Bonwill-Hawley arch demystified - A Comparative Mathematical Study Demonstrating the Consistent Symmetry of 42° in the Hawley Arch Across Varying Tooth Widths

Aim : To mathematically analyze and simplify the geometric Bonwill-Hawley method, optimizing it as an individualized approach to enhance diagnostic accu-racy. Methods : The geometric construction principles from Bonwill’s and Hawley’s original works were mathematically analysed and compared. Hawley’s method was translated into a simplified clinical application and compared with current studies of archforms. Results : The primary distinction between the two methods is that Bonwill’s approach is based on the measurement of 10,000 mandibles, whereas Hawley did not utilise any preliminary investigations. In Bonwill’s method, the widths of the six incisors are determined by his triangle, always resulting in a fixed width of 35.4 mm. In contrast, Hawley’s method incorporates individual incisor widths, which, paradoxically, consistently result in a constant angle of 42 degrees between the posterior segments. In contrast, Bonwill’s method would produce varying angulations when applied to individual tooth widths. In current studies on den-tal arches, Hawley’s 42 degrees can still be observed in both growing and non growing Class I subjects. Conclusion : Hawley’s arch form construction remains valid until today and should be regarded as a clear approximation of the individual treatment objective, serving as an additional key to optimal occlusion.