Euclid (c. 330-c. 260 BC)

Greek mathematician who wrote the Stoicheia/Elements in 13 books, nine of which deal with plane and solid geometry and four with number theory. His great achievement lay in the systematic arrangement of previous mathematical discoveries and a methodology based on axioms, definitions, and theorems.

Euclid's works, and the style in which they were presented, formed the basis for all mathematical thought and expression for the next 2,000 years. He used two main styles of presentation: the synthetic (in which one proceeds from the known to the unknown via logical steps) and the analytical (in which one posits the unknown and works towards it from the known, again via logical steps). Both methods were based on axioms (statements assumed to be true), and from which mathematical propositions, or theorems, were deduced.

In the Elements, Euclid incorporated and developed the work of previous mathematicians as well as including his own many innovations. He was rigorous about the actual detail of the mathematical work, attempting to provide proofs for every one of the theorems. The first six books deal with plane geometry (points, lines, triangles, squares, parallelograms, circles, and so on), and includes hypotheses such as Pythagoras' theorem, which Euclid generalized, and the theorem that only one line can be drawn through a given point parallel to another line.

Books 7 to 9 are concerned with arithmetic and number theory, including Euclid's proof that there are an infinite number of prime numbers. In book 10 Euclid treats irrational numbers, and books 11 to 13 discuss solid geometry, ending with the five Platonic solids (the tetrahedron, octahedron, cube, icosahedron, and dodecahedron).

Very little is known about Euclid. No record is preserved of his date or place of birth, his education, or his date and place of death. The influence of Plato is detectable in his work - so Euclid must either have been contemporary or later. Some commentators have suggested that he attended Plato's Academy in Athens but, if so, it is likely to have been after Plato's death. It has been established that Euclid went to the recently-founded city of Alexandria (now in Egypt) in 300 BC and set up his own school of mathematics there. His mathematical works survived in almost complete form because they were translated first into Arabic, then into Latin; before being translated into other European languages.

In Elements, Euclid used the synthetic approach; he used the analytical mode of presentation in his other important mathematical work, Treasury of Analysis.

Euclid

Town in Cuyahoga County, northeast Ohio; population (2000) 52,700. It is located 19 km/12 mi northeast of Cleveland. Founded in 1798, it is a residential centre that also manufactures such products as multigraphing and road machinery, aircraft parts, and office machines.