CHAPTER XIX.

HYPER-SPACE*.
I propose to devote the remaining pages to the considera- tion, from the point of view of a mathematician, of certain properties of space, time, and matter, and to a sketch of some hypotheses as to their nature. Philosophers tell us that space, time, and matter are the "categories under which phj^sical phenomena are concerned." They cannot be defined, and such
* On the possibility of the existence of space of more than three dimensions see C. H. Hinton, Scientific Romances, London, 1886, a most interesting work, from which I have derived much assistance in compiling the earlier part of this chapter; his later work, The Fourth Dimension, London, 1904, may be also consulted. See also G. F. Kodwell, Nature, May 1, 1873, vol. viii, pp. 8, 9 ; and E. A. Abbott, Flatland, London, 1884.
On Non-Euclidean geometry, see chapter xiii above. The theory is due primarily to N. I. Lobatschewsky, Geomctrische Untersuchtingen zur Theorie der Parallellinien, Berhn, 1840 (originally given in a lecture in 1826) ; to C. F. Gauss {ex. gr. letters to Schumacher, May 17, 1831, July 12, 1831, and Nov. 28, 1846, printed in Gauss's collected works); and to J. Bolyai, Appendix to the first volume of his father's Tcntamen, Maros-Vasarkely, 1832 ; though the subject had been discussed by J. Saccheri as long ago as 1733 : its development was mainly the work of G. F. B. Kiemann, Ueher die Hypothesen welche der Geo- metric zu Grunde liegen, written in 1854, Gottinger Abhandlungen, 1866-7, vol. xin, pp. 131 — 152 (translated in Nature, May 1 and 8, 1873, vol. viii, pp. 14—17, 36—37) ; H. L. F. von Helmholtz, Gottinger Nachrichten, June 3, 1868, pp. 193 — 221 ; and E.Beltrami, Saggio di Interpretazione della Geometria non-Euclidean Naples, 1868, and the Annali di Matematica, series 2, vol. ii, pp. 232 — 255 : see an article by von Helmholtz in the Academy, Feb. 12, 1870, vol. i, pp. 128 — 131. In recent years the theory has been treated by several mathematicians.
On hyper-space, see V. Schlegel, Enseignement Mathematique, Paris, vol. ii, 1900; and D. M. Y. Somerville, Bibliography of Non-Euclidean Geometry y St Andrews, 1911.
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explanations of them as have been offered involve difficulties of the highest order and are far from simplifying our conceptions of them. I shall not discuss the metaphysical theories that profess to account for the origin of our conceptions of them, for these theories rest on assertions which are incapable of definite proof — a foundation which does not commend itself to a scientific student. The means of measuring space, time, and mass, and the investigation of their properties fall within the domain of mathematics.
I devote this chapter to considerations connected with space, leaving the subjects of time and mass to the following two chapters.
I confine my remarks to two speculations which recently have attracted considerable attention. These are (i) the possi- bility of the existence of space of more than three dimensions, and (ii) the possibility of kinds of geometry, especially of tw^o dimensions, other than those which are treated in the usual text-books : some aspects of the latter question have been already considered in chapter xiil. These problems are related. The term hyper-space was used originally of space of more than three dimensions, but now it is often employed to denote also any non-Euclidean space. I attach the wider meaning to it, and it is in that sense that this chapter is on the subject of hyper-space.
In regard to the first of these questions, the conception of a world of more than three dimensions is facilitated by the fact that there is no difficulty in imagining a world confined to only two dimensions — which we may take for simplicity to be a plane, though equally well it might be a spherical or other surface. We may picture the inhabitants of flatland as moving either on the surflice of a plane or between two parallel and adjacent planes. They could move in any direction along the plane, but they could not move perpendicularly to it, and would have no consciousness that such a motion was possible. We may suppose them to have no thickness, in which case they would be mere geometrical abstractions; or, preferably, we may
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think of them as having a small but uniform thickness, in which case they would be realities.
Several writers have amused themselves by expounding and illustrating the conditions of life in such a world. To take a very simple instance* a knot is impossible in flatland, a simple alteration which alone would make some difference in the experience of the inhabitants as compared with our own.
If an inhabitant of flatland was able to move in three dimensions, he would be credited with supernatural powers by those who were unable so to move; for he could appear or disappear at will, could (so far as they could tell) create matter or destroy it, and would be free from so many constraints to which the other inhabitants were subject that his actions would be inexplicable to them.
We may go one step lower, and conceive of a world of one dimension — like a long tube — in which the inhabitants could move only forwards and backwards. In such a universe there would be lines of varying lengths, but there could be no geometrical figures. To those who are familiar with space of higher dimensions, life in line-land would seem somewhat dull. It is commonly said that an inhabitant could know only two other individuals; namely, his neighbours, one on each side. If the tube in which he lived was itself of only one dimension, this is true ; but we can conceive an arrangement of tubes in two or three dimensions, where an occupant would be conscious of motion in only one dimension, and yet which would permit of more variety in the number of his acquaintances and con- ditions of existence.
Our conscious life is in three dimensions, and naturally the idea occurs whether there may not be a fourth dimension. No inhabitant of flatland could realize what life in three dimensions
* It is obvious that a knot cannot be tied in space of two dimensions. As long ago as 1876, F. C. Klein showed that knots cannot exist in space of four dimensions ; see Mothematische Annalen, Leipzig, 1876, vol. ix, p. 478. It is not easy to give a definition of a knot in hyper-space, but, taking it in its ordinary sense, it would seem that it is only in space of three dimensions that knots can be tied in strings : see D. M. Y. Somerville, Messenger of Mathe- matics, N.S., vol. XXXVI, 1907, pp. 139—144.
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would mean, though, if he evolved an analytical geometry applicable to the world in which he lived, he might be able to extend it so as to obtain results true of that world in three dimensions which would be to him unknown and inconceivable. Similarly we cannot realize what life in four dimensions is like, though we can use analytical geometry to obtain results true of that world, or even of worlds of higher dimensions. Moreover the analogy of our position to the inhabitants of Hatland enables us to form some idea of how inhabitants of space of four dimensions would regard us.
Just as the inhabitants of flatland might be conceived as being either mere geometrical abstractions, or real and of a uniform thickness in the third dimension, so, if there is a fourth dimension, we may be regarded either as having no thickness in that dim.ension, in which event we are mere (geometrical) abstractions — as indeed idealist philosophers have asserted to be the case — or as having a uniform thickness in that dimension, in which event we are living in four dimensions although we are not conscious of it. In the latter case it is reasonable to suppose that the thickness in the fourth dimension of bodies in our world is small and possibly constant; it has been conjectured also that it is comparable with the other dimensions of the molecules of matter, and if so it is possible that the constitution of matter and its fundamental properties may supply experi- mental data which will give a physical basis for proving or disproving the existence of this fourth dimension.
If we could look down on the inhabitants of flatland we could see their anatomy and what was happening inside them. Similarly an inhabitant of four-dimensional space could see inside us.
An inhabitant of flatland could get out of a room, such as a rectangle, only through some opening, but, if for a moment he could step into three dimensions, he could reappear on the other side of any boundaries placed to retain him. Similarly, if we came across persons who could move out of a closed prison-cell without going through any of the openings in it, there might be some reason for thinking that they did it by
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passing first in the direction of the fourth dimension and then back again into our space. This however is unknown.
Again, if a finite solid was passed slowly through flatland, the inhabitants would be conscious only of that part of it which was in their plane. Thus they would see the shape of the object gradually change and ultimately vanish. In the same way, if a body of four dimensions was passed through our space, we should be conscious of it only as a solid body, namely, the section of the body by our space, whose form and appearance gradually changed and perhaps ultimately vanished. It has been suggested that the birth, growth, life, and death of animals may be explained thus as the passage of finite four-dimensional bodies through our three-dimensional space. I believe that this idea is due to Hinton.
The same argument is applicable to all material bodies. The impenetrability and inertia of matter are necessary conse- quences ; the conservation of energy follows, provided that the velocity with which the bodies move in the fourth dimension is properly chosen: but the indestructibility of matter rests on the assumption that the body does not pass completely through our space. I omit the details connected with change of density as the size of the section by our space varies.
We cannot prove the existence of space of four dimensions, but it is interesting to enquire whether it is probable that such space actually exists. To discuss this, first let us consider how an inhabitant of flatland might find arguments to support the view that space of three dimensions existed, and then let us see whether analogous arguments apply to our world. I commence with considerations based on geometry and then proceed to those founded on physics.
Inhabitants of flatland would find that they could have two triangles of which the elements were equal, element to element, and yet which could not be superposed. We know that the explanation of this fact is that, in order to superpose them, one of the triangles would have to be turned over so that its under- surface came on to the upper side, but of course such a movement would be to them inconceivable. Possibly however they might
CH. XIX] HYPER-SPACE . 429
have suspected it by noticing that inhabitants of one-dimensional space might experience a similar difficulty in comparing the equality of two lines, ABC and CB'A\ each defined by a set of three points. We may suppose that the lines are equal and such that corresponding points in them could be superposed by rotation round C — a movement inconceivable to the inhabitants — but an inhabitant of such a world in moving along from A to A' would not arrive at the corresponding points in the two lines in the same relative order, and thus might hesitate to believe that they were equal. Hence inhabitants of flatland might infer by analogy that by turning one of the triangles over through three-dimensional space they could make them coincide.
We have a somewhat similar difficulty in our geometry. We can construct triangles in three dimensions — such as two spherical triangles — whose elements are equal respectively one to the other, but which cannot be superposed. Similarly we may have two helices whose elements are equal respectively, one having a right-handed twist and the other a left-handed twist, but it is impossible to make one fill exactly the same parts of space as the other does. Again, we may conceive of two solids, such as a right hand and a left hand, which are exactly similar and equal but of which one cannot be made to occupy exactly the same position in space as the other does. These are difficulties similar to those which would be experi- enced by the inhabitants of flatland in comparing triangles; and it may be conjectured that in the same way as such difficulties in the geometry of an inhabitant in space of one dimension are explicable by temporarily moving the figure into space of two dimensions by means of a rotation round a point, and as such difficulties in the geometry of flatland are explicable by temporarily moving the figure into space of three dimensions by means of a rotation round a line, so such difficulties in our geometry would disappear if we could temporarily move our figures into space of four dimensions by means of a rotation round a plane — a movement which of course is inconceivable to us.
Next we may enquire whether the hypothesis of our exist- ence in a space of four dimensions affords an explanation of
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any difficulties or apparent inconsistencies in our physical science*. The current conception of the luminiferous ether, the explanation of gi'avity, and the fact that there are only a finite number of kinds of matter, all the atoms of each kind being similar, present such difficulties and inconsistencies. To see whether the hypothesis of a four-dimensional space gives any aid to their elucidation, we shall do best to consider first the analogous problems in two dimensions.
We live on a solid body, which is nearly spherical, and which moves round the sun under an attraction directed to it. To realize a corresponding life in flatland we must suppose that the inhabitants live on the rim of a (planetary) disc which rotates round another (solar) disc under an attraction directed towards it. We may suppose that the planetary world thus formed rests on a smooth plane, or other surface of constant curvature ; but the pressure on this plane and even its existence would be unknown to the inhabitants, though they would be conscious of their attraction to the centre of the disc on which they lived. Of course they would be also aware of the bodies, solid, liquid, or gaseous, which were on its rim, or on such points of its interior as they could reach.
Every particle of matter in such a world would rest on this plane medium. Hence, if any particle was set vibrating, it would give up a part of its motion to the supporting plane. The vibrations thus caused in the plane would spread out in all directions, and the plane would communicate vibrations to any other particles resting on it. Thus any form of energy caused by vibrations, such as light, radiant heat, electricity, and possibly attraction, could be transmitted from one point to another without the presence of any intervening medium which the inhabitants could detect.
If the particles were supported on a uniform elastic plane film, the intensity of the disturbance at any other point would vary inversely as the distance of the point from the source of
* See a note by myself in the Messenger of Mathematics, Cambridge, 1891, vol. XXI, pp. 20 — 24, from which the above argument is extracted. The question has been treated by Hinton on similar lines.
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disturbance ; if on a uniform elastic solid medium, it would vary inversely as the square of that distance. But, if the supporting medium was vibrating, then, wherever a particle rested on it, some of the energy in the plane would be given up to that particle, and thus the vibrations of the intervening medium would be hindered when it was associated with matter.
If the inhabitants of this two-dimensional world were sufficiently intelligent to reason about the manner in which energy was transmitted they would be landed in a difficulty. Possibly they might be unable to explain gravitation between two particles — and therefore between the solar disc and their disc — except by supposing vibrations in a rigid medium between the two particles or discs. Again, they might be able to detect that radiant light and heat, such as the solar light and heat, were transmitted by vibrations transverse to the direction from which they came, though they could realize only such vibrations as were in their plane, and they might determine experimentally that in order to transmit such vibrations a medium of great rigidity (which we may call ether) was necessary. Yet in both the above cases they would have also distinct evidence that there was no medium capable of resisting motion in the space around them, or between their disc and the solar disc. The explanation of these conflicting results lies in the fact that their universe was supported by a plane, of which they were necessarily unconscious, and that this rigid elastic plane was the ether which transmitted the vibrations.
Now suppose that the bodies in our universe have a uniform thickness in the fourth dimension, and that in that direction our universe rests on a homogeneous elastic body whose thick- ness in that direction is small and constant. The transmission of force and radiant energy, without the intervention of an intervening medium, may be explained by the vibrations of the supporting space, even though the vibrations are not them- selves in the fourth dimension. Also we should find, as in fact we do, that the vibrations of the luminiferous ether are hindered when it is associated with matter. I have assumed
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that the thickness of the supporting space is small and uniform, because then the intensity of the energy transmitted from a source to any point would vary inversely as the square of the distance, as is the case ; whereas if the supporting space was a body of four dimensions, the law would be that of the inverse cube of the distance.
The application of this hypothesis to the third difficulty mentioned above — namely, to show why there are in our universe only a finite number of kinds of atoms, all the atoms of each kind having in common a number of sharply defined properties — will be given later*.
Thus the assumption of the existence of a four-dimensional homogeneous elastic body on which our three-dimensional universe rests, affords an explanation of some difficulties in our physical science.
It may be thought that it is hopeless to try to realize a figure in four dimensions. Nevertheless attempts have been made to see what the sections of such a figure would look like.
If the boundary of a solid is (p {x, y, z) — 0, we can obtain some idea of its form by taking a series of plane sections by planes parallel to 2: = 0, and mentally superposing them. In four dimensions the boundary of a body would be and attempts have been made to realize the form of such a body by making models of a series of solids in three dimensions formed by sections parallel to t(; = 0. Again, we can represent a solid in perspective by taking sections by three co-ordinate planes. In the case of a four-dimensional body the section by each of the four co-ordinate solids will be a solid, and attempts have been made by drawing these to get an idea of the form of the body. Of course a four-dimensional body will be .bounded by solids.
The possible forms of regular bodies in four dimensions, analogous to polyhedrons in space of three dimensions, have been discussed by Stringhamf.
* See below, p. 475.
t American Journal oj Mathematics, 1880, vol. in, pp. 1 — 14.
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I now turn to the second of the two problems mentioned at the beginning of the chapter: namely, the possibility of there being kinds of geometry other than those which are treated in the usual elementary text-books. This subject is so technical that in a book of this nature I can do little more than give a sketch of the argument on which the idea is based.
The Euclidean system of geometry, with which alone most people are acquainted, rests on a number of independent axioms and postulates. Those which are necessary for Euclid's geometry have, within recent years, been investigated and scheduled. They include not only those explicitly given by him, but some others which he unconsciously used. If these are varied, or other axioms are assumed, we get a different series of propo- sitions, and any consistent body of such propositions constitutes a system of geometry. Hence there is no limit to the number of possible non-Euclidean geometries that can be constructed.
Among Euclid's axioms and postulates is one on parallel lines, which is usually stated in the form that if a straight line meets two straight lines, so as to make the sum of the two interior angles on the same side of it less than two right angles, then these straight lines being continually produced will at length meet upon that side on which are the angles whose sum is less than two right angles. Expressed in this form the axiom is far from obvious, and from early times numerous attempts have been made to prove it. All such attempts failed, and it is now known that the axiom cannot be deduced from the other axioms assumed by Euclid. I have already discussed this question in chapter xiii, and I do not propose to add here anything more. The conclusion was that three consistent systems of geometry could be constructed, termed respectively hyperbolic, parabolic or Euclidean, and elliptic. These are dis- tinguished from one another according as no straight line (that is, a geodetic line), or only one straight line, or a pencil of straight lines can be drawn through a point parallel to a given straight line.
To work out a body of propositions relating to figures on a surface (that is, a two-dimensional space) analogous to that given
B. R. 28
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by Euclid relating to figures drawn on a plane, it is necessary that it should be possible at any point on the surface to con- struct a figure congruent to a given figure ; this is equivalent to saying that if we take up a triangle drawn anywhere on the surface and move it to another part of the surface, it will lie flat on the surface there. This is so only if the measure of curvature at every point of the surface is constant. Such surfaces of constant curvature are spherical surfaces, where the product is positive; plane surfaces, where it is zero; pseudo- spherical surfaces, where it is negative. A tractroid, that is, a figure produced by the revolution of a tractrix about its asymp- tote, is an example of a pseudo-spherical surface ; it is saddle- shaped at every point. Hence on spheres, planes, and tractroids we can construct these systems of geometry. And these systems are respectively examples of hyperbolic, Euclidean, and elliptic geometries.
Moreover if any surface is bent without dilation or contrac- tion, the measure of curvature remains unaltered. Thus from these three surfaces we can form others on which congruent figures, and therefore consistent systems of geometry, can be constructed. For instance, a plane can be rolled into a cone or cylinder, and the system of geometry on a conical or cylindrical surface will be similar to that on a plane. Similarly a hemi- sphere can be rolled up into a sort of spindle, and the system of geometry on such a spindle will be similar to that on a sphere. In fact there are three kinds of surfaces of constant positive curvature, which are respectively spherical, spindle-shaped, and bolster-shaped, and on each of these a system of hyperbolic geometry can be constructed. So too there are three kinds of surfaces of constant negative curvature.
Throughout this discussion I have tacitly assumed that the measure of distance employed remains the same wherever it is employed. If this is not so, we may evolve in plane space non-Euclidean geometries which are not inconsistent with ex- perience. Suppose, to take one example, that a foot-rule shrunk as it w^as moved away from some point of the plane — as it might do by a fall of temperature. Then a distance, which we
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should describe as finite, micrht when measured by this rule appear to be infinite, since repeated applications of the ever- shortening rule would not cover it. Thus the boundary of what we should describe as a finite area round the point would to those who were confined to the area of this foot-rule appear to be infinitely distant from the point. If the law of shrinking be properly chosen, the geometry of figures in this area would be hyperbolic. The length of the foot-rule might also alter in such a way as to lead to an elliptic geometiy*.
Thus the conception of hyperbolic, Euclidean, and elliptic geometries can be reached from the theory of the measure of distances as well as from the theory of parallels and of con- gruent figures. This view has led to further discussion of the essential characteristics of space by F. C. Klein, S. Lie,