阿什凯隆的欧托基奥斯Eutocius of Ascalon

长期以来，人们认为阿什凯隆的欧托基奥斯出生于530年。看看这是怎么来的很有启发性，因为它表明历史研究中存在多少陷阱。阿什凯隆的欧托基奥斯为阿基米德的三部著作写了注释。他对On the Sphere and Cylinder第二卷的注释以这样的陈述结尾：-

……该版本由米利都的机械工程师、我们的老师伊西多罗斯修订。

由此人们认为阿什凯隆的欧托基奥斯是Isidorus的学生，他的年代也据此推断出来。然而，进一步的研究表明，这与阿什凯隆的欧托基奥斯在其他一些著作中的献词等其他信息相矛盾。随后人们意识到，阿什凯隆的欧托基奥斯对阿基米德的On the Sphere and Cylinder的注释末尾的那条评论，是由该著作后来的编者插入的，而此人确实是米利都的Isidorus的学生。人们认为阿什凯隆的欧托基奥斯对阿基米德的注释中的第一部写于约510年。

阿什凯隆的欧托基奥斯，即阿什凯隆的欧托基奥斯的出生地，有着悠久的历史，在《旧约》中被称为Askelon。它在公元前332年被亚历山大大帝征服后获得了希腊名称，城中有希律大帝建造的精美公共建筑。阿什凯隆的欧托基奥斯很可能离开阿什凯隆的欧托基奥斯前往亚历山大里亚求学。

保罗·塔内里令人信服地论证（见[5]），阿什凯隆的欧托基奥斯几乎可以肯定是在亚历山大的阿摩尼乌斯的学生，而且看来他在Ammonius之后成为亚历山大里亚学派的掌门人。Ammonius本人是普罗克洛的学生，阿什凯隆的欧托基奥斯把他对阿基米德的On the sphere and cylinder第一卷的注释献给了他。阿什凯隆的欧托基奥斯在序言中对Ammonius说，请他[1]：-

……如果他因年轻而犯错，请宽容他。他解释说，在他之前，他没有找到关于阿基米德的令人满意的注释，并承诺如果他的著作得到Ammonius的认可，将进一步阐明这位大师的学说。

当然，这读起来像是阿什凯隆的欧托基奥斯在对他老师说话，而保罗·塔内里的推论似乎是可靠的。人们必须假定阿摩尼奥斯确实认可了，因为阿什凯隆的欧托基奥斯接着写了关于阿基米德其他著作的评注，即Measurement of the circle和On plane equilibria。然而，Bulmer-Thomas在[1]中确信阿什凯隆的欧托基奥斯不知道阿基米德的某些其他著作，如求积 of a 抛物线和On spirals，因为他声称他会在其评注的某些自然位置提到它们，而不是在这些地方给出远不合适的参考文献。

阿什凯隆的欧托基奥斯还编辑并撰写了关于阿波罗尼奥斯的圆锥曲线前4卷的评注。托马斯·利特尔·希思写道[2]：-

阿什凯隆的欧托基奥斯对阿波罗尼奥斯的《圆锥曲线》的评注现存前四卷，很可能正是由于它们被阿什凯隆的欧托基奥斯评注过，以及它们比其余部分更初等，这四卷才单独以希腊文保存下来。

人们立刻看到，像阿什凯隆的欧托基奥斯这样的评注者在数学史上非常重要，许多重要著作仅由于评注者的工作才得以保存。

阿什凯隆的欧托基奥斯似乎没有做过任何原创性工作。然而，他的评注包含许多具有历史信息性质的无价内容，否则这些信息可能已完全丢失。托马斯·利特尔·希思列出了其中一些重要信息[2]：-

1. the account of the solutions of the problem of the duplication of the cube, or the finding of two mean proportionals, by Plato, Heron, Philon, Apollonius, Diocles, Pappus, Sporus, Menaechmus, Archytas, Eratosthenes, Nicomedes；
2. the fragment discovered by Eutocius himself containing the missing solution, promised by Archimedes in On the Sphere and Cylinder Book II. 4, of the auxiliary problem amounting to the solution by means of conics of the 三次方程 $(a - x) x^{2} = b c^{2}$。
3. the solutions (a) by Diocles of the original problem of II.4 without bringing in the cubic, (b) by Dionysodorus of the auxiliary cubic equation。

至于对天文学的贡献，阿什凯隆的欧托基奥斯确实为AlmagestⓉ第一卷写了一篇导论（主要论点：源自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语译本，后来被译为拉丁语“Magna Syntaxis”），但奥托·纽格包尔写道[3]：-

阿什凯隆的欧托基奥斯并没有写那种逐章跟随给定文本的普通类型的“评注”。……主要部分涉及六十进制计算的方法：乘法、除法、平方根等。另一章涉及isoperimetric problems，随后是一个关于地球形状和大小的简短部分，基于克劳狄乌斯·托勒密的赤道度数500斯塔德的标准。显然，从阿什凯隆的欧托基奥斯那里没有流传下任何真正具有天文学意义的东西。

Eutocius of Ascalon was for a long time thought to have been born in 530. It is instructive to see how this came about for it shows how many pitfalls there are in the study of history. Eutocius wrote commentaries on three works of Archimedes. His commentary on Book II of On the Sphere and Cylinder ends with the statement:-

... the edition was revised by Isidorus of Miletus, the mechanical engineer, our teacher.

From this it was thought that Eutocius was a pupil of Isidorus and his dates were deduced from this information. However, further investigation showed that this contradicted other information such the dedications that Eutocius makes in some of his other works. It was then realised that the comment at the end of Eutocius's commentary to Archimedes' On the Sphere and Cylinder was inserted by a later editor of the work who was indeed a pupil of Isidorus of Miletus. It is thought that the first of Eutocius's commentaries on Archimedes was written around 510.

Ascalon, where Eutocius was born, had a long history and is mentioned in the Old Testament as Askelon. It received its Greek name after it was conquered by Alexander the Great in 332 BC and the city had fine public buildings built by Herod the Great. It seems likely that Eutocius left Ascalon and went to Alexandria to study.

Paul Tannery has argued convincingly (see [5]) that Eutocius was almost certainly a pupil of Ammonius in Alexandria and it appears that he went on to become head of the Alexandrian School after Ammonius. Ammonius himself was a pupil of Proclus and Eutocius dedicates his commentary on Book I of Archimedes' On the sphere and cylinder to him. Eutocius addresses Ammonius in the preface asking him to [1]:-

... bear with him if he should have erred through youth. He explains that he found no satisfactory commentaries on Archimedes before his own time and promises further elucidation of the master if his work should meet with approval from Ammonius.

Certainly this reads as if Eutocius is addressing his teacher and Paul Tannery's deduction seems secure. One has to assume that indeed Ammonius did approve, for Eutocius went on to write commentaries on other works by Archimedes, namely Measurement of the circle and On plane equilibria. However, Bulmer-Thomas in [1] is convinced that Eutocius did not know of certain other works by Archimedes such as Quadrature of a parabola and On spirals for he claims that he would have referred to them at certain natural places in his commentaries rather than give much less suitable references at these points.

Eutocius also edited and wrote commentaries on the first 4 books of the Conics of Apollonius. Heath writes [2]:-

Eutocius's commentary on Apollonius's "Conics" is extant for the first four Books, and it is probably owing to their having been commented on by Eutocius, as well as to their being more elementary than the rest, that these four Books alone survive in Greek.

One sees immediately that commentators such as Eutocius are very important in the history of mathematics and many important works have only survived due to the work of the commentators.

Eutocius does not appear to have done any original work. However, his commentaries contain much that is invaluable in the nature of historical information which might otherwise have been completely lost. Heath lists some of these important pieces of information [2]:-

1. the account of the solutions of the problem of the duplication of the cube, or the finding of two mean proportionals, by Plato, Heron, Philon, Apollonius, Diocles, Pappus, Sporus, Menaechmus, Archytas, Eratosthenes, Nicomedes;
2. the fragment discovered by Eutocius himself containing the missing solution, promised by Archimedes in On the Sphere and Cylinder Book II. 4, of the auxiliary problem amounting to the solution by means of conics of the cubic equation $(a - x) x^{2} = b c^{2}$.
3. the solutions (a) by Diocles of the original problem of II.4 without bringing in the cubic, (b) by Dionysodorus of the auxiliary cubic equation.

As to contributions to astronomy, Eutocius did write an introduction to Book I of the Almagest Ⓣ but Neugebauer writes [3]:-

Eutocius did not write a "commentary" of the ordinary type that follows a given text chapter by chapter. ... the main part concerns methods of sexagesimal computation: multiplication, division, square roots etc. Another chapter concerns isoperimetric problems, followed by a short section about the shape and size of the earth, based on Ptolemy's norm of 500 stades for the equatorial degree. Obviously nothing of real astronomical interest has come down from Eutocius.