FEW branches of elementary mathematics have escaped reform in recent years; algebra, trigonometry, elementary geometry, statics, and the calculus have all been transformed to suit the practical needs of the time. But amid all this change, there have been few attempts to alter the methods employed in introducing students to the geometry of the conic. Dr. Filon, in his most interesting treatise on Projective Geometry published in 1909, points out that students learn the same facts about the conic practically three times over, (1) analytically, (2) through a course of what is called “geometrical conies” based on the focus-directrix definition, and (3) protectively ; and his book indicates a method of co-ordinating and uniting the last two systems of approach. Although it is undoubtedly the general custom in this country to start in the first place from the focus definition, it is worth while considering afresh whether this plan is really the most advantageous.

It is interesting to note that historically this was not the starting point of the early geometers. It is true that the focus-directrix property was known to Pappus, but so far from being regarded as fundamental, it was actually lost sight of altogether, until attention was called to it by Newton. The geometrical investigations of Apollonius were based on the conception of the conic as the plane section of a cone, the advantages of which plan may be enumerated roughly as follows: