Mathematics

Mathematics developed well from antiquity also in the Far East, with distinct traditions in India, China, Japan, Korea, and Vietnam. Arithmetic, geometry, and mechanics were again prominent; special features include a powerful Chinese method equivalent to solving a system of linear equations, a pretty theory of touching circles in Japanese "temple geometry," and pioneering work on number theory by the Indians. They also introduced the place-value system of numerals to base 10, of which we use a descendant that developed after several changes in adopted symbols.

This system of numerals was mediated into Europe by mathematicians working in medieval Islamic civilization, often though not always writing in Arabic. They became the dominant culture in mathematics from the ninth century and continued strongly until the fourteenth. They assimilated much Greek mathematics; indeed, they are our only source for some of it.

The first major author was al-Khwarizmi (fl. c. 800–847), who laid the foundations of algebra, especially the solution of equations. He and his followers launched the theory using words rather than special symbols to mark unknowns and operations. Other interests in geometry included attempts to prove Euclid's parallel axiom and applications to optics and trigonometry; an important case of the latter was determining the qibla (that is, the direction of Mecca) for any time and place at times of Muslim prayer. Their massive contributions to astronomy included theory and manufacture of astrolabes.