Abu Mahmud Hamid ibn al-Khidr Al-Khujandi

Born: about 940 in Khudzhand, Tajikistan
Died: 1000

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Few facts about al-Khujandi's life are known. What little we know comes through his writings which have survived and also some comments made by Nasir al-din al-Tusi. From al-Tusi's comments we can be fairly certain that al-Khujandi came from the city of Khudzhand. The city lies along both banks of the Syrdarya river, at the entrance to the fertile Fergana Valley, and it was captured by the Arabs in the 8th century. Al-Tusi says that al-Khujandi was one of the rulers of the Mongol tribe in that region so he must have come from the nobility.

Al-Khujandi was supported in his scientific work for most of his life by members of the Buyid dynasty. The dynasty came to power in 945 when Ahmad ad-Dawlah occupied the 'Abbasid capital of Baghdad. Members of Ahmad ad-Dawlah's family became rulers in different provinces and there was never a great deal of cohesion in the Buyid empire. Al-Khujandi received patronage from Fakhr ad-Dawlah who ruled from 976 to 997.

It was Fakhr ad-Dawlah who supported al-Khujandi in his major project to construct a huge mural sextant for his observatory at Rayy, which is near modern Tehran. It was believed by many Arabic scientists that the larger an instrument was, the more accurate were the results obtained. In fact al-Khujandi's mural sextant was his own invention and it did break new ground in having a scale which indicated seconds, a level of accuracy never before attempted.

During the year 994 al-Khujandi used the very large instrument to observe a series of meridian transits of the sun near the solstices. He used these observations, made on 16 and 17 June 994 for the summer solstice and 14 and 17 December 994 for the winter solstice, to calculate the obliquity of the ecliptic, and the latitude of Rayy. He described his measurements in detail in a treatise On the obliquity of the ecliptic and the latitudes of the cities.

From his observations he obtained 23° 32' 19" for the obliquity of the ecliptic. This value was lower than values obtained previously [1]:-

Al-Khujandi says that the Indians found the greatest obliquity of the ecliptic, 24° ; Ptolemy 23° 51' ; himself 23° 32' 19". These divergent values cannot be due to defective instruments. Actually the obliquity of the ecliptic is not constant; it is a decreasing quantity.

There is, however, an error in al-Khujandi's value for the obliquity of the ecliptic; it is about two minutes too low. The error was discussed by al-Biruni in his Tahdid where he claimed that the aperture of the sextant settled about one span in the course of al-Khujandi's observations due to the weight of the instrument. Al-Biruni is almost certainly correct in pinpointing the cause of the error. However, al-Khujandi's latitude for Rayy, 35° 34' 38.45", despite being calculated using his erroneous value for the obliquity of the ecliptic, is accurate to the nearest minute of arc.

It remains for us to discuss the claim that al-Khujandi discovered the sine theorem. The claim was made by al-Tusi who gives al-Khujandi's proof of the result for spherical triangles in his Shakl al-qatta. Although there is no reason to doubt al-Tusi that the proof he gives does indeed come from al-Khujandi there is quite a few reason to believe that one of Abu'l-Wafa or Abu Nasr Mansur was the original discoverer.

Both Abu'l-Wafa and Abu Nasr Mansur claim to have discovered the sine theorem while, as far as we are aware, al-Khujandi makes no such claim. Also al-Khujandi was more of a designer of astronomical instruments and an astronomical observer than he was theoretician. Finally, although this really proves little, the theorem appears many times in the writings of Abu Nasr Mansur: both his writings on geometry as well as those on astronomy.

We should make one final comment on the mathematical contributions of al-Khujandi. He stated Fermat's Last Theorem in the case n = 3 although, not surprisingly, his proof is wrong. Al-Khazin wrote:-

I demonstrated earlier ... that what Abu Muhammad al-Khujandi advanced - may God have mercy on him - in his demonstration that the sum of two cubic numbers is not a cube, is defective and incorrect.

It is certainly interesting that al-Khujandi, despite his practical rather than theoretical achievements, should be interested in this number theory result.

Article by: J J O'Connor and E F Robertson

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JOC/EFR November 1999
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