By the addition of an article the substantival nature of the infinitive is emphasised. This makes the infinitive the equivalent of a nomen actionis: τὸ μάχεσθαι = ἡ μάχη.
The articular infinitive may fulfil all the functions of a noun, including subject, object or adverbial. Unless the infinitive is in the genitive or dative or is accompanied by a preposition, the article is not obligatory.
Note that the articular infinitive does not lose its verbal nature: it may be accompanied by a subject (in the accusative), objects and adverbials: τὸ ἀποδιδόναι τὰς ἐπιστόλας = ἡ τῶν ἐπιστολῶν ἀπόδοσις. In contrast to a nomen actionis an infinitive expresses aspect and diathesis (even with the article) and, in the cast of the fut. inf., temporality.
Sometimes the article has anaphoric force: it refers back to an earlier verb.
The first certain examples of an infinitive with an article occur in the archaic lyrical poetry of the seventh century. These cases are best regarded as complex constructions involving a demonstrative pronoun with an infinitive in apposition. In Homer τό and τοῦτο are still largely interchangeable. The nominative/accusative is much more common than the genitive or the dative: 82,8 % in drama and 60,1 % in prose (Burguière, *apud* Duhoux).
The infinitive with article becomes current from the fifth century B.C.E., when articles in the accusative (τό), genitive (τοῦ) and dative (τῷ) appear. From Aeschylus (one instance) onwards prepositions are added.Around 400 B.C.E. the infinitive with the article is in common use, but from the Roman period its use decreases again.
The nominative/accusative is much more common than the genitive or the dative: 82,8 % in drama and 60,1 % in prose (Burguière, *apud* Duhoux).