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

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