CONSTRUCTION IX. THE CAPITAL 143

the angle d b c, and the depth d e. Let the single quantity b c be variable; let B be a capital and shaft which are found to be perfectly safe in proportion to the weight they bear, and let the weight be equally distributed over the whole of the abacus. Then this weight may be represented by any number of equal divisions, suppose four, as l, m, n, r, of brickwork above, of which each division is one fourth of the whole weight; and let this weight be placed in the most trying way on the abacus, that is to say, let the masses l and r be detached from m and n, and bear with their full weight on the outside of the capital. We assume, in B, that the width of abacus c f is twice as great as that of the shaft, b c, and on these conditions we assume the capital to be safe.

But b c is allowed to be variable. Let it become b₂ c₂ at C, which is a length representing about the diameter of a shaft containing half the substance of the shaft B, and, therefore, able to sustain not more than half the weight sustained by B. But the slope b d and depth d e remaining unchanged, we have the capital of C, which we are to load with only half the weight of l, m, n, r, i.e. with l and r alone. Therefore the weight of l and r, now represented by the masses l₂, r₂, is distributed over the whole of the capital. But the weight r was adequately supported by the projecting piece of the first capital h f c; much more is it now adequately supported by i h₂ f₂ c₂. Therefore, if the capital of B was safe, that of C is more than safe. Now in B the length e f was only twice b c; but in C, e₂ f₂ will be found more than twice that of b₂ c₂. Therefore, the more slender the shaft, the greater may be the proportional excess of the abacus over its diameter.

§ 15. (2.) The smaller the scale of the building, the greater may be the excess of the abacus over the diameter of the shaft. This principle requires, I think, no very lengthy proof; the reader can understand at once that the cohesion and strength of stone which can sustain a small projecting mass, will not sustain a vast one overhanging in the same proportion. A bank even of loose earth, six feet high, will sometimes

[Version 0.04: March 2008]