I found this interesting inasmuch as it shows the strong influence of Butler and 18th-century issues well into the 19th-century. From Apologia Pro Vita Sua ("History of My Religious Opinions up to 1833"):
It was about this date , I suppose, that I read Bishop Butler's Analogy; the study of which has been to so many, as it was to me, an era in their religious opinions. Its inculcation of a visible Church, the oracle of truth and a pattern of sanctity, of the duties of external religion, and of the historical character Revelation, are characteristics of this great work which strike the reader at once; for myself, if I may attempt to determine what I most gained from it, it lay in two points, which I shall have an opportunity of dwelling on in the sequel; they are the underlying principles of a great portion of my teaching. First, the very idea of an analogy between separate works of God leads to the conclusion that the system which is of less importance is economically or sacramentally connected with the more momentous, and of this conclusion the theory, to which I was inclined as a boy, viz. the unreality of material phenomena, is an ultimate resolution. At this time I did not make the distinction between matter itself and its phenomena, which is so necessary and so obvious in discussing the subject. Secondly, Butler's doctrine that Probability is the guide of life, led me, at least under the teaching to which a few years later I was introduced, to the question of the logical cogency of Faith, on which I have written so much. Thus to Butler I trace those two principles of my teaching, which have led to a charge against me both of fancifulness and of scepticism.
The last sentence, I think, is a sort of defensive argument. The Apologia Pro Vita Sua is an apology in the old sense, i.e., a defense; and by tracing his view to Butler, certainly someone that no one could reasonably consider either fanciful or skeptical, he effectively defangs these charges against himself.
Later, in discussing Keble's The Christian Year, which had an immense influence on the religious atmosphere of the time, he says,
Butler teaches us that probability is the guide of life. The danger of this doctrine, in the case of many minds, is, its tendency to destroy in them absolute certainty, leading them to consider every conclusion a doubtful, and resolving truth into an opinion, which it is safe to obey or to profess, but not possible to embrace with full internal assent. If this were to be allowed, then the celebrated saying, "O God, if there be a God, save my soul, if I have a soul!" would be the highest measure of devotion:--but who can really pray to a Being, about whose existence he is seriously in doubt?
This problem certainly who cannot be laid at the door of Butler, who insists that probability is only one consideration among many; but Newman seems clearly to recognize that this is not a Butlerian problem, but a problem that has arisen "in the case of many minds" when they have taken up emphasizing probability as the guide of life.
What is interesting, however, is how Newman goes on to fuse Butler with Keble:
I considered that Mr. Keble met this difficulty by ascribing the firmness of assent which we give to religious doctrine, not to the probabilities which introduced it, but to the living power of faith and love which accepted it. In matters of religion, he seemed to say, it is not merely probability which makes us intellectually certain, but probability as it is put to account by faith and love. It is faith and love which give to probability a force which it has not in itself. Faith and love are directed towards an Object; in the vision of that Object they live; it is that Object, received in faith and love, which renders it reasonable to take probability as sufficent for internal conviction. Thus the argument about Probability, in the matter of religion, became an argument from Personality, which in fact is one form of the argument from Authority.
This move, if it could be made, would be an immensely important one; but Newman recognized that there were unresolved issues brought up by it (as he says, "It was beautiful and religious, but it did not even profess to be logical"), so he began to try to work it out more rigorously in his University Sermons, Essay on Ecclesiastical Miracles, and Essay on Development of Doctrine, in other words, the major works of his late Anglican period that led up to his conversion to Catholicism. Although he doesn't say so here, this line of thought still seems to me to be prominent in his Catholic period, e.g., in the brilliant Essay in Aid of a Grammar of Assent. (I hope at some point in the future to look at this work and the critique of it in H. H. Price's interesting Gifford Lectures, Belief, and put up thoughts on it on this weblog.)