巴努·穆萨Banū Mūsā

巴纳巴努·穆萨有三个人穆萨：Jafar Muhammad ibn Musa ibn Shakir、Ahmad ibn Musa ibn Shakir和al-Hasan ibn Musa ibn Shakir。他们几乎无法区分，但我们确实知道，尽管他们经常合作，但他们确实有各自的专长领域。

上面的三个链接给出了每个穆萨的具体细节，但关于他们的大部分信息都在这个页面上。

Jafar Muhammad主要研究几何学和天文学，而Ahmad主要研究力学，al-Hasan主要研究几何学。要分别为这三兄弟写传记是相当不可能的，他们通常被称为穆萨 穆萨，我们不会尝试这样做。

穆萨是最早开始继承古希腊人开创的数学发展的一批数学家之一。因此，值得审视一下阿拉伯数学如何填补这一角色的背景。

Harun al-Rashid于786年9月14日成为阿拔斯王朝的第五任哈里发，不久之后，穆萨的父亲穆萨 ibn Shakir出生。Harun从首都巴格达的宫廷统治着从地中海延伸到印度的伊斯兰帝国。他将文化带入宫廷，并试图建立当时在阿拉伯世界尚未繁荣的智力学科。这种变化的一个例子体现在穆萨的父亲穆萨 ibn Shakir的生活中，他年轻时是个强盗，但转向科学，成为天文学方面非常精通的人。正是在al-Rashid统治期间，al-Hajjaj首次将欧几里得的Elements翻译成阿拉伯语。这是让希腊数学在伊斯兰帝国传播的最初步骤。

拉希德有两个儿子，长子是阿明，幼子是马蒙。哈伦·拉希德于809年去世，他的两个儿子之间发生了武装冲突。马蒙赢得了这场武装斗争，阿明于813年战败被杀。此后，马蒙成为哈里发，从巴格达统治帝国。甚至在此之前，穆萨·伊本·沙基尔就已成为马蒙的密友，当穆萨·伊本·沙基尔去世时，马蒙成为穆萨的监护人。穆萨在巴格达接受了最好的教育，学习几何学、力学、音乐、数学和天文学。

马蒙延续了他父亲开创的对学术的赞助，并创立了一所名为“智慧宫”的学院，在那里翻译希腊哲学和科学著作。他还建立了一个手稿图书馆，这是自亚历山大图书馆以来建立的第一个大型图书馆，从费隆收集重要著作。除了智慧宫，马蒙还建立了天文台，穆斯林天文学家可以在那里在早期民族所获得的知识基础上进一步发展。

马蒙为智慧宫招募了最有才华的人，并任命了他很快赏识其才能的穆萨。他还任命了花拉子米、肯迪和哈贾吉，后者是第一个将欧几里得的Elements翻译成阿拉伯语的人。侯奈因·伊本·伊斯哈格·伊本·侯奈因·伊本·伊斯哈格以及后来的塔比·伊本·库拉也与穆萨一起在智慧宫工作。穆罕默德成为侯奈因·伊本·伊斯哈格的密友。

833年，al-Ma'mun去世，其弟al-Mu'tasim继位。智慧宫在历任哈里发的统治下继续繁荣。Al-Mu'tasim于842年去世，al-Wathiq继位，后者又于847年由al-Mutawakkil继任哈里发。在这两位哈里发统治下，智慧宫的学者之间出现了内部争论和竞争，而穆萨肯定卷入了这场竞争。穆萨 穆萨 [4]：-

……成为著名哲学家肯迪的敌人，并导致他失去al-Mutawakkil的宠信，后者下令殴打他，并允许穆萨没收他的图书馆。

当然，此时文化和知识成就的氛围已相当成为过去，因为 al-Mutawakkil 迫害所有非正统和非穆斯林群体，同时他下令摧毁巴格达的犹太会堂和教堂。Muhammad 和 Ahmad 无疑受到 al-Mutawakkil 的青睐，他雇用他们从事与为 al-Djafariyya 新城修建运河相关的工作。

我们现在转向 穆萨 所作的重要数学贡献。正如 al-Dabbagh 在 [1] 中所写：-

穆萨 穆萨是最早研究希腊数学著作并奠定阿拉伯数学学派基础的阿拉伯科学家之一。他们可以被称为希腊数学的弟子，然而他们在某些方面偏离了古典希腊数学，这些偏离对某些数学概念的发展非常重要。

穆萨 穆萨所著研究最多的论著是Kitab marifat masakhat al-ashkal Ⓣ（《平面与球面图形测量之书》）。这部著作通过克雷莫纳的克雷莫纳的杰拉德翻译成拉丁文而广为人知，题为Liber trium fratum de geometria Ⓣ（《三位穆萨的几何书》）。该论著考虑的问题类似于阿基米德的两部文本，即On the measurement of the circle和On the sphere and the cylinder中所考虑的问题。

穆萨穆萨所使用的方法与阿基米德所使用的方法有许多相似之处。然而更重要的是，也存在许多差异，这些差异虽然乍看之下可能并不显得十分重要，却为通向数学的新方法迈出了第一步。穆萨穆萨应用了欧多克索斯发明并由阿基米德如此有效使用的穷竭法。然而，他们省略了该方法中涉及将具有$2^{k}$条边的多边形视为$k$趋于无穷大的那部分。相反，他们选择使用一个其证明本身就需要这种趋于无穷大的命题。这本身可能并不是向前迈出的一步，因为正如[2]的作者所暗示的，这可能是由于对希腊几何思维的精妙之处缺乏理解。正如穆萨穆萨所使用的那样，“穷竭法”失去了其大部分精妙和力量。

然而，在另一方面，穆萨 穆萨 确实向前迈出了明确的一步。希腊人没有把面积和体积看作数，而只是比较面积等的比。穆萨 穆萨 的数概念比希腊人的更宽广。例如，他们把 π 描述为 [2]：-

……这个量，当乘以圆的直径时，给出周长。

在文本中，面积被描述为线性量度的乘积，因此算术术语也许首次被应用于几何运算。穆萨 穆萨 还引入了涉及将几何对象视为运动的几何证明。特别是，他们使用运动学方法来解决经典的 trisecting an angle 问题。

在天文学中，穆萨 做出了许多贡献。他们受 al-Ma'mun 指示测量纬度一度，并在美索不达米亚北部的沙漠中进行了测量。他们还从巴格达对太阳和月亮进行了许多观测。Muhammad 和 Ahmad 测量了年的长度，得到了 365 天 6 小时的值。三位 穆萨 于 840-41、847-48 和 850-51 年在巴格达一座桥上的家中对轩辕十四星进行了观测。

The Bana Musa brothers were were three brothers: Jafar Muhammad ibn Musa ibn Shakir, Ahmad ibn Musa ibn Shakir and al-Hasan ibn Musa ibn Shakir. They are almost indistinguishable but we do know that although they often worked together, they did have their own areas of expertise.

The three links above give details specific to each of the brothers but most of the information about them is on this page.

Jafar Muhammad worked mainly on geometry and astronomy while Ahmad worked mainly on mechanics and al-Hasan worked mainly on geometry. It is quite impossible to write separate biographies of the three brother, who are usually known as the Banu Musa, and we shall not attempt to do so.

The Banu Musa brothers were among the first group of mathematicians to begin to carry forward the mathematical developments begun by the ancient Greeks. It is therefore worth looking at the background to how Arabic mathematics came to fill this role.

Harun al-Rashid became the fifth Caliph of the Abbasid dynasty on 14 September 786, not long after Musa ibn Shakir, the father of the Banu Musa brothers, was born. Harun ruled from his court in the capital city of Baghdad over the Islam empire which stretched from the Mediterranean to India. He brought culture to his court and tried to establish the intellectual disciplines which at that time were not flourishing in the Arabic world. An example of this change is seen in the life of Musa ibn Shakir, the father of the Banu Musa brothers, who was a robber in his youth but turned to science, becoming highly proficient in astronomy. It was during al-Rashid's reign that the first Arabic translation of Euclid's Elements into Arabic was made by al-Hajjaj. The first steps were being taken to allow Greek mathematics to spread through the Islam empire.

Al-Rashid had two sons, the eldest was al-Amin while the younger was al-Ma'mun. Harun al-Rashid died in 809 and there was an armed conflict between his two sons. Al-Ma'mun won the armed struggle and al-Amin was defeated and killed in 813. Following this, al-Ma'mun became Caliph and ruled the empire from Baghdad. Even before this time Musa ibn Shakir had become a close friend of al-Ma'mun and when Musa ibn Shakir died, al-Ma'mun became the guardian of the Banu Musa brothers. The brothers were given the best education in Baghdad, studying geometry, mechanics, music, mathematics and astronomy.

Al-Ma'mun had continued the patronage of learning started by his father and had founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated. He also built up a library of manuscripts, the first major library to be set up since that at Alexandria, collecting important works from Byzantium. In addition to the House of Wisdom, al-Ma'mun set up observatories in which Muslim astronomers could build on the knowledge acquired by earlier peoples.

Al-Ma'mun recruited the most talented men for the House of Wisdom and appointed the Banu Musa brothers whose talents he had quickly come to appreciate. He also appointed al-Khwarizmi, al-Kindi and al-Hajjaj the first translator of Euclid's Elements into Arabic. Hunayn ibn Ishaq and later Thabit ibn Qurra also worked in the House of Wisdom with the Banu Musa brothers. Muhammad became a close friend of Hunayn.

In 833 al-Ma'mun died and was succeeded by his brother al-Mu'tasim. The house of Wisdom continued to flourish under successive caliphs. Al-Mu'tasim died in 842 and was succeeded by al-Wathiq who, in turn, was succeed as Caliph in 847 by al-Mutawakkil. Under both these Caliphs internal arguments and rivalry arose between the scholars in the House of Wisdom and the Banu Musa brothers were certainly involved in this rivalry. The Banu Musa [4]:-

... became enemies of the famous philosopher al-Kindi and caused him to lose favour with al-Mutawakkil, who ordered him to be beaten and allowed the brothers to confiscate his library.

Certainly the atmosphere of culture and intellectual achievement was by now rather a thing of the past as al-Mutawakkil persecuted all non-orthodox and non-Muslim groups while he had synagogues and churches in Baghdad destroyed. Muhammad and Ahmad were certainly in favour with al-Mutawakkil who employed them in work related to the construction of canals for a new city of al-Djafariyya.

We now turn to the important mathematical contributions made by the Banu Musa brothers. As al-Dabbagh writes in [1]:-

The Banu Musa were among the first Arabic scientists to study the Greek mathematical works and to lay the foundation of the Arabic school of mathematics. They may be called disciples of Greek mathematics, yet they deviated from classical Greek mathematics in ways that were very important to the development of some mathematical concepts.

The most studied treatise written by the Banu Musa is Kitab marifat masakhat al-ashkal Ⓣ. This work became well known through the translation into Latin by Gherard of Cremona entitled Liber trium fratum de geometria Ⓣ. The treatise considers problems similar to those considered in the two texts by Archimedes, namely On the measurement of the circle and On the sphere and the cylinder.

There are many similarities in the methods employed by the Banu Musa and those employed by Archimedes. More significant, however, is the fact that there are also many differences which, although at first sight may not seem of major importance, yet were providing the first steps towards a new approach to mathematics. The Banu Musa apply the method of exhaustion invented by Eudoxus and used so effectively by Archimedes. However, they omitted that part of the method which involves considering polygons with $2^{k}$ sides as $k$ tends to infinity. Rather they chose to use a proposition which itself required this passage to infinity in its proof. This in itself may not have been a step forward for, as the author of [2] suggests, this may have been due to a lack of understanding of the finer points of Greek geometric thinking. As used by the Banu Musa the "method of exhaustion" loses most of its subtlety and power.

In another aspect, however, the Banu Musa made a definite step forward. The Greeks had not thought of areas and volumes as numbers, but had only compared ratios of areas etc. The Banu Musa's concept of number is broader than that of the Greeks. For example they describe π as [2]:-

... the magnitude which, when multiplied by the diameter of a circle, yields the circumference.

In the text areas as described as products of linear magnitudes, so the terminology of arithmetic is perhaps for the first time applied to the operations of geometry. The Banu Musa also introduce geometrical proofs which involve thinking of the geometric objects as moving. In particular they used kinematic methods to solve the classical problem of trisecting an angle.

In astronomy the brothers made many contributions. They were instructed by al-Ma'mun to measure a degree of latitude and they made their measurements in the desert in northern Mesopotamia. They also made many observations of the sun and the moon from Baghdad. Muhammad and Ahmad measured the length of the year, obtaining the value of 365 days and 6 hours. Observations of the star Regulus were made by the three brothers from their house on a bridge in Baghdad in 840-41, 847-48, and 850-51.