2i8 THE POPULAR SCIENCE MONTHLY 



able difference of opinion which there is among mathematicians regard- 

 ing the existence of a definable infinity as due to the difference in the 

 psychology of the two classes. One, the idealistic, feeling that every- 

 thing we define is due to the mind, and finite; the other, the realistic, 

 feeling that there is an external world which may well contain an in- 

 finity. The idealistic class, to which Poincare belonged, would consider 

 that these extensions to which we referred are in a sense creations. 



It is scarcely necessary to enumerate the creations of Poincare. 

 They are many, for he was gifted with extraordinary originality. The 

 account given above of the creation of the fuchsian functions is an ex- 

 ample of one of his most important. It opened an immense field of in- 

 vestigation. He created a type of arithmetic invariants expressible as 

 series or definite integrals, which opened a new field in theory of num- 

 bers. His investigations of ordinary differential equations which are 

 not linear, such as those in dynamics and the problem of N bodies, cre- 

 ated an extensive class of new functions which (I believe) are yet with- 

 out special names, as well as suggesting the existence of classes of func- 

 tions for which we have, as yet, even no means of expression. The in- 

 vestigations of asymptotic expansions opened paths to dizzy heights. 

 Fundamental functions in partial differential equations also open a re- 

 gion now under development. We may say that the most marvelous of 

 his creations rise from the general field of differential equations. We 

 might cite further his researches in analysis situs, the realm of the in- 

 variants of a battered continuity. His double residues and studies in 

 functions of many real variables are creations from which will spring a 

 noble progeny. Even the lectures in which he presented the results of 

 others scintillate with original thoughts. 



To generalize in mathematics and science it is not enough simply to 

 get together facts or ideas and to put them into new combinations. Most 

 of these combinations would be useless. The real investigator does not 

 form the useless combinations at all, but unconsciously rejects the un- 

 profitable combinations. It is as if he were an examiner for a higher 

 degree ; only the candidates who have passed the lower degrees ever ap- 

 pear before him at all. Often domains far distant furnish the useful 

 combinations, as in the account given of the genesis of the fuchsian 

 functions, the theory of arithmetic forms through the roundabout route 

 of non-euclidean geometry furnished the generalization of the first 

 fuchsian functions to the complete class. This was of the first type. 

 But how are those of the second type born ? 



We come thus to the heart of the matter. Merely to say that we dis- 

 cover laws is not sufficient. How do we discover extensions? How 

 devise new formulas? Make new constructions? The answer to this 

 question is, for Poincare, found in psychology. It is necessary to get 

 together many facts, but this does not insure that we shall build with 



