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Not Distant Origins?, The Arabic Innovations, European Developments To The Seventeenth Century, Developments With Equations From Descartes To Abel

The word algebra refers to a theory, usually mathematical, which is dominated by the use of words (often abbreviated), signs, and symbols to represent the objects under study (such as numbers), means of their combination (such as addition), and relationships between them (such as inequalities or equations). An algebra cannot be characterized solely as the determination of unknowns, for then most mathematics is algebra.

For a long time the only known algebra, which was and is widely taught at school, represented numbers and/or geometrical magnitudes, and was principally concerned with solving polynomial equations; this might be called "common algebra." But especially during the nineteenth century other algebras were developed.

The discussion below uses a distinction between three modes of algebraic mathematics that was made in 1837 by the great nineteenth-century Irish algebraist W. R. Hamilton (1805–1865): (1) The "practical" is an algebra of some kind, but it only provides a useful set of abbreviations or signs for quantities and operations; (2) In the "theological" mode the algebra furnishes the epistemological basis for the theory involved, which may belong to another branch of mathematics (for example, mechanics); (3) In the "philological" mode the algebra furnishes in some essential way the formal language of the theory.

Lack of space prevents much discussion of the motivations and applications of algebras. The most important were geometries, the differential and integral calculus, and algebraic number theory.

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