You're correct on both counts. Lawlor's Figure 4 shows the pendulum string coming down from between two upward curvature cycloids. But in the spirit of the antikytheras, I was trying to come up with something involving _gears_. At 06:39 PM 11/25/2012, Allan Wechsler wrote:

The oldest solution to this problem is to make the pendulum very long, and have it describe a very small angle, so that the system is locally linear and the period is independent of amplitude for all practical purposes. Furthermore, every clock will turn out to have a preferred pendulum amplitude, so it's really only important to get the time constant right for that amplitude.

There is another solution, which is to make the pendulum arm of a flexible material, like a thin wire or string, and to confine its upper reaches between two convex-downward guides which come to a cusp at the fulcrum. This has the effect of shortening the arm for large excursions. The correct guide shape will direct the pendulum bob to trace a cycloid. I don't remember the correct shape: it's something simple, like a circle or another cycloid. But it doesn't matter how complicated the guide shape is: it's fairly easy to carve or bend it to within tolerance by trial and error. It turns out not to be worth the effort in increased accuracy, though.