The bony labyrinth of vertebrates houses the semicircular canals. These sense rotational accelerations of the head and play an essential role in gaze stabilisation during locomotion. The sizes and shapes of the semicircular canals have hypothesised relationships to agility and locomotory modes in many groups, including birds, and a burgeoning palaeontological literature seeks to make ecological interpretations from the morphology of the labyrinth in extinct species. Rigorous tests of form-function relationships for the vestibular system are required to support these interpretations. We test the hypothesis that the lengths, streamlines and angles between the semicircular canals are related to body size, wing kinematics and flying style in birds. To do this, we applied geometric morphometrics and multivariate phylogenetic comparative methods to a dataset of 64 three-dimensional reconstructions of the endosseous labyrinth obtained using micro-computed tomography scanning of bird crania. A strong relationship between centroid size of the semicircular canals and body size indicates that larger birds have longer semicircular canals compared with their evolutionary relatives. Wing kinematics related to manoeuvrability (and quantified using the brachial index) explain a small additional portion of the variance in labyrinth size. We also find strong evidence for allometric shape change in the semicircular canals of birds, indicating that major aspects of the shape of the avian labyrinth are determined by spatial constraints. The avian braincase accommodates a large brain, a large eye and large semicircular canals compared with other tetrapods. Negative allometry of these structures means that the restriction of space within the braincase is intense in small birds. This may explain our observation that the angles between planes of the semicircular canals of birds deviate more strongly from orthogonality than those of mammals, and especially from agile, gliding and flying mammals. Furthermore, we find little support for relationships between labyrinth shape and flying style or wing kinematics. Overall, our results suggest that the topological problem of fitting long semicircular canals into a spatially constrained braincase is more important in determining the shape of the avian labyrinth than the specifics of locomotory style or agility. Our results tentatively indicate a link between visual acuity and proportional size of the labyrinth among birds. This suggests that the large labyrinths of birds compared with other tetrapods may result from their generally high visual acuities, and not directly from their ability to fly. The endosseous labyrinths of extinct birds and their close dinosaurian relatives may allow broad inferences about flight or vision, but so far provide few specific insights into detailed aspects of locomotion.

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