Sixteen Experimental Investigations from the Harvard Psychological Laboratory - Hugo Münsterberg (best life changing books txt) 📗
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bands persist as after-images while new ones are being generated. The
very oldest, however, disappear pari-passu with the generation of
the new. We have already seen (p. 169) how well these authors have
shown this, in proving that the number of bands seen, multiplied by
the rate of rotation of the disc, is a constant bearing some relation
to the duration of a retinal image of similar brightness to the bands.
It is to be noted now, however, that as soon as the rod has produced a
band and passed on, the after-image of that band on the retina is
exposed to the same stimulation from the rotating disc as before, that
is, is exposed to the fused color; and this would tend to obliterate
the after-images. Thus the oldest bands would have to disappear more
quickly than an unmolested after-image of the same original
brightness. We ought, then, to see somewhat fewer bands than the
formula of Jastrow and Moorehouse would indicate. In other words, we
should find on applying the formula that the ‘duration of the
after-image’ must be decreased by a small amount before the numerical
relations would hold. Since Jastrow and Moorehouse did not determine
the relation of the after-image by an independent measurement, their
work neither confirms nor refutes this conjecture.
What they failed to emphasize is that the real origin of the bands is
not the intermittent appearances of the rod opposite the lighter
sector, as they seem to believe, but the successive eclipse by the rod
of each sector in turn.
If, in Fig. 2, we have a disc (composed of a green and a red sector)
and a pendulum, moving to the right, and if P represents the
pendulum at the instant when the green sector AOB is beginning to
pass behind it, it follows that some other position farther to the
right, as P’, will represent the pendulum just as the last part of
the sector is passing out from behind it. Some part at least of the
sector has been hidden during the entire interval in which the
pendulum was passing from P to P’. Clearly the arc BA’ measures
the band BOA’, in which the green stimulation from the sector AOB
is thus at least partially suppressed, that is, on which a relatively
red band is being produced. If the illusion really depends on the
successive eclipse of the sectors by the pendulum, as has been
described, it will be possible to express BA’, that is, the width of
a band, in terms of the widths and rates of movement of the two
sectors and of the pendulum. This expression will be an equation, and
from this it will be possible to derive the phenomena which the bands
of the illusion actually present as the speeds of disc and rod, and
the widths of sectors and rod, are varied.
[Illustration: Fig 2.]
Now in Fig. 2 let the
width of the band (i.e., the arc BA’) = Z
speed of pendulum = r degrees per second;
speed of disc = r’ degrees per second;
width of sector AOB (i.e., the arc AB) = s degrees of arc;
width of pendulum (i.e., the arc BC) = p degrees of arc;
time in which the pendulum moves from P to P’ = t seconds.
Now
arc CA’
t = ––-;
r
but, since in the same time the green sector AOB moves from B to B’,
we know also that
arc BB’
t = ––-;
r’
then
arc CA’ arc BB’
––- = ––-,
r r’
or, omitting the word “arc” and clearing of fractions,
r’(CA’) = r(BB’).
But now
CA’ = BA’ - BC,
while
BA’ = Z and BC = p;
therefore
CA’ = Z-p.
Similarly
BB’ = BA’ + A’B’ = Z + s.
Substituting for CA’ and BB’ their values, we get
r’(Z-p) = r(Z+s),
or
Z(r’ - r) = rs + pr’,
or
Z = rs + pr’ / r’ - r.
It is to be remembered that s is the width of the sector which
undergoes eclipse, and that it is the color of that same sector which
is subtracted from the band Z in question. Therefore, whether Z
represents a green or a red band, s of the formula must refer to the
oppositely colored sector, i.e., the one which is at that time
being hidden.
We have now to take cognizance of an item thus far neglected. When the
green sector has reached the position A’B’, that is, is just
emerging wholly from behind the pendulum, the front of the red sector
must already be in eclipse. The generation of a green band (red sector
in eclipse) will have commenced somewhat before the generation of the
red band (green sector in eclipse) has ended. For a moment the
pendulum will lie over parts of both sectors, and while the red band
ends at point A’, the green band will have already commenced at a
point somewhat to the left (and, indeed, to the left by a trifle more
than the width of the pendulum). In other words, the two bands
overlap.
This area of overlapping may itself be accounted a band, since here
the pendulum hides partly red and partly green, and obviously the
result for sensation will not be the same as for those areas where red
or green alone is hidden. We may call the overlapped area a
‘transition-band,’ and we must then ask if it corresponds to the
‘transition-bands’ spoken of in the observations.
Now the formula obtained for Z includes two such transition-bands, one
generated in the vicinity of OB and one near OA’. To find the formula
for a band produced while the pendulum conceals solely one, the
oppositely colored sector (we may call this a ‘pure-color’ band and
let its width = W), we must find the formula for the width (w) of a
transition-band, multiply it by two, and subtract the product from the
value for Z already found.
The formula for an overlapping or transition-band can be readily found
by considering it to be a band formed by the passage behind P of a
sector whose width is zero. Thus if, in the expression for Z already
found, we substitute zero for s, we shall get w; that is,
o + pr’ pr’
w = ––- = ––
r’ - r r’ - r
Since
W = Z - 2w,
we have
rs + pr’ pr’
W = ––— = 2 ––,
r’ - r r’ - r
or
rs - pr’
W = ––— (1)
r’ - r
[Illustration: Fig 3.]
Fig. 3 shows how to derive W directly (as Z was derived) from the
geometrical relations of pendulum and sectors. Let r, r’, s, p, and
t, be as before, but now let
width of the band (i.e., the arc BA’) = W;
that is, the band, instead of extending as before from where P
begins to hide the green sector to where P ceases to hide the same,
is now to extend from the point at which P ceases to hide _any
part_ of the red sector to the point where it just commences again to
hide the same.
Then
W + p
t = ––- ,
r
and
W + s
t = ––- ,
r’
therefore
W + p W + s
––- = ––- ,
r r’
r’(W + p) = r(W + s) ,
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.”
In formula 3, W represents the width of a band, and s the width of
the oppositely colored sector. Therefore, if a disc is composed, for
example, of a red and a green sector, then
rs(green) - pr’
W(red) = –––––– ,
r’ ± r
and
rs(red) - pr’
W(green) = –––––– ,
r’ ± r
therefore, by dividing,
W(red) rs(green) - pr’
––– = ––––––- .
W(green) rs(red) - pr’
From this last equation it is clear that unless s(green) = s(red),
W(red) cannot equal W(green). That is, if the two sectors are
unequal in width, the bands are also unequal. This was the first
feature of the illusion above noted.
Again, if one sector is larger, the oppositely colored bands will be
larger, that is, the light-colored bands will be narrower; or, in
other words, ‘the narrower bands correspond in color to the larger
sector.’
Finally, if the sectors are equal, the bands must also be
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