The Notebooks of Leonardo Da Vinci (9 page)

The same idea is repeated in C. A. I a; I a, stated as follows:
Infra le cose d'equal grandeza quella si dimostra di minor figura
che sara pi� distante dall' ochio
.—]

96.

Why an object is less distinct when brought near to the eye, and why
with spectacles, or without the naked eye sees badly either close or
far off [as the case may be].

97.

Among objects of equal size, that which is most remote from the eye
will look the smallest.

98.

No second object can be so much lower than the first as that the eye
will not see it higher than the first, if the eye is above the
second.

And this second object will never be so much higher than the first
as that the eye, being below them, will not see the second as lower
than the first.

If the eye sees a second square through the centre of a smaller one,
that is nearer, the second, larger square will appear to be
surrounded by the smaller one.

Objects that are farther off can never be so large but that those in
front, though smaller, will conceal or surround them.

This proposition can be proved by experiment. For if you look
through a small hole there is nothing so large that it cannot be
seen through it and the object so seen appears surrounded and
enclosed by the outline of the sides of the hole. And if you stop it
up, this small stopping will conceal the view of the largest object.

The apparent size of objects defined by calculation (99-105)

99.

Linear Perspective deals with the action of the lines of sight, in
proving by measurement how much smaller is a second object than the
first, and how much the third is smaller than the second; and so on
by degrees to the end of things visible. I find by experience that
if a second object is as far beyond the first as the first is from
the eye, although they are of the same size, the second will seem
half the size of the first and if the third object is of the same
size as the 2nd, and the 3rd is as far beyond the second as the 2nd
from the first, it will appear of half the size of the second; and
so on by degrees, at equal distances, the next farthest will be half
the size of the former object. So long as the space does not exceed
the length of 20 braccia. But, beyond 20 braccia figures of equal
size will lose 2/4 and at 40 braccia they will lose 9/10, and 19/20
at 60 braccia, and so on diminishing by degrees. This is if the
picture plane is distant from you twice your own height. If it is
only as far off as your own height, there will be a great difference
between the first braccia and the second.

[Footnote: This chapter is included in DUFRESNE'S and MANZI'S
editions of the Treatise on Painting. H. LUDWIG, in his commentary,
calls this chapter "
eines der wichtigsten im ganzen Tractat
", but
at the same time he asserts that its substance has been so
completely disfigured in the best MS. copies that we ought not to
regard Leonardo as responsible for it. However, in the case of this
chapter, the old MS. copies agree with the original as it is
reproduced above. From the chapters given later in this edition,
which were written at a subsequent date, it would appear that
Leonardo corrected himself on these points.]

100.

A second object as far distant from the first as the first is from
the eye will appear half the size of the first, though they be of
the same size really.

If you place the vertical plane at one braccio from the eye, the
first object, being at a distance of 4 braccia from your eye will
diminish to 3/4 of its height at that plane; and if it is 8 braccia
from the eye, to 7/8; and if it is 16 braccia off, it will diminish
to 15/16 of its height and so on by degrees, as the space doubles
the diminution will double.

101.

Begin from the line
m f
with the eye below; then go up and do the
same with the line
n f
, then with the eye above and close to the 2
gauges on the ground look at
m n
; then as
c m
is to
m n
so
will
n m
be to
n s
.

If
a n
goes 3 times into
f b, m p
will do the same into
p g
.
Then go backwards so far as that
c d
goes twice into
a n
and
p
g
will be equal to
g h
. And
m p
will go into
h p
as often as
d c
into
o p
.

[Footnote: The first three lines are unfortunately very obscure.]

102.

Although the objects seen by the eye do, in fact, touch each other
as they recede, I will nevertheless found my rule on spaces of 20
braccia each; as a musician does with notes, which, though they can
be carried on one into the next, he divides into degrees from note
to note calling them 1st, 2nd, 3rd, 4th, 5th; and has affixed a name
to each degree in raising or lowering the voice.

103.

Let
f
be the level and distance of the eye; and
a
the vertical
plane, as high as a man; let
e
be a man, then I say that on the
plane this will be the distance from the plane to the 2nd man.

104.

The differences in the diminution of objects of equal size in
consequence of their various remoteness from the eye will bear among
themselves the same proportions as those of the spaces between the
eye and the different objects.

Find out how much a man diminishes at a certain distance and what
its length is; and then at twice that distance and at 3 times, and
so make your general rule.

105.

The eye cannot judge where an object high up ought to descend.

106.

If two similar and equal objects are placed one beyond the other at
a given distance the difference in their size will appear greater in
proportion as they are nearer to the eye that sees them. And
conversely there will seem to be less difference in their size in
proportion as they are remote from the eve.

This is proved by the proportions of their distances among
themselves; for, if the first of these two objects were as far from
the eye, as the 2nd from the first this would be called the second
proportion: since, if the first is at 1 braccia from the eye and the
2nd at two braccia, two being twice as much as one, the first object
will look twice as large as the second. But if you place the first
at a hundred braccia from you and the second at a hundred and one,
you will find that the first is only so much larger than the second
as 100 is less than 101; and the converse is equally true. And
again, the same thing is proved by the 4th of this book which shows
that among objects that are equal, there is the same proportion in
the diminution of the size as in the increase in the distance from
the eye of the spectator.

On natural perspective (107—109).

107.

The practice of perspective may be divided into … parts [Footnote
4:
in
…
parte
. The space for the number is left blank in the
original.], of which the first treats of objects seen by the eye at
any distance; and it shows all these objects just as the eye sees
them diminished, without obliging a man to stand in one place rather
than another so long as the plane does not produce a second
foreshortening.

But the second practice is a combination of perspective derived
partly from art and partly from nature and the work done by its
rules is in every portion of it, influenced by natural perspective
and artificial perspective. By natural perspective I mean that the
plane on which this perspective is represented is a flat surface,
and this plane, although it is parallel both in length and height,
is forced to diminish in its remoter parts more than in its nearer
ones. And this is proved by the first of what has been said above,
and its diminution is natural. But artificial perspective, that is
that which is devised by art, does the contrary; for objects equal
in size increase on the plane where it is foreshortened in
proportion as the eye is more natural and nearer to the plane, and
as the part of the plane on which it is figured is farther from the
eye.

And let this plane be
d e
on which are seen 3 equal circles which
are beyond this plane
d e
, that is the circles
a b c
. Now you
see that the eye
h
sees on the vertical plane the sections of the
images, largest of those that are farthest and smallest of the
nearest.

108.

Here follows what is wanting in the margin at the foot on the other
side of this page.

Natural perspective acts in a contrary way; for, at greater
distances the object seen appears smaller, and at a smaller distance
the object appears larger. But this said invention requires the
spectator to stand with his eye at a small hole and then, at that
small hole, it will be very plain. But since many (men's) eyes
endeavour at the same time to see one and the same picture produced
by this artifice only one can see clearly the effect of this
perspective and all the others will see confusion. It is well
therefore to avoid such complex perspective and hold to simple
perspective which does not regard planes as foreshortened, but as
much as possible in their proper form. This simple perspective, in
which the plane intersects the pyramids by which the images are
conveyed to the eye at an equal distance from the eye is our
constant experience, from the curved form of the pupil of the eye on
which the pyramids are intersected at an equal distance from the
visual virtue.

[Footnote 24:
la prima di sopra
i. e. the first of the three
diagrams which, in the original MS., are placed in the margin at the
beginning of this chapter.]

109.

This diagram distinguishes natural from artificial perspective. But
before proceeding any farther I will define what is natural and what
is artificial perspective. Natural perspective says that the more
remote of a series of objects of equal size will look the smaller,
and conversely, the nearer will look the larger and the apparent
size will diminish in proportion to the distance. But in artificial
perspective when objects of unequal size are placed at various
distances, the smallest is nearer to the eye than the largest and
the greatest distance looks as though it were the least of all; and
the cause of this is the plane on which the objects are represented;
and which is at unequal distances from the eye throughout its
length. And this diminution of the plane is natural, but the
perspective shown upon it is artificial since it nowhere agrees with
the true diminution of the said plane. Whence it follows, that when
the eye is somewhat removed from the [station point of the]
perspective that it has been gazing at, all the objects represented
look monstrous, and this does not occur in natural perspective,
which has been defined above. Let us say then, that the square
a b
c d
figured above is foreshortened being seen by the eye situated
in the centre of the side which is in front. But a mixture of
artificial and natural perspective will be seen in this tetragon
called
el main
[Footnote 20:
el main
is quite legibly written in
the original; the meaning and derivation of the word are equally
doubtful.], that is to say
e f g h
which must appear to the eye of
the spectator to be equal to
a b c d
so long as the eye remains in
its first position between
c
and
d
. And this will be seen to
have a good effect, because the natural perspective of the plane
will conceal the defects which would [otherwise] seem monstrous.

Six books on Light and Shade.

Linear Perspective cannot be immediately followed by either the
"prospettiva de' perdimenti"
or the
"prospettiva de' colori"
or
the aerial perspective; since these branches of the subject
presuppose a knowledge of the principles of Light and Shade. No
apology, therefore, is here needed for placing these immediately
after Linear Perspective.

We have various plans suggested by Leonardo for the arrangement of
the mass of materials treating of this subject. Among these I have
given the preference to a scheme propounded in No.
III,
because,
in all probability, we have here a final and definite purpose
expressed. Several authors have expressed it as their opinion that
the Paris Manuscript
C
is a complete and finished treatise on
Light and Shade. Certainly, the Principles of Light and Shade form
by far the larger portion of this MS. which consists of two separate
parts; still, the materials are far from being finally arranged. It
is also evident that he here investigates the subject from the point
of view of the Physicist rather than from that of the Painter.

The plan of a scheme of arrangement suggested in No.
III
and
adopted by me has been strictly adhered to for the first four Books.
For the three last, however, few materials have come down to us; and
it must be admitted that these three Books would find a far more
appropriate place in a work on Physics than in a treatise on
Painting. For this reason I have collected in Book V all the
chapters on Reflections, and in Book VI I have put together and
arranged all the sections of MS.
C
that belong to the book on
Painting, so far as they relate to Light and Shade, while the
sections of the same MS. which treat of the
"Prospettiva de'
perdimenti"
have, of course, been excluded from the series on Light
and Shade.