海什木Ibn al-Haytham

海什木有时被称为al-Basri，意思是来自伊拉克的巴士拉城，有时被称为al-Misri，意思是他来自埃及。他常被称为Alhazen，这是他名字“al-Hasan”的拉丁化版本。
特别是这个名字出现在以他最著名的问题命名中，即Alhazen问题：-

给定一个光源和一个球面镜，找到镜面上光线将反射到观察者眼睛的点。

我们将在给出一些传记细节之后讨论这个问题以及海什木的其他工作。与我们对许多阿拉伯数学家生平的缺乏了解相比，我们对海什木的生平有相当多的细节。然而，尽管这些细节彼此大体一致，但它们在几个方面相互矛盾。因此，我们必须设法确定哪些更可能是准确的。值得评论的是，海什木在1027年写的一部自传留存了下来，但它没有提及他生平的事件，而是集中讲述他的思想发展。

由于我们所知的海什木生平中的主要事件涉及他在埃及的时间，我们应该设定关于那个国家的背景。法蒂玛政治和宗教王朝得名于先知穆罕默德的女儿法蒂玛。法蒂玛王朝领导了一场致力于接管整个伊斯兰政治和宗教世界的宗教运动。因此，他们拒绝承认阿拔斯哈里发。法蒂玛哈里发在10世纪上半叶统治北非和西西里，但在多次未能征服埃及后，他们于969年开始大举进军该国，征服了尼罗河谷。他们建立了开罗城作为他们新帝国的首都。这些事件发生在海什木还是一个在巴士拉长大的小男孩的时候。

我们对海什木在巴士拉的岁月知之甚少。在他的自传中，他解释了他年轻时如何思考各种宗教运动相互冲突的宗教观点，并得出结论，它们中没有代表真理的。看来他年轻时并没有致力于数学和其他学术主题的研究，而是接受了可能最好描述为公务员工作的培训。他被任命为巴士拉及周边地区的部长。然而，海什木对他深入的宗教研究越来越不满，并决定完全致力于科学研究，他发现这在亚里士多德的著作中描述得最清楚。做出这个决定后，海什木在余生中坚持这一决定，将所有精力投入到数学、物理学和其他科学中。

海什木在决定放弃牧师职位并献身科学之后相当长一段时间才前往埃及，因为他在仍身处巴士拉时就已作为著名科学家而声名鹊起。我们确知，当伊本海什木抵达埃及时，al-Hakim是哈里发。Al-Hakim是第二位在埃及开始统治的法蒂玛哈里发；al-Aziz是第一位在埃及开始统治的法蒂玛哈里发。Al-Aziz于975年在其父al-Mu'izz去世后成为哈里发。他深度参与了叙利亚北部的军事和政治冒险，试图扩张法蒂玛帝国。在其20年统治的大部分时间里，他都为此目标而努力。Al-Aziz于996年在组织军队进军拜占庭人时去世，当时年仅十一岁的al-Hakim成为哈里发。

尽管al-Hakim是一位残忍的领袖，谋杀他的敌人，但他是科学的赞助人，雇用了顶尖科学家，如天文学家伊本伊本·优努斯。他对科学的支持可能部分源于他对占星术的兴趣。Al-Hakim非常古怪，例如他下令洗劫al-Fustat城，他下令杀死所有狗，因为它们的吠叫烦扰他，他还禁止某些蔬菜和贝类。然而，al-Hakim在他俯瞰开罗的家中保存了天文仪器，并建立了一座图书馆，其重要性仅次于150多年前的智慧宫。

我们对海什木与al-Hakim互动的了解来自多个来源，其中最重要的是al-Qifti的著作。我们得知，al-Hakim获悉海什木提出了一项调节尼罗河水流量的建议。他要求海什木来埃及实施他的建议，al-Hakim任命他领导一个工程团队来承担这项任务。然而，随着团队沿尼罗河向上游行进得越来越远，海什木意识到他用大型建筑调节水流量的想法行不通。

海什木带着他的工程团队返回，并向al-Hakim报告他们无法实现目标。Al-Hakim对伊本海什木的科学能力感到失望，任命他担任行政职务。起初伊本海什木接受了这一任命，但很快意识到al-Hakim是一个危险的人，他无法信任。看来伊本海什木假装发疯，结果被限制在家中，直到1021年al-Hakim去世后。在此期间，他从事科学工作，在al-Hakim去世后，他能够表明自己只是假装发疯。根据al-Qifti的说法，伊本海什木余生都在开罗的Azhar清真寺附近度过，撰写数学文本、教学并通过抄写文本赚钱。由于法蒂玛王朝于970年在这座清真寺的基础上创办了Al-Azhar大学，伊本海什木必定与这个学术中心有关联。

另一份报告称，在调节尼罗河的任务失败后，海什木逃离埃及前往叙利亚，并在那里度过了余生。然而这似乎不太可能，因为其他报告确实使海什木在1038年身处埃及这一点确定无疑。还有一个复杂之处是海什木在1027年写的一部著作的标题，题为Ibn al-Haytham's answer to a geometrical question addressed to him in Baghdad。有几种不同的解释可能，其中最简单的是他在返回埃及之前曾短暂访问巴格达。他也可能在叙利亚度过了一些时间，这可以部分解释故事的另一个版本。还有一个版本是海什木在仍身处巴士拉时假装发疯。

海什木的著作过于丰富，我们无法涵盖哪怕合理的数量。他似乎写了大约92部作品，其中引人注目的是，超过55部留存了下来。他写作的主要主题是光学，包括光的理论和视觉理论，天文学，以及数学，包括几何学和数论。我们将至少指出他在这些领域的贡献。

一部七卷本的光学著作Kitab al-Manazir被许多人认为是海什木最重要的贡献。它在1270年被译为拉丁文Opticae thesaurus Alhazeni。此前主要的光学著作是克劳狄乌斯·托勒密的AlmagestⓉ（主要论题：源自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语译本，后来译为拉丁语“Magna Syntaxis”），尽管海什木的著作影响不及克劳狄乌斯·托勒密的著作，但它仍必须被视为该领域的下一个重大贡献。该著作以一篇导论开始，海什木在其中说，他将开始“对原理和前提的探究”。他的方法将涉及“批判前提并在得出结论时保持谨慎”，同时他的目标是“秉持公正，不追随偏见，在我们所判断和批判的一切事情上都小心谨慎，寻求真理，而不被意见所左右”。

同样在第一卷中，海什木明确表示，他对光的研究将基于实验证据，而不是抽象理论。他指出，无论光源如何，光都是相同的，并举例说太阳光、火光或从镜子反射的光都具有相同的性质。他给出了关于视觉的第一个正确解释，表明光从物体反射进入眼睛。第一卷其余大部分内容都用于讨论眼睛的结构，但在这里他的解释必然有误，因为他没有透镜的概念，而理解眼睛功能的方式需要这一概念。然而，他对光学的研究确实促使他提出使用暗箱，并且他是第一个提到暗箱的人。

Optics的第二卷讨论视觉感知，而第三卷考察良好视觉所必需的条件以及视觉错误是如何产生的。从数学的角度看，第四卷是最重要的卷之一，因为它讨论了反射理论。海什木给出了[1]：-

……偶然光以及本质光的镜面反射的实验证明，反射定律的完整表述，以及对一种用于测量来自平面、球面、圆柱面和圆锥面镜（无论凸面还是凹面）的反射的铜制仪器的构造和使用的描述。

本文开头附近引用的Alhazen问题出现在第五卷中。尽管我们引用的是球面镜的问题，海什木也考虑了圆柱面和圆锥面镜。论文[36]详细描述了海什木在解决此问题时使用的六个几何引理。克里斯蒂安·惠更斯将问题重新表述为：-

给定与眼睛和可见物体相关的两个点，求球面镜（凸面或凹面）表面上的反射点。

克里斯蒂安·惠更斯找到了一个好的解法，Vincenzo Riccati以及后来的Saladini对其进行了简化和改进。

Optics的第六卷考察由于反射引起的视觉错误，而最后一卷，即第七卷，考察折射[1]：-

海什木并没有让人觉得他在寻找一条他未能发现的定律；但他对折射的“解释”确实构成了折射定律表述史的一部分。该解释基于这样一种观念：光是一种运动，其速度可变（在较稠密的物体中较小）……

海什木对折射的研究使他提出大气具有约15公里的有限深度。他通过太阳位于地平线以下不到19°时阳光的折射来解释曙暮光。

海什木 al-Qasim ibn Madan是一位天文学家，他向ibn 海什木提出问题，对克劳狄乌斯·托勒密关于物理现象的一些解释提出疑问。海什木写了一篇论文Solution of doubts，在其中给出了对这些问题的回答。这些问题在[43]中进行了讨论，其中问题以如下形式给出：-

我们应该如何看待克劳狄乌斯·托勒密在“Almagest”Ⓣ（主要论题：源自阿拉伯语‘al-majisti’——希腊语‘Mathematike Syntaxis’的阿拉伯语译本，后来译为拉丁语‘Magna Syntaxis’）I.3中关于天体大小（恒星及其相互距离）在地平线上视觉放大的论述？该论述显然暗示的解释是否正确，如果正确，在什么物理条件下？我们应该如何理解克劳狄乌斯·托勒密在同一处将这一天文现象与水中所见物体的视觉放大所作的类比？……

海什木与克劳狄乌斯·托勒密相关的工作中存在奇怪的对比。在Al-Shukuk ala Batlamyus（关于克劳狄乌斯·托勒密的疑问）中，海什木批评了克劳狄乌斯·托勒密的观点，然而在一部面向外行的通俗著作Configuration中，海什木完全不加质疑地接受了克劳狄乌斯·托勒密的观点。这与他在其Optics中所采取的方法非常不同，正如上面从引言中给出的引文所表明的那样。

海什木所攻克的数学问题之一是squaring the circle问题。他写了一部关于lunes面积的著作，即由两个相交圆形成的月形，（例如见[10]），随后又写了两部关于用月形化圆为方的论著中的第一部（见[14]）。然而，他似乎已经意识到自己无法解决这个问题，因为他所承诺的关于该主题的第二部论著从未问世。海什木是怀疑这个问题不可解，还是仅仅意识到自己无法解决它，这是一个永远不会有答案的有趣问题。

在数论中，海什木使用现在所谓的约翰·威尔逊定理解决了涉及同余的问题：-

若 p 是 prime，则 $1 + (p - 1) !$ 可被 p 整除。

在 Opuscula 中，海什木 考虑了一个同余方程组的解。用他自己的话来说（采用 [7] 中的译文）：-

求一个数，使得用它除以二余一；除以三余一；除以四余一；除以五余一；除以六余一；除以七则无余数。

海什木 给出了两种解法：-

这个问题是不定的，即它允许有许多解。有两种方法可以求出它们。其中一种是规范方法：我们将所提到的那些能整除所求数的数彼此相乘；在乘积上加一；这就是所求的数。

海什木 给出了一种一般解法，在特殊情形下，它给出的解是 (7 - 1)! + 1。利用 约翰·威尔逊 的定理，这个数可被 7 整除，并且显然当它被 2、3、4、5 和 6 除时都余 1。海什木 的第二种方法给出了所述类型的同余方程组的所有解（这当然是中国剩余定理的一个特殊情形）。

海什木对数论的另一贡献是他关于完全数的工作。欧几里得在Elements中已经证明：-

如果对于某个$k > 1, 2^{k} - 1$是素数，那么$2^{k-1}(2^{k} - 1)$是完全数。

这个结果的逆命题，即每个偶完全数都具有$2^{k-1}(2^{k} - 1)$的形式，其中$2^{k} - 1$是素数，是由莱昂哈德·欧拉证明的。Rashed（[7]、[8]或[27]）声称海什木是第一个陈述这个逆命题的人（尽管这一陈述并未明确出现在海什木的工作中）。Rashed考察了海什木在Analysis and synthesis中证明它的尝试，正如Rashed所指出的，这一尝试并不完全成功[7]：-

但这种部分失败不应掩盖本质：一次刻画完全数集合的有意尝试。

海什木在Analysis and synthesis中的主要目的是研究数学家用来解决问题的方法。古希腊人使用分析来解决几何问题，但ibn 海什木将其视为一种更普遍的数学方法，可以应用于其他问题，例如代数中的问题。在这部著作中，ibn 海什木意识到分析不是一种可以按照给定规则自动应用的算法，但他意识到这种方法需要直觉。更多细节见[18]和[26]。

Ibn al-Haytham is sometimes called al-Basri, meaning from the city of Basra in Iraq, and sometimes called al-Misri, meaning that he came from Egypt. He is often known as Alhazen which is the Latinised version of his first name "al-Hasan".
In particular this name occurs in the naming of the problem for which he is best remembered, namely Alhazen's problem:-

Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer.

We shall discuss this problem, and ibn al-Haytham's other work, after giving some biographical details. In contrast to our lack of knowledge of the lives of many of the Arabic mathematicians, we have quite a number of details of ibn al-Haytham's life. However, although these details are in broad agreement with each other, they do contradict each other in several ways. We must therefore try to determine which are more likely to be accurate. It is worth commenting that an autobiography written by ibn al-Haytham in 1027 survives, but it says nothing of the events his life and concentrates on his intellectual development.

Since the main events that we know of in ibn al-Haytham's life involve his time in Egypt, we should set the scene regarding that country. The Fatimid political and religious dynasty took its name from Fatimah, the daughter of the Prophet Muhammad. The Fatimids headed a religious movement dedicated to taking over the whole of the political and religious world of Islam. As a consequence they refused to recognise the 'Abbasid caliphs. The Fatimid caliphs ruled North Africa and Sicily during the first half of the 10th century, but after a number of unsuccessful attempts to defeat Egypt, they began a major advance into that country in 969 conquering the Nile Valley. They founded the city of Cairo as the capital of their new empire. These events were happening while ibn al-Haytham was a young boy growing up in Basra.

We know little of ibn al-Haytham's years in Basra. In his autobiography he explains how, as a youth, he thought about the conflicting religious views of the various religious movements and came to the conclusion that none of them represented the truth. It appears that he did not devote himself to the study of mathematics and other academic topics at a young age but trained for what might be best described as a civil service job. He was appointed as a minister for Basra and the surrounding region. However, ibn al-Haytham became increasingly unhappy with his deep studies of religion and made a decision to devote himself entirely to a study of science which he found most clearly described in the writings of Aristotle. Having made this decision, ibn al-Haytham kept to it for the rest of his life devoting all his energies to mathematics, physics, and other sciences.

Ibn al-Haytham went to Egypt some considerable time after he made the decision to give up his job as a minister and to devote himself to science, for he had made his reputation as a famous scientist while still in Basra. We do know that al-Hakim was Caliph when ibn al-Haytham reached Egypt. Al-Hakim was the second of the Fatimid caliphs to begin his reign in Egypt; al-Aziz was the first of the Fatimid caliphs to do so. Al-Aziz became Caliph in 975 on the death of his father al-Mu'izz. He was very involved in military and political ventures in northern Syria trying to expand the Fatimid empire. For most of his 20 year reign he worked towards this aim. Al-Aziz died in 996 while organising an army to march against the Byzantines and al-Hakim, who was eleven years old at the time, became Caliph.

Al-Hakim, despite being a cruel leader who murdered his enemies, was a patron of the sciences employing top quality scientists such as the astronomer ibn Yunus. His support for science may have been partly because of his interest in astrology. Al-Hakim was highly eccentric, for example he ordered the sacking of the city of al-Fustat, he ordered the killing of all dogs since their barking annoyed him, and he banned certain vegetables and shellfish. However al-Hakim kept astronomical instruments in his house overlooking Cairo and built up a library which was only second in importance to that of the House of Wisdom over 150 years earlier.

Our knowledge of ibn al-Haytham's interaction with al-Hakim comes from a number of sources, the most important of which is the writings of al-Qifti. We are told that al-Hakim learnt of a proposal by ibn al-Haytham to regulate the flow of water down the Nile. He requested that ibn al-Haytham come to Egypt to carry out his proposal and al-Hakim appointed him to head an engineering team which would undertake the task. However, as the team travelled further and further up the Nile, ibn al-Haytham realised that his idea to regulate the flow of water with large constructions would not work.

Ibn al-Haytham returned with his engineering team and reported to al-Hakim that they could not achieve their aim. Al-Hakim, disappointed with ibn al-Haytham's scientific abilities, appointed him to an administrative post. At first ibn al-Haytham accepted this but soon realised that al-Hakim was a dangerous man whom he could not trust. It appears that ibn al-Haytham pretended to be mad and as a result was confined to his house until after al-Hakim's death in 1021. During this time he undertook scientific work and after al-Hakim's death he was able to show that he had only pretended to be mad. According to al-Qifti, ibn al-Haytham lived for the rest of his life near the Azhar Mosque in Cairo writing mathematics texts, teaching and making money by copying texts. Since the Fatimids founded the University of Al-Azhar based on this mosque in 970, ibn al-Haytham must have been associated with this centre of learning.

A different report says that after failing in his mission to regulate the Nile, ibn al-Haytham fled from Egypt to Syria where he spent the rest of his life. This however seems unlikely for other reports certainly make it certain that ibn al-Haytham was in Egypt in 1038. One further complication is the title of a work ibn al-Haytham wrote in 1027 which is entitled Ibn al-Haytham's answer to a geometrical question addressed to him in Baghdad. Several different explanations are possible, the simplest of which being that he visited Baghdad for a short time before returning to Egypt. He may also have spent some time in Syria which would partly explain the other version of the story. Yet another version has ibn al-Haytham pretending to be mad while still in Basra.

Ibn al-Haytham's writings are too extensive for us to be able to cover even a reasonable amount. He seems to have written around 92 works of which, remarkably, over 55 have survived. The main topics on which he wrote were optics, including a theory of light and a theory of vision, astronomy, and mathematics, including geometry and number theory. We will give at least an indication of his contributions to these areas.

A seven volume work on optics, Kitab al-Manazir, is considered by many to be ibn al-Haytham's most important contribution. It was translated into Latin as Opticae thesaurus Alhazeni in 1270. The previous major work on optics had been Ptolemy's Almagest Ⓣ and although ibn al-Haytham's work did not have an influence to equal that of Ptolemy's, nevertheless it must be regarded as the next major contribution to the field. The work begins with an introduction in which ibn al-Haytham says that he will begin "the inquiry into the principles and premises". His methods will involve "criticising premises and exercising caution in drawing conclusions" while he aimed "to employ justice, not follow prejudice, and to take care in all that we judge and criticise that we seek the truth and not be swayed by opinions".

Also in Book I, ibn al-Haytham makes it clear that his investigation of light will be based on experimental evidence rather than on abstract theory. He notes that light is the same irrespective of the source and gives the examples of sunlight, light from a fire, or light reflected from a mirror which are all of the same nature. He gives the first correct explanation of vision, showing that light is reflected from an object into the eye. Most of the rest of Book I is devoted to the structure of the eye but here his explanations are necessarily in error since he does not have the concept of a lens which is necessary to understand the way the eye functions. His studies of optics did led him, however, to propose the use of a camera obscura, and he was the first person to mention it.

Book II of the Optics discusses visual perception while Book III examines conditions necessary for good vision and how errors in vision are caused. From a mathematical point of view Book IV is one of the most important since it discusses the theory of reflection. Ibn al-Haytham gave [1]:-

... experimental proof of the specular reflection of accidental as well as essential light, a complete formulation of the laws of reflection, and a description of the construction and use of a copper instrument for measuring reflections from plane, spherical, cylindrical, and conical mirrors, whether convex or concave.

Alhazen's problem, quoted near the beginning of this article, appears in Book V. Although we have quoted the problem for spherical mirrors, ibn al-Haytham also considered cylindrical and conical mirrors. The paper [36] gives a detailed description of six geometrical lemmas used by ibn al-Haytham in solving this problem. Huygens reformulated the problem as:-

To find the point of reflection on the surface of a spherical mirror, convex or concave, given the two points related to one another as eye and visible object.

Huygens found a good solution which Vincenzo Riccati and then Saladini simplified and improved.

Book VI of the Optics examines errors in vision due to reflection while the final book, Book VII, examines refraction [1]:-

Ibn al-Haytham does not give the impression that he was seeking a law which he failed to discover; but his "explanation" of refraction certainly forms part of the history of the formulation of the refraction law. The explanation is based on the idea that light is a movement which admits a variable speed (being less in denser bodies) ...

Ibn al-Haytham's study of refraction led him to propose that the atmosphere had a finite depth of about 15 km. He explained twilight by refraction of sunlight once the Sun was less than 19° below the horizon.

Abu al-Qasim ibn Madan was an astronomer who proposed questions to ibn al-Haytham, raising doubts about some of Ptolemy's explanations of physical phenomena. Ibn al-Haytham wrote a treatise Solution of doubts in which he gives his answers to these questions. They are discussed in [43] where the questions are given in the following form:-

What should we think of Ptolemy's account in "Almagest" Ⓣ I.3 concerning the visible enlargement of celestial magnitudes (the stars and their mutual distances) on the horizon? Is the explanation apparently implied by this account correct, and if so, under what physical conditions? How should we understand the analogy Ptolemy draws in the same place between this celestial phenomenon and the apparent magnification of objects seen in water? ...

There are strange contrasts in ibn al-Haytham's work relating to Ptolemy. In Al-Shukuk ala Batlamyus (Doubts concerning Ptolemy), ibn al-Haytham is critical of Ptolemy's ideas yet in a popular work the Configuration, intended for the layman, ibn al-Haytham completely accepts Ptolemy's views without question. This is a very different approach to that taken in his Optics as the quotations given above from the introduction indicate.

One of the mathematical problems which ibn al-Haytham attacked was the problem of squaring the circle. He wrote a work on the area of lunes, crescents formed from two intersecting circles, (see for example [10]) and then wrote the first of two treatises on squaring the circle using lunes (see [14]). However he seems to have realised that he could not solve the problem, for his promised second treatise on the topic never appeared. Whether ibn al-Haytham suspected that the problem was insoluble or whether he only realised that he could not solve it, in an interesting question which will never be answered.

In number theory al-Haytham solved problems involving congruences using what is now called Wilson's theorem:-

if p is prime then $1 + (p - 1) !$ is divisible by p .

In Opuscula ibn al-Haytham considers the solution of a system of congruences. In his own words (using the translation in [7]):-

To find a number such that if we divide by two, one remains; if we divide by three, one remains; if we divide by four, one remains; if we divide by five, one remains; if we divide by six, one remains; if we divide by seven, there is no remainder.

Ibn al-Haytham gives two methods of solution:-

The problem is indeterminate, that is it admits of many solutions. There are two methods to find them. One of them is the canonical method: we multiply the numbers mentioned that divide the number sought by each other; we add one to the product; this is the number sought.

Here ibn al-Haytham gives a general method of solution which, in the special case, gives the solution (7 - 1)! + 1. Using Wilson's theorem, this is divisible by 7 and it clearly leaves a remainder of 1 when divided by 2, 3, 4, 5, and 6. Ibn al-Haytham's second method gives all the solutions to systems of congruences of the type stated (which of course is a special case of the Chinese Remainder Theorem).

Another contribution by ibn al-Haytham to number theory was his work on perfect numbers. Euclid, in the Elements, had proved:-

If, for some $k > 1, 2^{k} - 1$ is prime then $2^{k-1}(2^{k} - 1)$ is a perfect number.

The converse of this result, namely that every even perfect number is of the form $2^{k-1}(2^{k} - 1)$ where $2^{k} - 1$ is prime, was proved by Euler. Rashed ([7], [8] or [27]) claims that ibn al-Haytham was the first to state this converse (although the statement does not appear explicitly in ibn al-Haytham's work). Rashed examines ibn al-Haytham's attempt to prove it in Analysis and synthesis which, as Rashed points out, is not entirely successful [7]:-

But this partial failure should not eclipse the essential: a deliberate attempt to characterise the set of perfect numbers.

Ibn al-Haytham's main purpose in Analysis and synthesis is to study the methods mathematicians use to solve problems. The ancient Greeks used analysis to solve geometric problems but ibn al-Haytham sees it as a more general mathematical method which can be applied to other problems such as those in algebra. In this work ibn al-Haytham realises that analysis was not an algorithm which could automatically be applied using given rules but he realises that the method requires intuition. See [18] and [26] for more details.