Dr. Kim H. Veltman
The Camera Obscura
2. Astronomical Context
3. Inversion of Images
5. Images all in all
6. Intensity of Light and Shade or Image
7. Contrary Motion
8. Size of Aperture
9. Shape of Aperture
10. Number of Apertures
11. Apertures and Interposed Bodies
12. Spectrum of Boundaries
13. Camera Obscuras and the Eye
Figs. 678-679: Pinhole apertures and camera obscuras in observations of the sun on Triv 6v and A20v.
Leonardo has commonly been credited with the invention of the camera obscura.1. This is not true. Unequivocal descriptions of the instrument go back at least until the ninth century AD2 and by the thirteenth century it had assumed an important function in astronomy.3 On the fourth of June, 1285, for instance, William of St. Cloud used a camera obscura to observe an eclipse of the sun.4 This use of the camera obscura develops in the fourteenth century and in the latter fifteenth century provides one source of Leonardo's interest in the instrument. He uses it, for example, to estimate the size and distance of the sun and moon.
The aperture of the camera obscura is, for Leonardo, analogous to the aperture of the pupil and this leads him to study various characteristics of apertures: how images passing through them are inverted, how they do not interfere with one another, how such images are "all in all and all in every part"; how they can vary their intensity, and how they move in a contrary direction beyond the aperture. He considers the effect of changing the size of the aperture and examines in some detail both the properties of a single aperture with a changing number of sides and the characteristics of multiple pinhole images. From a note on CA277va (1513-1514) cited above (see p. ), he clearly intended to adopt these findings for two additional books on light and shade.
Indeed many of the experiments that he had made with umbrous bodies in the open he repeats in combination with a camera obscura, now focussing on a particular phenomenon: how the boundaries between light and shade are actually a series of subtle graduations. These late studies of 1508-1510, as will be shown, have important consequences for his theories of vision and perception. But before considering these, we need to examine the details of his camera obscura studies.
2. Astronomical Context.
Leonardo's earliest extant reference to the use of an aperture in observing eclipses is on Triv.6v (fig. 678, 1487-1490):
Way of seeing the sun eclipsed without hurting the eye. Take a card and make the aperture with a needle and through these /pinhole/ apertures look at the sun.
In this case the image is seen directly and the aperture serves merely to screen off excessive light. On CA270vb (c. 1490) he describes a case where the image is seen indirectly under the heading:
How bodies...send...their form and heat and power beyond themselves.
When the sun, through eclipses, remains in the form of the moon, take a thin sheet of iron and in this make a little hole and turn the face of this sheet towards the sun...holding a piece of cardboard a 1/2 braccia behind this and you will see the similitudes of the sun come in a lunar shape, similar to its shape and colour.
Immediately following he offers a (cf. fig. 679).
This said sheet /of iron/ will also do the same at night with the body of the moon and also with the stars. But from the sheet /of iron/ to the cardboard there is by no means to be any other aperture other than this little hole and this is similar to a square box, of which the faces above and below and the two on the side of the card are of solid wood; that in front has the sheet /of iron/ and that behind, a thin white cardboard or paper pasted to the edges of the wood.
Finally he provides an illustration that simulates the effects produced by these natural phenomena:
Again take a candle of wax, which makes a long light and placed in front of this aperture, the said light will appear on the paper opposite in a long form and similar to the form of its cause, but upside down.
This example, described on CA270vb, is illustrated in diagrams on CA126ra (fig. 156),
CA125vb (fig. 694), and CU789 (fig. 704). The case of the moon is also considered on A64v (1492):
Because all the effects of luminous bodies are demonstrative of their causes, the moon in the form of a boat, having passed through the aperture, will produce at the object /i.e. the wall/ a boat shape.
On A61v (1492) he pursues this theme, now adding an illustration (fig. ):
That perforation of round quality which is half closed will appear in the form of ab and the part c will be the light and n will be the closed off part and this same happens to the luminous half-moon.
The problem of the moon's shape continues to perplex him. On CA243ra (1510-1515) he notes:
Since over a long distance a long luminous source makes itself round to us and /yet/ the horns of the moon do not observe this rule and even the light from nearby observes the demonstration of its point.
Figs. 680-681: Use of camera obscura for astronomy on A21r (1492) and by Mario Bettini (1642).
Figs. 682-683: Uses of camera obscura for astronomy on BM174v and CA243rb.
The use of the camera obscura in determining the diameter of the sun had been discussed by late mediaeval authors such as Levi ben Gerson5 (c.1370). This also interests Leonardo as is shown by a diagram on A20v (fig. 679, cf. A21r, fig. 680 and Mario Bettini, fig. 681) beneath which he adds:
Way of knowing how large the sun is. Make that from a /to/ b there are hundred braccia and make that the aperture where the solar rays pass is 1/16 of a braccia and note how much the ray has expanded in percussion.
The problem is not forgotten. On CA225rb (1497-1500) he reminds himself in passing of "the measurement of the sun promised me by Master Giovanni the Frenchman." On BM174v (1500-1505) Leonardo describes a more complex procedure involving a combination of camera obscura and mirror (fig. 682):
ab is the aperture through which the sun passes. And if you could measure the size of the solar rays at nm, you could see very well the true lines of the concourse of these solar rays, the mirror standing in ab, and then make the rays reflected at equal angles towards nm. But in order that you do not then distort (torse) it at nm, take it inside the aperture at cd, which can be measured in the percussion of the solar ray, and then place your mirror at the distance ab and there make fall the rays db /and/ ca and then rebound under equal angles towards cd. And this is the true method: but you need to operate such a mirror at exactly the same month, day and hour and you will do it better than at any other time because in such a distance of the sun, such a pyramid is caused.
The precise function of this procedure is not explained.
On Leicester 1r (1506-1509) he alludes to but again does not elaborate on a:
Record of how I at first demonstrated the distance of the sun from the earth and with one of its rays which have passed through an aperture in a dark place find its true quantity again and besides this, through the centre of the water to find the size of the earth.
To this problem of sizes and distances of the planets he returns on CA243rb (1510-1515) now providing a more detailed explanation:
If you have the distance of a body you will have the size of the visual pyramid which you will cut near the eye on the window (pariete) and then you remove the eye to that extent, such that the intersection is doubled, and note the space from the first to the 2nd intersection and say: if in so much...space the diameter of the moon increases so much above the first intersection, what will it do in all the space that there is from the eye to the moon? This will make the true diameter of this moon.
The method here described is identical to the surveying procedure used to demonstrate principles of linear perspective on CA42rc, (cf. vol. 1, fig. 122). On CA151va (1500-1506) he gives an alternative method of measuring the distance of the sun, this time using a staff again familiar from the surveying tradition. Lower down on CA243rb he draws a rough sketch of a camera obscura (fig. 683) beneath which he notes "Measure of the size of the sun, knowing the distance." On CA297va (1497-1500) he drafts a passage concerning the use of apertures in a meteorological context:
The solar rays, penetrating the apertures...which are interposed between the various...globosities of clouds, illuminate with their straightness all...the passage interposed...between the earth and this aperture...and tinge from themselves all the sites where they intersect.
Figs. 684-685: Apertures in clouds and camera obscuras on CU476 and CA248va.
This he drafts again directly below:
The solar rays penetrating the apertures interposed between the various globosities of clouds...make a straight and spreading course towards the earth where they are intersected...illuminating...with their...brightness all the penetrated air.
These drafts serve in turn as the basis for a passage on CU476 (fig. 684, TPL447, 1510-1515):
On the solar rays that penetrate the apertures of clouds.
The solar rays penetrating the apertures positioned between the various densities and globosities of clouds illuminate all the sites where they intersect and even illuminate the darknesses or tinge with themselves all the dark places that are behind them, which darknesses show themselves to be between the intervals of these solar rays.
Also in this late period, on CA248va (fig. 685, 1510-1515), he again mentions the relative intensity of the sun's illuminating in a camera obscura:
ab is brighter than cd. But the point t being illuminated by the narrow aperture by the part of the sun o will be that much less illuminated than being illuminated by the diameter ab to the extent that this o is less than this diameter ab.
These astronomical and meteorological uses of the camera obscura constitute but a small part of Leonardo's concern for this instrument. He is fascinated by the analogies that it offers with the pupil (see below pp. ) and therefore employs the camera obscura to demonstrate such optical principles as inversion of images, their non-interference, and their existence "all in all and all in every part." We shall examine each of these in turn.
Figs. 686-687: Use of a camera obscura to demonstrate the inversion of images on W19147v (K/P 22v): fig. 688, another example on C6r.
3. Inversion of Images
Leonardo believes that images are inverted in passing through the aperture of the pupil. As early as 1489-1490 he employs a camera obscura to demonstrate this principle in a passage on W19147v (K/P 22v):
But if the plane of this interruption has in it a small aperture which enters into a dark home /that is/ dark not by colour but through privation of light, you will see the lines enter through this said aperture /and/ carry on the second wall the entire form of its origin, both with respect to colour and form. But everything will be upside down....
A specific example follows (fig. 686):
Let ab be the origin of the lines. Let de be the first wall; let c be the aperture where the intersection of the lines is. Let fg be the last wall. On the last wall and percussion you will find that a remains below at the place g and g below rises above to the place f.
He pursues this eye-camera obscura analogy on C6r (fig. 688, 1490-1491):
All the things that the eye sees beyond little apertures are seen upside down by this eye and are known as right side up.
Let ad be the pupil (luce) that sees through the aperture n. The line eh is seen by the lower part of the eye; dh is seen by the upper part of the pupil.
Figs. 689-696: Demonstrations of inversion in camera obscuras. Fig. 689, Forst. III 29v; figs. 690-691, CA373rb; fig. 692, CA155rde; fig. 693, BM170v; fig. 694, CA125vb; figs. 695-696, CA345vb.
He illustrates this inversion principle in sketches without text on Forst III 29v (fig. 869, 1490-1493), CA155rde (fig. 692, 1495-1497) and BM170v (fig. 692, 1492); then mentions it again on BM232v (1490-1495):
The bases of inverted pyramids, if they are in a dark place, will show upside down the shape and cause of their source.
A further illustration occurs on CA125vb (fig. 694, 1492) accompanying which he drafts an explanation:
The sun and every luminous body, which sends its rays through an aperture smaller than it in size, will send the rays upside down behind this aperture and you will see the experience with a lighted candle, taking its rays beyond an aperture smaller than it.
On CA126ra (c. 1492) the problem is further illustrated (fig. 156). Thereafter, more than a decade passes before he broaches the problem again on CA345vb (figs. 695-696, 1505-1508):
And they /the images/ impress themselves on the wall opposite the said point, perforated in a thin wall, and for this reason the eastern part will impress itself in the western part of such a wall and the western in the eastern and likewise the northern in the southern and conversely, etc.
In the Manuscript D he discusses the problem of inversion at length. A first passage on D10r (c. 1508) is entitled:
Of the species of objects that pass through narrow apertures into a dark place.
It is impossible that the species of bodies that penetrate through apertures into a dark place do not reverse themselves. This is proved by the 3rd of this which states (the particles of each umbrous ray are always rectilinear).
Figs. 697-698: Demonstrations concerning inversion of images on D10r and D8r.
To demonstrate this he provides a concrete example (fig. 697):
Therefore the part b of the object ab, passing through the aperture n into the dark place oqpr, will impress itself on the wall pr on the site c and the opposite extremity a of the same object ab will impress itself on the wall cr /sic: pr/ at the point r /sic:c/ and thus the right extremity of such an object makes itself left and the elft makes itself right, etc.
On D8r he pursues this analogy between eye and camera obscura under the heading:
How the species of objects received by the eye intersect inside the albugineous humour.
The experience which shows that objects send their intersected species or similitudes inside the eye in the albugineous humour is shown when the species of illuminated objects pass through some small round aperture into a habitation that is very dark, then you will receive such species on a white piece of paper...placed in such a habitation somewhat near this aperture and you will see all the aforesaid objects on this paper with their proper shapes and colours but they will be smaller and inverted as a result of the said intersection. Which images if they originate in the eye illuminated by the sun appear properly depicted on this paper which would be very thin and seen from behind.
A concrete demonstration follows (fig. 698):
And let the said aperture be made in a very thin sheet of iron. Let a, b, c, d /and/ e be the said objects illuminated by the sun. Let or be the face of the dark habitation in which is the said aperture nm. Let st be this /piece of paper where the rays of the species of these objects are intersected upside down, /and/ because their rays are straight a /on the/ right makes itself left at k and e /on the/ left makes itself on the right at f and it does the same inside the pupil.
He again mentions this way in which images are inverted in passing on W19150v (K/P 118v(a), fig. 699, 1508-1510): "No image of however small a body penetrates the eye without being turned upside down....." On CA241vc (also 1508-1510) he produces two further drafts:
Every umbrous and luminous species which penetrates through the apertures...behind such a penetration turns (upside down after such a penetration) in contrary aspects all the parts of its size.
Every umbrous and luminous species interests after the penetration made behind the apertures, turning in contrary aspect every part of their size.
These he crosses out and reformulates:
The rays of umbrous and luminous species intersecting after the penetration made by them inside the apertures, turn in contrary aspect every part of their size.
The non-interference of images is another phenomenon which he demonstrates using the camera obscura. On A93r (BN2038 13r, 1492), for instance, he shows how a red, white and yellow light can intersect without interference (fig. 700). Similar demonstrations occur on CA256rc (figs. 701-702, c.1492) involving a red, green and yellow light and a red, white and green light. Accompanying these are a series of draft notes concerning the intensity of colour, light and shade passing through apertures:
Figs. 700-702: Demonstrations concerning non-interference of images. Fig. 700, A93r; figs. 701-702, CA256rc.
Figs. 703-705: Further demonstrations concerning non-interference of colours in a camera obscura on K/P 118r, CU789, K/P 118v.
That colour which is more illuminated will show itself better in the percussion made by its rays within the aperture.
The luminous rays will make the shadows of bodies greater which oppose themselves between the aperture and the percussion, which bodies are touched by a less luminous ray.
To the extent that the umbrous body is closer...to the percussion of the rays, the more its shadow will observe the form of its derivation.
The qualities of rays are 2, that is: luminous and umbrous.
The percussion of the luminous ray is surrounded by those images of things which surround the luminous body.
On 19112r (K/P 118r, 1508-1510) he returns to the theme of non-interference of colours in a camera obscura, this time using yellow and blue lights (fig. 703). Slightly more complex is a demonstration on CU789 (fig. 704, TPL707, 1508-1510) where he shows how light from a candle and from the air produces different colours on an interposed object:
Of the colours of lights illuminating umbrous bodies.
The umbrous body positioned between nearby walls in a dark place which is illuminated on one side by a minimal light of a candle and is illuminated on its opposite side by a minimal aperture of air, if it is white, then such a body will show itself yellow on one side and azure on the other, the eye standing in a place illuminated by the air.
On CU797 (TPL645, 1508-1510) he had used a similar demonstration to establish that the colour of shadows participates with the colour of surrounding objects (see above pp. ). On W19150v (fig. 705, K/P 118v, 1508-1510) he pursues both the themes of colour participating and non-interference of images under the heading:
Of the rays which carry the images of bodies through the air.
All the smallest parts of the images penetrate each other without occupying the space of one another.
And let r be one of the sides of the opening. Opposite this, let s be an eye which sees the lower extremity u of the line no, which extremity cannot send its similitude from itself to such an eye s such that it does not touch the extremity r and m, the middle of this line does the same and the same happens to the upper extremity n /going/ to the eye /at/ v. And if such an extremity is r, the eye v does not see the colour green of o at the edge of the aperture, but only the red of n by the 7th of this where it is stated that every similitude sends its species beyond itself by the shortest line which, by necessity, is straight etc.
On the recto of this same folio (K/P 118r) this phenomenon of the non-interference of images is discussed further in a passage entitled:
On the nature of the rays of which the images of bodies are composed and their intersections.
The straightness of the rays which carry through the air the shape and colour of the bodies whence they depart do not tinge the air with themselves nor, furthermore, can they tinge one another in the contact of their intersection.
Figs. 706-707: Three light sources through one aperture and through two apertures on W19150v (K/P 118v).
This claim he qualifies:
But they only tinge the place where they lose their being because such a place sees and is seen by the origin of these rays and no other object which surrounds this object can be seen from the place where such a ray, being cut off is destroyed, leaving there the spoil /i.e. the image/ carried by it.
And this is proved by the 4th on the colours of bodies where it is stated that the surface of every opaque body participates in the colour of its object. Therefore it is concluded that the place by means of which the ray which carries the images sees and is seen by the origin of such a species is tinged by the colour of that object.
5. Images All in All
The way in which Leonardo uses a camera obscura to establish that images are all in all and all in every part has been analysed previously (see above pp. ). This was chiefly in terms of passages on A9v (fig. 133, 1492) and W19150v (figs. 706-706, K/P 118v, 1508-1510). This principle is also implicit in a sketch on BM171r (fig. 708, 1492) beneath which he writes:
Every surface is the boundary of a dark body /and/ multiplies in that boundary the species of the things positioned opposite it and if they are bright it sends them inside as bright.
Figs. 708-711: Camera obscura demonstrations on BM171r, W12353v, CA91vb and CA112va.
Figs. 712-721: Camera obscura demonstrations. Figs. 712-719, CA238rb; fig. 720, CA133va; fig. 721, CA238vb.
Figs. 722-728: Further camera obscura demonstrations. Figs. 722-726, CA238rb; fig. 727, CA382vb; fig. 728, K/P 118v.
Figs. 729-732: Camera obscuras on CA155rd.
Figs. 733-736: Camera obscura demonstrations. Fig. 733, CA256rc; fig. 734, W12353; figs. 735-736, C14v.
It is equally implicit in other sketches on W12352v (fig. 709, 1494), CA91vb (fig. 710, C.1500), CA112va (fig. 711, 1505-1508), as well as in a series of sketches (figs. 712-726) on CA238rb, vb (1505-1508) accompanying which is only an interrupted text: "Every part of ab...is in every part...d, but more...illuminate...short ray...where the spe/cies.../ powerful occupy...the less powerful." On CA155rd (figs. 729-732, 1497-1500) he implicitly demonstrates this principles again in a series of four diagrams accompanying which he merely notes:
For the adversary...the long light ab would only illuminate the point c and experience illuminates it in des.
abfik is according the adversary; abcgh is mine.
This principle is illustrated roughly once more in a diagram without text on W12353 (fig. 734, 1508-1511).
6. Intensity of Light and Shade or Image
Images in the camera obscura may be "all in all and all in every part." Nonetheless, Leonardo is convinced that they can vary in their intensity, and he uses the camera obscura to demonstrate this. On C14v (1490-1491), for instance, he draws a rough diagram (fig. 735) which he then develops (fig. 736). Accompanying this he opens with a general comment:
That part of the air will participate more in its natural darkness which is percussed by a more acute luminous angle.
It is clearly comprehended that where /there is/ a lesser luminous...angle there will be less light because the pyramid of this angle has a smaller base and because of this smaller base, a smaller number of luminous rays concur to its point.
This is followed by a specific demonstration which refers back to the diagram (fig. 736):
The angle a has a larger base than the angle b, the base of a is...sf and that of b is gh. Therefore a has a base that is a quarter larger than b /and/ has a quarter more light. Again c /and/ d hold a similar difference amongst one another because c sees ik which is half of the light ef and from d it sees the quarter lm.
This theme of differing intensities of light within the camera obscura is mentioned in passing on CA256rc (fig. 733, 1492): "The luminous rays make the shadows of bodies greater which are opposite the aperture and the percussion, which bodies are touched by a less luminous ray." On CA238rb (1505-1508) following the all in all passage cited previously (see above p. ), he again takes up this theme of differing intensities of light and shade (fig. 723):
By the simple proof of the lines intersected at the boundary of the umbrous body all the percussion of the luminous species would be of equal brightness...such that ab, a quarter of the luminous body responds to gh;...a quarter of the percussion at bc with ih which is similarly a quarter of the luminous body and of the percussion and the other 2 quarters do likewise: whence kf would be of equal light. But experience does not confirm it, whence other....
He now draws a second diagram (fig. 724) which he explains:
Therefore experience showing how the percussion of luminous rays acquires degrees of darkness in every part of height and this not being concluded by the first figure the second concludes it, because all the light ae sees i and 3/4 of this light be sees h and half the light ce sees g and a quarter of the light...de sees f. Hence f is less luminous and 3/4 so than i.
Another diagram follows (fig. 726 cf. 727-728) accompanying which he notes: "When the motion is from n to m, the shadow will descend from a to b." On this single folio CA238rb (1505-1508) he has thus used a camera obscura to demonstrate that (1) images are all in all and all in every part (see above pp. ); (2) that images vary in their intensity and (3) that images inside a camera obscura are inverted mind these three demonstrations are closely related. It is no wonder, then, that the remaining preparatory sketches on this folio (figs. 715-719, 722, 725) which are without text have a certain ambiguity about them: they could serve to support any of or all three of these demonstrations.
On C12v (fig. 737, 1490-1491), in the course of his studies of light and shade, he had illustrated how a light source in front of two opaque bodies produces concentric rings of light and shade of different intensity. The text on C12v is closely related to a draft, possibly in another hand on BM101r (1490-1495):
That umbrous body of spherical That umbrous body of spherical
rotundity will make circular rotundity will make circular mixed
mixed shade which will be be- shade which has placed between it
tween it /and/ the sun a body and the sun an umbrous body of its
placed opposite it similar to quality.
its quality /and/ quantity.
This demonstration of concentric rings caused by opaque bodies in the open air is the more interesting because it is paralleled by further demonstrations involving camera obscuras. On CA242v (1497-1500), for instance, he draws (fig. 738) sunlight entering through an aperture which is intersected at various distances. Directly beneath he describes the first percussion:
The first percussion of the solar ray is illuminated toward its centre b by the simple solar body b and at ab and bc it is illuminated by the air opnm...which mixes such a percussion, which is not simple light of the sun and towards the extremities a /and c, it is not illuminated except from the sides which are toward the centre of the sun such that between the lack /of light/ in such a site and the light of the sun, through the light of the air these spaces ab and bc become considerably darker towards their extremities a /and c.
He then describes the second percussion
At the 2nd percussion of the solar ray ac sees the entire solar body and would be very luminous if the darkness of the lateral air...did not corrupt such brightness with its darkness; that is, bt etc is seen other than by the sight of sun by ohegm.
ab and cd lack the light of the sun in each degree of distance from its centre r and towards a /and/ d, the extremities are only illuminated by the centre of the sun as the triangle adr shows.
And the said space ab is seen by op and the space cd by nm which obscure it.
A description of the third percussion is given short schrift: "At the 3rd figure bc is seen by all the body of the sun."
Figs. 738-740: Camera obscuras and concentric rings of light and shade on CA272v, CA262ra and CA238rb.
He pursues this theme on CA262ra (1497-1500). Here he carefully redraws his diagram showing various percussions of sunlight within a camera obscura (fig. 739). Directly above this he drafts a general claim: "The solar ray which penetrates inside the apertures (of the eye) of houses, in each degree of its length changes quality as quantity." It is noteworthy how he here writes "apertures of the eye" which he then crosses out to write "apertures of houses." The correction is significant because on the same folio he also discusses different kinds of pupils (see pp. ). The camera obscura-eye analogy is very important for him. Not content with his first draft he crosses it out and begins afresh: "The solar ray which, through a narrow aperture made in a thin wall, penetrates a dark place, in each degree of its length m..." Here the text breaks off. Beneath the diagram he starts anew, now with a description of the 6th percussion.
In the 6th demonstration the sun is more powerful in ab than in cd because ab sees and is seen by the entire solar body and cd is seen by half.
The triangle cem carries the entire figure of the sun to m: whence there is the first degree of luminosity in this m. And the triangle del carries the half less light to the site 1 than in m, because only half of the sun shows itself there. In the triangle egk there is carried to this k all the luminosity of the sky eg and to i is carried the quality...of the base of the triangle fgi which is half of eg.
Figs. 741-748: Demonstrations of contrary motion using camera obscuras. Fig. 741, C3r; figs. 741-742, CA133va; figs. 744-748, CA357rb.
Figs. 749-751: Demonstrations of contrary movement of images in camera obscuras on W19149r (K/P 118r).
He returns to this theme in a sketch without text on CA238rb (fig. 740, 1505-15087). In the Manuscript F similar sketches becoming a starting point for theories of the pupil (figs. ** , see below pp. ** ).
7. Contrary Motion
Leonardo's use of the camera obscura to demonstrate the phenomenon of contrary motion can be traced back to a note on C3r (fig. 741, 1490-1491):
The movement of the percussion of the sun which passes through the aperture of a wall and repercusses on the other side will make its growth towards the bottom. And this occurs because with the sun rising....
If the sun bc sees all of ef when it has risen to ab it will see as far as fn and from this it arises that apertures of the sun grow towards the bottom.
It is impossible that in bifureated and mixed derived shade there is a part where the entire umbrous body can be seen.
He pursues this theme in a series of sketches without text on CA357rb (figs. 744-748, c. 1490). On CM171v (1492) the principle is again mentioned:
All the similitudes of things which pass through a window out of the open air into the constrained air of the wall are seen in a contrary site: that thing which in the open air moved from east to west will appear as a shadow in the illuminated wall of constrained air / to be/ of contrary movement.
Further sketches (figs. 742-743) without text follow on CA133va (1497-1500). Approximately a decade later he demonstrates this principle of contrary movement of images in a camera obscura by moving the edges of the aperture on W19149r (K/P 118r, figs. 749-751, 1508-1510, see above pp. ). In this same passage he also broaches the themes of aperture size, rectilinear propagation and the principle that images are all in all in every part.
Gis. 752-756: Experiments concerning contrary motion of images in a camera obscura on CA277va.
The principle of contrary movement is further examined on CA277va (1508-1510). Here his approach is experimental and systematic. He begins with a situation (fig. 752-753) where a stick, situated in a high position in front of a near wall, casts a shadow low down on the far wall. The accompanying text is headed:
Operation of compound shade.
The operations of compound shade are always of contrary motion.
That is, if it is touched by an opaque body, the concourse of the luminous rays,...before they come to their intersection, all the shadows of this interrupting body of the superior ray are demonstrated beyond such an intersection in the percussion of the inferior ray and just as the...superior ray...makes itself inferior after the intersection, so too the motions that the umbrous body makes inside such a superior ray will show themselves of contrary motion behind such an intersection. And this is shown in the intersection of compound shade on the pavement or on the wall percussed by the sun or other luminous body.
Having considered the contrary motion of this stick's shadow when the stick is positioned in front of the first wall (fig. 754), he explores what happens if the stick is positioned behind this first wall:
But if the luminous ray is...impeded by an opaque body...somewhat behind its intersection, then...the percussion of the derived shade of the opaque body will make a motion similar to the motion of its opaque body.
Fig. 757: Demonstration of contrary motion of images in a camera obscura on E2v.
Next he turns to a case where the stick is positioned in the same plane as the first wall (fig. 755, cf. fig. 756):
And if such rays are impeded at the actual site of their intersection, then the shadows of the opaque body will be two...and they will move with contrary motions with respect to one another, before they come to unite.
This passage ends with a general comment:
The compound derived shade is the cause that the percussion of the solar ray which passes through some angle does not impress...these angles but portions of...that much larger or sm/aller/ to the extent that such impressions are more remote or close to these angles.
He returns to this theme of contrary movement on E2v (1513-1514) in a passage headed:
On shadow and its movement.
Of two umbrous bodies which are one behind the other behind a window and a wall is interposed at a given space.... The shadow of the umbrous body that is close to the wall will move if the umbrous body closer to the window is in transverse motion /relative/ to this window.
This he demonstrates with a concrete example (fig. 757):
This is proved. And let the two umbrous bodies be a /and/ b interposed behind the window nm and let the wall be op interposed with some space interposed between them which is the space ab /he means bc/. I say that if the umbrous body a moves towards s that the shadow of the umbrous body b which is c will move towards...d.
Figs. 758-765: Demonstrations with camera obscuras having different sizes of apertures. Fig. 758, CA373rb; fig. 759, CA256rc; fig. 760, H227 inf. 47v-48v; fig. 761, a2r; fig. 762, CA373rb; fig. 763, CA256rc; fig. 764, H227 inf. 47v; fig. 765, A85r.
8. Size of Aperture
Concerned as he is with studying the variables of a given problem it is not surprising to find him exploring the role played by different sizes of aperture. On CA373rb (1490-1495) he merely makes two preliminary diagrams without text (figs. 758, 762). These he develops in two diagrams (figs. 759, 763) on CA256rc (1492) where he draws a thin and a thick wall alongside which he writes:
Among apertures of equal...size that which is in a larger wall will render a darker...and smaller percussion.
On H227 in f. 47v=48r he takes these ideas further. Here he draws a thick wall (fig. 760) accompanying which he notes: "The aperture that is positioned in a thick wall will give little light to the site where it reaches." He also draws a thin wall (fig. 764) with the comment: "That aperture that is positioned in a thinner wall will give more light to the place where it reaches." He pursues this theme on A2r (fig. 761, 1492) where he draws a relatively thin wall and considers changing intensities of light inside a camera obscura under the heading:
Quality of lights
To the extent that ab enters into cd so many times is it more luminous than cd and similarly as many times as the point e enters into cd so many times is it more luminous than cd. And this light is good for those /things/ which entail subtle work.
On A85r (BN2038 5r, 1492) he draws a related diagram (fig. 765) alongside which he drafts an explanation:
That air which is luminous penetrates through perforated walls and passes inside dark habitations will make the place that much less dark to the extent that this perforation enters into the walls that surround and cover the pavement, to the extent that this perforation is less than the walls that surround and cover the pavement.
Not content, he crosses this out and reformulates it under the heading of:
That luminous air which penetrates and passes through perforated walls into dark habitations will make the place that much less...dark, to the extent that this perforation enters into the walls that surround and cover their pavement.
He redraws the diagrams of A2r and A85r on CA262v (figs. 766-768, 1497-1500) this time with no accompanying text. About a decade later he takes up this theme of apertures of different sizes once more in W19152v (K/P 118v, 1508-1510), beginning with a general claim: "Images which pass through apertures into a dark place intersect their sides that much nearer to the aperture to the extent that this aperture is of lesser width." By way of illustration he draws three diagrams and first discusses the case (fig. 769) on the far left in which the opaque objects ab and ik produce shadows which pass through the aperture de:
This is proved: And let ab be the umbrous body which sends not its shade but the image of its darkness through the aperture de which is the width of this umbrous body. And its sides ab being rectilinear (as was proved) it is necessary that they intersect between the umbrous body and the aperture but that much closer to the aperture to the extent that this aperture is less wide than the umbrous body.
Gis. 769-771: Demonstrations of concerning different sizes of aperture in camera obscuras on W19152v (K/P 118v).
He proceeds to discuss the two figures to the right of this, namely, abc, which is on the far right, (fig. 771) and nmo which is on the near right (fig. 770, or as he puts it, to the left relative to abc):
as is shown on your...right side and /the figure/ to the left /of it/ in the two figures abc /and/ nmo where the aperture /of the figure on the/ right being equal in width to the umbrous body ab, which intersection of such an umbrous body makes itself in the middle between the aperture and the umbrous body in the point c, which the figure to the left /fig. 770/ cannot do, the aperture o being considerably smaller than the umbrous body nm.
He then considers further properties of the diagram at the far left (fig. 769):
It is impossible that the images of bodies can be seen between the bodies and the openings through which the images of these bodies penetrate. And this is apparent because where the air is illuminated such visible images are not generated.
This discussion of where image formation occurs leads him to mention where images are doubled:
Images doubled ;by the reciprocal penetration of each other always double their darkness. To prove this let such a doubling be deh which, although it sees only between the bodies bi, this does not stop its being seen from fg or from fm. It is composed of the images a, ik which are infused with one another in deh.
Here the physics of light and shade dovetails with the problem of image formation in a camera obscura which interests him because of its relevance to vision (see below pp. ). In Leonardo's mind one topic constantly leads to another.
Figs. 772-776: Demonstrations concerning sizes of apertures in camera obscuras on CA385va.
These interweaving analogies he develops on CA385vc (1510-1515) where he sketches two examples with two opaque bodies (figs. 772-773) and three cases with three such bodies at various distances (figs. 774-776). Other sketches on the same folio (figs.** ) make explicit the camera obscura-eye analogy and leave no doubt concerning the parallels intended between physics of light and shade and physiology of vision.
9. Shape of the Aperture
When light passes through an aperture does the resulting image on the wall resemble the shape of the aperture or the light source? Already in Antiquity this had been a problem as is evident from two questions posed in the Problemata attributed to Aristotle:
Why does the sun penetrating through quadrilaterals form not rectilinear shapes but circles, as for instance when it passes through wicker work?6
Why is it that during eclipses of the sun if one views them through a sieve or a leaf - for example, that of a plane tree or any other broad-leaved tree - or through the two hands with the fingers interlaced, the rays are crescent-shaped in the direction of the earth?7
Alhazen, in the eleventh century, had mentioned the problem of light passing through apertures.8 Witelo, in the thirteenth century had considered briefly light passing through square, round and angular apertures.9 With Pecham this phenomenon emerged as a serious problem. He devoted two of his longest propositions to the properties of images passing through triangular apertures.10
Figs. 777-778: What happens to round images passing through triangular apertures in Pecham's Perspectiva communis and Leonardo's A82v.
Figs. 779-782: Round images passing through triangular apertures. Fig. 779, CA144vb; figs. 780-782, CA236ra.
Leonardo stands clearly within this tradition. A diagram on A82v (fig. 778, 1492) bears comparison with standard diagram from Pecham's work (fig. 777). But whereas his predecessors had been content to consider isolated cases, Leonardo is more systematic. In his notebooks he illustrates a series of situations showing how, with greater distance, the image gradually loses the shape of the aperture and takes on the shape of the light source. When the light source, aperture and projection plane are very close, the light passing through a triangular aperture produces a triangular image. This limiting case (fig. 783) he considers in some detail a situation in which a triangular image begins to become curved (figs. 780-782, cf. fig. 779):
The exterior sides of compound derived shade are always seen by all the luminous body.
But none of the interior sides of compound derived shade sees any part of the luminous body.
The more that the converse pyramid, created by decomposed (derived) shade is removed from its angle the more it will see of the luminous body but it will never see half.
Or so much more than half to the extent that the luminous body is greater than the umbrous body.
The percussion of the luminous ray, which...penetrates inside the concavity of the angle will never impress this angle, but in place...of this it impresses a portion of a circle.
But if the angle is convex then the impression of the obtuse angle will make an acute angle.
(Figs. 783-788: Stages in the transformation of a triangular image to a round image. Fig. 783, Author's reconstruction: fig. 784, CA277va; fig. 785, C10v; fig. 786, Forst III 29v; fig. 787, H227 inf. 50v-51r; fig. 788, H227 inf. 49r.
How the shadow of an obtuse angle makes itself acute with curved sides.
Of the curved and acute sides the derived shadow of the...obtuse angle is made convex from rectilinear sides.
And this is proved. And let the luminous object be ah of which all its rays see the obtuse angle c of the straight sided triangle kgc and thus the left ray a passes the angle c and bends to the side.
As the distance increases each point of the triangle generates a circle in the form of the light source which results in a triangular configuration of three circles.
He illustrates this situation on CA277va(fig. 784, 1508-1510). On C10v (1490-1491) he shows (fig. 785) a next step in this process where the circles have begun to overlap. The accompanying text summarizes the phenomenon:
It is impossible that the ray born of a spherical...luminous source, can at a distance go /on/ conducting in its percussion the similitude of the quality of any angle in the angular aperture through which is passes.
On Forster III 29v (fig. 786, c. 1493) he demonstrates a next stage in this process of transformation. The distance is now greater and the three circles have begun to overlap more. The accompanying text is brief: "The angle is terminated in a point. In the point are intersected the images of bodies." A next stage, where the distance is greater and the circles overlap even more is recorded on Manuscript H 227 inf. 50v-51r (fig. 787) accompanying which is a thorough explanation:
If the entire body of the sun sees all the square aperture it is necessary that every minimal part of this aperture sees all the sun and transfers it all behind the first wall where it terminates the course of the solar rays. Therefore no angle can appear over a long distance...in the solar sphere.
The point of the triangle B is centre of the circle D. A is the centre of the circle E and similarly C comes to be centre of F and if make a triangular aperture in a plate of thin iron of similar size and you make the rays of the sun pass inside and receive them in an object that much distant from a similar triangle that when the rays dilate in the size of the circle CDF , you will see the little triangle make itself in a spherical form.
When the distance is greater still the image takes on entirely the shape of the original light source and is fully round, in spite of having passed through a triangular aperture.11 This situation he describes in draft on CA135va(1490-1492) and develops on H227 inf. 49r (fig. 788):
CA135va H227 Inf. 49r
|The further from the intersection||The further from the intersection|
|/that/ the second pyramid /is/, ...||that the base of the second|
|the more it expands at the object.||pyramid made the object is|
|Its circles created at the angle||generated, the more it expands.|
|of the aperture enter more into||And its circles created in the|
|the body (the one than the other)||angles of the apertures are|
|and to the extent that they||incorporated together more, and|
|incorporate one another, the more||the more that they are incorpora-|
|the solar ray remains round at||ted together, the rounder the|
||base of the solar pyramid remains.
|When the rays have made so long a|
|path that they enter four hundred times|
|into the greater diameter of the|
|aperture, the rays will carry to the|
|object a spherical body and the form|
|of the aperture will be lost.|
Figs. 789-790: Eye looking through a semi-circular and triangular aperture on A61v and H71/23/r.
Fig. 791: Shadow produced by a round light source striking a rectangular occlusion on C12r;
Fig. 792:Image produced by a rectangular light source passing through a round aperture on CA256rc.
Figs. 791-800: Transformation from a square image to a round image generated by a round light source and square aperture. Fig. 793, Author's reconstruction; fig. 794, Forst II 5v; fig. 795, H227 inf. 48r-48v; fig. 796, CA135va; fig. 797, H227 inf. 48v; ig. 798, CA135va; figs. 799-800, H227 inf.
Given his interest in the light-sight analogy (see above pp. ) it is not surprising to find him on H71 /23/(r) (fig. 790, 1494) examining what happens when an eye looks through a triangular aperture. "The eye," he concludes, "does not comprehend the nearby luminous object."
He is equally interested in the properties of square apertures and occluding objects. On C12r (1490-1491), for instance, he (fig. 791) asks: "What shadow a square umbrous body will make with a spherical luminous source?" On A20r (1492) he notes how: "the solar rays repercussed on the square mirror will rebound in a circular form on the distant object." On CA256rc(c. 1492) he considers a variant situation in which (fig. 792) a rectangular light source passes through a circular aperture and projects a rectangular image.
When a circular light source, a square aperture and a projection plane are close to one another, the projected image takes on the squareness of the aperture. This limiting case (fig. 793) Leonardo does not illustrate. As the distance increases each of the four corners of the aperture generates intersecting circles. This situation he illustrates (figs. 794-797) and describes at length in a passage recorded on H227v inf., 48r-48v:
It is possible that the sun, having passed through four apertures, composes a spherical body on the object at a long distance. Let the four apertures be A, B, C, /and/ D. When the circles created by the sun by the said four apertures have expanded so much over a long distance that each one intersects itself with the nearby one in such a way that N /and/ M touch one another, then the four circles compose a single circle. The first degree of the light that is in the four circles is at E, because there the four luminous circles are superimposed on one another. F, G, H /and/ I are one quarter less bright then E, because there only three circles are superimposed on one another. P, K, L /and/ O are the half less light than E, because there two circles are superimposed. Q, R, S /and/ T are 3/4 less light /i.e. 1/4 of the light/ than E, because there is only one circle.
The passage ends with a description of what happens when the distance is increased:
Which /circles/ in going a long distance are lost because they are converted to darkness and thus P, K, L /and/ O become rounded and complete the spherical body and finally over a long distance the square E is converted to a circle and all the other parts of less duplicated light are lost.
How these circles gradually come together as the distance increases he sketches roughly on CA135va (fig. 798, 1492), and then with more care in a diagram recorded on H227 inf. 48v (figs. 799-800) accompanying which he writes:
Proof in what way the square is made in the form of a sphere by the solar rays at the object.
The point A spreads to M /and/ N and at a greater distance it spreads to OR and it sees the point of the angle which is extended to AS and then at AT when it has reached AV. The line above RD is consummated and the summits of the spheres will touch one another through the intersections of the circles and then the square is reduced to a circle.
This description of how a square is transformed into a circle effectively provides a visual demonstration of the age old problem of quadrature of the circle. Which raises the question: did Leonardo perhaps see in these optical experiments a case study in principles of practical geometry? This would, for example, account for a striking resemblance between the diagrams (figs. 795-795) just analysed and a diagram on Forst II 5v (fig. 794, c. 1495-1497), above which he writes "True proof of the square" and below which he adds:
If 4 circles are situated on the he line of a single circle with their centres in such a way that the circumference line of each one is made on the centres of each one, it is certain that they are equal and the circle where such an intersection is made, remains divided in 4 equal parts and is one half /the size of/ each of the four circles and inside each circle a square of equal angles and sides is produced.
Whether or not he saw these connections with transformational geometry, he was clearly fascinated in studying various stages in the process of a square image becoming round. On CA135va (fig. 798, 1492), for instance, he records a further stage in which the four circles nearly overlap one another completely. Accompanying this he drafts another passage which again emerges in more polished form on H227 inf. 49r (fig. 799):
CA135va H227 inf. 49r
|It is a necessary thing that the||It is a necessary thing that the|
|intersections of...round||intersection of the round pyramid|
|pyramids is made in a single||is made in a single closed point|
|point which is closed...of a||of a non-transparent circumference.|
|non-transparent circumference.||Therefore the half-round pyramid|
|It is necessary that the inter-||will divide the point only|
|section made by the half-round||surrounded by the half part, which|
|pyramid is amde in a point||point you will find in the right|
|closed only...by the half part.||angle and if you approach it to|
|the angle of the eye it will|
|appear to you in the form of a|
On H227 inf. 50v (fig. 800) he considers a case where the distance is greater still and the image becomes completely round, losing all trace of the square aperture:
Of luminous bodies.
Because the effects have similitude with their causes the sun, being a spherical body, it is necessary that the solar rays over a long distance do not retain the form of any angular aperture whence they pass, but rather, that they demonstrate after this the form of their cause in the first percussion.
An understanding of this phenomenon how the image of a round light source passing through a square aperture is square at a close distance and round at a greater distance helps us, in turn, make sense of a passage on A64v (1492) that is puzzling if read out of context:
The intersection of luminous rays made at the faces of the square aperture is produced beyond the said faces. And the intersection made at the angles is made in the size of its angle.
(Every aperture carries its form to the object over a long distance.)
No aperture can transmute the concourse of luminous rays in such a way that over a long distance they do not bring to the object the similitude of their cause....
Figs. 801-808: Effects on images of slit-shaped apertures. Fig. 801, H227 inf. 51r-51v; figs. 802-804;, CA135va; fig. 805, H227 inf. 81r-51v; fig. 806, A64v; fig. 807, CA135va; fig. 808, , H227 inf. 49v.
Having considered the characteristics of triangular and square apertures, he explores slit-shaped apertures. On CA135va (1492), for example, he draws (fig. 802) a case where each of the two end-points of the slit produces a circle. He then redraws the slit (fig. 803) showing how the two circles overlap more at a greater distance. These diagrams are without text. When he pursues the problem on H227 inf. 51r-51v (fig. 801) he adds an explanation:
The solar ray passing through an angular aperture in the percussion made by it on the wall will not carry the true similitude of the aperture.
Proof how the angles are the cause of making that the apertures render on the object the solar rays in spherical form.
The aperture composed of four faces makes four angles A, B, N (and/ M and these angles are the cause that the pyramids which have passed through them expand over a long distance in such a way that, occupying the faces they will make in the object the spherical light.
On CA135va (fig. 804, 1492) he also sketches a related case where two rectangular faces are positioned such that they produce an open slit. This he again draws and describes on H277 inf. 51r-51v (fig. 805):
Where the aperture has its faces without angles as PQ shows the solar rays having passed through, this will make on the object precisely the shape of the aperture with two lines. Therefore, if the sun, having passed where there are angles, it is made round, and where there are not angles, /it is/ never /made/ so round. This clearly shows that the angles are cause.
Figs. 809-814: Steps in the transformation from a cross shape to a round shape. Fig. 809, C9r; fig. 810-812, CA135va; fig. 813, H227 inf. 49v-50r; fig. 814, C10v.
On A64r 91492) he sketches (fig. 806) and comments briefly on another variant in which one slit is placed above another: "It is impossible that the luminous rays which have passed through parallels demonstrate to the object the form of their cause." On CA135va (1492) he sketches yet another variant in which the two slits are side by side (fig. 807). This he develops on H227 inf. 49v (fig. 808), explaining:
The sun having passed through narrow apertures which are divided by a small interval, it is necessary that in their percussion it demonstrates the two heads of these apertures in the form of two half circles which intersect one another.
Having placed the two slits above one another and beside one another Leonardo explores the next logical combination in which the two slits intersect one another to form a cross-shaped aperture. On C9r (1490-1491) he illustrates (fig. 809) and describes a case where light source, aperture and projection plane are near one another, with the result that the image resembles the aperture:
The luminous ray which has passed through a small aperture and the stamp of its percussion having been interrupted at a nearby opposition will be more similar to the aperture...through which it passes than the luminous body whence it originates.
On CA135va (1492) he sketches how each of the ends of the cross shaped image acquires a rounded shape (fig. 810). Next he shows how, at a greater distance, four circles emerge (fig. 811) which, at a greater distance still overlap more (fig. 812). Accompanying this he indicates the distances involved in quantitative terms:
The transit of solar rays through an angular aperture is necessary in some space.
At the distance of 20 braccia it will lose the parts e, f, g /and/ h and at 30 /braccia there will be produced a spherical body by the parts a, b, c /and d because the part which is towards the centre is more powerful.
On H227 inf. 49v 49v-50r, he illustrates (fig. 813) a situation where the distance is more than 30 braccia and the four circles are nearly coincident with one another. This he discusses in detail beginning with a general statement:
If each aperture sees the entire body of the sun, it sees all of its parts, which parts are received by all the aperture, and through all /of it/ and all in the parts. Therefore the part of the aperture, even if it is acute and apt to give passage to the sum of the rays that have parted from all parts of the sun which compose opposite (...) in the first percussion a spherical form, similar to their cause.
A specific description of the diagram (fig. 813) follows:
C is the centre of KR; D is the centre of L; E is /the centre/ of MZ; F of HY:G of OX; H of PV; A of QF; B of IS such that the eight exterior angles of the cross /- shaped/ aperture, the solar rays passing inside them compose in the object a round brightness, which brightness is composed of eight circles which make themselves the centre of eight angles of the aperture and each circle has in it fourteen intersections made by its seven companions, which are ninety-eight intersections in all.
At a still greater distance the image loses all trace of the cross-shaped aperture and becomes perfectly round like the light source. This situation he demonstrates (fig. 814) on C10v (1490-1491) beneath which he adds:
Figs. 815-819: Intersecting circles produced by a square aperture, a slit, a triangular, cruciform and star shaped aperture. Figs. 815-818, CA187ra; fig. 819, C7v.
The...luminous ray which has passed through some aperture of a strange form, if the stamp of its percussion be lengthened, will be similar to the luminous body whence it originates.../rather/ than the aperture through which it passes.
Leonardo has analysed in detail the properties of images passing through apertures in the form of triangles, squares, slits, double-slits and crosses. But he is not content to stop here. On C7v (1490-1491) he alludes to a more complex situation:
If you wish to make the rays of the sun pass through an aperture in the form of a star, you will see beautiful effects of perspective...in the percussion made by the sun which has passed.
Accompanying this he sketches in rough form an eight sided star (fig. 819). The problem is not forgotten. Within two years he describes this eight-sided star at length in connection with eight apertures on CA187ra (c.1492):
Remember that you note the quality and quantity of the shadows.
If you wish to see the clear and well-defined boundary of separation of simple shadows from the mixed ones you will have /the equivalent of a/ cloths for seiving made of paper soaked in turpentine and oil, in which the light of the sun shines and on the inside...you place a thin-board perforated by equal apertures made in a circle in 8 parts equidistant from one another and let the diameter of this circle be one...1/2 braccia and at a half a braccio...from the centre of this circle you will place, away from it /and/ facing you, a dense spherical body. Then you will place between your eye and the said body a thin folio of stationery which touches the spherical body, which is an inch in diameter and looking at its shadow behind it on the paper, the shadow of this body will appear to you precisely in the form of the image shown.
And if you wish to see the simple shadows with all the minuteness of degrees make a star of 8 rays which are as exactly large at the extremities as at the beginning and set this facing the sun, placing behind it the spherical umbrous body and then the paper and then your eye as was said above.
This entire description is written in the margin surrounding a carefully drawn diagram beneath which he writes:
This shadow is made by a spherical umbrous body illuminated by a light made in /the form of a/ star, which has its rays of equal size.
In the lower right-hand part of this folio he has drawn (fig. 837) a draft of this octagonal star shaped aperture. Directly above this are four intersecting circles such as would result from a square aperture (fig. 815, cf. figs. 794-797). Beneath the principal diagram showing the shadows produced by the octagonal star (fig. 838) are three other diagrams of intersecting circles: two circles, as would result from a slit (fig. 816, cf. fig. 803), three intersecting circles, as would result from a triangular aperture (fig. 817, cf. figs. 784-785), and four intersecting circles as would result from a cross-shaped aperture (fig. 818, cf. figs. 811-812). In short, this octagonally shaped aperture marks the culmination of a series of experiments.
Figs. 820-823: What happens to a round light passing through a single aperture. Figs. 820-822, CA277va; fig. 823, CA241rd.
Figs. 824-825: What happens to a round light passing through two apertures on CA277va and CA241rd.
On CA256rc (c.1492) he drafts a summary of these results: "In the percussion of rays is demonstrated part of the nature of its cause." Which idea he then reformulates in a passage headed:
On the nature of apertures
An aperture is composed of a number of sides and that which has fewer will demonstrate the truth of things less.
That which has more is better and maximally when the parts of the sides are equidistant from the centre of the aperture.
10. Number of Apertures
If we return to read more carefully the passage on CA187ra (c. 1492), we find that he not only mentions an eight-sided aperture, but also eight apertures equidistantly arranged in a circle. This is not an oversight on his part. For just as he has been studying the properties of multi-sided apertures, so too has he been exploring the comparable properties of multiple pinhole apertures. On CA277va (1508-1510), for instance, where he outlines his new plan for arranging the work on light and shade (cf. Chart 10 above), he illustrates the image cast by one pinhole aperture (fig. 822), two (fig. 824), three (fig. 829), four pinhole apertures (fig. 834), and a rough sketch with the image cast by perhaps as many as eight pinhole apertures (fig. 838, although this could well represent an advanced stage in the rounding produced by three apertures, fig. 830, cf. 831).
These draft sketches on CA277va (figs. 822, 824, 829, 1508-1510) are developed on CA241rd (figs. 823, 825, 836, 1508-1510), this time accompanied by text. In the upper right-hand margin he notes in passing: "Many minimal lights in the long distance will continue and make themselves noticeable." In the main body of the text he drafts general rules of light and shade (see below, p. ). Alongside the drawings he discusses the:
Figs. 826-832: Intersecting circles produced by three apertures. Figs. 826-831, CA277va, fig. 832.
Figs. 833-836: Effects produced by a light source passing through four apertures. Fig. 833, A177rb; fig. 834, CA177ve; figs. 835-836, CA241rd.
Nature of the light which penetrates apertures.
Of the light which penetrates apertures it is to be doubted whether, with the dilation of its rays, it recomposes as much size of impressions behind such an aperture, as the width of the body causing these rays. And other than this, whether such a dilation is of a power equal to the luminous body.
To the first doubt it is replied that the dilation made by the rays behind their intersection, recomposes such a size behind the aperture,...as was that which it had in front of the aperture: the space from the luminous body to the aperture being that which /there/ is from the impression of these rays to this aperture. And this is proved by the rectilinearity of luminous rays, concerning which it was proved that there is such a proportion from size to size...as there is from distance to distance of their intersections.
But the power does not go with the same proportion...as was proved, where it is stated:...such is the proportion of heat to heat...and of brightness to brightness of the luminous rays in the same centre, as there is from distance to distance from their origin. Therefore it is proved that the luminous ray loses that much in heat and brightness, to the extent that it is removed from its luminous body.
It is true that compound...derived shadows, that originate from the edges of such apertures, break such a rule through their intersections and...this is treated fully in the second book of shade.
Figs. 837-838: Effects produced by a round light source passing through an eight sided aperture.
Having answered his first two questions he considers the proportions of light involved under the heading:
Of the proportion that the impressions of light have placed partly one above the other.
Such are the proportions of lights that are generated in the impressions of luminous rays...in part superimposed on one another as is that which the number of impressions have, which are superimposed among one another.
To demonstrate this he now describes his figures (figs. 823, 825, 835, 836):
This is proved in the 2nd: And let the luminous rays be mb and mc which penetrate through the aperture op to the impression bc, which impressions are superimposed in part at the space n. I say that...the illuminated space, n, will be doubly bright than the remainder of the two impressions b /and/ c because n is seen twice by the luminous source m, and b and c are seen a single time. And by the second of this: such is the number of its luminous sources illuminating it at equal distances. And the same recurs in the 3rd figure where qmop have one degree of brightness, dpgh have two, enci have three and a 4. Therefore we shall say that the degrees of light will be as many as the number of apertures.
Further examples of four apertures occur on CA177rb (fig. 833); CA177vc (fig. 840, 1508-1510) he makes a rough sketch involving perhaps as many as eight apertures. On CA187ra (cf. fig. 838, 1492) he explicitly describes the use of eight apertures and on CA385vc (fig. 841, 1510-1515) he carefully draws apertures and the eight circles thereby produced. On the same folio he sketches two other cases with 18 apertures (figs. 842-843). This theme of multiple apertures is developed on CA241vc (1508-1510). Here he draws three intersecting circles which frame 24 apertures (figs. 845). Beneath this he draws another diagram (fig. 846) with 24 apertures to show how these, at a greater distance, produce 24 interlacing circles.
Accompanying these diagrams on CA241vc is a text that develops the ideas of CA241rc:
Figs. 839-843: Effects of light produced by multiple apertures. Figs. 839-840, CA277va; figs. 841-843, CA385vc.
Of the proportion that there is from illuminated object to illuminated object by a same luminous light.
Such will be the proportion...of the brightnesses that the illuminated sites of a same luminous body have as is that of the number of apertures through which this luminous body illuminates the aforesaid site. This is proved by the third placed behind this folio at the foot.12
This leads to a second general rule:
Of multiplied brightness taken from a single luminous body.
The brightness of a same luminous body at an equal distance will make itself of more power, to the extent that the number of the apertures, whence it penetrates to its impression (onto a same place).
And this is proved in the 3rd, behind this face, etc.
But it is also proved with the 13th of the other book where it is stated: that part of a site will be more illuminated which is seen by a greater number of luminous bodies.
Figs. 844-846: Effects produced by 24 apertures on CA241vc.
On CA229vb (1505-1508) he takes this theme further. He begins with a rough sketch showing two circles inscribed within a larger one (fig. 847): this might represent a situation involving two apertures. He then draws (fig. 848) four apertures and the four circles thereby produced. There follow other examples which are multiples of four, beginning with a sketch (fig. 849) of 16 (4 x 4) apertures with a hint of the circles they produce. Next he draws (fig. 850) 12 points on a half-circle which would amount to 24 (4 x 6) apertures in all. A case (fig. 851) involving 21 (4 x 7) apertures follows. Finally he draws (fig. 852) a series of eight circles which span but a quarter of the circumference of a circle that would contain 32 (4 x 8) apertures. Of these various examples only the case involving 28 apertures (fig. 851) is accompanied with a text:
A piece of iron is perforated with the perforations of a sieve and it takes the rays of the sun in such a way that all the perforations become enlarged, as the circle an and in the middle a makes the multiplication of rays placed one over the other which occupy the space of the lesser...circle m which will be warm and lucid.
We have already noted Leonardo's implicit comparison between multiple sided apertures (triangles, squares, crosses, octagons) and multiple-apertures (1, 2, 3, 4, 8, 16, 24, 32 pinholes). On looking more closely at CA229rb, vb as a whole another, theme of comparison becomes apparent: he is analysing multiple-apertures on the same folio that he is exploring multiple shadows produced by a St. Andrew's cross. This is not a coincidence. His analyses of the multiple shadows produced by a St. Andrew's cross occur on CA37ra, 177rb, 177ve, 241rcd, and CA229rb, vb (see above pp. ). These are the very folios on which he also explores multiple aperture problems of light (see Chart 18).
Figs. 847-852: Effects of multiple apertures on CA229vb.
CODEX MULTIPLE APERTURES MULTIPLE-SIDED MULTIPLE
CA177rb 4 4
CA177ve 4 6
CA241rcd 1,2,3,4 2,4
CA241vc 24 1 (on multiple surfaces)
CA229rb,vb 4,16,24,32 2,4,6
CA277va 1,2,3,4, 3 1
CA37ra 1 (in various degrees) 2,4,6
CA385vc 8, 18 1,2,3,
CA187ra 8 2,3,4,8 2,3,4,8
Chart 18: Links between Leonardo's work on multiple apertures, multiple-sided apertures and multiple shadows.
Figs. 853-863: Draft sketches showing effects of light which passes through a slit and encounters a sphere. Figs. 853-862, CA187va; fig. 863, CA187rab.
Figs. 864-866: Development of a demonstration on CA187vab, A89v and A89r.
11. Apertures and Interposed Bodies
We have examined how Leonardo explores the properties of light when it passes through apertures of various shapes such as slits and crosses. A next stage in complexity would be to study such apertures in combination with various shaped opaque bodies. This he does on CA187va (c. 1492) where he makes a series of preliminary sketches to show what happens when light passing through a slight-shaped opening encounters a spherical object (figs. 853-854, 856-862). Above these diagrams he makes two drafts of an explanation:
These umbrous bodies will make their derived shade more or less, depending on whether they are more or less far from the light.
These bodies will make their derived shade more or less short, depending on whether they are closer or further (from the light of the window) from their light.
Unsatisfied, he crosses out these drafts. He turns the sheet ninety degrees and makes two further sketches showing how light, having passed through a slit, and encountering a spherical object, produces a combination of simple and mixed shadow (figs. 855, 864). In the left-hand margin he drafts a phrase: "That light which," then stops short. Alongside the lower diagram he notes: "This light is long and then." Directly beneath this he writes: "No separate shade can stamp on the wall the true form of the umbrous body, if the centre of the light is not equidistant from the extremities of this body." In the upper left-hand margin he now claims: "No long light will send the true form of the separate shadows (to the wall) from the spherical bodies to the wall." On CA187ra (1492) he redraws his sketch of light passing through a slit, which encounters a sphere and casts shadows on the wall (fig. 863). Alongside this figure he writes: "Remind yourself that you note the qualities and the quantities of the shadows." On A89v (BN 2038 9v, 1492) he redraws the situation (fig. 865) and on A89r (BN 2038 9r, 1492) he develops it into a beautiful diagram without text (fig. 866). As is so often the case, he expects that his visual statement will speak for itself.
Closer attention to the other sketches on CA187ra (1492) again reveals Leonardo's delight in playing with variables. Having shown what happens when a slit-shaped light encounters a spherical opaque body (fig. 863), he considers what occurs when a spherical light encounters a slit-shaped opaque body (fig. 867). Not content to stop here he lets light pass first through a round and then through a slit-shaped aperture (figs. 868-869) and contrasts this with light which passes first through a slit-shape and then through a round aperture (fig. 870). Next he replaces this slit-shape with a cross shape (figs. 871-875).
This final example can be seen as a starting point for his illustration more than fifteen years later on CA207ra (fig. 876, c. 1508-1510) to "make a crucifix enter a room." Here he takes a blank wall on which he marks a crucifix. Opposite this wall he positions an aperture which is in a room. The sunlight reflects the light of the wall, enters through the camera obscura and casts the image of a cross into the room, (cf. Kircher's later example, fig. 877).
Figs. 867-875: Effects of light and shade involving combinations of apertures and/or occluding objects on CA187ra.
Fig. 876: Reflection of a cross shape through a camera obscura on CA207ra. Fig. 877: Development of this principle in Athanasius Kircher's Ars magna lucis et umbra (1646).
Figs. 878-879: Apertures, occluding objects and shade on triv. 22v and C11r.
Figs. 880-882: Sunlight, bubbles in water and cross shaped images on F28v.
A more complex play with cross-shaped images is suggested on Triv. 22v (fig. 878, 1487-1490) which may be the basis of his diagram on C11r (fig. 879, 1490-1491) where light passes through a cross-shaped aperture, encounters a transparent sphere and then casts a rounded cross-shaped shadow. Accompanying this he notes:
The shape of the derived shade will always have conformity with the form of the original shade.
The light in the form of a cross is the cause why the umbrous body of spherical rotundity will cause from it shadows in the shape of a cross.
In 1508, he returns to this problem of cross-shaped images, now in an unexpected context. On F28v (fig. 880) he observes that:
The ray of the sun, having passed through the bubbles of the surface of the water sends to the bottom of this water an image of this bubble that has the form of a cross. I have not yet investigated the cause, but I believe...that it is because of the other little bubbles...which are joined to this larger bubble.
By way of illustration he makes two sketches to show how smaller bubbles13 surrounding the larger bubble (figs. 881-882) might serve to generate a cross-shape. On CA236rd (1508-1510) he makes a note: "on the shadows situated at the bottom of the water and which send their species to the eye through water and through the air," but proceeds to discuss refraction (see below p. ). He appears not to have pursued the problem of cross-shaped bubbles as he had hoped.
Figs. 883-887: Slit shaped apertures and shade on CA258va.
In the period 1508-1510 he does return, however, to problems of slit-shaped apertures and opaque bodies on CA258va. Here he begins (fig. 883) with light passing through a slit-shaped aperture which encounters a narrow opaque body and casts a shadow on the ground at ninety degrees to this. Directly beneath he explains:
When the light of a long shape generates derived shade in rectangular conjunction with primitive shade, then the derived shade, in every degree of its length...diminishes its first darkness.
And this is proved by the 4th of this, where it is stated: so much of the darkness of derived shade is lost to the extent that it makes itself remote from its primitive shade.
Next he considers a situation where this shadow is cast at more than ninety degrees (fig. 884):
To the extent that the angle which is generated at the conjunction of the derived shade with its primitive shade is larger, to that extent the boundary of the derived shade is ?. And this arises by the said 4th because: to the extent that the angle created by the conjunction of the derived shade...with the primitive shade is of greater size then the opposite extremities of such shade will be...more distant from one another and by the 4th, the derived shade will be of less darkness.
Immediately following he turns to the converse:
There follows the converse of the said.
The extremity of the derived shade will be that much darker...to the extent that the angle which is generated in the conjunction of the primitive shade with the derived shade is more acute. And this is proved by the converse of the fourth which states: ...that part of the derived shade will be darker which is closer to its primitive shade.
Fig. 888: A slit-shaped aperture, a slit-shaped object and its shade on CU630.
To illustrate this he moves an interposed stick through various degrees of obliquity (fig. 885). Finally he considers a case where a slit-shaped aperture, thin opaque body and the resulting shadow are all in the same plane (figs. 886-887)
Therefore, the more acute angle being the cause of making its sides closer, it is necessary that the primitive shade and derived shade of which the sides are composed are again much closer to one another.
This situation interests him and on CU630 (TPL627, 1508-1510) he examines it in more detail under the heading:
Of the derived shade created by light of a long shape which percusses an object similar to it.
When the light which passes through an aperture of a long and narrow shape percusses the umbrous body similar to it in shape and position, then the shade will have the shape of the umbrous body.
A specific demonstration follows (fig. 888):
This is proved. Let the aperture, through which the light penetrates into a dark place, be ab and let the columnar object equal to and of the same shape as the aperture be cd. And let ef be the percussion of the umbrous ray of the said object cd. I say that such a shade cannot be /either/ greater or less than this aperture at any distance, the light being conditioned in the said way. And this remains proved by the fourth of this which states that all umbrous and luminous rays are rectilinear.
Figs. 887-891: Combinations of apertures and occluding objects on C9v, W12352v and CA236rc.
Having studied in isolation the effects of different shapes of umbrous bodies and apertures, he examines various situations where these factors act in combination. On C9v (1490-1491), for instance, he draws a light source (fig. 889) the rays of which, on meeting an opaque body, cast a shadow which passes through an aperture. On the far side of this aperture are two further light sources g and h which cast rays intersecting this shadow. Directly beneath this diagram he adds a brief text:
The simple percussion of derived shade will not change its darkness even though its umbrous rays are changed and mix in the air with luminous rays. The figure on the right is well placed over this said proposition.
On W12352v (c. 1494) he draws another diagram (fig. 890) of a luminous body the rays of which meet an opaque body and cause shadows which then pass through an aperture. Here there is no accompanying text. But then on CA236rc (1508-1510) he redraws the diagram carefully (fig. 891) and adds a full explanation under the heading:
What difference there is between shadow and image.
The difference that there is between simple shade...of the opaque body to the image of this body is that such simple shade does not penetrate inside minimal apertures as...does the image of the same umbrous body.
This is proved. And let the umbrous body be cd and let the luminous body accompanying the umbrous body in the generation of shadows be ab and let the aperture be r through which the said...species penetrate into the dark place vmhn. I say that the simple derived shade, cdp is first intersected at this p which comes to the aperture r and spreads in such a way that it cannot penetrate through this aperture.
Figs. 891-894: Apertures and occluding objects in combination on CA216rb.
Figs. 895-901: Effects of spherical and rectangular occlusions on shade on CA238vb.
Meanwhile, on CA216rb (c. 1495) he had been exploring more complex variants of this situation. In a first diagram (fig. 892) a light source is left undrawn and an opaque object casts its shadow through two apertures onto a wall. In a second diagram (fig. 893) there are again two apertures, but now there is an opaque body in front of these apertures and a smaller opaque body behind them. Their shadows combine to produce a series of four intersecting circles. Finally there is a third diagram (fig. 894) which has the same elements and differs only in that relative sizes of the opaque bodies are changed. There is no text accompanying these diagrams.
On CA238vb (1505-1508) he takes a flat rectangular board and a round ball. These he places in near proximity to one another in order that they effectively function as an aperture. He then examines the effects produced by moving the light source and altering relative positions of the board and ball. Accompanying the series of diagrams that result (figs. 895-963) he drafts a number of only half intelligible notes which are here translated without comment:
The more [fig. 900] it is closer to the intersection of the rays, the more m makes the function than the circles and likewise it will do the converse the more it approaches the percussion of these rays.
Here [fig. 721] the solar rays make an intersection at the upper limit of the ball and at the lower limit of the axis.
But those which are interrupted...are those which are intersected at the board and, after such an intersection, are interrupted by the upper limit of the ball.
All [fig. 901] the luminous rays that are cut by n are lacking at m.
Figs. 902-909: Demonstrations with occluding surfaces. Figs. 902 -903, CA238v; fig. 904-909, CA133va.
(The umbrous body outside the window)
And such an interruption of rays are of those which are intersected at the upper limit of the ball.
(The solar rays)
(Here the shadow of the board carried by solar rays).
Of which it happens (that the)
(Here) many correlates, that is derivatives.
(Here the...solar rays which terminate the shade)
When the solar rays, after their intersection at the upper part of some sphere have to terminate the inferior shade of the figure of straight lines.
Of the circle.
The shadow /fig. 895/ n is always greater or less depending on...its direct (saetta) vicinity to the shadow of the ball and in its growing and diminishing there will always be beyond the...shadow of the ball in...the shape of a semi-circle as can be seen (/se/ en by him who interposes the eye to the ray of the sun when) by him who puts the shadow of the ball opposite him, near the eye, when the stick touches...the boundary, the shade...of the ball and from such a boundary the said stick slowly moves towards the sun, not deviating from the contact of the ball.
The umbrous body inside the window, the sun always....
The derived shade of the spherical body illuminated by a light equal to it...will diminish strongly if the percussion is at the limit of the shade of another spherical body (that) that touches the first spherical body.
ae /fig. 897/ is the lower limit of the shadow of the board.
Related diagrams are to be found on CA238rb (figs. 712-719, 1505-1508) and CA133va (figs. 904-909, 1497-1500. With two exceptions these are without text. Beneath one (fig. 908), he points out that "the line ab is the boundary of the luminous body." Below the largest diagram (fig. 909) he writes: "When n touches m, f will touch g." On such folios which represent an interim stage int eh development of his ideas, rough sketches suffice. Careful explanation is not yet necessary.
12. Spectrum of Boundaries
Leonardo's studies of a camera obscura in combination with opaque bodies lead him to abandon his early assumptions concerning clearly defined boundaries and to emphasize instead a spectrum of gradations between light and shade. This he does in terms of demonstrations involving a series of basic arguments: (1) that derived shade has less power to the extent that it is more distant from its primitive shade; (2) conversely, that derived shade has more power when it is closer to its source; (3) to what extent one can speak of uniformity of derived shade; (4) that primitive and derived shade mix with distance; (5) where primitive and derived shade are joined together; (6) where shade is greater; (7) where primitive and derived shade are not joined; (8) implications for the perception of backgrounds and (9) simplified gradations of shade.
We shall consider his demonstrations for each of these arguments in turn and show how these interests lead directly from the physics of light and shade to problems of vision and perception.
12.1 Derived Shade is Less Powerful When More Distant From Its Primitive Shade
The idea that derived shade loses strength with distance is clearly expressed on CA258va (1508-1510):
Derived shade is that...much less...powerful than primitive /shade/, to the extent that it is more distant from this primitive shade. There follows the converse. And derived shade is that much more similar to the primitive shade to the extent that such derived shade is closer to this primitive shade.
He mentions this idea in passing on CU705 (TPL553d, 1508-1510): "The darkness of the derived shade diminishes to the extent that it is more remote from the primitive shade." On CU707 (TPL561, 1508-1510) Leonardo returns to this problem in greater detail in a pass age entitled:
On compound derived shade.
Compound derived shade loses that much more of its darkness to the extent that it is more remote from simple derived shade. This is proved by the ninth which states: that shade will make itself of less darkness, which will be seen by a greater quantity of the luminous body.
A concrete demonstration (fig. 924) is cited in support:
Therefore let ab be the luminous source and lo the umbrous body and abf be the luminous pyramid and lok the pyramid of simple derived shade. I say that...g will be a quarter less illuminated than at f because at f one sees all the light ab and at g a quarter of the light ab is missing such that only cb which is three-quarters of the luminous body ab, is that which illuminates g. And at h one sees the half db of the luminous body ab. Therefore, h has half of the light f and at i one sees a quarter of this light ab, that is, eb. Therefore i is three-quarters less luminous than f. And at k one does not see any part of this light. Therefore there is a privation of light and the beginning of simple derived shade. And thus we have defined compound derived shade.
12.2 Derived Shade is More Powerful When Closer to Primitive Shade
On CA144va (c. 1492) he drafts this idea: "The closer that the derived shade is to the primitive to that extent is it darker...and its boundaries are less /than the/...luminous part that surrounds it." This he crosses out. On CU730 (TPL598, 1508-1510) he takes up this claim afresh under the heading:
Whether the derived shade is darker in one place than in another.
Derived shade will be that much darker to the extent that it is closer to its umbrous body, or closer to its primitive shade and for this reason it arises that its boundaries are better known in the /ir/ origin than in other parts distant from this origin.
He reformulates the idea on CU699 (TPL606, 1508-1510) under the heading:
Of the boundaries of derived shade.
That boundary of derived shade will be darker and better known which is closer to its primitive shade. This is proved by the 5th of this, which states: in the contact that derived shade has with its primitive shade, there...the conjunction of the simple with the compound shadows are not noticeable because, beginning in an angle, they begin in a point, as is proved in the definition of an angle where it is stated: the angle is the concourse of two straight lines in a same point.
Immediately following the objects of an adversary are mentioned and answered:
Even though the adversary says that such lines composing an angle can be curved, this is partly accepted and partly denied, because such lines could be of a same curvature and equally distant from the centre...of that circle surrounded by lines, whence...the contact of those two lines would make a single line and would be like the contact of two straight lines in a same directness which, also would not compose an angle but a single straight line. But let us say that an angle is the contact of two straight lines situated outside a same rigour. And of the curves let us say that a curvilinear angle is composed of curved lines with various distances from their circle.
12.3 Uniformity of Darkness
The above two demonstrations serve as basis for his comments on CA258va (1508-1510):
On the Uniformity of Derived Shade.
But the derived shade is of more uniform darkness, which has a...more uniform distance from its primitive shadow. And this is proved by the 4th and by the 5th of this which states in the 4th: that part of the derived shade is darker which is closer to the primitive shade and by the 5th: that part of the derived shade will be of lesser darkness which is more distant from the primitive shade. It follows, from these two contraries, that that which is of a uniform distance from such primitive shade is of uniform darkness.
12.4 Primitive and Derived Shade Mix With Distance
The concept that primitive and derived shade mix with distance is implicit in a statement on CA256rc (c. 1492): "To the extent that the umbrous body is closer to the percussion of the rays its shadow will observe the form of its derivation more." On CA144va (c. 1492) he drafts an idea: "(That part of the derived shade will mix itself less with its boundaries in the light that surrounds it which is closer to the primitive shade.)" This he crosses out and makes two further drafts:
To the extent that the derived shade is more distant from the primitive, to that extent will it mix its extremities more with the luminous body that surrounds it.
To the extent that the light is more distant from the luminous body, the extremities of its shadow and the light will be mixed together more.
One reason for this claim stems from everyday experience as is clear from a passage on A92v (BN2038, 2v, 1492): "How the shadows are confused over a long distance is proved in the shadows of the moon which are never seen." On CU636 (TPL438a, 1505-1510) he returns to the general problem in passing: "and the derived shade mixes itself the more with the light to the extent that it is more distant from the umbrous body." Which idea he reformulates on CU699 (TPL606, 1508-1510): "That shade is more distinct and defined which is closer to its origin, and the more distant is the least defined," and on CA371rb (1510-1515) he expresses it differently again:
The more distant that the derived shade is from the primitive shade the more it varies from this primitive shade with its boundaries.
12.5 How Primitive and Derived Shade are Joined Together
Related to the foregoing is a demonstration on CU697 (TPL562, 1508-1510) entitled:
How primitive and derived shade are joined together.
Derived shade is always joined with primitive shade. This conclusion is proved per se, because primitive shades makes the basis of derived shade but they only vary insomuch that primitive shade of itself tinges the body to which it is joined and the derived shade is spread through all the air penetrated by it.
By way of illustration he gives a concrete example (fig. 925):
This is proved and let the luminous body be f and let the umbrous body be aobc, and the primitive shade which is joined to such an umbrous body is the part abc. And the derived shade abcd originates together with the primitive and such a shade is said /to be/ simple in which no part of the luminous body can see.
This theme he pursues on CU708 (TPL563, 1508-1510) headed:
How simple shade is conjoined with compound shade.
Simple shade is always conjoined with compound shade. This is proved by the foregoing where it is stated. Primitive shade makes its base of derived shade and since simple and composed shade are born in a same body joined to one another, it is necessary that the effect participates of the cause. And because the compound shade in itself is nothing other than diminution of light and begins at the beginning of the luminous body and finishes together with the boundary of this luminous body it follows that such shade is generated in the middle between simple shade and simple light.
A demonstration without an illustration follows:
This is proved and let the luminous body body be abc and the umbrous body de and let the simple derived shade be def and let the compound derived shade be fek.
And leads to a second claim:
But the compound derived shade always sees a part of the luminous body, greater or less, depending on the greater or lesser distances that its parts have from the simple derived shade.
Which, in turn, is demonstrated, again without an illustration:
This is proved and let such a shade be efk which, with half of its size sees fk, that is, ik sees half of the luminous body ab which is ac, /and/ this is the brighter part of this compound shade. And the other darker half of the same compound /shade/ which is fi sees cb, the second half of this luminous body. And thus we have determined the two parts of the compound derived shade, the one brighter or less dark than the other.
12.6 Where the Shade is Greater
On Forst III 87v (c. 1493) Leonardo mentions how the extremities of shade are affected by light:
The luminous or illuminated object bordering on the shade intersects as much as it cuts.
As much of the extremities of the shade of bodies will be lacking as is touched by an illuminated or luminous object.
On H66/18/(r) (January 1494) he notes: "that part of the derived shade will be less dark which is more distant from its extremities." He returns to this idea in two drafts on CA190rb (1505-1508):
The boundaries of all colour which pass through apertures are more evident than their middles.
...The boundaries of the species of each colour that penetrates through a narrow aperture into a dark place, are always of a more powerful colour than its middle.
On this same folio he also drafts another phrase: "That object will make itself darker which is...seen by a greater amount of darkness." The way in which this and related themes are associated in Leonardo's mind is seen clearly on CA230rb (1505-1508) which opens with a series of general claims and a questions:
To the extent that the umbrous body is closer to the luminous body, to that extent is the maximal whiteness more remote from the maximal derived shade surrounded by it.
The boundaries of the maximal derived shade is darker than its middle.
That part of the derived shade will be darker which sees a greater sum of darkness and that will be of less darkness which sees a darkness of less quantity.
The surface of every opaque body participates in the colour of its object.
The medium of uniform transparency gives passage to some species of a given colour or shape without occupation of the site in this medium.
Why shades tinge dense bodies and not rare ones?
This is followed by a further question:
Figs. 910-915: Gradations of light and shade in camera obscuras on CA230rb.
Why the shadows intersected behind the maximal shade, lose more darkness to the extent that they approach such a maximal shade?
This is answered with the help of a demonstration (fig. 911):
Let aco be the maximal darkness, codn /and/ aobm are the shadows intersected on the maximal shadow cno. I say that such a shadow, in separating itself from the maximal shade, the further it is removed, the darker it becomes with some space. And this shade increases because the whiteness of the two simple lights and the darkness which proceeds sees the dark background mixed with the light surrounding such a background.
The accompanying diagram (fig. 911, cf. figs. 910, 912) recalls his studies of divergent shade (see above pp. ). Above this diagram he adds a brief caption: "To the extent that g /and/ i are less, to that extent are the whites surrounding maximal shade narrower." To the left of this he draws a further diagram (fig. 913, cf. figs. 912, 914-915), beneath which he explains:
The pyramid Ste is tinged by the colour of its objects and thus makes its background. And for this /reason/ this pyramid is, in itself, variable in its part with various darknesses, such that it sees where it is darker and where it is less dark than the dark object cb.
Which explanation continues in the next column to the left:
Ab sees the end of the shade cd and therefore the derived shade is of little darkness; ec sees all the shade cb and for this reason there is shade of much darkness; nm sees half the shade cb and for this reason it is shade of middle darkness.
And by such a demonstration we have proved that the maximal derived shade is darker in the extremities than in the centre.
Tresf are of the observed darkness because in every part of their length they see a same darkness cbho.
He restate this conclusion in passing on CU699 (TP606, 1508-1510); "The shade will show itself as darker towards the extremities than towards its centre," and sets out to demonstrate it afresh on CA195va (fig. 930, c. 1510):
And in their boundaries colours are more intensive and brighter than their parts.
This is proved by the 4th of this which states: The surface of every opaque body participates in the colour of its object. It follows that the line ap...tinges with itself the surface np and this given line carries with it the boundary of brightness ca with the dark ab and in this line one does not see any part of the dark line, but all brightness. But if you remove yourself more from the boundary of the derived shade, then such brightness gh is tinged by the shade ak and thus the illuminated part is corrupted by the percussion or the mixture of the umbrous image ak which a could not do in gp and maximally in p. And thus is proven our intent to show that the limit(s) of the bright image with the derived shade is brighter in act and not in appearance as the boundary wishes than in the remainder of the other background.
Later on the same folio he pursues this theme asking (fig. 930):
Why the boundary of derived shade remains intersected after the pyramid of maximal darkness...and why such an intersected shade is dark towards the angle of the umbrous pyramid and outside it is bright /?/.
Figs. 916-917: Gradations of light and shade in a camera obscura on CA297va.
The exterior limit of the derived shade which is intersected will be that much darker outside than towards the middle because in hx, the more...you remove yourself from x towards s, the more you will find light at al and the contrary you will find at...the opposite side, because the more you remove yourself from p towards o, the more...the darkness ab is demonstrated and this is said concerning the background of intersected derived shade.
Such investigations lead him to examine precisely where gradations of light and shade are brighter or darker. On CA297va (1497-1500), for instance, he makes a preliminary sketch (fig. 916) which he then redraws (fig. 917) and describes:
The line ed sees the luminous body in every part of its length and the line bk sees...the middle of the same luminous body.
And the 3rd line pq sees the entire umbrous body cp and all the luminous body ac.
By that which was said above...the space qg will be that much less dark to the extent that it comes closer to the line dg and the space gf will be that much less bright.
Roughly a decade later he takes up this theme afresh on CA37ra (1508-1510). He now draws two preliminary diagrams (figs. 918-919) and then a third (fig. 920, cf. figs. 921-922). As usual, the accompanying text opens with a general statement:
Speak first of the qualities of divided lights of compound shadow frbc born of particular light.
Figs. 918-922: Gradations of light and shade in camera obscuras. Figs. 918-920, CA37ra; fig. 921, CA385vc; fig. 922, CA277ra.
The compound shade frbc is conditioned in such a way that to the extent that it is more remote from its intrinsic side, to that extent does it lose its darkness.
A demonstration follows (fig. 920):
This is proven. Therefore let the luminous source be da and the umbrous body fa and let ae be one of the side walls of the window, that is, da. I say by the 2nd that...the surface of every body participates in the colour of its object. Hence the side rc which is seen by the darkness ae participates in this darkness. And similarly the extrinsic side, which is seen by the light da, participates in this light and thus we have demonstrated such an extremity.
He now writes a new heading: "Of the middle contained by the extremities." He is, however, unsatisfied and crosses out the entire passage. In the right-hand margin he begins afresh:
This divides itself into 4:
1st: of the extremities containing the compound shade.
2nd: of the compound shade within the extremities.
Again he breaks off and in the lower centre of the folio he notes in passing: "Where the shade is greater or less or equal to the umbrous body, its origin." He now turns the folio to the side, draws a considerably more complex diagram (fig. 923) and analyses it in a passage headed:
Fig. 923: Compound shade on CA37ra.
Of the shade bch.
This is proved because the shade opch is that much darker to the extent that it comes closer to the line ph and is that much brighter to the extent that it comes closer to the line oc and let the light ab be a window and let the dark wall where this window is positioned be bs, that is one of the sides of the wall.
Therefore we shall say that the line ph is darker than another part of the space opch because this line sees and is seen by all the umbrous space of the wall bs. But...the line oc is brighter than any other part of this space...opch because this line sees all the luminous space abe.
He pursues this theme of various gradations of brightness and darkness on CA258va (1508-1510) beginning with two demonstrations (figs. 956):
Abo is illuminated by the entire light cdo but more...at a where it sees all of cd, than at b where the same dc finishes its sight in which a suddenly finishing this sight of the light cd, there begins the sight of the darkness de and the background is tinged by these bright and dark images.
The space opas begins dark at ps...because it sees the darkness de and goes on becoming brighter towards s to that point where it always acquires a greater sight of the light dc and this brightness having finished, the background oan begins to become dark again, because this background is seen by the darkness d and it makes itself that much darker to the extent that it approaches on more and it does the same from the opposite side.
An interim paragraph follows in which he introduces the question of maximum brightness.
Having proved the cause of the shape and darkness of which the simple derived shade is composed and, other than this, having proved the shape and darkness of compound shade, surrounding this simple shade...it remains to prove the maximum brightness of the background surrounding this compound shade.../by means of/ which we shall also prove the necessity of the maximum brightness of the aforesaid background.
To this end a further demonstration follows:
Therefore let the line oa be the boundary of the compound shade oba which, as was said, is seen by all the light cd and is illuminated the more to the extent that it is closer to the line oa where it sees all the light cd and it is illuminated the less to the extent that it comes closer to the opposite side qs. Therefore the line oa is the brightest part of this derived shade because such a line is continuous with od, the boundary of the light cd, behind which line oa the remainder of the background begins to brighten again, that is, the background aon, which background acquires that much more darkness and removes itself more from that line of brightness. And this is proved because, through such distance one always sees a greater amount of the dark background to the side of the light cd, that is, the darkness de.
He returns to this theme on CU669 (TPL719, 1508-1510) under the heading:
Figs. 924-927: Demonstrations of compound shade in camera obscuras. Fig. 924, CU707; fig. 925, CU697; fig. 926, CU669 and fig. 927, K/P 178r.
Of the brightness of derived light
The most excellent brightness of derived light is where it sees all the luminous body with half of its right or left umbrous background.
The diagram for the demonstration that follows is reminiscent of earlier discussion in this context (fig. 926, figs. 924-925):
This is proved and let the luminous body body be bc and let its right and left umbrous field be dc and ab. And let the umbrous body less than the luminous body be nm and the wall ps is where the umbrous and luminous species are impressed.
I therefore say that on this wall ps at the point r there will be a more excellent brightness of light than in any other part of this pavement. This is shown because at r one sees all the luminous body bc with half of the dark background ad, that is cd, as the rectilinear concourses of the umbrous pyramid(s) cds and the luminous pyramid bcr show. Therefore, at r one sees as much quantity of the dark background cd as there is of the luminous source bc. But at the point s one sees the umbrous /part/ ab and one also sees the umbrous /part/ cd, which two dark spaces amount to double that of the luminous body bc. But the more you move from s to r, the more you will lose of the darkness /of/ ab. Therefore, from us to r the pavement sr will always brighten. Again the more you move from r to o the less you will see of the luminous source. And for this reason the pavement ro becomes darker the more one approaches o.
And through such a discourse we have proved that r is the brightest part of the pavement qs.
Figs. 928-929: Demonstrations where primitive and derived shade are not joined on CA258va and CA195va.
12.7 Where Primitive and Derived Shade are not Joined
On CA258va (1508-1510), having discussed conditions under which primitive and derived shadow are joined he considers (fig. 928):
Of the shadow that does not join the derived and the primitive.
This figure is that...which is described here below...and it is said of the part of the triangle hnp, that is, its sides qop and qnm seen by the light vxs and yor which is that much more or less illuminated to the extent that it is closer or further from the line go or, if you wish, qm.
The fourth which is lacking below at de is...the space psb which becomes that much brighter again to the extent that it removes itself from the angle p...and this proved by the sixth which states: that part of the umbrous body will be...of less darkness which is illuminated by a greater quantity of the luminous body. Therefore our proposition is concluded because, to the extent...that the sides of the triangle psb remove themselves from the...point p, to that extent do they see the light of cd more and to that extent are they seen by a greater sum of light etc.
A similar diagram and demonstration are found on CA195va (fig. 929, 1508-1510) where he observes:
In the triangle grt is the triangle aco which is luminous and it also sees the opposite luminous triangle enp. Therefore this triangle grt will be twice as luminous as in the two lateral triangles ogr and pgt where its light, even though it has the same derivation, is simple, and the other is composed of two lights. Therefore this illuminated triangle...is that which separates the two shadows opqr and optn from one another.
Figs. 930-931: Reconstruction of CA195va by Pedretti.
These diagrams are the more interesting because they return to a problem that had perplexed him in his earlier studies of light and shade, namely, what causes the shadow of an opaque body smaller than the light source to be divergent.
On CA195va (1510) he draws (fig. 930, cf. 931) a camera obscura in which the entering light encounters two opaque bodies and produces complex gradations of light and shade, which he describes briefly: "op sees and is seen by ab and is tinged by its colour and on the side p is seen the beginning of the brightness of the air which brightens the place where its image percusses." Hence this combination of camera obscura and opaque objects provides yet another demonstration for his "colour participates" argument (see above pp. ).
12.8 Implications for The Perception of Backgrounds
At the same time this demonstration serves as a starting point for a further argument.
Why black painted bordering on white does not show itself as blacker than where it borders on black, nor white shows itself more...white bordering on black as do the species which have passed through an aperture or through the limit of some opaque obstacle.
This arises because the species tinge the place they intersect with their colour and when the difform species see a same site, they make a mixture of their colours, which mixture participates more or less in one colour than another, to the extent that the one colour is a greater quantity than the other.
Figs. 932-937: Gradations of shade in camera obscuras on CA354rb.
This particular demonstration is of considerable interest because, as will be shown (see below pp.** ) he had made various experiments to establish the contrary, namely, that, white on a black background appears whiter and black on a white background appears darker. On this same folio he explicitly compares the effects of a camera obscura with those of the pupil in the eye. Problems relating to physics of light and shade, the physiology of vision and perception are all intimately connected in Leonardo's mind. As a result what had traditionally been philosophical and psychological questions of vision and perception now emerge as problems of physics. Problems of optics are no longer a matter of theoretical debate but open to practical verification by experiment. He returns to this situation of a sphere placed within a camera obscura once more on W19086r (K/P178r, fig. 927, c. 1513) where he notes that:
...Among bodies of equal size and distance that tinges the body positioned opposite more with its species which is more luminous. Of bodies of equal brightness that tinges the surface of its object more which is of larger form, all being of equal distance. Of bodies of equal brightness and size the closest tinges its object more.
12.9 Simplified Gradations of Shade
Parallel with these demonstrations is a further series which omits the interposed opaque sphere and reduces the problem of gradations of shade within the camera obscura to its essentials. Preliminary drawings (figs. 932-941) on this theme are found on CA345rb (1505-1508) amidst discussions of species being everywhere in the air (cf. pp.** ) and how things cannot be seen without apertures (cf. pp.** ). Among these ten drawings, only one is explained (fig. 941):
Figs. 938-941: Further gradations of shade in camera obscuras on CA345rb.
dACB /is a/ triangle, /which/ through the luminous base ab illuminates the angle c in the maximum degree of illumination. Dfe /is a/ triangle /which/ has that much less light in angle f than the light c, to the extent that de, the base of f is less than ab the base of c.
On CA190rb (1505-1508) this theme of gradations of light/shade within a camera obscura is developed. In the right-hand column he begins with a preliminary sketch (fig. 942), beneath which he draws a camera obscura with various gradations (fig. 943). To this diagram he adds six letters. These, however, are not explained. Beneath the diagram he merely notes: "That object will be darker which is seen by a greater sum of darkness." He now draws two further diagrams showing gradations of shade in a camera obscura (figs. 944, 947) and in the passage that follows describes the one on the right under the heading:
How and where the dark object mixes...its derived shade with the derived light of the luminous body.
The derived light of the dark walls...lateral with the brightness of the window are those which with their various darknesses are mixed with the derived light of this window and with various darkness except for the maximal light c.
A precise description of the figure now follows:
This is proved: and let da be the primitive shade, which sees all and makes the point e dark with its derived shade, as is demonstrated by the triangle aed of which the angle e sees all the dark base da and the point v is seen by the darkness as, part of ad and since the whole is more than the part, it will be darker than v which sees only a part. Applying the above conclusion to the figure, t.../this/ will be less dark than v, because the base of the triangle f is part of the base of the triangle t and...c is the limit of the derived shade and maximal beginning of the maximally illuminated part.
Here the right-hand column ends. In the upper left-hand column he drafts two further diagrams (figs. 945-946) beneath which he drafts an explanation of the left-hand figure:
The simple light...absees all in the point m and is not...in any other part of the space hs, as the rectitude of the sides...of which the triangle aem is composed, which are in contact with the limits of the aperture fg. L lacks a quarter of the light ab. Therefore it is seen by the remainder of the light bc. K lacks half of the light ar. Therefore it is seen by the other half and 1 is only seen by a quarter of the light ab, that is by de and h is seen by the limit of the light...e is the beginning of the maximal shade...er.
h sees the weak limit of the light and sees...the maximum darkness of the maximum shade such that in this h one sees entirely shade.
Here his manner of referring to different fractions of light and shade strikes us as familiar. We have encountered it on more than one occasion (see above pp. ). His references to maximal light and shade we have also encountered elsewhere (CA258va, CA230rb, CA345rb). But if the initial thoughts remain similar, their applications are, nonetheless, quite different.
This diagram in the upper left-hand margin is probably a draft for the left-hand diagram (fig. 947) in the right-hand column, which he describes after he has crossed out his draft:
Why the derived light that passes through an aperture into a dark place does not make percussion of uniform brightness.
Let ab be the primitive light of a window. Let rs be the aperture where the derived light penetrates the dark place xtov. Let oc be the percussion of the derived light on the dark wall ov or the pavement of this place. I say that in such a percussion...oc...made by the luminous ray...will not be illuminated by uniform brightness. And this is proved by the 4th which states: that thing will be more illuminated which over an equal distance is seen by a greater quantity...of luminous body. Therefore, being in the percussion of the luminous ray oc, the part c seen by all the luminous source ab, it is necessary that the point c will be maximally luminous and the more illuminated than the point e which is seen by db, part of this luminous body and likewise...the point g will be less luminous than t because it is illuminated by fb, part of db and similarly m will be less luminous than g because it is seen by ub part of fb, whence it follows that the point o is the limit of such...an illuminated /object/ and is...the beginning...of maximal darkness of the maximal derived shade because the point o, besides being the limit of the luminous object ab, as has been demonstrated, sees the entire umbrous body bp, etc.
It is proved how the point o receives in itself the percussion of the maximal shade, part of the darkness of the maximal shade.
Figs. 948-950: Gradations of light and shade in camera obscuras and the eye on CA190vb.
Here the text breaks off and he gives instructions to turn the "page" to CA190vb (1505-1508) which opens:
O mathematicians throw light on such error.
Spirit has no voice because where there is voice there is body and where there is body there is occupation of place which impedes the eye from seeing the things positioned beyond such a place. Therefore such a body fills of itself all the surrounding air, that is with its species.
This is reminiscent of a passage on CA345 (see above pp. ) which also occurs in connection with a camera obscura passage. The lower part of CA190vb contains various diagrams relating to the inversion of images within the eye (figs. ) to be discussed later in section three. Amidst these diagrams he draws another preliminary sketch of a camera obscura with its gradations of shade (fig. 948), beneath which he draws two more elaborate versions (figs. 949-950), the latter of which appears intended to serve as an imitation eye. Alongside this figure he adds a text which is interrupted:
The images of objects are of two natures of which the first...receives the true image of the real thing, the 2nd...retains the same but with confused boundaries of their shape and the first passes with parallel lines onto the surfaces of plane mirrors and the second passes through the apertures of thin walls in a dark place where it enters but...
Here the transition from physics of light and shade in a camera obscura to problems of vision and perception remains implicit.
Figs. 951-952: Gradations in a camera obscura and an eye on D10v.
13. Camera Obscuras and the Eye
On D10v (1508) this analogy is taken one step further. Here towards the centre of the right-hand column he draws a camera obscura with various gradations of shade (fig. 951). Above this he writes: "first." Above this, in turn he draws an eye in which various rays are being inverted at the pupil (fig. 952). This figure is headed: "second." Between these two figures he adds a brief marginal note:
The boundaries of bodies are little known because such boundaries are made in surfaces reduced to lines which being indivisible are imperceptible.
Lower down the same right-hand column this perceptual problem is pursued:
But the extremities of things drawn (because they are joined to the background where they are drawn, where they figure) are not subjected to this lack, and for this reason paintings that are close to the eye have to be painted with boundaries which are less known than the boundaries of these things that are distant and this you will recognize perceptibly in judging the upper boundary of an object near the eye and then removed from it.
Here the bridge between Leonardo's physics of light and shade and his physiology of vision is manifest. Indeed it is clear how his camera obscura studies which make him aware of differing gradations of light and shade influence both his theories of perception and painting. Leonardo returns to these themes briefly on CA195va (c. 1510) which, as has been noted, is another of those folios on which the camera obscura-eye analogy is explicit (see above pp. ). In the lower left-hand portion of this sheet is a rough sketch (fig. 953) of a camera obscura with five gradations. In the lower centre is a slightly more developed version (fig. 954) with seven gradations and near the bottom is an example with nine gradations (fig. 955). Each of these three possibilities is duly recorded in a brief note: "Make five or 9...or 7 spaces in ir in order that the white no stands in the middle." Beneath this is a further passage which partly explains the bottom diagram (fig. 955):
adsees...rm and the extremity of the light a sees r and it illuminates it little because in the extremities of the light there is little light but in n is seen all the light ad, simple light and yet it is enough light...m...sees ad, light and dc, shadow, begins...will corrupt.../the/ light....
Even if this text is interrupted, the accompanying diagrams remain of considerable interest because they reveal that Leonardo is trying to quantify gradations of shade. He wants, as far as possible, to measure what had previously been a purely subjective problem and thereby he brings the field of optics one step closer to its modern position as a branch of mathematical physics.
Although it is generally known that Leonardo worked with the camera obscura and compared the inversion of images in this instrument with those of the eye, scholars often refer to these facts as if they were only mentioned in passing in the notebooks. Our comprehensive study of the topic has shown that Leonardo devoted no less than 270 diagrams to the theme of camera obscuras and that these interests grow in part out of the astronomical tradition.
He uses the camera obscura to demonstrate not only the inversion of images but also that images passing through an aperture do not interfere with one another, that images are all in all and all in every part, that pinhole apertures produce different intensities of light and shade and that inverted images demonstrate a contrary motion.
Mediaeval optical writers had given considerable attention to the images of round light sources passing through triangular and other complex apertures. Leonardo studies the problem systematically in the case of triangular, square, octangular, slit-shaped and cross-shaped apertures. He demonstrates that whether the shape of the projection resembles the aperture or light source depends on the relative distance of these factors. He does not attempt to arrive at a formula for these relationships but he does give some quantitative references to his experiments.
In addition he studies situations with 1, 2, 3, 4, 8, 16, 24 and 32 pinhole apertures. He also studies the effects of light which passes through apertures of different sizes and encounters various interposed objects. Such experiences lead him to new studies of gradations of shade which prompt further analogies with problems of visual perception: why, for instance, the eye cannot perceive clearly the boundaries of nearby objects.
The great importance of these extensive studies of the camera obscura is that they bring various questions concerning the nature of light and shade and vision into the experimental domain of physics. Optics is no longer a problem for philosophical discussion: it is now a domain which requires scientific demonstration. In the section that follows we shall see how this mentality also leads Leonardo to make physical models of the eye. If the answers he finds are not always correct, the new kinds of answers he seeks are nonetheless important.
Last Update: July 2, 1999