The Greek mathematician Apollonius of Perga (active 210 B.C.) was known as the "Great Geometer." He influenced the development of analytic geometry and substantially advanced mechanics, navigation, and astronomy.

# Apollonius of Perga

Very little is known about the life of Apollonius, the last great mathematician of antiquity. He was born at Perga in Pamphylia, southern Asia Minor, during the reign (247-222 B.C.) of Ptolemy Euergetes, King of Egypt. When he was quite young, Apollonius went to study at the school in Alexandria established by Euclid.

Apollonius's fame in antiquity was based on his work on conics. His treatise on this subject consisted of eight books, of which seven have survived. Like most of the well-known Greek mathematicians, Apollonius was also a talented astronomer.

Apollonius had Euclid's great collection, the *Elements,* available and was thus able to draw upon the work of all previous major mathematicians. Also, Euclid's own work on conics, now lost, was a basis for Apollonius's further work.

## Conics of Apollonius

The *Conics* was written book by book over a long period of time. The general preface to the work is given in Book I. Apollonius next outlines the contents of the eight books. The first four books are an "elementary introduction," that is, elementary in that they include those properties that are necessary to any further specialization. These books are thus an extension of the earlier conics by other mathematicians such as Euclid. Since most of these results were already well known, one might expect Apollonius's presentation to be more concise and to attempt a greater logic and generality. Beginning with Book V, more advanced topics are taken up. Book V is perhaps the best of the latter four.

## Other Works

A number of other works by Apollonius are mentioned by ancient writers, but only one exists in its entirety today. The work, *Cutting-off of a Ratio,* was found in an Arabic version, and a Latin translation was published in 1706. It is concerned with the general problem: given two lines and a point on each of them, draw a line through a given point cutting off segments on the lines (measured from the fixed points on the lines) which have a given ratio to each other.

Another treatise, *Cutting-off of an Area,* was concerned with the same problem as the previous treatise except that the segments cut off were to contain a given rectangle or, in modern terms, have a given product.

Of a similar nature was the treatise *On Determinate Sections.* Here the general problem was: given a line with four points *A, B, C,* and *D* on it, determine a fifth point *P* on the line such that the product of lengths *AP* and *CP* is a given constant times the product *BP* and *DP.* The determination of
point *P* is equivalent to solving a quadratic equation and is no great challenge. But the treatise apparently included more elaborate considerations.

The treatise *On Contacts* (or *Tangencies*) was devoted to the general problem: given three things (points, straight lines, or circles) in position, draw a circle which passes through the points (if any) and is tangent to the lines and circles (if any). For example, if two points and a line are given, then the problem would be to draw a circle through the two points and tangent to the given line. There are ten possibilities; two of them were already in Euclid's *Elements.* Six cases were treated in Book I of *On Contacts,* and Book II dealt with the remaining two, including the most difficult case of three circles. To draw a circle tangent to three given circles became known as the Apollonian problem.

Another treatise was *On Plane Loci.* Restorations of this have been attempted by many geometers. It was presumably concerned with straight lines and circles only and with the problem of showing, given certain conditions on a point, that the point must lie on a straight line or a circle.

A work in applied geometry, *On the Burning-mirror,* was probably about the properties of a mirror in the shape of a paraboloid of revolution. Even though the property is not mentioned by Apollonius in his treatise, he probably knew that light entering such a mirror parallel to its axis is reflected to a single point, its focal point.

Apollonius was also known as a great astronomer. In the *Almagest,* the great astronomical work by Ptolemy (2d century A.D.), Apollonius is mentioned as having proved two important theorems. These theorems, dealing with epicycles and eccentric circles, enabled the points on the planetary orbits to be determined where the planets, as seen from the earth, appeared stationary.

## Further Reading on Apollonius of Perga

The standard English translation of Apollonius's principal work, with modern mathematical notation, is Thomas L. Heath, ed., *Apollonius of Perga: Treatise on Conic Sections* (1896). Apollonius's work is described and analyzed by Heath in *A Manual of Greek Mathematics* (1931) and by Bartel L. van der Waerden in *Science Awakening* (1950; trans. 1954). For Apollonius's place in the development of analytic geometry see Carl B. Boyer, *History of Analytic Geometry* (1956).