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The History of Perspective

Pre-Renessaince Perspective

Although supported by scarce evidence, it is held that attempts to develop a system of perspective began around the fifth century B.C. in ancient Greece, as part of an interest in illusionism allied to theatrical scenery. However, even though Hellenistic painters could create an illusion of depth in their works there is no evidence that they understood the precise mathematical laws which govern correct representation.

Second Style wall paintings in Rome and Campania (fig. 1) of the first century B.C. exhibit different types of projection simultaneously: convergent projection (typically found in the upper areas of the composition) and oblique projection (in the lower areas and minor details). Particularly striking are the perspectives of the architectural frescoes from the Villa of Publius Fannius Synistor at Boscoreale(see$$$$$), near Pompeii. Although they may violate the strict rules of one-point perspective, they nonetheless demonstrate a pragmatic understanding that lines parallel to the viewer's line of sight converge at some point on the picture plane, something that would have not likely arisen by accident or through naked eye measurement. In some cases orthogonals recede precisely to a single point, albeit only within localized areas.

Villa of P. Fannius Synistor Cubiculum M alcove, Panel with temple at east end of the alcove, the north end of the east wall, Middle of the first century B.C., Boscoreale (Pompeii), Italy fig. 1 Villa of P. Fannius Synistor
Cubiculum M alcove
Panel with temple at east end of
the alcove, the north end of the east wall
Middle of the first century B.C.
Boscoreale (Pompeii), Italy
Egyptian Wall Paintings From The New Kingdomfig. 2 Egyptian wall painting from the New Kingdom

Whatever its degree of sophistication in antiquity, the knowledge of perspective was lost until the fifteenth century. From the Duecento to the Cinquecento, after which art academies formally introduced the teaching of perspective, painters explored various techniques to evoke spatial depth on a flat surface. Progress was relatively uneven because painters did not always work in close contact with each other. Moreover, medieval painting was essentially a representation of religious, rather than human, experience. The importance of the figures was fixed by canonical tradition so that the most significant figure in the painting was the largest and that all other figures were portrayed in diminishing in size regardless of their position within the pictorial space, similar in concept to Egyptian art(see$$$$$). Important figures are often shown as the highest in a composition (fig. 2), also from hieratic motives, leading to the so-called "vertical perspective." Thus, for the medieval artist there was little impetus to devise a rational system by which the things of the world might be represented in scale on a two-dimensional surface, in obedience to the unvarying laws of geometry and optics. Painters experimented with what art historians refer to as "empirical perspective," ad hoc solutions devoid of consistent rules. Gothic painting slowly progressed in the naturalistic depiction of distance and volume, although these elements were never essential features of representation.

For a complete list of pre-1900 perspective manuals (with subsequent republishings) consult the Russell Light's excellent PERSPECTIVE RESOURCES, from which the list below was derived.

Click on the links below to access PDF files of the treastises.

ALBERTI, Leon (1435) - De Pictura.
* Italian translation - Della Pittura, 1436. First published editions: Latin - Basel, 1540; Italian - Venice, 1547; English (trans. from Italian) - Leoni, 1726.

FILARETI (c.1461–164) - Libro architettonico, (later referred to as the Trattato…)

P. DELLA FRANCESCSA (c.1470) - De Prospectiva Pingendi, critical edition ed. G Nicco-Fasola, Florence, 1942.

DA VINCI (c. 1500–1518) - Notebooks

VIATOR (Pèlerin, Jean) (23 June, 1505) - De Artificiali P(er)spectiva, Toul, Petrus Jacobi.

DÜRER Albrecht (1525) - Unterweisung in der Messung mit Zirkel und Richtscheit, (Measurement by Compass and Ruler), published?

SERLIO, Sebastiano. (1537–1547) - Tutte l'Opera d'Architectura et Prospettiva, Venice.

ARETINO, Pietro (1557) - Dialogo della Pittura di M. Lodovico Dolce initolato l'Aretino, Venice.

COUSIN, Jean (1560) - Livre de Perspective, Paris, Jean le Royer.

BARTOLI, Cosimo (1564) - Del Modo di Misurare le Distantie, le Superficie, i Corpi, le Piante, le Provincie, le Prospettiue, & Tutte le Altre Cose Terrene, Venice, Francesco Franceschi.

BARBARO, Daniele (1568) - La Practica della Perspettiva di Monsignor Daniele Barbaro Eletto Patriarca d'Aquileia, Opera Molto Utile a Pittori, a Scultori, & ad Architetti, Venice, Camillo and Rutilio Borgominieri.

JAMITZER, Wenzel (1568) - Perspectiva Corporum Regularium, Nurnberg, Gotlicher Hulff.

BASSI, Martini (1572) - Dispareri in Materia d'Architettura, et Perspettiva. Con Pareri di Eccellenti, et Famosi Architetti, chi li Risoluono, Brescia, Francesco and Pietro Maria Marchetti.

DU CERCEAU THE ELDER, Jacques Androuet (1576) - Leçons de Perspective Positive, Paris, Mamert Patisson.

VIGNOLA, Jacopo Barozzi da (1583) - La Due Regole della Prospettiva di M. Iacomo Barozzi da Vignola con i Comentarij del R.P.M. Egnatio Danti, Rome.

VILLAFANE, Ioan de Arphe y (1585) - De Varia Commensuracion para la Escultura, y Arquitectura, Seville, Andrea Pescioni y Ivan de Leon.

SIRIGATTI, Lorenzo (28 October, 1596) - La Practica di Prospettiva, Venice, Girolamo Franceschi. (Eng. ed., Issac Ware, 1756)

DEL MONTE, Guido Ubaldo (1600) - Perspectivae Libri Sex, Pesaro, Hieronymus Concordia.

DE VREIS, Hans Vredeman (1604–1605) - Perspectiva, id est Celeberrima ars Inspicientis aut Transpicientis Oculorum Aciei, in Pariete, Tabula aut Tela Depicta, The Hague, Leyden.

HONDIUS, Hendrik (1622) - Onderwysinge in de Perspective Conste, The Hague, Hondius.
* (1622) - Institutio Artis Perspectivae.
* (1625) - Instruction en la Science de Perspective.
* (1640) - Gondige Onderrichtinge in de Optica, oft Perspective Konst, Amsterdam.

ACCOLTI, Pietro (1625) - Lo Inganno de Gl'ochi, Prospettiva Practica, Florence, Pietro Cecconcelli.

VAULEZARD, I.L. de (1630) - Perspective Cilindrique et Conique; ou Traicté des Apparences Veuës par le Moyen des Miroirs Cilindrique et Conique, Paris, J. Jacquin.

DESARGUES, Girard (1636) - Example d'une des Manières Universelles, Paris, the author.

NICERON, Jean François (1638) - La Perspective Curieuse, ou Magie Artificielle des Effets Merveilleux de l'Optique…la Catoprique…la Dioptique, Paris, Pierre Bilain.

DUBREUIL, Jean (1642) - La Perspective Practique…par un Parisien, Religieux de la Compagnie de Iesus, Paris, Melchior and François Langlois.

ALÉAUME and MIGON (1643) - La Perspective Spéculative et Pratique du Sieur Aléaume, ed. by Etienne Migon, Paris.

BOSSE, Abraham (1648) - Manière Universelle de Mr Desargues pour Pratiquer la Perspective par Petit-Pied, comme le Géometral, Paris, the author.

LECLERC, Sébastien (1669) - Practique de la Géométrie sur le Papier et sur le Terrain, Paris, Thomas Jolly.

TROILI, Giulio (1672) - Paradossi per Pratticare la Prospettiva, senza Saperla, Fiori, per Facilitare l'Intelligenza, Frutti, per non Operare alla Cieca, Bologna, heirs of Peri.

POZZO, Andrea (1693–1700) - Perspectiva Pictorum et Architectorum Andreae Pozzo Putei e Societate Jesu', Rome, Joannis Komarek Bohemi.

LAMY, Bernard (26 February, 1701) - Traité de Perspective, ou sont Contenus les Fondamens de al Peinture, Paris, Anisson.

BIBIENA, Ferdinando Galli (1711) - L'architettura Civile Preparate su la Geometria, e Ridotta alle Prospettive, Parma, P. Monti.

TAYLOR, Brook (1715) - Linear Perspective: or, a New Method of Representing justly All Manner of Objects as They Appear to the Eye in all Situations, London, R. Knaplock.

Cone of Vision (COV): The area of vision that emanates from our eyes, about 60 degrees wide, before distortion begins to affect what we see. Outside of the 60-degree angle, objects begin to blur. In linear perspective, the Cone of Vision is indicated with a 60 degree angle beginning at the station point it is 30 degrees to the left and right of the line of sight.

Distance Points & Distance Lines:8 The two vanishing points on the horizon at which diagonal 45 degrees lines in the horizontal plane meet, are known as distance points. They are the same distance from the central vanishing point as the viewer is from the picture plane. If within a picture, a horizontal square parallel to the picture plane can be identified, extending the diagonals to the horizon will give the distance points. The distance of the viewer to the picture plane is then known, and it becomes possible, by working backwards, to create a plan of the space within the picture.

It is debatable whether the correct viewing distance was of any importance to the early users of perspective. In reality, however, there are paintings that show an approach that could not be considered to be purely Albertian. Many paintings show a floor grid with a recession that appears to be governed solely by the 45 degrees diagonals of the grid squares being drawn towards a point at eye level, often placed at the edge of the painting. This approach is often referred to as the 'distance point' method and these points are known as 'distance points' simply because the distance between them and the central vanishing point is the same as the distance between the viewer and the picture plane. It follows that if the vanishing point for the orthogonals is placed centrally, and the edge of the painting is used as a distance point, then the "correct" viewing distance is half the width of the painting. It also follows that the angle of view is 90 degrees. It has been generally assumed that these points have been placed at the edge of the paintings for completely practical reasons.

We do not know the precise moment at which the two lateral points received their theoretical explanation as the "point of distance." We do not know if Brunelleschi that their distance from the central vanishing point represented, according to the scale of the picture, the distance between the vantage point of an ideal spectator and the plane of the image.

Field of Vision (FOV). The area wider than the Cone of Vision, coming out from the viewer at 90˚, in which distortion begins.

Converging Lines: In perspective drawing, parallel lines that come together towards a single vanishing point.

Diminishing Forms or Diminutation: Refers to the apparent size of objects and how they become smaller when the distance between the object moves further away from the viewer/artist, a key tenant of linear perspective.

Foreshortening: Refers to the fact that although things may be the same size in reality, they appear to be smaller when farther away, and larger when close up. Foreshortening is often used in relation to a single object, or part of an object, rather than to a scene or group of objects.

An excellent example of this type of foreshortening in painting is The Lamentation over the Dead Christ (c.1470–1480, Pinacoteca di Brera, Milan), a work by Andrea Mantegna.

Ground Line (G): A line drawn to establish the surface on which an object or objects rests; it is used to determine accurate vertical measurements in perspective drawings. The base or lower boundry of a picture plane. The term may also be applied to a similar construction line used anywhere in the picture to measure off points or to determine the scale of a figure.

The ground line is always parallel to the horizon line. In perspective drawings that show top and side views, the side view of an object is placed on the ground line. It is usually the plane supporting the object depicted or the one on which the viewer stands.

Horizon, Apparent Horizon, Visible Horizon, Skyline: The line at which the sky and Earth appear to meet. For observers near sea level the difference between the geometrical horizon (which assumes a perfectly flat, infinite ground plane) and the true horizon (which assumes a spherical Earth surface) is imperceptible to the naked eye (for someone on a 1000-meter hill looking out to sea the true horizon will be about a degree below a horizontal line).

Horizon Line (HL): The actual horizon, where earth and sky appear to meet, excluding obstructions like hills or mountains. In perspective drawing, the horizon is at the viewer's eye-level. Artists tend to use the term "eye level," rather than "horizon" because in many pictures, the horizon is hidden by walls, buildings, trees, hills etc. In perspective drawing, the curvature of the Earth is disregarded and the horizon is considered the theoretical line to which points on any horizontal plane converge (when projected onto the picture plane) as their distance from the observer increases.

Lines above the horizon line always converge down to it; lines below alwats converge upward to it.

Line of Sight: An imaginary line traveling from the eye of the viewer to infinity. In all paintings with perspective substructures, the line of sight is parallel to the ground. Lines which travel parallel to the line of sight are called orhtogonals, which in a perceptive drawing converge at the vanishing point.

One-point Perspective: A drawing has one-point perspective when it contains only one vanishing point on the horizon line. This type of perspective is typically used for images of roads, railway tracks, hallways, or buildings viewed so that the front is directly facing the viewer. Any objects that are made up of lines either directly parallel with the viewer's line of sight or directly perpendicular (the railroad slats) can be represented with one-point perspective. These parallel lines converge at the vanishing point.

One-point perspective exists when the picture plane is parallel to two axes of a rectilinear (or Cartesian) scene—a scene which is composed entirely of linear elements that intersect only at right angles. If one axis is parallel with the picture plane, then all elements are either parallel to the picture plane (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the picture plane are drawn as parallel lines. All elements that are perpendicular to the picture.

Orthogonal: Orthogonal is a term derived from mathematics. It means "at right angles" and is related to orthogonal projection, a method of drawing three-dimensional objects. Orthogonal lines are imaginary lines which are parallel to the ground plane and the line of sight of the viewer. The are usually formed by the straight edges of objects. Orthogonal move back from the picture plane. Orthogonal lines always appear to intersect at a vanishing point on the horizon line, or eye level. Although we do not generally note the convergence of orthogonal lines in real life, sometimes they become apparent when standing in the middle of a road, train tracks or on a long straight urban street.

Parallel: Said of any two lines or surfaces that are always the same distance from each other.

Perpendicular: At a right, or 90 degree angle to a given line or plane. An absolutely vertical line and an absolutely horizontal line are perpendicular to each other.

Picture Plane (PP): In painting, photography, graphical perspective and descriptive geometry, a picture plane is an imaginary plane located between the "eye point" (or oculus) and the object being viewed and is usually coextensive to the material surface of the work. It is ordinarily a vertical plane perpendicular to the sight line to the object of interest. In painting, the surface of the artist's paper or canvas. The image that is created on the picture plane gives the impression that the subject is behind this surface.

Plane: In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. In colloquial language, any flat surface, such as a wall, floor, ceiling, or level field.

Prospettiva: from Latin perspicere, to "see distinctly."

Projection: From Latin proicere, "to throw ahead." A projection is a straight line drawn through different points of an object from some given point to an intersection with the plane of projection.

Receding: Moving away from the viewer. The opposite is Advancing.

Station Point (SP or S): The position of the artist's eye relative to the object he or she is drawing. Sometiems referred to as "eyepoint," "point of veiw," or "viewpoint."

Transversal: Transversal lines are lines that are parallel to the picture plane and to one another. They are always at right angles to the orthogonal lines.

Two-point Perspective: A drawing has two-point perspective when it contains two vanishing points on the horizon line. In an illustration, these vanishing points can be placed arbitrarily along the horizon. Two-point perspective can be used to draw the same objects as one-point perspective, rotated: looking at the corner of a house, or at two forked roads shrinking into the distance, for example. One point represents one set of parallel lines, the other point represents the other. Seen from the corner, one wall of a house would recede towards one vanishing point while the other wall recedes towards the opposite vanishing point.

Two-point perspective exists when the painting plate is parallel to a Cartesian scene in one axis (usually the z-axis) but not to the other two axes. If the scene being viewed consists solely of a cylinder sitting on a horizontal plane, no difference exists in the image of the cylinder between a one-point and two-point perspective.

Two-point perspective has one set of lines parallel to the picture plane and two sets oblique to it. Parallel lines oblique to the picture plane converge to a vanishing point, which means that this set-up will require two vanishing points.plane converge at a single point (a vanishing point) on the horizon.

Vanishing Point (VP): Imaginary points on the horizon line in one- and two-point perspective. A point at which orthogonal lines receding into space appear to converge.

The vanishing point acts on the visual field as a point of attraction, somewhat like an open drain of a water basin which draws all the water to it.

Brook Taylor, Linear Perspective: Or, a New Method of Representing Justly All Manner of Objects as They Appear to the Eye in All Situations (1715) is said to have been the first to use the phrase "vanishing point."

The Jesuit friar Andrea Pozzo, the author of Perspectiva Pictorum et Architectorum (1693–1700) and the monumental ceiling of Sant'Ignazio in Rome, was the first commentator to systematize use of the "vanishing distance"point (punctum distantiæ) in order to resolve a broad spectrum of perspective problems. He even anticipated the geometrical drawing technique, from descriptive geometry proper, by introducing the simultaneous use of plan and elevation to originate a detailed solution to architectural ornamentation of the classical orders.

Philip Steadman, Vermeer's Camera: Uncovering the Truth behind the Masterpieces. Oxford: Oxford University Press, 2001.

Philip Steadman, Vermeer's Camera. 2001.

Jørgen Wadum, "Vermeer in Perspective," in exh. cat. Johannes Vermeer. National Gallery of Art, Washington, D.C. Royal Cabinet of Pictures Mauritshuis, The Hague (1995–1996) 67–79.

Jørgen Wadum, "Vermeer and Spatial Illusion," in The Scholary World of Vermeer. Waanders Publishers, Zwolle, 1996, 31–50.

Robert Wald, "The Art of Painting': Observations on Approach and Technique," in Vermeer: Die Malkunst, edited by Sabine Haag, Elke Oberthaler and Sabine Pénot, Kunsthistorisches Museum, Vienna: Residenz, 2010, 314.

Gerhard Gutruf and Hellmuth Stachel. "The Hidden Geometry in Vermeer's 'The Art of Painting.'" Journal for Geometry and Graphics vol. 14, no. 2 (2010): 187–202.

Thomas O. Halloran, "Reconstructing the Space, in Vermeer's 'Officer and Laughing Girl.'" Anistorian: In Situ, vol. 8, September 2004.

Christopher Heuer, "Perspective as Process in Vermeer." Anthropology and Aesthetics no. 38 (Autumn, 2000) 82–99.

Daniel Lordick, "Parametric Reconstruction of the Space in Vermeer's Painting 'Girl Reading a Letter at an Open Window," Journal for Geometry and Graphics, Volume 16 (2012), No. 1, 69–79.

C. Richard Johnson, Jr. and William A. Sethares, with contributions by:
Michiel Franken, C. Richard Johnson, Jr, Petria Noble, William A. Sethares, Chris Stolwijk, Ige Verslype, Sytske Weidema and Arthur K. Wheelock, Jr, "Optcial Devices, Pinholes and Perspective Lines," Counting Vermeer: Using Weave Maps to Study Vermeer's Canvases. RKD Monographs, 2018.

Yoriko Kobayashi-Sato, "Vermeer and his Thematic Use of Perspective."Amsterdam. In his Milieu: Essays on Netherlandish Art, in Memory of John Michael Montias, 2009. 212.

Tomás García-Salgado, "Some Perspective Considerations On Vermeer's 'The Music Lesson,'" 2009.

Tomás García-Salgado, "The Music Lesson and its Reflected Perspective Image on the
Mirror." Art+Math Proccedings, University of Boulder Colorado, 2005, 156–160.

Tomás García-Salgado, "Modular Perspective and Vermeer's Room." Bridges London
(Conference Proceedings 2006, Editors: R. Sarhangi & J. Sharp)

Aditya Liviandi, "Reconstruction of Vermeer's 'Music Lesson': An application of Projective Geometry"

Lee Yiwei Christina and Chew Mei Ru Madeleine, "The Length of Vermeer's Studio."

Oriental Perspective

Until Dutch traders began commercing in Western artworks in the seventeenth century, Oriental painters had not discovered, and therefore made no use of, linear perspective, because, as Erwin Panofsky1 would point out, perspective is not only a direct transcription of the visual reality but a form of representation that originates within broader cultural needs.

Methods used by Chinese landscape painters to express the sensation of distance and three-dimensionality were uniquely suited to their artistic priorities, which were profoundly divergent from those of Western artists. The principal motifs of Chinese painters offered little impetus for devising a system of mathematically-based perspective. Rocks, mountains, mythical and human figures have no consistent straight lines to represent and spatial depth could be effectively achieved by other means. Moreover, a perspectival system that hinges on a single view point is both technically and expressively antithetical to the extended scroll form, which was one of the dominant artistic mediums. Chinese paintings might be as much as 10 meters long by one meter high, designed to be viewed one section at a time in the manner of reading a book. Given that Chinese landscape painters strove above all to create an impression of infinite space (fig. 3) opening up in front of the viewer, a single, fixed viewpoint would create an insurmountable obstacle, interfering with the spectator's freedom to wander about and engage himself with the vastness of nature.

A document cosigned by Johannes Vermeerfig. 3Cloudy Mountains
Mi Youren
1130 (Southern Song)
Handscroll, ink and color on silk, 43.7 x 192.6 cm. (overall: 45.5 x 646.8 cm.)
Cleveland Museum of Art, Cleveland
Bare Willows and Distant Mountains, Am Yuanfig. 4 Bare Willows and Distant Mountains
Ma Yuan
c. 1175–1200
Looking in a Mirror by an Ornamental Box, Wang Shên fig. 5 Looking in a Mirror by an Ornamental Box
Wang Shên (c. 1036–c. 1093)
Southern Sung dynasty
National Palace Museum of Taipei, Taipei

In Oriental art spatial depth was attained via overlap and what might be called "planar" perspective(see$$$$$), consisting essentially of distributing subject matter on three spatial planes (fig. 4). The foreground plane was associated with "earthly bound" objects like people, animals, buildings and forests. The middle plane often suggested emptiness (i.e., clouds, mist or water). The background plane generally represents "heavenly" elements such as hills, mountains and sky. The distance between each plane was accentuated by gradating hue, detail and tone (aerial perspective) creating extraordinary effects of atmosphere rarely achieved in Western painting. Architecture and geometric objects (fig. 5) amenable to linear perspective were, instead, rendered with oblique, or parallel, perspective(see$$$$$) which avoids vanishing points and uses oblique but parallel lines to suggest localized spatial recession.

The Birth of One-Point Perspective

The Birth of Saint John the Baptist: Predella Panel, Giovanni di Paolo, 1454, Egg tempera on wood, 30.5 x 36 cm., National Gallery, Londonfig. 6 The Birth of Saint John the Baptist: Predella Panel
Giovanni di Paolo
1454
Egg tempera on wood, 30.5 x 36 cm.
National Gallery, London

"It is significant for the visual characteristics of central [linear] perspective that it was discovered at only one time and place in man's entire history. The more elementary procedures for representing pictorial space, the two-dimensional 'Egyptian' method as well as isometric perspective [i.e., oblique projection] (fig. 6) , were and are discovered independently all over the world at early levels of visual conception. Central perspective, however, is so violent and intricate a deformation of the normal shape of things that it came about only as the final result of prolonged exploration and in response to very particular cultural needs."2 Curiously, the distortions imposed by perspective on the real, tactile world are so successful that they are noted by modern viewers only when they are pointed out. Despite the fact that each of the black and white floor tiles in Vermeer's The Art of Painting was perfectly square and identical in dimension, on the surface of the painting each tile has a measurably different shape and different dimension with respect to all the others—no two are equal. And yet, the illusion of geometric regularity and spatial recession that these deformations create is nearly impossible to perceptually override.

Polyptych of St. Anthony (detail), Piero della Francesca, 1470, Oil and tempera on panel, 338 x 230 cm., Galleria Nazionale dell'Umbria, Perugiafig. 7 Polyptych of St. Anthony (detail)
Piero della Francesca
1470
Oil and tempera on panel, 338 x 230 cm.
Galleria Nazionale dell'Umbria, Perugia

Linear perspective initially arose from the desire to represent in a convincing manner the exteriors and interiors (fig.7 & 8) of buildings, which are, perhaps, the most vital and inspiring of human products. Objects were thought of not only a single entities, but as occupants of a spatial arena. Before it was employed to portray actual buildings, perspective was used to create architectural fictions on which to stage narratives(see$$$$$). Perspective could be used to create more interesting compositions and scale figures among themselves: the viewer could sense space almost fiscally. One of the prime building blocks of perspectival construction was the geometric pavement (fig. 9). "A paved floor, road or piazza, were all ideal grounds on which to lay out a grid of intersecting lines, to establish the base for the correct diminution of forms receding into the pictorial distance. Perspective, therefore, made paintings more architectura.l"3

Annunciation (predella panel from the St. Lucy Altarpiece), Domenico Veneziano, c. 1442-1445, Tempera on panel, 54 x 27.3 cm., Fitzwilliam Museum, Cambridgefig. 8 Annunciation (predella panel from the St. Lucy Altarpiece)
Domenico Veneziano
c. 1442–1445
Tempera on panel, 54 x 27.3 cm.
Fitzwilliam Museum, Cambridge
The Ideal City, Attributed to Fra Carnevale, c. 1480-1484, Oil and tempera on panel, 77.4 x 220. cm., The Walters Art Museum, Baltimorefig. 9The Ideal City
Attributed to Fra Carnevale
c. 1480–1484
Oil and tempera on panel, 77.4 x 220. cm.
The Walters Art Museum, Baltimore
Christ Before Caiphas, Giotto, c. 1305, Fresco, 200 x 185 cm., Scrovegni (Arena) Chapel, Padua, Italyfig. 10 Christ Before Caiphas
Giotto
c. 1305
Fresco, 200 x 185 cm.
Scrovegni (Arena) Chapel, Padua, Italy

The birth of a true, geometrically-based perspective is unique to the Italian Renaissance, and its development spans over the fourteenth and fifteenth centuries. Various trecento artists, such as Duccio di Buoninsegna (c. 1255/1260–c. 1318/1319) and Giotto (c. 1267–1337), had intuited the effectiveness of convergent lines as a means of evoking spatial depth in architectonic features, but unsupported by geometrical consistency. One of the first examples of convergent perspective is considered Giotto's Christ Before the Caïf (1305) (fig. 10), painted 100 years before Fillipo Brunelleschi's perspectival demonstrations. Although the rafters in the ceiling do not converge perfectly at a single vanishing point they are too organized to be the result judgment by eye, as Martin Kemp would point out. Giotto's perspectival understanding was essentially that "lines and planes situated above eye-level should appear to incline downwards as they move away from the spectator; those below eye-level should incline upwards; those to the left should incline inwards to the right; those to the right should incline inwards to the left; there should be some sense of the horizontal division and the vertical division which mark the boundaries between the zones; and along those divisions the lines should be inclined little if at all."4

Last Supper, Duccio di Buoninsegna, c. 1308-1311, Tempera on wood, 50 x 53 cm., Museo dell'Opera Metropolitana del Duomo, Sienafig 11 Last Supper
Duccio di Buoninsegna
c. 1308–1311
Tempera on wood, 50 x 53 cm.
Museo dell'Opera Metropolitana del
Duomo, Siena

Even though the Last Supper(see$$$$$) (fig. 11) and the Death of the Virgin by Duccio exhibit concerted attempts to create a realistic space, in which tangible objects occupy a space that continues beyond the picture, the orthogonals converge at different points. In The Last Supper the recession of the rafters is designed with a wishbone system and the table is titled at a bizarre angle inconsistent with anything else in the image(see$$$$$). Despite these errors, Duccio's approach constitutes a fundamental step forward toward the representation of space of a flat surface.

In its mathematical form, linear perspective is generally believed to have been devised about 1415 by the architect Filippo Brunelleschi (1377–1446) and codified in writing by the architect and writer Leon Battista Alberti (1404–1472), in 1435 (De pictura [On Painting]). The construction worked out by Alberti became was based on the belief that no picture can resemble nature unless it is seen from a definite distance and location, and the diminution in size as a function of distance.

The Healing of the Cripple and Raising of Tabitha, Masolinofig. 12 The Healing of the Cripple and Raising of Tabitha
Masolino
1426–1427
Fresco, 255 x 598 cm. (full fresco)
Cappella Brancacci, Santa Maria del Carmine, Florence
The Annunciation, with Saint Emidius, Carlo Crivelli, 1486, Egg and oil on canvas, 207 x 146.7 cm., National Gallery, Londonfig. 13 The Annunciation, with Saint Emidius
Carlo Crivelli
1486
Egg and oil on canvas, 207 x 146.7 cm.
National Gallery, London

It was not until the mid-1420s that paintings fully designed according to the principles of perspective science began to appear. One of the first accurate employments of precise central convergence was in The Healing of the Cripple and Raising of Tabitha (1426–1427) (fig.12 ), by Masolino da Panicale (c. 1383–c. 1447). In contrast with contemporary empirical attempts to use convergent lines, the orthogonals of the foreground buildings on both sides of the street converge accurately at a single vanishing point. This work contains more than 20 horizontals that converge to an accurate vanishing point, although 4 other lines deviate from this center by a small amount. As other early quattrocento works show, the probability of finding this degree of convergence on the basis of intuitive construction alone is so small as to be negligible.5 Also revealing is the fact that the vanishing point is stationed at the eye level of the standing figures, an occurrence which implies that the viewer observes the scene as he stands within the pictured environment. While Italian paintings following the 1420s display a sense of enthusiastic engagement with perspective construction(see$$$$$) (fig. 13), by the beginning of the sixteenth century enthusiasm waned, with artists presenting more subdued versions of single point perspective, such as Parmigianino's Madonna with a Long Neck. Artists of the fifteenth and sixteenth centuries rarely broke away from simple perspective systems.

Herod's Banquet, Fra Filippo Lippi, between 1452 and 1465, Fresco Duomo, Prato (fig. 14) Herod's Banquet
Fra Filippo Lippi
between 1452 and 1465
Fresco Duomo, Prato

Despite the rapid diffusion of perspective among painters, the perspective of individual objects or figures was generally omitted from the procedure. "Artists could construct the perspective grid that defines the stage and the location on the stage of the actors and props, but they did not explicitly develop the images of objects (other than walls, tables, cornices, stairs and the like) using strict perspective methods. With few exceptions (such as Mantegna, Correggio and Tintoretto), painters throughout the early Renaissance handled figure perspective much more freely (or clumsily) than architectural perspective. In Filippo Lippi's Adoration of the Magii (c.1500) (fig. 14), for example, the front left figure is huge in comparison to those standing just a few feet behind, and the eyes of dancing Salome, in the white dress at left, are at the same height as the seated figures behind her. Even architectural features could be represented with multiple vanishing points. Sandro Botticelli seems sometimes to have done this for dramatic effect, and even emphasized the perspective disparities with strongly foreshortened walls or platforms."6

The School of Athens, Raphael Santi, 1509-1511, Fresco, 500 x 770 cm., Apostolic Palace, Vatican Cityfig. 15 The School of Athens
Raphael
1509–1511
Fresco, 500 x 770 cm.
Apostolic Palace, Vatican City

One of the most consummate examples of the one-point perspective system is Raphael's School of Athens (fig. 15) in the Stanza della Segnatura. Raphael (1483–1520), who himself made no contribution to the theory of perspective. Nonetheless, he brought the practice to its full potential as an artistic tool, and seems to have been one few artists of the time to intuit two-point perspective, in which the horizontals of objects set obliquely to the viewer recede to vanishing points in both directions. "The painter, architect writer and art historian Giorgio Vasari (1511–1574) commented that Bramante (1444–1514), who was the architect of St. Peter's Cathedral under construction at the time, 'instructed Raphael of Urbino in many points of architecture and sketched for him the buildings which he later drew in the perspective in the Pope's chamber, representing Mount Parnassus [i.e., The School of Athens]. Here Raphael drew Bramante measuring with a compass.' Despite this help, Raphael must have had considerable understanding of the construction to be able to execute the imposingly complex vaulting on the curved arches, which are in faultless perspective."7 The School of Athens has often been cited as an outstanding example of the use of a vanishing point to emphasize the significance of the composition. It falls just below the outstretched right hand of the central figure, the aging Plato.

Although comprehending the idea of a uniform space, Northern European painters did not formulate a mathematically based concept of space independently. They began to apply the linear perspective to their pictures only after it was introduced by painters who had traveled to Italy, such as Jan Goessart (c. 1478–1532). Goessart's St Luke Drawing the Virgin(see$$$$$) (fig. 16) demonstrates that by the early 1500s Flemish painters were capable of successfully applying linear perspective to scenes of exceptionally architectural complexity. Previously, Flemish Primitives had used optically based space privileging the physical and sensual representation of man and his environment. The technique of convergence was employed empirically, rather than rationally. This approach is typified by the Arnolfini Portrait by Jan van Eyck (c. 1390–1441), in which different vanishing points were used for the beams of the ceiling, for the window and the bed.

St Luke Drawing the Virgin, Jan Goessart, c. 1515, Oil on oak panel, 230 x 205 cm., Národní Galerie, Praguefig. 16 St Luke Drawing the Virgin
Jan Goessart
c. 1515
Oil on oak panel, 230 x 205 cm.
Národní Galerie, Prague

Albrecht Dürer (1471–1528) was the first Northern artist to embrace perspective whole-heartedly. Although he made no innovations, he was the first Northern European to treat visual representation in a scientific way. In addition to geometrical constructions, Dürer discusses in this last book of Underweysung der Messung (1525) various mechanisms for drawing in perspective from models and provides woodcut illustrations of these methods that were often reproduced in discussions of perspective.

For almost four hundred years after 1500, one-point perspective served as the standard technique for any painter who wished to create a systematic illusion of receding forms on a flat surface, be it canvas, wall or ceiling, although in many cases, perspective remained one of many strands woven into pictures of the time. It was no accident that Gian Paolo Lomazzo (1538–1588), best remembered for his writings on art theory, once asserted that he would rather die than disregard perspective.

Two-Point Perspective

De Artificiali perspectiva..., Jean Pélérin, 1505, printed by Toul, P. Jacques, Parifig. 17 De Artificiali perspectiva…
Jean Pélérin
1505
printed by Toul, P. Jacques, Paris

The elaboration of two-point perspective, necessary to render objects set at an oblique angle to the viewer, took another century to evolve. The first known diagram of the two-point perspective by Jean Pélérin(see$$$$$), in his De Artificiali perspectiva (1505), which was the first printed treatise on perspective.8 Pélérin, who is usually known by the name as "Viator," did not invent the method, but was evidently satisfied to transmit it. His most important statements are that the "central point" (vanishing point) and the two "tier points" (distance points) are located on a line at the level of the eye (horizon line) (fig. 17 & 18) . The major theorist of perspective in sixteenth-century France, Jean Cousin, perfected Viator's "tier point" technique (Livre de Perspective, 1560) and offered an accurate method for foreshortening solid bodies by means of perspective and simple methods to create foreshortening and anamorphic images. It is possible that Raphael was inspired by one of Viator's two-point perspective illustrations(see$$$$$) to elaborate his Coronation of Charlemagne (1516–1517; see image right). But in Raphaels' work there are a total eight different horizontal positions of the vanishing points where there should be two had the whole composition been based on a uniform oblique grid. It would appear that Raphael adopted Viator's particular construction for each part of the scene without understanding how they should be modified to form a coherent perspective projection.9

De Artificiali perspectiva..., Jean Pélérin, 1505, printed by Toul, P. Jacques, Parifig. 18 De Artificiali perspectiva…
Jean Pélérin
1505
printed by Toul, P. Jacques, Paris

"The remarkable feature of angular [two-point] perspective is that, although it was well-understood by geometers such as Viator and Vredemann de Vries (1605), it was avoided by virtually all artists until the middle of the seventeenth century. Aside from two paintings of doubtful attribution painted around 1440, the first successful use of full angular perspective was by Dutch artist Gerard Houckgeest (c. 1600–1661) in 1650. There was limited use of the angular construction in floor tiling throughout the period, but this could easily be achieved by connecting the corners of a one-point perspective grid, and did not require an understanding of the rules of two-point construction. Inspired to develop a radical design for his painting of the tomb of William the Silent, the king whose efforts united Holland in 1581, Houckgeest turned to Vredemann's architectural representational technique of the oblique construction for the interior of the church at Delft. This dramatic shift from the unremitting one-point perspectives of the church interiors of Pieter Jansz. Saenredam (1597–1665) and Pieter Neeffs the Elder (c. 1578–after 1656 before 1661) gained Houckgeest immense popularity in the Netherlands, but it was to be another half-century before the two-point construction appeared in Italy in the hands of Canaletto."10

Inspiring, perhaps, innovative painters such as Poussin, Canaletto and Piranesi, "the Italian theatrical scenery designer Ferdinando Bibiena (1657–1743) gave a new dimension to the renessaince central perspective with his invention of the scena veduta in angolo or prospettivo per angolo, using two or more vanishing points to the sides of the stage picture. This innovation afforded an escape from the symmetry and was picked up by a few Italian designers, but was ignored by neoclassically oriented designers to the north."11

A View of Rome, The Arch of Settimio Severo, Giovanni Battista Piranesi, 1772, Etching on paper, 46.7 x 70 cm. fig. 19 A View of Rome, The Arch of Settimio Severo
Giovanni Battista Piranesi
1772
Etching on paper, 46.7 x 70 cm.

Giovanni Battista Piranesi (1720–1778), who belonged to the group of artists known as the Vedutisti (view painters), revisited many famous views of Rome (fig. 19) that had been commonly interpreted with one-point perspective, replacing it with two-point perspective thereby creating a greater sense of compositional dynamism, widening and accentuating the illusion of reality.

Perspective in the Netherlands

Architectural Capriccio with Jephthah and His Daughter, Dirck van Delen, 1633, Oil on panel, Private collectionfig. 20 Architectural Capriccio with Jephthah and His Daughter
Dirck van Delen
1633
Oil on panel
Private collection

Differently from their southern colleagues, seventeenth-century Dutch artists showed scarce propensity for the theoretical debate. Nonetheless, a range of practical literature on perspective was accessible in the Netherlands by the time Vermeer began to paint. In 1539, the Netherlandish painter and architect Peiter Coeke van Aalst began to publish a Dutch edition of Sabastiano Serlio's Regole generale de Architettura, a key publication that helped to introduce renaissance architecture and perspectival principles to northern Europe. In 1560, Johannes Vredeman de Vries (1527–c. 1607) (fig. 21), the father of the Dutch Perspectivists, a group of painters renowned for their imaginary of palaces (fig. 20), gardens and church interiors, published the first of nine books on the subject, simultaneously in Dutch, Latin, French and German. Vredeman's writing was influential, but he made the mistake of shortening the interval between the central vanishing point and the distance points with the consequence that his architectural scenes give the impression of looking into a funnel(see$$$$$).

Perspective print from: Perspective, c'est a dire, le tresrenomme art du poinct oculaire d'une veue dedans ou travers regardante, estant sur une muraille unie, sur un tableau, ou sur de la toile, en laquelle il y ayt quelques edifices, soyt d'eglises, temples, palais, sales, chambres, galeries, places, allees, jardins, marches & rües..., Vredeman de Vries, Published: The Hague, 1604-1605fig. 21 Perspective print from: Perspective, c'est a dire, le tresrenomme art du poinct oculaire d'une veue dedans ou travers regardante, estant sur une muraille unie, sur un tableau, ou sur de la toile, en laquelle il y ayt quelques edifices, soyt d'eglises, temples, palais, sales, chambres, galeries, places, allees, jardins, marches & rües…
Vredeman de Vries
Published: The Hague, 1604–1605

Many Dutch interior painters made the same mistake, creating checkered-tiled floors that race amusingly away from the viewer toward the vanishing point, seemingly detached from the figures. Hendrick Hondius I (1573–1650), a print-maker and publisher, also produced a manuscript on perspective addressed principally to draftsmen. In 1604, the painter and art theorist Karl van Mander (1548–1606) devoted special attention to linear perspective, although like Hondius he advised those interested in the finer points of the argument to consult books on geometry, perspective and architecture.

To be sure, the Dutch term used for perspective comprises a range of artistic compositions, from see-through views (doorsien or doorsicht), like Vermeer's The Love Letter, to perspective boxes (perspectyfkas), or "peep-shows," as they are imprecisely called. Real and fantasy church interiors and exteriors were also regularly referred to as perspectives (see the works of Bartholomeus an Bassen (c. 1590–1652) (fig. 22) and Dirck van Delen (c. 1605–1671). Both Dutch painters allied perspective with more complex spatial configurations and atmospheric effects to increase the illusion of depth gotten by the earlier Netherlandish precursors, who, instead, had employed only simplistic local coloring and the power of one-point perspective producing, as Walter Liedtke pointed out, the sensation of "airless boxes."

Although Italian artists occasionally employed perspective to portray real buildungs, or parts of real buildings, the overwhelming majority of buildings were, however seemingly realistic, imaginary geometrical constructs, compositional constructs meant to provide a proper and interesting context for narratives, as well as, no doubt, showcase the painter's mastery of this highly esteemed disciplin On the other hand the "avid interest in perspective in the United Provinces most fully expressed itself…not in pictures which imitate the Italian mode but in representations which find a new way of expressing the geometry of perspective within the framework of the direct scrutiny of nature. The way in which Dutch artists from about 1630 succeed in integrating perspective with the direct portrayal of real structures may be seen as the realization of one of the potentialities of Brunelleschi's original invention, a potentiality which had remained largely dormant."12

Interior of a Catholic Church, Bartholomeus van Bassen (figures attributed to Esaias van de Velde), 1626, Oil on canvas, 61 x 83 cm., Gallery Prince Willem, The Haguefig. 22 Interior of a Catholic Church
Bartholomeus van Bassen (figures attributed to Esaias van de Velde)
1626
Oil on canvas, 61 x 83 cm.
Gallery Prince Willem, The Hague

In the Netherlands, linear perspective continued to be a source of great intellectual excitement and bred one of the most avidly collected categories of painting of the time, architectural painting. As an independent motif, architectural painting had its roots in fifteenth-century Flanders, but in the 1630s it burst into a full-fledged school that developed accentuated perspective paintings of townscapes, church exteriors, as well as domestic, renaissance and baroque-style fantasy interiors. The perspective of these works is generally so painstakingly crafted that it dominates all other pictorial concerns, even though contemporary viewers would have found their ornately decorated interior furnishings and delightfully rendered staffage highly attractive. Saenredam single-handedly revolutionized the motif producing light-filled church interiors (fig. 23) (see$$$$$) and exteriors of disarming simplicity, whose formal rigor and monastic atmosphere led a few early critics to claim a spiritual kinship with the interiors of Vermeer.

St Antoniuskapel in the St Janskerk, Utrecht, Pieter Jansz. Saenredam, 1645, 41.7 x 34 cm., Centraal Museum in Utrechtfig. 23 St Antoniuskapel in the St Janskerk, Utrecht
Pieter Jansz. Saenredam
1645
41.7 x 34 cm.
Centraal Museum in Utrecht

After a short walk from Vermeer's studio in Delft to the art collection of his patron Pieter van Ruijven, a Dutch Liefhebber van de Schilderkonst, or "art lover," would have beheld some of the most astonishing pictures of church interiors ever painted. In the works of Emmanuel de Witte (1617–1692) and Houckgeest the massive pillars and soaring arches of Delft's monumental Nieuwe Kerk (fig. 24) are so ingeniously composed and masterfully depicted that the spectator cannot escape sensing, almost physically, their cavernous depths. Both artist employed and bold new perspective stratagem. They exchanged the conventional placing of the vanishing point in the middle of the scene for oblique views relying on the distance-point method(see$$$$$). This stirs movement of the pictorial space and "invites the observer to stroll around in the interior assuming different, but equally important, points of view. As parts of the background are usually not at an equal distance from the picture plane, the sense of space is enlarged."13 Unlike the Italian painters, whose perspectival works tend to be evenly lit, De Witte and Houckgeest relished the momentary play of light and shade, which obscures the architectural logic. We stand outside the Italian views, admirers of the timeless perfection of the imaginary townscape; in de Witte's picture we are participants in the contingent experience of everyday life.14

Interior of the Oude Kerk, Delft, Emanuel de Witte, c. 1650, Oil on wood, 48.3 x 34.6 cm., Metropolitan Museum of Art, New Yorkfig. 24 Interior of the Oude Kerk, Delft
Emanuel de Witte
c. 1650
Oil on wood, 48.3 x 34.6 cm.
Metropolitan Museum of Art, New York

The late John Michael Montias documented that around 1650 the price for a "perspective" was fairly high, at an average of 25.9 guilders a piece compared to the 5.6 guilders for a landscape. A single perspective by the Delft architecture painter Hendrik van Vliet (1611/1612–1675) was valued at 190 guilders, a considerable amount of money for a painting (most likely about the price of a painting by Vermeer). Vermeer's patron, Pieter van Ruijven, owned various works by Delft church painters.

All evidence points to the fact that enthusiasm for perspectival space was as strong for mid-seventeenth century Dutch painters as it had been in the early Renaissance.

Perspective Manuals

De pictura by Alberti, (c. 1474–1475), De Prospectiva pingendi ("On the Perspective of Painting") by Piero della Francesca (c. 1474) and Leonardo da Vinci's Treatise on Painting, were not true manuals but a collection of loose writings in manuscript form, while the first treatise on perspective by a professional artist did not appear in print in Italy until Vignola's Le due regole della Prospettiva Pratica in 1583.

Following the publication of Alberti's De Pictura in France (1651), a number of books on perspective were published, and disagreement concerning the relationship between optics and perspective transformed the matter into a theoretical war. Girard Desargues (1591–1661) and Abraham Bosse (c. 1602–1604) were on one side, and Le Brun and Grégoire Huret on the other, each attempting to establish the principles of correct projection of objects on a two-dimensional surface.

In 1569, the Venetian humanist Daniele Barbaro (1514–1570) published La Practica della perspectiva in 1569. Barbaro's treatise was the first text that brought together in a single book subject matter which until then had been dispersed in works coming from numerous, sometimes unrelated disciplines, and of very different statuses. He complained that painters had stopped using perspective, but what he undoubtedly meant was that painters were no longer painting architectural scenes.

In retrospect, the considerations on perspective brought forth by Alberti and Niceron "were based upon the simplest kind of practical ingenuity, and in some respects were little more than clever carpenter's work. The two solutions were full of implicit mathematical relationships, but the men who used them were content with them as easy contrivances that worked. The mathematical analysis of the perspective problem, and of the special variety of geometry that was implicit in Alberti's novel method of projection and section, seems to have been first undertaken, just about two hundred years after Alberti wrote his treatise, by Desargues, who utilized an assumption by which parallel lines concur at a point at infinity."15 Although the debate led to greater awareness of the problems of rendering spatial depth with a rational system, it was of no use to the practicing painter who needed simple methods for creating a convincing spatial illusion.

In 1822, J. V. Poncelet (1788–1867) published his great classical Traité des proprietes projectives des figures: Ouvrage utile à ceux qui s'occupent des applicationsde la geometriedescriptive et d'operations géolnétriques sur le terrain, in which projective geometry was finally developed into a full-fledged mathematical discipline, free of its original practical function, without which, modern machinery and the industrial revolution could not exist. In effect, it became the technique by which inventions could be made.

In any case, by 1600, no Western European artist who hoped to compete on international scale could not do so without a sound grasp of linear perspective.

† FOOTNOTES †

  1. Erwin Panofsky, Perspective As Symbolic Form, New York: Zone Books, 1991.
  2. Arnheim, Rudolf. Art and Visual Perception: A Psychology of the Creative Eye. Berkeley CA: University of California Press, 2004, 283.
  3. Amanda Lillie, Building the Picturer, "Putting perspective into perspective." https://www.nationalgallery.org.uk/paintings/research/exhibition-catalogues/building-the-picture/constructing-the-picture/putting-perspective-into-perspective
  4. Kemp, Martin, The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat. New Haven: Yale University Press, 1992.
  5. Christopher Tyler and Michael Kubovy, "The Rise of Renaissance Perspective," WebExhibits Science and Art of Perspective. http://www.webexhibits.org/sciartperspective/raphaelperspective1.html
  6. Bruce MacEvoy, One-Point Perspective, 2015 https://www.handprint.com/HP/WCL/perspect2.html
  7. Christopher Tyler and Michael Kubovy, "The Rise of Renaissance Perspective," WebExhibits Science and Art of Perspective.http://www.webexhibits.org/sciartperspective/raphaelperspective1.html
  8. Wlliam M. Ivins Jr., "On the Rationalization of Sight, with an Examination of Three Renaissance Texts on Perspective."Papers: no. 8. Metropolitan Museum of Art, New York, 1938.
  9. Christopher Tyler and Michael Kubovy, "The Rise of Renaissance Perspective," WebExhibits Science and Art of Perspective. http://www.webexhibits.org/sciartperspective/raphaelperspective6.html
  10. Christopher Tyler and Michael Kubovy,"The Rise of Renaissance Perspective," WebExhibits Science and Art of Perspective.http://www.webexhibits.org/sciartperspective/raphaelperspective6.html
  11. Felicia Hardison Londré and Margot Berthold, The History of World Theater: From the English Restoration to the Present (A Frederick Ungar book). London: Continuum Intl Pub Group, 1999, 102.
  12. Kemp, Martin, The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat, New Haven: Yale University Press, 1992, 113.
  13. Klaas van Berkel (ed.), Jørgen Wadum, and K, Zandvliet, K., with contribution from Rudi Ekkart. The Scholarly World of Vermeer. Zwolle: Waanders Publishers, 1996. 45.
  14. Peter Galassi, "Before photography: painting and the invention of photography invention of photography," The Museum of Modern Art, New York Graphic Society, 1981, 13-14.aa
  15. Bruce MacEvoy, Two Point Perspective, 2015. https://www.handprint.com/HP/WCL/perspect3.html