''Cours d'Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique'' is a seminal textbook in
infinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of ari ...
published by
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. H ...
in 1821. The article follows the translation by Bradley and Sandifer in describing its contents.
Introduction
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
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, so that one need never rely on arguments drawn from the
generality of algebra."
Preliminaries
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
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
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''.
The notation
:lim
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
’Huilier 1787, p. 31 Cauchy wrote this as “lim.” in
auchy 1821, p. 13 The period had disappeared by
auchy 1897, p. 26"
Chapter 2
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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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.
Section 2.2
This section 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 mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two impor ...
.
Sum 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
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
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)
Bibliography
*
Free versionat
archive.org
*
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{{Infinitesimals
Mathematics of infinitesimals
Calculus
History of calculus
Mathematics textbooks