ON THE STRENGTH AND NATURE OF ARCHES, AND WHERE THEY ARE STRONG OR
WEAK; AND THE SAME AS TO COLUMNS.

That part of the arch which is nearer to the horizontal offers least
resistance to the weight placed on it.

When the triangle _a z n_, by settling, drives backwards the 2/3 of
each 1/2 circle that is _a s_ and in the same way _z m_, the reason
is that _a_ is perpendicularly over _b_ and so likewise _z_ is above
_f_.

Either half of an arch, if overweighted, will break at 2/3 of its
height, the point which corresponds to the perpendicular line above
the middle of its bases, as is seen at _a b_; and this happens
because the weight tends to fall past the point _r_.--And if,
against its nature it should tend to fall towards the point _s_ the
arch _n s_ would break precisely in its middle. If the arch _n s_
were of a single piece of timber, if the weight placed at _n_ should
tend to fall in the line _n m_, the arch would break in the middle
of the arch _e m_, otherwise it will break at one third from the top
at the point a because from _a_ to _n_ the arch is nearer to the
horizontal than from _a_ to _o_ and from _o_ to _s_, in proportion
as _p t_ is greater than _t n_, _a o_ will be stronger than _a n_
and likewise in proportion as _s o_ is stronger than _o a_, _r p_
will be greater than _p t_.

The arch which is doubled to four times of its thickness will bear
four times the weight that the single arch could carry, and more in
proportion as the diameter of its thickness goes a smaller number of
times into its length. That is to say that if the thickness of the
single arch goes ten times into its length, the thickness of the
doubled arch will go five times into its length. Hence as the
thickness of the double arch goes only half as many times into its
length as that of the single arch does, it is reasonable that it
should carry half as much more weight as it would have to carry if
it were in direct proportion to the single arch. Hence as this
double arch has 4 times the thickness of the single arch, it would
seem that it ought to bear 4 times the weight; but by the above rule
it is shown that it will bear exactly 8 times as much.

THAT PIER, WHICH is CHARGED MOST UNEQUALLY, WILL SOONEST GIVE WAY.

The column _c b_, being charged with an equal weight, [on each side]
will be most durable, and the other two outward columns require on
the part outside of their centre as much pressure as there is inside
of their centre, that is, from the centre of the column, towards the
middle of the arch.

Arches which depend on chains for their support will not be very
durable.

THAT ARCH WILL BE OF LONGER DURATION WHICH HAS A GOOD ABUTMENT
OPPOSED TO ITS THRUST.

The arch itself tends to fall. If the arch be 30 braccia and the
interval between the walls which carry it be 20, we know that 30
cannot pass through the 20 unless 20 becomes likewise 30. Hence the
arch being crushed by the excess of weight, and the walls offering
insufficient resistance, part, and afford room between them, for the
fall of the arch.

But if you do not wish to strengthen the arch with an iron tie you
must give it such abutments as can resist the thrust; and you can do
this thus: fill up the spandrels _m n_ with stones, and direct the
lines of the joints between them to the centre of the circle of the
arch, and the reason why this makes the arch durable is this. We
know very well that if the arch is loaded with an excess of weight
above its quarter as _a b_, the wall _f g_ will be thrust outwards
because the arch would yield in that direction; if the other quarter
_b c_ were loaded, the wall _f g_ would be thrust inwards, if it
were not for the line of stones _x y_ which resists this.