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The Collaborative International Dictionary
Axinomancy

Axinomancy \Ax*in"o*man`cy\, n. [L. axinomantia, Gr. ? ax + -mancy.] A species of divination, by means of an ax or hatchet.

Wiktionary
axinomancy

n. 1 A species of divination, by means of an ax or hatchet. Including: 2 # placing an agate stone, piece of jet, or some other precious or semi-precious stone upon the axe-head and heating the metal, the movement of the agate then being interpreted. 3 # interpreting their movement when balanced on a post. 4 divination by saws. Almost certainly an error, there is only one citation of its use, and those who copied it.

Wikipedia
Axinomancy

Axinomancy is one of several obscure methods of divination using an axe, hatchet, or (rarely) a saw. Most of the methods involve throwing an axe into the ground, or swinging it into a tree, and interpreting the direction of the handle or the quivering of the blade. A form of this is axiomancy, this is when the quivering of the blade of an axe that has been thrust into a wooden table is interpreted by the diviner.

Another interesting method is heating an axe-head in a fire until it glows, and then interpreting the colors and shapes. A variant, attributed to the ancient Greeks, who held it in good repute, is to balance a spherical piece of agate on the edge of the axe (held sharp edge up). The direction in which the agate rolls can be interpreted as needed.

Some sources claim that Psalm 74 refers to the use of axinomancy to predict the fall of Jerusalem, although in the text the reference to upright axes is not specifically for divination.