LINEAR PERSPECTIVE

Anonymous, The Garden of Nebammun, ca. 1400 BC, fresco.

Ma Yuan, Landscape in Moonlight, ca. 1200, hanging scroll, ink, and color on silk.

Anonymous, The Battles of Hogen and Heiji, Edo period, 17th century.

Anonymous, The Persian Prince Humay meets the Chinese Princess Humayun in her garden, ca. 1430-40, tempera on parchment.

Anonymous, Madonna and Child on a Curved Throne, ca. 1280, tempera on wood.

Altarpieces = symbolic world
Hierarchical scale - scale by importance, not proximity or physical size
Abstracted backround - gold = divine light, a spiritual dimension
Figures in architectural niche
Frontal planes sometimes overlapping
Increasing realism of figures

Anonymous, Christ as Ruler of the Universe, the Virgin, and Child, and Saints, ca. 1190, mosaic. Cathedral of Monreale, Sicily.

Giotto di Bondone, Frescoes, Arena Chapel, Padua. 1305-06.

View of the Arena Chapel, Padua, with Giotto’s frescoes.

Two of Giotto’s fresco panels in the Arena Chapel.

Giotto di Bondone, The Mourning of Christ, ca. 1305, fresco. The Arena Chapel, Padua.

By the 13th Century Giotto was painting scenes in which he was able to create the impression of
depth by using certain rules which he followed.

He inclined lines above eye-level downwards as they moved away from the observer, lines
below eye-level were inclined upwards as they moved away from the observer, and similarly
lines to the left or right would be inclined towards the centre.

Although not a precise mathematical formulation, Giotto clearly worked hard on how to represent
depth in space and examining his pictures chronologically shows how his ideas developed.

Some of his last works suggest that he may have come close to the correct understanding of
linear perspective near the end of his life.

- John J O'Connor and Edmund F Robertson, Mathematics and Art - Perspective.

Giotto di Baldone, Raising of Drusia, ca. 1325, fresco. Santa Croce, Florence.

The niche becomes a more complete spatial illusion.

Carefully painted transitions increasingly integrate architecture into illusion of the painted space.

An increase of oblique planes, gives a greater sense of form and dimension to things, although sometimes not spatially coherent, not reharmonized to 2-D surface, conflicting demands of realism vs surface pattern.

Forms, figures described by physical light and shadow.

A multiplicity of things, not quite a coherent system of shapes.

Ambrogio Lorenzetti, Allegory of Good Government, ca. 1340, fresco.Palazzo Publico, Siena.

Cimabue, San Matteo, (detail), 1280-83. Fresco.

Pietro Lorenzetti, Birth of the Virgin,
1342, tempera on wood panel.

The person who is credited with the first correct formulation of linear perspective is Filippo
Brunelleschi (1377-1466.) He understood that there should be a single vanishing point to which
all parallel lines in a plane, other than the plane of the canvas, converge.

Also important was his understanding of scale, and he correctly computed the relation between
the actual length of an object and its length in the picture depending on its distance behind the
plane of the canvas.

Using these mathematical principles, he drew two demonstration pictures of Florence on wooden
panels with correct perspective. One was of the octagonal baptistery of St John, the other of the
Palazzo de Signori.

To give a more vivid demonstration of the accuracy of his painting, he bored a small hole in the
panel with the baptistery painting at the vanishing point. A spectator was asked to look through
the hole from behind the panel at a mirror which reflected the panel.

In this way Brunelleschi controlled precisely the position of the spectator so that the geometry
was guaranteed to be correct. These perspective paintings by Brunelleschi have since been lost
but a "Trinity" fresco by Masaccio from this same period still exists which uses Brunelleschi's
mathematical principles.

- John J O'Connor and Edmund F Robertson, Mathematics and Art - Perspective.

Masaccio, Holy Trinity with the Virgin, St. John and Donors, ca. 1425-28, fresco.

Brunelleschi understood the mathematical rules involving the vanishing point that we have
described above, he did not write down an explanation of how the rules of perspective work. The
first person to do that was Leone Battista Alberti (1404 - 1472) in his treatise On Painting.

Alberti gives background on the principles of geometry, and on the science of optics. He then
sets up a system of triangles between the eye and the object viewed which define the visual
pyramid referred to above. He gives a precise concept of proportionality which determines the
apparent size of an object in the picture relative to its actual size and distance from the observer.

The most mathematical of all the works on perspective written by the Italian Renaissance artists
in the middle of the 15th century was by Piero della Francesca. In some sense this is not
surprising since as well as being one of the leading artists of the period, he was also the leading
mathematician writing some fine mathematical texts.

In Trattato d'abaco which he probably wrote around 1450, Piero includes material on arithmetic
and algebra and a long section on geometry which was very unusual for such texts at the time. It
also contains original mathematical results which again is very unusual in a book written in the
style of a teaching text. Is there a connection with perspective? Yes there is, for Piero illustrates
the text with diagrams of solid figures drawn in perspective.

- John J O'Connor and Edmund F Robertson, Mathematics and Art - Perspective.

Piero della Francesca, Ideal City, c. 1470. Tempera on panel.

Piero della Francesca, The Flagellation, c. 1455. Oil and tempera on panel.

First is sight, that is to say the eye; second is the form of the thing seen; third is the distance
from the eye to the thing seen; fourth are the lines which leave the boundaries of the object and
come to the eye; fifth is the intersection, which comes between the eye and the thing seen, and
on which it is intended to record the object.

- Piero della Francesca, Short Book on the Five Regular Solids, ca. 1460-1470.

PERSPECTIVE MACHINES

Leonardo Da Vinci, Draughtsman Using a Transparent Plane to Draw an Armillary Sphere,1510.

Illustration of Leonardo’s perspective grid.

Illustration from the book The Practice of Perspective, by Jean Dubreuil, 1642,
showing an artist using a perspective glass.

Albrecht Durer, Artist using a glass to take a portrait, 1525, woodcut.

Albrecht Durer, Draughtsman using a net to draw a nude figure in foreshortening, 1538, woodcut. 

Albrecht Durer, The painter studying the laws of foreshortening, 1525, woodcut. Draughtsmen plotting points for the drawing of a lute in foreshortening.

Ucello, The Battle of San Romano, ca. 1452, tempera on wood.

Illustration from the book The Practice of Perspective by Jean Dubreuil,
1642, demonstrating foreshortening.

Leonardo da Vinci, The Last Supper, 1498. Fresco.

The refectory of the monastery of St. Maria delle Grazie, Milan.

Analysis of the perspective of Leonardo’s Last Supper.

Leonardo da Vinci, The Last Supper, 1498. Fresco.

Giovanni Bellini, Madonna with Saints, 1505, oil on wood and canvas.

Correggio, The Assumptin of the Virgin, ca. 1530, fresco. Parma Cathedral.

Fra Andrea Pozzo, The Glorification of Saint Ignatius, 1691-94, fresco. St. Ignazio, Rome.

THE CAMERA OBSCURA

CAMERA OBSCURA DEVELOPMENTS
Camera = Latin for “room”. Obscura = Latin for “dark”

5th C. B.C. China -
References to pinholes in screens revealing an understanding of image formationtranslated as “collecting place”, “locked treasure room.”

The earliest known written evidence of a camera obscura can be found in Aristotle's
documentation of a device in 350 BC in Problemata" (Patti, 1993). Aristotle's apparatus
contained a dark chamber that had a single small hole to allow for sunlight to enter.

With this device, he made observations of the sun. He noted that no matter what shape the hole
was, it would still display the sun correctly as a round object.

Another observation that he made was that when the distance between the aperture (the tiny
hole) and the surface with the image increased, the image would become amplified.

Although no one is perfectly sure, many attribute the invention of the camera obscura to
Aristotle.
- Michael Lahanas, Greek Optics.

11C. A.D.. - Arabian scholar Alhazen in his work on optics, which later became the main source book of Roger Bacon and other European scholars, mentions the use of pinhole camera obscuras for studying solar eclipses.

- The earliest printed account is by Cesare Cesariano, a pupil of Leonardo da Vinci, described in an annotation to a manuscript in which the image of everything outside the room can be seen. Leonardo had already written two descriptions of the camera obscura in his notebooks

Close all shutters and doors until no light enters the camera except through the lens, and
opposite hold a piece of paper, which you move forward and backward until the scene appears
in the sharpest detail.

There on the paper you will see the whole view as it really is, with its distances, its colors and
shadows and motion, the clouds, the water twinkling, the birds flying.

By holding the paper steady you can trace the whole perspective with a pen, shade it and
delicately color it from nature.
- Leonardo Da Vinci

Cubicle-type camera obscura illustrated by Athanasius Kircher in 'Ars Magna Lucis et Umbrae',
1646.. The first cameras were enormous. Kircher described one which consisted of an outer
shell with lenses in the centre of each wall, and an inner shell containing transparent paper for
drawing; the artist needed to enter by a trapdoor.

Light travels in a straight line and when some of the rays reflected from a bright subject pass
through a small hole in thin material they do not scatter but cross and reform as an upside down
image on a flat surface held parallel to the hole.

Camera obscura room, 1752.

Camera obscura room, 1754.

In the mid sixteenth century Giovanni Battista della Porta (1538-1615) published what is
believed to be the first account of the possibilities as an aid to drawing.

It is said that he made a huge "camera" in which he seated his guests, having arranged for a
group of actors to perform outside so that the visitors could observe the images on the wall.

The story goes, however, that the sight of up-side down performing images was too much for
the visitors; they panicked and fled, and Battista was later brought to court on a charge of
sorcery!

Alexandre Saverien, Tent, Room, and Book Camera Obscuras, 1753, engraving.

CAMERA OBSCURA DEVELOPMENTS

1558 - The fullest and best description of the camera obscura was published by a Neapolitan scientist, Giovanni Battista della Porta, in which for the first time it was recommended as an aid in drawing.

- The first significant improvement to the camera obscura was the insertion of a biconvex lens in the aperture to form a brighter image. Its use was recommended by Girolamo Cardano, a Milanese physician.

- Danielo Barbaro, a Venetian nobleman, mentioned that by adding lens diaphragms of various sizes the image could be sharpened.

1573 - Egnatio Danti, a Florentine mathematician and astronomer made known a further improvement of adding a concave mirror to redress the hitherto inverted image.

Portable Camera Obscuras, 1685.

Robert Hooke, Camera Obscura, 1694.

Camera obscura with extended lens for close-up view.

Johann Zahn, Reflex camera obscura,1685.

A reflex camera obscura.

Camera obscura tent.

The term "camera obscura" was first used by the German astronomer Johannes Kepler in the
early 17th century. He used it for astronomical applications and had a portable tent camera for
surveying in Upper Austria.

All purpose sedan chair camera obscura, 1711.

Camera obscura for copying drawings, 1727.

Camera obscura used to copy views of skeletons for the book, Osteographia, 1753.

French salon camera obscura, ca. 1700.

Camera obscura table, n.d.

Book camera obscura, ca. 1750.

Camera obscura book, 1764.

Jurrian Andriessen, Camera Obscura, ca. 1800, ink & wash drawing.

Portable Camera Obscura, n.d.

The goblet camera obscura, 1685.

Above is a print from the 1877 Frank Leslie's Popular Monthly showing a scene inside
the Central Park camera obscura.

Giant Camera Obscura of San Francisco

Flyer, Giant Camera Obscura of San Francisco.

JOHANNES VERMEER
1632 - 1675

Jan Vermeer, The Art of Painting, ca. 1666-67, oil on canvas.

Jan Vermeer, The Art of Painting, detail, ca. 1666-67, oil on canvas.

Jan Vermeer, The Art of Painting, detail, ca. 1666-67, oil on canvas.

Jan Vermeer, View of Delft, ca. 1660-61, oil on canvas.

Jan Vermeer, View of Delft, detail, ca. 1660-61, oil on canvas.

Circles of confusion in the left image above.

Jan Vermeer, The Kitchen Maid, ca. 1600, oil on canvas.

Jan Vermeer, The Music Lesson, ca. 1662-63, oil on canvas.

Jan Vermeer, Girl with the Red Hat, ca. 1666-67, oil on canvas.

Jan Vermeer, The Girl with the Red Hat, detail, ca. 1666-67, oil on canvas.

It is impossible to express the beauty [of the image] in words. The art
of painting is dead by comparison, for this is life itself, or something
more elevated, if we could find a word for it.
- Constantijn Huygens, 1622

Above all there is the pleasure of seeing the movement of birds, men, or
other animals and the quivering of plants waving in the wind; for although
all that is reversed, nevertheless, this beautiful painting, beyond being
foreshortened in perspective, represents ingeniously well that which no
painter has ever been able to represent in his painting, to realize
movement continued from place to place.
J. Leurechon, ca. 1620.

I am certain that vision from these reflections in the dark can give no
small light to the sight of the young artists; because besides gaining
knowledge of nature, so one sees here what main or general
(characteristics) should belong to truly natural painting.
- Samuel van Hoogstraten, 1678

Vermeer’s Girl with a Red Hat is painted on a hard panel unlike his other
paintings.

His conscious attempt to exploit optical phenomena visible in a camera
obscura -- intense colors, accentuated contrasts of light and dark, and
halation of highlights -- suggests that Vermeer sought to re-create the
impression of such an image.

He may have decided to paint on a hard, smooth surface to lend to his
small study the sheen of an image seen in a camera obscura as it is
projected onto a ground glass or tautly stretched oiled paper, as was
done in the mid-seventeenth century.

Even though he must have referred to an image from a camera obscura
when painting Girl With the Red Hat and sought to exploit some of its
optical effects, it is most unlikely that he traced its image directly on the
panel. The possibilities that he traced his more complex compositions
are even more remote..

There is no essential difference, other than the physical materials with
which they are made, between this painting and a contemporary
photograph. They are both representations of the three dimensional
world on a two dimensional surface, an instantaneous moment of time
captured in a still image, containing the illusion of space, light, color,
form, and movement.

The most mathematical of all the works on perspective written by the Italian Renaissance artists
in the middle of the 15th century was by Piero della Francesca. In some sense this is not
surprising since as well as being one of the leading artists of the period, he was also the leading
mathematician writing some fine mathematical texts.

In Trattato d'abaco which he probably wrote around 1450, Piero includes material on arithmetic
and algebra and a long section on geometry which was very unusual for such texts at the time. It
also contains original mathematical results which again is very unusual in a book written in the
style of a teaching text. Is there a connection with perspective? Yes there is, for Piero illustrates
the text with diagrams of solid figures drawn in perspective.

Development of the Room and the Box Camera Obscuras.

Illustration of wide angle vision as explored by Leonardo.

Lateral recession of a wall in front of the observer as demonstrated by Leonardo.

Benozzo Gozzoli, The Journey of the Magi to Bethlehem, ca. 1459-63, fresco.

Paul and Jean de Limbourg, May, ca. 1410, tempera on parchment. Tres Riches Heures manuscript page.

Anonymous, The Bayeux Tapestry, ca. 1080, tapestry.

Anonymous, Still Life with Eggs and Thrushes, ca. 79 A.D., fresco. Pompeii.