The Hatchet Planimeter

The hatchet planimeter is something of an oddity. It does not correctly measure the area when the pointer traces the perimeter of a figure. You might think this would be a drawback for a planimeter.

The hatchet planimeter was invented by Holger Prytz, a Danish cavalry officer, in about 1875 [SCO 01]. He called it a stang planimeter, "stang" being Danish for "rod". Prytz planimeters were manufactured in Denmark, starting in 1887, by Cornelius Knudsen of Copenhagen [FOO 1]. The Science Museum in London has one of Knudsen's planimeters [BAX 01, item 543] but I don't recall seeing it on display.

The basic design is very simple. It consists of a metal rod, perhaps a quarter of an inch in diameter, bent in two right angles to form am arm about ten inches long and two shorter arms about three inches long. One short arm is sharpened to a point and the other to a convex knife edge in line with the point. For convenience (the length is used as a multiplier), the distance from the point to the middle of the knife edge is usually made equal to ten inches or 25 centimeters.

basic shape of hatchet planimeter

The planimeter measures the sum of the area and some other properties of the figure. Provided that certain rules are observed when it is used, the other factors will be minimised or eliminated and the planimeter will give an adequate estimate of the area. However, if the planimeter is misused the error can be significant.

Prytz's instructions for measuring the area of a figure are as follows (referring to the diagram below):

operation of hatchet planimeter

The theory underlying the behaviour of the hatchet planimeter is more complex than for linear and polar planimeters and it has spawned a number of papers. Several of them are listed below in the References.

Robert Foote has lots of information about hatchet planimeters in his web page including pictures and some nice animations, and also a brief description of how they work.

The special benefits of the basic Prytz planimeter are that it is easy to make (and hence cheap) and that it is very robust because it has no moving parts. However it wasn't long before efforts were made to overcome some of its perceived shortcomings, which include:

This resulted in proposals by Goodman (1890 and 1891), Coradi (1895), Scott (1896), Chollar (1897), Schierbeck (1904). Menzin (1906), Larrazabal y Fernandez (1926) and Hounsfield (1932).

John Goodman (1890, 1891)

The hatchet planimeter was popularised in England by John Goodman, who patented two "improved" models. The first model was a general purpose hatchet planimeter patented in the UK in 1890 [GOO 05] and in the US in 1893 [GOO 03].

U.S. patent No 500,202

The second model was his "Patent Averaging Instrument"

The following companies (that I know of) made Goodman planimeters in England.

T. R. Harding & Son, Leeds

Harding produced a model that was very similar to the 1890 patent.

Goodman planimeter by Harding

Goodman planimeter by Harding, detail

Jackson Bros, Ltd. Leeds

Jackson Bros., Limited, of 50 Call Lane, Leeds, produced a more elegant model of the Goodman Patent Planimeter.

Goodman planimeter, Jackson's model

This pattern is illustrated in Engineering, 21 August 1896, so it was remodeled soon after the patent date.

1896 drawing of Goodman planimeter


Jackson's price in 1897, for either the planimeter or the averaging instrument, was twelve shillings and six pence, as shown in their advertisement in Pickworth's slide rule book [PIC 01].

1897 Advertisement of Goodman planimeters

Reynolds & Branson Ltd

Reynolds & Branson Ltd, Scientific Instrument Makers, Leeds, made a similar model to Jackson Bros Ltd

Goodman planimeter made by Reynolds & Branson

The Science Museum in London has a Goodman's planimeter made by Reynolds & Branson which it describes as follows: "This instrument for measuring areas, which is an improvement on the hatchet planimeter, was patented by Professor John Goodman in 1890. It consists of a graduated beam, a tracing point and a hatchet, all in one piece. The beam is graduated in such a way as to obviate the calculations that have to be performed when using the hatchet planimeter" [BAX 01, item 199]. This would describe either style of the planimeter so I am not sure which model they have.

Goodman's Patent Averaging Instrument

logo from box

Goodman's Patent Averaging Instrument was devised especially for use with indicator diagrams. The only such instruments I have seen were made by Jackson Bros. Ltd., and there are two versions, shown below. That on the left belongs to Peter Hopp and appears to be the earlier (it more closely matches the patent). The other is mine.

Goodman's Averaging Instrument, Type A Goodman's Averaging Instrument, Type B

The illustration in Engineering, 21 August 1896 [GOO 01], is of the instrument in the left-hand picture.

Goodman's Averaging Instrument - 1896 illustration

G. Coradi (1895)

Prytz reported in 1896 [PRY 03] that: "Last year Coradi, in Zurich, showed me a stang planimeter with a wheel instead of a keel; the wheel could be moved along the stang, but he allowed that the original simple form was the best and surest."

I have no idea whether Coradi made and marketed this model or whether it was, perhaps, just a one-off. Nor do I know if this was related in any way to "Kleritz's form of 1894" mentioned in [BAX 01, item 198].

Scott's Planimeter (1896)

In an article in Engineering in 1896, Ernest Kilburn Scott described an "Improved Stang Planimeter" he had designed [SCO 01]. It seems that at least one of these was made. The Science Museum in London has a planimeter, made by Elliott Bros. and lent to them by E. Kilburn Scott in 1922, that seems to match the article's description.

Scott's improved planimeter

The Science Museum describes their planimeter as follows [BAX 01, item 198]:

"In this modified form of hatchet planimeter, which was designed by Mr E. Kilburn Scott in about 1897, the knife-edge takes the form of a sharp-edged wheel, as in Kleritz's form of 1894. The bar is in three portions which can be attached to one another by compass joints, and is divided in inches figured to 19 and subdivided to tenths of an inch. The knife-edge can be set to any point of the scale and secured by a screw clamp.

Instead of a tracing point at the other end, there is a piece of transparent celluloid with a small circular hole, and a needle point pivoted to the tracer portion of the bar can be brought down when necessary through the center of the hole to the paper.

A small gearing which rotates a divided circle is attached to the tracer end for reading the distance between the marks made by the knife-edge."

B. E. Chollar (1897)

Byron E. Chollar patented an improved hatchet planimeter [CHO 01] in 1897. It is not known if examples of this planimeter were made.

Chollar's Planimeter

R. Schierbeck (1904)

Rudolf Schierbeck patented a measuring instrument [SCH 01] in 1904, for use specifically in "the measuring of steam-engine diagrams". Although he called it a "diagrammeter" it looks suspiciously like a hatchet planimeter. He makes no reference to hatchet planimeters in his patent. It is not known if examples of this planimeter were made.

Schierbeck's diagrammeter

One would have to question the accuracy of this device. It ignores some of the good operating practice for hatchet planimeters:
. the distance between the wheel and the pointer is less than the length of the diagram,
. users are advised to start the trace on the perimeter of the diagram (as shown in Fig.3) rather than at the center of gravity of the diagram.

A. L. Menzin (1906)

A. L. Menzin wrote an article in Engineering News in 1906, describing "The Tractigraph, an Improved Form of Hatchet Planimeter" [MEN 01]. I have not yet sighted this article, and do not know if examples of the tractigraph were made.

L. Larrazabal y Fernandez (1926)

Luis Larrazabal y Fernandez, a citizen of Cuba, obtained a U.S. patent for an improved hatchet planimeter in 1926 [LAR 01]. It is not known if examples of this planimeter were made.

Larrazabal y Fernandez's Planimeter

L. H. Hounsfield (1932)

Leslie Haywood Hounsfield patented an improved hatchet planimeter [HOU 01] in 1932. This planimeter was subsequently made by (or for) Tensometer Ltd., 81 Morland Road, Croydon (England) for use with their testing equipment.

Hounsfield Planimeter
Detail of pointer, Hounsfield planimeter Close up of tracer, Hounsfield planimeter

Hounsfield Planimeter - patent drawing

Hounsfield Planimeter with box

References

Note: This list is limited mainly to English language papers. US patents can be viewed on line (as *.tif files of individual pages) by going here and typing in the patent number. Or download them as pdf files from the links I have given.

[ANO 01] Anon: "The Hatchet Planimeter", Engineering, 57 (25 May 1894), p. 687.

[ANO 02] Anon: "The Hatchet Planimeter (correction)", Engineering, 57 (1 June 1894), p. 725.

[BAR 01] Barnes, George: "Hatchet or Hacksaw Blade Planimeter", American Journal of Physics, 25 (1957),p. 25-29

[BAX 01] Baxandall, D. (revised and updated by Jane Pugh): "Calculating Machines and Instruments - Catalogue of the Collections in the Science Museum", publ. Science Museum ISBN 0 901805 14 9.

[CHO 01] Chollar, B. E.: U.S. patent No 575,105, "Planimeter", 12 January 1897. pdf

[CRA 01] Crathorne, Arthur Robert: "The Prytz Planimeter", The American Mathematical Monthly, 15 (1908), p. 55-57.

[FAR 01] Farthing, Simon: "Theory of the Hatchet Planimeter", Sadhana vol 8, part 4, (October 1985), p 351-359.

[FOO 01] Foote Robert L.: "Geometry of the Prytz planimeter", Reports on Mathematical Physics, 42 (1998),1-2, p. 249-271.

[GOO 01] Goodman, John: "Goodman’s hatchet planimeter", Engineering, 62 (1896), p. 255-256.

[GOO 02] Goodman, John: "Goodman’s hatchet planimeter", letter to the Editor of Engineering, 62 (1896), p. 377.

[GOO 03] Goodman, John: U.S. patent No 496,562, "Planimeter", 2 May 1892. pdf

[GOO 04] Goodman, John: U.S. patent No 500,202, "Planimeter", 27 June 1893. pdf

[GOO 05] Goodman, John: U.K. patent No 1890/6087, "Planimeter", 22 April 1890

[GOO 06] Goodman, John: U.K. patent No 1891/407, "Planimeter", 9 January 1891

[GOO 07] Goodall, H. D. E.: "The Hatchet Planimeter - Constructional Details and Method of Operation", Newnes Practical Mechanics, October 1947, p14. (thanks Peter Hopp)

[HIL 01] Hill, F. W.: "The hatchet planimeter", The Philosophical Magazine (5), 38 (1894), p. 265-269.

[HIL 02] Hill, F. W.: "The hatchet planimeter", Proceedings of the Physical Society, 13 (1894), p. 229-234

[HOU 01] Hounsfield, L. H.: U.K. patent No 377,514, "Improvements in Planimeters", 28 July 1932 pdf

[HOU 02] Hounsfield, L. H.: U.S. patent No 1,988,357, "Planimeter", 27 October 1932, pdf

[KRY 01] Krylov, Aleksei Nikolaevich: "On the hatchet planimeter", Bulletin de l’Academie imperiale des sciences de Saint-Petersbourg, 19 (1903), p. 221-227.

[LAR 01] Larrazabal y Fernandez, L.: U.S. patent No 1,572,132, "Planimeter", 9 February 1926 pdf

[MEN 01] Menzin A. L.: "The Tractigraph, an Improved Form of Hatchet Planimeter", Engineering News, 56(1906), p. 131-132.

[PED 01] Pedersen, Olaf: "The Prytz planimeter", in Berggren (J. Lennart) & Goldstein (Bernard Raphael),eds., From Ancient Omens to Statistical Mechanics. Essays on the Exact Sciences. Presented to Asger Aaboe, Copenhagen : University Library, 1987 (Acta historica scientiarum naturalium et medicinalium, 39), p. 259-270.

[PIC 01] Pickworth, C. N.: "The Slide Rule: a Practical Manual" 4th edition, 1897.

[PRY 01] Prytz, Holger: (Pseud. Z) Stangplanimetret, Den tekniske Forenings Tidsskrift, 10 (1886), p. 23-28

[PRY 02] Prytz, Holger: "The Hatchet Planimeter", Engineering, 57 (1894), p. 813.

[PRY 03] Prytz, Holger: "The Prytz planimeter", letter to the Editor of Engineering, 62 (1896), p. 347.

[PRY 04] Prytz, Holger: "The Prytz planimeter", second letter to the Editor of Engineering, 62 (1896), p. 347.

[PRY 05] Prytz, Holger: "The hatchet planimeter and the 'Tractigraph'", Engineering News, 57 (1907), p. 386

[SAT 01] Satterly, John W.: "The hatchet planimeter", The Journal of the Royal Astronomical Society of Canada, 15 (1921), p. 221-243.

[SCH 01] Schierbeck, R.: U.S. patent No 752,470, "Measuring Instrument", 16 February 1904. pdf

[SCO 01] Scott, Ernest Kilburn: "An improved stang planimeter", Engineering, 62 (14 August 1896), p. 205-206, 308.

[SPA 91] Sparks, Robert: "The hatchet planimeter", Journal of The Franklin Institute, 213 (1932), p. 661-668

Copyright David R Green. Last updated 10 June 2008.