**Jacques Binet** worked on the foundations of matrix theory. He discovered the familiar rule for matrix multiplication.

- Philippe Binet was born in Paris and a student at the Academy there.
- However, the French Revolution stopped work on the cathedral and Philippe Binet died before it recommenced.
- 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.
- Paul Binet was appointed to teach at the École Centrale de Rennes, then as professor at the Académie d'Orléans.
- Jacques Binet was educated first at Rennes.
- 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.
- Binet published his first two papers in the journal Correspondance sur l'École Polytechnique in 1809 and 1810.
- Binet's first paper "Des trois axes rectangulaires des surfaces du second degré, qui ont un centre" Ⓣ(The three rectangular axes of the surfaces of the second degree, which have a 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.
- Binet's second paper "Proposition de géométrie" Ⓣ(A proposition in geometry) 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.
- Around 1814 Binet submitted his memoir "Sur l'expression analytique de l'élasticité et de la raideur des courbes à double courbure" Ⓣ(On the analytical expression of the elasticity and stiffness of curves of double curvature) to be considered for publication.
- In 1814 Binet was appointed examiner of descriptive geometry at the École Polytechnique then, in 1815, he was appointed Examiner in Geodesy.
- 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" Ⓣ(Analytical mechanics) that was being prepared two years after Lagrange died.
- 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.
- 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.
- 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.
- However the revolution of July 1830 was unfortunate for Binet.
- 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.
- In this paper Binet introduced what today is called the Beta function B(m,n)B(m,n)B(m,n).
- It has been suggested that Binet chose the notation BBB and called it a beta function, because of the first letter of his own name.
- 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" Ⓣ(Reflections on the problem of determining the number of ways in which a straight face can be divided into triangles by its diagonals) was his contribution which in turn inspired further work on the problem by Eugène Catalan and Gabriel Lamé.
- 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" Ⓣ(Note on a new method for finding the greatest common divisor of integers or algebraic polynomials, and the application of this method to congruences of the first degree).
- Binet is remembered for 'Binet's form' of the Fibonacci numbers.
- Now although this is known as the Binet form, he was not the first to find this expression since de Moivre had discovered it much earlier in 1718.
- The mathematicians and physicists, including Binet, viewed the experiment of the pendulum swinging in Meridian Hall, slowly rotating over the Paris Meridian.
- 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.
- Foucault's experiment clearly showed that Poisson was wrong but Binet was reluctant to say so.
- 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.
- 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.
- 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.
- 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.

Born 2 February 1786, Rennes, Bretagne, France. Died 12 May 1856, Paris, France.

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Algebra, Astronomy

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive