What law does Bayard rely upon for his claim?
The Constitution and the rule of the common law.
What law do you rely on for your claim that it takes birth on US soil plus two citizen parents for a person to be a natural born citizen?
And don't say Vattel. Because Vattel's opinion is not in itself the law of the United States, unless the courts say it is. Vattel is an opinion of a foreign writer on the law of nations. Other writers on the law of nations had different opinions. And citizenship, the relationship between the individual member of our nation and our national government and community, was never a matter of international law. It was a matter of our own domestic law.