College Geometry: A Discovery Approach

The book was in great condition, and arrived just in the nick of time. This was the first time a book was ever sent on time!

I was looking for a concrete axiomatix approach to spherical geometry (NOT the so called elliptic geometry in which antipodal point are identified) in which any two lines intersect in TWO points (so called antipodal points) etc. A hint given in an appendix on elliptic geometry in Marvin J. Greenberg's (excelent) book "Euclidean and Non Euclidean Geometry (3rd ed. D H Freeman & Co, 1993), brought me to David C. Kay's book "College Geometry" published by Holt Rinehart and Winston, 1969. Fortunately I found the book in a library and I found there a very satisfactory, readable and interesting axiomatic treatment of spherical geometry, including an interesting proof of the triangle inequality. It is a peaty that this concrete geometry on the sphere, a golden mine for various non easy theorems and questions, is ignored by most textbooks (most of them concentrate on hyperbolic and/or euclidean geometry, where the hyperbolic has such vague models as Klein's or Poincare's, in contrast to the spherical geometry with its clear model on the sphere), and it is a good occasion to praise David Kay's work here, which although printed in 1969 (35 years ago) it is the best of its kind as far as I know. From other sourses (Judith Cederberg's book "A Course in Modern Geometries", Springer-Verlag, 1989, p. 67) I learnt that also David Gans' book "An Introduction to Non-Euclidean Geometry" (Academic Press, 1973) made a similar approach to spherical geometry, but I didn't check it yet.(See also David Gans' paper in the supplement to the American Mathematical Monthly 62(1955), pp. 66ff., where he expreses the need for an aximatic approach to spherical geometry.) Let me add that David C. Kay's approach here is not purely synthetic in the sense that he takes for granted the real numbers (for measuring distanses, built in into his system of axioms) but this can be made purely synthetic (like Greenberg's approach to hyperbolic geometry) by a competent geometer, and I think this chalenge should be taken by a future writer of a new book based on David C. Kay's scheme, in which everything will become purely synthetic (solely based on the appropriate incidence, betweeness and congruence axioms, with the additional natural axioms on the antipodal points).