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Manacles is another word for shackles. Martin Luther used the words manacles and discrimination to express how segregation and discrimination were restricting to African Americans in his time.
The metaphors that Dr. King uses in his speech to describe the condition of African- Americans at the time of the March on Washington have multiple layers.  One such dimension was the fact that Dr. King wants to clearly state that "the dream" of which he speaks is part of the vision that the Framers had for American identity.