We are sitting in class and I was wondering: who came up with the arithmetic mean?

Who proposed it or who mentioned it first in his/her work? Google showed the oldest mentions in 1750-1800 but...

We looked at Wikipedia but didn't find the answer.

Thank you for any lead, as even the teacher doesn't know.

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    $\begingroup$ Arithmetic mean is used from the times immemorial. We do not have sources from that times, thus the question has no definite answer. $\endgroup$ Commented Jan 22, 2018 at 20:49

1 Answer 1


The source is ancient Greek mathematics: at least the Pythagorean Archytas.

The early extant souce seems to be Fragment 2 of the lost work of On Music [cited by Porphyry, On Ptolemy’s Harmonics, 1.5] :

And Archytas speaking about the means writes these things:

“There are three means in music: one is the arithmetic [αριθμητικά], the second geometric [γεωμετρικά] and the third sub-contrary [, which they call “harmonic”].


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