ἐπί (cp. Lat. ob) upon, on, on the surface of; opposed to ὑπό under, and to ὑπέρ when ὑπέρ means above the surface of.
1. ἐπί with the Genitive
a. Local: upon: οὔτ᾽ ἐπὶ γῆς οὔθ᾽ ὑπὸ γῆς
neither upon the earth nor under the earth Menex. 246d, ἐπὶ θρόνου ἐκαθέζετο
he seated himself on a throne C. 6.1.6; of the vehicle (lit. or figur.) upon which: ἐπὶ τῶν ἵππων ὀχεῖσθαι to ride on horseback 4. 5. 58 (never ἐπί with dat.), ἐπὶ τῆς ἐμῆς νεώς
on my ship 21.6; in the direction of: ἐπὶ Σάρδεων ἔφευγε
he fled toward Sardis C. 7.2.1; in the presence of (cp. παρά with dat.): ἐπὶ μαρτύρων before witnesses Ant. 2. γ. 8. ἐπί is rarely used of mere proximity in poetry or standard prose.
N.—In expressions of simple superposition ἐπί with the gen. denotes familiar relations and natural position; whereas ἐπί with the dat. gives clear and emphatic outlines to statements of the definite place of an object or action, is used in detailed pictures, and marks the object in the dative as distinct from the subject of the verbal action. ἐπί with the gen. is colourless and phraseological, and often makes, with the verb or the subject, a compound picture. Even in contrasting two objects ἐπί with gen. is used since no special point is made of position. With (unemphatic) pronouns of reference (αὐτοῦ) ἐπί with gen. is much more frequent than ἐπί with dat. The distinction between the two cases is often the result of feeling; and certain phrases become stereotyped, now with the gen., now with the dat.
b. Temporal, usually with personal gen.: in the time of: ἐπὶ τῶν προγόνων
in the time of our ancestors 3.178, ἐπ᾽ ἐμοῦ
in my time 7.86, ἐπὶ τοῦ Δεκελεικοῦ πολέμου
in the Decelean war 22.15.
c. Other relations: μενεῖν ἐπὶ τῆς ἀνοίᾱς τῆς αὐτῆς
to persist in the same folly 8.14, ἃ ἐπὶ τῶν ἄλλων ὁρᾶτε, ταῦτ᾽ ἐφ᾽ ῡ̔μῶν αὐτῶν ἀγνοεῖτε
what you see in the case of others, that you ignore in your own case 8.114, ἐφ᾽ ἑαυτῶν ἐχώρουν
they proceeded by themselves A. 2.4.10, ἐπὶ τεττάρων four deep 1. 2. 15, οἱ ἐπὶ τῶν πρᾱγμάτων
the men in power 18.247.
2. ἐπί with the Dative
a. Local: on, by: οἰκοῦσιν ἐπὶ τῷ ἰσθμῷ
they dwell on the isthmus 1.56, τὸ ἐπὶ θαλάσσῃ τεῖχος the wall by the sea 7. 4. The dat. with ἐπί denotes proxim ity much more frequently than the gen. with ἐπί; but denotes superposition less often than the gen. with ἐπί.
b. Temporal (rare in prose): ἦν ἥλιος ἐπὶ δυσμαῖς
the sun was near setting A. 7.3.34.
c. Other relations: Succession, Addition: τὸ ἐπὶ τούτῳ γ᾽ ἀπόκρῑναι
answer the next question A. 27b, ἀνέστη ἐπ᾽ αὐτῷ
he rose up after him C. 2.3.7, ἐπὶ τῷ σῑ́τῳ ὄψον
relish with bread M. 3.14.2. Supervision: ἄρχων ἐπὶ τούτοις ἦν
there was a commander over them C. 5.3.56. Dependence: καθ᾽ ὅσον ἐστὶν ἐπ᾽ ἐμοί
as far as is in my power 6.8. Condition: ἐφ᾽ οἷς τὴν εἰρήνην ἐποιησάμεθα
on what terms we made the peace 8.5. Reason, motive, end, as with verbs of emotion (instead of the simple dative, 1517): πάντα ταῦτα θαυμάζω ἐπὶ τῷ κάλλει I am astonished at all these trees because of their beauty O. 4.21, οὐκ ἐπὶ τέχνῃ ἔμαθες ἀλλ᾽ ἐπὶ παιδείᾳ you learned this not to make it a profession but to gain general culture Pr. 312b. Hostility (less common in prose than in poetry; usually with accus.): ἡ ἐπὶ τῷ Μήδῳ ξυμμαχίᾱ
the alliance against the Medes 3.63. Price: ἐπὶ πόσῳ; for how much? A. 41a.
3. ἐπί with the Accusative
a. Local: of the goal: ἐξελαύνει ἐπὶ τὸν ποταμόν
he marches to the river A. 1.4.11, ἀφῑ́κοντο ἐπὶ τὸν ποταμόν they arrived at the river 4. 7. 18 (rarely the gen. with verbs of arrival), ἀνέβαινεν ἐπὶ τὸν ἵππον
he mounted his horse C. 7.1.1. Extension: ἐπὶ πᾶσαν Ἀσίᾱν ἐλλόγιμοι
famous over all Asia Criti. 112e.
b. Temporal: extension: ἐπὶ πολλᾱ̀ς ἡμέρᾱς
for many days 21.41.
c. Quantity, measure: ἐπὶ μῑκρόν a little, ἐπὶ πλέον still more, ἐπὶ πᾶν in general, πλάτος ἔχων πλεῖον ἢ ἐπὶ δύο στάδια wider than (up to) two stades C. 7.5.8.
d. Other relations: Purpose, object in view: πέμπειν ἐπὶ κατασκοπήν
to send for the purpose of reconnoitering C. 6.2.9, ἀπέστειλαν ἐπὶ χρήματα
they sent for money 6.74. Hostility: ἔπλεον ἐπὶ τοὺς Ἀθηναίους they sailed against the Athenians 2. 90. Reference: τὸ ἐπ᾽ ἐμέ (with or without εἶναι) as far as I am concerned (more commonly ἐπ᾽ ἐμοί); τό γε ἐπ᾽ ἐκεῖνον εἶναι 13.58.
N.—To express purpose ἐπί with accus. is generally used when the purpose involves actual or implied motion to an object; ἐπί with dat. is used when the purpose may be attained by mental activity.
4. ἐπί in Composition
Upon (ἐπιγράφειν write upon), over (ἐπιπλεῖν sail over), at, of cause (ἐπιχαίρειν rejoice over or at), to, toward (ἐπιβοηθεῖν send assistance to), in addition (ἐπιδιδόναι give in addition), against (ἐπιβουλεύειν plot against), after (ἐπιγίγνεσθαι be born after, ἐπισκευάζειν repair); causative (ἐπαληθεύειν verify); intensity (ἐπικρύπτειν hide; ἐπιβουλεύεσθαι further deliberate = reflect); reciprocity (ἐπιμείγνυσθαι ἀλλήλοις exchange friendly dealings).