DEVELOPMENT OF GEOMETRIC METHODS. 4*3 



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 intro- 

 duced 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 ' Geometrie 

 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 im- 

 portant researches have demonstrated the happy fertility. 



Moreover, beside the ' Geometrie descriptive ' we must not forget 

 to place that other master-piece, the ' Application de l'analyse a la 

 geometrie ' ; nor should we forget that to Monge are due the notion of 

 lines of curvature and the elegant integration of the differential equa- 

 tion of these lines for the case of the ellipsoid, which, it is said, 

 Lagrange envied him. To be stressed is this character of unity of the 

 work of Monge. 



The renewer of modern geometry has shown us from the beginning, 

 what his successors have perhaps forgotten, that the alliance of 

 geometry and analysis is useful and fruitful, that this alliance is per- 

 haps for each a condition of success. 



II. 



In the school of Monge were formed many geometers: Hachette, 

 Brianchon, Chappuis, Binet, Lancret, Dupin, Malus, Gaultier de 

 Tours, Poncelet, Chasles, etc. Among these Poncelet takes first rank. 

 Neglecting, in the works of Monge, everything pertaining to the analysis 

 of Descartes or concerning infinitesimal geometry, he devoted himself 

 exclusively to developing the germs contained in the purely geometric 

 researches of his illustrious predecessor. 



Made prisoner by the Eussians in 1813 at the passage of the 



