An articular infinitive, that is, an infinitive with an article, is a verbal noun. An article gives the doing word the same qualities of a noun, so an action can, for instance, be the subject or an object of a sentence. It's a bit like the gerund in Latin, and although it may be more difficult to spot, it's much easier and more logical to form! We even sometimes use infinitives like nouns in English, like when we say 'I love to dance'.

Thankfully, unlike with nouns, infinitives don't all have separate genders for you to worry about learning -they all take the neuter article (το, το, του, τῳ). So if you're reading a passage of Greek and you come across a lonely neuter article with no noun to match up with, you might want to look for an infinitive!

Just as nouns use difference cases (nominative, accusative, genitive or dative) for difference scenarios, so do articular infinitives. When the case changes, the infintive won't change, so you need to pay particular attention to the article to work out what case is being used.

For example, to say that someone loves to run (that is, they love the act of running), the act of running would be the object of the verb to love, and so would take the accusative:

eg. φιλεῖ το τρεχειν (in the same way that you might say φιλεῖ την γυναικα)

Using the same infinitive, if you wanted to say that act of running was being used for a certain purpose, then it would act like an instrument, the dative case therefore being used.

eg. ἐφυγε τῳ τρεχειν = he escaped by running (in the same way that you might say ἐφυγε τῳ δολῳ)

Following prepositions, the articular infinitive will take the case which that preposition would require from a noun.

eg. ἀντι του τρεχειν = instead of running (genitive after ἀντι, in the same way that you might say ἀντι του οἰνου)

Approach articular infinitives as you would any other noun, and you'll find they are actually quite simple.

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Consider this passage: Homer, Odyssey 11.467-567