#Fixing continuity calculus series#

If, beginning from one value of x x x lying between these limits, we assign to the variable x x x an infinitely small increment a a a, the function itself increases by the difference f ( x + a ) − f ( x ) f (x + a) - f (x) f ( x + a ) − f ( x ), which depends simultaneously on the new variable a a a and on the value of x x x. Let f ( x ) f (x) f ( x ) be a function of a variable x x x, and let us suppose that, for every value of x x x between two given limits, this function always has a unique and finite value. When the successive absolute values of a variable decrease indefinitely in such a way as to become less than any given quantity, that variable becomes what is called an infinitesimal. When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the limit of all the others. By determining these conditions and these values, and by fixing precisely the sense of all the notations I use, I make all uncertainty disappear. We must even note that they suggest that algebraic formulas have an unlimited generality, whereas in fact the majority of these formulas are valid only under certain conditions and for certain values of the quantities they contain.

#Fixing continuity calculus series#

Reasons for this latter approach, however widely they are accepted, above all in passing from convergent to divergent series and from real to imaginary quantities, can only be considered, it seems to me, as inductions, apt enough sometimes to set forth the truth, but ill founded according to the exactitude which is required in the mathematical sciences. Cauchy's approach to the calculus:Īs for my methods, I have sought to give them all the rigour which is demanded in geometry, in such a way as never to fall back on reasons drawn from what is usually described in algebra. In it he attempted to make calculus rigorous and to do this he felt that he had to remove algebra as an approach to calculus. Cauchy wrote Cours d'Analyse (1821) based on his lecture course at the École Polytechnique.