No need to go to France. Just read the first sentence of the our Declaration of Independence.
Note the reference to Natural Law in the first sentence of our Declaration of Independence.
It is crystal clear that the Founding Fathers used the Natural Law definition of 'natural born Citizen' when they wrote Article II. By invoking "The Laws of Nature and Nature's God" the 56 signers of the Declaration incorporated a legal standard of freedom into the forms of government that would follow.
President John Quincy Adams, writing in 1839, looked back at the founding period and recognized the true meaning of the Declaration's reliance on the "Laws of Nature and of Nature's God." He observed that the American people's "charter was the Declaration of Independence. Their rights, the natural rights of mankind. Their government, such as should be instituted by the people, under the solemn mutual pledges of perpetual union, founded on the self-evident truth's proclaimed in the Declaration."
The Constitution, Vattel, and Natural Born Citizen: What Our Framers Knew
The Laws of Nature and of Nature's God: The True Foundation of American Law
The Supreme Court of the United States has never applied the term natural born citizen to any other category than those born in the country of parents who are citizens thereof.
Under the references you provide, women are not even considered citizens,
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Natural law is another term for Noachide law, or God’s law as expressed to Noah.
It does not deal with citizenship.
Citizenship relates to Roman law; the only society that used the term before the US was born.
Roman citizens did not have to be born in Rome, but had to have allegiance to Rome.
The apostle Shaul (Paul in the Greek) was a Roman citizen, born in Syria.