阿布·萨赫勒·库希Abu Sahl al-Quhi

库希的名字在英语中有两种拼写，出现频率似乎大致相同，即库希和阿布·萨赫勒·库希。

我们可以从库希的名字推断他来自塔巴里斯坦的Quh村。他是在一个新王朝正在建立并将统治伊朗的时期长大的。布韦希伊斯兰王朝在阿拉伯与土耳其征服之间的时期，从945年到1055年统治伊朗西部和伊拉克。这一时期始于945年Ahmad Buyeh占领阿拔斯王朝首都巴格达之时。布韦希王朝的鼎盛时期是'Adud ad-Dawlah从949年到983年的统治期间。他从巴格达统治整个伊朗南部以及现今伊拉克的大部分地区。作为科学和艺术的伟大赞助人，'Adud ad-Dawlah在巴格达宫廷中供养了若干数学家，包括库希、阿布·瓦法和阿布·赛义德·西杰兹。

969年，库希、阿布·赛义德·西杰兹和其他科学家在设拉子对冬夏solstices进行了观测。这些冬夏至点的观测是在969/970年间由他们在设拉子完成的。

Sharaf ad-Dawlah 是 'Adud ad-Dawlah 的儿子，他于983年成为哈里发。他继续支持数学和天文学，因此库希留在巴格达的宫廷为新哈里发工作。Sharaf ad-Dawlah 要求库希对七颗行星进行观测，为此库希在巴格达宫殿的花园里建造了一座天文台。天文台内的仪器按照库希自己的设计建造，并在建筑完工后安装。Al-Quhi 被任命为天文台台长，天文台于988年6月正式启用。

许多科学家出席了启用仪式。其中一位特别值得一提，那就是著名数学家和天文学家阿布·瓦法。他也受雇于Sharaf ad-Dawlah的宫廷。另一位出席启用仪式的是Abu Ishaq al-Sabi。Al-Sabi是巴格达的一位高级官员，对数学感兴趣。我们在本文后面会提到库希与al-Sabi之间的通信。当时进行了一些精确的观测，但天文台在989年因Sharaf ad-Dawlah去世而停止工作。白益王朝此时已开始失去对帝国的控制。经济处于下行轨道，军队中的叛乱使统治者的生活变得艰难。像天文台这样的高雅文化活动被置于较低的优先地位。

我们对库希生平的描述突出了他在天文学方面的工作。然而，他在数学上更为著名，是伊斯兰世界中希腊高等几何复兴与延续的领军人物。库希研究的几何问题通常导致quadratic或三次方程。Nasir al-Din 纳西尔丁·图西描述了库希考虑的一个问题，写道（例如见[1]）：-

要构造一个球缺，其体积等于给定的球缺，其面积等于第二个球缺——这个问题与阿基米德解决的相关问题类似但更难——库希通过将等边双曲线与抛物线相交来构造两个未知长度，并严格讨论了问题可解的条件。

Al-Quhi对该问题的解答见[5]。这是一种经典风格的解答，使用了欧几里得的Elements、阿波罗尼奥斯的圆锥曲线和阿基米德的On the sphere and cylinder的结果。如果解存在，库希表明它将具有位于他所构造的特定直角双曲线上的坐标。当然，库希并没有用这些现代术语表达数学，而是用古希腊数学中通常的经典几何。接下来库希引入了“曲面锥”，经过许多推导后，表明解的坐标位于一条抛物线上。然后问题被漂亮地解决为两条曲线的交点。

在另一篇论文On the construction of an equilateral pentagon in a known square中，库希再次使用两条圆锥曲线的交点来解决标题中给出的问题，这次是两条双曲线。虽然不可能在正方形中内接正五边形，但等边五边形可以以两种方式内接。一种需要解二次方程，已由阿布·卡米勒在九世纪找到。另一种需要解四次方程，是库希提出的。这篇论文的细节见[6]（见[7]的更正和补充）和[8]。

库希还在论文On the perfect compass[1]中描述了一种圆锥曲线规，一种一条腿长度可变的圆规，用于绘制圆锥曲线：-

……他首先描述了用这种圆规作直线、圆和圆锥曲线的方法，然后论述了其理论。他得出结论，现在人们可以很容易地构造astrolabes、日晷和类似的仪器。

库希确实在On the construction of the astrolabe中考虑了构造星盘的问题。星盘是一种用于观测高度的仪器，它提供了一种机械手段，在赤道坐标系和基于地平线的坐标系之间转换天球坐标。这篇论著分为两卷，第一卷分为四章，第二卷分为七章。

在这部著作中，库希解决了许多困难的映射问题。特别是，他使用一种类似于画法几何的方法，将球面上的圆映射到赤道平面上。经过操作之后，它们又以令人惊叹的视觉化方式被映射回球面上。尽管这部著作表面上具有构建星盘的实际用途，但看来库希对数学本身的兴趣要大于编写实用手册的兴趣。

最后，我们应当提到上文提及的库希与al-Sabi之间的通信。已知他们至少交换了六封信，但只有四封信的详细内容留存下来。这些内容在[3]中以阿拉伯文和英文给出。所涉及的主题相当多样，从讨论“已知”意味着什么，到解决诸如以下具体问题：假设给定一个圆和两条相交直线$l$和$m$。假设在点$T$处圆的切线与$l$相交于$L$，与$m$相交于$M$。如何选择$T$，使得$TL : TM$等于一个给定的比？也许通信中最有趣的部分是库希给出的关于各种图形重心的六个定理。六个结果中有五个是正确的，但第六个是错误的。它声称半圆的重心将半径按3:7的比例分割。从这个错误的结果出发，库希推导出同样错误的结论，即$\pi = \large\frac{28}{9}\normalsize$。即使是最好的数学家也会犯错误！

There are two spellings of al-Quhi's name in English which seem to appear about equally often, namely al-Quhi and al-Kuhi.

We can deduce from al-Quhi's name that he came from the village of Quh in Tabaristan. He was brought up during the period that a new dynasty was being established which would rule over Iran. The Buyid Islamic dynasty ruled in western Iran and Iraq from 945 to 1055 in the period between the Arab and Turkish conquests. The period began in 945 when Ahmad Buyeh occupied the 'Abbasid capital of Baghdad. The high point of the Buyid dynasty was during the reign of 'Adud ad-Dawlah from 949 to 983. He ruled from Baghdad over all southern Iran and most of what is now Iraq. A great patron of science and the arts, 'Adud ad-Dawlah supported a number of mathematicians at his court in Baghdad, including al-Quhi, Abu'l-Wafa and al-Sijzi.

In 969 'Adud ad-Dawlah ordered that observations be made of the winter and summer solstices in Shiraz. These observations of the winter and summer solstices were made by al-Quhi, al-Sijzi and other scientists in Shiraz during 969/970.

Sharaf ad-Dawlah was 'Adud ad-Dawlah's son and he became Caliph in 983. He continued to support mathematics and astronomy so al-Quhi remained at the court in Baghdad working for the new Caliph. Sharaf ad-Dawlah required al-Quhi to make observations of the seven planets and in order to do this al-Quhi had an observatory built in the garden of the palace in Baghdad. The instruments in the observatory were built to al-Quhi's own design and installed once the building was complete. Al-Quhi was made director of the observatory and it was officially opened in June 988.

A number of scientists were present at the opening. One in particular, the famous mathematician and astronomer Abu'l-Wafa, is worthy of mention. He was also employed at the court of Sharaf ad-Dawlah. Another who was present at the opening was Abu Ishaq al-Sabi. Al-Sabi was a high ranking official in Baghdad who was interested in mathematics. We mention later in this article correspondence between al-Quhi and al-Sabi. Some accurate observations were made but the observatory ceased work in 989 on the death of Sharaf ad-Dawlah. The Buyid dynasty was by this stage beginning to lose control of the empire. The economy was on a downward path, and rebellions in the army made the ruler's life difficult. Fine cultural activities such as an observatory took a lower priority.

Our description of al-Quhi's life has highlighted his work in astronomy. However, it is in mathematics that he is more famous, being the leading figure in a revival and continuation of Greek higher geometry in the Islamic world. The geometric problems that al-Quhi studied usually led to quadratic or cubic equations. Nasir al-Din al-Tusi described one of the problems considered by al-Quhi writing (see for example [1]):-

To construct a sphere segment equal in volume to a given sphere segment and equal in area to a second sphere segment - a problem similar to but more difficult than related problems solved by Archimedes - Al-Quhi constructed the two unknown lengths by intersecting an equilateral hyperbola with a parabola and rigorously discussed the conditions under which the problem is soluble.

Al-Quhi's solution to the problem is given in [5]. It is a classical style of solution using results from Euclid's Elements, Apollonius's Conics and Archimedes' On the sphere and cylinder. If a solution exists, al-Quhi showed that it will have coordinates which lie on a particular rectangular hyperbola that he has constructed. Of course, al-Quhi does not express the mathematics in these modern terms but rather in the usual classical geometry of ancient Greek mathematics. Next al-Quhi introduces the "cone of the surface" which, after many deductions, leads to showing that the solution has coordinates lying on a parabola. The problem is then beautifully solved as the intersection of the two curves.

In another treatise On the construction of an equilateral pentagon in a known square al-Quhi solves the problem given in the title again using the intersection of two conic sections, this time two hyperbolas. Although it is impossible to inscribe a regular pentagon in a square, an equilateral pentagon can be inscribed in two ways. One, which requires the solution of a quadratic equation, had been found by Abu Kamil in the ninth century. The other, which requires the solution of a quartic equation, is the one presented by al-Quhi. Details of this treatise are given in [6] (see the corrections and additions of [7]), and [8].

Al-Quhi also described a conic compass, a compass with one leg of variable length, for drawing conic sections in the treatise On the perfect compass [1]:-

... he first described the method of constructing straight lines, circles, and conic sections with this compass, and then treated the theory. He concluded that one could now easily construct astrolabes, sundials and similar instruments.

Indeed al-Quhi did consider the problem of constructing astrolabes in On the construction of the astrolabe. The astrolabe was an instrument used to observe altitudes, and it provided a mechanical means to transform celestial coordinates between an equatorial system and one based on the horizon. This treatise is in two Books, the first being divided into four chapters, the second book into seven chapters.

There are a number of difficult mapping problems solved by al-Quhi in this work. In particular, using a method resembling descriptive geometry, he maps circles on the sphere into the equatorial plane. After manipulation, they are mapped back again onto the sphere in a remarkable piece of visualisation. Despite the appearance of the work being of practical use in constructing an astrolabe, it would appear that al-Quhi was more interested in the mathematics for its own sake than he was in giving a practical manual.

Finally we should mention the correspondence between al-Quhi and al-Sabi which we mentioned above. It is known that there were at least six letters exchanged but only details of four survive. These are given in both Arabic and English in [3]. Topics covered are quite varied, ranging from a discussion of what "known" means to solutions of specific problems such as the following Suppose we are given a circle and two intersecting straight lines $l$ and $m$. Suppose the tangent to the circle at a point $T$ meets $l$ at $L$ and $m$ at $M$. How can one choose $T$ so that $TL : TM$ is equal to a given ratio? Perhaps the most interesting parts of the correspondence are six theorems given by al-Quhi concerning the centres of gravity of various figures. Five of the six results are correct but the sixth is false. It states that the centre of gravity of a semicircle divides the radius in the ratio 3 : 7. From this false result al-Quhi deduces the equally false result that $\pi = \large\frac{28}{9}\normalsize$. Even the best mathematicians can make mistakes!