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The Polar Planimeter

A Book by J. Y. Wheatley, 1903.
Adapted and Maintained by Lawrence Leinweber, 2000.

Preface to the HTML Version

Chapter X has all the pictures and is probably the best place to start if you are unfamiliar with the Polar Planimeter.  Otherwise, try the Table of Contents.

All graphics are stored as rasterized, monochrome, 300 dots per inch, from the originals printed with black ink.    But browsers usually show graphics at 100 dots per inch.  So the images are actually reduced 9 to 1.  Originally, the shading was rendered in narrowly spaced parallel lines, and the reduction produces moiré patterns in a few cases.  But an image can be viewed in its full detail by selecting “view image” with your browser.

To some extent, this version is a pure exercise in HTML.  Dividing the book by sections into HTML pages was part of the adaptation and linking each (or at least the first) reference to another section was also the significance of HTML.  This form was not intended by the original author and although he seemed overly fond of the phrase “as previously mentioned” and words to that effect, it is not always clear where the “previously” occurred or how aptly it was “mentioned.”  In any case, the linking may be excessive or useful.

With the exception of this page, this site is intended as the original book reformatted for the digital medium.  Some errors were corrected from the book.  (No doubt others were introduced.)  Some archaic references were modernized especially those that were used inconsistently or were inconsistent with modern usage.  Other information is preserved for historical interest.

The author of these HTML pages is tagged as Lawrence Leinweber, an unfortunate automatic feature of the editing software.  The author of the book is not disputed or available for comment.  The information provided herein is for educational purposes, specifically to provide information about the polar planimeter, a goal shared by both the actual and alleged authors.

While it is common engineering practical to focus all attention on problems, often little but the attention itself is offered as praise.  That this work is preserved here will be used as license to complain about it.

The theory of the planimeter is glossed over.  While most of the book goes into greater detail and treats mathematical issues with greater rigor, the purely theoretical operation of the planimeter is introduced without comment.  Much of the detail is hidden in Equation 1 of section 9 of Chapter III, “Relation of Roll of Wheel to Area Traced.”  (Eventually, I should try writing an explanation.)

A non-trivial planimeter has an adjustable arm, whose length is proportional to the planimeter reading.  So the reading can be artificially scaled and this is exploited in “Problems Involving Averaging” and other places and is the reason for the tables: if the user knows the scale of a drawing and can get the right conversion factor, the arm can be adjusted before the planimeter is used so that the planimeter's final reading needs no scaling (or at most needs to be multiplied by 2 or 5 and a power of 10).  Today, the need for a single multiply to do the scaling is a trivial operation, but avoiding it was, evidently, a significant increase in productivity in the planimeter's day.

Some of the data in the tables were self evidently wrong.  The tables are derivable from a small number of input variables, and lead to some obviously inconsistent combinations in the tables as they were originally published.  The intent here is not so much to point out the errors as to reconstruct the original author's intent.  But ultimately the original input variables are probably lost.  The tables use 3 planimeters, none of which seem to be the planimeter sited in the text.  So with modern machinery and original output data, the tables were run in reverse and forwards again and any original but implausible outputs were changed, to be consistent if not correct.

Although the determination of the center of gravity is mentioned on the title page, the text never quite gets around to saying how it is done.

Apparently, the prismoidal formula is used to get a three dimensional integration from a series of two dimensional cross sections.  Probably it is a higher order approximation than the obvious one.  Presumably it is correct, but I have never seen anything like it.

Some years later, the book has been scanned.

Anyway, if you care to, let the information wash over you and if any of it is any good, maybe some of it will stick.

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