The exact solution to the problem of rectilinear motion was found in 1864 by Peaucellier and, published as letter to the journal Nouvelles Annales de Mathématiques, it went nearly unnoticed. In 1871, Lipkin independently found the same solution, and was greatly honoured by the Russian government for the supposed originality of his invention. Although belatedly, Peaucellier's merit was recognised and he received the important Montyon prize from the Institute of France. For a complete picture of this discovery, we must remember that as early as 1853 Sarrus had already described a three-dimensional solution.

Let us then describe Peaucellier's discovery. His mechanism is constituted by seven connecting rods. Four of these, of the same length, form a rhombus, two longer ones are connected to two opposed vertices of the rhombus and between us in a fixed point O. The apparatus thus described is known as the Peaucellier cell, and it has as a notable property that the P and Q points correspond in a circular centre inversion O, that is the distances OP and OQ are such that OP · OQ = constant.

A property of circular inversion, easy to demonstrate for anyone with a minimal familiarity with Euclidean geometry, is that the circles that pass through the inversion centre (in our case point O) are sent into straight lines. This explains the functioning of Peaucellier's mechanism: if one adds a seventh rod which forces P to draw a circle through O, then the point Q will draw a straight line. A true straight line, not an approximation.

Peaucellier's cell can be obtained from the two articulated quadrilaterals that appear in the figure on the side, to which tradition has given the names kite and arrow, superimposing them and identifying the longer rods. If the same operation is executed identifying the shorter rods, we obtain the mechanism in the next figure.
In this case OQ · PQ = constant and if Q is forced to draw a circle through O then P will move in a straight line.

" In this very compact form, the mechanism has been successfully applied to the machines used to ventilate [London's] Houses of Parliament. The smoothness of movement due to the absence of noise and friction is really remarkable. The machines were built on the basis of Peaucellier's mechanism by Mr. Prim, engineer of the Houses of Parliament, to whose courtesy I owe the chance of having seen them: I assure you they are worth a visit." Thus the English mathematician Kempe, wrote in 1877 in his interesting booklet How To Draw a Straight Line. Unfortunately these ventilation machines were destroyed, and it was not possible to recover any documentation.

An interesting application of Peaucellier's cell is found in autofocus photographic enlargers. If we consider a lens with a focal distance f, foci F1, F2, if d1, d2, are the distances of the lens from the figure to be enlarged and from the enlarged figure, one must have:

1/d1 + 1/d2 = 1/f

which is equivalent to:

(d1-f) (d2-f) = f2

This equation represents an inversion between the two segments of length d1-f and d2-f and must be respected for every height of the lens, if one wants the projected image to always be in focus. Then one can mount the lens and the negative holder on a vertical support, perpendicular to the plane on which the figure is projected. The alignment between the O, A, and B mechanisms is then ensured and is therefore sufficient to guarantee the inversion, one half of Peaucellier's mechanism composed by two rods OH and BH of equal length, connecting in the common point H with an AH rod. The fixed point O is placed at a f distance from the plane on which the image is formed, while the point B is fastened to a negative-holding plane placed at a f distance from point B itself. The condition above is verified if one dimensions the rods such that OH2-AH2 = f2.

A last trivia, almost insignificant for us who speak of mechanism but important in other contexts: in 1964 the English physicist Penrose found that to determine the length of the sides, Peaucellier's mechanism kite and arrow configurations of Figure 7 can constitute a non-periodic plane covering, that is one that is not transformed in itself by any rigid movement of the plane.
The study of these coverings has found an application in near-crystal theory.