Fr.: méthode de Biruni
A method devised by the Iranian astronomer Biruni (973-1048) to measure the Earth radius, using trigonometric calculations. In contrast to foregoing → Eratosthenes' method and → Mamun's method, which required expeditions to travel long distances, Biruni's method was on-site. He carried out the measurement when he was at Nandana Fort (at the southern end of the pass through the Salt Range, near Baghanwala in the Punjab). He first calculated the height of a hill (321.5 m). To do this he used the usual method of observing the summit from two places in a straight line from the hill top. He measured the distance, d, between the two places and the angles θ1 and θ2 to the hill top from the two points, respectively. He made both measurements using an astrolabe. The formula that relates these angles to the hill height is: h = (d. tan θ1 . tan θ2) / (tan θ2 - tan θ1). He then climbed to the hill top, where he measured the → dip angle (θ), that is the angle of the line of sight to the horizon. He applied the values he obtained for the dip angle and the hill's height to the following trigonometric formula to derive the Earth radius: R = h cosθ / (1 - cos θ). The result for the Earth radius was 12,851,369.845 cubits (or 6335.725 km, using favorable conversion units). Despite the fact that the method is very ingenious, such a precise value is only by chance, because of several drawbacks: The plane was not perfectly flat to serve as the smooth surface of the sea. A measuring instrument more accurate than the alleged 5 arc minutes was needed. And the method suffered from the → atmospheric refraction (See, e.g., Gomez, A. G., 2010, Journal of Scientific and Mathematical Research).
Abu Rayhân Mohammad Biruni (973-1048 A.D.), one of the greatest scholars of the medieval era, was an Iranian of the Khwarezm region; → method.