The reasoning is precisely the same for the IIIds and 3ds, considering the former as forming 4 distinct series of an octave each, beginning with C, C♯, D and E♭; and the latter as forming 3 distinct series of an octave each, beginning with C, C♯ and D. If the former be made all equal, each will be sharpened 343; if the latter be made equal, each will be flattened 392. In every system which renders none of the former flat, and none of the latter sharp, the sum of their temperaments will be 12 × 343, and 12 × 392, respectively. _Cor._ The demonstration holds equally true, whatever be the magnitude of _a_, _b_, _c_, &c.: only if they be such that the difference -_a_ + _b_, -_b_ + _c_, &c. of any two successive ones be greater than the temperament of the corresponding concord in the system of equal semitones, the temperament of that chord must be reckoned negative, and the _sum_, in the enunciation of the proposition, must be considered as the excess of those temperaments which have the same sign with those of the same concords in the system of equal semitones, above those which have the contrary sign. Hence it is universally true that the excess of the flat above the sharp temperaments of the Vths is equal to 12 × 49; that the excess of the sharp above the flat temperaments of the IIIds is equal to 12 × 343; and that the excess of the flat above the sharp temperaments of the 3ds is 12 × 392. Hence likewise we have a very easy method of _proving_ whether the temperaments of any given system have been correctly calculated. It is only to add those which have the same sign; and if the differences of the sums be equal to the products just stated, the work is right. PROPOSITION IX. If all the concords of the same name, in a scale of twelve intervals to the octave, were of equally frequent occurrence, the best system of temperament would be that of equal semitones.