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. He called it a stang planimeter (stang is the Danish word for rod). Prytz planimeters were manufactured in Denmark, starting in 1887, by Cornelius Knudsen of Copenhagen. The Science Museum in London has one of Knudsen's planimeters [Baxandall, item 543] but I don't recall seeing it on display, and their website only shows its box.

However, the Geodätisches Institut der Leibniz Universität, Hannover also has one, shown in page 17 of their catalogue of planimeters.

Cornelius Knudsen hatchet planimeter 1887
Cornelius Knudsen hatchet planimeter 1887

Figure 1. Cornelius Knudsen's hatchet planimeter. Adapted for nonprofit, educational
purposes from the Geodätisches Institut der Leibniz Universität, Hannover catalogue

Knudsen planimeter shown in Poulain 1895

Figure 2. This same instrument is illustrated in Poulain's 1895 paper.

It is rare to find a commercial planimeter that exactly matches Prytz's original design. The only example I can recall is the Stella-Polaris planimeter.

Stella Polaris planimeter

Figure 3. The Stella-Polaris planimeter listed in Rechnerlexikon as "Planimeter Stella Polaris von E. Willoch, Oslo". I am not sure whether this means the planimeter was made by E. Willoch, was sold by E. Willoch, or the photograph was provided by E Willoch.

The basic design is very simple. It is a metal rod, perhaps a quarter of an inch in diameter, bent in two right angles to form an 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 often 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 that for linear and polar planimeters. This has resulted in a number of papers. Some appeared soon after the planimeter was introduced: Hill 1894, Hammer 1895, Runge 1895, Lehmer 1899, Krylov 1903, Schreiber 1908. And the subject has been revisited right up until the present time, including by: Satterly 1921, Barnes 1957, Farthing 1985, Foote 1998. Mostly, the papers are very heavy going. More recently the analysis has turned to bicycle tire tracks geometry with hatchet planimeters as a special case (see Levi and Tabachnikov 2008 or Foote, Levi & Tabachnikov 2013).

An approachable explanation of the hatchet planimeter (and of polar and linear planimeters as well) can be found in the paper "Area Without Integration: Make Your Own Planimeter" by Robert Foote and Ed Sandifer (pdf). And Stuart Boersma provides another easy to follow explanation here, together with an interesting analysis of Foucault's Pendulum.

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

hatchet planimeter by W. Harling

The special benefits of the basic Prytz planimeter are that it is easy to make (and hence cheap) and that it is 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 gave rise to a surprising variety of hatchet planimeters, none of which seems to have been made in large quantities. There were published proposals by Goodman (1890 and 1891) (whose planimeters were made by the Leeds manufacturers T. R. Harding & Son, Jackson Bros, Ltd. and Reynolds & Branson Ltd.), Kleritj (1891), Coradi (1895), Scott (1896), Chollar (1897), Krylov (1903), Schierbeck (1904), Menzin (1906), E. O. Richter & Co (1907), von Sanden (1911), Larrazabal y Fernandez (1926) and Hounsfield (1932).

Several manufacturers made hatchet planimeters that do not appear to have been patented. They include W H Harling, Gebrüder Fromme, Eckert und Hamann, Nestler, Planimetro CAT, Russian Ministry of Instrumentation, Amsler-Coradi, GEOMATH, Unknown (Dutch?) manufacturer, Unknown manufacturer 2, two armed planimeter, Unknown manufacturer 3.

The design is so simple that many users have made their own versions. And if all else fails, use a penknife.

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 Great Britain in 1890 and in the United States in 1893.

U.S. patent No 500,202

Figure 4. Patent drawing of Goodman's design. The improvement was the provision of a scale to measure the arc length (rather than the chord length) between the start and end indentations, with the scale calibrated so as to give the reading in units of area.

The second model was his "Patent Averaging Instrument".

Goodman's planimeters were described in detail in the Scientific American Supplement, No. 1082, September 26, 1896. Project Gutenberg has a copy on-line. The planimeters were also reviewed by Ernst Hammer in Zeitschrift für Instrumentenkunde in October 1896.

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

T. R. Harding & Son, Leeds

Goodman planimeter by Harding

Goodman planimeter by Harding, detail

Figure 5. Harding produced a model that was very similar to Goodman's 1890 patent. .

Jackson Bros, Ltd. Leeds

Goodman planimeter, Jackson's model

Figure 6. Jackson Bros., Limited, of 50 Call Lane, Leeds, produced a more elegant model of the Goodman Patent Planimeter.

1896 drawing of Goodman planimeter

Figure 7. This pattern was illustrated in Engineering, 21 August 1896, so it was remodeled soon after the patent date.

1897 Advertisement of Goodman planimeters

Figure 8. Jackson's price in 1897, for either the planimeter or the averaging instrument, was twelve shillings and six pence, as shown in this advertisement in Pickworth's slide rule book.

Reynolds & Branson Ltd

Goodman planimeter made by Reynolds & Branson

Figure 9. Reynolds & Branson Ltd, Scientific Instrument Makers, Leeds, made a similar model to Jackson Bros Ltd. The instructions that accompanied their planimeters can be seen in this pdf.

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" [Baxandall, item 199, shown below].

Goodman planimeter made by Reynolds & Branson, at Science Museum

Figure 10. Goodman planimeter made by Reynolds & Branson Ltd.
The original picture is on the Science Museum site here

Goodman's Patent Averaging Instrument

logo from box

Goodman's Patent Averaging Instrument was devised especially for use with indicator diagrams.

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

Figure 11. The only such instruments I have seen were made by Jackson Bros. Ltd., although Reynolds & Branson advertised them for 15 shillings in their brochure. Jackson Bros. made two versions. The upper model closely matches the patent. The lower model is cruder and was perhaps an early, trial version.

Goodman's Averaging Instrument - 1896 illustration
Figure 12. This illustration in Engineering, 21 August 1896 shows the instrument in its patented form.

Ljubomir Kleritj, The Tractoriograph (1891)

In 1897 Kleritj published a paper in German titled (loosely translated) "The Tractoriograph and the construction of the transcendental numbers "pi" and "e", as well as construction of the n-sided, regular polygons inscribed in the circle."

He mentioned that "As early as 1891 I invented a very simple instrument ... I have called this instrument "Tractoriograph".

"The instrument is made in the "mechanischen Institut" of Oskar Leuner in Dresden and costs 22 M."

The Tractoriograph

Figure 13. This is clearly a form of hatchet planimeter. However, so far as I can see (not being conversant with German) he does not propose using it as such. His focus is on drawing the tractrix of various curves.

His description (roughly speaking) is: "It consists of a rod AB with a tracing pin DK at one end and a sharp-edged wheel which rotates about a horizontal axis T at the other. This axle is mounted in the fork-shaped lower part of a sleeve which slides on the rod AB and can be fixed by means of the screw P on the same.

"The tracing pin DK is arranged in a slightly rotatable manner in the frame CH, which rests on the adjustable feet m and n. The feet are screw nuts that can be adjusted so that the tip K gently touches the drawing plane.

"The instrument rests on the three support points m, n and t, the contact point of the wheel T with the plane of the drawing, and can therefore be set up in a stable manner.

"The plane of the wheel T must always go through the tracing point K. For this purpose, adjusting screw u is provided, which allows for the axis of rotation of the wheel to be aligned.

"To use the instrument, grasp the instrument lightly between two fingers and guide the tip of the the tracing pin along the given curve; the point of contact t of the wheel then describes the tractrix of the given curve with the constant tangent t = Kt, which is equal to the set length of the instrument.

"In order to record the tractrix on paper, a wheel F provided with a groove is suspended from the displaceable sleeve like a pendulum. In this groove there is a felt ring soaked in printing ink, which colours the sharp edge of the little wheel T, because the former is always in contact with it.

"It can be seen that when tracing a curve with the tip K, the wheel T records the corresponding tractrix if the plane of the wheel always goes through point K. Finally, the needle is attached to the sleeve in order to mark the length of the constant tangent for which the relevant tractrix is to be used."

G. Coradi (1895)

Prytz reported in 1896 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 marketed the model or if it was a one-off. Nor do I know if it was related in any way to "Kleritz's form of 1894" mentioned by Baxandall (item 198).

Many years later the Amsler-Coradi company did market a modernistic hatchet planimeter.

Scott's Planimeter (1896)

In an article in Engineering in 1896, Ernest Kilburn Scott described an "Improved Stang Planimeter" he had designed.

Scott's improved planimeter

Figure 14. Scott's "Improved Stang Planimeter" as illustrated in Engineering.

Scott's planimeter made by Elliott Bros. at Science Museum

Figure 15. At least one of them was made. The Science Museum in London has this one, made by Elliott Bros. and lent to them by E. Kilburn Scott in 1922. I have rotated the original picture to better fit it on this page.

The Science Museum describes their planimeter as follows [Baxandall, 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)

Chollar's Planimeter

Figure 16. Byron E. Chollar patented an improved hatchet planimeter in 1896. It is not known if examples of this planimeter were made.

A. Krylov (1903). A Russian Hatchet Planimeter.

In his 1903 paper, Krylov shows "that an exact and obvious geometrical explanation of what the hatchet planimeter really gives is not only possible, but can be obtained in a most elementary and simple manner."

He concludes by mentioning that "In order to obtain a more assured guiding of the point A, I have modified the construction of the hatchet planimeter, having replaced the chisel-edge by a little sharp edged wheel, the plane of which is adjusted in such a manner as to pass through the tracing point B, which is formed by the end of a pin freely revolving in its bearings."

Krylov's Planimeter

Figure 17. "Fig 7 represents in elevation the new instrument one quarter natural size as manufactured by M. R. Wetzer, mechanician in S. Petersburgh. For the use of the instrument a piece of copying paper is put under the wheel, then the curve of pursuit is sharply traced, the distance A0A1 is easily measured, and the limit of the error committed, when neglecting the areas contained by the parts of the leading line can be clearly seen and readily ascertained if necessary."

I haven't seen one of these yet but some may have survived.

R. Schierbeck (1904)

Schierbeck's diagrammeter

Figure 18. Rudolf Schierbeck patented a measuring instrument 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.

One would have to question the accuracy of this device, as described in the patent. It ignores some of the good operating practice for hatchet planimeters because:
a. the distance between the wheel and the pointer is less than the width of the diagram. Prytz recommends it should be at least double.
b. 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.

Schierbeck's diagrammeter

Figure 19. It is not known if this planimeter was made commercially, but at least two examples exist, probably made as patent models. This one is believed to be the patent agent's copy. It can be seen to be different from the patent. The change ensures that the distance from the point to the wheel can be set equal to the width of the indicator diagram. This goes some way to rectifying problem (a) but the length of the planimeter is still less than generally recommended.

Private correspondence with the grandson of a friend of Rudolf Schierbeck suggests there is another copy out there somewhere: "You wonder if a model was ever made. I can answer to the affirmative. I can remember as a kid handling it and wondering what in the world it did and how in the world it worked. ... The mechanism I played with in the fifties was the patent model. The paperwork was with it ... Your photos show what appears to be a different planimeter than the one my father had. The one I am familiar with was spotless and, as I recall, was in an intact leather case with no wear."

A. L. Menzin, "Tractrigraph" (1906)

A. L. Menzin wrote an article in Engineering News in 1906, describing "The Tractrigraph, an Improved Form of Hatchet Planimeter". (This is not to be confused with Kleritj's Tractoriograph).

He claimed: "The methods recommended for using the hatchet planimeter involve an error, whose magnitude is so uncertain that the usefulness of the instrument is greatly limited. ... The "Tractrigraph", a development of the hatchet planimeter permitting much greater accuracy of measurement, was recently designed ..."

Prytz was not amused and commented in 1907: "Sir: I have read with great interest the article of Mr A. L. Menzin about his "Tractrigraph" in your issue of Aug. 9 1906. I think Mr Menzin has been too severe against my poor instrument, at least I had believed hitherto that it worked quite as well as the hundreds of improvements it has suffered in various countries, when the hints given were followed; and I should be very obliged if you would tell your readers this and give them the hints appended below. ..."

Tractrigraph illustrated by Menzin

Figure 20. This illustration from Menzin's article appears to be the photograph of a real tractrigraph,
but I have never seen such a model

E. O. Richter & Co (1907).

Richter Advertisement 1914

Figure 21. E. O. Richter & Co of Chemnitz made a form of tractrigraph, which they advertised as the planimeter System Pregél.

The instrument was patented in Germany on 25 July 1907 (Nr.201785)

1908 illustration of Pregel planimeter

Figure 22. The instrument was reviewed by Professor Ernst Hammer of Stuttgart in 1908.

He wasn't particularly impressed.

"... E. O. Richter & Co. is bringing a planimeter to the market that was constructed at the suggestion of Prof. Th. Pregel in Chemnitz, which, of course, like so many newer instruments, only presents a modification of Prytz's hatchet planimeter. The price is higher than that of the Prytz instrument (24 marks in a case), corresponding to the finer version ..."

"Instead of the cutting wheel in several modifications of Prytz's instrument, there are two such wheels with separate axes. The tracing rod is attached to the metal block at the ends of which these two cutting wheels sit, not quite in the middle between the two wheels, because the tracing pin on the tracing rod has also moved out by that eccentricity distance ..."

"A theory of the instrument is neither hinted at in the prospectus enclosed by the factory, nor is it otherwise known to this reporter; in any case, the effect of the two cutting rollers instead of only one must first be examined theoretically."

Richter's System Pregel and box
Richter's System Pregel and box

Figure 23. Richter's planimeter #660. Usually they take this form.

In 1909, shortly after Hammer's review, there were two further publications about the planimeter; a book (in German) by Theodor Pregél, and an article by Albert Schreiber. I have been unable to view either of these documents but speculate that they outline the theory that Hammer found lacking.

Compact version

Figure 24. A compact version was also made, with the rod in two sections.
This one appeared on Ebay in 2017

von Sanden

Horst von Sanden of Göttingen introduced yet another two wheeled hatchet planimeter in 1911. The model is also discussed in Galle's book (pages 128-129).

Sanden's planimeter

Figure 25. Photograph of his planimeter, from Sanden's paper.

von Sanden noted that in order to calculate the area a2φ with the Prytz planimeter, one does not have a simple way to measure the angle φ between the start and end positions of the rod. He therefore modified the planimeter by replacing the hatchet at B with two wheels (with sharp edges) R1 and R2, sitting on an axis at right angles to the rod AB and at the same distance from the rod (see Fig. 26). The center point B between the two wheels, i.e. the end point of the rod, is then guided in the same way as a hatchet edge. (If the wheel radii are completely the same, B will remain in the same place when the rod is simply turned, since both wheels then run through the same arc in opposite directions).

Goodman planimeter by Harding

Figure 26. Schematic of Sanden's planimeter.

The contact points of the wheels describe two curves parallel to the curve described by B. If the arc element described by point B is ds = rdφ and the distance between the wheels is 2b, then the two wheels describe arc elements dσ1 = (r + b)dφ and dσ2 = (r - b)dφ. It follows that ∫(dσ1 - dσ2) = 2bφ.

In order to be able to read the difference ∫(dσ1 - dσ2) of the relative rotation of the wheels against each other, each wheel is rigidly connected to a drum in the middle of the axle. One drum is divided into 100 parts, the other engraved with two verniers that are 180° apart. The twist is read from the vernier that is just visible, which multiplied by a constant indicates the area.

small Sanden planimeter in Gronongen

Figure 27 Sanden's small planimeter. Adapted for nonprofit, educational
purposes from the Göttingen Collection of mathematical models and instruments

A planimeter with the following dimensions was made: wheel diameter: 40.00 mm.; distance between the wheels 150.21 mm.; length AB variable between 50 mm and 192 mm. The perimeter of the drum was divided into 100 parts, so that the vernier unit indicates 1/1000 of the circumference. Price: 54 M.

Sanden remarked that he planned to make another version of the planimeter "it is intended to divide the circumference of the drum into 200 parts and to increase the wheel spacing to 200 mm, as well as to increase the length of the rod, so that the accuracy of the new planimeter should approach that of the most accurate known while the price is about 80% lower." It seems he did this, because the Göttingen collection has both a small model (shown above) and a large model.

Sanden performed some experiments to measure the precision of his planimeter. With AB set to 191mm he obtained an error of 0.7%. With AB set to 101mm he obtained an error of 0.4%. For comparison, with a Coradi polar planimeter he obtained an error of 0.9%.

So better than the polar planimeter! This is a little surprising because the polar planimeter is an exact instrument, subject only to human error (not properly tracing the curve, not accurately reading the scale, etc), while all hatchet planimeters are inherently subject to theoretical errors as well as the human errors. Never-the-less, the device clearly worked well.

Russian Ministry of Instrumentation, Kharkovsky Factory

An interesting combined integrator/planimeter construction set made in Russia, model КИ-3 (KI-3) includes parts to assemble a hatchet planimeter similar to the tractrigraph.

Heading on Instruction Manual

The Google translation of this (which doesn't quite look right) is ...
Russian Ministry of Instrumentation, Means Automation and Control Systems
Kharkovsky Factory of Markshading Instruments

Russian hatchet planimeter

Figure 28. The Kharkovsky construction set includes components to assemble a hatchet planimeter similar to the tractrigraph, or alternatively to construct a polar planimeter, or a linear planimeter, or an integrimeter, or a coordinatograph, or (I think) an eidograph and more besides (if only I could read the manual).

The set comes with a comprehensive, 78 page text manual and a separate booklet containing 45 diagrams and photos. Unfortunately the paper is of poor quality and the ink has run and smeared much of the Cyrillic text. There is, particularly, a problem in trying to distinguish between the "square" symbols such as ж, и, к, м, н, п and ш. It makes translation a very frustrating task.

Illustration from Manual

Figure 29. How bad the smearing is can be seen here (Figure 31 of the hatchet planimeter set-up).

The real thing

Figure 30. Figure 29 is meant to look like this.

Section 2.14, Reproduction of the action of the hatchet planimeter, in pages 45-47, discusses the configuration as a hatchet planimeter. In part it says ...

The device is assembled in the form it is stored in the box. An extra pencil 15 with a weight 17 is added, and a twist-off needle 12 with an adjustable sleeve 9 or a twist-off sighting device 11, as shown in Fig. 31. The planes of the wheels of the carriage should be parallel to bar L (angle(γ)=0). The SIM (schetno-integpipyyushchiy mekhanizm, ie. counting-integrating mechanism) should be set so that angle(β) = 0.

Set up as hatchet planimeter

Figure 31 Assembly as hatchet planimeter.

To determine the area F of a figure, mark a point on it approximately coinciding with the center of gravity. Fig. 29 shows a circle, so the indicated point coincides with the center 0. Draw a straight line through point 0 to an arbitrary point A on the contour.

Place the instrument on the drawing so that the center of the sighting device is over the point 0, and rod L makes an angle of about 15-20° with straight line OA. Then trace from point O along the line OA to point A, trace a complete circuit of the contour and finally trace back along the line AO to the starting point O.

At the same time, the pencil draws a running line S0, S1, S2, S3, S4 with three points of return S1, S2, S3.

The area sought, F, is the projection length L = OS0 = OS4 (that is, the distance from the center of the sighting device to the sharp point of the pencil or to the imaginary point of contact of the curved blade) multiplied by the length of the arc S0S4 described by the radius L and the center 0, i.e.

F ≅ L × Ω         (47)         where ∠Ω = ∠S0OS4 is in radians.

When Ω ≤ 20°, the arc S0-S4 is sometimes approximated by a chord. In this case, the formula (47) takes the form

F ≅ L × S0→S4

The design of the KI-3 allows one not only to set the desired value L (since it is determined by the distance between the carriage 21 and the SIM 22), but also to measure the angle Ω. The carriage 21 must be fixed in the extreme left position and angle(β) set equal to 0.

The possibility is based on the property of the running line that the tangent at any point of it always coincides with the straight line passing through the trace of the pencil tip on the drawing planes and the trace of the geometric axis of the bypass sight. In the process of drawing a running line, the integrating wheel constantly registers the angle between successive positions of the device, as if the latter rotated around a fixed center that coincides with the trace of the pencil point.

In Fig. 29, a circle of radius R ≅ 56.4 mm was inscribed. Its area is F = 100 cm2, L = 150 mm.

After tracing around the circle, Ω = 1523 (the integrating wheel has passed 1.523 revolutions). This corresponds to 0.443 radians and therefore F = 0.443 × 152 = 99.68 cm2.

L. Larrazabal y Fernandez (1926)

Larrazabal y Fernandez's Planimeter

Figure 32. Luis Larrazabal y Fernandez, a citizen of Cuba, obtained a U.S. patent for an improved hatchet planimeter in 1926. It is not known if examples of this planimeter were made.

L. H. Hounsfield (1932)

Hounsfield Planimeter - patent drawing

Figure 33. Patent drawing. Leslie Haywood Hounsfield patented an improved hatchet planimeter in 1932 in the UK and in 1935 in America.

Tensometer Operator

Figure 34. The planimeter was subsequently made by (or for) Tensometer Ltd., 81 Morland Road, Croydon (England) for use with their testing equipment. Conducting a Tensometer test.

Hounsfield Planimeter with box
Hounsfield Planimeter

Figure 35. Hounsfield Planimeter and its box.

Detail of pointer, Hounsfield planimeter

Figure 36. Hounsfield Planimeter, details.

Tensometer Ltd. published a comprehensive manual for their Type W tensometer. It includes instructions for using the planimeter but these are the usual procedures and are not included here.

W. H. Harling

Harling planimeter

Figure 37. A Harling hatchet planimeter, together with the smallest Prytz planimeter I have seen. With a gap of just 64mm it is hard to imagine what it might have been used for.

Harling's leaflet

Figure 38. Harling's original instruction sheet.

Gebrüder Fromme

Prytz planimeter

Figure 39. Simona Fišnarová posted this photograph in Wikimedia, in 2015.

The instrument is stamped "Gebrüder Fromme Vien" so it may have been made before 1913. Planimetrica notes that: "Gebrüder Fromme, successors to Joseph Perfler (est. 1835), formed by brothers Adolf and Karl Fromme in 1884 with the workshop located at Hainburgerstraße 21. The company produced surveying instruments such as theodolites, levelling instruments, compasses used mainly in forestry. The company was renamed Gebrüder Fromme GmbH in 1913, with Adolf Fromme as the sole proprietor."

Eckert und Hamann

This planimeter, made by Eckert and Hamann in Friedenau near Berlin, was reviewed by Ernst Fischer in 1898.

He briefly describes the instrument as follows: Opposite a wheel, at the other end of the rod and perpendicular to it, is a pin f, which ends in a point and serves as a tracing pin. For better guidance and maintenance, a kind of bracket b is attached to f. The distance between the wheel and the tracing pin is the constant of the instrument and measures 20 cm. By means of a device by which the edge of the wheel is constantly refreshed with dye, you get a sharp line while tracing the curve, through which a precise reading from s1 to s2 is possible. The degree of accuracy here is somewhat inferior to that of the polar planimeters from Amsler and others, but can be disregarded because of the great simplicity, durability, and easy handling of the Instrument.

Sketch of Eckert und Hamann Planimeter

Figure 40. Fischer's simplified sketch of Eckert and Hamann's planimeter.

Drawing of Eckert und Hamann Planimeter

Figure 41. Korselt's drawing of Eckert and Hamann's planimeter.

Korselt, writing about Kleritj's Tractoriograph, compares it with the Eckert und Hamann planimeter as follows: "The tractoriograph from Kleritj differs only slightly from the bar planimeter, which also precisely draws the tractors with color, which the Eckert and Hamann company in Friedenau near Berlin brought on the market; but the price of the latter instrument (see Fig. 7) is lower (15 marks against 22 marks)."

Eckert und Hamann Planimeter

Figure 42. The Geodätisches Institut der Leibniz Universität, Hannover has the real thing, illustrated in page 16 of their catalogue and adapted here for nonprofit, educational purposes.


Nestler Planimeter

Figure 43. Nestler made a quite small hatchet planimeter, measuring just 10 cm. A pamphlet explaining its use is available on the rechnerlexikon site.

Planimetro CAT

Planimetro a scure-CAT

Figure 44. This photo of an Italian hatchet planimeter was uploaded to wikimedia in 2010.

Amsler-Coradi Schneidenplanimeter 2007

Amsler-Coradi Schneidenplanimeter

Figure 45. Illustration of Amsler-Coradi Schneidenplanimeter.

Details of Amsler-Coradi Schneidenplanimeter

Figure 46. Details of Amsler-Coradi Schneidenplanimeter.

Instructions for Amsler-Coradi Schneidenplanimeter


Sandra Trenovatz kindly brought to my attention a hatchet planimeter, the GEOMATH planimeter, possessed by the Vienna University of Technology in Austria. It is labeled "Schneidenplanimeter" (Schneiden = cutting or cut).

I had thought it might have been built by a former employee of the university, perhaps the Ing. Killian whose name appears on the box. But Prof. Dr. Joachim Fischer has advised me of two more of these instruments: "An identical instrument which came to the university in 1947, is conserved by the geodetic department of the Technical University Graz (Austria). Another one exists at the geodetic department of the Leibniz University in Hannover (Germany). The Graz copy is named "Artaker", probably a hint as to the seller? Maybe it was Wilhelm Artaker in Vienna, but seemingly no longer existing today". The Leibniz University planimeter is shown in page 16 of their catalogue.

Ing. Killian's planimeter

Figure 47. The GEOMATH planimeter, possessed by the Vienna University of Technology in Austria.

Further study reveals that Karl Killian (1903-1991) was a highly respected member of the Technische Universität Wien and that his career was celebrated by a memorial volume of some of his works. It also contains a diary of his life which includes the following note:

In 1946 Killian became scientific director of a precision mechanics workshop that was relocated to Lend. Only a small fraction of its machine tools (around 2%) and tools made it to Lend and only two workers from the company came to Lend. They waited in vain for 20 people who should have come. The company moved to a hall of the Lend aluminum works. Nobody knew what was to happen or what was to be created.

The machine tools required for the production of precision mechanical devices were missing. However, plenty of silumin (Si-Al alloy) was available, so Killian thought of making everyday objects by casting. Practitioners at the plant believed that thin objects could not be cast from silumin but after a few trials Killian managed to construct a steel mold for casting school compasses. The tip of one of the compass shanks was an old gramophone needle, as these were available in large quantities practically free of charge. It was placed in a recess in the mold and metal poured onto it. Over 10,000 compasses were cast with this mold.

Killian went on to make various other useful things including, and very relevant here, bar planimeters with tip and cutting edge made of steel and also cast in.

So it seems likely that the planimeter shown here was made by Killian after all, along with the other examples identified by Dr. Fischer. It certainly has the rough appearance of having been cast.

Unknown (Dutch?) manufacturer

unnamed planimeter

Figure 48. This unnamed planimeter was sold on Ebay in 2018. It was accompanied by instructions in Dutch so was, perhaps, of Dutch manufacture.

Unknown manufacturer 2, two armed planimeter

Two armed hatchet planimeter

Figure 49. This peculiar hatchet planimeter is in the Leibniz University collection, and is shown in page 18 of their catalogue.

Unknown manufacturer 3

unnamed planimeter

Figure 50. This neat, unnamed planimeter was sold on Ebay in 2015.

Science Museum planimeter, unknown manufacturer

Figure 51. The Science Museum in London also has one them, shown here. But they provide no additional information about it.

Home made hatchet planimeters

It is not difficult to make you own hatchet planimeter. An article in Newnes Practical Mechanics by H. D. E. Goodall describes how to do it. And the tinkerprojects website shows in great detail one actually being made.

1917 Planimeter Hatchet Hand Made

Figure 52. As an example, these two sets of planimeters made by G. W. Mc Pherson for his son in 1917 were sold on eBay in 2009.

Do-it-yourself hatchet planimeter

Figure 53. Robert Marík, in his website, mentions a Czech patent to adapt a pair of compasses as a hatchet planimeter.

Do-it-yourself hatchet planimeter

Figure 54. And in 2003 this great improvisation appeared in the Boatdesign blog.

One happy boat builder subsequently reported: "I've just built one, using this pic as my model. It is surprisingly accurate (certainly good enough for my modest needs). I used scraps that were lying around the shop -- walnut rods, small finishing brads, and an Xacto chisel blade. I put about twenty minutes' work into it, and spent precisely nothing. In my tests on known areas (circles and rectangles) error levels hovered around 1% ..."

Figure 55. Or why not use K'NEX, like Tina Cordon (

If all else fails, use a Penknife ...

A simple hatchet planimeter can be made from a two-bladed pocket knife.

Pete & Sheran Stanaitis have a nice web page where they demonstrate the use of a penknife. They say it was "an idea from someone in Finland".

In fact the idea has occurred to many people over the years. As long ago as 1899, Lehmer noted that "A pocket knife, with a blade at each end, opened up until the ends are at a distance L apart, answers very well". In 1932 Robert Sparks, in the Journal of The Franklin Institute, recommended the use of a penknife, and Manaran, in 1941, published "The use of the penknife for determining areas."

Swiss Army Knife hatchet planimeter

Figure 56. Many pocket knives are rather small for the job. This Swiss Army knife will extend comfortably to 16 cm.

Do-it-yourself hatchet planimeter

Figure 57. Some multi-purpose knives, like this Leatherman, will extend to 25 cm., equal to the largest commercial varieties.

... or even a spoon

as in Robert Marík's video.


Note: 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.

  1. Anon: "The Hatchet Planimeter", Engineering, 57 (25 May 1894), page 687.

  2. Anon: "The Hatchet Planimeter (correction)", Engineering, 57 (1 June 1894), page 725.

  3. Barnes, George: "Hatchet or Hacksaw Blade Planimeter", American Journal of Physics, 25 (1957), pp. 25-29.

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

  5. Boersma, Stuart: "Moving Forward by Traveling in Circles" 2018. pdf

  6. Chollar, B. E.: U.S. patent No 575,105, "Planimeter", 12 January 1897. pdf

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

  8. Dickerson, L. M. 1942. An inexpensive planimeter. Journal Forestry 40: 19-22.

  9. Doucet, E.: "Sur le planimètre de Pritz", Nouvelles annales de mathématiques série 4, tome 11 (1911), p. 193-200, pdf

  10. Eckel, L. H.: "The Hatchet Planimeter", The Forestry Chronicle, September 1953 (download).

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

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

  13. Foote R. L., M. Levi, S. Tabachnikov: "Tractrices, Bicycle Tire Tracks, Hatchet Planimeters, and a 100-year-old Conjecture", Amer. Math. Monthly 120 (2013), 199-216.

  14. Galle, A. W. G.: "Mathematische Instrumente", B.G. Teubner, 1912. (download)

  15. Gentilli, Amadeo: "Das Beil-Planimeter", Schweizeriche Bauzeitung, Band 28, August 1896, pages 61-64.

  16. Goodman, John: "Goodman's hatchet planimeter", Engineering, 62 (1896), p. 255-256.

  17. Goodman, John: "Goodman's hatchet planimeter", letter to the Editor of Engineering, 62 (1896), p. 377.

  18. Goodman, John: U.S. patent No 496,562, "Planimeter", 2 May 1892. pdf

  19. Goodman, John: U.S. patent No 500,202, "Planimeter", 27 June 1893. pdf

  20. Goodman, John: U.K. patent No 1890/6087, "Planimeter", 22 April 1890

  21. Goodman, John: U.K. patent No 1891/407, "Planimeter", 9 January 1891

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

  23. Hamann, Johann: Ueber das Stangenplanimeter, insbesondere ein Stangenplanimeter mit Rolle, 1896

  24. Hammer, Ernst: Das Stangenplanimeter von Prytz, Zeitschrift für Instrumentenkunde, 1895, pages 90-97.

  25. Hammer. "Referate: Goodman's Hatchet Planimeter", Zeitschrift für Instrumentenkunde, XXVIII. Jahrgang. Oktober 1896, page 309.

  26. Hammer. "Referate: Planimeter System Pregél", Zeitschrift für Instrumentenkunde, XXVIII. Jahrgang. Dezember 1908, page 373. here

  27. Heer, R.: "Sammlung des Geodätischen Instituts. Hilfsmittel und Instrumente zur Flächenberechnung und Kartenzeichnung". (Collection of Geodetic Institute. Aids and instruments for area calculation and map drawing). (Geodätisches Institut der Leibniz Universität, Hannover). here

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

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

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

  31. Hounsfield, L. H.: U.S. patent No 1,988,357, "Planimeter", 27 October 1932, pdf

  32. Tensometer Ltd.: "The Hounsfield Tensometer Type W, Instruction Manual" Publication No W1953, The Galleon Press, 1953.

  33. Jordan, Wilhelm: Stangenplanimeter Prytz, 1899

  34. Killian, Karl: "Zum Gedenken an Karl Killian "Ingenieur" und "Geodät" 1903 -1991", GEOWISSENSCHAFTLICHE MITTEILUNGEN Heft 47, Veröffentlichung der Fachgruppe Geowissenschaften Wien, im August 1998.
    ("In memory of Karl Killian "Ingenieur" and "Geodesist" 1903-1991", GEOSCIENCE COMMUNICATIONS Issue 47, Publication of the Geosciences Section, Vienna, August 1998)

  35. Kleritj, Ljubomir: Tractoriograph und Construction der transcendenten Zahlen "pi" und "e", sowie Construction der n-seitigen, dem Kreise eingeschriebenen regelm ässigen Polygone, 1897

  36. Korselt, Alwin Reinhold: Über den Tractoriographen von Kleritj und das Stangenplanimeter, Zeitschrift für Mathematik und Physik, 1898, 312-317.

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

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

  39. Lehmer, Derrick Norman: "Concerning the tractrix of a curve, with planimetric application", 1899

  40. Levi, Mark and Serge Tabachnikovy, "On bicycle tire tracks geometry, hatchet planimeter, Menzin's conjecture and oscillation of unicycle tracks". February 3, 2008

  41. Maffiotti, Giovanni Battista: Il planimetro a scure di H. Prytz: teoria e pratica, 1899

  42. Manaran. M. D. 1941. The use of the penknife for determining areas. Philippine Journal of Forestry 4, No. 3.

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

  44. Messa, Romeo: "Teoria e pratica del planimetro a scure del capitano H. Prytz", 1909

  45. 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.

  46. Pickworth, C. N.: "The Slide Rule: a Practical Manual" 4th edition, 1897.

  47. Poulain, Auguste: Les aires des tractrices et le stang-planimètre, Journal de Mathématiques Spéciales(2), 4 (1895), p.49-54. (Hathi Trust)

  48. Poulain, Auguste: Le stang-planimètre, Mathesis (2), 5 (1895), suppl. 1, p. 1-10. (Hathi Trust)

  49. Theodor Pregél, "Theorie des Präzisions-Stangenplanimeters : System Pregél"
    book published in Chemnitz by E. O. Richter and Co. 1909

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

  51. Prytz, Holger: "The Hatchet Planimeter", Engineering, 57 (1894), p. 813.

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

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

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

  55. Runge, Carl: Das Stangenplanimeter, 1895.

  56. Satterly, John W.: "The hatchet planimeter", The Journal of the Royal Astronomical Society of Canada, 15 (1921), pages 221-243.

  57. Schierbeck, R.: U.S. patent No 752,470, "Measuring Instrument", 16 February 1904. pdf

  58. Schleiermacher, Ludwig: "Das Stangenplanimeter von Prytz", Zeitschrift Für Vermessungswesen, Band 27, 1898, pages 408-411.

  59. Schreiber, Albert: "Zur Theorie des Stangenplanimeters", Zeitschrift Für Vermessungswesen 1908, pages 703-716.

  60. Schreiber, Albert: "Das Präzisions-Stangenplanimeter. System Pregél",
    Zeitschrift Für Vermessungswesen, Band 38, 1909, pages 401-414.

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

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

  63. Strong, C.L.:"An Excursion into the Problem of Measuring Irregular Areas" Scientific American, August 1958, pages 107-114. This was reprinted in "The Scientific American book of projects for the amateur scientist" by C.L. Stong A description of this is given in the Amateur Scientist column

  64. von Standen, Horst: Zeitschrift für Mathematik und Physik, Band 59, 1911, pages 314-318.

  65. Willers, Friedrich Adolf: "Mathematische Instrumente", 1943

David R Green. Last updated 18 May 2021.