Editing Euclidean geometry (section)

==Later history{{anchor|History}}==
{{see also|History of geometry|Non-Euclidean geometry#History}}

===Archimedes and Apollonius===
[[File:Archimedes sphere and cylinder.svg|thumb|right|A sphere has 2/3 the volume and surface area of its circumscribing cylinder. A sphere and cylinder were placed on the tomb of Archimedes at his request.]]
[[Archimedes]] ({{circa|287 BCE|212 BCE}}), a colorful figure about whom many historical anecdotes are recorded, is remembered along with Euclid as one of the greatest of ancient mathematicians. Although the foundations of his work were put in place by Euclid, his work, unlike Euclid's, is believed to have been entirely original.<ref>Eves, p.&nbsp;27.</ref> He proved equations for the volumes and areas of various figures in two and three dimensions, and enunciated the [[Archimedean property]] of finite numbers.

[[Apollonius of Perga]] ({{circa|240 BCE|190 BCE}}) is mainly known for his investigation of conic sections.

[[File:Frans Hals - Portret van René Descartes.jpg|thumb|left|René Descartes. Portrait after [[Frans Hals]], 1648.]]

===17th century: Descartes===
[[René Descartes]] (1596–1650) developed [[analytic geometry]], an alternative method for formalizing geometry which focused on turning geometry into algebra.<ref>Ball, pp. 268ff.</ref>

In this approach, a point on a plane is represented by its [[Cartesian coordinate system|Cartesian]] (''x'', ''y'') coordinates, a line is represented by its equation, and so on.

In Euclid's original approach, the [[Pythagorean theorem]] follows from Euclid's axioms. In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.

The equation
:<math>|PQ|=\sqrt{(p_x-q_x)^2+(p_y-q_y)^2} \, </math>
defining the distance between two points ''P'' = (''p<sub>x</sub>'', ''p<sub>y</sub>'') and ''Q'' = (''q<sub>x</sub>'', ''q<sub>y</sub>'') is then known as the ''Euclidean [[metric space|metric]]'', and other metrics define [[non-Euclidean geometry|non-Euclidean geometries]].

In terms of analytic geometry, the restriction of classical geometry to compass and straightedge constructions means a restriction to first- and second-order equations, e.g., ''y'' = 2''x'' + 1 (a line), or ''x''<sup>2</sup> + ''y''<sup>2</sup> = 7 (a circle).

Also in the 17th century, [[Girard Desargues]], motivated by the theory of [[Perspective (graphical)|perspective]], introduced the concept of idealized points, lines, and planes at infinity. The result can be considered as a type of generalized geometry, [[projective geometry]], but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.<ref>Eves (1963).</ref>

[[File:Squaring the circle.svg|right|thumb|Squaring the circle: the areas of this square and this circle are equal. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized [[compass and straightedge]].]]

===18th century===
Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763, at least 28 different proofs had been published, but all were found incorrect.<ref>Hofstadter 1979, p.&nbsp;91.</ref>

Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem of [[trisecting an angle]] with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until [[Pierre Wantzel]] published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include [[doubling the cube]] and [[squaring the circle]]. In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two,<ref>Theorem 120, Elements of Abstract Algebra, Allan Clark, Dover, {{ISBN|0-486-64725-0}}.</ref> while doubling a cube requires the solution of a third-order equation.

[[Leonhard Euler|Euler]] discussed a generalization of Euclidean geometry called [[affine geometry]], which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an [[equivalence relation]] between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint).

===19th century===
{{comparison_of_geometries.svg}}
In the early 19th century, [[Lazare Carnot|Carnot]] and [[August Ferdinand Möbius|Möbius]] systematically developed the use of signed angles and line segments as a way of simplifying and unifying results.<ref>Eves (1963), p.&nbsp;64.</ref>

====Higher dimensions====
In the 1840s [[William Rowan Hamilton]] developed the [[quaternion]]s, and [[John T. Graves]] and [[Arthur Cayley]] the [[octonion]]s. These are [[normed algebra]]s which extend the [[complex numbers]]. Later it was understood that the quaternions are also a Euclidean geometric system with four real Cartesian coordinates.{{Sfn|Stillwell|2001|p=18–21|ps=; In four-dimensional Euclidean geometry, a [[quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[William Rowan Hamilton|Hamilton]] did not see them as such when he [[History of quaternions|discovered the quaternions]]. [[Ludwig Schläfli|Schläfli]] would be the first to consider [[4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space. Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.}} Cayley used quaternions to study [[rotations in 4-dimensional Euclidean space]].{{Sfn|Perez-Gracia|Thomas|2017|ps=; "It is actually Cayley whom we must thank for the correct development of quaternions as a representation of rotations."}}

At mid-century [[Ludwig Schläfli]] developed the general concept of [[Euclidean space]], extending Euclidean geometry to [[Ludwig Schläfli#Higher dimensions|higher dimensions]]. He defined ''polyschemes'', later called [[polytope]]s, which are the [[Four-dimensional space#Dimensional analogy|higher-dimensional analogues]] of [[polygon]]s and [[polyhedron|polyhedra]]. He developed their theory and discovered all the regular polytopes, i.e. the <math>n</math>-dimensional analogues of regular polygons and [[Platonic solids]]. He found there are six [[Regular 4-polytopes|regular convex polytopes in dimension four]], and three in all higher dimensions.

{{Regular convex 4-polytopes}}

Schläfli performed this work in relative obscurity and it was published in full only posthumously in 1901. It had little influence until it was rediscovered and [[Regular polytopes (book)|fully documented in 1948]] by [[H.S.M. Coxeter]].

In 1878 [[William Kingdon Clifford]] introduced what is now termed [[geometric algebra]], unifying 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. The [[Clifford torus]] on the surface of the [[3-sphere]] is the simplest and most symmetric flat embedding of the Cartesian product of two circles (in the same sense that the surface of a cylinder is "flat").

====Non-Euclidean geometry====
{{Main|Non-Euclidean geometry}}
The century's most influential development in geometry occurred when, around 1830, [[János Bolyai]] and [[Nikolai Ivanovich Lobachevsky]] separately published work on [[non-Euclidean geometry]], in which the parallel postulate is not valid.<ref>Ball, p.&nbsp;485.</ref> Since non-Euclidean geometry is provably relatively consistent with Euclidean geometry, the parallel postulate cannot be proved from the other postulates.

In the 19th century, it was also realized that Euclid's ten axioms and common notions do not suffice to prove all of the theorems stated in the ''Elements''. For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself. The very first geometric proof in the ''Elements,'' shown in the figure above, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third [[wikt:vertex|vertex]]. His axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the [[Real number#Completeness|completeness]] property of the real numbers. Starting with [[Moritz Pasch]] in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of [[Hilbert's axioms|Hilbert]],<ref>* [[Howard Eves]], 1997 (1958). ''Foundations and Fundamental Concepts of Mathematics''. Dover.</ref> [[Birkhoff's axioms|George Birkhoff]],<ref>Birkhoff, G. D., 1932, "A Set of Postulates for Plane Geometry (Based on Scale and Protractors)", Annals of Mathematics 33.</ref> and [[Tarski's axioms|Tarski]].<ref name="Tarski 1951">Tarski (1951).</ref>

===20th century and relativity===
[[File:1919 eclipse negative.jpg|thumb|right|A disproof of Euclidean geometry as a description of physical space. In a 1919 test of the general theory of relativity, stars (marked with short horizontal lines) were photographed during a solar [[eclipse]]. The rays of starlight were bent by the Sun's gravity on their way to Earth. This is interpreted as evidence in favor of Einstein's prediction that gravity would cause deviations from Euclidean geometry.]]

[[Albert Einstein|Einstein's]] theory of [[special relativity]] involves a four-dimensional [[space-time]], the [[Minkowski space]], which is [[non-Euclidean geometry|non-Euclidean]]. This shows that non-Euclidean geometries, which had been introduced a few years earlier for showing that the [[parallel postulate]] cannot be proved, are also useful for describing the physical world.

However, the three-dimensional "space part" of the Minkowski space remains the space of Euclidean geometry. This is not the case with [[general relativity]], for which the geometry of the space part of space-time is not Euclidean geometry.<ref>Misner, Thorne, and Wheeler (1973), p.&nbsp;191.</ref> For example, if a triangle is constructed out of three rays of light, then in general the interior angles do not add up to 180 degrees due to gravity. A relatively weak gravitational field, such as the Earth's or the Sun's, is represented by a metric that is approximately, but not exactly, Euclidean. Until the 20th century, there was no technology capable of detecting these deviations in rays of light from Euclidean geometry, but Einstein predicted that such deviations would exist. They were later verified by observations such as the slight bending of starlight by the Sun during a solar eclipse in 1919, and such considerations are now an integral part of the software that runs the [[Global Positioning System|GPS]] system.<ref>Rizos, Chris. [[University of New South Wales]]. [http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm GPS Satellite Signals] {{Webarchive|url=https://web.archive.org/web/20100612004027/http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm |date=2010-06-12 }}. 1999.</ref>