Peaucellier and the exact answer
Charles-Nicolas Peaucellier published his linkage in 1864. Yom Tov Lipman Lipkin found it independently at about the same time. It solves exactly the problem Watt had solved approximately in 1784, and the gap between the two answers is not a matter of degree.
How it works
The mechanism is an inversor. Its two long arms and its rhombus together force
and that constant is arm² − rhombus², which for the proportions above is 5² − 3² = 16. Measured across the working range, the product drifts by 1.7 × 10⁻¹⁵ — arithmetic noise.
A map that holds constant is inversion in a circle about O, and inversion has one property that finishes the argument: it carries circles through the centre of inversion to straight lines.
So if Q can be made to travel on a circle that passes through O, then P travels on a straight line. The link CQ does exactly that: with , the circle Q runs on has O on it.
There is no approximation anywhere in that chain. The straightness is not a property of well-chosen proportions; it is a theorem about inversion, and any proportions satisfying the constraints give it.
Why it took eighty years
The problem was well known and generally believed to be insoluble. Sylvester, shown a working model, is reported to have said it was the most beautiful thing he had ever seen — a reaction the mechanism still gets, and one that says something about how unexpected the solution was.
The difficulty is that the natural approach is to look for a four-bar with a better coupler curve, and no four-bar coupler curve contains an exactly straight segment. Peaucellier’s answer needed eight links and an idea from geometry rather than from mechanism design.
Where it can go
The cell has a working range, and the limit is not a dead centre in the usual sense.
The rhombus has to close. With Q at distance q from O, P sits at (arm² − rhombus²)/q, and the rhombus spans half the difference between them — which is only possible while
Since Q runs on a circle through O, q varies from 0 to twice the crank length, so the reachable arc follows from the proportions directly.
That constraint is easy to violate. The first version of this mechanism on the site used an arm of 5 and a rhombus of 2, needing q ≥ 3 against a maximum of 3.2, and the default configuration put Q almost on top of O — where q is nearly zero and P is nearly at infinity. It assembled at none of the sampled angles. Arm 5 and rhombus 3 needs q ≥ 2, which the same crank reaches comfortably.
Mobility
Eight links: the frame, two long arms, four rhombus sides, and the crank. Ten pin joints, counting the ones where three links meet as two each.
The Jacobian agrees, which is worth noting because a mechanism with this many links and this much symmetry is exactly where a redundancy might hide — and here there is none. The cell is exactly constrained.
What exactness buys
Peaucellier’s cell was used in practice — in ventilating engines at the House of Commons, among other places — and it did not displace approximate linkages generally. It has eight links where Watt has four, ten joints where Watt has four, and every one of those joints has clearance and friction that this site does not model.
So the practical comparison is not straightforwardly in its favour, and the honest summary is that exactness bought less in engineering than it did in mathematics.
What it settled was the question. An approximation with a small error and an exact solution are different in kind, and until 1864 nobody knew the second existed.
Hart’s alternative
Hart found a second exact straight-line linkage in 1874, using six links rather than eight — the Hart inversor, built on a crossed four-bar rather than a rhombus.
It is a genuinely different mechanism producing the same result, and its existence changes the character of the 1864 discovery: what had looked like a single ingenious construction turned out to be one member of a family. Kempe later proved the general result that a linkage exists to trace any algebraic curve, which makes the straight line an unremarkable special case of something much larger — and which is easier to state than to use, since the linkages the construction produces are enormous.
What inversion does, in more detail
The claim that inversion carries circles through the centre to straight lines is the whole mechanism, so it is worth seeing why.
Inversion in a circle of radius k about O sends a point P at distance d to the point on the same ray at distance k²/d. Points inside go outside, points outside go inside, and points on the circle stay put.
Now take a circle passing through O. Every point on it has some distance from O, and those distances range from zero — at O itself — up to the diameter. Under inversion, the point at distance d goes to k²/d, so the points near O go very far away, and the point at O itself goes to infinity.
A curve whose image includes the point at infinity and which is otherwise a well-behaved conic is a line. Working it through in coordinates confirms it: the image of a circle through the origin is exactly a straight line, perpendicular to the diameter through the origin.
So the mechanism has two jobs and does them separately. The rhombus and the two long arms perform the inversion, holding |OP| · |OQ| at arm² − rhombus². The short link CQ constrains Q to a circle through O. Neither part knows about straight lines; the straight line is what the composition produces.
That separation is why the proportions are so free. Any arm and rhombus satisfying arm > rhombus give some inversion constant, and any |CQ| = |CO| gives a circle through O. The line moves and stays a line.
Kempe’s theorem, and what it does not give
Once an exact straight-line linkage exists, the obvious question is what else is reachable. Kempe answered it in 1876: for any algebraic plane curve there is a linkage that traces it.
The result is remarkable and the construction is not usable. It works by building mechanical adders and multipliers out of pantographs and parallelograms and composing them, and the linkages it produces for even simple curves have hundreds of links. Kempe’s original proof also had gaps — the construction can reach unwanted branches — which were only properly closed by Kapovich and Millson in 2002.
What it settles is the existence question that Peaucellier’s cell opened. A straight line turned out not to be a special case that happened to have an answer; it is one instance of a completely general fact, and the reason it took eighty years to find was that people were looking among four-bars, where no coupler curve contains an exactly straight segment.
What “exact” is being claimed, and what it is not
The measured straightness of the cell is 9.8 × 10⁻¹⁶ of the span, and it is worth being precise about what that number is and is not evidence for.
It is not a measurement of a physical mechanism. It is the departure of the solved positions from a straight line, in a model of rigid links and ideal pin joints, in double-precision arithmetic. The number is at the level of arithmetic noise, which is the correct outcome for a mechanism that is exactly straight in theory: the model’s error is dominated by the floating-point representation, not by the geometry.
That is a genuinely different claim from the ones made about Watt and Chebyshev, whose errors are 9% and 12% of the span. Those numbers do not shrink with better arithmetic; they are properties of the mechanisms. The cell’s number does shrink — it would fall further in higher precision — and that behaviour under refinement is what distinguishes an exact result from a very good approximation.
A real Peaucellier cell is not exact. It has seven pin joints, and clearance in each of them contributes an error that has nothing to do with the geometry. In practice a well-made approximate linkage with three joints can be straighter than a poorly-made exact one with seven, which is a large part of why the exact linkages never displaced the approximate ones in service.
The exactness is therefore a statement about the theory, and the theory is what the eighty-year search was for. Whether the resulting mechanism is the one to build is a separate question, and the answer was usually no.
Seven links, and why the joint count is the practical objection
Counting the parts makes the trade concrete. The cell has eight links counting ground and ten revolute joints. Watt’s linkage has four links and four joints. Chebyshev’s has four and four.
Each additional joint contributes clearance, friction and a wear site, and the contributions accumulate roughly linearly. Ten joints is two and a half times four, which is a fair first estimate of how much worse the cell’s real-world error and friction will be for the same quality of manufacture.
Against that, the geometric error the joints are competing with is zero rather than 9%. Which side wins depends entirely on the stroke and the accuracy required: over a short stroke Watt’s cubic error is tiny and the cell’s extra joints are pure cost, while over a long one the cell is the only option that does not degrade.
The mobility count says both are mechanisms with one degree of freedom, which is the count doing exactly what it is good for — confirming that adding six links and six joints to get exactness did not accidentally add a freedom. The Jacobian rank agrees at every configuration sampled, with a singular-value gap of six orders of magnitude, so the cell is exactly constrained rather than redundantly so — unlike the parallelogram, where the two routes part company.
Why the cell is the site’s calibration standard
The Peaucellier cell has a role here beyond its own interest: it is the mechanism against which the machinery is checked, because it is the only one whose correct answer is known in advance.
Every other measurement on this site is a number the solver produced and nothing independent confirms — the straightness of Watt’s linkage, the mechanical advantage at a toggle, the mobility of a redundant parallelogram. Those numbers are believable to the extent the solver is, and the solver’s credibility has to come from somewhere.
It comes from here. The cell’s straightness is exactly zero in theory, so any non-zero measurement is the machinery’s error and nothing else. Measuring 9.8 × 10⁻¹⁶ of the span therefore says something about the solver rather than about the mechanism: it says that Newton–Raphson on these constraints, with this Jacobian, at this tolerance, produces positions accurate to the last few bits of a double.
The inversion product provides a second, independent calibration on the same mechanism. |OP| · |OQ| must be constant at arm² − rhombus², and it is, to 1.7 × 10⁻¹⁵. That quantity is not the one being plotted, so it is a genuinely separate check on the same solved configurations — the pattern this site uses throughout, and the one that catches errors a single method cannot.
Having a calibration standard changes how the other numbers should be read. When the Watt linkage measures 9% error, that is not “9% plus unknown solver error”; the solver’s contribution is fifteen orders of magnitude smaller and can be ignored. When the four-bar’s worst residual is quoted at 2.4 × 10⁻¹⁴, the cell is the evidence that such a residual means what it appears to mean.
That is most of why a mechanism from 1864 with no practical use is worth a solver, a set of figures and an essay. It is the one place on the site where the right answer is known independently of the site, and everything else is measured with an instrument that this mechanism calibrated.
Sylvester, and the mechanism as an argument
The cell arrived in England through a lecture, and the reception is worth recording because it says something about what kind of result this was.
Sylvester presented it at the Royal Institution in 1874, and Kelvin — by that point the most eminent physicist in Britain, and a man who had spent a career on precision instruments — is said to have refused to give the model back, calling it the most beautiful thing he had ever seen. The reaction is out of proportion to any engineering use it had, and that is the point: what was beautiful was that the problem turned out to have an answer at all.
Eighty years of approximate linkages had left the impression that exactness was unavailable — that the straight line was intrinsically a limiting case that mechanisms could approach and not reach. The cell shows that the impression was wrong, and it shows it by construction rather than by argument.
That is a shape worth recognising. A long absence of a solution is weak evidence that none exists, and it reliably feels like strong evidence. The four-bar family genuinely cannot draw a straight line — no coupler curve of a four-bar contains a straight segment — so every attempt within that family was doomed, and the family was where everyone was looking. Adding three links changed the question.
The eventual generalisation, Kempe’s theorem, makes the straight line unremarkable: every algebraic curve is traceable, so of course the line is. But that is hindsight from a theorem that the cell made it reasonable to look for.
The site’s contribution is smaller and in the same direction: the exactness is not asserted from the theory but measured from solved configurations, at 9.8 × 10⁻¹⁶ of the span against 9% and 12% for the approximations. Three numbers on one scale, which is what the eighty years were arguing about.