xii AUTHOR’S PREFACES Another tendency has appeared in the teaching corps in the last few years, which we have had the bad manners not to applaud. We speak here—and I hope we begin to use it a bit—of the method of heuristics. The Note we have added in 1898 to our first edition (Note A) had exactly the goal of describing how, in our view, this method might be understood: how it might be understood, at least, from a theoretical point of view, since both are needed for the application of the heuristic method. I hope that this Note might now be of some use in indicating, at least, how these principles can be put to work. I have already explained (Preface to Solid Geometry) that the method described in Note C for tangent circles belongs in fact to M. Fouch´ e, or to Poncelet himself, and that a solution to the question of areas of plane figures, different, it is true, from that in Note D, is due to M. G´erard. I seize this occasion to add that an objection concerning the theory of dihedral angles has already been noted and refuted by M. Fonten´e. J. Hadamard Preface to the Eighth Edition The present addition contains no important changes from those that preceded it. We must note, however, that our ideas about the Postulate of Euclid have been modified considerably by recent progress in physics: I have had to recast the end of Note B to take into account this scientific evolution. A few modification have been made in the present edition, intended to give a bit more importance to properties of the most common articulated systems. J. Hadamard Preface to the Twelfth Edition This edition differs from the preceding only in the addition of several exercises. The elegant Exercise 421b is due to M. Daynac, a teacher in the French School of Cairo the simple proof of Morley’s theorem which is given by Exercise 422, and which brings into play only the first two Books, is due to M. Sasport`es 4 the supplement added to exercise 107 is borrowed from an article of M. Lapierre (Enseignement Scientifique, November 1934), with certain modifications intended to reduce the proof to properties of the inscribed angle Exercise 314b is from Japanese geometers. J. Hadamard Preface to the First Edition In editing these Lessons in Geometry, I have not lost sight of the very special role played by this science in the area of elementary mathematics. Placed at the entry point to the teaching of mathematics, it is in fact the simplest and most accessible form of reasoning. The importance of its methods, and their fecundity, are here more immediately tangible then in the relatively abstract theories of arithmetic or algebra. Because of this, geometry reveals itself capable of 4 An essentially equivalent proof was sent me by M. Gauthier, a student at the ´ Ecole Normale Sup´erieure.

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