How to draw a straight line, and why

But what is the purpose of drawing a straight line? Is it not easier to draw with a computer, without using either a ruler or a compass? The question makes sense, and it would be useless if the issue was just that of drawing. But this is not the only problem.

Let's look at any machine, a bicycle for example, or a blender. There are mobile parts in it, like the pedals or the wheels of the bicycle, or the blender's blades, that must move along predetermined trajectories, such as around a point. In this case there is no difficulty: it's enough to hinge the piece on the centre of rotation, so that it can only revolve around it. Every point of the piece thus traces a circumference around the hinge, which behaves like the finger holding the string to the table.

If, instead, the trajectory of the moving part is not circular, things become more difficult, such as in the rod we see coming out of the table, or in the piston's axis we see in the picture on the wall? We could certainly fix a ruler and make the rod slide on it, or, even better, go through two rings attached to the wall (and in this case the rod itself would act as a profile), but the movement would generate so much friction to make the functioning impossible.

In this situation, maybe even more than in the drawing, the difference between an instrument and a profile is fundamental. If we want the mechanism to work, we cannot use profiles, which always have sliding parts, but we must generate rectilinear motion with an instrument. The one we are seeing has been proposed by James Watt. It is a simple articulated quadrilateral, such that the midpoint of the smaller side - and thus the rod which is attached to it - moves up and down along a rectilinear trajectory.

But is it really a straight line? A table version of the same mechanism shows the opposite.The midpoint draws an 8-shaped curve, with two nearly rectilinear parts, or, anyway, straight enough for the task. Watt's mechanism uses these parts of the curve to maintain the rod constantly in a vertical position.