Feynman Lectures, Chapter 4, Fig 4-2 Chapter 4 of the Feynman Lectures
Feynman defines the following:

We imagine that there are two classes of machines, those that are not reversible, which includes all real machines, and those that are reversible, which of course are actually not attainable no matter how careful we may be in our design of bearings, levers, etc. We suppose, however, that there is such a thing—a reversible machine—which lowers one unit of weight (a pound or any other unit) by one unit of distance, and at the same time lifts a three-unit weight

So to my understanding per this definition, a reversible machine is a machine that can sustain perpetual motion.
We are then introduced with the scenario in Fig. 4-2 which is defined as a reversible machine. One of Feynman's conclusions is that "Now, if 3X exceeds one foot, then we can lower the ball to return the machine to the initial condition, (f), and we can run the apparatus again. Therefore 3X cannot exceed one foot, for if 3X exceeds one foot we can make perpetual motion."
I'm not really understanding what Feynman is trying to point out here, if we assume the correctness of the axiom that this is a reversible machine (that it can lift 3 balls by lowering 1 and vice versa) then it already is technically capable of perpetual motion, i.e we are already in the established hypothetical framework that allows this reversible machine to work. Is he attempting to prove the machine is incapable of perpetual motion while it already is established as capable? What is he trying to get at? What is the sense in that? What was the point then in postulating that this is a reversible machine?
 A: He is having a moment of inexact language.
If it's reversible, then 3X = 1 foot. If 3X > 1 foot, then it would be not a reversible machine but a perpetual battery. You could continually get more energy out of it than you put in.
The reversible machine IS capable of perpetual motion, by definition. He should have said something like:

Therefore 3X cannot exceed one foot, for if 3X exceeds one foot we can get perpetual new energy.

It was only a momentary wrong word.
A: You're incorrect that he defines a reversible machine as a machine capable of perpetual motion. He says:

"We suppose, however, that there is such a thing—a reversible
machine—which lowers one unit of weight (a pound or any other unit) by
one unit of distance, and at the same time lifts a three-unit weight."

Then he argues that if perpetual  motion is not allowed, it follows that the potential energy (weight times the height) must be constant. By lowering the single ball by 1 feet the height of the 3 balls can only increase by 1/3 feet. If not, one could reverse the "cycle" and achieve perpetual motion. Well he expressed it much better of course.
A: The reversible machine is like a car that coasts down a hill and up another to the same altitude without using any gas. It is a perpetual motion machine because there is no friction to stop it. It keeps the same amount of kinetic + potential energy forever. This does not violate conservation of energy, so it is not immediately obvious it is impossible. Feynman will show this is impossible because there is always some loss somewhere.
The machine lifts a weight > 3x is like a car that coasts to a height higher than its start. It gains energy from nothing, violating the conservation of energy law.
At this point, Feynman has shown that no self contained lift 3/lower 1weight raising/lowering machine can lift a weight higher than a revsible machine, and all such reversible machines lift 3 to the same height. But he hasn't shown what that height is. He now does so.
