I propose to consider adjustments which improve the teaching of two subjects that are related as are geography and mathematics. Obviously there is much in mathematics with which geography has no concern, and there is much in the geography course on which mathematics throws no light. But there is some material more or less common, and it is desirable to see in which department or by which master or mistress it should be introduced.

5. We shall now apply this method to a few typical cases.

(i) The equation y1=x2+y2

gives y2=2x+2y2(x2+y2).

Hence, y1 is always positive; y2 vanishes along the curve

r2=-cot θ,

in which x=r cos θ, y = r sin θ. Hence, in Fig. 5(a), the curves have the form) above the dashed curve (the locus of inflexion), and ( below this curve.

The section on “Ratio, Proportion and Variation” in the text-books is by far the least satisfactory part of the standard course of Algebra. And it is probably regarded by many readers of these books as of no particular importance. But, when seen in the right perspective and given proper treatment, this part of the subject may fairly be regarded as the most far-reaching in importance—a contention for which the reasons will appear in the following pages.