SUMS

Dr. Kim H. Veltman

The Fourth Book: On the Earth and Its Waters


1. Introduction
2. Chapter 1. The eye as Source of Astronomical Illusions
3. Chapter 2. Atmospheric Refraction Through Different Mediums
4. Chapter 3. Mirrors
5. Chapter 4. Reflection in Dense and Transparent Bodies
6. Chapter 5. Visual Pyramids
7. Chapter 6. Nature of the Elements
8. Chapter 7. Centres of the Elements and the World
9. Chapter 8. Light and Shade
10. Chapter 9. The Sun as Only Light Source
11. Chapter 10. Diminution of Earth's Light
12. Chapter 11. The Moon's Waters
13. Chapter 12. The Moon's Elements
14. Chapter 13. Earth and Moon
15. Chapter 14. The Earth as a Star
16. Conclusion

 

1. 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...and before that what /kind of/ thing is a luminous ray and how it is bent by various kinds of mediums. Then where the reflected ray is more powerful, whether it is on the side of an incident ray that is acute, at right angles or obtuse or whether it is on /surfaces that are/ convex, plane or concave, or from a body /that is/ dense or transparent. Besides this, how the solar rays which percuss the marine waves show themselves to the eye of such a size as the size of the waves at the end of the horizon and from this it follows that such solar brightness reflected from marine waves is of a pyramidal shape and consequently with every degree of distance it acquires degrees of size even if as far as our vision /is concerned/ it shows itself /as/ parallel.

    In modern terms this outline could read as chapter headings one to five in Chart 30. Other notes in Manuscript F help provide outline sketches of later chapters. A note on F41v, for instance, indicates that he plans to write a chapter on the earth's position at the centre of its elements (see chapter seven in Chart 30):

How the earth is not in the centre of the circle of the sun, nor in the centre of the world but in the centre of its elements which accompany it and are united to it. And for him who stands on the moon when it, along with the sun, are above us, this our earth, with its element of water would appear and perform the same function as the moon does to us.

    A discussion of the balance of the elements assumes a knowledge of their nature and weight and hence it is likely that he planned a chapter on this (see chapter six in Chart 30). On F77v (1508) he draws the sun's rays reflected from the wavy surface of the moon (fig. 1689) with a one line caption: "this will be preceded by the treatise on shadow and light," which must also have constituted a chapter in his book on astronomy (see chapter eight in Chart 30). Leonardo's theory is that the earth reflects the sun's light as do the moon and the stars. This requires that he must eliminate a contending theory that the moon has light of its own and establish that the sun is the sole source of light. He alludes to this on F4v while praising the sun: "All souls descend from it because the heat that lives in these animals comes from such souls and there is no other heat nor light in the universe as we shall show in the fourth book." If the reflection of sunlight depends on water, then the earth which was covered with water at the time of the Flood must have lost a considerable amount of its former light. This too is to be included in the treatise on astronomy (chapter ten in Chart 30) as is confirmed by a note on F69v (1508):

How the earth performing the function of the moon has lost a considerable amount of its former light in our hemisphere through the lowering of the waters as is proved in the fourth book on the moon and its waters.

    Having discussed how the earth's waters reflect light he intends to explain how the moon's waters also reflect light (chapter eleven in Chart 30). There is a contemporary theory which contends this, claiming that water being heavier than air would fall back onto the earth. To refute this Leonardo needs to show that the moon has its own elements which remain in equilibrium (see chapter twelve in Chart 30). If the nature of the moon is essentially the same as the earth's, it follows that the moon also has its days, months and seasons as does the earth or as he puts it on F63r (see chapter thirteen in Chart 30): "Define the earth with its long and its short days in the North and in the South and do the same for the moon and determine them accurately." Having established that the nature and function of the earth and moon are fully equivalent, he wishes to push the comparison further and argue that the earth from a great distance is like a star (chapter fourteen in Chart 30) or as he puts it on F56r:

Your entire discourse has to conclude that the earth is a star and practically the same as the moon and thus you will prove the nobility of our world and will make a discourse on the sizes of many stars according to the authors.

    Taken as a whole the above notes provide important clues concerning the structure of Leonardo's proposed treatise on astronomy. To gain some impression of the contents of the treatise, we need to consider what material exists for each of the hypothetical chapters listed in Chart 30.

 

Chapter 1.  The Eye as Source of Astronomical Illusions

    In the outline on BM94r Leonardo states that his book is to open with "the nature of the luminous ray." A passage on F25v clarifies what he means by this:

Order to prove that the earth is a star

First define the eye then show how the scintillation of each star comes to the eye and why the scintillation of these stars is more for one than for the other.

And how the rays of the stars arise in the eye.

    In other words he considers the nature of the "luminous ray" (which is his term for the light that causes twinkling of stars) to be an optical illusion. Another optical illusion in astronomy interests him also as becomes clear from F5r/4v where he discusses why Epicurus and others maintain that the sun is only as large as it appears:

I think that this reason/ing/ is taken from /the fact that when/ a light /is/ placed in this our air equidistant from the centre, he who sees it does not ever see it diminished in size at any distance and the reasons for its size and power I /shall/ reserve for the fourth book.

    These illusions interest him so much that he decides to make it the opening theme of his fourth book as he epxlains on F94v:

My book sets out to show how the ocean with the other seas, through the intermediary of the sun, makes our world shine like a moon and /that/ from a greater distance it appears like a star and this I prove.

First show how every light remote from the eye produces rays which appear to increase the shape of such a luminous body.

    From our earlier analysis (see above pp. ) we know that this passage marks the beginning of a treatise which continues on F95r, skips to F40r and then proceeds in revcerse until F28v. The chief themes of this treatise are (1) why lights further in the distance do not appear smaller and (2) the role fo the eye, and the pupil in particular, in producing these illusions, i.e., precisely the themes with which he proposes to open his book on astronomy. But this treatise within Manuscript F is, in turn a development of Manuscript D. Hence Leonardo's two principal works on optics are intended to serve as an introduction to his projected astronomical treatise.

 

Chapter 2.  Atmospheric Refraction Through Different Mediums

    The outline on BM94r also alludes to illusions from atmospheric refraction. A note on F25v clarifies what this chapter was to entail:

Then prove how the surface of the air at the boundaries of fire and the surface of fire at its boundary is that which, when penetrated by solar rays, carries the images of the celestial bodies large in their rising and setting and small when they are in the middle of the heavens.

    This problem, why planets appear larger at the horizon than overhead, had been mentioned by Ptolemy, Alhzen, Witelo and Pecham, Leonardo's interest in it can be traced back to A64v (1492) in a passage headed:

Proof of the growth of the sun in the west

Some mathematicians claim that the sun grows in the West because the eye always sees it through air of greater thickness, alledging that things seen in mist and water appear larger. To these I reply that this is not so, because things seen in mist are similar in colour to things far away and not being similar as regards diminution they appear of greater size. Again no object increases in level water and you will prove this in tracing /the aize of/ a board that is positioned under water. But the reason that the sun increases is that every luminoug body, the further away it is, the larger it appears.

(figure)

Figs. 1594-1595: Atmospheric refraction and apparent size of the sun on A64r and Mad.I5v.

Figs. 1596-1600: Atmospheric refraction. Fig. 1596, CA272va; fig. 1597, I34r; figs. 1598-1599, F25v; fig. 1600, F60r.

    On A64r he gives a succint explanation (fig. 1594)

Why the sun appears larger at the horizon than at midday which is closer.

Every body which is seen by a curved medium appears of a larger size than it is.

    He returns to this problem on Mad II 5v (fig. 1595):

I say that when the sun appears ont he horizon and even when it leaves the horizon, that it is of a much larger size than that which is overhead and it is more distant to the extent of half the diameter of the earth, as is shown in ab.

Now the reason for this is said by some to be on account of the medium, which is of a great variety of thicknesses and through this its rays /are said/ to be bent and curved outwards, moving further away from the central line which goes from the eye to the centre of the solar body. But for me this explanation does not hold for, if it were so, the sun would throw its rays beyond itself in the west before it appears on the horizon.

    On F25r he draws two further diagrams (figs. 1598-1599) and adds a caption for the first of these:

Let a be the earth. Let the surface of the air which borders on the sphere of fire be ndm. Let the course of the moon or, if you wish, the sun be hfg. I say that when the sun appears on the horizon g, that its rays are seen to pass through the surface of the air under unequal angles, i.e. om which is not /with-/ in /the radius/ dk and moreover it passes through a greater thickness of air. All of om is a thicker air.

    In 1492 he had rejected thickness of the air as a factor. Here on F25r he explicitly accepts it. On F60r (fig. 1600) he pursues the theme of atmospheric refraction:

Why the planets appear larger in the east than overhead which ought to be the contrary...being 3,500 miles closer to us when in the middle of the sky than when at the horizon.

All the degrees of the elements where the species of celestial bodies pass which come to the eye are curved and the angles of...the central line of such species which penetrates them are unequal...and the distance is greater as the excess of ab over ad and by the ninth of the sixth the size of these celestial bodies is proved.

    The proof to which he alludes my well have been an experimental demonstration using a model (see Appendix ).

 

Chapter 3.  Mirrors

    In his outline on BM 94r he mentions a further chapter on "mirrors plane, convex and concave and the conditions producing the most powerful reflection." There is a particular incentive for this interest: a contemporary theory claims that the moon is a /convex/ mirror. Leonardo therefore studies the properties of convex mirrors. On CA353rb (fig. 1674) for instance, he asks:

(figure)

Figs. 1601-1604: Reflection in mirrors. Figs. 1601-1603, CA251ra; fig. 1604, CA125vb.

Whether a gilded ball will be seen by the sun through the whole medium or in every part of the medium?

If you move from place to place you iwll be able to see the source of its light. If you move from point a you will see it at b, at c, at d and so on.

    On CA251ra he again broaches the question of reflection from convex surfaces under the heading "of the mirror between the centric line /of sight/ and the pyramid," this time with three diagrams (figs. 1601-1604) to which he alludes as proofs in the caption that follows:

A ball can never be seen from a single point, except as half or less, depending on whether it is less near. The proof is above. To the extent that point c sees the mirror, that amount of the mirror sees it and likewise it happens at the point f.

Hence the point c, the pyramidal lines departing from it make terminus and base at the points ab...and cannot see, nor be seen by the greater quantity of the spherical body. And this pyramid makes its centre at the centre of the mirror and the point f does the same, making /its/ base at the points d /and/ e.

    This "proof" is based purely on geometry and not on observation. Soon afterwards, however, observation brings him to a different conclusion. On CA125vb (c. 1492), for instance, he studies reflection in and from transparent glass spheres and discovers that an image is reflected in but a tiny part of its surface, hence (fig. 1604 cf. figs. 1605-1613): "In n, ab will appear...the size of rt."

(figure)

Figs. 1605-1613: Sunlight and convex mirrors. Fig. 1605, CA141ra; fig. 1606, CA237ra; fig. 1607, F85r; figs. 1608-1610, K/P 117v; figs. 1611-1613, BM107r.

    Over a decade later he alludes to the problem again in three diagrams on BM107r (figs. 1611-1613, 1506-1508). On CA131vb (figs. c. 1508) in a text cited elsewhere (see below p. ) he suggests that from a sufficient distance the reflection would appear larger than the surface of the object reflecting it. Further studies lead him to abandon this possibility. On F76r (1508), for instance, he simply claims: "The image of the sun in a convex mirror increases in going further from this mirror and the solar body would disappear in going /still/ further." On CA28ra (1505-1508), he makes a careful drawing (fig. 1614):

To find here the position and size of the image of the sun.

Given...a spherical mirror...and the position of the eye and object, it is asked /what is/...the size and position which the image of this object has?

    He uses effectively the same diagram on BM28r (fig. 1615, c. 1508) where he applies these mirror studies to speculations on the nature of the moon:

Either the moon has light of its own or not. If it has light of its own, why does it not shine without the help of the sun? And if it does not have light of its own, necessity makes it a spherical mirror and if it is a mirror, is it not proved in perspective that the image of a luminous object will never be equal to the part of that mirror which is illuminated by that luminous body. And if it be thus, as the figure in rs shows here, whence comes so great a quantity of splendour that the full moon has, which we see on the fifteenth /day/ of the moon/"s phase/.

    He restates this idea that the moon cannot be a spherical mirror on F93r (1508) in a passage headed:

(figure)

Figs. 1614-1615: Sun's image in a convex mirror on CA28ra and BM28r.

Of the moon and if it is polished and spherical.

The image of the sun is powerfully luminous in it and in a small part of its surface and you will see the proof in taking a polished gold sphere with a light source at a distance from it such that, although it lights about half of this sphere the eye does not see it except in a small part of its surface and all the rest of such a surface reflects the darkness that surrounds it and for this reason only the image of the light appears in it and all the rest remains invisible, when the eye stands at a distance from this sphere. The same would happen with the surface of the moon if it were polished, lustrous and dense as are the bodies which reflect.

    Hence his study of mirrors leads him to refute the possibility that the moon is a mirror. In the meantime his fascination with mirrors has become a study in itself, the details of which will be discussed elsewhere (see Appendix , pp. ).

 

Chapter 4.  Reflection in Dense and Transparent Bodies

    The outline on BM94r (c. 1508) suggests that a next chapter would consider whether reflection is greater in dense or transparent bodies. Although we know that he studied reflection in transparent spheres, spheres of polished gold and various mirrors, there is no record of his studying reflection in objects of different densities in a systematic fashion. Even so the reason for his interest in this problem can be reconstructed. There is a theory among his contemporaries that the spots on the moon are due to different degrees of density in the surface: as he notes on F85r (1508):

Of the spots of the moon

It has been said that the spots of the moon are produced in this moon from its being of various rarities and densities which, if it were so, the solar rays would penetrate through some parts of the aforesaid rarity in eclipses of the moon and since such an effect is not seen, such an opinion is false.

Others say that the surface of the moon, being terse and polished, that it receives the image of the earth in it like a mirror. This opinion is false because the earth not covered by water has different shapes from different points of view. Hence when the moon is in the east it would reflect other spots than when it is /directly/ above or where it is in the west. Yet the spots of the moon as one sees in the full moon never vary in the movement which they make in our hemisphere.

A 2nd reason is that the object reflected in its convexity occupies a small part of this mirror as was proved in perspective.

A 3rd reason is that during the full moon, the moon only sees half the sphere of the earth illuminated in which the ocean and other waters shine and the earth makes spots in this brightness and hence one would see the half of our earth girdled by the brightness of the sea illuminated by the sun and in the moon such an image would be a minimal part of this moon.

/A/ 4/th reason/ is that a bright object does not reflect in another bright object. Hence the sea /of the earth/ takes light from the sun - as does the moon - and such an earth could not be reflected in it /i.e. the moon/, and if it were reflected one would not particularly see the body of the sun and all the stars opposite it.

    On F84v, opposite he pursues this problem:

Concerning the spots of the moon.

Others claim that the moon is composed of more or less transparent parts, as if one part were of alabaster and another were of crystal or glass, whence it would follow that the sun, shining with its rays on the less transparent part, the light would remain on the surface and hence the denser part would remain illuminated and the transparent part would show the dark shadows of the depths and thus the quality of the moon is composed. And this opinion is favoured by many philosophers and especially by Aristotle and, nonetheless, it is a false opinion, because in the various aspects in which the moon and sun find themselves with respect to our eyes, we would see such spots vary and sometimes they would be dark and sometimes bright. They would be dark when the sun is in the west and the moon is in the middle of the sky, when the transparent concavity would take the shadows to the summit of the boundaries of this transparent concavity because the sun could not penetrate with its rays into the mouths of such a concavity which would appear bright in /the time of/ the full moon, where the moon is in the east when the sun is in the west. Hence the sun would illuminate to the depths of such transparent parts and, thus not generating shadows, the moon would not show the aforesaid spots at that time and likewise now more, now less, according to the change of the sun relative to the moon, and of the moon from our places as was stated above.

    Having dismissed existing theories concerning spots on the moon, he is challenged to propose his own explanation. On Leic 5r he mentions "how the spots of the moon are different from that which they once were as a result of the course of their waters." This idea he develops on CA112va: "If you look at an island surrounded by waves filled with images of the sun it appears to you as if you see one of the spots of the moon surrounded by its brightness." As early as 1490 he had considered another explanation concerning the haloes of the moon on CA349vc:

On the circles of the moon.

I find that those circles of various sizes and thicknesses which appear to surround the moon at night are caused by various qualities of thickness of humours, which are situated at various heights between the moon and our eyes. And the largest circle which is less red is lower than the said humours in its first part; the second, smaller one is higher and appears redder because it is seen through 2 humours and likewise the higher they are the smaller and redder they will appear because between the eye and this there are more solid humours and for this reason it is proved that where greater redness appears there is a greater sum of humours.

    On F84r (1508) he mentions the possibility that these vapours or clouds could also account for spots on the moon, but rejects the idea:

Spots of the moon

Some claim that there arise from this /moon/ vapours in the manner of clouds and that these interpose themselves between the moon and our eyes which, if it were thus, such spots would never be stable in either position or shape and seeing the moon in different aspects, even if the spots were not varied, they would change shape as does that thing which is seen from different sides.

    Nonetheless, he is prompted to make extended observations of the moon and record the results (figs. 1668-1673). Thereby he discovers that the spots of the moon do indeed vary considerably. Hence on BM19r (c. 1508) he adopts the theory which he had previously rejected:

If you continue to observe the details of spots of the moon you will often find great variety in these and I have experienced this myself in drawing them.

And this occurs from the clouds which raise themselves from the waters of this moon which interpose themselves between the moon and this water and with their shade they take the rays of the sun from such water whence this water, not being able to reflect the solar body, will remain dark.

 

Chapter 5.  Visual Pyramids

    The outline on BM94r refers to a further chapter:

how the solar rays which percuss the waves of the sea show themselves to the eye as the same size at the angle of the eye as at the last summit of the waves of the horizon and for this reason it does not fail that a solar brightness reflected by the waves of the sea is of a pyramidal shape and consequently it gains degrees in size with each degree of distance, even though it shows itself as parallel to our sight.

    Interest in the nature of visual pyramids was part of his Euclidean heritage (see above pp. ). But that which Euclid had studied purely geometrically in two planes Leonardo explores in terms of three-dimensional situations complete with interposed planes (figs. (1191-1193).

(figure)

Figs. 1618-1622: How the sun's image expands pyramidally although it appears parallel. Figs. 1616-1617, BM94r; figs. 1618-1621, BM62v; fig. 1622, BM94r.

    This leads to systematic diagrams, first rough as on BM62v (figs. 1618-1621), then more polished as on BM94r (figs. 1616-1617, 1622). On CA112ra, va he relates this three-dimensional pyramid to the sun's image in water (figs. 1547-1557). Frontal views of the same situation on F63r (figs. 1559-1560) CA237ra (fig. 1558), and CA243rb, vb (figs. 1537, 1700) go hand in hand with these. And as is clear from the passage on G20r (fig. 1524, see above p. ) he intended to analyse this pyramid quantitatively. Whether he intended to include the Euclidean comments concerning pyramids in this short chapter is not certain.

 

Chapter 6.  The Nature of the Elements

    Leonardo wished to establish that the earth is at the centre of its elements (see chapter seven in Chart 30). A preliminary discussion of the nature and weight of these elements was therefore fitting (chapter six in Chart 30). His interest in the elements earth, air, fire and water can be traced back to CA284va (c. 1497) where he mentions: "I believe that the air will have that proportion in resistance with fire that air will have with water," and discusses the balancing of weights of water and air. On CA180ra (c. 1505) he refers to the weights of elements in each other, a theme which he puruses on CA79va (1505-1508) where he makes a list of combinations: fire, earth > in water and in air; water, fire > in air and earth, air> in water. On CA79rb he makes mention of a fifth element, discusses the transformation of elements and conjectures concerning their relative weights: "one measure of fire weights 2 weights, and one measure of air weighs 4, one of water, 8, one of earth 16, and one of gold 32." On CA72va (1508-1510) this conjectural discussion of weights is pursued: "Let us posit that air has 4 of lightness being under water and 8 if it is under simple earth. Hence water has 4 of gravity between air and earth has 8." This leads, on CA72va and ra, to discussion of how elements interpenetrate one another posed in the form of problems such as:

I have air that weighs two and water 4 and earth 8.

Now I want that air and earth remain at the level of the water. And by the ninth of the fourth this cannot be done.

    A complex explanation describes what must be done to make this combination possible. On CA244va (1508) there is further discussion of fire in air, water in air, and air under water (cf. CA131rb and CA190rb, c. 1508) this time in connection with weights being attracted to the centre of the world. On F62v (1508) he draws a cubic piece of lead inside a spherical dew drop (fig. 1623) and explains that this is intended to simulate the relation of earth ot water:

In a dew drop which is well rounded, one can consider many cases of the function of the sphere of water, how it contains the body of the earth inside it without destruction of the sphericity of its surface. First let a circle of lead be taken, the size of a grain of panic /grass/ and with a very thin thread attached to it, let it be submerged within such a drop. And you will see that such a drop does not lose its first rotundity even if it be made larger to the extent of the cube enclosed within this dew drop.

    The convertibility of elements into another interests him increasingly. On CA172vc (1508), for instance, he claims:

The elements are equal if they are made with equal subtlety.

With a unity of earth is convertged in 10 it is similar to the density of water and when this earth is converted into hundred, it would be similar to air and air would do the same converted in 10. And when this air converted itself in thousand it would be similar to fire, and water would do the same converted in hundred and air converted in 10.

    This tenfold ratio of the elements is mentioned again on Leic. 35v (1506-1509). As might be expected Manuscript F, which contains numerous astronimical notes, develops this theme of the elements at some length. On F69v, for instance, there is a general discussion concerning their weight:

Earth is heavy in its sphere but the more so to the extent that it is in a lighter element.

Fire is light in its sphere and the more so to the extent that it is in a heavier element.

No simple element has weight or levity in its own sphere and if the vescicle full of air weighs more in the balances than...being empty this is because such air is condensed and the fire which is heavier than the air could condense, or equal to the air and perhaps heavier than the water and make itself equal to the earth.

    A specific discussion of the shape of the elements follows on F27v:

On the 5 regular solids.

They say that the earth is tetracedronic, that is cubic, that is a body of 6 bases and this they prove saying that there is no body among the regular bodies of less movement and more stable than is the cube. And to fire they attribute the tetrahedron, that is a pyramidal body which is more mobile (according to these philosophers) than is the earth. Hence they attribute this pyramid to fire and the cube to the earth which, if one had studied the instability...of the pyramidal body and compared it with the cube, /then one would find that/ without comparison this cube is more mobile than the pyramid and this is proved as follows. The cube has 6 sides and the regular pyramid has 4 in the margin in a /and/ b, a is the cube; b is the pyramid and in order to define such a proof I shall take one site of the cube and one side of the pyramid and they will be...c /and/ d. I say...that the cube c will be more apt to rotary movement than the pyramid d and let the beginning of this movement d be e /and/ f below. I say, in fact, that...if the base of the cube and the base of the pyramid are placed on a same plane that the pyramid will throw a third of its quantity to fall on its other side and the cube will throw a quarter of its circumference to change to the other side in order to make its base. It follows from these 2 demonstrations that one has concluded that the cube makes a complete turn with 4 of its sides on a same plane when the triangle or pyramid makes its entire turn with its three sides on a same plane and the pentagon will place all 5 of its sides and therefore the more sides the easier the movement is, because it approaches the sphere more. Hence I wish to infer that the triangle is of slower motion than the cube and consequently it is fitting to place this pyramid and not the cube for the earth.

That the cube makes its entire revolution with a same impetus along a same line and the pyramid not, as is seen.

    This passage is of great importance because it helps explain why Leonardo devotes so much attention to the centre of gravity of pyramids (see figs. 1625-1651 and below p. ). On F27r, he considers relative weights of the elements and their motion:

The weight of air, when it is in fire has the same proportion as water which is in air and such air falling from fire into another air would give the same proportion as the water which falls from the air into water.

How winds can arise from the motion of air in air as the motion of water in water and both are a motion of one element in itself which arises from the falling of a lighter element such as air fallen from fire, whence it would rain.

    He next discusses the:

Shape of the Elements.

On the shape of the elements and first against he who denies the opinion of Plato, who states that these elements enclosed in one another with the shapes that Plato posits, that a vacuum would be caused between them and this is not true and here I prove it but first it is necessary to propose some conclusions.

It is not necessary that any element which encloses another is of equal size in all its quantity among the parts which enclose it and that which is enclosed.

We see that the sphere of the water is clearly of different sizes from its surface to the bottom and that although it would cover...the earth if it were the shape of a cube, that is with 6 angles as Plato wishes, this covers the earth which has unnumerable angles of reefs covered with water and various globules and with cavities and there is no vacuum generated between the water and the earth. Again the air which covers the sphere of the water together with the mountains and valleys which cover this sphere, and there does not remain a vacuum between the earth and the air such that he who states that a vacuum is generated would have a sad story.

To Plato it is replied that the surfaces of the shapes which he proposes that the figures have, could not remain.

Every flexible and liquid element by necessity has its spherical surface. This is proved with the sphere of the water, but first it is necessary to propose some conceptions and conclusions.

    In 1515 he returns to the theme of elements. On CA200ra, for instance, he points out:

Water has its motion only through its gravity and levity and these are its accidents because this does not have gravity or levity or levity in itself, but it acquired gravity when it is above or in the lateral confines of air or some other liquid lighter than it and it acquires levity when in evaporating it becomes thinner through heat and then it stands above cold water.

    This idea of changing gravity and levity he pursues on CA219ra (1515):

The air and water and the earth are continually changeable in their levity and gravity through the heat of the sun which heating lightens that part of the element that is closer to it and it does the contrary in the opposite part of these elements.

(figure)

Figs. 1623-1626: Models of the earth, pyramids and gravity. Fig. 1623, F62v; figs. 1624-1626, Leic. 35v.

 

Chapter 7.  Centres of the Elements and the World

    If the earth's stability depends on its being in the centre of the world, then the moon's stability must depend on something else. If he can show, however, that the earth's stability arises through its being at the centre of its elements, he is then free to argue that the moon's stability also arises through its being at the centre of its elements. This would help support his further aim of showing that the earth and moon are equivalent to one another. His interest in these problems can be traced back to 1492. On A20v, for instance, he describes a method of measuring the distance from the surface of the earth to its centre. On A58v he ponders

On the centre of the ocean.

The centre of the sphere of water is the true centre of the rotundity of our world, which is composed of water and earth in rotund form. But if you wished to find the centre of the element of the earth, this is contained in a space equidistant from the surface of the ocean and not equidistant from the sruface of the earth, because it is clearly comprehended that this ball of the earth does not have anything of perfect rotundity except in that part where a sea or marshes or other waters are dead. And every part of the earth that rises above this sea is further from its centre.

    He develops these ideas in a series of drafts on CA153va (1495-1496) beginning with a general claim:

(figure)

Figs. 1627-1637: Pyramids and centres of gravity on BM72v.

Figs. 1637-1644: Centres of gravity. Figs. 1637-1639, BM111v; figs. 1640-1642, BM108r; figs. 1643-1644, BM124r.

<If> bodies were perfectly spherical and of equal material they would have a single centre. But this appears to be impossible, because matter is unequal.

    A long discussion of common centres and centres of true gravity follows which leads to the claim:

The common centre and the centre of the earth are not the same. Rather they are very different and of a different nature, because the common centre does not move, /since/ the site of the air and fire do not change and the centre of the earth is in constant motion, because it moves as often as the winds carry the water of the seas covering or uncovering various shores with their waves.

    He next demonstrates how a weight b will descend to a common centre a, and not rise up to the centre of the earth. This leads to a further distinction between different kinds of centres:

The centre of the universe is not the centre of any element, because the continuous revolution and various accidents that arise from the continuous celestial influxes, hold these elements in a continual change of site. For the earth, in large part surrounded by the sphere of water, makes a same body as these waters and as a ball stands suspended in the air and the centre of the sphere of the water is not the centre of its gravity and the centre of its gravity is not the centre of gravity of the earth. Rather it is a good distance away. Whence the surface of the sphere of water is not equidistant from its centre.

    He broaches the question of the earth's centre in passing on CA284v (1499) and CA120=rd (1504-1507). How to find these various centres becomes a practical question in the Codex Arundel. On BM111v (c. 1505), for instance, he notes:

(figure)

Figs. 1645-1647: Accidental and real centres of gravity on BM111v, 108r, 124r.

with the centre of natural gravity and with the centre of accidental gravity of the two parts in which a body is resolved, the accidental centre of the entire body is found.

With the centre of accidental gravity to find the centre of natural gravity of one of the 2 parts in which a body is resolved and the centre of accidental gravity of the other part.

    Beneath this he draws a geometrical diagram with a pyramid (corresponding to the element earth) which he lables (fig. 1645):

c is the centre of natural gravity of the cone: acrm
b is the centre of natural gravity of the bisected cone anec
d is the centre of natural gravity of the pyramid cnr.

    This diagram he develops on BM108r (fig. 1646, c. 1505) with a more developed caption:

abc is a conical body
zt is the centre of its size
op is the centre of its accidental gravity
de is the centre of its natural gravity
kh is the centre of natural gravity of the greater pyramid which
the cone has
gi is the accidental centre of the pyramid.

    He also drafts another method for finding various centres. On BM124r (c. 1505) this diagram is further developed (fig. 1647) and explained:

(figure)

Figs. 1648-1651: The earth's centres of gravity on BM72v.

Of the cone abc the centre of its magnitude is the line de.

The centre of its accidental gravity is in the line no. The centre of its natural gravity is in the line fg. The centre of the magnitude of the largest pyramid that can be found in this cone is in the line de.
The centre of accidental gravity of this pyramid is in the line rt.
The centre of gravity of such a pyramid is in the line hi.

    He then outlines another practical problem:

With knowledge of the centre of natural gravity of the cone and with knowledge of the centre of natural gravity of the greatest pyramid which can be imagined in this cone, I wish to find the centre of natural gravity of the aforesaid pyramid.

    This time he provides a solution as well (fig. 1644 cf. figs. 1640-1643):

Cde is a pyramid divided by the cone cdef; the centre of its natural gravity is at ab and such a pyramid cde is 1/3 of all the cone cdef. Gh is the centre of natural gravity of the entire cone. Hence being certain of this centre of natural gravity of the pyramid and similarly of the centre of gravity of the cone, and knowing that the pyramid is 1/3 of the gravity of the cone we shall make a converse proportion of 2 of space against 2 of weight and divide the space uag in 2 equal parts and place one of these spaces in 1k and have there the centre of natural gravity of the pyramid def.

    On BM72v (1505-1508) he pursues this theme, now providing a succinct description of the different centres:

(figure)

Figs. 1652-1563: The earth and its waters on BM236v and Leic. 36v.

In every heavy body there are found to be 3 centres of which one
is the centre of natural gravity, the 2nd accidental gravity, 3rd of
the size of the body.

    He goes on to relate these to the question of the earth's centre:

1/4 of 12 is 3/12/ and 1/3 of 12 is 4/12. Hence the difference that there is from 1/3 to 1/4 is 1/12. Now imagine that if the earth were not in its site and that this pyramid fell from on high to this central site, that the centre of gravity would be 4000 miles distant.

    On F27r (1508) Leonardo disagrees with Plato and argues that the shape of the earth must be pyramidal (see above p. ). Out of context this appears to be merely a philosophical quibble. This series of folios in the Codex Arundel reveal that he had studies the gravitational properties of his earth-pyramid very carefully and had related it to his general studies of different centres of weight. In the Codex Leicester he pursues the question of the earth's centre and the centre of the world. Following a preliminary passage on Leic. 36r (figs. 1653, 1657), he distinguishes, on Leic. 34v (figs. 1658-1659 cf. fig. 1651), between a universal and a particular centre of sphericity of water. This leads to a long discussion on Leic. 35v (figs. 1624-1626, 1660-1664) of the relation of this sphere of water to the centre of gravity of the earth and centre of the world, the conclusion of which is that there are only:

2 ways /that/ the gravity of the earth is concentric with the centre of the world.
That is either by being totally submersed in water or in having the opposite part of equal weight outside the water.

(figure)

Figs. 1654-1659: Gravity and the earth's centre. Fig. 1654, Forst.II136r; figs. 1655-1656, Leic. 35v; fig. 1657, Leic. 36r; figs. 1658-1659, Leic. 34r.

The centre of gravity of the waters and of the earth would be concentric with the centre of the world if the earth were perfectly spherical. Then the centre of the world would be the centre of the sphere of the earth, as the sphere of water. But this would not produce terrestrial animals.

    In other words the gravity of the earth could only be concentric with the centre of the world in hypothetical situations. In practice they are separate (fig. 1651). Conscious that this is a dramatic claim he adds: "It is in the power of the orders of Nature to make it that the earth stands by itself through its shape, outside the whole sphere of the water." He pursues these problems in a series of passages on F22v, 27r, 69r, 70r, and 83v which, as he explains on F41v, all have the purpose of showing:

How the earth is not at the centre of the circle of the sun nor at the centre of the universe but is rather, at the centre of its elements which accompany it and are united with them and if a person were on the moon when it, along with the sun is beneath us, this earth with its element of water would appear and would function as does the moon to us.

 

Chapter 8.  Light and Shade

    On F77v (1508) he draws a viewer looking at a wave filled moon reflecting sunlight (fig. 1689). Above this he writes: "This will have in front of it the treatise on light and shade." What this treatise entailed is not indicated, but almost certainly it would have included basic definitions and his distinction between explanding, contracting and parallel shade. It is likely that it would have included most of his first two books on light and shade (see above pp. ). Given the specialized problems of the later books it is improbable that he intended to include all of his work on light and shade as an intermediary chapter in his treatise on the earth and its waters.

(figure)

Figs. 1660-1664: Problems of gravity on Leic. 35v.

 

Chapter 9.  The Sun as Only Light Source

    In a eulogy of the sun on F4v (1508) he mentions that: "there is no other heat nor light than it in the universe as I shall show in the fourth book." On a number of occasions, A64r, BM28r, 104r and 94v, Leic 30r he merely states that the moon has no light of its own without further explanation. However, in the Codex Leicester, he examines the problem in greater detail, beginning with a draft passage on Leic 36v (1506-1509):

The adversary states that the light of the moon is if not entirely...,/then/ partly of itself and that it shows itself that much more or less illuminated depending on whether the eye sees its greater or less umbrous part, that is, if it is more eastern or western. Here in this part it is replied that if the

    At this point the text breaks off but he takes up the problem afresh on Leic 2r:

When the eye in the east sees the moon in the west near the setting...sun and it only sees it...with its umbrous part surrounded by its luminous part, of which the lateral and superior part derives from the sun and the inferior part derives from the western ocean which still receives the rays of the sun and reflects them to the inferior seas of the moon and again gives to the entire umbrous part of the moon so much brightness which is /equal to/ that which the moon gives to the earth at midnight and hence it does not remain entirely...dark and for this reason some have believed that the moon had, in part, light of its own rather than given by the sun, which light derives from our oceans illuminated by the sun for the aforesaid reason.

    He adds two further explanations:

Again it could be said that the circle of brightness which the moon makes when it is in the west with the sun derives entirely from the sun when this and the sun is situated with the eye in the way that is demonstrated above /fig. 1752/.

Some could say that the air..., element of the moon, taking light from the sun as does our sphere of air, would be that which furnishes the luminous circle at the body of the moon.

    He now reformulates the whole problem of the moon's light:

Some have believed that the moon has some light of its own which opinion is false because they have founded it on that brightness which is seen in the middle of the horns when the moon is new which at the boundary of the brightness appears dark and at the boundary of the darkness of the background it appears so bright that many believe it to be a circle of new light which finishes by surrounding where the points of the horns illuminated by the sun terminate their brightness and this variety of background arises because that part of this background which terminates with the luminous part of the moon, through such a comparison with brightness shows itself as darker than it is and that part above where it appears a part of a luminous circle of uniform size, it occurs that wehre the moon is brighter by half than the background in which it finds itself, through comparison with such darkness it shows itself as more luminous in that background than it is, which luminosity in such a time arises from our ocean with other mediterranean /seas/ which at this time is illuminated by the sun which is already below the horizon in such a way that the sea then fulfills the same function to the dark part of the moon that the moon does to us on its fifteenth day when the sun has set. And such is the proportion between the little light which the dark part of the moon has to the brightness of the illuminated part as is that....

(figure)

Figs. 1665-1665: Earthscapes on Leic. 36v.

Figs. 1668-1673: Moonscapes. Figs. 1668-1689, CA251rb; figs. 1670-1671, BM104r; figs. 1672-1673, CA112ra.

    Here the text breaks off once more but the thrust of the argument is clear. Leonardo claims that apparent differences in light intensity on the moon are due to contrast effects of light and dark backgrounds (see above pp. ). He concludes his discussion on Leic 2r with an outline of an experiment:

If you wish to see to what extent the umbrous part of the moon is brighter than the background where such a moon is found, occlude the luminous part of the moon with the hand or with some object more distant from the eye.

    Accompanying these passages are diagrams (figs. 1750-1752, 1754-1755) which demonstrate the same point visually. The largest of these (fig. 1752) serves, in turn, as the starting point for a more elaborate diagram on Leic 7r (fig. 1753), to which he adds the caption: " Here it is proved that in any part of the sky the umbrous part of the moon has some luminosity and that in no part of the heavens is it deprived of this light." In short, by visualising the various relationships between the sun, earth and moon, he can demonstrate how sunlight reflected from the earth accounts fully for all light on the moon (see below pp. ). That which applies to the moon applies equally to the stars, as he mentions in passing on D6r (1508):

And the light which you perceive in them /i.e. the stars/ is not their power, but is merely an image of the sun mirrored in them for these stars have no light themselves, but they do have a surface like the sphere of water suitable to receive and return the light of the sun mirrored in them.

    On F57r (1508) he pursues this problem:

Whether the stars have light from the sun or from themselves.

They say that they have light of their own alleging that if Venus and Mercury did not have light of their own when they interpose themselves between our eye and the sun, they would occlude this sun to the extent that they...cover our eye. And this is false because it has been proven how the umbrous body positioned in the luminous body is surrounded and completely covered by the lateral rays of the remainder of such a luminous body and hence it remains invisible. As is shown when the sun is seen through the ramifications of leaves without foliage /cf. figs. 432-434/. At a long distance these branches do not occlude any part of this sun from our eyes. The same happens with the aforesaid planets which, even if they were without light of their own, as we said, they do not occlude any part of the sun from our eyes.

    Another objection is now raised:

They say that the stars at night appear very bright to the extent that they are above us and if these did not have light of their own, that the shadow which the earth makes, which interposes itself between them and the sun would obscure them, since they do not see it...nor are they seen by the solar body.

    This objection he again counters (fig. 1727):

But these have not considered that the pyramidal shadow of the earth does not reach many of the stars and those which the pyramid reaches, it /the pyramid/ is so much diminished that it occupies little of the body of the star and the rest is illuminated by the sun.

    His passing comment on CA300rb (1508-1510) "The sun never sees shadow" (cf. W12700v) is probably intended as a further demonstration that the sun is the only light source in the universe.

 

Chapter 10.  Diminution of the Earth’s Light

    On F 69v (1508) he mentions a further problem to be dealt with in his treatise:

How the earth functioning like the moon has lost much of its ancient light in our hemisphere through the lowering of the waters as is proved in the fourth book on the earth and its waters.

    This may account for a series of passages on Leic 3r, 8v, 9v, 10v and 20r (1506-1509) as well as those on CA155rb and 92vc (c. 1515) in which he considers the evidence of different layers of shells in the mountains which suggest that the seas reached this height more than once. During the deluge he estimates that the water reached a point seven cubits (Leic 8v, 1506-1509) or ten cubits (CA155rb, c. 1515) above the highest mountains. This chapter on the earth's waters might also have included passages such as those on CA112ra 915050-1508) where he considers the refelctive properties of waves:

Give me a spherical and lustrous body which, positioned opposite the body of the sun, obstructs the entire image of the sun.

A great part of the earth is covered with water and makes a mirror of itself to the universe in which it receives as many images of the sun as are the waves which are seen by the eye and by the sun.

Each wave of the sphere of water, which refelcts the image of the sun to the eye, reflects with it that part of the universe which surrounds it, which sees and is seen by that wave.

The surface of the water which in itself covers a great part of the earth receives the image of the sun in it.

The mixture of the species created by the sun and by the sky which surround it, over the waves of the sphere of the water make a composition of bright and dark and render the brightness of the sun considerably diminished.

    It is possible that this chapter would have included his various demonstrations to show that waves of water function as cylindrical mirrors (see above pp. and figs. 1565-1573).

 

Chapter 11.  The Moon’s Waters

    Consideration of how the earth's waters reflect the sun's light leads to a discussion how the moon's waters do the same. That the moon has oceans he appears to assume from the outset of his writings. On CA80rb (c. 1490-1492), for instance, he mentions that "the moon cannot move the seas as it can move the lakes," without further explanation. On A64r (1492) he refers to the moon's ocean:

The moon when it is entirely illuminated to our sight we see its full day and when, through the reflection of rays of the sun percussed in it and thrown off to us, its ocean casts off less humidity and the less light it gives the more injurious it is.

    When he broaches this theme anew on Mad II 62v (c. 1503-1505) he draws (fig. 1699) the sun reflecting from a wave ruffled body which as the caption explains shows the "moon or if you wish the earth, that is, waves of water." Thus far he has discussed the oceans of the moon as if no one doubted their existence. On CA112va (c. 1505-1508), however, he reports a conflicting opinion:

The adversary says that there are no waves on the moon but minute globules, polished, capable of taking the images of the sun. Here a reply is made with the fourth of the observations of the moon where the variety of the brightness is shown to vary with grater and lesser waves.

    A heading on CA74va (1506-1508) indicates that he plans to write a chapter on these problems:

On the perspective of the reflection of waters with the principles of which it is proven that water is around and on the moon. 9m.

(figure)

Figs. 1674-1675: Demonstrations if the moon were a convex mirror on CA353rb and Leic. 1v.

    A series of passages in the Codex Leicester very probably constitute advanced drafts of this intended chapter. By of introduction he reminds himself on Leic 36v to list "all the contradictions of the adversary to say that in the moon there is no water." Chief among them is the notion that if the moon had water it would spill off and fall to the earth (see below p. ). On Leic 1r he begins with the idea that the moon is a mirror, rejects it and argues that it must have water with waves (fig. 1722):

Here it is shown how the moon, not having...any light of its own, that the light it takes from the sun, it could not take or reflect to us if it wre not a dense and lustrous surface as are the surfaces of mirrors and liquids. Whence being of the nature of a dense mirror and lustrous, this would give of the entire light nopm only the part op as if the eye which sees it were situated in a point which thing would make the moon very small. And if its lustre arises /from a/ liquid body /then the/ reflected rays /will again/ not /be/ of a large nature or brightness, but if they are wavy as we see occurs on the waters of the ocean, then the brightness will give itself to each wave all itself and then all together there will be a great quantity of brightness but not as powerful as at first because the umbrous parts of the wave which.

    Here the text is interrupted. Directly following is a further passage (fig. 1701):

(figure)

Figs. 1676-1677: Concerning the oceans of the moon on BM94r.

Here it is shown that even though the body of the moon is equidistant from the eye which sees it, its size will vary a good deal to this eye because here the moon, being in the east in its fullness, the eye does not see it illuminated except in the part between the lines b /and/ f.

    In the accompanying diagram is a further caption: "the part of the sun that regards the earth and the waves of the ocean and the other waters." On Leic 5r he pursues these problems more systematically beginning with a demonstration why the moon must have waves:

On the moon.

No opaque body is struck by the solar ray with equal illumination but the light is that much more powerful to the extent that this body receives the rays of the sun amidst more equal angles. Whence it remains unequally illuminated and here the sun, being a spherical and opaque body, remains in its parts capable of taking light of equal brightness and this is because that which was said above cannot be if there were water and being spherical water it would not take the solar ray and by reflection render it to our eyes except in an incident ray, that is tiny in comparison to the sun and lunar body. Whence by necessity marine waves are conceded and each in itself takes a solar ray and the darkness interposed between the peaks of the waves interposes itself amidst the luminous species and does not render such a brightness as would such a water were it without waves.

    He interjects, in draft form, what would happen if the moon's waters had no waves:

(figure)

Figs. 1678-1681: Oceans of the moon. Fig. 1678, BM94r; fig. 1679, CA120va; figs. 1680-1681, Leic. 30r.

If it were so, a part of the moon would be of so great a brightness as if it were the sun itself and it would do as one sees occurs in our waters without waves in which one sees the image of the sun reflected in such a way that it appears not to affect human eyes other than does the actual sun.

We see in water without waves a single image of the sun reflected not luminous in another way than the actual sun.

    Immediately following he returns to the idea that water with waves produces darker reflections and explains why:

And we see in an innumerable quantity of waves of water innumerable reflected images of the sun and through the innumerable intervals of the waves which do not receive such an image and remain dark, this multitude of such darkness mixes itself with the said images of the sun and they are confused together through their diminution made after a long distance from our eye /and/ they diminish in such a way that the shape of the umbrous and luminous wave is lost whence there remains such a brightness that the eye can better sustain it.

    An exception to this rule follows:

And why the waves of the sea towards the horizon do not let you see their umbrous concavity is because the tops of their peaks occlude them and one sees a much greater and more united brightness than one sees towards the middle of the moon, which would be the contrary if the moon were not water but a bright mirror, which object is more capable of receiving than its own image.

(figure)

Figs. 1682-1686: Reflection from the moon's oceans. Figs. 1682-1684, Leic. 7v; figs. 1685-1686, Leic. 30r.

    Which leads him to return to the question of the moon having light of its own (see above p. ):

And if it appears to you that the moon has light of its own you would certainly see it and that light which you see when it is new in the middle of its circle is that which sees our earth which receives the light of the sun and makes itself a moon on the fifteenth /day/ and it /the moon/ does the same by day when it has the sun above but the brightness of the sir removes it, as it does that of the stars.

    These passages on Leic 5r serve, in turn, as a draft for a still more comprehensive discussion on Leic 30r where he begins by eliminating the mirror hypothesis:

I say that the moon, not having light of its own..., being luminous, it is necessary that such light is caused by others. Being thus, it is of the nature of a spherical mirror and if it is spherical it takes light pyramidally, which pyramid makes its base at the sun and its angle terminates in the centre...of the body of the moon and intersecting...the surface of such a body, it only takes as much as is the intersection of this pyramid at its surface. And this...moon would only appear the size of such an intersection of the pyramid to human eyes. Whence there would follow with the light of the moon, an effect contrary to that which experience shows and this is that at the new moon, this moon gives with its entire luminous circle...which thing clearly shows that such a lumar body is illuminated more than half of its sphere which could not occur if it were a polished body, as are mirrors.

    This leads him to claim that the moon must have water with waves:

Whence through this we are forced to admit through the 5th of this that the surface of the moon is rough, which roughness could not occur, except in liquid bodies moved by the wind, as we have seen in the sea: the sun is reflected by a few waves near the eye and degree by degree this wave is illuminated more than 40 miles. Whence it is concluded that the luminous part of the moon is water which, if it did not move, it would not illuminate in such a quantity.

    He proceeds to explain why the light reflected by such waves is much dimmer than sunlight reflected in calm water:

But through the motion of this water stirred by the winds, this is filled by waves. And every wave takes the light of the sun and the great quantity of the innumerable waves reflect the solar body innumerable times. Which reflected sun would be as bright as the sun seen where the water does not move, which renders the sun to the eye in its proper brightness, as it is naturally. But there are also innumerable shadows with the waves which interpose themselves between wave and wave. And their species mix with the species of the solar images which are on the waves and the umbrous and luminous species are confused with one another and darken the luminous ray and make it weak as is manifestly demonstrated by the light of the moon. And when the sea of the moon is stormy by winds...the waves are larger and the greater shadows mix more with the weak images of the sun on the waves. And for this reason the moon is less luminous.

    As on 5r, exceptions to this rule again follows:

But when the moon is...full and positioned near the middle of our hemisphere, each wave shows the reflected sun and likewise in the middle of the valleys interposed between the waves, as occurs on the peaks of these waves. For this reason, the moon shows itself more luminous than ever through having doubled the light.

It also shows itself strongly luminous shortly after the new moon because the sun which stands above the moon...percussing the waves at their peaks, these peaks being nearby, the one and the other practically meet in the eye and from this position it occurs that the shadows interposed between the waves do not send their species mixed with luminous species to the eye and for this /reason/ the light of the moon is more powerful. And that which is proven for one luminary, is proven for all the rest.

    When he returns to this problem on Leic 2r he merely refers to

having proved that the part of the moon which shines in water...which reflects the brightness received from it and that if this water were without waves, that it /the image of sun/ would show itself as small, but of a brightness almost equal to the sun.

    Even so he is not yet satisfied. On BM104r (c. 1508) he again mentions why sunlight reflected from water with waves is less intense than sunlight seen directly:

The moon does not shine with its reflected light as does the sun, because the light of the moon does not take the light of the sun continuously in its surface, but in peaks and troughs of the waves of its water and since such a sun is confusedly mirrored in the moon through the mixture of the shadows which are there among the waves that produce lustre, hence its light is not lucid as is /that of/ the sun.

    On BM94v he devotes another folio to the problem of the moon's waves. As in the Codex Leicester (now Hammer) he begins with a claim that the moon has no light:

On the moon

The moon has no light of its own, but is illuminated to the extent that it sees the sun, of which luminosity we see as much as that which sees us. And its night receives as much brightness as is that which our waters take in reflecting the image of the sun which reflects itself in all those /bodies/ which see the sun and the moon.

    This leads to discussion of the moon's waves:

The skins or surfaces of the water of which the seas of the moon are composed and the sea of our earth are always rugged, either a little or considerably, or more or less and this roughness is the cause of spreading the innumerable images of the sun which are in the ridges and concavities and the sides and fronts of the innumerable waves, that is, in as many sites of each wrinkle as are the variety of the sites that the eyes have which see them.

    What would happen if the moon had no waves is now mentioned:

(figure)

Figs. 1687-1695: Sunlight reflected by the moon's oceans. Figs. 1687-1688, Leic.30r; figs. 1689, F77v; fig. 1690, CA112va; figs. 1691-1692, BM94r; fig. 1693, M80r; figs. 1694-1695, CA174vb.

This could not occur if the sphere of water which covers the moon in great part were of uniform sphericity because then the image of the sun would be only one to each and the spherical brightness would be as gold balls, positioned on top of high buildings, clearly show it.

    The effects of water with waves are again contrasted with this:

But if such gold balls were wrinkled or composed of globules like mulberries, a black fruit composed of tiny round globules, then each part of this globule, seen by the sun and the eye, will show to this eye the lustre generated by the image of this sun and hence in a same body one would see many tiny suns, which, on account of the long distance, are often united and appear continuous.

    The remainder of the passage is devoted to a comparison of light from the new moon and full moon:

And the lustre of the new moon is brighter and stronger than when it is full and this is caused because the angle of incidence is much more obtuse in the new moon than in the old, when those angles are most acute and the waves of the moon reflect the earth in their valleys as in their peaks and the sides remain dark. But in the sides of the moon the depths of the waves do not see the sun but only see the tops of these waves and for this reason these images are rarer and more mixed with the shadows of the valleys and this mixture of umbrous and luminous species, thus mixed together, comes to the eye with little brightness and in the extremities will be darker because the curvature of the sides of such waves will be insufficient to reflect to the eye the rays /which have been/ received.

(figure)

Figs. 1696-1701: Sunlight reflected by the oceans of the moon. Figs. 1696-1697, Leic.30r; fig. 1698, Leic.1r; fig. 1699, Mad.II 62v; fig. 1700, CA243va; fig. 1701, Leic.1r.

The new moon by nature reflects the solar rays more towards the eye through such extreme waves than through any other place as the figure [fig. 1698 cf. 1678] of the moon shows which, percussing with the ray a in the wave b reflects to bd where the eye is situated at d.

And this cannot happen in the full moon [fig. 1692] where the solar ray, standing in the west, percusses the extreme waves of the moon to the east from n to m, and it does not reflect to the eye in the west, but it results in the east, deviating slightly from the rectilinear course of the solar ray and hence the angle of incidence is extremely wide.

    A marginal note summarizes this discussion:

The innumerable images which, from the innumerable waves of the ocean, reflect the solar rays which are percussed in these waves are a cause of rendering continuous and very wide brightness on the surface of the sea.

    On F77v (1508) he draws another diagram (fig. 1689) of the eye looking at sunlight reflected in the full moon, this time adding only a brief caption: "The extremities of the moon will be that much more illuminated and will show themselves that much more luminous because in these there only appear the summits of the waves of its waters." He refers to the moon's waters once more on CA155rc (1516-1517, see above p. ) and on CA174vb (1516-1517) again discusses reflection from terse globulent surfaces:

Spherical bodies of globulent and terse surfaces are those of which the surface is composed of various globosity.

(figure)

Fig. 1702: The sun at different seasons on CA332vb.

When the moon is nearer the sun it has a lesser quantity of light from it and conversely.
The uniform sphericity of the terse body renders a single image to the single eye that sees it.

It follows that the terse and globulent surface renders as many images to the eye which sees it as are the globules seen by the real object and by the eye which stands in front of it.

So much greater or less is the number of images of a given object seen on the spherical bodies of globulent and terse surfaces as the eye...seeing such images is more remote or close to the aofresaid terse body.

    As a result of these discussions he has convinced himself that the moon has oceans like those of the earth and that their function in reflecting light is identical. This is a first step in his aim to show that earth and moon are interchangeable in their functions as planets.

 

Chapter 12.  The Moon’s Elements

    The claim that the moon had water contradicted the contemporary theory of cosmology which held that the earth is the centre of the universe. According to this theory, if there were water on the moon, it would fall back to earth, because the heavens were strictly the domain of air, fire and the empyrean. Leonardo does not broach this problem until K1r (1503-1505) where he asks: "The moon /is/...dense and heavy. How does the moon stay /up?/." In the next years he arrives at an ingenious answer. If the moon has water, it must also have elements and will therefore have its own centre of gravity around which it turns, or as he puts it on CA112va (1505-1508):

If the moon has waves and waves are not without wind and wind does not generate itself without terrestial vapours carried by the humidity drawn from the heat which is below the air, it is necessary that the body of the moon has earth, water, air and fire with the same conditions of motion as have our elements.

And if you said that weight is nothing other than the one element taken into the other then it follows that where there are no elements there is no weight. Hence the moon does not have weight in its elements and it cannot fall from its site.

    By 1508 the problem is playing on his mind. On BM94r, for instance, he jots down a series of prelininary thoughts:

No very light object is opaque.
No lighter object remains below a less light /object/.
If the moon has a site in the middle of its elements or not?
And if it has a particular site such as the earth has in its elements, why does it not fall to the centre of our elements?
And if the moon is not in the middle of its elements and it does not descend then it is lighter than another element.
And if the moon is lighter than another element, why is it solid and not transparent?

    On BM94v (fig. 1703, 1508) he describes an unexpected analogy to this phenomenon of the moon not falling from its position:

(figure)

Figs. 1703: Comparison between the earth and an egg-yolk on BM94r.

The yolk or yellow of an egg stands in the middle of its white without descending to any side and it is either lighter or heavier or equal to this white and if it is lighter it should rise above all the white and stop in contact with the shell of this egg. And if it is heavier it should descend and if it is the same /weight/ it could equally stand in one of the extremities as in the middle or below.

    On the same folio he decides that

The moon is an opaque and solid body and if, as according to the adversary it were transparent it would not receive light from the sun.

    In the Codex Leicester (1506-1509) the problem of the moon's elements is discussed in mored etail. On Leic 1r, for instance, he begins by describing a paradox of optics (fig. 1722):

Here there occurs an effect contrary to perspective, that is that those which are more remote from the solar body are seen less, that is, that those which are facing the two poles such as (g) f/and/ m, that the sun does not come except from the part an and the part mr.

    This passage prompts an adversary to challenge his theory concerning the moon's waters;

The adversary opposes

Here the adversary, who admits that the lunar body does not have light in itself, states that through the foregoing proofs he is constrained to admit it, but that he does not believe it to be a liquid body and that if it were liquid it would pour its waters onto the earth and consequently /it could not/ be wavy because there is no wind up there.

(figure)

Figs. 1704-1706: Earth and moon on CA300rb.

    The adversary's own solution to the problem is duly recorded:

But /he believes/ that this lustre is made in the manner of a dense mirror which even if it be in a small part of the moon it enlarges itself as do other lustrous objects /and/ at a long distance this enlargement occupies the other part of the moon at the eye and makes it luminous and consequently shows itself to be with great light.

    Which solution, in turn, is dismissed:

Now he contradicts himself through the 2nd of the foregoing because if it were thus when the place of reflection occured on the far side of the moon such a moon would remain dark and if the new /moon/ showed itself falcated when the place of reflection was so covered in the new moon, the entire moon would appear round and not falcated and such brightness would partly appear outside of the moon.

    On Leic 2v, Leonardo answers the problem with an appeal to common sense and experience:

Here it is disputed whether the ocean, which we have proved stands in the moon,...tends to the centre of the moon as does ours to the centre of the earth or not, and it appears to be so, and that the moon has its elements around it, as does our earth and that if it were not so, it /the water/ would spill itself from the moon and descend to cover the earth together with our ocean and not only the water of such a moon but the entire moon along with its water would descend as a heavy object to the centre of the world. Since it does not do this, by necessity it needs to have a stable site with its elements around it as was said.

(figure)

Figs. 1707-1709: Phases of the moon. Figs. 1707-1708, W12326v; fig. 1709, Leic. 29v.

    On Leic 2r under the heading "On the moon" he opens with the claim: "No solid is lighter than the air." He refers to how he has shown that the moon has waves (see above p. ) and then returns to the question of the moon's weight:

Now it is necessary to prove whether this moon is a heavy or light body, for if it were heavy, acknowledging that above the earth in every degree of height one acquires degrees of lightness, such that the water is lighter than the earth and the air than the water and fire than air and so on, successively. And it would appear that if the moon has density as it has, that it would have gravity, and having gravity that the space where it finds itself could not sustain it and consequently it would have to descend towards the centre of the universe and joint with the earth and if not it, then at least its waters would have to fall and spill from it and fall towards the centre /of the universe/ and leave the moon stripped of its /waters/ and without lustre. Whence, not following that which reason pormises him and is manifest to him, I note that if such a moon is covered with its elements, that is, with water, air and fire and if it likewise sustains itself in that space as does our earth with its elements in this other space,...that those heavy bodies perform that function in their elements which other leavy bodies perform in our elements.

    In short the moon, like the earth, has its own elements and centre of gravity, and as such is not in danger of falling to the centre of the universe. On Leic 36v he reformulates these ideas beginning with the adversary's objections:

(figure)

Figs. 1710-1712: Concerning the moon's orbits? Fig. 1710,BM52v; figs. 1711-1712, BM212r.

Of the moon: all the contradictions of the adversary to say that in the moon there is no water.

Every body...denser than air and heavier than this air,...cannot sustain another thing on it without some other thing. And the more it rises the less it is resisted from the middle. Hence if the water were on the moon it would spill itself from the moon and it would cover our earth because in this moon the water will always be above its air.

    Leonardo then gives his defence:

Here it is replied that if there is water on the moon, then there is also earth on which the water is sustained and consequently the other elements and these sustain water amidst the other 3 elements as here our water amongst the elements has its sequence. And if, according to the adversary, the water of the moon should fall, it should sooner be that the moon falls, as a body /that is/ heavier than water. Now since it does not fall, this is a manifest sign that the water on it and earth with its other elements are not sustained differently /on the moon/ than the elements heavy and light which are sustained here /on earth/ in their space heavier or lighter.

    When he returns to this theme on cA243va (c. 1515) he simply repeats his conclusion without further ado:

If the water of the moon weighed towards the centre of the world it would spill from the moon and fall on us. But the weight is at the centre of its sphere.

Having shown that the earth and moon both have their own elements and independent centres of gravity, he has established that they are effectively identical in nature. It only remains for him to show that they are also identical in function.

(figure)

Figs. 1713-1715: Phases of the moon. Figs. 1713-1714, CA208ra; Fig. 1715, CA243vb.

 

Chapter 13.  Earth and Moon

    As early as 1492 Leonardo had begun playing with the idea that the earth and moon have identical functions as planets. On A86v (BN 2038 16v), for instance, he mentions that: "it might be proved that the moon is another world identical to ours and that the part of it which shines is a sea which reflects the sun and the part which does not shine is earth." On A64r (1492) he takes this comparison further:

And if you were wehre the moon is, it would appear to you that this sun is reflected in as much sea as it illuminates by day. And the land /on earth/ would appear amidst the said water as the dark spots which are in the moon which, when looked at by men standing on our earth show themselves precisely as our earth would appear to men who live on the moon.

    To prove that the earth and moon are fully equivalent, he needs to show that the moon has its cycle of days, months and seasons as does the earth. This requires a systematic study of positions of the moon relative to the sun and earth. This leads him to study the phases of the moon with elementary sketches such as those on CA300rb (figs. 1704-1706, 1508-1510). On BM212r (1500-1505) he sketches eight phases of the moon (fig. 1712, cf. figs. 1710-1711). Further drafts follow on W12326v (1506-1508). Here he keeps constant the moon and systematically alters the position of the sun (fig. 1708) adding a caption which has been interrupted: "the moon in the east which from the...." In a further diagram (fig. 1707) he keeps the sun constant while systematically changing the position of the moon. This approach he develops on CA208ra (c. 1513) where he points out (fig. 1713):

(figure)

Figs. 1716-1720: The Sun, moon and earth. Figs. 17816-1719, BM104r; fig. 1720, CA303vb.

Every evening that the sun finds itself in the west it has opposite it the moon in one of its aspects.

The sun standing in the west, the full moon in the east and that part which it always shows is entirely illuminated.

    Beneath this he draws (fig. 1714) twenty phases of the moon, adding a caption which refers to more phases:

These are the 30 aspects whicht he moon has with the sun, the sun customarily standing in the west each evening.

Here I begin with the moon in the east and the sun in the west and I follow its variety each day until /both/ the moon and the sun are in the west, which are 15 days and proceeding thus until the moon in the east faces the sun in the west another time and thus is finished a lunar /month/.

    He draws the chief phases of the moon again on CA243vb (c. 1513) for which he drafts an explanation, which has been rendered almost incoherent through a mutilated left-hand corner of the folio:

Here is shown...in 15 days being consumed in night in...

This demonstration...as many moons as...days in the month, which are 30 and...12...30 through operating without interruption...the sun finding each...by as much variety it makes day by day...position and figure illuminated...such rule you see every evning of...of a month to the extent is the...ofthe moon from its full moon...in the west to the consummation of such a light in the west.

(figure)

Figs. 1721-1722: Sun's light and moon's shadow on Leic.7v and 1r.

Figs. 1723-1725: Shadows of the earth and moon are on BM100-r, CA28ra and F57r.

Figs. 1726-1727: Shadows of the earth and moon on BM104r.

    In spite of mutilation, the basic sense is clear: he is outlining the chief phases of the moon's thirty day cycle. In the moon's monthly cycle he sees an equivalent to the earth's yearly cycle, as he points out on CA303vb (1505-1508):

The moon has a summer and a winter every month.

And it has greater cold and greater heat and its equinoxes are colder than ours.

Or, as he puts it on CA208vb (c. 1513): "The moon has a year of 12 days and 12 nights."

    A preliminary diagram on CA303vb (fig. 1720) helps us understand his reasoning: each month the moon's orbit around the earth brings it closer to the sun than the earth, and these monthly summers can therefore be hotter than those of the earth. At the other extreme of its orbit, the moon is much further from the sun than the earth. These monthly winters can also be colder than those of the earth. The rough diagram on CA303vb (c. 1508 /-1510/) serves as a starting point for a series of sketches on BM104r (figs. 1716-1719) above which he writes:

Here you have to prove how the earth performs all the same functions with regard to the moon as does the moon with regard to the earth.

    In one series he draws the sun in the north with the moon and earth beneath (figs. 1716-1718), which configuration he draws in more elaborate form on Leic 7v (fig. 1721) and Leic 1r (fig. 1722) and again in slight diagrams on BM100r, CA28raand F57r (figs. 1723-1725). Complementary to these he draws two other diagrams on BM104r (figs. 1726-1727) in which the sun is in the south and the moon and earth are above. Such diagrams may have been a starting point for his plan on F63 (1508) to

(figure)

Figs. 1728-1729: The sun in the east and the moon in the west on CA300rb and BM104r.

Define the earth with its long and its short days in the north and in the south and do the same for the moon and terminate them accurately.

    Just as he studies the sun in the north and south, so too does he study it in the east and west. On CA300rb (1508 /-1510/), for instance, he shows (fig. 1728) the sun in the east and the moon in the west, with the caption: "If the moon is mirror of our earth, if it is in the fifteenth /day/, the earth will be ahlf dark and half illumined or perhaps more than half dark." Corresponding to this he draws a further sketch (fig. 1731) with the sun in the west* and the moon in the east.

* Leonardo who writes in mirror script also reverses west and east such that west is on the right and east on the left.

    A more complex version (fig. 1732) follows on BM104r (c. 1508) where he adds a caption both above: "the earth between the moon on its 15th /day/ and the sun" and to the side: "Here the sun is in the west and the moon in the east on its fifteenth /day." Beneath this he draws a reciprocal situation (fig. 1729) and again adds captions both overhead "moon between the earth on /its/ fifteenth /day/ and the sun" and to the side: "Moon between the earth on the fifteenth /day/ and the sun." This leads in turn to a systematic study of days and nights on the earth and moon. On F64v (1508), for example, he writes the heading:

Obscuration of the sun, moon and earth

And the moon has its days and nights as does the earth: night in the part which is not illumined and day in that /part/ which is illumined.

(figure)

Figs. 1730-1732: The sun in the west and the moon in the east. Figs. 1730-1731, CA300rb; fig. 1732, BM104r.

    Beneath this he draws (fig. 1733) a situation where the sun is again in the west and the moon in the east, with the caption:

Here the night of the moon sees the light of the earth obscured, that is, of its waters and the obscured water sees the obscurity of the sun and the night of the moon lacks the reflection of the solar rays which refelct to it from this earth.

    He redraws this diagram (fig. 1734) on CA208rb (1513) with the caption: "Here the night of the moon sees eclipsed the ocean of the earth (and the earth sees ecli) and the day of the earth sees the sun eclipsed." On CA208va (c. 1513) he draws the diagram a third time (fig. 1735), now adding a more elaborate introduction and caption:

The demonstration that the varieties of light and shade /of the moon/ make relative to the earth, is equal to that which the earth and water of our world make with respect to it.

    This is proved.

Let a be the sun, b the moon, c the earth. Here the earth does not see all the sun and the night of the moon does not take the rays from which the ocean usually reflects the solar rays behind it. Thwrefore in this aspect the earth sees the obscuration of the sun and the night of the moon sees the obscuration of our light or day.

    On F64v (1508) he also draws a complementary diagram (fig. 1737) showing the sun in the west and the moon in the east, with the caption:

(figure)

Figs. 1733-1736: Night of the moon and day of the earth. Fig. 1733, F64v; fig. 1734, CA208rb; figs. 1735-1736, CA208va.

Figs. 1737-1739: Night of the earth and day of the moon on F64v, CA208rb and 208va.

Figs. 1740-1743: Half day of the earth and moon. Fig. 1740, A64r; fig. 1741, CA208rb; figs. 1742-1743, CA208va.

In this other figure is shown the day of the moon being obscured and the night of the day remains deprived of solar rays reflected from the sun.

When the moon is in the east and the sun in the west...the whole day that the moon had when it was with the sun...in the west...is changed into night.

The day that the moon has which looks at the sun in the west will be entirely night when this moon is in the west with the sun.

    This diagram he redraws (fig. 1738) on CA208rb (c. 1513) with the comment: "Here the day of the moon sees eclipsed...the sun, and the night of the earth sees the moon eclipsed." On CA208va (c. 1513) he draws this a third time (fig. 1739), with almost the same caption: "Here the day of the moon sees the sun obscured and the night of the earth sees obscured the light, day of the moon." In addition to these situations he draws a case where the moon stands at right angles to the earth and sun, first on A64r (fig. 1740, 1492), again on CA208rb (fig. 1741, c. 1513) and a third time on CA208va (fig. 1742, c. 1513) to which he adds the caption:

Here the half...day of the earth sees the half...day of the moon and similarly the half...day of the moon sees the half...day of the earth and /both/ the earth and moon see the sun in the west.

Here the sun is in the west. Directly beneath, on CA208va (fig. 1743) he draws the reverse situation with the sun in the east:

(figure)

Figs. 1744-1745: Motions of the moon and sun on CA208rb and 208va.

The same demonstration would occur in the sun were in the east and the earth and moon were in the west, that is that it would do the same as the penultimate, /such/ that the half of the night of the moon sees the night of the earth and the half of the night of the earth similarly sees the midnight of the moon and the midday of the moon does the same with the midday of the earth.

    On CA208rb (c. 1513) he also draws a composite diagram (fig. 1744) showing the moon in four different positions relative to the earth and sun, explaining:

Here these 2 luminaries are without light of their own and to the extent /each/ sees the sun, to that extent it shiens and, standing in this aspect, the eastern evening of the earth sees the day of the moon.

    As an afterthought he interjects the question: "What difference does it make to the light of the moon /reflected/ from the ocean whether it is in storm or in fair weather?" Another composite diagram (fig. 1745) follows on CA208va (c. 1513) this time showing the sun in two different places, which leads him to conclude: "In approximately 30 days every part of the moon /has/ had sun and its night antipodal to the full moon requires 15 days in order to see the sun." His next step is to integrate drawings such as those on F64v, CA208rb and CA208va into synthetic diagrams such as those on CA208ra (figs. 1713-1714) to which he adds both general comments (see above p. ) and particular ones:

(figure)

Figs. 1746-1751: The moon between the sun and the earth. Figs. 1746-1747, F64r; fig. 1748, Leic. 7r; fig. 1749, F85r; figs. 1750-1751, Leic. 2r.

Every evening that the sun finds itself in the west it has the moon in front of it in one of its aspects.

The sun standing in the west, the full moon in the east and that part which it always shows is entirely illuminated.

    Such systematic studies convince himt hat the earth and moon are fully equivalent in their functions and hence he claims, on F94v (1508), for instance: "Our sea has the same influence on the moon as the moon has on us." Even so, he decides to study more carefully situations in which the moon is between the earth and the sun. On F64r (1508), for example, he draws the situation twice (figs. 1746-1747), and adds an unfinished caption:

When the intersection of solar rays which...pass the moon touching its extremities produces a right-angle with the intersection of the rays reflected from the earth which pass the moon touching its extremities then....

    This leads to a further sketch on F84r (fig. 1749, 1508) which he then develops dramatically on Leic 2r (figs. 1750-1751) and Leic 7r (fig. 1478) where he asdds a long explanation:

On the water of the moon.

Here it is proved that in some aspect of the sky the umbrous part of the moon has a luminous part and in some part of the sky it is deprived of this light.

Let ab be the site of the sun. Let en be the site of the moon. Let pq be that of the earth. I state that the dark part of the moon eo is seen and illuminated by the part...of our seas...which are seen by the sun in psq, which seas function...with respect to the seas...which cover this moon, as the seas of this moon function with respect to the seas...of the earth when that part of the sea shows itself and /when/ the moon is positioned in the east and the sun stands in the west....

Therefore the new moon positioned together with the sun in the west will show itself with two sorts of light, of which the one comes from the sun and the other is reflected from our seas, percussed and illuminated by solar rays. And if it were not so the comparison of the incident solar ray which is that much more powerful where it percusses the moon than is that of the solar day reflected in the moon from our seas, you would see the part of the moon surrounded by the illuminated circle illuminated by the sun appear to retain in itself some of the brightness which in itself is of so much lesser power as is the illuminated part of the sun when our seas, illuminated by the sun are of less brightness than the light of the sun. And if you wish to see the true comparison, hold in mind when, the moon and sun are in our hemisphere by day.

    He goes on to mention another situation:

But when the moon passes the aspects of the meridian towards...the east, then only that nocturnal part of the moon receives the solar ray reflected from our oceans which is facing the said oceans and the other part remains solely illuminated by the rays reflected by the stars which stand as its objects.

(figure)

Figs. 1752-1753: Sun, earth and moon on Leic. 2r and 7r.

Figs. 1754-1755: The new moon on Leic. 2r.

    He ends with a consideration when moonlight is brightest:

But when the moon passes above us, the sun standing at midday, this moon receives in its nocturnal seas a great light reflected from our seas, reflected from the sun, which /seas/ show themselves to this moon as does the moon to us on the fifteenth day in the middle of our night. And the greatest light is shown when it interposes itself between us and the sun, because then the reflected rays...of our seas, which reflect the sun are shorter and more powerful than in any other aspect of the moon in the sky. And that part of the brightness which the moon shows at this time arises from the sun which is not occluded by the moon and these reflect to the oceans of this moon and if you cover the part of the sun which advances beyond the moon you will see the night of the moon produce some brightness more than if you cover this sun behind it.

    In the late period he returns to this question of where moonlight is brightest on CA243va (c. 1513):

Why the moon surrounded by the illuminated part of the sun in the west has a greater brightness in the centre of such a circle than when this is eclipsed by the sun. This occurs because, in eclipsing the sun, it o'ershadows our ocean, which does not occur when it is in the west and the sun illuminates this ocean.

    Which leads him to ask:

Why when the centre of the moon is in the central line which extends itself between the centre of the sun and the eye, the moon does not show an illuminated circle around itself, it being seen by the sun on this side of the same diameter, and the eye even it should see it on this side of the diameter, it should see that which is seen by the sun on this side and does not see it? And this occurs because our sight is so much below the moon to the extent that one cannot see as far as the sun sees.

    This he demonstrates (fig. 1677):

This is proved. Let a be our eye. Let cdb be the umbrous pyramid of the moon clothed with the luminous rays of the sun which, terminating in the point b, is such that what is inside this pyramid does not see any part of the moon which is seen by the sun.

Hence it is concluded: a being inside this pyramid does not see any part of the moon which is illuminated by the sun.

 

Chapter 14.  The Earth as a Star

    Leonardo shown that at a certain distance the earth functions as a planet equivalent to the moon. But the ultimate aim of his treatise of astronomy is to show that at a great distance the earth functions as a star. This aim emerges gradually. On F5r (1508) where he describes looking at stars through a small aperture (see above p. ) and begins thinking about his future treatise, this aim is still implicit:

Now think of how our star would look at so great a distance and then consider how many stars, both in length and width one could put between the stars, which are seen in this dark space. I can never do other than blame many of the ancient authors who say that the sun does not have a size different from that which it shows. Among these [authors] was Epicurus. And I believe that the reason derives from the fact that a person who sees a light placed in our air equidistant from the centre, never sees its size diminished at any distance, and the reason for its size and power, I reserve for the fourth book.

    By F25r (1508) his aim is clear and hence his outline has the heading: "Order to prove that the earth is a star." This aim he restates on F93r:

Prove how, if you stood on the moon or a star, our earth would appear to have the same function with the sun as does the moon.

    On F94v, he puts it even more clearly:

My book aims to show how the ocean...makes our world shine in the fashion of /the/ moon and from a greater distance appears as a star.

    A slightly more detailed statement of purpose follows on F56r

In your discourse you have to show that the earth is a star nearly like the moon and thus you will prove the nobility of our world.

And likewise you will make a discourse on the size of many stars, according to the authors.

    Although no lists of the sizes of stars have come down to us, a few scattered notes give us hints concerning this intended final chapter of the treatise on astronomy. On CA112va (c. 1505-1508), for instance, he describes:

How the earth is a star.

The earth through the sphere of the water which covers it in great part which takes the image of the sun and brightens the universe as do the other stars, demonstrates that it too is a star.

The water which covers a great part of the earth with itself receives in its surface the image of the sun and with this it brightens the universe. It makes a star of itself with the same brightness that one sees the other stars make for us.

    He develops this idea on D6r:

How if the sphere of water diminished to the appearance of a star through long distance, the image of the sun would occupy it entirely.

...If through the long distance, which the eye had from the sphere of water, the sphere of water diminished to the size of a common image of the sun as is shown in perspective you would see the sphere. of this water when...seen by the sun

    Which leads him to raise a related question:

How distance makes stars many times larger than the earth appear minimal.

This is proved in perspective how things remote from the eye even if they are very large show themselves of a minimal size which thing, without too many demonstrations...if you raise...the eyes to the starry heavens you will see many stars in this...which are much larger than the earth and appear minimal through the long distance and the light which you see is not their own power but an image of the sun which is mirrored in them because in themselves these stars have no light but have a surface like the sphere of the water apt to receive and render the light of the sun mirrored in them.

(figure)

Figs. 1756-1759: Plans for overies. Fig. 1756, B21v; fig. 1757, B13r; figs. 1758-1759, CA8va.

    In the late period he returns to this phenomenon once more on CA208vb (fig. , c. 1513).

From a to b/c/hk is so remote from the eye that it diminishes and makes itself equal to the eye. I say this because I have already said that if the eye were like the world or the world diminished at a distance such that it made itself equal to the world, this would be seen all illuminated as /are/ the stars and the moon.

    Leonardo's quest to establish that the earth is a star may, in turn, have prompted his interest in overies. He drafts such an instrument on B21v (fig. 1756, 1490-1491) with the caption: "Instrument of the spheres." This he develops on B13r (fig. 1757, 1490). It is likely that his diagrams on CA8va (figs. 1758-1759, 1493-1495) also relate to an overy. Hence, just as he had built models in order to understand the human body (microcosm, see vol. one, part II.3), he also builds models to understand the planetary system (macrocosm).

 

Conclusion

    As in the case of Leonardo's work on light and shade, his studies of astronomy follow a plan which is sufficiently detialed to permit a reconstruction of his proposed treatise. This treatise confirms the scope of the work that he envisaged, ranging from illusions in the eye, the nature of basic elements, to relationships of earth, moon and sun and ultimately, the equivalence of planets and stars.

    The treatise opens (chapter one) with a demonstration that the eye is a source of astronomical illusions or, as he puts it on CU15 (1500-1505), that "the science of astronomy is born of the eye." A consideration of astronomical illusions due to atmosphere refraction follows (chapter two). Some of his contemporaries hold that the moon is either a convex mirror or a body with variable density and transparency. To counter these theories he examines the reflective properties of different kinds of mirrors (chapter three) as well as of dense and transparent bodies (chapter four). In this context he also considers the nature of visual pyramids (chapter five).

    The prevailing cosmology of the time assumes that the earth is at the centre of the universe and that if there were solid elements on the moon or other planets, these would fall from the heavens towards the centre of the earth which is also centre of the world. To confute this theory Leonardo makes his own study of the nature of the elements (chapter six) and concludes that the earth is not at the centre of the world but only at the centre of its elements. Similarly the moon and other planets are also at the centre of their own elements (chapter seven).

    An outline of the physics of light and shade (chapter eight) next leads to the claim that the sun is the only light source in the universe (chapter nine). From a distance the earth's light is due to sunlight reflected from its oceans. Because these waters have receded since the time of the flood, the earth's light is now less (chapter ten). In like manner, the moon also has oceans which reflect sunlight (chapter eleven), as well as elements (chapter twelve). Indeed, the functions of earth and moon, in terms of their having days, nights, summers, and winters are equivalent (chapter thirteen), and if seen from a still greater distance, the earth functions as a star (chapter fourteen).

    Leonardo's treatise On the Earth and its Waters thus emerges as a synthesis relating microcosm and macrocosm, beginning with the sight of a single eye and ending with a vision fo the universe. Vasari1 had a good reason to emphasize Leonardo's work in astronomy.

    In one late note Leonardo does on to claim that the sun does not move (K/P 127r, W12669v) but he does not pursue the idea.2 His concept of the universe remains geocentric. Even so he implicitly challenges traditional cosomology. He has abandoned a heirarchical notion of the elements with a chain of being (Lovejoy3) from the baseness of earth to the purity of the heavens. Earth and star are now equivalent. He has opened the way for an infinite universe, but it remains for Copernicus, Brahe, Kepler and Galileo to explore its dimensions.


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Last Update: July 11, 1999