Al-Farghani and the 'Short Degree'

Written by Paul Lunde

A marginal note in Columbus's own copy of Peter d'Ailly's Imago Mundi , now in the Columbina Library in Seville, reads: "Note: Sailing south from Lisbon to Guinea, I carefully noted the distance, as pilots and sailors do. Then I took the sun's elevation many times, using a quadrant and other instruments. I found myself in agreement with Alfraganus, that is to say, the length of a degree is 56⅔?miles. Thus this measurement must be accepted. As a result, we are able to state that the earth's cir?cumference at the equator is 20,400 miles...."

We know from another marginal note that an astronomer named Joseph, in the service of the king of Portugal, had calculated the latitude of Los Idolos Island, off the Guinea coast, as one degree five minutes north. The accepted lati?tude for Lisbon at the time was 40 degrees 15 minutes north. Columbus considered Lisbon and Los Idolos Island to be on the same meridian, and estimated the distance between the two places by dead-reckoning, probably comparing his own estimate with estimates made by the Portuguese navigators. By a simple calculation, he obtained the figure of 56 miles to the degree - close enough to Alfraganus's figure of 56⅔. To obtain the circumference of the earth at the equator, he simply multiplied 56⅔ by 360.

Columbus measured distance at sea by the Italian nautical mile, and thus, when he writes that the circumference of the earth is 20,400 miles, he is referring to Italian nautical miles. One Italian nautical mile is equivalent to 1480 meters (4856 feet), and, converted into modern units, Columbus's meas?ure of the circumference of the earth was thus 30,185 kilo?meters (18,756 miles), or about 25 percent less than the true value of 40,010 kilometers, or 24,861 miles.

His reading of Marco Polo and the Toscanelli letter and map had convinced Columbus that Asia extended much farther to the east than Ptolemy had thought and that, conse?quently, Cipangu lay about as far to the west of Spain as - in fact - the West Indies lie.

Columbus's argument for the feasibility of reaching the Spice Islands by sailing west hinged on this figure of 56⅔ miles to the equatorial degree. Since he was seeking royal support for his venture, he needed an authority of more weight than either Marco Polo or Toscanelli to underpin this crucial number; while they might both be dismissed as rather dotty fantasists, it was not so easy to dismiss Alfraganus, who carried all the authority of the Arab astronomical and mathematical tradition behind him. Columbus's claim to have verified Alfraganus's calculations must be seen in this light.

"Alfraganus" is the Latin version of the Arabic name al-Farghani, and refers to Abu al-'Abbas Ahmad ibn Muhammad ibn Kathir al-Farghani. He was one of the scholars associated with the Caliph al-Ma'mun's great efforts to produce Arabic versions of Greek scientific texts in early ninth-century Bagh?dad. He may well have himself taken part in the scientific expedition which, sometime between 820 and 833, set out to measure the actual length of one degree of a meridian.

This was probably the first attempt since the time of Eratos?thenes to measure the length of a degree. Although there are no surviving eyewitness accounts of the experiment, we know from later sources how it was done: Two locations were identified whose latitudes, determined astronomically, differed by one degree. A north-south baseline connecting them was carefully laid out by sighting along pegs, and the length of that baseline was measured. In the experiment in which al-Farghani took part, two pairs of locations were actually chosen, one pair in northern Iraq, on the plain of Sinjar, and the other near Kufah - both areas as flat and feature?less as possible. The results were then compared, and the length of a degree established as56⅔ miles.

Al-Farghani subsequently wrote a very influential little book on astronomy, a number of copies of whose Arabic text survive. The title can be translated Compendium of the Science of the Stars and Celestial Motions. This was twice translated into Latin in Spain during the Middle Ages, once by Gerard of Cremona and once by John of Seville, working under the auspices of Alfonso the Wise. A Hebrew translation also survives. The Compendium, in its Latin version, was widely circulated in Europe and remained a standard author?ity almost to the time of Galileo; it was first printed in 1493, the same year Columbus returned from his first voyage.

It is worth quoting al-Farghani's exact words, for they were of supreme importance to Columbus: "In that way we find that the value of a degree on the celestial sphere, taken on the circumference of the earth, is 56⅔ miles, each mile being equal to 4000 black cubits, as was ascertained during the time of al-Ma'mun - May God's grace be upon him! And on this point a large number of the learned are in agreement."

Yet the correct value for the length of a degree on the meri?dian is not 56⅔ but roughly 69 statute miles, of 60 nautical miles (by definition), or 111 kilometers and a fraction. How could competent astronomers, skilled in mathematics, have made an error of such magnitude?

The basic unit of measurement in the Arab world was the dhira', or cubit. Originally, this was the distance from the elbow to the tip of the middle finger, but a sophisticated cul?ture could not tolerate the variation implicit in this ancient unit of measurement, so the length of a cubit was standar?dized. The earliest standard cubit is known as the "legal cubit", so called because it is the one used in the holy law of Islam, the shar'iya . It is equivalent to 49.8 centimeters (19.6 inches). For surveying purposes, al-Ma'mun introduced another cubit, equivalent to 48.25 centimeters (19 inches). Finally, there is the "black cubit," the standard for which was indicated on the Nilometer on the island of Rawda, in the Nile River. This was equivalent to 54.04 centimeters (21.28 inches). Which cubit did al-Farghani use???

The obvious answer is that he used the "black cubit" of?54.04 centimeters, since he actually uses that term. But we know from other sources that the black cubit had not yet been introduced during the reign of al-Ma'mun, when the length of a degree was measured on the plain of Sinjar. So in spite of the terminology al-Farghani uses, his "black cubit" must in fact refer to either the "surveying cubit" of 48.25 centimeters, or to the legal cubit of.49.8 centimeters. The latter is the more likely, since we know that it was the most commonly used unit during al-Farghani's lifetime.

There are 4000 cubits in an Arab mile. If al-Farghani used the legal cubit as his unit of measurement, then an Arab mile was 1995 meters (6545 feet) long. A degree on the meridian would measure 113 kilometers (70.25 miles) - two kilometers greater than the true value, but well within acceptable limits of error. If he used al-Ma'mun's surveying cubit, then?a degree contained 109 kilometers (67.73 miles) - two kilo?meters less than the true value, but an equally respectable result under the circumstances.

In other words, al-Farghani's so-called "short degree" of 56⅔ miles was not short at all, but was very close to the true length of a degree of the meridian. The error was not al-Farghani's, but Columbus's. Unaware that an Arab mile was considerably longer than an Italian nautical mile, Columbus seized upon the figure of 56⅔ miles for the length of the degree and used it to justify the theory which - in all probabil?ity - he already held.

Historian and Arabist Paul Lunde, author of the whole issue of Aramco World , is a frequent Contributor to the magazines with some 50 articles to his credit over the past two decades, including special multi-article sections on Arabic-language printing and the history of the Silk Roads. His immediate research for this issue was carried out in Seville, Rome, London and Cambridge, and he wrote from his base in Seville?s Barrio do Santa Cruz, a stone?s throw from the city?s cathedral?once a mosque?and from Alc?zares Resales, the Moorish palace complex that remains today one of the residences of Spain?s Christian kings.