DEVELOPMENT OF GEOMETRIC METHODS. 433 



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

 polygons of 17 sides and analogous polygons. It is to it we owe the 

 demonstrations so long sought, of the impossibility of the quadrature 

 of the circle, of the impossibility of certain geometric constructions 

 with the aid of the ruler and the compasses. It is to it finally that 

 we owe the first rigorous demonstrations of the properties of maximum 

 and of minimum of the sphere. It will appertain 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, with its schools free from 

 all program 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 projective 

 geometry: but that geometry, however interesting it may be, is it 

 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 should 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 

 illustrious 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. 



A single objection can be made to studies of this sort, and was 

 already formulated by Poisson: the absence of all real foundation, of 

 all substratum permitting the presentation, under aspects visible and 

 in some sort palpable, of the results obtained. 



The extension of the methods of descriptive geometry, and above 

 all the employment of Pluecker's conceptions on the generation of 

 space, will contribute to take away from this objection much of its 

 force. 



I would have liked to speak to you also of the method of equi- 

 pollences, of which we find the germ in the posthumous works of Gauss, 

 of Hamilton's quaternions, of Grassmann's methods and in general 

 of systems of complex units, of the Analysis situs, so intimately con- 

 nected with the theory of functions, of the geometry called 'kinematic, 



