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Updated: May 25, 2025
Now since axiomatic geometry by itself contains no assertions as to the reality which can be experienced, but can do so only in combination with physical laws, it should be possible and reasonable whatever may be the nature of reality to retain Euclidean geometry.
'Metageometries' have been invented by Riemann and Lobatschewski as rivals to the assumptions of Euclid, and the brilliant writings of Poincaré have explained the human devices on which mathematical concepts rest. Euclidean geometry is reduced to a useful interpretation of the data of experience; it is not theoretically the only one.
The question whether the structure of this continuum is Euclidean, or in accordance with Riemann's general scheme, or otherwise, is, according to the view which is here being advocated, properly speaking a physical question which must be answered by experience, and not a question of a mere convention to be selected on practical grounds.
We must carefully bear in mind that our statement as to the growth of the disc-shadows, as they move away from S towards infinity, has in itself no objective meaning, as long as we are unable to employ Euclidean rigid bodies which can be moved about on the plane E for the purpose of comparing the size of the disc-shadows.
The construction is never finished; we can always go on laying squares if their laws of disposition correspond to those of plane figures of Euclidean geometry. The plane is therefore infinite in relation to the cardboard squares. Accordingly we say that the plane is an infinite continuum of two dimensions, and space an infinite continuum of three dimensions.
Observe the distinct description of how the relation between circumference and centre is inverted by the former becoming itself an 'indivisible centre'. In a space of this kind there is no Here and There, as in Euclidean space, for the consciousness is always and immediately at one with the whole space. Motion is thus quite different from what it is in Euclidean space.
Nor do we know whether it is only in the proximity of ponderable masses that its structure differs essentially from that of the Lorentzian ether; whether the geometry of spaces of cosmic extent is approximately Euclidean.
In fact the only objective assertion that can be made about the disc-shadows is just this, that they are related in exactly the same way as are the rigid discs on the spherical surface in the sense of Euclidean geometry.
Further, the spherical surface is a non-Euclidean continuum of two dimensions, that is to say, the laws of disposition for the rigid figures lying in it do not agree with those of the Euclidean plane. This can be shown in the following way. Place a paper disc on the spherical surface, and around it in a circle place six more discs, each of which is to be surrounded in turn by six discs, and so on.
Smith asserts that these definitions are false, and sustains his position by numerous demonstrations in the pure Euclidean style. He declares that every mathematical line has a definite breadth, which is as measurable as its length, and that every mathematical surface has a thickness, as measurable as the contents of any solid.
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