Notice the animated tangent to the parabola in the upper right hand corner. The parabola is undoubtedly the most studied curve in the history of mathematics. A treatise, Conic Sections, written by Euclid (ca. 300 B.C.), has been lost but is thought to have provided a foundation for Apollonius' first four books of the same name. Both the scholar and student can trace this work through the Alexandrian school. Hypatia (d. 415 A.D.), the first woman in the history of mathematics known by name, is said to have made the translation.

These manuscripts can then be traced to Western Europe via Arab conquest, first through North Africa, and then into Spain. Toledo, Cordova and Seville were outstanding centers of learning from the 9th to the llth centuries.

Later, Galileo (1564 - 1642) discovered that a cannonball follows a parabolic path. Scientists, monarchs and military leaders immediately took great interest! The aiming of a cannon became a function of measuring the precise angle of a trajectory, the "throw weight" and its momentum. With these experiments and especially the famous dropping of objects from the top of the Leaning Tower of Pisa, Galileo revolutionized science. He introduced the "scientific method" as a permanent contribution to civilization. Later his heliocentric theory of the universe almost cost him his life.

Descartes, in writing La Géométrie (1637), chose the parabola to illustrate his innovative analytic geometry. At this time in publishing history, all math figures were difficult to create and to print.

In 1992, R. A. Marcus of the California Institute of Technology won the Nobel Prize in Chemistry for his work showing that parabolic reaction surfaces can be used to calculate how fast electrons travel in molecules. His most famous theoretical result, an inverted rate-energy parabola, predicts electron transfer will slow down at very high reaction free energies.

Millions of students in recent centuries will remember the parabola as the introductory curve leading into study of the Calculus.

Glossary

An Informal Glossary of Common Terms
Term	Definition
Caustic	A caustic curve is the envelope of light rays emitted from a point, after reflection or refraction by a given curve. The caustics are either catacaustic as a result of reflection or diacaustic as a result of refraction.
Cusp	If a curve is traced by a moving point, a cusp-point is one where the moving point reverses its direction. See cusp-point and curve sketching.
Envelope	An envelope is a curve, or curves, touching every member of a system of lines or curves. See the illustration of an envelope and also the animation of Neile's Parabola.
Evolute	The envolute of a plane curve is the locus of centers of curvature of another curve, and therefore tangent to all its normals; so called because the other curve (called the involute) can be traced by the end of a string gradually being unwound from it. Moreover, the evolute may also be throught of as the envelope of the normals to the curve. The family of intersecting normals on the interior of a parabola form the evolute of the parabola and is commonly called a cissoid. Also, see the animation of Neile's Parabola.
Hermit Point	Please see the illustration under curve sketching.
Involute	One may think of the involute as the inverse operation of the evolute. Alfred Gray writes, "...the evolute is related to the involute in the same way that differentiation is related to indefinite integration." Thus, the original parabola becomes the involute of its evolute. See the example of an involute of a circle.
Node	Please see the illustration under curve sketching.
Normal	The term "normal" has many usages in mathematics. In curve tracing, a normal is the perpendicular to a tangent of a curve drawn at the point of tangency. Also, see the animation of Neile's Parabola.
Pedal and its Pole	A pedal curve represents the locus of the feet of perpendiculars let fall from a given point to tangent(s) on a given curve or curved surface.
Singularity	Please see the illustrations under curve sketching.

Useful Links and Books

We recommend the following books as having excellent, but varied information on curves and surfaces.
Some of these texts are certain to be in your library and all are available through interlibrary loan.

Eves, Howard. AN INTRODUCTION TO THE HISTORY OF MATHEMATICS, 6th ed., Saunders College Publishing, Harcourt Brace Jovanovich, 1990.
Frost, Percival. An Elementary Treatise on CURVE TRACING, 5th ed., Revised by R. J. T. Bell, Chelsea Publishing Company, 1960.
Gibson, C. G. Elementary Geometry of Algebraic Curves: An Undergraduate Introduction. Cambridge University Press, 1998.
Gray, Alfred. Modern Differential Geometry of Curves and Surfaces with MATHEMATICA® ,2nd ed., CRC Press, 1998.
Hilton, Harold. Plane Algebraic Curves, Oxford University Press, 1932.
Lawrence, J. D. A Catalog of Special Plane Curves, Dover Publications, 1972.
Lockwood, E. H. A Book of CURVES, Cambridge University Press, 1961.
Shikin, Eugene V. Handbook and Atlas of CURVES, CRC Press, 1995.
Yates, Robert. CURVES AND THEIR PROPERTIES, The National Council of Teachers of Mathematics, 1952.