The majority of text-books on Geometrical Conies do not give any proof of the fundamental property that one and only one conic passes through any five given points, no three of which lie on a straight line. Special attention is drawn to this omission by the Rev. Dr. C. Taylor in an article published in the Gazette for May, 1895. The following is a description of three methods which various attempts at obtaining an elementary proof of the theorem have suggested. It is, however, very desirable to have a simpler proof than any of these, if such can be given.