Planimeter

A two-bar linkage with a tracing point at one end and a measuring wheel mounted on the tracer arm. When the operator traces the boundary of an arbitrary shape, the wheel rolls only in the direction perpendicular to the tracer arm — the component encoding the integrand of Green's theorem (∮ x dy). The total wheel rotation equals the enclosed area regardless of path geometry. The polar planimeter (Amsler, 1854) requires no straight guide rail and works anywhere on a flat surface. Precision versions routinely achieve 0.1% accuracy. Historically used in cartography, engineering drawing, and medical imaging to measure irregular areas from printed plans. Speed: seconds to minutes per area measurement (tracing speed). Capacity: single scalar output (area); arbitrary curve complexity.