The use of diagrams in mathematics has traditionally been restricted to guiding intuition and communication. With rare exceptions such as Peirce's α and β systems, purely diagrammatic formal reasoning has not been in the mathematician's or logician's toolkit. This paper develops a purely diagrammatic reasoning system of ‘spiderdiagrams' that builds on Euler, Venn and Peirce diagrams.The system is known to be expressively equivalent to first-order monadic logic with equality. Two levels of diagrammatic syntax have been developed: an ‘abstract' syntax that captures the structure of diagrams,and a ‘concrete' syntax that captures topological properties of drawn diagrams. A number of simple diagrammatic transformation rules are given, and the resulting reasoning system is shown to be sound and complete.