From the entry "Linkages" in the Encyclopedia Brittanica, 1958.

The subject of three bar motion appears to have originated in 1784 with the invention by James Watt of his so-called "parallel motion". Writing to his partner, Bolton, in June of that year, he says: "I have started a new hare. I have got a glimpse of a method of causing a piston rod to move up and down perpendicularly by only fixing it to a piece of iron upon the beam without chains or perpendicular guides or untowardly frictions, arch heads or other pieces of clumsiness. I think it a very probable thing to succeed and one of the most ingenious simple pieces of mechanism I have contrived." ...

It was essentially a three bar motion, *ACDE* (fig.1), the links of which
were so adjusted in length and disposition as to cause a point *F* on the
traversing link *CD* to describe a figure of 8. The portion of the curve
near the point of inflection is nearly straight for an appreciable length.
If the two links *AC* and *DE* which rotate about the fixed points *A* and *E* are
of equal length and the point *F* is in the middle of *CD*, the figure of 8 is symmetrical; but if they are of unequal lengths, one limb of the curve is straighter
than the other and is straightest when the point *F* is taken so that *CF:FD* as *DE:AC*. If the head of the piston rod were fixed at *F* it is clear that the point *E*
would be an an inconvenient distance from the rest of the engine; so Watt added
two more links, *BG* and *DG*, forming a parallellogram, and it will be seen that
*ACDGB* forms a pantograph in which the usual tracing point and fixed centre have
changed places; and now if *AC:CB* as *CF:FD*,
the point *G* will describe a curve similar to that
described by *F*. The head of the piston rod was attached to *G*.

Several attempts were made in the early part of the 19th century to improve upon Watt's parallel motion (...) But it was reserved to Peaucellier, a lieutenant in the French army, to invent the first exact parallel motion. In 1864 he produced his famous Peaucellier cell, consisting of six links, which converts rotary motion into a truly straight line. It seems, however, to have completely escaped notice. Prof. Chebyshev of the University of St. Petersburg had been very interested in parallel motion without having arrived at any solution. But a pupil of his, named Lipkin, rediscovered the Peaucellier cell. The invention was introduced in England where it caused great admiration and interest, forming the subject of an address by Prof. Sylvester at the Royal Institution in 1874.

It consists of four equal bars, *FL, LM, MH, HF*, and two other longer
equal bars *KH* and *KL*, jointed together as shown in fig.2.
The outstanding
property of this simple mechanism is that during its deformation *KF.KM*
remains constant; so that if *K* is fixed, the curves traced out
by *F* and *M* are inverse curves. If *F* describes a circle, *M* will
describe the inverse circle; but if the centre of the *F* circle is at
a distance from *K* equal to its radius, then the radius of the *M* circle becomes
infinite; i.e. *M* describes a straight line.

This form of parallel motion is stated to have been used in an engine installed for the purpose of ventilating the Houses of Parliament before the introduction of electric fans.

For a LEGO model of the Peaucellier cell, click here, for a JAVA applet here.

After Sylvester's lecture, a period arose in England during which much attention
was paid to the subject of linkages, notable by Sir A.B. Kempe and H. Hart.
Chebyshev had proved to his own satisfaction that no linkage of five bars could
accurately convert circular motion into rectilinear motion; but in 1877
Hart produced one *(Proc. Lond. Math. Soc.)*, thus showing the danger of
trying to prove a negative. He had also previously invented one of four links
*(Messenger of Mathematics, 1875)*.

This mechanism consisted of a crossed
parallellogram *EAND* (fig. 2), formed by rotating the
triangle *ACD* about the diagonal *AD*. If any straight line *KF* be drawn
parallel to *AD* or *EN*, cutting the links *AN* and *ED* in *P* and *Q*, then *KP.KQ* is
a constant; so that if *K* is fixed, the point *P* and *Q* describe inverse curves.
If the link *AE* is fixed and *K* is in the midpoint of *AE*, *F* will descibe the inverse of a conic; and if we add a Peaucellier cell *KLMHF*, *M* will describe the
conic. If *ED:EA* as 1:sqrt(2), *F* describes a lemniscate and *M* a rectangular
hyperbola.

email: leo@science.uva.nl