Non-Euclidean Geometry
Max Marchini
In this essay I will investigate the development and ramifications of non-Euclidean geometry (any branch of the subject in which Euclid’s axioms do not hold) through the examinations of two distinct examples of it, Hyperbolic and Spherical geometry. Euclid was a Greek mathematician born in the 4 th century BCE. He was active in Alexandria and wrote his seminal work, Elements , there. 1 Euclid’s Elements is a compilation of 13 books dealing with Geometry. In Book 1 of Elements , Euclid lays out his 5 famous postulates: 1. Any two points can be joined by exactly one line segment. 2. Any line segment can be extended to exactly one line. 3. Given any point P and any length r , there is a circle of radius r with centre P . 4. Any two right angles are congruent. 5. If a straight line N intersects two straight lines L and M, and if the interior angles on one side of N add up to less than two right angles, then the lines L and M intersect on that side of N.
Euclid then uses these 5 postulates to derive a number of different theorems which form the basis for what is now known as Euclidean geometry. 2
Postulate number 5 (known colloquially as the parallel postulate) has historically garnered the most attention: it is notably more complex than the other four and this complexity has led many people to try and derive it from the other four axioms. 3 All of these attempts have ended in failure. In the context of the discovery and development of non-Euclidean geometry, this controversial 5 th postulate is the most important of the five. The first major milestone in the development of a formal alternative geometry comes in 1829-1830 and 1832, when Russian mathematician Nikolay Lobachevsky and Hungarian mathematician Janos Bolyai independently published separate treatises on hyperbolic geometry. 4 Upon being notified of this ‘ new ’ discovery by Bolyai’s father (a mathematician who spent much of his life attempting to prove the parallel postulate), Gauss (a distinguished German mathematician) claimed to have already come across these ideas, but decided against publishing them. 5 Lobachevsky and Bolyai developed their respective geometries in separate ways. Bolyai rejected the parallel postulate and all of its alternatives, thereby creating a geometry (which he dubbed ‘ absolute geometry ’ ) whose theorems were consistent with both Euclidean and Hyperbolic Geometry. This is because in both of those two subtopics of geometry, Euclid’s first four axioms remain consistent. 6
1 Taisbak 1998. 2 Gowers 2002. 3 Gardner 2001. 4 Henderson 1999; Gowers 2002.
5 Gowers 2002. 6 Gowers 2002.
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