I. To draw quickly a good figure of A, B, C, I, I1, I2, I3, etc. Draw a circle, and a chord BC. Mark L the middle point of the arc below BC by the “engineer's method,” viz., with B as centre and a radius as near BL as can be judged by the eye, make a mark on the arc, with C as centre and the same radius make another mark on the arc, judge by the eye the middle point of the arc between these marks; this is L. With centre L, radius LB or LC, describe a circle: I and I1 lie on this circle. Mark M the middle point of the arc above BC. With centre M, radius MB or MC, describe a circle: I2 and I3 lie on this circle. Now I and I1 lie on AL; I2 and I3 on AM. Mark A on the first circle, so that MA lies conveniently on the paper. The various collinearities and perpendicularities justify the figure to the eye; the properties of the mid-points of II1 etc., I2T3, etc., and the loci of I, I1 I2 I3 as A varies, are emphasised.