These chapters then, being 8-14 of that work, follow to test and prove this claim historically, wherein he posits exculpatory evidence against claims that 40 Popes had grievously erred in matters of faith.
Much as with the doctrine of Papal infallibility itself, St. Robert Bellarmine does not endeavor to show the impeccability of Popes, rather that in matters of faith, where the Popes are actually authoritative, they did not err. Some matters treated here are the objection of certain Protestants, while others are even of Catholics who are confused on the decrees or behavior of certain Popes.
These chapters were used as a blueprint at Vatican I by the fathers of that Council to further scrutinize these cases and be sure of the limits and nature of papal authority.
Bellarmine thus lays out four basic propositions; Two of these Catholics must believe with divine faith per the subsequent decree of Vatican I (which was no less incumbent upon the believer in Bellarmine's time, though then it were the universal teaching of all theologians), namely that the Pope is infallible when judging matters of Faith and Morals and defining these as matters that must be believed by all the faithful. This particular distinction is important, for the Pope, outside of this very narrow category, does not enjoy infallibility, thus in private letters, private teaching, their acts, behavior, etc., Popes can give scandal, they can give opinions that are in fact false, but they cannot teach the whole Church and bind it to believe error.