Charles Haros
Encyclopedia
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.
ping 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 ‘’règel des nombres moyens‘’. Today we call it the mediant. The mediant is the fraction between two fractions a/b and c/d whose numerator is the sum of the numerators, a+c, and whose denominator is the sum of the denominators, b+d. That is, the mediant of the fractions a/b and c/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 you start with the sequence of fractions
and just keep applying the rule, only keeping the result if the denominator is less than one-hundred, then you generate all 3,003.
, 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 fraction
s 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:
rather than perhaps more rightfully as either a Chuquet sequence or a Haros sequence.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
) in the French Bureau du Cadastre at the end of the eighteenth century and the beginning of the nineteenth century.
Haros' conversion table
One of the primary tasks of the Bureau du Cadastre was the accurate mapMap
A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....
ping of France
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan 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
French Revolution
The French Revolution , sometimes distinguished as the 'Great French Revolution' , was a period of radical social and political upheaval in France and Europe. The absolute monarchy that had ruled France for centuries collapsed in three years...
was to convert France to the metric system
Metric system
The metric system is an international decimalised system of measurement. France was first to adopt a metric system, in 1799, and a metric system is now the official system of measurement, used in almost every country in the world...
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
Ephemeris
An ephemeris is a table of values that gives the positions of astronomical objects in the sky at a given time or times. Different kinds of ephemerides are used for astronomy and astrology...
, Connaissance des Temps
Connaissance des Temps
The Connaissance des temps is an official astronomical yearly publication in France.- History :Connaissance des temps is the oldest such publication in the world, published without interruption since 1679 , after the astronomer Jean Picard obtained from the King the right to create the journal...
, 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
Institut de France
The Institut de France is a French learned society, grouping five académies, the most famous of which is the Académie française.The institute, located in Paris, manages approximately 1,000 foundations, as well as museums and chateaux open for visit. It also awards prizes and subsidies, which...
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
Irreducible fraction
An irreducible fraction is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent vulgar fraction...
with denominators less than 100. In order to make sure he got them all he harnessed an algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
elucidated by Nicolas Chuquet
Nicolas Chuquet
Nicolas Chuquet was a French mathematician whose great work, Triparty en la science des nombres , was unpublished in his lifetime...
some one-hundred and fifty years earlier. Chuquet called it his ‘’règel des nombres moyens‘’. Today we call it the mediant. The mediant is the fraction between two fractions a/b and c/d whose numerator is the sum of the numerators, a+c, and whose denominator is the sum of the denominators, b+d. That is, the mediant of the fractions a/b and c/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 you start 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 keep applying the rule, only keeping the result if the denominator is less than one-hundred, then you generate all 3,003.
A curious property
Roughly fifteen years later in EnglandEngland
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
, 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 fraction
Irreducible fraction
An irreducible fraction is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent vulgar fraction...
s 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
John Farey, Sr.
John Farey, Sr. was an English geologist and writer. However, he is better known for a mathematical construct, the Farey sequence named after him.-Biography:...
observed the mediant property in this table and mused in a letter to The Philosophical Magazine
Philosophical Magazine
The Philosophical Magazine is one of the oldest scientific journals published in English. Initiated by Alexander Tilloch in 1798, in 1822 Richard Taylor became joint editor and it has been published continuously by Taylor & Francis ever since; it was the journal of choice for such luminaries as...
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 sequenceFarey sequence
In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size....
rather than perhaps more rightfully as either a Chuquet sequence or a Haros sequence.
Publications
- Cauchy, Augustin Louis. ‘’Démonstration d'un Théorème Curieux sur Les Nombres‘’. Bulletin des Sciences, par la Société Philomatique de Paris, Vol. 3, No. 3 (1816), pp. 133–135.
- Farey, JohnJohn Farey, Sr.John Farey, Sr. was an English geologist and writer. However, he is better known for a mathematical construct, the Farey sequence named after him.-Biography:...
. ‘’On a Curious Property of Vulgar Fractions‘’. The Philosophical Magazine and Journal, Vol. 47, No. 3 (1816), pp. 385–386.
- Goodwyn, Henry. 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, Private Distribution, 18p, 1816.
- Haros, Charles. Comptes faits à la manière de Darême, sur les nouveaux poids et measures, aves les pris proportionnels, à l’usage et autres. Paris:Frimin Didot, 1806.
- Haros, Charles. ‘’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‘’. Journal de École Polytechnique, Vol. 6, No. 11 (1801), pp. 364–368.
- Haros, Charles. Instruction Abrégée sur les nouvelles Mesures qui dovient étre introduites dans toute république, au vendémiaire an 10; avec tables de rapports et reductions. Paris:Firmin Didot, 1801.
See also
- Ivor Grattan-GuinnessIvor Grattan-GuinnessIvor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...
has written a number of books and papers on mathematics in France in the eighteenth and nineteenth centuries.
- Gaspard De PronyGaspard de PronyGaspard 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.-Education and early works:...
set up the Bureau du Cadastre and lead the project to compute the great logarithmic and trigonometric tables, the Tables du cadastre
Further reading
- Guthery, Scott. A Motif of Mathematics: History and Application of the Mediant and the Farey Sequence. Boston:Docent Press, 2010. ISBN 1453810579
External links
- Mansuy, Roger. Les calculs du citoyen Haros. Les calculs du citoyen Haros. L’apprentissage du calcul décimal. http://www.dma.ens.fr/culturemath/
- Roegel, Denis. The great logarithmic and trigonometric tables of the French Cadastre: a preliminary investigation. http://www.loria.fr/~roegel/locomat.html.