Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

THE MATHEMATICS OF PSYCHOLOGY 383
sures or of weights or of sizes or of sounds and so on for
the other departments of sense.
A Remarkable Property of Sense " Continua."-Owing
to the presence of difference thresholds in sense departments, the sense continuum of any department possesses
a remarkable property. It is this: If Si, S2, Sa are sensations corresponding to three different amounts, R1, R2, Ra,
of stimulus, it may happen that no two of the S's are distinguishable from each other and it may happen that Si
is indistinguishable from S2, and that S2 is indistinguishable from S3 but that nevertheless Si and Ss are distinguishable from each other. So that in a given department we may have three S's (three in the sense of their
being produced by three different amounts of stimulus)
such that
(I) Sl=S2, S2-=S3, S1 =S3;
and we may have three such that
(2) S1=S2, S2=S3, S1 S3;
of course there are other possibilities. Thus if the sense
"continuum" be that of pressures, then, if the S's
correspond respectively to Io, Io0 and iI grams, they
satisfy relations (I); but if they correspond to 10, i and
I2 grams, they satisfy relations (2). The matter may,
of course, be exemplified in other sense "continua."
Thus in the sense " continuum " of lengths the sensations
produced by (say) three pencils of three different lengths
may satisfy relations (I) or (2).
Another Aspect of the Matter.-This matter has another
striking and important aspect. If sensation be allowed to
judge, then we should say that a stimulus Q and a stimulus
Q' (of the same kind as Q) are equal if the corresponding