Fig. XXIII.]

Sec. XIII. All the five capitals (which are indeed five orders with
legitimate distinction; very different, however, from the five orders as
commonly understood) may be represented by the same profile, a section
through the sides of _a_, _b_, _d_, and _e_, or through the angles of
_c_, Fig. XXII. This profile we will put on the top of a shaft, as at A,
Fig. XXIII., which shaft we will suppose of equal diameter above and
below for the sake of greater simplicity: in this simplest condition,
however, relations of proportion exist between five quantities, any one
or any two, or any three, or any four of which may change, irrespective
of the others. These five quantities are:

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.

Sec. 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_,