Editing Space (section)

=== Non-Euclidean geometry ===
{{main|Non-Euclidean geometry}}

[[Image:Sphere closed path.svg|thumb|150px|left|[[Spherical geometry]] is similar to [[elliptical geometry]]. On a [[sphere]] (the [[Surface (topology)|surface]] of a [[ball (mathematics)|ball]]) there are no [[parallel line]]s.]]Euclid's ''Elements'' contained five postulates that form the basis for Euclidean geometry. One of these, the [[parallel postulate]], has been the subject of debate among mathematicians for many centuries. It states that on any [[Plane (mathematics)|plane]] on which there is a straight line ''L<sub>1</sub>'' and a point ''P'' not on ''L<sub>1</sub>'', there is exactly one straight line ''L<sub>2</sub>'' on the plane that passes through the point ''P'' and is parallel to the straight line ''L<sub>1</sub>''. Until the 19th century, few doubted the truth of the postulate; instead debate centered over whether it was necessary as an axiom, or whether it was a theory that could be derived from the other axioms.<ref>Carnap, R. ''An Introduction to the Philosophy of Science''. p. 126.</ref> Around 1830 though, the Hungarian [[János Bolyai]] and the Russian [[Nikolai Ivanovich Lobachevsky]] separately published treatises on a type of geometry that does not include the parallel postulate, called [[hyperbolic geometry]]. In this geometry, an [[Infinity|infinite]] number of parallel lines pass through the point ''P''. Consequently, the sum of angles in a triangle is less than 180° and the ratio of a [[circle]]'s [[circumference]] to its [[diameter]] is greater than [[pi]]. In the 1850s, [[Bernhard Riemann]] developed an equivalent theory of [[elliptical geometry]], in which no parallel lines pass through ''P''. In this geometry, triangles have more than 180° and circles have a ratio of circumference-to-diameter that is less than [[pi]].

{| class="wikitable" style="margin: 1em auto 1em auto" style="text-align:center"
! style="width:100px" | Type of geometry || style="width:90px" | Number of parallels || style="width:100px" | Sum of angles in a triangle || style="width:120px" | Ratio of circumference to diameter of circle || style="width:90px" | Measure of curvature
|-
! Hyperbolic
| Infinite || < 180° || > π || < 0
|-
! Euclidean
| 1 || 180° || π || 0
|-
! Elliptical
| 0 || > 180° || < π || > 0
|}