According to Aristotle, logic is an argument in which certain things being laid down, other things necessarily follow. The subject of Chapter I was expressing mathematical facts in the language of logic. This is a necessary prerequisite to logic, but from Aristotle's definition, we see that it is not logic. In order to actually do logic we must construct arguments which show that some statements follow from other statements. What Aristotle discovered, and this discovery was the birth of logic, is that the individual steps in these arguments connecting hypotheses to conclusion can be made to conform to highly stylized mechanical rules. In this chapter it is my plan to give some examples drawn from number theory, algebra, and analysis of real proofs which illustrate some of these rules. I will first present the proofs as they might appear in an ordinary math text and then analyze them in terms of the theory we have built up in Chapter 1.