August 1 6, 1877J 



NA TURE 



313 



discussion. And when in this way some interpretation of the 

 experimental results has been arrived at, and it has been proved 

 that two or more physical quantities stand in a definite relation 

 to each other, the mathemntician is very often able to infer, 

 from the existence of this relation, that the quantities in question 

 also fulfil some other relation, that was previously unsus- 

 pected. Thus when Coulomb, combining the functions of 

 experimentalist and mathematician, had discovered the law of 

 the force exerted between two particles of electricity, it became 

 a purely mathematical problem, not requiring any further experi- 

 ment, to ascertain how electricity is distributed upon a charged 

 conductor, and this problem has been solved by mathematicians 

 in several cases. 



It thus happens that a very large part of our knowledge of 

 physics is due in the first instance to the mathematical discussion 

 of previous results, and is experimental only in the second, or 

 perhaps still more remote degree. 



Another way in which the mathematician co-operates in the 

 discovery of physical truths is almost exactly the converse of 

 that last-mentioned. In very many cases the most obvious and 

 direct experimental method of investigating a given problem 

 is extremely difficult, or for some reason or other untrustworthy. 

 In such cases the mathematician can often point out some 

 other problem more accessible to experimental treatment, the 

 solution of which involves the solution of the former one. 

 For example, if we try to deduce from direct experiments the 

 law according to which one pole of a magnet attracts or repels 

 a pole of another magnet, the observed action is so much 

 complicated with the effects of the mutual induction of the 

 magnets and of the forces due to the second pole of each 

 magnet, that it is next to impossible to obtain results of any 

 great accuracy. Gauss, however, showed how the law which 

 applied in the case mentioned can be deduced from the deflec- 

 tions undergone by a small suspended magnetic needle when 

 it is acted upon by a small fixed magnet placed successively in two 

 determinate positions relatively to the needle ; and being an 

 experimentalist as well as a mathematician, he showed likewise 

 how these deflections can be measured very easily and with 

 great precision. 



It thus appears not only that mathematical investigations 

 have aided at every step whereby the present stage in the 

 development of a knowledge of physics have been reached, 

 but that mathematics has continually entered more and more 

 into the very substance of physics, or, as a physiologist might 

 say, has been assimilated by it to a greater and greater extent. 



Another way of convincing ourselves how largely this process 

 has gone on would be to try to conceive the effect of some intellec- 

 tual catastrophe, supposing such a thing possible, whereby all 

 knov/ledge of mathematics should be swept away from men's 

 minds. Would it not be that the departure of mathematics 

 would be the destruction of physics ? Objective physical pheno- 

 mena would, indeed, remain as they are now, but physical 

 science would cease to exist. We should no doubt see the same 

 colours on looking into a spectroscope or polariscope, vibrating 

 strings would produce the same sounds, electrical machines 

 would give sparks, and galvanometer needles would be deflected ; 

 but all these things would have lost their meaning ; they would 

 be but as the dry bones — the disjecta membra — of what is now 

 a living and growing science. To follow this conception further, 

 and to try to image to ourselves in some detail what would be 

 the kind of knowledge of physics which would remain possible, 

 supposing all mathematical ideas to be blotted out, would be 

 extremely interesting, but it would lead us directly into a dim 

 and entangled region where the subjective seems to be always 

 passing itself off for the objective, and where I at least could 

 not attempt to lead the way, gladly as I would follow any one 

 who could show where a firm footing is to be found. But 

 without venturing to do more than look from a safe distance 

 over this puzzling ground, we may see clearly enough that 

 mathematics is the connective tissue of physics, binding what 

 would else be merely a list of detached observations into an 

 organised body of science. 



In my opinion, however, it would be a very serious miscon- 

 ception to suppose that on this account an elaborate apparatus 

 of technical mathematics is in general needful for the proper 

 presentation of physical truths. The ladders and ropes of 

 formula; are no doubt often essential during the building up of a 

 newly-discovered physical principle, but the more thoroughly the 

 building is finished, the more completely will these signs that it 

 is still m progress be cleared away, and easy ascents be 

 provided to all parts of the edifice. In an address delivered 



from the Chair of this Section four years ago. Prof. Henry 

 .Smith quoted the saying of an old French geometer, "that a 

 mathematical theory was never to be considered complete till 

 you had made it so clear that you could explain it to the first 

 man you met in the street." Very likely Prof Smith was right 

 to call this "a brilliant exaggeration," at any rate 1 know of no 

 reason for disputing his opinion, but I believe the exaggeration 

 would really be very small if the dictum were applied to the 

 theories cf physics instead of to those of pure mathematics. 

 When a physical principle or theory is grasped with thorough 

 clearness, I believe it is possible to explain it to the man in 

 the street ; only he must not be hurrying to catch a train ; and 

 it would, I think, be difficult to find a more wholesome maxim to 

 be kept in mind by those of us whose business it is to teach physics, 

 than that we should never think we understand a principle till we 

 can explain it to the man in the street. I do not say that our modes 

 of exposition should always be r.dapted to him, for as a rule, he 

 forms but a small part of our audience, but even when the 

 conditions are such that a teacher is free to avail himself to the 

 fullest extent of mathematical methods, I believe he would find 

 his mathematical discussions gain marvellously in freshness and 

 vigour if he had once made up his mind how he would treat his 

 ."subject supposing all use of mathematical technicalities denied 

 him. 



So far, in considering the mutual relations of mathematics 

 and physics, I have placed myself, as it was natural for me to 

 do, at a physical point of view, and, starting from the fact that 

 the existence and progress of the latter science are essentially 

 dependent upon help derived from the former, I have tried to 

 point out somef of the ways in which this help is rendered. If 

 we turn now to inquire in what light the relations between the 

 two sciences appear from the side of mathematics, we find that 

 mathematicians are not slow to admit the advantages which their 

 science derives from contact with physics. It was a saying of 

 Fourier that "a more attentive study of nature is the most 

 fruitful source of mathematical discoveries;" and Prof. Henry 

 Smith, in the Address I have already referred to, says that 

 ' ' probably by far the greater part of the accession 5 to our mathe- 

 matical knowledge have been obtained by the efforts of mathe- 

 maticians to solve the problems set to them by experiment." 

 V\'e may perhaps regard such expressions as equivalent to the 

 statement that the law of inertia is not without application even 

 to the mind of the mathematician, and that it, too, continues to 

 move in a straight line *' except in so far as it may be compelled 

 by impressed forces " to change its direction ; or, to put the 

 matter a little differently, may we not look upon the fact as 

 illustrating what is probably a general principle of mental action, 

 namely, that the human mind has no more power to create an 

 idea tlian the hand has to create matter or energy — our seemingly 

 most original conceptions being in reality due to suggestions 

 from without ? But however this may be, the fact remains that 

 the origin of many most important mathematical theorems, and 

 even entire departments of mathematics, can be distinctly traced 

 to the attempt to express mathematically the observed relations 

 among physical magnitudes. By way of illustration of this 

 statement, it may suffice to refer to the well-known cases of the 

 theory of Auctions, to Fourier's theorem and the doctrine of har- 

 monic analysis, to spherical harmonics, and to the theory of the 

 potential. 



The way in which physics reacts, so as to promote the advance- 

 ment of a knowledge of mathematics, finds in many respects a 

 close parallel in the influence exerted by the practical industrial 

 arts on the progress of physics. This influence shows itself very 

 distinctly, first, in the new conceptions and new points of view 

 which practical pursuits supply to scientific physics, and, secondly, 

 in the new subjects and opportunities which they offer for physical 

 investigation. 



A very remarkable and important example of the former kind 

 of influence is afforded by the idea of Work and the correlative 

 one of Energy. These ideas, which have been found to have a 

 most far-reaching significance, and have exerted a transforming 

 effect upon every branch of physics, owe their recognition, not 

 to the spontaneous growth of science, but to their having been 

 forced on the attention of physicists by tlie cultivators of prac- 

 tical mechanics.' Very much the same thing may also be said of 

 the modern conception of the nature of heat, and of the relation 

 between thermal phenomena and those of other branches of 

 physics. The notion of heat as a measurable magnitude, of 

 which definite quantities could be given to or taken aw.iy from 



' See, on this point, Diiliring. " Kiili^clie Geschichte der .'itlgemeinen 

 Principien der Mechaoik " (Btrlin, 1873), pp. 483-486. 



