Islamic heritage in Europe

"Recent research paints a new picture of the debt that we owe to Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the 16th, 17th, and 18th centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects, the mathematics studied today is far closer in style to that of Islamic mathematics than to that of Greek mathematics."

R. Rashed wrote in The development of Arabic mathematics: between arithmetic and algebra:

"Al-Khwarizmi's successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose."

There are three theories about the origins of Arabic Algebra. The first emphasizes Hindu influence, the second emphasizes Mesopotamian or Persian-Syriac influence and the third emphasizes Greek influence. Many scholars believe that it is the result of a combination of all three sources.

The Muslim Persian mathematician Muhammad ibn Mūsā al-khwārizmī was a faculty member of the "House of Wisdom" (Bait al-hikma) in Baghdad, which was established by Al-Mamun. Al-Khwarizmi, who died around 850 A.D., wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian Sindhind. One of al-Khwarizmi's most famous books is entitled Al-jabr wa'l muqabalah or The Compendious Book on Calculation by Completion and Balancing, and it gives an exhaustive account of solving polynomials up to the second degree.

Al-Jabr is divided into six chapters, each of which deals with a different type of formula. The first chapter of Al-Jabr deals with equations whose squares equal its roots (ax� = bx), the second chapter deals with squares equal to number (ax� = c), the third chapter deals with roots equal to a number (bx = c), the fourth chapter deals with squares and roots equal a number (ax� + bx = c), the fifth chapter deals with squares and number equal roots (ax� + c = bx), and the sixth and final chapter deals with roots and number equal to squares (bx + c = ax�).

'Abd al-Hamid ibn-Turk authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi's Al-Jabr and was published at around the same time as, or even possibly earlier than, Al-Jabr. The manuscript gives the exact same geometric demonstration as is found in Al-Jabr, and in one case the same example as found in Al-Jabr, and even goes beyond Al-Jabr by giving a geometric proof that if the determinant is negative then the quadratic equation has no solution. The similarity between these two works has led some historians to conclude that Arabic algebra may have been well developed by the time of al-Khwarizmi and 'Abd al-Hamid.

Al-Karkhi was the successor of Abu'l-Wefa and he was the first to discover the solution to equations of the form ax²ⁿ + bxⁿ = c. Al-Karkhi only considered positive roots.

Omar Khayy�m (c. 1050-1123) wrote a book on Algebra that went beyond Al-Jabr to include equations of the third degree. Omar Khayy�m provided both arithmetic and geometric solutions for quadratic equations, but he only gave geometric solutions for general cubic equations since he mistakenly believed that arithmetic solutions were impossible. His method of solving cubic equations by using intersecting conics had been used by Menaechmus, Archimedes, and Alhazen, but Omar Khayy�m generalized the method to cover all cubic equations with positive roots. He only considered positive roots and he did not go past the third degree. He also saw a strong relationship between Geometry and Algebra.

In the 12th century, Sharaf al-Din al-Tusi found algebraic and numerical solutions to cubic equations and was the first to discover the derivative of cubic polynomials.

J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:

"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before."

Abū al-Hasan ibn Alī al-Qalasādī (1412-1482) was the last major medieval Arab algebraist, who made the first attempt at creating an algebraic notation since Ibn al-Banna two centuries earlier, who was himself the first to make such an attempt since Diophantus and Brahmagupta in ancient times. The syncopated notations of his predecessors, however, lacked symbols for mathematical operations. Al-Qalasadi's algebraic notation was the first to have symbols for these functions and was thus "the first steps toward the introduction of algebraic symbolism." He represented mathematical symbols using characters from the Arabic alphabet