Author(s):

Robert E. Bradley (Adelphi University) and Salvatore J. Petrilli, Jr. (Adelphi University)

Servois' memoir is a long paper, some sections of which can be omitted without affecting understanding of the material that follows. Here we provide a thematic breakdown of the sections in Servois' paper, where page numbers refer to the pagination of the original article, which are given in square brackets in the English translation of Servois' "Memoir on Quadratures" (pdf).

- pp. 74–75: Presentation of notation and results about differential and integral operators, which are used to derive the Euler-MacLaurin formula.
- pp. 76–78: Composite Trapezoid and Midpoint Rules are presented with error terms. Also, left-hand and right-hand Riemann Sums are discussed.
- pp. 78–80: Servois engages in a digression on polar coordinates inspired by Adrien-Marie Legendre (1752–1833). This section can be omitted without affecting the logical flow of the remainder of the paper.
- pp. 81–84: Servois begins a discussion of the Newton-Cotes formulas as presented by Kramp [1815b].
- pp. 85–87: Servois suggests a better way to calculate the coefficients of the Newton-Cotes formulas, in response to [Kramp 1816] .
- pp. 87–89: Servois discusses [Kramp 1815a] and d'Obenheim's method; the first example presented, on page [88], is equivalent to d'Obenheim's procedure.
- pp. 89–94: Servois presents a second example, which vindicates [Bérard 1816].
- pp. 95–99: Servois addresses why the case of \(n=12\) in [Kramp 1815b] is incorrect. He then moves to contemplating complete parabolic curves, i.e., polynomials of degree \(n\). Servois then returns to a discussion of [Kramp 1815b], specifically, fitting the curve to a collection of points and integrating that equation. Finally, he demonstrates the equivalence of the methods in [Kramp 1815b] and [Bérard 1816].
- pp. 99–100: In this section, Servois explains what d'Obenheim actually did in his original paper and provides commentary on the connection between d'Obenheim's extrapolation and the integration of parabolic curves.
- pp. 100–103: Servois addresses objections to the use of complete parabolic curves.
- pp. 103–106: Servois engages in a discussion of convergent and divergent series, asserting that divergent series have no place in numerical approximation.
- pp. 107–112: In this section, Servois discusses the technique of fitting a parabolic curve to pass through given points in such a way that the derivatives of the fitted curve agree with the derivatives of the original curve at these points. This section may be omitted without loss of continuity, but it is very interesting to see Hermite Interpolation introduced long before the time of Charles Hermite (1822–1901).
- pp. 113–115: Here, Servois closes with three major notes: (i) a brief note about how Euler numerically approximated the area under the curve in his
*Institutiones calculi differentialis*; (ii) a note about using Taylor's Theorem to derive all the formulas in this paper in a uniform manner; (iii) a connection to the problem of estimating the perimeter of a circle by means of circumscribed or inscribed polygons that was raised by Gergonne in [Gergonne 1815].

Robert E. Bradley (Adelphi University) and Salvatore J. Petrilli, Jr. (Adelphi University), "Servois' 1817 "Memoir on Quadratures" – A Thematic Breakdown of the Memoir," *Convergence* (May 2019)