Dug sez, "I gather that you can do two things with the machine. First, the machine will produce various sine waves for you on paper after you set values for the amplitude and phase angle. Second, in a reversal of this process, you can trace a curve and use Fourier analysis to extract the phase and amplitude of the curve. All of this is done mechanically. There are tons of great pictures of the device on this site."

This isn't a vintage piece, either — it was built recently, and the makers provided extensive material documenting the process.

Link

(Thanks, Dug!)