# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation

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Many historians of mathematics consider the nineteenth century to be the Golden Age of mathematics. During this time period many areas of mathematics, such as algebra and geometry, were being placed on rigorous foundations. Another area that experienced fundamental change was analysis. Grabiner [1981] considers Joseph-Louis Lagrange (1736-1813) to be the first mathematician to treat the eighteenth century foundations of calculus as a serious mathematical issue. The publication of his Fonctions analytiques [1797] can be seen as the first attempt to resolve these foundational issues. However, many other mathematicians also contributed to the foundational debates of the eighteenth and nineteenth centuries. One such figure was François-Joseph Servois (1767-1847). Servois was a priest, artillery officer during the French Revolutionary period, professor of mathematics, and supporter of Lagrange’s algebraic formalism. This paper provides an examination and English translation of Servois’ “Essay on a new method of exposition of the principles of differential calculus” [1814a], in which Servois continued the work of Lagrange by attempting to place calculus on a foundation of algebraic analysis without recourse to infinitesimals.

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

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Many historians of mathematics consider the nineteenth century to be the Golden Age of mathematics. During this time period many areas of mathematics, such as algebra and geometry, were being placed on rigorous foundations. Another area that experienced fundamental change was analysis. Grabiner [1981] considers Joseph-Louis Lagrange (1736-1813) to be the first mathematician to treat the eighteenth century foundations of calculus as a serious mathematical issue. The publication of his Fonctions analytiques [1797] can be seen as the first attempt to resolve these foundational issues. However, many other mathematicians also contributed to the foundational debates of the eighteenth and nineteenth centuries. One such figure was François-Joseph Servois (1767-1847). Servois was a priest, artillery officer during the French Revolutionary period, professor of mathematics, and supporter of Lagrange’s algebraic formalism. This paper provides an examination and English translation of Servois’ “Essay on a new method of exposition of the principles of differential calculus” [1814a], in which Servois continued the work of Lagrange by attempting to place calculus on a foundation of algebraic analysis without recourse to infinitesimals.

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

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François-Joseph Servois was born on July 19, 1767, in the village of Mont-de-Laval, located in the Department of Doubs, near the Swiss border. He was ordained a priest at Besançon near the beginning of the French Revolution; however, his religious career was cut short due to the revolution. In 1793, Servois left the priesthood and joined the army to become a Foot Artillery officer. In his leisure time he studied mathematics. Due to poor health Servois requested a non-active military position, one in the field of academia. This was the beginning of his professional career in mathematics, although he would be called back into active duty several times before his retirement. Servois had several research areas, such as mechanics, geometry, and calculus; however, to the extent that his name is known at all, it is for introducing the words “distributive” and “commutative” to mathematics. His final position was as Curator of the Artillery Museum, located in the 7th Arrondissement of Paris. Servois retired in 1827 and lived with his sister in Mont-de-Laval. He died on April 17, 1847. Interested readers can refer to the biography by Petrilli [2010] for further information about Servois’ life and a review of his other mathematical works.

Servois was one of the first mathematicians to consider abstract functional equations such as $D\left[f(x)+g(x)\right] = D\left[f(x)\right] + D\left[g(x)\right]\quad (1) \quad \rm{and}$ $D\left[af(x)\right] = aD\left[f(x)\right]\quad\quad\quad (2).$ Modern readers are familiar with these expressions and recognize transformations with these properties as linear operators. Servois tried to use these expressions to give a satisfactory account of the foundations of calculus. As such, he was one of the pioneers of linear operator theory and indeed of all of “soft analysis,” or the use of algebraic notions and techniques to prove results in real or complex analysis.

Because he was breaking new ground with his research, Servois’ point-of-view was somewhat different from the one that we have inherited. For example, we understand that an operator like $$D$$ is an object of a different kind than the function $$f$$ upon which it operates. Servois made no such distinction. To him, $$D$$, $$f$$ and $$g$$ were all functions, so relation (1) is reminiscent of the distributive law. In Section 3 of his “Essay on a new method of exposition of the principles of differential calculus” [Servois 1814a], he called any function satisfying $\varphi(x+y+\cdots)=\varphi(x)+\varphi(y)+\cdots\quad\quad (3)$ distributive. However, the only example he gave of a distributive function that we would consider truly to be a function is $$\varphi(x) = ax$$, which he called the constant factor $$a$$. All other examples we would consider to be operators and not functions.

Servois was wrapped up in foundational issues for calculus and was one of many mathematicians contributing to the Golden Age of mathematics.

# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - The Golden Age of Analysis

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A key feature to understanding the history of a mathematical field is becoming acquainted with the progression of its development. The beginning of the nineteenth century was considered the Golden Age of mathematics, a time when many fields, such as abstract algebra and geometry, were given a rigorous foundation [Piccolino 1984]. Another area of mathematics that experienced fundamental change was analysis. The turning point for rigor in calculus might have been the publication of Théorie des fonctions analytiques [Lagrange 1797], in which Joseph-Louis Lagrange argued that calculus ought to be placed on a foundation of algebraic analysis. In fact, Lagrange gave this book the following subtitle “The Principles of the differential Calculus, freed from any consideration of the infinitely small or vanishing quantities, of limits or of fluxions, and reduced to the algebraic Analysis of finite quantities.”

Figure 1. Portrait of Joseph-Louis Lagrange (public domain).

During the early nineteenth century, mathematicians began to give a rigorous foundation to calculus; however, rigor does not have a definitive definition. Judith Grabiner described this undertaking as follows:

First, every concept of the subject had to be explicitly defined in terms of concepts whose nature was held to the already known …. Second, theorems had to be proved, with every step in the proof justified by a previously proved theorem, by a definition, or by an explicitly stated axiom. Third, the definitions chosen, and the theorems proved, had to be sufficiently broad to support the entire structure of valid results belonging to the subject [Grabiner 1981, p. 5].

# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Lagrange and Algebraic Analysis

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Figure 2. Title page of Lagrange’s Théorie des fonctions analytiques (public domain).

Mathematicians tried a variety of approaches in giving a rigorous foundation for calculus at the beginning of the nineteenth century. There were three competing notions: differentials, limits, and power series expansions. This last approach was due to Lagrange, who believed he could use algebraic analysis to derive a power series expansion of any function, without recourse to derivatives, limits or differentials. He described his program in the following passage:

In a memoir printed among those of the Academy of Berlin for 1772, I proposed that the theory of the expansion of functions into series contained the true principles of the differential calculus, freed from any consideration of the infinitely small, or of limits, and I proved the theorem of Taylor using this theory, which we may regard as the fundamental principle of this calculus and which had previously never been proven except with the assistance of this same calculus, or by consideration of infinitely small differences [Lagrange 1797, p. 5].

Servois was a disciple of Lagrange and supported his algebraic approach to explaining how calculus works, although he was also sympathetic to the value of limits to calculus. Lagrange’s contention that a function could always be expanded into a Taylor series and that this could be used as the fundamental basis of calculus was Servois’ guiding principle, a doctrine to which he would adhere with an almost religious fervor.

# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Lagrange's [i]fonction derivee[/i]

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What follows is a summary of Lagrange’s method of series expansion, which is contained in [Lagrange 1797, pp. 1-15]. Many of the details can be found in his Théorie des fonctions analytiques [Grabiner 1981, 1990] and [Katz 2009, pp. 633-636].

Lagrange began by taking $$f(x)$$ to be an arbitrary function of $$x$$. Then, if $$h$$ is an indeterminate quantity, he supposed he could form an infinite series in terms of $$h$$, $f(x + h) = f(x) + ph + qh^{2} + rh^3 + \cdots,\quad\quad (4)$ where $$p, q, r, \ldots$$ are new functions of $$x$$, independent of $$h$$, and are derived from the original function $$f(x)$$.

To find the exact terms of the power series, Lagrange wrote series (4) in the following form, $f(x + h) = f(x) + h\left[P(x, h)\right],$ where $$P(x, h)$$ represents the difference quotient, $P(x, h) = \frac{f(x + h) - f(x)}{h}.$ Lagrange argued that it is possible to separate from $$P$$, the part $$p$$, which does not vanish when $$h = 0$$. Therefore, $$p(x) = P(x, 0)$$ and $Q(x, h) = \frac{P(x, h) - p(x)}{h},$ or $$P = p + hQ$$. Thus, $$f(x + h) = f(x) + ph + h^{2}Q$$. Continuing similarly, we can let $$Q = q + hR$$, where $$q(x) = Q(x, 0)$$. Then $$f(x + h) = f(x) + ph + qh^2 + h^3R$$. The continuation of this process yields expansion (4).

The coefficients $$p, q, r, \ldots$$ are derived from $$f(x)$$ and Lagrange called them fonctions dérivées (this is where our modern term “derivative” comes from). Lagrange used the notation $$f^{\prime}(x)$$ for $$p$$ and then investigated the relationship among $$p$$, $$q$$, $$r$$, …. By considering the expansion of $$f(x + h + i)$$ in two different ways, where $$i$$ is another indeterminate increment, he showed that $$p = f^{\prime}(x)$$, $$2q = p^{\prime}$$, $$3r = q^{\prime}$$, …. Expressing all of these derived functions in terms of $$f(x)$$ gives series (4) the familiar form of the Taylor series: $f(x) + f^{\prime}(x)h + \frac{f^{\prime\prime}(x)}{2!}h^2 + \ldots.$

By taking $$h$$ to be sufficiently small, but still finite, Lagrange argued he could control the error in any approximate value of $$f(x + h)$$ based on finitely many terms in the series (4). In particular, he showed that $f(x + h) = f(x) + f^{\prime}(x)h + \frac{f^{\prime\prime}(x)}{2!}h^2 + \ldots + \frac{f^{(n)}(x)}{n!}h^{n} + \frac{f^{(n+1)}(x+i)}{(n+1)!}h^{n+1},$ for some value of $$i$$ satisfying $$0 < i < h$$. The term $\frac{f^{(n+1)}(x+i)}{(n+1)!}h^{n+1}$ is therefore called the Lagrange Remainder Term for the Taylor series.

# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Lagrange's Influence

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Lagrange’s definition of function in his Théorie des fonctions analytiques [Lagrange 1797] was much narrower than the one we use today. Article 1 of the book reads as follows:

We define a function of one or several variables to be any expression of calculation in which these quantities appear in any way whatsoever, combined or not with other quantities that we consider to be given and invariable values, whereas the quantities of the function may receive any possible value.

Thus, Lagrange considered only the sort of well-behaved functions that are given by formulas, the functions that we usually consider in a freshman calculus course. Therefore, it’s not so surprising that he assumed that the process he described for expanding functions into series could always be undertaken or that, like Newton and Taylor, he gave little consideration to the issue of convergence; see [Grabiner 1990, pp. 15-16]. Servois largely followed Lagrange’s lead in assuming that a function was essentially a formula involving algebraic expressions, as well as familiar logarithmic, exponential and trigonometric functions.

Although Lagrange’s goal was to create a foundation for calculus that did not depend on infinitesimals or limits, we can find embedded in his work the use of something like the limit concept. When he made the claim that (h) can be taken as small as one desires, he was essentially letting (h) approach zero in the difference quotient, which at that point is the only missing component of the modern definition of the derivative.

Finally, Lagrange had set the stage for other mathematicians to see the necessity for rigor in calculus. His work would influence many mathematicians of Servois’ generation, and also a later generation of students of the Ecole Polytechnique, including Augustin-Louis Cauchy (1789-1857) [Gillispie 2004].

Servois took up the challenge to expand upon Lagrange’s foundational work and in 1805 he submitted his first version of the “Essay on a new method of exposition of the principles of differential calculus” [Servois 1814a] to the mathematics section of the Institut National des Sciences et des Arts, which had been called the Académie des Sciences in pre-Revolutionary days. He followed this with a second part in 1809. The papers were received with approval in a report issued by Sylvestre Lacroix (1765-1843) and Adrien-Marie Legendre (1752-1833) in 1812. He subsequently revised the “Essay” and published it in the journal Annales de mathématiques pures et appliquées. At 48 pages, the “Essay” filled the entire October 1814 issue of the Annales. Furthermore, it was followed immediately in the November issue of the journal by another of Servois’ articles, “Reflections on the various systems of exposition of the principles of the differential calculus and, in particular, on the doctrine of the infinitely small” [Servois 1814b]. A translation and analysis of this second article is available in [Bradley and Petrilli 2010]. Servois had both articles published together in 1814 as a monograph.

Accompanying the present article is an English translation of Servois’ “Essay.” The remainder of this article consists of a reader’s guide to the “Essay” (pages 7-16), along with some suggestions for how you might use it with students of mathematics and of mathematics history (page 17).

# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - A Reader's Guide to Servois' 'Essay'

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This chapter is designed to be a friendly guide to understanding Servois’ journey through the differential calculus. We propose two different ways to read Servois’ “Essay”:

1. to gain a general understanding of Servois’ notation, algebraic foundations (which are the hallmark of the “Essay”), and the derivation of the rules of calculus, or
2. to examine the mathematical details in depth.

For those interested in a basic understanding of Servois’ “Essay,” we recommend reading Sections 1-12, in which Servois gives his notation and derives the algebraic properties of operators. Then, using this reader’s guide to make the transition, skip ahead to Section 18, where Servois develops the rules of the differential calculus. Readers who want to examine the full mathematical detail of Servois’ derivations will want to read Sections 13-17 and 19 in their entirety. This guide is designed to aid both audiences. Enjoy the journey!

Figure 3. Title page of Servois’ “Essay” (public domain).

Servois begins the “Essay” by giving vital definitions and notation. He denotes a function of $$z$$ by $$fz$$ and it follows that the composition of two functions of $$z$$ would be denoted by $$fgz$$. For simplicity, we use the modern notation (f(z)) and (f(g(z))) for function and function composition, respectively, throughout the reader’s guide. Except for this lack of parentheses, Servois’ basic definitions look familiar to modern readers. For example, he uses (f^2), (f^3), (ldots) for repeated composition of a function (f) and (f^{-1}) for the inverse function.

Following the notation of Arbogast [1800], Servois defines the varied state $E(z)=\varphi(x + \alpha, y + \eta, \ldots),$ and, based on it, $E^{-1}(z) = \varphi(x - \alpha, y - \eta, \ldots) \quad \mbox{and}$  $E^{n}(z) = \varphi(x + n\alpha, y + n\eta, \ldots),$ where $$z = \varphi(x,y, \ldots)$$ is a function of many variables and $$\alpha,\eta, \ldots$$ are “invariable” or constant increments. Using the varied state, Servois defines the difference of a function, $\Delta (z) = E(z) - z = \varphi(x + \alpha, y + \eta, \ldots) - \varphi(x, y, \ldots).$

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

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At this point it seems appropriate to observe that Servois does not make a distinction between “function" and “operator" as we do today. Sometimes he uses the word “function" in the same way that students of calculus do (i.e., to describe a relation between independent and dependent variables), but he also uses it to describe operators, functions of functions such as the varied state, and difference operators. One must keep in mind that the formal definition of the term “function" would not be introduced until 1837 by Dirichlet [Katz 2009]. Once again, for the sake of clarity, we will often distinguish functions from operators in this guide.

Servois defines the varied state and difference for multivariable functions. Generally, Servois calls a multivariable function a “complex.”  However, these should not be confused with complex numbers.  On the other hand, in the one instance where Servois does consider complex quantities later in the “Essay,” he uses the word “imaginary,” which was the standard term at the time.

Early within the “Essay” we witness Servois establishing a sort of group structure for operators, defining the identity operator, $$f^0(z)$$, and the inverse operator. He calls the inverse of his difference operator an integral, although “sum” might be more appropriate, and he uses the symbol $$\sum$$ and not $$\int$$. He observes that it takes an arbitrary additive complement, analogous to the constant of integration. From here Servois establishes some common inverse functions, such as $$\ln z$$ and $$e^z$$. Servois uses $$\mbox{L}$$ to denote the natural logarithm; however, we will use the modern “$$\ln$$” notation in this guide.

Servois uses the term polynomial for any function or operator $$F$$ of the form $F(z)=\mbox{f}(z) + f(z) + \varphi (z) + \ldots,$ where $$\mbox{f}$$, $$f$$, $$\varphi$$, $$\ldots$$ are the composing monomial functions. This literal use of the word “polynomial” is much broader than the modern use. Not only may the constituents be functions other than the familiar monomial functions $$ax^n$$, they may be operators, including the partial varied state, which increments only a single variable in a multivariable function, or the partial operator that multiplies a single variable by a constant. Furthermore, later in the paper, Servois considers “polynomial” functions with an infinite number of constituents.

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

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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.

# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Algebraic Properties

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These sections include the algebraic “closure" properties of distributive and commutative functions or operators. We will follow Servois in calling them all functions, even though in many cases the objects would be called operators today.

In Section 5, Servois proves that distributivity is closed under composition: if two functions $$\varphi$$ and $$\psi$$ are distributive, then the composition of $$\varphi$$ and $$\psi$$ is also distributive. It follows immediately that different orders of distributive functions are also distributive. In Section 6, Servois shows that distributivity is closed under addition.

Next, Servois tackles the properties of commutative operations. In Section 7, he shows that given a collection of $$n > 2$$ functions that are pairwise commutative, all of the $$n!$$ composed functions that can be formed from them are equal.  Servois gives an example of such a situation: if $$\mbox{f}, f,$$ and $$\varphi$$ are three operators, which are commutative between themselves, then we have $\mbox{f}(f(\varphi (z))) = f(\mbox{f}(\varphi (z))) = \mbox{f}(\varphi (f(z))) = \varphi (\mbox{f}(f(z))) = f (\varphi (\mbox{f}(z))) = \varphi (f(\mbox{f}(z))).$ Servois provides a proof by induction for this theorem. In these sections, we sometimes witness Servois laboring over intricate details of somewhat “obvious” theorems, illustrating his devotion to rigor.

In Section 8, Servois proves that if $${\mbox f}$$ and $$f$$ are commutative between themselves, then they are also commutative with their inverses. In Section 9 he deduces from Sections 7-8 that, given a set of commutative functions, the positive integral powers of these functions also commute amongst themselves.

In Sections 10-12, Servois considers collections of distributive functions that commute with each other. He shows that a sum $$F=\sum_i f_i$$ of such functions commutes with all of the constituent functions $$f_i$$, as do the powers $$F^n$$. As a consequence, two such functions $$F=\sum_i f_i$$ and $$G=\sum_i g_i$$ also commute. Servois does not use subscript and summation notation; his proofs might be easier for a modern reader to follow if he did. We also note that Servois seems to be including the case of infinite sums, although he does not say so explicitly. However, he will eventually define the differential operator $${\mbox d}$$ as an infinite sum of distributive operators, so it is clear that he means to include the case of infinite sums.

In Sections 5-12, Servois essentially shows that the class of invertible, distributive, and pairwise commutative functions forms a field with respect to the operations of addition and composition. He does not prove that these operations satisfy the associative laws, but it wasn’t until later in the nineteenth century that mathematicians paid attention to the associative property, although Carl Friedrich Gauss (1777-1855) did prove an associative law in [Gauss 1801, Section 240]. In addition, he assumes that all his functions are invertible, rather than proving the existence of inverses. It is important to bear in mind, however, that Servois wasn’t interested in abstract algebraic structures, although he influenced the British school of symbolical algebra [Allaire and Bradley 2002].

# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Expansion of Functions and Forward Differences

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In Sections 13-15, Servois describes what he calls “the general theory of the expansion of functions into series,” which includes both the Newton forward difference formula and Taylor series. In Section 13, Servois introduces his equation (25), a very general formula for the expansion of a function $$F$$ in terms of a sequence of auxiliary functions $$\varphi, \varphi^{\prime}, \varphi^{\prime\prime}, \ldots$$. In order to establish series (25) “analytically,” Servois uses a generalized notion of the difference quotient, as given in his formula (24). Then in Section 14, he uses similar techniques to derive the Newton forward difference formula, his (36).

In Section 15, Servois makes the transition from forward differences to powers of a variable. He lets $\varphi (x) = x - p, \; \varphi^{\prime} (x) = x - p - \alpha, \; \varphi^{\prime\prime} (x) = x - p - 2\alpha, \; \ldots$ for a fixed number $$p$$ and a fixed increment $$\alpha$$. At the top of his page [108], he observes, but does not clearly explain, that $\Pi_n = \frac{(x - p)(x - p - \alpha) \cdots (x - p - (n - 1)\alpha)}{\alpha^n}$ can be written in the form $\Sigma_n = a_1 \left(\frac{x - p}{\alpha}\right) + a_2 \left(\frac{x - p}{\alpha}\right)^2 + \ldots + a_n \left(\frac{x - p}{\alpha}\right)^n.$ Here, we have used subscript notation where Servois used $$A$$, $$B$$, $$C$$, $$\ldots$$. We have also introduced the notation $$\Pi_n$$ and $$\Sigma_n$$. It is easy to prove Servois’ claim by induction, because $\Pi_{n+1} = \Pi_{n} \left(\frac{x-p - n\alpha}{\alpha}\right)$ $\phantom{xxxxx}= \Sigma_n \left[\left(\frac{x-p}{\alpha}\right) - n \right],$ which has the form $$\Sigma_{n+1}$$.

Servois uses this observation to rewrite his series (33), which is a form of the Newton forward difference formula, in terms of the powers of $\frac{x - p}{\alpha}.$

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

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In Section 15, when Servois expands $$F(x)$$ in terms of $\frac{x - p}{\alpha} ,$ he finds the first order coefficient is $\Delta F (p) - \frac{1}{2}\Delta^{2} F (p) + \frac{1}{3}\Delta^{3} F (p) - \cdots.$ Lagrange’s work [1797] leads us to expect that the first order coefficient in a power series expansion should tell us about the derivative of the function. Indeed, combining this formula and his results of Section 14, Servois is led to define the differential $$\mbox d$$ of an arbitrary function $$z$$ as follows: ${\mbox d}z = \Delta z - \frac{1}{2}\Delta^{2}z + \frac{1}{3}\Delta^{3}z - \cdots.$ Then, he investigates the higher order powers of the differential and derives several forms of the Taylor series in his equations (45)-(48).

Of these, equation (48) is the easiest to recognize as a Taylor series: $F(x) = F(x_0) + \frac{x}{\alpha} {\mbox d} F (x_0) + \frac{x^2}{1 \cdot 2 \cdot \alpha^2}{\mbox d}^2 F(x_0) + \frac{x^3}{1 \cdot 2 \cdot 3 \cdot \alpha^3}{\mbox d}^3 F(x_0) + \cdots.$ If we let $$z$$ be the identity function $$f(x) = x$$ and $$\alpha$$ be the increment in $$x$$, then $$\Delta x = \alpha$$ and $$\Delta^n x = 0$$ for $$n > 1$$. Using Servois’ definition of the differential, we have $${\mbox d} x = \alpha$$. If we therefore replace $$\alpha$$ in (48) with $${\mbox d} x$$ and formally rearrange each term, we see that the coefficient of $$x^n$$ is $\frac{1}{n!} \frac{{\mbox d}^n F(x_0)}{{\mbox d} x^n} = \frac{F^{(n)}(x_0)}{n!},$ which is the familiar coefficient.

In eighteenth century infinitesimal calculus, differentials like $${\mbox d} x$$ and $${\mbox d} y$$ were thought of as infinitely small quantities. In definition (5), above, Servois has succeeded in defining the differential without recourse to the infinitely small, but in terms of an infinite series of finite differences. This provided him with a more satisfactory foundation for calculus; however, it made the derivation of the rules of calculus more difficult.

Definition (5) is easy to apply to polynomials. Consider the simple case of $$z = x^2$$. Then, $\Delta z = (x + \alpha)^2 - x^2 = 2x \alpha + \alpha^2 \quad \rm{and}$ $\Delta^2 z = [2(x + \alpha) \alpha + \alpha^2] - [2x \alpha + \alpha^2] = 2\alpha^2.$ All higher orders of the difference are zero. (It is actually an easy induction to show that if $$z$$ is a polynomial of degree $$n$$, then $$\Delta^{n+1} z$$, and higher differences, are 0.) Substituting the above differences into definition (5), we obtain $${\mbox d} z = 2x \alpha.$$ Understanding $$\alpha$$ to be $$dx$$, we have that $${\mbox d} z = 2x dx.$$ Definition (5) is not so easy to apply when $$z$$ is a transcendental function. If you attempt to apply the same procedure to $$\sin x$$, then you will notice that no amount of manipulation will simplify the differences. Therefore, Servois had to invent a new function to aid in evaluating trigonometric differentials (see his Section 18).

# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - The Natural Logarithm Series

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In Section 16, Servois applies his expansion formulas to $$F(x) = \varphi ^x(z)$$, where $$\varphi$$ is a distributive function that is commutative with constant factors. That is $$\varphi$$ and $$z$$ are given and what varies in $$F(x)$$ is the order of the function $$\varphi$$. It is clear that this means when $$x$$ is an integer, but Servois doesn’t tell us how to interpret $$F(x)$$ for other values of $$x$$. Letting $$\alpha$$ be the constant increment in $$x$$, Servois finds an expression for $$\Delta^n F(x)$$. By his definition of the differential in (39), it follows that ${\mbox d} F(x) = \Delta F(x) - \frac{1}{2}\Delta^{2}F(x) + \cdots\phantom{xxxxxxxxxxxxxx}$ $\phantom{xxxxxxxxxx}=\varphi^{x}\left[\left(\varphi^{\alpha} - 1\right)z - \frac{1}{2}\left(\varphi^{\alpha} - 1\right)^{2}z + \frac{1}{3}\left(\varphi^{\alpha} - 1\right)^{3}z - \cdots\right].$

Servois calls the expression in the brackets the logarithm of $$\varphi^{\alpha}$$ of $$z$$. The analogy between this formula and the usual power series for $$\ln (x)$$ (as opposed to $$\ln (1 + x)$$) is clear. However, Servois observes that the operator $$\ln \varphi^{\alpha}$$ is distributive and commutes with the function $$\varphi$$ and the constant factor, which is certainly not the case for the ordinary natural logarithm.

Servois then derives analogs of the familiar properties of the ordinary logarithm for this operator. Then, by looking at the inverse of this logarithm, Servois is able to derive a power series in his (62) that has the form for the usual power series for $$F(x) = e^x$$. Although no where in these formula arguments does Servois address the issue of non-integer values of $$x$$ and $$\alpha$$, his most important application in Section 18 will be when the function $$\varphi$$ is the constant factor $$a$$, which is a distributive function that certainly commutes with constant factors. In this case, as Servois observes at the end of Section 17, $$\ln \varphi^{\alpha} z$$ is the natural logarithm of $$a^{\alpha} z$$ and the inverse $$\ln ^{-1} \psi z$$ is nothing more than $$e^{\psi} z$$. In this case, Servois observes that the ordinary properties of logarithms follow from his formulas in Section 16.

# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Properties of Differentiable Functions

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Within Section 17, Servois explores the properties of what he calls differential functions, but what we would call differential operators. He begins with an arbitrary function $$z$$ having only $$x$$ and $$y$$ as variables and then explores the varied states, differences, and differentials of $$z$$, both total and partial, $Ez, \; \frac{E}{x}z, \; \frac{E}{y}z; \quad \Delta z, \; \frac{\Delta}{x}z, \; \frac{\Delta}{y}z; \quad {\mbox d} z, \; \frac{{\mbox d}}{x}z, \; \frac{{\mbox d}}{y}z.$ Servois had defined the partial varied state and partial difference in his equation (5) of Section 1. In this case, $$z = \varphi (x, y)$$ and we have $\frac{E}{x}z = \varphi (x + \alpha, y) \quad \mbox{and} \quad \frac{E}{y}z = \varphi (x, y + \beta).$ The partial differences and differentials are defined from the partial varied states analogously with the total difference and total differential. Thus, all the results in this section are derived from the properties of the varied states. The varied states are distributive and all powers of the varied states are commutative with constant factors. Since differences are derived from varied states and differentials are found using differences, Servois concludes that these operators inherit the distributive and commutative properties of the varied states. Following this investigation, Servois makes the following observations about his class of differential functions:

1. All differential functions are distributive.
2. All differential functions, and their various orders, are commutative, both among themselves and with constant factors.

It is clear that integrals share these properties, since they represent the inverses of the differential operators. Servois had already defined the inverse $$\Sigma$$ of the difference $$\Delta$$ in Section 1, and he implicitly defines the integral $$\int$$ here as the inverse of the differential operator $${\mbox d}$$. Thus, $\Sigma, \; \frac{\Sigma}{x}, \; \frac{\Sigma}{y}, \; \int, \; \frac{\int}{x}, \;{\rm and} \; \frac{\int}{y}$ can be added to the list of operators that are distributive and commutative with constant factors, as can their various orders. Servois observes that the results of this section are easily generalized for functions of more than two variables.

Servois concludes this section with the following algebraic theorem for differential functions: linear combinations of differential functions give rise to an infinite number of new differential functions, all of which are distributive and commutative among themselves and with constant factors. Therefore, Servois has shown that this large class of operators is linear, in the modern sense of items 1 and 2.

# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - The Rules for Differential Calculus

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In Section 18, Servois develops the rules of the differential calculus. He begins by listing the elementary simple functions of a single variable $$x$$: $x^m, \quad a^x, \quad \ln x, \quad \sin x, \quad \cos x,$ and the elementary composed functions: $\varphi (x) \cdot \psi (x), \quad (\varphi (x))^m, \quad a^{\varphi (x)}, \quad \ln (\varphi (x)), \quad \sin (\varphi (x)), \quad \cos (\varphi (x)).$ Before Servois can start developing the rules of the differential calculus he must first establish a well-known theorem, the chain rule; see Servois’ equation (81). Using the chain rule he finds the differentials of exponential functions and then derives the power rule and product rule. From a modern point-of-view, we would say that Servois used logarithmic differentiation in the derivation of the power and product rules. This is a natural progression for Servois, because he has observed that his differential and the natural logarithm behave similarly.

Servois then turns to differentiating the trigonometric functions $$\sin x$$ and $$\cos x$$. Unfortunately, the series for the differential is not easy to apply to trigonometric functions. Therefore, Servois introduced a new function, which he used to evaluate trigonometric differentials, $F(x) = \frac{\cos (\alpha x) + \sqrt{-1}\sin (\alpha x)}{\cos^{x} (\alpha)}, \quad\quad(6)$ where $$\alpha$$ is a constant and the increment in the variable $$x$$ is 1. This is the only place in the “Essay" where Servois uses complex numbers. He assumes the reader is familiar with their properties. He doesn’t give any motivation for definition (6), nor does he make any mention of De Moivre’s formula in the numerator. Throughout his papers Servois was always careful to provide credit to discoveries made by other mathematicians. Because Servois did not attribute (6) to anyone else, we suspect that it is original to him. Therefore, Petrilli [2009, p. 91] called this “Servois’ Function.”

Using the angle sum formula for sine and cosine, Servois shows that $\Delta^{m} F(x) = F(x) \cdot \left(\sqrt{-1}\tan \alpha\right)^{m},$ which gives him an expression for $${\mbox d} F(x)$$ in equation (90). Comparing this to the expression $${\mbox d} F(x)$$ given by the product rule and differentiating $$\cos^2 (\alpha x) + \sin^2 (\alpha x) = 1$$ implicitly to write $${\mbox d} \cos (\alpha x)$$ in terms of $${\mbox d} \sin (\alpha x)$$ allows him to give a simple expression for $${\mbox d} \sin (\alpha x)$$ in terms of a constant $$A$$, which depends only on $$\alpha$$. Although $$A$$ would appear to have an imaginary component, Servois shows that it is in fact real and proves that ${\mbox d} \sin x = \cos x \, {\mbox d} x \quad \mbox{and} \quad {\mbox d} \cos x = -\sin x \, {\mbox d} x.$

# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Influence of Servois' Work / Translation

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Influence of Servois’ Work

By the end of Section 18, Servois had finished his exposition of the rules of the single variable calculus. He included one further section, in which he considered some results from multivariable calculus. His equation (119), for example, is the two-variable version of Taylor’s theorem. The paper ends with some brief closing comments, which he concludes by setting the stage for his “Reflections” paper [Servois 1814b], which followed in the next issue of Annales de mathématiques pures et appliquées. In his “Reflections,” Servois examines the competing foundational notions for the calculus. Of course, he argues for the superiority of Lagrange’s formal series approach. He also argues forcefully against the use of the infinitely small in mathematics, but expresses some sympathy for the notion of limit.

It’s quite reasonable to imagine that Servois felt he had made a persuasive case for a Lagrangian foundation for calculus with the combined weight of the “Essay” and the “Reflections,” which he published together as a single small volume. However, the winds of change were probably already blowing in a direction that was favorable to the idea of limits. Jean Le Rond d’Alembert (1717-1783) had already championed the limit as the “true metaphysics of the differential calculus” in Diderot’s Encyclopédie [Diderot 1751]. Lacroix published a popular elementary calculus text that used the informal notion of limit as the basis for the calculus [Lacroix 1802]. Perhaps it’s telling that Legendre and Lacroix, the referees who evaluated the first version of the “Essay,” said that by “recalling to the differential calculus several methods, some of which don’t seem very appropriate to the current state of analysis, [the author] has done something that is very useful for the science.” They praised Servois’ work, but seemed to consider it to be already old-fashioned.

A decisive step in the development of the modern rigorous approach to calculus came with the publication of Cauchy’s Cours d’analyse [1821] just a few years after Servois’ “Essay.” Grabiner states: “After Cauchy, foundations had become an essential part of analysis, and Cauchy’s books and teaching were largely responsible" [1981, p. 15]. Cauchy led the way for later analysts, such as Weierstrass, Riemann, and Lebesgue, to complete a foundation for calculus based on limits.

Even though Servois was not successful in creating a foundation for calculus, his work was important to other mathematicians. For instance, his work spread to England and significantly influenced mathematicians such as Duncan Farquharson Gregory (1813-1844) and Robert Murphy (1806-1843) in their own efforts in the foundations of analysis. However, the real lasting influence of his work in England was in the development of abstract algebra and linear operator theory [Allaire and Bradley 2002].

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# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Recommendations for the Classroom

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The material discussed in this paper can aid teachers of both calculus and the history of mathematics. The history of mathematics provides the opportunity to illustrate how mathematics is a constantly evolving field and how the oversights of past mathematicians should not be viewed as carelessness or error, but as innovations appropriate to the context of their times [Petrilli 2009]. Servois’ “Essay” is an original source that provides insight into the competing notions of calculus in the early nineteenth century and can be used to illustrate the aforementioned statement.

This translation provides many opportunities for student research projects in the development of the rigorous calculus. The reader's guide in this paper provides only glimpses into the details of Servois’ calculus. Thus, there are opportunities for exploring much of Servois’ mathematics and its historical influences in depth. For instance, the following questions could be considered: After examining Sections 1-12, how close was Servois to discovering the ring structure? Who else was influenced by the work of Servois, besides Gregory and Murphy? To what extent was Servois influenced by the work of Louis François Antoine Arbogast (1759-1803) and Jacques Français (1775-1833)?

Teachers of the history of mathematics can incorporate some concrete problems and proofs into their course using Servois’ “Essay.” For example, students can use Servois’ definition (5) to find the differentials of simple polynomial functions, such as $${\mbox d} x$$, $${\mbox d} x^2$$, $${\mbox d} x^3$$, $$\ldots$$, $${\mbox d} x^n$$, as well as linear combinations of these functions. Then, students can compare Servois’ method of finding differentials to Newtonian fluxions and the modern limit-based method. Additionally, students can use their knowledge of mathematical induction to prove several statements that Servois did not formally prove. Also, it is possible to use some of Servois’ definitions to construct additional proofs by induction, such as: If $$z = F(x)$$ is a polynomial of degree $$n$$, then $$\Delta^k z=0$$ for $$k \ge n$$.

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

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[Allaire and Bradley 2002] Allaire, P. and R. Bradley (2002). “Symbolic Algebra as a Foundation for Calculus: D.F. Gregory’s Contribution,” Historia Mathematica 29, 395-426.

[Arbogast 1800] Arbogast, L. F. A. (1800). Du calcul des dérivations. Strasbourg: LeVrault Frères.

[Bradley and Petrilli 2010] Bradley, R. E. and S. J. Petrilli (2010). “Servois’ 1814 Essay on the Principles of the Differential Calculus, with an English Translation,” Loci: Convergence 7 (April 2010), DOI: 10.4169/loci003487.

[Cauchy 1821] Cauchy, A.-L. (1821). Cours d’analyse. Paris: de Bure. English translation by R. E. Bradley and C. E. Sandifer (2009). Cauchy’s Cours d’analyse: An Annotated Translation. New York: Springer.

[Diderot 1751] Diderot, D. (1751-1780). Encyclopédie (J. d’Alembert, co-editor). Paris: Royal Society.

[Gauss 1801] Gauss, C. F. (1801). Disquisitiones Arithmeticae. English Translation by Clarke et al, New York: Springer, 1986.

[Gillispie 2004] Gillispie, C. C. (2004). Science and Polity in France: The Revolutionary and Napoleonic Years. Princeton, NJ: Princeton University Press.

[Grabiner 1981] Grabiner, J. V. (1981). The Origins of Cauchy’s Rigorous Calculus. Cambridge: MIT Press. Reprinted New York: Dover Publications, 2005.

[Grabiner 1990] Grabiner, J. V. (1990). The Calculus as Algebra: J.-J. Lagrange, 1736-1813. New York: Garland Publishing.

[Grattan-Guinness 1990] Grattan-Guinness, Ivor (1990). Convolutions in French Mathematics, 1800-1840, 3 vols., Basel: Birkhäuser.

[Katz 2009] Katz, V. (2009). A History of Mathematics: An Introduction, 3rd ed. Boston, MA: Addison-Wesley.

[Lacroix 1802] Lacroix, S. F., 1802. Traité élémentaire de calcul différentiel et de calcul intégral. Paris: Duprat.

[Lagrange 1788] Lagrange, J. (1788). Méchanique analitique. Paris: Vve. Desaint. Second edition: Méchanique analytique, 2 vols., Paris: Vve. Courcier, 1811, 1815.

[Lagrange 1797] Lagrange, J. (1797). Théorie des fonctions analytiques. Paris: L’Imprimerie de la République. Second edition, Paris: Vve. Courciet, 1813.

[Petrilli 2009] Petrilli, S. J. (2009). A Survey of the Contributions of François-Joseph Servois to the Development of the Rigorous Calculus. Doctoral thesis: Columbia University Teachers College.

[Petrilli 2010] Petrilli S. J. (2010). “François-Joseph Servois: Priest, Artillery Officer, and Professor of Mathematics,” Loci: Convergence 7 (June 2010), DOI: 10.4169/loci003498.

[Piccolino 1984] Piccolino, A. V. (1984). A Study of the Contributions of Early Nineteenth Century British Mathematicians to the Development of Abstract Algebra and Their Influence on Later Algebraists and Modern Secondary Curricula. Doctoral thesis: Columbia University Teachers College.

[Servois 1814a] Servois, F. J. (1814). “Essai sur un nouveau mode d’exposition des principes du calcul différentiel," Annales de mathématiques pures et appliquées 5 (1814-1815), 93-140.

[Servois 1814b] Servois, F. J. (1814). “Réflexions sur les divers systèmes d’exposition des principes du calcul différentiel, et, en particulier, sur la doctrine des infiniment petits," Annales de mathématiques pures et appliquées 5 (1814-1815), 141-170.

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