The Compendious Book on Calculation by Completion and Balancing

The Compendious Book on Calculation by Completion and Balancing (, "al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah"; ), also known as Al-Jabr (), is an Arabic mathematical treatise on algebra written by the Polymath Muḥammad ibn Mūsā al-Khwārizmī around 820 CE while he was in the Abbasid capital of Baghdad, modern-day Iraq. "Al-Jabr" was a landmark work in the history of mathematics, establishing algebra as an independent discipline, and with the term "algebra" itself derived from "Al-Jabr".
The "Compendious Book" provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree. It was the first text to teach algebra in an elementary form and for its own sake. It also introduced the fundamental concept of "reduction" and "balancing" (which the term "al-jabr" originally referred to), the transposition of subtracted terms to the other side of an equation, i.e. the cancellation of like terms on opposite sides of the equation. Mathematics historian Victor J. Katz regards "Al-Jabr" as the first true algebra text that is still extant. Translated into Latin by Robert of Chester in 1145, it was used until the sixteenth century as the principal mathematical textbook of European universities.
Several authors have also published texts under this name, 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'Connor 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 algebra, establishing it as an independent discipline. The word "algebra" is derived from the name of one of the basic operations with equations described in this book, following its Latin translation by Robert of Chester.
Quadratic equations.
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:
Thus the equations are verbally described in terms of "squares" (what would today be ""x"2"), "roots" (what would today be "x") and "numbers" ("constants": ordinary spelled out numbers, like 'forty-two'). The six types, with modern notations, are:
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" ("forcing", "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), ""x"2 = 40"x" − 4"x"2" is transformed by "al-ğabr" into "5"x"2 = 40"x"". Repeated application of this rule eliminates negative quantities from calculations.
"Al-Muqabala" (, "balancing" or "corresponding") means subtraction of the same positive quantity from both sides: ""x"2 + 5 = 40"x" + 4"x"2" is turned into "5 = 40"x" + 3"x"2". 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. 
Subsequent parts of the book do not rely on solving quadratic equations.
Area and volume.
The second chapter of the book catalogues methods of finding area and volume. These include approximations of pi (π), given three ways, as 3 1/7, √10, and 62832/20000. This latter approximation, equalling 3.1416, earlier appeared in the Indian "Āryabhaṭīya" (499 CE).
Other topics.
Al-Khwārizmī explicates the Jewish calendar and the 19-year cycle described by the convergence of lunar months and solar years.
About half of the book deals with Islamic rules of inheritance, which are complex and require skill in first-order algebraic equations.