BIOGRAPHY: Menelaus of Alexandria

In Book I of sphaerica he created the basis for the treatment of spherical triangles as triangles Euclidean plane processing. He used arcs of great circles instead of arcs of parallel circles on the sphere. This marks a milestone in the development of spherical trigonometry. However, Menelaus seemed satisfied with the method of proof by reductio ad absurdum which Euclid uses frequently. Menelaus avoid this way of proving theorems, and consequently, it gives evidence of some theorems of Euclid where the evidence could easily be adapted to the case of spherical triangles by very different methods.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 hard enough. On the other hand we know that some works are commentaries on the previous comments so it is easy to see how the original was darkened. There are detailed discussions of these translations Arabic [6], [9] and [10]. There are other works of Menelaus, who are mentioned by Arab authors, but were lost in both Greek and their translation into Arabic.We have given a quote above from the registry of the 10th century recorded an Arabic book entitled Elements of Geometry has been in three volumes and has been translated into Arabic by Thabit ibn Qurra. He also recorded another work by Menelaus was entitled Book of Triangles and even if it did not survive fragments of an Arabic translation was found. Proclus refers to a geometric sequence of Menelaus, who does not appear in the work that has survived and is thought to 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):- 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.

Geometry, Our Cultural Heritage

The mathematics of the heavens and the earth, the early history of trigonometry

Greek science of the Hellenistic era, a sourcebook

The development of trigonometry from Regiomontanus to Pitiscus

A History of Mathematics