I learned from my textbook (From Alpha to Omega, Groton) about articular infinitives, in which a definite article is coupled with an infinitive, to form a phrase of many uses. At first glance, that seems to be the case in the bolded text (bolding mine) but on closer inspection, we see that the article τὰ is plural, so it more likely modifies δέοντα than πραχθῆναι.
τὸ ἄρχοντι καὶ πατρὶ ὑποταχθῆναι, ὃς ἔμελλε πάντα τὸν τῦφον ἀφαιρήσειν μου καὶ εἰς ἔννοιαν ἄξειν τοῦ ὅτι δυνατόν ἐστιν ἐν αὐλῇ βιοῦντα μήτε δορυφορήσεων χρῄζειν μήτε ἐσθήτων σημειωδῶν μήτε λαμπάδων καὶ ἀνδριάντων τοιῶνδέ τινων καὶ τοῦ ὁμοίου κόμπου, ἀλλ̓ ἔξεστιν ἐγγυτάτω ἰδιώτου συστέλλειν ἑαυτὸν καὶ μὴ διὰ τοῦτο ταπεινότερον ἢ ῥᾳθυμότερον ἔχειν πρὸς τὰ ὑπὲρ τῶν κοινῶν ἡγεμονικῶς πραχθῆναι δέοντα.
That I came under a ruler and a father, who intended to take away all of my conceit, and brought me to realize that it is possible to live in a court having need of neither bodyguards, nor fancy clothes, nor candelabra, and statues such as those and of similar vaunt, but that one can reduce oneself very close to the station of a private citizen and not have less dignity or vigor in a ruler’s need to competently effect the common good.
My reasoning might be wrong, and if so, please correct me. But by comparison with the first clause (τὸ ... ὑποταχθῆναι), which also has an aorist passive articular infinitive, I would expect to see the singular τὸ and not the plural τὰ (although perhaps the number is flexible).
But regardless, could somebody explain how this complex phrase actually works? Why is δέοντα neuter and plural, and what is the term for an article paired with a participle, such as τὰ ... δέοντα? Furthermore, what function and meaning does πρὸς have preceding this phrase?
Any other criticisms or improvements would be welcome. Thanks!