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"I'm no idealist to believe firmly in the integrity of our courts and in the jury system—that is no ideal to me, it is a living, working reality. Gentlemen, a court is no better than each man of you sitting before me on this jury. A court is only as sound as its jury, and a jury is only as sound as the men who make it up. I am confident that you gentlemen will review without passion the evidence you have heard, come to a decision, and restore this defendant to his family. In the name of God, do your duty." (20.51-52)
If Atticus had a car, it'd have a "Be the Change You Wish to See in the World" bumper sticker. While he says here that he's no idealist, he's been realistic throughout about his extremely low chances of winning this case. In his closing argument, he's acting as if the outcome he knows is impossible is actually the only possible one, in an attempt to make it so.
"First of all," he said, "if you can learn a simple trick, Scout, you'll get along a lot better with all kinds of folks. You never really understand a person until you consider things from his point of view […] until you climb into his skin and walk around in it." (3.85-87)
Atticus's advice "to climb into someone's skin and walk around in it" is a little more Silence of the Lambs than the typical advice to walk a mile in someone's shoes, but the idea is the same: compassion is based on sympathy, on being able to put yourself in the other person's place and understand why they act the way they do even if you don't agree with it.
Atticus was saying, "With people like us—that's our share of the bill. We generally get the juries we deserve. Our stout Maycomb citizens aren't interested, in the first place. In the second place, they're afraid. [..] Well, what if—say, Mr. Link Deas had to decide the amount of damages to award, say, Miss Maudie, when Miss Rachel ran over her with a car. Link wouldn't like the thought of losing either lady's business at his store, would he? So he tells Judge Taylor that he can't serve on the jury because he doesn't have anybody to keep store for him while he's gone. So Judge Taylor excuses him. Sometimes he excuses him wrathfully." (23.46-49)
Personal concerns are more important for the people of Maycomb than public duty. When only white men can serve on a jury, what happens to a "jury of one's peers"? And when your peers do everything they can to avoid serving, who's left? (Judging by the number of times Shmoop's been called to jury duty… not much.)