Muhammad ibn Musa al-Khwarizmi

How to use this course

Two everyday words of mathematics come from one ninth-century scholar. Algebra is the second word of the title of his book on equations, al-jabr. Algorithm is a Latin distortion of his own name, al-Khwarizmi, which opened the Latin version of his arithmetic. He also wrote astronomical tables used from Central Asia to Spain, and a list of more than two thousand geographical coordinates that corrected Ptolemy.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proofs and exercises with folded solutions. Al-Khwarizmi wrote entirely in words, with numbers spelled out. We translate into modern symbols and say so. Several of his works survive only in Latin adaptations made three centuries later, and we flag where the surviving text may differ from what he wrote. The biography rests on a few lines in medieval sources. Each claim below names its source.

Timeline

A name and a few lines of evidence

Muhammad ibn Musa al-Khwarizmi means Muhammad, son of Musa, from Khwarazm. Khwarazm was an oasis region on the lower Amu Darya, south of the Aral Sea, in today’s Uzbekistan and Turkmenistan. Arab armies brought it into the caliphate in the early eighth century. His native language was probably Khwarazmian, an Iranian language, but every work of his that survives was written in Arabic, the language of science at the Abbasid court.

What the medieval sources say

No contemporary wrote his biography. Three kinds of evidence remain.

• Ibn al-Nadim, a Baghdad bookseller, compiled the Fihrist (Catalogue) in 987–88. He gives the name, connects al-Khwarizmi with the House of Wisdom (Bayt al-Hikma), and lists his books: two editions of the Zij, the algebra, works on the astrolabe and the sundial, the geography and a history.
• Al-Tabari, the great historian (d. 923), names a “Muhammad ibn Musa al-Khwarizmi al-Majusi al-Qutrubbulli”. Al-Majusi suggests Zoroastrian ancestry. Al-Qutrubbulli points to Qutrubbul, a district near Baghdad.
• His own prefaces, which show a devout Muslim writing for the caliph al-Maʾmun.

Two readings of al-Tabari

Gerald Toomer, in the Dictionary of Scientific Biography, read al-Tabari’s phrase as describing one man: a family from Khwarazm settled near Baghdad, perhaps Zoroastrian a generation or two back. Toomer noted that the pious preface of the algebra shows al-Khwarizmi himself was an orthodox Muslim. Roshdi Rashed argued that a conjunction has dropped out of the manuscripts, so that al-Tabari really names two people, al-Khwarizmi and a second man called al-Majusi al-Qutrubbulli. The question cannot be settled on present evidence. Both readings agree that his career was in Baghdad.

Dates

The birth year c. 780 and the death year c. 850 are modern inferences. They fit a man active as an established scholar under al-Maʾmun (813–833) and still consulted by the court in 847. Nothing is known of his family, his teachers or his education.

Baghdad and the translation movement

The Abbasid caliph al-Mansur founded Baghdad in 762 as a round city on the Tigris. Within two generations it was the largest city in western Eurasia and the centre of a movement that translated Greek, Persian, Syriac and Sanskrit learning into Arabic. Paper, recently introduced from Central Asia, made books cheap. Officials, physicians and astrologers wanted reliable mathematics, medicine and astronomy.

Three traditions meet

Indian astronomy. In the early 770s an embassy from Sindh brought a Sanskrit astronomical treatise to al-Mansur. Muhammad ibn Ibrahim al-Fazari and others turned it into Arabic as the Zij al-Sindhind, from Sanskrit siddhanta, a system or treatise. Its parameters derive from Brahmagupta’s Brahmasphutasiddhanta of 628. With it came Indian sine tables and decimal place-value numerals.

Persian astronomy. The Sasanian royal tables, the Zij al-Shah, were known in Arabic and supplied other parameters.

Greek science. Under Harun al-Rashid and especially al-Maʾmun, translators produced Arabic versions of Euclid’s Elements and Ptolemy’s Almagest and Geography. Al-Hajjaj ibn Yusuf translated the Elements twice, once under each caliph.

Al-Khwarizmi’s work sits exactly where these streams meet. His astronomy is Indian in method with Persian and Greek corrections, his geography is Ptolemaic in form with new data, and his algebra uses geometric proofs in the style of Euclid’s Book II for rules whose arithmetic recalls Indian and older Mesopotamian practice.

At the court of al-Maʾmun

Al-Maʾmun defeated his brother al-Amin in a civil war and became caliph in 813. He ruled from Merv until 819, then from Baghdad. He sponsored translation, astronomical observation at Baghdad and Damascus, and a survey to measure the size of the Earth. Ibn al-Nadim connects al-Khwarizmi with the House of Wisdom, the palace library and translation office. How formal that institution was is disputed, and later claims that he headed it have no early source.

Work for the caliph

The preface of the algebra says that al-Maʾmun encouraged him to write a short book on calculation by completion and balancing. He was to keep to what is easiest and most useful in arithmetic, the kind of calculation people constantly need for inheritance, legacies, partition, lawsuits, trade, land measurement and the digging of canals. This is a practical programme, and the book follows it.

He dedicated the Zij to al-Maʾmun as well. The geography, finished in 833, may draw on the world map al-Maʾmun commissioned; Toomer suggests this but the map is lost. Al-Khwarizmi worked alongside the three sons of Musa ibn Shakir, the Banu Musa, who were also patrons and scholars at the court.

Two reported missions

Some modern accounts say al-Khwarizmi took part in al-Maʾmun’s measurement of a degree of meridian in the plain of Sinjar, and in an embassy to the Khazars north of the Caspian under al-Wathiq. Both are doubtful. The tenth-century geographer al-Muqaddasi names a “Muhammad ibn Musa” as envoy to the Khazars, and Toomer thought this was more likely Muhammad ibn Musa ibn Shakir of the Banu Musa. David King has questioned the received account of the geodetic survey itself.

Completion and balancing

For the general reader. Before al-Khwarizmi, people solved particular problems with particular tricks. He showed that every problem of a certain kind reduces, by two simple moves, to one of six standard equations, and that each standard equation has a fixed rule of solution. This shift from problems to equation types is the beginning of algebra as a subject.

The vocabulary

Al-Khwarizmi works with three kinds of quantity:

• jidhr (root), also shayʾ (thing): the unknown, our \(x\);
• mal (wealth, property): the square of the root, our \(x^2\);
• ʿadad mufrad (simple number), counted in dirhams: the constant term.

Coefficients are always positive numbers, and an equation sets a sum of positive terms equal to another sum of positive terms. There are no negative numbers and no zero coefficients. All of this is in words: “a mal and ten roots equal thirty-nine dirhams” is our \(x^2+10x=39\).

The two operations

In the book’s own examples, \(x^2=40x-4x^2\) becomes \(5x^2=40x\) by al-jabr, and \(50+3x+x^2=29+10x\) becomes \(21+x^2=7x\) by al-muqabala. A third step, dividing or multiplying to make the coefficient of the square equal to one, al-Khwarizmi calls reduction (radd) or completion (ikmal).

The six types

The rules for the compound types, translated into symbols, are

For type 5 he says that both values may serve, that if \((b/2)^2=c\) the root is half the roots, and that if \((b/2)^2<c\) the problem is impossible. In type 1 he discards the root \(x=0\), which is not a number for him. Each rule is stated with a worked numerical example, and the book’s first example for type 4 is \(x^2+10x=39\): halve the ten to get five, square it to get twenty-five, add thirty-nine to get sixty-four, take the root eight, subtract five, and the root is three.

Geometric proofs of the rules

For the general reader. A rule that works on examples is not yet a proof. Al-Khwarizmi justified his rules with pictures: the unknown square is a real square, the roots are rectangles stuck to its sides, and the rule is read off by filling in a missing corner. “Completing the square” is literally that.

Type 4: two figures

First figure. Draw the square \(x^2\). Split the ten roots into four rectangles of size \(x\times\tfrac{10}{4}\) and attach one to each side. The cross-shaped figure has area \(x^2+10x=39\). Its four corners are missing squares of side \(2\tfrac12\), total area \(4\cdot 6\tfrac14=25\). Adding them gives a square of area \(64\) and side \(x+5=8\), so \(x=3\).

Second figure. Attach two rectangles of size \(x\times5\) to two adjacent sides of the square. The missing corner is a single square \(5\times5\). Then

The hero figure at the top of this page is the second construction.

Proof, following the figure

Suppose first \(x<b/2\). Draw a rectangle of length \(b\) and width \(x\), area \(bx\). At its left end cut off the square of side \(x\). The remaining rectangle has area \(bx-x^2=c\). Mark the midpoint of the length \(b\); it lies in the remaining rectangle since \(x<b/2\). On the half-length \(b/2\) erect a square of area \((b/2)^2\). Rearranging the pieces, as in Euclid II.5, shows that this large square consists of the rectangle of area \(c\) (moved around) plus a small square of side \(b/2-x\). Hence

If \((b/2)^2<c\) the right side is negative, which no square can be: the problem is impossible. Otherwise \(x=\tfrac b2-\sqrt{(b/2)^2-c}\). The case \(x>b/2\) is the same picture with the midpoint inside the square, giving \(x-\tfrac b2=\sqrt{(b/2)^2-c}\). In modern language, the two cases are the two signs of the square root.

Euclid, Book II

Euclid’s Elements II.5 says: if a line is cut into equal and unequal segments, the rectangle on the unequal segments plus the square on the stretch between the cuts equals the square on half the line. With the line of length \(b\) cut at \(x\), this is \(x(b-x)+(b/2-x)^2=(b/2)^2\), exactly the identity above. II.6 plays the same role for types 4 and 6. Al-Khwarizmi does not cite Euclid, but Toomer took the figures as evidence that he knew Book II.

Problems, mensuration and legacies

For the general reader. Most of the algebra is not about equations in the abstract. It is a manual of problems: numbers to be split, fields to be measured, estates to be divided under Islamic law. These chapters show what the new method was for.

The structure of the book

Toomer describes three parts: algebra proper (the six types, operations on binomials such as \((10-x)(10-x)\), and solved problems), practical mensuration, and the long chapter on legacies, which takes about half the text. A short section on commercial transactions applies the rule of three.

“I have divided ten into two parts”

Many problems begin this way. The unknown is one part, \(x\), so the other is \(10-x\), and the algebra of binomials is needed before the six types can be applied. In one problem one part, multiplied by itself, equals the other taken eighty-one times: \(x^2=81(10-x)\), or \(x^2+81x=810\), type 4, with \(x=9\) and the parts 9 and 1.

Mensuration

For the circumference of a circle of diameter \(d\), the book gives a practical rule and two refinements:

The last, equal to \(3.1416\,d\), appears earlier in the Aryabhatiya of Aryabhata (499 CE). Al-Khwarizmi remarks that none of these is exact. The chapter also computes areas of triangles and quadrilaterals and volumes of solids including the cone and pyramid. Toomer notes close links with the Hebrew Mishnat ha-Middot, whose date and relation to al-Khwarizmi are disputed.

Legacies

Islamic law fixes the shares of heirs (Qurʾan 4:11–12): for example a husband receives a quarter of his wife’s estate if she leaves children, and a son receives twice a daughter’s share. A bequest to a non-heir may not exceed one third of the estate. Al-Khwarizmi’s problems combine these rules with bequests, debts and returned dowries. The unknown is usually the size of one share, and the equations are of the first degree.

Calculation with the Indian numerals

For the general reader. With nine digits and a sign for “nothing”, any number can be written, and the position of a digit tells its value. Al-Khwarizmi wrote one of the first Arabic books explaining this Indian system and how to add, subtract, multiply and divide with it. Its Latin version began “Dixit Algorizmi”, “so said al-Khwarizmi”, and the name of the author became the name of the method.

The text

The Arabic original is lost; its title was probably Kitab al-hisab al-hindi (Book of Indian calculation). It cites the algebra, so it was written after c. 820. A single Latin manuscript, Cambridge University Library Ii.6.5, begins Dixit Algorizmi. Boncompagni printed it in 1857 under the title Algoritmi de numero Indorum, and Kurt Vogel published a facsimile in 1963. It is incomplete and was adapted by its translator, sometimes attributed to Adelard of Bath. Other twelfth-century Latin works, such as the Liber Ysagogarum Alchorismi and the Liber Alchoarismi de practica arismetice, rework the same material. A method for square roots that was in the Arabic is missing from the Latin.

Content

The Latin text introduces nine figures and a small circle for an empty place, and explains that a figure is worth ten times more for each place it moves to the left. It describes addition, subtraction, doubling and halving (treated as separate operations, as in older practice), multiplication and division, then common fractions and the sexagesimal fractions used by astronomers. Calculation was done on a dust board (takht), where digits were written in sand or dust and erased as the work proceeded. Later in the tenth century al-Uqlidisi adapted the methods to pen and paper.

Proof of uniqueness. If \(\sum d_i10^i=\sum e_i10^i\) with \(d\ne e\), let \(j\) be the least index with \(d_j\ne e_j\). Divide the difference by \(10^j\): then \(d_j-e_j\equiv0\pmod{10}\), but \(|d_j-e_j|\le9\), a contradiction. Existence follows from repeated division by 10. The proof explains why zero is indispensable: without a sign for \(d_i=0\), the strings for 23, 203 and 2003 coincide.

Multiplication on the dust board

To multiply, the multiplicand is written above the multiplier, shifted so that the last digit of the multiplicand stands over the first digit of the multiplier. Each partial product is added into the row above, its digits erased and rewritten, and the multiplier slides one place right. The method is our long multiplication with erasure instead of paper.

The Zij: sines, mean motions, epochs

For the general reader. A zij is an astronomer’s handbook: instructions and tables for finding where the Sun, Moon and planets are on any date. Al-Khwarizmi’s Zij is the earliest Arabic zij whose contents survive in substance, though only in Latin. It brought Indian methods, including the sine function, into the astronomy of the Islamic world and later of Europe.

A text in layers

Al-Khwarizmi’s Arabic Zij is lost. In Cordoba around 1000, Maslama al-Majriti revised it, replacing the Persian era of Yazdegerd (epoch 16 June 632) with the Hijra era and adapting it for use in Cordoba. Adelard of Bath translated this revision into Latin in 1126. Heinrich Suter edited the Latin in 1914, and Otto Neugebauer published a translation and analysis in 1962. The Latin has 37 chapters and 116 tables for the Sun, Moon and five planets, eclipses, lunar visibility, trigonometry and astrology.

Sources

The mean motions come from the Indian Sindhind, ultimately Brahmagupta. Some equations come from the Persian Zij al-Shah. The solar declination and the obliquity of the ecliptic, \(23^\circ51'\), come from Ptolemy through Theon’s Handy Tables. The resulting mixture was criticized a century later by al-Farghani, whose critique was answered by the Andalusian Ibn al-Muthanna.

The sine

Ptolemy used chords. Indian astronomers used half-chords, the jya, and the Arabic transliteration jayb became Latin sinus. For a circle of radius \(R\), the Indian sine of an arc \(\theta\) is \(R\sin\theta\). Toomer reports that al-Khwarizmi’s original tables used \(R=150\), while the surviving Latin tables use \(R=60\).

Derivation. Put the Sun on the unit sphere at \((\cos\lambda,\sin\lambda,0)\) in ecliptic coordinates. Rotating by \(\varepsilon\) about the equinox direction (the \(x\)-axis) to equatorial coordinates gives \(z=\sin\lambda\sin\varepsilon\), and \(z=\sin\delta\).

Mean motion

Each body has a mean longitude growing uniformly: \(\bar\lambda(t)=\bar\lambda_0+n\,(t-t_0)\), with \(\bar\lambda_0\) its value at the epoch \(t_0\) and \(n\) the mean daily motion. A true position adds an equation (correction) read from a table as a function of the anomaly. The tables give \(\bar\lambda\) for whole years and months so that the user adds a few entries instead of multiplying.

Calendars and instruments

For the general reader. Astronomy at the Abbasid court served timekeeping: prayer times, the start of months, conversion between the calendars of Muslims, Persians, Jews and Christians. Al-Khwarizmi wrote a short treatise on the Jewish calendar and two on the astrolabe, the instrument that turned a model of the sky into a hand-held computer.

The Jewish calendar

The treatise Istikhraj taʾrikh al-yahud (Extraction of the Jewish era) explains how to find the weekday of the Jewish New Year, the mean longitudes of the Sun and Moon at the beginning of a year, and the place of a year in the 19-year cycle. A worked example uses the year 823–24, so it cannot be earlier. Toomer called it accurate and noted that it shows the present Jewish calendar was already in place in the ninth century. E. S. Kennedy analysed it in 1964.

The Jewish calendar takes the mean month as \(29\) days \(12\) hours and \(793\) parts, where one hour has \(1080\) parts.

The astrolabe

Ibn al-Nadim lists two astrolabe works by al-Khwarizmi, on construction and on use. Only extracts of the second may survive. A planispheric astrolabe projects the celestial sphere stereographically from the south pole onto the plane of the equator. Stereographic projection maps circles to circles and preserves angles, so the horizon, the ecliptic and the circles of altitude all appear as circles on the plates, and rotating the star map (the rete) over a plate simulates the turning sky.

The image of the Earth

For the general reader. Ptolemy’s Geography located thousands of places by latitude and longitude. Al-Khwarizmi rewrote it as a table of coordinates for 2402 localities, corrected and extended with information from the Islamic world. His Mediterranean is much closer to its true length, and his Indian Ocean opens onto the world sea.

The book

Kitab surat al-ard (The image of the Earth) was completed in 833. It survives in a single Arabic manuscript at the National and University Library of Strasbourg; Hans von Mžik edited it in 1926, and Carlo Alfonso Nallino’s study remains the classic analysis. The book lists cities, mountains, seas, islands and rivers, each with longitude and latitude, arranged by the seven climata (zones of latitude) and ordered by longitude within each. It contains no map; Hubert Daunicht later reconstructed maps from the coordinates.

Corrections to Ptolemy

Ptolemy made the Mediterranean span about \(63^\circ\) of longitude; al-Khwarizmi gives about \(50^\circ\). His zero meridian lies at the Fortunate Isles (the Canaries), as Ptolemy’s did, but his western coordinates are shifted by about \(10^\circ\). He shows the Atlantic and Indian Oceans as open bodies of water, where Ptolemy had closed the Indian Ocean with a southern land. For Europe he mostly follows Ptolemy; for the Islamic world, Africa and the East he adds much new material.

Proof. The unit vectors to the two points are \((\cos\varphi_i\cos\lambda_i,\cos\varphi_i\sin\lambda_i,\sin\varphi_i)\); their dot product is \(\cos\sigma\) and simplifies to the formula.

Last years and the Arabic tradition

Al-Maʾmun died on campaign in 833, the year the geography was completed. Al-Khwarizmi remained connected to the court. Al-Tabari reports that in 847 the caliph al-Wathiq, gravely ill, summoned astronomers, al-Khwarizmi among them, to cast his horoscope. They promised him fifty more years of life; he died within ten days. Ibn al-Nadim also lists a historical work, Kitab al-taʾrikh, written after 826 and known only through quotations by later chroniclers such as Elias of Nisibis. Al-Khwarizmi probably died around 850. No source records the date or place.

Successors in Arabic

• Abu Kamil of Egypt (c. 850–930) wrote his own Algebra, extending the method to irrational coefficients and systems, and explicitly defended al-Khwarizmi’s priority.
• Al-Karaji (fl. c. 1000) in Baghdad developed an arithmetic of polynomials, freeing algebra further from geometry.
• Omar Khayyam (1048–1131) classified cubic equations, as al-Khwarizmi had classified quadratics, and solved them by intersecting conics.
• The Zij was revised in Cordoba by Maslama al-Majriti (d. 1007) and his pupil Ibn al-Saffar. Some of its tables passed into the Toledan Tables of the eleventh century, and some were still being copied in Egypt in the nineteenth.

Algorismus in Europe

In the twelfth century, scholars in Spain, Sicily and southern Italy translated Arabic science into Latin. Al-Khwarizmi’s works were among the first.

• Astronomy. Adelard of Bath translated Maslama’s revision of the Zij in 1126. It introduced sine tables to Latin Europe.
• Arithmetic. The Dixit Algorizmi and its relatives taught the Indian numerals. Practitioners of the new method were called algorists, opposed to the abacists who used counting boards. Through later corruptions, algorismus became “algorism” and finally “algorithm”, now meaning any precise step-by-step procedure.
• Algebra. Robert of Chester translated the algebra at Segovia in 1145 as Liber algebre et almucabola; Gerard of Cremona translated it again in Toledo. Leonardo of Pisa drew on it in the Liber abaci (1202). The six types and their geometric proofs remained the core of algebra teaching in Europe until the sixteenth century, when Italian algebraists solved the cubic.
• Words. “Algebra” comes from al-jabr. The Latin radix (root) translates jidhr, and census translates mal.

Modern recovery

Frederic Rosen published the Arabic algebra with an English translation in 1831. The Zij (Suter 1914, Neugebauer 1962), the geography (von Mžik 1926), the arithmetic (Boncompagni 1857, Vogel 1963, Allard 1992) and the algebra again (Rashed 2007) have since received critical editions. His name is borne today by universities and institutes in Uzbekistan and elsewhere in Central Asia.

Sources and further reading

Biography

1. G. J. Toomer, al-Khwārizmī, Dictionary of Scientific Biography 7 (1973).
2. J. J. O’Connor and E. F. Robertson, Abu Ja’far Muhammad ibn Musa Al-Khwarizmi, MacTutor.
3. B. van Dalen, Khwārizmī, Biographical Encyclopedia of Astronomers, Springer, 2007.
4. Muhammad ibn Musa al-Khwarizmi, Wikipedia.
5. D. Gutas, Greek Thought, Arabic Culture: the Graeco-Arabic Translation Movement in Baghdad, Routledge, 1998.

Mathematics and astronomy

1. F. Rosen, The Algebra of Mohammed ben Musa, London, 1831 (Arabic text and English translation).
2. R. Rashed, Al-Khwārizmī: The Beginnings of Algebra, Saqi, 2009 (French edition 2007).
3. B. B. Hughes, Robert of Chester’s Latin Translation of al-Khwarizmi’s Al-jabr, Steiner, 1989.
4. The Compendious Book on Calculation by Completion and Balancing, Wikipedia.
5. K. Vogel, Mohammed ibn Musa Alchwarizmi’s Algorismus, Aalen, 1963; A. Allard, Muhammad ibn Mūsā al-Khwārizmī: Le Calcul indien, Blanchard, 1992.
6. Algoritmi de numero Indorum, Wikipedia.
7. O. Neugebauer, The Astronomical Tables of al-Khwārizmī, Copenhagen, 1962.
8. Zij al-Sindhind, Wikipedia.
9. E. S. Kennedy, Al-Khwārizmī on the Jewish calendar, Scripta Mathematica 27 (1964).
10. H. von Mžik, Das Kitāb ṣūrat al-arḍ des Abū Ǧaʿfar Muḥammad ibn Mūsā al-Ḫuwārizmī, Leipzig, 1926; Book of the Description of the Earth, Wikipedia.
11. J. L. Berggren, Episodes in the Mathematics of Medieval Islam, 2nd ed., Springer, 2016.