The aim of this book is to explain the shape of Greek mathematical thinking. It can be read on three levels: first as a description of the practices of Greek mathematics; second as a theory of the emergence of the deductive method; and third 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 mathematicans' 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. R N is a Research Fellow at Gonville and Caius College, Cambridge, and an Affiliated Lecturer in the Faculty of Classics.

Did science, as we know it, have to be? The article explores a possible response in the negative, organized around a specific contingency: that of Greek mathematics or, even more specifically, that of the mathematics of the generation of Archimedes. The argument is that (1) Greek mathematics, seen against a cross-cultural comparison, is an anomaly, (2) the scientific revolution, as it in fact unfolded, was directly shaped by the anomaly of Greek mathematics, and (3) it is not clear that, absent Greek mathematics, an equivalent scientific revolution would have taken place. The argument is developed in some detail concerning the history of ancient Greek science, but more is said on the inevitably philosophical questions of counterfactuals in history and on the specific question of the contingency of science.

The transformation of mathematics from ancient Greece to the medieval Arab-speaking world is here approached by focusing on a single problem proposed by Archimedes and the many solutions offered. In this trajectory Reviel Netz follows the change in the task from solving a geometrical problem to its expression as an equation, still formulated geometrically, and then on to an algebraic problem, now handled by procedures that are more like rules of manipulation. From a practice of mathematics based on the localized solution (and grounded in the polemical practices of early Greek science) we see a transition to a practice of mathematics based on the systematic approach (and grounded in the deuteronomic practices of Late Antiquity and the Middle Ages). With three chapters ranging chronologically from Hellenistic mathematics, through late Antiquity, to the medieval world, Reviel Netz offers an alternate interpretation of the historical journey of pre-modern mathematics.

Ancient Greek mathematics developed the original feature of being deductive mathematics. This article attempts to give a (partial) explanation f or this achievement. The focus is on the use of a fixed system of linguistic formulae (expressions used repetitively) in Greek mathematical texts. It is shown that (a) the structure of this system was especially adapted for the easy computation of operations of substitution on such formulae, that is, of replacing one element in a fixed formula by another, and it is further argued that (b) such operations of substitution were the main logical tool required by Greek mathematical deduction. The conclusion explains why, assuming the validity of the description above, this historical level (as against the universal cognitive level) is the best explanatory level for the phenomenon of Greek mathematical deduction.

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