  This explains why Socrates does not stop after offering his first “proof.” Many readers are puzzled about why he offers two more. After all, the geometer does not need to offer multiple proofs of his theorem. What might seem worse, the additional proofs concern pleasure, and thereby introduce—seemingly at the eleventh hour—a heap of new considerations for the ethics of the Republic. But as the considerations at the end of the  previous section show, these pleasure proofs are crucial.