The Perspective in Polyhedrons, Optics and Sundials

In the middle of the sixteenth century there were two opposing tendencies in science, both followed by Daniele Barbaro. One comes from Giovanni Benedetto Lampridio, who believed in the languages of the ancients, Latin and Greek. In his opinion these provided the only way to reach a profound understanding of the world. The second tendency found Italian to be an adequate tool for the expression of any idea, above all when dealing with new subjects, such as perspective. Perhaps inspired by the idea to facilitate readers in the understanding of perspectiva artificialis, Daniele Barbaro wrote a treatise in Italian entitled La pratica della perspettiva (Barbaro 1568), based on the work of his Venetian teacher, Giovanni Zamberti—of whom unfortunately little is known—and based on his mathematical studies while attending the University in Padua. He probably began to dedicate himself to this subject starting from the years in Padua, studying mathematics, as the dedicatory letter addressed to his classmate Matteo Macigni, suggests. The treatise of Barbaro looks like a collection of the best-known perspective theories of the time. Although it is not a complete summa, we can find in it intelligent, methodological indications, especially in reference to ichnographia and orthographia—pseudo-orthogonal projections—or plan and elevation of Platonic and Archimedean solids, which were particularly useful in the architectural practice.Footnote 1 Barbaro mainly focused on perspective of regular and semi-regular solids, but he also worked on very complex objects such as the mazzocchio, and wireframe and starred polyhedrons (Fig. 1).

Fig. 1
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Set of drawings for the perspective of a mazzocchio: Barbaro (1568: 118–125)

For all these objects he provided orthogonal projections, their shaded perspectives and their nets. At the very end of any exercise, he also invites the reader to create them himself, cutting and folding the net of a solid to generate a three-dimensional model; he then asks the reader to place his eye at the same point of view of the perspectival images of the treatise, to check if the retinal image of the real models matches the illustrations.

The solids provide the idea of a mathematical order of the universe: Plato considered the regular bodies at the base of the elements of the cosmos; Euclid studied their mathematical laws; and Archimedes established appropriate rules to arrive at more complex semi-regular solids. It was during the Renaissance, however, that polyhedrons turned into archetypes of perfection: they were considered ‘divine’ by philosophers, models of proportional harmony by mathematicians, explanatory examples by scientists, synonyms of virtuosity by the artists (Folicaldi and Folicaldi 2005). In this context it is necessary to remember that during the Renaissance the need to facilitate the reader in seeing the three-dimensional solids had been felt not only in Italy but throughout the rest of Europe. Indeed, when Henry Billingsley published his edition of Euclid’s Elements (1570) he enriched the illustrations, matching them with a series of glued-on pop-ups (Fig. 2).

Fig. 2
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Pop-ups of a tetrahedron and two pyramids (square and pentagonal bases): Billingsley (1570: 314)

In Book XIII of Euclid’s Elements we find the first rigorous geometric discussion concerning the Platonic solids. Euclid considered the regular bodies inscribed in a sphere of a specific diameter. He determined the relationship between their edge and this diameter, and demonstrated that there can be no more than five regular polyhedrons and he provided the formulas to calculate their surfaces and volumes. Euclid’s images are geometrically rigorous but less useful in describing the spatial configuration of the Platonic solids. They are images in pseudo-orthogonal projection which present a complex special rotation of dimensions; the octahedron and the cube are the only exceptions because they were represented in a pseudo-parallel projection view, but they never display the material consistency of the bodies, showing the edges of the solids in a sort of wireframe mode, a type of visualization used in the Renaissance and still widespread today in 3D modeling programs. Another important classical reference on polyhedrons relates to Pappus of Alexandria, who, in chapter V of a treatise entitled Mathematical Collections, quoted a work of Archimedes—now lost—written as a systematic and complete study of the thirteen semi-regular polyhedrons (Pappus 1588; Rose 1975: 203). To generate these solids Archimedes used the ‘truncation’ method, i.e., a section with a plane of the vertices of the polyhedron; this procedure allows to eliminate a pyramid, whose base constitutes a new face for the polyhedron. In terms of representation, in classical times and throughout the Middle Ages, these issues were so complicated that mathematicians reasoned abstractly based on horizontal sections of solids. It was in the Renaissance, thanks to the discovery of the mathematical-geometric rules of perspective, that Platonic and Archimedean solids were represented as bodies composed of matter. So, mathematicians and artists in Palladio’s time used the newest technology available, the perspectiva artificialis, to show, on the pages of the printed books, drawings of polyhedrons which convincingly referred to the third dimension. Going deeper in this subject, we have to remember that in the history of scientific images, and referring to the polyhedrons, it is precisely perspective that brought new impetus and innovation. Piero della Francesca displayed regular and semi-regular polyhedrons in his Trattato d’abaco and Libellus de quinque corporibus regolaribus. He used pseudo-orthographic projection views to explain their mathematical and geometric properties and pseudo-parallel projection and perspective, both wireframe, when he wanted to show the reader the spatiality of the solids (Daly 1977). These kinds of images reached their most expressive and refined peak at the beginning of Renaissance in De divina proportione by Pacioli (1509), a book that contained engravings based on perspectival drawings made by Leonardo da Vinci, who created a double illustrative version of the polyhedrons. The first solution considered the solids as real matter and, therefore, shows them completely full; the second one shows, echoing the tradition of mathematicians, emptied polyhedrons thanks to rods joined together in the vertices. The realistic rendering was also enhanced by Leonardo, who simulated the hanging of the solids on a virtual wall using wires and identification cards (Fig. 3).

Fig. 3
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Solid and wireframe dodecahedron, Pacioli, Ms. 170 (n.d.), Veneranda Biblioteca Ambrosiana/Mondadori Portfolio: Drawings XXVII and XXVIII

A further step in the representational strategy of the Platonic and Archimedean solids came from Albrecht Dürer, who in his treatise titled Underweysung der Messung (1525) considered the net of a polyhedron as a valid alternative to the double pseudo-orthogonal projection view; this way the reader, cuttings the shape from the page, could make three-dimensional models of solids by himself.

In Palladio’s time Daniele Barbaro, with the encyclopedic approach that characterizes all his works, gathered in a single treatise all these strategies of representation related to the regular and semiregular solids, summarizing the mathematical and geometrical knowledge about them. In La pratica della perspettiva Barbaro includes all the possible images that can be designed to show Platonic and Archimedean solids. But, even if this is a descriptive and collecting treatise, the author did not lack scientific rigor. Indeed, the book, though addressed to painters, sculptors and architects, respects a certain mathematical approach. The reason why Barbaro devoted so much energy to the pseudo-orthogonal and perspective representation of these solids can be traced back to the fascination that these polyhedrons held for Renaissance men (Andersen 2007: 155). These solids were seen as being part of a mathematical world that had something divine and magical in it; they became some of the most widespread decorative objects, but they also appeared in the artistic representations as emblems of the perspective ability of the painters. We said that Barbaro used pseudo-orthogonal projections of the polyhedrons to represent polyhedrons in perspective; this process works extraordinarily well when plan and elevation of an object is known, but during the Renaissance orthogonal projections still posed mathematical complexity and many scholars were not able to face these problems. The way in which the pseudo-orthogonal projections of the regular polyhedrons have to be drawn was usually obtained by scholars from their description in Book XIII of Euclid’s Elements, but it is interesting to note that Euclid’s discussion of the dodecahedron is not very useful in this sense. Nevertheless, Barbaro provided a correct solution in his book, so we must conclude that he had found an original way by himself for the dodecahedron or, at least, he shared information about this specific subject with the scientific community of his time.

Aside from the close link between perspective, mathematics and geometry, the pages of La pratica della perspettiva also show the debt that Renaissance physics owed to physiological vision. Barbaro states where the observer has to be located, considering the distance from the picture, thus introducing problems related to perspectiva naturalis or optics. He suggests that the best distance from the picture is the one that generates extreme visual rays, coming out from the eye, inclined 45° with the Albertian centric ray (the ray orthogonal to the pictorial plane) and the picture. Barbaro also warns the painter to keep in mind the physical limits of the place where the painting hangs, because a perspective can be correctly seen only from one point of view. This warning, which is an anticipation of anamorphosis, is explained by a very simple illustration (Fig. 4): having chosen the position to perceive a tower, in such a way that the painter can see a cross on its top, the painting of the scene can be appropriately perceived only by placing the eye at the same distance chosen in the physical world. Indeed, if an observer erroneously gets too close to the painting, in reference to the real point of view, even though ‘in the fiction of the picture’ the cross is still visible, in the real world the upper battlements of the tower would hide it.

Fig. 4
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Demonstration of the uniqueness of the point of view: Barbaro (1568: 22)

The reality and its representation no longer coincide and this situation, according to Barbaro, must be absolutely avoided. The problem of vision linked to objects placed at the top of buildings has been examined in greater depth in Barbaro’s treatise, where the author focuses on how these objects must be prospectively deformed so that, seen from below, they appear perfectly proportioned. The graphical method for this purpose comes, as Barbaro specifies, from Dürer’s treatise, in which the German artist brings up the question relating to the height of the letters written on the upper part of a building, so that they could appear perfectly recognizable to an observer placed below. Unlike Dürer, Barbaro applies the same construction to a line, an abstract geometric entity, thus underlining the universality of the method with the typical approach of mathematicians (Fig. 5).

Fig. 5
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Projection of objects that have to be seen from below: Barbaro (1568: 23)

The optical and geometric explanation of the reasons for which objects placed at the top must be deformed, so that they seem proportionate when seen from below, was an interesting subject to architects such as Palladio. In one of the three manuscripts on perspective preserved at the Biblioteca Nazionale Marciana in Venice (Barbaro n.d.), Barbaro wrote an example, not so well known because it was never published in La pratica della perspettiva. The episode tells that Daniele Barbaro witnessed the raising of the statue of the archangel Gabriel, which had been temporarily located at the base of the tower in San Marco square, Venice (Barbaro n.d.: 7r). Measuring its proportions Barbaro was astonished for its very elongated body, head, arms and legs; Jacopo Sansovino was standing by him and he said that even the architects of antiquity used to lengthen the proportions of the capitals on columns to make sure that their view is correct from below. Barbaro is not referring to the first installation of the statue in 1513, which he did not see personally, but to a second relocation which took place in 1542 (Telesio 1998). Indeed, due to the bad conditions of the first sculpture, a new one was located at the top of the tower. This episode helps us to date the manuscript, considering the 1542 a date post quem for which our author began working on La pratica della perspettiva, this however does not exclude the possibility that Barbaro’s interest in this subject arose during his university years.

Given this date, however, it is astonishing that there is no mention in either the printed book or the manuscripts of the beautiful perspectival paintings and trompe l’oeil that Paolo Veronese realized for the Villa Barbaro at Maser, built for Daniele and his brother Marcantonio, whose construction took place more or less a decade later. This circumstance is at least curious if the reader thinks that the building can be considered the manifesto of Venetian art and architecture in the Renaissance, above all in reference to the personalities that contributed to its realization: Daniele Barbaro, a humanist who wrote about architecture and painting; his brother Marcantonio, who may have himself sculpted the nymphaeum behind the Villa (Kolb and Beck 1997); Andrea Palladio, who designed the architecture of the Villa; Paolo Veronese, who focused on pictorial decoration; and Alessandro Vittoria, who worked on sculptures and stuccos.

Other connections emerge between scientific disciplines and the rules of perspectiva artificialis. Daniele Barbaro in La pratica della perspettiva, dealing with sundials in the first and second chapter of Book IX, wrote about: “many tools, and ways to set, and to commutate things into perspective”.Footnote 2 Indeed, Barbaro knew perfectly well that the abstract entities of Euclidean geometry, line and plane, are the basis of perspective and gnomonic rules. Thus, in Book IX he makes explicit that the physical element capable of exemplifying the geometric process of vision is light, even when light generates its most direct and opposite consequence: the shadow. Federico Commandino passed through similar considerations when he connected geography and gnomonic to conics and perspective in the Ptolomaei Planisphaerium (1558) and then in the Ptolomaei Liber De Analemmate (Commandino 1562; Rose 1975: 190). In the premise to the first mentioned work, after comparing the subject of the Greek treatise to the scaenographia of modern architects, the mathematician clearly states that the design of meridians and parallels, the curved lines useful for mapping both the earth and the sky, comes from the geometry that innervates perspective constructions of painters:

In our days at the workshops of far from disreputable painters and architects, a certain way of working is handed down that was of great help to me in following the thought of this little book.Footnote 3

In this context, the reader has to keep in mind that one of the synonyms of the word ‘gnomonics’ is photosciaterica, literally ‘writing with light’, a word that has a reference in the Vitruvian passage concerning the triad of drawing and, in particular, the definition of scænographia or perspective: “Scenography is the rendering [adumbratio] of the front and the sides in relief, with the convergence of all lines in the center”.Footnote 4 It is interesting to note that among the interpretations of this obscure definition (let’s remember that Vitruvius’s work had no images) some critics think that the Latin term adumbratio may correspond to the translation of the word skiagraphia, which in the Greek culture commonly indicates the illusionistic-perspective drawing or the representation of shadows (De Rosa 1997: 117–176).

The overlap between visual and light rays is again used by Barbaro in the Chapter V of La pratica della perspettiva in which he deals with: “a beautiful and secret part of perspective”,Footnote 5 that is, anamorphosis. Barbaro suggests piercing the lines of the drawing of a human head in order to create its anamorphosis. He asks the reader to expose the perforated sheet to the sun so that the rays of light can pass through it and reach the surface on which the anamorphosis will be outlined. Even if Barbaro’s exercise is not correct from a projective point of view (because the rays of the sun generate a parallel projection and not a conical one—as in the case of a candle, which would have better approximated the geometric procedure of sight) we can affirm that in Palladio’s time the comparison between light ray and visual ray was perfectly known and explicitly used. This comparison left a profound legacy in the art of gnomonics and in the science of perspective in the following centuries. About 80 years later, in the Proposition X of Book II of the Thaumaturgus Opticus, Jean François Niceron described a revolutionary machine, useful for creating anamorphic images (Niceron 1646: 165–169). This tool marked the physical and geometric link between propagation of sunlight and visual rays; it also proved in a certain way the interest, at the beginning of XVII century, of perspective scholars for time measurement. The ‘skiagraphic’ instrument, that Niceron described, worked: “through the shadow projection of a gnomon”.Footnote 6 The machine must be exposed to the parallel rays of the sun so that the shadow, generated by a gnomon, retraces an existing image. At the same time a second shadow, produced by another gnomon—of the same size as the first and oriented in the same way—can identify the projected image on any kind of surface. As for sundials, the rules of perspective, exemplified by light, generated a ‘writing’ with shadow, so that an eye located on the tip of the second gnomon could perceive an image equal to the initial drawing (Fig. 6).

Fig. 6
figure 6

Device that uses shadows to draw perspectives and anamorphoses: Niceron (1646: Pl. 32)

The extraordinary worth of this skiagraphic tool lies in the possibility to force nature to follow representative needs: the integral movement of the system exposed to the rays of the sun changes a parallel projection of light in a conic shadow projection of the gnomon. Coming back to Palladio’s time, in addition to Daniele Barbaro’s disquisition on sundials in Book IX of La pratica della perspettiva and Book IX of Vitruvius,Footnote 7 Commandino also translated and commented on the first four books of the Conics of Apollonius (Commandino 1566; Rose 1975: 203–204, 214). So, we can say that if mathematicians in Palladio’s time brought back to life the Greek treatises forgotten for over a millennium, it would be only in the following century that the theory was applied to scientific and technological needs. Indeed, Johannes Kepler, with his masterpiece Astonomia nova (Keplero 1609), brought conic sections into astronomical practice. By the Renaissance the analytical study of the curves already went beyond the theoretical treatment and they were used also in the geographical and military field: respectively to represent on the maps the spherical surface of the earth and to describe the trajectory of projectiles. Emblematic in this context is the treatise entitled Nova scientia by Niccolò Tartaglia (Tartaglia 1537; Rose 1975: 151–152), who gave renewed impulse to mathematics and to the scientific method studying curves and kinematics.

Theoretical and Practical Application of Perspective

The mathematical and geometrical principles that unify the conics of Apollonius and the laws of perspective, as highlighted by Daniele Barbaro and Federico Commandino, would be used the following century in Rome—not by chance—by a brother of Niceron, Emmanuel Maignan, who created two catoptric sundials in the papal capital, one in Palazzo Spada and one in the convent of Trinità dei Monti (Fig. 7).

Fig. 7
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Engraving of the sundial at Palazzo Spada in Roma: Maignan (1648: 341)

Indeed, these catoptric astrolabes show that it is possible to overcome a consistent geometric problem, common to gnomonics, as well as to the ‘curious’ applications of perspective: the tracing of the hour lines on a barrel vault. The difficulty to be solved is the projection onto surfaces of any shape and orientation, from a given point, of an object hovering in space. Considering the specific case of Maignan’s catoptric sundials, the question can be reformulated in geometrical terms as the intersection between two quadratic surfaces: one is the reflected cone, which has as its base the apparent path of the sun in the sky, while the lines that generate its lateral surface correspond to the rays of light. The second quadric surface is the cylinder of the barrel vault which has a circular base, while lines that form the lateral surface are parallel to a horizontal plane. The intersection between these two surfaces generates quartic curves, i.e., curves that cannot be represented on a plane; they correspond to the intricate system of hour lines. Figure 8 shows the net of the barrel vault of Palazzo Spada, in which the different hour lines (Babylonian hours, Italian hours, French hours, temporary hours) are distinguished by colors; the overlapping of the time systems was outlined by Emmanuel Maignan ideally intersecting the reflected cone of the sunlight with the cylinder of the inner vault.

Fig. 8
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Maignan, Palazzo Spada, net of the hour lines. Image: Salmaso (2014–2015)

The problem is clearly explained on p. 283 of Maignan’s treatise dedicated to catoptric sundials, emblematically entitled Perspectiva horaria sive de Horographia gnomonica (Maignan 1648). It shows a celestial sphere divided into two equal parts by the segment OB; the left half has a schematization of the sun on the arch of the sky vault and the outermost lines of a luminous cone, i.e., the incident sunrays. The right half shows the reflection of the sun and of the luminous cone in symmetrical positions referring to the left half of the image. A plane is interposed between the mirror, symbolized by the ellipse VX, and the symmetrical drawing of the sun; it intercepts the reflection of the luminous cone, generating on its surface a further image of the celestial body (Fig. 9).

Fig. 9
figure 9

Scheme indicating the reflection of solar rays on a circular mirror: Maignan (1648: 283)

The rules of perspective and gnomonic projection are perfectly interchangeable in this simple scheme realized by Maignan. The ‘virtual’ positions of the incident vertex (D) and the reflected vertex (E) of the cones can be considered both as disembodied eyes of a hypothetical observer and as projection centers; consequently, a similar double meaning can be associated to the two suns, on the left (AFG) and on the right (RST). If we compare Maignan’s scheme to the principles of perspective, the two suns can be intended as the bases of the visual cones whose vertices are in the point of observation D and E; these two visual cones, intercepted by the mirror and the plane, generate section images or perspectives. Otherwise, if we submit the same graphic composition to the rules of the gnomonic projection, the outlines, drawn on the mirror and the plane, projected from the centers D and E, generate cones whose intersections with the celestial sphere identify the two suns or hour lines.

There is also a final link to consider before concluding about perspective in Palladio’s time: the relationship between measurement tools and perspective (Beltrame 1973). The techniques and tools of measurement in the Renaissance were based on well-known geometric principles, widely disseminated as early as the Middle Ages. The measures were obtained per perspectivam, that is ‘through the sight’, basing on the theoretical support offered by Euclid’s Optics, which is substantially founded on the similarity of triangles. Surveying techniques were based on a strongly schemed vision process without epistemological implication on perception: it is mainly focused on the geometric relationships established between the real size of objects, their perspectival image and the position of the eye. On the basis of these considerations, the architects of the Renaissance, like Palladio, learnt to metrically evaluate the apparent dimensions in reference to the amplitude of the angles formed by the visual rays; indeed, the size of the images is proportional to the size of the object and the vision angle, as well as inversely proportional to the distance of the object from the eye. Thus, the calculation of an inaccessible dimension could be solved by knowing two factors: the visual angle and the distance of the observer from the object, reporting the comparison to the Euclidean proportion between similar triangles (Camerota 2006). Palladio, as a man of the Renaissance, is testimony to a scientific attitude that accords full confidence in the power of technology; the machines overcome the limits of the body and the senses to exceed the imitation of nature, producing the new forms conceived by the man’s mind. Once again, Daniele Barbaro makes explicit the synthesis assumed by the Renaissance perspective, linking the surveying practice to the theories of vision, so that the perspectival practice can be considered as a mathematical tool for measuring reality. The success and dissemination of perspective and topographic instruments must be interpreted under the light of the demonstrative truth of mathematics that allows it to overcome the uncertainty of senses (Massey 2007). It is not therefore surprising if in perspective treatises an important section is usually dedicated to the construction and use of the machines suitable for perspectival drawings, from Leon Battista Alberti’s veil to Dürer’s machines, up to scientific revolution, to the perspectograph of Ludovico Cardi (known as Cigoli). Daniele Barbaro is no exception and in La pratica della perspettiva he deals with a tool to draw views of cities invented by Baldassarre Lanci, superintendent of the fortifications of Siena on behalf of Cosimo de’ Medici (Barbaro 1568: 192). The device described by Barbaro consists of a circular disk, on which a portion of cylindrical surface rises, where a sheet of paper is applied for the perspective delineation. The drawing can be performed thanks to a pointing-tracking system, positioned at the center of the disc and anchored to a vertical shaft. To explain why a superintendent of fortifications designed and built an instrument for perspective, it is better to rely on the words of Egnazio Danti contained in Le due regole (Vignola 1583) who, relating to topographic measurements and military art (Fig. 10), wrote:

Fig. 10
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Rendering of the perspective tool by Baldassarre Lanci: Vignola (1583: 61)

…in addition to the many advantages it [perspective] brings to military art, it also brings much benefit to the siege, and defense of the fortresses, being able, with the instruments of this Art, to realize the drawing of any place without approaching it, obtaining not only the plan, but also the elevation with every detail; and the measures of its parts proportioned to the distance between our eyes and the thing we have represented.Footnote 8

The efficiency of Lanci’s device has been challenged by Danti, due to cylindrical support used for perspective, instead of the classical plan: “by many it is used and taken into account … but as the paper is stretched on a plane … everything appears altered and confused”.Footnote 9 But according to Maltese, on the cylindrical surface the angular amplitudes are shown faithfully, meaning that it is easy to quickly obtain a complete triangulation of a place to be measured (Maltese 1980: 419). It should however be emphasized that Lanci’s device, described by Barbaro, was used also in the centuries that followed, even though the cylindrical image does not match the rigorous mathematical criteria of linear perspective. Indeed, if the diameter of the cylinder is sufficiently large and the drawing is contained in a portion of a cylinder included within a 45° angle, it can be considered a good approximation. This is probably why, even in the nineteenth century, the tool was considered useful as a:

machine for finding points, upon a concave cylindric surface, which was used by Baldassarre Lanzi da Urbino… This apparatus may, however, be so modified, as to become useful in finding panoramic points (Ronalds 1828: 11).


In Palladio’s time, perspective, after having perfected its rules and instruments, becomes the means for a geometric-mathematical diffusion of reality that fits perfectly into the Renaissance concept of the universe, according to which knowledge is an exclusive prerogative of human intellect, a world separated from the celestial and terrestrial worlds, placed in the middle between the metaphysical and the sensible spheres. This middle world is considered separated because it originates from the reading that man performs autonomously on nature, outside of a divine intervention, with the only instrument at his disposal being the intellect, which makes use of a mathematical-geometric approach (Angelini 1999). The perspective that Palladio knew, also through Daniele Barbaro’s works, contributed to changing the way of perceiving the world, based on Euclidean geometry adopted by the scientific thought in the Renaissance as well as in following centuries. This idea was summarized in the title page of Niccolò Tartaglia’s Nova Scientia (Fig. 11), which shows Plato pointing upward to the world of ideas, while Aristotle points his finger to the earth or the sensible world. Euclid, on the other hand, opens the gate of the wall where all knowledge is enclosed. It follows that the motto: “Let none enter here who knows no geometry” at the entrance to the Academy of Athens is not to be related to Platonic mathematics but to Euclidean geometry, with which all aspects of perspective, as well as of the new Renaissance sciences, are unquestionably permeated (Chastel 1964: 494–495; Garin 1989: 171–181).

Fig. 11
figure 11

Niccolò Tartaglia, Nova Scientia (1537), title page