# Proof of the Number

# of

# Revolutions

Now, it is time to find and prove an algorithm for determining
the number of revolutions that a circle traveling around the outside
or the inside of the circle will make. Recall out picture.

Here, the radius of the original circle has been labeled m
and the radius of the inner and outer circles are n. So, now we
claim that the the number of revolutions traveled around the original
circle by the inner circle m/n -1 and that the number of revolutions
traveled by the outer circle is m/n + 1.

The key for us was to realize that the number of revolutions
hinges on the centers of the inner and outer circles. Although
the animations show the circles traveling around the original
circles, the centers of the circles are actually traveling around
different circles that can be seen as the dashed circles in the
next picture.

Now, we need to look at the ratio of the circumferences of
the dashed circles to the original circle and this will determine
the number of revolutions that the circles will make around the
original circle.

### Case One: The Outer Circle

For this case, we will use the above diagram with original
circle, o and the outer circle c1. The original circle has radius
m and the outer circle has a radius of n. It is known that the
ratio of the radii of two circles is equal to the ratio of their
circumferences. So, c1 will give us the number of revolutions
made by the outer circle to the ratio of the radius of the original
circle.

The radius of the of c1 is equal to m + n, the sum of the radius
of the original circle and the outer circle. Now, computing the
ratio of c1 to o gives us: (m + n)/n. This could also be written
as m/n + 1. This gives us our algorithm for working with the outer
circle.

### Case Two: The Inner Circle

This case works just like the case above except the radius
of the dotted circle is now m - n. So, when finding the ratio
of the original circle to the dotted circle, it is (m - n)/n.
Or, m/n - 1.

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