μετά: original meaning amid, among (cp. Germ. mit, Eng. mid in midwife). Hence properly only with plurals or collectives (so in Hom. with gen. and dat.). μετά denotes participation, community of action. πεδά (Lesb. and other dialects) agrees in meaning with μετά, but is of different origin.
1. μετά with the Genitive
Usually of persons and abstract nouns.
Local: among, together with, as καθήμενος μετὰ τῶν ἄλλων
sitting among the rest R. 359e, θῦσαι μετ᾽ ἐκείνων
to sacrifice in company with them C. 8.3.1; on the side of, as οἱ μετὰ Κῡ́ρου βάρβαροι
the barbarians in the army of Cyrus A. 1.7.10, μετὰ τῶν ἠδικημένων πολεμεῖν
to wage war on the side of the wronged 9.24, οὐ μετὰ τοῦ πλήθους
without the consent of the people 3.66; besides: γενόμενος μετὰ τοῦ ξυνετοῦ καὶ δυνατός
showing himself powerful as well as sagacious 2.15. Accompanying circumstances (concurrent act or state): μετὰκινδῡ́νων κτησάμενοι (τὴν τάξιν) having acquired their position amid dangers 3.36, λῡ́πη μετὰ φόβου
grief and terror 7.75. Joint efficient cause: μετὰ πόνων ἐλευθέρᾱν ἐποίησαν τὴν Ἑλλάδα by (amid) struggles they freed Greece 2.55. Conformity: μετὰ τῶν νόμων in accordance with the laws 3. 82.
2. μετά with the Dative (Locative)
Chiefly Epic (usually with the plural or with the collective singular of persons or things personified, or of the parts of living objects): μετὰ μνηστῆρσιν ἔειπεν
he spake amid the suitors ρ 467, μετὰ φρεσί
in their hearts Δ 245.
3. μετά with the Accusative
Local: into the midst of: νεκροὺς ἔρυσαν μετὰ λᾱὸν Ἀχαιῶν they dragged the dead into the midst of the host of the Achaeans E 573; with an idea of purpose: ἰέναι μετὰ Νέστορα to go after (in quest of) Nestor K 73. Extension over the midst of: μετὰ πληθῡ́ν throughout the multitude B 143. Phrase: μετὰ χεῖρας ἔχειν
to have in hand 1.138.
N.—From the use in μετ᾽ ἴχνια βαῖνε θεοῖο
he went after the steps of the goddess γ 30 is derived the prose use: after (of time or rank), as μετὰ τὰ Τρωϊκά
after the Trojan war 2.68, μετὰ θεοὺς ψῡχὴ θειότατον after the gods the soul is most divine P. L. 726. The range of μετά with acc. in Attic prose is not wide.
4. μετά in Composition
Among (μεταδιδόναι give a share), after, in quest of (μεταπέμπεσθαι send for). When one thing is among other things, it may be said to come after another, to succeed or alternate with it; hence of succession (μεθημερινός diurnus; cp. μεθ᾽ ἡμέρᾱν after daybreak), alteration or change (μεταγράφειν rewrite, μεταμέλειν repent i.e. care for something else).
When contrasted with σύν, μετά often denotes participation: ὁ μέτοχος the partner, ὁ συνών the companion. σύν often denotes something added. But μετά is usually the prose preposition for σύν, though it does not mean inclusive of.