Trace a closed curve with four instruments that measure area by rolling, sliding, or counting: the polar and linear planimeters, Prytz's hatchet, and an x–y encoder. Each reading comes with its error, split into where it comes from.
Polar planimeter (Amsler, 1854). A pole is pinned down. An arm of length a runs to an elbow, and an arm of length b runs to the tracer. A wheel on the tracer arm rolls only across the arm. By Green's theorem the area is A = b·w, where w is the distance the wheel rolled. If the pole sits inside the curve, the elbow goes all the way around and you add π(a² + b² − 2bc), where c is the wheel's distance from the elbow.
Linear planimeter. The elbow runs along a straight rail instead of a circle. The same Green's-theorem argument gives A = b·w exactly, with no correction term, and the curve can be as long as the rail.
Hatchet planimeter (Prytz, 1875). A rod of length L has a knife edge at one end and a point at the other. The knife can only move along the rod, like a bicycle's back wheel. Start at a point near the middle, go out to the curve, trace it, and come back. The rod has turned by an angle θ, and A ≈ L²θ. That is only approximate: the error shrinks roughly like 1/L² and depends on where you start. The knife's path is a tractrix, and the leftover rotation is holonomy, the same effect as the rolling ball.
X–Y encoder. Two wheels count steps in x and in y, and the area is the sum of x·Δy: Green's theorem done by counting. Its error is quantization: each count is ±½ step.
Where error comes from. Tremor is the hand wobbling off the line. Slip is the wheel skidding by a random percentage at each step. Reading is the resolution of the dial or vernier. Intrinsic is what the instrument does wrong even with a perfect hand: sampling for the exact instruments, and the L²θ approximation for the hatchet.
Units. The table is 25 cm × 17.5 cm with a 1 cm grid. The wheel has a 2 cm diameter, the vernier reads 1/1000 of a turn, and the dial alone reads 1/100.