Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (Arabic for "The Compendious Book on Calculation by Completion and Balancing", in Arabic script 'الكتاب المختصر في حساب الجبر والمقابلة'), also known under a shorter name spelled as Hisab al-jabr w’al-muqabala, Kitab al-Jabr wa-l-Muqabala and other transliterations) is a mathematical book written in Arabic in approximately AD 820 by the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī in Baghdad, the capital of the Abbasid Caliphate at the time.
The book was translated into Latin in the mid 12th century under the title Liber Algebrae et Almucabola. Today's term "algebra" is derived from the term al-jabr, or al-ğabr, in the title of this book. The book is considered a foundational text in the history of the development of algebra. The al-ğabr provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree, and introduced the fundamental methods of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. The "novelty of Al-Khwarizmi lies in his extremely systematic treatment, aiming at a general classification of linear and quadratic equations, and at general methods of solving them which are established with proofs." 
Several authors have also published texts under the name of Kitāb al-ğabr wa-l-muqābala, including Abū Ḥanīfa al-Dīnawarī, Abū Kāmil Shujā ibn Aslam, Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, 'Abd al-Hamīd ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr, and Šarafaddīn al-Ṭūsī.
J. J. O'Connor and E. F. Robertson wrote in the MacTutor History of Mathematics archive:
"Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra 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."
Father of algebraEdit
Al-Khwarizmi is widely considered "the father of algebra", though debate exists as to whether the Hellenized Babylonian mathematician Diophantus also deserves this title. Many agree that Al-Khwarizmi deserves this title most.
Those who support Diophantus point to the fact that the algebra found in Al-Jabr is more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical. However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to earlier Babylonian mathematics. For this, and other, reasons mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.”
Those who support Al-Khwarizmi point to the fact that he gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, introduced the fundamental methods of reduction and balancing, and was the first to teach algebra in an elementary form and for its own sake, whereas Diophantus was primarily concerned with the theory of numbers. The "novelty of Al-Khwarizmi lies in his extremely systematic treatment, aiming at a general classification of linear and quadratic equations, and at general methods of solving them which are established with proofs."  In addition, R. Rashed and Angela Armstrong write:
"Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus' Arithmetica. It no longer concerns a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."
The Muslim Persian mathematician Muhammad ibn Mūsā al-Khwārizmī was a faculty member of the "House of Wisdom" (Bait al-Hikma) in Baghdad, which was established by Al-Mamun. Al-Khwarizmi, who died around 850 CE, wrote more than half a dozen mathematical and astronomical works; some of which were based on the Indian Sindhind. One of al-Khwarizmi's most famous books is entitled Al-jabr wa'l muqabalah or The Compendious Book on Calculation by Completion and Balancing, and it gives an exhaustive account of solving polynomials up to the second degree. The book also introduced the fundamental concept of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which Al-Khwarizmi originally described as al-jabr.
Al-Jabr is divided into six chapters, each of which deals with a different type of formula. The first chapter of Al-Jabr deals with equations whose squares equal its roots (ax2 = bx), the second chapter deals with squares equal to number (ax2 = c), the third chapter deals with roots equal to a number (bx = c), the fourth chapter deals with squares and roots equal a number (ax2 + bx = c), the fifth chapter deals with squares and number equal roots (ax2 + c = bx), and the sixth and final chapter deals with roots and number equal to squares (bx + c = ax2).
In Al-Jabr, al-Khwarizmi uses geometric proofs. and with this new form of algebra they were able to find solutions to equations by using a process that they invented, known as "the application of areas". He also recognizes that the discriminant must be positive and described the method of completing the square. He makes use of lettered diagrams but all of the coefficients in all of his equations are specific numbers since he had no way of expressing with parameters what he could express geometrically; although generality of method is intended.
The book was a compilation and extension of known rules for solving quadratic equations and for some other problems, and considered to be the foundation of modern algebra, establishing it as an independent discipline. The word algebra is derived from the name of one of the basic operations with equations (al-ğabr) described in this book. The book was introduced to the Western world by the Latin translation of Robert of Chester entitled Liber algebrae et almucabola, hence "algebra".
Since the book does not give any citations to previous authors, it is not clearly known what earlier works were used by al-Khwarizmi, and modern mathematical historians put forth opinions based on the textual analysis of the book and the overall body of knowledge of the contemporary Muslim world. Most certain are connections with Indian mathematics, as he had written a book entitled Kitāb al-Jamʿ wa-l-tafrīq bi-ḥisāb al-Hind (The Book of Addition and Subtraction According to the Hindu Calculation) discussing the Hindu-Arabic numeral system.
The book reduces arbitrary quadratic equations to one of the six basic types and provides algebraic and geometric methods to solve the basic ones. Lacking modern abstract notations, "the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation (see History of algebra) found in the Greek Arithmetica or in Brahmagupta's work. Even the numbers were written out in words rather than symbols!" Thus the equations are verbally described in terms of "squares" (what would today be "x2"), "roots" (what would today be "x") and "numbers" (ordinary spelled out numbers, like 'forty-two'). The six types, with modern notations, are:
- squares equal roots (ax2 = bx)
- squares equal number (ax2 = c)
- roots equal number (bx = c)
- squares and roots equal number (ax2 + bx = c)
- squares and number equal roots (ax2 + c = bx)
- roots and number equal squares (bx + c = ax2)
The al-ğabr (in Arabic script 'الجبر') ("completion") operation is moving a negative quantity from one side of the equation to the other side and changing its sign. In an al-Khwarizmi's example (in modern notation), "x2 = 40x − 4x2" is transformed by al-ğabr into "5x2 = 40x". Repeated application of this rule eliminates negative quantities from calculations.
Al-Muqabala (in Arabic script 'المقابله') ("balancing") means subtraction of the same positive quantity from both sides: "x2 + 5 = 40x + 4x2" is turned into "5 = 40x + 3x2". Repeated application of this rule makes quantities of each type ("square"/"root"/"number") appear in the equation at most once, which helps to see that there are only 6 basic solvable types of the problem.
The next part of the book discusses practical examples of the application of the described rules. The following part deals with applied problems of measuring areas and volumes. The last part deals with computations involved in convoluted Islamic rules of inheritance. None of these parts require the knowledge about solving quadratic equations.
- ↑ Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second, John Wiley & Sons, Inc., 228. ISBN 0471543977.
"The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization – respects in which neither Diophantus nor the Hindus excelled."
- ↑ 2.0 2.1 2.2 (Boyer 1991, "The Arabic Hegemony" p. 229) "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation."
- ↑ Rasāla fi l-ğabr wa-l-muqābala
- ↑ Possibly.
- ↑ O'Connor, John J.; Robertson, Edmund F., "Al-Jabr", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Khwarizmi.html.
- ↑ 6.0 6.1 6.2 (Boyer 1991, "The Arabic Hegemony" p. 228) "Diophantus sometimes is called "the father of algebra," but this title more appropriately belongs to al-Khwarizmi. It is true that in two respects the work of al-Khwarizmi represented a retrogression from that of Diophantus. First, it is on a far more elementary level than that found in the Diophantine problems and, second, the algebra of al-Khwarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek Arithmetica or in Brahmagupta's work. Even numbers were written out in words rather than symbols! It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers."
- ↑ (Derbyshire 2006, "The Father of Algebra" p. 31) "Diophantus, the father of algebra, in whose honor I have named this chapter, lived in Alexandria, in Roman Egypt, in either the 1st, the 2nd, or the 3rd century CE."
- ↑ Harald Kittel, Übersetzung: ein internationales Handbuch zur Übersetzungsforschung, Volume 2 p. 1123, 1124
- ↑ (Boyer 1991, "The Arabic Hegemony" p. 230) "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions."
- ↑ Gandz and Saloman (1936), The sources of al-Khwarizmi's algebra, Osiris i, p. 263–277: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".
- ↑ (1994) The Development of Arabic Mathematics. Springer, 11–2. ISBN 0792325656. OCLC 29181926.
- ↑ (Boyer 1991, "The Arabic Hegemony" pp. 228–229) "the author's preface in Arabic gave fulsome praise to Mohammed, the prophet, and to al-Mamun, "the Commander of the Faithful.""
- ↑ (Boyer 1991, "The Arabic Hegemony" p. 227) "The first century of the Muslim empire had been devoid of scientific achievement. This period (from about 650 to 750) had been, in fact, perhaps the nadir in the development of mathematics, for the Arabs had not yet achieved intellectual drive, and concern for learning in other parts of the world had faded. Had it not been for the sudden cultural awakening in Islam during the second half of the eighth century, considerably more of ancient science and mathematics would have been lost. [...] It was during the caliphate of al-Mamun (809–833), however, that the Arabs fully indulged their passion for translation. The caliph is said to have had a dream in which Aristotle appeared, and as a consequence al-Mamun determined to have Arabic versions made of all the Greek works that he could lay his hands on, including Ptolemy's Almagest and a complete version of Euclid's Elements. From the Byzantine Empire, with which the Arabs maintained an uneasy peace, Greek manuscripts were obtained through peace treaties. Al-Mamun established at Baghdad a "House of Wisdom" (Bait al-hikma) comparable to the ancient Museum at Alexandria. Among the faculty members was a mathematician and astronomer, Mohammed ibn-Musa al-Khwarizmi, whose name, like that of Euclid, later was to become a household word in Western Europe. The scholar, who died sometime before 850, wrote more than half a dozen astronomical and mathematical works, of which the earliest were probably based on the Sindhad derived from India."
- ↑ (Boyer 1991, "The Arabic Hegemony" p. 228) "The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization—respects in which neither Diophantus nor the Hindus excelled."
- ↑ (Boyer 1991, "The Arabic Hegemony" p. 229) "in six short chapters, of the six types of equations made up from the three kinds of quantities: roots, squares, and numbers (that is x, x2, and numbers). Chapter I, in three short paragraphs, covers the case of squares equal to roots, expressed in modern notation as x2 = 5x, x2/3 = 4x, and 5x2 = 10x, giving the answers x = 5, x = 12, and x = 2 respectively. (The root x = 0 was not recognized.) Chapter II covers the case of squares equal to numbers, and Chapter III solves the cases of roots equal to numbers, again with three illustrations per chapter to cover the cases in which the coefficient of the variable term is equal to, more than, or less than one. Chapters IV, V, and VI are more interesting, for they cover in turn the three classical cases of three-term quadratic equations: (1) squares and roots equal to numbers, (2) squares and numbers equal to roots, and (3) roots and numbers equal to squares."
- ↑ 16.0 16.1 (Boyer 1991, "Europe in the Middle Ages" p. 258) "In the arithmetical theorems in Euclid's Elements VII–IX, numbers had been represented by line segments to which letters had been had been attached, and the geometric proofs in al-Khwarizmi's Algebra made use of lettered diagrams; but all coefficients in the equations used in the Algebra are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry."
- ↑ (Boyer 1991, "The Heroic Age" pp. 77–78) "Whether deduction came into mathematics in the sixth century BCE or the fourth and whether incommensurability was discovered before or after 400 BCE, there can be no doubt that Greek mathematics had undergone drastic changes by the time of Plato. [...] A "geometric algebra" had to take the place of the older "arithmetic algebra," and in this new algebra there could be no adding of lines to areas or of areas to volumes. From now on there had to be strict homogeneity of terms in equations, and the Mesopotamian normal form, xy = A, x ± y = b, were to be interpreted geometrically. [...] In this way the Greeks built up the solution of quadratic equations by their process known as "the application of areas," a portion of geometric algebra that is fully covered by Euclid's Elements. [...] The linear equation ax = bc, for example, was looked upon as an equality of the areas ax and bc, rather than as a proportion—an equality between the two ratios a:b and c:x. Consequently, in constructing the fourth proportion x in this case, it was usual to construct a rectangle OCDB with the sides b = OB and c = OC (Fig 5.9) and then along OC to lay off OA = a. One completes the rectangle OCDB and draws the diagonal OE cutting CD in P. It is now clear that CP is the desired line x, for rectangle OARS is equal in area to rectangle OCDB"
- ↑ (Boyer 1991, "The Arabic Hegemony" p. 230) "Al-Khwarizmi here calls attention to the fact that what we designate as the discriminant must be positive: "You ought to understand also that when you take the half of the roots in this form of equation and then multiply the half by itself; if that which proceeds or results from the multiplication is less than the units above mentioned as accompanying the square, you have an equation." [...] Once more the steps in completing the square are meticulously indicated, without justification,"
- ↑ Robert of Chester (1915). Algebra of al-Khowarizmi. Macmillan.
- ↑ Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
- Barnabas B. Hughes, ed., Robert of Chester's Latin Translation of Al-Khwarizmi's Al-Jabr: A New Critical Edition, (in Latin language) Wiesbaden: F. Steiner Verlag, 1989. ISBN 3-515-04589-9
- Boyer, Carl B. (1991). "The Arabic Hegemony", A History of Mathematics, Second, John Wiley & Sons, Inc.. ISBN 0471543977.
- R. Rashed, The development of Arabic mathematics: between arithmetic and algebra, London, 1994.
- Annotated excerpt from a translation of the Compendious Book. University of Duisburg-Essen.
- The Compendious Book on Calculation by Completion and Balancing In the Arabic original with an English translation (PDF)
- Ghani, Mahbub (5 January 2007). "The Science of Restoring and Balancing – The Science of Algebra". Muslim Heritage.