The Compendious Book on Calculation by Completion and Balancing is a book written approximately 830 CE. The book was written with the encouragement of the Caliph al-Ma'mun as a popular work on calculation and is replete with examples and applications to a wide range of problems in trade, surveying and legal inheritance.The term algebra is derived from the name of one of the basic operations with equations (al-jabr, meaning completion, or, subtracting a number from both sides of the equation) described in this book. The book was translated in Latin as Liber algebrae et almucabala by Robert of Chester,hence "algebra", and also by Gerard of Cremona. A unique Arabic copy is kept at Oxford and was translated in 1831 by F. Rosen. A Latin translation is kept in Cambridge.
This book is considered the foundational text of modern algebra. It provided an exhaustive account of solving 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.
