# The Compendious Book on Calculation by Completion and Balancing

The Compendious Book on Calculation by Completion and Balancing (Arabic: الكتاب المختصر في حساب الجبر والمقابلة‎, Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala; Latin: Liber Algebræ et Almucabola) is an Arabic treatise on mathematics written by Persian polymath Muḥammad ibn Mūsā al-Khwārizmī around 820 CE while he was in the Abbasid capital of Baghdad.Translated into Latin by Robert of Chester in 1145, it was used until the sixteenth century as the principal mathematical text-book of European universities.It also introduced the term "algebra" (الجبر, al-jabr) to European languages. The Compendious Book provided an exhaustive account of solving for the positive roots of polynomial equations up to the second degree.

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.

R. Rashed and Angela Armstrong write:

Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from the Diophantus' Arithmetica. It no longer concerns a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems.

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

Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before.