Various - Harvard Psychological Studies, Volume 1

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This is true, point for point, of the interception-bands of Fig. 7. It
is clear that the number of bands depends on the number of
intersections of _PP'_ with the several locus-bands _RR_, _GG_, _RR_,
etc. Since the two sectors are complementary, having a constant sum of
360 deg., their relative widths will not affect the number of such
intersections. Nor yet will the width of the rod _P_ affect it. As to
the speed of _P_, if the locus-bands are parallel to the line _A'C'_,
that is, of the disc moved _infinitely_ rapidly, there would be the
same number of intersections, no matter what the rate of _P_, that is,
whatever the obliqueness of _PP'_. But although the disc does not
rotate with infinite speed, it is still true that for a considerable
range of values for the speed of the pendulum the number of
intersections is constant. The observations of Jastrow and Moorehouse
were probably made within such a range of values of _r_. For while
their disc varied in speed from 12 to 33 revolutions per second, that
is, 4,320 to 11,880 degrees per second, the rod was merely passed to
and fro by hand through an excursion of six inches (J. and M., _op.
cit._, pp. 203-5), a method which could have given no speed of the rod
comparable to that of the disc. Indeed, their fastest speed for the
rod, to calculate from certain of their data, was less than 19 inches
per second.

The present writer used about the same rates, except that for the disc
no rate below 24 revolutions per second was employed. This is about
the rate which v. Helmholtz[4] gives as the slowest which will yield
fusion from a bi-sectored disc in good illumination. It is hard to
imagine how, amid the confusing flicker of a disc revolving but 12
times in the second, Jastrow succeeded in taking any reliable
observations at all of the bands. Now if, in Fig. 8 (Plate V.), 0.25
mm. on the base-line equals one degree, and in the vertical direction
equals 1[sigma], the locus-bands of the sectors (here equal to each
other in width), make such an angle with _A'C'_ as represents the disc
to be rotating exactly 36 times in a second. It will be seen that the
speed of the rod may vary from that shown by the locus _P'P_ to that
shown by _P'A_; and the speeds represented are respectively 68.96 and
1,482.64 degrees per second; and throughout this range of speeds the
locus-band of _P_ intercepts the loci of the sectors always the same
number of times. Thus, if the disc revolves 36 times a second, the
pendulum may move anywhere from 69 to 1,483 degrees per second without
changing the number of bands seen at a time.

[4] v. Helmholtz, H.: 'Handbuch d. physiolog. Optik,' Hamburg
u. Leipzig, 1896, S. 489.

And from the figure it will be seen that this is true whether the
pendulum moves in the same direction as the disc, or in the opposite
direction. This range of speed is far greater than the concentrically
swinging metronome of the present writer would give. The rate of
Jastrow's rod, of 19 inches per second, cannot of course be exactly
translated into degrees, but it probably did not exceed the limit of
1,483. Therefore, although beyond certain wide limits the rate of the
pendulum will change the total number of deduction-bands seen, yet the
observations were, in all probability (and those of the present
writer, surely), taken within the aforesaid limits. So that as the
observations have it, "The total number of bands seen at any one time
is approximately constant, howsoever ... the rate of the rod may
vary." On this score, also, the illusion-bands and the deduction-bands
present no differences.

But outside of this range it can indeed be _observed_ that the number
of bands does vary with the rate of the rod. If this rate (_r_) is
increased beyond the limits of the previous observations, it will
approach the rate of the disc (_r'_). Let us increase _r_ until _r_ =
_r'_. To observe the resulting bands, we have but to attach the rod or
pendulum to the front of the disc and let both rotate together. No
bands are seen, _i.e._, the number of bands has become zero. And this,
of course, is just what should have been expected from a consideration
of the deduction-bands in Fig. 8.

One other point in regard to the total number of bands seen: it was
observed (page 174, No. 5) that, "The faster the disc, the more
bands." This too would hold of the deduction-bands, for the faster the
disc and sectors move, the narrower and more nearly parallel to _A'C'_
(Fig. 7) will be their locus-bands, and the more of these bands will
be contained within the vertical distance _A'A_ (or _C'C_), which, it
is remembered, represents the age of the oldest after-image which
still contributes to the characteristic effect. _PP'_ will therefore
intercept more loci of sectors, and more deduction-bands will be
generated.

6. "The colors of the bands (page 175, No. 6) approximate those of the
two sectors; the transition-bands present the adjacent 'pure colors'
merging into each other. But _all_ the bands are modified in favor of
the moving rod. If, now, the rod is itself the same in color as one of
the sectors, the bands which should have been of the other color are
not to be distinguished from the fused color of the disc when no rod
moves before it."

These items are equally true of the deduction-bands, since a deduction
of a part of one of the components from a fused color must leave an
approximation to the other component. And clearly, too, by as much as
either color is deducted, by so much must the color of the pendulum
itself be added. So that, if the pendulum is like one of the sectors
in color, whenever that sector is hidden the deduction for concealment
will exactly equal the added allowance for the color of the pendulum,
and there will be no bands of the other color distinguishable from the
fused color of the disc.

It is clear from Fig. 7 why a transition-band shades gradually from
one pure-color band over into the other. Let us consider the
transition-band 2-3 (Fig. 7). Next it on the right is a green band, on
the left a red. Now at the right-hand edge of the transition-band it
is seen that the deduction is mostly red and very little green, a
ratio which changes toward the left to one of mostly green and very
little red. Thus, next to the red band the transition-band will be
mostly red, and it will shade continuously over into green on the side
adjacent to the green band.

7. The next observation given (page 175, No. 7) was that, "The bands
are more strikingly visible when the two sectors differ considerably
in luminosity." This is to be expected, since the greater the
contrast, whether in regard to color, saturation, or intensity,
between the sectors, the greater will be such contrast between the two
deductions, and hence the greater will it be between the resulting
bands. And, therefore, the bands will be more strikingly
distinguishable from each other, that is, 'visible.'

8. "A _broad_ but slowly-moving rod shows the bands lying over itself.
Other bands can also be seen behind it on the disc."

In Fig. 9 (Plate V.) are shown the characteristic effects produced by
a broad and slowly-moving rod. Suppose it to be black. It can be so
broad and move so slowly that for a space the characteristic effect is
largely black (Fig. 9 on both sides of _x_). Specially will this be
true between _x_ and _y_, for here, while the pendulum contributes no
_more_ photo-chemical unit-effects, it will contribute the newer one,
and howsoever many unit-effects go to make up the characteristic
effect, the newer units are undoubtedly the more potent elements in
determining this effect. The old units have partly faded. One may say
that the newest units are 'weighted.'

Black will predominate, then, on both sides of _x_, but specially
between _x_ and _y_. For a space, then, the characteristic effect will
contain enough black to yield a 'perception of the rod.' The width of
this region depends on the width and speed of the rod, but in Fig. 9
it will be roughly coincident with _xy_, though somewhat behind (to
the left of) it. The characteristic will be either wholly black, as
just at _x_, or else largely black with the yet contributory
after-images (shown in the triangle _aby_). Some bands will thus be
seen overlying the rod (1-8), and others lying back of it (9-16).

We have now reviewed all the phenomena so far enumerated of the
illusion-bands, and for every case we have identified these bands with
the bands which must be generated on the retina by the mere
concealment of the rotating sectors by the moving rod. It has been
more feasible thus far to treat these deduction-bands as if possibly
they were other than the bands of the illusion; for although the
former must certainly appear on the retina, yet it was not clear that
the illusion-bands did not involve additional and complicated retinal
or central processes. The showing that the two sets of bands have in
every case identical properties, shows that they are themselves
identical. The illusion-bands are thus explained to be due merely to
the successive concealment of the sectors of the disc as each passes
in turn behind the moving pendulum. The only physiological phenomena
involved in this explanation have been the persistence as after-images
of retinal stimulations, and the summation of these persisting images
into characteristic effects--both familiar phenomena.

From this point on it is permissible to simplify the point of view by
accounting the deduction-bands and the bands of the illusion fully
identified, and by referring to them under either name indifferently.
Figs. 1 to 9, then, are diagrams of the bands which we actually
observe on the rotating disc. We have next briefly to consider a few
special complications produced by a greater breadth or slower movement
of the rod, or by both together. These conditions are called
'complicating' not arbitrarily, but because in fact they yield the
bands in confusing form. If the rod is broad, the bands appear to
overlap; and if the rod moves back and forth, at first rapidly but
with decreasing speed, periods of mere confusion occur which defy
description; but the bands of the minor color may be broader or _may
be narrower_ than those of the other color.

VII. FURTHER COMPLICATIONS OF THE ILLUSION.

9. If the rod is broad and moves slowly, the narrower bands are like
colored, not with the broader, as before, but with the narrower
sector.

The conditions are shown in Fig. 9. From 1 to 2 the deduction is
increasingly green, and yet the remainder of the characteristic effect
is also mostly green at 1, decreasingly so to the right, and at 2 is
preponderantly red; and so on to 8; while a like consideration
necessitates bands from _x_ to 16. All the bands are in a sense
transition-bands, but 1-2 will be mostly green, 2-3 mostly red, and so
forth. Clearly the widths of the bands will be here proportional to
the widths of the like-colored sectors, and not as before to the
oppositely colored.

It may reasonably be objected that there should be here no bands at
all, since the same considerations would give an increasingly red band
from _B'_ to _A'_, whereas by hypothesis the disc rotates so fast as
to give an entirely uniform color. It is true that when the
characteristic effect is _A' A_ entire, the fusion-color is so well
established as to assimilate a fresh stimulus of either of the
component colors, without itself being modified. But on the area from
1 to 16 the case is different, for here the fusion-color is less well
established, a part of the essential colored units having been
replaced by black, the color of the rod; and black is no stimulation.
So that the same increment of component color, before ineffective, is
now able to modify the enfeebled fusion-color.

Observation confirms this interpretation, in that band _y-1_ is not
red, but merely the fusion-color slightly darkened by an increment of
black. Furthermore, if the rod is broad and slow in motion, but white
instead of black, no bands can be seen overlying the rod. For here the
small successive increments which would otherwise produce the bands
1-2, 2-3, etc., have no effect on the remainder of the fusion-color
plus the relatively intense increment of white.

It may be said here that the bands 1-2, 2-3, etc., are less intense
than the bands _x_-9, 9-10, etc., because there the recent or weighted
unit-effects are black, while here they are the respective colors.
Also the bands grow dimmer from _x_-9 to 15-16, that is, as they
become older, for the small increment of one color which would give
band 15-16 is almost wholly overridden by the larger and fresher mass
of stimulation which makes for mere fusion. This last is true of the
bands always, whatever the rate or width of the rod.

10. In general, equal sectors give equal bands, but if one sector is
considerably more intense than the other, the bands of the brighter
color will, for a broad and swiftly-moving rod, be the broader. The
brighter sector, though equal in width to the other, contributes more
toward determining the fusion-color; and this fact is represented by
an intrusion of the stronger color into the transition-bands, at the
expense of the weaker. For in these, even the decreased amount of the
stronger color, on the side next a strong-color band, is yet more
potent than the increased amount of the feebler color. In order to
observe this fact one must have the rod broad, so as to give a broad
transition-band on which the encroachment of the stronger color may be
evident. The process is the same with a narrow rod and narrow
transition-bands, but, being more limited in extent, it is less easily
observed. The rod must also move rapidly, for otherwise the bands
overlap and become obscure, as will be seen in the next paragraph.

11. If the disc consists of a broad and narrow sector, and if the rod
is broad and moves at first rapidly but more slowly with each new
stroke, there are seen at first broad, faint bands of the
minority-color, and narrow bands of the majority-color. The former
grow continuously more intense as the rod moves more slowly, and grow
narrower in width down to zero; whereupon the other bands seem to
overlap, the overlapped part being doubly deep in color, while the
non-overlapped part has come to be more nearly the color of the minor
sector. The overlapped portion grows in width. As the rate of the rod
now further decreases, a confused state ensues which cannot be
described. When, finally, the rod is moving very slowly, the phenomena
described above in paragraph 9 occur.

The successive changes in appearance as the rod moves more and more
slowly, are due to the factors previously mentioned, and to one other
which follows necessarily from the given conditions but has not yet
been considered. This is the last new principle in the illusion which
we shall have to take up. Just as the transition-bands are regions
where two pure-color bands overlap, so, when the rod is broad and
moves slowly, other overlappings occur to produce more complicated
arrangements.

These can be more compactly shown by diagram than by words. Fig. 10,
_a_, _b_ and _c_ (Plate VI.), show successively slower speeds of the
rod, while all the other factors are the same. In practice the
tendency is to perceive the transition-bands as parts of the broad
faint band of the minor color, which lies between them. It can be
seen, then, how the narrow major-color bands grow only slightly wider
(Fig. 10, _a_, _b_) until they overlap (_c_); how the broad
minor-color bands grow very narrow and more intense in color, there
being always more of the major color deducted (in _b_ they are reduced
exactly to zero, _z_, _z_, _z_). In _c_ the major-color bands overlap
(_o_, _o_, _o_) to give a narrow but doubly intense major-color band
since, although with one major, two minor locus-bands are deducted.
The other bands also overlap to give complicated combinations between
the _o_-bands. These mixed bands will be, in part at least,
minor-color bands (_q_, _q_, _q_), since, although a minor locus-band
is here deducted, yet nearly two major locus-bands are also taken,
leaving the minor color to predominate. This corresponds with the
observation above, that, '... the non-overlapped part has come to be
more nearly the color of the minor sector.'

A slightly slower speed of the rod would give an irreducible confusion
of bands, since the order in which they overlap becomes very
complicated. Finally, when the rod comes to move very slowly, as in
Fig. 9, the appearance suffers no further change, except for a gradual
narrowing of all the bands, up to the moment when the rod comes to
rest.

It is clear that this last principle adduced, of the multiple
overlapping of bands when the rod is broad and moves slowly, can give
for varying speeds of the rod the greatest variety of combinations of
the bands. Among these is to be included that of no bands at all, as
will be understood from Fig. 11 (Plate VII). And in fact, a little
practice will enable the observer so to adjust the rate of the (broad)
rod to that of the disc that no bands are observable. But care must be
taken here that the eye is rigidly fixated and not attracted into
movement by the rod, since of course if the eye moves with the rod, no
bands can be seen, whatever the rate of movement may be.

Thus, all the phenomena of these illusion-bands have been explained as
the result solely of the hiding by the rod of successive sectors of
the disc. The only physiological principles involved are those (1) of
the duration of after-images, and (2) of their summation into a
characteristic effect. It may have seemed to the reader tedious and
unnecessary so minutely to study the bands, especially the details
last mentioned; yet it was necessary to show how _all_ the possible
observable phenomena arise from the purely geometrical fact that
sectors are successively hidden. Otherwise the assertions of previous
students of the illusion, that more intricate physiological processes
are involved, could not have been refuted. The present writer does not
assert that no processes like contrast, induction, etc., come into
play to modify somewhat the saturation, etc., of the colors in the
bands. It must be here as in every other case of vision. But it is now
demonstrated that these remoter physiological processes contribute
nothing _essential_ to the illusion. For these could be dispensed with
and the illusion would still remain.

[Illustration: PSYCHOLOGICAL REVIEW. MONOGRAPH SUPPLEMENT, 17. PLATE VI.
Fig. 10.]

If any reader still suspects that more is involved than the
persistence of after-images, and their summation into a characteristic
effect, he will find it interesting to study the illusion with a
camera. The 'physiological' functions referred to belong as well to
the dry-plate as to the retina, while the former exhibits, presumably,
neither contrast nor induction. The illusion-bands can be easily
photographed in a strong light, if white and black sectors are used in
place of colored ones. It is best to arrange the other variable
factors so as to make the transition-bands as narrow as possible (p.
174, No. 4). The writer has two negatives which show the bands very
well, although so delicately that it is not feasible to try to
reproduce them.

VIII. SOME CONVENIENT DEVICES FOR EXHIBITING THE ILLUSION.

The influence of the width of sector is prettily shown by a special
disc like that shown in Fig. 12 (Plate VII.), where the colors are
dark-red and light-green, the shaded being the darker sector. A narrow
rod passed before such a disc by hand at a moderate rate will give
over the outer ring equally wide green and red bands; but on the inner
rings the red bands grow narrower, the green broader.

The fact that the bands are not 'images of the rod' can be shown by
another disc (Fig. 13, Plate VII.). In all three rings the lighter
(green) sector is 60 deg. wide, but disposed on the disc as shown. The
bands are broken into zigzags. The parts over the outer ring lag
behind those over the middle, and these behind those over the inner
ring--'behind,' that is, farther behind the rod.

Another effective variation is to use rods alike in color with one or
the other of the sectors. Here it is clear that when the rod hides the
oppositely-colored sector, the deduction of that color is replaced
(not by black, as happens if the rod is black) but by the very color
which is already characteristic of that band. But when the rod hides
the sector of its own color, the deduction is replaced by the very
same color. Thus, bands like colored with the rod gain in depth of
tone, while the other pure-color bands present simply the
fusion-color.

IX. A STROBOSCOPE WHICH DEPENDS ON THE SAME PRINCIPLE.

If one produce the illusion by using for rod, not the pendulum of a
metronome, but a black cardboard sector on a second color-mixer placed
in front of the first and rotating concentrically with it, that is,
with the color-disc, one will observe with the higher speeds of the
rod which are now obtainable several further phenomena, all of which
follow simply from the geometrical relations of disc and rod (now a
rotating sector), as discussed above. The color-mixer in front, which
bears the sector (let it still be called a 'rod'), should rotate by
hand and independently of the disc behind, whose two sectors are to
give the bands. The sectors of the disc should now be equal, and the
rod needs to be broader than before, say 50 deg. or 60 deg., since it is to
revolve very rapidly.

First, let the rod and disc rotate in the same direction, the disc at
its former rate, while the rod begins slowly and moves faster and
faster. At first there is a confused appearance of vague, radial
shadows shuffling to and fro. This is because the rod is broad and
moves slowly (cf. p. 196, paragraph II).

As the velocity of the rod increases, a moment will come when the
confusing shadows will resolve themselves into four (sometimes five)
radial bands of one color with four of the other color and the
appropriate transition-bands between them. The bands of either color
are symmetrically disposed over the disc, that is, they lie at right
angles to one another (if there are five bands they lie at angles of
72 deg., etc.). But this entire system of bands, instead of lying
motionless over the disc as did the systems hitherto described, itself
rotates rapidly in the opposite direction from disc to rod. As the rod
rotates forward yet faster, no change is seen except that the system
of bands moves backward more and more slowly. Thus, if one rotate the
rod with one's own hand, one has the feeling that the backward
movement of the bands is an inverse function of the increase in
velocity of the rod. And, indeed, as this velocity still increases,
the bands gradually come to rest, although both the disc and the rod
are rotating rapidly.

But the system of bands is at rest for only a particular rate of the
rod. As the latter rotates yet faster, the system of bands now
commences to rotate slowly forward (with the disc and rod), then more
and more rapidly (the velocity of the rod still increasing), until it
finally disintegrates and the bands vanish into the confused flicker
of shadows with which the phenomenon commenced.

[Illustration: PSYCHOLOGICAL REVIEW. MONOGRAPH SUPPLEMENT, 17. PLATE VII.
Fig. 11.
Fig. 12. Fig. 13.]

This cycle now plays itself off in the reverse order if the speed of
the rod is allowed gradually to decrease. The bands appear first
moving forward, then more slowly till they come to rest, then moving
backward until finally they relapse into confusion.

But let the rate of the rod be not decreased but always steadily
increased. The bands will reappear, this time three of each color with
six transition-bands. As before, the system at first rotates backward,
then lies still, and then moves forward until it is dissolved. As the
rod moves still faster, another system appears, two bands of each
color forming a diameter and the two diameters lying at right angles.
This system goes through the same cycle of movements. When the
increased velocity of the rod destroys this system, another appears
having one band of each color, the two lying on opposite sides of the
center. The system goes through the same phases and is likewise
dissolved. Now, at this point the rod will be found to be rotating at
the same speed as the disc itself.

The explanation of the phenomenon is simple. The bands are not
produced by a single interruption of the vision of a sector by a rod,
but each band is made up of successive superpositions on the retina of
many such single-interruption bands. The overlapping of bands has been
already described (cf. Fig. 10 and pp. 196-198); superposition depends
of course on the same principle.

At the moment when a system of four bands of either color is seen at
rest, the rod is moving just one fifth as rapidly as the disc; so
that, while the rod goes once around, either sector, say the green
one, will have passed behind it exactly four times, and at points
which lie 90 deg. apart. Thus, four red bands are produced which lie at
right angles to one another. But the disc is revolving at least 24
times in a second, the rod therefore at least 4.8 times, so that
within the interval of time during which successive stimuli still
contribute to the characteristic effect the rod will have revolved
several times, and with each revolution four red bands at right angles
to one another will have been formed. And if the rod is moving
_exactly_ one fifth as fast as the disc, each new band will be
generated at exactly that position on the disc where was the
corresponding band of the preceding four. The system of bands thus
appear motionless on the disc.