BROWSING for amusement in a book that I never saw at school, Casey's Sequel to Euclid, I was introduced to something new to me. It was my first encounter with harmonic polygons, and it was interesting to trace how the painstaking geometers built up a series of propositions applicable to such figures. What annoyed me, however, was the attempt to derive some of their properties by “inversion” of a regular polygon. Again it seemed illogical to build up a proof of general properties by proving a succession of mensurations. This feeling was intensified when it came to deal with the “Brocard” properties and I started to build up the theory from a new angle.