卡西Jamshīd al-Kāshī

贾姆希德·卡西的生平和著作的细节比这一时期的许多其他人更为人所知，尽管他生平的细节并不详尽。原因之一是他为许多著作标注了完成的准确日期，另一个原因是他写给父亲的一些信件保存了下来，提供了引人入胜的信息。

阿尔·卡西出生在卡尚，该地位于伊朗中部山脉东麓的沙漠中。在卡西成长的时代，帖木儿（常被称为Tamburlaine）正在征服大片地区。他于1370年在撒马尔罕宣称自己为蒙古帝国的君主和恢复者，并于1383年开始在波斯的征服，攻占了赫拉特。帖木儿于1405年去世，他的帝国被分给了他的两个儿子，其中之一是沙哈鲁。

当帖木儿进行军事征战之时，条件非常艰难，普遍贫困。卡西像当时的许多人一样生活在贫困之中，并在从一个城镇迁移到另一个城镇的同时致力于天文学和数学。当沙哈鲁在其父去世后接管政权时，情况明显改善。他为该地区带来了经济繁荣，并大力支持艺术和知识生活。随着氛围的改变，卡西的生活也明显改善。卡西生平中我们能准确确定日期的第一件事是他在1406年6月2日在卡尚进行的月食观测。

可以合理假设卡西留在卡尚，从事天文文本的研究。他肯定在1407年3月1日在他的家乡完成了Sullam Al-sama，该文本保存了下来。该著作的全名意为The Stairway of Heaven, on Resolution of Difficulties Met by Predecessors in the Determination of Distances and Sizes（天体的）。此时，科学家有必要从他们的国王、王子或统治者那里获得赞助。阿尔·卡西利用这一点为自己谋利，并在艺术和科学赞助变得流行的新时代中赢得了青睐。他在1410-11年间写的Compendium of the Science of Astronomy献给了统治的帖木儿王朝的一位后裔。

撒马尔罕位于乌兹别克斯坦，是中亚最古老的城市之一。该城市成为帖木儿帝国的首都，沙哈鲁让自己的儿子乌鲁伯格统治该城。乌鲁伯格本人是一位伟大的科学家，开始将该城建设成一个伟大的文化中心。阿尔·卡西将他的重要天文表著作Khaqani Zij献给了乌鲁伯格，该书基于纳西尔丁·图西的表格。在引言中，卡西说如果没有乌鲁伯格的支持，他无法完成它。在这部著作中，有三角函数表，给出正弦函数值到四位六十进制数字，每个度数参数，并附有每分钟要添加的差值。还有表格给出天球上不同坐标系之间的转换，特别是允许将黄道坐标转换为赤道坐标。关于这部著作的详细讨论，见[14]。

Khaqani Zij还包含[1]：-

……太阳、月亮和行星的纵向运动详细表格。卡西还给出了某些地理纬度下的纵向和横向视差表、食表以及月亮的可见性表。

Al-Kashi在乌鲁伯格那里确实找到了合适的赞助人，因为后者大约在1420年在撒马尔罕创办了一所研究神学和科学的大学，并寻找最优秀的科学家来协助他的项目。乌鲁伯格邀请Al-Kashi加入他在撒马尔罕的这所学府，同时还邀请了大约六十位其他科学家，包括Qadi Zada。毫无疑问，卡西是撒马尔罕领先的天文学家和数学家，同一世纪后期的一位历史学家称他为第二个克劳狄乌斯·托勒密。

卡西 用波斯文写给他住在卡尚的父亲的信件保存了下来。这些信件写于撒马尔罕，精彩地描述了那里的科学生活。1424 年，乌鲁伯格 开始在撒马尔罕建造一座天文台，尽管 卡西 的信件没有日期，但它们写于天文台开始建造的时期。其中一封信的内容直到最近才发表，见 [8]。

在信中，卡西赞扬了乌鲁伯格的数学才能，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和Qadi Zada之外的所有人来说都太难，有几次只有卡西成功了。显然，卡西是撒马尔罕乌鲁伯格最优秀的科学家和最亲密的合作者，尽管卡西不懂正确的宫廷礼仪且缺乏优雅的举止，他仍受到乌鲁伯格的高度尊重。Al-Kashi去世后，乌鲁伯格这样描述他（例如见[1]）：-

……一位杰出的科学家，世界上最著名的人物之一，他精通古代科学，为其发展做出了贡献，并能解决最困难的问题。

尽管卡西在加入撒马尔罕的乌鲁伯格之前已经做过一些出色的工作，但他最好的工作是在那座城市完成的。他在1424年7月完成了他的Treatise on the Circumference，在这部著作中，他将2π计算到九位六十进制小数，并将其转换为十六位十进制小数。这一成就远远超出了此前任何人所取得的成果，无论是古希腊人还是中国人（中国人在5世纪达到了六位小数）。将近200年后，鲁道夫·范·科伊伦才以20位小数超越了Al-Kashi的精度。

然而，Al-Kashi最令人印象深刻的数学著作是The Key to Arithmetic，他于1427年3月2日完成。这部著作是一部主要教材，旨在用于撒马尔罕的学生教学，特别是卡西试图为学习天文学、测量学、建筑学、会计学和贸易的学生提供必要的数学知识。[1]的作者们对该著作描述如下：-

就其内容的丰富性，以及在运用算术和代数方法解决各种问题（包括若干几何问题）方面，还有其表述的清晰与优雅而言，这部卷帙浩繁的教科书是整个中世纪文献中最好的之一；它既证明了作者的博学，也证明了他的教学能力。

Dold-Samplonius在[11]、[12]和[13]中讨论了卡西的Key to Arithmetic的几个方面。（另见[3]）。例如，muqarnas的测量指的是一种用于隐藏清真寺和宫殿等建筑边缘和接缝的装饰类型。这种装饰类似钟乳石，由三维多边形组成，有些具有平面，有些具有曲面。Al-Kashi在计算各类muqarnas的总表面积时使用了十进制分数。qubba是一位著名人物的陵墓穹顶。Al-Kashi找到了很好的方法来近似计算构成qubba穹顶的壳体的表面积和体积。

我们在上面提到了 卡西 对十进制分数的使用，正是通过他对这些分数的使用，他获得了相当大的名声。普遍认为 西蒙·斯蒂文 是第一个引入十进制分数的人，这一观点在 1948 年被证明是错误的，当时 P Luckey（见 [4]）表明，在 Key to Arithmetic 中，卡西 对十进制分数的描述与 西蒙·斯蒂文 一样清晰。然而，像许多数学家在 Luckey 的工作之后所做的那样，声称 卡西 是十进制分数的发明者，这与事实相去甚远，因为这个想法已经出现在 卡拉吉 学派的几位数学家的著作中，特别是 al-Samawal。

Rashed（见[5]或[6]）对卡西的重要贡献进行了恰当的评价。他表明卡西带来的主要进展是：-

(1) The analogy between both systems of fractions; the sexagesimal and the decimal systems。
(2) The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as π。

拉希德还写道（见[5]或[6]）：-

……卡西不能再被视为小数分数的发明者；尽管如此，在他的论述中，这位数学家远非简单的编纂者，而是比al-Samawal更进一步，代表了小数分数历史上的一个重要维度。

在卡西的著作中还有其他重要成果，这些成果由Luckey指出。他发现卡西有一种计算$n$次方根的算法，这是许多世纪后保罗·鲁菲尼和威廉·乔治·霍纳所给出方法的特例。在后来的著作中，拉希德表明（例如见[5]或[6]），Al-Kashi再次描述了卡拉吉学派数学家著作中存在的方法，特别是al-Samawal。

卡西的最后一部著作是The Treatise on the Chord and Sine，这部著作可能在他去世时尚未完成，之后由Qadi Zada完成。在这部著作中，卡西计算了sin 1°，其精度与他早期著作中计算π的精度相同。他还考虑了与trisecting an angle问题相关的方程，即一个三次方程。他并不是第一个研究该方程近似解的人，因为比鲁尼此前已经研究过它。然而，卡西提出的迭代方法是[1]：-

……中世纪代数的最佳成就之一。……但卡西的所有这些发现长期以来在欧洲不为人知，仅在十九世纪和二十世纪由……科学史家研究。……

让我们以对卡西在天文学方面工作的最后一点评论作为结束。我们前面提到了卡西制作的天文表Khaqani Zij。值得注意的是，乌鲁伯格也制作了天文表和正弦表，几乎可以肯定这些表是基于卡西的表，并且几乎可以肯定是在卡西的帮助下制作的。

Details of Jamshid al-Kashi's life and works are better known than many others from this period although details of his life are sketchy. One of the reasons we is that he dated many of his works with the exact date on which they were completed, another reason is that a number of letters which he wrote to his father have survived and give fascinating information.

Al-Kashi was born in Kashan which lies in a desert at the eastern foot of the Central Iranian Range. At the time that al-Kashi was growing up Timur (often known as Tamburlaine) was conquering large regions. He had proclaimed himself sovereign and restorer of the Mongol empire at Samarkand in 1370 and, in 1383, Timur began his conquests in Persia with the capture of Herat. Timur died in 1405 and his empire was divided between his two sons, one of whom was Shah Rokh.

While Timur was undertaking his military campaigns, conditions were very difficult with widespread poverty. al-Kashi lived in poverty, like so many others at this time, and devoted himself to astronomy and mathematics while moving from town to town. Conditions improved markedly when Shah Rokh took over after his father's death. He brought economic prosperity to the region and strongly supported artistic and intellectual life. With the changing atmosphere, al-Kashi's life also improved markedly. The first event in al-Kashi's life which we can date accurately is his observation of an eclipse of the moon which he made in Kashan on 2 June 1406.

It is reasonable to assume that al-Kashi remained in Kashan where he worked on astronomical texts. He was certainly in his home town on 1 March 1407 when he completed Sullam Al-sama the text of which has survived. The full title of the work means The Stairway of Heaven, on Resolution of Difficulties Met by Predecessors in the Determination of Distances and Sizes (of the heavenly bodies). At this time it was necessary for scientists to obtain patronage from their kings, princes or rulers. Al-Kashi played this card to his advantage and brought himself into favour in the new era where patronage of the arts and sciences became popular. His Compendium of the Science of Astronomy written during 1410-11 was dedicated to one of the descendants of the ruling Timurid dynasty.

Samarkand, in Uzbekistan, is one of the oldest cities of Central Asia. The city became the capital of Timur's empire and Shah Rokh made his own son, Ulugh Beg, ruler of the city. Ulugh Beg, himself a great scientist, began to build the city into a great cultural centre. It was to Ulugh Beg that Al-Kashi dedicated his important book of astronomical tables Khaqani Zij which was based on the tables of Nasir al-Tusi. In the introduction al-Kashi says that without the support of Ulugh Beg he could not have been able to complete it. In this work there are trigonometric tables giving values of the sine function to four sexagesimal digits for each degree of argument with differences to be added for each minute. There are also tables which give transformations between different coordinate systems on the celestial sphere, in particular allowing ecliptic coordinates to be transformed into equatorial coordinates. See [14] for a detailed discussion of this work.

The Khaqani Zij also contains [1]:-

... detailed tables of the longitudinal motion of the sun, the moon, and the planets. Al-Kashi also gives the tables of the longitudinal and latitudinal parallaxes for certain geographical latitudes, tables of eclipses, and tables of the visibility of the moon.

Al-Kashi had certainly found the right patron in Ulugh Beg since he founded a university for the study of theology and science at Samarkand in about 1420 and he sought out the best scientists to help with his project. Ulugh Beg invited Al-Kashi to join him at this school of learning in Samarkand, as well as around sixty other scientists including Qadi Zada. There is little doubt that al-Kashi was the leading astronomer and mathematician at Samarkand and he was called the second Ptolemy by an historian writing later in the same century.

Letters which al-Kashi wrote in Persian to his father, who lived in Kashan, have survived. These were written from Samarkand and give a wonderful description of the scientific life there. In 1424 Ulugh Beg began the construction of an observatory in Samarkand and, although the letters by al-Kashi are undated they were written at a time when construction of the observatory had begun. The contents of one of these letters has only recently been published, see [8].

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 Qadi Zada and on a couple of occasions only al-Kashi succeeded. It is clear that al-Kashi was the best scientist and closest collaborator of Ulugh Beg at Samarkand and, despite al-Kashi's ignorance of the correct court behaviour and lack of polished manners, he was highly respected by Ulugh Beg. After Al-Kashi's death, Ulugh Beg described him as (see for example [1]):-

... a remarkable scientist, one of the most famous in the world, who had a perfect command of the science of the ancients, who contributed to its development, and who could solve the most difficult problems.

Although al-Kashi had done some fine work before joining Ulugh Beg at Samarkand, his best work was done while in that city. He produced his Treatise on the Circumference in July 1424, a work in which he calculated 2π to nine sexagesimal places and translated this into sixteen decimal places. This was an achievement far beyond anything which had been obtained before, either by the ancient Greeks or by the Chinese (who achieved six decimal places in the 5th century). It would be almost 200 years before van Ceulen surpassed Al-Kashi's accuracy with 20 decimal places.

Al-Kashi's most impressive mathematical work was, however, The Key to Arithmetic which he completed on 2 March 1427. The work is a major text intended to be used in teaching students in Samarkand, in particular al-Kashi tries to give the necessary mathematics for those studying astronomy, surveying, architecture, accounting and trading. The authors of [1] describe the work as follows:-

In the richness of its contents and in the application of arithmetical and algebraic methods to the solution of various problems, including several geometric ones, and in the clarity and elegance of exposition, this voluminous textbook is one of the best in the whole of medieval literature; it attests to both the author's erudition and his pedagogical ability.

Dold-Samplonius has discussed several aspects of al-Kashi's Key to Arithmetic in [11], [12], and [13]. (see also [3]). For example the measurement of the muqarnas refers to a type of decoration used to hide the edges and joints in buildings such as mosques and palaces. The decoration resembles a stalactite and consists of three-dimensional polygons, some with plane surfaces, and some with curved surfaces. Al-Kashi uses decimal fractions in calculating the total surface area of types of muqarnas. The qubba is the dome of a funerary monument for a famous person. Al-Kashi finds good methods to approximate the surface area and the volume of the shell forming the dome of the qubba.

We mentioned above al-Kashi's use of decimal fractions and it is through his use of these that he has attained considerable fame. The generally held view that Stevin had been the first to introduce decimal fractions was shown to be false in 1948 when P Luckey (see [4]) showed that in the Key to Arithmetic al-Kashi gives as clear a description of decimal fractions as Stevin does. However, to claim that al-Kashi is the inventor of decimal fractions, as was done by many mathematicians following the work of Luckey, would be far from the truth since the idea had been present in the work of several mathematicians of al-Karaji's school, in particular al-Samawal.

Rashed (see [5] or [6]) puts al-Kashi's important contribution into perspective. He shows that the main advances brought in by al-Kashi are:-

(1) The analogy between both systems of fractions; the sexagesimal and the decimal systems.
(2) The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as π.

Rashed also writes (see [5] or [6]):-

... Al-Kashi can no longer be considered as the inventor of decimal fractions; it remains nonetheless, that in his exposition the mathematician, far from being a simple compiler, went one step beyond al-Samawal and represents an important dimension in the history of decimal fractions.

There are other major results in the work of al-Kashi which were pointed out by Luckey. He found that al-Kashi had an algorithm for calculating $n$th roots which was a special case of the methods given many centuries later by Ruffini and Horner. In later work Rashed shows (see for example [5] or [6]) that Al-Kashi was again describing methods which were present in the work of mathematicians of al-Karaji's school, in particular al-Samawal.

The last work by al-Kashi was The Treatise on the Chord and Sine which may have been unfinished at the time of his death and then completed by Qadi Zada. In this work al-Kashi computed sin 1° to the same accuracy as he had computed π in his earlier work. He also considered the equation associated with the problem of trisecting an angle, namely a cubic equation. He was not the first to look at approximate solutions to this equation since al-Biruni had worked on it earlier. However, the iterative method proposed by al-Kashi was [1]:-

... one of the best achievements in medieval algebra. ... But all these discoveries of al-Kashi's were long unknown in Europe and were studied only in the nineteenth and twentieth centuries by ... historians of science....

Let us end with one final comment on the al-Kashi's work in astronomy. We mentioned earlier the astronomical tables Khaqani Zij produced by al-Kashi. It is worth noting that Ulugh Beg also produced astronomical tables and sine tables, and it is almost certain that these tables were based on al-Kashi's tables and almost certainly produced with al-Kashi's help.