Lacroix and the Calculus pp 53-88 | Cite as

# The principles of the calculus

Chapter

## Abstract

In the late 18th century there were various competing foundational approaches for the differential calculus. In this section I will try to present them, drawing mainly upon works that were published (not necessarily for the first time) while Lacroix was preparing the first edition of his *Traité*, or that were then still widely used.

## Keywords

Power Series Late 18th Century Integral Calculus Algebraic Analysis Foundational Approach
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## References

- 1.“Leibnizian” here does not refer necessarily to adherence to Leibniz’s personal views, but rather to the “Leibnizian tradition”, which had other authors, among whom Jacob (I) and Johann (I) Bernoulli. Leibniz’s personal views on infinitesimals are a quite complicated subject [Bos
*1974*, 52–66].Google Scholar - 3.“We say that a quantity is infinitely great or infinitely small with regard to another, when it is not possible to assign any quantity sufficiently large or sufficiently small to express the ratio of the two, that is, the number of times that one contains the other” [Bézout
*1824*, 8].Google Scholar - 4.“we may always take one of the first differentials as a fixed term of comparison for the other first differentials” [Bézout
*1824*, 20]Google Scholar - 6.Euler did not use any particular name for the differential ratios. In [Bos
*1974*] they are called*differential coefficients*(opposed to*differential quotients*). But it seems that it was Lacroix who introduced the expression*differential coefficients*(see page 73 below). Therefore, here I will use the expression*differential ratios*when referring to Euler.Google Scholar - 10.Third in [Cousin
*1796*], after two introductory chapters on analytic geometry and the method of undetermined coefficients.Google Scholar - 13.Although the judges spoke in their report of his text not as the best, but as the least unsatisfactory of the entries to the prize [Acad. Berlin
*1786*].Google Scholar - 14.L’Huilier took these definitions from a small tract by Robert Simson (
*De Limitibus Quantitatum et Rationum Fragmentum*), published posthumously in [Simson*1776*].Google Scholar - 16.This was prompted from the study of the ratio of two decreasing quantities
*AX, CY*, with limits*AB, CD*, respectively;*AX*:*CY*may be made as close as wished to*AB*:*CD*, but it is not necessarily always greater or always smaller [l’Huilier*1795*, 16–17].Google Scholar - 21.
*Change*of notation*within*this memoir. The accent notation had already been used by Lagrange in 1770 and possibly 1759 [Cajori*1928–1929*, II, 208]. And also, very clearly, by Euler [*Integralis*, III, § 138]: “in designandis functionibus hac lege utemur, ut sit*d*.f:*v*=*dv*f′:*v*, sicque porro*d*.f′:*v*=*dv*f ″:*v*et*d*.f ″:*v*=*dv*f ″′:*v*etc.” (“we will use this rule in designating functions, so that*d*.f:*v*=*dv*f′:*v*, and so forth*d*.f′:*v*=*dv*f ″:*v*et*d*.f ″:*v*=*dv*f ″′:*v*etc.”). But most often Euler used*p, q*, etc.; and of course it was [Lagrange*Fonctions*] that made the accent notation popular.Google Scholar - 23.
- 26.There are two surviving manuscripts of the 1789 memoir, one kept at the
*Biblioteca Medicea Laurenziana*in Florence, and the other at the*École des Ponts et Chaussées*of Paris. Accounts of the memoir can be found in [Grabiner*1966*, 47–59], [Panza*1985*] and [Friedelmeyer*1993*, 69–131]. I have used them to write this passage and another in section 4.2.1.1 on contact of curves. Later, I was able to make some improvements thanks to photocopies of the Florence manuscript, kindly supplied to me by Marco Panza.Google Scholar - 27.It is likely, but not certain, that Lacroix had direct access to the 1789 memoir. He does not mention Arbogast at all while treating the principles of the calculus; he does allude to his memoir in passing in chapter 4 [Lacroix
*Traité*, I, 370], but only referring to the similarity between the ways in which Arbogast and Lagrange treated curves-he might know this from elsewhere, namely from Lagrange; while it is very clear that he had read another unpublished memoir by Arbogast, on “arbitrary functions” [Lacroix*Traité*, II, viii, 619], and that he was in contact with Arbogast already in 1794 [*Traité*, III, 543]. It is of course possible that Lacroix read Arbogast’s memoir but only in or after 1795, making that reading irrelevant for his development of the principles of the calculus, but in time for the reference in a later chapter.Google Scholar - 30.Grabiner [
*1966*, 142] argues that this proof, and particularly the “characterization of the continuity of*iP*” that Lagrange gives here, can be easily translated into algebra. But then, why did not Lagrange, the algebraist*par excellence*, do so? The fact is that Lagrange does not really*characterize*continuity here; he only uses a*property*of continuity. He did not have an algebraic characterization of continuity — continuity was a fundamentally geometrical property — and when he needed to appeal to continuity he had to resort to geometrical language.Google Scholar - 45.This was useful later in establishing higher-order differentials: the differential of
*dy*=*y′h*was of course*dy′ · h*=*y″h*^{2}=*y′″dx*^{2}, since*h*was a constant [Cauchy*1823*, 45]. The alternation between*dx*constant/variable according to*x*as independent/dependent variable followed [Cauchy*1823*, 48].Google Scholar - 46.This habit can be seen for instance in [Bossut
*1798*, II, 351]. Partial differential equations were usually called “equations in partial differences” [Condorcet*1770*; Lagrange*1772b*; Laplace*1773c*; Monge*1771*].Google Scholar - 50.This second deduction is the only one used in [Lagrange Calcul] (which does not have sections on applications) [Lagrange
*Calcul*, 62–68].Google Scholar - 53.[Lacroix
*Traité*, I, 233]: “Every time the function we want to expand is irrational in general, but ceases to be so by the substitution of a particular value of*x*, the irrationality must then fall upon the increment”.Google Scholar - 54.[Lacroix
*Traité*, I, 234]: “Nous offre [...] le moyen de l’établir sur des fondemens plus solides que l’induction dont nous l’avons déduite”. “Induction”, of course, is here used in the nonmathematical sense — see footnote 47.Google Scholar - 56.[Lacroix
*Traité*, I, 1]: “Any quantity the value of which depends on one or more other quantities is said to be a*function*of these latter, whether or not it is known which operations are necessary to go from them to the former.”Google Scholar - 58.[Lacroix
*Traité*, 1st ed, I, 6]: “Henceforth, we will call*limit, every quantity which a magnitude cannot surpass as it increases or decreases, or even that it cannot achieve, but which it can approach as close as one might wish*.”Google Scholar - 62.In the Introduction there is a situation in which he does verify that the extra condition holds and thus he can use Arbogast’s principle [Lacroix
*Traité*, 58–59]. This is in a deduction of a power series for the sine.Google Scholar - 65.[Lacroix
*Traité*, I, xxiv]: “The reconciliation of the Methods which you are planning to make, serves to clarify them mutually; and what they have in common contains very often their true metaphysics: this is why that metaphysics is almost always the last thing that one discovers”. This translation is taken from [Grattan-Guinness*1990*, I, 139]Google Scholar

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