500 ALGEBRA AND ANALYSIS 



its notation more and more perfected; thus was created this lan- 

 guage so admirably clear, which brings about for thought a veritable 

 economy and renders further progress possible. 



This is also the moment when distinct divisions are organized. 



Trigonometry, which, in antiquity, had been only an auxiliary of 

 astronomy, is developed independently; toward the same time the 

 logarithm appears, and essential elements are thus put in evidence. 



II 



In the seventeenth century, the analytic geometry of Descartes, 

 distinct from what I have just called the geometric algebra of the 

 Greeks by the general and systematic ideas which are at its base, 

 and the new-born dynamic were the origin of the greatest progress of 

 analysis. 



When Galileo, starting from the hypothesis that the velocity of 

 heavy bodies in their fall is proportional to the time, from this 

 deduced the law of the distances passed over, to verify it afterward 

 by experiment, he took up again the road upon which Archimedes 

 had formerly entered and on which would follow after him Cavalieri, 

 Fermat, and others still, even to Newton and Leibnitz. The integral 

 calculus of the Greek geometers was born again in the kinematic of 

 the great Florentine physicist. 



As to the calculus of derivatives or of differentials, it was founded 

 with precision apropos of the drawing of tangents. 



In reality, the origin of the notion of derivative is in the confused 

 sense of the mobility of things and of the rapidity more or less great 

 with which phenomena happen; 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 conducted to the statement 

 of the principle that the rapidity with which the dynamic state of 

 a system changes depends in a determinate manner on its static state 

 alone. In a more general way we reach the postulate that the in- 

 finitesimal changes, of whatever nature they may be, occurring in 

 a system of bodies, depend uniquely on the actual state of this 

 system. 



In what degree are the exceptions apparent or real? This is a ques- 

 tion which was raised only later and which I put aside for the 

 moment. 



From the principles enunciated becomes clear a point of capital 



