He was born in Cremona in Lombardy, then a part of the Austrian Empire, and now part of Italy. Beltrami first began studying mathematics at University of Pavia in 1853, but was forced to discontinue his studies in 1856 because of financial hardship. He was appointed to the University of Bologna as a professor in 1862, the year he published his first paper. Beltrami later taught at universities in Pisa, Rome and Pavia. He died in Rome in 1900.
In 1868, (in Essay on an interpretation of non-Euclidean geometry) Beltrami gave the first model of hyperbolic geometry. In Beltrami's model, lines of hyperbolic geometry are represented by geodesics on the pseudosphere. Thus, Beltrami provided the first proof that Euclid'sparallel postulate could not be derived from the other axioms of Euclidean geometry.
Beltrami first began studying mathematics at University of Pavia in 1853, but was forced to discontinue his studies in 1856 because of financial hardship.
Thus, Beltrami attempted to prove that Euclid's parallel postulate could not be derived from the other axioms of Euclidean geometry; this proof fails however since the pseudosphere is only a small portion of the hyperbolic plane.
Beltrami studied at Pavia from 1853 to 1856, and there he was taught by Brioschi who had been appointed as professor of applied mathematics at the University of Pavia the year before Beltrami began his studies.
Beltrami would have liked to continue his mathematical studies but he was suffering financial hardship so in 1856 he had to stop his studies and take up a job.
Beltrami showed that not all geodesics could be represented in this way and he then went on to consider the natural question of which surfaces had the property that geodesics on the surface could be represented as straight lines on the plane.
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