On a Fundamental Theorem of Geometry

If the projective axioms of the space are satisfied, then Desargues' theorem holds. If on the other hand in a plane the projective two-dimensional axioms are satisfied and Desargues' theorem holds, then the plane can be embedded into a space in which the projective axioms hold. The proof of the first proposition is easy and elementary; the second proposition has been proved by D. Hilbert by an ingenious arithmetical method which has the highest importance for the investigation of the fundamentals of Geometry. In this paper an alternative proof will be given which is elementary and geometrical; it applies a simple idea of descriptive geometry. Although descriptive geometry has occasionally been considered from an axiomatic point of view, its methods have-as far as 1 know-never been utilised for axiomatic purposes. As Descriptive Geometry is rather neglected in India, this paper is written in such a manner that the reader may follow it without references to that branch of mathematics.