This book presents a radically new interpretation of early Greek mathematics; its central theme is a mathematical and historical exploration of several different ways of characterizing ratio. To set the reconstruction in context, David Fowler describes both how early Greeks wrote and how their writings have then been transmitted to us. This leads to descriptions of taxation and land-measurement in the Greek administration of Egypt, of the development of continued fractions from the seventeenth century, including a recent algorithm for arithmetic with continued fractions, and many other topics.