Ancient Greek means

Knowing / By rag / February 14, 2024

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

$\frac{a-m}{m-b}=\frac{a}{a};$

(2) the geometric mean of a and b if

$\frac{a-m}{m-b}=\frac{a}{m}=\frac{m}{b};$

(3) the harmonic mean of a and b if

$\frac{a-m}{m-b}=\frac{a}{b};$

(4) the contraharmonic mean of a and b if

$\frac{a-m}{m-b}=\frac{b}{a};$

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

$\frac{a-m}{m-b}=\frac{b}{m};$

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

$\frac{a-m}{m-b}=\frac{m}{a};$

(7) the semicontraharmonic mean of a and b if

$\frac{a-m}{a-b}=\frac{b}{a};$

(8) the semicontrageometric mean of a and b if

$\frac{a-m}{a-b}=\frac{m}{a};$

(9) the semiharmonic mean of a and b if

$\frac{a-b}{m-b}=\frac{a}{b};$

(10) the semigeometric mean of a and b if

$\frac{a-b}{m-b}=\frac{m}{b}.$

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 M₁(a, b)

$\frac{a+b}{2};$

(2) the geometric mean M₂(a, b)

$(ab)^{1/2};$

Note that

$M_2=(M_1M_3)^{1/2}$

(3) the harmonic mean M₃(a, b)

$\frac{a \boxplus b}{2^{-1}}=\left ( \frac{a^{-1}+b^{-1}}{2} \right )^{-1}= \frac{2ab}{a+b}=\frac{(a+b)^2-(a^2+b^2)}{a+b};$

(4) the contraharmonic mean M₄(a, b)

$\frac{a^2+b^2}{a+b}=\frac{(a+b)^2-2ab}{a+b};$

(5) the first contrageometric mean M₅(a, b)

$\frac{a-b+\sqrt{(a-b)^2+4b^2}}{2};$

(6) the second contrageometric mean M₆(a, b)

$\frac{b-a+\sqrt{(b-a)^2+4a^2}}{2};$

(7) the semicontraharmonic mean M₇(a, b)

$\frac{a^2-ab+b^2}{a}=\frac{ab+(a-b)^2}{a};$

(8) the semicontrageometric mean M₈(a, b)

$\frac{a^2}{2a-b}=\frac{a}{2-b/a};$

(9) the semiharmonic mean M₉(a, b)

$\frac{2ab-b^2}{a}=\frac{a^2-(a-b)^2}{a};$

(10) the semigeometric mean M₁₀(a, b)

$\frac{b+\sqrt{4ab-3b^2}}{2}.$

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

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