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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.
638
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