Servois' 1814 Essay on the Principles of the Differential Calculus, with an English Translation - Criticism of Wronski in the 'Reflections'

Servois' purpose in writing his “Essay” had been to establish a rigorous foundation for analysis. In his opinion, calculus based on power series ascends from a firm foundation to higher levels of generality. Wronski's algorithmie, a term he used where we would use analysis, is a program which, according to Servois, descends from non-rigorous generality to particular cases. Servois used a substantial portion of his “Reflections” paper to refute Wronski's algorithmie and, in general, the use of the infinite in analysis. Although Wronski manipulated differences and differentials using Servois’ distributive property, he used the infinite loosely to change differences to differentials, which “might seem quite good to eyes afflicted with the infinitesimal squint” [Servois 1814b, p. 155].

Central to Wronski's philosophy of mathematics [Wronski 1811] and his later work was his “Absolute Law” for expanding a function \( F(x) \) as \( \sum^{\infty}_{n=0} A_n \Omega_n(x) \) for some class of “generating functions” \( \{\Omega_n(x)\} \). This law, which he inferred by informal induction, includes Taylor Series and other expansions as special cases. Servois discussed an important particular case of Wronski's formula on page 154 of the “Reflections,” in which the generating functions are,

\[ \varphi x, \; \varphi x \cdot \varphi (x + \xi), \; \varphi x \cdot (x + \xi) \cdot \varphi (x + 2\xi), \; \ldots; \]

for a particular function \( \varphi \) and a constant increment \( \xi \) in \( x \). Servois noted that this follows as a particular case of his equation (13) in the “Essay,” which is his version of Isaac Newton's (1643-1727) interpolation formula. Servois gives the details in an extended four-page footnote, which we include as an appendix at the end of our translation of the “Reflections.”

We close this survey of Wronski's work with a brief description of operators that he called grades and gradules. Grades and gradules are analogous to differences and differentials, respectively; however, the increments applied to the variables are exponential instead of additive [Montferrier 1856, pp. 96-103]. Grades and gradules are defined as follows: begin with a function \( y= \varphi(x) \) and suppose that the power of \( x \) receives an increment \( \mu x \). The function \( \varphi(x) \) becomes \( \varphi(x^{1+\mu x}) \). Wronski expressed the resulting change in \( y \) by

\[ y^{1+\mu y} = \varphi\left(x^{1+\mu x}\right). \]

We note the analogy with ordinary differences in the relation \( y+\Delta y = \varphi(x+\Delta x) \). When the increment \( \mu x \) is finite, Wronski called the quantity \( y^{\mu y} \) the grade of \( \varphi(x) \). When the increment \( \mu x \) is infinitely small, then it is called the gradule. Servois pointed out that grades and gradules are an unnecessary complication, because they can be expressed as

\[ \frac{{\Delta ^m \ln \varphi (x + \mu \xi )}}{{\ln \varphi (x)}} \quad \mbox{and} \quad \frac{{d^m \ln \varphi (x)}}{{\ln \varphi (x)}}, \quad \mbox{respectively}. \]