**Jacques Français**was the brother of François Français. The brothers were the children of Jacques Frédéric Français, who was a grocer in Saverne, and Maria Barbara Steib. Jacques entered the College at Strasbourg and excelled in his studies there.

He volunteered for the army in 1793. The reason why volunteers were needed in that year was that the French Revolution of 1789 had not gained universal support and there were attempted counter-revolutions in parts of France. In the Vendée region in the west of France there was an uprising which was sparked off by the introduction of conscription in February 1793. By the middle of March there was an alliance of Royalists and peasants with a fair size army. Jacques' brother François Français also joined the government army in 1793 which was being assembled to put down the rebellion.

In September 1794 Jacques Français became an assistant in the engineering corps. In the autumn of 1797, when François Français left the army, Jacques Français entered the École Polytechnique and in the spring of the next year he moved to the École du Génie. By 1801 he had reached the rank of first lieutenant and he was sent by Napoleon to Egypt in January of that year. Once back in France he was stationed at Toulon and promoted to captain of the sappers. Further promotions followed, and in November 1802 he became second in command at the staff headquarters of the engineering corps.

He went on to participate, under the command of Admiral Villeneuvre, in the naval battles of Cape Finisterre and Trafalgar. In 1807 he was stationed at Strasbourg and his commander there was Malus. We discuss below the mathematics which Jacques Français produced at this time due mainly to the encouragement of Malus. By 1810 Français had reached the position of first in command at the staff headquarters of the École d'Application in Metz and, in 1811, he was appointed professor of military art in Metz.

The first mathematics memoir which Jacques Français seems to have written was submitted in 1800. It was a work on the integration of first order partial differential equations, but the memoir had been lost so there are few details as to its precise contents. He then appears to have lost interest in mathematics until, under the command of Malus, he was encouraged to prepare his work on analytic geometry for publication. During 1807-08 he wrote works on the straight line and the plane in oblique coordinates, also considering transformations between systems of oblique coordinates. He applied his methods to the famous problem of finding a sphere tangent to four given spheres, publishing a number of notes on the topic between 1808 and 1812, and giving a complete solution in the 1812 paper which appeared in *Gergonne's Journal*.

Although with encouragement from Malus Jacques Français had begun to publish mathematics, a second boost to his mathematics occurred after the death of his brother François Français. He worked on his brother's manuscripts after his death, and published a mixture of his own work and his brother's over the next few years. It is fair to say that he was inspired by studying his brother's manuscripts.

In September 1813 Français published a work in which he gave a geometric representation of complex numbers with interesting applications. This was based on Argand's paper which had been sent, without disclosing the name of the author, by Legendre to François Français. Although Wessel had published an account of the geometric representation of complex numbers in 1799, and then Argand had done so again in 1806, the idea was still little known among mathematicians. This changed after Français' paper since a vigorous discussion between Français, Argand and Servois took place in *Gergonne's Journal*. In this argument Français and Argand believed in the validity of the geometric representation, while Servois argued that complex numbers must be handled using pure algebra.

After this burst of mathematical activity, Français appears to have given up mathematics at the end of 1815. Taton, writes in [1]:-

While not of the first rank, the mathematical activity of the Français brothers merits mention for its originality and diversity.

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

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