THE VALUE OF SCIENCE 177 



ately. Experiment is individual, the law deduced from it is general; 

 experiment is only approximate, the law is precise, or at least pretends 

 to be. Experiment is made under conditions always complex, the 

 enunciation of the law eliminates these complications. This is what is 

 called ' correcting the systematic errors.' 



In a word, to get the law from experiment, it is necessary to 

 generalize; this is a necessity imposed upon the most circumspect ob- 

 server. But how generalize? Every particular truth may evidently 

 be extended in an infinity of ways. Among these thousand routes 

 opening before us, it is necessary to make a choice, at least provisional; 

 in this choice, what shall guide us ? 



It can only be analogy. But how vague is this word ! Primitive 

 man knew only crude analogies, those which strike the senses, those of 

 colors or of sounds. He never would have dreamt of likening light to 

 radiant heat. 



What has taught us to know the true, profound analogies, those the 

 eyes do not see but reason divines? 



It is the mathematical spirit, which disdains matter to cling only 

 to pure form. This it is which has taught us to give the same name 

 to things differing only in material, to call by the same name, for 

 instance, the multiplication of quaternions and that of whole numbers. 



If quaternions, of which I have just spoken, had not been so 

 promptly utilized by the English physicists, many persons would doubt- 

 less see in them only a useless fancy, and yet, in teaching us to liken 

 what appearances separate, they would have already rendered us more 

 apt to penetrate the secrets of nature. 



Such are the services the physicist should expect of analysis :, but for 

 this science to be able to render them, it must be cultivated in the 

 broadest fashion without immediate expectation of utility — the mathe- 

 matician must have worked as artist. 



What we ask of him is to help us to see, to discern our way in the 

 labyrinth which opens before us. Now, he sees best who stands highest. 

 Examples abound, and 1 1 limit myself to the most striking. 



The first will show us how to change the language suffices to reveal 

 generalizations not before suspected. 



When Newton's law has been substituted for Kepler's, we still know 

 only elliptic motion. Now, in so far as concerns this motion, the two 

 laws differ only in form; we pass from one to the other by a simple 

 differentiation. And yet from Newton's law may be deduced by an 

 immediate generalization all the effects of perturbations and the whole 

 of celestial mechanics. If, on the other hand, Kepler's enunciation 

 had been retained, no one would ever have regarded the orbits of the 

 perturbed plants, those complicated curves of which no one has ever 

 written the equation, as the natural generalizations of the ellipse. The 



V>L, LXX. — 12. 



