**HARMONIC.** In acoustics, a harmonic is a secondary tone
which accompanies the fundamental or primary tone of a vibrating
string, reed, &c.; the more important are the 3rd, 5th, 7th,
and octave (see Sound; Harmony). A harmonic proportion
in arithmetic and algebra is such that the reciprocals of the
proportionals are in arithmetical proportion; thus, if *a*, *b*, *c*
be in harmonic proportion then 1/*a*, 1/*b*, 1/*c* are in arithmetical
proportion; this leads to the relation 2/*b*＝*ac*/(*a*＋*c*). A harmonic
progression or series consists of terms whose reciprocals
form an arithmetical progression; the simplest example is:
1＋12＋13＋14＋. . . (see Algebra and Arithmetic). The occurrence
of a similar proportion between segments of lines is the
foundation of such phrases as harmonic section, harmonic ratio,
harmonic conjugates, &c. (see Geometry: II. *Projective*). The
connexion between acoustical and mathematical harmonicals
is most probably to be found in the Pythagorean discovery that
a vibrating string when stopped at 12 and 23 of its length yielded
the octave and 5th of the original tone, the numbers, 1 23, 12
being said to be, probably first by Archytas, in harmonic proportion.
The mathematical investigation of the form of a
vibrating string led to such phrases as harmonic curve, harmonic
motion, harmonic function, harmonic analysis, &c. (see
Mechanics and Spherical Harmonics).