Help with translation of Greek text "The Cheiroballistra"

[sjsepp]

Hi everybody,

I'm translating into English the cheiroballistra text written by (Pseudo-)Heron in (Koine) Greek around ~50-100AD. I'm publishing the translation under a free Creative Commons License in the Greek and Roman artillery wiki on this page. The full original text is available in public domain, for example in this publication available on Google Books:

• Wescher, C. 1867. Poliocétique des Grecs. Traités Théoretiques. Récits Historiques. Paris.

I've attended two classical Greek courses in the University, so I should be able to translate most of it just fine, especially as I have other (non-free) English translations at hand. That said, I've already stumbled upon a few "nuances" in the text which I didn't quite understand and would need help with. I've tried finding the words using Perseus search tools but came up with nothing useful. Anyways, here's the issue:

Γεγονέτωσαν κανόνες δύο πελεχινωτοὶ οἱ ΑΒ ΓΔ, ἐν τετραγώνοις πελεχίνοις, ὧν θῆλυς μὲν ἔσω ὁ ΑΒ, ἄρρην δὲ ὁ ΓΔ.

I translated this as "Make two dovetailed beams, ΑΒ and ΓΔ, with four-sided dovetails, of which ΑΒ should be female and ΓΔ male". In practice this translation should be adequate, but I can't wrap my head around the two words marked in bold. Γεγονέτωσαν looks like some second perfect form of γίγνομαι / γίνομαι, base form of which is γέγονα. However, I'm confused about the rest of the word. Where does the -έτωσαν come from? I couldn't decipher the meaning of ὧν, either, although it does not seem important for understanding the sentence. Any clues?

Samuli

PS. If this is not the proper board (or even forum) to ask this kind of questions, please let me know.

[mingshey]

My humble layman's guess is like this:
It is used parallel to the 3rd person singular imperative γεγονέτω in a commentary to Euclid. So Ι think it may be intended to be the 3rd person plural perfect imperative, conjectured from ἔστωσαν.

See below:
---------------------------------------------------------------
Euclid, Elementa, Demonstrationes alterae, lib, 10
Ἄλλως τὸ αʹ θεώρημα
Ἐκκείσθω δύο μεγέθη ἄνισα τὰ ΑΒ, Γ· καὶ ἐπεὶ ἔλασσόν ἐστι τὸ Γ,
πολλαπλασιαζόμενον ἔσται ποτὲ τοῦ ΑΒ μεγέθους μεῖζον.
γεγονέτω ὡς τὸ ΖΜ καὶ διῃρήσθω εἰς [τὰ] ἴσα τῷ Γ,
καὶ ἔστω τὰ ΜΘ, ΘΗ, ΗΖ, καὶ ἀπὸ τοῦ ΑΒ ἀφῃρήσθω μεῖζον
ἢ τὸ ἥμισυ τὸ ΒΕ, καὶ ἀπὸ τοῦ ΕΑ μεῖζον ἢ τὸ ἥμισυ τὸ ΕΔ,
καὶ τοῦτο ἀεὶ γινέσθω, ἕως αἱ ἐν τῷ ΖΜ διαιρέσεις ἴσαι
γένωνται ταῖς ἐν τῷ ΑΒ διαιρέσεσιν.
γεγονέτωσαν ὡς αἱ ΒΕ, ΕΔ, ΔΑ, καὶ τῷ ΔΑ ἕκαστον
τῶν ΚΛ, ΛΝ, ΝΞ ἔστω ἴσον, καὶ τοῦτο γινέσθω, ἕως αἱ διαιρέσεις
τοῦ ΚΞ ἴσαι γένωνται ταῖς τοῦ ΖΜ.

[edit]
Smyth
466 Endings of the Imperative
...
3. 3Pl. -- For -ντων and -σθων we find -τωσαν and -σθωσαν in prose after Thucydides, in Euripides, and in inscriptions after 300 B.C.