Approximate integration and conceptions of the integral

Part of the Science Networks. Historical Studies book series (SNHS, volume 35)


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].


Taylor Series Arbitrary Constant Eighteenth Century Explicit Function Primitive Function 
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  1. 3.
    A simple (although perhaps not very faithful) rendition of that derivation could be: put y = ∫ X(x, a) dx, so that dxy = X(x, a) dx; from dxday = dadxy comes dxdaX(x, a) dx = daX(x, a) dx; integration (on x) gives daX(x, a) dx = ∫daX(x, a) dx. The original is in a very geometrical language [Engelsman 1984, 202–203].Google Scholar
  2. 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
  3. 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
  4. 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
  5. 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
  6. 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
  7. 15.
    Except in the few situations in which it was technically unavoidable (see footnote 6).Google Scholar
  8. 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
  9. 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
  10. 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
  11. 33.
    “On the mode of integrating by approximation and some uses of that method” [Bézout 1824, 106–119]Google Scholar
  12. 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
  13. 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
  14. 40.
    Tournès [2003, 458–463] indicates several geometrical antecedents of this method in its version for differential equations.Google Scholar
  15. 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
  16. 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
  17. 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
  18. 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
  19. 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
  20. 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
  21. 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
  22. 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 YbYa.Google Scholar
  23. 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
  24. 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
  25. 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

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