The Action of Two-joint Muscles : the Legacy of W . P . Lombard

In a five-page missive, W. P. Lombard (1903a) helped lay the foundation for a new and rational approach to understanding how muscles are coordinated. Although investigators dating to the time of Borelli (1685) had described the relationship between muscle, joint, and limb in a logical, intuitive, and physically meaningful manner, Lombard made an astonishing, counterintuitive claim. He asserted that muscles with apparently opposing actions, termed pseudoantagonists, can be used together in a productive way (see also Lombard, 1903b and 1907). This concept came from a remarkable intuition, one that has challenged us to state more quantitatively and objectively for nearly 100 years. Though his ideas were expressed qualitatively, the definitive nature of his statements implied that they could be translated into precise mathematical descriptions. The enduring influence of his work stems perhaps from its tantalizing character, a mix of biological description and bold prediction. We might classify Lombard’s theory of the pseudo-antagonist as a conjecture—a theorem waiting for definitive proof. In our attempts to find the proof, we have adopted the principles of mechanics, carving out a new field of biomechanics. In our review of Lombard’s influence, it will be seen that much of our as yet incomplete understanding of muscular function has been fed and driven from this remarkable source. As with any developing field, advances in biomechanics have come on a variety of fronts. Motor tasks have been differentiated in terms of upper and lower extremity, unconstrained and constrained, isometric and otherwise. In taking stock of Lombard’s impact on biomechanics, it is expedient to use a single mathematical notation to assess and compare the many contributions made by different researchers studying different motor tasks. Mathematics is a precise and unambiguous language, but it is also somewhat inaccessible. Fortunately, there is a geometric interpretation that will be both accessible and mathematical. We will use Lombard’s main thesis as a vehicle for introducing the mathematics and geometry, after which contributions by others will motivate appropriate extensions but all within a single framework.