Leonardo da Vinci

Here you will find a complete corpus of Professor Kim Veltman’s copious writings on the Italian polymath Leonardo Da Vinci.

Kim wrote five major treatises on Leonardo, each running to around one thousand pages (texts in English), plus he authored over a dozen major papers on Leonardo (texts in English, Italian, French etc). We provide pdf copies of all works here for your convenience (see below).

Plus, we plan to publish the Encyclopedia of Leonardo Da Vinci (approx. 5000 pages), which will unite all of Kim Veltman’s writing(s) on the great polymath.

To begin, we have publish below a complete version of Kim’s first article on Leonardo from the year 1982, entitled: ‘Visualisation and Perspective’.

Visualisation and Perspective

Kim H. Veltman

Published in: Leonardo e l’eta della ragione, eds. Enrico Bellone e Paolo Rossi, (Scientia, Milan, 1982), pp.185-210 (English), pp. 211-224 (Italian).


  • Introduction
  • Visual and Verbal Images
  • Perspective
  • Scales of Abstraction
  • Viewpoints, Levels and Models
  • Focus
  • Series
  • Systematic Play with Variables
  • Literal Visualisation
  • Observation, Experiment and Quantification Substance and Function
  • Universals and Particulars
  • Subject-Object
  • Conclusions


The 6,500 pages about of Leonardo da Vinci’s notebooks contain ca. 100,000 sketches, diagrams and drawings. No one in recorded history prior to him and possibly no one since, has ever produced such a wealth of visual records.

The first part of this paper will attempt to answer why Leonardo was so concerned with visual images. The significance of his perspectival studies will be emphasized and the role of six related factors will be examined: 1) scales of abstraction, 2) models, 3) levels of focus, 4) series, 5) systematic play with variables and 6) literal visualism. The second part of this essay will consider some of the consequences of Leonardo’s visual approach.

Visual and Verbal Images

In his notes Leonardo is very explicit concerning the supremacy of visual images over verbal ones. On W 19013 v (ca. I510-1514), for example, he writes:

«And you who want to demonstrate with words the figure of man in all aspects of his limb formation do away which such an idea, because the more minutely you describe the more you will confuse the mind of the reader, and the more you will remove him from knowledge of the thing described. Therefore it is necessary to make a drawing of it as well as describe it».

This idea of the supremacy of visual images is restated on W 19071 r (ca. 1513) in a note at the side of a drawing of the heart showing the arrangement of veins and arteries:

«O writer, with what words will you write such perfection of the complete configuration which this drawing here composes. This you describe confusedly having no knowledge and you leave little conception of the true shapes of things which you, deceiving yourself, believe capable of satisfying your auditors when you have to speak of the configuration of any corporeal thing surrounded by surfaces. But I remind you not to involve yourself in words unless you are speaking to the blind. If, however, you wish to demonstrate with words to the ears and not to the eyes of men, speak of the essence or nature of things and do not busy yourself with things belonging to the eyes by making them enter through the ears, for you will be far surpassed by the work of the painter».

Leonardo is not only convinced of the superiority of drawings over words. He even goes so far as to claim that drawings are more informative than the anatomical parts themselves (W 19070 v, 1509-1510; W 19013 v, ca. 1510-1514).

It is not surprising, therefore, to find that Leonardo often develops an argument (CA 2 va., ca. 1495), or series of demonstrations using only drawings without a word of text (e.g. Ms A 39 r – 40 v, 1492). Nonetheless, it would be misleading to see in Leonardo someone who sees the visual as opposed to the verbal, pictures as opposed to words, for his notebooks evidence him exploring various combinations of the.two. Sometimes, as on W 12701 (1508-1509), a drawing occurs with only an emblematic epithet. Sometimes as on CA 167 rab (1510-I515), the drawings have regular captions. On occasion the drawings, in turn appear in the margin, as visual captions to the text (e.g. Ms D 8 v, 1508). On other occasions, as on W 12701, 1508-1509, a drawing occurs with only an emblematic epithet. Sometimens as non CA 79 ra (ca. 1505-1506), words predominate and there are effectively no diagrams. For the purposes of this paper we shall be concerned primarily with those cases where the visual image dominates and attempt to explain why this should be so.


One of Leonardo’s basic incentives for making visual records stems from his study of linear perspective which he defines on Ms A 1 v (1492) as «nothing else than seeing a site behind a flat transparent pane, on the surface of which are marked all the things which are behind that glass and which can be conducted by means of pyramids to the point of the eye and these pyramids intersect the said pane».

On Ms A 3 r he provides another definition of this term: «Perspective is a demonstrative means, whereby experience confirms that all things send their similitudes to the eye by pyramidal lines and that those bodies equal in size will have their pyramids at a greater or lesser angle in accordance with the variety of distance that there is from the one to the other».

For Leonardo, perspective is not only based on experiental demonstrations. It involves careful quantitative records of the natural world. On CA 1 bis ra (1480-1482) he draws a picture of his interposed glass pane «pariete di vetro» without text, which he then de- scribes verbally some ten years later on Ms A 104 r (Ash I, 24 r, 1492). «Have a piece of glass as large as half a royal folio paper and set this firmly in front of your eyes, that is, between your eye and the thing you wish to draw; then place yourself at a distance of 2/3 of a braccio from the glass fixing your head with a device in such a way that you cannot move it at all. Then shut or cover one eye and with a brush or drawing-chalk draw upon the glass that you see beyond it; then trace it on paper from the glass. Afterwards transfer it onto good paper and paint it if you like, then using aerial perspective carefully». These basic definitions are important because they point to an experimental, quantitative approach to linear perspective that is lacking in earlier authors.

It is commonly assumed that linear perspective was simply a trick for making realistic pictures (re-)discovered by Filippo Brunelleschi.1 This view is simplistic. Linear perspective essentially involves a systematic play with three variables: eye, picture-plane and object (Chart 1). One can A) move the object, while keeping the eye and picture- plane stationary; B) move the picture-plane while keeping the eye and object stationary or C) move the eye while keaping the picture-plane and object stationary.

The inverse size-distance law of linear perspective applies to situations where the picture-plane and object-plane are parallel (type 1). But the picture-plane can also be at right angles to objects, either beneath the eye (type 2 i) or above the eye (type 2 ü). Or it can be oblique, tilting towards the viewer (type 3 i) or away from the viewer (type 3 ü) or it can be concave relative to the viewer (type 4 i) or convex relative to the viewer (type 4 ü).

Chart 1. Survey of basic types of linear perspective. Drawing by Ute Barner ABC

Object moved while eye and picture plane stationary

Picture plane moved while eye and object


Eye moved while picture plane and object stationary

Approached in these terms linear perspective is recognized as a complex phenomenon, the underlying relations of which were only gradually understood. As early as the 1390’s Biagio Pelacani da Parma2 intimates acquaintance with the principles of type 1, but he provides no quantitative demonstrations. Leon Battista Alberti, in Della pittura, the first extant treatise on perspective, intimates acquaintance with types IA and 2iA, but again gives no quantitative demonstrations. Filarete is equally vague. Piero della Francesca, in his De prospectiva pingendi, provides quantitative evidence,3 but a) he moves two variables at a time and, â) when he sets out to give an example of type 1, he provides an example of type 2.

Leonardo da Vinci’s notebooks contain the first extant evidence of someone who goes further. He systematically alters the three variables and arrives at the very inverse size- distance law4 that both Euclid5 and Piero della Francesca had denied existed.6 Where his predecessors and contemporaries had been qualitative, Leonardo is experimental and quantitative (Ms A 8 v, 1492). Where they had been purely verbal in their demon- strations, Leonardo provides visual demonstrations (Ms A 39 r ff., 1492).

Leonardo’s systematic exploration of the relations between eye, picture-plane and object extend to the domains of colour perspective as we learn from a passage on Ms A 102 v (Ash I, 22 v, 1492):

”Wishing to put into practice this perspective of variation and loss or diminution of the proper essence of colours, observe at every hundred braccia objects positioned one behind the other in the landscape such as trees, houses, men and sites and in front of the first tree you will have a good, steady plate of glass and your eye likewise good and steady and on this plate of glass you will draw a tree in the form of that original one. Then move it far enough to the side so that the real tree practically borders on the one that you have drawn. Then colour your drawing in such a way that in colour and in form the one matches the other and so that, if the one be closed, the two appear painted on the said window at an identical distance. And using this rule do the same for the second and the third tree, each in turn a hundred braccia further distant. And these imitation trees on the glass plane will serve as your authors and masters, always serving in your works where they belong and they will save a good deal of work».

Elsewhere (Ms A 94 v [Ash I, 14 v], 1492), Leonardo applies the same principles to loss of distinctness on which topic he writes no less than 50 passages. This leads him to speak of three perspectives: linear, colour and loss of distinctness (e.g. Ms A 98 r [Ash I, 18 r], 1492), and each of these categories provides him systematic means of recording visible aspects of the natural world.

Linear perspective is, for Leonardo, much more than a method of recording objects in the natural world. It serves equally to render three-dimensionally objects at different levels of abstraction. As such perspective serves to establish a spectrum ranging from the concreteness of the natural world to the abstractions of mathematics.

Scales of Abstraction

Leonardo’s use of perspective in visualising different levels of abstraction can be seen in terms of a threefold programme. First, perspective serves to create visual bridges between organic objects and their abstract geometrical equivalents (cf. Chart 5). Second, it serves to render three-dimensionally geometrical figures, thereby creating bridges between geometry and the natural world. Third, linear perspective serves to visualise three- dimensionally invisible concepts of nature such as force of wind or concentration of heat. Each stage of this threefold programme for creating bridges linking different levels of reality requires a brief explanation.

In the case of heart, for instance, Leonardo’s use of linear perspective helps to represent convincingly both the whole (e.g. W 19074 v, ca. 1513) and its various parts (e.g. W 19118 v, ca. 1508-1510; and W 19117 v, ca. 1508-1510) (Fig. 1). Perspective serves equally to represent model versions (Fig. 2) and more abstract variants (Fig. 3) that provide visual bridges to his purely geometrical figures (Fig. 4).

These geometrical versions relate, in turn, to Leonardo’s studies of transformational geometry. What may at first appear as a purely abstract play of forms, can therefore be recognized as an attempt to analyse the opening and closing of the heart’s lunules with the aid of geometrical lunules.

A second stage in Leonardo’s programme is to employ perspective in rendering the geo- metrical world naturally. Ever since Antiquity mathematicians had discussed the regular, Platonic bodies and the semi-Archimedean solids in abstract terms. Leonardo takes these forms and renders them three-dimensionally (cf. De divina proportione, Mss in Geneva, Milan) with a verisimilitude of which none of his contemporaries were capable, teste Pacioli.

Almost immediately afterwards they are copied. Giovanni da Verona uses them for his illusionistic intarsie in S. Maria in Organo (Verona). Luca Pacioli makes models of these bodies. This making of realistic versions of the Platonic solids becomes linked with cosmology (Pacioli,7 Jamnitzer8) and the way is prepared for Kepler’s use of these solids in his model of the universe (cf. Ms F 27 v, 1508).

Leonardo’s perspectival versions also help to explain why the regular and semi-regular solids should take up so much space in the cupboards of the Cabinet de physique of Bonnier de la Mosson in the eighteenth century. Once represented realistically these forms served as a concrete alphabet of geometry with which to read the book of Nature.

Leonardo does not stop at the regular solids, however. He uses linears perspective to produce three-dimensional versions of the Pythagorean principle. Faced with the problem of transforming a rectangular shape into a cylindrical one, he uses perspectival sketches to illustrate how clay can be used to enact this transformation physically (CA 283 v, 1500-1505, Figs. 5-6). Indeed, Leonardo uses perspective to change a traditionally abstract Euclidean geometry into a corpus of three-dimensional problems. Whence we find Leonardo, on For I, 40 v, (ca. 1489) referring to: «Geometry which extends itself to the transmutation of metallic bodies which are of a material apt to extend and to foreshorten themselves in keeping with the necessity of their observers (speculanti)».

A third stage of Leonardo’s programme is to use perspective in rendering concretely abstract problems of physics. His sketch showing pyramidal concentration of wind on Madrid II, 74 v (1503-1504) provides a good example of this. Here again Leonardo is concerned with producing visual bridges between different levels of abstraction. Hence, when referring to pyramids with respect to concentration of heat (Fig. 10) or acoustics (Figs. 11-12) he commonly shows them simply as two-dimensional triangles. These studies lead Leonardo in the direction of an inverse square law of light. He grapples with ideas later pursued by Mersenne and subsequently codified by Bouguer. In 1848 when Edgar Allan Poe9 sets out to summarize the Renaissance approach in his essay on the cosmos dedicated to Alexander von Humboldt, he unwittingly uses (Fig. 14) a Leonardian pyramidal diagram (Fig. 13).

A consequence of this threefold programme is that Leonardo regularly attacks problems at different levels of abstraction. In his studies of light and shade, for instance, he uses perspective to provide some sketches relating to everyday experience (e.g. CU 658, 1508- 1510) others relating to model situations (e.g. Ms C 22 r, 1490-1491) which still others render the same problem in abstract terms (e.g. Ms A 89 v [Ash I, 9 v], 1492).

Leonardo’s interest in these visible bridges between the natural and the geometrical, between the abstract and the concrete, deserves to be seen as one of his most significant achievements. Herein lies, moreover, a major reason for the abundance of his sketches and drawings. Whereas, his predecessors had been content to describe a heart verbally, or at best to provide a diagram in passing, Leonardo is commited to drawing naturalistic versions, model versions, more abstract versions and geometrical versions. This leads him, in the Windsor Anatomical Corpus, to devote no less than 181 representations relating to this organ alone.

It would be misleading, however, to suggest that Leonardo simply sat down one day and developed this threefold programme `out of the blue’. It grew gradually out of his struggles to represent perspectivally complex organic forms.

Viewpoints, Levels and Models

In the case of a regular geometrical shape such as a cube only a ground-plan and ele- vation are needed to provide the information necessary for an accurate perspectival drawing. Leonardo recognized that in the case of a hand or some other part of the body a more complex approach is needed. He therefore recommends a minimum of three viewpoints (e.g. W 190049 v, 1489 – ca. 1510) or four viewpoints (W 19054 v, 1489 – ca. 1510). With respect to the human body as a whole he prefers eight (W 19008 v, ca. 1509- 1510) and in the case of a hand he notes that the number of viewpoints is potentially in- finite (e.g. TPL 402, 1508-1510). As a result, instead of one picture to be drawn there are now at least four to eight drawings that are required.

Leonardo recognizes, moreover, that a detailed representation of the surface of any or- ganic object occludes everything eise, and he therefore evolves various methods of rep- resenting the layers hidden beneath the surface. Among these are the blow up (e.g. CA 8 vb, 1478-1480 or W 19007 v, 1508-1510), the cross-section (e.g. W 12609 v, 14851487), transparency (e.g. W 19054 v, 1489-1510) and the systematic rendering of levels involving both an additive method, starting with the bones and applying layer by layer

(e.g. W 19009 r, 1509-1510) and a subtractive method beginning, as on W 19013 v, 1509- 1510), with a complete shoulder and then stripping off layer by layer.

The number of these layers or levels varies. With respect to the hand Leonardo recommends eight on one occasion and ten on another (cf. W 19009 v, 1509-1510). Each hand is, moreover, to be seen from a minimum of four positions. Hence a single hand now requires at least 40 diagrams. In addition, he recommends that each series should be drawn in terms of a child, a young man and an old man. Now a single hand requires a minimum of 120 diagrams. What applies to the hand applies equally to every other organ of the body. Leonardo’s systematic representation of the body in its various levels thus leads to a dramatic increase in the scope of visualisation.

Leonardo discovers that a theory of levels is easier than the practice. He finds that he cannot simply draw the muscles of the hand by looking at an anatomical specimen and hence recommends an alternative (W 19009 r, 1509-1510): «When you have drawn the bones of the hand and wish to draw in this the muscles which are joined with these bones make threads instead of muscles. I say threads and not lines in order that one should know what muscle goes below or above another muscle; which cannot be done with simple lines. And having done this, then make another hand alongside this one where the true shape of these muscles is shown as is demonstrated above)). Leonardo pursues this theme in connection with the muscles of the foot on W 19017 r, 1509- 1510):

«You will never cause anything but confusion in the demonstration of muscles, their positions, origins and insertions unless you first make a demonstration of the fine muscles by making rows of threads. Thus you will be able to represent one upon the other as nature has placed them and thus you will be able to name them according to the part that they serve, that is to say, the mover of the tip of the great tos, of its bone in the middle or of its first bone.»

«And having given such knowledge, you will draw alongside the true shape, size and position of each muscle. But remember to make the threads which denote the muscles in the very same positions as the central lines of each muscle. And thus these threads will demonstrate the shape of the leg and their spacing is readily known».

Hence for a perspectival representation of muscles, nerves or veins, simple observation is not enough. One needs to make as model to determine whether one has understood correctly the interweaving of the various layers. In short, reconstruction is a prerequisite for representation of organic forms. Model-making is a consequence of linear perspective.

The use of models in representation becomes a basic aspect of Leonardo’s method. He uses such threads in representing the tendons of the neck (W 19015 r, 1509-1510). On occasion he uses cords to reconstruct the role of muscles (W 19008 r, 1509-1510). Sometimes he uses wires to reconstruct the course of arteries in the legs (W 12619 r, 1509-1510). Sometimes he goes further and reduces an arm to a mechanical model illustrating its essential characteristics in terms of weights and balances (W 19026 v, 1489-1510).

Leonardo’s preoccupation with models is partly to be understood in light of a long tra- dition of mechanical gadgets and automata:10 cf. Hero of Alexandria’s steam bird that simulated flight. Leonardo’s own mechanical lion that walked and then opened up to reveal lilies is very much in this tradition.

Yet there are fundamental differences between such examples and the anatomical models that Leonardo develops. Automata and mechanical models had traditionally been linked with illusionism, were an end in themselves, designed to produce a realistic effect superficially while concealing information concerning their inner structure and causes.

Leonardo’s models are not an end in themselves. They are designed to reveal the nature and causes of movement in other organic objects, by providing information that simu- lates levels below the surface. They are intended as aids in re-presentation and subsequent further abstraction.

Indeed we can safely go further. Leonardo’s anatomical models are a direct product of his attempts to render perspectivally complex organic objects.11 He recognizes that if he is to re-present the natural world realistically he must first re-construct it. Paradoxically the natural is accessible only through the artificial. Or to modify Gombrich’s term: model-making comes before matching.12

As a result the complex organic shapes of the human figure become transformed into their geometrical equivalents which serve as an intermediary step in the process of natu- ralistic representation. A few years later Dürer takes this process one step further and in the next generation robotoid-geometrical figures begin to people the pattern books. An understanding of the links between perspective and model-making,13 thus helps us to recognize the emergence of a new type of pattern book in the generation following Leonardo as a logical consequence of his perspectival studies, (cf. Gombrich’s psycho- logical analysis’14 which does not explain why these pattern books developed so dramat- ically in the 1530’s and ’40’s).

These diagrams and indeed the complete corpus of Leonardo’s work on light and shade have been analysed in detail in the author’s forthcoming book on Leonardo’s optics.
This tradition has again been dealt with in detail in the above mentioned book.


Intimately connected with questions of levels and models in anatomical representation is the problem of focus which Leonardo also explores to a considerable extent. In its simplest form this involves drawing a structure and then redrawing a part in detail as on W 19114 v, ca. 1506-1508 (Figs. 15-16) in the case of arteries in the arm.

Usually, the problem of focus is more difficult, however. For instance, with respect to internal organs such as the lung or heart, Leonardo recognized that it is not enough merely to represent each of them in isolation. Hence, on W 19054 v (1489-1510) he outlines a preliminary programme that involves drawing each organ separately and then in combination, in both cases, from four viewpoints.

He then proceeds, on the same folio, to outline a more elaborate scheme beginning with the lung on its own, then in relation to the tranchea, then to the heart alone, then the heart and its ramification of veins and so on (Chart 2). Each of these in again to be seen from four or possibly six viewpoints. Hence these two organs require a minimum of 28 and probably 42 illustrations. A commitment to clarifying relations between different organs is thus another factor that expands the scope of visualisation.

Wherever Leonardo sets out to illustrate one set of relations he inevitably omits others. For instance, when he is concerned with showing the interior of the heart, the lungs are subordinated to two shady lumps in the background (W 19112 r, ca. 1506-1508). When he is concerned with indicating the arteries leading from the heart to the lungs, he again provides no information about the inner structure of the lungs (W 19104 v, ca. 1506- 1508). On the other hand, where he decides to focus on the structure of the lungs, de- tails concerning other organs disappear (W 19054 v, 1489-1510, Figs. 17-18).

This applies equally to other organs. For example, when Leonardo draws the arteries and veins supplying the liver, he draws them both on their own (W 19051 v, 1489-1510, Fig. 19) and in context (Fig. 20), consciously reducing surrounding organs to outlines in order to achieve a transparency effect. The particular relation that he wishes to clarify now determines the focus and there are now as many illustrations as there are foci. These drawings of different levels and foci lead, in turn, to further representation. They do not pretend to show the object in some absolute form. They serve as visual hypotheses of how the organ might actually be and they thereby increase the challenge of matching quantitatively the next representation with actual organic objects in the natural world.

What is revolutionary about Leonardo’s anatomical corpus is that it is not a closed study: it is not merely a record of his claims visual and verbal concerning the body. The anatomical corpus is filled with instructions concerning procedure. It is a do-it- yourself manual in the deepest sense. The reader is not expected to accept passively Leonardo’s representations. The reader is challenged to take an active part personally in making his own illustrations, thereby expanding further the horizons of representation and visualisation.

This involves one drawing of each organ plus five others in various combinations, i.e. seven basic drawing Each of these is to be represented from at least four viewpoints, namely, front (1-7), behind (8-14), left side (15-21) and right side (22-28). This would amount to 28 drawings for the two organs.

Chart 2(i). First programme for anatomical illustration of lung and heart described on W19054 v (1489-I5/0)

Lung Front Heart Front Lung and Heart Front

Lung Behind Heart Behind Lung and Heart Behind

Lung Lung
Left Side Right Side Heart Heart
Left Side Right Side Lung and Lung and Heart Heart
Left Side Right Side

Lung Above Heart Above Lung and Heart Above

Lung Below Heart Below Lung and Heart Below


Front 1 2 Behind 8 9 LeftSide116 Right Side 2 23 Above 2 30 Below 3 37

34 10 11 1718 24 25 31 32 38 39

5 6 7 12 13 14 19 20 21 26 27 28 33 34 35 40 41 42

Chart 2(ii). A second programme for the systematic representation of the heart and lungs on W 19054 v (1489-1510).

Organ or combination of organLungLung and Ramifications of TracheaHeartHeart and Ramifications and Veins and ArteriesVeins and Arteries of Heart with Ramification of TracheaRamification of NervesHeart
and Lung Joined Together

Sometimes as on W19054 v (1489-1510) Leonardo insists on two further viewpoints, namely, above (29-35) and below (36-42). These, if applied, to each of the seven basic drawings would result in 14 further drawings with a total of 42 drawings just for the two organs.

Leonardo goes on to add that the same procedure should be extended to eight (nine) further organs, namely, liver, spleen, kidney, womb, testicles, head, bladder, stomach, (intestine). This would imply at least 140 and as many as 210 illustrations.


Leonardo is fascinated, at times nearly obsessed with showing series of drawings. This constitutes another basic factor underlying his concern with visual images. Whereas his predecessors had been content with a single drawing, Leonardo is intent on showing sequences. Hence he makes four numbered drawings of the spinal cord (W 190121 v; W 19040 r; W 19021 r, 1489-1510); six numbered drawings of a hand (W 19009 r; W 19012 v, 1509-1510) and literally dozens of drawings of balances (CA 153 vc, ca. 1500), not to mention his series of over 150 lunulae on CA 167rab (1510-1515). This concern with series is partly a result of his commitment to showing various viewpoints, levels and foci.

But it relates also to his studies of change and movement. Here Leonardo is working in a long tradition. Change had been an important concept among such Greeks as Heraclitus, Aristotle or Euclid, who had set out to analyse change and transformation geometrically in his Elements. The Romans had pursued this quest in their mosaics and in the Middle Ages this interest continued both at the level of theory in writings on De ludo geometrico and at the level of practice on the walls of Mediaeval churches such as Palermo, Monreale or indeed Santa Maria del Fiore.

As we have already suggested earlier, Leonardo takes this traditional game much further. He develops a system for transforming solids three-dimensionally on paper as well as physically with the aid of wires (W 19121 v, 1508-1510) or clay (Figs. 7-9). In these transformations Euclid’s approach in terms of books and propositions serves as Leonardo’s standard. Hence Codex Forster I, for instance, is arranged in three books with eleven, thirteen and twenty consecutive propositions respectively. In short, Leonardo’s series of sketches are partly a reflection of the propositional logic of Euclidian geometry.

There is, nonetheless, a fundamental difference. Euclid had been concerned strictly with an abstract geometrical analysis. Leonardo is intent on relating this abstract analysis to concrete inorganic and organic forms. He wants to use geometry not only in cataloguing visible manifestations of movement in general, but also to analyse through visual records ever smaller degrees of movement. He is seeking to achieve empirically what the calculus later codified mathematically.

Leonardo cannot, therefore, be content to make a single sketch of a working man. He must make a series (Figs. 21-25) that appears to us like a forerunner of the stills in an early motion picture, and he proceeds to make pages of such records (e.g. W 12643– 12646 v). Similarly in the case of a foetus: both a series out of context (W 19101 r, ca. 1510-1515, Figs. 26-28) and a series in context (W 19103 r; W 19102 r, ca. 1510-1515, Figs. 29-31).

In a sense the whole of Nature is now like a geometry book. No part can be studied in isolation. Each of its phenomena must be presented in sequence like propositions in order that relations among different appearances can be recognized as part of a single scheme of things created by the Divine Author (cf. W 19001 r, 1509-1510; W 19029r; W 19074r). And the logic of geometry, which acts as an incentive for Leonardo’s series of drawings serves, in turn, to expand the scope of visualisation.

Systematic Play with Variables

Indeed this mathematical context, concerned with systematic demonstrations and proofs helps account for a particular type of series involving a systematic play of variables, which Leonardo develops. Five folios in the Ms A, concerning linear perspective serve to illustrate this method.

On Ms A 39 r (1492) he begins by drawing a group-plan of a square and then shows the same in a perspectivally foreshortened version. He draws the same form from a position off to the side and from an inverted position. He then inscribes a triangle within the square and renders this perspectivally.

On Ms A 39 v (1492) this systematic study continues with a perspectivally foreshortened diamond, pentagon and hexagon. On Ms A 40 r, he proceeds to draw an octagonal shape and a circle followed by a cube and, on Ms A 40 v, a pyramid. Only after this thoroughgoing study of basic forms does Leonardo arrive at his version of the standard perspectival drawing on Ms A 41 v.

No less dramatic in their systematic approach are Leonardo’s explorations of light and shade. On CA 37 va (1508-1510), for example, he explores how one light source in front of one leaning stick produces one shadow, how two light sources produce two shadows and how three light sources produce three shadows (Figs. 32-34).

Elsewhere on CA 177 rb (1505-1508), Ar 248 v and CA 241 rc (1508-1510) he takes two candles in front of two leaning sticks in the form of a St. Andrew’s cross and shows how these produce four shadows (Fig. 35). On CA 241 re (1508-1510) he then systematically extinguishes first the left candle (Fig. 36) and then the right candle in order to show what differences this occasions (Fig. 37). On CA 229 rb (1508-1510) he considers the effect of three light sources. On Ar 243 r (1508-1510) he employs four

Chart 3. Examples of Leonardo’s systematic approach with respect to studies of light and shade.

Number of Light Sources Number of Opaque Bodies Codex 1 1C22r

2 1C22r 3 1C22r

1 1C21r 1 2C13r

1 3C13v

light sources, and on this same folio he explores what happens when a vertical stick is added to the Andrew’s cross formation”.

Leonardo uses the term simple shadow to describe case when only one light source and one opaque body are involved. He refers to cases where more than one light source and/or more than one opaque body are involved as compound shade. Here again he explores various basic combinations with a surprising thoroughness as is shown in Chart 3. Through this approach each new situation leads to representation of each of the variants within that situation. Systematic play of variables is, therefore, another impor- tant factor in expanding the horizons of visualisation.

Literal Visualisation

A majority of Leonardo’s visual images come from his explorations of the spectrum of visual phenomena ranging from concrete objects in Nature to abstract figures of geometry. On occasion, however, he also explores the visual potentials of verbal images through literal visualisation and visual literalism. In his rebus studies, for instance, this leads him to visualise phrases such as: «Thus if fortune makes me happy I shall show such a face» (W 12692 v, ca. 1487-1490). Whence it is no coincidence that Leonardo becomes one of the first to develop the emblem, which is essentially a witty visualisation of words and their meanings. This again expands the range of his visualisation and prepares the way for Alciati’s work in the next generation.

In the realm of science this approach has more dramatic consequences. It leads Leonar- do to put to a visual test traditional verbal similes. Ever since Antiquity there had, for example, been a topos that waves of light and sound spread throughout the air like waves produced by a pebble in water. This simile continued in the Mediaeval tradition, ever as a verbal image”. Leonardo transforms this verbal image by testing it visually. He throws a single pebble into water and studies its expanding waves (Madrid I, 126 v, ca. 1499-1500). He throws two pebbels into water and observes how their expanding waves can intersect without interference (Ms A 61 r, 1492). He proceeds to explore whether a triangular object can produce spherical waves (CA 199 vb, ca. 1500).

Elsewhere he makes experiments to determine whether these expanding spherical waves can propagate themselves through and beyond a small aperture (W 19106 v). This problem is later taken up by Isaac Newton in his Principia (41, Theorem 32). Leonardo

also examines what happens to expanding circular waves when the water is in motion (Lei 14 v, now Ham 14B, ca. 1506-1509). In short Leonardo’s play with variables in his visual testing of verbal images again leads him to increase the scope of visualisation. To answer the question why was Leonardo so concerned with linear perspective, a series of factors can therefore be named. Basic was his systematic study of linear perspective which led him to explore relation between eye, picture plane and objects in the natural world and to draw visible objects. His attempts to draw organic objects in the natural world made it clear to him that he needed to re-construct objects in terms of models before he could represent them perspectivally.

In the process the scope of his perspectival drawings expanded to include a whole spec- trum of levels of abstraction ranging from the concrete objects of Nature to the abstract figures of geometry. Leonardo concern with rendering systematically different viewpoints, levels and relations among objects, also increased the scope of his visual images. So too did his concern with series, his play with variables and. his interest in visualising verbal images. And taken together these factors serve to explain why Leonardo should have made in the order of 100,000 sketches, diagrams and drawings in his notebooks. What were the consequences of this fascination with visual images? This we shall now attempt to answer briefly.

Observation, Experiment and Quantification

We have suggested that linear perspective was, for Leonardo, much more than a means of simply recording objects in the natural world: it was ultimately a process that stimulated the building of visual bridge linking Nature and geometry. Implicit in this programme of relating different levels of abstraction visually is an active approach to the study of Nature, with built in directive for scientific discovery. Man is no longer a passive instrument of Nature. He now has the challenge to create models, picture and geometrical figures that serve as visual hypotheses in relation to objects in the natural world.

Leonardo’s approach in terms of different layers of abstraction (Chart 5) prompts us, moreover, to re-think a number of standard catchwords associated with the rise of early modern science. For instance, observation is, in this context, not merely a looking at the natural world but rather a seeking to record it in terms of pictures, diagrams or some other level of abstraction. Experiment is a translation of a verbal analogy, claim or hypothesis into visual terms and a confrontation of the same with objects in the natural world. Quantification is a measurement of a relation between a diagram or model and the original object(s).

Seen in this context observation, experiment and quantification emerge as aspects of a more complex story of how man gradually recognized that the ideal and natural are not opposed realities, but rather two extremes in a single spectrum and the challenge of knowledge lies in building visual bridges at different levels of abstraction between these two extremes. Hence Leonardo’s approach implicitly challenges us to re-think our ap- proach not only to the history of science but equally to contemporary science.

Substance and Function

Ernst Cassirer in his Substance and Function15 described a basic shift from a concern with definition of objects in isolation (substance) to a preoccupation with relations between different objects (function). He explained this shift in terms of a development from a finite to an infinite world-view. In Antiquity a finite world view had precluded a concept of mathematical space and had constrained thinkers to concentrate on questions of substance in isolation. In the Renaissance the development of an infinite world view demanded a new concept of mathematical space and concern with relations between objects. In short Cassirer believed that the study of systematic relations between objects could be explained simply in terms of a change in a philosophical superstructure, theory determining practice.

The evidence of Leonardo’s studies of perspective and visualisation suggests a different explanation, namely, that these relations among objects were brought to light by expe- rience at the level of practice.

As long as one is intent on representing only the outer form of an object, it remains possible to think of it in terms of its substance, its quiddity etc. But once one begins to explore the various layers of an object, as Leonardo did, this question of quiddity quickly goes out of focus. For this object now has as many layers or levels as one choo- ses to see and one can no longer represent the object as such in its totality: one can only draw aspects thereof and attempt to clarify the relations between aspects. In short, it bears consideration whether Leonardo’s practical visualisation was not more important for the shift from substance to function than any theoretical concepts. Indeed one is tempted to go further to ask whether such practical visualisation might not have also have played a significant role in the development of a concept of infinity. In which case what Cassirer assumed was the effect would turn out to be the cause and conversely.

Universals and Particulars

Leonardo’s development of a spectrum of visual images from concrete Nature to abstract geometry brought at least two basic innovations. First, his diagrams at various levels of abstraction served as markers in helping artists to represent particulars from the natural world. His studies of noses are an excellent case in point. A general classification of straight, convex and concave noses (TPL 288-289, 1508-1510) serves as a starting point in recording particular faces (cf. W 12491). Second, these markers permitted artists to endow abstract ideas from their imagination with a new realism (cf. W 12592 r, ca. 1506?).

In other words, linear perspective as developed by Leonardo is much more than a means of simply copying and not to be seen as a straight-jacket on the artist’s freedom (cf. Novotny,16 Gombrich17). It serves to expand the horizons of visualisation and the possibilities of art. In short linear perspective is a source of creativity.


Implicit in the spectrum of levels of abstractions developed by Leonardo (Chart 5) is also a fresh-approach to the whole subject-object problem. It is possible, of course, to interpret this problem in such a way that the discovery of linear perspective requires a strict


Chart 4 (i-ii). Different combinations of subject and object in the production of visual images

  1. i  Person (with unconscious verbal/ mental, visual preconceptions
  2. ii  Person
  3. iii  Person (with conscious ideas, ideals)
  4. iv  Person (with conscious ideals, anti- ideals)


Visual picture Visual picture Visual picture


Image plane

Image plane

Image plane

Object in Natural World on Object in Natural World on Object in Natural World on Object in Natural World

In Antiquity (i) subject and object remained intertwined. Renaissance perspective (ü) introduced the possibility of producing objective images geometrically even in the absence of an actual observer. It remained possible, however, to interpolate consciously into this objective image subjective ideals (iii) or even to strive for a purely subjective image (iv).

Chart 5. An alternative approach to the subjectlobject problem

Geometrical figures, visual images, models and physical objects are seen in terms of the accuracy of the fit between various elements and their potential reversibility. The subject’s role in this case becomes one of making models and geometrical figures as well as establishing relations. Hence subjectivity and objectivity emerge as interdependent activities. Algebraic formulae represent a part of the spectrum not considered by Leonardo.

relation between object and picture-plane independent of the viewer. In this approach objectivity is possible only through the exclusion of the subject (Chart 4). Whence it appears that scientific advance is possibly solely through the exclusion of the personal, as if science and art were truly two opposed cultures. Approached in terms of such schemata there is an apparent opposition between subjective and objective activities.

Approached in terms of the spectrum of abstractions (Chart 5), the subject-object problem emerges in a fully different light. For it now becomes clear that the models, pictures, diagrams and geometrical figures needed for any scientific analysis are all manmade and require the intervention of a person.

Subjectivity is, therefore, not opposed to science. It is a prerequisite for any scientific activity. Subjectivity is the conscious translation of objects in the natural world into different levels of abstraction. And when this activity is so accurate that it become mea- surable, reversible, repeatable, then this subjectivity gains objectivity. Or to put it in Leonardo’s own words (CA 108 rb, 1517-1518): «If a rule transmutes the shape of an area into another shape, and the same rule restores the first shape back to the area, cer- tainly such a rule is perfect. This is seen in arithmetic with numbers divided by another; then remultiplying by the number that divided it remakes the first number. For example 12/4 equals 3; multiplied by 4 equals 12».

«Certainty» writes Leonardo elsewhere (W 19084 r, ca. 1513), «is born from the integrated knowledge of those parts which, being united together, compose the whole of the thing». Leonardo’s approach thus assumes an integration of subjectivity and objectivity that is being called for by thinkers in our own day (cf. Foucault,18 Lorenz19).


In this paper we have examined some of the basic factors underlying Leonardo’s fasci- nation with visual images and we have touched briefly on some of the consequences of this visualisation.

The implications of this visual approach introduced through Leonardo’s perspectival studies affect not only man’s relationship to the natural world, but extend to the most fundamental questions of philosophy. Ever since Antiquity there had been an ongoing debate whether truth and reality exist only in the world of ideas or whether these were characteristics of the natural world. Truth, in Leonardo’s approach becomes primarily a question of developing relationships that can be visually tested, relationships among various levels of abstraction, between natural objects and models, between models and naturalistic drawings, between naturalistic drawings and geometrical figures and con- versely. For Leonardo science is the development of a spectrum.

Since the seventeenth century science has become ever more identified with the most abstract part of this spectrum. The data of the visual world were translated first into geometrical figures and then, increasingly, into algebraic formulae”. The abstract limit of the spectrum became so much emphasized that the spectrum itself became forgotten. Scientists concentrated so much on the results that they lost interest in showing how they got there.

A folio of Leonardo’s notebooks (Fig. 38), notwithstanding its superficial confusion, is refreshing and stimulating precisely because it makes manifest these relations between different layers of abstraction that are no longer clear in contemporary science. Quite simply, Leonardo’s visual approach has philosophical and scientific implications that deserve much further attention.

Lord Clark in his series on Civilisation noted: «People sometimes wonder why the Re- naissance Italians, with their intelligent curiosity, didn’t make more of a contribution to the history of thought. The reason is that the most profound thought of the time was not expressed in words but in visual imagery»20. Clark’s perceptive comment came in the hour that he devoted to the enlargement of man’s spirit through the visual image in which he discussed examples from painting and sculpture, focussing only on Michelangelo and Raphael. Perhaps he would now add that Leonardo’s role in this process was at least as significant and if one were more radical perhaps one might even assert that the visual thinking of the Renaissance artists was much more profound than the verbal thinking of the Ancients which they believed they were imitating.