In 1871, Felix Klein, by adapting a metric discussed by Arthur Cayley in 1852, was able to bring metric properties into a projective setting and was thus able to unify the treatments of hyperbolic, euclidean and elliptic geometry under the umbrella of projective geometry.
42.
The essential difference between the metric geometries is the nature of infinitely many lines through " A " not intersecting " ! ", while in elliptic geometry, any line through " A " intersects " ! ".
43.
Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry, such as proposition 1 from book I of the " Elements ", which states that given any line segment, an equilateral triangle can be constructed with the segment as its base.
44.
However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition that the triangle be right is replaced with the condition that two of the angles sum to the third, say " A " + " B " = " C ".
45.
An important geometry related to that of the sphere is that of the real projective plane; it is obtained by identifying antipodal points ( pairs of opposite points ) on the sphere . ( This is elliptic geometry . ) Locally, the projective plane has all the properties of spherical geometry, but it has different global properties.
46.
The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were " the first few theorems of the elliptic geometries . " These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the later development of non-Euclidean geometry.
47.
Interpreting Euclid's axioms in the spirit of this more modern approach, axioms 1-4 are consistent with either infinite or finite space ( as in elliptic geometry ), and all five axioms are consistent with a variety of topologies ( e . g ., a plane, a cylinder, or a torus for two-dimensional Euclidean geometry ).
48.
In a work titled " Euclides ab Omni Naevo Vindicatus " ( " Euclid Freed from All Flaws " ), published in 1733, Saccheri quickly discarded elliptic geometry as a possibility ( some others of Euclid's axioms must be modified for elliptic geometry to work ) and set to work proving a great number of results in hyperbolic geometry.
49.
In a work titled " Euclides ab Omni Naevo Vindicatus " ( " Euclid Freed from All Flaws " ), published in 1733, Saccheri quickly discarded elliptic geometry as a possibility ( some others of Euclid's axioms must be modified for elliptic geometry to work ) and set to work proving a great number of results in hyperbolic geometry.
50.
For example, Einstein's theory of general relativity has static solutions in which space containing a gravitational field is ( locally ) described by three-dimensional elliptic geometry, but the theory does not posit the existence of a fourth spatial dimension, or even suggest any way in which the existence of a higher-dimensional space could be detected . ( This is unrelated to the treatment of time as a fourth dimension in relativity . ) Metaphorically, we can imagine geometers who are like ants living on the surface of a sphere.
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