748 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1912. 



places by the new and odd theory of "quanta," to the construction 

 of which several physical discoveries have led. This supposes a cer- 

 tain physical discontinuity in the atomic phenomena which produce 

 radiation.^ Not wishing to go too much into detail on this subject, 

 I am gomg to make a somewhat bold comparison, but one which is 

 perhaps not wholly void of meaning: The hypothesis of quanta has 

 grown up side by side with that of continuity, just as in biology the 

 Larmarkian and Darwinian theories of slow and imperceptible evolu- 

 tion have come recently face to face with those of sudden and dis- 

 continuous mutation of the Dutch naturalist, De Vries. The latter, 

 by the new evidence which he has brought forward, has not destroyed 

 the older theory; he has merely enlarged it, shown its limitations, and 

 left it intact in its greater significations. Similarly, the theory of 

 quanta, it is probable, will not prevent the greater part, if not all, of 

 physical phenomena from being capable of representation by differ- 

 ential equations. The progress which the latter have brought, the 

 physical discoveries to which they have led, notably in optics, in 

 electricity, and in astronomy, are their guaranty. Accordingly, the 

 new functions discovered by Poincare will always remain one of the 

 most brilliant contributions which he brought through pure theory 

 to the study of external phenomena. 



If we study the characteristics of the method of Poincar^ and of his 

 mathematical genius, we find especially a wonderful faculty for gen- 

 eralization. Instead of starting, as do most students, with a study of 

 the minor details, he jumped to the very heart of his problem, neglect- 

 ing the intermediate details, like an audacious conqueror, who, with- 

 out preliminary skirmishing, makes his first onslaught upon the 

 master difficulty, the most impregnable fortress, inventing on the 

 spot the instruments for subduing it and then forcing its sm-render 

 without striking a blow. Then we would leave to others the investi- 

 gation and organization of the new province which he had just won 

 and pass immediately to other conquests. In that sense we may 

 speak of him as being "more a conqueror than a colonizer. "^ There 

 resulted that peculiar method of thought so noticeable in his philo- 

 sophical writings, so disconcerting at times to the novice and which 

 brought upon lum the reproach of being disconnected. True, Poin- 

 care's process of reasoning was not smooth and continuous; he pro- 

 ceeded by successive bounds which had more the effect of a broken 

 Ime. But the profile of a diamond is likewise made of broken lines, 

 from the very virtue of which its brilliancy results. Such a logical 

 method is not common, but, borrowing the words of M, Painleve, 



1 Poincard summarized excellently in the following words shortly before his death the conclusions to 

 which the theory of quanta led him: "A physical system can exist only in a finite number of distinct states: 

 it leaps from one of these states to another without passing through a continuous series of intermediate 

 states." 



* Borel: Revue du Mois, vol. 7, p. 362. 



