The theory of Amsler's polar planimeter, as commonly given, leads to the expression A= L h, where A is the area circumscribed, L is the length of the tracer arm, and h is the net distance of translation of the tracer arm in a direction perpendicular to its length. However, it can be shown that the area may also be given by A = L 2 ∅, where ∅ is the net angle through which the tracer arm has rotated about either of its ends. But as the tracer point passes around the area A, the tracer arm does not, in general, rotate simply about one end. At any particular instant it rotates about some point which may be situated anywhere along its length. However, about whatever single point the arm may be rotating, such rotation can be resolved into two simultaneous rotations about the two ends. Therefore for purposes of analysis we may consider that the tracer arm rotates only about the ends, and we may express the area in terms of that rotation, as already stated.
Proceedings of the Iowa Academy of Science
©1926 Iowa Academy of Science, Inc.
Davis, George E.
"A New Analysis of the Action of Amsler's Polar Planimeter,"
Proceedings of the Iowa Academy of Science, 33(1), 250-251.
Available at: https://scholarworks.uni.edu/pias/vol33/iss1/71