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Mathematics as we know it today was born in the Greek world of the sixth century BCE at the hands of Thales of Miletus and his contemporaries, who harnessed the power of logical deduction to discover truths about the mathematical world and to draw forward universal and irreproachable consent for the validity and certainty of these discoveries. Within this context, the first mathematical program was established: the determination, if possible, of methods for determining the areas of plane figures. This paper, adapted from a forthcoming textbook by the author for a course aimed at undergraduate students of the humanities, will investigate some of the work of geometers in the fifth century BCE, especially the advance of Hippocrates of Chios regarding the quadrature of lunes. We do this by commenting through hypertext links on a few important ancient texts that record this work.

Daniel E. Otero (Xavier University), "The Quadrature of the Circle and Hippocrates' Lunes," *Loci* (July 2010)