The Planimeter Bench

¶Commentary

Green's theorem. The area inside a closed curve can be read off the curve alone: A = ∮x dy = ½∮(x dy − y dx). Every instrument on this bench is a machine for one form of that integral. The tracer supplies the curve, and something that rolls, turns, or counts does the adding.

Why A = b·w. As the tracer arm moves, it sweeps area. If its elbow end moves sideways by dn and the arm turns by dφ, it sweeps b·dn + ½b²dφ. The wheel, at distance c from the elbow, rolls dw = dn + c·dφ. So the swept area is b·dw + b(½b − c)dφ. Round a closed curve the arm comes back to its old angle, so ∮dφ = 0, and the elbow, swinging back and forth on its arc, encloses nothing. What is left is A = b·w.

The pole inside. If the pole lies inside the curve, the elbow goes once round its circle of radius a and the arm turns once round, ∮dφ = 2π. The same bookkeeping gives A = b·w + πa² + 2πb(½b − c) = b·w + π(a² + b² − 2bc). The constant π(a² + b² − 2bc) is the area of the instrument's "zero circle", and it is usually given with the instrument.

The linear planimeter. Here the elbow runs on a straight line. It encloses no area, the arm returns to its angle, and A = b·w with no constant. Makers such as Coradi built them to run on a pair of heavy rollers instead of a rail, so that a long strip could be measured in one pass.

The hatchet. Prytz's rod is a different kind of machine. Its knife edge cannot move sideways, so the rod obeys a rule about direction rather than position, and a closed trip of the point need not bring the rod back to its old direction. The leftover angle is a holonomy, like the turn of a pendulum carried round the Earth. For a curve small compared with L, starting near its centroid, θ ≈ A/L², which is the rule A ≈ L²θ. The error falls as L grows, and grows as S moves off the centroid.

The radial planimeter. In polar coordinates about any centre, Green's theorem reads A = ½∮r² dθ. The instrument of Plate IV forms this by a disk-and-wheel integrator, of the kind James Thomson described and Kelvin used in his harmonic analyser, and Bush later in his differential analyser: a wheel resting on a disk at distance ρ from its centre rolls ρ dθ as the disk turns by dθ against it. A cam makes ρ = r²/2k, so the wheel's total is A/k. The design here is drawn to show the principle; I do not know of a maker who sold one in this form.

The counting frame. A cross-slide reads x and y separately, as a plotting table or a digitizer does. The area is then a sum, Σ x̄·Δy, taken step by step with the average x of each step. The only intrinsic error is the size of a count; the errors of the individual counts partly cancel round the curve.

Where error comes from. Tremor is the hand wandering off the line; it adds and removes slivers, which nearly cancel. Slip is the wheel skidding by a small random fraction at each step; for the hatchet it is the knife drifting sideways. Reading is the resolution of the dial: a vernier reads a thousandth of a turn, the dial alone a hundredth. Intrinsic is what the instrument gets wrong with a perfect hand: almost nothing for the exact instruments, and the whole L²θ approximation for the hatchet. The budget in each margin turns on one source at a time, with the same random draws.

Units. The sheet is 25 by 17.5 cm, ruled in centimetres. Every measuring wheel is 2 cm across, so one turn is 2π cm of rolling. The true area is computed from the curve by the shoelace formula.

Further. Amsler's own account of the polar planimeter (Zürich, 1856); the chapter on planimeters in Bryant and Sangwin, How Round Is Your Circle? (2008); and Foote, Levi, and Tabachnikov, "Tractrices, bicycle tire tracks, hatchet planimeters, and a 100-year-old conjecture", American Mathematical Monthly (2013), on the geometry of Prytz's hatchet.