impossible. It must mean that the sensation seems just as much like
two as it does like one, and he therefore describes it as half way
between. If we could discover any law governing this feeling of
half-way-between-ness, that might well indicate the threshold. But
such feelings are not common. Sensations which seem between one and
two usually call forth the answer 'doubtful,' and have a negative
rather than a positive character. This negative character cannot be
due to the stimulus; it must be due to the fluctuating attitudes of
the subject. However, if the doubtful cases could be classed with the
'more than one but less than two' cases and a law be found governing
them, we might have a threshold mark. But such a law has not been
formulated, and if it had been an analysis of the 'doubtful' cases
would invalidate it. For, since we cannot have half of a sensation or
half of a place as we might have half of an area, the subject regards
each stimulation as produced by one or by two points as the case may
be. Occasionally he is stimulated in such a way that he can regard the
object as two or as one with equal ease. In order to describe this
feeling he is likely to use one or the other of the methods just
mentioned.

We might say that when the sum of conditions is such that the subject
perceives two points, the points are above the threshold, and when the
subject perceives one point when two are given they are below the
threshold. This might answer the purpose very well if it were not for
the _Vexirfehler_. According to this definition, when the
_Vexirfehler_ appears we should have to say that one point is above
the threshold for twoness, which is a queer contradiction, to say the
least. It follows that all of the elaborate and painstaking
experiments to determine a threshold are useless. That is, the
threshold determinations do not lead us beyond the determinations