Barbier was born on March 18, 1839, in Saint-Hilaire-Cottes, Pas-de-Calais, in the north of France. He studied at the College of Saint-Omer, also in Pas-de-Calais, and then at the Lycée Henri-IV in Paris. He entered the École Normale Supérieure in 1857, and finished his studies there in 1860, the same year in which he published the paper containing his theorem on constant-width curves.
He began teaching at a lycée in Nice, but it was not a success, and he soon moved to a position as an assistant astronomer at the Paris Observatory. He left there in 1865, and in 1880 Joseph Louis François Bertrand found him in the Charenton asylum. Bertrand arranged for Barbier's support and encouraged him to return to mathematical publication.
In this later period of his work, he published ten more papers. He contributed to Bertrand's studies of combinatorics, and announced a generalization of Bertrand's ballot theorem. He was given the Francoeur Prize for his mathematical research by the French Academy of Sciences in multiple years.
- O'Connor, John J.; Robertson, Edmund F., "Joseph Émile Barbier", MacTutor History of Mathematics archive, University of St Andrews.
- Ren, De-lin, Topics in Integral Geometry, World Scientific, 1994, p. 7, ISBN 978-981-02-1107-3.
- Barbier, E. (1860), Note sur le problème de l’aiguille et le jeu du joint couvert, Journal de mathématiques pures et appliquées, 2e série (in French) 5: 273–286.
- Heyde, C. C.; Heyde, Eugene Seneta Editors C. C.; Seneta, Eugene (2001), Statisticians of the Centuries, Springer, p. 186, ISBN 978-0-387-95283-3.
- Addario-Berry, L.; Reed, B. A. (2008), "Ballot theorems, old and new", Horizons of combinatorics, Bolyai Soc. Math. Stud. 17, Berlin: Springer, pp. 9–35, doi:10.1007/978-3-540-77200-2_1, MR 2432525.
- Scientific News, The American Naturalist 18, 1884: 750–752, doi:10.1086/273730.
- Notes and News, Science 7 (154), January 15, 1886: 49–51.