Houston, we have another winner!
A Treatise on citizenship, by birth and by naturalization, with reference to the law of nations, Roman civil law, law of the United States of America, and the law of France; including provisions in the federal Constitution, and in the several state constitutions, in respect of citizenship; together with decisions thereon of the federal and state courts (1881)
"Greek philosophy emphasized the distinction between "nature" (physis, φúσις) on the one hand and "law", "custom", or "convention" (nomos, νóμος) on the other. What the law commanded varied from place to place, but what was "by nature" should be the same everywhere."
Heck, even Turbin Durbin recognizes (in 2008) that natural law was an important factor in our founding.
Since it's probably next to impossible, that the framers made up the term "Natural Born Citizen" on the spot and entered it into the Constitution as a requirement for POTUS without debate, we must determine from where they got their definition. That is, what/who were they familiar with at the time. What ideas where influencing them at that time, with all that was going on then. In my opinion, it's 100% clear they got their definition for NBC from Vattel. Someone they were very familiar with, and referenced often...during the task of raising a state and during the writing of and debating of the Constitution [1], [2] (for a few examples).
As we know, nobody is trying to make the argument that Vattel (or natural law) was it, period. That it was 100% the single only source they used to found the country and frame the Constitution. They used multiple sources, of course, to create their own unique system.
So far as I can tell, Vattel is the only source they would have been familiar with, that defined a "natural" born citizen.