The Cardan Gear
The arm of this ingenious mechanism traces out an exact straight line as the gears go through their circular motion.
The small gears must be half the size of the large gear. Also, the arm length must equal the distance from the center of the large gear to the center of the outer gear.
The Cardan mechanism was invented by the Italian mathematician, physician, chess player, and gambler Gerolamo Cardano (1501 – 1576). He also invented the universal joint.
Proving mathematically that the Cardan gear works as claimed makes for an interesting trigonometry problem.