With the fifth book of the Konika ancient mathematics reached a climax. In it Apollonius presents the first known theory of the maxima and minima lines, which was taken up again by mathematicians in the early 10th century, and above all in the 17th century. As is the case for books 6 and 7, the original Greek text of this book is lost, and it is only known from an Arabic translation made in Baghdad in the 9th century. This volume presents the reader with a truly critical edition of the fifth book of Apollonius' Konika, an exact translation that follows the original wording closely, as well as a broad and detailed historical and mathematical commentary, as befits the edition of such an important text.