548 GEOMETRY 



Preceded by innumerable researches on systems of straight lines, 

 due to Poinsot,Moebius, Chasles, Dupin, Malus, Hamilton, Krummer, 

 Transon, above all to Cayley, who first introduced the notion of the 

 coordinates of the straight, researches originating perhaps in statics 

 and kinematics, perhaps in geometrical optics, Pluecker's geometry of 

 the straight line will always be regarded as the part of his work where 

 are met the newest and most interesting ideas. 



Pluecker first set up a methodic study of the straight line, which 

 already is important, but that is nothing beside what he discov 

 ered. It is sometimes said that the principle of duality shows that 

 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 

 applied by their master to curves of double curvature, to lines of curv- 

 ature, to the generation of surfaces, notions expounded at least in 

 part in the Application de V Analyse a la Geometrie. 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: Les Dcveloppements de Geometrie pure, 

 issued in 1813, and Les Applications de Geometrie et de Mecanique, 

 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 



