The Notebooks of Leonardo Da Vinci/IV
IV. Perspective of Disappearance. 
_The theory of the_ "Prospettiva de' perdimenti" _would, in many important details, be quite unintelligible if it had not been led up by the principles of light and shade on which it is based. The word_ "Prospettiva" _in the language of the time included the principles of optics; what Leonardo understood by_ "Perdimenti" _will be clearly seen in the early chapters, Nos._ 222--224. _It is in the very nature of the case that the farther explanations given in the subsequent chapters must be limited to general rules. The sections given as_ 227--231 _"On indistinctness at short distances" have, it is true, only an indirect bearing on the subject; but on the other hand, the following chapters,_ 232--234, _"On indistinctness at great distances," go fully into the matter, and in chapters_ 235--239, _which treat "Of the importance of light and shade in the Perspective of Disappearance", the practical issues are distinctly insisted on in their relation to the theory. This is naturally followed by the statements as to "the effect of light or dark backgrounds on the apparent size of bodies"_ (_Nos._ 240--250). _At the end I have placed, in the order of the original, those sections from the MS._ C _which treat of the "Perspective of Disappearance" and serve to some extent to complete the treatment of the subject_ (251--262).
Definition (222. 223).
OF THE DIMINISHED DISTINCTNESS OF THE OUTLINES OF OPAQUE BODIES.
If the real outlines of opaque bodies are indistinguishable at even a very short distance, they will be more so at long distances; and, since it is by its outlines that we are able to know the real form of any opaque body, when by its remoteness we fail to discern it as a whole, much more must we fail to discern its parts and outlines.
OF THE DIMINUTION IN PERSPECTIVE OF OPAQUE OBJECTS.
Among opaque objects of equal size the apparent diminution of size will be in proportion to their distance from the eye of the spectator; but it is an inverse proportion, since, where the distance is greater, the opaque body will appear smaller, and the less the distance the larger will the object appear. And this is the fundamental principle of linear perspective and it follows:--every object as it becomes more remote loses first those parts which are smallest. Thus of a horse, we should lose the legs before the head, because the legs are thinner than the head; and the neck before the body for the same reason. Hence it follows that the last part of the horse which would be discernible by the eye would be the mass of the body in an oval form, or rather in a cylindrical form and this would lose its apparent thickness before its length--according to the 2nd rule given above, &c. [Footnote 23: Compare line 11.].
If the eye remains stationary the perspective terminates in the distance in a point. But if the eye moves in a straight [horizontal] line the perspective terminates in a line and the reason is that this line is generated by the motion of the point and our sight; therefore it follows that as we move our sight [eye], the point moves, and as we move the point, the line is generated, &c.
An illustration by experiment.
Every visible body, in so far as it affects the eye, includes three attributes; that is to say: mass, form and colour; and the mass is recognisable at a greater distance from the place of its actual existence than either colour or form. Again, colour is discernible at a greater distance than form, but this law does not apply to luminous bodies.
The above proposition is plainly shown and proved by experiment; because: if you see a man close to you, you discern the exact appearance of the mass and of the form and also of the colouring; if he goes to some distance you will not recognise who he is, because the character of the details will disappear, if he goes still farther you will not be able to distinguish his colouring, but he will appear as a dark object, and still farther he will appear as a very small dark rounded object. It appears rounded because distance so greatly diminishes the various details that nothing remains visible but the larger mass. And the reason is this: We know very well that all the images of objects reach the senses by a small aperture in the eye; hence, if the whole horizon _a d_ is admitted through such an aperture, the object _b c_ being but a very small fraction of this horizon what space can it fill in that minute image of so vast a hemisphere? And because luminous bodies have more power in darkness than any others, it is evident that, as the chamber of the eye is very dark, as is the nature of all colored cavities, the images of distant objects are confused and lost in the great light of the sky; and if they are visible at all, appear dark and black, as every small body must when seen in the diffused light of the atmosphere.
[Footnote: The diagram belonging to this passage is placed between lines 5 and 6; it is No. 4 on Pl. VI. ]
A guiding rule.
OF THE ATMOSPHERE THAT INTERPOSES BETWEEN THE EYE AND VISIBLE OBJECTS.
An object will appear more or less distinct at the same distance, in proportion as the atmosphere existing between the eye and that object is more or less clear. Hence, as I know that the greater or less quantity of the air that lies between the eye and the object makes the outlines of that object more or less indistinct, you must diminish the definiteness of outline of those objects in proportion to their increasing distance from the eye of the spectator.
When I was once in a place on the sea, at an equal distance from the shore and the mountains, the distance from the shore looked much greater than that from the mountains.
On indistinctness at short distances (227-231).
If you place an opaque object in front of your eye at a distance of four fingers' breadth, if it is smaller than the space between the two eyes it will not interfere with your seeing any thing that may be beyond it. No object situated beyond another object seen by the eye can be concealed by this [nearer] object if it is smaller than the space from eye to eye.
The eye cannot take in a luminous angle which is too close to it.
That part of a surface will be better lighted on which the light falls at the greater angle. And that part, on which the shadow falls at the greatest angle, will receive from those rays least of the benefit of the light.
OF THE EYE.
The edges of an object placed in front of the pupil of the eye will be less distinct in proportion as they are closer to the eye. This is shown by the edge of the object _n_ placed in front of the pupil _d_; in looking at this edge the pupil also sees all the space _a c_ which is beyond the edge; and the images the eye receives from that space are mingled with the images of the edge, so that one image confuses the other, and this confusion hinders the pupil from distinguishing the edge.
The outlines of objects will be least clear when they are nearest to the eye, and therefore remoter outlines will be clearer. Among objects which are smaller than the pupil of the eye those will be less distinct which are nearer to the eye.
On indistinctness at great distances (232-234).
Objects near to the eye will appear larger than those at a distance.
Objects seen with two eyes will appear rounder than if they are seen with only one.
Objects seen between light and shadow will show the most relief.
Our true perception of an object diminishes in proportion as its size is diminished by distance.
Why objects seen at a distance appear large to the eye and in the image on the vertical plane they appear small.
I ask how far away the eye can discern a non-luminous body, as, for instance, a mountain. It will be very plainly visible if the sun is behind it; and could be seen at a greater or less distance according to the sun's place in the sky.
[Footnote: The clue to the solution of this problem (lines 1-3) is given in lines 4-6, No. 232. Objects seen with both eyes appear solid since they are seen from two distinct points of sight separated by the distance between the eyes, but this solidity cannot be represented in a flat drawing. Compare No. 535.]
The importance of light and shade in the perspective of disappearance (235-239).
An opaque body seen in a line in which the light falls will reveal no prominences to the eye. For instance, let _a_ be the solid body and _c_ the light; _c m_ and _c n_ will be the lines of incidence of the light, that is to say the lines which transmit the light to the object _a_. The eye being at the point _b_, I say that since the light _c_ falls on the whole part _m n_ the portions in relief on that side will all be illuminated. Hence the eye placed at _c_ cannot see any light and shade and, not seeing it, every portion will appear of the same tone, therefore the relief in the prominent or rounded parts will not be visible.
When you represent in your work shadows which you can only discern with difficulty, and of which you cannot distinguish the edges so that you apprehend them confusedly, you must not make them sharp or definite lest your work should have a wooden effect.
You will observe in drawing that among the shadows some are of undistinguishable gradation and form, as is shown in the 3rd [proposition] which says: Rounded surfaces display as many degrees of light and shade as there are varieties of brightness and darkness reflected from the surrounding objects.
OF LIGHT AND SHADE.
You who draw from nature, look (carefully) at the extent, the degree, and the form of the lights and shadows on each muscle; and in their position lengthwise observe towards which muscle the axis of the central line is directed.
An object which is [so brilliantly illuminated as to be] almost as bright as light will be visible at a greater distance, and of larger apparent size than is natural to objects so remote.
The effect of light or dark backgrounds on the apparent size of objects (240-250).
A shadow will appear dark in proportion to the brilliancy of the light surrounding it and conversely it will be less conspicuous where it is seen against a darker background.
OF ORDINARY PERSPECTIVE.
An object of equal breadth and colour throughout, seen against a background of various colours will appear unequal in breadth.
And if an object of equal breadth throughout, but of various colours, is seen against a background of uniform colour, that object will appear of various breadth. And the more the colours of the background or of the object seen against the ground vary, the greater will the apparent variations in the breadth be though the objects seen against the ground be of equal breadth [throughout].
A dark object seen against a bright background will appear smaller than it is.
A light object will look larger when it is seen against a background darker than itself.
A luminous body when obscured by a dense atmosphere will appear smaller; as may be seen by the moon or sun veiled by mists.
Of several luminous bodies of equal size and brilliancy and at an equal distance, that will look the largest which is surrounded by the darkest background.
I find that any luminous body when seen through a dense and thick mist diminishes in proportion to its distance from the eye. Thus it is with the sun by day, as well as the moon and the other eternal lights by night. And when the air is clear, these luminaries appear larger in proportion as they are farther from the eye.
That portion of a body of uniform breadth which is against a lighter background will look narrower [than the rest].
 _e_ is a given object, itself dark and of uniform breadth; _a b_ and _c d_ are two backgrounds one darker than the other; _b c_ is a bright background, as it might be a spot lighted by the sun through an aperture in a dark room. Then I say that the object _e g_ will appear larger at _e f_ than at _g h_; because _e f_ has a darker background than _g h_; and again at _f g_ it will look narrower from being seen by the eye _o_, on the light background _b c_. [Footnote 12: The diagram to which the text, lines 1-11, refers, is placed in the original between lines 3 and 4, and is given on Pl. XLI, No. 3. Lines 12 to 14 are explained by the lower of the two diagrams on Pl. XLI, No. 4. In the original these are placed after line 14.] That part of a luminous body, of equal breadth and brilliancy throughout, will look largest which is seen against the darkest background; and the luminous body will seem on fire.
WHY BODIES IN LIGHT AND SHADE HAVE THEIR OUTLINES ALTERED BY THE COLOUR AND BRIGHTNESS OF THE OBJECTS SERVING AS A BACKGROUND TO THEM.
If you look at a body of which the illuminated portion lies and ends against a dark background, that part of the light which will look brightest will be that which lies against the dark [background] at _d_. But if this brighter part lies against a light background, the edge of the object, which is itself light, will be less distinct than before, and the highest light will appear to be between the limit of the background _m f_ and the shadow. The same thing is seen with regard to the dark [side], inasmuch as that edge of the shaded portion of the object which lies against a light background, as at _l_, it looks much darker than the rest. But if this shadow lies against a dark background, the edge of the shaded part will appear lighter than before, and the deepest shade will appear between the edge and the light at the point _o_.
[Footnote: In the original diagram _o_ is inside the shaded surface at the level of _d_.]
An opaque body will appear smaller when it is surrounded by a highly luminous background, and a light body will appear larger when it is seen against a darker background. This may be seen in the height of buildings at night, when lightning flashes behind them; it suddenly seems, when it lightens, as though the height of the building were diminished. For the same reason such buildings look larger in a mist, or by night than when the atmosphere is clear and light.
ON LIGHT BETWEEN SHADOWS
When you are drawing any object, remember, in comparing the grades of light in the illuminated portions, that the eye is often deceived by seeing things lighter than they are. And the reason lies in our comparing those parts with the contiguous parts. Since if two [separate] parts are in different grades of light and if the less bright is conterminous with a dark portion and the brighter is conterminous with a light background--as the sky or something equally bright--, then that which is less light, or I should say less radiant, will look the brighter and the brighter will seem the darker.
Of objects equally dark in themselves and situated at a considerable and equal distance, that will look the darkest which is farthest above the earth.
TO PROVE HOW IT IS THAT LUMINOUS BODIES APPEAR LARGER, AT A DISTANCE, THAN THEY ARE.
If you place two lighted candles side by side half a braccio apart, and go from them to a distance 200 braccia you will see that by the increased size of each they will appear as a single luminous body with the light of the two flames, one braccio wide.
TO PROVE HOW YOU MAY SEE THE REAL SIZE OF LUMINOUS BODIES.
If you wish to see the real size of these luminous bodies, take a very thin board and make in it a hole no bigger than the tag of a lace and place it as close to your eye as possible, so that when you look through this hole, at the said light, you can see a large space of air round it. Then by rapidly moving this board backwards and forwards before your eye you will see the light increase [and diminish].
Propositions on perspective of disappearance from MS. C. (250-262).
Of several bodies of equal size and equally distant from the eye, those will look the smallest which are against the lightest background.
Every visible object must be surrounded by light and shade. A perfectly spherical body surrounded by light and shade will appear to have one side larger than the other in proportion as one is more highly lighted than the other.
No visible object can be well understood and comprehended by the human eye excepting from the difference of the background against which the edges of the object terminate and by which they are bounded, and no object will appear [to stand out] separate from that background so far as the outlines of its borders are concerned. The moon, though it is at a great distance from the sun, when, in an eclipse, it comes between our eyes and the sun, appears to the eyes of men to be close to the sun and affixed to it, because the sun is then the background to the moon.
A luminous body will appear more brilliant in proportion as it is surrounded by deeper shadow. [Footnote: The diagram which, in the original, is placed after this text, has no connection with it.]
The straight edges of a body will appear broken when they are conterminous with a dark space streaked with rays of light. [Footnote: Here again the diagrams in the original have no connection with the text.]
Of several bodies, all equally large and equally distant, that which is most brightly illuminated will appear to the eye nearest and largest. [Footnote: Here again the diagrams in the original have no connection with the text.]
If several luminous bodies are seen from a great distance although they are really separate they will appear united as one body.
If several objects in shadow, standing very close together, are seen against a bright background they will appear separated by wide intervals.
Of several bodies of equal size and tone, that which is farthest will appear the lightest and smallest.
Of several objects equal in size, brightness of background and length that which has the flattest surface will look the largest. A bar of iron equally thick throughout and of which half is red hot, affords an example, for the red hot part looks thicker than the rest.
Of several bodies of equal size and length, and alike in form and in depth of shade, that will appear smallest which is surrounded by the most luminous background.
DIFFERENT PORTIONS OF A WALL SURFACE WILL BE DARKER OR BRIGHTER IN PROPORTION AS THE LIGHT OR SHADOW FALLS ON THEM AT A LARGER ANGLE.
The foregoing proposition can be clearly proved in this way. Let us say that _m q_ is the luminous body, then _f g_ will be the opaque body; and let _a e_ be the above-mentioned plane on which the said angles fall, showing [plainly] the nature and character of their bases. Then: _a_ will be more luminous than _b_; the base of the angle _a_ is larger than that of _b_ and it therefore makes a greater angle which will be _a m q_; and the pyramid _b p m_ will be narrower and _m o c_ will be still finer, and so on by degrees, in proportion as they are nearer to _e_, the pyramids will become narrower and darker. That portion of the wall will be the darkest where the breadth of the pyramid of shadow is greater than the breadth of the pyramid of light.
At the point _a_ the pyramid of light is equal in strength to the pyramid of shadow, because the base _f g_ is equal to the base _r f_. At the point _d_ the pyramid of light is narrower than the pyramid of shadow by so much as the base _s f_ is less than the base _f g_.
Divide the foregoing proposition into two diagrams, one with the pyramids of light and shadow, the other with the pyramids of light [only].
Among shadows of equal depth those which are nearest to the eye will look least deep.
The more brilliant the light given by a luminous body, the deeper will the shadows be cast by the objects it illuminates.