Al-Khwarizmi: Algebra and the Algorithm
The Islamic Golden AgeA Man We Can Barely See
He is one of the most consequential names in mathematics, and we can barely make out the man. Muhammad ibn Musa al-Khwarizmi worked in Baghdad under the caliph al-Ma'mun, who reigned from 813 to 833; the dates usually printed for him, around 780 to around 850, are estimates rather than records. His name means "from Khwarazm," the region south of the Aral Sea, though David King argued from a second epithet that the family came from Qutrubbull near Baghdad, and Roshdi Rashed thinks that reading rests on a copyist's slip. The one firmly dated glimpse of him is unflattering: al-Tabari places him among the astrologers called to the sickbed of the caliph al-Wathiq in 847, who read the horoscope and promised fifty more years. Al-Wathiq died in about ten days.
A Manual for Estate Disputes
His book is al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala, the Compendious Book on Calculation by Restoration and Balancing, written around 820 with al-Ma'mun's encouragement. It is often described as a research monograph. It is not. The preface promises the arithmetic men 'constantly require in cases of inheritance, legacies, partition, law-suits, and trade' — Rosen's rendering — and roughly half the book is exactly that: the punishing fractional arithmetic of dividing an estate under Islamic law. The two title words are procedures. Al-jabr, restoring, carries a subtracted quantity to the other side, where it turns positive. Al-muqabala, balancing, cancels matching terms on both sides. Together they force any problem into a standard shape.
Six Shapes, and Not One Symbol
There are six such shapes rather than one, because al-Khwarizmi would not write a negative coefficient or accept a negative answer. And there is no notation in the book at all. Everything is spelled out in ordinary sentences: mal, wealth, for the square, jidhr, root, for the unknown, dirham for a plain number. His showpiece reads, a square and ten of its roots equal thirty-nine dirhams. Halve the ten to get five; multiply it by itself, twenty-five; add thirty-nine, sixty-four; take the root, eight; subtract the five. The answer is three. Then he proves it, and the proof is a drawing: lay a rectangle two and a half wide along each side of the square, fill in the four corners, and you have a square of area sixty-four whose side must be eight. Completing the square was not yet a metaphor.
Two Words That Outlived Him
Al-jabr had an ordinary life outside mathematics: the reuniting of broken parts, bones included. In Spanish, algebrista long meant not a mathematician but a bonesetter, a trade often shared with the barber-surgeon, and the Spanish Royal Academy's first dictionary still defined it that way in the eighteenth century. A second book, on Indian numerals, gave the other word. Its Arabic is lost; it survives through Latin versions, one opening Dixit Algorizmi, "al-Khwarizmi said." Medieval algorismus meant one narrow thing, reckoning with the nine figures and zero instead of on a counting board. In Chaucer's Miller's Tale the Oxford student Nicholas keeps his augrym stones on a shelf by his bed. The broad modern sense, any step-by-step procedure, came much later.
What Came In, and What Is Still Argued
He received at least as much as he sent on. His astronomical tables were the Zij al-Sindhind, from the Sanskrit siddhanta, built on Indian and Persian work; his geography of 833 reworked Ptolemy with some 2,402 coordinates, cutting Ptolemy's overlong Mediterranean from about 63 degrees of longitude to nearly 50 (the true span is about 42, so he improved it without fixing it). Brahmagupta had handled negative numbers and a general quadratic rule two centuries earlier. A contemporary, Abd al-Hamid ibn Turk, left a chapter using the same geometric demonstrations, which Aydin Sayili argued may even predate the Algebra. Whether al-Khwarizmi founded algebra is genuinely disputed: Solomon Gandz said yes; Gerald Toomer rated the mathematics far lower.
Key Dates to Remember
- c. 820 — Al-Khwarizmi writes al-jabr wa'l-muqabala in Baghdad; the book can only be dated to al-Ma'mun's reign, 813-833
- 833 — Kitab surat al-ard, his reworking of Ptolemy's geography, lists some 2,402 coordinates and shortens the Mediterranean
- 847 — Al-Tabari names him among the astrologers at caliph al-Wathiq's sickbed - the last dated sighting of him alive
- 1145 — Robert of Chester translates the Algebra into Latin at Segovia, and al-jabr enters European languages as 'algebra'
- 1831 — Frederic Rosen prints and translates the unique surviving Arabic manuscript in London, putting the original text back in front of readers