On page 1 of the Introduction, Cauchy writes: "In speaking of the continuity of functions, I could not dispense with a treatment of the principal properties of infinitely small quantities, properties which serve as the foundation of the infinitesimal calculus." The translators comment in a footnote: "It is interesting that Cauchy does not also mention limits here."
Cauchy continues: "As for the methods, I have sought to give them all the rigor which one demands from geometry, so that one need never rely on arguments drawn from the generality of algebra."
On page 6, Cauchy first discusses variable quantities and then introduces the limit notion in the following terms: "When the values successively attributed to a particular variable indefinitely approach a fixed value in such a way as to end up by differing from it by as little as we wish, this fixed value is called the limit of all the other values."
On page 7, Cauchy defines an infinitesimal as follows: "When the successive numerical values of such a variable decrease indefinitely, in such a way as to fall below any given number, this variable becomes what we call infinitesimal, or an infinitely small quantity." Cauchy adds: "A variable of this kind has zero as its limit."
is introduced on page 12. The translators observe in a footnote: "The notation “Lim.” for limit was first used by Simon Antoine Jean L'Huilier (1750–1840) in [L’Huilier 1787, p. 31]. Cauchy wrote this as “lim.” in [Cauchy 1821, p. 13]. The period had disappeared by [Cauchy 1897, p. 26]."
This chapter has the long title "On infinitely small and infinitely large quantities, and on the continuity of functions. Singular values of functions in various particular cases." On page 21, Cauchy writes: "We say that a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge towards the limit zero." On the same page, we find the only explicit example of such a variable to be found in Cauchy, namely
On page 22, Cauchy starts the discussion of orders of magnitude of infinitesimals as follows: "Let be an infinitely small quantity, that is a variable whose numerical value decreases indefinitely. When the various integer powers of , namely
enter into the same calculation, these various powers are called, respectively, infinitely small of the first, the second, the third order, etc. Cauchy notes that "the general form of infinitely small quantities of order n (where n represents an integer number) will be
- or at least .
On pages 23-25, Cauchy presents eight theorems on properties of infinitesimals of various orders.
This is entitled "Continuity of functions". Cauchy writes: "If, beginning with a value of x contained between these limits, we add to the variable x an infinitely small increment , the function itself is incremented by the difference
and states that "the function f(x) is a continuous function of x between the assigned limits if, for each value of x between these limits, the numerical value of the difference decreases indefinitely with the numerical value of ." Cauchy goes on to provide an italicized definition of continuity in the following terms:
- "the function f(x) is continuous with respect to x between the given limits if, between these limits, an infinitely small increment in the variable always produces an infinitely small increment in the function itself."
On page 32 Cauchy states the intermediate value theorem.
In Theorem I in section 6.1 (page 90 in the translation by Bradley and Sandifer), Cauchy presents the sum theorem in the following terms.
When the various terms of series (1) are functions of the same variable x, continuous with respect to this variable in the neighborhood of a particular value for which the series converges, the sum s of the series is also a continuous function of x in the neighborhood of this particular value.
Here the series (1) appears on page 86: (1)
- Bradley, Robert E.; Sandifer, C. Edward Cauchy's Cours d'analyse. An annotated translation. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York, 2009. xx+411 pp. ISBN 978-1-4419-0548-2