Whilst the greatest effort has been made to ensure the quality of this text, due to the historical nature of this content, in some rare cases there may be minor issues with legibility. The present volume is an attempt to carry out the program out lined in the preface to Volume I. Unfortunately, Professor Young was obliged by the pressure of other duties to cease his collabora tion at an early stage of the composition of this volume. Much of the work on the first chapters had already been done when this hap pened, but the form of exposition has been changed so much since then that although Professor Young deserves credit for constructive work, he cannot fairly be held responsible for mistakes or oversights. Professor Young has kindly read the proof sheets of this volume, as have also Professors A. B. Coble and A. A. Bennett. Most of the drawings were made by Dr. J. W. Alexander. I offer my thanks to all of these gentlemen and also to Messrs. Ginn and Company, who have shown their usual courtesy and efficiency while converting the manuscript into a book. The second volume has been arranged so that one may pass on a first reading from the end of Chapter VII, Volume I, to the beginning of Volume II. The later chapters of Volume I may well be read in connection with the part of Volume II from Chapter V onward. I shall pass by the opportunity to discuss any of the pedagogical questions which have been raised in connection with the first vol ume and which may easily be foreseen for the second. It is to be expected that there will continue to be a general agreement among those who have not made the experiment, that an abstract method of treatment of geometry is unsuited to beginning students. In this book, however, we are committed to the abstract point of view. We have in mind two principles for the classification of any theorem of geometry: (a) the axiomatic basis, or bases, from which it can be derived, or, in other words, the class of spaces in which it can be valid; and (b) the group to which it belongs in a given space.

The present volume is an attempt to carry out the program out lined in the preface to Volume I. Unfortunately, Professor Young was obliged by the pressure of other duties to cease his collabora tion at an early stage of the composition of this volume. Much of the work on the first chapters had already been done when this hap pened, but the form of exposition has been changed so much since then that although Professor Young deserves credit for constructive work, he cannot fairly be held responsible for mistakes or oversights. Professor Young has kindly read the proof sheets of this volume, as have also Professors A. B. Coble and A. A. Bennett. Most of the drawings were made by Dr. J. W. Alexander. I offer my thanks to all of these gentlemen and also to Messrs. Ginn and Company, who have shown their usual courtesy and efficiency while converting the manuscript into a book. The second volume has been arranged so that one may pass on a first reading from the end of Chapter VII, Volume I, to the beginning of Volume II. The later chapters of Volume I may well be read in connection with the part of Volume II from Chapter V onward. I shall pass by the opportunity to discuss any of the pedagogical questions which have been raised in connection with the first vol ume and which may easily be foreseen for the second. It is to be expected that there will continue to be a general agreement among those who have not made the experiment, that an abstract method of treatment of geometry is unsuited to beginning students. In this book, however, we are committed to the abstract point of view. We have in mind two principles for the classification of any theorem of geometry: (a) the axiomatic basis, or bases, from which it can be derived, or, in other words, the class of spaces in which it can be valid; and (b) the group to which it belongs in a given space.