Lacroix and the Calculus pp 141-177 | Cite as

# Approximate integration and conceptions of the integral

## Abstract

It is well known that one of the first *innovations* introduced by the Bernoulli brothers on the Leibnizian differential calculus was the answer to “what is ∫ *y dx*?”. Leibniz originally meant this to be the ∫*um* of the infinitesimally narrow rectangles of sides *y* and *dx* (∫ is a typical 18th-century italic *s*) — and therefore the area under the curve represented by *y*. However, he later adopted the name *integral*, coined by Johann I Bernoulli but first proposed in print by his brother Jacob, suggestive of a different definition for the operation represented by ∫ : simply the inverse operation of differentiation [Bos *1974*, 20–22; Boyer *1939*, 205].

## Keywords

Taylor Series Arbitrary Constant Eighteenth Century Explicit Function Primitive Function## Preview

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## References

- 3.A simple (although perhaps not very faithful) rendition of that derivation could be: put
*y*=*∫ X(x, a) dx*, so that*d*_{x}*y*=*X(x, a) dx*; from*d*_{x}*d*_{a}*y*=*d*_{a}*d*_{x}*y*comes*d*_{x}*d*_{a}∫*X(x, a) dx*=*d*_{a}*X(x, a) dx*; integration (on*x*) gives*d*_{a}∫*X(x, a) dx*= ∫*d*_{a}*X(x, a) dx*. The original is in a very geometrical language [Engelsman 1984, 202–203].Google Scholar - 6.A more serious challenge was posed by Euler’s “isoperimetric rule”; Lagrange was able to derive it without resorting to integral-as-sum considerations only in 1806. It is almost certainly not a coincidence that isoperimetric problems were neglected in the meantime [Fraser
*1992*].Google Scholar - 9.The conception of the integral as sum also carried-in theory at least — the danger of more frequent appearances of infinitely large quantities of the form ∫
*y*, where*y*is finite [Bos*1974*, 22].Google Scholar - 11.As for [Newton
*Principia*], it was a very explicit attempt at writing in a synthetic and geometric style — soon very old fashioned; in his other writings it is the inverse relationship between fluxions and fluents that we see [Bos*1980*, 54–60; Boyer*1939*, 190–202, 206; Guicciardini*2003*, 78–84, 100–102].Google Scholar - 13.The American translation of this passage is not quite literal: “The method known by the name of
*Integral Calculus*is the reverse of the Differential Calculus. It has for its object to ascend from differential quantities to the functions from which they are derived” [Bézout*1824*, 74]. Note that the word “function” is only defined three paragraphs below.Google Scholar - 14.“To indicate the integral of a differential, the letter ∫ is written before this quantity; this letter is equivalent to the words
*sum of*, because,*to integrate*, or*take the integral*, is nothing but to sum up all the infinitely small increments which the quantity must have received, to arrive at a determinate, finite state.” [Bézout*1824*, 75]Google Scholar - 15.Except in the few situations in which it was technically unavoidable (see footnote 6).Google Scholar
- 19.Often Euler
*forgets*to include the constant of integration. Sometimes this is because*C*was previously set = 0 for the same or a similar integral. When that is not the case it might be interpreted as an implicit setting of*C*= 0, particularly if that would make the integral vanish for*x*= 0; this interpretation is weakened before a list of integrals such as in [*Integralis*, I, §7–78], all lacking a constant of integration, and having different values for*x*= 0. Whatever the case, often the integral is afterwards calculated for a specific value of the variable; this is what happens, for instance, in the title of [Euler*1774a*], which includes the expression “casu quo post integrationem ponitur*z*= 1” (“when after the integration*z*is set = 1”).Google Scholar - 26.There is at least one precedent for this sort of thing: in [Euler
*Integralis*, I, §304] Euler speaks of the formula \( \frac{{x dx}} {{\sqrt {1 - x^3 } }} \) in the interval*x*= 1 −*ω*to*x*= 1; he introduces the change*x*= 1 −*z*, so that the new bounds are*z*= 0 and*z*=*ω*. The context is that of approximating integrals (see section 5.1.3).Google Scholar - 28.This could be particularly cumbersome in the calculus of variations, where one tries to find the function
*y*of*x*for which “la fonction primitive de*f*(*x, y, y′, y″*...), fût un*maximum*ou un*minimum*, en supposant que cette fonction soit nulle lorsque*x*aura une valeur donnée*a*, et qu’elle devienne un*maximum*or a*minimum*lorsque*x*aura une autre valeur donnée*b*” (“the primitive function of*f*(*x, y, y′, y″*...) is a*maximum*or a*minimum*, supposing that that function is null when*x*has a given value*a*, and that it becomes a*maximum*or a*minimum*when*x*has a different given value*b*”) [Lagrange*Fonctions*, 201].Google Scholar - 33.“On the mode of integrating by approximation and some uses of that method” [Bézout
*1824*, 106–119]Google Scholar - 34.“The art of integrating by approximation, consists in converting the proposed quantity into a series of simple quantities whose value continually diminishes; each term is then easily integrated and it is sufficient to take a certain number of them, in order to obtain an approximate value for the integral” [Bézout
*1824*, 106].Google Scholar - 38.According to [Grabiner
*1981*, 149], Euler did impose monotonicity: “first, he [Euler] said, assume that the function is always increasing or always decreasing on the given interval”. I cannot locate any such passage in Euler’s text.Google Scholar - 40.Tournès [
*2003*, 458–463] indicates several geometrical antecedents of this method in its version for differential equations.Google Scholar - 41.Although the arithmetic mean between these
*upper*and*lower*estimates was used, namely by Carl Runge (1856–1927), to obtain an improved method [Chabert*1999*, 381–387].Google Scholar - 47.“in varying the arbitrary constants in the approximate integrals and then determining their values for a given time by integration.” [Gillispie
*1997*, 48]Google Scholar - 49.[Lagrange
*1776*] is nevertheless an important work, namely for the (pre-)history of Padé approximants [Brezinski*1991*, 137–139]. Also from that memoir Lacroix extracted a method for expanding functions into series, which he reported in chapter 2 and used in chapter 4 of [Lacroix*Traité*, I] (see page 108).Google Scholar - 55.This issue had already appeared, apropos of an expansion for \( \int {\frac{{x^m dx}} {{x^n + a^n }}} \) [
*Lacroix Traité*, II, 68–69]. Apparently Lacroix always*preferred*convergent series.Google Scholar - 62.Lagrange, on the other hand, in a passage equivalent to that referred to in footnote 65, decided to have −1 > −2, but he had to state this explicitly [Lagrange
*Fonctions*, 46].Google Scholar - 65.In case
*X*takes negative values somewhere in the interval,*m*must be the “greatest” of these — that is, the greatest in absolute value, what we would still call the smallest. Similarly, if*X*only takes negative values, then*M*must be the “smallest”, not the “greatest” value [Lacroix*Traité*, II, 142].Google Scholar - 72.It must have been clear enough for the textbook writer Jean-Guillaume Garnier, who reproduced it almost word for word in [Garnier
*1812*, 108].Google Scholar - 74.
*Sic*; not only is this not corrected in the errata as it is repeated in [Lacroix*1802a*, 288] and [Lacroix*Traité*, 2nd ed, II, 134] (but, curiously, it appears as*B — A*in [*1802a*, 2nd ed, 303] and subsequent editions). One can only assume that Lacroix is only concerned here with the*absolute*difference. Nevertheless, as we have seen above, he speaks further ahead of this difference as*Y*_{b}−*Y*_{a}.Google Scholar - 76.In this same year (
*an*VI ≈ 1798) “indefinite integral” made a fleeting appearance in [Bossut*1798*, I, 415], but “definite integral” does not seem to have accompanied it there.Google Scholar - 77.Grabiner is of course well aware of Lacroix’s “eclectic view” of the concepts of the calculus, but explains it on purely technical grounds: “Lacroix, like most mathematicians of the time, wanted to show how to solve problems; therefore his
*Traité*included whatever techniques were applicable to this end” [Grabiner*1981*, 79–80]. This interpretation of Lacroix’s motivations, while not at all wrong, is in my view too restrictive.Google Scholar - 83.Concerning the influence of Lacroix’s
*Traité*, it is also noteworthy that Cauchy’s first existence theorem derived from the same method of approximation [Cauchy*1981*, 39–66]. Gilain [*1981*, xxiv–xxv, xxxiii] compared Cauchy’s work with Lacroix’s*Traité*, but because he used only the second edition of the latter he missed Lacroix’s connection between the analytical version of this method and the “possibility” of differential equations.Google Scholar