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. 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
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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.”
1
Pappus 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 coordinates:
x2 y 2
− 2 = 1, (hyperbola)
a2
b
2
x
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
be described as finding the intersection of the parabola y = 12 x2 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.
2
Menaechmus (380 - 320 B.C.), see the footnote in Lecture 6.
Johannes Kepler (1571 - 1630) was a German mathematician, astronomer and astrologer, and key figure
in the 17th century scientific revolution.
3
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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
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