The Hatra ballista: a secret weapon of the past?

The mechanics and experimental work concerned with ancient catapults powered by torsion springs is described. Mathematical models are proposed both for the earlier Greek palintone, or stone thrower, and a supposedly more powerful machine based on the conjectured form of a notable find in the ruins of Hatra, a town in Iraq sacked in AD 241. For each of these machines an energy method is formulated together with differential equations governing their performance. Using a moment function derived in earlier work and allowing for internal friction, we solve the equations by numerical integration, assuming rigid straight arms and an inextensible sling. The ranges of projectiles launched from these machines are predicted, and the results are compared with observed ranges produced from physical models. Good agreement with experiment is found, confirming the superiority of the Hatra machine which is also more mechanically efficient. Comparison with some results of other workers on the bow is appropriate. In the course of this study it is found that the previously derived nonlinear relation between moment and angular displacement of a torsion spring gives much closer agreement between theory and experiment than a linear relation that was conjectured earlier.