Proving a Postulate
1733
 Girolamo Saccheri (1667-1733) was a Jesuit priest from Italy. First to try proving the parallel postulate using a proof by contradiction, he uncovered two new types of geometry.
Gauss
 Carl Friedrich Gauss (1777-1855) was a German mathematician who tried to prove the fifth postulate. He assumed that multiple parallel lines existed and thought that other types of geometry might exist. He developed hyperbolic geometry, but never went public for fear of controversy.
Bolyai
 Janos Bolyai (1802-1860) was a Hungarian Army officer and a mathematician. He assumed that there were more than one parallel line and published his findings in 1832. He wrote, "out of nothing I have created a strange new world."
Lobachevsky

Nikolai Lobachevsky (1792-1856) was a Russian mathematician, and the first to publish a complete book about this "imaginary geometry" in 1829. This year is considered the birth of Non-Euclidean geometry.

Riemann

Bernhard Riemann (1826-1866) was a German mathematician. He wrote his doctoral dissertation under the supervision of Gauss. His inaugural lecture in 1854 reconfigured the entire concept of geometry. His work was in spherical geometry.