_
VOLUME I - II. LINEAR PERSPECTIVE
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) 40.
ON PAINTING.
Perspective is the best guide to the art of Painting.
[Footnote: 40. Compare 53, 2.]
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) 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.
43.
A point is not part of a line.
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.]
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.
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) 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.
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.
The nature of the outline (49) 49.
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.
Definition of Perspective (50) 50.
[Drawing is based upon perspective, which is nothing else than a
thorough knowledge of the function of the eye. And this function
simply consists in receiving in a pyramid the forms and colours of
all the objects placed before it. I say in a pyramid, because there
is no object so small that it will not be larger than the spot where
these pyramids are received into the eye. Therefore, if you extend
the lines from the edges of each body as they converge you will
bring them to a single point, and necessarily the said lines must
form a pyramid.]
[Perspective is nothing more than a rational demonstration applied
to the consideration of how objects in front of the eye transmit
their image to it, by means of a pyramid of lines. The _Pyramid_ is
the name I apply to the lines which, starting from the surface and
edges of each object, converge from a distance and meet in a single
point.]
[Perspective is a rational demonstration, by which we may
practically and clearly understand how objects transmit their own
image, by lines forming a Pyramid (centred) in the eye.]
Perspective is a rational demonstration by which experience confirms
that every object sends its image to the eye by a pyramid of lines;
and bodies of equal size will result in a pyramid of larger or
smaller size, according to the difference in their distance, one
from the other. By a pyramid of lines I mean those which start from
the surface and edges of bodies, and, converging from a distance
meet in a single point. A point is said to be that which [having no
dimensions] cannot be divided, and this point placed in the eye
receives all the points of the cone.
[Footnote: 50. 1-5. Compare with this the Proem. No. 21. The
paragraphs placed in brackets: lines 1-9, 10-14, and 17--20, are
evidently mere sketches and, as such, were cancelled by the writer;
but they serve as a commentary on the final paragraph, lines 22-29.]
The perception of the object depends on the direction of the eye (51) IN WHAT WAY THE EYE SEES OBJECTS PLACED IN FRONT OF IT.
51.
Supposing that the ball figured above is the ball of the eye and let
the small portion of the ball which is cut off by the line _s t_ be
the pupil and all the objects mirrored on the centre of the face of
the eye, by means of the pupil, pass on at once and enter the pupil,
passing through the crystalline humour, which does not interfere in
the pupil with the things seen by means of the light. And the pupil
having received the objects, by means of the light, immediately
refers them and transmits them to the intellect by the line _a b_.
And you must know that the pupil transmits nothing perfectly to the
intellect or common sense excepting when the objects presented to it
by means of light, reach it by the line _a b;_ as, for instance, by
the line _b c_. For although the lines _m n_ and _f g_ may be seen
by the pupil they are not perfectly taken in, because they do not
coincide with the line _a b_. And the proof is this: If the eye,
shown above, wants to count the letters placed in front, the eye
will be obliged to turn from letter to letter, because it cannot
discern them unless they lie in the line _a b;_ as, for instance, in
the line _a c_. All visible objects reach the eye by the lines of a
pyramid, and the point of the pyramid is the apex and centre of it,
in the centre of the pupil, as figured above.
[Footnote: 51. In this problem the eye is conceived of as fixed and
immovable; this is plain from line 11.]
Experimental proof of the existence of the pyramid of sight (52-55) 52.
Perspective is a rational demonstration, confirmed by experience,
that all objects transmit their image to the eye by a pyramid of
lines.
By a pyramid of lines I understand those lines which start from the
edges of the surface of bodies, and converging from a distance, meet
in a single point; and this point, in the present instance, I will
show to be situated in the eye which is the universal judge of all
objects. By a point I mean that which cannot be divided into parts;
therefore this point, which is situated in the eye, being
indivisible, no body is seen by the eye, that is not larger than
this point. This being the case it is inevitable that the lines
which come from the object to the point must form a pyramid. And if
any man seeks to prove that the sense of sight does not reside in
this point, but rather in the black spot which is visible in the
middle of the pupil, I might reply to him that a small object could
never diminish at any distance, as it might be a grain of millet or
of oats or of some similar thing, and that object, if it were larger
than the said [black] spot would never be seen as a whole; as may be
seen in the diagram below. Let _a_. be the seat of sight, _b e_ the
lines which reach the eye. Let _e d_ be the grains of millet within
these lines. You plainly see that these will never diminish by
distance, and that the body _m n_ could not be entirely covered by
it. Therefore you must confess that the eye contains within itself
one single indivisible point _a_, to which all the points converge
of the pyramid of lines starting from an object, as is shown below.
Let _a_. _b_. be the eye; in the centre of it is the point above
mentioned. If the line _e f_ is to enter as an image into so small
an opening in the eye, you must confess that the smaller object
cannot enter into what is smaller than itself unless it is
diminished, and by diminishing it must take the form of a pyramid.
53.
PERSPECTIVE.
Perspective comes in where judgment fails [as to the distance] in
objects which diminish. The eye can never be a true judge for
determining with exactitude how near one object is to another which
is equal to it [in size], if the top of that other is on the level
of the eye which sees them on that side, excepting by means of the
vertical plane which is the standard and guide of perspective. Let
_n_ be the eye, _e f_ the vertical plane above mentioned. Let _a b c
d_ be the three divisions, one below the other; if the lines _a n_
and _c n_ are of a given length and the eye _n_ is in the centre,
then _a b_ will look as large as _b c. c d_ is lower and farther off
from _n_, therefore it will look smaller. And the same effect will
appear in the three divisions of a face when the eye of the painter
who is drawing it is on a level with the eye of the person he is
painting.
54.
TO PROVE HOW OBJECTS REACH THE EYE.
If you look at the sun or some other luminous body and then shut
your eyes you will see it again inside your eye for a long time.
This is evidence that images enter into the eye.
The relations of the distance points to the vanishing point (55-56) 55.
ELEMENTS OF PERSPECTIVE.
All objects transmit their image to the eye in pyramids, and the
nearer to the eye these pyramids are intersected the smaller will
the image appear of the objects which cause them. Therefore, you may
intersect the pyramid with a vertical plane [Footnote 4: _Pariete_.
Compare the definitions in 85, 2-5, 6-27. These lines refer
exclusively to the third diagram. For the better understanding of
this it should be observed that _c s_ must be regarded as
representing the section or profile of a square plane, placed
horizontally (comp. lines 11, 14, 17) for which the word _pianura_
is subsequently employed (20, 22). Lines 6-13 contain certain
preliminary observations to guide the reader in understanding the
diagram; the last three seem to have been added as a supplement.
Leonardo's mistake in writing _t denota_ (line 6) for _f denota_ has
been rectified.] which reaches the base of the pyramid as is shown
in the plane _a n_.
The eye _f_ and the eye _t_ are one and the same thing; but the eye
_f_ marks the distance, that is to say how far you are standing from
the object; and the eye _t_ shows you the direction of it; that is
whether you are opposite, or on one side, or at an angle to the
object you are looking at. And remember that the eye _f_ and the eye
_t_ must always be kept on the same level. For example if you raise
or lower the eye from the distance point _f_ you must do the same
with the direction point _t_. And if the point _f_ shows how far the
eye is distant from the square plane but does not show on which side
it is placed--and, if in the same way, the point _t_ show _s_ the
direction and not the distance, in order to ascertain both you must
use both points and they will be one and the same thing. If the eye
_f_ could see a perfect square of which all the sides were equal to
the distance between _s_ and _c_, and if at the nearest end of the
side towards the eye a pole were placed, or some other straight
object, set up by a perpendicular line as shown at _r s_--then, I
say, that if you were to look at the side of the square that is
nearest to you it will appear at the bottom of the vertical plane _r
s_, and then look at the farther side and it would appear to you at
the height of the point _n_ on the vertical plane. Thus, by this
example, you can understand that if the eye is above a number of
objects all placed on the same level, one beyond another, the more
remote they are the higher they will seem, up to the level of the
eye, but no higher; because objects placed upon the level on which
your feet stand, so long as it is flat--even if it be extended into
infinity--would never be seen above the eye; since the eye has in
itself the point towards which all the cones tend and converge which
convey the images of the objects to the eye. And this point always
coincides with the point of diminution which is the extreme of all
we can see. And from the base line of the first pyramid as far as
the diminishing point
[Footnote: The two diagrams above the chapter are explained by the
first five lines. They have, however, more letters than are referred
to in the text, a circumstance we frequently find occasion to
remark.]
56.
there are only bases without pyramids which constantly diminish up
to this point. And from the first base where the vertical plane is
placed towards the point in the eye there will be only pyramids
without bases; as shown in the example given above. Now, let _a b_
be the said vertical plane and _r_ the point of the pyramid
terminating in the eye, and _n_ the point of diminution which is
always in a straight line opposite the eye and always moves as the
eye moves--just as when a rod is moved its shadow moves, and moves
with it, precisely as the shadow moves with a body. And each point
is the apex of a pyramid, all having a common base with the
intervening vertical plane. But although their bases are equal their
angles are not equal, because the diminishing point is the
termination of a smaller angle than that of the eye. If you ask me:
"By what practical experience can you show me these points?" I
reply--so far as concerns the diminishing point which moves with you
--when you walk by a ploughed field look at the straight furrows
which come down with their ends to the path where you are walking,
and you will see that each pair of furrows will look as though they
tried to get nearer and meet at the [farther] end.
[Footnote: For the easier understanding of the diagram and of its
connection with the preceding I may here remark that the square
plane shown above in profile by the line _c s_ is here indicated by
_e d o p_. According to lines 1, 3 _a b_ must be imagined as a plane
of glass placed perpendicularly at _o p_.]
How to measure the pyramid of vision (57) 57.
As regards the point in the eye; it is made more intelligible by
this: If you look into the eye of another person you will see your
own image. Now imagine 2 lines starting from your ears and going to
the ears of that image which you see in the other man's eye; you
will understand that these lines converge in such a way that they
would meet in a point a little way beyond your own image mirrored in
the eye. And if you want to measure the diminution of the pyramid in
the air which occupies the space between the object seen and the
eye, you must do it according to the diagram figured below. Let _m
n_ be a tower, and _e f_ a, rod, which you must move backwards and
forwards till its ends correspond with those of the tower [Footnote
9: _I sua stremi .. della storre_ (its ends ... of the tower) this
is the case at _e f_.]; then bring it nearer to the eye, at _c d_
and you will see that the image of the tower seems smaller, as at _r
o_. Then [again] bring it closer to the eye and you will see the rod
project far beyond the image of the tower from _a_ to _b_ and from
_t_ to _b_, and so you will discern that, a little farther within,
the lines must converge in a point.
The Production of pyramid of Vision (58-60) 58.
PERSPECTIVE.
The instant the atmosphere is illuminated it will be filled with an
infinite number of images which are produced by the various bodies
and colours assembled in it. And the eye is the target, a loadstone,
of these images.
59.
The whole surface of opaque bodies displays its whole image in all
the illuminated atmosphere which surrounds them on all sides.
60.
That the atmosphere attracts to itself, like a loadstone, all the
images of the objects that exist in it, and not their forms merely
but their nature may be clearly seen by the sun, which is a hot and
luminous body. All the atmosphere, which is the all-pervading
matter, absorbs light and heat, and reflects in itself the image of
the source of that heat and splendour and, in each minutest portion,
does the same. The Northpole does the same as the loadstone shows;
and the moon and the other planets, without suffering any
diminution, do the same. Among terrestrial things musk does the same
and other perfumes.
61.
All bodies together, and each by itself, give off to the surrounding
air an infinite number of images which are all-pervading and each
complete, each conveying the nature, colour and form of the body
which produces it.
It can clearly be shown that all bodies are, by their images,
all-pervading in the surrounding atmosphere, and each complete in
itself as to substance form and colour; this is seen by the images
of the various bodies which are reproduced in one single perforation
through which they transmit the objects by lines which intersect and
cause reversed pyramids, from the objects, so that they are upside
down on the dark plane where they are first reflected. The reason of
this is--
[Footnote: The diagram intended to illustrate the statement (Pl. II
No. i) occurs in the original between lines 3 and 4. The three
circles must be understood to represent three luminous bodies which
transmit their images through perforations in a wall into a dark
chamber, according to a law which is more fully explained in 75?81.
So far as concerns the present passage the diagram is only intended
to explain that the images of the three bodies may be made to
coalesce at any given spot. In the circles are written,
giallo--yellow, biacho--white, rosso--red.
The text breaks off at line 8. The paragraph No.40 follows here in
the original MS.]
62.
Every point is the termination of an infinite number of lines, which
diverge to form a base, and immediately, from the base the same
lines converge to a pyramid [imaging] both the colour and form. No
sooner is a form created or compounded than suddenly infinite lines
and angles are produced from it; and these lines, distributing
themselves and intersecting each other in the air, give rise to an
infinite number of angles opposite to each other. Given a base, each
opposite angle, will form a triangle having a form and proportion
equal to the larger angle; and if the base goes twice into each of
the 2 lines of the pyramid the smaller triangle will do the same.
63.
Every body in light and shade fills the surrounding air with
infinite images of itself; and these, by infinite pyramids diffused
in the air, represent this body throughout space and on every side.
Each pyramid that is composed of a long assemblage of rays includes
within itself an infinite number of pyramids and each has the same
power as all, and all as each. A circle of equidistant pyramids of
vision will give to their object angles of equal size; and an eye at
each point will see the object of the same size. The body of the
atmosphere is full of infinite pyramids composed of radiating
straight lines, which are produced from the surface of the bodies in
light and shade, existing in the air; and the farther they are from
the object which produces them the more acute they become and
although in their distribution they intersect and cross they never
mingle together, but pass through all the surrounding air,
independently converging, spreading, and diffused. And they are all
of equal power [and value]; all equal to each, and each equal to
all. By these the images of objects are transmitted through all
space and in every direction, and each pyramid, in itself, includes,
in each minutest part, the whole form of the body causing it.
64.
The body of the atmosphere is full of infinite radiating pyramids
produced by the objects existing in it. These intersect and cross
each other with independent convergence without interfering with
each other and pass through all the surrounding atmosphere; and are
of equal force and value--all being equal to each, each to all. And
by means of these, images of the body are transmitted everywhere and
on all sides, and each receives in itself every minutest portion of
the object that produces it.
Proof by experiment (65-66) 65.
PERSPECTIVE.
The air is filled with endless images of the objects distributed in
it; and all are represented in all, and all in one, and all in each,
whence it happens that if two mirrors are placed in such a manner as
to face each other exactly, the first will be reflected in the
second and the second in the first. The first being reflected in the
second takes to it the image of itself with all the images
represented in it, among which is the image of the second mirror,
and so, image within image, they go on to infinity in such a manner
as that each mirror has within it a mirror, each smaller than the
last and one inside the other. Thus, by this example, it is clearly
proved that every object sends its image to every spot whence the
object itself can be seen; and the converse: That the same object
may receive in itself all the images of the objects that are in
front of it. Hence the eye transmits through the atmosphere its own
image to all the objects that are in front of it and receives them
into itself, that is to say on its surface, whence they are taken in
by the common sense, which considers them and if they are pleasing
commits them to the memory. Whence I am of opinion: That the
invisible images in the eyes are produced towards the object, as the
image of the object to the eye. That the images of the objects must
be disseminated through the air. An instance may be seen in several
mirrors placed in a circle, which will reflect each other endlessly.
When one has reached the other it is returned to the object that
produced it, and thence--being diminished--it is returned again to
the object and then comes back once more, and this happens
endlessly. If you put a light between two flat mirrors with a
distance of 1 braccio between them you will see in each of them an
infinite number of lights, one smaller than another, to the last. If
at night you put a light between the walls of a room, all the parts
of that wall will be tinted with the image of that light. And they
will receive the light and the light will fall on them, mutually,
that is to say, when there is no obstacle to interrupt the
transmission of the images. This same example is seen in a greater
degree in the distribution of the solar rays which all together, and
each by itself, convey to the object the image of the body which
causes it. That each body by itself alone fills with its images the
atmosphere around it, and that the same air is able, at the same
time, to receive the images of the endless other objects which are
in it, this is clearly proved by these examples. And every object is
everywhere visible in the whole of the atmosphere, and the whole in
every smallest part of it; and all the objects in the whole, and all
in each smallest part; each in all and all in every part.
66.
The images of objects are all diffused through the atmosphere which
receives them; and all on every side in it. To prove this, let _a c
e_ be objects of which the images are admitted to a dark chamber by
the small holes _n p_ and thrown upon the plane _f i_ opposite to
these holes. As many images will be produced in the chamber on the
plane as the number of the said holes.
General conclusions (67) 67.
All objects project their whole image and likeness, diffused and
mingled in the whole of the atmosphere, opposite to themselves. The
image of every point of the bodily surface, exists in every part of
the atmosphere. All the images of the objects are in every part of
the atmosphere. The whole, and each part of the image of the
atmosphere is [reflected] in each point of the surface of the bodies
presented to it. Therefore both the part and the whole of the images
of the objects exist, both in the whole and in the parts of the
surface of these visible bodies. Whence we may evidently say that
the image of each object exists, as a whole and in every part, in
each part and in the whole interchangeably in every existing body.
As is seen in two mirrors placed opposite to each other.
That the contrary is impossible (68) 68.
It is impossible that the eye should project from itself, by visual
rays, the visual virtue, since, as soon as it opens, that front
portion [of the eye] which would give rise to this emanation would
have to go forth to the object and this it could not do without
time. And this being so, it could not travel so high as the sun in a
month's time when the eye wanted to see it. And if it could reach
the sun it would necessarily follow that it should perpetually
remain in a continuous line from the eye to the sun and should
always diverge in such a way as to form between the sun and the eye
the base and the apex of a pyramid. This being the case, if the eye
consisted of a million worlds, it would not prevent its being
consumed in the projection of its virtue; and if this virtue would
have to travel through the air as perfumes do, the winds would bent
it and carry it into another place. But we do [in fact] see the mass
of the sun with the same rapidity as [an object] at the distance of
a braccio, and the power of sight is not disturbed by the blowing of
the winds nor by any other accident.
[Footnote: The view here refuted by Leonardo was maintained among
others by Bramantino, Leonardo's Milanese contemporary. LOMAZZO
writes as follows in his Trattato dell' Arte della pittura &c.
(Milano 1584. Libr. V cp. XXI): Sovviemmi di aver gia letto in certi
scritti alcune cose di Bramantino milanese, celebratissimo pittore,
attenente alla prospettiva, le quali ho voluto riferire, e quasi
intessere in questo luogo, affinche sappiamo qual fosse l'opinione
di cosi chiaro e famoso pittore intorno alla prospettiva . . Scrive
Bramantino che la prospettiva e una cosa che contrafa il naturale, e
che cio si fa in tre modi
Circa il primo modo che si fa con ragione, per essere la cosa in
poche parole conclusa da Bramantino in maniera che giudico non
potersi dir meglio, contenendovi si tutta Parte del principio al
fine, io riferiro per appunto le proprie parole sue (cp. XXII, Prima
prospettiva di Bramantino). La prima prospettiva fa le cose di
punto, e l'altra non mai, e la terza piu appresso. Adunque la prima
si dimanda prospettiva, cioe ragione, la quale fa l'effetto dell'
occhio, facendo crescere e calare secondo gli effetti degli occhi.
Questo crescere e calare non procede della cosa propria, che in se
per esser lontana, ovvero vicina, per quello effetto non puo
crescere e sminuire, ma procede dagli effetti degli occhi, i quali
sono piccioli, e percio volendo vedere tanto gran cosa_, bisogna che
mandino fuora la virtu visiva, _la quale si dilata in tanta
larghezza, che piglia tutto quello che vuoi vedere, ed_ arrivando a
quella cosa la vede dove e: _e da lei agli occhi per quello circuito
fino all' occhio, e tutto quello termine e pieno di quella cosa_.
It is worthy of note that Leonardo had made his memorandum refuting
this view, at Milan in 1492]
A parallel case (69) 69.
Just as a stone flung into the water becomes the centre and cause of
many circles, and as sound diffuses itself in circles in the air: so
any object, placed in the luminous atmosphere, diffuses itself in
circles, and fills the surrounding air with infinite images of
itself. And is repeated, the whole every-where, and the whole in
every smallest part. This can be proved by experiment, since if you
shut a window that faces west and make a hole [Footnote: 6. Here the
text breaks off.] . .
[Footnote: Compare LIBRI, _Histoire des sciences mathematiques en
Italie_. Tome III, p. 43.]
The function of the eye as explained by the camera obscura (70-71) 70.
If the object in front of the eye sends its image to the eye, the
eye, on the other hand, sends its image to the object, and no
portion whatever of the object is lost in the images it throws off,
for any reason either in the eye or the object. Therefore we may
rather believe it to be the nature and potency of our luminous
atmosphere which absorbs the images of the objects existing in it,
than the nature of the objects, to send their images through the
air. If the object opposite to the eye were to send its image to the
eye, the eye would have to do the same to the object, whence it
might seem that these images were an emanation. But, if so, it would
be necessary [to admit] that every object became rapidly smaller;
because each object appears by its images in the surrounding
atmosphere. That is: the whole object in the whole atmosphere, and
in each part; and all the objects in the whole atmosphere and all of
them in each part; speaking of that atmosphere which is able to
contain in itself the straight and radiating lines of the images
projected by the objects. From this it seems necessary to admit that
it is in the nature of the atmosphere, which subsists between the
objects, and which attracts the images of things to itself like a
loadstone, being placed between them.
PROVE HOW ALL OBJECTS, PLACED IN ONE POSITION, ARE ALL EVERYWHERE
AND ALL IN EACH PART.
I say that if the front of a building--or any open piazza or
field--which is illuminated by the sun has a dwelling opposite to
it, and if, in the front which does not face the sun, you make a
small round hole, all the illuminated objects will project their
images through that hole and be visible inside the dwelling on the
opposite wall which may be made white; and there, in fact, they will
be upside down, and if you make similar openings in several places
in the same wall you will have the same result from each. Hence the
images of the illuminated objects are all everywhere on this wall
and all in each minutest part of it. The reason, as we clearly know,
is that this hole must admit some light to the said dwelling, and
the light admitted by it is derived from one or many luminous
bodies. If these bodies are of various colours and shapes the rays
forming the images are of various colours and shapes, and so will
the representations be on the wall.
[Footnote: 70. 15--23. This section has already been published in the
"_Saggio delle Opere di Leonardo da Vinci_" Milan 1872, pp. 13, 14.
G. Govi observes upon it, that Leonardo is not to be regarded as the
inventor of the Camera obscura, but that he was the first to explain
by it the structure of the eye. An account of the Camera obscura
first occurs in CESARE CESARINI's Italian version of Vitruvius, pub.
1523, four years after Leonardo's death. Cesarini expressly names
Benedettino Don Papnutio as the inventor of the Camera obscura. In
his explanation of the function of the eye by a comparison with the
Camera obscura Leonardo was the precursor of G. CARDANO, Professor
of Medicine at Bologna (died 1576) and it appears highly probable
that this is, in fact, the very discovery which Leonardo ascribes to
himself in section 21 without giving any further details.]
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) 72.
In the practice of perspective the same rules apply to light and to
the eye.
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) 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.
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 inversion of the images (76) 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.
The intersection of the rays (77-82) 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_.]
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.
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.
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.
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.]
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) 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.
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.
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) 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.
87.
How every large mass sends forth its images, which may diminish
through infinity.
The images of any large mass being infinitely divisible may be
infinitely diminished.
88.
Objects of equal size, situated in various places, will be seen by
different pyramids which will each be smaller in proportion as the
object is farther off.
Opposite pyramids in juxtaposition (89) 89.
Perspective, in dealing with distances, makes use of two opposite
pyramids, one of which has its apex in the eye and the base as
distant as the horizon. The other has the base towards the eye and
the apex on the horizon. Now, the first includes the [visible]
universe, embracing all the mass of the objects that lie in front of
the eye; as it might be a vast landscape seen through a very small
opening; for the more remote the objects are from the eye, the
greater number can be seen through the opening, and thus the pyramid
is constructed with the base on the horizon and the apex in the eye,
as has been said. The second pyramid is extended to a spot which is
smaller in proportion as it is farther from the eye; and this second
perspective [= pyramid] results from the first.
On simple and complex perspective (90) SIMPLE PERSPECTIVE.
Simple perspective is that which is constructed by art on a vertical
plane which is equally distant from the eye in every part. Complex
perspective is that which is constructed on a ground-plan in which
none of the parts are equally distant from the eye.
The proper distance of objects from the eye (91-92) 91.
PERSPECTIVE.
No surface can be seen exactly as it is, if the eye that sees it is
not equally remote from all its edges.
92.
WHY WHEN AN OBJECT IS PLACED CLOSE TO THE EYE ITS EDGES ARE
INDISTINCT.
When an object opposite the eye is brought too close to it, its
edges must become too confused to be distinguished; as it happens
with objects close to a light, which cast a large and indistinct
shadow, so is it with an eye which estimates objects opposite to it;
in all cases of linear perspective, the eye acts in the same way as
the light. And the reason is that the eye has one leading line (of
vision) which dilates with distance and embraces with true
discernment large objects at a distance as well as small ones that
are close. But since the eye sends out a multitude of lines which
surround this chief central one and since these which are farthest
from the centre in this cone of lines are less able to discern with
accuracy, it follows that an object brought close to the eye is not
at a due distance, but is too near for the central line to be able
to discern the outlines of the object. So the edges fall within the
lines of weaker discerning power, and these are to the function of
the eye like dogs in the chase which can put up the game but cannot
take it. Thus these cannot take in the objects, but induce the
central line of sight to turn upon them, when they have put them up.
Hence the objects which are seen with these lines of sight have
confused outlines.
The relative size of objects with regard to their distance from the eye (93-98) 93.
PERSPECTIVE.
Small objects close at hand and large ones at a distance, being seen
within equal angles, will appear of the same size.
94.
PERSPECTIVE.
There is no object so large but that at a great distance from the
eye it does not appear smaller than a smaller object near.
95.
Among objects of equal size that which is most remote from the eye
will look the smallest. [Footnote: This axiom, sufficiently clear in
itself, is in the original illustrated by a very large diagram,
constructed like that here reproduced under No. 108.
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 piu 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.
PERSPECTIVE.
Among objects of equal size, that which is most remote from the eye
will look the smallest.
98.
PERSPECTIVE.
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.
PERSPECTIVE.
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.
PERSPECTIVE.
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.
PERSPECTIVE--PROPOSITION.
Objects that are farther off can never be so large but that those in
front, though smaller, will conceal or surround them.
DEFINITION.
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.
OF LINEAR PERSPECTIVE.
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.
OF THE DIMINUTION OF OBJECTS AT VARIOUS DISTANCES.
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.
OF THE DEGREES OF DIMINUTION.
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.
I GIVE THE DEGREES OF THE OBJECTS SEEN BY THE EYE AS THE MUSICIAN
DOES THE NOTES HEARD BY THE EAR.
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.
PERSPECTIVE.
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.
PERSPECTIVE.
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.
OF EQUAL OBJECTS THE MOST REMOTE LOOK THE SMALLEST.
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.
OF A MIXTURE OF NATURAL AND ARTIFICIAL PERSPECTIVE.
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. _