“To hang up one's 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. 14–17.) (See Ask.)
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.