The ancient Greek mathematicians did not use limits. We will examine their proofs of some propositions for which our proofs would involve limits. The discussion will show that their proofs helped the reader gain insight neither into how these theorems were discovered nor into how new results might be found. Some historians thought that calculus was being used to make the discoveries, even though it was not used in their proofs. Indeed, in about 1906, a copy of a book of Archimedes\'s book, \"The Method\"" was discovered. Archimedes revealed his process of