比鲁尼Al-Biruni

比鲁尼拉伊汉比鲁尼出生于花剌子模，这是毗邻咸海的一个地区，今称卡拉卡尔帕克斯坦。该地区的两座主要城市是卡特和朱尔贾尼耶。比鲁尼出生在卡特附近，他出生的小镇今天以这位伟大学者的名字命名为比鲁尼。他在成长过程中既住在卡特也住在朱尔贾尼耶，我们知道他很早就开始在著名天文学家和数学家阿布·纳斯尔·曼苏尔门下学习。到十七岁时，比鲁尼肯定已在从事严肃的科学工作，因为正是在990年，他通过观测太阳的最大高度计算出了卡特的纬度。

比鲁尼年轻时从事的其他工作则更具理论性。在995年之前（当时他22岁），他已撰写了许多短篇著作。其中流传下来的一部是他的Cartography，这是一部关于地图投影的著作。除了描述他自己把半球投影到平面上的方法外，比鲁尼还表明，到22岁时他已经博览群书，因为他研究了他人发明的各种地图投影，并在该论著中加以讨论。比鲁尼至此所过的相对平静的生活，即将突然结束。不妨设想一下，若不是995年的政治事件迫使他的人生发生改变，他的人生以及对学术的贡献会多么不同，这是很有意思的。

10世纪末和11世纪初是伊斯兰世界极度动荡的时期，比鲁尼所生活的地区发生了内战。花剌子模当时是萨曼帝国的一部分，该帝国以布哈拉为统治中心。该地区的其他国家有齐亚尔王朝，其首都位于里海沿岸的Gurgan。再往西，布韦希王朝统治着里海与波斯湾之间的地区以及美索不达米亚。另一个影响力迅速上升的王国是伽色尼王朝，其首都位于阿富汗的Ghazna，这个王国将在比鲁尼的一生中扮演重要角色。

比鲁尼 Nasr 阿布·纳斯尔·曼苏尔 是花剌子模地区的统治者，也是 比鲁尼 的老师，他是该家族的一位王子。995年，Banu Iraq 的统治在一场政变中被推翻。比鲁尼 在内战爆发时逃离，但他的老师 阿布·纳斯尔·曼苏尔 在这一阶段发生了什么则不太清楚。后来 比鲁尼 描述这些事件时写道[1]：-

在我刚刚安顿下来几年之后，时代之主允许我回到家乡，但我被迫参与世俗事务，这引起了愚人的嫉妒，却让智者对我感到怜悯。

比鲁尼 从花剌子模逃离后究竟去了哪里并不清楚。他此时可能去了 Rayy（靠近今天德黑兰城所在的地方），但可以肯定的是，在接下来的几年中他曾在那里待过。他写道，在 Rayy 时他没有赞助人，生活贫困。al-Khujandi 是一位天文学家，他使用自己建在 Rayy 上方山上的一个非常大的仪器来观测太阳在 solstices 附近的子午 中天。他在994年6月16日和17日观测夏至，在994年12月14日和17日观测冬至。根据这些数值，他计算了 黄赤交角 以及 Rayy 的纬度，但两者都不特别准确。

Al-Khujandi与比鲁尼讨论了这些观测和他的大型六分仪，后者后来在其Tahdid中报告了这些观测，并声称由于仪器的重量，六分仪的孔径在al-Khujandi的观测过程中移动了约一拃。比鲁尼几乎可以肯定正确地指出了al-Khujandi错误的根源。由于al-Khujandi于1000年去世，我们可以相当确定比鲁尼在995年至997年间的部分时间是在Rayy度过的。他也一定在这段时间的一部分待在吉兰，吉兰北临里海，因为大约此时他将一部著作献给了吉兰的统治者ibn Rustam，后者与齐亚尔王朝有联系。

我们确切知道比鲁尼生平中的某些日期，因为他在著作中描述了天文事件，这使得准确的日期和地点得以确定。他对997年5月24日月食的描述——他是在Kath观测到的——意味着他此时已经回到了祖国。这次月食也是在巴格达可见的事件，比鲁尼已安排与阿布·瓦法在那里观测。比较他们的计时使他们能够计算出两城之间的经度差。我们知道比鲁尼在此期间频繁迁移，因为到1000年时他在Gurgan，受到齐亚尔王朝统治者Qabus的支持。他大约在1000年将其著作Chronology献给Qabus，并且在1003年2月19日和1003年8月14日他仍在那里观测月食。我们应当记录，在Chronology中，比鲁尼提到了他此前所写的七部著作：一部关于十进制，一部关于星盘，一部关于天文观测，三部关于占星术，两部关于历史。

到1004年6月4日，比鲁尼已回到祖国，因为那天他在Jurjaniyya观测到了另一次月食。Ali ibn Ma'mun曾统治花剌子模，当他的兄弟比鲁尼'l Abbas Ma'mun继任统治者时，他仍留在宫廷。Ma'mun两兄弟都娶了来自加兹尼强大国家的统治者Mahmud的姐妹，该国最终将控制比鲁尼'l Abbas Ma'mun的王国。

Ali ibn Ma'mun和比鲁尼'l Abbas Ma'mun都是科学的赞助人，并在他们的宫廷中支持了多位顶尖科学家。到1004年，比鲁尼'l Abbas Ma'mun成为统治者，他为比鲁尼的科学工作提供了慷慨的支持。不仅比鲁尼在那里工作，他的前老师阿布·纳斯尔·曼苏尔也在那里工作，使两人得以重新合作。在比鲁尼'l Abbas Ma'mun的支持下，比鲁尼在Jurjaniyya建造了一台仪器来观测太阳子午线过境，并在1016年6月7日至1016年12月7日期间用该仪器进行了15次这样的观测。

该地区的战争扰乱了比鲁尼和阿布·纳斯尔·曼苏尔的科学工作，最终两人在大约1017年都离开了花剌子模。Mahmud正从他在加兹尼的基地扩大对该地区的影响力，并在1014年要求比鲁尼'l Abbas Ma'mun将他的名字插入周五祈祷中。这是一个信号，表明他想结束Ma'mun的统治，并试图让该地区归他控制。在Ma'mun至少部分同意了Mahmud的要求后，他被自己的军队杀害，因为他们认为这是背叛行为。此后，Mahmud率军进入该地区，并于1017年7月3日控制了Kath。比鲁尼和阿布·纳斯尔·曼苏尔都随胜利的Mahmud离开，也许是作为他的囚犯。

接下来是一段奇怪的时期，比鲁尼 自己的著作中有证据表明他遭受了极大的困苦，但他似乎也曾为一些科学工作得到 Mahmud 的支持。尽管 比鲁尼 从这位统治者那里得到的赞助有限，但一些关于 Mahmud 对 比鲁尼 残酷的报道可能仍有一定依据。这一时期的一些日期和地点又可以再次从 比鲁尼 记录的天文事件描述中推断出来。1018年10月14日他在 Kabul，尽管没有仪器可供观测，他还是用自己就地取材制作的一件巧妙仪器进行了观测。1019年4月8日，在 Kabul 以北的 Lamghan，他观测到一次日食，并写道[2]：-

……日出时我们看到大约三分之一的太阳被食，并且日食正在减弱。

在1018年至1020年间，在 Mahmud 的支持下，比鲁尼 从 Ghazna 进行了观测，从而能够准确确定其纬度。1019年9月17日，比鲁尼 从 Ghazna 观测到一次月食，并且[2]：-

他给出了在初切时刻各颗著名恒星精确高度的详细数据。

Mahmud与比鲁尼之间的关系很有意思。很可能比鲁尼实质上是Mahmud的囚徒，不能自由离开。然而Mahmud对印度的军事远征意味着比鲁尼被带到了那个国家，而比鲁尼能享有的经历恐怕很少有比这更令人愉快的了。他也许希望从Mahmud那里得到更好的待遇，但比鲁尼的科学工作无疑从中受益。大约从1022年起，Mahmud的军队开始成功控制印度北部地区，1026年他的军队推进到了印度洋。比鲁尼似乎只到过印度北部，我们不确定他去过多少次，但他在那里所做的观测使他能够确定旁遮普周边以及克什米尔边界十一个城镇的纬度。他最著名的著作India正是他在那个国家进行研究工作的直接成果。

India是一部巨著，涵盖该国的许多不同方面。比鲁尼描述了印度的宗教和哲学、种姓制度和婚姻习俗。然后他研究了印度的文字和数字系统，接着考察该国的地理。该书还考察了印度天文学、占星术和历法。

比鲁尼研读了印度文献原文，将若干梵文文本翻译成阿拉伯文。他还写了几篇专论，论述他特别感兴趣的印度天文学和数学的某些方面。比鲁尼博览群书，令人惊叹，通晓梵文文献中诸如占星术、天文学、年代学、地理学、语法、数学、医学、哲学、宗教以及度量衡等主题。更多细节见[65]。

马哈茂德于1030年去世，其长子马斯乌德继位，不过在此之前经历了一段艰难的政治局面，马哈茂德的两个儿子都试图继承父亲成为统治者。显然比鲁尼不确定谁会继位，因为他选择不在此时出版的India中题献。不题献总比选错人好！马斯乌德被证明是一位比其父对待比鲁尼更和善的统治者。如果比鲁尼此前实际上是囚徒，那么他现在似乎可以自由随意旅行了。马斯乌德于1040年被谋杀，其子毛杜德继位，统治了八年。此时比鲁尼已是老人，但他继续大量产出科学著作，直到去世。

比鲁尼一生中完成的著作总数令人惊叹。Kennedy在[1]中估计他写了大约146部著作，总计约13,000对开页（一对开页的篇幅大致相当于现代书籍的一印刷页）。我们在上文已提到其中一些著作，但比鲁尼著作的范围基本上涵盖了他那个时代的全部科学。Kennedy写道[1]：-

……他的志趣强烈地倾向于研究自然界和人类中可观察的现象。在各门科学之中，他被那些当时能够接受数学分析的领域所吸引。

我们在上文多次提到比鲁尼的天文观测。值得注意的是，他对误差的把握比克劳狄乌斯·托勒密更好。在[66]中，作者评论说克劳狄乌斯·托勒密的态度是挑选他认为最可靠的观测（这往往意味着与他的理论相合），而不告诉读者他正在舍弃哪些观测。另一方面，比鲁尼以更科学的方式处理误差，当他确实选择某些观测比其他的更可靠时，他也会给出被舍弃的观测。他还非常注意计算中的舍入误差，并且总是试图观测那些为得出答案所需操作最少的量。

在比鲁尼众多著作中最重要的之一是他被认为写于1021年左右的Shadows。Rosenfel'd对比鲁尼的这部著作写了大量文章（例如见[52]、[55]和[59]）。该著作的内容包括阴影和影子的阿拉伯语命名法、涉及影子的奇异现象、日晷术、正切和正割函数的历史、影子函数在星盘和其他仪器上的应用、用于解决各种天文问题的影子观测，以及由影子确定的穆斯林祈祷时间。Shadows是我们了解数学史、天文学史和物理学史的一个极其重要的资料来源。它还包含一些重要的思想，例如加速度与非匀速运动有关的思想、用三个直角坐标来定义三维空间中的点，以及一些人认为预示了极坐标引入的思想。

《5》一书详述了比鲁尼的数学贡献。这些贡献包括：理论与实用算术、级数求和、组合分析、三数法则、无理数数、比的理论、代数定义、解代数方程的方法、几何学、阿基米德的定理、trisection of the angle以及其他仅靠尺规无法解决的问题、conic sections、立体几何、球极投影、三角学、平面中的正弦定理，以及解球面三角形。

比鲁尼也对大地测量学和地理学做出了重要贡献。他引入了利用三角测量来测量地球及其上距离的技术。他求得地球半径为6339.6公里，这一数值直到16世纪才在西方获得（见[50]）。他的Masudic canon包含一张给出六百个地点坐标的表，其中几乎全部都是他直接了解的。然而，并非所有地点都是由比鲁尼本人测量的，有些取自花拉子米给出的一张类似的表。[27]的作者指出，比鲁尼似乎意识到，对于花拉子米和克劳狄乌斯·托勒密都给出的地点，由花拉子米得到的值更为准确。

比鲁尼还写了一篇关于计时的论著，写了几篇关于星盘的论著，并描述了一种机械历法。他对光速作了有趣的观察，指出光速与声速相比是巨大的。他还将银河描述为

……无数性质如星云般的恒星碎片的集合。

比鲁尼研究过的物理学主题包括流体静力学，并非常精确地测量了比重。他描述了金、汞、铅、银、青铜、铜、黄铜、铁和锡的密度之间的比值。比鲁尼将结果表示为整数与形如$\large\frac{1}{n}\normalsize , n = 2, 3, 4, ... , 10$的数的组合。

比鲁尼的许多想法是在与其他学者的讨论和争论中形成的。他与他的老师比鲁尼 Nasr 阿布·纳斯尔·曼苏尔有着长期的合作，彼此要求对方承担特定的工作来支持自己的研究。他与伊本·西那就热和光的本质进行了通信，语气相当对抗。在[4]中，给出了伊本·西那为回答比鲁尼提出的问题而寄给他的十八封信。这些信件涵盖哲学、天文学和物理学等主题。比鲁尼还与阿布·赛义德·西杰兹通信。论文[10]包含一封比鲁尼写给阿布·赛义德·西杰兹的信（在[63]中译为英文），其中包含正弦定理的平面和球面版本的证明。比鲁尼说这些证明归功于他的老师阿布·纳斯尔·曼苏尔。

最后，我们应当略微谈谈这位伟大学者的个性。与许多其他人的著作不同，我们从他的著作中能了解到大量关于比鲁尼的情况。尽管他的著作只有不超过五分之一留存下来，我们仍能清晰地看到这位伟大科学家的形象。我们看到一个人，他并非数学或其他领域原创理论的伟大创新者，而是一位细心的观察者，是实验方法的主要倡导者。他是一位伟大的语言学家，能够直接阅读当时存在的数量惊人的大量论著，并且他清楚地将科学的发展视为历史进程的一部分，他总是谨慎地将其置于恰当的语境中。因此，他的著作对科学史家具有极大的价值。

显然，尽管比鲁尼写了许多占星术著作，他并不相信这门“科学”，而是把它用作支持其严肃科学工作的一种手段。作为一名虔诚的穆斯林，他确实撰写了迎合其赞助人特定教派的宗教文本。他对不同宗教教派或种族没有偏见，但他对他们所犯的各种行为有严厉的言辞。例如，花剌子模的阿拉伯征服者毁掉了古代文本——对于像比鲁尼这样献身于学术和历史的学者来说，还有什么罪过比这更严重呢。关于基督教信仰，比鲁尼思考了宽恕的教义，在India[1]中写道：-

以我的生命起誓，这是一种高尚的哲学，但这个世界的人并不都是哲学家。……事实上，自从胜利者君士坦丁成为基督徒以来，刀剑和鞭子就一直被使用着。

为了表明他对那些他视为愚蠢之人所使用的讽刺，我们给出他对一位宗教人士的答复，这位宗教人士反对比鲁尼向他展示的一件用于确定祈祷时间的仪器上刻有拜占庭月份。比鲁尼在Shadows中记载，他对他说：-

拜占庭人也吃食物。那么在这件事上不要模仿他们！

Abu Rayhan al-Biruni was born in Khwarazm, a region adjoining the Aral Sea now known as Karakalpakstan. The two major cities in this region were Kath and Jurjaniyya. Al-Biruni was born near Kath and the town were he was born is today called Biruni after the great scholar. He lived both in Kath and in Jurjaniyya as he grew up and we know that he began studies at a very early age under the famous astronomer and mathematician Abu Nasr Mansur. Certainly by the age of seventeen al-Biruni was engaged in serious scientific work for it was in 990 that he computed the latitude of Kath by observing the maximum altitude of the sun.

Other work which al-Biruni undertook as a young man was more theoretical. Before 995 (when he was 22 years old) he had written a number of short works. One which has survived is his Cartography which is a work on map projections. As well as describing his own projection of a hemisphere onto a plane, al-Biruni showed that by the age of 22 he was already extremely well read for he had studied a wide selection of map projections invented by others and he discusses them in the treatise. The comparatively quiet life that al-Biruni led up to this point was to come to a sudden end. It is interesting to speculate on how different his life, and contribution to scholarship, might have been but for the change in his life forced by the political events of 995.

The end of the 10th century and beginning of the 11th century was a period of great unrest in the Islamic world and there were civil wars in the region in which al-Biruni was living. Khwarazm was at this time part of the Samanid Empire which ruled from Bukhara. Other states in the region were the Ziyarid state with its capital at Gurgan on the Caspian sea. Further west, the Buwayhid dynasty ruled over the area between the Caspian sea and the Persian Gulf, and over Mesopotamia. Another kingdom which was rapidly rising in influence was the Ghaznavids whose capital was at Ghazna in Afghanistan, a kingdom which was to play a major role in al-Biruni's life.

The Banu Iraq were the rulers of the Khwarazm region and Abu Nasr Mansur, al-Biruni's teacher, was a prince of that family. In 995 the rule by the Banu Iraq was overthrown in a coup. Al-Biruni fled at the outbreak of the civil war but it is less clear what happened to his teacher Abu Nasr Mansur at this stage. Describing these events later al-Biruni wrote [1]:-

After I had barely settled down for a few years, I was permitted by the Lord of Time to go back home, but I was compelled to participate in worldly affairs, which excited the envy of fools, but which made the wise pity me.

Exactly where al-Biruni went when he fled from Khwarazm is unclear. He might have gone to Rayy (near to where the city of Tehran stands today) at this time, but certainly he was there at some time during the following few years. He writes that he was without a patron when in Rayy, and lived in poverty. al-Khujandi was an astronomer who was working with a very large instrument he had built on the mountain above Rayy to observe meridian transits of the sun near the solstices. He made observations on 16 and 17 June 994 for the summer solstice and 14 and 17 December 994 for the winter solstice. From these values he calculated the obliquity of the ecliptic, and the latitude of Rayy but neither are particularly accurate.

Al-Khujandi discussed these observations, and his large sextant, with al-Biruni who later reported on them in his Tahdid where he claimed that the aperture of the sextant settled by about one span in the course of al-Khujandi's observations due to the weight of the instrument. Al-Biruni is almost certainly correct in pinpointing the cause of al-Khujandi's errors. Since al-Khujandi died in 1000, we can be fairly certain that al-Biruni spent part of the time between 995 and 997 at Rayy. He must also have spent part of this time in Gilan, which is bordered by the Caspian Sea on the north, for around this time he dedicated a work to the ruler of Gilan, ibn Rustam, who had connections with the Ziyarid state.

We know certain dates in al-Biruni's life with certainty for he describes astronomical events in his works which allow accurate dates and places to be determined. His description of an eclipse of the moon on 24 May 997 which he observed at Kath means that he had returned to his native country by this time. The eclipse was an event that was also visible in Baghdad and al-Biruni had arranged with Abu'l-Wafa to observe it there. Comparing their timings enabled them to calculate the difference in longitude between the cities. We know that al-Biruni moved around frequently during this period for by 1000 he was at Gurgan being supported by Qabus, the ruler of the Ziyarid state. He dedicated his work Chronology to Qabus around 1000 and he was still in Gurgan on 19 February 1003 and 14 August 1003 when he observed eclipses of the moon there. We should record that in the Chronology al-Biruni refers to seven earlier works which he had written: one on the decimal system, one on the astrolabe, one on astronomical observations, three on astrology, and two on history.

By 4 June 1004 al-Biruni was back in his homeland, for on that day he observed another eclipse of the moon from Jurjaniyya. Ali ibn Ma'mun had ruled over Khwarazm and he remained at the court when his brother Abu'l Abbas Ma'mun succeeded him as ruler. Both the Ma'mun brothers married sisters of the ruler Mahmud from the powerful state at Ghazna which would eventually take control of Abu'l Abbas Ma'mun's kingdom.

Both Ali ibn Ma'mun and Abu'l Abbas Ma'mun were patrons of the sciences and supported a number of top scientists at their court. By 1004 Abu'l Abbas Ma'mun was ruler and he provided generous support for al-Biruni's scientific work. Not only did al-Biruni work there but Abu Nasr Mansur, his former teacher also worked there, allowing the pair to renew their collaboration. With Abu'l Abbas Ma'mun's support al-Biruni built an instrument at Jurjaniyya to observe solar meridian transits and he made 15 such observations with the instrument between 7 June 1016 and 7 December 1016.

Wars in the region were to disrupt the scientific work of al-Biruni and Abu Nasr Mansur and eventually both left Khwarazm in about 1017. Mahmud was extending his influence over the region from his base in Ghazna and made a demand of Abu'l Abbas Ma'mun in 1014 to have his name inserted into the Friday prayers. This was a signal that he wanted an end to Ma'mun's rule and he was making a bid for the region to come under his control. After Ma'mun had at least partially agreed to Mahmud's demands, he was killed by his own army for what they considered to be an act of treachery. Following this Mahmud marched his army into the region and gained control of Kath on 3 July 1017. Both al-Biruni and Abu Nasr Mansur left with the victorious Mahmud, perhaps as his prisoners.

There follows a strange period during which there is evidence in al-Biruni's own writings that he suffered great hardships but he also seems to have been supported by Mahmud for some scientific work. Some reports that Mahmud was cruel to al-Biruni may have some basis despite the limited patronage al-Biruni received from the ruler. Some dates and places from this period can again be deduced from descriptions of astronomical events recorded by al-Biruni. He was in Kabul on 14 October 1018 but, despite having no instruments with which to observe, he was able to make an observation with an ingenious instrument he made from materials at hand. At Lamghan, north of Kabul, on 8 April 1019 he observed an eclipse of the sun, writing [2]:-

... at sunrise we saw that approximately one-third of the sun was eclipsed and that the eclipse was waning.

Between 1018 and 1020, supported by Mahmud, al-Biruni made observations from Ghazna which allowed an accurate determination of its latitude. On 17 September 1019 there was a lunar eclipse observed by al-Biruni from Ghazna and [2]:-

He gives precise details of the exact altitude of various well known stars at the moment of first contact.

The relationship between Mahmud and al-Biruni is interesting. It is likely that al-Biruni was essentially a prisoner of Mahmud and was not free to leave. However Mahmud's military excursions into India meant that al-Biruni was taken to that country, and there can have been few experiences that al-Biruni would have enjoyed more. He may have wished for better treatment from Mahmud but al-Biruni's scientific work certainly benefited. From around 1022 Mahmud's armies began to have success in taking control of the northern parts of India and in 1026 his armies marched to the Indian Ocean. Al-Biruni seems only to have been in the northern parts of India, and we are uncertain how many visits he made, but observations he made there enabled him to determine the latitudes of eleven towns around the Punjab and the borders of Kashmir. His most famous work India was written as a direct result of the studies he made while in that country.

The India is a massive work covering many different aspects of the country. Al-Biruni describes the religion and philosophy of India, its caste system and marriage customs. He then studies the Indian systems of writing and numbers before going on to examine the geography of the country. The book also examines Indian astronomy, astrology and the calendar.

Al-Biruni studied Indian literature in the original, translating several Sanskrit texts into Arabic. He also wrote several treatises devoted to certain aspects of Indian astronomy and mathematics which were of particular interest to him. Al-Biruni was amazingly well read, having knowledge of Sanskrit literature on topics such as astrology, astronomy, chronology, geography, grammar, mathematics, medicine, philosophy, religion, and weights and measures. See [65] for further details.

Mahmud died in 1030 and he was succeeded by his eldest son Mas'ud, although not before a difficult political situation in which the two sons of Mahmud each tried to follow their father as ruler. Clearly al-Biruni was unsure who would succeed for he chose not to give a dedication in his India which appeared at this time. Better to have no dedication than to choose the wrong one! Mas'ud proved to be a ruler who treated al-Biruni more kindly than his father had done. If al-Biruni had been a virtual prisoner before, he now seems to have become free to travel as he pleased. Mas'ud was murdered in 1040 and succeeded by his son Mawdud who ruled for eight years. By this time al-Biruni was an old man but he continued his enormous output of scientific works right up to the time of his death.

The total number of works produced by al-Biruni during his lifetime is impressive. Kennedy. writing in [1], estimates that he wrote around 146 works with a total of about 13,000 folios (a folio contains about the same amount as a printed page from a modern book). We have mentioned some of the works above, but the range of al-Biruni's works cover essentially the whole of science at his time. Kennedy writes [1]:-

... his bent was strongly towards the study of observable phenomena, in nature and in man. Within the sciences themselves he was attracted by those fields then susceptible of mathematical analysis.

We have mentioned al-Biruni's astronomical observations many time above. It is worth noting that he had a better feel for errors than did Ptolemy. In [66] the author comments that Ptolemy's attitude was to select the observations which he thought most reliable (often that meant fitting in with his theory), and not to tell the reader about observations that he was discarding. Al-Biruni, on the other hand, treats errors more scientifically and when he does chose some to be more reliable than others, he also gives the discarded observations. He was also very conscious of rounding errors in calculations, and always attempted to observe quantities which required the minimum manipulation to produce answers.

One of the most important of al-Biruni's many texts is Shadows which he is thought to have written around 1021. Rosenfel'd has written extensively on this work of al-Biruni (see for example [52], [55], and [59]). The contents of the work include the Arabic nomenclature of shade and shadows, strange phenomena involving shadows, gnomonics, the history of the tangent and secant functions, applications of the shadow functions to the astrolabe and to other instruments, shadow observations for the solution of various astronomical problems, and the shadow-determined times of Muslim prayers. Shadows is an extremely important source for our knowledge of the history of mathematics, astronomy, and physics. It also contains important ideas such as the idea that acceleration is connected with non-uniform motion, using three rectangular coordinates to define a point in 3-space, and ideas that some see as anticipating the introduction of polar coordinates.

The book [5] details the mathematical contributions of al-Biruni. These include: theoretical and practical arithmetic, summation of series, combinatorial analysis, the rule of three, irrational numbers, ratio theory, algebraic definitions, method of solving algebraic equations, geometry, Archimedes' theorems, trisection of the angle and other problems which cannot be solved with ruler and compass alone, conic sections, stereometry, stereographic projection, trigonometry, the sine theorem in the plane, and solving spherical triangles.

Important contributions to geodesy and geography were also made by al-Biruni. He introduced techniques to measure the earth and distances on it using triangulation. He found the radius of the earth to be 6339.6 km, a value not obtained in the West until the 16th century (see [50]). His Masudic canon contains a table giving the coordinates of six hundred places, almost all of which he had direct knowledge. Not all, however, were measured by al-Biruni himself, some being taken from a similar table given by al-Khwarizmi. The author of [27] remarks that al-Biruni seemed to realise that for places given by both al-Khwarizmi and Ptolemy, the value obtained by al-Khwarizmi is the more accurate.

Al-Biruni also wrote a treatise on time-keeping, wrote several treatises on the astrolabe and describes a mechanical calendar. He makes interesting observations on the velocity of light, stating that its velocity is immense compared with that of sound. He also describes the Milky Way as

... a collection of countless fragments of the nature of nebulous stars.

Topics in physics that were studied by al-Biruni included hydrostatics and made very accurate measurements of specific weights. He described the ratios between the densities of gold, mercury, lead, silver, bronze, copper, brass, iron, and tin. Al-Biruni displayed the results as combinations of integers and numbers of the form $\large\frac{1}{n}\normalsize , n = 2, 3, 4, ... , 10$.

Many of al-Biruni's ideas were worked out in discussions and arguments with other scholars. He had a long-standing collaboration with his teacher Abu Nasr Mansur, each asking the other to undertake specific pieces of work to support their own. He corresponded with Avicenna, in a rather confrontational fashion, about the nature of heat and light. In [4], eighteen letters which Avicenna sent to al-Biruni in answer to questions that he had posed are given. These letters cover topics such as philosophy, astronomy and physics. Al-Biruni also corresponded with al-Sijzi. The paper [10] contains a letter that al-Biruni wrote to al-Sijzi (translated into English in [63]) which contains proofs of both the plane and spherical versions of the sine theorem. Al-Biruni says were due to his teacher Abu Nasr Mansur.

Finally we should say a little about the personality of this great scholar. In contrast with the works of many others, we find out a lot about al-Biruni from his writings. Despite the fact that no more than one fifth of his works have survived, we get a clear picture of the great scientist. We see a man who was not a great innovator of original theories, mathematical or otherwise, but rather a careful observer who was a leading exponent of the experimental method. He was a great linguist who was able to read first hand an amazing number of the treatises that existed and he clearly saw the development of science as part of a historical process which he is always careful to put in proper context. His writings are therefore of great interest to historians of science.

It appears clear that, despite his many works on astrology, al-Biruni did not believe in the 'science' but used it as a means to support his serious scientific work. A devout Muslim, he did write religious texts to suit his patrons particular sect. He shows no prejudice against different religious sects or races, but he does have strong words to say about various acts they committed. For example the Arab conquerors of Khwarazm destroyed ancient texts - what sin could be worse than that to the scholar as dedicated to learning and history as was al-Biruni. On the Christian faith al-Biruni considered the doctrine of forgiveness, writing in India [1]:-

Upon my life, this is a noble philosophy, but the people of this world are not all philosophers. ... And indeed, ever since Constantine the Victorious became a Christian, both sword and whip have been ever employed.

An indication of the sarcasm that he employed against those he saw to be foolish we give the reply that he made to a religious man who objected to the fact that an instrument which al-Biruni was showing him to determine the time for prayers had Byzantine months engraved on it. Al-Biruni reports in Shadows that he said to him:-

The Byzantines also eat food. Then do not imitate them in this!