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.

As noted in the introduction, the two works on the mathematics of voting theory written by Borda and Condorcet, respectively, were essentially forgotten until the twentieth century. In this section, we consider the purely mathematical aspects of their works. We first briefly comment on their relative publication dates.

At first glance, Borda’s work had priority, by more than a decade, over that of Condorcet. Borda’s “Mémoire sur les élections au scrutin” (“Memoir on elections by ballot”) appeared in the Annual Proceedings of the French Academy of Sciences for the year 1781, together with an “Analysis” of that memoir that was likely written by Condorcet (in an editorial capacity derived from his position as Permanent Secretary). In a footnote to the title of his paper, Borda further asserted that he had already presented the ideas it contained to the Academy some 14 years earlier, on June 16, 1770—a claim repeated by the author of the “Analysis” of Borda’s “Mémoire” (again, likely Condorcet himself), but for which no archival evidence has yet emerged.^[14] In fact, based on the available archival documentation, Brian [2008, p. 2] has shown that:

The date of 1781 is a fake. The story really begins the 14th of July of 1784, exactly five years before a date of some importance in the history of democracy.

It was on this “Bastille Day” date in 1784 that Condorcet, then Permanent Secretary of the Academy, announced his intention to publish a book on the application of probability to the analysis of electoral decisions. To do so under the auspices of the Imprimerie Royale, both a positive report by two referees and a vote of approval by the members of the Academy was required. The two referees were chosen by Condorcet, and the required report was read to a meeting of the Academy just three days later—such an astonishingly short turnaround time given the length and complexity of Condorcet’s tome that historians generally believe the report was written by Condorcet himself.^[15] Regardless of who actually wrote that report, it sufficed to secure the vote of approval necessary to go forward with publication. A possible delay then arose when Borda read his own paper on the arithmetic of voting to the Academy on July 21, 1784, apparently as a means to establish priority. In response to Borda’s paper, Condorcet took quick action to avoid a potential stumbling block to his own publication plans: using his authority as Permanent Secretary, he arranged for Borda’s paper to be included in the Academy’s proceedings that were then in preparation, as those would appear in print prior to the intended publication date for his own book. To quote Brian [2008, p. 3] once more:

This is why a paper, supposedly read in 1770, actually presented in 1784, was published in an official academic volume for the year 1781—the one under press in July 1784.

Because Condorcet’s Essai—actually written, supposedly reviewed by referees and already approved for publication by July 1784—was not officially published until 1785, Borda thus technically published on voting theory a year before Condorcet. To see why Condorcet’s work is nevertheless the more celebrated of the two as a pioneering contribution to voting theory, we now turn to a brief overview of the technical contents of both works, beginning with that of Borda.

One of the standard voting methods studied in an undergraduate treatment of voting theory is named in honor of Borda’s 9-page paper [Borda 1784]. Under the Borda Count Method of Voting, candidates are assigned points based on their rankings across all ballots; for instance, 1 point for each last place vote, 2 points for each second-to-last place vote, and so on. The candidate with the highest total number of points is then selected as the winner of the election. Borda himself called this voting method an “election by order of merit,” and presented the following as an example.^[16]

Suppose again 21 voters and three presented subjects A, B, C and let

A  A  A  A  A   A  A  A  B  B   B  B  B  B  B  C  C  C C  C  C
B  C  C  C  C  C  C  C  C  C  C  C  C  C C  B  B  B  B  B  B
C  B  B  B  B   B  B  B  A  A   A  A  A  A  A  A  A  A  A  A  A

be the 21 election ballots. One will have by what has been already said, the comparative value of votes by multiplying the first-place votes by 3, the second-place votes by 2, and the third by 1, which will give the following results.

Votes for A \(\left\{\begin{array}{c}8 \mbox{ first-place votes, multiplied by  } 3 = 24\\13 \mbox{ third-place votes, multiplied by  } 1 = 8\end{array}\right \}\) 37.

Votes for B \(\left\{\begin{array}{c}7 \mbox{ first-place votes, multiplied by  } 3 = 21,\\7 \mbox{ second-place votes, multiplied by  } 2 = 14\\7 \mbox{ third-place votes, multiplied by  } 1 = 7\end{array}\right \}\) 42.

Votes for C \(\left\{\begin{array}{c}6 \mbox{ first-place votes, multiplied by  } 3 = 6\\14 \mbox{ second-place votes, multiplied by  } 2 = 28\\1 \mbox{ third-place votes, multiplied by  } 1 = 1 \end{array}\right \}\) 47.

From which one can see the superiority of votes will be in favor of subject C, that the second place will be given to subject B & the last to subject A.^[17]

Borda continued:

It may be remarked that if one conducted the election in the usual manner, one would have had the following result:

8 votes for A,
7 votes for B,
6 votes for C,

That is to say that the plurality would have been for subject A, who is last in the opinions of the voters, & that the subject C, who is really the first, would have had fewer of the votes than each of the other two.^[18]

Here, we hear Borda’s primary complaint about the Plurality Method of Voting: if only the first-place votes are considered in deciding the results of an election, then the wrong candidate could end up winning the election. He first expressed this complaint (and illustrated it by way of this same example) at the very start of his paper in order to illustrate how the Plurality Method of Voting could “induce an error” in elections involving more than two candidates.

It’s clear then that the subject A will have, in the collective opinion of the voters, a marked inferiority, as much with respect to B as with respect to C, because each of these latter [two], compared to the subject A has 13 votes, while the subject A only has 8; from this it clearly follows that the voice of the electors would exclude subject A.^[19]

Having noted that a candidate who loses to every other candidate in head-to-head comparisons (in this example, candidate A) could end up winning the overall election under the Plurality Method of Voting,^[20] Borda provided the following analogy to describe how this situation could come about:

In reflecting on the reported example, one sees that the subject A only has the advantage in the results of the [plurality] election, because the two subjects B & C, who are superior to him, almost equally divided between them the votes of the 13 electors [who did not place A in first place]. One could compare them fairly exactly to two athletes, who, after having spent their forces one against the other, were then defeated by a third [athlete] more feeble than either of them.^[21]  

Having provided an example to illustrate how election by order of merit (or the Borda Count Method of Voting) could be used to avoid such an error in at least some elections, Borda next considered a second method of voting which assigns point totals in a different fashion.

Let us suppose now that one wishes to use the method of special elections,^[22] & that there are once more three subjects A, B, C presented; as one can combine these three subjects taken two by two in three different manners, it would be necessary to have three special elections. Let the results of these elections be as follows.

1.^st election between A & B . . . \(\left\{\begin{array}{l}a \mbox{ votes for }A,\\b \mbox{ votes for } B,\end{array}\right .\)

2.^nd election between A & C . . .  \(\left\{\begin{array}{l}a' \mbox{ votes for }A,\\c \mbox{ votes for } C,\end{array}\right .\)

3.^rd election between B & C . . . \(\left\{\begin{array}{l}b' \mbox{ votes for }A,\\c' \mbox{ votes for } C,\end{array}\right .\)  . . . . ^[23]

Borda next discussed in general how the values of a,a',b,b',c,c' relate to the point totals of the candidates in an “election by order of merit,” showing that candidates A,B,C will have Borda count totals of a +a' + E, b + b' + E, c + c'+ E points respectively, where E is the number of voters casting ballots in the election. He then illustrated this idea by applying the “method of special elections” to his earlier example:

If one determines the values of a,a',b,b',c,c' under the supposition that the special elections are the result of the election by order of merit that reported above, one would find^[24]

 a = 8, b  = 13, c  = 13,
 a' = 8, b'  = 8, c  = 13,

            & consequently, one will have

the votes for A where a + a' = 16,
the votes for B where b + b' = 21,
the votes for C where c + c'; = 26,

which gives between the three votes the same differences that would have been found by the first type of election.^[25]

In short, Borda algebraically proved, and then illustrated via an example, that the two methods of voting he presented are equivalent: not only do they lead to the same election results, but the point totals assigned to the candidates by each method are readily converted to those assigned by the other method.^[26] Having established their equivalence, Borda further commented:

Besides [the fact that these two methods produce the same results], we will remark here that the second form of election of which we have spoken, would be awkward in practice, when a large number of candidates are presented, because the number of particular elections that it would be necessary to complete, will be very large. For this reason one should prefer the form of election by order of merit, which is much more expedient.^[27]

In the next section of this article, we consider how Condorcet made use of pairwise comparisons, despite their “awkwardness,” in a considerably different fashion than Borda proposed.

Borda described the purpose of his brief “Memoire” simply as an illustration that, in the case of elections with three or more candidates, the common belief that a plurality of (first-place) votes “always indicates the wish of the electorate” is mistaken [Borda 1784, p. 657]. In contrast, Condorcet began his 1785 book with a 191-page Discours Préliminaire,^[28] or Preface, in which he laid out the motivation behind and the general layout of his 304-page Essai. Voting theory itself occupied only a small part of Condorcet’s massive (495-page!) book [Condorcet 1785]. Nevertheless, his name is also connected to standard topics within an undergraduate treatment of voting theory. In fact, three related concepts are named in his honor: Condorcet Paradox, Condorcet Candidate and Condorcet Fairness Criterion. These ideas are related to a voting method that involves head-to-head comparisons of each pair of candidates, a procedure that Borda also considered in his treatment of “special elections.” In this and the next two sections, we examine excerpts from the Preface to Condorcet’s Essai that justify the attachment of his name (and not Borda’s) to these ideas.

Condorcet’s analysis of elections involving more than two candidates appeared relatively late in the Preface of his Essai, following several examples related to jury trials and legislative voting that illustrated his concerns about the Plurality Method of Voting.^[29] His reasons for seeking an alternative to the Plurality Method of Voting were, in fact, quite similar to those presented by Borda:

The method used in ordinary elections is defective. In effect, each voter is limited to naming the one that he prefers: thus in the example of three Candidates, someone that votes for A does not announce his view on the preference between B & C, and similarly for the others. However, there may result from this manner of voting a decision that is actually contrary to the plurality [choice].^[30]

Like Borda, Condorcet also illustrated how a plurality result could be “actually contrary to the plurality [choice]” with a specific example:

Suppose, for example, 60 Voters, of whom 23 are in favor of A, 19 are in favor of B, & 18 are in favor of C; suppose next that the 23 voters for A would have unanimously decided that C is better than B; that the 19 Voters for B would have decided that C is better than A; and finally that of the 18 Voters for C, 16 would have decided that B is better than A, & only 2 that A is better than B.

One would have therefore, 1° 35 votes for the proposition B is better than A, & 25 for the contradictory proposition.^[31]

2° 37 votes for the proposition C is better than A, & 23 for the contradictory proposition.

3° 41 votes for the proposition C is better than B, & 19 for the contradictory proposition.

We would therefore have the system of the three propositions that have the plurality, formed of the three propositions.

B is better than A,
C is better than A,
C is better than B,

            which implies a vote in favor of C.^[32]

In other words, A would win the election under the Plurality Method of Voting, since A has more first-place votes than the other two candidates, but in reality C should win, since C is preferred in every head-to-head comparison against the other candidates. Today, such a candidate is called a Condorcet Candidate, and the assertion that a Condorcet Candidate (when one exists) should win the election is known as the Condorcet Fairness Criterion.

Either of the two examples that we have shared so far (from Borda and Condorcet respectively) suffice to show that the Plurality Method of Voting violates the Condorcet Fairness Criterion: in both cases, there is a Condorcet Candidate who loses the election under the Plurality Method of Voting.^[33] Like Borda, Condorcet also used his example (the first of several) to set the stage for proposing an alternative to the Plurality Method of Voting. He continued his discussion of this particular example as follows:

What’s more, we would have two propositions that form a vote in favor of C.

C is better than A,
C is better than B,

one decided by a plurality of 37 against 23, the other by a plurality of 41 against 19.

The two propositions that form a vote in favor of B,

B is better than A,
B is better than C,

one decided by a plurality of 35 against 25, the other by a minority of 19 against 41.

Finally the two propositions that form a vote in favor of A.

A is better than B,
A is better than C,

decided by a minority, the one of 25 against 35, the other of 23 against 37.

So the one among these Candidates who would really have the preference of the plurality, would be precisely the one that, in following the ordinary method, would have the least numbers of votes.

Thus it is that A who, following the ordinary form, would have the most votes, is to the contrary found in reality to be the one who to be the furthest from being the wish of a plurality [of the voters].^[34]

Condorcet’s move to a consideration of propositions (rather than simply the head-to-head vote counts themselves) may seem an unnecessary complication,^[35] until one recalls the title of his text: Essay on the Application of the Analysis of Probabilities to Decisions Rendered by a Plurality of Votes. As previously noted, this work was part of a broader investigation into the problem of how individuals within a society could be provided with sufficient assurance that the decisions rendered by groups of individuals (e.g., trial juries, legislators, voters) are, in fact, correct. Naturally, the appraisals of these individuals could be mistaken concerning the issue at hand, even in the best of circumstances.^[36] By considering the likelihood that each individual vote is correct, Condorcet was able to use the relatively new mathematical field of probability^[37] as a tool for analyzing collective decision-making procedures under various conditions (e.g., the number of jurors or voters, the competence of the individual decision makers), with the goal of maximizing the probability that the final outcome is correct, or true.^[38] In the case of elections, formulating ballot results as propositions (each of which has a truth value), allowed Condorcet to consider the probability that a given proposition (e.g., A is better than candidate B) is actually true, as well as the probability that a compound system of such propositions is true. Condorcet was especially interested in the probabilities of the compound systems, as these represented an aggregate opinion of all voters from which the election winner could be chosen. In the previous example, for instance, the opinion “B is better than A; C is better than A; C is better than B” is the one with the highest probability of being correct, which in turn implies that C is the candidate who should be elected.

Since the connection to probability theory is not part of a standard undergraduate treatment of voting theory, we will not examine the details of how probability did (and did not) enter in Condorcet’s analysis of elections.^[39] Instead, we turn to the conclusions that Condorcet drew from the example presented above, which are relevant to share with students.

Having given a particular example to exhibit the inherent flaw that can result in election decisions based on the Plurality Method of Voting, Condorcet next noted that there is one circumstance in which the Plurality Method of Voting could suffice to select the correct choice among candidates, namely when there is a candidate with a majority of the first place votes:

One sees therefore already that one should reject the form of election generally adopted: if one wishes to conserve it, it will be possible [to do so] only in the case where one is not required to elect on the spot, & where one could require that only someone who would have gathered more than half of the votes would be looked upon as elected. In this same case, this form [of voting] would again have the disadvantage of agreeing to look upon as non-elected the one who would have in reality has a very great plurality.^[40]

Today, a candidate with a majority of the first-place votes is known as a Majority Candidate, and it is easy to see that a Majority Candidate is also necessarily a Condorcet Candidate. Similarly, it is straightforward to show that a Majority Candidate (if one exists) will necessarily win an election under both the Plurality and Borda Count Methods of Voting. This latter voting method was not, however, the one endorsed by Condorcet as an alternative to the former. Instead, he declared:

So one should in general substitute for this [plurality] form [of voting] that in which each Voter, [by] expressing the order according to which he places the Candidates, would pronounce at the same time on the preference that he would accord respectively to each.^[41]

In other words, election decisions should be based on the outcomes of the head-to-head comparisons of every pair of candidates, which in turn can readily be determined from voters’ ranked orderings of the candidates. Importantly, Condorcet’s use of head-to-head comparisons differed from the way such comparisons featured in Borda’s “special elections” method. In that method, recall that Borda summed the numbers of votes cast for each candidate in all of the various pairs in a way that rendered that method equivalent to the basic Borda Count Method of Voting. Given this equivalency together with the dramatic increase that occurs in the number of pairs as the number of candidates increases, Borda ultimately rejected using head-to-head comparisons as an avoidable inconvenience. In contrast, Condorcet’s plan tallies the actual number of votes cast for each candidate in the pair only to determine which of the two candidates in the pair wins that particular comparison: did the voters find that A is better than B, or that B is better than A? Once this is decided, the vote totals themselves are discarded.

Unlike Borda, Condorcet also seemed undaunted by the number of head-to-head comparisons that may need to be tallied.^[42] Nor was he daunted by a feature of his recommended voting method that is today known as Condorcet Paradox: the final results of the head-to-head comparisons can be non-transitive. Here is Condorcet’s first example of this situation.

Let us suppose indeed that in the example already chosen, where one has 23 votes for A, 19 for B, 18 for C, the 23 votes for A are for the proposition B is better than C; this proposition [B is better than C] will have a plurality of 42 votes against 18.

Let us next suppose that of the 19 votes in favor of B, there are 17 for B is better than C, & 2 for the contradictory proposition; this proposition C is better than A will have a plurality of 35 votes against 25. We suppose finally that of the 18 votes for C, 10 are for the proposition A is better than B, & 8 for the contradictory proposition, we will have a plurality of 33 votes against 27 in favor of the proposition A is better than B. The system that obtains the plurality will therefore be composed of three propositions,

A is better than B,
C is better than A,
B is better than C.

            This system is . . . one of the two [possible systems] that implies contradiction.^[43]

For many students (and others), the fact that individual transitive preferences (an assumption implicit in the use of rank orderings) can lead to a intransitive group decision (with A > B, B > C, but C > A) is indeed a paradoxical aspect of the mathematics of social choice. How can the group outcome be so irrational when each individual within the group has voted rationally? As noted earlier, however, Condorcet himself was not fazed by this fact—indeed, he did not even view it as an impediment to the viability of the Pairwise Comparison Method of Voting. Instead, he devoted several pages of his Essai to a discussion of how to proceed with voting in order to produce the winner with the highest probability of being the correct choice in the event that election results do result in one of these “contradictory systems” [Condorcet 1785, pp. clxx–clxxvii; pp. 118–136]. While his full description of this somewhat complicated procedure in Part 2 of his Essai was not completely clear,^[44] Condorcet’s concluding remarks about contradictory systems in his Preface provide a glimpse of the basic idea behind his procedure, both in its technical aspects and its societal implications:

Finally, let us observe that these contradictory systems cannot arise without indicating uncertainty in [the] opinions [of the voters], and they will not take place, neither if the votes being taken as usual, one of the Candidates has more than half of the votes, nor if one requires in order to admit the propositions which form the ballot, a plurality of one third.

There results from all the reflections that we have just made, this general rule, that whenever we are forced to elect, it is necessary to take successively all the propositions which have the plurality, beginning with those which have the largest, and pronounce according to the result formed by these first propositions, as soon as they form one, without having regard to the propositions less likely to follow them.

If by this means we disagree with the result least susceptible to error, or a result whose probability is greater than 1/2, and formed from two propositions which are more probable than their contradictories, we will at least have the one which does not oblige [us] to adopt the least probable proposals, and from which it results a lesser injustice between the Candidates, considered two by two.^[45]

Today’s standard undergraduate treatment of voting theory features a simplified version of Condorcet’s voting procedure that drops his provision for handling cases in which Condorcet Paradox occurs. Known as the Pairwise Comparison Method of Voting, this method simply uses the number of comparisons won by each candidate to determine the overall election winner.

As Condorcet noted, the Pairwise Comparison Method of Voting satisfies both the Majority and Condorcet Fairness Criteria. Of course, the Borda Count Method of Voting (Borda’s proposed alternative to the Plurality Method of Voting) also selects the Condorcet and  Majority Candidates as the overall winner in at least some cases (e.g., the two examples that we have seen thus far), and, because it is based on integer point totals (which always fall in transitive order), it avoids both the non-transitivity that can occur with Condorcet’s method, as well as the potentially tedious computations of pairwise comparison. Condorcet, having arranged for the publication of his paper, was well aware of Borda’s suggested alternative, which he described as follows:

A celebrated Geometer, who has observed before us the drawback of ordinary elections, has proposed a method, which consists of having each Voter give the order in which he ranks the candidates; of then giving to each first-place vote, the value of unity, for example; to each second-place vote a value less than unity; a value still smaller for each third-place vote, & so on, & of then choosing the candidate for whom the sum of these values, taken across all the voters, will be the largest.^[46]     

This method has the advantage of being very simple, & one can without doubt, by setting the law of decreases for these values,^[47] avoid many of the difficulties that the ordinary method has, of giving as the decision of the plurality one that is really contrary to it: but this method is not strictly protected from that drawback.

Although Condorcet did not say anything further about how “setting the law of decreases for these values” could lead to different election outcomes for the same set of preference ballots, his suggestion that the point values we assign for each place could change the election winner is one that can (and should) be explored by students. Also highly suitable for exploration by students is the election example that Condorcet gave in which the Condorcet Candidate loses the election under the Borda Count Method of Voting.  ^[48]

Having thus noted that Borda’s proposed method shares certain defects of the Plurality Method of Voting (e.g., violation of the Condorcet Fairness Criterion[^49]), in addition to certain drawbacks of its own (e.g., susceptibility to manipulation of election outcomes by varying point assignments), Condorcet concluded his discussion on Borda’s method with remarks that might be taken as politely-veiled insults—never actually naming Borda, but making it clear to their fellow Academicians who this “celebrated Geometer” was! Any response that Borda may have made in return (assuming he was even aware of Condorcet’s remarks, hidden away as they were in his voluminous text) was either never recorded or has not yet been found in written records. In the next section, we consider the turn of events in France that almost certainly distracted both men from further discussion of the mathematics of voting theory.

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.

And what of the two methods of voting that were respectively championed by Borda and Condorcet in their pre-revolutionary works?

Clearly, elections of all kinds were being held in France during the Revolution. And, within those elections, the use of ballots to select from a slate of three or more candidates was not unknown. For example, delegates to the First National Convention were chosen by some local assemblies through a two-stage balloting process similar to what is known today as the Plurality with Elimination Method of Voting: if a candidate received a majority of the votes in the first ballot, that individual served as the elected delegate; otherwise, the top two candidates in the first round of voting were placed on a second ballot and the winner of that round became the elected delegate.^[75] In other local assemblies, however, voting by voice (rather than ballot) served as an expedient way to select delegates that avoided counting ballots and was generally justified by a revolutionary-spirited appeal to its openness [Crook 1996, p. 88].

The use of ranked preference ballots that would require a more elaborate algorithm for determining the election outcome, such as those proposed by Borda and Condorcet, thus did not find its way into popular political elections. Not even in his rejected draft Constitution of 1792, which included a complicated electoral system for delegates to the Convention, did Condorcet mention the Pairwise Comparison Method of Voting that he concluded was the most likely to produce correct results in his 1785 Essai. Importantly for the later rediscovery of their works, however, the discussion of voting theory as it was developed by Borda and Condorcet did not die with the publication of their works in the 1780s. Within the Academy of Sciences, the outcomes of its various elections—which in turn depended on the method of election used—continued to have important consequences, both for the individual members of the Academy and the scientific community in general. Within this context, the voting methods proposed by both Borda and Condorcet did garner attention, both during and in the immediate aftermath of the French Revolution. Indeed, the Academy served as “a living laboratory where[in] to test, in the small, the virtues and the vices of different voting rules” [Barberà et al. 2020, p. 2].

Interestingly, the “voting laboratory” of the Academy of Sciences adopted the Borda Count Method of Voting for membership elections beginning in 1795, at the recommendation of author, historian and French statesman Pierre Daunou (1761–1840). In fact, this was the method in use when Napoléon Bonaparte^[76] was granted membership in the National Institute^[77] that replaced all Royal Academies between 1795 and 1815. Napoléon also served for a brief time as the Institute’s president, during which time his sole intervention in its running appears to have been his March 1800 request for a review of the balloting system—the same system by which he himself had been elected. The requested review was subsequently conducted by Daunou—the same person whose recommendation of Borda’s method led to its 1795 adoption.

In Daunou’s report on elections by ballot (presented in July 1801 and published in 1803), he reversed directions and expressed serious reservations about the point system proposed by Borda. In fact, his report was quite critical not only of the method itself, but also of Borda’s presentation of it in his 1784 “Memoire.”^[78] After recounting Condorcet’s observations on the defects of Borda’s proposed method, Daunou wrote:

But it suffices to attentively read the memoir of Borda himself, to be convinced that his method is based on inaccurate, incomplete observations, which lead to very false results.^[79]

He also offered the following comparison of Borda’s method with the method proposed by Condorcet.

Indeed, the results of this addition with regard to two subjects are variable according to the number of their other competitors, and according to the various ranks that these others obtain before, after or between the . . . two [given] subjects; while the comparisons between two candidates, or the preference given to one over the other, is, in the thought of each elector as in the general thought, a simple, constant, determined comparison, independent of all other relations.^[80]

While Daunou’s representation of Condorcet’s Pairwise Comparison Method of Voting must certainly have appealed to its members’ quest for uniformity, constancy and rationality, the Institute nevertheless continued to use the Borda Count Method of Voting for some (but not all) types of elections. The fact that it began to use a variety of methods in its elections does, however, show that one of Daunou’s recommendations was heeded; namely, that different voting methods are more or less suitable for different types of elections. This latter point is one that is well worth raising with students, alongside a discussion of the advantages and disadvantages of the various methods of voting. In the next (and final) section of this article, we consider how the eighteenth-century French connection to voting theory might be used to promote student understanding of and interest in these and other issues related to collective decision making.

In this article, we have attempted to show that the temporarily “lost” texts written by Borda and Condorcet in the 1780s merit the attachment of their names to certain ideas that are part of today’s mathematical treatment of voting theory. Certainly, the technical content of their works shows that the two men did indeed study and care deeply about the concepts that are now attributed to them. The historical and institutional contexts in which they wrote further help to explain why the other pre-Arrow discoveries of voting theory (by Lull and Cusa in the medieval period and Dodgson in the nineteenth century) remain relatively unknown even today. In the brief historical remarks added to the 1963 second edition of Social Choice and Individual Values, Arrow himself confessed to a “certain want of diligence in tracking down the historical origins of social choice” [Arrow 1963, p. 93]. He then applauded the “excellent history” of the topic given in The Theory of Committee and Elections [Black 1958], “beginning with the work of Borda, and including that of Condorcet, Laplace, Nanson, Galton and, most especially C.L. Dodgson (Lewis Carroll)” [Arrow 1963, p. 94]. But Black was really the first to take note of Dogdson’s work, and Lull’s and Cusa’s ideas were re-discovered even later. For all three of these authors, readership of their works at the time they were written was quite limited. In contrast, the ongoing and well-documented discussions of election methods within the French Academy of Sciences were bound to be noticed once Arrow’s Impossibility Theorem was announced in 1951, as indeed they were.

Our second objective in this article has been to share the story of Borda’s and Condorcet’s treatment of issues related to elections involving more than three candidates with instructors who teach this topic, but are unfamiliar with why their names came to be associated with them. We have included details of the non-mathematical background to their works because of the rich social and cultural connections that could be made by bringing the broader historical context into the classroom. The lives and ambitions of our two authors further offer students a wonderful window through which to glimpse the human side of mathematics. Especially within a high school setting, the voting theory works of Borda and Condorcet could also serve as a unifying theme for an interdisciplinary unit on the French Enlightenment and Revolutionary period.

As we hope has been made clear, the nature and content of Borda’s and Condorcet’s original writings further offer an excellent vehicle for teaching students the technical aspects of voting theory themselves. The student project “The French Connection: Borda, Condorcet and the Mathematics of Voting Theory” (pdf)  interweaves excerpts from these two primary sources with exercises intended to engage students with the mathematical ideas contained in those excerpts. Designed to be completed via a combination of individual advance reading, small-group work and whole-class discussion over the course of approximately two weeks, this project contains all the content from a standard textbook treatment of this topic, including the Plurality, Plurality with Elimination, Borda Count and Pairwise Comparison Methods of Voting; the Majority, Condorcet, Independence of Irrelevant Alternatives and Monotonicity Fairness Criteria^[82]; and the use of a Preference Schedule as a means to organize voter ballots. By drawing on Condorcet’s rich discussion of his own motivations for studying the problem of collective decision making, the project also goes beyond a standard textbook treatment in terms of its investigation of why Arrow’s Impossibility Theorem, and voting more generally, matters to their own lives.

We close by quoting the final paragraph from the student project, inspired by the writing of Condorcet, and offered as advice to students and instructors alike:

Voting is a privilege, one that is enjoyed by too few people in today’s world, and which can be too easily lost. Those of us who enjoy this privilege also bear the responsibility of helping to fulfill Condorcet’s vision of enlightened voters committed to the common good of society-at-large. Be sure to do your bit by becoming educated about the issues, identifying and setting aside your prejudices (we all have them, after all), and, most importantly, getting out there to vote.

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Janet Heine Barnett joined the faculty at Colorado State University – Pueblo in 1990, and now holds the rank of Professor Emerita of Mathematics. A 1995–1996 fellow at the MAA Institute for History of Mathematics and Its Use in Teaching (funded by the NSF), her scholarly interests have long included the history of mathematics and its use in promoting mathematical understanding and as a vehicle for promoting teacher reflection on pedagogical issues. She currently serves as a PI on the NSF-funded project TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS), and has now written twenty Primary Source Projects through that grant and its parent and grandparent grants. She also currently serves as an editor of Convergence, with Amy Ackerberg-Hastings.

An active member of the MAA Rocky Mountain Section, Janet received the Section Certificate of Meritorious Service Award in 2007. She has also been recognized at the University level, regionally and nationally for her excellence in teaching, including the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics in 2016. In 2019, she was awarded the President’s Medallion for Distinguished Service to Education by Colorado State University – Pueblo.

Janet shares her passions for mathematics and history, as well as dance and yoga, with her husband and traveling companion George Heine, whom she met while serving as a Peace Corps volunteer in the Central African Republic (1982–1984).