This is a preliminary draft of an outline for a learning unit which is proposed for the 9th grade of the Ross School, Spring term, 1998.

Conics

by

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

Abstract. For various reasons, the transmission of the Conics of the ancient Greeks (Euclid, Archimedes, and Apollonius) was associated with Perspectiva. Several artists took up the subject, and we begin to see conics represented in the paintings of the Renaissance (esp. Giotto, Fra Angelico, and Durer). But later, the art/math connection evaporates, and the transformation from the geometric to the dynamical mentality appears as a chemical reaction involving two different branches of math: algebra and geometry. In this unit we trace the reactants and their immediate product: analytic geometry. The explosion of the calculus from this point on is reserved for another unit.

CONTENTS
1. Introduction
2. Geometry and conics, 300 BC to 1600 AD
3. Algebra and geometry, 300 BC to 1600 AD
4. Analytic geometry, 1637 AD
5. Chronology
6. Some instructive paintings
Bibliography

1. Introduction.
The invention of conics is lost in the early history of Greek Geometry. Euclid wrote a treatise on the subject, but is was lost. We know of it from the later works of Archimedes and Apollonius, and their translations by the Arabs, which carried the theory 16 centuries through time to the European Renaissance, where it arrived in a package with Perspectiva. See (Clagett, 1980). 
From time to time there occurred a leakage from the seclusion of the mathematical specialists into the artistic community, who recorded visual evidence of the theory in their paintings. In the preceding unit we paid attention especially to these leaks, which stitch together the mathematical and the artistic, characteristically, in the Renaissance. In the context of Conics, the artists triggered a upsurge of interest in conics on the part of mathematicians, and then they fade from the story.
In this unit we follow primarily two branches of mathematics, geometry and algebra, which come together in 1637 in the analytic geometry of Fermat and Descartes. This is the essential background to understand the shift from the geometric to the dynamical mentality, the main event of the 9th grade of the spiral curriculum, which is the subject of the next unit.
Acknowledgments. Special thanks to Victor Katz, who suggested a unit on conics, and gave pointers to its history.

2. Geometry and conics, 300 BC to 1600 AD
Ancient Greek geometry leads naturally to conics. Conics are conic sections, that is, intersections of a cone and a plane. One motive for the development of the theory of conics was their usefulness in Optics. Geometry, conics, and optics: all were recorded by Euclid around 300 BC.
Euclid’s theory was improved by Apollonius, known as the Geometer in his own time. He gave us the three names: ellipse, parabola, and hyperbola, and obtained most of the theory as we have it today.
Beginning with the translations from Greek and Syriac into Arabic after 800 AD, the Arabs preserved and extended conics and optics. As in the unit on Perspectiva, it was the text of Alhazen which first brought conics to Medieval Europe. (See the chronology.) 
The Renaissance begins with the connection of mathematics and the arts. Conics made an early appearance in the paintings of Giotto, along with his rediscovery of natural (ie, intuitive) perspective. The ellipse in particular, is a circle seen obliquely. 
The theory of Apollonius became broadly known through the Perspectiva of Witelo, which became a very popular text after 1270 AD. A version of this, edited by Regiomontanus, was published as a book in 1544.
Suggested activities:
	• Construct a parabola by the pointwise method (Katz, 1993, p. 109)
	• Experiment with the optics of a parabolic mirror
	• Construct an ellipse by the method of a string and two pins
	• Trace curves from a Renaissance painting and see if they are conics
	• construct an ellipse by the method of Durer (Katz, 1993, p. 359), (Field, 1997, p. 182), (Kemp, 1990, p. 55)

3. Algebra and geometry, 300 BC to 1600 AD
Conics were probably discovered while trying to solve an algebraic problem geometrically, see (Katz, 1993, p. 108). To understand this, the best approach may be the geometric algebra of Euclid’s Elements, Books I and II. See (Katz, 1993, p. 64). Here problems of Babylonian algebra were solved geometrically. This literature, on arrival in Early Islam, triggered the development of algebra, see (Katz, 1993, p. 228). Imported into Europe by Leonardo of Pisa around 1200, the subject was radically improved by Viete around 1590. Perhaps the key to this development was a novel system of symbolization, similar to the role played by staff notation n the history of music at the same time. See (Crosby, 1997).
Note: A main principle of the cultural historical curriculum, the Sheldrake principle, asks for the introduction of new ideas in historical order. The historical precedence of geometry before algebra is a most important example, as the violation of historical order in the usual school math program is seen as a causative factor of math anxiety. So it is very important to reach this point in the inclusive math program before the Viete system of algebraic notation is introduced in the exclusive math program.
Suggested activities:
	• Solve a quadratic equation by the geometric method of the Babylonians. (Katz, 1993, p. 31).
	• Solve the same equation by the constructions of Euclid’s geometric algebra (Katz, 1993, p. 64)
	• Solve the same equation by the methods of Al-Khwarizmi (Katz, 1993, p. 231)
	• Solve the same equation by the methods of Descartes (Katz, 1993, p. 408)

4. Analytic geometry, 1637 AD
A coordinate system for Euclidean geometry is a means of locating a point in space by numerical data. The introduction of coordinates, together with the symbolic methods of Viete, empowered a completely new method for studying geometry. Fermat and Descartes, independently, developed this approach in 1637, which is now known as analytic geometry. Among other things, it enormously simplifies the study of conics, and makes the discoveries of Apollonius easy to understand and to use. 
Suggested activities:
	• How did Giotto determine the correct width of a hat?
	• Find the equation for an ellipse in a painting by Giotto (Edgerton, 1996)
	• Prove a theorem of Apollonius using analytic geometry
	• Trisect an angle or double a cube

5. Chronology 
Here is a chronology of the conics thread abstracted from (Clagett, 1980), combined with a chronology of algebra extracted from (Katz, 1993), and with some dates from our unit 9M-01, Perspectiva.

0300 BC, Euclid, Optica, Conics
0250 BC, Archimedes, On conoids and spheroids and other works
0200 BC, Apollonius, Conics
.........
0825 AD, Al-Khwarizmi, Al-Jabr
1039 AD, Alhazen, Kitab al-Manazir
1200 AD, Gerard of Cremona, translation of Alhazen from Arabic into Latin
1225 AD, John of Palermo, translation of an arabic tract on the hyperbola,
1269 AD, William of Moerbeke, translation of Archimedes from Greek into Latin
1270 AD, Witelo, Perspectiva, a popular manuscript on perspective, optics, and conics
1410 AD, Giotto, natural perspective used in paintings
1434 AD, Fra Angelico, natural conics used in paintings
1450 AD, Jacobus Cremonensis translation of Archimedes
1525 AD, Durer, book on conics for artists
1543 AD, Copernicus, De revolutionibus
1544 AD, Regiomontanus version of Jacobus appears as a printed book, popular
1591 AD, Viete, Introduction: algebra as we know it today
1596 AD, Kepler, regular solids
1604 AD, Kepler, conics and optics
1609 AD, Kepler, elliptical orbits
1637 AD, Fermat and Descartes independently develop analytic geometry
1638 AD, Galileo, parabolic trajectories
1684 AD, Leibniz’ calculus
1687 AD, Newton’s calculus

6. Some instructive paintings
Conics appear only rarely in paintings. The ellipse, as it represents a circle seen obliquely, is occasionally seen. Be on the lookout for hats with round brims, halos, and angel wings.
Giotto, Second Modillion Border, 1310
Van Eyck, Ghent Altarpiece, 1415
Fra Angelico, Crotona Annunciation, 1434
Durer, St. Jerome in His Study, 1514

BIBLIOGRAPHY
Boyer, Carl B., A History of Mathematics, Princeton: Princeton University Press, 1968.
Boyer, Carl B., The History of the Calculus and its Conceptual Development, New York: Dover Publications, 1949/1959.
Clagett, Marshall, Archimedes in the Middle Ages, Volume Four: A Supplement on the Medieval Latin Traditions of Conic Sections (1150-1566), Part I: texts and Analysis, Philadelphia: The American Philosophical Society, 1980.
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., 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.
Thomas, Brian, Geometry in Pictorial Composition, Newcastle: Oriel Press, 1971.