chapter Leonardo has written "_azurro_" (blue) where in the

facsimile I have marked _A_, and "_giallo_" (yellow) where _B_

stands.]

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

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

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

82.

An experiment showing that though the pupil may not be moved from

its position the objects seen by it may appear to move from their

places.

If you look at an object at some distance from you and which is

below the eye, and fix both your eyes upon it and with one hand

firmly hold the upper lid open while with the other you push up the

under lid--still keeping your eyes fixed on the object gazed at--you

will see that object double; one [image] remaining steady, and the

other moving in a contrary direction to the pressure of your finger

on the lower eyelid. How false the opinion is of those who say that

this happens because the pupil of the eye is displaced from its

position.

How the above mentioned facts prove that the pupil acts upside down

in seeing.

[Footnote: 82. 14--17. The subject indicated by these two headings is

fully discussed in the two chapters that follow them in the

original; but it did not seem to me appropriate to include them

here.]

Demostration of perspective by means of a vertical glass plane

(83-85).

83.

OF THE PLANE OF GLASS.

Perspective is nothing else than seeing place [or objects] behind a

plane of glass, quite transparent, on the surface of which the

objects behind that glass are to be drawn. These can be traced in

pyramids to the point in the eye, and these pyramids are intersected

on the glass plane.

84.

Pictorial perspective can never make an object at the same distance,

look of the same size as it appears to the eye. You see that the

apex of the pyramid _f c d_ is as far from the object _c_ _d_ as the

same point _f_ is from the object _a_ _b_; and yet _c_ _d_, which is

the base made by the painter's point, is smaller than _a_ _b_ which

is the base of the lines from the objects converging in the eye and

refracted at _s_ _t_, the surface of the eye. This may be proved by

experiment, by the lines of vision and then by the lines of the

painter's plumbline by cutting the real lines of vision on one and

the same plane and measuring on it one and the same object.

85.

PERSPECTIVE.

The vertical plane is a perpendicular line, imagined as in front of

the central point where the apex of the pyramids converge. And this

plane bears the same relation to this point as a plane of glass

would, through which you might see the various objects and draw them

on it. And the objects thus drawn would be smaller than the

originals, in proportion as the distance between the glass and the

eye was smaller than that between the glass and the objects.

PERSPECTIVE.

The different converging pyramids produced by the objects, will

show, on the plane, the various sizes and remoteness of the objects

causing them.

PERSPECTIVE.

All those horizontal planes of which the extremes are met by

perpendicular lines forming right angles, if they are of equal width

the more they rise to the level of eye the less this is seen, and

the more the eye is above them the more will their real width be

seen.

PERSPECTIVE.

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

of it.

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

86.

A simple and natural method; showing how objects appear to the eye

without any other medium.

The object that is nearest to the eye always seems larger than

another of the same size at greater distance. The eye _m_, seeing

the spaces _o v x_, hardly detects the difference between them, and

the. reason of this is that it is close to them [Footnote 6: It is

quite inconceivable to me why M. RAVAISSON, in a note to his French

translation of this simple passage should have remarked: _Il est

clair que c'est par erreur que Leonard a ècrit_ per esser visino _au

lieu de_ per non esser visino. (See his printed ed. of MS. A. p.

38.)]; but if these spaces are marked on the vertical plane _n o_

the space _o v_ will be seen at _o r_, and in the same way the space

_v x_ will appear at _r q_. And if you carry this out in any place

where you can walk round, it will look out of proportion by reason

of the great difference in the spaces _o r_ and _r q_. And this

proceeds from the eye being so much below [near] the plane that the

plane is foreshortened. Hence, if you wanted to carry it out, you

would have [to arrange] to see the perspective through a single hole

which must be at the point _m_, or else you must go to a distance of

at least 3 times the height of the object you see. The plane _o p_

being always equally remote from the eye will reproduce the objects

in a satisfactory way, so that they may be seen from place to place.

87.

How every large mass sends forth its images, which may diminish

through infinity.

The images of any large mass being infinitely divisible may be

infinitely diminished.

88.

Objects of equal size, situated in various places, will be seen by

different pyramids which will each be smaller in proportion as the

object is farther off.

89.

Perspective, in dealing with distances, makes use of two opposite

pyramids, one of which has its apex in the eye and the base as

distant as the horizon. The other has the base towards the eye and

the apex on the horizon. Now, the first includes the [visible]

universe, embracing all the mass of the objects that lie in front of

the eye; as it might be a vast landscape seen through a very small

opening; for the more remote the objects are from the eye, the

greater number can be seen through the opening, and thus the pyramid

is constructed with the base on the horizon and the apex in the eye,

as has been said. The second pyramid is extended to a spot which is

smaller in proportion as it is farther from the eye; and this second

perspective [= pyramid] results from the first.

90.

SIMPLE PERSPECTIVE.

Simple perspective is that which is constructed by art on a vertical

plane which is equally distant from the eye in every part. Complex

perspective is that which is constructed on a ground-plan in which

none of the parts are equally distant from the eye.

91.

PERSPECTIVE.

No surface can be seen exactly as it is, if the eye that sees it is

not equally remote from all its edges.

92.

WHY WHEN AN OBJECT IS PLACED CLOSE TO THE EYE ITS EDGES ARE

INDISTINCT.

When an object opposite the eye is brought too close to it, its

edges must become too confused to be distinguished; as it happens

with objects close to a light, which cast a large and indistinct

shadow, so is it with an eye which estimates objects opposite to it;

in all cases of linear perspective, the eye acts in the same way as

the light. And the reason is that the eye has one leading line (of

vision) which dilates with distance and embraces with true

discernment large objects at a distance as well as small ones that

are close. But since the eye sends out a multitude of lines which

surround this chief central one and since these which are farthest

from the centre in this cone of lines are less able to discern with

accuracy, it follows that an object brought close to the eye is not

at a due distance, but is too near for the central line to be able

to discern the outlines of the object. So the edges fall within the

lines of weaker discerning power, and these are to the function of

the eye like dogs in the chase which can put up the game but cannot

take it. Thus these cannot take in the objects, but induce the

central line of sight to turn upon them, when they have put them up.

Hence the objects which are seen with these lines of sight have

confused outlines.

The relative size of objects with regard to their distance from the

eye (93-98).

93.

PERSPECTIVE.

Small objects close at hand and large ones at a distance, being seen

within equal angles, will appear of the same size.

94.

PERSPECTIVE.

There is no object so large but that at a great distance from the

eye it does not appear smaller than a smaller object near.

95.

Among objects of equal size that which is most remote from the eye

will look the smallest. [Footnote: This axiom, sufficiently clear in

itself, is in the original illustrated by a very large diagram,

constructed like that here reproduced under No. 108.

The same idea is repeated in C. A. I a; I a, stated as follows:

_Infra le cose d'equal grandeza quella si dimostra di minor figura

che sara più distante dall' ochio_.--]

96.

Why an object is less distinct when brought near to the eye, and why

with spectacles, or without the naked eye sees badly either close or

far off [as the case may be].

97.

PERSPECTIVE.

Among objects of equal size, that which is most remote from the eye

will look the smallest.

98.

PERSPECTIVE.

No second object can be so much lower than the first as that the eye

will not see it higher than the first, if the eye is above the

second.

PERSPECTIVE.

And this second object will never be so much higher than the first

as that the eye, being below them, will not see the second as lower

than the first.

PERSPECTIVE.

If the eye sees a second square through the centre of a smaller one,

that is nearer, the second, larger square will appear to be

surrounded by the smaller one.

PERSPECTIVE--PROPOSITION.

Objects that are farther off can never be so large but that those in

front, though smaller, will conceal or surround them.

DEFINITION.

This proposition can be proved by experiment. For if you look

through a small hole there is nothing so large that it cannot be

seen through it and the object so seen appears surrounded and

enclosed by the outline of the sides of the hole. And if you stop it

up, this small stopping will conceal the view of the largest object.

The apparent size of objects defined by calculation (99-105)

99.

OF LINEAR PERSPECTIVE.

Linear Perspective deals with the action of the lines of sight, in

proving by measurement how much smaller is a second object than the

first, and how much the third is smaller than the second; and so on

by degrees to the end of things visible. I find by experience that

if a second object is as far beyond the first as the first is from

the eye, although they are of the same size, the second will seem

half the size of the first and if the third object is of the same

size as the 2nd, and the 3rd is as far beyond the second as the 2nd

from the first, it will appear of half the size of the second; and

so on by degrees, at equal distances, the next farthest will be half

the size of the former object. So long as the space does not exceed

the length of 20 braccia. But, beyond 20 braccia figures of equal

size will lose 2/4 and at 40 braccia they will lose 9/10, and 19/20

at 60 braccia, and so on diminishing by degrees. This is if the

picture plane is distant from you twice your own height. If it is

only as far off as your own height, there will be a great difference

between the first braccia and the second.

[Footnote: This chapter is included in DUFRESNE'S and MANZI'S

editions of the Treatise on Painting. H. LUDWIG, in his commentary,

calls this chapter "_eines der wichtigsten im ganzen Tractat_", but

at the same time he asserts that its substance has been so

completely disfigured in the best MS. copies that we ought not to

regard Leonardo as responsible for it. However, in the case of this

chapter, the old MS. copies agree with the original as it is

reproduced above. From the chapters given later in this edition,

which were written at a subsequent date, it would appear that

Leonardo corrected himself on these points.]

100.

OF THE DIMINUTION OF OBJECTS AT VARIOUS DISTANCES.

A second object as far distant from the first as the first is from

the eye will appear half the size of the first, though they be of

the same size really.

OF THE DEGREES OF DIMINUTION.

If you place the vertical plane at one braccio from the eye, the

first object, being at a distance of 4 braccia from your eye will

diminish to 3/4 of its height at that plane; and if it is 8 braccia

from the eye, to 7/8; and if it is 16 braccia off, it will diminish

to 15/16 of its height and so on by degrees, as the space doubles

the diminution will double.

101.

Begin from the line _m f_ with the eye below; then go up and do the

same with the line _n f_, then with the eye above and close to the 2

gauges on the ground look at _m n_; then as _c m_ is to _m n_ so

will _n m_ be to _n s_.

If _a n_ goes 3 times into _f b, m p_ will do the same into _p g_.

Then go backwards so far as that _c d_ goes twice into _a n_ and _p

g_ will be equal to _g h_. And _m p_ will go into _h p_ as often as

_d c_ into _o p_.

[Footnote: The first three lines are unfortunately very obscure.]

102.

I GIVE THE DEGREES OF THE OBJECTS SEEN BY THE EYE AS THE MUSICIAN

DOES THE NOTES HEARD BY THE EAR.

Although the objects seen by the eye do, in fact, touch each other

as they recede, I will nevertheless found my rule on spaces of 20

braccia each; as a musician does with notes, which, though they can

be carried on one into the next, he divides into degrees from note

to note calling them 1st, 2nd, 3rd, 4th, 5th; and has affixed a name

to each degree in raising or lowering the voice.

103.

PERSPECTIVE.

Let _f_ be the level and distance of the eye; and _a_ the vertical

plane, as high as a man; let _e_ be a man, then I say that on the

plane this will be the distance from the plane to the 2nd man.

104.

The differences in the diminution of objects of equal size in

consequence of their various remoteness from the eye will bear among

themselves the same proportions as those of the spaces between the

eye and the different objects.

Find out how much a man diminishes at a certain distance and what

its length is; and then at twice that distance and at 3 times, and

so make your general rule.

105.

The eye cannot judge where an object high up ought to descend.

106.

PERSPECTIVE.

If two similar and equal objects are placed one beyond the other at

a given distance the difference in their size will appear greater in

proportion as they are nearer to the eye that sees them. And

conversely there will seem to be less difference in their size in

proportion as they are remote from the eve.

This is proved by the proportions of their distances among

themselves; for, if the first of these two objects were as far from

the eye, as the 2nd from the first this would be called the second

proportion: since, if the first is at 1 braccia from the eye and the

2nd at two braccia, two being twice as much as one, the first object

will look twice as large as the second. But if you place the first

at a hundred braccia from you and the second at a hundred and one,

you will find that the first is only so much larger than the second

as 100 is less than 101; and the converse is equally true. And

again, the same thing is proved by the 4th of this book which shows

that among objects that are equal, there is the same proportion in

the diminution of the size as in the increase in the distance from

the eye of the spectator.

On natural perspective (107--109).

107.

OF EQUAL OBJECTS THE MOST REMOTE LOOK THE SMALLEST.

The practice of perspective may be divided into ... parts [Footnote

4: _in_ ... _parte_. The space for the number is left blank in the

original.], of which the first treats of objects seen by the eye at

any distance; and it shows all these objects just as the eye sees

them diminished, without obliging a man to stand in one place rather

than another so long as the plane does not produce a second

foreshortening.

But the second practice is a combination of perspective derived

partly from art and partly from nature and the work done by its

rules is in every portion of it, influenced by natural perspective

and artificial perspective. By natural perspective I mean that the

plane on which this perspective is represented is a flat surface,

and this plane, although it is parallel both in length and height,

is forced to diminish in its remoter parts more than in its nearer

ones. And this is proved by the first of what has been said above,

and its diminution is natural. But artificial perspective, that is

that which is devised by art, does the contrary; for objects equal

in size increase on the plane where it is foreshortened in

proportion as the eye is more natural and nearer to the plane, and

as the part of the plane on which it is figured is farther from the

eye.

And let this plane be _d e_ on which are seen 3 equal circles which

are beyond this plane _d e_, that is the circles _a b c_. Now you

see that the eye _h_ sees on the vertical plane the sections of the

images, largest of those that are farthest and smallest of the

nearest.

108.

Here follows what is wanting in the margin at the foot on the other

side of this page.

Natural perspective acts in a contrary way; for, at greater

distances the object seen appears smaller, and at a smaller distance

the object appears larger. But this said invention requires the

spectator to stand with his eye at a small hole and then, at that

small hole, it will be very plain. But since many (men's) eyes

endeavour at the same time to see one and the same picture produced

by this artifice only one can see clearly the effect of this

perspective and all the others will see confusion. It is well

therefore to avoid such complex perspective and hold to simple

perspective which does not regard planes as foreshortened, but as

much as possible in their proper form. This simple perspective, in

which the plane intersects the pyramids by which the images are

conveyed to the eye at an equal distance from the eye is our

constant experience, from the curved form of the pupil of the eye on

which the pyramids are intersected at an equal distance from the

visual virtue.

[Footnote 24: _la prima di sopra_ i. e. the first of the three

diagrams which, in the original MS., are placed in the margin at the

beginning of this chapter.]

109.

OF A MIXTURE OF NATURAL AND ARTIFICIAL PERSPECTIVE.

This diagram distinguishes natural from artificial perspective. But

before proceeding any farther I will define what is natural and what

is artificial perspective. Natural perspective says that the more

remote of a series of objects of equal size will look the smaller,

and conversely, the nearer will look the larger and the apparent

size will diminish in proportion to the distance. But in artificial

perspective when objects of unequal size are placed at various

distances, the smallest is nearer to the eye than the largest and

the greatest distance looks as though it were the least of all; and

the cause of this is the plane on which the objects are represented;

and which is at unequal distances from the eye throughout its

length. And this diminution of the plane is natural, but the

perspective shown upon it is artificial since it nowhere agrees with

the true diminution of the said plane. Whence it follows, that when

the eye is somewhat removed from the [station point of the]

perspective that it has been gazing at, all the objects represented

look monstrous, and this does not occur in natural perspective,

which has been defined above. Let us say then, that the square _a b

c d_ figured above is foreshortened being seen by the eye situated

in the centre of the side which is in front. But a mixture of

artificial and natural perspective will be seen in this tetragon

called _el main_ [Footnote 20: _el main_ is quite legibly written in

the original; the meaning and derivation of the word are equally

doubtful.], that is to say _e f g h_ which must appear to the eye of

the spectator to be equal to _a b c d_ so long as the eye remains in

its first position between _c_ and _d_. And this will be seen to

have a good effect, because the natural perspective of the plane

will conceal the defects which would [otherwise] seem monstrous.

_III._

_Six books on Light and Shade._

_Linear Perspective cannot be immediately followed by either the_

"prospettiva de' perdimenti" _or the_ "prospettiva de' colori" _or

the aerial perspective; since these branches of the subject

presuppose a knowledge of the principles of Light and Shade. No

apology, therefore, is here needed for placing these immediately

after Linear Perspective._

_We have various plans suggested by Leonardo for the arrangement of

the mass of materials treating of this subject. Among these I have

given the preference to a scheme propounded in No._ III, _because,

in all probability, we have here a final and definite purpose

expressed. Several authors have expressed it as their opinion that

the Paris Manuscript_ C _is a complete and finished treatise on

Light and Shade. Certainly, the Principles of Light and Shade form

by far the larger portion of this MS. which consists of two separate

parts; still, the materials are far from being finally arranged. It

is also evident that he here investigates the subject from the point

of view of the Physicist rather than from that of the Painter._

_The plan of a scheme of arrangement suggested in No._ III _and

adopted by me has been strictly adhered to for the first four Books.

For the three last, however, few materials have come down to us; and

it must be admitted that these three Books would find a far more

appropriate place in a work on Physics than in a treatise on

Painting. For this reason I have collected in Book V all the

chapters on Reflections, and in Book VI I have put together and

arranged all the sections of MS._ C _that belong to the book on

Painting, so far as they relate to Light and Shade, while the

sections of the same MS. which treat of the_ "Prospettiva de'

perdimenti" _have, of course, been excluded from the series on Light

and Shade._

[Footnote III: This text has already been published with some slight

variations in Dozio's pamphlet _Degli scritti e disegni di Leonardo

da Vinci_, Milan 1871, pp. 30--31. Dozio did not transcribe it from

the original MS. which seems to have remained unknown to him, but

from an old copy (MS. H. 227 in the Ambrosian Library).]

GENERAL INTRODUCTION.

Prolegomena.

110.

You must first explain the theory and then the practice. First you

must describe the shadows and lights on opaque objects, and then on

transparent bodies.

Scheme of the books on Light and shade.

111.

INTRODUCTION.

[Having already treated of the nature of shadows and the way in

which they are cast [Footnote 2: _Avendo io tractato._--We may

suppose that he here refers to some particular MS., possibly Paris

C.], I will now consider the places on which they fall; and their

curvature, obliquity, flatness or, in short, any character I may be

able to detect in them.]

Shadow is the obstruction of light. Shadows appear to me to be of

supreme importance in perspective, because, without them opaque and

solid bodies will be ill defined; that which is contained within

their outlines and their boundaries themselves will be

ill-understood unless they are shown against a background of a

different tone from themselves. And therefore in my first

proposition concerning shadow I state that every opaque body is

surrounded and its whole surface enveloped in shadow and light. And

on this proposition I build up the first Book. Besides this, shadows

have in themselves various degrees of darkness, because they are

caused by the absence of a variable amount of the luminous rays; and

these I call Primary shadows because they are the first, and

inseparable from the object to which they belong. And on this I will

found my second Book. From these primary shadows there result

certain shaded rays which are diffused through the atmosphere and

these vary in character according to that of the primary shadows

whence they are derived. I shall therefore call these shadows

Derived shadows because they are produced by other shadows; and the

third Book will treat of these. Again these derived shadows, where

they are intercepted by various objects, produce effects as various

as the places where they are cast and of this I will treat in the

fourth Book. And since all round the derived shadows, where the

derived shadows are intercepted, there is always a space where the

light falls and by reflected dispersion is thrown back towards its

cause, it meets the original shadow and mingles with it and modifies

it somewhat in its nature; and on this I will compose my fifth Book.

Besides this, in the sixth Book I will investigate the many and

various diversities of reflections resulting from these rays which

will modify the original [shadow] by [imparting] some of the various

colours from the different objects whence these reflected rays are

derived. Again, the seventh Book will treat of the various distances

that may exist between the spot where the reflected rays fall and

that where they originate, and the various shades of colour which

they will acquire in falling on opaque bodies.

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

112.

First I will treat of light falling through windows which I will

call Restricted [Light] and then I will treat of light in the open

country, to which I will give the name of diffused Light. Then I

will treat of the light of luminous bodies.

113.

OF PAINTING.

The conditions of shadow and light [as seen] by the eye are 3. Of

these the first is when the eye and the light are on the same side

of the object seen; the 2nd is when the eye is in front of the

object and the light is behind it. The 3rd is when the eye is in

front of the object and the light is on one side, in such a way as

that a line drawn from the object to the eye and one from the object

to the light should form a right angle where they meet.

114.

OF PAINTING.

This is another section: that is, of the nature of a reflection

(from) an object placed between the eye and the light under various

aspects.

115.

OF PAINTING.

As regards all visible objects 3 things must be considered. These

are the position of the eye which sees: that of the object seen

[with regard] to the light, and the position of the light which

illuminates the object, _b_ is the eye, _a_ the object seen, _c_ the

light, _a_ is the eye, _b_ the illuminating body, _c_ is the

illuminated object.

116.

Let _a_ be the light, _b_ the eye, _c_ the object seen by the eye

and in the light. These show, first, the eye between the light and

the body; the 2nd, the light between the eye and the body; the 3rd

the body between the eye and the light, _a_ is the eye, _b_ the

illuminated object, _c_ the light.

117.

OF PAINTING.

OF THE THREE KINDS OF LIGHT THAT ILLUMINATE OPAQUE BODIES.

The first kind of Light which may illuminate opaque bodies is called

Direct light--as that of the sun or any other light from a window or

flame. The second is Diffused [universal] light, such as we see in

cloudy weather or in mist and the like. The 3rd is Subdued light,

that is when the sun is entirely below the horizon, either in the

evening or morning.

118.

OF LIGHT.

The lights which may illuminate opaque bodies are of 4 kinds. These

are: diffused light as that of the atmosphere, within our horizon.

And Direct, as that of the sun, or of a window or door or other

opening. The third is Reflected light; and there is a 4th which is

that which passes through [semi] transparent bodies, as linen or

paper or the like, but not transparent like glass, or crystal, or

other diaphanous bodies, which produce the same effect as though

nothing intervened between the shaded object and the light that

falls upon it; and this we will discuss fully in our discourse.

Definition of the nature of shadows (119--122).

119.

WHAT LIGHT AND SHADOW ARE.

Shadow is the absence of light, merely the obstruction of the

luminous rays by an opaque body. Shadow is of the nature of

darkness. Light [on an object] is of the nature of a luminous body;

one conceals and the other reveals. They are always associated and

inseparable from all objects. But shadow is a more powerful agent

than light, for it can impede and entirely deprive bodies of their

light, while light can never entirely expel shadow from a body, that

is from an opaque body.

120.

Shadow is the diminution of light by the intervention of an opaque

body. Shadow is the counterpart of the luminous rays which are cut

off by an opaque body.

This is proved because the shadow cast is the same in shape and size

as the luminous rays were which are transformed into a shadow.

121.

Shadow is the diminution alike of light and of darkness, and stands

between darkness and light.

A shadow may be infinitely dark, and also of infinite degrees of

absence of darkness.

The beginnings and ends of shadow lie between the light and darkness

and may be infinitely diminished and infinitely increased. Shadow is

the means by which bodies display their form.

The forms of bodies could not be understood in detail but for

shadow.

122.

OF THE NATURE OF SHADOW.

Shadow partakes of the nature of universal matter. All such matters

are more powerful in their beginning and grow weaker towards the

end, I say at the beginning, whatever their form or condition may be

and whether visible or invisible. And it is not from small

beginnings that they grow to a great size in time; as it might be a

great oak which has a feeble beginning from a small acorn. Yet I may

say that the oak is most powerful at its beginning, that is where it

springs from the earth, which is where it is largest (To return:)

Darkness, then, is the strongest degree of shadow and light is its

least. Therefore, O Painter, make your shadow darkest close to the

object that casts it, and make the end of it fading into light,

seeming to have no end.

Of the various kinds of shadows. (123-125).

123.

Darkness is absence of light. Shadow is diminution of light.

Primitive shadow is that which is inseparable from a body not in the

light. Derived shadow is that which is disengaged from a body in

shadow and pervades the air. A cast transparent shadow is that which

is surrounded by an illuminated surface. A simple shadow is one

which receives no light from the luminous body which causes it. A

simple shadow begins within the line which starts from the edge of

the luminous body _a b_.

124.

A simple shadow is one where no light at all interferes with it.

A compound shadow is one which is somewhat illuminated by one or

more lights.

125.

WHAT IS THE DIFFERENCE BETWEEN A SHADOW THAT IS INSEPARABLE FROM A

BODY AND A CAST SHADOW?

An inseparable shadow is that which is never absent from the

illuminated body. As, for instance a ball, which so long as it is in

the light always has one side in shadow which never leaves it for

any movement or change of position in the ball. A separate shadow

may be and may not be produced by the body itself. Suppose the ball

to be one braccia distant from a wall with a light on the opposite

side of it; this light will throw upon the wall exactly as broad a

shadow as is to be seen on the side of the ball that is turned

towards the wall. That portion of the cast shadow will not be

visible when the light is below the ball and the shadow is thrown up

towards the sky and finding no obstruction on its way is lost.

126.

HOW THERE ARE 2 KINDS OF LIGHT, ONE SEPARABLE FROM, AND THE OTHER

INSEPARABLE FROM BODIES.

Of the various kinds of light (126, 127).

Separate light is that which falls upon the body. Inseparable light

is the side of the body that is illuminated by that light. One is

called primary, the other derived. And, in the same way there are

two kinds of shadow:--One primary and the other derived. The primary

is that which is inseparable from the body, the derived is that

which proceeds from the body conveying to the surface of the wall

the form of the body causing it.

127.

How there are 2 different kinds of light; one being called diffused,

the other restricted. The diffused is that which freely illuminates

objects. The restricted is that which being admitted through an

opening or window illuminates them on that side only.

[Footnote: At the spot marked _A_ in the first diagram Leonardo

wrote _lume costretto_ (restricted light). At the spot _B_ on the

second diagram he wrote _lume libero_ (diffused light).]

General remarks (128. 129).

128.

Light is the chaser away of darkness. Shade is the obstruction of

light. Primary light is that which falls on objects and causes light

and shade. And derived lights are those portions of a body which are

illuminated by the primary light. A primary shadow is that side of a

body on which the light cannot fall.

The general distribution of shadow and light is that sum total of

the rays thrown off by a shaded or illuminated body passing through

the air without any interference and the spot which intercepts and

cuts off the distribution of the dark and light rays.

And the eye can best distinguish the forms of objects when it is

placed between the shaded and the illuminated parts.

129.

MEMORANDUM OF THINGS I REQUIRE TO HAVE GRANTED [AS AXIOMS] IN MY

EXPLANATION OF PERSPECTIVE.

I ask to have this much granted me--to assert that every ray

passing through air of equal density throughout, travels in a

straight line from its cause to the object or place it falls upon.

FIRST BOOK ON LIGHT AND SHADE.

On the nature of light (130. 131).

130.

The reason by which we know that a light radiates from a single

centre is this: We plainly see that a large light is often much

broader than some small object which nevertheless--and although the

rays [of the large light] are much more than twice the extent [of

the small body]--always has its shadow cast on the nearest surface

very visibly. Let _c f_ be a broad light and _n_ be the object in

front of it, casting a shadow on the plane, and let _a b_ be the

plane. It is clear that it is not the broad light that will cast the

shadow _n_ on the plane, but that the light has within it a centre

is shown by this experiment. The shadow falls on the plane as is

shown at _m o t r_.

[Footnote 13: In the original MS. no explanatory text is placed

after this title-line; but a space is left for it and the text

beginning at line 15 comes next.] Why, to two [eyes] or in front of

two eyes do 3 objects appear as two?

Why, when you estimate the direction of an object with two sights

the nearer appears confused. I say that the eye projects an infinite

number of lines which mingle or join those reaching it which come to

it from the object looked at. And it is only the central and

sensible line that can discern and discriminate colours and objects;

all the others are false and illusory. And if you place 2 objects at

half an arm's length apart if the nearer of the two is close to the

eye its form will remain far more confused than that of the second;

the reason is that the first is overcome by a greater number of

false lines than the second and so is rendered vague.

Light acts in the same manner, for in the effects of its lines

(=rays), and particularly in perspective, it much resembles the eye;

and its central rays are what cast the true shadow. When the object

in front of it is too quickly overcome with dim rays it will cast a

broad and disproportionate shadow, ill defined; but when the object

which is to cast the shadow and cuts off the rays near to the place

where the shadow falls, then the shadow is distinct; and the more so

in proportion as the light is far off, because at a long distance

the central ray is less overcome by false rays; because the lines

from the eye and the solar and other luminous rays passing through

the atmosphere are obliged to travel in straight lines. Unless they

are deflected by a denser or rarer air, when they will be bent at

some point, but so long as the air is free from grossness or

moisture they will preserve their direct course, always carrying the

image of the object that intercepts them back to their point of

origin. And if this is the eye, the intercepting object will be seen

by its colour, as well as by form and size. But if the intercepting

plane has in it some small perforation opening into a darker

chamber--not darker in colour, but by absence of light--you will see

the rays enter through this hole and transmitting to the plane

beyond all the details of the object they proceed from both as to

colour and form; only every thing will be upside down. But the size

[of the image] where the lines are reconstructed will be in

proportion to the relative distance of the aperture from the plane

on which the lines fall [on one hand] and from their origin [on the

other]. There they intersect and form 2 pyramids with their point

meeting [a common apex] and their bases opposite. Let _a b_ be the

point of origin of the lines, _d e_ the first plane, and _c_ the

aperture with the intersection of the lines; _f g_ is the inner

plane. You will find that _a_ falls upon the inner plane below at

_g_, and _b_ which is below will go up to the spot _f_; it will be

quite evident to experimenters that every luminous body has in

itself a core or centre, from which and to which all the lines

radiate which are sent forth by the surface of the luminous body and

reflected back to it; or which, having been thrown out and not

intercepted, are dispersed in the air.

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.

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