Vattel repeats that there is little relationship to the place born and natural born, and that is can only be inherited from the father.
It is asked whether the children born of citizens in a foreign country are citizens? The laws have decided this question in several countries, and their regulations must be followed.(59) By the law of nature alone, children follow the condition of their fathers, and enter into all their rights (§ 212); the place of birth produces no change in this particular, and cannot, of itself, furnish any reason for taking from a child what nature has given him; I say "of itself," for, civil or political laws may, for particular reasons, ordain otherwise.
You may want to read through this paper..
http://www.federalistblog.us/2008/11/natural-born_citizen_defined/
The natives, or natural-born citizens, are those born in the country, of parents who are citizens.
Emer de Vattel. Law of Nations, Book 1 chapter 19.
I highlighted what I wanted you to see.