Two cases illustrating the history of algebraic expressions (Q2325881)

The author compares and contrasts two traditions from the early history of algebra before the emergence of the general concept of polynomials. In the Arabic school of the followers of al-Karajī in the 10th to 13th centuries, the rhetorical nature of algebra drew in terminology from linguistics. This school investigated the interactions between arithmetic and algebra considering that where the arithmetician manipulates known quantities, the algebraist manipulates unknowns. Algebraic expressions could involve arbitrary \(n\)th powers, both positive and negative, but algebraic equations were restricted to quadratics. Coefficients were treated as adjectives, so that \(3x^2\) represented three squares. In the German Cossic tradition following Stifel algebra was considered part of arithmetic. Powers of unknowns were restricted to positive degree but powers were given symbolic notation and cubic and quartic equations were considered. In the 17th century, ``cossic numbers'' gradually became the somewhat more abstract ``cossic quantities'' and the tradition embraced the notion of ``fictive'' or negative numbers inherited from Cardano.