Dr. Kim H. Veltman
The Simile of Percussion
3. Punctiform Propagation
5. Rectilinear Propagation
6. Speed of Light, Vision
7. Cone or Pyramid of Vision
We have shown that Leonardo's concepts of the four powers (and percussion in particular) involve physical analogies, which lend themselves to a purely mechanistic physics of light and shade. Nonetheless, his own conception of light is neither atomistic nor corpuscular. For Leonardo light is ultimately composed of mathematical points, which occupy no space as becomes clear from a detailed examination of his definitions of point, line and pyramid.
Leonardo's definitions of a point build directly on previous authors.1 One of his earliest passages devoted to this theme on CA253vd (c.1490-1491) is a direct translation of Alberti's Elementa picturae2, as is evident from the parallel texts below:
|Point, they say, is that which||Point (is), they say, is that which|
|cannot be divided in any part.||cannot be divided in any part.|
|(The) line they say is made||The line, they say, is made by|
|with a point drawn in length.||drawing the point in length.|
|Hence the length of the line||Hence the length of the line|
|will be divisible, but its||will be divisible, but the width|
|width in its entirety will be||in its entirety (is in-) will be|
|Surface, they say, is as if you||Surface, they say, is as if you|
|extended the width of a line,||extended the width of a line,|
|whence it is that its length||whence it is that its length|
|and its width can be divided.||and also its width can be divided.|
|But depth there will not be.||But depth there will not be.|
|But body, they affirm, is that||But body, they affirm, is that|
|of which length, width and||of which length, width and|
|depth is divisible.||depth is divisible.|
|This, in short, is what the||..........................|
|ancients used to say. We add||..........................|
|Body, I call, that which is||Body, I call, that which is|
|covered with a surface in||covered with a surface in|
|front of the eye/(s)/ and||front of the eye/(s)/ and|
|with light can be seen.||with light can be seen.|
|Surface I call the outer skin||Surface I call the outer skin|
|of a body which defines the||of a body which defines the|
|border.||form of the body and it's border|
|Border,I call, the extreme||Border,I call, the extreme|
|circuit of each surface seen,||circuit of each surface seen,|
|of which the terminus is the||of which the terminus is the|
Alberti also wrote an Italian version of this treatise. A careful analysis of the texts confirms, however, that Leonardo's passage is a precise translation from the Latin and not simply a copy from the Italian, another confirmation that the omo sanza lettere had direct access to and understood the learned traditions. Leonardo returns to this definition when he is drafting basic concepts of linear perspective on A3r (1492): "Point they say, is that which in no part can be divided."
Alberti, both in his Elementa picturae and De punctis et lineis apud pictores,3 had made a fundamental distinction between the theoretical point of mathematicians and the practical point of painters. Leonardo also makes this basic distinction and makes various attempts to define both kinds of point. With respect to the theoretical point of mathematicians Leonardo relies largely on Aristotle and Euclid. "A line," Aristotle had claimed in the Physics,4 "cannot be composed of points, the line being continuous, and the point indivisible." "A point," claims Leonardo, on Triv.34r (1487-1490), "is not part of a line." "A point," wrote Euclid at the beginning of the Elements,5 "is that which has no part." "A point," writes Luca Pacioli,6 citing Fibonacci (Leonardus Pisanus) as his source, "is that which has no part." "A point," echoes Leonardo in a draft on BM173v (1500-1505), "has no part." This statement he redrafts in several versions on BM173r and 176v (1500-1505) only to cross them out again. On BM173v (1500-1505) he arrives at a more polished version which he later crosses out.
In the course of these drafts on BH173v,r and 176v,r Leonardo explores an analogy between an instant, which has no time, and a point, which has no part. Aristotle had explored the same analogy in the Physics? Leonardo restates the idea that a point has no part no BM190v (1500-1505). On CA68rb he drafts another definition of a point as having no centre and being indivisible, which he restates on BM160r (1505-1508; cf. BM266r, 1505-1508; 267v, 1505-1508). On CA289ra (1505-1508) he devotes an entire folio to this theme. In the right-hand margin he writes related terms: terminus, contact, separation, conjunction which he crosses out. Next come draft phrases concerning line and point and then, not crossed out: "Nothing is the lack of the being or of the thing. The point is terminus of the being or of...the thing." These marginal notes come to an end with a final draft phrase, again crossed out. The main body of the text on cA289ra opens with a series of arguments raised by an adversary which he subsequently crosses out:
1st. The adversary says: either the point is in a site, or it is not in a site.
2nd. And if it is in a site, it is in being.
3rd. And if it is not in site, it does not exist in nature.
4th. And if the point exists in nature, either it is one, or there are many.
5th. And if it is a single one, either it is mobile, or it is immobile.
6th. And if it is mobile, motion describes a line composed of as many points, as are the changes of site made by the point.
Not crossed out is Leonardo's rebuttal in terms of a reductio ad absurdum:
Hence, the site, (of the point) occupied by the point, being equal to this point, the line is composed of sites left by the motion of the point, the continuation of which compose such a line.... And if there are more points, and the juncture of two points (occupies) in itself is divisible in two parts, which are two points. Hence the point is a part and this is contrary to the definition of a point.
Leonardo next replies to five of the adversary's objections in systematic fashion:
It is replied to the first that the point (has being) is in a site, without occupation of the site; to the 2nd, that the point (is is) exists in nature; third, that the points are infinite; 4th that the point is mobile along with the site where it resides; 5th (that it describes) that the motion of the point describes an imperceptible line, which is divisible infinitely, and its points are two.
Another objection is raised and disposed of:
The adversary says that the conjunction of the two points is in itself divisible with a single division, with which the 2 points are separated anew.It is replied. The two points joined together do not make a divisible line, but a single name, which is divided by a name, since the two points conclude the line within themselves.
Here Leonardo interjects a quick-thought:
Between the nothing and the thing there is an infinite proportion, because nothing is less than nothing and the thing in itself is divisible to infinity.
There follow two definitions of a point and one of a line:
The point has no centre and its boundaries is the contact of the front of two lines.
The point has no centre and its boundary is (the) nothing. The line has a middle in length, but not (in thickness) in width or depth, and the boundaries of its width are two lines.
Surface has no....
This folio ends with three further draft definitions of a point. In the case of key terms Leonardo appears to be inexhaustible in his search for an acceptable formulation. On CA176vc (c.1510) the drafting process continues. Again the objections of an adversary are considered and rejected.
The point has no centre and its boundary is nothing.
The adversary says that nothing is the vacuum and that the vacuum does not exist in the elements.
It is replied that (the) nothing (...) is that which does not occupy place and (the) vacuum is in place, which has to be, and consequently it is not nothing like the point, which is a place without occupation of place.
The drafting process continues on CA289vb (1505-1508) now under the heading of First Book. Here Leonardo drafts a definition of a point, the objections of an adversary, a reply to these objections and a conclusion; then a second draft of each of these: a third draft of the conclusion, definition, the adversary's objection, and finally arrives at a version which he does not cross out:
The nothing has no middle and its boundaries are the nothing.
The adversary says that the nothing and the vacuum is one and the same thing with two names, of which one speaks and they do not exist in nature.
It is replied that if the vacuum existed and if there were a place which surrounded it, and the nothing exists without occupation of place, it follows that the nothing and the vacuum are not similar because the one is infinitely divisible and the nothing does not divide itself, because no thing can be less; and if there existed parts of this, this part would be equal to all and all to the part.
In the last sentence of this passage Leonardo hints at a connection between his definition of a point and the concept of "all in all" which is of central importance for his optics (cf. below pp. ). He develops this connection between a point and things being "all in all" on BM159v (1505-1508):
The point is that than which nothing can be said to be smaller and it is common terminus of the nothing with the line and it neither nothing nor line, nor does it occupy a position between the nothing and the line. Hence the end of the nothing and the beginning of the line are in contact between themselves, but not joined and in this contact is the dividing point of the continuation of the nothing with the line. It follows that the point is less than nothing and if all the parts of nothing are equal to one, it could be concluded on the whole that all the points are equal to a single point and one point is equal to all.
This idea he repeats on BM204v (1505-1508) and then reformulates more succinctly on BM205v (1505-1508) under the heading:
The mathematical point is this
The point has no centre and its termini is nothing: it follows that the point is indivisible, nor is it part of any thing.
Hence all the points joined together are equal to one and one to all.
This simple conclusion is of the greatest significance for Leonardo's thought because it provides a theoretical justification for the concept that an image can be "all in all and all in very part" (cf. below Pt.1:4). Alberti in his Elementi di pittura 8 had also provided a more practical definition of a point: "The point we call in paint that little inscription, than which nothing can be smaller." As a painter Leonardo is likewise interested in a more practical definition as he explains on BM159r (1505-1508), where he arrives at a formulation very close to Alberti's:
The point is said not to have a part and by this it follows that it is indivisible and indivisible things have no middle and what has no middle is terminated by nothing. Hence the point is nothing and on nothing no science can be begun. And to flee such a beginning we shall say: the point is that than which no thing can be smaller.
In the Treatise of Painting he goes on to note (TPL 3) that: "the beginning of the science of painting is the point." On folio 27r of Francesco di Giorgio Martini's La praticha di gieometria9 (Ash. 361) where that author makes his own definitions of a point, line, etc., Leonardo writes in the margin:
The natural point
The smallest natural point is greater than all the mathematical points and this is proved because the natural point is a continuous quantity and every continuous quantity is divisible to infinity and the mathematical point is indivisible, because it is not a quantity.
He pursues this distinction between a mathematical and a natural point on CA200rb (1508-1510):
What thing is a mathematical point?
Mathematical point is that which has no middle and there nothing in nature that is less than it and for this [reason] it is indivisible.
What is the natural point?
The natural point is that...impression that the point of some [piece of] iron leaves of itself and this is divisible infinitely.
What difference is there between centre and mathematical point:
The centre is there between centre and mathematical point:
The centre and the mathematical point is one and the same thing, but only varies in the place where it is joined: insomuch that the centre is placed in the middle of the quantity or gravity of something, but the point is terminus of a line or some angle.
On BM204v (1505-1508) Leonardo pursues this discussion of natural and mathematical points:
Every continuous quantity is divisible to infinity. Hence the quantity of this (truly) divided will never lead to the print given (by the extre-) by the extremity of the line. It follows that the length and width of the natural line is divisible to infinity.
The times to divide things successively by half are equal insomuch that if, with the mind, you begin to divide the universe, it is the same as dividing the natural point with the mind in the same time.
Hence from the natural point to the mathematical there is an infinite proportion as there would be in dividing the infinite in half would not make two finite parts and thus you will make in the same time from the natural point because in the (the point) one and in the other division made in equal number successively one will never experience the mathematical point. Hence things of near infinite proportion have an equal (divisi-) number of divisions made in equal time with the mind because in actuality it cannot be done and there is no other difference except that the infinite has a greater number of parts than the natural point.
This passage is written around a sketch of an optical pyramid, which has its apex in a point (fig. 123). The pyramid and point are both a mathematical abstraction and something physically real. The significance of the passage written around the pyramidal figure thus becomes clear. Leonardo is trying to reconcile a traditional tension between mathematical (theoretical) and physical (practical) reality. How these opposites are to be reconciled he clarifies on BM132r (1505-1508).
Fig. 123: Illustration of point, line and pyramid on BM 204v.
Here he begins with definitions of a point and line in terms of limits: "The point is limit of the line and no other thing can be less...." He notes that "All the limits of things are not at all a part of these things because the limit of one thing is the beginning of another." This bears comparison with Aristotle's criticism of Plato in the Topics10 where he objects to his teacher's definition of a point as the extremity of a line. From this premise that a point is not the extremity of a line, Leonardo draws a striking conclusion:
Here, since the limits of things are not parts of these things nor of those things which touch them, these limits do not occupy anything and all things which occupy nothing are equal amongst themselves and all together equal to each of them and each of them [is] equal to all. (In) Whence in this case, it follows that the part will be equal to the all, and all to the art, and the divisible to the indivisible, and the finite to the infinite. Hence from what has been said the surface, the line and the point is nothing because it occupies nothing and because they occupy nothing they are each equal to all and all to one as is proved in arithmetic.
Two brief drafts follow before he restates this idea lucidly:
All things which occupy nothing are equal amongst themselves and all joined together will be equal to one and each in itself equal to all. (This demonstrates that the part is equal) to all and all to the part and the divisible to the indivisible, the finite to the infinite.
When we examine in detail Leonardo's concept of "all in all and all in every part" in the next chapter, the full import of the above "demonstration" will become clearer. Here, it is fascinating in itself to see the depth with which Leonardo approaches a problem conceptually.
On BM132r (1505-1508) he makes further preliminary definitions:
The point being indivisible occupies nothing. All things which occupy nothing are nothing.
The boundary of a thing is the beginning of another. That which is not part of any thing is said to be nothing and that which is not part of anything occupies nothing...what has no limit has no figure at all.
The limits of 2 bodies joined together are in turn the surface one of the other as is water with the air.
He begins a fresh draft: "All the points are equal to all an all to one." This he crosses out and then makes a further demonstration with accompanying sketches (fig.):
If in a circle there is only one point to which concur infinite lines, within every 2 lines 1 angle is included and each separate angle terminates in a point, hence many angles have many points which, returned in the circle are equal to a single point, centre of this circle, whence it is manifest that many points are equal to one and 1 to many, which thing cannot happen except in nothing. Hence (the point in ) it is said the point is nothing and nothing because it occupies nothing.
On BM131v (1505-1508), the folio opposite he reformulates these ideas in terms of basic tenets and numbers them. The order is not yet final. In the next chapter, when we examine in detail Leonardo's concept of "all in all and all in every part," the full import of the above "demonstration" will become clearer. On BM132r (1505-1508) he makes further preliminary definitions:
The point being indivisible occupies nothing. All things which occupy nothing are nothing.
The boundary of a thing is the beginning of another. That which is not part of anything is said to be nothing and that which is not part of anything occupies nothing...what has no limit has no figure at all.
The limits of 2 bodies joined together are in turn the surface of one another as is water with the air.
He begins a fresh draft: "All points are equal to all and all to one." This he crosses out and then makes a further demonstration with accompanying sketches (fig. 124):
If in a circle there is only one point to which concur infinite lines, within every 2 lines 1 angle is included and each separate angle terminates in a point, hence many angles have many points which, returned in the circle, are equal to a single point, centre of this circle, whence it is manifest that many points are equal to one and 1 to many, which thing cannot happen except in nothing. Hence (the point in) it is said that a point is nothing and nothing because it occupies nothing.
On BM131v (1505-1508), the folio opposite he reformulates these ideas and numbers them, although the order is not yet final.
Figs. 124-125: Punctiform propagation of light on BM132r and 131v.
1. The surface is limit of the body.
2. The limit of a body is not part of this body.
3. That is nothing, which is not part of any thing.
4. That is nothing, which occupies nothing.
5. The limit of a body is the beginning of another.
Below this he redraws three of the figures (fig. ) sketched on BM132r (1505-1508) and restates his fundamental concept.
If 1 single point placed in a circle can be the beginning of infinite lines and limit of infinite lines by such a point infinite points are separated, equally reduced, returning to one [point] it follows that the part is equal to the all.
This claim is of the greatest importance because it is implicitly a rejection of atomism. It explains why images can be all in "all and all in every part" and is the basis of his concept of punctiform propagation of light.
3. Punctiform Propagation
Punctiform propagation was by no means a new concept. It had been clearly formulated by Alhazen in the eleventh century.11 It was restated by Witelo.12 Pecham in the Perspectiva communis noted that "Any point of a luminous or illuminated object simultaneously illuminates the whole medium."13 The anonymous author of Della prospettiva restated the idea: "from every point of the visible thing infinite rays are multiplied terminating in various parts of the said medium or space."14
Fig. 126: Proof how every part of light makes a point on W12604r.
On MB232r (c.1490) Leonardo notes:
2. Every surface is full of infinite points.
3. Every point makes a ray.
4. The ray is made up of infinite separating lines.
He restates this principle on CA144va (c.1492): "every point of the luminous body makes itself the cause of infinite luminous pyramids." Behind this principle lie more than traditional explanations, however. As early as 12485 he attempts to visualize the nature of such a punctiform propagation (eg. CA353vb, fig. ). In the course of the next two decades these attempts continue (eg. W19148v, K/P 22v, 1489: CA144va, 1492; CA179rc, C.1505; CA345rb, 1505-1508, figs. ). By 1488 on W12604r (fig. 126) he offers a
Proof how every part of light makes a point.
Even though the balls a., b., c. have lights from a window nonetheless if you follow the lines of its shadows you will see that these make an intersection and point at the angle n.
Elsewhere on the same folio he alludes to such a proof as a given:
Because it has been proved that every limited light makes or appears to originate from a single point, that part illuminated by it will have its particles more luminous on which the luminous line will fall between two equal angles.
He explains why light has a single centre in more detail on W19147v (K/P 22v, figs. ,1489):
The reason why light has in itself a single centre is this. We know clearly that a large light is much greater than a small thing. Nonetheless, even if its rays surround it much more than half, the shadow always appears on the first wall and is always seen.Let us posit that cf. is the large light and that n. is the object opposite it which generates shadow on the wall and that ab. is the wall. It appears clear that the large light would not conduct the shadow n. to the wall. But since the light has in it a centre, I prove by experiment, that the shadow conducts itself to the wall as the figure motr.
This experiment he repeats on CA204ra (c.1492) and, elsewhere as will be seen later (cf. pp. figs. ). For the moment, however, we need to examine his definitions concerning lines and pyramids.
As noted above ( 2) one of Leonardo's sources for his definition of a line was Leon Battista Alberti whose Elementa picturae he copied. Another source was Aristotle who, in his De anima had written: "They say a moving line generates a surface and a moving point a line."15 "A line," claims Leonardo on BM173v (1500-1505), "is the transit made by a point." In the period 1500-1505 he repeats this definition on BM176v, 176r and again on 190r where it is accompanied by an alternative even closer to Aristotle's concept: "The line is made by the movement (moto) of the point." This he reformulates on BM159r (1505-1508): "The line is a length made by the movement of a point," and restates afresh on BM204v (1505-1508): "The line is a length born together with the motion of the point and terminated at the limit of the movement of this point than which line nothing can be said to be thinner".
Fig. 127: Demonstration concerning the nature of a line on W19151r (K/P 118r).
On CA289ra (1505-1508) Leonardo drafts a further variant: "The motion of the point describes a line composed of as many points as are the mutations of the sites made by the motion of the point." This he crosses out and defines anew: "The motion of the point describes an imperceptible line, which in itself is divisible infinitely, and its points are two." On W19151r (K/P 118r, 1508-1510), to clarify the nature of a line, he sets out to demonstrate that a (fig. 127):
Line cannot intersect itself.
This is proved by the motion of the line af. to ab. and of the line eb. to ef. which are the sides of the surface afeb. But if you move the line ab. and the line ef. with the front ends ae. to the position c. you will have moved the opposite ends fb. towards each other at the point d. And from the two lines you will have made the straight line cd. which resides in the middle of the intersection of these two lines in the point n. without any intersection. For if you imagine two such lines to be corporeal, through the said motion one will necessarily completely cover the other being equal to it without any intersection at the position cd. And this is enough to prove our proposition.
Fig. 128: Sketch on A113v relating to rectilinear propagation of light in the mountains.
5. Rectilinear Propagation
In Leonardo's physics of light a straight line defines the path of rays. This concept of rectilinear propagation of light and lines of sight was again a well established one. Euclid, in the first definition of his Optics had posited that "straight lines" emanate from the eye."16 The author of the Problemata had been even clearer in this respect: "the course of sight can take only one direction, namely, a straight line, as is shown by the rays of the sun and the fact that we can only see what is directly opposite us."17 Elsewhere in the same treatise, he wrote "light travels in a straight line only."18 Galen in The Usefulness of the Parts reported that the rectilinearity of light is established by the sun's passage through a narrow opening.19 Similar demonstrations occur in Alkindi20, Alhazen21 and Witelo.22 Like Euclid, Leonardo makes the rectilinearity of light a preliminary axiom (A4v, 1492):
Mention of the things which I demand be conceded me in the proofs of this my perspective.I demand that it be conceded me the affirmation that each ray passing through the air which is of equal subtlety, go along a straight line from their source to the object of percussion.
On A9v (1492) he restates this basic principle: "The concourse of the pyramidal lines caused by the objects and terminating at the eye must be rectilinear." On A113v (BN2038 32v, TPL747, fig. 128, 1492) he invokes this principle once more under the heading.
Way where the shadows made by objects must terminate
If the object be the mountain here drawn and the light be the point a., I say that from b., d. and similarly from c., f. there is no light, except for reflected light and this occurs because luminous rays do not adapt themselves except along straight lines and the second rays which are reflected will do the same.
On CA222va (1492) he alludes briefly to this principle of rectilinearity: "No species is carried to the eye passing through equal air, that is not along straight lines." On TPL11 (c.1492) he cites this principle as a reason why sight is a more dependable sense:
The eye at moderate distances and in moderate mediums is less deceived in its function than any other sense because it does not see except by straight lines which compose the pyramid which makes itself the base of the pyramid and conducts it to this eye, as I intend to show.
Leonardo paraphrases his earlier statement on A8v (1482) when he returns to this principle of rectilinearity on Mad.I 0r (1494): "I wish that it be conceded me that the line which goes from the object to the eye through the same quality of air be straight."
Figs. 129-131: Rectilinearity of light in a convex mirror, model eye and a camera obscura on W19120v (K/P117v); D10v and D10r.
On CA150ra (1500-1505) he writes: "All the luminous rays are straight which pass through an equal space." On W19120v (K/P 117v, c.1508-1510) he illustrates this principle (fig. 129) in connection with reflection in convex mirrors, adding the caption: "Every action of nature is made by the shortest way possible." Almost the same phrase recurs on BM85v (c.1505), this time in connection with concave mirrors: "Every action made by Nature is made in the shortest way." On D10v (1508) he identifies Aristotle23 as the source of this idea (fig. 130):
All vision made in the same quality of air is rectilinear. Therefore since it is possible to draw a straight line from the eye to each part of the air seen by this eye, this vision is rectilinear. And this is proved by that of Aristotle which says: every natural action is made in the briefest possible way etc. Therefore vision will be made through the shortest line, i.e. a straight [one].
On D10r Leonardo goes on to note that (fig. 131) "the dark or luminous particles of any given ray are always rectilinear," an idea which he restates on CU613 (1508-1510): "The boundaries of derivative shadows are rectilinear." These passages are particularly interesting because they point to a close link between his principle of rectilinear propagation of light and camera obscura demonstrations (see below pp. ).
By 1508 Leonardo refers to rectilinear propagation in terms of propositions. On D10v and r it is "the third proposition." On D4v he lists it as "the ninth of the first, in which it is stated that each act of vision is made by the eye in the same way and that this is accomplished by straight lines." On W19149r (K/P 118r, 1508-1510) he refers to "the second of this...which shows that all rays which convey the images of objects through the air are straight lines." On CU630 (1508-1510) he refers to "the fourth of this which states that all umbrous and luminous rays are rectilinear." On W19150v (K/P 118v, 1508-1510) he lists it as "the seventh of this where it is said: Every form projects images from itself by the shortest line, which is necessarily a straight line." These frequent references confirm that rectilinearity is a basic principle of his physics of light and shade.
6. Speed of Light, Vision
In Antiquity, there had been a debate whether the propagation of light is instantaneous or not. Aristotle, for instance, had argued that light was instantaneous and criticized Empedocles for holding a contrary view.24 This Aristotelian claim was in turn challenged by the Arabic optical thinker, Alhazen, who devoted a long chapter to how "light and colour in themselves are perceived in time."25 Witelo, who borrowed heavily from Alhazen, nonetheless, continued to argue that light is instantaneously propagated.26 Roger Bacon, on the other hand, reviewed conflicting theories on the subject27 and went on to assert that light is temporal, that it is much swifter than sound or smell, and that, even when its motion is imperceptible, light is still not instantaneous.28
Leonardo's position on this question is equivocal. He broaches the problem at least eleven times in the extant writings. In five cases he seems to agree with Aristotle that light is instantaneous. On CA204va (1490-1492), for instance, Leonardo claims that the eye, a ray of the sun and the mind are the most speedy motions which exist. He points out that, the moment the sun appears in the east, it immediately sends its rays to the west. The moment the eye opens it sees all the stars of the hemisphere. The mind, Leonardo adds, goes from east to west in "a wink" and all spiritual things are similar. Aristotle, in his De Anima had made a similar comment about light moving from extreme east to extreme west instantaneously.29 Leonardo makes an unequivocal statement concerning the instantaneous nature of light on A27r (1492):
Immediately the air is illumined, it is filled with infinite species, which are caused by various bodies and colours which are collected in front of it; of which species the eye acts as a target and magnet.
This recalls Pecham's proposition: "Any point of a luminous or illuminated object simultaneously illuminates the whole medium adjacent to it."30 On A81r (BN 2038 1r, 1492), in a discussion why the eye does not send out visual rays by extromission (cf. below pp. ), the simultaneous propagation of light is again implicitly suggested. In the Treatise of Painting (TPL22, c.1492) Leonardo contrasts the slowness of hearing with vision: "But the work of the painter is immediately comprehended by its regarders." On CA179rb (1497-1500) he restates the instantaneous nature of light: "The air is, in itself, capable of producing without time and leaving every species and similitude of anything seen by it."
In various other passages, however, he suggests that light and vision are not instantaneous. For instance, on C6v (14590), he mentions how the eye is always swifter than hearing, noting that a blow is always seen prior to its being heard, Roger Bacon, in a passage cited earlier (p. ), had made the same point. On A108r (BN 2038, 28r, TPL44, 1492) the temporal nature of sight is broached again: "We clearly know that sight is the most speedy operation that there is and in a point infinite forms are seen. Nonetheless only one things is understood at a time." By way of illustration he cites the example of how the eye cannot read an entire page at once, but must read it bit by bit.31
In three passages Leonardo alludes to the physiological transit of images. On CA90rb (1490-1492) he merely notes in passing that the eye is the swiftest sense "because it is the sense nearest the imprensiva." On CA250ra (1490-1495) he claims that "to the extent that the eye in its function is faster than the ear, to this extent it reserves the similitudes of things impressed in it." The ambiguity of Leonardo's position is most evident on CA203va (1492) where he discusses various branches of motion. He begins with temporal motion which, he claims, "embraces all others." Fourth on his list is the motion "of species of things spread through the air by rectilinear propagation." This he claims "appears not to be defined by time because it occurs in an indivisible time." A few lines later he adds that the motion of similitudes is swiftest and that of mind, second. The passage ends with a physiological remark:
Of the motion of the senses we will not make mention except for hearing because, applied to visible things, it is accomplished in time as demonstrated in...sound, voices....
Finally, in this regard may be mentioned a passage on BM176v (1500-1505). Here he considers and accepts the possibility of a potential infinite: cf. Aristotle in the Physics.32 But he then rejects infinite motion in practice because every medium exerts resistance and a vacuum does not exist or to use his own words "the medium does not concede it." Aristotelian physics thus leads Leonardo to reject Aristotle's idea that light has an infinite speed.
7. Cone or Pyramid of the Eye
The notion of a cone of vision is found in Euclid's Optics.33 Ptolemy used the term at least 46 times in the extant books of his Optics 34 and implicitly connected it with the notion that images are everywhere in the air.35 Ptolemy's work was translated into Arabic and when Eugene of Sicily translated it anew into latin in the twelfth century he chose to translate the Greek, cone, (konos**) with the Latin, pyramid, (piramis) in all but one case.36 Hence from the twelfth century onwards the visual cone becomes the visual pyramid. It is a standard term in Roger Bacon, Witelo, Pecham, Biagio Pelacani da Parma, Alberti and the anonymous author of Della prospettiva. Throughout these texts there is a convention of drawing the pyramid as a triangle. Leonardo often follows this convention. The evidence leaves no doubt, however, that the pyramid is conceived as being three-dimensional (e.g. fig. 123). Because the concept of the visual pyramid is so well established in the optical tradition, Leonardo only rarely defines this term as on BM176v (1500-1505):
A pyramidal body is that of which all the lines parting from the angles of its base concur in a point. And such a body can be invested with infinite angles and sides.
Euclid had primarily been concerned with a single cone of vision. Alhazen's concept of punctiform propagation had led him to claim that pyramids originated from every point of an object.37 He went on to show that: "between the visible object and a mirror innumerable pyramids are produced with alternating based and vertices."38 Witelo pursued this idea in his optical compendium:
As many as are the points on the surface of a mirror, so many are the pyramids to the entire surface of the form of the terminated body, which surface is the basis of all these pyramids and as many are the points in the whole surface of the body the form of which falls on the mirror, so many are the pyramids to the entire surface of the terminated mirror, which is the base of all these pyramids.39
Pecham repeated this concept40 which gave rise, in turn, to the third quaestio of Biagio Pelacani's treatise: "Whether the entire visible object and each of its parts terminates a radiant pyramid of its light or colour in every part of the medium."41 Notwithstanding doubts Biagio fully accepted the concept of infinite pyramids as is clear from a passage at the end of the seventh argument:
Since on the authority of the quaestio, at any point some part of the visible object terminates its pyramid, therefore in the eye there will be a pyramid of the whole and infinite pyramidsof infinite parts.42
In the third conclusion Biagio claimed that "in every sphere of light there are de facto infinite pyramids.43 That he was basing his ideas on Pecham is confirmed by the first conclusion to the second article:
at any given distance one sees the whole and any of its parts, however small, is seen. This is proved since, as is stated in the fourth conclusion of the perspective [i.e. proposition 4 of Pecham's perspectiva communis] every point of the luminous body terminates a pyramid in every part of the medium.44
Leonardo's concepts of pyramidal diffusion grow out of this tradition as is evidenced by his preliminary drafts on CA256rc (1492):
Every body makes rays
In each part (of) a given ray (is the concourse) concur (all the...) pyramidally the similitudes of that part of a body which faces this line.
Every part of a body which faces the ray (line), which results from it with pyramidal concourse is all in all of this line and all in every part of it.
These notes may well have served as a draft for a long compilation on BM232r (1490-1495), which opens with numbered definitions of body, surface and ray (1-4) and is followed by five consecutive propositions concerning pyramids (i.e. 6-11 because 5 is omitted):
6. In each point of the length of any line, lines parting from the points of the surfaces of bodies intersect and they produce pyramids.
7. Each line occupies the entire point from which it originates.
8. In the extremity of each pyramid lines parting from the whole and from the parts of the bodies intersect such that from this extremity one can see the whole and the parts.
9. The air that is found between bodies is full of intersections made by the radiating similitudes of these bodies.
10. And colours of each body transfer themselves from the one to the other by pyramids.
11. Each body fills the surrounding air by means of these rays with its infinite similitudes.
The last of these propositions is restated on A2v (1492): "Each body fills the surrounding air with its similitude, which similitude is all in all and all in every part." Immediately following these propositions on BM232r (1490-1495) he adds another six, which he again numbers:
5. The similitude of each point is caused by this point in the whole and in the part of this line.
6. Through its similitudes each point of one object is capable of the entire base of the other.
7. Each body makes itself the base of innumerable and infinite pyramids.
8. Each pyramid, which originates within more equal angles will give a truer similitude of the body from which it originates.
9. A same base is the cause of innumerable and infinite pyramids turned in various directions and of various lengths.
10. The point of each pyramid has in it the entire similitude of its base.
Not content he adds eight further unnumbered propositions regarding pyramids:
The centric line of pyramids is filled with infinite points of other pyramids
(The) One pyramid passes through the other without its confusion. The quality of the base is in every part of the length of the pyramid.
That point of the pyramid which includes within it all those which originate at the same angle will be less demonstrative of the body from which it departs than any other one enclosed within its.
The pyramid of [a] more subtle point will demonstrate less the true form and quality of the body from which it originates.
That pyramid will be more thinner, which has more disform angles of the base.That pyramid which is shorter will transform in greater variety the similar and equal parts of its base.
Infinite lengths of pyramids will originate from the same quality of angles.
The pyramid of thickest point will tinge the place percussed by it with the colour of the body whence it derives, more than any other.
Nor does he stop here. On BM232v (1490-1495) he outlines thirteen further propositions:
Pyramids which derive from spherical bodies will always be of equal angles in their origin.
If one pyramid finds itself originating on plane bodies of equal angles...
That pyramid which falls on bodies under more equal angles will tinge the place percussed by its more with the colour of its base.
Every pyramid in transit with the friction of the sides of the bodies opposite will produce an intersection after which it will cause opposite it a new and inverted pyramid.
In their percussion inverted pyramids render the species upside down.
The upside down (pyramids) species will be that much smaller than their origin to the extent that the inverted pyramid is shorter than the right side up [one].
The base of the inverted pyramid is infinite.
The bases of inverted pyramids if they are in a dark place will show the form and the colour of their origin upside down.
By the intersections of pyramids it is possible to know the true size of their bases.
If the angles of the intersection of the pyramid be equal to those of its base...each base will be proportionate, the one to the other as is the proportion of the length of the cut pyramid to the whole.
If they intersect the bases of inverted pyramids are redoubled in their quality.
The eye does not see except by pyramids.
The perspective of painters does not occur within pyramids.
These final two propositions confirm that, for Leonardo, the pyramid is essential to both vision and linear perspective. Hence the pyramid plays an important role in drafting his definition of linear perspective as knowing how to draw well the function of the eye (A3r, 1492):
which function extends itself solely in drawing of the forms and colours of all things placed opposite it by means of pyramids. By pyramids, I say, because there is no thing so minimal that it is not larger than the place where these pyramids are conducted in the eye. Hence if you take the lines at the extremities of each body and their concourse leads to one point, it is necessary that these lines are pyramidal.
Two drafts follow and then a definition in the final paragraph of A3r: "Pyramidal lines I intend to be those which depart from the superficial extremities of bodies and by distant concourse conduct themselves to a sole point." As we have shown elsewhere (vol. 1, Pt.I.2) such definitions on A3r are the starting point for his treatise on perspective (A36v-43v). Thus the concept of the pyramid is fundamental for both his linear perspective and his physics of light and shade. To understand the complexities of Leonardo's theories concerning pyramids we need to examine how he relates them to his claim that images are "all in all and all in every part." This is the concern of the next chapter.
Last Update: July 2, 1999