Euclidian geometry stood unchallenged until the 19th century, when the development of non-Euclidian geometries began. Non-Euclidian geometry may be thought of as any form of geometry different to that devised by Euclid. This development prompted a revolution in human knowledge – not least that Einstein’s Theory of Relativity only works in non-Euclidian space.
For around 2,000 years, no one seriously challenged Euclid’s theories. But his fifth postulate, concerning parallel lines, had long posed certain problems for some of the greatest mathematicians from the Western and Arabic traditions (Omar Khayyám, Nasir al-Din al-Tusi, and Giovanni Girolamo Saccheri to name a few) who attempted to formulate alternative theories, their work helping pave the way for the eventual discovery of non-Euclidian geometries.
The emergence of non-Euclidian geometry has allowed for innovations across a disparate array of mathematics and science disciplines, but it has also had philosophical significance. After millennia in which most mathematicians assumed Euclid had revealed a system of unassailable truth, his teachings were shown to be but one of many interpretations of space. While freeing Einstein to consider the cosmos in an entirely original way – and liberating generations of mathematicians, scientists and engineers to tackle problems with new tools – the full implications of the existence of non-Euclidian space remain to be discovered.
Hyperbolic and elliptic geometry
In the early 1830s, two mathematicians – the Russian Nikolai Lobachevsky and the Hungarian János Bolyai – published independent treatises on hyperbolic geometry, thus sharing credit for defining the first non-Euclidian geometry. Hyperbolic geometry (often known as Bolyai-Lobachevskian geometry) deals with ‘curved’ space and though it has much in common with Euclidian geometry, incorporates fundamental differences too. For instance, in hyperbolic geometry ultraparallels curve away from each other the further they get from the point of intersection with their shared perpendicular. It is within the realm of hyperbolic geometry that we may make sense of the General Theory of Relativity.
Then, in the mid-1850s, the German Bernhard Riemann introduced his theories of elliptic geometry, in which space is considered to be a sphere and lines as great circles. Again, Euclid’s fifth postulate required rewriting, since in this system the parallel lines curve towards each other until finally intersecting. Hyperbolic and elliptic geometry remain the classical non-Euclidian geometries, but a significant number of other self-consistent non-Euclidian systems have subsequently been discovered. In other words, space does not exist in a single, standardized form as suggested by Euclid’s geometry.