It cannot be said too often that each portion of a mathematical curriculum must be chosen with some definite purposes in mind, and that these must be the controlling influence in the scheme of development. Such purposes may be roughly divided into two classes : the first includes all desires to illustrate or elucidate general ideas and principles, and the second includes all desires to relate one part of the subject to another, or to some other domain of thought. It is not here implied that possible fulfilment of some such purposes enforces the inclusion of any particular branch of mathematics ; its difficulty may be too great for the students in question, and in any case only a limited amount can be included in a given time-table. But it is very definitely intended that no branch shall be included merely because it happens to be sufficiently easy or to tickle the fancy. There is ample real and human interest to be found without sacrifice of purpose ; and while undue difficulty is to be deprecated, it must always be remembered that definite achievement resulting from continued effort is of the very essence of the learning of mathematics.