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The Compendious Book on Calculation by Completion and Balancing

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 820 AD by the Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī.

The term "algebra" is derived from the al-ğabr in the title of this book, which is considered the foundational text of modern 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.

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.

Legacy

R. Rashed and Angela Armstrong write:

J. J. O'Conner and E. F. Robertson wrote in the MacTutor History of Mathematics archive:

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 almucabala, 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.

Footnotes

References

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). A History of Mathematics. Second Edition, John Wiley & Sons, Inc..
R. Rashed, The development of Arabic mathematics: between arithmetic and algebra, London, 1994.