The Notebooks of Leonardo Da Vinci

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Page 785 of 1565.
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ON THE STRENGTH OF THE ARCH IN ARCHITECTURE.

The stability of the arch built by an architect resides in the tie
and in the flanks.

ON THE POSITION OF THE TIE IN THE ABOVE NAMED ARCH.

The position of the tie is of the same importance at the beginning
of the arch and at the top of the perpendicular pier on which it
rests. This is proved by the 2nd "of supports" which says: that part
of a support has least resistance which is farthest from its solid
attachment; hence, as the top of the pier is farthest from the
middle of its true foundation and the same being the case at the
opposite extremities of the arch which are the points farthest from
the middle, which is really its [upper] attachment, we have
concluded that the tie _a b_ requires to be in such a position as
that its opposite ends are between the four above-mentioned
extremes.

The adversary says that this arch must be more than half a circle,
and that then it will not need a tie, because then the ends will not
thrust outwards but inwards, as is seen in the excess at _a c_, _b
d_. To this it must be answered that this would be a very poor
device, for three reasons. The first refers to the strength of the
arch, since it is proved that the circular parallel being composed
of two semicircles will only break where these semicircles cross
each other, as is seen in the figure _n m;_ besides this it follows
that there is a wider space between the extremes of the semicircle
than between the plane of the walls; the third reason is that the
weight placed to counterbalance the strength of the arch diminishes
in proportion as the piers of the arch are wider than the space
between the piers. Fourthly in proportion as the parts at _c a b d_
turn outwards, the piers are weaker to support the arch above them.
The 5th is that all the material and weight of the arch which are in
excess of the semicircle are useless and indeed mischievous; and
here it is to be noted that the weight placed above the arch will be
more likely to break the arch at _a b_, where the curve of the
excess begins that is added to the semicircle, than if the pier were
straight up to its junction with the semicircle [spring of the
arch].

AN ARCH LOADED OVER THE CROWN WILL GIVE WAY AT THE LEFT HAND AND
RIGHT HAND QUARTERS.

This is proved by the 7th of this which says: The opposite ends of
the support are equally pressed upon by the weight suspended to
them; hence the weight shown at _f_ is felt at _b c_, that is half
at each extremity; and by the third which says: in a support of
equal strength [throughout] that portion will give way soonest which
is farthest from its attachment; whence it follows that _d_ being
equally distant from _f, e_ .....

If the centering of the arch does not settle as the arch settles,
the mortar, as it dries, will shrink and detach itself from the
bricks between which it was laid to keep them together; and as it
thus leaves them disjoined the vault will remain loosely built, and
the rains will soon destroy it.

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