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The Compendious Book on Calculation by Completion and Balancing

Title page, 9th century
Author	Muhammad ibn Musa al-Khwarizmi
Original title	كتاب المختصر في حساب الجبر والمقابلة
Illustrator	Muhammad ibn Musa al-Khwarizmi
Language	Arabic
Subject	Algebra
Genre	Mathematics
Publication date	820
Publication place	Abbasid Caliphate
Original text	كتاب المختصر في حساب الجبر والمقابلة at Arabic Wikisource
Translation	The Compendious Book on Calculation by Completion and Balancing at Wikisource

Al-Jabr (Arabic: الجبر), also known as The Compendious Book on Calculation by Completion and Balancing (Arabic: الكتاب المختصر في حساب الجبر والمقابلة, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah; or Latin: Liber Algebræ et Almucabola), is an Arabic mathematical treatise on algebra written in Baghdad around 820 by the Persian polymath Al-Khwarizmi. It was a landmark work in the history of mathematics, with its title being the ultimate etymology of the word "algebra" itself, later borrowed into Medieval Latin as algebrāica.

Al-Jabr 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 elementary algebra, and the first to teach algebra 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. The 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 Abu Hanifa Dinawari, Abu Kamil, 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:

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.

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:

... 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!

— Carl B. Boyer, A History of Mathematics

Thus the equations are verbally described in terms of "squares" (what would today be "x"), "roots" (what would today be "x") and "numbers" ("constants": ordinary spelled out numbers, like 'forty-two'). The six types, with modern notations, are:

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

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 ("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 = 40x − 4x" is transformed by al-Jabr into "5x = 40x". Repeated application of this rule eliminates negative quantities from calculations.

Al-Muqābala (المقابله, "balancing" or "corresponding") means subtraction of the same positive quantity from both sides: "x + 5 = 40x + 4x" is turned into "5 = 40x + 3x". 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.

References

Notes

1. This book is the source of the word; see transliterated title.
2. The Arabic title is sometimes condensed to Hisab al-Jabr wal-Muqabalah or Kitab al-Jabr wal-Muqabalah or given under other transliterations.
3. "The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization – respects in which neither Diophantus nor the Hindus excelled."
4. "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".
5. "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation, which is evident in the treatise; the word muqabalah is said to refer to "reduction" or "balancing"—that is, the cancellation of like terms on opposite sides of the equation."
6. "The first true algebra text which is still extant is the work on al-jabr and al-muqabala by Mohammad ibn Musa al-Khwarizmi, written in Baghdad around 825."
7. "The Compendious Book on Calculation by Completion and Balancing" (Hisab al-Jabr wa H-Muqabala) on the development of the subject cannot be underestimated. Translated into Latin during the twelfth century, it remained the principal mathematics textbook in European universities until the sixteenth century"

Citations

1. ^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7.
2. Gandz; Saloman (1936). The sources of al-Khwarizmi's algebra. Vol. I. Osiris. pp. 263–277.
3. Katz, Victor J.; Barton, Bill (December 2006). "Stages in the history of algebra with implications for teaching". Educational Studies in Mathematics. 66 (2): 185–201. doi:10.1007/s10649-006-9023-7. See p. 190.
4. Philip Khuri Hitti (2002). History of the Arabs. Macmillan International Higher Education. pp. 379. ISBN 9780333631423.
5. Fred James Hill, Nicholas Awde (2003). A History of the Islamic World. Hippocrene Books. pp. 55. ISBN 9780781810159.
6. Overbay, Shawn; Schorer, Jimmy; Conger, Heather. "Al-Khwarizmi". University of Kentucky.
7. "Islam Spain and the history of technology". www.sjsu.edu. Retrieved 24 January 2018.
8. Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 11–12. ISBN 0-7923-2565-6. OCLC 29181926.
9. O'Connor, John J.; Robertson, Edmund F. (1999). "Arabic mathematics: forgotten brilliance?". MacTutor History of Mathematics Archive. University of St Andrews.
10. Robert of Chester (1915). Algebra of al-Khowarizmi. Macmillan. Archived from the original on 18 November 2018.
11. Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), p. 228
12. Berggren, J. L. (1986). "Algebra in Islam". Episodes in the Mathematics of Medieval Islam. Springer. pp. 99–126. doi:10.1007/978-1-4612-4608-4_4. ISBN 9781461246084. See p. 103
13. ^ B.L. van der Waerden, A History of Algebra: From al-Khwārizmī to Emmy Noether; Berlin: Springer-Verlag, 1985. ISBN 3-540-13610-X
14. David A. King (2003). "Mathematics applied to aspects of religious ritual in Islam". In I. Grattan-Guinness (ed.). Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Vol. 1. JHU Press. p. 83. ISBN 9780801873966.

Further reading

• Hughes, Barnabas B. ed., Robert of Chester's Latin Translation of Al-Khwarizmi's Al-Jabr: A New Critical Edition, (in Latin) Wiesbaden: F. Steiner Verlag, 1989. ISBN 3-515-04589-9
• Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7.
• Rashed, R. The development of Arabic mathematics: between arithmetic and algebra, London, 1994.

External links

• 19th Century English Translation at the Internet Archive
• Al-Khwarizmi
• Annotated excerpt from a translation of the Compendious Book. University of Duisburg-Essen.
• The Compendious Book on Calculation by Completion and Balancing – in the Arabic original with an English translation (PDF)
• Ghani, Mahbub (5 January 2007). "The Science of Restoring and Balancing – The Science of Algebra". Muslim Heritage.

v t e Mathematics in the medieval Islamic world
Mathematicians	9th century 'Abd al-Hamīd ibn Turk Sanad ibn Ali al-Jawharī Al-Ḥajjāj ibn Yūsuf Al-Kindi Qusta ibn Luqa Al-Mahani al-Dinawari Banū Mūsā brothers Hunayn ibn Ishaq Al-Khwarizmi Yusuf al-Khuri Ishaq ibn Hunayn Na'im ibn Musa Thābit ibn Qurra al-Marwazi Abu Said Gorgani 10th century Abu al-Wafa al-Khazin Al-Qabisi Abu Kamil Ahmad ibn Yusuf Aṣ-Ṣaidanānī Sinān ibn al-Fatḥ al-Khojandi Al-Nayrizi Al-Saghani Brethren of Purity Ibn Sahl Ibn Yunus al-Uqlidisi Al-Battani Sinan ibn Thabit Ibrahim ibn Sinan Al-Isfahani Nazif ibn Yumn al-Qūhī Abu al-Jud Al-Sijzi Al-Karaji al-Majriti al-Jabali 11th century Abu Nasr Mansur Alhazen Kushyar Gilani Al-Biruni Ibn al-Samh Abu Mansur al-Baghdadi Avicenna al-Jayyānī al-Nasawī al-Zarqālī ibn Hud Al-Isfizari Omar Khayyam Muhammad al-Baghdadi 12th century Jabir ibn Aflah Al-Kharaqī Al-Khazini Al-Samawal al-Maghribi al-Hassar Sharaf al-Din al-Tusi Ibn al-Yasamin 13th century Ibn al‐Ha'im al‐Ishbili Ahmad al-Buni Ibn Munim Alam al-Din al-Hanafi Ibn Adlan al-Urdi Nasir al-Din al-Tusi al-Abhari Muhyi al-Din al-Maghribi al-Hasan al-Marrakushi Qutb al-Din al-Shirazi Shams al-Din al-Samarqandi Ibn al-Banna' Kamāl al-Dīn al-Fārisī 14th century Nizam al-Din al-Nisapuri Ibn al-Shatir Ibn al-Durayhim Al-Khalili al-Umawi 15th century Ibn al-Majdi al-Rūmī al-Kāshī Ulugh Beg Ali Qushji al-Wafa'i al-Qalaṣādī Sibt al-Maridini Ibn Ghazi al-Miknasi 16th century Al-Birjandi Muhammad Baqir Yazdi Taqi ad-Din Ibn Hamza al-Maghribi
Mathematical works	The Compendious Book on Calculation by Completion and Balancing De Gradibus Principles of Hindu Reckoning Book of Optics The Book of Healing Book on the Measurement of Plane and Spherical Figures Encyclopedia of the Brethren of Purity Toledan Tables Tabula Rogeriana Zij
Concepts	Alhazen's problem Islamic geometric patterns
Centers	Al-Azhar University Al-Mustansiriya University House of Knowledge House of Wisdom Constantinople observatory of Taqi ad-Din Madrasa Maragheh observatory University of al-Qarawiyyin
Influences	Babylonian mathematics Greek mathematics Indian mathematics
Influenced	Byzantine mathematics European mathematics Indian mathematics
Related	Hindu–Arabic numeral system Arabic numerals (Eastern Arabic numerals, Western Arabic numerals) Trigonometric functions History of trigonometry History of algebra

v t e Mathematics in Iran
Mathematicians	Before 20th Century Abu al-Wafa' Buzjani Jamshid al-Kashi (al-Kashi's theorem) Omar Khayyam (Khayyam-Pascal's triangle, Khayyam-Saccheri quadrilateral, Khayyam's Solution of Cubic Equations) Al-Mahani Muhammad Baqir Yazdi Nizam al-Din al-Nisapuri Al-Nayrizi Kushyar Gilani Ayn al-Quzat Hamadani Al-Isfahani Al-Isfizari Al-Khwarizmi (Al-jabr) Najm al-Din al-Qazwini al-Katibi Nasir al-Din al-Tusi Al-Biruni Modern Maryam Mirzakhani Caucher Birkar Sara Zahedi Farideh Firoozbakht (Firoozbakht's conjecture) S. L. Hakimi (Havel–Hakimi algorithm) Siamak Yassemi Freydoon Shahidi (Langlands–Shahidi method) Hamid Naderi Yeganeh Esmail Babolian Ramin Takloo-Bighash Lotfi A. Zadeh (Fuzzy mathematics, Fuzzy set, Fuzzy logic) Ebadollah S. Mahmoodian Reza Sarhangi (The Bridges Organization) Siavash Shahshahani Gholamhossein Mosaheb Amin Shokrollahi Reza Sadeghi Mohammad Mehdi Zahedi Mohsen Hashtroodi Hossein Zakeri Amir Ali Ahmadi
Prize Recipients	Fields Medal Maryam Mirzakhani (2014) Caucher Birkar (2018) EMS Prize Sara Zahedi (2016) Satter Prize Maryam Mirzakhani (2013)
Organizations	Iranian Mathematical Society
Institutions	Institute for Research in Fundamental Sciences

• 9th-century Arabic-language books
• History of algebra
• Mathematical works of the medieval Islamic world
• Mathematics textbooks
• Scientific works of the Abbasid Caliphate