Measurements and Predictions

We may now turn to some actual arrows. Through the kindness of Dr. Cranston, I have been able to measure the stiffness of a number of reed arrows in the Pitt Rivers Museum, which were found together with an angular bow in an Egyptian tomb of the 26th Dynasty - the 7th Century B.C. (Balfour, 1890, 1896; origin of the bow discussed by McLeod, 1969). The arrows have a length around 32 in. (0.91 m), but of that some 20 in. (0.5 m) is reed, the remainder bein wooden foreshaft. Their weight is now between 0.36-0.43 oz. (10-12 g), nearly half of which is due to the foreshaft; we should expect the arrow to behave dynamically very much as if it were a 26 in. (0.35 m) reed with a concentrated mass at the tip quivalent to that of the shaft. In shooting, it would seem likely that some 3 to 4 inches of this foreshaft, including the microlithic head, projected beyond the bowstave, so that the length of the draw would be some 29-30 in. ( 0.74-0.76 m), bringing the nock just past the ear of a short man. In flight, a ballistic coefficient of about C = 0.06 seems probable on the basis of Rheingans' calculations (Rheingans, 1936). The reed has a stiffness which would produce a 'spine' of about 160 G.N.A.S. in a shat of that length. This is twice the deflection recommended for a 30 lbf bow in standard tables, and apparently suitable for a modern bow of 15 lbf.

One's first thought is that the reed must have weakened with age. However, although change in the strength properties of wood and reed over long periods of time have yet to receive systematic study, they are unlikely in this case to have been very great. The stiffness is similar to that of reeds of similar diameter which I have cut and measured in Greece, and neither the external nor the internal diameter can have changed very much, or it would have upset the binding and the fitting of the nock and foreshaft. It would seem that the reeds were fully dried out and shrunk before the arrows were made. Moreover, the elasticity of reed, like that of timber, seems to depend mostly on the helical arrangement of cellulose fibrils in the cell walls. It is difficult to see how it could alter, except through distortion of the cells or breakdown of the cellulose, either of which would have drastic results. Making the extreme assumption, that there might have been a shrinkage across the grain of 25%, and noting that the stiffness of a shafft or tube varies as the fourth power of its diameter, we might consider as a limiting case that the stiffness could have been halved with age, but the actual diminution is probably much less. Similarly the weight might originally have been up to 50% greater, due to moisture content, but again comparison with modern reeds suggests that the actual changes are likely to have been small.

The limits of performance of the arrows may therefore be calculated as follows. The maximum load supplied by the string must have been less than:

Fcrit = ((1+1)/(1+2))*(1,000,000/(160*26^2)) = 6 lbf (28.5 N)

(or up to twice that if we are allowing for shrinkage). A force of 6 lbf, if it were applied to the arrow for a distance of two feet (which is approximately the distance from the full draw of 29-30 in. suggested above to the point at which the string would become straight) would give the arrow a kinetic energy of 12 ft. lb. (17.3 Joules). In a practical bow, as we have seen, the load will vary during the shot, and the energy will be lower by between perhaps one-third and two thirds, depending on the shape of the bow, i.e. between 4 and 8 ft. lb. (5.8 and 11.5 J). It is easily shown that for arrows weighing 0.4 oz. (11.2 g) such values for the kinetic energy (K.E. = 0.5*m*V^2) correspond to velocities of 101-143 ft./sec. (31-44 m/s), and these in turn indicate maximum ranges between 85 yards (78 m) and 150 yards in the graphs drawn by Rheingans (1936, and A.T.S., p. 248) from Ingalls' Ballistic tables, if we assume the ballistic coefficient C = 0.06. The loss of energy in flight implied by such a coefficient is of the order of 20% over 40 yards (the loss over equal distances representing a constant proportion of the energy remaining) so that the maximum strike energy for the arrows in their present state should be:
at 40 yards (36 m) between 3.25 and 6.5 ft. lb. (4.7-9.4 J);
at 100 yards (92 m) - supposing that they reach that distance - between 2.5 and 5.0 ft. lb. (3.6-7.2 J)

If maximum allowance were made for shrinkage, the values for strike energy would be increased to just over twice the above, since in addition to the effect of stiffness, the extra mass of the arrow would raise the ballistic coefficient to around 0.077, implying a smaller loss of energy in flight. On these assumptions the range would be increased for each type of bow by about 25%


The values for the arrows 'in their present state' are remarkably low, in fact too low to be credible as maximum limits. For comparison, the energy with which a modern target arrow leaves a longbow can be estimated very approximately as number of ft. lb. equal to 40% of the number of lbf in the draw-weight of the bow, and it falls to the equivalent of about 30% over 100 yards. A heavy hunting arrow will do rather better. So, when hunting with a bow drawing between 40 lbf and 70 lbf (180-310 N) a modern archer will be achieving a strike energy of 12-23 ft. lb. (17-33 J). Conversely, the best strike energy predicted for the Egyptian arrows in their present state is limited to that which would be expected of a modern archer using a 15 lb. bow, and that assumes the use by the Egyptian of a reflex composite; the wooden longbows ought to have done even worse. So, either there as been some deterioration in the arrows, as already discussed, or there is some point about the dynamics of the system which we don't yet understand, and which was exploited by the ancients. A possible candidate might be the effect of the foreshaft on the dynamics of the arrow.

Whatever the reason, however, the range and strike energy still seem unlikely to have been greater than predicted 'assuming maximum shrinkage', i.e., a range between 110 yards (100 m) and 180 yards (166 m), and a strike energy at 40 yards (37 m) between 6.7 ft. lb. (9.7 J) and 13.4 ft. lb. (19.3 J), and as maximum limits these remain very low by modern standards.

The reason why arrows with such a low energy were effective weapons must lie in their narrow heads in which three small slivers of obsidian set in bitumen form a chisel-ended trapezoid, with a maximum breadth of 0.28 to 0.4 in. (7-10 mm). The breadth of these heads bears much the same relation to the predicted strike energy as that of the heads described by Pope (1962) to the energy of the bows in which those were shot. In each case it looks as though the proportion is very roughly half an inch of head for every lb. ft. of strike energy (or for every 27 lb. ft. of bow-weight) which is equivalent to about 1 mm per Joule. Such matching of the head to the strike energy would be likely because the aim must be to make the largest hole possible with the availablew energy; it will be governed by the breadth of the head in a simple proportion, because most of the energy goes into cutting, so that the work done will depend on the area of the cut. Assuming that every arrow is intended to reach the same depth - i.e., to pass right through the victim,but not to waste energy by coming out the other side, the area of the cut depends on its breadth. So on this score, the arrows are well designed, though there is one curious head of hammered bronze with a breadth of 0.57 in. (17 mm) which would seem inefficient.

On the other hand, the occurence of such arrows with bows apparently drawing about 40 lb. (as both the Balfour bow, found with these arrows, and the self-wood longbows in the British Museum appear likely to have done), must mean that the bows were inefficient, since a modern bow of that weight would almost certainly snap them. Reasons for the inefficiency of the longbows are not far to seek; they seem to have been so arranged that there was no stress in the limb when the bow was strung, and the limbs seem too heavy to be matched to such light arrows. However, there is also a factor which applies both to them and to the much more elegant composites, namely the weight and elasticity of the string. A modern string may weigh about half an ounce (14 g) and will be as stiff as possible. The Egyptian strings look as though they were about three times as heavy, and they seem to have been made of gut, which is very soft. Their weight alone would make a very considerable difference, since Hickman (1931) has shown that the effect on an increase in string weight on the velocity of an arrow is equivalent to adding one-third of the increase to the weight of the arrow. Thus, in a modern bow where the string weighs half an ounce and the arrow one ounce, the movement of the strings absorbs only one-seventh of the energy transmitted by the arms. In the Egyptian case, if the string weighs three times as much as the arrow, it will half the energy. Further energy will be absorbed in the stretching of the string, and it is quite credible that the efficiency should be between two thirds and one half that of a modern bow.

In view of the high standard of craftsmanship shown in the making of the composite bows and arrows, the low performance is somewhat surprising, and disappointing. Why did they not do better? Three possibilities may be suggested. Firstly, that the longbows always were inefficient, and could not be made more efficient by bracing (pre-stressing) for fear they would creep in the hot sun. Arrows which were suited to these bows became standard, and were not improved when better bows became available. Secondly, that there is a vicious circle by which the inefficiency of the bow leads to the use of both of a light arrow, in order to get the range, and of an elastic bowstring, in order to absorb the surplus energy which would otherwise damage the bow, and then both of these compound the inefficiency. Thirdly, that reeds are cheap, and good arrow timber expensive. The second and third of these factors may have influenced most of the ancient Near East.

The problem of the stability of arrows with a long foreshaft made of a material much stiffer than that of the rest of the shaft seems to merit furhter study and experiment; I should be interested to hear from anyone prepared to co-operate either in that, or in the development of this method of investigation to cover arrows of other periods.

St. Mary's College, Twickenham


The main part of the study on which this paper is based was done at the University of Reading, with the support of the Leverhulme Trust. I am grateful to several colleagues in the Engineering Department for discussions of the mechanical aspects.