OF KNOWLEDGE. 



323 



perfected by Descartes himself in the application of 

 Algebra — the general arithmetic of the Arabians — to 

 Geometry ; it changed the latter from a science which, 

 though rigorous, was somewhat casual, to a methodical 

 doctrine by which configurations in space could be 

 generally and exhaustively treated. It must indeed 

 have been a seductive prospect for those acquainted with 

 the great development of mathematical science which 

 followed the invention of the analytical and infinitesimal 

 methods to acquire in the uncertain regions of philo- 

 sophic thought the grasp and mastery exhibited by 

 the mathematical sciences. Nearly all the great Con- 

 tinental, notably the French, thinkers of the seven- 

 teenth and eighteenth centuries came more or less under 

 the spell of this idea. That it did not exert a similar 

 spell in this country was largely owing to the fact that 

 here the foremost mathematical genius, Newton, retained 

 in his immortal works the synthetic methods of the 

 ancients, which in the hands of all but the very greatest 

 mathematicians remained specific and did not rise to 

 abstract generality.^ 



The detailed arguments by which Descartes elaborated 

 the two main principles of his philosophy, viz., that 

 certainty can be found only in and by thinking, and 



^ The synthetic methods of the 

 ancients which were, following the 

 example of Newton, retained for a 

 long time in the teaching of higher 

 mathematics in this country, at- 

 tained nevertheless, under the 

 hands of French geometricians 

 (notably of Monge and Poncelet in 

 the beginning of the nineteenth 

 century) a systematisation equal in 

 importance to the analytical prin- 



ciple of Descartes. This was by 

 means of the ' Principle of Projec- 

 tion.' An analytical interpretation 

 of this principle led in the course 

 of the nineteenth century to an 

 approximation of the two methods 

 and in the sequel to an extra- 

 ordinary development of mathe- 

 matical thought and knowledge 

 (see vol. ii. of this History, p. 

 658 sqq.) 



