Page:Popular Science Monthly Volume 82.djvu/224

220 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 handling Dirichlet's problem on a Riemann 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 convergence 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 knowing 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 development. 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. Poincaré rarely did more than to acquaint himself with the present status of a problem he desired to consider. It is evident 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, Poincaré'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. Poincaré 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, Riemann in pictures and geometric constructions. Poincare is spoken of as belonging to the audile type, for he remembered sounds well.