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The Circular Points and Elementary Geometry
Published online by Cambridge University Press: 03 November 2016
Extract
Elementary geometry will be supposed to have its usual meaning as euclidean metrical real geometry of the plane. I propose to point out that, while “points at infinity” and “the line at infinity” may be properly introduced in conformity with the axioms, the “circular points” cannot be so introduced. They do not belong to this geometry; and those results which do properly belong to it, and are often made apparently to depend on the existence of the circular points, should therefore be expressed with no mention of these points.
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- Copyright © Mathematical Association 1937
References
page no 346 note * “Line” means “straight line”.
page no 347 note * This is merely an ad hoc term, and should not be confused with its use, sometimes made, for the Absolute Points or Circular Points.
page no 347 note † It can be proved by means of the theorems of Ceva and Menelaus.
page no 350 note * Views of a similar tenor, though not explicitly in regard to the “circular points”, were expressed by Professor Neville in a recent address on “Negative Squares in Real Geometry” to the Mathematical Association of N. Ireland. See also Robson, A., Gazette 20 (1936), 51 Google Scholar.
page no 351 note * An instance of a different and more blatant misuse of the circular points was shown to me by my colleague, Dr. R. Cooper. A student asserted “The circles |z| = a, |z| = b, in the complex plane, intersect in the circular points”!
page no 351 note † These include, and give a geometrical interpretation of, the rectangular hyperbolas found in one of the standard analytic methods of getting the foci, e.g. Sommerville, Analytical Conics (1933), 139, Equations (3),(4).