If vaulting were carried out on perfect theoretical principles, and with the level ridge, these plans would be _rectangular_. The mere substitution of circular curves, unless made a little segmental, softens the angle of the square, while any modification of these curves produces its effect, one way or another, on this half-height plan. Professor Willis has illustrated this in a very interesting manner from the variations found in the cloisters at Norwich; a work generally of uniform design, but which, having been carried out at several different periods, the habits of the masons had undergone changes which produced very curious effects in this respect. I may mention that where the ribs are of the same curve the ridges cannot be level; but that the use of the two-centred curve enabled them to repeat the same curve, for the springers, but slightly to change it above by varying the position of the centre, from which the upper part of the curve is struck, so as to make the ridges level where it was desired. This tendency to the repetition of the same curve led to the development of a most remarkable variety in vaulting, which especially characterised its later history in this country. I allude to that extraordinary form known as _Fan_ vaulting. It is self-evident that if a number of ribs of equal curvature spring at equal angles from a single pillar, the plan of the vault at any level will be a circle or a portion of a circle; and that they may be bounded at the level of their apex, or at any other level, by a circular moulding, forming the whole into a figure which I have had occasion to mention as having been generated at a very different date and from another cause, in the Norman vaulting of the circular Chapter-house at Worcester,[60] a figure which I have defined as a concave-sided conoid, and likened to the flower of a convolvulus. [Illustration: Fig. 389.] [Illustration: Fig. 390.] Now, the vaulting of any space, if set out in square compartments, and having ribs of similar curvature and at equal distances, may be formed into a number of portions of this figure by merely drawing semicircles and quadrants, as the case may be, from apex to apex of the surrounding arches (Fig. 389). A remainder of the ordinary vaulting, with rising ridges, exists, however, above this extemporised fan, and the whole has to be dealt with artistically on a system suited to the new form.