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From Book I Chapter 1 of The Fellowship of the Ring by J. R. R. Tolkien:

Mr. Drogo married poor Miss Primula Brandybuck. She was our Mr. Bilbo's first cousin on the mother's side (her mother being the youngest of the Old Took's daughters); and Mr. Drogo was his second cousin. So Mr. Frodo is his first and second cousin, once removed either way, as the saying is, if you follow me.

What is the meaning of the phrase in bold?

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‘Either way’ modifies ‘first and second cousin’, so that

first and second cousin, once removed either way

means

first cousin, once removed, and second cousin, once removed

See Wikipedia for an explanation of the relationship terminology.

There are Baggins, Took and Brandybuck family trees in Appendix C of The Return of the King, which allow us to confirm this passage as follows:

  1. Frodo and Bilbo are first cousins, once removed, on their mothers’ sides, because Bilbo’s mother Belladonna née Took was sister of Frodo’s grandmother Mirabella née Took. (Their father Gerontius was known as ‘the Old Took’.)

  2. Frodo and Bilbo are second cousins, once removed, on their fathers’ sides, because Bilbo’s grandfather Mungo was brother of Frodo’s great-grandfather Largo.

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Let's say you and X are first cousins.

X has a child, Y. You and Y are still "first cousins," but Y is one (generation) removed from X (in a downward direction). If you had a child, Z, Y and Z would be second cousins, both of them being children of two first cousins.

X also has a parent, A, who is a sibling of one of your parents, B. At this stage, A would be your uncle or aunt. But A is "once removed" from your first cousin, X, in an upward direction.

The "higher" of two generations controls, so A would be "avuncular" (root word, uncle) to you, not your "first cousin once removed." And you and Y would be "first cousins" once removed, not "second cousins" once removed, even though Y would be a "second cousin" to your child Z.

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