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Engel, E. A., Monserrat, B. & Needs, R. J. Anharmonic nuclear motion and the relative stability of hexagonal and cubic ice.

Anharmonic, an-har-mon′ik, adj. not harmonic: in geometry, a term applied to the section of a line by four points, A, B, C, D, when their mutual distances are such that AB divided by CB is unequal to AD divided by CD; the ratio between these two quotients being called the anharmonic ratio of AC.

The expression AC/CB : AD/DB has been called by Chasles the “anharmonic ratio of the four points A, B, C, D.”

That this formula is projective is rendered evident by observing that e−2iρ/k is the anharmonic ratio of the range consisting of the two points and the intersections of the line joining them with the fundamental quadric.

These were found in the anharmonic ratio, a device leading to the liberation of synthetic geometry from metrical relations, and in involution, which yielded rigorous definitions of imaginaries.