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BS. Arthur's father's naturalization records were a matter of public record (after all, your source got them from the Library of Congress), and known to Hinman, the lawyer the Democrats hired to discredit Arthur.

As previously pointed out, Vattel's 'The Law of nations' is not binding as U.S. law. According to the Supreme Court, in U.S. v. Wong Kim Ark, the language of the Constitution is the language of English common law. The majority opinion contains a long discussion of the meaning of 'natural-born' in English common law, concluding that it basically depends on where you are born. That's why Hinman's attack focused on the false accusation that Arthur was born in either Canada of Ireland.