Menelaus of Alexandria in al-Biruni

Continuing from yesterday…. A little more information about Menelaus of Alexandria can be got from Roshdi and Papadopoulos’ introduction, although not without effort.

On page 13 they tell us about the lost work of Menelaus:

The Book on the Elements of Geometry, translated by Thābit ibn Qurra, was quoted by other scholars, like al-Bīrūnī.37

With the deeply unhelpful footnote:

37. Istikhrāj al-awtār, in Rasā’il al-Bīrūnī, Hyderabad, 1948, p. 49.

One’s heart sinks, doesn’t it?  What’s the chances of getting hold of an Arabic work printed in Hyderabad in India?  And a Google search gives you nothing.  Anyway, I couldn’t read it.

Fortunately specialist J. P. Hogendijk has a webpage on al-Biruni, with bibliograph and links here, including PDFs!  The work in question turns out to be al-Biruni’s work on “Chords”.  The Hyderabad edition is there for download!  But even better:

German translation based on the Leiden ms.: H. Suter, Das Buch der Auffindung der Sehnen im Kreise, Bibliotheca Mathematica Dritte Folge, 11 (1910-11), 11-78, reprinted in IMA 35, pp. 39-106, reprinted in Suter, Beiträge vol. 2, pp. 280-347, scan,  another scan.

Yes, the links to “scan” are PDFs of the article!  And the volume of Bibliotheca Mathematica 11 (1910) is online at Europeana here.  In fact Hogendijk also tells us that the manuscript is “Leiden Or. 513, 108-129”, and provides a scan of photographs!  This is such a valuable site!

So we can now see what al-Biruni had to say about Menelaus of Alexandria.  It’s on p.31 of Suter’s article.

1. Es sollen aus zwei gegebenen Punkten zwei Gerade gezogen werden, die einen gegeben en Winkel miteinander bilden, und deren Summe gleich einer gegebenen Geraden ist, von mir.

Menelaus wollte im zweiten Satze des dritten Buches seines Werkes über die Elemente der Geometrie beweisen, wie in einem gegebenen Halbkreis eine gebrochene Linie gezeichnet werde gleich einer gegebenen Linie; er schlug aber hierzu einen sehr langen Weg ein.(1) Nachher behandelte sie (diese Aufgabe) Thäbit b. Kurra in seinem Kommentar dieses Buches (des Menelaus) auf einem ungefähr so langen Wege wie Menelaus selbst. Nachdem nun aber die Eigenschaften der gebrochenen Linie bekannt geworden sind (2), so ist die Behandlung dieser Aufgabe des Menelaus eine leichtere geworden, und sie erstreckt sich nun sogar allgemeiner auf alle Bögen eines gegebenen Kreises (nicht nur auf den Halbkreis).

1. Let us draw two straight lines from two given points, which form a given angle with each other, and whose sum is equal to a given line, by me.

In the second section of the third book of his work on the Elements of Geometry Menelaus wanted to prove how a broken line is drawn in a given semicircle, equal to a given line; but he took a very long journey for this purpose. (1) Afterwards Thabbit ibn Kurra treated (this proposition) in his commentary on this book (Menelaus) with a pathway as long as Menelaus himself. Now that the properties of the broken line have become known, (2) the treatment of this task of Menelaus has become easier and extends now even more generally on all arcs of a given circle (not only on the semicircle).

The footnotes are just references to studies.

Well, it doesn’t tell us very much in truth.  It confirms that Menelaus’ lost work on The Elements of Geometry did survive into the 9-10th century, and was in the hands of the translators in Baghdad, and was in at least 3 books.  We know from elsewhere that Thabit ibn Qurrah translated it into Arabic.

Still, it’s nice to get this far.  Even the baffling reference could be elucidated, with a bit of ingenuity and the use of the world-wide web.

Marvellous really, isn’t it?


You can call me al-… : Arabic sources on Menelaus of Alexandria

I ran out of time when doing yesterday’s post so I had to cut short my investigation of Arabic sources for Menelaus of Alexandria and just post what a secondary source said.

Today we only know the Sphaerica of Menelaus; but his Elements of Geometry were translated into Arabic by Thabit ibn Qurrah in the 9th century.

Apparently a certain al-Sijzī was familiar with the work in Thabit’s version, or so I learned from the article by Rashed and Papadopoulos.  I thought it might be interesting to find out more.

I’d never heard of al-Sijzī, although he is the author of some 20 astronomical works and 40 mathematical ones, most unpublished.[1]  But I discovered today that an English translation of one of his works is online in the Wayback When Machine in, here.[2]  Sadly this is not the one that mentions Menelaus.

Following R&P’s reference leads us to an unpublished source and an article inaccessible to me.[3]  So sadly we can’t see what al-Sijzi actually says.  Still, not bad going for a language that I don’t read.

While looking for references, I found that Roshdi Rashed has been extremely busy creating reference literature in French and English on Islamic science.  This is very praiseworthy.  We really do need good reference literature on Arabic texts.

  1. [1]Glen van Brummelen, “Sijzī: Abū Saʿīd Aḥmad ibn Muḥammad ibn ʿAbd al‐Jalīl al‐Sijzī” in Thomas Hockey et al. (eds.). The Biographical Encyclopedia of Astronomers, Springer Reference. New York: Springer, 2007, p. 1059, Online here.
  2. [2]Hogendijk, Jan P. (1996). Al-Sijzi’s Treatise on Geometrical Problem Solving (PDF). Tehran: Fatemi Publishing Co. ISBN 964-318-114-6.
  3. [3]Roshdi Rashed, Les mathématiques infinitésimales du IXe au XIe siècle, vol. 4, 2002, p.612, n.15: “Qawî Ahmad ibn Muhammad ‘Abd al-Jalil al-Sijzi fi khawàss al-a’mida al-wâqi’a min al-nuqta ai-mu’tà ila al-muthallath al-mutasawi al-adla’ al-mu’ta bi-tariq al-tahdid,: mss Dublin, Chester Beatty 3652, fol. 66r – 67r; – Istanbul, Reshit 1191, fol. 124v-125v. Cf. J. P. Hogendijk, “Traces of the Lost Geometrical Elements of Menelaus in Two Texts of al-Sijzi”, Zeitschrift fur Geschichte der arabisch-islamischen Wissenschaften, Band 13 (1999-2000), p. 129-164, p, 142 ff; et R Crozet, «Géométrie: La tradition euclidienne revisitée», dans Enciclopedia Italiana, forthcoming”.  The Rashed volume contains an excerpt from the al-Sijzi work.

Everything you ever wanted to know about Menelaus of Alexandria but were afraid to ask

I’ve just come across an ancient author who is completely unfamiliar to me.  His name was Menelaus, and he was a mathematical writer, and one of his books even survives today, his Spherics, although only just.

Let’s see what ancient sources say about him.

In the Almagest (or Syntaxis) VII.3, Ptolemy tells us that he was in Rome and making astronomical observations in the first year of Trajan; that is, in 98 AD.  Here are the two passages from that chapter of book 7.  Page numbers are from the Toomer translation.[1]

(p.336): [Thirdly] the geometer Menelaus says that the following observation was made [by him] in Rome. In the first year of Trajan, Mechir 15-16, when the tenth hour [of night] was completed. Spica had been occulted by the moon (for it could not be seen), but towards the end of the eleventh hour it was seen in advance of the moon’s centre, equidistant from the [two] horns by an amount less than the moon’s diameter.

(p.338): Similarly, Menelaus, who observed in Rome, says that in the first year of Trajan, Mechir 18/19, towards the end of the eleventh hour, the southern horn of the moon appeared on a straight line with the middle and the southernmost of the stars in the forehead of Scorpius, and its centre was to the rear of that straight line, and was the same distance from the middle star as the middle star was from the southernmost; it appeared to have occulted the northernmost of the stars in the forehead, since [this star] was nowhere to be seen.

Mechir is the Egyptian 6th month, by the way, which I am told is Feb. 8th – March 9th in our calendar.

Menelaus appears in the Moralia of Plutarch, in De facie in orbe lunae (On the face in the moon), chapter 17.[2]  This is a dialogue, most likely a Greek-style symposium, between various well-connected people.  A man who may be Plutarch’s brother is speaking:

“Yes, by Heaven,” said Lucius, “there was talk of this too”; and, looking at Menelaus the mathematician as he spoke, he said: “In your presence, my dear Menelaus, I am ashamed to confute a mathematical proposition, the foundation, as it were, on which rests the subject of catoptrics. Yet it must be said that the proposition, ‘all reflection occurs at equal angles,’ is neither self-evident nor an admitted fact….

Menelaus does not reply, nor speak.  I picture him as perhaps being busy with the food tray.

In Proclus’ Commentary on the first book of Euclid’s Elements, we find this under Proposition 25:[3]

But the proofs that others have produced for the same proposition we shall recount briefly, and first the proof discovered and set forth by Menelaus of Alexandria.

This indicates that Menelaus came from Alexandria, like so many other mathematical philosophers.  There is also something in Pappus, Mathematical Collection.[4]  In book 4 he says:[5]

Some of these curves have been found worthy of a deeper study, and one has even been labelled the “paradoxical curve” by Menelaus.

In book 6:[6]

In his Spherics, Menelaus calls such a figure a “trilateral”.


He says nothing about the setting of these arcs, because the reasoning of the demonstration falls back on the distinctions relative to their rising, and a work, on which we will later put forward some considerations, has already been written on this subject by Menelaus of Alexandria.

No such “considerations” appear in the text of Pappus as it has reached us, however.  But Pappus confirms that Menelaus is from Alexandria, and indicates knowledge of a lost work by Menelaus.

There is a little more information about Menelaus’ books from Arabic sources, much later.  By the 10th century scholars in Baghdad were translating as much of Greek technical literature as they could find.

Four works are listed in al-Nadim’s Fihrist or Catalogue, which has an entry on Menelaus.[8]  It reads:


He lived before the time of Ptolemy, who mentioned him in the book Almagest. Among his books there were:

Forms of Spherics [Menelai Alexandria sphaericorum], Knowledge of Quantity in Distinguishing Mixed Bodies [De cognitione quantatis discretae corporum permixtorum]—He wrote it for Emperor Domitian.[9] The Elements of Geometry [Elementa geometriae], which Thabit ibn Qurrah rendered in three sections; Triangles [De triangulis], a small part of which appeared in Arabic.

Rather later is al-Qifti who writes:[10]

Menelaus is among the guides of the geometers of his time, before the time of Ptolemy, the astronomical observer who mentions him in his book the Almagest. He was at the forefront for the benefit of his domain in the city of Alexandria – some said Memphis. His books were once translated into Syriac, then into Arabic. Among his writings is a book On the Knowledge of the Quantity of Specification of Mixed Bodies, dedicated to the King Ṭūmāṭiyānūs.

The Elements of Geometry translated by Thabit ibn Qurrah is known to other Arabic writers.  Al-Sijzī had a copy, and al-Biruni quotes from it.[11]  It is a lost book, however.

The only work of Menelaus to survive is his Spherics.  The original Greek text is lost.  Only a few quotations survive, which were mainly collected by A.A. Bjørnbo.[12]

However a number of translations were made into Arabic, from which the text became known in the west.  The Arabic tradition is complicated and is the subject of study at the moment.  One reason for the complexity is that Menelaus did not give complete proofs of his propositions, but rather outlines of how they might be proven.  Consequently the book was invariably expanded and “rectified” by translators or editors in Arabic, in order to make the book actually useful.  The result is that a number of versions exist, and the relationship between them is unclear.  Krause in the 1930’s edited and translated into German the edition by Abu Naṣr Manṣūr[13], and Rani Hermiz translated it into English[14].  I understand that a revision of this was printed in Hyderabad.[15].  A portion of a very early translation, differing from all the others, plus the edition of al-Māhānī/al-Harawī, has recently been edited and translated into English by Rashed and Papadopoulos.[16]

A Hebrew translation made from Arabic exists.  There are 8 manuscripts of this.  Two of them attribute the translation into Arabic to Hunain ibn Ishaq, the famous translator; the other six attribute it to Ishaq ibn Hunain, his son.  But the actual text of the Hebrew translation is based on more than one of the Arabic editions, not on whatever Hunain ibn Ishaq may have produced.[17]

There is also a medieval Latin translation by Gerard of Cremona, but this also is based on a mix of Arabic editions.

The Arabic version was translated for the first time into Latin by Francois Maurolyc, and printed at Messina in 1558, in a work containing also Latin versions of the Spherics of Theodore and the treatise “Of the movement of the sphere” by Autolycus.[18]  This was reprinted at Paris in 1644 in a work by Fr Mersenne, and then with the 2nd edition of the Greek text of Theodosius by Hunt, at Oxford in 1707.  The English astronomer Halley, the discoverer of Halley’s Comet, produced two Latin editions of the work at Oxford.  These were based upon the Hebrew and the Arabic versions, both of which were accessible in manuscripts in the Bodleian Library in Oxford.[19]

That’s all I have on Menelaus, perhaps a famous man in his own day and now hardly known.  Only his Spherics has survived.

Let us hope that perhaps a copy of the Elements of Geometry lingers somewhere, in some unexplored eastern library.  It was a “religious library” in Meshed in Iran that proved to contain an Arabic version of Galen’s On my own books, so discoveries may yet be made.

It is interesting to reflect that, when Menelaus was taking his observations in the capital of the world, and chatting with Plutarch, the apostle John was still alive.[20]  Doubtless neither Menelaus nor his important friends had ever heard of John.  But it was John who changed the world.  It’s a reminder to us, not to take the valuations of our world at face-value.

  1. [1]G. J. Toomer, Ptolemy’s Almagest, Duckworth, 1984.
  2. [2]The Loeb translation is online at Lacus Curtius here.  Beware: chapter 18 appears before chapter 17.
  3. [3]Proclus: A commentary on the first book of Euclid’s Elements, tr. Glenn R. Morrow, 1970, p.269.
  4. [4]Paul ver Eecke, “Pappus d’Alexandrie: La collection mathematique”, Paris, 1933.
  5. [5]p.208.
  6. [6]Proposition 1, p.371.
  7. [7]Proposition 56, p.459.
  8. [8]Bayard Dodge, The Fihrist of Al-Nadim: A Tenth-Century Survey of Muslim Culture, 1970, 2 vols. Chapter 7, p.638.
  9. [9]Lit. “King Ṭūmāṭiyānūs”, so R&P, ch. 1, p.11.
  10. [10]Quoted from R&P, p.12, rather than directly.
  11. [11]See R&P for details.
  12. [12]A.A. Bjørnbo, Studien über Menelaos’ Sphärik (Leipzig, 1902): pp.22-25.  Online at here.  A few more were located in the scholia to the Almagest : F. Acerbi, “Traces of Menelaus’ Sphaerica in Greek Scholia to the Almagest,” SCIAMVS 16 (2015): p.91-124; online at here.
  13. [13]M. Krause, Die Sphärik von Menelaos aus Alexandrien in der Verbesserung von Abū Naṣr Manṣūr b. ‘Alī b. ‘Irāq, Berlin: Weidmannsche Buchhandlung, 1936.
  14. [14]Rani Hermiz, “English Translation of the Sphaerica of Menelaus”, California State University San Marcos, MA diss., 2015.  Online here.
  15. [15]Naṣr al-Dīn al-Ṭūsī, Kitāb Mānālāwus, Taḥrīr, Hyderabad: Osmania Oriental Publications Bureau, 1940
  16. [16]Roshdi Rashed, Athanase Papadopoulos, Menelaus’ ‘Spherics’: Early Translation and al-Māhānī/al-Harawī’s Version. Scientia Graeco-Arabica 21.   Berlin; Boston:  De Gruyter, 2017.  Pp. xiv, 873.  ISBN 9783110568233. There is a Google Books preview here. I am indebted to the Bryn Mawr review of this by Nathan Sidoli of Waseda University for much of the above.  The review is here.
  17. [17]So R&P.
  18. [18]Theodosii Sphaericorum elementorum libri. III. ex traditione Maurolyci Messanensis. Online here.
  19. [19]Menelai Sphaericorum libri III, ed. Ed. Halleius, Oxonii 1758; and Menelai Sphaericorum lib. III, quos olim collatis mss. hebraicis et arabicis typis exprimandos curavit Ed. Halleius, praefationem addidit G. Costard.  Oxonii, 1758.  Online at Google Books here.
  20. [20]Irenaeus, Adversus Haereses, book 2.