perplexities in the vaulting and create strange problems, for the mere pleasure to be derived from their solution and the beauty of the puzzle when solved.[59] It has been argued that the Gothic vault is less refined than some of the previous forms, because less strictly mathematical; that a refined system of construction should in all cases possess an exact mathematical solution, though the builder may, when once master of the true theory, depart from it in execution; that the work, in short, though irregular in execution, should be perfect and mathematically accurate in its theoretical type. I agree with this doctrine in the main; but I hold that the Gothic vault complies with its conditions. The square groined vault, with semicircular arches, is perfect in its theory, and gives elliptical arches for its arris lines. The same, if vaulted with the pointed arch, is equally true in theory, for the diagonal ribs may be pointed arches, formed each of portions of two ellipses. The oblong vault, again, is perfect if the wide arch is a semicircle, the narrow one a vertical semi-ellipse, and the arrises horizontal semi-ellipses of the same height; but the ancients generally chose to stilt the narrow arch instead of using the vertical ellipse, and by doing so threw the diagonal arris out of the plane and out of shape; but the theoretical form remained, nevertheless, perfect. In like manner, if the same figure be vaulted across its widest span by a pointed vault, and if the narrow vault have a pointed arch composed of two portions of ellipses, and the intersections be of the same figure as resulting geometrically from the intersection of the two vaults, the theoretical form is perfect. Now, if in either case the architect thinks the elliptical pointed arches inferior in beauty to those composed of parts of circles, and by using ribs finds himself enabled to throw the error resulting from the substitution of the latter form into the vaulted surfaces where it will be invisible, surely he is only using that discretionary power of introducing irregularities upon a perfect theory which is claimed as his right; and this is exactly what the Gothic architects introduced.