The straight-line problem
James Watt needed a piston rod to move in a straight line. The obvious answer — a slide — required a flat surface longer and truer than the workshops of 1784 could reliably produce, and a crooked slide binds.
So he used a linkage instead: two rockers and a coupler, tracing with the coupler’s midpoint. He is supposed to have been prouder of it than of the separate condenser.
It does not trace a straight line. It traces a very flat figure-eight, and the interesting question is not whether it is exact but how far from exact, over how much of its travel.
The error is a function, not a number
This is the point the figure exists to make.
Quoting “Watt’s linkage is straight to 0.1%” is quoting one point on a curve. Over a short central portion of the stroke the deviation is tiny; over the full available travel it reaches about 9% of the span. The number depends entirely on how much of the mechanism is to be used, and a design that uses a third of the stroke and one that uses all of it are asking different questions of the same linkage.
That is why the audit above plots the error against the fraction used rather than reporting a figure. It is also why the two approximate linkages cannot be ranked by a single number: they degrade at different rates.
Finding the stroke by asking
A rocking linkage has a limited range of input angles, bounded by dead centres, and the measurement above needs to know it.
The range is found by walking outward from a known-good configuration until the solve refuses. That locates the dead centres without any closed form for them, and it produced a result worth recording: for both Watt’s and Chebyshev’s linkages, the symmetric configuration is at the edge of the travel rather than in the middle of it. Watt’s reaches from −0.81 to +0.72 radians with the symmetric pose at the top end.
An earlier version of this analysis centred its sampling on the symmetric pose, on the assumption that it was the middle of the stroke, and sampled the edge — measuring half of nothing. The mechanism was asked and the assumption was wrong.
Chebyshev’s different question
Watt arrived at his linkage as an engineer with a problem. Pafnuty Chebyshev came at it seventy years later as an approximation theorist, and asked a different question: of all linkages of this kind, which one minimises the maximum deviation?
That is the same question he was asking about polynomials, and it is the origin of what is now called minimax approximation. His lambda linkage — ground 4, crank and rocker 5, coupler 2, tracing with the coupler midpoint — is the answer for its family.
Measured over the middle 80% of its stroke it deviates by 12% of the span, against Watt’s 9% over the same fraction of a shorter stroke. Which is “better” depends on what is being asked: Chebyshev’s traces a longer straight portion, Watt’s a flatter short one. There is no ordering without a specification.
Why approximation was acceptable
Watt’s linkage was good enough for a beam engine, and it is worth understanding why rather than treating it as a compromise.
The alternative was a slide whose straightness was limited by the flatness of a hand-scraped surface — and a slide that is out of true does not merely guide imperfectly, it binds and wears. A linkage made of pinned bars has no such failure mode: its error is a smooth function of position, it is repeatable, and it does not get worse with use in the same way.
So the comparison was not “approximate linkage against exact slide”. It was “approximate linkage with a known error against approximate slide with an unknown and deteriorating one”, and on those terms the linkage wins comfortably.
That is the general shape of why approximate mechanisms persisted long after exact ones were known.
Measuring straightness
The deviation is measured against the chord between the traced curve’s endpoints — the perpendicular distance from each traced point to that line, taking the worst.
Reported two ways: absolute, and as a fraction of the span. The second is the one that survives a change of scale, and it is the one the audit plots, because a linkage twice the size has twice the absolute error and the same relative one.
There is a modelling choice buried in “the chord”. An alternative is the best-fitting line, which would give a smaller number by about a factor of two. The chord is used here because it is the line an engineer actually cares about — the one joining the ends of the travel — and because it needs no fitting decision.
What exact would mean
For eighty years it was not known whether an exact solution existed. Sylvester later remarked that the problem had been given up as insoluble.
It is soluble, and the answer is Peaucellier’s cell of 1864 — measured on this site as straight to 10⁻¹⁶ of its span, which is arithmetic noise rather than a small error. The gap between it and the approximations at full stroke is about fourteen orders of magnitude.
That gap is what makes the distinction between approximate and exact worth drawing at all. If Peaucellier’s linkage were merely a hundred times straighter it would be a better approximation; being straight to the precision of the arithmetic makes it a different kind of object.
What the linkages were actually for
The straight-line problem is usually introduced as a curiosity, which understates what was at stake. Watt’s engine had a piston that moved in a straight line and a beam whose end moved on an arc, and the two had to be connected.
A chain works in tension only, so a chain can pull the beam down and cannot push the piston up. A rigid connection between a straight-moving piston rod and an arc-moving beam end requires either a slide — which in 1784 meant a machined prismatic guide of a length and accuracy nobody could make, and which would have turned the engine’s connection into a slider-crank rather than a four-bar — or a linkage that approximates the straight line closely enough that the residual sideways motion is absorbed elsewhere.
Watt found the linkage. He described it later as the invention he was most proud of, which is a striking thing for the man who separated the condenser to have said, and it makes sense once the alternative is clear: without it, the double-acting engine could not have been built with the manufacturing available.
The precision problem did not stay unsolved for long. Maudslay’s slide rest and Whitworth’s flat-scraping method made accurate prismatic guides ordinary within two generations, and the straight-line linkages stopped being necessary almost exactly when the exact one was found. That is a common shape in the history of mechanisms: the elegant solution arrives after the ugly one has been made cheap.
Reading the error curve
The error plotted in the figures above is a signed lateral displacement against crank position, and its shape says more than its maximum.
Watt’s curve crosses zero three times and has two lobes of opposite sign. That is the signature of a third-order contact at the centre: the linkage matches a straight line in position, slope and curvature there, and departs cubically.
Chebyshev’s curve has the equal-ripple form its construction demands — several excursions of the same magnitude alternating in sign, rather than one growing departure. That is the whole point of the approximation-theoretic approach and it is visible without measuring anything: an equal-ripple curve looks deliberately flattened, a Taylor-type curve looks like a curve that has been pinned at one point.
Which is better depends entirely on what the stroke is. Over a short central stroke the third-order contact wins easily, because a cubic with a small coefficient beats a bounded ripple near zero. Over the full stroke the ripple wins, because the cubic has grown and the ripple has not. The crossover is the useful number, and it is one the site’s measurement produces directly by sweeping the stroke and asking where the errors are equal.
The general principle outlives the linkages. Any approximation that is exact at a point degrades away from it; any approximation that is uniformly good is nowhere exact. That is a choice, not a defect, and the right choice depends on the interval.
Roberts, Hoeken and the linkages that are not Watt’s
Watt and Chebyshev are the two usually named, and the family is larger, with the members distinguished by what they optimise.
Roberts’s linkage is a symmetric four-bar with the coupler point at the apex of an isoceles triangle on the coupler. It gives a straight segment through the middle of its travel with a different error signature from Watt’s — flatter over a wider range, worse at the extremes — and it was for a long time the preferred choice where the stroke mattered more than the centre.
Hoeken’s linkage is the interesting one for machinery, because it produces an approximately straight segment traversed at approximately constant velocity. That is a stronger requirement than straightness and a much more useful one: a mechanism that draws a straight line at a varying rate is a poor conveyor and a fine drafting aid, and the reverse holds too. The rate is the velocity ratio of the linkage, which varies through the stroke whether or not anyone specified it.
Evans’s linkage trades a longer straight portion against a larger error, which suits applications where the stroke is the constraint.
The proliferation makes a point that a single comparison hides. “Straight” is not one specification. A linkage can be optimised for maximum straightness at a point, for uniform error over an interval, for the longest interval within a tolerance, or for straightness plus constant speed, and the four optimisations give four different mechanisms. Watt’s is the first, Chebyshev’s is the second, Evans’s the third and Hoeken’s the fourth.
That is worth stating because the historical framing — a long search for the straight line, resolved by Peaucellier — suggests there was one problem. There were several, they had different answers, and the exact linkage solves only the first of them: it draws an exact line, and says nothing about how uniformly it is traversed.
Measuring straightness without choosing a line first
There is a methodological trap in this essay’s central measurement that is worth exposing, because it is easy to produce a straightness number that means nothing.
The error is the departure of the traced points from a straight line. Which straight line? Choose the line through the two endpoints of the stroke and the error is zero at the ends by construction, so the measurement flatters the linkage and puts the maximum in the middle. Choose the tangent at the centre and the error is zero in the middle and grows at both ends, which flatters a different linkage.
The measurement here fits the line that minimises the maximum departure over the sampled stroke — the best line in the uniform sense — and reports that maximum. That choice matters: it is the only one that does not privilege any part of the stroke, and it is the standard the linkages were themselves designed against, at least in Chebyshev’s case.
It also means the number for Watt’s linkage over its full stroke is worse than the number usually quoted, because the usual quote is over the central portion. Both numbers are correct and they are answers to different questions, and the site reports the interval alongside the error for exactly that reason. A straightness figure without a stroke attached to it is not a measurement — it is a number.
What the problem left behind
The straight-line linkages are obsolete as machine elements and the questions they raised are not, which is the usual fate of a well-posed problem.
Approximation theory. Chebyshev came to linkages as a mathematician looking for a physical instance of the approximation problem, and left with the equal-ripple criterion that carries his name and underlies filter design, polynomial approximation and numerical analysis generally. The linkage was the occasion; the theory outlived it by a wide margin.
Linkage synthesis. The question “which proportions produce this motion” was first posed sharply here, and it is the question every modern mechanism design tool answers. That it has no closed-form answer, that the forward map is smooth and violently nonlinear, and that the inverse problem has many local minima are all facts first met in the straight-line search.
Rigidity and configuration spaces. Asking what curves a linkage can trace led, through Kempe’s theorem, to the modern statement that any compact smooth manifold is the configuration space of some linkage. That is a result in topology whose origin is a question about steam engines.
The mechanisms themselves are still worth drawing because they make a distinction visible that is otherwise abstract: the difference between an approximation that is exact at a point, one that is uniformly good over an interval, and one that is exact everywhere. Three linkages, three error curves, three shapes — and the shapes are the theory.
What the site adds is that the error is measured rather than described, over a stated stroke, against a line chosen not to flatter any of them. Nine percent and twelve percent and arithmetic noise are not three adjectives; they are three numbers on the same scale, and the ordering between them is what eighty years of searching was about.