The science called ‘harmonics’ was one of the major intellectual enterprises of Greek antiquity. Ptolemy’s treatise seeks to invest it with new scientific rigour; its consistently sophisticated procedural self-awareness marks it as a key text in the history of science. This book is the first sustained methodological exploration of Ptolemy’s project. After an analysis of his explicit pronouncements on the science’s aims and the methods appropriate to it, it examines Ptolemy’s conduct of his complex investigation in detail, concluding that despite occasional uncertainties, the declared procedure is followed with remarkable fidelity. Ptolemy pursues tenaciously his novel objective of integrating closely the project’s theoretical and empirical phases and shows astonishing mastery of the concept, the design and (it is argued) the conduct of controlled experimental tests. By opening up this neglected text to historians of science, the book aims to provide a fresh point of departure for wider studies of Greek scientific method.