Heron Alexandrinus, Mechanica, 1999

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    <archimedes>
      <text>
        <body>
          <chap n="1">
            <pb n="23">
              <s id="A18-1.23.01">[23] Now as for the loads situated on inclined planes, they have the natural tendency also to move downward, as is the motion of all bodies.If that is not as mentioned, then we have also here to think of the reason mentioned before.</s>
              <s id="A18-1.23.02">So let us assume we want to move a load upward on an inclined plane.Let its bottom be smooth and even, the same also the part of the load it supports.</s>
              <s id="A18-1.23.03">For this purpose we have to attach a force or a weight to the other side so that it may be equal to the load, i.e., keep its balance equilibrium so that the surplus of force over it may be strong enough for the load, and lift it upward.</s>
              <s id="A18-1.23.04">In order that our claim is proven correct, we want to prove it with a given cylinder.Since a large part of the cylinder does not touch the ground, it has the natural tendency to roll downward.</s>
              <s id="A18-1.23.05">If we now imagine a plane that goes through the line that touches the ground and is perpendicular to this ground, then it turns out that this plane goes through the axis of the cylinder and divides it into two halves; for if a line touches a circle and if one erects a perpendicular at the point of contact, then this goes through the center of the circle.</s>
              <s id="A18-1.23.06">If we further put through the same line, namely the line of the cylinder, a plane perpendicular to the horizon, then it will not be the first positioned plane and will divide the cylinder into two different parts, the smaller of which lies to the top, the larger to the bottom.So the larger one has the superior weight over the smaller one, since it is larger, and the cylinder rolls.</s>
              <s id="A18-1.23.07">If we now imagine on the other side of the plane perpendicular to the horizon the amount of superior weight over the smaller [part] taken away from the larger part, then both parts are in balance and the weight of both of them remains on the line touching the ground, without inclining to any side, namely neither upward nor downward.</s>
              <s id="A18-1.23.08">Thus we need a force equivalent to this difference that withstands it.If however a small surplus is added to this force, then it gains superior weight over the load.</s>
            </pb>
          </chap>
        </body>
      </text>
    </archimedes>