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Imaginaries in Geometry, and Their Interpretation in Terms of Real Elements
Published online by Cambridge University Press: 03 November 2016
Extract
The principal ideas are due to Von Staudt and Luroth. There is also a paper by Prof. Mathews, though it does not go far enough; and a recent treatise in English on the subject makes no reference at all to the question. It may be useful therefore to put down a brief account of the leading ideas of the two writers first mentioned, together with some simple developments.
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- Copyright © Mathematical Association 1920
References
page note 129 * Geometrie der Lage, Nuremberg, 1856, page 76.
page note 129 † Math. Annalen, vol. 8, 1875, page 145.
page note 129 ‡ Proc. London Math. Soc. vol. 11, 1912.
page note 129 § Hatton, The Imaginary in Geometry, Cambridge, 1920.
page note 130 * When the conic is taken as a circle, the centre of the triangle as here defined becomes what is usually known as the Symmedian point.
page note 131 * These two exceptional cases correspond to the vanishing of two invariants.
page note 132 * Not equal to-ω or-ω2.
page note 132 † This is a very necessary qualification, as will be apparent shortly.
page note 132 ‡ We are considering only imaginary planes situated in a real space of three dimensions. There are obvious generalisations. For instance, there exist imaginary planes which contain only a single real point, but such planes do not exist in a real space of three dimensions, but only in higher space.