TRANSACTIONS OF THE SECTIONS. O 



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

 pletely will these signs that it is still in 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 quotes the saying of an old French 

 geometer, " that a mathematical theoiy 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 I know of no reason for disputing his opinion ; but I believe the exag- 

 geration would really be very small if the dictum were applied to the theories of 

 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 under- 

 stand 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 adapted 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 

 aud vigour if he had once made up bis mind how he would treat his subject suppo- 

 sing 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 de- 

 pendent upon help derived from the former, I have tried to point out some 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 

 larger part of the accessions to our mathematical knowledge have been obtained by 

 the efforts of mathematicians to solve the problems set to them by experiment." 

 We 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 

 than 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 fluxions, to Fourier's theorem and the doctrine of harmonic analysis, 

 to spherical harmonics, and to the theory of potential, out of which, in the hands of 

 Riemann, there grew a general and most remarkable theory of mathematical 

 functions. 



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

 jects and new opportunities which they offer for physical investigation. 



A very remarkable and important example of the former kind of influence is 



