102 SCIENTIFIC THOUGHT. 



interest) this new calculus, which in the course of the 

 eighteenth century had in the hands of Lagrange been 

 adapted to all the purposes and problems contained or 

 suggested in Newton's ' Principia.' 

 12. This leads me to a third and yet more important element 



Modern 



analytical o f scientific thought, which was peculiar to the Continental, 

 and especially to the French mathematicians, counting 

 among them Leibniz, who, though a German, was wholly 

 trained in the French school. This factor is the estab- 

 lishment of pure mathematics on an independent founda- 

 tion, and the cultivation of research into the abstract 

 relations of quantity, without reference either to geomet- 

 rical or mechanical problems and applications. It is 

 the modern analytical spirit introduced by the great 

 French algebraists of the seventeenth century, which 

 looks upon geometry, mechanics, and astronomy merely 

 as " questions d'analyse," and makes their solutions de- 

 pend upon the perfecting of an abstract calculus rather 

 than on the study of these individual problems them- 

 selves. Opposed to this spirit of analysis, which in 

 general seeks the solution of any given question by 

 looking upon it as a special case of a wider and more 

 abstract problem, is the method known to the ancients, 

 which never loses sight of the actual application, be it a 

 figure in geometry or a special arrangement of physical 

 forces, and is more interested in the peculiarities of the 

 individual case than in the abstract formula of which it 

 may be considered an application. This opposite view 

 regards the calculus and mathematics in general merely 

 as an instrument, the value of which lies solely in its 

 application to real physical problems. It is usually 



