Eugenio Beltrami (born November 16, 1835,Cremona,Lombardy, Austrian Empire [now in Italy]—died February 18, 1900,Rome, Italy) was an Italian mathematician known for his description ofnon-Euclidean geometry and for his theories of surfaces of constant curvature.

Following his studies at the University of Pavia (1853–56) and later in Milan, Beltrami was invited to join thefaculty at theUniversity of Bologna in 1862 as a visiting professor ofalgebra andanalytic geometry; four years later he was appointed professor of rational mechanics (the application of calculus to the study of the motion of solids and liquids). He also held professorships at universities in Pisa, Rome, and Pavia.

Influenced by the RussianNikolay Ivanovich Lobachevsky and the GermansCarl Friedrich Gauss andBernhard Riemann, Beltrami’s work on thedifferential geometry of curves and surfaces removed any doubts about the validity of non-Euclidean geometry, and it was soon taken up by the GermanFelix Klein, who showed that non-Euclidean geometry was a special case ofprojective geometry. Beltrami’s four-volume work,Opere Matematiche (1902–20), published posthumously, contains his comments on a broad range of physical and mathematical subjects, includingthermodynamics,elasticity,magnetism,optics, andelectricity. Beltrami was a member of the scientific Accademia dei Lincei, serving as president in 1898; he was elected to the Italian Senate a year before his death.

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