December 1958 CHARACTERISTICS OF A RETINAL RECEPTOR 911 wellian beam. From this data it is possible to compute the response to a self luminous disk centered on the axis of the receptor and to determine how much the response is amplified by the foam cone. It turns out that a model cone placed at the center of a Fraunhofer image and pointed at the center of the exit pupil yields about as much amplification as if it were exposed to an extended source having the same size, shape, and position as the exit pupil. It may be demonstrated in the case of an extended source in the plane of the pupil, that the amplification of the model cone increases as the size of the source decreases and this parallels what has been found in human vision in connection with the Stiles-Crawford effect. It would be interesting to know in the case of a focused beam what would be the effect of varying the size of the exit pupil. This possibility could have been checked by varying the size of the mirror or the object and image distances, but these things have yet to be done. It is possible that the directional sensitivity of cones might result in an improvement in resolving power when the eye is out of focus or is subject to spherical aberration. This possibility has not been tested. The model receptor has a greater amplification for an extended Maxwellian source than for an extended Fraunhofer source, but no measurement was made of the relative radiance values of the two sources required to produce equal responses. In the study of the response of the model cone to a focused beam the foam cone was placed at several positions in the diffraction pattern and directional sensitivity data were obtained. The direction of maximum amplification shifts as a function of the position of the receptor in the diffraction pattern. The results obtained in this study using two wavelengths for both types of irradiation of the detector indicate that the amplification produced by the ellipsoid is more pronounced with the shorter wavelength. The results indicate that the curves used to represent the Stiles-Crawford effect must vary from wavelength to wavelength and that the amplification that takes place in a cone ought to affect its relative luminosity for the different wavelengths. It was shown that one may treat the individual cone cell as an independent unit if its axis and the axes of neighboring cone cells are parallel. However, when the sides of two adjacent cones are nearly parallel and almost touching each other marked interaction effects occur. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 48, NUMBER 12 DECEMBER, 1958 Analysis of Experiments in Binocular Space Perception ALBERT A. BLANK Institute of Mathematical Sciences, New Fork University, and Mathematics Department, University of Tennessee, Knoxville, Tennessee (Received April 18, 1958) Experiments connected with Luneburg theory as developed by the author are analyzed with the purpose of making explicit their underlying assumptions. In particular, the role of ad hoc assumptions is explored in detail and minimized wherever possible. It is shown that the special assumptions under which much of the experimental work was executed may be considerably broadened thereby indicating how the theory may be more directly founded upon experiment. The principal problem is the determination of the sensory visual transformation between the geometry of the binocular perception and that of the stimulus, and, in particular, the determination of the visual radial distance function. Three principal techniques, the double circumhoropters, the Blumenfeld alleys, and the equipartitioned geodesics are discussed from this generalized point of view. The specific experimental material treated here consists of results obtained by Zajaczkowska, Shipley, and the Knapp Laboratory group at Columbia University. Some of these results appear for the first time. Theoretical material presented for the first time consists most notably of the analysis of the equipartitioned geodesics, the two-point experiments for the determination of Gaussian curvature, and the meta- theoretical discussions of the several experiments. 1. INTRODUCTION W HEN the predictions of a formal mathematical NV theory are to be compared with factual experimental results, as in this analysis, one may treat the theory as closed and not subject to modification in which case the entire deductive structure stands or falls as a whole in the face of any significant experiment. We shall view the theory of binocular space perception as open in the sense that the fundamental assumptions of the theory are not sufficient to specify the system categorically but only up to a point where the completion of the system may be left to properly conducted experiments. For present purposes, we take as fundamental the axiom frame developed by the author.1 In particular, we assume that binocular visual perception ' A. A. Blank, J. Opt. Soc. Am. 48,328 (1958). (a) Sec. 5. (b) Sec.4. 912 ALBERT A. BLANK Vol. 48 may be described in a compact convex metric space of three dimensions for which there exists a unique geodesic (visually straight) segment connecting each arbitrary pair of points. This postulation is strongly founded upon qualitative empirical observations concerned only with length orderings and alignments. Such a postulation is probably the indispensable minimum for the class of observers who can be said to have a well-defined space sense. There are an infinity of conceivable geometries which satisfy the stated basic assumptions. The principal problems are to specify categorically for each observer the intrinsic binocular visual geometry, and to discover the relation or visual transformation between the geometry of the physical stimulus and that of the visual perception. Ultimately, it would be desirable to relate the visual geometry to anatomical and physiological parameters and the theory has already been useful and suggestive in that respect.2 On the basis of simple qualitative tests on a number of observers it is postulated that the binocular visual space is adequately described as a Riemannian space of constant Gaussian curvature.' In order to categorize the intrinsic visual geometry it is then only necessary to determine the sign of the curvature, whether positive (spherical geometry), negative (hyperbolic), or zero (Euclidean). This problem and the problem of determining the visual transformation from physical space into sensory space are left to experiment. The greatest benefit of the systematic postulational development is that it permits analysis of experimental methods in a very general framework of weak assumptions. This is a notable advance over the initial form of the theory which was fettered by special a priori hypotheses. It has become easier to understand the assumptions underlying each empirical technique and to appreciate the character of the ambiguities and certainties in the inferences framed upon the resultant data. Moreover, in designing each experiment, we are freed in large measure from the obligation of deciding beforehand what are the relevant physical parameters in binocular space perception. As yet, no experimental program has made full use of this great freedom, but there can be little question of its eventual value. For the purpose of formal simplicity only we shall here further restrict application of the theory to observers with clinically normal binocular vision. No new difficulties are introduced in principle by the study of observers with anisometropia (binocular asymmetry) and, in particular the formal analysis of aniseikonia (asymmetry due to unequal magnification) appears to be especially simple.2 0') However, for the purpose of determining the visual transformation, each asymmetric observer is a law unto himself, and in order to obtain 2 A. A. Blank, Brit. J. Physiol. Optics 14, 154-169,222-235 (1957). (a) Sec. 4, Horopters of Fixed Radial Distance, etseq. (b) Sec. 5,The Mapping of Physical into Visual Space. some generality and simplicity in this discussion we confine our attention to the putative normal observer. As our principal sources of empirical information we shall utilize the published researches of the Knapp Laboratory of Physiological Optics,3 A. Zajaczkowska,4 and T. Shipley.5 We shall also utilize some hitherto unpublished information from the same sources. 2. SUMMARY OF THE THEORY For the purposes of this analysis we shall touch briefly upon the central points of the theory. More detailed presentations exist elsewhere.3'- It may be well to emphasize since Luneburg does not make it clear and usage differs,* that the subject matter of the theory is binocular vision in a stationary stimulus environment with freedom of fixation, but immobile erect head. Once motions of any kind are allowed, time automatically enters as a significant variable. Presumably, a completely adequate description of visual phenomena would require a space-time metric. To describe even what an observer with immovable head sees in a given stimulus configuration, it would be necessary to supply a record of his eye movements as well as the specification of the stimulus. Yet, as one might expect, if the question were not raised in such generality, the observer arrives in time at a perception which is principally dependent upon the stimulus, and not the exact history of his mode of observation. Time, in a manner of speaking, has been removed as a factor in the situation.t Observation under the condition of a time-stabilized response is perforce the subject matter of the theory because of the obvious demand for experimental practicability. It takes time to fully develop the sense of depth in a given stimulus configuration, and the many references in Luneburg to "immediate" sensations are inappropriate from this point of view. The properties of time-stabilized binocular perception with immobile 3 L. H. Hardy et al. and A. A. Blank et al., The Geometry of Binocular Space Perception (Knapp Memorial Laboratories, Institute of Ophthalmology, Columbia University College of Physicians and Surgeons, New York, 1953). (a) Fig. 25, p. 56. (b) Sec. II, 2b (i), p. 43. Also similar unpublished data of the FourPoint Experiment. (c) Calculated from Table VI, p. 54 and Table VII, p. 59. d. Sec. II, 2b (ii), p. 45. 4 A. Zajaczkowska, J. Opt. Soc. Am. 46,514 (1956). (a) Table IV, p. 523. (b) Fig. 11, p. 521. (c) Fig. 12 (left), p. 522. (d) Table II, p. 517. 5 T. Shipley, J. Opt. Soc. Am. 47, 795 (1957). (a) Figs. 10-12, pp. 810 ff. (b) Fig. 3, p. 807, ELC-OWR and BLC-OWR. 6A. A. Blank, J. Opt. Soc. Am. 43, 717 (1953). * For example, the comments of H. von Schelling [J. Opt. Soc. Am. 46, 309 (1956)] in connection with one of Luneburg's discussions do not apply to the fully formulated theory. t It is natural to wonder about the possibility of framing a theory of time-stabilized observations with moving head. The principal difficulty would be to determine the relevant physical parameters. A space-time theory which permitted motion of the stimulus viewed with immobile head would appear to be the simpler undertaking at this time. A second difficulty is that observations with moving head permit the observer to make judgments on the basis of motion parallax, which is primarily a monocular factor. December 1958 BINOCULAR SPACE PERCEPTION 913 head can easily be described in terms of the elemental assigned to the sagittally forward direction. The geometric relations which serve as a basis for the entire metric describing the visual distance between two development of the theory.' points PI=(rj,,;o) and P 2= (r2,(P2) may then be written Since the experiments considered here have been explicitly for each of the three possible geometries. conducted primarily in the eye level horizontal plane, If the visual transformation from the physical some few in a plane of elevation through the interocular coordinates of the stimulus to the visual coordinates axis, it will be sufficient for the sequel to remain within (r,p) is known, the visual metric gives a complete a plane geometry. characterization of the geometry of the resultant im- The categorical specification of the intrinsic visual pression. For the purpose of describing the visual geometry may be accomplished at one stroke by observ- transformation, Luneburg introduced bipolar coordi- ing the qualitative outcome of a trial of the Blumenfeld nates , defined for any point P in the eye level Alleys experiment. However, this experiment involves horizontal plane. Consider the circle passing through P new assumptions and, therefore, we shall proceed to and the ocular centers L, R (Fig. 1). Such a circle is this end in an epistemologically independent fashion by known as a Vieth-Mueller circle. The coordinate yis the first examining some of the properties of the visual angle at which the ocular axes meet when they cross transformation and then decide the issue between the at P. The Vieth-Mueller circle, itself, satisfies the euclidean, spherical, and hyperbolic geometries by a equation y= const; it is the locus of constant con- simple experiment of an entirely different kind. vergence passing through P. If A denotes the forward An observer acts as though his perceptions originate intersection of the circle with the median, the angle 0, from a single point of regard or egocenter and he usually the bipolar azimuth, is the angle subtended by the arc takes no conscious note of the double ocular source of PA at either eye. The equation O=const describes a his visual information. The perceptual egocenter rectangular hyperbola (Hillebrand) passing through P naturally suggests itself as the origin of a polar coordi- and the homolateral eye with center at the point 0 nate system for visual space.8 In the sensory horizontal midway between the ocular centers. We shall be some- plane through the egocenter, we may choose polar what inconsistent in specifying values of 4)and oyand coordinates r, s, where r describes sensed radial utilize degree measure for 4 and radian measure for y. distance in a properly chosen system of units, and o The use of radian measure for 7 is convenient since denotes sensory azimuth angle, the value =0 being physical radial distance from the observer is then roughly equal to the reciprocal of y multiplied by the distance between the ocular centers. The fundamental visual role of these coordinates in normal balanced A vision (definitely not in aniseikonial(a)) follows from the observation that the hyperbolas O=const correspond very nearly to sensory loci of fixed direction from the egocenter, the so-called radial horopters, and that the circles y= const approximate the loci of fixed distance from the egocenter, the so-called circumhoropters. In other terms, binocular vision transforms a hyperbola q5= const into a polar ray so= const, and a circley= const into an egocentered circle r= const. It may be inferred therefore, that in a fixed stimulus situation there are two numerical functions; ,P = f (.0), r = g (,Y), (2.1) FIG. 1. Bipolar coordinates, -y, . 1W. Blumenfeld, Z. Psychol. 65, 241 (1913). 8 A. A. Blank, "The Luneburg theory of binocular space perception," in Psychology, a Study of a Science, edited by S. Koch, Study I, Vol. 1, (McGraw-Hill Book Company, Inc., New York, 1958); cf. Part III, Sec. A.2. which describe the sensory geometry corresponding to any given stimulus. We expressly avoid the hypothesis that the functions f and g have an invariant form independent of the particular stimulus. The basic assumptions do not guarantee the possibility of arbitrary prolongation of segments beyond the convex hull of a given visual configuration.l 2(b)It may well happen that the extension of a stimulus configuration may change the geometric relations among the points already present and hence, the form of the transformation (2.1). Luneburg assumed explicitly that these relations remain constant and are independent of the particular stimulus configuration. It is important to 914 ALBETR A. BLANK Vol. 48 realize that no assumption of this kind is necessary. It is possible to determine the relations in various configurations by direct experiment. A number of considerations indicate that it is possible to define f(o), independently of the stimulus, by o= +)o, (2.2) where soo is an arbitrary constant.2 6 An over-all additive constant 0o has no effect on the metric relations and it may as well be assumed that )o= O.t If the relation (2.2) is accepted as a special hypothesis, it is unnecessary to assume anything about the functional dependence of r; in particular, we need not even assume r=g(-y). There exist techniques for the determination of r which depend only on (2.2). However, experiments do indicate that the circumhoropters, the physical loci which correspond to the perception of constant distance from the egocenter, are reasonably closely described in the central and paracentral binocular field by the equation y= const. More peripherally, the true circumhoropters appear to be somewhat flatter than the Vieth-Mueller circles. This effect is most marked in the proximal region. However, with this reservation in mind, it is convenient to adopt the hypothesis r=g(-y). 2a. Sign of the Gaussian Curvature On the basis of (2.2) alone it is possible to determine the sign of the Gaussian curvature by means of extremely simple techniques which involve the use of stimuli consisting of two points. Here we discuss two such techniques.§ In one technique, the equilateral triangle experiment, the observer is instructed to set two variable lights PI=(,4)), P2 (2,4)2) so that the visual distances from the two points to himself are equal to each other and to the visual distance between the two points. In effect, the observer is asked to place the egocenter and the two perceived points at the vertices of an equilateral triangle. The instruction is given in the more complicated way in order to assure that the observer carefully compares the lengths involved and is not prejudiced by any conceptions with regard to the vertex angles of an equilateral triangle. The datum measured is the angle = 142-4)1. If the angle is 60°, the geometry is euclidean; greater than 600, spherical; less than 600, hyperbolic. In a trial of this experiment by C. J. Campbell and the author, one observer yielded a mean angular setting of 39.5°, another, 37.8°. While the observers find the subjective comparison of distances from the egocenter to other distances a difficult one, and settings are more scattered than in other experiments, the simplicity of the method has much to commend it. Another simple experiment with two points is the isosceles right triangle experiment. Let a light be fixed at P1= (1,41). The observer is instructed to set a light P2 = (Y2,02) on a perpendicular at PI to the ray P1 , from the egocenter to P1 (Fig. 2) in such a way that the visual separation between P1 and P2 is equal to that between P and the egocenter 0. The geometry is euclidean if the angle 4= I42-4)l is 45°; if the angle is greater, the geometry is elliptic; if less, hyperbolic. No experimental determinations have been made by this technique. The interpretation of an angle defect in these two experiments as a hyperbolic effect is intuitively clear if it is recalled that in a Lobachevskian geometry the sum of the angles of a triangle is less than 1800, but in spherical geometry the sum is in excess. The limited experimental findings above are cited only to show with what simple empirical means it is possible to categorize the visual geometry among the three Riemannian geometries of constant curvature. The finding here is that the Gaussian curvature is negative and this is in agreement with the great preponderance of hyperbolic results by every method. 2b. Visual Metric Let us assume in this discussion that the visual geometry is hyperbolic.11 In that case the metric D(P1 ,P2 ) describing the visual distance between two mPa '2 Vr 0 FIG. 2. Isosceles right-triangle experiment to determine the curvature of visual space. TAccording to Shipley (see reference 5), Balasz and Walker give analytical indications of the reasonableness of the relation p=0 (in a paper to be published). § Unpublished results at the Knapp Laboratory (1952). 11Although this assumption has a restrictive appearance it actually is in no way delimiting. If it is in error, the experimental results will yield either vanishing (Euclidean) or imaginary (spherical) visual lengths. December 1958 BINOCULAR SPACE PERCEPTION 915 points P= (rso), P2 = (r2,s02) is given explicitly in certain units by coshD (P,,P2) = coshri coshr2 -sinhri sinhr2 cos(so2- sV). (2.3) This formula is cognate to the Euclidean law of cosines. Luneburg used a different polar coordinate frame p,5s where 2 r p= ~~tanh-; (2.4) (-K)' 2 K is a constant which is interpreted as Gaussian curvature. Luneburg's representation has the advantage that when p and so are plotted on a euclidean polar frame, the resultant map is a conformal representation of the Lobachevskian plane; that is, angles are the same in the map as in the visual plane. The advantages of the coordinates r, so lie in much greater simplicity of computation and intuitive understanding. Mathematically, the two systems are completely equivalent, the differences are purely formal. In the following, all relations will be written in the r, s system. It is interesting to note for the hyperbolic geometry in contrast to the Euclidean that there is an absolute unit of length and hence that it is possible to calculate the lengths of the sides in the triangle experiments of Sec. 2a from the measured angles. In the equilateral triangle experiment the side length s is given by cosh's= 2 cosecd4. (2.5) In the isosceles right triangle experiment the length r, of a leg is given by coshr,= cot4 (2.6a) and the length r2 of the hypotenuse by coshr2= cosh2ri. (2.6b) 2c. Radial Distance Function The possibility that the form of the function r= g(y) may depend upon the stimulus is an interesting one. Luneburg' originally considered the possibility that there exist certain transformations of the physical stimulus which do not result in changes in the visual perception, but he discarded the idea and it does not recur in his later work.",'"' The existence of these transformations would imply that g(-y) does change with the stimulus. Since the empirical determination of the radial distance function must automatically shed light on this question we need not assume more concerning the function g(y) than can be ascertained by direct experiment and this possibility may be left open. The work at the Knapp Laboratory3 concluded with a direct attack upon the problem of characterizing the dependency of the function g(-y) upon the parameters of the stimulus. Zajaczkowska 4 and Shipley' have not yet reached that stage in their investigations but some of Zajaczkowska's results bear directly upon the problem. 3. METHODS AND RESULTS We shall take up, in turn, each of the three principal techniques for the determination of r(y). In each case we shall examine the underlying assumptions as well as the experimental indications. 3a. Luneburg Double Circumhoroptersl This experiment is performed in two independent stages, the three-point and four-point experiments. In the three-point experiment two points, Po=(-yo,4o) and Pi=(,yo,oi) are fixed on the circle -y='yo. A third point, P2 = (7y1,k2) is variable on an inner Vieth-Mueller circle -y=-y, and is set by the observer so that the distance from P2 to Po is visually equated to the distance from P1 to Po (Fig. 3). The observer's task is repeated for a number of different settings of 01-+P. If it is assumed that the Vieth-Mueller circles are circumhoropters in a homogeneous geometry, then the cosines Under the preceding assumptions the complete geometric characterization of binocular perception for a normal individual is given by specifying the visual radial coordinate r in terms of the physical parameters. In particular, it is convenient for present purposes to accept as a good approximation the special assumption of (2.1) that the radial distance function r in a given stimulus depends only upon the convergence angle y. Most of the experimental efforts have been devoted to the empirical determination of the function r= g(-y), the principal methods being the Blumenfeld alleys, the Luneburg double circumhoropters, and our equipartitioned geodesics. The discussion is devoted primarily to these experiments; other experimental trials in visual mensuration such as the triangles of Sec. 2a or the Helmholtz geodesics yield some limited information concerning the radial distance function. Y= COS(01-0), X= COS(02-00) (3.1a) must satisfy a linear relation y= mx+b. (3.1b) Luneburg" has demonstrated the converse: among the Riemannian geometries, the homogeneous geometries 9R. K. Luneburg, MathematicalAnalysis of Binocular Vision (Princeton University Press, Princeton, 1947). (a) Formula (6.892), p. 72. (b) Sec. 4.6, p. 44f. 10R. K. Luneburg, "Metric Methods in Binocular Visual Perception," in Studies and Essays, CourantAnniversary Volume (Interscience Publishers, Inc., New York, 1948). R. K. Luneburg, J. Opt. Soc. Am. 40, 627 (1950). ¶ The author prefers this name to that of double Vieth- Mueller circles since the theorerical design of the experiment is valid in principle for true circumhoropters only and these are not Vieth-Mueller circles for asymmetric observers and possibly not for balanced observers in the proximal and lateral fields. 916 ALBERT A. BLANK Vol. 48 FIG. 3. Three-point double circumhoropters experiment. of the range <0j*--k<50'. (Zajaczkowska uses a more limited range and probably does not encounter this difficulty.) The value of n' in (3.2b) depends on the choice of the values of 7i' and yo' and may also depend upon changes in the form of the function g(-y) due to differences in the stimulus configuration. In order to couple the two experiments it is necessary to develop some additional hypothesis about the radial distance function. From the epistemological considerations it almost seems better to deal with the insensitive three-point experiment alone and contend with variability in the results.** It is of interest to consider the current hypotheses concerning g(y) in relation to the problem of coupling the two experiments. For certain reasons (discussed below in Sec. 3d) Luneburg' 0 assumed that the coordinate p of formula (2.4) is given in the form p= 2e--Y, (3.3) are the only ones for which the general outcome of the three-point experiment is a linear relation. The experimental results do generally follow a linear pattern up to the limits of measurement (the Knapp Laboratory3 reports angles 1-0o out to about 240, Zajaczkowska 4 to 21°). The values ro=g(yo) and r=g(r,) are determined by b2 cosh2ro= sinhr,=m sinhro. (3.1c) (1 -b)2- "g Analogous relations hold for the other two homogeneous geometries.6 The physical limitations of the three-point experiment do not permit a size match if yi-yo is made too large. It is generally true in these investigations that when the physical parameters are small the experiment is insensitive.(") For that reason the determination by the three-point experiments of the constants in and b is not sufficiently accurate.3 (,) In order to overcome the insensitivity of the threepoint experiment, Luneburg devised the four-point experiment for the determination of tn. Let Po= (o',qo) and P0 *= (yo',ko*) be fixed points on the circle y= yo'. On a smaller circle y= -yl with y'>yo' there are two variable lights, P1 = (yi',01) and P1 *= (1i',c,0*). The observer's task is to equate the visual distance between Pi and P1 * with that between Po and P0* for several different settings of 00*-0. Again, a linear relation is expected, namely, sin (po*-0o) = in' sin (0i*-0j), (3.2a) where a is a personal constant of the observer. In that case the radial distance function will be determined for each observer by the two parameters a. and K. It then becomes possible to calculate the value of a. from the three-point experiment and then, without any restriction on yo', 7yi', to calculate K from the data of the four-point experiment." We are not compelled, however, to make a categorical assumption of this type. We may entertain the more general hypothesis that r=g(y), where g is independent of the stimulus without postulating any special functional form. In that case, the two experiments may be matched by setting o'= yo, 7i'= 7i and it would follow that ro'=ro, r=ri, the slopes n and in' are equal, and hence that the Eqs. (3.1c) may be used to determine the values of r. The author's hypothesis' 6 that visual radial distance depends upon differences in convergence rather than convergence itself would require only that 'y'-yo'=,-y-,o. Trials of the double circumhoropters were made independently by Zajaczkowska"l and by the Knapp Laboratory.3 The experiments at the Knapp Laboratory cover the more extensive conditions, those of Zajaczkowska utilize only one condition (excluding pilot experiments), but for a considerable variety of observers. The Knapp Laboratory trials were executed in two series. The earlier series was based on Luneburg's design using the assumptions of (3.3). The results have received only partial publication.' At the time the data were obtained no attention was paid in calculation to the matching of conditions from the two stages of the experiment. While the data almost uniformly where, setting r'=g(,yl'), ro'=g(o'), we have sinhr1 '= in' sinhro'. (3.2b) In the Knapp Laboratory experiments, limited departures from exact linearity were found in the extremes **It is even possible that the three-point experiment can be made adequately sensitive by using the method of "doubling back," would that is, increase by the taking PI and sensitivity by P2 on the effectively same side of doubling the P0. This range of available angles. 12A. Zajaczkowska, Quart. J. Exptl. Psychol. VIII(2), 66 (1956). December 1958 BINOCULAR SPACE PERCEPTION 917. indicated negative Gaussian curvature they did not yield consistent values of a- and K under differing conditions.tt The results exhibited a significant measure of conformity with the idea that convergence disparity is the significant factor, rather than convergence (see Sec. 3d).3(b) In the second series of experiments, the restriction yo= yo', yl= yl' was imposed on each trial and the value of yo was fixed throughout. As an indicator of the validity of the assumption that the function r=g(,y) is the same in each experimental configuration we may observe whether g(yo) is constant. Values of g(yo) are given in Table I as calculated from the published settings of Knapp Laboratory observers. Despite the variability in the results, the difference between the two observers is evident and it is probably fair to say that g(-yo) is sufficiently constant for each observer to encourage the belief that the experiments are mutually consistent under the stated conditions. Zajaczkowska treats the double circumhoropter experiment by the same methods as the first Knapp Laboratory series and therefore also did not consider the problem of mutual consistency. In Zajaczkowska's work each of the DVMC experiments is performed under the following condition: the values of y are given in the three-point experiment by yo=0.05, yl=0.06, approximately; in the four-point experiment by -yo'= 0.02, yl'=0.06. From the data of these experiments, values of a and K are calculated by Luneburg's formulas. Under Luneburg's hypothesis (3.3) concerning the visual radial coordinate p, the quantities a-and K ought to be constants independent of the choice of yo, y1, yo', yl', but since the experiments are based on only one selection of values of these experimental variables, it is clear that the results obtained cannot verify or disprove Luneburg's hypothesis. The results of certain pilot experiments are cited as indicating that Luneburg's computation of a rests on a satisfactory assumption.' 2 In particular, Zajaczkowska reports linear plots of logp,/po against (y1--yo). This alone is not sufficiently specific since it includes many other possi- bilities such as p=F(,yo)e--Y where F(yo) is completely arbitrary. Perhaps it is also worth remarking that the exponential form is not essential since, as Shipley also notes, many other functional forms fit the data equally well. None of this is meant to preclude the possibility TABLE I. Values of r=g(-yo), yo-0.025, obtained from Knapp Laboratory settings of the double circumhoropters.a that other aspects of the pilot data may be more strongly indicative one way or the other. One other aspect of the matching problem may deserve mention. Some of the values of K obtained in the first series of Knapp Laboratory experiments were more negative than - 1. This result is anomalous in the sense of implying the possibility of perception of sizes and distances greater than infinity. Such results are probably due to the improper matching of conditions in the two stages of the experiment and were not obtained in the matched experiments of the second series. An observer E. K. with K= -1.06 is also reported 2 by Zajaczkowska.tT An alternative method of testing the hypothesis p= 2e70`Y is to use the calculated values of o-and K from the double circumhoropters to predict and compare with the outcomes of other experiments. For the purpose of this comparison, we shall consider in the following, the results of Zajaczkowska's experiments on the Helmholtz' geodesics and the Blumenfeld alleys. 3b. Blumenfeld Alleys The Blumenfeld alleys experiment suffers from the same fundamental weakness as the double circumhoropters in consisting of two independent stages which require the assumption that r=g(y) is the same function in both. Nonetheless, within the frame of this assumption, the Blumenfeld alleys present the most striking demonstration of the curvature of visual space. The Blumenfeld alleys compare the visual perceptions of equidistance and parallelism. In the most commonly executed version of the experiment two lights are fixed on the horizon at points (yobo) and (o, -4o) symmetric to the median. On each of a sequence of smaller ViethMueller circles y=yi, (i= 1, 2, 3, * , ) with ,yo