Lacroix and the Calculus pp 21-51 | Cite as

# An overview of Lacroix’s *Traité*

## Abstract

According to his own statement, Lacroix started collecting material for his *Traité* in 1787, while employed at the *École Royale Militaire* in Paris [*Traité*, I, xxiv]. This is confirmed by his correspondence: during his stay in Besançon (1788–1793) he wrote to mathematicians in Paris asking them to send him material or information on how to find it. In October 1789 Lacroix thanked Legendre for information on a work by Landen, and explained that he wished to use the tables of integrals included there for a project “dans lequel j’ai pour objet de rassembler dans un corps d’ouvrage les materiaux sur le calcul integral qui se trouvent dans les memoires des societes savantes”^{1} [Lacroix *IF*, ms 2397]; in 1792 he communicated the same intent to Laplace [Taton *1953b*, 353].^{2}

## Keywords

Transcendental Function Explicit Function Analytic Geometry Title Page Integral Calculus## Preview

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## References

- 4.“Elementary” here must be understood in the sense that they start with the first notions, the “elements” of the calculus, rather than assuming them and addressing original research straight away. After the educational reforms of the 1790’s and 1800’s, “elementary” would mean simple, or introductory — see for example [Lacroix
*Traité*, 2nd ed, I, xx], where the*Traité*is specifically opposed to “elementary books”; see also section 8.1.Google Scholar - 6.This sentence can be found in fl. 19v of Lacroix’s “Compte rendu [...] des progrès que les mathématiques ont faits depuis 1789 [...]”. See appendix B for the relation between the “Compte rendu” and [Delambre
*1810*].Google Scholar - 7.[Euler
*Introductio*] does include geometrical applications (analytic geometry); but they are missing from [Euler*Differentialis*] and [Euler*Integralis*].Google Scholar - 10.As an aside, it is curious to know that in 1818–1819 Lacroix took a course in Chinese by Rémusat (the first professor of Chinese at the
*Collège de France*) [Lacroix*IF*, ms 2402, fls 380–465].Google Scholar - 11.In [Domingues
*2005*, 281] I said that “a ‘weak’ version of the binomial theorem, stating (1 +*x*)^{n}= 1+*nx*^{n−1}+ etc. is proven (for ‘any*n*’; the full expansion is given for integer*n*)”. Apart from the fact that one should read “rational” instead of “integer”, this is misleading because Lacroix shows the recursive relation between the coefficients independently of*n*being integer or not [*Traité*, I, 19–22]. My mistake resulted from the physical separation between this and the general proof that the first two terms in the expansion of (1 +*x*)^{n}are 1 +*nx*[*Traité*, I, 49].Google Scholar - 14.This would change in the second edition, where the coverage of the second volume increases from one small paragraph [Lacroix
*Traité*, I, xxvii] to about six pages [Lacroix*Traité*, 2nd ed, I, xxxviii–xliv]. This is more than the three pages for the third volume (one page in the first edition), but still much less than the nineteen pages for the first volume (about three pages in the first edition).Google Scholar - 17.Very briefly, these methods relied on obtaining all possible forms for the solutions (or integrating factors) of differential equations, and then trying to adequate one of those to the equation to be solved (using the method of indeterminate coefficients) [Gilain
*1988*, 91–97].Google Scholar - 18.Consisting essentially in a method to solve systems of 1st-order linear equations using multipliers, and in the reduction of systems of higher-order equations to first order, considering new variables
*p*=_{dx}^{dy},*q*=_{dx}^{dp}, etc. Gilain stresses Lacroix’s role in the transmission of d’Alembert’s theory, which was not particularly well known by his contemporaries (still, it appears in [Cousin*1796*, I, 234–238]). Gilain focuses especially on the transmission through [Lacroix*1802a*], and especially to Lacroix’s student Cauchy, who would give it in [*1981*] an importance much greater than the marginal place it occupies in [Lacroix*1802a*] (and, it may be added, in [Lacroix*Traité*]).Google Scholar - 21.Kline [
*1972*, II, 535] also tells this story but, ignoring the existence of two manuscript copies of Charpit’s memoir [Grattan-Guinness & Engelsman*1982*], he still relies exclusively on Lacroix’s information (carefully adding not to “know whether Lacroix’s statement is correct”).Google Scholar - 22.Lacroix’s basic version [
*Traité*, II, 524–526] is as usual much clearer and/or easier to follow than Monge’s. Kline’s account [*1972*, II, 538–539], who claims to follow [Monge*Feuilles*] rather than [Monge*1784b*], in fact seems to draw on Lacroix. I also do not understand why Kline calls “nonlinear” these equations which are “linear only in the second derivatives”, while a few pages earlier he had used “linear” for first-order equations which are linear only in the derivatives.Google Scholar - 26.The differential calculus is introduced at the end as the infinitesimal case [Prony
*1795a*, IV, 543–551].Google Scholar - 27.Jordan [
*1947*, 100–101] calls this the “indefinite sum”.Google Scholar - 30.Thus, we do not find here the
*true*analogue of the definite integral, namely the modern*definite sum*S_{a}^{b}*f*(*x*) =*f*(*a*) +*f*(*a*+ 1)+... +*f*(*b*− 1) [Jordan*1947*, 116; Goldstine*1977*, 99].Google Scholar - 34.Notice that the Introduction of [Bézout
*1779*] is a short account of the direct and inverse calculus of differences.Google Scholar - 35.Much earlier, Moivre had determined the general term of recurrent sequences, which is equivalent to solving linear finite difference equations with constant coefficients. But apparently it was Lagrange who first made the connection, and treated them as difference equations [Laplace
*1773a*, 38].Google Scholar - 39.Both I [Domingues
*2005*, 277] and Grattan-Guinness [*1990*, I, 140] have been tricked, by the fact that the translation has two volumes, into thinking that it was a translation of the first*and second*volumes of Lacroix’s*Traité*.Google Scholar - 43.Phili [
*1996*, 305] also gives the alternative spelling*Καρανδίνος*. The online library*Hellinomnimon*<http://www.lib.uoa.gr/hellinomnimon/main.htms> (accessed on 23 January 2007) uses*Καραντηνός*. The title pages of his books available there seem to alternate between*Καρανδίνος*,*Καραντινός*, and*Καρανδηνός*.Google Scholar