This is an updated edition of an original and controversial book. As well as revising parts of the text and substantially updating the bibliography, in a new Appendix the author takes a more polemical stance and enters into a discussion of the nature and range of different interpretations. The book is divided into three parts; Interpretation, Evidence, and Later developments. The first part presents several new interpretations of the idea of ratio in early Greek mathematics and illustrates them in detailed discussions of several texts. Part Two focuses on the sources themselves, and questions the depth of modern knowledge of Plato’s Academy during his lifetime, the source of our text of Euclid’s Elements, and modern understanding of early Greek mathematics. The final part contrasts some of the evidence from early and late antiquity and then gives a historical account, since the seventeeth century, of the theory of continued fractions, our version today of the mathematics underlying the reconstruction. From reviews of the first edition: ‘…a real treat.’ Greece and Rome ‘…cites an impressive array of evidence…The result should be widely read by classicists and mathematicians as well as historians of mathematics.’ ISIS ‘…he enters into classical scholarship here with a really 'new reconstruction’ of early Greek mathematics.‘ Nature ’…this fascinating book…will arouse the interest and command the admiration of any historically minded lover of mathematics with a taste for the unorthodox.‘ Institute of Mathematics and its Applications 'This book, speculative in the best sense, engages the ancient material on its own terms in setting forth what the Greeks might have thought and done…While the book represents an important departure in historical research in its reaching beyond the spare formalism of surviving materials to an understanding of motivation and perception, its careful documentations and technical descriptions make it valuable in a more traditional way.’ Zentralblatt fur Mathematik