Indexing the archive…
Your Universe of Digital Possibilities
Draw a straight line using nothing but things that turn. James Watt needed one in 1784 to guide a piston rod from a rocking beam and found a linkage that is almost straight — straight to one part in thousands. For eighty years nobody did better; in 1864 Charles-Nicolas Peaucellier did it exactly. Three linkages on one bench, solved from the crank angle you set — and an auditor that prices each line to the last digit.
Walk round a four-bar linkage and you come home: crank and coupler, as complex vectors, add up to ground and rocker. Give the crank angle and the loop fixes the other two — one circle–circle intersection. Watt’s pen is the coupler’s midpoint; at his symmetric proportions its error grows only as the fifth power of the stroke.
The degree-n polynomial closest to f in the worst case is the one whose error E = ‖f − p*‖∞ is reached n + 2 times with alternating sign. Born from trying to straighten Watt’s linkage: spread the error evenly across the stroke instead of making it tiny in the middle.
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 first Heron machine, and the edition’s whole method on one bench: build the mechanism, drive it by hand, read the pen, then cross-examine. Watt’s linkage is approximate and honest about it — its residual is the size of the approximation. Peaucellier’s is exact and the residual sits at the float floor, which is the moral: the column of zeros. The same cell returns at the end of the cycle as the last gadget of Kempe’s universal linkage in The Signature, which draws every algebraic curve; what ruler and compass cannot do is The Square’s wall, and the fastest path between two points — not the straightest — is The Bead’s.