SUMS

Dr. Kim H. Veltman

The Sun's Image in Water


1. Introduction
2. Images Everywhere
3. Size of Sun's Image
4. Size of the Pupil
5. Water with Waves and Lunar Considerations

The Moon and the Book on the Earth and its Waters: Introduction

 

1. Introduction

    For Leonardo the everyday experience of the sun's image reflected in water acquires particular significance. At the outset it provides him with a further illustration that images are everywhere (..."all in all and all in every part," see above pp. ). He subsequently links this experience of the sun's reflection in water with his principles of linear perspective and his studies of pupil size. In his early studies, he considers the sun's image reflected from the surface of smooth, unruffled water. Later, he also considers situations where the water's surface is ruffled by waves. He treats waves as cylindrical mirrors, and studies the properties of such reflecting surfaces. His associative mind finds in these experiences a new explanation why the full moon reflects light as it does. An understanding of these themes will reveal how his optical writings are connected with problems in astronomy.

 

2. Images Everywhere

    Leonardo had used mirrors to illustrate that images are everywhere in the air (see above pp. ). The sun's image reflected in water is, for him, another instance of this mirror effect and hence it too can be used to demonstrate his "all in all" principle as on C17v (1490) in a passage headed:

Of the sun mirrored on the water

If the sun is seen by all the seas which have day...all these seas are seen by the sun. Hence all the luminous water makes images all in all this water and all in the part appears to the eye.

(figure)

Figs. 1515-1516: Image in a mirror and the sun's image mirrored in water. Fig. 1515, Witelo, Optics V.55; fig. 1516,

BM107r.

    The sun's reflection in water also poses a problem: "I therefore ask why when a ship sails /and/ the sun sees it, the eye does not see the sea all luminous and it does not always seem that a sun sails following the path of a ship." This problem will continue to play on his mind for the next twenty years but, already in the next paragraph he offers a first tenative answer (fig. ):

The sun makes as many pyramids as there are pinholes and apertures (where) through which it can penetrate with its rays. And as many are the eyes of the animals that look at it, so it will be found that the sun is always the base of each pyramid. The sun mirrored in water appears to the eye to be as much below this water as it is outside it and this mirrored sun makes itself a base of the pyramid which terminates in the eye and this mirrored sun is as large as the interaction of the cut pyramid is large, from the surface of the water at an. Let tr be the water; m, the sun; f, the sun mirrored in the water. Let shc be the pyramid of the mirrored sun /and/ let an be the above mentioned intersection of the pyramid.

    This long text passage does not, however, tell the whole story which his visual statement in diagram form presents (fig. ). The text mentions only an eye at c. But his diagram also shows a second viewing point at b and a third wh ich has been cut off by the upper limit of the folio.

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Figs. 1517-1518: Diagrams on C17v and author's reconstruction.

(figure)

Figs. 1519-1514: The sun's image in water. Fig. 1519, C17v; fig. 1520, CA351vb; fig. 1521, CA250rb; fig. 1522, 34v; figs. 1523, B25r; fig. 1524, G20r.

    The surface of the water here functions as the equivalent of the glass plane (parieta) which he uses in his perspectival studies (cf. vol. one, part I.3). Hence the further the eye moves back, the larger is the image on the plane. When the eye is at c, the intersection is tiny. When the eye is at b the intersection increases to na. Lower down the folio he adds four circles (fig. 1517). These represent cross-sections of the visual pyramid in the form of a ground-plan. The smaller circles correspond to intersections at the water's surface. The larger circles represent the sun's image under water (fig. 1518). He draws two sets to correspond with the experience of binocular vision. Two years later on A19v (1492) he again demonstrates how a mirror shows that images are "all in all and all in every part," and this leads once more to consideration of the sun's image in water:

Let us take the example of the sun which, if you walk along a bank of a river and you see the sun mirrored in that stream, the extent that you walk along that stream, to that extent will it appear that the sun walks with you and this because the sun is all in all and all in the part.

    This time he does not bother to add a diagram. More than a decade passes. Then in the Codex Leicester (c. 1506 possibly 1508-1509) he employs the example of the sun's image in water twice: first in passing on 1r and again on 7v in connection with pupil size (see below pp. ). On F39r (1508) he considers the problem afresh (fig. 1593):

Because the image of the sun is all in all of the sphere of water which sees it, the sun is all in every part of the aforementioned water.

The entire sky which sees the part of the sphere of water seen by the sun sees all this water occupied by the image of the sun and every part sees all.

The surface of the water without waves illumines equally the places percussed by the reflected rays of the images of the sun in water....

The image of the sun is one in the sphere of water seen by the sun which shows itself to the entire sky positioned opposite it and each part of this heaven sees in itself an image and that which is seen in one place is seen by the other in another place...in such a way that no part of the sky sees all.

That image of the sun will occupy more space of the surface of the water which will be seen by an eye more distant from it.

    In the accompanying diagram (fig. 1593) he marks the location of the sun, the waters of the ocean and adds 14 letters. None of these are mentioned in the text, but the problem continues to play on his mind. One of the phrases used on F39r ("The image of the sun is one in the sphere of water"), is adapted to introduce a subsequent passage on D6r (1508):

The image of the sun appears as only one in the whole of the sphere of water which sees and is seen by the sun, but appears divided into as many parts as there are eyes of animals which see the surface of the water from diverse places. This which is proposed is proved because, however far the eyes of seafarers carried by ships may move through the universe, they behold the image of the sun simultaneously through all the waters of their hemisphere in all the movements made in all aspects.

Figs. 1525-1528: The sun's image in water and motion of the eye. Fig. 1525, W12587v; figs. 1526-1528, CA243va.

    A draft passage on CA243ra (1510-1515) echoes these themes:

The image of the sun is all in all...the parts of the objects where its rays intersect it and all in every particle...

The sun will be seen in as many parts of the sea and...as are the eyes of...

The sun will be seen in as many parts of the sea and...as are the eyes of...

    On CA243va(1510-1515), amidst passages on the moon (see below p. ) he asks what occurs: "If the eye moves along the shore the length of a canal which has its axis facing west." A draft answer follows:

...The length of the line that the motion of the eye makes along that water...that sees the image of the sun in some stream and you walk along this image.

    This draft he crosses out and then draws three diagrams (figs. 1526-1528) beneath which he adds a caption:

If you...are moved towards the sun through that water which finds itself between that sun and its image...you will be navigated along a continuous image which will be the length of your motion.

    On CA243vb (1510-1515) he pursues the theme, beginning with a by now familiar formulation:

The image of the sun is all in all the water which sees it and is all in every minimal part of it.

This is proved because /there/ are as many images of the sun as /there/ are positions of the eyes which see part of it.

(figure)

Figs. 1529-1530): Sun's image in water and time factor on CA243vb.

This idea he restates in two visual statements (figs. 1529-1530). One diagram (fig. 1529) effectively develops his sketch on C17v drawn 25 years earlier (fig. 1519), and the accompanying text discusses the case of a boat also mentioned on C17v:

Again moving the eye,...carried along a line by a boat, one sees the image of the sun moving along the same line as the motion of such an eye, but it is not parallel because, with the sun moving towards the west, the line of the image moves in a curve towards the sun in such a way that ultimately it joins the image of the sun in appearance...when it is joined to the horizon.

    A more precise description follows:

If the motion of the ship occurs at noon and the sun is in the middle of the sky, the line of the image of this sun is curved and will always go enlarging itself in such a way that in the end it will join up with the sun at the horizon and the image appears euqal in size to that of this sun.

    To the right he draws a second diagram (fig. 1530) which he then explains:

Let abc be the middle of the course of the sun in our hemisphere. Let edc be the shape made by its image on the sea. Let fg be the maximal /mo/tion of the ship, which carries the viewer that much /away/ from the image.

    Even so his caption leaves much implicit. We are expected to realize that during the time when the ship moves from f to g, the sun moves from its noonday position at a to the horizon at c, while its corresponding reflection in the water grows as it describes an arc edc until it ends at c coincident with the setting sun itself. These verbal and visual statements on CA243vb (c. 1510-1515) constitute his last extant answer to questions first broached on C17v (1490-1491).

 

3. Size of the Sun’s Image

    One of the diagrams of C17v (fig. 1519) implicitly demonstrates a basic tenet of linear perspective: that as the eye is moved back, the image on the glass plane, here the water's surface, increases proportionately. On CA250rb (1490-1591) he carefully redraws this diagram (fig. 1521) letters it and adds a caption which broaches both the eye's size (see below pp. ) and distance:

If the eye or indeed the point were as large as the sun and at a distance equal to that of the sun from the sea, that sun which, mirrored in water appears the size of a platter, would appear to occupy all that part of the sea which can be seen by you.

    A more detailed explanation follows in the upper left-hand column:

This is clear...ly shown in the book of mirrors, how each body appears by images /from/ plane mirrors to the extent that these bodies are distant from the surface of these mirrors. Whence, for this reason, if you are close to the said surface with your eye, as appears at r, the sun will appear mirrored the size to, because so great will be the intersection of the pyramid made by the surface of the water; and if the eye he /moved/ to the point f, the image of the sun will occupy that much more water, as there is from t to n and if you raise yourself...to the point m, the reflected sun will occupy a greater/er/ sum of water and thus it would do step by step successively /.../ such that, if it were possible that you raise yourself to the moon when, during the day when it passes without light through our hemisphere; the sun, reflected in the water would occupy all the waters illuminated by it. It being so, /our/ world would function for him who was on the moon as the moon appears to us at night when it shines.

    This passage suggests that by 1491 Leonardo had written "books on mirrors" in which he had explored thoroughly the connection between perspectival intersections and mirror images. He had also begun contemplating the astronomical implications: that the earth seen from the moon must appear as the moon appears from the earth. This will become an important theme (see below p. ). On CA250rb he next considers the factor of the eye's size:

And if our eye were the size of the world, even if this /eye/ were close to this world, it would see all the waters of the sea, lakes and rivers luminous with the image of the sun. But since this eye is small, it operates through a small point, to which a small image /comes/. Nonetheless, that which confirms my opinion is that, walking along a stream, throughout the entire voyage, the sun, reflected in the water appears to walk along with the traveller. And this is because the sun is all in all the parts illuminated by it through its images.

    On A96v (BN 2185 16v, 1492) he pursues the consequences for astronomy, beginning with the phrase: "the particles must correspond to their parts and the parts /cor/respond to the whole," which may be a variation of his familiar principle of images being "all in all and all in every part." This is followed by a (fig. 1531):

Demonstration (pruova) how to the extent that you are closer to the cause of the rays of the sun, the larger the sun reflected in the sea will appear to you.

If the sun adopts its splendour with its centre fortified by the power of the entire body, it is necessary that its rays, the further that they go out from it, the more they open out. If it be so, you who are with the eye close to the water which reflects the sun, see a minimal part of the rays of the sun carry the form of that reflected sun on the surface of the water.

And if you were close to the sun, as would be /the case/ when the sun is at its zenith and the sea in the west, you will see the sun reflected on this sea in the largest form. /This is/ because, you being closer to the sun, /and/ your eye drawing the rays closer to the point, it draws more and hence there results a greater brightness and by this reason/ing/ one could prove that the moon is another world similar to ours and /that/ that part of it which shines is a sea which reflects the sun and that which does not shine is land.

    Here links with the moon are mentioned as a possibility. On A64r 91492) he pursues these connections in a more affirmative tone in a passage entitled

What /kind of/ thing is the moon?

And if you see the sun or the moon in water which is near you, it will appear to you in that water to be the size that appears to you in the sky. And if you go back a mile, it will appear 100 times larger. And if you see it reflected in the sea at sunset, the reflected sun will appear larger than 10 miles to you because in that reflecting it will occupy more than 10 miles of sea. And if you were where the moon is, it would appear to you that the sun is reflected in as much sea as it illuminates by day and in the evening there would appear in that water what appear like the dark spots which are in the moon which, standing on earth, show themselves as such to men, exactly as our world would do those men who live on the moon.

    Some three years later on CFA351vb (c. 1495) he draws a diagram (fig. 1520) showing the eye at different levels above water looking at the reflected image of the sun. On BM25r (c. 1508) he develops this diagram (fig. 1523) this time adding an explanation:

The image of the sun will occupy more space on the surface of the water which is more distant from the eye which sees it.

Let a be the sun; pq is the image of this sun; ab is the surface of the water where the sun is reflected; let r be the eye which sees this image occupying the space om on the surface of the water; c is the eye remoter from this surface of the water and thus /remoter/ from the image. Hence this image occupies a greater space of water to the extent of the sapce no.

    On CA243rb (c. 1510-1515) he makes explicit the connection between this intersection in water and his transparent plane (pariete) of linear perspective:

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 a transparent plane (pariete) and then you remove the eye to such an extent that the intersection is doubled and note the space from the first to the second intersection and state: if in so much...space the diameter of the moon increases by this much relative to the first intersection, how much will it grow in all the space which there is from the eye to the moon? It will make the true diameter of this moon.

    This passage suggests that the principles of linear perspective can be combined with the rule of three (cf. Mad II 51r) to determine quantitatively the size of the sun's image. On G20v (c. 1515) the develops this idea in a passage entitled (fig. 1524):

Explanation of the moon with the image of the sun.

If the sun f mirrored in the surface of the water nm appears to be at d, (i.e., going under water to the extent that this /eye/ is above water) and to the eye /at/ b it appears to be the size a and if in removing the eye from b to uc the image doubles, how much would the image grow if the eye removed itself from c to the moon /?/

Do /it/ with the rule of 3 and you will see that the light which the moon has on the fifteenth /day, i.e., when it is full/ can never be the light which this moon would receive if it were /a/ spherical /mirror/. Hence it is necessary that this moon is water.

    Some explanation is needed to understand how he arrives at this dramatic conclusion. The laws of perspective show that an image on the intersected plane, here the water's surface, increases in proportion as the eye is moved further back. If one knows the distance of the moon one can apply the rule of three to the perspectival pyramid and determine the size of the intersection when seen from the moon. All this assumes reflection in a plane mirror and Leonardo knows only too well that the surface of both the earth and moon is spherical. His experiments with convex spherical mirrors (see below pp. ) confirm, however, that these produce reflections much smaller than the original light source. Hence if the moon were a spherical mirror the sun's image would be reflected from only a small part of the moon's surface. Further study of the sun's image in water shows him that if there are waves, each of these can function as a cylindrical mirror and reflect the sun in every part of its surface. The moon, he reasons, must have oceans with such waves (see below pp. ).

 

4. Size of the Pupil

    The web of Leonardo's associative mind does not stop here: he also connects the sun's image in water with the problem of variable pupil size. If the pupil is tiny it can effectively be drawn as a point, and be forgotten, as he does in his early diagrams (figs. 1519-1524). But if the pupil were larger then its size would also affect apparent size. Leonardo considers this possibility in the passage on CA250rb (1495-1497 ciated above p. ), where he states that if the eye were as large as the sun, the sun's reflection would stretch across any sea in which it was seen. On Leic. 7v he broaches the problem again:

The sun reflects itself all in all and all in every part of the water placed opposite it, but the eye which sees its image is of minimal size and hence this image appears amll under water, as that of the sky. And if the eye were as large as the sphere of the water, this eye would see all the waters resplendent with the image of the sun.

    A demonstration follows:

(figure)

Figs. 1532-1534: Apparent size of the sun if the eye were smaller, the same size, or larger than the sun on C237ra.

And I prove it as follows. You see the sun mirrored in our seas when it is in the west and it does the same to all the antipodes which are on the horizons positioned on the circle which separates day from night on earth. Now imagine a combined visual power which terminates...its circle with the circle of the eyes of these antipodes. Hence there will be in this combined visual power all the power of the eyes which, in the hemisphere see the sun mirrored in its waters through diverse countries. But a single eye is of the nature of a point to which the rays...from the terminus of the image...converge, which concourse of pyramidal lines, being cut at the surface of our waters show to what extent such an image appears large in such an interseection of a pyramid, which intersection will be smaller to the extent that the eye which sees this image will be closer to the surface of the water where the sun is reflected.

    These ideas crystallize on CA257ra (1505-1508). Here he begins with the now familiar diagram of the sun's image in water (fig. 1532) and adds the caption: "Because the eye is small it cannot see the sun in image except as small." Directly beneath he draws a second diagram (fig. 1533) in which the eye is now as large as the sun and as the caption explains: "If the eye were equal to the sun, it would see in the waters, provided they were smooth, the image of the sun equal to the true body of the sun." He then adds a third variant, in which the eye is larger than the sun (fig. 1534), with the caption: "if the eye were larger than the thing mirrored it will see on the mirror the image larger than the said thing." He draws another enlarged eye on W12587v (fig. 1525) and explains:

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Figs. 1535-1538: Pyramids of light on F63r, CA237ra, CA243rb and CA243vb.

Figs. 1539-1542: Reflected light at sunset on CA208vb

If the eye were /as/ large in diameter /as/ all of ab, the image m would appear on the surface of the water...of such size that its diameter would be the whole of cd.

    On D6r (1508) he mentions how there are as many images of the sun in water as there are eyes seeing it and then pursues the theme of the eye's size:

If the eye were as large as the sphere of water, it would see the image of the sun occupying a great part of the ocean.

This can be demonstrated because if you move on a bridge, from which you see the image of the sun in the water of its stream and...move some 25 braccia to the side, you will see that the image of the sun moves on the surface of this water and thus he who puts together all the images seen in such a way that you would have made a single image in the form of a flaming beam. Now imagine making a circle of which this flaming beam is the diameter and that this entire circle is filled with images. Without doubt you would see an image of which the diameter would be 25 braccia. How you have to imagine that there is a pupil which has /as/ its diameter the same 25 braccia which, without moving would see in the same water, a reflection of the sun which would have a circumference of 78 and 4/7 braccia.

    He summarizes this discussion with a marginal note:

The small quantity of the pupil is the cause of making the image of the sun appear a minimal thing on the surface of the water.

(figure)

Figs. 1543-1546: Demonstrations that the sun's image is everywhere in water on CA208ra.

If the eye were as large as the sphere of the water, this water, when seen by the sun, would be all a single image of the sun.

    In the lower part of D6r he considers a further alternative: if the eye were so distant from the sphere of water that its size was reduced, then the entire sphere of water would appear as a single image of the sun. On CA208ra (fig. 1544, c. 1513) he illustrates how the sun's image in water produces a fiery beam such as that described on D6r. Beneath this diagram on CA208ra he notes that this is a: "proof how...the images of the sun in water are as many as...the sites of the eyes which see it in water." He then draws a second diagram (fig. 1545) with a draft caption "move many eyes" and a third diagram (fig. 1546) showing: "eyes placed in a circle in a pond where the sun is at the zenith," which leads to a:

Conclusion: if the eye were as the earth or if the earth through distance diminished as an eye...that our earth would appear a celestial body which was like the moon or another star.

    In Leonardo's mind the sun's image in water has become intimately linked with problems of astronomy. To understand why this is so we need to consider his studies of water with waves.

 

5. Water With Waves and Lunar Considerations

    Ptolemy in his Optics 1 had noted that waves increase the size of the sun's image in water. Leonardo's first extant comment concerning this phenomenon occurs on W12350 (c. 1493) where he points out (fig. 1590) that: "the sun will appear greater in moving and wavy water than in still water: example of the light seen on the chords of the monochord." On H76/28v/ (January 1494) he mentions it again under the heading of:

(figure)

Figs. 1547-1554: Pyramids of sight and light in water. Figs. 1547-1551, CA112ra; figs. 1552-1553, CA112va; fig. 1554, CA243rb.

Perspective

The shadow(s) or the thing(s) reflected in moving water, that is, like small waves, will always be larger than the thing outside where it originates.

    In the period 1505-1508 he explores this problem more thoroughly. On CA112va, for instance, he notes that: "The image of the sun on a wave of water increases to the extent that this wave diminishes in size through the long distance of the eye." This idea he reformulates on F38v (1508): "the image that is reflected from the wave to the object acquires size in every degree of its distance," which is followed by a diagram (fig. 1592) and caption:

The image f rebounds to cd and the image of the wave e rebounds to ab. These 2 percussions of reflected rays mutually superimpose themselves on one another in cb and there the light is doubly more luminous than in ac or bd.

    On F39r, opposite, he develops this principle to show how the sun is reflected over the entire ocean (fig. 1593) and not just a small stretch of waves. On F63v discussion of the distance factor continues with case where the eye is close to the water:

If you hold the eye as close to the surface of the sea as you can you will see the image of the sun in a single wave of the water which you can measure and which you will find to be very small.

    He then describes what happens if the eye is removed to a distance of several miles:

(figures)

Figs. 1555-1557: Rough sketches of the sun's image in water. Fig. 1555, CA112ra; figs. 1556-1557, CA112va.

 

Figs. 1558-1560: Sun's image in water at sunset. Fig. 1558, CA237ra; figs. 1559-1560, F63r.

If you hold your eye near the surface of the water of that sea or pond which interposes itself between your eye and the sun you will find the image of the sun on that surface showing itself /as/ very small. But if you remove yourself from that sea by a space of several miles you will see the image of the sun make itself an equal number of miles and if the first image retains the true shape and light of the sun, as do mirrors, this second one will reserve neither the shape nor the light of that sun but /rather/ a shape with interrupted and with diminished light relative to the first.

    Why the shape of the sun should become unclear, (a theme considered in draft on CA112ra, c. 1505-1508) is the subject of the next paragraph:

The shape of interrupted and confused boundaries will be generated by a composition of many images...of the sun reflected to your eye from many waves of the sea...And the diminished brightness derives because the umbrous and luminous images of the waves come to the eye mixed together, whence their light is altered by their shades /which/ cannot happen on the surface of a single wave which you have placed close to the eye.

    These thoughts are restated visually on F63r under the heading: "Every image of the sun grows in being removed from the eye which sees it." He draws one eye close to the waves (fig. 1575), a second further away (fig. 1574), and demonstrates (fig. ) that: "even if the waters are separated all their images run to the eye."

(figure)

Figs. 1561-1564: Crests of waves as mirrors on CA237ra, F63v, F62v and BM28r.

    He also draws (fig. 1535) an eye positioned well above the surface of the water, from which viewpoint it sees at least seven waves. This he redraws (fig. 1562) practically as a ground-plan, showing the waves as a series of circular forms presumably as equivalents of individual mirrors. An earlier version occurs on CA237ra (fig. 1567, c. 1500). On F62v this is further developed (fig. 1563) and explained.

If the shape of the waves were in the shape of a half sphere as are the spheres of water, the concourse of images of the sun which part from these waves and come to the eye would be of a very great angle when this eye is on the shore of this sea which interposes itself between it and the sun.

    On F62r he explores the alternative that the water's surface is hemicylindrical inshape. He draws a cylinder (fig. 1576) and adds the caption: "The ray of the luminous body makes its angle of incidence under 4 equal angles, that is, the axis of the angle." Beneath a second diagram (fig. 1572) he notes: "Here the angle acm, not being equal to one mcb opposite it, the eye o does not see n in c. The axis of the angle of incidence falls between four right angles." In the right-hand margin he makes two more drafts of a diagram (figs. 1569-1570), redraws this (fig. 1571) and adds an explanation:

Cad is the angle of incidence on the cylindrical object eg and the axis of such an angle ba terminates at a between 4 equal...angles, that is to say each of these angles is equal to its corresponding one, as daf is equal to cah, its corresponding one and likewise bae to bag.

    On F61v he pursues the problem (fig. 1573):

The angle of incidence always terminate among equal angles each corresponding to its own.

(figure)

Figs. 1565-1573: Reflection from cylindrical mirrors. Figs. 1565-1568, F61r; figs. 1569-1572, F62r; fig. 1563, F61r.

    Above this explanation on F61v the reason for these theoretical discussions becomes clear. He draws a diagram (fig. ) showing the sun at d, an incident ray, da, from the sun; a central ray ab; a reflected ray ae; an eye e as well as nine other points identified by letters. The curves are clearly meant to be waves which Leonardo assumes are equivalent to a cylindrical mirror and this provides him, as his caption states, with an

Explanation why waves of water well removed from the concourse of the rays reflected to the eyes from these waves nearer by do not render the image of the sun which illuminates them.

    In the upper right-hand corner of F61v he adds a further note: "In all places that the sun sees the water, the water (sees the water) sees it and can, in each part, render to the eye the image of the sun." This is clearly a restatement of an idea expressed earlier on C17v (1492): "If the sun is seen by all the seas which have day, all these seas are seen by the sun." On F61r (1508) he also draws four cylindrical objects (figs. ) this time with only a brief caption: "If the sun is seen by all the seas which have day, all these seas are seen by the sun."

Let the angle of incidence be onp. Let its axis by mn which terminates at n among infinite equal angles...each equal to its corresponding one. Here I take only 4 as examples and let the first be odn equal to its corresponding one mnc because these two are each equal to a curved and a straight side and the curves dn and cn are of equal curvature because they are placed on a columnar body of uniform size. The other two angles are...at mna and mnb which have their straight sides but are not equal because mnb is less than its corresponding mna. Therefore the angle will be lower.

Among the 4 angles abcd one can place infinite other angles and each will be equal to its corresponding one.

    In the upper portion of F61v the reason for these theoretical discussions becomes clear. He draws a diagram (fig. 1579) showing the sun at d, an incident ray, da, from the sun; a central ray ab; a reflected ray ae; an eye e as well as nine other points identified by letters. The curves are clearly meant to be waves which Leonardo assumes are equivalent to a cylindrical mirror and, as his caption states, this provides an:

Explanation why waves of water well removed from the concourse of the rays reflected to the eyes from these waves bearer by do not render the image of the sun which illuminates them.

    In the upper right-corner at F61v he adds a further note: "In all places that the sun sees the water, the water...sees it and can, in each part, render to the eye the image of the sun." This is clearly a restatement of an idea expressed earlier on C17v (1492): "If the sun is seen by all the seas which have day, all these seas are seen by the sun." On F61r (1508) he also draws four cylindrical objects (figs. 1565-1568) this time with only a brief caption: "if the sun is seen by all the seas which have day, all these seas are seen by the sun."

Among the 4 angles abcd one can place infinite other angles and each will be equal to its corresponding one.

    On BM28r (1508) he pursues this theme with a diagram (fig. ) and detailed explanation:

This demonstration of so many spherical bodies interposed between the eye and sun is made to show that, just as in each of these bodies one sees the image of the sun, so too can one see this image in each globosity of the waves of the sea. As in many of these spheres one sees many suns, so too in many waves does one see many lustres and, at a great distance, each lustre in itself makes itself large to the eye and since each wave does this, one sees the sapces interposed between the waves consumed and for this reason there appears to be a single continuous sun in the many suns mirrored in the many waves and the shaded parts mixed with the luminous spaces have the effect that this brightness is not lucid like that of the sun mirrored in these waves.

    On CA155rc (1516-1517) he describes how this principle of reflection can also be observed in a storm:

The lustres of clouds (little) will not show themselves in those places where the dark rains with their clouds reflect.

But where the flashes generated by lightening in the sky reflect, one sees as many lustres made by the images of their flashes as are the waves which can reflect to the eyes of those standing about.

The number of images made by the flashes of lightening on the waves will increase to the extent that the distance of the eyes of their viewers increases.

    And likewise this number of images diminishes to the extent that they approach the eyes which see them as is proved in the definition of the brightness of the moon and of our maritime horizon when the sun is reflected with its rays and the eye which receives this reflection is distant from the said sea.

    He consdiers the role of distance in this reflection from waves once more on CA174vb (1518) beginning with a comment that

the spherical bodies of blobulent and terse surfaces are those of which surfaces are composed of various globosities.

This demonstration of so many spherical bodies interposed between the eye and sun is made to show that, just as in each of these bodies one sees the image of the sun, so too can one see this image in each globosity of the waves of the sea. As in many of these spheres one sees many suns, so too in many waves does not see many lustres and, at a great distance, each lustre in itself makes itself large to the eye and since each wave does this, one sees the spaces interposed between the waves consumed and for this reason there appears to be a single continuous sun in the many suns mirrored in the many waves and the shaded parts mixed with the luminous spaces have the effect that this brightness is not lucid like that of the sun mirrored in these waves.

    On CA155rc (1516-1517) he describes how this principle of reflection can also be observed in a storm:

The lustres of clouds...will not show themselves in those places where the dark rains with their clouds reflect.

But where the flashes generated by lightening in the sky reflect, one sees as many lustres made by the images of their flashes as are the waves which can reflect to the eyes of those standing about.

The number of images made by the flashes of lightening on the waves will increase to the extent that the distance of the eyes of their viewers increases.

And likewise this number of images diminishes to the extent that they approach the eyes which see them as is proved in the definition of the brightness of the moon and of our maritime horizon when the sun is reflected with its rays and the eye which receives this reflection is distant fromt he said sea.

    He considers the role of distance in this reflection from waves once more on CA174vb (1518) beginning with a comment that

The spherical bodies of globulent and terse surfaces are those the surfaces of which are composed of various globosities.

 

(figure)

Figs. 1581-1586: Sunlight reflected by waves of water. Figs. 1580, CA120vd; figs. 1581-1582, BM104r; figs. 1583-1584, CA249rc; figs. 1585-1586, CA237ra.

    A smooth convex surface, he explains, would produce only one image hence:

It follows that a surface which is terse and globulent renders as many images to the eye which sees it as are the number of globules seen of the real thing and from the eye which stands before them.

A number of images of any object seen on the spherical bodies with a terse and globulent surface will be as many more or less as the eye...seeing such images is remoter or closer to the above mentioned terse body.

    The diagram (figs. ) and other texts on this folio again point to a connection with astronomy. This astronomical connection becomes clearer on BM104r (1506-1508) where he makes two sketches of the moon's surface (figs. ) and seven diagrams showing the relationship earth, sun, moon and earth (figs. see below pp. ) in the lower part of the folio. In the upper part, a further diagram (fig. ) shows the sun and moon and, as he explains in a draft, demonstrates:

How the sun cannot be reflected in the body of the moon it being a convex mirror in such a way that, to the extent that this sun illuminates, this moon reflects to that extent, unless such a moon already had a surface capable of reflection which was rough like the surface of the sea when it is partly moved by wind.

    Directly beneath this passage he makes a draft sketch (fig. 1581) of the sun's rays being reflected from such rough waves and adds the caption: "These waves produce at every line /an effect/ like the surface of a pine cone." Beside this he draws a second draft sketch (fig. 1582), which leads to a third and much larger draft (fig. 1587), with a caption: "These are 2 figures thus you will make the one different than the other, /one/ with wavy water and/ the other/ with level water."

(figure)

Figs. 1587-1588: Sun's image in smooth and rough waters on BM104r and BM25r.

    He has, in fact, taken the familiar diagram of the sun's image in water (cf. figs. 1519-1524) and has integrated it with a dynamic situation where the water is ruffled by waves. In a further caption he adds that "the waves of the water increase the image of the thing which is reflected." He redraws this diagram on Bm25r (fig. 1588) where he repeats the caption and adds a more detailed explanation:

Let a be the sun. Let mn be the water with waves, b the image of the sun when the water is without waves. Let f be the eye which sees the image in all the waves which are contained in the triangle cef. Hence the sun, which on the surface without waves occupies the water cd, on the surface of the water with waves now occupies all the water ce (as is proved in the fourth of my perspective) and it would occupy more of this water to the extent that this iamge is more distant from the eye.

    His reference here to a proof in "the fourth of my perspective" invites comparison with a reference on F69v to a "fourth book on the earth and water" and to a passage on F38v headed:

Perspective of solar rays

The solar rays reflected on the surface of the water with waves make the image of the sun appear to be continuous throughout all that water which is between the universe and the sun.

    This is illustrated by a diagram (fig. 1592) followed by a proof:

It is proved. Let a be the body of the sun. Let bc be the surface of the water with waves. Let dt be the universe which sees this water between it and the sun. By the 2nd of the 1st the image of the sun which comes from this sun to the wave m is necessarily reflected at t and not somewhere else and similarly the image (which) of the sun which comes from the sun to f rebounds to /d/ and not elsewhere and so too does every wave interposed between these 2 (extremities) said extremities. Whence by necessity the entire line dt (interposed in) positioned opposite these images is seen (by all) and illumined by all these images where the reflected rays are enlarged and it will, at some distance, be discontinuous in brightness as is shown here.

(figure)

Figs. 1589-1593: Sun's image ont he waves of the ocean(s). Fig. 1589, CA27ra; fig. 1590, W12350; figs. 1591-1592, F38v; fig. 1593, F39r.

    This is the proof to which he appeals on BM25r. Hence F38v is a draft for a treatise "on the earth and its waters," to which he also refers as a treatise "of perspective" because it deals with linear perspective, optics and the reflection of images from waves. This treatise, if it was ever written, is now lost. Nonetheless, Manuscript F and the Codices Arundel and Leicester contain a series of drafts which permit at least a rough reconstruction of his astronomical treatise.

 

The Moon and the Book on the Earth and its Waters

Introduction

    In the period 1505-1508 Leonardo begins to plan a treatise on the moon. On CA74va, for instance, he jots down two chapter headings:

On the water that is in the moon.

On the perspective of reflections of water with principles which prove that water is around and within the moon.

    In 1508, on BM94r, an outline follows:

On the moon.

Wishing to treat of the essence of the moon it is necessary first to describe the perspective of mirrors plane, concave and convex the image of the sun which comes from this sun to the wave m is necessarily reflected at t and not somewhere else and similarly the image...of the sun which comes from the sun to f rebounds to /d/ and not elsewhere and so too does every wave interposed between these 2...said extremities. Whence by necessity the entire line dt...positioned opposite these images is seen...and illumined by all these images where the reflected rays are enlarged and it will, at some distance, be discontinuous in brightness as is shown here.

    This is the proof to which he appeals on Bm25r. Hence F38v is a draft for a treatise "on the earth and its water," to which he also refers as a treatise "of perspective" because it deals with linear perspective, optics and the reflection of images from waves. This treatise, if it was ever written, is now lost. Nonetheless, Manuscript F and the Codices Arundel and Leicester contain a series of drafts which permit at least a rough reconstruction of this astronomical treatise. This we shall examine in the chapter that follows.


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Last Update: July 10, 1999