Greek Text and Annotated Translation -- Greek Text -- Annotated Translation of Collectio IV -- Commentary -- Plane Geometry, Apollonian Style -- Plane Geometry, Apollonian Style -- Plane Geometry, Archaic Style -- Plane Geometry, Archimedean -- Motion Curves and Symptoma-Mathematics -- Meta-theoretical Passage -- Angle Trisection -- General Angle Division -- Quadratrix, Rectification Property -- Analysis for an Archimedean Neusis.

Although not so well known today, Book 4 of Pappus’ Collection is one of the most important and influential mathematical texts from antiquity, both because of its content and because of its impact on early modern mathematics after 1600. As a kind of textbook in anthology format, the mathematical vignettes form a portrait of mathematics during the Hellenistic "Golden Age", illustrating central problems – for example, it discusses all three of the famous ancient problems in geometry: squaring the circle; doubling the cube; and trisecting an angle – varying solution strategies, and the different mathematical styles within ancient geometry. This volume provides an English translation of Collection 4, in full, for the first time, including: a new edition of the Greek text, based on a fresh transcription from the main manuscript and offering an alternative to Hultsch’s standard edition; notes to facilitate understanding of the steps in the mathematical argument; a commentary highlighting aspects of the work that have so far been neglected, and supporting the reconstruction of a coherent plan and vision within the work; bibliographical references for further study. Historians of mathematics will find it useful for scholarly work on ancient geometry and its reception in the early modern era and it will also serve as a source book for exemplary arguments in ancient geometry. Pappus himself probably intended Collection 4 to be an introductory survey of the classical geometrical tradition – from the point of view of mathematical methods and strategies – for readers that had a basic training in elementary geometry (Elements I – VI). Likewise, this edition can be used as a textbook in advanced undergraduate and graduate courses on the history of ancient geometry.