Heron Alexandrinus, Mechanica, 1999

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[5] If we imagine a third constructed circle which touches the circle with the center <b>, so we shall prove for the third circle what we mentioned about the first one.For if the first circle is moving in the direction opposite from the second one, the second one however moves opposite to the third, then the motion of the first circle is the same as that of the third.For if something is moving in the same manner as something else, this however moves in the opposite direction of a third thing, so is the first thing moving in the direction opposite to the third.If further a fourth circle is present, we proceed after the same method.In general, what ensues from the three circles will occur with all circles whose number is odd and what ensues from the two circles takes place with all circles whose number is even.But one not only sees with two and more circles that the motion is now equal, now opposite, but in one circle one sees the same point move now in one direction, now in its opposite.For when the moving point starts moving at any point, it does not stop moving in the same direction until it has run through a semicircle; when it now runs through the second semicircle it moves in the direction opposite to it.

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