Abu Rayhan (Muhammad ibn Ahmad) al-Biruni was born in Kath, capital of Khwarezm (the region of the Amu Darya delta), in modern Khiva, on 973 Sept 5 (362-12-03 AH), and died in Ghazni, Khorasan, on 1048 Dec 13 (440-07-04 AH).  He was a well-travelled Persian sage who spoke fluent Persian, Arabic, Turkish, Sanskrit, Greek, Hebrew, Syrian, and several other local languages in whose literatures he was also well versed. Not only did he possess ‘all the well known books on astronomy written within the area extending from the Mediterranean Sea to the Bay of Bengal, including all the Greek, Indian, and Muslim authors from Spain to Egypt’ (Canon 1:13, Ahmad 2009:171), but he also wrote prolifically on all branches of knowledge like cartography, geography, ethnography, history, philosophy, mathematics, astronomy, and science in general, a total of about 180 works, some sadly lost to us now (EI2 1:1237). Chief among these is his Masudic Canon, an encyclopaedia of astronomy he dedicated to Sultan Masud who, delighted with the sage’s accomplishment, presented him with an elephant-load of silver! To everyone’s amazement, Biruni refused it, pleading that he could not bear to abuse the generosity of someone who had already shown him more kindnesses than he felt he deserved (Minhaj Siraj Tabaqat 1975 2:343-4).
Of his many contributions to the good of humanity, it is hard to decide which is the most important and, certainly, it is not in the scope of a short article like this to do justice to the vastness of his genius. So it is as a matter of personal choice that I have decided to deal here with just one of these: the fact that he measured the earth with an unprecedented precision, not equalled in the West until the 16th century (Norhudzaev 1973). You may find I am rather critical sometimes for, in Biruni’s own teachings, it is the truth that we must seek and speak, even when it goes against us.
His value for the earth’s radius (6335.725 km) was the result of having solved a complex geodesic equation, which is the aim of this paper to explain. But to fully understand the whole story, we must know something about its background and about the early Islamic measures then (and herein) used: these were exactly those of ancient Mesopotamia, where ‘each mile was a third of a farsang, or 4000 cubits, called black in Iraq, each of which equalled 24 digits’ (Biruni Instr. Astrol. 208, tr. Wright 1934:119, Mercier 1994:178). That is, one farsang (5916 m) equalled three miles, one mile (1972 m) equalled 4000 cubits, and one (black) cubit (493 mm) equalled twenty-four digits. Note the connection with the Roman mile (1479 m) of 3000 cubits (each of 493 mm).
The story begins before Biruni, when Sultan al-Mamun ordered two teams of surveyors to measure the earth. They did so by departing from a place in the desert of Sinjad, nineteen farsangs from Mosul and forty-three from Samarra, heading north and south respectively, and both determining that the length of one degree of latitude is somewhat between 56 and 57 Arabic miles (Biruni Tahdid, tr. Ali 1967:178-80). Among the several extant accounts of this survey, Habash al-Hasib (tr. Langermann 1985:108-28) quotes at length from a direct account from Khalid:
“The Commander of the Faithful al-Mamun desired to know the size of the earth. He inquired into this and found that Ptolemy mentioned in one of his books that the girth of the earth is so and so many thousands of stades. He asked the commentators about the meaning of stade, and they differed about the meaning of this. Since he was not told what he wanted, he directed Khalid ibn Abd al-Malik al-Marwarrudhi, Ali bin Isa al-Asturlabi [from his surname, evidently an instrument maker], and Ahmad ibn al-Bukhturi al- Dhari [from his surname, the Surveyor] with a group of surveyors and skilled artisans, including carpenters and brass makers, who were to maintain the instruments they needed. He led them to a place, which he chose in the desert of Sinjar. From there, Khalid and his party headed for the North Pole of the Little Bear, and Ali and Ahmad and their party headed to the South Pole. They proceeded until they found that the height of the Sun at noon had increased (or differed) by one degree from the noon height they had taken at the place from which they had separated, after subtracting from it the sun’s declination along the path of the outward journey. They put arrows there. Then they returned to the arrows, testing the measurement a second time, and so found that one degree of the earth was 56 miles, of which one mile is 4000 black cubits.This is the cubit adopted by al-Mamun for the measurement of cloths, surveying of fields, and the distribution of way-stations.”
Another report is given by Ibn Yunus (Hakimite Tables 2), based on the accounts of Sind ibn Ali and Habash al-Hasib:
“Sind ibn Ali reports that al-Mamun ordered that he and Khalid ibn Abd al-Malik al-Marwarrudhi should measure one degree of the great circle of the earth’s surface. “We left together,” he says, “for this purpose.” He gave the same order to Ali ibn Isa al-Asturlabi and Ali ibn al-Bukhturi, who took themselves to another direction. Sind ibn Ali said, “I and Khalid ibn Abd al-Malik travelled to the area between Wamia and Tadmor, where we determined a degree of the great circle of the earth’s equator to be 57 miles. Ali ibn Isa and Ali ibn al-Bukhturi found the same, and these two reports containing the same measure arrived from the two regions at the same time.”
“Ahmad ibn Abdallah, named Habash, reported in his treatise on observation made at Damascus by the authors of the Mumtahan [Verified tables] that al-Mamun ordered the measurement of one degree of the great circle of the earth. He said that for this purpose they travelled in the desert of Sinjar until the noon heights between the two measurements in one day changed by one degree. Then they measured the distance between the two places, which was 56¼ miles of 4000 cubits, the black cubits adopted by al- Mamun’.” 
Biruni’s take on the matter (tr. Ali 1967:178-80) is that the figure that eventually became generally accepted as the length of 1º of latitude is 56⅔ miles (111.747 km), which is quite close to the actual value (110.95 km) for the latitudes involved (35º to 36º N). 360 times this number yields the earth’s girth (20400 mls), and from it the radius is easily deduced (6402.612 km). Mamun’s teams had got a nearly perfect hit!
Years later, Biruni wished to repeat the experiment, but was hindered by lack of support. ‘Who is going to help me in this venture?’ He says. ‘It requires strong command over huge tracts of desert, and extreme caution is needed from the dangerous treacheries of those spread over it. I once chose for this project the localities between Dahistan, in the vicinity of Jurjan, and the land of the Turks, but the findings were not encouraging and then the patrons who financed the project lost interest in it’ (Biruni Tahdid, Ali’s tr., p. 183).
Instead of being put off by difficulties, he thought up a new method for measuring the earth that ‘did not require walking in deserts’ (Ibid, p. 183, note 24). It only involved measuring the height of a mountain and taking the dip of the horizon from its top. That’s four measurements in all, as we will see. It also involved his having to work out a mathematical equation that related these four measurements and, of course, finding a suitable mountain to yield the size of the earth by this method! Let us now turn to the first of these requirements.
He was well acquainted with the mathematical procedure for measuring mountain heights, having himself measured all sorts of distances and heights from his very youth. He did this by the usual method of taking the summit from two places, that is, by measuring the distance d between two places (in a straight line from the mountain) and the angles θ from them to the mountaintop. The formula he used that relates these angles to the mountain height h is this (see Figure 1): 
Then, one day, while he was staying in the fort of Nandana  (at the southern end of the pass through the Salt Range, near Baghanwala in the Punjab),  he spotted ‘a high mountain standing west of the fort particularly suited for this project, for] it faces south to a wide flat plain whose flatness serves as the smooth surface of the sea’ (Canon 5.7, ed. 1954-6 2:530; Tahdid, Ali’s tr., p. 188).  He measured its height by the method described, and found it to be 652 cubits and 3′18″ (321.463 m) above the plain.This figure, which was expressed in the then customary mixture of decimal and sexagesimal systems, seems too precise to represent a physical reality. I originally thought it to be the result of averaging no fewer than ten measurements which yielded 652 as the average number of cubits and 3.3 as the average number of minutes, but now I think this is likely to be a self-fulfilling mathematical mirage that proves just the opposite: that he took a single measurement; but this raises the question: which one?
Biruni doesn’t describe the instrument he used for his observations, but an astrolabe has been traditionally assumed. Assuming it was big enough to take angles good to half a degree and assuming that Biruni chose an integer (or at least a not very complex fractional number) as the number of cubits between his points for ease of computation leads to just one mathematical probability: that of his taking the angles 5½º and 7½º from two points exactly 1819 cubits apart. If his instrument was good to a quarter of a degree, then another possibility arises: that of taking the angles 5¼º and 6½º from twopoints 1373¼ cubits apart. If his instrument did not take angles directly, but rather computed them instead, for example, by comparing the length of and distance between two sticks aligned with the mountaintop and sticking out of a portable trough of water which provided the necessary horizontal line, then the accuracy thus computed (though not necessarily real) could reach even seconds of a degree.
Allow for complex fractional cubit gaps between our angles, and several other possibilities are offered us, perhaps the most exotic of which is that of 1111⅓ cubits separating 4º40′ and 5º25′, or, for its smallness, 145⅞ cubits separating 4º35′ and 4º40′.
Once he had the mountain’s height, he climbed to the top and, with his instrument, took the angle of the line of sight to the horizon as it dips below the horizontal. He found it to be 0º34′. All that remained was to find a formula for the earth’s radius R relating the mountain height h and the dip angle θ he had taken. To this end he applied the law of sines most ingeniously as follows (see Figure 2) :
Exactly how exact?
Armed with his newly found formula and with the data he had taken at Nandana (652.055 cubits for the height, and 0º34′ for the dip), he stated that the earth’s radius was 12,851,369.845 cubits (6335.725 km).1 Had he had a calculator to hand, he would have got 13,331,728.352 cubits (6572.542 km)  instead but, in his time, calculations were performed in a rather mental way. Instead of using cos(0º34′), he used sin(90º – 0º34′), that is, sin(89º26′), which is pretty much the same, but for the fact that it was then also fashionable to express numbers in the old Babylonian way and, the value he gives in his Masudic Canon for this sine is 0º59′59″49′′′2′′′′28′′′′′, which is slightly inaccurate. The correct number (given by our modern computers and expressed in the same historical fashion) would have been 0º59′59″49′′′26′′′′9′′′′′; hence the disagreement. He then went on to find the earth’s girth, which he stated to be 80,780,039 cubits and 1′33″ (clearly, the result of rounding π to 22/7).
Let me readdress now the issue of the mountain. I have already mentioned how Mercier (1994:183) pinpointed it as likely to be one standing about 1.4 km south-southwest of the fort and 478 m above sea level. I haven’t found such a place. Instead, I found a mountain 780 m southwest of the fort (32º43′14.60″ N, 73º13′26.00″ E) that stands 533 m above sea level. Because Biruni said ‘west,’ I also checked another candidate 3.8 km west of the fort (32º43′46.08″ N, 73º11′20″ E) standing 677 m above sea level. From them, the plain to the south looks strikingly flat. It is hard to imagine a flatter thing, or one better suited for Biruni’s purpose. Yet, of course, it cannot be as flat as the sea.
Slowly and somewhat unevenly, it slopes away to the south getting higher and then lower than at the mountain’s foot. In Figure 3 you can see how the plain evolves as seen from the 533 m peak: it first sinks to about 202 m above sea level, then rises again to about 211 or even 212 m, and then resumes sloping away for as far as the eye can see. All this is important because, if you take the mountain’s height and subtract from it the higher of the plain’s levels, you find how much the mountain stands above these levels, that is, about 321 m or 652 cubits, which is Biruni’s figure exactly. But it wouldn’t be wise to measure a mountain from this far away. If you come closer and allow Biruni to take his measurements from the area between 2½ and 3½ km south of our peak and gently sloping from about 212 to 211 m above sea level, you get again 652 cubits exactly. The choice of this area has the advantage not only of being close enough to allow reliable measurements, but also, and above all, of being on a level with the apparent horizon, which is seen from it as the plain’s upper levels. Seemingly, this was Biruni’s logical way of finding the ‘plain’s [otherwise relative] level.’
Measuring the dip of the horizon from the Salt Range tops is easier said than done. As Rizvi (1979:619) reports, it takes a good deal of patience to wait for the right atmospheric conditions to get a clear view of the horizon. It is best to try after rain has cleared away the haze and dust from the air. It is possible that Biruni may have lived in rainier times for, after all, there must be some reason why the name Nandana means Paradise in Sanskrit. Even so, the dip he reports, 0º34′, though strikingly accurate, is not given to the seconds of a degree and, therefore, does not allow us to use mathematics this time to find out whether he averaged it out of several measurements or not. Perhaps the prevailing poor visibility forced him to be satisfied with fewer measurements, or perhaps he had some other reason for the figure he gave.
I found by a formula that the apparent dip θ of the horizon (that is, taking into account ray-bending at mean air conditions)  from this mountain is about 0º32′, which is almost the same as Biruni reports. Care must be taken to provide the formula with the right mountain height h for this problem, which is neither 533 m above sea level (we’ll consider this one later for sport) nor Biruni’s 321½ m above the plain’s top grounds, but 340 m above the farthest visible point on the horizon (which is about 72½ km from our peak and about 193 m above sea level). 0º32′ is close to Biruni’s figure, but not quite. At first sight, a difference of just a couple of minutes seems quite negligible, but it is hardly so, and this is one of the drawbacks of Biruni’s method: that it is so dependent on taking the dip angle with such hair splitting accuracy, that just a minute of a degree results in a difference of hundreds of kilometres for the computed earth-radius, not to mention the difficulty in taking such angle with any reliable accuracy at all!  Let us pay more attention to Biruni’s own words (Canon 5.7, ed. 1954-6 2:530; Tahdid, Ali’s tr., p. 188):
“I changed to another way owing to having found in a region in India a mountain peak facing toward a wide flat plain whose flatness served as the smooth surface of the sea. Then on its peak I gauged the intersection of heaven and earth [the horizon] in the prospect, and I found it by an instrument to incline from the East-West line [southern astr. hor.] a little less than ⅓¼ of a degree, and I took it as 0º34′. I derived the height of the mountain taking the summit in two places, and I found it to be 652¹⁄20 cubits.”
So he took the dip as ‘a little less than ⅓¼ of a degree’ and rounded it down to 0º34′. Yet, a little less than ⅓¼ of a degree should round up to 0º35′. Why 0º34′ and not 0º35′? There is just one minute of difference, but one minute that makes all the difference: 0º34′ got him closer to Mamun’s value than 0º35′ or 0º33′. Certainly, Mamun’s value was serving him as a reference all the time, and the difference between 32′, 33′, 34′, or 35′ is so tiny that he might indeed have believed in earnest that 34′ must be the right angle his instrument was reading. Had he chosen a different dip, he would have gone wrong by hundreds of kilometres. It is only by chance, therefore, that the compensating features of the lucky scenario chosen led him to a figure so close to the truth! Even his own measurements do lead to a value slightly worse than Mamun’s, should the cosine mistake alone be corrected! (Oh, by the way, note that the little ⅓¼ detail may lead us to suspect that his instrument was accurate to 5 minutes of a degree; I leave the guesswork to you!)
Scientifically speaking, Biruni’s method leads to results infected by ray bending when applied to other scenarios. For example, as mentioned above, in the completely imaginary case that our 533 m mountain looked south onto the sea (which, remember, it does not), the apparent (refracted) dip of the horizon would be about 0º40′22″. With these data, Biruni’s formula gives an earth-radius (for Nandana’s latitude) of 7734 km, which is too large by about 6/5 due to ray bending. His celebrated method, nevertheless, bears the immortal originality of saving a lot of walking through deserts and, provided we take accurate measurements and compensate for ray bending, it certainly does correctly yield the earth’s radius! (see Figure 4 below.)
To be fair to Biruni, it must be said that he was not just lucky. Clearly, he picked his landscape most carefully after testing his method on several places, noting the disparity of the results, and finally picking the one that best matched that of Mamun’s teams (see Figure 5). It is to his credit that he openly regarded his own method as a curious complement to their work, accepting their result as more reliable, because, as he said (Canon 5.7, tr. Mercier 1994:183), ‘Their instrument was more refined, and they took greater pains in its accomplishment.’
1. Some think that, according to the solar calendar, the Muslim date 440 AH should be taken to mean AD 1050 (Encyclopaedia of Islam 1979 1:1236, and Encyclopaedia of World Biography 1973 1:578).
2. Ibn Yunus, Hakimite Tables, Chapter 2, from a Manuscript in Paris, Bibliotheque Nationale, MS Arabe 2495, fols. 44r-v. Later in this manuscript, Yunus explains that ‘it is possible to keep the direction by means of three bodies spread out along the meridian, one of them hiding the others in line of sight. One advances by fixing the nearest one by sight, then the second, the third, and so on.’
3 Today we would use a slightly simpler formula, see the Appendix.
4 Nandana Fort’s coords are 32º43′33.52″ N, 73º13′45.16″ E, 404 m above the sea.
5 This is the pass Alexander the Great took to descend from Taxila into the Indian Plain, just before his famous battle with Raja Poros in 326 BC (Stein 1932:31-46).
6 Mercier (1994:183) says this peak is likely to be one situated 1.4 km south-southwest of the fort and standing 478 m above sea level, and about 265 m above the plain to the south. He mentions how Rizvi (1979), certainly having this peak in mind, gave its height as 547 m above see level, and 321½ m above the plain, in order to make it fit Biruni’s report exactly. I haven’t found such a place. Perhaps they meant the one situated 780 m southwest of the fort (32º43′14.60″ N, 73º13′26.00″ E) and standing 533 m above sea level. There is another candidate 3.8 km west of the fort (32º43′46.08″ N, 73º11′20″ E) and standing 677 m above sea level.
7. Or, as Biruni puts it, 12,851,369 cubits and 50′42″.
8 Mean air conditions are taken to be 15ºC of temperature, 1013.25 mb of pressure, 6.49ºC/km of lapse rate, 0% of relative humidity, and 450 ppm of CO2 content.
9 The main difficulties arise from the ever-changing air conditions and from the limitations of human eyesight.
Ahmad, R., ‘Al-Biruni, A Great Muslim Scientist, Philosopher and Historian,’ in Pakistan Vision, Vol. 10, No. 1, pp. 167-179, 2009.
Biruni, Al-Qanun al-Masudi [Masudic Canon], or An Encyclopaedia of Astronomical Sciences, 3 vols, Osmani Oriental Publications Bureau, Hyderabad, India, 1954-6.
Biruni, The Book of Instruction in the Elements of the Art of Astrology, ed. and tr. by Robert Ramsey Wright, London, Luzac, 1934.
Biruni, The Determination of the Coordinates of Positions for the Correction of Distances between Cities, tr. by Jamil Ali, Beirut, The American University of Beirut, 1967.
Encyclopaedia of Islam, New Ed. (EI2), Leiden, E. J. Brill, Vol. 1, p. 1236, 1979.
Encyclopaedia of World Biography, New York, McGraw-Hill, Vol. 1, p. 578, 1973.
Langermann, Y. T., ‘The Book of Bodies and Distances of Habash al-Hasib,’ Centaurus 28, pp. 108-28, 1985.
Mercier, R. P., ‘Geodesy,’ in The History of Cartography, ed. by J. B. Harley and D. Woodward, Chicago and London, University of Chicago Press, Vol. 2, Book 1, 1994.
Minhaj Siraj, Tabaqat-i-Nasiri, original in Persian translated into Urdu by Ghulam Rasool Mehr, Lahore,
Markazi Urdu Board, Vol. 2, 1975.
Norhudzaev, K., ‘Al-Biruni and the science of geodesy (Uzbek),’ in Collection dedicated to the 1000th anniversary of the birth of al-Biruni, Tashkent, pp. 145-158, 1973.
O’Connor, J. J. and E. F. Robertson, McTutor History of Mathematics Archive, Univ. of St. Andrews,Scotland.
Rizvi, S. S. H., ‘A Newly Discovered Book of al-Biruni, “Ghurrat-uz-Zijat” and al-Biruni’s Measurements of Earth’s Dimensions,’ in Al-Biruni Commemorative Volume, Proceedings of the International
Congress held in Pakistan on the occasion of the Millenary of Abu Rayhan Muhammed ibn
Ahmad al-Biruni (c.973 – AD 1051), from Nov 26 to Dec 12, 1973, ed. Hakim Mohammed Said,
Karachi, Times Press, pp. 605-80, 1979.
Stein, M. A., ‘The Site of Alexander’s Passage of the Hydaspes and the Battle with Poros,’ in Geographical Journal 80, pp. 31-46, 1932.
Yunus, Ibn, Hakimite tables, Chapter 2, from a Manuscript in Paris, Bibliotheque Nationale, MS
Arabe 2495, fols. 44r-v.
Taken from a research paper.