NIL 2-4 treats the PIE Etymon *bhah2g-ó- "beech". They mention that some scholars reconstruct long /a:/ and some (not always the same) scholars link it to the previously discussed *bhag-. In general I don't see any reasonable link between a tree name and a root meaning "share" etc. But there is a possible connection for the Germanic cognates meaning "book, letter".
NIL also mentions (FN 2 on p.3) that there are doubts about the relationship between the words meaning "book, letter" and the continuants of this root meaning "beech". The main formal problem is that the words meaning "letter / rune" seem to go back to a root noun, that actually is attested in Old English, while the beech words are eh2 stems (or n-stems derived from them). So we have derived morphology for what is supposed to be the original meaning ("beech") and a root noun for what is supposed to be the derived meaning ("book, letter").
Elmar Seebold, addressing this issue in his "Etymologie: eine Einführung am Beispiel der deutschen Sprache", (München : Beck 1981, quoted herafter as "Et."), also mentions that the proposed writing on beech tables that is supposed to be behind the change of the meaning from "beech" to "letter" is actually not attested, neither archaeologically, nor in written sources; Germanic runes are attested only on bark, stone, and various household objects (Et., pp. 290-291). It is also clear that the original meaning was "letter", not "book" - the oldest attestations mean "letter" in the singular and "document, book" in the plural, an obvious calque from Latin (littera - litterae) and Greek (gramma - grammata) (Et. p. 290). Seebold argues that the meaning "letter" is derived from the compound Norse bókstafr, Old Saxon
bōkstaf, OHG. buohstap, whose second member means "staff". The writing of runes on staffs is widely attested (here Seebold is undermining his previous argument somewhat, as these staffs could of course have been made of beech wood, but the written source he quotes actually mentions ash wood).
He then adduces a parallel from Welsh, coelbren "sign-wood, lot-wood", composed of coel "sign, omen" and bren "wood", designing a piece of wood covered with signs used to throw lots, a custom also attested for the Germanic people. He takes this parallel as an indication that the first element of bókstafr etc. originally meant "sign, omen, lot", and links it with our old acquaintance, the root *bhag-, reconstructing a root noun Proto-Germanic *bōk-s "lot, portion" (Et. pp. 291-292). That would mean that the word family of "book" is not related to "beech", but that the purported writing on beech tablets is only a folk etymology. (A shorter version of these arguments can also be found in Kluge(-Seebold), "Etymologisches Wörterbuch der deutschen Sprache", s.v. Buch).
In general, I find this argumentation attractive. A formal problem is that the proposed root noun is attested only in Vedic, as a second part of compounds, with an active meaning "enjoying" (NIL p.1), but not as an independent word with a resultative meaning "allotment, lot", and that it would be the only continuation of *bhag- in Germanic. The same problems woud also arise if, as I proposed, we eliminate the root bhag- and take its purported continuations as derived from the root bheg- "break"; that root also is not continued in Germanic (at least according to LIV p. 66/67 and NIL, p. 6; the forms with nasal infix mentioned in IEW p.115 look onomatopoetic, for which reason Pokorny himself states that they don't belong to *bheg- ). There is also no root noun formed from *bheg- attested in NIL (p. 6); if we eliminate *bhag-, we would have at least the Vedic root noun mentioned above, but still with the same problems. On the other hand, an isolated root noun from a root that otherwise doesn't have any cognates in Germanic is a perfect candidate for the kind of folk etymology discussed.
Questionable Sino-Mongolian toponymy
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