Euclid of Alexandria (Greek: Eukleides) (circa 365-275 BC) was a Greek mathematician who lived in the 3rd century BC in Alexandria. His most famous work is the Elements, a book in which he deduces the properties of geometrical objects and integers from a set of axioms, thereby anticipating the axiomatic method of modern mathematics. Although many of the results in the Elements originated with earlier mathematicians, one of Euclid's major accomplishments was to present them in a single logically coherent framework. The geometry of Euclid was known for ages as "the" geometry, but is nowadays referred to as Euclidean geometry.
The fifth postulate of the Euclidean geometry, called the Parallel Postulate, states that if a straight line (note: in Euclid's terminology a line may be finite) intersects two other straight lines, and the sum of the interior angles on one side of the line is less than 180 degrees (literally "two right angles"), then the two lines, if they are lengthened indefintely, will intersect on the same side on the line as the interior angles. Since this axiom is less obvious than the others, many mathematicians tried to derive it from the others. Then, in the 19th century, Janos Bolyai (and probably Carl Friedrich Gauss before him) realized that its negation leads to consistent non-euclidean geometries, which were later developed by Lobachevsky and Riemann.
In addition to a treatment of plane geometry, including proofs of the Pythagorean theorem and a version of the more general law of cosines, Euclid's book also contains the beginnings of elementary number theory, such as the notion of divisibility, the greatest common divisor and the Euclidean algorithm to determine it, and the infinity of prime numbers. Later chapters deal with three-dimensional geometry and the platonic solids. The book also contains proofs that the area of a circle is proportional to the square of its radius, and that the volume of a sphere is proportional to the cube of its radius.
While the Elements was still used in the 20th century as a geometry text book and has been considered a fine example of the formally precise axiomatic method, Euclid's treatment does not hold up to modern standards and some logically necessary axioms are missing. The first correct axiomatic treatment of geometry was provided by Hilbert in 1899.