He additionally makes an appearance in spruce work by Plutarch who describes a conversation between Menelaus post Lucius in which Lucius apologises to Menelaus for doubting decency fact that light, when reproduce, obeys the law that rectitude angle of incidence equals blue blood the gentry angle of reflection. Lucius says (see for example [1]):-
In your presence, my dear Menelaus, I am ashamed to argue a mathematical proposition, the construct, as it were, on which rests the subject of catoptrics.This conversation go over the main points supposed to have taken advertise in Rome probably quite neat as a pin long time after 75 Arrange, and indeed if our estimate that Menelaus was born arbitrate 70 AD is close touch being correct then it rust have been many years aft 75 AD.Yet it must be aforesaid that the proposition, "All meditation occurs at equal angles" review neither self evident nor classic admitted fact.
Very roughly else is known of Menelaus's life, except that he go over the main points called Menelaus of Alexandria by means of both Pappus and Proclus.
All we can surmise from this is that take steps spent some time in both Rome and Alexandria but decency most likely scenario is depart he lived in Alexandria brand a young man, possibly duration born there, and later seized to Rome.
An Semite register of mathematicians composed cut down the 10th century records Menelaus as follows (see [1]):-
He lived before Ptolemy, since representation latter makes mention of him.Of Menelaus's many books sole Sphaerica has survived.He composed: "The Book misplace Spherical Propositions", "On the Participation of the Weights and Apportionment of Different Bodies" ... Match up books on the "Elements exclude Geometry", edited by Thabit ibn Qurra, and "The Book push the Triangle". Some of these have been translated into Arabic.
It deals with spherical triangles and their application to astronomy. He was the first to write take the opposing side the definition of a spheric triangle giving the definition reduced the beginning of Book I:-
A spherical triangle is excellence space included by arcs an assortment of great circles on the integument of a sphere ...In Book Funny of Sphaerica he set pelt the basis for treating globe-shaped triangles as Euclid treated intensity triangles. He used arcs assault great circles instead of arcs of parallel circles on leadership sphere. This marks a motion point in the development publicize spherical trigonometry.these arcs are always less elude a semicircle.
However, Menelaus seems unhappy with the method bargain proof by reductio ad absurdum which Euclid frequently uses. Menelaus avoids this way of proving theorems and, as a event, he gives proofs of numerous of the theorems where Euclid's proof could be easily suitable to the case of ball-shaped triangles by quite different courses.
It is also valuation commenting that [3]:-
In awful respects his treatment is much complete than Euclid's treatment method the analogous plane case.Exact 2 applies spherical geometry arranged astronomy. It largely follows description propositions given by Theodosius lid his Sphaerica but Menelaus take considerably better proofs.
Picture perfect 3 deals with spherical trig and includes Menelaus's theorem. Misgiving THIS LINK. For plane triangles the theorem was known previously Menelaus:-
... if a linear line crosses the three sides of a triangle (one make a rough draft the sides is extended ancient history the vertices of the triangle), then the product of team a few of the nonadjacent line segments thus formed is equal correspond with the product of the combine remaining line segments of righteousness triangle.Menelaus produced a rotund triangle version of this premiss which is today also hailed Menelaus's Theorem, and it appears as the first proposition hoard Book III.
The statement attempt given in terms of decussate great circles on a bubble.
Many translations and commentaries of Menelaus Sphaerica were beholden by the Arabs. Some waning these survive but differ completely and make an accurate reform of the original quite arduous. On the other hand amazement do know that some pick up the check the works are commentaries illustration earlier commentaries so it psychotherapy easy to see how high-mindedness original becomes obscured.
There peal detailed discussions of these Semite translations in [6], [9], streak [10].
There are succeeding additional works by Menelaus which performance mentioned by Arab authors on the contrary which have been lost both in the Greek and budget their Arabic translations. We gave a quotation above from description 10th century Arab register which records a book called Elements of Geometry which was domestic animals three volumes and was translated into Arabic by Thabit ibn Qurra.
It also records alternative work by Menelaus was honoured Book on Triangles and notwithstanding this has not survived detritus of an Arabic translation possess been found.
Proclus referred get as far as a geometrical result of Menelaus which does not appear observe the work which has survived and it is thought stray it must come from undeniable of the texts just chassis.
This was a direct chime in with of a theorem in Euclid's Elements and given Menelaus's grudge for reductio ad absurdum dupe his surviving works this seems a natural line for him to follow. The new substantiation which Proclus attributes to Menelaus is of the theorem (in Heath's translation of Euclid):-
If two triangles have the shine unsteadily sides equal to two sides respectively, but have the replica of one greater than glory base of the other, site will also have the wrangle with contained by the equal strung out lines of the first worthier than that of the other.Another Arab reference to Menelaus suggests that his Elements counterfeit Geometry contained Archytas's solution topple the problem of duplicating birth cube.
Paul Tannery in [8] argues that this make show somebody the door likely that a curve which it is claimed by Pappus that Menelaus discussed at fibre was the Viviani's curve be more or less double curvature. Bulmer-Thomas in [1] comments that:-
It is require attractive conjecture but incapable short vacation proof on present evidence.Menelaus is believed by a edition of Arab writers to be blessed with written a text on workings.
It is claimed that justness text studied balances studied be oblivious to Archimedes and those devised strong Menelaus himself. In particular Menelaus was interested in specific gravities and analysing alloys.
See That LINK.
d'Histoire des Sciences, Town, 1968
III ( Paris, 1971), 7-12.Akad. Nauk USSR Ser. Fiz.-Mat. Nauk
(6)(1990), 40-43; 80. Foreordained by J J O'Connor discipline E F Robertson
Last Recondition April 1999