It took Biot some months to recover from the illness during which time he continued with his studies of mathematics. He took the entrance examinations for the École des Ponts et Chausées and was accepted in January 1794. The École Polytechnique was founded later in 1794 (actually named 'École centrale des travaux publics' for its first year) and Biot transferred there in November of that year. Gaspard Monge, one of the founders of l'École Polytechnique who taught the first intake of students, quickly realised Biot's potential. Biot, however, quickly became involved in student politics and was made a section leader. There was an attempted insurrection by the royalists against the Convention and Biot took part. He was captured by government forces and taken prisoner. Had it not been for Monge, who could not see someone with such talents remain in jail, or even die, pleading successfully for his release his promising career might have ended. It is not surprising that for a second time he had come close to a death sentence, giving the dangerous times in Revolutionary France through which he lived. He returned to his studies at the École Polytechnique, befriending a fellow student, Siméon-Denis Poisson, and graduated in 1797. He had found a patron in Sylvestre-François Lacroix who was highly influential in helping Biot develop his career.
He became Professor of Mathematics at the École Centrale de l'Oise at Beauvais in 1797. It might seems remarkable that even someone as able as Biot could move from being an undergraduate straight into a professorship. However, things had worked in his favour :-
The Convention, in a law of 25 February 1795 had called for a system of Écoles Centrales, one in each large town, to replace the Collèges of the Ancien Régime as seats of secondary education. The curricula of the new schools was practical and modern and included two years devoted to mathematics, experimental physics and chemistry. These subjects had never been taught before on a high school level, so, inevitably, there was a shortage of qualified teachers.Biot got to know of the vacant mathematics professorship through his friendship with Barnabé Brisson, the son of Antoine François Brisson de Beauvais, while studying at l'École Polytechnique. Through Barnabé Brisson, Biot had got to know his sister Gabrielle. Although Gabrielle was only sixteen years old, they married in 1797 soon after Biot took up his position at Beauvais. Biot taught Gabrielle mathematics and physics so that she might translate into French a German text by Ernst Gottfried Fischer. It was Claude Louis Berthollet who had asked Biot to make the translation which was published as Physique mécanique Ⓣ in 1806. Biot's wife was well educated and fluent in German, so her translation was excellent. However, in line with the practice of the time, the book records Biot himself as the translator rather than his wife. Biot and his wife had a son Édouard Constant Biot who was born in 1803.
It was largely through Lacroix that Biot had been appointed at Beauvais and Lacroix advised him frequently through the first years that he worked there. However, he also managed to get support in his career from Laplace. In fact in late 1799 Biot approached Laplace, who had taught him at l'École Polytechnique, and offered to proof-read the Mécanique céleste Ⓣ which at that time was with the publisher. He did not give up when Laplace said "No thanks" but persisted in a polite way and eventually Laplace agreed. Biot was, by this time, an entrance examiner at the École Polytechnique so was frequently in Paris. He wrote :-
From that time on, each time I went to Paris I brought my proof-reading work and personally presented it to M Laplace. He always received it with kindness, examined it and discussed it, and that gave me the opportunity to submit to him the difficulties that had stopped me. His willingness to explain them was boundless. But even he could not always do it without stopping to think for a long while. This usually occurred in places where he had used the expedient phrase 'it can easily be seen'.Certainly the effort required by Biot was enormous. He wrote near the end of 1799 :-
I spent all of my days at this work [proof-reading the 'Mécanique céleste'] which I felt to be important in several respects, and I hope I have succeeded. I worked like a devil to finish it on time, and 1 remained at the task eighteen hours in a row the last day, without eating or drinking. Finally, thank heaven, it is finished, and I can flatter myself that I understand the 'Mecanique celeste' Ⓣ. If this were to be the only benefit I would get from it, it would still be a lot.Laplace was also interested in the research on mathematics which Biot was undertaking and gave him advice both on the material and on getting it published. In 1800 he was appointed Professor of Mathematical Physics at the Collège de France, an appointment which was due mainly to the influence of Laplace. In 1800 Biot was elected an associate to the First Class of the Institute, replacing Jean Montucla who died in December 1799. Again Laplace's support was important to Biot in this election. In January 1803 the Institut was reorganised and Delambre became perpetual secretary, creating a vacancy in the Mathematics Section. Although Biot had produced excellent mathematics memoirs in the years up to 1800, he had concentrated on experimental physics from that time on. However, presumably to improve his chances of gaining a place in the Mathematics Section, he submitted a memoir on the axes of tautochronous curves just before the election was to take place. It was sufficient to keep his mathematical reputation high and he was elected on 11 April 1803.
The first balloon ascent made for scientific purposes was by Biot and Joseph-Louis Gay-Lussac from the garden of the Conservatoire des Arts on 24 August 1804. They achieved a height of 4000 metres and measured magnetic, electrical, and chemical properties of the atmosphere at various heights. In 1806, again with the support of Laplace, Biot was appointed as an assistant astronomer at the Bureau de Longitudes in addition to his other roles. On 3 September of that year he set out with François Arago to Formentera, in the Balearic Islands, to complete earlier work begun there on calculating the measure of the arc of the meridian. They were still undertaking measurements when, in May 1808, Napoleon declared his brother Joseph Bonaparte as Spanish ruler and the War of Independence began. Biot and Arago must have looked extremely suspicious; two Frenchmen with sophisticated measuring instruments working on Spanish territory. Biot fled back to France immediately. Later in 1808, together with Claude-Louis Mathieu, he embarked on a series of measurements of the length of the seconds pendulum at different points on the meridian, in particular at Bordeaux and at Dunkirk. Both Mathieu and Biot received a prize from the Académie des Sciences in 1809 for this highly accurate work, and in 1812 they received a second prize from the Academy for their achievements. In 1809 Biot was appointed Professor of Physical Astronomy at the Faculty of Sciences. He held this position for over fifty years.
Biot studied a wide range of mathematical topics, mostly on the applied mathematics side. He made advances in astronomy, elasticity, electricity and magnetism, heat and optics on the applied side while, in pure mathematics, he also did important work in geometry. He collaborated with Arago on refractive properties of gases. Biot, together with Felix Savart, discovered that the intensity of the magnetic field set up by a current flowing through a wire varies inversely with the distance from the wire. This is now known as Biot-Savart's Law and is fundamental to modern electromagnetic theory. Morris Kline, reviewing , writes:-
Biot depended entirely upon experiments to determine the interaction of a magnet and a straight line of electrical current and then inferred a mathematical law concerning the force between current and magnet.Light was the topic on which Biot devoted most time, making the major discovery of the laws of rotary polarization by crystalline bodies. Having discovered these laws he used them in analysis of saccharine solutions using an instrument called a polarimeter which he invented. For this work on the polarisation of light passing through chemical solutions he was awarded the Rumford Medal of the Royal Society of London in 1840. He had been elected as a foreign member of the Royal Society in 1815.
One of his important works was Mémoire sur la figure de la terre Ⓣ (1827) which describes the shape of the Earth. Among his other major works we mention: Analyse de la mécanique céleste de M Laplace Ⓣ (1801); Traité analytique des courbes et des surfaces du second degré Ⓣ (1802); Recherches sur l'intégration des équations différentielles partielles et sur les vibrations des surfaces Ⓣ (1803); Essai de géométrie analytique appliqué aux courbes et aux surfaces de second ordre Ⓣ (1806); Recherches expérimentales et mathématiques sur les mouvements des molécules de la lumière autour de leur centre de gravité Ⓣ (1814); Traité de physique experimentale et mathématique Ⓣ (1816); Precis de physique Ⓣ (1817); (with Arago) Recueil d'observations géodésiques, astronomiques et physiques exécutées en Espagne et Écosse Ⓣ (1821); Mémoire sur la vraie constitution de l'atmosphère terrestre Ⓣ (1841); Traité élémentaire d'astronomie physique Ⓣ (1805); Recherches sur plusieurs points de l'astronomie égyptienne Ⓣ (1823); Recherches sur l'ancienne astronomie chinoise Ⓣ (1840); Études sur l'astronomie indienne et sur l'astronomie chinoise Ⓣ (1862); Essai sur l'histoire générale des sciences pendant la Révolution Ⓣ (1803); Discours sur Montaigne Ⓣ (1812); Lettres sur l'approvisionnement de Paris et sur le commerce des grains Ⓣ (1835); Traite d'astronomie physique Ⓣ (1850); and Mélanges scientifiques et littéraires Ⓣ (1858).
He tried twice for the post of Secretary to the Académie des Sciences and to improve his chances for election to this post he wrote Essai sur l'Histoire Générale des Sciences pendant la Révolution. However he lost out in 1822 to Fourier for this post, then again when Fourier died in May 1830 he applied again for the post of Secretary, only to lose to Arago on this occasion. The rivalry between Arago and Biot was again evident in 1839 when different photographic processes were competing :-
Arago and Biot, France's leading authorities in the field of optics, had spent the past thirty years disagreeing about the ability of optical instruments to represent the world. Photography became one more opportunity for them to disagree, and they did not shrink from the task. The debate extended beyond questions of assigning priority and apportioning credit. At its heart was the question of what the photographic surfaces showed, what relation the visible inscription bore to the real world.Arago supported the Daguerre photographic process with silver plates while Biot championed an approach with paper soaked in a silver solution as developed by Henry Fox Talbot. This was not because he thought this process produced clearer photographs, rather it was because he believed that the images captured 'chemical radiation' invisible to the eye. We should note that the idea of 'chemical radiation' was widely believed at this time. Biot also believed that the photographic process should be one reserved for scientific use and not made available for public use. There was a period of collaboration between Biot and Talbot, the two exchanging letters frequently. Basically the photographic process became another tool for Biot to use in his investigations of light which had always been a major interest to him.
As well as interests in almost every branch of science, Biot was interested in the history of the subject. He published works on this topic such as: Essai sur l'histoire générale des sciences pendant la Révolution française Ⓣ (1803); Notice historique sur la vie et les ouvrages de Newton Ⓣ (1822); Recherches sur plusieurs points de l'astronomie égyptienne appliquées aux monuments astronomiques trouvés en Égypte Ⓣ (1823); Sur la manière de calculer les positions des étoiles relativement à l'équateur et à l'écliptique pour les époques anciennes Ⓣ (1823); Opuscule sur l'Astronomie ancienne des Chinois, des Indous et des Arabes Ⓣ (1840); Mémoire sur le Zodiaque de Denderah Ⓣ (1844); Précis de l'histoire de l'Astronomie planétaire Ⓣ (1847); and Études sur l'astronomie indienne et l'astronomie chinoise Ⓣ (1862).
In  St Beuve says that Biot was endowed to the highest degree with all the qualities of curiosity, finesse, penetration, precision, ingenious analysis, method, clarity, in short with all the essential and secondary qualities, bar one, genius, in the sense of originality and invention.
A contrasting comment by Olinthus Gregory in 1821 is:-
With regard to M Biot, I had an opportunity of pretty fully appreciating his character when we were together in the Zetland [= Shetland] Isles; and I do not hesitate to say that I never met so strange a compound of vanity, impetuosity, fickleness, and natural partiality, as is exhibited in his character.In addition to the honours we mentioned above, Biot was also honoured by being made Chevalier of the Legion of Honour in 1814 and Commander in 1849. He was elected to the to the Academy of Inscriptions and Belles-Lettre in 1841 and to the French Academy in 1856.
Article by: J J O'Connor and E F Robertson