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“That fondness for science, ... that affability and condescension which God shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the
removal of difficulties, has encouraged me to compose a short work on calculating by al-jabr and al-muqabala, confining it to what is easiest and most useful in arithmetic.”
Muhammad ibn Musa al-Khwarizmi (Arab Mathematician, 780-850)

   The handmaiden of the sciences, as mathematics is called, affects our early
school years in such a way that few of us afterwards want to have any give and
take with the subject. Yet, ironically, maths is all around us. One way or another, almost every aspect of our modern civilization is based on calculations. Be it in architecture, astronomy, medicine, hi-tech development and the Internet, or mere payment in the local store, we make use of mathematical reckoning throughout our lives. And, are we lucky to live in this advanced age, taking for granted all
historically accumulated knowledge in science, much of which is based on mathematics?

   Throughout history, different centres of civilization such as those of ancient India, Mesopotamia, Egypt, and Greece contributed greatly for the study of mathematics and all its practical functions. No less important was the role of the Arab mathematics for the development of the discipline, which made popular the Arabic numeration system, the concept of zero, geometry, and algebra.

The Arab Contribution to Mathematics

   In order to understand the history of mathematics, it is essential to review its development in the Muslim world in the period from the 9th to the 15th centuries AD.
To begin with, there was a relationship between early Muslim mathematics and the mathematics of Hellenistic and Sanskritic schools, and it looks like the Arabs found useful those earlier Greek and Hindu pre-algebra efforts.

   Learned Muslim men found interest in the mathematical questions. As a point in case, Thabit ibn Qurrah (836-901) translated the works of key Greek mathematicians for the time. As a typical scenario, ibn Qurrah studied and improved those Hellenistic compositions. One such work was Nicomachus of Gerasa's Arithmetic, which he revised and which led him to finding the rule for amicable numbers. These are a pair
of numbers such that each number of the pair is the sum of the set of proper divisors of the other number. The search for amicable numbers since has established a long-term fascination with them in the Muslim centers of study. Later in history, Kamal ad-Din al-Farisi in the 14th century found the pair 17,926 and 18,416 as an illustration of ibn Qurrah’s rule, and Muhammad Baqir Yazdi in the 17th century produced the pair 9,363,584 and 9,437,056.

Al-Khwarizmi – the Great Muslim Mathematician

   Among the rest of the early Arab scientists, there is but one name that deserves closer attention when the influence of Muslim mathematics comes under discussion.
It is that of the 9th century scientist called Muhammad ibn Musa al-Khwarizmi. Famous as the father of Algebra, he was born about 780AD near Baghdad, and lived until about 850AD. Al- Khwarizmi settled in Baghdad which was under the caliphates of al-Ma`mun and al-Mu'tasim in what is considered to be the first Golden Age of Muslim science He produced his most important work in about 830, and called it Kitab al-jabr wa’l-muqabala, or the Book of Restoring and Balancing. It dealt with "what is easiest and most useful." From the term “al-jabr,” translating as "restoring," in the title, we get “algebra” which was the way the term was translated into Latin in the 12th century. Restoring, in this case, referred to the method of taking a subtracted quantity from one side and placing it to the other side of an equation.

   Al-Khwarizmi’s book Kitab al-jabr wa’l-muqabala proposes a set of rules for arithmetical solutions of linear and quadratic equations, and for elementary geometry. It also resolves inheritance problems regarding the division of money according to proportions, which was in line with the complex requirements of Muslim religious law. The whole work built upon an extended tradition beginning with Babylonian mathematics of the 2nd century BC, going through stages of Greek, Hebrew, and Hindu development. The book served as the prime example for later scientists such
as the Egyptian Abu Kamil. Even millennia later, Kitab al-jabr wa’l-muqabala was still used as the standard mathematics text at universities in Europe until the 16th century.

   Working as an instructor in the academic institution known as the “House of Wisdom,” Al-Khwarizmi explicitly presented Indian influence in his works and produced a book on Hindu arithmetic. It was entitled The Book of Addition and Subtraction According to the Hindu Calculation. This volume was translated in Latin as Algoritmi de Numero Indorum, which means “Al-Khwarizmi Concerning the Hindu Art of Reckoning.” The “Algoritmi” in this translation of the title made popular the term “algorithm.” In the book, Al-Khwarizmi tackled and found solutions for specific algebra equations called “quadratic equations,” which are widely used in science today.

   As a sign of service to the Muslim faith, al-Khwarizmi's developed a method to calculate the time of visibility of the new moon, indicating the beginning of the Muslim month.

   Another of Al-Khwarizmi’s realizations is the arrangement of a system for quadratics. The Latin version of Kitab al-jabr wa’l-muqabala sets off with the positional rule for numbers and continues with the solutions in six chapters of six kinds of quadratics:

The 6 kinds of quadratics classified by al-Khwarizmi are:

1. Squares equal to roots (x² = square root of 2)

2. Squares equal to numbers (x² = 2)

3. Roots equal to numbers (square root of x = 2)

4. Squares and roots equal to numbers (x² + 3x = 25)

5. Squares and numbers equal to roots (x² + 1 = 9)

6. Roots and numbers equal to squares (3x + 4 = x²)


10th Century Arabic Mathematics

   The 10th millennium saw Muslim mathematical study concentrated in three main sub-disciplines. These were the ongoing progress in algebra, the development of arithmetic algorithms, and the increasing complexity in geometry. In addition, the introduction of the zero was destined to revolutionize mathematics as it allowed for key innovations. It was proposed by Muhammad Bin Ahmad in 967 AD. Zero arrived in the West much later, in 13th century.

   As a result of the work in arithmetic development, three numeration systems were invented. The “finger” arithmetic used was one of them. It relied on mental arithmetic and on the help of the fingers to memorize results in the course of calculation. The system found practical application for records of treasury administrators. The finger system was widely used throughout the 10th and 11th centuries, and leading mathematicians such as Abu al-Wafa (940-998) wrote on this system



Abu Al-Wafa

   The Arab mathematician Abu Al-Wafa Al-Buzajani translated and improved on the works of the Greek mathematicians Euclid and Diophantus and of forerunner Al-Khwarizmi. Al-Wafa composed Kitab fima yahtaj ilayh al-kuttab wa al-ummal min 'ilm al-hisab, which translates as A Book on What is Necessary from the Science of Arithmetic for Scribes and Businessmen, and Kitab fima yahtaj ilayh al-sani 'min al-a'mal al-Handasiyha, or A Book on What is Necessary from Geometric Construction for the Artisan. The exceptional achievements of Al-Wafa include the invention of a field in geometry, which deals with problems leading to equations in algebra of a higher degree than the second. His labor included work on the polyhedral theory and on the development of trigonometry, much of which he put to use in astronomy. Al-Wafa’s success in astronomy was marked by his creation of the first wall quadrant for studying the stars. From his observatory in Baghdad, he used mathematics in his lunar theory experiments, where he employed the tangent and cotangent trigonometric functions. In addition, Al-Wafa formulated the secant and cosecant functions, showed the generality of the sine theorem for spherical triangles, and thought of a technique of using sine tables.

   Al-Wafa’s case demonstrates the evolvement of 10th century Islamic algebra from Al-Khwarizmi's quadratic polynomials to the expressions algebra understanding, which comprised arbitrary positive or negative integral powers of the unknown. A movement occurred which dealt with the similarity between the rules for operating with powers of the unknown in algebra and with powers of 10 in arithmetic. This prompted an intertwined relationship between the growth of algebra and arithmetic from the 10th to the 12th century. Thus, by the 14th century symbolism in algebra was widely used in the western Muslim territories.  

   In the meantime, there was intense study in other branches of algebra too. The further improvement of earlier Hellenistic and Sanskritic works was under way. The basis of the ancient outcomes mixed well with the new inventions in algebra. Muslim mathematicians such as Abu Jafar Al-Khazan and Abu Kamil of the 10th century were involved in the investigation of the equations of Diophantine. They were also working to prove a special case of what later became popular as Fermat's last theorem. It maintains that rational solution to the equation x3 + y3 = z3 does not exist.

   Apart from the progress made in the disciplines of algebra and arithmetic, geometry as well developed significantly. Thabit ibn Qurrah’s grandson, Ibrahim ibn Sinan (908-946), was involved in the study of geometry and particularly in tangents to circles. He also investigated the apparent motion of the Sun and the geometry of shadows. With his grandfather, Ibrahim formulated a method for designing the curves needed for sundials.

   In pure mathematical terms, Ibrahim ibn Qurrah’s most valuable labor dealt with the quadrature of the parabola. As a result of his efforts, he produced a system of integration more general than the system of Archimedes. Admitedly, it was in fact Ibrahim’s grandfather Thabit ibn Qurra who had begun working with integration in a different manner than did Archimedes. Ibrahim, however, was the one to grasp that there were improvements on what his grandfather had reached.

   Ibrahim ibn Qurrah also deserves respect for being the most advanced Arab mathematician to be concerned with mathematical philosophy. The following passage illustrates his contemplation: “I have found that contemporary geometers have neglected the method of Apollonius in analysis and synthesis, as they have in most of the things I have brought forward, and that they have limited themselves to analysis alone in so restrictive a manner that they have led people to believe that this analysis did not correspond to the synthesis effected.”

Arabic Mathematics Worldwide

   In the 11th century, the Arab mathematical foundation was one of the strongest in the world. The Muslim mathematicians had invented geometrical algebra and had taken it to advanced levels, capable of solving third and fourth degree equations. The world witnessed a new stage in the development of mathematical science, driven by the numerous translated works from Arabic into European languages.

   Indubitably, Al-Khwarizmi was very influential with his methods on arithmetic and algebra which were translated into much of southern Europe. Again, these translations became popular as algorismi – a term which is derived from the name of Al-Khwarizmi. Not all went smoothly nonetheless. The Arabic numerals introduced by Al-Khwarizmi, like much of new mathematics, were not welcomed wholeheartedly. In fact, in 1299 there was a law in the commercial center of Florence forbidding the use of such numerals. Initially, only universities dared use them, but later they became popular with merchants, and eventually became commonly used.

   In time, Europe realized the great potential value of the Arab mathematical contributions and put into popular use all that seemed practical. The sciences, with mathematics as their essence, flourished and developed into the disciplines we know today. None would have been the same though, had it not been for that book on restoration, or had the zero not been invented, or had the Arabic numerals not made their way to Europe. That “fondness of science,” which inspired an early Arab mathematician to propose calculating by al-jabr and al-muqabala, did much to make the world run as we know it today.


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