ἐξ

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223. The preposition ἐξ (or ἐκ) usually expresses motion out from an object. It is not used purely adverbially, but there are many examples of tmesis.

ἐξ ἔρον ἕντο, ἐκ δέ οἱ ἡνίοχος πλήγη φρένας
his chariοteer lost (lit. was stuck out of) his wits

ἔκ τε καὶ ὀψὲ τελεῖ (Il. 4.161)
he brings it to pass (ἐκτελεῖ) late

With a genitive (ablatival) ἐξ is used of mοtiοn from or out of. Sometimes the idea of motion is implied.

Il. 13.301 ἐκ Θρῄκης Ἐφύρους μέτα θωρήσσεσθον
                   armed themselves to come from Thrace
                   after the Ephyri

Il. 14.129 ἔνθα δʼ ἔπειτʼ αὐτοὶ μὲν ἐχώμεθα δηϊοτῆτος
                  ἐκ βελέων
                    hold back from fighting (going) out of range

Cp. Il. 16.122, 678, 18.152.

So of direction

Il. 14.153 Ἥρη δʼ εἰσεῖδε . . . στᾶσʼ ἐξ Οὐλύμποιο
                    stοοd and looked from Olympus

Od. 21.420 (drew the bow) αὐτόθεν ἐκ δίφροιο καθήμενος
                      from the chair as he sat

Il. 19.375 ὅτʼ ἂν ἐκ πόντοιο σέλας ναύτῃσι φανήῃ
                   when a meteor appears to sailors at sea
                   (seeing it from the sea)

of choosing out of

Il. 15.680 ἐκ πολέων πίσυρας συναείρεται ἵππους

and similarly

Il. 18.431 ὅσσʼ ἐμοὶ ἐκ πασέων Κρονίδης Σεὺς ἄλγε ἔδωκε
                   to me (taken from, hence) more than all

ἐξ is also used of an agent as the source of action

Il. 5.384 τλῆμεν . . . ἐξ ἀνδρῶν
                 have endured at the hands of men

cp. Il. 22.280, Od. 7.70, 9.512: also

Il. 24.617 θεῶν ἐκ κήδεα πέσσει
                   endures heaven-sent troubles

Hes. Theog. 94 ἐκ γὰρ Μουσάων καὶ ἑκηβόλου Ἀπόλλωνος
                          ἄνδρες ἀοιδοὶ ἔασιν.

The meaning in consequence of (a thing) occurs in Il. 9.566 ἐξ ἀρέων μητρὸς κεχολωμένος, and in the Odyssey (3.135, 5.468, etc.).

Of time.

ἐκ τοῖο
from that time

ἐξ ἀρχῆς
from the first (Od. 1.188, etc)

ἐκ νεότητος (Il. 14.86).

With an abstract word

Il. 10.107 ἐκ χόλου ἀργαλέοιο μεταστρέψῃ φίλον ἦτορ

Note also: Il. 10.68 πατρόθεν ἐκ γενεῆς ὀνομάζων calling them by the father's name according to family; Il. 9.343 (486) ἐκ θυμοῦ from the heart, heartily (but Il. 23.595 ἐκ θυμοῦ πεσέειν to fall away from a person's favor).