费隆Philo of Byzantium

文献中只有少数几处提到费隆。罗马建筑师和工程师维特鲁威提到了他。维特鲁威（公元前1世纪）是著名论著De architectura（《论建筑》）的作者，在这部作品中他列出了十二位机械发明者，其中包括阿尔库塔斯（列表中的第二位）、阿基米德（列表中的第三位）、克特西比乌斯（列表中的第四位）和费隆（列表中的第六位）。

亚历山大的海伦提到了费隆的一部著作On automatic theatres，该著作实际上构成了他的Mechanics论著的一部分。阿什凯隆的欧托基奥斯也提到了费隆，并引用了他关于duplication of the cube的一部著作，这些材料同样包含在他的Mechanics论著中。也许关于费隆生平的最多信息，而且这确实非常少，来自他唯一幸存下来的著作（至少主要部分幸存下来）Mechanics。在这部论著中，他写到了最近由克特西比乌斯发明的弩炮，我们在上文提到过，在维特鲁威给出的发明者列表中，克特西比乌斯排在费隆之前。根据这些信息，我们可以相当准确地确定费隆的年代，并且我们知道他在公元前250年左右撰写了论著Mechanics。

在描述费隆的杰作Mechanics的内容之前，让我们给出一些关于费隆生平的细节，这些细节可以从他在该文本中所作的评论推断出来。当然，费隆描述了他前往罗得岛和亚历山大港研究弩炮的旅行。他似乎与亚历山大港的统治者讨论过弩炮的军事应用。这里的语气表明费隆是一个拥有独立财富的富人，能够为追求研究而旅行。另一方面，他可能被认为是那种在军事事务上应征求其建议的合适人选，他可能通过向军事统治者提供建议来谋生。

费隆的Mechanics论著中究竟写了什么？我们知道它共有九卷：1. 导论
2. 论杠杆
3. 论海港的建造
4. 论弩炮
5. 论气体力学
6. 论自动剧场
7. 论堡垒的建造
8. 论围攻与守城
9. 论计谋。第4、5、7、8卷的文本留存了下来，其余各卷已佚失。然而费隆在其著作中有充分交叉引用的习惯，因此通过研究留存下来的部分，我们可以了解到不少已佚失部分的内容。这部论著的风格相当不寻常，因为各卷由许多短章组成。例如第8卷包括两个部分，第一部分有75章，第二部分有111章。

这部论著不仅仅是一部我们今天会视为应用数学的著作。例如第8卷除了描述从陆地和海上防御城墙的方法外，还强调拥有一个好医生的重要性。费隆主张，那些在攻击中受重伤以致无法再工作的人应获得养老金，阵亡者的妻子应得到供养。

根据费隆的说法，要通过围城攻占一座城镇，必须恰当使用弩炮和其他战争器械等机器。此外，还必须设法让城中居民挨饿，贿赂合适的人来协助你，使用他的毒药配方来杀死居民，还要使用密码术来传递秘密信息。如果能了解他提出的密码术的细节会很有趣，但不幸的是费隆关于这一主题的著作已经失传。

费隆的一项重要数学贡献是解决倍立方问题。乍一看，这似乎与我们注意到的他论著中的主题相去甚远。然而并非如此，因为费隆考察了以下问题。给定一台弩炮，如何制造第二台弩炮，使其能发射比第一台重一倍的投射物。要做到这一点，必须构造一台机器，其线性尺寸增加的幅度恰好使其体积（线性尺寸的立方）加倍。

他倍立方的方法与海伦的方法类似。其解法实际上是通过一个圆与一个矩形双曲线的交点产生的。

Only a few references to Philon of Byzantium exist in the literature. He is mentioned by Vitruvius who was a Roman architect and engineer. Vitruvius (1st century BC) was the author of the famous treatise De architectura (On Architecture) and in this work he gives a list of twelve inventors of machines which include Archytas (second in the list), Archimedes (third in the list), Ctesibius (fourth in the list), and Philon of Byzantium (sixth in the list).

Heron of Alexandria mentions a work by Philon On automatic theatres which in fact forms part of his Mechanics treatise. Eutocius also mentions Philon and cites a work by him on the duplication of the cube and this material is again contained in his Mechanics treatise. Perhaps the most information about Philon's life, and this is very little indeed, comes from the only work of his which has survived (at least major parts have survived) Mechanics. In this treatise he writes about the catapult which was recently invented by Ctesibius, who we mentioned above as coming before Philon in the list of inventors given by Vitruvius. From this information we can date Philon fairly accurately and we know that he wrote his treatise Mechanics around 250 BC.

Before describing the contents of Philon's masterpiece Mechanics let us give some small details of Philon's life which can be deduced from comments which he makes in this text. Certainly Philon describes journeys he had made to Rhodes and to Alexandria to study catapults. He appears to have discussed military applications of catapults with the rulers of Alexandria. The tone here would suggest that Philon was a wealthy man of independent means able to travel in the pursuit of his studies. On the other hand it is possible that he was considered the right sort of person whose advice should be sought on military matters and he may have been earned his living advising military rulers.

What exactly was in Philon's Mechanics treatise? We know that it had nine books:

This treatise is not just a work on what we would consider today to be applied mathematics. For example Book 8, in addition to describing ways of defending town walls from both land and sea attack, also stresses how important it is to have a good doctor available. Philon argues that those badly injured in attacks so that they cannot work again should be awarded pensions, and that the wives of those killed should be provided for.

To capture a town through a siege one must, according to Philon, make proper use of machines such as catapults and other war engines. In addition one must try to starve the inhabitants of the town, bribe suitable people to assist you, use his poison recipes to kill the inhabitants, and also use cryptography to pass secret messages. It would be interesting to have details of his proposed cryptography but unfortunately Philon's work on this topic has been lost.

One important mathematical contribution by Philon was to the problem of duplicating the cube. At first sight this seems far removed from the topics we have noted that are in his treatise. However, this is not so for Philon examines the following problem. Given a catapult, how do you make a second catapult which can fire a missile twice as heavy as the first. To do this it is necessary to construct a machine whose linear dimensions are increased exactly the amount necessary for its volume (the cube of the linear dimension) to double.

His method of duplicating the cube is similar to that due to Heron. The solution is effectively produced by the intersection of a circle and a rectangular hyperbola.