Plates XI and XII. The Dome over a Square Substructure

Plates XI and XII. — These show the circular plan used over a square sub-structure, a combination of two elemental plan forms in a vertical direction of the first importance, developed originally by Byzantine dome builders. Accepted in all countries throughout the Renaissance period, this application of geometry to the solution of a structural problem frequently occurs in modern buildings. The introduction of a domical covering gives expression to a plan by determining a climax in internal perspective, and it may also be relied upon to ensure that an important junction appears dominant.

On Plate XII the diagrams Nos. I to VI show alternative solutions of the domical problem, but plan No. I applies to all of them. The diagrams Nos. VII, VIII and IX illustrate the theory of the “pendentive.” It is a property of the hemisphere that when cut by true planes, circles or segments of circles will be formed. If cut by four vertical planes, representing four supporting arches forming an inscribed square on plan, as shown in No. VII, the curves of four semi-circular arches will be produced, and the portions remaining at the four angles will constitute the pendentives. Thus the full semi-circle occurs only on the diagonal section, and a cross section gives a segment as seen in Nos. II and V. If the hemisphere is also cut horizontally at the level of the crown of the arches, nothing will remain of it beyond the four pendentives which, starting from the four angles of a square, form a circle on plan directly they unite. Upon this circle, once obtained, it is perfectly easy to place another hemisphere and to obtain a compound dome, as shown in No. VIII. The curve of the dome and of its pendentives is then no longer continuous, since the radius of the former is equal to half the side of the square instead of half the diagonal: the one, in fact, is invariably marked off from the other by a cornice or a series of mouldings, as in No. IV. The dome, instead of being placed directly upon the pendentives can now be raised above them upon a cylindrical wall or “drum” of the required height, as in Nos. III and VI. This device facilitates the admission of light without cutting into and weakening the dome about its springing where strength is most required, and the additional height so gained gives a greater prominence to the external design of the domical covering. In the finest examples, prior to this development, the dome though majestic and all sufficient internally had shown externally as little more than an excrescence.