E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
To hang up ones axe. To retire from business, to give over a useless project. The allusion is to the ancient battle-axe, hung up to the gods when the fight was done. All classical scholars will call to mind the allusion of Horace to a similar Roman custom. Being snubbed by Pyrrha, he says, He will hang up his axe upon her wall, or more literally, his drenched garments on the temple-walls of Neptune. (1 Odes, V. 1417.) (See ASK.)
To put the axe on the helve. To selve a difficulty. To hit the right nail on the head.
To send the axe after the helve. To spend good money after bad, or under the hope of recovering bad debts.
He has an axe to grind. Some selfish motive in the background; some personal interest to answer. Franklin tells of a man who wanted to grind his axe, but had no one to turn the grindstone. Going to the yard where he saw young Franklin, he asked the boy to show him how the machine worked, and kept praising him till his axe was ground, and then laughed at him for his pains.