Al-Biruni’s measurements: from the hill at Nandana to the weight of gold

In the 25th article of this series, we saw that Abu Rayhan al-Biruni measured the radius of the Earth at Nandana Fort in Jhelum district. This time we look at that measurement closely: what exactly he did, how he worked out the result, and where errors could have crept in. We then turn to another of al-Biruni’s lesser-known but equally remarkable measurements: the weight of metals and gems.

A peerless scholar

According to the MacTutor archive of St Andrews University, al-Biruni was born on 15 September 973 near Kath in Khwarazm (present-day Uzbekistan) and died in Ghazni (present-day Afghanistan) on 13 December 1048. Citing the historian Kennedy, MacTutor says he wrote about 146 books, totalling some 13,000 pages, but not even a fifth of them survives today.

Al-Biruni is known for his precision. According to MacTutor, in 990 he found the latitude of Kath by measuring the maximum height of the sun. He did the same for Ghazni between 1018 and 1020, and found the latitudes of eleven towns in Punjab and Kashmir. His book “Qanun-i-Masudi” lists the locations of about 600 places.

Nandana: the first step, the height of the hill

According to the astronomical dictionary of the Paris Observatory, al-Biruni’s method had two steps. First, he needed the exact height of a hill. For this he stood on the plain at two points in line with the summit of the hill and measured the distance between them. Then, using an astrolabe, he took the angle of elevation of the summit from each point.

From these two angles and the distance between the points, the height of the hill can be found by the tangent rule for triangles. According to the dictionary, the rule is: height equals the distance multiplied by the tangent of the first angle and the tangent of the second angle, divided by the difference of the two tangents. By this calculation, the height of the hill came to about 321.5 metres.

The second step: how far the horizon dips

Al-Biruni then climbed to the summit and looked towards the horizon of the distant level plain. If the Earth were flat, the horizon would lie exactly at eye level. But the Earth is round, so from a height the horizon appears slightly lower. The angle of this drop is called the “dip of the horizon”, and al-Biruni measured it.

Now imagine this. The line of sight from the summit to the horizon touches the Earth at only one point. The line from that point to the Earth’s centre, that is the radius, is perpendicular to the line of sight. This forms a right-angled triangle: one side is the radius, and the hypotenuse is the radius plus the height of the hill. The angle at the centre is the same as the dip of the horizon. According to the dictionary, this gives the rule: radius equals height multiplied by the cosine of this angle, divided by one minus the cosine.

The result and its limits

According to the Paris Observatory dictionary, al-Biruni’s answer was 12,851,369.845 cubits, which comes to about 6,335.725 kilometres with favourable units. MacTutor gives the same result as 6,339.6 kilometres. According to NASA, the mean radius of the Earth is 6,371 kilometres, so the difference is very small.

According to a 2013 research paper by the researcher Amelia Sparavigna, al-Biruni’s own instrument could measure angles to about ten minutes of arc. The key to the accuracy of the measurement therefore lay in the correct values of the sines of the angles, which he probably took from Indian mathematical sources. The Paris Observatory dictionary also says that such accuracy was partly a matter of chance, because the plain was not perfectly flat like the sea, the instrument was not that fine, and the bending of light in the air also affects the result. The method, however, was entirely sound.

Gold, mercury and gems

Al-Biruni measured not only the Earth but also the “weight per volume” of things, that is, their density. According to MacTutor, he compared the densities of gold, mercury, lead, silver, bronze, copper, brass, iron and tin.

According to Sparavigna, for this work he used a cone-shaped vessel with a spout near its neck. The vessel was filled with water until it began to flow out of the spout. The weighed object was then dropped in, and the water that flowed out was collected in a scale pan and weighed. From the weight of the object and the weight of the displaced water, the density was obtained. This vessel was later described by the 12th-century scholar al-Khazini in his “Kitab Mizan al-Hikma”. Al-Biruni advised that the larger the sample, the more accurate the result.

According to a table published by the Sindhi Adabi Board from his book on gems, “Kitab al-Jamahir”, al-Biruni found the density of gold to be 19.05 relative to water, that of mercury 13.58 and that of iron 7.74. The same table gives modern measurements from 1876 as 19.30, 13.568 and 7.79 respectively. According to Sparavigna, al-Biruni objected to identifying gems by colour alone and said that density, brilliance and hardness were the real properties.

Kitab al-Hind: a scientific observation

During his stay in the subcontinent, al-Biruni learned Sanskrit. According to MacTutor, his book “Kitab al-Hind” gives a detailed account of the writing methods, number systems, geography, astronomy and calendar of the region, and he translated several Sanskrit books into Arabic. Nandana Fort still stands today in the Salt Range of Punjab. The horizon seen from its summit is a reminder that a thousand years ago the whole Earth was measured from this very view.