History of geometry
Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, "The Elements" is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.
Foundations of mathematics
After many failed attempts to derive the parallel postulate from other axioms, the study of the still hypothetical hyperbolic geometry by Johann Heinrich Lambert (1728–1777) led him to introduce the hyperbolic functions and compute the area of a hyperbolic triangle (where the sum of angles is less than 180°). Then the Russian mathematician Nikolai Lobachevsky (1792–1856) established in 1826 (and published in 1829) the coherence of this geometry (thus the independence of the parallel postulate), in parallel with the Hungarian mathematician János Bolyai (1802–1860) in 1832, and with Gauss.
Later in the 19th century, the German mathematician Bernhard Riemann developed Elliptic geometry, another non-Euclidean geometry where no parallel can be found and the sum of angles in a triangle is more than 180°. It was proved consistent by defining point to mean a pair of antipodal points on a fixed sphere and line to mean a great circle on the sphere. At that time, the main method for proving the consistency of a set of axioms was to provide a model for it.