1. Speed describes how fast; velocity describes how fast and in what direction. People often refer to muzzle velocity when they mean muzzle speed.

2. The physiology of throwing is described by Chowdhary and Challis (2001); a physical model is provided by Cross (2003).

3. For Roman use of the pilum, see Webster (1980, p. 81) or DeVoto (1993, p. 132).

4. Both Strabo and Vegetius (a fourth-century Roman general) make this claim. See, e.g., Vegetius, De Re Militari, book 1; for a convenient online source, see the bibliography.

5. Of course, there are many other considerations that determine the outcome of battles, such as soldiers’ morale and determination, their physical condition, and the skill of their leaders. Here, though, I am concerned only with the influence of ballistic weapons and so will limit my discourse to factors relevant to these.

6. To delve deeper into ancient warfare, see, e.g., Montgomery (1972); Sabin, van Wees, and Whitby (2007, chap. 13); and Sidnell (2007). The Greek peltast of fig. 1.1 was an innovation to combat skirmishers; peltasts were somewhere between light and heavy infantry. Note that the peltast in the illustration carries a sword as well as javelins.

7. A slinger must exert effort throughout the entire internal ballistics phase, and so, to repeat a shot, he must reproduce exactly the motions of the previous shot. A bowman need only point his weapon in exactly the same direction as before, because once the bowstring is released, the weapon takes over. In other words, the slinger inputs energy throughout the internal ballistics phase of a shot, whereas the bowman inputs energy only before he aims his weapon.

8. The English longbow had an astonishingly heavy draw weight of around 100 lb. It required years of training to operate effectively, especially when the archer was part of a military formation. The fame of this longbow was not due to its range or accuracy (it was used militarily against dense formations) but instead was due to the training of its archers and the logistics of their deployment. At the Battle of Crécy in 1346 some 7,000 English archers decimated charging French heavy cavalry—previously considered invincible—by firing half a million arrows at them. At their maximum rate of fire, these archers would have dispatched their missiles at the rate of 2 tons per minute. See Fowler (1967, p. 108), Bradbury (1992), and Hardy (1993) for details of the English longbow and its use on medieval battlefields.

9. See Denny (2003). Those readers who are following the math may refer to my Web site (Denny 2010), where a copy of this paper is available. The math model I construct is approximate; bow dynamics is complex, not least because it involves beam theory, so my model involved simplifications. For the mathematically inclined reader, references in my paper will direct you to articles that present the detailed, full-blown calculations.

10. When the arrow leaves the bow, the string vibrates; and if the bowstring has mass, this vibration consumes energy.

11. The energy stored in the bow is approximately ½ Fd, where F is draw force and d is draw distance. The bow power is determined by dividing this energy by the time it takes to fire the bow. For the numbers provided in the caption to fig. 1.5, we obtain the figure of 10 kW.

12. For more on the history and technology of bows, see Denny (2007, chap. 1).

13. The crossbows of ancient Greece were known as gastraphetes (belly-shooters) because of the way they were spanned. Crossbows of classical antiquity are discussed by Landels (1978, chap. 5).

14. Two YouTube videos listed in the bibliography (2007c and 2008b) demonstrate the archer’s paradox more eloquently than any math analysis can. For further technical details, see Denny (2005, 2007).

15. My onager paper, available on my Web site (Denny 2010), contains a derivation of the equation for efficiency.

16. See Denny (2005) for a technical analysis. This paper is available online at Denny (2010). A less technical account of trebuchet dynamics, plus historical notes of the evolution and development of these fearsome machines, can be found in Chevedden et al. (1995) and in Denny (2007, chap. 3). Many trebuchets are still being built, for educational purposes in university engineering departments or simply for fun. See www.youtube.com/watch?v=-wVADKznOhY for a video of a large modern trebuchet throwing a flaming piano and a small car.