Appendix A     (Revised November 09, 2004)

The (New) Theory of Size, Distance and Visual Angle Perception.

The new theory of visual perception of linear size, distance, and the visual angle (McCready, 1965, 1983, 1985, 1986) still is controversial because it conflicts with well-entrenched basic assumptions of conventional theories of "size" and distance perception.
For instance, the theory that dominates the literature (the SDIH) does not include the idea that we perceive the visual angle. A few writers even state that the visual angle is not perceived.
In order to counteract those traditional assumptions, details of the new theory are presented here.

New Logical Rule.
Consider the relationships of the objective (physical) values for a viewed frontal extent (Figure A1) and the subjective (perceived) values for it (Figure A2).

Physical Values: Figure A1
An observer whose eye is at point O is looking at the base of a frontal extent (the vertical arrow, the "distal stimulus". It has a frontal linear size, S meters and is D meters from point O, the center of the eye's entrance pupil.
The direction of the extent's base from point O is, d0 , let's say "straight ahead" or "eye level to the horizon". The direction of the extent's upper end from point O is, d1 , toward some specific elevation.
Each of those direction lines in Figure A1 also indicates (in reverse) the path of a ray of light from an object's endpoint to and through eye point O. Each such ray is known as the chief ray in the center of the large bundle of the light rays from the object point which pass through the cornea, pupil and lens to focus as the optical image of the extent's endpoint on the retina.
The angular difference between the two chief rays indicated in Figure A1 is the visual angle, V degrees. Thus, the visual angle, V degrees, also is the difference ( d1 - d0 ) between the directions of the extent's endpoints from point O.



Equation A1 states the relationship between those external (distal stimulus) measures.

          tanV = S/D         (Equation A1)

An inverted real image of the object is formed on the retina. This retinal image (proximal stimulus) has a linear size, R millimeters given by the equation, R/n = tanV ,
in which n is the nodal distance, approximately 17 mm.
Consider next the response measures commonly referred to as the "perceived values."

Perceived Values: Figure A2
The subjective perceived frontal extent has a perceived linear size, S' meters.
Its perceived distance, is D' meters.
Point O' is an objectively measured point that researchers refer to as the effective locus of the subjective cyclopean eye: For present purposes, point O' may be considered to coincide with point O in Figure A1.

The two perceived directions from point O' are, d0' and d1'.
For instance, the object's base appears, let's say, "straight ahead" or "toward the horizon", and the tip appears toward "a higher elevation."
Those perceived directions differ by the perceived visual angle, V', degrees.

According to the new theory, the three perceived values relate to each other as stated by Equation A2.

      S'/D' = tan V'        (Equation A2, the perceptual invariance hypothesis)

Ross & Plug (2002) recently dubbed that equation (McCready, 1965, 1985) the Perceptual Invariance Hypothesis.

An additional equation is needed to relate the perceived angular size, V' deg, to the physical angle, V deg, and that relationship must involve the retinal image size, R mm.

Retinal Size and Perceived Angular Size
All modern researchers agree that we do not perceive or "sense" the retinal image (a proximal stimulus) or its properties, as such: Instead, perception is "distally focused." That is, we have (we create) visual images only of external objects and their properties. There certainly is no "sensation" which could be called the "perceived retinal image size", R' mm.
One reason why some researchers did not use the concept of the perceived visual angle, V' deg is because they felt that to use V' deg was tantamount to accepting the ancient, obsolete idea that a person somehow "senses" the retinal image's size.

However, in terms of the present theory, V' deg does not concern a linear size value in meters, it concerns, instead, one's perception of directions. For instance, many theorists, notably Helmholtz (1962/1910) and Hering (1942/1879) have argued, convincingly, that the visual direction of a viewed point (its subjective egocentric direction) is determined by a combination of factors, with its final value due to a process that necessarily combines the position of the point's image on the retina with information about the position of the eye with respect to the head (and body). And, the final perceived direction, d, for an object predicts the direction the eye should turn to in order to focus directly upon it: Also, d predicts the direction the head or body should be aimed in order to let the person examine the object more closely or to reach toward it.

For two points which have different egocentric directions, the person obviously is seeing the difference between those visual directions, and the magnitude by which they differ correlates primarily with the distance, R mm, between the points' retinal images.
That perceived difference between two subjective directions defines the perceived visual angle, V' deg, for their separation. Therefore, V' deg essentially is the perceptual (subjective) correlate of the "size" of the retinal separation, R mm, between the optical images of the points.

Technical Note: The Subjective (Mental) Dimensions:
Initial presentations of the 'new' theory (McCready, 1965, 1983, 1985) carefully differentiated between the private, subjective (mental) dimensions and the public, objective response measures of those mental entities. That is, the values, V' deg, S' meters and D' meters which a researcher obtains and writes down as the "perceived" values are physical behavioral response measures of the observer's subjective entities, v , s, and d. The present theory is that those (private) mental dimensions relate as illustrated below.

Point E represents the visual egocenter, the place in one's (subjective) body image of one's head from which one feels one is viewing the world. E also could represent the (subjective) cyclopean eye.
The subjective directions from E to the phenomenal object's endpoints are the visual egocentric directions, e0 and e1
The phenomenal visual angle, v, is the difference (e1 - e0 ) between the directions.
Those phenomenal directions may or may not agree with the actual directions from the face, thus v may or may not agree with the actual visual angle.
The phenomenal linear size, is s.
The phenomenal distance, is d.
(Please overlook this very temporary re-use of the small case symbol, d.)
The proposed rule is that, s/d = v.
From that theory is derived the testable perceptual invariance hypothesis, S' / D' = tan V' , which proposes how the response measures of the subjective values would be expected to relate.

However, the custom in the literature has been to let the term "perceived value" (also "apparent value") stand for both the subjective value and an objective response measure of it.
Accordingly, I have been following that traditional practice in this article.
(End of Technical Note.)

New Psychophysical Relationship
Next we need to express the psychophysical relationship between the perceived value V' deg and the physical value , V deg, as mediated by R mm. Any such rule must express the flexible relationship between V' deg and V deg which shows up in visual angle illusions.
For the moment, it is sufficient to say first that, in general, V' deg is a function (F) of R mm and 'other factors'. So, we can write,

             V' = F (R, and 'other factors' )

And, because R mm is directly valuated by V deg (in accord with, R/n = tanV), we can express the relationship between V' and V as,

             V' = F (V, and 'other factors' )

Consequently, to arrive at a complete explanation of visual angle illusions, including the moon illusion, researchers will have to fully identify and quantify those 'other factors' and write psychophysical equations much more detailed than those above. Indeed, quite a few equations have been published which use very complex mathematics (see recent textbooks on cognitive psychology, and, for instance, Indow, 1991).

Brain Models: Moreover, recent texts often offer complex models of (hypothetical) brain activities: Each model describes neurophysiological mechanisms invented to explain why the particular psychophysical equation the author is using fits published data. But, a problem with some of those complex models (e.g., Trehub, 1991) is that the equations they are trying to explain are based upon a logic (usually the SDIH) which does not differentiate between perceived visual angle, V' deg and perceived linear size, S' meters: Such models confuse those two qualitatively different concepts, and thus are very confusing.

Those 'Other Factors'.
As already noted, one set of 'other factors' that can change V' away from V deg, includes changes in the state of the oculomotor system. For instance, oculomotor micropsia illustrates that V' will become slightly less than V deg, when the oculomotor system adjusts to a closer distance. Some theories of that adjustment refer to hypothetical brain mechanisms (Enright, 1989).
A second set of 'other factors' includes changes in distance cues (which may or may not alter the oculomotor state). For instance, as the moon illusion illustrates, with V deg constant for a viewed object, there usually will be a slight increase in V' when major distance cue patterns signal an increase in the object's distance.

Visual Angle Contrast Theories as Evidence.
A third well-known set of 'other factors' includes changes in the size of the visual angles subtended by extents which appear close to the target object (simultaneous contrast), or changes in extents that are intently stared at before the target is viewed (successive contrast).
For instance, as previously discussed, the "size contrast" theory of the moon illusion (Baird, Wagner & Fuld, 1990; Restle, 1970) clearly addresses it as a visual angle-contrast illusion

Other theories of "size"-contrast illusions appeal to hypothetical interactions among nerve cells in the visual system beyond the retina, which supposed activities relate directly to the size of the retinal image, R mm: So, those activities are considered to be the biological precursors to V' deg (Oyama, 1977).
Yet other theories refer, instead, to changes among "spatial frequency detectors," and some others refer to an adaptation of "size channels" (best called, "angular size channels").
Both of those terms refer to hypothetical nerve cell structures in the brain whose activities are directly related to R mm. Therefore, they logically relate to the perceptual angular size dimension, V' degrees, rather than to the perceptual linear size dimension, S' meters.
The writers of those theories obviously accept that the perceptual correlate of R mm is not S' meters, but V' degrees.

Additional evidence that the visual angle is perceived can be found in discussions of visual acuity.

Visual Acuity.
One's basic skill of perceiving the different directions of two viewed points from oneself is presumed in all discussions of visual acuity: It refers to the accuracy with which one can distinguish the direction of one viewed point from the direction of another.
For, example, minimum separable visual acuity can be measured using a series of two small black bars printed on a bright white paper, like those below.


Typically, a person views such patterns from a distance of 20 feet, and notices which patterns have bars so small that they don't look like two bars, but like just dark blobs. The ability called "20/20 vision" means that the person is able to see two bars separated by a visual angle of one minute of arc, but cannot see as two bars the pairs separated by less than 1 minute.

Perception textbooks invariably describe visual acuity in their discussions of the optics of the eye, so those discussions obviously reveal that people perceive the visual angle. Yet, in later chapters the discussions of "size" perception and "size" illusions, rarely discuss direction perception and our perception of the difference between two directions (the visual angle and angular size perception) even though the most puzzling "size" illusions begin as visual angle illusions.

How the new theory describes valuations of the perceived values has been published in detail elsewhere (McCready, 1965 and 1985, especially pp. 333-334). This visual processing model is reviewed below.

Visual Processing: A Modified, Unconscious Inference Model.
To describe how the perceived metric values (S' and D' ) are valuated it is convenient to use a modified version of the unconscious inference model attributed to Helmholtz (1962/1910) and used by most theorists who have used the standard approach, especially Rock (1977, 1983). The present model differs from the standard model by including both the perceived visual angle, V' deg and the perceived linear size, S' meters.

The model uses Figure A2 and Equation A3 (below), which is a simpler version of Equation A2 obtained by using the small angle rule that,     tanV' = V' radians.

S'/D' = V'   rad   (Eqn. A3)

The model proposes that, for a viewed object, an initial activity (a pre-processing event) provides a specific perceived visual angle value, V' rad, as determined by the retinal image size, R mm, and 'other factors'. The values of S' meters and D' meters are obtained in a manner that is as if two "processing steps" had occurred almost simultaneously to furnish values which will agree with Equation A3 for the given value of V'.
Those processing activities occur outside conscious awareness. Modern theorists accept that such valuations are the result of visual system activities which furnish the physiological precursors (brain correlates) of the final subjective conscious qualities for the perceived object. So, as noted earlier, many different models of brain events have been published. None has been accepted yet.

Describing The Visual Processing of S' and D'
In processing Step 1, either S' meters or D' meters becomes scaled (valuated) directly.
In Step 2, the other metric value becomes valuated indirectly, as if it had been computed to conform with the rule, S'/D' = V', for the value of V', (as given by V' = F(R and 'other factors').
The term "unconscious inference" refers to Step 2. Step 2 also has been called a "taking-into-account" process (Epstein, 1973).
The initial scaling of S' or D' can occur by cuing or by hypothesizing.

Scaling of D'
D' m may be scaled by absolute cues to distance.
For instance, if we are touching an object, we feel how far it is from our face, and this perceived "haptic distance," can directly scale the visual D' for it.
And, as discussed earlier, the physiological brain activity pattern which correlates with the "effort" we are about to expend to focus and converge (binocularly) upon a target may directly valuate D'. For instance, Foley, (1980) called it the egocentric distance signal.

Or, D' can be hypothesized, a form of visual imagery in which the target appears at an assumed, presumed or suggested distance (Gibson, 1950).

Once D' is scaled, S'   becomes 'computed' (as Step 2) so the results agree with the rule, S' = V'D' meters.

Scaling of S'
The perceived linear size, S' , may be scaled by cues to linear size.
For instance, if we are holding an object, we feel a "haptic size" for it (in inches) which can directly scale the visual S' for it.
Also, Sí m for an object can be scaled directly by a cognitive linear size for it which illustrates the "familiar size" (or "known size") cue to linear size (Bolles & Bailey, 1956; Ono, 1969).

After all, our current visual percept of an object predicts what we would find upon closer inspection of it, and that prediction is based upon what we already have learned about such an object when we previously perceived it in sufficient detail to allow us to have learned most of its properties.
Those properties include such things as how large it feels (its haptic linear size), its volume, its weight, its hardness, its colors, its temperature, its smell, its taste, the sounds it emits when tapped, etc., etc. All those perceptual properties collectively make up the objectís cognitive identity. And, most objects have a name which by itself can call up the appropriate perceived linear size, S' for the object.

S' also can be scaled by hypothesizing, in which the target appears an assumed, presumed or suggested linear size (Coltheart, 1970; Hastorf, 1950).

Once S' is scaled, D'   becomes 'computed' (as Step 2) in accord with, D' = S' / V' meters.

Although D' or S' could be scaled to be almost any reasonable value, the computing step (or its logical equivalent) prevents any conflicting independent scalings of D' and S' from producing a ratio, S' / D' , that would be inconsistent with V' rad.
As discussed earlier, there may be conflicting distance cues for an object, and the final D' value for it may be a compromise among several potential values.

For this review I won't focus upon the differences between Steps 1 and 2, except to show that some of the traditional 'cues to distance' actually involve not cuing of D' , but a 'computation' of D' after S' has been scaled.

Two Targets
Consider next the processing steps when two (or more) targets are compared. Several likely types of outcomes were described earlier for the moon illusion. Two are reviewed below using a front view picture of two men.

An Equidistance outcome.
The picture easily leads to a percept of two men at the same perceived distance.
As Step 1, this equal distance scaling could result from a (cognitive) "equidistance assumption" (McCready, 1965, 1985) or an "equidistance tendency" (Gogel, 1965).
(Also, there might be pictorial details which act as cues to signal an equal perceived distance scaling.)

If so, in Step 2, the perceived linear height (S' m) of the man on the left becomes twice that of the man on the right because his perceived angular subtense (V' deg.) is twice as large.

An Equal Linear Size Outcome.
Now consider an example in which the two men in the picture were the same height and the one on the right was twice as far from the eye as the other. A side view of that arrangement appears below.
The front view (above) as seen by the person whose eye is at point O, is repeated below.

For the man on the left (in both pictures) the perceived values relate as stated by the equation, S1' /D1' = tan V1'
For the man on the right, the perceived values relate as stated by the equation, S2' /D2' = tan V2'

Now let scaling (Step 1) make the men's linear sizes look equal.
That could happen if the observer knows their heights, so the S' values could be scaled equal by the (cognitive) familiar size cue to linear size.
Or, if the men are unfamiliar, that scaling could occur due to an "equal linear size assumption " (McCready, 1965, 1985). Indeed, looking at that front view you can see both men as being the same height (and at different distances).

Thus. given S2' = S1' , Step 2 computing makes the ratio of the perceived distances equal to the inverse ratio of the perceived visual angles, as stated by D2' / D1' = V1' / V2'   .
As illustrated in the side view, the man on the left thereby looks half as far away as the other one.
That particular computation illustrates, of course, the relative angular size "cue" to distance.

Identity Constancy: The above diagrams and equations also can describe the simpler case in which a single viewed object is compared at time 1 and time 2, after the object's distance and visual angle have changed. Equal linear size scaling for many objects can be attributed quite simply to identity constancy. After all, we adults assume that most objects stay the same object from moment to moment when other things change, which illustrates identity constancy (see Piaget, 1954). Obviously, if one accepts that an object is the same unchanging object from time 1 to time 2, its perceived linear size, S', remains the same, hence, if its perceived visual angle changes, its perceived distance also must change.

That equisize scaling illustrates linear size constancy.

Linear Size Constancy.
An object typically appears to stay about the same linear size when its visual angle and retinal image size change as a result of a change in the object's distance from the eye. That illustrates linear size constancy.

Two Objects: Linear size constancy also refers to having two objects of the same linear size look the same linear size when they are at different distances. For instance, reconsider the example illustrated above, with the two men looking the same height, as illustrated in the side view.
Equations that describe the outcome are as follows:

Again we can use the small angle approximation, tan V' = V' rad, and substitute V' for tan V'.   Putting the two previous equations together thus yields,
        (S1' /S2' ) /(D1' /D2' ) = V1' /V2'         which rearranges to:
         S1' /S2' = (D1' /D2' ) (V1' /V2' )     Equation B1
For this example, V1' = 2V2',    so the equation becomes,
        S1' /S2' = 2D1' /D2'      Equation B2

Equal perceived linear size scaling here furnishes S1' = S2',  so Equation B2 becomes,
D2' = 2D1',    Thus, D1' = D2' / 2,   which means the man on the left looks half as far away as the man on the right, (and, again, that 'computing step' illustrates the relative perceived visual angle cue to distance).

Linear Size Constancy By Computing.
An alternative process is that, as Step 1, some distance cues could scale both perceived distances correctly, so the man on the left looks half as far away (D1' = D2 / 2 ). Thus Equation B2 becomes S1' /S2' = 1. In this case, the linear size constancy outcome is the result of computing (Step 2).

A jargon problem. is that the (ambiguous) term "size constancy scaling" (Gregory, 1963...1998) often refers not to a scaling step but to that computing step described by the general Equation B1.
A more serious jargon problem concerns both "cuing" and "relative size."

The So-Called "Relative Size Cue To Distance."
The ambiguous term "relative size" could refer either to a comparison of values or to a comparison of values.
And, of course, as a Step 1 scaling operation, cuing should not be confused with a Step 2 computing process.
Consequently, the common use of the term "relative size cue to distance" creates a problem.
As illustrated above, the processing steps being referred to yield a D' ratio, are described by Equation B1 rearranged as, D1' /D2' = (S1' /S2' ) / (V1' /V2' ),
and that equation rearranges into Equation B3.

         D1' /D2' = (V2' /V1' ) (S1' /S2' )     Equation B3

The desired ratio of perceived distances is obtained by the 'computing' step after the linear sizes have been scaled equal (linear size constancy), which makes the equation,

        D1' /D2' = V2' /V1'
Accordingly, the "relative size" relationship actually involved in this process is the ratio of the perceived visual angles. It certainly cannot be the ratio of the perceived linear sizes because, of course, they are equal (that is, S2'   is neither larger nor smaller, "relative to" S1' )
Conventional discussions, however, do not knowingly use the perceived visual angle concept! Therefore, those discussions of the "relative size cue to distance" cannot openly refer to the comparison of two perceived visual angles. So, their term "relative size" would have to refer to the linear size ratio, but that makes no sense because that S' ratio must be about 1.0 (linear size constancy) in order to have the desired perceived distance outcome occur.
In other words, standard discussions often unwittingly let the term "size" refer to the perceived visual angle,V' deg, although those same discussions use the SDIH which ignores the concept, V' deg.

The Distinction Is Not New, Just Neglected..
Perhaps the first clear discussion of the distinction between the perceived visual angle, V' deg and the perceived linear size S' m was published long ago by Joynson (1949; Joynson & Kirk, 1960).
The idea that we perceive the visual angle has been strongly advocated by Baird (1970; Baird, Wagner & Fuld, 1990).
In addition to McCready (1965, 1985, 1986), other articles which explicitly accept that we perceive both the visual angle and linear size for an object have been published by Rock & McDermott (1964), Ono (1970), and Komoda & Ono (1974). More recent supporting articles include Enright (1989); Reed, (1989); Higashiyama (1992) Higashiyama & Shimono, (1994) and Gogel & Eby, (1997).

The logic of new theory wholly replaces that of standard treatments, reviewed below.

Old Logic: The SDIH.
Traditional theories use the very different logical rule called the size-distance invariance hypothesis (SDIH) illustrated by Figure A3.
The geometry the SDIH uses is stated by the equation,

S'/D' = tan V         (The SDIH)

Here V deg is the physical visual angle.
D' is the perceived distance, in meters.
In SDIH sentences, S' is called simply the "apparent size" or "perceived size."
But notice that Figure A3 and the equation require that the measuring unit for S' must be the same unit used for D' , say meters: So, S' clearly has to be the perceived linear size, in meters.

Discussions based upon the SDIH (virtually all popular and textbook discussions) do not knowingly use the concept of the perceived visual angle, V' deg, and remain silent concerning the perception of directions.
Thus, by default, use of the SDIH logic implicitly suggests that the visual angle always will look equal to its actual value.
Consequently, the SDIH cannot logically be used to describe, predict, or explain a visual angle illusion, in which V' deg disagrees with the visual angle, V deg. That explains why conventional descriptions of "size" illusions that begin as visual angle illusions have been so puzzling and paradoxical.

More Jargon Problems: Much confusion in the perception literature is due to uncritical use of the ambiguous terms, "apparent size" and "perceived size." which could refer either to the perceived linear size, S' meters, or to the perceived visual angle, V' deg. When one reads standard discussions of "size" perception and "size" illusions one often cannot tell what the terms "apparent size" and "perceived size" refer to.

A very confused old suggestion was that the "apparent size" (or "perceived size") for a viewed object sometimes is valuated by factors directly related to its linear size, S meters, and sometimes valuated, instead, by factors directly related to its angular size, the subtended visual angle, V deg.
But the magnitude of that peculiar kind of "apparent size" would have to magically switch back and forth from being a certain number of meters to being a certain number of degrees, a clearly impossible transition.

The misuse of SDIH jargon occurs often in standard discussions of visual processing, as follows.

The SDIH Psychophysical Relationship.
Standard descriptions of visual processing have been using the SDIH equation as the basic psychophysical equation.
Many descriptions use it in its rearranged form as,

     S' = D' tanV.    SDIH Rule A.

Such descriptions can be proper only if S' clearly is the perceived linear size.

But, in some SDIH-based sentences the term "perceived size" (or "apparent size") unwittingly refers to the perceived visual angle, V' deg, which is not used as such. Those sentences thus actually imply that the SDIH psychophysical equation is, illogically,
   V' = D' tanV deg.
Such sentences suggest that an increase in perceived distance, D' , "magnifies" V' deg for a constant V deg!
That unintentional mistaken interpretation of SDIH Rule A is as close as standard discussions get to describing a visual angle illusion controlled by distance cues (such as the moon illusion). Of course, their logic (geometry) is wrong, and the magnitudes of the "magnification" they imply are far too large.
If the writers had properly interpreted the diagrams which illustrate the geometry (logic) they were trying to use, they would not have written those illogical sentences.

Retinal Size: Other discussions of visual processing use the SDIH restated in terms of the proximal stimulus value, the retinal image size, R mm. For instance, they use the rule,
R/n = tan V,   to write the SDIH as,

     S' = D' R/n    SDIH Rule B.

Processing sentences that use Rule B can be logical only if 'perceived size' clearly is the perceived linear size [because that rule also is, S' = D' (S/D) meters].
But some sentences are based, in effect, upon a mistaken logic expressed by a version of Rule B in which the constant, n, (the nodal distance of about 17 millimeters) is omitted, as follows:
     S' = D' R.
Those sentences thus imply that the perceptual correlate of the retinal image size would be expected to be S' meters, and that a visual processing step provides the 'perceived size' for an object by having the perceived distance for it somehow "expand" or "magnify" the tiny retinal size, R mm, up to the much larger "size" the object looks! That very ancient idea is illogical but still appears in some articles (see below).

All the illogical sentences discussed above are responsible for the incredibly misleading 'textbook' descriptions of size constancy, which were critiqued in Section II and are discussed in more detail below.

The Size Constancy Pseudoproblem.
Nearly all published definitions of size-constancy begin by posing a Ďproblemí which does not exist. This size constancy pseudoproblem is created by comparing two facts about "size" perception in a way that makes them appear to contradict each other, but they really don't. And, in one way or another, this (artificial) contradiction is said to be 'resolved" by the activity of a 'size constancy' mechanism, often described in terms of hypothetical brain activities.

In accord with the previous discussion of linear size constancy, the two facts are as follows:

Fact 1:   When an object of constant size (S meters) recedes from the eye, its distance ((D meters) increases, so its angular size (V degrees) decreases, hence the size (R mm) of its retinal image decreases.

Fact 2:   The perceived linear size (S' m ) for the object almost always remains constant.

Additional facts are, of course, the perceived visual angle (V' deg) decreases, and the perceived distance (D' m ) increases while (S' m ) remains constant, as shown, for example, by the side-view diagram of the two perceived men.

The pseudoproblem is created by defining 'size constancy' with phrases that use the ambiguous terms 'perceived size' and 'apparent size' , and merge Facts 1 and 2 in illogical ways which, unfortunately, sound reasonable to an uncritical reader.

For instance, a short example of the illogical combination of Facts 1 and 2 would be:
"The retinal size decreases, but the 'perceived size' usually remains constant (size constancy)."
The linking word, "but", obviously implies that an object's 'perceived size' normally would be a direct perceptual correlate of the object's retinal image size, R mm, however, a 'size constancy' process often keeps it constant when R changes.

A correct replacement for that illogical sentence is: "The retinal size decreases, and the perceived linear size (S' meters ) usually remains constant (linear size constancy) while the perceived angular size (V' deg) decreases, and the perceived distance (D' ) increases." That proper description does not suggest any problem that needs to be solved by a 'size constancy' mechanism'.

Some published sentences which can fairly represent virtually all "textbook" definitions of 'size constancy' are quoted below. (I have emphasized the words that create the illogical link between Facts 1 and 2.)

Some Influential Quotations.
"There is a well-known set of phenomena which certainly does involve perceptual modification of retinal images ---size constancy. This is the tendency for objects to appear much the same size over a wide range of distance in spite of the changes of the retinal images associated with the distance of the object. We may refer to the processes involved as constancy scaling." (Gregory, 1963, p. 2).

"Although all objects give smaller retinal images as they recede from the eye, this geometrical shrinking is generally compensated by the brain, to give 'size constancy'." (Gregory, 1965b, p. 17)

"It is a well-known fact that the apparent size of an object depends not only on the size of the retinal image or visual angle but on the distance as well. Within certain limits, objects do not appear to vary substantially in size when viewed from varying distances, despite the fact that the size of the optical image varies inversely with distance. This phenomenon is known as size constancy. It is as if the observer took the distance into account in perceiving the size of the object. " (Kaufman & Rock, 1962a, p.953.)
"In line with this same reasoning, where the visual angle remains constant but where the distances are registered as different, the apparent size will change. In the case of an afterimage projected on surfaces at different distances, the apparent size is a direct function of the distance of the surface, a relation known as Emmert's law. The moon illusion can be considered a special manifestation of Emmert's law..." (Kaufman & Rock, 1962a.)

Notice that all such definitions implicitly assume an objectís Ďperceived sizeí would be expected to get smaller when its retinal image size (R mm) and angular size (V deg) decrease, except that a 'size constancy' process 'corrects' it in a manner that is as if   V deg and R mm had been expanded (magnified).

The previous discussion of linear size constancy, however, clearly shows that the equal linear size outcomes for the perceived linear size, S' meters, certainly do not involve a 'correction' of the perceived visual angle, V' deg: Therefore, the decrease in V deg and R mm with the increased distance of an object does not create a "perceptual size" problem that needs to be solved.

Consider next how those standard descriptions create the pseudoproblem.

An Analysis.
Keep in mind that virtually all the standard definitions of 'size constancy' (and certainly those quoted above) are using the logic of the SDIH, stated by S'/D' = R/n. And, as shown below, they are illogical no matter what the term 'perceived size' (S' ) actually refers to.

Is S' the perceived linear size? Suppose, first, that 'perceived size' means 'perceived linear size', and the intention is to (correctly) define 'size constancy' as linear size constancy. In that case, in order to see what the SDIH suggests is the perceptual correlate of R mm, we can rearrange the equation to become,   nS'/D' = R mm.

So, according to the conventional logic, the direct perceptual correlate of R mm is the ratio of the perceived linear size to the perceived distance, multiplied by the nodal distance, n. In other words, according to the logic those definitions are using, the perceived linear size, S' , cannot be the direct perceptual correlate of R. That is, when R decreases, S' meters can remain constant without requiring any correction that would have the same effect as 'magnifying' R mm.

That pseudoproblem also is revealed by consulting the diagram repeated at the right.
It shows that , for the observer at O, the men look equally tall, let's say 6 feet. That equal linear size perception certainly is not as if a retinal image had changed size.
Indeed, if the proposed 'magnification' of the retinal size' for the more distant man had occurred (to yield the same value for both men) then the picture of the man on the right would have to be drawn twice as tall as it now is, which would indicate that the observer at O would be seeing him as about 12 feet tall. And, of course, that would destroy the linear size constancy (rather than creating it).

Is S' , instead, the perceived angular size? Now suppose the term 'perceived size' in the above definitions stands for the perceived visual angle, V' deg (even though this concept is not supposed to be used explicitly in discussions based upon the SDIH).

When reworded using V' deg, the quoted definitions would imply that the perceptual correlate of R mm is V' deg, and, of course, that agrees with the present 'new' theory. Those reworded definitions thus would begin by correctly implying that V' deg will decrease when R mm decreases, due an increase in D meters. But, unfortunately, the linked phrases then convert that perfectly normal result into a "problem" (a pseudoproblem) which supposedly must be solved in order to have the 'perceived size' (V' deg ) remain constant when R mm decreases. Hence the proposed 'size constancy' process would have the same effect as would a magnification of the visual angle and retinal image.

Those reworded sentences obviously would be defining 'size constancy' as an "angular size constancy", in which an objectís visual angle almost always appeared to remain the same when the objectís distance from the eye changed! That absurd idea suggests that we usually do not see the different directions of two fixed objects become less different when our distance from them increases. Such an outcome certainly would be maladaptive.

How illogical that idea is can easily be seen if you re-consult the diagram above.
Again, an author who uses properly interpreted diagrams, or equations with appropriate units of measure, can avoid writing those illogical verbal descriptions which can badly mislead an uncritical reader.

The conclusion is that the widespread standard definitions of 'size constancy' do not make sense, no matter what the ambiguous terms 'perceived size' and 'apparent size' refer to.

Those flawed definitions of 'size constancy', and the flawed descriptions of projected afterimages and Emmert's Law lie behind the standard, 'apparent distance' theories of the moon illusion (Gregory, 1963--1998; Kaufman & Rock, 1962a, 1962b; Kaufman & Kaufman, 2000). That familiar "textbook" approach also was used in a moon illusion explanation offered by Trehub (1991) discussed next.

Trehub's Model
In the book, "The Cognitive Brain", Trehub (1991) constructed a complex theoretical model of brain processes that might be involved in human cognition and perception. Part of his model is devoted to hypothetical brain structures and processes involved in visual spatial perception.
In a short section (p. 242-247) he describes how those supposed neural activities might explain the moon illusion. Ross and Plug (2002) offer a brief review of the brain structures and activities Trehub posits, but in order to understand the details one must study Trehub's book.

Of course, many models of presumed brain functions which might underlie visual space perception have been proposed (see recent texts on Cognitive Psychology). Some describe complex physiological mechanisms, and some are complex mathematical models (see Indow, 1991). Each model has taken into account, of course, what was known about brain physiology at the time the model was invented. And, published research on the physiology of the visual system is adding new knowledge so fast that one must be a specialist in the area just to keep up. Only those specialists can vote on the validity of a given model.

I do not know enough to usefully comment on the details of any of those many models. However, I sometimes can judge the validity of the description being offered for the visual phenomena (usually a 'size' illusion) the model was designed to explain.

So, although not fully understanding the intricate neural structures in Trehub's model, I can, nevertheless, examine (below) his definition of the moon illusion, and the logic he used to explain it.

Analysis of Trehub's Model
Trehub (1991) did not use both S' meters and V' degrees, and did not explicitly specify what the term 'perceived size' refers to. He did not explicitly refer to the SDIH or to the apparent distance theory, but, as shown below, he offered the same 'size constancy' and projected afterimage, and Emmert's Law descriptions critiqued above.

For instance, on page 92 the changes in the 'perceived size' of an afterimage projected to different distances are described. This use of Emmert's law (also specifically referred to on page 245) clearly indicates use of the SDIH logic stated by, S' = D' tanV, and S' = D' R/n.

The following quotations deal with the portion of the proposed model which would do the visual processing that results in 'size constancy' and Emmert's law. (Again I have emphasized the 'linking' words that create the pseudoproblem.)

Some Trehub Quotes
"The perceived constancy in the size of an object over a range of observer-object distance despite large variations in the retinal size of the object .... does not seem to depend on a process of iterative size adjustment like that performed by the size transformer. ........ Thus, retinal images that become smaller as a function of increasing object distance are magnified in compensatory fashion as they are mapped onto the space represented by the mosaic cell array." (Trehub, 1991, p. 89.)

And elsewhere, "This provides a neuronal circuit for magnifying (or reducing images in rough compensation for the change in retinal size at different viewing distances." (Trehub, 1991, p.244).

Moreover, some 'moon illusion' articles and newsgroup discussions on the internet include comments Trehub sent to them about his book. Those messages point out that what the proposed mechanism keeps constant is the "perceived intrinsic size of an object despite changes in its projected retinal size as the distance between the object and the observer changes (size constancy)".

Some of those messages also mention that the book offers the details of   "... an innate neuronal system that maintains size constancy by automatically expanding or contracting the brain's representation of the changing retinal image in compensatory fashion as an object's egocentric distance increases or decreases.

All those sentences are just as illogical as the others quoted and critiqued above.
However, if Trehub's confused 'size constancy' idea were ignored, his model could be interpreted to describe the moon illusion as part of the more general visual angle illusion in which distant objects in the horizon direction look a larger angular size than objects of the same angular size located in an elevated position, as reported by Higashiyama (1992; Higashiyama & Shimono, 1994).
That is, Trehub's model might be describing some hypothetical brain activities that would yield a modification of V' deg away from V deg (and, of course, away from R mm) as a function of factors related to global egocentric distances and directions. If so, that revised model would apply to the moon illusion, which cannot be explained by his model based upon the logic of Emmert's law and 'size constancy scaling'. That is, if changes in V' deg away from V is what Trehub meant to describe, then he defeated his argument by using the SDIH logic of Emmert's law and the flawed 'size constancy' approach, both of which do not refer to or use the concept of V' deg.

If Trehub's specific hypothetical mechanisms were redescribed in terms of perceived visual angles (rather than perceived linear sizes), they could apply to oculomotor micropsia/macropsia, for which some general models of brain mechanisms have already been proposed (Enright, 1989; Foley, 1980).

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