Medieval Arabic algebra is a good example of an artificial language.Yet despite its abstract, formal structure, its utility was restricted to problem solving. Geometry was the branch of mathematics used for expressing theories. While algebra was an art concerned with finding specific unknown numbers, geometry dealtwith generalmagnitudes.Algebra did possess the generosity needed to raise it to a more theoretical level—in the ninth century Abū Kāmil reinterpreted the algebraic unknown “thing” to prove a general result. But mathematicians had no motive to (...) rework their theories in algebraic form. Because it offered no advantage over geometry, algebra remained a practical art in both the Islamic world and in Europe until the scientific uphevals of the 17th–18th centuries. (shrink)

It is shown in this article that the two sides of an equation in the medieval Arabic algebra are aggregations of the algebraic “numbers” (powers) with no operations present. Unlike an expression such as our 3x + 4, the Arabic polynomial “three things and four dirhams” is merely a collection of seven objects of two different types. Ideally, the two sides of an equation were polynomials so the Arabic algebraists preferred to work out all operations of the enunciation to a (...) problem before stating an equation. Some difficult problems which involve square roots and divisions cannot be handled nicely by this basic method, so we do find square roots of polynomials and expressions of the form “A divided by B” in some equations. But rather than initiate a reconsideration of the notion of equation, these developments were used only for particularly complex problems. Also, the algebraic notation practiced in the Maghreb in the later middle ages was developed with the “aggregations” interpretation in mind, so it had no noticeable impact on the concept of polynomial. Arabic algebraists continued to solve problems by working operations before setting up an equation to the end of the medieval period. (shrink)

This paper presents the first edition, translation and analyse of al-Mns commentary of the Book X of Euclid one. For the first time, irrational numbers are defined and classified. The algebraisation of Elementsrizms Algebra, shows irrational numbers as solution to algebraic quadratic equations. The algebraic calculus makes here the first steps. On this occasion, negative numbers and their calculation rules appears. Simplifications imposed by the algebraic writings are sometimes in opposition with the conclusions of propositions conceived in a purely geometrical (...) framework, revealing a contradiction between geometrical and algebraic goals. It will be resolved by the independant way algebra will take with mathematicians belonging to the tradition of al-Karajal from the 11th-12th centuries on. (shrink)