On the Relation of Non-Euclidean Geometry to Extension Theory

Abstract

To the detriment of science, the entire presentation in SS15-23 still remains almost totally unnoticed. Neither Riemann in his Habilitationsschrift 1 of 1854, first published in 1867, nor Helmholtz,2 in his paper ``Über die Tatsachen, welche der Geometrie zur Grunde liegen'' (1868), nor even in his excellent lecture ``Über den Ursprung und die Bedeutung der geometrischen Axiome'' (1876) mention it, even though the foundations of geometry come into view much more simply than in those later publications. In extension theory the straight line is quite special and, in contrast to Euclid, is the foundation for geometric definitions. In S16 the plane is defined as a collection of parallels that intersect a straight line, and space as a collection of parallels that intersect a plane; geometry can proceed no further, but the abstract science is not so limited. Since all points of a straight line may be numerically derived from two of its points, the straight line appears as a simple elementary domain of second order, and correspondingly the plane as a simple elementary domain of third, and ultimately space as one of fourth order.3