The Compendious Book on Calculation by Completion and Balancing
Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala (Arabic: الكتاب المختصر في حساب الجبر والمقابلة, "The Compendious Book on Calculation by Completion and Balancing"), 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 language in approximately AD 820 by the Persian mathematician Muḥammad 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 (with algebrae and almucabola being simply Latinized corruptions of the words in the Arabic title). Today's term algebra is derived from the term الجبر al-ğabr in the title of this book. The al-ğabr provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree.
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.
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."
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."
The book 
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 Bringing_together and Separating According to the Hindu Calculation) discussing the Hindu-Arabic numeral system.
The book classifies quadratic equations to one of the six basic types and provides algebraic and geometric methods to solve the basic ones. Historian Carl Boyer notes the following regarding the lack of modern abstract notations in the book:
... 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!— Carl B. Boyer , A History of Mathematics
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)
Islamic mathematicians, unlike the Hindus, did not deal with negative numbers at all; hence an equation like bx + c = 0 does not appear in the classification, because it has no positive solutions if all the coefficients are positive. Similarly equation types 4, 5 and 6, which look equivalent to the modern eye, were distinguished because the coefficients must all be positive.
The al-ğabr (in Arabic script 'الجبر') ("forcing " or "restoring") operation is moving a deficient quantity from one side of the equation to the other side. 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"or "corresponding") 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, when restricted to positive coefficients and solutions.
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 ed.). John Wiley & Sons, Inc. p. 228. ISBN 0-471-54397-7.
"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."
- Rasāla fi l-ğabr wa-l-muqābala
- Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 11–2. ISBN 0-7923-2565-6. OCLC 29181926
- O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews.
- Robert of Chester (1915). Algebra of al-Khowarizmi. Macmillan.
- Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
Further reading 
- 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 ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7.
- R. Rashed, The development of Arabic mathematics: between arithmetic and algebra, London, 1994.
- 19th Century English Translation
- 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.