Non-euclidean Algebra

Euclidean geometry today is most naturally understood as affine geometry equipped with a suitable bilinearity as a basis for notions of length, angle, area, rectangles, circles, etc. In this talk we express the affine content of Euclid's Postulates 2, 5, and 1 algebraically in that order. In our account Euclid's Fifth or Parallel Postulate is expressed as an equation parametrized with the number 6. Parameters in the range 3 to 5 yield discrete notions of elliptical geometry while 7 and beyond do the same for hyperbolic geometry. Had the tools of the past half century of universal algebra been available to Euclid, the first of his thirteen books could have been both simpler and more general, and the conceptual obstacles to hyperbolic geometry overcome more easily.