The Constitution, which is our basic law, does not define the term natural-born citizen. But the First Congress, which included members of the Convention, chose to follow Lex Sanguinas and their definition therefore has great weight. The intent is to deny the Presidency to a foreign prince, such as the liberal Prince Henry of Prussia, and it does.
A real life example of the phenomenon was Maximilian I, Emperor of Mexico, a nation which lacked a constitution with that critical natural-born clause.
After a distinguished career in the Austrian Navy, Maximilian entered into a scheme with Napoleon III of France to rule Mexico. With the support of the French army and a group of Mexican monarchists, Maximilian traveled to Mexico where he declared himself Emperor of Mexico on 10 April 1864.
The Mexicans lacked a proper constitution, but thankfully they made up for their oversight with their impeachment procedure, which was even more cut and dried than that of Oliver Cromwell: