奥托里库斯Autolycus of Pitane

我们知道一些关于奥托里库斯生平的信息，但确实不足以准确确定他的年代。他是阿尔克西拉乌斯的老师，而阿尔克西拉乌斯生于公元前315年，因此奥托里库斯一定活到公元前300年之后。人们普遍认为他比欧几里得年长。正如托马斯·利特尔·希思在[3]中所写：-

他写作早于欧几里得这一点从以下事实可以清楚看出：欧几里得……使用了奥托里库斯中出现的定理，不过，正如这类情况中通常的那样，没有指明其来源。

然而，在我们接受托马斯·利特尔·希思的“清晰”论证之前，有理由提出来自奥托·纽格包尔[4]的反驳：-

支持奥托里库斯优先权的普遍接受的论证异常天真。欧几里得的《现象》中的定理2由四个命题组成，但只给出了其中三个的证明，而缺失的那个被替换为“这种情况已在别处证明”的备注；事实上，定理和证明在奥托里库斯的《旋转球》中作为定理10出现。这种备注在任何希腊数学论著中——无论是否欧几里得——都显得真实，在我看来完全不可信；我会假设显而易见的情况，即一个注释可能在一份损坏的副本中，用对众所周知定理的简单引用替换了四个证明中的第一个。事实上，我看不出为什么欧几里得（大概在亚历山大）和奥托里库斯（大概在雅典）不应该独立地，甚至可能同时地，撰写关于天文现象的数学理论。

Huxley在[1]中写作，同意托马斯·利特尔·希思，而论文[4]甚至标题为Autolycus of Pitane, predecessor of Euclid。优先权论证对于欧几里得和奥托里库斯相互依赖的重要性肯定不是不重要的，因为奥托里库斯以与欧几里得完全相同的一般风格撰写他的命题。这意味着奥托里库斯著作中的一个定理首先有一个一般陈述，然后是与特定图形相关的构造，图形中的点用字母表示，接下来是定理的证明，最后有时会得出与一般陈述相关的结论。尽管有上述论证，无论哪部论著先写，人们仍必须得出相同的结论，即这种今天被认为如此具有欧几里得特色的数学阐述风格，肯定不是他发明的。尽管这种风格被奥托里库斯使用，也没有人认为是他发明的。

通过与欧几里得的比较，我们走上了一条引人入胜的道路，但我们应该回到奥托里库斯生平中唯一由第欧根尼·拉尔修记载的其他细节，当时他叙述说阿尔克西拉乌斯在他的老师奥托里库斯陪同下前往萨迪斯。

关于奥托里库斯的另一个重要事实是，他的两部著作以希腊文原文保存下来，我们相信它们是现存最早的兩部数学著作。在这些著作中，On the Moving Sphere是一部关于球面几何的著作，这等同于一部数学天文学文本。第二部著作On Risings and Settings则更多地是一部关于观测天文学的著作。

比提尼亚的狄奥多西在200年后写了Sphaerics，一部类似的关于球面几何的著作，同样是为天文学提供数学背景而写。人们认为比提尼亚的狄奥多西的Sphaerics和奥托里库斯的著作On the Moving Sphere基于同一部现已失传的前欧几里得教科书。有人推测——不得不说证据相当少——欧多克索斯写了这部较早的文本。关于这一点的猜测似乎永远无法得到解决。

奥托里库斯在天文学观点上严重依赖欧多克索斯，这一点毋庸置疑。他是欧多克索斯的同心球理论的有力支持者，该理论由若干旋转的球面组成，每个球面绕穿过地心的轴旋转。这一理论有一个很快就被注意到的困难，即金星和火星的亮度会变化，而欧多克索斯的理论中没有对此的机制。此外，日食有时是全食，有时是环食，此时月亮看起来比太阳小，太阳的一圈光环在月亮周围可见。尽管奥托里库斯试图在欧多克索斯的理论内解释这些观测，但他对这些问题没有真正的答案。

On Risings and Settings是一部由两卷组成的著作。Schmidt的有趣文章[7]旨在表明，事实上这两卷并非一部两卷本著作的两个部分，而是同一部著作的两个版本。第二卷实际上是第一卷的修订扩充版，包含相当多的新内容。它也是一部结构更好的著作，看到奥托里库斯的著作在两个版本之间如何发展和改进是很有趣的。

We know some information about the life of Autolycus of Pitane, but not really enough to date him accurately. He was a teacher of Arcesilaus who was born in 315 BC so Autolycus must have lived until after 300 BC. It is generally assumed that he was older than Euclid. As Heath writes in [3]:-

That he wrote earlier than Euclid is clear from the fact that Euclid ... makes use of theorems appearing in Autolycus, though, as usual in such cases, giving no indication of their source.

However, before we accept Heath's 'clear' argument, it is reasonable to put a counter argument from Neugebauer [4]:-

The generally accepted argument in favour of Autolycus's priority is singularly naive. Theorem 2 of Euclid's Phaenomena consists of four propositions with proofs for only three of them while the missing one is replaced by the remark "that this is the case has been shown elsewhere"; indeed theorem and proof are found as Theorem 10 in Autolycus's 'Rotating Sphere'. That a remark of this kind should be genuine in any Greek mathematical treatise, Euclidean or not, seems to me utterly implausible; I would assume the obvious, i.e. that a scholion replaced, perhaps in a damaged copy, the first of four proofs by a simple reference to generally known theorems. In fact I see no reason why Euclid (presumably in Alexandria) and Autolycus (presumably in Athens) should not have written independently, and perhaps even simultaneously, on the mathematical theory of astronomical phenomena.

Huxley, writing in [1], agrees with Heath, and the paper [4] even has the title Autolycus of Pitane, predecessor of Euclid. The priority argument is certainly not an unimportant one for the interdependence of Euclid and Autolycus on each other is significant since Autolycus writes his propositions in exactly the same general style as Euclid. This means that a theorem in Autolycus's work has first a general statement, then a construction related to a particular figure with points in the figure denoted by letters, next comes the demonstration of the theorem, and finally a conclusion relating to the general statement is sometimes drawn. Despite the arguments above, one must still draw the same conclusion whichever treatise was written first, namely that this style of mathematical exposition accepted today as so characteristic of Euclid, was certainly not invented by him. Despite the style being used by Autolycus, nobody credits him with inventing it either.

We have been taken down a fascinating road by the comparisons with Euclid, but we should return to give the only other detail of Autolycus's life which is reported by Diogenes Laertius, when he relates that Arcesilaus was accompanied by his teacher Autolycus on a journey to Sardis.

Another important fact regarding Autolycus is that two of his books have survived in the original Greek and we believe that they are the earliest two mathematics works to have survived. Of these books, On the Moving Sphere is a work on the geometry of the sphere which is the same as being a mathematical astronomy text. The second work On Risings and Settings is a book more on observational astronomy.

Theodosius, 200 years later, wrote Sphaerics, a similar book on the geometry of the sphere, also written to provide a mathematical background for astronomy. It is thought that Theodosius's Sphaerics and Autolycus's work On the Moving Sphere are based on the same pre-Euclidean textbook which is now lost. It is conjectured, on rather little evidence one would have to say, that Eudoxus wrote this earlier text. There seems to be no way in which the speculation on this point can ever be settled.

That Autolycus relys heavily on Eudoxus for his view of astronomy is not in doubt. He is a strong supporter of Eudoxus's theory of homocentric spheres which consisted of a number of rotating spheres, each sphere rotating about an axis through the centre of the Earth. This theory had a difficulty which had been quickly noticed, namely that Venus and Mars varied in brightness and there was no mechanism for this in Eudoxus's theory. Again eclipses of the sun were sometimes total, sometimes annular where moon appears smaller than the sun and a ring of the sun is visible right round the moon. Despite Autolycus's attempts to explain these observations within Eudoxus's theory, he had no real answer to these problems.

On Risings and Settings is a work which consists of two books. Schmidt's interesting article [7] was written to show that in fact these are not two parts of a single two volume work but rather they were both versions of the same piece of work. The second book is actually a revised and expanded edition of the first, which contains quite a bit of new material. It is also a better constructed book and it is interesting to see how Autolycus's book has developed and improved between the two editions.