M. Choisy in his most interesting book shows that the chief consideration in the construction of the Byzantine vaults was to avoid wooden centring. With this view we entirely agree, but in the system explained in _L’Art de Bâtir_, the lines of construction would be arrived at by an elaborate system, which required fixed axes to the vaults and either a diagonal centre or a rod revolving in a vertical plane over the diagonal. Then two rods, forming an angle with its apex touching any given point in the diagonal curve and the ends resting on the axis of the vault as a base, revolved as a trammel for that course of the filling. This had to be repeated for a series of points. By the method we have suggested nothing was required except a single template to a fixed angle, the upper arm cut to the curve from the crown of the arch to the crown of the vault; we may suppose this to sweep round the generating arches like a trammel, but practically testing the work with it at the crown, as it gradually grew forward, was doubtless found sufficient (see Fig. 40). Thus the vault surfaces gave the conditions of the problem and the intersections found themselves. We did not notice the curious “curve of inflection” of which M. Choisy speaks; certainly it does not generally exist, although according to _L’Art de Bâtir_ “S. Sophia is the most curious example which remains of this singular conception, where the spirit of Greek logic did not hesitate before anomalies of form” (p. 55). We believe this curve is deduced only by the logic with which M. Choisy’s follows up his method of geometrical projection, which certainly generates such an inflected curve. We cannot say this without at the same time expressing our great admiration for _L’Art de Bâtir_; its freshness of sight, clearness, vitality, and logic are entirely delightful. Strzygowski and Forchheimer[342] follow Choisy’s demonstration; and give an elaborate and analytical explanation of the curve and its points of inflexion. One of the cisterns they say showed the inflected line in the axial sections of the vaults (p. 71). Now the cistern vaults are roughly built and some of them may have settled down; some indeed may have been designed so that the axial section is horizontal for some distance from the walls before the doming is commenced, especially in the long direction of parallelogramic compartments. The essential points are two. Did these vaults grow forward from the walls and the intersections find themselves, or was the curve of intersection first designed? Are horizontal sections through the intersection of two vault surfaces just above the springing obtuse or acute? The vaults at S. Sophia have the angles of intersection so obtuse that this first drew our attention to the subject.