The Mathematical Association existed first of all as a society for the improvement of geometrical teaching in the schools, and had as its avowed intention the infusion of a spirit of sweet-reasonableness into abstract geometry. Whether the Association aimed in the first instance at the unconditional abdication of Euclid is difficult to say, but at all events such was the final result. So successful were the labours of the Association in the realm of geometry that it resolved to extend its activities into the domains of Arithmetic and Algebra. To these were quite recently added the Calculus and Mechanics; but, broadly speaking, it must be said that its activities have been almost entirely confined within the somewhat narrow limits of the Secondary Schools. The contention of this communication is that great and clamant opportunities of service lie outside the walls of our more academic Secondary Schools, and it is proposed to deal with some of those newer spheres of activity.

No general agreement about a first-year syllabus in Geometry is possible unless there is some measure of agreement as to the aim we ought to have in teaching geometry at all. For to different aims there will correspond differences, not only in the manner of teaching the syllabus, but also in the actual matter taught. What, then, ought to be our aim?

The object of the paper is to call attention to some simple means of interpolating points in a cubic graph, and of indicating gradients without actual calculation.