János Bolyai

János Bolyai was a Hungarian mathematician born in 1802 in Kolozsvár, then part of the Kingdom of Hungary, who worked out absolute geometry, a body of results that holds in both Euclidean and hyperbolic geometry. Trained in military engineering at the academy in Vienna and in mathematics by his father Farkas Bolyai, he concluded that Euclid's parallel postulate cannot be proved from the other axioms and that a consistent geometry follows from denying it. His discovery, published as an appendix to his father's work, showed mathematicians that geometry need not describe physical space and helped open the study of abstract structures.