The Notebooks of Leonardo Da Vinci (12 page)

On the intensity of shadows as dependent on the distance from the
light (150-152).

150.

The smaller the light that falls upon an object the more shadow it
will display. And the light will illuminate a smaller portion of the
object in proportion as it is nearer to it; and conversely, a larger
extent of it in proportion as it is farther off.

A light which is smaller than the object on which it falls will
light up a smaller extent of it in proportion as it is nearer to it,
and the converse, as it is farther from it. But when the light is
larger than the object illuminated it will light a larger extent of
the object in proportion as it is nearer and the converse when they
are farther apart.

151.

That portion of an illuminated object which is nearest to the source
of light will be the most strongly illuminated.

152.

That portion of the primary shadow will be least dark which is
farthest from the edges.

The derived shadow will be darker than the primary shadow where it
is contiguous with it.

On the proportion of light and shade (153-157).

153.

That portion of an opaque body will be more in shade or more in
light, which is nearer to the dark body, by which it is shaded, or
to the light that illuminates it.

Objects seen in light and shade show in greater relief than those
which are wholly in light or in shadow.

154.

The shaded and illuminated sides of opaque objects will display the
same proportion of light and darkness as their objects [Footnote 6:
The meaning of
obbietti
(objects) is explained in no 153, lines
1-4.—Between the title-line and the next there is, in the
original, a small diagram representing a circle described round a
square.].

155.

The outlines and form of any part of a body in light and shade are
indistinct in the shadows and in the high lights; but in the
portions between the light and the shadows they are highly
conspicuous.

156.

Among objects in various degrees of shade, when the light proceeds
from a single source, there will be the same proportion in their
shadows as in the natural diminution of the light and the same must
be understood of the degrees of light.

157.

A single and distinct luminous body causes stronger relief in the
object than a diffused light; as may be seen by comparing one side
of a landscape illuminated by the sun, and one overshadowed by
clouds, and so illuminated only by the diffused light of the
atmosphere.

Definition of derived shadow (158. 159).

158.

Derived shadow cannot exist without primary shadow. This is proved
by the first of this which says: Darkness is the total absence of
light, and shadow is an alleviation of darkness and of light, and it
is more or less dark or light in proportion as the darkness is
modified by the light.

159.

Shadow is diminution of light.

Darkness is absence of light.

Shadow is divided into two kinds, of which the first is called
primary shadow, the second is derived shadow. The primary shadow is
always the basis of the derived shadow.

The edges of the derived shadow are straight lines.

[Footnote: The theory of the
ombra
dirivativa_—a technical
expression for which there is no precise English equivalent is
elaborately treated by Leonardo. But both text and diagrams (as Pl.
IV, 1-3 and Pl. V) must at once convince the student that the
distinction he makes between
ombra primitiva
and
ombra
dirivativa
is not merely justifiable but scientific.
Ombra
dirivativa
is by no means a mere abstract idea. This is easily
proved by repeating the experiment made by Leonardo, and by filling
with smoke the room in which the existence of the
ombra dirivativa
is investigated, when the shadow becomes visible. Nor is it
difficult to perceive how much of Leonardo's teaching depended on
this theory. The recognised, but extremely complicated science of
cast shadows—
percussione dell' ombre dirivative
as Leonardo
calls them—is thus rendered more intelligible if not actually
simpler, and we must assume this theory as our chief guide through
the investigations which follow.]

The darkness of the derived shadow diminishes in proportion as it is
remote from the primary shadow.

Different sorts of derived shadows (160-162).

160.

The forms of shadows are three: inasmuch as if the solid body which
casts the shadow is equal (in size) to the light, the shadow
resembles a column without any termination (in length). If the body
is larger than the light the shadow resembles a truncated and
inverted pyramid, and its length has also no defined termination.
But if the body is smaller than the light, the shadow will resemble
a pyramid and come to an end, as is seen in eclipses of the moon.

161.

The simple derived shadow is of two kinds: one kind which has its
length defined, and two kinds which are undefined; and the defined
shadow is pyramidal. Of the two undefined, one is a column and the
other spreads out; and all three have rectilinear outlines. But the
converging, that is the pyramidal, shadow proceeds from a body that
is smaller than the light, and the columnar from a body equal in
size to the light, and the spreading shadow from a body larger than
the light; &c.

Compound derived shadows are of two kinds; that is columnar and
spreading.

162.

Derived shadows are of three kinds of which one is spreading, the
second columnar, the third converging to the point where the two
sides meet and intersect, and beyond this intersection the sides are
infinitely prolonged or straight lines. And if you say, this shadow
must terminate at the angle where the sides meet and extend no
farther, I deny this, because above in the first on shadow I have
proved: that a thing is completely terminated when no portion of it
goes beyond its terminating lines. Now here, in this shadow, we see
the converse of this, in as much as where this derived shadow
originates we obviously have the figures of two pyramids of shadow
which meet at their angles. Hence, if, as [my] opponent says, the
first pyramid of shadow terminates the derivative shadow at the
angle whence it starts, then the second pyramid of shadow—so says
the adversary—must be caused by the angle and not from the body in
shadow; and this is disproved with the help of the 2nd of this which
says: Shadow is a condition produced by a body casting a shadow, and
interposed between this shadow and the luminous body. By this it is
made clear that the shadow is not produced by the angle of the
derived shadow but only by the body casting the shadow; &c. If a
spherical solid body is illuminated by a light of elongated form the
shadow produced by the longest portion of this light will have less
defined outlines than that which is produced by the breadth of the
same light. And this is proved by what was said before, which is:
That a shadow will have less defined outlines in proportion as the
light which causes it is larger, and conversely, the outlines are
clearer in proportion as it is smaller.

[Footnote: The two diagrams to this chapter are on Plate IV, No. 1.]

On the relation of derived and primary shadow (163-165).

163.

The derived shadow can never resemble the body from which it
proceeds unless the light is of the same form and size as the body
causing the shadow.

The derived shadow cannot be of the same form as the primary shadow
unless it is intercepted by a plane parallel to it.

164.

If the rays of light proceed, as experience shows, from a single
point and are diffused in a sphere round this point, radiating and
dispersed through the air, the farther they spread the wider they
must spread; and an object placed between the light and a wall is
always imaged larger in its shadow, because the rays that strike it
[Footnote: 7. The following lines are wanting to complete the
logical connection.] would, by the time they have reached the wall,
have become larger.

165.

Any shadow cast by a body in light and shade is of the same nature
and character as that which is inseparable from the body. The centre
of the length of a shadow always corresponds to that of the luminous
body [Footnote 6: This second statement of the same idea as in the
former sentence, but in different words, does not, in the original,
come next to the foregoing; sections 172 and 127 are placed between
them.]. It is inevitable that every shadow must have its centre in a
line with the centre of the light.

On the shape of derived shadows (166-174).

166.

The pyramidal shadow produced by a columnar body will be narrower
than the body itself in proportion as the simple derived shadow is
intersected farther from the body which casts it.

[Footnote 166: Compare the first diagram to No. 161. If we here
conceive of the outlines of the pyramid of shadow on the ground as
prolonged beyond its apex this gives rise to a second pyramid; this
is what is spoken of at the beginning of No. 166.]

167.

The cast shadow will be longest when the light is lowest.

The cast shadow will be shortest when the light is highest.

168.

Both the primary and derived shadow will be larger when caused by
the light of a candle than by diffused light. The difference between
the larger and smaller shadows will be in inverse proportion to the
larger and smaller lights causing them.

[Footnote: In the diagrams
A
stands for
celo
(sky),
B
for
cadela
(candle).]

169.

Among bodies of equal size, that one which is illuminated by the
largest light will have the shortest shadow. Experiment confirms
this proposition. Thus the body
m
n
is surrounded by a larger
amount of light than the body
p q
, as is shown above. Let us say
that
v c a b d x
is the sky, the source of light, and that
s t
is a window by which the luminous rays enter, and so
m n
and
p q
are bodies in light and shade as exposed to this light;
m n
will
have a small derived shadow, because its original shadow will be
small; and the derivative light will be large, again, because the
original light
c d
will be large and
p q
will have more derived
shadow because its original shadow will be larger, and its derived
light will be smaller than that of the body
m n
because that
portion of the hemisphere
a b
which illuminates it is smaller than
the hemisphere
c d
which illuminates the body
m n
.

[Footnote: The diagram, given on Pl. IV, No. 2, stands in the
original between lines 2 and 7, while the text of lines 3 to 6 is
written on its left side. In the reproduction of this diagram the
letter
v
at the outer right-hand end has been omitted.]

170.

The shadow
m
bears the same proportion to the shadow
n
as the
line
b c
to the line
f c
.

171.

Of different shadows of equal strength that which is nearest the eye
will seem the least strong.

Why is the shadow
e a b
in the first grade of strength,
b c
in
the second;
c d
in the third? The reason is that as from
e a b
the sky is nowhere visible, it gets no light whatever from the sky,
and so has no direct [primary] light.
b c
faces the portion of the
sky
f g
and is illuminated by it.
c d
faces the sky at
h k
.
c
d
, being exposed to a larger extent of sky than
b c
, it is
reasonable that it should be more lighted. And thus, up to a certain
distance, the wall
a d
will grow lighter for the reasons here
given, until the darkness of the room overpowers the light from the
window.

172.

When the light of the atmosphere is restricted [by an opening] and
illuminates bodies which cast shadows, these bodies being equally
distant from the centre of the window, that which is most obliquely
placed will cast the largest shadow beyond it.

173.

These bodies standing apart in a room lighted by a single window
will have derivative shadows more or less short according as they
are more or less opposite to the window. Among the shadows cast by
bodies of equal mass but at unequal distances from the opening by
which they are illuminated, that shadow will be the longest of the
body which is least in the light. And in proportion as one body is
better illuminated than another its shadow will be shorter than
another. The proportion
n m
and
e v k
bear to
r t
and
v x
corresponds with that of the shadow
x
to 4 and
y
.

The reason why those bodies which are placed most in front of the
middle of the window throw shorter shadows than those obliquely
situated is:—That the window appears in its proper form and to the
obliquely placed ones it appears foreshortened; to those in the
middle, the window shows its full size, to the oblique ones it
appears smaller; the one in the middle faces the whole hemisphere
that is
e f
and those on the side have only a strip; that is
q r
faces
a b
; and
m n
faces
c d
; the body in the middle having a
larger quantity of light than those at the sides is lighted from a
point much below its centre, and thus the shadow is shorter. And the
pyramid
g
4 goes into
l y
exactly as often as
a b
goes into
e
f
. The axis of every derivative shadow passes through 6 1/2
[Footnote 31:
passa per
6 1/2 (passes through 6 1/2). The meaning
of these words is probably this: Each of the three axes of the
derived shadow intersects the centre (
mezzo
) of the primary shadow
(
ombra originale
) and, by prolongation upwards crosses six lines.