214. The preposition διά seems to mean properly apart, in two. It is not used freely as an adverb; but the original sense appears in the combinations διαπρό, διαμπερές, and in tmesis and composition.
to stand apart
I cut asunder
διὰ κτῆσιν δατέοντο
divided the possession
From the notion of going through it means thoroughly, as in δια-πέρθω I sack utterly.
215. The accusative with διά is often used to denote the space through which motion takes place.
Il. 1.600 διὰ δώματα ποιπνύοντα
bustling through the palace1
Il. 14.91 μῦθον ὃν οὔ κεν ἀνήρ γε διὰ στόμα πάμπαν ἄγοιτο
= with which a man would not sully his mouth2
Od. 9.400 ᾤκεον ἐν σπήεσσι διʼ ἄκριας
dwelled in caves about (scattered through)
So Il. 2.40 διὰ κρατερὰς ὑσμίνας lasting through hard fights and διὰ νύκτα (chiefly in the Odyssey, and books 10 and 24 of the Iliad).
This use is distinctively Homeric. Sometimes also διά with the accusative is used in Homer to express cause or agency.
Il. 1.73 ἣν διὰ μαντοσύνην
(Calchas led the army) by virtue of his soothsaying
Od. 8.520 διὰ μεγάθυμον Ἀθήνην
(to conquer) by the help of Athene
216. The genitive with διά implies passing through something in order to get beyond it; especially getting through some obstacle.
Il. 4.135 διὰ μὲν ἂρ ζωστῆρος ἐλήλατο
So of a gate
Il. 3.263 διὰ Σκαιῶν ἔχον ἵππους
and of lower and upper air, etc.
διʼ ἠέρος αἰθέρʼ ἵκανεν
διʼ αἰθέρος οὐρανὸν ἷκε
πεδίονδε διὰ νεφέων
So again διὰ προμάχων, διʼ ὁμίλου etc. of making way through the press.
The accusative is used where we expect this genitive in
Il. 7.247 ἓξ δὲ διὰ πτύχας ἦλθε
went through six folds
but this may be partly due to the metrical impossibility of πτυχῶν. Conversely, in Il. 10.185 ὅς τε καθʼ ὕλην ἔρχηται διʼ ὄρεσφι the accusative would be right, and ὄρεσφι is perhaps a false archaism: but cp. § 158.