Servois' 1817 "Memoir on Quadratures" – The Differential

In Continental Europe before the 19th century, calculus was done in terms of differentials and not derivatives. For example, taking the differential of the equation $y=x^2$ would give rise to $dy = 2x \, dx$ by the rules of Gottfried Wilhelm Leibniz's (1646–1716) differential calculus. Today, we might formally divide this equation through by $dx$ to get $\frac{dy}{dx} = 2x$, the expression of the derivative in Leibniz' notation, but to mathematicians in Servois' time and earlier, the equation $dy = 2x \, dx$ was meaningful in itself, giving the relationship between an "infinitely small" increment in the $x$-direction and the corresponding increment in the $y$ direction—also infinitely small.

At the time Servois was writing his "Memoir on Quadratures,'' this point of view was gradually being replaced by the scheme we use today: finding the quotient of the finite differences $\Delta y$ and $\Delta x$, then taking the limit as $\Delta x \rightarrow 0$. For more on the foundation of calculus in Servois' time, and his contributions to the search for a foundation, see [Bradley and Petrilli 2010b].

Instead of taking limits, Servois defined the differential $dy$ to be the following alternating series:

$$dy = \Delta y - \frac{1}{2} \Delta^2 y + \frac{1}{3} \Delta^3 y - \ldots .\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \mbox{(IV)}$$

He gave a derivation of this series in [Servois 1814a]. We can get an idea of how it works by considering the following simple example.

Set $y=x^2$. Then $\Delta y = Ey - y = (x + \Delta x)^2 - x^2 = 2 x \Delta x + (\Delta x)^2$, and

$$\begin{align*}\Delta^2 y &= \Delta (\Delta y) \\&= \Delta \left(2 x \Delta x + (\Delta x)^2\right)\\&= \left[2(x + \Delta x)\Delta x + (\Delta x)^2\right] - \left[2 x \Delta x + (\Delta x)^2\right]\\&= 2(\Delta x)^2.\end{align*}$$

Furthermore, $\Delta^3 y = 0$ and so $\Delta^k y = 0$ for $k \geq 3$. Substituting these values into (IV), we thus have

$$dy = 2 x \Delta x + (\Delta x)^2 -\frac{1}{2}\left(2(\Delta x)^2\right) = 2x \Delta x,$$

or $dy = 2x \,dx$ upon interpreting $dx$ for $\Delta x$. This nominal equivalence between $dx$ and $\Delta x$ explains why Servois wrote $\omega = \Delta x = dx$ just above his equation (1).

Servois, and a number of mathematicians before him, noticed that the series (IV) has the same form as the MacLaurin series for $\ln(1 + x)$:

$$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + \ldots.$$

Servois used Log to denote the natural logarithm and wrote $dy = \mbox{Log}(1+ \Delta)y$ in his equation (3). Cancellation of the $y$’s in this equation then gives $d = \mbox{Log}(1+ \Delta)$, so that Servois' equation (3) can be formally expressed as $e^d y = (1 + \Delta)y = Ey$.

We note that Servois' equation (1) is incorrect as written, whether the error is typographical or otherwise. It should probably have been given as

$$E^n(y) = F(x+n\omega) \quad \mbox{and} \quad Ey = e^d y.$$