E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
Wrong End of the Stick (You have got hold of the).
You have quite misapprehended the matter; you have got the wrong sow by the ear. There is another form of this phrase which determines the allusion. The toe of the stick is apt to be fouled with dirt, and when laid hold of defiles the hand instead of supporting the feet.