  Section 10 includes a philosophically important result that has gone largely unnoticed in the literature on the Principia. Newton's argument that terrestrial gravity extends to the Moon depends crucially on Huygens's precise measurement of the strength of surface gravity. This theory-mediated measurement was based on the  isochronism[36] of the cycloidal pendulum under uniform gravity directed in parallel lines toward a flat Earth. But gravity is directed toward the center of the (nearly) spherical Earth along lines that are not parallel to one another, and according to Newton's theory it is not uniform. So, does Huygens's measurement cease to be valid in the context of the Principia? Newton recognized this concern and addressed it in Propositions 48 through 52 by extending Huygens's theory of the cycloidal pendulum to cover the hypocycloidal pendulum — that is, a cycloidal path produced when the generating circle rolls along the inside of a sphere instead of along a flat surface. Proposition 52 then shows that such a pendulum, although not isochronous under inverse-square centripetal forces, is isochronous under centripetal forces that vary linearly with the distance to the center. Insofar as gravity varies thus linearly below the surface in a uniformly dense sphere, the hypocycloidal pendulum is isochronous up to the surface, and hence it can in principle be used to measure the strength of gravity. A corollary to this proposition goes further by pointing out that, as the radius of the sphere is increased indefinitely, its surface approaches a plane surface and the law of the hypocycloidal asymptotically approaches Huygens's law of the cycloidal pendulum. This not only validates Huygens's measurement of surface gravity, but also provides a formula that can be used to determine the error associated with using Huygens's theory rather than the theory of the hypocycloidal pendulum.