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The Notebooks of Leonardo Da Vinci, Volume 1

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[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 ècrit_ 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.

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.

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.

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 più 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.

_III._

_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._

[Footnote III: This text has already been published with some slight
variations in Dozio's pamphlet _Degli scritti e disegni di Leonardo
da Vinci_, Milan 1871, pp. 30--31. Dozio did not transcribe it from
the original MS. which seems to have remained unknown to him, but
from an old copy (MS. H. 227 in the Ambrosian Library).]

GENERAL INTRODUCTION.

Prolegomena.

110.

You must first explain the theory and then the practice. First you
must describe the shadows and lights on opaque objects, and then on
transparent bodies.

Scheme of the books on Light and shade.

111.

INTRODUCTION.

[Having already treated of the nature of shadows and the way in
which they are cast [Footnote 2: _Avendo io tractato._--We may
suppose that he here refers to some particular MS., possibly Paris
C.], I will now consider the places on which they fall; and their
curvature, obliquity, flatness or, in short, any character I may be
able to detect in them.]

Shadow is the obstruction of light. Shadows appear to me to be of
supreme importance in perspective, because, without them opaque and
solid bodies will be ill defined; that which is contained within
their outlines and their boundaries themselves will be
ill-understood unless they are shown against a background of a
different tone from themselves. And therefore in my first
proposition concerning shadow I state that every opaque body is
surrounded and its whole surface enveloped in shadow and light. And
on this proposition I build up the first Book. Besides this, shadows
have in themselves various degrees of darkness, because they are
caused by the absence of a variable amount of the luminous rays; and
these I call Primary shadows because they are the first, and
inseparable from the object to which they belong. And on this I will
found my second Book. From these primary shadows there result
certain shaded rays which are diffused through the atmosphere and
these vary in character according to that of the primary shadows
whence they are derived. I shall therefore call these shadows
Derived shadows because they are produced by other shadows; and the
third Book will treat of these. Again these derived shadows, where
they are intercepted by various objects, produce effects as various
as the places where they are cast and of this I will treat in the
fourth Book. And since all round the derived shadows, where the
derived shadows are intercepted, there is always a space where the
light falls and by reflected dispersion is thrown back towards its
cause, it meets the original shadow and mingles with it and modifies
it somewhat in its nature; and on this I will compose my fifth Book.
Besides this, in the sixth Book I will investigate the many and
various diversities of reflections resulting from these rays which
will modify the original [shadow] by [imparting] some of the various
colours from the different objects whence these reflected rays are
derived. Again, the seventh Book will treat of the various distances
that may exist between the spot where the reflected rays fall and
that where they originate, and the various shades of colour which
they will acquire in falling on opaque bodies.

Different principles and plans of treatment (112--116).

112.

First I will treat of light falling through windows which I will
call Restricted [Light] and then I will treat of light in the open
country, to which I will give the name of diffused Light. Then I
will treat of the light of luminous bodies.

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