Algebras

Several branches of mathematics must have primeval, unknown, origins: for example, arithmetic, geometry, trigonometry, and mechanics. But algebra is not one of them. While Mesopotamian and other ancient cultures show evidence of methods of determining numerical quantities, the means required need only arithmetical calculations; no symbolism is evident, or needed. Concerning the Greeks, the Elements of Euclid (fourth century B.C.E.), a discourse on plane and solid geometry with some arithmetic, was often regarded as "geometric algebra"; that is, the theories thought out in algebraic terms. While it can easily be so rendered, this reading has been discredited as historical. For one reason among many, in algebra one takes the square on length a to be a times a but Euclid worked with geometrical magnitudes such as lines, and never multiplied them together. The only extant Greek case of algebraization is the number theory of Diophantus of Alexandria (fl. c. 250 C.E.), who did use symbols for unknowns and means of their combination; however, others did not take up his system.

A similar judgment applies to ancient Chinese ways of solving systems of linear equations. While their brilliant collection of rules can be rendered in terms of the modern manipulation of matrices, they did not create matrix theory.