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, i/c are in arithmetical proportion; this leads to the 'relation 2/b=acl(a+c). A harmonic progression or series consists of terms whose reciprocals form an arithmetical progression; the simplest example is: i + 2 + a + 4 + ... (See Algebra arid 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 z and a of its length yielded the octave and 5th of the original tone, the numbers, 13, 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).