CHAPTER IX

SUMMARY AND CONCLUSION

I will bring this work to a close by a brief recapitulation of its more salient points.

A dimension is defined as "an independent direction in space." A flat surface is two-dimensional and the space we know is three-dimensional. The direction of the fourth dimension must be at right angles to every direction which can be drawn in our space and four-dimensional space is such that through any point in it, four, and only four, lines can be drawn mutually at right angles.

From every point in our space a line can be drawn running off in the direction of four space.

Consequently every point in our space is absolutely accessible from the direction of the fourth dimension.

The best way of drawing conclusions as to the properties of four space is by means of the analogy of the two-dimensional world; since four space is to three space as the latter is to two space.

The fact that we cannot perceive four space, or picture its nature to ourselves, is no proof that it is non-existent.

I suggest as a working hypothesis that four space is a reality and that Man possesses at least one other vehicle of Consciousness--a four-dimensional one--besides his physical body. In this vehicle he is embodied after discarding the physical vehicle at death and also during temporary absences from the body during life.

This hypothesis is likely to prove of importance in two respects. First, it provides Psychic Research with a working hypothesis which may be essential to its development as a science. Secondly the adoption of some such hypothesis should go far to remove the principle cause of recent cleavage between Religious and Scientific thought.

The hypothesis is capable of throwing light on a number of "Psychic" phenomena which are otherwise very obscure. It affords us a means of conceiving a mode of existence which is real and yet imperceptible to our senses, thus surmounting one of the chief difficulties in the way of conceiving of post-mortem existence.

In the realm of Clairvoyance it enables us to form some idea of the nature of the faculty of internal vision. With regard to Clairvoyance in space, it also helps us to some slight extent, although this phenomenon presents special difficulties of its own.

Other varieties of "out of the body" experiences are much elucidated by its aid.

The phenomena with which it is most closely connected, however, are those known under the general title of "apparent penetration of matter by matter."

To these it affords by far the simplest and probably the only explanation and, if they are regarded as irrefutably established, it will be difficult to avoid the conclusion that four space is a reality.

The _locus classicus_ of such phenomena is the Slade-Zöllner investigation, but this is worthless as evidence. The literature of the subject abounds with records of similar occurrences.

The hypothesis also seems to offer a possible means of explaining the paradoxical rigidity of the impalpable structure discovered by Crawford.

The hypothesis may also have a certain significance, even in the realm of pure Philosophy. It enables us to conceive of the simultaneous existence of a series of three space simultaneities and, consequently, is of interest in the consideration of Time and of the possibility of Prevision.

It also works in well with a certain view of the nature of Vitality.

As regards its relation to ordinary physical science, we find nothing to conflict with it, but, on the contrary that there are a certain number of indications that four space is, as I suggest, more than a mere mathematical concept. It is possible that it may some day come to be recognised as having some significance in the theory of the nature of electrons and of ether, while recent views on "Relativity" strongly indicate that Physicists will soon regard the four-dimensionality of the Universe as a common place.

* * * * *

If the four-dimensional hypothesis is correct there should exist some sort of connecting link between the physical body and the four-dimensional vehicle.

The function of this link would be to deflect sensory impressions out of three space into four space thus enabling them to reach the Consciousness resident in the latter. Such a link must therefore be, in some way, intermediate between ordinary matter and four-dimensional matter.

That is to say, it must possess some degree of four-dimensional complexity. This may reasonably be supposed to endow it with peculiar properties.

If such a connecting link be found to exist in practice, it would tend to confirm the hypothesis.

The experiments of de Rochas, of Joire, of MacDougal, of Baraduc, of Kilner, and of Crawford seem to indicate that such a connecting link does, in fact, exist.

This is confirmed by the testimony of clairvoyants, which, though not of a nature to be rated too highly or accepted lightly, should be allowed some weight.

The attempt to formulate an hypothesis of this nature is not repugnant to Religion. Nor is there anything in this particular hypothesis which can be held to conflict with Religious doctrines.

On the other hand, the acceptance of such an hypothesis would cut the ground from under the feet of those who seek to maintain that matter is the only reality and that therefore Spirit and the Spiritual life are mere illusions.

* * * * *

No writer can expect to bring all his readers to his way of thinking. Indeed it would be unfortunate if he were to do so, as the effect would be to eliminate that element of critical discussion which is so fruitful a source of progress.

Consequently, I do not anticipate that every reader will agree with me. All I venture to hope is that I may have made good my contention that the four-dimensional concepts, in spite of the scorn poured on them as a result of the Zöllner fiasco, are worthy of very careful consideration as a tentative working hypothesis by those who are seeking to clear up the many obscure problems presented by Psychical Research.

If this little book is thought worthy of criticism, I shall welcome it. Its purpose will have been amply served if it succeeds in arousing interest in what will prove, I believe, a very fruitful field of speculation and research.

APPENDIX.

To illustrate how the analogy of the relation between two and three-dimensional space enables us to determine some of the properties of four-dimensional figures:

(1)

"Any figure in a space of a given dimensionality generates a corresponding figure in the next higher space, by moving in a direction at right angles to any direction that can be drawn within itself.[7] Or, in general, space of any dimensionality generates, by such a movement, the next higher space."

Thus, the lowest sort of space is space of zero dimensions, _i.e._, a mathematical point. If it moves a distance of one inch, it traces out a Line one inch long--that is to say a one space "figure." If this moves at right angles to itself for a distance of one inch, it traces out a two space figure, viz., a square of side one inch. If this again moves a distance of one inch in a direction at right angles to every direction that can be drawn within it, that is, in a direction perpendicular to itself, it traces out a cube of side one inch, _i.e._, a three space figure or "solid."

We must, therefore, conclude, from analogy, that if the cube were itself to move, a distance of one inch, in a direction at right angles to every direction that can be drawn in our space--in the unknown direction, that is, of the fourth dimension--it would generate a "higher solid" of side one inch. The higher solid thus generated is called a "Tesseract" and its properties are quite well known.

(2)

"Every figure, in a space of a given dimensionality, contains an infinite number of the 'corresponding' figures--see (1)--in the next lower space."

Since a point is defined as having "position but no magnitude," it follows that it would require an infinite number of points to make up a line.

Similarly a line has length, but no breadth or thickness, and it would therefore require an infinite number of lines laid side by side to make up a surface.

Again a surface has, theoretically, no thickness, and it would therefore require an infinite number of surfaces superimposed on one another to make up a solid.

We must therefore conclude, by analogy, that it would require an infinite number of solids to make up a "higher solid."

In particular, a Tesseract must be supposed to contain an infinite number of cubes, and, in general, four space must be conceived of as containing an infinite number of three spaces.

(3)

"The Boundaries of a figure in a space of any dimensionality are themselves figures in the next lower space."

Thus a Line (one space) is bounded by Points (zero space).

A surface (two space) is bounded by Lines (one space).

A solid (three space) is bounded by Surfaces (two space).

We must conclude therefore that "higher solids" (four space) are bounded by Solids (three space).

To take the special case with which we are already familiar. The line AB, is bounded by the points A and B. (Fig. 10). The square, A B C D, is bounded by four lines AB, BC, CD, DA. The cube, A B C D E F G H, is bounded by six surfaces, namely, ABCD, CDEF, EFGH, GHAB, ADEH, BCFG.

Similarly we must conclude that a tesseract is bounded by cubes.

We shall see later that there are eight of them.

(4)

We may put (3) in a slightly different way, by saying that:

"Two adjacent portions of space, of any dimensionality, are separated by a space of the next lower dimensionality."

The portions AB and BC of the line AC are separated by the point B. (Fig. 11.) The portions ABEF and BCDE of the fig. ACDF are separated by the line EB. The portions ABEFGHIM and BCDEMIKL of the solid ACDFGHKL are separated by the surface BIME.

Similarly we must suppose that any two adjacent portions of four space are separated by a three space figure.

Or, again, to alter it slightly, "any space is no more than a boundary between two adjacent portions of the next higher space." Whence it follows that the whole of our three space is but the boundary between two adjacent portions of four space.

(5)

"A tesseract, which is the four-dimensional analogue of the cube, is bounded by Eight cubes. It has Twenty-four plane square faces, Thirty-two linear edges, and Sixteen corner points."

This may at first sight seem difficult to grasp.

In reality however, it is quite simple.

We have only to remember that the tesseract is generated by the movement of a cube, in a direction at right angles to every direction that can be drawn in the cube, and that whenever a figure of a given dimensionality moves thus it generates a figure of the next higher dimensionality.

Thus every point in the cube will trace out a line, every line a surface, and every surface a solid, and, since the distance moved is equal to the length of the side of the cube, these surfaces will be squares and the solids will be cubes.

But let us first consider the analogous case of the generation of the cube by the movement of a square.

Let A B C D represent the original position of the square. It moves, a distance equal to one of its sides, in a direction at right angles to every direction that can be drawn within itself--at right angles, _i.e._, to every one of its sides--and finally comes to rest in the position E F G H.

Every side has traced out another square and we have, in addition, the old square ABCD, with which we started and the new square EFGH, with which we end.

Thus even if we had no idea how many sides, edges, and corners a cube had we could deduce them.

We should say:--

Every side of the original square has traced out a new square--that makes 4--and we also have the original square and the "final" square making a total of 6. A cube, therefore, must be bounded by 6 square surfaces.

Similarly we should reflect that the original square and the final square have each 4 linear edges, making 8, and that each of the 4 corner points of the original square would trace out a line, making new lines, and we would therefore conclude that a cube must have 8 + 4 = 12 edges.

Finally, since in a uniform motion no new points will be generated, we should expect the cube to have a total of 8 corner points, _i.e._, the four corners of the original square and the four corners of the final square.

Now let us apply the same methods to the generation of the tesseract by the movement of a cube.

Observe that just as in the case of the square generating the cube we had the original square to start with and what I called the "final" square to end up with, so, in this case, we shall start and end up with a cube.

In the process of the movement every face of the cube will generate a new cube--that means 6 new cubes, since the cube must have had 6 faces--and there will also be the original cube and the final cube, making a total of 8 cubes all told. A tesseract must therefore be bounded by 8 cubes.

Similarly each line of the original cube will trace out a square. This, since a cube has 12 edges, gives us 12 new squares plus 6 from the original and 6 from the final cube, or a total of 24. A tesseract therefore has 24 plane square faces. Again each point of the original cube will trace out a line, making 8 new lines, and there will also be 12 lines in the original and 12 in the final cube, making a total of 32.

Finally, there will be 8 points in the original cube and 8 in the final cube, but none will have been produced on the way. So a tesseract will therefore have 16 corner points.

There is no reason why this process should not be continued indefinitely. For a tesseract may be supposed to move, in distance equal to the length of one of its edges, in a direction not contained in itself and will generate a _five_ dimensional figure, bounded by ten tesseracts, and having in it 40 cubes, 80 squares, 80 lines, and 32 corner points. Thus a whole series of Higher Space figures may be produced. But these are of little practical interest, and I shall not deal with them here.

FOOTNOTES:

[Footnote 7: NOTE.--The figures thus produced are not necessarily the strict analogues of the figures which generate them. For instance a circle, moving in a direction perpendicular to itself, would generate a cylinder; whereas the three-dimensional analogue of a circle is a sphere.]

E. AUSTIN AND SON, LTD.,

PRINTERS,

-- CLIFTON, BRISTOL. --

INDEX

Anæsthesia, 146, 147

Apologetics, Christian, 177

Apport. Two-dimensional analogue of, 15, 16 Discussion of evidence for, 62, 73 sqq.

Astral plane, 35, 53

Aura, 142, 143, 161

Cantilever, Crawford's, 86-91, 145

Carrington, Hereward, 71, 74, 77, 113-141

Change--in a two dimensional world, 17, 20

Clairvoyance, 42, 48

Crawford, 86, 145, 147

Death. Loss of weight at, 143

Dimension. Definition of, 3

Direct Voice, 62

Disembodiment, cases of, 58, 150-154

Dreams, 54, 55

Electricity. Hinton's theories of, 127

Etheric double, 35, 62, 147-148

Energy, conservation of, 117, 120

Ether. Hinton's analogy, 127

Exteriorisation of Sensibility, 141

Faith and Reason, 169

Fatalism, 107-109

Flatland, 7

Geometry. Possible break down of, 124, 126

Hair-trigger theory, 116, 141

Hallucination, 50, 51

Hypothesis. Need of, 24-38 Valid, 29 True, 29

Hyslop, Dr., 77

Internal Vision, 46-49

Kilner, Dr., 142-143

Levitation, 86, 91, 145

Light. Theories, of 29, 30

Materialists, 32, 176, 177

Milan Committee, 83

Occultists 32, 34

One-dimensional space, 7

Palladino, Eusapia, 74, 83

Parallaxes, Negative, 126

Peters, Dr., 83

Phantasms, 52, 55, 57

Pogorelsky, 84

Poincaré, 124

Postvision, 42, 52

Prevision, 39, 42, 52, 103-107

Psycho-analysis, 55

Reason and Faith, 169

Reichenbach, 140

Relativity, 133-135

Religion, 32, 168-180

Richet, 31

Rotation in four space, 129

Sambor, 84

Secondary personality, 41

Sensibility. Exteriorisation of, 141

Slade, 64, 73 sqq.

Space. Objectivity of, 109-112

Spectrum, 43

Symmetry, 131-133

Telekinesis, 39, 86

Telepathy, 39, 41

Television, 46, 48, 111

Tesseract, 188, 189, 191

Theologians, 32

Theosophists, 35, 159, 160

Time, 92-103 Measurement of, 94 Bergson's views on, 96, 98 Subjective, 99

Two-dimensional world, analogy of, 7 sqq.

Ultra-violet light, 43, 144, 160

Vitality, 113-141

Vortices. Four-dimensional, 127

Will, 113-141

Zero-dimensional space, 7

Zöllner, 1, 62, 73 sqq.