Indexing the archive…
Your Universe of Digital Possibilities
Trace any boundary with a needle and a small wheel on the arm reads the area inside it. Jakob Amsler’s polar planimeter (1854) has no computer and no coordinates — only two hinged arms, a pole pinned to the paper and a wheel that can roll only sideways. It worked for surveyors, engineers and doctors for a century, and it works because of a theorem George Green had published in 1828. Draw a shape, trace it, and audit the brass against the shoelace.
The area inside a closed curve is a sum taken only along the curve. Walked counter-clockwise it is positive, clockwise negative; a figure-eight returns the difference of its lobes. On a polygon the sum is the shoelace formula.
The wheel on the tracer arm rolls only with the part of its motion perpendicular to the arm. Summed round a closed trace, that sideways roll ℓ times the arm length b is the area the tracer enclosed — so long as the pole arm swung out and came back without going round.
With the pole inside the outline both arms make a full turn, and the wheel misses the area of one special circle — the one the tracer could ride forever without the wheel turning. Here a is the pole arm, b the tracer arm and c the wheel’s place along the tracer arm from the hinge; a wheel set behind the hinge makes c negative.
The third Heron machine computes an integral with no symbols in it. Green’s theorem turns the area inside a curve into a sum taken along it; Amsler’s arms turn that sum into the roll of a wheel. The bench measures the roll from the arms’ own motion and the auditor checks it against the shoelace — residual at the float floor, or the honest size of the slip. The same wheel-on-a-surface idea, ganged into shafts, becomes Thomson’s integrator in The Integrator; the curves of constant width whose areas the gauge in The Roller compares are exactly what this wheel can read; and the boundary of The Set is one no needle could ever finish tracing.