The Notebooks of Leonardo Da Vinci

Page 49

The nature of the outline.

The boundaries of bodies are the least of all things. The
proposition is proved to be true, because the boundary of a thing is
a surface, which is not part of the body contained within that
surface; nor is it part of the air surrounding that body, but is the
medium interposted between the air and the body, as is proved in its
place. But the lateral boundaries of these bodies is the line
forming the boundary of the surface, which line is of invisible
thickness. Wherefore O painter! do not surround your bodies with
lines, and above all when representing objects smaller than nature;
for not only will their external outlines become indistinct, but
their parts will be invisible from distance.

Page 48

OF DRAWING OUTLINE.

Consider with the greatest care the form of the outlines of every
object, and the character of their undulations. And these
undulations must be separately studied, as to whether the curves are
composed of arched convexities or angular concavities.

Page 47

DEFINITION OF THE NATURE OF THE LINE.

The line has in itself neither matter nor substance and may rather
be called an imaginary idea than a real object; and this being its
nature it occupies no space. Therefore an infinite number of lines
may be conceived of as intersecting each other at a point, which has
no dimensions and is only of the thickness (if thickness it may be
called) of one single line.

HOW WE MAY CONCLUDE THAT A SUPERFICIES TERMINATES IN A POINT?

An angular surface is reduced to a point where it terminates in an
angle. Or, if the sides of that angle are produced in a straight
line, then--beyond that angle--another surface is generated,
smaller, or equal to, or larger than the first.

Page 46

The point, being indivisible, occupies no space. That which occupies
no space is nothing. The limiting surface of one thing is the
beginning of another. 2. That which is no part of any body is called
nothing. 1. That which has no limitations, has no form. The
limitations of two conterminous bodies are interchangeably the
surface of each. All the surfaces of a body are not parts of that
body.

Of the line (47-48).

Page 45

1, The superficies is a limitation of the body. 2, and the
limitation of a body is no part of that body. 4, and the limitation
of one body is that which begins another. 3, that which is not part
of any body is nothing. Nothing is that which fills no space.

If one single point placed in a circle may be the starting point of
an infinite number of lines, and the termination of an infinite
number of lines, there must be an infinite number of points
separable from this point, and these when reunited become one again;
whence it follows that the part may be equal to the whole.

Page 44

OF THE NATURAL POINT.

The smallest natural point is larger than all mathematical points,
and this is proved because the natural point has continuity, and any
thing that is continuous is infinitely divisible; but the
mathematical point is indivisible because it has no size.

[Footnote: This definition was inserted by Leonardo on a MS. copy on
parchment of the well-known _"Trattato d'Architettura civile e
militare"_ &c. by FRANCESCO DI GIORGIO; opposite a passage where the
author says: _'In prima he da sapere che punto e quella parie della
quale he nulla--Linia he luncheza senza apieza; &c.]

Page 43

A point is not part of a line.

Page 42

All the problems of perspective are made clear by the five terms of
mathematicians, which are:--the point, the line, the angle, the
superficies and the solid. The point is unique of its kind. And the
point has neither height, breadth, length, nor depth, whence it is
to be regarded as indivisible and as having no dimensions in space.
The line is of three kinds, straight, curved and sinuous and it has
neither breadth, height, nor depth. Hence it is indivisible,
excepting in its length, and its ends are two points. The angle is
the junction of two lines in a point.

Page 41

The art of perspective is of such a nature as to make what is flat
appear in relief and what is in relief flat.

The elements of perspective--Of the Point (42-46).

Page 40

ON PAINTING.

Perspective is the best guide to the art of Painting.

[Footnote: 40. Compare 53, 2.]

Page 39

The eye--which sees all objects reversed--retains the images for
some time. This conclusion is proved by the results; because, the
eye having gazed at light retains some impression of it. After
looking (at it) there remain in the eye images of intense
brightness, that make any less brilliant spot seem dark until the
eye has lost the last trace of the impression of the stronger light.

_II.

Linear Perspective.

We see clearly from the concluding sentence of section 49, where the
author directly addresses the painter, that he must certainly have
intended to include the elements of mathematics in his Book on the
art of Painting. They are therefore here placed at the beginning. In
section 50 the theory of the "Pyramid of Sight" is distinctly and
expressly put forward as the fundamental principle of linear
perspective, and sections 52 to 57 treat of it fully. This theory of
sight can scarcely be traced to any author of antiquity. Such
passages as occur in Euclid for instance, may, it is true, have
proved suggestive to the painters of the Renaissance, but it would
be rash to say any thing decisive on this point.

Leon Battista Alberti treats of the "Pyramid of Sight" at some
length in his first Book of Painting; but his explanation differs
widely from Leonardo's in the details. Leonardo, like Alberti, may
have borrowed the broad lines of his theory from some views commonly
accepted among painters at the time; but he certainly worked out its
application in a perfectly original manner.

The axioms as to the perception of the pyramid of rays are followed
by explanations of its origin, and proofs of its universal
application (58--69). The author recurs to the subject with endless
variations; it is evidently of fundamental importance in his
artistic theory and practice. It is unnecessary to discuss how far
this theory has any scientific value at the present day; so much as
this, at any rate, seems certain: that from the artist's point of
view it may still claim to be of immense practical utility.

According to Leonardo, on one hand, the laws of perspective are an
inalienable condition of the existence of objects in space; on the
other hand, by a natural law, the eye, whatever it sees and wherever
it turns, is subjected to the perception of the pyramid of rays in
the form of a minute target. Thus it sees objects in perspective
independently of the will of the spectator, since the eye receives
the images by means of the pyramid of rays "just as a magnet
attracts iron".

In connection with this we have the function of the eye explained by
the Camera obscura, and this is all the more interesting and
important because no writer previous to Leonardo had treated of this
subject_ (70--73). _Subsequent passages, of no less special interest,
betray his knowledge of refraction and of the inversion of the image
in the camera and in the eye_ (74--82).

_From the principle of the transmission of the image to the eye and
to the camera obscura he deduces the means of producing an
artificial construction of the pyramid of rays or--which is the same
thing--of the image. The fundamental axioms as to the angle of sight
and the vanishing point are thus presented in a manner which is as
complete as it is simple and intelligible_ (86--89).

_Leonardo distinguishes between simple and complex perspective_ (90,
91). _The last sections treat of the apparent size of objects at
various distances and of the way to estimate it_ (92--109).

General remarks on perspective (40-41).

Page 38

The pupil of the eye, in the open air, changes in size with every
degree of motion from the sun; and at every degree of its changes
one and the same object seen by it will appear of a different size;
although most frequently the relative scale of surrounding objects
does not allow us to detect these variations in any single object we
may look at.

Page 37

Experiment [showing] the dilatation and contraction of the pupil,
from the motion of the sun and other luminaries. In proportion as
the sky is darker the stars appear of larger size, and if you were
to light up the medium these stars would look smaller; and this
difference arises solely from the pupil which dilates and contracts
with the amount of light in the medium which is interposed between
the eye and the luminous body. Let the experiment be made, by
placing a candle above your head at the same time that you look at a
star; then gradually lower the candle till it is on a level with the
ray that comes from the star to the eye, and then you will see the
star diminish so much that you will almost lose sight of it.

[Footnote: No reference is made in the text to the letters on the
accompanying diagram.]

Page 36

ON PERSPECTIVE.

The eye which turns from a white object in the light of the sun and
goes into a less fully lighted place will see everything as dark.
And this happens either because the pupils of the eyes which have
rested on this brilliantly lighted white object have contracted so
much that, given at first a certain extent of surface, they will
have lost more than 3/4 of their size; and, lacking in size, they
are also deficient in [seeing] power. Though you might say to me: A
little bird (then) coming down would see comparatively little, and
from the smallness of his pupils the white might seem black! To this
I should reply that here we must have regard to the proportion of
the mass of that portion of the brain which is given up to the sense
of sight and to nothing else. Or--to return--this pupil in Man
dilates and contracts according to the brightness or darkness of
(surrounding) objects; and since it takes some time to dilate and
contract, it cannot see immediately on going out of the light and
into the shade, nor, in the same way, out of the shade into the
light, and this very thing has already deceived me in painting an
eye, and from that I learnt it.

Page 35

ON PAINTING.

If the eye, when [out of doors] in the luminous atmosphere, sees a
place in shadow, this will look very much darker than it really is.
This happens only because the eye when out in the air contracts the
pupil in proportion as the atmosphere reflected in it is more
luminous. And the more the pupil contracts, the less luminous do the
objects appear that it sees. But as soon as the eye enters into a
shady place the darkness of the shadow suddenly seems to diminish.
This occurs because the greater the darkness into which the pupil
goes the more its size increases, and this increase makes the
darkness seem less.

[Footnote 14: _La luce entrera_. _Luce_ occurs here in the sense of
pupil of the eye as in no 51: C. A. 84b; 245a; I--5; and in many
other places.]

Page 34

Why when the eye has just seen the light, does the half light look
dark to it, and in the same way if it turns from the darkness the
half light look very bright?