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.

[Illustration: Fig. XXIV.]

Sec. XV. 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
overhang its base a foot or two, as you may see any day in the gravelly
banks of the lanes of Hampstead: but make the bank of gravel, equally
loose, six hundred feet high, and see if you can get it to overhang a
hundred or two! much more if there be weight above it increased in the
same proportion. Hence, let any capital be given, whose projection is
just safe, and no more, on its existing scale; increase its proportions
every way equally, though ever so little, and it is unsafe; diminish
them equally, and it becomes safe in the exact degree of the diminution.

Let, then, the quantity _e d_, and angle _d b c_, at A of Fig. XXIII.,
be invariable, and let the length _d b_ vary: then we shall have such a
series of forms as may be represented by _a_, _b_, _c_, Fig. X