I have noticed that sometimes the definite article is sometimes used with adverbial accusatives and other times it is not. This seems to happens more often with το πρότερον, το δεύτερον.
If need be I can try and find some examples but this has been a question that has been floating around in my head for a while and I wanted to see if anyone has encountered this before or if I am just crazy.
I read the relevant section of Smyth but I did not notice any comment on this observation of mine.
Not sure exactly what you are asking. Here is a sample.
Euripides Trag., Fragmenta
“Tragicorum Graecorum fragmenta”, Ed. Nauck, A.
Leipzig: Teubner, 1889, Repr. 1964.
Fragment 382, line 5
σχεδὸν παρ’ αὐτοῖς κρασπέδοις Εὐρωπίας
ἐγὼ πέφυκα γραμμάτων μὲν οὐκ ἴδρις,
μορφὰς δὲ λέξω καὶ σαφῆ τεκμήρια.
κύκλος τις ὡς τόρνοισιν ἐκμετρούμενος,
οὗτος δ’ ἔχει σημεῖον ἐν μέσῳ σαφές·
τὸ δεύτερον > δὲ πρῶτα μὲν γραμμαὶ δύο,
ταύτας διείργει δ’ ἐν μέσαις ἄλλη μία·
τρίτον δὲ βόστρυχός τις ὣς εἱλιγμένος·
τὸ δ’ αὖ τέταρτον ἣ μὲν εἰς ὀρθὸν μία,
λοξαὶ δ’ ἐπ’ αὐτῆς τρεῖς κατεστηριγμέναι
εἰσίν· τὸ πέμπτον δ’ οὐκ ἐν εὐμαρεῖ φράσαι·
This question will have to be a work in progress. As I come across any occurrences I will post them here. I should have waited to ask the question until I had some examples that I have seen.
I was just curious if there was some rule out there governing when adverbial accusatives use the definite article.
Perhaps what I have thought was an adverbial accusative is actually not.
I think it’s more a matter of idiom than rules of grammar.
Adverbial accusative is a rather nebulous category, but you’re right, sometimes they have the article. Often it makes very little difference, but in some cases distinctions can be made. το προτερον is much less common than plain προτερον, which is simply an adverb, whereas there’s a bit more to το προτερον, something like “in the former case” perhaps. Similarly with e.g. το πριν, το νῦν (and τὰ νῦν), το πρωτον, το δευτερον, etc. etc.
το δευτερον in Stirling’s Euripides passage is not an adverbial accusative at all, but a nominative, the subject (sc. γράμμα). πρῶτα is the adverb there, used almost like πρῶτον (which wouldn’t scan)—but you didn’t ask about plurals. 
My first inclination was to point out the obvious that the article with any constituent can make it function as a substantive. With τὸ δεύτερον δὲ πρῶτα μὲν γραμμαὶ δύο you see δεύτερον being treated as a substantive. But πρῶτα w/o article is close by and that is what caught my eye. A contrast between πρῶτα and τὸ δεύτερον.
382
Herdsman
I am not acquainted with letters, but will tell you their shapes and identify them clearly: a circle such as is measured out with compasses, that has in its centre a conspicuous mark; > the second, first of all > a pair of lines, and another5 one holding these apart at their middles; third, something like a curly lock of hair, and then the fourth has one part standing upright, and three more that are fastened crosswise on it; the fifth is not an easy one to explain—there are two lines that begin from separate points, and these run together10 into a single base; and the last of all is similar to the third.1– Kovaks 2008
Louw & Nida
60.49 δεύτερος, α, ον; β: second in a series involving either time, space, or set — ‘second, in the second place, secondly.’
δεύτεροςa: ὁμοίως καὶ ὁ δεύτερος καὶ ὁ τρίτος ‘and likewise also the second and the third’ Mt 22:26; δευτέρα δὲ ὁμοία αὐτῇ ‘and the second is like it’ (referring to commandments) Mt 22:39; πάλιν ἐκ δευτέρου ἀπελθὼν προσηύξατο ‘again, a second time, he went away and prayed’ Mt 26:42.
β (occurring only in titles of NT writings): πρὸς Κορινθίους β ‘Second Letter to the Corinthians’; πρὸς Τιμόθεον β ‘Second Letter to Timothy.’60.46 πρῶτος, η, ον; α: first in a series involving time, space, or set — ‘first.’
πρῶτοςa: πρῶτος ἐξ ἀναστάσεως νεκρῶν ‘first to rise from the dead’ Ac 26:23; προσελθὼν τῷ πρώτῳ ‘he came to the first (son)’ Mt 21:28; ἤρξατο λέγειν πρὸς τοὺς μαθητὰς αὐτοῦ πρῶτον ‘he began to speak first to his disciples’ Lk 12:1; σκηνὴ γὰρ κατεσκευάσθη ἡ πρώτη ‘for the first tent was put up’ He 9:2. In He 9:2 πρῶτος must, however, be understood in the sense of the outer tent, which was the first one to which a person came in entering the sanctuary.
α (occurring only in titles of NT writings): πρὸς Κορινθίους α ‘First Letter to the Corinthians’; Ἰωάννου α ‘First Epistle of John.’60.47 πρότερος, α, ον: first, with the implication of emphasis, frequently in reference to time — ‘the first time.’ εὐηγγελισάμην ὑμῖν τὸ πρότερον ‘I announced to you the good news the first time’ Ga 4:13. It is also possible to understand πρότερος in this context as meaning ‘earlier, formerly’ (see 67.18).
60.48 πρώτως: first in a temporal sequence — ‘for the first time.’ χρηματίσαι τε πρώτως ἐν Ἀντιοχείᾳ τοὺς μαθητὰς Χριστιανούς ‘and it was in Antioch that the believers were first called Christians’ Ac 11:26.
I looked into Cooper to see what he said about this; vol 1, p409 50.5.10, “The neuter singular or plural articles are used with adverbs that count as substantives just as the substantivised adjectives … such adverbs are usually temporal or local.” Cooper suggests that cases outside of this are probably an article used to substantivise a larger constituent. (50.6.10-11.B)
RE: Cooper and others like him.
English speaking philologists tie themselves in verbal knots by switching between formal and functional categories. Adverbial Accusatives is a mixture of formal and functional categories. But the examples in the original question are formal and without context leading to an ambiguous situation. So lets assume that the form articular accusative in the sample must be demonstrated to be an example with the adverbial function.
Perhaps it would be best to provide a unambiguous example of a articular accusative adjective or adverb functioning adverbially. Then we could explore the options of how to understand the article in that context. Cooper appears to suggest that the scope of such an article might extend beyond the immediate constituent.
I was simply pointing out, for the benefit of anyone who might have been misled by Stirling’s first post, that τὸ δεύτερον in his Euripides passage, which he highlighted in red as if it were a case in point, is neither adverbial nor accusative. Perhaps you were aware of that, Stirling, but it certainly didn’t appear so. I can’t say that your follow-up posts do much to advance the matter.
Is there ever a difference between τολοιπόν and τὸ λοιπόν? The LSJ claims that it should always be τὸ λοιπόν, but I don’t think that was always the practice in printed Greek.
The above is 2 Th. 3:1 in Codex Vaticanus. I don’t know how late the accentuation was added.
No there’s no difference. There’s no justification for τολοιπόν as a single word, or no more than there is with any other prepositive (which of course the article is). Lexical automony is more a sliding scale than an either/or thing in ancient Greek, and our modern habit of inserting spaces between “words” was unknown. Deciding what qualifies as a word in its own right was (and is) a dilemma only for grammarians and lexicographers.
Both τὸ λοιπόν and plain λοιπόν are used adverbially in koine (λοιπον in particular gained semantic ground), and I guess some older editors took to printing adverbial τολοιπόν as a single word (unlike e.g. καὶ τὰ λοιπά, which is substantival).
Manuscripts with systematically written accents routinely accent the article too (except ὁ οἱ ἡ αἱ). The original writers won’t have used accents at all, nor word divisions.
Good example jeidsath. το λοιπον is definitely one of the instances I was thinking about.