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 century 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 solution of this important and difficult question.
Von Staudt has sometimes been called the Euclid of the nineteenth 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 equipollences, 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 connected with the theory of functions, of the geometry called kinematic,
