The Course described here has been evolved at Northallerton Grammar School during the past nine years. The field-work and plan-drawing are done during Geography lessons, while the calculation of area, height finding and resection are undertaken by the mathematics classes. Many mathematical ideas are involved throughout the Course, which acts as a connecting link between the two subjects. The ages given are only approximate.

In a recent number of the Mathematical Gazette (March 1920) I proposed the following problem for solution:

Some of the figures in the long division sum have been replaced by crosses, and it is required to restore them. Four solutions are possible—of course in scale ten. It is understood that the sum is to be worked as in elementary arithmetic and that there is nothing fictitious. There must be no line of zeros, neither must the left-hand figure of any line be zero. In the divisor the digits figure ought not to be zero (for such a zero would ordinarily be cancelled before working the sum), but it is not enough to assume this: proof must be possible from the data. In the Four Fours the proof is instantaneous, because there is no remainder and the right-hand figure of the dividend is 4.

The question of sign constitutes a real difficulty to the intelligent boy at the outset of his study of Coordinate Geometry. At the beginning of his Trigonometry he is told OP must be considered always positive, but later on he will find some authorities giving a point in the third quadrant as ( - r, θ), while others prefer (r, θ + π). The perpendicular distance of (h, k) from ax + by + c = 0 is given by ± (ah + bk +c)/(a2 + b2)½, and sign seems to matter, but usually the pupil is told that he only wants to know how far off (h, k) is, and he is advised to stick to the absolute value. But a little later on he wants the equations of the bisectors of the angles between two given lines, and then he is blamed for not remembering that signs matter a good deal.