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  1. Ocular Measurements, Trans. Sect. on Ophthalm. of A.M.A.; 1928. Size and Shape of Ocular Images. I. Methods of Determination and Physiologic Significance, Arch. of Ophthalm. 7, pp. 576–597; 1932. II. Clinical Significance, Arch. of Ophthalm. 7, pp. 720–738; 1932. III. Visual Sensitivity to Differences in the Relative Size of the Ocular Images of the Two Eyes, Arch. of Ophthalm. 7, pp. 904–924; 1932.
    [CrossRef]
  2. Optischer Raumsinn, Bethes Handbuch der Normalen u. Pathologischen Physiologie 12, Part 2, pp. 834–1000; 1930.
  3. A satisfactory description of the neurological arrangement was given by Verhoeff. He states the theory that “there is a separate cortical system for each eye [see a′, a″, Fig. 1] and that connecting the two systems and bringing them into relation with one another there is a third system, [see bFig. 1] composed of a more or less complicated arrangement of neurones, upon which the binocular perception of relief is dependent. This intermediate system may thus be regarded as the cerebral representative of the hypothetical cyclopean eye of Hering and others, and may be spoken of as the cyclopean center, and with it all ideas concerned with binocular relief would be associated It is further assumed that the monocular systems may act independently both as regards each other and as regards the cyclopean center.” (A Theory of Binocular Perspective etc., p. 11, Ann. Ophth., Apr.1902.) Since Dr. Verhoeff advanced this conception of the ocular nervous system as a theory in 1902, knowledge regarding the anatomical relations of those portions of the brain concerned with vision has increased considerably and has not only confirmed the correctness of Verhoeff’s theory but indicates that the cyclopean center is anatomically intermediate in position with relation to the monocular systems. Henschen, Arch. f. Ophthalm. 117, pp. 403, 419; 1926. See also Lloyd Etiology of Strabismus, Archives of Ophthalmology 7, pp. 934–953; 1932.
    [CrossRef]
  4. Nor can the exact position of the retinas be known. Until both the exact position of the path of optical projections and also the position of the retina is known there is no hope of ascertaining the exact anatomical position of corresponding retinal points.
  5. It can also be done monocularly as will be explained later.
  6. This relationship between the apparent size of objects, the size of the image formed on the retina and the extent of the projection fans, tends to be somewhat confusing. If an object forms images on corresponding retinal points of both eyes it will appear the same size to both eyes and both projection fans will have the same extent. If a converging lens is put before one eye it causes a larger image to be formed on the retina. This larger image will excite brain centers that give a subjective impression of more extended space values and the object will appear larger as seen by that eye. The ocular image of that eye can be said to be larger. The effective projection of corresponding retinal points of that eye, however, will have less extent than it did before the lens was put before it for the reasons above stated. Another way of explaining this is, that with the plus lens the nodal point of the combined optic systems of the lens and the dioptric system of the eye is moved forward. This is what causes the object to form a larger image on the retina. Also corresponding retinal points which of course have not changed their position, will, when projected through the advanced nodal point, form a less extended fan.
  7. Panum, Arch, f. O.G. 5, pp. 1–36; 1859, also (Kiel, 1858).
  8. That such is in fact the direction of projection of fused objects will be shown later.
  9. Reference 2, page 894.
  10. Determinations made here by the first criterion, i.e., of “median single vision” are described in Size and Shape of Ocular Images, Part III (reference 1) and later in this paper. It is believed that the findings indicate that this criterion may be somewhat more accurate than it has heretofore been considered.
  11. Tschermak, Pflügers Arch. 204, 177–202; 1924.
    [CrossRef]
  12. Reference 11, p. 185.
  13. See Ocular Measurements, (reference 1).
  14. The inclusion of these mirrors necessitates a correction of the data recorded since an unequal displacement of the apparent positions of the wires takes place across the visual field.
  15. The above described arrangement is a development and improvement of Tschermak’s nonius method. See F. P. Fischer: Fortgesetzte Studien über Binokularsehen. III. Mitt. Pflügers Arch. 204, pp. 234–246; 1924. It is in part similar to Tschermak’s method and in fact resulted from the use of that method. The differences between the two methods as well as the difference in results obtained will be described fully later.
  16. K. N. Ogle, An Analytical Treatment of the Longitudinal Horopter: Its Measurement and Application to Related Phenomena, especially to the Relative Size and Shape of the Ocular Images. (To be published.) It is of importance to know just how accurately the eyes must be positioned. A mathematical study was made of the nature of the variation of the horopter that would result if the eyes were not in the proper position. In general considerable displacement of both eyes equally either forward or back has only a second order effect on the position of the horopter. If however the eyes are not parallel with the frontal parallel plane, for example, if the head were slightly rotated about a vertical axis, a considerable change in the position of the horopter may result. The horopter will take a position as if rotated about a vertical axis the same number of degrees that the head is rotated. The displacement of one eye forward and the other eye back of one mm with a P.D. of 65 mm is equivalent to a rotation of 1.7° which would amount to an error in percentage size between the two eyes of about 0.47 percent at 40 cm, and 0.09 percent at 20 cm, and 0.03 percent at 6 cm. Great care must therefore be taken that the poles of the two corneas lie in a plane parallel to the fixation frontal plane and so remain throughout the determinations. Another type of variation is introduced by lateral displacement of the eyes. Such displacement has a similar effect to that of rotating the head. For a given displacement however the effect is much less, a lateral displacement of 1 mm being equivalent to a rotation of only about 0.12° at 40 cm.
  17. In detail, the method of handling the data is as follows. The empirical data give the average distances in mm from the frontal plane at which the wires on the various lanes are set, see Table 1. When so positioned every wire subtends a particular angle with the lines of fixation of each eye. If the wire is positioned on the Vieth-Müller circle these angles will be equal. If it is on the right-hand side of the field and farther away than the Vieth-Müller circle, the angle subtended to the O.D. will be larger than that subtended to the O.S., etc. The exact difference between the angles can be calculated from the known distance of each wire from the frontal plane and the angle of the particular lane. The formula for determining these relations for each lane, is(y-b)=(1/2P) {S+(S2+4Pa2)1/2}whereP=(tan ϕ)2-[e2+ea2+b2ab] tan (ϕ+2)S=(b2-a2)/band where (y−b) is the distance of a point from the fixation frontal plane, on a lane, ϕ degrees from the median plane taken positively for the right field and negatively for the left, for a difference between the longitudinal angles of 100e percent. b and a are the fixation (distance) and nodal point base line distances, respectively. To avoid the delay of such calculations, transformation graphs for the right and left field are used, see Fig. 28. These graphs are computed from the above formula. Distances nearer and farther than the frontal plane are marked on the abscissa axis, nearer distances minus (−) and farther distances plus (+). Percent differences in size of the angles subtended to each eye are marked on the ordinate axis, plus (+) if the angle subtended to the O.D. is larger than to the O.S., and minus (−) if it is smaller. The slanting lines marked 1° and 2°, etc., represent the lanes. An example of the transformation process is as follows: At 4° in the right field the average setting of the wire in the right field was 1.7 mm behind the frontal plane. Following the 4° lane on the graph for the right field to a point 1.7 mm to the right of the ordinate axis one refers to the ordinate axis and reads +2.1 percent. This indicates that the angle subtended to the O.D. by that particular point is 2.1 percent larger than that to the O.S. This is then plotted on the analytical graph, see Fig. 27, at 4° on the abscissa to the right of the vertical axis and at the 2.1 percent mark on the ordinate above the horizontal axis. The other data are plotted in the same way. Different interpolating graphs must be used for each different distance, and theoretically for every different interpupillary distance. Interpupillary differences of three mm each way introduce such slight variations, however, that they can be neglected. [Cf. reference 16.]
  18. The slope H is usually specified by the ratio of the percent difference of longitudinal angles (expressed in the decimal) relative to the tangent of the peripheral angle of the right eye, for theoretical reasons. The slope expressed otherwise would of course be approximately proportional over the ordinary angles used.

1930 (1)

Optischer Raumsinn, Bethes Handbuch der Normalen u. Pathologischen Physiologie 12, Part 2, pp. 834–1000; 1930.

1928 (1)

Ocular Measurements, Trans. Sect. on Ophthalm. of A.M.A.; 1928. Size and Shape of Ocular Images. I. Methods of Determination and Physiologic Significance, Arch. of Ophthalm. 7, pp. 576–597; 1932. II. Clinical Significance, Arch. of Ophthalm. 7, pp. 720–738; 1932. III. Visual Sensitivity to Differences in the Relative Size of the Ocular Images of the Two Eyes, Arch. of Ophthalm. 7, pp. 904–924; 1932.
[CrossRef]

1924 (2)

Tschermak, Pflügers Arch. 204, 177–202; 1924.
[CrossRef]

The above described arrangement is a development and improvement of Tschermak’s nonius method. See F. P. Fischer: Fortgesetzte Studien über Binokularsehen. III. Mitt. Pflügers Arch. 204, pp. 234–246; 1924. It is in part similar to Tschermak’s method and in fact resulted from the use of that method. The differences between the two methods as well as the difference in results obtained will be described fully later.

1859 (1)

Panum, Arch, f. O.G. 5, pp. 1–36; 1859, also (Kiel, 1858).

Fischer, F. P.

The above described arrangement is a development and improvement of Tschermak’s nonius method. See F. P. Fischer: Fortgesetzte Studien über Binokularsehen. III. Mitt. Pflügers Arch. 204, pp. 234–246; 1924. It is in part similar to Tschermak’s method and in fact resulted from the use of that method. The differences between the two methods as well as the difference in results obtained will be described fully later.

Ogle, K. N.

K. N. Ogle, An Analytical Treatment of the Longitudinal Horopter: Its Measurement and Application to Related Phenomena, especially to the Relative Size and Shape of the Ocular Images. (To be published.) It is of importance to know just how accurately the eyes must be positioned. A mathematical study was made of the nature of the variation of the horopter that would result if the eyes were not in the proper position. In general considerable displacement of both eyes equally either forward or back has only a second order effect on the position of the horopter. If however the eyes are not parallel with the frontal parallel plane, for example, if the head were slightly rotated about a vertical axis, a considerable change in the position of the horopter may result. The horopter will take a position as if rotated about a vertical axis the same number of degrees that the head is rotated. The displacement of one eye forward and the other eye back of one mm with a P.D. of 65 mm is equivalent to a rotation of 1.7° which would amount to an error in percentage size between the two eyes of about 0.47 percent at 40 cm, and 0.09 percent at 20 cm, and 0.03 percent at 6 cm. Great care must therefore be taken that the poles of the two corneas lie in a plane parallel to the fixation frontal plane and so remain throughout the determinations. Another type of variation is introduced by lateral displacement of the eyes. Such displacement has a similar effect to that of rotating the head. For a given displacement however the effect is much less, a lateral displacement of 1 mm being equivalent to a rotation of only about 0.12° at 40 cm.

Panum,

Panum, Arch, f. O.G. 5, pp. 1–36; 1859, also (Kiel, 1858).

Tschermak,

Tschermak, Pflügers Arch. 204, 177–202; 1924.
[CrossRef]

Arch, f. O.G. (1)

Panum, Arch, f. O.G. 5, pp. 1–36; 1859, also (Kiel, 1858).

Bethes Handbuch der Normalen u. Pathologischen Physiologie (1)

Optischer Raumsinn, Bethes Handbuch der Normalen u. Pathologischen Physiologie 12, Part 2, pp. 834–1000; 1930.

Fortgesetzte Studien über Binokularsehen. III. Mitt. Pflügers Arch. (1)

The above described arrangement is a development and improvement of Tschermak’s nonius method. See F. P. Fischer: Fortgesetzte Studien über Binokularsehen. III. Mitt. Pflügers Arch. 204, pp. 234–246; 1924. It is in part similar to Tschermak’s method and in fact resulted from the use of that method. The differences between the two methods as well as the difference in results obtained will be described fully later.

Pflügers Arch. (1)

Tschermak, Pflügers Arch. 204, 177–202; 1924.
[CrossRef]

Trans. Sect. on Ophthalm. of A.M.A. (1)

Ocular Measurements, Trans. Sect. on Ophthalm. of A.M.A.; 1928. Size and Shape of Ocular Images. I. Methods of Determination and Physiologic Significance, Arch. of Ophthalm. 7, pp. 576–597; 1932. II. Clinical Significance, Arch. of Ophthalm. 7, pp. 720–738; 1932. III. Visual Sensitivity to Differences in the Relative Size of the Ocular Images of the Two Eyes, Arch. of Ophthalm. 7, pp. 904–924; 1932.
[CrossRef]

Other (13)

Reference 11, p. 185.

See Ocular Measurements, (reference 1).

The inclusion of these mirrors necessitates a correction of the data recorded since an unequal displacement of the apparent positions of the wires takes place across the visual field.

K. N. Ogle, An Analytical Treatment of the Longitudinal Horopter: Its Measurement and Application to Related Phenomena, especially to the Relative Size and Shape of the Ocular Images. (To be published.) It is of importance to know just how accurately the eyes must be positioned. A mathematical study was made of the nature of the variation of the horopter that would result if the eyes were not in the proper position. In general considerable displacement of both eyes equally either forward or back has only a second order effect on the position of the horopter. If however the eyes are not parallel with the frontal parallel plane, for example, if the head were slightly rotated about a vertical axis, a considerable change in the position of the horopter may result. The horopter will take a position as if rotated about a vertical axis the same number of degrees that the head is rotated. The displacement of one eye forward and the other eye back of one mm with a P.D. of 65 mm is equivalent to a rotation of 1.7° which would amount to an error in percentage size between the two eyes of about 0.47 percent at 40 cm, and 0.09 percent at 20 cm, and 0.03 percent at 6 cm. Great care must therefore be taken that the poles of the two corneas lie in a plane parallel to the fixation frontal plane and so remain throughout the determinations. Another type of variation is introduced by lateral displacement of the eyes. Such displacement has a similar effect to that of rotating the head. For a given displacement however the effect is much less, a lateral displacement of 1 mm being equivalent to a rotation of only about 0.12° at 40 cm.

In detail, the method of handling the data is as follows. The empirical data give the average distances in mm from the frontal plane at which the wires on the various lanes are set, see Table 1. When so positioned every wire subtends a particular angle with the lines of fixation of each eye. If the wire is positioned on the Vieth-Müller circle these angles will be equal. If it is on the right-hand side of the field and farther away than the Vieth-Müller circle, the angle subtended to the O.D. will be larger than that subtended to the O.S., etc. The exact difference between the angles can be calculated from the known distance of each wire from the frontal plane and the angle of the particular lane. The formula for determining these relations for each lane, is(y-b)=(1/2P) {S+(S2+4Pa2)1/2}whereP=(tan ϕ)2-[e2+ea2+b2ab] tan (ϕ+2)S=(b2-a2)/band where (y−b) is the distance of a point from the fixation frontal plane, on a lane, ϕ degrees from the median plane taken positively for the right field and negatively for the left, for a difference between the longitudinal angles of 100e percent. b and a are the fixation (distance) and nodal point base line distances, respectively. To avoid the delay of such calculations, transformation graphs for the right and left field are used, see Fig. 28. These graphs are computed from the above formula. Distances nearer and farther than the frontal plane are marked on the abscissa axis, nearer distances minus (−) and farther distances plus (+). Percent differences in size of the angles subtended to each eye are marked on the ordinate axis, plus (+) if the angle subtended to the O.D. is larger than to the O.S., and minus (−) if it is smaller. The slanting lines marked 1° and 2°, etc., represent the lanes. An example of the transformation process is as follows: At 4° in the right field the average setting of the wire in the right field was 1.7 mm behind the frontal plane. Following the 4° lane on the graph for the right field to a point 1.7 mm to the right of the ordinate axis one refers to the ordinate axis and reads +2.1 percent. This indicates that the angle subtended to the O.D. by that particular point is 2.1 percent larger than that to the O.S. This is then plotted on the analytical graph, see Fig. 27, at 4° on the abscissa to the right of the vertical axis and at the 2.1 percent mark on the ordinate above the horizontal axis. The other data are plotted in the same way. Different interpolating graphs must be used for each different distance, and theoretically for every different interpupillary distance. Interpupillary differences of three mm each way introduce such slight variations, however, that they can be neglected. [Cf. reference 16.]

The slope H is usually specified by the ratio of the percent difference of longitudinal angles (expressed in the decimal) relative to the tangent of the peripheral angle of the right eye, for theoretical reasons. The slope expressed otherwise would of course be approximately proportional over the ordinary angles used.

A satisfactory description of the neurological arrangement was given by Verhoeff. He states the theory that “there is a separate cortical system for each eye [see a′, a″, Fig. 1] and that connecting the two systems and bringing them into relation with one another there is a third system, [see bFig. 1] composed of a more or less complicated arrangement of neurones, upon which the binocular perception of relief is dependent. This intermediate system may thus be regarded as the cerebral representative of the hypothetical cyclopean eye of Hering and others, and may be spoken of as the cyclopean center, and with it all ideas concerned with binocular relief would be associated It is further assumed that the monocular systems may act independently both as regards each other and as regards the cyclopean center.” (A Theory of Binocular Perspective etc., p. 11, Ann. Ophth., Apr.1902.) Since Dr. Verhoeff advanced this conception of the ocular nervous system as a theory in 1902, knowledge regarding the anatomical relations of those portions of the brain concerned with vision has increased considerably and has not only confirmed the correctness of Verhoeff’s theory but indicates that the cyclopean center is anatomically intermediate in position with relation to the monocular systems. Henschen, Arch. f. Ophthalm. 117, pp. 403, 419; 1926. See also Lloyd Etiology of Strabismus, Archives of Ophthalmology 7, pp. 934–953; 1932.
[CrossRef]

Nor can the exact position of the retinas be known. Until both the exact position of the path of optical projections and also the position of the retina is known there is no hope of ascertaining the exact anatomical position of corresponding retinal points.

It can also be done monocularly as will be explained later.

This relationship between the apparent size of objects, the size of the image formed on the retina and the extent of the projection fans, tends to be somewhat confusing. If an object forms images on corresponding retinal points of both eyes it will appear the same size to both eyes and both projection fans will have the same extent. If a converging lens is put before one eye it causes a larger image to be formed on the retina. This larger image will excite brain centers that give a subjective impression of more extended space values and the object will appear larger as seen by that eye. The ocular image of that eye can be said to be larger. The effective projection of corresponding retinal points of that eye, however, will have less extent than it did before the lens was put before it for the reasons above stated. Another way of explaining this is, that with the plus lens the nodal point of the combined optic systems of the lens and the dioptric system of the eye is moved forward. This is what causes the object to form a larger image on the retina. Also corresponding retinal points which of course have not changed their position, will, when projected through the advanced nodal point, form a less extended fan.

That such is in fact the direction of projection of fused objects will be shown later.

Reference 2, page 894.

Determinations made here by the first criterion, i.e., of “median single vision” are described in Size and Shape of Ocular Images, Part III (reference 1) and later in this paper. It is believed that the findings indicate that this criterion may be somewhat more accurate than it has heretofore been considered.

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Figures (29)

Fig. 1
Fig. 1

Schematic representation of the cortical connections and the optical projections of corresponding retinal points.

Fig. 2
Fig. 2

Schematic representation of the eyes in normal binocular vision.

Fig. 3
Fig. 3

Schematic representation illustrating the error introduced by a misplacement of the assumed mean nodal points.

Fig. 4
Fig. 4

Schematic representation of the concept of projection fans to corresponding points.

Fig. 5
Fig. 5

Schematic representation of the cortical connections and the optical projections of corresponding points when those of one eye are less separated than those of the other.

Fig. 6
Fig. 6

Illustration showing that the Vieth-Müller circle is the loci of the intersects of identical projection fans.

Fig. 7
Fig. 7

Illustration of the rotated position of the horopter trace when one of the projection fans has been extended overall.

Fig. 8
Fig. 8

Illustration of the asymmetrical form of the projection fans whose intersects lie on the fixation frontal plane when those fans are equal.

Fig. 9
Fig. 9

Illustration of the projection fans whose intersects lie on the Vieth-Müller circle, both of which are similarly asymmetric.

Fig. 10
Fig. 10

Illustration of the projection fans whose intersects lie in the fixation frontal plane, both of which are asymmetric in the same direction.

Fig. 11
Fig. 11

Schematic representation of a possible anatomical arrangement explaining the fusional areas.

Fig. 12
Fig. 12

Schematic representation of a possible anatomnical arrangement for explaining the directional value associated with fused disparate images.

Fig. 13
Fig. 13

Details of the near horopter apparatus.

Fig. 14
Fig. 14

The nonius method.

Fig. 15
Fig. 15

Plan of the horopter apparatus when grid-nonius method is used.

Fig. 16
Fig. 16

Screens used in the grid-nonius device.

Fig. 17
Fig. 17

Appearance of the wires in the horopter apparatus when adjusted in the nonius method.

Fig. 18
Fig. 18

The appearance of the wires of the horopter apparatus when not adjusted in the nonius method.

Fig. 19
Fig. 19

Photograph of the grid-nonius apparatus. (Rear view.)

Fig. 20
Fig. 20

Photograph of the near horopter apparatus, showing adjustment for 2.5 D., with grid-nonius device in place.

Fig. 21
Fig. 21

Details of the 6.1 meter horopter apparatus

Fig. 22
Fig. 22

Photograph of the distant horopter apparatus.

Fig. 23
Fig. 23

Graphical representation of schematic data in a spatial graph (cf. Tschermak).

Fig. 24
Fig. 24

Illustration of the iso-magnification curves for a fixation distance of 40 cm.

Fig. 25
Fig. 25

Illustration of the iso-magnification curves for a fixation distance of 6.1 meters.

Fig. 26
Fig. 26

Graphical representation of the same schematic data in a spatial graph on which the iso-magnification curves are shown.

Fig. 27
Fig. 27

Graphical representation of the same schematic data in an analytical form.

Fig. 28
Fig. 28

The transformation graphs for the right and left fields for the fixation distance of 40 cm.

Fig. 29
Fig. 29

Method of graphical representation of the monocular multiple partition experiments from which the relationship between the projection lines of the fans can be shown.

Equations (2)

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(y-b)=(1/2P){S+(S2+4Pa2)1/2}
P=(tanϕ)2-[e2+ea2+b2ab]tan(ϕ+2)S=(b2-a2)/b