538 GEOMETRY 



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 origin- 

 ality 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 

 without recognizing its power and fruitfulness. Moreover, all these 

 transformations 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 us 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 

 propositions in juxtaposition; and he had the idea of substituting 

 for Poncelet's demonstrations, which required an intermediary 

 curve or surface of the second degree, the famous "principle of 

 duality," of which the signification, a little vague at first, was suffi- 

 ciently cleared up by the discussions which took place on this subject 

 between Gergonne, Poncelet, and Pluecker. 



Bobillier, Chasles. Steiner, Lame", Sturm, and many others whose 

 names escape me, were, at the same time as Pluecker and Poncelet, 

 assiduous collaborators of the Annales de Mathematiques. Gergonne, 

 having become rector of the Academy of Montpellier, was forced to 

 suspend in 1831 the publication of his journal. But the success it had 

 obtained, the taste for research it had contributed to develop, had 

 commenced to bear their fruit. Que"telet had established in Belgium 

 the Correspondance mathematique et physique. Crelle, from 1826, 

 brought out at Berlin the first sheets of his celebrated journal, where 

 he published the memoirs of Abel, of Jacobi, of Steiner. 



A great number of separate works began also to appear, wherein 

 the principles of modern geometry were powerfully expounded and 

 developed. 



First came in 1827 the Barycentrische Calcul of Moebius, a work 

 truly original, remarkable for the profundity of its conceptions, the 

 elegance and the rigor of its exposition; then in 1828 the Analytisch- 

 geometrische Entwickelungen of Pluecker, of which the second part 

 appeared in 1831, and which was soon followed by the System der 

 analytischen Geometrie of the same author, published at Berlin in 

 1835. 



