Experiment 6: Distances to Afterimages on the Sky
It was considered possible that Ss might not only
see a difference in the sizes of afterimages projected
at different elevations in the sky (King & Gruber, 1962),
but they might also see a difference in apparent distances, thus providing data directly relevant to the
applicability of Emmert's law in this situation. Each
S was asked to fixate foveally and binocularly a small
white square, subtending 1.3 deg of visual arc, pasted
on a very dark blue field. He was then asked to observe
the dark afterimage of the square in the sky both over
the horizon and at 45 deg elevation (as defined by S).
The sky was a homogeneous gray-blue field with the
sun located over the shoulders on both days the data
were collected. The S was then asked in half of the
cases which image was larger, and then larger by what
percent. Following another fixation period, these Ss
were asked to determine which image was farther away,
and by what per cent. For the other half of the Ss
the two questions were reversed in order.
RESULTS
The index which seemed to provide the most comparability across the experiments was the ratio of
apparent vertical distance to apparent horizontal distance. For example, a mean apparent VIH distance ratio
of 1.65 would mean that, on the average, a particular
vertical distance seemed to be 65 percent greater
than a physically equal horizontal distance in that
situation. In Experiment 1 the' mean apparent v/n
ratio was given directly by (a) the distance of the chosen
ceiling plate from the wall, or (b) the distance the S
stood from the wall-divided by the distance from the
wall which would correctly match the distance to the
ceiling. In Experiments
2, 3, and 4, the S was not
presented with physically equal vertical and horizontal
distances so the data had to be corrected by the physical
ratio actually presented. In the cases of the slopes of
hills (Experiments 3 and 4), the estimated and physical
v/a ratios were defined by the tangents of the estimated
and correct angles, respectively. Finally, in Experiment 6 the ratio given could be called the mean E/H
ratio since it represented apparent distance to a 45 deg
elevated target as compared with the apparent distance
to a target just over the horizon; no vertical distance
was involved.
The results, according to the sex of the Ss, are given
in Table 1; the formulas for calculating the VIH ratios
are given in the footnotes. It can be seen that in every
case both males and females showed vertical overestimation, statistically significant in all but one
case which had too small an N. Furthermore, in
every case except that of the afterimages in the sky
(Experiment 6), the females showed a greater amount
of vertical overestimation, which was statistically
significant in Experiments 1 to 5. The percent of
judgments in all situations which were underestimations, correct, and overestimations were 4.2, 6.4,
and 89.4 for males, respectively, and 0.0, 2.5, and
97.5 for females, respectively.
In Experiments 1 and 2 (vertical distances upward
and downward) there remained the possibility that the
Ss were not using the eyes as a reference point for
vertical distance, although the instructions emphasized
this point. However, if the Ss used some point nearer
their feet as a reference point, it would reduce the
calculated illusion in Experiment I, but it would increase it in Experiment 2.
In order to interpret the data from Experiments 3
and 4, both involving estimates of slope,iUs necessary
Table 1. Mean Apparent Vertical/Horizontal Distance Ratios in Five Experimentsa
Males Females Signi!.
Experiment Situation N V/H Ratio N V/H Ratio t(sex) Level
10. Ceiling with paper plates 15 1.42** 15 1.52" 1.05 N.S.
b. Ceiling without plates 15 1.62** 15 1.94" 3.20 .01
2 22.9 ft. bridge b 4 1.28 12 1.69" 2.34 .05
3 34° hill slope c 66 1.65" 13 2.12" 3.53 .001
40. 25° streetC 78 1.71" 92 2.31" 5.04 .001
b. 17.5° street (SF)C 4 2.48* 6 4.46** 2.12 N.S.
6. Images in the skyd 7 1.48" 8 1.28* 2.07 N.S.
a. V/H ratios significantly different by t test from a ratio of 1.00 are marked * for the .05 level and" for
the .001 level.
b. In experiment 2 the mean estimated V/H ratio was divided by .80. the physical.v/H ratio presented to
the S.
c. In Experiments 3 and 4 the V/H ratio is the tangent of the mean estimated angle divided by the tangent
of the correct or maximum ang le of slope (34°. 25°, 17.5°).
d. The V/H ratio in Experiment 6 is the mean of the reported ratios of the distance to the 45° elevated
afterimage to the distance of the horizon atterimaoe .
Perception & Psychophysics, 1967, Vol. 2 (12) 587
Table 2. Reported Degrees of Angle for Six Drawn Angles a
(N = 20 males, 20 females)
Drawn Angles
Measure 10° 20° 35° 50° 70° 90°
Mean (males) 11.3- 23.5- 36.3 SO.5 73.5 90.0
Mean (females) 15.0-- 27.3-- 41.3- - 57.7- 79.0-- 90.0
S.D. (males) 2.36 4.33 5.90 4.33 8.27 0.00
S.D. (females) 3.94 3.94 4.33 10.22 3.15 0.00
VIH Ratio
(males)b 1.11 1.19 1.04 1.02 1.23
V/H Ratio
(females)b 1.51 1.41 1.25 1.33 1.82
t(sex diff.) 3.56 2.80 2.98 2.84 2.58
s gnif. level .001 .01 .01 .01 .05
a. Means of reported angles which are significantly different from
the correct angle by t test are marked - for the .05 level and - - for
the .001 level.
b. The V/H ratio refers to the tangent of the mean reported angle
divided by the tangent of the correct angle.
to examine the results of the control study on judgments of drawn angles. The results may be seen in
Table 2. It will be noted first that the data were stable
enough so that even small mean deviations from the
correct angle were often significant. In general, males
showed a slight overestimation but only for the smaller
angles, while females showed a consistent small overestimation of all angles (except 90 deg), That these
differences are not based simply on a greaterIack of
familiarity with angles by females is probable in that
females usually showed no greater variability of judgments than males.
Although both overestimation and the sex difference
held for drawn angles, the degree of illusion was substantially less than that found in Experiments 3 and 4.
For example, a 34 deg hill slope produced mean estimates of 48 deg and 55 deg for males and females,
respectively, but a drawn angle of 35 deg produced
mean estimates of only 36 deg and 41 deg, respectively.
Similarly, the estimated maximum streetslope of 25 deg
(with actual values perhaps in 15 deg to 20 deg range)
produced recall estimates of 38 deg and 47 deg, while
a drawn angle of 20 deg produced recall estimates
of only 23 deg and 27 deg for males and females,
respectively. All considered, the data seem to say that
the illusion holds to some degree even for drawn
angles, rather than seeming to say that the Ss simply
don't know how to use the concept of degrees of angle.
In Experiment 6 (afterimages in the sky) the size
[udgments, not appearing in Table I, revealed that the
images appeared larger on the horizon than at elevation, just as with the moon. The illusion for males,
with a horizontal-to-elevated size ratio of 1.44, was
comparable to that found by King and Gruber, but for
females, it was substantially less (1.17).
DISCUSSION
An obvious question is whether the various judgmental tasks used in this study involved the same perceptual processes. The judgmentof distances in Experi588
ments I, 2, and 6, involved comparisons of distances
along the line of sight with the observer's eyes as a
point of origin. In the judgments of slopes and drawn
angles the relevant distances were between external
points, may have varied from parallel to perpendicular
to the line of sight (as for a hill viewed from the front
versus from the side), and did hot necessarily involve
head tilt.
The results are tentatively placed together here because of the geometric consistency by analyzing slopes
into their vertical and horizontal components and because the sex difference was consistent in all situations except Experiment 6 (afterimages on the sky),
Given these considerations it is less easy to find an
acceptable alternative classification for the tasks. For
example, the overestimation of drawn angles could be
easily subsumed under the classical vertical-horizontal
illusion, but this would not explain the sex differential,
the virtual absence of angle overestimation by males,
and the dramatically greater amount of the illusion
in the other experimental situations.
The probable irrelevance of head tilt for the slope
and angle judgments is also not a compelling reason
to classify them separately, although head tilt may remain a unique, contributing factor. An alternative explanation of the illusion covering all of the cases
might be based on a transactional or behavioral coding
of the visual cues for distance and length.
For example,
it is possible that When, in the past, visual cues have
involved head tilt up or down or involved directly perceived slope, they have represented targets which
required more effort to reach and were thus coded in
motor-effort units as being farther away.
The Moon Illusion
The data clearly show that not Emmert's law but,
if anything, its reverse holds for afterimages in the
sky, Kaufman and Rock (1962) and Woodworth and
SChlosberg (1954) have admitted, while subscribing to
Emmert's law in the situation, that the moon itself is
typically experienced as not only larger, but also as
closer, in its horizon setting. This phenomenal finding
is explained away as a secondary effect,
a result of
the size illusion itself.
To understand this apparently perverse loyalty to
Emmert's law in the face of contradictory data, it
may be noted that the moon's image and the afterimage
are alike in two respects: they both have a constant
retinal size, and they both lack the ordinary visual
cues to distance (except for cues that should indicate
that they are not nearby). It is only parsimonious that
if Emmert's law describes size and distance judgments
for afterimages it should also cover size and distance
jUdgments for the moon,
On the other hand the demonstration that the larger
appearing horizon moon must also (unconsciously)
appear to be farther away as Emmert's law demands
has not been accomplished. What Kaufman and Rock
Perception & Psychophysics, 1967, Vol. 2 (12)
have shown, for example, is that a visual context of
Iearthly terrain contributes to an apparently larger
moon, but if and how this visual terrain produces a
hypothetically increased apparent distance remains to
be shown. The one point offered to support the flattened sky assumption is that observers typically aim
too low when asked to point out an elevation of n
degrees (from 0 to 90) above the horizon. Since this
fact can be seen to be simply another case of perceptual overestimation of vertical distance and slope,
it is obvious that several well chosen assumptions
are necessary to convert it into support for an opposite conclusion (see Kaufman & Rock, 1962).
There is an interesting alternative to Emmert's law
in understanding the size-distance interaction in these
cases which was first suggested to the E by the spontaneous statements of a few of the Ss participating
in the experiment on afterimages(Experiment 6). These
students commented to the effect that "Of course if
it is closer it should be larger," with the words
"closer" and "larger" sometimes interchanged. This
kind of statement is indeed a description of general
daily experience, but only if the observer shifts the
locus of his size judgments from external objects to
retinal images, for it is the retinal image which
becomes larger when an object is moved closer. To
make this kind of statement the observer must abandon
the assumption or construction of size constancy and
behave like an introspectionist, although his choice of
words hides this fact.
If an observer can report changes in his retinal
image size then it is also conceivable that he would
suffer from biasesin his[udgment of them, For example,
Perception & Psychophysics. 1%7, \'01 (1~)
if he assumed that the moon and sky afterimages were
like solid objects of constant physical size, then he
would expect a decrease in perceived distance to be
accompanied by an increase in (retinal) size. Thus
his verbal statement of the relationship might correspond with the perceptual expectancy mechanisms at
work. The advantage of considering this position isthat
it is consistent with the data and does not postulate
contrary unconscious processes, or involve the assumption that the sky is perceived as a surface, whose
distance is somehow biased by the visual terrain.
None of this rules out the possibility that both the
size and distance illusions could be simultaneous byproducts of some third factor, such as vestibular
stimulation. Until more data are collected, however,
it is difficult to specify how such simultaneous biases
might arise and operate.
References
Kaufman, L" & Rock, I. The moon illusion. Scient. American,
1962,207,120-130.
King, W. L., & Gruber, H. E. Moon illusion and Emmert's Law.
Science. 1962, 135, 1125-1126.
Thor, D. H.; & Wood, R. J. A vestibular hypothesis for the moon
illusion. Paper presented to the Annual Meeting of the Midwestern Psychological Association in Chicago. May 5, 196p.
Woodworth, R. S., & Scholosberg ,H. Experimental psychology.
Henry Holt, 1954.
Notes
1. Now on a one year's leave of absence to Bell Telephone Laboratories, Inc., Holmdel, New Jersey.
2. Emmert's law states that the apparent size of an afterimage is
directly proportional to its apparent distance from the observor.
(Accepted for publication August 16,1967.)