Other instruments to draw straight lines

Watt's mechanism, which because of its extreme simplicity is still in use today, solves in practice the problem of drawing a straight line, or at least a curve so near to a straight line to be practically indistinguishable in applications. After Watt, other instruments were found, certainly more complicated, that draw approximated lines; some of these can be seen and operated.

The theoretical problem still stands, however - is it possible to build an instrument that draws a true straight line, not just an approximation? A first positive answer is given by Sarrus' mechanism, in which the points on the upper plate all move along vertical lines. It is, however, a machine which is neither practical (Watt's mechanism is much simpler and more reliable) nor satisfying from the theoretical point of view, since it operates in three-dimensional space and not on the plane. Technically, this also means it takes up a lot of space.

The exact solution to the problem is given by a mechanism invented in 1864 by A. Peaucellier, based on the properties of a specific mathematical transformation: the inversion compared to a circumference. The mechanism is made of a number of rods hinged so that, whatever way they are moved, the product of the distances of the P and Q point from O is always the same. To use a more technical language, points P and Q correspond through inversion compared to a circle whose centre is O.

One of the properties of inversion is that when point P traces a circumference, its corresponding point Q also traces a circumference. Only one case provides an exception, which is what we need - when the circumference traced by P goes through the centre O, the corresponding point Q traces not a circumference, but a straight line. One then understands the role of the PR rod, whose extremity R is fixed to the table. It has nothing to do with inversion, but it ensures that point P, which now can only rotate around R, draws a circumference. If PR is equal to RO, this circumference will go through the centre O, and then the corresponding point Q will draw a rect, or, more precisely, a segment.

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