Servois' 1817 "Memoir on Quadratures" – Finite Differences and the Varied State

When tackling questions regarding numerical integration in his "Memoir on Quadratures,'' Servois used slightly different notation than the current notation that we introduced earlier in this article. He considered a function \(F\), fixed a constant difference \(\Delta x=\omega\), and then defined a sequence of \(n+1\) points

\[(a,F(a)), (a+\omega,F(a+\omega)), (a + 2\omega,F(a + 2 \omega)), \ldots  (x,F(x))\]

with equally-spaced \(x\)-coordinates (which Servois usually called "abscissas").

Although subscripts were not in general use during Servois' time, it will make our lives simpler to use them in discussing his memoir. Servois considered a sequence on the \(x\)-axis

\[x_k = a +k \omega; \; k=0, 1, 2, \ldots, n\]

and the corresponding \(y\)-values \(y_k = F(x_k)\). He denoted the left endpoint \((x_0,y_0)\) by \((a,v)\) and the right endpoint \((x_n,y_n)\) by \((x,y)\), which can be a little confusing, given that \(x\) and \(y\) are also the general names of the variables. Although he didn't use subscripts, Servois had access to a useful notation, due to the French mathematician Louis François Antoine Arbogast (1759–1803). In his influential textbook Du calcul des dérivations [Arbogast 1800], Arbogast defined an operator \(E\) called the varied state by the equation

\[E y = F(x + \Delta x),\]

which is \(F(x + \omega)\) in Servois' setting. The varied state can be iterated, so that¹

\[E^ny=F(x + n\omega),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{(III)}\]

which is the definition given by Servois in his equation (1).

The varied state also has an inverse \(E^{-1}\) which can also be iterated

\[E^{-n}y = F(x-n\omega).\]

In fact, Servois even considered expressions like

\[E^{\frac{1}{2}}y = F\left(x + \frac{1}{2} \omega\right);\]

thus, in equation (III) we could even think of \(n\) as representing a continuous parameter.

The varied state is useful for the differential calculus, because it can be used in the definition of the finite difference \(\Delta y\):

\[\Delta y = F(x + \Delta x) - F(x)= Ey - y.\]

Because \(Ey - y\) may be thought of as \(Ey - 1y\), Servois wrote

\[\Delta y = (E-1)y\]

in his equation (2), and even abstracted this to the equation \(\Delta = E -1\), defining the operator \(\Delta\) in terms of the varied state and the identity operator. This rather forward-looking kind of abstraction was at the core of what was called the “Calculus of Derivations” or the “Operational Calculus,'” a field pioneered by Arbogast to which Servois made important contributions in [Servois 1814a].

Notes:

1. Recall that equation numbers in this article are given in Roman numerals to distinguish them from the original equation numbers used by Servois in [Servois 1817]. There is no correlation between these Roman numerals and his Hindu Arabic numerals.