Mathematics, from the points of 
 
 view of the Mathematician 
 
 and of the Physicist 
 
 An address delivered to the Mathematical and Physical 
 Society of University College, London 
 
 by 
 
 E. W. HOBSON, Sc.D., LL.D., F.R.S. 
 
 Sadleirian Professor of Pure Mathematics in the 
 University of Cambridge 
 
 Cambridge 
 
 at the University Press 
 
 1912 
 
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MATHEMATICS, FROM THE POINTS OF VIEW 
 
 OF THE MATHEMATICIAN AND OF THE 
 
 PHYSICIST. 
 
 If we were to question a man of average education, or even one of 
 the majority of those who belong to the cultured class, as to what he 
 conceives to be the nature of Mathematical Science, and as to what 
 he thinks are the aims of those who cultivate it, we should probably 
 receive a somewhat vague answer, conveying his impression that 
 Mathematics is concerned with calculations carried on by means of 
 processes involving a copious use of symbols and diagrams entirely 
 unintelligible to the uninitiated. He might not improbably add that 
 such symbols and diagrams are of no interest to anyone except a few 
 individuals who have an unaccountable taste for such things, and 
 happen to be endowed with a peculiar transcendental faculty the 
 possession of which is quite unnecessary for other people, and which 
 he himself does not possess, and moreover that he is quite happy 
 without it. If pressed, our friend would probably admit that there 
 are subjects, such as Arithmetic and Mensuration, which are extremely 
 useful, and some knowledge of which is essential for various purposes, 
 but that such subjects are hardly Mathematics, and that anything 
 beyond them is of little or no concern to the world in general. If, as 
 is likely, he had himself, at school, studied a little Geometry and 
 Algebra, he would say that the former is largely concerned with 
 difficult and tedious proofs of things that are either obvious or 
 meaningless, and that the latter is concerned with juggling with 
 letters in an extremely troublesome manner, and for no ascertainable 
 object. Even if we obtained a somewhat less crude answer to our 
 question, that answer would probably be, in a greater or less degree, 
 an inadequate one as a characterization of the essential features of 
 Mathematical Science. In the scientific world in general, and even 
 
 1 
 
 308200 
 
2 :', IV^ATHEMM^lCS, FOR THE MATHEMATICIAN 
 
 am4^"ii^ .sfudentp.' Of il3LQ5e**'l5miiches of Science in which Mathematical 
 thought has played an exceptionally large part, decidedly vague and 
 narrow conceptions of the functions of Mathematical thinking are 
 current. 
 
 In such circles, the notion is extremely common that the sole 
 function of Mathematics is to provide the means of carrying out 
 whatever calculations may he necessary for the special purposes of 
 each department of Physical Science, theoretical or applied, and thus 
 that Mathematics plays in them a comparatively humble part analogous 
 to that of a mechanical tool. As a matter of fact Mathematical thinking 
 has done very much more than merely to provide methods of calculating ; 
 it has played a most important part in the formation of the concepts 
 with which the Physical Sciences work ; it has decided in many cases 
 for them not only how to calculate but what the things are that they 
 ought to calculate ; it has reduced the originally vague conceptions 
 which arise in connection with physical observation to precise forms 
 in which they can be exhibited as measurable quantities. 
 
 Mathematical thinking, in a more or less explicit form, pervades 
 every department of human activity. The grocer, when he weighs out 
 his sugar, makes use of Mathematical conceptions which were de- 
 veloped only by a long process of evolution ; when he enters his 
 receipts in his books, the notation he uses is a warrant of the 
 importance of Mathematical form, and embodies one of the triumphs 
 of our race as a mode of economizing thought. The Engineer, when 
 he makes the necessary graphical calculations of the stresses in the 
 bridge he is building, has to employ concepts and methods belong- 
 ing to a decidedly advanced state of Mathematical thought. The 
 Philosopher, in his reflections on spatial and temporal relations, on 
 number and quantity, on matter and motion, is in a region of thought 
 in which the boundary between his own domain and that of the 
 Mathematician is almost non-existent. The Epistemologist has 
 always to take Mathematical knowledge as a kind of touchstone on 
 which to test his theories of the nature of knowledge. The dominant 
 views in various departments of philosophical thinking have been 
 modified in important points by the results of recent Mathematical 
 research, and will, I think, in the future, not improbably be further 
 modified from the same quarter. 
 
 Mathematical thought is at once the most all-pervading and the 
 most highly specialized department of mental activity. 
 
 The origin of Mathematical thinking, and the reason of its ubiquity 
 
AND FOR THE PHYSICIST 3 
 
 in all departments of Science, and even of practical life, are not far to 
 seek. The physical world in which we live appears to us as a manifold 
 of objects extended in space, their spatial relations exhibiting at any 
 one time an endless variety, and varying at different times. The 
 primary Mathematical operations of counting and measuring, leading 
 up to the concepts of number and geometric form, took their first 
 beginnings in the practical interests of man living in that world. 
 These Mathematical concepts, and the methods of dealing with them, 
 underwent a gradual evolution in close connection with the practical 
 efforts required to grasp, classify, and characterize, and so far as 
 possible to dominate, the spatial relations of actual bodies, both in 
 their fixity and as changing in temporal succession. The more closely 
 men scrutinized natural phenomena, at first for practical reasons, and 
 later from intellectual curiosity, the more things and processes they 
 found to have aspects which are measurable, and the more they were 
 able to employ their developing Mathematical processes and concepts 
 for the precise characterization of various aspects of the world of 
 phenomena. But, as I hope to be able to show in some detail, the 
 natural development of Mathematical thought, starting as it did in 
 connection with the more obvious aspects of sensuous experience, 
 under the pressure of physical needs, brings it to a region reaching 
 far beyond that in which the primitive intuitions of time, space, and 
 matter formed the exclusive subject matter of the Science. 
 
 Mathematics is not merely a subject to be studied and developed 
 by a class of specialists; it has to play a part, and usually an im- 
 portant and indispensable part, in other departments of Science ; and 
 consequently there are various attitudes taken up with regard to 
 Mathematics, dependent on the relations of the subject to these 
 various departments. I have chosen for my subject this evening a 
 comparison of the ways in which the nature and functions of Mathe- 
 matics are regarded, and should be regarded, by the Mathematician 
 proper, and by the Physicist. I deal with the subject primarily as 
 a comparison of the attitudes of those who have been engaged in the 
 past, and of those who are now engaged, in furthering the progress of 
 Mathematics and Physics as growing branches of knowledge : but if it 
 be true, as I think it is, that the spirit in which a student of any 
 branch of Science pursues his work should approximate as much as 
 possible to that spirit which animates the researcher, the kind of 
 comparison which I here attempt may, I hope, be not without interest 
 or utility to the younger students of either branch of knowledge. For 
 
 12 
 
4 MATHEMATICS, FOR THE MATHEMATICIAN 
 
 many purposes, the Astronomer, the Engineer, the Statistician, and 
 the Biometrist might be grouped with the Physicist, as having analo- 
 gous, although by no means identical, dealings with the Mathematician. 
 For example, the Engineer, like the Physicist, has constantly to make 
 use of Mathematical methods ; but as his ultimate aim is to harness 
 the forces of nature and use them to obtain practical results, rather 
 than to bring their relations under general laws and concepts as the 
 theoretical Physicist does, he is perhaps less directly concerned than 
 the Physicist with the part which Mathematical Science has played in 
 the formation of the concepts with which the theoretical Physicist 
 works. He uses applied Mathematics, and applied Physics, and is apt 
 to take both of them more or less as ready-made products, although he 
 cannot do so beyond a certain point without grave danger to his 
 efficiency as a scientific engineer. 
 
 In former times the Mathematician and the Physicist were usually 
 one and the same man. Even as late as the eighteenth century this 
 was very generally the case ; it was in the nineteenth century that the 
 increasing complexity of both Sciences produced that separation of 
 the two departments which has become continually more marked, and 
 has reached its extreme point in our own time. To attempt to give a 
 complete explanation of this phenomenon would be to attempt to write 
 the history of both Sciences. I can only give, in the course of my 
 remarks, some brief indications of the causes, lying in the lines of 
 development of both Sciences, which have brought about this separation. 
 
 The typical Mathematician and the typical Physicist of to-day 
 have their centres of interest very differently located. This difference, 
 though inevitable in view of the exigencies of the two studies, and to 
 a large extent advantageous, has of course serious drawbacks. The 
 chief drawback is that each specialist, from lack of interest in, and 
 knowledge of, the progress of the other great department, is apt to 
 miss that large source of inspiration in his own study which is supplied 
 by the other one. The essential difference of the ways in which 
 Mathematics is contemplated by the Pure Mathematician and by the 
 Physicist sometimes comes strikingly into expression. I remember, 
 some quarter of a century ago, at a Board meeting at Cambridge, the 
 subject of Bessel's functions came into the discussion, I think in con- 
 nection with some proposal to include them in an examination syllabus. 
 Their utility in connection with Applied Mathematics having been 
 referred to, a very great Pure Mathematician who was present ejacu- 
 lated " Yes, Bessel's functions are very beautiful functions, in spite of 
 
AND FOR THE PHYSICIST 5 
 
 their having practical applications." It would have been interesting to 
 have heard what this great man would have said if he had known that 
 Professor Perry would one day propose the desecration of these beautiful 
 functions by recommending them as suitable playthings for young boys. 
 The typical Pure Mathematician is quite satisfied with a result 
 which is expressed with the requisite precision by the employment of 
 an apparatus of symbols each of which has been properly defined. To 
 the typical Physicist, on the other hand, a formula seems often to have 
 very little meaning unless he can interpret it in some intuitional 
 manner that gives it a meaning which in its purely abstract form it 
 does not possess for him. This divergence of mental habits appears in 
 a striking way in some remarks made by that ultra-Pure Mathematician 
 Sylvester about a treatment of the subject of Spherical Harmonics by 
 the theory of poles, that was given by Maxwell, a man with physical 
 instincts of the most pronounced order. Maxwell wrote of his own 
 theory : ''In numerical investigations I have often been perplexed on 
 account of the apparent want of definiteness of the idea of a Laplace's 
 coefficient or spherical harmonic. By conceiving it as derived by the 
 successive differentiation of 1/r with respect to i axes, and as expressed 
 in terms of the positions of its i poles on a sphere, I have made the 
 conception of the general spherical harmonic of any integral degree 
 perfectly definite to myself, and I hope also to those who may have 
 felt the vagueness of some other forms of the expression." Commenting 
 upon this, Sylvester writes : " The method of poles for representing 
 spherico-harmonics, devised or developed by Professor Maxwell, really 
 amounts to neither more nor less than the choice of an apt canonical 
 form for a ternary quantic, subject to the condition that the sum of 
 the squares of its variables (here differential operators) is zero ; and 
 I am quite at a loss to understand how it can at all assist in making 
 the conception of the general spherical harmonic of an integral degree 
 perfectly definite, or what want of definiteness apart from the use of 
 this canonical form can be said to exist in the subject." But the spirit 
 of the great Algebraist had been further grieved by Maxwell in this 
 matter, for he writes : " With all possible respect for Professor 
 Maxwell's great abihty, I must own that to deduce purely analytical 
 properties of spherical harmonics, as he has done, from Green's theorem 
 and the principle of potential energy, seems to me a proceeding at 
 variance with sound method, and of the same kind and as reasonable 
 as if one should set about to deduce the binomial theorem from the 
 law of virtual velocities, or make the rule for the extraction of the 
 
6 MATHEMATICS, FOR THE MATHEMATICIAN 
 
 square root flow as a consequence from Archimedes' law of floating 
 bodies." 
 
 Speaking to some extent from personal experience, one of the eff'ects 
 of prolonged study of some of the more abstract branches of Mathematics, 
 as for example the Theory of Functions, is that one begins to take the 
 greatest interest in, and to be most attracted by, just those aspects of 
 the subject which are most remote from the interests of the Physicist. 
 One gets into an attitude of mind in which the kind of well-behaved 
 functions, without abnormal singularities, that are likely to turn up in 
 Physical investigations, appear to have a somewhat bourgeois aspect, 
 in their comparatively uninteresting respectability. In the mind of 
 one who makes a minute and prolonged study of the peculiarities which 
 Fourier's series may present, a not improbable eff'ect of that study is 
 that the ordinary Fourier's series, which converge everywhere quite 
 normally, begin to acquire a certain tameness of aspect which deprives 
 them of interest. The failure of convergence of Taylor's series becomes 
 to some Mathematical students a matter of greater interest than that 
 presented by the series in the ordinary cases in which, by their regular 
 convergence, they are fitted for purposes of application. 
 
 I have already called attention to the fact that Mathematical 
 thinking has played a very important part in the formation of the 
 fundamental concepts of the Physicist ; very often this part has been 
 a dominant one. Many of these concepts could only have received a 
 precise meaning, and could only have taken definite forms, as the 
 result of the work of Mathematicians, and their formulation often 
 appeared as the result of a long train of previous Mathematical 
 thinking. For example, the conception of Energy, and the exact 
 meaning of the great generalization known to us as the law of the 
 Conservation of Energy, emerged as results of the development of the 
 abstract side of molar mechanics, which determined the mode in which 
 the kinetic energy of moving bodies and potential energy as work are 
 defined as measurable quantities. Only by the transference and 
 extension of these notions to the molecular domain did the conception 
 involved in the modern doctrine become possible. The doctrine of the 
 conservation of energy for the case of molar bodies had been established 
 before Joule and Mayer commenced their work, and was a necessary 
 presupposition of their further development. Joule was able to deter- 
 mine the mechanical equivalent of heat only owing to the fact that 
 mechanical work was already regarded as a measurable quantity, 
 measured in a manner which had been fixed in the course of the 
 
AND FOR THE PHYSICIST 7 
 
 development of the older Mathematical Mechanics. The notion of 
 Potential, fundamental in Electrical Science, and which every Physicist, 
 and every Electrical Engineer, constantly employs, was first developed 
 as a Mathematical conception during the eighteenth century in con- 
 nection with the theory of the attractions of gravitating bodies. It 
 was transferred to the electrical domain by George Green and others, 
 together with a good deal of detailed mathematics connected with it 
 which had previously been applied to the gravitational potential 
 function. 
 
 The ultimate aim of the Physicist, even of one whose work is 
 almost exclusively experimental, is much higher than that of attaining 
 to a merely empirical knowledge of facts. His real object is to classify 
 facts in such a way as to refer them to general laws which are of a 
 more or less abstract character, and which involve concepts of schematic 
 representations that require, for the fixing of their exact denotation, 
 the aid of the Mathematician. The man of true Physical instincts, 
 endowed with the great faculty of scientific imagination, possessed for 
 example by Lord Kelvin in a very remarkable degree, is for ever 
 imagining models which shall enable him by their working to represent 
 and depict the course of actual physical processes. The possibility 
 and consistency of such models require Mathematical Analysis for 
 their investigation. The Mathematician may also, by tracing the 
 necessary consequences of the postulation of a model of a particular 
 t3rpe, formulate crucial tests in accordance with which further experi- 
 ments will decide whether a particular type of model can be retained 
 at least provisionally, or whether it must be rejected as inadequate for 
 the representation of known facts, and must give place to some other 
 model of a different type. Perhaps the most striking example of the 
 services which have been rendered to Science by the contemplation of 
 various models, many or all of which have ultimately been found to be 
 inadequate for complete representation, is to be found in the history of 
 Optics. The various forms of the corpuscular theory, and of the wave 
 theory", of Light were all attempts to represent the phenomena by 
 models, the value of which had to be estimated by developing their 
 Mathematical consequences, and comparing these consequences with 
 the results of experiments. The adynamical theory of Fresnel, the 
 elastic solid theory of the ether developed by Navier, Cauchy, Poisson, 
 and Green, the labile ether theory developed by Cauchy and Kelvin, 
 and the rotational ether theory of MacCullagh were all efforts of the 
 kind I have indicated ; they were all successful in some greater or less 
 
8 MATHEMATICS, FOR THE MATHEMA.TICIAN 
 
 degree in the representation of the phenomena, and they all stimulated 
 Physicists to further efforts to obtain more minute knowledge of those 
 phenomena. Even such an inadequate theory as that of Fresnel led 
 to the very interesting observation by Humphry Lloyd of the pheno- 
 menon of conical refraction in crystals, as the result of the prediction 
 by Rowan Hamilton that the phenomenon was a necessary consequence 
 of the Mathematical fact that Fresnel's wave surface in a biaxal 
 crystal possesses four conical points. 
 
 Although the theoretical Physicist has for his real aim the formula- 
 tion of abstract schemes for the description and correlation of the 
 physical phenomena which he observes, with the Mathematician pro- 
 cesses of abstraction must go very much further than with the 
 Physicist. A strong tendency of Mathematics in its later developments 
 is to split up notions, originally undivided, into components, and to 
 proceed to deal with these components in isolation, and often in 
 separate branches of study. For example, during the last half-century, 
 number and measurable quantity have been separated from one another ; 
 the idea of number alone has been recognized as the foundation upon 
 which Mathematical Analysis rests, and the theory of extensive 
 magnitude is now regarded as a separate department in which the 
 methods of Analysis are applicable, but as no longer forming part of 
 the foundation upon which Analysis itself rests. For the purposes 
 of analysing the implications of the methods employed, and of pushing 
 those methods to the highest possible degree of development, this kind 
 of separation is indispensable, and has led to the very abstract form in 
 which much modern Mathematics is exhibited. For the Physicist on 
 the other hand, it is essential that abstraction should not go nearly so 
 far as the Mathematician must push it. Too much abstraction on the 
 part of the Physicist would entail the penalty that he would lose his 
 way in a field which is barren for his purposes, and would lead to 
 a loss of contact with the phenomenal world. To do what the 
 Mathematician does, and must do, in view of the essential interests of 
 his study, would be fatal to the Physicist, to whom above all things a 
 large degree of concreteness in his conceptions is indispensable. 
 
 Not only did Mathematical Science take its origin in the necessities 
 and interests of man living in the physical world, but at every stage of 
 its development the problems of Physics have been the source of the 
 ideas which have directed the Mathematician, and from which new 
 paths of investigation have been suggested to him. But every great 
 problem which has been placed before the Mathematician from the 
 
AND FOR THE PHYSICISf 9 
 
 physical side of things has given rise to a train of ideas which the law 
 of his being compelled him to pursue, and has started a host of 
 questions to which he was impelled to find the answer. These have 
 led him in most cases far beyond the original domain ia which the 
 problems originated. The most abstract branches of modern Mathe- 
 matics, the theory of functions, real and complex, the theories of 
 groups and of differential equations all arose originally from physical 
 beginnings, but have reached out into vast developments, apparently, 
 but not by any means always really, remote from the physical region. 
 At any moment one or other of these developments may become 
 urgently necessary for the purposes of Physics, and may thus be in a 
 position to pay back some of the debt they owe to the parent from 
 whose side they have wandered so far. 
 
 The question is often asked, in some form or other, why Mathe- 
 maticians cannot restrict themselves more to those aspects of their 
 Science which bring them in contact with Physics, and which are 
 concerned with what often receives the question-begging and am- 
 biguous name of reality ; a word that has an indefinite number of 
 shades of meaning, varying with every difference in philosophical 
 view, but which in this connection is generally associated with that 
 kind of reality supposed exclusively to appertain to the physical world. 
 Why, it is asked, do modern Mathematicians to so great an extent as 
 at present wander away from the source in which their Science had its 
 origin, and from which its ever-renewed inspiration has been received, 
 in order to lose themselves in a transcendentalism which, in its aloof- 
 ness from physical investigation, condemns them to an endless and 
 barren immersion in abstractions of their own creation ? Why has 
 modern Mathematics cut itself off from the roots of the Science? 
 Well, the answer to this often-recurring question is that the modern 
 abstract lines of thought of Mathematicians lie in the natural line of 
 progress for the minds of those who are unable to rest content when 
 questions force themselves upon their attention, without making the 
 most strenuous endeavours to find answers to those questions. To 
 stop short at a point dictated by considerations of applicability to 
 Physics is impossible to those to whom clear and thoroughly defined 
 conceptions are a desideratum, the lack of which in any department of 
 their study leaves them no rest. To attempt to confine the activities 
 of Mathematicians by imposing upon them a restriction of the nature 
 indicated in the above question would be to attempt to strangle the 
 Science as a progressive development. Mathematics can in the long 
 
 15 
 
10 MATHEMATICS, FOR THE MATHEMATICIAN 
 
 run be developed to the highest degree of perfection, not only from the 
 point of view of specialists within its own domain, but also as con- 
 stituting an essential component of the intellectual life and stock of 
 ideas of the world, onl}^ on the condition that it is allowed full freedom 
 of self-expression. The utilitarian notion, in this connection as in so 
 many others, has the fatal limitation that it attempts to assign limits 
 to what is, or may in the future become, useful, in accordance with a 
 more or less arbitrarily restricted standard of what constitutes utility. 
 
 When the exigencies of Physics suggest to the Mathematician some 
 process, or some special problem for solution, he is impelled to sparch 
 for some generalization, some law, under which a whole class of 
 analogous processes or problems can be subsumed. He is never 
 content with having obtained merely isolated results; general methods 
 and general laws of which the isolated results are but special cases are 
 what he invariably looks for. The Physicist also pursues the same 
 course; he too is really occupied in attempting to exhibit isolated 
 phenomena as particular cases of some general law under which a 
 whole class of phenomena can be subsumed. A narrow utiHtarianism 
 would be as fatal to the growth of Physics as to that of Mathematics. 
 The Physicists of the eighteenth century, when they examined the 
 phenomena of Electricity and of Magnetism, before any relation between 
 these two sets of phenomena had been discovered, were actuated solely 
 by intellectual curiosity, which impelled them to attempt the discovery 
 of laws for the correlation of the isolated facts known to them. Had 
 their spirit been utilitarian in the narrow sense, they would probably 
 have left the matter on one side, as holding out no prospect of useful 
 application ; and one result of such abstention would have been that 
 we should not now possess the electric telegraph, the telephone, or the 
 motor-car. 
 
 The most abstract Mathematical investigators, the most purely 
 experimental Physicist, and every investigator whose path lies between 
 the two extremes all are taking humble shares in the great enterprise 
 of bringing, in whatever measure may prove possible, processes not 
 only of what we call the physical world, but also of what we call the 
 psychical world, under the categories of conceptual thought. It would 
 be rash indeed to attempt to estimate with any accuracy the ultimate 
 values of the shares contributed by different kinds of investigators, with 
 all their varied shades of mental leanings, towards this great work. 
 Mathematicians of all shades of interest, and Physicists of all kinds as 
 regards their attitude towards the more abstract and the more concrete 
 
AND FOR THE PHYSICIST 11 
 
 aspects of their subject, are required as contributors to the supreme 
 work, the attainment of the underlying purpose of which demands the 
 sustained efforts of minds of the most varied types through the ages 
 of the history of the human race. 
 
 The Mathematical Physicist plays a part of supreme importance as 
 an intermediary and interpreter between the Pure Mathematician and 
 the experimental Physicist. In order that he may do his work effectually 
 he must follow with an alert eye the lines of progress both in Mathe- 
 matics proper, and in experimental Physics. The nature and the 
 amount of the work he can do vary much with the character of 
 the current physical investigations. It is probably true that, in spite 
 of some brilliant exceptions, the Mathematical Physicist does not just 
 at present take as prominent a part as was formerly the case, especially 
 during the nineteenth century, the age of Maxwell, Kelvin, Stokes, 
 Helmholtz. In attempting to assign reasons which may explain this 
 partial eclipse of the activities of the Mathematical Physicist, it 
 appears possible that a part of the explanation may be afforded by 
 the actual separation in education of the training of Physicists and 
 Mathematicians. Although this separation is probably to some extent 
 necessary in the highly specialized state of both departments which 
 now subsists, and as the differing tastes of students determine whether 
 their studies shall be in the main experimental or in the main 
 Mathematical, the separation has been in practice carried much too 
 far. No doubt more might be done thaii at present by our educational 
 institutions to co-ordinate Mathematical and Physical studies so as 
 to train up a race of Mathematical Physicists. I am inclined however 
 to think that the main ground of the present comparatively smaller 
 part played in scientific life by the Mathematical Physicist is to be 
 found in the present trend of Physical investigations, and that probably 
 the present state of things is only a temporary one. In earlier times, 
 when physical investigations were principally concerned with the 
 grosser and more obvious phenomena, when molar Mechanics, and 
 especially celestial Mechanics, occupied the centre of the interests 
 of physical investigation, the passage from the observation of concrete 
 phenomena to their abstract Mathematical representation was com- 
 paratively easy. The observational work was simpler and less technical 
 than at present; highly equipped physical laboratories had not yet 
 come into existence and were not then necessary. The Mathematical 
 Physicists and Astronomers of the eighteenth century were largely 
 engaged in working out the detailed implications of the law of 
 
12 MATHEMATICS, FOR THE MATHEMATICIAN 
 
 gravitation, and had commenced, largely under the influence of the 
 idea of action at a distance, to work out problems such as those 
 connected with the vibrations of strings and other bodies. In the 
 earlier part of the nineteenth century the centre of physical interest 
 passed on to such subjects as Hydrodynamics, the Conduction of Heat, 
 and Elasticity, in which an abstract representation of a body as a 
 continuous plenum was sufficient for many purposes, and made the 
 problems readily accessible to continuous Mathematical Analysis. 
 During this period much attention was given to the more obvious 
 phenomena of Electricity and Magnetism, and much of the Mathe- 
 matical Analysis which had been devised for the purpose of dealing 
 with problems of gravitational attractions, vibrations, etc., was found, 
 with further development, to be applicable to the new problems which 
 arose. Much of the work, such as that of Ampere in connection with 
 the ponderomotive forces due to electric circuits, was still carried out 
 under the influence of the idea of action at a distance, first brought 
 into prominence in connection with the Newtonian law of gravitation, 
 but the idea of the continuous medium gradually became the domi- 
 nating notion. The period in which Physical Mathematics was 
 applied with such great success to continuous media probably reached 
 its culmination in Maxwell's equations of Electrodynamics which are 
 now usually regarded as representing the average effects exhibited 
 when actual discreteness is smoothed out. In our own time the centre 
 of physical interest has transferred itself, in connection with Electro- 
 magnetism, to the molecular and sub-molecular domain, in which 
 discrete objects become the subject of scrutiny. The boundaries 
 between Physics and Chemistry have been broken down. In this 
 region of investigation the perseverance, skill, and ingenuity of 
 Physicists working in well-equipped laboratories have been rewarded 
 by the discovery, during the last two decades, of a crowd of remarkable 
 facts, probably destined to have the most far-reaching influence upon 
 our conceptions of the material world. The sifting and interpretation 
 of these facts is at present most incomplete, and will in the future 
 give rise to many problems in grappling with which work in abundance 
 will be provided for the Mathematician. As yet the statement of these 
 problems is in many cases not sufficiently precise and definite to allow 
 of translation into terms with which the Mathematician can readily 
 deal. It seems however highly probable that, when Mathematics gets 
 a grip of these problems, questions will arise which will tax to the 
 utmost the resources of Mathematical Science. It would be rash to 
 
AND FOR THE PHYSICIST 13 
 
 attempt to limit the kinds of Mathematical processes which may in 
 good time be made available for application to the new and difficult 
 problems which can be discerned at present as likely to arise in the 
 not very distant future. It has been said that the Theory of Numbers 
 is a subject which has never been soiled by any practical application. 
 Who can be absolutely sure that even so apparently transcendental 
 a /branch of thought as this will always remain undefiled by the 
 contaminating touch of physical application ? 
 
 In the history of Science it is possible to find many cases in which 
 the tendency of Mathematics to express itself in the most abstract 
 forms has proved to be of ultimate service in the physical order of 
 ideas. Perhaps the most striking example is to be found in the 
 development of abstract Dynamics. The greatest treatise which the 
 world has seen, on this subject, is Lagrange's Mecanique Analytique, 
 published in 1788. In order to characterize the spirit in which this 
 great work was conceived I cannot do better than quote the words of 
 Lagrange himself from the Preface. He writes : " We have already 
 various treatises on Mechanics, but the plan of this one is entirely 
 new. I intend to reduce the theory of this Science, and the art of 
 solving problems relating to it, to general formulae, the simple de- 
 velopment of which provides all the equations necessary for the solution 
 of each problem. I hope that the manner in which I have tried to 
 attain this object will leave nothing to be desired. No diagrams will 
 be found in this work. The methods that I explain require neither 
 geometrical, nor mechanical, constructions or reasoning, but only 
 algebraical operations in accordance with regular and uniform pro- 
 cedure. Those who love Analysis will see with pleasure that Mechanics 
 has become a branch of it, and will be grateful to me for having thus 
 extended its domain." 
 
 If ever a work was conceived in the purely abstract Mathematical 
 spirit it is surely this one. Well, let us see what came of it. Lagrange's 
 idea of reducing the investigation of the motion of a dynamical system 
 to a form dependent upon a single function of the generalized co- 
 ordinates of the system was further developed by Hamilton and 
 Jacobi into forms in which the equations of motion of a system 
 represent the conditions for a stationary value of an integral of a 
 single function. The extension by Routh and Helmholtz to the case 
 in which " ignored co-ordinates " are taken into account, was a long 
 step in the direction of the desirable unification which would be 
 obtained if the notion of potential energy were removed by means 
 
14 MATHEMATICS, FOR THE MATHEMATICIAN 
 
 of its interpretation as dependent upon the kinetic energy of concealed 
 motions included in the dynamical system. The whole scheme of 
 abstract Dynamics thus developed upon the basis of Lagrange's work 
 has been of immense value in theoretical Physics, and particularly in 
 statistical Mechanics, which is now a subject of enormous importance. 
 But the most striking use of Lagrange's conception of generalized 
 co-ordinates was made by Clerk Maxwell, who in this order of ideas, 
 and inspired on the physical side by the work of Faraday, conceived 
 and developed his dynamical theory of the Electromagnetic field, and 
 obtained his celebrated equations. The form of Maxwell's equations 
 enabled him to perceive that oscillations could be propagated in the 
 electromagnetic field with the velocity of light, and suggested to him 
 the Electromagnetic theory of light. Heinrich Herz, under the direct 
 inspiration of Maxwell's ideas, demonstrated the possibility of setting 
 up electromagnetic waves differing from those of light only in respect 
 of their enormously greater length. We thus see that Lagrange's 
 work, conceived in the spirit he has himself described, was an essential 
 link in a chain of investigation of which one result is of the kind which 
 gladdens the heart of the practical man, viz. wireless telegraphy. 
 
 The habits of thought of such men as Lagrange have however 
 their drawbacks, as I proceed to illustrate by means of a quotation 
 from Thomson and Tait's Natural Philosophy. "With reference to the 
 determination of the motions of a system disturbed from a position of 
 equilibrium, by means of an equation of which the roots are the fre- 
 quencies of the resulting vibrations, they write : " It is remarkable 
 that both Lagrange and Laplace fell into the error of supposing that 
 equality among roots necessarily implies terms in the solution of the 
 form te^^ (or tco^pt\ and therefore that for stability the roots must all 
 be unequal. This we find in the Mecanique Analytique, in the second 
 edition of 1811 published three years before Lagrange's death, and 
 repeated without change in the posthumous edition of 1853. It would 
 be curious if such an error had remained for twenty-three years in 
 Lagrange's mind. It could scarcely have existed even during the 
 writing and printing of the article for his last edition if he had been 
 in the habit of considering particular applications of his splendid 
 analytical work : if he had he would have seen that a proposition 
 which asserted that the equilibrium of a particle in the bottom of a 
 frictionless bowl is unstable, if the bowl be a figure of revolution with 
 its axis vertical, cannot be true." The correct theory of equal roots 
 was first given by Routh. That even physical instinct of the highest 
 
AND FOR THE PHYSICIST 15 
 
 order does not always preserve the mind from such errors is curiously 
 illustrated by the statement made a little further on by Thomson and 
 Tait, that "an error the converse of that of Laplace and Lagrange 
 occurred in page 278 of our first edition." 
 
 An interesting instance in which some very abstract departments 
 of Mathematics were largely dependent for their origin upon concrete 
 physical problems is the case of the problems connected with the vibra- 
 tion of strings and bars. These problems were discussed by D. Bernoulli, 
 Euler, and Lagrange, in the eighteenth century, and led to the attempt 
 to represent arbitrarily prescribed functions by means of trigonometrical 
 series. To the discussions as to the possibilities in this direction are 
 largely owing the progressive development of the notion of a Mathe- 
 matical function, culminating in the extreme generality of the notion 
 as it is at present accepted. In the hands of Fourier, the representa- 
 tion by trigonometrical series acquired the form which has become of 
 so much importance in the resolution of disturbances, such as the 
 tides, into harmonic components. But for those of enquiring mind it 
 was also the starting point of a long series of purely Mathematical 
 investigations, and it gave rise to all sorts of difficult questions of 
 theoretic interest. The writings of Dirichlet, Riemann, G. Cantor, and 
 more recently of Lebesgue, are witnesses to the very general nature of 
 the discussions which have arisen in this connection ; and which, to a 
 student of the history of the Science, appear as the natural evolution in 
 an order of ideas which had its origin in the problems of vibrations 
 and of the conduction of heat. The developments which arose in 
 this way have been a dominant factor in the modern transformation 
 of Mathematical Analysis. 
 
 Poincar^ has said that Mathematics is the art of giving the same 
 name to different things. We have here in epigrammatic form the 
 statement of a most fundamental characteristic of modern Mathematics; 
 and one which explains the high degree of abstractness that many of 
 its branches exhibit. It has frequently happened that processes which 
 have been employed at different times and by different investigators 
 in connection with things of very diverging characters have been per- 
 ceived to have an underlying identity that was in the earlier stages 
 quite unsuspected. When this perception has once been made, these 
 processes dealing with things that are superficially of the most varying 
 characters become, from the abstract point of view of the Mathematician, 
 identical ; the same names and the same language become applicable 
 to all of them. For example, the modern theory of groups of operations 
 
16 MATHEMATICS, FOR THE MATHEMATICIAN 
 
 is of this character. Whatever be the more concrete character of 
 a set of operations, from the point of view of the theory of groups, 
 the group alone is the one essential characteristic of these operations, 
 and all sets of operations, whatever be their nature, which have the 
 same group, may be regarded as identical. The analysis of the nature 
 of the abstract group itself furnishes results which are immediately 
 interpretable for operations of the most varied character, provided 
 they have one and the same group. The relation between operations 
 connected with the roots of a quintic equation and the operations 
 connected with the rotations of an icosahedron into coincidence with 
 itself is an example of this representation by one and the same abstract 
 group. The many methods of transformation by means of which a 
 single theorem once established is capable of translation so as to give 
 a whole series of theorems about objects that are in their nature quite 
 distinct from those with which the original theorem had to do, afford 
 examples of the identity of Mathematical form underlying differences 
 in every aspect except just that one which is of importance for the 
 specific purpose of the Mathematician. For him the most striking 
 differences between different species of objects may be all irrelevant 
 when he is concerned only with certain aspects of those objects which 
 are identical in the different species for his special purpose. The 
 importance which the Mathematician attaches to elegance and con- 
 ciseness in his results and formulae is not due merely to a species of 
 aesthetic taste, although such aesthetic instinct often acts as an 
 efficient guide. By attention to these details of Mathematical form he 
 is frequently led to the discovery of unsuspected relations between 
 different departments, and to a synthesis in which unity takes the 
 place of an earlier diversity, leading to a true economy of thought. 
 An examination of the mode in which those vast modern develop- 
 ments which we group under the name of Geometry have arisen, 
 illustrates in an instructive manner the natural growth of Mathematical 
 theories from the parent stem rooted in empirical observation of the 
 physical world, and it brings to light the inherent necessity which 
 causes such a branch of knowledge to take up forms of ever-increasing 
 grades of abstractness. In its origin, as a study of the forms and 
 relative positions of material bodies, Geometry is a physical Science 
 built up by observation and experiment ; on its practical side, as 
 applied Geometry, it still retains that character. But at an early 
 stage, with the Greeks, it became a purely rational Science, for the 
 further development of which empirical observation was no longer 
 
AND FOR THE PHYSICIST 17 
 
 a necessity. In rational Geometry, the objects dealt with, points, 
 straight lines, circles, conies, etc., are not physical objects, but are 
 abstractions from objects perceived by the senses; they are idealiza- 
 tions of such objects of perception, in that they are regarded as 
 possessing in perfection certain properties which we discern in the 
 corresponding physical objects as only approximately realized in 
 greater or less degree. Thus, for example, the straight line in rational 
 Geometry has in absolute perfection the properties of linearity and 
 straightness, whereas these properties are only imperfectly realized 
 in any physical object which we may take as affording a sensuous 
 image of a straight line. The precise properties of these idealized 
 objects are fixed by means of some scheme of definitions, axioms, and 
 postulates, the nature of which is to a large extent, but as we shall 
 see not wholly, determined by empirical observation of actual relations 
 in the physical domain. Only on the condition that sense data under- 
 went this process of idealization could Geometry as an exact rational 
 Science become possible. The procedure in other physical sciences is 
 not generically different from that in Geometry. Any one of them 
 becomes a rational Science when, and so far as, a schematic representa- 
 tion of the phenomena witli which it deals is set up. The peculiarity 
 of Geometry is that it became a purely rational Science earher, and by 
 more rapid stages, than could be the case in other departments of 
 physical investigation which have not yet emerged from the stage in 
 which empirical observations form an essential element in the process 
 of furthering our knowledge. The reason of the fact that Geometry, 
 having become a purely rational Science, is in a more advanced stage 
 than other Sciences, is to be found in the comparative obviousness of 
 the empirical observations on which it is based. The history of the 
 development of Geometry is of general interest because Geometry may 
 be regarded as the type to which every Science may be expected to 
 conform at the distant time when it has become completely rational. 
 When that distant consummation has been reached, laboratories will 
 be useful only for illustrative and didactic purposes, just as drawings 
 and models are still used in Geometry, but will no longer be essential 
 for purposes of research ; the progress of such a Science will, as is the 
 case for Geometry at presetit, consist of a detailed development of the 
 schematic representations of the phenomena. Just as Geometry has 
 no need of further empirical fact, completely rationalized Physics and 
 Chemistry, as ideal schemes, would contain within themselves every 
 element which could be supplied by physical observation, and would no 
 
18 MATHEMATICS, FOR THE MATHEMATICIAN 
 
 longer be dependent for their further progress upon the work of the 
 experimenter. It is of course true that the amount and nature of the 
 empirical data required by such Sciences as Physics and Chemistry are 
 of such a character that we can hardly conceive such a consummation 
 as the one I have spoken of as anything but an ideal ; how far that ideal 
 is capable of realization posterity alone will decide. For a long time 
 to come these Sciences can be only partly rational, but their line of 
 progress is such that the rational part of them must ever increase 
 relatively to their empirical part. 
 
 I have said that the properties and relations assigned to the ideal 
 objects of rational Geometry arose out of suggestions originating in 
 observations of spatial properties and relations in the physical world ; 
 there are however noteworthy limitations to the amount of leading 
 that can in this regard be obtained from our actual perceptions. In 
 the first place, all our spatial perceptions are affected by an essential 
 element of inexactitude, the amount of which varies with the precision 
 of the instruments we use for measurements, but which can never be 
 wholly eliminated. In the second place, all our actual observations 
 have reference to some more or less bounded, and certainly finite, portion 
 of what we call physical space, although we have the intuition that it 
 is always possible to pass beyond the boundaries of the space at any 
 time observed. In rational Geometry, we have in the scheme of axioms, 
 definitions, and postulates, not only to make precise statements as to 
 the possession by certain entities of certain precise properties, but we 
 have also to make statements which refer to what happens in every part 
 of unbounded space ; and to do this w^e have to pass in certain respects 
 beyond anything we can learn from physical observation. In one point, 
 namely in connection with what is usually known as the axiom of 
 parallels, this disability has proved to be of the greatest importance 
 in the development of the Science. Euclid's axiom of parallels, being 
 a statement of what happens in unbounded space, is essentially in- 
 capable of direct verification. Indirect verification, for example, by 
 observation of the sum of the angles of a triangle is indecisive, on 
 account of the essential inexactitude of our measurements, and on 
 account of the fact that the size of the triangles that can be observed 
 even by the Astronomer is limited. All attempts to prove the truth 
 of the axiom, that is to deduce it from the other axioms and postulates, 
 proved a failure in face of the most determined efforts to throw light 
 upon the matter ; it has now been demonstrated that this failure was 
 inevitable. It is absolutely necessary for the development of rational 
 
AND FOR THE PHYSICIST 19 
 
 Geometry either to postulate, as Euclid does, in some form or other, 
 the truth of the axiom of parallels, or else to substitute for it some 
 other postulate of a divergent character, but still not inconsistent with 
 our intuitions of actual spatial relations. During the last century, the 
 actual failure to prove the truth of Euclid's axiom within the rational 
 scheme, led to the investigation of the results of making a different 
 postulation, and at least two systems, that of Bolyai-Lobachewsky, and 
 that of Riemann, were invented. Each of these contains a substitute 
 for Euclid's axiom, and is inconsistent with it, and they are incon- 
 sistent with one another; but each is, when suitably interpreted, a 
 sufficient representation of our actual space perceptions, although not 
 so simple a representation as that of the Euclidean scheme. It has 
 been demonstrated that each of these schemes is logically self-consistent, 
 and thus the possibility of diverging conceptual schemes for the 
 representation of a single set of perceptual data has been established. 
 This is a result of the highest possible general interest. The view is 
 no longer tenable that the relations of time and space exist in the 
 mind as empty forms of thought, as in Kant's scheme, prior to all 
 perception, and as forms in which all our spatial and temporal per- 
 ceptions clothe themselves. Nor is it true that the conceptual schemes 
 for spatial and temporal relations are simple products of empirical 
 observation. They could never have arisen apart from our actual 
 experience of the external world, but they contain much more than 
 the raw data obtained from perception. The rise from the rough data 
 of sensuous perception to a rational scheme representing the relations 
 involved can only be accomplished by a process in which the reflective 
 activities of the mind contribute an essential element which is lacking 
 in the data themselves. I have now to refer to a most important fact 
 in this connection, namely that the mode in which the data of the 
 senses can be idealized and completed leaves a latitude in which there 
 is a large element of free choice left to the mind. In the case of 
 Geometry this element of free choice has been exercised in the con- 
 struction of various types of Geometry, all of which are logically 
 possible, and several of which are equally applicable for the repre- 
 sentation of our actual spatial experience. Thus Geometry has been 
 transformed into Geometries. This remarkable result suggests that 
 in other Sciences it is possible that the system of laws forming the 
 schematic representation of the phenomena may not necessarily be 
 unique ; that there may be in any given domain several or even an 
 unlimited number of different rational schemes which will all be 
 
20 MATHEMATICS, FOR THE MATHEMATICIAN 
 
 equally valid; although, as in the case of Geometry, they will not 
 be all equally simple. 
 
 The great discovery of Descartes which in our time has become 
 part of the mental furniture of multitudes of non-Mathematicians, in 
 its various applications to graphical representation of variables, gave 
 rise to a revolutionary increase in the scope and in the power of 
 geometrical Science, by reducing Geometrical investigation to a form 
 in which the relations of number are made to do the work, and in 
 which the powerful and expanding methods of Algebraical Analysis 
 are made available for research in the domains of the various 
 Geometries. Here again the mode in which geometrical objects 
 shall be correlated with numbers and number-systems leaves to the 
 mind a large element of free choice, and has led to the study of 
 schemes varying in accordance with the manner in which this free 
 choice is exercised. The correlation of the points of a straight line, 
 for example, with a scheme of numbers, leaves latitude of the kind 
 I have described ; we have, for example, not only the m.ore usual 
 schemes of correlation in which the existence of actually infinitesimal 
 segments of a straight line is denied, in the so-called Archimedean 
 systems, but there have been developed also non- Archimedean systems 
 in which quite different postulations are fundamental. The differences 
 in these various modes of correlation are in a region which is inac- 
 cessible to any kind of verification by sense-perceptions, and is of no 
 direct interest to the Physicist, but the contemplation of the various 
 possibilities implied in these schemes is forced upon the mind of the 
 Mathematician in the course of his endeavours to explore and complete 
 the rational foundations of the subjects which he studies. Time will 
 not allow me to refer in detail to the vast modern developments such 
 as projective Geometry, descriptive Geometry, line Geometry, etc., but 
 I will refer briefly to one further great step in abstraction or generaliza- 
 tion. All geometrical schemes may be regarded as specializations of 
 an abstract scheme in which the subject-matter consists of the possible 
 determinations which can be made within an ordered manifold, in 
 which the order is not necessarily spatial or temporal, but in which 
 the notion of order is purely abstract, and of which spatial and temporal 
 orders are merely specializations. The elements of such a manifold 
 are regarded as purely ideal objects, identified only by means of some 
 assumed correlation with a number-system which may have any number 
 of dimensions, or even an infinite number of dimensions ; the order of 
 the elements is assigned by means of the ordinal relations of the 
 
AND FOR THE PHYSICIST 21 
 
 number-systems with which the elements are correlated. A generaliza- 
 tion of this kind would appear to be the final abstraction, of which all 
 Geometries are specialized examples; from this final generalization 
 all remnants of specially spatial or specially temporal intuition have 
 been removed by abstraction, the relations of order alone remaining. 
 The advantages of the contemplation of such a purely abstract scheme 
 consist in its suitability as a means of investigation and classification 
 of all those schemes of a less abstract character which may be subsumed 
 under it as specialized instances. It may seem strange that the in- 
 vestigation of the relations within an ordered manifold of purely ideal 
 objects should still be sometimes spoken of as Geometry, although the 
 relations with which it deals have no necessary, but only an adventitious, 
 connection with what we call spatial relations. 
 
 I have already referred to the difficulties which occur in connection 
 with the extension of our intuitions of finite relations to the domain 
 of the infinite. I propose here to give one illustration which I think 
 exhibits strikingly the necessity for caution in this matter, the lack 
 of which has often led to generalizations which were afterwards found 
 to be untenable, at least in their original form. 
 
 1 2 3 
 
 H 4 
 
 Let us contemplate a straight line AB, say a mile long. If we 
 imagined that there were removed from this straight line a number 
 of segments, or portions, the total length of which is less than one 
 foot, what would remain ? Naturally, a number of segments whose 
 total length would be under a foot less than one mile. That is the 
 answer given by our intuition, and it is a perfectly correct one, but 
 only so long as the number of segments removed is finite. I propose 
 to show you that it is possible to conceive an infinite set of segments 
 removed from the straight line, the total length of which is less than 
 one foot, in such wise, that in what remains there are no segments at 
 all, but only points, so that nothing is left which can be said to have j U^) 
 a length. 
 
 Take the various proper fractions 
 1 12 123 1234 12345 
 
 2' 3'3' 4'4'4' 5'5'5'5' 6'6'6'6'6' 
 
 .... Here every proper fraction occurs at a definite place in the 
 
22 MA.THEMATICS, FOR THE MATHEMATICIAN 
 
 sequence, and in fact it occurs not only once, but an indefinite 
 number of times. Let us suppose that, corresponding to each frac- 
 tion, a point representing the corresponding fraction of the mile is 
 taken on AB. Remove a segment, of length half a foot, with its 
 
 centre at the point - . Then remove a segment, of length ^ of a 
 
 foot, with its centre at the point - ; then a segment, of length -^ foot, 
 
 2 
 
 with its centre at the point - ; and so on. We thus imagine removed 
 
 o 
 
 a segment, of length foot, with its centre at the point represented by 
 
 the ?^th fraction. We can disregard those fractions which are not 
 in their lowest terms, because the segments which have the corre- 
 sponding points for centres have been already removed. The total 
 length of all the segments which we conceive to have been removed 
 is, since some of them are contained in others, or overlap others, 
 
 certainly less than h + ^2 + ^+--- +h^+--- ^^^ foot, and is therefore 
 
 less than one foot. All the points corresponding to the rational 
 fractions exist on AB^ and there can be drawn no length qys. AB 
 which is devoid of these points. As we regard all these points as 
 existing, we may also regard all the segments with these points as 
 centres as existent, and to have been removed from AB. When these 
 segments have all been removed there can be no segment oi AB left, 
 because any such segment contains points of the above set, and thus 
 segments with such points as centres will have been removed. Nothing 
 can remain in J. ^except points which make up no single segment. 
 We have thus the apparently paradoxical result that, by removing 
 from a straight line of length one mile, a set of portions, the total 
 length of which is less than one foot, there are left no portions of the 
 straight line at all, but only single points. 
 
 Mathematics may perhaps be compared with a set of games which 
 Mathematicians occupy themselves in playing. In all the earlier 
 games of the set the rules were suggested by physical experience, 
 and were useful as a representation of the behaviour of actual bodies ; 
 like other species of games these were imitative in character. But it 
 was found necessary to give the rules complete precision in their form, 
 and it ultimately turned out that this could be done in a variety of 
 ways, and hence what was originally one game split up into a variety 
 
AND FOR tije; /K&tsi0$m.I'r. :'::/". 23 
 
 of different games played according to different rules. Those who 
 were primarily interested in the games for their own sake were 
 interested not only in those with the simplest sets of rules, but in 
 others of less simplicity. Those, viz. the Physicists, who are primarily 
 interested only in the imitative aspect of the games, confine their 
 attention to those species of games in which the rules are of the 
 simplest possible character compatible with their imitative purpose. 
 To the Mathematician it has always been a matter of supreme interest 
 to find out what would be the effect of altering or extending the rules 
 of a game in various ways, thus creating new games. Every now and 
 then it has been discovered that what appeared to be quite different 
 games with different sets of rules could be reduced to a single game 
 with a single set of rules which in a sense superseded the original 
 games. The only rules that are absolutely essential in all the games 
 are those known as the laws of logic ; the other rules always possess 
 a greater or less degree of arbitrariness. When the players become 
 too skilful, and the games become too complicated, onlookers have 
 sometimes a difficulty in comprehending the ardour of the players, 
 and are apt to think that the matter has been carried beyond all 
 reason. But ^ the Mathematical games are played with definite 
 purposes ; they are infinite in variety ; they call forth and develop 
 some of the highest faculties of the human mind. A true estimate 
 of their value can be made only if they are observed in a sympathetic 
 spirit, and in the light of a comprehensive view of the history of human 
 thought. If this is done the general verdict cannot fail to be a 
 favourable one. 
 
 I have endeavoured to point out, in a necessarily fragmentary and 
 incomplete manner, some of the main differences which distinguish the 
 view of Mathematics taken by the Mathematician from its aspect for 
 the Physicist. In conclusion, I would emphasize the fact that, although 
 the study of Mathematics which the Physicist or the Engineer must 
 make has rightly many points of difference from the study as pursued 
 for other purposes by the Mathematician proper, it must nevertheless 
 be of a serious and thorough character, if it is to be really effectual 
 for the purposes for which it is needed. Many branches of modern 
 Mathematics may and must be disregarded by those who study 
 Mathematics mainly with a view to its application in Physical Science, 
 and many details may be omitted which are of subordinate importance ; 
 but so far as it is studied at all it must be studied as a Science, not as 
 
24 JiAlHSM>xTrcS/F0?t, ;MXTHf3MATICIAN AND PHYSICIST 
 
 an Art. Its study must be pursued in a spirit of patience, without too 
 exclusive attention to those parts of it which can be immediately- 
 applied to practical problems. The instruments provided by Mathe- 
 matics are of so delicate a character that they are apt to break in 
 the necessarily unskilful hands of those that do not understand their 
 construction. 
 
 Cambridge: Printed at the University Press. 
 
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