I can't verify this, but I read long ago that the increments in the Fahrenheit Scale (the first widely-accepted temperature scale) were called "degrees" because Daniel Fahrenheit (1686–1736) initially established 2 points of reference -- the freezing and boiling points of water, which, being "opposite" states of water, were designated on his scale to be 180 degrees apart. So he borrowed the concept from geometry circle measure, and so a little circle is placed as a superscript before F when specifying degrees Farhrenheit.
The small circle is the symbol for degrees. These are both measurements of temperature, although the value of a degree in one is different than the other.
Freezing is 0 degrees Centrigrade and 32 degrees Fahrenheit. The boiling point of water is 100 degrees Centigrade and 212 degrees Fahrenheit. As you can see, one degree represents a greater change in C than in F. The actual formula is:
(F - 32) x 5/9 = C
Even though they are different scales, in both a degree is a measure of the change in temperature.
The small circle above letter C (Celsius) and K(Kelvin) stands for degree, and it is a degree for measurement for scientific purposes.
The purpose of the degree symbol in Centigrade works out to be some sort of a scale, you know for standardization so that the thermometer would be easier to use.