Frenet entered the École Normale Supérieure in 1840, then studied at the University of Toulouse. The University was an ancient establishment founded by Pope Gregory IX in 1229, being the first university founded by a Pope. However the University was suppressed during the French Revolution and by the time Frenet went there to study it had been broken up into separate faculties of law, theology, science, letters, and medicine. It was not reconstituted as a university until 1896.
At Toulouse Frenet undertook research in geometry and he wrote a doctoral thesis there which he submitted in 1847. His thesis was entitled Sur les fonctions qui servent à déterminer l'attraction des sphéroides quelconques. Programme d'une thèse sur quelque propriétés des courbes à double courbure Ⓣ and published in Toulose in 1847. In his thesis Frenet presented the idea of attaching to each point of an arbitrary curve in space a frame. As this frame moves along the curve, we can look at its rate of change to determine how the curve turns and twists, two ideas that actually describe the whole geometry of the curve. The part of Frenet's thesis which contains the theory of space curves gives the formulas now known as the Serret-Frenet formulas. Frenet gave only six formulas while Serret gave all nine. Frenet published this part of his thesis as the paper Sur quelque propriétés des courbes à double courbure Ⓣ in the Journal de mathematique pures et appliques in 1852. In a further paper Théorèmes sur les courbes gauches Ⓣ, published in the Nouvelles annales de mathématiqe in 1853, Frenet explained how the formulas could be applied. In  Godeaux suggests a publication of the first Frenet formula by a Belgian mathematician G M Pagani in (1847) may predates Frenet's publication of the formulas. It is likely that this was produced at about the same time as Frenet's work and that the two were independent.
Frenet was appointed as a professor at the University of Toulouse, then in 1848 he was appointed professor of mathematics at the University of Lyon, the most important educational centre outside Paris. He was also director of the astronomical observatory there and, in this capacity, he conducted meteorological observations. He published these observations in the Mémoires de l'Académie impériale de Lyon in a continuing series, first in 1853, next in 1856, and again in 1858. A Drian took over publishing the series of meteorological observations. Frenet retired at the age of 52 in 1868 and went to live on the family estate of Bayot in Périgueux. He had never married and, once back living on the family estate, he lived with his sister.
Frenet's exercise book on the calculus Recueil d'exercises sur le calcul infinitésimal Ⓣ, first published in 1856, ran to seven editions, the seventh being published in 1917 :-
It contains problems with full solutions and often historical remarks.Struik writes :-
Frenet was a man of wide erudition and a classical scholar who was respected in the community, but his mathematical production was limited.
Article by: J J O'Connor and E F Robertson