The semicircular arch requires a massive thickness for stability; the inverted chain can only be contained if the thickness exceeds about 10 per cent of the radius. A semicircular barrel vault to cover the 14 m nave of Amiens would need a thickness, with no factor of safety, of over 700 mm. In practice, circular vaults need not embrace a full semicircle, and thicknesses may then be reduced markedly moreover, the springings of the vaults, the haunches, may be filled with masonry rubble, adding further to the stability of the construction.

However, the major step in reducing structural weight comes from the use of barrels intersecting at right angles, to create a true three-dimensional vault rather than a repetition of parallel two-dimensional arches. The Romans had used the idea of intersecting barrels to form the groin vault; each concrete bay of the Basilica of Maxentius (AD 30) spans over 25 m. Such a vault needs support only at the corners, and windows may be introduced in the side walls; moreover, smaller thicknesses of vaulting material can be used, so that weights on supporting piers and thrusts on external buttresses are reduced.

Two equal cylindrical barrels will intersect to give a vault which is square on plan, and the diagonals of the square give the location of the groins. If the vault is made of stone (rather than Roman concrete), then geometrical difficulties will be encountered in cutting the stones meeting at the groins the art of stereotomy is concerned with...