536 GEOMETRY 



mental activity towards wholly new subjects. This need, as we 

 recognize, manifested itself with particular force at the epoch of La- 

 grange. At this moment, in fact, the programme of researches opened 

 to geometers by the discovery of the infinitesimal calculus appeared 

 very nearly finished up. Some differential equations more or less 

 complicated to integrate, some chapters to add to the integral 

 calculus, and one seemed about to touch the very outmost bounds 

 of science. 



Laplace had achieved the explanation of the system of the world 

 and laid the foundations of molecular physics. New ways opened 

 before the experimental sciences and prepared the astonishing 

 development they received in the course of the century just ended. 

 Ampere, Poisson, Fourier, and Cauchy himself, the creator of the 

 theory of imaginaries, were occupied above all in studying the appli- 

 cation of the analytic methods to molecular physics, and seemed to 

 believe that outside this new domain, which they hastened to cover, 

 the outlines of theory and science were finally fixed. 



Modern geometry, a glory we must claim for it, came, after the 

 end of the eighteenth century, to contribute in large measure to the 

 renewing of all mathematical science, by offering to research a way 

 new and fertile, and above all in showing us, by brilliant successes, 

 that general methods are not everything in science, and that even 

 in the simplest subject there is much for an ingenious and inventive 

 mind to do. 



The beautiful geometric demonstrations of Huygens, of Newton, 

 and of Clairaut were forgotten or neglected. The fine ideas introduced 

 by Desargues and Pascal had remained without development and 

 appeared to have fallen on sterile ground. 



Carnot, by his Essai sur les transversales and his Geometric de 

 position, above all Monge, by the creation of descriptive geometry 

 and by his beautiful theories on the generation of surfaces, came to 

 renew a chain which seemed broken. Thanks to them, the conceptions 

 of the inventors of analytic geometry, Descartes and Fermat, retook 

 alongside the infinitesimal calculus of Leibnitz and Newton the place 

 they had lost, yet should never have ceased to occupy. With his 

 geometry, said Lagrange, speaking of Monge, this demon of a man 

 will make himself immortal. 



And, in fact, not only has descriptive geometry made it possible 

 to coordinate and perfect the procedures employed in all the arts 

 where precision of form is a condition of success and of excellence for 

 the work and its products; but it appeared as the graphic translation 

 of a geometry, general and purely rational, of which numerous and 

 important researches have demonstrated the happy fertility. 



Moreover, beside the Geometric descriptive we must not forget 

 to place that other masterpiece, the Application de I'analyse a la 



