1. Between arithmetic, geometry and music there is a difference in
finding the means. In arithmetic in the first place you find it in
this way. You add the extremes and divide and find the half; as for
example, suppose the extremes are VI and XII, you add them and they
make XVIII. You divide and get IX, which is the mean of arithmetic
(_analogicum arithmeticae_), since the mean is surpassed by the last
by as many units as it surpasses the first. For IX surpasses VI by
three units, and XII surpasses it by the same number.
2. According to geometry you find it this way. The extremes
multiplied together make as much as the means multiplied, for
example, VI and XII multiplied make LXXII; the means VIII and IX
multiplied make the same.
3. According to music you find it in this way: The mean is exceeded
by the last term by the part by which it exceeds the first term, as
for example, VI is surpassed by VIII by two units, which is a third
part, and by the same part the mean VIII is surpassed by the last
term which is XII.