d spire, on the other hand, they found that this distinction of
light and shade was too little marked; it was vapid and effeminate, and
quite without that delicious crispiness of effect which they at once
obtained by cutting off the corners of the square spire, and reducing it
to an octagon. With very rare exceptions, as in the southwest spire of
Chartres Cathedral, this form was always used. Now it will be seen that
a difficulty arises in the beginning, how to unite the octagon of the
spire with the square of the tower. There are four triangular spaces at
the summit of the tower left uncovered by the superstructure; and how
best to treat these, simple as the task may seem, constitutes what may
be called the touchstone of architectural genius in spire-building.
There are several general ways of effecting this, each of them subject
to such modifications, in individual instances, as to give them an
ever-varying character.

Perhaps the earliest method was simply to occupy those triangular spaces
with pyramidal masses of masonry, sloping back against the adjacent
faces of the tower,--an expedient which Nature herself might have
suggested in the first snow-storm. Then they boldly cut the Gordian knot
by shaving off the corners of the tower at the top, thus creating there
an octagonal platform, to which the spire would exactly correspond.
Still oftener they chamfered the spire upwards from the corners of the
tower: in other words, they placed, as it were, a square spire on
their tower, occupying the whole of its summit, and then obtained the
necessary octangularity by shaving off the angles of the spire from the
apex to a certain point near the base, where the cutting was continued
obliquely to the corners of the tower. The latest method was to build
pinnacles on the triangular territory. In such cases the spire usually
stood wholly within the outer boundaries, and parapets assisted to
conceal the first springing of the spire.

The first of these methods is usually considered the most perfect and
beautiful, on account of its simplicity and candor. This is called the
broach; and it is the only form thus far spoken of wherein the tapering
surfaces rise directly from the tower-cornice, without mutilating the
tower or violating the pure outlines of the spire. The heavenward
aspiration, as it were, ascends without effort from the solidity of the
tower. It seems to typify a certain fitness and adaptability to heavenly
things even