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 treatises 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, eclipse, 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 Greek 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.

**Bibliography **

1. Boyer, Carl B. , The History of Analytic Geometry (1956) McGraw – Hill

2. Heath, Thomas L. , Manual of Greek Mathematics (1921; repr. 1981)

3. Van der Waerden, Bartel L., Science Awakening (1961).

## Leave a Reply