(no subject)
[Footnote: In the diagrams _A_ stands for _celo_ (sky), _B_ for
_cadela_ (candle).]
169.
ALL BODIES, IN PROPORTION AS THEY ARE NEARER TO, OR FARTHER FROM THE
SOURCE OF LIGHT, WILL PRODUCE LONGER OR SHORTER DERIVED SHADOWS.
Among bodies of equal size, that one which is illuminated by the
largest light will have the shortest shadow. Experiment confirms
this proposition. Thus the body _m_ _n_ is surrounded by a larger
amount of light than the body _p q_, as is shown above. Let us say
that _v c a b d x_ is the sky, the source of light, and that _s t_
is a window by which the luminous rays enter, and so _m n_ and _p q_
are bodies in light and shade as exposed to this light; _m n_ will
have a small derived shadow, because its original shadow will be
small; and the derivative light will be large, again, because the
original light _c d_ will be large and _p q_ will have more derived
shadow because its original shadow will be larger, and its derived
light will be smaller than that of the body _m n_ because that
portion of the hemisphere _a b_ which illuminates it is smaller than
the hemisphere _c d_ which illuminates the body _m n_.
[Footnote: The diagram, given on Pl. IV, No. 2, stands in the
original between lines 2 and 7, while the text of lines 3 to 6 is
written on its left side. In the reproduction of this diagram the
letter _v_ at the outer right-hand end has been omitted.]
170.
The shadow _m_ bears the same proportion to the shadow _n_ as the
line _b c_ to the line _f c_.
171.
OF PAINTING.
Of different shadows of equal strength that which is nearest the eye
will seem the least strong.
Why is the shadow _e a b_ in the first grade of strength, _b c_ in
the second; _c d_ in the third? The reason is that as from _e a b_
the sky is nowhere visible, it gets no light whatever from the sky,
and so has no direct [primary] light. _b c_ faces the portion of the
sky _f g_ and is illuminated by it. _c d_ faces the sky at _h k_. _c
d_, being exposed to a larger extent of sky than _b c_, it is
reasonable that it should be more lighted. And thus, up to a certain
distance, the wall _a d_ will grow lighter for the reasons here
given, until the darkness of the room overpowers the light from the
window.
172.
When the light of the atmosphere is restricted [by an opening] and
illuminates bodies which cast shadows, these bodies being equally
distant from the centre of the window, that which is most obliquely
placed will cast the largest shadow beyond it.
173.
These bodies standing apart in a room lighted by a single window
will have derivative shadows more or less short according as they
are more or less opposite to the window. Among the shadows cast by
bodies of equal mass but at unequal distances from the opening by
which they are illuminated, that shadow will be the longest of the
body which is least in the light. And in proportion as one body is
better illuminated than another its shadow will be shorter than
another. The proportion _n m_ and _e v k_ bear to _r t_ and _v x_
corresponds with that of the shadow _x_ to 4 and _y_.
The reason why those bodies which are placed most in front of the
middle of the window throw shorter shadows than those obliquely
situated is:--That the window appears in its proper form and to the
obliquely placed ones it appears foreshortened; to those in the
middle, the window shows its full size, to the oblique ones it
appears smaller; the one in the middle faces the whole hemisphere
that is _e f_ and those on the side have only a strip; that is _q r_
faces _a b_; and _m n_ faces _c d_; the body in the middle having a
larger quantity of light than those at the sides is lighted from a
point much below its centre, and thus the shadow is shorter. And the
pyramid _g_ 4 goes into _l y_ exactly as often as _a b_ goes into _e
f_. The axis of every derivative shadow passes through 6 1/2
[Footnote 31: _passa per_ 6 1/2 (passes through 6 1/2). The meaning
of these words is probably this: Each of the three axes of the
derived shadow intersects the centre (_mezzo_) of the primary shadow
(_ombra originale_) and, by prolongation upwards crosses six lines.
This is self evident only in the middle diagram; but it is equally
true of the side figures if we conceive of the lines 4 _f_, _x n v
m_, _y l k v_, and 4 _e_, as prolonged beyond the semicircle of the
horizon.] and is in a straight line with the centre of the primary
shadow, with the centre of the body casting it and of the derivative
light and with the centre of the window and, finally, with the
centre of that portion of the source of light which is the celestial
hemisphere, _y h_ is the centre of the derived shade, _l h_ of the
primary shadow, _l_ of the body throwing it, _l k_ of the derived
light, _v_ is the centre of the window, _e_ is the final centre of
the original light afforded by that portion of the hemisphere of the
sky which illuminates the solid body.
[Footnote: Compare the diagram on Pl. IV, No. 3. In the original
this drawing is placed between lines 3 and 22; the rest, from line 4
to line 21, is written on the left hand margin.]
174.
THE FARTHER THE DERIVED SHADOW IS PROLONGED THE LIGHTER IT BECOMES.
You will find that the proportion of the diameter of the derived
shadow to that of the primary shadow will be the same as that
between the darkness of the primary shadow and that of the derived
shadow.
[Footnote 6: Compare No. 177.] Let _a b_ be the diameter of the
primary shadow and _c d_ that of the derived shadow, I say that _a
b_ going, as you see, three times into _d c_, the shadow _d c_ will
be three times as light as the shadow _a b_. [Footnote 8: Compare
No. 177.]
If the size of the illuminating body is larger than that of the
illuminated body an intersection of shadow will occur, beyond which
the shadows will run off in two opposite directions as if they were
caused by two separate lights.
On the relative intensity of derived shadows (175-179).
175.
ON PAINTING.
The derived shadow is stronger in proportion as it is nearer to its
place of origin.
176.
HOW SHADOWS FADE AWAY AT LONG DISTANCES.
Shadows fade and are lost at long distances because the larger
quantity of illuminated air which lies between the eye and the
object seen tints the shadow with its own colour.
177.
_a b_ will be darker than _c d_ in proportion as _c d_ is broader
than _a b_.
[Footnote: In the original MS. the word _lume_ (light) is written at
the apex of the pyramid.]
178.
It can be proved why the shadow _o p c h_ is darker in proportion as
it is nearer to the line _p h_ and is lighter in proportion as it is
nearer to the line _o c_. Let the light _a b_, be a window, and let
the dark wall in which this window is, be _b s_, that is, one of the
sides of the wall.
Then we may say that the line _p h_ is darker than any other part of
the space _o p c h_, because this line faces the whole surface in
shadow of [Footnote: In the original the diagram is placed between
lines 27 and 28.] the wall _b s_. The line _o c_ is lighter than the
other part of this space _o p c h_, because this line faces the
luminous space _a b_.
Where the shadow is larger, or smaller, or equal the body which
casts it.
[First of the character of divided lights. [Footnote 14: _lumi
divisi_. The text here breaks off abruptly.]
OF THE COMPOUND SHADOW _F, R, C, H_ CAUSED BY A SINGLE LIGHT.
The shadow _f r c h_ is under such conditions as that where it is
farthest from its inner side it loses depth in proportion. To prove
this:
Let _d a_, be the light and _f n_ the solid body, and let _a e_ be
one of the side walls of the window that is _d a_. Then I
say--according to the 2nd [proposition]: that the surface of any
body is affected by the tone of the objects surrounding it,--that
the side _r c_, which faces the dark wall _a e_ must participate of
its darkness and, in the same way that the outer surface which faces
the light _d a_ participates of the light; thus we get the outlines
of the extremes on each side of the centre included between them.]
This is divided into four parts. The first the extremes, which
include the compound shadow, secondly the compound shadow between
these extremes.
179.
THE ACTION OF THE LIGHT AS FROM ITS CENTRE.
If it were the whole of the light that caused the shadows beyond the
bodies placed in front of it, it would follow that any body much
smaller than the light would cast a pyramidal shadow; but experience
not showing this, it must be the centre of the light that produces
this effect.
[Footnote: The diagram belonging to this passage is between lines 4
and 5 in the original. Comp. the reproduction Pl. IV, No. 4. The
text and drawing of this chapter have already been published with
tolerable accuracy. See M. JORDAN: "_Das Malerbuch des Leonardo da
Vinci_". Leipzig 1873, P. 90.]
PROOF.
Let _a b_ be the width of the light from a window, which falls on a
stick set up at one foot from _a c_ [Footnote 6: _bastone_ (stick).
The diagram has a sphere in place of a stick.]. And let _a d_ be the
space where all the light from the window is visible. At _c e_ that
part of the window which is between _l b_ cannot be seen. In the
same way _a m_ cannot be seen from _d f_ and therefore in these two
portions the light begins to fail.
Shadow as produced by two lights of different size (180. 181).
180.
A body in light and shade placed between two equal lights side by
side will cast shadows in proportion to the [amount of] light. And
the shadows will be one darker than the other in proportion as one
light is nearer to the said body than the other on the opposite
side.
A body placed at an equal distance between two lights will cast two
shadows, one deeper than the other in proportion, as the light which
causes it is brighter than the other.
[Footnote: In the MS. the larger diagram is placed above the first
line; the smaller one between l. 4 & 5.]
181.
A light which is smaller than the body it illuminates produces
shadows of which the outlines end within [the surface of] the body,
and not much compound shadow; and falls on less than half of it. A
light which is larger than the body it illuminates, falls on more
than half of it, and produces much compound shadow.
The effect of light at different distances.
182.
OF THE SHADOW CAST BY A BODY PLACED BETWEEN 2 EQUAL LIGHTS.
A body placed between 2 equal lights will cast 2 shadows of itself
in the direction of the lines of the 2 lights; and if you move this
body placing it nearer to one of the lights the shadow cast towards
the nearer light will be less deep than that which falls towards the
more distant one.
Further complications in the derived shadows (183-187).
183.
The greatest depth of shadow is in the simple derived shadow because
it is not lighted by either of the two lights _a b, c d_.
The next less deep shadow is the derived shadow _e f n_; and in this
the shadow is less by half, because it is illuminated by a single
light, that is _c d_.
This is uniform in natural tone because it is lighted throughout by
one only of the two luminous bodies [10]. But it varies with the
conditions of shadow, inasmuch as the farther it is away from the
light the less it is illuminated by it [13].
The third degree of depth is the middle shadow [Footnote 15: We
gather from what follows that _q g r_ here means _ombra media_ (the
middle shadow).]. But this is not uniform in natural tone; because
the nearer it gets to the simple derived shadow the deeper it is
[Footnote 18: Compare lines 10-13], and it is the uniformly gradual
diminution by increase of distance which is what modifies it
[Footnote 20: See Footnote 18]: that is to say the depth of a shadow
increases in proportion to the distance from the two lights.
The fourth is the shadow _k r s_ and this is all the darker in
natural tone in proportion as it is nearer to _k s_, because it gets
less of the light _a o_, but by the accident [of distance] it is
rendered less deep, because it is nearer to the light _c d_, and
thus is always exposed to both lights.
The fifth is less deep in shadow than either of the others because
it is always entirely exposed to one of the lights and to the whole
or part of the other; and it is less deep in proportion as it is
nearer to the two lights, and in proportion as it is turned towards
the outer side _x t_; because it is more exposed to the second light
_a b_.
[Footnote: The diagram to this section is given on Pl. V. To the
left is the facsimile of the beginning of the text belonging to it.]
184.
OF SIMPLE SHADOWS.
Why, at the intersections _a_, _b_ of the two compound shadows _e f_
and _m e_, is a simple shadow pfoduced as at _e h_ and _m g_, while
no such simple shadow is produced at the other two intersections _c
d_ made by the very same compound shadows?
ANSWER.
Compound shadow are a mixture of light and shade and simple shadows
are simply darkness. Hence, of the two lights _n_ and _o_, one falls
on the compound shadow from one side, and the other on the compound
shadow from the other side, but where they intersect no light falls,
as at _a b_; therefore it is a simple shadow. Where there is a
compound shadow one light or the other falls; and here a difficulty
arises for my adversary since he says that, where the compound
shadows intersect, both the lights which produce the shadows must of
necessity fall and therefore these shadows ought to be neutralised;
inasmuch as the two lights do not fall there, we say that the shadow
is a simple one and where only one of the two lights falls, we say
the shadow is compound, and where both the lights fall the shadow is
neutralised; for where both lights fall, no shadow of any kind is
produced, but only a light background limiting the shadow. Here I
shall say that what my adversary said was true: but he only mentions
such truths as are in his favour; and if we go on to the rest he
must conclude that my proposition is true. And that is: That if both
lights fell on the point of intersection, the shadows would be
neutralised. This I confess to be true if [neither of] the two
shadows fell in the same spot; because, where a shadow and a light
fall, a compound shadow is produced, and wherever two shadows or two
equal lights fall, the shadow cannot vary in any part of it, the
shadows and the lights both being equal. And this is proved in the
eighth [proposition] on proportion where it is said that if a given
quantity has a single unit of force and resistance, a double
quantity will have double force and double resistance.
DEFINITION.
The intersection _n_ is produced by the shadows caused by the light
_b_, because this light _b_ produces the shadow _x b_, and the
shadow _s b_, but the intersection _m_ is produced by the light _a_
which causes the shadow _s a_, and the shadow _x a_.
But if you uncover both the lights _a b_, then you get the two
shadows _n m_ both at once, and besides these, two other, simple
shadows are produced at _r o_ where neither of the two lights falls
at all. The grades of depth in compound shadows are fewer in
proportion as the lights falling on, and crossing them are less
numerous.
186.
Why the intersections at _n_ being composed of two compound derived
shadows, forms a compound shadow and not a simple one, as happens
with other intersections of compound shadows. This occurs, according
to the 2nd [diagram] of this [prop.] which says:--The intersection
of derived shadows when produced by the intersection of columnar
shadows caused by a single light does not produce a simple shadow.
And this is the corollary of the 1st [prop.] which says:--The
intersection of simple derived shadows never results in a deeper
shadow, because the deepest shadows all added together cannot be
darker than one by itself. Since, if many deepest shadows increased
in depth by their duplication, they could not be called the
_deepest_ shadows, but only part-shadows. But if such intersections
are illuminated by a second light placed between the eye and the
intersecting bodies, then those shadows would become compound
shadows and be uniformly dark just as much at the intersection as
throughout the rest. In the 1st and 2nd above, the intersections _i
k_ will not be doubled in depth as it is doubled in quantity. But in
this 3rd, at the intersections _g n_ they will be double in depth
and in quantity.
187.
HOW AND WHEN THE SURROUNDINGS IN SHADOW MINGLE THEIR DERIVED SHADOW
WITH THE LIGHT DERIVED FROM THE LUMINOUS BODY.
The derived shadow of the dark walls on each side of the bright
light of the window are what mingle their various degrees of shade
with the light derived from the window; and these various depths of
shade modify every portion of the light, except where it is
strongest, at _c_. To prove this let _d a_ be the primary shadow
which is turned towards the point _e_, and darkens it by its derived
shadow; as may be seen by the triangle _a e d_, in which the
angle _e_ faces the darkened base _d a e_; the point _v_ faces the
dark shadow _a s_ which is part of _a d_, and as the whole is
greater than a part, _e_ which faces the whole base [of the
triangle], will be in deeper shadow than _v_ which only faces part
of it. In consequence of the conclusion [shown] in the above
diagram, _t_ will be less darkened than _v_, because the base of the
_t_ is part of the base of the _v_; and in the same way it follows
that _p_ is less in shadow than _t_, because the base of the _p_ is
part of the base of the _t_. And _c_ is the terminal point of the
derived shadow and the chief beginning of the highest light.
[Footnote: The diagram on Pl. IV, No. 5 belongs to this passage; but
it must be noted that the text explains only the figure on the
right-hand side.]
FOURTH BOOK ON LIGHT AND SHADE.
On the shape of the cast shadows (188-191).
188.
The form of the shadow cast by any body of uniform density can never
be the same as that of the body producing it. [Footnote: Comp. the
drawing on PI. XXVIII, No. 5.]
189.
No cast shadow can produce the true image of the body which casts it
on a vertical plane unless the centre of the light is equally
distant from all the edges of that body.
190.
If a window _a b_ admits the sunlight into a room, the sunlight will
magnify the size of the window and diminish the shadow of a man in
such a way as that when the man makes that dim shadow of himself,
approach to that which defines the real size of the window, he will
see the shadows where they come into contact, dim and confused from
the strength of the light, shutting off and not allowing the solar
rays to pass; the effect of the shadow of the man cast by this
contact will be exactly that figured above.
[Footnote: It is scarcely possible to render the meaning of this
sentence with strict accuracy; mainly because the grammatical
construction is defective in the most important part--line 4. In the
very slight original sketch the shadow touches the upper arch of the
window and the correction, here given is perhaps not justified.]
191.
A shadow is never seen as of uniform depth on the surface which
intercepts it unless every portion of that surface is equidistant
from the luminous body. This is proved by the 7th which says:--The
shadow will appear lighter or stronger as it is surrounded by a
darker or a lighter background. And by the 8th of this:--The
background will be in parts darker or lighter, in proportion as it
is farther from or nearer to the luminous body. And:--Of various
spots equally distant from the luminous body those will always be in
the highest light on which the rays fall at the smallest angles: The
outline of the shadow as it falls on inequalities in the surface
will be seen with all the contours similar to those of the body that
casts it, if the eye is placed just where the centre of the light
was.
The shadow will look darkest where it is farthest from the body that
casts it. The shadow _c d_, cast by the body in shadow _a b_ which
is equally distant in all parts, is not of equal depth because it is
seen on a back ground of varying brightness. [Footnote: Compare the
three diagrams on Pl. VI, no 1 which, in the original accompany this
section.]
On the outlines of cast shadows (192-195).
192.
The edges of a derived shadow will be most distinct where it is cast
nearest to the primary shadow.
193.
As the derived shadow gets more distant from the primary shadow, the
more the cast shadow differs from the primary shadow.
194.
OF SHADOWS WHICH NEVER COME TO AN END.
The greater the difference between a light and the body lighted by
it, the light being the larger, the more vague will be the outlines
of the shadow of that object.
The derived shadow will be most confused towards the edges of its
interception by a plane, where it is remotest from the body casting
it.
195.
What is the cause which makes the outlines of the shadow vague and
confused?
Whether it is possible to give clear and definite outlines to the
edges of shadows.
On the relative size of shadows (196. 197).
196.
THE BODY WHICH IS NEAREST TO THE LIGHT CASTS THE LARGEST SHADOW, AND
WHY?
If an object placed in front of a single light is very close to it
you will see that it casts a very large shadow on the opposite wall,
and the farther you remove the object from the light the smaller
will the image of the shadow become.
WHY A SHADOW LARGER THAN THE BODY THAT PRODUCES IT BECOMES OUT OF
PROPORTION.
The disproportion of a shadow which is larger than the body
producing it, results from the light being smaller than the body, so
that it cannot be at an equal distance from the edges of the body
[Footnote 11: H. LUDWIG in his edition of the old copies, in the
Vatican library--in which this chapter is included under Nos. 612,
613 and 614 alters this passage as follows: _quella parte ch'e piu
propinqua piu cresce che le distanti_, although the Vatican copy
agrees with the original MS. in having _distante_ in the former and
_propinque_ in the latter place. This supposed amendment seems to me
to invert the facts. Supposing for instance, that on Pl. XXXI No. 3.
_f_ is the spot where the light is that illuminates the figure there
represented, and that the line behind the figure represents a wall
on which the shadow of the figure is thrown. It is evident, that in
that case the nearest portion, in this case the under part of the
thigh, is very little magnified in the shadow, and the remoter
parts, for instance the head, are more magnified.]; and the portions
which are most remote are made larger than the nearer portions for
this reason [Footnote 12: See Footnote 11].
WHY A SHADOW WHICH IS LARGER THAN THE BODY CAUSING IT HAS
ILL-DEFINED OUTLINES.
The atmosphere which surrounds a light is almost like light itself
for brightness and colour; but the farther off it is the more it
loses this resemblance. An object which casts a large shadow and is
near to the light, is illuminated both by that light by the luminous
atmosphere; hence this diffused light gives the shadow ill-defined
edges.
197.
A luminous body which is long and narrow in shape gives more
confused outlines to the derived shadow than a spherical light, and
this contradicts the proposition next following: A shadow will have
its outlines more clearly defined in proportion as it is nearer to
the primary shadow or, I should say, the body casting the shadow;
[Footnote 14: The lettering refers to the lower diagram, Pl. XLI,
No. 5.] the cause of this is the elongated form of the luminous body
_a c_, &c. [Footnote 16: See Footnote 14].
Effects on cast shadows by the tone of the back ground.
198.
OF MODIFIED SHADOWS.
Modified shadows are those which are cast on light walls or other
illuminated objects.
A shadow looks darkest against a light background. The outlines of a
derived shadow will be clearer as they are nearer to the primary
shadow. A derived shadow will be most defined in shape where it is
intercepted, where the plane intercepts it at the most equal angle.
Those parts of a shadow will appear darkest which have darker
objects opposite to them. And they will appear less dark when they
face lighter objects. And the larger the light object opposite, the
more the shadow will be lightened.
And the larger the surface of the dark object the more it will
darken the derived shadow where it is intercepted.
A disputed proposition.
199.
OF THE OPINION OF SOME THAT A TRIANGLE CASTS NO SHADOW ON A PLANE
SURFACE.
Certain mathematicians have maintained that a triangle, of which the
base is turned to the light, casts no shadow on a plane; and this
they prove by saying [5] that no spherical body smaller than the
light can reach the middle with the shadow. The lines of radiant
light are straight lines [6]; therefore, suppose the light to be _g
h_ and the triangle _l m n_, and let the plane be _i k_; they say
the light _g_ falls on the side of the triangle _l n_, and the
portion of the plane _i q_. Thus again _h_ like _g_ falls on the
side _l m_, and then on _m n_ and the plane _p k_; and if the whole
plane thus faces the lights _g h_, it is evident that the triangle
has no shadow; and that which has no shadow can cast none. This, in
this case appears credible. But if the triangle _n p g_ were not
illuminated by the two lights _g_ and _h_, but by _i p_ and _g_ and
_k_ neither side is lighted by more than one single light: that is
_i p_ is invisible to _h g_ and _k_ will never be lighted by _g_;
hence _p q_ will be twice as light as the two visible portions that
are in shadow.
[Footnote: 5--6. This passage is so obscure that it would be rash to
offer an explanation. Several words seem to have been omitted.]
On the relative depth of cast shadows (200-202).
200.
A spot is most in the shade when a large number of darkened rays
fall upon it. The spot which receives the rays at the widest angle
and by darkened rays will be most in the dark; a will be twice as
dark as b, because it originates from twice as large a base at an
equal distance. A spot is most illuminated when a large number of
luminous rays fall upon it. d is the beginning of the shadow _d f_,
and tinges _c_ but _a_ little; _d e_ is half of the shadow _d f_ and
gives a deeper tone where it is cast at _b_ than at _f_. And the
whole shaded space _e_ gives its tone to the spot _a_. [Footnote:
The diagram here referred to is on Pl. XLI, No. 2.]
201.
_A n_ will be darker than _c r_ in proportion to the number of times
that _a b_ goes into _c d_.
202.
The shadow cast by an object on a plane will be smaller in
proportion as that object is lighted by feebler rays. Let _d e_ be
the object and _d c_ the plane surface; the number of times that _d
e_ will go into _f g_ gives the proportion of light at _f h_ to _d
c_. The ray of light will be weaker in proportion to its distance
from the hole through which it falls.
FIFTH BOOK ON LIGHT AND SHADE.
Principles of reflection (203. 204).
203.
OF THE WAY IN WHICH THE SHADOWS CAST BY OBJECTS OUGHT TO BE DEFINED.
If the object is the mountain here figured, and the light is at the
point _a_, I say that from _b d_ and also from _c f_ there will be
no light but from reflected rays. And this results from the fact
that rays of light can only act in straight lines; and the same is
the case with the secondary or reflected rays.
204.
The edges of the derived shadow are defined by the hues of the
illuminated objects surrounding the luminous body which produces the
shadow.
On reverberation.
205.
OF REVERBERATION.
Reverberation is caused by bodies of a bright nature with a flat and
semi opaque surface which, when the light strikes upon them, throw
it back again, like the rebound of a ball, to the former object.
WHERE THERE CAN BE NO REFLECTED LIGHTS.
All dense bodies have their surfaces occupied by various degrees of
light and shade. The lights are of two kinds, one called original,
the other borrowed. Original light is that which is inherent in the
flame of fire or the light of the sun or of the atmosphere. Borrowed
light will be reflected light; but to return to the promised
definition: I say that this luminous reverberation is not produced
by those portions of a body which are turned towards darkened
objects, such as shaded spots, fields with grass of various height,
woods whether green or bare; in which, though that side of each
branch which is turned towards the original light has a share of
that light, nevertheless the shadows cast by each branch separately
are so numerous, as well as those cast by one branch on the others,
that finally so much shadow is the result that the light counts for
nothing. Hence objects of this kind cannot throw any reflected light
on opposite objects.
Reflection on water (206. 207).
206.
PERSPECTIVE.
The shadow or object mirrored in water in motion, that is to say in
small wavelets, will always be larger than the external object
producing it.
207.
It is impossible that an object mirrored on water should correspond
in form to the object mirrored, since the centre of the eye is above
the surface of the water.
This is made plain in the figure here given, which demonstrates that
the eye sees the surface _a b_, and cannot see it at _l f_, and at
_r t_; it sees the surface of the image at _r t_, and does not see
it in the real object _c d_. Hence it is impossible to see it, as
has been said above unless the eye itself is situated on the surface
of the water as is shown below [13].
[Footnote: _A_ stands for _ochio_ [eye], _B_ for _aria_ [air], _C_
for _acqua_ [water], _D_ for _cateto_ [cathetus].--In the original
MS. the second diagram is placed below line 13.]
Experiments with the mirror (208-210).
208.
THE MIRROR.
If the illuminated object is of the same size as the luminous body
and as that in which the light is reflected, the amount of the
reflected light will bear the same proportion to the intermediate
light as this second light will bear to the first, if both bodies
are smooth and white.
209.
Describe how it is that no object has its limitation in the mirror
but in the eye which sees it in the mirror. For if you look at your
face in the mirror, the part resembles the whole in as much as the
part is everywhere in the mirror, and the whole is in every part of
the same mirror; and the same is true of the whole image of any
object placed opposite to this mirror, &c.
210.
No man can see the image of another man in a mirror in its proper
place with regard to the objects; because every object falls on [the
surface of] the mirror at equal angles. And if the one man, who sees
the other in the mirror, is not in a direct line with the image he
will not see it in the place where it really falls; and if he gets
into the line, he covers the other man and puts himself in the place
occupied by his image. Let _n o_ be the mirror, _b_ the eye of your
friend and _d_ your own eye. Your friend's eye will appear to you at
_a_, and to him it will seem that yours is at _c_, and the
intersection of the visual rays will occur at _m_, so that either of
you touching _m_ will touch the eye of the other man which shall be
open. And if you touch the eye of the other man in the mirror it
will seem to him that you are touching your own.
Appendix:--On shadows in movement (211. 212).
211.
OF THE SHADOW AND ITS MOTION.
When two bodies casting shadows, and one in front of the other, are
between a window and the wall with some space between them, the
shadow of the body which is nearest to the plane of the wall will
move if the body nearest to the window is put in transverse motion
across the window. To prove this let _a_ and _b_ be two bodies
placed between the window _n m_ and the plane surface _o p_ with
sufficient space between them as shown by the space _a b_. I say
that if the body _a_ is moved towards _s_ the shadow of the body _b_
which is at _c_ will move towards _d_.
212.
OF THE MOTION OF SHADOWS.
The motion of a shadow is always more rapid than that of the body
which produces it if the light is stationary. To prove this let _a_
be the luminous body, and _b_ the body casting the shadow, and _d_
the shadow. Then I say that in the time while the solid body moves
from _b_ to _c_, the shadow _d_ will move to _e_; and this
proportion in the rapidity of the movements made in the same space
of time, is equal to that in the length of the space moved over.
Thus, given the proportion of the space moved over by the body _b_
to _c_, to that moved over by the shadow _d_ to _e_, the proportion
in the rapidity of their movements will be the same.
But if the luminous body is also in movement with a velocity equal
to that of the solid body, then the shadow and the body that casts
it will move with equal speed. And if the luminous body moves more
rapidly than the solid body, the motion of the shadow will be slower
than that of the body casting it.
But if the luminous body moves more slowly than the solid body, then
the shadow will move more rapidly than that body.
SIXTH BOOK ON LIGHT AND SHADE.
The effect of rays passing through holes (213. 214).
213.
PERSPECTIVE.
If you transmit the rays of the sun through a hole in the shape of a
star you will see a beautiful effect of perspective in the spot
where the sun's rays fall.
[Footnote: In this and the following chapters of MS. C the order of
the original paging has been adhered to, and is shown in
parenthesis. Leonardo himself has but rarely worked out the subject
of these propositions. The space left for the purpose has
occasionally been made use of for quite different matter. Even the
numerous diagrams, most of them very delicately sketched, lettered
and numbered, which occur on these pages, are hardly ever explained,
with the exception of those few which are here given.]
214.
No small hole can so modify the convergence of rays of light as to
prevent, at a long distance, the transmission of the true form of
the luminous body causing them. It is impossible that rays of light
passing through a parallel [slit], should not display the form of
the body causing them, since all the effects produced by a luminous
body are [in fact] the reflection of that body: The moon, shaped
like a boat, if transmitted through a hole is figured in the surface
[it falls on] as a boatshaped object. [Footnote 8: In the MS. a
blank space is left after this question.] Why the eye sees bodies at
a distance, larger than they measure on the vertical plane?.
[Footnote: This chapter, taken from another MS. may, as an
exception, be placed here, as it refers to the same subject as the
preceding section.]
On gradation of shadows (215. 216).
215.
Although the breadth and length of lights and shadow will be
narrower and shorter in foreshortening, the quality and quantity of
the light and shade is not increased nor diminished.
[3]The function of shade and light when diminished by
foreshortening, will be to give shadow and to illuminate an object
opposite, according to the quality and quantity in which they fall
on the body.
[5]In proportion as a derived shadow is nearer to its penultimate
extremities the deeper it will appear, _g z_ beyond the intersection
faces only the part of the shadow [marked] _y z_; this by
intersection takes the shadow from _m n_ but by direct line it takes
the shadow _a m_ hence it is twice as deep as _g z_. _Y x_, by
intersection takes the shadow _n o_, but by direct line the shadow
_n m a_, therefore _x y_ is three times as dark as _z g_; _x f_, by
intersection faces _o b_ and by direct line _o n m a_, therefore we
must say that the shadow between _f x_ will be four times as dark as
the shadow _z g_, because it faces four times as much shadow.
Let _a b_ be the side where the primary shadow is, and _b c_ the
primary light, _d_ will be the spot where it is intercepted,_f g_
the derived shadow and _f e_ the derived light.
And this must be at the beginning of the explanation.
[Footnote: In the original MS. the text of No. 252 precedes the one
given here. In the text of No. 215 there is a blank space of about
four lines between the lines 2 and 3. The diagram given on Pl. VI,
No. 2 is placed between lines 4 and 5. Between lines 5 and 6 there
is another space of about three lines and one line left blank
between lines 8 and 9. The reader will find the meaning of the whole
passage much clearer if he first reads the final lines 11--13.
Compare also line 4 of No. 270.]
On relative proportion of light and shadows (216--221).
216.
That part of the surface of a body on which the images [reflection]
from other bodies placed opposite fall at the largest angle will
assume their hue most strongly. In the diagram below, 8 is a larger
angle than 4, since its base _a n_ is larger than _e n_ the base of
4. This diagram below should end at _a n_ 4 8. [4]That portion of
the illuminated surface on which a shadow is cast will be brightest
which lies contiguous to the cast shadow. Just as an object which is
lighted up by a greater quantity of luminous rays becomes brighter,
so one on which a greater quantity of shadow falls, will be darker.
Let 4 be the side of an illuminated surface 4 8, surrounding the
cast shadow _g e_ 4. And this spot 4 will be lighter than 8, because
less shadow falls on it than on 8. Since 4 faces only the shadow _i
n_; and 8 faces and receives the shadow _a e_ as well as _i n_ which
makes it twice as dark. And the same thing happens when you put the
atmosphere and the sun in the place of shade and light.
[12] The distribution of shadow, originating in, and limited by,
plane surfaces placed near to each other, equal in tone and directly
opposite, will be darker at the ends than at the beginning, which
will be determined by the incidence of the luminous rays. You will
find the same proportion in the depth of the derived shadows _a n_
as in the nearness of the luminous bodies _m b_, which cause them;
and if the luminous bodies were of equal size you would still
farther find the same proportion in the light cast by the luminous
circles and their shadows as in the distance of the said luminous
bodies.
[Footnote: The diagram originally placed between lines 3 and 4 is on
Pl. VI, No. 3. In the diagram given above line 14 of the original,
and here printed in the text, the words _corpo luminoso_ [luminous
body] are written in the circle _m_, _luminoso_ in the circle _b_
and _ombroso_ [body in shadow] in the circle _o_.]
217.
THAT PART OF THE REFLECTION WILL BE BRIGHTEST WHERE THE REFLECTED
RAYS ARE SHORTEST.
[2] The darkness occasioned by the casting of combined shadows will
be in conformity with its cause, which will originate and terminate
between two plane surfaces near together, alike in tone and directly
opposite each other.
[4] In proportion as the source of light is larger, the luminous and
shadow rays will be more mixed together. This result is produced
because wherever there is a larger quantity of luminous rays, there
is most light, but where there are fewer there is least light,
consequently the shadow rays come in and mingle with them.
[Footnote: Diagrams are inserted before lines 2 and 4.]
218.
In all the proportions I lay down it must be understood that the
medium between the bodies is always the same. [2] The smaller the
luminous body the more distinct will the transmission of the shadows
be.
[3] When of two opposite shadows, produced by the same body, one is
twice as dark as the other though similar in form, one of the two
lights causing them must have twice the diameter that the other has
and be at twice the distance from the opaque body. If the object is
lowly moved across the luminous body, and the shadow is intercepted
at some distance from the object, there will be the same relative
proportion between the motion of the derived shadow and the motion
of the primary shadow, as between the distance from the object to
the light, and that from the object to the spot where the shadow is
intercepted; so that though the object is moved slowly the shadow
moves fast.
[Footnote: There are diagrams inserted before lines 2 and 3 but they
are not reproduced here. The diagram above line 6 is written upon as
follows: at _A lume_ (light), at _B obbietto_ (body), at _C ombra
d'obbietto_ (shadow of the object).]
219.
A luminous body will appear less brilliant when surrounded by a
bright background.
[2] I have found that the stars which are nearest to the horizon
look larger than the others because light falls upon them from a
larger proportion of the solar body than when they are above us; and
having more light from the sun they give more light, and the bodies
which are most luminous appear the largest. As may be seen by the
sun through a mist, and overhead; it appears larger where there is
no mist and diminished through mist. No portion of the luminous body
is ever visible from any spot within the pyramid of pure derived
shadow.
[Footnote: Between lines 1 and 2 there is in the original a large
diagram which does not refer to this text. ]
220.
A body on which the solar rays fall between the thin branches of
trees far apart will cast but a single shadow.
[2] If an opaque body and a luminous one are (both) spherical the
base of the pyramid of rays will bear the same proportion to the
luminous body as the base of the pyramid of shade to the opaque
body.
[4] When the transmitted shadow is intercepted by a plane surface
placed opposite to it and farther away from the luminous body than
from the object [which casts it] it will appear proportionately
darker and the edges more distinct.
[Footnote: The diagram which, in the original, is placed above line
2, is similar to the one, here given on page 73 (section 120).--The
diagram here given in the margin stands, in the original, between
lines 3 and 4.]
221.
A body illuminated by the solar rays passing between the thick
branches of trees will produce as many shadows as there are branches
between the sun and itself.
Where the shadow-rays from an opaque pyramidal body are intercepted
they will cast a shadow of bifurcate outline and various depth at
the points. A light which is broader than the apex but narrower than
the base of an opaque pyramidal body placed in front of it, will
cause that pyramid to cast a shadow of bifurcate form and various
degrees of depth.
If an opaque body, smaller than the light, casts two shadows and if
it is the same size or larger, casts but one, it follows that a
pyramidal body, of which part is smaller, part equal to, and part
larger than, the luminous body, will cast a bifurcate shadow.
[Footnote: Between lines 2 and 3 there are in the original two large
diagrams.]
_IV._
_Perspective of Disappearance._
_The theory of the_ "Prospettiva de' perdimenti" _would, in many
important details, be quite unintelligible if it had not been led up
by the principles of light and shade on which it is based. The word_
"Prospettiva" _in the language of the time included the principles
of optics; what Leonardo understood by_ "Perdimenti" _will be
clearly seen in the early chapters, Nos._ 222--224. _It is in the
very nature of the case that the farther explanations given in the
subsequent chapters must be limited to general rules. The sections
given as_ 227--231 _"On indistinctness at short distances" have, it
is true, only an indirect bearing on the subject; but on the other
hand, the following chapters,_ 232--234, _"On indistinctness at
great distances," go fully into the matter, and in chapters_
235--239, _which treat "Of the importance of light and shade in the
Perspective of Disappearance", the practical issues are distinctly
insisted on in their relation to the theory. This is naturally
followed by the statements as to "the effect of light or dark
backgrounds on the apparent size of bodies"_ (_Nos._ 240--250). _At
the end I have placed, in the order of the original, those sections
from the MS._ C _which treat of the "Perspective of Disappearance"
and serve to some extent to complete the treatment of the subject_
(251--262).
Definition (222. 223).
222.
OF THE DIMINISHED DISTINCTNESS OF THE OUTLINES OF OPAQUE BODIES.
If the real outlines of opaque bodies are indistinguishable at even
a very short distance, they will be more so at long distances; and,
since it is by its outlines that we are able to know the real form
of any opaque body, when by its remoteness we fail to discern it as
a whole, much more must we fail to discern its parts and outlines.
223.
OF THE DIMINUTION IN PERSPECTIVE OF OPAQUE OBJECTS.
Among opaque objects of equal size the apparent diminution of size
will be in proportion to their distance from the eye of the
spectator; but it is an inverse proportion, since, where the
distance is greater, the opaque body will appear smaller, and the
less the distance the larger will the object appear. And this is the
fundamental principle of linear perspective and it
follows:--[11]every object as it becomes more remote loses first
those parts which are smallest. Thus of a horse, we should lose the
legs before the head, because the legs are thinner than the head;
and the neck before the body for the same reason. Hence it follows
that the last part of the horse which would be discernible by the
eye would be the mass of the body in an oval form, or rather in a
cylindrical form and this would lose its apparent thickness before
its length--according to the 2nd rule given above, &c. [Footnote 23:
Compare line 11.].
If the eye remains stationary the perspective terminates in the
distance in a point. But if the eye moves in a straight [horizontal]
line the perspective terminates in a line and the reason is that
this line is generated by the motion of the point and our sight;
therefore it follows that as we move our sight [eye], the point
moves, and as we move the point, the line is generated, &c.
An illustration by experiment.
224.
Every visible body, in so far as it affects the eye, includes three
attributes; that is to say: mass, form and colour; and the mass is
recognisable at a greater distance from the place of its actual
existence than either colour or form. Again, colour is discernible
at a greater distance than form, but this law does not apply to
luminous bodies.
The above proposition is plainly shown and proved by experiment;
because: if you see a man close to you, you discern the exact
appearance of the mass and of the form and also of the colouring; if
he goes to some distance you will not recognise who he is, because
the character of the details will disappear, if he goes still
farther you will not be able to distinguish his colouring, but he
will appear as a dark object, and still farther he will appear as a
very small dark rounded object. It appears rounded because distance
so greatly diminishes the various details that nothing remains
visible but the larger mass. And the reason is this: We know very
well that all the images of objects reach the senses by a small
aperture in the eye; hence, if the whole horizon _a d_ is admitted
through such an aperture, the object _b c_ being but a very small
fraction of this horizon what space can it fill in that minute image
of so vast a hemisphere? And because luminous bodies have more power
in darkness than any others, it is evident that, as the chamber of
the eye is very dark, as is the nature of all colored cavities, the
images of distant objects are confused and lost in the great light
of the sky; and if they are visible at all, appear dark and black,
as every small body must when seen in the diffused light of the
atmosphere.
[Footnote: The diagram belonging to this passage is placed between
lines 5 and 6; it is No. 4 on Pl. VI. ]
A guiding rule.
225.
OF THE ATMOSPHERE THAT INTERPOSES BETWEEN THE EYE AND VISIBLE
OBJECTS.
An object will appear more or less distinct at the same distance, in
proportion as the atmosphere existing between the eye and that
object is more or less clear. Hence, as I know that the greater or
less quantity of the air that lies between the eye and the object
makes the outlines of that object more or less indistinct, you must
diminish the definiteness of outline of those objects in proportion
to their increasing distance from the eye of the spectator.
An experiment.
226.
When I was once in a place on the sea, at an equal distance from the
shore and the mountains, the distance from the shore looked much
greater than that from the mountains.
On indistinctness at short distances (227-231).
227.
If you place an opaque object in front of your eye at a distance of
four fingers' breadth, if it is smaller than the space between the
two eyes it will not interfere with your seeing any thing that may
be beyond it. No object situated beyond another object seen by the
eye can be concealed by this [nearer] object if it is smaller than
the space from eye to eye.
228.
The eye cannot take in a luminous angle which is too close to it.
229.
That part of a surface will be better lighted on which the light
falls at the greater angle. And that part, on which the shadow falls
at the greatest angle, will receive from those rays least of the
benefit of the light.
230.
OF THE EYE.
The edges of an object placed in front of the pupil of the eye will
be less distinct in proportion as they are closer to the eye. This
is shown by the edge of the object _n_ placed in front of the pupil
_d_; in looking at this edge the pupil also sees all the space _a c_
which is beyond the edge; and the images the eye receives from that
space are mingled with the images of the edge, so that one image
confuses the other, and this confusion hinders the pupil from
distinguishing the edge.
231.
The outlines of objects will be least clear when they are nearest to
the eye, and therefore remoter outlines will be clearer. Among
objects which are smaller than the pupil of the eye those will be
less distinct which are nearer to the eye.
On indistinctness at great distances (232-234).
232.
Objects near to the eye will appear larger than those at a distance.
Objects seen with two eyes will appear rounder than if they are seen
with only one.
Objects seen between light and shadow will show the most relief.
233.
OF PAINTING.
Our true perception of an object diminishes in proportion as its
size is diminished by distance.
234.
PERSPECTIVE.
Why objects seen at a distance appear large to the eye and in the
image on the vertical plane they appear small.
PERSPECTIVE.
I ask how far away the eye can discern a non-luminous body, as, for
instance, a mountain. It will be very plainly visible if the sun is
behind it; and could be seen at a greater or less distance according
to the sun's place in the sky.
[Footnote: The clue to the solution of this problem (lines 1-3) is
given in lines 4-6, No. 232. Objects seen with both eyes appear
solid since they are seen from two distinct points of sight
separated by the distance between the eyes, but this solidity cannot
be represented in a flat drawing. Compare No. 535.]
The importance of light and shade in the perspective of
disappearance (235-239).
235.
An opaque body seen in a line in which the light falls will reveal
no prominences to the eye. For instance, let _a_ be the solid body
and _c_ the light; _c m_ and _c n_ will be the lines of incidence of
the light, that is to say the lines which transmit the light to the
object _a_. The eye being at the point _b_, I say that since the
light _c_ falls on the whole part _m n_ the portions in relief on
that side will all be illuminated. Hence the eye placed at _c_
cannot see any light and shade and, not seeing it, every portion
will appear of the same tone, therefore the relief in the prominent
or rounded parts will not be visible.
236.
OF PAINTING.
When you represent in your work shadows which you can only discern
with difficulty, and of which you cannot distinguish the edges so
that you apprehend them confusedly, you must not make them sharp or
definite lest your work should have a wooden effect.
237.
OF PAINTING.
You will observe in drawing that among the shadows some are of
undistinguishable gradation and form, as is shown in the 3rd
[proposition] which says: Rounded surfaces display as many degrees
of light and shade as there are varieties of brightness and darkness
reflected from the surrounding objects.
238.
OF LIGHT AND SHADE.
You who draw from nature, look (carefully) at the extent, the
degree, and the form of the lights and shadows on each muscle; and
in their position lengthwise observe towards which muscle the axis
of the central line is directed.
239.
An object which is [so brilliantly illuminated as to be] almost as
bright as light will be visible at a greater distance, and of larger
apparent size than is natural to objects so remote.
The effect of light or dark backgrounds on the apparent size of
objects (240-250).
240.
A shadow will appear dark in proportion to the brilliancy of the
light surrounding it and conversely it will be less conspicuous
where it is seen against a darker background.
241.
OF ORDINARY PERSPECTIVE.
An object of equal breadth and colour throughout, seen against a
background of various colours will appear unequal in breadth.
And if an object of equal breadth throughout, but of various
colours, is seen against a background of uniform colour, that object
will appear of various breadth. And the more the colours of the
background or of the object seen against the ground vary, the
greater will the apparent variations in the breadth be though the
objects seen against the ground be of equal breadth [throughout].
242.
A dark object seen against a bright background will appear smaller
than it is.
A light object will look larger when it is seen against a background
darker than itself.
243.
OF LIGHT.
A luminous body when obscured by a dense atmosphere will appear
smaller; as may be seen by the moon or sun veiled by mists.
OF LIGHT.
Of several luminous bodies of equal size and brilliancy and at an
equal distance, that will look the largest which is surrounded by
the darkest background.
OF LIGHT.
I find that any luminous body when seen through a dense and thick
mist diminishes in proportion to its distance from the eye. Thus it
is with the sun by day, as well as the moon and the other eternal
lights by night. And when the air is clear, these luminaries appear
larger in proportion as they are farther from the eye.
244.
That portion of a body of uniform breadth which is against a lighter
background will look narrower [than the rest].
[4] _e_ is a given object, itself dark and of uniform breadth; _a b_
and _c d_ are two backgrounds one darker than the other; _b c_ is a
bright background, as it might be a spot lighted by the sun through
an aperture in a dark room. Then I say that the object _e g_ will
appear larger at _e f_ than at _g h_; because _e f_ has a darker
background than _g h_; and again at _f g_ it will look narrower from
being seen by the eye _o_, on the light background _b c_. [Footnote
12: The diagram to which the text, lines 1-11, refers, is placed in
the original between lines 3 and 4, and is given on Pl. XLI, No. 3.
Lines 12 to 14 are explained by the lower of the two diagrams on Pl.
XLI, No. 4. In the original these are placed after line 14.] That
part of a luminous body, of equal breadth and brilliancy throughout,
will look largest which is seen against the darkest background; and
the luminous body will seem on fire.
245.
WHY BODIES IN LIGHT AND SHADE HAVE THEIR OUTLINES ALTERED BY THE
COLOUR AND BRIGHTNESS OF THE OBJECTS SERVING AS A BACKGROUND TO
THEM.
If you look at a body of which the illuminated portion lies and ends
against a dark background, that part of the light which will look
brightest will be that which lies against the dark [background] at
_d_. But if this brighter part lies against a light background, the
edge of the object, which is itself light, will be less distinct
than before, and the highest light will appear to be between the
limit of the background _m f_ and the shadow. The same thing is seen
with regard to the dark [side], inasmuch as that edge of the shaded
portion of the object which lies against a light background, as at
_l_, it looks much darker than the rest. But if this shadow lies
against a dark background, the edge of the shaded part will appear
lighter than before, and the deepest shade will appear between the
edge and the light at the point _o_.
[Footnote: In the original diagram _o_ is inside the shaded surface
at the level of _d_.]
246.
An opaque body will appear smaller when it is surrounded by a highly
luminous background, and a light body will appear larger when it is
seen against a darker background. This may be seen in the height of
buildings at night, when lightning flashes behind them; it suddenly
seems, when it lightens, as though the height of the building were
diminished. For the same reason such buildings look larger in a
mist, or by night than when the atmosphere is clear and light.
247.
ON LIGHT BETWEEN SHADOWS
When you are drawing any object, remember, in comparing the grades
of light in the illuminated portions, that the eye is often deceived
by seeing things lighter than they are. And the reason lies in our
comparing those parts with the contiguous parts. Since if two
[separate] parts are in different grades of light and if the less
bright is conterminous with a dark portion and the brighter is
conterminous with a light background--as the sky or something
equally bright--, then that which is less light, or I should say
less radiant, will look the brighter and the brighter will seem the
darker.
248.
Of objects equally dark in themselves and situated at a considerable
and equal distance, that will look the darkest which is farthest
above the earth.
249.
TO PROVE HOW IT IS THAT LUMINOUS BODIES APPEAR LARGER, AT A
DISTANCE, THAN THEY ARE.
If you place two lighted candles side by side half a braccio apart,
and go from them to a distance 200 braccia you will see that by the
increased size of each they will appear as a single luminous body
with the light of the two flames, one braccio wide.
TO PROVE HOW YOU MAY SEE THE REAL SIZE OF LUMINOUS BODIES.
If you wish to see the real size of these luminous bodies, take a
very thin board and make in it a hole no bigger than the tag of a
lace and place it as close to your eye as possible, so that when you
look through this hole, at the said light, you can see a large space
of air round it. Then by rapidly moving this board backwards and
forwards before your eye you will see the light increase [and
diminish].
Propositions on perspective of disappearance from MS. C. (250-262).
250.
Of several bodies of equal size and equally distant from the eye,
those will look the smallest which are against the lightest
background.
Every visible object must be surrounded by light and shade. A
perfectly spherical body surrounded by light and shade will appear
to have one side larger than the other in proportion as one is more
highly lighted than the other.
251.
PERSPECTIVE.
No visible object can be well understood and comprehended by the
human eye excepting from the difference of the background against
which the edges of the object terminate and by which they are
bounded, and no object will appear [to stand out] separate from that
background so far as the outlines of its borders are concerned. The
moon, though it is at a great distance from the sun, when, in an
eclipse, it comes between our eyes and the sun, appears to the eyes
of men to be close to the sun and affixed to it, because the sun is
then the background to the moon.
252.
A luminous body will appear more brilliant in proportion as it is
surrounded by deeper shadow. [Footnote: The diagram which, in the
original, is placed after this text, has no connection with it.]
253.
The straight edges of a body will appear broken when they are
conterminous with a dark space streaked with rays of light.
[Footnote: Here again the diagrams in the original have no
connection with the text.]
254.
Of several bodies, all equally large and equally distant, that which
is most brightly illuminated will appear to the eye nearest and
largest. [Footnote: Here again the diagrams in the original have no
connection with the text.]
255.
If several luminous bodies are seen from a great distance although
they are really separate they will appear united as one body.
256.
If several objects in shadow, standing very close together, are seen
against a bright background they will appear separated by wide
intervals.
257.
Of several bodies of equal size and tone, that which is farthest
will appear the lightest and smallest.
258.
Of several objects equal in size, brightness of background and
length that which has the flattest surface will look the largest. A
bar of iron equally thick throughout and of which half is red hot,
affords an example, for the red hot part looks thicker than the
rest.
259.
Of several bodies of equal size and length, and alike in form and in
depth of shade, that will appear smallest which is surrounded by the
most luminous background.
260.
DIFFERENT PORTIONS OF A WALL SURFACE WILL BE DARKER OR BRIGHTER IN
PROPORTION AS THE LIGHT OR SHADOW FALLS ON THEM AT A LARGER ANGLE.
The foregoing proposition can be clearly proved in this way. Let us
say that _m q_ is the luminous body, then _f g_ will be the opaque
body; and let _a e_ be the above-mentioned plane on which the said
angles fall, showing [plainly] the nature and character of their
bases. Then: _a_ will be more luminous than _b_; the base of the
angle _a_ is larger than that of _b_ and it therefore makes a
greater angle which will be _a m q_; and the pyramid _b p m_ will be
narrower and _m o c_ will be still finer, and so on by degrees, in
proportion as they are nearer to _e_, the pyramids will become
narrower and darker. That portion of the wall will be the darkest
where the breadth of the pyramid of shadow is greater than the
breadth of the pyramid of light.
At the point _a_ the pyramid of light is equal in strength to the
pyramid of shadow, because the base _f g_ is equal to the base _r
f_. At the point _d_ the pyramid of light is narrower than the
pyramid of shadow by so much as the base _s f_ is less than the base
_f g_.
Divide the foregoing proposition into two diagrams, one with the
pyramids of light and shadow, the other with the pyramids of light
[only].
261.
Among shadows of equal depth those which are nearest to the eye will
look least deep.
262.
The more brilliant the light given by a luminous body, the deeper
will the shadows be cast by the objects it illuminates.
_V._
_Theory of colours._
_Leonardo's theory of colours is even more intimately connected with
his principles of light and shade than his Perspective of
Disappearance and is in fact merely an appendix or supplement to
those principles, as we gather from the titles to sections_ 264,
267_, and _276_, while others again_ (_Nos._ 281, 282_) are headed_
Prospettiva.
_A very few of these chapters are to be found in the oldest copies
and editions of the Treatise on Painting, and although the material
they afford is but meager and the connection between them but
slight, w
_cadela_ (candle).]
169.
ALL BODIES, IN PROPORTION AS THEY ARE NEARER TO, OR FARTHER FROM THE
SOURCE OF LIGHT, WILL PRODUCE LONGER OR SHORTER DERIVED SHADOWS.
Among bodies of equal size, that one which is illuminated by the
largest light will have the shortest shadow. Experiment confirms
this proposition. Thus the body _m_ _n_ is surrounded by a larger
amount of light than the body _p q_, as is shown above. Let us say
that _v c a b d x_ is the sky, the source of light, and that _s t_
is a window by which the luminous rays enter, and so _m n_ and _p q_
are bodies in light and shade as exposed to this light; _m n_ will
have a small derived shadow, because its original shadow will be
small; and the derivative light will be large, again, because the
original light _c d_ will be large and _p q_ will have more derived
shadow because its original shadow will be larger, and its derived
light will be smaller than that of the body _m n_ because that
portion of the hemisphere _a b_ which illuminates it is smaller than
the hemisphere _c d_ which illuminates the body _m n_.
[Footnote: The diagram, given on Pl. IV, No. 2, stands in the
original between lines 2 and 7, while the text of lines 3 to 6 is
written on its left side. In the reproduction of this diagram the
letter _v_ at the outer right-hand end has been omitted.]
170.
The shadow _m_ bears the same proportion to the shadow _n_ as the
line _b c_ to the line _f c_.
171.
OF PAINTING.
Of different shadows of equal strength that which is nearest the eye
will seem the least strong.
Why is the shadow _e a b_ in the first grade of strength, _b c_ in
the second; _c d_ in the third? The reason is that as from _e a b_
the sky is nowhere visible, it gets no light whatever from the sky,
and so has no direct [primary] light. _b c_ faces the portion of the
sky _f g_ and is illuminated by it. _c d_ faces the sky at _h k_. _c
d_, being exposed to a larger extent of sky than _b c_, it is
reasonable that it should be more lighted. And thus, up to a certain
distance, the wall _a d_ will grow lighter for the reasons here
given, until the darkness of the room overpowers the light from the
window.
172.
When the light of the atmosphere is restricted [by an opening] and
illuminates bodies which cast shadows, these bodies being equally
distant from the centre of the window, that which is most obliquely
placed will cast the largest shadow beyond it.
173.
These bodies standing apart in a room lighted by a single window
will have derivative shadows more or less short according as they
are more or less opposite to the window. Among the shadows cast by
bodies of equal mass but at unequal distances from the opening by
which they are illuminated, that shadow will be the longest of the
body which is least in the light. And in proportion as one body is
better illuminated than another its shadow will be shorter than
another. The proportion _n m_ and _e v k_ bear to _r t_ and _v x_
corresponds with that of the shadow _x_ to 4 and _y_.
The reason why those bodies which are placed most in front of the
middle of the window throw shorter shadows than those obliquely
situated is:--That the window appears in its proper form and to the
obliquely placed ones it appears foreshortened; to those in the
middle, the window shows its full size, to the oblique ones it
appears smaller; the one in the middle faces the whole hemisphere
that is _e f_ and those on the side have only a strip; that is _q r_
faces _a b_; and _m n_ faces _c d_; the body in the middle having a
larger quantity of light than those at the sides is lighted from a
point much below its centre, and thus the shadow is shorter. And the
pyramid _g_ 4 goes into _l y_ exactly as often as _a b_ goes into _e
f_. The axis of every derivative shadow passes through 6 1/2
[Footnote 31: _passa per_ 6 1/2 (passes through 6 1/2). The meaning
of these words is probably this: Each of the three axes of the
derived shadow intersects the centre (_mezzo_) of the primary shadow
(_ombra originale_) and, by prolongation upwards crosses six lines.
This is self evident only in the middle diagram; but it is equally
true of the side figures if we conceive of the lines 4 _f_, _x n v
m_, _y l k v_, and 4 _e_, as prolonged beyond the semicircle of the
horizon.] and is in a straight line with the centre of the primary
shadow, with the centre of the body casting it and of the derivative
light and with the centre of the window and, finally, with the
centre of that portion of the source of light which is the celestial
hemisphere, _y h_ is the centre of the derived shade, _l h_ of the
primary shadow, _l_ of the body throwing it, _l k_ of the derived
light, _v_ is the centre of the window, _e_ is the final centre of
the original light afforded by that portion of the hemisphere of the
sky which illuminates the solid body.
[Footnote: Compare the diagram on Pl. IV, No. 3. In the original
this drawing is placed between lines 3 and 22; the rest, from line 4
to line 21, is written on the left hand margin.]
174.
THE FARTHER THE DERIVED SHADOW IS PROLONGED THE LIGHTER IT BECOMES.
You will find that the proportion of the diameter of the derived
shadow to that of the primary shadow will be the same as that
between the darkness of the primary shadow and that of the derived
shadow.
[Footnote 6: Compare No. 177.] Let _a b_ be the diameter of the
primary shadow and _c d_ that of the derived shadow, I say that _a
b_ going, as you see, three times into _d c_, the shadow _d c_ will
be three times as light as the shadow _a b_. [Footnote 8: Compare
No. 177.]
If the size of the illuminating body is larger than that of the
illuminated body an intersection of shadow will occur, beyond which
the shadows will run off in two opposite directions as if they were
caused by two separate lights.
On the relative intensity of derived shadows (175-179).
175.
ON PAINTING.
The derived shadow is stronger in proportion as it is nearer to its
place of origin.
176.
HOW SHADOWS FADE AWAY AT LONG DISTANCES.
Shadows fade and are lost at long distances because the larger
quantity of illuminated air which lies between the eye and the
object seen tints the shadow with its own colour.
177.
_a b_ will be darker than _c d_ in proportion as _c d_ is broader
than _a b_.
[Footnote: In the original MS. the word _lume_ (light) is written at
the apex of the pyramid.]
178.
It can be proved why the shadow _o p c h_ is darker in proportion as
it is nearer to the line _p h_ and is lighter in proportion as it is
nearer to the line _o c_. Let the light _a b_, be a window, and let
the dark wall in which this window is, be _b s_, that is, one of the
sides of the wall.
Then we may say that the line _p h_ is darker than any other part of
the space _o p c h_, because this line faces the whole surface in
shadow of [Footnote: In the original the diagram is placed between
lines 27 and 28.] the wall _b s_. The line _o c_ is lighter than the
other part of this space _o p c h_, because this line faces the
luminous space _a b_.
Where the shadow is larger, or smaller, or equal the body which
casts it.
[First of the character of divided lights. [Footnote 14: _lumi
divisi_. The text here breaks off abruptly.]
OF THE COMPOUND SHADOW _F, R, C, H_ CAUSED BY A SINGLE LIGHT.
The shadow _f r c h_ is under such conditions as that where it is
farthest from its inner side it loses depth in proportion. To prove
this:
Let _d a_, be the light and _f n_ the solid body, and let _a e_ be
one of the side walls of the window that is _d a_. Then I
say--according to the 2nd [proposition]: that the surface of any
body is affected by the tone of the objects surrounding it,--that
the side _r c_, which faces the dark wall _a e_ must participate of
its darkness and, in the same way that the outer surface which faces
the light _d a_ participates of the light; thus we get the outlines
of the extremes on each side of the centre included between them.]
This is divided into four parts. The first the extremes, which
include the compound shadow, secondly the compound shadow between
these extremes.
179.
THE ACTION OF THE LIGHT AS FROM ITS CENTRE.
If it were the whole of the light that caused the shadows beyond the
bodies placed in front of it, it would follow that any body much
smaller than the light would cast a pyramidal shadow; but experience
not showing this, it must be the centre of the light that produces
this effect.
[Footnote: The diagram belonging to this passage is between lines 4
and 5 in the original. Comp. the reproduction Pl. IV, No. 4. The
text and drawing of this chapter have already been published with
tolerable accuracy. See M. JORDAN: "_Das Malerbuch des Leonardo da
Vinci_". Leipzig 1873, P. 90.]
PROOF.
Let _a b_ be the width of the light from a window, which falls on a
stick set up at one foot from _a c_ [Footnote 6: _bastone_ (stick).
The diagram has a sphere in place of a stick.]. And let _a d_ be the
space where all the light from the window is visible. At _c e_ that
part of the window which is between _l b_ cannot be seen. In the
same way _a m_ cannot be seen from _d f_ and therefore in these two
portions the light begins to fail.
Shadow as produced by two lights of different size (180. 181).
180.
A body in light and shade placed between two equal lights side by
side will cast shadows in proportion to the [amount of] light. And
the shadows will be one darker than the other in proportion as one
light is nearer to the said body than the other on the opposite
side.
A body placed at an equal distance between two lights will cast two
shadows, one deeper than the other in proportion, as the light which
causes it is brighter than the other.
[Footnote: In the MS. the larger diagram is placed above the first
line; the smaller one between l. 4 & 5.]
181.
A light which is smaller than the body it illuminates produces
shadows of which the outlines end within [the surface of] the body,
and not much compound shadow; and falls on less than half of it. A
light which is larger than the body it illuminates, falls on more
than half of it, and produces much compound shadow.
The effect of light at different distances.
182.
OF THE SHADOW CAST BY A BODY PLACED BETWEEN 2 EQUAL LIGHTS.
A body placed between 2 equal lights will cast 2 shadows of itself
in the direction of the lines of the 2 lights; and if you move this
body placing it nearer to one of the lights the shadow cast towards
the nearer light will be less deep than that which falls towards the
more distant one.
Further complications in the derived shadows (183-187).
183.
The greatest depth of shadow is in the simple derived shadow because
it is not lighted by either of the two lights _a b, c d_.
The next less deep shadow is the derived shadow _e f n_; and in this
the shadow is less by half, because it is illuminated by a single
light, that is _c d_.
This is uniform in natural tone because it is lighted throughout by
one only of the two luminous bodies [10]. But it varies with the
conditions of shadow, inasmuch as the farther it is away from the
light the less it is illuminated by it [13].
The third degree of depth is the middle shadow [Footnote 15: We
gather from what follows that _q g r_ here means _ombra media_ (the
middle shadow).]. But this is not uniform in natural tone; because
the nearer it gets to the simple derived shadow the deeper it is
[Footnote 18: Compare lines 10-13], and it is the uniformly gradual
diminution by increase of distance which is what modifies it
[Footnote 20: See Footnote 18]: that is to say the depth of a shadow
increases in proportion to the distance from the two lights.
The fourth is the shadow _k r s_ and this is all the darker in
natural tone in proportion as it is nearer to _k s_, because it gets
less of the light _a o_, but by the accident [of distance] it is
rendered less deep, because it is nearer to the light _c d_, and
thus is always exposed to both lights.
The fifth is less deep in shadow than either of the others because
it is always entirely exposed to one of the lights and to the whole
or part of the other; and it is less deep in proportion as it is
nearer to the two lights, and in proportion as it is turned towards
the outer side _x t_; because it is more exposed to the second light
_a b_.
[Footnote: The diagram to this section is given on Pl. V. To the
left is the facsimile of the beginning of the text belonging to it.]
184.
OF SIMPLE SHADOWS.
Why, at the intersections _a_, _b_ of the two compound shadows _e f_
and _m e_, is a simple shadow pfoduced as at _e h_ and _m g_, while
no such simple shadow is produced at the other two intersections _c
d_ made by the very same compound shadows?
ANSWER.
Compound shadow are a mixture of light and shade and simple shadows
are simply darkness. Hence, of the two lights _n_ and _o_, one falls
on the compound shadow from one side, and the other on the compound
shadow from the other side, but where they intersect no light falls,
as at _a b_; therefore it is a simple shadow. Where there is a
compound shadow one light or the other falls; and here a difficulty
arises for my adversary since he says that, where the compound
shadows intersect, both the lights which produce the shadows must of
necessity fall and therefore these shadows ought to be neutralised;
inasmuch as the two lights do not fall there, we say that the shadow
is a simple one and where only one of the two lights falls, we say
the shadow is compound, and where both the lights fall the shadow is
neutralised; for where both lights fall, no shadow of any kind is
produced, but only a light background limiting the shadow. Here I
shall say that what my adversary said was true: but he only mentions
such truths as are in his favour; and if we go on to the rest he
must conclude that my proposition is true. And that is: That if both
lights fell on the point of intersection, the shadows would be
neutralised. This I confess to be true if [neither of] the two
shadows fell in the same spot; because, where a shadow and a light
fall, a compound shadow is produced, and wherever two shadows or two
equal lights fall, the shadow cannot vary in any part of it, the
shadows and the lights both being equal. And this is proved in the
eighth [proposition] on proportion where it is said that if a given
quantity has a single unit of force and resistance, a double
quantity will have double force and double resistance.
DEFINITION.
The intersection _n_ is produced by the shadows caused by the light
_b_, because this light _b_ produces the shadow _x b_, and the
shadow _s b_, but the intersection _m_ is produced by the light _a_
which causes the shadow _s a_, and the shadow _x a_.
But if you uncover both the lights _a b_, then you get the two
shadows _n m_ both at once, and besides these, two other, simple
shadows are produced at _r o_ where neither of the two lights falls
at all. The grades of depth in compound shadows are fewer in
proportion as the lights falling on, and crossing them are less
numerous.
186.
Why the intersections at _n_ being composed of two compound derived
shadows, forms a compound shadow and not a simple one, as happens
with other intersections of compound shadows. This occurs, according
to the 2nd [diagram] of this [prop.] which says:--The intersection
of derived shadows when produced by the intersection of columnar
shadows caused by a single light does not produce a simple shadow.
And this is the corollary of the 1st [prop.] which says:--The
intersection of simple derived shadows never results in a deeper
shadow, because the deepest shadows all added together cannot be
darker than one by itself. Since, if many deepest shadows increased
in depth by their duplication, they could not be called the
_deepest_ shadows, but only part-shadows. But if such intersections
are illuminated by a second light placed between the eye and the
intersecting bodies, then those shadows would become compound
shadows and be uniformly dark just as much at the intersection as
throughout the rest. In the 1st and 2nd above, the intersections _i
k_ will not be doubled in depth as it is doubled in quantity. But in
this 3rd, at the intersections _g n_ they will be double in depth
and in quantity.
187.
HOW AND WHEN THE SURROUNDINGS IN SHADOW MINGLE THEIR DERIVED SHADOW
WITH THE LIGHT DERIVED FROM THE LUMINOUS BODY.
The derived shadow of the dark walls on each side of the bright
light of the window are what mingle their various degrees of shade
with the light derived from the window; and these various depths of
shade modify every portion of the light, except where it is
strongest, at _c_. To prove this let _d a_ be the primary shadow
which is turned towards the point _e_, and darkens it by its derived
shadow; as may be seen by the triangle _a e d_, in which the
angle _e_ faces the darkened base _d a e_; the point _v_ faces the
dark shadow _a s_ which is part of _a d_, and as the whole is
greater than a part, _e_ which faces the whole base [of the
triangle], will be in deeper shadow than _v_ which only faces part
of it. In consequence of the conclusion [shown] in the above
diagram, _t_ will be less darkened than _v_, because the base of the
_t_ is part of the base of the _v_; and in the same way it follows
that _p_ is less in shadow than _t_, because the base of the _p_ is
part of the base of the _t_. And _c_ is the terminal point of the
derived shadow and the chief beginning of the highest light.
[Footnote: The diagram on Pl. IV, No. 5 belongs to this passage; but
it must be noted that the text explains only the figure on the
right-hand side.]
FOURTH BOOK ON LIGHT AND SHADE.
On the shape of the cast shadows (188-191).
188.
The form of the shadow cast by any body of uniform density can never
be the same as that of the body producing it. [Footnote: Comp. the
drawing on PI. XXVIII, No. 5.]
189.
No cast shadow can produce the true image of the body which casts it
on a vertical plane unless the centre of the light is equally
distant from all the edges of that body.
190.
If a window _a b_ admits the sunlight into a room, the sunlight will
magnify the size of the window and diminish the shadow of a man in
such a way as that when the man makes that dim shadow of himself,
approach to that which defines the real size of the window, he will
see the shadows where they come into contact, dim and confused from
the strength of the light, shutting off and not allowing the solar
rays to pass; the effect of the shadow of the man cast by this
contact will be exactly that figured above.
[Footnote: It is scarcely possible to render the meaning of this
sentence with strict accuracy; mainly because the grammatical
construction is defective in the most important part--line 4. In the
very slight original sketch the shadow touches the upper arch of the
window and the correction, here given is perhaps not justified.]
191.
A shadow is never seen as of uniform depth on the surface which
intercepts it unless every portion of that surface is equidistant
from the luminous body. This is proved by the 7th which says:--The
shadow will appear lighter or stronger as it is surrounded by a
darker or a lighter background. And by the 8th of this:--The
background will be in parts darker or lighter, in proportion as it
is farther from or nearer to the luminous body. And:--Of various
spots equally distant from the luminous body those will always be in
the highest light on which the rays fall at the smallest angles: The
outline of the shadow as it falls on inequalities in the surface
will be seen with all the contours similar to those of the body that
casts it, if the eye is placed just where the centre of the light
was.
The shadow will look darkest where it is farthest from the body that
casts it. The shadow _c d_, cast by the body in shadow _a b_ which
is equally distant in all parts, is not of equal depth because it is
seen on a back ground of varying brightness. [Footnote: Compare the
three diagrams on Pl. VI, no 1 which, in the original accompany this
section.]
On the outlines of cast shadows (192-195).
192.
The edges of a derived shadow will be most distinct where it is cast
nearest to the primary shadow.
193.
As the derived shadow gets more distant from the primary shadow, the
more the cast shadow differs from the primary shadow.
194.
OF SHADOWS WHICH NEVER COME TO AN END.
The greater the difference between a light and the body lighted by
it, the light being the larger, the more vague will be the outlines
of the shadow of that object.
The derived shadow will be most confused towards the edges of its
interception by a plane, where it is remotest from the body casting
it.
195.
What is the cause which makes the outlines of the shadow vague and
confused?
Whether it is possible to give clear and definite outlines to the
edges of shadows.
On the relative size of shadows (196. 197).
196.
THE BODY WHICH IS NEAREST TO THE LIGHT CASTS THE LARGEST SHADOW, AND
WHY?
If an object placed in front of a single light is very close to it
you will see that it casts a very large shadow on the opposite wall,
and the farther you remove the object from the light the smaller
will the image of the shadow become.
WHY A SHADOW LARGER THAN THE BODY THAT PRODUCES IT BECOMES OUT OF
PROPORTION.
The disproportion of a shadow which is larger than the body
producing it, results from the light being smaller than the body, so
that it cannot be at an equal distance from the edges of the body
[Footnote 11: H. LUDWIG in his edition of the old copies, in the
Vatican library--in which this chapter is included under Nos. 612,
613 and 614 alters this passage as follows: _quella parte ch'e piu
propinqua piu cresce che le distanti_, although the Vatican copy
agrees with the original MS. in having _distante_ in the former and
_propinque_ in the latter place. This supposed amendment seems to me
to invert the facts. Supposing for instance, that on Pl. XXXI No. 3.
_f_ is the spot where the light is that illuminates the figure there
represented, and that the line behind the figure represents a wall
on which the shadow of the figure is thrown. It is evident, that in
that case the nearest portion, in this case the under part of the
thigh, is very little magnified in the shadow, and the remoter
parts, for instance the head, are more magnified.]; and the portions
which are most remote are made larger than the nearer portions for
this reason [Footnote 12: See Footnote 11].
WHY A SHADOW WHICH IS LARGER THAN THE BODY CAUSING IT HAS
ILL-DEFINED OUTLINES.
The atmosphere which surrounds a light is almost like light itself
for brightness and colour; but the farther off it is the more it
loses this resemblance. An object which casts a large shadow and is
near to the light, is illuminated both by that light by the luminous
atmosphere; hence this diffused light gives the shadow ill-defined
edges.
197.
A luminous body which is long and narrow in shape gives more
confused outlines to the derived shadow than a spherical light, and
this contradicts the proposition next following: A shadow will have
its outlines more clearly defined in proportion as it is nearer to
the primary shadow or, I should say, the body casting the shadow;
[Footnote 14: The lettering refers to the lower diagram, Pl. XLI,
No. 5.] the cause of this is the elongated form of the luminous body
_a c_, &c. [Footnote 16: See Footnote 14].
Effects on cast shadows by the tone of the back ground.
198.
OF MODIFIED SHADOWS.
Modified shadows are those which are cast on light walls or other
illuminated objects.
A shadow looks darkest against a light background. The outlines of a
derived shadow will be clearer as they are nearer to the primary
shadow. A derived shadow will be most defined in shape where it is
intercepted, where the plane intercepts it at the most equal angle.
Those parts of a shadow will appear darkest which have darker
objects opposite to them. And they will appear less dark when they
face lighter objects. And the larger the light object opposite, the
more the shadow will be lightened.
And the larger the surface of the dark object the more it will
darken the derived shadow where it is intercepted.
A disputed proposition.
199.
OF THE OPINION OF SOME THAT A TRIANGLE CASTS NO SHADOW ON A PLANE
SURFACE.
Certain mathematicians have maintained that a triangle, of which the
base is turned to the light, casts no shadow on a plane; and this
they prove by saying [5] that no spherical body smaller than the
light can reach the middle with the shadow. The lines of radiant
light are straight lines [6]; therefore, suppose the light to be _g
h_ and the triangle _l m n_, and let the plane be _i k_; they say
the light _g_ falls on the side of the triangle _l n_, and the
portion of the plane _i q_. Thus again _h_ like _g_ falls on the
side _l m_, and then on _m n_ and the plane _p k_; and if the whole
plane thus faces the lights _g h_, it is evident that the triangle
has no shadow; and that which has no shadow can cast none. This, in
this case appears credible. But if the triangle _n p g_ were not
illuminated by the two lights _g_ and _h_, but by _i p_ and _g_ and
_k_ neither side is lighted by more than one single light: that is
_i p_ is invisible to _h g_ and _k_ will never be lighted by _g_;
hence _p q_ will be twice as light as the two visible portions that
are in shadow.
[Footnote: 5--6. This passage is so obscure that it would be rash to
offer an explanation. Several words seem to have been omitted.]
On the relative depth of cast shadows (200-202).
200.
A spot is most in the shade when a large number of darkened rays
fall upon it. The spot which receives the rays at the widest angle
and by darkened rays will be most in the dark; a will be twice as
dark as b, because it originates from twice as large a base at an
equal distance. A spot is most illuminated when a large number of
luminous rays fall upon it. d is the beginning of the shadow _d f_,
and tinges _c_ but _a_ little; _d e_ is half of the shadow _d f_ and
gives a deeper tone where it is cast at _b_ than at _f_. And the
whole shaded space _e_ gives its tone to the spot _a_. [Footnote:
The diagram here referred to is on Pl. XLI, No. 2.]
201.
_A n_ will be darker than _c r_ in proportion to the number of times
that _a b_ goes into _c d_.
202.
The shadow cast by an object on a plane will be smaller in
proportion as that object is lighted by feebler rays. Let _d e_ be
the object and _d c_ the plane surface; the number of times that _d
e_ will go into _f g_ gives the proportion of light at _f h_ to _d
c_. The ray of light will be weaker in proportion to its distance
from the hole through which it falls.
FIFTH BOOK ON LIGHT AND SHADE.
Principles of reflection (203. 204).
203.
OF THE WAY IN WHICH THE SHADOWS CAST BY OBJECTS OUGHT TO BE DEFINED.
If the object is the mountain here figured, and the light is at the
point _a_, I say that from _b d_ and also from _c f_ there will be
no light but from reflected rays. And this results from the fact
that rays of light can only act in straight lines; and the same is
the case with the secondary or reflected rays.
204.
The edges of the derived shadow are defined by the hues of the
illuminated objects surrounding the luminous body which produces the
shadow.
On reverberation.
205.
OF REVERBERATION.
Reverberation is caused by bodies of a bright nature with a flat and
semi opaque surface which, when the light strikes upon them, throw
it back again, like the rebound of a ball, to the former object.
WHERE THERE CAN BE NO REFLECTED LIGHTS.
All dense bodies have their surfaces occupied by various degrees of
light and shade. The lights are of two kinds, one called original,
the other borrowed. Original light is that which is inherent in the
flame of fire or the light of the sun or of the atmosphere. Borrowed
light will be reflected light; but to return to the promised
definition: I say that this luminous reverberation is not produced
by those portions of a body which are turned towards darkened
objects, such as shaded spots, fields with grass of various height,
woods whether green or bare; in which, though that side of each
branch which is turned towards the original light has a share of
that light, nevertheless the shadows cast by each branch separately
are so numerous, as well as those cast by one branch on the others,
that finally so much shadow is the result that the light counts for
nothing. Hence objects of this kind cannot throw any reflected light
on opposite objects.
Reflection on water (206. 207).
206.
PERSPECTIVE.
The shadow or object mirrored in water in motion, that is to say in
small wavelets, will always be larger than the external object
producing it.
207.
It is impossible that an object mirrored on water should correspond
in form to the object mirrored, since the centre of the eye is above
the surface of the water.
This is made plain in the figure here given, which demonstrates that
the eye sees the surface _a b_, and cannot see it at _l f_, and at
_r t_; it sees the surface of the image at _r t_, and does not see
it in the real object _c d_. Hence it is impossible to see it, as
has been said above unless the eye itself is situated on the surface
of the water as is shown below [13].
[Footnote: _A_ stands for _ochio_ [eye], _B_ for _aria_ [air], _C_
for _acqua_ [water], _D_ for _cateto_ [cathetus].--In the original
MS. the second diagram is placed below line 13.]
Experiments with the mirror (208-210).
208.
THE MIRROR.
If the illuminated object is of the same size as the luminous body
and as that in which the light is reflected, the amount of the
reflected light will bear the same proportion to the intermediate
light as this second light will bear to the first, if both bodies
are smooth and white.
209.
Describe how it is that no object has its limitation in the mirror
but in the eye which sees it in the mirror. For if you look at your
face in the mirror, the part resembles the whole in as much as the
part is everywhere in the mirror, and the whole is in every part of
the same mirror; and the same is true of the whole image of any
object placed opposite to this mirror, &c.
210.
No man can see the image of another man in a mirror in its proper
place with regard to the objects; because every object falls on [the
surface of] the mirror at equal angles. And if the one man, who sees
the other in the mirror, is not in a direct line with the image he
will not see it in the place where it really falls; and if he gets
into the line, he covers the other man and puts himself in the place
occupied by his image. Let _n o_ be the mirror, _b_ the eye of your
friend and _d_ your own eye. Your friend's eye will appear to you at
_a_, and to him it will seem that yours is at _c_, and the
intersection of the visual rays will occur at _m_, so that either of
you touching _m_ will touch the eye of the other man which shall be
open. And if you touch the eye of the other man in the mirror it
will seem to him that you are touching your own.
Appendix:--On shadows in movement (211. 212).
211.
OF THE SHADOW AND ITS MOTION.
When two bodies casting shadows, and one in front of the other, are
between a window and the wall with some space between them, the
shadow of the body which is nearest to the plane of the wall will
move if the body nearest to the window is put in transverse motion
across the window. To prove this let _a_ and _b_ be two bodies
placed between the window _n m_ and the plane surface _o p_ with
sufficient space between them as shown by the space _a b_. I say
that if the body _a_ is moved towards _s_ the shadow of the body _b_
which is at _c_ will move towards _d_.
212.
OF THE MOTION OF SHADOWS.
The motion of a shadow is always more rapid than that of the body
which produces it if the light is stationary. To prove this let _a_
be the luminous body, and _b_ the body casting the shadow, and _d_
the shadow. Then I say that in the time while the solid body moves
from _b_ to _c_, the shadow _d_ will move to _e_; and this
proportion in the rapidity of the movements made in the same space
of time, is equal to that in the length of the space moved over.
Thus, given the proportion of the space moved over by the body _b_
to _c_, to that moved over by the shadow _d_ to _e_, the proportion
in the rapidity of their movements will be the same.
But if the luminous body is also in movement with a velocity equal
to that of the solid body, then the shadow and the body that casts
it will move with equal speed. And if the luminous body moves more
rapidly than the solid body, the motion of the shadow will be slower
than that of the body casting it.
But if the luminous body moves more slowly than the solid body, then
the shadow will move more rapidly than that body.
SIXTH BOOK ON LIGHT AND SHADE.
The effect of rays passing through holes (213. 214).
213.
PERSPECTIVE.
If you transmit the rays of the sun through a hole in the shape of a
star you will see a beautiful effect of perspective in the spot
where the sun's rays fall.
[Footnote: In this and the following chapters of MS. C the order of
the original paging has been adhered to, and is shown in
parenthesis. Leonardo himself has but rarely worked out the subject
of these propositions. The space left for the purpose has
occasionally been made use of for quite different matter. Even the
numerous diagrams, most of them very delicately sketched, lettered
and numbered, which occur on these pages, are hardly ever explained,
with the exception of those few which are here given.]
214.
No small hole can so modify the convergence of rays of light as to
prevent, at a long distance, the transmission of the true form of
the luminous body causing them. It is impossible that rays of light
passing through a parallel [slit], should not display the form of
the body causing them, since all the effects produced by a luminous
body are [in fact] the reflection of that body: The moon, shaped
like a boat, if transmitted through a hole is figured in the surface
[it falls on] as a boatshaped object. [Footnote 8: In the MS. a
blank space is left after this question.] Why the eye sees bodies at
a distance, larger than they measure on the vertical plane?.
[Footnote: This chapter, taken from another MS. may, as an
exception, be placed here, as it refers to the same subject as the
preceding section.]
On gradation of shadows (215. 216).
215.
Although the breadth and length of lights and shadow will be
narrower and shorter in foreshortening, the quality and quantity of
the light and shade is not increased nor diminished.
[3]The function of shade and light when diminished by
foreshortening, will be to give shadow and to illuminate an object
opposite, according to the quality and quantity in which they fall
on the body.
[5]In proportion as a derived shadow is nearer to its penultimate
extremities the deeper it will appear, _g z_ beyond the intersection
faces only the part of the shadow [marked] _y z_; this by
intersection takes the shadow from _m n_ but by direct line it takes
the shadow _a m_ hence it is twice as deep as _g z_. _Y x_, by
intersection takes the shadow _n o_, but by direct line the shadow
_n m a_, therefore _x y_ is three times as dark as _z g_; _x f_, by
intersection faces _o b_ and by direct line _o n m a_, therefore we
must say that the shadow between _f x_ will be four times as dark as
the shadow _z g_, because it faces four times as much shadow.
Let _a b_ be the side where the primary shadow is, and _b c_ the
primary light, _d_ will be the spot where it is intercepted,_f g_
the derived shadow and _f e_ the derived light.
And this must be at the beginning of the explanation.
[Footnote: In the original MS. the text of No. 252 precedes the one
given here. In the text of No. 215 there is a blank space of about
four lines between the lines 2 and 3. The diagram given on Pl. VI,
No. 2 is placed between lines 4 and 5. Between lines 5 and 6 there
is another space of about three lines and one line left blank
between lines 8 and 9. The reader will find the meaning of the whole
passage much clearer if he first reads the final lines 11--13.
Compare also line 4 of No. 270.]
On relative proportion of light and shadows (216--221).
216.
That part of the surface of a body on which the images [reflection]
from other bodies placed opposite fall at the largest angle will
assume their hue most strongly. In the diagram below, 8 is a larger
angle than 4, since its base _a n_ is larger than _e n_ the base of
4. This diagram below should end at _a n_ 4 8. [4]That portion of
the illuminated surface on which a shadow is cast will be brightest
which lies contiguous to the cast shadow. Just as an object which is
lighted up by a greater quantity of luminous rays becomes brighter,
so one on which a greater quantity of shadow falls, will be darker.
Let 4 be the side of an illuminated surface 4 8, surrounding the
cast shadow _g e_ 4. And this spot 4 will be lighter than 8, because
less shadow falls on it than on 8. Since 4 faces only the shadow _i
n_; and 8 faces and receives the shadow _a e_ as well as _i n_ which
makes it twice as dark. And the same thing happens when you put the
atmosphere and the sun in the place of shade and light.
[12] The distribution of shadow, originating in, and limited by,
plane surfaces placed near to each other, equal in tone and directly
opposite, will be darker at the ends than at the beginning, which
will be determined by the incidence of the luminous rays. You will
find the same proportion in the depth of the derived shadows _a n_
as in the nearness of the luminous bodies _m b_, which cause them;
and if the luminous bodies were of equal size you would still
farther find the same proportion in the light cast by the luminous
circles and their shadows as in the distance of the said luminous
bodies.
[Footnote: The diagram originally placed between lines 3 and 4 is on
Pl. VI, No. 3. In the diagram given above line 14 of the original,
and here printed in the text, the words _corpo luminoso_ [luminous
body] are written in the circle _m_, _luminoso_ in the circle _b_
and _ombroso_ [body in shadow] in the circle _o_.]
217.
THAT PART OF THE REFLECTION WILL BE BRIGHTEST WHERE THE REFLECTED
RAYS ARE SHORTEST.
[2] The darkness occasioned by the casting of combined shadows will
be in conformity with its cause, which will originate and terminate
between two plane surfaces near together, alike in tone and directly
opposite each other.
[4] In proportion as the source of light is larger, the luminous and
shadow rays will be more mixed together. This result is produced
because wherever there is a larger quantity of luminous rays, there
is most light, but where there are fewer there is least light,
consequently the shadow rays come in and mingle with them.
[Footnote: Diagrams are inserted before lines 2 and 4.]
218.
In all the proportions I lay down it must be understood that the
medium between the bodies is always the same. [2] The smaller the
luminous body the more distinct will the transmission of the shadows
be.
[3] When of two opposite shadows, produced by the same body, one is
twice as dark as the other though similar in form, one of the two
lights causing them must have twice the diameter that the other has
and be at twice the distance from the opaque body. If the object is
lowly moved across the luminous body, and the shadow is intercepted
at some distance from the object, there will be the same relative
proportion between the motion of the derived shadow and the motion
of the primary shadow, as between the distance from the object to
the light, and that from the object to the spot where the shadow is
intercepted; so that though the object is moved slowly the shadow
moves fast.
[Footnote: There are diagrams inserted before lines 2 and 3 but they
are not reproduced here. The diagram above line 6 is written upon as
follows: at _A lume_ (light), at _B obbietto_ (body), at _C ombra
d'obbietto_ (shadow of the object).]
219.
A luminous body will appear less brilliant when surrounded by a
bright background.
[2] I have found that the stars which are nearest to the horizon
look larger than the others because light falls upon them from a
larger proportion of the solar body than when they are above us; and
having more light from the sun they give more light, and the bodies
which are most luminous appear the largest. As may be seen by the
sun through a mist, and overhead; it appears larger where there is
no mist and diminished through mist. No portion of the luminous body
is ever visible from any spot within the pyramid of pure derived
shadow.
[Footnote: Between lines 1 and 2 there is in the original a large
diagram which does not refer to this text. ]
220.
A body on which the solar rays fall between the thin branches of
trees far apart will cast but a single shadow.
[2] If an opaque body and a luminous one are (both) spherical the
base of the pyramid of rays will bear the same proportion to the
luminous body as the base of the pyramid of shade to the opaque
body.
[4] When the transmitted shadow is intercepted by a plane surface
placed opposite to it and farther away from the luminous body than
from the object [which casts it] it will appear proportionately
darker and the edges more distinct.
[Footnote: The diagram which, in the original, is placed above line
2, is similar to the one, here given on page 73 (section 120).--The
diagram here given in the margin stands, in the original, between
lines 3 and 4.]
221.
A body illuminated by the solar rays passing between the thick
branches of trees will produce as many shadows as there are branches
between the sun and itself.
Where the shadow-rays from an opaque pyramidal body are intercepted
they will cast a shadow of bifurcate outline and various depth at
the points. A light which is broader than the apex but narrower than
the base of an opaque pyramidal body placed in front of it, will
cause that pyramid to cast a shadow of bifurcate form and various
degrees of depth.
If an opaque body, smaller than the light, casts two shadows and if
it is the same size or larger, casts but one, it follows that a
pyramidal body, of which part is smaller, part equal to, and part
larger than, the luminous body, will cast a bifurcate shadow.
[Footnote: Between lines 2 and 3 there are in the original two large
diagrams.]
_IV._
_Perspective of Disappearance._
_The theory of the_ "Prospettiva de' perdimenti" _would, in many
important details, be quite unintelligible if it had not been led up
by the principles of light and shade on which it is based. The word_
"Prospettiva" _in the language of the time included the principles
of optics; what Leonardo understood by_ "Perdimenti" _will be
clearly seen in the early chapters, Nos._ 222--224. _It is in the
very nature of the case that the farther explanations given in the
subsequent chapters must be limited to general rules. The sections
given as_ 227--231 _"On indistinctness at short distances" have, it
is true, only an indirect bearing on the subject; but on the other
hand, the following chapters,_ 232--234, _"On indistinctness at
great distances," go fully into the matter, and in chapters_
235--239, _which treat "Of the importance of light and shade in the
Perspective of Disappearance", the practical issues are distinctly
insisted on in their relation to the theory. This is naturally
followed by the statements as to "the effect of light or dark
backgrounds on the apparent size of bodies"_ (_Nos._ 240--250). _At
the end I have placed, in the order of the original, those sections
from the MS._ C _which treat of the "Perspective of Disappearance"
and serve to some extent to complete the treatment of the subject_
(251--262).
Definition (222. 223).
222.
OF THE DIMINISHED DISTINCTNESS OF THE OUTLINES OF OPAQUE BODIES.
If the real outlines of opaque bodies are indistinguishable at even
a very short distance, they will be more so at long distances; and,
since it is by its outlines that we are able to know the real form
of any opaque body, when by its remoteness we fail to discern it as
a whole, much more must we fail to discern its parts and outlines.
223.
OF THE DIMINUTION IN PERSPECTIVE OF OPAQUE OBJECTS.
Among opaque objects of equal size the apparent diminution of size
will be in proportion to their distance from the eye of the
spectator; but it is an inverse proportion, since, where the
distance is greater, the opaque body will appear smaller, and the
less the distance the larger will the object appear. And this is the
fundamental principle of linear perspective and it
follows:--[11]every object as it becomes more remote loses first
those parts which are smallest. Thus of a horse, we should lose the
legs before the head, because the legs are thinner than the head;
and the neck before the body for the same reason. Hence it follows
that the last part of the horse which would be discernible by the
eye would be the mass of the body in an oval form, or rather in a
cylindrical form and this would lose its apparent thickness before
its length--according to the 2nd rule given above, &c. [Footnote 23:
Compare line 11.].
If the eye remains stationary the perspective terminates in the
distance in a point. But if the eye moves in a straight [horizontal]
line the perspective terminates in a line and the reason is that
this line is generated by the motion of the point and our sight;
therefore it follows that as we move our sight [eye], the point
moves, and as we move the point, the line is generated, &c.
An illustration by experiment.
224.
Every visible body, in so far as it affects the eye, includes three
attributes; that is to say: mass, form and colour; and the mass is
recognisable at a greater distance from the place of its actual
existence than either colour or form. Again, colour is discernible
at a greater distance than form, but this law does not apply to
luminous bodies.
The above proposition is plainly shown and proved by experiment;
because: if you see a man close to you, you discern the exact
appearance of the mass and of the form and also of the colouring; if
he goes to some distance you will not recognise who he is, because
the character of the details will disappear, if he goes still
farther you will not be able to distinguish his colouring, but he
will appear as a dark object, and still farther he will appear as a
very small dark rounded object. It appears rounded because distance
so greatly diminishes the various details that nothing remains
visible but the larger mass. And the reason is this: We know very
well that all the images of objects reach the senses by a small
aperture in the eye; hence, if the whole horizon _a d_ is admitted
through such an aperture, the object _b c_ being but a very small
fraction of this horizon what space can it fill in that minute image
of so vast a hemisphere? And because luminous bodies have more power
in darkness than any others, it is evident that, as the chamber of
the eye is very dark, as is the nature of all colored cavities, the
images of distant objects are confused and lost in the great light
of the sky; and if they are visible at all, appear dark and black,
as every small body must when seen in the diffused light of the
atmosphere.
[Footnote: The diagram belonging to this passage is placed between
lines 5 and 6; it is No. 4 on Pl. VI. ]
A guiding rule.
225.
OF THE ATMOSPHERE THAT INTERPOSES BETWEEN THE EYE AND VISIBLE
OBJECTS.
An object will appear more or less distinct at the same distance, in
proportion as the atmosphere existing between the eye and that
object is more or less clear. Hence, as I know that the greater or
less quantity of the air that lies between the eye and the object
makes the outlines of that object more or less indistinct, you must
diminish the definiteness of outline of those objects in proportion
to their increasing distance from the eye of the spectator.
An experiment.
226.
When I was once in a place on the sea, at an equal distance from the
shore and the mountains, the distance from the shore looked much
greater than that from the mountains.
On indistinctness at short distances (227-231).
227.
If you place an opaque object in front of your eye at a distance of
four fingers' breadth, if it is smaller than the space between the
two eyes it will not interfere with your seeing any thing that may
be beyond it. No object situated beyond another object seen by the
eye can be concealed by this [nearer] object if it is smaller than
the space from eye to eye.
228.
The eye cannot take in a luminous angle which is too close to it.
229.
That part of a surface will be better lighted on which the light
falls at the greater angle. And that part, on which the shadow falls
at the greatest angle, will receive from those rays least of the
benefit of the light.
230.
OF THE EYE.
The edges of an object placed in front of the pupil of the eye will
be less distinct in proportion as they are closer to the eye. This
is shown by the edge of the object _n_ placed in front of the pupil
_d_; in looking at this edge the pupil also sees all the space _a c_
which is beyond the edge; and the images the eye receives from that
space are mingled with the images of the edge, so that one image
confuses the other, and this confusion hinders the pupil from
distinguishing the edge.
231.
The outlines of objects will be least clear when they are nearest to
the eye, and therefore remoter outlines will be clearer. Among
objects which are smaller than the pupil of the eye those will be
less distinct which are nearer to the eye.
On indistinctness at great distances (232-234).
232.
Objects near to the eye will appear larger than those at a distance.
Objects seen with two eyes will appear rounder than if they are seen
with only one.
Objects seen between light and shadow will show the most relief.
233.
OF PAINTING.
Our true perception of an object diminishes in proportion as its
size is diminished by distance.
234.
PERSPECTIVE.
Why objects seen at a distance appear large to the eye and in the
image on the vertical plane they appear small.
PERSPECTIVE.
I ask how far away the eye can discern a non-luminous body, as, for
instance, a mountain. It will be very plainly visible if the sun is
behind it; and could be seen at a greater or less distance according
to the sun's place in the sky.
[Footnote: The clue to the solution of this problem (lines 1-3) is
given in lines 4-6, No. 232. Objects seen with both eyes appear
solid since they are seen from two distinct points of sight
separated by the distance between the eyes, but this solidity cannot
be represented in a flat drawing. Compare No. 535.]
The importance of light and shade in the perspective of
disappearance (235-239).
235.
An opaque body seen in a line in which the light falls will reveal
no prominences to the eye. For instance, let _a_ be the solid body
and _c_ the light; _c m_ and _c n_ will be the lines of incidence of
the light, that is to say the lines which transmit the light to the
object _a_. The eye being at the point _b_, I say that since the
light _c_ falls on the whole part _m n_ the portions in relief on
that side will all be illuminated. Hence the eye placed at _c_
cannot see any light and shade and, not seeing it, every portion
will appear of the same tone, therefore the relief in the prominent
or rounded parts will not be visible.
236.
OF PAINTING.
When you represent in your work shadows which you can only discern
with difficulty, and of which you cannot distinguish the edges so
that you apprehend them confusedly, you must not make them sharp or
definite lest your work should have a wooden effect.
237.
OF PAINTING.
You will observe in drawing that among the shadows some are of
undistinguishable gradation and form, as is shown in the 3rd
[proposition] which says: Rounded surfaces display as many degrees
of light and shade as there are varieties of brightness and darkness
reflected from the surrounding objects.
238.
OF LIGHT AND SHADE.
You who draw from nature, look (carefully) at the extent, the
degree, and the form of the lights and shadows on each muscle; and
in their position lengthwise observe towards which muscle the axis
of the central line is directed.
239.
An object which is [so brilliantly illuminated as to be] almost as
bright as light will be visible at a greater distance, and of larger
apparent size than is natural to objects so remote.
The effect of light or dark backgrounds on the apparent size of
objects (240-250).
240.
A shadow will appear dark in proportion to the brilliancy of the
light surrounding it and conversely it will be less conspicuous
where it is seen against a darker background.
241.
OF ORDINARY PERSPECTIVE.
An object of equal breadth and colour throughout, seen against a
background of various colours will appear unequal in breadth.
And if an object of equal breadth throughout, but of various
colours, is seen against a background of uniform colour, that object
will appear of various breadth. And the more the colours of the
background or of the object seen against the ground vary, the
greater will the apparent variations in the breadth be though the
objects seen against the ground be of equal breadth [throughout].
242.
A dark object seen against a bright background will appear smaller
than it is.
A light object will look larger when it is seen against a background
darker than itself.
243.
OF LIGHT.
A luminous body when obscured by a dense atmosphere will appear
smaller; as may be seen by the moon or sun veiled by mists.
OF LIGHT.
Of several luminous bodies of equal size and brilliancy and at an
equal distance, that will look the largest which is surrounded by
the darkest background.
OF LIGHT.
I find that any luminous body when seen through a dense and thick
mist diminishes in proportion to its distance from the eye. Thus it
is with the sun by day, as well as the moon and the other eternal
lights by night. And when the air is clear, these luminaries appear
larger in proportion as they are farther from the eye.
244.
That portion of a body of uniform breadth which is against a lighter
background will look narrower [than the rest].
[4] _e_ is a given object, itself dark and of uniform breadth; _a b_
and _c d_ are two backgrounds one darker than the other; _b c_ is a
bright background, as it might be a spot lighted by the sun through
an aperture in a dark room. Then I say that the object _e g_ will
appear larger at _e f_ than at _g h_; because _e f_ has a darker
background than _g h_; and again at _f g_ it will look narrower from
being seen by the eye _o_, on the light background _b c_. [Footnote
12: The diagram to which the text, lines 1-11, refers, is placed in
the original between lines 3 and 4, and is given on Pl. XLI, No. 3.
Lines 12 to 14 are explained by the lower of the two diagrams on Pl.
XLI, No. 4. In the original these are placed after line 14.] That
part of a luminous body, of equal breadth and brilliancy throughout,
will look largest which is seen against the darkest background; and
the luminous body will seem on fire.
245.
WHY BODIES IN LIGHT AND SHADE HAVE THEIR OUTLINES ALTERED BY THE
COLOUR AND BRIGHTNESS OF THE OBJECTS SERVING AS A BACKGROUND TO
THEM.
If you look at a body of which the illuminated portion lies and ends
against a dark background, that part of the light which will look
brightest will be that which lies against the dark [background] at
_d_. But if this brighter part lies against a light background, the
edge of the object, which is itself light, will be less distinct
than before, and the highest light will appear to be between the
limit of the background _m f_ and the shadow. The same thing is seen
with regard to the dark [side], inasmuch as that edge of the shaded
portion of the object which lies against a light background, as at
_l_, it looks much darker than the rest. But if this shadow lies
against a dark background, the edge of the shaded part will appear
lighter than before, and the deepest shade will appear between the
edge and the light at the point _o_.
[Footnote: In the original diagram _o_ is inside the shaded surface
at the level of _d_.]
246.
An opaque body will appear smaller when it is surrounded by a highly
luminous background, and a light body will appear larger when it is
seen against a darker background. This may be seen in the height of
buildings at night, when lightning flashes behind them; it suddenly
seems, when it lightens, as though the height of the building were
diminished. For the same reason such buildings look larger in a
mist, or by night than when the atmosphere is clear and light.
247.
ON LIGHT BETWEEN SHADOWS
When you are drawing any object, remember, in comparing the grades
of light in the illuminated portions, that the eye is often deceived
by seeing things lighter than they are. And the reason lies in our
comparing those parts with the contiguous parts. Since if two
[separate] parts are in different grades of light and if the less
bright is conterminous with a dark portion and the brighter is
conterminous with a light background--as the sky or something
equally bright--, then that which is less light, or I should say
less radiant, will look the brighter and the brighter will seem the
darker.
248.
Of objects equally dark in themselves and situated at a considerable
and equal distance, that will look the darkest which is farthest
above the earth.
249.
TO PROVE HOW IT IS THAT LUMINOUS BODIES APPEAR LARGER, AT A
DISTANCE, THAN THEY ARE.
If you place two lighted candles side by side half a braccio apart,
and go from them to a distance 200 braccia you will see that by the
increased size of each they will appear as a single luminous body
with the light of the two flames, one braccio wide.
TO PROVE HOW YOU MAY SEE THE REAL SIZE OF LUMINOUS BODIES.
If you wish to see the real size of these luminous bodies, take a
very thin board and make in it a hole no bigger than the tag of a
lace and place it as close to your eye as possible, so that when you
look through this hole, at the said light, you can see a large space
of air round it. Then by rapidly moving this board backwards and
forwards before your eye you will see the light increase [and
diminish].
Propositions on perspective of disappearance from MS. C. (250-262).
250.
Of several bodies of equal size and equally distant from the eye,
those will look the smallest which are against the lightest
background.
Every visible object must be surrounded by light and shade. A
perfectly spherical body surrounded by light and shade will appear
to have one side larger than the other in proportion as one is more
highly lighted than the other.
251.
PERSPECTIVE.
No visible object can be well understood and comprehended by the
human eye excepting from the difference of the background against
which the edges of the object terminate and by which they are
bounded, and no object will appear [to stand out] separate from that
background so far as the outlines of its borders are concerned. The
moon, though it is at a great distance from the sun, when, in an
eclipse, it comes between our eyes and the sun, appears to the eyes
of men to be close to the sun and affixed to it, because the sun is
then the background to the moon.
252.
A luminous body will appear more brilliant in proportion as it is
surrounded by deeper shadow. [Footnote: The diagram which, in the
original, is placed after this text, has no connection with it.]
253.
The straight edges of a body will appear broken when they are
conterminous with a dark space streaked with rays of light.
[Footnote: Here again the diagrams in the original have no
connection with the text.]
254.
Of several bodies, all equally large and equally distant, that which
is most brightly illuminated will appear to the eye nearest and
largest. [Footnote: Here again the diagrams in the original have no
connection with the text.]
255.
If several luminous bodies are seen from a great distance although
they are really separate they will appear united as one body.
256.
If several objects in shadow, standing very close together, are seen
against a bright background they will appear separated by wide
intervals.
257.
Of several bodies of equal size and tone, that which is farthest
will appear the lightest and smallest.
258.
Of several objects equal in size, brightness of background and
length that which has the flattest surface will look the largest. A
bar of iron equally thick throughout and of which half is red hot,
affords an example, for the red hot part looks thicker than the
rest.
259.
Of several bodies of equal size and length, and alike in form and in
depth of shade, that will appear smallest which is surrounded by the
most luminous background.
260.
DIFFERENT PORTIONS OF A WALL SURFACE WILL BE DARKER OR BRIGHTER IN
PROPORTION AS THE LIGHT OR SHADOW FALLS ON THEM AT A LARGER ANGLE.
The foregoing proposition can be clearly proved in this way. Let us
say that _m q_ is the luminous body, then _f g_ will be the opaque
body; and let _a e_ be the above-mentioned plane on which the said
angles fall, showing [plainly] the nature and character of their
bases. Then: _a_ will be more luminous than _b_; the base of the
angle _a_ is larger than that of _b_ and it therefore makes a
greater angle which will be _a m q_; and the pyramid _b p m_ will be
narrower and _m o c_ will be still finer, and so on by degrees, in
proportion as they are nearer to _e_, the pyramids will become
narrower and darker. That portion of the wall will be the darkest
where the breadth of the pyramid of shadow is greater than the
breadth of the pyramid of light.
At the point _a_ the pyramid of light is equal in strength to the
pyramid of shadow, because the base _f g_ is equal to the base _r
f_. At the point _d_ the pyramid of light is narrower than the
pyramid of shadow by so much as the base _s f_ is less than the base
_f g_.
Divide the foregoing proposition into two diagrams, one with the
pyramids of light and shadow, the other with the pyramids of light
[only].
261.
Among shadows of equal depth those which are nearest to the eye will
look least deep.
262.
The more brilliant the light given by a luminous body, the deeper
will the shadows be cast by the objects it illuminates.
_V._
_Theory of colours._
_Leonardo's theory of colours is even more intimately connected with
his principles of light and shade than his Perspective of
Disappearance and is in fact merely an appendix or supplement to
those principles, as we gather from the titles to sections_ 264,
267_, and _276_, while others again_ (_Nos._ 281, 282_) are headed_
Prospettiva.
_A very few of these chapters are to be found in the oldest copies
and editions of the Treatise on Painting, and although the material
they afford is but meager and the connection between them but
slight, w