Opening the text

Harvard Psychological Studies, Volume 1: Containing Sixteen Experimental Investigations from the Harvard Psychological Laboratory.

Science

Harvard Psychological Studies, Volume 1: Containing Sixteen Experimental Investigations from the Harvard Psychological Laboratory.

Psychology, Experimental

W (r' - r) = rs - pr' ,
and, again,
                          rs - pr'
                      W = -------- .
                           r' - r

Before asking if this pure-color band _W_ can be identified with the
bands observed in the illusion, we have to remember that the value
which we have found for _W_ is true only if disc and pendulum are
moving in the same direction; whereas the illusion-bands are observed
indifferently as disc and pendulum move in the same or in opposite
directions. Nor is any difference in their width easily observable in
the two cases, although it is to be borne in mind that there may be a
difference too small to be noticed unless some measuring device is
used.

From Fig. 4 we can find the width of a pure-color band (_W_) when
pendulum and disc move in opposite directions. The letters are used as
in the preceding case, and _W_ will include no transition-band.

[Illustration: Fig. 4]

We have

           W + p
       t = -----,
             r
and
           s - W
       t = -----,
             r'

       r'(W + p) = r(s - W) ,

       W(r' + r) = rs - pr' ,

           rs - pr'
       W = -------- .           (2)
            r' + r

Now when pendulum and disc move in the same direction,

            rs - pr'
       W = --------- ,          (1)
            r' - r

so that to include both cases we may say that

           rs - pr'
       W = -------- .           (3)
            r' ± r

The width (W) of the transition-bands can be found, similarly, from
the geometrical relations between pendulum and disc, as shown in Figs.
5 and 6. In Fig. 5 rod and disc are moving in the same direction, and

        w = BB'.

Now
             W - p
        t = ------- ,
              r'

             w
        t = --- ,
             r'

        r'(w-p) = rw ,

        w(r'-r) = pr' ,


                pr'
         w = ------- .          (4)
               r'-r


[Illustration: Fig. 5]

[Illustration: Fig. 6]

In Fig. 6 rod and disc are moving in opposite directions, and

         w = BB',

              p - w
         t = ------- ,
                r

              w
         t = --- ,
              r'

         r'(p - w) = rw ,

         w(r' + r) = pr' ,

                pr'
         w = -------- .
              r' + r            (5)

So that to include both cases (of movement in the same or in opposite
directions), we have that

                pr'
         w = -------- .
              r' ± r            (6)



VI. APPLICATION OF THE FORMULAS TO THE BANDS OF THE ILLUSION.


Will these formulas, now, explain the phenomena which the bands of the
illusion actually present in respect to their width?


1. The first phenomenon noticed (p. 173, No. 1) is that "If the two
sectors of the disc are unequal in arc, the bands are unequal in
width; and the narrower bands correspond in color to the larger
sector. Equal sectors give equally broad bands."

Public-domain text, read in full here on John Shaqi.

Opening the text

Science

Harvard Psychological Studies, Volume 1: Containing Sixteen Experimental Investigations from the Harvard Psychological Laboratory.

Psychology, Experimental

W (r' - r) = rs - pr' ,
and, again,
                          rs - pr'
                      W = -------- .
                           r' - r

Before asking if this pure-color band _W_ can be identified with the
bands observed in the illusion, we have to remember that the value
which we have found for _W_ is true only if disc and pendulum are
moving in the same direction; whereas the illusion-bands are observed
indifferently as disc and pendulum move in the same or in opposite
directions. Nor is any difference in their width easily observable in
the two cases, although it is to be borne in mind that there may be a
difference too small to be noticed unless some measuring device is
used.

From Fig. 4 we can find the width of a pure-color band (_W_) when
pendulum and disc move in opposite directions. The letters are used as
in the preceding case, and _W_ will include no transition-band.

[Illustration: Fig. 4]

We have

           W + p
       t = -----,
             r
and
           s - W
       t = -----,
             r'

       r'(W + p) = r(s - W) ,

       W(r' + r) = rs - pr' ,

           rs - pr'
       W = -------- .           (2)
            r' + r

Now when pendulum and disc move in the same direction,

            rs - pr'
       W = --------- ,          (1)
            r' - r

so that to include both cases we may say that

           rs - pr'
       W = -------- .           (3)
            r' ± r

The width (W) of the transition-bands can be found, similarly, from
the geometrical relations between pendulum and disc, as shown in Figs.
5 and 6. In Fig. 5 rod and disc are moving in the same direction, and

        w = BB'.

Now
             W - p
        t = ------- ,
              r'

             w
        t = --- ,
             r'

        r'(w-p) = rw ,

        w(r'-r) = pr' ,


                pr'
         w = ------- .          (4)
               r'-r


[Illustration: Fig. 5]

[Illustration: Fig. 6]

In Fig. 6 rod and disc are moving in opposite directions, and

         w = BB',

              p - w
         t = ------- ,
                r

              w
         t = --- ,
              r'

         r'(p - w) = rw ,

         w(r' + r) = pr' ,

                pr'
         w = -------- .
              r' + r            (5)

So that to include both cases (of movement in the same or in opposite
directions), we have that

                pr'
         w = -------- .
              r' ± r            (6)



VI. APPLICATION OF THE FORMULAS TO THE BANDS OF THE ILLUSION.


Will these formulas, now, explain the phenomena which the bands of the
illusion actually present in respect to their width?


1. The first phenomenon noticed (p. 173, No. 1) is that "If the two
sectors of the disc are unequal in arc, the bands are unequal in
width; and the narrower bands correspond in color to the larger
sector. Equal sectors give equally broad bands."

Public-domain text, read in full here on John Shaqi.

Harvard Psychological Studies, Volume 1: Containing Sixteen Experimental Investigations from the Harvard Psychological Laboratory.

Harvard Psychological Studies, Volume 1: Containing Sixteen Experimental Investigations from the Harvard Psychological Laboratory.

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