The geometrical construction is that of the pendentives of all domes. The four supporting arches intersect a hemispherical surface whose diameter is equal to the diagonal of the supporting square. The pendentives produce at the crown line of the arches a circular plan which is filled in by a saucer dome of the same radius as the pendentives, constructed of circular brick rings, the joints of which radiate to the centre. If the space to be covered is not square the broader arches intersect at a higher level, while the narrow arches are not stilted, but kept down so as to receive the dome surface, and in this case the narrow arches are not semicircular, but segmental. Where the difference in size between the two sides was not great, the difficulty presented was easily overcome by the Byzantine builder, who in the later buildings, at any rate, rarely built anything within four inches of its geometrical position. Where the difference was too great it was frankly accepted, and we find segmental arches at the narrow ends. [Illustration: FIG. 11.] The vaulting of the outer narthex of S. Saviour in the Chora illustrates this fully (Fig. 11). Though some of the bays of that narthex are oblong and others almost square all are covered with dome vaults. The almost square bays, although their sides vary considerably, are covered precisely as if their sides were exactly equal. But in two of the oblong bays, which are nearly three times as long as they are broad, such a method could not be applied. Longitudinal arches (AA) were accordingly thrown between the transverse arches (CC) and made to rest on their spandrils. The oblong form of the intervening space was thus very much reduced, and over it flat domes are thrown. Their rings are true circles, and as the space they cover is still somewhat oblong they descend lower, with additional segments of rings (BB), at the ends than at the sides. In the remaining two oblong bays of the narthex, the result of introducing the longitudinal arches is to convert a decidedly oblong space in one direction into a slightly oblong space in the opposite direction, an additional proof, if any were needed, that the exact shape of plan with this form of vault was a matter of comparative indifference to the builder. In S. Sophia the vault springs from the intrados of the transverse arches, that is, from the lower edge. In SS. Sergius and Bacchus it springs from a point so slightly raised as to be hardly noticeable. In the later vaults, however, the transverse arches, when present, are boldly shown, and the vault springs from the extrados or outer edge (_e.g._ S. Saviour in the Chora, S. Theodore).