The Transmission of the Works of Apollonius of Perge

Apollonius of Perge (or Perga; Apollonius Pergaeus) was an ancient mathematical writer who lived in the Hellenistic era, ca. 200 BC.  His reputation is based upon his Conics (or Conic Sections), one of the great works of ancient mathematics.  The Conic Sections deals with the shapes that are seen when a cone is intersected by a plane, i.e., when it is sliced open. You can get one of four shapes when that happens, including a circle, an ellipse, a parabola, or a hyperbola.

Seven works, including the Conics, are mentioned by Pappus of Alexandria (fl. A.D. 320) in his Collections book 7:^[1]

3.  The order of the books of the Domain of Analysis alluded to above is this: Euclid, Data, one book; Apollonius, Cutting off of a Ratio, two; Cutting off of an Area, two; <Determinate Section>, two; Tangencies, two; Euclid, Porisms, three; Apollonius, Neuses, two; by the same, Plane Loci, two; Conics, eight; Aristaeus, Solid Loci, five; Euclid, Loci on Surfaces, two; Eratosthenes, On Means [two]. These make up 32 books. I have set out epitomes of them, as far as the Conics of Apollonius, for you to study, with the number of the dispositions and diorisms and cases in each book, as well as the lemmas that are wanted in them, and there is nothing wanting for the working through of the books, I believe, that have I left out.^[2]

Pappus then goes on to summarise the contents of each work.  Listing them:

1. Conic sections (τομαι τῶν κωνικῶν), in 8 books
2. Cutting off a ratio (λόγου ἀποτομή), in 2 books
3. Cutting off an area (χωρίου ἀποτομή), in 2 books
4. Determinate section (διωρωσμένη τομή), in 2 books
5. Tangencies (ἐπαφαί), in 2 books
6. Plane Loci (τόποι ἐπίπεδοι), two books
7. Neuseis / Inclinations (νεύσεις), two books

Another six works are referenced by other ancient authors, a list of which can be found in the old Encyclopedia Britannica article here.

The Conic Sections

The only material that has reached us in Greek is the first four books of the Conics, although this is not the original text, but as incompetently revised by Eutocius in the 6th century AD. Books 5, 6 and 7 are not preserved in Greek.  The fate of book 8 is unclear, but it does not seem to have existed much later than Pappus in the 3rd century AD, if indeed it existed then.

In the Heiberg edition of 1891^[3] the manuscripts are listed as:

• V – ms. Vatican. gr. 206 (12-13th c.), on fol. 1-160.
• v – ms. Vatican. gr. 203 (13th c.), contains an extract copied from V on fol. 56-84.
• c – ms. “Constantinopolitanus palatii veteris” – i.e. in the Topkapi Palace Library – 40 (13-14th c.), fol. 349-516.  Badly damaged by damp.
• p – ms. Paris BNF gr. 2342 (13th c.).  Heavily (“impudently”) interpolated by someone who knew a lot about Greek mathematics.

Other mss also exist, which the editor dismisses, but without saying why.  28 mss are listed in the Pinakes database, mostly renaissance or later.

The Conics is also preserved in a translation of books 1-7 into Arabic.  This was made in the 9th century as part of the official Translation Movement, undertaken to translate the whole of Greek science into Arabic.  Indeed the Conics was translated into Arabic, not once, but twice.^[4]  This consists of the first seven books, and seems to be taken from a copy of Apollonius’ own text, prior to the revisions of Eutocius.^[5]

The Arabic translation is preserved in the following manuscripts, listed in Rashed’s edition with a detailed description of their history:^[6]

• A – Istanbul, Sulemaniye library, Aya Sofia no. 2762 (1024 AD).  Written by the mathematician Ibn al-Haytham.
• B – Oxford, Bodleian Library, Marsh 667 (1070 AD).
• S – Mashhad, Astan Quds museum library, no. 5619.  Copied from B.  Wrongly catalogued as a commentary on the Conics.
• M – Tehran, Milli Library no. 3597 (1290 AD).  Does not derive from any other manuscript, but is of the same general family as B, S and D.
• V – Tehran, Sepahsalar mosque library no. 557.  Copied from M.
• D – Mashhad, Astan Quds museum library, no. 5391 (1235 AD).  Not descended from any other manuscript, but related to B.
• T – Tehran, Milli library no. 867 (1860 AD).  A recent copy of D.
• L – Leiden, Bibliothèque de l’Université, or. 14 (1627).  A partial copy of B.
• K – Oxford, Bodleian Library, Thurston 1 (1668). Western copy of the last three books of the Conics.

There are also various derivative texts derived more or less directly from Apollonius which may preserve readings.

Rashed gives the following stemma of the manuscripts:

There are two families; A, and M,B,D,S,V.

The Cutting off of a ratio (De rationis sectione)

It was not only the Conics that was translated into Arabic.  The Cutting off of a ratio does exist today in Arabic.  The manuscripts are:

• I – Istanbul, Suleymaniye library, Ayasofia 4830 (1228 AD), fols. 2v-52v.  Copied in Damascus.
• B – Oxford, Bodleian Library, Arch. Seld. A. 32 (before 1235-6 AD).  Part of the Selden collection.  Fols. 2v-81r.

A Latin translation was made by Edmund Halley and published in 1706, from B, the Selden manuscript.^[7]

The work was edited for the first time, with a French translation, by Roshdi Rashed in 2010.^[8]

There is also a rather curious English translation, made without preparing a text: Apollonius of Perga, On cutting off a ratio, An attempt to recover the original argumentation through a critical translation of the two extant medieval Arabic manuscripts, translated by E. M. Macierowski, edited by Robert H. Schmidt, The Golden Hind Press, Fairfield CT (1988).

Other works

Many of the other works by Apollonius are referenced by Arabic writers. Ibn al-Nadim in his Fihrist tells us that the Cutting off of an area, the Determinate Sections, and the Tangent Circles were translated into Arabic.  There are fragments of the Plane Loci and the Neuseis quoted by two 10th century Arabic mathematical writers, which suggests that these works also may have been translated.  No manuscript is currently known of any of these texts.^[9]  However Arabic manuscripts remain very understudied, and it is quite possible that these works are extant in a copy in some library, but simply unknown to scholarship.  Let us hope so!

1. ^[1]Jan P. Hogendijk, “Arabic Traces of Lost Works of Apollonius”, Archive for History of Exact Sciences 35 (1986) pp.187-253.  JSTOR.↩
2. ^[2]Alexander Jones (tr.), Pappus of Alexandria: Book 7 of the Collection, part 1. Springer: New York (1986), p.84.↩
3. ^[3]I. L. Heiberg, Apollonii Pergaei quae graece exstant cum commentariis antiquis, 2 vols, Leipzig (1891, 1893).  Online. Also contains the ancient commentaries, including Eutocius.↩
4. ^[4]Roshdi Rashed, “Arabic versions and reediting Apollonius’ Conics,” in: Écrits d’histoire et de philosophie des sciences: Volume II Géométries. De Gruyter, Scientia Graeco-Arabica 36.2 (2023), p.351-362.↩
5. ^[5]So Rashed in “Arabic versions and reediting Apollonius’ Conics,” pointing out errors in the Greek proofs of theorems which are not present in the Arabic.↩
6. ^[6]Roshdi Rashed, Apollonius de Perge: Coniques. Tome 1.1, Livre 1.  De Gruyter (2008).↩
7. ^[7]E. Halley, Apollonii Pergæi De sectione rationis libri duo ex Arabico Msto. Latine versi. Accedunt ejusdem De sectione spatii libri duo restituti… Praemittitur Pappi Alexandrini Praefatio ad 7.mum collectionis mathematicae, nunc primum Graece edita: cum lemmatibus ejusdem Pappi ad hos Apollonii libros. Opera & studio Edmundi Halley apud Oxonienses geometriae professoris Saviliani.  Oxford (1706).  Online here.  The first page of the unnumbered praefatio indicates the text, but this is not printed.↩
8. ^[8]Roshdi Rashed (Editor), Hélène Bellosta (Editor), Apollonius de Perge, La section des droites selon des rapports: Commentaire historique et mathématique, édition et traduction du texte arabe (Scientia Graeco-Arabica, 2). De Gruyter (2010)↩
9. ^[9]Hogendijk, “Arabic Traces…”, p.189.↩