Numbers, Magnitudes, Ratios, and Proportions in Euclid'sElements: How Did He Handle Them?

Abstract

For a century or so much Greek mathematics has been interpreted as algebra in geometric and arithmetical disguise. But especially over the last 25 years some historians of mathematics have raised objections to this interpretation, finding it to be misleading and anachronistic, and even wrong. Accepting these criticisms, I consider Euclid'sElementsin this context: if it cannot be read in this algebraic manner, how did he conceive and handle his various types of quantity? The question is not merely of historical interest, for it raises issues about basic relationships between algebra, arithmetic, and geometry.