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
“in which my goal is to assemble in a single work the materials on integral calculus that are found in the memoirs of learned societies”
Presumably “calcul integral” is to be read here as short for “differential and integral calculus”.
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“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.
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].
[Euler Introductio] does include geometrical applications (analytic geometry); but they are missing from [Euler Differentialis] and [Euler Integralis].
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].
In [Domingues 2005, 281] I said that “a ‘weak’ version of the binomial theorem, stating (1 + x)n = 1+nxn−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].
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).
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].
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 = dydx , q = dpdx , 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]).
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”).
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.
The differential calculus is introduced at the end as the infinitesimal case [Prony 1795a, IV, 543–551].
Jordan [1947, 100–101] calls this the “indefinite sum”.
Thus, we do not find here the true analogue of the definite integral, namely the modern definite sum S ba f(x) = f(a) + f(a + 1)+... + f(b − 1) [Jordan 1947, 116; Goldstine 1977, 99].
Notice that the Introduction of [Bézout 1779] is a short account of the direct and inverse calculus of differences.
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].
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.
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.
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(2008). An overview of Lacroix’s Traité. In: Lacroix and the Calculus. Science Networks. Historical Studies, vol 35. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8638-2_3
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