塔比·伊本·库拉Thābit ibn Qurra

塔比·伊本·库拉是哈兰人，也是萨比教派的成员。萨比教派是来自哈兰的星辰崇拜者，常被与曼达教徒混淆（正如在1中那样）。当然，作为星辰崇拜者意味着有强烈的动机去研究天文学，该教派产生了许多优秀的天文学家和数学家。该教派与希腊有密切联系，在更早的时期就采用了希腊文化，其成员讲希腊语很常见，尽管在萨比教徒被伊斯兰征服后，他们改讲阿拉伯语。在土耳其东南部还讲另一种语言，即叙利亚语，它基于埃德萨的东阿拉米方言。这种语言是塔比·伊本·库拉的母语，但他能流利地讲希腊语和阿拉伯语。

有些记载说，塔比·伊本·库拉年轻时是个货币兑换商。这很有可能，但一些历史学家并不同意。他肯定继承了一大笔家族财产，必定来自社区中地位很高的家庭。Muhammad ibn Musa ibn Shakir曾访问哈兰，对塔比·伊本·库拉的语言知识印象深刻，并意识到这个年轻人的潜力，说服他去巴格达，从他和他兄弟（巴努·穆萨）那里学习数学。

在巴格达，塔比·伊本·库拉接受了数学训练，也接受了医学训练，这在当时的学者中很常见。他回到哈兰，但他的自由哲学导致他出庭受审，不得不放弃他的“异端邪说”。为了逃避进一步的迫害，他离开哈兰，被任命为巴格达的宫廷天文学家。在那里，塔比·伊本·库拉的赞助人是哈里发al-Mu'tadid，他是最伟大的“阿拔斯王朝哈里发之一。

此时有许多赞助人雇用有才华的科学家将希腊文本翻译成阿拉伯语，而塔比·伊本·库拉凭借其卓越的语言能力和数学能力，翻译并修订了许多重要的希腊著作。欧几里得的Elements最早的两个译本是由al-Hajjaj完成的。除了一些残篇外，这些译本已经失传。然而，有许多手稿版本的第三个阿拉伯语译本，它是由侯奈因·伊本·伊斯哈格 ibn 侯奈因·伊本·伊斯哈格完成的，并由塔比·伊本·库拉修订。今天对欧几里得的Elements阿拉伯语译本复杂故事的认识表明，所有后来的阿拉伯语版本都源自塔比·伊本·库拉的这一修订本。

事实上，许多希腊文本今天之所以能保存下来，仅仅是因为这种将希腊学术引入阿拉伯世界的努力。然而，我们不能认为像塔比·伊本·库拉这样的数学家仅仅是希腊知识的保存者。远非如此，塔比·伊本·库拉是一位杰出的学者，做出了许多重要的数学发现。

尽管塔比·伊本·库拉在许多领域都有贡献，但他最重要的工作是在数学方面，他[1]：-

……在为诸如将数的概念扩展到（正）实数、积分、球面三角学定理、解析几何以及non-euclidean geometry等重要数学发现铺平道路方面发挥了重要作用。在天文学中，塔比·伊本·库拉是托勒密体系最早的改革者之一，在力学中，他是静力学的奠基人。

我们将更详细地考察塔比·伊本·库拉在这些领域的工作，特别是他在数论中关于亲和数的工作。假设用现代记号，$S(n)$表示$n$的真因数之和，即其真商之和。完全数是那些满足$S(n) = n$的$n$数，而如果$S(n) = m$且$S(m) = n$，则$m$和$n$是亲和数。在Book on the determination of amicable numbers中，塔比·伊本·库拉声称毕达哥拉斯开始了对完全数和亲和数的研究。这一说法几乎肯定是错误的，它可能最早由扬布利科斯在其公元三世纪所写的毕达哥拉斯传记中提出，其中给出了亲和数220和284。然而，塔比·伊本·库拉随后相当正确地指出，尽管欧几里得和尼各马科研究了完全数，并且欧几里得给出了确定它们的规则（[6]或[7]）：-

……这两位作者都没有提到或表现出对[亲和数]的兴趣。

塔比·伊本·库拉继续（[6]或[7]）：-

由于完全数的问题浮现在我的脑海中，并且我已经为它们推导出了一个证明，我不想在没有完美证明的情况下写下这个规则，因为它们被欧几里得和尼各马科忽视了。因此，我将在引入必要的引理后证明它。

在给出九个引理后，塔比·伊本·库拉陈述并证明了他的定理：对于$n > 1$，令$p_{n} = 3.2^{n} -1$和$q_{n} = 9.2^{2^{n-1}} -1$。如果$p_{n-1}, p_{n}$，且$q_{n}$是素数，那么$a = 2^{n}p_{n-1}p_{n}$和$b = 2^{n}q_{n}$是完全数，而$a$是abundant，$b$是deficient。注意，一个过剩数$n$满足$S(n) > n$，一个亏数$n$满足$S(n) < n$。更多细节见[9]，作者们推测塔比·伊本·库拉可能如何发现了这个规则。在[13]中，Hogendijk表明塔比·伊本·库拉可能是第一个发现完全数对17296, 18416的人。

塔比·伊本·库拉工作的另一个重要方面是他关于比的合成的书。在这本书中，塔比·伊本·库拉处理了应用于几何量之比的算术运算。希腊人处理了几何量，但没有像对待可以应用通常算术规则的数那样对待它们。[22]和[23]的作者强调，通过对先前被视为几何的、非数值的量引入算术运算，塔比·伊本·库拉开创了一种趋势，最终导致了数概念的推广。

塔比·伊本·库拉将毕达哥拉斯的定理推广到任意三角形（帕普斯也是如此）。他还讨论了抛物线、三等分角和magic squares。塔比·伊本·库拉关于抛物线和抛物面的工作特别重要，因为它是迈向发现积分学的步骤之一。这里的一个重要考虑是塔比·伊本·库拉是否熟悉阿基米德的方法。大多数作者（例如见[29]）认为，尽管塔比·伊本·库拉熟悉阿基米德关于抛物线求积的结果，但他没有阿基米德关于该主题的两篇论文。事实上，塔比·伊本·库拉有效地计算了$√x$和[1]的积分：-

计算本质上基于上下积分和的应用，证明是通过穷竭法完成的：在那里，首次将积分段分成不相等的部分。

塔比·伊本·库拉还写了天文学著作，写了Concerning the Motion of the Eighth Sphere。他（错误地）认为二分点的运动是振荡的。他还发表了太阳的观测。事实上，塔比·伊本·库拉的八篇完整天文学论文保存了下来，文章[20]描述了这些。 [20]的作者写道：-

当我们在九世纪巴格达科学运动开始的背景下考虑这一系列工作时，我们看到塔比·伊本·库拉在天文学作为一门精确科学（方法、主题和纲领）的建立中发挥了非常重要的作用，这沿着三条路线发展：观察与理论关系的理论化，天文学的“数学化”，以及对“数学”天文学与“物理”天文学之间冲突关系的关注。

塔比·伊本·库拉的一部重要著作Kitab fi'l-qarastun（关于杆秤的书）是关于力学的。它由克雷莫纳的克雷莫纳的杰拉德译成拉丁文，成为一部广受欢迎的力学著作。在这部著作中，塔比·伊本·库拉证明了杠杆平衡原理。他证明，两个相等的载荷与第三个载荷平衡时，可以用它们的和放在两者中点处来代替而不破坏平衡。在给出推广之后，塔比·伊本·库拉接着考虑了均匀分布的连续载荷的情形，并求出了重梁平衡的条件。当然，阿基米德考虑过重心理论，但在[14]中，作者认为塔比·伊本·库拉的工作并非基于阿基米德的理论。

最后，我们应当评论一下塔比·伊本·库拉在哲学和其他主题上的工作。塔比·伊本·库拉有一个学生Abu Musa Isa ibn Usayyid，他是来自伊拉克的基督徒。Ibn Usayyid向他的老师塔比·伊本·库拉提出了各种问题，现存有一份塔比·伊本·库拉所作回答的手稿，这份手稿在[21]中得到了讨论。塔比·伊本·库拉的数概念沿袭了柏拉图，他论证说数存在，无论是否有人知道它们，并且它们与可数的事物相分离。在其他方面，塔比·伊本·库拉批评了柏拉图和亚里士多德的观念，特别是关于运动的观念。看来在这里，他的观念基于接受在其几何论证中使用关于运动的论证。

塔比·伊本·库拉还撰写了关于[1]的内容：-

……逻辑学、心理学、伦理学、科学分类、叙利亚语语法、政治学、柏拉图的《理想国》的象征体系……宗教以及萨比教徒的风俗。

本档案包含关于塔比·伊本·库拉家族其他成员的信息。他的儿子Sinan ibn Thabit和孙子Ibrahim ibn Sinan ibn Thabit都是杰出学者，对数学的发展作出了贡献。然而，两人都未达到塔比·伊本·库拉的数学高度。

Thabit ibn Qurra was a native of Harran and a member of the Sabian sect. The Sabian religious sect were star worshippers from Harran often confused with the Mandaeans (as they are in [1]). Of course being worshipers of the stars meant that there was strong motivation for the study of astronomy and the sect produced many quality astronomers and mathematicians. The sect, with strong Greek connections, had in earlier times adopted Greek culture, and it was common for members to speak Greek although after the conquest of the Sabians by Islam, they became Arabic speakers. There was another language spoken in southeastern Turkey, namely Syriac, which was based on the East Aramaic dialect of Edessa. This language was Thabit ibn Qurra's native language, but he was fluent in both Greek and Arabic.

Some accounts say that Thabit was a money changer as a young man. This is quite possible but some historians do not agree. Certainly he inherited a large family fortune and must have come from a family of high standing in the community. Muhammad ibn Musa ibn Shakir, who visited Harran, was impressed at Thabit's knowledge of languages and, realising the young man's potential, persuaded him to go to Baghdad and take lessons in mathematics from him and his brothers (the Banu Musa).

In Baghdad Thabit received mathematical training and also training in medicine, which was common for scholars of that time. He returned to Harran but his liberal philosophies led to a religious court appearance when he had to recant his 'heresies'. To escape further persecution he left Harran and was appointed court astronomer in Baghdad. There Thabit's patron was the Caliph, al-Mu'tadid, one of the greatest of the 'Abbasid caliphs.

At this time there were many patrons who employed talented scientists to translate Greek text into Arabic and Thabit, with his great skills in languages as well as great mathematical skills, translated and revised many of the important Greek works. The two earliest translations of Euclid's Elements were made by al-Hajjaj. These are lost except for some fragments. There are, however, numerous manuscript versions of the third translation into Arabic which was made by Hunayn ibn Ishaq and revised by Thabit. Knowledge today of the complex story of the Arabic translations of Euclid's Elements indicates that all later Arabic versions develop from this revision by Thabit.

In fact many Greek texts survive today only because of this industry in bringing Greek learning to the Arab world. However we must not think that the mathematicians such as Thabit were mere preservers of Greek knowledge. Far from it, Thabit was a brilliant scholar who made many important mathematical discoveries.

Although Thabit contributed to a number of areas the most important of his work was in mathematics where he [1]:-

... played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry. In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.

We shall examine in more detail Thabit's work in these areas, in particular his work in number theory on amicable numbers. Suppose that, in modern notation, $S(n)$ denotes the sum of the aliquot parts of $n$, that is the sum of its proper quotients. Perfect numbers are those numbers $n$ with $S(n) = n$ while $m$ and $n$ are amicable if $S(n) = m$, and $S(m) = n$. In Book on the determination of amicable numbers Thabit claims that Pythagoras began the study of perfect and amicable numbers. This claim, probably first made by Iamblichus in his biography of Pythagoras written in the third century AD where he gave the amicable numbers 220 and 284, is almost certainly false. However Thabit then states quite correctly that although Euclid and Nicomachus studied perfect numbers, and Euclid gave a rule for determining them ([6] or [7]):-

... neither of these authors either mentioned or showed interest in [amicable numbers].

Thabit continues ([6] or [7]):-

Since the matter of [amicable numbers] has occurred to my mind, and since I have derived a proof for them, I did not wish to write the rule without proving it perfectly because they have been neglected by [Euclid and Nicomachus]. I shall therefore prove it after introducing the necessary lemmas.

After giving nine lemmas Thabit states and proves his theorem: for $n > 1$, let $p_{n} = 3.2^{n} -1$ and $q_{n} = 9.2^{2^{n-1}} -1$. If $p_{n-1}, p_{n}$, and $q_{n}$ are prime numbers, then $a = 2^{n}p_{n-1}p_{n}$ and $b = 2^{n}q_{n}$ are amicable numbers while $a$ is abundant and $b$ is deficient. Note that an abundant number $n$ satisfies $S(n) > n$, and a deficient number $n$ satisfies $S(n) < n$. More details are given in [9] where the authors conjecture how Thabit might have discovered the rule. In [13] Hogendijk shows that Thabit was probably the first to discover the pair of amicable numbers 17296, 18416.

Another important aspect of Thabit's work was his book on the composition of ratios. In this Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. The authors of [22] and [23] stress that by introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalisation of the number concept.

Thabit generalised Pythagoras's theorem to an arbitrary triangle (as did Pappus). He also discussed parabolas, angle trisection and magic squares. Thabit's work on parabolas and paraboliods is of particular importance since it is one of the steps taken towards the discovery of the integral calculus. An important consideration here is whether Thabit was familiar with the methods of Archimedes. Most authors (see for example [29]) believe that although Thabit was familiar with Archimedes' results on the quadrature of the parabola, he did not have either of Archimedes' two treatises on the topic. In fact Thabit effectively computed the integral of $√x$ and [1]:-

The computation is based essentially on the application of upper and lower integral sums, and the proof is done by the method of exhaustion: there, for the first time, the segment of integration is divided into unequal parts.

Thabit also wrote on astronomy, writing Concerning the Motion of the Eighth Sphere. He believed (wrongly) that the motion of the equinoxes oscillates. He also published observations of the Sun. In fact eight complete treatises by Thabit on astronomy have survived and the article [20] describes these. The author of [20] writes:-

When we consider this body of work in the context of the beginnings of the scientific movement in ninth-century Baghdad, we see that Thabit played a very important role in the establishment of astronomy as an exact science (method, topics and program), which developed along three lines: the theorisation of the relation between observation and theory, the 'mathematisation' of astronomy, and the focus on the conflicting relationship between 'mathematical' astronomy and 'physical' astronomy.

An important work Kitab fi'l-qarastun (The book on the beam balance) by Thabit is on mechanics. It was translated into Latin by Gherard of Cremona and became a popular work on mechanics. In this work Thabit proves the principle of equilibrium of levers. He demonstrates that two equal loads, balancing a third, can be replaced by their sum placed at a point halfway between the two without destroying the equilibrium. After giving a generalisation Thabit then considers the case of equally distributed continuous loads and finds the conditions for the equilibrium of a heavy beam. Of course Archimedes considered a theory of centres of gravity, but in [14] the author argues that Thabit's work is not based on Archimedes' theory.

Finally we should comment on Thabit's work on philosophy and other topics. Thabit had a student Abu Musa Isa ibn Usayyid who was a Christian from Iraq. Ibn Usayyid asked various questions of his teacher Thabit and a manuscript exists of the answers given by Thabit, this manuscript being discussed in [21]. Thabit's concept of number follows that of Plato and he argues that numbers exist, whether someone knows them or not, and they are separate from numerable things. In other respects Thabit is critical of the ideas of Plato and Aristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments.

Thabit also wrote on [1]:-

... logic, psychology, ethics, the classification of sciences, the grammar of the Syriac language, politics, the symbolism of Plato's Republic ... religion and the customs of the Sabians.

This archive contains information on other members of Thabit's family. His son, Sinan ibn Thabit, and his grandson Ibrahim ibn Sinan ibn Thabit, both were eminent scholars who contributed to the development of mathematics. Neither, however, reached the mathematical heights of Thabit.