Servois' 1814 Essay on the Principles of the Differential Calculus, with an English Translation - Analogies and Algebraic Properties

Servois then pointed to yet another disadvantage of the method of limits and differentials: that of leaving “veiled in mystery” the analogies between differential expressions and powers [Servois 1814b, p. 151]. For example, the general form of the product rule

\[ d^n(uv) = \sum^n_{k=0} {{n}\choose{k}} d^k u \, d^{n-k} v \]

is entirely analogous to the binomial expansion

\[ (a+b)^n = \sum^n_{k=0} {{n}\choose{k}} a^k \,b^{n-k}. \]

He noted that in refereeing his “Essay,” the Commissioners of the Institut de France had said that by “showing that it is to their nature of being distributive, in general, and commutative amongst themselves and with the constant factor, that varied states, differences and differentials owe their properties, and the analogies with powers, [Servois] gives their true origin” [Servois 1814b, p. 151]. These two algebraic properties are at the heart of Servois' work on the differential calculus. Servois further argued that these two properties imply “that the Leibnizian notation for the differential calculus ought to be preserved” [Servois 1814b, p. 154].

From this point onwards, the “Reflections” consists largely of a detailed and spirited criticism of the mathematics and philosophy of Wronski.