Islamic Civilization

Algebra

Writing on Algebra , O’Connor and Robertson explain that it:

‘Was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before.’[1]

The word algebra was used for the first time by al-Khwarizmi, and the science named after the title of his Kitab al Mukhtassar fi'l hisab al jabr wa'l muqabalah (Compendious Book of Calculation by Completion and Balancing). The work of al-Khwarizmi is the first in which that word appears in the mathematical sense.[2] ‘Algebra  means in Arabic ‘restoration,’ that is the transposing of negative terms of an equation to the opposite side.[3] Algebra, as the form of the name indicates, is an Arabic word: al-jabr, which means the restoration of something broken, the amplifying of something incomplete.[4] More often jabr is associated with muqabala, the balancing of the two sides of an equation.[5] Sabra explains that the title of the treatise, ‘al jabr wal muqabala,' referred to the two operations used by al-Khwarizmi in the process of solving linear and quadratic equations, namely those of eliminating negative quantities and reducing positive quantities of the same power on both sides of the equation.[6] Al-Khwarizmi's treatise thus started something new, its systematic approach represented by its reduction of the treated problems to canonical forms provided with proofs, and ‘can be said to have impressed its character on subsequent algebraic works, even when these (like the treatises of al Karaji and Omar Khayyam) went far beyond it.’[7]

Al-Khwarizmi ’s Algebra , including its pious introduction was rendered into modern English in 1831 by Frederick Rosen,[8] who says:

‘When I looked at Kitab al-Jabr… a textbook written over eleven hundred years ago, it was among the most lucid and useful I’d ever seen. It is straight forward and practical, full of examples, and al-Khwarizmi himself stated, deals with ‘what is easiest and most useful in mathematics.’[9]

Rosen observes that Al-Khwarizmi  intended to teach:

‘What is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.’[10]

This does not sound like the contents of an algebra text, O’Connor and Robertson point out, however it is important to realise that the book was intended to be highly practical and that algebra was introduced to solve real life problems that were part of everyday life in the Islamic world at that time.[11] Which leads to those fundamental sources of Islamic science: faith and practicality frequently encountered.

A number of Muslim scholars followed in the wake of al-Khwarizmi in dealing with the subject. Al-Mahani (born 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.[12] Abu Kamil (b. 850) solved complicated quadratic equations with irrational coefficients.[13] Despite not using symbols, but writing powers of x in words, he had begun to understand what we would write in symbols as xn+xm = x (m+n).[14] He forms an important link in the development of algebra between al-Khwarizmi and al-Karaji. Al-Karaji (b. 953) is seen by many as the first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today.[15] He treated various properties of quadratic irrationals, which Euclid had proved geometrically, and he also discussed cubic irrationalities, and explained the extraction of roots of a polynomial (which is assumed to be a perfect square).[16] He was first to define the monomials, ... and to give rules for products of any two of these.[17] He started a school of algebra which flourished for several hundreds of years, of which  Al-Samawal, nearly 200 years later, was an important member. Al-Samawal (b. 1130) was the first to give the new topic of algebra a precise description when he wrote that it was concerned:

‘ ... with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.’ [18]

Under the Muslims algebra became an independent science having its own fields of study and methods of research, which is obvious in the algebraic treatise of Omar  Khayyam.[19] Knowing the importance of cubic equations for the construction of trigonometrical tables and for the solution of geometrical problems, the detailed geometrical theory of cubic equations was constructed and used to determine upper and lower limits for their roots.[20] Later an analogous study was made of quadratic equations by al-Kashi (1380-1429), and together with the geometrical theory  number of approximate methods were invented (by al-Biruni), al-Kashi and others.[21]

Islamic algebra, like other sciences, was transferred to Europe primarily in the 12^th and 13^th centuries. Sarton offers chronological and geographical pointers to follow its passage.[22] He also provides a rich bibliography that can give answers to the inquisitive mind.[23] There is also good information provided by Van Der Waerden,[24] Sesiano,[25] and Kaunzner.[26]

The particular transfer al-Khwarizmi’s algebra to Europe is well outlined by Mahoney. He explains that Al-Khwarizmi ’s Kitab al-mukhtasar… (Compendious..) became in Latin  simply Liber algebre, but only two of its three parts were transmitted to Europe, and they were separately.[27] It was translated into Latin twice during the 12^th century, by Gerard of Cremona who retained the Arabic title: De jebra et almucabala, and by Robert of Chester, who gave an exact Latin rendering of it Liber restaurationis et oppositionis numeris.[28] The term "algebra" itself was first applied only to the first part translated by Robert of Chester in 1145, and by Gerard of Cremona several decades later. It set forth solution paradigms for six types of problems expressed in modern symbolism by the equations (1) ax² = bx, (2) ax² = b, (3) ax = b, (4) ax² + bx = c (5) ax² + c = bx, and (6) bx + c = ax², where a, b, and c are all positive rational numbers.[29]

The second part of al-Khwarizmi's treatise was transmitted through an expanded version of it by Abraham bar Hiyya (Savasorda in Latin ), translated by Plato of Tivoli in 1145 as the Liber embadorum (Book of Areas). It adds algebra to the techniques of the mensurator by using it to solve problems of dividing areas and volumes into parts having a given relation, and of determining lengths from various combinations of dimensions.[30] The third part of al-Khwarizmi's algebra deals with the division of inheritances and is partly found in the writings of Leonardo Fibonacci, the section as a whole remaining un-translated.[31]

The place of Islamic algebra in modern mathematics has been studied by modern sources. There is a good summary by O'Connor and Robertson already referred to. Rashed is possibly the best source of information for either French or English readership.[32] Particular Islamic algebraists such as Abu Kamil have been studied by many, including Weinberg[33] and Levey.[34] Al-Uqlidisi’s algebra is accessible via an excellent translation by Saidan.[35] The algebra of al-Khayyam is explored conjointly by Rashed and Djebbar.[36] Rashed also sheds light on Sharaf Eddin al-Tusi’s algebra,[37] whilst Suter[38] and Sezgin[39] offer an overall, but thorough, picture of the science and its leading figures. King on the other hand shows the connection of algebra with other sciences.[40]