Euclid based his work on five postulates (today they are called axioms):
These postulates reflect the constructive geometry Euclid, along with his contemporary Greeks, was interested in. The first three postulates basically describe the constructions one can carry out with a compass and an unmarked straightedge or ruler.
The last, socalled "parallel postulate" seems less obvious than the others and many geometers tried in vain to prove it from them. By the mid19th century, it was shown that no such proof exists, because one can construct nonEuclidean geometries where the parallel postulate is false, while the other postulates remain true. Mathematicians say that the parallel postulate is independent of the other postulates. Two alternatives are possible: either an infinite number of parallel lines can be drawn through a point not on a straight line (hyperbolic geometry, also called Lobachevskian geometry), or none can (elliptic geometry, also called Riemannian geometry). That other geometries could be logically consistent was one of the most important discoveries in mathematics, with vast implications for science and philosophy. Indeed, Einstein's theory of general relativity shows that the "real" space in which we live can be nonEuclidean. That Euclid recognized the independence of the parallel postulate long before other mathematicians accepted it is a testament to Euclid's dedication to a logical development from as few assumptions as possible.
One criticism that arose as mathematicians investigated Euclid's system is that Euclid's five axioms are incomplete[?], meaning that they are insufficient to produce the results one would like to be true in Euclidean geometry. Euclid made some hidden assumptions, which were made explicit by later mathematicians. For example, one of his theorems is that any line segment is part of a triangle, which he constructs in the usual way, by drawing circles around both endpoints[?] and taking their intersection and them as three corners. However, his axioms do not guarantee that the circles actually do intersect. David Hilbert gave a revised list containing no fewer than 23 separate axioms. As Gödel proved, all axiomatic systems  excepting the very simplest  are either incomplete or contradict themselves, and this is no exception.
Although Elements is a geometric work, it also includes results that today would be classified as number theory. The contents of the work are as follows:
Books 1 through 4 deal with plane geometry:
Books 5 through 10 introduce ratios and proportions[?]:
Books 11 through 13 deal with spatial geometry:
Search Encyclopedia

Featured Article
