The ancient Greeks defined ten means in terms of the following proportions (see *here*):

Let *a* > *m* > *b* > 0. Then *m* represents

(1) the arithmetic mean of *a* and *b* if

(2) the geometric mean of *a* and *b* if

(3) the harmonic mean of *a* and *b* if

(4) the contraharmonic mean of *a* and *b* if

(5) the first contrageometric mean of *a* and *b* if

(6) the second contrageometric mean of *a* and *b* if

(7) the semicontraharmonic mean of *a* and *b* if

(8) the semicontrageometric mean of *a* and *b* if

(9) the semiharmonic mean of *a* and *b* if

(10) the semigeometric mean of a and b if

The names for the last four means are mine.

The algebraic expressions for the ten means *M*_{n}(*a*, *b*) are

(1) the arithmetic mean *M*_{1}(*a*, *b*)

(2) the geometric mean *M*_{2}(*a*, *b*)

Note that

(3) the harmonic mean *M*_{3}(*a*, *b*)

(4) the contraharmonic mean *M*_{4}(*a*, *b*)

(5) the first contrageometric mean *M*_{5}(*a*, *b*)

(6) the second contrageometric mean *M*_{6}(*a*, *b*)

(7) the semicontraharmonic mean *M*_{7}(*a*, *b*)

(8) the semicontrageometric mean *M*_{8}(*a*, *b*)

(9) the semiharmonic mean *M*_{9}(*a*, *b*)

(10) the semigeometric mean *M*_{10}(*a*, *b*)

Only the first four means are symmetric. All means are reflexive.