHAPTER VIII

IMAGINARY NUMBERS (Continued )
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
(α) the result is another ordered couple, (β) the operation is commutative, so that

(x, y) × (x , y ) = (x , y ) × (x, y),

(γ) the operation is associative, so that

{(x, y) × (x , y )} × (u, v) = (x, y) × {(x , y ) × (u, v)},

(δ) 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),

or by

(x,

y)

=

(c, d) (a, b)

.

( ) Furthermore the law involving both addition and multiplication, called the distributive law, must be satisﬁed,

INTRODUCTION TO MATHEMATICS

81

namely
(x, y) × {(a, b) + (c, d)} = {(x, y) × (a, b)} + {(x, y) × (c, d)}.
All these conditions (α), (β), (γ), (δ), ( ) can be satisﬁed by an interpretation which, though it looks complicated at ﬁrst, is capable of a simple geometrical interpretation.
By deﬁnition we put
(A) (x, y) × (x , y ) = {(xx − yy ), (xy + x y)}.
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 righthand side of equation (A) is not altered by simultaneously interchanging x with x , and y with y . Hence conditions (α) and (β) are evidently satisﬁed. The proof of the satisfaction of (γ), (δ), ( ) 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 x2 = −3, to which no solutions could be assigned in terms of positive and negative real numbers. We then found that all our diﬃculties would vanish if we could interpret the equation x2 = −1, i.e., if we could so deﬁne (−1) that (−1) × (−1) = −1.

IMAGINARY NUMBERS

82

Now let us consider the three special ordered couples∗ (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. Finally consider (0, 1): this will interpret for us the symbol (−1). The symbol must therefore possess the characteristic property that (−1) × (−1) = −1. Now by the law of multiplication for ordered couples
(0, 1) × (0, 1) = {(0 − 1), (0 + 0)} = (−1, 0).
∗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).

INTRODUCTION TO MATHEMATICS

83

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 ± (−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 −1. Accordingly the ordered couples (0, 1) and (0, −1) are the interpretations of ± (−1) in terms of ordered couples. But which corresponds to which? Does (0, 1) correspond to + (−1) and (0, −1) to − (−1), or (0, 1) to − (−1), and (0, −1) to + (−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” 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

IMAGINARY NUMBERS

84

after noting three yet more special cases. Thirdly, we multiply the “real” couple (a, 0) by the imag-
inary (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, b ) and obtain
(0, b) × (0, b ) = (−bb , 0).
We now turn to the geometrical interpretation, beginning ﬁrst with some special cases. Take the couples (1, 3) and (2, 0) and consider the equation
(2, 0) × (1, 3) = (2, 6).

In the diagram (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).

INTRODUCTION TO MATHEMATICS

85

Y

N′

Q

M′ P R

N1

X′

X

M1

OMN

Fig. 11.

The vector ON1, corresponds to (0, 2) and the vector OR to (−6, 2). Thus OR which 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 ON1 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 ON1. 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 ON1 as modiﬁed by the vector OP . The

IMAGINARY NUMBERS

86

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 (i.e. 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 ON1 and OP ; the new direction for the enlarged ON1 (i.e. OR) is found by rotating ON in the anti-clockwise direction of rotation through an angle equal to the angle XOP , namely, the angle N1OR is equal to the angle XOP .
The general rule for the geometrical representation of multiplication can now be enunciated thus:

Y

R

Q P

O

X

Fig. 12.

INTRODUCTION TO MATHEMATICS

87

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 (i.e. the angle QOR = the angle XOP ), or the vector OQ as making the vector OP rotate through the angle XOQ (i.e.the angle P OR = the angle XOQ).
We do not prove this general law, as we 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 (γ) 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

IMAGINARY NUMBERS

88

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 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.
The nett result of our investigations is that 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 x2 − 2x + 4 = 0; but there are an endless number of other equations, for ex-

INTRODUCTION TO MATHEMATICS

89

ample, x3 − 2x + 4 = 0, x4 + x3 + 2 = 0, and so on without limit. Have we got to make a 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.

CHAPTER IX
COORDINATE GEOMETRY
The methods and ideas of coordinate geometry 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 Y OY , 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 (cf. ﬁg. 13) x is the length OM and y is the length P M . 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 already said) to the philosopher 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 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

INTRODUCTION TO MATHEMATICS

91

Y

P

y

x

X′

O

M

X

Y′
Fig. 13.
knowledge of it are entirely worthless, being either trivial or wrong. The fact is a curious one; since the ultimate ideas of mathematics 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 im-

COORDINATE GEOMETRY

92

munity 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, being born about 330 b.c.; 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 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

INTRODUCTION TO MATHEMATICS

93

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.
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

COORDINATE GEOMETRY

94

then see how they are related by Descartes’ method of coordinates. Take 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, 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 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 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

INTRODUCTION TO MATHEMATICS

95

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.
Again we generalize x2 + y2 = 1 into ax2 + by2 = c, or still further into ax2 + 2hxy + by2 = c, or, still further, into ax2 + 2hxy + by2 + 2gx + 2f y = 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 of ﬁrst thoughts on the subject—which was the cause of the comparative sterility of the study of the science during so many

COORDINATE GEOMETRY

96

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.
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 at all points exhibits the character-