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Prepositions with the genitive: that by which one swears
ἀτὰρ σὺ τί θηρίον ποτ’ εἶ πρὸς τῶν θεῶν;
‘And what kind of an animal are you, by the gods?’ (Aristoph. Birds 69)
The prepositions κατά and πρός with the genitive signal the person or object by which one swears an oath.
‘by’, ‘in the name of’
ὀμνύντων τὸν ὅρκον κατὰ ἱερῶν τελείων
Let them swear the oath by fullgrown sacrificial animals. [provisional translation]