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Euclid is one of the best known and most influential of classical Greek mathematicians but almost nothing is known about his life. He was a founder and member of the academy in Alexandria, and may have been a pupil of Plato in Athens. Despite his great fame Euclid was not one of the greatest of Greek mathematicians and not of the same caliber as Archimedes.

Euclid's most celebrated work is the Elements, which is primarily a treatise on geometry contained in 13 books. The influence of this work not only on the future development of geometry, mathematics, and science, but on the whole of Western thought is hard to exaggerate. Some idea of the importance that has been attached to the Elements is gained from the fact that there have probably been more commentaries written on it than on the Bible. The Elements systematized and organized the work of many previous Greek geometers, such as Theaetetus and Eudoxus, as well as containing many new discoveries that Euclid had made himself. Although mainly concerned with geometry it also deals with such topics as number theory and the theory of irrational quantities. One of the most celebrated number theoretic results is Euclid's proof that there are an infinite number of primes. TheElements is in many ways a synthesis and culmination of Greek mathematics. Euclid and Apollonius of Perga were the last Greek mathematicians of any distinction, and after their time Greek civilization as a whole soon became decadent and sterile.

Euclid's Elements owed its enormously high status to a number of reasons. The most influential single feature was Euclid's use of the axiomatic method whereby all the theorems were laid out as deductions from certain self-evident basic propositions or axioms in such a way that in each successive proof only propositions already proved or axioms were used. This became accepted as the paradigmatically rigorous way of setting out any body of knowledge, and attempts were made to apply it not just to mathematics, but to natural science, theology, and even philosophy and ethics.

However, despite being revered as an almost perfect example of rigorous thinking for almost 2000 years there are considerable defects in Euclid's reasoning. A number of his proofs were found to contain mistakes, the status of the initial axioms themselves was increasingly considered to be problematic, and the definitions of such basic terms as ‘line’ and ‘point’ were found to be unsatisfactory. The most celebrated case is that of the parallel axiom, which states that there is only one straight line passing through a given point and parallel to a given straight line. The status of this axiom was long recognized as problematic, and many unsuccessful attempts were made to deduce it from the remaining axioms. The question was only settled in the 19th century when Janos Bolyai and Nicolai Lobachevski showed that it was perfectly possible to construct a consistent geometry in which Euclid's other axioms were true but in which the parallel axiom was false. This epoch-making discovery displaced Euclidean geometry from the privileged position it had occupied. The question of the relation of Euclid's geometry to the properties of physical space had to wait until the early 20th century for a full answer. Until then it was believed that Euclid's geometry gave a fully accurate description of physical space. No less a thinker than Immanuel Kant had thought that it was logically impossible for space to obey any other geometry. However when Albert Einstein developed his theory of relativity he found that the appropriate geometry for space was not Euclid's but that developed by Georg Riemann. It was subsequently experimentally verified that the geometry of space is indeed non-Euclidean.

In mathematical terms too, the discovery of non-Euclidean geometries was of great importance, since it led to a broadening of the conception of geometry and the development by such mathematicians as Felix Klein of many new geometries very different from Euclid's. It also made mathematicians scrutinize the logical structure of Euclid's geometry far more closely and in 1899 David Hilbert at last gave a definitively rigorous axiomatic treatment of geometry and made an exhaustive investigation of the relations of dependence and independence between the axioms, and of the consistency of the various possible geometries so produced.

Euclid wrote a number of other works besides the Elements, although many of them are now lost and known only through references to them by other classical authors. Those that do survive include Data, containing 94 propositions, On Divisions, and the Optics. One of his sayings has come down to us. When asked by Ptolemy I Soter, the reigning king of Egypt, if there was any quicker way to master geometry than by studying the ElementsEuclid replied “There is no royal road to geometry.”