SUMS

Dr. Kim H. Veltman

Optimal and Minimal Conditions of Vision


1. Introduction
2. Central Ray
3. Visual Fields
4. Occlusions
5. Objects too near
6. Objects too far
7. Objects too small
8. Diplopia
9. Excessive Light
10. After Images
11. Monocular and Binocular Vision
12. Displacement of Eyeball
13. Conclusions

 

1. Introduction

    Leonardo devotes considerable attention to optimal and minimal conditions of vision. He emphasizes the importance of the central ray, explores the range of the visual field and considers the problems with objects which are either too near the eye or too small. This leads him to study problems of diplopia. He is aware of the effects of excessive light. In addition he is interested in comparing monocular and binocular vision. Each of these will be considered in turn.

 

2. Central Ray

    Ptolemy, in his Optics,1 stated that objects along the central line of sight are seen more clearly. In Alhazen's Optics2 this idea was developed. Witelo adopted it.3 Alberti, in his On Painting goes further:

It remains to speak of the centric ray. The centric ray is that sole one which strikes the quantity directly and of which each angle is equal to the other. This /is/ a ray, among all the others the most active and strong, /and/ acts such that no quantity ever appears greater than when it hits it. One could say more things about this ray, but this one will suffice: closely surrounded by the other rays it is the last to abandon the object seen. Whence with merit it can be called prince of rays.4

    Leonardo is equally enthusiastic in his praise of this central ray. On A103v (BN 2038 23v, 1492) and again on D8v (1508) he refers it as the master" ray. In an early note on W19148v (fig. 1300, K/P 22v, 1489-1490) he claims that only those objects which are seen along the central line are seen clearly:

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Figs. 1295-1300: Demonstrations of the central ray of vision on CA270rb, CA353vb, A103v, CA120rb, CA85va and K/P 22v.

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Figs. 1301-1303: Further demonstrations of the eye and centric ray. Figs. 1301-1302, CA138vb; fig. 1303, CA345vb.

I say that the eye carries with it infinite lines which are applied to or united with those which part from the things seen and only the line in the middle of this sense organ is that which recognizes and judges the objects and colours, /while/ all the rest are false and deceptive.

    He returns to this theme on CA138vb (1490):

Of the eye

The eye is an...instrument of a spherical surface which is medium between the object and the senso commune and this surface takes in itself the images of all the things positioned opposite it and those (species) which are found in the middle of the others are comprehended with less error by the sense and those which are more distant are comprehended less.

It is clearly understood that the eye does not recognize objects unless the species come to it along a straight line and that thing will be less understood which departs more from that said line.

    By way of illustration he gives a concrete example (fig. 1302):

Hence if the depth of the eye f be the first degree of judgment /and/ the surface...n be the last, and that object which finds itself in line with fn will be judged more clearly by the sense, as /in the case of/ bn and likewise nc and na are badly recognized because the straightness of...of the lines of their species terminates at n, which is an obtuse angle and the lat degree of the judgment of the imprensiva, fn.

    On CA270rb (c. 1490) he again mentions the importance of the central ray in a discussion of the visual process (see above pp. ). On CA144vb (c. 1490) he expresses its importance in more general terms: "the centre is the most powerful and noble part of spherical bodies, because to this respond equally all the extremities of bodies." On A78 (1482) he notes its importance once again: "That object is seen better than the others which falls under a straight line on the middle of each eye under equal angles. On H2 33r (1494) he restates this idea: "No surface will show itself as perfect if the eye regarding this is not equally distant from its extremities." This principle is implicit also in diagrams on CA120rb (fig. 1298, 1497-1498) and CA37rb (figs. 1293-1294, 1497-1498). On CA85v (c. 1503-1504) he illustrates this anew (fig. 1299) now adding a fuller explanation:

And know that the pupil does not transmit anything perfectly to the intellect or senso comune except when the things given to it from the cornea (luce) are directed along the line ab as you see the line ca and although the lines m, n, f /and/ g are seen by the pupil they are not considered because they are not directed with the line ab.

    As evidence he cites the example of individual letters in a text which can only be discerned clearly when seen along this central line (see A108r, BN 2038 28r, 1492 above pp. ). He pursues this problem on W19152r (K/P 118r, 1508-1510) under the heading:

Of the central line of the eye.

There is only one line among the species which penetrate to the visual power which does not intersect itself and this line does not have a power that can be sensed because it is a mathematical line which has its origins from the mathematical point which has no middle.

    This he illustrates with a diagram (fig. 1300), beneath which he mentions the objection of an adversary:

According to the adversary necessity wishes that the central line of all the species which enter through thin and narrow apertures in a dark place be turned upside down together with all the other species of the bodies which enclose it.

    This claim he refutes (see above pp. ). On D8v (1508) he pursues his discussion of the central ray (fig. 1317, cf. figs. 1314-1316):

It is true then that object is less noted which impresses itself more distantly from the middle of the cornea (luce) where the front of the median line terminates, which always directs itself to all those objects of which it knows that it has in this way a true and certain indication of its shape and such a line is straight without any intersection and is the matter of the other lines whence this one is always moved in order to determine what the others /i.e. lines/ see and do not comprehend and this is the line kg positioned in the middle of so many lines as are those of the species that come to the eye point by point.

    A draft passage follows:

In what way the species of objects come to the eye.

The eye has in it a single line positioned in the middle of infinite other lines adherent to this which is called central.

    Here he breaks off and begins afresh:

(figure)

Figs. 1304-1305: Rough sketches of the eye or light and rays on CA144vb.

The eye...has in it a single line which is said /to be/ central and all the species of the objects that come to the eye along this line are perfectly seen if the (too) long distance does not impede it. Around this line are infinite /others/ adherent to it, which are of so much greater or lesser value to the extent that they are closer or /more/ remote...from this central /one/.

    These drafts on D8v are closely related to a more polished version on W19117r (K/P 115r, 1508-1510):

The eye has a single central line and all objects, which come to the eye along this line are well seen. Around this line are infinite other lines adhering to this central /one/ which (obey the former less to the extent that they are born more remote fromthis, less to the extent) are of lesser value to the extent that they are more remote from this central one.

    He considers the central ray as an example of a more general physical law as he notes on D1r:

Why nature did not make.../the/ power...in the visual power equal.

Nature did not make /the/ power equal in the visual power but gave to that power /(virtu) so much more power (potentia) as it is nearer to its centre (of such a power) and this it did in order not to break the laws of all the powers (potentie) which are the more so to the extent that they approach this centre and this is seen at the side of the percussion of some body and in the supports of the arms of a balance where the weight, in coming closer, diminishes its gravity. It is seen also/ in columns, walls and pilasters and it is seen in heat and all natural powers.

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Figs. 1306-1308: Whether the visual field is limited to 90o. Fig. 1306, CA222ra; figs. 1307-1308, CA204rb.

    For Leonardo the central ray is, therefore, another manifestation of his concept of the four powers (see above pp. ).

 

3. The Visual Field

    Euclid, in his Optics, did not attempt to measure quantitatively the limits of the visual field. According to Damianus it was Ptolemy who introduced a proof that the maximal visual cone was ninety degrees.5 In the latter Middle Ages this idea was modified slightly. Witelo, for example, claimed, that the maximal visual angle is "nearly a right angle."6 Pecham7 and Bacon8 make the same claim. Leonardo is aware of this traditional claim and attacks it through an appeal to experience on CA204rv (fig. 1307, c. 1490):

On the sight of the eye.

These our mathematicians wish that the ball of the eye be divided into four, as appears above in kief and that a quarter, that is, bef, be filled with the crystalline humour, which refers to the angle b, which appears in the triangle bcd.

And I say that experience says that if you lean /your/ back against the middle of the wall of a room and you look with fixed eyes towards the middle of the other /wall/, that along the 3 faces /of the walls/, that is, /to the/ right, left and directly in front, there will be no movement that is not seen, as appears in g...hcd and this is a sign that the crystalline humour stands as tsr.

    This visual field of 180o, shown in geometrical form on CA204rv (fig. 1307), he illustrates again on CA144vb (fig. 1310, 1490) and CA125ra (fig. 1311, 1490-1492) each time without text.

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Figs. 1309-1312: The visual field and refraction. Figs. 1309-1310, CA144vb; fig. 1311, CA125ra; fig. 1312, I46r; fig. 1313, CA42vab.

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Figs. 1314-1323: Central line and limits of the visual field. Fig. 1314, F34r; figs. 1315-1316, D1r; fig. 1317, D8v; figs. 1318-1321, K/P 115r; figs. 1322-1323, K/P 118r.

    He subsequently revises his claim. On I46r (fig. 1312, 1497-1499) he indicates a visual field of approximately 240o. This he redraws without explanation on W19117r (K/P 115r, figs. 1318-1321, 1508-1510) and again on W19152r (K/P 118r, figs. 1322-1323, 1508-1510) where he adds: "That which sees the cornea of the eye is seen by this cornea and that which the cornea sees is seen by this pupil. This idea he develops on D8v (fig. 1317, 1508):

It is shown why the eye sees objects in the lateral spaces behind itself

The eye...sees the motion of two lights...in contact with the wall wehre the middle of the neck of the observer leans.

The reason is that the cornea sees all those places where it is seen and all those objects, which see such a light impress its similitude. Since it is more eminent than any other part of the eye, the aperture or opening of the eye sees and is seen by all the parts of such a pupil and hence it can very well receive within itself that which the cornea of the eye shows.

    He draws related diagrams on F34r (fig. 1314, 1508) D8r (fig. 1177, 1508) and D3r (fig. 1174). On D1r (figs. 1315-1316), he again adds an explanation:

Why Nature made the pupil convex, that is, raised like part of a ball.

Nature made the surface of the cornea positioned in the eye of a convext shape in order that the surrounding objects can impress their similitudes with greater angles which could not occur if the eye were flat.

    In late sketches on CA385vc (1324-1328, 1513) he indicates a visual field of somewhat less than 180o.

(figure)

Fiugs. 1324-1328: Sketches concerning the visual field on CA385vc.

    Leonardo is also interested in how the eye perceives objects positioned off to the side. On E4r (fig. 1329, 1513-1514), for instance, he broaches this problem under the heading:

Case of perspective

Whether the wall with 4 sides and 4 /right/ angles will show itself to the eye with upper and lower boundaries /as/ rectilinear or curvilinear.

By the second of this, such parallel sides will show a hexagonal figure, that is, of six sides and six angles, even though, in reality, it has only 4 angles and 4 sides. And this is proved by means of the second which says: Among things of equal size the more remote will demonstrate itself that much smaller to the extent that it is more distant, /from which/ it follows that on the wall rtop, the length op is less than the length ab by the extent to which the line pf is longer than the line bf, which line pf exceeds the line bf by the space pm, which is a third of the space pf. Thus op will be that much less than ab, i.e. 1/3, and this proportion is inverse because the greater distance makes the thing seen smaller and the lesser distance increases /the apparent size of/ the thing seen.

    This leads to a conclusion (fig. 1330):

What is proved above leads one to conclude in the second figure how the two straight lines ro /and/ tp are divided into 4 straight lines rp, ts, rq and th and these are rectilinear and thus it is proved that the eye which stands in the centre /in front/ of the rectilinear wall with 4 straight sides and four right angles sees 6 straight sides and six angles of which 2 are obtuse and 4 are right.

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Figs. 1329-1331: Problems of lateral vision. Figs. 1329-1330, E4r; fig. 1331, G32r.

    He returns to this problem on G32v (1510-1515) in a passage entitled: "Of this bi-angular figure, the one obtuse angle will be greater than the other to the extent that the eye is closer to the one than the other." This he illustrates with a diagram (fig. 1331) and then explains:

Here follows what is missing in the margin at the foot of the facing folio /folio 31v missing/. Hence I say that the angle mentioned above is not the maximal of the obtuse angles because this angle increases itself by the extent to which the eye which sees it is further away from it and will diminish to the extent that the eye is close to it. Whence it is concluded that /if/ the eye is positioned centrally in front of some parallel, then this parallel will show itself as a bisangular figure with curved sides, but in truth this figure will have 4 angles of which two are in the centre namely, one below and one above, and the other two are on the other two opposite extremities as has been conceded by necessity on the facing page.

 

4. Occlusions

    On four folios Leonardo examines how small apertures and their occlusions affect the visual field and perception.9 On CA347va (c. 1490), for instance, he begins with a case of two eyes in front of a small hole (fig. 1332): "With straight lines, with an aperture smaller than the sides of the visual pyramid it is impossible to comprehend the object positioned beyond this aperture clearly with the eye."

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Figs. 1332-1336: Occlusion problems in binocular vision on CA347va

Even though the eyes a/and/ b have in front of them the aperture (t)rs opened with respect to n, nonetheless, n will not be comprehended because the master pyramid cannot lead to this n.

    He next considers a case (fig. 1333) where one eye is off to the side relative to the aperture:

Eye

The object seen through an aperture smaller than the base of the visual pyramid will be seen through a traverse line and the object on the right will go to the left eye and cannot be seen by the two eyes at the same time and if it be seen it is poorly comprehended.

The eye n, when it accords with the eye m to focus the angle of the visual pyramid on c, cannot pass through the aperture tr, whence it cannot comprehend this n, whence the image of this c is reflected and presents itself to the eye m through a transverse line.

    In the left-hand column of CA347va he examines a case (fig. 1334) where two eyes can see clearly in spite of occluding objects:

Function of the Eye

Even though an object interposes itself between the eyes and the object, /and/ even though it is not transparent, it will not remove the form of this object from the eye, provided that this body is less than the space which is found between one pupil and the other.

Let a be the object. Let p /and/ q be the two eyes. Let mn be the body interposed between the eyes and the object.

    Directly below he considers a case (fig. 1335) where an object smaller than the distance between the eyes can, nonetheless, occlude the visual pyramid:

On the eye

That body which is interposed between the object and the eyes, even though it is smaller than the space which there is between one pupil and the other, it will, nonetheless, occlude from the eye the object which finds itself in such a part of the visual pyramid which, with its extremities, exceeds the width of this pyramid.

Let rca be the visual pyramid. Let df be the place.

    Next he considers a case (fig. 1336) where an interposed body occludes vision from one eye:

Eye

If the object interposed between the eye and the object occludes one of the 2 sides of the visual pyramid the body seen will appear of less brightness to the extent that it is seen by a lesser quantity of light. Let thn be the /visual/ pyramid, let nh be the intersected side, let ab be the body which is interposed and which removes the sight of the luminous object positioned at n from the eye h. And if the two eyes see this light in an aspect of brightness, the eye sees it the half darker, because with the two eyes operating on the luminous object with the same function, these refer a luminous object to the sense equally. But if one eye carries to the one sense a dark object and to the other, a luminous object, then one mixes with the other and the shadows appear half less dark than before and the light half less bright than before, and you will see this by experience, interposing a finger between one of the eyes and the light.

    The experiment to which he alludes in the last sentence he later describes more carefully on W19042r (K/P 45r, c. 1508) in connection with his studies on light intensity (see below p. ).

(figure)

Figs. 1337-1343: Problems in binocular perception involving apertures. Figs. 1337-1341, CA347ra; figs. 1342-1343, CA112ra.

 

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Figs. 1344-1345: Occlusion problems in binocular vision on D9r.

    In the right-hand column on CA347va he also drafts two further diagrams: one showing how binocular vision remains unaffected when looking through a larger aperture (fig. 1337); a second demonstrating how a small aperture interferes with binocular vision (fig. 1338). Accompanying this latter diagram is a brief text:

The inverted pyramids of uncertain vision do not carry to the eye the true image of some figure and they are called inverted because they make their base along the width of the object.

    On CA347ra he pursues this theme, beginning with a case where a large aperture does not interfere with binocular vision (fig. 1339):

On the Eye

When two eyes focus the visual pyramid on an object, this object is seen by the eyes and is well understood.

Let the pyramids which are focussed on the object be rsn; let n be the object. The place which does not interrupt it is the aperture tu.

    Directly following he describes a situation (fig. 1340) where the aperture is smaller and occludes vision:

On the Eye

It is possible that the eyes, having an object in front of them and directing themselves to this with all their power, that this object is not seen by the eyes. Even though it varies from the background in colour and /though/ it be an evident body, the way that is free and open appears closed and the part /that is/ closed, appears open.

    This he illustrates with a concrete example (fig. 1340, cf. fig. 1338):

Let s and e be the eyes. Let r be the object positioned in front of the eye. Let tu be the way or aperture where the eye wishes to understand r. I say that the right side of the aperture pt is carried or it appears in the left part at rc and uq on the left is transported to the right side at mr and the entire part mc appears the colour of the wall where the aperture is and although it be distant from this wall, it nonetheless appears continuous at the same distance. The aperture appears in 2 places of the object: tg goes from the right to the left and it appears that the aperture co is judged by the eye emqf and likewise from the eye /at/ s, it appears that the wall is interrupted at hp and that, from there, one sees the object am

    He also draws a third diagram (fig. 1341) which he does not explain. At least fifteen years pass before he returns to the problem in terms of two further diagrams (figs. 1342-1343) without text on CA112ra (c. 1505-1508). On D9r (1508) he pursues this theme, now exploring in detail how a small aperture affects the visual field (fig. 1344):

What part of the background the eyes see looking through a /small/ aperture.

Let there be two eyes which look at the background ac through the aperture dc. I say that these two eyes will not see other than the space b of such a background and that the rest of the space ab /on/ the right will be seen by the left eye g and the remainder of the other space bc /on/ on the left will only be seen by the right eye f.

    He then considers a situation where the aperture is smaller (fig. 1345):

Where the two eyes will not see a background entirely freely through a given aperture

The right eye a sees the background eg and it sees all the rest ef occluded by the wall sh. And the eye b only sees the background freely and it sees the remainder dg occluded by the wall it and the triangular space cd is neither seen by one eye nor the other because the eye a occludes this boundary with the rim of the aperture h and the eye b covers the boundary d with the rim of the aperture i. Hence it is concluded that even though the two eyes, although each on its own sees a part of the background, the other eye tinges it /i.e. throws it into shadow/ carrying it onto one of the sides of the interposed plane.

    This leads to an interjection concerning light intensity:

And for this /reason/ it can be concluded that with a single eye the object appears less bright than with two. Because if one eye is closed it sees darkness and if the other is open it sees light which light is mixed with darkness in the visual power and it does not allow simple light nor simple darkness to appear but it only comprehends a mixture composed of darkness and light.

    Immediately following he considers the deceptions of sight arising from this situation:

And for this /reason/ one understands that the right eye, even if it sees the left object fromt he left side it appears to see it with the left eye and the sense does not know that it is deceived and similarly with the left eye which sees a background on the right side, it appears to the sense /which is/ again deceived, that it has seen such a background with the right eye etc...

    In the next paragraph that follows he relates this deception to that of a style which appears to move in a contrary direction (see above pp. ). In Leonardo's approach one problem continually has cross-references to other problems.

 

5. Objects too Near

    Aristotle had noted that objects touching the eye could not be seen.10 Alhazen11 and Witelo12 repeated this claim. Biagio Pelacani da Parma adds that objects too near the eye are seen unclearly whereas those at a moderate distance are seen distinctly.13 Leonardo's interest in minimal conditions of vision leads him to consider this problem in greater detail. Among his earliest remarks on the problem in a passage on W19148v (K/P 22v, 1487-1490):

and if you place 2 objects at half an arm's length's distance from one another and such that the first is close to the eye, the surface of the first will remain much more confused than the second. The reason is that the first is overcome by a greater number of false lines than the second and so it is dubious.

    On CA138vb (c. 1490) he estimates the distance for accurate vision:

If the eye has to see an object, which is too close, it cannot judge it well as happens to one who wants to see the point of the nose. Whence, as a general rule, Nature teaches that the object will never be seen perfectly if the interval, which is found between the eye and the object seen is not at least as large as the size of the face.

 

(figure)

Figs. 1346-1349: Experiments on D6v concerning visual power, images everywhere in the eye, and why nearby objects are not seen.

    He offers a slightly different estimate of this minimum distance on CA250rb (c. 1490):

Every body which is larger than the distance between the pupils, which is to be judged by the eye will be at a distance 4 times its size.

And if it is less than the distance between the pupils, no closer object will be...comprehended by the eye, if /a distance/ 4 times the distance of the interval of the pupils is not interposed between the eye and the object seen by it.

    Why this should be so begins to concern him as is evident from a draft passage on CA298va (fig. 1402, c. 1490):

Why this happens more from nearby than from afar

(The boundaries) The true extremities of the object positioned (opposta) between the eye and the object are never entirely terminated or understood by the pupil in the object.

    A preliminary explanation follows on CA144vb (c. 1490):

Why the object, the closer it comes to the eye, the less it is understood...It does not see from nearby because the authors of the transverse lines, going to the common concourse of the pyramid of vision are two oblique.

    Intimately connected with this problem why nearby objects cannot be seen clearly, are a series of experiments which are also intended to demonstrate that images are spread throughout the pupil and have already been analysed in this connection (see above pp. ). One set of these experiments on D6v (figs. 1346-1349, 1508) involves sieves made of horse hairs. Another series involves pinhole images and styles which are moved back and forth (figs. 1350-1362). A variant of these experiments occurs on F31r (figs. 1363-1364, 1508):

(figure)

(figure)

Figs. 1350-1362: Experiments concerning images "all in all the eye" and why nearby objects are not seen clearly. Figs. 1350-1351, D2v; fig. 1352, CA112ra; fig. 1353, CA222vc; figs. 1354-1356, D9r; figs. 1357-1358, D4r; fig. 1359, D4v; fig. 1360, K126/46/r, fig. 1361, K127/47/r; fig. 1362, K127/47/v.

terminate. I say that such an eye cannot see the boundaries of such an object in this background clearly and distinctly because, as was proposed, the visual power is spread throughout the entire pupil of the eye. Hence the part, fe, (the part) of this pupil sees the upper part of the background occupied in gh; oe sees hi of the object occupied (and d); d sees the top of the object in the background i; c sees it in k; b in n and a in m and m downwards nothing is seen of the

    Here the folio ends and the text continues on F30v:

background. Whence it follows that the part g is has little shade (p) because the summit of the object p occludes the view of such a site from the part f of the eye. But it does not follows, however, that the entire remainder of such a pupil does not see this site g, but in h it iwll appear more obscure because its smaller part of the pupil sees it than at g, and lwss of the pupil sees i and less /of the pupil sees/ k and the entire height of such a pupil goes on consuming itself successively in this sight, through which the object p darkens its background at each degree to such an extent that in the end the entire colour of such an object remains wholly obscured.

    He returns to these themes on D10v (1508) in a passage entitled:

How the eye does not recognize the boundaries of any body.

The eye will never be capable /of seeing/ the true boundary that the shapes of some body have bordering against a remote place. This is to be proved and /so/ let the pupil of the eye be ab (and the extremity of the object) and let cp be the body positioned opposite the eye, of which the superior extremity c.

The pupil of the eye adopts the visual power in every part of its size but that much less to the extent that the part which functions is of smaller quantity.

This is proved by the 5th which defines the background for every object interposed between the eye and this background. Let rs be the background where the eye atd sees the object pq terminate. I say that such an eye cannot see the boundaries of such an object in this background clearly and distinctly because, as was proposed, the visual power is spread throughout the entire pupil of the eye. Hence the part, fe, (the part) of this pupil sees the upper part of the background occupied in gh; oe sees hi of the object occupied...; d sees the top of the object in the background i; c sees it in k; b in n and a in m and from m downwards nothing is seen of the

    Here the folio ends and the text continues on F30v (fig. 1364):

background. Whence it follows that the part g is has little shade...because the summit of the object p occludes the view of such a site from the part f of the eye. But it does not follow, however, that the entire remainder of such a pupil does not see this site g, but in h it will appear more obscure because a smaller part of the pupil sees it than at g, and less of the pupil sees i and less /of the pupil sees/ k and the entire height of such a pupil goes on consuming itself successively in this sight, through which the object p darkens its background at each degree to such an extent that in the end the entire colour of such an object remains wholly obscured.

    He returns to these themes on D10v (1508):

How the eye does not recognize the boundaries of any body.

(figure)

Figs. 1363-1366: Demonstrations why nearby objects are not seen clearly. Figs. 1363-1364, F31r; fig. 1365, D10v; fig. 1366, E15r.

is noted

extremity is to be seen by this eye. I say /with/ the extremity of such a background it will not be noted in what part of the background it is terminated and this is proven with the aid of the 33rd of this /cf. D4r cited above p. /, which states that the visual power the painters of perspective would wish but is all in all the pupil where the species of the within the eye in a greater space than is the pupil, but the images are more noted to the extrent that they are closer to the centre of the power positioned in this space and less so the more remote that they are from this centre.

    Immediately following he offers a concrete example

Hence if the power ab strikes the extremity of the object c, the central line of the visual power, r, sees c in part f of the background and the superior extremity of this power, that is s, sees c in the background /at/ h and the lower part of the power sees c in the background /at/ d and thus it goes disseminating throughout the entire background dh and for this reason such an extremity is not recognized by the eye because the sense of the visual power is infused throughout all this power which sends to the judgment (existimatium) a confused boundary of this extremity and the more or less so as it is closer or remoter from this and the more or less so as it is more remote or closer to the eye.

    He ***???

The eye will never be capable /of seeing/ the true boundary which the shapes of some body have, bordering against a remote place. This is to be proved and /so/ let the pupil of the eye be ab (and the extremity of the object) and let cp be the body positioned opposite the eye, of which the superior extremity c is noted...and let nm be the background in which this extremity is to be seen by this eye. I say...that /with/ the extremity of such a background it will not be noted in what part of the background it is terminated and this is proven with the aid of the 33rd of this /cf. d4v cited above p. /, which states that the visual power...is not in a point as the painters of perspective would wish but is all in all the pupil where the species of the...objects penetrate within the eye in a greater space than is the pupil, but the images are more noted to the extent that they are closer to the centre of the power positioned in this space and less so the more remote that they are from this centre.

    Immediately following he offers a concrete example (fig. 1365):

Hence if the power ab strikes the extremity of the object c, the central line of the visual power, r, sees c in part f of the background and the superior extremity of this power, that is s, sees c in the background /at/ h and the lower part of the power sees c in the background /at/ d, and thus it goes disseminating throughout the entire background dh and for this reason such an extremity is not recognized by the eye...because the sense of the visual power is infused throughout all this power which sends to the judgment (existimatium) a confused boundary of this extremity and the more or less so as it is closer or remoter from this...central line of this power and the more or less so as it is more remote or closer to the eye.

    He adds a marginal note concerning the boundaries of paintings:

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Figs. 1367-1369: Visual power and the perception of boundaries. Fig. 1367, CA298va; figs. 1368-1369, BM188r.

There follows what was lacking below.

But the boundaries of things drawn...are not subject to this lack and for this reason the paintings which are close to the eye have to be drawn with boundaries less known than the boundaries of remote objects and for this reason you will recognize sensibly with your judgment the upper boundary of an object brought near the eye and /then/ further away.

    A passage in Melzi's hand on BM Arundel 188r (figs. 1368-1369, cf. 1367, c. 1510) summarizes Leonardo's findings concerning the unclear boundaries of nearby objects:

The boundaries of that object which is closer to the eye will be less noted. It follows that more remote boundaries will be better noted. Among objects smaller than the pupil of the eyes that will be less noted which is closer to that pupil.

    On E15r (1513-1514) Leonardo takes up this theme once more (fig. 1366):

The boundaries of that body placed in front of the pupil of the eye will show themselves less clearly the closer they are to this pupil.

This is shown by the extremity of the body n placed in front of the eye d, which pupil, in seeing this boundary, also sees the entire space ae which is beyond this boundary and the species which come from this space are mixed with the species of these boundaries and hence the one species confounds the other and such a confusion deprives the pupil of the information (notitia) of such a boundary.

    On E15v, cited earlier (see above p. ) the connection is again evident between his demonstrations that images are all in all the pupil and his claim that nearby objects are not seen clearly.

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Figs. 1370-1372: Demonstrations why the eye does not see boundaries clearly on CU805, 806, and 808.

    Related to the foregoing series of demonstrations involving an interposed stick are three further examples in the Treatise of Painting involving an interposed sphere positioned close to the eye. One of these on CU805 (TPL742, 1508-1510) simply makes the point that different parts of the pupil see opaque objects differently (fig. 1370):

How the boundaries of umbrous bodies seen by a same pupil are not in a same site in this body.

The boundaries of opaque bodies, seen by a same pupil are never seen in a same site in this body.

This is proved and let it be that the pupil ab sees the upper part of an opaque body n. I say that the lower part b of such a pupil will see the boundary of this body at the point d terminated along the wall or at the point ue. And the upper part a of the pupil will see this opaque body at the point c terminated at f in this wall. Hence since c and d are not in a same site of such an opaque body we have proved our intent.

    On CU806 (TPL741, 1508-1510) he relates this demonstration both to the claim that nearby boundaries are unclear and to the principle that the visual power is everywhere in the eye (fig. 1371):

The true boundaries of opaque bodies iwll never be seen with clear definition. And this occurs because the visual power is not caused in a point as is proved in the 3rd of the 5th of perspective, where it is said: the visual power is infused in all the pupil of the eye.

Hence the pupil being abc, which sees the boundary of the body n at the extremity m occupying the entire space def on the wall, because the upper part, a, of the pupil, sees the boundary m of the object at the point dd and the middle of the pupil b sees another boundary /from/ lower down at the point e, which is higher than d and the lower part of the pupil, c sees another boundary /from/ lower down, which is carried higher on the said wall. And thus is proved the cause of the confusion of boundaries which opaque bodies have.

    In a third example on CU808 (TPL743, 1508-1510) he again uses the same basic demonstration, this time to emphasize the confused boundaries of nearby obejcts (fig. 1372):

How that body has its boundaries more confused which is closer to the eye that sees it.

That which is proposed is proved through showing that the pupil ab sees the boundaries of the body e at c /and/ ud, very much apart from one another and for this reason they remain confused. And it sees the boundaries of the body f, which is more remote, as being closer, that is, /at/ n /and/ o and consequently it comes to see them more closely than those of the body e.

 

6. Objects Too Far

    In the case of objects too far from the eye, Leonardo explores further problems of perception. Concerning objects in shade, for instance, there is draft, possibly in another hand on BM Arundel 101r (1490-1495): "Many umbrous bodies very close to one another, being seen in the luminous air at a long distance appear separated by a long interval."This idea he restates clearly on C144 (1490-1491): "If many umbrous bodies very close to one another are seen against a luminous backgrouind, at a long distance they appear separated by a large interval."

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Figs. 1373-1376: Perception of light sources at a distance on C6r, F94v, 36r, and 35v.

    The converse, that luminous bodies tend to merge at a large distance, is expressed in a draft, again possibly in another hand, on BM Arundel 101r

    Many luminous bodies very close to one another, the distant lights appear to the eye to be united and attached together.

    He restates this on C14v (1490): "If many luminous bodies are seen from afar, even though they are separated from one another they will appear united and conjoined together." Elsewhere in the same manuscript on C6r (fig. 1373, 1490) he describes a related phenomenon, how a luminous object seen from afar appears larger:

If the eye looks at the light of a candle 400 braccia distant, this light will necessarily appear to its observer to be many times its true quantity but if you place a stick in front of this /candle/ which is somewhat larger than this large light, this stick occupies this light which appears 2 braccia large. Hence this error comes from the eye which takes the luminous species not only through the point of the light but equally through all the light and for this /reason/ it appears considerably larger in the other eye.

    On A64v (1492) he integrates his description of both of these phenomena in a:

Proof how luminous bodies appear larger than they are from a distance.

If you place 2 lighted candles, the one half a braccio from the other, and you go back 200 braccia away from these, you will see through the increase that together they /appear to/ form a single luminous body of two lights and they will appear to be a single light a braccio large.

    On Forst. III 35v (1493) he cites the case of a sieve which appears without apertures at a distance. As will be seen below this case is equally relevant with respect to objects which are too small. This paradox that light sources appear to increase in size with distance has a special bearing on his astronomical interests. He suspects that the apparent size of stars might be affected by optical illusions, and therefore explores the problem in detail in the Manuscript F (see below pp. ).

 

7. Objects Too Small

    The minimal size of objects which can be seen clearly was a well established problem in the optical tradition. It had been broached by Galen14 and discussed by Alhazen15, Witelo16, Biagio Pelacani da Parma17 and the anonymous author of Della prospettiva.18 Perhaps the earliest of Leonardo's extant notes on this problem occurs on C27r (c. 1490-1491) beginning with a general statement:

If the eye looks at the object less than its pupil (luce) this object will not occlude any object that is beyond it in the eye. But if the lids of the eye are closed in such a way that the aperture of these lids is less than this first object, /then,/ you will see this one occupy the second object to the extent that is convenient.

    To support this he cites the example of a sieve:

And if you wish to see this proof clearly look at an object behind a seive with the eye completely open and with the eye nearly closed...the iron strands of the seive will appear to grow and occupy the object and if the eye is well opened as usual, the iron strands of the seive will not occlude anything of the object.

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Figs. 1377-1380: Perception of objects smaller than the eye. Figs. 1377-1378, C19v; figs. 1379-1380, F28v.

    He cites this example again on Forst III 35v (c. 1493): "A seive through which the luminous air penetrates, at a long distance will appear without apertures and entirely luminous." On C19v (1490-1491) he studies the nature of this occlusion or non-occlusion of nearby objects in detail (fig. 1377 cf. fig. 1379):

The object positioned in front of the eye which is smaller than its pupil will occupy in transparent occlusion as much of its background as the size of the base made by the pyramid which is caused behind the intersection...found between the eye and the object.

Let the transparent occlusion made by the object against its background be between f and p. Let the pyramid made by the base fp /have its apex/ at r, produced by the line bp and the line cf at the intersection which is found between the eye and the object.

    Immediately following he adds a further note:

The transparent occlusion made by the object less than this pupil of this eye against its background behind this object will be of diverse qualities of obscurity.

The body de will occlude the entire part of the background no from the half uab /of the/ pupil.

    As for objects larger than this minimal size, he points out, on C10r (1490-1491):

All umbrous objects of a size larger than the pupil which interpose themselves between the eye and the luminous body iwll show themselves of a dark quality.

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Figs. 1381-1388: Concerning the perception of objects smaller than the eye. Figs. 1381-1387, CA298va; fig. 1388, CA290rb.

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Figs. 1389: Experiment concerning objects smaller than the eye on CA290rb.

    This problem of the occlusion of nearby objects becomes intimately connected with Leonardo's theories concerning the visual process. His reasoning is as follows: if vision occured through images converging to a point then a small object near the eye would occlude everything behind it. If small objects do not produce occlusion then images must be inverted at the pupil and the visual power must be "all in all and all in every part." As early as 1490 he alludes to this connection between the perception of small objects and the nature of the visual process in a draft passage on CA298va (fig. ):

Why the object, the closer it approaches the eye, the less it is recognized and why spectacles and why the eye does not see well from nearby or from afar.

It does not see /well/ from nearby because the authors of the lines traversed, going to the common concourse of the pyramids of sight, are too oblique.

    On CA290rb (c. 1490) he pursues this problem with two detailed diagrams (figs. ) one without text, the other with the accompanying passage:

If the eye were the circle hk and the pupil were fr, it would be necessary that the bodies 1, 2, 4, /and/ 5 would not be seen, nor any which makes a small pyramid behind it. And at a certain distance the eye diminishes objects without confusion, to the extent that they are further removed. And of the object in front of this /certain distance/ to the extent that the object is closer to the eye the more it diminishes and this happens when the object seen is less than the pupil of the eye. But a similar effect can never occur when the object seen is equal to the pupils except when it is to the side of the eye as shown above in /the case of/ the object marked 3.

    This problem of the occlusion of nearby objects becomes intimately connected with Leonardo's theories concerning the visual process. His reasoning is as follows: if vision occured through images converging to a point then a small object near the eye would occlude everything behind it. If small objects do not produce occlusion then images must be inverted at the pupil and the visual power must be "all in all and all in every part." As early as 1490 he alludes to this connection between the perception of small objects and the nature of the visual process in a draft passage on CA144vb (fig. 1305):

Why the object, the closer it approaches the eye, the less it is recognized and why spectacles and why the eye does not see well from nearby or from afar.

It does not see /well/ from nearby because the authors of the lines traversed, going to the common concourse of the pyramids of sight, are too oblique.

    He explores the problem of objects smaller than the eye in a series of sketches (figs. 1381-1387) and a text on CA298va (1490):

That is, that the pupil will see beyond the object placed in front of it which is smaller than itself....

The object interposed between the eye and the object which is of lesser quantity than the pupil is the reason that the species come together upside down in this pupil.

 

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Figs. 1390-1398: Central ray and the perception of objects smaller than the eye. Figs. 1390-1391, C27r; fig. 1392, A10r; fig. 1393, A92v; fig. 1394, A10r; fig. 1395, BM112r; fig. 1396, A103v; figs. 1397-1398, K/P 118v.

Why the object near the eye leaves its boiundaries indiscernible.

/In the case of/ of those objects opposite the eye, which are too close to this /eye/ it will happen that their boundaries are /too/ confused to discern, as occurs with objects which are close to the light which make a large and confused shadow and it does this because in the judgment of outside objects linear perspective is, in all cases, identical to light.

    To complete the explanation he appeals to his concept of the central ray (see above p. ):

And the reason is that the eye makes a master line which at a distance emcompasses and embraces with true cognition large objects from afar as little objects from nearby. But since the eye sends multitudes of lines which surround the principal one in the middle which, finding themselves mroe distant from the centre in this circulation, they are less powerful in recognizing the true /outlines/. Whence it happens that the object placed near the eye, not being at that distance so close to the master line capable of comprehending the boundaries of that object, it befits these boundaries to fall into lines of weak comprehension which are to the function of the eye as the hounds to the chase which flush the prey and cannot catch it. Likewise these cannot catch /the image/, but they are the cause that the master line turns itself to the things that have been flushed by these lines.

To the extent that an object smaller than the pupil interposes itself between the eye and the object at a distance greater than /the size of/ this eye, to that extent is it necessary to increase the distance of the small object if it is not to be seen.

    On CA290rb (1490) he pursues this problems with two detailed diagrams (figs. 1388-1389) one without, one with text:

If the eye were the circle hk and the pupil were fr, it would be necessary that the bodies 1, 2, 4, /and/ 5 would not be seen, nor any which makes a small pyramid behind it. And at a certain distance the eye diminishes objects without confusion, to the extent that they are further removed. And of the object in front of this /certain distance/ to the extent that the object is closer to the eye the more it diminishes and this happens when the object seen is less than the pupil of the eye. But a similar effect can never occur when the object seen is equal to the pupils except when it is to the side of the eye as shown above in /the case of/ the object marked 3.

    On A103v (fig. 1398, BN 2038 23v, 1492) he provides another explanation:

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Figs. 1399: Demonstration on A77 whether one sees an object smaller than the eye.

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Figs. 1400-1403: Occlusion of objects smaller than the eye. Figs. 1400-1401, Forst. III 36r; figs. 1402-1403, CA298va.

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Figs. 1404-1409: Concerning objects smaller than the eye. Figs. 1404-14-5, CA237ra; figs. 1406-1407, CA112ra; figs. 1408-1409, CA250va.

    On A77 (figs. 1399, 1492) he analyses this problem in connection with his demonstrations that vision does not occur at a point. Here a connection between his notion that images are "all in all and all in every part of the pupil" remains implicit. In subsequent passages on BM112r and CA250va (see above pp. ) he develops these connections between (a) his theory that images do not terminate at a point in the eye and b) problems of perception of objects smaller than the eye. By 1500 on CA237ra he draws two diagrams of the eye looking at objects smaller than the pupil (figs. 1404-1405) and merely adds alongside:

The species of the object smaller than the eye do not converge pyramidally in this eye.

The visual power is infused with equal power in all the pupil of the eye, whence the visual operation is all in all in every part of this.

    Here he takes for granted the connection between his theory of vision in terms of images being "all in all..." the eye and the perception of objects smaller than the eye. On CA112ra (1506-1508) he alludes to this connection in two diagrams without text (figs. ). By 1508 the connection is explicit. Hence his experiments involving horse-hairs (figs. 1346-1349) and styles (figs. 1350-1362) close to the eye are used a) to refute the print theory of vision (see above pp. ) and b) to explain why the eye cannot see nearby objects clearly. In Leonardo's associative mind there is also a connection between the quantity seen of nearby objects (see above pp. ) and the quality with which they are seen due to occlusion. This connection is implicit in a passage on Forst III 36r (figs. 1400-1401, cf. 1402-1403, c. 1493):

Which object is better seen.

Among the walls of equal distance and quality which are seen beyond the extremities of an opaque body positioned opposite, that part of this body will appear more illuminated which is seen by a greater sum of a pupil.

    On K125/45/r (figs. 1412-1413, after 1504) this connection becomes more explicit:

The object less than the pupil placed in front of the object will not occlude any remote object in this pupil.

No spherical object less than the pupil will ever be seen by a single pupil which is not seen more than half and this is at any distance it may happen to be.

And that much more than half to the extent that it is closer and that much less to the extent that it is far from the eye that sees it.

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Figs. 1410-1413: Demonstrations concerning objects smaller than the eye. Figs. 1410-1411, K124/44/v; figs. 1412-1413, K125/45/r.

    On K124/44v/ (figs. 1410-1411) he pursues the question of the quantity of objects seen (see above pp. ). In 1508 he returns to the theme of occlusion on F28v (figs. 1379-1380, cf. figs. 1377-1378):

If the object interposed between the background and the eye is smaller than the pupil of this eye no part of this background will be occluded by such an object.

Let ep be the pupil (luce) of the eye. Let q be the image interposed between the background and the eye. I say that such an object will not occlude any part of the background.

    Which demonstration leads to a further attack on the point theory of vision (fig. 1377; see above pp. ):

I say that if the object sent its image to the point a that no part of this background ds could be seen by such an eye because here the object less than the pupil occludes the whole background at the point a.

    On D9r (fig. 1356, 1508) he again considers the transparency of objects smaller than the pupil:

Why when the point of the style is placed in front of the eye it makes a larger shadow on the object.

The point of the style placed crosswise in front of the pupil of the eye (- of which the diameter of its size is considerably smaller than the diameter of this pupil) will occlude that much more or less space in other objects to the extent that this is more or less space in other objects to the extent that this is more or less clsoe to the eye which occlusion will obscure and not prohibit the transit of the species of the aforesaid objects (to the eye).

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Fig. 1414: Perceptionof real objects and perspectival images on A65.

    Leonardo also relates these problems of occlusion in the case of objects smaller than the eye to both his study of linear perspective and the camera obscura, as is evidenced by an early passage on A65 (fig. 1414, 1492):

The more that the eye approaches the image of the object placed opposite it, the more /the image/ diminishes. And the more /the eye/ approaches the true and proper object, the larger /the object/ appears. That is, at the line mn, the line cb appears the size of ad. And if you bring the line cb towards the eye as far as mn it will appear the size of mn. And where cb occludes the line st, the line mn occludes ru.

    On H71/23/r (1493) he again relates this problem of nearby objects to his camera obscura studies (see above p. ) when he notes that (fig. 790): "the eye does not comprehend the nearby luminous angle."

 

Objects smaller than the distance between two eyes

    Euclid, in his Optics had broached the question of how much is seen of objects smaller than the distance between the eyes. Leonardo also examines such questions (see above pp. ) but in addition explores perceptual problems not considered by Euclid. On W12351r (c. 1493-149 ), for instance, he discovers that objects smaller than the distance between the two eyes contradict a basic law of linear perspective wehreby projected size varies inversely with distance (fig. 1415):

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Figs. 1415-1417: Paradox how nearby objects appear smaller on W12351r, CA120vd and CU804.

If you look at the rectangular figure lmik on the interposed plane ab, you will find that the more distant part ik will appear larger on this plane at cd, that the closer part lm at ef, looking with the eyes g /and/ h.

    On CA120vd (c. 1504) he draws a similar situation (fig. 1416) except that the rectangular figure is seen from the side. No text accompanies this draft. On CU804 (TPL821, 1505-1510) he explains the phenomenon:

Of Perspective

When, with two eyes one sees two equal objects, each of which, per se, is less than the interval between the pupils of these eyes then the second object will appear greater than the first.

    This he illustrates with an example (fig. 1417):

The pyramid ac embraces the first object and the pyramid bd embraces the second object. Now m will appear larger than n to the extent that the width of the pyramid bd is greater than ac.

    Meanwhile he had also been studying what effects of occlusion occur with sucj objects smaller than the distance between the two eyes. On C23r (1490-1491), for instance, he notes:

The object which is opposite the eye which is, in itself, less than the interval which is found between the one and the other pupil of these eyes cannot occlude as much of a wall on which it borders, as is its proper size when the eyes are looking at these backgrounds.

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Figs. 1418-1420: Demonstrations concerning objects smaller than the distance between the eyes. Fig. 1418, C27r; figs. 1419-1420, C23r.

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Figs. 1421-1423: Demonstrations concerning objects smaller than the distance between the eyes. Figs. 1421-1422, CA175ra; fig. 1423, I43r.

    This he illustrates (fig. 1419, cf. fig. 1418) and then describes:

Let r /and/ s be the eyes that look at the above mentioned object.

Let pt be the object /which is/ being looked at. Let mk be the wall, or if you wish the plane where the extremities...of the object makes known the shape of the object. The left eye looking at this plane sees and recognizes the part of the wall na and it finds the part ac occluded by the interposed object pt. The right eye sees that part of the wall...ac which the left eye cannot see, although it cannot see everything and cannot see the part bc.

    In this example only the space ab is occluded from both eyes. Directly beneath this he draws a second example (fig. 1420) in which the object is smaller still and consequently occludes none of the background. No text accompanies this diagram. Some two years later he returns to this problem on BM115r (fig. c. 1492):

If you place an opaque body in front of the eyes at a space of 4 fingers and if this be less than is the distance from one to the other pupil, it iwll not occlude the sight of any object which is beyond this.

    He then restates the principle in more general terms: "No object situated behind an object seen by the eye can be occluded by this /first/ object if it is less than the space that stands between the pupils." Leonardo qualifies this general claim in a note on CA347va (earlier 1490-1495):

That object which interposes itself between the object and the eyes, even if it be less than the space which stands between (the) one pupil and the other will, nonetheless, occlude the object at the eye, if it finds itself in that part of the visual pyramid which, with its extremities, passes through the width of this pyramid.

    This he again illustrates with an example (fig. 1335):

Let rc be the visual pyramid. Let df be the place where the above mentioned pyramid intersects the interposed body. Let df be the body larger than the intersection of the pyramid.

    Further consideration of the problem leads him to claim, on CA250rb (c. 1490) that if objects smaller than the interval between the two eyes are to be seen clearly, a minimum distance four times that between the two eyes is required (fig. , see above p. ). On CA175ra (c. 1493-1494) he pursues this theme under the heading (figs. 1421-1422, 1424-1428, cf. figs. 1418-1420):

    It is impossible that the opaque body appears of that perfect rotundity as the plane circle.

No opaque body of spherical shape which is seen by the 2 eyes will appear to these of perfect rotundity.

You will see this experience by taking a sphere the diameter of which is less than the interval...between one eye and the other. And you stand in a place which is only illuminated by a single window and turning the face to the air, interposing this sphere between your face and the air near the centre of your two opens eyes, and look at the centre of this sphere and you will see this to be as I say.

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Figs. 1424-1428: Further demonstrations concerning the perception of small objects on CA175ra.

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Figs. 1429-1433: Diplopia experiments. Fig. 1429, Ptolem;y, Optics; fig. 1430, Witelo, Optics IV 107; fig. 1431, CVA396rb; fig. 1432, C19v; fig. 1433, CA190vb.

    To explain this phenomenon he again appeals to his concept of the central ray (see above pp. );

The reason for this is that eacy eye per se produces infinite visual rays which, in vision, are of so much greater power, as they are nearer the centric line which is in the first degree of visual power. Hence these lines spread out in circles from this centric line and operate on the powers of the species and images of objects which are positioned in front of the eyes.

    He describes a related phenomenon on I43r (fig. 1423, c. 1497-1499):

No opaque body of spherical shape seen by two eyes will ever show itself of perfect roundness.

a is the position of your right eye; b is the position of the left /eye/. If you close the right eye you will see your spherical body around the centre b and if you close the left eye then the said body will surround the centre a.

 

8. Diplopia

    Ptolemy, in his Optics,18 had explored the problem of diplopia in some detail. Some of these experiments were recorded by Alhazen.19 Witelo20 included additional cases in his compilation. Two of Witelo's examples have a parallel in Leonardo's notes (figs. 1430, 1438, cf. figs. 1431, 1439-1444). There is no firm evidence, however, that Leonardo drew directly on these mediaeval sources. His observations concerning diplopia may well have developed from his own detailed studies of objects smaller than the eye. An early example of this problem how a single object can appear double occurs on CA396rb (c. 1492) when Leonardo notes (fig. 1431): "it appears to be 2 because the pyramids do not intersect at it as at a /and/ o. On C19v (fig. 1432, 1490-1491) and CA190vb (fig. 1433, c. 1508) he draws related diagrams without text.

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Figs. 1434-1437: Diplopia experiments on CA125rb, A2v, K/P 34r and D8v.

    On CA125rb (c. 1490-1492) he draws two eyes with two objects in front of them: one object is seen normally; the other, double (fig. 1434):

Only the centric lines of 2 eyes are those through which the object seen by them at b is carried as a single /image/ to the sense m. The others all show to this sense a double object, such as the object n which, seen by the eyes a /and/ c appears to be 2 because it occupies 2 places as appears in d and f.

    This phenomenon he indicates again in a diagram on A2v (fig. 1435, 1492) and a rough sketch on W19096v (K/P 34r, fig. 1436, c. 1493) both without text. Some fifteen years later he analyses the problem in greater detail on D8v (fig. 1437, 1508):

The function of the central lines in the concourse of the visible.

The concourse of the two central lines is always in a point where an angle is generated /which is/ so much greater or lesser in size to the extent that the object seen...is at a lesser or greater distance from the eye. If two central lines focus on the object x the adherent inferior lines sv and ry will see the object t occupy two places on the wall nm, that is, at v /and/ y. But if these central /lines/ terminate at t, then the object x will be seen by the two adherent exterior /lines/, that is rx and sx, because the right eye sees /it/ with the right adherent lines and the left eye sees it with the left adherent lines.

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Figs. 1438-1444: Diplopia experiments. Fig. 1438, Witelo, Optics IV. 105; fig. 1439, C19v; figs. 1440-1441, CA125rb; figs. 1442-1444, K/P 115r.

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Figs. 1445-=1447: Diplopia demonstrations on C19v, CA347va and C19v.

    In the foregoing examples one object is positioned flush with the wall or interposed plane and the object which is seen double is nearer the eyes. In another series (fig. 1438) both objects are positioned away from the wall. He draws preliminary sketches of this situation on CA125rb (figs. 1440-1441, c. 1490-1492) and C19v (fig. 1439, 1490-1491). These he develops on W19117r (figs. 1442-1444, K/P 1125r, 1508-1510) accompanying which he explains:

Many objects placed one behind another in front of the 2 eyes with clear and expedite spaces, all appear double except that one which is seen best. And the space interposed between these duplications appears that much greater to the extent that such an object is closer to the eye in looking at the farthest /one/. But if you look at the first one the said spaces appear that much less to the extent that the object is closer to the eyes.

    On W19147v (K/P 22v, 1489-1490) he considers a situation in which two eyes have three objects in front of them two of which are doubled, (fig. 1451). This puzzles him. Hence he asks: "why in two /eyes/ or in front of two eyes /when/ three objects are represented 2 /of them/ are /doubled/. He redraws this diagram on A2r (fig. 1452, 1492) this time adding only the captions "distinct" for the single image and "indistinct" for the double image.

    On A2r (fig. 1453, 1492) he also draws a smaller diagram illustrating that each of the three objects in turn can appear double depending on where the eye is focussed. This composite diagram he also presents in three separate diagrams on CA125vb (figs. 1448-1450, c. 1490-1492). In a first example (fig. 1448) the eyes focus on the furthest of the three objects and the two nearest objects are both doubled. In a second instance the eyes focus on the middle object (fig. 1449) as a result of which both the nearest and the furthest object appear doubled. In a third case (fig. 1450) the eyes focus on the nearest object, whence both the middle and the furthest object appear doubled. Some fifteen years after he returns to this theme in a diagram (fig. 1454) on W19117r (K/P 115r, 1508-1510) again showing two eyes, now marked a and b in front of three objects c, d, and e respectively. The nearest object c appears double when the eye focusses alternatively on d or e.

(figure)

Figs. 1448-1455: Complex examples of diplopia. Figs. 1448-1450, CA125vb; fig. 1451, K/P 22v; figs. 1452-1453, A2r; figs. 1454-1455, K/P 115r.

    On CA347ra (c. 1490) he considers another alternative (fig. 1446), namely, how two eyes a and b, focussed on the point m, perceive the two objects c and d as the four objects e, f, g and h respectively. On C19v (1490-1491) he redraws this diagram, using identical letters (fig. 1447). He also draws a related situation (fig. 1445) in which the objects in front of the eye are somewhat larger. In this case the two objects r and s are perceived as three, namely, t, n and u. These studies lead him, on W19117r (K/P 115r, 1508-1510), to propose an explanation and a general rule concerning the number of apparent images seen in any situation (fig. 1455):

Here the objects are doubled to the visual power appearing /as/ 10 minus one and thus they do for every number, that is, always redoubling and omitting one from each sum which results from such doubling. And one begins with two, which doubled and 1 taken away from the sum leaves 3. Hence the twofold seen by two eyes will appear threefold. And if you put 100, double it and remove 1, there remains 199. And thus 100 objects placed one behind the other in front of the eyes will appear /as/ 199.

And the cause which gets this one taken away and leaves an uneven number is that...the angle of the two central lines...which focus on one of the objects /which is/ well seen and understood, does not intersect the lines beyond the creation of this angle and it terminates the 2 lines on the same object and to the 2 eyes one object...does not appear 2 as happens in objects seen by each...pair of non-central visual lines.

(figure)

Fig. 1456: How a bright object obscures an opaque one on BM171v.

 

9. Excessive Light

    The phenomenon that a greater light overcomes and disperses a lesser one had been considered by Aristotle.21 Galen mentions it in an optical context in De usu partium.22 In the Arabic tradition Alhazen23 discussed the problem, whence it was subsequently noted by authors such as Pecham.24 Leonardo alludes to this problem in passing on A100r (BN 2038 20r, 1492): "on the eye which through a greater light cannot discern the lesser ones." This idea he restates in a draft on BM171v (fig. 1456, c. 1492): "When the eye...has luminous objects (in front of it/,..., opaque objects appear tenebrous." He broaches the matter again on F57r (1508) in answering claims that the stars have light of their own:

This is false because it has been proved how the opaque body placed in the luminous body is surrounded by the lateral rays from the remainder of such a luminous body and for this reason remains invisible.

    On CU692 (TPL626, 1508-1510) he describes more clearly this phenomenon:

Of light which converts itself into shade.

The site illumined by the air iwll make itself shady if it is surrounded by the percussion of solar rays. And this occurs because the greater light makes the light of a lesser light source appear dark.

    A converse experience is described on CU693 (TPL625, 1508-1510);

Of the shade which is converted into light.

The site shaded by the sun will remain illuminated by the air after the setting of the sun because a lesser light is always the shadow of a greater light.

Excessive brightness hurts the eye

    That excessive brightness hurts the eye was mentioned by Aristotle25 and subsequent authors in the optical tradition including Galen,26 Alhazen,27 Witelo28 and Pecham.29 Leonardo discusses this problem in connection with variations in pupil size on C15r (1490-1491, see above pp. ). He mentions the problem again on Mad II 23v (1503-1504), this time in connection with background illusions involving glowing irons (see above p. ):

And this occurs because the spirits spread through the visual power, being overcome by excessive light, restrict all the pores, /either/ through the entire pupil (luce) or through part, in whatever part of this pupil it might be.

    On D7r (1508) he returns to this theme in connection with varying pupil size:

Excess light injures the eye and in order to protect it from being injured in this way the visual power turns to the kind of help one gets who shuts part of a window in order to lessen the excessive brightness generated by the sun in one's home.

 

10. After Images

    This visual phenomenon can be traced back clearly to Aristotle's De somniis.30 Ptolemy discusses the problem in his Optics31 as do later authors such as Witelo32 and the anonymous author of Della prospettiva.33 Leonardo's earliest extant reference to after-images occurs on Triv. 43r (1487-1490) in connection with the four powers:

Concerning violence

I say that every body moved or struck retains in itself the nature of that blow or movement for a certain time, and retains it more or less depending on how much greater or less is the power in that force or movement.

    To support this general claim he cites examples from acoustics and optics:

Example.

Note: a blow given to a bell, - how much it keeps in itself the sound of the repercussion. Note a rock or stone which has issued from a mortar - how much it preserves the nature of its movement. The blow given to a dense body - will preserve the sound longer than in a a less dense body and will be longer lasting which will be given to a body which is suspended and slight. The eye keeps in itself the image of luminous bodies for some time.

    On C7v (1490-1491) he seeks to explain after-images in terms of a principle of opposites:

The eye will take and reserve in itself the images of luminous things to a greater extent than umbrous ones. The reason is that the eye, in itself, is of maximal obscurity and since the similar is not distinguished between by the similar, hence night or other dark things cannot be reserved or recognized by the eye. Light is entirely contrary and divides to a greater extent and is a great detriment and variety to the customary darkness of the eye, whence it leaves impressed its image of itself.

    He returns to this explanation on CA203ra (1489-1490):

The pupil, operating in seeing converse things, retains the species of these to a certain extent. This conclusion is proved by the effects since the sight, in seeing light, retains it somewhat. Also, after the glance, images of the intense object remain in the eye and make the place of lesser light look dark, since the vestige of the impression of the greater light is retained by the eye.

    On CA204va (c. 1490) he mentions the phenomenon anew: "The eye reserves within it the images of luminous things which represent themselves to it," and on CA204ra he cites such after-images as evidence to support the intromission theory of vision (see above pp. ):

Proof how objects come to the eye.

Looking at the sun or some other luminous body and then closing the eyes you will see it similarly within the eye for a long interval of time and this is a sign that the species enter within.

    A more vivid description of the consequences of looking at the sun follows on CA369vd (c. 1490): "The dark place is seen inseminated with luminous round spots and the luminous /place/ with dark /round spots/ by the eye which many times and quickly has just look at at the body of the sun." On CA250va (c. 1490) he offers a rule concerning the duration of these after-images: "Those images which are born from a more luminous body will be reserved in the eye to a greater extent." Nearly eighteen years later he uses this phenomenon to explain the characteristics of the crystalline lens on D5v (1508):

Whether the eye sees bright and dark things at the same time?

The crystalline humour inside the pupil becomes denser when it encounters bright objects and more rarified when it encounters dark objects and that this is true is shown when the eye closes itself, for the species retained of what were bright objects are seen as dark and the dark objects show themselves as bright, which occurs more in weak eyes than in strong ones. Of this, I shall speak more fully...in its place.

    He mentions the problem once more on G73r (c. 1513), where, having provided various definitions of impetus, he adds:

Every impression tends to permanence or desires permanence.

This is proved...in the impression...made by the sun in the eye...of this viewer and in the sound made by the hammer of such a percussing bell.

    He had used the same examples in his earliest extant note on after-images on Triv. 43r (147-1490). To illustrate the phenomenon of after-images Mediaeval optical writers such as Biagio Pelacani da parma and the anonymous author of Della prospettiva had also cited the instance of a firebrand which, when revolved, appeared to be a flaming circle.34 Leonardo adapts this example on A26 (1492) in a passage entitled:

Perspective and motion

Every body which moves with speed, appears to tinge its path with the image of its colour.
This proposition is seen through experience since, when moving a firebrand beneath dark clouds, through the speed of its serpentine flight, its entire path appears like a luminous snake. And similarly, if you move a flaming stick in a circular movement, its entire path will appear to you to be a flaming circle. And this is because the imprensiva is swifter than the judgment.

    He pursues this theme in the Manuscript K (post 1504) where he establishes that this phenomenon is dependent on two factors rapid movement (a) of the eye and (b) of the object. He discusses the first of these on K120/40/r (fig. 1263) under the heading:

If the eye which looks at the star turns quickly in the opposite direction, it appears that this star is composed of a flaming curved line.

Let abc be the cornea (luce) of the eye which looks at the star ud. I say that, if the cornea moves the part a to c quickly, /then/ bv, in coming to the place a, will impress itself in a continuous line of the colour of the star and this occurs because the eye retains the image of the thing that shines for some interval /of time/ and since such an impression of the brightness of the star is more permanent in the pupil than was the limit of its motion which such an impression endures together with the motion in all the sites through which it passes facing the star.

    On K119/39/v, he considers a second faxtor: movement of the object:

It is as much to move the eye while keeping the luminous object stationary as /it is/ to move the object keeping the eye stationary.

That which is said in the first part is proved by the foregoing and the second part I shall prove with the aid of the foregoing, because in keeping the eye stationary and waving a flaming stick in a circle or from below the eye to above it, this stick appears to be a burning line...which rises from below to above, and this stick is only in one place at a time along this line. And hence with this stick standing firm and moving the eye from above to below it appears that this stick mounts in a continuous line from below to above.

    He develops his ideas on the forebrand on CA207ra (1508-1510) in the form of claims:

The motion of a stick of fire makes such a length of tail in calm air which does not move, as does this stick standing still while the air moves.

And by the eighth, the motion of fire in the air which moves with movement equal to that of this fire...

(figure)

Figs. 1457-1458: Flaming sticks in connection with after-images on CA207ra.

If the motion of this fire and the air which occludes it is uniform then such fire will remain without a tail.

    This stick illustrates in two sketches (figs. 1457-1458) adding the captions:

Let a be the stick, b the wind, uc the tail of fire.

Let sr be the air without motion, c the tail of fire which moves against the air. I say that mr will have such a tail of fire as is nh.

    On the W19117r (K/P 115r) he cites another example of after images:

And the drop which rains as illuminated by the sun according to the sight of the eye and in its passage it appears continuous in so much space as it shows all the colours of the rainbow and this it makes larger or smaller depending on the distance.

    This illustration of after-images is alluded to again on G6v (1510-1515) under the heading. "Description of the flood" where he refers to "the lines which drops of water make in descending" and once more on CA79rc (1515-1516): "The stone thrown through the air leaves in the eye that sees it the impression of its motion and the drops of water which descend from the clouds when it rains, do the same." Meanwhile, he had cited all three of these illustrations of after-images: namely, the sun's image, the waving firebrand and the apparent lines of rain, on CA360ra (c. 1504) opening, as usual with a general statement:

Every impression is retained for some time in its sensible object and that /one/ is more retained in its object, which is of greater power. And similarly the less by the less powerful. In this case I term that object as "sensible" which, through some impression is moved from that which it was at first. An insensible object is that which even if it is moved from its first being, it does not retain in itself any impression of the thing which moved it.

    After giving examples from acoustics, he turns to after-images in vision:

(figure)

Figs. 1459-1461: Binocular vision on Mad. II 24r, 24v and 25v.

Again the brightness of the sun or some other luminous body remains for some time in the eye after it has been seen and

a circle, makes this circle appear to be a continuous and equal flame. Drops of water, while raining, appear as a continuous thread which descends from hence through this one shows that the impression of things shown /to and/ seen by the eye are reserved in it.

 

11. Monocular and Binocular

    Leonardo examines how monocular and binocular vision affect the apparent brightness, clarity and relief of objects. With respect to brightness, for instance, he notes on H91/43/v (c. 1494) that "the luminous body appears larger and more luminous with two eyes than with one." On Mad II 24r (1503-1505) he develops this idea:

A same /intensity of/ brightnes or darkness will appear of as many different degrees of brightness or darkness as the eyes which see it are various. And this is manifested because, closing one eye, the object seen by the other appears darker than it appears with two eyes.

    This claim he reformulates, now adding an example

Two eyes which lead to a single imprensiva see double the brightness than a single eye. This is proved. The eye b sees the entire air ce and the eye a sees the entire air cd. I say that the quantity of the air cd is judged by the imprensiva to be twice as bright as the air dede because ucd is seen by the eye b and the eye a and the air dee is only seen by the eye b and the eye a does not see it.

(figure)

Fig. 1462: Monocular and binocular vision on K/P 45r.

    On Mad II 25v he restates this idea (fig. 1461):

The light seen with one eye is the half less powerful and large than the light seen with two eyes.

This is proved. Let a be the imprensiva to which the eye bears luminous objects. I say that billuminates this imprensiva through a single degree of light. Adding b /he means c/, this imprensiva receives 2 degrees of light. And because 2 degrees of light are in double proportion to one degree we find this imprensiva being doubly illuminated by these two lights and by a hundred or, a hundredfold more. That place will be more illuminated which is percussed by a greater sum of light. And hence it will be less illumined which is seen by a lesser light.

    Beneath the diagram (fig. 1461) showing a human imprensiva, he adds the caption "a is the imprensiva for seeing luminous objects." He describes a related experiment on W19042r (K/P 45r, fig. 1462, c. 1508):

It follows that in closing one eye, the visual power is diminished by half and this test is made with luminous bodies such as the sun, moon and stars and also with a light or fire.

This diminution of brightness is seen without closing one of the eyes. But instead of closing it, let your hand or finger be interposed in front of one of the pupils between the air and the eye and you will see a quantity of air with 2 pupils which will have the same boundary as the air seen by a single pupil and that which is seen by only one pupil will be that much more /i.e. twice/ as dark as that seen by 2 pupils. And the reason is that which the figure shows.

    These demonstrations recall those in which he studies the effects of various sizes of pupils (see above pp. ). Aristotle, in the Problemata 35 had noted that objects seen by two eyes are seen more clearly than with one eye. Ptolemy36 amd Witelo37 also mention this. Leonardo refers to this implicitly on several occasions and explicitly on CA347ra (1490) under the heading:

Of the eye

When the two eyes conduct the visual pyramid on the object, this object is seen by the two eyes and well understood.

    The way in which binocular vision contributes to effects of relief interests him far more because it raises the question whether a painting achieved with a monocular vanishing-point can ever fully imitate the perceptual effects of objects seen with two eyes. His notes on this problem have been analysed elsewhere (see above vol. 1, part III, 4). As noted in the introduction (see above p. ) this particular aspect of his interest in binocular vision served in 1839 as a starting point for Wheatstone's classic essay on the stereoscope.

 

12. Displacement of Eyeball

    Abnormalities of sight and the pathology of vision constitute a domain to which Leonardo devotes very little attention in the extant notes. An exception is the displacement of the eyeball. The phenomenon as such was well known. Aristotle, for instance, had considered it in his Metaphysics.38 Leonardo's first extant reference to this problem is in the form of a brief question on CA125rb (c. 1490-1492): "Why the eye, pushed by a finger, sees itself, which appears, /as/ a circle of fire." On BM115v (figs. 1468-1469, c. 1492) he broaches the problem again:

(figure)

Figs. 1463-1466: Problems of binocular vision. Fig. 1463, BM115v; fig. 1464, CA138vb; figs. 1465-1466, CA204rb.

Of the light which appears in the eye in its movement.

The eye which is pushed with a finger from below to above will make a contrary movement of its posterior parts where the imprensiva is caused and it will appear to this imprensiva that, raising the eye in front, things seen go down and this appears with the one eye standing firm and the other is moved by a finger.

    These notes on BM115v may well have been a draft for a passage on A81r (fig. 1467, 1492) where he discusses this problem in greater detail:

That eye which, with a finger is pushed from below above will, from its centre within make a contrary movement, which appears to move stable and firm things from their place and carry them from above to below.

This effect occurs because the eye which is moved by a finger from below to above or from above to below moves moving or pushing its lids which are joined inside their origins with the surface skin of the eye. And if you push the lower lid upwards it will be moved inside the eye and will lift the cornea (luce) upwards and the part of the eye which is being the centre will make the contrary movement and a contrary movement will be made by the images of the objects impressed in this part of the back of the eye where the power of the imprensiva is based. And this effect occurs holding both eyes open because the eye which is not moved sees the object in its site and that which is moved, moves the image of the impressed object in its imprensiva. And the object seen by the moved eye is never of that clear shape as is that seen by the eye which is not moved, because in being moved /the eye/ does not see the image by that centric line through which things are better judged, but rather it is seen by those parts which are around this centre and because they are of a less transparent humour things are seen more confusedly.

(figure)

Figs. 1467-1469: Concerning displacement of the eyeball. Fig. 1467, A81r; figs. 1468-1469, BM115v.

    On CA204rb (c. 1490) he cites this as the most important demonstration to confirm the intromission theory of vision (see above p. ). In the late period he returns once more to this phenomenon on W19117r (K/P 115r, figs. 1470-1473, 1508-1510):

The image.

Why the object seen by the eye which is turned /by/ being pushed sideways by one of the fingers in any direction...makes a movement contrary to that which is made by this eye. This appears to arise as the figure above shows /fig. 1471/, that is, if the concourse of the two central lines rt /and/ st terminate in the angle t and the non-central lines sx and sv remain apart, and if you push the eye s downwards you displace the master line st from its place into which the line sx moves. And because the object moves as much towards the line, as the line towards the object, therefore, moving the line sx towards the object t, it appears that this object t moves to the position ux where the line sx /is/.

    To illustrate this he drafts a rough diagram (fig. 1470) which he crosses out and draws again (fig. 1471) showing in composite fashion both the central rays rt and st and the non-central lines sx and sv. Next he draws a sketch showing only the central lines (fig. 1472) and a further sketch with the two central lines and one of the non-central lines (fig. 1473). These final two diagrams he does not discuss in his text.

(figure)

Figs. 1470-1473: Experiments with displacement of the eyeball on W19117r (K/P 115r).

 

13. Conclusion

    Euclid had focussed on ordinary visual conditions in his Optics. In the mediaeval period authors such as Alhazen, Witelo, Pecham and Biagio Pelacani da Parma had devoted more attention to problem conditions of vision. Leonardo develops these interests. He devotes attention to the role of the central ray, but is equally concerned with determining the limits of the visual field. His experiments in this connection lead him to reject the mediaeval notion that the maximal viewing angle is 90 degrees. Further experiments in this connection involve the use of occlusions and small apertures. He does not, however, perform Scheiner's experiment.

    Leonardo writes extensively on the perceptual problem of objects too near the eye, too far from the eye and too small to be seen clearly. He develops models and demonstrations for these purposes. He also studies diplopia and develops a rule for the number of images seen. The problem of excessive light he considers briefly. The question of after-images concerns him more. He has some notes on the distinction between monocular and binocular vision. Displacement of the eyeball also concerns him because he believes that it offers conclusive evidence for the intromission theory of vision. Notably absent from his writings are studies of pathological conditions of vision. Even fairly common disorders such as squinting or cataract are not mentioned in his extant work.


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Last Update: July 9, 1999