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Pascal's theorem in n–space*

Published online by Cambridge University Press:  09 April 2009

Sahib Ram Mandan
Sahib Ram Mandan
Affiliation:
Indian Institute of Technology, Kharagpur.
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An analogue in a solid of the well known Pascal's theorem (Baker, [1], p. 219) for a conic is established by Baker ([2], pp. 53–54, Ex. 15) after Chasles [6] and by Salmon ([2], p. 142). The same is discussed in detail by Court [8]. The purpose of this paper is to extend it to a projective space of n dimensions or briefly to an n-space Sn. To prove it, we introduce here once again the idea of a set of n+1 associated lines in Sn as indicated in an earlier work (Mandan, [12]) in analogy with a set of 5 associated lines in S4 (Baker, [4], p. 122), and make use of the method of induction.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 5 , Issue 4 , November 1965 , pp. 401 - 408
Copyright
Copyright © Australian Mathematical Society 1965

References

[1],[2],[3].Baker, H. F., Principles of geometry, Vols. 2, 3, 4, (Cambridge 1922, 1923, 1925).Google Scholar
[4]Baker, H. F., Polarities for the nodes of a Segre' cubic primal in space of four dimensions, Proc. Camb. Phil. Soc. 32 (1936), 507–520.CrossRefGoogle Scholar
[5]Beatty, S., Advanced problem 4079, Amer. Math. Monthly 50 (1943), 264.Google Scholar
[6]Chasles, M., Aperçu historique 32 (1837), 400.Google Scholar
[7]Court, N. A., Modern pure solid geometry (New York, 1935).Google Scholar
[8]Court, N. A., Pascal's theorem in space, Duke Math. J. 20 (1953), 417–420.CrossRefGoogle Scholar
[9]Coxeter, H. S. M., The real projective plane (Cambridge, 1955).Google Scholar
[10]Coxeter, H. S. M., and Todd, J. A., Solution of advanced problem 4079, Amer. Math. Monthly 51 (1944), 599–600.Google Scholar
[11]Mandan, S. R., Cevian simplexes, Proc. Amer. Math. Soc. 11 (1960), 837–845.CrossRefGoogle Scholar
[12]Mandan, S. R., Polarity for a quadric in n-space, Istanbul Üniv. Fen Fak. Mec. Ser. A 24 (1959), 21–40.Google Scholar
[13]Mandan, S. R., Altitudes of a general simplex in 4-space, Bull. Calcutta Math. Soc., 1958, Supplement, 34–41.Google Scholar
[14]Mandan, S. R., Semi-isodynamic and -isogonic tetrahedra, Rend. Mat. e Appl. (5) 19 (1960), 401–415.Google Scholar
[15]Mandan, S. R., Altitudes of a simplex in four dimensional space, Bull. Calcutta Math. Soc., 1958, Supplement, 8–20.Google Scholar
[16]Mandan, S. R., Semi-orthocentric and orthogonal simplexes in 4-space, Bull. Calcutta Math. Soc., 1958, Supplement, 21–29.Google Scholar
[17]Mandan, S. R., Uni- and demi-orthocentric simplexes, J. Indian Math. Soc. (N.S.) 23 (1961), 169–184.Google Scholar
[18]Mandan, S. R., Dandelin's figure in n-space, Casopic Propestovani Mathematiky 90 (1965), 58–64.Google Scholar
[19]Mandan, S. R., Orthogonal hyperspheres, Acta Math. Acad. Sci. Hungar. 13 (1962), 25–34.CrossRefGoogle Scholar
[20]Mandan, S. R., Altitudes of a simplex in n-space, J. Australian Math. Soc. 2 (1961/1962), 403–424.CrossRefGoogle Scholar
[21]Mandan, S. R., Isodynamic and isogonic simplexes, Ann. Mat. pura e appl. (4) 53 (1961) 45–55.CrossRefGoogle Scholar
[22]Salmon, G., Analytical geometry of three dimensions, Vol. 1 (New York, 1927).Google Scholar