64 CARNOT. 
finite quantities. At each transformation of the formule. 
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 tlte modest title of Reflections on the 
