This section elaborates the new descriptions of the moon illusion.
It carefully defines the relevant perceptual dimensions, describes how they relate to each other and compares these perceptual magnitudes for the horizon moon and zenith moon.

It emphasizes that for most people the illusion starts with the constant angular size (visual angle) of 0.52 degrees looking larger for the horizon moon than for the zenith moon. Consequently, either the perceived linear size (in meters) or the perceived distance (in meters) also changes, or else both of those perceived metric values appear to change along with the change in the perceived angular size.
Some logical rebalancings of the perceived metric values for the moon due to the change in the perceived angular size are described and illustrated.
Many facts about the moon illusion are reviewed, especially the strong correlation between changes in the perceived angular size and changes in patterns of distance cues.
Finally, some classic flat-pattern "size" illusions are reviewed that also begin as angular size illusions controlled by distance cues.

THE BASIC DIMENSIONS
Consider first the physical measures.

The Physical (Optical) Dimensions.
The two drawings below represent the optical facts for the moon.
For the side-view we are looking due north at an observer at point O who is looking east (to the far right) at the extremely distant full moon at the horizon. The moon rises on a circular arc to a zenith position above the observer's head.
The dark half of each circle indicates the unseen back hemisphere of the full moon.

At the left is a front view that goes with that side-view. It illustrates how a camera at point O would record the rising full moon and three trees on the ground at a great distance east of the observer. In the side-view the tree image represents the third tree, which obscures the other two in this view.
Keep in mind that these diagrams do not illustrate the perceptual experience of the person at point O.

Linear Size and Optical Distance:  The moon's linear diameter is approximately 2160 miles and it's distance from point O averages about 238,800 miles.
[Such diagrams rarely are drawn to exact scale, but they accurately portray the relative relationships. Also, for present purposes the diagram overlooks the fact that the horizon moon is about 2% farther away from point O than the overhead zenith moon due to the added radius of the earth, so the angular size of the horizon moon actually measures about 2% smaller than the overhead zenith moon's.]

Angular Size or Visual Angle: For each 'moon' in the side-view the arrows indicate the optical directions of its top and bottom edges from the observer's eye at point O. The difference between those two optical directions can be measured directly using a theodolite, as the angular size for the moon's diameter.
Again, at all positions in the sky, the moon subtends about 0.52 degrees.

Vision researchers prefer to call this measured angle the visual angle, (the more "scientific" synonym for angular size). It is a physical measure, so the adjective "visual" may be misleading. It probably should be called, instead, the "ocular angle" or the "optical angle".

Some confusion once existed in the moon illusion literature because astronomers use the term "apparent size" for this objective (physical) angle they measure for the moon's diameter. But psychologists use the adjective "apparent" (and also "perceived") only for a subjective value (or else for a research measure of a subjective value).

Technical Note:
The visual angle can be calculated, of course, using the simple rule, TangentV = S/D, in which S is the frontal linear size and D is the distance from the eye.
So, the ratio of the moon's linear diameter (3475 km) to the viewing distance (384,400 km) equals 0.009, which is the tangent of 0.52 degrees.
In optical terms, the ray of light from a diameter's endpoint to the center of the eye pupil is the chief ray of the bundle of light rays that focus to a small point on the retina to form there the optical image of that diameter's endpoint. Likewise for the opposite end of that diameter.
The angle between those two chief rays is the visual angle, V deg.
The optical image of the moon formed on the retina is just like the real image formed on the film in a camera. The diameter of this retinal image, R millimeters, is determined directly by the angular size, V degrees, in accord with the simple rule, R = nTan V with n equal to about 17 mm. (See Figure A1 in the Appendix.) So, the diameter of the moon's circular retinal image is about 0.15 mm. [End of Technical Note.]

By the same token, the front of any object will subtend a visual angle of 0.52 degrees (so its retinal image size is 0.15 mm) whenever the distance to that frontal extent is 111 times its linear size. For example, 0.52 degrees would be the visual angle for a barn 47.5 feet wide located one mile away, for a USA penny (19 mm in diameter) held 2.1 meters from the eye, and for this printed capital letter, O, viewed from a distance 111 times its measured diameter.

In order to properly describe the moon illusion, we must use unambiguous terms for the perceptual (subjective) dimensions, so let's define the required terms by considering the absolute moon illusions.

Absolute Moon Illusions.
Perceived Linear Size.
The moon obviously doesn't look 2160 miles wide.
A person might say, for instance, that its width looks about 1 mile, or 100 yards, or 30 feet. [Interestingly, Ross and Plug (2002) cite many old reports that people (including some scientists) had said the moon looked only about 10 to 30 centimeters in diameter! ]
A researcher would record such reports as perceived linear size values (synonym, apparent linear size).
Most people have difficulty stating how large the moon's linear size looks. But our main interest is in the relative comparison: And people easily can say that the horizon moon's linear size looks either larger, the same, or smaller, than the zenith moon's without having to state an absolute perceived linear size value for either moon.

Perceived Distance.
The moon obviously doesn't look 240,000 miles away.
A person might say it looks 4 miles away, or 2 kilometers away or 100 yards. Those reports would be recorded as perceived distance values (synonym, apparent distance).

Most people have difficulty stating how far away the moon looks, or even how far away a distant terrestrial object looks. For instance, even professional golfers do not always trust their expert distance perception, so they pace the remaining distance to the green or else consult their list of prior measurements.

Perceived Visual Angle (Perceived Angular Size).
Remember, the angular size experience has everything to do with one's perception of the directions of viewed points from oneself.
The crucial observation here is that one sees that the moon's left edge lies in some direction from oneself; it has a certain perceived direction. For example, let's say the direction the moon's left edge looks happens to be "due east" from oneself.
One also sees, of course, that the moon's right edge lies in a given direction from oneself, a direction that obviously appears to differ from the direction in which the left edge appears. Let's say the right edge looks simply "a wee bit in the southeasterly direction."
The amount by which those two perceived directions differ (in degrees) is the perceived visual angle for the moon. Unambiguous synonyms are, apparent visual angle, perceived angular size, and apparent angular size)
[An old term, "perceived extensity" (Rock & Kaufman, 1962a) was used by Plug & Ross (1989) but it lacks an angular connotation, so they switched to using the term "perceived angular size" (Plug & Ross, 1994; Ross & Plug, 2002).]

Evaluating The Perceived Visual Angle.
A well-trained observer might say, for instance, that the horizon moon looks 2 degrees wide, or 1 degree, or 3/4 degree, or 1/2 degree (no illusion). Each such report would be recorded as the perceived visual angle, in degrees.
However, most of us cannot reliably say how many degrees a given viewed object appears to subtend. We don't practice doing that because it usually isn't necessary.
Instead, to indicate to someone else the magnitude of an angular subtense we are seeing, we often use a sweeping gesture (Ono, 1970): For instance, we point a finger in the direction of place 1, then at place 2, and the other person observes the change in the direction our finger is aiming: That method provides the other person with a value for the difference (angle) we are seeing between the directions those two places appear to lie from us.

Indeed, more sophisticated pointing (aiming) devices have been used to measure absolute perceived visual angles for viewed objects (Gogel & Eby, 1994; Komoda & Ono, 1974; Ono, Muter and Mitson, 1974).
A research problem, however, has been that the moon's subtended angle is too small to obtain an objective measure of the visual angle experience for it.
For instance, we could point our nose from the moon's left edge to its right edge, and the angle our head turns through would be a measure of the perceived angular size. But that tiny rotation would move the tip of the nose only about 1.3 mm, because the head's rotation axis is about 130 mm (5 inches) behind the nose.
Besides, the important prediction is that the initial head rotation angle would be about 1.5 times greater for the horizon moon than for the zenith moon. So the difference between those two movements of the nose tip that would reveal that relative illusion would be only about 0.65 mm, much too small to measure.

No objective measures of the perceived angular size have been published for the moon.

Of course, because the perceived visual angle usually is larger for the horizon moon than for the zenith moon, there must be an absolute angular size illusion under some (or most) viewing conditions. [Yet, at some elevation of the moon the perceived angular size might "correctly" equal 0.52 degrees.]

At any rate, people can compare the perceived angular sizes of the horizon and zenith moons without having to provide an absolute value for either moon. And the relative illusion can be objectively measured, as follows.
In the simplest direct experiments, observers view the natural horizon moon and compare its "size' with the "size' of a distant artificial 'moon' image located at a high "zenith" elevation. That surrogate moon's angular size then is changed until it looks the "same size" as the horizon moon. Its final "looks equal" angular subtense typically is larger than 0.52 degrees. The ratio of its angular size to the horizon moon's value of 0.52 degrees thus is an objective measure of the relative moon illusion.
Experiments like that have provided the many published ratios that range from 1.0 to over 2.0.

Side-View Descriptions.
To describe what a person perceives when looking at an object we begin with an optical diagram of the physical arrangement, and then create a diagram of the perceptual values the person is reporting.

Optical Drawing.
For example, the side-view below has an observer's eye at point O looking at a small sphere located directly in line with a larger cube that is just far enough away so that the height of the cube and the diameter of the sphere subtend the same visual angle, V deg, at point O.
The cube's height is its linear size, S meters. Its front surface is D meters from the eye. The geometrical relationship is given by the simple equation, tanV = S/D.
(The dimension lines for the smaller S and D values for the sphere are not shown.)

In reverse, the upper arrow indicates the chief (central) ray of the bundle of light rays that head toward the eye from the objects' upper edges. Likewise, the lower arrow indicates, in reverse, the chief ray for each object's bottom edge. Those ray bundles eventually form the physical (optical) images of those edges on the retina, to generate the retinal images of the sphere and the cube face.
The diagram illustrates, as well, the optical situation for a camera with its lens at point O.

Front Views.
The "YES" front view is what the camera's picture would look like.
That "YES" diagram also shows what the optical images on the retina would be like.

The "NO" front view is not what the camera would record, and not what the optical images on the retina would be like.
Of course, that "NO" front view might appear on an engineer's drawing to go with the side-view. But that orthographic projection method is inappropriate for visual science.
Now consider what the person at point O may perceive.

Perceived Values Drawings.
The diagrams below illustrate an arrangement of perceptual values for the observer whose eye is at point O. For this example we'll assume the perception is correct (no illusion), so the cube looks the same angular size as the sphere, also a larger linear size and farther away.
Of course the person sees only the front surfaces of the sphere and cube, so the unseen rear portions are indicated with dotted lines in the side-view.
The arrows from the eye to the top and bottom of the perceived sphere indicate the perceived directions of those edges. They coincide with the perceived directions of the top and bottom edges of the perceived cube.
The difference between the two perceived directions is, V' deg, the perceived visual angle. It is the same for both objects here.

The dimension, S' is the perceived linear size (in meters) for the cube's height. And, D' is its perceived distance (in meters).
[The smaller S' and D' values for the sphere are not indicated.

The "YES" front view shows the reader what the person at point O is seeing.
That is, it resembles what an artist would draw to accurately portray what he or she sees when looking at the sphere and cube from point O.

The "NO" front view certainly does not imitate what that person sees.

Notice, however, that if readers were given only that side-view, it would be easy for them to think (mistakenly) that the small circle and larger square in that side-view could imitate what the front view would be like. That is, without having the appropriate front view, a viewer might be misled into assuming that what he or she would see from point O would resemble the "NO" front view with the cube looking a larger angular size than the sphere.

As cautioned in the earlier critique of the 'sky dome' idea it is important not to misread a side-view diagram in that manner.

The Perceptual Invariance Hypothesis.
The diagram illustrates the theory that those three perceptual values relate to each other in accord with the "new" simple rule, S'/D' = tan V'. This rule (McCready, 1965, 1985) recently was dubbed the perceptual invariance hypothesis by Ross and Plug (2002).

Psychophysical Rule.
An additional rule is needed, of course, which will relate the subjective visual angle value, V' deg, to the objective visual angle, V deg, by way of the size of the retinal image of the viewed object. That additional psychophysical rule is discussed later (in Section III).
However, for present purposes, it is sufficient to say merely that "certain factors" can make V' deg, not equal V deg. The moon illusion will become fully explained when all those 'certain factors' are identified and understood. As already noted, the most important "certain factors" seem to be changes in the patterns of distance cues.

The Relative Moon Illusions
How the experiences of perceived angular size, perceived linear size and perceived distance for the majority moon illusions relate to each other was described in the Introduction using a front view. They now can be described using a side-view as well.

The side-view (at the right below) portrays that we are looking north toward an observer at point O who is facing east (way off to our far right). The perceived full moons are drawn as semi-circles because a person obviously doesn't see the rear hemisphere of the moon.
[Any side-view diagram that purports to illustrate the moon illusion cannot logically use a uniformly filled circle (or full disc) to represent a perceived moon. However, virtually all the side-views in other articles on the moon illusion have used full circles. That simple mistake undoubtedly has helped side-view diagrams like the sky dome diagram fool some readers.]

The lower semi-circle represents a potential perceived horizon moon. The other three semi-circles represent potential perceived zenith moons, all at the same perceived elevation angle, but at three different perceived distances.

Side-views with the perceived angular sizes drawn unequal, were included in my moon illusion lectures (McCready, 1964 - 1992) but have been formally published in just a few places (McCready, 1983, 1986). So, most readers probably are seeing them for the first time.

Perceived Angular Sizes.
The arrows from point O through the top and bottom of each semi-circle indicate the perceived directions of the top and bottom edges of that moon. For this example the perceived angular size for the horizon moon is assigned, arbitrarily, a value of 1.5 angular units simply to illustrate an illusion with it looking 1.5 times angularly larger than the zenith moon's perceived angular size of 1.0 unit.

The front-view (at the left) portrays what the person at point O is seeing. For this front view we are looking east and see a portrayed 'horizon' moon whose angular size looks about 1.5 times larger than the 'zenith' moon's. Our angular size experience for those two circles thus imitates all of the angular size experiences the side-view diagram is describing for the person at O.
That is, for every outcome illustrated by the side-view, the person at O says "the horizon moon looks angularly larger than the zenith moon."

Perceived Linear Sizes.
The number 30 next to the perceived horizon moon represents the perceived linear size in some unit of linear measure. For instance, the person at point O might have said the horizon moon's diameter looks about 30 cm, or 30 in., or 30 ft., or 30 yards, or 30 meters, or 30 stories tall.

The numbers beside the perceived zenith moons (20, 25, 30) indicate their perceived linear sizes, for comparison with the horizon moon's perceived linear size of 30.

Perceived Distances.
In the side-view the distance of each semi-circle from point O represents the perceived distance of that apparent moon.
Some observers report that the just rising horizon moon looks only slightly beyond the most remote looking horizon objects (Hershenson, 1982). The three trees represent such perceived horizon objects, and tree on the right is pictured in the side view, where it obscures the other two perceived trees.

Some Common Outcomes.
As previously discussed, the most popular versions of the moon illusion seem to be a same-distance outcome, a same-linear size outcome and an intermediate outcome.
Those outcomes and some others are described below.

The Same Distance Outcome:
This outcome is illustrated by 'horizon moon'-30 and 'zenith moon'-20. Both moons look about the same radial distance away, so the horizon moon's linear size (30) necessarily looks larger than the zenith moon's (20) by the same proportion that its angular size looks larger. The person's typically abbreviated report is that the horizon moon "looks larger and about the same distance away" as the zenith moon.

Accordingly, if the moon looks like a sphere, the horizon moon appears to have a greater volume than the zenith moon despite one's knowledge that the moon's volume remains constant. That is, the moon looks somewhat like a round balloon that becomes deflated as it rises.
Factors that can lead to a 'same-distance' perception include the so-called "equidistance tendency" (Gogel, 1965) or an "equal-distance assumption" (McCready, 1965, 1985) which could be due simply to one's knowledge that the distance to the moon remains essentially the same.

The Same Linear Size Outcome:
This outcome is illustrated by 'horizon moon'-30 and 'zenith moon'-30 which necessarily appears 1.5 times farther from point O than the 'horizon moon' does.
The horizon moon's angular size again looks larger than the zenith moon's. But the moon appears to remain the same linear size of 30 units, so that makes the horizon moon look closer than the zenith moon because it looks angularly larger.

Most observers who experience this outcome seem content to say merely that the horizon moon "looks larger and closer" than the zenith moon, without mentioning that the linear sizes appear equal.

That outcome illustrates linear size constancy. Factors that can lead to it include an "equal linear size assumption" (McCready 1965, 1985) which can be due simply to one's knowledge that the moon remains the same moon (identity constancy) so its physical size would not appear to change.

The "looks closer" illusion here illustrates the powerful, 'relative perceived visual angle cue to distance'.
It also illustrates that the distance-cue patterns responsible for increasing the perceived visual angle for the horizon moon are being overruled by that other cue. (See a later discussion of this cue conflict.)

Apparent Initial Retreat.
In this linear size constancy outcome, the rising moon logically would at first appear to move farther away, retreating to the east before it appears to move overhead toward the zenith.

An Intermediate Outcome:
Sometimes, all three relative illusions occur at the same time, as shown by 'horizon moon'-30 and 'zenith moon'-25.
The observer's full report is that, compared with the zenith moon, "the horizon moon looks angularly larger, linearly larger and closer."

The common abbreviated report, "looks larger and closer" thus is ambiguous. It could refer either to the same linear size outcome or to an intermediate outcome.

Vertical Ascent.
Notice that in this intermediate outcome, 'zenith moon'-25 appears almost directly above the 'horizon moon'. Indeed, a characteristic of most intermediate outcomes is that the rising moon first appears to move away (a bit more eastward) and then almost straight up before it appears to come forward to start its long trip overhead toward the zenith. That logical result of the new description presently is illustrated much better in Figure 6.

It is important to point out, again, how a side-view diagram can mislead hasty readers.

Potential Misreadings.
One possible misreading of the side-view concerns the three upper 'zenith moon' semi-circles. For the reader they correctly appear to have different angular sizes, but what they clearly illustrate is that, for the person at point O, all three zenith moons have the same perceived angular size, as illustrated by the front view at the left.

Also, the reader correctly sees that the 'horizon' semi-circle-30 and the 'zenith' semi-circle-30 subtend the same visual angle, but what those two illustrate, of course, is that the person at O is seeing a horizon moon that looks 1.5 times angularly larger than that zenith moon, as shown by the front view.
A reader who doesn't take those facts into account will miss a main point of the diagrams.

Survey Data.
Many anecdotal reports published over the last 100 years have indicated that a majority of people say simply that the horizon moon either "looks larger and closer" than the zenith moon, or "looks larger and at the same distance."
That also has been the finding of many published 'moon illusion' experiments in which participants were asked for a distance comparison.

Some surveys indicate that the three outcomes described above account for the moon illusions of at least 90% of the population.
For instance, at least annually, from 1964 to 1992, just before beginning my lecture on the moon illusion (McCready, 1964-1982), and without giving hints about what I expected, I asked the class (or audience) to recall their usual distance comparison for the horizon moon versus the zenith moon, and "vote" for either "farther" or "same distance" or "closer" on a 'ballot' I provided. Then they "voted" for their usual 'size' comparison ("larger" or "same size" or "smaller").
Each person exchanged ballots with a nearby person who then reported that other person's set of choices in the show of hands that provided the totals I wrote on the blackboard
Over those 28 years, more than 800 participants took part.
In a large audience, at least 75% (and often 90%) chose both "larger" and "closer."
About 5% to 15% chose both "larger" and "same distance."
Only about 5% chose both "larger" and "farther" (McCready, 1983, 1986).
It must be noted that some people report no relative "size" illusion for the moon.

Hershenson (1989) has reported similar survey results.

Of course, reports about memories are not reliable evidence, but those numbers do not disagree with published results from experiments.

Notice that the most popular simple report, "looks larger and closer," is incomplete, hence ambiguous. It can refer to either a same linear size outcome or an intermediate outcome. In order to obtain a full report an experimenter must explain to the observer the distinction between the linear size and the angular size, which is not easy to do.
At any rate, after my lectures, many people said they know the moon remains the same physical size, but the horizon moon usually appears a larger physical size (and volume) than the zenith moon, as well as closer. That may indicate that the intermediate outcome is the most common one.

A fourth possible outcome needs to be described.

A Larger and Farther Outcome:
For the relatively few observers who say the horizon moon "looks larger and farther away" than the zenith moon, it could be that for some of them, "looks larger" refers both to the angular size and to the linear size.
That perception is shown in the different side-view below.

Here the 'horizon moon' looks 1.5 times angularly larger than the 'zenith moon' and farther away, so its perceived linear size of 30 units looks more than twice as large as the 'zenith moon's' of 13 units.
The report, "looks farther away," at least partially agrees with the distance-cue patterns responsible for the illusory increase in the perceived angular size

Two more possible outcomes are illustrated below.

Two Outcomes With No Angular Size Illusion:
If the moon appears to remain the same angular size, there are two likely outcomes.

1. No Moon Illusion:
"Horizon moon'-20 and 'zenith moon'-20, indicate the result for those few people who have no relative moon illusion. Both moons have the same perceived visual angle, the same perceived linear size and the same perceived distance.

2. The Apparent Distance Theory Description.
Among the approximately 5% of people who say the horizon moon "looks larger and farther away" than the zenith moon, it could be that for some of them the ambiguous phrase "looks larger" refers only to the perceived linear size, while the angular sizes look equal.
That outcome is illustrated by 'horizon moon-30' and 'zenith moon-20'.
This is the only moon illusion the apparent distance theory can describe.
It proposes that the horizon moon and zenith moon look the same angular size (as shown by the front view), so if the horizon moon "looks larger" it is because it looks farther away than the zenith moon.
That also is the only illusion the sky dome diagram illustrates.

Again, don't let the fact that 'horizon' semi-circle 30 looks angularly larger than 'zenith' semi-circle 20 mislead you. The diagram clearly is constructed to show that the observer at point O sees all the moons as having the same angular size.

Previously published articles (McCready, 1983, 1986) included a more elaborate side-view diagram of the new descriptions. Ross and Plug (2002) recently republished it as their Figure 10.8. A similar version is Figure 6, below.

A More Complete Side-View: Figure 6.
Many of the anecdotal reports in the scientific literature seemed "strange" because they contradicted the dominant, apparent distance theory (creating the unresolved size-distance paradox). However, most of those reports now can be seen to be not paradoxical but quite sensible. And they can be used to help construct Figure 6.

For the Figure 6 side-view the observer at point O is looking eastward at the horizon moon and watching it rise to the zenith. The distance from point O to each perceived moon represents its perceived distance. The 'horizon moon' appears just beyond point H, where the most-distant terrestrial object appears to be.

The number to the right of an ascending 'moon' specifies its perceived linear size in some arbitrary metric unit. For the 'horizon moon' it is 16 units.

The 'moons' along the arc at the same perceived distance from point O are drawn at elevation intervals of five degrees. The number to the left of each 'moon' on that arc (such as < 1.6 > for the horizon moon) refers to the perceived visual angle for that moon and for the other 'moons' at that same elevation angle.

The front view approximates, in part, what the observer at point O is seeing.

How fast the perceived visual angle decreases while the moon's elevation steadily increases, evidently varies among observers. Figure 6 makes use of several published reports which indicate that the visual angle appears to decrease more rapidly during the first part of the moon's upward journey than it does later. For this example, the perceived visual angles begin with a value of 1.6 for the horizon moon, and decrease in an irregular manner up to 1.0 for the zenith moon at an elevation of about 45 degrees, after which point the perceived visual angle remains 1.0 up to the 90-degree zenith pole.

Figure 6 is a carefully scaled drawing: Each 'moon' is drawn at the distance from point O specified by the chosen perceived visual angle and perceived linear size values for it. Accordingly, for each type of outcome the diagram accurately predicts the path the rising moon would appear to take as it climbs into the zenith sky. The three most common outcomes are described again with each description now specifying the distinctive route the moon would appear to take as it rises.

The same perceived distance outcome is represented by the 'moons' strung along the circular arc at the same radial distance from point O. Accordingly, while the perceived visual angle decreases from 1.6 units to 1.0 units, the perceived linear size decreases from 16 to 10 units.

The same perceived linear size outcome is represented by the outermost arc of 'moons'. They have the same perceived linear size of 16 units while their perceived visual angles decrease from 1.6 units to 1.0 unit.

This linear size constancy outcome is especially noteworthy because the geometry specifies that the horizon moon initially appear to rapidly retreat eastward and then appear to ascend in an almost vertical path, as if it were a balloon rising straight up.

An intermediate outcome example is illustrated by 'moons' drawn along a path of dots between the other two outcomes. It includes a slight "bulge" to the east followed by an almost vertical apparent ascension during the first part of the path. The path ends in the overhead zenith moon of perceived linear size, 13 units. Countless other intermediate outcome paths obviously are possible.

That Initial Upward Path and the "Bulge".
In the linear size constancy and intermediate outcomes, that "bulging" path the geometry requires was a very pleasant surprise when it first showed up (in 1981) in the first carefully scaled diagram made for a moon illusion article (McCready, 1983).
In the first place, that predicted apparent movement of the moon to a greater distance just after it rises, followed by the almost vertical ascension, agrees with my own moon illusion experience. Many students (like me in 1949) have found that textbooks do not describe their moon illusion.
Secondly, that "bulge" became mentioned in the literature by Hershenson (1982, p. 438) for the setting sun. Specifically, as the sun appears to drop down in the west, its perceived visual angle typically enlarges, and some people say it appears to come forward (toward them) in the moments just before it slips from view below the horizon.

Likely Zenith Paths.
The route the moon appears to follow through the higher elevations seems to vary greatly among observers. Researchers have not yet agreed upon any particular description of the path through the upper sky. Consider, however, some likely paths Figure 6 can predict.

The zenith path for the same perceived distance outcome follows the circular arc to the 90-degree zenith pole.

The zenith path for the same linear size outcome is shown by the string of 'moons' of perceived linear size 16. The ones higher than an elevation of 45 degrees are arbitrarily on a circular arc at a perceived distance 1.6 times greater than those on the arc for the same perceived distance outcome.

The zenith path for an intermediate outcome arbitrarily is drawn as the string of dots about halfway between those two other paths: It ends with circle 13 at the 90-degree elevation.

The Sky Dome, Perhaps.
Those three paths from horizon to zenith obviously differ greatly from the traditional "sky dome" description. For instance, for many people that hypothetical sky surface at the horizon is far beyond the apparent horizon moon.
Yet part of a sky dome illusion could be appealed to for higher elevations, as indicated in Figure 6 by the dashed horizontal line at the altitude of the perceived 'zenith moon'-13.

Indeed, Gilinsky (1980) proposed a compromise version, in which the horizon moon looks considerably in front of the sky, but rises to eventually meet the sky dome at a higher elevation. For example, as Figure 6 can suggest, the rising moon might appear to ascend along the equal-perceived-linear size path until it reaches the sky dome and then it could appear to follow that more-or-less flat path to the zenith pole, and through it.
In reverse, that predicts a perceived path for the sun moving from zenith to the western horizon.

Established Facts About the Moon Illusion.
The publications mentioned so far, along with many others, have provided information that any theory of the moon illusion must explain. The essential facts are reviewed below.

Fact 1: It Is A General Illusion.
The 'moon' illusion is not limited to celestial bodies.
The apparent angular magnification and minification controlled by distance cues exists for other objects, as well.
For terrestrial objects this is a logical result illustrated by the diagrams below.

In each diagram, the two outer trees subtend at your eye the same angular subtense as the horizon moon's diameter.
The diagram at the left can mimic what a person would see if there were no moon illusion: The horizon moon looks a particular angular size, and the zenith moon looks the same angular size.
The diagram at the right can mimic the most common moon illusion. When compared with the diagram on the left it shows the horizon moon's visual angle looking 1.5 times larger than it would if there were no illusory magnification. The main point being made here is that the angular separation between the two outer trees also looks 1.5 times larger than it would if there were no illusion.
That has to be true, because those two outer trees logically must appear to bracket the just risen moon, no matter what size it looks.
In other words, as a general rule, the apparent enlargement of visual angles applies to distant terrestrial objects (and their separations) that appear far away near a distant horizon, such as trees, barns, and hills (Higashiyama, 1992, Higashiyama & Shimono, 1994, Roscoe, 1989)

For instance, one old 'explanation' was that the horizon moon looks big because it looks as big as, say, a huge barn seen near it. Well, if that barn were magically hoisted up to a high elevation while being kept at the same viewing distance, it would "look smaller," just as the zenith moon does.

This same generic "moon illusion" also occurs for objects indoors. Therefore, investigators have measured it using surrogate 'moons' objects (such as golf balls) presented in viewing conditions that simulate horizon and zenith vistas.

Fact 2: Distance-Cues Control The Illusion.
Long ago, the experiments by Rock & Kaufman (1962a, Kaufman & Rock, 1962a 1962b) convincingly showed that the changes in the moon's perceived angular size correlate most strongly with changes in distance-cue patterns in the moon's vista.
Recently, Kaufman and Rock (1989) reviewed their research as well as the abundant research of others who demonstrated that role of distance-cues.
How far away an object looks from oneself is determined almost entirely by distance-cues. For example, binocular cues include stereoscopic depth perception, the most accurate source of 3D perception for distances up to, say, 100 meters. Convergence of the eyes also is listed. Movement parallax is extremely accurate.
Monocular distance-cues include linear perspective and texture gradients, often responsible for the depth illusions in pictures. Both are combinations of the basic distance cue of 'relative angular size' combined with linear size constancy.

For instance, a pictorial illusion for the picture at the left includes some trees and a horizon moon that look much farther away than the clipped cornstalks in the foreground.

A texture gradient is illustrated by the individual images of the short stalks whose angular sizes appear to decrease toward the middle of the picture: If the stalks realistically look about the same linear size (in inches) they appear to be in a snowy plane tilted toward a distant horizon.

Linear perspective is illustrated by the converging rows of cornstalk images. If the lateral separations (in inches) between those pictured rows appear to be the same linear size, so they look linearly parallel, those rows appear to recede toward the distant pictorial horizon because their angular separations appear to progressively decrease from the bottom to the middle of the picture.

Regarding the moon illusion:
The horizon moon looks its largest when distance-cue patterns in its vista are indicating that terrestrial objects, such as the trees, hills, or buildings which appear on or near the horizon moon, are very much farther away than nearby objects.
That is, the distance-cue patterns are indicating large perceived depths. [Perceived depth is the perceived distance between a near object and a farther object.   And, for an object, perceived depth is the perceived distance from a front edge to a back edge.]

When changes in the available distance cue patterns in the moon's vista 'signal' that it is even farther away from us, the distant-looking moon will look angularly larger.
Such cue patterns often exist for the horizon moon. (And as already noted, the now "larger-looking" horizon moon typically does not appear to move farther away because some other distance cues dominate the final outcome.)

Evidence.
If available distance cues are artificially reduced for a large-looking moon it will look smaller. For instance, one can hide the distance cues in the terrain under the horizon moon by using one's hand, or by looking at the horizon moon through a small tube. [A complication here is that looking through a tube is known to induce oculomotor micropsia.]
Another way is to reduce the effectiveness of the distance-cue patterns for great depth. For instance, if one bends over and looks at the horizon scene with the head upside-down, the horizon moon looks much smaller than it did before the inversion (Washburn, 1894).

On the other hand, when a change in distance-cues indicates that the moon has moved to a shorter distance, or else when there are very few distance cues, the moon's constant angular size appears to decrease (minify). This latter condition typically occurs for the zenith moon, especially when there is a large empty sky, a vista that lacks many distance-cue patterns which would create a great perceived depth between nearby viewed objects and far ones

Fact 3: Flat Pictures Offer Angular Size Illusions Due To Distance Cues.
A picture rich in distance cues can create an angular size illusion. 

For instance, in the picture below, the three circles are the same size.
Observers typically say the middle one looks larger than the lower one. But, there are two quite different linear size illusions here.

First there is a pictorial outcome: A horizon sphere' looks much farther away than a lower 'sphere' sitting between the cornrows, so it looks a much larger linear size (and volume) than the lower one. [As with any picture, viewing it with one eye, especially through a tiny hole (pinhole) can enhance the illusion.] That illusion illustrates the apparent distance theory if the two 'spheres' look the same angular size.

The other linear size illusion here concerns the circles themselves: They correctly appear on the same page, so they have the same perceived distance, and the middle circle looks slightly larger, in millimeters, than the lower circle, let's say 5% larger.
This linear size illusion occurs, of course, because the perceived visual angle is slightly larger for the middle circle than for the lower one.
This small visual angle illusion is the most interesting one, because it cannot be described or explained by the apparent distance theory.

Going back to the pictorial illusion, the far-looking horizon sphere also looks, say, about 5% larger angularly larger than the nearby 'cornrow' sphere in addition to looking a very much larger linear size.

Now compare the upper and middle circles. They also can illustrate a small visual angle illusion, which partially mimics the moon illusion: The 'horizon moon' looks, say, about 5% angularly larger than the 'zenith moon'.
Indeed, researchers have used pictures like that and measured that small 'moon illusion' (Enright, 1987a, 1987b). Others, however, have reported their results only in millimeters, and, guided by the apparent distance theory, did not mentioned the visual angle illusion that lies behind that linear size illusion (see Coren & Aks, 1990). Yet, that angular size illusion is precisely the one that needs to be explained.

Even in very realistic pictures the angular size illusions are much smaller than the natural moon illusion, but the distance-cue patterns for the real moon are, of course, more complex.

********* NOTE Added June 19, 2006 **********
The crude sketch at the right slightly resembles the much more realistic picture used in the experiment by Murray, et al. (2006) previously discussed in the Introduction and Summary Section.
As they point out, it was in the same category as the Ponzo illusion (below).
And, of course, that category includes Figures like those described above and the ones used in Enright's studies of the moon illusion in pictures.
Again, the Murray, et al. discovery was that the sizes of the activity patterns in cortical area V1 that corresponded with the equal retinal images of the two disks , were not equal; and the measured size difference there correlated almost perfectly with the measured perceived visual angle difference for the two disks (which was at least a 17% V-illusion for their very realistic picture).
Thus, as noted earlier, it seems that in all such 'classic' flat pattern 'size" illusions, the distance cue patterns are having their influence before the neural activity in the retina reaches Area V1.
(The Murray, et al. results are analyzed in Appendix B.)
***************End of June 19 Note ********

Fact 4. Very Simple Patterns Also Illustrate Angular Size Illusions.
The Ponzo Illusion.
The two horizontal lines are the same linear size, so they subtend the same angular size at the reader's eye.

Popular descriptions emphasize the pictorial illusion that the two converging lines can appear to portray the edges of a road going into the distance. In that linear perspective outcome, the upper line can appear to portray a 'bar' lying on the road about twice as far away as the 'bar' the lower line portrays; thus the farther-looking 'bar' looks about twice as long (its length in meters) than the closer-looking 'bar'.
Some articles suggest that the moon illusion is "merely the Ponzo illusion upside down."
But, again, the report here that one object "looks larger and farther away" than the other does not describe the majority moon illusion.

On the other hand there is the only 'Ponzo illusion' that has seriously interested researchers. Namely, all the lines appear on the page, at the same perceived distance, and the upper horizontal line looks a slightly larger linear size (say about 5% larger) than the lower one because it looks a slightly larger angular size. (A more detailed version is in Section II).

The Classic Ebbinghaus illusion (Titchner's circles).

In the patterns at the left, the two central circles are the same size, but their linear sizes on the page look unequal, because the angular size for the circle surrounded by smaller circles looks slightly larger (say about 10% larger) than the angular size for the circle surrounded by larger circles (McCready, 1985).

The Angular Size Contrast Theory.
Again, because the apparent distance theory cannot explain these classic flat pattern illusions, the best-known alternative has been the angular size contrast theory.

It proposes that an observer is comparing each target image's angular size to the average angular size for the images in its immediate context, which images subtend larger or smaller visual angles than the target. And that somehow makes each target's angular size look even smaller or larger than its actual visual angle.

The theory uses the fact that for many of these illusions (including the moon illusion) those changes in the visual angle subtenses of context images for the targets happen to be the bases of linear perspective and texture gradients, the distance-cue patterns so often correlated with visual angle illusions.
Some explanations for this well-known contrast effect are discussed in detail in Section II.
But, again, researchers have not yet accepted any one explanation.

Any theory of 'size' perception that applies to the moon illusion must deal with all the facts listed above.

The Two "Sizes" Controversy.
A few researchers still question how the perceived visual angle concept differs from the perceived linear size.
The distinction between these qualitatively different "sizes" was pointed out long ago by R. B. Joynson (1949; Joynson & Kirk, 1960). But, until recently, few researchers have used Joynson's revelation. Recognizing the distinction is crucial to understanding the new theory (McCready, 1965, 1985, 1986).

The "Perceived Size" Problem.
A major impediment to recognizing the distinction has been the common use of just one "size" concept called simply "perceived size" or "apparent size".
In some articles it consistently refers only to the perceived linear size. But some other articles use the amazing idea that the "perceived size" sometimes correlates with an object's linear size in meters, and sometimes correlates with the object's angular subtense in degrees! The reader often must try to figure out which concept the ambiguous "size" term refers to at the moment!

Some of the most confusing treatments of illusions invoke the concepts ambiguously called "size constancy" and "size constancy scaling" (Gregory, 1963. 1965, 1970, 1998). The concept properly refers to constancy of the perceived linear size. But in many discussions, "size constancy" is defined as if it were not constancy of perceived linear size, but constancy of perceived angular size, which, in the present view, doesn't make sense and would be quite maladaptive.
This "size-constancy" problem is discussed near the end of Section II.

Ross & Plug (2002) accept the concept of perceived angular size, but question how it differs from the perceived linear size. They evidently are reluctant to abandon the most influential general theory of human spatial perception that uses only one "perceived size" (or "apparent size") concept (Gregory, 1963, 1965, 1966, 1968, 1970, 1998).

Moreover, a few researchers, notably Kaufman, L. & Kaufman, J. (2000) still do not accept that a person can have an angular size experience along with the linear size experience!

The Main Task.
The angular size illusion, however, is the basic "size" illusion in many classic illusions, and precisely the one that has defied explanation for so long.
In order to move toward an explanation for it, it is crucial to keep in mind that it is controlled by changes in distance-cue patterns.

Once that angular size illusion is explained, it becomes relatively easy to explain the linear size illusions and distance illusions that accompany it.

That task is addressed in Section III, after the critique of conventional theories in Section II.

Index Page.
Introduction and Summary.
Section I. New Description of the Moon Illusion
Section II. Conventional Versus New Descriptions
Section III. Explaining the Moon Illusion
Section IV. Explaining Oculomotor Micropsia
Bibliography and McCready VITA
Appendix A. The (New) Theory
Appendix B. Analysis of the Murray, Boyaci & Kersten (2006) Experiment