Abstract
Drawing on work done by Helmholtz, I argue that Reid was in no position to infer that objects appear as if projected on the inner surface of a sphere, or that they have the geometric properties of such projections even though they do not look concave towards the eye. A careful consideration of the phenomena of visual experience, as further illuminated by the practice of visual artists, should have led him to conclude that the sides of visible appearances either look straight, in which case their angles appear to approximate Euclidean measures, or their angles do not appear to approximate Euclidean measures for straight line figures, in which case their sides do not look straight.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
von Helmholtz (1925) 159.
Reid (1785a). Hereafter cited as Inquiry with chapter and section numbers. Pagination and line numbering are as in the critical edition.
I henceforth drop the qualification “immediately.” The visible appearances and visible figures Reid was concerned with are appearances in immediate visual perception, so unless otherwise specified, all references to perception or to the appearance or the look of things are to immediate perception.
AUL MS2131 5i18.
Unless otherwise specified, figures are offered only as aids to understanding what is being described. They should not be taken to constitute evidence for the claims they illustrate. Any projection on a flat surface runs the risk of begging the question against someone who wishes to maintain that the immediate objects of vision are seen as if projected on a concave surface. And many of the diagrams include projective lines that appeal to acquired perceptual abilities, further begging the question against someone who wants to make a claim about immediate visual experience.
One of Reid’s sources, Smith (1738), seems to have thought otherwise. See book 1, ch. 5, plate 20 figure 271 (misprinted 171), printed between pp. 54 and 55, and the corresponding explanation over p. 63. But this isolated example aside, an internet search for 18th century paintings and drawings of landscapes or seascapes or sunsets shows that visual artists of the time represented the horizon is a straight line. According to Reid (Inquiry 6.3: 81–82), they are the ones most qualified to describe the character of what immediately appears to us in vision.
A similar point has been made by Meadows (2009) 417, though Meadows applies the point in a different context and without reference to a perceiver’s sense of motion.
This is not to say that the person would perceive depth or distance outwards in any direction. Perceiving fore and aft differences to the right and the left does not entail perceiving them to the front and the back and perceiving right and left differences to the front and back does not entail being able to perceive them on either side. Imagine a being surrounded by a cylindrical screen. Being able to see forward images as disposed to the left and the right and sideward images as disposed to the front and the back does not by itself entail being able to perceive how far away the screen is in any direction.
I say “closed curve” rather than circle because, following on the previous note, the being would have no conception of how far out any point on the curve is, and so no commitment to their all being equally far out.
Porterfield drew on experiments conducted by Scheiner (1619).
Because the integration is governed by an innate law, the integration would be a feature of immediate visual perception and not acquired.
For human beings, corresponding points are the center points of each eye and points at same distances in same directions from the center points. Reid considered the empirical evidence to readily establish that in “perfect human eyes” images focused on corresponding points are in fact seen to be in the same place (Inquiry 6.13: 136/28–137/9). The bulk of his efforts were devoted to establishing that this is due to an innate law of our constitution rather than being a consequence of experience or of the physical joining of retinal nerves (Inquiry 6.19).
This is “because such a constitution in them could serve no other purpose but to exhibit false appearances” (Inquiry 6.14: 139/22–23).
Reid did not consider either of these accounts to be a mere hypothesis. He maintained that both are based on “analogy” with what he had established for human vision. The analogy is established by considering that nature’s purpose in giving a sense of vision to animals is to enable them to tell the direction in which objects lie, and then suitably modifying the account established for human vision to achieve this purpose in light of the differences between our eyes and those of animals. See Inquiry 6.14: 139/9–12 and 139/24–27.
Reid would have wanted our test subjects to be “Idomenians,” that is, beings who lack any other sense and so would report just on what they immediately perceive. Given that there are no such subjects in reality, we can attempt to control as far as possible for the known factors leading subjects to have acquired perceptions, or make use of thought experiments. We do the best we can, and one possibility we must be open to is that it may turn out that some of these things are not immediately perceived.
von Helmholtz (1925), 172.
This is just one crooked path. There are likewise infinitely many curved paths different from the one I mention. Controversially, I maintain in what follows that there is only one straight path, and it is only half way straight.
Accepting that the longitudinal lines appear this way resolves the violation of continuity that otherwise occurs when we transition from considering the semicircular path taken by the longitudinal line defining the outer periphery of the visual field to considering any of the other suggested paths for longitudinal lines rising just a short distance to the right of the left periphery. Instead of an abrupt change, we have lines that start off being visibly straight at the centre point of the horizon and that become increasingly concave as we move away from that point on either side until, at the periphery, the lines appear as quarter circles.
Making the image so large it cannot all be seen at once is not an option because the question here concerns what is capable of being contained on the visual field of a being with 180 degree peripheral vision.
What I have said would hold true not only of lines marked on the inner surface of a sphere, but of the visual appearance of wires stretched from the zenith to points on the equator of a dome as seen by an eye at the centre of the dome. (These wires would, after all, project onto the inner surface of the dome as longitudinal lines and so would be indistinguishable from the longitudinal lines I have been discussing.) The fact that the wires are stretched taut does not entail that they must appear straight to the eye. Lines do not always appear to have the shapes they really have, and this is the case here. See the following note.
“When we consider a straight line, like the edge of a ruler, say, and try to determine by the eye whether it is really straight or curved, we find … that our judgment will depend on the direction of the eye in the head. When the ruler is held horizontal and too low down [relative to the fixation point] the edge seems to be concave upward; when it is held up too high, it appears to be concave downward.” von Helmholtz (1925), 172.
With apologies to Hannes for applying his work to a point he would likely resist. Hannes was in no position to take the photograph from the ideal position, at the centre of the dome, but even had he been so, a camera that works by projecting an image on a flat surface does not provide the best evidence against someone who believes that we see images projected on a concave surface. I appeal to the picture only as an illustration of what I am trying to describe.
This is a concrete instance of what I meant in my note in Section III referring to controlling for the factors responsible for acquired perceptions. Perhaps the most striking piece of introspective evidence is the checkerboard pattern (von Helmholtz 1925, 181), reproduced here as Fig. 12. If Helmholtz was right that lines that are really straight appear increasingly concave to the center in a peripheral view, then a checkerboard made of lines that are really increasingly convex to the centre ought to appear as a regular grid. And so it does. Helmholtz wanted the diagram to be blown up to the point where the line is 20 cm long, hung on a wall, and viewed with one eye fixated on the centre point from a distance of 20 cm, though I am able to see the lines as straight in a smaller image at closer distance.
Op. cit. In passing they have also noted one possible, though contentious explanation for the convex contours in classical architecture: ancient peoples, being less accustomed to the view of long, straight lines, which do not exist in nature, were more inclined to see obliquely viewed straight lines as concave. Hence Vitruvius wrote that “we must counteract the ocular deception by an adjustment of proportions.” Cited by Oomes (2009, 1286).
This may be one way of understanding the “alarming fallacy” that Van Cleve (2002) 380, has detected in Reid’s argument for a non-Euclidean visible geometry, though Van Cleve associates the fallacy with a different passage.
When justifying the claim Reid said no more than that “mathematicians know” this (Inquiry 6.9: 104/14–15). But mathematicians are not perceptual psychologists, and the question of the apparent size of an angle is a question of perceptual psychology. The mathematical view of spheres, lines, and angles is a view from nowhere, and because we can’t take the view from nowhere, we instead take the view from some ideal perspective, such as when the angle is viewed directly, to the neglect of how it appears when viewed peripherally.
Given that Lucas asked the reader to sketch the corner and that Reid claimed that visual artists are those most qualified to describe visible appearances [Inquiry 6.3: 82–83], it may not be inappropriate to comment on how visual artists sketch the insides of rooms. Visual artists from the most different eras and schools have drawn the lines defining the walls and ceilings of rooms in “one point perspective” by taking the “vanishing point” to be the centre point of a wall, drawing the lines defining the wall as all visibly perpendicular, and drawing the ceiling lines at 135° to these right angles. The angles comprising the back wall of the room in Van Eyck’s 14 century Flemish realist “Arnolfini wedding portrait,” and Van Gogh’s nineteenth century Dutch impressionist “bedroom at Arles,” are alike perpendicular, and the floor/ceiling lines radiating from these perpendicular angles are alike drawn at 135°. But this does not mean that Van Eyck and Van Gogh considered the angles comprising the apparently quadrilateral floor and ceiling to be greater than 360°. It takes four lines and four angles to make a quadrilateral and only two of four ceiling (floor) angles and three of four ceiling (floor) lines appear in their paintings, with two of the lines only appearing a portion of the way back. Van Eyck, forced by his subject to take on the challenge of painting the back portions of the room as well as the front, curved some of his lines and made others crooked, turning the quadrilateral of the ceiling into a hexagon (as seen in the mirror image he painted at the centre of the room), and Van Gough would have been forced to do the same had he taken on the same challenge.
These considerations raise the question of whether it is even intelligible to claim, as Reid did, that the visible lines appear to intersect at an angle that is as large as the angle made by the corresponding great circles really is. Real angles are measured by juxtaposition with protractors or sextants or rulers whereas visible angles cannot be measured with these instruments but at best with visible appearances of these instruments, and those appearances will not yield consistent results when one considers that a six inch ruler will appear to take up different portions of the visual field depending on the distance from which it is viewed. I have chosen not to make anything of this problem in what follows. If the spherical geodesics intersect at right angles, but the angles around the point of intersection of the corresponding visible lines do not appear equal, then I will say that the visible lines appear to intersect at a larger or smaller angle than the real angle. A more precise means of determining equality or divergence will not be called for.
“For further light on why P2 [the claim that the visible angle made by two great circles = the real angle made by those circles] is true, I now present an alternative argument for it.” (2002, 391).
The notations A5, W and Y are reference markers for a larger argument Van Cleve means to prosecute. From A5 Van Cleve goes on to argue that the angles made by the great circles of the sphere and the angles made by any visibly straight lines that coincide with those circles must likewise appear to be as large as the tangent angles really are. These further aspects of the argument can be ignored. The problem is with A5.
All great circles contain the centre point of the sphere, so the radius from the centre point to a point of intersection of any two great circles is a common radius of both circles and marks the line of intersection of the planes containing those circles. Lines tangent to the circles at their point of intersection will be individually contained in the plane of the circle they are tangent to and both contained in the plane tangent to the sphere at that point, and as that plane is perpendicular to the radius from the centre point, and so to the line of intersection of the planes of the great circles, the tangent lines are perpendicular to the line of intersection of the planes.
I say “close to being” because the points Z and E are not taken as being the zenith and due east, but only as being as close to the zenith and due east as they can be while still fitting on the visual field while fixating N. However, to reduce clutter, I drop this qualification in what follows and simply speak of the spherical lines as if they were perpendicular and the cables as if they were really intersecting at 60-degree angles.
This leads to the conjecture that for a grid of lines to be drawn to look straight to a single, stationary eye, increasingly peripheral lines must be drawn convex to the centre, following the ancient architectural principle of Vitruvius alluded to in note 15 above, and giving rise to the checkboard pattern with “pin cushion distortion” depicted earlier from von Helmholtz (1925, 181).
In physical space I can move a rigid rod end-over-end to measure a line, but in visual space the analogue would be something like forming a visual after-image of one interval on the line and then rotating the eye to superimpose that image on another part of the line, which begs the question of whether intervals marked off in this way continue to appear to be the same size when the eye turns the after-image away from them. Being fixed to a part of the visual field, after-images move with the eye and cannot be used to compare different parts of the field. Moving a rigid rod to mark off equal intervals on a tape, and then stretching the tape along the line would similarly beg the question of whether the physically equal intervals marked on the tape are visually equal. Recall that visual equality is defined as what would be judged to be not clearly different in size by test subjects asked to fixate a point in circumstances that control as far as possible for the factors known to prejudice judgments of lateral distance.
It might be objected that Reid intended his lateral distance law to be purely definitional. But this would be to trivialize his position. It is an empirical question whether test subjects asked to mark off equal intervals along a line while fixating a point on that line will mark off intervals that subtend equal angles at the centre of the eye. Taking the distance law to be definitional merely legislates a meaning for terms; it does not decide what test subjects see.
Reid himself suggested that since straight lines in visual space supposedly return to themselves, measurements might be based on “an infinite right line, which … returns to itself, and hath no limits, but bears a finite ratio to every other line” (Inquiry 6.9: 109/25–28) and that a visible figure “will bear the same ratio to the whole of visible space, as the part of the spherical surface which represents it, bears to the whole spherical surface” (Inquiry 6.9: 104–105). But this begs the question of how we determine the ratios in question. On what basis do we judge that two remotely situated sections of the infinite line or the whole spherical surface bear the same proportion to the whole, when the two are not comparable by juxtaposition or superposition?
In Reid’s day, the properties of rubber were only beginning to be investigated (by his antagonist, Priestley). But the case of the diaphanous sheets makes the same point.
Something like this happens with the transformations brought about by the eye motions discussed in previous sections. When raising an eye causes central parts of an equatorial line to drop lower than more peripheral parts, and so take on an appearance of concavity, it is not as if the central parts of the line drop down to reveal previously occluded objects lying behind them. Their surroundings drop down as well, so that same objects continue to stand in same betweenness relations to one another notwithstanding that what previously appeared straight now appears curved.
For details see von Helmholtz (1925) 162–178.
Alberti spoke variously of a “cut” or cross-section of the visual pyramid, a window, and a veil. I.12: 34; I.19: 39; II.31.51. I have been told that Da Vinci attempted to work out a perspective theory for projection onto concave surfaces, but that he gave up because he found the enterprise too complicated.
Thanks to Giovanni Grandi for identifying this manuscript.
So it is false that “if two [visible] lines be parallel, that is, every where equally distant from each other, they cannot both be straight” Inquiry 6.9: 105/18–19.
See Van Cleve 373–375 and 396–401 for discussion of the problem. Recent attempts at a resolution include (Grandi 2006; Wilson 2013; Quilty-Dunn 2013; Nichols 2007, ch. 4), which reference much of the earlier literature. I would be less than forthright not to acknowledge that my own earlier (2004) includes an effort in this same direction (103–108).
Reid was admittedly invested in drawing a distinction between angular or “apparent” magnitude and tangible or “real” magnitude. That distinction plays an important role in his later response to Hume’s “diminishing table” argument for the theory of ideas. See Reid (1785b), Essay II, Chapter 14, 180–183 in the pagination of the critical edition. This work will hereafter be cited as Intellectual Powers. But it is not obvious that Reid’s response to Hume requires that angular magnitude be immediately perceived. And he might have done better to attempt to make his case by appeal to a distinction between real magnitude and magnitude as determined by superposition.
Daniels (1972) esp. 227 and (1974) 47–53, has speculated that Reid wanted to rebut Berkeley’s claim that immediately visual objects are not the objects of geometry. But it is not clear why Reid would have thought it important to make such a point. Daniels’ attempt to link it with Reid’s realism is unconvincing given that (1) Berkeley did not consider the fact that tangibles are the objects of geometry to stand in the way of his idealism; (2) Reid elsewhere declared that the differences between the geometries of visibles and tangibles constitute “one of the strongest arguments by which [Berkeley’s] system is supported” (Intellectual Powers II.19: 224/37–38 but see all of 222/24–225/8); and (3) Reid’s gave unqualified praise of the “force of genius” exerted by Berkeley in discussing what notions an intelligent being who could see but not touch would form of the objects of sense (Intellectual Powers II.10: 140–141). Had criticizing Berkeley’s views on this topic been Reid’s purpose, he would have said so. He was not one to miss an opportunity to digress on how any aspect of his views might rebut the errors of his predecessors.
References
Alberti LB (1435/36) Della Pittura. Modern translation. In: Sinnisgali R (ed) Leon Battista Alberti: on painting: a new translation and critical edition. Cambridge University Press: Cambridge, 2011
Angell RB (1974) The geometry of visibles. Noûs 8:87–117
AUL MS 2131 Aberdeen University Library manuscript, Birkwood collection. http://www.abdn.ac.uk/diss/historic/collects/reid/reid2.htm
Berkeley G (1733) The theory of vision, or visual language, shewing the immediate presence and providence of a deity, vindicated and explained. London: J Tonson. Modern edition in Ayers MR (ed) Philosophical works, including the works on vision. London: Dent, 1975
Daniels N (1972) Thomas Reid’s discovery of a non-euclidean geometry. Philos Sci 39:219–234
Daniels N (1974) Thomas Reid’s inquiry. Stanford University Press, Stanford
Falkenstein L (2004) Reid and Smith on vision. J Scott Philos 2:103–118
Grandi G (2006) Reid’s direct realism about vision. Hist Philos Q 23:225–241
Lucas JR (1969) Euclides ab omni naevo vindicatus. Br J Philos Sci 20:1–11
Meadows PJ (2009) Contemporary arguments for a geometry of visible experience. Eur J Philos 19:408–430
Nichols R (2007) Thomas Reid’s theory of perception. Oxford University Press, Oxford
Oomes AHJ, Koenderink JJ, van Doorn AJ, de Ridder H (2009) What are the uncurved lines in our visual field? A fresh look at Helmholtz’s checkerboard. Perception 38:1284–1294
Porterfield W (1759) A treatise on the eye, the manner and phaenomena of vision, vol. 2. A Millar, London
Quilty-Dunn J (2013) Was Reid a direct realist? Br J Hist Philos 21:302–323
Reid T (1785a) An inquiry into the human mind on the principles of common sense, 4th edition. London: J Bell and W Creech. Critical edition: Brookes DR (ed) University Park, Pa: The University of Pennsylvania Press, 1997
Reid T (1785b) Essays on the intellectual powers of man. London: J Bell. Critical edition: Brookes DR (ed) Edinburgh University Press: Edinburgh, 2002
Scheiner C (1619) Oculus: hoc est, fundamentum opticum. np: Oeniponti
Van Cleve J (2002) Reid’s geometry of visibles. Philos Rev 111:373–416
von Helmholtz H (1925) Helmholtz’s treatise on physiological optics, vol 3. Menasha, Wisconson, Optical Society of America. Electronic edition: University of Pennsylvania, 2001. http://poseidon.sunyopt.edu/BackusLab/Helmholtz/
Wilson K (2013) Reid’s direct realism about visible figure. Philos Q 63:783–803
Wood P (1998) Reid, parallel lines, and the geometry of visibles. Reid Stud 2:27–41
Smith R (1738) A compleat system of opticks. Cornelius Crownfield, Cambridge
Acknowledgments
Thanks to James Van Cleve, Hannes Matthiessen, Giovanni Grandi, and anonymous referees for raising objections that inspired major revisions to this paper, and to J. Michael Nuttall for producing the illustrations. Special thanks to Jan Koenderick, who urged me to consider Helmholtz’s views on this topic, and whose work, with that of his colleagues, has been an inspiration.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Falkenstein, L. Reid’s Account of the “Geometry of Visibles”: Some Lessons from Helmholtz. Topoi 35, 485–510 (2016). https://doi.org/10.1007/s11245-015-9337-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11245-015-9337-0