D. Hilbert, M. L. Gerard, A. N. Whitehead, and others.

The above remarks refer only to space of two dimensions. Naturally there arises the question whether there are different kinds of non- Euclidean space of three or more dimensions. Kiemann showed that there are three kinds of non-Euclidean space of three dimensions having properties analogous to the three kinds of non-Euclidean space of two dimensions already discussed. These are differentiated by the test whether at every point no geodetical surface, or one geodetical surface, or a fasciculus of geodetical surfaces can be drawn parallel to a given surface : a geodetical surface being defined as such that every geodetic line joining two points on it lies wholly on the surface. It may be added that each of the three systems of geometry of two dimensions described above may be deduced as properties of a surface in each of these three kinds of non-Euclidean space of three dimensions.
It is evident that the properties of non-Euclidean space of three dimensions are deducible only by the aid of mathematics, and cannot be illustrated materially, for in order to realize or construct surfaces in non-Euclidean space of two dimensions we think of or use models in space of three dimensions ; similarly
* See A. Cayley, Collected Mailieinatical Fa^era, Cambridge, 1396, vol. xi, p. -iiio et seq.
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the only way in which we could construct models illustrating non-Euclidean space of three dimensions would be by utilizing space of four dimensions.
We may proceed yet further and conceive of non-Euclidean geometries of more than three dimensions, but this remains, as yet, an unworked field.
Returning to the former question of non-Euclidean geome- tries of two dimensions, I wish again to emphasize the fact that, if the axioms enunciated in the usual books on elementary geometry are replaced by others, it is possible to construct other consistent systems of geometry. For instance, just as one kind of non-Euclidean geometry has been constructed by assuming that Euclid's parallel postulate is not true, so D. Hilbert and M. Dehn of Gottingen have elaborated another kind, known as non- Archimedian geometry. Archimedes had assumed as axiomatic that if A and B are magnitudes of the same kind and order, it is possible to find a multiple of A which is gTeater than B, which implied that the geometrical magnitudes considered are continuous. If this be denied, Hilbert and Dehn showed* that it is still possible to construct consistent systems of geometry closely analogous to that given by Euclid. Assuming that in these a pencil of straight lines can be drawn through a point parallel to a given straight line, then in one form, known as the non-Legendrian system, the angle-sum of a triangle is greater than two right angles, while in another form, termed the semi- Euclidean system, the sum-angle is equal to two right angles. I do not however concern myself here further with these systems, for the methods and results appeal only to the professional mathematician. On the other hand the elliptic, parabolic, and hyperbolic systems described in chapter X have a special interest, from the somewhat sensational fact that they lead to no results necessarily inconsistent with the properties, as far as we can observe them, of the space in which we live ; we are not at present acquainted with any other systems which are con- sistent with our experience. We may, however, fairly say that of these systems the Euclidean is the simplest. • M. Dehn, Matheinatischen Annalen, Leipzig, 1900, vol. Lvn, pp. 404—439.
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If we go a step further and ask what is meant by saying that a geometry is true or false we land ourselves in an inter- minable academic dispute. Some philosophers hold that certain axioms are necessarily true independent of all experience, or at any rate are necessarily true as far as our experience extends. Others agree with Poincare, that the selection of a geometry is really a matter of convenience, and that that geometry is the best which enables us to state the known physical laws in the simplest form ; or, more generally, that it is desirable to choose axioms and to define quantities so as to permit the expression in as simple a way as possible of all observed laws and facts in nature. But for practical purposes the conclusion is immaterial, and at any rate the discussion belongs to metaphysics rather than mathematics.
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