There is no one who has studied the subject who is not aware how excellent, as a constructive expedient, the pointed arch is as applied to intersecting vaults, but it is not so generally understood why it was equally necessary in the construction of domes. So long as these rested on drums rising from the ground the circular form sufficed; but when it became necessary to rest them on pendentives in the angles of square or octagonal buildings, the case was widely different. The early Byzantine architects—in Sta. Sophia, for instance—did fit pendentives to circular arches, but it was with extreme difficulty, and required very great skill both in setting out and in execution. But the superiority of the pointed form was perceived at an early date; and the Saracens, who were trammelled by no traditions, adopted it at once as a doming expedient and adhered to it as exclusively as the Gothic architects did in the construction of their vaults—and for the same reason—simply because it was the best mode of construction. It is easy to explain why this should be so. In the diagram on the preceding page, fig. 1 represents the pendentives of a dome resting on circular arches. At A they become evanescent, and for some distance from the centre are so weak that it is only by concealed construction that they can be made to do their work. When the pointed arch is introduced, as in fig. 2, not only is great freedom obtained in spacing, but the whole becomes constructively correct; when, as in fig. 3, an octagonal arrangement is adopted, the whole becomes still more simple and easy, and very little adjustment is required to fit a dome to an octagon: and if the angles are again cut off, so as to form a polygon of 16 sides, all the exigencies of construction are satisfied. [Illustration: 546. Section of Church at Carcassonne, with the outer aisles added in the 14th century. No scale.]