Part II. has been written from a hurried sketch, which he considered very inadequate, and which we have consequently corrected and supplemented. Chapter XI. of this part has been entirely re-written by us, and has thus not had the advantage of his supervision. This remark also applies to Appendix E, which is an elaboration of a theorem he suggested. Appendix H, and all the exercises have, in accordance with his wish, been written solely by us. It will be apparent to the reader that Appendix H is little more than a brief introduction to a very large subject, which, being concerned with tessaracts and solids, is really beyond treatment in writing and diagrams. This difficulty recalls us to the one great fact, upon which we feel bound to insist, that the matter of this book _must_ receive objective treatment from the reader, who will find it quite useless even to attempt to apprehend it without actually building in squares and cubes all the facts of space which we ask him to impress on his consciousness. Indeed, we consider that printing, as a method of spreading space-knowledge, is but a “pis aller,” and we would go back to that ancient and more fruitful method of the Greek geometers, and, while describing figures on the sand, or piling up pebbles in series, would communicate to others that spirit of learning and generalization begotten in our consciousness by continuous contact with facts, and only by continuous contact with facts vitally maintained. ALICIA BOOLE, H. JOHN FALK. N.B. Models.--It is unquestionably a most important part of the process of learning space to construct these, and the reader should do so, however roughly and hastily. But, if Models are required as patterns, they may be ordered from Messrs. Swan Sonnenschein & Co., Paternoster Square, London, and will be supplied as soon as possible, the uncertainty as to demand for same not allowing us to have a large number made in advance. Much of the work can be done with plain cubes by using names without colours, but further on the reader will find colours necessary to enable him to grasp and retain the complex series of observations. Coloured models can easily be made by covering Kindergarten cubes with white paper and painting them with water-colour, and, if permanence be desired, dipping them in size and copal varnish. TABLE OF CONTENTS. PART I. PAGE INTRODUCTION 1-7 CHAPTER I. Scepticism and Science. Beginning of Knowledge 8-13 CHAPTER II. Apprehension of Nature. Intelligence. Study of Arrangement or Shape 14-20 CHAPTER III. The Elements of Knowledge 21-23 CHAPTER IV. Theory and Practice 24-28