Perception &Psychophysics1974, Vol. 16 (2), 213-221

Assessing Poggendorff effects via collinearity,perpendicularity, parallelism, and

Oppel (distance) experiments*

DANIELl. WEINTRAUBt and LILLIAN TONGHuman Performance Center, UniveT'SityofMichigan, Ann Arbor, Michigan 48104

By adjusting the orientation of, and separation between, two free-standing dots, Ss indicated directions anddistances associated with the Poggendorff display (a transversal interrupted by parallel lines). Judgeddistance between parallels (with transversal absent) increased slightly when additional interior parallelswere added; this Oppel effect can be interpreted as contour repulsion. Errors in judging the orientation of anactual transversal segment were too small to account for the Poggendorff effect. The usual large errorsoccurred for estimates of the orientation of the missing transversal segment between the parallel lines.Cognitive mistracking adequately describes the Poggendorff effect. Mistracking is a function of the anglesubtended between transversal and parallels, and of the orientation of the entire display.

The traditional Poggendorff display consists of twoparallel lines interrupting a transversal; thetransversal segments do not appear collinear.Contour-interaction theories claim that perception isdistorted by the displacement of the neuralrepresentation of the contours forming the physicalstimulus. For example, the Poggendorff effect islogically explained by postulating that parallelsappear too close, i.e., through lateral interactions,neural contours attract one another (discussed byTong & Weintraub, 1974). Alternatively, the acuteangle between transversal and parallels might appeartoo large, an instance of contour repulsion(Blakemore. Carpenter, & Georgeson, 1970; Burns &Pritchard, 1971).

The initial experiments are direct offshoots of ourprevious work (Tong & Weintraub, 1974), attempting

.to assess the parallels-attract hypothesis applied to thePoggendorff effect. If parallels between transversalsegments are represented in experience as too closetogether, then the attached transversals will seem tobe vertically misaligned when they are actuallycollinear. Additional parallel lines were addedbetween the original parallels under the assumptionthat the additional parallels would cause increasingcontour attraction and therefore an increasingPoggendorff effect. However, the data clearly showedthat extra interior parallels reduced the Poggendorfferror (Tong & Weintraub, 1974). The Oppel effect(Robinson, 1972), the overestimation of filled space

• Support was provided by a United States Publich HealthService Research Scientist Development Award <K2-MH-35,253) toD. J. Weintraub. Research funds were provided by National ScienceFoundation Grant GB8181.

tAddress: Human Performance Center, Perry Building, 330Packard Road, Ann Arbor, Michigan 48104.

relative to unfilled space, seemed able to account forthe findings. The Poggendorff display containingextra parallels is directly analogous to an Oppeldisplay containing many interior line segmentsparallel to end segments defining an interval. The firsttwo experiments continue the investigation of the roleof the parallels.

EXPERIMENT I

MethodThe paid Ss were 40 students at the University of Michigan,

tested one at a time. Each S received eight white stimulus sheets,21.6 x 27.9 em (81/ , x II in.) containing Multilithed black lines anddots. Vertical lines, .4 mm wide, were drawn parallel to the longeredges of a sheet. The separation between outer parallels was either32 or 42'13 mm. (These choices were dictated by the separationsemployed in the Tong & Weintraub, 1974, experiments.) At each ofthe two separations, 2, 3, 4, or 5 evenly spaced parallel lines weredrawn. A dot on the left parallel and a dot on the right paralleldefined a direction through the field of parallels at an angleof +45 deg from horizontal. Each dot was black and 1.5 mm indiam. The 5-line stimulus display is depicted in Fig. 1.

Procedure. The S sat at a desk with the eight stimuli face down ina randomized pile to hisleft. He was instructed to turn the top sheetface up in front of himself, with the short (8'/, in.) edge flush withthe front edge of the desk. In this position, the parallel linesproceeded away from S. With the pencil provided, S was instructedto place a small dot on an extension ofthe imaginary line defined bythe two dots lying on the parallels, the same distance from the rightdot as the right dot was from the left dot. Stated differently, the Swas asked to place a third dot to the right of the two dots on thesheet, so that the three dots were collinear and the S's dot was thesame distance from the middle dot as the left dot was from themiddle dot. The instructions were illustrated by referring to asample stimulus sheet. The S was encouraged to erase andreposition his dot until he was satisfied. The S circled his finaljudgment (to avoid any scoring ambiguity), turned the sheet over ina pile to his right and proceeded to the next sheet. Head tilting,drawing lines, etc., were not permitted. Subjects were allowed onlyto viewthe display and position a dot. Eye-to-stimulus distance was

213

214 WEINTRAUB AND TONG

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free-standing dot representing S's judgment. Averidical judgment would provide a slope of 1.0 (tan45 deg = 1.0). For a two-line display, neither functionis significantly different from a tangent of 1.0, butboth functions decreased as interior parallels wereadded. [For these tangent data at the bottom -ofFig. 2, with an .01 level test, number of lines wassignificant, F(3,117) = 10.15; objective distancebetween outer parallels was significant, F(3,117) =10.15; objective distance between outer parallels wassignificant, FO,39) = 11.04; the interaction was not.]The outcome is consistent with the hypothesis thatfilled space is perceived as more extensive thanunfilled space. If the dots lying on the parallels areperceived as farther apart, then the slope of theimaginary line between them should decrease.

The disturbing aspect of the outcome is that themean distance judgments shown in Fig. 1 areprimarily underestimates. In what sense can filledspace be considered overestimated when the obtaineddistance judgments are underestimates? At thisjuncture, two attributes of the stimulus displaysbecame obvious to us. First, the unfilled space wasdirectly adjacent to the filled space, and theadditional parallels used to fill the space might wellhave influenced the unfilled region from which themeasurements were obtained. (The same criticism canbe leveled at any study of the Oppel effect employingadjacent regions.) Second, the stimulus display couldbe considered a degraded variation of the Poggendorffdisplay in which segments of the transversal had beenreplaced by dots. This particular variant has beenshown to give a small Poggendorff effect (Krantz &

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Results and DiscussionIn Fig. I, note that, for either separation between

parallels, mean judged slant distance, labeled D,increased as more lines were added. On all figures,vertical bars about each mean depict ±1 standarderror of the mean (Si). [At a = .01, the number oflines produced a statistically significant main effect,F(3,117) = 4.16; it was obvious that the objectivedistance between outer parallels would do likewise,F(1,39) 1,041.98. The interaction was notsignificant.] An indirect estimate of the perceiveddistance between parallels can be derived from theslant distance between dots by taking the horizontal(X) component of a judgment. These mean data areplotted in the lower half of Fig. I, and they alsoincreased with an increase in the number of parallellines. Subjects might have produced the increasingslant distance by increasing the slope of theircollinearity judgments while holding the horizontalcomponent constant. However, the data shown inFig. 1 support the interpretation that adding interiorparallels increased perceived distance between theouter .parallels, of which X is an indirect measurederived from the judged slant distance, D.

Adding interior lines reduced the perceived angle ofinclination as shown by the lower functions inFig. 2. The measure plotted is the mean slope of theline between the dot on the right parallel and the

POGGENDORFF ILLUSION 215

Fig. 3. Mean distance judgments for Experiment II, with astimulus display depicted at the upper right. (The true slantdistance, D, Is 45.25 mm; distances X and Y are 32 mm). Derivedvertical distances, Y, are also shown for Experiment I.

components in order to facilitate a comparison withExperiment II. In Experiment I (Fig. 1), the portionof the display requiring S's attention during thedot-setting aspect of the judging (the comparison­stimulus portion so to speak) is bounded by a dot lyingon the right parallel and a dot not on a line, the dotproduced by S. In Experiment II (Fig. 3), thecomparison stimulus is composed of two unattacheddots, one produced by S. Therefore, the comparison­stimulus conditions differ between experiments. But alink exists. The comparison-stimulus conditions ofExperiment I are duplicated by a I-line display inExperiment II. Thus, the line-and-a-dot comparison­stimulus configuration of Experiment I can beevaluated in Experiment II with respect to the two-dotconfiguration. Relative to two free-standing dots, thedistance between a pair of dots is underestimatedwhen one lies on a line, and the underestimation ismore pronounced when each dot of a pair lies on a line(as can be seen in Fig. 3, solid-line functions). Onecan account for Experiment I in terms of the outcomeof Experiment II. In Experiment I, S set hisfree-standing dot closer to the right parallel because,when both distances were objectively equal, thedistance between the dots lying on the parallelsseemed smaller than the distance between the dot onthe right parallel and a free-standing dot.

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MethodA new sample of 40 paid Ss, students at the University of

Michigan. was tested. Multilithed stimulus sheets, like those ofExperiment I. contained dots and vertically oriented parallel lines.A stimulus sheet always contained a pair of dots (each 1.5 mm indiam) lying either horizontally with respect to one another, orobliquely, i.e., tilted at +45 deg from horizontal. The horizontaldistance. X, between a pair of dots was fixed at 32 mm. Verticalparallel lines (.4 mm thickness) were drawn in the space betweenthe pair of dots as well as through the dots. A given displaycontained from 0 through 5 vertical lines. The O-Iine displaycontained only the dot pair. There were two different f-Iinedisplays, one with a vertical line through the left dot, the other witha vertical line through the right dot. The 2-line display contained aline through each dot. Displays with 3 or more lines containedevenly spaced lines added to the interior of the 2-line display. Eachof the 14 displays appeared to the left of center on a stimulus sheet.Included on every sheet was a 1.5-mm black dot placed randomly tothe right of the display with the restriction that it not be at the sameheight as either dot on the left. (The diagram in Fig. 3 depicts twodots on the Tight parallel, but only one appeared in any givendisplay.)

Procedure. Experimental procedures were like those in the firststudy except that 5 was now asked for a judgment of parallelism.Again, head tilting was not allowed. He was told to observe thedistance and direction between the pair of dots on the left, and tocomplete a pair on his right to match in distance and direction. Thesingle dot (to S's right) represented the left dot of the pair to becompleted. The instructior:s were: "Look at the two dots to the left,then look to the right, and try to recreate the relationship."

EXPERIMENT n

Weintraub, 1973, Stimulus 4,Experiment VII). Analternative to a collinearity judgment is a parallelismjudgment. The stimulus used as an assessment devicecan be removed from the region where perturbinginfluences are presumed to be operating.

Results and DiscussionThe function labeled X in Fig. 3 represents the

mean judged distance between a pair of horizontaldots. With no lines present, the true width betweendots of 32 mm was judged as essentially 32 mm.Adding a line through either the left or right dotdecreased judged distance. A line through both dotsled to a further decrease. [As an illustration ofstatistical significance, the differences amongdistance judgments for displays of 0, 1 (left), and 2lines for the X function of Fig. 3 produced F(2,78) =19.11, an outcome well beyond the .Ol-Ievel criterion.]The addition of one or more interior lines (designatedas 3, 4, and 5 lines in Fig. 3) produced a statisticallysignificant (a = .01) increase in judged distance over2 lines, which are the outer parallels alone. Forjudgments of slant distance, the function labeled D,adding parallels through the dots again causedjudgments to decrease significantly (a = .01).However, there was no rise with the addition ofinterior lines.

These data resolve the dilemma posed by the firstexperiment that' nearly all distances were under­estimated. The displays of Experiment I can beconsidered hybrids. Let us logically separate their

216 WEINTRAUB AND TONG

With the addition of interior lines in Experiment I,judged distances increased (i.e., underestimatesdecreased), the usual Oppel effect. For a similarOppel effect in Experiment II, the additional interiorlines should have produced a consistent increase injudged distance relative to two lines. For a horizontaldistance, X, the judgments did increase when thethird line was added. For slant distance, D, there wasno change with the addition of interior lines. Is theOppel effect most salient when adjoining distances arebeing judged as in Experiment I? Overall, theevidence from both experiments plus the Tong­Weintraub (1974) data point toward an Oppel effect:Filled space appears more extensive than unfilledspace. The contours that bound the distances beingjudged cannot be ignored because they exert aconsiderable influence on the outcome.

The derived functions, X (from D) and Y (from D),in Fig. 3 represent the X,Y coordinates of the judgedslant distance, D. Derived X is everywhere less than Xjudged directly. Adding lines produced a greaterchange in derived X than in derived Y. Therefore, Sscan be presumed to vary their slope judgments mainlyby varying dot displacement horizontally.

Judged slopes for Experiment II are shown inFig. 2. The dominant finding is that the slope betweendots of 1.0 is significantly overestimated whenever anylines are present in the display. (Note, in Fig. 2, thatthese particular means lie more than 1.96 SDs abovethe objective value of 1.0.) The result is not easy tointerpret. The fact that it is in the opposite directionfrom the outcome of Experiment I is not surprising,because collinearity judgments were linked to theperceived distance between dots being judged, whilethe parallelism judgments utilized in Experiment IIwere not. We had anticipated that judged slopeswould rise as the number of lines increased from a to2, reflecting an increasing slope with decreasingperceived distance, and then decrease slowly out tothe 5-line display, reflecting the Oppel effect. Thedata from the 2-line display are clearly too discrepantto support such a hypothesis. An important findingconcerning slope estimations for dots· lying onparallels is that the slope errors (overestimations) liein the opposite direction from errors that could leadto a Poggendorff effect.

In summary, the evidence from the first twoexperiments leads to several important conclusions. Ifobjective distances are equal, then the perceiveddistance between a free-standing dot pair is greatest,followed by a dot pair of one free dot and one dot on aline, followed by a dot pair with each dot on a line.With respect to the Oppel effect itself, the marks usedto delineate the boundaries of distances beingcompared should be identical, in order not tointroduce the confounding effects of differentialboundary markings. The Oppel effect is evidently notlarge under the experimental conditions employed,but the data taken in the aggregate do imply that

adding interior parallels increased the judged distancebetween outer parallels. Slopes between two dots at a45-deg angle and lying on parallels were shown to beoverestimated.

In what sense can measurements of the referencestimulus, free-standing dots, be said to "measure"perceived orientation or perceived distance? Theimplied preface to every statement about data is"relative to the manner in which the free-standingdots are perceived." The proviso is not merely atechnical nicety. For example, do parallels attract?The questions cannot be answered directly. Relativeto two free-standing dots, parallels are perceived ascloser together. But free-standing dots might repelone another, and parallels might also repel, but lessstrongly. From the converging evidence provided byadding interior parallels, we are willing to infer thatparallels do indeed repel rather than attract, whichwould lead to an anti-Poggendorff effect. Thus,although our data agree with those of Quina andPollack (1973), who used very similar stimuliconsisting of parallel lines and dot pairs, we areunwilling to conclude with them that parallels attract.Furthermore, in the absence of a transversal,directional errors while tracking in the space betweenparallels were also anti-Poggendorff errors. Clearly,then. since the displays did not contain a transversal,the Poggendorff effect depends upon the presence of atransversal.

EXPERIMENT III

The dot-setting method used in the first twoexperiments was again employed with a modifiedPoggendorff display consisting of a pair of parallellines with a transversal segment attached to the leftparallel only. The modified display has been usedbefore (Weintraub & Krantz, 1971; Pressey &Sweeney, 1972; Tong & Weintraub, 1974). It leads toPoggendorff errors oflarge magnitude, comparable tothose produced by the traditional display containing atransversal segment attached to each parallel, and it iswell suited to dot-setting procedures. Administrative­ly, two separate experiments were executed usingseparate sets of Ss. The tasks required of Ss weredifferent. Stimulus materials were identical.

MethodThe first group of Ss consisted of 40, the second group of 41,

University of Michigan paid volunteers, who were testedindividually. The Ss in the first group performed the first two taskslisted below, C and W, and the second group of Ss undertook theremainder. Multilithed in black were lines and dots on white sheets,as described in the first two experiments. Each stimulus display,consisting of parallels and an attached transversal segment,appeared to the left of center on the sheet. Parallels were a pair oflines (always at least twice the length of the transversal), 30 mmapart. A transversal, 70 mm in length, was attached to the leftparallel at an angle of 22'/2, 45, or 67'/2 deg. Each display wasoriented in one of two positions on the sheet: parallel lines uprightor transversal upright (see the diagrams in Fig. 6). Included on

POGGENDORFFILLUSION 217

Fig. 4. Mean directional errors, In degrees, for judgments onupright displays In Experiment IH. (Counterclockwise errors aredesignated as positive.)

Fig. S. Mean directional errors, In degrees, for judgments ontilted displays In Experiment HI. (Counterclockwise errors aredesignated as positive.)

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Procedure. The S sat at a desk covered by a white cloth andplaced each sheet in turn in front of himself with the long (11 in.)edge flush with the front edge of the desk. The stimulus displaysappeared to S as depicted in Fig. 6 (one to a sheet). He was askedto perform one of several geometric operations, .

The free-standing dot to S's right represented the intersection oftransversal segment and left parallel. The S was required to place asecond dot with respect to the given dot such that attributes of thePoggendorff display, as specified below, were matched in bothdistance and direction. Repositionings, after erasing the previousjudgment. were encouraged, S was requested to circle his finaljudgment. Symbols for each task listed below correspond to thoseappearing in Figs. 4-7.

Judge collinearity, C: The task required S to make a Poggendorffjudgment with respect to the display, i.e .. to estimate a point on theright parallel collinear with the transversal. and then to transferthat estimate to free-standing dots. No mark was permitted on theright parallel. In essence. the requirement was to duplicate acollinearity judgment via a parallelism judgment.

Judge the width between parallels, W: The S positioned a dotat the right so that the free-standing dots duplicated the perceiveddistance and direction of the width between the parallels measuredperpendicular to the parallels at the point of the transversalintersection.

Judge perpendicularity, 1: From the transversal intersection. Sestimated the perpendicular (to the transversal) lying between theparallels. transferring his judgment to free-standing dots. No markwas perm itted on the right parallel.

Judge the transversal, T: The S judged the length and orientationof the transversal segment by means of free-standing dots.

Judge collinearity on the display. [C]: The S marked the point onthe right parallel representing the intersection of an imagineryextension of the transversal (the usual Poggendorff task).

Judge perpendicularity on the display [1]: From the transversalintersection, S estimated the perpendicular (to the transversal) lyingbetween the parallels and marked its point of intersection on theright parallel itself.

The two preceding tasks, [C], and [1] constitute direct estimateson the display without the encumbrance of a concomitantparallelism judgment. Note that the "direct" assessment procedurecannot be applied to an existing line like the transversal.

Results and DiscussionThe data most pertinent to an explanation of the

Poggendorff effect are errors in directionality (slope),so these will be treated first. Figure 4, uprightdisplays, and Fig. 5, tilted displays, present errors indegrees for the various tasks. Conforming withconvention, counterclockwise angular errors aredesignated as positive errors. A plausible and widelyheld theory concerning the Poggendorff effect is thatthe transversal orientation is misperceived becauseacute angles are overestimated (Blakemore,Carpenter, & Georgeson, 1970; Burns & Pritchard,1971). The explanation implies that contour repulsionexists between a transversal and attached parallelsuch that the transversal would appear tilted too farclockwise (a negative angular error). The transversal,T, data do not reveal a consistent negative error. Themost striking feature of the transversal data is that theerrors are close to zero.

Unlike the orientation of the transversal segment,collinearity (the Poggendorff effect> can bedetermined not only indirectly by a parallelism

218 WEINTRAUB AND TONG

judgment, but also directly on the display. Comparethe two types of judgments, C vs [C], in Figs. 4 and 5.An important finding is that, although there are cleardifferences, the two types of judgment producedsimilar amounts of error. In addition, collinearityjudgments, in contrast to judgments of transversalorientation, produced the expected Poggendorffeffect. The Poggendorff effect was expected to be afunction of the angle between transversals andparallels. The Weintraub-Krantz (1971) formula forupright displays (the data of Fig. 4) predicts errors forC and [C] of -3.80, -5.04, and -3.36 deg for transversalangles of 221/ 2, 45, and 671/ 2 deg, respectively. Thenew data did not confirm the expectation of an errorcloser to zero at 221;2 deg than at 45 deg, but are other­wise consistent. For any given acute angle betweentransversal and parallels, the Poggendorff effect wasexpected to be greatly attenuated when the transversalcoincided with the horizontal or vertical axis of theobserver's visual field. (See Experiment V ofWeintraub & Krantz, 1971.) Figures 4 and 5corroborate the expectation, as' collinearity errorswere always smaller for the same display when tilted.Therefore, collinearity errors measured by twomethods generally confirmed expectations withrespect to angle subtended between transversal andparallel, and tilt of display.

A general feature of the data can be illustrated byjudgments of the width between parallels, W. For anupright display, the free-standing dots were sethorizontally without appreciable orientation error (W,Fig. 4). For tilted displays, width between parallels isdefined along an oblique line and orientation errorswere large (W, Fig. 5). The same trends can be seenfor collinearity, C, and perpendicularity, 1,judgments. The greatest directional errors occurredwith oblique tracking. When judging the orientation(If a line physically present, the transversal, T, alldirectional errors were small regardless of theorientation of the transversal. From these findings,and from the similarities between judgments of C and[C], 1 and [1], we conclude that the perceptualtasks of determining collinearity and perpendicularityare themselves affected by a change in orientation ofthe physical stimulus rather than orientation affectingonly the perceptual parallelism operation used intransferring a judgment from display to free­standing dots: Appelle (1972) has suggested the termoblique effect, for the pervasive finding that sensitivitydeclines and veridicality deteriorates for stimulioriented obliquely.

A new tack in checking the adequacy ofPoggendorff theories was to investigate apparentr\rpendicularity to determine the correspondenceb .ween perpendicularity and collinearity errors. Thetasks involved, in one case, imagining an extension ofthe transversal and, in the other case, imagining a line90 deg to the transversal. Thus, if the transversalorientation is misperceived and all errors follow as a

consequence, then the same angular error shouldoccur in both tasks. With upright displays (Fig. 4),errors for collinearity and perpendicularity were notthe same. The noncorrespondence of angular errorsconstitutes evidence against the hypothesis that thecause of the Poggendorff effect lies in misperceivingthe orientation of the transversal.

Judged distances were influenced only modestly bythe orientation of the display as shown by the data ofFigs. 6 and 7. The size of the angle betweentransversal and parallel exerted a minor influence onthe judged length of T and the distance betweenparallels, W (Fig. 6). Judged W is significantly lessthan the objective width between parallels of 30 mm.From the data on upright parallels in Experiment II,it can be calculated that to match a horizontaldistance between parallels, free-standing dots shouldbe placed too close together physically by 9.4 %(computed from the value of the X function for 2 linesin Fig. 3). Under the assumption that the presence orabsence of a transversal does not affect judged W,applying the correction to the 30-mm width leads to apredicted judged width of 27.2 mm, in excellentagreement with the upright-display data of Fig. 6.The derived Ws shown as dotted lines in Fig. 6 aremean judged widths between the parallels obtainedindirectly by computation from judgments ofcollinearity and perpendicularity. A derived W is notequivalent to a width between parallels judgeddirectly, grounds for rejecting again the hypothesisthat the parallels are misperceived in separation andall errors follow accordingly. The widths derived fromcollinearity estimates are underestimates; the data ofExperiment II permit an independent estimate of thiserror at 45 deg. In Fig. 3, the function of X (from D)at 2 lines shows an underestimation of 15.2%. Whenapplied to a 30-mm width, the prediction is 25.4 mm.This last number is to be compared with W (fromcoIlinearity) of an upright display at an acute angle of45 deg in Fig. 6. The agreement is excellent. We haveno explanation to offer for the overestimation shownby the W derived from perpendicularity at67 1/ 2 deg.

The large discrepancies between true and judgeddistances for collinearity and perpendicularityjudgments are plotted in Fig. 7. The graph wasdesigned to emphasize the finding that distance errorswere related to the length of the path to betraversed. Judged distances are plotted as afunction of the angular deviation of the true trackfrom the shortest possible path-the shortest is thepath beginning at the transversal intersection andproceeding perpendicularly across the space betweenthe parallels. All but one distance was under­estimated. Any distance estimate associated with aperpendicularity judgment always showed consider­ably less underestimation than a collinearity tracktraversing the same objective distance. Much of theerror in distance judgments can be accounted for by

POGGENDORFF ILLUSION 219

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distance between two free-standing dots lying at a45-deg angle is the same as that between twofree-standing horizontal dots, will the equal slant andhorizontal distances appear to be equal? Since suchdata were not gathered, discrepancies between judgeddistances made along differently tilted tracks must beinterpreted conservatively. It is not possible todetermine the influence of the various tilts of thefree-standing dots comprising the comparisonstimulus.

GENERAL DISCUSSION

Psychophysical null-match methods, those basedupon judgments of equality between stimuli, have highface validity for investigating neurophysiologicalhypotheses. Judging distance can be cast into thenull-match mold: Adjust one distance until it appearsequal to another. Parallelism is clearly a null-matchoperation, judging when one line has the sameorientation as another. Collinearity can be interpretedas a special case of parallelism, parallel lines set end toend. What of perpendicularity? When line segmentscross, adjacent angles at the intersection can beequated, and, since the angles are supplementary,each will be 90 deg. Oftentimes, in the preceding

Fig. 7. Mean slant·dlstance judgments along tracks determinedby perpendicularity and collinearity estimates. (Deviation unitswere used for the abscissa In order to faJ.U1tate comparisons amongjudged lengths based on the same true dlstance.)

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applying two assumptions discussed previously.Assume that parallels appear closer together than dofree-standing dots. In Fig. 6, for upright displays theunderestimation of W, averaged across the threeacute angles, was 8.1 %; for tilted displays, it was5.0%. Assume. further, that because S mistracks hetherefore intercepts the right parallel too early or toolate. In other words, an adjusted "true" distanceagainst which to compare Ss' mean distance judgmentis the true distance along the track actually taken bySs. Using the mean directional error for eachcondition as an estimate of the track, the adjusted"true" distance can be found by measuring from thestart of the track to the intercept of the track with theright parallel. These distances were then corrected bythe percentages given in the first assumption. Thepredictions are given in parentheses next to eachplotted mean in Fig. 7. For collinearity judgments,the lower two functions in Fig. 7, the predictionsaccount for a large portion of the underestimateddistances. (Even the residual might have beenanticipated by noting, from Experiment II, that the45-deg slant distance between parallels, Fig. 3,showed a greater percent of underestimation than didseparation between parallels. W.) When the samecorrections are applied to distance associated withperpendicularity judgments, three of the six obtainedvalues are substantially larger than the predictedvalues (Fig. 7). Again, a problem arises concerningthe evaluation of discrepant outcomes. If the objective

220 WEINTRAUB AND TONG

experiments, more than one null-match operation wasrequired for performing the task.

Brindley (1960) referred to null matches as"Class A" observations, and argued that a simplepostulate supported the leap from null-matchpsychophysics to neurophysiological theory, namely," ... that whenever two stimuli cause physicallyindistinguishable signals to be sent from the senseorgans to the brain, the sensations produced by thesestimuli as reported by the subject in words, symbols,or actions must be indistinguishable [p, 144]."Indistinguishability means indistinguishability in allrespects. For example, in colorimetry, a metameric(look alike) match is required between a given colorand one generated by S from a set of primaries.However, the criterion of ind istinguishability in allrespects demands a further experimental test: In atwo-alternative forced-choice temporal or spatialdiscrimination experiment, the S must be unable todetect any difference between the look-alike colors.(Compare this treatment of Class A observations withthat of Cornsweet & Pinsker, 1965, and of Sekuler,1974.)

Unfortunately, complete indistinguishability isimpossible in most experiments because the stimuliare indelibly tagged by their contexts. For instance, inan experiment concerning simultaneous colorcontrast, a stimulus in a chromatic (colored) surroundmight be matched by a stimulus in an achromatic(neutral) surround. After the color match has beenachieved, the stimuli are still distinguishable by theirrespective surrounds. Since stimuli to be matched arenot often context free, of necessity the null matchesrequired are null with respect to selected attributesonly. The Poggendorff effect occurs in the presence ofparallels, the context tag, while parallels are absentfrom a comparison stimulus. The S was asked to"disregard" context differences in performing a nullmatch. These less-than-ideal null matches are the bestthat can be done.

One research strategy is to investigate differentnull-match operations employing the samePoggendorff display to discover whether the resultingdata coincide or whether differences can be resolvedby a theory. Requiring that various geometricrelationships (distance, collinearity, perpendicularity)be duplicated by a null match for distance andorientation using a pair of free-standing dots provedfruitful. First, it was established that the distancebetween a pair of parallel lines is underestimatedrelative to the dots. After allowing for this effect ofmarking the boundaries of the intervals differently,adding interior lines between parallels gave evidenceof a modest Oppel effect, filled space seeming moreextensive than unfilled space. From it, one mayconclude that if the neurophysiological represent­ations of contours interact, then they repel. Sincerepulsion will lead to an anti-Poggendorff effect, adescriptive statement of the form, parallels attract, is

not an admissible explanation for the Poggendorffeffect.

In principle, the experimental technique permits Sto judge and set the orientation between a pair of free­standing dots independently of their separation. Suchfreedom does not exist with the usual judgments madedirectly on the Poggendorff display, so the causes ofmisalignment may be hidden. Misperceived orienta­tion of the transversal cannot explain the Poggendorffeffect. These errors are small and do not conform tochanges in the magnitude of the effect with either achange in the angle subtended between transversaland parallel or a change in the orientation of theentire display. The data support the contention thatthe primary perceptual anomaly leading to thePoggendorff effect is misperceived colIinearity, thatis, extending the transversal between the parallels isnot performed veridically.

Although both C and [C] judgments involveperceptual colIinearity, the C judgment requires S toperform an additional parallelism operation whentransferring the judgment to free-standing dots. TheC and [C] judgments (and 1 and [1] judgments) didnot coincide, but they were in reasonable agreement.One may assume that the perceptual collinearity (orperpendicularity) information is not destroyed by theoperation transferring it to free dots. Granted that thetransversal may be misperceived in orientation, theerrors, as assessed by the same parallelism techniqueemployed for Poggendorff collinearity, correspondedin neither direction nor amount to the collinearityerrors. The Poggendorff collinearity judgmentsproduced predictably large directional errors; errorsin transversal orientation were small. Corroborativeevidence comes from Hotopf and Ollerearnshaw(1972a, b), who obtained a similar finding via themethod of constant stimuli, with a free-standing lineserving as comparison stimulus. Finally, the anomalyis a collinearity misjudgment associated with thetransversal itself, because the perceived orientation oftwo dots lying obliquely to one another on theparallels in the absence of a transversal showed ananti-Poggendorff effect (Fig. 2, upper function). Wetherefore feel very confident in rejecting any theorybased upon misperceiving the orientation of thetransversal. A transversal is necessary, nevertheless.

The data of Experiment III substantiate theinfluence of two characteristics of Poggendorffdisplays. The first is the angle subtended betweentransversal and parallel. The weight of evidence fromprevious work singles out the obtuse angle as the mostcritical stimulus feature (Day, 1973; Green & Hoyle,1964; Imai, 1973; Krantz & Weintraub, 1973;Weintraub & Krantz, 1971). A second criticalvariable is the orientation of the display in the visualfield (Green & Hoyle, 1964; Leibowitz & Toffey,1966; Weintraub & Krantz, 1971), which conforms tothe oblique effect (Appelle, 1972).

Since transversal misorientation is not an

acceptable explanation, the neurophysiologicalhypotheses leaning upon contour interactionsproducing overestimation of acute angles (Blakemore,Carpenter, & Georgeson, 1970; Burns & Pritchard,1971) cannot apply. Versions of Gregory's (1963)misapplied-constancy scaling theory (Gillam, 1971;Green & Hoyle, 1964) might be modified bypostulating that, in order for a parallelism judgmentto be feasible for S, the free-standing dots are firstunconsciously scaled in depth to the same extent asthe transversal segment being evaluated. The dotjudgments will show no orientation error because bothtransversal and dots are subject to identicalperturbations in depth, and will look equal inorientation when they are physically equal. AlthoughYarbus (1967, p. 204) has demonstrated aPoggendorff-like eye-tracking error in the presence ofa parallel, eye movements are not required to obtain aPoggendorff effect (Pritchard, 1958). The efferent­readiness theory of Festinger, Burnham, Ono, andBamber (1967), and especially its modifications(Virsu, 1971; Robinson, 1972), are interestingproposals but ones difficult to submit to empiricaltest.

We suggest that, since the perceptual anomalyinvolved in the Poggendorffeffect is a misjudgment ofcollinearity, cognitive mistracking is a fitting label fora theory. The label does not deny a possible role foreye movements in the development of Poggendorfferrors. But, since eye movements are not necessary toobtain a Poggendorff effect with adults, mistrackingis presumed to be a cognitive activity. The label doesnot deny that contour-interaction processes are atwork. Indeed, a critical stimulus feature for aPoggendorff effect is an obtuse angle produced by theactual intersection of transversal and parallel. Thelabel does imply, however, that contour interactionsultimately disturb higher-order stimulus processing.The mistracking interpretation is quite compatiblewith Pressey's assimilation theory, which he employedto account for the Poggendorff effect (Pressey, 1971;Pressey & Sweeney, 1972). We cannot bring ourselvesto endorse that theory, because it is based upon animprecisely specified hypothetical construct, theattentive field. Contradictory predictions seem to beobtainable by modifying the locus, size, or shape ofthe attentive field.

Pinning down the surplus meaning conveyed by thelabel, cognitive mistracking, is a goal for the future.For the present, we can claim only to have added tothe evidence differentiating among broad classes oftheory. Our prediction is that, given the diverseindependent variables already known to be operating,the ultimate solution to the Poggendorff puzzle willnot be captured in an elegant theory grounded on asingle principle.

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(Received for publication October 24, 1973;revision received April 15, 1974.)