A Comparative Study On Two Algebraic Works Between 9～(th) And 10～(th) In Islamic World

Al-Khwarizml and Abu Kamil both are the famous algebraists between 9' and 10th in Islamic world. In Al-Khwarizmi's Algebra, it first gave all the solutions to the quadratic equations. And from then on, the solution to the equations is considered to be the basic characteristic of algebra. It provided the direction of algebra. Abu Kamil is the first algebraist after Al-Khwarizm I. His Algebra is a connecting link between the preceding and the following for the development of algebra. This paper makes the compare of the two works form the source, content and the influence. The writer points that Al-Khwarizml learnt wildly from other' s strong points, in his Algebra we can find the influence of Babylon, India, Syria and Persia; Al-Khwarizml excluded the Euclidian logistic deduction method, but accepted Helen' s practical geometrical method. Al-Khwa rizmi' s geometry seems different from "geometrical algebra" of Euclid in essence, but is similar to Chinese ancient mathematical method— out-in complementary-like principle. Abu Kamil' s Algebra is based on Al-Khwarizmi' s Algebra, there are 15 examples same or similarly in the two works. In some degree, the latter is considered to be the commendatory of the former. But different from of Al-Khw arizmi,when the coefficient of quadratic equations is irrational number, Abu Kamil abandoned the geometry demonstration. It shows the trend of arithmetization. Besides Abu Kamil gives the more general, more systemic methods of the solution to quadratic equation, more various examples. Both of the works exerted an influence on the Arabian mathematicians, such as Al-Karaji, Omar Khayyam, Samaw' al and so on . They were also introduced to Europe. Most of their contents were brought into Practica Geometriae by Fibonacci, so they come down widely.