Beltrami, Eugenio (1835–1899)

Italian mathematician who pioneered modern non-Euclidean geometry. His work ranged over almost the whole field of pure and applied mathematics, but especially theories of surfaces and space of constant curvature.

Beltrami was born in Cremona, Lombardy, and studied mathematics at Pavia. He held academic posts at Bologna, Pisa, Pavia, and Rome.

In 1862 he published his first paper, an analysis of the differential geometry of curves, to which he would return in his most important paper, ‘Saggio di interpretazione della geometria non-euclidia’ (1868). This advanced a theory of hyperbolic space that laid the analytical base for the development of non-Euclidean geometry.

Beltrami demonstrated that the concepts and formulae of Russian mathematician Nikolai Lobachevsky's geometry are realized for geodesics on surfaces of constant negative curvature. He showed also that there are rotation surfaces of this kind – he called them ‘pseudospherical surfaces’. He also demonstrated the usefulness of employing differential parameters in surface theory, thereby beginning the use of invariant methods in differential geometry.

After 1872 Beltrami switched his attention to questions of applied mathematics, especially problems in elasticity and electromagnetism.