The Hatchet Planimeter
by: L. M. Dickerson
copy from Walter C. Scott files (1940s)

One of the problems of any technician using maps is to determine quickly and accurately the numerical value of given areas.  The Polar Planimeter is usually used, but for the field man, has the disadvantage of being rather cumbersome, expensive, and subject to breakage.  Neither of  these objections occur in the "Hatchet Planimeter", a simple instrument, which can be constructed in less than an hour from materials costing less than one dollar.
The use of the Hatchet Planimeter was demonstrated at the recent meeting of the Ohio Academy of Science in Colombus, by Professor Andrew R. Weber and a group of his students from the University of Dayton.  The students used the instrument freely in their class work.  Promptly upon my return home, after watching the demonstration, a bar of drill rod steel was obtained and a Hatchet Planimeter made.  On my first attempt to use it on a conventional figure (triangle) the results were within less than one percent of error.  Subsequent trials have shown less than two percent error.  Other members of the regional office staff (see footnote) have tried this instrument and have obtained results equal to or better than the Polar type planimeters' usual performance.
The Hatchet Planimeter was invented by a German Mathematician named Prytz.  The mathematical proof according to theory is complex and involves the use of infinite series.  A shorter proof of the theory has been worked out by John W. Satterly, associate professor of Physics, University of Toronto.
This instrument should be useful to many of our technicians.  Its simplicity is deceptive.  Probably that is the reason it is so little known.  The following instructions should enable any technician to make and operate the Hatchet Planimeter.
Figure 1. (This drawing was on the back of the type written page in pencil. Figure 1?) shows the planimeter more clearly than it can be described in equal space.  It is extremely simple to make.  The only critical points to be considered are that the instrument should be rigid, the two legs should be approximately in the same plane and the "hatchet" edge should be kept sharp, razor sharp for best results.  Drill rod, three sixteenths inch in diameter, is a convenient and satisfactory material.  It can be tempered and is not too difficult to work.
The distance AB can be any convenient length.  Ten inches is satisfactory for ordinary use and simplifies computations of the moving of the decimal point.  It is best to heat the drill rod to a dull red in order to bend it.  Too much heat will cause it to crack.  To operate the Hatchet Planimeter proceed as follows:
Place the figure to be planimetered on a flat surface covered with paper having an unglazed surface.  The flared point of the instrument rides freely over the unglazed paper, while the operator traces the outline with the opposite point of the planimeter.

In sequence the detailed manipulations are as follows:  (Refer to figure 2) (figure 2 is missing)

1.  Locate, by inspection, the approximate centroid of the figure to be planimetered, and draw through it a pair of lines at right angles of each other, and intersecting the perimeter to be measured.
2.  Place the instrument in the initial position extending approximately perpendicular to one of the coordinates previously drawn (preferably the longer one), and the tracing point on the intersection (O).
3.  Hold the planimeter lightly so that both ends will move freely over the paper, the flared end resting on the unglazed paper surface.  The instrument should never be allowed to tilt.
4.  Mark point L by pressing on the planimeter, forcing the sharp edge lightly through the paper.
5.  Trace along line OA and around the perimeter in a clockwise direction returning to O along AO.
6   Mark point M by pressing on the flared end of the planimeter.
7.  With the hatchet at M and the point held firmly on O, rotate the figure through 180 degrees.  The instrument should not be allowed to shift its position,  nor should point O change its relations to M and L.
8.  Trace the figure a second time along OA, but this time in a counter clockwise direction, returning  to point O as before.
9.  Mark point N by pressing on the planimeter as before (4 and 6).  Points L and N will be close together, the distance between them depending upon how accurately the tracing has been done.  If points L and N coincide exactly it indicates that the tracing has been exact.  This rarely occurs but the area is corrected by measuring the distance from M to point half way between L and N.  A steel scale calibrated to one hundredths gives acceptable results.
10.  Multiply the distance obtained in operation 9 by the distance between the points of the planimeter (in this case 10 inches) to get the area of the figure measured.  For greater accuracy the distance should be measured on an arc with a radius equal to the distance between the planimeter points.  When ML is small the error due to use of a straight scale will be negligible, however.
It is advisable to practice on conventional figures such as squares, triangles and parallelograms with known dimensions until one has developed some skill in holding the instrument lightly and tracing accurately the outlines and figures.  In practice it is advisable to use as a corrected distance the average of three or more readings.  This reduces the affect of mechanical errors in holding the instrument, tilt, and etc.

footnote: (J Scott 1998) The "regional office staff probably refers to the U.S. Soil Conservation Service Civilian Conservation Corps where Walter worked from April 37 to Jan 42.  Walter owned a nice issue of the hatchet planimeter which I remember him showing to visitors to the farm.  He showed me how he used it on aerial photographs to measure acreage of farmland.