In the plane geometry of Vol. ii we have seen that the ordinary metrical relations are particular cases of the relation of the figure to two arbitrary Absolute points, or, more generally, to an arbitrary Absolute Conic; the recognition of this adds greatly to the breadth of view obtained, without increasing the difficulty of proof. A similar gain is found in the geometry of three dimensions. Here we consider an arbitrary Absolute Conic. In a later Volume we shall consider an arbitrary Absolute Quadric.

When we use coordinates we shall most generally suppose the points of reference, A, B, C, D, so taken that A, B, C form a self-polar triangle in regard to this Absolute Conic, and suppose, therefore, that the equations of this conic have the forms t = 0, x2 + y2 + z2 = 0. This is immaterial; but it will enable the reader easily to make comparison with metrical formulae which may be familiar.

Parallel lines and planes. Middle point. Lines and planes at right angles. Two lines which meet the plane of the absolute conic in the same point, not themselves lying in this plane, may be said to be parallel, relatively to this plane; two planes which meet this absolute plane in the same line may similarly be said to be parallel; a line and a plane may be said to be parallel if they have a common point lying on the absolute plane.