Bouma's law of crowding predicts an uncrowded central window through which we can read and a crowded periphery through which we cannot. The old discovery that readers make several fixations per second, rather than a continuous sweep across the text, suggests that reading is limited by the number of letters that can be acquired in one fixation, without moving one's eyes. That “visual span” has been measured in various ways, but remains unexplained. Here we show (1) that the visual span is simply the number of characters that are not crowded and (2) that, at each vertical eccentricity, reading rate is proportional to the uncrowded span. We measure rapid serial visual presentation (RSVP) reading rate for text, in both original and scrambled word order, as a function of size and spacing at central and peripheral locations. As text size increases, reading rate rises abruptly from zero to maximum rate. This classic reading rate curve consists of a cliff and a plateau, characterized by two parameters, critical print size and maximum reading rate. Joining two ideas from the literature explains the whole curve. These ideas are Bouma's law of crowding and Legge's conjecture that reading rate is proportional to visual span. We show that Legge's visual span is the uncrowded span predicted by Bouma's law. This result joins Bouma and Legge to explain reading rate's dependence on letter size and spacing. Well-corrected fluent observers reading ordinary text with adequate light are limited by letter spacing (crowding), not size (acuity). More generally, it seems that this account holds true, independent of size, contrast, and luminance, provided only that text contrast is at least four times the threshold contrast for an isolated letter. For any given spacing, there is a central uncrowded span through which we read. This uncrowded span model explains the shape of the reading rate curve. We test the model in several ways. We use a “silent substitution” technique to measure the uncrowded span during reading. These substitutions spoil letter identification but are undetectable when the letters are crowded. Critical spacing is the smallest distance between letters that avoids crowding. We find that the critical spacing for letter identification predicts both the critical spacing and the span for reading. Thus, crowding predicts the parameters that characterize both the cliff and the plateau of the reading rate curve. Previous studies have found worrisome differences across observers and laboratories in the measured peripheral reading rates for ordinary text, which may reflect differences in print exposure, but we find that reading rate is much more consistent when word order is scrambled. In all conditions tested—all sizes and spacings, central and peripheral, ordered and scrambled—reading is limited by crowding. For each observer, at each vertical eccentricity, reading rate is proportional to the uncrowded span.

*critical print size*(CPS), reading rate is nearly flat, independent of letter size. The curve finally descends due to perspective compression at large visual angles. The critical print size and maximum reading rate depend on the viewing conditions, but the curve shape—steep cliff and wide plateau—is universal for central, peripheral, static, and rapid serial visual presentation. This basic result is well established but unexplained.

*crowding*(Stuart & Burian, 1962). Crowding is excessive feature integration, inappropriately including extra features that spoil recognition of the target object. An early preattentive bottleneck in the object recognition process, crowding is characterized by a critical spacing that depends on eccentricity (distance from fixation) and little else. Critical spacing is the smallest distance between letters (center-to-center) that avoids crowding. We have previously studied the effects of crowding on identification of single letters, words, and faces (Martelli, Majaj, & Pelli, 2005; Pelli, Palomares, & Majaj, 2004). Here we examine reading, focusing on the identification of words in the context of a sentence. The sentence context normally helps in identifying each word but can be abolished by scrambling the word order, revealing unexpectedly high consistency among observers (4).

*isolation field,*a region over which the visual system integrates features (Fig. 3). The isolation field defined by the critical spacing is the smallest isolation field at that location. We suppose that isolation fields larger than this are also available at that location and are used to identify larger letters. We have wondered whether there is a maximum isolation field size and what consequences this might have, but that is not relevant here. Our experiments keep our observers at the threshold spacing for letter identification, so they are always using their smallest available isolation field. The minimum isolation field size increases with eccentricity, but is independent of target size (and type, Martelli et al., 2005).

*visual span,*the number of characters in a line of text that can be read without moving one's eyes. Here, we consider a stronger version of this conjecture, which we also attribute to Legge, namely, that RSVP reading rate is proportional to visual span. We return to this in the Discussion section. Legge et al. define the visual span operationally: They measure letter recognition for triplets (random strings of three letters) as a function of position in the visual field. This is a slight variation on Bouma's (1970) classic crowding paradigm (Fig. 3). (Bouma assessed accuracy of reporting the middle letter of the triplet; Legge et al. assess average accuracy across all three letters.) Visual span is typically about 10 characters, extending slightly further to the right than to the left (Legge et al., 2001). That rightward bias for English, which is read from left to right, is reversed in Hebrew, which is read from right to left (Pollatsek, Bolozky, Well, & Rayner, 1981).

*word identification span*(or area from which words can be identified on a given fixation) is smaller than the total perceptual span (McConkie & Zola, 1987; Rayner et al., 1982; Underwood & McConkie, 1985) and generally does not exceed 7–8 letter spaces to the right of fixation.” Similarly, “When the first 3 letters of the word to the right of fixation were available and the remainder of the letters were replaced by visually similar letters, reading rate was not too different from when the entire word to the right was available” (Rayner, 1998, p. 381).

^{1}

^{2}However, the results presented in Figure 6 allow us to rule out all alternatives to crowding.

*lateral masking*is the effect of a nonoverlapping irrelevant pattern and includes a variety of phenomena and many models, but none, except crowding and surround suppression, are strongly dependent on eccentricity. Thus, the strong eccentricity dependence rules out all known forms of lateral masking except crowding and surround suppression. Surround suppression is similar to crowding in many ways, but we can rule it out because it only occurs when the flankers have higher contrast than the target (Chubb, Sperling, & Solomon, 1989; Petrov & McKee, 2006; Snowden & Hammett, 1998; Xing & Heeger, 2001; Zenger-Landolt & Koch, 2001), whereas here the target and flankers have equal contrast. We show that performance on the classic flanked letter task and the Legge et al. (2001) variation is determined solely by the ratio of actual to critical spacing (predicted by Bouma's law), independent of size, spacing, and eccentricity per se. Theories other than crowding say those variables matter, but in fact they do not matter (Fig. 6), so we reject those theories. Only spacing matters (relative to the critical value, which depends only on location and direction in the visual field). Thus, identification of flanked letters is limited by crowding.

*uncrowded span,*the number of uncrowded character positions in a line of text (Figs. 5 and 7).

*φ*is the eccentricity (following Bouma),

*s*

_{0}is the critical spacing at zero eccentricity (about 0.1° or 0.2°), and

*b*is a proportionality constant that we name after Bouma. Characters spaced more widely than

*s*are uncrowded.

*b,*in fact, depends somewhat on the proportion correct that we take as “critical,” 80%. However, assuming an abrupt transition greatly simplifies the initial development of the model. As explained in 3, we add dither to the simple model to obtain graded psychometric functions that match those of human observers.)

*b*provides a good fit to data from some observers, but, as you will see below, to fit all the observers, from our laboratory and others, it is necessary to allow

*b*to have a linear dependence on eccentricity,

*b*

_{1}and

*b*

_{2}are observer-specific constants, independent of eccentricity

*φ*. The effect of Equation 2 on Equation 1 is a further generalization of Bouma's law, from linear to quadratic, to accommodate individual differences. (Happily, this extension for differences among normals also allows the model to fit amblyopes as well, Levi et al., 2007.) This generalization bends Bouma's line but does not impact his fundamental observation that critical spacing is determined by eccentricity: It is the site, not the signal, that matters.

*ordinary text,*we suppose reasonable (center to center) spacing, enough to prevent overlap of neighboring letters, which is known to slow reading down (Chung, 2002), and no more than twice normal (for the font), since arbitrarily large spacing would reduce reading to identification of isolated letters. For well-corrected fluent readers with adequate light, when one shrinks ordinary text (e.g. by increasing viewing distance), it becomes unreadable, due to crowding, before the acuity limit (for isolated letters) is reached.

*ordinary conditions*(well-corrected fluent observers reading ordinary text at moderate luminance), we will see that the visual span is the uncrowded span over a wide range of text size (400:1 in Fig. 1; 5:1 in Levi et al. 2007). That is a large territory in which only spacing matters, but it has limits. With fixed letter spacing, reading eventually fails if contrast, (letter) size, or luminance is greatly reduced. (Reading slows at sunset, as afternoon fades into night.) We are inclined to attribute these failures to something other than crowding, because we suspect that the uncrowded span is independent of size, contrast, and luminance. Reading is impossible when text contrast falls below threshold for an isolated letter. Threshold contrast rises as size and luminance are reduced, so reducing contrast, letter size, or luminance reduces the ratio of text contrast to threshold contrast. Legge et al. (2007) show that reading rate is practically independent of contrast, provided contrast is at least four times threshold (0.04). Below 0.04, both visual span and reading rate gradually drop, reaching zero at threshold contrast (0.01). To believe that the reduced visual span found at near-threshold contrast is still the uncrowded span, we would need evidence that crowding is worse (critical spacing is greater) at low contrast. In fact, Pelli et al. (2004) found that critical spacing is independent of contrast over the range 0.1 to 1, and found no crowding at all at the contrasts they tested below 0.1. Thus, the visual span at contrasts less than four times threshold is limited by contrast, not spacing (crowding). It seems that the effects of reducing contrast, size, or luminance might all be described by one rule: The visual span is equal to the uncrowded span if text contrast is at least a factor of four above threshold for an isolated letter, and gradually shrinks to zero as text contrast approaches threshold.

*b*is constant, independent of eccentricity. 2 shows that the uncrowded span

*u*is 1 + 2/

*b,*which is constant if

*b*is constant. Legge et al. (2001) find that visual span (in characters) falls with eccentricity and suggest that this accounts for the falling reading rate. Bouma's and Legge's claims are incompatible. In terms of Equation 2, Bouma would have said that

*b*

_{2}is zero, whereas Legge et al. would have said that

*b*

_{2}is large and positive. Surely they cannot both be right. It is an empirical issue.

*b*depends on eccentricity ( Fig. 9). We find that some individuals conform to Bouma's prediction and others to Legge's, but that most lie in between, conforming to neither prediction. In Figure 9, observers EK (green) and STC (red) have the zero and steep slopes claimed by Bouma and Legge, respectively, but most observers are intermediate between these extremes.

*b*does grow with eccentricity (

*b*

_{2}> 0), demanding a generalization of Bouma's law ( Eq. 2), but the growth is typically too small to account for much of the large drop in reading rate with eccentricity. RSVP reading rate drops with eccentricity, reaching one sixth the foveal rate at an eccentricity of 20° (Legge et al., 2001). Excluding STC and EK as outliers, for the other four observers in Figure 9,

*b*roughly doubles as eccentricity increases from 0 to 20°, which roughly halves the uncrowded span

*u*= 1 + 2/

*b*. That goes in the right direction, but falls far short of accounting for the sixfold drop in reading rate.

*s*(threshold spacing for 80% correct). We then calculated Bouma's factor (roughly

*b*≈

*s*/

*φ,*see Methods). In Figure 9, we plot

*b*at the radial eccentricity

*φ*for that condition. We did a linear regression ( Eq. 2) for each observer in Figure 9 to describe how his or her

*b*depends on eccentricity

*φ*. This fit has two degrees of freedom: intercept

*b*

_{1}and slope

*b*

_{2}. For each observer, we used the fitted Equation 2 to model critical spacing for each condition (i.e., for each psychometric function). Finally, in Figure 8 we plot the proportion correct as a function of the ratio of the actual spacing to the model's critical spacing.

*b*to have a linear dependence on eccentricity. The only contribution of the model to Figure 8 is to provide the model's critical spacing, which slides the psychometric function right or left.

*u*and write the conjecture as

*r*is the reading rate (characters per second) and

*ρ*is a proportionality constant with a value on the order of 10 Hz. (As a mnemonic, think of

*ρ*as the rate of glimpses and

*u*as the number of letters harvested per glimpse.) We define

*r*as characters per second, but we measure and report the traditional word/min. For English text, with an average of five letters plus a space per word, 1 word/min equals 0.1 character per second. For casual reading of a static page, typical values might be 280 word/min (i.e.,

*r*= 28 character/s), a

*ρ*of 4 Hz, and a span of 7. For central RSVP reading, participants striving to read as quickly as possible reach a rate of 910 word/min (i.e.,

*r*= 91 character/s) with a

*ρ*of 13 Hz and a span of 7. (We return to this comparison in the Discussion section.) The uncrowded span

*u*depends solely on the spacing, the critical spacing constant

*b,*and the eccentricity. In writing Equation 3, one anticipates that the proportionality constant

*ρ*will be independent of most experimental variables. Its variation among observers and with text difficulty was expected, but it is surprising that

*ρ*falls with increasing eccentricity.

*φ*

_{v}. The letter spacing is fixed, but the observer's critical spacing increases with the horizontal eccentricity ( Fig. 6). This happens partly because the radial eccentricity increases ( Eq. 1) and partly because the orientation of the elliptical isolation fields (always aligned with fixation) becomes less favorable ( 1). Figure 10 shows how Bouma's law determines the uncrowded window. Starting from the midline and proceeding to greater horizontal eccentricity to the right or left, eventually the critical spacing, increasing with eccentricity, grows to exceed the given spacing of the text. This is the edge of the

*uncrowded span*. Beyond that span, spacing will be less than critical and the letters will be crowded ( Fig. 5). The proportion‐correct criterion for “critical” is to some extent arbitrary. Bouma used 100%. We use 80%, which results in a smaller value of

*b*. 1 and 2 work out the geometry to derive an expression for

*u,*the width of the span ( Eq. B10). We use that expression, which is exact, in the rest of our plots, but in Figure 11 we present a simple approximation that retains the important features of the exact expression:

*b*is Bouma's constant,

*φ*

_{v}is the vertical eccentricity, and

*s*is the center-to-center letter spacing. For simplicity, this approximation, based on Equation B7, assumes that the ellipse is a circle (

*ɛ*= 1), ignores the perspective transformation that compresses the angular spacing of eccentric letters, neglects the minimum critical spacing found at small eccentricity (

*s*

_{0}= 0), and omits the +1 conversion from breadth to span ( 2). Combining Equations 3 and 4, the predicted reading rate is approximately

*ρ*/

*b*at large spacing (where the square root term approaches 1) and drops abruptly to zero as spacing

*s*is reduced to the critical spacing

*bφ*

_{v}at the vertical midline. Assuming only Bouma's law, 1 calculates the cliff ( Eq. A6) and 2 calculates the plateau ( Eq. B8). In general, for arbitrary ellipticity

*ɛ*and nonzero minimum critical spacing

*s*

_{0}, the curve is characterized by its horizontal and vertical asymptotes: maximum reading rate (1 + 2/

*b*)

*ρ*( Eqs. 3 and B8) and critical spacing for reading

*s*

_{0}+

*bφ*

_{v}/

*ɛ*( Eq. A6).

*b*and

*ρ,*for each curve (Fig. 12).

^{3}

^{,}

^{4}

*r*is the product of

*ρ*and

*u*(Eq. 2). The graphs in Figure 13 have consistent vertical scales (0.9 log unit per inch) and identical horizontal scales. Thus, the slopes of log

*ρ*(Fig. 13b) and log

*u*(Fig. 13c) must sum to the slope of log

*r*(Fig. 13a). As Legge et al. note, reading rate drops with eccentricity, approximately a straight line in these log-linear coordinates, falling sixfold from 0° to 20°. In effect, Legge et al. supposed that the proportionality constant

*ρ*is fixed, independent of eccentricity, and that

*u*shrinks with eccentricity. Their attempt to test this eccentricity conjecture was inconclusive because the bounds on their model's predicted performance were too broad. Contrary to what they supposed, Figure 13 shows that the span

*u*of these six observers hardly changes with eccentricity. The uncrowded span model fits the measured rates by reducing

*ρ,*not

*u*.

*r*with increasing eccentricity, Legge et al. (2001) predicted that

*u*would drop and

*ρ*would be flat, but we find instead that

*ρ*drops and

*u*is flat. This invalidates the Legge et al. claim that changes in

*u*account for slow peripheral reading, but we are still at a loss to explain why

*ρ*drops with increasing eccentricity.

^{5}

*determines*reading rate rather than simply being an independent consequence of the manipulation of visibility. We return to this in the Discussion section. We have just seen that attempts to confirm proportionality, by looking for corresponding changes in visual span and maximum reading rate with eccentricity, have failed (Fig. 13). For some observers (e.g., STC), the link seems to hold, but for most observers only a small part of the drop in reading rate with eccentricity is accounted for by reduced visual span 1 + 2/

*b*. In terms of Equation 3, the drop in reading rate is accounted for mostly by reducing the rate parameter

*ρ,*which has no theoretical basis.

*b*and

*ρ*. Both affect reading rate, so it might seem that we could attribute changes in reading rate to either parameter. However,

*b*is the critical spacing constant and is thus determined by the position of the cliff, leaving only

*ρ*to absorb any independent variation of reading rate.

*u*= 1 + 2/

*b*. We do this in three ways and at several eccentricities ( Fig. 14). At each eccentricity, every method indicates that reading rate is proportional to the uncrowded span.

*h*of the center of the word relative to fixation ( Fig. 15). Reading rate is highest at zero offset and declines monotonically, reaching zero at an offset of five to eight letters, depending on the vertical eccentricity. The rectilinear curve through the data represents our model: Reading rate is proportional to the number of characters within the observer's uncrowded span,

*u*/2,

*u*/2) and the four-letter word centered at position

*h*has bounds (

*h*− 2,

*h*+ 2). The best-fitting value of

*u*is 8.1 at vertical eccentricity 0°, 7.4 at −5°, and 5.3 at −20°.

^{6}has space between words, so the end letters are exposed, each flanked on only one side. These end letters are less crowded. We avoided this complication in the data we collected for Figures 15 16 17 18– 19 by adding flankers, x, at the beginning and end of each word in the RSVP presentation. This makes it more reasonable to expect equality of the word- and letter-based estimates of the uncrowded span.

*ρ*depends on eccentricity. Even so, we can still ask, for any given eccentricity (and

*ρ*), does the uncrowded span determine the reading rate? We do not know any way to increase the uncrowded span, but we devised a way to effectively reduce it.

*u*and that letters displayed outside that span are crowded. The experimenter defines an

*unsubstituted*span

*U,*on the display, within which letters are displayed normally. Letters outside that span are substituted. In modeling reading under these conditions, we suppose that substituting crowded letters has no effect on reading, because it is a “silent” substitution, invisible to the observer. Any measured effect must be due to substitution of uncrowded letters. With both spans centered at fixation, we suppose that reading is limited by whichever is narrower,

*r*is the reading rate,

*r*

_{0}is the residual reading rate when all the letters are substituted (i.e.,

*U*= 0),

*ρ*is the rate parameter,

*u*is the observer's uncrowded span, and

*U*is the display's unsubstituted span.

*r*

_{0},

*ρ,*and

*u*. The best-fitting value of

*u,*indicated by the horizontal gray bar, ranges from 4 to 6.

*ρ*is still the rate parameter and is the slope of the rise in Figure 17.

*U*

_{R}), and a left curtain exposes a window at the right with bounds (

*U*

_{L}, ∞). We measured reading rate as a function of edge position for both right and left curtains.

*u*

_{L},

*u*

_{R}) and the unsubstituted span has bounds (

*U*

_{L},

*U*

_{R}). We use Equation 8 to make one fit to all the data in one panel (right and left curtain for one observer at one vertical eccentricity). We plot the fit as two curves. The right curtain has

*U*

_{L}= −∞ and the left curtain has

*U*

_{R}= ∞. The estimated

*u*is 5.3 at vertical eccentricity 0°, 3.6 at −5°, and 2.6 at −20° for JF; 3.8 at 0°, 3.6 at −5°, and 4.5 at −20° for EK; 6.9 at 0°, 7.7 at −5°, and 4.9 at −20° for KAT.; and 7.1 at 0°, 7.4 at −5°, and 4 at −20° for NB.

*ρ*across the span. The left- and right-curtain data provide independent estimates of the span, and they agree.

*b*depends on eccentricity for each observer. From

*b*we now calculate the uncrowded span for letter identification ( Eq. B10) for each observer at each eccentricity. Figure 19 is a scatter diagram, plotting span for reading versus span for letter identification. Each point represents one observer at one eccentricity assessed by one of the three reading experiments. All the data points lie near the line of equality, across four observers, three vertical eccentricities (0°, −5°, and −20°), and three methods of measuring reading span.

^{7}The idea is that the effects of many sensory manipulations on reading rate are mediated by changes in the visual span. These papers find a correlation between log reading rate and visual span, and note that such correlation is consistent with the visual span hypothesis. Anything less than a strong correlation would decisively reject the visual span as mediator. However, a strong correlation only weakly endorses the visual span's role as mediator because changes in reading rate and visual span could be independent consequences of the same sensory manipulation. Furthermore, finding a strong correlation is much weaker than demonstrating proportionality. Proportionality means

*y*=

*ax,*which has only one degree of freedom,

*a*. Correlation merely indicates that the data can be fit with a line

*y*=

*ax*+

*b,*which has two degrees of freedom,

*a*and

*b*. In this special issue, Legge et al. (2007) go further, observing that five experiments yield the same slope of log reading rate versus visual span (0.14 log unit per character or 0.03 log unit per bit).

*ρ*is a fitted constant. The recent Legge conjecture is that log reading rate is linearly related to span,

*α*and

*β*are fitted constants. If we insist that the model predict zero reading rate when the span is zero, then we must reject the recent conjecture because it predicts a nonzero reading rate when the span

*u*is zero. Setting that problem aside, we combined all the rates from Figure 5 of Legge et al. (2007). To fit Legge's recent model, we plot their data as log

*r*versus

*u*(not shown), and a linear regression yields log

*r*= 0.121

*u*+ 0.90,

*R*

^{2}= 0.89, which agrees with their fits.

^{8}To fit Legge's original model, we plot log reading rate versus log span. Fitting with a unit-slope line yields log

*r*= 0.90 + log

*u,*

*R*

^{2}= 0.80. This is proportionality:

*r*= 8

*u*(Eq. 9). The recent model accounts for slightly more variance (0.89 versus 0.80) but has two degrees of freedom instead of one, and, as noted above, erroneously predicts a useful reading rate (79 word/min) at zero span. On balance, this favors the original Legge conjecture:

*r*=

*ρu*.

*ρ*) drop with eccentricity. It is a burning issue, partly because of the practical consequences for readers with central field loss. It might seem that Legge's visual span had solved it. Legge et al. (2001) say, “We conclude … that shrinkage of the visual span results in slower reading in peripheral vision,” but in fact they only showed a correlation between reduced visual span and reduced reading rate. Our results replicate theirs in finding a tendency for visual span to shrink at greater eccentricity, but there are large individual differences, and for only one of Legge's and none of Chung's or our observers was the shrinkage sufficient to account for the slower reading at greater eccentricity (Figs. 9 and 13). As noted above, Legge et al. (2007) find a consistent slope of log reading rate versus visual span in five experiments, but this slope is too shallow, by a factor of five, to account for the effect of eccentricity on reading rate shown in their Figure 2.

*b*and

*ρ,*which control the critical spacing and maximum reading rate. Legge and his collaborators have suggested that reading farther out in the periphery reduces visual span enough to account for the reduction in reading rate. Here we show that the visual span is the uncrowded span

*u*= 1 + 2/

*b*. We document an unexpected dependence of

*b*on eccentricity and striking variation among observers. We find that

*b*grows with eccentricity, differently for each observer, but rarely grows enough to account for much of the sixfold reduction in maximum reading rate from 0° to −20° vertical eccentricity, disappointing the hope expressed by Legge. For most observers, most of the drop in reading rate with eccentricity is accounted for by the rate parameter

*ρ,*not the uncrowded span

*u*.

^{2}. Target (middle) and flanker letters were displayed at 90% contrast. Letters were light on a dark background. Three letter sizes, scaled for radial eccentricity, were tested at each horizontal position ( Table 1). The observer clicked the mouse to begin each trial. Triplets were displayed at vertical eccentricities of 0° or −5° (lower visual field) at horizontal eccentricities of 0°, 6°, or 12° (right visual field). The viewing distance was 120 cm for stimuli presented at horizontal eccentricities of 0° and 6° and 80 cm at 12°.

Vertical eccentricity | Horizontal eccentricity | Radial eccentricity | Letter sizes (deg) | Triplet orientations (deg) |
---|---|---|---|---|

0° | 6° | 6° | 0.3, 0.43, 0.55 | 0, 45, 90, 135 |

0° | 12° | 12° | 0.55, 0.83, 1.2 | 0, 45, 90, 135 |

−5° | 0° | 5° | 0.3, 0.43, 0.55 | 0, 90 |

−5° | 6° | 7.8° | 0.43, 0.55, 0.83 | 0, 40, 90, 130 |

−5° | 12° | 13° | 0.83, 1.2, 1.5 | 0, 23, 45, 90, 113, 130 |

*x*and

*x*+ 180° are identical, by definition, and we plot the measured threshold at both points.

Vertical eccentricity | Horizontal eccentricity | Radial eccentricity | Letter spacing (deg) |
---|---|---|---|

−5° | 0° | 5° | 2.2, 2.75 |

−5° | 2.5° | 5.6° | 2.2, 2.75 |

−5° | 5° | 7.1° | 2.2, 2.75 |

−5° | 10° | 11.2° | 2.2, 2.75 |

−10° | 0° | 10° | 3.85, 4.4, 5.5 |

−10° | 5° | 11.2° | 3.85, 4.4, 5.5 |

−10° | 10° | 14.1° | 3.85, 4.4, 5.5 |

−10° | 20° | 22.4° | 3.85, 4.4, 5.5 |

−20° | 0° | 20° | 5.5, 8.8, 11 |

−20° | 5° | 20.6° | 5.5, 8.8, 11 |

−20° | 10° | 22.4° | 5.5, 8.8, 11 |

−20° | 20° | 28.3° | 5.5, 8.8, 11 |

*b*is approximately the ratio of radial critical spacing to radial eccentricity (see Eq. 1). Thus, each curve in Figure 8 yields one point in Figure 9, the value of

*b*at a particular eccentricity. Figure 9 shows linear regression lines (

*b*vs. radial eccentricity

*φ*) for each observer. Each line corresponds to Equation 2 of the generalized Bouma law,

*b*=

*b*

_{1}+

*b*

_{2}

*φ,*and is specified in the lower right of each graph in Figure 8. Figure 8 plots proportion correct as a function of spacing relative to the spacing predicted by the Bouma law, using the given

*b*formula to specify how that observer's

*b*depends on eccentricity.

*u*for each observer and vertical eccentricity. For observers JF, EK, and KAT,

*u*was calculated from that observer's

*b*formula in Figure 8, as follows. As explained in 2, for a given vertical eccentricity, the uncrowded span is defined by the letter positions, −

*φ*

_{h}and

*φ*

_{h}, at the critical spacing. Initially we take the vertical eccentricity as a rough approximation of the radial eccentricity and calculate an approximate

*b*( Eq. 2). Using this approximate

*b*and the given values for vertical eccentricity and letter spacing, we calculate an approximate horizontal eccentricity

*φ*

_{h}for the right end of the uncrowded span ( Eq. B3), from which we calculate an approximate radial eccentricity

*φ*( Eq. A2), from which we re-calculate

*b*accurately ( Eq. 2). We then retrace our steps, using this accurate

*b*to obtain an accurate

*φ*

_{h}from which we calculate the uncrowded span

*u*( Eq. B10). (Note that the simpler Eq. B4 gives practically the same answer as Eq. B10.)

*u*= 1 + (

*φ*

_{R}−

*φ*

_{L})/

*s*.

*u*was estimated by fitting Equation 6 to her data.

*Loves Music, Loves to Dance*by Mary Higgins Clark (1991). The text has a 7.5 Fog index and 5.5 Fleish–Kincaid Index. The text was not altered prior to experimental manipulations. No observer read the same passage twice.

Original | Substitutes | Original | Substitutes |
---|---|---|---|

a | a e | A | A |

b | b h | B | B P R |

c | c e | C | C G |

d | d | D | D |

e | a c e | E | E F |

f | f | F | E F |

g | g | G | C G |

h | b h | H | H |

i | i | I | I L |

j | j | J | J |

k | k | K | K |

l | l | L | I L |

m | m | M | M |

n | n r | N | N |

o | c o | O | O Q |

p | p q | P | B P R |

q | p q | Q | O Q |

r | n r | R | B P R |

s | s | S | S |

t | t | T | T |

u | u | U | U |

v | v | V | V |

w | w | W | W |

x | x | X | X |

y | y | Y | Y |

z | z | Z | Z |

*unsubstituted span*were subject to substitution, as specified in Table 3.

*u*was estimated by fitting Equation 7 to the results.

*Loves Music, Loves to Dance*. Observers who also participated in the substitution experiment did not re-read any passages they had already read. Letter substitutes were drawn from Table 4. Here we used a large unsubstituted window, so large that only one edge was in the display, and measured performance as a function of the edge position. We call this a “curtain.” A right curtain exposes a window at the left with bounds (−∞,

*U*

_{R}), and a left curtain exposes a window at the right with bounds (

*U*

_{L}, ∞). We measured reading rate as a function of edge position for both right and left curtains.

Original | Substitutes | Original | Substitutes |
---|---|---|---|

a | a e | A | A |

b | b h | B | B P R |

c | c e | C | C G |

d | b d h | D | D |

e | a c e | E | E F |

f | f t | F | E F |

g | g | G | C G |

h | b h | H | H |

i | i | I | I L |

j | j | J | J |

k | k | K | K X |

l | l | L | I L |

m | m | M | M |

n | n r | N | N |

o | c o | O | O Q |

p | p q | P | B P R |

q | p q | Q | O Q |

r | n r | R | B P R |

s | s | S | S |

t | f t | T | T |

u | u | U | U |

v | v | V | V X Y |

w | w | W | W |

x | x | X | K V X Y |

y | y | Y | V X Y |

z | z | Z | Z |

*U*

_{R}tested were −∞ (all letters substituted), −2, −1, 0, 1, 2, 3, and ∞ (no letters substituted). Values of

*U*

_{L}tested were −∞ (no letters substituted), −2, −1, 0, 1, 2, 3, and ∞ (all letters substituted). In some cases, we also measured thresholds at

*U*

_{R}= −3 and

*U*

_{L}= −3. Conditions were run in blocks, and blocks were run in random order. For each observer at each eccentricity, the uncrowded span

*u*was estimated by fitting Equation 8 to the results.

*φ*

_{h}, vertical eccentricity

*φ*

_{v}, and radial eccentricity

*φ*. Figure A1 shows, in gray, the fixation cross (upper left) and the location of the target (lower right). The target is at the center of an ellipse that represents the critical spacing for neighboring flankers. The experimenter typically chooses a fixed vertical eccentricity and measures the horizontal critical spacing

*s*as a function of the horizontal eccentricity. (This approach omits in-out asymmetry, as explained in Footnote 9.)

*s*

_{0}is the critical spacing at zero eccentricity. Using a low threshold criterion, Toet and Levi found

*s*

_{0}to be about 0.06°; using a higher criterion we find it to be about 0.2°. Pythagoras's theorem relates the radial eccentricity to the horizontal and vertical eccentricities.

^{9}The ellipticity

*ɛ*(ratio of length to width) is about 2. Assuming such an ellipse, we can calculate the ratio of the major and horizontal radii,

*s*

_{r}and

*s,*as a function of the angle

*θ*between them,

*ɛ*= 1).

*ɛ*= 1 (thin gray) and 2 (thick blue). The axes represent position (deg) in the visual field, relative to fixation.

*φ*

_{h}to +

*φ*

_{h}. For a given vertical eccentricity

*φ*

_{v}, we select the horizontal eccentricity

*φ*

_{h}so that the given spacing

*s*is the horizontal critical spacing at that location (

*φ*

_{h},

*φ*

_{v}). First we solve Equation A5 for (squared) radial eccentricity

*φ*

^{2}.

*φ*

^{2}.

*φ*

_{h}versus

*s*produces a rising curve (not shown) that descends to the left, asymptotically vertical, and ascends to the right, asymptotically proportional to

*s*. The curve is characterized by the two asymptotes. When the spacing is large, the window's extent is asymptotically proportional to the spacing,

*φ*

_{h}shrinks to zero when the horizontal spacing is reduced to the horizontal critical spacing at the vertical midline, which is the critical spacing for reading ( Eq. A6).

*φ*

_{h}versus

*φ*

_{v}( Eq. B2) traces out the boundary of the uncrowded window ( Fig. B1). When there is no ellipticity (

*ɛ*= 1) this boundary is a circle, as in Figure 5, and the gray circle in Figure B1. Increasing the ellipticity (

*ɛ*= 2) makes the isolation fields narrower ( Fig. A2), better able to isolate letters above and below fixation, so the uncrowded window extends further up and down, as shown by the thick blue hourglass boundary in Figure B1.

*φ*

_{h}/

*s,*

*φ*

_{h}/

*s*), with a

*breadth,*left to right, of 2

*φ*

_{h}/

*s*. The span is slightly larger. By tradition,

*span*refers to the number of letters acquired. Our interval extends from the center of the leftmost uncrowded letter to the center of the rightmost uncrowded letter. The span includes the whole of those letters and thus extends half a letter further to the left and right, so the span equals the interval's breadth plus one. (Except, of course, that when no letter is uncrowded the span and breadth are both zero.) Thus, the uncrowded span

*u*

_{o}is

*φ*

_{h},

*s*≫

*s*

_{0}then the span depends on spacing and eccentricity solely through the ratio of actual to critical spacing

*s*/(

*bφ*

_{v}),

*s*≫

*s*

_{0}+

*bφ*

_{v}, the uncrowded span ( Eq. B5) asymptotically approaches a constant value, independent of spacing,

*b*is typically around 0.4, so the added 1 is only a small fraction of the span value. The span

*u*

_{o}shrinks gradually to 1 and then suddenly to zero when the horizontal spacing is reduced to the horizontal critical spacing at the midline ( Eq. A6).

*s*

_{0}, of the reading rate curve for zero vertical eccentricity produced by this equation (after multiplying by

*ρ*) is less steep than that produced by Equation B6 for nonzero vertical eccentricity. The lesser steepness is discernable in the curves plotted in Figure 12 for zero and nonzero vertical eccentricity.

*u*

_{o}for Equation B4 above, we assumed a uniform angular spacing of the letters at the observer's eye. However, most text displays, including the printed page, instead maintain a uniform spacing in the plane of the display, which we take to be orthogonal to the line of sight at fixation. This uncrowded span

*u*is

*φ*

_{h}), tan

*φ*

_{h}≃

*φ*

_{h}so Equation B10 converges on Equation B4 and thus

*u*≃

*u*

_{o}.

*b*( Eq. B8). Thus, the known eccentricity dependence of critical spacing and the geometry of horizontal text together predict an uncrowded span that is asymptotically independent of spacing at large spacing and drops quickly at small spacings, hitting zero when the spacing is less than the critical spacing at the vertical midline.

*u,*will be uncrowded, and all those beyond will be crowded ( Fig. 5). However, ordinary text, today

^{6}, has space between words, which may relieve crowding of the initial and final letters of each word (Bouma, 1973). It is not surprising that these end letters are recognizable beyond the uncrowded span calculated here for uniformly spaced letters.

*Dither*is an engineering trick in which adding a perturbation to the input of a highly nonlinear system (e.g. a threshold cut-off) extends the range of stimuli to which it gives a graded response.

*s*

_{0.8}corresponding to 80% correct identification, our model for reading rate is

_{ P}is the expected value (i.e., average) over all values of

*P*= 0.01, 0.02,…, 0.99,

*r*is defined by Equation 3, and

*s*

_{ P}is the

*P*-th quantile of the critical spacing

*s,*which is now a random variable. For the uncrowded span

*u*in Equation 3, our software implementation of the uncrowded span model offers three choices: the simple Equation 4, the uniform-angle Equation B5, and the perspective-corrected Equation B10 (used in all plots, except the didactic Fig. 11).

^{10}

*s*

_{0},

*b,*and

*ρ*. (Fits at zero vertical eccentricity hold

*b*fixed and optimize

*s*

_{0}and

*ρ*. Fits at nonzero vertical eccentricity hold

*s*

_{0}fixed and optimize

*b*and

*ρ*.) The critical spacing is

*s*=

*s*

_{0}+

*bφ*

_{v}/

*ɛ*( Eq. A6). The maximum reading rate is (1 + 2/

*b*)

*ρ*( Eqs. 1 and B8). Footnote 4 explains how to estimate critical print size from critical spacing.

^{2}. The display resolution was 1024 × 768 pixels at 75 Hz, 28 pixels/cm. The viewing distance was 23 cm for peripheral trials and 200 cm for foveal trials.

*Reader's Digest*(average Flesch–Kincaid grade level 11.0; Kincaid, Fishburn, & Chissom, 1975) and were shown in order or in scrambled word order. We will refer to text that preserves the original word order as

*ordered*and to text with scrambled word order as

*unordered*. See Table D1.

Ordered | Unordered |
---|---|

It used to be that when I felt that way I'd just drink some coffee and presto energy | Just way coffee I'd to used presto I some that energy felt used be and that It when |

Vertical eccentricity | Size (deg) | Spacing (deg) |
---|---|---|

0° | 0.1 | 0.1 |

0° | 0.16 | 0.07, 0.09, 0.14, 0.16, 0.28, 0.42, 0.56 |

0° | 0.4 | 0.4 |

0° | 1.6 | 1.6 |

−5° | 0.5 | 0.5 |

−5° | 0.7 | 0.7 |

−5° | 1.6 | 1.6 |

−5° | 3.5 | 3.5 |

−10° | 1.4 | 0.7, 1.0, 1.4, 4.2, 5.6, 7.0 |

−10° | 2.0 | 2.0 |

−10° | 3.0 | 3.0 |

*Journal of Vision*in 2004. The editor, Gordon Legge, invited us to deal with the Legge et al. (2001) account of the peripheral reading rate. This paper (draft 145) is the result. Some of these results were presented at the Vision Sciences Society meetings in Sarasota, FL, May 2003 and May 2006. This project was supported by National Institutes of Health grant EY04432 to Denis Pelli.

^{1}In principle, the much larger rightward extent of the eye-movement-based perceptual span could represent effects on eye movements of distant text features, such as word breaks or word shape, that are insufficient for letter and word identification. However, the evidence suggests that such effects do not extend far. The most common location of the first fixation in a word is the third letter in the word, independent of word length (McConkie, Kerr, Reddix, & Zola, 1988). Alternatively, the large perceptual span might be an artifact of the perceptually obvious substitutions used to measure it. Commenting on Rayner's studies, Underwood and McConkie (1985) note that finding an effect (on eye movements) of substitution by a square-wave grating or X's may reflect “interference” by these “perceptually obvious” insertions, “disruptive to normal reading for reasons other than the removal of normal textual information.”

^{2}To this, Legge (personal communication) adds that mislocation errors (reporting the right letter in the wrong place in the triplet) “certainly play some role.” In his new book, Legge (2007, Section 3.7) calls this “decreasing accuracy of position signals in peripheral vision.” Legge et al. (2001) required observers to report all three letters in the triplet in the correct order. Strasburger (2005) too also reports that observers often mislocate numerical digits under conditions that produce crowding. Mislocating, i.e., reporting characters in the wrong order, could happen in at least two ways. We can imagine a cognitive account in which the observer mixes up the letter order in forming the report. However, a crowding account seems equally plausible. Crowding is the inappropriate integration of features into an object in which they do not belong. This can produce “feature migration” and illusory conjunction and would be expected to occasionally change the identity of a letter to that of a neighbor (Pelli et al., 2004). The cognitive and crowding accounts can be distinguished by comparing results in the left and right visual field. We would expect the cognitive effect to depend on the letter sequence, read left to right. We would expect the crowding effect to depend on radial eccentricity (distance from fixation). Legge et al. report that, on both sides of the vertical midline, the outermost letter of the triplet is identified much more accurately than the inner two, which are closer to the midline. These opposite order effects in left and right visual field are consistent with the crowding account and utterly unexpected in a cognitive account. Thus, any mislocation seems attributable to crowding, not a cognitive limitation.

^{3}Chung et al. (1998) did their two-line fits by eye. We redid them by computer, minimizing RMS log error. With three degrees of freedom the two-line fit yields the same RMS error as the uncrowded span model, which has only two degrees of freedom. The average log-log slope of the cliff in the two-line fits was 3. Fixing that slope at 3 and fitting the two-line model with only two degrees of freedom yields an RMS error that was, on average, 1.3 ± 0.1 times that of the fit by the uncrowded span model.

^{4}In comparing studies, some readers may wish to convert critical spacing, as estimated here (Eq. A6), to critical print size, as estimated by Chung et al. (1998) and others from a two-line fit. For the 36 fits (6 observers × 6 eccentricities) in Figure 12, the mean ± standard deviation of the ratio of critical print size to critical spacing is 0.9 ± 0.2. In converting other data, we suggest two steps. First, take the critical spacing as an estimate of what one might call “critical print spacing,” and then calculate the size from the spacing, based on the actual text's metrics. For Chung's Times Roman text, we estimate the letter size to be 0.9 × spacing. The fitting software we provide fits both models (uncrowded span and the two-line fit) to any reading-rate data.

^{5}As we mentioned briefly at the beginning, Legge et al. (2001, p. 726) list many possible determinants of visual span, including crowding, and note that some of those causes would predict a plateau: “A consequence of the linear scaling laws that apply to both peripheral letter acuity and crowding … is that the size of the visual span is roughly constant when measured in letter spaces over a moderate range of angular character size.” (O'Regan, 1990, made the same point.) However, we take the Legge et al. comment as a hypothetical aside, not their preferred account, because the linear scaling laws also make the visual span independent of vertical eccentricity, contradicting the central thesis of their paper, namely, that shrinkage of the visual span accounts for slower reading at greater eccentricity.

^{6}Noordzij (2005) suggests that word breaks first appeared in written documents sometime in the years 600–650 AD, initially in Ireland, and quickly spread to the rest of Europe.

^{7}These papers mostly report visual span in bits, instead of characters, but you can convert from bits to characters by dividing by the number of bits (4.7) per reliably recognized character (Legge et al., 2007).

^{9}Bouma (1970) noted an in-out asymmetry in crowding. Testing a peripheral target with a single flanker reveals that the critical spacing is smaller toward fixation than away from fixation. See Motter and Simoni (2007) for an explanation. That asymmetry is not captured by testing with symmetrically placed flankers (Fig. 6 and Toet & Levi, 1992), but the in-out asymmetry is present in Bouma's (1978, p. 24) sketch of the isolation fields. For simplicity, our elaboration of Bouma's law omits the in-out asymmetry. The ellipticity is needed to accurately account for reading rate data, but the in-out asymmetry is not.

^{10}Levi et al. (2007) set the dither range to 0.55, slightly less than the 0.7 used here. Here we are trying to test a simple model. Such a test is more stringent, more convincing, if it has fewer parameters left free to accommodate the fit to the data. So here we estimated the dither range, 0.7, in advance, from the slope of the psychometric functions in Figure 8. Levi et al. took the model as established (for normals) and were interested in what it could say about amblyopia. Taking range of dither as a degree of freedom in optimizing their fits, they estimated the dither range to be 0.55. This difference is small and does not affect the fits much, but we recommend that future studies use the value 0.55.