556 GEOMETRY 



with so much success; and I leave to him the task of developing it 

 with all the amplitude which it assuredly merits. 



Thave just spoken of the elements of geometry. They have received 

 in the last hundred years extensions which must not be forgotten. 

 The theory of polyhedrons has been enriched by the beautiful dis- 

 coveries of Poinsot on the star polyhedrons and those of Moebius 

 on polyhedrons with a single face. The methods of transformation 

 have enlarged the exposition. We may say to-day that the first book 

 contains the theory of translation and of symmetry, that the second 

 amounts to the theory of rotation and of displacement, that the 

 third rest on homothety and inversion. But it must be recognized 

 that it is due to analysis that the Elements have been enriched by 

 their most beautiful propositions. 



It is to the highest analysis that we owe the inscription of regular 

 polygons of seventeen sides and analogous polygons. To it we owe 

 the demonstrations, so long sought, of the impossibility of the quad- 

 rature of the circle, of the impossibility of certain geometric con- 

 structions with the aid of the ruler and the compasses; and to it finally 

 we owe the first rigorous demonstrations of the properties of maxi- 

 mum and of minimum of the sphere. It will belong to geometry to 

 enter upon this ground where analysis has preceded it. 



What will be the elements of geometry in the course of the cen- 

 tury which has just commenced? Will there be a single elementary 

 book of geometry? It is perhaps America, w r ith its schools free from 

 all programme and from all tradition, which will give us the best solu- 

 tion of this important and difficult question. 



Von Staudt has sometimes been called the Euclid of the nine- 

 teenth century; I would prefer to call him the Euclid of protective 

 geometry ; but is projective geometry, interesting though it may be, 

 destined to furnish the unique foundation of the future elements? 



XV 



The moment has come to close this over-long recital, and yet there 

 is a crowd of interesting researches that I have been, so to say, forced 

 to neglect. 



I would have loved to talk with you about those geometries of 

 any number of dimensions of which the notion goes back to the first 

 days of algebra, but of which the systematic study was commenced 

 only sixty years ago by Cayley and by Cauchy. This kind of researches 

 has found favor in your country and I need not recall that our illus- 

 trious president, after having shown himself the worthy successor 

 of Laplace and Le Verrier, in a space which he considers with us as 

 being endowed with three dimensions, has not disdained to publish, 

 in the American Journal, considerations of great interest on the 

 geometries of n dimensions. 



