Mathematics Behind Planimeters

Abstract

Original master thesis project: By studying the literature, collect and write a survey paper on the mathematics of the planimeters. Planimeter is a somewhat forgotten ingenious device which was invented in 1814 to satisfy the area computational need of land surveyors and other real life applications. The project also asked for lling in details of published proofs and asked for including, if possible, some new statements and generalizations. On the importance of planimeters: The history of approximating and computing areas goes back to 3000 BC, when the ancient Egyptians approximated the area of circles. A great deal of knowledge on areas was summarized by Euclid around 300 BC in his book entitled Elements. Undoubtedly the discovery of modern time calculus by Newton and Leibnitz around 1660 was the biggest advancement in area computation. At the beginning of the 18th century, practical mechanical tools, called planimeters, were patented for determining the area closed regions. The following are the outcomes of the thesis: 8 papers ([1], [2], [3], [4], [5], [6], [7], [8])were included in this review. The mathematics of both of the linear and the polar planimeters were studied. All arguments were based on Green's theorem, on the Area Di erence Theorem, and on the theorems concerning sweeping line segments. These theorems are explained in the rst half of the thesis. It turned out that there are two basic approaches at proving the correctness of planimeters. The section which explains the indirect approach (using Green's theorem without computing the involved integrals) is based on a work of B. Casselman [4]. The section which explains the direct approach (using Green's theorem with computing the involved integrals) is based on the work of Ronald W. Gatterdam, [1]. The explicit approach in the paper of Gatterdam was explained when the two arms of the polar planimeter had equal length. Using the method of Gatterdam, I veri ed the correctness of the polar planimeter in case the arms had di erent lengths.