Three positive numbers $x$ , $m$ , $y$ are in contraharmonic proportion, if the ratio of the difference of the second and the first number to the difference of the third and the second number is equal the ratio of the third and the first number, i.e. if

(1)

The contraharmonic proportion has very probably been known in the proportion doctrine of the Pythagoreans, since they have in a manner similar to (1) described the classical Babylonian means: $$\frac{m-x}{y-m} = \frac{x}{x} \qquad (\mbox{arithmetic mean }m)$$ $$\frac{m-x}{y-m} = \frac{x}{m} \qquad (\mbox{geometric mean }m)$$ $$\frac{m-x}{y-m} = \frac{x}{y} \qquad (\mbox{harmonic mean }m)$$

The contraharmonic mean $m$ is between $x$ and $y$ . Indeed, if we solve it from (1), we get

(2)

Example. The integer 5 is the contraharmonic mean of 2 and 6, as well as of 3 and 6, i.e. 2, 5, 6, are in contraharmonic proportion, similarly are 3, 5, 6: $$\frac{2^2+6^2}{2+6} = \frac{40}{8} = 5, \quad \frac{3^2+6^2}{3+6} = \frac{45}{9} = 5$$

Note. Generalising (2) one defines the contraharmonic mean of several positive numbers: $$c(x_1,\,\ldots,\,x_n) := \frac{x_1^2+\ldots+x_n^2}{x_1+\ldots+x_n}$$ There is also a more general type of mean: $$c^m(x_1,\,\ldots,\,x_n) := \frac{x_1^{m+1}+\ldots+x_n^{m+1}}{x_1^m+\ldots+x_n^m}$$

Bibliography

1

DIDEROT & D'ALEMBERT: Encyclopédie. Paris (1751-1777). Electronic version in L'Encyclopédie de Diderot et d'Alembert.

2

HORST HISCHER: ``Viertausend Jahre Mittelwertbildung''. -- mathematica didactica 25 (2002). See also this.