The MSS. which formed the basis of this book were committed to us by the author, on his leaving England for a distant foreign appointment. It was his wish that we should construct upon them a much more complete treatise than we have effected, and with that intention he asked us to make any changes or additions we thought desirable. But long alliance with him in this work has convinced us that his thought (especially that of a general philosophical character) loses much of its force if subjected to any extraneous touch.

This feeling has induced us to print Part I. almost exactly as it came from his hands, although it would probably have received much rearrangement if he could have watched it through the press himself.

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]
CHAPTER V.
Knowledge: Self-Elements[29]-[34]
CHAPTER VI.
Function of Mind. Space against Metaphysics. Self-Limitation and its Test. A Plane World[35]-[46]
CHAPTER VII.
Self Elements in our Consciousness[47]-[50]
CHAPTER VIII.
Relation of Lower to Higher Space. Theory of the Æther[51]-[60]
CHAPTER IX.
Another View of the Æther. Material and Ætherial Bodies[61]-[66]
CHAPTER X.
Higher Space and Higher Being. Perception and Inspiration[67]-[84]
CHAPTER XI.
Space the Scientific Basis of Altruism and Religion[85]-[99]
PART II.
CHAPTER I.
Three-space. Genesis of a Cube. Appearances of a Cube to a Plane-being[101]-[112]
CHAPTER II.
Further Appearances of a Cube to a Plane-being[113]-[117]
CHAPTER III.
Four-space. Genesis of a Tessaract; its Representation in Three-space[118]-[129]
CHAPTER IV.
Tessaract moving through Three-space. Models of the Sections[130]-[134]
CHAPTER V.
Representation of Three-space by Names and in a Plane[135]-[148]
CHAPTER VI.
The Means by which a Plane-being would Acquire a Conception of our Figures[149]-[155]
CHAPTER VII.
Four-space: its Representation in Three-space[156]-[166]
CHAPTER VIII.
Representation of Four-space by Name. Study of Tessaracts[167]-[176]
CHAPTER IX.
Further Study of Tessaracts[177]-[179]
CHAPTER X.
Cyclical Projections[180]-[183]
CHAPTER XI.
A Tessaractic Figure and its Projections[184]-[194]
APPENDICES.
A.100 Names used for Plane Space[197]
B.216 Names used for Cubic Space[198]
C.256 Names used for Tessaractic Space[200]-[201]
D.List of Colours, Names, and Symbols[202]-[203]
E.A Theorem in Four-space[204]-[205]
F.Exercises on Shapes of Three Dimensions[205]-[207]
G.Exercises on Shapes of Four Dimensions[207]-[209]
H.Sections of the Tessaract[209]-[217]
K.Drawings of the Cubic Sides and Sections of the Tessaract (Models 1-12) with Colours and Names[219]-[241]