克莱奥迈季斯Cleomedes

克莱奥迈季斯仅通过他的书On the Circular Motions of the Celestial Bodies而为人所知，这是一本平淡无奇的天文学教科书。然而，正如我们将在下面讨论的，这本书中有许多有趣之处。

我们首先应讨论克莱奥迈季斯生活时期的这个令人困惑的问题。这里唯一确定的是On the Circular Motions of the Celestial Bodies详细讨论了波希多尼的工作，因此显然写于公元前一世纪中叶之后。事实上，On the Circular Motions of the Celestial Bodies以这样的话结束（例如见[3]）：-

前述教导并非作者本人的观点，而是从较早或较近的摘要中收集而来；其中大部分取自波希多尼。

很难从这些话中估计作者克莱奥迈季斯是在波希多尼之后多久写作的。

托马斯·利特尔·希思[2]倾向于公元前一世纪中叶的日期。他还指出：-

由于克莱奥迈季斯似乎对克劳狄乌斯·托勒密的著作一无所知，他几乎不可能……生活在公元二世纪初之后。

然而，奥托·纽格包尔不同意托马斯·利特尔·希思的这些结论，并提出克莱奥迈季斯大约在公元370年撰写了他的文本。他的论证基于克莱奥迈季斯在文本中的一条评论，其中他指出有两颗亮星（毕宿五和心宿二），使得一颗升起和另一颗落下同时发生。克莱奥迈季斯声称这些星位于其星座的15°处。利用克劳狄乌斯·托勒密给出的在AlmagestⓉ（主要论题：源自阿拉伯语'al-majisti'——希腊语'Mathematike Syntaxis'的阿拉伯语翻译，后来译为拉丁语'Magna Syntaxis'）成书时这些恒星的位置，以及克劳狄乌斯·托勒密的岁差每100年1°的值，奥托·纽格包尔得出了克莱奥迈季斯著作的公元371年这一日期，对此奥托·纽格包尔估计最大误差为前后50年。

从天文学上讲，奥托·纽格包尔的计算当然是完全正确的。然而，由于许多其他原因，它们是可疑的。首先，On the Circular Motions of the Celestial Bodies中的数据似乎并非来自克莱奥迈季斯，而是来自各种来源。当然，接受这一论点会使克莱奥迈季斯的日期更晚。其次，克莱奥迈季斯中的数据准确性差异很大。有些非常好，而其他数据有20%的误差。第三，毕宿五落下和心宿二同时升起的实际天文事件永远无法观测到。

托马斯·利特尔·希思关于克莱奥迈季斯对克劳狄乌斯·托勒密的作品一无所知的评论也不像乍看起来那么确定。克莱奥迈季斯正在写一本基础教科书，而在低级别教科书中提到最近的研究当然并不总是如此。例如，许多应用数学的基础教科书在阿尔伯特·爱因斯坦给出改进的引力理论100年后仍然使用艾萨克·牛顿的引力方法（并且有充分的理由）。

在1中，Dicks提出克莱奥迈季斯最可能的年代是公元一世纪，我们已将其作为根据已知信息所能得到的最佳折衷方案。这并不是说奥托·纽格包尔的年代不可能。事实上，文本中还有其他特征倾向于支持公元四世纪这一年代，尽管缺乏对克劳狄乌斯·托勒密的引用。其中不容忽视的一点是，这一时期正是许多此类二流教科书被撰写的时代，其风格与其它公元四世纪文本并无不同，其中一些给出了与克莱奥迈季斯相同的天文数据。

On the Circular Motions of the Celestial Bodies是一部两卷本的著作。我们在上文评论过，它由于若干原因而重要。最重要的原因当然是，它为我们提供了关于波希多尼一部文本的内容和风格的最佳指示。正如托马斯·利特尔·希思所评论的[2]：-

……第二卷非常长的第一章（几乎占该卷的一半）……似乎大部分是从波希多尼原封不动地抄来的。

但这并不是克莱奥迈季斯撰写该文本的主要目的。写它是为了攻击伊壁鸠鲁学派，后者除其他古怪信念外，还相信太阳和它看起来一样大，即一英尺宽。克莱奥迈季斯在文本中花了很多时间表明这是错误的，但当他将伊壁鸠鲁派与老鼠、爬行动物和蠕虫进行不利比较时，他似乎确实走极端了。克莱奥迈季斯自己的哲学观点表明他是一个斯多亚派。

由于On the Circular Motions of the Celestial Bodies是从若干来源编纂而成，其质量参差不齐，在许多地方，当各来源相互矛盾时，该书未能保持一致。每当一段文本被认为出自克莱奥迈季斯本人之手时，就有大量证据表明他对该主题的理解非常有限，而且，若不是其来源的质量，人们会觉得克莱奥迈季斯在幼稚程度上不会比伊壁鸠鲁派好到哪里去。

文本中对月食的描述很好，地球阴影的锥形显示出对（至少是克莱奥迈季斯来源的）理解的相当深度。他还正确地将太阳和月亮都位于地平线以上时所见月食的报告解释为由于折射所致。

奥托·纽格包尔给出了克莱奥迈季斯中混杂的合理与荒谬之处的样本[3]：-

克莱奥迈季斯 指出，没有一颗恒星的视直径小于一指宽（这相当荒谬夸张），而金星的视直径应为两指宽，即月球或太阳直径的 $\large\frac{1}{6}\normalsize$。有一点值得注意，即恒星的绝对大小可能达到甚至超过太阳……[还]有人说，从太阳上看地球，最多不过是一颗很小的恒星。

克莱奥迈季斯 的著作还有一个引人之处，即正是在 On the Circular Motions of the Celestial Bodies 中我们得知了 埃拉托色尼 测量地球周长的方法。这是早期数学天文学最为著名的成就之一，我们感谢 克莱奥迈季斯 记述了这一方法。当然，并非所有人都相信 埃拉托色尼 测量的故事是真实的，但尽管如此，它仍被广泛接受。

最后我们应当指出，奥托·纽格包尔根据对On the Circular Motions of the Celestial Bodies中某些天文数据的研究提出，克莱奥迈季斯生活在黑海地区的赫勒斯滂，并暗示是利西马基亚城。奥托·纽格包尔承认利西马基亚城于公元前144年被毁，这似乎与他为克莱奥迈季斯所定的公元370年这一年代相矛盾，但他能够表明，尽管公元前144年遭遇灾难，该城的记录确实延续到公元四世纪。奥托·纽格包尔论证的弱点必定在于，克莱奥迈季斯的文本和数据几乎全部取自他人的著作，因此奥托·纽格包尔的论证似乎只能有力地证明克莱奥迈季斯的某一来源曾在利西马基亚写作。

Cleomedes is known only through his book On the Circular Motions of the Celestial Bodies which is an uninspiring astronomy textbook. There are a number of points of interest in this book, however, as we shall discuss below.

We should first discuss the perplexing question of the period in which Cleomedes lived. The only certainty here is that On the Circular Motions of the Celestial Bodies discusses the work of Posidonius at length and so is clearly written after the middle of the first century BC. In fact On the Circular Motions of the Celestial Bodies ends with the words (see for example [3]):-

The preceding teachings are not the author's own opinion but collected from older or more recent summaries; much of it is taken from Posidonius.

It is hard to estimate from these words how long after Posidonius the author, Cleomedes, is writing.

Heath [2] favours a date in the middle of the first century BC. He also points out:-

As [Cleomedes] seems to know nothing of the works of Ptolemy, he can hardly... have lived later than the beginning of the second century AD.

Neugebauer, however, disagrees with these conclusions of Heath and proposes that Cleomedes wrote his text around 370 AD. His argument is based on a comment by Cleomedes in the text where he remarks than there are two bright stars (Aldebaran and Antares) such that the rising of one and the setting of the other take place at the same time. These stars Cleomedes claims lie at 15° of their sign. Using Ptolemy's positions for the stars at the time the Almagest Ⓣ was written and Ptolemy's value of 1° per 100 years for precession, Neugebauer gets his date of 371 AD for Cleomedes writings, to which Neugebauer estimates a maximum error of 50 years on either side.

Astronomically Neugebauer's calculations are of course perfectly correct. However they are suspect for a number of other reasons. Firstly the data in On the Circular Motions of the Celestial Bodies does not seem to be due to Cleomedes but to a variety of sources. Of course accepting this argument would make Cleomedes dates later still. Secondly the data in Cleomedes is of widely differing degrees of accuracy. Some is very good, while other data has errors of 20%. Thirdly the actual astronomical event of Aldebaran setting and Antares rising at the same instant can never be observed.

Heath's comment that Cleomedes knows nothing of the works of Ptolemy is also less certain than it might at first appear. Cleomedes is writing an elementary textbook and it is certainly not always the case that one mentions recent research in a low level textbook. For example many elementary textbooks on applied mathematics still use Newton's gravitational methods (and for good reason) 100 years after Einstein gave an improved theory of gravitation.

In [1] Dicks suggests that the most likely date for Cleomedes is the first century AD and we have taken that as the best available compromise from what is known. That is not to say that Neugebauer's date is impossible. In fact there are other features of the text which would tend to support the fourth century AD as a date, despite the lack of references to Ptolemy. Not least of these is the fact that this was a period when many second rate textbooks of this nature were written and the style is not unlike that of other fourth century AD texts, some of which give the same astronomical data as Cleomedes.

On the Circular Motions of the Celestial Bodies is a work in two volumes. We commented above that it was important for a number of reasons. The most important certainly is that it gives us the best indication that we have of the contents and the style of a text by Posidonius. As Heath comments [2]:-

... the very long first chapter of Book II (nearly half of the Book) ... seems for the most part to be copied bodily from Posidonius.

But this is not Cleomedes' main aim in writing the text. It is written to attack the Epicureans who believed among other odd beliefs, that the sun was as large as it looked, namely one foot across. Cleomedes spends much time in his text showing that this is false, but it does seem as if he is going to extremes when he compares Epicureans unfavourably with rats, reptiles and worms. Cleomedes' own philosophical views show that he is a Stoic.

Since On the Circular Motions of the Celestial Bodies is compiled from a number of sources, there is a fair variation in the quality, and in many places the book fails to be consistent where the various sources disagree. Whenever a piece of text is thought to be due to Cleomedes himself, there is much evidence that his understanding of the topic was very limited and, but for the quality of his sources, one feels that Cleomedes would not fare any better than the Epicureans for naiveté.

Lunar eclipses are described well in the text, and the conical shape of the earth's shadow shows an interesting depth of understanding (at least of Cleomedes' source). He also correctly explains the reports of lunar eclipses seen when both the sun and moon are above the horizon as being due to refraction.

Neugebauer gives samples of the sense and nonsense which are mixed in Cleomedes [3]:-

Cleomedes states that no fixed star has an apparent diameter less than one finger (a rather absurd exaggeration) while the apparent diameter of Venus should be two fingers, i.e. $\large\frac{1}{6}\normalsize$ of the lunar or solar diameter. Of some interest is the remark that the absolute size of fixed stars may reach, or even surpass, the sun ... it is [also] said that the earth, seen from the sun, would appear at best a very small star.

One further interest in Cleomedes' work is that it is in On the Circular Motions of the Celestial Bodies that we learn of Eratosthenes method of measuring the circumference of the earth. This is one of the best known of the achievements of early mathematical astronomy and we are indebted to Cleomedes for relating the method. Of course not everyone believes that the story of Eratosthenes' measurement is authentic but, despite this, it is widely accepted.

Finally we should remark that Neugebauer suggests from a study of certain astronomical data given in On the Circular Motions of the Celestial Bodies that Cleomedes lived in the Hellespont on the Black Sea, suggesting the city of Lysimachia. Neugebauer admits that the city of Lysimachia was destroyed in 144 BC which seems at odds with his own date of 370 AD for Cleomedes but he is able to show that despite the disaster of 144 BC records of the city certainly extend up to the fourth century AD. The weakness of Neugebauer's argument must surely be that almost all of Cleomedes' text and data is taken from the works of others so Neugebauer's arguments seem only to give strong evidence for one of Cleomedes' sources having written in Lysimachia.