The French Connection: Borda, Condorcet and the Mathematics of Voting Theory – Borda and Condorcet in Later Years

Janet Heine Barnett (Colorado State University – Pueblo)

A landmark event that is often hailed as the official start of the French Revolution took place exactly five years after Condorcet announced his intention to publish on voting theory to the Academy, when the Bastille prison was destroyed by a group of Parisians on July 14, 1789. The events of the next decade significantly shaped the lives and works of all French citizens, including Borda and Condorcet. In this section, we complete our biographical sketch of their professional lives during that decade, by the end of which both had died. We again begin with some general historical context.

While the causes and events behind the French Revolution are numerous and complex, these certainly included the citizenry’s discontent with both the French monarchy in general and the fiscal crisis in which the monarchy found itself in particular. As a result of that crisis, the government of Louis XIV proposed the enactment of new tax measures in May 1789, as a means to stave off bankruptcy. Circumstances related to that proposal in turn led to the formation of a National Assembly[50] (later renamed the National Constitutional Assembly) by a group of individuals who demanded governmental restructuring, and, further, that the new structure be guaranteed by a written constitution. Grudgingly acknowledged by Louis XVI in June 1789, the National Assembly began to function as a governing body on July 9, 1789. Meanwhile, popular insurgencies began to arise throughout France, including the Paris uprising that culminated in the storming of the Bastille. On August 26, 1789, the Assembly approved the Declaration of the Rights of Man and of the Citizen,[51] premised on the belief that “­the ignorance, neglect, or contempt of the rights of man are the sole cause of public calamities and of the corruption of governments.” The “natural, unalienable and sacred rights of man” set forth in the Declaration’s seventeen articles[52] served as a set of guiding principles for the aspirations of the revolutionaries. The Declaration later became the preamble of the constitution[53] that was ratified by the Assembly on September 3, 1791, thereby establishing France as a constitutional monarchy with sovereignty residing effectively in the Assembly.

Simultaneously with the development of a new constitution for France, the National Constitutional Assembly began to institute reforms aimed at addressing a range of national concerns. Among these was a problem to which both Borda and Condorcet contributed their technical expertise: the development of a national system of weights and measures. At the time of the revolution, in France and elsewhere, there existed a profusion of different systems of weights and measures, with units varying both from town to town and within the different trade guilds. In addition to hindering trade and tax collection, this disorganized state of affairs drew frequent complaints from French citizens concerned about the potential it created for unfair manipulation of weights and measures (e.g., landowners collecting more than their share of the harvest from the peasants farming their lands).[54] In May 1790, the Assembly turned to the Academy of Sciences for a resolution of this problem, and provided it with funding for that effort.[55] The Commission on Weights and Measures appointed to study the issue was chaired by Borda, with Cordorcet and three other mathematicians serving as its other members.[56] Although the commission gave serious consideration to the advantages of a duodecimal system (e.g., greater ease in the partitioning of goods into fractional parts), they ultimately proposed a decimal system with the unit of length to be based on a fractional arc of a quadrant of the Earth's meridian (eventually defined to be one ten millionth of distance from North Pole to Equator measured along the meridian through Paris). The Assembly approved the proposed plan in March 1791, and charged the Academy with its implementation. Based on a proposal by Borda, five separate groups were established to complete different aspects of the necessary scientific work.[57] This included in particular a survey of the meridian arc that defined the meter, an undertaking that took over six years and considerable expense[58] to complete; the construction of the necessary equipment (including three copies of Borda’s repeating circle[59]) alone took one full year. The proposed definitions (and names) of the new units of the metric system were adopted into French law in 1793, although the official standard for the meter stick itself was not recognized until 1800 and the widespread use of the metric system across France took even longer.[60]

Figure 5: One of two remaining provisional meter sticks
(located in the Place Vendome and at 36 rue de Vaugirard respectively)
of the sixteen meter sticks that were originally mounted throughout Paris in 1796–1797
to provide the public with a means of learning the new measurement system. Photo by the author.

Borda’s contributions to the development of the metric system essentially constituted the full extent of his participation in revolutionary activities, as one might expect given his conservative political leanings. In contrast, Condorcet’s ever-increasing involvement in the political events of the time led to the (non-mathematical) works and the writings for which he is best remembered by the world-at-large today. In light of his Enlightenment values, Condorcet’s enthusiastic support of the ideals set forth in the Declaration of the Rights of Man and of the Citizen comes as no surprise. Echoes of his belief in the universality of equal rights can be heard in his assertion (as quoted in [Alder 2002]):

A good law ought to be good for all men,
as a good proposition [in geometry] is good for all men.[61]

In fact, Condorcet embraced this ideal beyond a literal interpretation of the phrase “all men.” He was ahead of his time, for instance, in supporting the abolition of slavery, as well as equal political rights for Jews, Protestants and women.[62] And, his commitment to social equality was much more than academic. When the National Constitutional Assembly was disbanded in September 1791 (its work in drafting a new constitution having been completed) and the Legislative Assembly took its place, Condorcet was elected as one of the delegates representing Paris on the new Assembly, where he quickly established himself as a torch bearer for the revolution. He served, for instance, as the secretary of the Assembly and led the efforts of the Assembly’s Committee on Education by drafting a plan for state-supported education[63] based on the principles of meritocracy and equality, in keeping with his belief that social progress required an enlightened citizenry.

It was also Condorcet who composed the Legislative Assembly’s declaration that justified the later suspension of the monarchy. Despite Louis XVI’s reluctant acceptance of France’s new constitutional monarchy, growing fears of counter-revolutionaries and mistrust of the King’s intentions led to the full arrest of the Royal family in 1792.[64] In September of that year, the Legislative Assembly was replaced by the National Convention and the (First) Republic of France was declared. Condorcet was again elected as a delegate to the Convention, and served as both its secretary and vice-president. He also chaired the committee that was charged with drafting the new republic’s constitution. Although Condorcet promptly began drafting a version of that constitution, its completion was delayed by his scholarly perfectionism and his extensive involvement in other political activities. In the interim, Louis XVI was condemned to death by the Convention and executed by guillotine in Paris in January 1793.[65] By the time Condorcet presented his careful and thorough draft constitution to the Convention, the relatively moderate political group with whom he was associated (the Girondists) was beginning to lose power. For this and other reasons,[66] Condorcet’s proposed constitution was rejected in June 1793, in favor of a hastily-drafted “Constitution of Year I” that was put forward by a far more radical group (the Jacobins). With the passage of the Jacobins’ constitution, a 9-member Committee of Public Safety[67] gained unlimited executive power and the infamous period of the revolution known as the Reign of Terror began soon thereafter. Between September 1793 and July 1794, Queen Marie-Antoinette and thousands of other French citizens from all social classes and professions were accused of counterrevolutionary actions and met their fate, as had Louis XVI, at the blade of the guillotine.[68

Although Condorcet himself escaped the guillotine, he did not outlive the Reign of Terror. After openly and passionately accusing the Committee of Public Safety of using fear tactics to gain passage of their version of the constitution, a warrant for his arrest was issued by the Convention in July 1793. Condorcet became a fugitive, and hid in the Parisian home of Madame Vernet[69] for the next eight months. During this time, Condorcet wrote his most famous philosophical work, Esquisse d'un tableau historique des progrès de l'esprit humain (Sketch of a historical picture of the progress of the human mind),[70] a declaration of his enlightenment belief in the power of rationality and perfectibility of human society. Concerned about the peril in which he was placing his protector, Condorcet fled Paris on foot on March 25, 1794, but was denounced to local authorities by an innkeeper in the nearby village of Clamart-de-Vignoble. Placed under arrest, he was transferred to a guarded house[71] located in the village of Bourg-la-Reine[72] on March 29, 1794. The following day, Condorcet was found dead in his makeshift cell; whether he took his own life, died of natural causes, or was murdered is unknown to this day.[73]

Figure 6: The gaoler opening the door of the prison cell to find Condorcet
lying dead on his bed. Stipple engraving by G. Aliprandi after J.H. Fragonard.
Wellcome Collection. Attribution 4.0 International (CC BY 4.0).

As for Borda, after retiring to his family estate during the Reign of Terror, he returned to Paris and resumed his work on the metric system. He died[74] after a long illness on February 20, 1799, at age 65, just months before the mètre des Archives, a platinum bar selected as the first prototype of the meter based on the results of the meridional survey effort that Borda coordinated, was placed in the National Archive.

The final years of Borda’s life, from 1796 to 1799, also witnessed the military rise of Napoléon Bonaparte (1769–1821), which in turn signaled the end of the revolutionary period in France. In the next section, we return to the history of the Academy of Sciences, where Napoléon exerted his influence as well.


[50] The formation of the National Assembly was instigated by delegates of the Third Estate who were summoned to Versailles by Louis XVI for an emergency meeting of the General Estates, a parliamentary entity that was convened (for the first time since 1614) with the express purpose of securing the necessary approval to enact his government’s tax proposals. Traditionally, each of the three Estates voted as a single group, despite the fact that the Third Estate represented a far greater proportion of the population than either the First or Second Estates, which represented the clergy and the aristocracy respectively. Although members of the Third Estate are generally referred to as “commoners,” its representatives at the Estates General predominantly belonged to the bourgeoisie, particularly the legal professions. After their demand that each deputy instead be given a vote was overruled by a declaration from the First Estate, the representatives of the Third Estate constituted itself as a National Assembly on June 17, 1789; they were later joined in their commitment to create a constitutional monarchy by members of the other two estates.

[51] Like the Declaration of Independence of the United States of America, written some 13 years earlier, primarily by Thomas Jefferson, the Declaration of the Rights of Man and of the Citizen clearly embodies Enlightenment ideals. It begins with a preamble that describes the Assembly’s goal in setting forth a set of “simple and incontestable principles” as an effort to combat the “ignorance, forgetfulness or contempt of the rights of man [that are] the only causes of public misfortunes and the corruption of Governments.” By serving as a unceasing reminder to “all members of the body politic . . . of their rights and their duties,” its authors asserted the desire that “the acts of the legislative power and those of the executive power . . . may . . . be the more respected” and that “the demands of the citizens . . . may always be directed toward the maintenance of the Constitution and the happiness of all.” Jefferson, who was then living in Paris as the US ambassador to France, served as a consultant to those authors, Emmanuel-Joseph Sieyès (1748–1836) and the Marquis de La Fayette (1737–1834), the latter of whom had served as a military consultant to the US during its war of independence from Britain. Unlike the US document, which served in a sense as a rationale for that country’s declaration of war aimed at ousting an oppressor, the French document was initially intended by its authors to serve as a rationale for a transition from an absolute to a constitutional monarchy, rather than an attempt to overthrow the monarchy per se.

[52] The seventeen articles of the Declaration were individually considered and approved by the National Assembly between August 20 and August 26, 1789.

[53] The preamble of the current French constitution still asserts the constitutional status of the Declaration of the Rights of Man and of the Citizen, by declaring that “The French people solemnly proclaim their attachment to the Rights of Man and the principles of national sovereignty as defined by the Declaration of 1789 . . .” [Constitution of the Fifth Republic of France, adopted October 4, 1958].

[54] These complaints were quite literally registered in the Cahiers de Doléances (Complaints Books) that were filed with King Louis XVI (at his invitation) by individuals, trade guilds and legislative units across France in preparation for the 1789 meeting of the General Estates. For example, of the 50 topics found among the complaints submitted by the Third Estate, weights and measures ranked fourteenth [Shapiro et al. 1998, p. 381]. Other frequently mentioned topics of complaint included personal liberties, taxes and the judiciary system.

[55] Condorcet was influential in convincing the National Assembly to approve funding for the Academy in connection with metric reform, although the proposal itself was put before the Assembly by Bishop Charles Maurice de Talleyrand, a former member of the First Estate who aligned himself with the revolutionary cause early on.

[56]According to both [Mascart 1919, p. 497] and [Heilbron 1990, p. 224], the other three members of the Commission on Weights and Measures were initially Jean-Louis Lagrange (1736–1813), the chemist Antoine Lavoisier (1743–1794) and the botanist Mathieu Tillet (1714–1791). After rendering an initial report to the Assembly on October 27, 1790, in which a decimal system was recommended, Lavoisier and Tillet were replaced by Pierre-Simon Laplace (1749–1827) and Gaspard Monge (1746–1818).

[57] The assigned tasks and original working group members were: triangulations and latitude determinations (Cassini IV, Méchain, Legendre), baselines (Meusnier, Monge), pendulum of Paris (Borda, Coulomb), weight of water (Lavoisier, Haüy), and comparison of old and new measures (Tillet, Brisson, Vandermonde). Responsibility for overall supervision of the endeavor was assigned to Borda, Condorcet, Lagrange, and Lavoisier. Two of these four (Condorcet and Lavoisier) died during the Reign of Terror that took place from July 1793 to September 1794. Several members of the original working groups also left the project due to “death and disinclination” [Heilbron 1990, p. 224]. See also [Mascart 1919, pp. 497–500].

[58] Heilbron [1989, p. 991] reports that estimates of the actual amount spent by the Academy to complete the survey range from 300,000 to millions of livres. An additional 500,000 livres in funding was also granted to the Committee of Public Instruction by the National Assembly in order “to defray all expenses relative to the establishment of the new measures, as well as the advances which are indispensable for the success of such work,” as stipulated in Article 21 of a National Assembly decree approved on April 7, 1795.

[59] Some have suggested that the Commission’s decision not to instead define the meter as the length of a pendulum with a half-period of one second (called a seconds pendulum) was due to an interest within the Academy in seeing a demonstration of the accuracy of Borda’s repeating circle. See, for instance, [Hahn 1971, p. 164].

[60] The full text of the legislative acts related to the official adoption of the metric system in France can be found in a report on the question of adopting the metric system in the United Kingdom that was later commissioned by Queen Victoria [Airy et al. 1869].The French decree of August 1, 1793, that legally adopted the metric system as that nation’s official system of weights and measures also mandated its obligatory use beginning July 1, 1794. In a new decree approved on April 7, 1795, the latter provision was rescinded indefinitely, due in part to the reluctance of the public to abandon its use of the existing units of measurements. (That same decree also indefinitely suspended the 1793 provision requiring the use of the decimal division of the day and the parts thereof.) Historian of science Kenneth Alder has offered a convincing explanation for why there was such resistance to actually using the metric system, despite the fact that it was complaints from the French people themselves that helped bring the new system into existence [Alder 1995; Alder 2002]. His book, The Measure of all Things: The Seven Year Odyssey and Hidden Error that Transformed the World [Alder 2002], also provides a riveting account of the surveying project behind the definition of the meter.


Figure 7: Woodcut dated 1800 illustrating the new decimal units which became the legal norm in France on November 4, 1800, five years after the metrical system was first introduced.

In the captions, each one of these six new units are followed by the old French unit in brackets.These were for length (metre), area (are = 100 m2), solid volume (stère = 1 m3), liquid volume (litre = 1 dm3), mass (gramme = the mass of 1 cm3 of water) and currency (franc).


[61] In the original French: “Une bonne loi doit être bonne pour tous les hommes, comme une proposition vrai est vrai pour tous.” Condorcet wrote his “Observations de Condorcet sur le vingt-neuvième livre de "L'esprit des lois" (”Observations on the twenty ninth book of “Spirit of Laws”) in 1780 as a commentary on a highly influential political theory treatise by Charles de Secondat, Baron de Montesquieu (1689–1755), De l'esprit des loix (The Spirit of the Laws), originally published anonymously in 1748. The publication details of Condorcet’s essay are not fully clear. It was not included in the 1804 edition of Condorcet’s collected works, but does appear in Volume 1 of the 1847 edition [Condorcet 1847, pp. 363–388]. In the intervening years, Condorcet’s essay appeared in its entirety in Thomas Jefferson’s 1811 English translation of de Tracy's Commentaire sur l'Esprit des Lois de Montesquieu par M. le comte de Destutt de Tracy (A Commentary and Review of Montesquieu's Spirit of Laws) [de Tracy 1811], and in the later French editions of that same work [de Tracy 1817; de Tracy 1819]. Antoine Louis Claude Destutt, comte de Tracy (1754–1836) was a French philosopher who is credited with coining the term "ideology" while being held in prison during the Reign of Terror.

[62] Condorcet wrote several essays advocating the abolition of slavery, the first of which, “Reflexions sur l’esclavage des nègres” (“Reflections on the enslavement of negros”), was published well before the revolution, in 1781. He was also a founding member, along with the military commander La Fayette, of the Société des Amis des Noirs (Society of the Friends of Negros), established in 1788. His 1790 essay, “Sur l’admission des femme au droit de Cité(“On the admission of women to the right of Citizenship”), further argued the ability to reason, a human attribute shared by men and women of all races, justifies granting equal rights to all:

The rights of men stem exclusively from the fact that they are sentient beings, capable of acquiring moral ideas and of reasoning upon them. Since women have the same qualities, they necessarily also have the same rights. Either no member of the human race has any true rights, or else they all have the same ones; and anyone who votes against the rights of another, whatever his religion, colour or sex, automatically forfeits his own.

[63] The plan for state-supported education that Condorcet helped to draft formed the basis of the national educational system that was eventually adopted, which in turn led to the creation of the École normale supérieure and the École centrale des travaux publics (later called the École polytechnique).

[64] The Royal family had been essentially under house arrest at the Palace of the Tuileries in Paris since the so-called Women's March on Versailles took place on October 6, 1789, in part as a reaction to the chronic hunger and increasing frustration brought about by the costs and scarcity of bread in the city. Concerns about Louis XVI’s intentions became more pronounced following his family’s aborted attempt to flee the country during the night of June 20–21, 1791. Nevertheless, the National Constitutional Assembly was willing to allow Louis to retain the throne under a constitutional monarchy at that time. According to the nineteenth-century historian François Auguste Marie Mignet (1796–1884), after Louis XVI examined the Assembly’s 1791 constitution, he sent a letter to the Assembly in which he wrote: “I accept the constitution. I engage to maintain it at home, to defend it from all attacks from abroad; and to cause execution by all the means it places at my disposal. I declare, that being informed of the attachment of the great majority of the people to the constitution, I renounce my claim to assist in the work, and that being responsible to the nation alone, no other person, now that I have made this renunciation, has a right to complain” [Mignet 1846, p. 110]. The Assembly is then reported to have subsequently invited Louis XVI to address their final session on September 29, 1789, where the King pronounced: “Tell [your fellow citizens] that the king will always be their first and most faithful friend; that he needs their love; that he can only be happy with them and by their means; the hope of contributing to their happiness will sustain my courage, as the satisfaction of having succeeded will be my sweetest recompence” [Mignet 1846, pp. 110–111].

[65] Although opposed to the death penalty on principle, Condorcet did not vote against the death penalty for Louis XVI, but instead chose to abstain. The abolition of the death penalty was, however, one of the 370 articles in Condorcet’s proposed version of the new constitution. In his 1791 “Opinion sur le jugement de Louis XVI” (“Opinion on the trial of Louis XIV”), a copy of which was owned by Thomas Jeffereson, Condorcet also presented his probabilistic view of voting to argue against executing the King.

[66] While the politics of the French revolution are too complex to recount here, the description of Condorcet’s abilities (or lack thereof) as a politician offered by historian John Herivel suggest another possible factor in the failure of Condorcet’s constitution [Herivel 1975, p. 247]:

Condorcet was no politician. His uncompromising directness of manner and inability to suffer illogical windbags in silence made him many enemies and few friends. His weak voice, lack of oratorical powers, and tendency to bore the Convention by the excessive height of his arguments was one of the tragedies of the Revolution.

[67] The Committee of Public Safety was actually created in April 1793, to replace the previous Committee of General Defense, an administrative body that was charged with protecting the new republic from internal and external attacks. In July 1793, Robespierre (1758–1794) became a member of the restructured Committee of Public Safety and served as its de facto leader until his own arrest and subsequent death by guillotine a year later essentially brought the Reign of Terror to a close.

    Interestingly, the mathematician and military engineer Lazare Carnot (1753–1823) served as a member of the Committee of Public Safety from September 1793 until it was disbanded in March 1795. As a relatively conservative member of the committee, Carnot concentrated primarily on issues related to education and military operations. From 1795 to 1797, he continued to provide military leadership for the country as an elected member of the Directory, the five-member executive branch of the re-organized government that ruled France immediately following the dissolution of the Committee of Public Safety. As the Director in charge of military affairs, it was Carnot who appointed Napoléon Bonaparte (1769–1821) as general-in-chief of the Army of Italy. When Napoleon declared himself emperor of France in 1804, the republican-minded Carnot retired from public life. He lived out his final days in exile from France, devoting his energies to writing works on engineering and geometry.

[68] Estimates of the number of individuals that were executed in France during the Reign of Terror by the guillotine or other means range upwards of 17,000, excluding those who died at the hands of vigilantes or in prison while awaiting their trial. Lynn, who sets the number of official executions at 40,000, further estimates that the number of individuals who were arrested accounted for about 1 in every 50 French citizens, or approximately 300,000 individuals in all [Lynn n.d.]. Members of the clergy and aristocrats were disproportionately represented both among those arrested and those executed by the state.

[69]Located on Paris’ left bank on the Rue des Fossoyeurs (now 15 Rue Servandoni), Condorcet’s refuge for these months is situated just behind the beautiful Église Saint Étienne de Mont, where Blaise Pascal (1623–1662) is buried. Coincidentally, Olympe de Gouges (1745–1793), a playwright and author of the Declaration des Droites de Femmes et Citoyennes (Declaration of the Rights of Women and Citizenesses), lived on this same street, within half a block of Condorcet’s hiding place. Although it is unlikely that they associated with each other during this time, the two were almost certainly acquainted, given De Gouges’s attendance at the salon of Condorcet’s wife Sophie de Condorcet, née de Grouchy, at the Hôtel des Monnaies. Condorcet only met his protector Madame Vernet, the widow of the sculptor Louis-François Vernet (1744–1784), when mutual friends brought him to her home seeking refuge.

[70] Condorcet’s Esquisse was published posthumously by his wife, Sophie de Condorcet, née de Grouchy, in 1795. In 1799, she also published his Éloges des academicians (Eulogies for the Academicians). Although de Grouchy had filed for divorce while Condorcet was in hiding, she did so with his secret consent as a move to prevent the seizure of her property by the revolutionary government due to her marital connection to the fugitive Condorcet. This strategy failed when he died before the divorce was finalized. De Grouchy thereafter earned a living for herself and their daughter Eliza by painting miniature portraits. She also continued to edit Condorcet’s collected works, and remained active in France’s intellectual culture through her re-established salon. Although de Grouchy wrote many articles that were published anonymously, both before and after the revolution, the only one of her works to appear in her own name was the book Huit Lettres sur la Sympathie (The Letters on Sympathy), which was published alongside her translation of Adam Smith’s Theory of Moral Sentiment and Origins of Language in 1798.

[71] Located at 49 Grande Rue (now 81 Ave Le Clerc), the lower floor of the house in which Condorcet spent his last night was being used as a hair salon in 1999 when the author took the following photo (Figure 8).

[72] At the time, the town was called “Bourg-de-l’Égalité.” Bourg-la-Reine is also familiar to students of the history of mathematics as the birthplace of Évariste Galois (1811–1832), known both for his contributions to algebra and as a staunch supporter of the French republicans during the post-Napoléonic era.

[73] Although the location of his remains is also unknown to this day, memorials of Condorcet’s life and work include plaques at his final “residences” in Paris and Bourg-la-Reine, a statue at 15 Quai de Conti in Paris (between the Musée de Monnaie and the Institut du France), and the symbolic interment in the Pantheon in Paris of ashes taken (in 1989) from the cemetery in Bourg-la-Reine.

[74] Memorials of Borda’s life and work include a statue in the Place de la Halle in his birthplace of Dax, and a series of French naval vessels harboured in Brest (and nicknamed The Borda) that have successively served as the site of French Naval School since its founding in 1830. The current collection of scientific instruments and other objects in the Borda Museum in his hometown of Dax was founded on Borda’s own collection of curiosities. Borda’s name (but not Condorcet’s) is also listed, together with 71 other scientists, on the first floor of the Eiffel Tower. Both men have Paris streets named after them (Borda in the 3rd Arrondisement and Condorcet in the 9th).