These two modes of proving an identity are so closely allied that it is doubtful how far we can detect any difference in their powers and instances of application. The first method is perhaps more convenient when the phenomena to be compared are rare. Thus we prove that all the musical concords coincide with all the more simple numerical ratios, by showing that each concord arises from a simple ratio of undulations, and then showing that each simple ratio gives rise to one of the concords. To examine all the possible cases of discord or complex ratio of undulation would be impossible. By a happy stroke of induction Sir John Herschel discovered that all crystals of quartz which cause the plane of polarization of light to rotate are precisely those crystals which have plagihedral faces, that is, oblique faces on the corners of the prism unsymmetrical with the ordinary faces. This singular relation would be proved by observing that all plagihedral crystals possessed the power of rotation, and *vice versâ* all crystals possessing this power were plagihedral. But it might at the same time be noticed that all ordinary crystals were devoid of the power. There is no reason why we should not detect any of the four propositions A = AB, B = AB, *a* = *ab*, *b* = *ab*, all of which follow from A = B (p. 115). Sometimes the terms of the identity may be singular objects; thus we observe that diamond is a combustible gem, and being unable to discover any other that is, we affirm-- Diamond = combustible gem. In a similar manner we ascertain that Mercury = metal liquid at ordinary temperatures, Substance of least density = substance of least atomic weight. Two or three objects may occasionally enter into the induction, as when we learn that Sodium ꖌ potassium = metal of less density than water, Venus ꖌ Mercury ꖌ Mars = major planet devoid of satellites. *Induction of Partial Identities*.