irect relation of cause and effect.

It may occur to the reader that we ought also to distinguish Qualitative
and Quantitative Variations as two orders of phenomena to which the
present method is applicable. But, in fact, Qualitative Variations may
be adequately dealt with by the foregoing methods of Agreement, Double
Agreement, and Difference; because a change of quality or property
entirely gets rid of the former phase of that quality, or substitutes
one for another; as when the ptarmigan changes from brown to white in
winter, or as when a stag grows and sheds its antlers with the course of
the seasons. The peculiar use of the method of Variations, however, is
to formulate the conditions of proof in respect of those causes or
effects which cannot be entirely got rid of, but can be obtained only in
greater or less amount; and such phenomena are or course, quantitative.

Even when there are two parallel series of phenomena the one
quantitative and the other qualitative--like the rate of air-vibration
and the pitch of sound, or the rate of ether-vibration and the
colour-series of the spectrum--the method of Variations is not
applicable. For (1) two such series cannot be said to vary together,
since the qualitative variations are heterogeneous: 512: 576 is a
definite ratio; but the corresponding notes, C, D, in the treble clef,
present only a difference. Hence (2) the correspondence of each note
with each number is a distinct fact. Each octave even is a distinct
fact; there is a difference between C 64 and C 128 that could never have
been anticipated without the appropriate experience. There is,
therefore, no such law of these parallel series as there is for
temperature and change of volume (say) in mercury. Similar remarks apply
to the physical and sensitive light-series.

We may illustrate the two cases of the method thus (putting a dash
against any letter, A' or _p_', to signify an increase or decrease of
the phenomenon the letter stands for): Agreement in Variations (other
changes being admissible)--

A B C A' D E A'' F G
_p q r_ _p' s t_ _p'' u v_

Here the accompanying phenomena (_B C q r, D E s t, F G u v_) change
from time to time, and the one thing in which the instances agree
throughout is that any increase of A (A' or A'') is followed or
accompanied by an increase of _p (p' or p'')_: whence it is argued that
A is the cause of _p_, according to Prop. III. (a) (