PREFACE TO THE FIRST EDITION xiii exercising an undeniable influence on the activity of the mind. I have, first of all, sought to develop this influence in awakening and assisting the student’s initiative. Thus it seemed to me necessary to increase the number of exercises as much as the framework of the work would allow. This requirement has been, so to speak, the only rule guiding me in this part of my work. I believed that I must pose questions of very different and gradually increasing diﬃculty. While the exercises at the end of each chapter, and especially the first few chapters, are very simple, those which I have inserted after each book have solutions which are less immedi- ate. Finally, I have postponed to the end of the volume the statement of problems which are relatively diﬃcult. Certain questions have been borrowed from some important theories—among these we note problems related to the theory of inver- sion and to systems of circles, many of which come from the note On the relations between groups of points, of circles, and of spheres in the plane and in space, of M. Darboux.5 Others, on the contrary, have no pretensions other than to train the mind in the rules of reason. I have been no less eclectic in the choice of sources I have drawn on: alongside classic exercises which are immediate applications of the theory, and whose absence in this sort of book would be almost astonishing, can be found exercises which are borrowed from various authors and periodicals, both French and foreign, and also a large number which are original. I have also included, at the end of the work, a note in which I seek to sum- marize the basic principles of the mathematical method, methods which students must begin to understand starting from the first year of instruction, and which we find poorly understood even by students in our schools of higher education. The dogmatic form which I have had to adopt is not, it must be admitted, the one that fits this topic best: this sort of subject is best taught through a sort of dialogue in which each rule intervenes at exactly the moment when it applies. I believed, despite all, that I had to attempt this exposition, hoping to find readers who are indulgent of the fact that it is presented imperfectly. Let this essay, imperfect as it is, perform a number of services and contribute to the infusion into the classroom of ideas on whose importance we must not tire of insisting. The other notes, also placed at the end of the volume, are more special in character. Note B concerns Euclid’s postulate. The ideas of modern geometers on this subject have assumed a form which is clear and well enough defined that it is possible to give an account of them even in an elementary work. Note C concerns the problem of tangent circles. As M. Koenigs has noted,6 the known solution of Gergonne, even when completed by the synthesis neglected by the author, leaves something to be desired. It is this gap that I seek to fill. Finally, Note D is devoted to the notion of area. The usual theory of area presents, as we know, a serious logical fault. It supposes a priori that this quantity is well-defined and enjoys certain properties. The theory that I give in the note in question, and in which we do without this postulatum , must be preferred, especially if one realizes that it applies to space geometry without any significant change. In the text itself, various classical arguments might be modified to advantage, sometimes to support more rigor, and sometimes in the interests of simplicity. 5 Annales Scientifiques de l’Ecole ´ Normale Superieure, 2nd series, Vol. I, 1876. Exercise 401 (the construction of tangent circles) was provided to me by M. G´ erard, a teacher at the Lyc´ee Amp`ere 6 Le¸ cons de l’agr´ egation classique de Math´ ematiques, p. 92, Paris, Hermann, 1892.

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