The Division of the Circle—Concluded

Extract

To establish the order of the locus we must note that besides the above mentioned n intersections with the circumference, the point D also belongs to the curve, so that the latter cuts the circle in n+1 points. The order of the curve is therefore Let n be even.

The ratio is not an integer, so that a complete rotation of the straight line BC must take place before the two generating straight lines can be in their initial mutual position; the line revolving round D will be, after a semi-rotation of BC, parallel to BC in DE, and we may say that all the loci for n = 2p are asymptotic in the direction BC, on account of the initial position of BC and DA.