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Note on the extension to higher space of a theorem of Wallace

  • J. P. Gabbatt (a1)
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The theorem is attributed to Wallace that, in a euclidean plane, the circumcircles of the triangles determined by four lines, of general position, meet at a point. It is further known that, in euclidean space of n dimensions, the circumhyperspheres of the simplices determined by n + 2 flats, of general position, meet at a point, if and only if n be even.

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* Scoticus’, Leybourn's Math. Repos., N.S., 1 (1806), 170;see Mackay, , Proc. Edin. Math. Soc. 9 (1891), 87.

Grace, , Trans. Camb. Phil. Soc., 16 (1898), 153190 (163);Kühne, , Crelle, 119 (1898), 186195 (corrected by Baker, H. F., Proc. Camb. Phil. Soc., 22 (1924), 2833);Haskell, , Arch. d. Math. u. Phys. (3), 5 (1903), 278281.

For two dimensions, Miquel, , Liouville, 3 (1838), 485487; for three dimensions, Roberts, S., Proc. Lond. Math. Soc., 12 (1881), 102, 117–120, ibid. 25 (1894), 306–314; for four dimensions, Grace, , loc. cit., (168); for n dimensions, Haskell, , oc. cit., Meyer, W. F., Arch. d. Math. u. Phys. (3), 5 (1903), 282287.

§ See e.g. Scott, and Mathewes, , Theory of Determinants, ed. 2, Cambridge (1904), 9296.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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