Four straight-line mechanisms, on a log scale

Watt's and Chebyshev's linkages approximate a straight line and are honest about it: their deviation over the full stroke is 9% and 12% of the travel. Peaucellier's cell is exact by an inversion argument and measures 9.8 × 10⁻¹⁶. The Sarrus linkage is exact for a different reason — its two chains constrain the plate to two perpendicular planes, and two planes meet in a line — and measures 1.8e-14 of its travel over 60 solved positions. Fourteen orders of magnitude separate the two kinds, and no amount of care with an approximate linkage crosses it.