Conic Sections

To Mr. Miller, my high school math teacher, who first interested me in conics.

The theory of conic sections used to be taught in schools. It still forms part of the classical liberal education at great books schools where the Conic Sections of Apollonius is included in the curriculum. In mainstream education, however, the study of conics has fallen by the wayside.

This is unfortunate. The theory of conics has a long and colorful history: it began, they say, with a religious quandary on the Holy Isle of Delos, and has been contributed to or employed by Archimedes, Apollonius, Kepler, Galileo, Pascal, Newton, and others, in works that are readily available at little or no cost. In addition, there's a wealth of applications in optics, architecture, celestial mechanics, and so forth. But most of these are left to the side in the modern classroom, which admits conics only as the graphs of equations like y = x² and xy = 1, and ignores their most beautiful properties.

What follows is a primer on the basic properties of conics. It should be approachable to an advanced high school student who has taken geometry, but it has also been used in college-level topics courses. The treatment completely eschews the use of coordinates, using synthetic rather than analytic reasoning, and silencing what one mathematician referred to as "the clatter of the coordinate mill."

Complete Table of Contents

Resources for Teachers

The Altar at Delos

An anonymous tragic poet of Greek antiquity once represented the legendary king Minos as erecting a tomb. Minos, dissatisfied with the size of the tomb, which measured 100 feet each way, decided to double its volume by doubling each of the dimensions. No one today knows who this poet was; his work has not survived. The only reason we know about him is that he became notorious among the Greek mathematicians for the error in his reasoning. For, if the height, breadth, and depth of the tomb were all doubled, then the size of the tomb would be octupled, not doubled, because 2 ⋅ 2 ⋅ 2 = 8.

Later on, the same problem of "doubling the cube" arose in a religious quandary. The tiny island of Delos was held as sacred by all the Greeks; during historical times, it was revered as the birthplace of the god Apollo. According to the story, a terrible plague afflicted the people of the island, and they sent representatives to Delphi, the oracle of Apollo on the Greek mainland, to ask the god's advice. In order to end the plague, the oracle replied, the island's craftsmen had only to double the size of the cubical altar of Apollo.

This advice, while easy to state, was not so easy to carry out. The problem was to find a new side length that exactly doubled the volume of the cube. The Delians weren't sure how to solve it, so they sent to the philosopher Plato (c.428 BC – c.348 BC) for help. Plato explained that the oracle's purpose wasn't so much to double the size of the altar as to shame the Greeks for their ignorance of geometry. He then handed the problem over to his colleagues at the Academy. Archytas, Eudoxus, and Menaechmus each provided independent solutions.

The solution of Menaechmus involved mean proportionals. Suppose that the dimensions of the altar are a × a × a. Suppose further that we can find mean proportionals x and y between a and 2a so that a < x < y < 2a and a : x :: x : y :: y : 2a. This reduction of the problem to finding mean proportionals was due to an earlier mathematician, Hippocrates of Chios.

Cross-multiplying, we obtain x² = ay, y² = 2ax, and xy = 2a². Combining these, we find that x³ equals axy, which equals 2a³. So, if x is the side of the new altar, then the volume of the new altar is twice the volume of the old altar. The goal, then, is to find x and y so that x² = ay and y² = 2ax. Taking a = 1, we have x² = y and y² = 2x. Our modern eyes recognize these as the equations of two parabolas in the plane. The value of x is given by their intersection point. It is important to remember, though, that the Greeks had no coordinate geometry; the Cartesian or xy-plane as we know it was developed by René Descartes (1596 – 1650). Our Greek consultants resorted to mechanical means to construct the length, only to earn the censure of Plato for having introduced physical methods into the purely intellectual science of geometry.

Thus began the study of conic sections. The theory was systematized by Apollonius of Perga (c.262 BC – c.190 BC) several generations later. His approach to conic sections was to view them precisely as that—as sections of cones. The alternate focus-and-directrix definition wasn't known for several centuries after his work appeared.

Our exploration of conic sections will use the latter definition as being more in the spirit of high school geometry.

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