Planimeter

Earn the instrument by suffering the alternative.

1. Draw an irregular closed shape on graph paper.
2. Count the whole squares inside; estimate the partial ones.
3. Record your area and how long it took.

One wheel with one very particular property.

1. Mount a small wheel so its axle is fixed rigidly to an arm.
2. Roll the arm along the wheel's rolling direction — the wheel turns.
3. Slide the arm SIDEWAYS, along the axle direction — the wheel skids and does not turn.

Two arms: one anchored to the table, one carrying the tracer.

1. Pin a pole arm to a fixed point on the paper so it can only pivot.
2. Hinge a tracer arm to its far end, with the measuring wheel on it.
3. Put the tracing point at the end of the tracer arm.

Calibrate on a shape whose area you already know.

1. Zero the wheel and trace a rectangle of known area, clockwise, returning exactly to the start.
2. Note the wheel reading and compute area per unit reading.
3. Now trace your irregular shape and convert.
4. Trace it the other way round and note the sign reverses.

The result depends on area alone, and you can prove it.

1. Trace a shape and record the reading.
2. Trace a completely different shape of the same area — a long thin rectangle against a square.
3. Compare the readings.

Jacob Amsler (1823-1912) was a Swiss mathematician who gave up an academic career to make instruments in Schaffhausen. Earlier planimeters existed — Johann Martin Hermann's from 1814, and others by Gonnella and Maxwell — but they were expensive and awkward. Amsler's polar planimeter of 1854 was simple, small, and cheap enough to be standard equipment. His workshop reportedly made tens of thousands of them.

The killer application was the steam engine. An indicator diagram plots pressure inside a cylinder against piston position through a cycle, and the AREA of that closed loop is the work done per cycle. Every engine being tested produced these diagrams, and every one needed its area measured. A planimeter turned an afternoon of square-counting into a minute's tracing, and it did it more accurately.

What it demonstrates about mathematics is more interesting than what it measures. Green's theorem relates an integral over a region to an integral around its boundary. That sounds abstract, and here it is a small brass linkage that a surveyor used without knowing the theorem's name. The same boundary-to-interior relationship underlies Stokes' theorem, the divergence theorem, and a great deal of physics.

Superseded but not obsolete: digitising tablets and image analysis do this work now. Planimeters are still used where a physical drawing must be measured without being digitised, and they remain the clearest teaching demonstration of an integral theorem that exists.