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Atticus' closing statement is considered a masterpiece of courtroom literature. He covers virtually every angle of the case, reminding the jury that there was not "one iota of medical evidence" that proved Tom guilty. He mentions the conflicting testimony of the Ewells and that it had been "flatly contradicted" by the defendant. He asks the jury treat his client as an equal and tells them to "do your duty."
I don't see how his statement hurt his case in any way. He tells Jem later that the fact that the jury took several hours to determine what most people expected to take five minutes was "the shadow of a beginning." His statement made enough of an impact on one juror--one of the Cunninghams--that the man held out until he was finally convinced to change his vote to guilty.