Servois' 1817 "Memoir on Quadratures" – Dispute in Gergonne's \(\scriptsize Annales\)

Joseph-Diez Gergonne's (1771–1859) Annales de mathématiques pures et appliquées had its fair share of debates and disputes within its pages. Servois was involved in what was probably the most famous of these, regarding the use of geometry to represent complex numbers, along with Jean-Robert Argand (1768–1822) and Jacques Français (1775–1833).

Figure 2: Portrait of Joseph-Diez Gergonne. Public domain.

Servois' “Quadratures” paper was inspired by another debate, which occurred from 1815 to 1817. Servois [1817] stated:

For my part, I confess that I read with great satisfaction, in the Annales de mathématiques, the detailed expositions of three new methods of approximation that came from good sources, because they belong to professors Dobenheim¹, Kramp, and Bérard. I have participated, with more or less the aptitude of an interested party, in the debates that they have initiated [p. 73].

In 1815 Christian Kramp (1760–1826) published a paper in Gergonne's Annales, in which he gave an approximation for the area under a curve using the Trapezoidal Rule and an ad hoc extrapolation method that was first proposed by Alexandre-Magnier (Magnus) d'Obenheim (1753–1840), who was a professor at various French artillery schools from 1801 to 1835.

In particular, Kramp provided approximations to the area using the  Composite Trapezoidal Rule for \(n =\) 1, 2, 3, 4, 6, and 12 subdivisions. His extrapolation procedure used these cruder approximations to generate a much better approximation for the area under the curve. He showed that his method of approximation is very accurate when used to approximate the values of \(\ln 2\) and \(\frac{\pi}{4}\).

Shortly after Kramp's article was published, Gergonne wrote a follow-up piece stating that Kramp's idea was truly remarkable, but the approximation it produced was not very useful in a practical sense without an error term. In the examples of \(\ln 2\) and \(\frac{\pi}{4}\), these values are well-known, but there is no way to assess the accuracy of an approximation without an error bound when the true value is unknown. Gergonne argued that we should not adopt methods of numerical integration that sacrifice precision for the sake of speed.

Kramp wrote a follow-up paper to Gergonne's rebuttal. In this paper, Kramp abandoned his original idea, which was based on the work of d'Obenheim. Instead, he took on the problem of expressing a curve as a Taylor Series and integrating it term-by-term. He derived what are now known as the Newton-Cotes formulas for several cases; however, there was an error in his formula for calculating the approximation in the case of \(n=12\) subdivisions.

Joseph-Balthazard Bérard (1763–1844?) entered the debate in 1816. In his paper, Bérard discussed issues with the d'Obenheim-Kramp method of finding the area under a curve and the improved method that Kramp introduced in his second article. Bérard asserted that the method in Kramp's first article was arbitrary and that he only succeeded in making accurate approximations by chance. He noted that Gergonne's proposed improvement was just as arbitrary and gave less accurate approximations. In fact, he claimed that adapting Gergonne's method to \(n= 16\) subdivisions gave rise to a poorer approximate of \(\ln 2\) than the case of \(n=12\). Instead, he proposed to follow the methods of Kramp's second paper and gave much more efficient ways of calculating the coefficient formulas needed for the approximation. He was even able to calculate the formulas for \(n = 24\) and \(n = 120\) subdivisions. Along the way, he turned up an error in Kramp's formula for the case of \(n = 12\) subdivisions.

Finally, later in 1816, Kramp wrote a third paper that defended the method he introduced in his second paper and critiqued Bérard's paper. Kramp argued that his formula for \(n = 12\) subdivisions was the correct one, even though Bérard's formula gave a slightly more accurate approximate value of \(\ln 2\). There is little if any new mathematics in this paper, which is more of a polemical review of Bérard's work.

In his “Memoir on Quadratures,” Servois attempted to give an overview of numerical integration that encompassed the points of view expressed in these works of Kramp, Gergonne, and Bérard. Let us now explore Servois' insights into this debate. We begin by presenting an overview (in today's notation) of the numerical integration formulas discussed by Servois: the Trapezoidal Rule and the Newton-Cotes Formulas. We then provide a section-by-section thematic breakdown of Servois' “Quadratures” paper.  We next provide introductions to the finite difference notation that Servois himself employed and to his general approach to questions in analysis, sometimes called the operational calculus. Finally, we examine two specific topics from Servois' memoir in more detail: his use of Bernoulli Numbers and his derivation of the Euler-MacLaurin formula, which served as his primary tool for resolving the debate on numerical integration.

Notes:

1. In [Servois 1817], Servois used this alternate spelling of d'Obenheim.