176 POPULAR SCIENCE MONTHLY 



This is why I do not hesitate to say that mathematics deserve to be 

 cultivated for their own sake, and that the theories inapplicable to 

 physics should be so as well as the others. Even if the physical aim and 

 the esthetic aim were not united, we ought not to sacrifice either. 



But more: these two aims are inseparable and the best means of 

 attaining one is to aim at the other, or at least never to lose sight of it. 

 This is what I am about to try to demonstrate in setting forth the 

 nature of the relations between the pure science and its applications. 



The mathematician should not be for the physicist a mere pur- 

 veyor of formulas; there should be between them a more intimate 

 collaboration. Mathematical physics and pure analysis are not merely 

 adjacent powers, maintaining good neighborly relations ; they mutually 

 interpenetrate and their spirit is the same. This will be better under- 

 stood when I have shown what physics gets from mathematics and 

 what mathematics, in return, borrows from physics. 



II 



The physicist can not ask of the analyst to reveal to him a new 

 truth; the latter could at most only aid him to foresee it. It is a 

 long time since one still dreamt of forestalling experiment, or of con- 

 structing the entire world on certain premature hypotheses. Since all 

 those constructions in which one yet took a naive delight it is an age, 

 to-day only their ruins remain. 



All laws are therefore deduced from experiment; but to enunciate 

 them, a special language is needful; ordinary language is too poor, it 

 is besides too vague, to express relations so delicate, so rich, and so 

 precise. 



This therefore is one reason why the physicist can not do without 

 mathematics; it furnishes him the only language he can speak. And 

 a well-made language is no indifferent thing ; not to go beyond physics, 

 the unknown man who invented the word heat devoted many genera- 

 tions to error. Heat has been treated as a substance, simply because it 

 was designated by a substantive, and it has been thought indestructible. 



On the other hand, he who invented the word electricity had the 

 unmerited good fortune to implicitly endow physics with a new law, 

 that of the conservation of electricity, which, by a pure chance, has been 

 found exact, at least until now. 



Well, to continue the simile, the writers who embellish a language, 

 who treat it as an object of art, make of it at the same time a more 

 supple instrument, more apt for rendering shades of thought. 



We understand, then, how the analyst, who pursues a purely esthetic 

 aim, helps create, just by that, a language more fit to satisfy the 

 physicist. 



But this is not all : law springs from experiment, but not immedi- 



