乌鲁伯格Ulugh Beg

乌鲁伯格是征服者Timur的孙子，Timur通常被称为Tamerlane（来自Timur-I-Leng，意为跛子Timur，是他的波斯敌人使用的蔑称）。尽管在本档案中我们主要关注乌鲁伯格在数学和天文学方面的成就，但我们需要考察该地区的历史，因为它对乌鲁伯格的生活产生了如此重大的影响。

乌鲁伯格的祖父来自突厥Barlas部落，这是一个生活在Transoxania（今基本为乌兹别克斯坦）的蒙古部落。他在自己的领导下联合了几个突厥-蒙古部落，并率领他的骑射手军队开始征服现在伊朗、伊拉克和土耳其东部所占据的地区。

他的孙子乌鲁伯格出生后不久，Timur入侵印度，到1399年他已控制了德里。Timur从1399年到1402年继续他的征服，向西扩展他的帝国，在叙利亚战胜了埃及Mamluks，并在安卡拉附近的一场战斗中战胜了Ottomans。Timur于1405年率领他的军队进入中国时去世。

Timur死后，他的帝国在他的儿子们之间争夺。乌鲁伯格的父亲Shah Rukh是Timur的第四个儿子，到1407年，他已经获得了帝国大部分地区的总体控制权，包括伊朗和Turkistan，重新控制了Samarkand。Samarkand曾是Timur帝国的首都，但尽管他的孙子乌鲁伯格在Timur的宫廷中长大，他却很少在那座城市。当Timur不在他的军事战役中时，他带着他的军队从一个地方移动到另一个地方，他的宫廷，包括他的孙子乌鲁伯格，也随他一起旅行。

1409年，沙哈鲁决定将呼罗珊的赫拉特（今阿富汗西部）作为他的新都城。沙哈鲁在此统治，使其成为贸易和文化中心。他在那里建立了一座图书馆，并成为艺术的赞助人。然而沙哈鲁并没有放弃撒马尔罕，而是决定将其交给他的儿子乌鲁伯格，后者更感兴趣的是将这座城市打造成文化中心，而非政治或军事征服。尽管当父亲让他掌管撒马尔罕时，乌鲁伯格只有十六岁，但他成为了父亲的副手，并成为河中地区的统治者。

乌鲁伯格主要是一位科学家，特别是数学家和天文学家。然而，他当然没有忽视艺术，写诗和历史，并研究《古兰经》。1417年，为了推进天文学研究，乌鲁伯格开始建造一所伊斯兰学校，这是高等教育的中心。这所伊斯兰学校面向撒马尔罕的雷吉斯坦广场，于1420年完工，随后乌鲁伯格开始任命他能找到的最优秀的科学家担任那里的讲师。

乌鲁伯格邀请卡西加入他在撒马尔罕的伊斯兰学校，以及大约六十位其他科学家，包括Qadi Zada。毫无疑问，除了乌鲁伯格本人之外，卡西是撒马尔罕领先的天文学家和数学家。卡西写给他父亲的信件保存了下来。这些信件是从撒马尔罕写的，精彩地描述了那里的科学生活。其中一封信的内容最近才出版，见[5]。

在信中，卡西赞扬了乌鲁伯格的数学能力，但在撒马尔罕的其他科学家中，只有Qadi Zada赢得了他的尊重。乌鲁伯格主持科学会议，自由讨论天文学问题。通常这些问题对除卡西以外的所有人都太难了，信件证实卡西是乌鲁伯格在撒马尔罕伊斯兰学校最密切的合作者。

除了伊斯兰学校，乌鲁伯格还在撒马尔罕建造了一座天文台，始建于1428年。这座天文台呈圆形，有三层。直径超过50米，高35米。天文台的主任是阿里·库奇，一位穆斯林天文学家。卡西和其他被任命到伊斯兰学校的数学家和天文学家也在乌鲁伯格的天文台工作。

为天文台特别建造的仪器中有一个象限仪，其巨大到必须移除部分地面才能放入天文台。还有一座大理石六分仪、一个三角仪和一个浑天仪。科学家们在乌鲁伯格的指导下并在与他的合作中在天文台取得的成就，在[4]中有详细讨论。这本优秀的书记录了主要成就，包括：给出三次方程精确近似解的方法；与二项式定理的工作；乌鲁伯格精确到八位小数的正弦和正切表；球面三角学公式；以及特别重要的，乌鲁伯格的Catalogue of the stars，这是自克劳狄乌斯·托勒密以来的第一部综合性星表。

这份星表，即Zij-i Sultani，为此类著作树立了标准，直到十七世纪。它出版于1437年，给出了992颗恒星的位置。这份星表是天文台众多人员共同努力的成果，其中包括乌鲁伯格、卡西和Qadi Zada。除了天文台观测的表格外，这部著作还包含历法计算和三角学成果。

三角学成果包括按1°间隔给出的正弦表和正切表。这些表显示出很高的精度，至少精确到小数点后8位。这一计算建立在准确确定sin 1°的基础上，乌鲁伯格通过证明它是三次方程的解，然后用数值方法求解该方程而解决了这一问题。他得到

sin 1° = 0.017452406437283571

正确的近似值是

sin 1° = 0.017452406437283512820

这表明乌鲁伯格达到了惊人的精度。

在天文台所作的观测揭示出克劳狄乌斯·托勒密计算中的若干错误，这些计算在此之前一直被毫无疑问地接受。来自他天文台的数据使乌鲁伯格能够将一年的长度计算为365天5小时49分15秒，这是一个相当准确的值。他给出了关于太阳、月球和行星的数据。他关于行星一年内运动的数据，如同他许多工作一样，非常准确[1]：-

……乌鲁伯格的数据与现代关于[土星、木星、火星、金星]的数据之间的差异落在二至五秒的范围内。

乌鲁伯格的政治才能不及他的科学才能，而且在1447年他父亲去世后，尽管他是独子，却无法保住权力。他最终在撒马尔罕因自己的儿子'Abd al-Latif的煽动而被处死。他的陵墓于1941年在撒马尔罕由帖木儿建造的陵墓中被发现。人们发现乌鲁伯格被埋葬时穿着自己的衣服，这已知表明他被视为殉道者。当他的遗体被检查时，他所受的伤害显而易见[1]：-

……第三颈椎被一件锋利的器具切断，以致身体的主要部分和该椎骨的一段弧被干净利落地切开；这一击从左侧打来，还切穿了下颌骨的右角和它的下缘。

Ulugh Beg was the grandson of the conqueror Timur, who is often known as Tamerlane (from Timur-I-Leng meaning Timur the Lame, a title of contempt used by his Persian enemies). Although in this archive we are primarily interested in Ulugh Beg's achievements in mathematics and astronomy, we need to examine the history of the area since it had such a major impact on Ulugh Beg's life.

Timur, Ulugh Beg's grandfather, came from the Turkic Barlas tribe which was a Mongol tribe that was living in Transoxania, today essentially Uzbekistan. He united several Turko-Mongol tribes under his leadership and set out on a conquest, with his armies of mounted archers, of the area now occupied by Iran, Iraq, and eastern Turkey.

Shortly after his grandson Ulugh Beg was born, Timur invaded India and by 1399 he had taken control of Delhi. Timur continued his conquests by extending his empire to the west from 1399 to 1402, winning victories over the Egyptian Mamluks in Syria and the Ottomans in a battle near Ankara. Timur died in 1405 leading his armies into China.

After Timur's death his empire was disputed among his sons. Ulugh Beg's father Shah Rukh was the fourth son of Timur and, by 1407, he had gained overall control of most of the empire, including Iran and Turkistan regaining control of Samarkand. Samarkand had been the capital of Timur's empire but, although his grandson Ulugh Beg had been brought up at Timur's court, he was seldom in that city. When Timur was not on one of his military campaigns he moved with his army from place to place and his court, including his grandson Ulugh Beg, travelled with him.

In 1409 Shah Rukh decided to make Herat in Khorasan (today in western Afghanistan) his new capital. Shah Rukh ruled there making it a trading and cultural centre. He founded a library there and became a patron of the arts. However Shah Rukh did not give up Samarkand, rather he decided to give it to his son Ulugh Beg who was more interested in making the city a cultural centre than he was in politics or military conquest. Although Ulugh Beg was only sixteen years old when his father put him in control of Samarkand, he became his father's deputy and he became ruler of the Mawaraunnahr region.

Ulugh Beg was primarily a scientist, in particular a mathematician and an astronomer. However, he certainly did not neglect the arts, writing poetry and history and studying the Qur'an. In 1417, to push forward the study of astronomy, Ulugh Beg began building a madrasah which is a centre for higher education. The madrasah, fronting the Rigestan Square in Samarkand, was completed in 1420 and Ulugh Beg then began to appoint the best scientists he could find to positions there as lecturers.

Ulugh Beg invited al-Kashi to join his madrasah in Samarkand, as well as around sixty other scientists including Qadi Zada. There is little doubt that, other than Ulugh Beg himself, al-Kashi was the leading astronomer and mathematician at Samarkand. Letters which al-Kashi wrote to his father have survived. These were written from Samarkand and give a wonderful description of the scientific life there. The contents of one of these letters has only recently been published, see [5].

In the letters al-Kashi praises the mathematical abilities of Ulugh Beg but of the other scientists in Samarkand, only Qadi Zada earned his respect. Ulugh Beg led scientific meetings where problems in astronomy were freely discussed. Usually these problems were too difficult for all except al-Kashi and the letters confirm that al-Kashi was the closest collaborator of Ulugh Beg at his madrasah in Samarkand.

In addition to the madrasah, Ulugh Beg built an observatory at Samarkand, the construction of this beginning in 1428. The Observatory, which was circular in shape, had three levels. It was over 50 metres in diameter and 35 metres high. The director of the Observatory was Ali-Kudschi, a Muslim astronomer. Al-Kashi and other mathematicians and astronomers appointed to the madrasah also worked at Ulugh Beg's Observatory.

Among the instruments specially constructed for the Observatory was a quadrant so large that part of the ground had to be removed to allow it to fit in the Observatory. There was also a marble sextant, a triquetram and an armillary sphere. The achievements of the scientists at the Observatory, working there under Ulugh Beg's direction and in collaboration with him, are discussed in detail in [4]. This excellent book records the main achievements which include the following: methods for giving accurate approximate solutions of cubic equations; work with the binomial theorem; Ulugh Beg's accurate tables of sines and tangents correct to eight decimal places; formulae of spherical trigonometry; and of particular importance, Ulugh Beg's Catalogue of the stars, the first comprehensive stellar catalogue since that of Ptolemy.

This star catalogue, the Zij-i Sultani, set the standard for such works up to the seventeenth century. Published in 1437, it gives the positions of 992 stars. The catalogue was the results of a combined effort by a number of people working at the Observatory including Ulugh Beg, al-Kashi, and Qadi Zada. As well as tables of observations made at the Observatory, the work contained calendar calculations and results in trigonometry.

The trogonometric results include tables of sines and tangents given at 1° intervals. These tables display a high degree of accuracy, being correct to at least 8 decimal places. The calculation is built on an accurate determination of sin 1° which Ulugh Beg solved by showing it to be the solution of a cubic equation which he then solved by numerical methods. He obtained

sin 1° = 0.017452406437283571

The correct approximation is

sin 1° = 0.017452406437283512820

which shows the remarkable accuracy which Ulugh Beg achieved.

Observations made at the Observatory brought to light a number of errors in the computations of Ptolemy which had been accepted without question up to that time. Data from his Observatory allowed Ulugh Beg to calculate the length of the year as 365 days 5 hours 49 minutes 15 seconds, a fairly accurate value. He produced data relating to the Sun, the Moon and the planets. His data for the movements of the planets over a year is, like so much of his work, very accurate [1]:-

... the difference between Ulugh Beg's data and that of modern times relationg to [Saturn, Jupiter, Mars, Venus] falls within the limits of two to five seconds.

Ulugh Beg's politics were not up to his science and, after his father's death in 1447, he was unable to retain power despite being an only son. He was eventually put to death at Samarkand at the instigation of his own son 'Abd al-Latif. His tomb was discovered in 1941 in the mausoleum built by Timur in Samarkand. It was discovered that Ulugh Beg had been buired in his clothes which is known to indicate that he was considered a martyr. The injuries inflicted on him were evident when his body was examined [1]:-

... the third cervical vertebra was severed by a sharp instrument in such a way that the main portion of the body and an arc of that vertebra were cut cleanly; the blow, struck from the left, also cut through the right corner of the lower jaw and its lower edge.