Borda´s vote method was used by the French Academy from 1796 to 1803 when Napoleon got it abolished. In the laws and regulations of the Academy there are stipulations regarding the election of new members. The practice is demonstrated by some examples from the protocols. In fact, Borda presented two vote methods which he showed would give the same conclusion. The method which has not been in the limelight proves to be the same one as is used for selecting the best player in chess tournaments.
A historical note
When one is only allowed to vote for a single proposal, which has hitherto been the usual practice, those who share similar views, if each puts forward his own proposal, risk being defeated by those who have contrary views and cooperate. By Borda´s point procedure, however, no risk is involved in presenting related proposals, because then it is possible to rank them jointly above a contrary proposal, in which case partners do not have the above-mentioned reason for not presenting related proposals. Special arrangements might then have to be made to limit the number of proposals, so that their number does not cause trouble, but Borda did not discuss whether the right of proposal must be limited in his statement on voting by a point settlement (Borda 1784).
In the laws and regulations of the French Academy there is a provision on the election of new members. There one can find the only known case during the first two centuries after Borda´s method was published by the Academy in the year 1784, where his method was applied. The stipulations were in force from 1796 to 1803 (L`institut de France 1889: 20-4).
At that time the total number of members was 144, divided in three classes: there were 60 members in the first class (science), 36 in the second class (moral and political science) and 48 in the third class (literature and fine art). The classes were divided into academic sections of six members. Borda was a member of the mathematics section of the first class from 1768 till he died in 1799. The stipulations regarding the election of new members were, as follows: When a vacant seat was to be occupied the section concerned should present a list to the class with the names of at least five people. At a class meeting it was possible to add names to the list by at least two members proposing a name. The class meeting determined by a special vote whether each individual name could be added to the list. Then the class meeting voted on the list by means of Borda´s point procedure. The names that received the three highest point totals (the names could be more than three, if the point totals were equal) were put on a list that was presented to an Academy meeting. There they were voted on by means of Borda´s point procedure. If votes were equal, the names that were equal should be voted on a month later by means of a point settlement.—There were provisions both stipulating a minimum attendance at meetings in order that voting could take place and regarding the deadline for elections.
In the protocols of the Academy it is generally not possible to find the candidate lists of sections and classes or point totals resulting from class votings, but the point totals of the names that were voted on at its meetings are registered there. I will give one example from the meeting of nivose 5 in the year 5 according to the calendar of the revolution (25 December 1796). There were 92 present at the meeting. Three candidates in astronomy received 223, 204 and 123 points respectively (total 550). At another meeting the point totals were 317, 292 and 232 (total 841) for a seat in the classical section, 237, 219 and 186 (total 642) for a seat in the mathematics section, and 185, 170 and 124 (total 479) for a seat in the anatomy section. In this case the participation is remarkably variable but a full attendance of the 144 members on the electoral register produces 864 points with the point settlement 3+2+1 per list.
There were special rules about the making of a list of names at the election of foreign members. For that purpose there was a special committee for each section, with one representative from each section of the class. At the last-mentioned Academy meeting the point totals of the lists of names of foreign members in literature were 136, 121 and 117 (total 374) and in linguistics 133, 132 and 97 (total 362).
Furthermore, there were special provisions for the election of those persons whom the Academy, according to law on the organization of public education, was to appoint to be responsible for agricultural research. At an Academy meeting a list with three times more names than the vacant seats was to be presented. For setting up the list there was a special committee with a representative of each section of the first two classes, but the meeting voted on the names by means of Borda´s point procedure.
"The Citizen" Napoleon was introduced at an Academy meeting on frimaire 5 in the year 6 (25 November 1797) because of class 1 (mathematics and physics) together with two other candidates in the mechanics section, Napoleon receiving 471 points at the class election, and the others 371 and 367 points. The figures show that no fewer than 8 names have been on the list of the class. A month later, on nivose 5 in the year 6 (on Christmas day), he was elected by 305 points, while the others received 166 and 123 points respectively, which makes a total of 624 points, according to the protocol (594 is, in fact, the correct sum). There must be an error there, unless this simply proves that Napoleon has managed to do what no one else has been able to do yet; the top candidate of three cannot, in fact, gain more points than the others in all, even if he is at the top on all the lists.—That autumn the French had conquered countries, such as Belgium, under the leadership of Napoleon, which had been ratified by a peace treaty on 17 October.
Mascart (1919: 133-4) considers Napoleon has caused Borda´s method to be abandoned and it seems to him that the reasons are mysterious. Bonaparte became president of the first class of germinal 1 in the year 8 according to the calendar of the revolution, i.e. 22 March 1800, thirteen months after Borda´s death, but three months earlier he had become president of the state council. Five days later, when he became class president, he criticized with unknown arguments very much the method that was applied in the election of members and proposed it to be altered. His proposal was discussed by the class for a long time, and finally a committee was formed by the three classes to deal with the issue. The committee presented a proposal that harmonized with Napoleon´s views at a meeting on the 25 April. The issue was concluded with the reorganization of the Academy in four classes, and new election rules that took effect on the 19 of March 1803. At that time the French Institute published an article, where Borda´s method was criticized (Daunou 1803). The author criticizes Borda´s method mainly for not being emphatic enough, as it is not clear whether A is considered thousand times better than B and B only a little bit better than C. Secondly, he adopted Condorcet´s views on majority rule. He made more critical points, but this is no place to argue them. The arguments of the issue and such criticism must have been discussed thoroughly ever since Borda presented his theory for the first time in 1770 (Borda: 1784, 657). Borda presents, among other things, a different view from that of Condorcet in his essay and shows (pp. 664-5) with a formula how those who, by traditional voting, are in the majority can be put in a difficult position by his point procedure.
A monarch had expressed his view. That view was bound to become law.
Borda´s argumentation is remarkable. He presents two methods and proves that they in fact give the same conclusion, the only difference being a matter of presentation. The one method is the one that is generally referred to and according to which points are given in a series, the lowest name gaining one point, the next name one point more, and so forth. Borda argued that the conclusion does not depend on which number is assigned to the candidate that a voter ranks the lowest. The only necessary condition is that there be an equal numerical distance between the ranked candidates. The other method, he said, is to separate the alternatives (the candidates) in pairs and let all meet in this way. The point total is, in fact, not the same as when the lowest candidate in an order of preference gains points, which does not change the order, as he demonstrates. Borda could well have pointed out that by giving the lowest ranked name 0 the methods would not only have produced the same order, but also the same point total. Behind the latter method lies, in fact, the thought which I have dealt with, with reference to a chess tournament where everybody plays against everybody (Stefansson and ).
Borda points out that the first method is easier in practice. (Certainly, computer processing of the votes reduces this difference). It is not known that scholars have paid attention to the other method. Two figure errors on p. 663 may confuse those who would scrutinize the original text, where Borda demonstrates that both methods give the same conclusion. In the upper table there stands b` = 13 instead of 8, and in the lower table there stands `les suffrages de B ou b + b` = 12 instead of 21.
1. Borda, Jean-Charles de (1784). “Mémoire sur les Élections au Scrutin”, . Année M. DCCLXXXI. Avec les Mémoires de Mathématique et de Physique, pour la même Année. Paris 1784.
2. Daunou, P.Cl.F. (1803). “Mémoire sur les élections au scrutin” . Paris.
3. L`institut de France (1889), . Paris.
4. Mascart, Jean (1919). . Lyon.
5. Stefansson, Björn S. (1982). "Group choice between three or more alternatives", 16: 433-54.
6. Stefansson, Björn S. (1991). "On irrelevant and infeasible alternatives", 25: 297-306.