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I've seen many times in journals and books, they (the writers) omitted the possessive noun in a name of something (methods, rules, problems, etc.).

Look at these examples:

  • Coombs method/Coombs' method.

  • Trapezoidal rule/Trapezoidal's rule.

  • Cauchy problem/Cauchy's problem.

  • Newton method/Newton's method.

I could say so, because some of them wrote the possessive form as well. Also, actually at first I'm not really sure either whether it's a possessive form or not. In terms of methods' names, do we really need aphostrope and "s"?

For example, the second of the examples is abstract and the inventor of the method is not mentioned (Trapezoidal is a term related to a shape that has four sides) . Whereas the first, the third and the fourth have real people' names on it (Coombs, Cauchy, and Newton are people).

Is there a general rule to write "something's name" like this? I couldn't find related topics so far.

For further explanation, those examples are nouns. The third example, "problem" is meant as a phenomenon in mathematics not an actual somebody's problem.

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As you point out, "trapezoidal" is not someone's name. So "trapezoidal's rule" is never correct.

For terms that do use someone's name, this is how I have seen it, and what sounds correct to my ear:

  • The Coombs method, the Cauchy problem
  • Coombs' method, Cauchy's problem

That is: Use the possessive form when you do not use an article in front of the term, and use the non-possessive (or descriptive) form when you do use an article in front of the term.

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