INFINITIVE

In essence the infinitive is a substantive noun introducing the performing of the action (being subjected to the action, etc.), including phase and genus (i.e. A/Md/P) into the sentence. As examples a complete set of infinitives of the verb ποιέω (Ι do, Ι make) will be given with preferred translations:

dur A: ποιεῖν “to make, to be making”

dur Md/P: ποιεῖσθαι “to make/be making (for) oneself, to be made, to be (in the process of) being made”

aor A: ποιῆσαι “to have made, to make”

aor Md: ποιήσασθαι “to have made (for) oneself, to make (for) oneself”

aor P: ποιηθῆναι “to have been made, to be made”

fut A: ποιήσειν "to be going to make"

fut Md: ποιήσεσθαι "to be going to make (for) oneself"

fut P: ποιηθήσεσθαι "to be going to be made"

pf A: πεποιηκέναι “to have made, to be in the position of having made”

pf Md/P: πεποιῆσθαι “to have made (for) oneself, to be in the position of having made (for) oneself/ to have been made, to be in the position of having been made.”

[The inadequacy of some of these translations has not prevented me from using them; when translated from the the Greek they will hardly ever come out in this form]

Take note of the following:

Φημὶ αὐτὸν τοῦτο ποιεῖν. I assure that he is doing this (usually does this). Φημὶ αὐτὸν τοῦτο ποιῆσαι. I assure that he has done this. (The purpose of these examples is to mark the opposition between dur and aor inf). But when an aor inf depends on a verb such as βούλομαι (= to want to) or ἐπιθυμέω (= to desire to), it carries no connotation of past tense: Βούλει ἀναγνῶναι τὰ τοῦ Ὁμήρου ἔπη; Do you want to read Homer’s epics? Ἐπιθυμοῦμεν βοηθῆσαί σοι. We desire to help you.

Although a substantive, the inf has no case forms; often an article is added, which of course then takes the case form that expresses the function of the inf in the sentence, e.g. ἡ τοῦ πεφευγέναι ἡδονὴ. Τhe joy of having fled/being free.

The infinitive can have any function that other substantives take, but some occur more frequently than others. Examples: Εὖ ἐσθίειν ἡδύ ἐστιν. (To be) eating well is agreeable. Ἡ τοῦ εὖ ἐσθίειν ἡδονὴ μειζών ἐστιν ἢ τοῦ ἔρωτος. The pleasure of eating well is greater than that of love. Ἀδικεῖν κακιόν ἐστι ἢ ἀδικεῖσθαι. To do wrong is worse than to be done wrong by. Ἐσθίειν ἡδιόν ἐστιν ἢ ἐδηδοκέναι. To be eating is more pleasant than to have eaten.

The infinitive is not so completely abstract as suggested in the above; because it is a verb form, the idea of a subject is always near and is often expressed in the form of an Accusative (a subject Accusative): perhaps derived from sentences like Διδάσκω παῖδας λέγειν. I teach children to speak, where “children” is surely conceived of as subject of “speak”, a combination of Accusative and infinitive arises which can serve as object or subject of such verbs as admit a verb with accessory subject in that position, e.g. to wish, to tell,  to know, to delight etc. Examples: Herodotus tells us that the Greeks have defeated the Persians. Ἡρόδοτος λέγει ἡμῖν τοὺς Ἕλληνας νικῆσαι τοὺς Πέρσας. [This sentence could of course just as well be interpreted Herodotus tells us that the Persians have defeated the Greeks.] Ἥδιστόν ἐστι τοὺς παῖδας ἀμαχητὶ παίζειν. It is very pleasant that the children are peacefully playing. 
This turn of phrase is even more frequent in Greek than it is in English and has it’s own name: Accusative-with-infinitive.

An inf is of course just as much a verb as it is a substantive and introduces a (grammatically) secondary verb into a sentence; where there are two verbs the matter of temporal relationship arises: which action is first in time? When both actions are contemporaneous (I know he is sleeping) a dur inf is used; when the action of the inf comes after that of the principal verb (I think we will sleep well.), a fut inf is used; when the action of the inf precedes that of the principal verb (I know he has slept.), an aor inf occurs: Φημὶ ὑμῖν αὐτὸν οὐκ ἀποθνᾑσκειν. I assure you that he is not dying. Φημὶ ὑμῖν αὐτὸν οὐκ ἀποθανεῖσθαι. I assure you that he will not die. Φημὶ ὑμῖν αὐτὸν οὐκ ἀποθανεῖν. I assure you that he has not died. [The function of the perf inf: this is used when a perf phase is contemporaneous with the principal verb: Φημὶ ὑμῖν αὐτὸν οὐ τεθνάναι. I assure you that he is not dead.]

The best way of further acquainting oneself with the use of the inf is to study examples:

1. Inf as subject: Εὖ ἐσθίειν ἡδύ ἐστιν. To eat well is pleasant. Τὸ ὑμᾶς ἀδικεῖν μᾶλλον ἀλγύνει με ἢ τὸ ἀδικεῖσθαι. That you do injustice hurts me more than that you suffer injustice.

2. Inf as object: Μανθάνωμεν ᾄδειν. Let us learn to sing. Μισῶ τὸ μάχεσθαι, ἀλλὰ τὸ νικῆσαι στέργω. I hate fighting, but I love to win. Βουλόμεθα προσεθῆναι. We want to be admitted.

3. Inf as complement of preposition: Περὶ τοῦ πείθειν ἐκκλησίαν διδάξω ὑμᾶς πάντα. I will teach you all about persuading an assembly. Εἰς τὸ πλείω κτᾶσθαι χρήματα οἱ Ἀθηναῖοι πάντα δρῶσιν. To get more money the Athenians (will) do anything.

4. Inf as adjunct denoting means or cause: Τῷ ἰσχυρότερον μάχεσθαι τοὺς πολέμιους θᾶττον ἐνικήσαμεν. By fighting more powerfully we beat the enemy sooner.

5. Inf as a Genitive of comparison (see The Genitive, I,c,ii): Ἀδικεῖσθαι ἄμεινόν ἐστιν τοῦ ἀδικεῖν. To be injured is better than to injure. 

6. Inf as “Accusative” of limitation (see The Accusative, II,a): Οὐ δεινός εἰμι λέγειν. I am not very competent in speaking.

7. Inf as complement of adjectives: Οἱ δυνατοὶ βαδίζειν  οὐ φέρονται. Those able to walk are not carried.

8. Inf in adnominal Genitive functions: Ἡ τοῦ πείθειν τέχνη πασῶν τῶν ἀλλῶν προέχει. The ability to persuade surpasses all others.

9. Ac c Inf: see preceding paragraph.

The subordinate conjunctions ὥστε (sometimes ὡς in later prose), πρίν (and a few others of similar meaning) are often followed by an (Accusative cum) inf instead of a finite verb: Πρὶν θανεῖν τὴν Νεάπολιν ἰδεῖν ἐπιθυμῶ. Before dying I wish to see Naples. Τὰς Ἀθήνας πολιορκοῦμεν, ὥστε τοὺς Ἀθηναίους πεινῆν. We are beseiging Athens, so that the Athenians are suffering from hunger.

Some less frequent turns of phrase entailing infinitives: infinitivus finalis: Ἐπιτρέπω ὑμῖν τὴν γῆν ἁρπάζειν ὅποσα ἂν βούλησθε. I give you the land to plunder as much as you want. Δός μοι οἶνον πιεῖν. Give me wine to drink. Also the G of an inf-with-article is used as an adjunct of purpose; this might be explained by the ellipsis of some such word as ἕνεκα: Τοῦ εὐλογεῖν τὸν Κύριον ἀφικόμεθα. We have come to praise the Lord.

Inf exclamationis: (Τὸ) ἐμὲ τοῦτο παθεῖν! That I (should have to) experience this!  

Absolute inf: Τὸ νῦν εἶναι οἱ Λακεδαίμονες οὐκ εἰσὶ πολέμιοι. As for now the Spartans are not belligerent.

In epic (Homer) an inf instead of an imperative (inf pro imperativo) is not infrequent. Θαρσῶν νῦν, Διόμηδες, ἐπὶ Τρώεσσι μάχεσθαι. Without qualms now, Diomede,  fight among the Trojans.