Euclidian geometry depends on a number of axioms, most important of which concerns the property of parallel lines. Non-Euclidian geometry changed Euclidian axioms. It was to have application in physics, particularly Albert Einstein’s theory of relativity, when it enabled a concept of “the fourth dimension.”

Carl Friedrich Gauss (1777–1855) was the first to figure out the principles of non-Euclidian geometry, although because he did not publish his ideas, the credit was given to Janos Bolyai (1802–1860) and Nikolai Lobachevsky (1792–1856), who were working independently. They rejected the Euclidian assumption that could not be proved in which only one line passes through a point in a plane that is parallel to a separate coplanar line. In their new system, a line can have more than one parallel and the sum of the angles of a triangle may be less than 180 degrees.

By the middle of the nineteenth century, Bernhard Riemann (1826–1866) developed a geometry in which straight lines always meet, thereby having no parallels, and in addition allowing for the sum of the angles of a triangle to be greater than 180 degrees. (In Euclidian geometry, parallel lines never meet and the sum of the angles of a triangle is always 180 degrees.) Reimann also went on to distinguish between the unboundedness of space as part of its extent, and the infinite measure over which distance could be taken that is related to the curvature of the same space. Riemann returned to Gauss’ now-published work and explained the new ideas of distance first introduced by Loybachevski and Bolyai in terms of trigonometry. The bottom line was that “arc length” could be understood as the shortest distance between two points on a surface, without reference to the geometric properties or applicable geometry of that in which the surface itself was imbedded.

In 1868, Eugenio Beltrami (1835–1899) demonstrated a model of a Bolyai-type two-dimensional space, inside a planar circle. This proved that the consistency of non-Euclidian geometry depended on the consistency of Euclidian geometry, thus reassuring skeptics that non-Euclidian geometry was valid.