DEVELOPMENT OF GEOMETRIC METHODS 537 



geometric ; nor should we forget that to Monge are due the notion 

 of lines of curvature and the elegant integration of the differential 

 equation 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 Morige. 



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 

 perhaps 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, et al. 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 Russians in 1813 at the passage of the Dnieper 

 and incarcerated at Saratoff, Poncelet employed the leisure captivity 

 left him in the demonstration of the principles which he has developed 

 in the Traite des proprietes projectives des figures, issued in 1822, 

 and in the great memoirs on reciprocal polars and on harmonic 

 means, which go back nearly to the same epoch. So we may say the 

 modern geometry was born at Saratoff. 



Renewing the chain broken since Pascal and Desargues, Poncelet 

 introduced at the same time homology and reciprocal polars, putting 

 thus in evidence, from the beginning, the fruitful ideas on which the 

 science has evolved during fifty years. 



Presented in opposition to analytic geometry, the methods of Ponce- 

 let were not favorably received by the French analysts. But such 

 were their importance and their novelty, that without delay they 

 aroused, from divers sides, the most profound researches. 



Poncelet had been alone in discovering the principles; on the 

 contrary, many geometers appeared almost simultaneously to study 

 them on all sides and to deduce from them the essential results which 

 they implicitly contained. 



At this epoch, Gergonne was brilliantly editing a periodical which 

 has to-day for the history of geometry an inestimable value. The 

 Annales de Mathematiques, published at Nimes from 1810 to 1831, 

 was during more than fifteen years the only journal in the entire 

 world devoted exclusively to mathematical researches. 



Gergonne, who, in many regards, was a model editor for a scienti- 

 fic journal, had the defects of his qualities; he collaborated, often 



