Question

how did carton know that the switch had worked
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Answer 1

First of all, Carton gets Darnay acquitted because witnesses cannot tell the two men apart. Second of all, both men are in love with Lucie Manette, but Carton uses this to switch places with Darnay before he is executed.Of course, Carton dies famously—saying it is a far better thing he's doing than he has ever done.