Analytic philosophy, as we recognize it today, has its origins in the work of Gottlob Frege and Bertrand Russell around the turn of the twentieth century. Both were trained as mathematicians and became interested in the foundations of mathematics. In seeking to demonstrate that arithmetic could be derived from logic, they revolutionized logical theory and in the process developed powerful new forms of logical analysis, which they employed in seeking to resolve certain traditional philosophical problems. There were important differences in their approaches, however, and these approaches are still pursued, adapted, and debated today. In this paper I shall elucidate the origins of analytic philosophy in the work of Frege and Russell and explain the revolutionary significance of their methods of logical analysis.