DEVELOPMENT OF MATHEMATICAL THOUGHT. 



663 



Of this principle of projection, which Poncelet at once 25- 

 introduces in the more general form as conical or central projection, 

 projection, two signal applications existed in the treatises 

 on Conic Sections handed down from antiquity, and in 

 the practical methods and Eules of Perspective invented 

 by Lionardo da Vinci and further developed by various 

 geometricians. The results, which lay scattered in many 

 books and memoirs, Poncelet collected in a systematic 

 form, bringing them, by the application of the law of 

 continuity, under a few general and eminently useful 

 points of view or principles. By the method of projec- 

 tion or perspective he " transformed figures which are 

 very general into others which are particular, and vice 

 versa." He established the principle of " homology " in 

 figures, and by showing how figures apparently very 

 different could be described by the process of projection 

 from the same original figure, he showed that there 

 existed a peculiar relation among figures — viz., their 

 " reciprocity." ^ 



et par uiie uiarche uniforme, on 

 ne tarde pas h reconnaitre que cela 

 tient uuiquement ii I'usage qu'elles 

 font cle la projection." 



' The properties of figures, called 

 by Poncelet " liomology " and "I'e- 

 ciprocity," refer to the correspond- 

 ence of certain elements of one 

 figure to those of another figure. 

 In the case of " homology," we 

 have to do with corresponding 

 points or corresponding lines— ix., 

 with the correspondence of the 

 same elements. In the case of 

 "reciprocity," we have to do with 

 correspondence of points or lines 

 in the one figure, with lines or 

 points in the other — i.e., with 

 the correspondence of different 

 elements. The idea of placing 



figures in an homologous rela- 

 tion was got by the device of 

 making two planes, which con- 

 tained figures in perspective, fall 

 together into one plane ; upon 

 which the section of the two orig- 

 inal jilanes became the " axis," 

 and the ej-e-point the " centre " 

 of homology — all situated in one 

 and the same plane. Poncelet had 

 already conceived of the possibil- 

 ity of reducing the two planes in 

 Monge's ' Descriptive Geometry,' 

 which represent the plan and ele- 

 vation of a figure in one plane, 

 on which the elevations were 

 marked by what are now called 

 " contour lines." The idea of the 

 correspondence of figures by what 

 is called " reciprocity " was sug- 



