The student must thoroughly understand the meaning and the philosophy, so to speak, of each mathematical concept presented to him. It takes a long time before he familiarizes himself so thoroughly with the conceptions of any mathematical subject so that he gets their significance, meaning and spirit, and the ability and facility to apply them readily. Few students have a clear understanding of the quantitative meaning and significance of the theorems of proportion, such as: “A line drawn parallel to a side of a triangle divides the other sides proportionally,” or, “Similar triangles are to each other as the squares of the homologous sides.” In all my experience I have not received from a pupil a satisfactory explanation of the truth that one divided by infinity is equal to zero, a conception used in secondary mathematics. The same is true of college mathematics. Few students who have studied Analytical Geometry are able to give the true meaning, for instance, of the equation of the straight line, y = sx + m, i. e., that the ordinate of any point of the straight line is m greater than s times the abscissa of the point. These instances may be multiplied to quite an extent.