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.