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The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Borda and Condorcet in Pre-Revolutionary France

Janet Heine Barnett (Colorado State University – Pueblo)

Like all human beings, Borda and Condorcet were products of the place and time in which they lived, as were their texts on voting theory.[1] Both men were born in France in the first half of the eighteenth century: Borda in 1733 and Condorcet in 1743. This situates their lives and works firmly within the period of the French Enlightenment, and the political revolutions which that movement helped to spur in the latter part of the eighteenth century. In this section, we focus on events related to their public lives, particularly the publication of their works on voting, in the period leading up to the 1789 start of the French Revolution. We begin with some general historical context.

While the starting date of the Enlightenment itself is debated by historians, its epicenter is universally recognized to be Paris. It was in the City of Lights that the artists, writers, politicians, scientists, and philosophers of the day began to regularly meet in intellectual circles, or salons, held in the homes of the (generally well-to-do) women, or salonnières, who organized and directed these gatherings.[2] Among the philosophes, as these thinkers came to be called, science and its successes over the preceding two centuries served as an important model of how knowledge can be generated through human reason and the evidence of our senses. Indeed, the name of the Enlightenment period (Siècle des lumières in French) itself grew out of the metaphor of bringing light to the dark ages that predated the Scientific Revolution. During the Enlightenment, human reason came to be seen not only as a means to shed light on our understanding of the universe in which we live, but as a means to improve our living conditions as social beings inhabiting that universe. This belief in the power of human reason as the primary source of authority led in turn to widespread criticism of existing political, religious and social institutions and the limitations they imposed on individual liberty and happiness.[3]

Figure 3: A Reading in the Salon of Mme Geoffrin, 1755.
By Anicet Charles Gabriel Lemonnier - Web Gallery of Art. Public Domain.

Unsurprisingly, one institution that escaped the criticism of the Enlightenment thinkers was the French Royal Academy of Sciences.[4] Founded in 1666 as one of the earliest learned societies in Europe, the Academy was initially organized around six sections (mathematics, mechanics, astronomy, chemistry, botany, and anatomy); members of the Academy belonged to one particular section based on their recognized accomplishments within that field of study. Importantly for the story we tell in this article, membership in the Academy was determined by election. The rules and regulations governing these elections were originally quite complicated, and Academy commissions were appointed to study the problem of reaching majority consensus in Academy elections as early as 1699 [Urken, 2009]. In the period immediately preceding the French Revolution, the usual process was for the current membership to produce, via voting, a list of ranked recommendations for new members that was forwarded to the King for his consideration; the King himself was not bound to accept the recommended ranking, or to even choose new members from the Academy’s list. The Academy itself, on the other hand, was expected to remain neutral in terms of politics, religion and social issues.

Both of our authors were elected to the Academy while still in their twenties, Borda in 1756 and Condorcet in 1769, based on the quality of their early works in mathematics. They also both began their mathematical educations under the Jesuits, Borda at the college in La Flèche and Condorcet first privately and then at the college in Reims. Despite the explicitly religious nature of the Society of Jesus itself, the goals behind the educational opportunities that it provided extended to the secular in that the Society sought to provide boys with the skills necessary to “take their place in that world even more strongly” and “be in a position to serve the common good” [Grendler 2014, p. 9]. Many of the boys who attended Jesuit colleges were of lower- and middle-class backgrounds, and the education they received was important simply for the sake of earning a living. Others—including both Borda and Condorcet—came from families with connections to the nobility[5] who were interested in seeing their sons trained for their future careers and to assume leadership positions within society.

In keeping with the job-oriented training that a Jesuit education sought to provide its students, mathematics was often taught in Jesuit colleges with an emphasis on its practical applications. The Jesuit college at La Flèche offered a course of study that was especially well-suited for the training of military engineers. Upon completing his studies at La Flèche in 1748, Borda entered the French army as a military mathematician. During this period of his military career, he wrote his first mathematical papers: a 1753 paper on geometry that brought Borda to the attention of Jean le Rond d’Alembert[6] (1717–1783), and a paper on the theory of projectile motion that secured his 1756 election to the Academy. Soon thereafter, in 1758, Borda enrolled at the School of Engineering at Mézières, taking just one year to complete the two-year course of study. He then entered the French navy, where he served in a variety of leadership capacities. These included participation in maritime campaigns during the American Revolutionary War between 1777 and 1778, appointment as the Major General of the Naval Army in 1781, and an appointment as the French Inspector of Naval Shipbuilding in 1784. In 1782, Borda was briefly held by the English after being captured while returning to France from Martinique, but he was soon released on his word of honor, perhaps due to his reputation as a scientist.

Ironically, Borda’s family had wanted him to pursue a career as a magistrate, while Condorcet’s family had wanted him to pursue a military career. In fact, both families had a long tradition of military service. Just days after his birth, Condorcet’s cavalry captain father was killed during an Austrian attack on the town of Neuf-Brisach.[7] Yet Condorcet himself rejected a military life, and instead transferred from the Jesuit Collège in Reims to the prestigious Collège de Navarre in Paris in 1758. He then studied at the equally prestigious College de Mazarin, also in Paris. At the age of 16, he successfully defended a dissertation on mathematical analysis (written in Latin) to a committee whose members included d’Alembert. Following this, Condorcet established himself as an independent scholar, living in Paris on a small allowance provided by his mother. As an independent scholar, Condorcet continued his study of mathematics, writing several works, including his Essai sur le calcul intégral [Condorcet 1765], which helped to secure his 1769 election to the Academy of Sciences.

Condorcet also fully immersed himself in Parisian Enlightenment society, attending the most important salons of the period and establishing friendships with d’Alembert[8] and other important philosophes. One philosophe friend in particular influenced Condorcet to become increasingly involved in public life: the reform economist Jacques Turgot (1727–1781). Turgot served as the national Minister of Finance under Louis XVI from 1774 to 1776, during which time he promoted policies aimed at mitigating the economic condition of the French people. After appointing Condorcet as the General Inspector of the French Mint in 1775, Turgot later persuaded him to remain in that post even after Turgot’s own dismissal from all governmental duties by Louis XVI. Condorcet remained head of the French Mint until 1790, serving for a time under Turgot’s political nemesis, Jacques Necker (1732–1804).

At least on the surface, then, our two authors shared a number of similarities: noble family backgrounds (both with long military traditions), initial training in the Jesuit tradition (but with no religious commitment of their own), early recognition of their mathematical talents (for which each earned the praise of d’Alembert), and prolonged service to their nation (Borda as a military man and Condorcet as an administrator). There were, however, marked differences between the two.

With regard to academic orientation, Borda’s interests were quite pragmatic, as one might expect of an engineer and military man. In addition to his study of applied topics such as projectile motion and fluid mechanics, he developed a number of utilitarian instruments, most notably the repeating circle, a celestial navigational device that employed a rotating telescope and a system of repeated observations to greatly reduce measurement error.[9] Borda’s other contributions in the area of marine navigation and cartography included development of a series of trigonometric tables, conduct of chronometer tests in the Caribbean, and creation of maritime charts of the Azores and Canary Islands.

Figure 4: Borda measuring the Peak of Ténériffe, Canary Islands.
Painting by Pierre Ozanne, 1775–1776. Wikimedia Commons.

Condorcet’s commitment to Enlightenment ideals, on the other hand, significantly influenced the direction of his scholarly work towards questions related to social issues. His interest in probability in the early 1780s, for example, was motivated in part by his vision of the role that probability could play in understanding and ameliorating social and economic conditions. Increasingly, however, Condorcet’s energies and interests were diverted away from considerations of how mathematics could promote social change, and towards active involvement in social reform efforts themselves. The extent to which this had occurred even before the start of the Revolution is highlighted by the author of Condorcet’s entry in the Dictionary of Scientific Biography, who remarked that “after 1787 Condorcet’s life is scarcely of interest to the historian of science” [Granger 1970–1990].

Borda and Condorcet also differed with regard to their general views on the nature of science and mathematics. Borda considered himself a disciple of Georges-Louis Leclerc, count de Buffon (1707–1788), a fellow Academician whose views about mathematics and science often came into conflict with those of d’Alembert.[10] As d’Alembert’s protégé, Condorcet held views about mathematics, science and its correct conduct that thus frequently ran opposite to those of Borda and Buffon. Within the political realm, the differences in their ideological beliefs were even more pronounced. A staunch monarchist, Borda supported the traditionalist views of Necker, Turgot’s political rival. Condorcet’s alignment with Turgot’s politics, on the other hand, placed him among the “modernists” who believed there was a need for constitutional restructuring in France.

Taken together, these differences were sufficiently divisive that Borda and Condorcet may have been seen as (and possibly were) rivals within the Academy. While Condorcet (in his letters to Turgot and elsewhere) was openly critical of many of Borda’s ideas, we will see later in this article that the two did work together on at least one occasion. And, as the economist Duncan Black (1908–1991) has noted, “during Borda’s prolonged absences from Paris [the two] corresponded on scientific matters” [Black 1958, p. 169].[11]

Whatever the personal relationship between the two men, Borda’s military and scientific travels did indeed often take him away from Paris and the day-to-day workings of the Academy. In contrast, Condorcet became intimately involved in the administrative work of the Academy relatively early, serving first as its Assistant Secretary (1773–1776) and then as its Permanent Secretary[12] (1776–1793). As Permanent Secretary, he was responsible for assembling the Academy committees that prepared responses to government requests for scientific studies related to proposed public works (e.g., prisons, canals) and composed reports on evaluations of new scientific discoveries.[13] His administrative role also placed Condorcet in the position of helping to organize the Academy’s various elections, including a vote to allow the publication his own Essai on voting theory [Condorcet 1785] by the Imprimerie Royale.

In the next section, we turn our attention to the mathematical details of both Borda’s and Condorcet’s texts on the topic of elections, a topic that was quite naturally of interest to all members of the Academy.


[1] Biographical information for Borda in this and later sections of this article is drawn from [Fairclough 2018], [Gillmour 1970–1990], [Mascart 1919], [Noguès n.d.], [O'Connor and Robertson 2003] and [Tietz 2017]; and, for Condorcet, from [Acton 2020], [Encyclopedia Britannica 1911], [Fonseca n.d.], [Granger 1970–1990], [Landes 2018] and [O'Connor and Robertson 1996]. Additional information about the relationship between the two men is drawn from [Black 1958], [Brian 2008] and [Young 1998].

[2] References consulted in connection with the Enlightenment and Parisian salon culture include [Brewer 2020], [Bristow 2017], [Haag 2020] and [Lougee 2020].

[3] One hears clear echoes of this, for example, in the Declaration of Independence of the United States, whose writers were strongly influenced by Enlightenment ideals.

[4] After the French Revolution, this was known simply as the French Academy of Sciences.

[5] Recall that Cordorcet’s actual name was Marie-Jean-Antoine-Nicolas de Caritat. The Caritats were an ancient family that had long resided in the town of Condorcet (located not far from Marseilles), from which he derived his title “Marquis de Condorcet.” This was a courtesy title (versus a title associated with land ownership) which had no legal significance, but served to indicate that Condorcet’s parents were members of the nobility. Borda came from a family of lesser untitled nobility that resided in the town of Dax, in the southwest region of France. His father held the rank of Lord (in French: Seigneur), while Borda himself held the rank of Knight (in French: Chevalier).

[6] In addition to his work in mathematics and physics, d’Alembert was a contributor to and co-editor (with the philosophe Denis Diderot (1713–1784)) of the Encyclopédie, the famous Enlightenment project aimed at spreading light across French society by way of educating its citizens. (For a student project based on one of d’Alembert’s contributions to the Encyclopédie, and more about his biography, see the Convergence article, “Investigations Into d'Alembert's Definition of Limit: A Mini-Primary Source Project for Students of Real Analysis and Calculus 2,” by Dave Ruch.)

[7] Neuf-Brisach is a fortified town in Alsace, near the Rhine River and the German border. It was designed by Sébastien Le Prestre de Vauban (1633–1707), a French military engineer whose principles for applying rational and scientific methods in fortification design were widely used for nearly a century. Neuf-Brisach is considered to be Vauban’s masterpiece and is now a UNESCO World Heritage site.

[8] Through his friendship with d’Alembert, Condorcet was recruited to write mathematical articles for supplements to Diderot’s famous Enlightenment project, the Encyclopédie. Their friendship was so close that, upon his death in 1783, d’Alembert left his property to Condorcet, thereby ending the financial difficulties that Condorcet faced up to that time. Soon thereafter, in 1786, Condorcet married the influential salonnière Sophie de Grouchy, whose salon at the Hôtel des Monnaies (which also housed the offices of the national Mint) was attended by foreign dignitaries such as Thomas Jefferson, who served as the American Minister to France from 1785 to 1789. The intellectual partnership between Condorcet and de Grouchy played a critical role in the social reform efforts in which Condorcet engaged during the last decade of his life, which we discuss in a later section of this article. We note here that, in contrast to Condorcet, Borda neither married nor participated in salon culture.

[9] Borda also published, in 1778, a pre-chronometer method for computing the longitude using lunar distances that is regarded as the best of several similar mathematical procedures that were available at that time. It was used, for example, by the 18031806 Lewis and Clark Expedition.

[10] See, for example, the discussion of the views held by Buffon and d’Alembert regarding rigor in mathematical proof in [Richards 2009].

[11] Black opened this sentence by claiming that “throughout their lives, the two were close friends” [Black 1958, p.179]. However, this claim is not repeated (or even mentioned) by any other historian that the author of this article has consulted, and it seems unlikely given the vast differences in their personal values and beliefs. Black provided little evidence for this assertion beyond providing a reference to a letter that is described and quoted in part by Borda’s biographer, French astronomer and mathematician Jean Mascart (1872–1935). According to Mascart [1919, p. 96], Borda wrote this letter to Condorcet as a response to the latter’s criticisms of the former’s work on fluid dynamics, and concluded it by writing (again, as quoted by Mascart):

Here is a response to your profession of faith: I am angry that you are not of my belief, but we will not both go to Paradise, except to argue about fluids. I embrace you with all my heart.

Whether the closing salutation [in French: Je vous embrasse de tout coeur] was merely a standard courtesy of the day or a genuine expression of fondness, the existence of the correspondence itself does suggest that relations between the two men were of a sufficiently cordial professional nature, even if not as personally close as Black suggested.

[12] Although his personal friendship with Turgot led to preferments (e.g., Turgot’s earmarking of a significant portion of the government’s funding for the Academy specifically to support Condorcet's research) that complicated Condorcet’s early relations with other Academicians, his 1776 election as the Academy’s Permanent Secretary was unanimous.

[13] Another of Condorcet’s responsibilities as Permanent Secretary was the preparation of official eulogies for Academicians and other important mathematicians and scientists. In all, he composed over fifty such eulogies, including those of his friend d'Alembert and his enemy Buffon, often inserting comments about mathematics, science and their role in society that were reflective of his own views, rather than those held by the subject of the eulogy.

Janet Heine Barnett (Colorado State University – Pueblo), "The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Borda and Condorcet in Pre-Revolutionary France," Convergence (September 2020)