Many early windows and window-like openings--such as those with the triforium galleries of churches[37]--are divided into two or more portions by pillars and small arches in the inner plane or order; the outer order or orders embracing the whole, and the plane of the inner or functional order forming a second wall-space over the heads of these subordinate arches. Thus the triforium at St. Bartholomew’s is divided into _four_ subordinate arches. This window plane, as it may be called, is often ornamented in different ways, and occasionally even in Norman work, is pierced. At a later stage this piercing becomes systematic, and has received the name of “plate tracery,” the _plate_ being the window plane or order. It is simply the piercing of this plane of the functional order of the window arch; and as it is clear that this piercing developed itself into window-tracery, so is it equally manifest that the plane thus pierced originated in the division of the window-arch into receding orders; and, consequently, that traceried windows were a natural result of the conditions of arcuated architecture. The subject of windows being quite sufficient to occupy a separate lecture, I leave it for the present to go on with the more elementary questions resulting from the conditions I laid down at the outset. You will have noticed that, having in those prescribed conditions divided my subject into two great natural heads,--viz. the _arching_ over of openings in walls, and between piers; and the _vaulting_ over of the spaces enclosed by walls or ranges of piers,--I have hitherto dealt exclusively with the former; and that, as the forms of piers and clustered columns are influenced as much by the requirements of the _vaulting_ as of the _arches_ they have to support, I have been obliged to leave my description of their forms imperfect; and as it is my wish to treat of vaulting as systematically as I am able, I must beg you to allow this incompleteness to remain till it is incidentally filled up as we proceed with this, the second great elementary division of arcuated architecture. It must be clear, even on the most superficial glance, that the vaulting over of extended areas is a matter of far greater intricacy, and requiring vastly more thought and contrivance, than the mere arching over of an opening in a wall; and though its primary elements are simple, I must beg you to follow me over easy ground,--and ground already trodden in my previous lectures,--because these early and simple steps are needful to the due appreciation of the more advanced and complex ones which we shall presently have to consider. The simplest elements of vaulting are--first, the covering over of a rectangular space enclosed between parallel walls by means of a semi-cylindrical vault, usually known as a “_barrel vault_;” and secondly, the covering over a space enclosed by a circular wall, by means of a hemispherical vault or _dome_. [Illustration: Fig. 313.]