On the occasion of the GAUSS-WEBER celebration in 1899 HILBERT published an important memoir, Grundlagen der Geometrie, in which he devoted chapter V to the consideration of DESARGUES's theorem. He summarized the results obtained in this chapter as follows :t The necessary and sufficient condition that a plane geometry fulfilling the plane axioms 1 1-2, II, III may be a part of (or set in) a spatial geometry of more than two dimensions fulfllling the axioms I, II, III, is that in the plane geometry Desargues's theorem shall be fulfllled. The proofs of the necessity and of the sufficiency of the condition are given by HILBERT in ? 22 and ?? 24-29 respectively. In ? 23 he proves that DDESARGUES'S theorem is not a consequence of the axioms I 1-2, II, III,t and states (theorem 33) that it cannot be proved even though the axioms IV 1-5 and V be added. His method is to exhibit a non-desarguesian geometry fulfilling the axioms in question. His example is of a somewhat complicated nature, involving in its description the intersections of an ellipse and a system of circles (euclidean) which are defined so that no circle intersects the ellipse in more than two real points. The demonstration that the geometry fulfills the axioms in question, the details of which are not given by HILBERT, depends upon the real solutions of simultaneous quiadratic equations. Moreover, HILBERT's example does not fulfill all of the axioms enumerated, the exception being IV 411 -which, in connection with the definition which precedes it, requires that the angles (h, Ik) and (k, h) shall be congruent, while the angles in HILBERT'S geometry whose vertices are on the ellipse depend upon the order in which their arms are taken for their non-desarguesian congruence relations. HILBERT'S final theorem, stated at the beginning of this paper, does not involve IV 4 and was completely