From: Henry Baker <hbaker1@pipeline.com> cycloidal escapement for a clock.

The problem is that the rolling circle has to roll on the _ceiling_, not on the floor, according to Lawlor's Brachistochrone paper.

Ok, so we put a "rack" (a linear gear) on the ceiling & parallel to the ceiling.

Then we _hang_ a gear wheel from a rail which runs parallel to the ceiling. The hanger has a small rolling mechanism similar to that in a "sliding" door.

The gear teeth of the hanging gear wheel engages the teeth in the linear gear rack on the ceiling.

Now, if we push the entire gear wheel slide forward, the engaged gear teeth will rotate the gear wheel.

So far, so good.

We now make the wheel extremely light with as low a moment of inertia as possible -- e.g., magnesium.

--is this necessary? To be elegant you want it to work even for a nonzero-mass wheel. Would that be mathematically possible?

We also make the sliding hanger mechanism as light as possible, with as little friction as possible.

--unfortunately, in practice I think the friction would be far larger than just using a plain pendulum. Ditto for the hanging rope (rope confined by cycloidal walls) scheme.

We then mount a lead weight on the gear wheel.

With small-to-medium displacements of the wheel, the lead weight will now move in a cycloidal path, similar to Huygens's path.

--why "small to medium"? Why not "all"?

We can now arrange for some other mechanism (light actuated, e.g.) which notices when the amount of motion becomes too small, and gives the wheel a small additional kick to add energy.

This is what I came up with in a few minutes.

--another attempt to rebuild obsolete technology. Somebody in England actually did build what he claimed was the world's most accurate pendulum clock, using pendulum swinging low-amplitude in vacuum, photosensor "escapement" with electromagnet kicker, old-invar rod. Whole thing built inside a special hut with heavy concrete foundation, and active temperature regulation using "box inside a box" 2-level temperature control. Pointless exercise, really, but he enjoyed it and offered evidence from comparison with atomic clocks he'd indeed achieved greater accuracy than any previous pendulum clock, and also he was able using it to detect the motion of the moon, which might make a good sci-fi story -- you know, the alien race trapped on planet shrouded in perpetual fog, nevertheless detects the existence of moon via pendulum clocks. Oh wait, good sci-fi stories also are obsolete. QUESTION WHICH ACTUALLY HAS A SLIGHT CHANCE OF MATTERING: Could this kind of detector be used to detect gravity waves?