al-Biruni

How to use this course

Abu Rayhan al-Biruni is often remembered as the author of the finest medieval book about India. That book is one part of an output of about 146 works, roughly two-thirds of them in the exact sciences. He measured the Earth from a hilltop, determined longitudes from eclipse timings, built trigonometry into the foundation of astronomy, and weighed metals and gems with a precision that impressed nineteenth-century chemists. He also argued with Ibn Sina about Aristotle, and he insisted, more firmly than most of his contemporaries, that a claim about nature must answer to observation.

The sources for his life are better than for most medieval scientists. His own prefaces and his bibliography of 1036 date many works, and later historians, notably Bayhaqi and Nizami ʿAruzi, add episodes, some of them anecdotal. The lessons say which source supports each claim and mark later tradition as such. All dates are CE; “c.” marks an approximate date. Most of his works are in Arabic, the language of science in his world, though his mother tongue was Khwarezmian, an Eastern Iranian language. This course is a companion to the course on Ibn Sina, his contemporary and sparring partner.

Timeline

Kath and Khwarazm, 973–995

Al-Biruni was born on 4 September 973 in Khwarazm, the oasis of the lower Amu Darya south of the Aral Sea. His name is usually derived from the Persian birun, “outside”: he came from the outer district of Kath, the capital. He later said little about his family, and nothing reliable is known of his parents. He spent his first twenty-five years or so in Khwarazm.

Khwarazm was then divided between two dynasties: the old Afrighid shahs at Kath and the rising Maʾmunids at Gurganj (Urgench) to the north. The Afrighid family produced al-Biruni’s teacher, Abu Nasr Mansur ibn ʿIraq, a mathematician of real distinction. Abu Nasr worked on spherical trigonometry, and al-Biruni later credited him with proofs of the sine theorem (Lesson 05). Several of Abu Nasr’s treatises are addressed to his pupil.

First observations

As a young man al-Biruni was already observing. Using a meridian ring, he measured the Sun’s noon altitudes at Kath to find the latitude of the town. This was the start of a lifelong preference for measurement: wherever he lived, he set up instruments, and his later books keep records of dated observations.

Wandering years, 995–c. 1004

In 995 the Maʾmunid ruler of Gurganj overthrew the Afrighids. Al-Biruni left Khwarazm. He seems to have gone to Rayy, near modern Tehran, where he met the astronomer Abu Mahmud al-Khujandi and discussed the giant mural sextant al-Khujandi had built to measure the obliquity of the ecliptic. Al-Biruni later reported those observations, and their instrumental problems, in his Tahdid.

He was back in Kath for the lunar eclipse of 997. By arrangement, Abu’l-Wafaʾ al-Buzjani observed the same eclipse in Baghdad. Because an eclipse of the Moon is seen at the same physical moment everywhere, the difference between the local times of the two observations gives the difference in longitude. It is one of the earliest documented observations coordinated between distant cities (Lesson 08).

Bukhara and Gurgan

He found patrons in Bukhara under the Samanid ruler Mansur II (997–999), and then at Gurgan, south-east of the Caspian, under the Ziyarid amir Qabus ibn Wushmgir. For Qabus he wrote al-Athar al-baqiya ʿan al-qurun al-khaliya, “the remaining traces of past centuries”, around 1000. It is a comparative study of the calendars, eras and festivals of Persians, Sogdians, Khwarazmians, Jews, Christians, Arabs and others, and the subject of Lesson 04.

The questions to Ibn Sina

Around 1000 al-Biruni sent eighteen questions on natural philosophy to the young Ibn Sina, ten of them aimed at Aristotle’s On the Heavens. Why must celestial motion be circular, and why could the heavens not move in an ellipse? Why is a vacuum impossible? Why could there not be other worlds? Why does ice float, when it is solid water? Ibn Sina answered for the Aristotelians; al-Biruni objected sharply; and Ibn Sina’s pupil Abu Saʿid al-Maʿsumi wrote a final reply. The tone on both sides is impatient. The exchange is our best evidence of al-Biruni’s habit of testing doctrine against experience.

Gurganj and the conquest, c. 1004–1017

Around 1004 al-Biruni made peace with the Maʾmunids and returned to Khwarazm, now ruled from Gurganj. He served Abu’l-Hasan ʿAli ibn Maʾmun and then his brother Abu’l-ʿAbbas Maʾmun (ruled 1009–1017). According to the historian Bayhaqi, he was a companion and political adviser for seven years under the second ruler. Ibn Sina also served at this court until about 1012. His teacher Abu Nasr Mansur was there too, and the two men continued their trigonometric work.

The court lay in the shadow of Mahmud of Ghazna, the most powerful ruler of the eastern Islamic world. In 1017, after the murder of Abu’l-ʿAbbas Maʾmun in a revolt, Mahmud invaded Khwarazm and annexed it. He carried off its scholars to Ghazna (modern Ghazni, Afghanistan), among them al-Biruni and Abu Nasr. Whether al-Biruni went as a prisoner, a hostage or a reluctant courtier is debated; the early years at Ghazna were hard.

A twelfth-century collection of anecdotes by Nizami ʿAruzi tells that Mahmud once asked al-Biruni to predict by which gate he would leave a pavilion, and then broke through a wall to defeat him, only to find the prediction written down correctly. The story is later tradition and cannot be checked; it shows how al-Biruni’s reputation as an astrologer outlived his own scepticism about astrology.

Calendars and the Chronology

For the general reader. To compare the histories of different peoples, you must convert dates between their calendars. Some count days by the Sun, some by the Moon, some by both, and each starts from a different era. Al-Biruni’s Chronology (c. 1000) gathered the rules for all the calendars he could find and turned the conversion between them into arithmetic.

Three kinds of calendar

A solar calendar tracks the year of the seasons (the tropical year, about 365.2422 days). The Julian calendar does this with a year of 365 days and a leap day every fourth year. The old Persian calendar had 12 months of 30 days and 5 extra days, 365 in all, with no regular intercalation, so its new year drifted through the seasons. A lunar calendar, such as the Islamic one, counts 12 synodic months (about 29.5306 days each) and ignores the seasons. A lunisolar calendar, such as the Jewish one, inserts a thirteenth month 7 times in 19 years.

The Islamic calendar uses a 30-year cycle of 360 months in which 11 years have 355 days and 19 have 354, giving \(10631\) days. The mean month is then \(10631/360=29.530\overline5\) days, very close to the true mean synodic month.

Counting days

To convert a date, count the days from a fixed epoch in each calendar. Al-Biruni tabulated the epochs of the eras: the Seleucid era (“era of Alexander”), the era of Yazdegerd III, the Hijra, and others. The weekday of any day number \(N\) counted from a known epoch follows from \(N\bmod 7\).

The chessboard

In the Chronology, al-Biruni computed the number of grains on a chessboard when one grain is placed on the first square and the number doubles on each square. He gave the result in full.

One way to compute it, close to al-Biruni’s, is repeated squaring: \(2^{8}=256\), \(2^{16}=65536\), \(2^{32}=4\,294\,967\,296\), and \(2^{64}=(2^{32})^2\).

Sines, tangents and shadows

For the general reader. Greek astronomers computed with chords of circles. Indian astronomers replaced the chord by the half-chord, our sine, and astronomers of the Islamic world added the shadow functions, our tangent and cotangent, read off the shadow of a vertical stick. Al-Biruni’s generation turned these into a systematic trigonometry, with clean theorems that replaced Ptolemy’s cumbersome configurations. His Exhaustive Treatise on Shadows (c. 1021) is the fullest medieval account of the shadow functions.

Medieval functions

Medieval tables used a base circle of radius \(R\), commonly \(R=60\). The sine of an arc \(\theta\) was the half-chord \(\mathrm{Sin}\,\theta=R\sin\theta\). For a gnomon (vertical stick) of length \(g\) and Sun altitude \(h\), the horizontal shadow is \(g\cot h\) and the shadow on a wall perpendicular to the Sun’s azimuth is \(g\tan h\). Gnomons were divided into 12 digits or 60 parts.

Proof. In the circumcircle, side \(a\) is a chord subtending the central angle \(2A\), by the inscribed angle theorem. A chord subtending \(2A\) in a circle of radius \(\rho\) has length \(2\rho\sin A\). The same holds for \(b\) and \(c\).

Al-Biruni sent proofs of the plane and spherical sine theorems to the geometer al-Sijzi and credited them to Abu Nasr Mansur. Abu’l-Wafaʾ also gave a proof, and priority was disputed at the time. The theorem let astronomers solve spherical triangles directly, replacing Menelaus’s theorem on the complete quadrilateral.

Proof sketch (spherical). Put the triangle on a unit sphere with centre \(O\). Drop the perpendicular from vertex \(C\) to the plane \(OAB\), meeting it at \(P\). Dropping perpendiculars from \(P\) to the lines \(OA\) and \(OB\) and computing \(|CP|\) in two ways gives \(|CP|=\sin b\sin A=\sin a\sin B\).

Chords, the broken chord and the nonagon

For the general reader. Every sine table needs a starting point, and the hardest is the sine of 1°. Euclid’s constructions give the chords of 36°, 60°, 72° and 90°, and halving and subtracting give all multiples of 3°. Getting to 1° requires trisecting an angle, which ruler and compass cannot do. Al-Biruni’s Book on Chords (c. 1027) gathered many proofs of a theorem he attributed to Archimedes, and his Masʿudic Canon reduced the trisection to a cubic equation and solved it numerically.

The broken chord

Proof. Choose \(E\) on \(AC\) with \(|AE|=|CB|\). Since \(M\) is the midpoint of the arc, \(|MA|=|MB|\). The inscribed angles \(\angle MAC\) and \(\angle MBC\) stand on the same arc \(MC\), so they are equal. Hence triangles \(MAE\) and \(MBC\) are congruent (SAS) and \(|ME|=|MC|\). Triangle \(MEC\) is isosceles, so the foot \(D\) of the perpendicular from \(M\) is the midpoint of \(EC\). Therefore \(|AD|=|AE|+|ED|=|CB|+|DC|\).

Al-Biruni used the theorem to derive trigonometric rules. With chords \(\mathrm{crd}\,\theta=2R\sin(\theta/2)\) on a unit circle and arcs \(AC=2\alpha\) and \(CB=2\beta\), the theorem becomes the sum-to-product rule \(\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}2\cos\frac{\alpha-\beta}2\) (Exercise 6.1), which is equivalent to the addition formula for sines.

The chord of 40° and the nonagon

The side of a regular nonagon subtends 40°. With the triple-angle formula \(\cos3\theta=4\cos^3\theta-3\cos\theta\) and \(\theta=20^\circ\), \(\cos60^\circ=\tfrac12\), the number \(x=2\cos20^\circ\) satisfies

In the Canon (Book III, chapter 3), al-Biruni set up the cubic \(x^3=1+3x\) in the course of finding the chord of 40° and found its root by an iterative numerical procedure, correct to three sexagesimal places. Non-constructibility, proved by Pierre Wantzel in 1837, explains why such a numerical method was unavoidable.

Measuring the Earth

For the general reader. Earlier measurements of the Earth, by Eratosthenes and by the astronomers of the caliph al-Maʾmun, needed long journeys across flat desert to measure a degree of latitude. Al-Biruni found a way to do it from one place. Climb a hill of known height, look at the horizon, and measure how far it lies below the horizontal. He carried this out at Nandana, a fort in the Salt Range of the Punjab, while in Ghaznavid service, and reported it in the Masʿudic Canon.

Step 1: the height of the hill

From two points on the plain, a distance \(d\) apart and in line with the summit, measure the angles of elevation \(\theta_1<\theta_2\). If \(x\) is the distance from the nearer point to the foot of the summit, then \(h=x\tan\theta_2=(x+d)\tan\theta_1\), so

Step 2: the dip

Proof. The line of sight to the horizon touches the Earth at a point \(T\), so \(OT\perp\) the sight line, where \(O\) is the centre. In the right triangle \(O T P\), with \(P\) the observer, \(|OP|=R+h\) and \(|OT|=R\). The horizontal at \(P\) is perpendicular to \(OP\), so the dip \(\alpha\) equals the angle \(\angle TOP\). Hence \(\cos\alpha=R/(R+h)\). Solve for \(R\). For small \(\alpha\), \(1-\cos\alpha\approx\alpha^2/2\).

His numbers

As reported in the Canon, the height was \(652;3,18\) cubits (about 652.06) and the dip \(0;34^\circ\), that is 34 arcminutes. His stated radius is about 12.8 million cubits (12,803,337 cubits in the usual reading). Converting cubits to kilometres is uncertain, because the length of his cubit is not known exactly; reasonable conversions give values within a few per cent of the modern mean radius of 6371 km. Two cautions: refraction bends the line of sight and makes the dip appear smaller, and the plain at Nandana is not a sea horizon. The close agreement owes something to luck.

Longitudes, the qibla and the Tahdid

For the general reader. Latitude is easy to measure from the Sun or the pole star. Longitude is hard, because it requires knowing the time at two places at once. A lunar eclipse supplies a common moment. In the Tahdid nihayat al-amakin (“determination of the coordinates of places”, 1025), al-Biruni combined eclipse timings, travellers’ distances and spherical trigonometry to compute the longitude of Ghazna. The motive was partly religious: Muslims pray facing Mecca, and the direction, the qibla, is a problem in spherical trigonometry.

Al-Biruni computed the longitude difference between Baghdad and Ghazna as \(24;20^\circ\), within about 18 arcminutes of the modern value.

Spherical distance and direction

Both follow from the spherical triangle formed by the pole and the two places, whose sides are the colatitudes \(90^\circ-\varphi_1\), \(90^\circ-\varphi_2\) and the arc \(\sigma\), with angle \(\Delta\lambda\) at the pole. Al-Biruni gave several methods for the qibla, some by projection onto the plane, some by successive applications of the sine theorem.

The Masʿudic Canon: tables and astronomy

For the general reader. Al-Qanun al-Masʿudi, dedicated around 1030 to the sultan Masʿud, is al-Biruni’s astronomical summa, covering the ground of Ptolemy’s Almagest in eleven books. It is not a copy. It defines \(\pi\) as the ratio of circumference to diameter, builds trigonometric tables with second-order interpolation, shows that the Sun’s apogee moves independently of precession, and gives a star catalogue of 1029 stars.

Structure

Book III treats trigonometry, with tables of sines and tangents. Book IV covers spherical astronomy, Book V geodesy and mathematical geography with a table of coordinates of about 600 places, Books VI–VII the Sun and Moon, Book VIII eclipses, Book IX the fixed stars, Book X the planets, and Book XI astrology. In the star catalogue he added \(13^\circ\) to Ptolemy’s longitudes to account for precession.

Interpolation

Proof. Write \(p(t)=f_0+bt+ct^2\). The conditions \(p(1)=f_1\) and \(p(-1)=f_{-1}\) give \(b+c=\Delta_+\) and \(-b+c=-\Delta_-\), so \(b=\tfrac12(\Delta_-+\Delta_+)\) and \(c=\tfrac12(\Delta_+-\Delta_-)\). The error bound is the standard Lagrange remainder. Brahmagupta had used a related second-difference rule; al-Biruni knew Brahmagupta’s Khandakhadyaka but developed his own method.

The solar apogee

Ptolemy held that the Sun’s apogee, the direction where the Sun is farthest from Earth, was fixed relative to the equinoxes. Astronomers of the ninth century had already found it in a different place. Al-Biruni concluded that the apogee has a motion of its own, distinct from precession. In modern terms, the apsidal line of the Earth’s orbit advances slowly relative to the stars because of planetary perturbations.

Does the Earth turn?

Al-Biruni knew that Aryabhata and, in his own world, the astronomer Abu Saʿid al-Sijzi had proposed that the Earth rotates daily. He noted that the hypothesis does not change any astronomical computation and judged that the question belonged to natural philosophy rather than mathematics. He seems to have leaned against rotation without claiming a proof; his treatise devoted to the question is lost.

Specific gravity and the mineral world

For the general reader. How can you tell gold from a cheaper alloy without destroying it? Archimedes’ answer was to compare its weight with the weight of the water it displaces. Al-Biruni built an instrument for doing this precisely, a conical flask with a spout, and measured the specific gravities of metals and precious stones. His results are preserved in his Kitab al-Jamahir on gems and metals and in the Book of the Balance of Wisdom by al-Khazini a century later.

The instrument

The flask was filled with water to the level of a lateral spout. A sample dropped in displaced exactly its own volume of water, which ran out of the spout into a pan and was weighed. The ratio of the sample’s weight to the weight of displaced water is its specific gravity. Al-Khazini noted a practical difficulty: drops of water clinging inside the spout spoil the reading.

Proof. Replace the body by fluid of the same shape. That fluid is in equilibrium, so the pressure on its boundary supports exactly its weight. The pressure depends only on depth, not on what is inside, so the same net force acts on the body. The apparent loss of weight \(W-W'\) is therefore the weight of the displaced water, and \(W/(W-W')\) is the ratio of the body’s density to water’s.

Ghazna and India, 1017–c. 1050

At Ghazna al-Biruni began again with almost nothing. He recorded observations near Kabul in 1018, improvising instruments, and in the following years he accompanied Mahmud’s campaigns into northwest India. He learned Sanskrit, worked with pandits, and read Indian astronomy, mathematics, philosophy and religion in the original. He translated works in both directions, including Patanjali’s Yoga Sutras into Arabic, and he intended to translate Brahmagupta’s Brahmasphutasiddhanta.

The book on India

His Tahqiq ma li-l-Hind, usually called India, was completed in 1030, shortly after Mahmud’s death. It describes Indian religion, philosophy, literature, geography, calendars, astronomy and astrology, with long accounts of Aryabhata, Brahmagupta and Varahamihira. Al-Biruni states in his preface that he will report what Indians say in their own terms and compare it with Greek ideas, without polemic. Modern scholars often call him the first Indologist. He is also frank: he criticizes Indian astronomers for mixing exact computation with myth, and he records the destruction caused by the Ghaznavid conquests.

The last works

Mahmud’s son Masʿud treated al-Biruni well, and the Masʿudic Canon repays him with a dedication. The Tafhim, an introduction to geometry, arithmetic, astronomy and astrology in question-and-answer form, appeared in Arabic and Persian around 1029. In 1036 he compiled a bibliography of the physician al-Razi and appended a list of his own 103 completed works. Under the sultan Mawdud he wrote the Kitab al-Jamahir on gems and metals. His last book, on pharmacology (Kitab al-Saydana), says he was over eighty and failing in sight and hearing, and that he worked with a helper. Uzbek scholars date his death to December 1048; the Saydana evidence suggests a date after 1050. He died at Ghazna.

Legacy

Of roughly 146 works, about 22 survive. The losses include his book on the Earth’s rotation, his book on root extraction and most of his astronomical instruments treatises. The survivors were read mainly in the eastern Islamic world. His student tradition is thin, but al-Khazini preserved his specific gravities in the twelfth century, and parts of the Canon passed into later astronomy: an irregularity in its Mercury tables reappears in the Chinese Huihui li of 1384.

Modern recognition began with the German orientalist Eduard Sachau, who edited and translated the Chronology (1879) and India (1888). In the twentieth century, Edward S. Kennedy and his students translated the Shadows (1976) and analysed the Tahdid and the Canon. The millennium of his birth in 1973 was marked by conferences and a Soviet stamp; a town in Karakalpakstan is named Beruniy, and the lunar crater al-Biruni and the asteroid 9936 al-Biruni carry his name.

Assessment

• Trigonometry: systematic use of sines, tangents and the sine theorems in astronomy; the broken chord as a source of identities; numerical solution of the trisection cubic.
• Geodesy: the single-station dip method for the Earth’s radius; eclipse longitudes and spherical-trigonometric coordinates.
• Astronomy: second-order interpolation in tables, the independent motion of the solar apogee, a new star catalogue.
• Method: repeated measurement, attention to instrumental error, and comparison of sources across languages and cultures.

Connections

People linked to al-Biruni across this collection. A name without a link has no page here yet. See al-Biruni in the network of all 337 lives →

Teachers and advisors

• Abu Nasr Mansur ibn 'Iraq

Influenced by

• Archimedes
• Aryabhata
• Brahmagupta
• Claudius Ptolemy
• Muhammad ibn Musa al-Khwarizmi

Influenced

• al-Khazini

Colleagues and collaborators

• Abu Mahmud al-Khujandi
• Abu Sa'id al-Sijzi
• Abu'l-Wafa' al-Buzjani 997

Rivals

• Ibn Sina c. 1000

Also mentioned in this life

• Aristotle
• Eratosthenes
• Euclid
• Johannes Kepler
• Joseph-Louis Lagrange
• Isaac Newton

Fields

• astronomy 50 lives
• trigonometry 9 lives
• geodesy 12 lives
• mineralogy 3 lives

On CoursesHub

• Astrophysics for Beginners

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Abu Arrayhan Muhammad ibn Ahmad al-Biruni, MacTutor History of Mathematics.
2. Al-Biruni, Wikipedia.
3. D. J. Boilot et al., Biruni, Abu Rayhan: Life, and Mathematics and astronomy, Encyclopaedia Iranica.
4. al-Biruni, Biographical Encyclopedia of Astronomers (via MacTutor).
5. E. S. Kennedy, Al-Biruni, in Dictionary of Scientific Biography, vol. 2, Scribner, 1970.
6. S. F. Starr, Lost Enlightenment: Central Asia’s Golden Age from the Arab Conquest to Tamerlane, Princeton, 2013.

Works and mathematics

1. al-Biruni, The Chronology of Ancient Nations, trans. E. Sachau, London, 1879.
2. al-Biruni, Alberuni’s India, trans. E. Sachau, London, 1888.
3. al-Biruni, The Exhaustive Treatise on Shadows, trans. and comm. E. S. Kennedy, 2 vols., Aleppo, 1976.
4. al-Biruni, The Determination of the Coordinates of Positions for the Correction of Distances between Cities (Tahdid), trans. J. Ali, Beirut, 1967.
5. J. L. Berggren, Episodes in the Mathematics of Medieval Islam, 2nd ed., Springer, 2016.
6. G. Van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton, 2009.
7. D. Joyce, al-Biruni’s measurement of the Earth (worked solution), Clark University.
8. Biruni’s method, Etymological Dictionary of Astronomy and Astrophysics, Paris Observatory.
9. Al-Biruni’s pyknometer, Chemistry World.