British Journal of Mathematical and Statistical Psychology (1997), 50, 187-203 




Helen E. Ross

Department of Psychology, University of Stirling,

Stirling FK9 4LA, Scotland, UK.



Running Head: Discriminability and apparent magnitude






Some psychophysicists seek a unified theory in which the scaling of apparent sensory magnitude and the discrimination of differences in intensity can be encompassed. When contextual factors are held constant, there is a positive correlation over ma ny sensory continua between the exponent of the power function for sensory scaling and the inverse of the Weber fraction. Variations in neural efficiency also show a similar correlation with apparent magnitude and discrimination. The role of other factors (such as contrast, adaptation and size scaling) remains unclear. Both Weber and Fechner considered aspects of these issues. Weber held that, for the same physical stimulus intensity, apparent magnitude and discriminatory ability increased with neural eff iciency. Fechner held that changes in apparent magnitude had no effect on discrimination: a stable differential threshold was predicted by the Parallel Law (Weber's Law applied to internal sensations). Evidence is considered for the tactile sense and weig ht perception, and for visual size. The relation between apparent heaviness and weight discrimination is complex, varying with the state of adaptation and with neural efficiency. There is some evidence that both tactile and visual apparent size increase w ith a finer underlying neural structure; but it is unclear whether other types of increase in apparent size lead to increased spatial acuity, or to differences in the Weber fraction for line length. The variety of values of apparent magnitude and discrimi nation, and the lack of a monotonic relation between them, makes it unlikely that all known relationships could be encompassed in a unified psychophysical theory.



1. Introduction


Much has been written on the history of psychophysics. There have been excellent recent review papers by Krueger (1989) and Murray (1993). Murray argues that psychophysics is dividing into two camps, one concerned with the relation between psychophysic al and neurophysiological data (e.g. Laming, 1986), and the other attempting to integrate psychophysical scaling into measurement theory (e.g. Falmagne, 1985). These two branches correspond approximately to Fechner's distinction between `inner psychophysi cs' and 'outer psychophysics' (Scheerer, 1992; Murray, 1993). A contrary trend is the search for a unified psychophysical law, which attempts to unite these approaches (e.g. Baird, 1985, 1997; Krueger, 1989; Link, 1992; Norwich, 1993; Ward, 1995, 1996; Wa rd et al., 1996). This paper aims to outline some of the issues concerning discrimination and apparent magnitude that would have to be encompassed in a unified theory, and to give an account of some representative topics of historical interest.

A persistent theme in the history of psychophysics is the idea that there must be some relation between discriminability and the apparent magnitude of a sensation. Fechner (1801-1887) is usually credited with this idea (1860, 1966), but Weber (1795 -1878) predated him in 1834. Many authors have discussed the relation between discriminability and apparent magnitude when only physical stimulus intensity is varied; but some have also discussed the issue when a change in apparent magnitude is caused by some other factor (such as context, adaptation, fatigue or neural structure). Weber and Fechner differed in some aspects of their interpretation, particularly when discussing the effects of fatigue or neural structure.

The first topic is the subject of mainstream psychophysics: it has been much reviewed and will be dealt with relatively briefly here. The second topic has been the subject of many unrelated studies in perception and neuropsychology: it is a very co mplex area, but one that must be encompassed by proponents of a unified psychophysical theory. Examples could be taken from many areas of perception; but this paper will concentrate on some topics of historical interest in the tactile and weight senses an d in size perception.


2. The relation between apparent magnitude and discrimination when the physical magnitude of the stimulus is varied.


The first requirement for a unified theory is that there exist stable numbers that represent the relation between apparent and physical magnitude, and that between discrimination and physical magnitude. This has always been open to question (e.g. Bartl ett, 1940). If such numbers exist, it should be possible to show a correlation between them - though a mathematical formula would not determine the direction of causation.

2.1 Apparent and physical magnitude

Apparent magnitude could, in principle, have several different relationships to physical magnitude. The relation could be linear, a logarithmic function, a power function, or some more complex function. Weber did not discuss the issue. Fechner (1860, 1966) proposed a logarithmic function, of the form: S=k logI (where S is sensation intensity, k is a constant, and I is physical intensity). However, he did not attempt to measure apparent magnitude directly; he believed in a necessary relation be tween apparent magnitude and discrimination, and relied on the validity of Weber's Law and on mathematical reasoning. Dissent soon followed. Plateau (1801-1883) argued for a power law (1872), of the form: S=k(In) (where n is the exponent of th e power function). He based this on the results of a bisection experiment in which artists produced a grey that appeared midway between white and black: the chosen grey was the same despite variations in illumination. Delboeuf (1831-1896) used more sophis ticated apparatus, and found that the bisection values changed considerably with changes in illumination. He argued (1873) that the data fitted a logarithmic function related to a contrast ratio between the stimulus intensity and some background value. He modified Fechner's equation to:

S=k log[(I+c)/c] (where c is the state of internal excitation). Delboeuf believed that increments in sensation magnitude were in fact increments in successive degrees of contrast. Titchener (1867-1927) credited him with replacing Fechner's `sensation m agnitudes' with what he called `sense-distances' (1905). Delboeuf's views on sense-distances were known and discussed in the 1930s (Bartlett, 1940; Boring, 1942), but were then neglected till recently. The history of these early arguments can be found in Brysbaert (1992), Murray (1993), and Nicolas and Murray (1998).

Later authors used various techniques such as magnitude estimation, category ratings and ratio matching, and produced several different functions. Stevens (1957) argued for a power law, and maintained that this followed the underlying neural firing rate. The functions that occur seem to depend very much on the measurement technique, and the different biasses they produce (Poulton, 1989). There are also difficulties in how the physical stimulus is measured: weights, for example are measured on a lin ear scale (kilograms), but sound is measured on a logarithmic scale (decibels). The units of the scale have a profound influence on the resulting function (Weiss, 1981; Myers, 1982). The relation between stimulus intensity and the rate of neural firing is also controversial (see review by Lipetz, 1971): the latter varies with the site at which it is measured (peripheral or more central), the state of adaptation, the sense modality, and many other factors. In some modalities intensity is not coded by the r ate of firing, but by the number of neurons recruited. Other modalities are more qualitative than quantitative. Various authors have made a distinction between additive, prothetic or intensive dimensions (such as heaviness, loudness o r brightness) and substitutive, metathetic or extensive dimensions (such as pitch and position). Stevens and Galanter (1957) claimed that the former produce subjective magnitude scales that can be fitted by power functions while the l atter do not. It has also been claimed (e.g. Stevens, 1939; Postman, 1946) that the former are susceptible to the time-order error (in which the second stimulus usually appears more intense than the first), while the latter are not. However, the distincti on between the two is often blurred: length of line, for example, gives a linear function rather than a power function (Poulton, 1989), but it is often described as a prothetic dimension (Pitz, 1965). Thus Stevens' hope (1975) of finding a clear relation between stimulus intensity, the rate of neural firing and apparent intensity, seems bleak. Discussions of the possible laws can be found in Treisman (1964a, 1964b), Warren (1981), Falmagne (1985), Laming (1988), Luce and Krumhansl (1988), Poulton (1989), Krueger (1989), Bolonowski and Gescheider (1991).

As mentioned above, Delboeuf (1873) pointed out that the same stimulus intensity gives different apparent magnitudes depending on the state of the observer, and many authors have noted that we normally perceive complex sensations rather than pure intensities. Treisman (1970) argued that we perceive brightness contrast and not simple intensity, and that the relation between apparent magnitude and the physical stimuli can be described by a power function. Lockhead (1992) similarly argued that in man y modalities (particularly brightness and loudness) perceived magnitude depends on contrast with the background, or with change of intensity over time. Thus the primary stimulus may be relational, with the absolute value being a secondary derivation. Few of the commentators on Lockhead's article were as pessimistic as him about the value of exploring simple stimulus intensity, and many found the power function a useful descriptive tool.

If power functions are valid descriptors, the exponent of the power function can be used as a simple measure to describe the growth of apparent magnitude in a given sensory domain. However, there is little agreement about unique exponents for the d ifferent modalities.

2.2 Discriminability and physical magnitude

It is also difficult to obtain a unique measure of discriminability as a function of stimulus magnitude. Discrimination is often measured by the differential threshold (DL or jnd), but the various test methods may give different values (Laming, 1986). The DL may also have several possible relationships to the physical intensity of the stimulus, such as Weber's Law, a square root law, or some other function (Laming, 1986). The most common of these is Weber's Law, which is of the form: DL/I=k. It states that the DL increases linearly with the physical stimulus magnitude. Weber did not produce much evidence to support his law, and neither did he discuss the possible reasons for its form. It was Fechner who named the law after Weber, and gave it ma thematical form. He developed experimental methods to measure thresholds, and produced data on the validity of the law. He was concerned at the breakdown of the law at low intensities, and speculated that the rise in the fraction was due to what we now ca ll background noise. He therefore produced an improved version of Weber's Law, which can be described by the formula DL/(I+n)=k (where n is a small constant). It is generally accepted that Weber's Law is an approximation, and that it applies to intensive rather than extensive sense modalities. In so far as it holds, the steady state Weber fraction at moderate intensities can be used as a single measure of the discriminatory ability of a sense modality.

It has often been pointed out the DL is not a pure measure of discrimination, since it is contaminated by constant errors or bias (e.g. Bartlett, 1940). Usually the effects of various biasses are reduced or removed by counterbalancing test orders, or by the use of forced choice tasks. Signal Detection Theory (Tanner and Swets, 1954) allows a more precise separation of sensitivity (d') and one type of bias (b ). It is now generally accepted that two parameters are needed to describe discrimination thresholds. The relation between d' and the Weber fraction is discussed by Laming (1986), who lists standardised Weber fractions and discriminability parameters for different sense modalities and types of stimuli.

2.3 Discriminability and apparent magnitude

Many authors have argued that there is a causal link between discrimination and sensory magnitude, the latter depending on the former. Fechner (1860, 1966) was the first to formulate this idea clearly, maintaining that sensory magnitude was dep endent on the sum of the jnds, all of which were subjectively equal: if Weber's Law holds, the jnds increase in proportion to the stimulus intensity, and the apparent magnitude increases logarithmically. Fechner's views were similar in some respects to th ose of Weber (1834, 1996), who argued that increased neural discrimination was accompanied by increased apparent stimulus magnitude. However, Weber was concerned with neurophysiological differences in different parts of the body rather than differences in physical stimulus magnitude; his arguments are therefore discussed later (sections 3.3,4.1 and 4.2).

Stevens (e.g. 1975) denied that there was any relation between sensory magnitude and discrimination, and a similar position was taken by Gescheider (1985). Those authors who believe that magnitude scales are arbitrary must also deny any link betwee n the scale and discrimination. However, many modern authors have supported Stevens in accepting some form of power function for the relation between the subjective magnitude of sensations and the physical stimulus intensity; and some have sought a relati on between the exponent of the power function and the DL or Weber fraction (e.g. Teghtsoonian, 1971; Norwich, 1993; Zwislocki, 1994; Hellman & Hellman, 1995; Ward, 1995, 1996). Ward (1996) and Ward et al. (1996) argue that the equations used by these authors are fundamentally similar: if the intensity discriminability of sensations (d' for sensations) is considered in addition to the intensity discriminability of stimuli (Weber fraction), there is a clear relation with the exponent of the power functi on. Ward argued that the equation could be simplified to the approximate general form of m1/m2=WF2/WF1 where the subscripts refer to two different sensory continua, m is the exponent of the power function, and WF is the Weber Fraction. Taking the data for pairs of sensory continua from Table 1 of Teghtsoonian (1971), he plotted the ratio of the exponents against the inverse ratio of the Weber fractions: the data were very well fitted by a straight line with a s lope of 0.91. This relationship confirms the idea that those sensory continua with the higher exponent also have the better discrimination. It also lends some support to Weber and Fechner's belief that there is a necessary relation between discrimination and subjective magnitude, though it does not specify the underlying cause. For Weber, the underlying neurophysiological structure had an effect (imprecisely quantified) on both measures; while for Fechner, subjective magnitude bore a defined and quantitat ive relation to discrimination.

This result is encouraging, and it appears to account for a large part of the variance. However, it must be noted that the formula only holds because most contextual factors have been held constant and many parameters have been eliminated. We shall consider the importance of some of these factors in the next section.


3. The relation between apparent magnitude and discrimination when the apparent but not the physical magnitude of the stimulus is varied.


There can be little doubt that changes in physical intensity are the largest cause of changes in apparent magnitude. Nevertheless, quite large changes in apparent magnitude can be caused by contrast, adaptation and other factors, the size of the effect s varying in different modalities and test conditions. The effects of such factors on discrimination are more controversial, since the changes are often small and difficult to measure. Indeed, Laming (1984, 1988) claimed that response variance in scaling tasks is 100 times greater than in threshold discrimination tasks. It is nevertheless theoretically interesting to see whether there are any relationships that could be incorporated in a unified psychophysical theory.

3.1 Other factors and apparent magnitude

There are many factors that cause changes in apparent magnitude when the physical intensity of the main stimulus is unchanged. Some factors are clearly neurophysiological: these include natural variations in the underlying neural structure, or damage to the sense organ or brain. Damage, or an insensitive nervous structure, generally causes a reduction in apparent magnitude. However, damage to the motor system may cause an increased sense of effort, which may increase the apparent heaviness of l ifted weights under some circumstances (see review by Jones, 1986).

Some other factors have a less clear neurophysiological basis, and are often hard to separate. They include adaptation, contrast, assimilation, constancy mechanisms, perceptual illusions, cognitive factors and various context and bias effects. Some bias effects are quite large (see Poulton, 1989). Lockhead (1992) reported that a change in the stimulus range could change judgements by a factor of 6, and manipulation of the response range could change the slope of the power function by a factor of 3. The order of presentation of stimuli is of considerable importance for magnitude estimates and category judgements, as these are made relative to the previous stimulus (Laming, 1984). The time-order error is also of importance in discrimination tasks, th ough its effect is normally cancelled out when measuring thresholds. However, it is a topic of interest in its own right, and various attempts have been made to measure it (e.g. Guilford, 1954; Ross, 1964; Hellström, 1977; Saito, 1994). Whatever meas ures are used, this error is relatively small; for example, Ross (1964) found constant errors in weight discrimination (the difference between the point of subjective equality and the standard) lying between 1 and 7% of the value of the standard.

Perceptual illusions are generally of intermediate magnitude. In the realm of size perception, geometrical illusions and figural aftereffects do not usually exceed a factor of 1.3; the moon illusion (which is related to size-constancy) is larger, u sually between 1.5 and 2.0; while full size-constancy can give a very much larger effect, the overestimation of the angular size of distant objects reaching a factor of about 4.0 in relation to that of close objects (Ross and Plug, 1997). Regarding weight perception, the size-weight illusion gives a factor of up to 1.4 (Ross, 1969); and mass-constancy (when effective weight is varied through changes in acceleration in the human centrifuge or parabolic flight) a factor of about 1.5 (Ross, 1981).

Some of the factors described above are small or present measurement difficulties (e.g. response biasses and time-order errors); but others, such as perceptual illusions, are fairly straightforward to measure, and should be large enough to reveal a ny relation with discrimination.

3.2 Other factors and discriminability

There appears to be no simple answer as to how the above factors affect discrimination. Neurophysiological impairment generally causes a deterioration of discrimination, the extent of the deterioration varying with the impairment. Cold hands, f or example, can raise the DL for lifted weights by a factor of about 1.5-2.0 in comparison with warm performance (see Ross, 1981). Adaptation to a level higher or lower than that of the test intensity is similar to neurophysiological impairment in that it impairs discrimination, while adaptation to the appropriate intensity improves discrimination. The range of differences in the DL varies with the sense modality and the type of adaptation. If the standard is continually varied rather than constant, the w eight DL rises by a factor of about 1.3; and if the subject is previously adapted to heavy weights, the DL for lighter weights may rise by a factor of 2-6 (see Ross, 1981). Similar effects occur in other modalities, and may reach a factor of about 6 for b rightness discrimination (see Keidel, Keidel and Wigand, 1961). The effects of other perceptual and cognitive factors are smaller and more controversial: for example, the size-weight illusion may raise the DL by a factor of up to 1.2 for objects that are apparently too heavy or too light for their size (see Ross, 1981).

3.3 Apparent magnitude and discriminability

It should be clear from the above examples that there is no simple relation between the size or direction of changes in apparent magnitude and changes in discrimination. However, there is an early history of discussion of the issue. Weber (1834 , 1996) stated that there was a positive and necessary correlation between good discrimination and apparent magnitude, the same neural mechanisms affecting both aspects of sensation. Weber used the Latin term subtilitas to mean both `acuity' (for d ifference thresholds), and `sensitivity' (for the absolute threshold and for the apparent intensity of suprathreshold stimuli). Perhaps the use of the same word encouraged him to believe that these factors were intrinsically related. He did not usually ma ke a clear distinction between the meanings, though he worked towards it in some passages (1834; 1996, pp. 82-83). He also used the term tactus to cover a wide range of cutaneous and tactile senses, and again this may have encouraged him to believe that acuity or sensitivity in one tactile sense might be correlated with that in another.

Fechner, on the other hand, thought that Weber's Law could not hold unless the `Parallel Law' also held (Murray and Ross, 1988). The Parallel Law states that Weber's Law applies to apparent intensity (the internal realm) as well as to phy sical intensity (the external realm): thus a change only in apparent rather than physical intensity should leave the DL unchanged, since all subjective values and differences are changed in the same proportion. (If, of course, Weber's Law does not hol d, one could still argue that the effect of a change in apparent intensity should be the same as that of a change in physical intensity. But this was not Fechner's argument.)

Recent work in the field of apparent auditory intensity seems to support Weber's viewpoint rather than Fechner's. Zwislocki and Hertig (1995) reviewed the literature and concluded that intensity jnds were correlated with apparent loudness rather th an with sound pressure level or other variables. The experiments were conducted with subjects who had normal hearing in one ear and a hearing loss in the other ear, or with noise masking in the second ear, so that loudness equality required different SPLs in the two ears. Given loudness equality, DLs were approximately the same in the two ears. Zwislocki (1994) argued that the correlation was a unique property of power functions, given that the internal noise controlling detection of intensity increments originates in the sensory periphery.

This last example is clearly based on neurophysiology as the causal factor; it is consistent with the relation between apparent magnitude and discrimination described in section 2.3 for various sensory continua. However, there are many examples tha t are inconsistent with that formulation. We shall now consider in more detail the evidence concerning these questions in some modalities that interested Weber and Fechner: the tactile and weight senses, and visual size perception. These will serve as exa mples of the questions that can be raised in many modalities.



4. Tactile and weight perception


4.1 Tactile acuity, neural structure and apparent length

Weber (1834; 1996, p.46) stated that two points feel further apart in areas of the skin with better tactile acuity. The statement is supported by tables of two-point thresholds in different parts of the body, and by a verbal description of what is now known as Weber's illusion: if two compass points with a fixed suprathreshold separation are moved over different parts of the body, the distance between the points seems to shrink or grow in those areas where the two-point threshold is larger or small er. Weber made many similar statements about other factors affecting the two-point threshold, including the orientation of the compass points in relation to the body axes (longitudinal discrimination is poorer than transverse, and the separation seems les s); and he speculated on the underlying neurophysiology (1996, pp. 110-112). Weber's illusion is well supported (Goudge, 1918), and it seems plausible that apparent separation should increase with spatial acuity.

There appear to be no experiments on the reverse question - whether an increase in apparent separation (caused, for example, by a tactile geometrical illusion) would cause an increase in tactile acuity within that space.

It could also be asked whether the Weber fraction for line length varies in different parts of the body (as Weber would predict) or remains the same (as Fechner would predict); but there appear to be no data on this question.

4.2 Tactile acuity, static weight discrimination and apparent heaviness

Weber moved into less secure ground when he discussed weight discrimination and apparent heaviness, and their possible relation to tactile acuity (two-point threshold). He argued that static weight perception should be related to tactile acuity , while active (muscular) weight perception need not.

Weber stated (1834; 1996, pp. 67-69), without presenting any data, that weights seemed to press more heavily on the left than the right hand. He speculated that the underlying neural structure was more sensitive on the left side. He believed that s tatic weight discrimination should also be better when weights were placed on different fingers of the left hand than when placed on the right hand - but he reported that his experiments showed no difference.

Weber measured static weight thresholds in different parts of the body by successive presentations of two weights to left and right equivalent parts, increasing the difference until it could be discriminated. He argued that the smaller the DL, the more acute the touch sense, and the heavier the impression of the weights (1834; 1996, p.70). The lowest thresholds were obtained for the palmar surface of the fingers, the sole of the foot, and the forehead (all equal); the shoulder and heel were middlin g; and poor discrimination was found for the back of the head, the frontal surface of the chest, the back and other places.

Weber also used another method, which he took to be equivalent: he placed weights simultaneously on two different parts of the body, and varied one weight until the two felt equal or slightly different (1834; 1996, p.72): "If equal weight are place d on various organs ... we do not feel equal pressure on both organs. A weight pressing on the forehead seems less than the same weight on the lips, because the forehead has poorer tactile sensitivity than the lips. Now if we increase the weight on the fo rehead until the pressure seems equal on both places, the difference in weight gives numerical expression to the difference in sensitivity between the two places."

In this experiment Weber calculated the point of subjective equality rather than the difference threshold, thus treating the data as a magnitude match. He then used the results to compare `tactile sensitivity' on the palmar surface of the fingers w ith various other parts of the body. His arithmetic is approximate and hard to follow, but he concluded that the rank order of sensitivity for different body parts was much the same as for the two-point threshold (1834; 1996, p. 119). Details of the rank orders are given in Ross (1996), and the correlation is only moderately convincing.

The static weight matching method seems to confuse magnitude matching with discrimination. It tells us about relative heaviness in different parts of the body, but does not give a direct test of discrimination. For that, two weights on the equivale nt part of the body are needed, as in Weber's bilateral experiment. Unfortunately Weber did not give enough information on the same body parts to make a good comparison - but, again, the correlation is not very convincing.

As stated earlier, Fechner was concerned to uphold the Parallel Law. He argued that body areas that differ in `absolute sensitivity' (as measured by weight matching or magnitude estimates) should nevertheless have the same objective differential s ensitivity. Fechner (1860, 1966, pp. 267-268) examined Weber's data, and came to the conclusion that there was no correlation between `absolute' and 'differential' sensitivity for weight on different parts of the body. The comparison seems to involve only 5 parts of the body (Ross, 1996), so it is hard to reach any conclusion. Fechner argued that the two methods were not strictly comparable. Nevertheless, he took the absence of a correlation to mean that there was in fact no real difference in discriminat ion in different body parts, and took this as evidence in favour of the Parallel Law. Fechner appears to have drawn large conclusions from a very small amount of data. Modern work on correlations between different tactile abilities gives some support to Weber. J.C. Stevens (1979) found that sensitivity to weights placed on the body (as measured by magnitude estimates) gave the following rank order: palm, abdomen, forearm, upper arm, thigh, back. This order correlates well with Weinstein's (1968) data on point localization and two-point threshold, but not with punctate threshold or vibration threshold; nor with sensitivity to roughness (Stevens, 1990). However, there appear to be no adequate data on whether weight magnitude estimates correlate with wei ght discrimination in different body parts.

4.3 Adaptation and other factors affecting the heaviness and discrimination of lifted weights

Fechner further investigated the Parallel Law (that discrimination is unaffected by extraneous changes in apparent weight) by looking at adaptation, since adaptation alters sensitivity. He fatigued his arm, making lifted weights feel heavier, a nd again claimed that the DL was unaffected. However, he suffered pain from these experiments, and perhaps failed to obtain satisfactory results.

Weber's Law does not hold exactly for lifted weights, the Weber fraction rising for lighter weights (see Laming, 1986). It could, then, be predicted that (for light weights) a reduction in apparent weight should raise the Weber fraction, and an inc rease in apparent weight should lower it. A different prediction was made by Seashore (1896) and some later authors: they argued that the DL (measured in physical units) should be proportional to the apparent rather than the physical stimulus intensity, s o that an increase in apparent weight should raise the Weber fraction. Neither of these predictions was supported by the experimental results. Modern experiments on increases or decreases in apparent weight caused by adaptation show that discrimination de teriorates when the arm is unadapted to a change of weight, or when affected by various weight illusions (see reviews by Ross, 1981; Jones, 1986). They give no support to Fechner's Parallel Law, nor to Weber's assumption that an improvement in discriminat ion is always accompanied by an increase in apparent intensity. However, there is some evidence that a peripheral reduction in sensitivity (as brought about by cooling the arm) causes both a reduction in discrimination and a reduction in apparent weight ( Ross, 1995).


5. Size perception


5.1 The effect of neural structure on acuity, contrast sensitivity, and apparent length.

Several authors have asked whether there is a relation between apparent length and spatial acuity within that length. Weber's illusion is well established for the tactile perception of length, with lines feeling longer over the more acute skin areas; a nd it might be expected that the same would be true for the relation between retinal acuity and apparent visual length (see Carr, 1935, pp. 368-381). The question can be examined for naturally occuring variations in retinal acuity, and for neurological da mage. The latter gives a clear answer - supporting Weber's position - and will be discussed first.

The effects of neurological damage have been studied from the retina to the brain. Distortion of the retina has clear phenomenal effects (Duke-Elder, 1934; Critchley, 1953). The phenomenal size is larger (retinal macropsia) when an image falls on c ompressed retinal tissue, and smaller (retinal micropsia) when it falls on stretched retina. The latter case may occur in elongated eyeballs (a common cause of myopia), and may cause aniseikonia (e.g. Winn et al., 1988). These findings are consistent wit h the view that the retinal cells continue to send information about their original location before damage occurred; or that size is coded by the number of retinal cells stimulated, or by the number of their cortical counterparts. Some authors have consid ered the effects of brain damage. Bender and Teuber (1949) reported that localised injuries (gunshot wounds) in the calcarine visual cortex may cause the visual field to become anisotropic: in general, objects in the disturbed region appear too small (mic ropsia) and too distant (teleopsia).

The answer is not so clear concerning anisotropy in the visual field for subjects with normal vision. Anisotropy could be related to optical blur, or to neural factors such as retinal receptor density, receptive field size, or cortical mapping. Vis ual acuity and contrast sensitivity decrease in peripheral vision, owing to adverse changes in these factors (see Olzak & Thomas, 1986). There are also nasotemporal differences: cone receptor density is slightly less in the nasal than the temporal hem iretina (in addition to the blind spot in the nasal side), but the rods are more numerous in the nasal than the temporal hemiretina. If apparent size depends on retinal density, it might be expected that with monocular cone vision objects in the outer vis ual field should appear slightly smaller, while with monocular rod vision they should appear larger (and vice versa for the inner visual field). Thus the direction of the effect might vary with luminance and other factors. James (1890; 1950, pp. 14 0-144) and Helmholtz (1910; 1962, pp. 179-180) both noted that with monocular vision objects appeared smaller in the periphery than the fovea. Other early work generally showed that objects appeared larger in the upper than the lower part of the visual fi eld, and in the right than the left part; but the evidence on peripheral and foveal apparent size was considered equivocal by Carr (1935, pp. 368-381). More recent work confirms that objects in the middle to upper part appear larger than those in the lowe r part (e.g. Piaget, 1969, pp. 152-163; Hotopf et al. 1983), and vertical lines appear longer in the right than the left parts (Hotopf et al, 1983). It has also been found (Chukova and Gusev, 1993) that squares appear wider in the right than the left visu al field, whether viewed with the left or right eyes. The authors explained this as due to differences in the sizes of receptive fields in the two hemispheres, since the effect was not correlated with nasotemporal asymmetry in cone density at the retinal level. Several modern authors have found that objects appear smaller in peripheral than foveal vision (e.g. Newsome, 1972; Schneider et al., 1978). A few have claimed that objects appear larger in peripheral vision, and Bedell and Johnson (1984) have show n that the direction of the effect depends on luminance, high luminance targets appearing larger and low luminance targets smaller.

Many modern authors have used spatial frequency gratings as test stimuli, partly because sine wave gratings are unaffected by optical blur. With such stimuli it is unclear whether apparent size should refer to the fineness of the grating lines or t o the total width of the stimulus patch. Gratings generally appear finer (higher spatial frequency) when viewed in the periphery (e.g. Georgeson, 1980; Davis et al., 1987), perhaps because of inappropriate labelling of spatially tuned channels in the peri phery (Georgeson, 1980). Wink and Harris (1993) found that gratings appeared to have lower contrast and finer grain in the periphery; however, paradoxically, high contrast grating patches appeared to have both finer grain and to cover a greater total wid th. They argued (following Bedell and Johnson, 1984) that the outer edges of the patch were affected by blur: at low contrasts the blurred edges fall below the detection threshold, but at high contrasts they add to the apparent width. We may conclude that apparent size (width) effects are affected by luminance and neural density, while the apparent texture of gratings may depend only on spatially tuned channels. Other lines of research (Stuart et al., 1993) suggest that size perception does not depend on spatial frequency detectors, but rather on broadly tuned layers of size detectors.

It remains uncertain how far the various retinal location effects are due to anisotropy of the image, anisotropy of the retinal or cortical architecture; or, in the case of up-down anisotropy, to perceptual learning about the anisotropy of typical scenes as represented in the retinal image. However, a strong case can be made for a role of retinal and cortical density, similar to Weber's illusion in the tactile sense.

5.2 The effect of apparent size on acuity and length discrimination

While there may be some grounds for supposing that an increase in neural density leads both to an increase in visual acuity and to an increase in apparent size, it does not necessarily follow that an increase in apparent size leads to an increa se in visual acuity. There are several possible causes of an increase in apparent size, and a gain in acuity would be expected only if enlarged size was encoded through the involvement of a greater number of cortical cells (e.g. Schwartz, 1980) or through some some other type of enhanced efficiency.

There is a conflicting literature on whether changes in apparent rather than retinal size can affect visual acuity. A factor that can change the former is viewing distance, since under full-cue viewing conditions size-constancy scaling enlarges the apparent size of more distant objects to compensate for the decrease in retinal size. There is some evidence that visual acuity in the fovea is slightly better at far than near viewing distances (Freeman 1932a, 1932b; Luckiesh and Moss, 1933, 1941; Giese , 1946; White & Jorve, 1956; Pigg & Kama, 1961; McCready, 1963; Chapanis and Scarpa, 1967), and similarly for stereoscopic acuity (Amigo, 1963; Brown et al, 1965). Sloan (1951) reviewed the earlier literature and concluded that, for adequately con trolled conditions, acuity is almost independent of distance, though it decreases at distances of less than two metres for optical reasons. Olzak and Thomas (1986) concluded that there was some variation with viewing distance owing to accommodation errors at both far and near distances, but the loss is not eliminated when errors are corrected (Johnson, 1976). Size-constancy scaling normally produces a large change in apparent angular size; but any effect on acuity is hard to demonstrate, since well-contro lled viewing situations entail reduced contextual cues, and this also reduces size constancy. However, the reported effects on acuity seem to be much to small to be consistent with the simple substitution of apparent for retinal size.

Apparent size can change independently of retinal size in parts of various perspective pictures and geometrical illusions. This may be because of size contrast between adjacent parts, or because perspective features lead to the enlargement of ty pically more distant parts, or for other reasons. The Oppel-Kundt illusion of filled and empty space is sometimes explained in the same way as Weber's tactile illusion of length, the increased number of discretely stimulated units leading to greater appar ent length. There is some evidence that increased apparent size is accompanied by a slight increase in visual acuity (McFadden, 1940; Alluisi, 1955; Ross, 1965). Taylor argued that this was true for figural after-effects (1962a) and certain geometrical il lusions (1962b). Contrast sensitivity is not the same ability as high spatial acuity; but it is interesting that Lockhead et al. (1996) found that low contrast lines with Müller-Lyer arrows on the ends were more detectable in the apparently longer fo rmat.

If the apparent length of a line is increased by size-constancy scaling or some illusion, how might this affect the Weber fraction? If Weber's Law holds for length of line, then, as Fechner argued, the Parallel Law should hold: a change in apparent length should have no effect on relative discrimination. As Weber pointed out (1834; 1996, pp.121-122), the law does not apply if the lines are presented close together and aligned at one end: in this arrangement the test is one of visual acuity, and the jnd is independent of the length of the lines. However, if the lines are presented successively, or some distance apart, the law holds approximately. As in most modalities, the Weber fraction rises for smaller line lengths (e.g. Henmon, 1906; Ono, 1967). Since the fraction is not constant for all line lengths, it can be asked whether it follows retinal or apparent length more closely. Ono (1967) used lines subtending angles of about 0.5 to 6.0 deg at viewing distances of 1.5 to 4.5 m, thus contrasting re tinal and apparent size through size-constancy scaling: he found that the Weber fraction decreased with line length, but the relationship lay midway between retinal and apparent size. This analysis assumes the latter to have been equivalent to physical li near size, which may not have been the case. However, on any analysis, apparent size had the expected direction of effect on the Weber fraction.

In summary, there is good evidence that neural structure affects both visual acuity and apparent size; but little evidence that other changes in apparent size have a reciprocal and quantitatively similar effect on visual acuity, or on length discri mination.


6. Conclusions


There is evidence from several modalities that the rate at which apparent magnitude increases with physical magnitude is correlated with the discriminatory ability of the sense modality. There is also evidence from some modalities that, without any cha nge in physical magnitude, an increase in neural efficiency is correlated with increases both in discrimination and in the apparent magnitude of the stimulus. In these examples there seems to be a monotonic relation between discrimination and apparent mag nitude, and it is legitimate to postulate that there is an underlying latent variable - in this case, neural efficiency.

However, there appears to be no monotonic relation between discrimination and apparent magnitude when the latter changes for reasons other than changes in neural efficiency. In many sense modalities apparent magnitude can change in either direction through adaptation or contrast effects; but discrimination is best when the observer is adapted to the test background level, and is poorer when adapted higher or lower. Apparent visual size can change through size-constancy scaling or through geometrica l illusions, but it is unclear how far such changes affect visual acuity or length discrimination. Any effects are small, and very precise experiments are needed to measure them.

Where there are non-monotonic relationships, it is difficult to discover any latent variables (Macdonald, 1993). Progress can only be made if the direction and size of the various effects are measured more accurately. Even then it may not be possib le to construct a psychophysical theory that embraces all aspects of apparent magnitude and discrimination. The argument will continue into the next millenium.



I should like to thank Pat Lovie, Ranald Macdonald, Sandy MacRae and David Murray for helpful comments on this paper, and for suggestions to the literature.




Alluisi, E.A. (1955) Measured visual acuity as a function of phenomenal size. USAF WADC Tech.Rep. No. 55-384, iv.

Amigo, G. (1963) Variation of stereoscopic acuity with observation distance. Journal of the Optical Society of America, 53, 630-635.

Baird, J.C. (1985) Complementarity theory and the power law. In Falmagne, J.-C. Elements of psychophysical theory. Oxford University Press.

Baird, J.C. (1997) Sensation and judgment: Complementarity theory of psychophysics. Mahwah, N.J.: Lawrence Erlbaum Associates.

Bartlett, R.J. (1940) Measurement in psychology. Advancement of Science, 1, 422-441.

Bedell, H.E. & Johnson, C.A. (1984) The perceived size of targets in the peripheral and central visual fields. Ophthalmological and Physiological Optics, 4, 123-131.

Bender, M.B. and Teuber, H-L. (1949) Disturbances in visual perception following cerebral lesions. The Journal of Psychology, 28, 223-233.

Bolanowski, S.J., Jr, and Gescheider, G.A. (Eds.) (1991) Ratio scaling of psychological magnitudes. Hillsdale, N.J.: Lawrence Erlbaum Associates.

Boring, E.G. (1942) Sensation and perception in the history of experimental psychology. New York: Appleton-Century-Crofts.

Brown, J.P., Ogle, K.N. & Reiher, L. (1965) Stereoscopic acuity and observation distance. Investigative Ophthalmology, 4, 894-900.

Brysbaert, M. (1992) Accounting for an old inconsistency in the psychophysics of Plateau and Delboeuf. Behavioral and Brain Sciences, 15, 562-563.

Carr, H.A. (1935) An introduction to space perception. New York: Lomgmans, Green & Co.

Chapanis, A. and Scarpa, L.C. (1967) Readability of dials at different distances with constant visual angle. Human Factors, 9, 419-425.

Chukova, S., Gusev, S. (1993) Size discrimination of contour images: Horizontal illusion. Perception, 22 (Supplement), 121.

Critchley, M. (1953) The parietal lobes. Edward Arnold, London.

Davis, E.T., Yager, D. & Jones, B.D. (1987) Comparison of perceived spatial frequency between the fovea and the periphery. Journal of the Optical Society of America. A, 4, 1606-1611.

Delboeuf, J. (1873) Etude psychophysique: Recherches théoriques et expérimentales sur la mesure des sensations at spécialement des sensations de lumière et de fatigue. Mémoires couronnés et autres mém oires publiés par l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique, vol.23. Bruxelles: Hayez.

Duke-Elder, W.S. (1934) Textbook of Ophthalmology. Vol.1. St Louis: Mosby.

Falmagne, J.C. (1985) Elements of psychophysical theory. Oxford Psychology Series No.6. Oxford University Press.

Fechner, G.T. (1860, 1966) Elemente der Psychophysik. Leipzig: Breitkopf und Härtel. Vol. 1 translated as Elements of Psychophysics by H.E. Adler, New York: Holt, Rinehart & Winston, 1966.

Freeman, E. (1932a) Anomalies of visual acuity in relation to stimulus distance. Journal of the Optical Society of America, 22, 285-292.

Freeman, E. (1932b) Accommodation, pupillary width and stimulus distance. Journal of the Optical Society of America, 22, 729-734.

Georgeson, M.A. (1980) Spatial frequency analysis in early visual processing. Philosophical Transactions of the Royal Society of London, B290, 11-22.

Gescheider, G.A. (1985) Psychophysics: Method and theory. (2nd ed). Hillsdale, N.J: Erlbaum.

Giese, W.J. (1946) The interrelationship of visual acuity at different distances. Journal of Applied Psychology, 30, 91-106.

Goudge, M.E. (1918) A qualitative and quantitative study of Weber's illusion. American Journal of Psychology 29, 81-119.

Guilford, J.P. (1954) Psychometric methods. New York: McGraw-Hill.

Hellman, R.P. and Hellman, W.S. (1995) Relating the Weber fraction to loudness. In C-A. Possamai (Ed) Fechner Day 95. Cassis, France: International Society for Psychophysics, pp.67-72.

Hellström, A. (1977) Time errors are perceptual. Psychological Research, 39, 345-388.

Helmholtz, H. von (1910; 1962) Helmholtz's treatise on physiological optics. Vol.III. Translated from the 3rd German edition and edited by J.P.C. Southall. New York: Dover.

Helson, H. (1964) Adaptation level theory. New York: Harper & Row.

Henmon, V.A.C. (1906) The time of perception as a measure of differences in sensation. Archives of Philosophy, Psychology, and Scientific Method, No.8, 46-62.

Hotopf, W.H.N., Hibberd, M.C. & Brown, S.A. (1983) Position in the visual field and spatial expansion. Perception, 12, 469-476.

James, W. (1890; 1950) Principles of psychology. Vol.2. New York: Dover.

Johnson, C.A. (1976) Effects of luminance and stimulus distance on accommodation and visual resolution. Journal of the Optical Society of America, 66, 138-142.

Jones, L.A. (1986) Perception of force and weight: Theory and research. Psychological Bulletin, 100, 29-42.

Keidel, W.D., Keidel, U.O. & Wigand, M.E. (1961) Adaptation: Loss or gain of sensory information? In Rosenblith, W.A. (Ed) Sensory communication. London: Wiley, pp. 319-338.

Krueger, L.E. (1989) Reconciling Fechner and Stevens: Toward a unified psychophysical law. Behavioral and Brain Sciences, 12, 251-320.

Laming, D. (1984) The relativity of 'absolute' judgements. British Journal of Mathematical and Statistical Psychology, 37, 152-183.

Laming, D. (1986) Sensory analysis. London: Academic Press.

Laming, D. (1988) Sensory discrimination and sensory judgment: An introduction to modern psychophysics. University of Cambridge: Department of Experimental Psychology.

Link, S.W. (1992) The wave theory of difference and similarity. Hillsdale, N.J.: Lawrence Erlbaum Associates.

Lipetz, L.E. (1971) The relation of physiological and psychological aspects of sensory intensity. In Loewenstein, W.R. (Ed) Handbook of sensory physiology Vol.1. Principles of receptor physiology. Berlin: Springer-Verlag, pp.191-225.

Lockhead, G., Dragoi, V. & Huettel, S. (1996) Context and judgment. In Masin S.C. (Ed) Fechner Day 96. Proc. 12th Annual Meeting of the International Society for Psychophysics. Padua, Italy: The International Society for Psychophysics, pp. 1 49-154.

Lockhead, G.R. (1992) Psychophysical scaling: Judgments of attributes or objects? Behavioral and Brain Sciences, 15, 543-558.

Luce, R.D. and Krumhansl, C.L. (1988) Measurement, scaling and psychophysics. In R. Atkinson, R. Herrnstein, G. Lindzey & R.D. Luce (Eds.) Stevens's handbook of experimental psychology. 2nd ed. New York: Wiley Interscience, pp. 3-74.

Luckiesh, M. and Moss, F.K. (1933) The dependence of visual acuity upon stimulus distance. Journal of the Optical Society of America, 23, 25-29.

Luckiesh, M. and Moss, F.K. (1941) The variation in visual acuity with fixation distance. Journal of the Optical Society of America, 31, 594-595.

Macdonald, R.R. (1993) Qualitative generalizations from quantitative analyses. British Journal of Mathematical and Statistical Psychology

46, 49-61.

McCready, D.W. (1963) Visual acuity under conditions that induce size illusions. University Microfilms, Ann Arbor, Michigan. Doctoral Dissertation, Univ. of Michigan.

McFadden, H.B. (1940) The dependence of the resolving power of the eye upon the dynamics of the test field and upon practice effect. Ph.D. Thesis, Ohio State University. (Cited by Alluisi, 1955).

Murray, D.J. (1993) A perspective for viewing the history of psychophysics. Behavioral and Brain Sciences, 16, 115-186.

Murray, D.J. and Ross, H.E. (1988) E.H. Weber and Fechner's psychophysics. In Bro_ek, J. and Gundlach, H. (Eds) Passauer Schriften zur Psychologiegeschichte Nr 6. G.T. Fechner and Psychology. Passau: Passavia Universitätsverlag, pp.79-86.

Myers, A.K. (1982) Psychophysical scaling and scales of physical stimulus measurement. Psychological Bulletin, 92, 203-214.

Newsome, C.R. (1972) Visual angle and apparent size of objects in peripheral vision. Perception and Psychophysics, 12, 300-304.

Nicolas, S. & Murray, D.J. (1998) The psychophysics of J.-R.-L. Delboeuf (1831-1896). Perception (in press).

Norwich, K.H. (1993) Information, sensation and perception. San Diego: Academic Press.

Olzak, L.A. & Thomas, J.P. (1986) Seeing spatial patterns. Ch 7. in Boff, K.R., Kaufman, L., Thomas, J.P. (Eds.) Handbook of perception and human performance. Vol.1. Sensory processes and perception.

Ono, H. (1967) Difference threshold for stimulus length under simultaneous and nonsimultaneous viewing conditions. Perception and Psychophysics, 2, 201-207.

Piaget, J. (1969) The mechanisms of perception. London: Routledge & Kegan Paul.

Pigg, L.D. & Kama, W.N. (1961) The effect of transient weightlessness on visual acuity. USAF WADD Tech. Rep. 61-184.

Plateau, J. (1872) Sur la mesure des sensations physiques, et sur la loi qui lie l'intensité de ces sensations à l'intensité de la cause excitante. Bulletins de l'Académie Royale des Sciences, des Lettres et des Beaux-Art s de Belgique, 34 (2d serie), 141-142.

Poulton. E.C. (1989) Bias in quantifying judgments. Hove: Lawrence Erlbaum Associates.

Pitz, G.F. (1965) Magnitude scales of line length. Psychonomic Science, 2, 213-214.

Postman, L. (1946) The time-error in auditory perception. American Journal of Psychology, 59, 193-219.

Ross, H.E. (1964) Constant errors in weight judgements as a function of the size of the differential threshold. British Journal of Psychology, 55, 133-141.

Ross, H.E. (1965) The effect of apparent versus physical stimulus magnitude upon sensory discrimination. Ph.D. Thesis, University of Cambridge.

Ross, H.E. (1969) When is a weight not illusory? Quarterly Journal of Experimental Psychology, 21,346-355.

Ross, H.E. (1981) How important are changes in body weight for mass perception? Acta Astronautica, 8, 1051-1058.

Ross, H.E. (1995) Weber on temperature and weight perception. In Possamaï, C.-A. (Ed.) Fechner Day 95, Cassis, France: International Society for Psychophysics, pp. 29-34.

Ross, H.E. (1996) Weber and Fechner on the relation between discriminability and subjective magnitude for constant physical intensity. In Masin S.C. (Ed) Fechner Day 96. Proc. 12th Annual Meeting of the International Society for Psychophysics. P adua, Italy: The International Society for Psychophysics, pp. 387-392.

Ross, H.E. & Plug, C. (1997) The history of size constancy and size illusions. In Walsh, V. and Kulikowski, J.J. (Eds) Visual constancies: Why things look as they do. Cambridge: Cambridge University Press (in press).

Saito, T. (1994) Psychological scaling of the asymmetry observed in comparative judgement. British Journal of Mathematical and Statistical Psychology 47, 41-62.

Scheerer, E. (1992) Fechner's inner psychophysics: its historical fate and present status.In Geissler, H.-G., Link, S.W. and Townsend, J.T. (Eds). Cognition, information processing, and psychophysics: Basic issues. Hove: Lawrence Erlbaum Associa tes, pp. 3-21.

Schneider, B., Ehrlich, D.J., Stein, R., Flaum, M. & Mangel, S. (1978) Changes in the apparent lenths of lines as a function of degree of retinal eccentricity. Perception, 7, 215-223.

Schwartz, E.L. (1980) Computational anatomy and functional architecture of striate cortex: A spatial mapping approach to perceptual coding. Vision Research, 20, 645-669.

Seashore, C.E. (1896) Weber's Law in illusions. Studies from the Yale Psychological Laboratory. No.4

Sloan, L.L. (1951) Measurement of visual acuity. American Medical Association Archives of Ophthalmology. 45, 704-725.

Stevens, J.C. (1979) Thermal intensification of touch sensation: Further extensions of the Weber phenomenon. Sensory Processes 3, 240-248.

Stevens, J.C. (1990) Perceived roughness as a function of body locus. Perception and Psychophysics, 47, 298-304.

Stevens, S.S. (1939) On the problem of scales for the measurement of psychological magnitudes. Journal of Unified Science, 9, 94-99.

Stevens, S.S. (1957) On the psychophysical law. Psychological Review, 64, 153-181.

Stevens, S.S. (1975) Psychophysics: Introduction to its perceptual, neural, and social prospects. New York: Wiley.

Stevens, S.S. & Galanter, E.H. (1957) Ratio scales and category scales for a dozen perceptual continua. Journal of Experimental Psychology 54, 377-411.

Stuart, G.W., Bossomaier, T.R.J. & Johnson, S. (1993) Preattentive processing of object size: implications for theories of size perception. Perception, 22, 1175-1193.

Tanner, W.P. Jr. & Swets, J.A. (1954) A decision-making theory of visual detection. Psychological Review 61, 401-409.

Taylor, M.M. (1962a) Figural after-effects: a psychological theory of the displacement effect. Canadian Journal of Psychology, 16, 247-282.

Taylor, M.M. (1962b) Geometry of a visual illusion. Journal of the Optical Society of America, 52, 565-569.

Teghtsoonian, R. (1971) On the exponents in Stevens' law and the constant in Ekman's law. Psychological Review, 78, 71-80.

Titchener, E.B. (1905) Experimental psychology: A manual of laboratory practice. Vol.II Quantitative experiments. Part II Instructor's manual. New York: Macmilan.

Treisman, M. (1964a) Sensory scaling and the psychophysical law. Quarterly Journal of Experimental Psychology, 16, 11-22.

Treisman, M. (1964b) What do sensory scales measures? Quarterly Journal of Experimental Psychology, 16, 387-391.

Treisman, M. (1970) Brightness contrast and the perceptual scale. British Journal of Mathematical and Statistical Psychology 23, 205-224.

Ward, L.M. (1995) On the role of discriminability in psychophysics. In Possamaï, C.-A. (Ed.) Fechner Day 95, Cassis, France: International Society for Psychophysics, pp. 61-66.

Ward, L.M. (1996) On the relationship of fundamental concepts in psychophysics. In Masin S.C. (Ed) Fechner Day 96. Proc. 12th Annual Meeting of the International Society for Psychophysics. Padua, Italy: The International Society for Psychophysic s, pp. 417-422.

Ward, L.M., Armstrong, J & Golestani, N. (1996) Intensity resolution and subjective magnitude in psychophysical scaling. Perception and Psychophysics, 58, 793-801.

Warren, R.M. (1981) Measurement of sensory intensity. Behavioral and Brain Science, 4, 175-189.

Weber, E.H. (1834, 1996) De tactu. Annotationes anatomicae et physiologicae. Leipzig: Koehler. Translated in Ross, H.E. and Murray, D.J. (Eds) E.H. Weber on the tactile senses. 2nd edition. Hove: Erlbaum (UK) Taylor & Francis, 1996.

Weinstein, S. (1968) Intensive and extensive aspects of tactile sensitivity as a function of body part, sex, and laterality. In Kenshalo, D.R. (Ed.) The skin senses Springfield, Ill.: Thomas, pp. 195-222.

Weiss, D.J. (1981) The impossible dream of Fechner and Stevens. Perception, 10, 431-434.

White, W.J. & Jorve, W.R. (1956) The effects of gravitational stress upon visual acuity. USAF WADC Tech. Rep.No 56-247.

Wink, B. & Harris, J.P. (1993) The perceived spatial extent of peripherally viewed gratings. Perception 22 (Supplement), 124-125.

Winn, B., Ackerley, R.G., Brown, C.A., Murray, F.K., Prais, J. & St John, M.F. (1988) Reduced aniseikonia in axial anisometropia with contact lens correction. Ophthalmological and Physiological Optics, 8, 341-344.

Zwislocki, J.J. (1994) Differential intensity sensitivity in relation to subjective magnitude: Experimental results and mathematical theory. In L.M. Ward (Ed.) Fechner Day 94. Vancouver, Canada: International Society for Psychophysics, pp. 1-10.

Zwislocki, J.J. and Hertig, L.Y. (1995) On the relationship of differential intensity sensitivity to loudness level. In Possamaï, C.-A. (Ed.) Fechner Day 95, Cassis, France: International Society for Psychophysics, pp. 49-54.