MATHEMATICS

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 sources--the 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 Ibrahim-al-Fazari
in 772-773 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 Ibn-uz-Zaya al-Misri 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 works--Tahir
al-Majisty and Kitab al-Kamil. The last of the Arab translators and commentators of Greek
works was the eminent Arab mathematician Al-Buzjani who died in 998 A.D. He commented upon
and simplified the works of Euclid, Ptolemy and Diophantus.

Arabic translations of the well-known 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 so-called
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 al-Khwarizmi (780--850 A.D.) a native of Khwarizm, who lived in the reign of
Mamun-ar-Rashid, 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 Kitab-ul 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.

Al-Nasavi is the author of Abnugna Fil Hissab Al-Kindi 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.

Al-Karkhi was primarily responsible for popularising Hindu numerals before the advent of
Arabic ones. His book 'Al-kafi fil Hissab was translated into German by Hochhevin and
published at Halle in 1878--80.

Abu Zakariya Muhammad Al-Hissar who probably lived in the 12th century A. D. is the author
of Kitab-us-sagh ir-Jil-h issab . One of its important sections was translated and
published by H. Suter in 1901. AL-Hissar 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 al-Banna, gained much popularity and was published in French by
A. Narre in 1864 and reprinted in Rome in 1865.

Nasir-ud-din 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)
Nasir-uddin has written Al-mutawassat 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.

Al-Khwarizmi the celebrated mathematician is also the author ofHisab Al-Jabr 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 Al-Khwarizmi is lucid and well-arranged. 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 Al-Khwarizmi 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
Mushkilat-i-Hissab, 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 equations--as 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 Al-Buzjani, (940--997, 998 A. D.) is the author of Kitab al-Hindusa 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."'

Nasir-ud-din Toosi, a great intellectual giant of Islam wrote Oawaid-ul-Hindasiya 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. Al-Isfahani 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.

Al-Battani (Latin Albategnius). The nation of trigonometrical ratios, which is now
prevalent, owes its birth to the mathematical talents of Al-Battani. 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 (939--997, 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 tables--the 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 (971--1029 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.

Nasir-ud-din 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 al-Qita 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."' Nasir-ud-din'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.

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