Understanding Feynman's argument about Stevinus

The ball-chain is intended to represent uniform weight distribution along to two upper sides of the triangle; if they were 10 times smaller you would have 30 on the vertical face, and 50 on the ramp. The hanging loop completes the chain.

Does the chain begin to rotate on its own? If it does, we have found a mechanical perpetual motion machine! So no, it doesn't move.

What about the forces at the two ends of the hanging loop? The tension must be the same, or the hanging loop would be pulled to one side -- and the chain would start to move.

Since the tension is the same on both sides, we can cut off the hanging loop, and the rest of the chain will remain stationary, for it was already balanced.

Also see History of Mechanics of the Inclined Plane, and the many footnotes.

Looking at the diagram, you can see there are five "units of weight" along the diagonal, and three units along the vertical. We need to prove that these two sets of weight are in equilibrium; because, if they are, then "five units along the diagonal" is in equilibrium with "three units along the vertical", and thus if I have 1 unit along the diagonal (1/5th of the original), then I need 3/5 of a unit along the vertical.

So - why is this in equilibrium? Let's look at the chain as drawn. Should it move? If it would freely move clockwise, then I can put a very small generator on it to extract power - because as it moves, weights travel up one side as they come down the other, and this means that after it's moved one "click" (one unit of weight spacing) we have the same situation as before. Ditto if it moved anti-clockwise by itself. We conclude that the whole system is in equilibrium, or we have a perpetuum mobile. And we don't...

Would it still be in equilibrium without the bottom part? Well, the chain below the triangle is symmetrical. That means that the tension applied on the left must equal the tension on the right. And if these are the same, then removing the chain at the bottom shouldn't change the equilibrium.

And there you have it. This means that without the chain at the bottom, the thing is still in equilibrium: five along the diagonal equals three along the vertical.

Clever Stevinus.