| 28 | Any Way You Slice It Conic Sections |
This is a tale of how pursuit of a problem posed by a Greek oracle unlocked a secret of the solar system and helped to start the Scientific Revolution. It is a case study in how a mathematical tool, once discovered, can prove useful in unexpected ways.
A legend from ancient Greece tells how the people of the island of Delos, afflicted by a plague, appealed to the oracle of Apollo. The oracle told them to double the size of Apollo's cubical altar. They built a new altar — some say by putting two cubes together, others say by doubling the length of each side. The plague continued, and the Delians realized that the oracle wanted a cubical altar of exactly twice the volume of the original. When they turned to Plato for advice, he told them that "by this oracle [Apollo] enjoined all the Greeks to leave off war and contention, and apply themselves to study, and... to live peacably with one another, and profit the community."1 That is how the problem of "duplicating the cube" — finding a geometric construction for what we would call — became known as the Delian Problem.
Greek mathematicians attacked this problem in various ways. Before Plato's time, Hippocrates of Chios had shown how the Delian Problem reduces to finding line segments that form two mean proportionals. Specifically, if a is the side length of the original altar and if there are segments of lengths x and y such that the three ratios a : x, x : y, and y : 2a are equal, then x will be the side length of the doubled altar.
The next step came half a century later. In the last half of the 4th century B.C., the Macedonians under Alexander conquered all the eastern Mediterranean from Greece to Egypt and as far east as central India. Many of those regions came to share in the unifying influence of a common language, Greek. Many ancient mathematicians we now call Greek came from various parts of that sprawling empire. Menaechmus. from an area that is now part of Turkey, was one of several mathematicians to solve the problem of finding two mean proportionals. He did it using curves obtained from three-dimensional geometry, by slicing a cone. Each of the curves represented the equality of two of the ratios, so intersecting two of them yielded the desired line segment.
The person most closely tied to curves formed by slicing cones is Apollonius of Perga (in Turkey), the "Great Geometer" of the third century B.C., who built on the ideas of Menaechmus, Aristaeus, and Euclid to unify these curves in a single, elegant theory. Apollonius began with a double cone formed by rotating two intersecting lines around an axis that bisects their angle of intersection. When the cone is cut by a plane that does not pass through the tip of the cones, the points of intersection on the plane form a curve called a conic section, or simply a conic.
By rotating the plane so that its angle with the axis changed from perpendicular to parallel, Apollonius produced all four kinds of conic sections — circle, ellipse, parabola, and hyperbola. The last three of these names were derived from earlier work on properties of areas. Loosely translated, they imply "too little," "just right," and "too much."2 (Shades of Goldilocks!)
Here is why Apollonius chose those names. For each type of curve, he looked at the line segment formed by arbitarily choosing a point P on it and dropping a perpendicular to the major axis. He compared the area of a square built on that segment with a rectangle formed by a segment equal in length to what we now call the latus rectum and the segment of the major axis determined by the end of the perpendicular from P and the nearest vertex of the curve. If the areas are exactly equal, the curve is a parabola; if the square area is too small or too large, the curve is an ellipse or a hyperbola, respectively, and the exact shape could be described in terms of how much smaller or larger the area was.
Parabola; areas equal
Apollonius's Conics was a long, detailed account of conic sections and their uses. It encompassed earlier work, but it went far beyond what others had done. During the following several centuries of Greek scholarship, others expanded the theory of conics and applied it to various problems. Notable among them was Pappus of Alexandria, a mathematician of the early 4th century A.D. who used properties of the focus and directrix of a hyperbola, to find a method for trisecting an angle.3 But Apollonius's Conics remained the definitive work on conic sections for a very long time. Four of its eight books have survived directly from the Greek versions, and three others have come down to us by way of translations from Arabic; one book has been lost.
It is important to keep in mind that the Greeks' treatment of conics, and of geometry in general, was entirely "synthetic." That is, it dealt with lines, squares, and various other shapes, not with numbers or coordinates. The algebra and coordinate geometry that we take for granted were nearly 2000 years in the future. Comparisons of lengths, areas, and the like involved proportionality, not explicit measurement. In hindsight, one can see in Apollonius's work some powerful antecedents of a coordinate treatment of conics, but the machinery of analytic geometry simply didn't exist then.
In the several centuries following the collapse of the Graeco-Roman Empire and the spread of Islam around the Mediterranean Sea and eastward, many Greek mathematical writings were translated into Arabic. In the 9th and 10th centuries, Arab4 and Persian mathematicians began to develop the algebra of equations. This early algebra, done entirely in words, began by listing and solving quadratic equations. The next step was the search for ways to solve cubic equations (which are, in fact, extensions of the Delian Problem). A prominent player in this game was 'Umar al-Khāyammī,'5 a 12th-century Persian scholar whose writings spanned poetry, mathematics, science, and philosophy. Building on the work of several Arab mathematicians of the previous century, al-Khāyammī solved several different forms of cubic equations using the method of Menaechmus, i.e., by intersecting two conic sections.
There is some evidence that Apollonius's Conics was known in Europe as far back as the 13th century, when Erazmus Witelo used conics in his book on optics and perspective. Nevertheless, there was little new work on the subject for a long time. Perhaps it was simply a matter of Apollonius being hard to read and giving the impression that he had basically done it all already.
The Renaissance of European art and learning began in earnest during the 15th century. When Constantinople fell in 1453, some Greek scholars brought their manuscripts and knowledge to the cities and new universities of the West. Movable-type printing was invented, making it much easier to disseminate new ideas. And there were lots of new ideas, including Nicolaus Copernicus's new theory of the solar system, published early in the 16th century. This was a watershed period for early European science and mathematics. Algebra was becoming systematically symbolic (though not yet standardized) and decimal representation of fractions was spreading rapidly in the scientific community. Napier was developing logarithms (see Sketch 27) and Galileo was using the parabola to describe projectile motion.
German astronomer Johannes Kepler was living in Graz in the summer of 1600, when there was a solar eclipse. He assembled in the city square a large wooden instrument that he used to observe the eclipse. It was a kind of large-size pinhole camera, designed to allow him to see and measure what was happening without hurting his eyes. Now, light can do funny things when it goes through a pinhole, and Kepler's observation led him to wonder about some questions of optics. After all, astronomical observations were all done by looking at the sky, and weird optical effects might affect the precision of those observations.
Soon after that, Kepler had to leave Graz. He ended up in Prague, where he worked at the Imperial Observatory. At first he was Tycho Brahe's assistant, but Brahe died in 1601 and Kepler inherited both his job and an immense amount of observational data that Brahe had collected over many decades. Using this scientific treasure-trove, he set out to try to work out the orbits of the planets. But that work went slowly, and he decided to write out his work on optics first.
He started by reading a book that contained two famous works on optics, one by Ibn al-Haytham (also known as Alhazen) and the other by Witelo. Perhaps it was because they both quoted Apollonius that Kepler decided he needed to read that, as well. Luckily, a good Latin translation by Commandino had been published in 1566. It was hard work. In 1603, he wrote to a friend that "All the Conics of Apollonius had to be devoured first, a job which I have now nearly finished."6 Kepler's Optics was finally published in 1604.
They say that chance favors the prepared mind. As Kepler was doing all this, he was also studying the planet Mars. A convinced Copernican, Kepler assumed at first, like everyone else, that the orbit had to be a circle with the Sun somewhere near the center. But after examining Tycho Brahe's observations, Kepler realized that the orbit of Mars did not fit a circular pattern: it was flatter than a circle, some kind of oval. Of course, he thought of the conic sections! By further careful observation and measurement, he concluded that the orbit must be an ellipse with the sun at one focus. He asserted that all planets have such orbits and set out three "laws" that described the orbits in detail. He also argued that there must be some sort of force exerted by the Sun on the planets that caused them to move along such orbits.
The 17th century was a time of great progress in understanding motion. As a result, it became possible to work out whether a central force would produce orbits that followed Kepler's laws. Kepler had suggested that the force would get smaller with distance. By the 1680s, the more common guess was that the force should get smaller with the square of the distance, but no one knew how to relate the nature of the force to the shape of the orbit.
In 1684, British astronomer Edmund Halley visited Isaac Newton to discuss this problem. Newton told him that he knew that an inverse-square central force would lead to Kepler's elliptical orbits. Halley was surprised and asked him how he new. Newton's answer was simply "I have calculated it." It was Halley's request that Newton write up this calculation that led. after a few years, to the publication of Newton's masterpiece, Philosophiae Naturalis Principia Mathematica.7 Kepler's laws were thereby confirmed, and thus the conic sections were inscribed in the heavens!
Of course, there had been progress between 1600 and 1680. Mostly, it had to do with two radically new methods. The first was the coordinate geometry invented by Descartes and Fermat. (See Sketch 16.) As Fermat showed, every equation of degree 2 in two variables describes a conic section. His proof boils down to showing that the curve defined by such an equation always had one of the properties that Apollonius called the "symptoms" of the conics.
The other new method was projective geometry. Girard Desargues and Blaise Pascal extended the principles of perspective drawing to create a new geometry. In Euclidean geometry, two figures are congruent if one is the rigid-motion image of the other. In projective geometry, one figure is allowed to be transformed into another by a "projection." Think of images projected onto a screen from a light source passing through a film. If a circle is on the film, the rays passing through the circle from the point of light form a cone. If the screen is parallel to the film, the image is a (bigger) circle. As you tip the screen, the image changes into an ellipse, then a parabola, then a hyperbola. Thus, in projective geometry, all four kinds of conic sections are essentially "the same" because you can get from one to another by a projective transformation. (See Sketch 20.)
Coordinate geometry and projective geometry come together in what became known as "algebraic geometry." the geometry of figures defined by polynomial equations. The conic sections — the family of all curves described by two-variable polynomials of degree 2 — are the simplest algebraic curves. They serve as model, motivation, and source of questions for the study of other algebraic curves and surfaces.
The modern world has seen increasingly sophisticated applications of conic sections — parabolic reflectors and optical lenses, elliptical satellite orbits, hyperbolic radio-wave navigation — undoubtedly with more to come. Amid that impressive diversity, there is a striking unity to this powerful quartet of curve families. Emerging from a single problem, they are unified by the geometry of their constructions and the algebra of their descriptions. Moreover, their utility is a testament to the value of pure curiosity. In the words of the 19th-century British mathematician J. J. Sylvester,
"But for this discovery [of conic sections], which was probably regarded in Plato's time and long after him, as the unprofitable amusement of the speculative brain, the whole course of practical philosophy of the present day... might have run in a different channel; and the greatest discovery that has ever been made in the history of the world, the law of universal gravitation, ... might never to this hour have been elicited."8
For a Closer Look: Apollonius's Conics and Kepler's Optics are both available in English translations. Coolidge's [31] is showing its age, but remains a good source of references to original work. All the textbooks include discussions of the ancient work, but tend to get thinner on the later material.
1 This version of the story is told by Plutarch in De Genio Socrates 579. See [113, p. 399]
2 For more on the etymology of these names, see [10, Capsule 61].
3 This was another of the three "Problems of Antiquity," closely related to the Delian Problem. See page 21.
4 As it was with "Greek", the term "Arab" refers to someone who lived somewhere in the Islamic Empire and wrote in Arabic.
5 Perhaps better known these days as Omar Khayyám, author of the Rubáiyát.
7 Literally, ''mathematical principles of natural philosophy." What Newton's title is saying is that he has developed a mathematical theory of the physics of motion.
8 from "A Probationary Lecture on Geometry," [175, vol. 2. p. 7]