This is revision 2 of an outline for a learning unit which is in development for the 9th grade of the Ross School, Fall term 1997.

Perspectiva

by

Ralph H. Abraham, abraham@vismath.org
Visual Math Institute, http://www.vismath.org
POB 7920, Santa Cruz, CA, USA-95061-7920

Abstract. This outline traces a thread from ancient geometry and optics, through medieval perspectiva (geometrical optics, physiology and psychology of vision) to the rediscovery of linear perspective in the Renaissance, and the crucial interaction of mathematics, science, and the visual arts. It is intended as a learning theme for an integrated course at the 9th grade level.

CONTENTS
1. Introduction
2. Ancient geometry and optics
3. Medieval perspectiva
4. Renaissance perspective
5. Chronology and correlates
6. Some instructive paintings
Bibliography

1. Introduction.
This thread is among the most important in the history of art, science, math, and their interconnections. Like archeoastronomy, it is a crucial thread. Regarding this importance, some quotations and paraphrases:
	• The geometrization of space distinguishes modern science from its precedents. (Needham, p. 11)
	• The application of algebra to geometry in the 17th C. was the greatest single step ever taken in the progress of the exact sciences. (Mahoney, p. 15)
	• Columbus was motivated by the first printed Ptolemaic map of the world. (p. 17)
	• In every historical age, the current concept of physical and metaphysical space is manifest in its visual arts. (Panofsky, p. 35)
	• There was a revolutionary shift in methods of scientific inquiry after the year 1100, coeval with the shift in the visual arts from two- to three-dimensional space. (Panofsky, p. 36)
	• The Elements and Optics of Euclid were translated into Latin in the 12th C. (p. 42)
All of these are taken from (Edgerton, 1991; Ch. 1) on the pages given above.
	• It is essential to ask of artistic periods and regions not only whether they have perspective, but also which perspective they have. (Panofsky, 1997, p. 41)
Panofsky makes clear the differences between several kinds of perspective: natural, angle, curved, linear, etc.
	• The epochal shift from qualitative to quantitative perception in Western Europe during the late Middle Ages and Renaissance made modern science, technology, business practice, and bureaucracy possible. (Crosby, 1997, frontis)
Crosby analyzes this epochal paradigm shift, under the heading visualization, in three parts: music, painting, and bookkeeping. This unit on perspective focuses on one of these: painting.
Acknowledgements. It is a pleasure to thank Andra Akers and Paul Lee for their extensive help with this outline.

2. Ancient geometry and optics
Our thread begins in remote prehistorical times, in the conception of space and its representation in the visual arts of the early cultural ecologies. But we will pick it up in ancient Greece, where we find its first historical record in the Elements and Optics of Euclid, and later works by Ptolemy, Apollonius, and others. The minimum background of this material required to understand the evolution of Greek math, science, and art into the Renaissance and modern times may be considered to be one construction from each of the first four books of Euclid’s Elements. My recommendation would be:
	• Construction #12 (from Book I, Prop. 44: application of areas)
	• Construction #15 (Book II, Prop. 11: golden section)
	• Construction #18 (Book III, Prop. 17: draw a tangent to a circle)
	• Construction #33 (Book IV, Prop. 11: inscribe a pentagon in a circle)
These are all from plane geometry. The minimum of solid geometry might be a single construction from any of the books XI, XII, or XIII. My recommendation:
	• Construction #57 (Book XIII, Prop. 13: inscribe a pyramid in a sphere)
These need only include the constructions (including all subroutines) as found on my website, and not the proofs. More ambitious students should also know some proofs, especially that of the final proposition of Book XIII (there only five Platonic solids), and the three conic sections.
Texts: (Abraham, 1997), (Katz, 1993)

3. Medieval perspectiva
The sequence of development here may be regarding as beginning with Ibn al-Haytham (965-1039, also known as Alhazen in the Latin corpus) who advanced the subject substantially beyond the Greek sources in his long and abtruse work, Kitab al-Manazir, 1039 A.D., translated into Latin as De aspectibus or Perspectiva around the year 1200. (Perspectiva, from the Latin perspicere, “to see through”, or better, “to see clearly”) The subject matter includes:
	• the theory of vision
	• the anatomy and physiology of the eye
	• the psychology of perception
	• the formation of images by refraction and reflection
	• the theory of the rainbow
	• experimental science
From Alhazen’s appearance in Latin translation around 1200 begins the European thread with:
	• Robert Grosseteste, various, 1235
	• Roger Bacon, Perspectiva, 1262
	• Witelo, Perspectiva, 1269
	• John Pecham, Perspectiva communis, 1277
Pecham’s text was a simple instructional manual, and was enormously popular throughout the 14th and 15th C. It was primarily geometrical, and refers frequently to Euclid and Aristotle. It was used as a text in courses of Perspectiva in the main universities: Vienna, Paris, Prague, Leipzig, Alcacla, Salamanca, Cracow, and Wurzburg. Most likely this was the trigger for the next evolutionary leap, in the Renaissance. An interesting development concerns the change from extromission (visual rays emitted by the eye) to intromission (light rays focussed by the eye) which is still controversial to this day (eg, recent results of Rupert Sheldrake).
Artists involved in the early application of perspectiva to natural perspective include:
	• Giotto, 1290 
	• Duccio, 1301
	• Lorenzetti, 1342
Texts: For full details, see all of the items in the Bibliography by David Lindberg, also (Edgerton, 1991; esp. p. 58), and (Panofsky, 1977).

4. Renaissance perspective
The usual account of the rediscovery of linear perspective (Edgerton, 1975) gives credit to Brunelleschi (Filippo di Ser Brunellesco, 1377-1446) in 1413. For a competing theory, see (Kubovy, 1986, p. 38) who credits Alberti with the rediscovery in 1435. Here, for simplicity, we follow the conventional account, in which the main steps are:
	• Brunelleschi, two demonstrations, 1413: Perspective drawing superimposed on an actual view of the Baptistry in Florence by a mirror, rediscovered the linear perspective of Ptolemy. (See Edgerton, 1991; p. 34).
	• Donatello, 1417: among the earliest applications to painting
	• Mosaccio, 1425: ditto.
	• Alberti, De pictura, 1435; Della Pittura, 1436: first texts on perspective drawing.
	• Ghiberti, writings, 1435: refers to Alhazen, Bacon, Witelo, Pecham.
	• Piero della Francesca, De prospectiva pingendi, 1474: advanced treatment including the application of algebra (abaco) to analyze the geometry. Luca Pacioli was his student.
	• Leonardo da Vinci, Annunciation, 1472 (and later paintings) and various books, from 1470, in one of which he included the following quote from Pecham:
Among the studies of natural causes and laws, it is light that most delights its students. Among all the great branches of mathematics, the certainty of its demonstrations pre-eminently elevates the minds of its investigators. Perspective, therefore, should be preferred above all man’s discourses and disciplines. In this subject the visual rays are elucidated by means of demonstrations which derive their glory not only from mathematics but also from physics; the one is adorned equally with the flowers of the other. 
Texts: Most texts on the history of mathematics touch briefly on the discovery of perspective by Renaissance artists. For example, see (Katz, 1993, p. 357), or (Boyer, 1968, p. 322). For a more complete account see (Edgerton, 1975), (Edgerton, 1991), (Kemp, 1990), and (Wright, 1983). Piero’s work on perspective is well described and illustrated in (Kemp, 1990). For a study of Piero’s other contributions to mathematics, see (Davis, 1977) and (Rose, 1975). The theory of Panovsky (1924) on the relativity of linear perspective is disputed in (Edgerton, 1975) and defended in (Kubovy, 1986, Ch. 10).

5. Chronology and correlates
Here is a chronology of the optics thread abstracted from (Edgerton, 1975. p. xv), which gives a bit more detail.
	• 300 BC, Euclid’s Optica
	• 25 BC, Vitruvius’ De architectura
	• 140 AD, Ptolemy’s Optica, and Geographia
	• 175 AD, Galen’s De usu partium
	• 1000 AD, Alhazen’s Perspectiva
	• 1260 AD, Roger Bacon’s Opus Majus
	• 1270 AD, John Pecham’s Perspectiva communis
	• 1390 AD, Blasius’ Quaestiones perspectivae
	• 1400 AD, first arrival of Ptolemy’s Geographia in Florence
	• 1424 AD, return of Toscanelli to Florence
	• 1425 AD, Brunelleschi’s demonstrations, Masaccio’s Trinity
	• 1435 AD, Alberti’s text
Edgerton describes in much detail the role of perspectiva in cartography, and thus, in the Voyages of Discovery.

6. Some instructive paintings
This list of works is taken from (Kemp, 1990), who shows the early works and analyses their geometry, except as noted.
	• Giotto?, Second Modillion Border, 1290 (See Edgerton, 1991, for this one)
	• Duccio, Last Supper, 1301 (Panofsky, 1977)
	• Duccio, Temptation of Christ on the Temple, 1308
	• Giotto, Confirmation of the Rule of St. Francis, 1325
	• Lorenzetti, Birth of the Virgin, 1342
	• Brunelleschi, two demonstration panels, 1413 (rediscovery of linear perspective)
	• Jan van Eyck, Office of the Dead, 1415 (Panofsky, 1997)
	• Donatello, St. George and the Dragon, 1417
	• Donatello, Feast of Herod, 1423
	• Masaccio, Trinity, 1426 (first surviving painting using linear perspective)
	• Ghiberti, Story of Jacob and Esau, 1435 (sculpture)
	• Piero della Francesca, Flagellation of Christ, 1460
	• Uccello, The Selling of the Host, 1468
These paintings and the accompanying drawings could be scanned for ready use by students.
Texts: See (Kemp, 1990), (Edgerton, 1991), (Field, 1997), (Panofsky, 1997).

BIBLIOGRAPHY
Abraham, Ralph H., The Euclid Project, www.vismath.org/euclid, 1997.
Boyer, Carl B., The Rainbow: From Myth to Mathematics, Princeton: Princeton University Press, 1959/1987.
Boyer, Carl B., A History of Mathematics, Princeton: Princeton University Press, 1968.
Crosby, Alfred W., The Measure of Reality: Quantification and Western Society, 1250 — 1600, Cambridge: Cambridge University Press, 1997.
Edgerton, Samuel Y., The Renaissance Rediscovery of Linear Perspective, New York: Harper & Row, 1975.
Edgerton, Samuel Y., The Heritage of Giotto’s Geometry: Art and Science on the Eve of the Scientific Revolution, Ithaca: Cornell University Press, 1991.
Davis, Margaret Daly, Piero della Francesca's Mathematical Treatises, Ravenna: Longo, 1977.
Field, Judith V., The Invention of Infinity: Mathematics and Art in the Renaissance, Oxford: Oxford University Press, 1997.
Kemp, Martin, The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat. New Haven: Yale University Press, 1990.
Kubovy, Michael, The Psychology of Perspective and Renaissance Art, Cambridge: Cambridge University Press, 1986.
Lindberg, David C., John Pecham and the Science of Optics, Perspectiva Communis, Madison: University of Wisconsin Press, 1970.
Lindberg, David C., Late thirteenth-century synthesis in optics, in: Edward Grant, ed., A Source Book in Medieval Science, Cambridge: Harvard University Press, 1974; pp. 392-434.
Lindberg, David C., Theories of Vision from al-Kindi to Kepler, Chicago: University of Chicago Press, 1976.
Lindberg, David C., The science of optics, in: David C. Lindberg, ed., Science in the Middle Ages, Chicago, University of Chicago Press, 1978.
Lindberg, David C., Roger Bacon’s Philosophy of Nature, Oxford: Oxford University Press, 1993.
Lindberg, David C., Roger Bacon and the Origins of Perspectiva in the Middle Ages, Oxford: Oxford University Press, 1996.
Panofsky, Erwin, Perspective as Symbolic Form, New York: Zone Books, 1997.
Pirenne, M. H., Optics, Painting, and Photography, Cambridge: Cambridge University Press, 1970.
Rose, Paul Lawrence, The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from Petrarch to Galileo, Geneve: Droz, 1975.
Shlain, Leonard, Art and Physics: Parallel Visions in Space, Time, and Light, New York: Morrow, 1991.
Thomas, Brian, Geometry in Pictorial Composition, Newcastle: Oriel Press, 1971.
Wright, Lawrence, Perspective in Perspective, London: Routledge & Kegan Paul, 1983.