1. The height of the shaft, _a b_; 2. Its diameter, _b c_; 3. The length of slope of bell, _b d_; 4. The inclination of this slope, or angle _c b d_; 5. The depth of abacus, _d e_. For every change in any one of these quantities we have a new proportion of capital: five infinities, supposing change only in one quantity at a time: infinity of infinities in the sum of possible changes. It is, therefore, only possible to note the general laws of change; every scale of pillar, and every weight laid upon it admitting, within certain limits, a variety out of which the architect has his choice; but yet fixing limits which the proportion becomes ugly when it approaches, and dangerous when it exceeds. But the inquiry into this subject is too difficult for the general reader, and I shall content myself with proving four laws, easily understood and generally applicable; for proof of which if the said reader care not, he may miss the next four paragraphs without harm. § XIV. 1. _The more slender the shaft, the greater, proportionally, may be the projection of the abacus._ For, looking back to Fig. XXIII., let the height _a b_ be fixed, the length _d b_, 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 _e 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 h_, _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.