亚历山大的席恩Theon of Alexandria

亚历山大的席恩在亚历山大的席恩担任数学和天文学教师。从他自己的著作中我们知道，他于364年6月16日在亚历山大的席恩观测到一次日食，并于364年11月25日在亚历山大的席恩观测到一次月食。我们还知道，他编制了一份罗马执政官名单，一直持续到372年。

在Suda Lexicon（10世纪希腊词典编纂者的一部著作）中提到，亚历山大的席恩生活在狄奥多西一世皇帝（在位379至395年）统治下。因此这些日期是一致的。Suda还指出，亚历山大的席恩是博物馆的成员，该博物馆是公元前300年在亚历山大的席恩建立的高等教育机构。这同样是可能的，但博物馆肯定没有在亚历山大的席恩时代之后存在多久，如果它在他时代确实存在的话。总的来说，接受他是其最后成员之一是合理的。

亚历山大的席恩是希帕蒂娅的父亲，而且确实似乎在她于415年被谋杀之前他就去世了。除了这些他肯定在工作的时间的少数迹象外，似乎没有其他证据可以让我们更准确地猜测他的出生和死亡日期。

亚历山大的席恩以其对许多著作的评注而闻名，例如克劳狄乌斯·托勒密的AlmagestⓉ（《大论》：源自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语译本，后来译为拉丁语“Magna Syntaxis”）以及欧几里得的著作。这些评注是为他的学生写的，有些甚至被认为是学生在听他的讲座时所做的笔记。对一部著作，他写了两部评注，在第二部的序言中，他解释说，他给出一个更初等的论述，因为他的大多数学生无法理解几何证明。这再次证实，他的教学机构已经失去了活力，而它似乎吸引来的学生素质低下，这可能是博物馆关闭的一个关键因素（如果我们上面所评论的，Suda将其作为他的机构是正确的）。

亚历山大的席恩是一位称职但缺乏独创性的数学家。亚历山大的席恩版本的欧几里得的Elements（带有文本改动和一些增补）被认为是在其女儿希帕蒂娅的协助下写成的，并且是已知唯一的Elements希腊语文本，直到19世纪末在梵蒂冈发现了一个更早的版本。然而，既然梵蒂冈手稿已被发现，就有可能确切地看到亚历山大的席恩在其版本中所做的改动。

亚历山大的席恩似乎采取的方法，是试图改进较早的手稿，而不是试图通过评注来准确地复制。因此，他纠正了他在数学中发现的错误，但不幸的是，并非所有他未能理解的地方都是错误，有些是完全正确的。亚历山大的席恩还试图规范欧几里得的写作方式，所以当亚历山大的席恩遇到一个与常规略有不同的表述时，他就用标准的表述形式来替换它。

然而，从积极的一面来看，亚历山大的席恩在认为某个论证过于简略时，会扩充欧几里得的文本，有时添加命题以使初学者更容易阅读。在这一点上他是成功的，以至于他的版本成为标准版本，几乎所有早期版本都已失传。托马斯·利特尔·希思在写到Theon的Elements版本[2]时说道：-

……虽然对《几何原本》的内容只做了微不足道的增补，但他努力消除学习者在研读此书时可能感到的困难，就像现代编辑为学校使用而编辑经典教科书时所做的那样；毫无疑问，他的版本得到了他在亚历山大的席恩的学生们的认可，因为这是为他们而写的，也得到了后来几乎完全使用该版本的希腊人的认可……

亚历山大的席恩还对欧几里得的其他著作进行了评注。他确实对欧几里得的Optics及其Data进行了评注。亚历山大的席恩对Data的评注写得相对高深，在其中亚历山大的席恩倾向于缩短欧几里得的证明而非扩充它们。另一方面，Optics是初等的，写作风格完全不同，一些历史学家推测它实际上是亚历山大的席恩某位学生的课堂笔记。手稿中多次出现诸如“他说”之类的短语，人们认为确实是某位学生在记录“亚历山大的席恩所说”的内容。

Catoptrica的情况则相当不同，因为这里有一部表面上声称由欧几里得所写的著作。然而这是不可能的，因为其内容是欧几里得时代的著作与远晚于欧几里得时代的著作的混合。该著作的风格和初等性质使得亚历山大的席恩为作者的可能性很大。如果情况如此，那么他再次是为他的薄弱学生而写作。

亚历山大的席恩还撰写了关于克劳狄乌斯·托勒密天文学著作的大量评注，既有关于AlmagestⓉ（《大论》：源自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语译本，后来译为拉丁语“Magna Syntaxis”）的，也有关于Handy tables的。同样，他的女儿希帕蒂娅协助他撰写了关于AlmagestⓉ（《大论》：源自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语译本，后来译为拉丁语“Magna Syntaxis”）的评注，这是亚历山大的席恩最重要的著作。

在其对AlmagestⓉ的评注的序言中（主要论题：来自阿拉伯语“al-majisti”——希腊语“Mathematike Syntaxis”的阿拉伯语翻译，后来译为拉丁语“Magna Syntaxis”），亚历山大的席恩写道，他的意图是改进前人的评注（例如参见[1]）：-

……他们声称只会省略较为明显的要点，但实际上却被证明省略了最困难的部分。

然而，正如Toomer在1中指出的，这正是亚历山大的席恩本人接下来所做的。

亚历山大的席恩写了两部对克劳狄乌斯·托勒密的Handy Tables的评注。小评注只解释如何使用表格，而大评注则解释其构造。大评注最近由Tihon在[5]和[6]中出版。尽管亚历山大的席恩肯定是在大评注之后写的小评注，因为他在小评注的序言中提到了大评注。然而，Tihon发现，现存最古老的手稿，一部可追溯到9世纪的梵蒂冈手稿，表明亚历山大的席恩从未完成其大评注的文本。这部梵蒂冈手稿是从亚历山大的席恩文本的早期抄本制作的，该抄本于463年在叙利亚的阿帕梅亚使用。

至于亚历山大的席恩对克劳狄乌斯·托勒密的Syntaxis的评注，托马斯·利特尔·希思写道[2]：-

这篇评注并不足以让我们对亚历山大的席恩的数学才能给予很高的评价，但它因提供了若干历史记载而有价值，我们得益于它对希腊人处理六十进制分数的方法的有用叙述，该叙述以乘法、除法以及通过近似方式求非平方数的平方根的例子加以说明。

Theon of Alexandria worked in Alexandria as a teacher of mathematics and astronomy. We know from his own writings that he observed a solar eclipse on 16 June 364 at Alexandria and a lunar eclipse, again in Alexandria, on 25 November 364. We also know that he made a list of Roman consuls which he continued to make until 372.

There is a reference in the Suda Lexicon (a work of a 10th century Greek lexicographer) which states that Theon of Alexandria lived under the Emperor Theodosius I (who reigned from 379 to 395). These dates are therefore consistent. The Suda also states that Theon was a member of the Museum, which was an institute for higher education set up in Alexandria in 300 BC. Again this is possible, but the Museum certainly did not exist much beyond the time of Theon if indeed it existed in his time. On balance it seems reasonable to accept that he was one of its last members.

Theon was the father of Hypatia and it certainly seems to be the case that he died before she was murdered in 415. There does not seem to be any other evidence which would let us give a more accurate guess of the dates of his birth and death other than these few indications of times when he was certainly working.

Theon is famed for his commentaries on many works such as Ptolemy's Almagest Ⓣ and the works of Euclid. These commentaries were written for his students and some are even thought to be lecture notes taken by students at his lectures. On one work he gave two commentaries and in the preface to the second he explains that he is giving a more elementary account for the majority of his students are unable to understand geometrical proofs. This again confirms that the vigour had gone out of his teaching establishment and indeed the poor quality of students it seemed to be attracting could have been a telling factor in the closure of the Museum (if as we commented above the Suda is right in giving that as his institution).

Theon was a competent but unoriginal mathematician. Theon's version of Euclid's Elements (with textual changes and some additions) is thought to have been written with the assistance of his daughter Hypatia and was the only Greek text of the Elements known, until an earlier one was discovered in the Vatican in the late 19th century. However, now that the Vatican manuscript has been discovered it is possible to see exactly the changes that Theon made in his version.

The approach that Theon makes appears to make is to try to improve the earlier manuscript rather than to try to reproduce an accurate reproduction with comments. So he corrected mistakes which he spotted in the mathematics, but unfortunately not all the points that he fails to understand are mistakes, some are perfectly correct. Theon also tried to standardise the way that Euclid writes, so when Theon came across an expression which was somewhat different from the norm, he replaced it by the standard form of expression.

On the positive side, however, Theon amplified Euclid's text whenever he thought that an argument was overly brief, sometimes adding propositions to make the text more easily read by beginners. In this he was successful, so much so in fact that his became the standard edition and almost all earlier editions have been lost. Heath writes of Theon's edition of the Elements [2]:-

.. while making only inconsiderable additions to the content of the "Elements", he endeavoured to remove difficulties that might be felt by learners in studying the book, as a modern editor might do in editing a classical text-book for use in schools; and there is no doubt that his edition was approved by his pupils at Alexandria for whom it was written, as well as by later Greeks who used it almost exclusively...

Theon also produced commentaries on other works of Euclid. Certainly he produced a commentary on Euclid's Optics and on his Data. Theon's commentary on the Data is written at a relatively advanced level and in it Theon tends to shorten Euclid's proofs rather than to amplify them. The Optics on the other hand is elementary and written in a totally different style and some historians conjecture that it is really a set of lecture notes by one of Theon's students. Many times the manuscript contains a phrase such as "he said" and it is thought that a student is indeed writing down what "Theon said".

The Catoptrica is a rather different case for here we have a work which on the face of it claims to be written by Euclid. This however is impossible since the contents are a mixture of work dating from Euclid's time together with work which is much later than Euclid's time. The style and elementary nature of the work make authorship by Theon a distinct possibility. If this is the case then again he is writing for his weak students.

Theon also wrote extensive commentaries on the astronomical works of Ptolemy, both on the Almagest Ⓣ and the Handy tables. Again his daughter Hypatia assisted him in the commentary on the Almagest Ⓣ and this is Theon's most major piece of work.

In the preface to his commentary on the Almagest Ⓣ Theon writes that his intention is to improve on previous commentators (see for example [1]):-

... who claim that they will only omit the more obvious points, but in fact prove to have omitted the most difficult.

However, as Toomer points out in [1], this is exactly what Theon himself goes on to do.

Theon wrote two commentaries on Ptolemy's Handy Tables. The small commentary only explains how to use the tables while the large commentary explains their construction. The larger commentary has been published recently by Tihon in [5] and [6]. Although Theon certainly wrote the small commentary after the larger one, since he refers to the larger commentary in the preface to the smaller. However, Tihon discovered that the oldest manuscript which has been preserved, a Vatican manuscript dating from the 9th century, suggests that Theon never completed the text of his large commentary. This Vatican manuscript is made from an earlier copy of Theon's text which was being used in the year 463 in Apamea in Syria.

As to Theon's commentary on Ptolemy's Syntaxis Heath writes [2]:-

This commentary is not calculated to give us a very high opinion of Theon's mathematical calibre, but it is valuable for several historical notices that it gives, and we are indebted to it for a useful account of the Greek method of operating with sexagesimal fractions, which is illustrated with examples of multiplication, division, and the extraction of the square root of a non-number by way of approximation.