SUMS

Dr. Kim H. Veltman

"Images All in All and All in Every Part"


1. Introduction
2. Classical and Mediaeval Sources
3. Mirror Demonstrations
4. Camera obscura Demonstrations
5. Further Drafts and Statements
6. Philosophical and Theological Dimensions
7. A Systhesis of Circular and Pyramidal Propagation
8. Conclusions

 

1. Introduction

    The concept that images are "all in all and all in every part" can be traced back to Antiquity and is found in various mediaeval sources. Leonardo is not content merely to cite such sources. He develops demonstrations using both mirrors and camera obscuras. He drafts numerous versions of the concept exploring its philosophical and theological implications. His efforts lead him to a synthesis of circular and pyramidal modes of propagation: a visualization of the ubiquity of images.

 

2. Classical and Mediaeval Sources

The idea that images are everywhere in the air can be traced back to Lucretius' Of the Nature of Things:

For every outside streams away
From off all objects since discharge they may...
From things there must be borne, in many modes
To every quarter round, upon the moment,
The many images of things, because
Unto whatever face of things we turn
The mirror, the things of form and hue the same respond.1

    Lucretius' reference to a mirror in this connection is particularly interesting. It recurs in Alhazen2, and Leonardo. Two paragraphs later Lucretius returns to this theme:

To such degree from all things is each thing
Borne streamingly along and sent about.
To every region round; and nature grants
Nor rest nor respite of the onward flow.3

 

(figure)

Fig. 132: Punctiform propagation of light in a seventeenth century interpretation of Lucretius.

    In the paragraph that follows he pursues the idea:

Now these same films I name are borne about
And tossed and scattered into regions all.
But since we do perceive alone through eyes
It follows hence that witherso we turn
Our sight, all things do strike against it there with form and hue.4

    The significance of these passages in Lucretius becomes apparent through a seventeenth century optical writer such as Zahn5 (fig. 132) who cites the Roman author to illustrate the concept which Leonardo expresses with his description of images as "all in all and all in every part." Lucretius refers to both this concept and to a cone of vision but does not relate the two. Ptolemy's approach in the Optics is comparable. In the eleventh century Alhazen builds on this notion of a cone or pyramid and emphasizes how "light and colour are reflected from every point of a polished surface."8 In the thirteenth century, Witelo goes further to argue that what applies to light and colour applies equally to the forms of objects: "The surfaces of terse polished bodies of whatever shape they may be, reflect the lights, colours and forms of things opposite from each point [of their surface] along straight lines."9 Witelo's version of Alhazen's proposition concerning reflection in mirrors and pyramidal diffusion is therefore somewhat different:

    Between the points of forms falling on the surface of any mirror and the surface opposite the mirror, it is necessary to draw infinite pyramids, cones and bases...

...Therefore the entire form of the body will be in any one point of the mirror and the form of any point of the body in the entire surface of the mirror. Hence as many as are the points on the surface of the mirror, as many are the pyramids of the form of the object terminating at the entire surface.10

    Whether this form is a mental image or a physical one troubles neither Witelo, nor his younger contemporary, John Pecham, who is more intent on emphasizing that pyramids are propagated from every point of objects:

Any point of a luminous or illuminated object simultaneously illumines the whole medium adjacent to it (Prop.3).... The pyramid of light originating from the whole luminous or illuminated object terminates at any point of the medium (Pro.4).... Pyramids proceed from pyramids at every point of an illuminated medium (Prop.5).... Radiant pyramids proceeding from the same or different surfaces differ in effective strength and weakness at different points in the medium (Prop.6)....11

    To illustrate this paradoxical concept that pyramids and images stem from every point of objects and that they are everywhere in the air, Alhazen had used the example of images in a mirror.12 Witelo13 and Pecham14 also use this image in passing. Leonardo da Vinci writing in the 1490's, develops it.

 

3. Mirror Demonstrations

    One of Leonardo's earliest appeals to a mirror to demonstrate that images are "all in all and all in every part" (`9V, 1492) has already been cited in connection with percussion (cf. above pp. ). This is followed by a second example involving mirrors taken from everyday experience:

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

    On CA138rb (1492) he considers an example with two mirrors positioned opposite one another:

Perspective
The air is filled with infinite similitudes of the things, which are distributed beneath it and all are represented in all and all in one and all in each, whence it happens that if there were 2 mirrors turned in such a way that by a straight line they faced one another, the first would be mirrored in the second and the second in the first. The first which is mirrored in the second carries with it all the similitudes of itself with all the similitudes that are represented in front of it, among which is the species of the second mirror and from similitude to similitude they go into infinity in such a way that each mirror has in front of it mirrors, one smaller than the other and one in front of another.

    Following a digression on vision, to be examined later (see pp. ) he gives further examples of mirrors to demonstrate that images are "all in all and all in every part":

That the species of all things are [dis]seminated through the air is seen by the example of many mirrors in a circle, which mirror one another an infinite number of times and joining one another it leaps ahead to its cause and there diminishing it again leaps to the object and then returns and thus it does an infinite number of times.
If you place a night light between two flat mirrors which have an interval of one braccio you will see in each of these mirrors infinite lights, the one less than the other.
If at night you place a light beneath the wall of a room, all the parts of this wall will become tinted by the similitudes of this light. And all these which will be seen by the light, the light will see similarly, that is, when there is no opposition betwixt them which [dis]rupts the concourse of the species.
This same example appears better in the concourse of solar rays which, all in all and each in itself carries to its object the similitude of its cause.... That each body by itself fills all the counterposed air with its...similitudes and that this same air is capable at the same time, of receiving in itself the species of infinite other bodies which are before it, is clearly demonstrated by these examples. And each body appears all in all of said air and all in every minimal part of it, all in all and all in every minimal part, each in all and all in the part.

    He again uses two mirrors opposite one another to illustrate the principle that images are "all in all and all in every part" in a rough draft on BM Arundel 186r (c.1492):

All bodies have their similitudes and species infused and mixed being luminous
Hence all and the similitudes of bodies has
Hence all and the part of the species of bodies appears all in all and in the part of the air opposite it and the body of the air appears in all and in the part of the surface of these bodies.
Whence we can clearly state that each body is all in part in each part and in all the opposed bodies as is demonstrated by mirrors opposite one another.

    This he crosses out and re-writes on CA179vc (C.1500):

All bodies have infused and mixed all their species and similitudes in all the quantity of the air opposite it.
The species of every point of corporeal surfaces are in each part of that air
All species of bodies are in each point of the air and part of the similitude of air is in each point of the surface of bodies. Thus part and all of the species of a body appear in all and in part of the air opposite it. And the body of air appears in all and part of the surface.
Whence we can clearly say: the similitude of each body is all and in part in each part and in all of the opposite bodies reciprocally, as is seen in mirrors, one opposite the other.

    On Leic.7v (1504-1508) he again mentions this mirror demonstration: "The sun is mirrored all in all and all in every part of the water opposite it." As will be shown later (Pt.4:3) this principle has important consequences for his theories of astronomy. In the late period Leonardo mentions the mirror demonstration once more in connection with problems of visual perception on W19076r (K/P 167r, c.1513):

Describe how no object is terminated by itself in a mirror, but the eye which sees it inside such a mirror, terminates it. For if you represent your face on a mirror the part is similar to the whole, such that the part is all in all the mirror and is all in every part of the same mirror and the same occurs with the entire image of the entire object placed opposite this mirror etc.

 

4. Camera Obscura Demonstrations

    In the eleventh century Alhazen had used a pinhole in a darkened room to demonstrate that light and colour do not mix when passing through an aperture.15 From this he concluded that light comes "from each point of each body."16 Witelo used the same demonstration.17

    Leonardo builds on this tradition. In the passage on A9v (1492) cited earlier (p. ) he begins by comparing the waves of a stone in water and the waves of sound in air. These he compares to light which, he claims, appears "all in all." Next he draws a sketch (fig. ) and notes: "this is proved by experience because if you close off a window facing the west and make an aperture...." The meaning of this unfinished phrase can be reconstructed with aid of diagrams (figs. 136-138). An object such as a tree (AB) sends forth its images in circular waves (C) which pass through a pinhole (D), continuing in the shape of a now inverted pyramid (E) to appear inverted on the wall opposite (B,A).

 

(figure)

Figs. 133-138: Demonstrations of the "all in all" principle using a camera obscura. Fig. 133, A9v; figs. 134-135, CA373rb; figs. 136-138 Author's reconstructions.

 

(figure)

Fig. 139: Illustration how images are all in all and all in every part from Mario Bettini's Apiaria universae, (1646).

    Leonardo is fascinated how a pinhole, effectively a single point, conveys the images of an entire object. Indeed this point conveys the images of all surrounding objects, which send out circular waves passing through one another without interference, like the waves produced by pebbles thrown into water. Leonardo's sketch shows a pinhole in a shut off window (A. in fig. 137) through which pass two laterally positioned objects (C. and D. in fig. 136) which then appear inverted on the two side walls (as C., and D. in fig. 137).

    With Leonardo, a sketch that is difficult to interpret is usually presented more clearly elsewhere. In this case we need to return to the draft for A9v on CA373rb (1490-1495). Here he draws two sketches. One is a simple line drawing showing the inversion principle in a camera obscura (fig. 134). The second sketch (fig. 135) shows two objects, analogous to these on A0v, intersecting one another without interference in a camera obscura. A seventeenth century illustration from Mario Bettini's Apiaria universae demonstrates the basic principle more dramatically (fig. 139). Leonardo pursues the problem some sixteen years later on W19151r-19152r (K/P 118r, 1508-1510), beginning with a sketch (fig. 140) to which he adds letters, which he subsequently forgets to mention in the text:

How we conclude that a surface reduces itself in a point.
The angular surface reduces itself into a point when it terminates itself in its angle, or if the
sides of such an angle are produced in a straight line, then, behind that angle another surface
will be generated [that is] less or equal [to] or greater than the first.

 

(figure)

Figs. 140-141: Camera obscuras to demonstrate that images are "all in all and all in every part" on W19151r-19152r (K/P 118r).

    He then draws a second sketch (fig. 141) which shows two light sources of different colours, yellow and blue, passing through the point of an aperture without mixing. Alongside he explains:

How innumerable rays of innumerable species can reduce themselves to a single point.

Just as in a point all the lines pass without occupying one another through being incorporeal, so too can all the species of the surface pass through it, and just as every given point sees every opposite object, every object sees the natural point opposite. The diminished rays of such species can also pass through this point, beyond which passage they are re-formed and the size of such species again grows. But their impressions will be reversed as is proved in the first of the above, where it is said that every species is intersected at the entrance of the narrow apertures made in material of minimal size.

    A note directs us to: "read opposite in the margin":

In accordance with the inversion of the central line necessity requires that all the species which enter by thin and narrow apertures in dark places are turned upside down along with all the species of bodies which invert them.

    Comparison of this late passage on W19152r (K/P 118r, 1508-1510), with the early passage on A9r (1492) points to the continuity of Leonardo's thought. Consideration of further camera obscura demonstrations confirms this. On A93r (BN 2038 13r, 1492), for instance, he begins with a general proposition:

all bodies together and each in itself fills the surrounding air with its infinite similitudes which are all in all and all in the part carrying with them the (essential) quality of the body, the colour and shape of their source.

    He then draws (fig. ) three light sources, red, white and yellow respectively which penetrate the aperture of a camera obscura without interference, beneath which he adds:

That bodies with their similitudes are all in all the surrounding air and all in the part in body, shape and colour is clearly demonstrated by the species from many various bodies which are produced to a single perforated point where with intersected lines, through contrary pyramids they carry things inverted to the first dark wall, the reason for this is...

`Here the text breaks off and no reason is given. But on CA135vb (c.1492) Leonardo produces two further drafts of this camera obscura demonstration; repeating the phrase "the reason is," and this time providing an answer. The draft opens with a familiar phrase: "Every body is all in part of the air and each in all the opposite air, all in part and all in all." This he restates: "Every body appears by similitudes all in every part of the (opposed air) counterposed air and ag-(appears)-ain appears." He goes on to draft a related idea:

But since (the air) each body appears by similitudes (in each part of the) air positioned opposite, that is all the object in all in the part, all the objects in all the air and all in the part, speaking of this air which is capable of receiving in itself the straight (lines of rays ra-) and radiant lines of the species sent by the objects, whence for this [reason] (in) [it] appears (to be) necessary to confess it to be the nature of this air (which attracts like a lodestone to itself the similitudes of the things positioned in front of it.

    Immediately following Leonardo turns to his camera obscura demonstration (which Pedretti dates c.1508) under the heading:

Proof how all the objects positioned in a site are all in all and all in a part.
I say that if a face of a building or piazza or landscape (campagna), which is illumined by the sun has opposite it a habitation and in this face [of a wall] which does not see the sun a little round aperture be made, that all of the illumined objects sidestep their similitudes through this aperture and appear inside the habitation on the opposite wall, which should be white, and they will be there exactly (a punto) and inverted and if in many points of this face [of the wall] you made similar apertures, there would be a similar effect in each.
Hence the species of the illumined things are all in all of this face [of the wall] and all in every minimal part of it.

    Why this should be so he now proceeds to explain

If the object positioned...opposite the eye sends to it from itself the similitude (to it), the eye also sends its similitude to the object. If from the object and through the departed similitudes nothing streams off from any cause neither to the eye nor to the object (the eye does the same), then we can sooner (say) believe (it) to be the nature and power of this luminous air (whi) which attracts and catches in itself the species of things which are in front of it, rather than the nature of the objects which sends species through this air.

    This he redrafts in terms of his "all in all" concept:

If an object positioned opposite the eye (the eye) sends to it a similitude from itself, the eye will have to do the same to the object, whence it would be convenient that these...species were spiritual powers (virtu spirituali). Being thus...it would be necessary that each thing would be diminished immediately because (the air) each body appears by similitudes (in each part of the) in the air positioned opposite, that is, all the object in all the air and all in the part; all the objects in all the air and all in the parts, speaking of this air which is capable of receiving in itself the straight (lines of rays ra-) and radiant lines of the species sent by objects. Whence for this [reason]...it appears (to be) necessary to confess it to be the nature of this air...which attracts like a lodestone to itself the similitudes of the things positioned in front of it.

    His camera obscura demonstration (which Pedretti dates c.1508) follows:

Proof how all the objects positioned in a site are all in all and all in a part.
I say that if a face of a building or piazza or landscape, which is illumined by the sun has opposite it a habitation and in this face [of a wall], which does not see the sun, a little round aperture be made, that all of the illumined objects send their similitudes through this aperture and appear inside the habitation on the opposite wall, which should be white, and they will be there exactly and inverted and if in many points of this face [of the wall] you made similar apertures, there would be a similar effect in each [cf. fig. 139]. Hence the species of the illumined things are all in all of this face [of the wall] and all in every minimal part of it.

    Why this should be so he now explains:

The reason is [that] we know clearly that this aperture must send some light into this habitation and the light which renders it a mediator is caused by one...or many luminous bodies. If these bodies are of various colours and various kinds, the rays of the species will be of various colours and various kinds and the representations on the dark wall will be of various colours and kinds....

    This example of various apertures in a room again illustrates how Leonardo adapts and transforms his sources. Both Alhazen18 and Witelo19 had mentioned that there would be as many images as there are candles in front of an aperture. They had also made the proviso that one should have a white background.20 Nonetheless, it is Leonardo's idea to use this as a demonstration that the images of objects are everywhere in the air:

The reason is [that] we know clearly that this aperture must send some light into this habitation and the light, which renders it a mediator is caused by one or many luminous bodies. If these bodies are of various colours and various kinds, of various colours and various kinds will be the rays of the species and of various colours and kinds will be the representations on the dark wall.

    The example of various apertures in a room here cited again illustrates how Leonardo adapts and transforms his sources. Both Alhazen18 and Witelo19 had mentioned that there would be as many images as there are candles in front of an aperture. They had also made the proviso that one should have a white background.20 Nonetheless, the idea of using experience to demonstrate that the images of objects are everywhere in the air is Leonardo's. It is typical for him to consider various alternatives. Hence having described how three lights pass through one aperture on A93r (1492), he proceeds on W19150v (K/P 118v, 1508-1510) to consider a case in which three objects pass through two apertures to produce six images (fig. ). This time he letters the diagram and the accompanying text serves partially as a caption:

The species of the bodies are all infused through the air which sees them and all in every part of it. To test it let a, c, [and] e. [be] objects, the species of which penetrate a dark place through the apertures m. [and] p. and impress themselves on the wall f.; opposite these apertures, which impressions will be made in as many places on this wall as will be the number of the above-mentioned apertures.

    On W19149r (K/P 118r, 1508-1510) he explores the role of aperture size in this phenomenon:

Principle how the species of bodies intersect themselves at the edges of apertures penetrated by them.
What difference is there in the penetration of species that pass through narrow apertures and those that pass through large apertures? On those that pass along the sides of umbrous bodies?

    He interjects a subtitle before answering these questions:

On the movement of the species of immobile objects.

The species of immobile objects are moved, moving the lips of that aperture through which the rays of the species penetrate and this occurs through the 9th which states that the species of any body are all in all and all in every part of the site surrounding them. It follows that moving one of the lips of the aperture, where such species penetrate a dark place, it permits the rays of species which were in contact with it and joined themselves with other rays of these species which were remote etc....

    In previous passages he had used the camera obscura principle to confirm that images are "all in all." Here he uses the "all in all" concept to confirm principles of the camera obscura. Beneath this passage he draws a diagram (fig. ) showing four objects passing through an aperture. He adds nine letters to the diagram but these receive no caption. Diagram and text function independently. He now writes a further heading: "on the motion of right or left, superior or inferior edge" of the aperture, which introduces a more detailed analysis of the phenomenon:

If you move the right side of the aperture, then the impression will move left of the object [on the] right which penetrates through this aperture and all the other sides of such an aperture will do the same and this is proved by the second of this which states that all the rays which carry the species of bodies through the air are rectilinear. Hence, with species of very large bodies which need to pass through minimal apertures and after such an aperture to recompose to their maximal dilation, it is necessary to generate the intersection.

    Beneath this he in turn draws three diagrams (figs. ) of which only one is lettered and none is explained.

 

5. Further Drafts and Statements

    In addition to these demonstrations involving mirrors and camera obscuras, there are a number of other drafts and statements concerning the idea that images are "all in all and all in every part." These are primarily of interest because they reveal how painstakingly he juggles with various combinations of a key word term. On BM186v (c.1492), for example, he makes a very tentative draft:

the all the quantities of bodies send all the quanti.
the pure air is passage is capable of receiving in itself similitudes fromwithout interval of time. Every...corporeal quality which...is percussed by solar rays or by the cause from an illuminated or other luminous body and all the parts of this are capable of all the things seen by this.
the entire quantity of the pure air positioned opposite bodies sees the part

Entirely and see [n] by the bodies positioned opposite it. Every particle of a point of this body sees all the said air and all this air sees each point and each point of the air is capable of all.

    Immediately following comes a second draft:

Every part of the air receives in itself all the species of bodies which see it and are seen by them at the same time. Every point of the air is capable of all the species of bodies seen by it (and each point of the bodies is capable of receiving in itself the entire similitude of the air positioned opposite it (the entire (air) sum of the air sees each
all the pure air is capable of all...the bodies...
and the species of these (and each body is capable of the similitude of the air seen by it.

    On CA396rb (1492) a clearer formulation emerges:

The air is all in all and all by similitudes in the parts positioned opposite it.
If there is not an opaque body int he air, all of this is capable of all and of the part and the part of the part and of the all.

Whence we shall say that the air is all in all...interwoven and filled with infinite rays of species of bodies which are situated in it and this air is filled with infinite points and every point is indivisible and all the parts of this indivisible and all the species are capable of the parts of the bodies opposite each other and in this point they are entirely united and entirely divided and distinct without interference (confusion) by one another.
And the pyramids of species are in all this air without occupation of one another and each in itself is divided all in all and united. And although the species go pyramidally to the eye, the eye does not recognize [them] if it does not...make contrary pyramids facing the thing seen.

    These drafts are not always long-winded. For instance, on CA138vb (1492) there is a short version:

On air.
The air is (filled with that) all filled with species (equal) caused by the objects which surround them.
(Again) I say that the objects fill the air with their similitudes.

    On CA120rb (c.1505) there is another short draft, this time interrupted: "The air is all in all and all in every part of this filled with the similitudes of bodies which in themselves include...." The drafting process continues into the late period as witnessed by a series of crossed out statements on CA345rb (1505-1508):

(The ... species of any visible object are all infused in all the counterposed air and are all separate in each part of the same air.)
All (the species see) the objects seen from a single point are seen again from the same point.
The species of the objects which, confusedly with their mixture fill of themselves the air opposite are all in all this air and are all in every part of it.
The species of the object are all in all the air opposite these objects and all in every part of this air.
The objects fill their species in the air seen by these objects.
All that air sees each object, ... which is seen by that object.

    For Leonardo drafting is obviously not just a matter of crossing out a troublesome word. It is a process of reformulating an entire phrase, transforming its elements with a thoroughness that recalls the systematic manner in which he plays with perspectival variables. Immediately following these drafts on CA345rb he interjects a passage on light and shade (cf. p. below) and then continues:

No thing is seen which does not send its species through the air
Since it is visible...
One cannot see a spirit in the countryside that is other than what one sees.
Therefore no spiritual or transparent thing can see anything positioned opposite it, because it is necessary [that there is] in it a dense and opaque object; and if it be thus, it is not called spirit.
Proof how no thing can be seen except through an aperture through which passes the air full of the species of objects, which intersect one another ... on the dense and opaque sides of the aforesaid apertures; and for this [reason] nothing that does not have a body can see either the shape or the colour of any object, such that it is necessary that there is a dense and opaque object through which the species of objects impress their colours and shapes.

    Throughout the mediaeval tradition it had remained an open question whether the images of sight are spiritual (psychological) or corporeal (physical). With statements such as the above Leonardo brings the problem of vision into the domain of physics. We are a significant step closer to Kepler's concern with the pictura rerum which he distinguishes from the imagines rerum. On the reverse of the same folio (i.e. CA345vb, fig. , 1505-1508) he launches into a eulogy of the aperture in the camera obscura:

Since the species of objects are all in all...the air positioned opposite them and are all in a point of it, it is necessary that the species...of our hemisphere...enter and pass with all the celestial bodies through the natural point, in which they are infused and mix in the penetration and intersection of one another, as the other of the one in which the species of the moon in the east and the species of the sun in the west are united and infused together with our hemisphere at such a natural point. O marvelous necessity, thou, who with reason constrain (with supreme reason) all the effects to participate in their causes, and with supreme and irrevocable laws...every natural action obeys thee with the shortest (law) operation. Who would believe that this...very smallest of spaces would be capable of the species of the entire universe? O mighty action, what ingenuity could penetrate such a nature? What tongue would there be who could unravel so great a wonder? Certainly none. This it is that guides human discourse to divine things, etc.

And [they] imprint themselves in the...wall, opposite the above mentioned perforated point in a thing wall, and by this means the eastern part will impress itself on the western part of this wall and the western on the eastern and likewise the northern on the southern and conversely.

    A recent commentator has claimed that the section between the markers...constitutes an "unconnected" digression from "physical phenomena into meditations on philosophical matters."21 That Leonardo himself would not have considered this an "unconnected" digression becomes clear if we turn to the philosophical and theological traditions.

 

6. Philosophical and Theological Dimensions

    The philosophical conotations of the "all in all" concept can be traced back in Antiquity. Lucretius reported it as the view of Anaxagoras and attacked it.22 In the fifteenth century Nicholas of Cusa in his Docta Ignorantia returns to Anaxagoras' notion that "everything is everywhere" and uses it as starting point for his own concept of God who constitutes "all in all and everything in every point" (omnia in omnibus esse constat et quodlibet in quolibet).23 Nicholas of Cusa uses the example of two mirrors positioned opposite one another to demonstrate this.24 An edition of Cusa's Docta ignorantia was published in Milan in 1502. Luca Pacioli, in the introduction to his Divina proportione (15-09) adapts Cusa's concept of God when listing reasons for the title of his book:

the fourth reason is that just as God cannot change and is all in all and all in every part (tutto in tutto e tutto in ogni parte) so too does our proportion present it always in every quantity continuous and discrete.25

    Pacioli's words to describe God are precisely those, which Leonardo uses to describe the nature of images. Pacioli and Leonardo were colleagues in Milan (1496-1499) and later in Florence Leonardo lived in Pacioli's house in Florence.26

    These links are not restricted to optics and theology. On Forst.II 128r (1495-1497) Leonardo notes that "Gravity is all in all the length of its support and all in every part of this" (cf. M. 40v, 1499-1500). For Leonardo the concept of "all in all" thus involves connections with various branches of physics, and even metaphysics as evidenced on CA385vc (1510-1515) where he credits Anaxagoras with the idea that:

Everything comes from everything and everything becomes everything and everything turns into everything because that which is in the elements is made from these elements.

    In this context the passage on CA345vb (1505-1508) emerges as anything but a digression. His physical demonstrations of the "all in all" concept using mirrors and camera obscuras can be seen as ways of visualizing the metaphysical ideas of Anaxagoras. At the same time these physical demonstrations serve as visible experiments mirroring the invisible Creator, a subtle form of natural theology. By shifting from the physical to the metaphysical on CA345vb he thus establishes a synthesis between natural and divine, microcosm and macrocosm.

 

7. A Synthesis of Circular and Pyramidal Propagation

    Writers on optics in Antiquity had been familiar with both circular (cf. above p. ) and pyramid 1 (cf. above p. ) concepts of rays, but did not consider possible contradictions between these modes of explanation. This is probably because Ancient authors never attempted to visualize the progress in terms of three-dimensional diagrams. Granted Euclid had used figures throughout his Optics but these remained abstractions, always viewer-centred and usually amounting only to variations on the theme of visual angles. The optical writings of Aristotle as they have come down to us are virtually devoid of illustrations. In Ptolemy and Galen diagrams were minimal and even in Alkindi and Alhazen they remained at a premium.

 

(figure)

(Figs. 142-143: Pyramidal propagation of light in Witelo's optical treatise (II.37-38).

    Witelo's optical compendium (c.1270-1280) marked a significant advance in this respect. It contained illustrations of complex optical phenomena independent of a vanishing point at the observer's eye. In book two27, for instance, Witelo described the properties of rays passing through a pinhole aperture in a camera obscura. Accompanying this he attempted to reconcile pyramidal and circular modes of propagation with the help of two diagrams (figs. 142-143).

    A fourteenth century manuscript copy of Pecham's Perspectiva communis 28 went one step further (fig. 144). It showed a circle circumscribed by pyramids, some of which in turn are framed by larger pyramids. In the latter fourteenth century Blasius of Parma wrote a commentary on Pecham's treatise with a third quaestio asking "whether the entire visible object and any point in it terminates a radiant pyramid of its light and colour in every part of the medium"29 The third argument of this quaestio turned on the contradiction between pyramidal and spherical modes of explanation:

Any luminous body (and as I shall say briefly) any natural agent acts spherically, or in the mode of a sphere: hence none such acts pyramidally. The argument holds since the pyramid and the sphere are bodies of various kinds (rationum) and the foregoing is evident by experiment, reason and authority....30

    Attempts to reconcile spherical and pyramidal explanations continued. A fifteenth century manuscript of Pecham's treatise contains a diagram31 (fig. 145) analogous to the earlier example just cited, but with one significant distinction: although more roughly drawn each of the circumscribing pyramids is here the source of a larger pyramid. In Bacon's Tractatus de speculis 32, there is a related diagram (fig. 146).

 

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Figs. 144-146: Pyramidal propagation of light from mediaeval works by Pecham and Bacon.

 

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Figs. 147-150: Pyramidal propagation of light in Leonardo's notebooks. Fig. 147, C20r; figs. 148-149, CA101vb; fig. 150, A86v.

 

    Leonardo builds on this tradition. On A2v (1492) he formulates his "all in all" principle:

Every body fills the surrounding air with its similitudes which similitude is all in all and all in the part.
The air is filled with infinite lines rectilinear and radiant intersecting and joined together without occupation of one another and will present from each object the true form of their cause.

    This he reformulates on A27r

Perspective
Immediately that the air is illumined, it is filled with infinite species which are caused by various bodies and colours which are arranged beneath it, of which species the eye makes itself a target and lodestone.

    Pecham, it will be recalled, had written: "Any point of a medium or illuminated object simultaneously illumines the whole medium adjacent to it."33 On A86v (BN2038 6v, 1492) Leonardo pursues this theme, now attempting to picture how pyramidal diffusion takes place. He draws (fig. 150) by drawing a circle around which he subscribes a series of pyramids in all directions in a manner that recalls the manuscript illustrations in Pecham (figs. 144-145). But unlike these, in which each of the pyramids is extended, Leonardo has extended only one of the pyramids and in order to demonstrate how it becomes narrower as it goes further from its source. He adds letters to this extended pyramid, but does not mention them in the text.

 

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Figs. 151-153: Drafts of pyramidal diagrams on CA101vb. Cf. figs.148-150.

    Even so the text offers another instructive glimpse into how his associative mind deals with a problem. In the right-hand column he continues with the theme of pyramids of light:

Every pyramid composed of a long concourse of rays, contains within it infinite pyramids and each has power for all and all for one.
The equidistant circuit of the pyramidal concourse will give to its objects equality of angles and the thing received from the object will be of equal size.

    In the left-hand column he balances this with a discussion of pyramids of shade (cf. Pt.2.1):

Every umbrous body fills the surrounding air with its infinite similitudes, which through the infinite pyramids infused through it, represent this body all in all and all in every part.

    He begins to draft a another phrase: "Every radiant pyramid from afar," stops and crosses it out. Whereupon he decides to return to an idea which he had begun drafting on CA101vb (c.1492). The first of these drafts had been very rough indeed:

Perspective...body of the air is filled with infinite radiant lines, representing (carries) from the objects the form of their cause, in[ter]sected and intermixed by cause of...courses.
(Every body) which...by straight lines going through the air, with disunited concourse...give to the first object cause to recreate in it the similitude...as a result of the said rays.
A second draft on CA101vb (figs. 148-149, 151-153, cf. 150) directly below this had been considerably more coherent:
The body of the air is filled with infinite radiant pyramids caused by the object positioned in it, which, intersecting and mixing, without occupation of one another, with disunited concourse infuse themselves with equal power and all are able as much as each and each as much as all and by these the similitude of the body is carried all in all and all in every part and each in itself receives in every minimal part it entire cause.

    A second draft on CA101vb (figs. 148-149, 151-153, cf. 150) directly below this had been considerably more coherent:

The body of the air is filled with infinite radiant pyramids caused by the object positioned in it, which, intersecting and mixing, without occupation of one another, with disunited concourse infuse themselves with equal power and all are able as much as each and each as much as all and by these in every part and each in itself receives in every minimal part it entire cause.

    This draft becomes the starting point for the lower paragraph on A86v (1492):

The body of the air is filled with infinite pyramids composed of radiant and straight lines which are caused by the superficial extremities of umbrous bodies positioned in this air and the more they go further from their cause the more they make themselves acute. And even though their concourse is intersected and intermixed nonetheless they do not interfere with one another (nonsi confondano) and with disunited concourse they go amplifying and infusing themselves through all the surrounding air and among themselves they are equal in power and all as each and each as all and through this the similitude of the body is carried all in all and all in the part and each pyramid receives in every minimal part of it, the entire form of its cause.

    This passage clarifies the integral connection between the "all in all" concept and the principle of pyramidal propagation. It offers no hint, however, as to how circular and pyramidal modes are to be reconciled. A first step towards such a reconciliation is taken on CA126ra (c.1487-1490), here Leonardo draws (fig. 154) a candle, which has its pyramidal rays, mn, blocked by a wall. This is to show: "how the lines, or indeed luminous rays, do not pass through other than diaphanous bodies." A second diagram (fig. 155) shows the pyramidal properties of reflected light or, as he puts it:

How the base...xo, being illuminated by the point p, generates a pyramid which finishes at the point c and recauses another base at rs, which receives upside down that which is at xo.
Next, he draws a box (fig. 156) with an aperture at both ends in front of which candles have been placed in order to demonstrate "how the point is cause of the base, and [if you] put a coloured glass in front of each light, you will see the base tinted in this." Implicit here is a demonstration how pyramids pass through one another without interference.

    He now turns to consider sound. He draws (fig. 157) a hammer hitting an anvil, shows how sound spreads pyramidally and "how the lines of the blow pass through every wall," without the pyramid being interrupted. A further diagram (fig. 158) shows "how, finding an aperture, many lines are caused, each weaker than the first ab." Below this he illustrates (fig. 159) how the pyramidal reflection of sound passes through "the mouth of the echo."

 

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Figs. 154-165: Studies of pyramidal propagation of light and sound, magnetism and odour. Figs. 159-164, CA146ra; fig. 165, C20r.

    In the following diagram (fig. 160) he applies his pyramidal principles to magnetism to show "How the lines of the magnet and those of iron pass through the wall, but that which is lighter is attracted by the heavier." In a second example devoted to magnetism he illustrates (fig. 161) how "being of equal weight the magnet and the iron will be drawn in one way." A final example in this series shows (fig. 162) two rooms joined by an aperture to illustrate "how odour does the same as the blow."

    This series of diagrams on CA126ra (1492) illustrates how light, sound, magnetism and odour all follow the same pyramidal laws and thus eiptomize Leonardo's potentially mechanistic view of physics (cf. Pt.I.2). Directly beneath these nine diagrams on CA126ra he draws (fig. 163) a base al., on which he draws a series of seven pyramids, each having their apex along the circumference of a semicircle. He marks various intersections with letters and numbers, does not discuss them, however, and only notes that "every base fills the air with infinite pyramids."

    He has so simplified his diagram that it takes an effort to appreciate the complexity of the physical model which it implies. The half-circle which he draws (fig. 164) needs to be interpreted three-dimensionally as a sphere, the circumference of which represents a circular wave-front, simultaneously a circular bundle of pyramids, which is constantly expanding.

 

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Figs. 166-169: Author's reconstruction of various expanding and contracting pyramids implicit in Leonardo's diagram. Fig. 170: Leonardo's diagram on C20r (fig. 115) doubled to produce a complete circle.

    This expansion can be visualized in various ways. The original centre is the apex for a series of pyramids which expand with distance (fig. 166). Then expansion can also be seen as a series of double pyramids (i.e. squares) and circles, which increase with distance (fig. 167). Each of these pyramids and circles can be seen as a base for other pyramids which become successively narrower with distance (fig. 168, cf. fig. 150). The apexes of these pyramids in turn become the bases of other pyramids whose apexes lie along a common circumference (fig. 169, cf. fig. 163 where he notes that "every point causes infinite bases").

    On C20r (fig. 165, cf. 170) Leonardo integrates these contracting and expanding pyramids within a single picture. He draws a line abc. This serves as diameter of an imaginary half-circle with centre b. Along this diameter he marks 41 points some of which he marks with letters. If we begin at a., for instance, then the first point is m., the twelfth is d., the twenty-first is e., the twenty-ninth is n. and the thirty-second is f. He joins each of these 41 points to each of the points a, b and c producing an interplay of pyramids which are at once expanding and converging. For the point b. is the apex of 42 expanding pyramids while the diameter ac. is the base of 41 converging pyramids with their apex at the circumference of the semi-circle which represents half of a three dimensional sphere. Along the line be., which bisects the semi-circle, he marks the ninth intersection as 1. and then numbers each successive intersection until the final one at 13.

 

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Figs. 171-173: Punctiform pyramidal propagation of light. Fig. 171, CA144vb; figs. 172-173, CU2.

    In the accompanying text only some of these letters and features are explained:

Universally all points causing the extreme points of pyramidal species of things are continually all in all the air together with these and joined without any interval.
Necessity makes that nature orders or indeed has ordered [that] in all the points of the air all the species of the things positioned opposite concur by means of a pyramidal concourse of rays which have parted from these things. And if this were not so the eye would not (recognize) see in every point of the air which is found between itself and the object seen, the form and quality of the thing positioned opposite it.
Let us say that ac. is the thing seen. Let e. be the eye seeing this thing. You see that towhatever point the eye moves along the circle adefc. that the eye always finds the intersection where the entire base a.c. can be seen.

    Well over a decade later he returns briefly to these ideas on CU2 (1500-1505) under the heading "Beginning of the science of painting" (cf. pp. ). here the diagrams (figs. 172-173) are so simplified that, if seen out of context, one would hardly guess the complex synthesis of circular and pyramidal propagation which they imply.

 

8. Conclusions

    We have shown that Leonardo's physics of light and shade has its roots in Ancient thought; that his concept of percussion and other powers can be traced back to Antiquity as can his basic definitions of line, point and pyramid. What distinguishes Leonardo from this tradition in his use of experiment and diagrams to render visible complex three-dimensional situations. Hence the circles produced by pebbles thrown into water (figs. 79-81; 113-114) visualize in one plane the spherical and pyramidal propagation of light in the air.

    The unending reflections of objects positioned between two plane mirrors illustrate the principle that images are all in all and all in every part. Images which intersect without interference at the point of the aperture of a camera obscura provide a further demonstration of the same. These experiments so intrigue him that he becomes almost obsessed with his "all in all" phrase. As we have shown, however, there are also deeper reasons for this. Through Anaxugoras the phrase had philosophical implications and through Nicholas of Cusa it had religious connotations.

    As we have shown Leonardo's response to this tradition is an ingenious synthesis. He uses Euclidean concepts of mathematical points lines and pyramids as a basis for his physics and thus develops a concept of non-spatial light which avoids later problems of Huygen's mechanistic wave model (see pp. ) and Newton's corpuscular mode.34 We have also shown how Leonardo uses the four powers (percussion, force, gravity and movement) to provide a theoretical framework for this mathematical physics of light. Ironically, his experiments and demonstrations concerning these powers are so vivid and concrete that they invite precisely the mechanistic wave and corpuscular interpretations which he was at pains to avoid. With this understanding of the historical context and the principles underlying Leonardo's approach, we shall examine the details of his physics of light and shade in part two.


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Last Update: July 2, 1999