As the bow is drawn, the limbs are caused to bend in arcs more or less uniformly circular. Their resistance to bending, or stiffness, determines the drawing force. The stiffness depends on (a) the characteristics of the wood and (b) the dimensions of the limb. The former question has been fully discussed in Archery Review, Vol. II, September, 1932, p. 8; repetition at this point is therefore unnecessary. The latter has also been discussed from various angles in several articles, but not "from the bottom up," as it is here proposed to do.
Because the bow limb of rectangular section, with uniform thickness, and bending in a true circular arc, has been proved so definitely superior to the limb of traditional design, and because it has, during the past few years become so well known, let us apply our considerations to this form of limb.
To begin with, we perform, in imagination, an experiment. Clamp the handle in a vise, and apply a force to the tip. The stiffness is measured by the force required to bend the limb so that the tip moves through some arbitrary distance, say an inch. This will be an approximate gage of the drawing force of the bow, because the greater this force, the greater also will be the force at full draw. The first fundamental principle is that the stiffness, so measured, is proportional to (1) the width of the limb. (2) the cube of its thickness, and (3) the reciprocal of its length. Let us picture the meaning of this statement more clearly. By "width" in this case we might mean average width, or width at midlimb, or greatest width; any one will do so long as we always mean the same thing by the designation "width," and arc talking about similar limbs. Let us arbitrarily take greatest width, or width at the dip. Now imagine a limb of certain dimensions, and find what happens as we change them, one at a time, while keeping the other two dimensions constant.
The force needed to produce a deflection of the tip by an inch increases in the same ratio as the width; double the width, and the force is doubled; increase it 10%, and the force is increased the same amount. Here thickness and length are unchanged.
Next keep length and width constant, and vary the thickness. Double the thickness, and the stiffness increases by a factor of two cubed (23) or 8. Increase it 10%, and the stiffness increases to (1.1)3, or 1.33, i.e. by 33%. This shows why taking a small amount of material off the belly or back of the bow affects the drawing weight so much more than does taking it off the sides.
Finally, with thickness and width held constant, change the length. Increase the length 10%, and the stiffness becomes 1/1.1, or .91 as great, i.e., it decreases 9%. This, we note, is the inverse effect from that produced by a change in width. Shorten the limb 10%, and its stiffness becomes 1/.9 or 1.11 times as great, an increase of 11%.