While Vatican I is commonly regarded as the council of papal infallibility, and much time and energy were indeed devoted to that matter, both inside and outside the Council, the key to the doctrine, and the real stumbling-block for Christian unity, surely lies in the third chapter of the Constitution “Pastor Aeternus”. This is the text which commits Catholics in communion with the Roman see to the belief that the bishop of Rome as pope is endowed ex officio with “full and supreme power of jurisdiction over the universal Church, not only in things which relate to faith and morals but also in matters relating to the discipline and government of the Church spread throughout the world”. Our immediate concern at this stage of our enquiry must be to make sense of the notion of universal jurisdiction, but it should be stated now that the doctrine of papal infallibility needs to be situated in the perspective of the doctrine of papal jurisdiction. The right, and the duty, of having, on occasion, at last to define a matter of doctrine affecting the Church as a whole, must be treated as an implication, or an example of the exercise, of the claim for the successor of St Peter of an all-embracing pastoral care for the universal Church. “The supreme power of teaching”, in the words with which chapter 4 of “Pastor Aeternus” opens, “is included in the apostolic primacy which the Roman pontiff, as the successor of Peter, chief of the Apostles, possesses over the whole Church”.