aha ted CR rt pa Brrr v A fo tt 2% Pot i oi adn ath Chart 3 eran Soi ’ Sr SRST A fio ed 0 Lo Te tc ai! edd 3 2 an THE STORY OF MATHEMATICS - By DENHAM LARRETT B.A.(Cantab.) With a Foreword by C. G. DARWIN, F.R.S. Tait Professor of Natural Philosophy in the University of Edinburgh Late Fellow and Lecturer of Christ's College, Cambridge LONDON : ERNEST BENN LTD. 8 BOUVERIE STRERT, E.C... 1926 Knowledge and thought are at once the delight and the prerogative of man, but they are also a part of the wealth of nations and often afford to them an abundant indemni- fication for the more sparing bestowal of natural riches. VoN HUMBOLDT. As the eye rejoices to receive the light, the ear to hear sweet music ; so the mind, which is the man, rejoices to discover the secret works, the varieties and beauties of nature. Bacon. Printed in Great Britain by The Riverside Press Limited Edinburgh FOREWORD OxE of the hardest tasks that an expert in any subject can undertake is to try to explain to the layman what his subject deals with, and why he makes such a fuss about it. This is nowhere more true than in the Exact Sciences. It is easy to be- lieve that the botanist has undertaken a long task, for after all is he not expected to know all about every part of every plant in every region of the world ; so too in medicine we can understand that there is a good deal to learn both about the dis- eases of the human body and about the effect of drugs on it—for these are, so to speak, sciences of the collector. But how can this be so in physics, for the physicist soon tells us that there are only two kinds of thing in the world, protons and elec- trons—and surely a few pages should dispose of such a simple business! It is even harder to ex- plain to the layman why there can exist so many vast tomes on mathematics, and the best way is to give him a history of the subject. This is what Mr Larrett has done, and the reader of his work will realise how it comes about that there is so much to say. The author takes us through the surveying of the Egyptians to the geometry of the Greeks, and describes how these were blended with the algebra of the Arabs to produce the great edifice of modern mathematics. It would of course be impossible to explore all the ramifications of the subject, but he has made a very judicious selection which will bring the reader into 5 6 FOREWORD contact with most of its leading branches. In particular he presents both the fields in which physics has made its most striking recent advances —mnamely, atomic theory and relativity. To rela- tivity he wisely devotes some considerable space, for it is thoroughly typical of the whole development of mathematics. It is essentially an applied” subject arising out of experiment, and yet a great part of it was developed in advance by the ¢ pure ” mathematician, Riemann, as a speculation in the realms of abstract thought. In an era when applied science is entering more and more into his circle of interests, the ordinary man will realise the fundamental position of mathe- matics, and may well wish to gain some idea of its origins. Mr Larrett’s book provides for this desire in a very adequate and enjoyable way, and will no doubt stimulate a few to explore more deeply into a noble and fascinating subject. C.G.D PREFACE Trae mere mention of the word mathematics to the majority of people conjures up visions of dreary text-books and “sums” innumerable that have to be done. It has to be admitted that over- coming the technical side of the subject is a task calling for serious and close application, and it is perhaps unfortunate that the inner significance of mathematics is only realised to the full when this is accomplished. Yet there is more than technique in mathematics. There is a story also, and in this little book I have tried to tell something of this story in a manner that should at least give a meaning to this inner side of the subject and at the same time spare the reader the terrors of symbols, formule and the other intricacies that beset the path of the mathematical student. After all, mathematics is at the root of all our activities. We are all mathematicians from the day we receive our first allowance of pocket-money until we render our last income-tax return. Com- merce, Engineering, Building Construction and a thousand other activities are touched on in the purview of life that mathematics takes. ~ To-day advanced mathematics is approaching Philosophy and Metaphysics. Even the psych- ologist has called to his aid the services of the mathematician, and, as will be seen later, this service is providing a basis for a complete and final union of the sciences for a fuller description 7 - 8: PREFACE of the universe. These are aspects of the sub- ject which both the layman and the expert can appreciate. ~ I must thank Prof. C. G. Darwin, F.R.S., for very kindly contributing the Foreword. To him and many other brilliant men I owe a deep debt of gratitude for creating a love for this fascinating subject, and, if this volume in any way helps to widen public interest in the subject, to them the credit is due. DL. November 1925. TABLE OF CONTENTS PAGE FOREWORD . : ‘ . «5 PREFACE . . . . . i Time CHART : . ‘ riots. Just as the trade guilds attained to some measure of self-government, regulating the conduct of their members, supervising the conditions of their industry and receiving the privileges of the Law Merchant, so these early guilds of students sought similar privileges and were in due time incorporated, Oxford receiving official recognition in 1214, Cambridge in 1231. Although these guilds were THE MEDIEVAL UNIVERSITIES 39 not ecclesiastical foundations, they were not infre- quently under the protection of the neighbouring bishop or abbot, who sometimes, in the early days, filled the office of chancellor. We must now examine the position of science at this time. The Middle Ages were the fabulous era of science. Alchemy was the study that held men’s minds, and the search for the philosopher’s stone and the elixir of life were the objects of scientific devotion. These experiments were con- ducted in mysterious fashion and the results recorded in most enigmatic language. In such an age, the charlatan had ample scope for his activities, and although there may have been a few genuine seekers after truth, there were so many impostors that both Church and State had to put down this growing evil. The alchemists, like the seekers after perpetual motion, failed in their quest, but in their search for the impossible made many discoveries of considerable importance. With the revival of learning came a healthier out- look. Fairies and demons alike were cast out, and more rational ideas adopted. The course of study in these days was a curious one. The “trivium” of Latin grammar, rhetoric and logic formed the foundation of all studies. Some passable acquaintance with these satisfied the requirements of most of those who had any desire to learn. A few then devoted themselves, either to the study of one of the ¢ faculties?” of theology, law or medicine, in preparation for one of the learned professions, or else to the 40 THE STORY OF MATHEMATICS ¢quadrivium ” of arithmetic, geometry, music and astronomy. The standard demanded in these subjects was very low. Music was little more than the chanting of the church services, and astronomy was almost confined to the calculation of Easter. Mathematical work at this time con- sisted mainly of keeping the accounts of the monastery or similar foundation and the preparation of almanacs showing the different feasts of the Church. Roger Bacon, a Franciscan monk who lectured at Oxford in the late thirteenth century, tells us that few of his pupils read further than the first five propositions of Euclid’s Elements. He, himself, was one of the few men of this age who had a real scientific outlook. ¢ The knowledge dis- played by Roger Bacon and by Albertus Magnus, even in the mixed mathematics, under every dis- advantage from the imperfection of instruments and the want of recorded experience, is sufficient to inspire us with regret that their contemporaries were more inclined to astonishment than to emu- lation. These inquiries indeed were subject to the ordeal of fire, the great purifier of books and men; for if the metaphysician stood a chance of being burned as a heretic, the natural philosopher was in not less jeopardy as a magician.” It will be seen that in such an atmosphere mathematical studies could scarcely reach a very high level. Even Roger Bacon, who so clearly laid down the lines which scientific investigation should take, was infected with some of the super- stitious notions of the age. He believed, with his THE MEDIEVAL UNIVERSITIES 41 contemporary astrologers, that the stars exercised an influence on the human body. Again, in one of his books, he publishes the opinion—probably well accepted at the time—that it is only a suc- cession of accidents which prevents Nature from continually making gold and these create the baser metals which are found in the earth. In such an age of superstition, when pure scholarship was at so low an ebb, we must indeed be thankful for the few whose brilliance shines up in contrast with the ignorance of their fellows. It was before the invention of printing, when means of communication were none too easy, and consequently the works of the Arabs penetrated Europe only very slowly. Nevertheless, with increasing commerce the merchant classes began to recognise the advantages of the new notation, and the fifteenth century saw it in common use. It is interesting to note that at this time the merchants of Florence were beginning to use that method of book-keeping known in these days as the ‘“double-entry” system. Already, bills of exchange and similar negotiable instruments were in common use, and modern business pro- cedure can be said to have commenced. When the clouds of ignorance were at last dispelled before the enthusiasm of the Renaissance there came a new outlook; the work of the ancients was studied with a new interest and the founda- tions of science began to be laid on the surer bases of experiment and reason rather than speculation and intuition, CHAPTER VII BRIGHTER DAYS—THE RENAISSANCE Science is the great antidote to the poison of enthusiasm and superstition. Apam Smith. THe last centre of Greek learning came to an end in 1453, when the Turks captured Constantinople. The refugees fled westwards, bringing with them originals and translations of the great Greek writers, thus giving a fresh impetus to the cause of science. At the same time, the invention of printing rendered it possible to publish the discoveries of the period at a much quicker rate. These two factors were largely re- sponsible for that great advance in thought usually described as the Renaissance. As far as progress in mathematics is concerned, the characteristic of the fifteenth century is the advance made in arithmetic and algebra. This was the period that saw the introduction of our familiar plus and minus signs. How these signs came into use is a matter of considerable con- jecture, and several very interesting theories have been put forward. The one which seems to find most favour is usually called the ¢ warehouse” theory. It ascribes the origin of these signs to the practice of merchants of the time in signifying the excess weight of their chests of merchandise by a +, and using — to denote any deficiency, the sign in each case being followed by the actual 42 BRIGHTER DAYS—THE RENAISSANCE 43 amount of the surplus or shortage. The con- venience of this method led to its introduction into the accounts of the merchants and ultimately into general scientific work. Prior to this, various means had been adopted for signifying the pro- cesses of addition and subtraction, but these soon gave way to the simpler new method. There are in this century some rather interest- ing personages that we should be well to consider. The best-known is probably Leonardo da Vinci, the famous artist. His mathematical work is mainly concerned with mechanical questions, and the application of mathematics to problems con- cerning levers, friction, etc. Since the days of Archimedes, such problems had scarcely been considered, so that this renewed interest in applied mathematics is noteworthy. Leonardo made no very serious contribution to the subject, but the suggestiveness of his studies kept it alive in people’s minds. Two rather interesting characters in the early sixteenth century must claim our attention for a moment. The first is Stifel, the author of a small algebra book. He was originally an Augustine monk, who renounced his faith and joined with Martin Luther. Un- fortunately, Stifel believed that mathematical analysis could be applied to religious questions, and he was rash enough to publish some of his interpretations of the Scriptures, including a prophecy of the date when the world was to come to an end. The latter nearly cost him his life. 44 THE STORY OF MATHEMATICS The second remarkable mathematician of the age is Cardan, whose name students will always associate with a particular method of solving the cubic equation. Although a very brilliant man, as his published works clearly show, Cardan was a very remarkable example of the alliance between genius and madness. The solution of the cubic he stole from his contemporary Tartaglia, with whom a bitter controversy subsequently developed, and, after a life of debauchery, he committed suicide, having just been appointed astrologer to the Papal Court. The close of the Renaissance saw two very important developments of our subject. The first was the discovery of logarithms by John Napier in 1617, and the preparation of the first tables of logarithms by John Briggs, who was a professor at Oxford from 1620 to his death in 1631. It is impossible to exaggerate the importance of the work of these two men. Tedious calculations could now be performed rapidly, and it is not sur- prising, therefore, to see how quickly this new system of calculation was accepted throughout Europe. Before we close this chapter, brief mention must be made of Galileo (1564-1642) and Kepler (1571-1630), although it is rather on account of their astronomical discoveries that they are well known. The former was for a time professor of mathematics at Pisa, and from its famous lean- ing tower conducted a number of experiments on falling bodies, thus laying the foundation of the BRIGHTER DAYS—THE RENAISSANCE 45 modern study of dynamics. The latter was the first to discover the laws of planetary motion, and to establish the fact that the planets described elliptic orbits round the sun. It was largely on the work of these two that Newton built up the modern theories on these subjects. CHAPTER VIII THE FOUNDATIONS OF MODERN MATHEMATICS— NEWTON AND HIS AGE “I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”’ Sir Isaac Newton. » THE seventeenth and eighteenth centuries form a most important epoch in the history of Mathe- -matics. There is no clear-cut line of demarcation between the work done in the preceding century and this era, such as we noticed in the case of the Greek schools. We must, however, consider the progress of our subject during this period in a chapter apart, since it was in this age that the foundations of modern mathematics were laid. Let us, however, cast our eyes backwards and see how we stand at present. It will be re- membered that during the previous few centuries advances have been made in arithmetic and algebra, and, although the Greek geometry was familiar, practically no advances in this subject had been made at all. The stage has now been reached when these become amalgamated and the study of a new subject—analytical geometry—commenced. This may be described, in a general way, as the application of algebraical methods to geometrical problems. The method is quite well known, appearing as it does in all graphical representa- 46 MODERN MATHEMATICS 47 tions of varying quantities such as temperatures, pressures, and statistical results of all kinds. Two straight lines of indefinite length are taken, usually at right angles to each other, and by means of suit- able scales on these “axes,” points at appropriate distances from each are joined by a curve and a “graph” obtained. Easily drawn and producing accurate results, such methods find wide applica- tion in these days, not only to pure academic studies, but also to the practical problems confronting engineers and architects. The use of this method is generally attributed to René Descartes (1596-1650), an officer in the French Army, who on his retirement devoted himself to philosophical and mathematical studies. Not only did he use the means of graphical representation, but also he was the first to show that the geometrical properties of the curve he thus obtained could be described by an algebraical equation, and conversely, the algebraical relation- ships of different points on the curve could be interpreted geometrically. The accompanying figure (p. 48) will perhaps illustrate this. The two curves that are drawn are represented by the algebraical equations wy=6 and x+y=3. (It should be noted that a “curve” in its mathe- matical sense includes the straight line.) Speak- ing algebraically, we say that these equations are satisfied by the values ¥=2 or 3 and y=3 or 2. Speaking geometrically, we say, that the two curves represented by these equations intersect in two points A and B such that, the “co-ordinates” of 48 THE STORY OF MATHEMATICS A are 2 and 3 (ie. A is a point 2 units from the y-axis and 3 units from the x-axis), and the co- ordinates of B are 3 and 2. The extension of methods such as these have found wide applica- tion in all branches of mathematics, and thus it is impossible to exaggerate the importance of Descartes’ work, although we must add that his Géométrie was not remarkable for its lucidity of expression. i IN xe Cr id : eo 7 2 2 ~AXi. 3 frm mit mie emer «0 Fic. 3 Having briefly mentioned the development ot this new branch of our subject we must pass on to one who is perhaps the greatest of all scientists —Sir Isaac Newton (1642-1726). He was the son of a small farmer in Lincolnshire, and was originally destined to follow in his father’s footsteps, but the great mechanical ingenuity he displayed whilst a boy at school at Grantham so impressed his family, that it was resolved to send him to Trinity MODERN MATHEMATICS 49 College, Cambridge. He quickly mastered the books prescribed for his study and his great ability was soon recognised. A few years later, his tutor, Barrow, resigned the Lucasian Professorship of Mathematics in favour of his more brilliant pupil. For some eighteen years Newton lectured to ever-increasing classes of students and conducted numerous researches into every branch of science. There was scarcely a subject in which he was not interested and very few to which he did not con- tribute. In optics, he discovered that a ray of light on passing through a glass prism was decomposed into its component colours, forming a spectrum.” This led to his explanation of rainbows. His interest in the motions of the stars and the study of the universe led to the construction of a new type of telescope by which he observed the motion of the celestial bodies. Newton devoted consider- able time to such questions, and we shall do well to examine some of the results he obtained. There are two main problems to consider. First, how is the motion of the celestial bodies to be explained ; second, how does the light from the sun and stars reach the earth. It was upon the consideration of these two main questions that the majority of Newton’s work was done, the first leading to an investigation into gravitation and its effects, the second to Newton’s theory of light. This latter theory was later shown to be unsound, but it occupied a central place in science for many years. According to it, any body from which light was emitted, such as the sun, sent out a stream D 50 THE STORY OF MATHEMATICS of particles, or corpuscles as they were usually called, and these, on hitting the eye, produced the phenomenon described as light. On this hypothesis, Newton worked out all the well-known laws of optics, including reflection, refraction and the decomposition of light into the spectrum. Such a theory overcame a very great difficulty that faced thinkers of this age. Either light was produced by actual material corpuscles or else was caused by an undulatory or wave motion in an ther or fictitious substance which was supposed to fill all space. Huygens had in 1690 developed this idea in detail, and these two schools of thought were rivals for a long time, favour at first being largely for Newton, but towards the end of the eighteenth century the researches of Young and Fresnel led to the general adoption of the wave theory and hence the corpuscular theory was superseded. Newton’s works on the theory of gravitation and its various applications to astronomy and dynamics were collected into his famous work Principia. When it is remembered that this remained a standard book, and its study was enforced upon all mathematical students until the nineteenth century, the reader will quickly recognise what a marvellous contribution to science it was. Starting with the assumption that two material bodies attract one another with a force which is inversely proportional to the square of the distance between them, he showed that Kepler was correct in believing that the planets described elliptic orbits about the sun, and then proceeded to investigate MODERN MATHEMATICS 51 planetary motion in detail, including the tides and motion of the moon. But his work did not stop here. As so frequently happens, advances in one branch of the subject bring advances in another, and thus we find him increasing our knowledge of the processes of algebra by the discovery of many interesting results in the theory of equations. It was these researches that led to the invention of the calculus or, as he termed it, the “Method of Fluxions.” This was one of the greatest achievements of the century, for it provided scientists of the future with their strongest weapon of attack on any problem which faces them. Though the fundamentals of the calculus were familiar when he wrote the Principia, Newton’s treatment of the subjects therein is largely geo- metrical, and we may note here that such methods were used so exclusively after his death by his countrymen, that the study of the calculus did not proceed so rapidly as one would have thought. The basic ideas of the calculus are not difficult to understand. Suppose two quantities vary so that a change in one produces a corresponding change in the other, then we say that one is a function of the other. The following example will probably make this clearer. We will consider the radius of a circle and its area, then, as we change the radius, we shall change the area. In other words, the area of a circle is a function of its radius. The ratio of the rates at which these two quantities change is called the differential coefficient, and the process by which it is calculated is called differentiation. v 52 THE STORY OF MATHEMATICS The inverse process is integration—that is, knowing the differential coefficient, finding how the original quantities varied. The calculus, therefore, is a method by which we can study quantities that are continually changing, and, as can be seen at once, there are innumerable examples that can be investigated by this means. The claim to priority of discovery of the method of calculus led to one of the famous quarrels in the history of science. Newton was familiar with the principles as early as 1666, but did not publish any account of them until 1687. But in 1684 a Hanoverian philosopher, named Leibnitz, published a statement of the method and claimed the dis- covery as his own. Leibnitz (1646-1716) was a brilliant student of Leipzig, who passed into the diplomatic service of the Elector of Maine and afterwards became a devoted servant of the Hanoverians. Although largely a philosopher, he was a mathematician of some repute, and it is generally conceded that the methods of notation that he adopted for the calculus (and which are in use to-day) are superior to those of Newton. The acrimonious controversy lasted for many years and embittered the lives of both men. In his Life of Newton, the verdict given by Sir David - Brewster is as follows :— (1) That Newton was the first inventor of the method of fluxions in 1666; that the method was incomplete in its notation, and that the funda- mental principle of it was not published to the world till 1687. MODERN MATHEMATICS 53 (2) That Leibnitz communicated to Newton in 167% his differential calculus, with a complete system of notation, and that he published it in 1684. This opinion is the one generally held by scientists to-day. : Newton was returned as Member of Parliament for the University of Cambridge in 1689 and sat for this constituency until the dissolution in the following year. He did not seek re-election as his health was beginning to fail. In 1695 he left Cambridge to become Warden of the Mint, and with the exception of a little work in connection with the moon, and which, unfortunately, led to a disagreement with Flamsteed, the Astronomer Royal, he did little more scientific work. In 1703 he was knighted by Queen Anne, and it is in- teresting to note that the ceremony was performed in Trinity College, Cambridge, where the Queen and the Prince Consort were staying on their way to Newmarket. The entertainment was on so sump- tuous a scale that the University was afterwards compelled to borrow [500 to meet the expenses incurred! Newton died in 1726 and was buried with great state in Westminster Abbey, where a handsome monument has been erected to his memory. The seventeenth century saw a scientific in- vestigation pursued with eagerness and enthusiasm. In 1666 the Royal Society was founded in London, and similar institutions were established in Paris and in Italy about the same time. Men of science 54 THE STORY OF MATHEMATICS began to correspond with one another, and the discussions on their work were fashionable pur- suits at the time. This was the age in which the germs of true scientific inquiry were laid. A little over a century later the deadening influence of medieval thought and ideas had at last faded away and the methods of modern times come into use. CHAPTER IX THEORY AND PRACTICE Nothing great was ever achieved without enthusiasm. EMERSON. IT took some fifty years or so for the world to digest all Newton’s work. Unfortunately, the controversy between him and Leibnitz played a far more important part than it deserved, and hence we find English workers following in the lines he suggested, whilst Continental students used the more powerful methods of Descartes’ analytical geometry and the more advantageous notation of Leibnitz. It was not until the early nineteenth century that a freer interchange of ideas took place and the bitterness of the quarrel subsided. One who undertakes to tell the story of Mathe- matics during the late eighteenth and the nine- teenth centuries is confronted by a great difficulty. The subject now becomes very complex and technical, and it is impossible to give any succinct account of it without using the symbolism and methods of the science. We must therefore con- fine ourselves to examining the characteristics of the century and thus obtaining a general idea of what happened. It will facilitate our task if we view the subject from a somewhat different standpoint. Hitherto, it has been considered as worthy of study only as an academic pursuit, and comparatively little attempt made to apply it to the phenomena of Nature. During the century we are considering 55 56 THE STORY OF MATHEMATICS now, men adopted a somewhat wider outlook. Their efforts were directed along two main lines: (1) The perfection of the subject itself, thereby making it a more powerful weapon of attack on any problem which it was desired to solve. (2) The application of mathematics to any kind of subject which could possibly lend itself to treat- ment by its methods. We may describe the first as pure,” and the second as “applied,” mathematics, if thereby we do not destroy the unity of the subject. There is a further point we must consider. This period saw an increase—and a very consider- able increase too—in educational facilities, thus making the study of mathematics much wider. At the same time, the creation of a Press and the establishment of journals and magazines gave greater publicity to the results that science was obtaining. Thus we find many more persons giving serious attention to these subjects than before. We must, however, remember that the great mathematicians of this age contributed to both pure and applied mathematics. For example, Euler (1707-1783) extended our knowledge of the processes of algebra and the methods of mathe- matical analysis, as well as investigating the motion of liquids—a subject known as hydrodynamics, and which of recent years has attracted consider- able attention. Further, he did much important astronomical work, including the construction of lunar tables, for which the English Parliament voted him an honorarium. We must also mention THEORY AND PRACTICE 57 at this stage the two great French mathematicians, Lagrange (1736-1813) and Laplace (1794-1827), although a detailed account of their work cannot be given here. Both were very accomplished mathe- maticians who made noteworthy contributions to astronomy, dynamics and kindred subjects. In addition to the development of mathematics, during this period we must also note the effect which it was beginning to have on other subjects. Hitherto, other sciences had progressed, each along its own lines and independently of the others. Gradually, and probably unconsciously, the con- venience of mathematical symbolism and methods became recognised, and slowly almost all branches of physical and chemical science came under their influence. The result is, to-day there is scarcely a subject with any pretensions to being called a science which does not sooner or later reduce its objects to mathematical investigation. Even those proverbially ¢dry-as-dust” statistics of births, deaths and marriages, in the hands of the mathe- matician yield interesting facts, which throw an important light on the social and economic life of a nation. Just as in other spheres of activity, this rapid expansion brought with it a certain amount of laxness of expression, and the latter part of the nineteenth century saw many inquirers into the fundamental truths and processes of the subject, which by now had reached considerable import- ance. It was a necessary overhauling of a machine that was being worked very hard. Cauchy 58 THE STORY OF MATHEMATICS (4 M1 ped (1759-18 iii his contemporaries, with a patience that was not always appreciated, examined the foundations of the calculus and the methods of its application, thereby creating a new branch of mathematics—the theory of functions and the methods of mathematical analysis. Before this chapter is closed, mention must be made of the discovery of the planet Neptune. The presence of some star in the vicinity of Uranus had been suspected for some time, owing to the disturb- ances in the orbit of the last-named planet, and the simultaneous discovery of its cause, by Leverrier in France and by Adams in England, attracted considerable attention at the time. Mathematical investigation of this puzzling phenomenon resulted in conclusions which permitted a telescope to be pointed at a definite spot in the heavens and at a definite hour, to put to the test the theoretical calculation. Itis common knowledge that the star duly made its appearance as theory had dictated. This marvellous application of mathematics raised it in popular esteem to a position it had never attained before. Although this was a century that saw enormous progress when compared with the generations which preceded it, we must not lose sight of an important change in scientific outlook that was taking place. At last the meaningless dogmas and idle specula- tions of the Middle Ages were banished and the ingenious hypotheses advanced by gold-hunting alchemists finally cast out. This is the reason why the history of this period is so important. THEORY AND PRACTICE 59 As one writer puts it, “It is a great lesson to all subsequent ages. It shows that truth in physical science is not to be sought for in dogma, nor can a system of nature be built up from our inner consciousness, but that it must be sought for earnestly and honestly by patient observation and skilled experiment.” When the shackles of a ° superstitious past were finally shaken off, real progress could be made. It is not so much a question of the intellectual abilities of workers in the nineteenth century being on a higher level than those of other centuries. It is much more a question of the nature of their surroundiags. In an atmosphere of free thought, true scientific inquiry can flourish, but in an age of mysticism and superstition the spirit of inquiry is soon choked out of existence. CHAPTER X ENERGY Science is true judgment in conjunction with reason. Prato. OnE of the most important achievements of the nineteenth century was the formulation in a con- cise form of the doctrine of energy. Although the scientists before this era undoubtedly had some knowledge of the laws by which energy is governed, it is remarkable that some statement on the subject had not been published before. Whatever may have been the cause of this timidity, the reasons which prompted the inquiry into the nature of energy at this time are not hard to find. The beginnings of the century had seen the commence- ment of the Industrial Revolution. Old methods of manufacture were giving way, factories were being erected and machinery replacing the slow handicraft efforts of the olden times. The industry of the villages was fading away before the in- dustrialism of the new towns. So profound a revolution affected scientific work just as much as it did the social and economic life of the people. The most common form of energy is heat. The important factor in the industrial revolution was the introduction of the steam-engine. The con- nection between the two will now be apparent, for the problem which is almost immediately suggested is to find out what relation there is 6o ENERGY 61 between the heat supplied by the furnace and the amount of work that the engine will perform. Before we go into this question in any detail we must clear our minds of any hazy ideas on energy and work. Everyone has some idea as to what we mean by these terms. The expenditure of energy and the performance of work are matters of every- day occurrence, but ill-defined notions are of little use in science. Energy is best defined as the capacity for doing work, and all kinds of energy are ultimately measured in terms of work which is performed. The common unit of work is the foot-lb.—that 1s, the amount of work which is done in lifting a weight of 1 lb. through a dis- tance of one foot. Let us consider an example which will show this connection between work and energy. Everyone has seen large masses of masonry raised from the ground to the tops of buildings by means of systems of pulleys and similar mechanical devices. By turning the handle of the windlass the workman is able to lift stone- work which it would be impossible to move under other conditions. It is a simple matter to calculate how much work is done in raising such a weight: it is proportional to the weight (in lbs.) and the distance (in feet) through which it is moved. This amount of work, calculated in foot-lbs., is a measure of the muscular energy supplied by the workman. All energy is not alike, and various methods of classifying the different kinds have been used. The above example immediately suggests the 62 THE STORY OF MATHEMATICS division into available” energy and ¢diffuse” energy. Available energy is that which can be used at once for mechanical effects; diffuse energy is that which cannot be so used and is for practical purposes inapplicable. All the energy which the workman puts into his task is not used in raising the stone; some of it is lost in overcoming the friction of different parts of the mechanism, and is wasted in heating the bearings, making creak- ing sounds and so on. This is the diffuse energy. Another classification which is frequently used is that of “kinetic and “potential.” Kinetic energy is that possessed by a body by reason of its motion —for example, the energy of a steam crane used in moving heavy articles. On the other hand potential energy is that possessed by a body in virtue, either of its position or of its nature—for example, a watch-spring that is wound up possesses potential energy which will be used in keeping the watch going. A further example is a mixture of chemicals—such as gunpowder— which can be exploded. This explosion is the manifestation of the potential energy which the mixture possessed. Such a classification as this need not be stressed unduly, for, although it may be a commoner one than Lord Kelvin’s of available and diffuse energy, yet it is doubtful whether it is as important. It is quite easy to change kinetic energy into poten- tial and wice versa, as the following example will show: when we wind up a clock, kinetic energy is supplied and performs the work of tightening the cord. This stores the energy in potential ENERGY 63 form, only to be released later to drive the mechanism. These changes in energy are governed by the great law of the Conservation of Energy, one of the great basic principles of science. According to this, the energy of the universe can neither be increased nor decreased in amount. It is con- tinually being transformed, producing changes and phenomena, being concentrated here and dissipated there, but always remaining invariable in amount. It has been said of this law that ¢ the principle, as it now stands, has come to be by far the most fruitful generalisation of modern physics, and its truth is supported by every experiment and ap- plication of physical principles. There is no de- partment of physical science with which it does not deal.” Newton had some knowledge of this law, and so had several others of his age, but it was not until 1842 that Mayer published an account of it in the most general form. But the law of the conservation of energy is not of itself sufficient to account for all natural processes; we must now examine the second great principle that is governing all changes which are taking place in the universe. This was formulated in 1852 by Lord Kelvin, and is usually described as the Principle of the Dissipation of Energy. In 1840 James Prescott Joule of Manchester showed that heat and energy are of the same nature and that all forms of energy can be trans- formed into heat. Further, he was able to demon- strate the most important fact, that there is a 64 THE STORY OF MATHEMATICS definite equivalence between a quantity of energy and the amount of heat it is able to produce. This is usually called the mechanical equivalent of heat. But the curious point about this is, that al- though the whole of the energy of any mechanical system can be converted into heat, the inverse process cannot be performed, and it is impossible to utilise the whole of a given quantity of heat in doing mechanical work. The proportion of energy of a system which can be used for work is called the “availability ” of the energy. Now let us see how these ideas fit in with the principle of the con- servation of energy. The total amount of energy in the universe is constant and will remain so, but with the innumerable transformations that are taking place, and the continuous tendency for all kinds of energy to degenerate into heat and be dissipated by reason of the friction of rough surfaces, resistance in various forms, etc., we see that the availability of the energy of the universe is steadily diminishing. Lest such a statement should create anxious fears, we will add that this process is a slow one, and it is not likely that any discomfort will be felt in the world for many generations by reason of this Dissipation of Energy. We have now to see how this transference of energy takes place. Until recent years it was believed to be a continuous process, but the work which has been done on the structure of the atom has necessitated some alteration of this idea. The atom is believed to be made up of a central nucleus as ll ENERGY 65 of positive electricity around which a number of negatively charged particles, called electrons, travel in circular or elliptical orbits, in much the same way as the planets revolve about the sun. (A detailed account of this will be found in The Story of the Atom, of the same series as this book.) Unlike the planets, some of these electrons have alternate orbits along which they may move, and furthermore, they occasionally *“jump” from one orbit to another. Each ¢jump” is accompanied by the radiation of energy and, what is more im- portant, the amount of energy radiated is always a multiple of some definite unit. This is called a quantum of energy, and it seems that just as matter is made up of indestructible units called atoms, so energy is likewise ‘atomistic’ in nature, always being transferred as a definite number of quanta. Such a conception as this has necessitated many changes in the outlook of science, and although there is ample experimental and other evidence to support it, the whole subject is bristling with many questions which are still waiting to be answered, and with this we must leave it. The two outstanding principles that we have already mentioned appear to be well established, but whether these of themselves are sufficient to ac- count for a// the processes of nature it is impossible to say. Scientific discoveries follow one another with such rapidity that it is by no means unlikely that in the near future we shall have a much clearer idea of the actual mechanism by which each transference of energy is effected, E ~ 1 a CHAPTER XI ¢ : Pr] w is the sixteenth letter of the Greek alphabet. LippeLL anp Scott. Berore considering the mathematics of the twen- tieth century and the new ideas that find so important a place in this era, it would be as well to round off our consideration of the earlier work with a reference to that well-known quantity used by mathematicians called Pi, and usually denoted by the letter of the Greek alphabet =. It is gener- ally defined as the ratio of the circumference of a circle to its diameter, although in many branches of higher mathematics it occurs in questions that have nothing at all to do with circles. Pi is one of the most interesting, as well as famous, of numbers, and as it is so important we must say something about it and its history. A learned professor once compared our numbers with a nation. In some respects they are all alike—or rather, they have many characteristics in common, like the different individuals that all go to make a nation. On the other hand each has its own little peculiarity, some point of difference from the others, some ¢ personality” which marks it out as unlike its neighbours. Just as some persons are interesting, so are some numbers, and one of theseis = The reader will have seen from the preceding 66 PI 67 pages that geometry was one of the earliest of the mathematical studies, and that the ancients had a remarkable knowledge of the subject. Amongst the simple figures with which they were familiar was the circle, and in the course of their investiga- tions they became acquainted with some of the properties of =, although the use of the present symbol was not introduced until many centuries later. The Babylonians and the Jews estimated the circumference of a circle as three times that of the diameter, thus giving = the value three. It appears from the papyrus which Ahmes has left us that, in the course of his work on the pyramids, he gave = the value of 31408 which is practically the correct value. To-day, for rough approximations, it is frequently sufficient to put ==, although for more accurate work it is usual to put = = 3141592. . «+ . The following rhyme appeared in a letter to Nature, and the number of letters in each word, taken in order, gives the value of = to thirty-one places of decimals :— Sir, I send a rhyme excelling In sacred truth and rigid spelling Numerical sprites elucidate For me, the lesson’s dull weight. If Nature gain, Not you complain Tho’ Dr Johnson fulminate. To return to our historical consideration of this curious number, the reader will recall that one of the three famous problems of antiquity was “Squaring the Circle ”—that is, the construction 68 THE STORY OF MATHEMATICS of a square equal in area to that of a given circle. This is practically equivalent to determining by geometrical means the value of =. Although the Greeks never solved the original problem, Archi- medes showed by geometry the limits within which «= must lie. Further, Ptolemy took ==3-1416, which is certainly very nearly its true value. The Hindoo mathematicians were not so fortunate. From their works it seems that the ,/10 was as near as they could get. Later, many attempts were made to determine accurately the value of the mysterious number, and the inventions of logarithms by Briggs and Napier greatly facili- tated the work of those industrious computators, who devoted themselves with great courage to the task, although it must be admitted that the evalution of 7 to dozens of decimal places seems a rather useless proceeding. It was left to Lambert (1728-1477) to prove that their labour was in vain and that = is an incommensurable number—that is, its value cannot be finally determined. The use of the symbol = is first found in text- books in this century—the eighteenth—being used by Euler (1707-1783). The history of m closes in 1882 when Lindemann showed that it could not be the root of an algebraical equation. - With this short account of one of the curiosities of mathematics we will now continue our story. CHAPTER X11 NEW CONCEPTIONS Science is organised knowledge. HERBERT SPENCER. Tue story of Mathematics would not be com- plete without some account of the Theory of Relativity, the great discovery of the twentieth century. This is no place for an examination of its philosophical basis—important as it is—nor can we delve deeply into the purely mathematical aspect of the theory, owing to the great difficulty and complexity of the subject. We must there- fore content ourselves with a short summary of some of its more salient features, and any reader who desires a more intimate acquaintance will find many books devoted entirely to the theory, and a more detailed study of it will be amply repaid by its intrinsic interest. In the first place we must go right back to one of our earliest conceptions—namely, that of length. We are all familiar with the word and its meaning. When we wish to purchase ten yards of cloth, our minds immediately conjure up a vision of the draper’s shop that is going to supply it, the assistant who will, in our presence, select the roll of our choice and measure it out by means of the brass ruler affixed to the shop counter. We are then in possession of a length of cloth which we call ten yards, measured by 69 470 THE STORY OF MATHEMATICS means of a strip of brass which Parliament has decreed is one yard in length. Furthermore, thanks to the kindly thought of our Government, Inspectors of Measures periodically check that strip of brass by comparing it with a standard yard, which is deposited in London, to see that customers are not defrauded. Now the question we have to consider is, whether that brass strip is always the same length. Our eyes can detect no change, nor can any microscope, so that for practical purposes the question is rather superfluous, but from a purely theoretical point of view it is of interest to ask if our yard length is really akways exactly the same length. We can put the question another way. Suppose we take a rod and, holding it in the position North—South, cut it off to be exactly one yard long, and then turn it into the position East—West. Is it still the same length? Un- _ hesitatingly we answer “ Yes,” but simply because we cannot detect a change by any means that we possess. But because we cannot detect a change, it does not necessarily follow that there is #0 change. Suppose our rod were moving through space very rapidly, in the direction of its length, is it possible that its length might alter? This is not so fanciful a question as would appear at first sight, when we consider that the rod could easily be placed in the direction of the earth’s motion and with the earth moved through space at a very considerable speed. It is now believed, as a result of a famous experiment by Michelson NEW CONCEPTIONS 71 and Morley in 1884 (designed, we may add, for quite a different purpose), that a moving body contracts slightly in the direction of its motion. Now, this Fitzgerald contraction, as it is called, is extremely minute—being only a matter of an inch or so in the case of the earth’s diameter as it moves through space—and so again quite negligible for all practical purposes, but not so when we are considering, either the enormous distances that exist in the universe, or the high velocities that are possessed by the satellite electrons in an atom. Reverting therefore to our “yard” rod, we notice that its length in the direction of the earth’s motion will be different from that when placed in any other position. Having cast doubt on the constancy of length, now let us consider another fundamental measure- ment—namely, time.” What we mean by this measurement has been well understood for cen- turies,. We are well able to perceive a time- difference between two events. For example, if we place a kettle of cold water on a gas ring, we can appreciate a definite time-interval between the first event—lighting the gas—and the second event—seeing the kettle boiling. If we wish to record the results of our observations carefully, we measure this time-interval by means of a clock. Let us imagine the earth to be stationary and fixed in space, and on it an observer provided with a kettle of cold water, a gas ring and a clock. He will then measure the time it takes to boil the kettle. Now, suppose a second kettle of the 79 THE STORY OF MATHEMATICS same size and containing an equal quantity of water, a similar gas ring and a similar clock to move away from the earth with a definite velocity. Then it has been shown that our observer, who is fixed, will notice that the time-interval in this second case will be less than the first, suffering a “contraction” in just the same way as the measuring rod we have already discussed. As the reader will see at once, these two new ideas strike home at the very fundamentals of mathematics, and therefore it is not surprising that when these views first obtained acceptance, it was thought the whole theory of dynamics created by the Newtonian School needed drastic revision. We must point out that revision is obviously necessary in view of the changes which the Fitzgerald contraction causes, but not revision so drastic as some were inclined to believe. This contraction, we would emphasise, is very minute, and only can become of importance when either enormous distances or high velocities are to be considered. By “high velocity” we mean a velocity which approximates to that of light— namely, 186,000 miles per second. With this in mind, the reader will see that these changes in our ideas will have no appreciable effect on the prac- tical affairs of daily life, and that the dynamics of Newton and his followers—the great achievement of the “classical” era of English Mathematics —1s still of the utmost importance. CHAPTER XI11 RELATIVITY The learned is happy nature to explore ; The fool is happy that he knows no more. Pork. WE are now in a position to use some of the ideas we have just considered and see how they fit in together and form what is usually called the theory of relativity. In doing so, we shall have occasion to use the word “system.” This, in its mathe- matical sense, has a meaning somewhat more definite than when used colloquially. It consists of every- thing that is concrete and of importance to the person making an experiment or measurement of any kind. In the last chapter, we referred to a fixed observer heating a kettle of water and noticing how long it would take to boil; here the observer, his gas ring, his kettle of water, his clock and any other means of measurement he possesses, all form a system. For convenience we might call this * System A.” Now, let us imagine an observer, whom we will . call Mr Jones, provided with a measuring rod and a clock, and engaged in the study of a definite phenomenon of a certain system, which we will call S. We will further suppose that he is situ- ated at some definite fixed point anywhere in space and able to record the results of his observations. Let a second experimenter, whom we will call Mr Smith, be similarly equipped and studying the 73 74 THE STORY OF MATHEMATICS same phenomenon in another system, St. Further, let Mr Smith and his system be moving through space, without being aware of it, such as would happen if Mr Smith and his system St! were situ- ated on the earth. The records made in both systems would be identical and neither observer would be able to determine from the results he obtained whether he were in motion or not. Now let us go a step further and suppose that Mr Jones, from his fixed position, can watch the moving system S! and Mr Smith, similarly the fixed system S. Here, again, the impressions that each would obtain are identical. Both would be able to appreciate that there was a relative velocity between the two systems S and Si, but neither Mr Jones nor Mr Smith would be able to deter- mine from his results which observer was moving, unless he knew beforehand which system was the fixed one. But we on the earth have no knowledge of any fixed point in space to which we can refer our measurements. Therefore we can have no knowledge of our motion other than that which is relative to bodies in space outside the earth. This is sometimes put in another way by saying that absolute motion is undeterminate. This conclusion, reached as above by philo- sophical speculation, was the basis of Einstein’s famous paper in 1905, when he proved for mov- ing electric systems that the contraction assumed by Fitzgerald, as a result of the Michelson-Morley experiment, could actually exist. The publication of his results received little attention at the time, RELATIVITY 75 but of recent years the whole question has been investigated in considerable detail, with the result that the conceptions we have possessed for so long have found interpretation in a new and strange form. This theory has been applied to many physical questions and also to the great problem of gravitation. In a short book like this, a de- tailed discussion of these applications cannot be given, and so we must conclude this chapter with merely a brief mention of some of the more im- portant results that have been obtained. It has long been the object of scientists to find an explanation of the several forces that act in the universe, and their inquiry has centred round electrical forces and gravitation. The nineteenth- century physicists explained the former by filling all space with an hypothetical substance called the “ether,” which, when in a state of strain, gave rise to the forces we describe as electric and magnetic. Of course nobody ever isolated this substance—it is doubtful if anyone tried—but by endowing it with certain properties, a very complete explanation of electrical phenomena was given by Faraday, Clerk-Maxwell and their followers, and although several strange contradictions crept into the subject, everyone had to be satisfied with it until something better was available. One of the most serious drawbacks created by the introduc- tion of this hypothetical ther” was the fact that the ether, which was capable of giving an explanation of electricity and magneticism, failed to explain gravitation, and conversely, an ther that 76 THE STORY OF MATHEMATICS. could explain gravitation failed to account for electrical forces. This difficulty has been overcome by the re- lativist by sweeping away ethers and looking elsewhere for a solution. Arising out of the Einstein theory, Minkowski constructed in 1908 a “four-dimensional geometry of space—that is, the three dimensions of length, breadth and height, together with time—in which it is impossible to separate the space from the time in any absolute manner.” At this stage we must remind the reader that it is quite impossible to visualise such a space. To the accomplished mathematician, it can exist in terms of his symbols and his equa- tions, but any attempt to form a mental picture of the ¢ Minkowski world,” as it is generally called, must end in disaster. With this interpre- tation of his earlier work, Einstein was able to publish, in 1915, his general theory of relativity. From this these physical phenomena of electro- magnetism and gravitation follow as geometrical necessities. The reader will find a more detailed account of this in the appropriate chapter in The Story of Electricity, in the series uniform with this volume; and as it is rather to the domain of Electricity that this belongs, we will not consider it in any further detail. One of the early successes of the theory of relativity was to afford an explanation of one of the curiosities of astronomy—namely, the motion of the orbit of the planet Mercury. It had been shown many years previously by Leverrier that the RELATIVITY 79 orbit of this star was slowly turning, and the only explanation which could be given was the possi- bility of some other unknown star in the vicinity causing these disturbances in the orbit of Mercury. The Einstein theory, however, afforded the solution of the difficulty and, what is perhaps more import- ant, the rotation of the orbit as calculated has been confirmed by actual observation. Another deduction which has been made from the Einstein theory is, that rays of light when passing close to large masses are bent towards the attracting mass. In other words, rays of light from a distant star passing near to the sun are curved on their way to the earth, and consequently the star appears slightly displaced in a direction away from the sun. This result was confirmed by observations during the solar eclipse in 1919, and again in 1922, and so this revolutionary idea can now be regarded as well established. It does not seem likely that the theory of re- lativity as it is understood at present will remain unmodified, and although there are not wanting those—even including scientists of distinction— who believe that the riddle of the universe has now been solved, we should do well to hesitate before considering that the quest for Nature’s secrets is nearing an end. The history of science abounds with theories, accepted with eagerness only to be rejected later, and it is much too easy to forget Herschel’s famous dictum, er Rn et HOME USE CIRCULATION DEPARTMENT MAIN LIBRARY This book is due on the last date stamped below. 1-month loans may be renewed by calling 642-3405. 6-month loans may he recharged by bringing hooks to Circulation Desk. Renewals and recharges may he made 4 days prior to due date. ALL BOOKS ARE SUBJECT TO RECALL 7 DAYS AFTER DATE CHECKED OUT. — ARNRcR.4 75, B 3 29 JR Ls 1053 NOV 14 1986 ~~ recdcirc. FEB 17 1983 MAY 41984 “REC CIRMAY 14 1984 MAY 11 1986 RECEIVEL BY smi 1 ATIOSAL _REDT CIRCULATION DEPT. WIA BL * LD2 atom] Te General Library (S2700L) University of California Berkeley DrLIBCI-Jy jversity Univ ge T(J401s10)4768 163 _aom1} LD 252610) grow Sxeley - GENERAL LIBRARY - U.C. 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