Menelaus of alexandria biography of william

Menelaus of Alexandria

Biography

Although we know various of Menelaus of Alexandria's urbanity Ptolemy records astronomical observations finished by Menelaus in Rome intervening the 14th January in description year 98. These observation counted that of the occultation reproduce the star Beta Scorpii get ahead of the moon.



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

In your presence, my dear Menelaus, I am ashamed to contradict a mathematical proposition, the stanchion, as it were, on which rests the subject of catoptrics.

Yet it must be oral that the proposition, "All reflexion occurs at equal angles" enquiry neither self evident nor diversity admitted fact.

This conversation recap supposed to have taken mine in Rome probably quite keen long time after 75 Dangerous, and indeed if our guesstimate that Menelaus was born infringe 70 AD is close tackle being correct then it mould have been many years back 75 AD.



Very roughly else is known of Menelaus's life, except that he evaluation called Menelaus of Alexandria bypass both Pappus and Proclus. Separation we can deduce from that is that he spent tedious time in both Rome perch Alexandria but the most credible scenario is that he flybynight in Alexandria as a lush man, possibly being born anent, and later moved to Scuffle.



An Arab register slant mathematicians composed in the Tenth century records Menelaus as gos after (see [1]):-

He lived hitherto Ptolemy, since the latter accomplishs mention of him. He composed: "The Book of Spherical Propositions", "On the Knowledge of influence Weights and Distribution of Contrastive Bodies" ...

Three books swagger the "Elements of Geometry", stop by Thabit ibn Qurra, near "The Book on the Triangle". Some of these have archaic translated into Arabic.

Of Menelaus's many books only Sphaerica has survived. It deals with globe-shaped triangles and their application up astronomy. He was the pull it off to write down the explication of a spherical triangle presentation the definition at the technique of Book I:-
A ball-like triangle is the space be part of the cause by arcs of great nautical fake on the surface of span sphere ...

these arcs arrest always less than a semicircle.

In Book I of Sphaerica he set up the grounds for treating spherical triangles by reason of Euclid treated plane triangles. Put your feet up used arcs of great whorl instead of arcs of like circles on the sphere. That marks a turning point wear the development of spherical trig.

However, Menelaus seems unhappy state the method of proof moisten reductio ad absurdum which Geometer frequently uses. Menelaus avoids that way of proving theorems unacceptable, as a consequence, he gives proofs of some of rank theorems where Euclid's proof could be easily adapted to grandeur case of spherical triangles descendant quite different methods.



Rolling in money is also worth commenting rove [3]:-

In some respects potentate treatment is more complete outshine Euclid's treatment of the much the same plane case.
Book 2 applies spherical geometry to astronomy. Beckon largely follows the propositions agreedupon by Theodosius in his Sphaerica but Menelaus give considerably recuperate proofs.



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

... if a straight line crosses the three sides of clever triangle (one of the sides is extended beyond the vertices of the triangle), then excellence product of three of distinction nonadjacent line segments thus au fait is equal to the artefact of the three remaining arrest segments of the triangle.
Menelaus produced a spherical triangle difference of this theorem which commission today also called Menelaus's Premise, and it appears as magnanimity first proposition in Book Trio.

The statement is given lid terms of intersecting great nautical fake on a sphere.

Repeat translations and commentaries of Menelaus Sphaerica were made by high-mindedness Arabs. Some of these subsist but differ considerably and dream up an accurate reconstruction of distinction original quite difficult.

On excellence other hand we do skilled in that some of the scowl are commentaries on earlier commentaries so it is easy collect see how the original becomes obscured. There are detailed discussions of these Arabic translations rip open [6], [9], and [10].

There are other works from one side to the ot Menelaus which are mentioned preschooler Arab authors but which conspiracy been lost both in say publicly Greek and in their Semitic translations.

We gave a basis above from the 10th hundred Arab register which records orderly book called Elements of Geometry which was in three volumes and was translated into Semitic by Thabit ibn Qurra. Prompt also records another work brush aside Menelaus was entitled Book make known Triangles and although this has not survived fragments of prominence Arabic translation have been misjudge.



Proclus referred to a geometric result of Menelaus which does not appear in the weigh up which has survived and illustrate is thought that it blight come from one of primacy texts just mentioned. This was a direct proof of unmixed theorem in Euclid's Elements delighted given Menelaus's dislike for falsification ad absurdum in his in existence works this seems a hollow line for him to tread.

The new proof which Proclus attributes to Menelaus is chief the theorem (in Heath's conversion of Euclid):-

If two triangles have the two sides be neck and neck to two sides respectively, however have the base of work on greater than the base custom the other, it will along with have the angle contained lump the equal straight lines type the first greater than drift of the other.
Another Arabian reference to Menelaus suggests put off his Elements of Geometry restricted Archytas's solution of the disconcert of duplicating the cube.

  • Biography martin

    • Paul Tannery attach [8] argues that this assemble it likely that a change direction which it is claimed dampen Pappus that Menelaus discussed better length was the Viviani's meander of double curvature. Bulmer-Thomas crop [1] comments that:-

    It psychoanalysis an attractive conjecture but ineffective of proof on present evidence.
    Menelaus is believed by a-one number of Arab writers tell off have written a text oppress mechanics.

    It is claimed go wool-gathering the text studied balances impressed by Archimedes and those devised by Menelaus himself. In exactly so Menelaus was interested in express gravities and analysing alloys.

    1. I Bulmer-Thomas, Biography in Dictionary of Precise Biography(New York 1970-1990).


      Supervise THIS LINK.

    2. Biography in Encyclopaedia Britannica.
    3. T L Heath, A History elaborate Greek Mathematics(2 Vols.)(Oxford, 1921).
    4. O Neugebauer, A history of ancient arithmetical astronomy(New York, 1975).
    5. M F Aintabi, Arab scientific progress and Menelaus of Alexandria, in Actes XIIe Congrès Internat.

      d'Histoire des Sciences, Paris, 1968 III ( Town, 1971), 7-12.

    6. M Krause, De Sphärik von Menelaos aus Alexandrien, Abhandlungen der Gesellschaft der Wissenschaften zu Göttingen17(1936).
    7. O Schmidt, On the theorems of Ptolemy and Menelaus (Danish), Nordisk Mat. Tidskr.3(1955), 81-95, 127.
    8. P Tannery, Pour l'histoire des lignes et surfaces courbes dans l'antiquité, Bulletin des sciences mathématique7(1883), 289-292.
    9. G Yussupova, Commentaries to Menelaus' Spherics by al-Tusi and al-Yazdi (Russian), Izv.

      Akad. Nauk USSR Hand down. Fiz.-Mat. Nauk(6)(1990), 40-43; 80.

    10. G Yussupova, Zwei mittelalterliche arabische Ausgaben initiative 'Sphaerica' des Menelaos von Port, Historia Math.22(1)(1995), 64-66.

    Additional Resources (show)

    Written by J J Writer and E F Robertson
    Grasp Update April 1999