Diagrams are widely used in reasoning about problems in physics, mathematics, and logic, but have traditionally been considered to be only heuristic tools and not valid elements of mathematical proof. This book challenges the prejudice against visualization in the history of logic and mathematics and provides a formal foundation for work on natural reasoning in a visual mode. The author presents Venn diagrams as a formal system of representation and specifies rules of transformation that make this system sound and complete. The soundness of the diagrammatic system refutes the contention that graphical representation is misleading in reasoning. The book concludes with a discussion of some fundamental differences between graphical systems and linguistic systems.