CHAPTER XIV.[6]A RECAPITULATION AND EXTENSION OF THE PHYSICAL ARGUMENT

[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.

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.

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 findout 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 movementon 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 wehave 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 tohim 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, ifxrotates to take the place ofy, andyturns so as to point to negativex, 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.

Fig. 1 (129).

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. LetAx,Aybe two axes andABCDa square. As far as movements in the plane are concerned, the square can rotate about a pointA, for example. It cannot rotate about a side, such asAC.

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. LetAzbe the axis, which stands perpendicular to his plane atA. He can draw in his plane two lines to represent the two axes,AxandAz. Let Fig. 2 be this drawing. Here thezaxis has taken the place of theyaxis, and the plane ofAxAzis represented in his plane. In this figure all that exists of the squareABCDwill be the lineAB.

Fig. 2 (130).

The square extends from this line in theydirection, but more of that direction is represented in Fig. 2. The plane being can study the turning of the lineABin this diagram. It is simply a case of plane turning around the pointA. The lineABoccupies intermediate portions likeAB1and after half a revolution will lie onAxproduced throughA.

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 atA´, one alongA´B´, the other perpendicular to his plane. He will obtain a figure precisely similar to Fig. 2, and will see that, asABcan turn aroundA, soA´C´aroundA.

In this turningABandA´B´would not interfere with each other, as they would if they moved in the plane around the separate pointsAandA´.

Hence the plane being would conclude that a rotation round a line was possible. He could see his square as itbegan to make this turning. He could see it half way round when it came to lie on the opposite side of the lineAC. 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 pointA, moving to the right, traces out the lineAC. The lineAC, moving away in a new direction, traces out the squareACEGat the base of the cube. The squareAEGC, moving in a new direction, will trace out the cubeACEGBDHF. 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.

Fig. 3 (131).

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 axesx,y,zdenoting its three dimensions. Letwrepresent 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 axesx,z,w. This is a three-dimensional space determined by two of the axes we have drawn,xandz, and in place ofythe fourth axis,w. We cannot, keepingxandz, have bothyandwin our space;so we will letygo and drawwin its place. What will be our view of the cube?

Fig. 4 (132).

Evidently we shall have simply the square that is in the plane ofxz, the squareACDB. The rest of the cube stretches in theydirection, and, as we have none of the space so determined, we have only the face of the cube. This is represented infig. 4.

Now, suppose the whole cube to be turned from thexto thewdirection. Conformably with our method, we will not take the whole of the cube into consideration at once, but will begin with the faceABCD.

Fig. 5 (133).

Let this face begin to turn. Fig. 5 represents one of the positions it will occupy; the lineABremains on thezaxis. The rest of the face extends between thexand thewdirection.

Now, since we can take any three axes, let us look at what lies in the space ofzyw, and examine the turning there. We must now let thezaxis disappear and let thewaxis run in the direction in which thezran.

Fig. 6 (134).

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 ofxyz. In the space ofxyzwe have merely the base of the cube lying in the plane ofxy, as shown infig. 6.

Now let thextowturning take place. The squareACEGwill turn about the lineAE. This edge will remain along theyaxis and will be stationary, however far the square turns.

Fig. 7 (135).

Thus, if the cube be turned by anxtowturning, both the edgeABand the edgeACremain stationary; hence the whole faceABEFin theyzplane remains fixed. The turning has taken place about the faceABEF.

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

Fig. 8 (136).

We can think of this turning as a turning of the faceABCDaboutAB, and a turning of each section parallel toABCDround the vertical line in which it intersects the faceABEF, 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 fromProf. 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 motionsui 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 infig. 3, which turns aboutthe face ofABFE, we see that any line in the face can take the place of the vertical and horizontal lines we have examined. Take the diagonal lineAFand the section through it toGH. The portions of matter which were on one side ofAFin this section infig. 3are on the opposite side of it infig. 8. They have gone round the lineAF. 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, supposeAandBare two parallel lines in thexyplane, and letCDandEFbe two rods crossing them. Now, in the space ofxyzif the rods turn round the linesAandBin the same direction they will make two independent circles.

Fig. 9 (137).

When the endFis going down the endCwill be coming up. They will meet and conflict.

But if we rotate the rods about the plane ofABby theztowrotation these movements will not conflict. Suppose all the figure removed with the exception of the planexz, and from this plane draw the axis ofw, so that we are looking at the space ofxzw.

Here,fig. 10, we cannot see the linesAandB. We see the pointsGandH, in whichAandBintercept thexaxis, but we cannot see the lines themselves, for they run in theydirection, and that is not in our drawing.

Now, if the rods move with theztowrotation they willturn in parallel planes, keeping their relative positions. The pointD, for instance, will describe a circle. At one time it will be above the lineA, at another time below it. Hence it rotates roundA.

Fig. 10 (138).

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.

Fig. 11 (139).

Letfig. 11represent a spherical surface, on each side of which a layer of matter exists. The thickness of the matter is represented by the rodsCDandEF, extending equally without and within.

Now, take the section of the sphere by theyzplane we have a circle—fig. 12. Now, let thewaxis be drawn in place of thexaxis so that we have the space ofyzwrepresented. In this space all that there will be seen of the sphere is the circle drawn.

Fig. 12 (140).

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 atEandCto be separated as they are atFandD, they can rotate round to the positionDandF, and a similar motion is possible for all other particles. There is no matter or obstacle to prevent them from moving out in thewdirection, and then on round the circumference as an axis. Now, what will hold for one section will hold forall, 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 infig. 12above 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 theAmerican 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 inthe 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 thexaxis changes into theyaxisand all parallel sections move in a similar manner, then the point will describe a circle. If, now, in addition to the rotation by which thexaxis changes into theyaxis the body has a rotation by which thezaxis turns into thewaxis, 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,xtoyandztow, 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 ofxtoy, 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 axesx,w, andz. Infig. 13we have the view of a four-dimensional sphere in the space ofxyz. Fig. 13 shows all that we can see of the four sphere in the space ofxyz, for it represents all the points in that space, which are at an equal distance from the centre.

Let us now take thexzsection, and let the axis ofwtake the place of theyaxis. Here, infig. 14, we have the space ofxzw. 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 thexzwspace, thexzcircle seen in the space ofxzw. But now, taking the view in the space ofxzw, we have a sphere in that space also. In a similar manner, whichever set of three axes we take, we obtain a sphere.

Showing axes xyzFig. 13 (141).

Showing axes xwzFig. 14 (142).

Infig. 13, let us imagine the rotation in the directionxyto be taking place. The pointxwill turn toy, andptop´. The axiszz´remains stationary, and this axis is all of the planezwwhich we can see in the space section exhibited in the figure.

Infig. 14, imagine the rotation fromztowto be taking place. Thewaxis now occupies the position previously occupied by theyaxis. This does not mean that thewaxis can coincide with theyaxis. 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 infig. 14we are looking atxzw.

The only part that is identical in the two diagrams is the circle of thexandzaxes, which axes are contained in both diagrams. Thus the planezxz´is the same in both, and the pointprepresents the same point in bothdiagrams. Now, infig. 14let thezwrotation take place, thezaxis will turn toward the pointwof thewaxis, and the pointpwill move in a circle about the pointx.

Thus infig. 13the pointpmoves in a circle parallel to thexyplane; infig. 14it moves in a circle parallel to thezwplane, indicated by the arrow.

Now, suppose both of these independent rotations compounded, the pointpwill 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 pointpwill 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 ofxtoycan be accompanied with the rotation ofztow. Call this the A kind.

But also the rotation ofxtoycan be accompanied by the rotation, of notztow, butwtoz. 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 anxtoy,ztowrotation would beytox,wtoz. The semi-negative isxtoyandwtoz.

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 ofzw, the other about the plane ofxy. 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 beingxtoyandztow, and the second beingxtoyandwtoz, when they are put together theztowandwtozrotations neutralise each other, and we obtain anxtoyrotation only, which is a rotation about the plane ofzw. Similarly, if we take a B rotation,ytoxandztow, we get, on combining this with the A rotation, a rotation ofztowabout thexyplane. In this case the plane of rotation is in the three-dimensional space ofxyz, and we have—what hasbeen 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.

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