Cours d'Analyse

Cours d'Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821. The article follows the translation by Bradley and Sandifer in describing its contents.

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."

On page 10, Bradley and Sandifer confuse the versed cosine with the coversed sine. Cauchy originally defined the sinus versus (versine) as siv(θ) = 1-cos(θ) and the cosinus versus (what is now also known as coversine) as cosiv(θ) = 1-sin(θ). In the translation, however, the cosinus versus (and cosiv) are incorrectly associated with the versed cosine (what is now also known as vercosine) rather than the coversed sine.

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