Harmonic Proportion — Chambers's Encyclopaedia

Harmonic Proportion. Three numbers are said to be in harmonic proportion when the first is to the third as the difference between the first and second is to the difference between the second and third; otherwise, harmonic proportion is that which subsists between the reciprocals of numbers which are in arithmetical proportion. Thus, 3, 5, 7, &c., being in arithmetical proportion, $\frac{1}{3}$ , $\frac{1}{5}$ , $\frac{1}{7}$ , &c., are in harmonic proportion. In geometry, a line

\begin{array}{c} \overline{A \quad C \quad B \quad D} \end{array}

AB is said to be harmonically divided when two points are taken, one in the line and the other in the line produced, as C and D, such that $AC : CB :: AD : DB$ . When the line is thus divided, AD, CD, and BD are in harmonic proportion. A harmonic progression is a series of numbers in harmonic proportion, as the series formed by the reciprocals of numbers forming an arithmetical series.