How did al-Biruni measure the size of the Earth from the top of a mountain?

With one mountain, a clear horizon and trigonometry. Al-Biruni (c. 973 to c. 1050), an Iranian Khwarazmian polymath of the Islamic Golden Age, devised a novel method: measure the height of a hill, then measure the dip of the horizon from its top. From those two numbers, trigonometry gives the radius of the Earth. He carried it out at Nandana in Pind Dadan Khan, in present-day Pakistan.

Why does it work? When the observer is elevated, the horizon lies below the level: its zenith angle can be greater than 90 degrees. Al-Biruni compared the angle of the horizon seen from the mountain top with that from a nearby plain, and trigonometry turns that angle plus the hill's height into the radius.

The earlier way to measure the Earth was Eratosthenes' method, which compares the midday Sun at two places a known north-south distance apart; around 830 the caliph al-Ma'mun's astronomers measured a degree of latitude between Palmyra and Raqqa. Al-Biruni's method instead uses a single place.

How good was it? His estimate was 12,803,337 cubits, and the exact length of a cubit is not clear, so how close he came to the modern value cannot be pinned down. One real problem: he did not know about atmospheric refraction, which bends light near the horizon and can shift the measured dip by about one sixth, so his result is accurate to within only about 20%.