Diagrams are ubiquitous in scientific discovery and scientific reasoning. The uses of diagrams are studied by examining their role in Galileo’s work on kinematics and the computational benefits of reasoning with diagrams are investigated by comparing computational models of conventional and diagrammatic approaches to discovery. Diagrams provide a convenient form of representation for experimental setups that preserve valuable topological information about the setups. Diagrams allow the ready-made and powerful inference machinery of Euclidean geometry to be used for reasoning. When suitable standardization techniques exist, diagrams furnish a basis for reasoning about the magnitudes of dissimilar properties. Diagrams can be deliberately organized to provide perceptual clues to aid the making of inferences, and to act as structured records of the available information. The consequences of this are computational savings in the processes of search and recognition. Finally, the efficiency of inferences can also be improved by replacing expensive forms of reasoning with cheaper perceptual inferences.