Persia, a Museum of Materials · Hall 9 · Paper & Ink

Measure the Earth from a hill

Around the year 1020 the scholar Abu Rayhan al-Biruni stood on a hill above the plain of the Punjab with an astrolabe. He sighted the horizon, read how far it dipped below the level, and from that one angle and the hill’s height worked out the radius of the Earth: 6,339.6 km, half a percent short of today’s value. Tilt the sight until it rests on the horizon and do the same.

Your reading

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Drag the sight down to the horizon

Radius of the Earth

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Today’s value: 6,371 km · Al-Biruni’s: 6,339.6 km

Height of the hill500 m

Scholar: Abu Rayhan al-Biruni, 973–1048
Born: Kath, in Khwarazm
Where: Nandana, a fort above the Punjab plain
Instrument: An astrolabe, for the dip of the horizon
Result: 6,339.6 km, against 6,371 km today

cos α = R / (R + h)

R = h · cos α / (1 − cos α)

Try this

Drag the sight down until it rests on the horizon. The reading is the dip; the radius appears.
Raise the hill to 3,000 m. The dip grows, but the Earth stays the same size: the method works from any height.
Leave the sight a few arcminutes off the horizon and watch the radius jump. That is why Biruni needed an instrument that read minutes of arc.

The view is drawn from the real geometry of a sphere of radius 6,371 km, so a perfect reading returns today’s value; the bending of light in the air, which Biruni did not know about and which shifts the horizon by a few arcminutes, is left out. The hill’s height and the arcminute scale are yours to set; Biruni measured his hill first, with the same astrolabe, from two points on the plain. The astrolabe drawn here is a simplified one.

Written on paper

Paper reached Samarkand in the eighth century and Baghdad had paper mills by the 790s; by the end of the ninth century it had replaced papyrus across the Muslim world. Cheap pages made big books possible. Five of them, in the order they were finished:

c. 820KhwarizmiThe Compendious Book on Calculation by Completion and BalancingThe book that gave “algebra” its name; his own name became “algorithm”.
1010FerdowsiShahnamehAbout 50,000 couplets, finished after some 33 years of work in Tus.
1025Ibn SinaThe Canon of MedicineFive books, begun in Gorgan (Jurjan) and finished in Hamadan; a textbook in Europe for centuries.
c. 1030BiruniIndiaWhat he learned of India’s sciences and beliefs; he also measured the Earth.
1079Khayyam and the astronomers of IsfahanThe Jalali calendarA solar calendar begun on 15 March 1079, the ancestor of the calendar Iran uses today.

A hill in the Punjab

In 1017 Mahmud of Ghazni took Biruni from Khwarazm to his court, and the sultan’s campaigns carried the scholar into northern India, where he observed and wrote between about 1022 and 1026. At Nandana, a fort above the plain, he first measured the height of a hill with his astrolabe, then climbed it and sighted the horizon. The dip, and a little trigonometry, gave him the size of the planet.

Paper before printing

Baghdad’s House of Wisdom, richest under the caliph al-Ma’mun in the 820s, gathered translators and mathematicians such as Khwarizmi; by the second half of that century it held the largest collection of books in the world. When the Mongols took the city in 1258 the books were thrown into the Tigris, which was said to have run black with ink. What survived did so because paper had made copies cheap.

Words that travelled

Khwarizmi’s book on solving equations was called al-Jabr, and Latin readers kept the word: algebra. His name, in Latin, became algorithm. Ibn Sina’s Canon of Medicine, finished in Hamadan in 1025, was still taught in European universities in the 1700s. And Biruni, who wrote about 146 works, measured a world that his readers in the West would not measure so well for another five centuries.

Episode 9 of 12

Still alive today

Every time someone says “algorithm” they are saying al-Khwarizmi’s name, and every algebra lesson repeats al-Jabr, the short title of his book.

Questions people ask

How does the method work?

From a height h above a sphere of radius R, the horizon lies below the horizontal by the dip angle α, and cos α = R / (R + h). Turn that round and R = h · cos α / (1 − cos α). One height and one angle are enough.

How did he measure the hill?

With the same astrolabe, from the plain: he measured the angle to the hill top from two points a known distance apart, and the two angles fix the height. Everything he needed was a sighting instrument and a way to pace out a distance.

Why was his answer a little small?

Reading a dip of about half a degree to the arcminute is hard, and the hill’s height had its own error. He also did not know that air bends light and lifts the horizon slightly. Even so, 6,339.6 km against 6,371 km is remarkable; no measurement in the West came so close until the sixteenth century.

Who measured the Earth before him?

Eratosthenes, in Egypt around 240 BC, compared the sun’s angle at two towns a known distance apart. Biruni’s method needs no second town and no sun: one person, one hill, one instrument.

What is an astrolabe?

A flat brass instrument for measuring angles in the sky: a disc with a scale round its rim and a rotating sight, the alidade, across its back. Medieval astronomers used it to tell the time, find the direction of Mecca and measure the height of stars, hills and horizons.

Why “paper and ink”?

Because the age of these scholars is the age of paper. Papermaking came to Samarkand and then Baghdad in the eighth century, and by 900 paper was cheaper and more common than papyrus. The books in this hall exist because paper was cheap enough to copy them.

Continue your visit

In the next episode…

Geometry climbs onto the walls: with a kit of just five tile shapes, the craftsmen of Isfahan make a pattern in 1453 that, two physicists argue, never quite repeats, a kind of order science would name only five centuries later.
Hall 10 · Glazed Tile · Isfahan →

Next hallHall 10 · Glazed Tile · Build a girihEnter the hall →EarlierHall 6 · Grey Stone · PersepolisEnter the hall →The museumThe entrance · all twelve halls, oldest firstBack to the entrance →

Sources: Wikipedia, “Al-Biruni”, “History of geodesy”, “Al-Khwarizmi”, “The Canon of Medicine”, “Shahnameh”, “Jalali calendar”, “History of paper” and “House of Wisdom”; MacTutor History of Mathematics, “Al-Biruni”.

The sight and the astrolabe are drawn with code by Bridge Labs; the geometry is that of a sphere of radius 6,371 km.