The Compendious Book On Calculation By Completion and Balancing - The Book

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. 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:

1. squares equal roots (ax2 = bx)
2. squares equal number (ax2 = c)
3. roots equal number (bx = c)
4. squares and roots equal number (ax2 + bx = c)
5. squares and number equal roots (ax2 + c = bx)
6. 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.