His fifth postulate, called the *Parallel Postulate*, states that for any line and any point not on that line, there exists a unique line passing through the point and never intersecting the line. It was long assumed to follow from the other axioms, but in the 19-th 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 Lobatchevsky? and Riemann.

In addition to a treatment of geometry, Euclid's book also contains the beginnings of elementary number theory, such as the notion of divisibility, the greatest common divisor and Euclid's algorithm to determine it, and the infinity of prime numbers.

While the *Elements* was still used in the 20th century as a geometry text book and has been called the hailmark 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.