Reference / Paper · 2018/2026

The Polar Planimeter of Jakob Amsler (van der Salm)

A history and vector-analytic treatment of Amsler's polar planimeter and its precursors.

Author: W. van der Salm
Year: 2018/2026
Type: Reference / Paper
Credit: W. van der Salm, rekeninstrumenten.nl.

A Mathematical Explanation of the Polar Planimeter of Jakob Amsler-Laffon Simon A. M. van der Salm Independent Researcher of Historical Calculating Instruments, Hilversum, The Netherlands Retired Lecturer in Applied Mathematics, Intop Zorgsector College for Biomedical Technology E-mail: [email protected] First edition November 2018/Second edition with minor revisions and additions June 2026 ABSTRACT The Amsler polar planimeter is a classical mechanical instrument designed to determine the area enclosed by a planar curve through the tracing of its boundary. Invented in the nineteenth century by Jakob Amsler-Laffon, the device constitutes one of the earliest realizations of analog computation. Its operation is rooted in the geometry of constrained motion and in the theory of line integrals. In particular, the planimeter mechanically evaluates an integral that corresponds to the area formula arising from Green’s theorem. The following descriptive study examines the mathematical foundations of the Amsler polar planimeter, including its kinematic structure, differential-geometric interpretation, and its relation to some classical integral formulae. KEYWORDS Polar planimeter, Amsler, Stokes’ and Green’s theorem, vector analysis, area measurement. Science Museum Group Collection, London, License CC BY-NC-SA 4.0 1 Simon A. M. van der Salm Amsler’s Polar Planimeter CONTENTS ABSTRACT ....................................................................................................................................... 1 KEYWORDS ...................................................................................................................................... 1 1. INTRODUCTION ........................................................................................................................... 4 1.1. Some historical context ....................................................................................................... 4 1.2. Explanation of its functioning by Green ............................................................................. 5 1.3. Object of this study .............................................................................................................. 6 2. CONSTRUCTION OF THE POLAR PLANIMETER ....................................................................... 7 2.1. Amsler’s drawings ................................................................................................................ 7 2.2. Explanatory drawings........................................................................................................... 8 3. THE MEASUREMENT PRINCIPLE OF THE POLAR PLANIMETER ............................................ 9 3.1. Tracing the contour .............................................................................................................. 9 3.2. Positive and negative wheel movements ........................................................................... 9 4. MEASURING THE AREA WITH THE WHEEL ............................................................................ 11 4.1. Translation and rotation .................................................................................................... 11 4.2. The measuring wheel ......................................................................................................... 12 4.3. Amsler’s formula ................................................................................................................ 12 5. A BRILLIANT INSIGHT: THE HIDDEN ROLE OF THE POLAR ARM ......................................... 13 5.1. Sweeping the region R ....................................................................................................... 13 5.2. Straight-line segments ....................................................................................................... 13 5.3. The pivot point S ................................................................................................................ 14 5.4. Not a unique construction ................................................................................................. 14 6. VECTORIAL INTERPRETATION OF THE POLAR PLANIMETER .............................................. 16 6.1. An approach using Green’s Theorem ................................................................................ 16 6.2. Analysis of the measuring wheel ...................................................................................... 17 7. A PROOF USING GREEN’S THEOREM ...................................................................................... 19 7.1. Some historical context ..................................................................................................... 19 7.2. What is Green’s theorem? .................................................................................................. 20 7.3. Proof of the Amsler planimeter by Green’s theorem ....................................................... 21 8. CONCLUSIONS ......................................................................................................................... 24 8.1. Amsler’s geometrical explanation .................................................................................... 24 8.2. Explanation by Green’s theorem........................................................................................ 24 9. ACKNOWLEDGEMENTS............................................................................................................ 25 10. REFERENCES ........................................................................................................................... 25 11. APPENDIX: VECTOR ANALYSIS CONCEPTS USED .............................................................. 27 11.1. Rotation (curl) of a vector field ....................................................................................... 27 2 Simon A. M. van der Salm Amsler’s Polar Planimeter 11.2. Stokes' theorem ............................................................................................................... 28 11.3. Green’s theorem ............................................................................................................... 29 11.4. Relation to the Amsler planimeter .................................................................................. 31 3 Simon A. M. van der Salm Amsler’s Polar Planimeter 1. INTRODUCTION 1.1. Some historical context In 1854, the Swiss mathematical physicist Jakob Amsler-Laffon1 (1823-1912), see the photograph in Figure 1, constructed a polar planimeter with which the area AR of an arbitrarily shaped region R in a plane can be measured exactly (i.e. not approximated by the area of a finite number of basic shapes like rectangles), simply by tracing the contour C enclosing that region2. See references [1] and [2]. Figure 2 shows the Amsler polar planimeter by Albert Ott3, about 100 years younger than Amsler’s, but a variant operating according to the same principle he discovered. Amsler produced over 50,000 units across six variants by the late 19th century, driving widespread adoption in surveying, cartography, and mechanical engineering by the 1860s. See [21]. Various variants of Amsler’s polar planimeter can be found on the website [22]. Also, David Green's website [8] provides a detailed overview of various types of planimeters; the site includes specific information on Amsler planimeters. Figure 1 Jakob Amsler(-Laffon). See biography [10]. It is remarkable that, almost simultaneously with and independently of Amsler, another scientist, Albert Miller von Hauenfels, professor at the Leoben Mining Academy in Graz, Austria, devised a similar planimeter, patent granted 13 November 1855, so, somewhat later than Amsler. See [20]. The instrument was manufactured in Vienna by the company Starke, makers of surveying and astronomical instruments. Hence, the device became known as the Miller–Starke planimeter. One can find a photo of a Starke & Kammerer planimeter on the website [18]. Detlef Zerfowski and John Vossepoel wrote an important study of the history of planimeters [17]. The authors examine more than thirty lesser-known area-measuring instruments developed 1 Amsler-Laffon studied first theology and later mathematics and physics at the University of Königsberg, where he was influenced by the physicist Franz Ernst Neumann. Among his fellow students was the later famous physicist Gustav Kirchhoff. He founded a precision instrument company in Schaffhausen and developed scientific and engineering instruments, including testing machines and hydrometric devices. See Wikipedia. 2 That the area of a region can be determined merely by tracing its contour was already well known in Amsler’s time. In his 1856 paper [2], Amsler refers, among others, to Johannes Oppikofer (1783–1864), who in 1827 constructed a conical planimeter that determines the area by tracing the contour. Amsler notes that this idea is even older, having been discovered in 1814 by the German land surveyor Johann Martin Hermann (1785–1841), but that it subsequently fell into oblivion. Barbera Haeberlin has created a beautiful digital simulation of the construction and operation of Hermann’s conical planimeter, which can be viewed online: https://www.youtube.com/watch?v=XQXO6edF6sA. In 2014, an exhibition on Hermann’s work was held at the Deutsches Museum in Munich. Reference [6] primarily describes the history of Hermann and his conical planimeter, but places this history in the broader context of the historical development of various types of planimeters devised and constructed after 1814. 3 Albert Ott was a 19th-century German precision instrument maker based in Kempten who became one of the main industrial producers of planimeters. While the polar planimeter was originally invented by Jakob Amsler-Laffon, Ott’s workshop played a key role in manufacturing, refining, and widely distributing Amsler-type planimeters for practical use in engineering and cartography. 4 Simon A. M. van der Salm Amsler’s Polar Planimeter between 1814 and 1884. Their work reveals that the evolution of the planimeter was far more diverse than is commonly assumed4. Figure 2 Polar planimeter, type 31, no. 89475, manufactured by A. Ott, Kempten, Bavaria, distributed in the Netherlands by N.V. Wed. J. Ahrend & Zoon. Property of the author. It is mentioned, the great physicist James Clerk Maxwell, see [15, p. 20], who at roughly the same time also designed a planimeter, stared in astonishment at Amsler’s polar planimeter. He must undoubtedly have wondered what everyone wonders when seeing a polar planimeter for the first time: how can such a simple instrument measure the area AR of arbitrarily shaped plane figures exactly? Which ingenious mathematical idea is incorporated in this instrument? Therefore, the main questions we try to answer in this study will be: on the basis of which mathematical principle does an arbitrary closed curve C in the plane determine the area of the enclosed region R and how is this principle applied in Amsler’s polar planimeter? 1.2. Explanation of its functioning by Green However relatively simple the basic construction of a polar planimeter may be, answering these questions requires a rather complex mathematical analysis. In this essay we analyse that remarkable mathematical functioning of Amsler’s polar planimeter. Anyone searching books or the internet for an explanation of its operation will often find the elegant explanation using Green’s theorem for the plane, which is also discussed subsequently in this essay. However, we do not know for certain whether Amsler was familiar with this theorem. Green’s theorem became widely known in the professional mathematical community only around the same time of Amsler’s invention. Although Amsler was a professional mathematician it is 4 “A Contribution to the History of Planimeters – Lesser-Known Instruments and Variants” by Detlef Zerfowski and John Vossepoel [17] is an important contribution for anyone interested in the early history of planimeters. Published in Slide Rule Gazette, Issue 23 (Autumn 2023, pp. 61–102), it offers a comprehensive survey of many little-known area-measuring instruments that are often overlooked in standard historical accounts. Most histories of the planimeter focus on the well-known figures like Amsler and Miller. Zerfowski and Vossepoel take a different approach. Rather than concentrating on the famous instruments, they document a wide range of lesser-known devices and variants developed before and alongside the classic planimeters. The article describes more than thirty relatively obscure area-measuring instruments dating from approximately 1814 to 1884. 5 Simon A. M. van der Salm Amsler’s Polar Planimeter probable he didn’t know yet of its belated republication in the scientific literature5. In any case, he based the operation of his planimeter on a different, more intuitive geometrical idea rather than on the vectorial rotational principle of Stokes that underlies Green’s theorem. 1.3. Object of this study So, in short: the object of this study is to investigate the mathematical principles underlying the planimeter: (1) To gain insight into the brilliant idea of Amsler, we first discuss his classical description (of which many variants exist), using a combination of translation and rotation of a line segment, and the conclusion drawn from the surprisingly simple formula with which that rather difficult explanation ends. (2) Subsequently shall we present a proof of Amsler’s theorem based on concepts from vector analysis, a method that is likely to be more accessible to contemporary historians of mathematical instruments. In the appendix of this study the reader may find the concepts of vector analysis that are used in this proof. 5 Green's theorem originated in George Green's 1828 essay An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Green's original 1828 essay was circulated only in a small local edition and remained largely unknown for more than two decades. Its significance became widely recognized only after William Thomson (later Lord Kelvin) rediscovered the work and promoted its republication during the 1850s. The essay was later reprinted in Crelle's Journal (Journal für die reine und angewandte Mathematik) in three instalments, published in 1850, 1852, and 1854. These publications brought Green's work to the attention of the wider European mathematical community and contributed significantly to the dissemination of Green's theorem and related concepts. 6 Simon A. M. van der Salm Amsler’s Polar Planimeter 2. CONSTRUCTION OF THE POLAR PLANIMETER 2.1. Amsler’s drawings Figures 3a, 3b, and 3c below show the construction drawings published by Amsler in 1856 in his article [2] on the newly designed polar planimeter. Figure 4 shows additional details of another drawing by Amsler, in which original Figure 2 is particularly important for understanding the mathematical description that follows below. Figure 5 shows a simplified instantaneous configuration of the planimeter while tracing contour C of region R, with the most relevant components, but without the measuring wheel W. Point P is the pole, a fixed reference point during measurement, usually, but not necessarily chosen outside region R. Figure 3a Amsler’s original Figures 1 and 4, showing the tracing arm A, polar arm B, pole E, and tracing point F, as referenced in [10]. Figure 3b Amsler’s original side view Figure 2 of the polar planimeter, showing the measuring wheel D, transmission to a horizontal dial with vernier scale, and spindle C. Using Figure 5, one can easily identify the components in Amsler’s Figures 3a – 3c. In those Figures, A is the tracing arm ST, B the polar arm PS, C the pivot S, D the measuring wheel W, E the pole P, and F the tracing point T. The contour C of region R is traced by the tracing point T of a hinged arm system PST, consisting of polar arm PS (red) and tracing arm ST (green), with ST ≤ PS. The polar arm rotates around pole P, and the tracing arm rotates around pivot S. The pole P is 7 Simon A. M. van der Salm Amsler’s Polar Planimeter preferably chosen such that the angle θ between PS and ST in Figure 5 does not deviate too much from 90°. In any case, θ is kept below 180°, so that the tracing arm cannot align with the polar arm. Figure 3c (Left) Top view (original Figure 3) showing the measuring wheel D and dial displaying total angular displacement. Figure 4 Explanatory detail from Amsler’s construction drawing Figure 2. See the rotation at L and perpendicular translation from D to D’. See the American Mathematical Society website [5]. 2.2. Explanatory drawings Figure 5 (Left) Stylized situation. The region R is enclosed by the closed contour C. P is the pole. S is the pivot point. T is the tracing point. PS is the polar arm, and ST is the tracing arm. The tracing point T traverses the contour C once, here in the counterclockwise direction. Figure 6 (Right) The contour C is traversed counterclockwise. As the tracing point T moves from lowest tracing point T1 to highest tracing point T2, via contour point A, the arm system PST, consisting of the polar arm PS and the tracing arm ST, sweeps out the by dotted lines shaded area. A measuring wheel W measures this total area, up to a scale factor. See figure 7. Mind, this total area is larger than the area of region R, we want to measure. 8 Simon A. M. van der Salm Amsler’s Polar Planimeter 3. THE MEASUREMENT PRINCIPLE OF THE POLAR PLANIMETER 3.1. Tracing the contour That Amsler’s polar planimeter remained an important instrument until around 1980 is shown by the well-known mathematics textbook [3], which was used in the first school year of many Dutch colleges for higher technical education (Hogere Technische School, HTS) during the 1960s and 1970s. For an initial answer to the questions raised in the introduction of this essay, we essentially follow [3, pp. 236–239] from that textbook. Figure 6 above shows the situation in which the tracing point T follows the contour C. First, T moves along the right-hand side of C, via point A, from lowest tracing point T1 to highest tracing point T2. The articulated arm system PST then sweeps out the shaded area shown with straight lines in Figure 6, i.e. oblique dotted lines from the lower right to the upper left. This area is clearly considerably larger than the area AR of the region R to be determined. This initially too large area is measured by a small measuring wheel W mounted perpendicular to the tracing arm ST, indicated in Figure 7 by a small yellow arrow. When tracing point T traverses the contour C via point A, between the instantaneous tracing points T1 and T2, the measuring wheel W rotates in the forward direction, which corresponds to the usual positive orientation in mathematics and physics. Interestingly, however, Amsler assumes a negative orientation, in which the contour C is traversed in the clockwise direction. For the final result, namely the measuring the area AR of the region R, this of course makes no essential difference6. 3.2. Positive and negative wheel movements Figure 7 The tracing point T traverses the contour C, via T3, from T2 down to T1. When the right-hand side of the contour C was traversed, the measuring wheel W rotated forward, as indicated by the yellow arrow in the tracing arm S1T1. When the tracing arm reaches the position S2T2, the measuring wheel begins to rotate in the reverse direction. As noted, when the tracing point T follows the righthand side of the contour C, the measuring wheel W rotates forward. See the yellow arrow in the initial position S1T1 of the tracing arm ST in Figure 7. 6 In Amsler’s polar planimeter, the contour C is traversed in the clockwise direction, contrary to what is customary in mathematics nowadays. For understanding the planimeter, this direction is not essential. However, we shall here adopt the usual mathematically positive direction, i.e. counterclockwise, particularly because the vector-analytical concepts of circulation and curl, which play an underlying role, are based on this orientation. The positive counterclockwise direction corresponds to the positive orientation expressed in the right-hand screw rule or right-hand rule used in electrical engineering. 9 Simon A. M. van der Salm Amsler’s Polar Planimeter While traversing the right-hand side of the contour C, the measuring wheel continues to rotate forward, causing W to measure an increase in the area swept by the arm system PST. When the tracing arm ST reaches the highest instantaneous position S2T2, the measuring wheel W begins to rotate backward, which is indicated by a yellow double arrow at the tracing arm S2T2 in Figure 7. Then W measures a decrease in area. Figure 7 also shows the situation at the moment when the tracing arm ST reaches the intermediate instantaneous position S3T3. During the traversal of left side of the contour C from T2 to T3, the measuring wheel W rotates backward; the arm system PST has swept out the doubly shaded area, with an extra second set of oblique dotted lines, now from upper right to lower left. This area is subtracted, due to the reverse rotation of the measuring wheel W, from the previously measured, too large area. When, after tracing point T having completely traversed contour C, the tracing arm ST returns to its initial position S1T1. Then instantaneous point S3 coincides with S1 and T3 coincides with T1. The resulting maximum doubly shaded area of the region PS1T1T2S2P is subtracted from the total area that was originally counted by W when the tracing point T followed the right-hand side of the contour C (from T1 via point A to T2). So, only the area of the region R to be measured remains as the difference between the two areas. The result is therefore that the measuring wheel W ultimately determines, up to a scale factor, the area AR of R. A key question that then arises is: why does the measuring wheel W measure the area AR at all (up to a scale factor)? 10 Simon A. M. van der Salm Amsler’s Polar Planimeter 4. MEASURING THE AREA WITH THE WHEEL Figure 8 During the tracing of an infinitesimally small segment T1T2 of the right-hand side of the contour C, the tracing arm ST performs two motions: a rotation about the pivot point S1 and a translation over a distance dh perpendicular to S1T1. The articulated system consisting of the polar arm PS and the tracing arm ST sweeps out an area equal to the sum of the areas of the sectors PS1S2 and S1T1T2’, and of the parallelogram S1T2’T2S2. The small region T1T2’T2T1 outside the contour C that makes (in this case) this sum too large will be ignored in the next calculation. See also in Figure 4, Amsler’s original Figure 2 in [2]. 4.1. Translation and rotation We broadly follow the reasoning of reference [3, pp. 236–239]. In Figure 8, the tracing point T moves over an infinitesimal segment of right-hand side of the contour from point T1 to a neighbouring point T2. The pivot S moves along an infinitesimally small arc S1S2, in Figure 8 approximated by a straight-line segment. The arc radius of that sector equals the length of the polar arm PS. We may regard the motion of tracing point T from T1 to T2 as a rotation of the tracing arm ST about the pivot point S1, followed by a translation of the tracing arm ST over the distance dh, perpendicular to ST. This produces an imaginary auxiliary tracing point T2’, which in this case lies outside the contour C, although it may also lie on or inside it. The area of the region PS1T1T2S2P, that is, the area swept out by the articulated arm system PST, is, in linear approximation, equal to the sum of (1) the area of the sector PS1S2, (2) the area of the sector S1T1T2’, and (3) the area of the parallelogram S1T2’T2S27. 7 Here lies the weakness of this explanation from [3], as well as of many similar variants. Because we make use of differentials, interpreted here as infinitesimal small quantities, various function differences are approximated linearly, and it is tacitly assumed that second- and higher-order terms of these differences are negligible compared to the first-order terms (Taylor expansion). Owing to this simplified line of reasoning, we assume, for example, that the planar region T1T2'T2 outside the contour C (see Figure 8) plays no role of importance. However, it remains questionable whether this assumption is actually justified. 11 Simon A. M. van der Salm Amsler’s Polar Planimeter We notice, (1) the area of the sector PS1S2 is 1 2 PS d ; (2) the area of the sector S1T1T2’ is 2 1 2 ST d ; and (3) the area of the parallelogram S1T2’T2S2 is ST dh , so that, in linear approxima2 tion, the area of the infinitesimally small region PS1T1T2S2P can be approximated by: 1 1 dAR  PS2 d + ST 2 d + STdh 2 2 (1) 4.2. The measuring wheel As with the original Amsler polar planimeter, in Figure 8 a small measuring wheel W is mounted to the left of the pivot point S, on the extension of tracing arm ST. The distance from the measuring wheel W to the pivot point S is denoted by w. As can be noticed in Figure 8 the length of w is constant, but the wheel position changes continuously while tracing point T traverses the contour of the region R. See the wheel positions W1, W2 and W2’. While T tracing the segment of the contour from T1 to T2, the motion of the measuring wheel W can be decomposed into a displacement from W1 to W2', with approximate signed value w  (−d ) , followed by a displacement from W2') to W2, with length dh. If r is the radius of the measuring wheel W and α is the angle traversed by the rotating W, then it follows that: r d = − w d + d h → d h = w d + r d (2) So, the wheel’s motion is decomposed into a translational and rotational contribution, leading to equation (2). 4.3. Amsler’s formula Combination of formulae (1) and (2) , as well as integration yields the following equation (3): 1 1  AR  PS2  d +  ST 2 + wST   d + rST  d 2 2  (3) The polar arm PS doesn’t make a complete revolution, so eind = begin and  d = 0 . The same reasoning applies to the tracing arm ST:  eind =  begin , so  d = 0 . Hence, because neither arm completes a full rotation, the boundary terms of the integrals vanish, resulting in the remarkably simple formula: AR  rST (4) So, Jakob Amsler-Laffon essentially discovered that the area AR of the region R is, to first order, directly proportional to the total angular displacement of the measuring wheel W, after the tracing of the contour C that encloses R. Of course, an obvious objection is that this proof neglects all higher-order (Taylor-) terms. 12 Simon A. M. van der Salm Amsler’s Polar Planimeter 5. A BRILLIANT INSIGHT: THE HIDDEN ROLE OF THE POLAR ARM 5.1. Sweeping the region R It is noteworthy that formula (4) does not include the length of the polar arm PS, but only the length of the tracing arm ST8. Apparently, we could even leave the polar arm PS entirely out of consideration! This is also suggested in Figure 4 with Amsler’s original drawing: there, the polar arm PS is drawn as an insignificant little line pointing downward, while the pole P is entirely absent. Yet the polar arm PS clearly performs an essential function. Against the explanation in the previous section, mathematical objections can be raised (based on linear approximations only), but it has the advantage that it directly refers to the fundamental reasoning that Amsler himself also followed. See again his article [2], as well as the commentaries [1] and [4]. Apparently, those objections are not decisive, since the final result turns out to be correct. Amsler’s brilliant insight is that the area of a plane figure can be determined exactly by sweeping the region R, whose area AR is to be determined, with a single line segment of constant length (in this case the tracing arm ST), which is what the extremely simple formula (4) expresses. Anyone who studies the operation of predecessors of Amsler’s polar planimeter, such as Zobel & Muller’s (approximate) area-measuring machine, Hermann’s conical planimeter (1814), Oppikofer’s conical planimeter (1827), or Maxwell’s design of a spherical planimeter (1855), see [6], all of which are mechanically and mathematically much more complicated than the polar planimeter, cannot help but be impressed by Amsler’s brilliant insight. And one is left wondering how he came up with that idea. 5.2. Straight-line segments A revealing description of the idea implemented by Amsler in his polar planimeter can be found in [6, pp. 134–137]. Amsler came up with the idea of thinking of a region R in the plane as being composed of infinitely many straight-line segments. A subsequent line segment, at an infinitesimally small distance from a previously determined one, is always obtained via a rotation and a translation from that earlier line segment, as we likewise saw in a previous section. See also Figure 4 with Amsler’s original Figure 2. This leads to the problem that the region R can be thought of as being constructed from line segments in infinitely many different ways, and that these segments will moreover differ in length. See, for example, line segments L1 and L2 in Figure 9. Amsler then had the idea that a tracing arm ST of constant length could be used as well, which is also reflected in formula (4). See Figure 10. 8 In the polar planimeter the length of the tracing arm ST can usually be adjusted. At the maximum length ST = STmax, the inaccuracy is approximately 0.1 cm² at scale 1:1, and therefore 0.1a² at scale 1:a. If ST = b STmax, then it follows from formula (4) that the measuring wheel, for the same area AR, indicates a value that is 1/b times as large; consequently, the inaccuracy becomes a factor b smaller. 13 Simon A. M. van der Salm Amsler’s Polar Planimeter Figure 9 (Left) A planar region R constructed from line segments. A measuring wheel W ensures a positive orientation when the contour C of the region R is traversed counterclockwise. From line L1 to line L2, the area measured by W therefore increases. Figure 10 (Right) The area swept out by the tracing arm ST (green from bottom to top (+); red from top to bottom (–)) as the tracing point T traverses the contour C once counterclockwise. In doing so, the pivot point S traverses the contour C’ of an imaginary region R’ once. The yellow arrows indicate the direction of rotation of the measuring wheel, perpendicular to ST. 5.3. The pivot point S Up to this point, we have focused on the tracing point T of ST, but let us now also consider the pivot point S, the other end of the tracing arm ST. While in Figure 10 the tracing point T follows the right-hand side of the contour C, from bottom to top, the tracing arm ST (green) initially sweeps out region R, plus some region between R’ and R, to the left of R. The orientation of the area swept out by ST is positive. The measuring wheel W registers an increase in area as the infinite number of green line segments ST with infinitesimally small mutual distances moves from bottom to top. When the tracing arm ST reaches the top of contour C and the left side of the contour is subsequently traced, the tracing arm (now shown in red) moves downward and the orientation of the measuring wheel becomes negative. The measuring wheel W therefore registers a decrease in area because its direction of rotation has reversed. This ensures that the pivot point S forms a closed contour around an imaginary region R' during one tracing of the contour. The area shaded in both green (+) and red (−) between the region to be measured, R, and the fictitious region R′ is counted once positively and once negatively by the measuring wheel W. The positive and negative contributions cancel each other out, and therefore the doubly shaded area does not appear on the dial connected to W. Because the lines passing through region R are shown in green (+), the area of region R is measured positively. Because the lines passing through the fictitious region R′ are shown in red (−), the area of R′ is measured negatively. Consequently, after contour C has been traced, the measuring wheel W indicate, apart from a scale factor, the area of R minus the area of R′. 5.4. Not a unique construction It should be realized that the construction described above is not unique. There are infinitely many ways to construct Figure 10. For the given region R, neither the fictitious region R′ nor its contour C′ is uniquely determined. The shape of the fictitious region R′ can therefore be chosen arbitrarily. Amsler came up with the brilliant idea of choosing the area of R′ to be zero. He did this by taking R’, and therefore also its contour C′, to be a circular arc centred at the pole P with radius equal to the polar arm PS, and by restricting the pivot S to move only along that circular arc. 14 Simon A. M. van der Salm Amsler’s Polar Planimeter As a result, the shading shown in Figure 10 becomes uniquely defined, the area of R′ equals zero, and the measuring wheel W indicates, apart from a scale factor, the exact area of R , after its boundary C has been traced once. Evidently, nothing more than the articulated arm system PST is required to determine the area of R exactly. 15 Simon A. M. van der Salm Amsler’s Polar Planimeter 6. VECTORIAL INTERPRETATION OF THE POLAR PLANIMETER 6.1. An approach using Green’s Theorem Anyone seeking a mathematical explanation of the operating principle of Amsler’s polar planimeter will find that, in most cases, the device is analysed using Green's theorem. The geometric argument presented above, based on geometry, is somewhat elaborate and tends to obscure Amsler’s original intuitive insight, making it less immediately apparent why the instrument functions as it does. By contrast, the vector-analytic treatment based on Green’s theorem provides a considerably more elegant mathematical framework. However, this approach likewise fails to convey the intuitive geometric understanding of the planimeter’s operation that appears to have motivated Amsler’s original design. In the Green’s theorem approach, the analysis proceeds as follows. Let a Cartesian coordinate  system and corresponding position vector r be introduced as shown in Figure 11. Figure 11 From the pole P=(0,0) extends the polar arm PS, which is free to rotate about P. From the pivot point S=(a,b) extends the tracing arm ST, which is free to rotate about S. The region R is enclosed by the curve C, which is traced in the counterclockwise direction by the tracing point T=(x,y). The point T serves both as the endpoint of the tracing arm ST and as the endpoint of the radius vector originating from the pole P. A measuring wheel W (shown as a yellow line segment), is mounted perpendicular to the tracing arm ST; in the figure, it is located to the left of the pivot point S. The pole is chosen as P=(0,0), so as the origin of a Cartesian coordinate system, typically chosen outside the region R, such that the entire region R is contained within the annular domain bounded by two concentric circles of radii PS – ST and PS + ST, respectively. The coordinates of the pivot point S are denoted by (a,b), while the tracing point T on the contour of region R has coordinates (x,y). During operation, the pivot point S is constrained to move along a circular arc centred at P with radius PS. Figure 12  The unit tangent vector T is tangent to the curve C at the tracing point  T=(x,y). The unit normal vector N is perpendicular to the tracing arm ST at T. The projection of the tangent vector onto the normal vector determines the distance travelled by the measuring wheel W, as the curve C is traversed. Figure 12 shows an arbitrarily chosen tracing point T=(x,y) on the curve C. Also shown are the unit   tangent vector T to contour C, at tracing point T, as well as the unit normal vector N , perpendicular to the tracing arm ST. 16 Simon A. M. van der Salm Amsler’s Polar Planimeter 6.2. Analysis of the measuring wheel Because the measuring wheel W can rotate only in the direction perpendicular to the tracing arm ST, the displacement recorded by the wheel is determined by the projection of the tangent vector  onto the normal vector, that is, by the inner product N  T . See Figure 12. If the tracing point T moves through an infinitesimal arc length ds, then the corresponding distance travelled by the measuring wheel W, in the direction perpendicular to ST, is given by:  N  T ds = N  d r (5) When the tracing point T completes a single traversal of the contour C, the measuring wheel W describes a closed curve CW of length:  N  T ds =  N  d r CW (6) CW In Figure 11, the measuring wheel W is located to the left of the pivot point S on the tracing arm ST extended in the leftward direction. However, it is essentially immaterial where along the tracing arm the measuring wheel is mounted. Figure 13 Two successive positions of the polar arm PS (red), the tracing arm ST (green), and the measuring wheel W(yellow), as the tracing point T, in positive counterclockwise orientation, slides along the curve from point T, via U, to neighbouring point T', thereby traversing an infinitesimally small segment ds along the contour C. In doing so, the measuring wheel W undergoes a combined motion consisting of a translation dl, perpendicular to the tracing arm ST, resulting from the parallel displacement of the tracing arm ST to S'U, and a rotation through an angle dθ about the pivot point S’ ≈ S, arriving at the position W'. If w denotes the distance between S and W, then the infinitesimal displacement ds in (6) can be decomposed into a translational component dl), perpendicular to the tracing arm ST, and a rotational component wdθ, arising from the rotation of ST about the pivot point S. This idea has already been encountered above, but is illustrated more clearly in a slightly different manner in Figure 13. The left-hand integral in (6) can therefore be split into two integrals:  N  T ds =  dl +  wd  CW CW (7) CW Since the pole P lies outside the contour C, the cumulative angular change θ after a single traversal of C) is equal to 0 radians. Consequently, the second integral on the right-hand side of (7) is equal to zero, so that: 17 Simon A. M. van der Salm Amsler’s Polar Planimeter  N  T ds =  dl CW (8) CW Note: The distance w (from wheel W to pivot point S) does not appear in formula (8), and therefore the line integral is independent of the position of the measuring wheel W, on the tracing arm ST. The wheel W may even be placed (in the limiting case) at the tracing point T, in which case the measuring wheel W also traces the contour C. In that ultimate case:  N  T ds =  N  T ds CW C 18 (9) Simon A. M. van der Salm Amsler’s Polar Planimeter 7. A PROOF USING GREEN’S THEOREM  What relationship exists between the length  N  T ds =  N  d r in (6) measured by the wheel CW CW W and the area AR of the region R, enclosed by contour C? Figure 14 Green’s mill in Sneiton, Nottinghem. Source: https://en.wikipedia.org/wiki/Green%27s_Mill, 7.1. Some historical context The physicist and mathematician George Green (1793–1841) was self-taught, the exceptionally gifted son of a miller who had received no more than a few years of primary education. While working in his father’s grain mill, he succeeded in mastering the most advanced mathematics and physics, especially electrodynamics, of his time. See biography [14] and see [16]. No portraits of George Green are known to exist; however, his mill still stands today and now serves as a science centre. See Figure 14. Figure 15 Mathematical physicist and telegraph engineer Oliver Heaviside, like