The concepts of "natural law" and the phrase "Laws of Nature" (of which Law of Nations is built upon) are found within the Declaration of Independence itself:
When in the Course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume among the powers of the earth, the separate and equal station to which the Laws of Nature and of Nature's God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation. We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.--That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governedThose (& others) are clearly NOT derived from English law, but rather from natural law concepts (which can be found in Vattel's Law of Nations).
When the Decleration of Independence and the Constitution were writen, English subjects could not renounce their alligience to the one soverign, the crown:
"Natural allegiance is such as is due from all men born within the king's dominions immediately upon their birth. For, immediately upon their birth, they are under the king's protection; at a time too, when (during their infancy) they are incapable of protecting themselves. Natural allegiance is therefore a debt of gratitude; which cannot be forfeited, cancelled, or altered, by any change of time, place, or circumstance, nor by any thing but the united concurrence of the legislature. An Englishman who removes to France, or to China, owes the same allegiance to the king of England there as at home, and twenty years hence as well as now. For it is a principle of universal law, that the natural-born subject of one prince cannot by any act of his own, no, not by swearing allegiance to another, put off or discharge his natural allegiance to the former: for this natural allegiance was intrinsic, and primitive, and antecedent to the other; and cannot be devested without the concurrent act of that prince to whom it was first due. Indeed the natural-born subject of one prince, to whom he owes allegiance, may be entangled by subjecting himself absolutely to another; but it is his own act that brings him into these straits and difficulties, of owing service to two masters; and it is unreasonable that, by such voluntary act of his own, he should be able at pleasure to unloose those bands, by which he is connected to his natural prince."William Blackstone, Commentaries 1:354, 357--58, 361--62
How, then, did they declare their Independence from the King if they were not permitted to do so under English Common law?
They used the principles found in natural law. Vattel states, in the Law of Nations (which is based on natural law), that citizens may renounce (or quit) their citizenship and allegience to a country:
"Chapter XIX: Of Our Native Country And Several Things That Relate To ItThe founders could never have writen our founding document, the Declaration of Independance, had they stuck with or adhered to, English Common law.
§ 220. Whether a person may quit his country.... 1. The children are bound by natural ties to the society in which they were born; they are under an obligation to show themselves grateful for the protection it has afforded to their fathers … They ought, therefore, to love it, as we have already shown, to express a just gratitude to it, and requite its services as far as possible, by serving it in turn … But every man is born free; and the son of a citizen … may examine whether it be convenient for him to join the society for which he was destined by his birth. If he does not find it advantageous to remain in it, he is at liberty to quit it ..."
Doctor Benjamin Franklin writes to M. Dumas, Philadelphia, December 19, 1775
I am much obliged by the kind present you have made us of your edition of Vattel, It came to us in good season, when the circumstances of a rising State make it necessary frequently to consult the law of nations. Accordingly, that copy which I kept (after depositing one in our own publick library here, and sending the other to the College of Massachusetts-Bay, as you directed) has been continually in the hands of the members of our Congress now sitting, who are much pleased with your notes and preface, and have entertained a high and just esteem for their author.
During the ratification period of our Constitution...
October 18, 1787 - James Madison wrote to George Washington, N. York:
"Since the Revolution every State has made great inroads & with great propriety in many instances on this monarchical code.[Edit: Englands "Common Law"] The "revisal of the laws" by a Committe of wch. Col. Mason [Edit: George Mason] was a member, though not an acting one, abounds with such innovations. The abolition of the right of primogeniture, which I am sure Col. Mason does not disapprove, falls under this head.. What could the Convention have done? If they had in general terms declared the Common law to be in force, they would have broken in upon the legal Code of every State in the most material points: they wd. have done more, they would have brought over from G.B. a thousand heterogeneous & anti-republican doctrines, and even the ecclesiastical Hierarchy itself, for that is a part of the Common law."
June 18, 1788 - George Mason, In Convention, Richmond (Debates in the Several State Conventions on the Adoption of the Federal Constitution), states:
"We have it in our power to secure our liberties and happiness on the most unshaken, firm, and permanent basis. We can establish what government we please. But by that paper we are consolidating the United States into one great government, and trusting to constructive security. You will find no such thing in the English government. The common law of England is not the common law of these states."
Just as the founders had done more than a decade earlier by declaring their independance from the crown, here (during the critical ratification period) we see the father of the Constitution, and the father of the Bill of Rights rejecting the English Common Law as being the American Common Law.
Furthermore, if the framers believed a "natural born Citizen" (natural law) was essentially the same thing as a "natural born Subject" (English common law) then the grandfather clause in AII,S1,C5 would not have have been necessary as they were all (save for Hamilton) natural born Subjects. They would all have meet the "natural born Citizen" requirement. No need for the "Or" grandfather clause.
The English common law provided that an alien naturalized is “to all intents and purposes a natural born subject.” Co. Litt. 129 (quoted and cited in Rhodes, 27 F.Cass. at 790). With such recognition, a naturalized citizen would have been eligible to be President of the new Republic."
http://puzo1.blogspot.com/2009/08/law-of-nations-and-not-english-common.html
Arnold Schwarzenegger is not POTUS eligible.
What? Did the Founders just pull their definition of natural-born citizen out of their asses?
Of course not, the lawyers in the colonies of the day were either trained in England, or universities in the colonies, OR they read for the law under a colonist who was a lawyer. IN ANY CASE, they were trained in English Common and English Statute Law.
When the Founders came up with the phrase, natural-born citizen, they applied their knowledge of said law to come up with their definition. They ALSO examined the writings of Grotius and Vattel as well in order to arrive at their conclusion.
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That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed ...
If you bothered to REALLY read the passages I posted, you would have found the following:
"... For the constitution of kingdoms and states, ordinances of cities and commonweals, and the liberties and freedoms thereof, are not by nature, but come by the consent of men and mens laws."
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The English common law provided that an alien naturalized is “to all intents and purposes a natural born subject.” Co. Litt. 129 (quoted and cited in Rhodes, 27 F.Cass. at 790). With such recognition, a naturalized citizen would have been eligible to be President of the new Republic."
The Rhodes case misinterpreted English Common Law. Firstly, naturalization granted all the legal rights of citizenship except political rights [e.g. holding office] and required an act of Parliament be passed. Alternatively, denization allowed a person to gain the rights of citizenship other than political rights. Denization was granted by letters patent, and was granted by the monarch as an exercise of royal prerogative. Denization was therefore an exercise of executive power, whereas naturalization was an exercise of legislative power.
To quote Blackstone:
"... A DENIZEN is an alien born, but who has obtained ex donatione regis letters patent to make him an English subject: a high and incommunicable branch of the royal prerogative. A denizen is in a kind of middle state between an alien, and natural-born subject, and partakes of both of them. He may take lands by purchase or devise, which an lien may not; but cannot take by inheritancee: for his parent, through whom he must claim, being an alien had no inheritable blood, and therefore could convey none to the son. And, upon a like defect of hereditary blood, the issue of a denizen, born before denization, cannot inherit to him; but his issue born after, may. A denizen is not excused from paying the alien's duty, and some other mercantile burthens. And no denizen can be of the Privy Council, or either House of Parliament, or have any office of trust, civil or military, or be capable of any grant from the crown."
NATURALIZATION cannot be performed but by act of parliament: for by this an alien is put in exactly the same state as if he had been born in the king's ligeance; except only that he is incapable, as well as a denizen, of being a member of the Privy Council, or Parliament ..."
THEREFORE, with such recognition, as you quoted in Rhodes, a naturalized citizen WOULD NOT be eligible to be President of the United States.