The author of the article reports that it is quite mysterious how the circular cheeks and the pendulum work together to compensate for differences in air pressure. And indeed, most clockmakers and virtually all books get it wrong. Conventional wisdom is to have a very heavy pendulum (so small vibrations won't affect it as much) and a very small swinging arc--so that the small angle approximation (sin(t) = t) holds.

Harrison threw this all out.

The problem being solved is that when the air pressure is low, there is less friction resistance to the swinging pendulum--and when the air pressure is high, the denser air gives more resistance to the swinging pendulum.

Therefore, in denser air, the pendulums swinging arc will be fewer degrees. For true simple harmonic motion, changing the swing arc wouldn't change the period of the pendulum Unfortunately, pendulums don't exactly swing as simple harmonic oscillators--and for clocks as accurate as Harrison's, this makes a difference.

So Harrison paid attention to the aerodynamic properties of his pendulum. He made it lighter than usual, so that its movement would be more sensitive to barometric pressure differences, and he gave it a very wide swinging arc, so that there was more room for the circular cheeks at the top of the pendulum to correct for the differences. And instead of making it slice as smoothly as possible through the air, he deliberately shaped it to have more drag, so it could "sense" the air pressure more accurately.

The length this guy went to for precision was incredible. And the craftsmanship! He could make clocks out of wood which can (and have) been keeping very accurate time for centuries.