Reference / Paper · 2025
A Treatise on Planimeters: Theory, History and Different Varieties (Frost, 2025)
Lecture slides from the Oughtred Society's 2025 international meeting: the history, theory and many varieties of planimeter.
Reference / Paper · 2025
Lecture slides from the Oughtred Society's 2025 international meeting: the history, theory and many varieties of planimeter.
Planimeters 1 The mechanical measurement of Areas, Moments of Inertia, and the like. Planimeter, n.: An instrument with a movable pointer used for mechanical measurement of the area of an (irregular) plane figure. (oed.com, 2013) Integrator, n: One who or that which integrates; spec. an instrument for indicating or registering the total amount or mean value of some physical quantity, as area, temperature, etc. A = ò dA A Several mechanical gadgets were invented in the 1800’s. The most successful was invented by Swiss mathematician Jakob Amsler in 1854. He founded a company to manufacture the planimeter, and then invented gadgets to determine more complicated integrals. Computing before Electronic Calculators Thayer School of Engineering, 2024F © H.J. Frost, 2018-2024 Brute Force Measurement of Area 2 Counting Squares: Area A 7 8 1 9 2 . 3 . 4 . 5 6 dA The more laborious, the more accurate. More conveniently, Counting Dots Area A 5 6 7 1 8 2 9 3 . 4 . . dA Even more conveniently, dots with a coarser grid. A few different papers in the early 20th Century have presented the idea of counting dots on a transparent overlay. AreaArea A by Measuring Line Segments 5 dx Li dA A = å Li dx i Linear Scale for Area Measurement 6 (Cooke, Troughton & Simms, area measuring scale) A grid of straight lines on a transparent (e.g. celluloid) sheet is placed over the area to be measured. The scale is then placed successively along each of the lines. For each line the hairline marker is moved across the area to add successive increments of length into a running sum. (photo: www.mathsinstruments.me.uk/page 34.html) Scale for Measuring Area in Surveyor’s Units 7 Designed for units of rods, roods and acres. 1 rod = 16.5 feet # roods # acres # square rods 4 roods = 1 acre 40 square rods = 1 rood This ruler with its fixed scale is designed for one particular scale of map. (sketch from W. F. Stanley, Mathematical Drawing Instruments, e.g. 8th ed. 1925, pp. 247-251) Hair Planimeter designed by A. Beuviére 8 Maker: Niggl, Paris (1855) Musée des Arts et Métiers – CNAM, Paris Figure is placed under the scribed glass slide; Carriage with measuring disc is moved repeated over the figure. https://irem.univ-reunion.fr/calculsavant/Exposition/ArtsetMetiers/06710-0000-e.html 9 Hair Planimeter Another view. system Beuvière, 1844. Maker Niggl, Paris, Inv. 06245-0000. Musée des Arts et Métiers - CNAM, Paris, photo M. Favareille Modern Measurement of Area 10 (by software) https://apps.apple.com/us/app/planimeter-measure-land-area/id423492040#?platform=ipad DL 5-23-2021 Area as the integral of rectangular increments 11 Recorded increments of area must have magnitude proportional to y as we scan over x. Early designs recorded the integral on a measuring dial on which an indicator pointer moved according to y times increments of x. The idea was to make a linkage between the y value and the rate of dial rotation. Early designs used a cone or a disk for this purpose. Several Inventors Tackled the Problem 12 • Johann Martin Hermann (1814) Bavarian Land Surveyor–never published–created for his own use. No example survives, but we have his design figure. * Tito Gonnella (1824) Prof. at Univ. of Florence. Published – did not commercialize. Hermann Wetli-Starke • Johannes Oppikofer (1827) Swiss Inventor. Mfg. in France by Ernst (1836+); then Claire. * Kaspar Wetli (1848) Swiss Engineer – Austrian patent - replaced cone with disc. Mfg. by Georg Christoph Starke of Vienna. * John Sang (1851) Sang Scottish Inventor. Cone based rolling “platometer” . * Hermann Ausfeld (1851) • Jakob Amsler (1854) Swiss Professor of Mathematics. Founded Amsler & Son. • James Clerk Maxwell (1855) Ausfeld Scottish Mathematical Physicist. *Displayed at the Crystal Palace Exhibition, Kensington, London,1851. Earliest Planimeters were Complicated Gadgets 13 Cone planimeter by Johannes Oppikofer, Germany, and Ernst, Paris, 1827 - 1835 H x y p Position of wheel G on cone C depends on y value as given by position of pointer p. Dial H records the rotation of wheel G. Sang Planimeter of 1851 14 Another complicated Cone-Based design K KD D R x y A Position of wheel R on cone K depends on y value as given by position of pointer A. Dial D records the rotation of wheel R. (Source of photograph uncertain.) Wheel on Disk Planimeter 15 Wetli-Starke Planimeter, patented 1849. Tracer point Disc rotates as rod extends in the x-direction. Readout Dial x As the disc moves, due to the ± y motion of the carriage, the gear ratio between the disc and the takeup wheel changes. y Carriage moves in the y direction, carrying the rotating disc. Photograph from Technik Museums der Technischen Universität Delft. http://www.history.didaktik.mathematik.uni-wuerzburg.de/ausstell/planimet/starke.html (2017) See article by Charles Care, A Chronology of Analog Computing, The Rutherford Journal – The New Zealand Journal for the History of Science and Technology Disk M moves along track S as the pointer B moves in the Y direction. Early Planimeters 16 Wetli-Starke Planimeter, patented 1849. Disk M rotates as pointer B moves in X direction because gear w (hidden underneath M) connects the pointer arm L to disk M. Wheel m rotates at a rate proportional to the Y distance of pointer B and proportional to the X speed of point B. Mechanical Integrators including the various forms of Planimeters, by Prof. Henry S. H. Shaw, D. Van Nostrand, 1886. Another Early Planimeter Invented by James Clerk Maxwell in 1855 while he was at Trinity College, Cambridge. (The Maxwell of Maxwell's Equations of electromagnetism, etc., etc.) Maxwell had not yet heard of Amsler’s invention, and abandoned the project after learning about Amsler's simpler and more practical device. from website of Prof. Robert L. Foote, Wabash College, Crawfordsville, Indiana http://persweb.wabash.edu/facstaff/footer/Planimeter/PreAmsler.htm 17 Amsler "Polar" Planimeter of 1854 Hinge 18 Indicator Dial Measuring Arm Fixed Pole Tracer Two new features: • Indicator Dial that rolls in one direction activated by rolling over the paper and slips in the other. • Linkage between the tracer point at the end of the measuring arm and a fixed pole, held in place by a needle point, forcing the hinge to travel along an arc of a circle. 19 Early Amsler Planimeter Serial Nr. 4107, circa 1866. J. Amsler, Uber das Polar-Planimeter Dinglers polytechnisches Journal, 1856, Band 140, Nr. LXXIII, (S. 321-327) 20 Tracer Pivot Allbrit Linear Rolling Planimeter This device dates from the early 20th century. The rolling/sliding dial mechanism this planimeter uses is derived from Amsler's mechanism, invented half a century earlier. It is shown here because its operation is simple to understand. (photograph courtesy of Douglas Fraser, 2017) Dial Wheel with Rolling & Sliding as Recording Mechanism. 21 The first novel feature of Amsler's design was the use of a readout by a measuring wheel that rolls and slides as it moves while a pointer traces the outline of the shape to be measured. In the linear geometry it looks like this: y = L sin a Recording x sin is the same as recording (y/L) x = ∆A/L dial roller slide Vernier scale counter Reading shown = 3.584 Vernier Scale 22 Simple method to reach an extra digit of precision. Invented by French Mathematician Pierre Vernier (1580-1637) dial roller Top View slide Vernier scale counter Reading shown = 3.584 10 increments left hand scale match 9 increments right hand scale. Hidden below the counter disc is a spiral thread on the axle for the dial roller that advances the counter by one unit for each full rotation of the dial roller. (Carpenter &Diederichs, Experimental Engineering and Manual for Testing, Fig. 5) Simple Planimeter with rolling & sliding dial wheel. y = L sin a Recording ∆x sin is the same as recording (y/L) ∆x = ∆A/L 23 24 Other Linear Planimeter mechanisms to measure increments of y · x ("Linear" in the sense that on the rollers the mechanism moves along a straight line.) Recording on a rotating disk. Recording on a rotating spherical cap. Coradi Rolling Planimeters (sketches from Keuffel & Esser catalog) 25 Coradi Rolling Sphere Planimeter Linear planimeter mechanism to measure increments of y · x Recording through a spherical cap geared to rotate with the rotation of the roller wheels. Recording Cylinder Roller Wheels Spherical Cap (Keuffel & Esser catalog) Computing Gadgets for the Late Victorian Engineering School Prof. Robert Fletcher Coradi Rolling Planimeter Sextant 26 Thacher's Calculating Instrument (Photograph from Dartmouth College Archives) 27 Thayer School's Rolling Sphere Planimeter Coradi Rolling Disc Planimeter 28 Linear planimeter mechanism to measure increments of y · x Recording on a rotating disc that is geared to rotate with the roller wheels. The recording wheel slips and rolls on the rotating disc according to its location that changes (from Keuffel & Esser catalog) according to the angle of the tracing arm. Roller Wheel Rotating Disc Recording Wheel Tracing Point (Planimeter was sold in the U.S. as K&E model 4262; photo from smgco-images.s3.amazonaws.com/media/W/P/A/large_1912_0201_0001_.jpg) Rotating Disc Measurement Mechanism 29 DA = LDxsina L = Length of Pointer Arm Dq = kDx k = gearing constant for disc rotation Rotating Disc Measurement Geometry 30 y x R = r 2 + s 2 - 2rs cosa (Law of Cosines) ∆a = increment of measuring wheel travel. sin a sin g (Law of Sines) R = s Polar Planimeters 31 The second novel feature of Amsler’s invention the simple linkage of two arms hinged together: The rolling/sliding measurement system works for any shape of the track the pivoting end of the measuring arm must follow. Hinge Measuring Arm Fixed Pole Tracer Jakob Amsler-Laffon (1823 – 1912) Swiss Mathematician and Industrialist Inventor of the Polar Planimeter Animation: http://persweb.wabash.edu/facstaff/footer/Planimeter/ Polar&Linear.htm; by Prof. Robert Foote Figure from Keuffel & Esser catalog, 1908, shown in Wikipedia, Planimeter, (2017). J. Amsler Über die mechanische Bestimmung des Flächeninhaltes,..insbesondere über einen neuen Planimeter 1856 (Portrait: Wikipedia, Jakob Amsler-Laffon, 2017) Amsler Polar Planimeters 32 (http://www.history.didaktik.mathematik.uni-wuerzburg.de/ausstell/planimet/amsler_7139.html) (Wikipedia, Planimeter, Creative Commons, creator Nol Aders) The planimeters became a standard office instrument. U.S. Dept. of Commerce, Bureau of the Census, 1940-41 From the National Archives. https://www.flickr.com/photos/usnationalarchives/6789717264/ Operation of the Polar Planimeter 34 Decompose the area traversed by the measuring arm into • area increments due to . • area increments due to . dA = 12 L2df + Ldn Distance measured by the roller: ds = dn + Kdf dn = ds - Kdf dA = 12 L2df + Lds - LKdf (The increments give no net contribution.) Area Difference Theorem 35 A moving line segment sweeps out area, called positive as the line moves in the direction of its normal vector N, and negative if it moves in the opposite direction. AR AL Positive area is shown in blue, negative area in red. The left-hand region (of area AL) is swept in the negative direction. The right-hand region (of area AR) is swept in the positive direction. The region between the two areas is swept both positive and negative directions. The total signed area swept out by the line segment is AR – AL . For a planimeter the left edge of the line moves back and forth along a fixed track: ⇒ AL = 0. Animation by Prof. Robert Foote from http://persweb.wabash.edu/facstaff/footer/Planimeter/AreaDiffThm.htm Area Difference Theorem 36 A moving line segment sweeps out area, called positive as the line moves in the direction of its normal vector N, and negative if it moves in the opposite direction. Constrained track of the other end of the measuring arm. AL Tracer Point + [+] - AR Area to be measured Positive area is shown in blue, negative area in red. The left-hand region (of area AL) is swept in the negative direction. The right-hand region (of area AR) is swept in the positive direction. The region between the two areas is swept both positive and negative directions. The total signed area swept out by the line segment is AR – AL . For a planimeter the left edge of the line moves back and forth along a fixed track: ⇒ AL = 0. Animation from Prof. Robert Foote http://persweb.wabash.edu/facstaff/footer/Planimeter/AreaDiffThm.htm Green's Theorem for Planimeters 37 The validity of the planimeter may be verified by applying Green's Theorem*. Consider a region D surrounded by closed curve C . Let P(x,y) and Q(x,y) be functions with continuous first derivatives over D. (We may think of (P,Q) as a vector field.) ¶Q ¶P If we pick P and Q such that: = 1 then the integral will give the area, A. ¶x ¶y P and Q must correspond to the accumulated measurement of the measuring wheel as we circuit the perimeter. Direction of rolling is perpendicular to the measuring arm. Measuring arm spans from point (a,b) to point on perimeter (x,y). Vector of the measuring arm length is [x-a, y-b]. Perpendicular direction that wheel will roll: [-(y-b), (x-a)]. Moving the tracer by an increment [x, y] increments the rolling by the projection of (x, y) in the rolling direction. * George Green (1793 – 1841), English Baker, Miller and Mathematician. Green's Theorem for Sliding/Rolling-Wheel Planimeters 38 The validity of the planimeter may be verified by applying Green's Theorem*. (We may think of (P,Q) as a vector field.) ¶Q ¶P If we pick P and Q such that: = 1 then the integral will give the area, A. ¶x ¶y P and Q must correspond to the accumulated measurement of the measuring wheel as we circuit the perimeter. Direction of rolling is perpendicular to the measuring arm. Measuring arm spans from point (a,b) to point on perimeter (x,y). Vector of the measuring arm length is [x-a, y-b]. Perpendicular direction that wheel will roll: [-(y-b), (x-a)]. 1 Unit Vector in the rolling direction: L éë- ( y - b) , ( x - a)ùû Moving the tracer by an increment [x, y] increments the rolling by the projection of [ x, y] in the rolling direction: (Vector dot product) 1é ë- ( y - b) × d x + ( x - a) × d yùû L The scale on the rolling wheel readout is adjusted to compensate for the factor (1/L), so it reads: Green's Theorem for the Linear Planimeter 39 Pick P and Q such that: ¶Q ¶P =1 ¶x ¶y For the linear planimeter, we use: P ( x, y) = -y ; Q ( x, y) = x - a (Special Case of b = 0.) ¶Q ¶P ¶( x - a) ¶( -y) ¶x ¶a ¶y ¶a ¶a = = - + = 1- +1 = 2 ¶x ¶y ¶x ¶y ¶x ¶x ¶y ¶x ¶x 2 Constraint: ( x - a) + y2 = L2 ; æ ¶a ö Þ ç ÷ =1 è ¶x øy ( x - a) = ( L2 - y2 2 2 a = x L y ; ( ) ) ¶Q ¶P ¶a Þ = 2 - = 2 -1 = 1 ¶x ¶y ¶x Green's Theorem for the Polar Planimeter 40 Pick P(x,y) and Q(x,y) to match the operation of the measuring wheel, and then show that: ( ) ( ¶Q ¶P = 1 (or a constant), meaning that we measure area. ¶x ¶y Use: P x, y = - y - b ); Q ( x, y) = x - a Note that a and b, coordinates of the hinge location, are not constants, but depend on x and y. 2 2 x a + y b = L2 ( ) ( ) Constraints: a + b = M ; 2 2 2 (Point (a,b) is constrained to a circle of radius M.) ¶a ¶b x + y = x-a ¶x ¶x ¶a ¶b x + y = y-a ¶y ¶y Solving 4 ¶b a ( x - a ) ¶a -b ( x - a) ¶a -b ( y - b) ¶b a ( y - a ) = = = equations for = ; ; ; ¶x ay - bx ¶y ay - bx ay - bx ¶y ay - bx 4 derivatives: ¶x From these: ¶a ¶b ¶ a + b =0 ; : ¶x ¶x ¶x ¶a ¶b ¶ : a +b = 0 ; ¶y ¶y ¶y é -b ( x - a) a ( y - b) ù é ay - bx ù ¶Q ¶P ¶( x - a) ¶éë- ( y - b)ùû ¶a ¶b = = 2 - - = 2 -ê + ú = 2 -ê ú=1 ¶x ¶y ¶x ¶y ¶x ¶y ay - bx û ë ay - bx û ë ay - bx Ordinary Vintage Planimeters 41 Keuffel & Esser Model 4240 Early 1950’s Adjustable arm length allows for different scales Calibration Device. Beam pinned at one end allows the tracer point to be taken around a circle of known radius/area. Collection of Clark McCoy http://mccoys-kecatalogs.com/PlanimeterModels/PlanimeterModels.htm Another Vintage Planimeter Keuffel & Esser Model 620022 1962 – 1972 Collection of Clark McCoy http://mccoys-kecatalogs.com/PlanimeterModels/PlanimeterModels.htm 42 Modern Digital Planimeters Might possibly be purchased new. (Available at e.g. https://tamaya-technics.com/en/planix/) 43 Sources of Error • Measuring wheel slippage. • Incorrect arm length. • Incorrect measuring wheel diameter. • Misalignment of measuring-wheel axis. ⇒ “Compensating” polar planimeter. 44 Planimeter Application for Steam Engine Testing The power is energy/stroke ´ strokes/unit time. The energy is given in terms of p dV ò p = Steam Pressure. V = Volume of Steam Cylinder. (given by distance of stroke) Recording pen Rotating cylinder holds the recording card. Connection to Pressure 45 Using a Steam Engine Indicator The Steam Engine by John Perry, 1902, McMillan and Co., on Intl. Slide Rule Museum website. Planimeter Models specific for Steam Engine Testing 47 Coffin Averaging Instrument A linear planimeter P×V record Tracer Point Groove Weight Post (under weight) slides in groove. Measuring Wheel (Carpenter & Diederichs, Experimental Engineering and Manual for Testing, Fig. 13) The Willis Planimeter 48 Disc W slides along the scale as the tracer point follows the perimeter. Fixed Pole Tracer point The Willis planimeter reverses the roles of rolling and sliding. (Carpenter & Diederichs, Experimental Engineering and Manual for Testing, Fig. 15) Willis Planimeter and Lippincott Planimeter Reverse the roles of sliding and rolling: Sliding is used for the measurement; Rolling is neglected. Willis Lippincott From “American Planimeters” by Bob Otnes, J. Oughtred Society, Vol. 11, No. 2, Fall 2002, pp. 59-64 49 Hatchet Planimeter 50 Invented in ~1875 by Danish Cavalry Officer Holgar Prytz Prytz looked for a very cheap alternative to the well-known Amsler planimeter. (C) (B) A ≈ L· • Trace around the shape with the pointer B. • Let the blade (C) follow whatever path it chooses. • Measure the distance moved by the blade during the circuit (). • Compute the area as A ≈ L· Circuit sketch from Robert Foote, The Geometry of the Prytz Planimeter, Hatchet Planimeter in action 51 A ≈ L· Animations from Robert Foote, The Geometry of the Prytz Planimeter, Variety of Hatchet Planimeter Designs (U.S.A.) 52 John Goodman (1891) Ernest Kilburn Scott (1896) G. Coradi (1895) Rudolf Schierbeck (1904) Leslie Haywood Hounsfield (1932) Luis Larrazabal y Fernandez (1926) Radial Planimeters 53 To measure the average distance from the center (pole, P) on circular charts. Paper disc printed for 24 hour duration. A knob at pole P fits into a groove in the underside of tracer arm A, so that the tracer point T moves easily towards or away from the pole as it makes its circuit. (sketch taken from the Keuffel & Esser catalog, 36 th Edition, 1921, p. 252; https://archive.org/details/CatalogueOfKeuffelAndEsserCo36Edition1921/page/n289/mode/2up) Square-Root Planimeters and Flow Meters 54 Flow meters often measure the pressure difference (∆p = p1 – p2) across an orifice. qv = CA2 2 ( p1 - p2 ) r Volumetric flow rate (m3/s): C = orifice flow coefficient, dimensionless. A2 = cross-sectional area of the orifice hole, m2. = density, kg/m3. T. Leise (2007). As the planimeter’s wheel rolls, The College Mathematics Journal 38(1):24-31. 55 Daily Chart for Pressure Difference Record Reference: book on flow meter engineering Daily Chart for Pressure Difference Record Center Post Reference: book on flow meter engineering Following Tanya Leise, As the Planimeter’s Wheel Turns… 56 Operation on a Circular Chart 57 Use chevron brackets notation for vectors: x, y º xi + yj c(q ) = ( f (q ) + a) cosq ,sinq c'(q ) = ( f (q ) + a) -sinq ,cosq + df cosq ,sinq dq unit vector for wheel rolling direction: w(q ) = cos (q + b ) ,sin (q + b ) The increment recorded by the wheel rolling during an increment of tracing along the curve: é ù df w(q )× c'(q )dq = cos (q + b ) ,sin (q + b ) × ê( f (q ) + a) -sinq ,cosq + cosq ,sinq ú dq ë û dq é ù df w(q )× c'(q )dq = ê( f (q ) + a) sin b + cos b ú dq ë û dq Following Tanya Leise, As the Planimeter’s Wheel Turns… Average Flow Calculation 2p f (0)+a 0 f (2 p )+a Total Wheel Roll: W = ò w(q )× c'(q )dq + The first integral has two terms: ò 2p 58 cos b (r)dr 2p df cos b dq 0 0 dq The second term cancels the second integral: ò ( f (q ) + a) sin b dq + ò 2p f (0)+a df ò cos b (r)dr = - ò cos b dq f (2 p )+a 0 dq Leaving: 2p W = ò ( f (q ) + a) sin b dq 0 2p To get our planimeter to calculate ò 0 we need: r sin b ( r ) = r - a ; f (q ) dq æ r-a ö b ( r ) = arcsin ç ÷ è r ø Radial Track Planimeters 59 p=1 p = 1/2 æ r-a p ö ( ) ÷ b ( r ) = arcsin ç ç r ÷ è ø p = 1/2 p=1 From Tanya Leise, As the Planimeter’s Wheel Turns… Amsler’s Radial Square-Root Planimeter 60 ( An adequate description of how this gadget works is not yet available.) Photograph from: http://www.mathinstruments.ch/en/gallery/inv-3013.html Linear Square-Root Planimeter 61 by Geo Kent Ltd. London & Luton. Planimeter intended to run along a rail that is not shown. A Geo Kent ‘Venturi’ water flow rate meter in Kew Bridge Steam Museum. Planimeter and flow meter as shown on website http://www.mathsinstruments.me.uk/page38.html Foxboro-Style Radial Planimeter Tracer Point Rotate Circular Chart Post (Figures from U.S. Patent 1,650,490, Filed Sept. 13, 1926, by W. C. Brown and L. K. Spink) Linear Scale Foxboro Radial Planimeters Square-Root Scale 63 From “American Planimeters” by Bob Otnes, J. Oughtred Society, Vol. 11, No. 2, Fall 2002, pp. 59-64 American Meter Co. - Square Root Planimeter 64 A description of this instrument is given in “American Planimeters” by Bob Otnes, J. Oughtred Society, Vol. 11, No. 2, Fall 2002, pp. 59-64 (photographs from eBay listing, Nov. 2018) Summary for Planimeters 65 • Early Planimeter designs, such as Cone and Disc Mechanisms. • Jacob Amsler’s invention of the rolling + sliding wheel. • Operation of the Linear Planimeter. • Mathematics of the Polar Planimeter. • Steam Engine Indicators. • Prytz (Hatchet) Planimeters. • Flow Meters and Square Root Planimeters. Next Lecture: Amsler Integrators and other derivative gadgets.