The 'Railroad Track' Illusion:  
In this popular version of the Ponzo illusion, the dark rectangles are the same linear size on the page, so they subtend the same angular size at the reader's eye. Most people say the upper one "looks larger" than the lower one

The Pictorial Illusion
Some moon illusion articles repeat the apparent distance theory from beginning psychology texts (for instance, Wenning, 1985) and suggest that the moon illusion is "merely the railroad track illusion upside down." But that analogy fails because it uses only the pictorial illusion of tracks going into the distance, with the rectangles portraying perceived objects lying on the roadbed. [To enhance the illusion, view it with one eye.]
For instance, the object portrayed by the upper rectangle can appear to be about three times farther away than the object the lower rectangle portrays. The viewer says, "the upper object looks about three times farther away and about three times as long (in feet) as the lower object." That rather large linear size illusion illustrates the SDIH and the apparent distance theory.
But, even upside down, the appearance that one object looks farther away and a larger linear size than the other object which subtends the same angular size does not imitate the majority moon illusion.

At any rate, that pictorial illusion is not the one that has interested scientists.

The 'Paradoxical' Ponzo Illusion.
The quite different illusion yielded by the Ponzo picture is that the two equal rectangles can correctly look like flat bars printed on the same flat page, so they correctly appear at the same viewing distance, but the upper bar looks a slightly bigger linear size (in millimeters) on the page than the lower bar does, let's say about 10% longer.
This linear size illusion clearly illustrates an underlying angular size illusion; the upper bar appears to subtend an angular size about 10% larger than the lower bar does.

This very small, combined angular size and linear size illusion has interested researchers much more than the mundane pictorial illusion, for several reasons:
1. It cannot be described or explained by the apparent distance theory and SDIH.
2. It illustrates an angular size illusion induced by changes in distance-cues, so, to a small degree it imitates the angular size illusion for the moon .
3. The linear perspective and texture gradient cues here are the changes in the visual angles of the context images for the bars. So, the angular size contrast theory can be applied to this illusion.
4. This illusion also can be explained as yet another example of oculomotor micropsia/macropsia (as discussed in Section III).

Afterimages and Emmert's law.
Another analogy, offered by King, & Gruber, (1962) concerns the "size" an afterimage looks when it is "projected' to different distances.

For instance, you could stare for about 15 seconds at the center of the black disc from a distance of, say, 20 inches, and then look at another place on this screen, where you will see a bright white afterimage that properly appears the same linear size and at the same distance as the black disc. But if you move back from this screen to, say, 40 inches, that afterimage on the screen now will look twice as far away as it did and twice the linear diameter.
That happens because the afterimage has, in effect, the same angular size the black disc had when you stared at it; therefore, in accord with the SDIH, as   S'/D' = tan V,  the afterimage's perceived linear size will increase by the same proportion that its perceived distance ("projected distance") increases.
Psychologists call that important rule, Emmert's law.

Some currently popular "moon illusion explanations" (e.g., Wenning, 1985) merely repeat the analogy (without attribution) that an afterimage of a small disc projected onto the horizon surface of the illusory sky dome would look farther away, hence look a larger linear size than when projected on the sky dome's apparently closer zenith surface.
But, once again, the analogy fails because hardly anyone says the horizon moon "looks larger and farther away" than the zenith moon.

Although the apparent distance theory of the majority moon illusion can be rejected its two main versions are reviewed next. One appeals to a supposed illusion that the sky appears like a surface: The other appeals to changes in distance-cues.

Alhazen's Ceiling and the Sky Dome.
According to Plug & Ross (1989; Ross & Plug, 2002) the first scientist to clearly recognize that there is no physical basis for the moon illusion was Ibn al-Haytham, the 11th-century astronomer known to us as Alhazen. That is, he observed that the moon's angular size remained constant. (So, the atmosphere does not somehow create the illusion.)
He then proposed that the sky looks like the flat ceiling of an enormous room, and the rising moon appears to move along it in a more or less flat trajectory, so the moon would look closer at the zenith thus it would look a smaller linear size than it did at the horizon.

The Sky Dome Again.  
A similar old idea was the sky dome illusion, described in the Introduction, and repeated below using Dagram A. The rising moon supposedly appears to glide along this illusory sky surface. For the person at point O, the 'H moon' thus looks farther away than a 'z moon', therefore it must look a larger linear size than a z moon in order to keep their angular sizes equal.

The angle, V deg, is the same for all 'moons' because in standard discussions it starts out being the constant physical value (0.52 deg). However, the moment a discussion turns to describing how the person at point O perceives the moons, all those equal angles become, by default, perceived angular size values, (V' deg). In other words, for the person at point O, all the perceived zenith moons look the same angular size as the perceived horizon moon.
The sky dome diagram obviously fails to describe most peoples' moon illusion.

Once again, filled circle H in the diagram correctly looks angularly larger than each z circle. That imitates the majority moon illusion, but does not imitate what the diagram is illustrating for the person at point O. For us to experience what that person is seeing, we must look at the front view at the left.

Other Problems.
For many people the horizon sky appears to extend far beyond the horizon moon (see Gilinsky, 1980).

Another difficulty is that research studies of the apparent shape of the sky do not confirm the flattened dome model for all observers (Baird & Wagner, 1982).
Similar disconfirming data is reviewed by Ross & Plug (2002).

Finally, even researchers who still advocate the apparent-distant theory have pointed out that one does not need to appeal to a sky surface illusion in order to 'explain' why the horizon moon would look farther away than the zenith moon (Rock & Kaufman, 1962a, Kaufman & Rock, 1989, Kaufman & Kaufman, 2000)
They appeal, instead, to the changes in distance-cue patterns.
In a sense, the sky dome theory is obsolete.

The Size-Distance Paradox.
Some researchers long ago pointed out that the size-distance paradox which results from trying to apply the SDIH and apparent distance theory to various illusions cannot be resolved. (Boring, 1962).
But, what rarely has been pointed out is that the 'paradox' completely vanishes when one uses the perceived angular size concept in addition to perceived linear size (McCready, 1965, 1985).

Nevertheless, there have been two major attempts to resolve the 'paradox' while still clinging to the SDIH and using a single 'size' concept called merely "perceived size" or "apparent size." One attempt uses a hypothetical "registered distance" concept.
The other refers to two hypothetical levels of "size" scaling.
However, as shown below, both attempts are illogical.

The Registered Distance Idea.
The registered distance proposal invents a three-Stage perceptual process (Kaufman and Rock, 1962, 1989; Kaufman L. & Kaufman, J. 2000).
In Stage 1, certain factors, such as distance-cues, establish a greater "registered distance" for the horizon moon than for the zenith moon. But this registered distance remains out of consciousness.

In Stage 2, the increased 'registered distance' operates at an "unconscious level" to make the conscious "perceived size" greater for the horizon moon, in accord with the SDIH (stated by, S'/D' = tanV  ). This "size" is, of course, the perceived linear size. The logic of this second stage is illustrated in the conventional diagram by the two 'moons' of perceived linear sizes, 15 units and 10 units.

Next is stage 3, which is described with a statement such as, "because the horizon moon now looks larger than the zenith moon, it looks closer."
Notice, however,this proposed stage 3 is impossible according to the very logic (the SDIH) being used,
As the diagram illustrates, when the moon's perceived visual angle remains the same, if the moon's perceived linear size is increased from 10 units to 15 units, the geometry requires that the moon then look about 1.5 times farther away than it did. The increase in perceived linear size cannot result in a decrease in perceived distance.
The registered distance idea does not resolve the 'paradox'.

The difficulty here, of course, is that every reader clearly understands the phrase "it looks larger, so it looks closer," which is how Stage 3 is worded. This phrase describes our many everyday experiences for objects whose visual angle increases, usually because the objects come closer to the eyes. It also describes our common experience that the object we 'see' in a movie appears to approach us when its image on the screen enlarges (zooms) to create the perceptual event called "looming," a popular term for a large increase in the perceived visual angle.

In other words, the "size" we invariably are referring to when we say an object "looks larger and closer" is not the object's linear size, but the visual angle it subtends. The object's linear size typically appears to remain the same (linear size constancy). Indeed, this linear size constancy must occur in order for the increase in the perceived visual angle to yield the shorter perceived distance as an example of the relative angular size cue to distance.

Because the sentence that describes the illogical Stage 3 "sounds" good it creates a false impression that the paradox has been resolved.

Two "Size" Scalings Proposal.
A slightly different attempt to resolve the size distance paradox proposes that the scaling (valuation) of "perceived size" and distance occurs at two different perceptual levels (Gregory (1963, 1965, 1966, 1968, 1970, 1998).
The proposal uses the logic of the SDIH and apparent distance theory, so the scaled "perceived size" has to be only the perceived linear size, in meters.
Applied to the moon illusion, the proposal is that the distance-cues for a greater distance cause a primary scaling of the horizon moon's 'perceived size' larger than the 'perceived size' for the zenith moon (which is yielded by the primary size scaling for it).

Then a secondary scaling occurs, described only by a phrase such as "the horizon moon's larger 'perceived size' makes it look closer than the zenith moon."
Again, that sentence is illogical according to the very logic (the SDIH) the proposal uses.

Other difficulties abound in moon illusion discussions that use a "size-constancy" concept.

The Flawed, Size-Constancy” Approach
The opening sentences in 'size-constancy' discussions point out that it refers to the very common observation that an object’s ‘perceived size’ (S’ ) appears to stay the same when, for example, an increase in the object's distance from the eye decreases the angular size (V deg) the object subtends, hence the size of the object’s retinal image decreases.
Other sentences point out that, when S’  remains constant, the object’s perceived distance (D’ meters) must increase: That is, in accord with the SDIH equation rearranged as, S’ = D’ tanV, when V decreases, S’, will remain the same only if D’ increases in inverse proportion to the decrease in V deg.
Because the SDIH equation specifies that S’  is the perceived linear size, in meters, that perceptual constancy must be more precisely called, linear size constancy.

Much confusion arises in 'size-constancy' discussions whenever the term ‘perceived size’ is used in a manner which makes it refer not to the perceived linear size, S’ m, but also to the perceived visual angle, V’ deg, which the SDIH doesn't include.
In other words, many sentences unwittingly fail to distinguish between S’ m and V’ deg, so from one phrase to another the term ‘perceived size’ can change from referring to S’ meters to referring to V’ degrees.

Consequently, many discussions of “size” constancy end up defining it as constancy of the perceived visual angle, V’ deg, when the visual angle, V deg , changes!
For instance, Trehub's (1991) theory of the moon illusion uses a single "perceived size" concept and the SDIH, to propose a hypothetical model of brain processes that might underlie 'size constancy' and Emmert's law. But, the proposal ends up treating 'size constancy' as constancy of perceived angular size instead!

********* NOTE Added June 20, 2006 **********
As already noted, Murray, et al. (2006) explicitly state that they measured the perceived angular size illusion.
So, the results do not (cannot) support the standard, SDIH, approaches.
However, in the article's discussion section, the interpretations of those results often use the SDIH logic, and confuse perceived visual angle (V' deg) and perceived linear size, S' cm, (called ‘perceived behavioral size’).

For instance, it is suggested that, for a viewed target that has a constant retinal image size, R mm, an increase in the target's perceived distance, D' cm, elicits a supposed “scaling” of some entity called the viewed object's ‘retinal projection” to yield a larger “perceived behavioral size” for the object, "whereby retinal size is progressively removed from the representation" (p.422).
But that suggestion states the SDIH logic used by Emmert’s Law, by "misapplied size-constancy scaling," by the apparent distance theory, and by the "registered distance" argument.
So, it cannot apply to the visual angle illusion that was measured!
It actually suggests that the (flexible) perceptual correlate of the extent. R mm, between two stimulated retinal points is the perceived linear size, S' cm, rather than the perceived visual angle, V' deg.
It also implies that 'size constancy' is a constancy of the perceived angular size, V' deg, which would be terribly maladaptive.

As was easily predictable, other articles already are mis-interpreting the Murray, et al. experiment in that same manner.
***************End of June 20 Note ********

That common mistake generally has gone unrecognized.
It is discussed in detail at the end of Appendix A.

Obviously, many 'size' illusions have made it necessary to reject the SDIH and apparent distance theory.
Also, the visual angle contrast theory doesn't go far enough to explain the moon illusion. The oculomotor micropsia theory offers another alternative, discussed in Section III.

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