Various - Harvard Psychological Studies, Volume 1

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Now, similarly, at _p^{B}_ the sector _B_ has come around and begins
to pass behind _P_. It in turn will emerge at some point to the right,
say _p^{C}_. And so the process will continue. From _p^{A}_ to _p^{B}_
the pendulum covers some part of the sector _A_; from _p^{B}_ to
_p^{C}_ some part of sector _B_; from _p^{C}_ to _P^{D}_ some part of
_A_ again, and so on.

[Illustration: Fig. 1.]

If, now, the eye which watches this process is kept from moving, these
relations will be reproduced on the retina. For the retinal area
corresponding to the triangle _p^{A}Op^{B}_, there will be less
stimulation from the sector _A_ than there would have been if the
pendulum had not partly hidden it. That is, the triangle in question
will not be seen of the fused color of _A_ and _B_, but will lose a
part of its _A_-component. In the same way the triangle _p^{B}OpC_
will lose a part of its _B_-component; and so on alternately. And by
as much as either component is lost, by so much will the color of the
intercepting pendulum (in this case, black) be present to make up the
deficiency.

We see, then, that the purely geometrical relations of disc and
pendulum necessarily involve for vision a certain banded appearance of
the area which is swept by the pendulum, if the eye is held at rest.
We have now to ask, Are these the bands which we set out to study?
Clearly enough these geometrically inevitable bands can be exactly
calculated, and their necessary changes formulated for any given
change in the speed or width of _A_, _B_, or _P_. If it can be shown
that they must always vary just as the bands we set out to study are
_observed_ to vary, it will be certain that the bands of the illusion
have no other cause than the interception of retinal stimulation by
the sectors of the disc, due to the purely geometrical relations
between the sectors and the pendulum which hides them.

And exactly this will be found to be the case. The widths of the bands
of the illusion depend on the speed and widths of the sectors and of
the pendulum used; the colors and intensities of the bands depend on
the colors and intensities of the sectors (and of the pendulum); while
the total number of bands seen at one time depends on all these
factors.

V. GEOMETRICAL DEDUCTION OF THE BANDS.

In the first place, it is to be noted that if the pendulum proceeds
from left to right, for instance, before the disc, that portion of the
latter which lies in front of the advancing rod will as yet not have
been hidden by it, and will therefore be seen of the unmodified, fused
color. Only behind the pendulum, where rotating sectors have been
hidden, can the bands appear. And this accords with the first
observation (p. 167), that "The rod appears to leave behind it on the
disc a number of parallel bands." It is as if the rod, as it passes,
painted them on the disc.

Clearly the bands are not formed simultaneously, but one after another
as the pendulum passes through successive positions. And of course the
newest bands are those which lie immediately behind the pendulum. It
must now be asked, Why, if these bands are produced successively, are
they seen simultaneously? To this, Jastrow and Moorehouse have given
the answer, "We are dealing with the phenomena of after-images." The
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 equal.

So far, then, the bands geometrically deduced present the same
variations as the bands observed in the illusion.

2. Secondly (p. 174, No. 2), "The faster the rod moves the broader
become the bands, but not in like proportions; broad bands widen
relatively more than narrow ones." The speed of the rod or pendulum,
in degrees per second, equals _r_. Now if _W_ increases when _r_
increases, _D_{[tau]}W_ must be positive or greater than zero for all
values of _r_ which lie in question.

Now
rs - pr'
W = --------- ,
r' +- r
and
(r' +- r)s [+-] (rs - pr')
D_{[tau]}W = -------------------------- ,
(r +- r')

or reduced,
r'(s +- p)
= -----------
(r' +- r) squared

Since _r'_ (the speed of the disc) is always positive, and _s_ is
always greater than _p_ (cf. p. 173), and since the denominator is a
square and therefore positive, it follows that

D_{[tau]}W > 0

or that _W_ increases if _r_ increases.

Furthermore, if _W_ is a wide band, _s_ is the wider sector. The rate
of increase of _W_ as _r_ increases is

r'(s +- p)
D_{[tau]}W = -----------
(r' +- r) squared

which is larger if _s_ is larger (_s_ and _r_ being always positive).
That is, as _r_ increases, 'broad bands widen relatively more than
narrow ones.'

3. Thirdly (p. 174, No. 3), "The width of The bands increases if the
speed of the revolving disc decreases." This speed is _r'_. That the
observed fact is equally true of the geometrical bands is clear from
inspection, since in

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

as _r'_ decreases, the denominator of the right-hand member decreases
while the numerator increases.

4. We now come to the transition-bands, where one color shades over
into the other. It was observed (p. 174, No. 4) that, "These partake
of the colors of both the sectors on the disc. The wider the rod the
wider the transition-bands."

We have already seen (p. 180) that at intervals the pendulum conceals
a portion of both the sectors, so that at those points the color of
the band will be found not by deducting either color alone from the
fused color, but by deducting a small amount of both colors in
definite proportions. The locus of the positions where both colors are
to be thus deducted we have provisionally called (in the geometrical
section) 'transition-bands.' Just as for pure-color bands, this locus
is a radial sector, and we have found its width to be (formula 6, p.
184)
pr'
W = --------- ,
r' +- r

Now, are these bands of bi-color deduction identical with the
transition-bands observed in the illusion? Since the total concealing
capacity of the pendulum for any given speed is fixed, less of
_either_ color can be deducted for a transition-band than is deducted
of one color for a pure-color band. Therefore, a transition-band will
never be so different from the original fusion-color as will either
'pure-color' band; that is, compared with the pure color-bands, the
transition-bands will 'partake of the colors of both the sectors on
the disc.' Since
pr'
W = --------- ,
r' +- r

it is clear that an increase of _p_ will give an increase of _w_;
_i.e._, 'the wider the rod, the wider the transition-bands.'

Since _r_ is the rate of the rod and is always less than _r'_, the
more rapidly the rod moves, the wider will be the transition-bands
when rod and disc move in the same direction, that is, when

pr'
W = --------- ,
r' - r

But the contrary will be true when they move in opposite directions,
for then

pr'
W = --------- ,
r' + r

that is, the larger _r_ is, the narrower is _w_.

The present writer could not be sure whether or not the width of
transition-bands varied with _r_. He did observe, however (page 174)
that 'the transition-bands are broader when rod and disc move in the
same, than when in opposite directions.' This will be true likewise
for the geometrical bands, for, whatever _r_ (up to and including _r_
= _r'_),

pr' pr'
---- > ----
r'-r r'+r

In the observation, of course, _r_, the rate of the rod, was never so
large as _r'_, the rate of the disc.

5. We next come to an observation (p. 174, No. 5) concerning the
number of bands seen at any one time. The 'geometrical deduction of
the bands,' it is remembered, was concerned solely with the amount of
color which was to be deducted from the fused color of the disc. _W_
and _w_ represented the widths of the areas whereon such deduction was
to be made. In observation 5 we come on new considerations, _i.e._, as
to the color from which the deduction is to be made, and the fate of
the momentarily hidden area which suffers deduction, _after_ the
pendulum has passed on.

We shall best consider these matters in terms of a concept of which
Marbe[3] has made admirable use: the 'characteristic effect.' The
Talbot-Plateau law states that when two or more periodically
alternating stimulations are given to the retina, there is a certain
minimal rate of alternation required to produce a just constant
sensation. This minimal speed of succession is called the critical
period. Now, Marbe calls the effect on the retina of a light-stimulation
which lasts for the unit of time, the 'photo-chemical unit-effect.'
And he says (_op. cit._, S. 387): "If we call the unit of time
1[sigma], the sensation for each point on the retina in each unit of
time is a function of the simultaneous and the few immediately
preceding unit-effects; this is the characteristic effect."

[3] 'Marbe, K.: 'Die stroboskopischen Erscheinungen,' _Phil.
Studien._, 1898, XIV., S. 376.

We may now think of the illusion-bands as being so and so many
different 'characteristic effects' given simultaneously in so and so
many contiguous positions on the retina. But so also may we think of
the geometrical interception-bands, and for these we can deduce a
number of further properties. So far the observed illusion-bands and
the interception-bands have been found identical, that is, in so far
as their widths under various conditions are concerned. We have now to
see if they present further points of identity.

As to the characteristic effects incident to the interception-bands;
in Fig. 7 (Plate V.), let _A'C'_ represent at a given moment _M_, the
total circumference of a color-disc, _A'B'_ represent a green sector
of 90 deg., and _B'C'_ a red complementary sector of 270 deg.. If the disc is
supposed to rotate from left to right, it is clear that a moment
previous to _M_ the two sectors and their intersection _B_ will have
occupied a position slightly to the left. If distance perpendicularly
above _A'C'_ is conceived to represent time previous to _M_, the
corresponding previous positions of the sectors will be represented by
the oblique bands of the figure. The narrow bands (_GG_, _GG_) are the
loci of the successive positions of the green sector; the broader
bands (_RR_, _RR_), of the red sector.

In the figure, 0.25 mm. vertically = the unit of time = 1[sigma]. The
successive stimulations given to the retina by the disc _A'C'_, say at
a point _A'_, during the interval preceding the moment _M_ will be

green 10[sigma],
red 30[sigma],
green 10[sigma],
red 30[sigma], etc.

Now a certain number of these stimulations which immediately precede
_M_ will determine the characteristic effect, the fusion color, for
the point _A'_ at the moment _M_. We do not know the number of
unit-stimulations which contribute to this characteristic effect, nor
do we need to, but it will be a constant, and can be represented by a
distance _x_ = _A'A_ above the line _A'C'_. Then _A'A_ will represent
the total stimulus which determines the characteristic effect at _A'_.
Stimuli earlier than _A_ are no longer represented in the after-image.
_AC_ is parallel to _A'C'_, and the characteristic effect for any
point is found by drawing the perpendicular at that point between the
two lines _A'C_ and _AC_.

Just as the movement of the disc, so can that of the concealing
pendulum be represented. The only difference is that the pendulum is
narrower, and moves more slowly. The slower rate is represented by a
steeper locus-band, _PP'_, than those of the swifter sectors.

We are now able to consider geometrically deduced bands as
'characteristic effects,' and we have a graphic representation of the
color-deduction determined by the interception of the pendulum. The
deduction-value of the pendulum is the distance (_xy_) which it
intercepts on a line drawn perpendicular to _A'C'_.

Lines drawn perpendicular to _A'C'_ through the points of intersection
of the locus-band of the pendulum with those of the sectors will give
a 'plot' on _A'C'_ of the deduction-bands. Thus from 1 to 2 the
deduction is red and the band green; from 2 to 3 the deduction is
decreasingly red and increasingly green, a transition-band; from 3 to
4 the deduction is green and the band red; and so forth.

We are now prepared to continue our identification of these
geometrical interception-bands with the bands observed in the
illusion. It is to be noted in passing that this graphic
representation of the interception-bands as characteristic effects
(Fig. 7) is in every way consistent with the previous equational
treatment of the same bands. A little consideration of the figure will
show that variations of the widths and rates of sectors and pendulum
will modify the widths of the bands exactly as has been shown in the
equations.

The observation next at hand (p. 174, No. 5) is that "The total number
of bands seen at any one time is approximately constant, howsoever the
widths of the sectors and the width and rate of the rod may vary. But
the number of bands is inversely proportional (Jastrow and Moorehouse)
to the time of rotation of the disc; that is, the faster the disc, the
more bands."

[Illustration: PSYCHOLOGICAL REVIEW. MONOGRAPH SUPPLEMENT, 17. PLATE V.
Fig. 7. Fig. 8. Fig. 9.]