Part Four, Chapter 2: 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

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

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. ):

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.

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:

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):

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):

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

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:

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

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

A more precise description follows:

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

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:

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

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:

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):

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?

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:

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

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.

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:

A demonstration follows:

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:

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:

He summarizes this discussion with a marginal note:

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:

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 Optics¹ 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:

Perspective

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:

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:

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

(figures)

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:

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."

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.

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:

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

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

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

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."

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:

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."

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

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

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

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

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

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

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:

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."

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:

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

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

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.

In 1508, on BM94r, an outline follows:

On the moon.

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.

Notes

1. Ptolemy - Optics ↩︎