860 



SCIENCE. 



[N. S. Vol. XX. No. 521. 



or less great with which phenomena hap- 

 pen; this is well expressed by the words 

 fluents and fluxions, which Newton used, 

 and which one might suppose borrowed 

 from old Heraclitus. 



The points of view taken by the founders 

 of the science of motion, Galileo, Huygens 

 and Newton, had an enormous influence on 

 the orientation of mathematical analysis. 



It was with Galileo an intuition of genius 

 to discover that, in natural phenomena, the 

 determining circumstances of the motion 

 produce accelerations: this must have con- 

 ducted to the statement of the principle 

 that the rapidity with which the dynamic 

 state of a system changes depends in a de- 

 terminate manner on its static state alone. 

 In a more general way we reach the postu- 

 late that the infinitesimal changes, of what- 

 ever nature they may be, occurring in a 

 system of bodies, depend viniquely on the 

 actual state of this system. 



In what degree are the exceptions ap- 

 parent or real? This is a question which 

 was raised only later and which I put aside 

 for the moment. 



From the principles enunciated becomes 

 clear a point of capital importance for the 

 analyst : Phenomena ar^ ruled by differen- 

 tial equations which can be formed when 

 observation and experiment have made 

 known for each category of phenomena cer- 

 tain physical laws. 



We understand the unlimited hopes con- 

 ceived 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 dis- 

 tracted by the elaboration of pure mathe- 

 matics, could turn their meditations ex- 

 clusively 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 ad- 

 vance, 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 present- 

 ing things under a form so analytic. 

 Geometry was intermingled in all this 

 progress. Huygens, for- example, fol- 

 lowed always by preference the ancients, 

 and his 'Horologium oscillatorium ' rests 

 at the same time on infinitesimal geometry 

 and mechanics; in the same way, in the 

 'Principia' of Newton, the methods fol- 

 lowed 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 Archi- 

 medes. 



The eighteenth century showed the ex- 

 treme fecundity of the new methods. That 

 was a strange time, the era of mathematical 

 duels where geometers hurled defiance, 

 combats not always without acrimony, 

 when Leibnitzians and Newtonians encoun- 

 tered in the lists. 



From the purely analytic point of view, 

 the classification and study of simple func- 

 tions is particularly interesting; the func- 



