GENERAL METEOROLOGICAL OPTICS 63 
physical distortion owing to astronomical refraction. The 
size variation can also be explained by the general 
overestimation of angular elevations of terrestrial and 
celestial objects. On the left side of Fig. 3 the flat are 
of the sky is divided into six equal segments correspond- 
ing to 15°-intervals of estimated elevations (slant num- 
bers). The true elevations show by comparison that 
the relative overestimation is > 100 per cent for small 
elevation angles and decreases to zero at 90°. The abso- 
lute overestimation reaches a maximum at true eleva- 
tions between 30° and 40° [23, 42]. The right side of Fig. 
3, showing true 15°-intervals and their estimated equiv- 
alents (in brackets), indicates that a given angular 
subtense is overestimated when below roughly 30° ele- 
vation, underestimated when above 30°. Accordingly, 
the 14° - subtense of sun or moon appears larger near 
the horizon, smaller at higher elevations. 
The overestimated height and steepness of moun- 
tains [22, 42] and the apparent ellipticity of circular 
halos [23, 34] are also related to this phenomenon, as 
is the incorrect estimation of the amount of clouds. The 
latter is of practical significance, because an observer 
tends to underestimate the amount of clouds overhead 
and to overestimate the amount of clouds near the 
horizon. This fact is qualitatively known, and in the 
observer’s manual, measurements. of angular elevations 
of clouds are suggested to eliminate this error for ad- 
vancing or receding cloud layers and those surround- 
ing the station. However, no such expedient is avail- 
able for estimations of the more frequent nonuniform 
states of sky. This problem can be summarized as fol- 
lows: The estimates of cloud covers of 0 and 10 tenths 
are usually correct. For all other cloud amounts, the 
error made by the observer is a function of subjective 
factors (experience, etc.), of the amount and type of 
the clouds, the distance of the terrestrial horizon, and 
the general brightness level (Tables I-IV). 
Theories and Problems. All attempts to formulate 
an all-inclusive theory of the apparent shape of the sky 
have thus far been unsuccessful, chiefly because of the 
simultaneous involvement of physical and psychophysi- 
ological factors. Although physical and geometrical vari- 
ables modify our impression of the sky shape, they do 
not suffice, in themselves, to explain the observed facts. 
The simple consideration of the geometric properties 
of a cloud layer at 2000 m, for example, would result in 
an a = 1°, whereas a is actually observed between 20° 
and 30°, that is, the sky does not appear as flat as it 
should. A similar discrepancy arises for the clear sky 
[88]. From a physical standpoint, Dember and Uibe [9] 
attribute the sky shape to the distribution of sky bright- 
ness. They assume that the maximum distance from 
which scattered light reaches the observer is propor- 
tional to the square root of brightness. However, their 
theory not only implies that the brighter objects appear 
to be farther away, which is not true [42] (e.g., brighter 
stars appear nearer), but also precludes any variations 
of the sky shape with the distance of the terrestrial 
horizon. Similarly, Humphreys [21] believes that the 
greater haziness near the horizon produces the impres- 
sion of greater distance than is the case at more elevated 
regions of the sky. While this effect is undoubtedly 
present, its magnitude has been shown in Table IV to 
be of secondary order only. Purely psychophysiological 
explanations are similarly unsatisfactory. Gauss and 
others [c. 42] were of the opinion that, because our line 
of sight is habitually horizontal and normal to the body 
axis, the illusion of a variable moon size should disap- 
pear when the direction of sight is changed by mirrors 
or the body orientation altered. Not all results of perti- 
nent experiments supported this theory. Also, the ef- 
fects of physical variables would remain unexplained. 
Various attempts at a combination of geometric and 
psychophysical explanations have also been made. For 
example, v. Sterneck based his transformation of physi- 
cal into visual space on the underestimation of dis- 
tances according to a hyperbolic function, while Witte 
assumed that a straight line in physical space also ap- 
pears rectilinear in visual space [c. 42]. According to 
Exner [42], the moon’s visible area is compared to that 
of the fovea when the moon is high, but at low moon the 
respective linear dimensions are compared because of 
the attention commanded by the horizon line. This 
hypothesis, however, is incapable of explaining the sky 
shape or the variations in the moon’s apparent size 
caused by external physical factors. Isimaru [23] com- 
bined, in his theory of the shape of the cloudy sky, the 
underestimation of distances with the overestimation 
of elevation angles. The same idea was followed by 
Hiuittenhain [22] who reverted to v. Sterneck’s theory. 
Both authors [22, 23] make the implicit assumption 
that the overestimation of angular elevations is in- 
dependent of the sky shape, because it can also be ob- 
served in rooms [22]. However, since both phenomena 
are due to the properties of our visual space, they can- 
not be independent. 
Fundamentally, all of the theoretical approaches to 
the general problem have hitherto been based on the 
assumption that the visual space is Euclidean and can 
be obtained by some unique transformation of the 
three-dimensional manifold of the physical space. How- 
ever, Luneburg [30] has recently shown that the sensa- 
tions produced by binocular vision represent a Rie- 
mannian manifold. From the analysis of certain optical 
illusions he concluded that the geometry of our visual 
space is the hyperbolic geometry of Lobachevski as 
had already been indicated by Skreb [49]. With this 
theory a differential equation for the apparent size of 
a line element in the horizontal plane was developed, 
which is, however, not immediately applicable to the 
problem of the apparent shape of the sky; at any rate, 
an extension of this theory to include this group of 
phenomena appears desirable. The major physical fac- 
tors whose effects must be explained by such a theory 
may be tabulated as follows: 
1. General brightness of the visual field and bright- 
ness distribution over the sky. 
2. Cloud types and amounts. 
3. Distance of terrestrial horizon, and facilities for 
estimating this distance. 
In addition, there are several phases of the problem 
that have been insufficiently explored or are entirely 
