The Notebooks of Leonardo Da Vinci (14 page)

[Footnote: The diagram on Pl. IV, No. 5 belongs to this passage; but
it must be noted that the text explains only the figure on the
right-hand side.]

On the shape of the cast shadows (188-191).

188.

The form of the shadow cast by any body of uniform density can never
be the same as that of the body producing it. [Footnote: Comp. the
drawing on PI. XXVIII, No. 5.]

189.

No cast shadow can produce the true image of the body which casts it
on a vertical plane unless the centre of the light is equally
distant from all the edges of that body.

190.

If a window
a b
admits the sunlight into a room, the sunlight will
magnify the size of the window and diminish the shadow of a man in
such a way as that when the man makes that dim shadow of himself,
approach to that which defines the real size of the window, he will
see the shadows where they come into contact, dim and confused from
the strength of the light, shutting off and not allowing the solar
rays to pass; the effect of the shadow of the man cast by this
contact will be exactly that figured above.

[Footnote: It is scarcely possible to render the meaning of this
sentence with strict accuracy; mainly because the grammatical
construction is defective in the most important part—line 4. In the
very slight original sketch the shadow touches the upper arch of the
window and the correction, here given is perhaps not justified.]

191.

A shadow is never seen as of uniform depth on the surface which
intercepts it unless every portion of that surface is equidistant
from the luminous body. This is proved by the 7th which says:—The
shadow will appear lighter or stronger as it is surrounded by a
darker or a lighter background. And by the 8th of this:—The
background will be in parts darker or lighter, in proportion as it
is farther from or nearer to the luminous body. And:—Of various
spots equally distant from the luminous body those will always be in
the highest light on which the rays fall at the smallest angles: The
outline of the shadow as it falls on inequalities in the surface
will be seen with all the contours similar to those of the body that
casts it, if the eye is placed just where the centre of the light
was.

The shadow will look darkest where it is farthest from the body that
casts it. The shadow
c d
, cast by the body in shadow
a b
which
is equally distant in all parts, is not of equal depth because it is
seen on a back ground of varying brightness. [Footnote: Compare the
three diagrams on Pl. VI, no 1 which, in the original accompany this
section.]

On the outlines of cast shadows (192-195).

192.

The edges of a derived shadow will be most distinct where it is cast
nearest to the primary shadow.

193.

As the derived shadow gets more distant from the primary shadow, the
more the cast shadow differs from the primary shadow.

194.

The greater the difference between a light and the body lighted by
it, the light being the larger, the more vague will be the outlines
of the shadow of that object.

The derived shadow will be most confused towards the edges of its
interception by a plane, where it is remotest from the body casting
it.

195.

What is the cause which makes the outlines of the shadow vague and
confused?

Whether it is possible to give clear and definite outlines to the
edges of shadows.

On the relative size of shadows (196. 197).

196.

If an object placed in front of a single light is very close to it
you will see that it casts a very large shadow on the opposite wall,
and the farther you remove the object from the light the smaller
will the image of the shadow become.

The disproportion of a shadow which is larger than the body
producing it, results from the light being smaller than the body, so
that it cannot be at an equal distance from the edges of the body
[Footnote 11: H. LUDWIG in his edition of the old copies, in the
Vatican library—in which this chapter is included under Nos. 612,
613 and 614 alters this passage as follows:
quella parte ch'e piu
propinqua piu cresce che le distanti
, although the Vatican copy
agrees with the original MS. in having
distante
in the former and
propinque
in the latter place. This supposed amendment seems to me
to invert the facts. Supposing for instance, that on Pl. XXXI No. 3.
f
is the spot where the light is that illuminates the figure there
represented, and that the line behind the figure represents a wall
on which the shadow of the figure is thrown. It is evident, that in
that case the nearest portion, in this case the under part of the
thigh, is very little magnified in the shadow, and the remoter
parts, for instance the head, are more magnified.]; and the portions
which are most remote are made larger than the nearer portions for
this reason [Footnote 12: See Footnote 11].

The atmosphere which surrounds a light is almost like light itself
for brightness and colour; but the farther off it is the more it
loses this resemblance. An object which casts a large shadow and is
near to the light, is illuminated both by that light by the luminous
atmosphere; hence this diffused light gives the shadow ill-defined
edges.

197.

A luminous body which is long and narrow in shape gives more
confused outlines to the derived shadow than a spherical light, and
this contradicts the proposition next following: A shadow will have
its outlines more clearly defined in proportion as it is nearer to
the primary shadow or, I should say, the body casting the shadow;
[Footnote 14: The lettering refers to the lower diagram, Pl. XLI,
No. 5.] the cause of this is the elongated form of the luminous body
a c
, &c. [Footnote 16: See Footnote 14].

Effects on cast shadows by the tone of the back ground.

198.

Modified shadows are those which are cast on light walls or other
illuminated objects.

A shadow looks darkest against a light background. The outlines of a
derived shadow will be clearer as they are nearer to the primary
shadow. A derived shadow will be most defined in shape where it is
intercepted, where the plane intercepts it at the most equal angle.

Those parts of a shadow will appear darkest which have darker
objects opposite to them. And they will appear less dark when they
face lighter objects. And the larger the light object opposite, the
more the shadow will be lightened.

And the larger the surface of the dark object the more it will
darken the derived shadow where it is intercepted.

A disputed proposition.

199.

Certain mathematicians have maintained that a triangle, of which the
base is turned to the light, casts no shadow on a plane; and this
they prove by saying [5] that no spherical body smaller than the
light can reach the middle with the shadow. The lines of radiant
light are straight lines [6]; therefore, suppose the light to be
g
h
and the triangle
l m n
, and let the plane be
i k
; they say
the light
g
falls on the side of the triangle
l n
, and the
portion of the plane
i q
. Thus again
h
like
g
falls on the
side
l m
, and then on
m n
and the plane
p k
; and if the whole
plane thus faces the lights
g h
, it is evident that the triangle
has no shadow; and that which has no shadow can cast none. This, in
this case appears credible. But if the triangle
n p g
were not
illuminated by the two lights
g
and
h
, but by
i p
and
g
and
k
neither side is lighted by more than one single light: that is
i p
is invisible to
h g
and
k
will never be lighted by
g
;
hence
p q
will be twice as light as the two visible portions that
are in shadow.

[Footnote: 5—6. This passage is so obscure that it would be rash to
offer an explanation. Several words seem to have been omitted.]

On the relative depth of cast shadows (200-202).

200.

A spot is most in the shade when a large number of darkened rays
fall upon it. The spot which receives the rays at the widest angle
and by darkened rays will be most in the dark; a will be twice as
dark as b, because it originates from twice as large a base at an
equal distance. A spot is most illuminated when a large number of
luminous rays fall upon it. d is the beginning of the shadow
d f
,
and tinges
c
but
a
little;
d e
is half of the shadow
d f
and
gives a deeper tone where it is cast at
b
than at
f
. And the
whole shaded space
e
gives its tone to the spot
a
. [Footnote:
The diagram here referred to is on Pl. XLI, No. 2.]

201.

A n
will be darker than
c r
in proportion to the number of times
that
a b
goes into
c d
.

202.

The shadow cast by an object on a plane will be smaller in
proportion as that object is lighted by feebler rays. Let
d e
be
the object and
d c
the plane surface; the number of times that
d
e
will go into
f g
gives the proportion of light at
f h
to
d
c
. The ray of light will be weaker in proportion to its distance
from the hole through which it falls.

Principles of reflection (203. 204).

203.

If the object is the mountain here figured, and the light is at the
point
a
, I say that from
b d
and also from
c f
there will be
no light but from reflected rays. And this results from the fact
that rays of light can only act in straight lines; and the same is
the case with the secondary or reflected rays.

204.

The edges of the derived shadow are defined by the hues of the
illuminated objects surrounding the luminous body which produces the
shadow.

On reverberation.

205.

Reverberation is caused by bodies of a bright nature with a flat and
semi opaque surface which, when the light strikes upon them, throw
it back again, like the rebound of a ball, to the former object.

All dense bodies have their surfaces occupied by various degrees of
light and shade. The lights are of two kinds, one called original,
the other borrowed. Original light is that which is inherent in the
flame of fire or the light of the sun or of the atmosphere. Borrowed
light will be reflected light; but to return to the promised
definition: I say that this luminous reverberation is not produced
by those portions of a body which are turned towards darkened
objects, such as shaded spots, fields with grass of various height,
woods whether green or bare; in which, though that side of each
branch which is turned towards the original light has a share of
that light, nevertheless the shadows cast by each branch separately
are so numerous, as well as those cast by one branch on the others,
that finally so much shadow is the result that the light counts for
nothing. Hence objects of this kind cannot throw any reflected light
on opposite objects.

Reflection on water (206. 207).

206.

The shadow or object mirrored in water in motion, that is to say in
small wavelets, will always be larger than the external object
producing it.

207.

It is impossible that an object mirrored on water should correspond
in form to the object mirrored, since the centre of the eye is above
the surface of the water.

This is made plain in the figure here given, which demonstrates that
the eye sees the surface
a b
, and cannot see it at
l f
, and at
r t
; it sees the surface of the image at
r t
, and does not see
it in the real object
c d
. Hence it is impossible to see it, as
has been said above unless the eye itself is situated on the surface
of the water as is shown below [13].

[Footnote:
A
stands for
ochio
[eye],
B
for
aria
[air],
C
for
acqua
[water],
D
for
cateto
[cathetus].—In the original
MS. the second diagram is placed below line 13.]

Experiments with the mirror (208-210).

208.

If the illuminated object is of the same size as the luminous body
and as that in which the light is reflected, the amount of the
reflected light will bear the same proportion to the intermediate
light as this second light will bear to the first, if both bodies
are smooth and white.

209.

Describe how it is that no object has its limitation in the mirror
but in the eye which sees it in the mirror. For if you look at your
face in the mirror, the part resembles the whole in as much as the
part is everywhere in the mirror, and the whole is in every part of
the same mirror; and the same is true of the whole image of any
object placed opposite to this mirror, &c.