301 extrinsic resistance: but since it is perhaps impossible to find such mobiles in the realm of matter, let no one be astonished, in putting these things to the test {2}, if the experiment is disappointing, and a large sphere, even if it is on an horizontal plane, cannot be moved by a minimal force. For to the causes already mentioned {1}, there is added this: namely, that a plane cannot be really parallel to the horizon. For the surface of the Earth is spherical, to which a plane cannot be parallel: that is why, since a plane touches a sphere in only one point, if we recede from such a point, it is necessary to go up: that is why the sphere, with reason, will not be able to be removed from such a point by any minimal force.
And from the things that have been demonstrated, it will be easy to obtain the solution of certain problems: such as the following. First: given two inclined planes, whose right descent is the same, to find the ratio of the swiftnesses of the same mobile on them.
{1} Cf. noteFor let the right descent be ab, and the plane of the horizon bd, and let ac, ad be the oblique descents: it is now asked, what ratio the swiftness on ca has to the swiftness on ad. And, since the slowness on ad is to the slowness on ab as the line da is to the line ab, as was shown above {1}; and just as line ab is to line ac, so the slowness on ab is to the slowness on ac; it will follow, ex aequali, that just as the slowness on ad is to the slowness on ac, so line da is to line ac: hence as the swiftness on ac is to the swiftness on ad, so line da is to line ac. It is therefore certain that, the swiftnesses of the same mobile on different inclines are to each other inversely as the lengths of the oblique descents, provided that these hold equal right descents. Furthermore, we can find inclined planes on which the same mobile observes a given ratio in its swiftnesses.
For let the given ratio be that which line e has to f; and let da to ac in the previous figure be made as e to f: what was asked will then be resolved. Also other similar problems {1} can be resolved: as, being given two mobiles of different genus, equal in size, to set up a plane inclined in such a way that the mobile which in vertical motion was moved faster than the other goes down on this plane at the same speed as that of the other in its vertical motion. But since these things and others similar can easily be found by those who