Died: about 130

Although we know little of **Menelaus of Alexandria**'s life Ptolemy records astronomical observations made by Menelaus in Rome on the 14^{th} 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 10^{th} 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. 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 10^{th} 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.

**Article by:** *J J O'Connor* and *E F Robertson*

**April 1999**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Menelaus.html]