/johnny
Johnny, good question. It's not unlike asking "under the Constitution, who has lawful authority to write the Constitution." The Constitution is our legal foundation, and since we are a nation based upon laws, and not, as a monarchy, upon men. The Constitution, ratified by the sovereign states that United to form our nation, acquired its authority from the nine states who ratified the Constitution in 1787-1788 - "...deriving their just powers from the consent of the governed."
If you know about axiomatic systems, that is what is how I regard the articles and subsequent amendments. Axioms are statements you agree to accept as the foundation of a system, which our states did (but not all of them, just nine to begin with). You've heard of Euclid's Axioms, which have changed little over 2000+ years, and are the basis for, no surprise, Euclidean Geometry, the Geometry most of us have wrestled with, polygons and congruency. All of numerical mathematics can be derived from two amazingly simple notions, those of order, and the notion of belonging. Axioms are notions which seem reasonable to those who use them. Our Constitution and Declaration are based upon "Laws of Nature and Natures God", notions with which our founders agreed.
All of our laws should have been derived from our Articles and Amendments, but, as Obama told us, there are things he would like to do, but is prevented from doing by the Constitution. It is a collection of axioms of negative rights written to protect us from government.
The authority that deemed a natural born citizen a requirement for our president was all states who ratified the Constitution. Virtually all of our founders and framers used Vattel as their primary reference, and Jefferson based our Declaration upon Law of Nations and assigned it as our first law book. Hamilton cited Vattel as his most authoritative reference in letters to Washington.
One of Vattel's greatest influences was a philosopher known more as a mathematician, Gottlieb Leibniz, credited, along with Newton, for the invention of The Calculus. I perceive the influence of the axiomatic methodology of Leibniz in Vattel's writings. Vattel, as Leibniz, is very familiar with, and cites frequently, the Greek philosophers, including Aristotle, who explains natural born citizenship - 'native citizen'. This is not a new idea. Our founders and framers were almost all steeped in classical philosophy, and most of them were proficient in Latin, many in Greek, and most in French.
That is what constitutes common-law, not English common-law or Blackstone's common-law, but U.S. common-law. Constitutional definitions, as Madison explained, and Morrison Waite repeated, depend upon the common-law and language familiar to its framers.