Charles Haros

Last updated
Charles Haros
Born1700s
Died1800s
Scientific career
Fields Mathematics

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.

Contents

Haros' conversion table

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).

In his paper Haros demonstrated that the mediant is always irreducible and, more importantly for this purposes, if one starts with the sequence of fractions

1/99, 1/98, 1/97, ..., 1/4, 1/3, 1/2, 2/3, 3/4, 5/6, ..., 96/97, 97/98, 98/99

and just keeps applying the rule, only keeping the result if the denominator is less than one-hundred, then they generate all 3,003.

A curious property

Roughly fifteen years later in England, Henry Goodwyn set out to create a much more ambitious version of Haros’ table. In particular, Goodwyn wanted to tabulate the decimal values for all irreducible fractions with denominators less than or equal to 1,024. There are 318,963 such fractions. As a warm up and a test of the commercial market for such a table in 1816 he published for private circulation The First Centenary of a Series of Concise and Useful Tables of all the Complete Decimal Quotients, which can arise from dividing a unit, or any whole Number less than each Divisor by all Integers from 1 to 1024.

John Farey observed the mediant property in this table and mused in a letter to The Philosophical Magazine and Journal as follows:

"I am not acquainted, whether this curious property of vulgar fractions has been before pointed out?; or whether it may admit of any easy or general demonstration ?; which are points on which I should be glad to learn the sentiments of some of your mathematical readers; ..."

(Mis)naming of the Farey sequence

Augustin Cauchy read Farey’s letter and published a paper "Démonstration d’un Théorème Curieux sur les Nombres" reproving Haros’ results without acknowledgement. In his paper Cauchy referred to the mediant as "a remarkable property of ordinary fractions observed by M. J. Farey." Thus, an ordered sequence of all vulgar fractions with denominators less than a given value became known as a Farey sequence rather than perhaps more rightfully as either a Chuquet sequence or a Haros sequence.

Publications

See also

Further reading

Related Research Articles

<span class="mw-page-title-main">Augustin-Louis Cauchy</span> French mathematician (1789–1857)

Baron Augustin-Louis Cauchy was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He (nearly) single-handedly founded complex analysis and the study of permutation groups in abstract algebra.

<span class="mw-page-title-main">Gaspard de Prony</span> French mathematician and engineer (1755–1839)

Baron Gaspard Clair François Marie Riche de Prony was a French mathematician and engineer, who worked on hydraulics. He was born at Chamelet, Beaujolais, France and died in Asnières-sur-Seine, France.

<span class="mw-page-title-main">Farey sequence</span> Increasing sequence of reduced fractions

In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions, either between 0 and 1, or without this restriction, which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.

<span class="mw-page-title-main">Jean-Antoine Letronne</span> French archaeologist (1787-1848)

Jean Antoine Letronne was a French archaeologist.

A vinculum is a horizontal line used in mathematical notation for various purposes. It may be placed as an overline or underline above or below a mathematical expression to group the expression elements. Historically, vincula were extensively used to group items together, especially in written mathematics, but in modern mathematics its use for this purpose has almost entirely been replaced by the use of parentheses. It was also used to mark Roman numerals whose values are multiplied by 1,000. Today, however, the common usage of a vinculum to indicate the repetend of a repeating decimal is a significant exception and reflects the original usage.

<span class="mw-page-title-main">Arnaud Denjoy</span> French mathematician

Arnaud Denjoy was a French mathematician.

<span class="mw-page-title-main">Fraction</span> Ratio of two numbers

A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator, displayed above a line, and a non-zero integer denominator, displayed below that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3/4 of a cake.

<span class="mw-page-title-main">Sylvestre François Lacroix</span> French mathematician (1765–1843)

Sylvestre François Lacroix was a French mathematician.

<span class="mw-page-title-main">David-Augustin de Brueys</span> French theologian (1641–1723)

David-Augustin de Brueys was a French theologian and playwright. He was born in Aix-en-Provence. His family was Calvinist, and he studied theology. After writing a critique of Jacques-Bénigne Bossuet's work, he was in turn converted to Catholicism by Bossuet in 1681, and later became a priest.

<span class="mw-page-title-main">Decimal time</span> Representation of the time of day using decimally related units

Decimal time is the representation of the time of day using units which are decimally related. This term is often used specifically to refer to the French Republican calendar time system used in France from 1794 to 1800, during the French Revolution, which divided the day into 10 decimal hours, each decimal hour into 100 decimal minutes and each decimal minute into 100 decimal seconds, as opposed to the more familiar standard time, which divides the day into 24 hours, each hour into 60 minutes and each minute into 60 seconds.

Continuation of the Séminaire Nicolas Bourbaki programme, for the 1950s.

<span class="mw-page-title-main">Louis Abel Beffroy de Reigny</span> French dramatist and man of letters

Louis Abel Beffroy de Reigny was a French dramatist and man of letters.

<span class="mw-page-title-main">Grave (unit)</span> Predecessor to the kilogram

The grave, abbreviated gv, is the unit of mass used in the first metric system which was implemented in France in 1793. In 1795, the grave was renamed as the kilogram.

Marie Henri Andoyer was a French astronomer and mathematician.

Roch Le Baillif, Sieur de la Riviere was a French physician, influenced by Paracelsianism. He was prosecuted for practicing medicine contrary to the official teaching of the Sorbonne.

<span class="mw-page-title-main">Antoine-François Delandine</span> French writer

Antoine-François Delandine, was a French writer.

<span class="mw-page-title-main">Louis Carré (mathematician)</span> French mathematician

Louis Carré was a French mathematician and member of the French Academy of Sciences. He was the author of one of the first books on integral calculus.

Nicolas-Étienne Framery was a French music theorist, critic and lyric writer associated with opera, especially opéra comique. He wrote and adapted librettos. His work became more academic and abstract and he eventually became surintendant de la musique for the Comte d'Artois,.

Pierre Nicolas Chantreau, called don Chantreau, was an 18th-century French historian, journalist, grammarian and lexicographer.

Undecimal is a positional numeral system that uses eleven as its base. While no known society counts by elevens, two are purported to have done so: the Māori and the Pañgwa. The idea of counting by elevens remains of interest for its relation to a traditional method of tally-counting practiced in Polynesia. During the French Revolution, undecimal was briefly considered as a possible basis for the reformed system of measurement. Undecimal numerals have applications in computer science, technology, and the International Standard Book Number system. They also occasionally feature in works of popular fiction. In undecimal, a capital letter or the digit ↊ is typically used as a transdecimal symbol to represent the number 10.