Lacroix and the Calculus pp 321-341 | Cite as

# The second edition of Lacroix’s *Traité*

Chapter

## Abstract

We do not know the print-run of the first edition of Lacroix’s *Traité*, but it must have sold well. [Lacroix *1805*] includes, just before the table of contents, a list of other works by Lacroix “that can be found in the same bookstore” (Courcier). We find the several textbooks in his *Cours de Mathématiques*, with their respective prices, and the large *Traité*. But the latter does not have a price; instead, it carries the indication “rare et épuisé”^{1}.

## Keywords

Arbitrary Function Arbitrary Constant General Integral Analytic Geometry Derivation Process
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## References

- 3.As is well known, Gauss gave four proofs of the Fundamental Theorem of Algebra [Kline
*1972*, 598–599]. The fourth proof appeared only in 1850, and the first proof was given in his doctoral dissertation, which Lacroix probably did not know. But the second and third proofs [*1814–1815b*;*1814–1815c*] appeared in the same volume of the Royal Society of Göttingen as a paper on approximation of integrals [Gauss*1814–1815a*] which was cited by Lacroix [*Traité*, 2nd ed, III, xii].Google Scholar - 4.It was Ampère who reported Binet’s proof to the
*Société Philomatique*, saying that Binet proposed to demonstrate this theorem “d’une manière plus simple qu’on ne l’a fait jusqu’à présent” (“more simply than what has been done until now”) [Binet*1809*, 275].Google Scholar - 5.He does cite it later apropos of its other subject: the remainder of Taylor’s series [
*Traité*, 2nd ed, I, 388, III, 399–400].Google Scholar - 6.This draft is kept at [Lacroix
*IF*, ms 2400], and is transcribed in [Grattan-Guinness*1990*, III, 1325–1329]. The letter was not sent, because meanwhile Français died (in October 1810).Google Scholar - 7.There is even a new application of this: the determination of infinite branches of curves [Lacroix
*Traité*, 2nd ed, I, 408–413]. New, that is, in the sense that it was not in the first edition — Lacroix attributes the procedure he reports to du Séjour and Goudin.Google Scholar - 8.Lacroix adds four pages on series with decreasing exponents [
*Traité*, 2nd ed, I, 417–421].Google Scholar - 9.We have seen in section 8.6 that polar coordinates is the only topic of analytic geometry appearing in [Lacroix
*1802a*] instead of [Lacroix*1798b*].Google Scholar - 11.In spite of the improvement, there is a clear editorial flaw: the equation of the sphere is derived twice [Lacroix
*Traité*, 2nd ed, I, 508, 519] — and this is not a sign of*encyclopédisme*, since the derivation is precisely the same.Google Scholar - 12.Ivory is cited because of his 1809 paper on attractions of spheroids, a paper that caused a sensation among Parisian mathematicians, although for much more than analytic geometry [Grattan-Guinness
*1990*, I, 418–422].Google Scholar - 14.This finishes with a reference to a couple of papers (or a couple of versions of a paper) on optics by étienne Louis Malus [Grattan-Guinness
*1990*, I, 473; Struik*1933*, 115], which do not appear in the table of contents (the version submitted to the*Institut*had received a favourable report by Lacroix).Google Scholar - 15.These new details are not necessarily
*new*— that is, not necessarily posterior to 1810; even d’Alembert is cited [*Traité*, 2nd ed, III, 671, 672–673].Google Scholar - 17.This contradicts Grabiner’s assertion that “Lacroix did not try to prove that the true value of the integral of an arbitrary function differs from the approximating sums by less than any given quantity for sufficiently small subintervals” [Grabiner
*1981*, 152]. She seems to have read the section on approximation only in the first edition of Lacroix’s*Traité*, and to have assumed that it was unchanged in the second.Google Scholar - 18.Asimilar result had been proved by other means in [Lacroix
*Traité*, 2nd ed, I, 382].Google Scholar - 24.And modifies the introductory article to the chapter [Lacroix
*Traité*, 2nd ed, II, 373]: now particular solutions “paraissent d’abord de deux sortes” (“appear at first to be of two kinds”), instead of “sont de deux sortes” (“are of two kinds”) [Lacroix*Traité*, 1st ed, I, 389].Google Scholar - 27.In the case of second-order equations, Lacroix gives also a construction using osculating circles, simpler than the one resulting from Euler’s “general method”, which involves osculating parabolas. Tournès [
*2003*, 469] remarks that although the determination of centres of curvature and osculating circles had long been an important problem, he has not found any instance of this kind of inverse problem prior to the second edition of Lacroix’s*Traité*.Google Scholar - 32.Binet had submitted a memoir with this result to the
*Institut*in August 1814 [Acad. Sc. Paris*PV*, V, 385]. Lacroix and Poisson had been charged with reporting on it, but Binet had withdrawn it “for perfecting”. It appears to have never been published.Google Scholar

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