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# Measuring the earth's radius 1000 years ago

Tuesday 10 November 2015

This past weekend I was reading a review of a new history book, The Silk Roads by Oxford historian Peter Frankopan, in which the reviewer states that the Islamic mathematician and scientist al-Biruni (973-1048) was the first to compute the radius of the earth. Two things struck me – firstly, I had never heard of al-Biruni (full name Abu Arrayhan Muhammad ibn Ahmad al-Biruni); and, secondly, I thought that the Greek mathematician Eratosthenes (276-194 BC) had the distinction of being the first to accurately measure the circumference (and, consequently the radius) of the earth. After some research, I realized that I really should have heard of al-Biruni. Even though little-known in the Western world, al-Biruni was one of the greatest scientists of all time. He wrote approximately 150 books on a wide range of subjects – including mathematics, astronomy, physics, geography and history; and was an accomplished cartographer, linguist and traveller. He studied the original works of several ancient Greek scientists and mathematicians. George Sarton (1884-1956), widely considered to be the founder of the discipline of history of science, wrote that al-Biruni was “one of the very greatest scientists of Islam, and, all considered, one of the greatest of all times”. At a time when religious fanaticism was rooted in medieval Europe, al-Biruni was well ahead of the state of European scientific thought.

In trying to learn more about al-Biruni – and especially his feat of measuring the radius of the earth – I came across an excellent 3-part BBC documentary made by the accomplished British theoretical physicist Jim Al-Kahili, entitled Science and Islam. In the 2nd episode, The Empire of Reason, al-Kahili spends about 10 minutes (15:00-24:30) of the 60-minute episode discussing and demonstrating al-Biruni’s method for measuring the earth’s radius. Although Eratosthenes was the first to accurately measure the earth’s circumference, al-Biruni applied some pure mathematics to devise a much more efficient, and potentially more accurate, method. Eratosthenes’ method relied on knowing the distance between two fairly distant locations – Eratosthenes used the cities of Alexandria and Syene (modern day Aswan) which are about 800 km apart. Al-Biruni’s method did not require the laborious and inaccurate task of measuring the long distance between two sites (probably by walking and counting paces), but could be performed by a single person measuring three angles (with an astrolabe) and a reasonably short distance (between 500 to 1000 meters in al-Biruni’s case).

Al-Biruni measured the horizontal distance d between two points and the angle of elevation from each of the points to the top of a nearby mountain (Figure 1). Using trigonometry and algebra, al-Biruni derived the following formula for the height h of the mountain in terms of only the two angles of elevation ${\text{θ}}_{1}$ and ${\text{θ}}_{2}$ and the horizontal distance d between the points where ${\text{θ}}_{1}$ and ${\text{θ}}_{2}$ were measured.

The third required angle measured by al-Biruni was the ‘dip angle’ $\text{φ}$ – the angle of depression from the top of the mountain to the distant horizon. Al-Biruni then imagined a very large right triangle (Figure 2) with its three vertices being the top of the mountain, the center of the earth, and the point on the horizon that was sited from the top of the mountain.

Using this triangle, al-Biruni again used some pure mathematics – algebra and trigonometry – to derive a formula for the radius of the earth R, expressed only in terms of the height h of the mountain and the dip angle $\text{φ}$ .

Al-Biruni computed the radius of the earth to be about 6339 km (converting from cubits – the distance measure that al-Biruni used) which gives the earth’s circumference to be about 39830 km which is more accurate that Eratosthenes’ calculation of about 39690 km. At the equator, the earth’s radius and circumference are 6378.1 km and 40075 km. The earth is not a perfect sphere (radius to the poles is shorter than to the equator). The mean radius and circumference are 6371.0 km and 40030 km. Eratosthenes had to deal with significant measurement inaccuracies – in measuring the angle of inclination of the sun and especially in measuring the distance between Alexandria and Syene. Although the one distance required for al-Biruni’s method was inherently more accurate it was difficult to measure angles with a great deal of accuracy. However, al-Biruni was carrying out his method a little more than 1000 years after Eratosthenes and the astrolabe that al-Biruni used would certainly have been more accurate than whatever angle measuring device that Eratosthenes had used. Historians have surmised that al-Biruni’s astrolabe was probably able to measure angles up to 1 minute of an arc which is 1/60 of a degree.

See the student handout Radius of the Earth in the set of Challenge Questions on the site (in Assessment section). I gave this handout to my students to work through and then had a discussion with them about the significant progress achieved by Islamic mathematicians and scientists during a time when scientific thought was at a low point in medieval Europe. After completing the handout and watching the 10-minute portion in the documentary that covered al-Biruni’s method of measuring the radius of the earth, I asked my students what “tools” were critical to the development and execution of the method. Obvious answers included an astrolabe (or giant protractor) and counting paces to measure distance, but most students did not go beyond those two. It did not take much to convince them that algebra was also a necessary “tool” that was absolutely critical for finding the two formulas that were central to al-Biruni’s method. But then I suggested that there was another necessary “tool” they had overlooked. I pointed out that it was necessary to evaluate trigonometric functions in order to apply the two formulas. How would al-Biruni have evaluated a trigonometric function (e.g. evaluate $\mathrm{tan}{0.57}^{\text{\hspace{0.17em}}\circ }$ )? It’s now been more than 30 years since students routinely used tables (and interpolation when necessary) to evaluate trigonometric functions. Of course, al-Biruni would have valued his trigonometric tables just as much – if not more so – than his precisely crafted astrolabe.