In the Parthenon, for instance, he found that the entire building is set out with the minutest accuracy, by the application of a few ratios which involve no higher number than 16, and in no case have a higher difference between them than 5. The greatest ingenuity and refinement were exercised in embracing the entire design in a network of proportional relations, in such a way that every division had a special dependence upon some other that was particularly contrasted or connected with it; and at the same time every member was implicated in more than one such comparison by what might seem happy accident, were it not that on trial it is proved how much study is required to effect such a result. At the same time, when the clue is once gained, it is easy to see how study was competent to effect it. Among the proportional applications affecting the present subject, which may be considered axiomatic are these:— The establishment of proportions of low numbers between— 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.