Kim H. Veltman
Linear perspective evolved in the 15th century as a branch of optics, as a practical science of representation as opposed to theoretical optics and vision. Brunelleschi, Alberti, Piero della Francesca, Filarete and Francesco di Giorgio Martini were early pioneers and authors in this field. Even so their work tended to be either theoretical or practical. Leonardo da Vinci’s studies on perspective marked a milestone in offering an approach that was both theoretical and practical. Where his predecessors had been content with isolated demonstrations, Leonardo introduced a more systematic approach that revealed underlying laws of perspective.
One initial motivation for these studies came from his studies of astronomy and optics, fueled by a desire to write a treatise on cosmology, entitled The Earth and its Waters, which was to show the correlations between man as microcosm and the universe as macrocosm. Another motivation came through a tradition that linked cosmology with the regular solids. Depicting the regularity of the solids thus seemed a key to understanding the underlying harmonies of nature and the universe as a whole.
Leonardo was probably introduced to perspective while he was a student of Verrocchio in Florence. We have evidence of careful perspectival studies for the Adoration of the Magi in 1481, about the time he was moving to the court of Ludovico Sforza, il Moro in Milan. Even so, it is eleven years before he writes a first mini-treatise on perspective in the Ms A. As he studied the details of linear perspective he soon realized that it was of limited use for painterly goals. Although he used it masterfully in his Last Supper, most of his painterly attentions turned to the potentials of aerial and colour perspective, which played a central role in his later paintings and became an important part in his perspective treatise that history has remembered as his Treatise on Painting.
Meanwhile, Leonardo, who had begun his studies of perspective in connection with practical optics and painting. gradually discovered its deeper dimensions. The pyramidal forms connecting the eye with objects and intersected by transparent planes, which demonstrated the transformations of 3-D shapes in a 2-D form on the picture plane, were also connected with his studies of the geometrical game (De ludo geometrico). Indeed, they applied also to percussion, force, motion and weight. Leonardo called these the four powers of nature. Thus perspective, which seemed to many as little more than a visual trick in representation, became for Leonardo an underlying key to the laws of Nature. Ironically, the consequences of this insight were that a method, which served to visualize spatial aspects of the visible world in art also became a key to understanding the underlying proportions and laws of the invisible world of nature that we now call science.
Ancient Greece developed 1) geometry (ge metron, measurement of the earth and geography, description of the earth) and 2) a science of optics (optike). While this was largely a study of the illusions of vision, four propositions in Euclid’s Optics dealt with problems of surveying and thus implicitly with measurement. In the context of astronomy, the ancient world also studied 3) astronomy with elementary laws of projection using the planisphere. The organization of the quadrivium meant that these remained three independent traditions. Accordingly, Ptolemy wrote treatises on all three domains as three completely separate subjects.
The Latin West inherited these three traditions and gradually integrated them. In the early Middle Ages, the term optike was translated as perspectiva. Through the Arabic tradition, the science of optics became a study of certification of vision. Already in the 9th century with Alkindi, optics and surveying became linked, thus preparing the way for perspective becoming an instruction in measurement (cf. Dürer’s Unterweysung der Messumg and Rodler’s Perspectiva…Kunst der Messung). But this process of integration took more than half a millennium. By the 12th century manuscripts on optics and surveying (as practical geometry) were increasingly bound together. By the 13th century, manuscripts on the sphere, planisphere and astrolabe (practical astronomy) were added to these bundles of knowledge as early versions of encyclopaedias of science.  By the late 14th century, Biagio Pelacani da Parma was exploring planispheric projections as part of lectures on optics (perspectiva) in Padua. Prosdocimo de Beldemandis  is said to have been among his students, who in turn is said to have been a teacher of Leon Battista Alberti.
The innovation of 15th and 16th century Italy was to integrate these three traditions of vision (perspectiva, prospettiva), measurement (geometria practica) and representation (prospettiva prattica). As a result the study of optics extended to both a) specific measurements in terms of inches and feet (oncie and braccia) and b) general proportions as in planispheres, astrolabes and other scientific instruments. The process was slow. Brunelleschi (1415-1425), is credited with a first demonstration, which is lost. This was effectively practice without theory. Alberti is credited with a first text (1434), which lacked diagrams in the earliest known manuscripts. This was effectively theory without practice. Even so, in a more practical vein, the same Alberti also wrote two treatises related to surveying (Descriptio urbis Romae and Ludi matematici). Piero della Francesca remained focused on theoretical and especially mathematical dimensions of perspective. With respect to regular solids he wrote a booklet on the five regular solids. With respect to his De perspectiva pingendi, he was content to demonstrate principles through a practical example, rather than explore these principles systematically. Filarete’s Treatise of Architecture (c. 1465), and Francesco di Giorgio Martini’s Architettura civile e militare (c. 1483) marked next steps in linking optical theory, perspective practice and surveying practice. Their work remained primarily practice, little theory and no systematic theory-practice.
Francesco di Giorgio and Leonardo da Vinci worked together in the early 1490’s. Indeed, Leonardo made seven pages of notes in one of Francesco di Giorgio’s manuscripts. Leonardo thus began in the Renaissance tradition of artist engineers. A casual browsing through the pages of his Codice Atlantico  could readily leave us with the impression he was merely an artist-engineer throughout his life. To understand Leonardo da Vinci’s contributions to perspective we need to understand his sources, his studies in optics, regular solids, surveying, measurement and transformation. These led him to a pyramidal law and to a vision of four powers of nature. His achievement lay in a first systematic approach integrating the theory and practice of geometry, optics and representation (prospettiva [prattica]). Inherent in these studies was a new approach that led directly to a new vision of science as codified by Galileo. Inherent in these same studies was also a curious paradox. Studying the theoretical and practical methods for representing theories of vision and representation, i.e. the keys to the concrete, visual world., led simultaneously to a world of mathematical measurements and proportions that was abstract and ultimately a-visual. The details of this story have been the subject of longer studies  and cannot be replicated within the confines of a short article. Hence, our paper is more in the direction of a sketch that outlines major themes in Leonardo’s work on perspective.
Leonardo da Vinci was the author of various manuscripts with notes on perspective. Chief among those extant were the Manuscripts A, E, G (now Paris, c.1492), the Codice Atlantico (Milan) and the Trattato della pittura. Cellini in his Trattati dell'Orifieria referred to a manuscript which, among other things contained "a discourse on perspective, the most beautiful which was ever found by anyone in the world". Comolli (1791,189-190) referred to a Libro delle ombre e dei lumi, which is no longer extant. In terms of practice, he used perspective for regular solids, for his stage design and for his paintings, of which the most famous example is his Last Supper (Milan, Santa Maria delle Grazie, 1495-1497).
Unlike the Sienese, Florentine, and Milanese artist-engineers, who were his contemporaries, Leonardo was fascinated with scholarly traditions of knowledge. In 1494, he had a private collection of 119 books. He called himself a man without letters (omo sanza lettere), but then Cicero had modestly said that he was also without letters (sine litteris, sine ingeniis). How well he knew Latin remains a matter of debate, but when he went to Milan c.1481, we find lists of Latin nouns and conjugations of verbs in his notebooks (e.g. Trivulzio). With respect to Optics, there is direct evidence that Leonardo was familiar with images from Euclid’s Optics, and Ptolemy’s Optics, either directly or via compilations. The link with Ptolemy is particularly with respect to diplopia studies. Leonardo shows familiarity with the work of Alhazen, probably via the mediaeval compiler, Witelo, whom he names. Leonardo copied an opening passage from Pecham’s Optics. Pecham’s study of light passing through a triangular aperture is a theme that Leonardo develops considerably in camera obscura studies, especially Manuscript C. Leonardo has well over 200 images on the theme of pinhole images. 
Ancient authors on optics were particularly concerned with illusions of vision. Mediaeval authors on optics were increasingly concerned with criteria for certification of sight, whereby one could overcome these illusions, or at least understand their role. Leonardo continued this tradition with a more practical goal. He planned to write a book on astronomy and cosmology and wanted clear criteria for observing both terrestrial and celestial phenomena. A tentative name for the book was The Earth and its Waters. Although never finished, this book required, by its very nature, an integration of theory and practice in cosmology, geometry, astronomy and optics. As such it was one of the important contexts for Leonardo’s studies of perspective.
Already in Antiquity there was a fascination with regular solids as a basis for understanding and explaining cosmology.  Pythagoras was a pioneer and was followed by Plato in the Timaeus. It is said that Plato commissioned Euclid, to write the Elements to provide a theoretical basis for the five Platonic Solids. Mediaeval and Renaissance commentators obviously believed this and thus the number of books in the Elements gradually increased from 12 to 16.
In the 1260’s, there were two major strands in these traditions: one neo-Platonic and Plotinian, championed by Franciscans; the other, Aristotelian, and Dominican, championed by Thomas Aquinas, the student of Albertus Magnus. It is quite possible that Pope Clement IV commissioned Roger Bacon to write his Opus Maius as an attempt to find a middle road between the seeming opposites of Plato vs. Aristotle. Bacon acknowledged three kinds of knowledge, namely, authority, reason, experience, and suggested that attention should be focused on this third kind of knowledge: experience. Bacon’s vision in the latter 13th century became increasingly important throughout the 15th and 16th centuries.
Through this tradition, concrete study of the regular solids in order to understand abstract principles of the universe became a concern. In the period 1460-1476, the mathematician, Regiomontanus (Johannes Müller) composed a treatise specifically addressing the problem of how these solids occupied space.  These concerns were probably a motivation for Piero della Francesca who, in addition to his treatises on the perspective of painting and the abacus, also wrote a treatise on the five regular solids.  Piero’s work was studied, translated and published by the Dominican friar, Luca Pacioli, who explored a new kind of theoretico-practical mathematics. In 1494, Pacioli acknowledged his debt to Piero when he published his Summa de arithmetica, geometria, proportioni e proportionalitàwhich, in addition to being the first printed book on accountancy, also contained a small section of 8 pages that effectively made it the first published treatise on perspective.
The same Luca Pacioli went on to write De Divina Proportione (c. 1496-1497, published Venice, 1509). This was in large part a translation of Piero’s Libellus  as Pacioli again acknowledged. Pacioli was friends with Leonardo and is said to have asked Leonardo to do the drawings of regular solids for this book. The case is complex. There are references to models that were made, possibly three sets. We have Luca Pacioli's complaint that after Leonardo left Milan in 1499, he was unable to find anyone who could draw the other semi-regular solids for his Divina proportione in proper perspective. This explains why the actual woodcuts in the book are too primitive to be directly by Leonardo. There must have been an intermediary woodcutter. Images in the main manuscript, Milan, Ambrosiana Ms. Et. 170 sup. also seem to be the work of a copyist, rather than directly by Leonardo. Images in a second manuscript, Geneva, Bibliothèque universitaire, Ms. 1.e.210 are again too primitive to be directly by Leonardo, and are almost certainly a copy not based directly on the Ambrosiana manuscript. All this is important because it demonstrates that at the turn of the 16th century, even in Milan, there were few real masters of perspective.
Meanwhile, there are drawings in the Codice Atlantico, which are undoubtedly related to the De Divina Proportione. A manuscript or collection of drawings appears to have gone missing. Pacioli published the theological and cosmological significance of the regular solids and gave a sermon on the subject in Venice in 1508. This included discussion of the 72 sided polygon, which is also found in the mediaeval tradition of Euclid. Meanwhile, Leonardo and Pacioli’s contemporary, the Benedictine monk, Fra Giovanni da Verona,  active in the first published edition of Vitruvius,  also set about a different kind of “publication”, adapting Leonardo’s images for his intarsia work in the Monastery of Monte Olivetto Maggiore (near Siena) and in the church of Santa Maria in Organo, Verona. Among his many remarkable shapes were also mazzocchi, a form that inspired Uccello and to which Leonardo also dedicated very detailed drawings, which he copied via a pin-prick method. One of the preparatory drawings for a complex mazzocchio shape has an accompanying text: “made by Leonardo da Vinci, disciple of experience.”
Leonardo had other incentives for his fascination with regular solids. One was geographical map projection. In the Codice Atlantico we find sketches of a globe being unfolded that is not far from Waldseemüller’s pseudo-Ptolemaic, cordiform projection (1507).  Another was astronomy. Leonardo’s contemporaries, Bottticelli and Carpaccio  both depicted Saint Augustine  studying an image of the sphere. Leonardo also deals with the theme of the sphere in the Codice Atlantico, but with a difference. Leonardo depicts a man drawing a sphere in perspective which is significantly, the first extant drawing we have of the window or intersected plane (velo) that the 15th century discussed and Dürer made famous. Such examples illustrate vividly how the study of regular solids, which began as a theme of abstract philosophy, metaphysics and theology, increasingly became a practical topic linked with drawings and physical models. The regularity of solids seemed a key to the structure of the visible universe, ranging from the micro-structure of snowflakes and crystals to the macro-structure of the universe. Ultimately, when Kepler discovered that the theory did not fit, he abandoned the model of the regular solids, but the quest for regularity became a cornerstone of early modern science.
As noted earlier, from the mid-13th century onwards, the themes of astronomy (sphere, planisphere), optics (perspective) and practical geometry (surveying), became increasingly intertwined in manuscript bundles. Leonardo is fully within this tradition and hence his Codice Atlantico moves seamlessly between these subjects. Even the drawings at Windsor, focused on the natural world, have some images devoted to these themes. Some of these drawings show surveying of nearby buildings using simple geometry directly in the tradition of Euclid. Others show long distance surveying that indicates effects of the earth’s curvature. They also show the use of surveying instruments and more significantly the use of measurements. Sometimes these are proportional, sometimes these are numerical.
Previous scholars had typically dismissed Leonardo’s work as belonging to the realm of mere thought experiments. This claim had overlooked Leonardo’s explicit distinctions between cases which were experimented and those which were not yet tested (non sperimentata). An initial study of these measurements in 1974 by Keele and Veltman at the Wellcome Institute (London), revealed that Leonardo’s numbers were more precise than the seemingly random nature of the notes might at first suggest. It was decided to reconstruct some of Leonardo’s basic instruments and repeat his perspectival/surveying experiments. Careful study confirmed the accuracy of proportions and measurements noted by Leonardo. It became clear that Leonardo’s description of himself as a disciple of experience was much more than an attractive slogan. There was an experimental basis to Leonardo’s work.
Essentially, linear perspective entails three elements: observer, picture plane and object. Leonardo’s predecessors had focused on a single demonstration. Leonardo explored the experimental basis of perspective under a variety of conditions. A treatise within the Manuscript A (1492) records Leonardo’s steps. Following initial comments in the early pages, the treatise proper begins on folio 36 and ends on folio 42. The approach is simple. He shows a ground plan and then the perspectival situation. He begins with a square, then a diamond shape, then a pentagon, a hexagon, an octagon, a circle, a cube and then ends with what may be the first official drawing of what history remembers as the costruzione legittima.  Leonardo’s predecessors had focused on a single experience/experiment. By contrast, Leonardo insisted on testing and demonstrating that the principle applied to each of the basic regular shapes. He explored this further in dozens of drawings and sketches, especially in the Codice Atlantico.
Perspective thus became linked with a systematic method. What had begun as a form of practical or applied optics, a means of representation in art, now became linked with a quest for laws of science. This quest led in two seemingly opposed directions: one, towards ever greater visualization of the natural world; another, towards an ever greater abstraction with respect to nature. Leonardo pursued both these goals and ultimately attempted a synthesis between visualization and abstraction.
Leonardo as he is popularized, is associated with visualization, as a man who could draw anything and seemed to draw everything. Visualization was undoubtedly an important tool in his studies. We noted that he was probably the first to record a painter recording an object. In this sense, he seems to introduce the notion of copying and photographic realism avant la lettre. On closer inspection, however, we discover that in many cases his drawings have a different goal. He is concerned with illustrating the principles underlying the objects he depicts. He typically omits as much as he shows in order that we can discern underlying structure and function. He is not concerned with a pretty picture of a clock’s surface. He wants to show us the inner mechanisms in order that we can literally understand how and why the clock ticks. This is especially visible in the clock mechanisms of Madrid Codices. Visualization and abstraction are thus twin features of his drawings. In his anatomical studies is less concerned with the challenges of drawing in the sense of copying and more concerned with the methods of organizing the results of drawing. Interestingly enough, in his anatomical studies, he uses the methods for organizing the world used in Ptolemy’s Geography, as a starting point for his treatment of man, as microcosm.
The consequences of perspective for his painting practice were paradoxical. In his Annunciation (1475-1480), perspectival effects in the architecture are evident, but the focus is on the angel Gabriel, the Virgin and the landscape. In another of his early paintings more than a decade before his detailed perspectival studies in the Ms. A (1492), he began with elaborate perspectival, preliminary sketches, dominated by architectural ruins.  Leonardo soon recognized that rectilinear surfaces, when seen from close by, produced excessive foreshortening.. This helps explain why, in his finished painting,  the Adoration of the Magi (Florence, Uffizi, c. 1481), the ruins remained, but as a background feature. In the decades that followed, his fascination with the potentials of strictly linear perspective culminated in his masterful treatment of the Last Supper (1495-1497). The topic was hardly new. There were over 80 depicted Last Suppers in Florence alone.  Predictably, Leonardo treated the space of the room masterfully. To avoid excessive foreshortening he chose a viewpoint above a regular visitor’s eye-level and a distance point that was beyond the fresco. In his later precepts, and as his friend Cesariano reported in the first illustrated edition of Vitruvius (Como, 1521), Leonardo recommended that the distance of objects should be at least 3.5. times the painter’s/ viewer’s distance from the (imaginary) picture plane. Perspective thus had some contexts where it was more effective than in others – a theme that would resurface in the 17th century debates between Bosse, Huret and the French Academy.
Meanwhile, the true genius of the Last Supper lies less in Leonardo’s treatment of the spatial context than in the spatial treatment of Christ and the Apostles. This is difficult to see in the original fresco which is effectively a ruin, but clear in the two main copies in Tongerlo  and London, Royal Academy. Linear perspective is excellent for rectilinear objects. Human subjects require something else. This explains the paradox why Leonardo, who studied linear perspective so masterfully, seems to abandon it in his paintings after 1497.
In the popular imagination Leonardo was a universal painter who could paint anything, anywhere, anytime. This should have led to hundreds of paintings In fact, Leonardo produced a very small number of paintings: the number unequivocally attributed to him varies from four to around thirty, although there are of course hundreds linked with his school. Leonardo focused on human subjects. Hence, as he progressed, Leonardo turned his attentions increasingly to the effects of light and shade (chiaroscuro) and the potentials of colour perspective and aerial perspective. After 1500, his major paintings can be seen as attempts to address one fundamental challenge: how to create effects of three-dimensional relief under carefully controlled conditions. Speaking anachronistically, he was trying to create auto-stereoscopic effects in paintings.
In Mona Lisa (Louvre, 1503-1506) these efforts centred on a single person and although the spatial effects of the landscapes involved distances of many miles, the focus of attention was on the face and hands, which involved a mastery of less than 30 cm. in spatial depth. Already in 1499-1500, Leonardo had explored a more ambitious possibility in his cartoon ofThe Virgin and Child with St Anne and St John the Baptist (London, National Gallery, previously Burlington House), which entailed almost a meter in spatial depth. He returned to this theme in The Virgin and Child with St Anne (Louvre, 1508), now adding remarkable landscapes with aerial perspective in the background, although the transition from the near space of the persons to the trees in the background remains (deliberately?) ambiguous. When he explores this challenge further in his Bacchus (Louvre, 1510-1515), the transition between the near space of the saint and the deer in the near distance is improved. In his Saint John the Baptist (Louvre, 1513-1516), Leonardo returned to a situation where there is solely a near space, possibly because he intended this to be seen mainly by candlelight in order to increase the spatial effects of face, torso and outstretched hand.
Tourist guides in the Louvre often draw attention to the phenomenon that the eyes of Mona Lisa “follow” us as we move from a frontal position, to lateral positions on the right or the left of the painting. If we look attentively, we discover that this is not all that happens. The spatial effects of the La Gioconda’s face become more convincing as we move to the side, in this case especially the right side. This applies to all his paintings. Hence, Leonardo’s goal of three-dimensional relief, led not simply to mastery of a single distance-point and viewpoint, but extended to multiple viewpoints as observers looked at the paintings from a range of positions. The effects of extreme viewpoints are now called anamorphoses – a topic whose invention is linked with both Piero della Francesca and Leonardo. But, paradoxically, in this larger sense, all of Leonardo’s paintings are explorations in anamorphoses in the sense of visual transformations. Contemporary psychologists such as Kubovy refer to this as the robustness of perspective.
5.4 Anamorphosis and Transformation
Ancient Greek science was focused on nature or essential character (phusis), wherein growth and motion were intertwined. An achievement of late Mediaeval and Renaissance science lay in separating cases where change could be measured as movement (kinematics); cases where change could be visualized, and those where this was not possible. This went hand in hand with a separation in the arts between visible changes and those which were purely imaginary (metamorphoses). By the turn of the 17th century this became Kepler’s distinction between imagines rerum (subjective images) and picturae rerum (objective images on walls, etc.) which, as Cassirier explored, led to new oppositions between subject and object. Leonardo, working a century earlier had a more organic approach: a fascination with the process of change, how one form became transformed into another. This was an underlying incentive for Leonardo’s studies of anamorphosis.
Vasari tells us how as a boy, when Leonardo began his career, he painted a head of Medusa, which was positioned strategically in a wooden cabin such that as a person entered, light fell on the painting to give a maximal effect. This was case, where one viewpoint was privileged. The anamorphic portrait drawing of Francis I, attributed to Leonardo, is again a case where the full effects of the picture were only seen from one viewpoint. In his studies Leonardo explored the possibilities of different and multiple viewpoints.
Leonardo was fascinated how such transformations affected the human body and head in particular. As usual there were precedents. Piero della Francesca explored this theme in book three of De Prospectiva pingendi.  Piero’s approach reduces the human head to a formal array of mathematical points. Leonardo is more informal. A sketch in Windsor 12565r shows a head in frontal view, which becomes a head tilting downwards in the figure below and a head seen in profile in the right margin of the drawing. In the Codice Atlantico he pursues this theme, subsequently developed by Pomponius Gauricus (1504) and by the author of the Codex Huygens (c.1570). Leonardo’s sketches occur on a folio which also deals with architecture and transformational geometry.
As Leonardo progressed also he became increasingly fascinated with the effects of contexts on change. There are stories of how he would take persons with interesting faces into a room, tell them jokes and stories and then draw their faces. His caricature studies (called caractères in the edition of the Comte de Caylus) were thus simultaneously studies of underlying nature (phusis as in physiognomy) and also explorations of how the outer manifestations thereof changed in extreme cases of surprise, fright, anger, themes that would evolve throughout the Mannerist and Baroque periods and emerge as treatises on the passions in 17th century France.
Leonardo saw these changes (morphoses, anamorphoses and metamorphoses) as much more than methods for recording the natural world. If one could understand the elements of change, they could be recombined to create new objects and images. The artist as craftsman (homo faber), was thus himself transformed into a creative artist: man not just in the image of God, but man as an image of God as creator.
In the Middle Ages and Renaissance, the idea of God as creator was intimately linked with the image of God as Geometer.  Geometrical games thus became much more than pastimes. They were activities in re-creation as much as they were recreational. Alberti’s Ludi matematici  (c.1452), Nicholas of Cusa’s De ludo globi  (1463) and Leonardo’s studies on De ludo geometrico  were thus further expressions of man in the image of God, man as a human creator imitating a Divine example. The actual subject matter of these “games” varied tremendously. For Alberti the ludi were largely exercises in surveying. For Cusa they were games with a globe, sphere or ball as expressions of homo ludens. For Leonardo they began as geometrical exercises. For instance, on CA 99vb Leonardo defined the geometrical game as giving "a process of infinite variety of quadratures of surfaces of curved sides." For Leonardo the real quest of De ludo geometrico became to provide a key to all systematic transformations of forms.  It thus provided a mathematical version of the changes and transformations metamorphoses of forms that fascinated him. Although he did not use the term, it was a mathematico-morphoses, a vision of 2-D and 3-D morphing avant la lettre.
The 2-D version was based on simple geometry. In Codex Forster, we find a simple diagram showing the Pythagorean Thereom. Elsewhere he develops a 3-D version of the same theorem. To take another example, in the Codice Atlantico, we find him beginning with a circle inscribed within a set of 36 squares. One side of the square is then extended to produce a rectangle twice as long as it is wide. Within this rectangle, what had been a circle, becomes an ovoid shape. In a first example we see a rough sketch, with no interest in precise drawing. Elsewhere in the same collection we find another example, where the diagrams are now carefully drawn with ruler and compass. Elsewhere again we see how this two-dimensional ovoid could readily become a three-dimensional egg-like shape. This, of course, is exactly what Piero had described in his Perspective of Painting, an idea which he had depicted so brilliantly in his Pala di Brera.  Whereas Piero has one example, Leonardo has a number of images devoted to what might be called “perspectival eggs.”
Leonardo saw these transformations of shapes as a key to understanding the building blocks of nature. In retrospect, some aspects of this progamme, such as squaring of the circle, did not lead directly towards modern science. Even so, there were at least three fundamental contributions. First, it united the methods of geometry and arithmetic, which had been separated by the ancient Quadrivium. Second, it linked mathematics with visual diagrams and models and thereby with the natural world. Third, it employed instruments, such as ruler, compass, and proportional compass in making this challenge a measurable quest. Mathematics, which was purely theoretical in much of the Greek tradition, now became linked with both the visible world and visible representations of that world of nature. The way was set for Galileo’s famous claim that the book of nature was written in the language of mathematics. 
By the time Leonardo began writing his mini-treatise in the Ms. A, Leonardo was fully aware that the theoretical visual pyramid extending from the eye to the object produced a series of regular intersections on the transparent plane. Every movement of the object away from the plane, had the same effect as moving a picture plane further from the eye and closer to the object. He illustrated this principle in a 2-D drawing on folio 37r . As he studied these principles further, it became clear that this diminution with distance entailed a square being inscribed with a circle, inscribed by a square, inscribed by a circle etc. This was exactly the sequence he was also exploring in his transformational geometry. Hence, the transformations of 2-D shapes on the transparent plane in his perspective studies followed the same principles as the transformation of 2-D and 3-D shapes in his studies of the geometrical game. Hence we find him drawing pyramids in a whole range of ways, abstract geometrical pyramids, pyramidal shapes, pyramidal shapes as weights. He spoke of pyramidal rules (regole), which amounted to pyramidal laws.  These pyramidal laws, which were basic to his perspective become the opening illustrations in Jean Cousin’s Livre de perspective (1560), a generation later.
Leonardo’s systematic study of perspective thus revealed that perspective was much more than an explanation of visual impressions or a demonstration of spatial representation. It entailed an inverse size/distance law or pyramidal law which applied to other branches of nature. He was fascinated by percussion. He observed that when a hammer or similar object hits a surface with grains of sand, the sand rises to form a miniature pyramid. In preliminary drawings, he literally drew the pieces of sand. Subsequently, in the Madrid Codices, we find him illustrating this as a geometrical principle of a pyramid --seen laterally as a triangle – inscribed within a circle or sphere. He drew similar diagrams for force, motion and weight.
Gradually he saw his pyramidal rules, his pyramidal law as a basis for of force, motion, percussion and weight. He called these the four powers of Nature. Perspective thus offered a gateway to his vision of science, which could be visually rendered, and at the same time had underlying measurable, proportions, which were typically invisible to the naked eye. Thus the same quest that led to optically delightful landscapes of the physical world, led also to mathematical proportions and gradually algebraic equations, which were the abstract range of a single spectrum of reality.
Leonardo eludes easy categories. The early 20th century depicted him as an artist or a universal genius. Leonardo was not just an artist. Nor was he a universal genius in the sense that he dealt with everything. His work was focused on specific aspects and dimensions of Nature. The mid-20th century tried to categorize him as an artist-engineer (e.g. Gilles). This risked reducing him to a technician. He was much more: unlike his contemporaries who were content with exemplary demonstrations, Leonardo insisted on a systematic study and demonstration of the underlying principles. He was a 15th and early 16th century genius, who helped lay the foundations that subsequently made possible the achievements of Galileo, Descartes, Leibniz Newton and other giants of early modern science. Leonardo’s studies of perspective are a key to understanding the deeper dimensions of Leonardo as a proto-modern scientist.
The author’s studies on Leonardo were made possible by the late Dr. Kenneth D. Keele, M.D., F.R.C.P. As a leading authority on the scientific and medical contributions of Leonardo he was the first to understand the full scale of Leonardo’s science. I am ever profoundly grateful to Dr. Keele for his inspiration and encouragement. These studies were also made possible through the generous support of a number of foundations: Wellcome Trust, Volkswagen Stiftung, Alexander Von Humboldt Stiftung, Thyssen Stiftung, Gerda Henkel Stiftung. I am deeply grateful for their financial and moral support. Finally I thank, my colleague and friend, Professor Joaquim Garriga, for generously inviting me to the Granada conference.
1. See, for instance, the work of Jordanus de Nemore who wrote treatises, De Sphaera and Demonstratio de plana spera (planisphere), “a specialised work on geometry, which studies stereographic projection.” See: http://www-history.mcs.st-and.ac.uk/Biographies/Jordanus.html. For an edition see Rocco Sinisgalli (2000). For details re: this and other perspective treatises see the present author’s Bibliography on Perspective at http://www.sumscorp.com/ via Demo.
2. Prosdocimo de Beldemandis. See: http://www.asu.cas.cz/~had/prosdoc.html#astrolabes
3. To understand Leonardo it is important to recognize that he had very different kinds of notebooks. Some were literally pocketbooks (8o or smaller) which served to take field notes. In his “office”, he appears to have had two main collections (folio size): one on the natural world (now the collection of Her Majesty the Queen at Windsor Castle) and the other on the man-made world of machines and instruments (now the Codice Atlantico in the Ambrosiana). These became collections of work in progress. As his ideas developed he used manuscripts (mainly 4o), to develop individual themes. With respect to perspective the most important example is the Manuscript A (Institut de France). For more on this theme see the author’s survey in the epilogue to Leonardo’s optics: http://www.sumscorp.com/books/contin/p5_epi.htm#3
4. Re: the pyramidal law see: Kenneth D. Keele, Leonardo da Vinci and the art of science, Hove: Wayland, 1981 and Leonardo Da Vinci's elements of the science of man, New York ; London : Academic, 1983. Re: perspective see the author’s: Leonardo da Vinci Studies 1: Linear Perspective and the Visual Dimensions of Science and Art, Munich: Deutscher Kunst Verlag, 1986. Re: optics see the unpublished second and third volumes available in the author’s website at: http://www.sumscorp.com/books/contin/title.html.
The Introduction to Leonardo in the author’s Database on Perspective (http://www.sumscorp.com/develop/) offers a brief summary of the Literature:
Cited by Pélerin (1521, 1r) and Caporali (1536), mentioned by Vasari (II,156-158, 160, 163, 165, IV, 28), Danti (1583, preface), Lomazzo (1585,336, 100101; 1590, 17,52,149), Comolli (1791, 189-201) and Poudra (1864,I,126), modern study began with Jordan (1873), Ravaisson Mollien's edition of the manuscripts at the Institut de France (1891), Ludwig's edition of the Treatise on Painting (1882) and Richter's anthology (1883). Nielsen (1897) devoted a first monograph to Leonardo's perspective in the context of Raphael, Bramante and Giulio Romano. Mesnil (1922) emphasized scientific aspects of his perspective. Ivins (1938) drew attention to Leonardo's diagram for the legitimate construction. Bassoli (1938) drew attention to CA 35va and claimed that Leonardo had invented anamorphosis. The following year Bassoli (1939) drew attention to CA i bis va with its perspectival window. Panofsky (1940) discussed a comination of Euclidean optics and linear perspective in the Codex Huygens. White (1949-1951) claimed that such curvilinear methods represented a major trend in the Renaissance and owed much to Leonardo. By way of reply, Pirenne (1952) in an important artice demonstrated the scientific basis of Leonardo's linear perspective.
Luporini (1953), Francastel (1953) made general claims. Pedretti (1953) attributed to Leonardo an anamorphic drawing of Francis I. Wittkower (1953) examined the problem of proportionality and re-assessed his contribution in relation to Piero della Francesca. Castelfranco (1954) discussed the Treatise on Painting. Nicco Fasola (1954) linked his perspective with cosmography and geography. Agostini (1954) collected basic quotations on light and perspective. Pedretti (1956, 1957) examined his anamorphosis in relation to Lomazzo. White (1957) pursued his earlier claims about Leonardo's use of spherical perspective which he termed synthetic perspective, which Gioseffi (1957) challenged, argueing that Leonardo's only contribution was in physiology and psychology of vision. Pedretti (1958) drew attention to folios at Oxford. Rzepinska's (1961) dissertation examined Leonardo's pictorial science. Bovi (1962) focussed on his light and shade. Maltese (1962) reviewed recent claims. Brion Guerry (1962) emphasized his Albertian heritage and (1963) asked whether he had known the distance point construction. Pedretti (1963) revived White's hypothesis concerning synthetic perspective. Brizio (1968) reviewed literature 1952-1968. Chastel (1972) drew attention to Madrid II 15v with respect to curvilinear perspective. Pedretti (1973) drew attention to the Zaccolini manuscripts. Maltese (1978, 1980) returned to questions of curvilinear perspective. Pedretti1978) offered idealized reconstructions of the perspective lines in his paintings. Naumann (1979), Polzer (1980) and Kemp (1981) offered reconstructions of his Last Supper. Veltman (1986) gave a survey of this literature and provided a first complete study of all Leonardo's perspectival writings.
5. Cf. the author’s :"Leonardo and the Camera Obscura," Studi Vinciani in memoria de Nando de Toni, (Brescia: Ateneo di scienze lettere et arti. Centro ricerche Leonardiane, 1986), pp. 81-92.
6. See, for instance, the author’s, Geometric Games: A Brief History of the Not so Regular Solids.
7. Johannes Müller, De quinque corporibus aequilateris, quae fulgó regularia nuncupantur, quae videlicet eorum locum impleant naturalem et quae non, contra commentatotorem Aristotelis, Averroem. This work is cited by Doppelmayr, 1730, p. 19.
8. Piero della Francesca, Libellus de Quinque Corporibus Regularibus . This theme continued to inspire Kepler, when he made his famous model of the universe over a century later. See: Johannes Kepler, “Harmonies of the World”, translated by Charles Glenn Wallis, Great Books of the Western World, Vol. 16, (Encyclopaedia Britannica, 1952), pp. 1017-18. The Science of the Harmony of the World (1619). Cited in: http://www.mlahanas.de/Greeks/PlatoSolid.htm
9. Pacioli’s Summa contained at least 105 propositions found in Piero’s Trattato and Pacioli’s Divina porportione was a translation of the Libellus that lacked the clarity of the original. This led Vasari to accuse Pacioli of plagiarism, an accusation that later scholars have challenged. Cf. an article on Pacioli in the DSB:
10. Fra Giovanni’s Intarsia Polyhedra: http://www.georgehart.com/virtual-polyhedra/intarsia.html
11. Lucia A. Capponi, “Fra Giocondo da Verona and His Edition of Vitruvius,” Journal of the Warburg and Courtauld Institutes, London, 1984: http://www.jstor.org/pss/751439
12. Cf. Carte de Martin Walseemüller: http://cristobal-colon.net/carto/C05p1.htm
13. Carpaccio: http://pr.caltech.edu/commencement/03/images/carpaccio.jpg
14. Botticelli: Saint Augustine: http://nibiryukov.narod.ru/nb_pinacoteca/nb_pinacoteca_painting/
15. The details of these experiments are discussed in the author’s Leonardo Studies 1.
16. Now Paris, Louvre: http://en.wikipedia.org/wiki/File:Leonardo_Magi_study_1.jpg and Florence, Uffizi: http://en.wikipedia.org/wiki/File:Leonardo_Magi_study_2.jpg
17. It is frequently claimed that the Adoration was a largely unfinished painting. By regular daylight it seems so. But if viewed by candlelight the painting takes on a very different air, looks more spatial and looks very finished.
18. See:Luisa Vertova, I Cenacoli Frorentini, Florence, 1965: http://www.comune.firenze.it/servizi_pubblici/turismo/cenacoli/Cenacoli.htm.
Cf. Maria Baciocchi del Turco, I cenacoli firentini, Florence: Tipografia Barbera, 1904.
19. Da Vinci Museum, Tongerlo; http://www.tongerlo.org/da_vinci/davinci_index.htm
20. Book III. Of Heads and Capitals, Bases, Irregular Objects Bounded by Several bases and Other Objects in Various Positions: www.philipresheph.com/a424/study/piero.doc
21. God As Geometer: http://en.wikipedia.org/wiki/File:God_the_Geometer.jpg. See: Teun Koetsier, Luc Bergmans, Mathematics and the Divine, Amsterdam: Elsevier Science, 2005.
22. Alberti, Ludi: http://www.eiris.it/eiris_numeri/eiris_2/
Alberti_final_EIRIS_2_pag_15-47_mercanti.pdf. There is evidence leonardo studied this work. See: Arundel 66r.
23. Nicholas Cusanus, De Ludo globi: http://books.google.com/books?id=QQT76WUNzJAC&pg=PA191&lpg
24. See: James Edward McCabe, Leonardo da Vinci's De Ludo Geometrico, Ph.D. dissertation, UCLA, 1972. Some have seen the Ludi geometrici as a graphic study of the religious symbolism of the Divina Proporzione. See: http://www.amicisantandrea.com/Libro/
25. For a further discussion, see the author’s Leonardo’s Method: http://www.sumscorp.com/books/leometh.htm
27. See, William Desmond, Being and the Between, Albany: Suny Books, 1995: http://books.google.com/books?id=gbbKIcJx_5cC&pg=PA96&lpg=PA96&
28. A variant on this principle inscribed a circle on one side of the square led to another series of squares and circles that became the standard diagram for the volumetric measurement of (wine) barrels, a tradition called Visierruthen in Germany and jeaugage in French. This became intimately connected with the tradition of the proportional compass which was developed via Jobst Bürgi in Kassel and Michel Coignet in Antwerp before being brought into prominence by Galileo with his publication of the Compasso militare (1606).