Amsler & Sons · Instrument Makers · Schaffhausen, est. 1854
Trace the edge of anything. The wheel will do the calculus.
The polar planimeter is Green's theorem cast in steel: two arms, a rolling wheel, and a dial that integrates while your hand goes around. The instrument below is in working order — wind it round a specimen plate, or draw a figure of your own, and compare the wheel against the arithmetic.
Clockwork trace
…or press on the paper and trace any figure clockwise, back to where you began.
Wheel
0.000rev
Reading — L × roll
0.00cm2
Exact — coordinates
—cm2
Deviation
—
Fit for survey · A&S
The instrument is at rest. The wheel reads zero.
Day book — measurements taken at this bench
| № | Figure | Wheel, cm² | Exact, cm² | Deviation |
|---|---|---|---|---|
| No measurements yet. The ledger waits. | ||||
The method · Green's theorem, 1854
How a wheel learns area
No gears compute. Nothing stores a number. The whole theorem lives in where the wheel is allowed to roll and where it is forced to skid.
01
The linkage
One arm is anchored to a pole weighted off the sheet; the second carries the tracer. Wherever the tracer goes, the elbow between them settles by geometry alone. The instrument never knows where it is — only how it is bent.
02
The wheel
A knife-edged wheel rides the tracer arm. Motion square to the arm rolls it; motion along the arm merely skids it, and a skid records nothing. Of all your hand's travel, the wheel keeps exactly one component.
03
The cancellation
Follow any closed circuit and return. Every roll the wheel picked up on the way out, it gives back on the way home — except the part owed to the area enclosed. Divide the surplus by the arm's length, and the figure's area sits on the dial.
Trace anticlockwise and the wheel runs backwards — the dial reads negative, the magnitude still true. Cross your own path and the instrument counts the doubly-wound region twice. That is not a fault. That is exactly what the theorem says it must do.
Certificate of adjustment
Tested against figures of known area
Every instrument leaves the bench traced around figures whose areas are known to the arithmetic. Yours was tested in this very window, the moment the page finished loading.
Instrument № 44 · polar pattern
Bench test — this session
| Figure | Perimeter run | Exact, cm² | Wheel reads, cm² | Deviation |
|---|---|---|---|---|
| Square of six centimetres | — | — | — | — |
| Circle of four radius | — | — | — | — |
| Sinuous figure, splined | — | — | — | — |
In adjustment · A&S
From the pattern book · 1889
Three instruments, one theorem
№ 2
Polar Planimeter
The pattern Jakob drew in 1854 and never much improved, because it didn't need it. Steel arms, hardened wheel, dovetailed boxwood case. For surveyors, actuaries, and anyone who must put a number to a lake.
145 francs · wheel to 1⁄1000 rev
№ 4
Compensating Planimeter
Traces with the elbow thrown either way; take both readings and the mounting error averages out to nothing. The choice of cadastral offices from Bern to Bombay.
210 francs · double reading
№ 6
Rolling Planimeter
For indicator diagrams and soundings by the fathom: the pole exchanged for a heavy roller, unlimited travel along the strip, the same wheel arithmetic across it.
260 francs · unlimited travel