The Notebooks of Leonardo Da Vinci (54 page)

[Footnote: Inside the large figure on the righi is the note:
Da
pesare la forza dell' archo
.]

781.

The arch which throws its pressure perpendicularly on the abutments
will fulfil its function whatever be its direction, upside down,
sideways or upright.

The arch will not break if the chord of the outer arch does not
touch the inner arch. This is manifest by experience, because
whenever the chord
a o n
of the outer arch
n r a
approaches the
inner arch
x b y
the arch will be weak, and it will be weaker in
proportion as the inner arch passes beyond that chord. When an arch
is loaded only on one side the thrust will press on the top of the
other side and be transmitted to the spring of the arch on that
side; and it will break at a point half way between its two
extremes, where it is farthest from the chord.

782.

A continuous body which has been forcibly bent into an arch, thrusts
in the direction of the straight line, which it tends to recover.

783.

In an arch judiciously weighted the thrust is oblique, so that the
triangle
c n b
has no weight upon it.

784.

I here ask what weight will be needed to counterpoise and resist the
tendency of each of these arches to give way?

[Footnote: The two lower sketches are taken from the MS. S. K. M.
III, 10a; they have there no explanatory text.]

785.

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

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].

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.

786.

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.

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.

787.

Here it is shown how the arches made in the side of the octagon
thrust the piers of the angles outwards, as is shown by the line
h
c
and by the line
t d
which thrust out the pier
m
; that is they
tend to force it away from the centre of such an octagon.

788.

An Experiment to show that a weight placed on an arch does not
discharge itself entirely on its columns; on the contrary the
greater the weight placed on the arches, the less the arch transmits
the weight to the columns. The experiment is the following. Let a
man be placed on a steel yard in the middle of the shaft of a well,
then let him spread out his hands and feet between the walls of the
well, and you will see him weigh much less on the steel yard; give
him a weight on the shoulders, you will see by experiment, that the
greater the weight you give him the greater effort he will make in
spreading his arms and legs, and in pressing against the wall and
the less weight will be thrown on the steel yard.

789.

The first and most important thing is stability.

As to the foundations of the component parts of temples and other
public buildings, the depths of the foundations must bear the same
proportions to each other as the weight of material which is to be
placed upon them.

Every part of the depth of earth in a given space is composed of
layers, and each layer is composed of heavier or lighter materials,
the lowest being the heaviest. And this can be proved, because these
layers have been formed by the sediment from water carried down to
the sea, by the current of rivers which flow into it. The heaviest
part of this sediment was that which was first thrown down, and so
on by degrees; and this is the action of water when it becomes
stagnant, having first brought down the mud whence it first flowed.
And such layers of soil are seen in the banks of rivers, where their
constant flow has cut through them and divided one slope from the
other to a great depth; where in gravelly strata the waters have run
off, the materials have, in consequence, dried and been converted
into hard stone, and this happened most in what was the finest mud;
whence we conclude that every portion of the surface of the earth
was once at the centre of the earth, and _vice_versa_ &c.

790.

The heaviest part of the foundations of buildings settles most, and
leaves the lighter part above it separated from it.

And the soil which is most pressed, if it be porous yields most.

You should always make the foundations project equally beyond the
weight of the walls and piers, as shown at
m a b
. If you do as
many do, that is to say if you make a foundation of equal width from
the bottom up to the surface of the ground, and charge it above with
unequal weights, as shown at
b e
and at
e o
, at the part of the
foundation at
b e
, the pier of the angle will weigh most and
thrust its foundation downwards, which the wall at
e o
will not
do; since it does not cover the whole of its foundation, and
therefore thrusts less heavily and settles less. Hence, the pier
b
e
in settling cracks and parts from the wall
e o
. This may be
seen in most buildings which are cracked round the piers.

791.

The window
a
is well placed under the window
c
, and the window
b
is badly placed under the pier
d
, because this latter is
without support and foundation; mind therefore never to make a break
under the piers between the windows.

792.

A pillar of which the thickness is increased will gain more than its
due strength, in direct proportion to what its loses in relative
height.

If a pillar should be nine times as high as it is broad—that is to
say, if it is one braccio thick, according to rule it should be nine
braccia high—then, if you place 100 such pillars together in a mass
this will be ten braccia broad and 9 high; and if the first pillar
could carry 10000 pounds the second being only about as high as it
is wide, and thus lacking 8 parts of its proper length, it, that is
to say, each pillar thus united, will bear eight times more than
when disconnected; that is to say, that if at first it would carry
ten thousand pounds, it would now carry 90 thousand.

793.

That angle will offer the greatest resistance which is most acute,
and the most obtuse will be the weakest.

[Footnote: The three smaller sketches accompany the text in the
original, but the larger one is not directly connected with it. It
is to be found on fol. 89a of the same Manuscript and there we read
in a note, written underneath,
coverchio della perdicha del
castello
(roof of the flagstaff of the castle),—Compare also Pl.
XCIII, No. 1.]

794.

If the beams and the weight
o
are 100 pounds, how much weight will
be wanted at
ae
to resist such a weight, that it may not fall
down?