Means of Learning to Count Surely and with Ease (Moyens d’apprendre à compter sûrement et avec facilité) – Jean-Antoine-Nicolas de Caritat de Condorcet

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Moyens d’apprendre à compter sûrement et avec facilité (English: Means of Learning to Count Surely and with Ease) is a short elementary arithmetic textbook by the French mathematician, philosopher and revolutionary politician Jean-Antoine-Nicolas de Caritat, Marquis de Condorcet (1743–1794). It was published posthumously in Paris in the Year VII of the Republic (1798–1799) and is generally regarded as the last book Condorcet wrote. Designed for the primary schools of the French Republic and accompanied by extensive notes for teachers, it is known above all for its method of teaching number and for its proposal of a fully regular, “purely decimal” way of naming numbers.

Author and Context

Condorcet is best known for his pioneering work in probability and statistics, his analysis of voting procedures (the Condorcet method, the “jury theorem” and the Condorcet paradox), and his philosophical and political writings. He is also remembered for his action against slavery and the death penalty and for equality of the sexes, both before and during the Revolution. As a mathematician he had been admitted to the circle of Jean le Rond d’Alembert and later became permanent secretary of the Academy of Sciences.

Education was a central concern of his political career. His plan for public instruction was presented to the Legislative Assembly in April 1792. It proposed five levels of instruction, beginning with primary schools and culminating in a National Society of Arts and Sciences charged with the general direction of education. In his first report on public education (1791), Condorcet argued that a man who knows the rules of arithmetic needed in everyday life is not dependent on the scholar of mathematical genius. The idea that basic calculation was a civic competence, enabling citizens to manage their affairs and to exercise their rights without being at the mercy of experts, underlies the textbook.

The book belongs to the final period of Condorcet’s life, when he was proscribed during the Terror and died in custody at Bourg-la-Reine in March 1794. It responded to a directive of the Convention of 8 January 1794 (9 Pluviôse, Year II) concerning elementary instruction. Some modern sources date the work itself to 1794, the year of its composition, whereas its publication followed several years later.

Publication History

The work appeared as a posthumous publication. The title page of the original edition describes it as a posthumous work by Condorcet, sold in Paris by the bookseller Moutardier, in the Year VII of the Republic. Specialist booksellers describe it as the posthumous first edition, established from a manuscript held by Madame de Condorcet. The editor was Condorcet’s widow, Sophie de Grouchy, who preserved and published much of his work; she is credited with a preface to the book, with Garat, in the Moutardier edition.

The volume is preceded by an “avertissement” of eleven pages by Dominique Joseph Garat, a philosopher and politician close to the author. In it, Garat insists on the authenticity of the work. He notes that those who doubted or pretended to doubt its authorship could examine at the widow’s home the original manuscript, written entirely in Condorcet’s own hand.

Bibliographic descriptions record some curiosities of the first printing. The edition is recorded as Paris, Moutardier, Year VII (1798–1799), with a reissue under the Bachelier imprint dated 1800. Certain copies carry two distinct title pages: the original, bearing a fictitious mention of a “second edition” (Moutardier, Year VII), and another added when the book was put back on sale, giving the address “Paris, Bachelier, Imprimé en 1800”. The format is a small duodecimo volume of roughly 130 pages of text.

Later publications include a nineteenth-century “nouvelle édition” (dated 1854 in the Bibliothèque nationale de France’s catalogue records), which is digitised on Gallica and has been reprinted on demand. In the twentieth century, a critical edition with studies, notes, commentaries and bibliography by Charles Coutel, Nicole Picard and Gert Schubring appeared from Art, Culture, Lecture (ACL-Éditions) in 1989 as a facsimile of the Moutardier edition. The ACL – Les Éditions du Kangourou reissue of 1988 includes a facsimile of the original edition supplemented with didactic, historical, philosophical and bibliographical commentary. The text was also reproduced in the journal Enfance in 1989.

Content and Method

The book is a manual of elementary arithmetic intended for primary schools, accompanied by abundant notes for teachers. Condorcet treats the teaching of elementary mathematics, in particular calculation. He adopts a methodical and rational approach, concerned with making learning accessible to all by means of suitable pedagogical tools. The work addresses pupils, teachers and the general public.

Garat’s introduction explains the philosophy behind the book. He observes that rules and formulae allow the mind to dispense with painful attention and to reach correct results almost mechanically, a benefit that carries the danger of reducing arithmetic to routine. According to Garat, the only remedy is the one Condorcet sought, found and taught in this little treatise on calculation. The aim is to make the reasons for each operation so clear that understanding accompanies the procedure. Among the means he identifies is a careful development of how numbers are formed in tens, hundreds and thousands, with attention to the basis of each unit in the lower orders.

This pedagogical ideal fits with Condorcet’s wider epistemology. What calculation taught its practitioners, in his view, was what he called the “exactitude of the mind” (justesse d’esprit). The historian of Enlightenment thought Lorraine Daston has noted that, in the textbook, Condorcet used the simplest arithmetic identities, such as “three plus four equals seven”, to teach children the meaning of self-evidence and justified belief. Arithmetic is thus presented as an introduction to reasoning and to the basis of certainty, not merely a set of practical techniques.

The Revolutionary System of Numeration

The most distinctive feature of the book, and the one that has kept it in discussion among teachers and historians, is its proposal for the spoken names of numbers. Before turning to the four operations, Condorcet sets out “the system of numeration currently in use in France,” the one issuing from the Revolution. This was a new system, which the author evidently wished to see adopted by introducing it first in the schools. In the same spirit as the metric system, of which Condorcet was one of the architects and which simplified and rationalised existing units, his system of numeration is a rationalisation of ordinary number words.

The irregular words “onze” to “seize” are replaced by the regular “dix-un” to “dix-six”. The vigesimal and mixed forms of ordinary French are also regularised. Examples cited include “duante-cinq” (25), “quarante-un” (41), “cent duante” (120) and “sept cent nonante-un” (791). For large numbers, digits are grouped by threes on the short scale, with names such as million, “dillion” (10⁹, replacing “milliard”), trillion and quadrillion. A guiding principle often quoted from the book is that the analogy of ideas must always be marked by the analogy of words.

The scheme sought to remove the memorisation burden imposed by irregular number names. Modern advocates of a regular numeration argue that, in conventional French, a list of one hundred numbers of which fifty-six are irregular can only be learned “by heart”, a point they trace back to Condorcet’s intuition.

Reception and Legacy

The Revolution’s educational plans were never fully carried out, and Condorcet’s own project met the same fate. He died just after writing the book, and his project was not carried out. The reformed number names did not enter common usage, and French retained its traditional forms such as soixante-dix and quatre-vingt-dix, although Belgian and Swiss French use septante and nonante.

The book has nonetheless remained a reference in the history of mathematics education. French didactic research on the teaching of large numbers still cites Condorcet’s definitions of the thousand and the higher units, which build on the principles of position and decimality. Studies of numeration teaching in primary schools cite the 1988 reissue as a historical starting point for how the knowledge of decimal numeration has been defined since his time. In the 2020s, French associations concerned with numeracy have revived interest in his proposals, claiming that Condorcet’s numeration, combined with a teaching abacus, drastically shortens the time needed to learn numeration.

Within Condorcet’s oeuvre, the book is a counterpart to his better-known Esquisse d’un tableau historique des progrès de l’esprit humain, written in the same final period. Both works express the Enlightenment belief that sound instruction, beginning with the clearest possible elementary principles, is essential to human progress and to equal citizenship.

Editions

• Moyens d’apprendre à compter sûrement et avec facilité, Paris: Moutardier, An VII (1798–1799), with Garat’s avertissement; reissued under the Bachelier imprint, 1800.
• “Nouvelle édition”, 1854, edited from the posthumous publication by Madame de Condorcet, with Garat’s avertissement; digitised on Gallica.
• Critical edition with study and commentary by Charles Coutel, Nicole Picard and Gert Schubring, Paris: ACL, 1988–1989.
• Enfance, vol. 42, no. 4 (1989), pp. 59–90, reproducing the text.