The difficulty which every student beginning the study of solid geometry finds in interpreting diagrams representing solid figures may be largely met, and an interest in the subject may be aroused at the outset, by a few lessons in drawing the simplest regular solids and in the study of the drawings made, before the formal propositions of Euclid, Book XI. are attacked.

The following variation seems an improvement on the usual method of tracing a refracted ray. In the usual method when an incident ray OP is given, any radius OP is taken and a circle described. PA is drawn perpendicular to the surface AB. OA is then sub-divided into the number of units expressed in the numerator of the fraction μ, and OB is measured backwards, the number of units expressed by the denominator. BQ is then drawn perpendicular to AB to meet the circle in Q. OQ is the refracted ray.

It is proposed to trace the curves represented by the equation

for the values of a (i) a = 2, (ii) a = 1.

Let O be the centre of a given circle of radius a; and OA, OB the containing radii of the are which is to be trisected.