xiv AUTHOR’S PREFACES Among these, for example, are the proof at the beginning of the first book that a perpendicular can be erected to a line from a point on that line. The considerations of continuity usually raised at this point can be set aside, as long as one assumes, without proof, that a segment or an angle can be divided into two equal parts. The consideration of the sense of rotation of an angle has permitted me to give the statements of theorems in the second book, as well as several following, all their cleanness and all their generality without rendering them less simple or less elementary. The theories described in the Complements to the Third Book are those that, while not included in the elements of geometry as set forth by Euclid, have not taken a lesser place in education in a definitive way. I have limited myself to the elements of these theories and I have systematically eliminated those without real importance. In any case, this work is edited so that these complements, as well as several passages printed in small characters, can be passed over in a first reading without losing the coherence of the rest. M. Darboux, who has given me the honor of trusting me with the editing of this work, has rendered the task singularly easy by the valuable advice which he has not ceased to give me for its composition. I would not want to end this preface without offering to him the homage of my recognition. Jacques Hadamard

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