Birthers often say this. No one has yet to produce a single civics textbook defining Natural Born Citizen in this fashion. It would sure be a coup if someone could.
Blackstone's definition of "Natural Born" in English Common Law (the source of our legal system), would make Obama eligible.
Vattel's definition is in dispute because there are multiple conflicting translations, and no one can accurately determine the translation referenced in the writings of the founders.
The Federalist Papers, as far as I know, do not go into any detail about the term while arguing in favor of the Constitution. Neither, as far as I know, do any arguments against the Constitution published at the time.
I don't consider myself a birther either. I did a lot of research on the subject during the primary campaign when Phil Berg brought it up. I understand the arguments on both sides neither of which have been definitely decided one way or the other in a court of law.
While there have been cases to point to and try to extrapolate from, there has not been one that exactly hits the nail on the head so to speak in my opinion. Nor is there likely to be one, which was my point.
The courts will continue to duck this issue if they can, because they do not want to be in the position of overturning an election.
My own theory is that it is likely that the founders were thinking that a natural born citizen would be one where the father was a citizen, and the child was born and raised in country, because Naturalization laws in the early days often hinged on the father's citizenship.
Congress only a decade or so later argued over whether a child born on foreign soil to American citizens should be considered natural born, with some pointing out they were more of a citizen than someone who was born in America to parents who were not citizens of America, so I really would not expect current day Americans to be on the same page either.
In some of the court cases, I find myself agreeing with the minority opinion particularly with respect to the 14th amendment. I wonder, when the Supreme Court decides cases, if they call each other pejorative names when they disagree or do they simply explain their opinion to each other?
At any rate, when I run across threads with birther and anti-birthers going at it, I usually skip them now. It's always the same old rehash and then the name calling. To me, it just looks like the pot calling the kettle black.