普罗克洛Proclus

普罗克洛的父亲Particius和母亲Marcella是吕基亚社会地位很高的公民。Particius是费隆法庭的高级法律官员。普罗克洛在吕基亚南海岸的克桑托斯长大，并在那里上学。

原本打算让普罗克洛追随父亲进入法律行业。怀着这个目标，他被送到亚历山大里亚，但在学业中途，他访问了费隆，并确信自己人生的使命是研究哲学。他回到亚历山大里亚，此时在Olympiodorus the 大阿里斯泰俄斯门下学习哲学，尤其深入研究了亚里士多德的著作。他还在亚历山大里亚学习数学，这门学科的教师是Heron（不是那位著名的数学家，Heron在当时是个常见的名字）。

普罗克洛对他在亚历山大里亚接受的哲学教育并不完全满意，所以还在十几岁时，他就从亚历山大里亚搬到雅典，在柏拉图学园师从哲学家雅典的Plutarch和叙利亚努斯（Plutarch的学生）。他从科学院的学生成长为那里的教师，随后在Syrianus去世后，普罗克洛成为科学院的负责人。此时他被授予普罗克洛的头衔，这个词的意思是继任者。

在科学院，普罗克洛似乎相当富有，并在经济上帮助过朋友和亲戚。他从未结婚，过着某种程度上有别于毕达哥拉斯所提倡的生活。他不吃肉，努力过一种宗教生活，创作献给诸神的赞美诗。他的赞美诗显然受到高度评价，因为其中七首保存了下来，如今被认为具有相当的文学价值。普罗克洛一直担任科学院的负责人直到去世。

普罗克洛学识渊博，深受同时代人的崇敬。他遵循新柏拉图主义者哲学，该哲学由普罗提诺创立，并由波菲利和扬布利科斯在公元300年左右发展。这些思想的其他发展者还有普鲁塔克和叙利亚努斯，他们是普罗克洛的老师。托马斯·利特尔·希思写道[4]：-

他是一位敏锐的dialectician，在学识广度上于同时代人中出类拔萃；他是一位有能力的数学家；他甚至是一位诗人。同时，他相信各种神话和秘仪，并虔诚崇拜希腊和东方的神祇。他与其说是数学家，不如说是哲学家。

当然，正如人们可能预料的那样，他对许多宗教说法的信仰意味着他在许多科学问题上的观点带有高度偏见。例如，他提到阿里斯塔克斯提出的太阳位于行星中心的假说，但立即予以拒绝，因为它与一位迦勒底人的观点相矛盾，而他说不信仰这位迦勒底人的观点是不合法的。

普罗克洛写了Commentary on Euclid，这是我们关于希腊几何学早期历史的主要资料来源。这本书无疑是他于科学院教学的产物。这部著作并未受到他宗教信仰的影响，19世纪中叶写作的Martin说（例如见[4]）：-

……对普罗克洛来说，“欧几里得的《原本》”有幸既未与迦勒底神谕相矛盾，也未与新老毕达哥拉斯学派的思辨相矛盾。

普罗克洛能够接触到如今已失传的书籍，而另一些在普罗克洛时代就已失传的著作，则是根据普罗克洛所能获得的其他书籍中的摘录来描述的。特别是，他肯定使用了罗德岛的欧德摩斯所著的History of Geometry，该书现已失传，同样失传的还有他也使用过的杰米纽斯的著作。托马斯·利特尔·希思在描述普罗克洛的Commentary on Euclid时写道[4]：-

普罗克洛按顺序对所有的定义、公设和公理进行了历史性的和批判性的处理。在对公设和公理的注释之前，有一段关于几何学原理、假设、公设和公理及其相互关系的总体讨论；在这里，一如往常，普罗克洛引用了所有重要权威的意见。

普罗克洛评注中另一个有趣的部分是他对几何学批评者的讨论。他写道：-

……正是针对[几何学原理]，大多数几何学批评者提出了反对意见，试图表明这些部分并未牢固确立。在这一群体中，其论点已变得臭名昭著的人里，有些如怀疑论者，会废除一切知识……而另一些如伊壁鸠鲁学派，则只是试图贬低几何学原理。然而，另一群批评者承认这些原理，但否认在原理之后出现的命题能够被证明，除非他们承认某些不包含在原理中的东西。这种争论方法为Zeno of Sidon所遵循，他属于伊壁鸠鲁学派，波希多尼曾针对他写了整整一本书，表明他的观点完全站不住脚。

Morrow在[1]中证实了我们对普罗克洛所欠下的巨大恩情，尤其是他的Commentary on Euclid，他在[1]中写道：-

普罗克洛并非一位有创造力的数学家；但他是一位敏锐的阐释者和批评家，对数学方法有透彻的把握，并对从泰勒斯到他自己的时代这一千年间的希腊数学有详细的了解。

最近出版的7很好地描述了普罗克洛在欧几里得的Elements第一卷注释中的著作。7这本书是对普罗克洛哲学，特别是其数学哲学研究的重要贡献。

普罗克洛 还写了 Hypotyposis，这是对 喜帕恰斯 和 克劳狄乌斯·托勒密 天文学理论的介绍，其中他基于 epicycles 和 eccentrics 描述了行星的数学理论。他结合了自己的几何技能和天文学知识，给出了一个几何证明，表明行星的本轮理论等价于偏心理论。在本轮理论中，地球位于一个圆的中心，较小的圆绕其圆周旋转。在偏心理论中，行星在圆心不与地球重合的圆上运动。

这里没有任何原创内容，普罗克洛 证明的是 喜帕恰斯 和 克劳狄乌斯·托勒密 首先给出的结果。然而，尽管 普罗克洛 认为这一理论应该由他在 科学院 的学生研究，但他并非不加批判，他认为这一理论过于复杂，而且是一个特设的理论，没有理由解释其各个部分。

在他的天文学著作中，普罗克洛 描述了如何使用 海伦 发明的漏壶来测量太阳的视直径。普罗克洛 的方法可以在 二分点 使用。当太阳升起时，从漏壶中收集水到一个容器中。太阳一升起，水就被收集到另一个容器中，这种测量持续到第二天日出。然后，两个容器中水的重量之比就给出了太阳的视直径。

普罗克洛的许多著作包括Liber de causis（《原因之书》）、Institutio theologica（《神学要义》）、对形而上学的简明阐述、Elements of Physics（主要表达亚里士多德的观点），以及In Platonis theologiam（《柏拉图神学》，阐述柏拉图的形而上学）。他的贡献在1中得到了很好的总结，如下所示：-

普罗克洛值得被铭记……因为他拥有在任何时代都极为罕见、在他那个时代几乎独一无二的品质：他思想的逻辑清晰与坚定，他分析的敏锐，他渴望理解并乐于呈现前人在争议问题上的观点，他长篇论述的持续连贯性，以及他思想所活动于其中的广阔视野——其广度如同整个存在。

Proclus's father, Particius, and his mother, Marcella, were citizens of high social position in Lycia. Particius was a senior law official in the courts at Byzantium. Proclus was brought up at Xanthus, on the south coast of Lycia, where he attended school.

It was intended that Proclus should follow his father and enter the legal profession. With this aim in mind he was sent to Alexandria but, while in the middle of his studies, he visited Byzantium and he became convinced that his calling in life was the study of philosophy. He returned to Alexandria where now he studied philosophy under Olympiodorus the Elder, in particular making a deep study of the works of Aristotle. He also learnt mathematics in Alexandria and in this subject his teacher was Heron (not the famous mathematician, Heron was a common name at this time).

Proclus was not entirely satisfied with the education he was receiving in philosophy in Alexandria so, while still a teenager, he moved from Alexandria to Athens where he studied at Plato's Academy under the philosophers Plutarch of Athens and Syrianus (a pupil of Plutarch). He progressed from being a student at the Academy to teaching there then, on the death of Syrianus, Proclus became head of the Academy. The title Diadochus was given to him at this time, the meaning of the word being successor.

At the Academy Proclus appears to have been well off and to have helped his friends and relations financially. He never married and lived a life which was, in certain respects, not unlike that proposed by Pythagoras. He did not eat meat and tried to live a religious life, composing hymns to the gods. His hymns were clearly highly thought of since seven of them have been preserved and are seen today as having considerable literary merit. Proclus was to remain as head of the Academy until his death.

A man of great learning, Proclus was regarded with great veneration by his contemporaries. He followed the neoplatonist philosophy which Plotinus founded, and Porphyry and Iamblichus developed around 300 AD. Other developers of these ideas were Plutarch and Syrianus, the teachers of Proclus. Heath writes [4]:-

He was an acute dialectician and pre-eminent among his contemporaries in the range of his learning; he was a competent mathematician; he was even a poet. At the same time he was a believer in all sorts of myths and mysteries, and a devout worshipper of divinities both Greek and Oriental. He was much more a philosopher than a mathematician.

Of course, as one might expect, his belief in many religious sayings meant that he was highly biased in his views on many issues of science. For example he mentions the hypothesis that the sun is at the centre of the planets as proposed by Aristarchus but rejects it immediately since it contradicted the views of a Chaldean whom he says that it is unlawful not to believe.

Proclus wrote Commentary on Euclid which is our principal source about the early history of Greek geometry. The book is certainly the product of his teaching at the Academy. This work is not coloured by his religious beliefs and Martin, writing in the middle of the 19th century, says (see for example [4]):-

... for Proclus the "Elements of Euclid" had the good fortune not to be contradicted either by the Chaldean Oracles or by the speculations of Pythagoreans old and new.

Proclus had access to books which are now lost and others, already lost in Proclus's time, were described based on extracts in other books available to Proclus. In particular he certainly used the History of Geometry by Eudemus, which is now lost, as is the works of Geminus which he also used. Heath, describing Proclus's Commentary on Euclid writes [4]:-

Proclus deals historically and critically with all the definitions, postulates and axioms in order. The notes on the postulates and axioms are preceded by a general discussion of the principles of geometry, hypotheses, postulates and axioms, and their relation to one another; here as usual Proclus quotes the opinions of all the important authorities.

Another interesting part of Proclus's commentary is his discussion of the critics of geometry. He writes:-

... it is against [the principles of geometry] that most critics of geometry have raised objections, endeavouring to show that these parts are not firmly established. Of those in this group whose arguments have become notorious some, such as the Sceptics, would do away with all knowledge ... whereas others, like the Epicureans, propose only to discredit the principles of geometry. Another group of critics, however, admit the principles but deny that the propositions coming after the principles can be demonstrated unless they grant something that is not contained in the principles. This method of controversy was followed by Zeno of Sidon, who belonged to the school of Epicurus and against whom Posidonius has written a whole book and shown that his views are thoroughly unsound.

Morrow in [1] confirms the great debt that we owe to Proclus, and in particular his Commentary on Euclid when he writes in [1]:-

Proclus was not a creative mathematician; but he was an acute expositor and critic, with a thorough grasp of mathematical method and a detailed knowledge of the thousand years of Greek mathematics from Thales to his own time.

The recent book [7] gives a good description of the writings of Proclus found in his commentary on Book I of Euclid's Elements. The book [7] is an important contribution to the study of the philosophy of Proclus and in particular his philosophy of mathematics.

Proclus also wrote Hypotyposis, an introduction to the astronomical theories of Hipparchus and Ptolemy in which he described the mathematical theory of the planets based on epicycles and on eccentrics. He combined his geometrical skills and his knowledge of astronomy to give a geometrical proof that the epicycle theory for the planets is equivalent to the eccentric theory. In the epicycle theory the Earth is in the centre of a circle which has smaller circles rotating round its circumference. In the eccentric theory the planets move round in circles whose centres do not coincide with the Earth.

Nothing here is original and Proclus is proving results first given by Hipparchus and Ptolemy. However, although Proclus believed that this theory should be studied by his students at the Academy, he was not uncritical, suggesting that the theory was overly complicated and also that it was an ad hoc theory with no reason to explain its various parts.

In his astronomical writings, Proclus described how the water clock invented by Heron could be used to measure the apparent diameter of the Sun. Proclus's method can be used at the equinox. Water is collected from the clock in a container while the sun rises. As soon as the Sun has risen the water is collected in another container and this measurement continues until sunrise the following day. Then the ratio of the weights of water in the two containers gives the apparent diameter of the Sun.

Among Proclus's many works are Liber de causis (Book of Causes), Institutio theologica (Elements of Theology), a concise exposition of metaphysics, Elements of Physics, largely giving Aristotle's views, and In Platonis theologiam (Platonic Theology) giving Plato's metaphysics. His contribution is well summarised in [1] as follows:-

Proclus deserves to be remembered ... for the qualities he possessed that are exceedingly rare in any age and were almost unique in his: the logical clarity and firmness of his thought, the acuteness of his analyses, his eagerness to understand and readiness to present the views of his predecessors on controversial issues, the sustained coherence of his lengthy expositions, and the large horizon, as broad as the whole of being, within which his thinking moved.