The second was of a more intricate nature, and requires to be explained more in detail. When the two intersecting vaults of a groin are similar and equal in their section, or when the section of one is the mathematical resultant of that of the other, the line of intersection falls in a _plane_. When vaulting, however, became general, all sorts of irregularly-formed spaces would have to be so covered, and would present problems of considerable difficulty, in which it would be impossible in all cases that the vaulting surfaces should be portions of cylinders or regular cylindroids, and in which the intersecting lines could not, without much twisting of the surfaces, be brought to fall into planes. The introduction of the diagonal rib met both of these difficulties. It strengthened the weak angle and gave it a substantial footing; and it at the same time gave to the lines of intersection a certain degree of independence of the vaulting surfaces; so that, instead of the surfaces governing the intersection, they were thenceforth governed by the ribs, and the latter could be made to fall into planes, and to avoid unsightly forms even in vaulting spaces of the most irregular and abnormal forms. The substitution of the rib for the arris worked as great a revolution in the principles of vaulted construction as did the pointed arch itself. Nothing in the way of vaulting was now impracticable or unsightly; the architect was absolutely master of his work, and could do what he liked with it. The facilities it offers are quite marvellous in the eyes of the modern practical man when once they are opened to them. I have myself found one of the most practical men I ever met with, who had for years taken the leading management of the business of the greatest builder of our day, though hitherto uninitiated in Gothic construction, almost in ecstasies at finding a difficult problem in vaulting he had been puzzled over for days and making models of in vain, solved in an instant by seeing the absolute liberty of action exercised in a similar case in Westminster Abbey. The old builders themselves perfectly luxuriated in their newly-discovered liberty: not only could they vault spaces of any conceivable plan, every dimension of it varying, and the difficulties increased by the necessity of pushing up windows in its sides in all kinds of difficult positions, but they could make the result so pleasing and apparently so straightforward and natural, that not one observer out of a thousand ever finds out that there was any difficulty to be got over at all. Sometimes, indeed, we find them rejoicing so much in their freedom as to set themselves needless puzzles for the very luxury of solving them. There is a most remarkable instance of this in the crypt under Glasgow Cathedral, where the pillars which support the floor have been placed in a variety of intricate positions for no reason, apparently, but to produce curious