Page:Popular Science Monthly Volume 66.djvu/436

432 In continuing the study of these special transformations, Lie was led to construct progressively his masterly theory of continuous groups of transformations and to put in evidence the very important role that the notion of group plays in geometry. Among the essential elements of his researches, it is proper to signalize the infinitesimal transformations, of which the idea belongs exclusively to him.

Three great books published under his direction by able and devoted collaborators contain the essential part of his works and their applications to the theory of integration, to that of complex units and to the non-Euclidean geometry.

By an indirect way I have arrived at that non-Euclidean geometry of which the study takes in the researches of geometers a place which grows greater each day.

If I were the only one to talk with you about geometry, I would take pleasure in recalling to you all that has been done on this subject since Euclid or at least from Legendre to our days.

Envisaged successively by the greatest geometers of the last century, the question has progressively enlarged.

It commenced with the celebrated postulatum relative to parallels; it ends with the totality of geometric axioms.

The 'Elements' of Euclid, which have withstood the action of so many centuries, will have at least the honor before ending of arousing a long series of works admirably enchained which will contribute, in the most effective way, to the progress of mathematics, at the same time that they furnish to the philosophers the points of departure the most precise and the most solid for the study of the origin and of the formation of our cognitions.

I am assured in advance that my distinguished collaborator will not forget, among the problems of the present time, this one, which is perhaps the most important, and with which he has occupied himself with so much success; and I leave to him the task of developing it with all the amplitude which it assuredly merits.

I have 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 discoveries 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 rests on homothety and inversion. But it must be recognized that it is thanks to analysis that the 'Elements' have been enriched by their most beautiful propositions.