  By recasting metrical geometry in a purely algebraic setting, both Euclidean
and non-Euclidean geometries can be studied over a general field with an
arbitrary quadratic form. Both an affine and a projective version of this new
theory are introduced here, and the main formulas extend those of rational
trigonometry in the plane. This gives a unified, computational model of both
spherical and hyperbolic geometries, allows the extension of many results of
Euclidean geometry to the relativistic setting, and provides a new metrical
approach to algebraic geometry.
