It is also clear that any irregularity of form may render these lines shapeless and unpleasing, and it is an obvious gain from an artistic point of view, to adopt a system which will at once render them _pronounced_ and _regular_. While, then, the introduction of the angular rib was in many cases a departure from geometrical accuracy, it was a vast gain both in strength and beauty. In that form of vaulting, which I have defined as that with the _raised ridges_, no geometrical inaccuracy would arise, the angles of intersection being semicircular, and in vertical planes; but in the more ordinary form of vaulting, where these lines are elliptical, that curve being unpleasing, two courses offered themselves for choice: the use of segments of circles for the diagonal ribs, or the bringing down the springing to a lower level than that of the vault. In either case the true geometrical figure has to be departed from, and the error has to be thrown into the vaulting-surfaces--a course which subsequently became so thoroughly adopted as a principle, that it may be received as an axiom that in ribbed vaulting, where the ridges are not raised, the ribs are made of such forms as will satisfy the eye, and the vaulting surfaces made to fit themselves to them as best they may, apart from geometrical accuracy,--a principle which, though it may at first sight offend the mathematical mind, has proved in practice so wonderfully useful, and to offer so many facilities, as to be a sort of _Magna Charta_ to the art of vaulting. This step once taken, round-arched vaulting seems to have completed its work. Square and oblong spaces were vaulted either with mathematical accuracy on the raised-ridge principle, or with deliberate departure from such accuracy on the level-ridge principle. Irregular spaces were covered over by expedients which satisfied the eye, and met practical conditions tolerably well, and many beautiful works were the results. The diagonal ribs, too, became a new source of decoration, not only by means of their own mouldings or enrichments, and through the bosses now sometimes placed at their point of intersection, but also because they were suggestive of additional colonnettes, and thus added more richness and intricacy to the piers; and sometimes they were carried upon sculptured corbels, as in the cathedral at Oxford. Among the richest specimens of this vaulting may be mentioned the gateway and the Chapter-house of Bristol Cathedral, the chancel of St. Peter’s Church at Oxford, etc. We have now arrived at a stage of our investigation when we must pause for the sake of asking ourselves what need or requirement yet remained unsatisfied which was essential to the perfecting of our arcuated developments, and what means remained--hitherto unused--by which such need might be met.