One example is where rotary motion is given, and linear motion is needed: water pushes a water wheel around in a circle, and a saw blade is required to move back and forth. Another example is the other way around: a piston drives a shaft back and forth, and a wheel is required to roll, as in the steam train.

Industrial design was satisfied with only approximate solutions and ones using guiding supports, such as those found on a steam locomotive's wheels.

But mathematicians continued search for a solution, using only a linkage, until 1864, when French army general Nicholas Peaucellier produced this surprisingly simple example:



The dotted line represents the line that the far right joint traces out, when the bar connected to the anchor on the right is swiveled. To better understand what this linkage is capable of, we'll need to understand a bit about complex numbers.

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