The early history of advanced geometry is little known, but hotly debated. The deductive-axiomatic method typical of Greek geometry appears to have developed in both mathematics and philosophy and it was probably practiced by Eudoxus in the late fourth century B.C.E. The first complete axiomatic work that has been preserved, however, is Euclid's (c. 325–270 B.C.E.) Elements. Euclid presents geometry as a deductively ordered sequence of propositions derived from a set of indemonstrables. It is known to contain material from earlier mathematicians and its aim was probably to systematize known material rather than to present original work.
With Archimedes (287–212 B.C.E.) the geometrical approach is developed and extended. Archimedes used the axiomatic method to explore new areas such as curvilinear figures; in Plane Equilibria and On Floating Bodies he also made the physical phenomena of statics and hydrostatics accessible to mathematical analysis. While Archimedes' work presents a series of rigorous geometrical proofs, he shows, in the Method, how many of the results were first found through a mechanical method.
Late antiquity was dominated by a mathematical tradition based on commentaries, which produced new classifications, systematizations, and definitions based on earlier work. Despite the dependence on earlier work, the treatises of mathematicians such as Pappus (fl. 320 C.E.) and Proclus (410–485 C.E.) cannot be described as merely derivative.
At any time, the community of advanced practitioners was probably very small. Mathematics as a whole, however, was not a minority pursuit or isolated from the world. Mathematics included disciplines such as optics, mechanics, harmonics, and astronomy, and professions such as builders, astrologers, land measurers, tax collectors, and traders used and displayed mathematical knowledge for a variety of purposes.