Crystal surface diffusion refers to the way in which atoms on the surface of a crystal redistribute to eventually settle into a configuration with minimal surface energy. Along with epitaxy, or crystal growth, crystal surface diffusion is important to study due to the role it plays in the production of thin films, which have wide-ranging applications in microelectronics. For example, the deformation of a crystal surface to an equilibrium state plays a central role in fuel cells that rely on thin crystal films, as the conversion efficiency of chemical energy to electricity depends on the surface configuration of the film. As is characteristic of large microscopic systems, we can gain more insight into the nature of the dynamics of surface diffusion by studying it at the macroscopic level than at the level of individual atoms. Since the physical process is microscopic, however, a faithful mathematical model of the diffusion should describe it with microscopic dynamics. Given a model of the microscopic dynamics, then, we are presented with the challenge of deriving macroscopic dynamics in the limit as the number of particles approaches infinity. This is known as a hydrodynamic, or scaling, limit; it is particularly appealing from the modeling perspective because the input is the true, microscopic dynamics, while the output is a much easier to analyze continuum equation. The main goal of this work is to derive such a scaling limit for a specific dynamics governing the microscopic process of crystal surface diffusion.