2 2o THE POPULAR SCIENCE MONTHLY 



and their solutions, he points out that one often has to content himself 

 with the guidance he can get by physical considerations. An example 

 of this was the use made by Klein of electrical considerations in hand- 

 ling Dirichlet's problem on a Eiemann surface. In the physical aspect 

 of the problem this would usually be sufficient, for the physical data are 

 at the best only approximate. The mathematical necessities of converg- 

 ence demand, however, that the problems be handled purely analytically 

 and deductively. In one of his lectures he compares the process with 

 the formation of a sponge. When we find it fully formed it is only a 

 delicate lace-work of needles of silica. But the really interesting thing 

 is the form it has taken, and this can be fully understood only by know- 

 ing the life-history of the sponge which has impressed its form, its will, 

 so to speak, on the silica. In the same way a logical development of a 

 theorem can really be understood only through a study of its living de- 

 velopment. Need we point out the significance of this to the research 

 student? Just as a painter who would become great must sit at the 

 feet of a master and see his creations grow on the canvas, so the student 

 does well to watch a master at work on scientific creations. This is the 

 good he gets at the university. No compendium of results of the great 

 creators will suffice. Nor is a too detailed study of the history of a 

 problem, or too extensive a list of its bibliography, of assistance to the 

 intuition. These might assist the later logical development, but not the 

 inventive power. Poincare rarely did more than to acquaint himself 

 with the present status of a problem he desired to consider. It is evi- 

 dent too that the intuition is sui generis, and guidance of it in the 

 seminar must simply stimulate, not undertake to determine its form. 

 The investigator must set his own problem and work it out in his own 

 way. The director of research should furnish favorable surroundings 

 and set forth the matter of his lectures in as genetic form as possible, 

 as for example, Poincare' s and Klein's masterly courses. But he should 

 not prescribe forms of development, nor methods of attack for the 

 novitiate. 



The types of intuition are numerous. We leave to the psychologist 

 their enumeration and description. For example, we should expect a 

 visualist to think in pictures, for in this direction his imagination would 

 be vivid. Such a mind would make use of diagrams and mechanical 

 forms to embody his ideas. We think at once of Faraday and his lines 

 of force, of Kelvin and his models of the ether. Poincare compares 

 Bertrand and Hermite, schoolmates educated at the same time in the 

 same way. Bertrand when speaking was always in motion, apparently 

 trying to paint his ideas. Hermite seemed to flee the world, his ideas 

 were not of the visible kind. Weierstrass thought in artificial symbols, 

 Eiemann in pictures and geometric constructions. Poincare is spoken 

 of as belonging to the audile type, for he remembered sounds well. 



