746 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1912. 



beings of strange beauty. The pure mathematician has at his beck 

 intimate dehghts of such esthetic quahty that it often comes to pass 

 that he no longer finds interest in the exterior world, lost in a kind 

 of grand mysticism. Poincare, however, was not of that kind, 

 although his researches in geometry and analysis made him the 

 greatest mathematician of our times. "Experience," he said, "is the 

 sole source of all truth." And those words acquired a singular force 

 coming from the mouth of the greatest theorist of our epoch. That 

 was why among mathematical problems Poincare attacked especially 

 those which physics brought before him. That was why he passed 

 so readily from pure analysis to mathematical physics and then to 

 celestial mechanics. And, finally, that was why he came to reflect 

 upon the very foundations of our knowledge, upon the past and the 

 future of our world, upon the value of our thoughts as we pass to 

 the Hmits of what we can know to the borders of that abyss wliich 

 separates physics and metaphysics and into which abyss most of us 

 can not glance except with dizziness. It tore from Pascal many 

 superb sighs of grief, yet Poincare could look at such matters as he 

 looked at all other things, not with a useless despair, but without 

 prejudice and foohsh illusions, with simple, clear, and profound good 

 sense; he knew how to look at them and after a glance with his eagle 

 eye to sum up all in a word. 



II. POINCARE, THE MATHEMATICIAN. 



"My daily mathematical studies," said Poincare — " how shall I ex- 

 press myself? — are esoteric and many of my hearers would revere 

 them more from afar than close to." That is what he said one day 

 to excuse himself for speaking on a mathematical subject. When- 

 ever he commenced one of his profound lectures, in which he charmed 

 his listeners, he felt the need of thus excusing himself. Thus by his 

 modesty he knew how to make us pardon his genius. However tliat 

 may be, you will permit me to appropriate that remark for the pres- 

 ent occasion that I may not beyond measure speak of the purely 

 mathematical researches of Poincare. It would require a dozen years 

 of prehminary mathematical study for the curious reader to be able 

 to know them, and if he were familiar with the elements such as he 

 would get in the ordinary coUege course he might take a glance at 

 them. 



Were I to characterize in a few words what Poincare brought new 

 into the divers processes of calculus and which won for liim the title 

 "Princeps Mathematicorum," which unanimous consent has given to 

 no other man since Gauss, I would proceed thus: In algebra and in 

 arithmetic, where he introduced the new and fertile idea of arith- 

 metical invariants and in the general theory of functions, his dis- 



