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## Mathematics Behind Planimeters

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##### Date

2013-06-18##### Author

Yardimci, Osman

##### Type of Degree

thesis##### Department

Mathematics and Statistics##### Metadata

Show full item record##### 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.

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