Abu-Mahmud Khujandi

关于al-Khujandi的生平，已知的事实很少。我们所知道的少量信息来自他留存下来的著作，以及Nasir al-din纳西尔丁·图西的一些评论。从纳西尔丁·图西的评论中，我们可以相当确定al-Khujandi来自苦盏城。该城位于锡尔河两岸，地处肥沃的费尔干纳谷地入口处，8世纪时被阿拉伯人攻占。纳西尔丁·图西说al-Khujandi是该地区蒙古部落的统治者之一，因此他必定出身贵族。

Al-Khujandi一生中大部分时间的科学工作都得到了白益王朝成员的支持。该王朝于945年掌权，当时Ahmad ad-Dawlah占领了阿拔斯王朝的首都巴格达。Ahmad ad-Dawlah的家族成员成为不同省份的统治者，白益帝国从未有过高度的凝聚力。Al-Khujandi得到了Fakhr ad-Dawlah的赞助，后者于976年至997年在位。

正是Fakhr ad-Dawlah支持al-Khujandi进行他的重大工程，即在雷伊（靠近现代德黑兰）的天文台建造一座巨大的墙式六分仪。许多阿拉伯科学家相信，仪器越大，获得的结果就越精确。事实上，al-Khujandi的墙式六分仪是他自己的发明，它确实开辟了新天地，其刻度可以指示秒，这是前所未有的精度水平。

在994年，al-Khujandi使用这个非常大的仪器观测了太阳在solstices附近的一系列子午线中天。他利用这些观测——分别于994年6月16日和17日夏至以及994年12月14日和17日冬至进行——来计算黄赤交角和雷伊的纬度。他在一篇题为On the obliquity of the ecliptic and the latitudes of the cities的论文中详细描述了他的测量。

根据他的观测，他得出黄赤交角为23° 32' 19"。这个值低于先前获得的值[1]：-

Al-Khujandi说印度人发现黄赤交角最大为24°；克劳狄乌斯·托勒密为23° 51'；他自己为23° 32' 19"。这些不同的值不可能由于仪器缺陷所致。实际上，黄赤交角并非恒定；它是一个递减的量。

然而，al-Khujandi关于黄赤交角的值存在一个误差；它大约低了2角分。比鲁尼在其Tahdid中讨论了这个误差，他声称由于仪器的重量，六分仪的孔径在al-Khujandi的观测过程中移动了大约一拃。比鲁尼几乎可以肯定地准确指出了误差的原因。然而，al-Khujandi给出的雷伊纬度35°34'38.45"，尽管是用他错误的黄赤交角值计算得出的，却精确到最接近的角分。

我们还需要讨论al-Khujandi发现了正弦定理这一说法。这一说法由纳西尔丁·图西提出，他在其Shakl al-qatta中给出了al-Khujandi关于球面三角形这一结果的证明。尽管没有理由怀疑纳西尔丁·图西所说的他给出的证明确实来自al-Khujandi，但有相当多的理由相信阿布·瓦法或Abu Nasr 阿布·纳斯尔·曼苏尔才是最初的发现者。

阿布·瓦法和阿布·纳斯尔·曼苏尔都声称发现了正弦定理，而据我们所知，al-Khujandi并未作出这样的声称。此外，al-Khujandi与其说是理论家，不如说是天文仪器的设计者和天文观测者。最后，尽管这实际上证明不了什么，但该定理在阿布·纳斯尔·曼苏尔的著作中多次出现：无论是他的几何学著作还是天文学著作。

关于al-Khujandi的数学贡献，我们应当作最后一点评论。他在$n = 3$的情形下陈述了费马大定理，尽管毫不奇怪，他的证明是错误的。Al-Khazin写道：-

我先前已经证明……阿布Muhammad al-Khujandi——愿真主怜悯他——在他关于两个立方数之和不是立方数的论证中所提出的内容，是有缺陷且不正确的。

al-Khujandi尽管其成就偏重实用而非理论，却对这个数论结果感兴趣，这无疑很有趣。

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 Khujand. 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.