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Euclid

geometry space mathematics euclidean

[yoo klid] (lived c .300 BC ) Greek mathematician: recorded, collated and extended mathematics of the ancient world.

Euclid offers strange contrasts: although his work dominated mathematics for over 2000 years, almost nothing is known of his life and personality. One alleged remark survives, his reply to Ptolemy Soter, King of Egypt, who hoped for an easy course of tuition: ‘in geometry there is no straight path for kings’. Working in Alexandria, then a new city but a centre of learning, Euclid brought together previous work in mathematics and his own results and recorded the whole in a systematic way in 13 ‘books’ (chapters), entitled Elements of Geometry . Others are lost.

The system attempted to be fully rigorous in proving each theorem on the basis of its predecessors, back to a set of self-evident axioms. It does not entirely succeed, but it was a noble attempt, and even the study of its deficiencies proved profitable for mathematicians. His work was translated into Arabic, then into Latin and from there into all European languages. Its style became a model for mathematicians and even for other fields of study. Six of the chapters deal with plane geometry, four with the theory of numbers (including a proof that the number of primes is infinite) and three with solid geometry, including the five Platonic solids (the tetrahedron, octahedron, cube, icosahedron and dodecahedron–Euclid finally notes that no other regular polyhedrons are possible).

Only in the 19th-c was it realized that other kinds of geometry exist. This arose from the fact that, while most of the Euclidean postulates are indeed self-evident (eg ‘the whole is greater than the part’), the fifth postulate (‘axiom XI’) is certainly not so. It states that ‘if a point lies outside a straight line, then one (and only one) straight line can be drawn in their plane which passes through the point and which never meets the line’. Then in the 19th-c it was accepted that this certainly cannot be deduced from the other axioms, and and others explored geometries in which this ‘parallel axiom’ is false. In the 20th-c, found that his relativity theory required that the space of the universe be considered as a non-Euclidean space; it needed the type of geometry devised by . For all everyday purposes, Euclidean space serves us well and the practical differences are too small to be significant.

Euclid’s achievement was immense. He was less talented than but for long-lived authority and influence he has no peer. Within the limits of his time (with its inadequate concepts of infinity, little algebra and no convenient arithmetic) his attempt at an unflawed, logical treatment of geometry is remarkable.

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