CHAPTER XIV.[6

A RECAPITULATION AND EXTENSION OF THE PHYSICAL ARGUMENT

There are two directions of inquiry in which the research for the physical reality of a fourth dimension can be prosecuted. One is the investigation of the infinitely great, the other is the investigation of the infinitely small.

[6] The contents of this chapter are taken from a paper read before the Philosophical Society of Washington. The mathematical portion of the paper has appeared in part in the Proceedings of the Royal Irish Academy under the title, “Cayley’s formulæ of orthogonal transformation,” Nov. 29th, 1903.

By the measurement of the angles of vast triangles, whose sides are the distances between the stars, astronomers have sought to determine if there is any deviation from the values given by geometrical deduction. If the angles of a celestial triangle do not together equal two right angles, there would be an evidence for the physical reality of a fourth dimension.

This conclusion deserves a word of explanation. If space is really four-dimensional, certain conclusions follow which must be brought clearly into evidence if we are to frame the questions definitely which we put to Nature. To account for our limitation let us assume a solid material sheet against which we move. This sheet must stretch alongside every object in every direction in which it visibly moves. Every material body must slip or slide along this sheet, not deviating from contact with it in any motion which we can observe.

The necessity for this assumption is clearly apparent, if we consider the analogous case of a suppositionary plane world. If there were any creatures whose experiences were confined to a plane, we must account for their limitation. If they were free to move in every space direction, they would have a three-dimensional motion; hence they must be physically limited, and the only way in which we can conceive such a limitation to exist is by means of a material surface against which they slide. The existence of this surface could only be known to them indirectly. It does not lie in any direction from them in which the kinds of motion they know of leads them. If it were perfectly smooth and always in contact with every material object, there would be no difference in their relations to it which would direct their attention to it.

But if this surface were curved—if it were, say, in the form of a vast sphere—the triangles they drew would really be triangles of a sphere, and when these triangles are large enough the angles diverge from the magnitudes they would have for the same lengths of sides if the surface were plane. Hence by the measurement of triangles of very great magnitude a plane being might detect a difference from the laws of a plane world in his physical world, and so be led to the conclusion that there was in reality another dimension to space—a third dimension—as well as the two which his ordinary experience made him familiar with.

Now, astronomers have thought it worth while to examine the measurements of vast triangles drawn from one celestial body to another with a view to determine if there is anything like a curvature in our space—that is to say, they have tried astronomical measurements to find out if the vast solid sheet against which, on the supposition of a fourth dimension, everything slides is curved or not. These results have been negative. The solid sheet, if it exists, is not curved or, being curved, has not a sufficient curvature to cause any observable deviation from the theoretical value of the angles calculated.

Hence the examination of the infinitely great leads to no decisive criterion. If it did we should have to decide between the present theory and that of metageometry.

Coming now to the prosecution of the inquiry in the direction of the infinitely small, we have to state the question thus: Our laws of movement are derived from the examination of bodies which move in three-dimensional space. All our conceptions are founded on the supposition of a space which is represented analytically by three independent axes and variations along them—that is, it is a space in which there are three independent movements. Any motion possible in it can be compounded out of these three movements, which we may call: up, right, away.

To examine the actions of the very small portions of matter with the view of ascertaining if there is any evidence in the phenomena for the supposition of a fourth dimension of space, we must commence by clearly defining what the laws of mechanics would be on the supposition of a fourth dimension. It is of no use asking if the phenomena of the smallest particles of matter are like—we do not know what. We must have a definite conception of what the laws of motion would be on the supposition of the fourth dimension, and then inquire if the phenomena of the activity of the smaller particles of matter resemble the conceptions which we have elaborated.

Now, the task of forming these conceptions is by no means one to be lightly dismissed. Movement in space has many features which differ entirely from movement on a plane; and when we set about to form the conception of motion in four dimensions, we find that there is at least as great a step as from the plane to three-dimensional space.

I do not say that the step is difficult, but I want to point out that it must be taken. When we have formed the conception of four-dimensional motion, we can ask a rational question of Nature. Before we have elaborated our conceptions we are asking if an unknown is like an unknown—a futile inquiry.

As a matter of fact, four-dimensional movements are in every way simple and more easy to calculate than three-dimensional movements, for four-dimensional movements are simply two sets of plane movements put together.

Without the formation of an experience of four-dimensional bodies, their shapes and motions, the subject can be but formal—logically conclusive, not intuitively evident. It is to this logical apprehension that I must appeal.

It is perfectly simple to form an experiential familiarity with the facts of four-dimensional movement. The method is analogous to that which a plane being would have to adopt to form an experiential familiarity with three-dimensional movements, and may be briefly summed up as the formation of a compound sense by means of which duration is regarded as equivalent to extension.

Consider a being confined to a plane. A square enclosed by four lines will be to him a solid, the interior of which can only be examined by breaking through the lines. If such a square were to pass transverse to his plane, it would immediately disappear. It would vanish, going in no direction to which he could point.

If, now, a cube be placed in contact with his plane, its surface of contact would appear like the square which we have just mentioned. But if it were to pass transverse to his plane, breaking through it, it would appear as a lasting square. The three-dimensional matter will give a lasting appearance in circumstances under which two-dimensional matter will at once disappear.

Similarly, a four-dimensional cube, or, as we may call it, a tesseract, which is generated from a cube by a movement of every part of the cube in a fourth direction at right angles to each of the three visible directions in the cube, if it moved transverse to our space, would appear as a lasting cube.

A cube of three-dimensional matter, since it extends to no distance at all in the fourth dimension, would instantly disappear, if subjected to a motion transverse to our space. It would disappear and be gone, without it being possible to point to any direction in which it had moved.

All attempts to visualise a fourth dimension are futile. It must be connected with a time experience in three space.

The most difficult notion for a plane being to acquire would be that of rotation about a line. Consider a plane being facing a square. If he were told that rotation about a line were possible, he would move his square this way and that. A square in a plane can rotate about a point, but to rotate about a line would seem to the plane being perfectly impossible. How could those parts of his square which were on one side of an edge come to the other side without the edge moving? He could understand their reflection in the edge. He could form an idea of the looking-glass image of his square lying on the opposite side of the line of an edge, but by no motion that he knows of can he make the actual square assume that position. The result of the rotation would be like reflection in the edge, but it would be a physical impossibility to produce it in the plane.

The demonstration of rotation about a line must be to him purely formal. If he conceived the notion of a cube stretching out in an unknown direction away from his plane, then he can see the base of it, his square in the plane, rotating round a point. He can likewise apprehend that every parallel section taken at successive intervals in the unknown direction rotates in like manner round a point. Thus he would come to conclude that the whole body rotates round a line—the line consisting of the succession of points round which the plane sections rotate. Thus, given three axes, _x_, _y_, _z_, if _x_ rotates to take the place of _y_, and _y_ turns so as to point to negative _x_, then the third axis remaining unaffected by this turning is the axis about which the rotation takes place. This, then, would have to be his criterion of the axis of a rotation—that which remains unchanged when a rotation of every plane section of a body takes place.

There is another way in which a plane being can think about three-dimensional movements; and, as it affords the type by which we can most conveniently think about four-dimensional movements, it will be no loss of time to consider it in detail.

We can represent the plane being and his object by figures cut out of paper, which slip on a smooth surface. The thickness of these bodies must be taken as so minute that their extension in the third dimension escapes the observation of the plane being, and he thinks about them as if they were mathematical plane figures in a plane instead of being material bodies capable of moving on a plane surface. Let A_x_, A_y_ be two axes and ABCD a square. As far as movements in the plane are concerned, the square can rotate about a point A, for example. It cannot rotate about a side, such as AC.

But if the plane being is aware of the existence of a third dimension he can study the movements possible in the ample space, taking his figure portion by portion.

His plane can only hold two axes. But, since it can hold two, he is able to represent a turning into the third dimension if he neglects one of his axes and represents the third axis as lying in his plane. He can make a drawing in his plane of what stands up perpendicularly from his plane. Let A_z_ be the axis, which stands perpendicular to his plane at A. He can draw in his plane two lines to represent the two axes, A_x_ and A_z_. Let Fig. 2 be this drawing. Here the _z_ axis has taken the place of the _y_ axis, and the plane of A_x_ A_z_ is represented in his plane. In this figure all that exists of the square ABCD will be the line AB.

The square extends from this line in the _y_ direction, but more of that direction is represented in Fig. 2. The plane being can study the turning of the line AB in this diagram. It is simply a case of plane turning around the point A. The line AB occupies intermediate portions like AB_{1} and after half a revolution will lie on A_x_ produced through A.

Now, in the same way, the plane being can take another point, A´, and another line, A´B´, in his square. He can make the drawing of the two directions at A´, one along A´B´, the other perpendicular to his plane. He will obtain a figure precisely similar to Fig. 2, and will see that, as AB can turn around A, so A´C´ around A.

In this turning AB and A´B´ would not interfere with each other, as they would if they moved in the plane around the separate points A and A´.

Hence the plane being would conclude that a rotation round a line was possible. He could see his square as it began to make this turning. He could see it half way round when it came to lie on the opposite side of the line AC. But in intermediate portions he could not see it, for it runs out of the plane.

Coming now to the question of a four-dimensional body, let us conceive of it as a series of cubic sections, the first in our space, the rest at intervals, stretching away from our space in the unknown direction.

We must not think of a four-dimensional body as formed by moving a three-dimensional body in any direction which we can see.

Refer for a moment to Fig. 3. The point A, moving to the right, traces out the line AC. The line AC, moving away in a new direction, traces out the square ACEG at the base of the cube. The square AEGC, moving in a new direction, will trace out the cube ACEGBDHF. The vertical direction of this last motion is not identical with any motion possible in the plane of the base of the cube. It is an entirely new direction, at right angles to every line that can be drawn in the base. To trace out a tesseract the cube must move in a new direction—a direction at right angles to any and every line that can be drawn in the space of the cube.

The cubic sections of the tesseract are related to the cube we see, as the square sections of the cube are related to the square of its base which a plane being sees.

Let us imagine the cube in our space, which is the base of a tesseract, to turn about one of its edges. The rotation will carry the whole body with it, and each of the cubic sections will rotate. The axis we see in our space will remain unchanged, and likewise the series of axes parallel to it about which each of the parallel cubic sections rotates. The assemblage of all of these is a plane.

Hence in four dimensions a body rotates about a plane. There is no such thing as rotation round an axis.

We may regard the rotation from a different point of view. Consider four independent axes each at right angles to all the others, drawn in a four-dimensional body. Of these four axes we can see any three. The fourth extends normal to our space.

Rotation is the turning of one axis into a second, and the second turning to take the place of the negative of the first. It involves two axes. Thus, in this rotation of a four-dimensional body, two axes change and two remain at rest. Four-dimensional rotation is therefore a turning about a plane.

As in the case of a plane being, the result of rotation about a line would appear as the production of a looking-glass image of the original object on the other side of the line, so to us the result of a four-dimensional rotation would appear like the production of a looking-glass image of a body on the other side of a plane. The plane would be the axis of the rotation, and the path of the body between its two appearances would be unimaginable in three-dimensional space.

Let us now apply the method by which a plane being could examine the nature of rotation about a line in our examination of rotation about a plane. Fig. 3 represents a cube in our space, the three axes _x_, _y_, _z_ denoting its three dimensions. Let _w_ represent the fourth dimension. Now, since in our space we can represent any three dimensions, we can, if we choose, make a representation of what is in the space determined by the three axes _x_, _z_, _w_. This is a three-dimensional space determined by two of the axes we have drawn, _x_ and _z_, and in place of _y_ the fourth axis, _w_. We cannot, keeping _x_ and _z_, have both _y_ and _w_ in our space; so we will let _y_ go and draw _w_ in its place. What will be our view of the cube?

Evidently we shall have simply the square that is in the plane of _xz_, the square ACDB. The rest of the cube stretches in the _y_ direction, and, as we have none of the space so determined, we have only the face of the cube. This is represented in fig. 4.

Now, suppose the whole cube to be turned from the _x_ to the _w_ direction. Conformably with our method, we will not take the whole of the cube into consideration at once, but will begin with the face ABCD.

Let this face begin to turn. Fig. 5 represents one of the positions it will occupy; the line AB remains on the _z_ axis. The rest of the face extends between the _x_ and the _w_ direction.

Now, since we can take any three axes, let us look at what lies in the space of _zyw_, and examine the turning there. We must now let the _z_ axis disappear and let the _w_ axis run in the direction in which the _z_ ran.

Making this representation, what do we see of the cube? Obviously we see only the lower face. The rest of the cube lies in the space of _xyz_. In the space of _xyz_ we have merely the base of the cube lying in the plane of _xy_, as shown in fig. 6.

Now let the _x_ to _w_ turning take place. The square ACEG will turn about the line AE. This edge will remain along the _y_ axis and will be stationary, however far the square turns.

Thus, if the cube be turned by an _x_ to _w_ turning, both the edge AB and the edge AC remain stationary; hence the whole face ABEF in the _yz_ plane remains fixed. The turning has taken place about the face ABEF.

Suppose this turning to continue till AC runs to the left from A. The cube will occupy the position shown in fig. 8. This is the looking-glass image of the cube in fig. 3. By no rotation in three-dimensional space can the cube be brought from the position in fig. 3 to that shown in fig. 8.

We can think of this turning as a turning of the face ABCD about AB, and a turning of each section parallel to ABCD round the vertical line in which it intersects the face ABEF, the space in which the turning takes place being a different one from that in which the cube lies.

One of the conditions, then, of our inquiry in the direction of the infinitely small is that we form the conception of a rotation about a plane. The production of a body in a state in which it presents the appearance of a looking-glass image of its former state is the criterion for a four-dimensional rotation.

There is some evidence for the occurrence of such transformations of bodies in the change of bodies from those which produce a right-handed polarisation of light to those which produce a left-handed polarisation; but this is not a point to which any very great importance can be attached.

Still, in this connection, let me quote a remark from Prof. John G. McKendrick’s address on Physiology before the British Association at Glasgow. Discussing the possibility of the hereditary production of characteristics through the material structure of the ovum, he estimates that in it there exist 12,000,000,000 biophors, or ultimate particles of living matter, a sufficient number to account for hereditary transmission, and observes: “Thus it is conceivable that vital activities may also be determined by the kind of motion that takes place in the molecules of that which we speak of as living matter. It may be different in kind from some of the motions known to physicists, and it is conceivable that life may be the transmission to dead matter, the molecules of which have already a special kind of motion, of a form of motion _sui generis_.”

Now, in the realm of organic beings symmetrical structures—those with a right and left symmetry—are everywhere in evidence. Granted that four dimensions exist, the simplest turning produces the image form, and by a folding-over structures could be produced, duplicated right and left, just as is the case of symmetry in a plane.

Thus one very general characteristic of the forms of organisms could be accounted for by the supposition that a four-dimensional motion was involved in the process of life.

But whether four-dimensional motions correspond in other respects to the physiologist’s demand for a special kind of motion, or not, I do not know. Our business is with the evidence for their existence in physics. For this purpose it is necessary to examine into the significance of rotation round a plane in the case of extensible and of fluid matter.

Let us dwell a moment longer on the rotation of a rigid body. Looking at the cube in fig. 3, which turns about the face of ABFE, we see that any line in the face can take the place of the vertical and horizontal lines we have examined. Take the diagonal line AF and the section through it to GH. The portions of matter which were on one side of AF in this section in fig. 3 are on the opposite side of it in fig. 8. They have gone round the line AF. Thus the rotation round a face can be considered as a number of rotations of sections round parallel lines in it.

The turning about two different lines is impossible in three-dimensional space. To take another illustration, suppose A and B are two parallel lines in the _xy_ plane, and let CD and EF be two rods crossing them. Now, in the space of _xyz_ if the rods turn round the lines A and B in the same direction they will make two independent circles.

When the end F is going down the end C will be coming up. They will meet and conflict.

But if we rotate the rods about the plane of AB by the _z_ to _w_ rotation these movements will not conflict. Suppose all the figure removed with the exception of the plane _xz_, and from this plane draw the axis of _w_, so that we are looking at the space of _xzw_.

Here, fig. 10, we cannot see the lines A and B. We see the points G and H, in which A and B intercept the _x_ axis, but we cannot see the lines themselves, for they run in the _y_ direction, and that is not in our drawing.

Now, if the rods move with the _z_ to _w_ rotation they will turn in parallel planes, keeping their relative positions. The point D, for instance, will describe a circle. At one time it will be above the line A, at another time below it. Hence it rotates round A.

Not only two rods but any number of rods crossing the plane will move round it harmoniously. We can think of this rotation by supposing the rods standing up from one line to move round that line and remembering that it is not inconsistent with this rotation for the rods standing up along another line also to move round it, the relative positions of all the rods being preserved. Now, if the rods are thick together, they may represent a disk of matter, and we see that a disk of matter can rotate round a central plane.

Rotation round a plane is exactly analogous to rotation round an axis in three dimensions. If we want a rod to turn round, the ends must be free; so if we want a disk of matter to turn round its central plane by a four-dimensional turning, all the contour must be free. The whole contour corresponds to the ends of the rod. Each point of the contour can be looked on as the extremity of an axis in the body, round each point of which there is a rotation of the matter in the disk.

If the one end of a rod be clamped, we can twist the rod, but not turn it round; so if any part of the contour of a disk is clamped we can impart a twist to the disk, but not turn it round its central plane. In the case of extensible materials a long, thin rod will twist round its axis, even when the axis is curved, as, for instance, in the case of a ring of India rubber.

In an analogous manner, in four dimensions we can have rotation round a curved plane, if I may use the expression. A sphere can be turned inside out in four dimensions.

Let fig. 11 represent a spherical surface, on each side of which a layer of matter exists. The thickness of the matter is represented by the rods CD and EF, extending equally without and within.

Now, take the section of the sphere by the _yz_ plane we have a circle—fig. 12. Now, let the _w_ axis be drawn in place of the _x_ axis so that we have the space of _yzw_ represented. In this space all that there will be seen of the sphere is the circle drawn.

Here we see that there is no obstacle to prevent the rods turning round. If the matter is so elastic that it will give enough for the particles at E and C to be separated as they are at F and D, they can rotate round to the position D and F, and a similar motion is possible for all other particles. There is no matter or obstacle to prevent them from moving out in the _w_ direction, and then on round the circumference as an axis. Now, what will hold for one section will hold for all, as the fourth dimension is at right angles to all the sections which can be made of the sphere.

We have supposed the matter of which the sphere is composed to be three-dimensional. If the matter had a small thickness in the fourth dimension, there would be a slight thickness in fig. 12 above the plane of the paper—a thickness equal to the thickness of the matter in the fourth dimension. The rods would have to be replaced by thin slabs. But this would make no difference as to the possibility of the rotation. This motion is discussed by Newcomb in the first volume of the _American Journal of Mathematics_.

Let us now consider, not a merely extensible body, but a liquid one. A mass of rotating liquid, a whirl, eddy, or vortex, has many remarkable properties. On first consideration we should expect the rotating mass of liquid immediately to spread off and lose itself in the surrounding liquid. The water flies off a wheel whirled round, and we should expect the rotating liquid to be dispersed. But see the eddies in a river strangely persistent. The rings that occur in puffs of smoke and last so long are whirls or vortices curved round so that their opposite ends join together. A cyclone will travel over great distances.

Helmholtz was the first to investigate the properties of vortices. He studied them as they would occur in a perfect fluid—that is, one without friction of one moving portion or another. In such a medium vortices would be indestructible. They would go on for ever, altering their shape, but consisting always of the same portion of the fluid. But a straight vortex could not exist surrounded entirely by the fluid. The ends of a vortex must reach to some boundary inside or outside the fluid.

A vortex which is bent round so that its opposite ends join is capable of existing, but no vortex has a free end in the fluid. The fluid round the vortex is always in motion, and one produces a definite movement in another.

Lord Kelvin has proposed the hypothesis that portions of a fluid segregated in vortices account for the origin of matter. The properties of the ether in respect of its capacity of propagating disturbances can be explained by the assumption of vortices in it instead of by a property of rigidity. It is difficult to conceive, however, of any arrangement of the vortex rings and endless vortex filaments in the ether.

Now, the further consideration of four-dimensional rotations shows the existence of a kind of vortex which would make an ether filled with a homogeneous vortex motion easily thinkable.

To understand the nature of this vortex, we must go on and take a step by which we accept the full significance of the four-dimensional hypothesis. Granted four-dimensional axes, we have seen that a rotation of one into another leaves two unaltered, and these two form the axial plane about which the rotation takes place. But what about these two? Do they necessarily remain motionless? There is nothing to prevent a rotation of these two, one into the other, taking place concurrently with the first rotation. This possibility of a double rotation deserves the most careful attention, for it is the kind of movement which is distinctly typical of four dimensions.

Rotation round a plane is analogous to rotation round an axis. But in three-dimensional space there is no motion analogous to the double rotation, in which, while axis 1 changes into axis 2, axis 3 changes into axis 4.

Consider a four-dimensional body, with four independent axes, _x_, _y_, _z_, _w_. A point in it can move in only one direction at a given moment. If the body has a velocity of rotation by which the _x_ axis changes into the _y_ axis and all parallel sections move in a similar manner, then the point will describe a circle. If, now, in addition to the rotation by which the _x_ axis changes into the _y_ axis the body has a rotation by which the _z_ axis turns into the _w_ axis, the point in question will have a double motion in consequence of the two turnings. The motions will compound, and the point will describe a circle, but not the same circle which it would describe in virtue of either rotation separately.

We know that if a body in three-dimensional space is given two movements of rotation they will combine into a single movement of rotation round a definite axis. It is in no different condition from that in which it is subjected to one movement of rotation. The direction of the axis changes; that is all. The same is not true about a four-dimensional body. The two rotations, _x_ to _y_ and _z_ to _w_, are independent. A body subject to the two is in a totally different condition to that which it is in when subject to one only. When subject to a rotation such as that of _x_ to _y_, a whole plane in the body, as we have seen, is stationary. When subject to the double rotation no part of the body is stationary except the point common to the two planes of rotation.

If the two rotations are equal in velocity, every point in the body describes a circle. All points equally distant from the stationary point describe circles of equal size.

We can represent a four-dimensional sphere by means of two diagrams, in one of which we take the three axes, _x_, _y_, _z_; in the other the axes _x_, _w_, and _z_. In fig. 13 we have the view of a four-dimensional sphere in the space of _xyz_. Fig. 13 shows all that we can see of the four sphere in the space of _xyz_, for it represents all the points in that space, which are at an equal distance from the centre.

Let us now take the _xz_ section, and let the axis of _w_ take the place of the _y_ axis. Here, in fig. 14, we have the space of _xzw_. In this space we have to take all the points which are at the same distance from the centre, consequently we have another sphere. If we had a three-dimensional sphere, as has been shown before, we should have merely a circle in the _xzw_ space, the _xz_ circle seen in the space of _xzw_. But now, taking the view in the space of _xzw_, we have a sphere in that space also. In a similar manner, whichever set of three axes we take, we obtain a sphere.

In fig. 13, let us imagine the rotation in the direction _xy_ to be taking place. The point _x_ will turn to _y_, and _p_ to _p´_. The axis _zz´_ remains stationary, and this axis is all of the plane _zw_ which we can see in the space section exhibited in the figure.

In fig. 14, imagine the rotation from _z_ to _w_ to be taking place. The _w_ axis now occupies the position previously occupied by the _y_ axis. This does not mean that the _w_ axis can coincide with the _y_ axis. It indicates that we are looking at the four-dimensional sphere from a different point of view. Any three-space view will show us three axes, and in fig. 14 we are looking at _xzw_.

The only part that is identical in the two diagrams is the circle of the _x_ and _z_ axes, which axes are contained in both diagrams. Thus the plane _zxz´_ is the same in both, and the point _p_ represents the same point in both diagrams. Now, in fig. 14 let the _zw_ rotation take place, the _z_ axis will turn toward the point _w_ of the _w_ axis, and the point _p_ will move in a circle about the point _x_.

Thus in fig. 13 the point _p_ moves in a circle parallel to the _xy_ plane; in fig. 14 it moves in a circle parallel to the _zw_ plane, indicated by the arrow.

Now, suppose both of these independent rotations compounded, the point _p_ will move in a circle, but this circle will coincide with neither of the circles in which either one of the rotations will take it. The circle the point _p_ will move in will depend on its position on the surface of the four sphere.

In this double rotation, possible in four-dimensional space, there is a kind of movement totally unlike any with which we are familiar in three-dimensional space. It is a requisite preliminary to the discussion of the behaviour of the small particles of matter, with a view to determining whether they show the characteristics of four-dimensional movements, to become familiar with the main characteristics of this double rotation. And here I must rely on a formal and logical assent rather than on the intuitive apprehension, which can only be obtained by a more detailed study.

In the first place this double rotation consists in two varieties or kinds, which we will call the A and B kinds. Consider four axes, _x_, _y_, _z_, _w_. The rotation of _x_ to _y_ can be accompanied with the rotation of _z_ to _w_. Call this the A kind.

But also the rotation of _x_ to _y_ can be accompanied by the rotation, of not _z_ to _w_, but _w_ to _z_. Call this the B kind.

They differ in only one of the component rotations. One is not the negative of the other. It is the semi-negative. The opposite of an _x_ to _y_, _z_ to _w_ rotation would be _y_ to _x_, _w_ to _z_. The semi-negative is _x_ to _y_ and _w_ to _z_.

If four dimensions exist and we cannot perceive them, because the extension of matter is so small in the fourth dimension that all movements are withheld from direct observation except those which are three-dimensional, we should not observe these double rotations, but only the effects of them in three-dimensional movements of the type with which we are familiar.

If matter in its small particles is four-dimensional, we should expect this double rotation to be a universal characteristic of the atoms and molecules, for no portion of matter is at rest. The consequences of this corpuscular motion can be perceived, but only under the form of ordinary rotation or displacement. Thus, if the theory of four dimensions is true, we have in the corpuscles of matter a whole world of movement, which we can never study directly, but only by means of inference.

The rotation A, as I have defined it, consists of two equal rotations—one about the plane of _zw_, the other about the plane of _xy_. It is evident that these rotations are not necessarily equal. A body may be moving with a double rotation, in which these two independent components are not equal; but in such a case we can consider the body to be moving with a composite rotation—a rotation of the A or B kind and, in addition, a rotation about a plane.

If we combine an A and a B movement, we obtain a rotation about a plane; for, the first being _x_ to _y_ and _z_ to _w_, and the second being _x_ to _y_ and _w_ to _z_, when they are put together the _z_ to _w_ and _w_ to _z_ rotations neutralise each other, and we obtain an _x_ to _y_ rotation only, which is a rotation about the plane of _zw_. Similarly, if we take a B rotation, _y_ to _x_ and _z_ to _w_, we get, on combining this with the A rotation, a rotation of _z_ to _w_ about the _xy_ plane. In this case the plane of rotation is in the three-dimensional space of _xyz_, and we have—what has been described before—a twisting about a plane in our space.

Consider now a portion of a perfect liquid having an A motion. It can be proved that it possesses the properties of a vortex. It forms a permanent individuality—a separated-out portion of the liquid—accompanied by a motion of the surrounding liquid. It has properties analogous to those of a vortex filament. But it is not necessary for its existence that its ends should reach the boundary of the liquid. It is self-contained and, unless disturbed, is circular in every section.

If we suppose the ether to have its properties of transmitting vibration given it by such vortices, we must inquire how they lie together in four-dimensional space. Placing a circular disk on a plane and surrounding it by six others, we find that if the central one is given a motion of rotation, it imparts to the others a rotation which is antagonistic in every two adjacent ones. If A goes round, as shown by the arrow, B and C will be moving in opposite ways, and each tends to destroy the motion of the other.

Now, if we suppose spheres to be arranged in a corresponding manner in three-dimensional space, they will be grouped in figures which are for three-dimensional space what hexagons are for plane space. If a number of spheres of soft clay be pressed together, so as to fill up the interstices, each will assume the form of a fourteen-sided figure called a tetrakaidecagon.

Now, assuming space to be filled with such tetrakaidecagons, and placing a sphere in each, it will be found that one sphere is touched by eight others. The remaining six spheres of the fourteen which surround the central one will not touch it, but will touch three of those in contact with it. Hence, if the central sphere rotates, it will not necessarily drive those around it so that their motions will be antagonistic to each other, but the velocities will not arrange themselves in a systematic manner.

In four-dimensional space the figure which forms the next term of the series hexagon, tetrakaidecagon, is a thirty-sided figure. It has for its faces ten solid tetrakaidecagons and twenty hexagonal prisms. Such figures will exactly fill four-dimensional space, five of them meeting at every point. If, now, in each of these figures we suppose a solid four-dimensional sphere to be placed, any one sphere is surrounded by thirty others. Of these it touches ten, and, if it rotates, it drives the rest by means of these. Now, if we imagine the central sphere to be given an A or a B rotation, it will turn the whole mass of sphere round in a systematic manner. Suppose four-dimensional space to be filled with such spheres, each rotating with a double rotation, the whole mass would form one consistent system of motion, in which each one drove every other one, with no friction or lagging behind.

Every sphere would have the same kind of rotation. In three-dimensional space, if one body drives another round the second body rotates with the opposite kind of rotation; but in four-dimensional space these four-dimensional spheres would each have the double negative of the rotation of the one next it, and we have seen that the double negative of an A or B rotation is still an A or B rotation. Thus four-dimensional space could be filled with a system of self-preservative living energy. If we imagine the four-dimensional spheres to be of liquid and not of solid matter, then, even if the liquid were not quite perfect and there were a slight retarding effect of one vortex on another, the system would still maintain itself.

In this hypothesis we must look on the ether as possessing energy, and its transmission of vibrations, not as the conveying of a motion imparted from without, but as a modification of its own motion.

We are now in possession of some of the conceptions of four-dimensional mechanics, and will turn aside from the line of their development to inquire if there is any evidence of their applicability to the processes of nature.

Is there any mode of motion in the region of the minute which, giving three-dimensional movements for its effect, still in itself escapes the grasp of our mechanical theories? I would point to electricity. Through the labours of Faraday and Maxwell we are convinced that the phenomena of electricity are of the nature of the stress and strain of a medium; but there is still a gap to be bridged over in their explanation—the laws of elasticity, which Maxwell assumes, are not those of ordinary matter. And, to take another instance: a magnetic pole in the neighbourhood of a current tends to move. Maxwell has shown that the pressures on it are analogous to the velocities in a liquid which would exist if a vortex took the place of the electric current: but we cannot point out the definite mechanical explanation of these pressures. There must be some mode of motion of a body or of the medium in virtue of which a body is said to be electrified.

Take the ions which convey charges of electricity 500 times greater in proportion to their mass than are carried by the molecules of hydrogen in electrolysis. In respect of what motion can these ions be said to be electrified? It can be shown that the energy they possess is not energy of rotation. Think of a short rod rotating. If it is turned over it is found to be rotating in the opposite direction. Now, if rotation in one direction corresponds to positive electricity, rotation in the opposite direction corresponds to negative electricity, and the smallest electrified particles would have their charges reversed by being turned over—an absurd supposition.

If we fix on a mode of motion as a definition of electricity, we must have two varieties of it, one for positive and one for negative; and a body possessing the one kind must not become possessed of the other by any change in its position.

All three-dimensional motions are compounded of rotations and translations, and none of them satisfy this first condition for serving as a definition of electricity.

But consider the double rotation of the A and B kinds. A body rotating with the A motion cannot have its motion transformed into the B kind by being turned over in any way. Suppose a body has the rotation _x_ to _y_ and _z_ to _w_. Turning it about the _xy_ plane, we reverse the direction of the motion _x_ to _y_. But we also reverse the _z_ to _w_ motion, for the point at the extremity of the positive _z_ axis is now at the extremity of the negative _z_ axis, and since we have not interfered with its motion it goes in the direction of position _w_. Hence we have _y_ to _x_ and _w_ to _z_, which is the same as _x_ to _y_ and _z_ to _w_. Thus both components are reversed, and there is the A motion over again. The B kind is the semi-negative, with only one component reversed.

Hence a system of molecules with the A motion would not destroy it in one another, and would impart it to a body in contact with them. Thus A and B motions possess the first requisite which must be demanded in any mode of motion representative of electricity.

Let us trace out the consequences of defining positive electricity as an A motion and negative electricity as a B motion. The combination of positive and negative electricity produces a current. Imagine a vortex in the ether of the A kind and unite with this one of the B kind. An A motion and B motion produce rotation round a plane, which is in the ether a vortex round an axial surface. It is a vortex of the kind we represent as a part of a sphere turning inside out. Now such a vortex must have its rim on a boundary of the ether—on a body in the ether.

Let us suppose that a conductor is a body which has the property of serving as the terminal abutment of such a vortex. Then the conception we must form of a closed current is of a vortex sheet having its edge along the circuit of the conducting wire. The whole wire will then be like the centres on which a spindle turns in three-dimensional space, and any interruption of the continuity of the wire will produce a tension in place of a continuous revolution.

As the direction of the rotation of the vortex is from a three-space direction into the fourth dimension and back again, there will be no direction of flow to the current; but it will have two sides, according to whether _z_ goes to _w_ or _z_ goes to negative _w_.

We can draw any line from one part of the circuit to another; then the ether along that line is rotating round its points.

This geometric image corresponds to the definition of an electric circuit. It is known that the action does not lie in the wire, but in the medium, and it is known that there is no direction of flow in the wire.

No explanation has been offered in three-dimensional mechanics of how an action can be impressed throughout a region and yet necessarily run itself out along a closed boundary, as is the case in an electric current. But this phenomenon corresponds exactly to the definition of a four-dimensional vortex.

If we take a very long magnet, so long that one of its poles is practically isolated, and put this pole in the vicinity of an electric circuit, we find that it moves.

Now, assuming for the sake of simplicity that the wire which determines the current is in the form of a circle, if we take a number of small magnets and place them all pointing in the same direction normal to the plane of the circle, so that they fill it and the wire binds them round, we find that this sheet of magnets has the same effect on the magnetic pole that the current has. The sheet of magnets may be curved, but the edge of it must coincide with the wire. The collection of magnets is then equivalent to the vortex sheet, and an elementary magnet to a part of it. Thus, we must think of a magnet as conditioning a rotation in the ether round the plane which bisects at right angles the line joining its poles.

If a current is started in a circuit, we must imagine vortices like bowls turning themselves inside out, starting from the contour. In reaching a parallel circuit, if the vortex sheet were interrupted and joined momentarily to the second circuit by a free rim, the axis plane would lie between the two circuits, and a point on the second circuit opposite a point on the first would correspond to a point opposite to it on the first; hence we should expect a current in the opposite direction in the second circuit. Thus the phenomena of induction are not inconsistent with the hypothesis of a vortex about an axial plane.

In four-dimensional space, in which all four dimensions were commensurable, the intensity of the action transmitted by the medium would vary inversely as the cube of the distance. Now, the action of a current on a magnetic pole varies inversely as the square of the distance; hence, over measurable distances the extension of the ether in the fourth dimension cannot be assumed as other than small in comparison with those distances.

If we suppose the ether to be filled with vortices in the shape of four-dimensional spheres rotating with the A motion, the B motion would correspond to electricity in the one-fluid theory. There would thus be a possibility of electricity existing in two forms, statically, by itself, and, combined with the universal motion, in the form of a current.

To arrive at a definite conclusion it will be necessary to investigate the resultant pressures which accompany the collocation of solid vortices with surface ones.

To recapitulate:

The movements and mechanics of four-dimensional space are definite and intelligible. A vortex with a surface as its axis affords a geometric image of a closed circuit, and there are rotations which by their polarity afford a possible definition of statical electricity.[7]

[7] These double rotations of the A and B kinds I should like to call Hamiltons and co-Hamiltons, for it is a singular fact that in his “Quaternions” Sir Wm. Rowan Hamilton has given the theory of either the A or the B kind. They follow the laws of his symbols, I, J, K.

Hamiltons and co-Hamiltons seem to be natural units of geometrical expression. In the paper in the “Proceedings of the Royal Irish Academy,” Nov. 1903, already alluded to, I have shown something of the remarkable facility which is gained in dealing with the composition of three- and four-dimensional rotations by an alteration in Hamilton’s notation, which enables his system to be applied to both the A and B kinds of rotations.

The objection which has been often made to Hamilton’s system, namely, that it is only under special conditions of application that his processes give geometrically interpretable results, can be removed, if we assume that he was really dealing with a four-dimensional motion, and alter his notation to bring this circumstance into explicit recognition.

APPENDIX I

THE MODELS

In Chapter XI. a description has been given which will enable any one to make a set of models illustrative of the tesseract and its properties. The set here supposed to be employed consists of:—

1. Three sets of twenty-seven cubes each. 2. Twenty-seven slabs. 3. Twelve cubes with points, lines, faces, distinguished by colours, which will be called the catalogue cubes.

The preparation of the twelve catalogue cubes involves the expenditure of a considerable amount of time. It is advantageous to use them, but they can be replaced by the drawing of the views of the tesseract or by a reference to figs. 103, 104, 105, 106 of the text.

The slabs are coloured like the twenty-seven cubes of the first cubic block in fig. 101, the one with red, white, yellow axes.

The colours of the three sets of twenty-seven cubes are those of the cubes shown in fig. 101.

The slabs are used to form the representation of a cube in a plane, and can well be dispensed with by any one who is accustomed to deal with solid figures. But the whole theory depends on a careful observation of how the cube would be represented by these slabs.

In the first step, that of forming a clear idea how a plane being would represent three-dimensional space, only one of the catalogue cubes and one of the three blocks is needed.

APPLICATION TO THE STEP FROM PLANE TO SOLID.

Look at fig. 1 of the views of the tesseract, or, what comes to the same thing, take catalogue cube No. 1 and place it before you with the red line running up, the white line running to the right, the yellow line running away. The three dimensions of space are then marked out by these lines or axes. Now take a piece of cardboard, or a book, and place it so that it forms a wall extending up and down not opposite to you, but running away parallel to the wall of the room on your left hand.

Placing the catalogue cube against this wall we see that it comes into contact with it by the red and yellow lines, and by the included orange face.

In the plane being’s world the aspect he has of the cube would be a square surrounded by red and yellow lines with grey points.

Now, keeping the red line fixed, turn the cube about it so that the yellow line goes out to the right, and the white line comes into contact with the plane.

In this case a different aspect is presented to the plane being, a square, namely, surrounded by red and white lines and grey points. You should particularly notice that when the yellow line goes out, at right angles to the plane, and the white comes in, the latter does not run in the same sense that the yellow did.

From the fixed grey point at the base of the red line the yellow line ran away from you. The white line now runs towards you. This turning at right angles makes the line which was out of the plane before, come into it in an opposite sense to that in which the line ran which has just left the plane. If the cube does not break through the plane this is always the rule.

Again turn the cube back to the normal position with red running up, white to the right, and yellow away, and try another turning.

You can keep the yellow line fixed, and turn the cube about it. In this case the red line going out to the right the white line will come in pointing downwards.

You will be obliged to elevate the cube from the table in order to carry out this turning. It is always necessary when a vertical axis goes out of a space to imagine a movable support which will allow the line which ran out before to come in below.

Having looked at the three ways of turning the cube so as to present different faces to the plane, examine what would be the appearance if a square hole were cut in the piece of cardboard, and the cube were to pass through it. A hole can be actually cut, and it will be seen that in the normal position, with red axis running up, yellow away, and white to the right, the square first perceived by the plane being—the one contained by red and yellow lines—would be replaced by another square of which the line towards you is pink—the section line of the pink face. The line above is light yellow, below is light yellow and on the opposite side away from you is pink.

In the same way the cube can be pushed through a square opening in the plane from any of the positions which you have already turned it into. In each case the plane being will perceive a different set of contour lines.

Having observed these facts about the catalogue cube, turn now to the first block of twenty-seven cubes.

You notice that the colour scheme on the catalogue cube and that of this set of blocks is the same.

Place them before you, a grey or null cube on the table, above it a red cube, and on the top a null cube again. Then away from you place a yellow cube, and beyond it a null cube. Then to the right place a white cube and beyond it another null. Then complete the block, according to the scheme of the catalogue cube, putting in the centre of all an ochre cube.

You have now a cube like that which is described in the text. For the sake of simplicity, in some cases, this cubic block can be reduced to one of eight cubes, by leaving out the terminations in each direction. Thus, instead of null, red, null, three cubes, you can take null, red, two cubes, and so on.

It is useful, however, to practise the representation in a plane of a block of twenty-seven cubes. For this purpose take the slabs, and build them up against the piece of cardboard, or the book in such a way as to represent the different aspects of the cube.

Proceed as follows:—

First, cube in normal position.

Place nine slabs against the cardboard to represent the nine cubes in the wall of the red and yellow axes, facing the cardboard; these represent the aspect of the cube as it touches the plane.

Now push these along the cardboard and make a different set of nine slabs to represent the appearance which the cube would present to a plane being, if it were to pass half way through the plane.

There would be a white slab, above it a pink one, above that another white one, and six others, representing what would be the nature of a section across the middle of the block of cubes. The section can be thought of as a thin slice cut out by two parallel cuts across the cube. Having arranged these nine slabs, push them along the plane, and make another set of nine to represent what would be the appearance of the cube when it had almost completely gone through. This set of nine will be the same as the first set of nine.

Now we have in the plane three sets of nine slabs each, which represent three sections of the twenty-seven block.

They are put alongside one another. We see that it does not matter in what order the sets of nine are put. As the cube passes through the plane they represent appearances which follow the one after the other. If they were what they represented, they could not exist in the same plane together.

This is a rather important point, namely, to notice that they should not co-exist on the plane, and that the order in which they are placed is indifferent. When we represent a four-dimensional body our solid cubes are to us in the same position that the slabs are to the plane being. You should also notice that each of these slabs represents only the very thinnest slice of a cube. The set of nine slabs first set up represents the side surface of the block. It is, as it were, a kind of tray—a beginning from which the solid cube goes off. The slabs as we use them have thickness, but this thickness is a necessity of construction. They are to be thought of as merely of the thickness of a line.

If now the block of cubes passed through the plane at the rate of an inch a minute the appearance to a plane being would be represented by:—

1. The first set of nine slabs lasting for one minute.

2. The second set of nine slabs lasting for one minute.

3. The third set of nine slabs lasting for one minute.

Now the appearances which the cube would present to the plane being in other positions can be shown by means of these slabs. The use of such slabs would be the means by which a plane being could acquire a familiarity with our cube. Turn the catalogue cube (or imagine the coloured figure turned) so that the red line runs up, the yellow line out to the right, and the white line towards you. Then turn the block of cubes to occupy a similar position.

The block has now a different wall in contact with the plane. Its appearance to a plane being will not be the same as before. He has, however, enough slabs to represent this new set of appearances. But he must remodel his former arrangement of them.

He must take a null, a red, and a null slab from the first of his sets of slabs, then a white, a pink, and a white from the second, and then a null, a red, and a null from the third set of slabs.

He takes the first column from the first set, the first column from the second set, and the first column from the third set.

To represent the half-way-through appearance, which is as if a very thin slice were cut out half way through the block, he must take the second column of each of his sets of slabs, and to represent the final appearance, the third column of each set.

Now turn the catalogue cube back to the normal position, and also the block of cubes.

There is another turning—a turning about the yellow line, in which the white axis comes below the support.

You cannot break through the surface of the table, so you must imagine the old support to be raised. Then the top of the block of cubes in its new position is at the level at which the base of it was before.

Now representing the appearance on the plane, we must draw a horizontal line to represent the old base. The line should be drawn three inches high on the cardboard.

Below this the representative slabs can be arranged.

It is easy to see what they are. The old arrangements have to be broken up, and the layers taken in order, the first layer of each for the representation of the aspect of the block as it touches the plane.

Then the second layers will represent the appearance half way through, and the third layers will represent the final appearance.

It is evident that the slabs individually do not represent the same portion of the cube in these different presentations.

In the first case each slab represents a section or a face perpendicular to the white axis, in the second case a face or a section which runs perpendicularly to the yellow axis, and in the third case a section or a face perpendicular to the red axis.

But by means of these nine slabs the plane being can represent the whole of the cubic block. He can touch and handle each portion of the cubic block, there is no part of it which he cannot observe. Taking it bit by bit, two axes at a time, he can examine the whole of it.

OUR REPRESENTATION OF A BLOCK OF TESSERACTS.

Look at the views of the tesseract 1, 2, 3, or take the catalogue cubes 1, 2, 3, and place them in front of you, in any order, say running from left to right, placing 1 in the normal position, the red axis running up, the white to the right, and yellow away.

Now notice that in catalogue cube 2 the colours of each region are derived from those of the corresponding region of cube 1 by the addition of blue. Thus null + blue = blue, and the corners of number 2 are blue. Again, red + blue = purple, and the vertical lines of 2 are purple. Blue + yellow = green, and the line which runs away is coloured green.

By means of these observations you may be sure that catalogue cube 2 is rightly placed. Catalogue cube 3 is just like number 1.

Having these cubes in what we may call their normal position, proceed to build up the three sets of blocks.

This is easily done in accordance with the colour scheme on the catalogue cubes.

The first block we already know. Build up the second block, beginning with a blue corner cube, placing a purple on it, and so on.

Having these three blocks we have the means of representing the appearances of a group of eighty-one tesseracts.

Let us consider a moment what the analogy in the case of the plane being is.

He has his three sets of nine slabs each. We have our three sets of twenty-seven cubes each.

Our cubes are like his slabs. As his slabs are not the things which they represent to him, so our cubes are not the things they represent to us.

The plane being’s slabs are to him the faces of cubes.

Our cubes then are the faces of tesseracts, the cubes by which they are in contact with our space.

As each set of slabs in the case of the plane being might be considered as a sort of tray from which the solid contents of the cubes came out, so our three blocks of cubes may be considered as three-space trays, each of which is the beginning of an inch of the solid contents of the four-dimensional solids starting from them.

We want now to use the names null, red, white, etc., for tesseracts. The cubes we use are only tesseract faces. Let us denote that fact by calling the cube of null colour, null face; or, shortly, null f., meaning that it is the face of a tesseract.

To determine which face it is let us look at the catalogue cube 1 or the first of the views of the tesseract, which can be used instead of the models. It has three axes, red, white, yellow, in our space. Hence the cube determined by these axes is the face of the tesseract which we now have before us. It is the ochre face. It is enough, however, simply to say null f., red f. for the cubes which we use.

To impress this in your mind, imagine that tesseracts do actually run from each cube. Then, when you move the cubes about, you move the tesseracts about with them. You move the face but the tesseract follows with it, as the cube follows when its face is shifted in a plane.

The cube null in the normal position is the cube which has in it the red, yellow, white axes. It is the face having these, but wanting the blue. In this way you can define which face it is you are handling. I will write an “f.” after the name of each tesseract just as the plane being might call each of his slabs null slab, yellow slab, etc., to denote that they were representations.

We have then in the first block of twenty-seven cubes, the following—null f., red f., null f., going up; white f., null f., lying to the right, and so on. Starting from the null point and travelling up one inch we are in the null region, the same for the away and the right-hand directions. And if we were to travel in the fourth dimension for an inch we should still be in a null region. The tesseract stretches equally all four ways. Hence the appearance we have in this first block would do equally well if the tesseract block were to move across our space for a certain distance. For anything less than an inch of their transverse motion we should still have the same appearance. You must notice, however, that we should not have null face after the motion had begun.

When the tesseract, null for instance, had moved ever so little we should not have a face of null but a section of null in our space. Hence, when we think of the motion across our space we must call our cubes tesseract sections. Thus on null passing across we should see first null f., then null s., and then, finally, null f. again.

Imagine now the whole first block of twenty-seven tesseracts to have moved tranverse to our space a distance of one inch. Then the second set of tesseracts, which originally were an inch distant from our space, would be ready to come in.

Their colours are shown in the second block of twenty-seven cubes which you have before you. These represent the tesseract faces of the set of tesseracts that lay before an inch away from our space. They are ready now to come in, and we can observe their colours. In the place which null f. occupied before we have blue f., in place of red f. we have purple f., and so on. Each tesseract is coloured like the one whose place it takes in this motion with the addition of blue.

Now if the tesseract block goes on moving at the rate of an inch a minute, this next set of tesseracts will occupy a minute in passing across. We shall see, to take the null one for instance, first of all null face, then null section, then null face again.

At the end of the second minute the second set of tesseracts has gone through, and the third set comes in. This, as you see, is coloured just like the first. Altogether, these three sets extend three inches in the fourth dimension, making the tesseract block of equal magnitude in all dimensions.

We have now before us a complete catalogue of all the tesseracts in our group. We have seen them all, and we shall refer to this arrangement of the blocks as the “normal position.” We have seen as much of each tesseract at a time as could be done in a three-dimensional space. Each part of each tesseract has been in our space, and we could have touched it.

The fourth dimension appeared to us as the duration of the block.

If a bit of our matter were to be subjected to the same motion it would be instantly removed out of our space. Being thin in the fourth dimension it is at once taken out of our space by a motion in the fourth dimension.

But the tesseract block we represent having length in the fourth dimension remains steadily before our eyes for three minutes, when it is subjected to this transverse motion.

We have now to form representations of the other views of the same tesseract group which are possible in our space.

Let us then turn the block of tesseracts so that another face of it comes into contact with our space, and then by observing what we have, and what changes come when the block traverses our space, we shall have another view of it. The dimension which appeared as duration before will become extension in one of our known dimensions, and a dimension which coincided with one of our space dimensions will appear as duration.

Leaving catalogue cube 1 in the normal position, remove the other two, or suppose them removed. We have in space the red, the yellow, and the white axes. Let the white axis go out into the unknown, and occupy the position the blue axis holds. Then the blue axis, which runs in that direction now will come into space. But it will not come in pointing in the same way that the white axis does now. It will point in the opposite sense. It will come in running to the left instead of running to the right as the white axis does now.

When this turning takes place every part of the cube 1 will disappear except the left-hand face—the orange face.

And the new cube that appears in our space will run to the left from this orange face, having axes, red, yellow, blue.

Take models 4, 5, 6. Place 4, or suppose No. 4 of the tesseract views placed, with its orange face coincident with the orange face of 1, red line to red line, and yellow line to yellow line, with the blue line pointing to the left. Then remove cube 1 and we have the tesseract face which comes in when the white axis runs in the positive unknown, and the blue axis comes into our space.

Now place catalogue cube 5 in some position, it does not matter which, say to the left; and place it so that there is a correspondence of colour corresponding to the colour of the line that runs out of space. The line that runs out of space is white, hence, every part of this cube 5 should differ from the corresponding part of 4 by an alteration in the direction of white.

Thus we have white points in 5 corresponding to the null points in 4. We have a pink line corresponding to a red line, a light yellow line corresponding to a yellow line, an ochre face corresponding to an orange face. This cube section is completely named in Chapter XI. Finally cube 6 is a replica of 1.

These catalogue cubes will enable us to set up our models of the block of tesseracts.

First of all for the set of tesseracts, which beginning in our space reach out one inch in the unknown, we have the pattern of catalogue cube 4.

We see that we can build up a block of twenty-seven tesseract faces after the colour scheme of cube 4, by taking the left-hand wall of block 1, then the left-hand wall of block 2, and finally that of block 3. We take, that is, the three first walls of our previous arrangement to form the first cubic block of this new one.

This will represent the cubic faces by which the group of tesseracts in its new position touches our space. We have running up, null f., red f., null f. In the next vertical line, on the side remote from us, we have yellow f., orange f., yellow f., and then the first colours over again. Then the three following columns are, blue f., purple f., blue f.; green f., brown f., green f.; blue f., purple f., blue f. The last three columns are like the first.

These tesseracts touch our space, and none of them are by any part of them distant more than an inch from it. What lies beyond them in the unknown?

This can be told by looking at catalogue cube 5. According to its scheme of colour we see that the second wall of each of our old arrangements must be taken. Putting them together we have, as the corner, white f. above it, pink f. above it, white f. The column next to this remote from us is as follows:—light yellow f., ochre f., light yellow f., and beyond this a column like the first. Then for the middle of the block, light blue f., above it light purple, then light blue. The centre column has, at the bottom, light green f., light brown f. in the centre and at the top light green f. The last wall is like the first.

The third block is made by taking the third walls of our previous arrangement, which we called the normal one.

You may ask what faces and what sections our cubes represent. To answer this question look at what axes you have in our space. You have red, yellow, blue. Now these determine brown. The colours red, yellow, blue are supposed by us when mixed to produce a brown colour. And that cube which is determined by the red, yellow, blue axes we call the brown cube.

When the tesseract block in its new position begins to move across our space each tesseract in it gives a section in our space. This section is transverse to the white axis, which now runs in the unknown.

As the tesseract in its present position passes across our space, we should see first of all the first of the blocks of cubic faces we have put up—these would last for a minute, then would come the second block and then the third. At first we should have a cube of tesseract faces, each of which would be brown. Directly the movement began, we should have tesseract sections transverse to the white line.

There are two more analogous positions in which the block of tesseracts can be placed. To find the third position, restore the blocks to the normal arrangement.

Let us make the yellow axis go out into the positive unknown, and let the blue axis, consequently, come in running towards us. The yellow ran away, so the blue will come in running towards us.

Put catalogue cube 1 in its normal position. Take catalogue cube 7 and place it so that its pink face coincides with the pink face of cube 1, making also its red axis coincide with the red axis of 1 and its white with the white. Moreover, make cube 7 come towards us from cube 1. Looking at it we see in our space, red, white, and blue axes. The yellow runs out. Place catalogue cube 8 in the neighbourhood of 7—observe that every region in 8 has a change in the direction of yellow from the corresponding region in 7. This is because it represents what you come to now in going in the unknown, when the yellow axis runs out of our space. Finally catalogue cube 9, which is like number 7, shows the colours of the third set of tesseracts. Now evidently, starting from the normal position, to make up our three blocks of tesseract faces we have to take the near wall from the first block, the near wall from the second, and then the near wall from the third block. This gives us the cubic block formed by the faces of the twenty-seven tesseracts which are now immediately touching our space.

Following the colour scheme of catalogue cube 8, we make the next set of twenty-seven tesseract faces, representing the tesseracts, each of which begins one inch off from our space, by putting the second walls of our previous arrangement together, and the representation of the third set of tesseracts is the cubic block formed of the remaining three walls.

Since we have red, white, blue axes in our space to begin with, the cubes we see at first are light purple tesseract faces, and after the transverse motion begins we have cubic sections transverse to the yellow line.

Restore the blocks to the normal position, there remains the case in which the red axis turns out of space. In this case the blue axis will come in downwards, opposite to the sense in which the red axis ran.

In this case take catalogue cubes 10, 11, 12. Lift up catalogue cube 1 and put 10 underneath it, imagining that it goes down from the previous position of 1.

We have to keep in space the white and the yellow axes, and let the red go out, the blue come in.

Now, you will find on cube 10 a light yellow face; this should coincide with the base of 1, and the white and yellow lines on the two cubes should coincide. Then the blue axis running down you have the catalogue cube correctly placed, and it forms a guide for putting up the first representative block.

Catalogue cube 11 will represent what lies in the fourth dimension—now the red line runs in the fourth dimension. Thus the change from 10 to 11 should be towards red, corresponding to a null point is a red point, to a white line is a pink line, to a yellow line an orange line, and so on.

Catalogue cube 12 is like 10. Hence we see that to build up our blocks of tesseract faces we must take the bottom layer of the first block, hold that up in the air, underneath it place the bottom layer of the second block, and finally underneath this last the bottom layer of the last of our normal blocks.

Similarly we make the second representative group by taking the middle courses of our three blocks. The last is made by taking the three topmost layers. The three axes in our space before the transverse motion begins are blue, white, yellow, so we have light green tesseract faces, and after the motion begins sections transverse to the red light.

These three blocks represent the appearances as the tesseract group in its new position passes across our space. The cubes of contact in this case are those determinal by the three axes in our space, namely, the white, the yellow, the blue. Hence they are light green.

It follows from this that light green is the interior cube of the first block of representative cubic faces.

Practice in the manipulations described, with a realization in each case of the face or section which is in our space, is one of the best means of a thorough comprehension of the subject.

We have to learn how to get any part of these four-dimensional figures into space, so that we can look at them. We must first learn to swing a tesseract, and a group of tesseracts about in any way.

When these operations have been repeated and the method of arrangement of the set of blocks has become familiar, it is a good plan to rotate the axes of the normal cube 1 about a diagonal, and then repeat the whole series of turnings.

Thus, in the normal position, red goes up, white to the right, yellow away. Make white go up, yellow to the right, and red away. Learn the cube in this position by putting up the set of blocks of the normal cube, over and over again till it becomes as familiar to you as in the normal position. Then when this is learned, and the corresponding changes in the arrangements of the tesseract groups are made, another change should be made: let, in the normal cube, yellow go up, red to the right, and white away.

Learn the normal block of cubes in this new position by arranging them and re-arranging them till you know without thought where each one goes. Then carry out all the tesseract arrangements and turnings.

If you want to understand the subject, but do not see your way clearly, if it does not seem natural and easy to you, practise these turnings. Practise, first of all, the turning of a block of cubes round, so that you know it in every position as well as in the normal one. Practise by gradually putting up the set of cubes in their new arrangements. Then put up the tesseract blocks in their arrangements. This will give you a working conception of higher space, you will gain the feeling of it, whether you take up the mathematical treatment of it or not.

APPENDIX II

A LANGUAGE OF SPACE

The mere naming the parts of the figures we consider involves a certain amount of time and attention. This time and attention leads to no result, for with each new figure the nomenclature applied is completely changed, every letter or symbol is used in a different significance.

Surely it must be possible in some way to utilise the labour thus at present wasted!

Why should we not make a language for space itself, so that every position we want to refer to would have its own name? Then every time we named a figure in order to demonstrate its properties we should be exercising ourselves in the vocabulary of place.

If we use a definite system of names, and always refer to the same space position by the same name, we create as it were a multitude of little hands, each prepared to grasp a special point, position, or element, and hold it for us in its proper relations.

We make, to use another analogy, a kind of mental paper, which has somewhat of the properties of a sensitive plate, in that it will register, without effort, complex, visual, or tactual impressions.

But of far more importance than the applications of a space language to the plane and to solid space is the facilitation it brings with it to the study of four-dimensional shapes.

I have delayed introducing a space language because all the systems I made turned out, after giving them a fair trial, to be intolerable. I have now come upon one which seems to present features of permanence, and I will here give an outline of it, so that it can be applied to the subject of the text, and in order that it may be subjected to criticism.

The principle on which the language is constructed is to sacrifice every other consideration for brevity.

It is indeed curious that we are able to talk and converse on every subject of thought except the fundamental one of space. The only way of speaking about the spatial configurations that underlie every subject of discursive thought is a co-ordinate system of numbers. This is so awkward and incommodious that it is never used. In thinking also, in realising shapes, we do not use it; we confine ourselves to a direct visualisation.

Now, the use of words corresponds to the storing up of our experience in a definite brain structure. A child, in the endless tactual, visual, mental manipulations it makes for itself, is best left to itself, but in the course of instruction the introduction of space names would make the teachers work more cumulative, and the child’s knowledge more social.

Their full use can only be appreciated, if they are introduced early in the course of education; but in a minor degree any one can convince himself of their utility, especially in our immediate subject of handling four-dimensional shapes. The sum total of the results obtained in the preceding pages can be compendiously and accurately expressed in nine words of the Space Language.

In one of Plato’s dialogues Socrates makes an experiment on a slave boy standing by. He makes certain perceptions of space awake in the mind of Meno’s slave by directing his close attention on some simple facts of geometry.

By means of a few words and some simple forms we can repeat Plato’s experiment on new ground.

Do we by directing our close attention on the facts of four dimensions awaken a latent faculty in ourselves? The old experiment of Plato’s, it seems to me, has come down to us as novel as on the day he incepted it, and its significance not better understood through all the discussion of which it has been the subject.

Imagine a voiceless people living in a region where everything had a velvety surface, and who were thus deprived of all opportunity of experiencing what sound is. They could observe the slow pulsations of the air caused by their movements, and arguing from analogy, they would no doubt infer that more rapid vibrations were possible. From the theoretical side they could determine all about these more rapid vibrations. They merely differ, they would say, from slower ones, by the number that occur in a given time; there is a merely formal difference.

But suppose they were to take the trouble, go to the pains of producing these more rapid vibrations, then a totally new sensation would fall on their rudimentary ears. Probably at first they would only be dimly conscious of Sound, but even from the first they would become aware that a merely formal difference, a mere difference in point of number in this particular respect, made a great difference practically, as related to them. And to us the difference between three and four dimensions is merely formal, numerical. We can tell formally all about four dimensions, calculate the relations that would exist. But that the difference is merely formal does not prove that it is a futile and empty task, to present to ourselves as closely as we can the phenomena of four dimensions. In our formal knowledge of it, the whole question of its actual relation to us, as we are, is left in abeyance.

Possibly a new apprehension of nature may come to us through the practical, as distinguished from the mathematical and formal, study of four dimensions. As a child handles and examines the objects with which he comes in contact, so we can mentally handle and examine four-dimensional objects. The point to be determined is this. Do we find something cognate and natural to our faculties, or are we merely building up an artificial presentation of a scheme only formally possible, conceivable, but which has no real connection with any existing or possible experience?

This, it seems to me, is a question which can only be settled by actually trying. This practical attempt is the logical and direct continuation of the experiment Plato devised in the “Meno.”

Why do we think true? Why, by our processes of thought, can we predict what will happen, and correctly conjecture the constitution of the things around us? This is a problem which every modern philosopher has considered, and of which Descartes, Leibnitz, Kant, to name a few, have given memorable solutions. Plato was the first to suggest it. And as he had the unique position of being the first devisor of the problem, so his solution is the most unique. Later philosophers have talked about consciousness and its laws, sensations, categories. But Plato never used such words. Consciousness apart from a conscious being meant nothing to him. His was always an objective search. He made man’s intuitions the basis of a new kind of natural history.

In a few simple words Plato puts us in an attitude with regard to psychic phenomena—the mind—the ego—“what we are,” which is analogous to the attitude scientific men of the present day have with regard to the phenomena of outward nature. Behind this first apprehension of ours of nature, there is an infinite depth to be learned and known. Plato said that behind the phenomena of mind that Meno’s slave boy exhibited, there was a vast, an infinite perspective. And his singularity, his originality, comes out most strongly marked in this, that the perspective, the complex phenomena beyond were, according to him, phenomena of personal experience. A footprint in the sand means a man to a being that has the conception of a man. But to a creature that has no such conception, it means a curious mark, somehow resulting from the concatenation of ordinary occurrences. Such a being would attempt merely to explain how causes known to him could so coincide as to produce such a result; he would not recognise its significance.

Plato introduced the conception which made a new kind of natural history possible. He said that Meno’s slave boy thought true about things he had never learned, because his “soul” had experience. I know this will sound absurd to some people, and it flies straight in the face of the maxim, that explanation consists in showing how an effect depends on simple causes. But what a mistaken maxim that is! Can any single instance be shown of a simple cause? Take the behaviour of spheres for instance; say those ivory spheres, billiard balls, for example. We can explain their behaviour by supposing they are homogeneous elastic solids. We can give formulæ which will account for their movements in every variety. But are they homogeneous elastic solids? No, certainly not. They are complex in physical and molecular structure, and atoms and ions beyond open an endless vista. Our simple explanation is false, false as it can be. The balls act as if they were homogeneous elastic spheres. There is a statistical simplicity in the resultant of very complex conditions, which makes that artificial conception useful. But its usefulness must not blind us to the fact that it is artificial. If we really look deep into nature, we find a much greater complexity than we at first suspect. And so behind this simple “I,” this myself, is there not a parallel complexity? Plato’s “soul” would be quite acceptable to a large class of thinkers, if by “soul” and the complexity he attributes to it, he meant the product of a long course of evolutionary changes, whereby simple forms of living matter endowed with rudimentary sensation had gradually developed into fully conscious beings.

But Plato does not mean by “soul” a being of such a kind. His soul is a being whose faculties are clogged by its bodily environment, or at least hampered by the difficulty of directing its bodily frame—a being which is essentially higher than the account it gives of itself through its organs. At the same time Plato’s soul is not incorporeal. It is a real being with a real experience. The question of whether Plato had the conception of non-spatial existence has been much discussed. The verdict is, I believe, that even his “ideas” were conceived by him as beings in space, or, as we should say, real. Plato’s attitude is that of Science, inasmuch as he thinks of a world in Space. But, granting this, it cannot be denied that there is a fundamental divergence between Plato’s conception and the evolutionary theory, and also an absolute divergence between his conception and the genetic account of the origin of the human faculties. The functions and capacities of Plato’s “soul” are not derived by the interaction of the body and its environment.

Plato was engaged on a variety of problems, and his religious and ethical thoughts were so keen and fertile that the experimental investigation of his soul appears involved with many other motives. In one passage Plato will combine matter of thought of all kinds and from all sources, overlapping, interrunning. And in no case is he more involved and rich than in this question of the soul. In fact, I wish there were two words, one denoting that being, corporeal and real, but with higher faculties than we manifest in our bodily actions, which is to be taken as the subject of experimental investigation; and the other word denoting “soul” in the sense in which it is made the recipient and the promise of so much that men desire. It is the soul in the former sense that I wish to investigate, and in a limited sphere only. I wish to find out, in continuation of the experiment in the Meno, what the “soul” in us thinks about extension, experimenting on the grounds laid down by Plato. He made, to state the matter briefly, the hypothesis with regard to the thinking power of a being in us, a “soul.” This soul is not accessible to observation by sight or touch, but it can be observed by its functions; it is the object of a new kind of natural history, the materials for constructing which lie in what it is natural to us to think. With Plato “thought” was a very wide-reaching term, but still I would claim in his general plan of procedure a place for the particular question of extension.

The problem comes to be, “What is it natural to us to think about matter _qua_ extended?”

First of all, I find that the ordinary intuition of any simple object is extremely imperfect. Take a block of differently marked cubes, for instance, and become acquainted with them in their positions. You may think you know them quite well, but when you turn them round—rotate the block round a diagonal, for instance—you will find that you have lost track of the individuals in their new positions. You can mentally construct the block in its new position, by a rule, by taking the remembered sequences, but you don’t know it intuitively. By observation of a block of cubes in various positions, and very expeditiously by a use of Space names applied to the cubes in their different presentations, it is possible to get an intuitive knowledge of the block of cubes, which is not disturbed by any displacement. Now, with regard to this intuition, we moderns would say that I had formed it by my tactual visual experiences (aided by hereditary pre-disposition). Plato would say that the soul had been stimulated to recognise an instance of shape which it knew. Plato would consider the operation of learning merely as a stimulus; we as completely accounting for the result. The latter is the more common-sense view. But, on the other hand, it presupposes the generation of experience from physical changes. The world of sentient experience, according to the modern view, is closed and limited; only the physical world is ample and large and of ever-to-be-discovered complexity. Plato’s world of soul, on the other hand, is at least as large and ample as the world of things.

Let us now try a crucial experiment. Can I form an intuition of a four-dimensional object? Such an object is not given in the physical range of my sense contacts. All I can do is to present to myself the sequences of solids, which would mean the presentation to me under my conditions of a four-dimensional object. All I can do is to visualise and tactualise different series of solids which are alternative sets of sectional views of a four-dimensional shape.

If now, on presenting these sequences, I find a power in me of intuitively passing from one of these sets of sequences to another, of, being given one, intuitively constructing another, not using a rule, but directly apprehending it, then I have found a new fact about my soul, that it has a four-dimensional experience; I have observed it by a function it has.

I do not like to speak positively, for I might occasion a loss of time on the part of others, if, as may very well be, I am mistaken. But for my own part, I think there are indications of such an intuition; from the results of my experiments, I adopt the hypothesis that that which thinks in us has an ample experience, of which the intuitions we use in dealing with the world of real objects are a part; of which experience, the intuition of four-dimensional forms and motions is also a part. The process we are engaged in intellectually is the reading the obscure signals of our nerves into a world of reality, by means of intuitions derived from the inner experience.

The image I form is as follows. Imagine the captain of a modern battle-ship directing its course. He has his charts before him; he is in communication with his associates and subordinates; can convey his messages and commands to every part of the ship, and receive information from the conning-tower and the engine-room. Now suppose the captain immersed in the problem of the navigation of his ship over the ocean, to have so absorbed himself in the problem of the direction of his craft over the plane surface of the sea that he forgets himself. All that occupies his attention is the kind of movement that his ship makes. The operations by which that movement is produced have sunk below the threshold of his consciousness, his own actions, by which he pushes the buttons, gives the orders, are so familiar as to be automatic, his mind is on the motion of the ship as a whole. In such a case we can imagine that he identifies himself with his ship; all that enters his conscious thought is the direction of its movement over the plane surface of the ocean.

Such is the relation, as I imagine it, of the soul to the body. A relation which we can imagine as existing momentarily in the case of the captain is the normal one in the case of the soul with its craft. As the captain is capable of a kind of movement, an amplitude of motion, which does not enter into his thoughts with regard to the directing the ship over the plane surface of the ocean, so the soul is capable of a kind of movement, has an amplitude of motion, which is not used in its task of directing the body in the three-dimensional region in which the body’s activity lies. If for any reason it became necessary for the captain to consider three-dimensional motions with regard to his ship, it would not be difficult for him to gain the materials for thinking about such motions; all he has to do is to call his own intimate experience into play. As far as the navigation of the ship, however, is concerned, he is not obliged to call on such experience. The ship as a whole simply moves on a surface. The problem of three-dimensional movement does not ordinarily concern its steering. And thus with regard to ourselves all those movements and activities which characterise our bodily organs are three-dimensional; we never need to consider the ampler movements. But we do more than use the movements of our body to effect our aims by direct means; we have now come to the pass when we act indirectly on nature, when we call processes into play which lie beyond the reach of any explanation we can give by the kind of thought which has been sufficient for the steering of our craft as a whole. When we come to the problem of what goes on in the minute, and apply ourselves to the mechanism of the minute, we find our habitual conceptions inadequate.

The captain in us must wake up to his own intimate nature, realise those functions of movement which are his own, and in virtue of his knowledge of them apprehend how to deal with the problems he has come to.

Think of the history of man. When has there been a time, in which his thoughts of form and movement were not exclusively of such varieties as were adapted for his bodily performance? We have never had a demand to conceive what our own most intimate powers are. But, just as little as by immersing himself in the steering of his ship over the plane surface of the ocean, a captain can lose the faculty of thinking about what he actually does, so little can the soul lose its own nature. It can be roused to an intuition that is not derived from the experience which the senses give. All that is necessary is to present some few of those appearances which, while inconsistent with three-dimensional matter, are yet consistent with our formal knowledge of four-dimensional matter, in order for the soul to wake up and not begin to learn, but of its own intimate feeling fill up the gaps in the presentiment, grasp the full orb of possibilities from the isolated points presented to it. In relation to this question of our perceptions, let me suggest another illustration, not taking it too seriously, only propounding it to exhibit the possibilities in a broad and general way.

In the heavens, amongst the multitude of stars, there are some which, when the telescope is directed on them, seem not to be single stars, but to be split up into two. Regarding these twin stars through a spectroscope, an astronomer sees in each a spectrum of bands of colour and black lines. Comparing these spectrums with one another, he finds that there is a slight relative shifting of the dark lines, and from that shifting he knows that the stars are rotating round one another, and can tell their relative velocity with regard to the earth. By means of his terrestrial physics he reads this signal of the skies. This shifting of lines, the mere slight variation of a black line in a spectrum, is very unlike that which the astronomer knows it means. But it is probably much more like what it means than the signals which the nerves deliver are like the phenomena of the outer world.

No picture of an object is conveyed through the nerves. No picture of motion, in the sense in which we postulate its existence, is conveyed through the nerves. The actual deliverances of which our consciousness takes account are probably identical for eye and ear, sight and touch.

If for a moment I take the whole earth together and regard it as a sentient being, I find that the problem of its apprehension is a very complex one, and involves a long series of personal and physical events. Similarly the problem of our apprehension is a very complex one. I only use this illustration to exhibit my meaning. It has this especial merit, that, as the process of conscious apprehension takes place in our case in the minute, so, with regard to this earth being, the corresponding process takes place in what is relatively to it very minute.

Now, Plato’s view of a soul leads us to the hypothesis that that which we designate as an act of apprehension may be a very complex event, both physically and personally. He does not seek to explain what an intuition is; he makes it a basis from whence he sets out on a voyage of discovery. Knowledge means knowledge; he puts conscious being to account for conscious being. He makes an hypothesis of the kind that is so fertile in physical science—an hypothesis making no claim to finality, which marks out a vista of possible determination behind determination, like the hypothesis of space itself, the type of serviceable hypotheses.

And, above all, Plato’s hypothesis is conducive to experiment. He gives the perspective in which real objects can be determined; and, in our present enquiry, we are making the simplest of all possible experiments—we are enquiring what it is natural to the soul to think of matter as extended.

Aristotle says we always use a “phantasm” in thinking, a phantasm of our corporeal senses a visualisation or a tactualisation. But we can so modify that visualisation or tactualisation that it represents something not known by the senses. Do we by that representation wake up an intuition of the soul? Can we by the presentation of these hypothetical forms, that are the subject of our present discussion, wake ourselves up to higher intuitions? And can we explain the world around by a motion that we only know by our souls?

Apart from all speculation, however, it seems to me that the interest of these four-dimensional shapes and motions is sufficient reason for studying them, and that they are the way by which we can grow into a fuller apprehension of the world as a concrete whole.

SPACE NAMES.

If the words written in the squares drawn in fig. 1 are used as the names of the squares in the positions in which they are placed, it is evident that a combination of these names will denote a figure composed of the designated squares. It is found to be most convenient to take as the initial square that marked with an asterisk, so that the directions of progression are towards the observer and to his right. The directions of progression, however, are arbitrary, and can be chosen at will.

Thus _et_, _at_, _it_, _an_, _al_ will denote a figure in the form of a cross composed of five squares.

Here, by means of the double sequence, _e_, _a_, _i_ and _n_, _t_, _l_, it is possible to name a limited collection of space elements.

The system can obviously be extended by using letter sequences of more members.

But, without introducing such a complexity, the principles of a space language can be exhibited, and a nomenclature obtained adequate to all the considerations of the preceding pages.

1. _Extension._

Call the large squares in fig. 2 by the name written in them. It is evident that each can be divided as shown in fig. 1. Then the small square marked 1 will be “en” in “En,” or “Enen.” The square marked 2 will be “et” in “En” or “Enet,” while the square marked 4 will be “en” in “Et” or “Eten.” Thus the square 5 will be called “Ilil.”

This principle of extension can be applied in any number of dimensions.

2. _Application to Three-Dimensional Space._

To name a three-dimensional collocation of cubes take the upward direction first, secondly the direction towards the observer, thirdly the direction to his right hand.

These form a word in which the first letter gives the place of the cube upwards, the second letter its place towards the observer, the third letter its place to the right.

We have thus the following scheme, which represents the set of cubes of column 1, fig. 101, page 165.

We begin with the remote lowest cube at the left hand, where the asterisk is placed (this proves to be by far the most convenient origin to take for the normal system).

Thus “nen” is a “null” cube, “ten” a red cube on it, and “len” a “null” cube above “ten.”

By using a more extended sequence of consonants and vowels a larger set of cubes can be named.

To name a four-dimensional block of tesseracts it is simply necessary to prefix an “e,” an “a,” or an “i” to the cube names.

Thus the tesseract blocks schematically represented on page 165, fig. 101 are named as follows:—

2. DERIVATION OF POINT, LINE, FACE, ETC., NAMES.

The principle of derivation can be shown as follows: Taking the square of squares the number of squares in it can be enlarged and the whole kept the same size.

Compare fig. 79, p. 138, for instance, or the bottom layer of fig. 84.

Now use an initial “s” to denote the result of carrying this process on to a great extent, and we obtain the limit names, that is the point, line, area names for a square. “Sat” is the whole interior. The corners are “sen,” “sel,” “sin,” “sil,” while the lines are “san,” “sal,” “set,” “sit.”

I find that by the use of the initial “s” these names come to be practically entirely disconnected with the systematic names for the square from which they are derived. They are easy to learn, and when learned can be used readily with the axes running in any direction.

To derive the limit names for a four-dimensional rectangular figure, like the tesseract, is a simple extension of this process. These point, line, etc., names include those which apply to a cube, as will be evident on inspection of the first cube of the diagrams which follow.

All that is necessary is to place an “s” before each of the names given for a tesseract block. We then obtain apellatives which, like the colour names on page 174, fig. 103, apply to all the points, lines, faces, solids, and to the hyper-solid of the tesseract. These names have the advantage over the colour marks that each point, line, etc., has its own individual name.

In the diagrams I give the names corresponding to the positions shown in the coloured plate or described on p. 174. By comparing cubes 1, 2, 3 with the first row of cubes in the coloured plate, the systematic names of each of the points, lines, faces, etc., can be determined. The asterisk shows the origin from which the names run.

These point, line, face, etc., names should be used in connection with the corresponding colours. The names should call up coloured images of the parts named in their right connection.

It is found that a certain abbreviation adds vividness of distinction to these names. If the final “en” be dropped wherever it occurs the system is improved. Thus instead of “senen,” “seten,” “selen,” it is preferable to abbreviate to “sen,” “set,” “sel,” and also use “san,” “sin” for “sanen,” “sinen.”

We can now name any section. Take _e.g._ the line in the first cube from senin to senel, we should call the line running from senin to senel, senin senat senel, a line light yellow in colour with null points.

Here senat is the name for all of the line except its ends. Using “senat” in this way does not mean that the line is the whole of senat, but what there is of it is senat. It is a part of the senat region. Thus also the triangle, which has its three vertices in senin, senel, selen, is named thus:

Area: setat. Sides: setan, senat, setet. Vertices: senin, senel, sel.

The tetrahedron section of the tesseract can be thought of as a series of plane sections in the successive sections of the tesseract shown in fig. 114, p. 191. In b_{0} the section is the one written above. In b_{1} the section is made by a plane which cuts the three edges from sanen intermediate of their lengths and thus will be:

Area: satat. Sides: satan, sanat, satet. Vertices: sanan, sanet, sat.

The sections in b_{2}, b_{3} will be like the section in b_{1} but smaller.

Finally in b_{4} the section plane simply passes through the corner named sin.

Hence, putting these sections together in their right relation, from the face setat, surrounded by the lines and points mentioned above, there run:

3 faces: satan, sanat, satet 3 lines: sanan, sanet, sat

and these faces and lines run to the point sin. Thus the tetrahedron is completely named.

The octahedron section of the tesseract, which can be traced from fig. 72, p. 129 by extending the lines there drawn, is named:

Front triangle selin, selat, selel, setal, senil, setit, selin with area setat.

The sections between the front and rear triangle, of which one is shown in 1b, another in 2b, are thus named, points and lines, salan, salat, salet, satet, satel, satal, sanal, sanat, sanit, satit, satin, satan, salan.

The rear triangle found in 3b by producing lines is sil, sitet, sinel, sinat, sinin, sitan, sil.

The assemblage of sections constitute the solid body of the octahedron satat with triangular faces. The one from the line selat to the point sil, for instance, is named selin, selat, selel, salet, salat, salan, sil. The whole interior is salat.

Shapes can easily be cut out of cardboard which, when folded together, form not only the tetrahedron and the octahedron, but also samples of all the sections of the tesseract taken as it passes cornerwise through our space. To name and visualise with appropriate colours a series of these sections is an admirable exercise for obtaining familiarity with the subject.

EXTENSION AND CONNECTION WITH NUMBERS.

By extending the letter sequence it is of course possible to name a larger field. By using the limit names the corners of each square can be named.

Thus “en sen,” “an sen,” etc., will be the names of the points nearest the origin in “en” and in “an.”

A field of points of which each one is indefinitely small is given by the names written below.

The squares are shown in dotted lines, the names denote the points. These points are not mathematical points, but really minute areas.

Instead of starting with a set of squares and naming them, we can start with a set of points.

By an easily remembered convention we can give names to such a region of points.

Let the space names with a final “e” added denote the mathematical points at the corner of each square nearest the origin. We have then for the set of mathematical points indicated. This system is really completely independent of the area system and is connected with it merely for the purpose of facilitating the memory processes. The word “ene” is pronounced like “eny,” with just sufficient attention to the final vowel to distinguish it from the word “en.”

Now, connecting the numbers 0, 1, 2 with the sequence e, a, i, and also with the sequence n, t, l, we have a set of points named as with numbers in a co-ordinate system. Thus “ene” is (0, 0) “ate” is (1, 1) “ite” is (2, 1). To pass to the area system the rule is that the name of the square is formed from the name of its point nearest to the origin by dropping the final e.

By using a notation analogous to the decimal system a larger field of points can be named. It remains to assign a letter sequence to the numbers from positive 0 to positive 9, and from negative 0 to negative 9, to obtain a system which can be used to denote both the usual co-ordinate system of mapping and a system of named squares. The names denoting the points all end with e. Those that denote squares end with a consonant.

There are many considerations which must be attended to in extending the sequences to be used, such as uniqueness in the meaning of the words formed, ease of pronunciation, avoidance of awkward combinations.

I drop “s” altogether from the consonant series and short “u” from the vowel series. It is convenient to have unsignificant letters at disposal. A double consonant like “st” for instance can be referred to without giving it a local significance by calling it “ust.” I increase the number of vowels by considering a sound like “ra” to be a vowel, using, that is, the letter “r” as forming a compound vowel.

The series is as follows:—

CONSONANTS.

0 1 2 3 4 5 6 7 8 9 positive n t l p f sh k ch nt st negative z d th b v m g j nd sp

VOWELS.

0 1 2 3 4 5 6 7 8 9 positive e a i ee ae ai ar ra ri ree negative er o oo io oe iu or ro roo rio

_Pronunciation._—e as in men; a as in man; i as in in; ee as in between; ae as ay in may; ai as i in mine; ar as in art; er as ear in earth; o as in on; oo as oo in soon; io as in clarion; oe as oa in oat; iu pronounced like yew.

To name a point such as (23, 41) it is considered as (3, 1) on from (20, 40) and is called “ifeete.” It is the initial point of the square ifeet of the area system.

The preceding amplification of a space language has been introduced merely for the sake of completeness. As has already been said nine words and their combinations, applied to a few simple models suffice for the purposes of our present enquiry.

_Printed by Hazell, Watson & Viney, Ld., London and Aylesbury._