Is natural law a type of common law?
No, Natural law is not a type or subset of Common Law
Natural Law is derived from what some person thinks is a logical and obvious rule, or what some person thinks is God's Law. There are many versions and varieties of natural law, and there is no wide agreement on what it does or does not specify.
Common law is derived from the judgements of the Royal Courts in England (later Great Britain, still later the UK) roughly during the period 1250-1850. (After that it continued to develop as case law, but major changes were usually statutory, although not always as a comment by Martin Bonner points out.)
It started with a rough distillation of then widespread customary law, ultimately traceable to Germanic and Norman traditional law. Supposedly, common law had been in force "since the memory of man runs not to the contrary" and judges did not create it but merely "discovered" it.
Common law was then expanded and altered over the centuries by the decisions of judges of the common-law courts in individual cases, primarily the court of the King's Bench, the court of the Exchequer, and the court of Common Pleas. It also included some rules derived from statutes, particularly early ones, such as those of King Henry II. Later statutes could and did alter the rules of common law. Technically such altered rules became "statutory law" not strictly part of the common law, but where they dealt with matters handled by the common-law courts, they were often considered to be part of the common law in an informal sense, as they were laws taught to the lawyers (barristers) who practiced in the common-law courts.
Common law was distinguished from several other kinds of law:
- Local customary law: the law used in local courts, particularly Hundred courts and Shire courts, but also the courts of local lords. This law varied from place to place, and was largely derived from "ancient local custom" supplemented by Royal and noble charters.
- Church law: the law used in church courts, which had jurisdiction over wills and land owned by the church, laws on Sunday observance and several other things, as well as the discipline of priests and other clerics.
- Admiralty law the law of ships and seagoing matters. It covered salvage, disposition of captured enemy ships and cargo (prize law), maritime insurance, mutinies and crimes aboard ships, and other nautical matters. In the seventeenth century Lord Justice Coke fought to reduce its jurisdiction in favor of common-law courts. Justice Story, of the US Marshall Court (1800-1835), fought to reverse Coke's changes on this point in US law.
- Law Merchant: law specifically for cases between merchants. It dealt with specifically commercial issues, and was eventually merged into the common law.
- Piepowder law: Special law for the dealings of foot peddlers. (The name comes from a Norman French term meaning "dusty feet".) It also was eventually merged into the common law.
- Prerogative law: law derived from the direct authority of the King, particularly as exercised by the Court of the Star Chamber. It was widely regarded as tyrannical and was largely abolished by the English Civil War and its aftermath.
- Feudal law: Law dealing with the holding and inheritance of land under feudal tenure (largely knight's service, but other kinds too) and with other aspects of feudal rights, privileges, and obligations. It slowly became largely obsolete as less and less land was so held.
- Equity: Law developed by the Lords Chancellor of England (later Great Britain), intended to modify injustice worked by inflexible parts of the Common Law, but eventually becoming rather stiff itself. It was merged into the common law at different times in the US and the UK, but generally during the 1800s. The modern remedy of injunction is derived from equity.
- Statute Law Law directly derived from acts of the legislature -- Acts of Parliament in England/Great Britain/the UK.
- Civil Law: Law largely in force in Europe, partly derived from Roman law or Roman legal concepts, mixed with various local customary laws, and later much influenced by the Napoleonic Code. Modern EU law is considered a part of Civil Law. Civil law varies between different countries, and is at least as large and complex a subject as common law. It is also a subject I know less about.
- International Law: Law derived from treaties and other intentional agreements, and the traditional customs of dealing between countries (mostly in Europe).
None of the above are "natural law" although in some cases natural law concepts influenced legislators and judges.
There is some conceptual similarity between natural law and early common law, in that both looked to a supposedly pre-existing but unwritten law But natural law generally attempted to apply over-arching, top-down logical or theological rules, while common law was built up from low-level, decisions on individual cases, and then derived rules from these individual decisions.
A version of natural law concepts, derived from Enlightenment philosophers, particularly Locke, influenced the drafters of the Constitution. (See Gary Wills, Explaining America; See also the Wikipedia article on Natural Law, linked above). But that was quite different in style and content from recent versions of natural law, which are usually aimed at imposing theological rules and abolishing secular rights which would hinder this process. (See How acceptable (mainstream/marginal) is the discussion about replacing the human rights with the natural law and to delete human rights chapters? and the link in the question for examples.) Moreover, different authors who advocate natural law do not agree on what that law is or should be.
The above is only a very brief summery. Anyone interested should read more widely. I recommend The Law of the Land: The Evolution of Our Legal System by Charles Rembar (1980) e-book version by Open Road Media. It is a history of law written by a once-prominent practicing US lawyer, intended for those who are neither historians nor lawyers. In my view it is particularly well-written. There are, however, quite a few books available on the history of the common law.
No. Natural law is a philosophical or theological concept. There is obviously quite a bit of debate about what it exactly means, and what those laws actually are, but it generally holds that all of mankind is inherently subject to certain laws.
Common law, on the other hand, is a legal system that began in England, spread throughout the British Empire, and generally continues to govern former commonwealth countries.
Although the development of common law realied heavily on natural-law principles, they can't be the same thing, as natural law purports to apply to everyone, while the common law generally applies only to common-law jurisdictions.
One could make an argument, therefore, that common law is (or should be) a branch or form of natural law, but I don't think there's a coherent argument to be made that natural law is a branch of common law.