Lecture 11. Apollonius and Conic Sections

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. Hismajor mathematical work on the theory of conic sections had a very great influence on thedevelopment 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 thanEuclid, who in antiquity earned the title the Great Geometer. He also made contributionto 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’ssuccessors. As far as we know he remained in Alexandria and became an associate amongthe great mathematicians who worked there. We do not know much details about his life. Hischief work was on the conic sections but he also wrote on other subjects. His mathematicalpowers were so extraordinary that he became known in his time. His reputation as anastronomer was also great.

Apollonius’ mathematical works Apollonius is famous for his work, the Conic, whichwas spread out over eight books and contained 389 propositions. The first four books were

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in the original Greek language, the next three are preserved in Arabic translations, whilethe last one is lost. Even though seven of the eight books of the Conics have survived, mostof his mathematical work is known today only by titles and summaries in works of laterauthors. Only 2 of 11 his works have survived. According to Pappus 1, Apollonius’ otherworks 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 sectionsappeared before Apollonius’. Apollonius, however, collected the knowledge of all relevantwork and fashioned them systematically. Besides being comprehensive, his Conic containshighly original materials which is ingenious and excellently organized. As an achievement, itpractically 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 byintersecting 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 canunderstand why these curves were called conical sections. According to Greek philosopherProclus in his Commentary, Menaechmus, who was a pupil of Eudoxus and a member ofPlato’s Academy, discovered these curves around 350 B.C. By the way, it was Apolloniuswho 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.

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Today we can express the conic sections in terms of the equations in Cartesian coordi-nates:

x2

a2− y2

b2= 1, (hyperbola)

x2

a2+

y2

b2= 1, (ellipse)

y = ax2, (parabola).

More generally, any second-degree equation represents a conic section or a pair of straightlines, which was a result proved by Descartes (1637). At Apollonius’ time, there wereno coordinates, no equations and no analytic geometry. It is hard for us to image howApollonius could discover and prove hundreds of beautiful and difficult theorems withoutusing coordinates, equations and analytic geometry.

The invention of conic sections is attributed to Menaechmus2. He used conic sections togive a very simple solution of the problem of doubling the cube. In analytic notion, this canbe described as finding the intersection of the parabola y = 1

2x2 with the hyperbola xy = 1.

Then

x · 1

2x2 = 1, or x3 = 2.

As we know from above, without using equations or coordinates, all the theoretical facts onecould wish to know about conic sections had already been worked out by Apollonius. Alsoit 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 perfectcircles or sphere. It was one of the Greek astronomy’s basic principles. However, twothousand years later, Kepler 3(1609) came to discover that the orbits of the planets shouldbe ellipses, not circles.

Kepler convinced himself that Copernicus’s theory represented the correct system of theworld. To work out a correct version of the theory in complete detail, he had to accessto the observations of Tycho Brahe. Brahe (1546 - 1601), a Danish nobleman known forhis accurate and comprehensive astronomical and planetary observations, was the premiereastronomer of his day. Brahe did devise a model of the universe “intermediate” betweenthat of Ptoley and Copernicus, but he was unable to discover the rules. Kepler worked ashis 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.

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After several years of pursuing false trails, Kepler began to realize that the orbits of theplanets 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 ofthe five regular polyhedra (see Figure 11.3). Finally he discovered his three laws of planetarymotion. 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

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