Non-Euclidean Geometries pp 109-114

Part of the Mathematics and Its Applications book series (MAIA, volume 581)

An Absolute Property of Four Mutually Tangent Circles

  • H.S.M. Coxeter


When Bolyai János was forty years old, Philip Beecroft discovered that any tetrad of mutually tangent circles determines a complementary tetrad such that each circle of either tetrad intersects three circles of the other tetrad orthogonally. By careful examination of a new proof of this theorem, one can see that it is absolute in Bolyai’s sense. Beecroft’s double-four of circles is seen to resemble Schläfli’s double-six of lines.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H. F. Baker, A Geometrical Proof of the Theorem of a Double Six of Straight Lines, Proc. Royal Soc. A 84 (1911), p. 597.MATHGoogle Scholar
  2. [2]
    H. F. Baker, The General Cubic Surface, Principles of Geometry, Vol 3, Solid Geometry, Cambridge University Press (1934), pp. 159 and 225.Google Scholar
  3. [3]
    P. Beecroft, The Concordent Circles, The Lady’s and Gentleman’s Diary, The Company of Stationers, London (1843).Google Scholar
  4. [4]
    H. S. Carslaw, The Elements of Non-Euclidean Plane Geometry and Trigonometry, Longmans, London (1916).MATHGoogle Scholar
  5. [5]
    H. S. M. Coxeter, Introduction to Geometry (2 nd ed.), Wiley, New York (1969).MATHGoogle Scholar
  6. [6]
    H. S. M. Coxeter, Inversive Geometry, in Educational Studies in Mathematics, Vol 3 (1971), pp. 310–321.MATHCrossRefGoogle Scholar
  7. [7]
    H. S. M. Coxeter, A Geometriák Alapjai, Műszaki könyvkiadó, Budapest (1973).Google Scholar
  8. [8]
    H. L. Dorwart, The Schäfli Double-Six Configurations, C.R. Math Rep. Acad. Sci. Canada, Vol 15 (1993), pp. 54–58.MATHMathSciNetGoogle Scholar
  9. [9]
    J. Dougall, The Double-Six of Lines and a Theorem, in Euclidean Plane Geometry, Proc. Glasgow Math. Assoc., Vol 1 (1952), pp. 1–7.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    A. Ichida, A Simple Proof of the Double-Six Theorem, Tohoku Math. Journ., Vol 32 (1929), pp. 52–53.MATHGoogle Scholar
  11. [11]
    R. J. Lyons, A Proof of the Theorem of the Double-Six, Proc. Cambridge Philos. Society, Vol 37 (1941) pp. 433–434.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    L. Schäfli, Theorie der vielfachen Kontinuität, Gesammelte Mathematische Abhandlungen, Band I, Verlag Birhäuser, Basel (1953).Google Scholar
  13. [13]
    L. Schäfli, An attempt to determine the twenty-seven lines upon a surface of the third order, and to devide such surfaces into species in reference to the reality of the lines upon the surface, Gesammelte Mathematische Abhandlungen, Band II, Verlag Birhäuser, Basel (1953).Google Scholar
  14. [14]
    B. Segre, Sulla costruzione delle bisestuple di nette, Rend. Acad. Naz. Lincei (6) Vol II (1930), pp. 448–449.Google Scholar
  15. [15]
    C. Yamashita, An Elementary and Purely Synthetic Proof for the Double-Six Theorem of Schäfli, Tohoku Math. Journ. (2), Vol 5 (1954), pp. 215–219.MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • H.S.M. Coxeter
    • 1
  1. 1.University of TorontoTorontoCanada

Personalised recommendations