How to draw a straight line without a straightedge

With the recent distribution by Datum of the E-Book "How to draw a straight line: A lecture on linkages" by A. B. Kempe, we learned of many other 'artifacts' that can be used to draw straight lines besides the usual ruler, and the draftsman's T-Square. In it we find some mechanical artifacts than can substitute the common straightedge in some cases.

After a brief introduction about how easy it is to make a circle, Kempe asks:

But the straight line, how are we going to describe that? Euclid defines it as “lying evenly between its extreme points.” This does not help us much. Our text-books say that the first and second Postulates postulate a ruler. But surely that is begging the question. If we are to draw a straight line with a ruler, the ruler must itself have a straight edge; and how are we going to make the edge straight? We come back to our starting-point.

With this linkage, devised by M. Peaucellier, we can draw straight lines without a straightedge.

To make circle, we do not need to have the 'straightness' constraint, that is, the 'straightedge' restriction of the ruler. All we need to make a circle is to have constant distance between the center and the circumference; this can be attained by many mechanical means, or by the natural properties of the materials.

One the linkages, described by Kempe, that can be used to draw straight lines is shown in the figure above (Figure 5 in his book).

M. Peaucellier’s apparatus is shown in Fig. 5. It has, as you see, seven pieces or links. There are first of all two long links of equal length. These are both pivoted at the same fixed point; their other extremities are pivoted to opposite angles of a rhombus composed of four equal shorter links. The portion of the apparatus I have thus far described, considered apart from the fixed base, is a linkage termed a “Peaucellier cell.” We then take an extra link, and pivot it to a fixed point whose distance from the first fixed point, that to which the cell is pivoted, is the same as the length of the extra link; the other end of the extra link is then pivoted to one of the free angles of the rhombus; the other free angle of the rhombus has a pencil at its pivot. That pencil will accurately describe a straight line.

For more information about Kempe's works and other similar linkages, see the following links:

Cornell University Digital Library of Kinematics
Here you can see a drawing by Leonardo da Vinci showing a device with two machine mechanisms, an endless screw coupled to a "slider-crank."
The Leonardo da Vinci's slider-crank converts rotary motion into alternating linear motion.

The Peaucellier's linkage to produce straight-line motion. Part of the exhibition at Cornell University Library of Kinematics

Da Vinci said about this mechanism: "This movement is most praiseworthy, both for the ease of motion and the compactness of design.

How round is your circle?
"Here we shall examine a number of linkages which are arranged so that the geometry guarantees an exact straight line movement.

Mechanism collection (1 – 35), “white board” models
This is a showcase of many linkages; many of them to produce straight lines.

Moving in a Line (Chapter 39)
This is very instructional website with many animations where the visitor can see in real-time the drawing mechanisms. Quoting the author:

When you want to draw a straight line, what do you do? We use rulers, straight-edges, sides of notebooks, etc. to make sure our lines are straight. That is, we use an object that we already presume to be straight with which to make another straight object. (No question is usually made of whether the factory producing rulers is actually making straight ones. With which ruler will they measure their products?) It is much different when it comes to the question of drawing a circle. Although you might use a glass or a roll of tape to draw a circle, a pre-existing circle shape is not necessary: how circular do a compass, or a pen tied to a piece of string, look? We are able to create circular motion without starting from an already-existing circular shape.

Now, does there exist a "compass" for drawing straight lines? Far from being a useless question, the creation of mechanisms to move in straight lines was essential for unleashing the potential of heat-powered machines launched by Leibniz.

"The slider-crank has become one of the most ubiquitous mechanisms in the world today, owing to its application to perhaps a billion internal combustion engines in the 20th century. A cross-section of a 1950s Chevrolet engine shows two slider-crank mechanisms in the form of pistons, connecting rods, and cranks. This engine cutaway is also part of Cornell's Collection of Kinematic Models and Machines."

This deserves to be repeated:  "... we use an object that we already presume to be straight with which to make another straight object. (No question is usually made of whether the factory producing rulers is actually making straight ones. With which ruler will they measure their products?"

Download the EBook " How to draw a straight line: A lecture on linkages " by A. B. Kempe.

If you like his book, you may also want to take a look to another fully illustrated EBook: 507 Mechanical Movements by Henry T. Brown.