The Use of Coordinates

  • H. S. M. Coxeter
  • George Beck


In Chapter 11 we saw how a system of coordinates is inherent in synthetic geometry. In the present chapter we shall reverse the process, building up the analytic geometry from first principles, and deriving the theorems (including the axioms) from properties of numbers. We shall find that the analytic method enables us to solve some problems more easily. On the other hand, it would be a grave mistake to abandon the synthetic method, which is far more stimulating to one’s geometrical ingenuity.


Conjugate Point Analytic Geometry Analytic Proof Affine Plane Primitive Concept 
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  1. 1.
    Cf. Veblen and Young 1910, pp. 1–6. The relation between synthetic and analytic geometry has been very ably described by Robson (1940, Chapter 8; 1947, Chapter 19).Google Scholar
  2. 2.
    Graustein 1930, p. 70, Ex.5. The idea of using capital letters for line coordinates is due to G.T. Bennett.Google Scholar
  3. 3.
    1950, pp. 120, 136.Google Scholar
  4. 4.
    Hesse 1897 seems to have been the first to write the equation for a conic in the form (math).Google Scholar
  5. 5.
    Robson 1947, p.91.Google Scholar
  6. 6.
    Salmon 1879, p. 11.Google Scholar
  7. 7.
    Salmon 1854, p. 59.Google Scholar
  8. 8.
    Altshiller-Court 1952, p.267.Google Scholar
  9. 9.
    Baker 1943, p. 114; Altshiller-Court 1952, p. 273.Google Scholar
  10. 10.
    Altshiller-Court 1952, p.270.Google Scholar
  11. 11.
    Robson 1940, p.286.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • H. S. M. Coxeter
    • 1
  • George Beck
    • 2
  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.TorontoCanada

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