1. The length and breadth of the basement, either upon its upper or lower step, or both.
2. The breadth of front and full height of the building; in most cases, also, the length of flank and full height.
3. The length and breadth of any other conspicuous rectangle, such as in the present case would be the plans of the cella, of the pyramid, of the base or pedestal of the statue.
4. The division of the grand height of the structure into a pair of well-contrasted parts, having a ratio to each other of which the terms differ by unity, as 2 to 3, 3 to 4, &c. The further subdivision of these parts is effected again by definite proportions, and a favourite scheme here, as elsewhere, is for an intermediate section of a vertical line to have a simple proportion to the joint dimensions of sections above and below it, these upper and lower sections being then proportioned independently. Thus in the entablature of the Mausoleum the frieze is just half the joint height of architrave and cornice; that is, one-third of the height is given to the frieze.
5. The lower diameter of the Ionic column has usually a ratio to the upper diameter expressible in low numbers with a difference of unity. In the Mausoleum the ratio is 5 to 6, the same as at Priene. In the columns at Branchidæ, which were more than double the height, the difference is slighter, viz., 7 to 8.
6. The height of the column is usually, but by no means invariably, commensurable with the lower diameter, or at least semi-diameter, and the columns are spaced in one or other of the schemes that supply a symmetry with their height; that is to say, the height of the column will be found invariably to measure off a space laterally that coincides with centre and centre of columns, centre and margin, or margin and margin of the foot of the shaft or base. This symmetry was of more importance than the commensurability of height by diameter.
7. In the architecture of temples, at least, the height either of the shaft or of the full column compares with the complementary height of the order, or of the front, in a ratio of which the terms differ by unity, and the larger term pertains to the columns. For example, the height of the Parthenon column is two parts out of three into which the full height of the order at the flank of the temple is divisible; the remaining part being divided between the entablature and the steps.[8]
Mr. Lloyd first publicly explained his theory of the system of proportions used in Greek architecture in a lecture he delivered at the Institute of British Architects in June, 1859, and he afterwards added an appendix to Mr. Cockerell’s work on Egina and Bassæ, explaining specially the proportions of those temples; but the full development of his views, and particularly their relation to the Parthenon, which it appears surpassed all known works in refined and exact application of the system, still unfortunately remains in manuscript.
The more direct application of this theory to the design of the Mausoleum will be explained as we proceed, but in the meanwhile it may be asserted that without it many of the dimensions of this celebrated monument might for ever have remained matters of dispute. With its assistance there is scarcely one that may not be ascertained with almost absolute certainty.
Another and quite distinct set of ratios was discovered by Colonel Howard Vyse and his architect Mr. Perring, in their explorations of the Pyramids of Egypt. They found, for instance, in the Great Pyramid that the distance