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More x than you can shake a stick at is used to indicate an abundance of something, but why would anyone want to shake a stick at anything? And what’s it got to do with quantity?
The origins are unfortunately uncertain and there are a few theories, bit one seems more logical than most.
The phrase is certainly an Americanism and the first known use of it to mean an abundance was in 1818, in the Lancaster Journal of Pennsylvania; “We have in Lancaster as many Taverns as you can shake a stick at”.
To illuminate things, it is useful to note that ‘shake’ has a meaning other than to move rapidly, it is also a term for wood that is split into pieces – it is essentially the most basic form of shingle; a log is chopped into thin pieces which are used as a cladding material for buildings. In pre-1818 America shakes would have been the most common way of cladding a building’s roof or walls so ‘shake’ would have been widely understood in this context at the time our phrase was most likely formed.
It can’t be said with certainty but knowing that the amount of shakes you can get from a log is large and the thinner the better, I think it’s likely that the phrase meant something similar to ‘More taverns than there are pieces you could split a log into’.