Charles Haros

Charles Haros was a geometer (mathematician) in the French Bureau du Cadastre at the end of the eighteenth century and the beginning of the nineteenth century. One of the primary tasks of the Bureau du Cadastre was the accurate mapping of France for the purpose of taxation but from time to time the bureau also provided computational services to other parts of the government. One of the changes instituted by the French revolution was to convert France to the metric system and this necessitated changing from a fractional to a decimal representation of rational numbers. While Haros was involved many computation projects at the Bureau du Cadastre including the computation of de Prony’s tables of logarithms and the preparation of the French ephemeris, Connaissance des Temps, he is best known for a small table he prepared to convert fractions to their decimal equivalents. Haros’ conversion table appeared in a tract, Instruction Abrégée sur les nouvelles Mesures qui dovient étre introduites dans toute république, au vendémiaire an 10; avec tables de rapports et reductions, that was presented to the Mathematics Section of the Institut de France and subsequently abstracted in Journal de l'École Polytechnique under the title "Tables pour évaluer une fraction ordinaire avec autant de decimals qu’on voudra; et pour trouver la fraction ordinaire la plus simple, et qui approche sensiblement d’une fraction décimale." In preparing his table, Haros needed to create the list of all 3,003 irreducible (vulgar) fractions with denominators less than 100. In order to make sure he got them all he harnessed an algorithm elucidated by Nicolas Chuquet some one-hundred and fifty years earlier. Chuquet called it his "règle des nombres moyens". Today, we call it the mediant. The mediant is the fraction between two fractions a/c and b/d whose numerator is the sum of the numerators, a+b, and whose denominator is the sum of the denominators, c+d. That is, the mediant of the fractions a/c and b/d is the fraction (a+b)/(c+d).