Thus in Fig. 40 the dome springing out of the angle requires the height _a_, the radius being equal to half the diameter; but it was wished to flatten this to _b_, and yet for the vault to rise everywhere from the arched line _e_, _c_. Now if the vault conforms to the surfaces generated by the revolution of the arc _d_, _f_, _b_, about the axis _o_, _d_, intersecting with a similarly generated surface at right angles, we get a mean between the domed and cylindrical forms--a domical vault. The intersections, instead of being everywhere square on plan as at _x_, _x_, and rising just to the crown of the vault, as would be the case with cylindrical penetrations, will be obtuse as at _i_, _i_, and not rising so high will practically leave a large concave surface unbroken at the crown of the vault. This is the principle of the vaults of S. Sophia; the gradations being gentle and the means less obvious, the forms are more like those found in nature, and the result is extremely beautiful. The forms are further softened by every edge of arch and vault being rounded, so that the mosaic completely envelops the whole like a vast embroidered gold tissue. There would be no difficulty in construction, for the vault falls everywhere on an arch in the angle _e_, _f_, _b_ that is in planes which are radii to the arch. The vaulting of the narthex is made up of a series of compartments, much narrower than the span, divided by plain arched bands. To meet the requirements of such oblong spaces two gauges would be needed. The “winding” of the lines of intersection was not to be feared, as they were so soon lost in the more domical surface of the upper part of the vault. After the above was written we found the geometrical and practical construction of these vaults explained in _L’Art de Bâtir chez les Byzantins_, in a manner which differs from that here given. M. Choisy’s method is first of all to design the curve of the intersection over the diagonal of the plan as a segment of a circle: then he considers all sections of each compartment of the vault, taken parallel to its arch, and therefore perpendicular to its axis, to be also segments of circles springing from a series of points on the diagonals, their centres being on the axis of each vault. We cannot agree with this, for, although theoretically the vault so conceived differs immaterially from the solution we have proposed, yet practically its erection would be full of difficulty. M. Choisy’s method is that proposed by M. Viollet-le-Duc for the later Romanesque vaults, in which, the materials being poor rubble, centring must have been required. In these Viollet-le-Duc thinks that diagonal centres were used, and then planks were placed from them to the generating arches, and the additional height of a domical vault made up by a layer of earth. It is to be noticed that diagonal centres in this case almost immediately produced diagonal stone ribs.