Καλησπέρα,
τὴν ἀρχήν is the accusative singular of ἀρχήχή while τὰς ἀρχὰς is the accusative plural. Κατά requires accusative when it is not used in the sense of "against", this I understand. I also understand that αρχή, in Modern Greek, has a different meaning, e.g. start, authority, principle...