§ XV. 1. If the group at _d3_, Fig. XIV., be taken from under its loading, and have its centre filled up, it will become a quatrefoil; and it will represent, in their form of most frequent occurrence, a family of shafts, whose plans are foiled figures, trefoils, quatrefoils, cinquefoils, &c.; of which a trefoiled example, from the Frari at Venice, is the third in Plate II., and a quatrefoil from Salisbury the eighth. It is rare, however, to find in Gothic architecture shafts of this family composed of a large number of foils, because multifoiled shafts are seldom true grouped shafts, but are rather canaliculated conditions of massy piers. The representatives of this family may be considered as the quatrefoil on the Gothic side of the Alps; and the Egyptian multifoiled shaft on the south, approximating to the general type, _b_, Fig. XVI.
§ XVI. Exactly opposed to this great family is that of shafts which have concave curves instead of convex on each of their sides; but these are not, properly speaking, grouped shafts at all, and their proper place is among decorated piers; only they must be named here in order to mark their exact opposition to the foiled system. In their simplest form, represented by _c_, Fig. XVI., they have no representatives in good architecture, being evidently weak and meagre; but approximations to them exist in late Gothic, as in the vile cathedral of Orleans, and in modern cast-iron shafts. In their fully developed form they are the Greek Doric, _a_, Fig. XVI., and occur in caprices of the Romanesque and Italian Gothic: _d_, Fig. XVI., is from the Duomo of Monza.
§ XVII. 2. Between _c3_ and _d3_ of Fig. XIV. there may be evidently another condition, represented at 6, Plate II., and formed by the insertion of a central shaft within the four external ones. This central shaft we may suppose to expand in proportion to the weight it has to carry. If the external shafts expand in the same proportion, the entire form remains unchanged; but if they do not expand, they may (1) be pushed out by the expanding shaft, or (2) be gradually swallowed up in its expansion, as at 4, Plate II. If they are pushed out, they are removed farther from each other by every increase of the central shaft; and others may then be introduced in the vacant spaces; giving, on the plan, a central orb with an ever increasing host of satellites, 10, Plate II.; the satellites themselves often varying in size, and perhaps quitting contact with the central shaft. Suppose them in any of their conditions fixed, while the inner shaft expands, and they will be gradually buried in it, forming more complicated conditions of 4, Plate II. The combinations are thus altogether infinite, even supposing the central shaft to be circular only; but their infinity is multiplied by many other infinities when the central shaft itself becomes square or crosslet on the section, or itself multifoiled (8, Plate II.) with satellite shafts eddying about its recesses and angles, in every possible relation of attraction. Among these endless conditions of change, the choice of the architect is free, this only being generally noted: that, as the whole value of such piers depends, first, upon their being wisely fitted to the weight above them, and, secondly, upon their all working together: and one not failing the rest, perhaps to the ruin of all, he must never multiply shafts without visible cause in the disposition of members superimposed:[41] and in his multiplied group he should, if possible, avoid a marked separation between the large central shaft and its satellites; for if this exist, the satellites will either appear useless altogether, or else, which is worse, they will look as if they were meant to keep the central shaft together by wiring or caging it in; like iron rods set round a supple cylinder,--a fatal fault in the piers of Westminster Abbey, and, in a less degree, in the noble nave of the cathedral of Bourges.