This is the first edition of the original text of the advanced part of the most important work on conic sections written in antiquity and one of the most influential works in mathematics. It is also the first literal English translation of it ever to be published. The purpose of the work is to make available, to those interested in the history of science and to mathematicians, a version of the work as close to the original as possible. This part of Apollonius' Conics is lost in the original Greek, and only an Arabic translation made in the 9th century survives. This text has never been published previously, and all "editions" of this part of Apollonius' work are based on the Latin translation from the Arabic published by Edmund Halley in 1710, which suffers from Halley's insufficient knowledge of Arabic and his use of a single manuscript. The present edition is based on all known manuscripts. Its other improvements over Halley's edition are: 1) the Arabic text with a full critical apparatus; 2) an accurate English translation (until now only a loose paraphrase, based on Halley's translation, has been available in English); 3) a commentary to elucidate both mathematical and historical difficulties. This book will replace Halley's edition and all its derivatives as the standard edition of this part of Apollonius' work.