The Notebooks of Leonardo Da Vinci — Complete
Arched cracks, wide at the top and narrow below are found in
walled-up doors, which shrink more in their height than in their
breadth, and in proportion as their height is greater than their
width, and as the joints of the mortar are more numerous in the
height than in the width.
The crack diminishes less in _r o_ than in _m n_, in proportion as
there is less material between _r_ and _o_ than between _n_ and _m_.
Any crack made in a concave wall is wide below and narrow at the
top; and this originates, as is here shown at _b c d_, in the side
figure.
1. That which gets wet increases in proportion to the moisture it
imbibes.
2. And a wet object shrinks, while drying, in proportion to the
amount of moisture which evaporates from it.
[Footnote: The text of this passage is reproduced in facsimile on
Pl. CVI to the left. L. 36-40 are written inside the sketch No. 2.
L. 41-46 are partly written over the sketch No. 3 to which they
refer.]
773.
OF THE CAUSES OF FISSURES IN [THE WALLS OF] PUBLIC AND PRIVATE
BUILDINGS.
The walls give way in cracks, some of which are more or less
vertical and others are oblique. The cracks which are in a vertical
direction are caused by the joining of new walls, with old walls,
whether straight or with indentations fitting on to those of the old
wall; for, as these indentations cannot bear the too great weight of
the wall added on to them, it is inevitable that they should break,
and give way to the settling of the new wall, which will shrink one
braccia in every ten, more or less, according to the greater or
smaller quantity of mortar used between the stones of the masonry,
and whether this mortar is more or less liquid. And observe, that
the walls should always be built first and then faced with the
stones intended to face them. For, if you do not proceed thus, since
the wall settles more than the stone facing, the projections left on
the sides of the wall must inevitably give way; because the stones
used for facing the wall being larger than those over which they are
laid, they will necessarily have less mortar laid between the
joints, and consequently they settle less; and this cannot happen if
the facing is added after the wall is dry.
_a b_ the new wall, _c_ the old wall, which has already settled; and
the part _a b_ settles afterwards, although _a_, being founded on
_c_, the old wall, cannot possibly break, having a stable foundation
on the old wall. But only the remainder _b_ of the new wall will
break away, because it is built from top to bottom of the building;
and the remainder of the new wall will overhang the gap above the
wall that has sunk.
774.
A new tower founded partly on old masonry.
775.
OF STONES WHICH DISJOIN THEMSELVES FROM THEIR MORTAR.
Stones laid in regular courses from bottom to top and built up with
an equal quantity of mortar settle equally throughout, when the
moisture that made the mortar soft evaporates.
By what is said above it is proved that the small extent of the new
wall between _A_ and _n_ will settle but little, in proportion to
the extent of the same wall between _c_ and _d_. The proportion will
in fact be that of the thinness of the mortar in relation to the
number of courses or to the quantity of mortar laid between the
stones above the different levels of the old wall.
[Footnote: See Pl. CV, No. 1. The top of the tower is wanting in
this reproduction, and with it the letter _n_ which, in the
original, stands above the letter _A_ over the top of the tower,
while _c_ stands perpendicularly over _d_.]
776.
This wall will break under the arch _e f_, because the seven whole
square bricks are not sufficient to sustain the spring of the arch
placed on them. And these seven bricks will give way in their middle
exactly as appears in _a b_. The reason is, that the brick _a_ has
above it only the weight _a k_, whilst the last brick under the arch
has above it the weight _c d x a_.
_c d_ seems to press on the arch towards the abutment at the point
_p_ but the weight _p o_ opposes resistence to it, whence the whole
pressure is transmitted to the root of the arch. Therefore the foot
of the arch acts like 7 6, which is more than double of _x z_.
II.
ON FISSURES IN NICHES.
777.
ON FISSURES IN NICHES.
An arch constructed on a semicircle and bearing weights on the two
opposite thirds of its curve will give way at five points of the
curve. To prove this let the weights be at _n m_ which will break
the arch _a_, _b_, _f_. I say that, by the foregoing, as the
extremities _c_ and _a_ are equally pressed upon by the thrust _n_,
it follows, by the 5th, that the arch will give way at the point
which is furthest from the two forces acting on them and that is the
middle _e_. The same is to be understood of the opposite curve, _d g
b_; hence the weights _n m_ must sink, but they cannot sink by the
7th, without coming closer together, and they cannot come together
unless the extremities of the arch between them come closer, and if
these draw together the crown of the arch must break; and thus the
arch will give way in two places as was at first said &c.
I ask, given a weight at _a_ what counteracts it in the direction
_n_ _f_ and by what weight must the weight at _f_ be counteracted.
778.
ON THE SHRINKING OF DAMP BODIES OF DIFFERENT THICKNESS AND WIDTH.
The window _a_ is the cause of the crack at _b_; and this crack is
increased by the pressure of _n_ and _m_ which sink or penetrate
into the soil in which foundations are built more than the lighter
portion at _b_. Besides, the old foundation under _b_ has already
settled, and this the piers _n_ and _m_ have not yet done. Hence the
part _b_ does not settle down perpendicularly; on the contrary, it
is thrown outwards obliquely, and it cannot on the contrary be
thrown inwards, because a portion like this, separated from the main
wall, is larger outside than inside and the main wall, where it is
broken, is of the same shape and is also larger outside than inside;
therefore, if this separate portion were to fall inwards the larger
would have to pass through the smaller--which is impossible. Hence
it is evident that the portion of the semicircular wall when
disunited from the main wall will be thrust outwards, and not
inwards as the adversary says.
When a dome or a half-dome is crushed from above by an excess of
weight the vault will give way, forming a crack which diminishes
towards the top and is wide below, narrow on the inner side and wide
outside; as is the case with the outer husk of a pomegranate,
divided into many parts lengthwise; for the more it is pressed in
the direction of its length, that part of the joints will open most,
which is most distant from the cause of the pressure; and for that
reason the arches of the vaults of any apse should never be more
loaded than the arches of the principal building. Because that which
weighs most, presses most on the parts below, and they sink into the
foundations; but this cannot happen to lighter structures like the
said apses.
[Footnote: The figure on Pl. CV, No. 4 belongs to the first
paragraph of this passage, lines 1-14; fig. 5 is sketched by the
side of lines l5--and following. The sketch below of a pomegranate
refers to line 22. The drawing fig. 6 is, in the original, over line
37 and fig. 7 over line 54.]
Which of these two cubes will shrink the more uniformly: the cube
_A_ resting on the pavement, or the cube _b_ suspended in the air,
when both cubes are equal in weight and bulk, and of clay mixed with
equal quantities of water?
The cube placed on the pavement diminishes more in height than in
breadth, which the cube above, hanging in the air, cannot do. Thus
it is proved. The cube shown above is better shown here below.
The final result of the two cylinders of damp clay that is _a_ and
_b_ will be the pyramidal figures below _c_ and _d_. This is proved
thus: The cylinder _a_ resting on block of stone being made of clay
mixed with a great deal of water will sink by its weight, which
presses on its base, and in proportion as it settles and spreads all
the parts will be somewhat nearer to the base because that is
charged with the whole weight.
III.
ON THE NATURE OF THE ARCH.
779.
WHAT IS AN ARCH?
The arch is nothing else than a force originated by two weaknesses,
for the arch in buildings is composed of two segments of a circle,
each of which being very weak in itself tends to fall; but as each
opposes this tendency in the other, the two weaknesses combine to
form one strength.
OF THE KIND OF PRESSURE IN ARCHES.
As the arch is a composite force it remains in equilibrium because
the thrust is equal from both sides; and if one of the segments
weighs more than the other the stability is lost, because the
greater pressure will outweigh the lesser.
OF DISTRIBUTING THE PRESSURE ABOVE AN ARCH.
Next to giving the segments of the circle equal weight it is
necessary to load them equally, or you will fall into the same
defect as before.
WHERE AN ARCH BREAKS.
An arch breaks at the part which lies below half way from the
centre.
SECOND RUPTURE OF THE ARCH.
If the excess of weight be placed in the middle of the arch at the
point _a_, that weight tends to fall towards _b_, and the arch
breaks at 2/3 of its height at _c e_; and _g e_ is as many times
stronger than _e a_, as _m o_ goes into _m n_.
ON ANOTHER CAUSE OF RUIN.
The arch will likewise give way under a transversal thrust, for when
the charge is not thrown directly on the foot of the arch, the arch
lasts but a short time.
780.
ON THE STRENGTH OF THE ARCH.
The way to give stability to the arch is to fill the spandrils with
good masonry up to the level of its summit.
ON THE LOADING OF ROUND ARCHES.
ON THE PROPER MANNER OF LOADING THE POINTED ARCH.
ON THE EVIL EFFECTS OF LOADING THE POINTED ARCH DIRECTLY ABOVE ITS
CROWN.
ON THE DAMAGE DONE TO THE POINTED ARCH BY THROWING THE PRESSURE ON
THE FLANKS.
An arch of small curve is safe in itself, but if it be heavily
charged, it is necessary to strengthen the flanks well. An arch of a
very large curve is weak in itself, and stronger if it be charged,
and will do little harm to its abutments, and its places of giving
way are _o p_.
[Footnote: Inside the large figure on the righi is the note: _Da
pesare la forza dell' archo_.]
781.
ON THE REMEDY FOR EARTHQUAKES.
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.
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.