Euclid scholium Bk XIII

Greek Learning Greek
1

I would be very grateful for any help translating the last section (17 words) of the following scholium from the start of Book XIII of Euclid’s elements (from Ευκλείδου δὲ onwards):
’Εν τούτῳ τῷ βιβλίῳ, τουτέστι τῷ ιγʹ, γράφεται τὰ λεγόμενα Πλάτωνος ε σχήματα, ἃ αύτοῦ μὲν ούκ ἔστιν, τρία δὲ τῶν προειρημένων ε σχημάτων τῶν Πυθαγορείων ἐστίν, ὅ τε κύβος καὶ ἡ πυραμὶς καὶ τὸ δωδεκάεδρον, Θεαιτήτου δὲ τό τε ὀκτάεδρον καὶ τὸ εἰκοσάεδρον. την δὲ προσωνυμίαν ἔλαβεν Πλάτωνος διὰ τὸ μεμνῆσθαι αὐτόν ἐν τῷ Τιμαίῳ περί αὐτῶν· Εὐκλείδου δὲ ἐπιγράφεται καὶ τοῦτο τὸ βιβλίον διὰ τὸ στοιχειώδη τάξιν ἐπιτεθεικέναι καὶ ἐπὶ τούτου τοῦ στοιχείου.

My attempt at the first lines are ‘In this 13th book, the so-called Platonic solids are discussed, although they are not actually from him. Three of them, the cube, pyramid, and dodecahedron, are from Pythagoras, while the octahedron and icosahedron are from Theaetetus. The nick-name Platonic is due to Plato’s discussion of the solids in the Timaeus.’

A translation along those lines can be found in LOEB Greek Mathematical Works, Thales to Euclid p. 379, but it does not give any translation for the final line, my attempt for which is ‘the order in which Euclid writes in this book of the Elements is based on the order of this element [i.e. the order in the Timaeus]’.

I have recently found a translation of the last line in the book ‘Polyhedra’ by Peter Cromwell which reads ‘This book also carries Euclid’s name because he embodied it in the Elements’. This is rather different to my translation so I’d be very grateful for any thoughts on why my translation isn’t right. (I am new to Greek so apologies in advance if I’ve made some very basic errors).

I am taking ἐπιγράφεται to be the main verb of the sentence - is that sensible? Also, I am taking ἐπιτεθεικέναι to be the infinitive of ἐπιτίθημι - is that correct? I’m taking the final στοιχείου to be referring to the Timaeus - while the word ‘element’ seems to have several possible meanings, is this plausible?

One problem with my interpretation is that the order in which the Platonic solids are discussion in Euclid’s Book XIII is the tetrahedron (proposition 13), octahedron (prop 14), cube (prop 15), icosahedron (prop 16) then dodecahedron (prop 17) whereas in the Timaeus (55A-C) the order is the tetrahedron, octahedron, icosahedron, cube and dodecahedron, Plato’s ordering being largely due to his theory of the elements/physics in making up the solids. Could it simply be that the author of the scholium got it slightly wrong, misremembering the exact order in the Timaeus?

Any thoughts / suggestions on any of my translations, either the first few lines but most particularly the last, would be very welcome and appreciated.

2

Here’s how I’d translate the scholium: “This book, bk.13, includes the so-called 5 schemas of Plato, which are not his, but three of the said 5 schemas are the Pythagoreans’—the cube, the pyramid and the dodecahedron—, while the octahedron and the dodecahedron are Theaetetus’; but they took the name of Plato on account of his having made mention of them in the Timaeus. — But this book too is registered as Euclid’s [i.e. is ascribed to Euclid on its title page] on account of its having applied the elements’ organization in the case of this element too.”
(This last bit doesn’t make much sense on the face of it.)

Notes:

  • σχηματα here applies to three-dimensional figures, of course. In other contexts it carries somewhat different meanings.
  • “Theaetetus” obviously refers to the mathematician himself, not to the Platonic dialogue named after him.
  • I take the subject of την δὲ προσωνυμίαν ἔλαβεν Πλάτωνος to be “the so-called 5 schemas of Plato” (neuter plural, so ἔλαβεν singular). προσωνυμία is an appellation, what something is called.
  • Scholia often get things slightly wrong, and often more than slightly; and they tend to get garbled in transmission.

I don’t know if this helps at all.

3

Thank you - yes, this is very useful, particularly your translation of the final phrase and your feeling that it doesn’t make obvious sense at a first reading. I assume this is why LOEB leaves the phrase out, both in the Greek and the translation. I will keep thinking about what it might be trying to say but I’ll bear in mind that that things can get lost in transmission so in the end it may not be possible to do so. Thank you again for your help and suggestions.