What is Atticus trying to show the judge in chapter 18 of To Kill a Mockingbird?
As well as going through the actual events of the alleged rape, Atticus in this chapter also, as Scout remarks, is 'building up a picture of the Ewells' home life'. This is almost as crucial as determining the actual details about the attack on Mayella because, by highlighting the dismal and sordid conditions of Mayella's home, he also highlights the fact that Mayella is actually an unreliable witness who lives in fear of her father.
By a series of questions, Atticus elicits from Mayella such details as the fact that she lives a life of drudgery, has scarcely been to school, and has a tribe of younger siblings to look after as well as doing practically all the chores around the house. In all of this, he draws out the essential fact that Mayella is a very lonely individual who has no-one to turn to. This comes to light when he asks her if she has any friends. She seems frankly 'puzzled' by such a question, and then lashes out:
'You making fun of me agin, Mr Finch?'
Atticus let her question answer his.
What this means is that Mayella is so friendless that when asked if she has any friends, she automatically assumes that she is being mocked. From this Atticus is able to lead up to the most important point of all:
“Do you love your father, Miss Mayella?” was his next (question)
“Love him, whatcha mean?”
“I mean, is he good to you, is he easy to get along with?”
“He does tollable, ‘cept when—”
“Except when?” Mayella looked at her father, who was sitting with his chair tipped against the railing. He sat up straight and waited for her to answer.
“Except when nothin‘,” said Mayella. “I said he does tollable.” Mr. Ewell leaned back again.
“Except when he’s drinking?” asked Atticus so gently that Mayella nodded.
By way of skillful questioning, Atticus is able to catch Mayella off her guard and admit that she is vulnerable and that there is a problem with her father. When he asks her about his father's drinking, he does it so 'gently' that she seems almost unaware of what he's asking and automatically comes out with the truth: that there have been occasions when her father has been less than 'tollable', or tolerable, to her. In this way Atticus is trying to show the judge that Mayella's own father, not Tom, could have been her attacker.
Of course, she goes on to retract this and to say 'firmly' that her father has never attacked her, but,thanks to Atticus's subtle and gradual approach, not a little doubt has been cast on her testimony that it was Tom who attacked her.
When Atticus Finch questions his witness, Mayella Ewell, he asks her to recount what has happened on the day in question. After she testifies, Atticus repeats what she has said, then he states that he wants to be sure that she has the right man. "Will you identify the man who raped you?"
"I will. That's him right yonder."
Atticus asks Tom Robinson to stand. When he does, it becomes apparent that his left arm is withered and useless. Atticus, then, continues to question her about saying that Tom had choked and beaten her. He asks Mayella if she wants to reconsider any of her testimony. Defiantly, Mayella asks Atticus is he is trying to get her to say something that did not happen. So, again Atticus asks her if Tom choked her; she says he did. Then, he asks her if Tom hit her; she replies, "I said I did." Atticus asks Mayella,
"He blacked your left eye with his right fist?"
"I ducked and it--it glanced, that what it did. I ducked and it glanced off." Mayella had finally seen the light (Scout narrates)
The"light" that Mayella finally sees is the fact that Tom would have had to use his one arm for everything that he did since his left arm is useless. If he were to punch Mayella, he would hit her in her right arm, most likely, not the left. So, Mayella tries to make her lie seem plausible by saying that "it glanced off."
Atticus tries with his questioning to demonstrate to Judge Taylor that Mayella has falsely charged Tom Robinson. Since he has previously had Bob Ewell demonstrate that he is left-handed, Atticus also leads the judge to understand that Bob Ewell most likely struck Mayella, not Tom.