Al-Jabr


The Concise Book of Calculation by Restoration and Balancing (, ;\\

Content

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", "roots" and "numbers". The six types, with modern notations, are:
  1. squares equal roots
  2. squares equal number
  3. roots equal number
  4. squares and roots equal number
  5. squares and number equal roots
  6. roots and number equal squares
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.
Al-Jabr operation is moving a deficient quantity from one side of the equation to the other side. In an al-Khwarizmi's example, "x2 = 40x − 4x2" is transformed by al-Jabr into "5x2 = 40x". Repeated application of this rule eliminates negative quantities from calculations.
Al-Muqābala 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 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, given three ways, as,, and. This latter approximation, equalling 3.1416, earlier appeared in the Indian Āryabhaṭīya.

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.

Legacy

R. Rashed and Angela Armstrong write:
J. J. O'Connor and E. F. Robertson wrote in the MacTutor History of Mathematics Archive: