Again M. Poincaré the distinguished French Physicist has said "The characteristic property of space, that of having three dimensions is only ... a property residing, so to speak, in human intelligence."
Mathematical physicists also find that certain experimental anomalies are resolved if they refer phenomena to four interchangeable axes involving homogeneous co-ordinates instead of to three space axes and one time axis. If this is not dealing in four-dimensional space it is first cousin to it.
M. Poincaré also pointed out that the postulates of Euclid are not experimentally verifiable facts and as a matter of fact much work has been done in the elaboration of non-Euclidean geometries. This is too mathematical a subject to be dealt with in detail here, but I can indicate the general drift of it, so far as it is relevant to the present discussion by means of the time honoured analogy of the two-dimensional world.
Most of my readers will know what are meant by the terms "latitude" and "longitude" and that the lines of longitude are "great circles" which pass through the poles and cut the earth's equator at right angles. It is also a matter of common knowledge that if on a plane surface two lines are drawn each of which cuts another line at right angles these two lines will be parallel—that is to say they will never meet however far they may be produced. This holds good provided that the surface in which they are drawn is truly plane—i.e., flat. But it breaks down, as we see in the case of the "great circles" of longitude, if the lines are drawn on a sphere. Now imagine two-dimensional beings, having no conception of the existence of a third dimension, living on the surface of a very large sphere. They might discover this principle about parallel lines and all would go well until they began making measurements over very large distances. Then their Geometry would begin to go wrong. They would find that lines drawn in their surface which ought not to meet however far produced would begin to show a tendency to do so. This would be an indication to them that there was such a thing as a third dimension of space and that their two-dimensional world was curved in this third dimension.
Now if a two-dimensional space can be curved in three dimensions there is no sort of reason why three-dimensional space should not be curved in four and in a precisely similar way three-dimensional geometry would, if such were the case, begin to "go wrong" where very large measurements were involved. Now, the largest measurements we ever make are astronomical measurements and as a matter of fact, according to Mr. Bragdon, there does seem to be a tendency for Geometry to go wrong in certain cases. He says that the number of negative parallaxes of stars is larger than would be expected having regard to the probable experimental errors. The parallax of an object is the angle which it subtends at two different points of observation, and so long as it is at a finite distance from these two points—which in the case of a star are the two opposite ends of the earth's orbit—this angle must be positive. That is to say the lines drawn in the observed direction of the star from the two points must converge.
If, as in certain cases seems to happen, they diverge, then one of three things must be the case; either the observations are wrong or else light does not, as is commonly believed, travel in straight lines (for after all what we call a straight line in astronomy is only the path of a ray of light) or else our geometry is breaking down and we must suppose that our space is curved, which would necessitate the acceptance of the existence of a fourth dimension.
It must be admitted that the explanation of negative parallaxes is more likely to be found in one or both of the two first alternatives than in the third.
Mr. Hinton has a good deal to say in his books about various four-dimensional theories of electricity involving four-dimensional vortices. These are highly ingenious but there does not seem to be any considerable reason for supposing them to be anything more and I shall therefore not describe them here. Two of his ideas however are so striking, although for different reasons, that I think a brief outline will not be out of place.
In his book "A new Era of Thought" he points out the remarkable analogy which exists between the properties of ether as postulated by physicists and those which a perfectly smooth solid sheet would present to the intelligence of two-dimensional beings living on it.