The Notebooks of Leonardo Da Vinci (13 page)

This is self evident only in the middle diagram; but it is equally
true of the side figures if we conceive of the lines 4
f
,
x n v
m
,
y l k v
, and 4
e
, as prolonged beyond the semicircle of the
horizon.] and is in a straight line with the centre of the primary
shadow, with the centre of the body casting it and of the derivative
light and with the centre of the window and, finally, with the
centre of that portion of the source of light which is the celestial
hemisphere,
y h
is the centre of the derived shade,
l h
of the
primary shadow,
l
of the body throwing it,
l k
of the derived
light,
v
is the centre of the window,
e
is the final centre of
the original light afforded by that portion of the hemisphere of the
sky which illuminates the solid body.

[Footnote: Compare the diagram on Pl. IV, No. 3. In the original
this drawing is placed between lines 3 and 22; the rest, from line 4
to line 21, is written on the left hand margin.]

174.

You will find that the proportion of the diameter of the derived
shadow to that of the primary shadow will be the same as that
between the darkness of the primary shadow and that of the derived
shadow.

[Footnote 6: Compare No. 177.] Let
a b
be the diameter of the
primary shadow and
c d
that of the derived shadow, I say that
a
b
going, as you see, three times into
d c
, the shadow
d c
will
be three times as light as the shadow
a b
. [Footnote 8: Compare
No. 177.]

If the size of the illuminating body is larger than that of the
illuminated body an intersection of shadow will occur, beyond which
the shadows will run off in two opposite directions as if they were
caused by two separate lights.

On the relative intensity of derived shadows (175-179).

175.

The derived shadow is stronger in proportion as it is nearer to its
place of origin.

176.

Shadows fade and are lost at long distances because the larger
quantity of illuminated air which lies between the eye and the
object seen tints the shadow with its own colour.

177.

a b
will be darker than
c d
in proportion as
c d
is broader
than
a b
.

[Footnote: In the original MS. the word
lume
(light) is written at
the apex of the pyramid.]

178.

It can be proved why the shadow
o p c h
is darker in proportion as
it is nearer to the line
p h
and is lighter in proportion as it is
nearer to the line
o c
. Let the light
a b
, be a window, and let
the dark wall in which this window is, be
b s
, that is, one of the
sides of the wall.

Then we may say that the line
p h
is darker than any other part of
the space
o p c h
, because this line faces the whole surface in
shadow of [Footnote: In the original the diagram is placed between
lines 27 and 28.] the wall
b s
. The line
o c
is lighter than the
other part of this space
o p c h
, because this line faces the
luminous space
a b
.

Where the shadow is larger, or smaller, or equal the body which
casts it.

[First of the character of divided lights. [Footnote 14:
lumi
divisi
. The text here breaks off abruptly.]

The shadow
f r c h
is under such conditions as that where it is
farthest from its inner side it loses depth in proportion. To prove
this:

Let
d a
, be the light and
f n
the solid body, and let
a e
be
one of the side walls of the window that is
d a
. Then I
say—according to the 2nd [proposition]: that the surface of any
body is affected by the tone of the objects surrounding it,—that
the side
r c
, which faces the dark wall
a e
must participate of
its darkness and, in the same way that the outer surface which faces
the light
d a
participates of the light; thus we get the outlines
of the extremes on each side of the centre included between them.]

This is divided into four parts. The first the extremes, which
include the compound shadow, secondly the compound shadow between
these extremes.

179.

If it were the whole of the light that caused the shadows beyond the
bodies placed in front of it, it would follow that any body much
smaller than the light would cast a pyramidal shadow; but experience
not showing this, it must be the centre of the light that produces
this effect.

[Footnote: The diagram belonging to this passage is between lines 4
and 5 in the original. Comp. the reproduction Pl. IV, No. 4. The
text and drawing of this chapter have already been published with
tolerable accuracy. See M. JORDAN: "
Das Malerbuch des Leonardo da
Vinci
". Leipzig 1873, P. 90.]

Let
a b
be the width of the light from a window, which falls on a
stick set up at one foot from
a c
[Footnote 6:
bastone
(stick).
The diagram has a sphere in place of a stick.]. And let
a d
be the
space where all the light from the window is visible. At
c e
that
part of the window which is between
l b
cannot be seen. In the
same way
a m
cannot be seen from
d f
and therefore in these two
portions the light begins to fail.

Shadow as produced by two lights of different size (180. 181).

180.

A body in light and shade placed between two equal lights side by
side will cast shadows in proportion to the [amount of] light. And
the shadows will be one darker than the other in proportion as one
light is nearer to the said body than the other on the opposite
side.

A body placed at an equal distance between two lights will cast two
shadows, one deeper than the other in proportion, as the light which
causes it is brighter than the other.

[Footnote: In the MS. the larger diagram is placed above the first
line; the smaller one between l. 4 & 5.]

181.

A light which is smaller than the body it illuminates produces
shadows of which the outlines end within [the surface of] the body,
and not much compound shadow; and falls on less than half of it. A
light which is larger than the body it illuminates, falls on more
than half of it, and produces much compound shadow.

The effect of light at different distances.

182.

A body placed between 2 equal lights will cast 2 shadows of itself
in the direction of the lines of the 2 lights; and if you move this
body placing it nearer to one of the lights the shadow cast towards
the nearer light will be less deep than that which falls towards the
more distant one.

Further complications in the derived shadows (183-187).

183.

The greatest depth of shadow is in the simple derived shadow because
it is not lighted by either of the two lights
a b, c d
.

The next less deep shadow is the derived shadow
e f n
; and in this
the shadow is less by half, because it is illuminated by a single
light, that is
c d
.

This is uniform in natural tone because it is lighted throughout by
one only of the two luminous bodies [10]. But it varies with the
conditions of shadow, inasmuch as the farther it is away from the
light the less it is illuminated by it [13].

The third degree of depth is the middle shadow [Footnote 15: We
gather from what follows that
q g r
here means
ombra media
(the
middle shadow).]. But this is not uniform in natural tone; because
the nearer it gets to the simple derived shadow the deeper it is
[Footnote 18: Compare lines 10-13], and it is the uniformly gradual
diminution by increase of distance which is what modifies it
[Footnote 20: See Footnote 18]: that is to say the depth of a shadow
increases in proportion to the distance from the two lights.

The fourth is the shadow
k r s
and this is all the darker in
natural tone in proportion as it is nearer to
k s
, because it gets
less of the light
a o
, but by the accident [of distance] it is
rendered less deep, because it is nearer to the light
c d
, and
thus is always exposed to both lights.

The fifth is less deep in shadow than either of the others because
it is always entirely exposed to one of the lights and to the whole
or part of the other; and it is less deep in proportion as it is
nearer to the two lights, and in proportion as it is turned towards
the outer side
x t
; because it is more exposed to the second light
a b
.

[Footnote: The diagram to this section is given on Pl. V. To the
left is the facsimile of the beginning of the text belonging to it.]

184.

Why, at the intersections
a
,
b
of the two compound shadows
e f
and
m e
, is a simple shadow pfoduced as at
e h
and
m g
, while
no such simple shadow is produced at the other two intersections
c
d
made by the very same compound shadows?

Compound shadow are a mixture of light and shade and simple shadows
are simply darkness. Hence, of the two lights
n
and
o
, one falls
on the compound shadow from one side, and the other on the compound
shadow from the other side, but where they intersect no light falls,
as at
a b
; therefore it is a simple shadow. Where there is a
compound shadow one light or the other falls; and here a difficulty
arises for my adversary since he says that, where the compound
shadows intersect, both the lights which produce the shadows must of
necessity fall and therefore these shadows ought to be neutralised;
inasmuch as the two lights do not fall there, we say that the shadow
is a simple one and where only one of the two lights falls, we say
the shadow is compound, and where both the lights fall the shadow is
neutralised; for where both lights fall, no shadow of any kind is
produced, but only a light background limiting the shadow. Here I
shall say that what my adversary said was true: but he only mentions
such truths as are in his favour; and if we go on to the rest he
must conclude that my proposition is true. And that is: That if both
lights fell on the point of intersection, the shadows would be
neutralised. This I confess to be true if [neither of] the two
shadows fell in the same spot; because, where a shadow and a light
fall, a compound shadow is produced, and wherever two shadows or two
equal lights fall, the shadow cannot vary in any part of it, the
shadows and the lights both being equal. And this is proved in the
eighth [proposition] on proportion where it is said that if a given
quantity has a single unit of force and resistance, a double
quantity will have double force and double resistance.

The intersection
n
is produced by the shadows caused by the light
b
, because this light
b
produces the shadow
x b
, and the
shadow
s b
, but the intersection
m
is produced by the light
a
which causes the shadow
s a
, and the shadow
x a
.

But if you uncover both the lights
a b
, then you get the two
shadows
n m
both at once, and besides these, two other, simple
shadows are produced at
r o
where neither of the two lights falls
at all. The grades of depth in compound shadows are fewer in
proportion as the lights falling on, and crossing them are less
numerous.

186.

Why the intersections at
n
being composed of two compound derived
shadows, forms a compound shadow and not a simple one, as happens
with other intersections of compound shadows. This occurs, according
to the 2nd [diagram] of this [prop.] which says:—The intersection
of derived shadows when produced by the intersection of columnar
shadows caused by a single light does not produce a simple shadow.
And this is the corollary of the 1st [prop.] which says:—The
intersection of simple derived shadows never results in a deeper
shadow, because the deepest shadows all added together cannot be
darker than one by itself. Since, if many deepest shadows increased
in depth by their duplication, they could not be called the
deepest
shadows, but only part-shadows. But if such intersections
are illuminated by a second light placed between the eye and the
intersecting bodies, then those shadows would become compound
shadows and be uniformly dark just as much at the intersection as
throughout the rest. In the 1st and 2nd above, the intersections
i
k
will not be doubled in depth as it is doubled in quantity. But in
this 3rd, at the intersections
g n
they will be double in depth
and in quantity.

187.

The derived shadow of the dark walls on each side of the bright
light of the window are what mingle their various degrees of shade
with the light derived from the window; and these various depths of
shade modify every portion of the light, except where it is
strongest, at
c
. To prove this let
d a
be the primary shadow
which is turned towards the point
e
, and darkens it by its derived
shadow; as may be seen by the triangle
a e d
, in which the
angle
e
faces the darkened base
d a e
; the point
v
faces the
dark shadow
a s
which is part of
a d
, and as the whole is
greater than a part,
e
which faces the whole base [of the
triangle], will be in deeper shadow than
v
which only faces part
of it. In consequence of the conclusion [shown] in the above
diagram,
t
will be less darkened than
v
, because the base of the
t
is part of the base of the
v
; and in the same way it follows
that
p
is less in shadow than
t
, because the base of the
p
is
part of the base of the
t
. And
c
is the terminal point of the
derived shadow and the chief beginning of the highest light.