BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 281 



triangles, polygons, circles, etc., in the Euclidean manner, is based 

 on a consideration of all points of a straight line, of all lines through 

 a common point and of the possible effects of setting up an orderly 

 one-to-one correspondence between them. In particular, Des- 

 argues makes a comparative study of the different plane sections 

 of a given cone, deducing from known properties of the circle anal- 

 ogous results for the other conic sections. 



In his chief work Desargues enunciates the propositions : 



1. A straight line can be considered as produced to infinity 

 and then the two opposite extremities are united. 



2. Parallel lines are lines meeting at infinity and conversely. 



3. A straight line and a circle are two varieties of the same 

 species. 



On these he bases a general theory of the plane sections of a cone. 



Desargues contented himself with enunciating general princi- 

 ples, remarking : " He who shall wish to disentangle this prop- 

 osition will easily be able to compose a volume." He met 

 Descartes while employed by Cardinal Richelieu at the siege of 

 Rochelle, and they with others met regularly in Paris for the 

 discussion of the new Copernican theory and other scientific 

 problems. 



He says ' I freely confess that I never had taste for study or re- 

 search either in physics or geometry except in so far as they could 

 serve as a means of arriving at some sort of knowledge of the proxi- 

 mate causes .... for the good and convenience of life, in maintaining 

 health, in the practice of some art, . . . having observed that a good 

 part of the arts is based on geometry, among others the cutting of 

 stones in architecture, that of sun-dials, that of perspective in 

 particular.' 



Perceiving that the practitioners of these arts had to burden them- 

 selves with the laborious acquisition of many special facts in 

 geometry, he sought to relieve them by developing more general 

 methods and printing notes for distribution among his friends. 



An interesting theorem bearing his name and typical of pro- 

 jective geometry is as follows : If two triangles ABC and A'B'C' 



