

4 i4 POPULAR SCIENCE MONTHLY. 



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 :iearly to the same epoch. So we may say the 

 modern geometry was born at Saratoff. 



Eenewing 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 con- 

 trary, 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 Mathemaiiques, 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 scientific journal, had the defects of his qualities; he col- 

 laborated, often against their will, with the authors of the memoirs 

 sent him, rewrote them, and sometimes made them say more or less 

 than they would have wished. Be that as it may, he was greatly 

 struck by the originality and range of Poncelet's discoveries. 



In geometry some simple methods of transformation of figures 

 were already known; homology even had been employed in the plane, 

 but without extending it to space, as did Poncelet, and especially with- 

 out recognizing its power and fruitfulness. Moreover all these trans- 

 formations were punctual, that is to say they made correspond a point 

 to a point. 



In introducing polar reciprocals, Poncelet was in the highest degree 

 creative, because he gave the first example of a transformation in which 

 to a point corresponded something other than a point. 



Every method of transformation enables its to multiply the num- 

 ber of theorems, but that of polar reciprocals had the advantage of 

 making correspond to a proposition another proposition of wholly 

 different aspect. This was a fact essentially new. To put it in evi- 

 dence, Gergonne invented the system, which since has had so much 

 success, of memoirs printed in double columns with correlative 



