Experimental and analytical methods are presented which enable one to examine the local rheological properties of biological tissues which can be captured as flat sheets between matching pressure manifolds and deformed under experimentally prescribed hydrostatic loading conditions. In spite of the fact that most biological tissues, including arteries, are nonlinearly elastic when considered over wide ranges of strain, it was found that the deformation of swine and canine arterial wall specimens in the physiological range of wall strain can be approximated by an isotropic, linearily elastic membrane model. In view of this, the elastic behavior was characterized approximately by an incremental modulus over the range of 0.45 to 0.65 strain. The incremental modulus in both species was shown to increase by a factor of three along the descending thoracic aorta from the ductus scar to the celiac orifice.