Dr. Kim H. Veltman
I Definitions and Origins
1. Introduction
2. Pseudo-perspectival Methods
3. Parallel-Perspective
4. Standard Methods
5. Picture Plane
6. Angular, Conic and Cylindrical Planes
7. Spherical Planes and Surfaces
8. Perspective and Anamorphosis
9. Geometry and the Distance Point
10. Optics and the Legitimate Construction
11. Practice and Theory
12. Architecture and Ruins
13. Surveying and Topography
14. Problems of Definition
15. Conclusions
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 may 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.
The evidence of the extant frescoes at
Pompeii (pl. 1.4), 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. 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, pl. 1.3-4). 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 (pl. 1.1) or to make a higher object larger in order that it appear the same size
(pl. 1.2.). These are again pseudo-perspectival methods variously termed negative
perspective, optical adjustments or visual angles methods (fig. 2a). 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. Drer 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 B1C1, D1E1, and F1G1 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 G1F1 is projected the same size
as B1C1.
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. His student, Abraham
Bosse, the first professor of perspective at the Acad‚mie 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.
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
(pl. I.5-6). 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. 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 (cf. below, pp. ).
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.
Multiview projection may involve first angle (also termed first quadrant) projection,
commonly used in Britain (fig. 3.3 cf. fig. 3.2), where one looks out at the projection:
or third angle projection, commonly used in the United States (fig. 3.5), where one looks
in at the projection as if it were in a transparent glass box (fig. 3.7). Axonometric
projection (fig. 4.1) is subdivided into isometric, dimetric and trimetric projection. In
isometric projection all three faces are equally oblique (fig. 4.2). In dimetric, two
faces are equally oblique (fig. 4.3). In trimetric, only one face is equally oblique (fig.
4.4). In addition, there are two other oblique parallel projections known as cavalier
projection, where the object is at 45o relative to the picture plane (fig. 4.5) and
cabinet projection, where the object is at arc tan 2 relative to the picture plane (fig.
4.6), and some include a third, military projection (fig. 4.7).
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 (cf. below, p. ) In addition, perspective is used to describe
practical conventions in Oriental art which appear to be without a theoretical basis.
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. 5).
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 P‚lerin, 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.
One of the distinguishing characteristics of linear perspective is the principle of the window or picture plane (pl. 58. 1-2 cf.pl. 30, 54) 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.
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. 6).
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. 7.1). 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. 7.2). Corresponding cylindrical shapes both convex and concave were also explored by the author of the Codex Huygens, Marolois, Nic‚ron and Schott (fig. 7.5-6, cf. pl. 67). 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. 7.3-4, 7-8).
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. 8.1, cf. pl. 52.2). The theory of spherical perspective (fig. 8.2), 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.
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. 8.3-6).
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 (cf. pl. 75.3 and 49.3). 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. 8.3-4). 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. 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. 9.1). 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. 9.2). Subsequent thinkers chose positions slightly further removed: Clarke at 1.35 radii from the centre of the sphere (fig. 9.3); James at 1.367 radii (fig. 9.4) and La Hire at 1.71 radii (fig. 9.5). Yet another alternative was to place the point of projection at infinity which resulted in an orthographic, parallel or (Juan de) Rojas projection (fig. 9.6), named after a Spanish contemporary of Gemma Frisius and Mercator.
Not all projections were rectilinear. Mercator projected the sphere onto a cylinder to arrive at his now famous grid system (fig. 11.1-2). Already in the second century Ptolemy had explored another possibility: projecting the sphere onto the inside of a cone (fig. 10.1-2) as well as a modified version thereof (fig. 10.3-4). 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 (pl. 27.3), was clearly a source for Drer's perspectival drawings (pl. 27.4). In the case of another of Drer's globes (pl. 27.1) 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.12).
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 (pl. 24.5). The same Danti also studied sundials. This combination of interests in perspective and dialling was subsequently pursued by Desargues (1636) and Maignan (1648) (cf. pl. 48-49).
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. 6) 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. 13.1) or, in extreme cases, as perspectival distortions (fig. 13.2). 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. 8.3), the sphere in various cartographic projections (fig. 9) 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.
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. 13.3-4), a flat projection plane at right angles to an original cylindrical (fig. 13.5-6), conic (fig. 13.7) or pyramidal plane (fig. 13.8). 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-Fran‡ois Nic‚ron 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. 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 (pl. 28.2). Later examples by Serlio (pl. 28.3), Barbaro (pl. 28.4), P‚lerin (pl. 29.1-2), Androuet du Cerceau (pl. 29.3) or even Galli-Bibiena (pl. 29.4) are 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. 14) 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. 14 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. 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 P‚lerin 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. 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 problems 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.
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 (pl. 2.3) is a case in point. Duccio had convincingly rendered a frontal view of this form in his Maest… (pl. 2.1-2). 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 (pl. 3.2) provides another example. This shape had been mastered by Giotto in the Scrovegni chapel in Padua at the beginning of the fourteenth century (pl. 3.1).
The same holds true for diagrams in other treatises. The barrel vaults in Barozzi's The two rules (pl. 3.4) also have a precedent in Giotto, this time in his Sanctioning of the Rule in the Upper Church at Assisi (pl. 3.3). Even a much later example such as the oblique building in Vaulezard's Abridgement (1631, pl. 3.6) had been used in a simpler form in P‚lerin's treatise and earlier still in painting practice in Christ and the Apostles in the Temple attributed to Andrea di Giusto (pl. 3.5).
Examination of Duccio's Maest… (pl. 2.1) 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. 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. Masaccio used it in his Trinity (pl. 5.1), 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 (pl. 11.5); less famous such as, Foppa and even obscure artists such as the Ferrarese Master (pl. 6.3). It is used in drawings by Bellini (pl. 6.1-2) and a preparatory drawing by Donatello (pl. 6.4). It is used equally in other media: in a marble alter by Desiderio da Settignano (pl. 5.2); in a stone facade by Pietro Lombardo in Venice (pl. 5.3), by Alberti in the facade of Sant'Andrea in Mantua and by Bramante in his famous illusionistic choir (pl. 5.4). Borromini's use of a variant nearly two centuries later in the Palazzo Spada (pl. 5.5) attests to the enduring fascination of this illusionistic form.
A cumulative process marks a next stage in development. Hence, Piero della Francesca, having mastered the dome shape (pl. 3.2) 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 (pl. 7.1). Artists at all levels were involved in this process. In his doors for the Baptistery at Florence, Ghiberti had represented a cross vault (pl. 4.4). This form also appeared in the treatises of Piero della Francesca (pl. 7.3) and Sebastiano Serlio (pl. 7.4). 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 (pl. 7.2).
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 (pl. 4.5).
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 (pl. 4.3). It is noteworthy that trompe l'oeil versions of such portals also date back to the twelfth century (pl. 4.2).
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 (pl. 8.1), Rogier van der Weyden's Christ Appearing to his Mother (pl. 8.3) and his Beheading of St. John (pl. 8.4). Another of Van der Weyden's paintings, The Eucharist with Christ on the Cross (pl. 8.2) 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 (9.3-4). 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 (pl. 9.5) subsequently adopted for his own purposes, as did Veronese (pl. 9.6) 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 (pl. 9.2), which in turn bears comparison with an idealized ruin from Androuet du Cerceau (pl. 9.1) 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 (pl. 4.1), and the Gothic cathedral of Notre Dame (pl. 4.3) to Brunelleschi's facade to the Pazzi Chapel (pl. 9.2) and Benedetto da Maiano's Santa Maria delle Grazie in Arezzo (pl. 15.1).
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 (pl. 8.1). Fouquet adapted it slightly in his Hours of Etienne Chevalier (pl. 10.1). The Master of the Burgundy Hours used it more dramatically in a miniature now in Vienna (pl. 10.2). In 1505 Jean P‚lerin 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 (pl. 11.4) and Raphael's School of Athens (pl. 11.5) and, as the sixteenth century progressed, connections with Roman ruins also became more apparent through engravings such as those of Androuet Du Cerceau (pl. 11.1) and Cock (pl. 11.3). 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).
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 (pl. 12, 7.1-4). 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 (pl. 13). 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 (pl. 11.2-4) 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 (pl. 8.2) gained in importance in the sixteenth century. Michael Pacher used it in his St. Wolfgang Altar (pl. 16.1), as did Albrecht Altdorfer in his rendering of the Jewish synagogue at Regensburg (pl. 16.2) and other church interiors. The method was used in the Luther Bible (pl. 16.3), and Rodler also employed it several times in his treatise on perspective (pl. 16.4). The same method was frequently used in combination with another basic perspectival form, the colonnade as in both Cesariano's Vitruvian commentary (pl. 80.2, 82.1 cf. pl. 80-83), and his marquetry work in St. Alessandro. Later sixteenth century examples include a drawing by Scamozzi (pl. 14.1) or, to take northern analogues, the engravings of Vredeman de Vries (pl. 14.2-3) and Cornelius Loos (pl. 14.4).
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 (p.l5.2). 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 (pl. 15.3) as well as northern
parallels, such as those in the treatises of Vredeman de Vries (pl. 15.4-5).
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 (pl. 17.1), as did Steenwyck in his version
of a Gothic Church (pl. 17.2) and Pieter Neefs in his painting of Onze Lieve Vrouwe
Kathedraal in Antwerp (pl. 20.1) which invites comparison with actual photographs of the
same building (pl. 20.2).
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 (pl. 20.3-6). The same process
occured in the representation of exteriors (pl. 21.1-3). 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 (pl. 21.1) 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 (pl. 22.1-2, pl. 23.1) and others looking in the opposite direction (e.g. pl. 23.2).
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, Drer, 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 P‚lerin (pl. 10.3-4), Androuet du Cerceau (pl. 11.3) or Vredeman de Vries (pl. 14.2-3, 15.4-5), 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 (see below 2.4). 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 (see below p. ). 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 playeda role in this reinterpretation of Vitruvius, which was continued by Cesariano (1521), Caporali (1536) and Ryff (1547). 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 (pl. 96.6). 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 P‚relle, 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. In 1575, Etienne Du P‚rac, 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 P‚rac subsequently went to France (1587) and introduced there the idea of perspectival gardens which had been developed in Tuscany through Buontalenti and others (cf. below 2.4) and which led ultimately to Le N“tre's work at Versailles (pl. 93.1).
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 (cf. below 2.4), 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. 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.
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 (see below 2.2).
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). 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 (see below 2.2) 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.
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.
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 (pl. 10.1) and other rooms (pl. 53.3). Fouquet's contemporary, Konrad Witz, also produced both perspectival interiors (pl. 53.4) and exteriors with fully realistic landscapes as, for instance, in Christ walking on water (Geneva, Mus‚e de l'art), where he depicted the westernmost shores of Lake Geneva with the peaks of the M“le and Mont Blanc in the background.
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 (pl. 13.2) 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 (pl. 26.1) in the fifteenth and sixteenth 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 (pl. 26.2). 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 seventeenth 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. 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 Drer who wrote on perspective and did interiors of rooms showing Saint Jerome, also did townscapes (pl. 60.1-2), views of earth from a nearby viewpoint (pl. 27.1), a viewpoint further away (pl. 27.4) and even maps of the stars (pl. 27.6). Similarly, the same Egnazio Danti who wrote the commentary to Giacomo Barozzi, il Vignola's Two rules of Biagio Pelacani da Parma Giovanni Fontana Leon Battista Alberti Jacopo Bellini Filippo Brunelleschi Paolo Uccello Domenico Veneziano Filarete Piero della Francesca Luca Pacioli Francesco di Giorgio Martini Leonardo da Vinci Fig. 14. Lines of influence among the chief theoreticians (1390-1500). Practical perspective (1583), was cosmographer to the Medici, produced the systematic series of maps in the hall of the globes in the Palazzo Vecchio and also produced star maps in the form of astrolabes (cf.pl. 27.5).
In the course of the sixteenth and
seventeenth centuries it became possible not only to compare different views of a scene in
one scale (pl. 22-23), but also different views of a scene in different scales, as for
instance, the bridge in Zurich (pl. 26.1-2), or the orphanage in Amsterdam (pl. 24.3-4).
For the latter of these there exist also detailed pictures of the gate to the inner
courtyard (pl. 24.2) and front gables of the house to the left in front of it (pl. 24.1).
Atlases of the time by Mercator (pl. 25.1-4) and Ortelius invite a similar comparison of a
given land in different scales. Theoretician Name of Treatise Manuscripts Written
(Published) 1. L.B. Alberti On painting 9 1434 (1540) Elements of painting 6 c.1436 (1890)
2. Jacopo Bellini Louvre Sketchbook 1 c.1430-1450 (1910-1912) London Sketchbook 1
c.1430-1450 (1910-1912) 3. Filarete Treatise on architecture 8 c.1460-1470 (1880) 4. Piero
della Francesca On painters' perspective 6 c.1480 (1880) Book of abacus 1 c.1480 (1972)
Book of 5 regular solids 1 c.1480 (1916) 5. F. di Giorgio Martini Practice of geometry 6
c.1480-1490 (1841) 6. Leonardo da Vinci Codex Atlantico 1 c.1480-1519 (1894-1904)
Manuscript A 1 c.1492 (1881) 7. Luca Pacioli Compendium of arithmetic 0 c.1493 (1494)
Divine proportion 2 c.1496-1499 (1508)
Fig. 15. Fifteenth century theoreticians and their works. Excluded from this list are possible works by Fontana, Mantegna, Bramante, Bramiantino,Foppa, Butinone and Zenale.
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 (pl. 25.5) 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.
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.
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. Drer 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 (see below p. ), 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, Lomazzo35 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.
Looking at the fifteenth century as a whole we find there were only seven authors whose works are extant (fig. 15). 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.
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 Drer'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 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 (Drer), and architecture (Serlio) or simply included in encyclopaedic works (Reisch, Ringelbergius, Ryff) rather than as an independent topic of its own (see below 1.4)
Jean P‚lerin'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 (pl. 86.3-4). 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.
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 Drer 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
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