42° as a reference for modern orthodontic arch forms-a mathematical analysis of the Bonwill and Hawley methods

To date, no consensus exists on an optimal orthodontic arch form. The aim of this study was to mathematically analyze and simplify the geometric constructions of the Bonwill and Hawley arch forms.

The original constructions by Bonwill and Hawley were systematically reconstructed and mathematically analyzed. The Hawley method was translated into a simplified clinical application and compared with current arch form studies.

The Bonwill method produced a 49° angle between the posterior segments, the Hawley method produced a consistent 42° angle. The Bonwill method used a fixed 35 mm width for lower incisors and canines, whereas Hawley accommodated individual widths. Consequently, Bonwill's posterior angle varied with individual cases, whereas Hawley's angle remained constant. The 42° Hawley angle closely matched empirical data from recent studies on untreated mandibular arch forms.

The Hawley method produces a consistent 42° posterior segment angle, regardless of tooth width. This angle aligns with natural mandibular arch forms and offers a simple, individualized reference for analog or digital orthodontic setups.