The questions concerning the nature and philosophy of mathematics and which are familiar to all acquainted with the history of speculative opinion, form, it is probable, but a small part of those which have at different times occurred to earnest students without perhaps ever passing out of the silent region of thought into the outward world of controversy. The common notion of mathematics that it consists of a few clear sharp axioms, capable of being applied in infinitely various combinations and involving in the application no higher intellectual difficulty than that of arranging the order of those combinations with a view to particular ends, was never quite true and is now very widely removed from truth. Beside the metaphysical questions which form a dark background to the whole subject and which seem to have chiefly attracted the attention of those who have surveyed it from without there are others relating to the logic and philosophy of mathematical processes of which only those who study it from within can feel the difficulty and importance. Of the questions which have been most prominent whether from their intrinsic interest or from accidential association some have so changed with the progress of human thought as either to have lost all significance or to seem to us but as examples of that power of anticipation which gifted minds sometimes possess. The reality of a connection between Number and the constitution of the Universe is now among the most established truths — but we can never think of that connection in the almost material form in which it was [B123.1] thought of by the followers of Pythagoras