This chapter serves as a text for a course using dual-number methods as well as a manual for the reader to develop his or her abilities for the design of machinery or evaluation of mechanical systems. A mechanism is comprised of members called "links" whose connections are called "joints." A link is considered as a rigid body defining the relationship between two neighboring joint axes. The two joint axes of a link are each a line in space. The coordinate-transformation matrix modeling each link of a mechanism will be considered the product of a matrix specified by the displacements at the proximal joint of the link and a matrix specified by the dimensions of the link. A ball joint permits motion with respect to all three of a set of mutually-perpendicular axes. A plane joint features a plane surface on one link held in contact to form a bearing with a plane surface on another link.