Heron Alexandrinus - Mechanica

Heron Alexandrinus, Mechanica, 1999

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[3] After this has been made clear in this introduction, let us rotate two equal circles, namely <hekd> and <zgqe>, around their centers <a>, <b>, while they touch at point <e>.If they now move from point <e> for the same time for half their extant, point <e> in this time runs through the arc <ehd> and reaches the point <d> by moving like the point <g> on the arc <gqe>.Then it can occur that points move in the same direction and that they move in opposite directions.The ones positioned on the same side move in opposite directions, the ones opposed to each other move in the same direction.It may occur however, that points that are described as being in opposite motion go in the same direction (both upward or both downward).For, when points move and their motion starts from one point, namely the point <e>, and we imagine two lines <zaq> and <hbk> perpendicular to the line <gd>, then the motion on the arc <ez> is the opposite of the motion on the arc <eh>, since the one goes to the right, the other to the left side.The motion can also occur in the same direction, if we imagine the distance of the points staying the same from <zh> (text <zk>).Likewise when the motion on the arc <zg> and <hd> towards <g> and <d> is balanced.We also have to assume the same for the arcs <gq>, <dk> and for the arcs <qe> and <ke>.We further say that they can move in the same direction.For we say that the points <de> move in the same direction (this time to the left), when point <e> moves on the arc <ezg> and point <d> on the arc <dke>, and their distance from points <z>, <k> as well as their approach to them remains the same, so the motion is still called opposite (because <e> moves up, then down, <d> down, then up).Therefore the same and the opposite are just complementary and in any motion one has to distinguish between the same and the opposite.This our explanation has to be observed with equal circles.As for different circles, we shall demonstrate it in the following.

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