Pre-Amsler Planimeters

During the first half of the Nineteenth Century, the need for an instrument that would accurately measure area inspired several ingenious devices.  Planimeters previous to Amsler's invention were either inaccurate, difficult to use, or cumbersome.

Cone Planimeters

One of the early designs had the wheel rolling on a spinning cone. Mathematically this is a direct application of the definition of the definite integral  ∫ y dx: the larger the y value the higher the wheel is on the cone, and so the faster the wheel turns.

Schematic of design by Johann Martin Hermann, Germany, 1814.
Taken from Mathematische Instrumente by A. Galle, Druck und Verlag, Leipzig und Berlin, 1912.

From planimeter history web page, Universität Würzburg
http://www.didaktik.mathematik.uni-wuerzburg.de/history/ausstell/planimet/index.html

The Science Museum in London

Maxwell's Planimeter

 The inventors were in good company -- one of them was J. C. Maxwell, of electromagnetic theory fame. This planimeter consists of a fixed hemisphere and a sphere of the same radius.  The motion of the tracer point causes the sphere to roll on the hemisphere.  Exactly how this translates into area is far from obvious!  The details (including this picture) are in Maxwell's paper Description of a New Form of Platometer, an Instrument for measuring the Areas of Plane Figures drawn on Paper (Maxwell's Collected Papers, v. I, p. 230).  Maxwell learned of Amsler's planimeter after this article was accepted, but before it appeared.  Without even seeing a working model, Maxwell immediately realized the significance of Amsler's invention and had the editor of the journal add the following to the end of the article: Since the design of the above instrument was submitted to the Society of Arts, I have met with a description of an instrument combining simplicity of construction with the power of adaptation to designs of any size, and at the same time more portable than any other instrument of its kind. I think that its simplicity, and the beauty of the principle on which it acts, render it worth the attention of engineers and mechanists, whether practical or theoretical.                              -- J. C. Maxwell, 1856