Desargues’ Theorem

Abstract

In his Foundations of Geometry, Hilbert has shown “the impossibility of demonstrating Desargues’ Theorem for plane geometry without the help of the axioms of congruence.” But it must be remembered that his demonstration depends entirely on the form which he has given to the other axioms, or definitions as I should prefer to call them, and in particular on the way he defines “straight line,” namely “two distinct points always determine completely a straight line,” and “any two distinct points of a straight line completely determine that line.” What he has shown then is that in plane geometry this is an inadequate definition of “straight line.” He has not shown that that definition cannot be amended otherwise than by the introduction of the conception of congruence, or by that of three-dimensional space.One of the most interesting results of the theory of Order as defined by Boundaries is that it explains this apparent paradox. I propose to add to the papers already published in the Gazette on this theory a brief note on this point. To aid the imagination of my readers I shall not attempt to state the problem in general terms, but shall merely state it in homely geometrical language, and for the sake of brevity I shall not pretend to give formal demonstrations of each successive point in the argument, but shall leave obvious deductions to the reader.