The mathematicians of ancient Greece laid many of the foundations for the subsequent development of mathematics over the next 1500-2000 years. Perhaps the most important advance made by the Greeks was in their methods rather than specific result–the approach we now call “the axiomatic method.” The axiomatic method was codified by Euclid of Alexandria in his Elements, a 13-book treatise compiled around 300 BC. At first sight it appears to be solely about geometry, but a closer study reveals that Euclid was also using geometric language to describe ideas that we now consider to be part of algebra, number theory, and the notion of an irrational number. Other mathematicians in turn extended his ideas, and we study three of them: Apollonius, Archimedes, and Ptolemy. We will read selected material from these authors’ works, focusing on ideas and results that were crucial for the subsequent development of mathematics and astronomy (in ancient times these subjects were closely related, and in some ways they still are). We will also compare the way the Greek writers thought about mathematics with more modern approaches. And we will see that these four Greek geometers were very careful logicians who understood their subject in considerable depth.