The aim of this book is to explain the shape of Greek mathematical thinking. It can be read on three levels: as a description of the practices of Greek mathematics; as a theory of the emergence of the deductive method; and as a case-study for a general view on the history of science. The starting point for the enquiry is geometry and the lettered diagram. Reviel Netz exploits the mathematicians' practices in the construction and lettering of their diagrams, and the continuing interaction between text and diagram in their proofs, to illuminate the underlying cognitive processes. A close examination of the mathematical use of language follows, especially mathematicians' use of repeated formulae. Two crucial chapters set out to show how mathematical proofs are structured and explain why Greek mathematical practice manages to be so satisfactory. A final chapter looks into the broader historical setting of Greek mathematical practice.

The transformation of mathematics from its ancient Greek practice to its development in the medieval Arab-speaking world is approached by focusing on a single problem proposed by Archimedes and the many solutions offered. From a practice of mathematics based on the localized solution (originating in the polemical practices of early Greek science), we see a transition to a practice of mathematics based on the systematic approach (grounded in the deuteronomic practices of Late Antiquity and the Middle Ages). A radically new interpretation is accordingly offered of the historical trajectory of pre-modern mathematics.

"The key methodological commitment of this book, then, is to the study of literary culture as a whole. From this follow certain other methodological consequences. Studying literary culture as a whole implies an attention to overall patterns more than to individual details. I thus regularly offer statistics and maps. While I do make many qualitative pronouncements, these are, with a few exceptions, generalizing and impressionistic, and not based on close readings. Studying the regularities of literary culture as a whole also implies adopting a perspective which need not have been available to the ancient actors themselves (they had pursued their own practices, without necessarily pausing to consider their literary culture as a whole). For this reason, I make no effort to identify the authors' concepts and am content to deploy my own observer's concepts throughout. The result is a book very different from traditional classical philology. The complex footnote with its plethora of primary and secondary sources is almost entirely avoided (my footnotes, instead, merely point to the key, recent studies with which the reader, wishing to pursue a point of detail, may begin her research ). Only rarely do I offer close readings of individual passages or reconstructions of the meanings of original terms. Fortunately for me, traditional philology now has few champions and I will not spend time arguing against the straw man of the philological critic. Instead, I will apologize to him "--