In what follows we will comment on a statement made by Hamilton: 12
"The function of the "ears", or siyahs, is well known today and no one can question the superiority of the type of bow which still holds the world record of shooting an arrow 972 yards (Klopsteg 2 )."

Hamilton continues:
"The siyah contributes in three ways to improve cast in the arrow.
( 1 ) It provides leverage for the bowstring so the bow can be designed to approach maximum weight earlier in the draw allowing more energy to be stored for the cast. "

This statement is in agreement with our results. The static quality coefficient of the PE-bow is larger than that of the straight-end KL-bow. In Figure 4 the static and dynamic force draw curve are shown for the PE-bow. The line indicated with F shows a bend at the place where the string leaves the bridges. The TU-bow stores even more energy in the fully drawn position, obviously because of the recurve of the working part of the limbs. So the good static performance of flight bows may result only partly from the use of the stiff ears.
"(2) Upon release, the bowstring imparts its energy to the arrow more uniformly and at a higher and more sustained rate of thrust than in an ordinary bow without siyahs."
This statement is not supported by the results obtained with the model. Because of the relatively heavy ears, there is a sudden decrease in the force in the string and, by implication, in the acceleration force upon the arrow. The result of this is oscillatory behaviour as shown in Figure 4.

Static (F) and dynamic (E) force-draw curves for the static-recurve PE-bow

Consequentially the efficiency of static-recurve bows is rather low. The amplitude of the oscillations depends largely on the modulus of elasticity of the string and the mass of the arrow relative to the mass of the ears.

"(3) When the bow string reaches the bridges it is in effect shortened, increasing the tension again on the bowstring and giving the arrow a final snap as it leaves the bow."

The dynamic force draw curve (E in Figure 4 ) shows that the acceleration of the arrow is rather large when the string has contact with the bridges.

Notwithstanding this, the efficiency n of the PE-bow and certainly that of the TU-bow, is rather low. This implies that the initial velocity v is not as large as one would expect on the basis of the static performance. This is caused by the relative heavy ears. These considerations demonstrate why these bows can, inherently, not be better than long straight-end bows. A large part of the available energy remains in the vibrating limbs and string after arrow exit.

This holds even to a larger extent for the ER-bow. The string cannot slow down the now light ends of the limbs during the final part of the acceleration of the arrow when the bow is close to its braced situation again.

The modern WR-bow seems to be a good compromise between the non-recurve bow and the static-recurve bow. The recurve yields a good static quality coefficient and the light tips of the limbs give a reasonable efficiency.

Construction of the bow

But what made the Turkish flight bow a superb type of bow for flight shooting? Until now we dealt with the mechanics of the bow but not with its construction. The efficiency is greatly affected by the relative mass of the arrow relative to that of the limbs. For a fixed mass of the arrow, the lighter the limbs the better the efficiency. This is the item where technology becomes important. The minimum mass of the limbs for a fixed weight and draw i determined largely by the appropriateness of the material to store energy.

In the past man used bows which differ not only in shape but also in the materials applied. Simple bows made out of one piece of wood, straight and tapering towards the ends have been used by primitives in Africa, South America and Melanesia. In the famous English longbow the different properties of sapwood and heartwood were deliberately put to use. Eskimoes used wood together with cords plaited of animal sinews and lashed to the wooden core at various points. The Angular bow found in Egypt and Assyria are examples of composite bows. In these bows more than one material was used. In Asia the bow consisted of wood, sinew and horn; "Just as man is made of four component parts (bone, flesh, arteries and blood) so is the bow made of four component parts. The wood in the bow corresponds to the skeleton in man, the horn to the flesh, the sinew to the arteries, and the glue to the blood ." These bows were used by the Mongolian races of Eastern Asia. They reached their highest development in India, in Persia and in Turkey. In modern bows maple and glass or carbon fibres, embedded in strong synthetic resin are used .

In Table 2 indications of the mechanical properties are given for some materials used in making bows. From this table we conclude that it is possible to store much more energy per unit of mass in the materials of the composite bow, sinew and horn, than in wood, the material of the old simple bow. Moreover, in the composite bows not only better materials were used, but they were also used in a more profitable manner. Sinew is very strong in tension. It is therefore used on the back side. Horn withstands compression very well; it is applied to the belly side of the limbs. Hence, a composite bow with the same mass as a simple wooden bow can have a much larger weight. This explains the good performance of the composite flight bow in flight shooting.

In Table 3 we give values for the weight, draw, mass and length, for a number of bows described in the literature. The longbow is the replica of the Mary Rose bow. The calculated weight of this bow called MRA1 was 102.4 Ibs. If the same values are used for the material properties of the Mary Rose bow called A812 the estimated weight becomes 108 Ibs. W.F. Paterson13 14 also investigated this bow. Dr. Clark calculated that the draw weight would be about 76 lb, depending on the modulus of elasticity of the yew 14 The late Paterson in a letter to the author informs that Dr. Clarke: "admits an error by the factor of two in his calculations. His estimate should now read 153 lb." Unfortunately no value for the elastic modulus is mentioned, 1314 but the final discrepancy is probably caused by a difference in this mechanical property. We decided to use a value of .75 105 kgf/cm2. Parenthetically, the spread in the modulus of elasticity of yew yields makes the predictions of the weight (almost proportional to the modulus of elasticity of yew) of the Mary Rose bows uncertain.

The quantity denoted by bv is proportional to the amount of energy stored in the bow per unit of mass. It equals the weight times draw length divided by the mass of one limb. When materials are used to their full extend bv divided by about 4 should equal Dbv.

We saw that because of the stiff ears or a recurve of the working parts of the limbs, much energy is stored in the static-recurve bow. In a recurved bow the amount of energy in the braced position is already large. This implies that the limbs must be relative heavy in order to store this extra and not usable energy, in addition to the recoverable energy. This is the price paid for a larger static quality coefficient. On the other hand, sinew and horn are relatively tough and flexible materials, see Table 2. This explains why the use of these materials fits well with the recurved shape of the unstrung bow. The values in Table 3 show that the Turkish bow is very strong but also light. This indicates why it permits one to shoot a light arrow a long distance. A short bow is moreover easier in operation and is suited for the use on horse back. In a letter 18 to the author E. McEwen informs that: "Pope did not properly test his larger 'Tartar' (actually Manchu-Chinese) bow. He only drew it 36 inches and bows of this type and size are made to draw as much as 40 inches." Pope only mentions the weight for a draw length of 29 inches. If the weight of this bow with a draw length of 101.6 cm is 70 kgf, we have bv = 9700 kgf cm/kg. This value is still rather low and this means that the materials of this bow are used only partly. This supports McEwen's 18 view that: "this bow was probably a 'test' bow used for exercise and for military examinations and not meant for actual shooting." The values obtained for the straight-end bows look very realistic. In the modern bow there is a surplus of material near the riser section. This affects the efficiency only slightly For this part of the limb hardly moves and the;efore the involved kinetic energy is small In this bow there is also a rather large amount of unrecoverable energy in the braced position. This puts a constraint on the amount of recurve With respect to this, it is perhaps more important that the efficiency of working-recurve bows decreases with increasing recurve. The mechanical properties of the materials of these bows however, are much better than those of the ancient composite bows. Indeed, the modern bow now holds the longest flight shooting record .

Additional features were added to improve the performance especially for target shooting; relative immunity of the mechanical properties to temperature and humidity variations, no tendency to follow the string, use of stabilisers, sculptured long centre-shot riser section, bow sights and last but not least stronger materials for bowstrings. Finally an improved arrow design adds to the steadiness of the equipment.


We conclude that these results indicate that the initial velocity is about the same for all types of bow under similar conditions. So, within certain limits, the design parameters which determine the mechanical action of a bow arrow combination appear to be less important than is often claimed. We would endorse a view one could call holistic. It is not always possible to isolate a single feature and state that it solely accounts for a good or bad performance of the whole bow, as Hamilton 12 did . Rausing 19 studies the development of the composite bow. According to him, the fact the static quality coefficient of the short static recurve bow to be larger than that of the short straight bow, disposes of the statement of Pitt Rivers, Balfour and Clark: "the composite bow has no inherent superiority over the wooden self-bow, so long as the latter was made from the most favourable kinds of timber and expertly used". The results obtained with the mathematical model suggest that, if the word inherent has the meaning within the context we used it in Section 4, their statement is true. A combination of many technical factors made the composite flight bow better for flight shooting .

The quality coefficients of the modern bow are only slightly better than those of the other types of bow. Materials used in modern working-recurve bows can store more deformation energy per unit of mass than the materials used in the past. Moreover the mechanical properties of these materials are more durable and much less sensitive to changing weather conditions. This contributes most to the improvement of the modern bow.

We hope that we have shown that mathematical modelling can be a helpful tool in the research on archery, not only for the design of new bow equipment but also for understanding the development of the bow in the past.