Lecture 11. Apollonius and Conic Sections

Home , Apollonius of Perga, Cone, Doubling the cube, Ellipse, Focus (geometry), Hyperbola

Figure 11.1 Apollonius of Perga

Apollonius of Perga Apollonius (262 B.C.-190 B.C.) was born in the Greek city of Perga, close to the southeast coast of Asia Minor. He was a Greek geometer and astronomer. His major mathematical work on the theory of conic sections had a very great inﬂuence on the development of mathematics and his famous book Conics introduced the terms parabola, ellipse and hyperbola. Apollonius’ theory of conics was so admired that it was he, rather than Euclid, who in antiquity earned the title the Great Geometer. He also made contribution to the study of the Moon. The Apollonius crater on the Moon was named in his honor. Apollonius came to Alexandria in his youth and learned mathematics from Euclid’s successors. As far as we know he remained in Alexandria and became an associate among the great mathematicians who worked there. We do not know much details about his life. His chief work was on the conic sections but he also wrote on other subjects. His mathematical powers were so extraordinary that he became known in his time. His reputation as an astronomer was also great. Apollonius’ mathematical works Apollonius is famous for his work, the Conic, which was spread out over eight books and contained 389 propositions. The ﬁrst four books were

69 in the original Greek language, the next three are preserved in Arabic translations, while the last one is lost. Even though seven of the eight books of the Conics have survived, most of his mathematical work is known today only by titles and summaries in works of later authors. Only 2 of 11 his works have survived. According to Pappus 1, Apollonius’ other works include: Cutting of a Ratio, cutting of an Area, determinate Section, tangencies, inclinations, plane Loci. Each of these was divided into two books.

There were some previous work, including Euclid’s and Archimedes’, on conic sections appeared before Apollonius’. Apollonius, however, collected the knowledge of all relevant work and fashioned them systematically. Besides being comprehensive, his Conic contains highly original materials which is ingenious and excellently organized. As an achievement, it practically closed the subject to later thinkers, at least from a purely geometrical standpoint.

Figure 11.2 Conic sections

What are conic sections ? Conic sections (or conics) are the curves obtained by intersecting a circular cone by a plane: hyperbolas, ellipses (including circles), and parabolas. By imaging the meeting of a cone of light from a lamp with the planes of wall, we can understand why these curves were called conical sections. According to Greek philosopher Proclus in his Commentary, Menaechmus, who was a pupil of Eudoxus and a member of Plato’s Academy, discovered these curves around 350 B.C. By the way, it was Apollonius who gave the conical sections their names:

• “hyperbola”, from Greek “hyper”, meaning “some added.”

• “ellipse,” from Greek for “something missing.”

• “parabola,” from Greek word “oaros” for “same.”

1Pappus of Alexandria (290 - 350 B.C.) was one of the last great Greek mathematicians of Antiquity.

70 Today we can express the conic sections in terms of the equations in Cartesian coordinates: x2 y2 − = 1, (hyperbola) a2 b2 x2 y2 + = 1, (ellipse) a2 b2 y = ax2, (parabola). More generally, any second-degree equation represents a conic section or a pair of straight lines, which was a result proved by Descartes (1637). At Apollonius’ time, there were no coordinates, no equations and no analytic geometry. It is hard for us to image how Apollonius could discover and prove hundreds of beautiful and diﬃcult theorems without using coordinates, equations and analytic geometry.

The invention of conic sections is attributed to Menaechmus2. He used conic sections to give a very simple solution of the problem of doubling the cube. In analytic notion, this can 1 2 be described as ﬁnding the intersection of the parabola y = 2 x with the hyperbola xy = 1. Then 1 x · x2 = 1, or x3 = 2. 2 As we know from above, without using equations or coordinates, all the theoretical facts one could wish to know about conic sections had already been worked out by Apollonius. Also it was Apollonius who gave the ellipse, the parabola, and the hyperbola their names.

Kepler’s discovery The Greek theory of harmony said that everything moved in perfect circles or sphere. It was one of the Greek astronomy’s basic principles. However, two thousand years later, Kepler 3(1609) came to discover that the orbits of the planets should be ellipses, not circles.

Kepler convinced himself that Copernicus’s theory represented the correct system of the world. To work out a correct version of the theory in complete detail, he had to access to the observations of Tycho Brahe. Brahe (1546 - 1601), a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations, was the premiere astronomer of his day. Brahe did devise a model of the universe “intermediate” between that of Ptoley and Copernicus, but he was unable to discover the rules. Kepler worked as his assistant for the ﬁnal two years of Brahe’s life and continued to work after Brahe’s death.

2Menaechmus (380 - 320 B.C.), see the footnote in Lecture 6. 3Johannes Kepler (1571 - 1630) was a German mathematician, astronomer and astrologer, and key ﬁgure in the 17th century scientiﬁc revolution.

71 After several years of pursuing false trails, Kepler began to realize that the orbits of the planets were not the circles demanded by Aristotle and assumed implicitly by Copernicus. Kepler initially tried to build his theory explaining the distances of the planets in terms of the ﬁve regular polyhedra (see Figure 11.3). Finally he discovered his three laws of planetary motion. The ﬁrst law says: “The orbit of every planet is an ellipse with the sun at a focus.”

Figure 11.3 Kepler’s diagram of polyhedra