ἀεί + θέω + ἀήρ
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Aristotle, De Caelo 270b22-25 and Meteor. 339b24-27, takes over the first part of this etymology, ἀεὶ θεῖν, dropping the "air" part, and considers Anaxagoras wrong because he states that the etymology is from "αἴθειν" (light up, kindle): αἰθέρα προσωνόμασαν τὸν ἀνωτάτω τόπον, ἀπὸ τοῦ θεῖν ἀεὶ τὸν ἀΐδιον χρόνον θέμενοι τὴν ἐπωνυμίαν αὐτῷ. Ἀναξαγόρας δὲ καταχρῆται τῷ ὀνόματι τούτῳ οὐ καλῶς· ὀνομάζει γὰρ αἰθέρα ἀντὶ πυρός (De Caelo 270b22-25) "they named aithera the upmost place, giving it its name after the fact that eternal time always runs. And Anaxagoras uses this noun improperly, for he uses "ether" as an equivalent of "fire"."
Afterwards ἀεὶ θεῖν became an aristotelician etymology, quoted with reference to Aristotle, mainly in commentaries to Aristotle but not only. It is repeated in Achilles Tatius, Isagoga excerpta 21, 56-57 together with the competing etymology deriving it from αἴθειν (αἰθὴρ δὲ λέγεται ἀπὸ τοῦ αἴθεσθαι (πυρώδης γάρ ἐστιν) ἢ ἀπὸ τοῦ ἀεὶ θεῖν καὶ ὁρμᾶν, ἐπειδὴ ἀεὶ περιφέρεται); Nicomachus, Harmonicum enchiridion 3.1 (ἔνθεν γὰρ καὶ τοῦ ἀστὴρ ὀνόματος τέτευχεν ἕκαστος οἷον στάσεως ἐστερημένος καὶ ἀεὶ θέων, παρ’ ὃ καὶ θεὸς καὶ αἰθὴρ ὠνοματοπεποίηται); Joannes Philoponus, in Aristotelis Meteorologicorum librum premium commentarium 14/1, p. 16 (τὸ γὰρ ἀεὶ σῶμα θέον ἅμα καὶ θεῖόν τι τὴν φύσιν ἐοίκασιν ὑπολαβεῖν, καὶ διώρισαν ὀνομάζειν αἰθέρα τὸ τοιοῦτον ὡς ὂν οὐδενὶ τῶν παρ’ ἡμῖν τὸ αὐτό); ibid., p. 17 (οἱ πρὸ ἡμῶν ὑπολαμβάνοντες αἰθέρα τοῦτο τὸ σῶμα προσαγορεύουσιν ἐκ τοῦ ἀεὶ θέειν καὶ παῦλαν τῆς κινήσεως μὴ ἔχειν καὶ θεῖον εἶναι οὕτως αὐτὸν ὀνομάσαντες); ibid., p. 18 (κατανοήσαντες δὲ τὰ οὐράνια καὶ ἀεὶ ταῦτα κινούμενα βλέποντες αἰθέρα καὶ θεὸν αὐτὰ ὠνόμασαν, τὸ μὲν παρὰ τὸ αἴθειν, τὸ δὲ παρὰ τὸ ἀεὶ θέειν); Olympiodorus, In Aristotelis meteora commentaria p. 20 (αἰθέρα γὰρ αὐτὸν ἐκάλεσαν παρὰ τὸ ἀεὶ θεῖν. ἀεὶ γὰρ κινοῦνται τὰ οὐράνια καὶ κατὰ ῥυθμόν. Ἀναξαγόρας δὲ ἐχρήσατο μὲν τούτῳ τῷ ὀνόματι, πλὴν οὐ καλῶς ἐτυμολογίᾳ. αἰθέρα γὰρ αὐτὸν ἔφησε καλεῖσθαι παρὰ τὸ αἴθειν καὶ καίειν); Eustathius, Comm. Il. 2, 524 (ἐπεὶ καὶ ὁ αἰθὴρ διχῇ οὕτω λέγεται. ἢ γὰρ παρὰ τὸ αἴθειν, ὅ ἐστι καίειν, πυρὸς δίκην κατά τινας, ἢ παρὰ τὸ αἴθειν, ἤγουν λάμπειν, ὅθεν καὶ ὁ ἀστήρ, εἰ καὶ μάλιστα τοῖς σοφοῖς ἀρέσκει παρὰ τὸ αἰεὶ θεῖν αἰθὴρ λελέχθαι, διὰ τὸ κατ’ αὐτὸν ἀεικίνητον συνεσφαιρωμένον τῷ οὐρανῷ); ibid. 3, 629 (καὶ παρὰ τὸ ἀεὶ θέειν αἰθὴρ καλεῖται); Etym. Magnum, Kallierges p. 32 (Αἰθήρ: Παρὰ τὸ ἀεὶ θεῖν κυκλοφορικῶς· φησὶν Ἀριστοτέλης περὶ Κόσμου); G. Pachymeres, in Aristotelis De caelo commentarium (liber iii) 1.1.3 (καὶ τὸν μὲν αἰθέρα, ἤτοι τὸν οὐρανόν (ἐκ τοῦ ἀεὶ θέειν) […]); Scholia in Pindarum, O. 1.10 (Εἴρηται δὲ παρὰ τὸ ἀεὶ θεῖν, ὥς φησιν Ἀριστοτέλης περὶ Κυρηναίων)
Th etymology is implicit in Oracula chaldaica 221 (Τίπτε μ’ ἀεὶ θείοντος ἀπ’ αἰθέρος ὧδε χατίζων / θειοδάμοις Ἑκάτην με θεὴν ἐκάλεσσας ἀνάγκαις;)
Comment
As often, the word is parsed as a compound. Plato often "recognizes" ἀεί "always" in words with initial [ai] (see αἰσχρός). The link established between -θηρ and θέω "to run" is in agreement with the idea of permanent motion in the sensible world. In fact, Plato even sees in the compound a third element, ἀήρ "air". It is of high interest that Aristotle (see below) borrows the etymology from the Cratylus and disapproves of the correct etymology given by Anaxagoras, although in general terms he does not accept Cratylus' view on language, as he declares the conventional character of human vocal sounds in his On Interpretation. Eventually for Aristotle there is no discrepancy between arguing that language is conventional and accepting Cratylus' aetymologies: the fact that not even Aristotle questioned the etymologies given in the Cratylus can tell us a lot about their status in the ancient etymological tradition.