Cours d'analyse

Title page
Author	Augustin-Louis Cauchy
Original title	Cours d'analyse de l’École royale polytechnique; I.re Partie. Analyse algébrique
Language	French
Subject	Calculus
Publisher	Debure frères
Publication date	1821
Publication place	Paris, France

Cours d'analyse de l’École royale polytechnique; I.re Partie. Analyse algébrique ("Analysis Course" in English) is a seminal textbook ininfinitesimal calculus published byAugustin-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 thecontinuity offunctions, I could not dispense with a treatment of the principal properties ofinfinitely 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 mentionlimits here."

Cauchy continues: "As for the methods, I have sought to give them all therigor which one demands fromgeometry, so that one need never rely on arguments drawn from thegenerality 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 thelimit of all the other values."

On page 7, Cauchy defines aninfinitesimal 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 callinfinitesimal, or aninfinitely small quantity." Cauchy adds: "A variable of this kind has zero as its limit."

On page 10, Bradley and Sandifer confuse theversed cosine with thecoversed sine. Cauchy originally defined thesinus versus (versine) as siv(θ) = 1 − cos(θ) and thecosinus versus (what is now also known ascoversine) as cosiv(θ) = 1 − sin(θ). In the translation, however, thecosinus versus (and cosiv) are incorrectly associated with theversed cosine (what is now also known asvercosine) rather than thecoversed sine.

The notation

lim

is introduced on page 12. The translators observe in a footnote: "The notation “Lim.” for limit was first used bySimon 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 becomesinfinitely 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

${\displaystyle \frac{1}{4},\frac{1}{3},\frac{1}{6},\frac{1}{5},\frac{1}{8},\frac{1}{7},\dots }$

On page 22, Cauchy starts the discussion of orders of magnitude of infinitesimals as follows: "Let${\displaystyle \alpha }$ be an infinitely small quantity, that is a variable whose numerical value decreases indefinitely. When the variousinteger powers of${\displaystyle \alpha }$, namely

${\displaystyle \alpha ,{\alpha }^2,{\alpha }^3,\dots }$

enter into the same calculation, these various powers are called, respectively, infinitely small of thefirst, thesecond, thethird order, etc. Cauchy notes that "the general form of infinitely small quantities of ordern (wheren represents an integer number) will be

${\displaystyle k{\alpha }^n}$ or at least${\displaystyle k{\alpha }^n(1\pm \epsilon )}$.

On pages 23-25, Cauchy presents eight theorems on properties of infinitesimals of various orders.

This section is entitled "Continuity of functions". Cauchy writes: "If, beginning with a value ofx contained between these limits, we add to the variablex aninfinitely small increment${\displaystyle \alpha }$, the function itself is incremented by the difference

${\displaystyle f(x+\alpha )-f(x)}$"

and states that

"the functionf(x) is a continuous function ofx between the assigned limits if, for each value ofx between these limits, the numerical value of the difference${\displaystyle f(x+\alpha )-f(x)}$ decreases indefinitely with the numerical value of${\displaystyle \alpha }$."

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 theintermediate 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 theneighborhood 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)${\displaystyle u_0,u_1,u_2,\dots ,u_n,u_{n+1},\dots }$

• Cauchy, Augustin-Louis (1821).Cours d'analyse de l'Ecole royale polytechnique. Paris: L'Imprimerie Royale, Debure frères, Libraires du Roi et de la Bibliothèque du Roi.
• Bradley, Robert E.; Sandifer, C. Edward (2009).Cauchy's Cours d'analyse: An Annotated Translation.Cauchy, Augustin-Louis. New York: Springer.doi:10.1007/978-1-4419-0549-9.ISBN 978-1-4419-0548-2.
• Grabiner, Judith V. (1981).The Origins of Cauchy's Rigorous Calculus. Cambridge: MIT Press.ISBN 0-387-90527-8.

• Mathematics of infinitesimals
• Calculus
• History of calculus
• Mathematics textbooks

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