The Notebooks of Leonardo Da Vinci (15 page)

210.

No man can see the image of another man in a mirror in its proper
place with regard to the objects; because every object falls on [the
surface of] the mirror at equal angles. And if the one man, who sees
the other in the mirror, is not in a direct line with the image he
will not see it in the place where it really falls; and if he gets
into the line, he covers the other man and puts himself in the place
occupied by his image. Let
n o
be the mirror,
b
the eye of your
friend and
d
your own eye. Your friend's eye will appear to you at
a
, and to him it will seem that yours is at
c
, and the
intersection of the visual rays will occur at
m
, so that either of
you touching
m
will touch the eye of the other man which shall be
open. And if you touch the eye of the other man in the mirror it
will seem to him that you are touching your own.

Appendix:—On shadows in movement (211. 212).

211.

When two bodies casting shadows, and one in front of the other, are
between a window and the wall with some space between them, the
shadow of the body which is nearest to the plane of the wall will
move if the body nearest to the window is put in transverse motion
across the window. To prove this let
a
and
b
be two bodies
placed between the window
n m
and the plane surface
o p
with
sufficient space between them as shown by the space
a b
. I say
that if the body
a
is moved towards
s
the shadow of the body
b
which is at
c
will move towards
d
.

212.

The motion of a shadow is always more rapid than that of the body
which produces it if the light is stationary. To prove this let
a
be the luminous body, and
b
the body casting the shadow, and
d
the shadow. Then I say that in the time while the solid body moves
from
b
to
c
, the shadow
d
will move to
e
; and this
proportion in the rapidity of the movements made in the same space
of time, is equal to that in the length of the space moved over.
Thus, given the proportion of the space moved over by the body
b
to
c
, to that moved over by the shadow
d
to
e
, the proportion
in the rapidity of their movements will be the same.

But if the luminous body is also in movement with a velocity equal
to that of the solid body, then the shadow and the body that casts
it will move with equal speed. And if the luminous body moves more
rapidly than the solid body, the motion of the shadow will be slower
than that of the body casting it.

But if the luminous body moves more slowly than the solid body, then
the shadow will move more rapidly than that body.

The effect of rays passing through holes (213. 214).

213.

If you transmit the rays of the sun through a hole in the shape of a
star you will see a beautiful effect of perspective in the spot
where the sun's rays fall.

[Footnote: In this and the following chapters of MS. C the order of
the original paging has been adhered to, and is shown in
parenthesis. Leonardo himself has but rarely worked out the subject
of these propositions. The space left for the purpose has
occasionally been made use of for quite different matter. Even the
numerous diagrams, most of them very delicately sketched, lettered
and numbered, which occur on these pages, are hardly ever explained,
with the exception of those few which are here given.]

214.

No small hole can so modify the convergence of rays of light as to
prevent, at a long distance, the transmission of the true form of
the luminous body causing them. It is impossible that rays of light
passing through a parallel [slit], should not display the form of
the body causing them, since all the effects produced by a luminous
body are [in fact] the reflection of that body: The moon, shaped
like a boat, if transmitted through a hole is figured in the surface
[it falls on] as a boatshaped object. [Footnote 8: In the MS. a
blank space is left after this question.] Why the eye sees bodies at
a distance, larger than they measure on the vertical plane?.

[Footnote: This chapter, taken from another MS. may, as an
exception, be placed here, as it refers to the same subject as the
preceding section.]

On gradation of shadows (215. 216).

215.

Although the breadth and length of lights and shadow will be
narrower and shorter in foreshortening, the quality and quantity of
the light and shade is not increased nor diminished.

[3]The function of shade and light when diminished by
foreshortening, will be to give shadow and to illuminate an object
opposite, according to the quality and quantity in which they fall
on the body.

[5]In proportion as a derived shadow is nearer to its penultimate
extremities the deeper it will appear,
g z
beyond the intersection
faces only the part of the shadow [marked]
y z
; this by
intersection takes the shadow from
m n
but by direct line it takes
the shadow
a m
hence it is twice as deep as
g z
.
Y x
, by
intersection takes the shadow
n o
, but by direct line the shadow
n m a
, therefore
x y
is three times as dark as
z g
;
x f
, by
intersection faces
o b
and by direct line
o n m a
, therefore we
must say that the shadow between
f x
will be four times as dark as
the shadow
z g
, because it faces four times as much shadow.

Let
a b
be the side where the primary shadow is, and
b c
the
primary light,
d
will be the spot where it is intercepted,
f g
the derived shadow and
f e
the derived light.

And this must be at the beginning of the explanation.

[Footnote: In the original MS. the text of No. 252 precedes the one
given here. In the text of No. 215 there is a blank space of about
four lines between the lines 2 and 3. The diagram given on Pl. VI,
No. 2 is placed between lines 4 and 5. Between lines 5 and 6 there
is another space of about three lines and one line left blank
between lines 8 and 9. The reader will find the meaning of the whole
passage much clearer if he first reads the final lines 11—13.
Compare also line 4 of No. 270.]

On relative proportion of light and shadows (216—221).

216.

That part of the surface of a body on which the images [reflection]
from other bodies placed opposite fall at the largest angle will
assume their hue most strongly. In the diagram below, 8 is a larger
angle than 4, since its base
a n
is larger than
e n
the base of
4. This diagram below should end at
a n
4 8. [4]That portion of
the illuminated surface on which a shadow is cast will be brightest
which lies contiguous to the cast shadow. Just as an object which is
lighted up by a greater quantity of luminous rays becomes brighter,
so one on which a greater quantity of shadow falls, will be darker.

Let 4 be the side of an illuminated surface 4 8, surrounding the
cast shadow
g e
4. And this spot 4 will be lighter than 8, because
less shadow falls on it than on 8. Since 4 faces only the shadow
i
n
; and 8 faces and receives the shadow
a e
as well as
i n
which
makes it twice as dark. And the same thing happens when you put the
atmosphere and the sun in the place of shade and light.

[12] The distribution of shadow, originating in, and limited by,
plane surfaces placed near to each other, equal in tone and directly
opposite, will be darker at the ends than at the beginning, which
will be determined by the incidence of the luminous rays. You will
find the same proportion in the depth of the derived shadows
a n
as in the nearness of the luminous bodies
m b
, which cause them;
and if the luminous bodies were of equal size you would still
farther find the same proportion in the light cast by the luminous
circles and their shadows as in the distance of the said luminous
bodies.

[Footnote: The diagram originally placed between lines 3 and 4 is on
Pl. VI, No. 3. In the diagram given above line 14 of the original,
and here printed in the text, the words
corpo luminoso
[luminous
body] are written in the circle
m
,
luminoso
in the circle
b
and
ombroso
[body in shadow] in the circle
o
.]

217.

[2] The darkness occasioned by the casting of combined shadows will
be in conformity with its cause, which will originate and terminate
between two plane surfaces near together, alike in tone and directly
opposite each other.

[4] In proportion as the source of light is larger, the luminous and
shadow rays will be more mixed together. This result is produced
because wherever there is a larger quantity of luminous rays, there
is most light, but where there are fewer there is least light,
consequently the shadow rays come in and mingle with them.

[Footnote: Diagrams are inserted before lines 2 and 4.]

218.

In all the proportions I lay down it must be understood that the
medium between the bodies is always the same. [2] The smaller the
luminous body the more distinct will the transmission of the shadows
be.

[3] When of two opposite shadows, produced by the same body, one is
twice as dark as the other though similar in form, one of the two
lights causing them must have twice the diameter that the other has
and be at twice the distance from the opaque body. If the object is
lowly moved across the luminous body, and the shadow is intercepted
at some distance from the object, there will be the same relative
proportion between the motion of the derived shadow and the motion
of the primary shadow, as between the distance from the object to
the light, and that from the object to the spot where the shadow is
intercepted; so that though the object is moved slowly the shadow
moves fast.

[Footnote: There are diagrams inserted before lines 2 and 3 but they
are not reproduced here. The diagram above line 6 is written upon as
follows: at
A lume
(light), at
B obbietto
(body), at
C ombra
d'obbietto
(shadow of the object).]

219.

A luminous body will appear less brilliant when surrounded by a
bright background.

[2] I have found that the stars which are nearest to the horizon
look larger than the others because light falls upon them from a
larger proportion of the solar body than when they are above us; and
having more light from the sun they give more light, and the bodies
which are most luminous appear the largest. As may be seen by the
sun through a mist, and overhead; it appears larger where there is
no mist and diminished through mist. No portion of the luminous body
is ever visible from any spot within the pyramid of pure derived
shadow.

[Footnote: Between lines 1 and 2 there is in the original a large
diagram which does not refer to this text. ]

220.

A body on which the solar rays fall between the thin branches of
trees far apart will cast but a single shadow.

[2] If an opaque body and a luminous one are (both) spherical the
base of the pyramid of rays will bear the same proportion to the
luminous body as the base of the pyramid of shade to the opaque
body.

[4] When the transmitted shadow is intercepted by a plane surface
placed opposite to it and farther away from the luminous body than
from the object [which casts it] it will appear proportionately
darker and the edges more distinct.

[Footnote: The diagram which, in the original, is placed above line
2, is similar to the one, here given on page 73 (section 120).—The
diagram here given in the margin stands, in the original, between
lines 3 and 4.]

221.

A body illuminated by the solar rays passing between the thick
branches of trees will produce as many shadows as there are branches
between the sun and itself.

Where the shadow-rays from an opaque pyramidal body are intercepted
they will cast a shadow of bifurcate outline and various depth at
the points. A light which is broader than the apex but narrower than
the base of an opaque pyramidal body placed in front of it, will
cause that pyramid to cast a shadow of bifurcate form and various
degrees of depth.

If an opaque body, smaller than the light, casts two shadows and if
it is the same size or larger, casts but one, it follows that a
pyramidal body, of which part is smaller, part equal to, and part
larger than, the luminous body, will cast a bifurcate shadow.

[Footnote: Between lines 2 and 3 there are in the original two large
diagrams.]

Perspective of Disappearance.

The theory of the
"Prospettiva de' perdimenti"
would, in many
important details, be quite unintelligible if it had not been led up
by the principles of light and shade on which it is based. The word
"Prospettiva"
in the language of the time included the principles
of optics; what Leonardo understood by
"Perdimenti"
will be
clearly seen in the early chapters, Nos.
222—224.
It is in the
very nature of the case that the farther explanations given in the
subsequent chapters must be limited to general rules. The sections
given as
227—231
"On indistinctness at short distances" have, it
is true, only an indirect bearing on the subject; but on the other
hand, the following chapters,
232—234,
"On indistinctness at
great distances," go fully into the matter, and in chapters
235—239,
which treat "Of the importance of light and shade in the
Perspective of Disappearance", the practical issues are distinctly
insisted on in their relation to the theory. This is naturally
followed by the statements as to "the effect of light or dark
backgrounds on the apparent size of bodies"
(
Nos.
240—250).
At
the end I have placed, in the order of the original, those sections
from the MS.
C
which treat of the "Perspective of Disappearance"
and serve to some extent to complete the treatment of the subject
(251—262).