Hair planimeters.

Last updated 23 December 2023, David Green work-in-progress

1. Introduction.

The mechanical hair planimeter (Section #2) was patented in 1861 by Swedish land surveyor Nils Fredrik Liedbeck. But it's use does not appear to have extended far beyond the borders of Sweden. It seems rather that before the introduction of polar planimeters different countries adopted quite different approaches to area measurement

Thus in Austria the instrument of choice seems to have been the hair planimeter (Section #4), while in England the computing scale was predominant(Section #5), and in France their own version of a mechanical hair planimeter was patented (Section #6).

All the planimeters discussed above assume the slices are parallel slices. But they don't have to be. The slices can instead be arcs or sectors of a circle. A few planimeters use this alternative(Section #7), for example Edwards and Greenhill 1904, Frikart 1908, Hawthorne 1915, Norton & Gregory 1925, Schnöckel 1925 and Smith 1938. Procedures based on adding the areas of slices work well if the area is a respectable shape: indicator diagrams and the like. But some areas are more difficult to calculate: they include many small regions and/or stringy shapes that have to be measured and added together. Here a statistical method can work better (Section #8).

2. Mechanical Hair Planimeters.

The mechanical hair planimeter (in Swedish ytberäknaren or sometimes simply kärra) was patented in 1861 by Swedish land surveyor Nils Fredrik Liedbeck (1815-1889). (Patent X1556; "å tillverkning af ett slags instrument att begagnas vid beräkning af ytor" [to manufacturing of a kind of instrument to be used in the calculation of surfaces]; N. F. Liedbeck, (Patentee); granted 1861-01-14)

Considering how few of these planimeters still survive, it is surprising how many firms were making them. I am aware of just eighteen examples, yet there were at least four makers: O. J. Sköldin, Anton Ludvig Köhler, Axel Ljungströms and Fr. J. Berg, and possibly also P.M. Sörensen (Figure 17) and an unknown other (Figure 21). All these gentlemen were Swedish - it seems to have been a singularly Swedish instrument.

The earliest known illustration of Liedbeck's planimeter is in Anderson's book: Geodetisk mätningskunskap, Bonniers, Stockholm 1876. It is quite likely that Anderson's drawing is taken from the patent, but we can't be sure. Full information about Swedish patents became available only in 1885 and thereafter.

One made by O. J. Sköldin (his number #133) closely resembles Andersson's drawing. It was sold on Ebay in 2002 for $150.

Figure 1. Sköldin's number #133 compared with Andersson's drawing.

Figure 2. Sköldin's name on planimeter number #133.

The only mention of Sköldin located so far is in Nya Dagligt Allehanda dated 1866-08-23.

Sköldin gets a bum rap in this 1866 article. Of six people mentioned he is the only one not addressed as "Herr". And his planimeter is "of not particularly inviting exterior" while the others "seem to have been carefully executed". Wow. At least this places him as active in 1866 and contemporary with Liedbeck.

A second example closely resembling Andersson's drawing is in Medelpads Fornminnesförening (Medelpad's Antiquities Association located at Norra Berget in Sundsvall, Sweden) identified as MFF4874. No makers name is recorded.

Figure 3. Early model held by Medelpad's Antiquities Association.

A third example resembling Andersson's drawing is in the Gotlands Museum in Visby, Sweden. Again, no maker's name is recorded. The dial is rather curious, the only one I have seen that uses Roman numerals.

Figure 4. Early model in Gotlands Museum.

Very similar to the above is this planimeter (even including the Roman numerals), also in the Gotlands Museum:

Figure 5. Another Early model in Gotlands Museum.

The other planimeter known to have been produced by O. J. Sköldin, number #184, has rather different features. The maker's name is engraved on the instrument. Rather than having the "clarinet keys" to engage the chain, it has a rocker. It also includes a lined glass plate in the box, to assist with the measurements.

Figure 6. Later model by O. J. Sköldin (number #184), in box.

Figure 7. Later model by O. J. Sköldin (number #184), with its glass plate.

Examples made by Köhler are relatively more common. Happily, Köhler put his name on the dial of his instruments.

One appeared on Reddit in about 2020. (The post is now deleted). It appears to be incomplete - the wheels missing.

Figure 8. Köhler planimeter on Reddit.

I have one that is identical to it, and complete:

Figure 9. Köhler planimeter - David Green collection.

A third one, again seemingly identical, is in the Tekniska Museet, Sweden's Technical Museum in Stockholm:

Figure 10. Köhler planimeter the Tekniska Museet.

A fourth, in the Planimetrica Collection, has a rather more elaborate dial showing some constants.

Figure 11. Different Köhler planimeter dials.

A fifth, with a similar dial, was sold in an auction in 2014:

Figure 12. Köhler planimeter auctioned in 2014.

A sixth is held in the Gotlands Museum, Strandgatan 14, Visby, Sweden.

Figure 13. Köhler planimeter in the Gotlands Museum.

And what might be a seventh ("Probably a planimeter") is in the Tekniska Museet. No makers name is recorded. It is hard to see, but the dial appears to be similar to the Köhler style but to have a different type of needle.

Figure 14. Köhler planimeter in the Tekniska Museet.

It is harder to identify the makers of the remaining planimeters.

An example by Axel Ljungströms Fabriks AB, Stockholm was offered on Ebay for many weeks in 2015 and eventually withdrawn. It was very rusty. The planimeter was unnamed but the box bears the maker's name.

Figure 15. Ljungströms planimeter with plate in box.

Figure 16. Ljungströms planimeter with its glass plate.

I have a model that looks exactly like it, but has no maker's name. I assume it was also made by Ljungströms.

The vendor remarked that "The instrument is unsigned but was part of a collection of drafting instruments furnished to the Commission for Land Measurement by Carl Johan Kruuse, a prominent Swedish surveyor and mapmaker. We have found no evidence that Kruuse was an instrument builder, and suspect that he ordered the instrument custom made (quite likely as a "one-off") to his specifications. These would include configuring the scales so percentages of area could be expressed in tunnland, a traditional Swedish unit of land area equal to 56,000 kvadratfot (Stockholm feet) or 4936.4 square meters. The term originated from the area that could be planted with one tunn (barrel) of seed. Given the arc of Kruuse's career and the style of the instrument we suspect it dates from the 1880s. The exceptional workmanship, extensive use of lacquered brass, design of the pulley mechanism and finely crafted ebony slide handle are all hallmarks of the workshop of P.M. Sörensen, chief instrument maker to the Royal Swedish Academy of Sciences in the last decades of the 19th century. Of course since it is unsigned that is just speculation, but it is certainly in the distinctive Sörensen style. A superb example of 19th century instrument making that is exceptionally attractive as a display piece." So, possibly Sörensen, but I suspect Ljungströms.

Figure 17. Ljungströms planimeter, David Green Collection.

Figure 18. Ljungströms planimeter.

Detlef Zerfowski has a mechanised hair planimeter which looks like my probable Ljungströms.

Figure 19. Ljungströms(?) planimeter, Detlef Zerfowski Collection.

Also in the Tekniska Museet, with maker not named but very similar to the above, and perhaps made by Ljungströms. They describe it as: "Roll planimeter of brass, iron and glass. Stored in a wooden case containing an instrument with a graduated disc, a rectangular glass plate in a brass frame and a bag of mica discs. Box length: 265 mm, width: 165 mm, height: 54 mm, total weight: 1.7 kg. Rusty."

Figure 20. Ljungströms(?) planimeter, Tekniska Museet.

There is one other model in the Tekniska Museet; maker not named; "Planimeter, in wooden case. In the case there is also a glass disc, length: 135 mm, width: 100 mm, with a grid pattern and some note sheets." It is quite unlike the ones listed above, so perhaps by a fifth, unknown maker.

Figure 21. Planimeter in Tekniska Museet, unknown maker.

Last, but not least, is a superb example made by Fr. J. Berg, Stockholm and held in the MM Instruments collection.

Figure 22. Berg Planimeter in the MM Instruments collection.

3. Before Mechanical Hair Planimeters.

It can be seen that glass grid plates were part of the apparatus for using these planimeters. Some of the planimeters (figures 6, 7, 12(?), 15, 16, 20) were even supplied with their own matching plate.

In fact, glass grid plates were in use before Liedbeck's invention in 1861. I have a grid plate made by Färngren in Stockholm (below) that can be dated to 1856.

Figure 23. Glass grid plate made by Färngren of Stockholm in 1856.

4. Hair Planimeters.

5. Computing Scales.

Alan Williams has provided a very comprehensive article on computing scales here. Following are pictures of some of my scales.

In "The Student's Column" of it's 29 October 1887 edition, The Builder explains the use of the computing scale in simple terms:

"THE annexed illustration shows a portion of a computing scale usually made to 3 chains to 1 inch. The large figures denote acres, and the subdivisions numbered 1, 2, 3 indicate roods. The perches are engraved upon the ivory scale attached to a movable metal frame, the use of which will be understood upon reference to the drawing of a complete instrument, as shown below, which illustrates the type of computing scale used at the Tithe Commission Office."

"The application to a plan is explained in note 1. Four scales are shown upon this instrument (see note 2). The example given supposes the plan to be drawn to a scale of 4 chains to 1 in., in which case the calculated distance between the parallel lines upon the tracing-paper is seen to be ¼ in., when the upper scale marked 1 is to be employed, and a length of 2½ in. is seen to measure an acre. The wire line C in the frame A B is first so set that the frame rests against the stop-piece F. It is then placed upon the tracing paper over the area to be calculated so as to start from zero at the line M, with the edge of the scale parallel to the lines upon the tracing-paper, and after carefully moving the frame so that the line C traverses from M to N the instrument is then lifted up and replaced with its edge parallel to the lower lines upon the tracing paper, so that the wire C starts from O at the same position on the scale as it indicated when at N. The frame is traversed over each rectangle successively from M to N, O to P, Q to R, S to T, &c., by means of the handle D (see section upon the line YZ) and thus by a series of mechanical additions the area can be read off the scale in acres, roods, and poles. With the use of the upper scale, as shown, when the frame reaches the stop piece L it indicates that five acres have been traversed."

Figure 24. Illustration of a computing scale in the Builder, 29 October 1887.

"In Merrett's improved computing scale the screw in the metal frame is made to act as a clamp, and different scales are supplied to fit the same metal frame, instead of the ivory pieces employed to cover the scales which are not required for use in the instrument adopted by the Tithe Commission Office. The reading edge is also bevelled off against the scale."

I have three of these scales. The one below is the simplest design, with a scale of 1/2500 chains. The top half of the rule shows 0-6 acres, the bottom half 6-12 acres. Moving the cursor from left to right, then back from right to left, one complete application of the rule measures 12 acres. The rule is 23 inches (580cm) long. There is no maker's name on the rule.

Figure 25. Computing scale - simplest design.

Figure 26. Cursor of Figure 25 compared with Bourns' 1843 illustration.

The first improvement was to provide two different scales on the one stock, as in this rule by Thornton.

Figure 27. Computing scale with two scales.

My second rule is a larger version of Figure 25 and is 35 inches (890cm) long. The rule is signed "Elliott Bros., 30 Strand". Alan quotes Clifton's SIS article, noting that Elliott Bros were at 30 Strand from 1858-63. The Ordnance Survey's 25-inch "County" series plans (1:2500 scale) were begun in 1854. Alan suggests that the extra length of this rule was to ensure that features in these larger maps would still fit within one length of the rule.

Figure 28. 35 inch and 23 inch versions.

My third rule is the so-called Stanley Universal Computing Scale. This version of the rule was introduced by Stanley in about 1865-1868 but mine was made much later, I think. The box has the grey label marked Stanley, Great Turnstile so the rule was made some time after that was introduced. The box has 12 slots to store the removable scales although I have only eight: ones for 1, 2, 3, 4, 5, 6 chains to the inch, and 6 inches and 5 feet to the mile. However, as some catalogues from the 1800's advertise only those 8 slides, it seems my set is complete and the extra slots were for optional extras.

Figure 29. Stanley Universal Computing Scale in box.

Figure 30. Stanley Universal Computing Scale components.

An interesting variation of the computing scale, Williams Proportional Plotting Scale, was patented by Peter Lloyd Armstrong Williams in 1905. The patent for it is available at Williams called it a computing scale but, unlike those shown above, it does not function as a planimeter. This, and the "Best" scale which follows, are more closely related to pantographs.

Figure 31. Williams Proportional Plotting Scale patent.

The patent is six pages of almost impenetrable officialese which I think boils down to this:

The object of the invention is to provide rules or computing scales for the use of surveyors, engineers and other persons by which measurements can be increased or decreased in any required ratios or proportions in an easy, speedy manner with a minimum of labour and a maximum of certainty.

Along the surface of a [wooden] rule A, parallel with its operative edge B, is a groove C into which can be placed a [wooden] slip D. This slip is inscribed with lines described and explained later.

Parallel with the groove C is another groove E in which can travel a block F on which is mounted a square frame G which extends over the face of the slip D and over the rule A. Near to edge B of the rule is another groove H which constrains a lug H1 on the frame G to keep the frame steady as it slides along the rule. Over the edge B of the rule, which is bevelled off, is a projecting tongue K on the frame G, fitting close upon the bevelled edge B, and on this tongue K is one or more marks K1.

The frame G can be moved by hand along the rule A by suitable knobs M. In the frame G is a bar N which is parallel with the edge B of the rule. The ends of this bar N can slide along guides fitted in the edges of the frame G, propelled by the knob P. Midway along the bar N is a mark N1.

Figure 32. Williams Proportional Plotting Scale slides.

Slip D is marked with a number of lines W parallel with edge B and a number of lines X crossing them. The lines X are radial lines emanating from a common centre situated perpendicular to the midpoint of the rule and about as far therefrom as is the length of the rule.

The longitudinal lines W are labeled at the ends with certain values: the central longitudinal line W is called the normal line and on the left is labeled "0"; the next line W on the side nearer to the supposed centre is labeled "-1" and on the side further from the supposed centre "+1" and so on, signifying either 1 per cent. increase or decrease or it may be 10 per cent. and so forth. Similarly the longitudinal lines W are labeled on the right with any appropriate "degrees of slope" as is shown in Figure 1A, to be able to make the due allowance in any particular case for the slope of the ground.

The transverse lines X may also be labeled with certain values or quantities or proportions as is shown in Figures 1, 1A and 2.

The instrument can be used as follows. Presume a number of points are delineated on a sheet of paper. To increase or decrease the distance between any two of the points at a certain predetermined ratio, place the rule A on the paper so that, with the frame G placed in the centre of the rule A opposite to the line X marked "500", the mark K1 is opposite one of the points on the paper and the bar N is on the normal line W marked "0" exactly in line therewith. Then slide the frame G along the rule A either to the right or to the left in order to bring the mark K1 opposite to the other point on the paper and move the bar N in the frame G upwards or downwards until it is on the longitudinal line W indicating the required amount of increase or decrease and the mark N1 on the bar N is brought exactly on the point where the appropriate transverse line X intersects the aforesaid longitudinal line W. This can be effected either with the knobs M or more minutely with the thumb screw L1.

The mark Kl will then indicate by how much the location of the point on the paper is to be changed. When on the other hand the amount of increase or decrease is not known the instrument could be used mainly in the general manner as has previously been described but the bar is moved upwards or downwards from the normal line when the frame and the rule have been placed in their proper required positions until the mark on the bar comes to the intersection of the transverse line and the longitudinal line and that particular longitudinal line gives the information concerning the ratio by which the particular mark or position on the paper is to be altered.

Figure 33. Early Williams Proportional Plotting Scale.

I have two of these rules and neither one exactly matches the patent. This one seems to be very early, possibly even a dummy run. The percent labels on the left of the rule are absent. The slope labels on the right have a different format and appear to have been hand-written in Indian ink. As they are right where you would normally hold the rule with your thumb, they are rather worn and hard to read. Also the rule is not quite complete - the knob marked P in the patent is missing. There is no maker's name on the rule, and no box which might have provided a name.

Figure 34. Modified Williams Proportional Plotting Scale, components.

Figure 35. Modified Williams Proportional Plotting Scale set up.

The other rule is considerably newer and incorporates improvements to the patent. It was made by J. H. Steward. I have not come across the instructions for using this version.

Another rule that has appeared on Ebay a couple of times is the "Best Enlarging and Reducing Scale". Alan Williams has placed some pictures of one offered on Ebay in 2018 in the picture gallery at This rule was made by Halden.

Figure 36. Halden "Best" Enlarging & Reducing Scale.

Another example offered more recently on Ebay was made by Reynolds. The vendor didn't provide a good picture of the whole unit, but did provide some excellent close ups, and the pamphlet:

Figure 37. Reynolds "Best" Enlarging & Reducing Scale details.

Figure 38. Reynolds "Best" Enlarging & Reducing Scale pamphlet.

7. "Polar" Hair Planimeters.

The "normal" way to calculate an area was to split it into parallel, linear slices and add up the slices. All the planimeters mentioned above were designed to accomplish that. But you can use arcs or sectors of circles instead and add them up. Several ideas were patented but only one, the Adisco, modeled on Hawthorne's 1915 US patent, seems to have been produced in any numbers. The Adisco Area Measurer pops up on Ebay from time to time.

7.1. A Radial Area-scale, Edwards 1904.

Figure 39. Radial Area-scale, Edwards 1904.

This idea was presented to the Royal Society in 1904 and published in their proceedings (Ref 9). Edwards described it as follows:

"This is a contrivance for finding the area of a plane figure by means of a transparency. The design in the transparency consists of number of radiating lines. Each of these lines is graduated. There are various patterns of this design, and their respective claims to convenience and accuracy form a wide field for discussion. In the accompanying transparency (reproduced in the figure), which is fairly simple and effective, there are eleven straight lines radiating from a point at equal angles. The way in which the transparency is used is as follows :- The figure whose area is to be found is placed under the transparency in close contact with it, so that its contour lies just between the two outside lines of the transparency, i.e. so that each outside line touches the contour, or passes through a cusp or angular point, or contains some rectilineal portion of the contour. Each of the radiating lines thus becomes a tangent or transversal, or contains a side of the figure. The graduations of the right-hand points of intersection of these transversals are read and added together; then the graduations of the left-hand points of intersection are read and added together. The second sum is subtracted from the first; and the result records the number of square inches in the figure."

"It will be seen that if each of the outside lines touches, or passes through an angular point of the figure, there will be eighteen graduations to be read - those on the outside lines cancelling each other. If one of the outside lines contains a rectilineal portion of the contour, there will be twenty graduations to read; if both outside lines do so, twenty-two graduations must be recorded."

"If a quicker use of the area-scale, with less chance of accuracy, is desired, the figure to be quadrated may be placed so as to lie just between the first and ninth, or the first and seventh, or the first and fifth lines, in which cases fourteen, or ten, or six graduations will be read respectively."

"If the figure is too large to be included between the outside lines, it may be divided into two parts by a straight line drawn across it, or into three parts by a pair of straight lines inclined to one another at the same angle as the outside lines, which may be done by means of the cardboard slip accompanying the diagram. The second of these methods of dividing up the figure may be also employed when it is desired to obviate the inaccuracy that may result from the two outside lines otherwise being tangent to the figure."

A couple of pages of mathematical justification follows in the report.

There is no record of whether these transparencies were made at the time, but some time later, in 1929, it was reported (Ref 10) that they had been introduced by G Cussons, Ltd. The report includes this additional explanation.

Figure 40. Cussons Radial Area-scale, 1929.

"This scale, which is made by Messrs G. Cussons, Ltd., Technical Works, Lower Broughton, Manchester, consists of a radial arrangement of graduated lines engraved on a sheet of transparent celluloid, about 8 in. by 6 in., and may be used for the approximate determination of the areas of plane figures of regular or irregular shape. For an explanation of the principles involved see Edwards (Ref 11)."

"In use the scale is placed over the figure to be determined so that no radiating line cuts its boundary more than twice, and
(a) So that the outer lines 1 and 11 (see figure) touch the curved boundary of the figure, as at I; or
(b) So that one of the outer lines, 1, contains a straight portion of the boundary, and the other, 11, touches the boundary, as at II; or
(c) So that the outer lines 1 and 11 contain sharp points of the boundary, as at III ; or
(d) So that the outer lines 1 and 11 both contain straight portions of the boundary, as at IV."

"The graduations are read at the points where the boundary of the figure cuts the lines; the eleven left-hand readings are added together and subtracted from the sum of the eleven right-hand readings. The difference gives the approximate area of the figure."

"It will be found on the scale that two sets of graduations are provided to lines 2, 3, 9 and 10, the inner to be used in cases where lines 1 and 11 are tangential to the figure, the outer in other cases."

7.2. Eugen Frikart, 1908.

Figure 41. Frikart's planimeter, 1908.

SWISS PATENT NUMBER 41786 April 9, 1908 Eugen FRIKART, Mülhausen (Germany).

The drawing shows the top view of an example of the planimeter, which consists of a circular plate 1 with a central pivot point 3 and an arm 2. The right edge of the arm lines up with 3. The edge of the plate and the right edge of the arm are graduated.

The planimeter is fixed, within the area to be measured, by a pin through an opening at 3. By rotating the instrument over the surface to be measured, the surface area can be broken down into small triangles that are equiangular at the pivot point. The number of these triangles is determined using the divisions of the circular plate and their area is read from the scale on the arm.

The scale on the arm indicates the content of the triangles, which is larger, the farther the intersection of the arm and the surface edge is from 3. Starting, say, at 6, you read the content of the triangle corresponding to the intersection point, then rotate the planimeter around point 3 step by step, in the direction of 7, taking readings until the entire area has been covered. The sum of all the readings is the area of the surface.

For more or less accurate measurement, the plate contains three circle divisions. The closest division on circle 8 gives the most accurate result: the readings on the arm give the contents directly. The readings corresponding to circle 9, where the plate is rotated through twice the angle, must be multiplied by 2, and the readings corresponding to circle 10 must be multiplied by 4.

7.3. Hawthorn 1915.

Figure 42. Hawthorne's planimeter.

7.4. Norton and Gregory 1925.

Figure 43. Norton and Gregory's planimeter, patent drawing.

Figure 44. Norton and Gregory's planimeter, actual.

7.5. Schnöckel 1925.

Figure 45. Schnöckel's planimeter, patent drawing.

Patented in the German Empire from March 15, 1925.

The invention is a vector planimeter of the known type, in which a rotatable, transparent, rectangular plate is inscribed with a numbered group of arcs and has a number of drilled holes h [the holes are used to insert a pencil for use as a compass]. By dividing a surface into sectors using the instrument and measuring them by adding up the angular rotations required for the individual sectors, the desired area is obtained.

The planimeter has a counting wheel k that adds up the rotations. The wheel is attached along with a plate d on the transparent plate e, f, g, h, i of the vector planimeter. The planimeter consists of the plate mentioned, which can be rotated against a round disk a, a1, a2, a3, b with graduations on the edge. There are g engraved circles on the plate at varying distances along a straight line, forming progressively narrower sectors of equal area.

The disc axis tapers downwards in the card paper and enables rotation against the plate, so that the latter is divided into parallel strips like a harp by a pendulum-like movement of the latter over the area to be calculated. Here, in the manner of a folded-down addition compass, the contents of the strips are temporarily transferred to the disk by locking the disk with finger pressure on the card and can be read from their division.

The new counter wheel k makes it possible to check how many revolutions the disk has made when calculating larger areas, so that gross errors that would question the value of the instrument are avoided. This wheel k, which is divided into 10 parts, has 10 teeth, sits on the transparent plate and is moved after each full revolution by a retaining pin that engages in the teeth and is located on the disk axis.

A second new device, the plate d in which the axle journal of the disk sits, allows the position of the disk center to be regulated in relation to the plate, so that the planimeter can be adjusted for the paper shrinkage of the card and temperature, etc. caused changes to the plate. This plate d is attached to the transparent plate by screws and can be removed from it as required.

7.6. Smith 1938.

Figure 46. Smith's planimeter, patent drawing.

This planimeter consists of a single sheet of transparent material on which a sequence of concentric circles is indelibly marked. Fig. 1 is a plan view of the planimeter, placed on an area 10 to be measured. Fig. 2 is a vertical cross section of the device.

11 is a circular disc of transparent and suitably flexible material, on which concentric circles 12 around the center point 13 are inscribed. The radius of each circle is such that the areas of the rings between the circles are all equal. For example, if the innermost circle covers an area of one square inch, the second outwards from the center an area of two square inches, and the third an area of three square inches, then the ring between the first and second circles and the ring between the second and third circles each has an area of one square inch.

The outermost circle is graduated to indicate the angular distance around the center of the rings. As shown, it is divided into 10 equal arcs and the numbers 1 to 10 are shown next to the ends of those arcs. Depending on the area of the rings and the scale of the map, the gradations can be determined so that the reading is obtained directly in terms of the actual area that the map represents, e.g. in square miles.

The center of the circles can be held in position by a pin that is pushed through a small hole 13 in the center. The disc is then rotated about the center point, so that the point a on the smallest circle that intersects the area moves from its position over the perimeter of the area on one side to a point a', over the perimeter of the area on the opposite side. To bring about the movement of the disc, an instrument with a sharp tip can be placed against the disc at point a and used to move point a to point a'. The instrument can then be placed at point b and the rim turned, until point b reaches point b'. (This will completely move point a away from the area 10). The same procedure is followed with each of the circles that intersect the area to be measured.

The center 13 of the circles can be placed anywhere. The more circles the perimeter of the area 10 intersects, the more accurate the determination of the numerical value of the area 10 generally becomes. When the disc turns, it is immaterial which circle is first moved from one side to the other of the area to be measured. It is only necessary that to complete the operation, all the circles that intersect the area should have been moved from one side of the area to the other in the same direction.

Before measurement commences, the zero point of the gradations is at some point A. When measurement is completed, the point A is near some other point of the gradations, e.g. the number 2. This means that the disc has been turned 2/10 of a turn and that the measured area equals 2/10 of the area that each ring occupies. If the inner circle is equal to one square inch, then the area measured is equal to 2/10 of a square inch.

Friction between the disc and the map over which it rotates is seen as a problem. Fig. 2 shows a modification to reduce this friction.

7.6. Adisco.

Figure 46. Adisco Area Measure.

Figure 47. Adisco instructions.

8. Bruning Areagraph Chart.

Procedures based on adding the areas of slices work well if the area is a respectable shape: indicator diagrams and the like. But some areas are more difficult to calculate: the area of water weeds in a lake [Reference 1], the proportion of voids in samples of cloth [Reference 2], the areas of glaciers [Reference 3], the exposure of mud flats at low tide [Reference 4]. These tend to include several small regions and/or stringy shapes that have to be measured and added together. They commonly arise in aerial photography or in the detail in microphotographs [Reference 2].

Here a statistical method can work better. A collection of random dots is plotted on a transparent sheet which will cover the photo. The transparent sheet is laid over the photo and the number of dots covering the features to be measured is counted. The sum of these dots, times the weight of each dot, is the area of the features. The weight of a dot is the area covered divided by the number of random dots on the sheet.

Transparent sheets with random dots already plotted were produced: the "Bruning Areagraph Chart" manufactured by the Bruning Division of Addressograph Multigraph Corporation of Cleveland, Ohio. Their No. 4850 Chart is shown in Figure 42.

Figure 48. Bruning Areagraph Chart No. 4850.

The chart mentions that a patent was applied for by Jack Lessinger in about 1957. One was granted to him in Canada as CA649436 in 1962.

Just how many versions of the sheets were available I don't know. A No. 4849 chart was shown in Long 1990 (see below). "An 85% Bruning areagraph" was mentioned elsewhere [Reference 6]. The #4850 has ten dots in each rectangle and 100 dots per square inch. Chart 4849 also has 10 dots per rectangle, but at a density of 30 dots per square inch. The patent suggests three versions would be adequate for all contingencies, having 10, 30 or 100 dots per square inch.

Reference 3 gave unqualified approval of the device: "The manufacturer's claim that the degree of precision for areas greater than twelve square inches was better than 97 per cent was supported in tests against other standard planimeters and square counting techniques. Experiments carried out by the Glaciology Subdivision have proved that operator variance with the random dot planimeter is small compared to other methods. The use of a random dot planimeter provided a considerable saving in time for the large number of area measurements that had to be made. For areas of a few square kilometres a millimetre grid was used so that the same level of accuracy could be maintained."

Reference 3 also noted that "Glacier areas were calculated using a Bruning Areagraph random dot overlay and a mechanical dot counter." Such a counter isn't mentioned in the patent - I can find no information about it.

Figure 49. Bruning Areagraph Chart No. 4849 from Reference 7.

9. References.

1: "AERIAL SURVEY TECHNIQUES TO MAP AND MONITOR AQUATIC PLANT POPULATIONS - FOUR CASE STUDIES"; Elba A. Dardeau, Jr.; Jan 1983; U. S. Army Engineer Waterways Experiment Station, Environmental Laboratory, P. 0. Box 631, Vicksburg, Hiss. 39180

2: Canadian patent CA1090515A, "Skin cleansing product having low density wiping zone treated with a lipophilic cleansing emollient"

3: "Glacier Inventory of Canada - Axel Heiberg Island, Northwest Territories"; C.S.L. OMMANNEY; 1969; INLAND WATERS BRANCH, DEPARTMENT OF ENERGY, MINES AND RESOURCES, OTTAWA, CANADA,

4: "Environmental Screening for Wharf Repairs at Tiverton, Digby County, Nova Scotia"; SMALL CRAFT HARBOURS BRANCH, MARITIMES REGION; Project 307381


6: Jan 1974; page1301 (

7: Long, Katherine S.; "Site Characterization for Radar Experiments"; 1990; US Army Corps of Engineers; Technical Report EL-90-8;

8: Swedish patent 92159; Title Planimeter; Application Date 1935-10-07; Granted 1938-02-24; Aerial Explorations Inc (Patentee) - New York City (US), Smith, H. G. (Inventor)

9: R. W. K. Edwards and Alfred George Greenhill, 31 July 1904, Proc. Roy. Soc. 73 (1904) p292 (

10: "A New Area-computing Scale", Anon, Journal of Scientific Instruments, vol 6, 1929

11: R. W. K. Edwards, Messenger of Mathematics, New Series," No. 404.