Amsler & Sons · Build notes · Instrument № 44
Build notes for Amsler & Sons — Instruments of Measure, a fictional Schaffhausen instrument house after the real Jakob Amsler-Laffon. The page has one job: put a working polar planimeter under your hand and let you watch a Victorian mechanism integrate the area of a figure — correctly, by Green's theorem — because you moved the tracer around its edge.
A polar planimeter is two hinged arms, a knife-edged wheel, and a counting dial. Wind the tracer once around any closed figure and the dial reads its area — no formula entered, nothing sampled. The site is that instrument, faithfully simulated on a single canvas: trace the specimen plates by clockwork, or press on the paper and draw a figure of your own. Every reading is computed live and set beside the exact coordinate area, so the wheel is always caught being right.
Argued entirely from the object: a lacquered case, its steel arms, the verdigris of aged brass, and the cream specimen sheet it rests on. No accent that a real bench wouldn't own.
Trace the edge
Spectral — a quiet, slightly bookish serif set light. It carries the prose and the italics the way an instrument-maker's catalogue would; the display line runs at 300 weight so the size does the work, not the ink.
28.27 cm²
Spline Sans Mono — the readout face. Tabular figures keep the dial, the day book and the calibration certificate in tidy columns, and its uppercase tracking does the engraved labels on the specimen sheet.
The instrument is a canvas 2D scene at a fixed 1120 × 660 design space, scaled to
the container and DPR-capped at 2. The linkage is solved by geometry: the pole
is fixed off the sheet, and for any tracer position the elbow is the intersection of two circles
(pole-arm radius R, tracer-arm radius L) — so the arms bend exactly as
steel would, and the reach stops glow when you hit the instrument's limit.
The signature: the measuring wheel does the calculus. As you move the tracer,
the engine subdivides the motion into sub-pixel steps and, at each one, projects the wheel's
displacement onto the normal of the tracer arm — the component square to the
arm rolls the wheel; the component along it merely skids and records nothing. Accumulate that net
roll s around a closed loop and the enclosed area is simply area = L × s.
That is Green's theorem made mechanical: the line integral the wheel keeps is exactly
∮ x dy. Alongside it the engine also keeps the honest shoelace
area of the traced polygon, so the panel can show wheel-versus-arithmetic and a deviation to two
decimals — and on load, three figures of known area are traced invisibly to print a calibration
certificate that agrees to four decimal places.
The rest is drawn, never photographed: the knurled pole, the vernier drum and rim graduations
that turn with the accumulated roll, the counting dial's needle, and a tracer lens that truly
magnifies the specimen sheet beneath it by re-drawing the paper at 1.9× inside a clip.
No images, no libraries — one self-contained HTML file, Google Fonts aside.
Trace anticlockwise and the wheel runs backwards; the reading goes negative but its magnitude stays true. Cross your own path and the doubly-wound region is counted twice. Neither is a bug — it is exactly what the theorem says must happen, so the instrument is left free to say it.
Rendered the specimen sheet and caught two flaws. The Plate III crescent
was wound anticlockwise, so it printed a negative exact −5.52 cm² — a negative
printed area reads as an error and contradicted the sheet's own “trace clockwise” instruction.
Reversed its point order so all three plates wind the same way, and the printed label now shows
the figure's magnitude. Its caption also collided with the SHEET № 44 · SCHAFFHAUSEN
footer at the sheet's foot; moved it above the crescent, clearing both.
Tied the numeric panel to the survey stamp: when a closed trace lands inside tolerance
(deviation under 0.5% on a figure over 4 cm²), the deviation readout now turns
verdigris — the same “in adjustment” signal as the stamp, but on the number itself, so a
second read of the panel is rewarded. Refined the clockwork motion with a short ease-in ramp so
the tracer settles onto the paper over the first centimetre instead of snapping to full
speed — it now reads like a pen being set down.
Re-checked the layout at 375 px (no horizontal overflow; the readout grid folds to two
columns), the reduced-motion path (plate traces resolve instantly to a settled frame, reveals
show, the rAF loop still parks on document.hidden), and focus rings on every
control. For the Chanel rule, removed the 6.5 px maker's-mark micro-text engraved along the
moving arm: illegible at size and colliding with the sheet labels wherever the steel crossed
them. The arm reads cleaner for losing it.