1. Straight lines and circles

What can be done with a piece of string?

The visit begins with a marker and a simple piece of string, with which you are invited to draw a straight line and a circumference.

Walking on the street, you can sometimes see road workers about to dig a trench, that first draw up its outline by pulling a string between two stakes. We can also draw a straight line (note the verb draw) by pulling the string with two fingers and trying to follow it with the marker. If, instead, we want to draw a circle, we will roll the string around the marker, and we will make it go round by pinning the string's other end with a finger.

The result in the two cases is very different. While usually the arcs of circle which we can draw (arcs - to draw a full circle one needs more precaution) are quite accurate, the segments of a straight line are usually disappointing.

The reason for this difference in behaviour is in the different function of the string. In the case of the circle, it is a tool. In the case of the straight line, it is a profile. With a profile one draws what is already there. We can draw a straight line because a string pulled by two fingers creates a straight line. The accuracy of the result in this case depends on the precision of the profile, and on the possibility of following it with the marker, and that is precisely what is difficult in our case.

Conversely, when drawing a circle, the string does not take up a circular form to follow the marker. We take advantage of a mathematical property of circumference, that is that all its points have the same distance from its centre. The string, pulled taut between the marker and with the end pinned to the table, ensures this very equidistance.

The same happens if, instead of using a string, we use more appropriate objects, like the ruler and the compass. Precision improves greatly with both, but there is no substantial change: the ruler is a profile, the compass an instrument. And although straight lines made with a ruler are always better than those made with a string (but circles are also rounder), the precision of an instrument will always be, at the same degree of complexity, better than the one of a profile.

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.

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.

Articulated quadrilaterals

Among the many connecting rod mechanisms that solve problems of a practical interest, the simplest is the articulated quadrilateral, which, because of its very simplicity and versatility, is the basis for many simple instruments that we see every day, some of which can be seen in the panel, from scales to Venetian blinds, to windscreen wipers, cranes, but also in some more sophisticated mechanisms like amputees' prostheses.

Four sides are the minimum to have a mobile mechanism. The triangle is a rigid,non-transformable figure, which because of this stability is used to build stable structures, such as towers, bridges and roofs. On the contrary, a square maintains a certain freedom of movement even when one fixes one if its sides, a freedom which makes it a very effective instrument to draw curves, or if you prefer, to have a piece move along a predetermined path.

Normally one of the sides of the quadrilateral is fixed, e.g. to the table, and remains immobile: thus it is possible to avoid putting it in, and just like in Watt's mechanism, the quadrilateral is reduced to three interconnected rods, of which the first and last are fixed to the table by one extremity, around which they can only rotate. Notwithstanding the extreme simplicity of the mechanism, articulated quadrilaterals are very versatile, and they have numerous applications. In particular, they are very useful to convert oscillatory movements in circular ones and viceversa, as it happens e.g. in a bicycle, where the alternated movement of the cyclist's legs generates the pedals' circular motion, or in the sewing machine, where the pedal's oscillating movement makes the machine's wheel spin.

If then you add two more rods to the quadrilateral, creating a triangle with the middle one, it is possible to draw a great many curves, even rather irregular ones, by properly adjusting the length of the added rods. In the machine shown, the additional rods are substituted with a sheet of Plexiglass, whose holes correspond to the vertices of the added triangle. Depending on the position of the hole, the mechanism describes curves, even very different ones.

Only with compasses

Although recently it has been almost completely superseded by the computer, the drawing board was, for several centuries, one of the main work tools of the architect and of the project manager. On the drawing board, the difference in precision between ruler and compasses is still quite noticeable, since neither Peaucellier's mechanism, nor the even less accurate mechanisms of by Watt and his followers, could be used to draw.

Thence derives the problem of using the ruler as little as possible, or even of eliminating it completely, and to execute the drawing with just the compasses. Of course we are not speaking of the actual drawings, since no compass could ever trace a straight line, but to the construction preliminary to the drawing, when it's time, for example, to find two points through which a straight line must pass. In these constructions, the use of the ruler entails a much lower precision than that of the compasses.

t the end of the Eighteenth century, the problem of ruler-less constructions was solved by Lorenzo Mascheroni, who demonstrated that all the constructions that can be effected with ruler and compasses can also be effected with only the compass. In particular, Mascheroni applies his method to the division of circumference in equal parts, an essential operation in building astronomy apparatuses.

Later it was discovered that Mascheroni had been preceded by a Seventeenth century Danish mathematician, Georg Mohr, whose work, since it had no applications at the time, had remained virtually unknown.

n the computer, we can see some of the main geometric constructions, executed with the compass only.