<s id="A18-1.07.01">[7] Sometimes however the motion of the smaller and the larger circles can be equally fast, even when the circles are attached to the same center and rotate around it.</s>
<s id="A18-1.07.02">Let us assume two circles attached to the same center <a> and let a tangent to the larger circle be given, namely the line <bb'>.</s>
<s id="A18-1.07.03">If we further connect the points <a>, <b>, then line <ab> is perpendicular to line <bb'> and line <bb'> is parallel to line <gg'>; then line <gg'> is a tangent of the smaller circle.</s>
<s id="A18-1.07.04">If we further draw through point <a> a line that is parallel to these lines, namely line <aa'>, then if we imagine the larger circle rolling on line <bb'>, the smaller circle will roll by running through the line <gg'>.</s>
<s id="A18-1.07.05">When now the larger circle has made one rotation, we see that the smaller one also has made one rotation, so that the position of the circles is the position of those circles whose center is at <a'> and the position of line <ab> is that which is taken by line <a'b'>.Therefore line <bb'> equals line <gg'>.</s>
<s id="A18-1.07.06">Line <bb'> however is the line on which the larger circle rolls when it makes one rotation and line <gg'> is the line on which the smaller circle rolls when it makes one rotation; thus the motion of the smaller circle is equally fast as that of the larger one, because line <bb'> equals line <gg'>.</s>
<s id="A18-1.07.07">Things that run through the same distance in the same time, however, have equal speed and equal motion.</s>
<s id="A18-1.07.08">One might think this sentence is absurd, since it is impossible that the circumference of the larger circle should equal the circumference of the smaller one.</s>
<s id="A18-1.07.09">We now say that not only the circumference of the smaller circle has rolled on line <gg'>, but that the smaller circle also runs through the path of the larger one, thus we see that the smaller circle through two motions reaches the same speed as the larger one; then, if we imagine the larger circle rolling, the smaller one, however, not rolling, but only attached to the point <g>, then it will in the same time cover line <gg'>; then the center <a> covers in this time line <aa'>.</s>
<s id="A18-1.07.10">This however equals the lines <bb'> and <gg'>; thus the continuous rolling of the smaller circle does not make any difference in the motion and as a consequence the length of the distance of the larger circle is the same as that covered by the small circle; for we see that the center, without rolling, covers the same distance, due to the motion the large circle is in.</s>
