From the Greek axine manteia, literally ax divination.
Axinomancy is a now rare way of telling the future or learning the unknown by either heating an axehead and then observing the results or balancing an axe and watching where it falls. It was most commonly practiced in ancient greece.
Research suggests the most common means of using one's axe was to find a thief or treasure.
To find a thief (I've tried this at home...it really works! I may be setting up a 1(800) number soon, along with a web site), "cast" your axe into the ground with the handle up, so that it is balanced--maybe by cast they meant to say, "carefully place". Anyway, now you and your friends dance around it in a circle (I'm picturing the KKK rally from O Brother Where Art Thou?, but YMMV. When the axe falls, look for the robber in the direction indicated by the handle--you and your friends should now go tearing off where the handle points, preferably with torches and farming implements.
Another way to find ne'er-do-wells is by heating the axe until it is red hot, and then putting an agate on the cutting edge. When it falls, it will fall in the direction of someone guilty of a crime.
This method can also be used to find treasure. If the agate does not fall, and stays balanced, then there is no treasure around (I imagine ancient greece must have been positively thick with treasure). This had to be repeated three times, and if the agate fell in the same direction every time, then treasure lay in that direction. If it fell off in a different direction every time, then the ancient greeks moved on to a different spot and tried again.
One can also watch how the ax quivers when whacked into a tree, but I couldn't find how to interpret this--maybe you and all the suspects could dance around the tree, and the person the handle points to when it stops wriggling is the guilty party. That's how I'd do it anyway. You can try it however you want, but I wouldn't use this particular method indoors--drywall will not make an ax quiver.
Supposedly, this how the diviners predicted the fall of Jerusalem in Psalm LXXIV. Honestly though, I think this is a somewhat optimistic reading of the psalm. Here is the part I believe they're referring to:
They seemed as men that lifted up
Axes upon a thicket of trees.
And now all the carved work thereof
They break down with hatchet and hammers.
They have set thy sanctuary on fire;
They have profaned the dwelling-place of thy name by casting it to the ground.
Personally, I think they mean they profaned the thicket by wrecking it, not that they profaned the thicket by casting axes on the ground there, but I could see where someone might have a different interpretation.
Anyway, enjoy your axinomancy, and remember to take proper precautions whenever working with hot coals or sharp objects.
References
http://www.occultopedia.com/a/axinomancy.htm
http://www.themystica.com/mystica/articles/a/axinomancy.html
http://www.bibliomania.com/2/3/255/1166/19712/1/frameset.html
http://www.bartleby.com/44/3/74.html