64 CARNOT. 



finite quantities. At each transformation of the formulae 

 it might be possible, according to this hierarchy, to dis- 

 embarrass one's self of fresh quantities ; and, nevertheless, 

 one was obliged to believe, to admit, that the definitive 

 results were rigorously exact ; that the infinitesimal cal- 

 culus was not merely a mere method of approximation. 

 Such was, considering the whole thing, the origin of the 

 strong and tenacious opposition which the new calculus 

 raised up at its birth ; such was also the difficulty which 

 a man equally celebrated as a geometer and a theologian, 

 Berkeley, bishop of Cloyne, had in view when he ex- 

 claimed, addressing himself to the incredulous in matters 

 of religion, " Look at the science of mathematics ; does 

 it not admit mysteries more incomprehensible than those 

 of religion ? " 



These mysteries at the present day, exist no longer 

 for those who desire to become initiated in the knowl- 

 edge of the methods which constitute the differential 

 calculus in Newton's theory of fluxions ; in a paper 

 wherein D'Alembert introduces the consideration of the 

 limits towards which the ratios of the finite differences 

 of functions converge ; or, indeed, in Lagrange's Theory 

 of Analytical Functions. Nevertheless, Leibnitz's course 

 has prevailed, because it is more simple, easier to recol- 

 lect, and more convenient in practice. It is, then, im- 

 portant to study it in itself, to penetrate into its essence, 

 and to assure one's self of the perfect exactness of the 

 rules which it furnishes, without the necessity of cor- 

 roborating them by the results of the calculus of fluxions, 

 or of limits, or of functions. That task, I mean the 

 search for the true spirit of differential analysis, forms 

 the principal object of the book which Carnot published, 

 in 1799, under the modest title of Reflections on the 



