From the 'Introductory Considerations' of The Plane Geometry of the Point in Point-Space of Four Dimensions, by C. J. Keyser. 1. As is well known, the dimensionality (in Riemann's sense) of any given space depends upon the element chosen for its construction; and in accordance with the Plucker principle of counting constants, any given space may be made to assume any prescribed dimensionality k by merely taking for element a configuration for whose determination within that space k independent data are necessary and sufficient - a configuration, in other words, whose general analytical representation in the given space involves exactly k parameters. A space being assumed, there are, in general, infinitely many possible choices of element for which the space will have a previously assigned dimensionality. Of such possible choices the great majority would be inexpedient as not leading to interesting results. Of all elements, in case of any given space, those are, in general, most practicable which present themselves in pairs of reciprocals, as in the familiar examples of the point and line in the plane, the line and plane in the sheaf, and the point and plane in ordinary space.
A space that is n-dimensional in points is also n-dimensional in point-spaces of n - 1 dimensions. It has 2(n - l) dimensions both in lines and in point- spaces of n - 2 dimensions; and, in general, its dimensionality is p (n - p + l) if the point-space either of p - 1 or of n - p dimensions be taken as element. Not only, however, do the two last mentioned elements furnish the same dimensionality, which is a necessary though not a sufficient condition for reciprocity, but they are indeed reciprocal elements in n-fold point-space; for the same system of equations, which on proper interpretation defines one of the elements, admits a second (dual) interpretation defining the other. It thus appears that by taking as elements the various point-spaces of less than n dimensions for the construction of n-fold point-space, there arise n geometries of this space; or, if we regard two reciprocal theories as but two phases of one geometry, the elements in question yield n / 2 or n - 1 / 2 + 1 geometries according as n is even or odd, the element having n - 1 / 2 dimensions being, in case of n odd, its own reciprocal, or self-reciprocal.
Like considerations hold for spaces of n dimensions in other elements than points. It will be convenient, however, and sufficient to conduct this discussion for space supposed n-fold in points.show more