Herbert Weir Smyth, A Greek Grammar for Colleges

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κατά

[*] 1690. κατά down (cp. κάτω), opposed to ἀνά. With the genitive (the genitive proper (of the goal) and the ablatival genitive) and the accusative. With the genitive, the motion is perpendicular; with the accusative, horizontal.

1. κατά with the Genitive

a. Local: down from, down toward, under: ““ἁλάμενοι κατὰ τῆς πέτρα_ς” having leapt down from the rock” X. A. 4.2.17, κατ᾽ ἄκρα_ς utterly, completely (down from the summit) P. L. 909b, ““ψυ_χὴ κατὰ χθονὸς ᾤχετο” his soul went down under the earth” Ψ 100, μύρον κατὰ τῆς κεφαλῆς καταχέαντες having poured myrrh (down) over their heads P. R. 398a; rarely of rest: ““ὁ κατὰ γῆς” the man under the earth” X. C. 4.6.5.

b. Temporal (very rare): κατὰ παντὸς τοῦ αἰῶνος for all eternity Lyc. 7.

c. Other relations: against, as ““κατ᾽ ἐμαυτοῦ ἐρεῖν” to speak against myself” P. A. 37b; rarely in a favourable or neutral sense, as ““οἱ κατὰ Δημοσθένους ἔπαινοι” the eulogies on Demosthenes” Aes. 3.50, ““κατὰ πάντων λέγειν” to speak with regard to all” X. C. 1.2.16; by (with verbs of swearing), as ὀμνύντων τὸν ὅρκον κατὰ ἱερῶν τελείων let them swear the oath by (lit. down over) fullgrown victims T. 5.47.

2. κατά with the Accusative

a. Local: ““ἔπλεον κατὰ ποταμόν” they sailed down-stream” Hdt. 4.44, ““κατὰ τὰ_ς εἰσόδους ἐφεπόμενοι” following to the entrances” X. C. 3.3.64. Extension: καθ᾽ ὅλην τὴν πόλιν throughout the entire city Lyc. 40, ““κατὰ γῆν” by land” L. 2.32, ““διώκοντες τοὺς καθ᾽ αὑτούς” pursuing those stationed opposite themselves” X. A. 1.10.4.

b. Temporal (post-Homeric): ““κατὰ πλοῦν” during the voyage” T. 3.32, κατ᾽ ἐκεῖνον τὸν χρόνον at that time 1. 139, ““οἱ καθ᾽ ἑαυτόν” his contemporaries” D. 20.73.

c. Other relations: Purpose: ““κατὰ θέα_ν ἧκεν” came for the purpose of seeing” T. 6.31. Conformity: ““κατὰ τούτους ῥήτωρ” an orator after their style” P. A. 17b, ““κατὰ τοὺς νόμους” according to the laws” D. 8.2. Ground on which an act is based: ““κατὰ φιλία_ν” owing to friendship” T. 1.60. Comparisons: μείζω ἢ κατὰ δάκρυα πεπονθότες having endured sufferings too great for (than according to) tears 7. 75 (cp. maior quam pro). Manner: ““καθ᾽ ἡσυχία_ν” quietly” T. 6.64. Distribution: ““κατ᾽ ἔθνη” nation by nation” T. 1.122, ““δέκα δραχμαὶ κατ᾽ ἄνδρα” ten drachmae the man” Aes. 3.187, ““κατὰ σφᾶς αὐτούς” per se” T. 1.79. Approximate numbers: ““κατὰ πεντήκοντα” about fifty” Hdt. 6.79.

3. κατά in Composition

Down from above (καταπί_πτειν fall down), back (καταλείπειν leave behind), against, adversely (καταγιγνώσκειν condemn, decide against, καταφρονεῖν despise), completely (καταπετροῦν stone to death, κατεσθίειν eat up), often with an intensive force that cannot be translated. An intransitive verb when compounded with κατά may become transitive (1559).