As will be explained in the sequel, we find that, by the application of the formula of simple ratios, we are enabled to fix the dimensions of almost every part of the Mausoleum with almost absolute certainty; and at the same time it is found that the Mausoleum is one of the most complete and interesting examples of a building designed wholly on a scheme of simple definite ratios. Thus the very science which assists materially in solving the problem, is at the same time illustrated and confirmed by the discoveries it aids in making. The first attempt to explain the peculiarities of buildings by a scheme of definite ratios seems to be that expounded by Cæsar Cæsarini, in his edition of Vitruvius, published in 1521. In this work he shows by diagrams how a series of equilateral triangles explains all the dimensions and peculiarities of design in Milan Cathedral; and in this he probably was right, for, being a foreign work, it is very probable that the Italian architects, not understanding the true principles of the art, squeezed the design into this formal shape and so spoiled it. The success of this attempt of Cæsarini, however, has induced numberless other architects to apply the same principle to other Gothic Cathedrals, but without success in a single instance. Those which approach nearest to it are such buildings as Westminster Abbey,—a French church built in England; Cologne Cathedral, which is a French example in Germany; and in like manner all foreign examples approximate to definite proportions; but it may safely be asserted that no truly native example of Gothic art was so arranged. It has, however, long been suspected that the Greeks proceeded on a totally different principle; but materials did not exist for a satisfactory elucidation of the question till Mr. Penrose published his exquisite survey of the Parthenon and other buildings at Athens made for the Society of Dilettanti, and Mr. Cockerell the result of his explorations at Bassæ and Egina. In the first-named work, its author pointed out with sufficient clearness some of the principal ratios of that celebrated building, which his survey enabled him to verify, and for others he supplied dimensions which for completeness and accuracy left nothing to be desired. With these new materials, Mr. Watkiss Lloyd undertook the investigation, and by a long and careful series of comparisons he has proved that the time-honoured doctrine of the Vitruvian school—that the lower diameter of a column was the modulus of every other part of a building—had no place in Greek art; on the contrary, that every part of a Greek building was proportioned to those parts in juxtaposition or analogy to it, in some such ratio as 3 to 4, 4 to 5, 5 to 6, and so on,—not by accident, but by careful study; and the whole design was evolved from a nexus of proportions as ingenious in themselves as they were harmonious in their result.