It will be seen from the above that the pointed arch was not introduced into Mediæval structures from mere caprice--merely from seeing it elsewhere and taking a fancy to its form,--but from the necessities of construction, from its increased strength and diminished thrust. It was at first used for the main arch only of the greater vault. The same reason soon led to its introduction wherever great weight was to be carried, as under towers, etc.; but for all small arches the semicircle was long retained. I have alluded to the very beautiful theory that it was introduced for the side arches of oblong groins, simply as a means of obtaining arches of equal height with only half the span with those of the main vault. True it is, that, at a later date, it became most useful for this purpose. But a careful study of the monuments in which it is first systematically used clearly shows that its introduction was from _statical_, and neither geometrical nor merely æsthetical motives; for in the face of that theory we find the narrower arch or wall-rib remaining round long after the wider arch had become pointed (Fig. 11). Such is the case in nearly all the earlier of the French transitional churches, as at Noyon and at St. Germain des Pres, and we see the same at Canterbury. In most of these buildings the narrow arch is stilted and the crown of the cross vault raised up as before described, thus losing a part of the clerestory wall, a disadvantage obviated when the pointed arch became more frankly acknowledged. Although, however, the pointed arch was _actually adopted_ from simple necessities of construction, its advantages in all points of view soon became apparent. In an essentially arcuated style it becomes necessary not only to have the command of a form of arch capable of carrying the greatest weights and of requiring the least abutment, but it is essential to have at command an arch of _variable_ proportions. It carries absurdity on the very face of it that, while able to give our piers a greater or a less degree of height at pleasure we should have no such power over the arch they sustain; not to mention the numerous cases in which we have to bring together arches of unequal span, and which nevertheless demand an equal height. The rules of harmony imperatively demand that the arch should be equally capable of modification in its proportions of height to width, with all other features of the architecture. [Illustration: Fig. 12.]