Editing History of geometry (section)

===The 18th and 19th centuries===

====Non-Euclidean geometry====
The very old problem of proving Euclid's Fifth Postulate, the "[[Parallel Postulate]]", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. Though Omar Khayyám was also unsuccessful in proving the parallel postulate, his criticisms of Euclid's theories of parallels and his proof of properties of figures in non-Euclidean geometries contributed to the eventual development of [[non-Euclidean geometry]]. By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. [[Saccheri]], [[Johann Heinrich Lambert|Lambert]], and [[Adrien-Marie Legendre|Legendre]] each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, [[Carl Friedrich Gauss|Gauss]], [[János Bolyai|Johann Bolyai]], and [[Nikolai Lobachevsky|Lobachevsky]], each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. By 1854, [[Bernhard Riemann]], a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for [[Albert Einstein|Einstein]]'s [[theory of relativity]].
[[Image:Newton-WilliamBlake.jpg|thumb|left|[[William Blake]]'s "Newton" is a demonstration of his opposition to the 'single-vision' of [[scientific materialism]]; here, [[Isaac Newton]] is shown as 'divine geometer' (1795).]]
It remained to be proved mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by [[Eugenio Beltrami|Beltrami]] in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry.

While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense (interstellar, not earth-bound) distances. With the development of relativity theory in physics, this question became vastly more complicated.

====Introduction of mathematical rigor====
All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms, now known as [[Hilbert's axioms]], were given by [[David Hilbert]] in 1894 in his dissertation ''Grundlagen der Geometrie'' (''Foundations of Geometry'').

====Analysis situs, or topology====
In the mid-18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a [[metric space]] was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus- and analysis-related concepts came to be known as analysis situs, and later as [[topology]]. The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and non-Euclidean geometry. Topology soon became a separate field of major importance, rather than a sub-field of geometry or analysis.

====Geometry of more than 3 dimensions====
{{see also|Euclidean geometry#19th century}}

The 19th century saw the development of the general concept of Euclidean space by [[Ludwig Schläfli]], who extended Euclidean geometry beyond three dimensions. He discovered all the [[Four-dimensional space#Dimensional analogy|higher-dimensional analogues]] of the [[Platonic solids]], finding that there are exactly six such [[Regular 4-polytopes|regular convex polytopes in dimension four]], and three in all higher dimensions.

In 1878 [[William Kingdon Clifford]] introduced what is now termed [[geometric algebra]], unifying [[William Rowan Hamilton]]'s [[quaternions]] with [[Hermann Grassmann]]'s algebra and revealing the geometric nature of these systems, especially in four dimensions. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modeled to new positions.