An Introduction to Mathematics (1911)

HANDBOUND AT THE UNIVERSITY OF TORONTO HOME UNIVERSITY LIBRARY OF MODERN KNOWLEDGE No. 15 Editors: THE RT. HON. H. A. L. FISHER, M.A., F.B.A. PROF. GILBERT MURRAY, LiTT.D., LL.D., F.B.A. PROF. SIR J. THOMSON, ARTHUR M.A. PROF. WILLIAM M.A. T. BREWSTER, A complete classified list of the volumes of HOME UNIVERSITY LIBRARY THE already published will be found at the back of this book. INTRODUCTION TO MATHEMATICS BY A. N. WHITEHEAD Sc.D., F.R.S. AUTHOR OF "UNIVERSAL ALGEBRA** NEW YORK HENRY HOLT AND COMPANY LONDON THORNTON BUTTERWORTH COPYRIGHT, 1911, BT HENRY HOLT AND COMPANY Q Pi W5 u t/. THE UNIVERSITY 5. A. PRESS, CAMBRIDGE, U.S.A. CONTENTS PAGE CHAP. I II III THE ABSTRACT NATURE OF MATHEMATICS VARIABLES . . 7 15 METHODS OF APPLICATION 25 IV DYNAMICS V THE SYMBOLISM OF MATHEMATICS VI GENERALIZATIONS OF NUMBER 42 58 71 VII IMAGINARY NUMBERS VII IMAGINARY NUMBERS (CONTINUED) CO-ORDINATE GEOMETRY 112 CONIC SECTIONS 128 IX X XI XII FUNCTIONS 145 PERIODICITY IN NATURE 164 XIII TRIGONOMETRY XIV SERIES XV 173 : THE DIFFERENTIAL CALCULUS XVI GEOMETRY XVII 87 101 . 217 236 245 QUANTITY NOTE ON BOOKS INDEX 194 . 251 . 353 AN INTRODUCTION TO MATHEMATICS CHAPTER I THE ABSTRACT NATURE OF MATHEMATICS THE study of mathematics is apt to com The important in disappointment. 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 Tis here, tis tis to grasp it there, and what we do see does not suggest gone" the same excuse for illusiveness as sufficed for the ghost, that it is too noble for our show of violence," if gross methods. ever excusable, may surely be "offered" to the trivial results which occupy the mence " "A 7 8 INTRODUCTION TO MATHEMATICS pages of some elementary mathematical treatises. The reason for this failure of the science to reputation is that its funda mental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their live up to its 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 Without a by any general conception. 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 nec essary 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 learn 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 allu sion in what follows to detailed deductions in any part of the science will be inserted ing. NATURE OF MATHEMATICS 9 merely for the purpose of example, and care will be taken to make the general argument comprehensible, even if here and there some technical process or symbol which the reader does not understand is cited for the purpose of illustration. The first acquaintance which most people have with mathematics is through arithmetic. That two and two make four is usually taken as the type of a simple mathematical pro position 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, thejjrst noticeable fact about arith metic 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 per fectly indifferent, of all things it is true that two and two make four. Thus we write down as the leading characteristic of mathe matics that it deals with properties and ideas which are applicable to things just be cause they are things, and apart from any particular feelings, or emotions, or sensa tions, in any way connected with them. This is what is meant by calling mathe matics an abstract science. The result which we have reached deserves It is natural to think that an attention. 10 INTRODUCTION TO MATHEMATICS abstract science cannot be of much import ance in the affairs of human life, because it has omitted from its consideration every thing of real interest. It will be remembered that Swift, in his description of Gulliver s voyage to Laputa, is of two minds on this He describes the mathematicians of that country as silly and useless dreamers, whose attention has to be awakened by point. Also, the mathematical tailor measures his height by a quadrant, and de duces his other dimensions by a rule and compasses, producing a suit of very illOn the other hand, the fitting clothes. mathematicians of Laputa, by their marvel lous 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 trans formed 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 flappers. an idea of its importance. It 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 arriving at is NATURE OF MATHEMATICS 11 Let us try to make clear thought. to ourselves why explanations of the order events necessarily tend to become of for 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 The progress of science consists in these mterconnecfions and in show observing ing 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 parpresent. ticular and whaLJs permanent in what is transitory is the aim of scientific thought. In the eye of science, the fall of an apple, the of a planet round a sun, and the cling ing 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 exam ples of permanent laws is the controlling idea motion of modern thought. Now us think of the sort of laws which in order completely to realize this scientific ideal. Our knowledge of the par ticular facts of the world around us is gained let we want 12 INTRODUCTION TO MATHEMATICS from our sensations. We see, and hear, and and smell, and feel hot and cold, and and rub, and ache, and tingle. These push, taste, are just our own personal sensations: my toothache cannot be your toothache, and my sight cannot be your sight. Butwe ascribe the origin of these sensations to relations" be tween the things which form the external Thus the dentist extracts not the world. 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 an other 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 sensa tions, 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 NATURE OF MATHEMATICS 13 sensations, the most serviceable part from its clearness, definiteness, and universality of what is left is composed of our general ideas of the abstract formal properties of things; in fact, the abstract mathematical ideas mentioned above. Thus it comes about that, step by step, and not realizing the full meaning of the process, mankind has been led to search for a mathematical description of the properties of the universe, because in this way only can a general idea of the course of events be formed, freed from reference to particular persons or to particular types of For example, it might be asked sensation. at dinner: "What was it which underlay my sensation of sight, yours of touch, and his of taste and smell?" the answer being "an But in its final analysis, science apple." seeks to describe an apple in terms of the and motions of molecules, a de 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 positions scription 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 pro claimed that number was the source of all things. In modern times the belief that the 14 INTRODUCTION TO MATHEMATICS 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 mathe matical 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 par ticular things. These propositions were first enunciated by the Greeks for geometry; geometry was the great Greek mathematical science. After the rise of geometry centuries passed away before and, accordingly, made a really effective start, despite faint anticipations by the later Greek algebra some mathematicians . The ideas of any and of some are intro duced into algebra by the use of letters, in stead 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. 2, we Again, in the place of saying that 3 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 > > 15 16 is INTRODUCTION TO MATHEMATICS of vital importance, both to philosophy for by it the notion of and to mathematics; is introduced. Perhaps it required the introduction of the arabic numerals, by which the use of letters as standing for defi nite numbers has been completely discarded in mathematics, in order to suggest to mathe maticians the technical convenience of the use of letters for the ideas of any number infinity and some number. The Romans would have stated the number of the year in which this is written in the form whereas we write it 1910, thus leaving the letters for the other usage. But this is merely a specu After the rise of algebra the differ lation. ential 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 con cerned; and it was not till within the last few years that it has been realized how fun damental any and some are to the very nature of mathematics, with the result of opening out still further subjects for mathe matical exploration. Let us now make some simple algebraic statements, with the object of understanding exactly how these fundamental ideas occur. MDCCCCX, (1) (2) (3) For any number x, x+%=%+x; For some number x, x +2 =3; 3. For some number x, x+% > VARIABLES 17 The first point to notice is the possibilities contained in the meaning of some, as here used. Since x -f-2 = 2 + x for any number x, it is true for some number x. Thus, as here exclude does not some used, 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 3. Hence there are an infinite num ber of numbers which answer to the some number in this case. Thus some may be anything between any and one only, includ x +2 > ing both these limiting cases. It is natural to supersede the statements (2) and (3) by the questions: For what number x is x +2 =3; For what numbers x is x +2 3. x =3 is an +2 Considering (2 ), equation, and (2 (3 it is ) ) easy to see that When we > its solution is #=3 2 = 1. have asked the question implied in the statement of the equation #4-2=3, x is called the unknown. The object of the solu tion of the equation is the determination of the unknown. Equations are of great im portance in mathematics, and it seems as though (2 ) exemplified a much more thor ough-going and fundamental idea than the original statement (2). This, however, is a mistake. The idea of the undetercomplete 18 INTRODUCTION TO MATHEMATICS mined "variable" as occurring in the use of is the really important one in mathematics; that of the "unknown" in an equation, which is to be solved as "some" or "any" 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 pre occupation of the text-books with the solu tion of equations. The same remark applies to the solution of the inequality (3 ) as com pared to the original statement (3). But the majority of interesting formulae, especially when the idea of some is present, For ex involve more than one variable. the consideration of the pairs of num ample, bers x and y (fractional or integral) which = 1 involves the idea of two cor satisfy x +y related variables, x and y. When two varia bles are present the same two main types For example, (1) for of statement occur. any pair of numbers, x and y, x+y=y-\-x, and (2) for some pairs of numbers, x and y, The second type of statement invites con sideration of the aggregate of pairs of num bers which are bound together by some fixed relation in the case given, by the relation x+y l. One use of formulae of the first type, true for any pair of numbers, is that by them formulae of the second type can be VARIABLES thrown into lent forms. =1 is 19 an indefinite number of equiva For example, the relation x+y equivalent to the relations and so on. Thus a skilful mathematician uses that equivalent form of the relation under consideration which is most conve nient 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 For example, when x and y determined. 2 = 4, y can be =*=2, thus, satisfy y =x, if x for any positive value of x there are alter Also in the relation native values for y. 1, when either x or y is given, an x-\-y indefinite number of values remain open for > the other. Again there is another important point to be noticed. If we restrict ourselves to posi tive numbers, integral or fractional, in con sidering the relation x-\-y = l, then, if either y be greater than 1, there is no positive number which the other can assume so as to satisfy the relation. Thus the "field" of the relation for x is restricted to numbers less than 1, and similarly for the "field" open to y. Again, consider integral numbers only, positive or negative, and take the relation x or INTRODUCTION TO MATHEMATICS 20 2 pairs of such numbers. 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 y =x, satisfied by Then whatever In the first place x must be positive, in the second place, since y is to be in tegral, x must be a perfect square. Accord ingly, the "field" of x is restricted to the set of integers I 2 , 2 2 , 3 2 , 4 2 , 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 ways. and as follows: X M I A X Fig. 1. OX and OF at right angles; any number x be represented by x units Draw two lines let VARIABLES scale) of length along (in any ber ybyy units (in 21 OX, any num any scale) of length along be x units in OF. Thus if OM, along 9 length, and ON, along OF, be y units in OX completing the parallelogram 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 corre sponds 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 conve nient 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 l, or y2 =x, then the points lie on a line, which is straight in the former case and curved in the latter. For example, consider ing only positive numbers, the points whose coordinates satisfy x +y = 1 lie on the straight length, by OMPN we find line AB Thus where OA = 1 and OB = 1. segment of the straight line AB in Fig. 1, this gives a pictorial representation of the proper ties of the relation under the restriction to positive numbers. Another example of a relation between two variables is afforded by considering the varia tions in the pressure and volume of a given mass of some gaseous substance such as air 22 INTRODUCTION TO MATHEMATICS or coal-gas or steam Let v be the at a constant tempera of cubic feet in its volume and p its pressure in Ib. weight per square inch. Then the law, known as Boyle s law, expressing the relation between p and v as both vary, is that the product ture. number 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 = \ (the exact number on the right-hand side of the equation makes no essential difference). Then in Fig. 2 we take two lines, 0V and OP, at right angles and draw OM along 0V to represent v units of volume, and ON along VARIABLES 23 OP to represent p units of pressure. Then the point Q, which is found by completing the parallelogram MONQ, represents the state of the gas when its volume is v cubic feet and its pressure is p Ib. weight per square inch. If the circumstances of the portion of gas con sidered are such that pv = I, then all these points Q which correspond to any possible state of this portion of gas must lie on the curved line ABC, which includes all points for which p and v are positive, and jw = l. Thus this curved line gives a pictorial repre sentation 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 en gineer or a physicist may want to know the particular pressure corresponding to some Then we have definitely assigned volume. the case of determining the unknown p when v is a known number. But this is only in In considering generally particular cases. the properties of the gas and how it will be have, he has to have in his mind the general form of the whole curve and its general In other words the properties. really funda mental idea is that of the pair of variables ABC 24 INTRODUCTION TO MATHEMATICS satisfying the relation pv = l. This example how the idea of variables is funda mental, both in the applications as well as in the theory of mathematics. illustrates CHAPTER III METHODS OF APPLICATION THE way in which the idea of variables satisfying a relation occurs in the applica tions 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 Is. per cubic 1. Then in all foot and that 20s. make 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 comple tion, 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 %Qy=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 25 26 INTRODUCTION TO MATHEMATICS particular man. They are stated in an ab stract general way, with complete indiffer ence to the owner s state of mind when he has to pay the bill. Now think a bit further as to what all this The building of a house is a com circumstances. It is im plicated possible to begin to apply the law, or to test it, unless amid the general course of events it is possible to recognize a definite set of occurrences as forming a particular instance of the building of a house. In short, we must means. set of know a house when we see it, and must events which belong to its Then amidst these events, thus building. 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 recognize the the law true? Anyone who has had 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 But is much another point which must be made clear. While we are making mathematical calcula tions connected with the formula %Qy = x, it is indifferent to us whether the law be true or METHODS OF APPLICATION 27 In fact, the very meanings assigned and y, as being a number of cubic feet and a number of pounds sterling, are in different. During the mathematical investi false. to x gation 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 num ber of fish caught, so that the assumed law is that on the average each fisherman catches The mathematical certainty of fish. the investigation only attaches to the results considered as giving properties of the corre lation %Qy=x between the variable pair of twenty numbers x and y. There is no mathematical certainty whatever about the cost of the actual building of any house. The law is not quite true and the result it gives will not be quite accurate. In fact, it may well be hope lessly wrong. Now all this no doubt seems very obvious. in truth with more complicated instances there is no more common error than to assume But that, because prolonged and accurate mathe matical 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 mathe matical calculations about the course of 28 INTRODUCTION TO MATHEMATICS 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 cer tainly it is not precisely accurate; and thus even at the best the result, precisely as calcu But then we lated, is not likely to occur. 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 This law of Newton and the Law of Gravity. states that any two bodies attract one an other with a force proportional to the product of their masses, and inversely proportional to the square of the distance between them. are the masses of the two and Thus if bodies, reckoned in Ibs. say, and d miles is the distance between them, the force on either body, due to the attraction of the other and m M H/f directed towards it, is proportional to -J2~ thus this force can be written as equal to = a2 , where k is a definite number depending on the absolute magnitude of this attraction and also on the scale by which we choose to measure forces. It is easy to see that, if we METHODS OF APPLICATION 29 wish to reckon in terms of forces such as the weight of a mass of 1 lb., the number which k represents must be extremely small; for and d are each put equal to when m and M 1, ~ a 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. F, it is F= k-p-, we call this force giving the correlation be tween the variables F, know If ra, how M, and d. We all 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 It may be that the final formu his mind. lation of the law occurred to him in an orchard, as well as elsewhere and he must have been somewhere. But for our purposes it is more instructive to dwell upon the vast amount of preparatory thought, the product of many minds and many centuries, which was necessary before this exact law could be formulated. In the first place, the mathe matical habit of mind and the mathematical procedure explained in the previous two chapters had to be generated; otherwise Newton could never have thought of a for mula representing the force between any two the story of it 30 INTRODUCTION TO MATHEMATICS 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 dis tances 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 man who had thoroughly mastered the first true general principles of Dynamics. Throughout the middle ages, under the in fluence of Aristotle, the science was entirely misconceived. Newton had the advantage of 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 He completed their work. Then, finally, it. having the ideas of force, mass, and distance coming METHODS OF APPLICATION 31 clear and distinct in his mind, and realizing their importance and their relevance to the fall of an apple and the motions of the planets, he hit upon the law of gravitation and proved it to be the formula always satisfied in these various motions. The vital point in the application of mathe matical formulae 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 im pressed 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 cere monies 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 grow sort of way in which physical sciences 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 m INTRODUCTION TO MATHEMATICS and fantastic hypotheses, though it may be doubted whether our modern scientific dis coveries 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 In would attract light and dry bodies. 1600 the A.D., first scientific Dr. Gilbert, of Colchester, published work on the subject method is followed. in which any a He made 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 Electrical century knowledge advanced. machines were made, sparks were obtained from them; and the Leyden Jar was in vented, by which these effects could be in tensified. Some organized knowledge was but still no relevent mathe obtained; being matical ideas had been found out. Franklin, in the year 1752, sent a kite into the clouds and proved that thunderstorms were elec trical. 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 METHODS OF APPLICATION 33 pursued for its own sake. The use of the com pass 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 prac tical bias of Europeans, but to the fact that in the West electrical and magnetic phe nomena were studied by men who were dom inated 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 phenomena for investigation. The world had now three separate, though allied, groups of occurrences on hand series of scientific the effects of "statical" electricity aris ing from frictional electrical machines, the magnetic phenomena, and the effects due From the end of the to electric currents. eighteenth century onwards, these three lines of investigation were quickly inter-connected 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 French man, proved that magnetic poles attract or repel each other, in proportion to the inverse square of their distances, and also that the INTRODUCTION TO MATHEMATICS 34 same law holds electric charges laws of to that curiously analogous gravitation. In 1820, Oersted, a Dane, discovered that electric currents exert a force on magnets, and almost immediately afterwards the mathematical law of the force was correctly for formulated by Ampere, a Frenchman, who also proved that two electric currents exerted forces on each other. "The experimental in vestigation by which Ampere established the law of the mechanical action between electric currents is one of the most brilliant achieve ments in science. The whole, theory and experiment, seems as if it had leaped, fullgrown and full armed, from the brain of It is perfect the Newton of Electricity. 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." The momentous laws of induction between currents and between currents and magnets were discovered by Michael Faraday in 183132. Faraday was asked: "What is the use He answered: "What is discovery?" 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 this * Electricity and Magnetism, Clerk Maxwell. Vol. II., ch. iii. METHODS OF APPLICATION 35 of electricity. Faraday also reorganized the whole theoretical conception of the science. His ideas, which had not been fully under stood 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 recog nized 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 Also Herz, a great science of electricity. German, in 1888, following on Maxwell s ideas, succeeded in producing electric vibra tions by direct electrical methods. His experiments are the basis of our wireless telegraphy. In more recent years even more funda mental discoveries have been made, and the science continues to grow in theoretic import ance 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 them selves suggesting fresh experiments, a whole mass of isolated and even trivial phenomena are welded together into one coherent science, in which the results of abstract mathematical deductions, starting from a few simple as- 36 INTRODUCTION TO MATHEMATICS sumed 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 In the first chapter of scientific thought. in what the barest outlines is the story place, 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 out line, then as now, a regularity of events was patent. But no minute tracing of their inter connection was possible, and there was no knowledge how even to set about to construct such a science. Detached speculations, a few happy or un happy shots at the nature of things, formed the utmost which could be produced. Meanwhile land-surveys had produced ge- METHODS OF APPLICATION 37 ometry, 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 hydrostatics and optics. In deed, 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 Syra cuse, the great Greek city of Sicily. When Romans besieged the town (in 210 to 212 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 repu tation which he had gained among his con temporaries for his knowledge of optics. At the end of this siege he was killed. According to one account given to Plutarch, in his life of Marcellus, he was found by a Roman soldier absorbed in the study of a geometrical dia gram 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 the worthy of his genius for mathematical and INTRODUCTION TO MATHEMATICS 38 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 quan tity 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 Hiero remainder with some baser metal. sent the crown to Archimedes and asked him to test it. In these days an indefinite num ber 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 (I have palace, shouting Eureka ! Eureka ! 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. immersed He saw that a body when 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, Ib. be the weight of the crown, as weighed if W in METHODS OF APPLICATION w 39 be the weight of the water displaces when completely immersed, w would be the extra upward force necessary to sustain the crown as it hung in and in air, which W Ib. it water. Now, this upward force can easily be ascer tained by weighing the body as it hangs in water, as shown in the annexed figure. If Weights The crown Fig. 3. the weights in the right-hand scale come to then the apparent weight of the crown in water is F Ib.; and we thus have F Ib., and thus and w W (A) where and F are determined by the easy, and fairly precise, operation of weighting. 40 INTRODUCTION TO MATHEMATICS Hence, by equation (A), W w is W w is known. But the ratio of the weight of the crown to the weight of an equal volume of water. This ratio is the same for any lump of metal of the same material it is now called the specific gravity of the material, and depends only on the intrinsic nature of the substance and not on its shape or quantity. Thus to test if the : crown were of gold, Archimedes had only to take a lump of indisputably pure gold and find its specific gravity by the same process. If the two specific gravities agreed, the crown was pure; if they disagreed, it was debased. This argument has been given at length, because not only is it the first precise example of the application of mathematical ideas to physics, but also because it is a perfect and simple example of what must be the method and spirit of the science for all time. The discovery of the theory of specific gravity marks a genius of the first rank. 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 super seded in the leadership of the European world by the practical Romans. Lord Beaconsfield, in one of his novels, has defined a practi cal man as a man who practises the errors of : METHODS OF APPLICATION 41 The Romans were a great but they were cursed with the sterility which waits upon practicality. They did not improve upon the knowledge of their fore fathers, 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 ab sorbed in the contemplation of a mathe matical diagram. his forefathers. race, CHAPTER IV DYNAMICS THE world had to wait for eighteen hundred till the Greek mathematical physicists found successors. In the sixteenth and seven years teenth 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. Mean while 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 ex perience, 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 prac It was an act eminently tically discovered. characteristic of the age that Galileo, a 42 DYNAMICS 43 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 im pulses to thought with impulses to action. This matter of the dropping of weights from the tower marks picturesquely an essen tial step in knowledge, no less a step than the first attainment of correct ideas on the science of dynamics, the basal science of the whole subject. The particular point in dis pute 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 Galileo affirmed that they would quicker. 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 follow ing Newton we now enunciate it, is Every body continues in its state of rest or of uni- 44 INTRODUCTION TO MATHEMATICS form motion a straight line, except so far compelled by impressed force to change that state. This law is more than a dry formula: it is also a psean of triumph over defeated heretics. The point at issue can be understood by deleting from the law the phrase 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 as it in is "or of rest except so far as it is compelled by impressed force to change that state." In this last false formula it is asserted that, apart from force, a body continues in a state of rest; and accordingly that, if a body is moving, a force is required to sustain the motion; so that when the force ceases, the motion ceases. The true Newtonian law takes diametrically the opposite point of view. The state of a body unacted on by force is that of uniform motion in a straight line, and no external force or influence is to be looked for as the cause, or, if you like to put it so, as the invariable accompaniment of this uniform rectilinear motion. Rest is merely a particular case of such motion, merely when the velocity is and remains zero. Thus, when a body is moving, we do not seek for any external influence except to explain changes in the rate of the velocity or changes in its direction. So long as the body is moving DYNAMICS 45 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 well seen by reference to the theory of the motion of the planets. Copernicus, a Pole, born at Thorn in West Prussia (born view is 1473, died 1543), showed how much simpler Force (on False hypothesis) Fig. 4. 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, it 46 INTRODUCTION TO MATHEMATICS held by Kepler, the actual velocity itself re quired 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 thus depart entirely from the sun. Newton, therefore, had to search for a force which would bend the motion Planer Fig. 5. round into its elliptical orbit. This he showed must be a force directed towards the sun, as in In fact, the force is the (5). 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 next figure the mechanical advantage obtained by the DYNAMICS 47 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 ac count shows, partly with the view of explain ing 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 cen turies past. But, according to modern scien tific theory, that greyness, which so heightens our sense of the immobility of the building, of 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 tempera ture, so symbolic of the quiet repose of the But this feeling of temperature building. simply marks our sense of the transfer of 48 INTRODUCTION TO MATHEMATICS 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 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 is DYNAMICST 49 anyone in particular, or to any special modes The result is that the chair of perception. 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 But the point is scientific ideas describe it. that science reduces the chair to things moving in space and influencing each other s Then elements or of circum stances, as thus conceived, are merely the motions. factors the various which enter into a set 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 to say, velocities, angular and suchlike things. Accordingly, mathematical physicc deals with correlations between variable numbers which are supposed to represent the correlations is velocities, accelerations, which exist in nature between the measures of these geometrical elements and of their rates of change. But always the mathe matical 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 50 INTRODUCTION TO MATHEMATICS thus conceived in this abstract way through out 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 suffi cient to "explain" our sensations. When we hear a sound, the molecules of the air have been agitated in a certain way: given the agitation, or air-waves as they are called, all normal people hear sound; and if there are no air- waves, there is no sound. And, simi larly, a physical cause or origin, or parallel event (according as different people might like to phrase it), underlies our other sensa Our very thoughts appear to corre spond to conformations and motions of the brain; injure the brain and you injure the thoughts. Meanwhile the events of this phys tions. 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 mathe matical physics to our emotions, sensations, and thoughts; and a great deal of contro versy has been occasioned by it and much ink We need only make one remark. spilled. The whole situation has arisen, as we have seen, from the endeavour to describe an ex ternal world "explanatory" of our various individual sensations and emotions, but a ical DYNAMICS 51 world, also not essentially dependent upon any particular sensations or upon any par Is such a world merely but one huge fairy tale? But fairy tales are ticular individual. fantastic and arbitrary: if in truth there ought to submit itseli to an exact description, which determines accurately its various parts and their mutual be such a world, it Now, to a large degree, this 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 relations. scientific here we have an inductive verification of assumption. It must be admitted that no inductive proof is conclusive; but if the whole idea of a world which has ex istence independently of our particular per ceptions of it be erroneous, it requires careful explanation why the attempt to characterize it, in terms of that mathematical remnant of our ideas which would apply to it, should issue in such a remarkable success. It would take us too far afield to enter into 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 mathe matical physics and to pure mathematics: these are the ideas of vector quantities and the parallelogram law for vector addition. our initial 52 INTRODUCTION TO MATHEMATICS 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 dis tinct elements to be settled before it is com pletely determined, namely its magnitude Now (i.e. the length AC) and its direction. is which like this transference, anything, completely given by the determination of a magnitude and a direction is called a vector. For example, a velocity requires for its defini tion the assignment of a magnitude and of a direction. It must be of so many miles per hour in such and such a direction. The ex istence 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 DYNAMICS 53 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 but this is a point which It is sufficient here to say that a force acts on a body with a certain magnitude in a certain direction. accelerations; we need not go into. Now all vectors can be graphically repre sented 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 corre sponding 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 rep resentation of vectors is of the first import ance. By its aid we can enunciate the famous "parallelogram law" for the addition of vectors of the same kind but in different directions. 54 INTRODUCTION TO MATHEMATICS AC in figure 6 as repre Consider the vector sentative of the changed position of a body to C: we will call this the vector of from 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 Now the final transportation single type. to C is equally well effected by a from and a transporta to transportation from tion from B to (7, or, completing the parallelo to gram ABCD, by a transportation from and a transportation from to C. These transportations as thus successively applied are said to be added together, ^is is simply a definition of what we mean bjikhe addition of transportations. Note further that, con sidering parallel lines as being lines drawn in the same direction, the transportations B to C and to may be conceived as the same transportation applied to bodies in the two A A A B A D D D A initial positions B and A. With this con we may talk of the transportation ception as applied to a body in any position, to for example at B. Thus we may say that the transportation to C can be conceived as the sum of the two transportations to and to applied in any order. Here we have the parallelogram law for the ad dition of transportations: namely, if the A D A B A D A DYNAMICS 55 B A A and to D, to transportations are then the and parallelogram ABCD, complete the sum of the two is the diagonal AC. All this at first sight may seem to be very But artificial. that nature itself it must be observed presents us with the idea. For example, a steamer is moving in the direction (cf. fig. 6) and a man walks across its deck. If the steamer were still, in one minute he would arrive at B\ but on during that minute his starting point the deck has moved to D, and his path on to DC. So the deck has moved from that, in fact, his transportation has been from to C over the surface of the sea. It is, however, presented to us analysed into the sum of two transportations, namely, one from to relatively to the steamer, and one from to which is the transportation of AD A AB A A B A D the steamer. By taking into account the element of time, this diagram of the man s namely one minute, AC transportation represents his velocity. if represented so many feet of trans portation, it now represents a transportation of so many feet per minute, that is to say, it Then represents the velocity of the man. AC For AD AB and 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 56 INTRODUCTION TO MATHEMATICS diagrams and definitions concerning trans portations are turned into diagrams and defi nitions concerning velocities by conceiving the diagrams as representing transportations per unit time. Again, diagrams and defini tions concerning velocities are turned into Fig. 7. diagrams and definitions concerning accelera tions by conceiving the diagrams as repre senting velocities added per unit time. Thus by the addition of vector velocities and of vector accelerations, we mean the addition according to the parallelogram law. Also, according to the laws of motion a force is fully represented by the vector acceleration it produces in a body of given mass. Accordingly, forces will be said to be added when their joint effect is to be reckoned according to the parallelogram law. DYNAMICS Hence for the 57 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 parallelo law. far the simplest type of parallelogram is a rectangle, and in pure mathematics it is to the the relation of the single vector and AD, at right two component vectors, angles (cf. fig. 7), which is continually re curring. Let x, y, and r units represent the units lengths of AB, AD, and AC, and let 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 con tinually recurring topic of pure mathematics; and the results are of the type required for application to the fundamental vectors of gram By AC AB m mathematical physics. This diagram is the chief bridge over which the results of pure mathematics pass in order to obtain applica tion to the facts of nature. CHAPTER V THE SYMBOLISM OF MATHEMATICS WE now return to pure mathematics, and 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 consider by more 0,1,2,..., 9, 10, 11,... 100, 101,... and This notation was introduced into Europe through the Arabs, but they appar ently obtained it from Hindoo sources. The first known work* in which it is systematic ally explained is a work by an Indian mathe matician, 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 so on. * For the detailed matics, I historical facts relating to pure mathe am chiefly indebted to A Short History of Mathematics, by W. W. R. Ball. SYMBOLISM OF MATHEMATICS 59 purposes, however, the history of the notation The interesting point to notice is a detail. is the admirable illustration which this numeral system affords of the enormous im portance of a good notation. By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race. Before the introduction of the Arabic notation, multipli cation was difficult, and the division even of integers called into play the highest mathe matical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest num bers. This fact would have seemed to him a sheer impossibility. The consequential ex tension of the notation to decimal fractions was not accomplished till the seventeenth Our modern power of easy reck century. oning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. Mathematics is often considered a diffi cult and mysterious science, because of the numerous symbols which it employs. Of is more incomprehensible than course, nothing 60 INTRODUCTION TO MATHEMATICS 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 intro duced to make things easy. So in mathe matics, granted that we are giving any serious attention to mathematical ideas, the sym is invariably an immense simplifica It is not only of practical use, but is of great interest. For it represents an analy sis of the ideas of the subject and an almost bolism tion. pictorial representation of their relations to each other. If any one 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: x+y=y+x (x+y)+z=x + (y+z) xXy=yXx (xXy)Xz=xx(yXz) xx(y+x)=(xxy}+(xxz) (1) (2) (3) (4) (5) Here (1) and (2) are called the commutative and associative laws for addition, (3) and (4) SYMBOLISM OF MATHEMATICS 61 are the commutative and associative laws for multiplication, and (5) is the distributive law relating addition and multiplication, For ex ample, 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 sym bolism, we can make transitions in reasoning almost mechanically by the eye, which other wise would call into play the higher faculties of the brain. It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are The precise opposite is the case. doing. Civilization advances by extending the num ber of important operations which we can perform without thinking about them. Opera tions of thought are like cavalry charges in a battle they are strictly limited in num ber, 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 we cannot place sym rapidly written. bols more concisely together than by placing them in immediate juxtaposition. In a good symbolism therefore, the juxtaposition of im- Now 62 INTRODUCTION TO MATHEMATICS portant 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 choice as to what shall be denoted by Now the two most their juxtaposition xy. are those of addition on hand ideas important and multiplication. Mathematicians have chosen to make their symbolism more concise by defining xy to stand for xXy. Thus the laws (3), (4), and (5) above are in general make a written, xy=yx, (xy)z=x(yz) 9 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 vari able: we write Sx for 3 Xx, and 30z for 30 Xx. It is evident that in substituting definite numbers for the variables some care must be taken to restore the X, 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 X 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 em phatic presentation for an idea, often a very SYMBOLISM OF MATHEMATICS 63 subtle idea, and by its existence make it easy to exhibit the relation of this idea to all the complex trains of ideas in which it occurs. For example, take the most modest of all symbols, namely, 0, which stands for the 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 be found in philosophic works. in real truth, more difficult or will quantity Zero is not, 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 opera tions 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 ren dered by the symbol 0, which stands for the number zero. The symbol developed in connection with the Arabic notation for numbers of which it is an essential part. For in that notation the 64 INTRODUCTION TO MATHEMATICS 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 intro duced for this purpose had no definite con ception 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 re present 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. SYMBOLISM OF MATHEMATICS 65 was to make the Thus the first use of Arabic notation possible no slight service. We can imagine that when it had been intro duced for this purpose, practical men, of the sort who dislike fanciful ideas, deprecated the silly habit of identifying it with a number But they were wrong, as such men zero. always are when they desert their proper function of masticating food which others have prepared. For the next service per formed by the symbol essentially depends upon assigning to it the function of repre senting 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 = l. This can be represented in an indefinite number of ways; for example, x = l y, y = Ix, %x+3y 1 =x+%y, and so on. But the important way of stating it is Similarly the important way of writing the equation # = 1 is # 1=0, and of representing the equation 3x-2 = 2x 2 is 2x*-3x+2 = 0. The point is that all the symbols which repre sent variables, e.g. x and y, and the symbols 66 INTRODUCTION TO MATHEMATICS 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 The first man to do this is said to zero. have been Thomas Harriot, born at Oxford in 1560 and died in 1621. But what is the importance cedure? It of this simple symbolic pro made possible the growth of the modern conception of algebraic form. This is an idea to which we shall have con tinually to recur; it is not going too far to say that no part of modern mathematics can be properly understood without constant re currence to it. The conception of form is so general that it is difficult to characterize in abstract terms. At this stage we shall do better merely to consider examples. Thus the equations %x S = 0, x 1 = 0, 5x 6 = 0, are all equations of the same form, namely, it equations involving one unknown x, which is not multiplied by itself, so that x 2 #3 etc., do not appear. Again 3x 2 - Zx + 1 = 0, x 2 = Sx +2 = 0, x2 4=0, are all equations of the same form, namely, equations involving one un known x in which xXx, that is z 2 appears. These equations are called quadratic equa tions. Similarly cubic equations, in which a3 appears, yield another form, and so on. Among the three quadratic equations given above there is a minor difference between the , , , SYMBOLISM OF MATHEMATICS 67 x2 4 = 0, and the preceding two to the fact that x (as distinct due equations, from x 2 ) does not appear in the last and does in the other two. This distinction is very unimportant in comparison with the great fact that they are all three quadratic last equation, equations. Then further there are the forms of equation stating correlations between two variables; for example, 2x+3y-8=Q, 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 x+y-l=0, such equations by a straight line. Then there are other forms for two variables for ex ample, the quadratic form, the cubic form, and so on. But the point which we here in sist upon is that this study of form is facili and, indeed, made possible, by the standard method of writing equations with on the right-hand side. the symbol There is yet another function performed by in relation to the study of form. Whatever tated, number x may be, OX# = 0, and x+Q=x. By means of these properties minor differ ences of form can be assimilated. Thus the difference mentioned above between the quad ratic equations #2 =3a;+2 = 0, and x 2 4=0, can be obliterated by writing the latter 68 INTRODUCTION TO MATHEMATICS 2 = equation in the form x + (0 X x) 4 0. For, 2 x stated the laws + (QXx) 4 = above, by 2 4 = #* 4. Hence the equation x 2 4 a; + = 0, is merely representative of a particular class of quadratic equations and belongs to the same general form as does x 2 3x+% =0. For these three reasons the symbol 0, rep resenting the number zero, is essential to modern mathematics. It has rendered pos sible types of investigation which would have been impossible without it. The symbolism of mathematics is in truth the outcome of the general ideas which domi nate 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 We its character, but none the less effective. have seen that an equation involving two variables, x and y, represents a particular cor relation between the pair of variables. Thus x+y \=Q represents one definite correla 5=0 represents another tion, and 3x+%y definite correlation between the variables x and y\ and both correlations have the form what we have called linear correlations. But now, how can we represent any linear of correlation between the variable numbers x and y? Here we want to symbolize any linear correlation; just as x symbolizes any SYMBOLISM OF MATHEMATICS 69 number. This is done by turning the numbers which occur in the definite correlation 3x+%y 5=0 into letters. We obtain ax + by c = 0. Here a, 6, 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 be tween x and y while a, 6, and c have un changed values. We do not determine what the values of a, 6, 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 ?/. But when we have obtained the properties of this correlation, we note that, because a, 6, and c have not in fact been deter mined, we have proved properties which must belong to any such relation. Thus, by now varying a, 6, and c, we arrive at the idea that represents a variable linear ax+by c = correlation between x and y. In comparison with x and y, the three variables a, 6, 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 their equations, by using letters at the end of the alphabet for the "variable" variables, and letters at the beginning of the alphabet for 70 the INTRODUCTION TO MATHEMATICS "constant" or parameters. naturally about the variables, The two systems meet 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 con fusion is caused by a procedure which seems so lax. The result of this continual elimination of definite numbers by successive layers of para meters is that the amount of arithmetic per formed 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 "alge braic form" commence their sway. CHAPTER VI GENERALIZATIONS OF NUMBERS 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., B.C. and 1100 probably a copy of a much older It deals largely with the properties of and work. it is It appears, therefore, that this concept was developed very early in the his tory 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 civiliza tions were familiar with the idea of twofractions. 71 72 INTRODUCTION TO MATHEMATICS thirds, first and with Thus as the number we place the The Greeks thought of allied notions. generalization of concept of fractions. 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 nota tion 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 this subject rather in the multiplier. In connection with the theory of ratio, or Greeks made a great discovery, which has been the occasion of a large amount of philosophical as well as mathematical thought. They found out the existence of "incommensurable" ratios. They proved, in fact, during the course of their geometrical investigations that, starting with a line of any fractions, the 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 can not be expressed as any fraction of the side of the same square; in our modern notation the length of the diagonal is V% times the length of the side. But there is no fraction which exactly represents V&. We can approximate GENERALIZATIONS OF NUMBERS to V2 as closely as exactly reach its we like, but we never 49 For example, value. 73 is /vt/ just less than 2, that V% lies g and - is greater than 2, so 73 between - and - But the best A 5 systematic way of approximating to A/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 14 is 1, -, 1414 141 , , and so on. Ratios of this sort are called by the Greeks incommensurable. They have excited from the time of the Greeks onwards a great deal and the difficulties connected with them have only recently been of philosophic discussion, cleared up. 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 We always think of the real numbers." numbers as arranged in order of magnitude, starting from zero and going upwards, and becoming proceed. indefinitely larger and larger as we real numbers are conveniently The INTRODUCTION TO MATHEMATICS 74 represented by points on a line. Let OMANBPCQD any line bounded at OX be X and stretching away Take any indefinitely in the direction OX. convenient point, A, on it, so that OA repre sents the unit length; and divide off lengths AB, BC, CD, and so on, each equal to OA. Then the point represents the number 0, the number 1, the number 2, and so on. In fact the number represented by any point is the measure of its distance from 0, in terms of the unit length OA. The points between and represent the proper frac tions and the incommensurable numbers less A B A than 1 that of ~, the middle point of ; OA represents -, AB represents 3-, that of BC represents 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, and moving regularly in the starting from to X, represents the real from numbers as arranged in an ascending order direction GENERALIZATIONS OF NUMBERS 75 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 0, A, B, C, D, etc. Here there is a first point 0, a definite next point, A, and each point, such as A or B, has one definite immediate predecessor and one definite immediate suc cessor, with the exception of 0, which has no predecessor; also the series goes on indefi This sort of order is nitely without end. called the type of order of the integers; its essence is the possession of next-door neigh bours on either side with the exception of No. 1 in the row. Again consider the integers and fractions together, omitting the points which correspond to the incommensurable ratios. The sort of serial order which we now obtain is quite different. There is a first term 0; but no term has any immediate pre decessor or immediate successor. This is seen to be the for between case, easily any two fractions we can always find another fraction intermediate in value. One very of is this to add the simple way doing fractions together and to halve the result. For example, between f and f , the fraction 2 (f I tnat is ||, lies; and between f and + )> 76 INTRODUCTION TO MATHEMATICS lies; ), that is ff J| the fraction J(f + and so on indefinitely. Because of this prop erty the series is said to be "compact." There is no end-point to the series, which in creases indefinitely without limit as we go along the line OX. It would seem at first sight as though the type of series got in this way from the fractions, always including the integers, would be the same as that got from all the real numbers, integers, fractions, and incommensurables taken together, that is, from all the points on the line OX. All that we have hitherto said about the series of , fractions applies equally well to the series of 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 end points to certain classes. Thus, consider all real the incommensurable V%. In the series of real numbers this stands between all the numbers whose squares are less than 2, and all the numbers whose squares are greater than 2. But keeping to the series of fractions alone and not thinking of the incommensur ables, so that we cannot bring in V 9 there 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 is have their squares less than 2, and on the Hence in the other side greater than 2. GENERALIZATIONS OF NUMBERS 77 a quasi-gap where ought to come. This presence of quasigaps 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. series of fractions there is V% It is to avoid this difficulty that recourse is had to the incommensurables, so as to ob tain a complete series with no gaps. There is another even more fundamental difference between the two series. We can rearrange the fractions in a series like that of the integers, that is, with a first term, and such that each term has an immediate suc cessor and (except the first term) an imme We can show how this diate predecessor. can be done. Let every term in the series of fractions and integers be written in the frac tional form by writing y for 1, f 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, f, f, f, T8^, etc., are all reckoned as distinct. Now group the frac tions 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 78 INTRODUCTION TO MATHEMATICS index. Thus 7 is the index of -J, and also of Let the fractions in each class f and of f be all fractions which have some specified index, which may therefore also be called . , the class index. Now arrange these classes in the order of magnitude of their indices. The first class has the index 2, and its only member is j; the second class has the index the third 3, and its members are | and f class has the index 4, and its members are J, f f the fourth class has the index 5, and its ; , ; members are \, f |- { ; and so 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 index. Thus for the class of index n, the members appear in the order. , , ~* i 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: 1121 nn 31234 2 3 d 4 3 2 1 1 1 I"" GENERALIZATIONS OF NUMBERS w-2 I and so 1 2 n-1 3 n-l n-2 n-3" 79 I ~T~ n 9 on. Now we can get rid of fractions of the repetitions of striking they appear after their all same value by simply them out whenever occurrence. In the few initial terms written down above, f which is enclosed above in square brackets is the only fraction not in its lowest terms. It has occurred before as Thus this must be struck out. But the \. series is still left with the same properties, namely, (a) there is a first term, (6) 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 of mathematical ideas. are here in fact touching on the fringe of the great problems of the meaning of continuity first We 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 80 INTRODUCTION TO MATHEMATICS the statement 2+3=5. We add 3 to 2 and obtain 5. Think of the operation of adding 3: = let this 1. be denoted by +3. Think Again 43 of the operation of subtracting let this be denoted by -3. Thus instead of considering the real numbers in themselves, we consider the operations of adding or sub tracting them: instead of A/2, we consider 3: + A/2 and -A/2, namely the operations of adding A/2 and of subtracting A/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 commutative. Mathematicians have a habit, which is puzzling to those engaged in tracing out meanings, but is very convenient in practice, of using the same symbol in different though allied senses. The one essential requisite for a symbol in their eyes is that, whatever its possible varieties of meaning, the formal laws In for its use shall always be the same. is GENERALIZATIONS OF NUMBERS 81 accordance with this habit the addition of operations is denoted by + as well as the addition of numbers. Accordingly we can write where the middle -f- 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 addi 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 ob taining the number 4. Any interpretation which is possible is always correct. But the tion, or as the only interpretation which is always possible, under certain conditions, is that of operations. The other interpretations often give non sensical results. This leads us at once to a question, which must have been rising insistently in the m INTRODUCTION TO MATHEMATICS reader s mind: What is the use of all this elaboration? At this point our friend, the practical man, will surely step in and insist on sweeping away all these silly cobwebs of the brain. The answer is that what the mathe matician 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 mathe matical 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. the equation # + 1=3; the solution is Take a; =2. Here we can interpret our symbols as mere numbers, and the recourse to "operations" is entirely unnecessary. But, if # is a mere number, the equation x+S = l 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 GENERALIZATIONS OF NUMBERS general equation of the This equation is 83 as x 4- 1 = 3. its solution is same form x+a = b, and x = ba. Here our difficulties become acute; for this form can only be used for the numeri cal 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 6, which we must drag like a chain throughout all our reasoning. Really prolonged mathe matical investigations would be impossible under such conditions. Every equation would at least be buried under a pile of limita tions. But if we now interpret our symbols as "operations," all limitation vanishes like magic. The equation x + I =3 gives x= +2, the equation #+3 = 1 gives x = 2, the equa which is an opera tion x +a = b gives x = tion 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 ba algebra. Our object is merely to make plain the fundamental ideas which guide the forma tion of the science. Accordingly we do not further explain the detailed rules by which the "positive and negative numbers" are INTRODUCTION TO MATHEMATICS 84 We have multiplied and otherwise combined. explained above that positive and negative numbers are operations. They have also been called "steps." Thus +3 is the step 3 is the by which we go from 2 to 5, and step backwards by which we go from 5 to 2. Consider the line divided in the way ex plained in the earlier part of the chapter, so that its points represent numbers. Then OX / D C B A -3 -2 -1 +1 +2 +3 A B C D E the step from to J5, or from A to C, or the divisions are taken backwards along to B , and so ) from C to A , or from on. Similarly 2 is the step from to J5 , or from B to from C , or from B to 0, or to A. We may consider the point which is reached by a step from 0, as representative of that is (if D OX D A represents +1, B represents f 1, B represents 2, and represents so on. It will be noted that, whereas previ ously with the mere "unsigned" real numbers step. +2, Thus A the points on one side of only, namely along OX, were representative of numbers, now with steps every point on the whole line is representative stretching on both sides of of a step. This is a pictorial representation of the superior generality introduced by the positive and negative numbers, namely the GENERALIZATIONS OF NUMBERS 85 operations or steps. These "signed" num bers are also particular cases of what have been called vectors (from the Latin veho, I draw or carry). For we may think of a to A, or from particle as carried from to B. In suggesting a few pages ago that the practical man would object to the subtlety involved by the introduction of the positive A 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 practi cal man himself who first employed the actual Their origin is not very symbols + and 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 pub . They seem mathematics by a German mathematician, Stifel, in a book lished at Leipzig, in A.D. 1489. first to have been employed in 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 86 INTRODUCTION TO MATHEMATICS the ideas which these symbols have gener ; ated are much too important for the wel fare of humanity to have come from the sea or from the land. The possibilities of application of the posi and negative numbers are very obvious. one direction are represented by a positive number, those in the opposite di rection are represented by negative numbers. tive If lengths in 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 posi If vitreous tive, an overdraft is negative. electrification is positive, resinous electrifica tion is negative. Indeed, in this latter case, the terms positive electrification and nega tive electrification, considered as mere names, have practically driven out the other terms. An endless series of examples could be given. idea of positive and negative numbers has been practically the most successful of The mathematical subtleties. CHAPTER VII IMAGINAKY 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 succes de Not only the practical man, but scandale. also men of letters and philosophers have expressed their bewilderment at the devo tion of mathematicians to mysterious entities name are confessed to be which by their very imaginary. At this point it may be useful to observe that a certain type of minor in tellect 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 Now, it cannot be too clearly understood that, in science, technical terms questions. are names arbitrarily assigned, like Christian 87 88 INTRODUCTION TO MATHEMATICS names of the to children. There can be no question names being may be judicious sometimes be so remember, or so important ideas. right or wrong. They or injudicious; for they can arranged as to be easy to as to suggest relevant and 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, pay them extra and make them mean what I So we will not bother as to "I like." 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 nega In exactly the same way it tive numbers. is due to the three great mathematical ideas of the variable, of algebraic form, and of generalization. numbers arose The positive and negative from the consideration of # + 1=3, x+3 = I, and the equations like general form x+a = b. Similarly the origin of imaginarv numbers is due to equations like # 2 + l=3, z 2 +3 = l, and x*+a = b. Exactly the same process is gone through. The equa tion a;2 + l =3 becomes x 2 ==%, and this has two V%. The solutions, either x = -\-V%, or x = statement that there are these alternative IMAGINARY NUMBERS usually written x = =*= A/2. So far plain sailing, as it was in the previous solutions all is 89 is case. But now an analogous difficulty arises. 2 and For the equation x2 +S = 1 gives x 2 = 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 equa is in fact nonsense. Thus, finally taking the general form x 2 +a = b, we find the pair tion =*= of solutions x V(b a), when, and only when, b is not less than a. Accordingly we cannot say unrestrictedly that the "con stants" a and b may be any numbers, that is, a and b are not, as they ought to be, independent unrestricted "vari and so again a host of limitations ables"; and restrictions will accumulate round our the "constants" as we proceed. The same task as before therefore awaits us: we must give a new interpretation to our work symbols, so that the solutions =t V(b a) for the equation x z -}-a = b always have meaning. In other words, we require an interpretation of the symbols so that Va always has mean Of ing whether a be positive or negative. course, the interpretation must be such that all the ordinary formal laws for addition, sub traction, multiplication, and division hold good; and also it must not interfere with the INTRODUCTION TO MATHEMATICS 90 generality which we have attained by the use of the positive and negative numbers. In fact, it must in a sense include them as special cases. c2 for write When it, a is so that c 2 negative is we may Then positive. we can so interpret our symbols that has a meaning, we have attained our 1) has come to be looked object. Thus V( on as the head and forefront of all the Hence, V( if 1) imaginary quantities. This business of finding an interpretation for V( 1) is a much tougher job than the 1. In fact, analogous one of interpreting while the easier problem was solved almost instinctively as soon as it arose, it at first hardly occurred, even to the greatest mathe maticians, that a problem existed which was perhaps capable of solution. Equations like x2 3, when they arose, were simply ruled aside as nonsense. it came to be gradually perceived the eighteenth century, and even during 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 However, IMAGINARY NUMBERS 91 algebraic laws of transformation; and it was seen that a whole world of interesting results could be attained, if only these symbols might legitimately be used. Many mathematicians were not then very clear as to the logic of their procedure, and an idea gained ground that, in some mysterious way, symbols which appropriate manip mean nothing can by ulation 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 It was during this epoch careless writer. that the epithet "imaginary" came to be applied to V( 1). What these mathema ticians had really succeeded in proving were a series of hypothetical propositions, of which this is the exist for traction, blank form: If interpretations and for the addition, sub and division of multiplication, V( 1) which make the ordinary algebraic (e.g. x+y=y+x, etc.) to be satisfied, then such and such results follow. It was natural that the mathematicians should not which ought always appreciate the big to have preceded the statements of their re V( 1) rules "If," sults. As may be expected the interpretation, 92 INTRODUCTION TO MATHEMATICS when found, was a much more elaborate affair numbers and the reader s attention must be asked for some We have careful preliminary explanation. already come across the representation of a point by two numbers. By the aid of the than that of the negative Fig. 8. positive and negative numbers we can now represent the position of any point in a plane by a pair of such numbers. Thus we take and the pair of straight lines , at right angles, as the "axes" from which we start all our measurements. Lengths mea sured along and are positive, and and measured backwards along are negative. Suppose that a pair of numbers, written in order, e.g. (+3, -j-1), so that there XOX OX YOY OY OX OY IMAGINARY NUMBERS 93 a first number (+3 in the above example), and a second number ( + 1 in the above ex ample), represents measurements from along XOX for the first number, and along YOY 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 towards X, and a direction, that is from length +1 measured along YOY in the posi towards F. tive direction, that is from Similarly in (3, +1) the length of 3 units towards and of is to be measured from Also in ( 3, 1 unit from towards Y 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 Then, from the two couple." numbers 1 and 3, eight ordered couples can be generated, namely is X , . + 1, +3), (-1, +3), (-1, -3), ( + 1, -3), (+3, +1), (-3, +1), (-3, -1), (+3, -1). ( Each of these eight "ordered couples" directs XOX and a process of measurement along YOY which is different from that directed by any of the others. The processes of measurement represented by the last four ordered couples, mentioned above, are given pictorially in the figure. The lengths and ON together correspond OM 94 INTRODUCTION TO MATHEMATICS OM to (+3, +1), the lengths and ON to and gether correspond to (-3, +1), and together to (-3, -1), and together to (+3, -1). But by com pleting the various rectangles, it is easy to see that the point P completely determines and is determined by the ordered couple ON ON OM OM (+3, +1), the point P by (-3, +1), the point by (-3, -1), and the point P More generally in the pre by (+3, 1). " P" vious figure (8), the point P corresponds to the ordered couple (x, y), where x and y in the figure are both assumed to be positive, the point P corresponds to (# y), where x in the figure is assumed to be negative, to f P and to Thus an ordered , 9 (x y ) (x, y ). , P" " IMAGINARY NUMBERS 95 couple (x, y), where x and y are any positive or negative numbers, and the corresponding point reciprocally determine each other. It is convenient to introduce some names at this In the ordered couple (x, y) the juncture. 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 The idea of de "coordinates" of the point. a the of point by its "co position termining ordinates" was by no means new when the was being formed. imaginaries theory of It was due to Descartes, the great French mathematician and philosopher, and appears in his Discours published at Ley den in 1637 The idea of the ordered couple as a A.D. 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 in fig. 9 is the couple (+3, 0), the point f is the couple (0, +1), the point point the couple the couple (3, 0), the point the the 0). (0, -1), point couple (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," 0, of a certain " " M N N M INTRODUCTION TO MATHEMATICS 96 length and in a certain direction. The line OP may be called the vector line from to P, or the step from 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 f couples (x, y) and (x , y ). The interpreta tion must, (a) make the result of addition to be another ordered couple, (6) make the operation commutative so that (x, y)-\f (x y )=(x y )+(x 9 y), (c) make the opera tion associative so that , , make the result of subtraction unique, when we seek to determine the un known ordered couple (x, y) so as to satisfy (d) so that the equation (x 9 y)+(a, b) = (c, d), is one and only one answer which we can represent by there (a:, y)=(c, d)-(a, b). IMAGINARY NUMBERS 97 All these requisites are satisfied by taking (x, y)+(x , y ) to mean the ordered couple (x+x 9 y+y ). Accordingly by definition we put Notice that here we have adopted the mathe matical habit of using the same symbol + in + different senses. The of the equation has the on the left-hand side new meaning of -fwhich we are just defining; while the two 4- s on the right-hand side have the meaning of the addition of positive and negative num bers (operations) which was defined in the No last chapter. practical confusion arises this double use. from As examples of addition we have (+3, +l)+(+2, +6) =(+5, +7), (+3, -!)+(-, -6)=( + l, -7), (+3, +l)+(-3, -1)=(0, 0). The meaning for us. We (a?, of subtraction find that y) is now settle - (u, v)=(x-u,y -v). Thus (+3, +2)-(+l, +l)=(+2, +1), and (+1, -2) -(+2, -4)=(-l, +2), and (-1, -2) -(+2, +3) =(-3, -5). 1 INTRODUCTION TO MATHEMATICS 98 It is easy to see that (x, y) - (u, v) = (x, y)+(-u, -v). Also (0,0). Hence (0, 0) is to be looked on as the zero For example ordered couple. + (0, Cr,2/) 0)=(*, y). The pictorial representation of the addition of ordered couples is surprisingly easy. Y Fig. 10. Let OP OM=x so that (a?, y) represent (xi, yj so that i =yi. OMi =Xi and Complete the paral and lelogram OPRQ by the dotted lines QR, then the diagonal OR is the ordered and represent PM =y; let OQ QM couple (x+Xi, y+yi). PR For draw PS parallel IMAGINARY NUMBERS to OX; then and MM PRS evidently the triangles are in all respects equal. =PS=x l9 and RS^QM,; Hence and there fore OM = OM +MM = x +xi, RM =SM +RS = Thus OR required. OP represents the ordered couple as This figure can also be drawn and OQ in other quadrants. 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 with It is 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 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 100 INTRODUCTION TO MATHEMATICS One of the most fascinating characteristics of mathematics is the sur prising way in which the ideas and results of different parts of the subject dovetail into each other. During the discussions of this velocities. and the previous chapter we have been guided merely by the most abstract of pure mathe matical 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 en gineer 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 of the multiplication of guided by exactly the same considerations as is that of their addition. The interpretation of multiplication must be such that (a) the result is another ordered couple, (ft) the operation is commutative, so that definition ordered couples (x 9 y) (7) X (x 9 y the operation \(x 9 ) = (x y ) X (x 9 is 9 y), associative, so that y)x(x ,y )}x(u,v) = 0, (8) is 2/)x{(z , y )x(u, v)} 9 must make the result [with an of division unique exception for the case of the zero couple (0, 0)], so that when we seek to de termine the unknown couple (x y) so as to satisfy the equation 9 0, 2/)x(a, 6)=(c, d) one and only one answer, which we 9 there is can represent by (x, y) = (c, d) + (a, 6), 101 or by (x 9 y) = g^j- 102 INTRODUCTION TO MATHEMATICS Furthermore the law involving both (e) addition and multiplication, called the dis tributive law, must be satisfied, namely = \(x,y)X(a,b)\+\(x,y)x(c,d)\. All these conditions (a), (), (7), (8), (e) can be satisfied by an interpretation which, though it looks complicated at first, is capable of a simple geometrical interpretation. By (x, y) definition X This symbol (* is , 2/0 we put - {(** - 2/2/0, f (xy + x y] \ (A) the definition of the meaning of the it is written between two X when ordered couples. It follows evidently from this definition that the result of multiplica is another ordered couple, and that the value of the right-hand side of equation (A) is not altered by simultaneously interchang Hence condi ing x with x and y with y tion f , . tions (a) and (/3) are evidently satisfied. proof of the satisfaction of (7), (S), The (e) is equally easy when we have given the geo metrical interpretation, which we will pro ceed to do in a moment. But before doing 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 z2 = 3, to which no solutions could be this it will IMAGINARY NUMBERS 103 assigned in terms of positive and negative numbers. We then found that all our difficulties would vanish if we could interpret the equation x 2 = 1, i.e., if we could so de real that xV(-l) = -1. the three special or Now let us consider dered couples * (0,0), (1,0), and (0,1). We have already proved that fine V(-l) VpT) (*,)+ (0,0) =(*,?). Furthermore we now have (*,y)x(0,0)-(0,0). Hence both for addition and for multiplica tion the couple (0,0) plays the part of zero in elementary arithmetic and algebra; com pare the above equations with x-{-Q=x and 9 Again consider (1, 0): this plays the part of 1 in elementary arithmetic and algebra. In these elementary sciences the special char acteristic of 1 is that xXl = x, for all values of x. by our law of multiplication Now (*, y) X(l, 0) ={(*-0), (y+0)} =(*, y). Thus (1, 0) is * the unit couple. For the future we follow the custom of omitting the wherever possible, thus f 1,0) stands for (+1,0) and 4- sign (0,1) for (O.+ l). 104 INTRODUCTION TO MATHEMATICS Finally consider (0,1): for us the symbol V( this will interpret The symbol must 1). therefore possess the characteristic property that V(~l)X V/(-l)= law of multiplication (0,1) X (0,1) ={(0-1), But (1,0) is Now by the ordered couples -1. for (0+0)} =(-1, 0). the unit couple, and (-1, 0) the negative unit couple; so that (0,1) has the desired property. There are, however, two roots of 1 to be provided for, namely is =*=V( 1). Consider membering that (0, 2 ( 1) 1); = 1, we here again re find, (0, -1) X(0, -1)=(-1,0). Thus (0, 1) is the other square root of V( 1). Accordingly the ordered couples (0,1) and 1) (0, are the interpretations of terms of ordered couples. But =*=V( which corresponds to which? Does (0,1) correspond to + V( 1) and (0, 1) to 1) in -A/(-l),or (0,1) to -V(-l), and (0, -1) 1)? The answer is that it is per fectly indifferent which symbolism we adopt. The ordered couples can be divided into to -f A/( three types, (i) the "complex imaginary" type (x,y\ in which neither x nor y is zero; (ii) the type (s,0); (iii) the "pure Let us consider the imaginary" type (0,y). relations of these types to each other. First "real" multiply together the "complex imaginary" IMAGINARY NUMBERS couple (x,y) and the "real" 105 couple (a,0) we find (a,0) X (x,y) = (ax, ay). Thus the effect is merely to multiply each of the couple (x,y) by the positive or negative real number a. term Secondly, multiply together the "complex couple (x,y) and the "pure imaginary" couple (0,6), we find imaginary" Here the effect is more complicated, and is best comprehended in the geometrical inter pretation to which we proceed after noting three yet more special cases. Thirdly, we multiply the couple (a,0) with the imaginary (0,6) and obtain "real" (a,0)X(0,6)=(0, a6). Fourthly, we multiply the two couples (a,0) and (a , 0) and obtain (a,0)x(a ,0)=(aa Fifthly, couples" (0,6) We now tation, ,0). we multiply the two (0,6) and (0, 6) "real" "imaginary and obtain x (0,6 ) =( -66 , 0). turn to the geometrical interpre beginning first with some special 106 INTRODUCTION TO MATHEMATICS Take the couples (1,3) and (2,0) and consider the equation cases. (2,0) X (1,3) =(2,6) (fig. 11) the vector OP represents (1, 3), and the vector repre sents (0,2), and the vector OQ represents Thus the product (2,0) X (1,3) is (2,6). 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 In the diagram product X (1,3), we have (0, 2) X (1,3) =(-6, ON (0,2) 2) The vector ON, corresponds to the vector OR to (-6,2). *Thus and which (0, 2) OR IMAGINARY NUMBERS 107 represents the new product is at right angles to OQ and of the same length. Notice that we have the same law regulating the length of OQ as in the previous case, namely, that its length is the product of the lengths of the two vectors which are multiplied to gether; but now that we have ONi along the "ordinate" axis OF, instead of along the "abscissa" axis OX, the direction of OP ON 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 ONi. We shall get a clue to the general law for the direction by inverting the way of thought, and by think ing of the vectors ON and ONi as modified by the vector OP. The law for the length re mains 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-clock wise) direction of rotation from OX towards OF through an angle equal to the angle POX: it is an accident of this particular case that this rotation makes OQ lie along the line OP. Again consider the product of ONi and OP; the new direction for the enlarged ONi (i.e. OR) is found by rotating ON in the anti clockwise direction of rotation through an angle equal to the angle POX, namely, the angle NiOR is equal to the angle POX. INTRODUCTION TO MATHEMATICS 108 The general rule for the geometrical repre sentation of multiplication can now be enunciated thus: Fig. 12. of the two vectors OP and a vector OR, whose length is the pro duct of the lengths of OP and OQ and whose direction OR is such that the angle is to the sum of the and equal angles QOX. Hence we can conceive the vector OP as making the vector OQ rotate through an angle (i.e. the angle ROQ=the angle POX), or the vector OQ as making the vector OP rotate through the angle QOX (i.e. the angle ROP = the angle QOX). We do not prove this general law, as we The product OQ is ROX POX POX IMAGINARY NUMBERS 109 should thereby be led into more technical processes of mathematics than falls within the design of this book. But now we can im mediately see that the associative law [num bered (7) 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. We So much for multiplication. 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. have not considered the technical de tails of all these processes because it would lead us too far into mathematical details; but we have shown the general mode of protedure. When we are interpreting our alge braic symbols in this way, we are said to be employing "imaginary quantities" or "com These terms are mere plex quantities." details, and we have far too much to think about to stop to enquire whether they are or are not very happily chosen. The net result of our investigations is that We 110 INTRODUCTION TO MATHEMATICS like any equations x +3 = 2 or (#+3) 2 = 2 can now always be interpreted into terms of In vectors, and solutions found for them. 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 respectively (w, v) two equations become and {(u,v) + (3,0) = (2,0), We have now completely solved the initial 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 addi difficulties tion. But, before we can congratulate our 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 selves on 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 2 %x +4=0; but there are equation like x an endless number of other equations, for 3 &r+4=0, a^+a: 3 +2=0, and so example, x on without limit. Have we got to make a IMAGINARY NUMBERS new science whenever a 111 new equation ap pears? 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 itself as far as its fundamental ideas are concerned. It was receiving its final form about the same time as when the steam complete in engine was being perfected, and will remain a great and powerful weapon for the achieve ment 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 geo metry have already been employed in the previous chapters. It is now time for us to consider them more closely for their own sake; and in doing so we shall strengthen our hold on other ideas to which we have attained. In the present and succeeding chapters we will go back to the idea of the positive and negative real numbers and will ignore the imaginaries which were introduced in the last two chapters. We have been perpetually using the idea that, by taking two axes, XOX and YOY in a plane, any point P in that plane can be determined in position by a pair of positive or negative numbers x and y, where (cf. fig. 13) x is the length OM and y is the length , PM This conception, simple as it looks, is the main idea of the great subject of co Its discovery marks a ordinate geometry. momentous epoch in the history of mathe It is due (as has been matical thought. . 112 COORDINATE GEOMETRY 113 already said) to the philosopher Descartes, and occurred to him as an important mathe*matical 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 P y Y Fig. 13. science with some of its best ideas. On the other hand it must be said that, with hardly an exception, all the remarks on mathematics made by those philosophers who have pos sessed but a slight or hasty and late-acquired knowledge of it are entirely worthless, being either trivial or wrong. The fact is a curious one; since the ultimate ideas of mathematics INTRODUCTION TO MATHEMATICS 114 i 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 seem, after abstract things, and a long training is neces sary to secure even a partial immunity from error as soon as we diverge from the beaten track of thought. The discovery of coordinate geometry, and also that of projective geometry about the same time, illustrate another fact which is being continually verified in the history of knowledge, namely, that some of the greatest discoveries are to be made among the most well-known topics. By the time that the seventeenth century had arrived, geometry had already been studied for over two thou sand 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 genera tion of mathematicians laboured at the im Nor did the provement of the subject. from that fatal to progress, suffer bar subject 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 COORDINATE GEOMETRY 115 through the minds of Egyptians and Greeks, 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 ele mentary geometry without feeling the lack of some guiding method. Every proposition has to be proved by a fresh display of in genuity; and a science for which this is true of lacks the great requisite of scientific thought, namely, method. Now the especial point of coordinate geometry is that for the first time it introduced method. The remote deductions of a mathematical science are not of primary theoretical importance. The science has not been perfected, until it consists in essence of the exhibition of great allied methods by which information, on any desired topic which falls within its scope, can easily be obtained. The growth of a science is not primarily in bulk, but in ideas; and the more the ideas grow, the fewer are the deductions which it is worth while to write down. Un fortunately, mathematics is always encum bered by the repetition in text-books of numberless subsidiary propositions, whose im portance has been lost by their absorption into the role of particular cases of more general truths and, as we have already in sisted, generality is the soul of mathematics. 116 INTRODUCTION TO MATHEMATICS Again, coordinate geometry illustrates another feature of mathematics which has already been pointed out, namely, that mathe matical sciences as they develop dovetail into each other, and share the same ideas in com mon. It is not too much to say that the various branches of mathematics undergo a perpetual process of generalization, and that become generalized, they coalesce. Here again the reason springs from the very as they 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 co ordinate geometry lies in the fact that it relates together geometry, which started as the science of space, and algebra, which has its origin in the science of number. Let us now recall the main ideas of the two sciences, and then see how they are related by Descartes method of coordinates. Take will not trouble algebra in the first place. ourselves about the imaginaries and will think merely of the real numbers with posi tive or negative signs. The fundamental idea is that of any number, the variable number, We which is denoted by a letter and not by any definite numeral. then proceed to the consideration of correlations between vari ables. For example, if x and y are two vari- We COORDINATE GEOMETRY 117 we may conceive them as correlated by x+y = \, or by x +?/ = !, or in of an one indefinite number of other ways. any ables, the equations This at once leads to the application of the idea of algebraic form. We think, in fact, of some interesting type, thus rising from the initial conception of vari able numbers to the secondary conception of variable correlations of numbers. Thus we into the generalize the correlation #+2/ = 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 any correlation of l> + correlation; and the correlation, when deter mined, correlates the variable numbers x and Variables, like a, 6, and c above, which y. 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 con cerned with the relation between the variables x and y, after a, &, c are supposed to have been determined. So in a sense, relatively to x and y, the "constants" a, b, and c are con stants . Thus ax+by = c stands for the general example of a certain algebraic form, that is, for a variable correlation belonging to a cer tain class. 118 INTRODUCTION TO MATHEMATICS 2 2 2 Again we generalize # +2/ = l into ax -}- by2 = c, C9 or further into ax 2 +%hxy -\-by 2 2 2 further, into ax -\-hxy-\- by -}-%gx still or, still +%fy=c. Here again we are led to variable correla tions which are indicated by their various algebraic forms. Now let us turn to geometry. The of the science at once recalls to our name minds the thought of figures and diagrams exhibiting 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 com triangles mence on shapes, and squares, which he has cut out with scissors. What, however, is a tri angle? 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 his geometrical reasoning like triangles 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 COORDINATE GEOMETRY 119 of first thoughts on the subject which was the cause of the comparative sterility of the study of the science during so many centuries. Coordinate geometry, and Descartes its in ventor, 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 direc tions. 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 funda mental 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 pro duced 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 immedi ately apparent to the senses, was very char acteristic of the Greeks in all their many activities. It is enshrined in the difference between Greek architecture and Gothic archi tecture, and between the Greek religion and the modern religion. The spire on a Gothic cathedral and the importance of the un bounded straight line in modern geometry are both emblematic of the transformation of the modern world. 120 INTRODUCTION TO MATHEMATICS The is straight line, considered as a whole, accordingly the modern geometry root idea from which But then and we arrive starts. other at the sorts of lines occur to us, of the curve which at complete conception every point of it exhibits some uniform char acteristic, 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 characteristic of being at a given distance from its centre, and again there is the ellipse, which is an oval curve, such that the sum of the two distances of any point on it from two fixed points, called its foci, is constant for all points on the curve. It is evident that a circle is merely a particu lar case of an ellipse when the two foci are superposed in the same point; for then the sum of the two distances is merely twice the radius of the circle. The ancients knew the properties of the ellipse and the circle and, of course, considered them as wholes. For example, Euclid never starts with mere seg ments (i.e., bits) of circles, which are then pro longed. He always considers the whole circle It is unfortunate that the as described. circle is not the true fundamental line in geometry, so that his defective consideration of the straight line might have been of less consequence. This general idea of a curve which at any COORDINATE GEOMETRY 121 point of it exhibits some uniform property is expressed in geometry by the term "locus." locus is the curve (or surface, if we do not confine ourselves to a plane) formed by points, A of which possess some given property. every property in relation to each other which points can have, there corresponds some locus, which consists of all the points In investigating possessing the property. the properties of a locus considered as a whole, we consider any point or points on the locus. Thus in geometry we again meet with the fundamental idea of the variable. Furthermore, in classifying loci under such headings as straight lines, circles, ellipses, etc., we again find the idea of form. Accordingly, as in algebra we are con cerned with variable numbers, correlations between variable numbers, and the classifica tion of correlations into types by the idea of algebraic form; so in geometry we are con cerned with variable points, variable points satisfying some condition so as to form a locus, and the classification of loci into types by the idea of conditions of the same form. Now, the essence of coordinate geometry is the identification of the algebraic corre lation with the geometrical locus. The point on a plane is represented in algebra by its two coordinates, x and y, and the condition satisfied by any point on the locus is repreall To 122 INTRODUCTION TO MATHEMATICS sented the corresponding correlation Finally to correlations y. expressible in some general algebraic form, such as ax+by = c there correspond loci of some general type, whose geometrical con ditions are all of the same form. We have thus arrived at a position where we can effect a complete interchange in ideas and Each results between the two sciences. science throws light on the other, and itself by between x and 9 It is im gains immeasurably in power. possible not to feel stirred at the thought of the emotions of men at certain historic moments of adventure and discovery Columbus when he first saw the Western shore, Pizarro when he stared at the Pacific Ocean, Franklin when the electric spark came from the string of his kite, Galileo when he first turned his telescope to the heavens. Such moments are also granted to students in the abstract regions of thought, and high among them must be placed the morning when Descartes lay in bed and in vented the method of coordinate geometry. When one has once grasped the idea of coordinate geometry, the immediate ques tion which starts to the mind is, Wliat sort of loci correspond to the well-known algebraic forms? For example, the simplest among the general types of algebraic forms is ax by = c. The + sort of locus which corresponds COORDINATE GEOMETRY to this is a straight line, 123 and conversely to every straight line there corresponds an equa It is fortunate that the tion of this form. simplest among the geometrical loci should correspond to the simplest among the al gebraic forms. Indeed, it is this general correspondence of geometrical and algebraic simplicity which gives to the whole subject It springs from the fact that the its power. connection between geometry and algebra is not casual and artificial, but deep-seated and The equation which corresponds essential. to a locus is called the equation of (or the locus. Some examples of equations of straight lines will illustrate the subject. "to") Fig 14 124 INTRODUCTION TO MATHEMATICS Consider y # = 0; here the a, b and c, of the general form have been replaced by 1, 1, and respectively. This line passes through the "origin," 0, in the diagram and bisects the angle XOY. It is the line L OL of the diagram. The fact that it passes through the origin, 0, is easily seen by observing that the equation is satisfied by putting x = Q and y = simultaneously, but and are the co ordinates of 0. In fact it is easy to generalize and to see by the same method that the equation of any line through the origin is of the form ax+by = Q. The locus of equation = also passes through the origin and y-\-x bisects the angle OY: it is the line i of the diagram. Consider y x 1 the corresponding locus does not pass through the origin. We there fore seek where it cuts the axes. It must cut the axis of x at some point of coordinates x and 0. But putting y = in the equation, we get x = 1 ; so the coordinates of this 1 and 0. Similarly the point point (A) are and (B) where the line cuts the axis OY are 1. The locus is the line in the figure and is parallel to LOL Similarly y+x = l is the equation of line AJ3 of the figure; and the 9 X L&L : AB . is parallel to LiOL i. It is easy to prove the general theorem that two lines represented by equations of the forms ax+by = Q and locus ax+by = c are parallel. COORDINATE GEOMETRY 125 which we next come important to deserve a chapter to themselves. But before going on to them we will dwell a little longer on the The group upon are main of loci sufficiently ideas of the subject. The position of any point P is determined by arbitrarily choosing an origin, 0, two axes, OX and OF, at right-angles, and then by noting its coordinates x and y i.e. OM and PM. Also, as we have seen in the last 9 chapter, P can be determined by the "vec OP, where the idea of the vector in cludes a determinate direction as well as a tor" From an abstract determinate length. mathematical point of view the idea of an arbitrary origin may appear artificial and clumsy, and similarly for the arbitrarily drawn axes, OX and OF. But in relation to the application of mathematics to the events of the Universe we are here symbolizing with direct simplicity the most fundamental fact respecting the outlook on the world afforded We each of us refer to us by our senses. our sensible perceptions of things to an origin which we call "here": our location in a particular part of space round which we is the essential fact group the whole Universe of our bodily existence. We can imagine beings who observe all phenomena in all space with an equal eye, unbiassed in favour of any With us it is otherwise, a cat at our part. 126 INTRODUCTION TO MATHEMATICS feet claims more attention than an earth quake at Cape Horn, or than the destruction of a world in the Milky Way. It is true that in making a common stock of our knowledge with our fellowmen, we have to waive some thing of the strict egoism of our own in We substitute "nearly dividual "here." for "here"; thus we measure miles here" from the town hall of the nearest town, or from the capital of the country. In measuring the earth, men of science will put the origin astronomers even at the earth s centre; rise to the extreme altruism of putting their origin inside the sun. But, far as this last origin may be, and even if we go further to some convenient point amid the nearer fixed stars, yet, compared to the immeasurable infinities of space, it remains true that our first procedure in exploring the Universe is to fix upon an origin "nearly here." Again the relation of the coordinates and (i.e. x and y) to the vector OP is an instance of the famous parallelogram law, as can easily be seen (cf. diagram) by completing the parallelogram OMPN. The idea of the "vector" OP, that is, of a directed magni tude, is the root-idea of physical science. Any moving body has a certain magnitude of velocity in a certain direction, that is to say, its velocity is a directed magnitude, a MP vector. OM Again a force has a certain magni- COORDINATE GEOMETRY tude and has a definite direction. when 127 Thus, analytical geometry the ideas of the "origin," of "coordinates," and of are introduced, we are studying "vectors" the abstract conceptions which correspond to the fundamental facts of the physical world. in CHAPTER X CONIC SECTIONS WHEN the Greek geometers had exhausted, as they thought, the more obvious and inter esting properties of figures made up of straight lines and circles, they turned to the study of other curves; and, with their almost infallible instinct for hitting upon things worth thinking about, they chiefly devoted themselves to conic sections, that is, to the curves in which planes would cut The man the surfaces of circular cones. who must have the credit of inventing the study is Menaechmus (born 375 B.C. and died 325 B.C.); he was a pupil of Plato and one of the tutors of Alexander the Great. Alexander, by the by, is a conspicuous ex ample of the advantages of good tuition, for another of his tutors was the philosopher We may suspect that Alexander Aristotle. found Menaechmus rather a dull teacher, for it is related that he asked for the proofs 128 CONIC SECTIONS 129 to be made shorter. It was to this request that Menaechmus replied: "In the country there are private and even royal roads, but in geometry there is only one road for This reply no doubt was true enough in the sense in which it would have been imme But if diately understood by Alexander. Menaechmus thought that his proofs could not be shortened, he was grievously mis taken; and most modern mathematicians would be horribly bored, if they were com pelled to study the Greek proofs of the prop erties of conic sections. Nothing illustrates better the gain in power which is obtained by the introduction of relevant ideas into a science than to observe the progressive shortening of proofs which accompanies the all." There is a cer of richness in idea. tain type of mathematician who is always rather impatient at delaying over the ideas of a subject: he is anxious at once to get on to the proofs of "important" problems. The history of the science is entirely against him. There are royal roads in science; but those who first tread them are men of genius and growth not kings. The way in which conic sections first pre sented themselves to mathematicians was as follows: think of a cone (cf. fig. 15), whose vertex (or point) is V, standing on a circular base STU. For example, a conical shade to 130 INTRODUCTION TO MATHEMATICS an electric light is often an example of such a surface. Now let the "generating" lines which pass through V and lie on the surface be all produced backwards; the result is a double cone, and PQR is another circular V cross section on the opposite side of to the cross section STU. The axis of the cone f passes through all the centres of these circles and is perpendicular to their planes, which are parallel to each other. In the diagram the parts of the curves which are supposed to lie behind the plane of the paper CVC are dotted lines, and the parts on the plane or in front of it are continuous lines. Now suppose this double cone is cut by a plane not perpendicular to the axis CVC or at least not necessarily perpendicular to it. Then three cases can arise: (1) The plane may cut the cone in a closed oval curve, such as E which lies entirely on one of the two half -cones. In this case the plane will not meet the other half -cone at all. Such a curve is called an is it an oval A particular curve. ellipse; case of such a section of the cone is when the plane is perpendicular to the axis CVC , then the section, such as or PQR, is a circle. Hence a circle is a particular case of the ellipse. (2) The plane may be parallel to one of the "generating" lines of the cone, as for 9 AEA STU CONIC SECTIONS 131 example the plane of the curve DiAiDi in the diagram is parallel to the generating line VS; the curve is still confined to one of the half-cones, but it is now not a closed oval curve, it goes on endlessly as long as the generating lines of the half-cone are pro duced away from the vertex. Such a conic section (3) is called a parabola. The plane may cut both the half- cones, so that the complete curve consists of two detached portions, or "branches" as they are called; this case is illustrated by two branches G 2A 2 G 2 and LiAJLJ which together make up the curve. Neither branch is closed, each of them spreading out the endlessly as the away from the is two half -cones are prolonged vertex. Such a conic section called a hyperbola. There are accordingly three types of conic sections, namely, ellipses, parabolas, and hyperbolas. It is easy to see that, in a sense, parabolas are limiting cases lying between ellipses and hyperbolas. They form a more special sort and have to satisfy a more par ticular condition. These three names are apparently due to Apollonius of Perga (born about 260 B.C., and died about 200 B.C.), who wrote a systematic treatise on conic sections which remained the standard work till the sixteenth century. It must at once be apparent how awkward INTRODUCTION TO MATHEMATICS 132, and the investigation of the proper these curves must have been to the Greek geometers. The curves are plane curves, and yet their investigation involves difficult ties of the drawing in perspective of a solid figure. in the diagram given above we have practically drawn no subsidiary lines and yet the figure is sufficiently complicated. The Thus CONIC SECTIONS 133 curves are plane curves, and it seems obvious that we should be able to define them without B going beyond the plane into a solid figure. At the same time, just as in the "solid" M Fig. 17. definition there is one uniform method of definition namely, the section of a cone by INTRODUCTION TO MATHEMATICS 134 which yields three cases, so in any definition there also should be one uniform method of procedure which falls into Their shapes when drawn on three cases. their planes are those of the curved lines in a plane "plane" the three figures 16, 17, and 18. The points and A in the figures are called the ver- A Fig. 18. and the line AA the major axis. It be noted that a parabola (cf. fig. 17) has * that only one vertex. Apollonius proved tices will the ratio of PM to AM.M (i.e. .. \ - A AM.MA 4 remains constant both for the hyperbola * (figs. Cf. Ball, loc. Pappus. ellipse and the 16 and 18), and that the ratio cit.t for this account of Apollonius and CONIC SECTIONS of of PM on 2 to 17; fig. AM is constant for the parabola and he bases most this fact. We 135 of his work are evidently advancing towards the desired uniform definition which does not go out of the plane; but have not yet quite attained to uniformity. In the diagrams 16 and 18, two points, S and S will be seen marked, and in diagram 17 one points, S. These are the foci of the curves, and are points of the greatest importance. Apollonius knew that for an ellipse the sum of SP and S P (i.e. SP+S P) is constant, as P moves on the curve and is equal to AA 1 Similarly for a hyperbola the difference S , . P SP is constant, and equal to AA when P is SP on one branch, and the difference, SP is constant and equal to AA when P is on y But no corresponding the other branch. point seemed to exist for the parabola. Finally 500 years later the last great Greek geometer, Pappus of Alexandria, discovered the final secret which completed this line of thought. In the diagrams 16 and 18 will be and in diagram and seen two lines, 17 the single line, XN. These are the direc trices of the curves, two each for the ellipse and the hyperbola, and one for the parabola. Each directrix corresponds to its nearer focus. The characteristic property of a focus, S, and XN its XN corresponding directrix, of the three types of curve, , XN, is for any one that the ratio 136 INTRODUCTION TO MATHEMATICS t. e. SP to PN ft. is constant, where PN is the perpendicular on the directrix from P, and P is any point on the curve. Here we have finally found the desired property of the curves which does not require us to leave the plane, and is stated uniformly for all three curves. than 1, For ellipses, the for parabolas it equal to SP -= is less 1, and for greater than 1. Pappus had finished his investiga hyperbolas When is ratio it is he must have felt that, apart from minor extensions, the subject was practically exhausted; and if he could have foreseen the history of science for more than a thousand years, it would have confirmed his belief. Yet in truth the really fruitful ideas in con nection with this branch of mathematics had not yet been even touched on, and no one had guessed their supremely important ap No more impressive plications in nature. warning can be given to those who would confine knowledge and research to what is apparently useful, than the reflection that conic -sections were studied for eighteen hun dred years merely as an abstract science, without a thought of any utility other than to satisfy the craving for knowledge on the part of mathematicians, and that then at the end of this long period of abstract study, they tions, CONIC SECTIONS 137 were found to be the necessary key with which to attain the knowledge of one of the most important laws of nature. Meanwhile the entirely distinct study of astronomy had been going forward. The great Greek astronomer Ptolemy (died 168 A.D.) published his standard treatise on the subject in the University of Alexandria, ex plaining the apparent motions among the fixed stars of the sun and planets by the con ception of the earth at rest and the sun and the planets circling round it. During the next thirteen hundred years the number and the accuracy of the astronomical observa tions increased, with the result that the de scription of the motions of the planets on Ptolemy s hypothesis had to be made more and more complicated. Copernicus (born 1473 A.D. and died 1543 A.D.) pointed out that the motions of these heavenly bodies could be explained in a simpler manner if the sun were supposed to rest, and the earth and planets were conceived as moving round it. However, he still thought of these motions as essentially circular, though modified by a set of small corrections arbitrarily superimposed on the primary circular motions. So the matter stood when Kepler was born at Stutt gart in Germany in 1571 A.D. There were two sciences, that of the geometry of conic sections and that of astronomy, both of which 138 INTRODUCTION TO MATHEMATICS had been studied from a remote antiquity without a suspicion of any connection be tween the two. Kepler was an astronomer, but he was also an able geometer, and on the subject of conic sections had arrived at ideas advance of his time. He is only one of examples of the falsity of the idea that success in scientific research demands an ex clusive absorption in one narrow line of study. Novel ideas are more apt to spring from an unusual assortment of knowledge not necessarily from vast knowledge, but from a thorough conception of the methods and ideas of distinct lines of thought. It will be re membered that Charles Darwin was helped to arrive at his conception of the law of evolution by reading Malthus famous Essay on Population, a work dealing with a dif ferent subject at least, as it was then in many thought. Kepler enunciated three laws of planetary motion, the first two in 1609, and the third ten years later. They are as follows: (1) The orbits of the planets are ellipses, the sun being in the focus. (2) As a planet moves in its orbit, the radius vector from the sun to the planet sweeps out equal areas in equal times. (3) The squares of the periodic times of the several planets are proportional to the cubes of their major axes. CONIC SECTIONS 139 These laws proved to be only a stage to wards a more fundamental development of ideas. Newton (born 1642 A.D. and died 1727 A.D.) conceived the idea of universal gravitation, namely, that any two pieces of matter attract each other with a force pro portional to the product of their masses and inversely proportional to the square of their This sweeping distance from each other. general law, coupled with the three laws of motion which he put into their final general shape, proved adequate to explain all astro nomical phenomena, including Kepler s laws, and has formed the basis of modern physics. Among other things he proved that comets might move in very elongated ellipses, or in parobolas, or in hyperbolas, which are nearly parabolas. The comets which return such as Halley s comet must, of course, move in ellipses. But the essential step in the proof of the law of gravitation, and even in the sug gestion of its initial conception, was the veri fication of Kepler s laws connecting the motions of the planets with the theory of conic sections. From the seventeenth century onwards the abstract theory of the curves has shared in the double renaissance of geometry due to the introduction of coordinate geometry and In protective geo cluster round ideas the fundamental metry of projective geometry. 140 INTRODUCTION TO MATHEMATICS the consideration of sets (or pencils, as they are called) of lines passing through a common Now point (the vertex of the "pencil"). (cf. fig. 19) if A, B, C, D, be any four fixed points on a conic section and P be a variable point on the curve, the pencil of lines PA, Fig. 19. PB, PC, and PD, has a known as the constancy of special property, cross ratio. It will suffice here to say that cross ratio is a fundamental idea in projective geometry. For projective geometry this is really the definition of the curves, or some analogous property which is really equivalent to it. It its CONIC SECTIONS 141 will be seen how far in the course of ages of study we have drifted away from the old original idea of the sections of a circular cone. We know now that the Greeks had got hold of a minor property of comparatively slight importance; though by some divine good fortune the curves themselves deserved all the attention which was paid to them. This unimportance of the "section" idea is now marked in ordinary mathematical phrase ology by dropping the word from their As often as not, they are now names. named merely "conies" instead of "conic sections." we come back to the point at coordinate geometry in the last chapter. We had asked what was the type of loci corresponding to the general algebraic form ax-{-by=c 9 and had found that it was the class of straight lines in the plane. had seen that every straight line possesses an equation of this form, and that every equation of this form corresponds to a straight line. We now wish to go on to the next general type of algebraic forms. This is evidently to be obtained by introducing terms involv 2 2 Thus the new general ing x and xy and y form must be written Finally, which we left We . ax 2 +2hxy +by 2 +2gx +2fy +c = What does this represent? The answer is 142 INTRODUCTION TO MATHEMATICS that it always represents a conic section, and, furthermore, that the equation of every conic section can always be put into this shape. The discrimination of the particular sorts of conies as given by this form of equation is very easy. It entirely depends upon the con 2 sideration of where a, 6, and h, are , the "constants" as written above. If ab h 2 is a positive number, the curve is an abh ellipse; if bola: and curve is abh if ab 2 = Q, h2 is the curve is a para a negative number, the a hyperbola. For example, put a = 6 = 1, c= = 0. We h=g=f = Q, then get the equation # 2 +2/ 2 4 It is easy to prove that this is the equa tion of a circle, whose centre is at the origin, 2 and radius is 2 units of length. Now becomes 1 Xl O 2 that is, 1, and is therefore Hence the circle is a particular positive. case of an ellipse, as it ought to be. Genera lizing, the equation of any circle can be 2 2 put into the form a(x +y ) +2gx+2fy+c=Q. 2 Hence becomes a2 0, that is, a 2 which is necessarily positive. Accordingly all circles satisfy the condition for ellipses. The general form of the equation of a para bola is 4. abh , abh , 2 (dx+ey) -Hfy so that the terms of the second degree, as CONIC SECTIONS 143 they are called, can be written as a perfect square. For squaring out, we get so that by comparison a d2 9 h=de, b=e2 9 2 = d 2e 2 (de) 2 = 0. Hence and therefore the necessary condition is automatically satis The equation %xy 4=0, where a = b fied. abh = 0=j* = 0, h = l, c= bola. I2, 4, represents a hyper For the condition abk 2 becomes that is, 1, which is negative. Some exceptional cases are included in the general form of the equation which may not be immediately recognized as conic sections. By properly choosing the constants the equa tion can be made to represent two straight lines. may Now fairly two intersecting straight lines be said to come under the Greek idea of a conic section. For, by referring to the picture of the double cone above, it will be seen that some planes through the vertex, V, will cut the cone in a pair of straight lines intersecting at V. The case of two parallel straight lines can be included by considering a circular cylinder as a particular case of a Then a plane, which cuts it and is cone. parallel to its axis, will cut it in two parallel straight lines. Anyhow, whether or no the ancient Greek would have allowed these special cases to be called conic sections, they 144 INTRODUCTION TO MATHEMATICS are certainly included among the curves re presented by the general algebraic form of the second degree. This fact is worth noting; for it is characteristic of modern mathematics general forms all sorts of particular cases which would formerly have received special treatment. This is due to to include its among pursuit of generality. CHAPTER XI FUNCTIONS THE mathematical use of the term function has been adopted also in common life. For example, "His temper is a function of his the term exactly in this digestion," uses mathematical sense. It means that a rule can be assigned which will tell you what his temper will be when you know how his Thus the idea of a digestion is working. "function" is simple enough, we only have to see how it is applied in mathematics to variable numbers. Let us think first of some concrete examples If a train has been travel ling at the rate of twenty miles per hour, the distance (s miles) gone after any number of hours, say t, is given by s=%QXt 9 and s is called a function of t. Also 20 X t is the func tion of t with which s is identical. If John is one year older than Thomas, then, when Thomas is at any age of x years, John s age l; and y is a (y years) is given by y=x function of x, namely, is the function # + 1. In these examples t and x are called the : + 145 146 INTRODUCTION TO MATHEMATICS "arguments" of the functions in which they appear. Thus t is the argument of the func tion 20 Xt, and x is the argument of the func tion x + 1. If s=20X/, and y=x + l, then s and y are called the "values" of the functions 20 Xt and x+l respectively. Coming now to the general case, we can define a function in mathematics as a corre lation between two variable numbers, called respectively the argument and the value of the function, such that whatever value be assigned to the "argument of the function" the value of the "value of the function" is The definitely (i.e. uniquely) determined. converse is not necessarily true, namely, that when the value of the function is determined the argument is also uniquely determined. Other functions of the argument x are y = x 2 , y=%x2 +3x + l, y=x,y=logx, y=sin x. The group will be who understand those readily recognizable by a little algebra and trigonometry. It is not worth while to delay now for their explana tion as they are merely quoted for the sake last two functions of example. Up to this point, of this though we have defined what we mean by a function in general, we have only mentioned a series of special func But mathematics, true to its general tions. methods of procedure, symbolizes the general idea of any function. It does this by writing FUNCTIONS 147 9 etc., for any function of F(x),f(x) 9 g(x), where the argument x is placed in a bracket and some letter like F,f, g, etc., is prefixed to the bracket to stand for the function. This notation has its defects. Thus it ob viously clashes with the convention that the single letters are to represent variable num bers; since here F, /, g, $, etc., prefixed to a bracket stand for variable functions. It would be easy to give examples in which we can only trust to common sense and the context to see what is meant. One way of evading the confusion is by using Greek letters (e.g. as above) for functions; an other way is to keep to / and F (the initial letter of function) for the functional letter, and, if other variable functions have to be symbolized, to take an adjacent letter like g. With these explanations and cautions, we write y f(x) to denote that y is the value of some undetermined function of the argument <t>(x) x, <, <f> , where f(x) may stand for anything such 2 &r-f-l, sin x, log x, or merely for +1, x x itself. The essential point is that when x is given, then y is thereby definitely deter mined. It is important to be quite clear as to the generality of this idea. Thus in y x; as we may determine, if we choose, f(x) to mean that when x is an integer, f(x) is zero, and when x has any other value, f(x) is 1. f(x), Accordingly, putting y =f(x), with this choice 148 INTRODUCTION TO MATHEMATICS or 1 accord for the meaning of /, y is either ing as the value of x is integral or otherwise. Thus /(I) = 0, /(2) = 0, /( J) = 1, /( V$) = 1, and This choice for the meaning of f(x) gives a perfectly good function of the argu ment x according to the general definition of a function. function, which after all is only a sort of correlation between two variables, is rep resented like other correlations by a graph, that is in effect by the methods of coordinate geometry. For example, fig. 2 in Chapter II so on. A is the graph of the function - where v is the argument and p the value of the function. In this case the graph is only drawn for posi tive values of v, which are the only values possessing any meaning for the physical ap plication considered in that chapter. Again in fig. 14 of Chapter IX the whole length of the line AB, unlimited in both directions, is the graph of the function x + I, where x is the argument and y is the value of the function; and in the same figure the unlimited line AiB is the graph of the function 1 x, and the line LOU is the graph of the function x, x being the argument and y the value of the function. These functions, which are expressed by simple algebraic formulae, are adapted for But for some representation by graphs. FUNCTIONS 149 functions this representation would be very misleading without a detailed explanation, or might even be impossible. Thus, consider the function mentioned above, which has the value 1 for all values of its argument x, except those which are integral, e.g. except for x = 0, x = l, x = %, etc., when it has the value 0. Its appearance on a graph would be that of the straight line drawn parallel to the ABA B Cf C2 C3 C4 12345 Fig. 20. axis XOX length. at a distance from But the points d, it (7 2 , ol 1 unit of C C 3, 4, etc., corresponding to the values 1, 2, 3, 4, etc., of the argument x, are to be omitted, and in stead of them the points BI, B 2) s B^ etc., on the axis OX, are to be taken. It is easy to find functions for which the graphical representation is not only inconvenient but Functions which do not lend impossible. themselves to graphs are important in the B , 150 INTRODUCTION TO MATHEMATICS higher mathematics, but we need not concern ourselves further about them here. The most important division between func tions is that between continuous and discon tinuous functions. function is continuous A when its value only alters gradually gradual alterations of the argument, for and is when it can alter its value by sudden jumps. Thus the two functions x + 1 discontinuous and Ix, whose graphs are depicted as Chapter IX, are con straight lines in fig. 14 of tinuous functions, and so is the function -, v depicted in Chapter II, if we only think of But the function de positive values of v. picted in fig. 20 of this chapter is discontinu ous since at the values x = 1, x = 2, etc., of its argument, its value gives sudden jumps. Let us think of some examples of functions presented to us in nature, so as to get into our heads the real bearing of continuity and discontinuity. Consider a train in its journey along a railway line, say from Euston Station, the terminus in London of the London and North- Western Railway. Along the line in order lie the stations of Bletchley and Rugby. Let t be the number of hours which the train has been on its journey from Euston, and s be the number of miles passed over. Then s is a function of t, i.e. is the variable value cor responding to the variable argument t. If FUNCTIONS 151 we know the circumstances of the train s run, we know s as soon as any special value of t is given. Now, miracles apart, we may confidently assume that s is a continuous function of t. It is impossible to allow for the contingency that we can trace the train continuously from Euston to Bletchley, and that then, without any intervening time, how ever short, it should appear at Rugby. The idea is too fantastic to enter into our calcula tion: it contemplates possibilities not to be found outside the Arabian Nights; and even in those tales sheer discontinuity of motion hardly enters into the imagination, they do not dare to tax our credulity with anything more than very unusual speed. But unusual speed is no contradiction to the great law of continuity of motion which appears to hold in nature. Thus light moves at the rate of about 190,000 miles per second and comes to us from the sun in seven or eight minutes; but, in spite of this speed, its distance travelled is always a continuous function of the time. It is not quite so obvious to us that the velocity of a body is invariably a continuous function of the time. Consider the train at any time t, it is velocity, say zero is definite per hour, where v is at rest in a station and when the train is when the train Now is backing. cannot v that allow change its readily negative we moving with some v miles 152 INTRODUCTION TO MATHEMATICS value suddenly for a big, heavy train. The train certainly cannot be running at forty miles per hour from 11.45 A.M. up to noon, and then suddenly, without any lapse of time, commence running at 50 miles per hour. at once admit that the change of velocity We will be a gradual process. But how about sudden blows of adequate magnitude? Sup pose two trains collide; or, to take smaller objects, suppose a man kicks a football. It certainly appears to our sense as though the football began suddenly to move. Thus, in the case of velocity our senses do not revolt at the idea of its being a discontinuous func tion of the time, as they did at the idea of the train being instantaneously transported from Bletchley to Rugby. As a matter of fact, if the laws of motion, with their conception of mass, are true, there is no such thing as discontinuous velocity in nature. Anything that appears to our senses as discontinuous change of velocity must, according to them, be considered to be a case of gradual change which is too quick to be perceptible to us. It would be rash, however, to rush into the generalization that no discontinuous func tions are presented to us in nature. man who, trusting that the mean height of the A land above sea-level between London and Paris was a continuous function of the dis tance from London, walked at night on FUNCTIONS 153 s Cliff by Dover in contempla Milky Way, would be dead before he had had time to rearrange his ideas as Shakespeare tion of the to the necessity conclusions. It is of caution in scientific very easy to find a discontinuous if we confine ourselves to the function, even X Y Fig. 21. simplest of the algebraic formulae. ample, take the function of y = -, For ex which we have already considered in the form where v was confined to positive p= values. -, But 154 INTRODUCTION TO MATHEMATICS now let x have any value, positive or negative. The graph of the function is exhibited in fig. Suppose x to change continuously from a large negative value through a numerically decreasing set of negative values up to 0, and thence through the series of increasing posi tive values. Accordingly, if a moving point, starts at the M, represents x on extreme left of the axis and succes etc. 2 sively moves through MI, The corresponding points on the function are It is easy to see that PI, P2, PS, P4, etc. there is a point of discontinuity at x = 0, i.e. at the origin 0. For the value of the function on the negative (left) side of the origin be comes endlessly great, but negative, and the function reappears on the positive (right) side as endlessly great but positive. Hence, there is a however small we take M%, s finite jump between the values of the func tion at 1/2 and 3 Indeed, this case has the peculiarity that the smaller we take 2 so long as they enclose the origin, the 3 bigger is the jump in value of the function between them. This graph brings out, what is also apparent in fig. 20 of this chapter, that for many functions the discontinuities only occur at isolated points, so that by restrict ing the values of the argument we obtain a continuous function for these remaining values. Thus it is evident from fig. 21 that 21. XOX M XOX , M , M^ M^ M M M , . , M FUNCTIONS in y= -, x if we keep and exclude the 155 to positive values only origin, we obtain a continuous function. Similarly the same function, if we keep to negative values only, excluding the origin, is continuous. Again the function which is graphed in fig. 20 is continuous be tween B and Ci, and between Ci and C 2 and between Cz and Cs, and so on, always in each case excluding the end points. It is, how , ever, easy to find functions such that their discontinuities occur at all points. For example, consider a function /(#), such that when x is any fractional number /(#)=!, and when x is any incommensurable number /(#)=2. This function is discontinuous at all points. Finally, we will look a little more closely at the definition of continuity given above. have said that a function is continuous when its value only alters gradually for gradual alterations of the argument, and is We when it can alter its value by sudden jumps. This is exactly the sort of definition which satisfied our mathematical forefathers and no longer satisfies modern mathematicians. It is worth while to spend some time over it; for when we understand the modern objections to it, we shall have gone a long way towards the understanding of the spirit of modern mathematics. The discontinuous 156 INTRODUCTION TO MATHEMATICS whole difference between the older and the newer mathematics lies in the fact that vague half -metaphorical terms like "gradually" are no longer tolerated in its exact statements. Modern mathematics will only admit state ments and definitions and arguments which exclusively employ the few simple ideas about number and magnitude and variables on which the science is founded. Of two num bers one can be greater or less than the other; and one can be such and such a multi ple of the other; but there is no relation of between two numbers, and "graduality" hence the term is inadmissible. Now this may seem at first sight to be great pedantry. To this charge there are two answers. In the first place, during the first half of the nineteenth century it was found by some great mathematicians, especially Abel in Sweden, and Weierstrass in Germany, that large parts of mathematics as enunciated in the old happy-go-lucky manner were simply wrong. Macaulay in his essay on Bacon contrasts the certainty of mathematics with the uncertainty of philosophy; and by way of a rhetorical example he says, "There has been no reaction against Taylor s theorem." He could not have chosen a worse example. For, without having made an examination of English text-books on mathematics contem porary with the publication of this essay, the FUNCTIONS 157 assumption is a fairly safe one that Taylor s theorem was enunciated and proved wrongly in every one of them. Accordingly, the anxious precision of modern mathematics is necessary for accuracy. In the second place it is necessary for research. It makes for clearness of thought, and thence for boldness of thought and for fertility in trying new combinations of ideas. When the initial statements are vague and slipshod, at every subsequent stage of thought, common sense has to step in to limit applications and to in creative thought explain meanings. common sense is a bad master. Its sole criterion for judgment is that the new ideas shall look like the old ones. In other words it can only act by suppressing originality. Now In working our way towards the precise definition of continuity (as applied to func tions) let us consider more closely the state ment that there is no relation of "graduality between numbers. It may be asked, Cannot " one number be only slightly greater than another number, or in other words, cannot the difference between the two numbers be small? The whole point is that in the ab stract, apart from some arbitrarily assumed application, there is no such thing as a great A million miles is a or a small number. small number of miles for an astronomer investigating the fixed stars, but a million 158 INTRODUCTION TO MATHEMATICS is a large yearly income. Again, onequarter is a large fraction of one s income to give away in charity, but is a small fraction of it to retain for private use. Examples can be accumulated indefinitely to show that great or small in any absolute sense have no abstract application to numbers. We can say of two numbers that one is greater or smaller than another, but not without speci fication of particular circumstances that any one number is great or small. Our task therefore is to define continuity without any mention of a "small" or "gradual" change in value of the function. In order to do this we will give names to pounds some ideas, which will also be useful when we come to consider limits and the differen tial calculus. An "interval" of values of the argument x of a function f(x) is all the values lying between some two values of the argument. For example, the interval between x = 1 and x = 2 consists of all the values which x can take lying between 1 and 2, i.e. it consists of all the real numbers between 1 and 2. But the bounding numbers of an interval need not be integers. An interval of values of the argument contains a number a, when a is a member of the interval. For example, the interval between 1 and 2 contains f f J, and , so on. , FUNCTIONS 159 A set of numbers approximates to a num ber a within a standard k, when the numerical difference between a and every number of the set is less than k. Here k is the "standard of approximation." Thus the set of num bers 3, 4, 6, 8, approximates to the number 5 within the standard 4. In this case the standard 4 is not the smallest which could have been chosen, the set also approximates to 5 within any of the standards 3-1 or 3*01 or 3 001. Again, the numbers, 3*1, 3*141, 3-1415, 3-14159 approximate to 3-13102 within the standard -032, and also within the smaller standard -03103. These two ideas of an interval and of ap proximation to a number within a standard are easy enough; their only difficulty is that they look rather trivial. But when combined with the next idea, that of the "neighbour hood" of a number, they form the founda tion of modern mathematical reasoning. What do we mean by saying that something is true for a function f(x) in the neighbour hood of the value a of argument x ? It is this fundamental notion which we have now got to make precise. The values of a function f(x) are said to possess a characteristic in the "neighbour when some interval can be found, hood of which (i) contains the number a not as an end-point, and (ii) is such that every value a" 160 INTRODUCTION TO MATHEMATICS of the function for arguments, other than a, lying within that interval posessses the char acteristic. The value /(a) of the function for the argument a may or may not possess the characteristic. point Nothing is decided on this by statements about the neighbourhood of a. For example, suppose we take the par x2 Now in the neighbour hood of 2, the values of x 2 are less than 5. For we can find an interval, e.g. from 1 to 2*1, which (i) contains 2 not as an end-point, and (ii) is such that, for values of x lying within it, x 2 is less than 5. Now, combining the preceding ideas we know what is meant by saying that in the ticular function . neighbourhood of a the function f(x) approxi mates to c within the standard k. It means that some interval can be found which (i) in cludes a not as an end-point, and (ii) is such that all values of f(x), where x lies in the inter ^ val and is not a, differ from c by less than k. For example, in the neighbourhood of 2, the function ^/x approximates to 1 41425 within the standard -0001. This is true because the square root of 1-99996164 is 1-4142 and the square root of 2-00024449 is 1-4143; hence for values of x lying in the interval 1-99996164 to 2-00024449, which contains 2 not as an end-point, the values of the function +Jx all lie between 1*4142 and 1-4143, and FUNCTIONS 161 they therefore all differ from 1 -41425 by less than 0001. In this case we can, if we like, standard of approximation, fix a smaller namely -000051 or -0000501. Again, to take another example, in the neighbourhood of 2 the function x 2 approximates to 4 within the standard -5. For (l-9) 2 = 3-61 and (2-l) 2 = 4-41, and thus the required interval 1-9 to 2-1, containing 2 not as an end-point, has been found. This example brings out the fact that statements about a function f(x) in the neighbourhood of a number a are distinct from statements about the value of f(x) when x = a. The production of an interval, through out which the statement is true, is required. Thus the mere fact that 2 2 =4 does not by itself justify us in saying that in the neigh bourhood of 2 the function x 2 is equal to 4. This statement would be untrue, because no interval can be produced with the required 2 property. Also, the fact that 2 =4 does not by itself justify us in saying that in the 2 neighbourhood of 2 the function x approxi mates to 4 within the standard *5; although as a matter of fact, the statement has just been proved to be true. If we understand the preceding ideas, we understand the foundations of modern mathematics. We shall recur to analogous ideas in the chapter on Series, and again in the chapter on the Differential Calculus. 162 INTRODUCTION TO MATHEMATICS Meanwhile, we are now prepared to define A function f(x) at a value a of its argu ment, when in the neighbourhood of a its values approximate to /(a) (i.e. to its value at a) within every standard of ap "continuous is functions." "continuous" proximation. This means that, whatever standard k be chosen, in the neighbourhood of a f(x) ap proximates to /(a) within the standard k. For example, x 2 is continuous at the value 2 of its argument, x 9 because however k be chosen we can always find an interval, which (i) contains 2 not as an end-point, and (ii) is such that the values of x 2 for arguments lying within it approximate to 4 (i.e. 2 2 ) within the standard k. Thus, suppose we choose the standard -1; now (l-999) 2 = 3 -996001, and (2-01) 2 =4-0401, and both these numbers differ from 4 by less than *1. Hence, within the interval 1-999 to 2-01 the values of x 2 approximate to 4 within the standard !. Similarly an interval can be produced for any other standard which we like to try. Take the example of the railway train. Its velocity is continuous as it passes the signal box, if whatever velocity you like to assign (say one-millionth of a mile per hour) an in terval of time can be found extending before and after the instant of passing, such that at all instants within it the train s velocity FUNCTIONS 163 differs from that with which the train passed the box by less than one-millionth of a mile per hour; and the same is true whatever other velocity be mentioned in the place of one-millionth of a mile per hour. CHAPTER XII PERIODICITY IN NATURE THE whole life of Nature is dominated by the existence of periodic events, that is, by the existence of successive events so analogous to each other that, without any straining of language, they may be termed recurrences of the same event. The rotation of the earth produces the successive days. It is true that each day is different from the preceding days, however abstractly we define the meaning of a day, so as to exclude casual phenomena. But with a sufficiently abstract definition of a day, the distinction in properties between two days becomes faint and remote from practical interest; and each day may then be conceived as a recurrence of the phenome non of one rotation of the earth. Again the path of the earth round the sun leads to the yearly recurrence of the seasons, and imposes another periodicity on all the operations of Another less fundamental perio nature. dicity is provided by the phases of the moon. In modern civilized life, with its artificial light, these phases are of slight importance, but in 164 PERIODICITY IN NATURE 165 ancient times, in climates where the days are burning and the skies clear, human life was apparently largely influenced by the existence of moonlight. Accordingly our divisions into weeks and months, with their religious associ ations, have spread over the European races from Syria and Mesopotamia, though inde pendent observances following the moon s phases are found amongst most nations. It is, however, through the tides, and not through its phases of light and darkness, that the moon s periodicity has chiefly influenced the history of the earth. Our bodily life is essentially periodic. It is dominated by the beatings of the heart, and the recurrence of breathing. The presupposition of periodicity is indeed fundamental to our very conception of life. We cannot imagine a course of nature in which, as events progressed, we should be unable to say: "This has happened before." The whole conception of experience as a guide Men would to conduct would be absent. always find themselves in new situations possessing no substratum of identity with anything in past history. The very means of measuring time as a quantity would be ab Events might still be recognized as sent. occurring in a series, so that some were earlier and others later. But we now go beyond this bare recognition. We can not only say that 166 INTRODUCTION TO MATHEMATICS three events, A, B C, occurred in this order, came before B, and B before C; so that but also we can say that the length of time between the occurrences of and B was twice as long as that between B and C. Now, 9 A A quantity of time is essentially dependent on observing the number of natural recurrences which have intervened. We may say that the length of time between A and B was so many days, or so many months, or so many years, according to the type of recur rence to which we wish to appeal. Indeed, at the beginning of civilization, these three modes of measuring time were really distinct. It has been one of the first tasks of science among civilized or semi-civilized nations, to into one coherent measure. The extent of this task must be grasped. It is necessary to determine, not merely what number of days (e.g. 365-25. .) go to some one year, but also previously to determine that the same number of days do go to the successive years. We can imagine a world in which periodicities exist, but such that no two are coherent. In some years there might be 200 days and in others 350. The determina tion of the broad general consistency of the more important periodicities was the first step This consistency arises in natural science. from no abstract intuitive law of thought; it is merely an observed fact of nature fuse them full . PERIODICITY IN NATURE 167 guaranteed by experience. Indeed, so far is from being a necessary law, that it is not even exactly true. There are divergencies in every case. For some instances these diver gencies are easily observed and are therefore immediately apparent. In other cases it re it the most refined astronomical accuracy to quires observations and make them appar Broadly speaking, all recurrences de pending on living beings, such as the beatings of the heart, are subject in comparison with ent. other recurrences to rapid variations. The great stable obvious recurrences stable in the sense of mutually agreeing with great accuracy are those depending on the motion of the earth as a whole, and on similar motions of the heavenly bodies. We therefore assume that these astronomi cal recurrences mark out equal intervals of But how are we to deal with their time. discrepancies which the refined observations of astronomy detect? Apparently we are reduced to the arbitrary assumption that one or other of these sets of phenomena marks out equal times e.g. that either all days are of equal length, or that all years are of equal This is not so: some assumptions must be made, but the assumption which length. underlies the whole procedure of the astrono mers in determining the measure of time is that the laws of motion are exactly verified. 168 INTRODUCTION TO MATHEMATICS Before explaining how this is done, it is in teresting to observe that this relegation of the determination of the measure of time to the astronomers arises (as has been said) from the stable consistency of the recurrences with which they deal. If such a superior con sistency had been noted among the recur rences characterisitc of the human body, we should naturally have looked to the doctors of medicine for the regulation of our clocks. In considering how the laws of motion come into the matter, note that two incon sistent modes of measuring time will yield different variations of velocity to the same body. For example, suppose we define an hour as one twenty-fourth of a day, and take the case of a train running uniformly for two hours at the rate of twenty miles per hour. Now take a grossly inconsistent measure of time, and suppose that it makes the first hour to be twice as long as the second hour. Then, according to this other measure of duration, the time of the train s run is divided into two parts, during each of which it has tra versed the same distance, namely, twenty miles; but the duration of the first part is twice as long as that of the second part. Hence the velocity of the train has not been uniform, and on the average the velocity during the second period is twice that during the first period. Thus the question as to PERIODICITY IN NATURE 169 whether the train has been running uniformly or not entirely depends on the standard of time which we adopt. Now, for all ordinary purposes of life on the earth, the various astronomical recurrences may be looked on as absolutely consistent; and, furthermore assuming their consistency, and thereby assuming the velocities and changes of velocities possessed by bodies, we find that the laws of motion, which have been considered above, are almost exactly But only almost exactly when we verified. come some of the astronomical phenomena. however, that by assuming slightly different velocities for the rotations and motions of the planets and stars, the laws would be exactly verified. This assumption is then made; and we have, in fact thereby, adopted a measure of time, which is indeed defined by reference to the astronomical phenomena, but not so as to be consistent with the uniformity of any one of them. But the broad fact remains that the uniform flow of time on which so much is based, is itself dependent on the observation of periodic We to find, events. Even phenomena, which on the surface seem casual and exceptional, or, on the other hand, maintain themselves with a uniform persistency, may be due to the remote influ ence of periodicity. Take, for example, the 170 INTRODUCTION TO MATHEMATICS principle of when two resonance. Resonance arises connected circumstances have the same periodicities. It is a dynami cal law that the small vibrations of all bodies when left to themselves take place in definite times characteristic of the body. Thus a pendulum with a small swing always vibrates in some sets of definite time, characteristic of its shape and distribution of weight and length. A more complicated body may have many ways of vibrating; but each of its modes of vibration will have its own peculiar "period." Those periods of vibration of a body are called its periods. Thus a pendulum has but one period of vibration, while a sus pension bridge will have many. We get a musical instrument, like a violin string, when "free" the periods of vibration are all simple submultiples of the longest; i.e. if t seconds be the longest period, the others are \t, \t, and so on, where any of these smaller periods may be absent. Now, suppose we excite the vibra tions of a body by a cause which is itself peri odic; then, if the period of the cause is very nearly that of one of the periods of the body, that mode of vibration of the body is very violently excited; even although the magni tude of the exciting cause is small. This phenomenon is called "resonance." The general reason is easy to understand. Any one wanting to upset a rocking stone will PERIODICITY IN NATURE 171 tune" with the oscillations of the as always to secure a favourable so stone, moment for a push. If the pushes are out of tune, some increase the oscillations, but others check them. But when they are in tune, after a time all the pushes are favour push "in The word "resonance comes from con siderations of sound: but the phenomenon extends far beyond the region of sound. The " able. laws of absorption and emission of light de it, the "tuning" of receivers for wire less telegraphy, the comparative importance of the influences of planets on each other s motion, the danger to a suspension bridge as troops march over it in step, and the excessive vibration of some ships under the rhythmical beat of their machinery at certain speeds. This coincidence of periodicities may produce steady phenomena when there is a constant as sociation of the two periodic events, or it may produce violent and sudden outbursts when the association is fortuitous and temporary. Again, the characteristic and constant pend on periods of vibration mentioned above are the underlying causes of what appear to us as steady excitements of our senses. work for hours in a steady light, or we listen to a steady unvarying sound. But, if modern science be correct, this steadiness has no counterpart in nature. The steady light is due to the impact on the eye of a countless We 172 INTRODUCTION TO MATHEMATICS number of periodic waves in a vibrating ether, and the steady sound to similar waves in a vibrating air. It is not our purpose here to explain the theory of light or the theory of sound* We have said enough to make it evident that one of the first steps necessary to make mathematics a fit instrument for the investigation of Nature is that it should be able to express the essential periodicity of If we have grasped this, we can things. understand the importance of the mathe matical conceptions which we have next to consider, namely, periodic functions. CHAPTER XIII TRIGONOMETRY TRIGONOMETRY did not take its rise from the general consideration of the periodicity of nature. In this respect its history is analo gous to that of conic sections, which also had their origin in very particular ideas. Indeed, a comparison of the histories of the two sciences yields some very instructive analogies and contrasts. Trigonometry, like conic sec had its origin among the Greeks. Its inventor was Hipparchus (born about 160 B.C.), a Greek astronomer, who made his observations at Rhodes. His services to tions, astronomy were very great, and it left his hands a truly scientific subject with important results established, and the right method of progress indicated. Perhaps the invention of trigonometry was not the least of these services to the main science of his study. The next man who extended trigonometry was Ptolemy, the great Alexandrian astronomer, whom we have already mentioned. We now 173 174 INTRODUCTION TO MATHEMATICS see at once the great contrast between conic sections and trigonometry. The origin of trigonometry was practical; it was invented because it was necessary for astronomical re search. The origin of conic sections was purely theoretical. The only reason for its initial study was the abstract interest of the ideas involved. Characteristically enough conic sections were invented about 150 years earlier than trigonometry, during the very best period of Greek thought. But the im portance of trigonometry, both to the theory and the application of mathematics, is only one of innumerable instances of the fruitful ideas which the general science has gained from its practical applications. We will try and make clear to ourselves what trigonometry is, and why it should be generated by the scientific studyjof astronomy. In the first place: What are the measure ments which can be made by an astronomer? They are measurements of time and measure ments of angles. The astronomer may adjust a telescope (for it is easier to discuss the familiar instrument of modern astronomers) it can only turn about a fixed axis pointing east and west; the result is that the telescope can only point to the south, with a greater or less elevation of direction, or, if turned round beyond the zenith, point to the north. This is the transit instrument, the so that TRIGONOMETRY 175 great instrument for the exact measurement of the times at which stars are due south or due north. But indirectly this instrument measures angles. For when the time elapsed between the transits of two stars has been noted, by the assumption of the uniform rotation of the earth, we obtain the angle through which the earth has turned in that period of time. Again, by other instruments, the angle between two stars can be directly measured. For if is the eye of the astrono- E Fig. 22. mer, and EA and EB are the directions in which the stars are seen, it is easy to devise instruments which shall measure the angle AEB. Hence, when the astronomer is form ing a survey of the heavens, he is, in fact, measuring angles so as to fix the relative directions of the stars and planets at any in stant. Again, in the analogous problem of 176 INTRODUCTION TO MATHEMATICS land-surveying, angles are the chief subject of measurements. The direct measurements of length are only rarely possible with any accuracy; rivers, houses, forests, mountains, and general irregularities of ground all get in the way. The survey of a whole country will depend only on one or two direct measure ments of length, made with the greatest elaboration in selected places like Salisbury Plain. The main work of a survey is the measurement of angles. For example, A, 9 and C will be conspicuous points in the dis- B Fig. 23. surveyed, say the tops of church towers. visible each from the others. Then it is a very simple matter at A to measure the angle BAC, and at B to measure the angle ABC, and at C to measure the angle BCA. Theoretically, it is only necessary to measure two of these angles; for, by a wellknown proposition in geometry, the sum of the three angles of a triangle amounts to two trict These points are TRIGONOMETRY 177 right-angles, so that when two of the angles are known, the third can be deduced. It is better, however, in practice to measure all three, and then any small errors of observa tion can be checked. In the process of map- making a country is completely covered with triangles in this way. This process is called triangulation, and is the fundamental process in a survey. Now, when all the angles of a triangle are known, the shape of the triangle is known that is, the shape as distinguished from the We here come upon the great principle size. The idea is very of geometrical similarity. familiar to us in its practical applications. are all familiar with the idea of a plan drawn to scale. Thus if the scale of a plan be an inch to a yard, a length of three inches in the plan means a length of three yards in the original. Also the shapes depicted in the plan are the shapes in the original, so that a right-angle in the original appears as a rightangle in the plan. Similarly in a map, which We only a plan of a country, the proportions of the lengths in the map are the proportions of the distances between the places indicated, and the directions in the map are the direc tions in the country. For example, if in the map one place is north-north-west of the other, so it is in reality; that is to say, in a is map the angles are the same as in reality. 178 INTRODUCTION TO MATHEMATICS Geometrical similarity Two may be defined thus: figures are similar (i) if to any point in one figure a point in the other figure corresponds, so that to every line there is a corresponding line, and to every angle a corresponding angle, and (ii) if the lengths of corresponding lines are in a fixed propor tion, and the magnitudes of corresponding angles are the same. The fixed proportion of the lengths of corresponding lines in a map (or plan) and in the original is called the scale of the map. The scale should always be indicated on the margin of every map and plan. It has already been pointed out that two triangles whose angles are respectively equal are similar. Thus, if the two triangles B C Fig. 24. ABC and DEF have the angles at A and D equal, and those at B and E and those at C and F, then DE is to AB in the same propor9 TRIGONOMETRY tion as But EF is to EC, and as FD 179 is to CA. not true of other figures that simi larity is guaranteed by the mere equality of angles. Take, for example, the familiar cases of a rectangle and a square. Let ABCD be a square, and ABEF be a rectangle. Then But all the corresponding angles are equal. it is C B Fig. 25. AB of the square is equal to of the rectangle, the side BC of the side the square is about half the size of the side Hence it is not true of the rectangle. that the square ABCD is similar to the rect angle ABEF. This peculiar property of the triangle, which is not shared by other recti linear figures, makes it the fundamental figure in the theory of similarity. Hence in surveys, triangulation is the fundamental process; the hence also arises the word "tri- whereas the side AB BE 180 INTRODUCTION TO MATHEMATICS gonometry," derived from the two Greek words trigonon, a triangle, and metria, meas urement. The fundamental question from which trigonometry arose is this Given the magnitudes of the angles of a triangle, what can be stated as to the relative magnitudes of the sides. Note that we say "relative magnitudes : since by the theory of similarity the proportions of the sides which only In order to answer this ques are known. tion, certain functions of the magnitudes of an angle, considered as the argument, are in In their origin these functions troduced. were got at by considering a right-angled tri angle, and the magnitude of the angle was defined by the length of the arc of a circle. In modern elementary books, the funda mental position of the arc of the circle as de fining the magnitude of the angle has been pushed somewhat to the background, not to the advantage either of theory or clearness of explanation. It must first be noticed that, in relation to similarity, the circle holds the same fundamental position among curvi linear figures, as does the triangle among rectilinear figures. Any two circles are simi of the sides," it is they only differ in scale. The the circumferences of two circles, of lengths and AiPiAi in the fig. 26 are such as in proportion to the lengths of their radii. lar figures; APA Furthermore, if the two circles have the same TRIGONOMETRY 181 centre 0, as do the two circles in fig. 26, then and AiPi intercepted by the the arcs arms of any angle A OP, are also in propor tion to their radii. Hence the ratio of the AP Fig. 26. length of the arc radius OP, that is is AP -^? to the length of the radius 7T^ OP is a number which quite independent of the length the same as the fraction r \^-. radms OPi . OP, and is This f rac- tion of "arc divided by radius" is the proper theoretical way to measure the magnitude of 182 INTRODUCTION TO MATHEMATICS an angle; for it is dependent on no arbitrary unit of length, and on no arbitrary way of dividing up any arbitrarily assumed angle, Thus the such as a right-angle. fraction represents the magnitude of the angle AP A OP. Now draw PM perpendicularly to OA. Then PM the Greek mathematicians called the line the sine of the arc AP, and the line the cosine of the arc AP. They were well aware that the importance of the relations of these various lines to each other was dependent on the theory of similarity which we have just expounded. But they did not make their definitions express the properties which arise from this theory. Also they had not in their heads the modern general ideas respecting functions as correlating pairs of variable num bers, nor in fact were they aware of any OM modern conception and algebraic was natural to analysis. Accordingly, them to think merely of the relations between certain lines in a diagram. For us the case of algebra it different: we powerful ideas. wish to embody our more is Hence, in modern mathematics, instead of considering the fraction the AP -, arc which same for all lengths of considering the lines AP, we is a number the OP; and, PM consider and instead of we con- OM , TRIGONOMETRY 183 ,OM - j *u fractions t 4.sider the and , , , . which again are numbers not dependent on the length of OP, i.e. not dependent on the scale of our diagrams. Then we to be the sine of the number - number define the number PA -, PM and the to be the cosine of the number These fractional forms are clumsy to print; so let us put u for the fraction AP which represents the magnitude of the angle AOP, and put for the fraction v for -y^ . the fraction Then u 9 v, pi/ , w 9 are and w num we are talking of any angle are variable numbers. But a they correlation exists between their magnitudes, so that when u (i.e. the angle AOP) is given the magnitudes of v and w are definitely deter mined. Hence v and w are functions of the have called v the sine of argument u. wish to adapt u, and w the cosine of u. the general functional notation y=f(x) to these special cases: so in modern mathe- bers, and, since AOP, We We for "/" when we want 184 INTRODUCTION TO MATHEMATICS to indicate the special function of "sine," for when we want to indicate and the special function of "cosine." Thus, with "cos" "/" the above meanings for u, v = sin u, and w, v, w = cos u we get 9 where the brackets surrounding the x in f(x) are omitted for the special functions. The meaning of these functions sin and cos as correlating the pairs of numbers u and v, and u and w is, that the functional relations are to 26) an angle divided by OP" is equal to u, and that then v is the number given by "PM divided by OP" and w is the number given by "OM divided by OP." It is evident that without some further defi nitions we shall get into difficulties when the number u is taken too large. For then the arc may be greater than one-quarter of the circumference of the circle, and the point and A and not (cf. fig. 26) may fall between and A. Also P may be below between the line AOA and not above it, as in fig. 26. In order to get over this difficulty we have recourse to the ideas and conventions of co ordinate geometry in making our complete Let one definitions of the sine and cosine. arm OA of the angle be the axis OX, and produce the axis backwards to obtain its Draw the other axis . negative part be found by constructing AOP, whose measure AP " (cf. fig. AP M OX TRIGONOMETRY YOY 185 P f perpendicular to it. Let any point at a distance r from have coordinates x and y. These coordinates are both positive in the first "quadrant" of the plan, e.g. the coordinates x and y of in fig. 27. In the P other quadrants, either one or both of the 1 coordinates are negative, for example, x and y for for P f P P" P", # and f and x and y for and # and $ for , 2/ , in and x and # 27, where fig. are both negative numbers. The is the arc divided positive angle AP POA 77 by r, its sine is - and T* its cosine is -; the posi- INTRODUCTION TO MATHEMATICS 186 tive angle P OA is the arc ABP its sine is 77^ sine is - f angle by P" r, its r, 1* - and cosine - the arc is P"OA divided by / 7/ and OA sine its ; divided by r, its T^ cosine is - ; the positive P" the arc is is ABA the positive angle - and its r ABA B cosine P" divided is -. r But even now we have not gone far enough. For suppose we choose u to be a number greater than the ratio of the whole circum ference of the circle to its radius. Owing to the similarity of all circles this ratio is the same for all circles. It is always denoted in mathematics by the symbol ^TT, where TT is the Greek form of the letter p and its name in the Greek alphabet is It can be proved that TT is an incommensurable number, and that therefore its value cannot be expressed by any fraction, or by any "pi." terminating or recurring decimal. Its value to a few decimal places is 3 14 159; for many purposes a sufficiently accurate approximate 22 value is - Mathematicians can easily cal culate just as TT to of accuracy required, can be so calculated. Its value actually given to 707 places of Such elaboration of calculation is V% has been decimals. any degree TRIGONOMETRY 187 merely a curiosity, and of no practical or theoretical interest. The accurate deter one of the two parts of the famous problem of squaring the circle. The other part of the problem is, by the theoretical methods of pure geometry to mination of TT is describe a straight line equal in length to the circumference. Both parts of the problem are now known to be impossible; and the insoluble problem has now lost all special practical or theoretical interest, having be come absorbed in wider ideas. After this digression on the value of TT, we now return to the question of the general definition of the magnitude of an angle, so as to be able to produce an angle corresponding to any value u. Suppose a moving point, Q, to start from A on (cf. fig. 27), and to rotate in the positive direction (anti-clock wise, in the figure considered) round the cir cumference of the circle for any number of times, finally resting at any point, e.g. at or or P or Then the total length of the curvilinear circular path traversed, divided by the radius of the circle, r, is the OX P P" P" . generalized definition of a positive angle of any size. Let x 9 y be the coordinates of the point in which the point Q rests, i.e. in one of the four alternative positions mentioned in fig. 27; x and y (as here used) will either x and 2/, or x and y, or x and y 9 or x and y". INTRODUCTION TO MATHEMATICS 188 Then the sign of this generalized angle is T /M and its cosine With is these definitions r the functional relations v = sin u and w = cos u, are at last defined for all positive real values of u. For negative values of u we simply take rotation of Q in the opposite (clockwise) direction; but it is not worth our while to elaborate further on this point, now that the general method of procedure has been explained. These functions of sine and cosine, as thus defined, enable us to deal with the problems concerning the triangle from which Trigono metry took its rise. But we are now in a position to relate Trigonometry to the wider idea of Periodicity of which the importance was explained in the last chapter. It is easy to see that the functions sin u and cos u are For consider the periodic functions of u. position, P (in fig. 27), of a moving point, Q, which has started from A and revolved round the circle. This position, P, marks the angles arc arc arc , AP and 6 TT AP -- -j , and so on r Now, all these angles have the 7/ cosine, namely, - and AP . indefinitely. same sine and 3* Hence it is easy to TRIGONOMETRY 189 u be chosen to have any value, the arguments u and %TT+U, and 47r+w, and 6-Tr-fw, and STT+U and so on indefinitely, have all the same values for the correspond ing sines and cosines. In other words, see that, if sn u =sn ?rw =sn ?rw =sn = etc. cos u = cos (2?r + u) = cos (4-Tr +u) = = etc. ; This fact is expressed by saying that sin u and cos u are periodic functions with their period equal to %TT. The graph of the function y = sin x (notice that we now abandon v and u for the more familiar y and x) is shown in fig. 28. We take on the axis of x any arbitrary length at pleasure to represent the number TT, and on the axis of y any arbitrary length at pleasure The numerical to represent the number 1. values of the sine and cosine can never ex The recurrence of the figure ceed unity. This after periods of &TT will be noticed. of the periodic graph represents simplest style function, out of which all others are con The cosine gives nothing funda structed. different from the sine. For it is mentally easy to prove that cos x it = sin ( x+ ^ ; hence J can be seen that the graph of cos x is simply 28 modified by drawing the axis of OF fig. INTRODUCTION TO MATHEMATICS 190 through the point on of drawing it OX marked , instead in its actual position on the figure. It is easy to construct a sine function in /A\tt2X/r\ \3ft I \/ 6 _r\V 4jtf \y Fig. 28. which the period has any assigned value For we have only to write a. and then gm L_^ i a =Sm Thus the period J of this a new ^TT L = Sm function is a now a. Let us now give a general definition of what TRIGONOMETRY 191 we mean by a periodic function. The func tion f(x) is periodic, with the period a, if (i) for any value of x we have f(x) =f(x+a), and (ii) there is no number b smaller than a such that for any value of x, f(x) =f(x+b). The second clause is put into the definition because when we have sin it is , a not only periodic in the period a, but also in the periods 2a and 3a, and so on; this arises since . sin -= %7r(x+3a) i a . sin -\ a f%7TX \ , , \-Qir ( } - = sin %7TX . . a ) So it is the smallest period which we want to get hold of and call the period of the function. The greater part of the abstract theory of periodic functions and the whole of the appli cations of the theory to Physical Science are dominated by an important theorem called Fourier s Theorem; namely that, if f(x) be a periodic function with the period a and if f(x) also satisfies certain conditions, which prac tically are always presupposed in functions suggested by natural phenomena, then f(x) can be written as the sum of a set of terms in the form CQ +d sin ( ~ . +c 3 sin +ei\ +c 2 -a /6-Tnr , ( [-03 \ sin f - \ + , 1 J etc. - +e z J 192 INTRODUCTION TO MATHEMATICS this formula c , Ci, Cz, c 3 , etc., and also 2, e s , etc., are constants, chosen so as to suit the particular function. Again we have In 0i, to ask, How many terms have to be chosen? here a new difficulty arises: for we can prove that, though in some particular cases And a definite number will do, yet in general all we can do is to approximate as closely as we like to the value of the function by tak more and more terms. This process of gradual approximation brings us to the con sideration of the theory of infinite series, an essential part of mathematical theory which we will consider in the next chapter. The above method of expressing a periodic function as a sum of sines is called the "har monic analysis" of the function. For ex ample, at any point on the sea coast the tides Thus at a point rise and fall periodically. near the Straits of Dover there will be two daily tides due to the rotation of the earth. The daily rise and fall of the tides are com plicated by the fact that there are two tidal waves, one coming up the English Channel, and the other which has swept round the North of Scotland, and has then come south ward down the North Sea. Again some high ing tides are higher than others: this is due to the fact that the Sun has also a tide-generat ing influence as well as the Moon. In this way monthly and other periods are intro- TRIGONOMETRY 193 We leave out of account the excep tional influence of winds which cannot be The general problem of the har foreseen. monic analysis of the tides is to find sets of terms like those in the expression on page 191 above, such that each set will give with approximate accuracy the contribution of the tide-generating influences of one "period" to the height of the tide at any instant. The argument x will therefore be the time reckoned duced. from any convenient commencement. Again, the motion of vibration of a violin string is submitted to a similar harmonic analysis, and so are the vibrations of the ether and the air, corresponding respectively to waves of light and waves of sound. We are here in the presence of one of the funda mental processes of mathematical physics namely, nothing less than its general method of dealing with the great natural fact of Periodicity. CHAPTER XIV SERIES No part of Mathematics suffers more from the triviality of its initial presentation to beginners than the great subject of series. Two minor examples of series, namely arith metic and geometric series, are considered; these examples are important because they are the simplest examples of an important general theory. But the general ideas are never disclosed; and thus the examples, which exemplify nothing, are reduced to silly trivialities. The general mathematical idea of a series that of a set of things ranged in order, that This meaning is accurately is, in sequence. represented in the common use of the term. Consider, for example, the series of English Prime Ministers during the nineteenth cen tury, arranged in the order of their first tenure of that office within the century. The series commences with William Pitt, and ends with is Lord Rosebery, who, appropriately enough, the biographer of the first member. We is 194 SERIES 195 might have considered other serial orders for the arrangement of these men; for example, according to their height or their weight. These other suggested orders strike us as trivial in connection with Prime Ministers, and would not naturally occur to the mind; but abstractedly they are just as good orders as any other. When one order among terms is very much more important or more obvious than other orders, it is often spoken of as the order of those terms. Thus the order of the integers would always be taken to mean their order as arranged in order of magnitude. But of course there is an indefinite number of other ways of arranging them. When the number of things considered is finite, the number of ways of arranging them in order is called the number of their permutations. The number of permutations of a set of n things, where n is some finite integer, is rcX(rc-l)X(ra-2)X(ra-3)X...X4X3X2Xl that is to say, it is the product of the first n this product is so important in mathematics that a special symbolism is used for it, and it is always written Thus, 2!=2X1=2, and 3!=3X2Xl=6, and 41=4X3X2X1=24, and 5!=5X4X3X2Xl integers; "ft!" = 120. As n increases, the value of n\ in creases very quickly; thus 100! is a hundred times as large as 99! 196 INTRODUCTION TO MATHEMATICS It is easy to verify in the case of small values of n that nl is the number of ways of arranging n things in order. Thus con sider two things a and b; these are capable of the two orders ab and 6a, and 2! =2. Again, take three things a, 6, and c; these are capable of the six orders, abc, acb, bac, bca, cab, cba, and 3! = 6. Similarly for the twenty-four orders in which four things a, b, c, and d, can be arranged. When we come to the infinite sets of things like the sets of all the integers, or all the fractions, or all the real numbers for instance we come at once upon the complications of the theory of order-types. This subject was touched upon in Chapter VI in considering the possible orders of the integers, and of the The fractions, and of the real numbers. whole question of order-types forms a com paratively new branch of mathematics of great importance. We shall not consider it any further. All the infinite series which we consider now are of the same order-type as the integers arranged in ascending order of magnitude, namely, with a first term, and such that each term has a couple of nextdoor neighbours, one on either side, with the exception of the first term which has, of course, only one next-door neighbour. Thus, if m be any integer (not zero), there will be always an mth term. A series with a finite SERIES 197 of terms (say n terms) has the same characteristics as far as next-door neighbours are concerned as an infinite series; it only differs from infinite series in having a last term, namely, the nth. The important thing to do with a series of numbers using for the future "series" in number the restricted mentioned is sense which has just been to add its successive terms together. are respec if ui, u 2 u 3 , Un . the nth, 1st, 2nd, 3rd, 4th, tively terms of a series of numbers, we form succes sively the series Ui, Ui+u 2 , Ui+u 2 +u 3 , Ui W2+w 3 -hW4, and so on; thus the sum of the Thus , . . . . . . . . . . + 1st n terms may be written. the series has only a finite number of terms, we come at last in this way to the sum of the whole series of terms. But, if the series has an infinite number of terms, this process of successively forming the sums of the terms never terminates; and in this sense there is no such thing as the sum of an If infinite series. But why is it important successively to add the terms of a series in this way? The answer is that we are here symbolizing the funda mental mental process of approximation. This is a process which has significance iar 198 INTRODUCTION TO MATHEMATICS of mathematics. Our limited intellects cannot deal with compli cated material all at once, and our method of arrangement is that of approximation. The beyond the regions statesman in framing his speech puts the dominating issues first and lets the details naturally into their subordinate places. of course, the converse artistic is, method of preparing the imagination by the presentation of subordinate or special details, and then gradually rising to a crisis. In either way the process is one of gradual fall There summation of effects; and this is exactly what is done by the successive summation of the terms of a series. Our ordinary method of stating numbers is such a process of gradual summation, at least, in the case of large numbers. Thus 568,213 presents itself to the mind as 500,000+60,000 +8,000 +200 + 10 +3 In the case of decimal fractions this Thus 3-14159 more avowedly. 3 +lV +TOT +T0W +TO Also, 3 0~0~0 is so is +TToVoiT and 3 + TV, and 3 + TV + T fo and 3 + TV and S+yV+Tw+T^W+Ttflors , +T^+TFO~O> are successive approximations to the complete result 3-14159. from right to If we read 568,213 backwards left, starting with the 3 units, SERIES 199 we read it in the artistic way, gradually pre paring the mind for the crisis of 500,000. The ordinary process of numerical multi plication proceeds by means of the summa Consider the computation tion of a series. 658 2736 1710 2052 225036 Hence the three lines to be added form a which the first term is the upper This series follows the artistic method line. of presenting the most important term last, not from any feeling for art, but because of the convenience gained by keeping a firm hold on the units place, thus enabling us to omit some O s, formally necessary. But when we approximate by gradually adding the successive terms of an infinite The series, what are we approximating to? in no is that series has the difficulty the straightforward sense of the word, because the operation of adding together its terms can never be completed. The answer is that we are approximating to the limit of the summation of the series, and we must now series of "sum" INTRODUCTION TO MATHEMATICS 200 proceed to explain what the series "limit" of a is. The summation of a series approximates to a limit when the sum of any number of its terms, provided the number be large enough, is as nearly equal to the limit as you care to approach. But this description of the mean ing of approximating to a limit evidently will not stand the vigorous scrutiny of modern mathematics. What is meant by large by nearly equal, and by care to All these vague phrases must be approach? explained in terms of the simple abstract ideas which alone are admitted into pure enough, and mathematics. Let the successive terms of the MI, u2 , Us, u*, . . nth term of the sum of the 1st ., un , series. series u so that etc., Also let sn be the be the is n terms, whatever n may be. So that , Then the terms form sn 81, s 2 s s and the formation of this series the process of summation of the original a new is and , , . . . , . . . series, Then the "approximation" of the original series to a "limit" means the "approximation of the terms of this new series to a limit." And we have series. summation of the SERIES now to explain 201 what we mean by the approxi mation to a limit of the terms of a series. Now, remembering the definition (given in Chapter XII) of a standard of approxima the idea of a limit means this: / the limit of the terms of the series s\ 9 tion, is $2* ,...#,..., if, corresponding to each taken as a standard of approximation, a term sn of the series can be found so that all succeeding terms (i.e. sn+i, sn+z, etc.) approximate to I within that standard of approximation. If another smaller standard k 1 be chosen, the term sn may be too early in the series, and a later term sm with the above property will then be found. If this property holds, it is evident that as you go along to series s l9 s z **,... ,4* ... from left to right, after a time you come to terms all of which are nearer to / than any number which you may like to assign. In other words you approximate to I as closely The close connection of this as you like. definition of the limit of a series with the definition of a continuous function given in Chapter XI will be immediately perceived. real mumber k, . , Then coming back to the original series u\ 9 Uz 9 u s , Un, , the limit of the terms of sn9 the series s i9 s 2 s S9 ., is called the "sum to infinity" of the original series. But it is evident that this use of the word , . . . . , . . . . . 9 . . 202 INTRODUCTION TO MATHEMATICS is very aritficial, and we must not assume the analogous properties to those of "sum" the ordinary sum of a finite number of terms without some special investigation. Let us look at an example of a "sum to Consider the recurring decimal infinity." 1111. This decimal is merely a way of symbolizing the "sum to infinity" of the . series ing . . -001, -0001, etc. -1, -01, series The correspond is Si = -l, etc. The limit found by summation = -n, s 3 =-lll, * 4 = -llll, of the terms of this series is ^; this see by simple division, for s2 i = -1 Hence, + -fa = -11 + if is ^7 <nh) is easy to = -HI + 9 oV o = etc. given (the k of the definition), and all succeeding terms differ from ^ by 3 less than T y if loVo is given (another choice for the k of the definition), -111 and all suc ceeding terms differ from ^ by less than I ^Vo5 and so on, whatever choice for k be made. 1 ; It is evident that nothing that has been said gives the slightest idea as to how the of a series is to be "sum to infinity" have merely stated the condi found. tions which such a number is to satisfy. In deed, a general method for finding in all cases the sum to infinity of a series is intrinsic ally out of the question, for the simple reason that such a "sum," as here defined, does not always exist. Series which possess a sum to We SERIES 203 infinity are called convergent, and those which infinity are called do not possess a sum to divergent. An obvious example of a divergent series i.e. the series of in ., n 1, 2, 3, For order of in their magnitude. tegers whatever number I you try to take as its sum to infinity, and whatever standard of approximation k you choose, by taking enough terms of the series you can always is make . their . . sum . differ . from / by more than Again, another example of a divergent series is 1, 1, 1, etc., i.e. the series of which each term is equal to 1. Then the sum of n terms is n, and this sum grows without limit as n increases. Again, another example of a divergent series is 1, -1, 1, -1, 1, -1, etc., i.e. the series in which the terms are alternately 1 and -1. The sum of an odd number of terms is 1, and of an even number of terms is 0. Hence the terms of the series Si, s 2 s s ... sn) ... do not ap proximate to a limit, although they do not increase without limit. It is tempting to suppose that the condi tion for Wi, u 2 ... Wn, ... to have a sum to infinity is that u n should decrease inde k. , , , n increases. Mathematics would be a much easier science than it is, if this were the case. Unfortunately the supposition is not true. finitely as INTRODUCTION TO MATHEMATICS 204 For example the series 111 1 4 n 2 3 divergent. It is easy to see that this is the case; for consider the sum of n terms A term. These n beginning at the (tt-f-l) is 111 are n of Hence , , them and their i.e. is 2ra 1 - terms are - 2n sum is is sum . . the least . 2 ., : there among them. greater than greater than altering the , , n times Now, without to infinity, if it exist, we can add together neighbouring terms, and obtain the series is, by what has been said above, a series whose terms after the 2nd are greater than that those of the series, 1, }, J, i, etc., where all the terms after the first But this series is divergent. original series is divergent. This question of divergency careful we must be in arguing are equal. Hence the shows how from the pro- SERIES 205 perties of the sum of a finite number of terms to that of the sum of an infinite series. For the most elementary property of a finite number of terms is that of course they possess a sum: but even this fundamental not necessarily possessed by an This caution merely states that we must not be misled by the suggestion of the technical term "sum of an infinite It is usual to indicate the sum of series." the infinite series property is infinite series. Ui, u z u s ... , , ... by Un, We now pass on to a generalization of the idea of a series, which mathematics, true to its method, makes by use of the variable. we have only contemplated series which each definite term was a definite number. But equally well we can generalize, and make each term to be some mathematical Hitherto, in expression containing a variable x. we may consider the series 1, x, x2 , x3 9 xn, , and the series . . Thus . . . , . 9 ?! L ?L 2 3 n In order to symbolize the general idea of any such function, conceive of a function of x, fn (x) say, which involves in its formation a variable integer n, then, by giving n the INTRODUCTION TO MATHEMATICS 206 values we in succession, 1, 2, 3, etc., get the series MX), MX), MX), . . .,/(*), . . . Such a series may be convergent for some values of x and divergent for others. It is, in fact, rather rare to find a series involving a variable x which is convergent for all values at least in any particular instance it is to assume that this is the case. unsafe very For example, let us examine the simplest of all instances, namely, the "geometrical" of x, series 1, The sum sn Now Now line X X 9 of 9 X ) n terms . is .,27, . . . . given by = I+x+x*+x*+. . . +x\ multiply both sides by x and we get subtract the last line from the upper and we get Sn (l and hence -x)=Sn -XSn =l- X n+1 (if n x be not equal to l-x 9 1) l-xx Now if x be numerically than 1, for suffin x is always less ciently large values of n, A ~~ X less -- SERIES 207 than Jc, however k be chosen. Thus, if x be than 1, the series 1, x, x 2 , ... x n , ... is less convergent, and -- - J. ment JL ~ is X is its limit. This state- symbolized by X But x numerically greater than 1, or numerically equal to 1, the series is divergent. In other words, if x lie between 1 and +1, the series is convergent; but if x be equal 1 or to -f 1, or if x lie outside the interval to 1 to +1, then the series is divergent. Thus the series is convergent at all "points" within the interval 1 to exclusive of the end-points. At this stage of our enquiry another ques tion arises. Suppose that the series if is +1> /l(*) +/!(*)+/(*)+ - - +/(*)+ is convergent for all values of x lying within the interval a to 6, i.e. f(x) is convergent for any value of x which is greater than a and than b. Also, suppose we want to be sure that in approximating to the limit we add together enough terms to come within less some standard of approximation k. Can we always state some number of terms, say n, such that, if we take n or more terms to form the sum, then whatever value x has 208 INTRODUCTION TO MATHEMATICS within the interval we have satisfied the desired standard of approximation? Sometimes we can and sometimes we can not do this for each value of k. When we can, the series is called uniformly convergent throughout the interval, and when we cannot do so, the series is called non-uniformly con vergent throughout the interval. It makes a great difference to the properties of a series whether it is or is not uniformly convergent through an interval. Let us illustrate the matter by the simplest example and the simplest numbers. Consider the geometric series convergent throughout the interval ==!. excluding the end values x But it is not uniformly convergent through out this interval. For if sn (x) be the sum of n terms, we have proved that the difference 1 x n+1 between sn (x) and the limit is l l x x Now suppose n be any given number of terms, say 20, and let k be any assigned standard of approximation, say -001. Then, by taking x near enough to +1 or near enough to 1, 21 x we can make the numerical value of to l x be greater than -001. Thus 20 terms will It is 1 to -f-1, - SERIES 209 not do over the whole interval, though it is more than enough over some parts of it. The same reasoning can be applied what ever other number we take instead of 20, and whatever standard of approximation in stead of -001. Hence the geometric series 1 + x + x2 + Xs -fis non-uni+ xn + - formly convergent over 1 to +1. convergence whole interval of if we take any smaller interval lying at both ends within the 1 to +1, the geometric series is interval For ex uniformly convergent within it. ample, take the interval to + TV- Then any its But x n+l value for n which makes less than k at these limits for all values of x between these - x 1+1 it X numerically for x also serves A ~~ so happens that - l limits, since diminishes in numeri- x cal value as x diminishes in numerical value. For example, take k =-001; then, putting we find: #= V> for for n-1,--L X 1 5- = ^ = -0111 T n-8, Thus three terms will do for the whole in- 210 INTRODUCTION TO MATHEMATICS terval, though, of course, for some parts the interval it is more than Notice that, because 1 + x is of necessary. + x 2 -f- ... n ... is x + + convergent (though not uni 1 to +1, formly) throughout the interval for each value of x in the interval some num ber of terms n can be found which will satisfy a desired standard of approximation; but, as we take x nearer and nearer to either end value +1 or 1, larger and larger values of n have to be employed. It is curious that this important distinction between uniform and non-uniform conver gence was not discovered till 1847 by Stokes afterwards, Sir George Stokes and later, in dependently in 1850 by Seidel, a German mathematician. The critical points, where non-uniform con vergence comes in, are not necessarily at the limits of the interval throughout which con vergence holds. This is a speciality belonging to the geometric series. In the case of the geometric series \-\-x a simple algebraic x + 2 + ... + x n + . expression ^JL ~~ X . . , can be given for its limit in interval of convergence. But this is not always the case. Often we can prove a series to be convergent within a certain interval, though we know nothing more about its limit except that it is the limit of the seriesits SERIES But this is a very good way of defining a func as the limit of an infinite conver and is, in fact, the way in which most functions are, or ought to be, defined. Thus, the most important series in ele tion; viz. gent series, mentary analysis is where n\ has the meaning defined earlier in This series can be proved to be convergent for all values of x, and to be uniformly convergent within any interval which we like to take. Hence it has all the comfortable mathematical properties which a series should have. It is called the ex ponential series. Denote its sum to infinity by expo;. Thus, by definition, this chapter. expo; is called the exponential function. It is fairly easy to prove, with a little knowledge of elementary mathematics, that (expz) X (expy) = exp(x+y) In other words that (expz) X (expy) = (x+y) n! . . . (A) INTRODUCTION TO MATHEMATICS is an example of what an addition-theorem. When any function [say /(#)] has been defined, the first thing we do is to try to express f(x+y) in terms of known functions of x only, and known functions of y only. If we can do so, the result is called an addition-theorem. This property (A) is called Addition-theorems play a great part in mathematical analysis. Thus the additiontheorem for the sine is given by sin (x-\-y) and =sin x cos ^+cos x sin for the cosine cos (x-\-y) y, by =cos x cos y sin x sin y. As a matter fining sin of fact the best ways of de x and cos x are not by the elaborate geometrical methods of the previous chapter, but as the limits respectively of the series z3 x7 ar> "" ^"" andl " * 1 ~2! + 4!~6! +etC so that , , * ---- * we put Xs X5 X 7 sm x =x - ^+ - + , i - 6 2 X - , *> tf /: X* etc. . . ., SERIES 213 These definitions are equivalent to the geo metrical definitions, and both series can be proved to be convergent for all values of x, and uniformly convergent throughout any These series for sine and cosine have a general likeness to the exponential interval. series given above. They are, indeed, inti mately connected with it by means of the theory of imaginary numbers explained in Chapters VII and VIII. X Z Fig. 29. The graph of the exponential function is given in fig. 29. It cuts the axis OF at the point 2/ = l, as evidently it ought to do, since when cc = every term of the series except the first is zero. The importance of the ex ponential function is that it represents any changing physical quantity whose rate of increase at any instant is a uniform per centage of its value at that instant. For INTRODUCTION TO MATHEMATICS example, the above graph represents the size at any time of a population with a uniform birth-rate, where the x corresponds to the time reckoned from any convenient day, and the y represents the population to the proper The scale must be such that OA re scale. presents the population at the date which is taken as the origin. But we have here come upon the idea of "rates of increase" which is the topic for the next chapter, An important function nearly allied to the exponential function is found by putting x2 for x as the argument in the exponential 2 We thus get exp. function. The ). graph 2/ = exp. (x (x 2 ) is given in fig. 30. Fig. 30. The hat, is curve, which is something like a cocked normal error. Its called the curve of SERIES 215 corresponding function is vitally important to the theory of statistics, and tells us in many cases the sort of deviations from the average results which we are to expect. Another important function is found by combining the exponential function with the sine, in this way: y = exp( - ex) X sin Fig. 31. Its graph is given in fig. 31. The points A, B, 0, C D, E F, are placed at equal in tervals %p and an unending series of them should be drawn forwards and backwards. 9 9 9 This function represents the dying away of vibrations under the influence of friction or of "damping" forces. Apart from the friction, the vibrations would be periodic, with a period p; but the influence of the friction 216 INTRODUCTION TO MATHEMATICS makes the extent of each vibration smaller than that of the preceding by a constant per centage of that extent. This combination of the idea of "periodicity" (which requires the sine or cosine for its symbolism) and of "constant percentage" (which requires the exponential function for its symbolism) is the reason for the form of this function, namely, its form as a product of a sine-function into an exponential function. CHAPTER XV THE DIFFERENTIAL CALCULUS THE invention of the differential calculus crisis in the history of mathematics. The progress of science is divided between periods characterized by a slow accumulation of ideas and periods, when, owing to the new material for thought thus patiently collected, some genius by the invention of a new method or a new point of view, suddenly transforms the whole subject on to a higher level. These contrasted periods in the progress of the history of thought are compared by Shelley to the formation of an avalanche. marks a The sun-awakened avalanche! whose mass, Thrice sifted by the storm, had gathered there Flake after flake, in heaven-defying minds As thought by thought is piled, till some great truth Is loosened, and the nations echo round, The comparison The final burst of bear some pressing. sunshine which awakens the avalanche is not necessarily beyond com parison in magnitude with the other powers of nature which have presided over its slow will 217 218 INTRODUCTION TO MATHEMATICS The same is true in science. The genius who has the good fortune to produce the final idea which transforms a whole region of thought, does not necessarily excel all his predecessors who have worked at the preliminary formation of ideas. In consider ing the history of science, it is both silly and ungrateful to confine our admiration with a gaping wonder to those men who have made the final advances towards a new epoch. In the particular instance before us, the subject had a long history before it as sumed its final form at the hands of its two inventors. There are some traces of its methods even among the Greek mathe maticians, and finally, just before the actual production of the subject, Fermat (born 1601 A.D., and died 1665 A.D.), a distinguished French mathematician, had so improved on previous ideas that the subject was all but created by him. Fermat, also, may lay claim to be the joint inventor of coordinate formation. geometry in company with his contemporary and countryman, Descartes. It was, in fact, Descartes from whom the world of science received the new ideas, but Fermat had cer tainly arrived at them independently. We need not, however, stint our admira tion either for Newton or for Leibniz. New ton was a mathematician and a student of physical science, Leibniz was a mathema- DIFFERENTIAL CALCULUS 219 and a philosopher, and each of them own department of thought was one of the greatest men of genius that the world The joint invention was the has known. tician in his occasion of an unfortunate and not very creditable dispute. Newton was using the methods of Fluxions, as he called the sub ject, in 1666, and employed it in the com position of his Principia, although in the work as printed any special algebraic notation is avoided. But he did not print a direct state ment of his method till 1693. Leibniz pub He was lished his first statement in 1684. accused by Newton s friends of having got it from a MS. by Newton, which he had been privately. Leibniz also accused New ton of having plagiarized from him. There is now not very much doubt but that both should have the credit of being independent The subject had arrived at a discoverers. shown stage in which it was ripe for discovery, and there is nothing surprising in the fact that two such able men should have indepen dently hit upon it. These joint discoveries are quite common Discoveries are not in general before they have been led up to by the previous trend of thought, and by that time many minds are in hot pursuit of the important idea. If we merely keep to dis coveries in which Englishmen are concerned, in science. made 220 INTRODUCTION TO MATHEMATICS the simultaneous enunciation of the law of natural selection by Darwin and Wallace, and the simultaneous discovery of Neptune by Adams and the French astronomer, Leverrier, at once occur to the mind. The disputes, as to whom the credit ought to be given, are often influenced by an unworthy The really inspiring spirit of nationalism. reflection suggested by the history of mathe matics is the unity of thought and interest among men of so many epochs, so many Indians, Egyp Greeks, Arabs, Italians, nations, and so many tians, Assyrians, races. Frenchmen, Germans, Englishmen, and Rus sians, have all made essential contributions to the progress of the science. Assuredly the jealous exaltation of the contribution of one particular nation is not to show the larger spirit. The importance of the differential calculus of the subject, which is the systematic consideration of the rates of increase of functions. This idea is immediately presented to us by the study of nature; velocity is the rate of increase of the arises from the very nature distance travelled, and acceleration is the rate of increase of velocity. Thus the funda mental idea of change, which is at the basis of our whole perception of phenomena, imme diately suggests the enquiry as to the rate of The familiar terms of "quickly" change. DIFFERENTIAL CALCULUS 221 and "slowly" gain their meaning from a tacit reference to rates of change. Thus the differ ential calculus is concerned with the very key of the position from which mathematics can be successfully applied to the explana tion of the course of nature. This idea of the rate of change was certainly in Newton s mind, and was embodied in the T N M jf Fig. 32. language in which he explained the subject. It may be doubted, however, whether this point of view, derived from natural pheno mena, was ever much in the minds of the preceding mathematicians who prepared the subject for its birth. They were concerned with the more abstract problems of drawing tangents to curves, of finding the lengths of curves, and of finding the areas enclosed by curves. The last two problems, of the recti- INTRODUCTION TO MATHEMATICS and the quadrature of named, belong to the In tegral Calculus, which is however involved in the same general subject as the Differential fication of curves curves as they are Calculus. The introduction of coordinate geometry makes the two points of view coalesce. For be any curved line and let (cf. fig. 32) let be the tangent at the point on it. Let the axes of coordinates be and and let y=f(x) be the equation to the curve, = x, and so that y. Now let Q be any on the with coordinates curve, moving point AQP PT P OX OY PM OM Xi,yi; then?/i=/(i). And let Q be the point on the tangent with the same abscissa Xi\ suppose that the coordinates of Q are Xi and moves along the Now suppose that y axis OX from left to right with a uniform velocity; then it is easy to see that the ordinate y of the point Q on the tangent TP also increases uniformly as Q moves along the tangent in a corresponding way. In fact N . f it is easy to see that the ratio of the rate of to the rate of increase of increase of Q to TN, which is the same is in the ratio of Q But the at all points of the straight line. rate of increase of QN, which is the rate of increase of f(xi), varies from point to point of the curve so long as it is not straight. As Q passes through the point P, the rate of in crease of f(xi) (where Xi coincides with x for N N ON DIFFERENTIAL CALCULUS 223 is the same as the rate of in on the tangent at P. Hence, if the moment) crease of y we have a general method of determining the rate of increase of a function f(x) of a variable x, we can determine the slope of the tangent at any point (x, y), on a curve, and thence can draw it. Thus the problems of drawing tangents to a curve, and of deter mining the rates of increase of a function are really identical. It will be noticed that, as in the cases of Conic Sections and Trigonometry, the more artificial of the two points of view is the one in which the subject took its rise. The really fundamental aspect of the science only rose into prominence comparatively late in the day. It is a well-founded historical generali zation, that the last thing to be discovered in any science is what the science is really about. Men go on groping for centuries, guided merely by a dim instinct and a puzzled curiosity, till at last "some great truth is loosened." Let us take some special cases in order to familiarize ourselves with the sort of ideas which we want to make precise. A train is in motion how shall we determine its velocity at some instant, let us say, at noon? We can take an interval of five minutes which includes noon, and measure how far the train has gone in that period. Suppose we find it to be five INTRODUCTION TO MATHEMATICS we may then conclude that the train of 60 miles per hour. But five miles is a long distance, and we cannot be sure that just at noon the train was moving at this pace. At noon it may miles, was running at the rate have been running 70 miles per hour, and afterwards the break may have been put on. It will be safer to work with a smaller interval, say one minute, which includes noon, and to measure the space traversed during that But for some purposes greater period. accuracy may be required, and one minute may be too long. In practice, the necessary inaccuracy of our measurements makes it useless to take too small a period for measure ment. But in theory the smaller the period the better, and we are tempted to say that for ideal accuracy an infinitely small period The older mathematicians, in is required. particular Leibniz, were not only tempted, but yielded to the temptation, and did say Even now it is a useful fashion of speech, it. provided that we know how to interpret it It is into the language of common sense. curious that, in his exposition of the founda tions of the calculus, Newton, the natural scientist, is much more philosophical than and on the other hand, Leibniz provided the admirable nota tion which has been so essential for the pro gress of the subject. Leibniz, the philosopher, DIFFERENTIAL CALCULUS Now take another example within the region of pure mathematics. Let us proceed to find the rate of increase of the function x2 for have not any value x of its argument. defined what we mean by rate of really yet will try and grasp its meaning increase. in relation to this particular case. When x increases to x+h, the function x 2 increases to 2 (x+h) ; so that the total increase has been 2 x 2 , due to an increase h in the argu (x+h) ment. Hence throughout the interval x to (x +h) the average increase of the function per We We unit increase of the argument is ^ x nI But and therefore 2 (x+h) -x* ~h~ Thus 2hx+h?_^ ljt ~T~ %x-\-h is the average increase of the function x 2 per unit increase in the argument, the average being taken over by the interval x to x+ h. But %x+h depends on h, the size of the interval. We shall evidently get what we want, namely the rate of increase at the value x of the argument, by diminishing h more and more. Hence in the limit when h INTRODUCTION TO MATHEMATICS has decreased indefinitely, we say that %x is the rate of increase of x 2 at the value x of the argument. Here again we are apparently driven up against the idea of infinitely small quantities in the use of the words "in the limit when h has decreased indefinitely." Leibniz held that, mysterious as it may sound, there were actu ally existing such things as infinitely small quantities, and of course infinitely small num bers corresponding to them. Newton s lan guage and ideas were more on the modern lines; but he did not succeed in explaining the matter with such explicitness as to be evidently doing more than explain Leibniz s ideas in rather indirect language. The real explanation of the subject was first given by Weierstrass and the Berlin School of mathe maticians about the middle of the nineteenth But between Leibniz and Weier century. strass a copious literature, both mathematical and philosophical, had grown up round these mysterious infinitely small quantities which mathematics had discovered and philosophy Some philosophers, proceeded to explain. Bishop Berkeley, for instance, correctly denied the validity of the whole idea, though for reasons other than those indicated here. But the curious fact remained that, despite all criticisms of the foundations of the subject, there could be no doubt but that the mathe- DIFFERENTIAL CALCULUS 227 matical procedure was substantially right. In fact, the subject was right, though the ex planations were wrong. It is this possibility of being right, albeit with entirely wrong ex planations as to what is being done, that so often makes external criticism that is so far as it is meant to stop the pursuit of a method singularly barren and futile in the progress of science. The instinct of trained observers, and their sense of curiosity, due to the fact that they are obviously getting at something, are far safer guides. Anyhow the general effect of the success of the Differential Cal culus was to generate a large amount of bad philosophy, centring round the idea of the in finitely small. The relics of this verbiage may still be found in the explanations of many elementary mathematical text-books on the Differential Calculus. It is a safe rule to apply that, when a mathematical or philoso phical author writes with a misty profundity, he is talking nonsense. Newton would have phrased the question by saying that, as h approaches zero, in the %x+h becomes %x. It is our task so to explain this statement as to show that it does not in reality covertly assume the existence of Leibniz s infinitely small quantities. In reading over the Newtonian method of state ment, it is tempting to seek simplicity by limit INTRODUCTION TO MATHEMATICS saying that %x-\-h is &r, when h is zero. But this will not do; for it thereby abolishes the interval from x to x+h, over which the aver age increase was calculated. The problem is, how to keep an interval of length h over which to calculate the average increase, and at the same time to treat h as if it were zero. New ton did this by the conception of a limit, and we now proceed to give Weierstrass s expla nation of its real meaning. In the first place notice that, in discussing %x+h, we have been considering x as fixed in value and h as varying. In other words x has been treated as a "constant" variable, or parameter, as explained in Chapter IX; and we have really been considering %x-\-h as a function of the argument h. Hence we can generalize the question on hand, and ask what we mean by saying that the function f(h) tends to the limit I, say, as its argument h tends to the value zero. But again we shall see that the special value zero for the argument does not belong to the essence of the subject; and again we generalize still further, and ask what we mean by saying that the f unction f(h) as h tends to the value a. to the Weierstrassian ex planation the whole idea of h tending to the value a, though it gives a sort of metaphorical picture of what we are driving at, is really off the point entirely. Indeed it is fairly obvious tends to the limit Now, according I DIFFERENTIAL CALCULUS 229 that, as long as we retain anything like as a fundamental idea, we are tending to really in the clutches of the infinitely small; for we imply the notion of h being infinitely "h a," near to This a. is just what we want to get rid of. Accordingly, we shall yet again restate our phrase to be explained, and ask what we mean by saying that the limit of the function (fh) at a is I. The limit of f(h) at a is a property of the neighbourhood of a, where "neighbourhood" is used in the sense defined in Chapter XI during the discussion of the continuity of The value functions. of the function f(h) at but the limit is distinct in idea from the value, and may be different from it, and may exist when the value has not been defined. We shall also use the term a is /(a); of approximation" in the sense is defined in Chapter XI. In fact, in the definition of "continuity" given towards the end of that chapter we have practically defined a limit. The definition of "standard in which a limit is it : A function f(x) has the limit I at a value a of its argument x, when in the neighbour hood of a its values approximate to / within every standard of approximation. Compare this definition with that already given for continuity, namely: 230 INTRODUCTION TO MATHEMATICS A function f(x) is continuous at a value a of its argument, when in the neighbourhood of a its values approximate to its value at a within every standard of approximation. It is at once evident that a function is con tinuous at a when (i) it possesses a limit at a, and (ii) that limit is equal to its value at a. Thus the illustrations of continuity which have been given at the end of Chapter XI are illustrations of the idea of a limit, namely, they were all directed to proving that /(a) was the limit of f(x) at a for the functions considered and the value of a considered. It is really more instructive to consider the limit at a point where a function is not con tinuous. For example, consider the function of which the graph is given in fig. 20 of Chap ter XI. This function f(x) is defined to have the value 1 for all values of the argument except the integers 1, 2, 3, etc., and for these integral values it has the value 0. Now let us think of its limit when x = 3. We notice that in the definition of the limit the value of the function at a (in this case, a = 3) is ex cluded. But, excluding /(3), the values of f(x), when x lies within any interval which (i) contains 3 not as an end-point, and (ii) does not extend so far as 2 and 4, are all equal to 1; and hence these values approxi mate to 1 within every standard of approxi mation. Hence 1 is the limit of f(x) at the DIFFERENTIAL CALCULUS value 3 of the argument x, 231 but by definition . This is an instance of a function which possesses both a value and a limit at the value 3 of the argument, but the value is not equal to the limit. At the end of Chapter XI the function x 2 was considered at the value 2 of the argument. Its value at % is 2 2 , i.e. 4, and it was proved that its limit is also Thus here we have a function with a value and a limit which are equal. Finally we come to the case which is essen tially important for our purposes, namely, to a function which possesses a limit, but no defined value at a certain value of its argu ment. We need not go far to look for 4. such a function, Now in %x - - will serve our purpose. any mathematical book, we might find the equation, hesitation or 2x x = comment. culty in this; for when x 2, written without But is there zero, is = x a diffi -; and u - has no defined meaning. Thus the value %x at # = x has no defined of the function 232 INTRODUCTION TO MATHEMATICS But meaning. for every other value of x 9 the value of the function limit of at x Qx - - = 0. at x = is 2, 2x x is 2. and it Similarly the limit of a whatever a may has no value x2 at x = a x But the value is 0. takes the form -, meaning. We now rr 2 of x at x = which has no defined Thus the function but no value at is be, so that the limit of 2 Xat x = x Thus the x2 has a limit x 0. come back which we started to the problem from this discussion on the nature How of a limit. are we going to define the rate of increase of the function x 9 at any value x of its argument. Our answer is that this rate of increase is the limit of the func- at the value zero for its argument h. (Note that x is here a "con Let us see how this answer works stant." DIFFERENTIAL CALCULUS 235 in the light of our definition of a limit. We have Now h(2x+h) h h h in finding the limit of - 7 - n - at the of the argument h, the value (if any) value is excluded. But for of the function at h = all values of h, except /i = 0, we can divide through by h. Thus the limit of at is the same as that of %x+h at h = Q. Now, whatever standard of approximation k we choose to take, by considering the interval from %k to +|fc we see that, for values of h which fall within it, %x+h differs from %x by less than %k, that is by less than k. This is true A= any standard k. Hence in the neighbour hood of the value for h,%x+h approximates to %x within every standard of approxima tion, and therefore %x is the limit of %x+h at /& = 0. Hence by what has been said above for c is the limit of h at the value It follows, therefore, that %x is what called the rate of increase of x2 at Thus this the value x of the argument. method conducts us to the same rate of in- for h. we have 234 INTRODUCTION TO MATHEMATICS crease for x 2 as did the Leibnizian way of making h grow "infinitely small." The more abstract terms "differential co or "derived function," are gener used for what we have hitherto called the "rate of increase" of a function. The definition is as follows: the differ general efficient," ally ential coefficient of the function f(x) limit, of the if it exist, of the f unction is ^x +h argument h at the value the "^ ] h of its argu ment How have we, by this definition and the a limit, really managed definition of subsidiary to avoid the notion of "infinitely small num which so worried our mathematical For them the difficulty arose because on the one hand they had to use an interval x to x+h over which to calculate the average increase, and, on the other hand, they finally wanted to put h = 0. The result was they seemed to be landed into the notion of an existent interval of zero size. Now how do we avoid this difficulty? In this bers" forefathers? we use the notion that corresponding to any standard of approximation, some in terval with such and such properties can be found. The difference is that we have grasped the importance of the notion of "the way variable," and they had not done so. Thus, DIFFERENTIAL CALCULUS 235 at the end of our exposition of the essential notions of mathematical analysis, we are led back to the ideas with which in Chapter II we commenced our enquiry that in mathe matics the fundamentally important ideas are those of "some things" and "any things." CHAPTER XVI GEOMETRY GEOMETRY, like the rest of mathematics, is abstract. In it the properties of the shapes and relative positions of things are studied. But we do not need to consider who is observ ing the things, or whether he becomes ac quainted with them by sight or touch or In short, we ignore all particular hearing. sensations. Furthermore, particular things such as the Houses of Parliament, or the terrestrial globe are ignored. Every pro position refers to any things with such and such geometrical properties. Of course it helps our imagination to look at particular examples of spheres and cones and triangles But the propositions do not merely apply to the actual figures printed in the book, but to any such figures. Thus geometry, like algebra, is dominated and squares. ideas of "any" and "some" things. Also, in the same way it studies the inter relations of sets of things. For example, con by the sider any two triangles ABC and DEF. 236 GEOMETRY What relations must exist 237 between some of the parts of these triangles, in order that the triangles may be in all respects equal? This is one of the first investigations undertaken in all elementary geometries. It is a study a c F Fig. 33. of a certain set of possible correlations be triangles. The answer is that the triangles are in all respects equal, if: Either, (a) Two sides of the one and the in cluded angle are respectively equal to two sides of the other and the included angle: Or, (6) Two angles of the one and the side tween the two joining them are respectively equal to two angles of the other and the side joining them: Or, (c) Three sides of the one are respec tively equal to three sides of the other. This answer at once suggests a further en quiry. What is the nature of the correlation between the triangles, when the three angles of the one are respectively equal to the three This further inves angles of the other? tigation leads us on to the whole theory 238 INTRODUCTION TO MATHEMATICS (cf. Chapter XIII), which is another type of correlation. Again, to take another example, consider the internal structure of the triangle ABC. Its sides and angles are inter-related the greater angle is opposite to the greater side, and the base angles of an isosceles triangle of similarity are equal. If we proceed to trigonometry this correlation receives a more exact deter mination in the familiar shape sin a? = b 2 -f c2 A sin QbccosA, B sin with C two similar formulae. Also there is the still simpler correlation between the angles of the triangle, namely, equal to two right angles; and between the three sides, namely, that the sum of the lengths of any two is greater than the length of the third. Thus the true method to study geometry is to think of interesting simple figures, such as the triangle, the parallelogram, and the circle, and to investigate the correlations between their various parts. The geometer has in his mind not a detached proposition, but a figure with its various parts mutually inter-depend ent. Just as in algebra, he generalizes the triangle into the polygon, and the side into that their sum is GEOMETRY the conic section. 239 Or, pursuing a converse route, he classifies triangles according as they are equilateral, isosceles, or scalene, and polygons according to their number of sides, and conic sections according as they are hyperbolas, ellipses, or parabolas. The preceding examples illustrate how the fundamental ideas of geometry are exactly the same as those of algebra; except that algebra deals with numbers and geometry with lines, angles, areas, and other geo metrical entities. This fundamental identity is one of the reasons why so many geo metrical truths can be put into an algebraic dress. Thus if A, B, and C are the numbers of degrees respectively in the angles of the triangle ABC, the correlation between the angles is represented by the equation = 180 and if a, 6, c are the number of feet respec tively in the three sides, the correlation be tween the sides is represented by a b -f c, Also the trigonometrical 6 c -f- a, c<a+b. formulae quoted above are other examples of the same fact. Thus the notion of the vari able and the correlation of variables is just as essential in geometry as it is in algebra. < < parallelism between geometry and be pushed still further, owing to can algebra the fact that lengths, areas, volumes, and But the 240 INTRODUCTION TO MATHEMATICS angles are all measurable; so that, for exam the size of any length can be determined by the number (not necessarily integral) of times which it contains some arbitrarily ple, known and similarly for areas, volumes, trigonometrical formulae, given above, are examples of this fact. But it receives its crowning application in analytical and unit, angles. The This great subject is often mis as Analytical Conic Sections, thereby fixing attention on merely one of its sub It is as though the great science divisions. of Anthropology were named the Study of Noses, owing to the fact that noses are a prominent part of the human body. Though the mathematical procedures in geometry and algebra are in essence identical and intertwined in their, development, there is necessarily a fundamental distinction between the properties of space and the properties of number in fact all the essential difference between space and number. The and the numerosity "spaciness" of space of number are essentially different things, and geometry. named " " directly apprehended. None of the applications of algebra to geometry or of geometry to algebra go any step on the road to obliterate this vital distinction. One very marked difference between space and number is that the former seems to be so much less abstract and fundamental than the must be GEOMETRY 241 The number of the archangels can be counted just because they are things. When we once know that their names are Raphael, Gabriel, and Michael, and that these distinct latter. names represent distinct beings, we know without further question that there are three All the subtleties in the world of them. about the nature of angelic existences can not alter this fact, granting the premisses. But we are still quite in the dark as to their relation to space. Do they exist in at all? it is space Perhaps equally nonsense to say that they are here, or there, or any where, or everywhere. Their existence may simply have no relation to localities in space. Accordingly, while numbers must apply to all things, space need not do so. The perception of the locality of things would appear to accompany, or be involved in many, or all, of our sensations. It is in dependent of any particular sensation in the sense that it accompanies many sensations. But it is a special peculiarity of the things which we apprehend by our sensations. The direct apprehension of what we mean by the positions of things in respect to each other is a thing sui generis, just as are the appre hensions of sounds, colours, tastes, and smells. At first sight therefore it would appear that mathematics, in so far as it includes geometry in its scope, is not abstract in the sense in INTRODUCTION TO MATHEMATICS which abstractness is ascribed to it in Chapter I. This, however, is a mistake; the truth being that the "spaciness" of space does not enter into our geometrical reasoning at all. It enters into the geometrical intuitions of mathematicians in ways personal and pecu But what enter into liar to each individual. the reasoning are merely certain properties of things in space, or of things forming space, which properties are completely abstract in the sense in which abstract was defined in Chapter I; these properties do not involve any peculiar space-apprehension or spaceintuition or space-sensation. They are on exactly the same basis as the mathematical properties of number. Thus the space-intui tion which is so essential an aid to the study of geometry is logically irrelevant: it does not enter into the premisses when they are properly stated, nor into any step of the rea soning. It has the practical importance of an example, which is essential for the stimulation of our thoughts. Examples are equally neces sary to stimulate our thoughts on number. When we think of "two" and "three" we see strokes in a row, or balls in a heap, or some other physical aggregation of particular The peculiarity of geometry is the things. and overwhelming importance of the fixity one particular example which occurs to our GEOMETRY minds. 243 The abstract logical form of the fully stated is, any of things have such and such propositions collections when "If abstract properties, they also have such and such other abstract properties." But what appears before the mind s eye is a collection of points, lines, surfaces, and volumes in the this example inevitably appears, and the sole example which lends to the propo sition its interest. However, for all its over space: is whelming importance, Geometry, viewed it is but an example. as a mathematical science, is a division of the more general It may be called the science of order. science of dimensional order; the qualifica tion "dimensional" has been introduced because the limitations, which reduce it to only a part of the general science of order, are such as to produce the regular relations of straight lines to planes, and of planes to the whole of space. It is easy to understand the practical im portance of space in the formation of the scientific conception of an external physical world. On the one hand our space-percep tions are intertwined in our various sensa nor tions and connect them together. mally judge that we touch an object in the same place as we see it; and even in ab normal cases we touch it in the same space We as we see it, and this is the real fundamental INTRODUCTION TO MATHEMATICS 244 which ties together our various sensa Accordingly, the space perceptions are in a sense the common part of our sensa tions. Again it happens that the abstract fact tions. properties of space form a large part of what ever is of spatial interest. It is not too much to say that to every property of space there corresponds an abstract mathematical statement. To take the most unfavourable instance, a curve may have a special beauty of shape: but to this shape there will cor respond some abstract mathematical prop erties which go with this shape and no others. Thus to sum up: (1) the properties of space which are investigated in geometry, like those of number, are properties belong ing to things as things, and without special reference to any particular mode of appre hension; (2) Space-perception accompanies our sensations, perhaps all of them, certainly many; but it does not seem to be a necessary quality of things that they should all exist in one space or in any space. CHAPTER XVII QUANTITY IN the previous chapter we pointed out that lengths are measurable in terms of some unit length, areas in term of a unit area, and volumes in terms of a unit volume. When we have a set of things such as lengths which are measurable in terms of any one of them, we say that they are quantities of the same kind. Thus lengths are quantities of the same kind, so are areas, and so are volumes. But an area is not a quantity of the same kind as a length, nor is it of the same kind as a volume. Let us think a little more on what is meant by being measurable, taking lengths as an example. Lengths are measured by the foot-rule. By transporting the foot-rule from place to place we judge of the equality of lengths. Again, three adjacent lengths, each of one foot, form one whole length of three feet. Thus to meas ure lengths we have to determine the equality of lengths and the addition of lengths. When some test has been applied, such as the trans porting of a foot-rule, we say that the lengths are equal; and when some process has been 245 246 INTRODUCTION TO MATHEMATICS applied, so as to secure lengths being con tiguous and not overlapping, we say that the lengths have been added to form one whole length. But we cannot arbitrarily take any test as the test of equality and any process as the process of addition. The results of operations of addition and of judgments of equality must be in accordance with certain preconceived conditions. For example, the addition of two greater lengths must yield a length greater than that yielded by the addition of two smaller lengths. These pre conceived conditions when accurately formu lated may be called axioms of quantity. The only question as to their truth or falsehood which can arise is whether, when the axioms are satisfied, we necessarily get what ordinary people call quantities. If we do not, then the name "axioms of quantity" is ill-judged that is all. These axioms of quantity are entirely ab stract, just as are the mathematical proper ties of space. They are the same for all quantities, and they presuppose no special of perception. The ideas associated with the notion of quantity are the means by which a continuum like a line, an area, or a volume can be split up into definite parts. Then these parts are counted; so that num bers can be used to determine the exact prop erties of a continuous whole. mode QUANTITY 247 Our perception of the flow of time and of the succession of events is a chief example of the application of these ideas of quantity. We measure time (as has been said in con sidering periodicity) by the repetition of similar events the burning of successive inches of a uniform candle, the rotation of the earth relatively to the fixed stars, the rotation of the hands of a clock are all ex amples of such repetitions. Events of these types take the place of the foot-rule in rela tion to lengths. It is not necessary to assume that events of any one of these types are exactly equal in duration at each recurrence. What is necessary is that a rule should be known which will enable us to express the relative durations of, say, two examples of some type. For example, we may if we like suppose that the rate of the earth s rotation is decreasing, so that each day is longer than the preceding by some minute fraction of a second. Such a rule enables us to compare the length of any day with that of any other day. But what is essential is that one series of repetitions, such as successive days, should be taken as the standard series; and, if the various events of that series are not taken as of equal duration, that a rule should be stated which regulates the duration to be assigned to each day in terms of the dura tion of any other day. 248 INTRODUCTION TO MATHEMATICS What then are the requisites which such ought to have? In the first place it should lead to the assignment of nearly equal durations to events which common sense a rule judges to possess equal durations. which lengths, made days of and which made A rule different the speeds of ap violently parently similar operations vary utterly out of proportion to the apparent minuteness of their differences, would never do. Hence the first requisite is general agreement with com mon sense. But this is not sufficient abso lutely to determine the rule, for common sense is a rough observer and very easily satisfied. The next requisite is that minute adjustments of the rule should be so made as to allow of the simplest possible statements of the laws of nature. For example, astron omers tell us that the earth s rotation is slow ing down, so that each day gains in length by some inconceivably minute fraction of a second. Their only reason for their assertion (as stated more fully in the discussion of periodicity) is that without it they would have to abandon the Newtonian laws of motion. In order to keep the laws of motion simple, they alter the measure of time. This is a perfectly legitimate procedure so long as it is thoroughly understood. What has been said above about the ab stract nature of the mathematical properties QUANTITY 249 space applies with appropriate verbal changes to the mathematical properties of time. A sense of the flux of time accompa nies all our sensations and perceptions, and practically all that interests us in regard to time can be paralleled by the abstract mathe matical properties which we ascribe to it. Conversely what has been said about the two requisites for the rule by which we determine the length of the day, also applies to the rule for determining the length of a yard measure namely, the yard measure appears to retain the same length as it moves about. Accord ingly, any rule must bring out that, apart from minute changes, it does remain of in variable lengths. Again, the second requisite is this, a definite rule for minute changes shall be stated which allows of the simplest For ex expression of the laws of nature. re second the with in accordance ample, to are measures the supposed yard quisite expand and contract with changes of tem perature according to the substances which they are made of. Apart from the facts that our sensations are accompanied with perceptions of locality and of duration, and that lines, areas, volumes, and durations, are each in their way quanti ties, the theory of numbers would be of very subordinate use in the exploration of the laws As it is, physical science of the Universe. of 250 INTRODUCTION TO MATHEMATICS reposes on the main ideas of number, quan The mathematical tity, space, and time. sciences associated with them do not form the whole of mathematics, but they are the substratum of mathematical physics as at present existing. BIBLIOGRAPHY NOTE ON THE STUDY OF MATHEMATICS THE difficulty that beginners find in the study of this due to the large amount of technical detail which has been allowed to accumulate in the elementary text books, obscuring the important ideas. The first subjects of study, apart from a knowledge of arithmetic which is presupposed, must be elementary geometry and elementary algebra. The courses in both subjects should be short, giving only the necessary ideas; the algebra should be studied graphically, so that in prac tice the ideas of elementary coordinate geometry are also being assimilated. The next pair of subjects should be elementary trigonometry and the coordinate geometry of the straight line and circle. The latter subject is a short one; for it really merges into the algebra. The student is then prepared to enter upon conic sections, a very short course of geometrical conic sections and a longer one of science is analytical conies. But in all these courses great care should be taken not to overload the mind with more detail than is necessary for the exemplification of the fundamental The ideas. differential calculus and afterwards the integral same system. A good teacher will already have illustrated them by the consideration of special cases in the course on algebra and coordinate geometry. Some short book on three dimensional geometry must also be read. calculus now remain to be attacked on the This elementary course of mathematics is sufficient for It is also the necessary of professional career. preliminary for any one wishing to study the subject for its intrinsic interest. He is now prepared to commence on a more extended course. He must not, however, hope to be some types 251 BIBLIOGRAPHY 252 able to master it as a whole. The science has grown tosuch vast proportions that probably no living mathe matician can claim to have achieved this. Passing to the serious treatises on the subject to be read after this preliminary course, the following may be men tioned: Cremona s Pure Geometry (English Translation, Clarendon Press, Oxford), Hobson s Treatise on Trigono metry, Chrystal s Treatise on Algebra (2 volumes), Salmon s Conic Sections, Lamb s Differential Calculus, and some book on Differential Equations. The student will probably not desire to direct equal attention to all these subjects, but will study one or more of them, according as his interest He will then be prepared to select more ad dictates. vanced works for himself, and to plunge into the higher he on the theory of Fractions or the Complex Variable; if he prefers to specialize in Geometry, he must now proceed to the standard treatises on the Analytical Geometry,! of three parts of the subject. should If his interest lies in analysis, now master an elementary treatise dimensions. But at this stage of his career hi learning he will not require the advice of this note. I have deliberately refrained from mentioning any elementary works. They are very numerous, and of various merits, but none of such outstanding superiority as to require special mention by name to the exclusion of all the others. INDEX Abel, 156 Abscissa, 95 Cantor, Georg, 79 180 etseqq. Circular Cylinder, 143 Clerk Maxwell, 34, 35 Columbus, 122 Circle, 120, 130, Abstract Nature of Geo metry, 242 et seqq. Abstractness (defined), 9, 13 Adams, 220 Addition Theorem, 212 Ahmes, 71 Alexander the Great, 128, 129 Algebra, Fundamental Laws of, 60 Ampere, 34 Analytical Conic Sections, 240 Apollonius of Perga, 131, 134 Approximation, 197 et seqq. Arabic Notation, 58 et seqq. Archimedes, 37 et seqq. Argument of a Function, 146 Aristotle, 30, 43, 128 Astronomy, 137, 173, 174 Axes, 125 Axioms of Quantity, 246 et seqq. Compact Series, 76 Complex Quantities, 109 Conic Sections, 128 et seqq. Constants, 69, 117 Continuous Functions, 150 et seqq.; 162 (defined) Convergent, 203 et seqq. Coordinate Gemeotry, 112 et seqq. Coordinates, 95 Copernicus, 45, 137 Cosine, 182 et seqq. Coulomb, 33 Cross Ratio, 140 Darwin, 138, 220 Derived Function, 234 Descartes, 122, 218 Differential 95, 113, Calculus, 116, 217 et seqq. Differential Coefficient, Directrix, 135 Bacon, 156 W. W. R., 58 Beaconsfield, Lord, 40 Berkeley, Bishop, 226 Discontinuous 150 et seqq. Distance, 30 Divergent, 203 Ball, Bhaskara, 58 253 234 Functions, ei seqq. INDEX 254 Dynamical Explanation, 47 et seqq. Dynamics, 30, 43 et seqq. 13, 14, Electric Current, 33 Electricity, 32 et seqq. 31 Electromagnetism, et 45, Imaginary Quantities, 109 Incommensurable Ratios, 211 Infinitely Small Quantities, 72 Exponential Series, seqq Form, Algebraic, 66 226 et seqq. Kepler, 45, 46, 137, 138 Kepler et seqq. t 117 Fourier s Theorem, 191 82, Fractions, 71 et seqq. Franklin, 32, 122 Function, 145 et seqq. 42 et seqq., 122 Galvani, 33 Generality in Mathematics, 82 Geometrical et seqq. Integral Calculus, 222 Interval, 158 et seqq. Faraday, 34 Fermat, 218 Fluxions, 219 Focus, 120, 135 Force, 30 Galileo, 30, et et 130 120, Euclid, 114 et Imaginary Numbers, 87 seqq. seqq. Ellipse, Herz, 35 Hiero, 38 Hipparchus, 173 Hyperbola, 131 et seqq. s Laws, 138 Laputa, 10 Laws of Motion, 167 et seqq. 248 Leibniz, 16, 218 et seqq. Leonardo da Vinci, 42 Leverrier, 220 Light, 35 Limit of a Function, 227 et seqq. Limit of a Series, 199 et seqq. 206 Series, et seqq. Geometry, 36, 236 et Gilbert, Dr., 32 Graphs, 148 et seqq. Limits, 77 Locus, 121 et seqq., 141 seqq. Gravitation, 29, 139 Halley, 139 Harmonic Analysis, 192 Harriot, Thomas, 66 Macaulay, 156 Malthus, 138 Marcellus, 37 Mass, 30 Mechanics, 46 Menaechmus, 128, 129 Motion, First Law, of 43 INDEX Neighbourhood, 159 et seqq. Newton, 10, 16, 30, 34, 37, 38, 43, 46, 139, 218 et seqq. Non-Uniform Convergence, 208 Relations between ables, 18 et seqq. Resonance, 170, 171 Rosebery, Lord, 194 Vari et seqq. Normal Error, Curve of, 214 Oersted, 34 Order, 194 et seqq. Order, Type of, 75 Parabola, 131 et seqq. Parallelogram Law, 51 seqq., 99, 126 Parameters, 69, 117 Pencils, 140 Period, 170, 189 et seqq. et et seqq., William, 194 Pizarro, 122 Plutarch, 37 Positive and Negative Num et seqq. Projective Geometry, 139 Ptolemy, 137, 173 Pythagoras, 13 from), et seqq., 237 159 229 Steps, 79 et Stifel, 85 tion, et seqq., seqq., et seqq. seqq., 201 et 96 Stokes, Sir George, 210 Sum to Infinity, 201 Surveys, 176 Swift, 10 et et seqq. Tangents, 221, 222 Theorem, 156, Taylor s 157 Time, 166 et seqq., 247 et seqq. et seqq. of, et seqq. Triangle, 176 et seqq., 237 Triangulation, 177 Trigonometry, 173 et seqq. Uniform Convergence, 208 et seqq. Real Numbers, 73 Rectangle, 57 177 Transportation, Vector Rate of Increase of Func tions, 220 et seqq. Ratio, 72 (quotation et 217 54 Quantity, 245 194 seqq., Sine, 182 et seqq. Specific Gravity, 41 Squaring the Circle, 187 Standard of Approxima Pappus, 135, 136 bers, 83 et Similarity, et seqq. Origin, 95, 125 164 Map, 178 210 74 Shelley et seqq., Ordered Couples, 93 Ordinate, 95 Periodicity, 188, 216 Scale of a Seidel, Series, 196 Pitt, 255 et seqq. et seqq. Unknown, The, 17, 23 256 INDEX Value of a Function, 146 Variable, The, 18, 24, 49, 82, 234, 239 Variable Function, 147 Vectors, 51 et seqq., 85, 96 Vertex, 134 Volta, 33 Wallace, 220 Weierstrass, 156, 226, 228 Zero, 63 et seqq., 103 THE HOME UNIVERSITY MODERN LIBRARY OF KNOWLEDGE EDITED BY PROFESSOR SIR J. ARTHUR THOMSON PROFESSOR GILBERT MURRAY THE RT. HON. PROFESSOR W. / H. A. T. L. FISHER BREWSTER titles listed in the Home University Library 1pHE * are not written for are reprints: they well-known especially by recognized authorities in their respective fields. Some of the most distin guished names in America and England will be found among the authors of the following books. 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H. Scott, LL.D., F.R.S., President of the Linnean Society of London. The story of the development of flowering plants, from the earliest zoological times, unlocked from technical language. 12. THE ANIMAL WORLD. By Prof. F. W. Gamble, F.R.S. 14. 15. EVOLUTION. By Prof. Sir J. Arthur Thomson and Prof. Patrick Geddes. Explains to the layman what the title means to the scientific world. INTRODUCTION TO MATHEMATICS. By Pro fessor A. N. Whitehead, D.Sc., F.R.S., author of versal 17. "Uni Algebra." CRIME AND INSANITY. By F.R.C.P., F.R.C.S., author of Dr. C. A. Mercier, and Criminals," "Crime etc. 21. AN INTRODUCTION TO SCIENCE. By Prof. Sir J. Arthur Thomson, LL.D., Science Editor of the Home University Library. For those unacquainted with the scientific volumes in the series this should prove an excellent introduction. 23. ASTRONOMY. By A. R. Hinks, Chief Assistant at the Cambridge Observatory. "Decidedly original in sub stance, and the most readable and informative little book on modern astronomy we have seen for a long time." Nature. 24. 37. 41. PSYCHICAL RESEARCH. By Sir W. F. Barrett, F.R.S., formerly President of the Society for Psychical Research. ANTHROPOLOGY. By R. R. Marett, D.Sc., F.R.A.I., Reader in Social Anthropology, Oxford. Seeks to plot out and sum up the general series of changes, bodily and mental, undergone by man in the course of his So enthusiastic, so clear and witty, and tory. "Excellent. so well adapted to the general reader." American Library Association Booklist. PSYCHOLOGY, THE STUDY OF BEHAVIOUR. Professor William McDougall, F.R.S., Reader in Mental Philosophy, Oxford University. A well-digested By summary of literary 42. the essentials of the science put in excellent form by a leading authority. THE PRINCIPLES OF PHYSIOLOGY. By Prof. A compact statement by the G. McKendrick. Emeritus Professor at Glasgow, for uninstructed readers. J. 43. MATTER AND ENERGY. By F. Soddy, F.R.S., Professor of Inorganic and Physical Chemistry in the Can hardly be sur University of Oxford. "Brilliant Sure to attract attention." New York Sun. passed. 53. ELECTRICITY. By Gisbert Kapp, Late Professor of Electrical Engineering, University of Birmingham. 54. THE MAKING OF THE EARTH. By J. W. Gregory, F.R.S., Professor of Geology, Glasgow Uni versity. 38 maps and figures. Describes the origin of the earth, the formation and changes of its surface and struc ture, and 56. geological history, the first appearance of influence upon the globe. its its life, MAN: A HISTORY OF THE HUMAN BODY. Sir A. Keith, F.R.S., Hunterian Professor, Royal College of Surgeons of England. Shows how the human body developed. By 63. THE ORIGIN AND NATURE OF LIFE. By Pro fessor Benjamin Moore. 68. DISEASE AND ITS CAUSES. By W. T. Council man, M.D., LL.D., Professor of Pathology, Harvard University. 71. 74. 85. PLANT LIFE. By Sir J. B. Farmer, D.Sc., F.R.S., Professor of Botany in the Imperial College of Science, London. This very fully illustrated volume contains an account of the salient features of plant form and function. NERVES. By David Fraser Harris, M.D., Professor of Physiology, Dalhousie University, Halifax. Explains in nontechnical language the place and powers of the nervous system. SEX. By Geddes, 90. Profs. Sir J. joint authors of CHEMISTRY. in. "The Evolution of Sex." By Raphael Meldola, F.R.S., Late Finsbury Technical College. Revised by Alexander Findlay, D.Sc., F.I.C., Profes Pre sor of Chemistry in the University of Aberdeen. sents the way in which the science has developed and the stage it has reached. Professor 107. Arthur Thomson and Patrick of Chemistry, AN INTRODUCTION TO THE STUDY OF HEREDITY. By E. W. MacBride, D.Sc., Professor of Zoology in the Imperial College of Science and Tech nology, London. BIOLOGY. By Patrick Geddes. Profs. Sir J. Arthur Thomson and BACTERIOLOGY. ii2. By Prof. Car! H. Browning, F.R.S. MICROSCOPY. By 115. Robert M. Neill, Aberdeen Uni Microscopic technique subordinated to results of investigation and their value to man. versity. EUGENICS. 116. By Professor A. M. Carr-Saunders. Biological problems, together with the facts and theories of heredity. AND GAS GASES. By R. M. Caven, D.Sc., F.I.C., Professor of Inorganic and Analytical Chemistry in the The chemical and Royal Technical College, Glasgow. physical nature of gases, both in their scientific and his 119. torical aspects. BIRDS, AN INTRODUCTION TO ORNITHOL OGY. By A. L. Thompson, O.B.E., D.Sc. A general 122. account of the characteristics, mainly of habit and be havior of birds. SUNSHINE AND HEALTH. By 124. Macfie, M.B.C.M., LL.D. Light and Ronald Campbell its relation to man treated scientifically. INSECTS. 125. By Frank Balfour-Browne, F.R.S.E., Professor of Entomology in the Science and Technology, London. Imperial College of TREES. By Dr. MacGregor Skene, D.Sc., F.L.S. Senior Lecturer on Botany, Bristol University. A concise study of the classification, history, structure, architecture, growth, enemies, care and protection of trees. Forestry and economics are also discussed. 126. 138. THE LIFE OF THE CELL. By David Lands- B.Sc., Ph.D., Lecturer in Biochem istry, McGill University. borough Thomson, 142. 35. VOLCANOES. By G. W. Tyrrell, A.R., C.Sc., Ph.D., F.G.S., F.R.S.E., Lecturer in Geology in the University of Glasgow. PHILOSOPHY AND RELIGION THE PROBLEMS OF PHILOSOPHY. By The Hon. Bertrand Russell, F.R.S., Lecturer and Late Fel low, Trinity College, Cambridge. 44. BUDDHISM. By Mrs. Rhys Davids, Lecturer on Indian Philosophy, Manchester. 46. ENGLISH SECTS: A HISTORY OF NONCON FORMITY. By The Rev. W. B. Selbie, Principal of Mansfield College, Oxford. 50. 52. 55. THE MAKING OF THE NEW TESTAMENT. By B. W. Bacon, D.D., LL.D., Professor of New Tes tament Criticism, Yale. An authoritative summary of the results of modern critical research with regard to the origins of the New Testament. ETHICS. By Professor G. E. Moore, D.Litt., Lec turer in Moral Science, Cambridge. Discusses what is right and what is wrong, and the whys and wherefores. MISSIONS: MENT. By THEIR RISE AND DEVELOP Mrs. Mandell Creighton, author of The author sions have done more to civilize human agency. tory of 60. 65. England." "His seeks to prove that mis the world than any other COMPARATIVE RELIGION. By Prof. J. Estlin Carpenter, LL.D. "One of the few authorities on this subject compares all the religions to see what they have to offer on the great themes of religion." Christian Work and Evangelist. THE LITERATURE OF THE OLD TESTA MENT. By George F. Moore, Professor of the His tory of Religion, Harvard University. popular work of the highest order. Will be profitable to anybody who cares enough about Bible study to read a serious book on the subject." American Journal of Theology. "A 69. A HISTORY OF FREEDOM OF THOUGHT. By John B. Bury, M.A., LL.D., Late Regius Professor of Modern History in Cambridge University. Summarizes the history of the long struggle between authority and reason and of the emergence of the principle that coercion of opinion is a mistake. 88. RELIGIOUS DEVELOPMENT BETWEEN OLD AND NEW TESTAMENTS. By The Ven. R. H. Charles, D.D., F.B.A., Canon of Westminster. Shows religious and ethical thought between 180 B.C. and 100 A.D. grew naturally into that of the New Testament. how 96. A HISTORY OF PHILOSOPHY. By Clement 130. C. J. Professor Webb, F.B.A. JESUS OF NAZARETH. By The Rt. Rev. Charles Gore, D.D., formerly Bishop of Oxford. SOCIAL SCIENCE i. PARLIAMENT. TION, ITS HISTORY, AND PRACTICE. By CONSTITU Sir Courtenay P. Ilbert, G.C.B., K.C.S.I., late Clerk of the House of Commons. THE STOCK EXCHANGE. By F. W. Hirst, for merly Editor of the London Economist. Reveals to the nonfinancial mind the facts about investment, speculation, and the other terms which the title suggests. 6. IRISH NATIONALITY. By Mrs. J. R. Green, D.Litt. A brilliant account of the genius and mission of the Irish people. "An entrancing work, and I would ad vise everyone with a drop of Irish blood in his veins or a vein of Irish sympathy in his heart to read New York Times Review. (Revised Edition, 1929.) SOCIALIST 10. By The Rt. 5. it." THE Hon. 11. MOVEMENT. J. Ramsay Macdonald, M.P. THE SCIENCE OF WEALTH. By of Poverty/ A J. A. Hobson, study of the struc ture and working of the modern business world. 16. LIBERALISM. By Prof. L. T. Hobhouse, LL.D., author of "Democracy and Reaction." masterly phil osophical and historical review of the subject. author of "Problems A 23. THE EVOLUTION OF INDUSTRY. MacGregor, Drummond Professor D. H. Economy, By in Political University of Oxford. An outline of the recent changes that have given us the present conditions of the working classes and the principles involved. 29. ELEMENTS OF ENGLISH LAW. By W. M. Geldart, B.C.L., Vinerian Professor of English Law, Oxford. Revised by Sir William Holdsworth, K.C., D.C.L., LL.D., Vinerian Professor of English Law, Uni versity of Oxford. A simple statement of the basic prin ciples of the English legal system on which that of the United States is based. 32. 49. THE SCHOOL: AN INTRODUCTION TO THE STUDY OF EDUCATION. By J. J. Findlay, M.A., formerly Professor of Education, Manchester. Presents the history, the psychological basis, and the theory of the school with a rare power of summary and suggestion. ELEMENTS OF POLITICAL ECONOMY. By Sir S. J. Chapman, late Professor of Political Economy and Dean of Faculty of Commerce and Administration, University of Manchester. 77. CO-PARTNERSHIP AND PROFIT-SHARING. By Aneurin Chairman, Executive Com Explains the various types of co-partnership and profit-sharing, and the details of in now force in gives arrangements many of Williams, late mittee, International Co-operative Alliance, etc. the great industries. 79. UNEMPLOYMENT. By A. C. Pigou, M.A., Pro fessor of Political Economy at Cambridge. The meaning, measurement, distribution and effects of unemployment, its relation to wages, trade fluctuations and disputes, and some proposals of remedy or 80. COMMON SENSE relief. IN LAW. By Prof. Sir Paul Vinogradoff, D.C.L., LL.D. Social and Legal RulesLegal Rights and Duties Facts and Acts in Law Legis lationCustomJudicial Precedents Equity The Law of Nature. 91. THE NEGRO. By W. E. of "Souls man 98. of Black in Africa, Folks," Burghardt DuBois, author etc. A history of the black America and elsewhere. POLITICAL THOUGHT: FROM HERBERT SPENCER TO THE PRESENT DAY. By Pro fessor Ernest Barker, D.Litt., LL.D. 99. POLITICAL THOUGHT: THE UTILITARIANS, FROM BENTHAM TO J. S. William L. Davidson, LL.D. 103. MILL. By Professor ENGLISH POLITICAL THOUGHT. From Locke to Bentham. By Harold J. Laski, Professor of Politi cal Science in the London School of Economics. 113. 118. ADVERTISING. By Sir Charles Higharn. BANKING. By Dr. Walter Leaf, late President, In Bankers; President, International Chamber Commerce. The elaborate machinery of the financing stitute of o: o: industry. 123. COMMUNISM. By Harold J. Laski, Professor o Political Science at the University of London. The autho: tries to state the communist "theses" in such a way tha even its advocates will recognize that an opponent cai summarize them 131. fairly. INDUSTRIAL PSYCHOLOGY. Edited by Dr Na Charles S. Myers, G.B.E., F.R.S., Director of the tional Institute of Industrial Psychology in England. Th only comprehensive study of the human factor in industry 133. THE GROWTH THOUGHT. By OF INTERNATIONAI F. Melian Stawell. 139- LIQUOR CONTROL. An 140. By George E. G. Catlin. impartial and comprehensive study of the subject. RACES OF AFRICA. By F.R.S. C. G. Seligman, F.R.C.P. QA W53 1911 Whitehead, Alfred North An introduction to mathematics. PLEASE CARDS OR DO NOT REMOVE SLIPS UNIVERSITY FROM THIS OF TORONTO POCKET LIBRARY