The zygomatic arch of mammals is usually considered a phylogenetic relic of the fenestrations of the skull roof which may be observed in morphological sequences of primitive vertebrate skulls. If this concept is correct, the element is comparable (though not homologous) to the jugal arches of diapsid reptiles. Two major questions then remain unanswered: why different elements are maintained in reptiles and mammals during evolution, and why the arches are maintained as relics of ancestral forms. It is tempting to respond to the latter question with a very simple answer, namely that the elements function in order to sustain mechanical stresses. In this paper, we raise the questions which quality of stresses occurs in a primate skull within the zygomatic arches and what relationship these stresses hold to the morphology of these bony elements. An answer has been sought by means of finite element stress analysis. We found that the zygomatic arch in primate skulls represents a structure which carries, under all biologically relevant conditions, either compressive or tensile stresses. In a very simple model of the human skull under bite forces, a strip of stresses occurs lateral to the orbits, which seems roughly comparable to the zygomatic arch. Once such a structure exists and is used as an insertion of adductor muscles, it will be exposed to bending stress in side view and in frontal view. Morphological details of the zygomatic arch (curvature, profile, suture) are well suited to sustain the evoked stresses by a minimum of material.