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The Notebooks of Leonardo Da Vinci, Complete
131.
THE RAYS WHETHER SHADED OR LUMINOUS HAVE GREATER STRENGTH AND EFFECT
AT THEIR POINTS THAN AT THEIR SIDES.
Although the points of luminous pyramids may extend into shaded places
and those of pyramids of shadow into illuminated places, and though
among the luminous pyramids one may start from a broader base than
another; nevertheless, if by reason of their various length these
luminous pyramids acquire angles of equal size their light will be
equal; and the case will be the same with the pyramids of shadow; as
may be seen in the intersected pyramids _a b c_ and _d e f_, which
though their bases differ in size are equal as to breadth and light.
[Footnote: 51--55: This supplementary paragraph is indicated as being
a continuation of line 45, by two small crosses.]
The difference between light and lustre (132--135).
132.
Of the difference between light and lustre; and that lustre is not
included among colours, but is saturation of whiteness, and derived
from the surface of wet bodies; light partakes of the colour of the
object which reflects it (to the eye) as gold or silver or the like.
133.
OF THE HIGHEST LIGHTS WHICH TURN AND MOVE AS THE EYE MOVES WHICH SEES
THE OBJECT.
Suppose the body to be the round object figured here and let the light
be at the point _a_, and let the illuminated side of the object be _b
c_ and the eye at the point _d_: I say that, as lustre is every where
and complete in each part, if you stand at the point _d_ the lustre
will appear at _c_, and in proportion as the eye moves from _d_ to
_a_, the lustre will move from _c_ to _n_.
134.
OF PAINTING.
Heigh light or lustre on any object is not situated [necessarily] in
the middle of an illuminated object, but moves as and where the eye
moves in looking at it.
135.
OF LIGHT AND LUSTRE.
What is the difference between light and the lustre which is seen on
the polished surface of opaque bodies?
The lights which are produced from the polished surface of opaque
bodies will be stationary on stationary objects even if the eye on
which they strike moves. But reflected lights will, on those same
objects, appear in as many different places on the surface as
different positions are taken by the eye.
WHAT BODIES HAVE LIGHT UPON THEM WITHOUT LUSTRE?
Opaque bodies which have a hard and rough surface never display any
lustre in any portion of the side on which the light falls.
WHAT BODIES WILL DISPLAY LUSTRE BUT NOT LOOK ILLUMINATED?
Those bodies which are opaque and hard with a hard surface reflect
light [lustre] from every spot on the illuminated side which is in a
position to receive light at the same angle of incidence as they
occupy with regard to the eye; but, as the surface mirrors all the
surrounding objects, the illuminated [body] is not recognisable in
these portions of the illuminated body.
136.
The relations of luminous to illuminated bodies.
The middle of the light and shade on an object in light and shade is
opposite to the middle of the primary light. All light and shadow
expresses itself in pyramidal lines. The middle of the shadow on any
object must necessarily be opposite the middle of its light, with a
direct line passing through the centre of the body. The middle of the
light will be at _a_, that of the shadow at _b_. [Again, in bodies
shown in light and shade the middle of each must coincide with the
centre of the body, and a straight line will pass through both and
through that centre.]
[Footnote: In the original MS., at the spot marked _a_ of the first
diagram Leonardo wrote _primitiuo_, and at the spot marked
_c_--_primitiva_ (primary); at the spot marked _b_ he wrote
_dirivatiuo_ and at _d deriuatiua_ (derived).]
Experiments on the relation of light and shadow within a room
(137--140).
137.
SHOWS HOW LIGHT FROM ANY SIDE CONVERGES TO ONE POINT.
Although the balls _a b c_ are lighted from one window, nevertheless,
if you follow the lines of their shadows you will see they intersect
at a point forming the angle _n_.
[Footnote: The diagram belonging to this passage is slightly sketched
on Pl. XXXII; a square with three balls below it. The first three
lines of the text belonging to it are written above the sketch and the
six others below it.]
138.
Every shadow cast by a body has a central line directed to a single
point produced by the intersection of luminous lines in the middle of
the opening and thickness of the window. The proposition stated above,
is plainly seen by experiment. Thus if you draw a place with a window
looking northwards, and let this be _s f_, you will see a line
starting from the horizon to the east, which, touching the 2 angles of
the window _o f_, reaches _d_; and from the horizon on the west
another line, touching the other 2 angles _r s_, and ending at _c_;
and their intersection falls exactly in the middle of the opening and
thickness of the window. Again, you can still better confirm this
proof by placing two sticks, as shown at _g h_; and you will see the
line drawn from the centre of the shadow directed to the centre _m_
and prolonged to the horizon _n f_.
[Footnote: _B_ here stands for _cerchio del' orizonte tramontano_ on
the original diagram (the circle of the horizon towards the North);
_A_ for _levante_ (East) and _C_ for _ponete_ (West).]
139.
Every shadow with all its variations, which becomes larger as its
distance from the object is greater, has its external lines
intersecting in the middle, between the light and the object. This
proposition is very evident and is confirmed by experience. For, if _a
b_ is a window without any object interposed, the luminous atmosphere
to the right hand at _a_ is seen to the left at _d_. And the
atmosphere at the left illuminates on the right at _c_, and the lines
intersect at the point _m_.
[Footnote: _A_ here stands for _levante_ (East), _B_ for _ponente_
(West).]
140.
Every body in light and shade is situated between 2 pyramids one dark
and the other luminous, one is visible the other is not. But this only
happens when the light enters by a window. Supposing _a b_ to be the
window and _r_ the body in light and shade, the light to the right
hand _z_ will pass the object to the left and go on to _p_; the light
to the left at _k_ will pass to the right of the object at _i_ and go
on to _m_ and the two lines will intersect at _c_ and form a pyramid.
Then again _a_ _b_ falls on the shaded body at _i_ _g_ and forms a
pyramid _f_ _i_ _g_. _f_ will be dark because the light _a_ _b_ can
never fall there; _i_ _g_ _c_ will be illuminated because the light
falls upon it.
Light and shadow with regard to the position of the eye (141--145).
141.
Every shaded body that is larger than the pupil and that interposes
between the luminous body and the eye will be seen dark.
When the eye is placed between the luminous body and the objects
illuminated by it, these objects will be seen without any shadow.
[Footnote: The diagram which in the original stands above line 1 is
given on Plate II, No 2. Then, after a blank space of about eight
lines, the diagram Plate II No 3 is placed in the original. There is
no explanation of it beyond the one line written under it.]
142.
Why the 2 lights one on each side of a body having two pyramidal sides
of an obtuse apex leave it devoid of shadow.
[Footnote: The sketch illustrating this is on Plate XLI No 1.]
143.
A body in shadow situated between the light and the eye can never
display its illuminated portion unless the eye can see the whole of
the primary light.
[Footnote: _A_ stands for _corpo_ (body), _B_ for _lume_ (light).]
144.
The eye which looks (at a spot) half way between the shadow and the
light which surrounds the body in shadow will see that the deepest
shadows on that body will meet the eye at equal angles, that is at the
same angle as that of sight.
[Footnote: In both these diagrams _A_ stands for _lume_ (light) _B_
for _ombra_ (shadow).]
145.
OF THE DIFFERENT LIGHT AND SHADE IN VARIOUS ASPECTS AND OF OBJECTS
PLACED IN THEM.
If the sun is in the East and you look towards the West you will see
every thing in full light and totally without shadow because you see
them from the same side as the sun: and if you look towards the South
or North you will see all objects in light and shade, because you see
both the side towards the sun and the side away from it; and if you
look towards the coming of the sun all objects will show you their
shaded side, because on that side the sun cannot fall upon them.
The law of the incidence of light.
146.
The edges of a window which are illuminated by 2 lights of equal
degrees of brightness will not reflect light of equal brightness into
the chamber within.
If _b_ is a candle and _a c_ our hemisphere both will illuminate the
edges of the window _m_ _n_, but light _b_ will only illuminate _f g_
and the hemisphere _a_ will light all of _d e_.
147.
OF PAINTING.
That part of a body which receives the luminous rays at equal angles
will be in a higher light than any other part of it.
And the part which the luminous rays strike between less equal angles
will be less strongly illuminated.
SECOND BOOK ON LIGHT AND SHADE.
Gradations of strength in the shadows (148. 149).
148.
THAT PORTION OF A BODY IN LIGHT AND SHADE WILL BE LEAST LUMINOUS WHICH
IS SEEN UNDER THE LEAST AMOUNT OF LIGHT.
That part of the object which is marked _m_ is in the highest light
because it faces the window _a d_ by the line _a f_; _n_ is in the
second grade because the light _b d_ strikes it by the line _b e_; _o_
is in the third grade, as the light falls on it from _c d_ by the line
_c h_; _p_ is the lowest light but one as _c d_ falls on it by the
line _d v_; _q_ is the deepest shadow for no light falls on it from
any part of the window.
In proportion as _c d_ goes into _a d_ so will _n r s_ be darker than
_m_, and all the rest is space without shadow.
[Footnote: The diagram belonging to this chapter is No. 1 on Plate
III. The letters _a b e d_ and _r_ are not reproduced in facsimile of
the original, but have been replaced by ordinary type in the margin.
5-12. The original text of these lines is reproduced within the
diagram.--Compare No 275.]
149.
The light which falls on a shaded body at the acutest angle receives
the highest light, and the darkest portion is that which receives it
at an obtuse angle and both the light and the shadow form pyramids.
The angle _c_ receives the highest grade of light because it is
directly in front of the window _a b_ and the whole horizon of the sky
_m x_. The angle _a_ differs but little from _c_ because the angles
which divide it are not so unequal as those below, and only that
portion of the horizon is intercepted which lies between _y_ and _x_.
Although it gains as much on the other side its line is nevertheless
not very strong because one angle is smaller than its fellow. The
angles _e i_ will have less light because they do not see much of the
light _m s_ and the light _v x_ and their angles are very unequal. Yhe
angle _k_ and the angle _f_ are each placed between very unequal
angles and therefore have but little light, because at _k_ it has only
the light _p t_, and at _f_ only _t q_; _o g_ is the lowest grade of
light because this part has no light at all from the sky; and thence
come the lines which will reconstruct a pyramid that is the
counterpart of the pyramid _c_; and this pyramid _l_ is in the first
grade of shadow; for this too is placed between equal angles directly
opposite to each other on either side of a straight line which passes
through the centre of the body and goes to the centre of the light.
The several luminous images cast within the frame of the window at the
points _a_ and _b_ make a light which surrounds the derived shadow
cast by the solid body at the points 4 and 6. The shaded images
increase from _o g_ and end at 7 and 8.
[Footnote: The diagram belonging to this chapter is No. 2 on Plate
III. In the original it is placed between lines 3 and 4, and in the
reproduction these are shown in part. The semi circle above is marked
_orizonte_ (horizon). The number 6 at the left hand side, outside the
facsimile, is in the place of a figure which has become indistinct in
the original.]
On the intensity of shadows as dependent on the distance from the
light (150-152).
150.
The smaller the light that falls upon an object the more shadow it
will display. And the light will illuminate a smaller portion of the
object in proportion as it is nearer to it; and conversely, a larger
extent of it in proportion as it is farther off.
A light which is smaller than the object on which it falls will light
up a smaller extent of it in proportion as it is nearer to it, and the
converse, as it is farther from it. But when the light is larger than
the object illuminated it will light a larger extent of the object in
proportion as it is nearer and the converse when they are farther
apart.
151.
That portion of an illuminated object which is nearest to the source
of light will be the most strongly illuminated.
152.
That portion of the primary shadow will be least dark which is
farthest from the edges.
The derived shadow will be darker than the primary shadow where it is
contiguous with it.
On the proportion of light and shade (153-157).
153.
That portion of an opaque body will be more in shade or more in light,
which is nearer to the dark body, by which it is shaded, or to the
light that illuminates it.
Objects seen in light and shade show in greater relief than those
which are wholly in light or in shadow.
154.
OF PERSPECTIVE.
The shaded and illuminated sides of opaque objects will display the
same proportion of light and darkness as their objects [Footnote 6:
The meaning of _obbietti_ (objects) is explained in no 153, lines
1-4.--Between the title-line and the next there is, in the original, a
small diagram representing a circle described round a square.].
155.
OF PAINTING.
The outlines and form of any part of a body in light and shade are
indistinct in the shadows and in the high lights; but in the portions
between the light and the shadows they are highly conspicuous.
156.
OF PAINTING.
Among objects in various degrees of shade, when the light proceeds
from a single source, there will be the same proportion in their
shadows as in the natural diminution of the light and the same must be
understood of the degrees of light.
157.
A single and distinct luminous body causes stronger relief in the
object than a diffused light; as may be seen by comparing one side of
a landscape illuminated by the sun, and one overshadowed by clouds,
and so illuminated only by the diffused light of the atmosphere.
THIRD BOOK ON LIGHT AND SHADE.
Definition of derived shadow (158. 159).
158.
Derived shadow cannot exist without primary shadow. This is proved by
the first of this which says: Darkness is the total absence of light,
and shadow is an alleviation of darkness and of light, and it is more
or less dark or light in proportion as the darkness is modified by the
light.
159.
Shadow is diminution of light.
Darkness is absence of light.
Shadow is divided into two kinds, of which the first is called primary
shadow, the second is derived shadow. The primary shadow is always the
basis of the derived shadow.
The edges of the derived shadow are straight lines.
[Footnote: The theory of the _ombra_ dirivativa_--a technical
expression for which there is no precise English equivalent is
elaborately treated by Leonardo. But both text and diagrams (as Pl.
IV, 1-3 and Pl. V) must at once convince the student that the
distinction he makes between _ombra primitiva_ and _ombra dirivativa_
is not merely justifiable but scientific. _Ombra dirivativa_ is by no
means a mere abstract idea. This is easily proved by repeating the
experiment made by Leonardo, and by filling with smoke the room in
which the existence of the _ombra dirivativa_ is investigated, when
the shadow becomes visible. Nor is it difficult to perceive how much
of Leonardo's teaching depended on this theory. The recognised, but
extremely complicated science of cast shadows--_percussione dell'
ombre dirivative_ as Leonardo calls them--is thus rendered more
intelligible if not actually simpler, and we must assume this theory
as our chief guide through the investigations which follow.]
The darkness of the derived shadow diminishes in proportion as it is
remote from the primary shadow.
Different sorts of derived shadows (160-162).
160.
SHADOW AND LIGHT.
The forms of shadows are three: inasmuch as if the solid body which
casts the shadow is equal (in size) to the light, the shadow resembles
a column without any termination (in length). If the body is larger
than the light the shadow resembles a truncated and inverted pyramid,
and its length has also no defined termination. But if the body is
smaller than the light, the shadow will resemble a pyramid and come to
an end, as is seen in eclipses of the moon.
161.
OF SIMPLE DERIVED SHADOWS.
The simple derived shadow is of two kinds: one kind which has its
length defined, and two kinds which are undefined; and the defined
shadow is pyramidal. Of the two undefined, one is a column and the
other spreads out; and all three have rectilinear outlines. But the
converging, that is the pyramidal, shadow proceeds from a body that is
smaller than the light, and the columnar from a body equal in size to
the light, and the spreading shadow from a body larger than the light;
&c.
OF COMPOUND DERIVED SHADOWS.
Compound derived shadows are of two kinds; that is columnar and
spreading.
162.
OF SHADOW.
Derived shadows are of three kinds of which one is spreading, the
second columnar, the third converging to the point where the two sides
meet and intersect, and beyond this intersection the sides are
infinitely prolonged or straight lines. And if you say, this shadow
must terminate at the angle where the sides meet and extend no
farther, I deny this, because above in the first on shadow I have
proved: that a thing is completely terminated when no portion of it
goes beyond its terminating lines. Now here, in this shadow, we see
the converse of this, in as much as where this derived shadow
originates we obviously have the figures of two pyramids of shadow
which meet at their angles. Hence, if, as [my] opponent says, the
first pyramid of shadow terminates the derivative shadow at the angle
whence it starts, then the second pyramid of shadow--so says the
adversary--must be caused by the angle and not from the body in
shadow; and this is disproved with the help of the 2nd of this which
says: Shadow is a condition produced by a body casting a shadow, and
interposed between this shadow and the luminous body. By this it is
made clear that the shadow is not produced by the angle of the derived
shadow but only by the body casting the shadow; &c. If a spherical
solid body is illuminated by a light of elongated form the shadow
produced by the longest portion of this light will have less defined
outlines than that which is produced by the breadth of the same light.
And this is proved by what was said before, which is: That a shadow
will have less defined outlines in proportion as the light which
causes it is larger, and conversely, the outlines are clearer in
proportion as it is smaller.
[Footnote: The two diagrams to this chapter are on Plate IV, No. 1.]
On the relation of derived and primary shadow (163-165).
163.
The derived shadow can never resemble the body from which it proceeds
unless the light is of the same form and size as the body causing the
shadow.
The derived shadow cannot be of the same form as the primary shadow
unless it is intercepted by a plane parallel to it.
164.
HOW A CAST SHADOW CAN NEVER BE OF THE SAME SIZE AS THE BODY THAT CASTS
IT.
If the rays of light proceed, as experience shows, from a single point
and are diffused in a sphere round this point, radiating and dispersed
through the air, the farther they spread the wider they must spread;
and an object placed between the light and a wall is always imaged
larger in its shadow, because the rays that strike it [Footnote: 7.
The following lines are wanting to complete the logical connection.]
would, by the time they have reached the wall, have become larger.
165.
Any shadow cast by a body in light and shade is of the same nature and
character as that which is inseparable from the body. The centre of
the length of a shadow always corresponds to that of the luminous body
[Footnote 6: This second statement of the same idea as in the former
sentence, but in different words, does not, in the original, come next
to the foregoing; sections 172 and 127 are placed between them.]. It
is inevitable that every shadow must have its centre in a line with
the centre of the light.
On the shape of derived shadows (166-174).
166.
OF THE PYRAMIDAL SHADOW.
The pyramidal shadow produced by a columnar body will be narrower than
the body itself in proportion as the simple derived shadow is
intersected farther from the body which casts it.
[Footnote 166: Compare the first diagram to No. 161. If we here
conceive of the outlines of the pyramid of shadow on the ground as
prolonged beyond its apex this gives rise to a second pyramid; this is
what is spoken of at the beginning of No. 166.]
167.
The cast shadow will be longest when the light is lowest.
The cast shadow will be shortest when the light is highest.
168.
Both the primary and derived shadow will be larger when caused by the
light of a candle than by diffused light. The difference between the
larger and smaller shadows will be in inverse proportion to the larger
and smaller lights causing them.
[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.