Contribution of Khwârazmî to Mathematics and Geography

Contribution to Mathematics

Mûsâ Al-Khwârazmî is one of the greatest scientific minds of the medieval period and most important Muslim mathematician who was justly called the 'father of algebra'. He wrote the Kitâb al-Jem wa'l Tafrîq bi Hisâb al-Hind also called Kitâb Hisâb al-adad al-Hindî on arithmetic in which he used Indian numerals including zero in place of depicting numbers by the letters of the alphabet and the decimal notations or numeration by position" for the first time. It deals with the four basic operations of addition, subtraction, multiplication and division as well as with both common and sexagesimal fractions and the extraction of the square root. The original Arabic text of the book is lost and its only Latin translation is available.

Other mathematical writings of Mûsâ Al-Khwârazmî are also not known. His best known classical work on algebra is the Kitâb al-Mukhtasar fî Hisâb al-Jabr wa'l Muqâbala. It was also translated into Latin in the Middle Ages and holds an eminent place in the history of mathematics, firstly, in the words of Galal S.A. Shawki, because it defined algebra as an independent disciple in mathematics, and secondly because it accelerated the introduction of the Arabic place value numbering into the West. The book is devoted to finding solutions to practical problems which Muslims encountered in daily life concerning matters of inheritance, legacies, partition, lawsuits and commerce, with over eight hundred examples. The original work in Arabic was written in 820 A.D. and was translated into Latin in the twelfth century. It is worth remarking that the term al-jabr, in the Latinized form of algebra, has found its way into the language of Europe, whiles the old mathematical term, algorism, is a distortion of al-Khwârazmî's name.

The meaning of the Arabic word Al-Jabr is restoration by transposing negative quantities to the other side of the equation to make them positive and the term Al-Muqâbalah refers to the process of eliminating identical quantities from the two sides of the equation. But the best translation for Hisâb al-Jabr wa'l-Muqâbala, according to John K. Baumgart, is 'the science of equations'. The algebra of Mûsâ al-Khwârazmî was rhetorical in form. Al-Khwârazmî had given the rules for the solution of quadratic equations which are supported in a number of cases by geometrical proofs also. The unknown quantity, in the words of Galal S.A. Shawki, was termed the "thing" (shay) or "root" (jazr); the latter means in Arabic the origin or base, also the root of a tree, hence, the use of the expression "root of an equation" is derived from the Arabic concept. Mûsâ al-Khwârazmî had used the Arabic word for root to denote the first degree term of a quadratic equation. Explaining in detail he says, "The following' is an example of squares equal to roots, a square is equal to 5 roots. The root of the square then is 5, and 25 forms its square, which of course equals 5 of its roots." For the second power of a quantity he employs mâl (wealth, property) which is also used to mean only "quantity" and dirham is used as unit of coinage.

The bases of algebra, first viewed as an independent mathematical discipline , were laid down by Mûsâ al-Khwârazmî who carefully formulated analytical solutions of the various forms of quadratic equation and profusely illustrated his method of solution by practical examples. He was quite aware of the existence of two roots of the quadratic equation, though he cared for positive, real roots only.
His Hisâb al-Jabr wa'l-Muqâbalah is actually on applied mathematics. Its first part discusses the equations of the first and second degrees. All his proposed problems can be reduced to one of the six standard forms. He gives rules for the solution of each of the six forms and explains how to reduce any given problem to one of these standard forms with examples. The second part of the book deals with practical mensuration by giving rules for finding the area of various plane figures including the circles, and for finding the volume of a number of solids including cones and pyramids. The third part concerns legacies as well as inheritance and is the longest. It consists entirely of solved problems which arise out of legacies.

Contribution to Geography

Mûsâ al-Khwârazmî had also contributed to the science of geography. As the book of Geography of Claudius Ptolemy (2nd century A.D.) was translated several times into Arabic he had a model for writing his book in this field of knowledge. His book on geography entitled Kitâb Sûrat al-Ard (Book of the image of the earth) consists almost entirely of lists of longitudes as well as latitudes of localities and gives in a tabulated form the coordinates of the places like cities, mountains, seas, rivers, islands etc. The book is arranged according to the Greek system of the seven climes (aqâlim) giving contemporary data but the knowledge acquired by the other Muslims are also incorporated into it. The first section lists cities, the second, mountains (giving the coordinates of their extreme points and their orientation); the third, seas (giving the coordinates of salient point on their coastlines and a rough description of their outlines); the fourth, islands (giving the coordinates of their centres, and their length and breadth); the fifth, the central points of various geographical regions; the sixth, rivers (giving their salient points and towns on them). This book had served as a basis for later works and stimulated geographical studies and the composition of original treatises. It is said that his Kitâb Sûrat al-Ard was also accompanied by regional maps of each of the climes and by a single world map called "al-Sûrat al-Ma'muniyya" but have been lost. It is also said that his map of the world was the first map of the heavens and the world drawn by Muslims. But the editor of the Kitâb Sûrat al-Ard Hans von Mzik, has produced only four maps. These four maps, in the words of S. Maqbul Ahmad, seem to be later recessions of the original maps. But Ibrahim Shawkat reasons that since Al-Khwârazmî wrote a brief work on geography, he did not draw a complete map of the world but confined himself to draw only the four maps as an illustration. His source of inspiration might possibly have been the mappa mundi constructed for Caliph Al-Ma'mun by a team of geographers in which Al-Khwârazmî himself would have been included.

Figure SourcesFigure (front). Mûsâ Al-Khwârazmî. The drawing of Khwârizmî on the stam. The stamp reads: Post USSR 1983, 1200 Years, Mukhammad al-Korezmi.Figure (top). A mathematical figure from Ali Kuscu's book Risâla al-Fathiyya fi ilmi'l-Hay'a. Süleymaniye Library, Ayasofya 2733.Figure (bottom). The world map of al-Idrisi., Ahmad b. Sahl Al-BAlhî, Akâlim al-buldân, Suleymaniye Library, Ayasofya 2577.
by: FSTC Limitied, Wed 27 December, 2006 rewrite by Kurnia25