The common theory about the origin of the beidana I find quite dubious. Sure, valdese/vaudois "heretics" of the Piemont did seem to use them, at least there is no good reason to question this, but the weapon, or at least an extremely similar one, existed before that, and outside that community. The proof can be found in these miniatures from the Romance of Alexander, attributed to Jehan de Guise, 1338-44, Ms. 264 of the Oxford Bodelian Library (at least that's the reference I found).
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We can see very clearly that, except for the shape of the tip which ressembles the Invalides falchion, it is obviously the same construction with the "gancio" forming a branch guard, and the two scales protruding past the tang, in a very classical fashion on Italian roncole. But it is a relatively immediate descendant of this kind of thing:
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With these two beidane, we can see the proximity with both the Lombardian cleavers and the Romance of Alexander falchions:
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and it still bears a strong link with cousin tools that are common in Italy, and also in Piemont-Savoy:
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By the way, the "belt hook" is a common feature on both French and Italian billhooks (pictures from my own collection, 2 French from the early/middle 20th century, and one Italian from 2016 or so - construction with two riveted scales largely fell out of fashion during the late 19th early 20th, after over two millenia of use, at least as far as industrial scale production goes):
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Now a brief point about vocabulary: in Italian, full size single edge billhooks are called roncola (roncole plur.), cleaver-like choppers ares called manaresso (manaressi plur.), and double edge, vine-prunning type of billhooks are called pennato (pennati plur.). A meat cleaver is called a mannaia (mannaie plur.), and manaresso and mannaia seem to sometimes be used interchangeably. The etymology of beindana is unclear, but it seems to me it may be related with the French word bédane, now a type of strong wood chisel mostly used for mortising, but for which etymological dictionnaries give completly flawed explanation (bec d'âne, donkey's beak). The latin name for a billhook, of the roncola/pennato type, is "falx". Fauchon (fr, used to mean falchion and short chrub scythe), falcione (it, for flachion and glaive), falchion (en, though from romance languages), facón (es, large knife), facão (po, large knife or machete) are augmentatives of the latin word falx.
And in the end, I just drop my bomb to where this whole tree of tools and weapons started from:
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Roman billhook found in the ruins of Pompei
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Roman billhook found in Moregine, close to Pompei
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And last but not least, a Roman beidana/machete/manaresso/falchion thing, date unknow, origin unknown, found in a facebook gallery of pictures taken by a tourist in the Museo Nazionale della Siritide, showing the Hamburg/Conyers/Cluny falchions blade mounted on a transitionnal hilt between the Pompei and Moregin billhooks, and the Lombardian cleavers, with the beak starting to transform into a real "gancio", and the whole tang announcing the handle protruding past the gancio.