**Menelaus** was one of the later Greek geometers who applied spherical geometry to astronomy. He is best known for the so-called Menelaus's theorem.

- 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.
- 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.
- Of Menelaus's many books only Sphaerica has survived.
- 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 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.
- 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.
- Many translations and commentaries of Menelaus Sphaerica were made by the Arabs.
- 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.
- 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.
- Another Arab reference to Menelaus suggests that his Elements of Geometry contained Archytas's solution of the problem of duplicating the cube.
- 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.

Born about 70, (possibly) Alexandria, Egypt. Died about 130.

View full biography at MacTutor

African, Analysis, Ancient Arab, Ancient Greek, Ancient Indian, Astronomy, Geometry, Origin Egypt, Puzzles And Problems

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive