Robert William Billings' investigations of rose windows

These investigatons were part of Billings' investigations into the geometry of Gothic architecture, but he devised hundreds of designs - mostly flamboyant - and some of these must have found their way into the designs of many Victorian and other nineteenth century revival Gothic buildings. Here are some of them:-

His geometrical analysis is not always easy to follow. He gives no explanation, so the reader has to deduce the logic for oneself! For example, the following design and the structural analysis he gives is all we have to go on:-

Some of the curves follow logcally, but not all of them. Presumably the circle centred at 1 on the circumference is supposed to be on one of the 15 degree radials, some of which are drawn in. However taking a compass and measuring for oneself shows this not quite to be the case!