On this page
Through late antiquity and the Middle Ages, star positions came from Ptolemy's catalogue. The first comprehensive catalogue based on fresh measurements came from Samarkand in 1437, made by a ruler who was a better astronomer than politician. This post covers Ulugh Beg, his observatory, how accurate his catalogue really was, and his famous value of sin 1°, with checks you can do on a calculator.
Who was Ulugh Beg?
The MacTutor biography gives his birth as 22 March 1394 at Soltaniyeh in Persia, and his death on 27 October 1449 at Samarkand. He was a grandson of the conqueror Timur. His father, Shah Rukh, gave him Samarkand when he was sixteen.
He used it to build a centre of learning. A madrasah on the Registan square was begun in 1417 and completed in 1420, and he invited about sixty scientists, among them the mathematician al-Kashi and Qadi Zada. Ulugh Beg himself led their discussions of problems in astronomy.
What was the observatory like?
MacTutor reports that construction began in 1428. The building was circular, had three levels, and was over 50 metres in diameter and 35 metres high. Its instruments included a quadrant so large that part of the ground had to be removed for it, a marble sextant, a triquetrum and an armillary sphere. The remains are part of the UNESCO World Heritage Site Samarkand, Crossroad of Cultures, inscribed in 2001.
Large instruments meant widely spaced scale divisions, the same reason Sawai Jai Singh II later built in stone, as our post on Jantar Mantar explains. Jai Singh's own tables named Ulugh Beg as the last to take on such a task.
What did the observatory produce?
The main result was the Zij-i Sultani, with tables of the Sun, Moon and planets and a star catalogue. MacTutor gives 992 stars, published in 1437, as the work of a team including Ulugh Beg, al-Kashi and Qadi Zada. The tables also included sines and tangents at 1° intervals.
Ulugh Beg's politics did not match his science. After his father died in 1447 he lost power, and he was put to death in 1449 at the instigation of his own son.
How accurate was the star catalogue?
In 2012, F. Verbunt and R. H. van Gent compared both catalogues with the modern Hipparcos catalogue in Astronomy & Astrophysics. Their abstract gives typical error widths of:
Ptolemy: about 27′ in longitude and 23′ in latitude.
Ulugh Beg: about 22′ in longitude and 18′ in latitude.
Stars wrong by more than 150′ made up about 2 per cent of Ptolemy's catalogue and 0.1 per cent of Ulugh Beg's.
Worked example: what those numbers mean
Convert to degrees. 22′ = 22/60 = 0.37°. 27′ = 0.45°.
Compare with something you can see. The NASA Moon fact sheet gives the Moon's apparent diameter as 1896″ = 31.6′. A typical Ulugh Beg position was off by about two-thirds of a Moon's width, using only the naked eye.
Count the big errors. In 1000 stars, 2 per cent is 20 badly placed stars; 0.1 per cent is 1.
Worked example: Ulugh Beg's sin 1°
MacTutor reports that Ulugh Beg found sin 1° by solving a cubic equation numerically, getting 0.017452406437283571. The modern value is 0.017452406437283512820. The two agree for the first 16 decimal places.
Now compare with the small-angle rule from your physics book, sin θ ≈ θ in radians:
1° = π/180 = 0.0174533 rad.
sin 1° = 0.0174524.
Difference = 0.0000009, about 5 parts in 100,000.
So for 1°, sin θ ≈ θ is excellent. That is why the approximation works for a simple pendulum at small angles.
Which mistakes does this help you avoid?
Using degrees in sin θ ≈ θ. sin 1° is not 1. The rule needs θ in radians.
Mixing arcminutes and minutes of time. 22′ here is an angle, a sixtieth of a degree each.
Treating "accuracy" as a single number. A catalogue has a spread of errors; the abstract gives its width and the share of outliers.
For earlier links in this chain, see our posts on Hipparchus's star catalogue and Ptolemy's Almagest.
What can a student learn from Ulugh Beg?
He did not accept an old catalogue because it was famous. He built better instruments, gathered strong colleagues and measured again. Modern checks show the effort paid off. In your own work, rechecking a standard result yourself is one of the surest ways to understand it.
If radians and small angles still trip you up, the Fermi Concept Library covers every concept in physics and mathematics, so you can review them before your next pendulum problem.
Try it now: compute sin 5° and 5° in radians, and find the percentage difference.
