Apollonius of Perga was a Greek geometer and astronomer who influenced the development of analytic geometry and substantially advanced mechanics, navigation, and astronomy. He became famous for his astronomical studies in the time of Ptolemy Philopator, who reigned from 221 to 205 B.C. He was known by his contemporaries as “the Great Geometer,” whose treatise Conics is one of the greatest scientific works from the ancient world.
Background
Very little is known of the life of Apollonius. The surviving references from antiquity are meager and in part untrustworthy. He is said to have been born around 262 B.C. at Perga (Greek nepyrj), a small Greek city in southern Asia Minor, when Ptolemy Euergetes was king of Egypt (i.e., between 246 and 221 B.C.).
Education
Little credence can be attached to the statement in Pappus that he studied for a long time with the pupils of Euclid in Alexandria. The best evidence for his life is contained in his own prefaces to the various books of his Conics. From these it is dear that he was for some time domiciled at Alexandria and that he visited Pergamum and Ephesus.
Career
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.
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.
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.
Views
Quotations:
In the preface to the second edition of Conics Apollonius addressed Eudemus:
If you are in good health and things are in other respects as you wish, it is well; with me too things are moderately well. During the time I spent with you at Pergamum I observed your eagerness to become aquatinted with my work in conics.
Apollonius explains in his preface how he came to write his famous work Conics:
... I undertook the investigation of this subject at the request of Naucrates the geometer, at the time when he came to Alexandria and stayed with me, and, when I had worked it out in eight books, I gave them to him at once, too hurriedly, because he was on the point of sailing; they had therefore not been thoroughly revised, indeed I had put down everything just as it occurred to me, postponing revision until the end.