纳西尔丁·图西Nasir al-Din al-Tusi

虽然通常被称为 纳西尔丁·图西，但他的本名是 Muhammad ibn Muhammad ibn al-Hasan 纳西尔丁·图西。事实上，纳西尔丁·图西 一生中以许多不同的名字为人所知，例如 Muhaqqiq-i Tusi、Khwaja-yi Tusi 和 Khwaja 纳西尔丁·图西。

Al-Tusi出生于图斯，该地位于伊朗东北部，靠近马什哈德，地处卡沙夫河谷的高处。他出生在一个世纪的开始，这个世纪将见证整个伊斯兰世界的征服，从东部的靠近中国到西部的欧洲。这是蒙古人庞大的军事力量横扫伊斯兰世界广大地区的时代，他们对伊斯兰表现出强烈的敌意，残忍地屠杀人民。这是一个伟大scholars难以追求其作品的和平与安宁的时期，纳西尔丁·图西不可避免地卷入了席卷其国家的冲突。

纳西尔丁·图西的父亲是十二伊玛目派的法学家。十二伊玛目派是什叶派穆斯林的主要教派，纳西尔丁·图西接受教育的学校主要是一个宗教机构。然而，在图斯学习期间，纳西尔丁·图西由他的叔叔教授其他科目，这些科目对他的智力发展有重要影响。这些科目包括逻辑、物理和形而上学，同时他还与其他老师学习数学，特别是代数和几何。

1214年，当纳西尔丁·图西13岁时，蒙古领袖成吉思汗放弃在中国的征服，开始迅速向西推进。不久之后，纳西尔丁·图西就会看到这些征服对他自己地区的影响，但在那之前，他能够学习更高级的科目。从图斯，纳西尔丁·图西去了图斯以西75公里的尼沙普尔。尼沙普尔是纳西尔丁·图西完成教育的好选择，因为它是一个重要的学习中心。在那里，纳西尔丁·图西学习了哲学、医学和数学。特别是，他由Kamal al-Din ibn 伊本·优努斯教授数学，后者本人是萨拉夫·丁·图西的学生。在尼沙普尔期间，纳西尔丁·图西开始获得杰出学者的声誉，并在整个地区闻名。

蒙古入侵在 1220 年前后到达图斯地区，造成了大量破坏。成吉思汗再次将注意力转向东方，把他的将领和儿子们留在西方继续他的征服。在该地区频繁的战斗中，有一些和平的避风港吸引了 纳西尔丁·图西。阿萨辛派奉行一种智识形式的极端什叶派教义，控制着厄尔布尔士山脉中的阿拉穆特城堡，以及山中其他类似的难以攻克的堡垒。当伊斯玛仪派统治者 纳西尔丁·图西 ad-Din 'Abd ar-Rahim 邀请他加入阿萨辛派的服务时，纳西尔丁·图西 接受了，并成为伊斯玛仪派宫廷中备受尊敬的一员。如果他愿意，他是否能够离开，并不完全清楚。然而，纳西尔丁·图西 在辗转于不同据点期间完成了他一些最好的工作，并且在这一时期他写下了关于逻辑学、哲学、数学和天文学的重要著作。这些著作中的第一部，Akhlaq-i nasiri，写于 1232 年。这是一部关于 伦理学 的著作，纳西尔丁·图西 将其献给了伊斯玛仪派统治者 纳西尔丁·图西 ad-Din 'Abd ar-Rahim。

1256年，当纳西尔丁·图西在阿拉穆特城堡时，它被蒙古领袖Hulegu的军队攻击，Hulegu是成吉思汗的孙子，当时正致力于在伊斯兰地区扩展蒙古势力。有些人声称纳西尔丁·图西将阿拉穆特的防御出卖给了入侵的蒙古人。当然，Hulegu的军队摧毁了阿拉穆特，而Hulegu本人对科学感兴趣，他对纳西尔丁·图西非常尊重。可能纳西尔丁·图西确实觉得他被违背意愿地关在阿拉穆特，因为他似乎热衷于加入胜利的蒙古人，他们任命他为科学顾问。他还被负责宗教事务，并在1258年蒙古军队在Hulegu领导下攻击巴格达时随行。

巴格达最后一位阿拔斯哈里发Al-Musta'sim是个软弱的领袖，当蒙古军队进攻巴格达时，他根本不是Hulegu率领的蒙古军队的对手。蒙古人围城之后，于1258年2月攻入城中，Al-Musta'sim连同他的300名官员一起被杀。Hulegu的军队打赢一场战斗后，对城市毫无怜悯之心，于是纵火劫掠，杀害了许多居民。就自身安全而言，纳西尔丁·图西无疑做出了正确的选择，而且他也将因改换效忠对象而在科学上获益。

Hulegu对征服巴格达非常满意，也很高兴像纳西尔丁·图西这样杰出的学者加入了他。因此，当纳西尔丁·图西向Hulegu呈上建造一座精美天文台的计划时，Hulegu欣然同意。Hulegu已将马拉盖定为他的首都。马拉盖位于伊朗西北部的阿塞拜疆地区，天文台就建在马拉盖。天文台于1259年在马拉盖以西动工，至今那里仍能看到它的遗迹。

马拉盖天文台于1262年投入使用。有趣的是，波斯人在天文台的建造和运行中得到了中国天文学家的协助。天文台拥有各种仪器，例如一座4米长的铜制墙象限仪，以及纳西尔丁·图西本人发明的方位角象限仪。纳西尔丁·图西还为天文台设计了其他仪器，而天文台远不止是一个天文学中心。它拥有一座出色的图书馆，藏书涵盖广泛的科学主题，同时科学、数学和哲学方面的研究工作也在那里蓬勃开展。

阿尔图西充分利用了他的天文台，制作了非常精确的行星运动表。他经过12年的观测后，出版了 Zij-i ilkhani（伊尔汗天文表），最初用波斯语写成，后来翻译成阿拉伯语。这部作品包含计算行星位置的表格，还包含一个星表。这并不是 纳西尔丁·图西 在天文学方面产生的唯一重要作品。可以说，直到 尼古拉·哥白尼 时代 heliocentric model 发展之前，纳西尔丁·图西 对 克劳狄乌斯·托勒密 的行星系统模型做出了最重大的发展。在 纳西尔丁·图西 的主要天文学论著 al-Tadhkira fi'ilm al-hay'a（天文学回忆录）中，他[17]：-

……设计了一种新的月球运动模型，本质上不同于 克劳狄乌斯·托勒密 的模型。他废除了偏心圆和 prosneusis 中心，完全基于八个均匀旋转球体的原理建立模型，从而成功地以与“天文学大成” Ⓣ（主要论题：来自阿拉伯语 'al-majisti'——希腊语 'Mathematike Syntaxis' 的阿拉伯语翻译，后来翻译成拉丁语为 'Magna Syntaxis'）相同的精确度表示了月球运动的不规则性。他声称两种理论之间经度的最大差异达到10，这完全正确。在他的模型中，纳西尔丁·图西 在天文学历史上首次使用了他自己发明的定理，250年后，这个定理再次出现在 尼古拉·哥白尼 的“论革命”III 4中。

这段引文中提到的定理涉及著名的“图西双圆”，它将线性运动分解为两个圆周运动之和。纳西尔丁·图西 得出这个结果的目标是去除 克劳狄乌斯·托勒密 系统中所有不基于均匀圆周运动原理的部分。许多历史学家声称，尼古拉·哥白尼 在阿尔图西的作品中发现图西双圆结果后使用了它，但并非所有人都同意；例如参见[38]，其中声称 尼古拉·哥白尼 从 普罗克洛 的 Commentary on the first book of Euclid 中获取了这个结果，而不是从 纳西尔丁·图西 那里。

在对天文学的众多其他贡献中，纳西尔丁·图西计算出岁差的值为51'。他还撰写了关于天文仪器的著作，例如关于构造和使用星盘的著作。

在逻辑学方面，纳西尔丁·图西 遵循 伊本·西那 的教导。他写了五部关于这个主题的作品，其中最重要的是关于推理的一部。在[33]中，Street 这样描述：-

图西，一位13世纪用阿拉伯语写作的逻辑学家，使用两个逻辑连接词来构建分子命题：“如果-那么”和“要么-要么”。通过参考二分树，图西展示了如何相对于析取项中的术语选择适当的析取。他还讨论了从条件命题中得出的析取命题。

Al-Tusi写了许多对希腊文本的评注。其中包括对奥托里库斯、阿里斯塔克斯、欧几里得、阿波罗尼奥斯、阿基米德、Hypsicles、比提尼亚的狄奥多西、梅涅劳斯和克劳狄乌斯·托勒密作品的修订阿拉伯语版本。特别是，他写了一篇对梅涅劳斯的Spherics的评注（详情见[41]），以及对阿基米德的On the sphere and cylinder的评注（详情见[21]）。在后一部作品中，纳西尔丁·图西讨论了早期数学家对比较直线和曲线长度的反对意见。Al-Tusi认为比较是合理的，尽管有人反对说，作为不同的实体，它们是不可比较的。

克劳狄乌斯·托勒密的AlmagestⓉ（主要论题：来自阿拉伯语的“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语译本，后来被译为拉丁语“Magna Syntaxis”）是阿拉伯科学家潜心研究的著作之一。1247年，纳西尔丁·图西写了Tahrir al-Majisti（对《天文学大成》的评注），其中他引入了各种三角学技术来计算正弦表；详情见[5]。正如在Zij-i Ilkhahi中那样，纳西尔丁·图西给出了正弦表，其条目对自变量的每半度计算到三位六十进制小数。

纳西尔丁·图西最重要的数学贡献之一是将三角学创建为一门独立的数学学科，而不仅仅是天文应用的工具。在Treatise on the quadrilateral中，纳西尔丁·图西首次现存地阐述了平面和球面三角学的整个体系。正如[1]所述：-

这部作品确实是历史上第一部将三角学作为纯数学独立分支的著作，也是第一部阐述了直角球面三角形所有六种情况的著作。

这部作品还包含了著名的平面三角形正弦公式：

$\Large\frac a {\sin A}\normalsize = \Large\frac b {\sin B}\normalsize = \Large\frac c {\sin C}$。

另一项数学贡献是纳西尔丁·图西的手稿，日期为1265年，涉及计算整数的$n$次方根；关于1413年制作的手稿副本的详情，见[6]。纳西尔丁·图西的这部作品几乎肯定不是原创的，而是他对卡拉吉学派所发展方法的版本。在手稿中，纳西尔丁·图西确定了expansion of a binomial任意次幂的系数，给出了二项式公式以及二项式系数之间的帕斯卡三角形关系。

我们应该简要提及纳西尔丁·图西做出贡献的其他领域。他写了一部关于矿物的著名著作，其中包含一种基于黑白混合的有趣颜色理论，并包括关于宝石和香水的章节。他还写了医学著作，但他的医学著作是他最不重要的著作之一。更为重要的是纳西尔丁·图西对哲学和伦理学的贡献。特别是在哲学中，他提出了关于空间本质的重要问题。

纳西尔丁·图西有许多学生，其中较为知名的是Nizam al-a'Raj，他也写了一篇对AlmagestⓉ的评注（主要论题：来自阿拉伯语'al-majisti'——希腊语'Mathematike Syntaxis'的阿拉伯语翻译，后来翻译成拉丁语为'Magna Syntaxis'）。他的另一个学生Qutb ad-Din ash-Shirazi首次对彩虹给出了令人满意的数学解释。纳西尔丁·图西的影响通过这些学生延续下来，在[1]中总结如下：-

纳西尔丁·图西的影响，尤其是在伊斯兰东部，是巨大的。很可能，如果我们将所有领域都考虑在内，他对伊斯兰科学复兴的贡献比任何其他个人都要大。他在马拉盖聚集了如此众多有能力的学者和科学家，不仅导致了数学和天文学的复兴，也导致了伊斯兰哲学乃至神学的更新。

Although usually known as Nasir al-Din al-Tusi, his proper name was Muhammad ibn Muhammad ibn al-Hasan al-Tusi. In fact al-Tusi was known by a number of different names during his lifetime such as Muhaqqiq-i Tusi, Khwaja-yi Tusi and Khwaja Nasir.

Al-Tusi was born in Tus, which lies close to Meshed in northeastern Iran high up in the valley of the Kashaf River. He was born at the beginning of a century which would see conquests across the whole of the Islamic world from close to China in the east to Europe in the west. It was the era when the vast military power of the Mongols would sweep across the vast areas of the Islamic world displaying a bitter animosity towards Islam and cruelly massacring people. This was a period in which there would be little peace and tranquillity for great scholars to pursue their works, and al-Tusi was inevitably drawn into the conflict engulfing his country.

In Tus, al-Tusi's father was a jurist in the Twelfth Imam School. The Twelfth Imam was the main sect of Shi'ite Muslims and the school where al-Tusi was educated was mainly a religious establishment. However, while studying in Tus, al-Tusi was taught other topics by his uncle which would have an important influence on his intellectual development. These topics included logic, physics and metaphysics while he also studied with other teachers learning mathematics, in particular algebra and geometry.

In 1214, when al-Tusi was 13 years old, Genghis Khan, who was the leader of the Mongols, turned away from his conquests in China and began his rapid advance towards the west. It would not be too long before al-Tusi would see the effects of these conquests on his own regions, but before that happened he was able to study more advanced topics. From Tus, al-Tusi went to Nishapur which is 75 km west of Tus. Nishapur was a good choice for al-Tusi to complete his education since it was an important centre of learning. There al-Tusi studied philosophy, medicine and mathematics. In particular he was taught mathematics by Kamal al-Din ibn Yunus, who himself had been a pupil of Sharaf al-Din al-Tusi. While in Nishapur al-Tusi began to acquire a reputation as an outstanding scholar and became well known throughout the area.

The Mongol invasion reached the area of Tus around 1220 and there was much destruction. Genghis Khan turned his attention again towards the east leaving his generals and sons in the west to continue his conquests. There was, amid the frequent fighting in the region, peaceful havens which attracted al-Tusi. The Assassins, who practised an intellectual form of extremist Shi'ism, controlled the castle of Alamut in the Elburz Mountains, and other similar impregnable forts in the mountains. When invited by the Isma'ili ruler Nasir ad-Din 'Abd ar-Rahim to join the service of the Assassins, al-Tusi accepted and became a highly regarded member of the Isma'ili Court. Whether he would have been able to leave, had he wished to, is not entirely clear. However, al-Tusi did some of his best work while moving round the different strongholds, and during this period he wrote important works on logic, philosophy, mathematics and astronomy. The first of these works, Akhlaq-i nasiri, was written in 1232. It was a work on ethics which al-Tusi dedicated to the Isma'ili ruler Nasir ad-Din 'Abd ar-Rahim.

In 1256 al-Tusi was in the castle of Alamut when it was attacked by the forces of the Mongol leader Hulegu, a grandson of Genghis Khan, who was at that time set on extending Mongol power in Islamic areas. Some claim that al-Tusi betrayed the defences of Alamut to the invading Mongols. Certainly Hulegu's forces destroyed Alamut and, Hulegu himself being himself interested in science, he treated al-Tusi with great respect. It may be that indeed al-Tusi felt that he was being held in Alamut against his will, for certainly he seemed enthusiastic in joining the victorious Mongols who appointed him as their scientific advisor. He was also put in charge of religious affairs and was with the Mongol forces under Hulegu when they attacked Baghdad in 1258.

Al-Musta'sim, the last Abbasid caliph in Baghdad, was a weak leader and he proved no match for Hulegu's Mongol forces when they attacked Baghdad. After having laid siege to the city, the Mongols entered it in February 1258 and al-Musta'sim together with 300 of his officials were murdered. Hulegu had little sympathy with a city after his armies had won a battle, so he burned and plundered the city and killed many of its inhabitants. Certainly al-Tusi had made the right move as far as his own safety was concerned, and he would also profit scientifically by his change of allegiance.

Hulegu was very pleased with his conquest of Baghdad and also pleased that such an eminent scholar as al-Tusi had joined him. So, when al-Tusi presented Hulegu with plans for the construction of a fine Observatory, Hulegu was happy to agree. Hulegu had made Maragheh his capital . Maragheh was in the Azerbaijan region of northwestern Iran, and it was at Maragheh that the Observatory was to be built. Construction of the Observatory began in 1259 west of Maragheh, and traces of it can still be seen there today.

The observatory at Maragheh became operational in 1262. Interestingly the Persians were assisted by Chinese astronomers in the construction and operation of the observatory. It had various instruments such as a 4 metre wall quadrant made from copper and an azimuth quadrant which was the invention of Al-Tusi himself. Al-Tusi also designed other instruments for the Observatory which was far more than a centre for astronomy. It possessed a fine library with books on a wide range of scientific topics, while work on science, mathematics and philosophy were vigorously pursued there.

Al-Tusi put his Observatory to good use, making very accurate tables of planetary movements. He published Zij-i ilkhani (the Ilkhanic Tables), written first in Persian and later translated into Arabic, after making observations for 12 years. This work contains tables for computing the positions of the planets, and it also contains a star catalogue. This was not the only important work which al-Tusi produced in astronomy. It is fair to say that al-Tusi made the most significant development of Ptolemy's model of the planetary system up to the development of the heliocentric model in the time of Copernicus. In al-Tusi's major astronomical treatise, al-Tadhkira fi'ilm al-hay'a (Memoir on astronomy) he [17]:-

... devised a new model of lunar motion, essentially different from Ptolemy's. Abolishing the eccentric and the centre of prosneusis, he founded it exclusively on the principle of eight uniformly rotating spheres and thereby succeeded in representing the irregularities of lunar motion with the same exactness as the "Almagest" Ⓣ. His claim that the maximum difference in longitude between the two theories amounts to 10 proves perfectly true. In his model Nasir, for the first time in the history of astronomy, employed a theorem invented by himself which, 250 years later, occurred again in Copernicus, "De Revolutionibus", III 4.

The theorem referred to in this quotation concerns the famous "Tusi-couple" which resolves linear motion into the sum of two circular motions. The aim of al-Tusi with this result was to remove all parts of Ptolemy's system that were not based on the principle of uniform circular motion. Many historians claim that the Tusi-couple result was used by Copernicus after he discovered it in Al-Tusi's work, but not all agree; see for example [38] where it is claimed that Copernicus took the result from Proclus's Commentary on the first book of Euclid and not from al-Tusi.

Among numerous other contributions to astronomy, al-Tusi calculated the value of 51' for the precession of the equinoxes. He also wrote works on astronomical instruments, for example on constructing and using an astrolabe.

In logic al-Tusi followed the teachings of ibn Sina. He wrote five works on the subject, the most important of which is one on inference. In [33] Street describes this as follows:-

Tusi, a thirteenth century logician writing in Arabic, uses two logical connectives to build up molecular propositions: 'if-then', and 'either-or'. By referring to a dichotomous tree, Tusi shows how to choose the proper disjunction relative to the terms in the disjuncts. He also discusses the disjunctive propositions which follow from a conditional proposition.

Al-Tusi wrote many commentaries on Greek texts. These included revised Arabic versions of works by Autolycus, Aristarchus, Euclid, Apollonius, Archimedes, Hypsicles, Theodosius, Menelaus and Ptolemy. In particular he wrote a commentary on Menelaus's Spherics (see [41] for details), and Archimedes' On the sphere and cylinder (see [21] for details). In the latter work al-Tusi discussed objections raised by earlier mathematicians to comparing lengths of straight lines and of curved lines. Al-Tusi argues that comparisons are legitimate, despite the objections that, being different entities, they are incomparable.

Ptolemy's Almagest Ⓣ was one of the works which Arabic scientists studied intently. In 1247 al-Tusi wrote Tahrir al-Majisti (Commentary on the Almagest) in which he introduced various trigonometrical techniques to calculate tables of sines; see [5] for details. As in the Zij-i Ilkhahi al-Tusi gave tables of sines with entries calculated to three sexagesimal places for each half degree of the argument.

One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. As stated in [1]:-

This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.

This work also contains the famous sine formula for plane triangles:

$\Large\frac a {\sin A}\normalsize = \Large\frac b {\sin B}\normalsize = \Large\frac c {\sin C}$.

Another mathematical contribution was al-Tusi's manuscript, dated 1265, concerning the calculation of $n$-th roots of an integer; see [6] for details of a copy of this manuscript made in 1413. This work by al-Tusi is almost certainly not original but rather it is his version of methods developed by al-Karaji's school. In the manuscript al-Tusi determined the coefficients of the expansion of a binomial to any power giving the binomial formula and the Pascal triangle relations between binomial coefficients.

We should mention briefly other fields in which al-Tusi contributed. He wrote a famous work on minerals which contains an interesting theory of colour based on mixtures of black and white, and included chapters on jewels and perfumes. He also wrote on medicine, but his medical works are among his least important. Much more important were al-Tusi's contributions to philosophy and ethics. In particular in philosophy he asked important questions on the nature of space.

Al-Tusi had a number of pupils, one of the better known being Nizam al-a'Raj who also wrote a commentary on the Almagest Ⓣ. Another of his pupils Qutb ad-Din ash-Shirazi gave the first satisfactory mathematical explanation of the rainbow. Al-Tusi's influence, which continued through these pupils, is summed up in [1] as follows:-

Al-Tusi's influence, especially in eastern Islam, was immense. Probably, if we take all fields into account, he was more responsible for the revival of the Islamic sciences than any other individual. His bringing together so many competent scholars and scientists at Maragheh resulted not only in the revival of mathematics and astronomy but also in the renewal of Islamic philosophy and even theology.