…in terms of historic grammar, at least.
Adverbs, like prepositions and conjunctions, were originally case forms, made from the stems of nouns and pronouns. Some of these nominal and pronominal stems have gone out of common use, so that only petrified forms are left in the adverbs. Some of these words were still felt to be live cases; in others no consciousness of their origin survived. Many adverbs show old suffixes joined to the stem or to a case form (342). It is sometimes uncertain whether we should speak of adverbs or of nouns with local endings.
Smyth then gives a list of the different cases that have forms fossilized in adverbs. First, four of the five that everyone accepts (excepting the vocative):
Nominative (rare): πύξ with clenched fist, ἅπαξ once, ἀναμίξ pell-mell.
Genitive: ἕνης day after to-morrow, ἑξῆς next, ποῦ, οὗ where, αὐτοῦ in the very place, ἐκποδών out of the way (ἐκ ¨ ποδῶν); by analogy, ἐμποδών in one’s way.
Dative: δημοσίᾳ at public cost, λάθρᾳ in secret, κοινῇ in common, etc. (1527 b), ἄλλῃ otherwise, πῇ how.
Accusative: very common, especially such adverbs as have the form of the accusative of neuter adjectives, as πολύ much, μι_κρόν a little, πρῶτον at first, τήμερον to-day, πολλά often.
And then things get crazy. Three other case forms are preserved as adverbs:
Locative: οἴκο-ι at home (οἶκο-ς house), Ἰσθμο-ῖ at the Isthmus, ποῖ whither, and all adverbs in -οι. The -ι of the consonantal declension is properly the ending of the locative, as in Μαραθῶν-ι at Marathon; -οισι (234) in O stems, in contrast to -οις; -α_σι (-ησι) in Ā stems (215): θύρα_σι at the doors, Πλαταιᾶσι at Plataea, Ἀθήνησι at Athens; further in πάλαι long ago, ἐκεῖ there, πανδημεί in full force.
Instrumental: ἄνω above, κάτω below, οὔπω not yet, ὧ-δε thus (but the forms in -ω may be ablatives); κρυφῆ and λάθρα_ in secret.
Ablative: all adverbs in -ως, as ὡς as, οὕτως thus, ἑτέρως otherwise. Here, e.g. original ἑτερωδ (cp. Old Lat. altoōd, abl. of altus) became ἑτερω (133), which took on -ς from the analogy of such words as ἀμφίς parallel to ἀμφί.
Bam. Eight cases.
Not only does Smyth’s schema provide a very good explanation of why Greek adverbs have such wildly varying forms (for which I’m incredibly grateful, and wish I’d have found out before now), it also provides a strong argument that Greek originally had eight cases, three of which were lost: the locative and instrumental being absorbed into the dative, and the ablative being expressed by the genitive or dative with prepositions.
Now, I don’t think I’m ready to go all the way and say that ancient Greek had eight viable cases; that is, I think I’d still explain a Greek locative or instrumental as a locatival dative or instrumental dative. However, I’m not willing anymore to say that eight-casers are deluded, led astray by the seduction of Latin grammar.