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Published online by Cambridge University Press: 24 October 2008
1. Considerable advantage has resulted from the postulation of unreal elements in projective geometry. In the first place these unreal elements were defined in terms of points represented by complex coordinates, and their use in purely geometrical reasoning had become well established before any serious attempt was made to justify this use, independently of algebraic considerations, by providing real representations of the unreal elements. The first successful attempt was that of von Staudt, who represented an unreal element by an elliptic involution associated with an order. In this system an ordered set of four real points is required to specify an unreal point. The system is comparatively simple to deal with in a single real plane, more complicated in a single real [3], and rapidly increases in complexity as the number of dimensions of the real field is increased.
* Staudt, Von, Beiträge zur Geometric der Lage (1856).Google Scholar
† Lüroth, , Math. Annalen, 11 (1877), 84.CrossRefGoogle Scholar
‡ Vahlen, , Abstrakte Geometrie, Leipzig (1903), 163–9.Google Scholar
§ Baker, H. F., Principles of Geometry, 1 (1922), 165–75.Google Scholar For a complete bibliography in this connection see p. 182.
* Clarendon symbols will be used to denote unreal entities.
* Two conjugate point-series represent unreal points whose complex coordinates are conjugate.
† Baker, loc. cit., 167.
‡ The general line-series represents von Staudt's ‘imaginary line of the second kind,’ which contains no real point. It is equivalent to the ‘skew linear series’ of the treatment given by Baker, loc. cit., 165–73.
* This represents von Staudt's ‘imaginary line of the first kind,’ which contains one real point and lies in one real plane. It is equivalent to the ‘line-series’ of Baker's treatment.
* This can be proved geometrically, without making use of cross-ratio.
* In the Euclidean field, if k be a circle and u the line at infinity, the triangles ABC of which u is the Hessian line, in regard to k, are equilateral.
