THE MATHEMATICAL SCIENCES 159 



geometry, it is equivalent to a construction. Thus to 

 superpose a triangle B on another triangle A in such a 

 way as to be able to compare them, comes to construct- 

 ing the triangle B on the triangle A according to the 

 conditions stated in the enunciation. 



We see that plane geometry avoids the direct use of 

 the methods of displacement, especially of turning over, 

 and the reason for this must be sought in the fear of 

 giving a hold to the arguments of Zeno respecting 

 motion and infinity. 



It was also for this same reason, we think, that the 

 Greek philosophers avoided the geometrical infinity 

 in the same way as they rejected the direct use 

 of numerical infinity in their methods of integration. 

 They possessed, however, since the works of Apollonius, 

 the essential elements (points of involution, anharmonic 

 ratio) for reaching, by generalization, to geometrical 

 infinity. But on this question they remained faithful 

 to the teaching of Aristotle, who considered real space, 

 and therefore geometrical space, to be finite. Conse- 

 quently, the conception of points, straight lines, and 

 planes, removed to infinity, is not only obscure from 

 a logical point of view, but contrary to experience. 

 Therefore it would not be possible, even as a convenient 

 symbolism, to appeal to geometrical infinity and make 

 it the starting-point of new methods. For want of 

 searching in this direction and from loyalty to its 

 logical ideal, Greek geometry was obliged to resort to 

 a complicated kind of demonstration, the application 

 of which rendered difficult the linking of theorems in 

 correct sequence. It was an event of outstanding 

 importance when Desargues, in the seventeenth cen- 

 tury, made a direct use of geometrical infinity. The 

 simplifications wrought by this act were so great that 

 they struck the contemporaries of the great geometer. 

 Speaking of Desargues, the engraver Bosse says that 

 the work which he has published on conic sections, one 



