sections of seven smaller ones, each one of which is so narrow that the pillar is felt as the repeated unit instead of the arch; or if the arch be taken, the narrow span prevents it from too great outward thrust. Thirty of the arch-series are on façades of buildings or in structures by themselves, as gates and triumphal arches, where the central arch is larger than the other, thereby emphasizing the middle point and drawing attention to it away from the ends. This centralizing a series or balancing it as a whole may be accomplished in various ways. Two examples make the central arch larger instead of smaller. Six make the end arches smaller while four make them larger. It will be readily seen that just which one of these variations is chosen for the series depends on the function of the series. The central arch is wider, with only one exception, when the series is of arched doors and the central door is the main entrance; while the end arches are more apt to be varied when the series is purely decorative and serves no function. The central balance may be further gained by differences of level. In the decorations of many façades, especially the early Romanesque, rows of arches go obliquely into the point of the roof and by this strong pointing toward the centre create an inward tendency. Six of the illustrations have the central arch accented by decoration; seven have heavier piers around the central and end arches; six have the end arches brought out into a nearer plane which effectually finishes the series. All these examples illustrate the necessary disposition of arches on a flat wall or façade where the series in the field of vision must end suddenly, that is, cannot gradually fade away around a corner. The variety and yet invariability of these devices shows the need felt for some finish at the end, some balance of the whole with the central accent, which need, apparently, is not felt for pillars and lintels. When the arch-series is on a circular structure, such as apses, porches, and the like, even when it does not entirely surround it, as an arena or spire, the regular diminishing of the series on either side, owing to the curve, supplies the finish necessary, and the size and arrangement of the arches need not vary otherwise. Twelve of the examples illustrate such a use of the arch, and although in some cases, Morano Cathedral, Nomantala Church, the arches are continued into the transepts gradually tapering in size, or are modified in size growing narrower from the centre, as in the Bergamo Church, such a treatment is not necessary for finished effect. The difference in proportion resulting from a curved series, or even on arches carried around a square corner (as in porches on Goslar and Braunschweig Rathäuser), where the series is open enough to clearly see its continuity as it runs into the main building, will suffice to make a series finished without modifications of the arch-units.