The general idea of such treatment of a fan or conoid may be said to be parallel to that of a circular or rose window. The ribs are viewed as radiating mullions, and are made to multiply in number as the circle expands, and usually to terminate round the outer boundary with a series of tracery forms like the heads of two-light windows. A good rose-window of late date, if imagined to be elastic, and drawn out from its centre into a conoidal form, would make a good compartment of fan groining; or a groining fan compressed into a plane would make a good rose-window. The interstices between the fans are filled up in various ways; either by circles of somewhat similar design, which sometimes drop down in little pendent fans, like stalactites from the roof of a cavern; or with a number of circles fitted together; or by continuing the diagonal ribs to their intersection with the ridges, and filling in the triangular spaces with tracery. It is not, however, essential that the compartments so dealt with should be _square_ in plan. If _oblong_, the semicircles or quadrants are drawn from the apex of the narrower and lower arches; and from the same centres portions of circles are drawn from the apex of the wider and higher arches, assuming the form of an additional or outer zone of a rose-window, and intersecting on the line of the cross ridges, and thus forming a portion cut out of a great fan (Fig. 390). Or the roof may be described as being formed of very large fans intersecting one another. Such is the famous roof of King’s College Chapel, at Cambridge, but the frustums of fans are there bisected by vast transverse ribs, which were, no doubt, required for strength, owing to the great scale of the vaulting (Fig. 391). [Illustration: Fig. 391. King’s College, Cambridge.] The earliest instance which is known of fan vaulting is in the cloisters at Gloucester, dating from the beginning of the fifteenth century; indeed, Professor Willis thinks the invention due to a school of masons there. It, contrary to the usual practice, is formed with simple arches, instead of four centres. Each fan has but the transverse and wall ribs at its springing. At the next stage it forms the diagonal, then two intermediate ribs, and finally four more on each side of the transverse rib, and ends in little two-light window-like heads, while the space above is filled with four large cusped arches, and as many pear-shaped figures (Fig. 392). [Illustration: Fig. 392.--Cloisters, Gloucester Cathedral.] The limits of my lecture will not permit me to follow up this part of my subject as its importance demands; and my means of producing illustrations (for which I have heartily to thank several zealous and talented gentlemen engaged in my office, as well as one of my sons) have exhausted themselves before this laborious phase is reached. I have already referred to the Gloucester cloisters and King’s College Chapel.