DEVELOPMENT OF MATHEMATICAL ANALYSIS 501 



importance for the analyst: Phenomena are ruled by differential 

 equations which can be formed when observation and experiment 

 have made known for each category of phenomena certain physical 

 laws. 



We understand the unlimited hopes conceived from these results. 

 As Bertrand says in the preface of his treatise, "The early successes 

 were at first such that one might suppose all the difficulties of science 

 surmounted in advance, and believe that the geometers, without 

 being longer distracted by the elaboration of pure mathematics, 

 could turn their meditations exclusively toward the study of the 

 natural laws." 



This was to admit gratuitously that the problems of analysis, to 

 which one was led, would not present very grave difficulties. 



Despite the disillusions the future was to bring, this capital point 

 remained, that the problems had taken a precise form, and that a 

 classification could be established in the difficulties to be surmounted. 



There was, therefore, an immense advance, one of the greatest 

 ever made by the human mind. We understand also why the theory 

 of differential equations acquired a considerable importance. 



I have anticipated somewhat, in presenting things under a form 

 so analytic. Geometry was intermingled in all this progress. Huy- 

 gens, for example, followed always by preference the ancients, and 

 his Horologium oscillatorium rests at the same time on infinitesi- 

 mal geometry and mechanics; in the same way, in the Principia 

 of Newton, the methods followed are synthetic. 



It is, above all, with Leibnitz that science takes the paths which 

 were to lead to what we call mathematical analysis; it is he who, 

 for the first time, in the latter years of the seventeenth century, 

 pronounces the word function. 



By his systematic spirit, by the numerous problems he treated, 

 even as his disciples James and John Bernoulli, he established in a 

 final way the power of the doctrines to the edification of which had 

 successively contributed a long series of thinkers from the distant 

 times of Eudoxus and of Archimedes. 



The eighteenth century showed the extreme fecundity of the new 

 methods. That was a strange time, the era cf mathematical duels 

 where geometers hurled defiance, combats not always without 

 acrimony, when Leibnitzians and Newtonians encountered in the 

 lists. 



From the purely analytic point of view, the classification and study 

 of simple functions is particularly interesting; the function idea, on 

 which analysis rests, is thus developed little by little. 



The celebrated works of Euler hold then a considerable place. 

 However, the numerous problems which present themselves to the 

 mathematicians leave no time for a scrutiny of principles; the 



