It is known that the Islamic civilization in the past is an advanced civilization both in the field of culture and science. In the mathematical sciences in particular countries or Islamic empire also contribute to the development of science, especially the Persian, the Middle East, Central Asia, North Africa, Iberia, and in parts of India. They make a significant contribution to mathematics.
Here are some Muslim figure who contribute to the development of mathematics:
a. Muhammad ibn Musa Al-Khawarizmi
Muhammad ibn Musa Al-Khawarizmi is a Persian mathematican in the 9th century. He wrote several important books on the Hindu-Arabic numerals and on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of Al-Kindi, were instrumental in spreading Indian mathematicsand India numerals to the West. The word algorithm is derived from the Latinization of his name, Algoritmi, and the word algebra from the title of one of his book, Al-kitab al-mukhtasar fi hisab al-gabr wa'l-muqbala (The Compendious Book on Calculation by Completion and Balancing). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, and he was the first to teach algebra in an elementary form and for its own sake. He also discussed the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as al-jabr. His algebra was also no longer concerned "with a series of problem to be resolved, but an exposition which starts with primitive terms in which the combination must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems.
b. Al-Karaji
Al-Karaji made further development in algebra in his book al-fakhri. On that book he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorm, pascal triangle, and the sum of integral cubes. The historian of mathematics, F. Woepcke, praised Al-Karaji for being "the first who introduced the theory of algebraic calculus"
c. Abul Wafa
Abul wafa (10 June 940 – 15 July 998) was a Persian mathematician and astronomer who worked in Baghdad. He made important innovations in spherical trigonometry, and his work on arithmetis for businessmen contains the first instance of using negative numbers in a medieval Islamic text.
He is also credited of compiling tables of sines and tangents at 15' intervals. He also introduced the sec and cosec and studied the interrelations between the six trigonometric lines associated with an arc. His Almagest was widely read by medieval Arabic astronomers in the centuries after his death. He is known to have written several other books that have not survived.
d. Ibn Al-Haytham
Ibn Al-Haytham was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume, and was able to generalize his result for the integrals of polynomial up to the fourth degree. He thus came close to finding a general formula for the integral of polynomials, but he was not concerned with any polynomials higher than the fourth degree.
e. Omar Khayyam
In the late 11th century, Omar Khayam wrote Discussions of the Difficulties in Euclid, a book about what he perceived as flaws in Euclid's Elements, especially the parallel postulate. He was also the first to find the general geometric solution to cubic equations.
f. Nasir al-Din Tusi
In the 13th century, Nasir al-Din Tusi (Nasireddin) made advances in spherical trigonometry. He also wrote influential work on Euclid's parallel postulate.
g. Ghiyath al-Kashi
In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots, which was a special case of the methods given many centuries later by Ruffini and Horner.
So many achievment of Muslim mathematicians during this period, but during the time of the Ottoman Empire from the 15th century, the development of Islamic mathematics is stagnant.
Source : http://en.wikipedia.org/wiki/History_of_mathematics
