Algebra (al-jabr wal-muqabala as a discipline in al-Khwarizmi's work)

Algebra, al-jabr wal-muqabala, is the art of finding an unknown number from known ones through an equation. Muhammad ibn Musa al-Khwarizmi set it out in Baghdad under the caliph al-Ma'mun in a concise book, al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, for people's needs in inheritance, bequests, trade and land measurement ¹. He reduced every problem to three kinds of term and six types of equation, each with a rule and a geometric proof ². Ibn Khaldun names him the first to write in this art ³, and the science still carries the name of his book in the world's languages ⁴.

Ibn Khaldun counts algebra among the branches of the science of number: a craft by which the unknown is drawn out of the known when there is a relation between them that requires it. Its terms are number, the "thing" (the root) and "wealth" (the square), and the equations between them end in six problems ³.

Al-Khwarizmi begins in the same way. The numbers needed in reckoning by al-jabr and al-muqabala, he says, are of three kinds: roots, squares and simple number that belongs to neither ⁵. Two operations give the art its name. Al-jabr, "restoring", removes a subtracted quantity from one side by adding it to both; al-muqabala, "balancing", cancels like terms that appear on both sides. To these TDV adds radd, dividing by the coefficient of the square, and ikmal, completing it to a whole square ⁶. In al-Khwarizmi's worked problems the restoring step appears again and again: a quantity "less ten things" is restored and the ten things are added to the other side ⁷.

People had solved equations long before al-Khwarizmi. The Cairo editors of his book open with that history. The Egyptian Ahmes papyrus of about 1700 BCE, the oldest surviving school book, already has a first-degree equation in one unknown with a special sign for the unknown ⁸. The Babylonians made tables of squares and cubes, and Euclid has problems that amount to geometric solutions of quadratic equations ⁹. Heron of Alexandria gives the first analytic solution one can be sure of ¹⁰, and Diophantus knew how to solve quadratics with positive coefficients but did not arrange their types in order as al-Khwarizmi did ¹¹.

In India, Aryabhata must have known the solution of quadratics, Brahmagupta in the seventh century gave a rule in words for solving them, and Mahavira, al-Khwarizmi's contemporary, also gave rules ¹². TDV adds that the methods of double false position and of working backwards, and some types of word problem, entered Islamic algebra from Indian books, and that Qusta ibn Luqa translated Diophantus's Arithmetica into Arabic under the title Sina'at al-jabr, "the art of algebra" ⁶.

In the early ninth century al-Ma'mun gathered scholars in Baghdad. The Cairo editors quote Ibn al-Nadim's Fihrist: al-Khwarizmi devoted himself to al-Ma'mun's treasury of wisdom, was one of the astronomers, and made the two astronomical tables known as the Sindhind ¹³.

In his preface al-Khwarizmi praises God and asks blessings on the Prophet ﷺ Muhammad, then says what moved him to write. The imam al-Ma'mun, commander of the faithful, loved learning, kept the learned close and helped them make clear what had been obscure, and this encouraged him to compose a concise book on reckoning by al-jabr and al-muqabala ¹⁴. He limited it to what is subtle and noble in reckoning and what people need constantly: inheritance, bequests, the division of estates, judgements, trade and all their dealings with one another, land measurement, digging canals, engineering, and other matters ⁵. The book has no year, but al-Ma'mun reigned from 813 to 833 CE ¹³, so it was written in those years.

A copy survives in the Bodleian Library at Oxford, written in Cairo in Muharram 743 AH. Frederic Rosen published it with an English translation in London in 1831, and the Cairo edition of 1939 was prepared from that manuscript ¹⁵.

Abu Abd Allah (or Abu Ja'far) Muhammad ibn Musa al-Khwarizmi was an astronomer and mathematician at al-Ma'mun's court in Baghdad. The Cairo editors describe him as learned in arithmetic, geography and astronomy, and as rightly regarded as the founder of algebra ¹⁶. In the preface to his algebra he writes as a Muslim: he praises God, asks blessings on the Prophet and on the prophets before him, and calls al-Ma'mun the imam and commander of the faithful ¹⁴.

Al-Khwarizmi first gives three simple types, in which one kind of term equals another: squares equal roots, squares equal a number, and roots equal a number. He then gives three compound types: squares and roots equal a number, squares and a number equal roots, and roots and a number equal squares ¹⁷.

His best-known example is "a square and ten roots equal thirty-nine dirhams". The rule: halve the roots, which gives five; multiply it by itself, twenty-five; add this to thirty-nine, sixty-four; take its root, eight; and subtract half the roots, five, which leaves three. That is the root, and the square is nine ¹⁸. He then proves the rule with a drawing. A square stands for the unknown square, and rectangles on its sides stand for the ten roots; the corner left to complete the large square is five by five, or twenty-five, and the whole square is sixty-four with a side of eight ¹⁹.

He also notes the case with no solution: when the square of half the roots is smaller than the number joined to the square, "the problem is impossible". The editors add that the name lasted among mathematicians until the study of imaginary quantities began at the end of the eighteenth century ²⁰. His algebra was wholly in words, with no symbols, and it accepted only positive roots ⁶. He ends the theory with the rule of the whole book: every problem worked by al-jabr and al-muqabala must lead to one of the six types ²¹.

The book was studied, commented on and extended across the Islamic world. The Fihrist names at least three men who wrote commentaries on it ²². Ibn Khaldun says that many Andalusians wrote fine commentaries on it, the best of them al-Qurashi's ²³. Robert of Chester translated it into Latin, and L. C. Karpinski published an English version of that Latin translation in 1915 ¹⁵.

Al-Khwarizmi's algebra drew on a long history of equation-solving: first-degree equations in the Ahmes papyrus, Babylonian tables, Greek geometric and analytic solutions, and Indian rules in words ²⁴. TDV also notes the Indian influence on its methods and word problems ⁶. What he added was order: three kinds of term, six types, a rule for each, and a proof for each.

Abu Kamil Shuja' ibn Aslam, about 925 CE, names al-Khwarizmi plainly as one of his sources ¹⁶. Ibn Khaldun puts him next after al-Khwarizmi ³, and TDV says he grounded algebra in Euclidean geometry, worked with irrational numbers and solved geometric problems by algebra. Al-Karaji then made algebra arithmetical, freeing it from geometry, and worked on the expansion of (x + 1) to the power n. His follower al-Samaw'al al-Maghribi wrote numbers and quantities with letters in an abstract style. Umar al-Khayyam did for third-degree equations what al-Khwarizmi had done for the first and second degrees: he sorted them into thirteen types and solved them with conic sections ⁶. The Cairo editors add that al-Khayyam quotes Ibn Musa as a known authority without needing to name him ¹⁶. Ibn Khaldun had heard that a mathematician of the East took the equations past the six types to more than twenty, with geometric proofs ²⁵.

The book was written for daily needs. Al-Khwarizmi says it was meant for inheritance, bequests, the division of estates, trade, land measurement and digging canals ⁵. A long Book of Bequests in the same work, beginning at scan p. 83, applies its rules to shares of estates and bequests ²⁶.

Solving number problems became a science, with its own kinds of term, its chapters and its proofs: every problem is brought by al-jabr and al-muqabala to one of the six types ²¹. The Cairo editors write that his equations, such as "a square and ten roots equal thirty-nine", were repeated in every algebra book from his time to the start of the modern age, and that some are still found in algebra books today ¹⁶.

The museum records al-Khwarizmi's work as an improvement. He did not invent the solving of equations, which is older than him, but he was the first known writer to make it an independent science.

Sunni view (Ibn Khaldun). Ibn Khaldun says the first to write in this art was Abu Abd Allah al-Khwarizmi, and after him Abu Kamil Shuja' ibn Aslam. He calls al-Khwarizmi's book on the six problems one of the best written on the subject, and says many Andalusians wrote commentaries on it ²³.

Academic view (TDV). TDV reports the general agreement of historians that algebra became an independent science with al-Khwarizmi, and that later mathematicians such as Sinan ibn al-Fath acknowledged him as first. The one dissenter in his own era was Abu Barza, who claimed the honour for his grandfather Abd al-Hamid ibn Wasi' ibn Turk; Abu Kamil firmly rebutted him. TDV also sets out why some modern historians doubt that the book was the first: it calls itself a summary, al-Khwarizmi does not claim to be first, its terms are already mature, and there was a book by Ibn Turk and a lost book by Sanad ibn Ali on the subject ⁶.

Both views agree that his book is the oldest surviving work to set out algebra as a science of its own.

Algebra is today a basic branch of mathematics taught in schools, and it writes equations in symbols. Al-Khwarizmi's algebra was wholly in words and accepted only positive roots; al-Samaw'al later began to write numbers and quantities with letters ⁶. The name of the science in all the world's languages comes from the word al-jabr that al-Khwarizmi used as the title of his book, and from his own name come the Latin "algorismus" and the English word "algorithm" ⁴.

The book survives in a manuscript in the Bodleian Library at Oxford, copied in Cairo in Muharram 743 AH ¹⁵. The image on this page shows two pages of a Bodleian manuscript of the book, with its proofs drawn as squares in red ink ²⁷.

The names of al-Khwarizmi and of his book passed into the languages of the world. Among Muslims, Ibn Khaldun names him the first to write on algebra, and Zakariyya al-Qazwini says he was the first to bring algebra to the Muslims. In English his name became "algorithm", a set way of solving problems, and Chaucer used "augrim" for zero; until the early eighteenth century the numerals were called "algorismus" in Latin, and the Spanish word for a digit is "guarismo". His book's title became "algebra" ⁴.

About twelve hundred years ago a scholar in Baghdad named al-Khwarizmi wrote a book teaching people how to find a number they did not know, so they could share out inheritances, measure land and work out trade. He sorted the problems into six kinds and explained each with drawings of squares. From his book's name, al-Jabr, comes the word algebra that pupils learn all over the world ^{28 4}.

• Muhammad ibn Musa al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Ali Mustafa Mosharafa and Muhammad Mursi Ahmad, Cairo 1939: editors' introduction, scan pp. 13–26; author's preface and text, scan pp. 29–43, 51–53. ²⁹
• Abd al-Rahman ibn Khaldun, al-Muqaddima, pp. 495–496, section "al-Jabr wal-muqabala". ²³
• İhsan Fazlıoğlu, "CEBİR", TDV Encyclopedia of Islam, vol. 7 (1993), pp. 195–201, online edition. ⁶
• Wikimedia Commons, "Bodleian MS. Huntington 214 roll332 frame36.jpg", public domain, Bodleian Libraries. ²⁷