This essay covers many of the basic ideas of Lakatos' method of proofs and refutations with discussion of 19th century mathematical topics. In a nutshell Lakatos' philosophy of mathematics in Proofs and Refutations is a way of describing the mathematical present--meaning contemporary methods of proof and conceptions of theoremhood--as an historical development solving various problems of rigor and content. Lakatos' focus is entirely on the 19th century and does not include, for example, the emergence of the concept of proof in Greek mathematics, a topic studied by Lakatos' teacher Arpad Szabo.