The notion of geometries of n dimensions began to suggest itself to mathematicians about the middle of the last century. Cayley, Grassmann, Rieman, Clifford, and some others introduced it into their mathematical investigations. Then from time to time different mathematicians took it up in different ways. Thus the first volume of the American Journal of Mathematics begins with an article in which Professor Newcomb shows that a sphere may be turned inside out in space of four dimensions without tearing, and in the third volume of the same journal Professor Stringham has given us a full account of the regular

   Geometry of Four Dimensions is not only of importance to the mathematician, but it is also of interest in certain other lines of study. Thus it involves questions of space which concern the philosopher; efforts to understand it call into exercise our space perceptions and so attract the attention of the psychologist; and attempts to utilize the theories of hyperspace in the explanation of physical and other phenomena serve to bring the subject under the notice of those working in other branches of science. Moreover, the many curious forms and relations that appear in its development excite popular interest; for example, the relation of symmetrical forms as one of position only, a form being changeable into its symmetrical by mere rotation; the plane as an axis of rotation, and the possibility that two completeplanes may have only a point in common; the possibility that a flexible sphere may be turned inside out without tearing, that an object may be passed out of a closed box or room without penetrating the walls, that a knot in a cord may be

   These curious features of space of four dimensions, while exciting our interest, baffle us in our study. Not only the possibility of such things but the facts themselves seem beyond our comprehension. In Plane and Solid Geometry we can draw figures and construct models; we are constantly seeing the things themselves and therefore, even when they are complicated, we can readily picture them in our minds. Geometry of Four Dimensions, however, in its ordinary application, deals with things which no one has known in experience or can imagine. Its very words seem to have no meaning. This is especially true at first, and any facility in perceiving the relations of these words, if acquired at all, must come slowly and of itself. In our efforts to understand the subject we naturally desire a perception of these things at the beginning. All that we should try to do, however, is to remember the various relations and to become familiar with them. In time they may perhaps acquire some of the vividness of the conceptions of three dimensional geometry. If we expect too much when we begin this study we shall be disappointed and discouraged. If we understand at the outset how little we should expect, we shall be in an attitude toward the subject that will be most conducive to success in its mastery.

   It follows that we shall not find this subject an easy one to understand. It is something that we have to read a little at a time, to read repeatedly and to think over. We have to look at it from different points of view and to examine different ways of expressing it. Thus there are distinct advantages in having the subject presented in several short essays by differet

   The essays in this book are all non-mathematical or popular in their treatment. It will assist us, therefore, if we understand the limitations of this form of presentation. From a comparison of the lower dimensional geometries we derive analogies for the Geometry of Four Dimensions and the analogies are so complete that the subject can be very fully explained in a non-mathematical way. The analogies are a guide, even to the mathematician, but the geometry does not depend on these analogies. As a system of theorems and proofs it is built up from its axioms by a process of logical reasoning just as the lower geometries are built up. If we wish to be convinced of the consistency of this geometry, of its truth as a mathematical system, we should study it mathematically. A non-mathematical exposition should be received solely as an explanation of the geometry itself, and the reader should understand clearly that it is designed not to convince him even of the possibility of such a geometry, but only to show him what it is.

   The adoption of such an attitude on the part of the reader will be a long step toward accomplishing all that can be achieved through a non-mathematical treatment of the subject. If, however, the analogies are viewed as arguments, a person of skeptical mind will be apt to suspect that there is some fatal defect beneath their plausible exterior. Even if a philosophical writer wishes to use the analogies as well as the consistency of this geometry as an argument for the actual existence of four-dimensional space, such a consideration of

   There is another way in which the principle of analogy may be used. By imagining two-dimensional beings living in a plane and unable to perceive anything of a third dimension we get a vivid idea of our own relation to four-dimensional space. A consideration of what ought to be their attitude toward any conceptions of a space of three dimensions makes clearer what should be our attitude toward conceptions of a higher space. This point of view is made more interesting by presentation in story form of a picture of life as it might be supposed to exist in a two-dimensional world. It is not necessary for such a presentation to go into all the details of the two-dimensional existence. A too minute description of such an existence would overburden the narrative with tedious explanations that would cause us to lose sight of its main purpose. But a story written so as to bring out skillfully a few of these relations does very much to help us in understanding what should be our attitude toward the higher geometry.⁶

   The Geometry of Four Dimensions based on a suitable

   In thus supposing the existence of two-dimensional beings it would be interesting in itself to see how far we can go in these details. Thus we may suppose that what we call two-dimensional matter is really three-dimensional, and that the two-dimensional beings are really three-dimensional, either with a slight thickness in the third dimension, or at least with a thickness which the beings themselves are unable to recognize. But we may also suppose them all to be really two-dimensional,

   When we come to consider a two-dimensional space and a three-dimensional space together, the two-dimensional space lying within the three-dimensional, we have a considerable choice as to the nature of matter in these spaces, and any apparent difficulties may be ignored without affecting the usefulness of these suppositions for purposes of analogy. We may, however, be interested in the question for its own sake and try to see how far we can carry the details of such a combination

   A study of the laws of four-dimensional matter, Four-dimensional Physics, would be very interesting,but we can give some idea of the various forms which occur, and the possible motions of things, without going too carefully into the theory or using the terms of science with great exactness. Our object is to give some idea, something as near a picture as we can, of the space of four dimensions, and we shall impose limitations upon the beings which we describe, or remove limitations, according to the course which seems best adapted to our object.

   We observe the forms and positions of objects very largely by sight. Now the organs of sight of a being confined to some particular space may be supposed suited to the dimensions of his space. The picture formed in the retina of our eye is two-dimensional, the retina is a surface. A two-dimensional being, unable to perceive anything outside of his plane would have a one-dimensional retina, or at least his picture of an object in his world would be a mere line, different pictures being distinguished by the lengths, colors, and shading of these lines. The retina of a four-dimensional being would be three-dimensional if he is to

   It is not easy for us to imagine such pictures, and so we can attempt to get an impression of the shapes of objects by supposing that a three-dimensional being, a person like ourselves, could pass through a series of parallel three-spaces (three-dimensional spaces) and in each three-space examine that portion of the object which lies in this space, that section of the object. This is just as we might suppose a two-dimensional being able to pass through a series of planes and in each plane to see the section of an object made by that plane. The section which we should see of a four-dimensional object would be a solid whose surface forms a part of the three-dimensional bundary of the object. This way of studying four-dimensional objects is discussed quite fully in Essay VII. (See also Essay V, page 85.)

   There is another somewhat similar way of studying an object that we may find quite useful. We can imagine ourselves turning from one three-space into another perpendicular three-space. That is, by discarding one of the directions in our space we can suppose that we take into view the fourth direction, which goes away from our space, and so get its relation to two of our directions. We shall describe the section of an object made by any three-space as what we can see in that three-space. We shall do this particularly with reference to the different sections of an object obtained at any point by taking different perpendicular three-spaces.

   One of the first things, for example, that we consider in studying Geometry of Four Dimensions is the line perpendicular to a three-space; such is the line which goes out from a point in our space in a new fourth direction perpendicular to all the lines of our space through that point.⁷ If we can let go of one of the dimensions of our space, keeping only that part which lies in a certain plane, and take into view the new fourth dimension, we shall see a plane and a line going out from it, perpendicular to all the lines of it, something with which we are perfectly familiar.

   As another example consider two absolutely perpendicular planes. If we take a plane through a point O and the line which is perpendicular to the plane at O all in our space, and then take the line through O in the fourth direction perpendicular to al the lines through O in our space, we shall have a plane through O and two lines both perpendicular to the plane and perpendicular to each other. These two lines themselves determine a plane every line of which through O is perpendicular to the first plane. The two planes are said to be absolutely perpendicular. (See Essay I, page 45, where the expression completely perpendicular is used.) The most that we could see in any three-space of two absolutely perpendicular planes would be one, of the planes and a single line of the other plane, a line passing through O perpendicular to the plane that we see. The other plane cuts through the space along this line. These planes meet only at the point O. Indeed, two planes which do not lie entirely in one

   If two planes are absolutely perpendicular to a third at two points O and O' they lie in a single three-space. In this three-space we should see them completely, and only a single line of the third plane. The line passes through O and O' and we see it as perpendicular to the two planes. On the other hand, in a three-space containing the third plane we can see all of it but only a single line of each of the two planes absolutely perpendicular to it.