One of the greatest mathematicians, Euclid lived under Ptolemy I and taught at Alexandria. His Elements contain thirteen books: six on plane geometry, three on the theory of numbers, one on irrationals, and three on solid geometry. Euclidean geometry is the greatest example of the pure axiomatic method, and as such had incalculable philosophical influence as a paradigm of rational certainty. It had no competitor until the 19th century when it was realized that the fifth axiom of his system (parallel lines never meet) could be denied without inconsistency, leading to Riemannian spherical geometry. The fifth chapter of Euclid’s Elements is attributed to the mathematician Eudoxus, and contains a precise development of the real numbers, work which remained unappreciated until rediscovered in the 19th century. See also geometry.