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Euclidean Geometry

  Euclidean geometry, in mathematics, is the term used to describe the whole of classical geometry, that is, everything that was known about the relationships between points and lines until the 18th and 19th centuries. It was called Euclidean because only insignificant advances had been made since the time that Euclid (3rd century  BCE) catalogued all that was known in his time in the field in his Elements, a book which became the standard for rigorous mathematics until the 19th century. In it, Euclid derived the results known in his time (such as Pythagoras\' theorem) from a small number of rules (axioms) and definitions (see axiomatization).

Although most of Euclid\'s axioms were accepted without quibble, the axiom of parallels was not. A parallel is a line which passes through a given point and does not cross a given line; Euclid\'s axiom states that for any line and any point not on the line, there is exactly one parallel to the line through the point. The reason it was not accepted is that in some situations it does not seem to be true in the real world: a pair of railway lines, for example, appear to meet on the horizon. Even Euclid does not seem to be convinced of the truth of the axiom, for he derived all the results he possibly could from the other axioms before the axiom of parallels was used at all.

For 2,000 years after the production of the Elements, mathematicians attempted to show that the parallel axiom could in fact be deduced from the other axioms proposed by Euclid, without success: it turned out to be an impossible task. Several mathematicians from the 16th century onwards tried to do this by assuming the opposite (that there were no parallels or at least two) and trying to derive a contradiction to show that the supposition was absurd. In the case where there were assumed to be no parallels, a contradiction was found, but in the other case, none was. None of the mathematicians who studied this took the next step of announcing the discovery of non-Euclidean geometry, being afraid of the derision they would receive from other mathematicians. In the end, it was left to two obscure mathematicians, Johann Bolyai (1802 - 1860) and , Nicholas Lobachevsky (1792 - 1856), to publish their conclusions: that assuming the existence of at least two parallels was equally valid to assuming exactly one. Mathematicians slowly began to accept this work, as models were devised which had exactly the properties of Bolyai-Lobachevsky geometry. As the 20th century began, the position of Euclidean geometry as that which governed the universe was displaced, especially after Einstein used non-Euclidean geometry to formulate his general theory of relativity.

Greek geometry had also come up with three problems, which no-one could solve. Every construction in the Elements can be carried out with a straight edge and compasses; no measuring implement was allowed for either angle or distance. The first is known as squaring the circle; it is to construct (using only compasses and straight-edge) a square of the same area as a given circle. The second is, given an arbitrary angle, to construct (using only compasses and straight-edge) the angle which is one third the size of the given one. The third is, given a cube, to construct (using only compasses and straight-edge) a new cube of exactly double the volume. Using the tools of Galois theory, it is now known that all these problems are impossible within the rules; unfortunately that does not stop mathematical societies from receiving hundreds of incorrect constructions every year. SMcL



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