PRESENT PROBLEMS OF GEOMETRY 561 



lui demande." As the field of research widens, as we proceed from 

 the simple and definite to the more refined and general, we naturally 

 cease to picture our processes and even our results. It is often neces- 

 sary to close our eyes and go forward blindly if we wish to advance 

 at all. But admitting the inevitableness of such a change in the 

 spirit of any science, one may still question the attitude of the geo- 

 meter who rests content with his blindness, who does not at least 

 strive to intensify and enlarge the intuition. Has not such an inten- 

 sification and enlargement been the main contribution of geometry 

 to the race, its very raison d'etre as a separate part of mathematics, 

 and is there any ground for regarding this service as completed? 



From the point of view here referred to, a problem is not to be 

 regarded as completely solved until we are in position to construct 

 a model of the solution, or at least to conceive of such a construction. 

 This requires the interpretation, not merely of the results of a geo- 

 metric investigation, but also, as far as possible, of the intermediate 

 processes -- an attitude illustrated most strikingly in the works of 

 Lie. This duty of the geometer, to make the ground won by means 

 of analysis really geometric, and as far as possible concretely intui- 

 tive, is the source of many problems of to-day, a few of which will 

 be referred to in the course of this address. 



The tendency to generalization, so characteristic of modern geo- 

 metry, is counteracted in many cases by this desire for the concrete, 

 in others by the desire for the exact, the rigorous (not to be con- 

 fused with the rigid). The great mathematicians have acted on the 

 principle "Devinez avant de demontrer," and it is certainly true 

 that almost all important discoveries are made in this fashion. But 

 while the demonstration comes after the discovery, it cannot there- 

 fore be disregarded. The spirit of rigor, which tended at first to the 

 arithmetization of all mathematics and now tends to its exhibition 

 in terms of pure logic, has always been more prominent in analysis 

 than in geometry. Absolute rigor may be unattainable, but it can- 

 not be denied that much remains to be done by the geometers, judg- 

 ing even by elementary standards. We need refer only to the loose 

 proofs based upon the invaluable but insufficient enumeration of 

 constants, the so-called principle of the conservation of number, and 

 the discussions which confine themselves to the "general case." 

 Examples abound in every field of geometry. The theorem announced 

 by Chasles concerning the number of conies satisfying five arbitrary 

 conditions was proved by such masters as Clebsch and Halphen be- 

 fore examples invalidating the result were devised. Picard recently 

 called attention to the need of a new proof of Noether's theorem that 

 upon the general algebraic surface of degree greater than three every 

 algebraic curve is a complete intersection with another algebraic 

 surface. The considerations given by Noether render the result 



