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In the context of pictorial and scenic art the term ‘perspective’ may refer to any graphic method, geometrical or otherwise, that is concerned with conveying an impression of spatial extension into depth, whether on a flat surface or with form shallower than that represented (as in relief sculpture and theatre scenery). (See also aerial perspective.) Perspective representation or composition results when the artist adopts a visual approach to drawing and consequently portrays perspective phenomena such as the diminution in size of objects at a distance and the convergence of parallel lines in recession from the eye. Western painting started to develop along optical lines first in Greece and received a geometrical bias from its early association with the optics of classical Antiquity. The illusion of the dimension of depth has bee n its distinguishing characteristic and to this end geometrical perspective one of its principal aids. The idea of the single static focus of Renaissance perspective and the characteristic system of central convergence (see below) owed much to the recurrence of themes demanding the representation of interiors. The only other highly developed pictorial tradition in which spatial values are paramount, i.e. the Chinese, evolved on the other hand principally from landscape and took for granted a travelling eye. Hence Chinese artists with rare exceptions adopted the convention of parallelism when they represented buildings, a system which, although denying the optical principle of the convergence of receding parallels, yet had the virtue of allowing the eye to glide easily from scene to scene.

2. The meaning of the word

The word ‘perspective’ derives from the Latin (ars) perspectiva, a term adopted by the Roman philosopher Boethius (d. AD 524) when translating Aristotle to render the Greek optiki (optics). In the 15th century ‘perspective’ came to mean seeing through a transparent plane on which the scene is traced from a single fixed eye-point. It then became in Latin perspectiva artificialis or perspectiva pingendi to distinguish it from the older science perspectiva naturalis or communis.

3. The scientific basis of the perspective image

Scientific perspective, known variously as central projection, central perspective, or picture plane or Renaissance or linear or geometrical perspective, may be regarded as the scientific norm of pictorial representation. The story of its development belongs as much to the history of geometry as to that of painting. It is the perspective of the pin-hole camera and (with certain reservations as regards lens dist ortion) of the camera obscura and the photographic camera. It derives from geometrical optics and shares with that science its basis in physics: the rectilinear propagation of light rays. In central perspective the picture surface is regarded as a transparent vertical screen (the picture plane), placed between the artist and his subject, on which he traces the outlines as they appear from a single fixed viewpoint. Ideally such a picture, suitably coloured and seen with one eye from the correct point, should evoke the illusion of the real scene viewed through a window. ‘Perspective’, wrote Leonardo, ‘is nothing else than seeing a place (sito) behind a pane of glass, quite transparent, on the surface of which the objects which lie behind the glass are to be drawn. They can be traced in pyramids to the point in the eye and these pyramids are intersected by the glass plane.’ The ‘pyramids’ are composed of the light rays which link the visible surfaces of objects to the eye by straight lines called visual rays. Their points of intersection with the transparent plane (the picture plane) form the perspective image.

4. The psycho-physiological basis of perspective illusion

The eyeball is a little dioptric camera and receives projected images of the outside world upon its interior surface. This is a light-sensitive membrane called the retina. Images are formed by the projective action of light. When light passes through a small hole the rays behave like straight lines intersecting in a point. The images of the pin-hole camera, the camera obscura, the photographic camera, and the eye all depend on this phenomenon. The image-screen intercepts the rays. No point of it receives light from more than a single point of the object. When the hole is enlarged to admit more light and give brighter images, a divergent beam of light from each object point enters the camera. It is the function of the lens to correct this divergence and to concentrate the beam onto the screen or retina.We may assume for our purpose that each object point is linked to its representation on the retina by one straight line, called a direction line or line of sight. Although in normal circumstances the eye moves continuously in its orbit, when a number of object points are involved all the direction lines may for practical purposes be considered to cross at a nodal point within the eye, very near the centre of rotation. This point is the ‘eye’ or centre of projection of perspective.

When a luminous point A moves along a direction line, its perspective on the picture plane (A’) and its retinal image (a) all remain in coincidence provided the head is kept still, even though the retinal image continuously moves over the surface of the retina. Thus the point A’ on the picture plane is the perspective of every point in the same direction line AA, and all points on AA give equal stimuli and hence give rise to identical perceptions. This is why the eye can be deceived and is the basis of perspective illusion. There is ambiguity inherent in all images. In order that a three-dimensional object be correctly represented to the beholder he must know what the object is.

5. Clues for the perception of depth

The retinal image is two-dimensional and can convey to the brain only directional messages about the location of objects in space. The points of stimulation can tell us nothing directly about distance or absolute size. But in binocular vision the fusion of two dissimilar images presents to consciousness a three-dimensional pattern that gives us very accurate judgements of distance quite impossible of achievement with one eye. Stereoscop ic perception, however, ceases at a certain distance from the eyes. In monocular vision, and in binocular vision at distances in excess of about 15 m (50 ft), the perception of depth depends upon indirect factors. These are clues for the third dimension, which may be grouped into nine classes. Five of them are used by painters to create an impression of depth by their flat pictures: overlapping contours; linear perspective; aerial perspective; distributions of light and shade (the direction of light being known); interpretation of size (the judgements of absolute size and distance are interrelated). Four other clues work against the impression of depth in pictures: parallactic movements; muscular efforts of accommodation and of convergence; stereoscopic influence of dissimilar images.

6. Some facts and definitions for the theory of central perspective, given for the vertical position of the picture plane

In Fig. 4:E is the eye: the centre of perspective or projection.

PP (i.e. the plane AZRT) is the picture plane. It is of indefinite extent (as are the other principal planes), and in its usual position is perpendicular to the ground plane.

GP is the horizontal ground plane. The ‘height’ and ‘distance’ of the eye are measured from these two planes.

GL, the ground line, is the intersection line in which the picture plane and the ground plane meet.

HL, the horizon line, is the line in which the horizontal plane containing the eye meets the picture plane. It is the vanishing line of the ground plane and of all other horizontal planes and is the locus of the vanishing points of all horizontal lines.

C is the central or principal vanishing point: the point of con vergence of the perspectives of all orthogonals, i.e. horizontal parallel straight lines of indefinite extent that meet the picture plane at right angles, such as L, Q, M, N. (Orthogonals are parallels that are objectively perpendicular to the picture plane. Their perspectives converge to a point exactly opposite the eye of the artist.)

S is the station point.

EC is the central ray from the eye drawn perpendicular to the picture plane and therefore parallel to the orthogonals.

C and D are examples of vanishing points, of which there can be an indefinite number. They are found by drawing a line from the eye parallel to the given line or system of parallels. Thus EC and ED are drawn parallel to AQ and AB respectively.

In order to draw the perspective of a given line of indefinite extent, it is sufficient to know its vanishing point, such as C or D, and the point in which it meets the picture plane, such as A, Y, Z. These points are called the intersection points or near ends of the given lines. In perspective each group of parallels is drawn to a single, particular vanishing point.

DD are distance points. They are the vanishing points of horizontal straight lines that make angles of 45° with the picture plane, such as the line AB, whose perspective is contained in AD, and are the same distance from C as the eye: hence their name. They are traditionally used for measuring in perspective along orthogonals and may be regarded as the earliest measuring points.

7. The perspective constructions of Brunelleschi and Alberti

Brunelleschi is generally acknowledged to have been the originator of the first construction of scientific perspective. Of his actual construction, which must date from about 1420, there is no detailed contemporary record. From Vasari’s life of Brunelleschi we learn only that he invented an ingenious system that included a plan and elevation and an intersection and that by the aid of this he painted the two famous panels (now lost) representing the Florentine baptistery of S. Giovanni and the Signoria. This was presumably similar to the construction given in Fig. 5. It is a mechanical system based exclusively on the principle that visual rays are straight lines. These are represented in plan and elevation by the lines joining selected points of the object to the eye. The points of intersection which form the image of the visual rays with the picture plane are first found on the plan and the side elevation and from there are transferred to some part of the paper representing the picture surface by the method of rectangular co-ordinates. The mechanical principle of Brunelleschi’s construction is exemplified in Dürer’s woodcut Draughtsmen Drawing a Lute (1525). The ring in the wall is the ‘eye’ and the mobile thread serves for the visual rays. While one artist directs the ray to the selected points on the lute, his companion notes the points in which the ray passes through the frame, i.e. the picture plane, and transfers them one by one to the drawing board. The plan and elevation construction with visual rays was particularly suitable for drawing individual architectural or geometrical forms of some complexity and was used for this purpose notably by Uccello and Piero della Francesca. The latter applied it also to the drawing of heads. It did not include the use of vanishing points.5
Fig. 5

Shortly after Brunelleschi made his perspective demonstrations his fellow architect Alberti devised a perspective construction for the special use of painters, which he described in detail in his famous treatise Della pittura (1436), written first in Latin (1435) and then immediately in Italian for his friend Brunellesc hi and the avant-garde artists of Florence (Fig. 6). This is the first known written account of a fully scientific perspective construction. He chose for demonstration a simple squared floor or chequerboard pavement that became in perspective the ground plane of the picture and could be extended, he wrote, ‘almost to infinity’. Furthermore, it could easily be elaborated into a three-dimensional grid for measuring heights in perspective. Its primary aim was to enable the history painter to locate and measure in perspective all the figures and objects in his picture, which would then appear to the spectator placed at the predetermined viewpoint as a real scene viewed through an open window: for ‘he who looks at a picture done as I have described’, wrote Alberti, ‘will see a certain cross-section of the visual pyramid’. Alberti’s method included the use of a vanishing point for the orthogonals, his ‘centric point’ (C). This enabled him to dispense with a ground plan. The artist could se t to work straightaway on his panel, mark in the squares along the base line, and join up the orthogonals to the centric point, placed centrally at the height of a man represented in the picture standing on the base line. For regulating the measurements in depth, i.e. for spacing his transverse parallels, he drew, like Brunelleschi, a separate side elevation with visual rays and vertical intersection. Finally he proved his perspective to be correct by drawing a diagonal across the foreshortened grid to show that the diagonals of the individual squares formed into continuous straight lines (Fig. 6), just as in a real chequerboard. For if an artist drew the foreshortening by some arbitrary method, such as that of making each successive space between transversals a constant fraction of the preceding one (Fig. 7), the diagonals would not lie in straight lines. ‘Those who would do thus would err,’ wrote Alberti. Perhaps this is why Alberti’s system is usually known as the costruzione legittima, a term of modern Italian origin. Some authors, however, apply this term also to Brunelleschi’s construction. Alberti’s construction is most commonly found in the abbreviated form in which the two diagrams are combined into one (Fig. 8). (For an example of curved diagonals see the tiled floor in Giovanni di Paolo’s painting The Presentation (c.1448; Siena, Pin.).)

Fig. 6

Fig. 7

Fig. 8

8. Principal types of perspective, scientific and otherwise

 (a) Parallel perspectiveThe term may refer to parallelism, i.e. the representation of parallels as parallel (Fig. 9), which is a non-scientific form and when found in early painting may be regarded as an ideoplastic element. In combination with the high viewpoint it is a traditional convention of Chinese painting. It is also a convention of modern stereometry. The term may also refer to the case in scientific perspective when the picture plane is parallel to a principal surface of the object (Fig. 10), as is commonly found in Renaissance pictures. Then none of the parallels of the frontal surface converge, but if the object is rectangular, for example a house or room, the parallels in depth become orthogonals and converge to a central vanishing point.

Fig. 9

Fig. 10

 (b) Angular or oblique perspective

A term of scientific perspective used when a rectangular form is represented at an angle to the plane of the picture such that its horizontal parallels recede into depth to the left and the right, thus requiring two vanishing points, but its verticals remain parallel to the picture plane (Fig. 11).

Fig. 11

 (c) Three-point or inclined picture plane perspective

A term of scientific perspective used with regard to a rectangular form placed so that none of its sides is parallel to the picture plane and the three groups of parallels vanish each to its respective vanishing point (Fig. 12). Three-point perspective may also refer to the distance point construction.

Fig. 12

 (d) Axial perspective, also called vanishing or vertical axis perspective

This is the earliest systematic form of punctual convergence and is much in e vidence before the invention of scientific perspective. In its original form it is simply the symmetrical convergence of parallels to points on a central vertical axis taken mirror-like right and left in pairs, and as such is a form of parallelism (Fig. 13a). Early examples are to be seen in the drawing of the parallel beams of ceilings. They are first found on Apulian Greek vases (See Greek art, ancient) of the 4th century BC. There are variations of this construction which persist through Antiquity and the Middle Ages and even survive the Renaissance. In many cases the central pair of beams meets unrealistically on the posterior border of the ceiling in a V, and sometimes artists disguised this with a cartouche or a nimbus. Often the central meeting point was placed out of sight and some degree of convergence was given to the outside parallels (Fig. 13b). But always the spacing of the central section was unrealistic. The same principle was applied to th e drawing of walls and floors, and led in this case to an inconvenient encroachment upon one another of neighbouring planes.

Fig. 13a

Fig. 13b

Why was this method universally preferred to the simpler device of the single vanishing point? Indeed, central convergence is virtually never found before the 15th century except in one brief period of Pompeian painting and then only in the upper parts of the pictures. The answer to this question may be found in the way in which we actually perceive parallels. Recent experiments have proved that we do not normally perceive more than two objective parallels as though they are directed towards a common point. The more a pair of parallels is situated towards the left or right the less the degree of apparent convergence will be, a fact that ag rees with modifications of the axial construction as found in some pictures. This principle was presumably known to artists in a general way through direct observation, and it deterred them from using one single convergence point for the whole group of parallels until central convergence was demonstrated with the aid of a projection plane and a fixed eyepoint and shown to be ‘scientific’. There may also have been a theoretical justification for the axial construction in the Euclidean theory of visual angles (see below). If the visual rays are projected onto subtended arcs and dimensions transferred from the arcs or their chords to the picture plane, the resulting perspective of a centrally viewed rectangular interior approximates to a vertical axis construction (Fig. 14).

Fig. 14

 (e) Inverted perspective

For the representation of rectangular foreground objects ‘inverted perspective’ is the common rule in pre-Renaissance painting (Fig. 15). Although it contradicts scientific perspective and seems wrong to modern eyes, there is a basis for it in experience. The fact that we do not easily see convergence in foreground objects but rather parallelism or even divergence of parallels can easily be verified by observation. But experience may have been reinforced by a naive interpretation of the visual ray theory. The divergent construction is abundantly exemplified in an Ottonian early 11th-century Last Judgement (Fig. 16). (Munich, Staatsbib.). The universal acceptance of central convergence belongs essentially to the age when science acquired overriding authority.

Fig. 15

Fig. 16. Last Judgement. Tracing of an Ottonian miniature from the Book of Pericopes of Henry II (Bayerische Staatsbibliothek, Munich, early 11th century). Dotted perspective lines added

 (f) Negative perspective

A term used to describe the application of lines of sight to the adjustment of proportions in large-scale decorations, paintings, or statuary in order to counteract the perspective effect of the more remote parts. In Dürer’s example (Fig. 17) all the letters will appear equal to an observer placed at the viewpoint because they have been designed to subtend equal angles at the eye. Plato refers to such a procedure employed by Greek artists of his time (Sophistes 236).

Fig. 17. Illustration from Underweyssung der Messung (1525), Albrecht Dürer

 (g) Bifocal perspective

This is a term for an empirical construction that uses two vanishing points placed symmetrically on the margins of the pic ture for the purpose of drawing a diagonal floor-grid (Fig. 18). It is a system without a central vanishing point and is thought to be a workshop tradition associated with the Giottesque perspective of the trecento. It is the construction given by Pomponius Gauricus in his De sculptura (printed in Florence, 1504). A famous example may be seen in the sinopia (in Frescoes from Florence, exhib. cat. 1969, London, Hayward) of Uccello’s Nativity fresco at Florence (S. Martino della Scala). It may also refer to the examples of angular perspective given by Viator (Jean Pélerin) in his De artificiali perspectiva (1505), the first treatise on perspective printed in Europe and called by him perspectiva cornuta or diffusa.

Fig. 18
Fig. 18

9. Measuring in perspective

There can be no measuring in perspective in the strict sense of the term (i.e. the establishment of a two-way metrical relationship between a given object and its representation on the picture plane) unless the relative positions of the eye, the picture plane, and the object to be represented are given. In Fig. 19 the problem is set out in two dimensions. E is the eye, AB the object, EB and EA are visual rays, and Ab is the perspective measurement of AB. It is clear that this situation could not exist before the invention of the picture plane (by Brunelleschi, as far as is known). Examples have been found, however, both in Pompeian painting and in the work of Giotto, in which the representation of a coffered ceiling appears to have been correctly foreshortened (not the same as measured in the strict sense). The artist may have arrived at his result from a drawing in plan by means of a simple intuitive development. This could proceed in four steps (similar to Fig. 20) onc e he had decided to converge his orthogonals to a central point. Such a drawing might appear to be the result of a distance point construction, since the protracted diagonal would meet the horizon line in a point coincident with a distance point, given a certain position for the eye. But there is no evidence to justify the conclusion that the perspective of either Giotto or the Pompeian artists was other than intuitive.Fig. 19
Fig. 19

Fig. 20

There are basically two methods of measuring in perspective, (i) by the visual ray and section system of Brunelleschi and Alberti; and (ii) by the modern method of measuring points, a simple geometrical device of great utility which first appears in print in La Perspective spéculative et pratique du Sieur Aléaume, edited by Étienne Migon (1643). Both Migon and Aléaume were professional mathematicians. Distance points are the measuring points for orthogonals and are a particular case of the general rule as described in all modern handbooks.

10. Marginal distortions

In geometrical optics the apparent sizes of objects are shown to be proportional to the visual angles they subtend at the eye. Because of this objects appear to decrease in size in all directions as they recede from the eye. An angle is measured by its subtending arc, but perspective produces flat projections on a plane. This inevitably leads to discrepancies between angular size and projected size. But when a flat projection is viewed from the projection centre the angular dimensions are restored by the natural foreshortening of the picture plane. This is the principle of anamorphosis. Two well-known paradoxes arise from this situation. Planes parallel to the picture plane have unforeshortened perspectives however far they recede from the eye. Suppose A and B (Fig. 21) to be a plan or elevation of two equal windows in a wall parallel to the picture plane, with the eye at E. B is seen under a smaller angle than A, yet the perspective projection B is equal to that of A, because it forms similar triangles. This is why the façade of a building drawn frontally in perspective is a true elevation. The second paradox concerns solids. A and B (Fig. 22) are two solids seen from E. B produces the greater intersection yet it is seen under a smaller angle than A. Such discrepancies are particularly noticeable in the case of objects of known regularity: par excellence the sphere, which in perspective retains its familiar circular outline only when its centre coincides with the central vanishing point. If A and B were two spheres the perspective of B would be elliptical in outline. Some painters have been in doubt whether they should draw in conformity with the pla ne projections or with the visual angles. Figure 23 is from a 17th-century French manual which insists that in order to avoid a double foreshortening of the image the former should be the case (A. Bosse, Traité des pratiques géométrales et perspectives, 1665). In Raphael’s School of Athens, a famous example, the architecture is drawn from a central viewpoint while the figures have in each case been drawn from a viewpoint in front of them, in fact possibly from life studies. This shifting of viewpoint is particularly observable in the case of the sphere held by the astronomer Ptolemy which, although eccentrically placed, shows no sign of marginal distortion. This method is a practical solution to the perspective problem of large-scale wall paintings.21
Fig. 21

Fig. 22

Fig.23. Illustration from Traité des pratiques géometrales et perspectives (1665), A. Bosse

In order to keep discrepancies within tolerable limits it has always been the practice amongst artists to work within a narrow angle of vision whenever possible: ideally between 30° and 60° or so. Most paintings are designed to be viewed from a distance of not less than their major dimension, giving a viewing angle of about 60°. When the angle of vision is small every visual ray will be nearly perpendicular to the picture plane. Then the perspective projections will most nearly correspond to the subtending arcs and marginal distortions be minimized.

11. The theory and the rule of vanishing points and Desargues’s Theorem

The perspective of a given line is normally considered to be contained between its inte rsection point or ‘near end’ (the point which is its own image) and its vanishing point. The validity of these two points is established respectively in two theorems of theoretical solid geometry: the Vanishing Point Theorem and Desargues’s Theorem. These form the basis of perspective geometry.The theory of vanishing points may be explained in simple terms as follows: In Fig. 24 the eye is looking through a sheet of glass at two parallels. The upper parallel produced through the glass passes through the eye and therefore appears as a point at V on the glass. A plane containing our two parallels will obviously intersect the glass in the line VA and if we take a series of points B, C, D, etc., on the lower line, these will appear as a series of points b, c, d, etc., where the lines joining B, C, D, etc., to the eye meet AV. As the points on the lower line recede from A they will appear to approach V on the glass and if we conceive of a point on our line at a very great dist ance from A it will appear on the glass so inconceivably close to V as to be coincident with it, so that V is called the vanishing point of that line. And similarly for any other line parallel to A and V.

Fig. 24

From this we have the useful rule: the vanishing point of a given line is that point where the line from the eye drawn parallel to the given line meets the picture plane. ‘Vanishing point’ is a modern term and a misleading one, for it is not the point on the picture plane that vanishes but the point that it represents. In general usage the term has varied meanings. It may refer to a mere operational point of convergence in a picture. Furthermore there is no separate term for the ‘natural’ vanishing point—that imaginary point in the distance towards which receding parallels may appear to converge, or that ‘point of diminution’ (Leonardo’s term) to which remote objects tend to diminish.

Brook Taylor, fellow of the Royal Society and famed amongst mathematicians for Taylor’s Theorem, coined the English term ‘vanishing point’, which first appears in his Linear Perspective (1715). This is the first perspective handbook in the English language to be entirely founded on the principles of projective geometry. During the 18th century some excellent manuals were written based on ‘Dr Brook Taylor’s Method’.

Before 1600 there was no general theory and no general term for vanishing points. The points of convergence used by the Renaissance pioneers of perspective were operational points. They were without a theoretical proof and were used as postulates. Arguments in support of their validity were based on the direct experience of the eyes or on the visual ray theory: as an object recedes it subtends an ever smaller angle at the eye. For ‘vision makes a triangle’, wrote Alberti, and ‘from this it is clear that a very distant quantity seems to be no larger than a point’. The results of the constructions using vanishing points were proved to be correct by empirical demonstration, for example by using the drawing frame or geometrically by an elaborate application of the theory of similar triangles.

The 15th century named only the centre point, to which the 16th century added distance points and particular or accidental points. The true antecedent of the modern vanishing point is the punctum concursus (point of concurrence) of Marquis Guidobaldo del Monte of Pesaro. This distinguished man of science, pupil of Federigo Commandino, the famous translator of Euclid, was first to formulate the general rule (Perspectivae libri sex, 1600). To Guidobaldo goes the distinction of being first to see that the line through the eye parallel to a given line is in the same plane with the given line and is the common line of intersection of all the planes containing the eye and th e given parallels. But his exposition was repetitive and diffuse. It was the French mathematician and engineer Girard Desargues of Lyon, friend of Descartes, who first stated this important theory in clear, axiomatic terms. It was first printed in his Méthode universelle (1636). At the end of this pamphet (which describes an ingenious but mathematically unimportant perspective construction), Desargues appended in the manner of an afterthought a few paragraphs addressed not to the artists but to ‘les contemplatifs’ in which the vanishing point theory is expressed in terms that make a break with the past and herald the projective geometry of the modern age. The following extract deals with the case in which the given parallels are not parallel to the picture plane: ‘When the given lines are parallel and the line from the eye is not parallel to the picture plane, the perspectives of the given lines all tend towards the point in which the line from the eye meets the picture plane, inasmuch as each of these lines is in a same plane with the line from the eye in which all these planes intersect as in their common axis and all these planes are cut by another plane, the picture plane’ (Fig. 25).

Fig. 25

Shortly after this Desargues enunciated the theorem that bears his name. And again, in 1639, he broke new ground when he defined a group of parallels as a pencil of lines whose vertex is at infinity. Desargues’s Theorem states that when plane figures are in perspective corresponding lines are either parallel or meet in a straight line (Traité de la section perspective, 1636). Corresponding lines are pairs of lines of which the one is the perspective of the other. The points in which they meet are called their intersection points and the line in which they meet is the intersection line of the picture plane and the object plane, i.e. the plane contai ning the plane figure (e.g. the ground plane) (see Fig. 26).

Fig. 26

Desargues’s Theorem and its converse are of the first importance to mathematicians by reason of their complete generality. But in the limited field of pictorial perspective its conclusions tend to be self-evident and the perspective practitioner can afford to ignore it. It is, of course, foreshadowed in the type of construction that shows the ground plane ‘hinged` to the picture plane along the ground line: for example in Guidobaldo’s construction (see below), although in fact Desargues’s Theorem had not yet been formulated. Guidobaldo’s book contains the earliest constructions to be effectively based on these two theorems.

Fig. 27

Fig. 27 is Guidobaldo’s First Method and shows how a given trian gle may be drawn in perspective by means of vanishing points and intersection points. The ground plane, containing the plan of the given triangle, has been rotated into the plane of the picture as though hinged on the ground line. The eye, and the parallels drawn from the eye that determine the vanishing points, are also represented in plan. The vanishing points in plan are transferred to the picture plane by elevating them to eye level. Lines are then drawn from the intersection points to the respective vanishing points. The intersections of these lines form the perspective (shaded area) of the original triangle.

Both the theory of vanishing points and Desargues’s Theorem are printed in A. Bosse’s Manière universelle de M. Desargues pour pratiquer la perspective (1648). This book does not, however, make use of the general vanishing point construction in its practical examples. The theoretical advances in perspective did not reach the popular, vernacular handbooks until the 18th century, the age in which the classical ideal of beauty based on a mathematical order was already in the process of being challenged and the authority of perspective undermined. The quasi-empirical methods of the 15th and 16th centuries (summed up in Vignola’s Le due regole della prospettiva pratica, 1583), lacking, as they did, the general rule for vanishing points, were entirely satisfactory for the needs of painters. And it is very doubtful if the Baroque owed anything at all to Desargues and Guidobaldo. For example, the great trompe-l’œil ceiling of Andrea Pozzo in the church of S. Ignazio at Rome was designed by the aid of 15th-century methods clearly described by the artist in hisPerspectiva pictorum et architectorum (1693).

12. The distance point or three-point construction

This is historically the most important construction after Alberti’s costruzione legittima and is by far the best known. It is an interesting case of practice preceding theory and provides an easy way of measuring in depth along the orthogonals and of drawing a chequerboard of squares in parallel perspective.A distance point is the vanishing point of lines that make angles of 45° with the picture plane. When a perspective line is drawn to a distance point so that it cuts across a pair of orthogonals (Fig. 28) the points of intersection define two diagonally opposed corners of a square drawn in parallel perspective. This is the key to a useful system for measuring into depth (Fig. 29) and is the easiest way of drawing a chequerboard.

Fig. 28

Fig. 29

In each vanishing line, of which the horizon line is the principal example, there are two distance poi nts placed symmetrically about its centre (Fig. 30). They are the same distance from this point as the eye; hence the origin of the term. Because the position of the distance point predetermines the distance of the eye and vice versa, the artist must choose the distance with discretion as it affects the character of the perspective drawing (Figs. 31a and b).

Fig. 30

Fig. 31a

Fig. 31b

The modern analysis shows that the lines from the eye, EC, ED (given in plan in Fig. 32), which determine the vanishing points of the orthogonals and diagonals of the given square, form right-angled isosceles triangles with the picture plane, making EC=CD. But without knowledge of the vanishing-point theory it is not possible to furnish this simple proof. Hence Renaissance theorists were in difficulty when they wished to show that the distance-point rule was in accordance with geometry.

Fig. 32

The distance-point construction is of uncertain origin. It first appears in literature in Piero della Francesca’s De prospectiva pingendi (c.1480), briefly and only once, as a control for another construction, and the reader is referred to a costruzione legittima diagram for an explanation. Some authorities regard this isolated example as having been added by a later hand.

The widespread popularity of the distance-point construction started in France. There Jean Pélerin, called Viator, in his De artificiali perspectiva (1505) presented it as the basic construction. Viator was a canon of the Benedictine abbey church at Toul and secreta ry to Louis XI. Being an amateur of perspective drawing he dispensed with geometrical proofs. His book, written in parallel French and Latin texts, was illustrated with many fine woodcuts and went into four editions in his lifetime. In Vignola’s words ‘it was more copious in drawings than in text’ and his rule ‘more difficult to understand than to execute’.

Viator (‘The Traveller’) called his distance points ‘tiers points’. They should be placed ‘equidistant from the centre: closer in near and further in distant views’. He presumably came across this construction in the course of his many travels north of the Alps, where it may have become established as a workshop tradition. It does not appear in Italian texts before Viator’s time, except in the isolated example mentioned above. Yet it is unlikely that the Italians were previously unaware of its existence. Their reticence on the subject is perhaps accountable to their greater concern for theory than the more empirical northerners.

A confusion arose amongst theorists over the identity of the operative diagonals. Were they graphic lines of the same nature as the orthogonals, or were they visual rays? They may have been prompted by the experiment of observing the orthogonals in a mirror which are seen to converge towards the reflection of the observer’s eye (Fig. 33). Filarete recommends using a mirror for examining the convergence of parallels (Trattato di architettura, 1461-4). If the diagonals were visual rays, the construction could be shown to be sound, being basically the same as Alberti’s construction. For this to be so, diagonal and visual rays should coincide and this happened only in the special case where the centre point and the vertical intersection were brought together as in Fig. 34; and there were practical disadvantages in this arrangement. In the normal arrangements of the costruzione legittima the centre point and the vertical intersection are apart and then the diagonal and the ray do not coincide (Fig. 35). To some this was evidence enough that the distance-point construction was faulty. Yet the greater economy of this construction made it popular with practitioners.

Fig. 33

Fig. 34

Fig. 35

Vignola set out in his treatise Le due regole della prospettiva pratica (printed in Rome, 1583) to demonstrate that both the distance-point construction and Alberti’s construction were equally sound. But he could only succeed in showing that if a square, frontally placed, is drawn in perspective by means of the costruzione legittima, its protracted diagonals meet in the distance points and that these points are the same distance from the centre point as the ‘eye’ of the construzione legittima is from the vertical line of intersection (Fig. 36). In the absence of a general theory of vanishing point, this quasi-empirical demonstration was the nearest Vignola could come to a geometrical proof.

Fig. 36

Dürer, who journeyed to Bologna in October 1506 to receive instruction in the art of ‘secret perspective’ and who afterwards at home must have come across the 1509 Nuremberg pirate edition of Viator, was inevitably confused over the relationship between the Italian and the French constructions. When he came to write his perspective treatise, printed first in 1525 as a section of his Underweyssung der Messung, he gave in addition to the Italian plan and elevation construction his own version of the distance-point rule, which he called his ‘näherer Weg’ or ‘shorter way’. This was a costruzione legittima with the centre moved over to the vertical line, an arrangement which causes the Albertian and the French constructions to coincide. It has the practical disadvantage of doubling the total angle of vision and of increasing the marginal distortions. In Dürer’s S. Jerome engraving, 1514, the total angle is about 108°, which results from placing the centre of vision at the side instead of near the centre. A close examination of Dürer’s works reveals that apart from the S. Jerome engraving and the Melancolia bearing the same date, he never again used a consistent perspective construction throughout a picture.

13. Spherical perspective

The idea that apparent magnitudes depend exclusively on the visual angles so that their relation has to be expressed in terms of arcs, not in terms of sections on a straight line, has led some people to maintain that our visual field appro ximates to a projection on a concave spherical surface centred in the eye. On this basis some theorists have held that plane perspective is unsatisfactory not only because of the wide-angle distortions inherent in every plane projection but also because it consistently represents objective straight lines as straight whereas they maintain that in natural vision some of them would appear curved.There are some strong arguments against this view. Vision is by no means exclusively conditioned by the geometry of light rays. Visual perception is partly an acquired faculty in which pre-knowledge and recognition play an important role. Most people do not see objective straight lines as curved except in unusual circumstances. If they habitually do, they will also see the picture plane as curved. When an impression of curvature is experienced it can usually be attributed to the movements of the eyes. For example, when the eyes are made to travel along a high horizontal straight line , the impression is very distinct that the point of vision glides along a line which is not straight but concave to the eye. On the other hand, as soon as the eye is arrested and caused to gaze steadily at some point on the line, the portion of the line in the vicinity of the point of fixation will appear to be horizontal and straight as it really is.

Partisans of the curved visual space theory draw support from the ancient Greeks. Euclid wrote that ‘planes elevated above the eye appear to be concave’ (Optics, Prop. 10). And Vitruvius, writing on Greek architecture, stated that the architect must take steps to counteract the ‘false judgements of the eyes’: for example, the apparent concavity of a horizontal stylobate (De architectura 3. 4, 5. 9, 6. 2). The well-known entasis of columns (also explained by Vitruvius), and other subtle curvatures built into Doric temples, also testify to the sensitivity of the ancient Greeks to subjective curvatures.

The concavity of the retina is another fact adduced in favour of the theory that our visual space is curved. There are, of course, no straight lines in the retinal image. But as we do not see our retinal images, this fact is quite irrelevant. The retinal images have to be left out of account in localizing objects; they are only means by which the rays of light from one object point are focused on one nervous fibre.

Practising artists may run up against the problem of a curved visual field when they measure the scene they are drawing by the traditional method of holding a pencil or brush handle at arm’s length at right angles to the line of vision and sighting across it with one eye. The measurements thus gained are the angular proportions of the scene and theoretically they are incapable of being projected onto a plane surface (in practice this method works very well when the total angle of view is fairly small). It may be objected that the attempt to transfer these measur ements made on the arc to the flat picture will produce a double foreshortening of the image because of the natural foreshortening of the picture plane.

It is a mathematical truism that a spherical surface cannot be developed into a plane. Nevertheless, over the years, several systems have been devised that set out to give an approximate solution to this problem. Guido Hauck’s system is the best known (Die subjektive Perspektive und die horizontalen Curvaturen des dorischen Styls, 1879). It has been claimed that Leonardo was first to invent such a system and that before him the first artist in post-classical times to represent optical curvatures in his paintings was the 15th-century French painter Jean Fouquet. But Leonardo does not appear to have used curved perspective in his paintings.

Methods of perspective incorporating a curved projection surface may be of some help to an artist who is confronted with the problem of representing a wide-angled view. But they have never had many followers. They disrupt the system of vanishing points and tend to be complicated to use. Furthermore, they produce curved images of objective straight lines (unless the lines pass through the centre of vision).

The multiplicity of systems of curvilinear perspective is witness to the impossibility of arriving at a completely satisfactory solution. In practice artists have generally preferred to employ the much simpler plane perspective and to make the necessary adjustments by eye.

14. Perspective in Antiquity

In the Greek red-figure vases of about 500 BC (See Greek art, ancient) we find expressed for the first time the beginning of a completely new approach to drawing: a gradual break from the conceptual or ideoplastic towards a visual or optical attitude. This change in art is part of the general breakaway from age-long habits of thought that th e Greeks achieved in the 5th century BC. The beginnings of perspective belong to this period.Perspective in ancient Greece and Rome (See Roman art, ancient) was known as skenographia, a term that covered all devices to regulate the effects of space between the observer and the thing seen and included the application of the rules of optics to painting, sculpture, and architecture. It was first applied in the Theatre of Dionysus at Athens in the second half of the 5th century BC, when drama began to require more elaborate scenic arrangements. Vitruvius writes that the painter Agatharchus of Samos was the inventor of scenography and that he first applied it when he ‘made the scene’ for a tragedy by Aeschylus (525-456 BC) given at Athens, probably at a revival in the 430s, and that he wrote a commentary on his method. Furthermore certain scientist-philosophers took up the subject and also wrote about it, working out rules of perspective. They showed how ‘if a fixed centre is taken for the outward glance of the eyes and the projection of the rays, we must follow these lines in accordance with a natural law, such that from an uncertain object certain images may give the appearance of buildings in the scenery of the stage, and how what is figured on vertical and plane surfaces can seem to recede in one part and to project in another’. We can only gather from this description that optical rules were applied to illusionistic scenery from the start, though we cannot deduce precisely in what manner. None of these treatises on scenography survives. But they were perhaps the prototypes of Euclid’s Optics. Here, as in many other fields, practical recipes preceded theoretical investigations.

Although virtually nothing remains of classical Greek painting, the figured vases give some indication of development in practical perspective. By c.500 BC artists were beginning to expe riment with foreshortening and with drawing untypical aspects—a shield from the side, a foot from the front. Soon too artists drawing architectural forms employed a rudimentary perspective in the representation of planes and lines in depth. Convergence was still unsystematic and was applied only, if at all, to individual planes and never to the picture as a whole. The receding parallel edges of solid objects, such as tables in the foreground, were mostly drawn parallel or divergently. Central convergence is used fairly consistently only in one short phase of the Antique painting that has come down to us: in the illusionistic wall decorations of the Architectural (or so-called second) Pompeian style dating from about 80-30 BC), examples of which have been excavated in and near Rome and Pompeii, the best preserved being from Boscoreale. These corroborate Vitruvius’ statements on wall decoration in his own time as well as his brief and somewhat obscure definition of sceno graphy as used by architects: ‘Scenography is the sketching of the front and the retreating sides and the correspondence of all the lines to the centre of a circle.’ This seems to confirm that some kind of functional system of central convergence was in use during the 1st century BC, at least amongst architects. In the following generation the fashion changed to a more decorative and fanciful mode in which the perspective was less systematic.

The question remains, did the Greeks have a theory of central perspective? Did they at any time conceive of a painting as a transparent projection plane with a theoretically fixed eye-point? It used to be thought that they did not, until H. G. Beyen discovered that some mural paintings in Pompeii, Rome, and Boscoreale showed a construction with a point that functionally corresponded with the vanishing point of central perspective. The lower parts of these paintings do not, however, show the same regard for central convergence and are more haphazard in their perspective treatment. It has been suggested that this may be because the artists who drew these pictures were not original masters, but copied more or less accurately pictures they had seen on the stage. As on the raised platform the lower part of such a construction was missing, they therefore had to fill in this deficiency from their own resources and made use of the more common parallel perspective.

Our familiar pair of railway lines that ‘vanish’ to a point have an equivalent in ancient literature. Lucretius (De rerum natura 4) describes how a colonnade may appear to vanish into the obscure point of a cone. But the geometrical principle of vision did not receive universal acknowledgement in Antiquity. It had a popular rival in the Epicurean theory of images, according to which objects are continually throwing off eidola or surface films of very fine texture that traverse the air at infinite speed. These impinge upon our eyes and give us vision by direct contact. This theory of vision led to the paradoxical belief that the heavenly bodies are the size they appear. Perhaps Euclid wrote his Optics to combat such views. We may believe that the ancients possessed a system of perspective which had close resemblance in its effects to Renaissance perspective, but it is not certain that this was a system of central projection. The first complete rationalization of the picture space seems to have been the achievement of the 15th century AD.

15. Classical optics: Euclid on perspective phenomena

The oldest extant book on geometrical optics is by the great Greek mathematician Euclid of Alexandria, written about 300 BC. It consists of 58 theorems (20 belong to optics proper and 38 deal with perspective phenomena) preceded by twelve definitions or postulates. Euclid’s Optics laid the foundations upon which geometrical optics could be developed to its present high level. It exerted a great influence upon medieval writers on optics and indirectly was of great consequence in the evolution of Renaissance perspective. The following are the first four definitions. Let it be assumed
(i) That straight lines proceeding from the eye diverge through a space of great extent. (Euclid adhered to the centrifugal theory of visual rays.)
(ii) That the form of space included within our vision is a cone with its apex in the eye and its base at the limits of what is seen.
(iii) That those things upon which the visual rays fall are seen and that those things upon which they do not fall are not seen.
(iv) That those things seen within a larger angle appear larger and those things seen within a smaller angle appear smaller, and those things seen within equal angles appear to be of the same size.The purpose of the Optics is to express in geometrical propositions the exact relation between the real quantities found in objects and the apparent quantities that constitute our visual image. Euclid connected, in pairs, object points with image points as is done in modern constructions. He shows the apparent size of objects to be directly proportional to the visual angle subtended at the eye. He deals geometrically with common deceptions of vision on the basis of the visual angle theory. The following are some of the propositions which have a bearing upon perspective.

Prop. 6. Parallel lines seen from a distance seem to be an unequal distance apart (Fig. 37). The proof follows logically from Def. iv. BD is seen within a smaller angle than PN; the angle PKN is smaller than the angle XKL. The intervals then between parallels do not appear equal but unequal.

Fig. 37

Prop. 10. Of a horizontal plane situated below the observer’s eye those parts which are further away appear to be the more elevated (Fig. 38).

In the proof the eye at B is connected with Z, D, and G of the horizontal ground plane. The connecting lines intersect the perpendicular EH. H is more elevated than L, etc.

Fig. 38

Similar proofs deal with planes above eye level that appear to descend as they recede, with planes on the left that appear to move to the right, and planes on the right that appear to move to the left.

Prop. 36 deals with an example of foreshortening, explaining why chariot wheels appear oval when seen obliquely.

Of particular interest is Prop. 10 since it shows that Euclid knew the principle that lies at the foundation of central perspective, namely that the objects are projected by the eye upon the picture plane. Yet there is no evidence to show that Euclid ever thought of his proof in this way. He wrote about perspective phenomena, but the etymological synonymity of optics and perspective has often led to the misconception that he wrote about perspective constructions.

It is noteworthy that in Prop. 6 Euclid does not conclude that parallels if produced indefinitely will ever appear to meet in a point. The 13th-century writer on optics, Witelo, who followed similar lines in discussing this theorem, concluded that parallel lines are never seen to meet in a point because the interspace between them will always subtend some angle at the eye. Nowadays one would say that at infinity the subtended angle is zero. But in the 13th century the concept of limit was not yet accepted by mathematicians in their proofs. Witelo’s qualification may indicate the existence of two opposing schools of thought as regards the geometrical validity of vanishing points and go some way towards explai ning the tardy adoption of the central vanishing point for orthogonals by the artists of that time.

16. Euclid’s Proposition 8

Something must be said about Euclid’s Prop. 8, which has often caused confusion in the minds of those who study the history of perspective. This theorem states that ‘equal and parallel magnitudes unequally distant from the eye do not appear (inversely) proportional to their distances from the eye’. Euclid wished to discover whether there existed a simple geometrical proportionality between the apparent size of equal and parallel lines and their distances from the eye. He found that this simple relationship did not exist. E is the eye; A and G are the two parallel magnitudes. A is twice as far from the eye as G but the angle it subtends is appreciably greater than one-half of the angle subtended by G (Fig. 39).39
Fig. 39

The confusion has arisen because Leonardo wrote: ‘A second object as far away from the first as the first is from the eye will appear half the size of the first, though they be the same size really.’ In other words, if you double the distance you halve the apparent size: hence apparent size and distance are inversely proportional. The apparent contradiction does not in fact exist because Euclid is speaking about the visual angles of natural perspective, which are measured by their subtending arcs, while Leonardo has the picture plane in mind, which produces straight projections and forms a series of similar triangles with the visual rays.

By considering the projections on the picture plane with regard to their proportional relationships Leonardo (and Alberti and Piero della Francesca before him) was introducing the classical concept of proportionality, with its cosmological and aesthetic connotations, into the theor y of perspective. Leonardo goes on to affirm that the same proportional intervals obtain in perspective space as in music. (See proportion.)

17. Perspective in Byzantine art

The post-classical age was a period of regression for perspective. After the division of the Roman Empire and the break-up of the West, Constantinople became the artistic centre of Christendom and conserved for 1,000 years and more much of the classical heritage of drawing, though in a much schematized form, while in the West the taste for abstraction and ornament led to the complete transformation of classical models and the breakdown of perspective schemata. The Byzantine artist, in common with the rest of the post-classical world, did not follow any consistent system of perspective. To start with, Constantinople copied 3rd-century official Roman art, the tendency of which was already to turn away from classicism and illusionis m. But whereas the art of the West went through many changes between the collapse of the empire and the Renaissance, Byzantine art soon settled into a comparatively rigid style. Official Byzantine art was from the start hieratic and anti-illusionistic. Yet artists never ceased to draw upon classical sources. Hellenistic illuminated manuscripts provided them with a rich quarry of classical forms, which at times they imitated freely, as for example the Joshua Rotulas (Vatican Mus.), executed at Constantinople in the 10th century, in which we find a use of shadow and aerial perspective almost unique in medieval art.The Optics of Euclid was current in the Byzantine world through Pappus and Theon of Alexandria, and Proclus (AD 412-85), in commenting on Euclid, describes scenography as the branch of optics which ‘shows how objects at various distances and of various heights may be so represented that they will not appear out of proportion and distorted in shape’. Byzantine painting never became a window opening onto a world outside in the manner of illusionistic Hellenistic art and the Renaissance conception initiated by the stil nuovo of Giotto. At most the pictures create the appearance of a boxlike space or niche in the wall which they occupy. Byzantine wall painting was primarily a functional adjunct to architecture and some authorities have maintained (though not all accept this interpretation) that the picture space was characteristically that of the church or room in which the picture was placed (and in which the observer stood), the perspective opening out into the actual room space. One device sometimes used for suggesting depth of picture space is an ‘inverted perspective’ in which the viewpoint of the picture might be taken to be behind the scene, not in front of it, figures being smaller in proportion to their distance from the viewpoint. This system of perspective fits naturally to the Euclidean theory of vi sion, which supposes rays to proceed from the eye to the object. On the assumption that equal sizes subtend equal angles at the eye, an artist who had misunderstood Euclid might assume that a more remote figure or object should be drawn larger than a nearer one since it would span a wider part of the visual pyramid. In the later period a high viewpoint is often used so that the scene is visualized as it would appear from above. Very often different systems or different viewpoints are combined in one composition and the same structure is made to carry planes belonging to successive acts of vision. The idiosyncrasies of Byzantine perspective have been attributed to the fact that Byzantine art derived from two irreconcilable traditions: Greek illusionism and oriental schematic abstraction. Furthermore in the 8th century it suffered the doctrinal severities of iconoclasm. In the last phase of Byzantine art there is evidence of cross-fertilization between the East and the Gothic West and of a movement towards spatial unity: witness the mosaics and frescoes in the church of Christ in Chora, Istanbul. It is only during the late 12th to the early 15th centuries that the representation of functional space became a dominant concern of Byzantine artists both in panel and wall paintings and in mosaics.

18. Perspective in post-classical and medieval Western painting

The shift of art to the transcendental and the symbolic that began during the break-up of the Roman Empire produced in the West many vicissitudes of style. Between the early Christian and the late Gothic periods regional differences were at times very marked, but everywhere perspective was in low esteem. In Italy, much under Byzantine influence, painting preserved more of the classical tradition than elsewhere, but on the borders of the Roman Empire the classical heritage was submitted to the disintegrating pressure of barbaric t raditions. Artists trained in a style of abstract linear pattern and who knew nothing of narrative painting could not immediately interpret and assimilate the representational forms of Mediterranean origin which they tried to copy. In certain phases of medieval painting the third dimension was virtually eliminated. Examples may be found in Celtic, Mozarabic, Ottonian, and high Romanesque illumination. The Carolingian attempt to revive the classical tradition was short-lived and it was not until the Gothic period that Western art turned finally towards naturalism. In Gothic illumination figures and objects again acquired relief and were set in a shallow stage space.Spatial inconsistencies are a common feature in medieval painting. They testify to its disregard for material values. To modern eyes they are a part of its charm. One of the most characteristic is the drawing of pillars as discontinuous. This was done to avoid overlapping in the interest of the overriding demand for narrative clarity. In certain cases such spatial non sequiturs may have been intended to underline the universal validity of the religious scene by introducing transcendental values. In medieval painting there was often a return to ideoplastic elements. Table tops were drawn as though vertical with the utensils sliding off them. Disproportion between figures and their surroundings was the rule. Symbols were preferred to descriptive realism. Knowledge of geometry and geometrical optics virtually died in the West until the revival of the 12th century. A unique and important document in the history of medieval drawing is the sketchbook (c.1235) of Villard de Honnecourt. This includes four pages of ‘the elements of portraiture’ with the preface: ‘Here begins the art of the elements of drawing, as the discipline of geometry teaches it, so explained as to make the work easy.’ But the precepts do not include any instruction in perspective and when the author himself draws buildings he has seen on his travels (and there are several in his sketchbook) the lack of such knowledge is visibly a handicap. Figure 40 shows his drawing of a clocktower, which although admirably observed in all its parts appears unstable because the horizontals are not related to an eye level. It shows how necessary was the seemingly naive perspective rule given by the Italian Cennino Cennini in his Il libro dell’arte, written in the 1390s: ‘And you put the buildings in by this uniform system: that the mouldings that you make at the top of the building should slant downwards… the moulding in the middle should be quite level and even; the moulding at the base must slant upward, in the opposite sense to the upper moulding.’ Cennini’s book was a late product of the Byzantine tradition—the maniera greca—which survived in Italy right through the Middle Ages. Cennini epitomizes the medieval attitude when, on the first page of his book, he states that the purpose of pai nting is to ‘discover things not seen, hiding themselves under the shadow of natural objects’. These words make a vivid contrast with one of the opening statements of another famous book on painting written a generation later by that pioneer of the Renaissance movement Leon Battista Alberti: ‘No one would deny that the painter has nothing to do with the things that are not visible. The painter is concerned solely with representing what can be seen.’

Fig. 40. A clock-tower. Pen-and-ink drawing by Villard d’Honnecourt. (Bib. nat., Paris, c.1235)

19. The revival of perspective

The beginnings of a return in the West to three-dimensional realism and to the drawing of architectonic space are to be found in 13th-century Rome, whose early churches still preserved many pictorial examples of the late Antique style , for example the 5th-century mosaics of S. Maria Maggiore. The greatest of the Roman early realists was Pietro Cavallini. His mosaics in S. Maria in Trastevere from the late 1290s such as theAnnunciation, the Birth of the Virgin, and the Presentation show that he possessed a firm grasp of vertical axis perspective. Nearly all of his frescoes have been destroyed. He used a unified focus for each separate structure or main architectural part, though not for the whole picture. For example, the two interior rooms that form the background of the Birth of the Virgin are each drawn to a separate central axis, implying two distinct viewpoints. The three aedicules in the Presentation are represented from high and low viewpoints alternately, as in the traditional Byzantine manner. Cavallini was by no means without immediate predecessors in Rome. Elaborate architectural backgrounds can still be glimpsed in the much-damaged 11th-century frescoes in the lower basilica of S. Clemente, by an unknown hand. And the fresco cycles in the portico of S. Lorenzo fuori le mura, painted between the 1260s and 1280s by painters only known as ‘Paulus and his son Filippus’, constitute a veritable pattern book of architectural forms. It was in Rome through the art of late Antiquity that the Italian painters first rediscovered perspective space.The two greatest monuments of the rapid advance made in spatial realism in Italy at the turn of the century are the great fresco cycles of the upper church of S. Francesco at Assisi and of the Arena chapel at Padua. The clerestory frescoes of the south wall of the nave at Assisi are of the Roman school and show some of the earliest convincing interiors. (Early interiors are always seen from the outside. The real interior seen from within does not appear in painting before the 1440s.) The 29 frescoes of the S. Francis cycle at Assisi are traditionally ascribed to Giotto, although doubts have been cast on this attribution; they were painted no later than 1307 and perhaps before 1300. Giotto painted the frescoes of the Arena chapel, Padua, in or about 1306. Striking examples are to be found there of empirical perspective. The two little Gothic chapels painted at eye level on the walls of the chancel arch, on the right- and the left-hand side as one looks towards the altar, are drawn towards a common vanishing point, indicating that Giotto aimed at an illusionistic piercing of the walls. His fresco of the Suitors Praying has a ceiling that appears to have been drawn to a central point and to an empirical distance point. Nevertheless Giotto’s perspective, in common with the whole of the trecento, was intuitive and practical rather than scientific.

The floor-plane, which was to become the key feature of the Renaissance picture space, was the last to be brought under perspective control during this empirical phase. The Sienese of the trecento made a habit of intro ducing tiled floors into their designs. Two early examples may be compared from Duccio’s Maestà (1311) (Siena, Mus. dell’Opera del Duomo): the scene representing Christ amongst the Doctors has a floor whose squares are barely foreshortened, while the Temptation of Jesus on the Temple displays a strip of chequered floor that leads the eye deep into the interior of the building. In the final phase before the formulation of scientific perspective the works of the Lorenzetti brothers are outstanding. The architecture in Ambrogio Lorenzetti’s big fresco of The Effects of Good Government on the City and Country (1338; Siena, Palazzo Pubblico) is a strikingly realistic portrait of the town. No less remarkable for its perspective is Pietro Lorenzetti’s Birth of the Virgin (1342; Siena, Mus. dell’Opera del Duomo) with its three architectural bays drawn to a single focus, while Ambrogio’s Annunciation (1344; Siena, Pin.) contains what is said to be th e first example of a floor drawn to a central vanishing point. The foreshortening of the tiles, however, is not geometrically controlled. After a period of standstill caused by the Black Death, further advances were made in the last quarter of the century at Padua by Altichiero, and Giusto de’Menabuoi.

20. Medieval optical studies

Euclid’s studies in optics had been carried further by Greek and Arab mathematicians in Egypt, notably by Claude Ptolemy (active 2nd century AD) and Alhazen d. 1038. By about 1200 some of the treatises had become available to the West through Latin translations made from the Arabic in Spain and Sicily and these stimulated great interest in the subject, besides providing material for further treatises by the scholastics in the 13th century (Robert Grosseteste, Bishop of Lincoln; Roger Bacon; John Peckham, Archbishop of Canterbury; Witelo) whose works were consulted up to the time of Kepler. This revival of optics in the West coincides with the return of naturalism in art. The two events were not directly connected yet both were part of the widespread movement that combined the revival of learning with a growing interest in the natural world and a belief in the importance of mathematics and experimental science.

21. Brunelleschi

In Florence of the 15th century and elsewhere these optical treatises were much studied (witness Lorenzo Ghiberti’s Commentarii, the third of which is composed of extracts taken from the better-known optical works). It seems that Filippo Brunelleschi was already acquainted with them when he made his famous perspective experiments early in the century. His first demonstrations are described by his biographer, Antonio Manetti (Vita di Filippo Brunnelleschic.1480; English trans. H. Saalman, 1970). Brunelleschi p laced himself just inside the central doorway of the Cathedral of Florence and from there he made a picture of the baptistery ‘showing as much as could be seen at one glance’. It was a panel about 30 cm (1 ft) square, painted with the precision of a miniature, and the sky was represented by burnished silver that was to reflect the real sky and the passing clouds. Having completed his panel, Brunelleschi bored a hole through it at that point in the view which had been exactly opposite his eye when he painted it: that is at the centre of vision. The spectator was instructed to look through the hole from the back, at the same time holding up a mirror on the far side in such a way that the painting was reflected in it. ‘When one looked at it thus,’ says Manetti, ‘the burnished silver, the perspective of the piazza, and the fixing of the point of vision made the scene absolutely real.’This was the first recorded centrally projected image in the history of painting. The questio n remains: how did Brunelleschi make it? He might have used a burnished silver ground as a mirror and traced the lines of the reflected buildings on its surface, thus producing a reversed image. (This would explain why the spectator was asked to look at the painting in a mirror.) If this was Brunelleschi’s procedure he may have started from a suggestion found in Ptolemy’s Optics, where an experiment of marking the reflection of an object on the surface of a mirror is described and accompanied by a geometrical diagram. Perhaps the next stage was to make diagrams of the whole experiment, complete with plan and side elevation, showing where the visual rays intersect the picture plane. Such an order of events would reconcile the opinion of the 15th-century Florentine architect Antonio Filarete, who believed that Brunelleschi discovered perspective while considering reflections in a mirror, with Vasari’s statement that Brunelleschi proceeded with the aid of a plan and elevation ‘and by means of the intersection’.

The earliest painting in which the architectural perspective is exactly based on Brunelleschi’s rules is the fresco of the Trinity in S. Maria Novella, Florence, by his friend Masaccio, painted c.1425. But Brunelleschi’s influence is already apparent in the reliefs of S. George and the Dragon (Florence, Or San Michele, c.1415-20) and Salome (Siena baptistery, c.1423-5), both by Donatello who was his close friend and perhaps the first to share the discovery. It is also strong in other works of the time: Uccello’s fresco of Sir John Hawkwood (Florence Cathedral, 1436), and panels of Ghiberti’s Gates of Paradise (1425-52) for the Florence baptistery. Shortly after Brunelleschi made his perspective demonstrations Alberti devised his perspective construction for the special use of painters (see above), which he described in his famous treatise Della pittura (1436). In the same work he explained the drawing frame which he claimed as his own invention.

22. Perspective in the North

The new Italian discoveries took some time to reach the North. There the advance towards realism was achieved without central perspective. Jan van Eyck’s Virgin and Child in a Church (Berlin, Gemäldegal.), is not drawn in central perspective, though it was perhaps painted about the same time as Masaccio’s Trinity. Nor is his Arnolfini Portrait (London, NG) drawn to a central point, but ceiling, floor, and walls each to separate points (Fig. 41). Evidently it is not easy to tell without a minute examination whether a systematic construction has been used. None of the Flemings used central convergence consistently, i.e. for the whole picture, until Petrus Christus as seen for instance in the Virgin and Child with S. Francis and S. Jerome (1457; Frankfurt, Städelsches Kunstinst.). Geometrically controlled foreshortening did not appear north of the Alps until the next century, after the treatises by Viator (1505) and by Dürer (1525) had become widely known.41
Fig. 41. The perspective of Jan van Eyck’s Arnolfini and his Wife (N.G., London, 1434). Illustration from Die Grundzüge der Linear-Perspektivischen Darstellung (1904), G.J. Kern

23. Perspective in modern times

Over the ages many great names in art and science have been linked with the story of perspective. Perhaps the last great painter to be learned in this subject was J. W. M. Turner, who was professor in perspective to the RA (See under London) from 1807 to 1828. Today most artists who paint representational pictures acquire their perspective empirically and possess only the rudiments of theory. In 1827 Delacroix was using the perspectivist Thénot to work on The Death of Sardanapalus (Paris, Louvre). Already in the 1860s no less a painter than Jean-François Millet was enquiring after a professional ‘perspecteur’ to help him with the perspective of his ceiling painting for the Hôtel Thomas. And Degas told Sickert that he had to employ a ‘perspecteur’ for his Miss Lola (London, NG) hanging by her teeth from a trapeze near the ceiling (marginal note in Sickert’s hand in his copy of A. Sensier’s book on Millet (1881), now in the library of University College, London). Through photography we become so habituated to the perspective image that artists acquire a sense of perspective almost unconsciously, and even make efforts to find new ways of representation. Recent investigations made by experimental psychologists suggest that geometrical perspective provides an inadequate account of our visual perceptions, which are found to be strongly influenced by size, shape, and colour constancy. Moreover, objects in close proximity to the eye do not follow the laws of scientific perspective—a fact that has been exploited by many modern artists since Cézanne and was no doubt observed by artists before the rule of science took over. Nevertheless it is wrong to call perspective a ‘convention’. It may be a convention to paint pictures according to strict geometrical perspective, but perspective itself is not a convention: it is a part of the theory of central projection. And it is a fallacy to suppose that the ‘visual truth’ may be represented by a flat image, if by ‘visual truth’ is meant the way we see the world through our two moving eyes forming successive perceptions conditioned by ‘constancy’. The perception of representational paintings—or photographs—is usually a very different process from the perception of the actual scenes in depth. Finally, pictures may be regarded as an arrangement of symbols for reality. Central perspective is unique in so far as it enables the picture to send to the eye of the beholder, who respects its conditions, the same distribution of light as the objective scene.Harold Osborne

Abbott, W., The Theory and Practice of Perspective (1950).

Doesschate, G. ten, Perspective (1964).

Helmholtz, H. van, Handbuch der physiologischen Optik (1867; English trans. J. Southall, repr. 1962).

Panofsky, E., Die Perspektive als ‘symbolische Form’ (1927).

Panofsky, E., Idea (1924, edn. 1960).

Schuritz, H., Die Perspektive in der Kunst Albrecht Dürers (1919).

White, J., The Birth and Rebirth of Pictorial Space (1957).

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