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Your Universe of Digital Possibilities
In 1876 Alfred Kempe proved that for any algebraic curve there is a linkage of bars and pins that traces it. Pick a curve, or draw one, and the bench builds that linkage: it writes the curve’s equation as a sum of cosines of an arm’s two angles, computes each term with bars, adds them, and makes the sum obey the equation with one last gadget — Peaucellier’s straight-line cell, the first machine of this edition. Then drag the pen, and the whole machine draws the curve back.
Put a two-bar arm’s pen into the curve’s polynomial and products of cosines become sums of cosines. Each term is a rod at angle mα + nβ + γ, a point at radius c on it; add the points and the sum’s x-coordinate is f − C. Hold it on x = −C and the pen is on the curve.
Two similar contraparallelograms share a rod, the first’s corner pinned to the second’s long bar: the third rod is the mirror of the first in the second. Chained from a fixed rod it multiplies an angle; two of them through a bisecting rod add two.
In the cell, P and Q stay on one ray from the fixed pivot O and the product of their distances never changes: Q is P inverted in the circle of radius k. Inversion sends every circle through O to a straight line — so drive P round a circle through O, and Q walks a line, exactly.
The tenth Heron machine closes the cycle where it began. Put the pen’s coordinates into the curve’s polynomial and it becomes a sum of cosines of the arm’s angles; reversors made of contraparallelograms multiply and add angles, parallelograms carry the terms end to end, and the sum is held on a line by the Peaucellier cell of The Straight — the edition’s first machine is its last joint. Walk back round the ring and every bench is a machine that computes with its shape. The Straight turned a circle into a line by inversion; The Roller rolled a width that is not a circle; The Planimeter read an area off a wheel by Green’s theorem; The Integrator solved a differential equation with a disk; The Governor held an engine’s speed until it hunted; The Chariot measured the curvature of the ground with a gear; The Arch let gravity find a shape; The Fold folded a sheet on one degree of freedom; The Soap let a film find the shortest network it could. Kempe says the bars alone could have done any of it, for any algebraic curve — in principle. Honesty belongs at the end too: Kempe’s own argument had a gap, where flat contraparallelograms let the linkage flip, closed by bracing in Kapovich and Millson’s 2002 proof; and the linkages are hopeless instruments, amplifying any play without limit at a singular point. Turning curves into sums of cosines is The Spectrum’s trade; what ruler and compass cannot construct is The Square’s.