What was Atticus trying to prove with the Beating of Mayella? Who caught on to this...Scout or Jem?

From the balcony Jem watched the proceedings with glee, certain that Atticus had pinned Ewell and proved Tom innocent. Scout, however, was doubtful. She thought Jem was getting a little ahead of himself because Tom could just as easily be left-handed as well. Jem knew that Tom did not have the use of his left hand.