Obama is apparently NOT a natural born US citizen.
So what is your point?
My point is that for "natural burn US citizen" to have any legal meaning, there must be an accepted legal definition of the term.
A single book by some long-dead French guy does not constitute an accepted legal definition. One could no doubt find other, different, definitions.
The Ark Decision (1898) goes into some detail on the matter, and even refers to Mr. Vattel's work on the topic. However, the decision does so apparently only to dismiss it!
It's a fine quote and all that, but the Supreme Court seems already to have dismissed it.