IN the evolution of the teaching of mathematics, many thoughtful teachers frequently examine critically the basic concepts of the subject in order to make quite sure that the edifice they are constructing is sound and stable. As Prof. Bell1 points out: it may seem strange "to build up vast systems of knowledge without seeing first whether the foundations will bear the superstructure, and mathematics did precisely that". This is especially true of the calculus, and experience has often shown that, in the case of non-specialist technical students, the calculus is sometimes regarded as a system of purely formal operations. Indeed, Prof. Murnaghan declares in the preface of the book under review : "Many teachers seem to feel, and have no hesitation in expressing their feeling, that it is impossible to teach calculus correctly. The best one can do, they claim, is to give some idea of what the subject is about and to impart, by repeated drill and practice, proficiency in the manipulative details of the subject." It may be remarked here that this refers obviously to the teaching of calculus in America. Is such an experience shared by English teachers? Undoubtedly, many cases could be cited where practical students become fairly expert in manipulative technique without a real understanding of the meaning and necessity for the operations involved. The subject thus becomes purely formal instead of an intelligent aid to calculation. Differential and Integral Calculus Functions of One Variable. By Prof. Francis D. Murnaghan. Pp. x + 502. (Brooklyn : Remsen Press, 1947.) n.p.