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In this section, we make direct references to Servois' “Essai" [1814a]. Readers can find a translated version of the “Essai” in Bradley and Petrilli's paper [2010].

In his “Essai,” Servois attempted to provide a rigorous foundation for the calculus through algebra. In light of what we know today, Servois did not fully succeed in putting calculus on a mathematically correct foundation. It was Cauchy [1821] who, through his approach to calculus by means of limits and inequalities, moved the subject into the modern age. Although Servois' efforts were ultimately unsuccessful, we find several ideas associated with abstract algebra in his “Essai.”

**Figure 6.** Title page of Servois' “Essai” (public domain).

In Sections 1-4 of the “Essai,” Servois presented several definitions that would be crucial to his work. He began by introducing his notation for a function as \(f\,z\), where a modern reader would understand this as \(f(z)\). The formal definition of a function that we use today would not be introduced until 1837 by Lejeune Dirichlet (1805-1859). A reader will notice that Servois used the term function not only to describe ordinary functions of an independent variable, but also to describe operators, such as the difference and differential operators. After he presented his preliminary definitions, Servois introduced two functions or operators with special properties, namely the identity \(f^0\) and inverse \(f^{-1}\). Undergraduate students will notice that these operators and properties are analogous to the modern day notions of an identity element and inverse element. For instance, Servois explained that when the identity function or operator is applied to \(z\), “\(z\) does not undergo any modification” [Servois 1814a, p. 96]. Additionally, Servois provided many examples of inverse functions. For instance, he considered the inverse of the sine function, noting that \[z = \mbox{sin}(\mbox{sin}^{-1} (z)) = \mbox{sin}(\mbox{arcsin}(z)).\] For the difference operator \(\Delta\), he noted that \[\Delta^n (\Delta^{-n} (z)) = \Delta^{-n} (\Delta^n (z)) = z.\]

In Section 3, Servois defined a function or operator \(\varphi\) to be *distributive* if it satisfied \[\varphi (x + y + ...) = \varphi(x) + \varphi(y) + \cdots.\] He presented several examples of functions and operators that are distributive and others that are not. One such distributive function was \(f(x) = ax\), because \[a(x + y + \cdots) = ax + ay + \cdots.\] Later, Cauchy [1821] showed that this is the only continuous function that satisfies this property. Conversely, Servois provided as a function that does not satisfy the distributive property \(f(x) = \mbox{ln}\,x\). He also demonstrated that the differential and integral operators are distributive.

Actually, undergraduates are exposed to specific examples of Servois' distributive property in a first-year calculus course, namely differentiation and integration under the operation of addition. Additionally, students see a generalized version of Servois' distributive property in a beginning linear algebra course. One of the properties of a linear transformation is that it preserves the addition operation.

Finally, in Section 4, Servois stated that two functions or operators \(f\) and \(\varphi\) are *commutative between themselves* if \[f(\varphi (z)) = \varphi (f (z)).\] For example, he stated that \(z\) commutes with any constants \(a\) and \(b\) because \[abz = baz;\] however, the sine function is not commutative with any constant \(a, a\not= -1,0,1\), because \[\mbox{sin}(az) \neq a \mbox{sin}(z).\]

If we consider the commutative operators \(f(z)\) and \(\varphi (z) = kz\), where \(k\) is any scalar, \[f(kz) = kf(z),\] then we have the familiar scalar multiplication property that undergraduates would see in a first-year calculus course for vectors or beginning linear algebra course for linear transformations.

In modern day mathematics, we speak of an algebraic structure (ring, field, etc.) as being commutative when \(a \cdot b = b \cdot a\) and distributive when \(a \cdot (b + c) = a \cdot b + a \cdot c\), for all elements \(a\), \(b\), and \(c\) in the structure. The hallmark of Servois' calculus was his examination of the set of all functions and operators that satisfy these properties. Interestingly, the words “commutative” and “distributive” were medieval legal terms [Bradley 2002] and Servois was the first to use them in a modern mathematical sense.

In Sections 5-9 of his “Essai,” Servois examined the “closure” properties of distributive and commutative functions or operators. Servois demonstrated that distributivity is closed under composition and addition, and if \(f\) and \({\rm f}\) are commutative functions or operators, then each commutes with the inverse of the other. An examination of Servois' proof of the latter theorem reveals a law from abstract algebra. By the definition of the inverse function, we have \[f( \mbox{f(f}^{-1}(z))) = \mbox{f(f}^{-1}(f(z))),\] and by virtue of the commutativity of \(f\) and \({\rm f}\), we get \[f( \mbox{f(f}^{-1}(z))) = \mbox{f}(f(\mbox{f}^{-1}(z))).\] Now, substitute \(\mbox{f}(f(\mbox{f}^{-1}(z)))\) for \(f( \mbox{f(f}^{-1}(z)))\) in equation (1), and we get \[\mbox{f}(f(\mbox{f}^{-1}(z))) = \mbox{f(f}^{-1}(f(z))).\] Finally, apply \(\mbox{f}^{-1}\) to both sides of equation (2) and we arrive at the desired result that: \[f(\mbox{f}^{-1}(z)) = \mbox{f}^{-1}(f(z)).\] Essentially, when Servois applied \(\mbox{f}^{-1}\) to both sides of equation (2), he invoked a familiar theorem from group theory, the left-hand cancellation law.

After he considered the properties of commutativity and distributivity separately, Servois examined the collection of functions or operators that satisfy both of these properties. He used this as a launching point to introduce his theory for the differential calculus.

From a modern standpoint, the first twelve sections of Servois' “Essai” constitute the creation of an algebraic structure. In them, he showed that the set of invertible, distributive, and pairwise commutative functions or operators forms a field with respect to the operations of addition and composition. Servois never discussed the associative property with respect to these two operations. However, the importance of associativity was being uncovered during the nineteenth century. For instance, Carl Friedrich Gauss (1777-1855) did prove an associative law in 1801 [Gauss 1801, Section 240] and Hamilton stated the importance of associativity in 1843 after his discovery of the quaternions [Crilly 2006, p. 102]. Hamilton’s statement was actually the first appearance of the term [Miller 2010]. Additionally, Servois assumed the existence of inverses for all of his functions. Again, a reader must keep in mind that Servois worked only with functions that were well-behaved and he did not examine the issue of domain.

Anthony J. Del Latto (Adelphi University) and Salvatore J. Petrilli, Jr. (Adelphi University), "Algebraic Formalism within the Works of Servois and Its Influence on the Development of Linear Operator Theory - Algebraic Structure within Servois' 'Essai' ," *Convergence* (January 2012), DOI:10.4169/loci003802