Measuring the hill first
From the plain, he sighted the summit from two places in line with it and measured the distance between them. Two angles and that distance give the height, even when the hill’s foot cannot be reached.
26 / ANTIQUITY TO THE RENAISSANCE
Abu Rayhan al-Biruni
From the top of a hill, the horizon lies a little below level. Can that one small angle tell you the size of the whole Earth?
Explore the mathematics ↓
01 / THE DIP OF THE HORIZON
One hill, one angle, the whole Earth.
Stand on a hilltop and sight the horizon. The taller the hill, the further the horizon dips below level.
The sightline just touches the Earth, at right angles to its radius. That right triangle links the hill, the dip and the Earth’s radius.
Whose hill:
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THE IDEAThe dip at the summit equals the angle at the Earth’s centre. With the hill’s height h, the right triangle gives R = h × cos(dip) ÷ (1 − cos(dip)).
The person behind the ideas
- Studied astronomy, geography and the sciences of India.
- Worked in Khwarazm and later in Ghazna.
- Connected careful measurements with geometric calculations.
More of the mathematics
A tiny angle, a big effect
His dip was just over half a degree. Reading it 1′ differently changes the answer by about 6%, so the instrument had to be read with great care.