The next set of examples which show the necessity of a geometrical system will be found in the construction of the intermediate ribs of vaults, and in the management of the curvature of the ribs generally. The limits of this paper will not allow me to enter fully into the description of the different classes of vaults in the decorative sense, neither is it necessary for my present purpose ; I shall therefore briefly state the different steps by which they appear to have been led on from the simple cross-ribbed vault to the fan tracery. The plain cross vault, Roman in arrangement, but pointed and with the addition of ribs upon the groins, is to be found at Salisbury, Gloucester nave, Canterbury choir, Wells nave, Beverley, Westminster choir, and in all the French cathedrals. Simple intermediate ribs were first added between the wall ribs and diagonal ribs, and between the transverse ribs and diagonal ribs. Thus, in fig. 9, a vault with intermediate ribs is represented in a diagram upon the principle of delineation, which was first employed by Mr. Ware, in his admirable treatise on this subject.

A B K L are the points whence the ribs spring, and between which is given the plan of the vault in a kind of diagonal perspective.

AC, BC, KC, LC are the diagonal ribs or great cross springers (termed croisee d’ogives by De l’Orme).

Ae Af Ag are the intermediate ribs of that spandrel of the vault which lies nearest to the eye in the diagram. These intermediate ribs are termed the tiercerons by De l’Orme, which being a very convenient word, I shall employ.

AD will, therefore, be the transverse rib of the vault…

…and AE the rib which lies next the wall of the clerestory or the wall rib, as I have ventured to call it, the formeret of De l’Orme.

In this figure I have shown one tierceron between the transverse and diagonal ribs, and two between the wall rib and diagonal rib; but the number varies in different examples, The figure agrees with the vaults of the choir of Lichfield and the south transept of Hereford.

One tierceron in each space DCA and CAE is to be found in Lichfield nave and Lady Chapel, Norwich cloister, Exeter side aisles, Lincoln nave, Westminster nave and cloisters, and in the vault at the intersection of the nave and transept at Amiens Cathedral.

Sometimes three and one are employed, as in Exeter nave,

or three and two, as in Norwich nave.

Much of the effect and character of these vaults depends upon the curvature of these tiercerons, and also upon that of the diagonal and transverse ribs between which they are placed; and even in the simpler vaults, which have only the diagonal and transverse ribs, this curvature governs the character of the vault by determining the form of the spandrel solid.