HENRI POINCAEE AS AN INVESTIGATOR 215 



stone of the whole problem was stubborn. He was compelled to go away 

 again to perform military duty, and his mind was full of other things. 

 But one day while crossing the boulevard the solution of the last diffi- 

 culty suddenly appeared and upon verification was found to be correct. 

 In this account of the birth and growth of mathematical develop- 

 ment, which he assures us is practically the same as for all such de- 

 velopments, it is obvious that the central notion is that of generaliza- 

 tion. Elliptic, abelian and theta functions are in turn generalized into 

 a new class of transcendents. Inversion of differentials is generalized 

 into inversion of differential equations. This notion of generalization 

 we need to inspect a little closely. Mathematical generalization consists 

 of two types of thought, often not discriminated, and often scarcely to 

 be discriminated from each other. One type consists in so stating a 

 known theorem that it will be true of a wider class than in its first state- 

 ment, and the predicate asserted has a wider significance. In such gen- 

 eralization the first statement of the theorem becomes a mere particular 

 case of the second statement. Examples will occur readily to every one. 

 There are two forms of this type : in one, many known cases are brought 

 together under one law; in the other form, the law thus found is made 

 to apply to other known cases, perhaps never before suspected to be re- 

 lated to the first set. It is the guiding threads of analogy that usually 

 bring about these forms of generalization. This kind of generalizing 

 power Poincare had in high degree. In his memoir on " Partial Dif- 

 ferential Equations of Physics " 6 he says : 



If one looks at the different problems of the integral calculus which arise 

 naturally when he wishes to go deep into the different parts of physics, it is 

 impossible not to be struck by the analogies existing. Whether it be electro- 

 statics, or electrodynamics, the propagation of heat, optics, elasticity or hydro- 

 dynamics, we are led always to differential equations of the same family; and 

 the boundary conditions though different, are not without some resemblances. 

 . . . One should therefore expect to find in these problems a large number of 

 common properties. 



Also in his " Nouvelles Methodes de la Mecanique Celeste " he says : 



The ultimate aim of celestial mechanics is to solve the great question 

 whether Newton's law alone will explain all astronomical phenomena. 



In his address awarding Poincare the gold medal of the Eoyal Astro- 

 nomical Society, G. H. Darwin 7 said: 



The leading characteristic of M. Poincare 's work appears to be the immense 

 wideness of the generalizations, so that the abundance of possible illustrations 

 is sometimes almost bewildering. This power of grasping abstract principles is 

 the mark of the intellect of the true mathematician; but to one accustomed 

 rather to deal with the concrete the difficulty of completely mastering the argu- 

 ment is sometimes great. 



8 Amer. Jour. Math., Vol. 12. 



T " Scientific Papers," Vol. 4, p. 519. 



