In English law there is no guestion as to who can ascend to the throne but that would not be the case in the U. S. That fact caused the founders great consternation as to how they were to prevent anyone with divided loyalties to become president and they ultimately agreed on Vattel's work to guide them. Vattel defines NBC as a person born in the country to parents who are themselves citizens.
The problem is Vattel's Laws of Nations was somewhat obscure at the time and is often confused with the philosopher Wolffe's general statement of 'the natural laws of nations' as a statement on sovereignty. This is one of those cases where the blogsphere recently found a reference to 'laws of nations' and then 1. made the assumption that it was referring specifically to Vattel's work. 2. Applied to this specific case. 3. Would be considered the over-riding law of the land.
None of the founders referred to Vattel's work in the Federalist Papers (at the time, Ben Franklin had brought just one copy into the US for the Library of Congress and Washington's own Library didn't even have a copy recorded until 1789, well after the Constitution phrase was written), at that, when referring the specific issue of NBC in the Federalist Papers, the founders themselves completely contradicted Vattel on the issue.
And finally, if we are going to use Vattel as the source meaning for that, it opens up a whole other can of worms because Vattel's Laws of Nations as a whole, is not compatible with the US system whatsoever. It is actually highly feudal and oppressive in nature. Give it a read and see if you think it was a guide for our founders.
http://www.constitution.org/vattel/vattel_01.htmI understand you are completely sold on this.. I had a lot of rows during the birther wars and I know once you are set on an idea, it is hard to shake it. But I do suggest reading that work as a whole (it isn't very long) and decide if you want that to be in any way used as a guide for what the Constitution means.