4 REPORT 1877. 



The consequence is that from the very outset of his investigations the physicist 

 has to rely constantly on the aid of the mathematician ; for, even in the simplest 

 cases, the direct results of his measuring operations are entirely without meaning 

 until they have been submitted to more or less of mathematical 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 mathematician 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 unsuspected. 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 experiment, to ascertain how electricity is dis- 

 tributed upon a charged conductor ; and this problem has been solved by mathema- 

 ticians 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 cooperates in the discovery of physical 

 truths is almost exactly the converse of that last mentioned. In very many cases t he 

 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 expeii- 

 mental 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 applies in the case mentioned can be deduced from 

 the deflections 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 has 

 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 intellectual catastrophe, supposing 

 such a thing possible, whereby all knowledge 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 phenomena would, indeed, 

 remain as they are now, but physical science would cease to exist. We should no 

 doubt still 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 meaniug ; the)' 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 know- 

 ledge 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 organized body of science. 



