The Notebooks of Leonardo Da Vinci

Page 86

A simple and natural method; showing how objects appear to the eye
without any other medium.

The object that is nearest to the eye always seems larger than
another of the same size at greater distance. The eye _m_, seeing
the spaces _o v x_, hardly detects the difference between them, and
the. reason of this is that it is close to them [Footnote 6: It is
quite inconceivable to me why M. RAVAISSON, in a note to his French
translation of this simple passage should have remarked: _Il est
clair que c'est par erreur que Leonard a ecrit_ per esser visino _au
lieu de_ per non esser visino. (See his printed ed. of MS. A. p.
38.)]; but if these spaces are marked on the vertical plane _n o_
the space _o v_ will be seen at _o r_, and in the same way the space
_v x_ will appear at _r q_. And if you carry this out in any place
where you can walk round, it will look out of proportion by reason
of the great difference in the spaces _o r_ and _r q_. And this
proceeds from the eye being so much below [near] the plane that the
plane is foreshortened. Hence, if you wanted to carry it out, you
would have [to arrange] to see the perspective through a single hole
which must be at the point _m_, or else you must go to a distance of
at least 3 times the height of the object you see. The plane _o p_
being always equally remote from the eye will reproduce the objects
in a satisfactory way, so that they may be seen from place to place.

Page 85

PERSPECTIVE.

The vertical plane is a perpendicular line, imagined as in front of
the central point where the apex of the pyramids converge. And this
plane bears the same relation to this point as a plane of glass
would, through which you might see the various objects and draw them
on it. And the objects thus drawn would be smaller than the
originals, in proportion as the distance between the glass and the
eye was smaller than that between the glass and the objects.

PERSPECTIVE.

The different converging pyramids produced by the objects, will
show, on the plane, the various sizes and remoteness of the objects
causing them.

PERSPECTIVE.

All those horizontal planes of which the extremes are met by
perpendicular lines forming right angles, if they are of equal width
the more they rise to the level of eye the less this is seen, and
the more the eye is above them the more will their real width be
seen.

PERSPECTIVE.

The farther a spherical body is from the eye the more you will see
of it.

The angle of sight varies with the distance (86-88)

Page 84

Pictorial perspective can never make an object at the same distance,
look of the same size as it appears to the eye. You see that the
apex of the pyramid _f c d_ is as far from the object _c_ _d_ as the
same point _f_ is from the object _a_ _b_; and yet _c_ _d_, which is
the base made by the painter's point, is smaller than _a_ _b_ which
is the base of the lines from the objects converging in the eye and
refracted at _s_ _t_, the surface of the eye. This may be proved by
experiment, by the lines of vision and then by the lines of the
painter's plumbline by cutting the real lines of vision on one and
the same plane and measuring on it one and the same object.

Page 83

OF THE PLANE OF GLASS.

Perspective is nothing else than seeing place [or objects] behind a
plane of glass, quite transparent, on the surface of which the
objects behind that glass are to be drawn. These can be traced in
pyramids to the point in the eye, and these pyramids are intersected
on the glass plane.

Page 82

An experiment showing that though the pupil may not be moved from
its position the objects seen by it may appear to move from their
places.

If you look at an object at some distance from you and which is
below the eye, and fix both your eyes upon it and with one hand
firmly hold the upper lid open while with the other you push up the
under lid--still keeping your eyes fixed on the object gazed at--you
will see that object double; one [image] remaining steady, and the
other moving in a contrary direction to the pressure of your finger
on the lower eyelid. How false the opinion is of those who say that
this happens because the pupil of the eye is displaced from its
position.

How the above mentioned facts prove that the pupil acts upside down
in seeing.

[Footnote: 82. 14--17. The subject indicated by these two headings is
fully discussed in the two chapters that follow them in the
original; but it did not seem to me appropriate to include them
here.]

Demostration of perspective by means of a vertical glass plane
(83-85).

Page 81

HOW THE INNUMERABLE RAYS FROM INNUMERABLE IMAGES CAN CONVERGE TO A
POINT.

Just as all lines can meet at a point without interfering with each
other--being without breadth or thickness--in the same way all the
images of surfaces can meet there; and as each given point faces the
object opposite to it and each object faces an opposite point, the
converging rays of the image can pass through the point and diverge
again beyond it to reproduce and re-magnify the real size of that
image. But their impressions will appear reversed--as is shown in
the first, above; where it is said that every image intersects as it
enters the narrow openings made in a very thin substance.

Read the marginal text on the other side.

In proportion as the opening is smaller than the shaded body, so
much less will the images transmitted through this opening intersect
each other. The sides of images which pass through openings into a
dark room intersect at a point which is nearer to the opening in
proportion as the opening is narrower. To prove this let _a b_ be an
object in light and shade which sends not its shadow but the image
of its darkened form through the opening _d e_ which is as wide as
this shaded body; and its sides _a b_, being straight lines (as has
been proved) must intersect between the shaded object and the
opening; but nearer to the opening in proportion as it is smaller
than the object in shade. As is shown, on your right hand and your
left hand, in the two diagrams _a_ _b_ _c_ _n_ _m_ _o_ where, the
right opening _d_ _e_, being equal in width to the shaded object _a_
_b_, the intersection of the sides of the said shaded object occurs
half way between the opening and the shaded object at the point _c_.
But this cannot happen in the left hand figure, the opening _o_
being much smaller than the shaded object _n_ _m_.

It is impossible that the images of objects should be seen between
the objects and the openings through which the images of these
bodies are admitted; and this is plain, because where the atmosphere
is illuminated these images are not formed visibly.

When the images are made double by mutually crossing each other they
are invariably doubly as dark in tone. To prove this let _d_ _e_ _h_
be such a doubling which although it is only seen within the space
between the bodies in _b_ and _i_ this will not hinder its being
seen from _f_ _g_ or from _f_ _m_; being composed of the images _a_
_b_ _i_ _k_ which run together in _d_ _e_ _h_.

[Footnote: 81. On the original diagram at the beginning of this
chapter Leonardo has written "_azurro_" (blue) where in the
facsimile I have marked _A_, and "_giallo_" (yellow) where _B_
stands.]

[Footnote: 15--23. These lines stand between the diagrams I and III.]

[Footnote: 24--53. These lines stand between the diagrams I and II.]

[Footnote: 54--97 are written along the left side of diagram I.]

Page 80

AS TO WHETHER THE CENTRAL LINE OF THE IMAGE CAN BE INTERSECTED, OR
NOT, WITHIN THE OPENING.

It is impossible that the line should intersect itself; that is,
that its right should cross over to its left side, and so, its left
side become its right side. Because such an intersection demands two
lines, one from each side; for there can be no motion from right to
left or from left to right in itself without such extension and
thickness as admit of such motion. And if there is extension it is
no longer a line but a surface, and we are investigating the
properties of a line, and not of a surface. And as the line, having
no centre of thickness cannot be divided, we must conclude that the
line can have no sides to intersect each other. This is proved by
the movement of the line _a f_ to _a b_ and of the line _e b_ to _e
f_, which are the sides of the surface _a f e b_. But if you move
the line _a b_ and the line _e f_, with the frontends _a e_, to the
spot _c_, you will have moved the opposite ends _f b_ towards each
other at the point _d_. And from the two lines you will have drawn
the straight line _c d_ which cuts the middle of the intersection of
these two lines at the point _n_ without any intersection. For, you
imagine these two lines as having breadth, it is evident that by
this motion the first will entirely cover the other--being equal
with it--without any intersection, in the position _c d_. And this
is sufficient to prove our proposition.

Page 79

OF THE CENTRAL LINE OF THE EYE.

Only one line of the image, of all those that reach the visual
virtue, has no intersection; and this has no sensible dimensions
because it is a mathematical line which originates from a
mathematical point, which has no dimensions.

According to my adversary, necessity requires that the central line
of every image that enters by small and narrow openings into a dark
chamber shall be turned upside down, together with the images of the
bodies that surround it.

Page 78

Necessity has provided that all the images of objects in front of
the eye shall intersect in two places. One of these intersections is
in the pupil, the other in the crystalline lens; and if this were
not the case the eye could not see so great a number of objects as
it does. This can be proved, since all the lines which intersect do
so in a point. Because nothing is seen of objects excepting their
surface; and their edges are lines, in contradistinction to the
definition of a surface. And each minute part of a line is equal to
a point; for _smallest_ is said of that than which nothing can be
smaller, and this definition is equivalent to the definition of the
point. Hence it is possible for the whole circumference of a circle
to transmit its image to the point of intersection, as is shown in
the 4th of this which shows: all the smallest parts of the images
cross each other without interfering with each other. These
demonstrations are to illustrate the eye. No image, even of the
smallest object, enters the eye without being turned upside down;
but as it penetrates into the crystalline lens it is once more
reversed and thus the image is restored to the same position within
the eye as that of the object outside the eye.

Page 77

THE PRINCIPLE ON WHICH THE IMAGES OF BODIES PASS IN BETWEEN THE
MARGINS OF THE OPENINGS BY WHICH THEY ENTER.

What difference is there in the way in which images pass through
narrow openings and through large openings, or in those which pass
by the sides of shaded bodies? By moving the edges of the opening
through which the images are admitted, the images of immovable
objects are made to move. And this happens, as is shown in the 9th
which demonstrates: [Footnote 11: _per la 9a che dicie_. When
Leonardo refers thus to a number it serves to indicate marginal
diagrams; this can in some instances be distinctly proved. The ninth
sketch on the page W. L. 145 b corresponds to the middle sketch of
the three reproduced.] the images of any object are all everywhere,
and all in each part of the surrounding air. It follows that if one
of the edges of the hole by which the images are admitted to a dark
chamber is moved it cuts off those rays of the image that were in
contact with it and gets nearer to other rays which previously were
remote from it &c.

OF THE MOVEMENT OF THE EDGE AT THE RIGHT OR LEFT, OR THE UPPER, OR
LOWER EDGE.

If you move the right side of the opening the image on the left will
move [being that] of the object which entered on the right side of
the opening; and the same result will happen with all the other
sides of the opening. This can be proved by the 2nd of this which
shows: all the rays which convey the images of objects through the
air are straight lines. Hence, if the images of very large bodies
have to pass through very small holes, and beyond these holes
recover their large size, the lines must necessarily intersect.

[Footnote: 77. 2. In the first of the three diagrams Leonardo had
drawn only one of the two margins, et _m_.]

Page 76

The inversion of the images.

All the images of objects which pass through a window [glass pane]
from the free outer air to the air confined within walls, are seen
on the opposite side; and an object which moves in the outer air
from east to west will seem in its shadow, on the wall which is
lighted by this confined air, to have an opposite motion.

Page 75

If the judgment of the eye is situated within it, the straight lines
of the images are refracted on its surface because they pass through
the rarer to the denser medium. If, when you are under water, you
look at objects in the air you will see them out of their true
place; and the same with objects under water seen from the air.

The intersection of the rays (76-82).

Page 74

The lines sent forth by the image of an object to the eye do not
reach the point within the eye in straight lines.

Page 73

The object which is opposite to the pupil of the eye is seen by that
pupil and that which is opposite to the eye is seen by the pupil.

Refraction of the rays falling upon the eye (74. 75)

Page 72

In the practice of perspective the same rules apply to light and to
the eye.

Page 71

HOW THE IMAGES OF OBJECTS RECEIVED BY THE EYE INTERSECT WITHIN THE
CRYSTALLINE HUMOUR OF THE EYE.

An experiment, showing how objects transmit their images or
pictures, intersecting within the eye in the crystalline humour, is
seen when by some small round hole penetrate the images of
illuminated objects into a very dark chamber. Then, receive these
images on a white paper placed within this dark room and rather near
to the hole and you will see all the objects on the paper in their
proper forms and colours, but much smaller; and they will be upside
down by reason of that very intersection. These images being
transmitted from a place illuminated by the sun will seem actually
painted on this paper which must be extremely thin and looked at
from behind. And let the little perforation be made in a very thin
plate of iron. Let _a b e d e_ be the object illuminated by the sun
and _o r_ the front of the dark chamber in which is the said hole at
_n m_. Let _s t_ be the sheet of paper intercepting the rays of the
images of these objects upside down, because the rays being
straight, _a_ on the right hand becomes _k_ on the left, and _e_ on
the left becomes _f_ on the right; and the same takes place inside
the pupil.

[Footnote: This chapter is already known through a translation into
French by VENTURI. Compare his '_Essai sur les ouvrages
physico-mathematiques de L. da Vinci avec des fragments tires de ses
Manuscrits, apportes de l'Italie. Lu a la premiere classe de
l'Institut national des Sciences et Arts.' Paris, An V_ (1797).]

The practice of perspective (72. 73).