# Francois-Joseph Servois: Priest, Artillery Officer, and Professor of Mathematics

## Introduction

Who was the mathematician François-Joseph Servois (1767-1847)? To the extent that his name is known at all, it is for introducing the words “distributive” and “commutative” to mathematics (see page 5). Servois was ordained a priest near the beginning of the French Revolution. Had it not been for the revolution, it seems likely he would have remained a priest and become a successful mathematician. With the outbreak of the revolutionary wars, he joined the armed forces and followed a military career while also pursuing mathematics during his leisure time. His mathematical career flourished once he was appointed professor of mathematics at the French artillery schools. This paper is a survey of the life and subtle mathematical contributions of Servois.

## The Early Years

Note on sources: Much of the biographical information found in this paper is drawn from [Petrilli 2009, 2010]. Primary sources include [Boyer 1895a, 1895b] and [Taton 1972].

François-Joseph Servois was born on July 19, 1767, in the small village of Mont-de-Laval, located in the Départment of Doubs in the Franche-Compté region of eastern France, close to the Swiss border (too small to be shown in the map in Figure 1, but situated north of Morteau). The only son of Jacques-Ignace Servois, a merchant, and Jeanne-Marie Jolliot, Servois had one sibling, a sister about whom little is known except that he spent his retirement years with her. It would seem appropriate here to include a portrait of Servois; however, there are no known paintings or photographs of him.

Figure 1. The Départment of Doubs (public domain)

As he grew up, Servois attended different religious schools in Mont-de-Laval and Besançon, the capital of Doubs, with the intention of becoming a priest. He was ordained at Besançon near the beginning of the French Revolution. Servois' religious career, however, was destined to be a short one.

## Military Career

In 1793, Servois left the priesthood and joined the army to become an officer in the Foot Artillery, sometimes referred to as Heavy Artillery. On March 5, 1794, he officially entered the artillery school at Châlons. He was commissioned as Second Lieutenant in the First Foot Artillery Regiment on November 13 of the same year.

While in the army, Servois was actively involved in battle many times. He participated, for instance, in the crossing of the Rhine and the battle at Neuwied (1796-97), and the final battle in March of 1814 to defend Paris from the Austrian and Prussian armies. Figure 4 (page 5) displays a painting of the 1814 Battle of Paris.

While on active duty, Servois would spend his leisure time studying mathematics. He demonstrated his mathematical abilities when he made improvements to one of the cannons, significantly increasing its firing range. Servois suffered from poor health during his years serving as an officer. These circumstances prompted him to seek a non-active military position as a professor of mathematics.

Figure 2. The Artillery School in Besançon (public domain)

On July 7, 1801, Servois was assigned to his first academic position, as a professor at the artillery school located in Besançon, thanks to a recommendation from Adrien-Marie Legendre (1752-1833). During the Napoleonic period, Servois was on faculty at several artillery schools, including Besançon (1801), Châlons (March 1802 - December 1802), Metz (December 1802 - February 1808, 1815-1816), and La Fère (February 1808-1814, 1814-1815).

## Early Publications

While at Metz, Servois began publishing and presenting his mathematical research. His first paper, written in 1805, was a treatise on expanding functions into power series. This paper was presented to the Institut National des Sciences et des Arts* and would undergo several revisions to become his celebrated “Essay” [Servois 1814a] on the foundations of the differential calculus. (The next section describes this paper in more detail.)

Figure 3. Figures from Servois' geometry text (public domain). The left-hand figure was used to construct a parallel to an inaccessible line. The right-hand figure was used to prolong a line beyond an obstacle. The dotted circles in these figures are not part of the constructions, but rather used to verify the validity of the constructions.

Additionally, Servois published his first and only book, titled Solutions peu connues de différents problèmes de géométrie-pratique [Servois 1804] (Little-known Solutions to Various Problems in Practical Geometry) in 1804.** Servois' geometry text was intended to be a reference on applied geometry for military officers, presenting constructions in ruler-geometry that could be used on the battlefield at any time [Bradley 2002]. Taton [1972] stated that the work was well received by the public and Jean-Victor Poncelet (1788-1867) declared Servois' book to be

a truly original work, notable for presenting the first applications of theory of transversals to the geometry of the ruler or surveyor's staff, thus revealing the fruitfulness and utility of this theory [Poncelet 1865, p. 44].

The geometry textbook is divided into two sections after a brief introduction: Theory and Practice. There is an index of the practical section, errata, and a table of figures. It is also bound with a 28-page article, a “Letter from S … to F …, Professor of Mathematics” who is possibly François Joseph Français (1768-1810). The military applications of the geometry text can be seen in the following index of the practical section. Note that all of these constructions are performed with straightedge, but no compass.

• To prolong a line beyond an obstacle.
• To find a point aligned with the two invisible points of intersection of two pairs of lines given in direction.
• From a given point, to draw a line to the invisible point of intersection of two lines given in direction.
• At a given point, to draw a line parallel to two other parallels given in direction.
• To draw two parallel lines on the ground.
• From a given point, to draw a parallel to an accessible line.
• From a given point, to draw a parallel to an inaccessible line.
• To divide a line into two equal parts.
• To divide a line into any number of equal parts.
• To divide an accessible angle into two equal parts.
• To divide an inaccessible angle into two equal parts.
• To erect a perpendicular on a line at a given point.
• From an inaccessible point, to drop a perpendicular to an accessible line.
• From a given point, to drop a perpendicular to an inaccessible line.
• To measure a line, one of whose extremities is inaccessible.
• To measure an inaccessible line.
• To measure the distance from a point to an inaccessible line.

Interestingly, Poncelet consulted Servois for his expertise on geometry several times during the writing of the 1822 edition of the Traité des propriétés projectives [Taton 1972].

Servois was eventually transferred to the artillery school at La Fère, where he presented papers on the elements of dynamics and on cometary and planetary orbits to the Institut. However, these papers were never published [Taton 1972]. Then in 1810, he wrote and published his “De principio velocitatum virtualium commentatio,” a paper elaborating Joseph-Louis Lagrange's (1736-1813) notion of “virtual velocities.” The paper was entered in a prize competition sponsored by the Academy of Turin. Curiously, his memoir was the only entry the Academy received and, because Servois missed the deadline, nobody won the prize. However, the paper was deemed worthy, so the Academy published it and elected him a corresponding member [Bradley 2002].

________________________

*After the French Revolution, the Royal Academy of Sciences of Paris, along with other royal societies, were incorporated into the Institut National des Sciences et des Arts, renamed the Institut de France in 1806.

**The year of publication is given as 1805 in [Poncelet 1865, Taton 1972], but is actually year XII of the French Revolutionary calendar, which translates to 1804 or late 1803.

## Battles for Paris and for the Foundations of Calculus

At the start of 1814, Servois was called back into active duty, one last time, to defend Paris from the Austrian and Prussian armies [Boyer 1895a]. Not only was France involved in a battle that would shape its destiny, but calculus also was in a similar state of upheaval. Mathematicians knew there were problems with the foundations of calculus, but Augustin-Louis Cauchy (1789-1857) had not yet begun his revision of those foundations.

Figure 4. The Battle of Paris, 1814, as painted by Bogdan Willewalde in 1834 (public domain)

During his time in the military, Servois had become acquainted with Joseph Diaz Gergonne (1771-1859). Servois' friendship with Gergonne was particularly helpful in building his mathematical publication record. Gergonne was the editor of the Annales des mathématiques pures et appliquées, often called Annales de Gergonne. Servois' most celebrated paper, “Essai sur un nouveau mode d'exposition des principes du calcul différential” [Servois 1814a] (Essay on a New Method of Exposition of the Principles of Differential Calculus), his primary work on the foundations of the differential calculus, appeared late in 1814. Following Lagrange, his work proposed using power series expansion as fundamental to the differential calculus. He defined the differential operator as an infinite series, which he called an infinitinôme, in powers of the difference operator: $dz = \Delta z - \frac{1}{2} \Delta^2 z + \frac{1}{3} \Delta^3 z - \frac{1}{4} \Delta^4 z + \ldots.$

Using this operator, he demonstrated how to find differentials of functions and was able to discover the basic laws of the differential calculus, such as the power and product rules. This paper truly is remarkable because, at the heart of Servois' arguments, lies the use of the linear properties of operators in proving theorems about the calculus. Servois didn't use the terms “linear” or “operator,” but rather he used the words commutative and distributive for the first time in their mathematical sense. In reviewing the contents of this paper, the Commissioners of the Institut de France praised him for having “done something that is very useful for the science [of analysis]” [Servois 1814a, p. 140].

Figure 5. The Institut de France (photograph by Benh Lieu Song, 2007, licensed under Creative Commons Attribution-Share Alike 3.0 Unported license)

Servois' second work on the foundations of calculus was “Réflexions sur les divers systèmes d'exposition des principes du calcul différentiel, et, en particulier, sur la doctrine des infiniment petits” [Servois 1814b] (Reflections on the Various Systems of Exposition of the Principles of the Differential Calculus and, in particular, on the Doctrine of the Infinitely Small). This work gives us insight into Servois' personal views of the history of calculus and into his philosophical positions, particularly on placing calculus on an algebraic foundation. This paper began with a historical overview of calculus from its birth to the time of Lagrange. Servois discussed the contributions made by well-known mathematicians, such as Newton, Leibniz, Taylor, Euler, and Lagrange, to the differential calculus. In addition, he pointed out where, in his opinion, some mathematicians had erred. For instance, he stated that the use of infinitesimals is difficult for mathematicians to justify as a basis for calculus. The remainder of the work is devoted to his justification of development of calculus by means of power series and to a harsh criticism of the mathematical philosophy of Josef-Maria Hoëné-Wronski (1776-1853).

Editor’s note: For a more detailed analysis and an English translation of Servois’ “Reflections,” see the Convergence article, “Servois’ 1814 Essay on the Principles of the Differential Calculus, with an English Translation,” by the author and Robert E. Bradley.

## The Mathematics of Servois: Other Works

During the years 1811 to 1817, Servois had the majority of his work published in Gergonne's Annales des mathématiques pures et appliquées. Some of his works were either solutions to problems posed by other mathematicians, or his remarks on other articles in the journal. In 1813, Servois published his “Calendrier perpétuel” [Servois 1813] (Perpetual Calendar). This work was not mathematically significant; however, Jacques Frédéric Français (1775-1833, brother of François Joseph Français) called it an “ingenious table” [Français 1813].

Figure 6. The Mathematical Philosophy section of the Table of Contents of volume 5 of Annales des mathématiques pures et appliquées (public domain). Note Servois' article [Servois 1814b], Argand's response to Servois' letter [Servois 1814c], and two pieces by the editor.

The majority of Servois' contributions to the field of mathematics can be described by the term “algebraic formalism.” This idea was very important to Servois, and he severely criticized individuals who followed paths he considered to be non-rigorous. One such incident occurred within the pages of the Annales, in which a heated debate took place among Servois, Jean Robert Argand (1768-1822), and Jacques Français. Français published a paper in 1813, based on the work of Argand, in which he viewed complex numbers geometrically. This view of the complex plane is now commonly known as the Argand Diagram or the Argand Plane. Servois criticized the work of these two mathematicians, saying: “I had long thought of calling the ideas of Messrs. Argand and Français on complex numbers by the odious qualifications of useless and erroneous …” [Servois 1814c, p. 228].

Servois published two more articles in the Annales before his retirement in 1827. In 1817 he published his “Mémoire sur les quadratures” [Servois 1817] (Memoir on quadratures). Then he published his final work in 1826, called “Sur la théorie des paralleles” [Servois 1826] (On the theory of parallels). These last two papers were not very influential. Bradley [2002] states that at the “courtesy” of an old friend, Gergonne placed this final paper as the first article in volume 16, number 7.

## Curator and Early Retirement

Servois' final position was as Curator of the Artillery Museum, located in the 7th Arrondissement of Paris. Currently, it is part of the Museum of the Army. Figure 7 shows the Museum of the Army as it stands today. Servois was officially assigned the position on May 2, 1817.

Figure 7. The Museum of the Army (photograph by Rama, 2006, licensed under Creative Commons Attribution-Share Alike 2.0 France license)

On August 17, 1822, Servois was made a Knight of Saint-Louis. The Order of Saint-Louis was founded in 1696 and later served as a model for Napoleon's Légion d'honneur. Officers were awarded this knighthood for distinguishing themselves through their honor, bravery, and loyalty to the military. Servois was one of 661 officers to receive this honor in 1822 [Herman 1992].

On June 1, 1827, Servois retired to his hometown of Mont-de-Laval and lived twenty years longer. He never married, but retired with his sister and his two nieces. Servois died on April 17, 1847, in Mont-de-Laval, where “his legacy remains legendary” [Boyer 1895a, p. 313].

## References

[Boyer 1895a] Boyer, J. (1895). “Le mathématicien franc-comtois François-Joseph Servois,” Mémoires de la Société d’émulation de Doubs 9, 305-313.

[Boyer 1895b] Boyer, J. (1895). “Pièces Justicatives et Notes Diverses” (appendix to the foregoing, including documents from Servois’ military dossier and notes by Abbé Filsjean), Mémoires de la Société d’émulation de Doubs 9, 314-328.

[Bradley 2002] Bradley, R. E. (2002). “The Origins of Linear Operator Theory in the Work of François-Joseph Servois,” Proceedings of Canadian Society for History and Philosophy of Mathematics 14, 1–21.

[Français 1813] Français, J. F. (1813). “Chronologie. Solution directe des principaux problèmes du calendrier,” Annales de mathématiques pures et appliquées 4, 273-276.

[Herman 1992] Herman, C. (1992). Knights and Kings in Early Modern France: Royal Orders of Knighthood, 1469-1715. New York: AMS Press.

[Poncelet 1865] Poncelet, J. V. (1865). Traité des propriétés projectives des figures (2nd ed.), Paris: Gauthier-Villars.

[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). “Monsieur François-Joseph Servois” (unpublished manuscript).

[Servois 1804] Servois, F. J. (1804). Solutions peu connues de différents problèmes de géométrie-pratique; pour servir de supplément aux Traités connues de cette Science. Metz: Chez Devilly/Paris: Chez Bachelier.

[Servois 1811] Servois, F. J. (1811). “De principio velocitatum virtualium commentatis, in responsum quastioni ab illustrissima Academia Taurinensi, pro anno 1810, propositae, conscripta," in Mémoires de l'Académie impériale des sciences de Turin 18, pt. 2 (1809-1810), 177-244.

[Servois 1813] Servois, F. J. (1813). “Calendrier perpétuel,” Annales de mathématiques pures et appliquées 4, 84-90.

[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.

[Servois 1814c] Servois, F. J. (1814). “Sur la théorie des imaginaries, Lettre de M. Servois," Annales de mathématiques pures et appliquées 4 (1814-1815), 228-235.

[Servois 1817] Servois, F. J. (1817). “Mémoire sur les quadratures,” Annales de mathématiques pures et appliqués 8, 73-115.

[Servois 1826] Servois, F. J. (1826). “Lettre sur la théorie des paralleles,” Annales de mathématiques pures et appliquées 16, 233-238.

[Taton 1972] Taton, R. (1972). “Servois," in Dictionary of Scientific Biography, 1972, C. C. Gillespie, Ed., New York: Scribner, XII.325-326.

For additional information and resources about the life of Servois, see the biography of François-Joseph Servois in the MacTutor History of Mathematics Archive.

## Acknowledgments and About the Author

Acknowledgments

The author is extremely grateful to the referees for their many helpful suggestions and corrections. In addition, the author expresses deep gratitude to Robert E. Bradley, professor of mathematics at Adelphi University, for graciously dedicating so much of his time, patience, and knowledge in guiding me through the writing of this article.