The Lunes of Hippocrates

My thesis will be divided into three sections. My first section will begin by discussing the development of ancient Greek mathematics from the earliest natural philosophers up to Archimedes. Careful attention will be paid to those mathematicians most likely to have had a direct influence on Archimedes and, in particular, on his work The Method. I will then examine the historical circumstances and influences leading up to Newton’s Analysis by Equations of an Infinite Number of Terms. Next, I will compare and contrast the intellectual environments in which each of the mathematicians developed his theories. The second section will consist of a mathematical comparison between the two methods of finding the area under a curve. I will explain how each method works and discuss the differences in the approaches taken by the two mathematicians. In the last section I will discuss Archimedes in the wider context of Greek mathematical and scientific thought. Using ancient Greek philosophers, natural philosophers, scientists, and mathematicians, I will demonstrate how mathematics was an integral part of Greek culture, both practical and intellectual. By comparison with the work of the later, English mathematician Newton, I will show how Archimedes and in particular The Method are firmly rooted in the mathematical tradition of the ancient Greeks and demonstrate how this tradition differed from the tradition on which Newton drew to formulate his Analysis