The elastic constants of the mineral anhydrite (CaSO4), which belongs to the orthorhombic crystal system, were unknown prior to this investigation. It was the main object of this thesis to investigate and measure these elastic constants which are required in the basic research being conducted by the Geological Sciences Department of the University of Saskatchewan. By the existence of strain energy potential and the morphologic symmetry, the elastic constants of anhydrite were reduced in number from 36 to 9. Six prismatic specimens were cut from natural single crystals of anhydrite, each according to theoretically specified directions. Foil type strain gauges were employed with auxiliary equipment to measure the associated strains during simple compression tests.