The fact is that, besides its unpleasing form, especially when the major axis is vertical, the use of the ellipse entails such an annoying series of difficulties as greatly to increase the trouble and consequent cost of execution. The constant change of curvature, the troublesome methods of striking it, and of finding the true lines of the arch-joints, not to mention the mathematical fact that the same joint line is never true both for the extrados and intrados, and that, if the rib-mould remains unchanged in depth, the extrados and intrados cannot be both true ellipses at all; all these furnish quite sufficient practical reasons for its rejection in cases where not only is there no necessity but an abstract mathematical idea to be satisfied by its use, but the beauty of the work is greatly improved by dispensing with it. Though the pointed arch was introduced from purely constructive reasons, there was another of a more æsthetical nature, which rendered its adoption more general when once introduced. It was a double one; not only did the general tendency towards lofty proportions render it necessary to make use of an arch more in harmony with the general feeling of the architecture, but the rejection of a fixed code of proportions for pillars and other parts demanded for the arch an equal power of varying its own proportions. The semicircular arch is absolute and invariable, and though the use of smaller segments would meet the case in one direction, there were no means of proportioning it to features of _increasing_ height. This was attempted both in Romanesque and Byzantine works by the expedient of _stilting_, but this is, after all, more a semblance than a reality. As in cases already cited, the mathematical solution of the problem is the ellipse; but only imagine anything so unpleasing as a series of elliptical arches placed the length-way upwards! Good taste would not suffer it. But the pointed arch at once met the difficulty. To illustrate my meaning, I will beg you to take an internal bay of a Norman cathedral (Fig. 146), and to suppose yourselves to have to increase its height throughout in the ratio of one-third (Fig. 147). [Illustration: Fig. 146.] [Illustration: Fig. 147.]