In many of the recent splendid discussions of the mechanics of the bow, there is a gratifying tendency towards mathematical analysis. Sound results should come from it, as there is no reason why the external action of a bow should not yield to analysis once the correct assumptions are made. It is not likely that we can do much with analyzing the internal or elastic action of the bow limbs because of the infinite variety of woods, their complicated cellular laminated structure and lack of uniformity. Certain relationships, demonstratable mathematically, may be of interest as not having been simply presented in previous articles.

Bows must not break. Stress is what breaks a bow and stress depends on two things—thickness and radius of curvature—varying directly as the former and inversely as the latter. Another factor—elasticity—enters into the strength of any given bow and plays an important part, if not the most important, in that indefinable quality of a wood called "cast." Strength and weight are probably the two other features in "cast." These features may be ignored in the basic analysis of bow design but enter later.

Bows have the least amount of bend or the largest radius of curvature for a given draw when they bend in the arc of a circle. This radius for a 28-inch draw cannot, under any circumstances, be greater than the following:

Any deviation from the circular arc means an increase of the curvature at some point. If the thickness is constant this results in a higher stress, localizes the ultimate break or at least lessens the safety of the bow.