Non-Euclidean Geometry
mathematical treatise) and Theodosius of Bythnia (whose treatise ‘Spherics’ is often considered a supplement to ‘Elements’, allowing its application in astronomy). Its development seems to have arisen along with the understanding that the world is a sphere and the coinciding desire to better navigate our planet. 20
In spherical geometry ‘ lines ’ take the form of the aforementioned great circles. A great circle is any intersection of a sphere that passes through its centre and therefore creates two equal hemispheres. As these are lines, the shortest distance between two points on the surface of a sphere lies along a great circle. This interpretation of a line also violates Euclid’s fifth postulate because as each great circle intersects the sphere at its centre and so must also intersect every other great circle at two points (or one if going by elliptic geometry’s formalized axioms). 21
Visual representation of a great circle of a sphere, where circles a and b are so- called ‘great circles’ and therefore intersect, whilst circle c does not.
There are more substantial changes between Euclidean and spherical geometry. For example, in spherical geometry the sum of the angles of a triangle is greater than 180 degrees. This property is neatly illustrated by the following example: beginning at the north pole, walk approximately 6000 miles south in a straight line (roughly the distance of the north pole from the equator) before turning 90 degrees to the left and walking a further 6000miles (also approximate). Now turn another 90 degrees to the left and walk 6000miles in a straight line. You should arrive back at your starting point at the north pole (assuming you haven’t gotten lost) and your route has constructed an equilateral triangle with sides ~6000 miles and 3 90 degree angles.
Your painstakingly constructed spherical triangle
One of the primary uses of spherical geometry, and the driving force behind its development, is in navigation. Similarly to Euclidean geometry, the shortest route between two points on a sphere is a line which in the context of spherical geometry lies on a corresponding great circle. 22 Some confusion can arise when viewing this shortest route from the perspective of a two-dimensional atlas or map; as the map is a flat representation of curved space one would intuitively assume that Euclidean geometry applies and that (what appears to be) a straight line is shortest route between any two given points. This is not the case. The shortest route is still that which follows the great circle and therefore appears to be a longer, curved line on an atlas.
20 Henderson 1999; Whittlesey 2020. 21 Gowers 2002. 22 Ibid.
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