Ancient mathematicians are a unique tribe. Exploring the works of Nicomachus in Algebra shows us that he and his comrades were thinking deeply, discussing thoroughly, and contributing consistently. Take the simple mean. Most students learn the calculation of the mean of a group of numbers as the sum of the group divided by the quantity of values. In simple lists, this mean suffices to give us a generally reliable measurement of the midpoint of data. Do Nicomachus and his friends stop there though? Of course not, they don't simply dwell on this arithmetic mean, but also seek to discover a mean in music, called the harmonic mean, and a mean in geometry, the geometric mean.
Suffice it to say, this is where most of my student's eye's glaze over and we walk away just thankful to have understood the first part. Bear with me, as I give you these definitions. There is goodness in the geometric mean. The geometric mean is found by taking the square root of the product of the numbers.A relatively simple concept for an algebra student. It's even not that difficult to see why it's more clearly a communication of the middle of the data. We just need to play with a few numbers... (did you think that I was going to start getting all numerical on you? Some other time I guess.)
While there is truth in the arithmetic mean, and goodness in the geometric mean, there is beauty in the harmonic mean. The harmonic mean is found in the 4/3 ratio in music is called the perfect fourth. 4 is the harmonic mean between 3 and 6, I would explain it to you, but that would require me to understand it more fully myself. Truly, my mind is not as advanced in these ideas as Nicomachus, Pythagoras, and their contemporaries.
What is my point of this long ramble of Nicomachus' famous three means? My point is simple: the school year ends, students graduate, but there is always something new and worthwhile to learn. Where will your next educational journey take you?