Dr. Kim H. Veltman
The Simile of Percussion
2. Ancient and Mediaeval Precedents
3. Percussion in Leonardo's Optics and Acoustics
4. Percussion and Leonardo's Physics in General
5. Percussion and the Four Powers
6. Percussion in Solid and Fluid Particles
7. Percussion in Water and Air: Mediaeval Precedents
8. Leonardo's Wave Theory
The starting point of Leonardo's physics of light and sound is the principle that they consist of impeded movement, producing the equivalent of a blow in the context of the four elements: earth, air, fire and water. This impact Leonardo terms either blow (colpo) or percussion (percussione). "Blow," writes Leonardo on A27v (c.1492) "I say to be the terminus of speedy motion made in bodies of resisting objects." This he restates on A32r: "Blow is motion interrupted by a resisting object." On BM138v (c.1505) he drafts an other formulation which he presents on BM Arundel 90r (c.1505-1508): "Percussion is the terminus of incident motion and the beginning of reflected motion achieved in an indivisible speed, time and position." For Leonardo percussion constitutes a major problem. As early as c.1497 we find him on CA384ra alluding to a seventh conclusion concerning percussion. On CA29rb (c.1500) he notes that:
In percussion you need to consider four things, namely, the power that moves the percussor, the nature of this percussor, (and) the nature of the percussed object and the object which sustains this object that has been struck.
On Mad II 137v (c.1503-1505) he distinguishes between simple and compound percussion.
In a note on CA354va (c.1505) he asks himself: why percussion on water makes more waves? Drafts for percussion of heavy bodies on CA252ra (c.1505) and heavy spherical bodies on CA354va (c. 1505) follow. Shortly afterwards as on CA74vb (c.1505-1508) he is making lists of different kinds of percussion:
Of the percussion of waters of various sizes
Of the percussion of waters wide and smooth
Of the percussion of waters narrow and rough
Of percussion in dense objects
Of the percussion in water of objects with apertures
Expanded versions of this list recur on CA79ra, 79vb and 65va (c.1505-1508). On CA241ra (c.1508-1510) Leonardo outlines further plans to organize his material:
You will divide percussion into books of which the first will be of two bodies of which the one percussor moves the immobile percussed object; in the second the percussor and the percussed move one another reciprocally; the third is of liquid materials; the fourth is of flexible bodies; fifth...
A further note on CA241vb (c.1508-1510) makes it clear that these books on percussion formed part of a larger work on the four powers of nature: "The book of impetus will go before this and before impetus goes motion." In Leonardo's mind, weight, force, motion and percussion are all interdependent. By way of introduction we shall, nonetheless, concentrate on the historical roots of percussion, before explaining its function in both Leonardo's physics and optics.
2. Percussion: Ancient and Mediaeval Precedents
The concept of percussion in optics can be traced back to Democritus (c.460-370 B.C.) who, according to Theophrastus held that:
the air between the eye and the object of sight is compressed by the object and the visual organ and thus becomes imprinted; since there is always an effluence of some kind arising from everything. Thereupon this imprinted air, because it is solid and is of a hue contrasting < with the pupil > is reflected in the eyes which are moist.1According to this Democritean theory of imprints "the air is moulded like wax that is squeezed and pressed."2
Aristotle developed this theory with respect to his concept of the senses in De anima:
By a 'sense' is meant what has the power of receiving into itself the sensible forms of things without the matter. This must be conceived of as taking place in the way in which a piece of wax takes on the impress of a signet-ring without the iron or gold...3
Elsewhere in the De anima when describing acoustics Aristotle expressed this imprint theory in terms of a blow:
What is required for the production of sound is an impact of two solids against one another and against the air. The latter condition is satisfied when the air impinged upon does not retreat before the blow, i.e. is not dissipated by it. That is why it must be struck by a sudden sharp blow if it is to sound...An echo occurs, when, a mass of air, having been unified, bounded and prevented from dissipation by the containing walls of a vessel, the air originally struck by an impinging body and set in motion by it rebounds from this mass of air like a ball from a wall.4
This analogy between an echo and a ball bouncing from a wall is taken up by Leonardo (cf. figs. 7-10). Aristotle went on to compare these effects of sound with these of light:
What happens here must be analogous to what happens in the case of light; light is always reflected - otherwise it would not be diffused and outside what was directly illuminated by the sun there would be a blank darkness; but this reflected light is not always strong enough, as it is when reflected from water, bronze, and other smooth bodies to cast a shadow, which is the distinguishing characteristic by which we recognize light.5
Implicit in Aristotle's analogy is a comparison between sound which is bounced from a wall like a ball and light which is reflected/bounced from a wall like an image in a mirror: a comparison that the author of the Problemata pursued, beginning with a question: "Why is it that objects which fall to the earth and rebound describe similar angles to the earth's surface on either side of the point at which they touch the earth's surface?"6 As if unsatisfied with his first answer he posed the question anew this time concluding with an analogy that made explicit Aristotle's implicit comparison in De anima:
As then, in a mirror, the image appears at the end of the line along which the sight travels, so the opposite occurs in moving objects, for they are repelled at an angle of the same magnitude as the angle at the apex (for it must be observed that both the angle and the impetus are changed) and in these circumstances it is clear that moving objects must rebound at similar angles.7
Hence if Aristotle borrowed the image of a bouncing ball from kinematics to explain principles of optics and acoustics, the author of the Problemata in turn borrowed the principle that the angle of incidence equals the angle of reflection from catoptrics to illustrate properties of kinematics. Lucretius in his On the Nature of Things continued this analogy between light and the blow caused by an object.
The peacock's tail, filled with copious light,
Changes its colour likewise, when it turns,
Wherefore, since by some blow of light begot,
Without such blow these colours can't become.
And since the pupil of the eye receives
Within itself one kind of blow, when said
To feel a white hue, then another kind,
When feeling a black or any other hue...8
To describe this blow caused by light Lucretius used the verb percutior, to percuss. Pliny in his Natural History also described reflection in mirrors in terms of percussion: "Still, the property of reflecting images is marvelous; it is generally agreed that it takes its place owing to the repercussion of the air which is thrown back into the eyes."9 Alexander Aphrodisias, in his commentary on Aristotle's Meteorology10 also referred to percussion in a similar context. The comparison made in the Problemata between the reflection of images in a mirror and objects bouncing from polished surfaces was repeated by Ptolemy11 and Hero of Alexandria.12 In the eleventh century this comparison was taken up afresh by the great Arabic optical writer Alhazen:
And since it [the image in a mirror] preserves in itself the force and nature of its prior motion, it is reflected in the direction from which it came, and along lines having the same position [i.e. slope] with the prior ones. Moreover, we can see something similar to this in natural motions and also in accidental ones. If we permit a heavy spherical body to descend from some altitude perpendicularly onto a polished surface we shall see it reflected along the [same] perpendicular [as the one] by which it descended.13
Elsewhere in his text Alhazen briefly compared sight and hearing14, again describing hearing in terms of percussion and suggesting that this was analogous to the action of images reflected in a mirror. For Alhazen, however, these were images in passing. Percussion did not play an important role in his optical explanations. This was also the case for Witelo and Pecham. For Roger Bacon, an elder contemporary of these two, percussion played a more significant role as we learn from his Optics:
In addition it can be expressed otherwise that there is a different ratio for light [than for] both sound and odour, for light is carried more swiftly by far through the air than these, as [when] we see [a person] somewhere from afar striking (percutiente) with a hammer or a staff [and] we see him strike before we hear the sound generated. For we perceive the second percussion with the eye before the sound of the first percussion comes to the ear.15
Implicit here was the notion that hearing and vision both involve some corporeal action, which the anonymous author of Della prospettiva, in early fifteenth century made a starting point to his treatise:
The first thing to be noted in the first part, is proposed in the form of a conclusion - It is not possible that any incorporeal thing, that is, non corporeal, be seen, speaking of human vision and of other animals...163.
3. Percussion in Leonardo's Optics and Acoustics
Leonardo stands clearly within this tradition. He believes that objects can only be heard and seen by means of corporeal instruments, but that sound and light themselves involve incorporeal energy, residing in space without occupying it. For this reason the mathematical point (cf. below pp. ) and not the atom is the basis of his physics of light and shade. From the outset percussion and the other powers of nature play a significant role in Leonardo's conception, as is evidenced by a passage concerning acoustics on B4v (c.1487-1490):
There can be no voice where there is not movement and percussion of the air; there can be no percussion of this air where there is not an instrument. There can be no incorporeal instrument. This being so a spirit can have neither voice, nor form, nor force and if you take it to be a body, it cannot penetrate nor enter where the exits are closed. Beware the precepts of those speculators whose reasons are not confirmed by experience.
And if someone were to say: by [means of] air congregated and restricted together, the spirit produces the bodies of various forms and by this instrument it speaks and moves with force, to this side I reply that, where there is not nerve and bone, there cannot be force operated in any movement made by the imagined spirits.
The need for a corporeal instrument in perceiving this incorporeal energy Leonardo discusses further on CA345rb (C.1505-1508):
one cannot see a spirit in the countryside which the other cannot see also.
Hence no spiritual or transparent object can see anything positioned opposite it, because a dense and opaque object is necessary in it and if it be thus it does not require a spirit.
This serves, in turn, as a:
Proof that no object can be seen except through an aperture through which passes the air filled with the species of the objects which intersect in the dense and opaque sides of the aforesaid apertures. And for this [reason] no object which does not have body can see either the figure nor the colour of any object. Whence it is necessary that it is a dense and opaque instrument through the aperture of which the species of objects impress their colours and figures.
In other words the incorporeal energy of sound cannot speak, nor that of light shine without a corporeal instrument of percussion. This theme Leonardo takes up again on W19048v (KP 49v, c.1508-1509), continues throughout W19048r (KP 49r) and into W19047vz (KP48v, c.1481-1510) in a passage entitled:
Whether the Spirit can Speak or Not.
Wishing to show, whether a spirit can speak or not it is necessary to define first what thing a voice is and how it generates itself and we shall say in this way: voice is [a] movement of the air rubbed together in a dense body, or a dense body rubbed together in the air which is the same, which rubbing of dense with rare condenses the rare and makes resistance and again the swift rare in the slow rare condense one another on contact and make sound and very great uproar. And the sound or murmur made by the rare which moves in the rare with mediocre movement, like the great flame generating the sound in the air and the greatest uproar made by the rare with the rare and when the swift rare penetrates the immobile rare, like the flame of the fire issuing from the cannon and is percussed in the air and again the [like] the flame issuing from a cloud percusses the air in the generation of (the) bolts. Hence we shall say that the spirit cannot generate noises without movement of air and air in it is not, nor can it emit from itself what it has not and if it wishes to move that in which it is infused it is necessary that the spirit multiplies and multiply it cannot if it have not quantity. And by the 4th which says: no rare [body] moves if it does not have a stable spot whence ittakes its motion and maximally having to move the element in the element which does not move of itself, if not by uniform evaporation at the centre of the evaporated thing, as happens in the sponge squeezed in the hand which stands under water, from which the water flees every which way with equal movement through the fissures interposed between the fingers of the hand in which it is squeezed.
The passage ends with a volley of questions:
Whether the spirit has an articulated voice and whether the spirit can be heard and what thing is hearing and seeing and how the wave of the voice goes through the air and how the species of objects pass to the eye?
If Leonardo clearly has difficulties in formulating his concepts of incorporeal energy there can be little doubt that his ideas on percussion build directly on a well established tradition. On Manuscript B 90v (c.1487-1490), for example, he sets out to give an explanation relating to acoustics:
The voice, having parted from the man and having repercussed on the wall, will escape upwards. If you have an overhang above this wall at right angles [to it] the upper face will send the voice to its source, as the voice of the echo must do which, for everything that you say, will be replied to you in many voices.
A diagram (fig. 7) follows which is then described:
[There are] 150 braccia from one wall to the next. The voice that issues from the horn is formed on the opposite wall and from there it bounces to the second and from the second [back] to the first, as a ball which bounces between 2 walls which diminishes its bounces and so too diminishes the voice.
Leonardo's analogy between echoing sound and bouncing ball is the same as that found in Aristotle's De anima II 8 (419b 20ff.). The difference between the Stagirite and Leonardo is one of emphasis. What had functioned as an image in passing for Aristotle becomes, for Leonardo, a recurrent motif that involves a systematic concept.
Figs. 7-10: Visual demonstrations of sound involving percussion or a blow. Note the hammer striking the ear, in fig. 10. Fig. 7, B90v; fig. 8, C6v; fig. 9. C 6v; fig. 10, C16r.
Figs. 11-15: Analogies between percussion, reflection, sound, light and sight. Figs. 11-13, A19r; fig. 14, A19v; fig. 15, A113r.
Having mentioned the image of bouncing balls on B90v, Leonardo illustrates it on C6v (c.1490, fig. 8), following which he makes a note which recalls Bacon's idea cited earlier:
On the sound made by percussion
Sound cannot be heard so close to the ear that the eye has not first seen the contact of the blow.
Accompanying this text is a diagram showing a hammer striking a bell (fig. 9). Alongside this diagram is an X. Near the top of 16r in the same treatise there are two other X's and lower down on this folio there is another diagram of a hammer striking a bell (fig. 10). Immediately following this is a text entitled:
Of corporeal movements
I say that the voice of the echo is reflected by (the) percussion at the ear as at the eye (the) percussions are made in mirrors by minds of objects. In the same way that the resemblances fall from the thing to the mirror and from the mirror to the eye under equal angles, so too will fall and rebound, at equal angles, the voice in the concavity of the first percussion at the ear.
Here Aristotle's loose analogy between acoustics and optics has become an explicit comparison between the percussion of sound in echoes and the percussion of images in mirrors. Some two years later on Manuscript A 19r (c.14592) Leonardo develops this comparison, beginning with a general statement: "It is possible with the ear to know the distance of thunder seeing its flash of lightning, through the similitude of the voice of the echo". Below this he draws a bouncing ball (fig. 11) which he explains in a caption alongside: "The line of percussion and that of the ball are set in the middle at equal angles." This he reformulates: "Every blow hitting the object rebounds back at an angle similar to that of the percussion." This is effectively a paraphrase of the analogy made in the Problemata, a text that Leonardo possessed under the title Problema D'Arisstotile.17 But whereas his predecessor had been content to use the analogy in passing, Leonardo explores it in detail, turning first to the case of sound, and again using Aristotle's image of the bouncing ball as a starting point for his explanation:
This proposition appears clearly for if you strike a ball on a wall it will rebound back at an angle equal to that of its percussion, that is, if the ball b., is thrown to c., it will return (back) by the line bc. because it is constrained to leave equal angles on the wall fg.; and if you throw it along the line bd., it will return back along the line de. and thus the line of percussion and the line of the ball will make one angle on the wall fg. situated in the middle of two equal angles as appears in the middle of fmn.
Hence, if one stands at b. and shouts, his voice is all in all the line fg. and all in (each) part. Hence, whoever stands, as was said, at b. and shouts, it will appear to him that he hears his voice in c. and comes to his ear by the line bc. And if, at the same time, one were at e., it would appear to him that he hears the voice b. in the place d., and [that] it comes by the line de.
Leonardo now formulates a general principle:
The voice is all in all and all in [each] part of the wall where it percusses, and that part which is formed in such a way that it is apt to send back its percussion, renders the voice in so many various particles of itself as are various the positions of the hearers.
This principle that the voice is "all in all and all in each part" is, as shall be seen presently, of great importance both for his acoustics and optics.
Immediately following this statement of a general principle Leonardo offers another specific example in diagram form (fig. 12) below which he writes another caption:The voice made at n. will percuss at the angles a,b,c. [and] d. and for every voice made at n., a,b,c,d. will send back the one [as] four.
Leonardo draws another diagram (fig. 13) this time with a longer caption:
If the person who stands at m. (and) shouts, his voice will be returned to him from r. and he who stands at n. will hear the echo from r., so near the percussion that he will confuse the one with the other, and will not be able to discern the original from the echo.
He ends with another brief note on acoustics: "The ear receives the species of sounds (voce) by straight, curved and bent lines and yet no bending can interrupt its operation." At the top of A19v Leonardo considers colours reflected in mirrors and provides a diagram of images in a mirror (fig. 14). That this is not a digression becomes clear from the proposition which follows in which he develops the comparison between images in a mirror and percussion of sound:
The voice percussed at the object will [re] turn to the air by a line of such obliquity as is the line of incidence, that is, the line which carries the voice from its source to the place where this voice is apt to be reformed and this voice acts like the similitude of a thing seen in a mirror which is all in all the mirror and all in [each] part. That is, let us say that the mirror is ab. and that the object mirrored is at c., [then] just as c. sees all the parts of the mirror, so too (do) all the parts of the mirror see c. Hence c. is all in all the mirror, since it is all in its parts and it is all in [each] part, because it is seen in as many various parts as are various the positions of the viewers. Hence if the object c. be in n. it appears as far inside as it is outside. Hence c. will be seen at d. and that which is at f., seeing the object d., will see it along a straight line i. hence, [he will see] the object d. at the part of the mirror e. and he who is at m. will see the object d. at t.
Thus both verbal and visual images are percussed; both are reflected and possess the property that they are "all in all and all in each part." On A94v (BN 2038 14v, TPL156, C.1492), this time without a diagram, Leonardo returns to his comparison between a bouncing ball and the percussion of light:
Reverberations are caused by bodies of a clear quality, of flat and semidense surfaces that are percussed by the light which, like the bounce of a ball, repercusses it at the first object.
Nineteen folios later on A113r (BN 2038 32r, c.1492) Leonardo takes up once again his comparison between percussing light and bouncing ball in a passage entitled (fig. 15):
How one is to understand which part of the body must be more or less luminous than the other.
If f. be the light and the head be the body illuminated by it, (and) that part of this head which receives above it the ray between more equal angles will be more illuminated and that part which will receive the rays under less equal angles will be less luminous. And this light in its function is like a blow, insomuch that a blow that falls under equal angles will be in a first degree of power and when it falls under unequal [angles] it will be that much less powerful than the first, by the extent to which the angles are more disform.
For example: if you throw a ball at a wall, the extremities of which are equidistant from you, the blow will fall between equal angles and if you throw the ball at that wall, standing at one of its extremities, the ball will fall under unequal angles and the blow will not remain.
Once again it is instructive to compare Leonardo's passages with tradition. Aristotle, had considered the blow of sound with a bouncing ball and had implied that this was comparable with the action of light. The author of the Problemata had, in turn, compared the blow of light on a mirror with a projectile rebounding from a polished surface. But each of these comparisons had served as isolated examples. By contrast, in the Manuscript A passages just cited (19rv, 94v and 113r), Leonardo has made explicit the connections between these comparisons and his integrated them into a coherent framework such that the percussion/blow of sound and light are equivalent to the percussion/blow of a bouncing ball or similar projectile. At the same time he has stressed how all percussion obeys the fundamental law of catoptrics, whereby the angle of incidence equals the angle of reflection.
4. Percussion and Leonardos Physics in General
In order to appreciate how these examples of percussion in light and sound fit into Leonardo's concept of physics a digression is useful (a) to consider other examples of percussion and b) to examine how percussion fits into a larger mechanistic scheme. The author of the Problemata had been content to suggest in passing that projectiles, like images in mirrors, had their angle of incidence equal to angle of reflection. Leonardo, by contrast, examines this comparison in detail. On C28r (c.1490) he draws three diagrams showing balls in percussion (fig. 17), two of which illustrate the reflection principle. The caption accompanying one diagram, entitled "Blow," is in general terms. It is followed by a specific description:
The angle [of incidence] caused by the percussion of equal spherical bodies is always equal to that of reflection.
If ef. were a wall, the ball s., departing from b., and knocked against this wall, would rebound to a. and likewise, the ball t., departing from fc., will rebound to d., after its percussion in the ball s.
A series of fourteen diagram ons A 8r (fig. 16, c.1492) illustrates Leonardo's determination to explore all the variants of this principle. On A22r he draws five further sketches of reflected projectiles (fig. 18). On A24r he compares the distance covered by a bouncing ball with that of one hurled directly through the air (fig. 19) a problem to which he returns on 161r (c.1497-1499, fig. 20) this time in answer to a question.
Fig.15: Visual list of fourteen types of percussion, A8r
Figs. 17-18: Further examples of reflected percussion. Fig. 17, C28r; Fig. 18, A22r.
Figs. 19-21: Comparison between direct and reflected percussion Fig. 19, A24r; fig. 20, 161r; fig. 21, 122r.
"I ask if the movement made by the stone in a continual line is equal to movement which will be in a reflected line, that is to say before the bound and after the bound?" Another diagram (fig. 21) on 122r answers a related question:
I ask in [the case of] an equal quantity of motion made by 2 equally heavy bodies which will give the greater percussion to its object, the direct motion ab. or the reflected, or genuflected motion ac?
On I128r (c.1497-1499) he poses a related question and outlines an experiment:
About the bounce
If the first bounce is ten braccia, tell me how much will the second be? Hold the ball in such a way that it marks the place where it strikes the marble or other hard surface and follow this rule for all successive rules and thus make the general rule.
Leonardo pursues this problem on I14v, this time using the example of a man on stairs, in a passage headed:
On movement and percussion
If one descends from step to step making a jump from one to the other, (then) if you joined together all the powers of the percussions and weight which such a man would give when he fell by a straight line perpendicularly from the head to the foot of those stairs from the height.
Again if this man fell from a height hitting from degree to degree objects which bent like a spring in such a way that the percussion from one to the other is small, you would find that such a man at the last part of his descent would have diminished his percussion as much had he fallen by a free and perpendicular line as in adding up all the percussions that were made at each degree of the said descent on these said springs.
Figs. 22-24: Visual demonstrations that the angle of incidence equals the angle of reflection in the percussion of objects. Fig. 22, L42r; fig. 23, K1v; fig. 24, E28v.
On Forster II 45v (c.1495) this problem recurs. This phenomenon of a problem being raised in one passage and then answered elsewhere, perhaps in a different manuscript, may strike us as irregular, but with Leonardo it is almost a norm. Hence the basic problem of projectiles being reflected like images in a mirror, described on C28r (fig. 17, c.1490-1491) is repeated on L42r (fig. 22, c.1497-1503), K1v (fig. 23, c.1503-1050), F22v (fig. 25, c.1508) and E28v (fig. 24, c.1513-1514). Similarly, the question of what occurs when two unequally sized balls collide, mentioned in passing on A8r (c.1492), is posed again in quantitative terms on I76(r) (fig. 26, c.1497-1499 cf. fig. 53). The question of percussion produced by a falling ball, broached on A22r, is posed afresh on I41(v) (c.1497) as a quantitative experiment. And the percussion of cannon-balls, discussed in isolated examples on A26r, 43v, 44r (c.1492) is considered more systematically on L43v (fig. 27, c.1497-1503).
Figs. 25-27: Reflected percussion of projectiles. Fig. 25, F22v; fig. 26, I76r; fig. 27, L43v.
Figs. 28-31: Simple examples of percussion with cannon-balls. Fig. 28, BM192v; fig. 29, BM85r; figs. 30-31, BM91r.
Figs. 32-39: Systematic demonstrations that the angle of incidence equals the angle of reflection. Fig. 32, BM90r; fig. 33, BM83v; figs. 34-35, BM82v; fig. 36, BM81v; fig. 37, BM93r; fig. 38, Mad I 147r; fig. 39, BM18r.
Figs. 40-44 Systematic studies of percussion with cannon-balls. Figs. 45-49 Systematic studies of percussion with water. Fig. 40, BM36r; fig. 41, BM92v; figs. 42-43, BM128r; fig. 44, BM228v; fig. 45, CA81ra; fig. 46, Mad I 134v; fig. 47, Mad I 151r; fig. 48, F53v; fig. 49, F16v.
In the Codex Arundel this problem of the percussion of balls becomes an independent theme. Here we find a whole spectrum ranging from simple sketches on Arundel 192v (fig. 28, c.1505-1508) and 85r (fig. 29, c.1505, in which the percussion of a weight is compared directly with percussion in a mirror) or elementary geometrical diagrams on Arundel 95r (figs. 30-31, c.1505), to a series in which he approaches the problem more systematically: Arundel 90r (fig. 32, C.1505); 83v (fig. 33, c.1505), 82v (figs. 34-35, c.1505); 81v (fig. 36, c.1505), 93r (fig. 37, c.1505) in which the results are quantified (cf. Mad I147r, fig. 38, c.1499-1500) and BM Arundel 18r (fig. 39, c. 1508). This series culminates in five comparative diagrams: 36r (fig. 40, c.1505); 128r (fig. 41, c.1505); 92v (figs. 42-43, c.1505) and 226v (fig. 44, c.1500-1505). Leonardo is intent on applying his bouncing ball image wherever possible. On A63v (c.1492), for example, he uses it to explain how the percussion of water erodes the banks of rivers with its blows:
And this is seen clearly and it is understood that the waters which strike the banks of rivers act like balls percussed from walls which depart from these at angles similar to those of (the) percussion and are going to batter the side of the wall opposite.
On I115(v) he proceeds to illustrate graphically this comparison between a bouncing ball and bouncing water (fig. 50) a theme to which he returns on K1v (fig. 23, c.1503-1505) in the form of a question.
Figs. 50-51: Percussion of bouncing balls and bouncing water. Fig. 50, I115(v); fig. 51, K99r.
Figs. 52-53: Reflected percussion in water and wind. Fig. 52, I114v; fig. 53, BM276v.
Whether the stone or water struck by the incident mobile object follows (the) reflected motion in the way the incident mobile [object] would follow by itself after its percussion or not [?]
On K99r (c.1503-1505) he makes further sketches concerning the reflection of bouncing water (fig. 51). This leads to comparative studies (figs. 45-49) analogous to those involving cannon balls (figs. 40-44). On BM Arundel 276 this principle of reflected percussion is applied also to wind (fig. 53 cf. fig. 52). Thus an image that had traditionally been mentioned in passing, Leonardo uses systematically with respect to light, sound, projectiles, water and wind. In his De sensu Aristotle had used the image of the bell with respect to the production of sound.18 For Leonardo the image of the bell and hammer also serves to illustrate his concept of percussion. Two early examples have already been cited (figs. 9-10). On A22v (c.1492) he returns to this image under the heading:
Of the blow
The blow on a bell leaves behind it its similtude impressed [on it], like the sun in the eye or odour in the air. But it is to be seen if the similitude of this blow remains in the bell or in the air and this you will learn [by] placing your ear on the surface of the bell after this blow.
Immediately following is another passage in which the resonance accompanying sound is discussed again under the heading:
Of the blow
The blow given to the bell will [make] respond and move somewhat another bell similar to it and the cord of a lute which is sounded will [make] respond and move another similar cord of like mouth in another lute and this you will see by putting a piece of straw on the cord similar to the one that is sounded.
In the late writings Leonardo returns to the bell and hammer comparison on G73r (c.1510-1515) under the heading:
Every impression tends to permanence or desires permanence.
One proves it with the impression made by the sun in the eye of the spectator and in the impression of sound made by the hammer that strikes the bell.Every impression desires permanence ... (and the eye which looks at the sun proves it) as is shown to us [by] the image of movement impressed on the mobile [object].
Many other examples illustrating the physical nature of percussion could be cited. On A31r (c.1492), for example, he writes a draft, crosses it out, then adds an interjection concerning method:
I remind you that you make your propositions and that you adduce the aforementioned things with examples and not by propositions, which would be too simple and you will say as follows:
A general proposition is now given under the heading:
A blow given in some dense and heavy body passes naturally through this body and injures whatever finds itself in the surrounding dense or rare bodies (that there are).
This he illustrates with a specific example
(and) let there be many fish in (a) water, which enter(s) under a rock and if you give a big blow to this rock, all the fish which find themselves below or to the side of this rock will come, as if dead, to the surface of this water....
Why Leonardo is so concerned with percussion is explained, in part, by his belief that this concept can be quantitatively tested. In the Manuscript A (c.1492) he outlines several demonstrations/experiments on this theme. On A32v, for instance, he notes that 100 blows on a glass vase with a needle would not break the vase, yet one blow with a needle 100 times as heavy would break it. On A4r Leonardo proposes to compare the effects of a one pound hammer falling 100 times from a height of one braccio, with a 100 pound hammer falling one time from a height of one braccio. A related experiment is described on A23r:
On the blow
Whether 10 blows of one pound by a blow that has fallen on a place, falling from one braccio [in] height fixing a nail of one braccio a given amount [will fix it] as much as would a combined weight of 10 pounds? This shows [that they do] not, for if you wished to fix a nail with the weight of another similar nail, this would be impossible, for [even] if you beat on this ten thousand similar blows, all would be nothing. And if you took a weight 20 times as much, its blow will be in proportion to the nail that you wish to fix.(figure)
Figs. 54-55: Quantitative experiments concerning percussion of hammers and of light. Fig. 54, M83v; fig. 55, Forst III 58v.
In the next paragraph he compares combined percussion in voices. But he does not forget the problem. On Forster II.2 74r (c.1495-1497) he notes that "100 pounds given in one blow gives a greater percusssion that a million [blows] of one pound given one after the other." The problem is pursued on M83v (c.1499-1500) in the form of an experiment (fig. 54):
If the hammer of 10 pounds drives a nail into a [piece of] wood in one blow, a hammer of one pound will not drive then such a nail entirely into the said wood in 10 blows. And a nail less by a tenth part will not be the more driven in by the said hammer of one pound in a single blow even if it is in equal proportion to the first given, because what is lacking is that the hardness of the wood does not diminish the properties of its resistance, that is to say, that it is hard as before.
If you wish to treat of the proportions of movement of things which penetrate wood driven by the power of a blow, you need to consider the nature of the weight which strikes and the place where the percussed thing is fixed.
A passage on A3v (c.1492) suggests that such experiments were intended to test a more general hypothesis: "The powers (potenzie) separated will not have, all at one time and in one operation that power (virtu) and effect (alturita) as when they are united." On A3v, Leonardo offers further test cases involving separate and combined voices, forces, supports and, to return to the theme of optics, lights:
Many little luminous bodies joined together will be of greater power in themselves than they would be being separate. The proof you will see if you take many lights in a straight line and you stand at a certain distance facing the centre of this line and you note the quality of the light made by these lights and then join them together. You will see that the place where you stood is more luminous than before. Again it is known that the stars are of an equal light to that of the moon and if it were possible to join them together, which would compose a body much larger than that of the moon, and nonetheless, even though it be calm and all [the stars are] shining, if the moon be not in our hemisphere our part of the world remains dark.
Leonardo pursues this theme of separate and combined lights on Forster III 58v (fig. 55, c.1493) under the heading:
On the Duplication of Lights.
If one light is of 4 ounces (and) it appears that joined together two of these lights make 8 ounces.
Again, the problem whether the combined light of two candles is more intense than that of one candle on its own, is not new with Leonardo. It is raised in a contemporary commentary on Aristotle's de Generatione under the quaestio: "Whether a like can act on a like."19 Nonetheless, the context is fundamentally different. The Aristotelian commentator had been concerned with the example as an isolated instance of an abstract philosophical concept, Leonardo is interested in the example to test a principle in his physics of percussion. For Leonardo percussion represents a basic law of nature by means of which he can account for the physics of light, sound, wind, water and various solid projectiles.
Percussion, as we have already mentioned earlier, is a complex concept which involves all four of his natural powers. This is implicit in a brief definition that he drafts on A27r (c.1492): "Blow (colpo) is terminus of the motion caused by force and operated by bodies in resisting objects in indivisible time". On A27v (c.1492) this relationship with the other powers becomes clearer in the course of three longer drafts, headed:
Blow I say to be the terminus of speedy motion made by bodies in resisting objects. This same is the cause of all sounds, breaker and transmuter of various things, recauser of second motion. No thing is shorter, nor of greater power and it goes diversifying itself by means of the causes.
Blow is the terminus of speedy motion caused by force and operated by weight in an object, caused by sounds, transmuter of its effect. And no thing is of shorter operation or of greater power. Its result is of greatest velocity and penetration in every counterposed variety of object and departing from its source through circular movement and it is all in all and all in [each] part.
Blow is the terminus of speedy motion, caused by force and operated by bodies in the resisting objects. From this derive the sounds, from this the breakings, and no thing is of shorter operation, nor of greater power. Its result is of the greatest velocity and penetration in every counterposed variety of object.
In attempting to define blow, Leonardo refers to the related powers of force, movement and weight. These four powers dominate the succeeding twelve folios as Leonardo explores various definitions and examples. Then it becomes clear that these four powers are fully interdependent and belong to a larger system.
5. Percussion and the Four Powers
The terms violence, weight, force, motion and blow are all familiar from he Aristotelian tradition. But on A35r (c.1492) Leonardo combines them in a fresh way:
Violence is composed of 4 things, that is, of weight, force, motion and blow (colpo); and some say that violence is composed of 3 passions, that is, force, motion and blow and that which is more powerful has less life, that is, (the) blow; second, is (the) force; third in weakness would be motion and if weight be accepted in this number, it is the weakest and most eternal of any of the above mentioned.
In the paragraph following Leonardo turns to weight and on the verso of the folio he continues the discussion under the heading: "Of weight, force, motion and blow." The interdependence of the powers he notes in passing on CA173vb (c.1490-1495): "Motion is not without percussion. Percussion is not without body and weight." As Leonardo's thought matures the "four things" which began as components of violence, assume an independent identity as accidental powers. These powers are "spiritual" and "incorporeal" (B63r, c.1490). That which he had at first termed blow, he now terms percussion. On BM Arundel 181r (c.1497-1500) he drafts a revised definition:
Gravity and force (are) together with percussion are 3 accidental powers which are nonetheless to be called generators of motion (which [is] created) generated by this.Gravity, (the) force along with motion are nonetheless to be said to be generators of motion which is daughter of this. (For from the motion is again caused motion. This motion is generated by it. Whence it is concluded that the one without the other without the one [does] not.) For these are again caused by motion; motion without these cannot be...
These versions he crosses out. The next version he finds acceptable:
Gravity and force which are interchangeably daughters and mothers of motion and sisters of impetus and of percussion always combat their cause which life and (if force and gravity lose their being) conquer one another and kill [themselves].
On BM Arundel 151v (c.1495-1497) he pursues the theme:
Material motion along with gravity, force and percussion are the four (powers) accidental powers with which all the works of mortals have their being and their death.
On the recto of the same fol. 151 he again modifies the definition: "Force, along with material motion, weight along with percussion are the four accidental powers with which all the works of mortals have their being and their death." This he repeats almost verbatim on Forster II2 116v (c.1495-1497): "Gravity, force, accidental motion along with percussion are the four accidental powers with which all the evident works of mortals have their being and their death." In his anatomical writings Leonardo develops this concept of the four powers into a basic principle of organization. On W19060r, (KP153r, c.1509-1510), for example, he notes:
Why nature cannot give motion to animals without mechanical instruments as is shown byme in this book on the motive works of this nature made in animals and for this [reason] I have composed the laws (reghole) in the 4 powers of nature, without which nothing can by itself, give local motion to these animals.
Hence I shall first describe local motion and how it gives birth to and is born from each of the other three powers. Then I shall describe natural weight, even if no weight can be said to be other than accidental, but it is preferred to name it thus in order to separate it from force which is of the nature of weight in all its operations and for this [reason] it is named accidental weight and this force is placed as the 3rd power of nature or naturated (because) the fourth and ultimate power is said [to be] percussion, that is, terminus or impediment of motion. And I shall say first of all that every insensible local motion is generated by a sensible motor, as in a clock, the counter-weight, lifted on high by man, its motor.
Here the four powers have become a crucial theme. As Keele,20 has so lucidly shown, the powers are a key to the whole of his science. For our purposes it is important that we remain conscious of this framework, as we focus on the concept of percussion which, as is clear from a late passage on G62v (c.1510-1515), Leonardo came to regard as the most significant of the powers:
Among the accidental powers of nature, percussion exceeds by a great excess each of the others which are produced by the motors of heavy bodies in equal time with various movement, weight and force. This percussion is divided into simple and composed. Simple is that [in] which the motor is joined with the mobile percussor at the junction of the percussed place. Composed is that for which the mobile which strikes does not terminate its movement at the place of its impression, as (is) the hammer which strikes the coin which impresses coins. And this composed percussion is much feebler than simple percussionbecause if the mouth of the hammer had attached [itself] to the coin which it is to press, which it had struck on the die of the impression and [such] that in this head of the hammer there had been engraved the concavity opposite the money, then the impression would be more expedite and clean-cut on its side percussed by simple motion than on the side of composed percussion, as is the money which remains struck in the corner where the descent of the hammer has struck it and the percussion is reflected and reverberated against the front of the hammer.
Whether he terms it colpo in the early period or percussione in the late period, his concept of percussion remains consistent, inviting a mechanistic interpretation.
6. Percussion in Solid and Fluid Particles
Thus far we have concentrated on instances of percussion involving solid objects. But Leonardo was also interested in the percussion of both solid and fluid particles. For example, on C22v (fig. 56, c.1490) he considers what happens when water falls on water:
Water or another thing which falls on water makes that this water which receives the blow enlarges under this blow and surrounds it and the cause of this blow being overcome it passes above this in pyramidal form:
The reason for this is that, [with] a drop of water, falling from a roof on other water, that part of the water which receives this blow, cannot have place, nor flee behind other water with that velocity with which it was assailed because it would need be that it raises too much weight in order to enter beneath such a quantity of water. Hence, needing to obey with its own flight being chased by the thing which chases it from its site and finding the nearby water which does not receive the blow (which) is not prepared to make a similar flight, this first cannot penetrate it and hence it seeks the shortest way and runs to the thing that produces the least resistance for it, that is, the air. And this first circle which surrounds the percussed place, closing it furiously, because it stood raised outside the common surface of the water, reduces the water which flees upwards in a pyramidal form. And if you believed that the water which falls was that which leaps up, make the water fall on a pebble and you will see the water likewise leaping up and not the pebble.
Figs. 59-61: Connections between percussion, pyramids and quadrature of the circle. Figs. 59-60, Mad I 126v; fig. 61, BM188r.
Leonardo develops a similar conclusion regarding the pyramidal qualities of small dust particles. In a note on A32v (c.1492) this idea is implicit when he mentions that "if you beat on a flat surface you will see the dust that is there reduce itself to small heaps." On Forster II1 16r (c.1495-1497) he proposes a general rule for the pyramids thus formed by percussion (fig. 57): "Every hillock of sand either in a plane or on a slope will be twice as wide in its base as its axis is long." On Forster II2 68v (c.1495-1497) he refers to experiments with circular motion and then adds: "I want to do the same with dust that is beaten." The accompanying diagram shows a hammer and a little pyramid of dust (fig. 58). On Mad I 126v he describes this phenomenon in greater detail (fig. 59-60):
Figs. 62-65: Pyramidal and inverted pyramidal percussion. Fig. 62, I 60v; fig. 63, L64v; figs. 64-65, L1r.
If you beat the board [with] dust, this dust will reduce [itself] to various little heaps, which little heaps will always pour such pulver through the point of their pyramid and descend to its base. Then having re-entered below it will pass through its centre and will fall again through the summit of this hillock. And so it will go passing again through the orthogonal triangle amn., as many times as such percussion follows.
Alongside is another passage describing the pyramidal hillock that has been formed (fig. 59):
The hillock made by many particles, of some solid material, which falls from a single aperture always has its width twice its height.
Let h. be the aperture where grains or dust are poured; let an. be the height of this "mountain, om. its width. You thus see that om. enters twice into an.
The geometrical form of the accompanying diagram (fig. 60 cf. 61) brings to light links with quadrature of the circle problems. On L64v (c.1497-1504) Leonardo pursues this theme asking: "What difference is there between the percussion of the thing that is united and the thing that is disunited?" His reply is in the form of a sketch (fig. 63 cf.62) and a brief text: "Grain thrown in the air with a sieve leaps pyramidally". On L1r (c.1497-1504) he considers the related problem of water striking a flat surface which he again illustrates (figs. 64-65):
The water which falls pyramidally by a perpendicular line will rebound up and will finish the apex towards the base of such a pyramid and will then intersect itself and pass beyond and fall below.
On F61r (c.1508) he returns to the example of hillocks produced by striking a table and suggests that this principle accounts for the sand dunes along the Po and in Libya. From such examples a connection between percussion and pyramidal shapes becomes clear which, as we shall see presently, explains aspects of his physics of light.
7. Percussion in Water and Air: Classical and Mediaeval Precedents
Leonardo's studies of percussion extended equally to water and air. Here again he was working within a well-established tradition, which it will be well to consider by way of introduction. Aristotle, in his De anima had mentioned analogies between water and air.21 He had also explained that the effect of any blow or contact was dependent on the medium involved:
Thus if an object is dipped into wax the movement goes on until submersion has taken place, and in stone it goes no distance at all, while in water the disturbance goes far beyond the object dipped: in air the disturbance is propagated furthest of all...22
The Roman architect Vitruvius developed this comparison between different mediums. Speaking of the voice he wrote:
It moves in an endless number of circular rounds, like the innumerably increasing circular waves which appear when a stone is thrown into smooth water, and which keep on spreading indefinitely from the centre unless interrupted by narrow limits, or by some obstruction which prevents such waves from reaching their end in due formation.23
The Stoic philosopher Seneca in his Natural Questions, in trying to account for haloes around the sun, took this analogy of the stone in water and applied it to light:
When a stone is thrown into a pond, the water is observed to part in numerous circles, which, very narrow at first, gradually widen out more and more until the impulse disappears, lost in the surface of the smooth water beyond. Let us suppose something of the same kind to occur in the atmosphere. When condensed, it is capable of receiving an impact: the light of the sun, moon, or any heavenly body encountering it forces it to recede in the form of circles. Moisture, be it observed, and air, and everything else flat takes shape from a blow, is driven into the same form as that possessed by the object that strikes it. Now every kind of light is round. Therefore, the air when struck by light will assume this form. Accordingly the Greeks gave the name Threshing-floor (i.e. Halo) to a brightness of this kind, because spaces set aside for threshing corn were, as a rule, round.24
For our purposes this passage is significant for at least two reasons. First, it introduces the analogy between the circular waves caused by a stone thrown into water and waves of light in the atmosphere. Second, it relates this analogy to the concept of a blow. Leonardo, as will be shown, developed both of these ideas. In the third century Diogenes Laertius referred to the analogy between waves in water and waves in air while discussing Zeno in his Lives of the Eminent Philosophers (VII:158):
We hear when the air between the sonant body and the organ of hearing suffers concussion, a vibration which spreads spherically and then forms waves and strikes upon the ears, just as the water in a reservoir forms wavy circles when a stone is thrown into it.25
In the thirteenth century, John Pecham, the Archbishop of Canterbury, used this same analogy in his standard textbook on optics, the Perspectiva communis:
Although, as should be known, all pyramids in a single body of illumination constitute essentially one light, they nevertheless differ virtually, that is, in efficacy. In the same way, when a stone is thrown into water, different circles are generated which, nonetheless, do not divide the water, so that it is in some way not all separate. From any (given) point of a luminous body a ray of light departs powerfully and the more direct it is, the more strength it has.26
8. Percussion in Water and Air. Leonardo's Contributions
On Mad II 2v (c.1494) there is a "record of the books that I keep locked up in the chest." Among these in Pecham's Prospettiva commune.27 As early as 1490, and probably while he was staying in Pavia with Fazio Cardan, editor of the first printed version of Pecham's work, Leonardo translated, or at least copied out, on CA203ra a translation of the opening paragraph of Pecham's treatise. It is therefore very likely that Pecham's comparison between waves in water and waves in air served as a direct source for Leonardo. On CA373rb(c.1492-1497) Leonardo drafts a passage, on circular diffusion of images in the air:
Every (opaque) body, placed in the luminous air fills circularly (this air) the infinite parts of this (air) with its similitudes, and is all in all and all in the part and goes diminishing itsspecies by the equidistant space surrounding it like the...
Of the 4 elements and 2...
Immediately following this he explores the traditional analogy between waves in water and waves in air in a series of eight more drafts which reveal how laboriously he reformulates an idea:
(Just as) The stone thrown in the water fills this (with waves) of circles makes itself the centre of various circles, which have for their centre the percussed place.
And the air similarly fills itself with circles, of which (they make themselves the centre of the vo[ice] their centres are sounds and voices made in this.
As the water with various circles surrounds the place percussed by the stone.
(As ) The stone, where the summit of the water percusses, causes circles around it, which go expanding (themselves) to such an extent that they disappear, and also (and) the air, percussed by a voice or by a noise, (by) similarly departing circularly, is gradually lost, such that the nearest is better heard and the far away less heard. a[s].
Just as the stone thrown in the water makes a centre (in various circles) is cause of various circles, so too, the sound made in the air, sends to the equidistant ears equal[lly], circularly spreads its (sound) voice. Just as (that) the air (which is re) percussed by the voice (is that) the water (which is percussed) by the stone, goes in a circular movement (spreading out and fleeing from their source) showing their source. Which circles make themselves centre of the percussed place and the more they become distant.
Just as the water and the air [are] percussed, the one by the voice and the other by the stone, you will see at the water by various circles demonstrate the percussed place, thus you will feel at an equal distance the sound of the voice made in the air
As (in bodies)
As circularly the species of bodies are spread beneath the infinite parts of the nearby air....
To the modern reader Leonardo's almost obsessive reformulation of ideas is often boring. Nonetheless, it offers a clue why there are so few instances where he merely copies: his transformative mind is too active. These drafts on CA373rb (c.1492) lead to a clear formulation on A9v (c.1492):
Just as the stone thrown in the water produces a centre and causes various circles, sound made in the air, spreads circularly.(figure)Figs. 66-69: Percussion, circular propagation and non-interference of waves. Fig. 66, A61r; fig. 67, Forst III 76r; fig. 68, H69v; fig. 69, Leic. 23r.
Figs. 70-73: Circles representing waves of water or air inter-secting without interference on CA300rb.
Figs. 74-75: Non-interference of circular wave patterns in Jan van Eyck's Ghent Altapiece and in Leonardo's treatise BM135r.
Thus every body placed between the luminous air spreads circularly and fills the surrounding parts with its infinite similitudes and appears all in all and all in every part. There follows a diagram (fig. 133) and a brief text: "This is proved by experience, because if you close off a window facing the West and make an aperture," which then breaks off. Here he has linked the concept of circular wave motion in water and air with (a) the principle that images are "all in all and all in every part" and (b) the camera obscura. (We shall return to this remarkable nexus of ideas, central to his physics of light, cf below pp. ). The topic of circular wave motion in water, which just broached on A9v (c.1492), is developed on A61r (fig. 66, c.1492):
Even though voices which penetrate this air part with circular movements from their sources, nevertheless, the circles springing from different origins meet together without any interference and penetrate and pass [through] one another always maintaining their causes through [their] centre.
There follows an assumption: "Since in all cases of motion, water has great conformity with the air I shall add to the above (mentioned) proposition with an example." This assumption is because it explains why Leonardo uses the example of a stone thrown in to water to illustrate effects of sound and light in the air, as for instance on A73 (c.1492), where he compares turbulent water and turbulent air. It also explains why he sometimes draws intersecting circles in contexts where they could equally represent waves of water or waves of air as on CA300rb (figs. 70-73, 1508-1510). In addition it accounts for a basic aspect of structure in Leonardo's notes: why, for instance, in the Manuscript C which is devoted primarily to light and shade, Leonardo should discuss water together with optics (cf. C6v, 22r-28v; figs. ); a combination of themes that dominates the Manuscript F and recurs throughout his writings.
Figs. 76-77: Demonstrations that circular waves can contract as well as expand. Fig. 76, CA126vb; fig. 77, Mad I 126v.
In his example on A61r (C1492) Leonardo examines how two spreading waves in water, - and by implication, in air - can meet and cross without interference (fig. 66, cf. figs. 67-75):
I say: if at the same time you throw 2 small stones somewhat distant from one another on an expanse of water without motion, you will see caused around these two said percussions. 2 separate quantities of circles, which quantities growing, will come to meet one another and then incorporate one another, the one circle intersecting the other, always maintaining as centre the places percussed by the stones. And the reason is that even if there appears some demonstration of movement, the water does not depart from its site, because the aperture made by the stones immediately closes itself again and this motion, made by the sudden opening and closing of the water makes in it a certain stir which can sooner be called a tremor than a movement and, in order that what I say be made more manifest to you, call to mind those fescue grasses which through their lightness stand above the water, which by the waves made beneath them by the advent of the circles will nevertheless not part from their first site. Now this stir of the water being a tremor, rather than a movement, they [the waves] cannot, in meeting, break one another, since the water, having all its parts of a same quality, it is necessary that the parts convey (apichino) this tremor from one to the other without moving from their place, for the water, staying in its place, one can readily take this tremor from nearby parts and send it to neighbouring parts, always diminishing its power until the end.
Figs. 78-81: Circular waves in a bowl and transformation from a triangular waves to circular waves. Fig. 78, Leic. 12v; Figs. 79-81, Mad I 95v.
Figs. 82-85: Transformations from triangular waves to circular waves. Fig. 82, CA173rb; fig. 83, CA199vb; figs. 84-85, Leic. 14v.
Leonardo wishes to demonstrate that these circular waves can contract as well as expand, and hence describes an experiment on CA126vb (fig. 76, c.1490-1492):
Here one gives an exit to the water close to the surface and one demands what part of the surface of the water will take more motion, speedily or more slowly in taking the water to such an exit. And in order to make a rule you will put [into the water] particles of things which remain noticeable, which are equal, as are some minute seeds of herbs, and place them in a circle equidistant from the exit at rmf and note how that which first reaches the mouth closes the water and retains the circle.
On Mad I 126v (fig. 77, c.1493-1495) he describes another demonstration of the same principle:
If a bowl, full of water, is beaten on one of its sides, it is certain that the water will begin circular motions which will diminish at the centre of its surface. But if the water which is contained in the bowl be struck at its centre, then circular movement will be caused which, going from minimal circles, will terminate in very large circles.
He pursues this problem on Mad I 95v (figs. 79-81, c.1499-1500), under the heading:
Of percussion and motion in water
Water, which is contained in some bowl, if it be percussed in its centre, will cause circles, which will make a common centre from that percussed place. But if the bowl be percussed from the outside, then the circles will be caused by the contact which the water has with the bowl and will flee diminishing towards the centre of this bowl. And the percussion having been made at this centre, they will return back, increasing until the contact of the bowl, and then they will redo this to the centre of the circle so many times that they lose themselves.
If the percussion made on the water in the middle of its bowl, be of a triangular shape, this, in long movement, will make itself nearly circular and will return, after the percussion made at the banks, towards the centre and then to the banks. In such a way that at its end, such motion of the water will be of a nearly perfect rotundity. And this motion will only be in that part which borders with the air.
This problem of a triangular shape mentioned in the last paragraph is again one which can be traced with some precision. On CA173rb (c.1500) Leonardo makes a quick sketch of a circle within which he inscribes three simple dots (fig. 82). In the caption alongside he asks: "Why the triangular percussion makes a round wave in the water, if the circular waves penetrate one another in their encounters?" In a further note on CA199vb (also c.1500) he restates the problem: "Test if the triangle thrown in still water in the end makes its wave of perfect circularity." The diagram alongside shows a triangle clearly inscribed within a circle (fig. 83). The diagrams (figs. 84-85) and text on mad I 95v (c.1499-1500) clearly represent the next step.
In his camera obscura studies Leonardo also experiments with triangular apertures (figs. ; cf. below pp. ) while exploring how light rays take on a rounded shape with distance. This parallel is no coincidence. He considers water as a medium that helps render visible corresponding effects in air and as such, water is his equivalent to a modern slow motion camera, allowing him to study more closely the transformations of waves from one shape to another.
In volume one (pp. ) it was noted that Leonardo's concern with transformational geometry was linked with his studies of the movement of the heart. In these transformations of waves of water and/or light it can be seen that Leonardo's fascination with De ludo geometrico had other links with physical reality (figs. ). For him it is far more than an abstract geometrical game; transformational geometry is a key to natural phenomena. Whence he claims that the quadrature of curvilinear surfaces is the goal of the geometrical sciences, and equates science with the ability of transforming one shape into another and back again. Meanwhile he has become preoccupied with another problem related to circular wave motion, which he posed as a question on Forster II 76r (fig. 67, c.1493): "Why do the circles of water not break when they intersect?" His preliminary answer is an experiment on H31v (fig. 87, March 1494):
The stone thrown into dead water will make equal circulation of motion, the water being of equal profundity.
If two stones be thrown [into water] the one near the other by an interval of one braccia, the circles of the water will grow equally the one body into the body of the other without the one causing the other interference (sanza guastamento lunu dellaltro)But if the base be not equal the circulation [i.e. production of circular waves] will not be equal except at the surface.
On H 69v (c.1494) he again draws two non-interfering wave patterns (fig. 67). Below this he draws four diagrams in which he compares the non-interference of streams of water and wine or tinted water. Accompanying this is a brief text: "[Streams of] water of equal course and descent which move(s) - counter to one another pass(es) through one another without changing natural course." On H 67 (c.1494) he compares waves in water, with waves in other mediums:
Water percussed by water makes circles with regard to the place percussed. Over a long distance the voice in air [does the same], [over a] longer [distance] in fire. More so the mind in the universe. But why does the finite not extend to the infinite?
In another passage on CA175ra (c.1494) he again considers the circular waves of vision:
... each eye in itself (has a cen[tre]) causes infinite visual lines which are in the eye of so much greater power to the extent that they are closer to the centric line which is in the firstdegree of visual power. Now these lines spread out circularly from this centric line and are adopted in the powers of the species and similitudes of the things which are placed in front of the eyes.
Figs. 86-87: Circular seats in the Roman theatre at Aspendos and circular waves of light/sight in Caesariano's commentary on Vitruvius (1521).
(Figs. 88-91: Intersecting wave fronts on Ca83vb.
Figs. 92-100: Waves in agitated water. Fig. 92, H31v; fig. 93, Leic. 14v; fig. 94, I87r; fig. 95, I86v; figs. 96-97, I87r; figs. 98-100, Leic. 14v.
This passage bears comparison with a diagram in Caesariano's commentary on Vitruvius23 (fig. 87). On CA83vb (c.1508) Leonardo devotes four more sketches to the problem (figs. 88-91) this time with only a brief caption: "Every part of the wave which percusses on the other waves is reflected towards the centres of their centres." Meanwhile he has also been reconsidering the possible interference of expanding circular waves on M I86(v) (fig. 95, c.1497):
I ask whether a circle, which has its growth, which meets with the growth of another enters with its wave penetrating the wave of the other, as n. passes in c. at the same time that n. passes in d. or, truly, if at their percussion they bounce back under equal angles as when c. entering n. jumps to d. and likewise d., hitting n., bouncing back on c.
This is a beautiful and subtle test.
Not content with a static situation, on the following folio, I87(r), he examines what happens in the case of flowing water (fig.s 96-97):
If the stone is thrown in motionless water, its circles are equidistant from its centre. But if the stream moves the figures will form long figures, nearly ovoid and will go with their centre, from the place where it was created, following the course of the [stream].
Leonardo considers further aspects of the physics of wave motion on Leic. 14v (figs. 84-85, 98-100, c.1500-1516):
More speedy (is the wave) is the motion of the wave than the motion of the water which generates it. This is seen in throwing a stone in a dead [pool of] water which generates around the place of percussion circular motion, which is speedy, (which) and the water, which (it) produces such circular inundation, does not move from its situation, nor even the things which sustain themselves above the water. For what reason are the reverse sides of the waves of the water more circular than tided with various faces? This arises because the water which moves itself in a curved way at the beginning cannot direct its course in a straight line if it does not find water of less resistance than the first and, not finding this, it needs be that it observes in the middle and the end of motion, that which from the beginning was given it.
Elsewhere is the same treatise on Leic. 23v he again compares (cf. A61v above) the equivalence between waves in various elements
The wave of the air takes the same role under the element of fire as the wave of water under air or the wave of sand, that is, earth under water and their motions are in proportion as their motive forces are amongst themselves.
The problem continues to fascinate him. On CA77vb (c.1505, cf. CA264rh, C.1500-1505 where he speaks of the sphere of water seeking perfect rotundity) Leonardo notes that the voice "acts like a circular wave in the sea." On BM Arundel 204r(c.1505-1508) he develops his air/water analogy:
Figs. 101-105: Percussion and circular waves beyond an aperture-like channel. Figs. 101-103, Leic. 14v; Figs. 104-105, CA281ra.
Figs. 106-109 CA281ra Lunule studies on CA281ra. Cf. figs. 104-105.
More speedy (is the wave) is the motion of the wave than the motion of the water which generates it. This is seen in throwing a stone in a dead [pool of] water which generates around the place of percussion circular motion, which is speedy, (which) and the water, which (it) produces such circular inundation, does not move from its situation, nor even the things which sustain themselves above the water. For what reason are the reverse sides of the waves of the water more circular than sided with various faces? This arises because the water which moves itself in a curved way at the beginning cannot direct its course in a straight line if it does not find water of less resistance than the first and, not finding this, it needs be that it observes in the middle and the end of motion, that which from the beginning was given it.
On this same folio (Leic. 14v) he draws three sketches of water in a very narrow canal, with waves extending beyond this in a half circle (figs. 101-103). He draws related diagrams on CA281ra (figs. 104-105, 1516-1517) where they appear in connection with lunule studies (figs. 106-109). In Leonardo's mind there is a connection between the (theoretical) intersections and transformations of circles in geometry and the (practical) intersections and transformations of circles in water. His associative mind compares waves in various elements as on A61v cited earlier, or on Leic. 23r:
The wave of the air takes the same role under the element of fire as the wave of water under air or the wave of sand, that is, earth under water and their motions are in proportion as their motive forces are amongst themselves.
Fig. 110 Circular sound waves in the open air on A43r;
Fig. 111 Experiment with circular waves in water passing through an aperture (cf. Newton).
Such comparison fascinates him. On CA77vb (c.1505) he notes that the voice "acts like a circular wave in the sea." On BM204r (c.1505-1508) he develops his analogy between air and water:
The air is a liquid body invested with a spherical surface and is penetrated by solar rays which restrict themselves in their concourse and the more they are heated the more they are restricted at the point of their concourse.
He repeats the comparison between circular expansion in water and air on CA112vb (c.1508-1510) and CA251rb (c.1510-1515). On CA84va (c.1510-1515) he returns once more to the non-interference of waves:
If you throw a stone into a body of water (pelago) with different sides, all the waves that strike these sides reflect towards the percussion, and in encountering other incident [waves] they never impede the course of one another.... All the impressions of percussions made on the water can penetrate one another without their destruction.
Meanwhile, on CA77vb (c.1507) Leonardo had considered a special case in the analogy between waves of air and water, namely, when the waves are broken or interrupted:
...The note of the echo is either continuous or intermittent
...The note of the echo is intermittent when the place which produces it is broken and interrupted. It is single when it is produced in one place only. Accompanied is when it is generated in several briefly. Long is when it turns round in the percussed bell or in [a] cistern or other concavity, or clouds in which the voice extends itself in degrees of space with degrees of time and always diminishing, the medium being uniform and it acts like the circular wave in a pond m.
Figs. 112-114: Experiments with circular waves of water passing through apertures on W19106v (K/P 126r).
On W19106v (K/P 126r, c.1510) he describes an experiment concerning such interrupted waves (figs. 111-114):
Go in a boat and make the enclosed place nmope and inside it put two pieces of board sr and tr and give percussion to a and see if the interrupted wave passes with its convenient part to bc and that which you experiment with the wave cut from the circular wave ... of the water, such you will conceive as having experimented on the part of the wave of the air which passes through the tiny aperture where the human voice, enclosed in a box, passes [through],m as I heard at Campi with someone enclosed in a cask open at the bunghole.
More than a century and a half later Newton repeated this experiment in his Principia.28 Leonardo has crossed out this passage on W19106v (K/P126r, c.1510). Immediately to the left he draws a diagram (fig. 113) without text: it shows a rectangular enclosure, three sides of which contain an aperture centrally positioned. An object has been dropped into the centre of this enclosure and has produced circular waves. These continue beyond the apertures and maintain their characteristic circularity. To the North and South of this rectangular enclosure two other objects have been dropped into the water and the circles which emanate from these objects, having passed through the apertures, move towards the centre of the enclosure, crossing the outgoing waves without interference. To the left is a tiny sketch showing two circles of waves meeting without interference (fig. 112).
Figs. 115-118: Four examples of percussion. In Leonardo's mind the blow of an axe, falling water, the blow of a hammer and light striking the eye are all examples of a single concept. Percussion, motion, gravity, and force constitute his four powers by means of which he believes that he can account for the whole of reality. Figs. 115-116, C22v; figs. 117-118, A1v.
On this far left and slightly lower down is another passage that has been crossed out: "test of throwing the object in an expanse of water and you see the wave, where it is interrupted, which is what it does in fo." Below this Leonardo has drawn a final diagram (fig. 114) showing three sides of a right-angled container, the side mn of which has an aperture in the middle. From a central point within the container emanate circular waves. These pass through the aperture and go beyond it to form the arc mban. Meanwhile, waves coming from outside are coming through the aperture in the opposite direction as they move or, as Leonardo might say, their tremors go towards the centre of the container. A one line caption accompanies the sketch: "ab. is the voice from the aperture cd." This confirms that he is again concerned with waves of sound akin to those produced by the man in the cask referred to in the first passage cited from this folio.
In this chapter we have stressed the continuity between classical ideas and Leonardo's approach. We have shown that his concept of percussion can be traced back at least to Aristotle, and that his simile comparing sight with waves of a stone thrown into water has precedents in both Vitruvius and Seneca. Leonardo's late experiments to determine what happens when circular waves pass of water or sound pass through small apertures (W19106v) could even be seen as demonstrations of a Vitruvian passage23 cited earlier (p. , cf. figs. 86-87).
Figs. 119-120: Percussion of circular waves of water on Mad II 126r.
Figs. 121-122: Percussion of circular waves of light in Christiaan Huygens Treatise of Light (1690).
While Leonardo's debts to tradition cannot be denied, his innovative qualities must not be overlooked. Similes, which his ancient and mediaeval predecessors had mentioned in passing, Leonardo uses systematically. That which they had used in a loose figurative sense, he adopts literally. This taking literal of the figurative again stands in a tradition - it is an outgrowth of nominalism - and it points in turn to the increased emphasis on literal interpretation and the veracity of verbal images that was to take place in the next generation with Luther and Calvin.
Distinctive in Leonardo's approach is how he renders visual the process of taking the figurative literally. He is not just interpreting words, he is trying to illustrate them as pictures. Breughel builds on this principle when he paints the Netherlandish Proverbs (Berlin, Dahlen). It is easy at this point to slip into a Hegelian mentality which insists on a complete harmony linking new ideas in science, religion and art. The details of history are always richer than such generalizations, varying from town to town, and ultimately from one person to the next. But if we keep these differences in mind it is of interest to see that underlying the contrasts between Netherlandish and Italian art; between science in the North and science in Italy and even between Protestantism and Catholicism there was a new relationship between pictures and words, between visual images and verbal images to which the whole of Europe was (re)acting.
Leonardo's visualisation of traditional verbal images concerning circular wave motion is of particular interest (cf. Argentieri, 1939) because it points to the work of Christiaan Huygens. Parallels between the two thinkers are striking. Huygens specifically mentions that light "spreads by spherical waves, like the movement of Sound."29 In describing the foundations of light and sound Huygens refers to "Laws of Percussion,"30 describes propagation in terms of "agitation"31 and speaks of waves as being "struck by blows,"32 images familiar from Leonardo's theories of percussion and the four powers. Leonardo's principle of new waves propagated from points along a wave front, recurs in Huygens treatise33 (figs. 121-122, cf. figs. 119-120). From such examples an extraordinary continuity comes into focus. Certain, concrete, observable situations are mentioned by Aristotle by way of similes. Leonardo takes these literally, tests them experimentally and demonstrates them visually. This prepares the way for the radically mechanistic approach of Huygens who claims:
It is inconceivable to doubt that light consists in the motion of some sort of matter.... This is assuredly the mark of motion, at least in the true Philosophy, in which one conceives the causes of all natural effects in terms of mechanical motions. This, in my opinion, we must necessarily do, or else renounce all hopes of ever comprehending anything in Physics.34
Hence there is a direct continuity between Aristotle's similes, Leonardo's visualization and Huygen's mechanism. Why then does the standard view hold that the mechanistic world picture came about through a rejection of Aristotelianism? Historically speaking there were effectively two Aristotles. Aristotle, number one, has an organic framework of knowledge, based on concepts of growth (generation) and decay (corruption) which are basically qualitative. Aristotle, number two uses specific similes are potentially quantifiable. Leonardo concentrates on Aristotle number two and begins to express his similes visually and quantitatively. Leonardo's successors explore the implications of this quantitative approach only to find that it contradicts the organic, qualitative framework of Aristotle number one. Galileo accordingly finds it convenient to make Aristotle number one, along with Ptolemy and the Church, into straw men, symbolic of everything qualitative and opposed to everything quantitative, modern and scientific.
Galileo's rhetorical picture of Aristotle overshadows and effectively eliminates the historical role of Aristotle number two. The course of the continuity is thereby forgotten and the way is prepared for a historiography that can interpret the mechanistic science of the seventeenth century strictly as a rejection of the past. This trend continues today; witness the view of Aristotle found in Dyksterhuis' Mechanization of the World Picture:
Aristotelian physics thus has the advantage over classical mechanics in that it deals with concrete, observable situations constantly encountered. But from a scientific point of view this very advantage constitutes its weakness, for those situations are so complicated (the reader needs only think of a vehicle drawn through the air along a rough road, or of a body of any form thrown upwards) that even with the aid of perfected classical mechanics they can be treated mathematically only by approximation and at the expense of comparatively arbitrary suppositions.35
If we are right then it is precisely these concrete, observable situations of Aristotle that hold the key to understanding how seventeenth century mechanism became possible. At the very least, such instances in Aristotle explain how Leonardo's physics of light and shade can be both Aristotelian and open to a mechanistic interpretation. To understand why Leonardo's physics is not mechanistic in the seventeenth century sense will require a more detailed analysis of his basic definitions.
Last Update: July 2, 1999