What were the philosophical foundations of the differential calculus at the turn of the nineteenth century? There were three competing notions: differentials, limits, and power series expansions. François-Joseph Servois (1768-1847), a disciple of Lagrange, supported the power series formalism and was sympathetic to a foundation based on limits. On the other hand, he claimed that the use of infinitesimals in mathematics would "one day be accused of having slowed the progress of the mathematical sciences, and with good reason." In this paper, we provide an analysis and an English translation of Servois' philosophical paper "Reflections on the various systems of exposition of the principles of the differential calculus."

Download the authors' English translation of Servois' “Reflections.”

What were the philosophical foundations of the differential calculus at the turn of the nineteenth century? There were three competing notions: differentials, limits, and power series expansions. François-Joseph Servois (1768-1847), a disciple of Lagrange, supported the power series formalism and was sympathetic to a foundation based on limits. On the other hand, he claimed that the use of infinitesimals in mathematics would "one day be accused of having slowed the progress of the mathematical sciences, and with good reason." In this paper, we provide an analysis and an English translation of Servois' philosophical paper "Reflections on the various systems of exposition of the principles of the differential calculus."

Download the authors' English translation of Servois' “Reflections.”

François-Joseph Servois (1768-1847) lived during times of great upheaval. (See [Petrilli 2010] for more biographical information on Servois.) Politically, the French Revolution broke out when he was a young priest of only twenty-one years. As a result of the new social order and the many upheavals and wars that followed the revolution, he chose to become a soldier. Later in his military career, he became a professor of mathematics at various military academies.

**Figure 1.** The Storming of Tuileries Palace in Paris, 1792. Painting by Jean Duplessis-Bertaux (public domain).

Mathematically, these were also revolutionary times. By the early 19th century, the differential and integral calculus was more than a century old, but mathematicians knew that there were still significant problems with the foundations of the subject. Augustin Louis Cauchy (1789-1857) devised a rigorous, limit-based analysis in the 1820s, which eventually settled the foundational question of the calculus. However, during Servois' period as an active researcher, there were at least three competing foundational notions for the calculus and no clear indication as to which of them would eventually give rise to a satisfactory account.

Calculus on the European continent began with Gottfried Wilhelm von Leibniz (1646-1716), who conceived of it as the manipulation of infinitely small increments, called differentials, of the variables in an equation. An infinitely small increment in a variable \( x \) is denoted \( dx \) and the rules of Leibniz' calculus allowed him to conclude, for example, that if \( y=x^2 \), then \( dy = 2x dx \). Modern day readers will notice that they can extract the derivative of \( y \) from this equation by formally dividing both sides by \( dx \) in order to get \( \frac{dy}{dx} = 2x \), but mathematicians did not speak of derivatives during the decades following the birth of the calculus. To them, the differentials \( dx \) and \( dy \) were both objects in their own right, variable quantities of infinitely small size, related to one another so that at any point \( (x,y) \) on the parabola, an infinitely small increment \( dx \) in \( x \) results in a corresponding increment of \( 2xdx \) in the variable \( y \).

**Figure 2. **Jean le Rond d'Alembert. Pastel drawing by de la Tour (public domain).

In the 1750s, the French mathematician Jean le Rond d'Alembert (1717-1783), wrote that the limit concept was the "true metaphysics of the differential calculus" [Calinger 1995, pp. 482-485] in Denis Diderot's (1713-1784) influential *Encyclopédie* [Diderot 1751]. D'Alembert was not the first person to use limits; for example, Colin Maclaurin (1698-1746) discussed them at some length in his 1742 book *A Treatise of Fluxions* [Maclaurin 1742]. However, d'Alembert gave limits a high profile by championing them in the widely-read French encyclopedia. Later in the 18th century, the Swiss mathematician Simon Antoine Jean Lhuilier (1750-1840) won the 1786 Berlin Academy prize for an essay [Lhuilier 1785] in which he gave a systematic account of the calculus using derivatives and limits. By this point, about a century after Leibniz' invention of the differential calculus, the subject had assumed a form that would be more or less recognizable to modern readers. However, the late 18th century limit concept was an informal one, relying on the sort of intuitive arguments still made in most freshman calculus courses. The \( \varepsilon\)-\( \delta \) formulation was still to come, so at this time, differentials and limits were simply competing informal notions.

It was Joseph-Louis Lagrange (1736-1813) who defined the *derived function* or derivative \( f'(x) \) of a function \( f(x) \). Derivatives can be defined in terms of limits, as we do today. Furthermore, derivatives and differentials are to a great extent interchangeable when talking about functions \( y=f(x) \). On one hand, if you know \( dy \) in terms of \( dx \), then you can solve for \( f'(x) \) by dividing the expression for \( dy \) by \( dx \). On the other hand, if you know how to find \( f'(x) \), then \( dy=f'(x)dx \), a familiar relation from integration by substitution.

**Figure 3.** Joseph-Louis Lagrange (print in public domain).

However, Lagrange promoted a third path towards the foundations of calculus involving power series expansions. If \( f(x) \) has a Taylor series expansion at \( x=x_0\), then the coefficient of the increment \( x - x_0 \) in the series expansion is \( f'(x_0)\). Lagrange used this property to *define* the derivative. This approach seems hopelessly circular to modern readers, because we think of power series expansions as given by Taylor's theorem, and therefore presupposing not only the first derivative \( f'(x) \) but also all the higher derivatives. However, through the course of the 17th and 18th centuries, mathematicians were able to give power series expressions for all of the familiar functions of calculus without the explicit use of derivatives. As an example, the geometric series

\[ \frac{1}{1-x} = 1 + x + x^2 + x^3 + \ldots \]

can be derived by long division, or by considering the telescoping expansion of \( (1-x)(1 + x + x^2 + \ldots + x^{n-1}) \).

For more examples, see [Euler 1748], where Leonhard Euler (1707-1783) demonstrates various tricks for deriving the power series for trigonometric, exponential, and logarithmic functions. If an arbitrary function \( f(x) \) has a power series expansion at every point \( x_0 \) in its domain, Lagrange could define \( f'(x) \) without recourse to either infinitely small increments or limits. In his *Theory of Analytic Functions* [Lagrange 1797], he gave a general method for deriving the power series of a large class of analytic functions. For more on Lagrange's foundational program, see Victor Katz's *History of Mathematics* [2009, pp. 633-636].

With hindsight, Lagrange's scheme is the weakest of the three competing notions. We know that limits eventually did provide a solid foundation for the calculus. Furthermore, when nonstandard analysis was invented in the 1960s, mathematicians were able to give a satisfactory account of Leibniz's differentials. Lagrange's program has some obvious weaknesses. In particular, we do not want the notion of function to be limited to only those functions that have power series expansions. Furthermore, there are tricky questions of convergence that were glossed over by Lagrange. Nevertheless, Lagrange was very influential, especially in the French mathematical community, and his program had many adherents. One notable example is Silvestre François Lacroix's (1765-1843) *Traité du calcul différentiel et du calcul intégral* [Lacroix 1797]. In this monumental three-volume reference work describing the state of the differential and integral calculus at the dawn of the 19th century, Lacroix follows Lagrange in basing his calculus on formal power series, although he also describes the history and use of both differentials and limits.

Servois was in many ways a disciple of Lagrange. He certainly believed that the Lagrangian scheme was the correct approach to the principles of the calculus. In 1814, he published two important papers in the *Annales de mathématiques pures et appliquées,* the first academic journal devoted entirely to mathematics. The first of these articles, “Essay on a new method of exposition of the principles of the differential calculus” [Servois 1814a], concerned general techniques for determining power series expansions of functions. He also derived properties of the derivative based on the algebraic properties of the difference operator and the difference quotient. He coined the terms* distributive* and *commutative* in this paper to describe those algebraic properties. Servois defined a function ( f ) to be *distributive* if ( f(x + y) = f(x) + f(y) ) and two functions ( f ) and ( phi ) to be *commutative between themselves* if ( fphi(x) ) = ( phi f(x) ). We note that he did not make a distinction between function and operator.

Servois' second paper, a philosophical essay entitled “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], followed immediately on the “Essay” in the pages of the *Annales* for 1814. Accompanying the present article is the first ever English translation of Servois' “Reflections.”

During the remainder of this introductory article, we provide a reader's guide to Servois’ “Reflections” [Servois 1814b], and then conclude with some recommendations as to how our translation may be used by teachers and students of mathematics and of the history of mathematics.

Servois began the “Reflections” with a philosophical justification of the research he had conducted in his “Essay” [Servois 1814a]. He began by stating that it is not necessary to consider only one primary foundation for calculus; however, the foundation selected must be rigorous. Servois was primarily speaking against the use of infinitesimals as a foundation for the differential calculus. The infinitesimal calculus, according to Servois, cannot rigorously answer the following questions:

- What is a differential?
- When and how do differentials arise naturally?
- With which analytic functions do they maintain intimate connections, not just simple analogies?

Servois argued that these questions can be answered using the Lagrangian method. He stated that the idea of using series as a foundation for calculus is natural because students are exposed to series early in their education when performing long division or extracting roots, in which the idea of series is an underlying principle. Thus, “series and the *differential calculus,* therefore ought to arise together” [Servois 1814b, p. 142]. In his “Essay” [Servois 1814a], Servois had defined the differential ( dz ) by the series: \[ \Delta z - \frac{1}{2} \Delta^2 z + \frac{1}{3} \Delta^3 z - \frac{1}{4} \Delta^4 z + \ldots, \] a definition he recapitulated in the “Reflections.” By analogy, Servois observed that the differential is *like* the logarithm of the varied state, i.e. \( \ln(z + \Delta z) \). (In his “Essay,” Servois had defined the *varied state* of \( \phi(x, y, \ldots) \) to be \( \phi(x + a, y + b, \ldots) \), for arbitrary constants \( a, b, \ldots \) [Servois 1814a, p. 94].) Servois then claimed that all the rules of the differential calculus can be *proven* using this series and the distributive and commutative properties of the varied state, the difference operator, and multiplication by a constant. Indeed, he had deduced all of these properties in his “Essay."

**Figure 4.** Title page of Servois’ “Reflections” (public domain).

After reviewing the material found in his “Essay,” Servois undertook a brief historical examination of the origins of the differential calculus, examining the three competing foundational notions for the calculus. He described the limit concept in a sympathetic way, tracing it all the way back to Isaac Newton and even finding Leibniz to be sympathetic to a sort of limiting argument. On the other hand, he expressed no sympathy for the use of the infinitely small.

**Figure 5.** Immanuel Kant (1724-1804) (print in public domain).

Servois then quoted at length from an author whom he did not identify by name, who argued against the exclusion of the idea of the infinite from mathematics [Servois 1814b, pp. 144-145]. The unidentified author was Josef-Maria Hoëné-Wronski (1776-1853), whom we will discuss in more detail shortly.

Wronski was a Kantian and Servois began his philosophical refutation of Wronski's position by exhibiting his own knowledge of Immanuel Kant's philosophy, attempting to use it to buttress his position on the inappropriateness of the use of infinite in mathematics. He further challenged his readers to examine the principle that if \( dz \) is an infinitely small quantity of the first order, then \( d^{2}z \) is one of the second order. Servois claimed that it is easy to demonstrate this by using Taylor series, but impossible with the infinitely small. Servois continued his critique, eventually concluding that infinitesimals “will one day be accused of having slowed the progress of the mathematical sciences, and with good reason” [Servois 1814b, p. 148].

Having rejected infinitesimals, Servois then explained that he has “always found several inconveniences in deducing [the exposition of the calculus] from the consideration of derived function or, in general, of limits” [Servois 1814b, p. 148]. He then described the advantages of using series expansions in foundational matters. “The first expansions into series that we encounter are the results of successive transformations applied to equations of identity” [Servois 1814b, pp. 148-149]. As an example, Servois considered the identity \[ \frac{1}{1+a} = \frac{1}{1+a}, \] then performed division on the right hand side to derive the geometric series.

**Figure 6.** Augustin-Louis Cauchy. Painting by Jean Roller, *ca.* 1840 (public domain).

In a second example, he began with the identity \[ \frac{1}{a-b} = \frac{1}{a+x} + \frac{b+x}{(a+x)(a-b)}, \] then transformed it into a series from a paper by Nicole [1727]. He observed that such expansions always have remainder terms, so that these expansions can be stopped at any term, and the identity can be preserved by means of this term. Servois called the remainder term of a series a “complement.” According to Servois, his method has the additional benefit that we need never assume that we know the form of the desired series.

Seven years before Cauchy's *Cours d'analyse* [Cauchy 1821], Servois observed that convergence is an issue which needs to be considered when manipulating series. He noted that the remainder term of a series can be omitted if the series converges, since the remainder can be shown to tend to zero. However, divergent series “may only be used with their complements, and in this way we have long ago happily resolved the paradox presented by the expansion of the fraction \( \frac{1}{1+1} \)” [Servois 1814b, p. 150]. The paradox of which Servois spoke is that a careless application of the geometric series suggests that \( \frac{1}{2}= 1 - 1 + 1 - 1 + \ldots \).

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.

Josef-Maria Hoëné-Wronski was born in Poland in 1776. Like Servois, he lived through a revolution and served in the military, first with the Polish revolutionaries of 1794 and later in the Russian army. (See [Dobrzycki 1978] or [Grattan-Guinness 1990, pp. 219-221] for more biographical information on Wronski.) Afterwards, he studied philosophy at several German universities and then moved to Marseilles in 1800, where he eventually became a French citizen. For about ten years, he worked in the mathematical sciences.

**Figure 7.** Sketch of Wronski by Félix Valloton (public domain).

In 1810, Wronski moved to Paris where he presented his paper “Premier principe des méthodes analytiques” to the *Institut. * Dobrzycki [1978] reports that the paper received a “rather sketchy review” from Lacroix and Lagrange. Grattan-Guinness [1990, p. 221] explains that Lacroix praised the generality of Wronski's results but expressed reservations about the lack of proof. Wronski published a revised version of his essay [Wronski 1811], which was intended to be his foundation for analysis based on infinitesimals. He followed this with a criticism of Lagrange's foundational program [Wronski 1812].

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

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

The French Revolution encouraged a move away from the categorical thinking of the Enlightenment and towards the idea of rigor in mathematics [Gillispie 2004]. Inspired by this movement and by the work of Lagrange, François-Joseph Servois believed that the only way to make the foundation of the differential calculus truly rigorous was to base it on the method of power series. Formally, Servois introduced calculus through series in his “Essay” [Servois 1814a]. His “Reflections” paper, published in the same year, was intended to be a manifesto regarding his philosophical beliefs on the foundations of the differential calculus.

Servois’ main argument was that there is no way to rigorously prove theorems using the idea of infinitesimals. The method of calculus through power series gave students of mathematics a rigorous foundation upon which to learn calculus. The dangers of using infinitesimals are best explained by Servois:

In a word, I am convinced that the infinitesimal method does not nor cannot have a theory, that in practice it is a dangerous instrument in the hands of beginners, that it necessarily imprints a long-lasting character of awkwardness and pusillanimity upon their work in the course of applications [Servois 1814b, p. 148].

Interestingly, Servois was mistaken in his claim that the method “cannot have a theory.” Some 150 years later, Robinson and Laugwitz invented nonstandard analysis, with which they were able to give a rigorous foundation to the idea of infinitesimals [Medvedev 1998].

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, with warring factions using non-mathematical arguments to bolster their beliefs in the superiority of certain methods or definitions. Besides providing a readable account of the history of the calculus, this translation of Servois’ “Reflections” is an original source for investigating philosophical ideas at an important turning point in the rigorization of calculus.

Furthermore, this translation provides many opportunities for student research projects in the development of the rigorous calculus. For example, Servois names dozens of mathematicians who contributed to the development of calculus [Servois 1814b, pp. 144-150]. Many are well-known, but some are absent from the standard biographical reference works. For example, who was Nicole [Servois 1814b, p. 150]? What did he write, other than the paper mentioned by Servois? In the same pages, Servois refers to many of the ideas of these mathematicians, without further elaboration. What, for example, were the “incomprehensibilities of Sturmius” or the “Subtleties of Guido Grandi” [Servois 1814b, p. 147]? Finally, for those with an interest in philosophy, there are many possible avenues for further research. What, for example, are Kant's four antinomies [Servois 1814b, p. 145]? Do Wronski and Servois really seem to understand Kant's doctrine of thesis and antithesis?

Download the authors' English translation of Servois' “Reflections.”

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**Acknowledgments**

The authors are extremely grateful to the referees for their many helpful suggestions and corrections.**About the Authors**

Rob Bradley is a professor in the department of mathematics and computer science at Adelphi University. With Ed Sandifer, he wrote *Cauchy's Cours d'analyse: An Annotated Translation* and edited *Leonhard Euler: Life, Work and Legacy.* He is chairman of HOM SIGMAA (the History of Mathematics Special Interest Group of the MAA) and past-president of CSHPM (the Canadian Society for History and Philosophy of Mathematics).

Salvatore J. Petrilli, Jr. is an assistant professor at Adelphi University. He has a B.S. in mathematics from Adelphi University and an M.A. in mathematics from Hofstra University. He received an Ed.D. in mathematics education from Teachers College, Columbia University, where his advisor was J. Philip Smith. His research interests include history of mathematics and mathematics education.