DEVELOPMENT OF GEOMETRIC METHODS. 425

the plane as well as the point may be considered as a space element. That is true; but in adding the straight line to the plane and point as possible space element, Pluecker was led to recognize that any curve, any surface, may also be considered as space element, and so was born a new geometry which already has inspired a great number of works, which will raise up still more in the future.

A beautiful discovery, of which we shall speak further on, has already connected the geometry of spheres with that of straight lines and permits the introduction of the notion of coordinates of a sphere.

The theory of systems of circles is already commenced; it will be developed without doubt when one wishes to study the representa- tion, which we owe to Laguerre, of an imaginary point in space by an oriented circle.

But before expounding the development of these new ideas which have vivified the infinitesimal methods of Monge, it is necessary to go back to take up the history of branches of geometry that we have neglected until now.

X.

Among the works of the school of Monge, we have hitherto con- fined ourselves to the consideration of those connected with finite geometry; but certain of the disciples of Monge devoted themselves above all to developing the new notions of infinitesimal geometry ap- plied by their master to curves of double curvature, to lines of curva- ture, to the generation of surfaces, notions expounded at least in part in the ' Application de l'Analyse a la Geometric' Among these we must cite Lancret, author of beautiful works on skew curves, and above all Charles Dupin, the only one perhaps who followed all the paths opened by Monge

Among other works, we owe to Dupin two volumes Monge would not have hesitated to sign : the ' Developpements de Geometrie pure,' issued in 1813 and the ' Applications de Geometrie et de Mechanique/ dating from 1822.

There we find the notion of indicatrix, which was to renovate, after Euler and Meunier, all the theory of curvature, that of conjugate tangents, of asymptotic lines which have taken so important a place in recent researches. Nor should we forget the determination of the surface of which all the lines of curvature are circles, nor above all the memoir on triple systems of orthogonal surfaces where is found, together with the discovery of the triple system formed by surfaces of the second degree, the celebrated theorem to which the name of Dupin will remain attached.

Under the influence of these works and of the renaissance of syn- thetic methods, the geometry of infinitesimals re-took in all researches the place Lagrange had wished to take away from it forever. vol. lxvi.— 28.

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