In "To Kill a Mockingbird" how does Mr. Gilmer try to convince the jury of Tom's guilt?

Mr. Gilmer doesn't really prove anything. He relies on circumstantial evidence and the inherent bigotry of the Jury. He humiliates Tom Robinson calling him "boy" in the court and using stereotypes to prove his case. Atticus sums it up best when he addresses the court,

"The state has not produced one iota of medical evidence to the effect that the crime Tom Robinson is charged with ever took place." (ch 20)