The History of Mathematics: A Brief Course

4. APOLLONIUS^305

readable. On the other hand, Apollonius' work is no longer current research, and

from the historian's point of view this kind of tinkering with the text only makes

it harder to place the work in proper perspective.

In contrast to his older contemporary Archimedes, Apollonius remains a rather

obscure figure. His dates can be determined from the commentary written on the

Conies by Eutocius. Eutocius says that Apollonius lived in the time of the king

Ptolemy Euergetes and defends him against a charge by Archimedes' biographer

Heracleides that Apollonius plagiarized results of Archimedes. Eutocius' informa-

tion places Apollonius reliably in the second half of the third century BCE, perhaps

a generation or so younger than Archimedes.

Pappus says that as a young man Apollonius studied at Alexandria, where he

made the acquaintance of a certain Eudemus. It is probably this Eudemus to whom

Apollonius addresses himself in the preface to Book 1 of his treatise. From Apol-

lonius' own words we know that he had been in Alexandria and in Perga, which

had a library that rivaled the one in Alexandria. Eutocius reports an earlier writer,

Geminus by name, as saying that Apollonius was called "the great geometer" by his

contemporaries. He was highly esteemed as a mathematician by later mathemati-

cians, as the quotations from his works by Ptolemy and Pappus attest. In Book 12

of the Almagest, Ptolemy attributes to Apollonius a geometric construction for lo-

cating the point at which a planet begins to undergo retrograde motion. From these

later mathematicians we know the names of several works by Apollonius and have

some idea of their contents. However, only two of his works survive to this day, and

for them we are indebted to the Islamic mathematicians who continued to work

on the problems that Apollonius considered important. Our present knowledge of

Apollonius' Cutting Off of a Ratio, which contains geometric problems solvable by

the methods of application with defect and excess, is based on an Arabic manu-

script, no Greek manuscripts having survived. Of the eight books of Apollonius'

Conies, only seven have survived in Arabic and only four in Greek.

4.1. History of the Conies. The evolution of the Conies was reported by Pap-

pus five centuries after they were written in Book 7 of his Collection.

By filling out Euclid's four books on the conies and adding four oth-

ers Apollonius produced eight books on the conies. Aristaeus... and

all those before Apollonius, called the three conic curves sections of

acute-angled, right-angled, and obtuse-angled cones. But since all

three curves can be produced by cutting any of these three cones,

as Apollonius seems to have objected, [noting] that some others

before him had discovered that what was called a section of an

acute-angled cone could also be [a section of] a right- or obtuse-

angled cone... changing the nomenclature, he named the so-called

acute section an ellipse, the right section a parabola, and the ob-

tuse section a hyperbola.

As already mentioned, the first four books of Apollonius' Conies survived in

Greek, and seven of the eight books have survived in Arabic; the astronomer Ed-

mund Halley (1656-1743) published a Latin edition of them in 1710.

4.2. Contents of the Conies. In a preface addressed to the aforementioned

Eudemus, Apollonius lists the important results of his work: the description of

the sections, the properties of the figures relating to their diameters, axes, and