638 SCIENTIFIC THOUGHT. 



troduction of algebra or general arithmetic, in the applica- 

 tion of this to geometry and dynamics, and in the invention 

 of the infinitesimal methods, through which the rigorous 

 theorems of the older geometricians which referred to the 

 simpler figures such as straight lines, circles, spheres, 

 cones, &c. became applicable to the infinite variety of 

 curves and surfaces in which the objects and phenomena 

 of nature present themselves to our observation. Logic- 

 ally speaking, it was a grand process of generalisation, 

 based mostly on inference and induction, sometimes 

 9. merely on intuition. 1 Such a process of generalisation 

 generaiisa- has a twofold effect on the progress of science. 



The first and more prominent result was the greatly 

 increased power of dealing with special problems which 

 the generalised method affords, and the largely increased 

 field of research which it opened out. We may say that 

 the century which followed the inventions of Descartes, 

 Newton, and Leibniz, was mainly occupied in exploring 

 the new field which had been disclosed, in formulating 

 and solving the numberless problems which presented 

 themselves on all sides ; also, where complete and 

 rigorous solutions seemed unattainable, in inventing 

 methods of approximation which were useful for prac- 

 tical purposes. In this direction so much had to be done, 

 so much work lay ready to hand, that the second and 

 apparently less practical effect of the new generalisations 

 receded for a time into the background. We may term 



1 " On se reportait inconsciem- claire et rigoureuse, mais par une 



inent au modele qui nous est sorte d'iutuition et d'obscur in- 



fourni par lea fonctions cousiderees stinct" (Poincare, " L'ccuvre math, 



eu mecanique et on rejetait tout de Weiers trass," ' Acta Mathema- 



ce qui s'ecartait de ce modele ; on tica.' vol. xxii. p. 4). 

 n'etait pas guide par une definition 



