Muslims have made immense contributions to almost all branches of the sciences and arts, but mathematics was their favourite subject and its development owes a great deal to the genius of Arab and persian scholars. The advancement in different branches of mathematical science commenced during the Caliphate of Omayyads, and Hajjaj bin Yusuf, who was himself a translator of Euclid as well as a great patron of mathematicians.
Translations
Whatever mathematical knowledge Arabs inherited came from two sourcesthe Hindus and the Greeks. The scholars of the Darul Hukarna of Mamun did the largest amount of work for the advancement of the sciences and arts by the Arabs. Abu Abdulla Muhammad IbrahimalFazari in 772773 A.D. translated Sidhanta from Sanskrit into Arabic, which, according to G. Sarton provided "possibly the vehicle by means of which the Hindu numerals were transmitted from India to Islam". The works of Greek mathematicians which were translated during the Abbasid Caliphate and served as the starting point for Arab mathematicians were those of Euclid, Ptolemy, Antolyscus, Aristarchos and Archimedes. Hajjaj bin Yusuf was the first to translate Euclid's Elements into Arabic while Abdur Rahman and Muhammad Ibn Muhammad Baqi wrote commentaries on the IOth book of Euclid. The latter's contribution was translated into Latin by Gerard of Cremona and edited by H. Suter in 1907. Ibrahim IbnuzZaya alMisri who died in 912 A.D. has written commentaries on Ptolemy's Centiloquim and Proportions, which influenced modern thought immensely. Abul Abbas Nairizi wrote commentaries on the works of Ptolemy and Euclid, which also were later translated into Latin by Gerard of Cremona. Abul Wafa, the celebrated mathematician, included a simplified version of Ptolemy's Almagest in his well known worksTahir alMajisty and Kitab alKamil. The last of the Arab translators and commentators of Greek works was the eminent Arab mathematician AlBuzjani who died in 998 A.D. He commented upon and simplified the works of Euclid, Ptolemy and Diophantus.
Arabic translations of the wellknown mathematical works of those times gave the Arabs the sources to develop the science of mathematics to an admirably high degree and later scientists owe much to the Arab genius. Writing in The Spirit of Islam, Ameer Ali says, "Every branch of higher mathematics bears tracts of their genius. The Greeks are said to have invented algebra, but among them, as Oelsner has justly remarked, it was confined to furnishing amusement 'for the plays of the goblet'. The Muslims applied it to higher purposes, and thus gave it a value hitherto unknown. Under Mamun they had discovered the equations of the second degree, and very soon after they developed the theory of quadratic aquations and the binomial theorem. Not only alalgebra geometry and arithmetic, but optics and mechanics made remarkable progress in the hands of the Muslims. They invented spherical trigonometry; they were the first to apply algebra to geometry, to introduce the tangent, and to substitute the sine for the arc in trigonometrical calculations. Their progress in mathematical geography was no less remarkable". Even the socalled enlightened west which has at times taken pains to minimise the greatness of Muslim achievements in furthering the cause of human civilization, had to admit, though half heartedly, the outstanding part played by the Arabs. "For with this limited ambition", says Carra De Vaux in Legacy of Islam, "the Arabs have really achieved great things in science; they taught the use of ciphers, although they did not invent them, and thus became the founders of arithmetic of every day life; they made algebra an exact science and developed it considerably and laid the foundations of analytical geometry; they were indisputably the founders of plane and spherical trigonometry which, properly speaking, did not exist among the Greeks"." Thus Muslims were pioneers in the development of mathematical sciences in mediaeval times.
Arithmetic
Arabs were the founders of every day arithmetic and taught the use of ciphers to the world.
Musa alKhwarizmi (780850 A.D.) a native of Khwarizm, who lived in the reign of MamunarRashid, was one of the greatest mathematicians of all times. He composed the oldest Islamic works on arithmetic and algebra which were the principal source of knowledge on the subject for a fairly long time. George Sarton pays glowing tribute to this outstanding Muslim mathematician and considers him "one of the greatest scientists of his race and the greatest of his time".' He systematised Greek and Hindu mathematical knowledge and profoundly influenced mathematical thought during mediaeval times. He championed the use of Hindu numerals and has the distinction of being the author of the oldest Arabic work on arithmetic known as Kitabul Jama wat Tafriq. The original version of this work has disappeared but its Latin translation Trattati a" Arithmetic edited by Bon Compagni in 1157 at Rome is still in existence.
AlNasavi is the author of Abnugna Fil Hissab AlKindi short extracts of which were published by F. Woepeke in the journal Asiatique in 1863. His arithmetic explains the division of fractions and the extraction of square and cubic roots in an almost modern manner. He introduced the decimal system in place of sexagesimal system.
AlKarkhi was primarily responsible for popularising Hindu numerals before the advent of Arabic ones. His book 'Alkafi fil Hissab was translated into German by Hochhevin and published at Halle in 187880.
Abu Zakariya Muhammad AlHissar who probably lived in the 12th century A. D. is the author of Kitabussagh irJilh issab . One of its important sections was translated and published by H. Suter in 1901. ALHissar was the first mathematician who started writing fractions in their present form with a horizontal line. A commentary on his treatise on arithmetic, written by Ibn alBanna, gained much popularity and was published in French by A. Narre in 1864 and reprinted in Rome in 1865.
Nasiruddin Toosi, a versatile genius, who was a prolific writer and has written more than hundreds of valuable books to his credit, has the distinction of being one of the greatest scientists and mathematicians of Islam. Born in 1201 A. D. in Toos (Persia) Nasiruddin has written Almutawassat and a short but concise book on arithmetic which is available both in Arabic and Persian.
Arabic numerals including zero were the greatest contributions made by the Arabs to the mathematical science. The outstanding quality of Arabic numerals lies in the fact that they possess an absolute value. Huroful Ghubar was a novel form of numerals adopted in Spain by 950 A. D. The most significant numeral invented by the Arabs was zero which according to Carra De Vaux "was used by the Arabs at least 250 years before it became known in the west". Before the introduction of the zero it was necessary to arrange all figures in columns to differentiate between tens, hundreds, thousands, etc. The earliest use of the zero is given as 873 A. D.
Algebra
Is a word derived from the Arabic source AlJabar and is the product of Arabic genius.
AlKhwarizmi the celebrated mathematician is also the author ofHisab AlJabr Wal Muqabla, an outstanding work on algebra which contains analytical solutions of linear and quadratic equations. Khwarizmi has the distinction of being one of the founders of algebra who developed this branch of science to an exceptionally high degree. He also gives geometric solutions of quadratic equations, e.g., x2+10x=39 an equation which was repeated by later mathematicians. Robert Chester was the first to translate this book into Latin in 1145 A. D. which introduced Algebra into Europe. Later on this book was translated by Gerard of Cremona also. The Algebra written by AlKhwarizmi is lucid and wellarranged. After dealing with equations of the second degree, the learned mathematician discussed algebraic multiplications and divisions. Writing in The Legacy of Islam Carra De Vaux says, "In the 18th century Leonardo Fibonacci of Pisa, an algebraist of considerable importance says he owed a great deal to the Arabs."' He travelled in Egypt, Syria, Greece and Sicily and learned the Arabic methods there, recognised it to be superior to the method of Pythagoras and composed a liber Abaci in 15 chapters, the rest of which deals with algebraic calculations. Leonardo enumerates the six cases of the quadratic equations just as AlKhwarizmi gives them. The translation by Robert Chester of Khwarizmi's algebra marks the beginning of the era of the introduction and advancement of this branch of science in Europe. "The importance of Robert's Latin translation of Khwarizmi's algebra", says a modern orientalist, "can hardly be exaggerated because it marked the beginning of European Algebra."
Omar Khayyam, the celebrated poet, philosopher, astronomer and mathematician has left behind an excellent book on algebra. His works on algebra were translated in 1851, while his Ruhaiyat were first pubished in 1859. The manuscripts of his principal works exist in Paris and in the India Office London; Mosadrat, researches on Euclid's axioms, and MushkilatiHissab, dealing with complicated arithmetical problems, have been preserved in Munich (Germany). According to V· Minorsky, "He was the greatest mathematician of mediaeval times." His primary contribution is in algebra in which he has registered much advance on the work of the Greeks. His algebra is an advance on that of Khwarizmi too in the degree of equationsas the greater Part of Omar's book is devoted to the cubic equations only. His algebra deals with the geometric and algebraic solution of equation of the second degree and includes an admirable classification of equations based on the number and different terms which they include. He recognises thirteen different forms of cubic equations. His solution of cubic and quadratic equations by the conic section method is probably the most advanced work of Arabic mathematics that has survived. "His skill as a geometer" says Carra De Vaux, "is equal to his literary erudition and reveals real logical power and penetration."
Abul Kamil improved upon the algebra of Khwarizmi. He dealt with quadratic equations, multiplication and division of algebraic quantities, addition and subtraction of radicals and the algebraic treatment of' pentagons and decagons.
Abu Bakr Karkhi, who adorned the court' of Fakhrul Mulk in the beginning of 11th century wrote an outstanding treatise on algebra known as AlFakhri. This is one of the best books on the subject left by a Muslim mathematician and was published by Woepeke in Paris in 1853 A.D.
Geometry, Like other branches of mathematics, geometry made much headway in the hands of Muslims. The three famous brothers Muhammad, Ahmad and Hassan, sons of Musa bin Shakir, wrote an excellent work on geometry which was translated into Latin by Gerard of Cremona. This was later translated into German by M. Kurtaza.
Abul Wafa AlBuzjani, (940997, 998 A. D.) is the author of Kitab alHindusa which was rendered into Persian by one of his friends. "It had a large number of" says H. Suter, "geometrical problems for the fundamental construction ofplane geometry to the constructions of the corners of a regular polyhedron on the circumscribed sphere of special interest is the fact that a number of these problems are solved by a single span of the compass, a condition which we find for the first time here."'
Nasiruddin Toosi, a great intellectual giant of Islam wrote OawaidulHindasiya a book of geometry. Arabs were much in advance of Hindus and Greeks in the development and use of arithmetic, algebra and geometry.
Thabit bin Qurra is universally recognised as the greatest Muslim geometer. He was born in Harran and knew Greek and Syraic languages very well, so that he could read books of these countries in original. He wrote a number of short treatises on astronomy and mathematics. His treatise on Balance was translated into Latin by Gerard of Cremona. AlIsfahani has contributed to conics. Isfahani also translated Greek works on Conics.
Trigonometry
It has been universally acknowledged that plane and spherical trigonometry were founded by Muslims who developed it considerably. The Greeks and other advanced nations of the ancient world were ignorant of this essential branch of mathematics.
Khwarizmi, the Muslim mathematician has made valuable contributions to this branch of mathematics also..His trigonometrical tables which deal with the sine and tangent were translated into Latin in 1126 A. D. by Adelard of Bath.
AlBattani (Latin Albategnius). The nation of trigonometrical ratios, which is now prevalent, owes its birth to the mathematical talents of AlBattani. The third chapter, of his astronomical work, dealing with trigonometry, was several times translated into Latin and Spanish languages.
Jabir Bin Afiah is the author of the celebrated book Kitab Elahia which deals with astronomy and trigonometry. "His book Kitab Elahia says H. Suter, "is noteworthy for preparing the astronomical part with a special chapter on trigonometry. In his spherical trigonometry he takes the rule of the four magnitudes as the foundation for the deviations of his formulae and gives for the first time the fifth main formula of the right angled triangle."' His work was translated in Latin by Gerard of Cremona.
Abul Wafa (939997, 998 A.D.) born at Buzjan in Khorasan later on established in Iraq was one of the greatest mathematicians that Islam has produced. He devoted himself to the researches in mathematics and astronomy. His Zijush Shamil (consolidated tables) are distinguished for their accurate observation and he introduced as well as popularised the use of the secant and tangent in trigonometry. "But this was not all" says Sedillot; "struck by the imperfection of the lunar theory of Ptolemy, he verified the ancient observations, and discovered, independently of the equation of the centre and the eviction, a third inequality, which is no other than the variation determined six centuries later by Tycho Brahe."' Abul Wafa was also an outstanding geometer who studied the quadrature of parabola and the volume of paraboloid. Writing in the Legacy of Islam, Carra de Vaux says, "Abul Wafa's services to trigonometry are indisputable. With him trigonometry becomes all the more explicit."" G. Sarton pays glowing tribute to the genius of this Muslim mathematician when he say’s, "Abul Wafa contributed considerably to the development of trigonometry. He was probably first to show the generality of sine theorem relative to spherical triangles. He gave a new method of constructing sine tablesthe value of sine 30 being correct to the 8th decimal place......He made a special study of tangent; calculated a table of tangents; introduced the secant and cosecant; knew those simple relations between the six trigonometric lines, which are now often used to define them.""
Abul Hasan Koshiyar (9711029 A. D.) was a Persian mathematician who wrote his works in Arabic. He played a dominant role in the development of trigonometry. His main subject was the elaboration and explanation of the tangent.
Nasiruddin Toosi, a versatile genius, played no mean part in the development of trigonometry. His works on trigonometry mark the culmination of the advancement on the subject. He is the author of the Kitab shakl alQita in which trigonometry has been treated independently of astronomy. The book is very comprehensive and rather the best work on the subject written in mediaeval times. It was translated into French and edited by Alexandre Cara Theodory Pasha in 1891. Carra de Vaux says "Trigonometry, plane or spherical, is now well established and finds in this book its first methodically developed and deliberate expression."' Nasiruddin's book remained to be the greatest work of its kind until De triangulurs of Regiomontenus two centuries later.
Such were the great mathematical giants which the Muslim world produced, who were not only the pioneers of mathematical science during mediaeval times, but are considered to be authorities on several mathematical problems even during the modern age. The development of mathematics owes a great deal to the genius of these Muslim luminaries.
Comment received on this article:
you write that the Mulsim were the discoverers of Spherical trigonometry and that the Greek (and other nations) were ignorant of this branc of trigonometry.
I am sorry to say, but that is not true!
The foundation of spherical trigonometry was laid by the Greek Hipparchus, who needed spherical trigonometry to study the a triangle projected by three stars on the celestial sphere.
Hipparchus was also credited to be generally the father of trigonometry because he made an actual table of ratio's what we would now refer to as a trig table although earlier remnants of the meaning of sine and cosine go back to pioneers before him
It is true that the Muslims have advanced mathematics (certainly spherical trig w.r.t. prayer timings and Kibla direction) to such an extent that all great mathematicians from other nations have learned from "us" in order to receive credit for what they have discovered later on.

