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I am not sure this is the right place to ask, but I give it a try anyway.

I am reading a book about how to write math. At some point the author shows the grammar rule for the possessive case: e.g. one usually writes 'Stokes's theorem', 'Bliss's book', 'van der Corput's lemma', and so on. He does not mention the so called 'Jacobi identity'. According to the general rule, I would write 'Jacobi's identity'. Is there a specific reason why we do not do it?

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    When Jacobi is part of the name of this identity (note the article in the Jacobi identity) it is adjectival in nature and doesn't refer to Jacobi the person any longer, at least not directly. It is not the same as "Fermat's Last Theorem" where Fermat refers directly to the mathematician. – Tᴚoɯɐuo Jun 1 '18 at 14:09
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    Once the name becomes widely used, it is difficult to say how people who use the name would understand the possessive there. And there is no clear demarcation when the name becomes a name and the possessive falls out of use; sometimes both forms exist side-by-side, as in the Heisenberg principle and Heisenberg's principle. – Tᴚoɯɐuo Jun 1 '18 at 14:18
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    I wrote the new Bliss's book. Note the article. The combination of the article the with the possessive name Bliss's is ungrammatical. – Tᴚoɯɐuo Jun 1 '18 at 14:26
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    No, you would not combine the and Joe's. the Joe's hat is not grammatical. You'd say something like "The hat Joe wears" or simply "Joe's hat". When there is a possessive proper name like Joe's, do not use the article. – Tᴚoɯɐuo Jun 1 '18 at 15:02
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    Nobody ever refers to Hawking's radiation either. It's not obvious to me there's any useful / learnable rule in play here. The name of a thing is whatever it's called - grammar and syntax hardly get a lookl-in. – FumbleFingers Jun 1 '18 at 15:26
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The general rule is to use the possessive as in "Euler's identity" but particular usage may overrule this in many situations. In particular, when using "the" we tend to use the name attributively without forming the possessive:

Euler's identity was in fact first proved by Bernoulli.

The Euler identity relates five fundamental constants.

In the case of Jacobi, it seems many of the things named after him use his name attributively: "The Jacobi symbol", "The Jacobi identity". However, we still speak of "Jacobi's formula".

So no simple rule can be derived for why and you just can follow established use.

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