In his preface the author expresses his desire that those who criticise his work shall “consider categorically” certain questions relating to his theory of definition, to the definitions and axioms prescribed by him, to his proofs of propositions and to the “objective applications” of his three axioms. Geometry may be studied for two distinct purposes, neither of which necessarily involves the other. Unless the aim is mainly the discipline of the logical faculty, it is plainly a poor method of study to pore over the definitions, axioms, postulates, theorems, problems, and demonstrations of Euclid or any similar text-book. Practical resources and geometrical information can be acquired much better and more rapidly by a course of mechanical drawing with here and there a more or less loose explanation of the grounds and reasons that warrant the geometrical doctrines, than by means of the Euclidian course. Under such a method of instruction the student would rarely feel any real doubt as to the truth of his geometrical knowledge. But where the paramount aim is the training of the reason the Euclidian rigor is all important. Hence the perfection of that method by the discovery and certification of the ultimate grounds on which, and the principles by which, it may be unfolded systematically and in necessary and sufficient sequence without presumption or fallacy, is an object of the most momentous concern to science, to philosophy, and to culture in general. For it is well known that however good an account elementary geometry may give of its superstructure the reports given of its foundations are all very far from satisfactory. Repeated and strenuous efforts have been made, and by the most competent of our race, to discover and certify the true state of the case in respect to the geometrical foundations, in order that the whole edifice of that science shall display throughout the same thorough-going necessity and sufficiency that distinguishes it in general. The author of the work under review is persuaded that he is now able to perform this so desirable service. He avers his belief that the system of geometry he “has set forth in this book is _logically sound_ and that consequently the more it is discussed and criticised, the more firmly will it become established.” He takes his stand upon two fundamental concepts, _position_ and _direction_, which he defines not explicitly but “implicitly.” This leads us to consider his first question and his theory of definition.