Optical Perspective

In this section we reprint the introductory theory chapter from Kim Veltman’s Encyclopedia of Perspective (Vol.1)—also known as the Sources of Perspective.

Kim’s treatise deals with the subject of Optical Perspective; whereby the Encyclopedia of Perspective looks at associated sub-categories including Natural (or Visual), Mathematical, Graphical, Instrument, Forced and Media Perspective types. A large part of Kim’s treatise deals with a form of Technical Perspective known as Linear Perspective; which accordingly has links to all of the other aforementioned perspective subcategories.

Kim begins by exploring the origins of Graphical Perspective, or copying reality to produce accurate drawings, paintings, models, maps and projections of dimensional space. Put simply, Kim considers problems related to making faithful representation(s) of the physical world.

Accordingly, in the sections below, Kim deals with the early history of Optical Perspective in general, and with Linear Perspective in particular.

Note on Use of Term: Optical Perspective

Concerning perspective, the need for clarity and strict definitions is especially desirable, which is why we have developed Perspective Category Theory (PCT).

Within the introductory text below, Kim Veltman correctly identifies the primary difference between Visual Perspective (of 2nd type or human vision) and Linear Perspective (a form of Graphical Perspective). As Kim explains, Linear Perspective has a strict geometrical basis and is based on the use of a perspective window, and not on visual angles as with the optics of the human eye. This fundamental difference between Linear and Visual Perspective explains the significant variation between visual images produced by the two kinds of perspective, variations that become apparent under specific circumstances.

We acknowledge the possibility that Kim (and other experts on perspective) would object to how we have employed the term Optical Perspective within PCT, and because we have adopted the term for a broader range of meanings than purely ‘optical’ phenomena such as (for example) photography. In particular, we have included Natural, Mathematical, Graphical, Instrument, Forced, and Media Perspective categories under the Optical Perspective branch. 

But why have we classified all Technical/Non-Technical forms of Perspective as falling under Optical Perspective? Isn’t it true that Optical Perspective applies only to methods that directly employ optical instruments (and ‘eyes’) to produce related images/views? We have done so simply because PCT theory concerns all forms of Visual Perspective that use, or purport to use, light to produce Visual Images of a spatial reality (including represented/modelled images).  

Perspective Category Theory is based on a hierarchy of concepts. We begin by identifying Visual and Symbolic types of Perspective. On the Visual Perspective branch, we have visual images produced by two primary methods, Optical Processes (Optical Perspective) and Contact Processes (Contact Perspective). We have subsequently identified five kinds of Optical Processes or physical/simulated/modelled processes that give rise to Visual Images; namely: Optical Imaging, Mathematical Modelling, Graphical Modelling; plus Optical and Computational Modelling. 

Noteworthy is that we have defined Optical Perspective to operate wherever a real, simulated, or modelled optical view/image of a spatial reality is produced (by whatever method). Hence we include the generation of optical views/images by eyes/vision, optical imaging, graphical representation, geometrical analysis, computer-generated methods, and VR (Virtual Reality) systems, etc. QED.


By Kim Veltman


1. Introduction

Perspective is a mathematical method of representation, which demonstrates how images change size with distance and change shape when they intersect planes of different shapes in various positions. It is sometimes used in a broader sense to mean all systematic, mathematical methods of representation, including various branches of parallel perspective, where distance plays no role. Hence, we need to distinguish at the outset between perspective, which is quantitative and objective, and pseudo-perspectival methods, which are qualitative and subjective.

By way of introduction we shall also define the standard branches of perspective, identify some properties of rectilinear picture planes and examine the effects of angular, conic, cylindrical and spherical planes in order to explain the reciprocal relation between perspective and anamorphosis (trick-perspective).

We shall then reconsider the two chief Renaissance methods of perspective and the question of their origins. The role of astronomy, geography, geometry, optics, surveying, topography, architecture and archaeology will be mentioned, as will the relation between early practice and theory. Finally some questions of definition will be raised concerning fifteenth and sixteenth century treatises on perspective.

2. Pseudo-Perspectival Methods

Vitruvius, in the introduction to book seven of his De architectura reports that Agatharcus, a contemporary of Aeschylus painted a scene and left a commentary about it:

This led Democritus and Anaxagoras to write on the same subject, showing how, given a centre in a definite place, the lines should naturally correspond with due regard to the point of sight and the divergence of the visual rays, so that by this deception a faithful representation of the appearance of buildings might be given in painted scenery, and so that, though all is drawn on a vertical flat facade, some parts seem to be withdrawing into the background, and others to be standing out in front.

Those who have interpreted these lines as proof of perspective in Antiquity have ignored the wider context of the discussion. In the same book (VII, Chapter V), Vitruvius laments the decadence of contemporary fresco paintings noting that, whereas the ancients required realistic pictures of real things, subsequent artists represented “the forms of buildings and of columns and overhanging pediments” as well as the facades of scenes in tragic, comic or satyric style.

Vitruvius adds that:

Those subjects which were copied from actual realities were scorned in those days of bad taste. We now have fresco paintings of monstrosities rather than of truthful representations of definite things.. Such things do not exist and cannot exist and never have existed.

If linear perspective had been involved Vitruvius, as a pragmatic architect, should have emphasized the practical applications of these methods for architecture and their significance in recording the natural world quantitatively. His deliberate opposition between truthful representation of the natural world and the unreal objects produced in these scene paintings and frescoes confirms that something else was involved.

Figure 1. Roman mural painting from the Villa of Publius Fannius Synistoin,
showing proto-perspective effects. Corinthian Oecus, in Oplontis.
(Buried, preserved in eruption of Vesuvius in 79 AD).

The evidence of the extant frescoes at Pompeii, Herculaneum and Oplontis supports this conclusion. Most of the buildings represent imaginary inventions, which are architectural impossibilities. In most cases the depth represented involves only a few feet. The scenes serve to close spaces rather than to open them. Nor do all the lines converge to a single point. In the rare cases where most lines observe this rule, the rest still converge to other points along an axis.

Figure 2. Three kinds of pseudo-perspectival methods: a. optical adjustments, b. fishbone, c. inverted perspective

This suggests that Vitruvius’ description of scene painting probably involved a pseudo-perspectival method variously known as axial, vanishing vertical axis or fish-bone perspective (fig, 2b). In which case, the centre which Vitruvius mentions, refers to an axis running through the central point of a Greek theatre, an axis being involved in order to accommodate the different heights of the viewers.

Other evidence has also been cited to claim that the Greeks were acquainted with the laws of perspective. Greek optical theory asserted that visual angles govern apparent size. This theory, when applied to their practice of representation, invited two solutions: to represent objects higher up as smaller (fig. 2) or to make a higher object larger in order that it appear the same size.

Figure 3. Pseudo-perspectival methods: visual angles in the Bedford Hours

These are again pseudo-perspectival methods variously termed negative perspective, optical adjustments or visual angles methods (figs. 2, 4). According to Pennethorne, the method of making objects higher in order that they appear the same size was used in the temple of Thebes in the 13th century B.C. and with the letters on the temple of Priene in the 5th century B.C.

In Antiquity such a method probably inspired Plato’s complaints against sculpture in the Sophist. The method remained popular throughout the Renaissance. Dürer described it in his Instruction of Measurement (1525). Michelangelo used it in the figures above the altar of the Sistine Chapel, and Serlio described it in his first book on architecture, as did Barbaro in his Practice of perspective (1568).

Most Renaissance thinkers were so convinced that the visual angles principles taken from Euclid’s Optics provided a theoretical basis for linear perspective that they overlooked a basic contradiction between their practice of perspective and theory of vision, namely, that perspective deals with planes and not with angles. Imagine (fig. 2a) a viewer at A looking at three equally sized objects BC, DE and FG. As long as the interposed plane HI is parallel with BG, the projected size of B’C’, D’E’, and F’G’ will be equal even though the angles subtended at the eye become smaller. In other words, although Euclid’s theory of vision predicts that GF appears smaller than BC, Euclid’s geometry predicts that G’F’ is projected the same size as B’C’.

Desargues (1636) recognized this contradiction between planes and angles and hence when Dubreuil continued to espouse optical adjustments methods in his Practice of perspective (1642), Desargues tried to clarify the issue.

Editor’s Note: Contradiction between geometrical planes and visual angles

The contradiction between planes and angles discussed here relates to the difference between the projected size of an image onto a spherical retina (visual projection); contrasted with the projected size of an image onto a plane located normal to the optical axis or direction of vision (geometrical projection). Both situations have a strict geometrical basis (the eye operates essentially as a pin-hole camera in terms of image/object magnification).

Another way of stating this problem, is that the eye deals with ‘visual angles’, or inherently measures the apparent angular size of an object (projected image angular extent on a spherical retina); wheres perspective deals with the size of an object (or image) created by an entirely different method, being projection onto a plane perspective window (represented on a picture plane).

A spherical retina causes objects that subtend smaller visual angles, that is, identically sized objects located at the same distance along the optic axis, with each object located at a greater width/height laterally from the optic axis, to create correspondingly smaller images. If the  human eye had a different form, for example with a planar (and non-moving) retina, then such objects would all project onto the retina with an identical size, and the apparent contradiction between planes and visual angles would not occur.

Alan Radley (3.12.23)

Desargues student, Abraham Bosse, the first professor of perspective at the Académie Royale, went further and set out “to prove that one must not draw or paint as the eye sees.” Bosse’s colleagues did not understand his plea to distinguish the objective laws of perspectival planes based on geometry from subjective theories of vision based on psychological optics and they conveniently hid their incomprehension by expelling him from the Academy.

Fig 4. Abraham Bosse Instructional Drawing on Perspective (1665).
(Depicting problem of contradiction of planes and angles)

Although more than three centuries have passed Bosse’s distinction continues to be overlooked. Even highly educated individuals when, under extreme conditions, they discover discrepancies between their subjective visual impressions and the objective laws of perspectival representation, assume that perspective must be a simple convention. Some also continue to associate optical adjustments methods based on visual angles with perspective.

Underlying optical adjustments is a principle of compensation: one adds to the object’s physical size, an amount that it would otherwise have lost in apparent size. But this addition usually occurs only in a vertical plane. In a third pseudo-perspectival method, inverted perspective, (fig. 2c), one applies this principle to both the vertical and horizontal planes simultaneously such that parts further away are both higher and wider (Fig. 5). However, the extent of this adjustment is not fixed and it is used so subjectively in many pre-literate cultures that it is difficult to think of it as a systematic method as Shegin claimed.

Figure 5. Inverted perspective in a detail from a Chinese handscroll in the British Library (17th century)

While creating a sense of depth this method also remained subjective. In all three of these pseudo-perspectival methods there is no way of studying the picture in order to determine, post facto, the original distance of the viewer. Others have been exploring spherical or cylindrical methods of perspective which they hope will objectively record their subjective impressions.

3. Parallel Perspective

Parallel perspective is another method in which the original distance of the viewer cannot be determined because the viewer’s position is taken to be at infinity. In parallel perspective further distinctions are now made between orthographic or axonometric projection where the faces of an object are oblique or tilted relative to the picture plane and multiview projection when the face of an object remains parallel to the picture plane. These categories are in turn subdivided.

Figure 6. Parallel perspective is divided into orthogonal and oblique methods. Orthogonal methods are divided into multiview and axonometric. Multiview considers the plane in terms of four quadrants or angles (6.1-2) of which only the first and the third are used for practical purposes. First quadrant methods are standard in Britain (6.2-3).
Third quadrant methods are standard in the United States (6.3-4-6) where this method has been developed to produce a transparent box providing projections on all six of its sides (fig.6.7-8).

Multiview projection may involve first angle (also termed first quadrant) projection, commonly used in Britain (fig. 6.3 cf. fig. 6.2), where one looks out at the projection: or third angle projection, commonly used in the United States (fig. 6.5), where one looks in at the projection as if it were in a transparent glass box (fig. 6.7).

Axonometric projection (fig. 7.2) is subdivided into isometric, dimetric and trimetric projection. In isometric projection all three faces are equally oblique (fig. 7.2). In dimetric, two faces are equally oblique (fig. 7.3). In trimetric, only one face is equally oblique (fig. 7.4). In addition, there are two other oblique parallel projections known as cavalier projection, where the object is at 45 degrees relative to the picture plane (fig. 7.5) and cabinet projection, where the object is at arc tan 2 relative to the picture plane (fig. 7.6), and some include a third, military projection (fig. 7.7).

Figure 7. Besides Multiview (7.1) the other type of orthogonal parallel projection is axonometric
(7.2), which is subdivided into isometric (7.3), dimetric (7.4.), and trimetric (7.5.).
Oblique parallel perspective, the second main division, is subdivided into cavalier, (7.6),
cabinet 7.7) and military (7.8) perspective.

Parallel perspective is often a misleading term because historically these technical distinctions did not apply. In the seventeenth century, for example, cavalier and military perspective were often interchangeable, and usually referred to rough and ready methods involving a bird’s eye view of a fortification. In addition, perspective is used to describe practical conventions in Oriental art which appear to be without a theoretical basis.

4. Standard Methods: Linear Perspective

Linear perspective, according to the Oxford English Dictionary is an application of projective geometry in which the drawing is such as would be made upon a transparent vertical plane (plane of delineation) interposed in the proper position between the eye and the object, by drawing straight lines from the position of the eye (point of sight) to the several points of the object, their intersections with the plane of delineation forming the corresponding points of the drawing.

Linear perspective is generally accepted as being synonymous with central, plane, Brunelleschian or Albertian perspective. A distinction has arisen between one-, two- and three-point perspective. In one-point or central perspective only one dimension (depth) is not parallel to the picture plane. In two-point perspective two dimensions (depth and breadth) are not parallel to the interposed plane. In three-point perspective all three dimensions (depth, breadth, and height) are not parallel with the picture plane (fig. 8).

Figure 8. Standard methods of linear perspective:
a) one-point, b) two-point, and c) three-point perspective.

Fifteenth century authors dealt almost exclusively with one-point perspective and hence this method is sometimes treated as synonymous with Renaissance perspective. One-point perspective is also occasionally treated as synonymous with linear perspective but this is misleading. Linear perspective includes one, two and three point methods.

Two-point perspective was made popular in the early sixteenth century by Jean Pélerin, le Viateur (1505) and Joachim Fortius Ringelbergius (1531). This method is synonymous with angular perspective. In England two-point and oblique perspective are also synonymous. In America, by constrast, oblique perspective is used as a synonym for three-point perspective. In both countries three point and inclined picture plane perspective are synonymous.

Because parallel perspective as well as one- and two-point perspective all have their frontal plane or height dimension parallel to the picture plane, these three methods are classed by some under a more general heading of orthogonal perspective. Nonetheless, mathematicians continue to distinguish between orthogonal, parallel and central projections.

5. Picture-Plane

One of the distinguishing characteristics of linear perspective is the principle of the window or picture plane whereby a transparent plane is used in arriving at a perspectival foreshortening. Alberti, in the Latin version of his On Painting, (1435) claims to have invented this principle. In the Italian version, dedicated to Brunelleschi, this claim is carefully avoided, which makes it likely that Brunelleschi was actually the first to use it when he made his famous rendering of the Baptistry of San Giovanni between 1420 and 1425.

Figure 9. If the object is at right angles to the picture plane the more distant parts of the
square are perspectivally foreshortened. The diminution is no longer a simple inverse size-distance law.
A series of such foreshortened squares produces the familiar pavements of renaissance paintings.

The inverse size/distance law of linear perspective applies in the case of objects positioned parallel to this picture-plane. Hence, when an object is twice as far away from the picture plane as the distance from the eye to the picture plane, its size on the picture plane is one half its original size. When an object is three times as far away, its size on the picture plane is one third. When it is four times as far away it is one fourth and so on.

In this process there are three variables: eye, picture plane and object. Normally two variables are kept constant while a third is moved systematically. If the object is moved away from the eye the object’s projected size becomes proportionately smaller. If the picture plane is moved away from the eye, the object’s projected size becomes proportionately larger. If the eye is moved away from the object and picture plane, the object’s projected size becomes proportionately larger, while other objects at right angles to the picture plane become increasingly foreshortened (fig. 2).

Figure 10. Linear perspective involves three basic variables eye, picture plane and object.
If two of these are kept constant while a third is moved, the inverse size distance law becomes apparent:
i.e. if the object is twice as far from the eye as the picture plane, its size on the picture plane is one half (10.2).
If it is three times as far, its size is one third (10.3) and so on 10.4-5).
The principle can also be shown by moving the picture plane systematically (10.6-7).

In retrospect, all this is eminently simple. But it did not seem so to fifteenth century thinkers. Neither Alberti, Filarete nor Francesco di Giorgio Martini was aware of the inverse size/distance law. Piero della Francesca considered the idea but did not distinguish clearly between objects parallel with and objects at right angles to the picture plane and therefore denied the existence of any simple inverse proportion. Leonardo first discovered this principle around 1492. It took another 144 years before Desargues formulated this principle in mathematical terms and another 20 years beyond that for Bosse to popularize them.

6. Angular, Conic and Cylindrical Planes

In the meantime thinkers explored the characteristics of various other types of planes. The simplest of these was a concave V-shaped projection plane consisting of two converging rectilinear planes (fig. 11.1). Marolois (1614), described this possibility which was actually used in a peep show of a church interior now in the National Gallery at Copenhagen. Bosse also considered the converse: a convex V- shaped projection plane again consisting of two rectilinear planes (fig. 11.2). Corresponding cylindrical shapes both convex and concave were also explored by the author of the Codex Huygens, Marolois, Nicéron and Schott (fig. 11.3-4). Other variants, probably based on concepts of visual pyramids, involved rectilinear and curvilinear pyramids or cones. Dubreuil (1649) considered frontal projections onto both their exteriors and interiors (fig. 11.5-6).

Figure 11. Renaissance examples of perspective involving alternative picture planes.
In this diagram, we see warped geometry of picture planes, with picture plane shown on right for each diagram (11.1-2: folded picture planes, 11.3-4: cylindrical picture planes, 11.5-6: pyramidical and conical picture planes).

Figure 12. One motivation for interest in alternative picture planes was a concern with
anamorphic effects (trick perspective), namely, how one could produce complex images that
would appear normal on the picture plane when seen from a pre-ordained viewpoint.
In this diagram, we see warped geometry of picture planes, with picture plane shown on right for each diagram (12.1-2: folded picture planes, 12.3-4: cylindrical picture planes, 12.5-6: pyramidical and conical picture planes). Notice in theses examples that the object geometry is also warped, and further that the objects are located in a non-planar fashion relative to the picture plane, or tilted plane, and in order to produce an optically negated image warping effect in the associated picture plane.

Figure 13. Further Renaissance examples of alternative picture planes
using pyramids and as illustrated by Dubreuil (1642-1649).

Figure 14. Sundials using cylindrical and conical planes. 

Fig 15: Renaissance sundial projection. 

Fig 16: Projections of the equator, tropics of Cancer and Capricorn, as seen from space, as projected onto a universal sundial and as projected onto the pavement by an obelisk used as a sundial in the forum at Rome.

7. Spherical Planes and Surfaces

The practical projection of rectilinear surfaces onto spherical planes evolved in Flemish painting practice of the fifteenth century when it became fashionable to depict scenes reflected in convex spherical mirrors (fig. 14.1). The theory of spherical perspective (fig. 14.2), which has received much attention since the 1870’s due to the analogies with the retina of the eye, was not considered in the fifteenth or sixteenth centuries. On the other hand, the reverse case of projecting spherical surfaces onto rectilinear planes received much attention.

Figure 17. The use of spherical planes, which have recently become popular in attempts to
simulate effects of visual perception effectively reverse the methods which evolved in ancient
astronomy and cartography. To simulate visual perception Renaissance theorists tended to use cylindrical projections. However, artists in Northern Europe explored these spherical principles in terms of images reflected in convex mirrors.

Ptolemy had considered this problem in the second century in his Planisphere when he treated the South Pole as the position of a viewer and the equator as a projection plane onto which he projected both the circles of Cancer and Capricorn. This was essentially a first demonstration of the principles of linear perspective, but under very limited conditions, where only scale was important and measured distance played no role. Projections of the astrolabe involved a direct extension of this principle: the lines of longitude and latitude corresponding to a particular place on earth now also being projected onto the equatorial plane (fig. 16.3-6).

Figure 18. In a perspectival treatment of the same problem the equator serves as a
picture plane onto which are projected the circles of Cancer (15.1-2) and Capricorn (15.3-4). In an astrolabe also involved are various lines of latitude and longitude corresponding to the observer’s viewpoint (15.5-6).

In the 1390’s, Blasius of Parma, a professor of optics at Padua, wrote commentaries on Euclid, Alhazen, Witelo and Peckham, confronting optical theory with practical surveying methods. He also used Ptolemy’s Planisphere as a textbook. By way of demonstration he employed an armillary sphere, a three-dimensional model of the earth reduced to circles of the poles and the circles of Cancer, Equator and Capricorn along with the ecliptic. Using candles he projected these circles onto the walls of a darkened room.

Biagio had two chief students. One was Paolo Nicoletto d’Udine (Paul of Venice), who had studied at Oxford and thus brought an awareness of Bradwardine, the Oxford calculators and the particular associations between theology, geometry and optics developed by Grosseteste, Bacon and Peckham. The other was Prosdocimo da Beldomandi, who had strong interests in mathematics and astronomy who in turn taught Fontana, Toscanelli, Cusa and possibly the young Alberti. Hence there were close links between those concerned with projection methods in astronomy and the pioneers of linear perspective.

Through study of the planisphere and astrolabe, thinkers became aware that perspective works in two directions: 1) to record images backwards onto the picture plane as with the tropic of cancer or 2) to project images forwards onto the picture plane as with the tropic of capricorn (fig. 15). The second of these effects could be achieved using methods analogous to those of Blasius of Parma: by substituting a candle for the viewpoint at the South pole and projecting the circle of capricorn onto the equatorial plane in its enlarged form.

Systematic study of these possibilities came only gradually. It was not until the sixteenth century that the rectilinear projection plane was shifted to a position at right angles to the equator, which led to further developments in cartography.

Figure 19. Projections of spherical surfaces onto flat planes was also a problem in cartography. Here a number of solutions evolved. One method, termed gnomonic or central projection, treated the earth’s centre as the point of projection (16.1). A second, termed horizontal or Gemma Frisius after its inventor, treated the earth’s circumference as the the point of projection (16.2). Cartographers have also considered other projection points measured as radii of the earth: Clark chose 1.35 radii (16.3), James chose 1.367 (16.4), while Rojas chose a point at infinity (16.6).
This final alternative is also called orthographic or parallel (cartographic) projection.

The position of the (sometimes imaginary) candle varied. One alternative was to position it at the centre of the earth, which resulted in a central or gnomonic projection (fig. 16.1). Or it was positioned on the circumference of the equator at a point opposite the hemisphere being projected, which resulted in a projection known as horizontal, stereographical or Gemma Frisius (the teacher of Mercator) (fig. 16.2). Subsequent thinkers chose positions slightly further removed: Clarke at 1.35 radii from the centre of the sphere (fig. 16.3); James at 1.367 radii (fig. 16.4) and La Hire at 1.71 radii (fig. 16.5). Yet another alternative was to place the point of projection at infinity which resulted in an orthographic, parallel or (Juan de) Rojas projection (fig. 16.6), named after a Spanish contemporary of Frisius and Mercator.

Not all projections were rectilinear. Mercator projected the sphere onto a cylinder to arrive at his now famous grid system (fig. 17.1-2).

Figure 20. A further alternative was developed in the sixteenth century by Mercator,
a cartographer active in Louvain. A Mercator map involved projecting the earth’s
sphere onto the inner surface of a cylinder. 
Variants of this method are termed transverse Mercator and oblique Mercator.

Figure 21. To deal with the problem of spherical surfaces in cartography,
 Ptolemy (c.150 A.D.) considered two forms of conic projection. In the eighteenth century, 
Lambert developed a related projection involving a secant cone.

Fig 22: The Fra Mauro map is a map of the world made around 1450 by the Italian cartographer Fra Mauro, which is “considered the greatest memorial of medieval cartography”. It is a circular planisphere or analogue computing instrument in the form of two adjustable disks that rotate on a common pivot. Drawn on parchment and set in a wooden frame the map measures over two by two meters. It includes Asia, the Indian Ocean, Africa, Europe, and the Atlantic. It is oriented with south at the top. Whilst the authors of such maps recognised that the world was in fact a globe, early maps of this kind were limited in accuracy due to a lack of necessary perspective measuring devices and position finding equipment plus perspective theoretical methods which would enable accurate calculation of longitude and latitude for ships that undertook mapping expeditions.
Fig 23: Johann Stabius’s 1515 world map, whose engraving is credited to Albrecht Dürer.
A planisphere globe surrounded by the twelve winds, equivalent to Ptolemy’s third projection. 

Fig 24: An armillary sphere (perspective measuring/modelling instrument); with variations are known as the spherical astrolabe, armilla, armil, or huntianyi (traditional Chinese: 渾天儀; simplified Chinese: 浑天仪) is a model of objects in the sky. The concept of the celestial sphere was fundamental to astronomy from Antiquity through the Middle Ages and into the early Modern era.  At the centre of the sphere is the Earth. As the Earth is stationary in this model, it is the celestial sphere which rotates about it and acts as a reference system for locating the celestial bodies – stars, in particular – from a geocentric perspective.

Already in the second century Ptolemy had explored another possibility: projecting the sphere onto the inside of a cone (fig. 18). Recent discussions of his having had a third method, which was perspectival are unfounded. It is true, however, that in his seventh book he described an eye looking at the earth. The diagram associated with this, effectively a proto-perspectival drawing of an armillary sphere, was clearly a source for Dürer’s perspectival drawings . In the case of another of Dürer’s globes it has been suggested that he actually used a model of the earth tilted at 23 degrees (as in the angle of the ecliptic) and drew it with the help of a perspectival window. A full analysis of cartographical methods is beyond the scope of this essay, but is an area deserving much more attention.

Fig 25: Position-finding by ‘shooting the sun’
from Ph. Lansbergen, Verclaringhe Vande platter Sphaere, Middelburg, Z. Roman, 1628.

Notwithstanding interplay between astronomy, geography and perspective, it was not until 1558 that Commandino published a formal study of correspondences between planisphere projection and perspective, a problem which also interested his student, Guidobaldo del Monte. Egnazio Danti in his commentary on Barozzi’s The two rules (1583) noted correspondences between perspective and geographical projection–a topic obviously of interest to one who was cosmographer to the Duke of Tuscany and author of the magnificent maps in the room of the globes in the Palazzo Vecchio. The same Danti also studied sundials. This combination of interests in perspective and dialling was subsequently pursued by Desargues (1636) and Maignan (1648).

8. Perspective and Anamorphosis

We have already mentioned that perspective works in two directions: to record an image backwards onto a picture plane and to project it forwards. In either case, as long as the object and picture are in the same plane (fig. 10) the image remains undistorted (or isometric), and varies in size only.

When the object and picture plane are parallel to one another, the perspectival image changes shape. In the case of images recorded onto the picture plane these changes are usually unwanted and are referred to as perspectival foreshortenings (fig. 12) or, in extreme cases, as perspectival distortions.

By contrast, anamorphosis involves deliberate changes in shape produced in the case of images projected forward onto a plane. The principle of anamorphosis is thus identical with that of projecting the tropic of capricorn in the astrolabe (fig. 15.3-4), the sphere in various cartographic projections (fig. 16-17) and of shadow projections in sundials. That those interested in anamorphosis were often also concerned with projections in astronomy, cartography and sundials is therefore no coincidence and of considerable importance because it reminds us that the development of perspective and its variants was considerably more than an artistic phenomenon: it was intimately connected with the rise of the mathematical sciences in the Renaissance.

Fig 26: Natural anamorphosis: ‘An Artist Drawing a Nude’ by Albrecht Durer (1525).
Anamorphosis is a component of natural vision, whereby warping/distortion
of an image happens (to a greater or lesser degree) according to object viewing position/angle.

Fig 27: Artificial anamorphosis (form concealment): Hans Holbein the Younger’s ‘The Ambassadors’ (1533).
The anamorphic skull in the foreground continues to delight and surprise viewers, and to inspire artists;
whereby when the painting is looked at from a side aspect (acute angle) then the anamorphic skull becomes
visually decoded to appear once again in its true aspect, and with its original geometrical form.

Fig 28: Artificial anamorphosis (form illusion): Andrea Pozzo’s painted ceiling and trompe-l’oeil dome (Apse) on the Church of St.Ignazio (1690). Pozzo’s masterpiece produces illusory perspectives in frescoes of the dome,the apse and the ceiling of Rome’s Jesuit church of Sant’lgnazio. The architecture of the trompe-l’œil dome seems to erase and raise the ceiling with such a realistic impression that it is difficult to distinguish what is real or not.

Although there exist a near infinite number of possible projections from a plane of one shape onto a plane of another shape, sixteenth and seventeenth century practitioners concentrated on a surprisingly small number of alternatives: a flat projection plane at right angles to the original (fig. 6), a flat projection plane at right angles to an original cylindrical (fig. 11.3-4), conic (fig. 11.6) or pyramidal plane (fig. 11.5). In the cases of cylinders, cones and pyramids a mirror was frequently positioned in the plane of the original object such that the anamorphic forms could be transformed back to their original shape. Anamorphosis thus demonstrated the principles of transformation and reversibility basic to linear perspective.

The origins of anamorphosis can be traced with some precision. At an empirical level problems of anamorphic distortion had arisen in trying to portray Christ, the Pantokrator, in the rounded dome above the altar in mediaeval byzantine churches. At the level of theory, Piero della Francesca, was the first to consider anamorphosis in his On perspective of painting (c. 1480). Leonardo da Vinci explored various aspects of anamorphosis.Jean-François Nicéron was the first to devote an entire treatise to the subject (1638).

The origins of linear perspective are not so readily summarized. Guidobaldo del Monte (1600), reminds us that there were twenty three competing methods at the turn of the seventeenth century. Other sources including Benedetti (1580), Egnazio Danti in his edition of Barozzi, il Vignola (1583) and Piero della Francesca (c. 1480) emphasize two principal methods: one based on geometry, the other on practical demonstrations.

9. Linear Perspective: Geometry and the Distance Point Construction

In his Elements, Euclid had explored basic properties of ratios and proportions of lines and surfaces as well as their equivalents and transformations. In the 13th century interest in these problems was revived by Leonard of Pisa (Fibonacci), who introduced them into the curriculum of the abaco school.

In the 1430’s Leon Battista Alberti applied these geometrical principles to perspective in his Elements of painting (pl. 28.1). Piero della Francesca (c.1480) developed this approach, devoting the first two books of On perspective of painting to these geometrical demonstrations based on proportional diminution. Later examples by Serlio, Barbaro, Pelerin , Androuet du Cerceau or even Galli-Bibiena re effectively logical extensions of this approach.

In one of his propositions, Piero della Francesca mentioned the possibility of confirming these principles with physical demonstrations. In the course of the 1480’s and 1490’s, individuals such as Francesco di Giorgio Martini sought to carry this out by reconstructing the geometrical principles in terms of actual surveying situations. In all likelihood it was Francesco who first explored the principle of the distance-point.

In determining the distance point one begins by extending the converging sides of a foreshortened square (fig. 19) until they meet at a central vanishing point. Through this point a horizon line, parallel to the base is drawn. Through the foreshortened square one also draws a diagonal which is extended until it meets the horizon line at the distance point, so-called because the space from this point to the central vanishing point marks in scale the original viewer’s distance from the picture plane which produced the foreshortening in question.

Fig.29. Frontal and three dimensional diagram to illustrate the principle of the distance point. ABC1D1 is the foreshortened version of ABCD as seen by a viewer at F1. The viewer’s distance from the picture plane F1E is equal to distance FE. The distance point F can be found by simply extending diagonal BD1 until it meets the horizon line.

This ability to work backwards from the foreshortened square to reconstruct the original viewpoint which caused it has been termed the reversibility principle of perspective (linear). This only functions in the case of regular squares (or cubes) positioned at right angles to the picture plane. That perspectival drawings tend to feature regular geometrical and idealized architectural shapes is therefore no coincidence.

Jean Pélerin gave the first published account of the distance point in 1505. In Italy the method did not appear in print until Danti’s edition of Barozzi’s Two rules (1583). As a result its Italian origins in geometrical principles were gradually overlooked and modern scholars generally assumed that it stemmed from practical workshop traditions in the North.

10. Linear Perspective: Optics and the Legitimate Construction

The development of the second major method was closely tied with the history of optics. Euclid’s Optics had dealt primarily with what would today be termed psychological optics, study of subjective aspects of vision. But the treatise also contained four propositions devoted to surveying and thereby the accurate perception and measurement of distance became part of the optical heritage.

By the ninth century thinkers in the Arabic tradition such as Al-Farabi could define optics in terms of measuring the heights of mountains and even distances of stars. Through this tradition there evolved an overlap between the ideals of optics and those of surveying. In the optical treatises of Alhazen (early 11th c.) and Witelo (c. 1280) the concept of measured distance acquired new significance. By the fourteenth century treatises on optics frequently appeared together with those on surveying or practical geometry.

One important consequence of this interplay between optics and surveying was that theoretical propositions in optics were increasingly tested in terms of practical demonstrations. Euclid, for instance, had claimed that visual angles do not vary inversely with distance. Blasius of Parma, in the 1390’s, tested this experimentally, just as he used candles to test experimentally the projections of armillary spheres. Brunelleschi’s picture-plane or window (c. 1415-1425) was probably a direct outgrowth of this tradition: a practical demonstration of the visual pyramids and other principles of optical theory.

Alberti, in his On painting, described the principles of this method verbally, thus providing a first theoretical formulation of what he termed the best method (modo optimo), now remembered as the legitimate construction (costruzione legittima. Even so, he saw the window, or veil (velo), as the practical equivalent of this method and insisted on its fundamental importance:

Nor will I hear what some may say, that the painter should not use these things…I do not believe that infinite pains should be demanded of the painter, but paintings which appear in good relief and a good likeness of the subject should be expected. This I do not believe can ever be done without the use of the veil.

Alberti assumed that optics provided the theory for both his verbal demonstration of the legitimate construction, and for its practical equivalent, which used the window. In the next generation, Francesco di Giorgio Martini and Luca Pacioli also assumed this, although they classed perspective under practical geometry and surveying.

Even in the latter sixteenth century perspective continued to be seen in terms of practice as is witnessed by titles such as Barbaro’s Practice of perspective, Barozzi’s Two rules of practical perspective and Sirigatti’s Practice of perspective. Because authors continued to assume that Euclid’s Optics and Elements provided such theory as was necessary for their subject, there was no theory of perspective as such at the time.

11. Practice and Theory

Indeed when we actually look at the fifteenth and sixteenth century treatises we find that they are very different from what we might have imagined. The early treatises are not repertories of elaborate spatial structures, which serve as harbingers for a revolution in the treatment of space. The earliest extant manuscripts of Alberti’s On painting have no diagrams at all. Later versions have only a few diagrams. By contrast, Piero della Francesca’s On perspective of painting may contain eighty diagrams, but these are only of isolated objects and the most impressive of these shapes had already been mastered at least a century earlier in painting practice.

Piero’s diagram of an octagonal building is a case in point. Duccio had convincingly rendered a frontal view of this form in his Maestà. Thereafter it had become a frequent theme in fourteenth century art and had served as the subject of Brunelleschi’s first perspectival picture. The interior of an apse in Piero’s treatise provides another example. This shape had been mastered by Giotto in the Scrovegni chapel in Padua at the beginning of the fourteenth century (Fig. 20).

Figure 30: Life of the Virgin by Giotto (1304)

Fig 31: Feast of Herod By Giotto (1320)

The same holds true for diagrams in other treatises. The barrel vaults in Barozzi’s The Two Rules also have a precedent in Giotto, this time in his Sanctioning of the Rule in the Upper Church at Assisi. Even a much later example such as the oblique building in Vaulezard’s Abridgement had been used in a simpler form in Pélerin’s treatise and earlier still in painting practice in Christ and the Apostles in the Temple attributed to Andrea di Giusto.

Examination of Duccio’s Maestà (Fig 21) offers some insight into the process that took place. At first sight the altar consists of a bewildering complexity of spatial scenes depicting the life of Christ. But on closer scrutiny it becomes apparent that the nearly axonometric roof and the three columns shown in panel 7 recur in panels 8, 14, 15, 23 and 28. Similarly a variant of this roof which appears in panel 13 recurs in panels 22 and 27. A further variant in panel 19, showing a type of axial perspective in the beams of the ceiling, recurs in panels 24 and 25. These shapes recur in Giotto, and indeed throughout the fourteenth century.

Fig 32. The Maestra by Duccio (1308-1311),
Example of how recurrent use of spatial scenes serves to connect different episodes in a story. 

The Florentine hat or mazzocchio offers another case in point. Uccello painted it in his frescoes long before it appeared in the treatises of Picro della Francesca, Leonardo da Vinci, Daniele Barbaro and their successors. Even the perspectival lines underlying Uccello’s sinopia appeared in practice long before they appeared in theoretical literature. Hence the spatial revolution, such as it was, lay in the gradual mastery of a small number of these basic forms in practice. The early treatises on perspective subsequently summarized these in mathematical terms. Thus, rather than offering new visions of that which practice might explore, the early treatises codified what practice had already achieved.

The development of perspective is too often associated specifically with painting. It is important to emphasize that it affected all media of expression as is witnessed by Donatello’s use of proto-perspectival methods in his sculpture of St. George (Florence, Or San Michele, 1417) or Ghiberti in his bronze doors of the Baptistery-particularly the Gates of Paradise (1435). In many cases perspective merely served to represent spatially models available from classical Roman architecture. The cassetted vault was, for instance, well known from the Roman temple of Maxentius and other buildings. It became one of the great themes of humanistic architecture.

Fig 33: Gilded bronze doors of Florence’s Baptistery of San Giovanni by Ghiberti (1452),
named as the Gates of Paradise by Michelangelo.

Masaccio used it in his Trinity (Fig 22), generally accepted as the first extant work in perspective. Thereafter it occurs in literally hundreds of examples: in paintings by the most famous artists, Mantegna and Raphael; less famous such as, Foppa and even obscure artists such as the Ferrarese Master. It is used in drawings by Bellini and a preparatory drawing by Donatello. It is used equally in other media: in a marble alter by Desiderio da Settignano; in a stone facade by Pietro Lombardo in Venice, by Alberti in the facade of Sant’Andrea in Mantua and by Bramante in his famous illusionistic choir. Borromini’s use of a variant nearly two centuries later in the Palazzo Spada attests to the enduring fascination of this illusionistic form.

Fig 34. Trinity by Masaccio (1426 – 1428)

A cumulative process marks a next stage in development. Hence, Piero della Francesca, having mastered the dome shape and the cassetted vault, produced an inverted dome in the form of a scallop and combined this with a cassetted vault in his famous Brera Altar (Fig .23). Artists at all levels were involved in this process. In his doors for the Baptistery at Florence, Ghiberti had represented a cross vault. This form also appeared in the treatises of Piero della Francesca and Sebastiano Serlio. A derivative painter such as Cima da Conegliano in turn combined a cross vault with a cassetted barrel vault in his St. Peter Martyr and Saints.

Fig 35: The Bierra Madonna by Piero della Francesca (1472)

Fig 36: Annunciation with St Emidius by CRIVELLI, Carlo (1495)

Perspective, as thinkers such as Vredeman de Vries noted, involves looking into or looking through objects. The number of objects which produce such an effect is surprisingly limited. The vault is one. Another is the door or portal, which is closely related to the vault. Fouquet used this perspectivally, for instance, in his Hours of Etienne Chevalier.

The case of the portal is particularly interesting because, long before painters represented it perspectivally, architects had begun to construct it spatially, as if receding towards a vanishing point, as witnessed clearly in the Romanesque example of St. Pierre, in Aulnaye La Santage in the South of France (pl. 4.1) and later, more dramatically in Gothic examples such as Notre Dame. It is noteworthy that trompe l’oeil versions of such portals also date back to the twelfth century.

Fig 37: The Church of Saint-Pierre d’Aulnay (1120 – 1140)

By the fifteenth century the portal had become a motif in Northern proto-perspectival painting such as Vranck van der Stock’s Altar of the Redemption (Fig 25), Rogier van der Weyden’s Christ Appearing to his Mother (pl. 8.3) and hisBeheading of St. John. Another of Van der Weyden’s paintings, The Eucharist with Christ on the Cross extended this effect of the portal until it became flush with the nave of the church. Technically speaking the perspective in these northern examples was not correct. In terms of details they were also very different from Italian drawings of roughly the same period found in Jacopo Bellini’s Sketchbooks.

Fig 38: Vranck van der Stock’s Altar of the Redemption (1455 – 1459)

There were sharp contrasts between the Gothic architecture of the North and the humanistic architecture of Italy with its emphasis on classical examples, on measure and proportion. But in terms of general approach to space Bellini also relied on portals to create perspectival effects in his drawings, a principle which Domenico Veneziano subsequently adopted for his own purposes, as did Veronese a century later.

Bellini’s portals, it will be noted, also involved the, by now familiar, vault form, and moreover, had their parallels in actual buildings of the time such as Brunelleschi’s Pazzi Chapel, which in turn bears comparison with an idealized ruin from Androuet du Cerceau over a century later. In this context, it is very tempting to see a logical progression from the spatial effects in the entrance to the Romanesque church of St. Pierre, and the Gothic cathedral of Notre Dame to Brunelleschi’s facade to the Pazzi Chapel and Benedetto da Maiano’s Santa Maria delle Grazie in Arezzo.

We have already noted Roger van der Weyden’s extension of the portal principle to produce a full view into a church, as if the entrance wall had been removed or rather made into the equivalent of a window such that the entire nave functioned as a cross section. Fouquet adapted it slightly in his Hours of Etienne Chevalier. The Master of the Burgundy Hours used it more dramatically in a miniature now in Vienna. In 1505 Jean Pélerin used it in his On artificial perspective. Once again theory followed practice.

Meanwhile, the theme continued to develop in Italy as witnessed by Domenico Ghirlandaio’s Feast of Herod and Raphael’s School of Athens (Fig 26) and, as the sixteenth century progressed, connections with Roman ruins also became more apparent through engravings such as those of Androuet Du Cerceau and Cock. The logic of looking into involved in perspective was such that it transcended regional differences. For all their stylistic variations Rogier van der Weyden, Fouquet and Bellini in the early fifteenth century, and Raphael, Du Cerceau and Cock in the sixteenth century, had a common approach to space such that one can speak in a new way of a European phenomenon.

Indeed, it is probably no coincidence that the words Europe, Renaissance and perspective have become so unconsciously linked in our minds. Perspective gave to Europe a unifying logic of space, which pointed simultaneously to a diversity of expressions, the opposite, as it were, of the later American ideal of making the many into one (e pluribus unum).

Fig 39: School of Athens by Raphiel (1509-1511)

That which occured with representations of sacred interiors happened equally to visualizations of secular interiors. Here again it became customary to treat one wall as if it were a transparent window permitting a clear view of the other walls. One variant, favoured in the North, was to produce oblique views, emphasizing the right walls or, as if in mirror versions of these, emphasizing the left walls. More frequently there was a frontal view of an interior, the real wall of which in turn contained a window or a larger opening such that one could see into the distance. In Jan van Eyck’s Madonna and the Chancellor Rolin the columns served to frame a landscape in essentially the same way that they did in Piero del Pollaiuolo’s Annunciation. Again the details might differ, but North and South share a common approach.

Such examples are particularly interesting because they show that the practice of representing windows to frame spatial views had become customary at just about the time that the principle of the perspectival window, as an analytical device, was establishing itself in practice in the 1430’s: this time a case of practice and theory developing almost simultaneously.

Meanwhile, the method of treating the front wall as a window gained in importance in the sixteenth century. Michael Pacher used it in his St. Wolfgang Altar, as did Albrecht Altdorfer in his rendering of the Jewish Synagogue at Regensburg and other church interiors. The method was used in the Luther Bible, and Rodler also employed it several times in his treatise on perspective. The same method was frequently used in combination with another basic perspectival form, the colonnade as in both Cesariano’s Vitruvian commentary, and his marquetry work in St. Alessandro. Later sixteenth century examples include a drawing by Scamozzi or, to take northern analogues, the engravings of de Vries and Cornelius Loos.

Already in the fifteenth century, Brunelleschi, one of the discoverers of perspective, was almost certainly aware of the perspectival effects of actual colonnades when he designed his Ospedale degl’Innocenti, as must have been the case with the later designers of the new market in Florence and Benedetto de Maiano when he designed Santa Maria delle Grazie in Arezzo.

By the end of the century, artists such as Bramante realized that the representation of colonnades was particularly suited for perspectival purposes because these permitted one not only to look into but even look straight through a space. Later sixteenth century examples included Vignola’s plan for an open loggia as well as northern parallels, such as those in the treatises of Vredeman de Vries (Fig 27).

Fig 40: Christ Healing the Blind by El Greco (1570)

Fig 41: Palastarchitektur mit Badenden by Hans Vredeman de Vries (1596)

These developments did not, however, undermine the method of treating the front wall as a window, which continued its popularity in the seventeenth century. Hondius used it, for instance, in his engraving of a modern temple, as did Steenwyck in his version of a Gothic Church and Pieter Neefs in his painting of Onze Lieve Vrouwe Kathedraal in Antwerp (Fig 28), which invites comparison with actual photographs of the same building.

Fig 42: Onze Lieve Vrouwe Kathedraal in Antwerp by Pieter Neef (1640 – 1675)

In the next generation, with Saenredam, this tendency towards what appears in retrospect like photographic realism increased. But the actual process of arriving at this result became considerably more complex. It became customary to make preliminary drawings which were then adjusted in arriving at the finished painting. The same process occured in the representation of exteriors. Physical models and printed engravings became increasingly important as exemplars. Ironically as paintings truly came to look like windows to the natural world, the number of versions or filters separating preliminary sketch and finished work increased.

Something else also happened. There was no longer one obvious point of view from which one represented a building or place. It is noteworthy that Saenredam’s preparatory drawing shows us two views of the same building complex. This tendency towards multiple viewpoints is even more obvious in treatments of the central square at Haarlem where we have various views looking towards the Grote Kerk and others looking in the opposite direction.

Fig 43: View of the main market place and Grote Kerk at Haarlem by Berckhyde (1696)

The spatial qualities of these paintings make them appear as epitomes of perspective. In retrospect they seem and may even be a logical consequence of the story in which perspective is linked with the conquest of realism.

What needs to be emphasized, however, is that the early treatises on perspective were not crucial to this story. The basic texts by Alberti, Dürer, Barbaro, Barozzi (il Vignola), or Accolti were sometimes too primitive, and invariably too abstract to serve as models in this process. Even texts such as those by Plerin, Androuet du Cerceau or Vredeman de Vries, which were effectively albums of handy examples, codified images from painting practice.

To understand better this story of the conquest of realism we need a catalogue of basic spatial forms in order to follow their gradual mastery within the practical tradition and their subsequent integration into the theoretical treatises. Theatre also played a role and will be discussed later. Two other aspects of this story deserve mention: architectural ruins and topographical surveying. Each of these will be considered briefly in turn.

12. Architectural Ruins and Plans

It is well known that the key figures in the early development (both in terms of practice and treatises) were architects, notably Brunelleschi, Alberti, Filarete, Francesco di Giorgio Martini, Leonardo da Vinci, Bramante, Raphael, Baldassare Peruzzi, Serlio and Palladio. Brunelleschi also spent considerable time studying architectural ruins in Rome. Indeed his biographer, Manetti, credits him with new methods in recording these.

Hence, the same individual who was at the frontiers of representing modern buildings, was also at the forefront of measuring ancient ones, a combination of interests which we now think of as typical to humanism. This combination of interests gains in significance when we realize that it applies equally to Alberti. The author of On painting and Elements of painting was also the author of Description of the city of Rome and On architecture. It applied also to Francesco di Giorgio Martini whose treatises dealt with practical perspective and the measurement of ruins and modern buildings alike.

Rome was the centre of these activities, due in part to new patronage, which came through the rise of papal power with Alexander VI and Leo X. In the early sixteenth century Raphael, in his famous letter to Pope Leo X, examined the use of ground plan and elevation with respect to both ruins and architectural representation. He also wrote his own commentary on Vitruvius. Fra Giocondo (1511) also played a role in this reinterpretation of Vitruvius, which was continued by Cesariano (1521), Caporali (1536) and Ryff (1547).

Fig 44: Engraving of the Pantheon by Antonio Lafreri of the mid-sixteenth century, showing the exterior with a section cut away to reveal the interior. The attention Brunelleschi showed to such Roman structures was the beginning of a new,
intense study of ancient architecture.

Serlio’s Works of architecture marked an obvious next step. It was the first published text in this tradition: his treatise on perspective (Bk. 2) effectively serving as an introduction to subsequent sections on both architectural ruins and contemporary architecture. In the next generation, this combination of interests continued with Palladio, who codified the uses of perspective in architectural design, in creating an artist’s conception of a projected work, and Androuet du Cerceau who, in one of his books, added modern structures such as San Pietro in Montorio. At the same time, a greater specialization also set in. Androuet du Cerceau wrote three different kinds of books: one devoted to the principles of perspective, a second applying it to ancient ruins, and a third to contemporary architecture.

Of these the third was the most interesting because it showed various French chateaux in the context of their gardens and surrounding landscapes. Many of these engravings, including Aret and Fontainebleau, were based on existing structures, while others are idealized projections. Androuet’s work nonetheless, pointed the way to later books, such as Pérelle, and Decker (pl. 93.2) which showed contemporary buildings in context as “perspectives.”

Meanwhile, other modern artists were adding this element of context to what had hitherto been studies of single columns, individual architectural elements or, at best, isolated monuments. The sketchbooks of Maarten Heemskerck and Francisco de Hollanda gave perspectival views, or vedute, of Roman ruins in their original settings. Hieronymous Cock was another Northerner who probably visited Rome in the period 1546-1548. His Some of the principal monuments of ruins (1550) was one of the first of these collections of views (vedute) to be published. A decade later they were reproduced, without acknowledgement, by Pittoni.

Fig 45: Roman Ruins by Hieronymous Cock (1551)

In 1575, Etienne Du Pérac, who had been living in Rome, published the first work in which these views were connected with perspective in the title: Vestiges of the antiquity of Rome, collected and drawn in perspective. The same Du Pérac subsequently went to France (1587) and introduced there the idea of perspectival gardens which had been developed in Tuscany through Buontalenti and others and which led ultimately to Le Nôtre’s work at Versailles.

Fig 46: Chateau de Versailles – Linear perspective applied to the whole environment (1608)

In Italy the tradition of perspectival views (vedute or prospettive) led on the one hand to Piranesi’s visionary scenes of terrifying dungeons and phantastic cityscapes, and on the other hand to Piranesi’s remarkable engravings in his Architectures and perspectives (1743 etc.) which, as Herschel Levit has so effectively shown, bear careful comparison with modern photographs of the sites.

Fig 47: Various Works of Architecture, perspectives, grotesques, and antiquities;
designed and etched by Giambattista Piranesi, Venetian Architect)

Hence, by the eighteenth century there existed those links between perspective and, —what we in retrospect—, see as a type of photographic realism. In these developments, the fascination with ancient monuments, and the tradition of measuring and surveying them, had played as much a role as formal treatises on perspective.

As a focal point for Italian, Flemish and French practitioners and theoreticians, Rome thus played a special role in the development of classicism in architecture, but also in perspectival views and the so-called conquest of realism. It is important to recall, however, that what occured in Rome was a manifestation of a deeper trend that affected the whole of Europe.

13. Surveying and Topography

Ever since Vasari it has become customary to see Giotto as a key figure in the reemergence of realism as a goal in Western art. Precisely why this happened remains a matter of debate. Some have pointed to a new interest in the natural world inspired by the Franciscan movement. Gombrich has connected this interest, in turn, with a new emphasis on narrative, such that paintings were involved with stories in cycles rather than isolated topics. Hence Giotto’s realism at Assisi, Padua and Florence was partly a function of his telling a story in many episodes.

In addition, it was almost certainly also a function of Giotto’s other activities, particularly his military concerns with surveying and topography which arose from his position as superintendent of fortifications in Padua. Giotto’s younger contemporary, Simone Martini, who worked in Naples and Avignon as well as Siena, had similar professional cross-appointments. As an artist he too had military connections which makes us look afresh at the fortresses in the background of his famous portrait of Guidoriccio de Fogliano (Siena, Palazzo pubblico, 1328).

Fig 48: Guidoriccio da Fogliano by Martini (1328)

Our concern here is not to enter into the lively debates whether these fortresses can be identified specifically as Montemassi and Sassoforte; whether they are merely part of a symbolic landscape or both, but simply to note that, notwithstanding connections between art and the military, fourteenth century paintings contained few stategic landmarks or recognizable buildings. Some would say that this was because the concept of mimesis, in a new sense had not yet reestablished itself.

In the fifteenth century this changed. Brunelleschi in addition to recording public sites such as the Piazza della Signoria perspectivally, was also secretly engaged in military reconnaissance involving surveying and topographical views. But the contexts of this new realism were not only military. Fra Angelico included a clear view of Lake Trasimeno in his Cortona polyptych.

Meanwhile, the Limbourg brothers in the North were producing the Very rich hours of the Duke of Berry with spatially convincing picture-postcard like miniatures of St. Michael’s Mount and a number of the duke’s great chateaux. Here patronage was almost certainly a factor. As artists began working for individual potentates, it became politic and even necessary, to include topographical views of their chateaux and other estates in the backgrounds.

Fig 49: The very rich hours of the Duke of Berry by Herman, Paul and Jean de Limbourg, ink on vellum (Musée Condé, Chantilly) (1409).  This work is widely regarded as the peak of late medieval book illumination, and is possibly the most valuable book in the world.

In the case of the Limbourg brothers, this concern with topographical views or exterior landscapes appeared in combination with interior churchscapes. A similar combination of interests is found further north in the work of Roger Campin, Roger van der Weyden and Jan van Eyck. Hence the same Van Eyck who did interiors of churches and rooms which were perspectivally convincing, although not yet completely accurate technically, includes the tower of a church at Utrecht in the midst of the landscape in the central panel of his Ghent altarpiece. Indeed an entire book has been written to show that landscapes such as those in the Madonna with the Chancellor Rolin represent the area near Maastricht where he spent his youth, although ambiguity remains about the extent to which these landscapes are indeed real or imaginary.

Fig 50: The Ghent Altarpiece by Van Eyck (1432)

In the next generation this ambiguity disappeared. Jean Fouquet, probably inspired by his visit to Rome, produced landscapes in his Hours of Etienne Chevalier, in which the cathedral of Notre Dame in Paris was clearly recognizable. In the same book of hours, Fouquet also did a convincing perspectival interior of the same church and other rooms. Fouquet’s contemporary, Konrad Witz, also produced both perspectival interiors and exteriors with fully realistic landscapes as, for instance, in Christ walking on water (Geneva, Musée de l’art), where he depicted the westernmost shores of Lake Geneva with the peaks of the Moule and Mont Blanc in the background.

Fig 51: Christ walking on water by Konrad Witz (1444)

In Old and New Testament scenes, the inclusion of geographical features, with local landscapes and townscapes in particular, soon became the fashion. The town of Florence in the background of Pollaiuolo’s Annunciation was a typical Italian example. But the phenomenon was by no means limited to Italy, as is confirmed by Meckseper’s excellent study of Renaissance German cities which gives dozens of examples from Germany, Austria and Switzerland in the 15-16th centuries.

There was a parallel development in secular landscapes. By the 1570’s, with Braun and Hohenberg, this had become systematic and included all the major cities of Europe from Constantinople, Buda, Pest, Prague, Cracow, Moscow, Riga and Stockholm at the peripheries to the familiar centres of Rome, Paris, London, Ghent, Amsterdam and Zurich. In the next generation with Merian, this systematic approach was further developed.

Hand in hand with this systematic approach was a new awareness of scale, a consciousness that one could show the same view at different levels of abstraction as is illustrated vividly in the sixteenth century hall of maps in the Vatican showing regions in one scale with inserts for cities and fortresses in a larger scale (pl. 26.3), much as modern highway maps do today (Editor note: Related also to modern Google/Apple map applications with zoomable views).

Fig. 52: Maps of different scales together in the Gallery of Geographical Maps in the Vatican

It is important to recall that the same individuals were frequently involved in the mastery of these different scales of reality. Hence the same Albrecht Dürer who wrote on perspective and did interiors of rooms showing Saint Jerome, also did townscapes, views of earth from a nearby viewpoint, a viewpoint further away and even maps of the stars.

Similarly, the same Egnazio Danti who wrote the commentary to Giacomo Barozzi, il Vignola’s Two rules of practical perspective, was cosmographer to the Medici, produced thesystematic series of maps in the hall of the globes in the Palazzo Vecchio and also produced star maps in the form of astrolabes.

Fig. 53: The Vatican is home to The Gallery of Maps, a stunning set of large-scale paintings that
showcase Italy through wonderfully rich and vibrant topographic maps.
They were drawn by geographer Ignazio Danti as part of a Commission by Pope Gregory XIII in 1580. 

Fig. 54: Map by Ignazio Danti as part of a Commission by Pope Gregory XIII in 1580. 

Fig 55: Astrolabe by Ignazio Danti

In the course of the sixteenth and seventeenth centuries it became possible not only to compare different views of a scene in one scale, but also different views of a scene in different scales, as for instance, the bridge in Zurich, or the orphanage in Amsterdam. For the latter of these there exist also detailed pictures of the gate to the inner courtyard and front gables of the house to the left in front of it. Atlases of the time by Mercator and Ortelius invite a similar comparison of a given land in different scales.

In this context we are able to look afresh at Vermeer’s Allegory of Painting (Vienna, Kunsthistorisches). On the surface, it is an epitome of perspectival realism applied to an interior scene. On closer inspection, however, the map on the wall reveals that we are looking at the Netherlands in a scale very similar to that found in Mercator. The left and right borders of the map show us cityscapes at another scale. The painter’s easel records a third scale. The model in the background stands for the original scale and, at the same time, because she is painted, again represents another scale. Hence Vermeer’s Allegory is a testament of a systematic integration of various scales of abstraction within a single painting. The sources of this achievement lie much more in the tradition of perspectival practice than in the so-called theoretical literature on perspective.

Fig 56: Johannes Vermeer The Allegory of Painting, 1666.

What then were the themes and functions of these theoretical texts? These are questions to which we shall return in the third and fourth chapters of our survey. Here it is important to note, by way of introduction, that the very question of what constitutes a theoretical text on perspective is itself problematic.

Fig. 57. Lines of influence among the chief theoreticians (1390-1500).

Fig. 58. Fifteenth century theoreticians and their works. Excluded from this list are possible works by Fontana, Mantegna, Bramante, Bramiantino, Foppa, Butinone and Zenale.

14. Problems of Definition

We have already mentioned that sixteenth century authors such as Barbaro, Barozzi and Sirigatti saw themselves as authors of books on practical perspective on the assumption that Euclid’s Elements and particularly his Optics, provided them with a theoretical basis. But the problem goes deeper. Alberti is regularly cited as author of the first extant treatise on perspective. Yet the title of that work is On Painting and it contains only a few paragraphs devoted to technical aspects of perspective. Dürer is another case. His work is entitled Instruction in measurement and again contains only a few pages on perspective.

The truth is that we know far too little about early perspectival theory. We know that a number of works have been lost. Giovanni Fontana, may have written the earliest treatise on perspective, although it is likely that he only dealt generally with effects relating to colour and aerial perspective. Paolo Uccello was certainly much concerned with perspectival principles. We have his famous sinopia for the Nativity (Florence, San Martino alla Scala), but we have no clear record of his having written a treatise. The same is true of Mantegna although, in this case, Lomazzo alludes to perspectival drawings he did. Both Lomazzo and Cellini refer to a now lost book by Leonardo da Vinci. Lomazzo also refers to treatises by Bramante, Bramantino and Foppa of which no trace remains.

Fig 59: Vase in Perspective by Paulo Uccello (1430)

Looking at the fifteenth century as a whole we find there were only seven authors whose works are extant (fig. 44). All their fifteenth century manuscripts together amount to 34. Published material was limited to seven pages in Luca Pacioli’s Compendium of arithmetic, geometry, proportions, and proportionality (1494). Only one treatise, Piero della Francesca’s On perspective of painting actually had perspective in its title. All the authors were Italian. There were four main centres: Florence, Venice, Milan and Rome. Connected with these were other cities: Padua, Mantua, Bologna, Pisa, Siena, Urbino, Perugia and Naples. The theoreticians moved with surprising freedom. Alberti worked in Venice, Mantua, Bologna, Florence and Rome. Luca Pacioli sojourned in Venice, Urbino, Milan, Perugia, Florence, Rome and Naples. Leonardo worked in Florence, Milan, the Romagna, Rome and Amboise.

This applied equally to practitioners. Masolino, for instance, worked in Florence, Prato, Rome, Castiglione d’Olona from whence he accompanied Cardinal Branda Castilione to Hungary and worked for King Matthias Corvinus. All this helps to explain the lack of manuscripts. Artists invariably worked together in workshops (botteghe) and generally would have learned their theory by word of mouth from the travelling experts. So, although perspective was technically limited to a dozen theoreticians moving between as many cities, the impact was larger. In retrospect, however, it is necessary to keep reminding ourselves just how small was the scale of the phenomenon if we are to appreciate, for instance, Luca Pacioli’s complaint that after Leonardo left Milan in 1499 he was unable to find anyone who could draw the semi-regular solids for his Divine proportion in proper perspective.

Fig 60: Practice of perspective by Barbaro (1568)

Fig 61:  Geometria et Perspectiva, Lorenz Stoer, (1567)

North of the Alps, technical knowledge of perspective was restricted to rare individuals with Italian contacts such as Jean Fouquet and Petrus Christus. In this context Dürer’s letter to Pirckheimer, in which he wrote that he hoped to learn the secret of perspective, makes sense. Knowledge of the laws of perspective evolved gradually and secretively during the fifteenth century in what was effectively a closed shop. Yet, as we have shown, the topics with which it dealt were part of a much larger phenomenon involving both the construction and representation of basic spatial forms which affected the Low Countries, France and Germany as well as Italy such that we can properly speak of a European phenomenon.

In the sixteenth century, this evolution gathered momentum. Between 1500 and 1600 there were thirty further authors who produced approximately 140 printed texts on perspective. Of these 70% were published North of the Alps. What had begun in Italy, spread to the major cities of Europe. Nevertheless, basic problems of definition remained, partly because perspective continued to be classified under other topics such as painting (Alberti), sculpture (Gauricus), optics (Ringelbergius), geometry (Hirschvogel), measurement (Dürer), and architecture (Serlio) or simply included in encyclopaedic works (Reisch, Ringelbergius, Ryff) rather than as an independent topic of its own.

Fig 62: Leonardo da Vinci Perspective Drawing: 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 vast majority of the images created by Leonardo employed perspective principles; and he often  used the mathematical techniques of linear perspective, aerial perspective, movement and multi-view perspective, plus transparency etc to: A) enhance the illusion of reality for depicted scenes; and B) to accurately record reality, and to enhance his own understanding of structure, change, movement and the form plus dimensionality of happenings. Leonardo’s images were cinematic centuries before cinema existed (multi-views that recorded reality at different viewpoints and times etc).

Jean Pélerin’s On artificial perspective (1505) in Toul was the first published text dedicated specifically to perspective. In Germany, Rodler’s A beautiful, useful booklet (1531) was the first such treatise. In the Low Countries, it was Vredeman de Vries’ Scenography or perspective (1560). In Italy, Barbaro’s Practice of perspective (1568) was the first published treatise to deal specifically with perspective.

The problem of definition became complicated in the mid-sixteenth century with the appearance of texts designed mainly for architects, which were primarily collections of perspectival images, serving as practical model books with no theoretical explanation. The case of Androuet du Cerceau is particularly interesting in this regard. One of his books, Contains optics which they call perspective (1551) alludes directly to perspective in its title. However, many of the engravings in this work are based directly on his Fragments of old structures (1550), in which perspective is not included in the title. Does this mean that Fragments is not a perspectival text whereas Contains optics which they call perspective is? At the risk of offending purists, I have included both.

Fig 63: The Book of Perspective by Jan Vredeman De Vries 1604-05

Fig 64: Garden view as stage scenery.:Ballet de la Princesse Elide (1664) 

Fig 65: Actual stage set in Torelli (1644)

Indeed, I have consciously chosen to err on the side of including too much rather than too little for reasons which should by now be obvious. A search for books on perspective in the narrow technical sense described at the outset would have excluded Alberti, Filarete, Francesco di Giorgio Martini, Leonardo da Vinci, Luca Pacioli and even Albrecht Dürer on the grounds that they dealt primarily with other subjects. In addition it would have excluded Jacopo Bellini, Androuet du Cerceau and Vredeman de Vries on the grounds that they contained only examples and no theory. For this reason another strategy was taken. All titles found in the 35 standard bibliographies thus far (Index I.A.) were included, as were a number of borderline works overlapping with architecture, optics, surveying and roman ruins.

15. Conclusions

This chapter opened with careful modern definitions to serve as tools in analyzing the texts and closes with a plea that they not be used too directly in determining the boundaries of the field. For to do so would be to miss the whole phenomenon of historical development, of the ambiguities that existed before a new term had found its own place in an established system of classification; of the changes in meaning that came as thinkers slowly turned from the practical effects to the theoretical foundations of the method and gradually recognized that the basis thereof lay in geometry and not in optics.



Kim Henry Veltman (1980 - 2017).
Alan Stuart Radley (2023).


Veltman, K.H. (1994) 'The Sources of Perspective' - published as an online book (no images). Later published with images as 'The Encyclopaedia of Perspective' - Volumes 1, 2 - (2020) by Alan Stuart Radley at the Perspective Research Centre.

Veltman, K.H. (1994) 'The Literature of Perspective' - published as an online book (no images). Later published with images as 'The Encyclopaedia of Perspective' - Volumes 3, 4 - (2020) by Alan Stuart Radley at the Perspective Research Centre.

Veltman, K.H. (1980s-2020) 'The Bibliography of Perspective' - began as a card index system in the 1980s; before being transferred to a dBASE-3 database system on an IBM PC (1990s). Later the bibliography was made available on the web on the SUMS system (2002-2020). In 2020 the Bibliography of Perspective was published as part of'The Encyclopaedia of Perspective' - Volumes 6, 7, 8 - by Alan Stuart Radley at the Perspective Research Centre.

Copyright © 2020-23 Alan Stuart Radley.
All rights are reserved.