Herbert Weir Smyth, A Greek Grammar for Colleges

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ἀνά

[*] 1682. ἀνά (Lesb. ὀν, Lat. an- in anhelare, Eng. on): originally up to, up (opposed to κατά). Cp. ἄνω.

1. ἀνά with the Dative

Local only (Epic, Lyric, and in tragic choruses): ἀνὰ σκήπτρῳ upon a staff A 15.

2. ἀνά with the Accusative

Up along; over, through, among (of horizontal motion). Usually avoided by Attic prose writers except Xenophon (three times in the orators).

a. Local: To a higher point: ““ἀνὰ τὸν ποταμόν” up stream” Hdt. 1.194 (cp. κατὰ τὸν ποταμόν). Extension: ἀνὰ στρατόν through the camp A 10, ἀνὰ πᾶσαν τὴν γῆν over the whole earth X. Ag. 11. 16, βασιλῆας ἀνὰ στόμ᾽ ἔχων having kings in thy mouth B 250 (cp. διὰ στόματος ἔχειν).

b. Extension in Time: ““ἀνὰ νύκτα” through the night” Ξ 80. See c.

c. Other relations: Distributively: ““ἀνὰ ἑκατὸν ἄνδρας” by hundreds” X. A. 3.4.21, ““ἀνὰ πᾶσαν ἡμέρα_ν” daily” X. C. 1.2.8. Manner: ἀνὰ κράτος with all their might (up to their strength) X. A. 1.10.15 (better Attic κατὰ κράτος), ““ἀνὰ λόγον” proportionately” P. Ph. 110d.

3. ἀνά in Composition

Up (ἀνίστασθαι stand up, ἀναστρέφειν turn upside down), back (ἀναχωρεῖν go back, ἀναμιμνῄσκειν remind), again (ἀναπνεῖν breathe again, ἀναπειρᾶσθαι practise constantly), often with a reversing force force (ἀναλύ_ειν unlcose).