Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Distributive and Commutative

In Section 3, Servois calls a function or operator \(\varphi\) distributive if it satisfies \[\varphi(x + y + z + \ldots) = \varphi(x) + \varphi(y) + \varphi(z) + \cdots.\] He gives the varied state \(E\) as an example of a distributive operator and both the sine and the natural logarithm as examples of non-distributive functions. In addition, he observes that \[a(x + y + \ldots) = ax + ay + \cdots.\] Modern readers can interpret this in one of two ways. On the one hand, this can be interpreted as saying that the function \(f(x)=ax\) is distributive in Servois’ sense. In fact, it is the only continuous single-variable function with this property, as was undoubtedly well-known to Servois and was published seven years later by Cauchy [1821]. On the other hand, the operation of multiplying may be thought of as an operator that maps \(z\) to \(az\).

In Section 4, Servois says that two functions \(f\) and \(\varphi\) are commutative between themselves if, \[f(\varphi(z)) = \varphi(f(z)).\] In modern usage, we speak of a collection (group, ring, field, etc.) being commutative when \(a \cdot b= b\cdot a\) for any two elements in the collection. Servois wants to consider the entire class of functions and operators that are of use in calculus and so he needs to distinguish pairs of elements within this collection that have the property. So for example, because \(aEz = Eaz\) for any number \(a\), the varied state and the multiplication operator are commutative between themselves. The same is true for any pair of multiplication operators. Servois points out other pairs that are not commutative, such as \[\sin az \ne a \sin z \quad \mbox{and} \quad Exz \ne xEz.\]

Servois’ “Essay” is the first place where the words “distributive” and “commutative” were used in their modern mathematical sense. The words were medieval legal terms that can be traced back to Aristotle.