''If one would understand the Greek genius fully it would be a good plan to begin with their geometry.'' As early as the sixth century BCE Thales of Miletus used geometrical principles to calculate distance and height. Within a few hundred years Euclid had produced his seminal Elements which was still used as a textbook when this two-volume work was first published in 1921. A distinguished civil servant as well as an expert on ancient Greek mathematics Sir Thomas Little Heath (18611940) includes here sufficient detail for a modern mathematician to grasp ancient methodology alongside explanatory sections aimed at classicists. This remains a rigorous and essential exposition of a vast topic. Volume 1 includes an introduction that touches on the conditions which made possible the rapid development of philosophy and science in ancient Greece. The coverage begins with Thales and ends with Euclid.