THE VALVE OF SCIENCE 179 



Thus mathematical analogies not only , may make us foresee phys- 

 ical analogies, but besides do not cease to be useful when these latter 

 fail. 



To sum up, the aim of mathematical physics is not only to facilitate 

 for the physicist the numerical calculation of certain constants or the 

 integration of certain differential equations. It is besides, it is above 

 all, to reveal to him the hidden harmony of things in making him see 

 them in a new way. 



Of all the parts of analysis, the most elevated, the purest, so to 

 speak, will be 'the most fruitful in the hands of those who know how 

 to use them. 



Ill 



Let us now see what analysis owes to physics. 



It would be necessary to have completely forgotten the history of 

 science not to remember that the desire to understand nature has had 

 on the development of mathematics the most constant and happiest 

 influence. 



In the first place the physicist sets us problems whose solution he 

 expects of us. But in proposing them to us, he has largely paid us in 

 advance for the service we shall render him, if we solve them. 



If I may be allowed to continue my comparison with the fine arts, 

 the pure mathematician who should forget the existence of the exterior 

 world would be like a painter who knew how to harmoniously combine 

 colors and forms, but who lacked models. His creative power would 

 soon be exhausted. 



The combi nations which numbers and symbols may form' are an 

 infinite multitude. In this multitude how shall we choose those which 

 are worthy to fix our attention? Shall we let ourselves be guided solely 

 by our caprice? This caprice, which itself would besides soon tire, 

 would doubtless carry us very far apart and we should quickly cease 

 to understand each other. 



But this is only the smaller side of the question. Physics will doubt- 

 less prevent our straying, but it will also preserve us from a danger 

 much more formidable; it will prevent our ceaselessly going around in 

 the same circle. 



History proves that physics has not only forced us to choose among 

 problems which came in a crowd ; it has imposed upon us such as we 

 should without it never have dreamed of. However varied may be the 

 imagination of man. nature is still a thousand times richer. To follow 

 her we must take ways we have neglected, and these paths lead us often 

 to summits whence we discover new countries. What could be more 

 useful ! 



It is with mathematical symbols as with physical realities; it is in 

 comparing the different aspects of things that we are able to compre- 



