Xlvi THE MODERN OKOMKTUY. 



From this period geometry, for a long time, served merely as an aid to 

 analysis, interpreting graphically its results (V.). From this union the 

 riv:itest advantages were derived, as analysis led to the infinitesimal cal- 

 culus of Newton and Leibnitz, and geometry to the analytical geometry of 

 Descartes (1596-1650). 



But the extension and generality which geometrical truths received by 

 this great creation of Descartes was essentially due to analysis. Desarguca 

 (1593-1663) and Pascal (1623-1662) extended pure geometrical considera- 

 tions, and made the first step towards the modern geometry when they 

 regarded the conic sections as projections of the circle, and deduced the 

 properties of the first from those of the last. Then De la Hire (1640-1718), 

 Le Poivre (1704) and Huygens (1629-1695) occupied themselves with geo- 

 metrical investigations. While the two first developed the methods of 

 Desargues and Pascal, Huygens and, later, Newton (1642-1727) applied 

 pure geometry in optics and mechanics. Soon, however, the Calculus of 

 Newton and Leibnitz (1684 and 1687) showed itself so wonderfully fertile 

 in analytical geometry, that geometry proper was put in the background. 

 Only a few, as Lambert (1728-1777), still regarded it with favor. 



Then appeared Monge (1728-1777), and gave the impulse to a complete 

 revolution in geometrical views, and to the reconstruction of the science 

 upon a new basis. In his Lecons de Geometric descriptive (Paris, 1788), all 

 those problems previously treated in a special and uncertain manner in 

 stereotomy, perspective, gnomonics, etc., were referred back to a few gen- 

 eral principles, and, without the aid of analysis, the most important prop- 

 erties of lines and surfaces were deduced. While descriptive geometry 

 taught the relations between bodies in space and drawn figures, it strength- 

 ened the power of abstraction ; introducing into geometry the transforma- 

 tion of figures, it gave to its deductions an advantage till then possessed 

 only by analysis ; and while, finally, it owed its comprehensive results to 

 the application of projections, it pointed the way for the further develop- 

 ment of geometry itself. 



Meanwhile, in the field of analytical geometry, the conclusion had been 

 reached that the desired truths admitted of a still more general compre- 

 hension. All properties had been obtained only with respect to and by 

 means of a determinate co-ordinate system. But already Godin (1704- 

 1760) had announced " que I 1 art de decouvrir les proprietes dea courbea eat 

 d proprement parler, Vart de clianger le ayateme de co-ordonnees" (Traite des 

 proprietes communes a toutes lea courbea). This idea Oarnot seized upon 

 (1753-1823), and in the sixth chapter of his Geometric de position (Paris, 

 1803) he sought to obtain a more general comprehension of figures by 

 analysis, and to avoid the indeterminancy of this last by the introduction 

 of the idea of position, and by many solutions after the method already 

 pointed out by Liebnitz and d'Alembert. 



Now began a veritable race in the condensation and promulgation of 

 geometrical truths, in which the pure geometrical method obtained the 

 palm. The scholars of Mongc BriancTwn, Servois, Chasles, Poncelet, Ger- 

 gonne working with him and in his spirit, filled the Annales dca mathe- 

 matiques and the Correapondance aur Vecole polytechnique with new re 



