only one circumstance in common, the
circumstance in which alone all the instances agree is the
cause (or effect) of the given phenomenon.

Herschel's statement, on which this canon is founded, runs as follows:
"Any circumstance in which all the facts without exception agree, may
be the cause in question, or if not, at least a collateral effect
of the same cause: if there be but one such point of agreement, the
possibility becomes a certainty".

All the instances examined must agree in one circumstance--hence the
title Method of Agreement. But it is not in the agreement merely that
the proof consists, but the agreement in one circumstance combined
with difference in all the other circumstances, when we are certain
that every circumstance has come within our observation. It is the
singleness of the agreement that constitutes the proof just as it is
the singleness of the difference in the Method of Difference.[3]

It has been said that Mill's Method of Agreement amounts after all
only to an uncontradicted _Inductio per enumerationem simplicem_,
which he himself stigmatised as Induction improperly so called. But
this is not strictly correct. It is a misunderstanding probably caused
by calling the method that of agreement simply, instead of calling it
the Method of Single Agreement, so as to lay stress upon the process
of elimination by which the singleness is established. It is true that
in the course of our observations we do perform an induction by simple
enumeration. In eliminating, we at the same time generalise. That is
to say, in multiplying instances for the elimination of non-causes, we
necessarily at the same time multiply instances where the true causal
antecedent, if there is only one possible, is present. An antecedent
containing the true cause must always be there when the phenomenon
appears, and thus we may establish by our eliminating observations a
uniformity of connexion between two facts.

Take, for example, Roger Bacon's inquiry into the cause of the colours
of the rainbow. His first notion seems to have been to connect the
phenomenon with the substance crystal, probably from his thinking of
the crystal firmament then supposed to encircle the universe. He found
the rainbow colours produced by the passage of light through hexagonal
crystals. But on extending his observations, he found that the passage
of light through other transparent mediums was also attended by the
phenomenon. He found it