**Jacques Binet**'s parents were Philippe Binet (1743-1815) and Renée Marie Jollivet (1760-1836). Philippe Binet was born in Paris and a student at the Academy there. He was awarded two medals and a prize for architecture. He spent several years in Italy, returned to Paris where he was involved in the design of the School of Medicine, and then went to Rennes where he was involved in many projects including the cathedral. However, the French Revolution stopped work on the cathedral and Philippe Binet died before it recommenced. The Jollivet family was in the pottery business in Rennes. Philippe and Renée were married in Rennes on 1 July 1777 and had five children: Paul René Binet (born 1779); Joseph Philippe Binet (1783-1796); Jacques Philippe Marie Binet (the subject of this biography, 1786-1856); Philippe Thomas Binet (1788-1855); and Adélaide Binet (1792-1810). The family were devoted Roman Catholics and staunch supporters of the King.

Paul René Binet was, as we see from the above dates, seven years older than his brother Jacques, the subject of this biography. Since Paul René Binet also became a mathematician, his career was clearly a highly significant influence on Jacques as he grew up. Therefore before giving details of Jacques we look briefly at Paul René Binet. After schooling in Rennes, he entered the École Polytechnique in 1798 in the same class as Siméon-Denis Poisson. He graduated in 1801 while Poisson had graduated in the previous year. Paul Binet was appointed to teach at the École Centrale de Rennes, then as professor at the Académie d'Orléans. He moved back to Paris and was appointed to the Lycée Bonaparte. He was, in addition to his position at the Lycée Bonaparte, a répétiteur at the École Polytechnique from 1808 to 1831. We note also that Jacques' younger brother Philippe Thomas Binet was admitted to the École Polytechnique on 28 September 1809. He graduated in 1811 and then worked for the Construction des Vaisseaux.

Jacques Binet was educated first at Rennes. He entered the École Polytechnique in Paris on 22 November 1804 in the same class as Augustin Jean Fresnel. After graduating in 1806, he was appointed as a student engineer at the Department of Bridges and Roads of the French government. This, however, was not to his liking and he quickly decided that he wanted to make a career as a teacher of mathematics. He left the Department of Bridges and Roads and took up a position at the Lycée Bonaparte, but he only taught there for a short time.

Jacques Binet became a teacher at École Polytechnique in 1807 and, one year later, he was appointed as a répétiteur to assist the professor of applied analysis and descriptive geometry. He held this position from 1808 to 1814. During this time the professors of descriptive geometry were Gaspard Monge (who held this professorship 1794-1809), Jean Nicolas Pierre Hachette (who held this professorship 1799-1815), and François Arago (who held this professorship 1810-1816).

Binet published his first two papers in the journal *Correspondance sur l'École Polytechnique* in 1809 and 1810. He was assisted in this by Hachette who was the editor of the journal. Binet's first paper *Des trois axes rectangulaires des surfaces du second degré, qui ont un centre* Ⓣ appears in the *Correspondance* with a heading that gives M Binet as the author while in the index the paper is attributed to J P M Binet. However, although the paper gives the proof of a theorem as stated and proved by J P M Binet, the paper itself was written by Jean Nicolas Pierre Hachette, the editor of the *Correspondance sur l'École Polytechnique*. In the introduction to the paper when it states that the paper gives an improvement on a theorem in the author's 1801 *Mémoire* it is referring to a *Mémoire* written by Hachette. Binet's second paper *Proposition de géométrie* Ⓣ appears in the *Correspondance* without any author in its heading and the paper not listed in the index. However, it also was written by Hachette who states that the paper consists of a theorem as stated and proved by J P M Binet.

Binet became a member of the *Société philomathique* on 14 March 1812. The Academy of Sciences had been closed in 1793 and many of the members of the Academy had joined the *Société philomathique* giving it a rather special status. In 1795 the Institut National replaced the Academy but the *Société philomathique* continued to have an important role in the scientific life of Paris. Around 1814 Binet submitted his memoir *Sur l'expression analytique de l'élasticité et de la raideur des courbes à double courbure* Ⓣ to be considered for publication. It was assigned to Lazare Carnot and Gaspard de Prony who were asked to report on it. Here is an extract from their report [3]:-

For a longer extract from this report see THIS LINK.After a purely geometric introduction, containing some new formulas, relating to polygons whose sides do not all lie in the same plane, and to curves of double curvature, the author considers problems of equilibrium which are the particular object of the paper. He successively introduces the effect of the action of forces on a polygon of the type just described and on curves of double curvature. He established in both systems a theory which is applicable both to the case of stiffness and to the case of elasticity. ... In general the analysis is managed with much skill; and the geometric introduction, which would itself be an interesting memoir, ought to confirm, and even increase, the good opinion which has been formed of his scientific merit, from the different works which he has earlier submitted to the judgement of the Class.

For a complete list of Binet's papers see THIS LINK.

In 1814 Binet was appointed examiner of descriptive geometry at the École Polytechnique then, in 1815, he was appointed Examiner in Geodesy. His work in this role covered descriptive geometry and 'graphic arts,' but sometimes he had to deal with physics questions. He succeeded Siméon-Denis Poisson as professor of mechanics in 1815. In 1816 not only did Binet become an inspector of studies at the École Polytechnique but he also became an editor of the edition of Lagrange's *Mécanique analytique* Ⓣ that was being prepared two years after Lagrange died. Binet's paper *Sur l'expression analytique de l'élasticité et de la raideur des courbes à double courbure* Ⓣ which we mentioned above did much to add to Lagrange's work as the reporters Lazare Carnot and Gaspard de Prony noted (see [3] or our longer extract of the report at THIS LINK):-

[The Collège Royal in Paris was renamed in 1795 as the Collège de France. Although there were changes in this new College, nevertheless it continued to offer students the chance of attending free lecture courses. Jean-Baptiste-Joseph Delambre was appointed as the professor of astronomy at the College in 1807 and from around 1815 Binet was appointed as his assistant. Delambre died in August 1822 and, in the following year, Binet was appointed to fill the vacant chair of astronomy at the Collège de France.Binet]was right to announce that his researches might serve as an explanation and supplement of several chapters of the 'Mécanique analytique'.

In 1820 Binet had married Marie Eléonore Ménard Dubourg (1798-1860). They had a son Paul Jacques Binet (1822-1878) who married Marie Anne Cramail (1829-1903) in 1856. They had a son Paul Jacques Marie Binet (1865-1940) and we give these details of Binet's son and his grandson since both maintained the family tradition and were students at the École Polytechnique. Binet's son entered in 1842 while his grandson entered in 1884.

Augustin-Louis Cauchy had been appointed as professor of analysis and mechanics at the École Polytechnique in 1816. Although Cauchy had an outstanding reputation as a researcher, he was renowned for giving lectures which were both too difficult and too pure in nature [3]:-

In 1825 François Arago, who was professor of applied analysis, and Binet, who was inspector of studies, wrote a joint letter in which they complained about the lack of knowledge of practical mathematics attained by Cauchy's students. This was followed up by critical comments about Cauchy's teaching by Gaspard de Prony. As with Binet's criticisms, there was no suggestion that what Cauchy taught was anything but top quality, rather the complains were that it was not appropriate for students at the École Polytechnique. We should note that on a personal level Binet and Cauchy were friends both being staunch Royalists.Cauchy's effectiveness as a teacher, however, is another matter. During the1820s the school earnestly asked him to modify his teaching of mathematical analysis, but without success.

However the revolution of July 1830 was unfortunate for Binet. The revolution started after Charles X published restrictive ordinances on 26 July which were contrary to the spirit of the Charter of 1814. There were protests, demonstrations and fighting on July 27, 28 and 29. Binet was a strong supporter of Charles X so it was bad news for him when Charles X abdicated on August 2 and, a week later, Louis-Philippe was proclaimed King of France. Binet was far too much associated with the previous regime to be acceptable to that of Louis-Philippe and he was dismissed as inspector of studies on 13 November 1830.

Binet investigated the foundations of matrix theory which was to set the scene for later work by Cayley and others. He discovered the rule for multiplying matrices in 1812 and it is almost certainly for this that he will be remembered rather than his other work.

He did, however, write a number of important papers which were influential in the development of mathematics, in particular he wrote *Mémoire sur les intégrales définies Eulériennes et sur leur application à la théorie des suites; ainsi qu'à l'évaluation des fonctions des grands nombres* Ⓣ in 1839. In this paper Binet introduced what today is called the Beta function *B*(*m*,*n*). It has been suggested that Binet chose the notation *B* and called it a *beta function*, because of the first letter of his own name. However, there is no evidence to support this claim. In the same year he published *Réflexion sur le problème de déterminer le nombre de manières dont une figure rectiligne peut être partagée en triangles au moyen de ses diagonales* Ⓣ in the *Journal de Mathématiques Pures et Appliquées*. This journal, known as Liouville's Journal, had been set up by Joseph Liouville in 1836 and he actively sought papers for the journal by encouraging mathematicians to write them. In 1838 Olinde Rodrigues had published a paper on the number of ways one can divide a polygon. Liouville thought the problem deserved further work and approached Binet encouraging him to investigate. Binet's *Réflexion sur le problème de déterminer le nombre de manières dont une figure rectiligne peut être partagée en triangles au moyen de ses diagonales* Ⓣ was his contribution which in turn inspired further work on the problem by Eugène Catalan and Gabriel Lamé. The numbers that result from the problem of dissecting a polygon into triangles by means of non-intersecting diagonals are now called the 'Catalan numbers' although they could equally well be named after any of a number of mathematicians who investigated them including Leonhard Euler and Jan Segner in addition to those just mentioned.

The 1841 Binet wrote on number theory, making a contribution to the theory of the Euclidean algorithm in his paper *Note sur une nouvelle méthode pour trouver le plus grand commun diviseur des nombres entiers, ou des polynômes algébriques, et sur l'application de cette méthode aux congruences du premier degré* Ⓣ. This particular piece of work is discussed in [6] where Shallit writes:-

Binet is remembered for 'Binet's form' of the Fibonacci numbers. He discovered this formula in 1843 which gives a direct value of the Fibonacci numberBinet's analysis[of the algorithm]is surprisingly modern in presentation.

*F*

_{n}without using the recurrence relation. Suppose

*a*and

*b*are the roots of the equation

*x*

^{2}-

*x*- 1 = 0 where

*a*-

*b*= √5. Then Binet gave the expression

*F*

_{n}= (

*a*

^{n}-

*b*

^{n})/(

*a*-

*b*).

On 3 February 1851 Léon Foucault invited members of the Academy of Sciences to come to the Paris Observatory to see his pendulum experiment which demonstrated the rotation of the earth. The mathematicians and physicists, including Binet, viewed the experiment of the pendulum swinging in Meridian Hall, slowly rotating over the Paris Meridian. On the same day François Arago read a paper by Foucault to a meeting of the Academy of Sciences. In this paper Foucault claimed that the time taken for the pendulum to complete one revolution was ^{24}/_{sin q} hours, where *q* is the latitude at which the experiment is performed. This meant that, in Paris, it should take just under 32 hours which was experimentally verified. However, Foucault was not an expert mathematician and he had no mathematical derivation of his formula. The first to address the Academy on the topic following the events of 3 February was Binet whose written presentation was read to the Academy on 10 February. Binet began (quoted in [1]):-

He then described various references to a pendulum that had been made by previous mathematicians. In particular he quoted Poisson who had written in an article published in theFoucault has asked me in what way the mechanical result he had obtained accords with mathematical theories.

*Journal de l'École Polytechnique*in 1837 (quoted in [1]):-

Foucault's experiment clearly showed that Poisson was wrong but Binet was reluctant to say so. He tried to give Poisson the benefit of the doubt by writing (quoted in [1]):-The force acting on the plane of oscillation of the pendulum is too small to move the pendulum perceptibly, to have any apparent influence on its movement.

He then explained what mathematics would be necessary to prove Foucault's sine law but said more time would be needed to complete the proof. He did so in a further report to the Academy one week later on 17 February, when he gave a complete mathematical derivation of Foucault's sine law.But the passage just cited permits a doubt: Poisson doesn't report a calculation of the force of which he speaks, and thus it is insufficient to allow us to know whether the perturbing force is very small, to conclude that it will only produce an imperceptible effect after a large number of oscillations.

Binet's other contributions were in the fields of physics and astronomy, and since he held the chair of astronomy at the Collège de France for over 30 years this is not surprising. In all he wrote over 60 papers.

We have attempted to give a complete list at THIS LINK.

Among the honours which Binet received for his work was Chevalier de la Légion d'Honneur on 1 May 1821 and election to the Académie des Sciences in 1843, succeeding Sylvestre Lacroix who died in May of that year. In 1856 he was elected as president of the Academy and, as president, chaired the weekly meeting on 28 April of that year. However, at the beginning of the meeting on Monday, 12 May, it was announced to the members that Binet was very seriously ill and, before the meeting ended, his death was reported to those present. He was buried in Montparnasse Cemetery on 14 May 14 with members of his family and many of his colleagues present. Gabriel Lamé and Augustin Cauchy gave an eulogy. In September 1994 the University of Rennes named its association for doctoral students for Jacques Binet. A street in the Cleunay district of Rennes bears Jacques Binet's name.

**Article by:** *J J O'Connor* and *E F Robertson*