On the other hand, the change was attended with the loss of geometrical accuracy. Hitherto we have dealt with none but perfectly correct geometrical figures; but the moment the pointed arch is introduced, the pendentives lose this exactitude, and have to be adapted by what is vulgarly called “rule of thumb” to conditions not precisely suited to their forms. A pendentive between pointed arches has, it is true, a geometrical form of its own, but this is so awkward in its sections that it has only to be seen to be rejected; for, instead of its central section being a regular arched curve, suited to a domical surface, it is a curve of _double flexure_, its lower part concave (as seen from within), and its upper part convex--in short, _an ogee_. This being inadmissible, the curve has to be accommodated the best way we can, so as to avoid this weak and unpleasing form. We have, in fact, to determine, according to the best of our judgment, what shall be the vertical section of the pendentive, and adopt such horizontal curves for the courses of masonry as will make it reach the extrados of the supporting arches in the easiest manner we are able. This was really done so successfully by the French architects, whose works I shall shortly have to describe, that, for myself, I must say I never found out the difficulty from seeing them, and was unaware of it till I worked out the profiles geometrically. After all, however, it is only parallel to what we have to do in filling in the spaces between the ribs of Gothic vaulting. The pointed arch, though beautiful and practically excellent, is no regular geometrical figure, but the union of portions of two; its use, consequently, induces irregularities which would be at once avoided by the substitution of an ellipse. But then our geometrical accuracy would be purchased by the sacrifice of beauty. All the sections of a sphere being circles, the supporting arches of a true pendentive hemispherical dome are semicircular arches, and in the same manner those of an elliptical spheroidal dome would be semi-ellipses; but there is no regular solid figure, more than one of whose sections are pointed arches, so it is natural that when they are used some part should have to be accommodated to fit them.