Introduction: pt.I. The earlier history of conic sections among the Greeks. 1. The discovery of conic sections; Menaechmus.2. Aristeaus and Euclid. 3. Archimedes.--pt.II. Introduction to the conics of Apollonius. 1. The author and his own account of the conics. 2. General characteristics. 3. The methods of Apollonius. 4. The construction of a conic by means of tangents. 5. The three-line and four-line locus. 6. The construction of a conic through five points. Appendix: Notes on the terminology of Greek geometry. The conics of Apollonius.