then, we show that the mobile at point s is less heavy than at point d, it will then be manifest that its motion will be slower on line gh than on ef: and if, again, we show that at r the mobile is even less heavy than at point s, it will then be manifest that the motion will be slower on line nt than on gh. </s>
<s id="id.1.2.1.04.09">Now it is already manifest, that the mobile at point r exerts less weight than at point s; and less in s, than in d. </s>
<s id="id.1.2.1.04.10">For the weight at point d weighs equally with the weight at point c, since the distances ca and ad are equal: but the weight at point s does not weigh equally with the weight in c. </s>
<s id="id.1.2.1.04.11">For if a line is drawn from point s perpendicular to cd, the weight at s, as compared with the weight at c, is as if it were suspended from p; but the weight at p exerts less weight than the weight at c, since the distance pa is smaller than the distance ac. </s>
<s id="id.1.2.1.04.12">And, similarly, a weight at r exerts less weight than a weight at s: which will likewise be evident if a perpendicular is drawn from r on ad, which will intersect this same ad between points a and p. </s>
<s id="id.1.2.1.04.13">Consequently it is manifest that the mobile will go down with a greater force on line ef than on line gh. and on gh than on nt. </s>
<s id="id.1.2.1.04.14">But with how much greater force it is moved on ef than on gh will be known thus: if line ad is extended up to outside the circle, it stands to reason that it intersects line gh at point q. </s>
<s id="id.1.2.1.04.15">And since the mobile goes down more easily on line ef than on line gh, by as much as it is heavier at point d than at point s; and it is heavier at point d than at s, by as much as line da is longer than line ap; hence the mobile will go down more easily on line ef than on gh, by as much as line da is longer than this same pa. </s>
<s id="id.1.2.1.04.16">Therefore the swiftness along ef will have the same ratio to the swiftness along gh, as line da has to line pa. </s>
<s id="id.1.2.1.04.17"> Now as da is to ap, so is qs to sp, that is the oblique descent to the right descent: it is hence certain that the same weight is pulled by force with a smaller force upward along an inclined ascent than along a right one, by as much as the right ascent is smaller than the oblique; and, consequently, the same heavy thing goes down with a greater force along a right descent than along an inclined one, by as much as the inclined descent is greater than the right one. </s>
<s id="id.1.2.1.04.18">But it must be understood of this demonstration that there exists no accidental resistance (roughness either of the mobile or of the inclined plane; or because of the shape of the mobile): but it must be presupposed that the plane is somehow incorporeal, or at least very carefully smoothed and hard, so that, while the mobile exerts weight on the plane, it may not cause the plane to bend, and somehow come to rest on it, as in a trap.</s>
<s id="id.1.2.1.04.19">It is also necessary that the mobile be perfectly smooth, and of a shape which does not resist motion, like </s>
