300 moved on a plane parallel to the horizon, by a force smaller than on a plane inclined above the horizon.
Thus let the plane parallel to the horizon be along line ab, to which bc is at right angles; and let the mobile be the sphere e; and let f be any force whatever: I say that the sphere e, having no extrinsic and accidental resistance, can be moved on plane ab by a force smaller than force f. Let n be the force that can pull weight e upward by force; and as the force n is to the force f, thus let line ad be to line db
Then from the things that have been demonstrated above {1}, the sphere e will be able to be pulled by force upward on the plane ad by the force f: hence on plane ab, the sphere e will be moved by a force smaller than f.
Which was to be demonstrated. {1} On the other hand, I am not unaware that someone here may object that I have presupposed as true for these demonstrations what is false: namely, that the weights suspended from the balance maintain right angles with the balance; even though the weights, since they tend to the center, would be convergent. To these people I would answer that I cover myself with the protecting wings of the superhuman Archimedes (whom I never mention without admiration).For he has presupposed the same thing in his Quadrature of the Parabola {1}; and this, perhaps, in order to show that he was so far ahead of the others that he could draw the true from the false: and yet, one must not be in doubt, as if he had concluded falsely, since he had proven the same conclusion previously by another geometric demonstration. That is why, either one must say that the suspended weights really maintain right angles with the balance, or it is of no importance whether or not they maintain right angles, but it suffices that they are equal; which perhaps will be more probable: unless we wish to say rather that this is a case of geometric licence; as when the same Archimedes presupposes that surfaces have heaviness, and that one is heavier than another, even though in fact they are all without heaviness. And the things we have demonstrated, as we have said above {1}, must be understood concerning mobiles free from all