Apollonius of Perga

Apollonius was a great mathematician, known by his contempories as " The Great

Geometer, " whose treatise Conics is one of the greatest scientific works from the ancient world.

Most of his other treatise were lost, although their titles and a general indication of their contents

were passed on by later writers, especially Pappus of Alexandria.

As a youth Apollonius studied in Alexandria ( under the pupils of Euclid, according to

Pappus ) and subsequently taught at the university there. He visited Pergamum, capital of a

Hellenistic kingdom in western Anatolia, where a university and library similar to those in

Alexandria had recently been built. While at Pergamum he met Eudemus and Attaluus, and he

wrote the first edition of Conics. He addressed the prefaces of the first three books of the final

edition to Eudemus and the remaining volumes to Attalus, whom some scholars identify as King

Attalus I of Pergamum.

It is clear from Apollonius' allusion to Euclid, Conon of Samos, and Nicoteles of Cyrene

that he made the fullest use of his predecessors' works. Book 1-4 contain a systematic account

of the essential principles of conics, which for the most part had been previously set forth by

Euclid, Aristaeus and Menaechmus. A number of theorems in Book 3 and the greater part of

Book 4 are new, however, and he introduced the terms parabola, eelipse, and hyperbola. Books

5-7 are clearly original. His genius takes its highest flight in Book 5, in which he considers

normals as minimum and maximum straight lines drawn from given points to the curve

( independently of tangent properties ), discusses how many normals can be drawn from

particular points, finds their feet by construction, and gives propositions determining the center

of curvature at any points and leading at once to the Cartesian equation of the evolute of any

conic.

The first four books of the Conics survive in the original Grrek and the next three in

Arabic translation. Book 8 is lost. The only other extant work of Apollonius is Cutting Off of a

Ratio ( or On Proportional Section ), in an Arabic translation. Pappus mentions five additional

works, Cutting off an Area ( or On Spatial Section ) , On Determinate Section, Tangencies, and

Plane Loci.

Tangencies embraced the following general problem : given three things, each of which

may be a point, straight line, or circle, construct a circle tangent to the three. Sometimes known

as the problem of Apollonius, the most difficult case arises when the three given things are

circles.

Of the other works of Apollonius referred to by ancient writers, one, On the Burning

Mirror, concerned optics. Apollonius demonstrated that parallel light rays striking a spherical

mirror would not be reflected to the center of sphericity, as was previously believed. The focal

properties of the parabolic mirror were also discussed. A work on entitled On the Cylindrical

Helix is mentioned by Proclus. Apollonius also wrote Comparison of the Dodecahedron and the

Icosahedron, considering the case in which they are inscribed in the same sphere. According to

Eutocius, in Apollonius' work Quick Delivery, closer limits for the value of Pi than the

3 1/7 and 3 10/71 of Archimedes were calculated. In a work of unknown title Apollonius

developed his system of tetrads, a method for expressing and multiplying large numbers. His On

Unordered Irrationals extended the theory of irrationals originally advanced by Eudoxus of

Cnidus and found in Book 10 of Euclid's Elements.

Lastly, from references in Ptolemy's Almagest, it is known that Apollonius introduced the

systems of eccentric and epicyclic motion to explain planetary motion. Of particular interest

was his determination of the points where a planet appears stationary.