梅涅劳斯Menelaus of Alexandria

尽管我们对梅涅劳斯的生平知之甚少，克劳狄乌斯·托勒密记录了梅涅劳斯于98年1月14日在罗马进行的天文观测。这些观测包括月亮对Beta Scorpii星的掩星。

他也在普鲁塔克的一部著作中出场，该著作描述了梅涅劳斯与卢修斯之间的一段对话，卢修斯向梅涅劳斯道歉，因为他曾怀疑光在反射时遵循入射角等于反射角这一定律。卢修斯说（例如见[1]）：-

亲爱的梅涅劳斯，在你面前，我羞于反驳一个数学命题，即反射光学主题所赖以成立的基础。然而必须指出，命题“所有反射均以等角发生”既非自明，也非公认的事实。

这段对话据推测发生在罗马，很可能是在公元75年之后相当长的一段时间；事实上，如果我们关于梅涅劳斯生于公元70年的猜测接近正确，那么它必定是在公元75年之后许多年。

关于梅涅劳斯的生平，其他所知甚少，除了帕普斯和普罗克洛都称他为梅涅劳斯。我们能从中推断的只是他曾在罗马和梅涅劳斯待过一段时间，但最可能的情况是他年轻时住在梅涅劳斯，可能在那里出生，后来搬到罗马。

一部于10世纪编纂的阿拉伯数学家名录对梅涅劳斯记载如下（见[1]）：-

他生活在克劳狄乌斯·托勒密之前，因为后者提到了他。他撰写了：《球面命题之书》、《论不同物体的重量与分布的认识》……三部关于《几何原本》的书，由塔比·伊本·库拉编辑，以及《三角形之书》。其中一些已被翻译成阿拉伯文。

在梅涅劳斯的众多著作中，只有Sphaerica流传了下来。它讨论球面三角形及其在天文学中的应用。他是第一个写下球面三角形定义的人，该定义出现在第一卷的开头：-

球面三角形是由球面上的great circles的弧所围成的空间……这些弧总是小于半圆。

在Sphaerica的第一卷中，他为处理球面三角形奠定了基础，正如欧几里得处理平面三角形那样。他使用大圆的弧而不是球面上平行圆的弧。这标志着球面三角学发展的一个转折点。然而，梅涅劳斯似乎对归谬法的证明方法感到不满，而欧几里得经常使用这种方法。梅涅劳斯避免了这种证明定理的方式，因此，对于某些定理，他给出了证明，而欧几里得的证明本可以通过完全不同的方法轻松地适用于球面三角形的情形。

同样值得评论的是[3]：-

在某些方面，他的处理比欧几里得对类似平面情形的处理更为完整。

第2卷将球面几何应用于天文学。它很大程度上遵循了Theodosius在其Sphaerica中给出的命题，但梅涅劳斯给出了好得多的证明。

第3卷讨论球面三角学，并包含梅涅劳斯定理。参见THIS LINK。对于平面三角形，该定理在梅涅劳斯之前就已为人所知：-

……如果一条直线与三角形的三条边相交（其中一条边延长到三角形的顶点之外），那么由此形成的三条不相邻线段之积等于三角形其余三条线段之积。

梅涅劳斯给出了该定理的球面三角形版本，今天也称为梅涅劳斯定理，它作为第III卷的第一个命题出现。该陈述是用球面上相交的大圆来给出的。

阿拉伯人制作了许多梅涅劳斯Sphaerica的译本和评注本。其中一些留存了下来，但差异很大，使得准确重建原著相当困难。另一方面，我们确实知道有些作品是对更早评注的评注，因此很容易看出原著是如何变得模糊不清的。在[6]、[9]和[10]中对这些阿拉伯语译本有详细的讨论。

梅涅劳斯还有其他著作被阿拉伯作者提及，但无论是希腊文原著还是其阿拉伯文译本都已失传。我们在上文引用了10世纪阿拉伯书目中的一条记载，其中记录了一部名为Elements of Geometry的书，共三卷，由塔比·伊本·库拉译成阿拉伯文。该书目还记录了梅涅劳斯的另一部著作，题为Book on Triangles，尽管此书未能留存下来，但已发现了阿拉伯文译本的残篇。

普罗克洛提到了梅涅劳斯的一个几何结果，该结果未出现在留存下来的著作中，人们认为它必定来自刚才提到的文本之一。这是对欧几里得的Elements中一个定理的直接证明，鉴于梅涅劳斯在其留存著作中对归谬法的厌恶，这似乎是他会遵循的一条自然路线。普罗克洛归于梅涅劳斯的新证明是关于该定理的（见托马斯·利特尔·希思对欧几里得的译本）：-

如果两个三角形有两边分别等于两边，但其中一个三角形的底大于另一个三角形的底，那么前一个三角形中由相等直线所夹的角也大于后一个三角形中的对应角。

另一处阿拉伯文献提到梅涅劳斯，表明他的Elements of Geometry包含了阿尔库塔斯对倍立方问题的解答。保罗·塔内里在[8]中论证说，这使人们很可能认为，帕普斯所声称梅涅劳斯曾详细讨论过的那条曲线就是温琴佐·维维亚尼的双曲率曲线。Bulmer-Thomas在[1]中评论道：-

这是一个有吸引力的猜想，但根据目前的证据无法证明。

若干阿拉伯作家相信梅涅劳斯写过一部力学著作。据称该著作研究了阿基米德所研究的平衡以及梅涅劳斯本人所设计的平衡。特别是梅涅劳斯对比重和分析合金感兴趣。

Although we know little of Menelaus of Alexandria's life Ptolemy records astronomical observations made by Menelaus in Rome on the 14th January in the year 98. These observation included that of the occultation of the star Beta Scorpii by the moon.

He also makes an appearance in a work by Plutarch who describes a conversation between Menelaus and Lucius in which Lucius apologises to Menelaus for doubting the fact that light, when reflected, obeys the law that the angle of incidence equals the angle of reflection. Lucius says (see for example [1]):-

In your presence, my dear Menelaus, I am ashamed to confute a mathematical proposition, the foundation, as it were, on which rests the subject of catoptrics. Yet it must be said that the proposition, "All reflection occurs at equal angles" is neither self evident nor an admitted fact.

This conversation is supposed to have taken place in Rome probably quite a long time after 75 AD, and indeed if our guess that Menelaus was born in 70 AD is close to being correct then it must have been many years after 75 AD.

Very little else is known of Menelaus's life, except that he is called Menelaus of Alexandria by both Pappus and Proclus. All we can deduce from this is that he spent some time in both Rome and Alexandria but the most likely scenario is that he lived in Alexandria as a young man, possibly being born there, and later moved to Rome.

An Arab register of mathematicians composed in the 10th century records Menelaus as follows (see [1]):-

He lived before Ptolemy, since the latter makes mention of him. He composed: "The Book of Spherical Propositions", "On the Knowledge of the Weights and Distribution of Different Bodies" ... Three books on the "Elements of Geometry", edited by Thabit ibn Qurra, and "The Book on the Triangle". Some of these have been translated into Arabic.

Of Menelaus's many books only Sphaerica has survived. It deals with spherical triangles and their application to astronomy. He was the first to write down the definition of a spherical triangle giving the definition at the beginning of Book I:-

A spherical triangle is the space included by arcs of great circles on the surface of a sphere ... these arcs are always less than a semicircle.

In Book I of Sphaerica he set up the basis for treating spherical triangles as Euclid treated plane triangles. He used arcs of great circles instead of arcs of parallel circles on the sphere. This marks a turning point in the development of spherical trigonometry. However, Menelaus seems unhappy with the method of proof by reductio ad absurdum which Euclid frequently uses. Menelaus avoids this way of proving theorems and, as a consequence, he gives proofs of some of the theorems where Euclid's proof could be easily adapted to the case of spherical triangles by quite different methods.

It is also worth commenting that [3]:-

In some respects his treatment is more complete than Euclid's treatment of the analogous plane case.

Book 2 applies spherical geometry to astronomy. It largely follows the propositions given by Theodosius in his Sphaerica but Menelaus give considerably better proofs.

Book 3 deals with spherical trigonometry and includes Menelaus's theorem. See THIS LINK. For plane triangles the theorem was known before Menelaus:-

... if a straight line crosses the three sides of a triangle (one of the sides is extended beyond the vertices of the triangle), then the product of three of the nonadjacent line segments thus formed is equal to the product of the three remaining line segments of the triangle.

Menelaus produced a spherical triangle version of this theorem which is today also called Menelaus's Theorem, and it appears as the first proposition in Book III. The statement is given in terms of intersecting great circles on a sphere.

Many translations and commentaries of Menelaus Sphaerica were made by the Arabs. Some of these survive but differ considerably and make an accurate reconstruction of the original quite difficult. On the other hand we do know that some of the works are commentaries on earlier commentaries so it is easy to see how the original becomes obscured. There are detailed discussions of these Arabic translations in [6], [9], and [10].

There are other works by Menelaus which are mentioned by Arab authors but which have been lost both in the Greek and in their Arabic translations. We gave a quotation above from the 10th century Arab register which records a book called Elements of Geometry which was in three volumes and was translated into Arabic by Thabit ibn Qurra. It also records another work by Menelaus was entitled Book on Triangles and although this has not survived fragments of an Arabic translation have been found.

Proclus referred to a geometrical result of Menelaus which does not appear in the work which has survived and it is thought that it must come from one of the texts just mentioned. This was a direct proof of a theorem in Euclid's Elements and given Menelaus's dislike for reductio ad absurdum in his surviving works this seems a natural line for him to follow. The new proof which Proclus attributes to Menelaus is of the theorem (in Heath's translation of Euclid):-

If two triangles have the two sides equal to two sides respectively, but have the base of one greater than the base of the other, it will also have the angle contained by the equal straight lines of the first greater than that of the other.

Another Arab reference to Menelaus suggests that his Elements of Geometry contained Archytas's solution of the problem of duplicating the cube. Paul Tannery in [8] argues that this make it likely that a curve which it is claimed by Pappus that Menelaus discussed at length was the Viviani's curve of double curvature. Bulmer-Thomas in [1] comments that:-

It is an attractive conjecture but incapable of proof on present evidence.

Menelaus is believed by a number of Arab writers to have written a text on mechanics. It is claimed that the text studied balances studied by Archimedes and those devised by Menelaus himself. In particular Menelaus was interested in specific gravities and analysing alloys.