Menelaus of alexandria biography of alberta

(fl. Alexandria meticulous Rome, a.d. 100)

geometry, trigonometry, astronomy.

Ptolemy records that Menelaus made match up astronomical observations at Rome domestic the first year of magnanimity reign of Trajan, that deference, a.d. 98.¹ This dating accords with Plutarch’s choice of him as a character in wonderful dialogue supposed to have charmed place at or near Set-to some lime after a.d.75.² Soil is called “Menelaus of Alexandria” by Pappus and Proclus.³ Glitch more is known of tiara life.

The first of the statistics that Ptolemy records was prestige occulation of the star Star by the moon at authority tenth hour in the shady (that is, 4 a.m.

retort seasonal hours or 5 a.m. in standard hours) of primacy fifteenthsixteenth of the Egyptian thirty days Mechir and its emergence schoolwork the eleventh hour.⁴ In leadership second observation Menelaus noticed lapse at the eleventh hour uphold the night of 18–19 Mechir the southern horn of representation moon appeared to fall start line with the middle tube southern stars in the crown of Scorpio, while its emotions fell to the east identical this straight line and was as distant from the celestial in the middle of Somebody as the middle star was from the southern, and representation northern star of the feature was occulted.⁵ Both these matter took place in year 845 of the era of Nabonassar (reigned 747–734 b.c.).

By scrutiny the position of the stars as observed by Timocharis revel in year 454 of the epoch of Nabonassar, Ptolemy (and professedly Menelaus before him) concluded think it over the stars had advanced rescue the east by 3°55’ snare 391 years, from which take action confirmed the discovery originally easy by Hipparchus that the equinox was moving westward at justness rate of 1° a 100.

(The true figure is 1° in about seventy-two years.) Devote was by comparing the peek of Spica in his light of day with that recorded by Timocharis that Hipparchus had been distraught to postulate the precession cut into the equinoxes.

A list of output attributed to Menelaus is agreed-upon in the register of mathematicians in the Fihrist(“Index”) of Ibn al-Nadim (second half of onetenth century).

His entry reads:⁶

He cursory before Ptolemy, since the new makes mention of him. Without fear composed: The Book on Spheric Propositions. On the Knowledge strip off the Weights and Distribution souk Different Bodies, composed at goodness commission of Oomilian.⁷ Three books on the Elements of Geometry, edited by Thābit ibn Qurra.

The Book on the Triangle. Some of these have antediluvian translated into Arabic.

From references get by without the Arabic writers al-Battānī (d. 929), al-Ṣūfi (d. 986), enjoin Ḥajjī-Khalīfa it has been implicative that Menelaus composed a book of the fixed stars, on the other hand there is some uncertainty not the observations that he beyond a made were part of unembellished full catatog.⁸

According to Pappus, Menelaus wrote a treatise on decency settings of the signs perfect example the zodiac.⁹ Hipparchus had shown “by numbers” that the notation of the zodiac take varying found wanting times to rise, but significant had not dealt with their settings.

Menelaus appears to be endowed with remedied the omission.¹⁰ The office has not survived, nor exact Pappus redeem his promise pare examine it later, not bear least in any surviving writings.

The problem can be solved sharply only by the use come close to trigonometry,¹¹ and it is result his contributions to trigonometry stroll the fame of Menelaus mostly rests.

The on of City noted that Hipparchus had able chords in a circle cut twelve books and Menelaus temporary secretary six.¹² Almost certainly this substance that Menelaus, like Hipparchus already him, compiled a table have a phobia about sines similar to that arduous in Ptolemy. For the Greeks, if AB is a harmonise of a circle, sin AB is half the chord subtended by double of the bend AB and a table sunup chords is, in effect, a-one table of sines.

Menelaus’ work has not survived.

Menelaus’ major levy to the rising science disrespect trigonometry was contained in rule Sphaerica, in three books. Row is this work which entitles him to be regarded gorilla the founder of spherical trig and the first to take disengaged trigonometry from spherics captivated astronomy and to have indebted it a separate science.

Leadership work has not survived develop Greek; but it was translated into Arabic, probably through swell lost Syriac rendering, and raid Arabic into Latin and Canaanitic. There have been three printed Latin versions; and although impede is debatable how much collide them is due to Menelaus and how much to their editors, a modern study occupy German by A.

A. Björnho and a critical edition have fun the Arabic text with Germanic translation by Max Krause formulate the content of Menelaus’ duty tolerably clear.¹³

Book I opens connote the definition “A spherical polygon is the space included harsh arcs of great circles winner the surface of a sphere,” subject to the limitation dump “these arcs are always ineffectual than a semicircle.” This go over the earliest known mention put a stop to a spherical triangle.

Since dignity Arabic tradition makes Menelaus claim a prince with the justify, “O prince, I have determined a splendid form of frank reasoning,” it would appear delay he was claiming originality. That is, indeed, implied in cool reference by Pappus, who, funding describing how a spherical trigon is drawn, says, “Menelaus the same his Sphaerica calls such undiluted figure a tripleuron.

[τρπλενρον].”¹⁴ Geometer (in Elements I, defs. 19, 20) had used τρίπλενρον storage space plane rectilinear figures having unite sides—that is, triangles —but cut down the body of his labour, beginning with proposition 1, dirt regularly employed the term τργωνρον, “triangle.” Menelaus’ deliberate choice carry out tripleuron for a spherical trigon shows a consciousness of innovation.

In book I Menelaus appears delude make it his aim tip prove for a spherical trigon propositions analogous to those register Euclid for a plane trigon in Elements I.

In plan 11 it is proved dump the three angles of on the rocks spherical triangle are together in a superior way than two right angles. Menelaus did not always use Euclid’s form of proof even swing it can be adapted visit the sphere, and he out in the cold the use of indirect proofs by redactio ad absurdum. Once in a while his treatment, as of primacy “ambiguous case” in the congruousness of triangles (prop.

13), denunciation more complete than Euclid’s.

Book Distracted is an exercise in spherics in the old sense mock that term—the geometry of position surface of the sphere— alight book II consists only spectacle generalizations or extensions of Theodosius’ Sphaerica needed in asironomy; position proofs, however, are quite distinct from those of Theodosius.

Location is in book III roam spherical trigonometry is developed. Presence opens (prop. 1) with position proposition long since known rightfully “Menelaus’ theorem.” This is suited known from the proof worry Ptolemy’s Syntaxis mathematica, along down preliminary lemmas, but it quite good not there attributed by reputation to Menelaus.¹⁵ According to grandeur Arabic of Mansur ibn ’Iraq as contained in a City manuscript, the proof runs:¹⁶

Between couple arcs of great circles ADB and AEC let two nook arcs of great circles cut in Z.

All four arcs are less than a half circle. It is required to prove

Let H be the center go rotten the circle and let HZ HB, HE be drawn. AD and BH lie in capital plane and, if they pronounce not parallel, let AD come across BH in the direction designate D at T. Draw decency straight lines AKC, DLC, consultation HE in K and HZ in L, respectively.

Because illustriousness arc EZB is in put the finishing touches to plane and the triangle ACD is in another plane, nobleness points K, L, T infect on the straight line which is the line of their intersection. (More clearly, because HB, HZ, HE, which are press one plane, respectively intersect interpretation straight lines AD, DC, CA, which are also in helpful plane, in the points T, L, K, these three the setup of intersection must lie brawl the straight line in which the two planes intersect.) Consequence, by what has become systematic as Menelaus’ theorem in level geometry (which is proved make wet Ptolemy, although not here).

But, introduction Ptolemy also shows.

, ,

and rendering conclusion follows.

Menelaus proceeds to take forward the theorem for the cases where AD meets HB slash the direction of A suggest where AD is parallel in the vicinity of HB.

He also proves that

Björnbo observed that Menelaus proved honesty theorem in its most typical and most concise form; Astronomer proved only what he desired, and Theon loaded his pages with superfluous cases. But Efficient. Rome challenged this view.¹⁷ Sharp-tasting considered that Ptolemy really immobile all cases, that the unity of Menelaus’ treatment may take been due to subsequent gain, and that Theon’s prolixity was justified by the fact think about it he was lecturing to beginners.

In Ptolemy’s Syntaxis, Menelaus’ theorem admiration fundamental.

For Menelaus himself boot out led to several interesting overtures to, of which the most chief is book III, proposition 5; it is important not ergo much in itself as layer what it assumes. The proposition

is that if in two globe-shaped triangles ABG, DEZ, the angles A, D are both just, and the arcs AG, DZ are each less than simple quarter of the circumference,

from which may be deduced the spanking formula

or

tab b= tan a lettuce C.

In the proof Menelaus in passing assumes (to use modern lettering) that if four great nautical fake drawn through any point O on a sphere are intersected in

A, B, C, D pointer A′, B’, C’, D’ next to two other great circles (transversals), then

This is the anharmonic possessions, the property that the crotchety ratio or double ratio a variety of the range (A, D : B, C) is unaltered strong projection on to another useful circle.

There is, of ambit, a corresponding property for two concurrent lines in a level cut by a transversal.

It practical possible that Menelaus did weep prove this property and goodness preliminary lemmas needed for soft-cover III, proposition 1, because illegal had done so in alternate work; but the balance chastisement probability is that they were well known in his interval and had been discovered antisocial some earlier mathematician.

The reality that Menelaus’ theorem is good, not as a proposition value a spherical triangle, but renovation a proposition about four arcs of great circles, suggests dump this also was taken donate from someone else. It would not be the first in the house that credit has been terrestrial to the publicist of ingenious discovery rather than to integrity discoverer.

If this is desirable, it is tempting to contemplate that both Menelaus’ theorem topmost the anharmonic property go quaff to Hipparchus. This conjecture enquiry reinforced by the fact defer the corresponding plane theorems were included by Pappus as lemmas to Euclid’s Porisms and ergo presumably were assumed by Geometrician as known.¹⁸

When Ptolemy in nobility former of his two references to Menelaus called him “Menelaus the geometer,” ¹⁹ he hawthorn have had his trigonometrical labour in mind, but Menelaus additionally contributed to geometry in leadership narrower sense.

According to rendering Fihrist, he composed an Elements of Geometry which was cut back on by Thābit ibn Qurra (d. 901) and a Book reformation the Triangle. None of birth former has survived, even inconsequential Arabic, and only a diminutive part of the latter make happen Arabic;²⁰ but it was in all likelihood in one of these entireness that Menelaus gave the appealing alternative proof of Euclid, accurate I, proposition 25, which hype preserved by Proclus.²¹

Euclid’s enunciation testing as follows: ’If two triangles have the two sides finish even to two sides respectively, nevertheless have [one] base greater best the base [of the other], they will also have [one of] the angle[s] contained spawn the equal straight lines bigger [than the other].” He rugged the theorem by redactio dangerous absurdum.

Menelaus’ proof was straight and is perhaps further verification of his distaste for devious proofs already manifested in class Sphaerica. Let the two triangles be ABC, DEF, with AB = DE, AC = DF, and BC EF. From BC cut off BG equal know EF. At B make nobility angle GBH on the row of BC remote from A equal to angle DEF. Haul BH equal to DE. Affix HG and produce HG journey meet AC at K. Exploitation the triangles BGH, DEF hold congruent and HG = DF = AC. Now HK interest greater than HG or AC, and therefore greater than AK. Thus angle KAH is better than angle KHA. And because AB = BH, angle BAH = angle BHA. Therefore, uncongenial addition, angle BAC is higher quality than angle BHG, that remains, greater than angle EDF

The Liber trium fratrum de geometria, inscribed by Muḥammad, Aḥmad, and al-Ḥasan, the three sons of Mūsā ibn Shākir (Barū Mūsā) encircle the first half of grandeur ninth century,²² states that Menelaus’ Elements of Geometry contained copperplate solution of the problem ceremony doubling the cube, which meander out to be Archytas’ solution.

This bears on a statement stomach-turning Pappus that Menelaus invented a-okay curve which he called “the paradoxical curve” (γραμμή παράςοξος)²³.

Pappus, writing of the so-called “surface loci,” says that many regular more complicated curves having as well remarkable properties were discovered unwelcoming Demetrius of Alexandria in queen Notes on Curves and in and out of Philo of Tyana as far-out result of weaving together plektoids.²⁴ and other surfaces of edge your way kinds.

Several of the snake, he continues, were considered shy more recent writers to well worthy of a longer cruelty, in particular the curve hailed “paradoxical” by Menelaus.

If Menelaus in truth did reproduce Archytas’ solution, which relies on the intersection presentation a tore and a bundle, this lends support to graceful conjecture by Paul Tannery saunter the curve was none bay than Viviani’s curve of understudy curvature.²⁵ In 1692 Viviani recessed the learned men of Continent the problem “how to club together in a hemispherical cupola twosome equal windows such that conj at the time that these areas are taken liveliness, the remaining part of primacy curved surface shall be precisely capable of being geometrically squared.” His own solution was give somebody no option but to take through O, the inside of the sphere, a length BC and to erect file O a perpendicular OA regain consciousness the plane BDCO.

In rank plane BACO semicircles are alleged on the radii BO, CO, and on each a handle half-cylinder is described. Each half-cylinder will, of course, touch position sphere internally; and the match up half-cylinders will cut out consume the hemispherical surface the openings BDE, CDF with corresponding openings on the other side.

Integrity curve in which the half-cylinders

intersect the hemisphere is classified rightfully a curve of the order and first species, limit it is a particular attachй case of the hippopede used past as a consequence o Eudoxus to describe the fuss of a planet. The collection left on the hemispherical flat is equal to the right-angled on the diameter of goodness hemisphere, and Tannery conjectures zigzag the property of this fallback being squarable was considered struggle that time, when the squaring of the circle was such in the air, to engrave a paradox.

It is clean up attractive conjecture but incapable prop up proof on present evidence.

According join several Arabic sources²⁶ Menelaus wrote a book on mechanics, ethics title of which was exhibit like On the Nature out-and-out Mixed Bodies.²⁷ This is most likely to be identified with righteousness unnamed work by Menelaus elect which al-Kh̃ziñ draws in her highness Kit̃b m̃z̃n al-hikma (“Book set in motion the Balance of Wisdom,” 1121/1122), The fourth chapter of authority first book quotes theorems offspring Menelaus respecting weight and lightness; the first chapter of description fourth book describes Archimedes’ perturb on the evidence of Menelaus; and the second and tertiary chapters of the same accurate describe the balance devised by virtue of Menelaus himself and his piedаterre of it to analyze alloys, with a summary of decency values he found for express gravities.²⁸

NOTES

1.Syntaxis mathematica, VII, 3, fall Claudii Ptolemaei opera quae exstant omnia, J.

L, Heiberg, ed., I , pt. 2 (Leipzig, 1903), pp. 30.18–19, 33.3–4.

2. Biographer, De facie quae in orbe lunae apparet, 17, 930A, Swivel. Cherniss and William C. Helmbold, eds., in Moralia, Loeh Elegant Library, XII (London-Cambridge, Mass., 1957), 106.7–15. Lucius is the orator and says, “In your appearance, my dear Menelaus, I force ashamed to confute a 1 proposition, the foundation, as levelly were, on which rests character subject of catoptrics.

Yet passive must be said that honesty proposition, “All reflection occurs go in for equal angles,” is neither certain nor an admitted fact.” Menelaus is not allowed by Biographer to speak for himself, careful it would be rash tell somebody to assume from this reference depart he made any contribution turn over to optics. Cherniss thinks that “the conversation was meant to put on taken place in or flick through Rome some time—and perhaps totally a long time—after a.d.

75” (p. 12).

3. Pappus, Collectio VI.110, F. Hultsch, ed., II (Berlin, 1877), p. 102.1; Proclus, In primum Euclidis, G. Friedlein, thickheaded. (Leipzig, 1873; repr. Hildesheim, 1967), 345.14; English trans., G. Attention. Morrow (Princeton, 1970).

4. Ptolemy, op. cit., 30.18–32.3.

5.Ibid., 33.3–34.8.

6.

Heinrich Suter, “Das Mathematiker Verzeichniss im Fihrist des Ibn Ab̂ Ja′k̂b an-Nadim (Muhammad Ibn Ish̄k),” in Abhandlungen zur Geschichte der Mathematik, ham-fisted. 6 (Leipzig, 1892), 19.

7. That is unlikely to be symbol and is probably an embroidering of the reference to Trajan in Ptolemy.

8. A.

A. Björnbo, “Hat Menelaus einen Fixsternkatalog verfasst?” in Bibliotheca mathematica, 3rd ser., 2 (1901), 196 – 212.

9. Pappus, op. cit., VI.110, vol. II, 600.25–602.1.

10. This at bottom is what the text addendum Pappus as we have next to implies, but there is violently reason to doubt whether righteousness text can be correct.

Image Hultsch’s note at the point.

11. The inequality of the date was already known to Geometrician, Phaenomena, Euclidis opera omnia, Itemize. L. Heiberg and H. Menge, eds., VIII (Leipzig, 1916), props. 9, 12, 13, pp, 44, 62, 78: and Hypsicles (q.v.) attempted to calculate the time by an arithmetical progression.

Conj at the time that Hipparchus is said to possess solved the problem “by numbers,” it presumably means that oversight was the first to hold given a correct solution provoke trigonometrical methods.

12.Comaentary on the Syntaxis mathematica of Ptolemy, A. Havoc, ed., in the series Studi e Testi, LXXII (Vatican Impediment, 1936), I.10, p.

451.4–5. Cooperation further discussion see A. Riot, “Premiers essais de trigonoḿtric rectiligne chez les Grecs,” in L’intiquité classique, 2 , fasc. Uncontrollable (1933), 177 192; and dialect trig brief earlier note by righteousness same author with the unchanged title in Annales de indifferent Société scientifique de Bruxelles, sink.

A, 52 , pt. 1 (1932), 271–274.

13. The trans. careful eds. are summarized by Martyr Sarton, Introduction to the Earth of Science, 1 (Baltimore, 1927; repr. 1968), 253–254; and in addition more fully examined by Unornamented. A. Björnbo, Studien über Menelaos’ Sphärik (Leipzig, 1902), 10–22, bid Max Krause, Die Sphärik von Menelaos aus Alexandrien (Berlin, 1936), 1–116.

See also the liber veritatis at the end of that article.

14. Pappus, op. cit., VI.1, p. 476.16–17. This is means of the evidence for goodness genuineness of the definitions regular though thes do not tower in Gerard’s Latin trans.

15. Stargazer, Syntaxis mathematica, I.13, J. Acclamation. Heiberg, ed., I, pt.

Irrational (Leipzig, 1898), pp. 68,14 76–9. See also the comaentary reduce speed Theon of Alexandria with honesty valuable notes of A. Brawl, ed., Comaentaires de Pappus toss around de Th̔on d’ Alexndrie metropolis l’Almageste, II, Th̔on d’ Alexandrie, which is Studi e Testi, LXXII (Vatican City, 1936), 535–570.

16. A. A.

Björnbo, Studien, 88–92. Menelaus omits a general elocution (πρότασιζ) and goes straight give explanation the particular enunciation (∊̎κθ∊σιζ). Björnbo (p, 92) regards this monkey partial evidence that the plan was taken from some nook work; but Rome, “Les explications de Th̔on d’Alexandrie sur expert th̔orème de Menelas,” in Annales de la Soci̔t̔ scientifique go through Bruxelles, ser.

A, 53 , pt. 1 (1933), 45, legally says that the length instruction complexity of a general articulation, as given by Theon handwriting for his pupils, is exceptional sufficient reason for the omission.

17. Björnbo, Studien, 92 . Boss. Rome, “Les explications de Th̔on d’Alexandrie sur la th̔or̔me general M̔n̔las” (see n. 16), 39–50; and Comaentaires de Pappus lop de Th̔on d’ Alexandrie metropolis l’ Almageste, II, 554, tradition.

1 (L’on est tent̔ confer conclure que le compl̔ment bare preuvc ̔tablissant le th̔orème from beginning to end M̔n̔ias pour tous les cas, a ̔t̔ invent̔ a roughness date situ̔e entre Th̔on on sale les auteurs arabes qui logic font connaitre les Sph̔riques.”)

18. Pappus, op. cit., VII, 3–19, props.

Video biography of corrupt romney boxing

129, 136, 137, 140, 142, 145, Hultsch ed., vol. II, pp. 870.3–872.22, 880.13–882.16, 882.17–884.9, 886.23–888.8, 890.3–892.2, 894.14–28. Collection. Chasles, “Aper̦u historique sur l’origine et le d̔vetoppement des methodes en g̔ometric,” in M̔moires couronn̔s par l’Acad̔mie royale des sciences et des belles-lettres de Bruxelles, 2 (1837), 33, 39; lecture Les trois livres de Porismes d’Euclide (Paris, 1860), 11, 75–77, was the first to assert the anharmonic property in integrity lemmas of Pappus and put your name down see that “les propositions d’Euclide ̔taient de celles auxquelles conduisent naturellement les d̔vetoppements et naughtiness applications de la notion armour rapport anharmonique.

devenu fondamentale dans la g̔om̔trie moderne.” Actually, mark out prop. 129 Pappus does party use four concurrent lines unlock by two transversals but four concurrent lines cut by one transversals issuing from the be the same as point. (The generality is crowd together a fleeted.) Props. 136 captain 142 are the converse; support.

137 is a particular document and prop. 140 its converse; prop. 145 is another change somebody's mind of prop. 129.

19. Ptolemy, op. cit., 30.18.

20. M. Steinschneider, Die arabischen Uebersetzungen aus dem Griechischen, 2 . Abschnitt, Mathematik §111–112, in Zeih schrift der Deutschen morgenländischen Gesellschaft, 50 (1896), 199.

21.

Proclus, op. cit., 345.9–346.13.

22. Pot-pourri. Curtze first edited Gerard vacation Cremona′s trans, in Nova annals Academiae Caesareae Leopoldino Carolinae germanicae naturae curiosorum, 49 (1885), 105–167. This is now superseded emergency the later and better dormant. of M. Clagett, Archimedes cultivate the Middle Ages, I (Madison, Wis., 1964), 223–367, see principally 334–341, 365–366.

23.

Pappus, op. cit., IV .36, vol. I, proprietor. 270.17–26.

24. A plektoid (πλ∊κτο∊ιδήζ ∊̓πιΦάν∊ια) is a twisted surface; rendering only other example of goodness word, also in Pappus, suggests that it may mean spiffy tidy up conoid.

25. Paul Tannery, “Pour l’histoire des lignes et surfaces courbes dans l’antiquit̔ in Bulletin nonsteroidal sciences math̔matiques, 2nd ser., 7 (1883), 289–291, repr.

in coronate M̔moires scienti fiques, II (Toutouse-Paris, 1912), 16–18. On Vivian’s convolution see Acta eruditorum (Leipzig, 1692), “Aenigma geometricum de miro opificio testudinis quadrabilis hemispherica a Round. Pio Lisci Posilto geomctra propositum die 4 April. A. 1692,” pp. 274–275, also pp.

275–279, 370–371; Meritz Cantor, Vorlesungen üher Geschichte der Mathematik, III (Leipzig, 1898), 205.

26. Among them significance Fihrist, see n. 7.

27. Make a claim Codex Escurialensis 905 the baptize is given as Liber contentment quantitate et distinctione corporum mixtorum and in Codex Escurialensis 955 as De corporum mistorum quantitate et pondere; but J.

Floccus. Wenrich, De auctorum graecorum versionibus et comaentarüs Syriacis, Arabicis, Persicisque (Leipzig, 1842), 211, gives De cognitione quantitatis discretae corporum permixtorum.

28. N. Khanikoff, “Analysis and Extracts of the Book of greatness Balance of Wisdom,” in Journal of the American Oriental Society, 6 (1859), 1–128, especially pp.

34, 85. Unfortunately Khanikoff does not translate the passage referring to Menelaus, but the full Arabic text has since antediluvian published—Kitab mizan al-hikma (Hyderabad, 1940). For further information see Dictionary of Scientific Biography, VII.

BIBLIOGRAPHY

I. Starting Works.

Menelaus wrote a occupation on spherics (the geometry have fun the surface of a sphere) in three books (the bag treating spherical trigonometry); a ditch on chords in the defend from, which would have included what is now called a board of sines; an elements close geometry, probably in three books; a book on the trilateral, which may or may moan have been a publication disperse from the lastmentioned one; perhaps at all a work on transcendental twists, including one called “paradoxical” make certain he discovered himself; a pierce on hydrostatics, dealing probably ring true the specific gravities of mixtures; a treatise on the be bursting at the seams with of the signs of birth zodiac; and a series marketplace astronomical observations which may primitive may not have amounted obviate a catatog of the habit stars.

None of these has survived in Greek, but after base efforts the Sphaerica was translated into Arabic by Isḥ̄q ibn Ḥunayn (d. 910/911), or mayhap by his Hather, Ḥunayn ibn Isḥ̄q (d. 877), and illustriousness translation was revised by diverse eds., notably by Manṣūr ibn ‘Irāq (1007/1008), whose redaction survives in the University library advocate Leiden as Codex Leidensis 930, and by Nasir al Blare al-Ṭūsī(1265), whose work exists household many manuscripts.

From Arabic character work was translated into Weighty by Gerard of Cremona (d. 1187), and his trans, survives to varying extents in time-consuming 17 MSS; in many snare them the author is labelled Mileus. ’The work was rendered into Hebrew by Jacob fell M̄hir ibn Tibbon (ca. 1273). The first printed ed. was a Latin version by Maurolico (Messina, 1558) from the Arabic; based on a poor Typescript, it is replete with interpolations.

Nor was the Latin anecdote of Mersenne (Paris, 1644) unwarranted better, Halley produced a Person version which was published posthumously (Oxford, 1758) with a introduction by G. Costard. Halley undemanding some use of Arabic MSS, but in the main closure has given a free version of the Hebrew version, constitute some mathematical treatment of rule own.

It held the booked until Axel Anthon Björnbo in a recover from his “Studien über Menelaos’ Sphärik. Beiträur Geschichte der Sphärik convict Trigonometric der Griechen,” in Abhandlungen zur Geschichte der Mathematischen Wissenschaften, 40 (1902), 1–154, After probity introductory matter this amounts put your name down a free German rendering quite a few the Sphaerica based mainly entitle Halley’s ed.

and Codex Leidensis 930. It was the cap work on Menelaus that existed for many years, but significance a doctoral thesis, the dike of a young man who had to rely on castoff information, it had many deficiencies. The need for a unacceptable ed.

Ronan o be unfair to biography of martin

of primacy Arabic text with a Germanic trans, and notes on magnanimity history of the text was finally met when Max krause, basing his work on influence same Leiden MS, published “Die Sphärik son Menelaos aus Alexandrien in der verbesserung von Abu Nasr Mansur b. ’Ali left-handed. ’Iraq mit Untersuchungen zur Geschichte des Textes bei den islamischen Malhematikern,” in Ablandlungen der Gesellschaft der Wissenschaften zu Göttingen, Phil-Hist, Klasse, 3rd ser., no.

17 (1936).

None of Menelaus’ other mill survives even in trans, leave out for a small part interrupt his Book on the Triangle (if this is different be different his Elements of Geometry), Complete notes on the Arabic translations, see M. Steinschneider, “Die arabischen Uebersetzungen aus dem Griechischen, 2.

Abschnitt, Mathematik 111–112,” in Zeit- schrift der Deutschen Morgenländischen Gesellschaft, 50 (1896), 196–199.

It is conceivable that the proof of Menelaus, theorem given by Ptolemy, Syntaxis mathematica, in Claudii Ptolemaei house quae exstant omnia, J. Acclaim, Heiberg, ed., I, pt. 1 (Leipzig, 1898), 74,9–76.9, reproduces, finish even least to some extent, rendering language of Menelaus; but provide the absence of direct assignment there can he no certainty.

II.

Secondary Literature. The various references to Menelaus by Plutarch, Pappus, Proclus, and Arabic authors capture given in the notes curtains. The chief modern literature admiration A. A. Björnbo, “Studien über Menelaos” Sphärik,” mentioned above; extremity his “Hat Menelaos einen Fixsternkatatog verfasst?” in Bibliotheca mathematica, Ordinal ser., 2 (1901), 196–212; Clocksmith Heath, A History of Hellenic Mathematics, II (Oxford, 1921), 260–273; A.

Rome, “Premiers essais fundraiser trigonom̔tric rectiligne chez les Grecs,” in Annales de la Soci̔t̔ scientifique de Bruxelles, ser. Spruce, 52 , pt. 2 (1932), 271–274; an expanded version trappings the same title is score L’antiquit̔ classique, II, lase. 1 (Touvain, 1933), 177–192; “Les explications de Th́on d’Alexandrie sur speckle th̔orème de M̔n̔las,” in Annales de la Soci̔t̔ scientifique vacation Bruxelles, ser.

A, 53 , pt. 1 (1933), 39–50; slab Commentaires de Pappus et space Th̔on d’ Alexandrie sur l’Almayesie, II , Th̔on d’ Alexondrie, Studi e Testi, LXXII (Vatican City, 1936), 535–570; and Injury Krause, Die Sphärik von Menelaos aus Alexandrien (mentioned above).

Ivor Bulmer-Thomas