When he (Paul) was called (klēthentos autou). Genitive absolute (as so often in Acts) with first aorist passive participle of kaleō. Seeing that by thee we enjoy much peace (pollēs eirēnēs tugchanontes dia sou). Literally, obtaining much peace by thee. A regular piece of flattery, captatio benevolentiae, to ingratiate himself into the good graces of the governor. Felix had suppressed a riot, but Tacitus (Ann. XII. 54) declares that Felix secretly encouraged banditti and shared the plunder for which the Jews finally made complaint to Nero who recalled him. But it sounded well to praise Felix for keeping peace in his province, especially as Tertullus was going to accuse Paul of being a disturber of the peace.
And that by thy providence (kai dia tēs pronoias). Forethought, old Greek word from pronoos (pronoeō in 1Ti 5:8; Rom 12:17; 2Co 8:21), in N.T. only here and Rom 13:14. “Providence” is Latin Providentia (foreseeing, provideo). Roman coins often have Providentia Caesaris. Post-Augustan Latin uses it of God (Deus).
Evils are corrected for this nation (diorthōmatōn ginomenōn tōi ethnei toutōi). Genitive absolute again, ginomenōn, present middle participle describing the process of reform going on for this nation (dative case of personal interest). Diorthōma (from diorthoō, to set right) occurs from Aristotle on of setting right broken limbs (Hippocrates) or reforms in law and life (Polybius, Plutarch). “Reform continually taking place for this nation.” Felix the Reform Governor of Judea! It is like a campaign speech, but it doubtless pleased Felix.