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Thomas Reid's geometry of visibles.

In a brief but remarkable section of the Inquiry into the Human Mind, (1) Thomas Reid argued that the visual field is governed by principles other than the familiar theorems of Euclid--theorems we would nowadays classify as Riemannian. On the strength of this section, he has been credited by Norman Daniels, (2) R. B. Angell, (3) and others (4) with discovering non-Euclidean geometry over half a century before the mathematicians--sixty years before Lobachevsky and ninety years before Riemann. I believe that Reid does indeed have a fair claim to have discovered a non-Euclidean geometry. However, the real basis of Reid's geometry of visibles is subtle and easy to misidentify. My main aim in what follows is to make clear what the real basis is, separating it from several fallacious or irrelevant considerations on which Reid may seem to be relying. A secondary aim is to air the worry that Reid's case for his geometry can succeed only at the cost of compromising the achievement for which Reid is best known--his direct realist theory of perception.

1. Reid's Direct Realism

In Reid's own opinion and the opinion of many since, the chief merit of his philosophy lies in its opposition to the "common theory of ideas." In place of the theory of ideas, Reid advocated a form of direct realism about perception. Since one of the main claims to be investigated here is that the geometry of visibles threatens to undermine direct realism, it behooves me to say a bit about what is involved in Reid's direct realism.

Hume claimed that the "universal and primary opinion of all men" that they perceive external objects directly is "destroyed by the slightest philosophy, which teaches us, that nothing can ever be present to the mind but an image or perception." He elaborated as follows:
   The table, which we see, seems to diminish as we remove further from
   it; but the real table, which exists independent of us, suffers no
   alteration. It was therefore nothing but its image which was present
   to the mind. (5)


Hume's slight bit of philosophy may be encapsulated in the following syllogism:

1. What I see diminishes in magnitude as I retreat from it.

2. The table itself does not diminish in magnitude as I retreat from it.

3. Therefore, what I see is not the table itself (but only an image or idea).

In his chapter in the Essays on the Intellectual Powers of Man entitled "Reflections on the Common Theory of Ideas," (6) Reid undertakes to refute this argument among others for the theory of ideas. He contends that Hume's premises are true only if we restate them as follows (see EIP 2.14, 224ff.):

1. What I see diminishes in apparent magnitude as I retreat from it.

2. The table itself does not diminish in real magnitude as I retreat from it.

3. Therefore, what I see is not the table (but only an image or idea).

Here Reid is appropriating for his own purposes Berkeley's distinction between tangible and visible magnitude, or as Reid also styles it, real and apparent magnitude. As Reid develops the distinction, the real magnitude of an object (for example, the edge of a table) is an intrinsic property of it, measured in inches or feet, whereas the apparent magnitude of an object is a relation between the object and a perceiver, measured by the angle the object subtends at the eye. It is easy to see that apparent magnitude varies with the distance between object and perceiver (objects subtending smaller angles when further away) while real magnitude does not. Once we record these facts correctly as in Reid's version of the syllogism, we see that the conclusion no longer follows. "I admit both the premises in this syllogism, but I deny the conclusion. The syllogism has what the logicians call two middle terms" (EIP 2.14, 226).

It is important to note that for Reid's purposes, apparent magnitude must be regarded as a relation that is irreducibly dyadic (or perhaps I should say, unexpandably dyadic), involving just the object and the observer (or his point of view). It is not to be analyzed into a triadic relation involving the object, the observer, and an intervening item (for example, an image) with a certain intrinsic size--an item that gets smaller (or gets succeeded by smaller items) as we move away from the table. He must refuse to say that "the table now looks smaller" means "the table now presents me with a smaller image or percept than it did when I was a step closer," since that would take us back to the theory of ideas.

It will be useful in what follows to have a definition of direct perception at our disposal. Though direct perception is a tricky concept that requires further refinement, the following definition by George Pappas will be adequate for our purposes:
   A person S directly perceives an object O at a time t = (1) S
   perceives O at t, and (2) it is false that: S would perceive O at t
   only if S were to perceive R at t, where R [not equal to] O, and
   where R is not a part of O nor is O of R. (7)


Pappas says that clause (2) is supposed to capture the idea of "nondependence on perceived intermediaries": I do not perceive something directly if I perceive it only by perceiving something else. More accurately, I do not perceive something directly if I perceive it only by perceiving something else that is not a part of it. If I perceive an elephant only by perceiving a side of it, I still perceive the elephant directly. But if I perceive Hume's table only by perceiving an image of it (which is not part of it, but something existing only in my mind), I do not perceive the table directly. Indeed, Reid would say that in that case I do not properly perceive the table at all, but only infer its existence. For if I perceived the table as well as its image,
   it would follow, that, in every instance of perception, there is a
   double object perceived: that I perceive, for instance, one sun in
   the heavens, and another in my own mind. But [this] contradicts the
   experience of all mankind. (EIP 2.7, 125)


A definition of direct perception (specifically, of direct seeing) similar in spirit to Pappas's has been proposed by Frank Jackson. It amounts to this: S sees x directly iff S sees x and there is no y other than x such that S sees x by (or in virtue of) seeing y. (8) Jackson is operating with the same basic notion as Pappas--that you see directly those things that you do not see by perceiving other things. But unlike Pappas, Jackson does not try to capture the "by" relation in terms of counterfactual dependence. Jackson's may well be the better course, but for our purposes here, we need not go into the matter further. (9)

2. Reid's Overall Argument for the Geometry of Visibles

According to Reid, Euclidean geometry is indeed the correct geometry for the objects we touch in the space around us. But it is not the correct geometry for those objects that are the immediate objects of sight:
   When the geometrician draws a diagram with the most perfect
   accuracy; when he keeps his eye fixed upon it, while he goes through
   a long process of reasoning, and demonstrates the relations of the
   several parts of his figure; he does not consider, that the visible
   figure presented to his eye, is only the representative of a
   tangible figure, upon which all his attention is Fixed; he does not
   consider that these two figures have really different properties;
   and that what he demonstrates to be true of the one, is not true of
   the other. (Inq. 6.8, 102-3)


He goes on to say, "This perhaps will seem so great a paradox, even to mathematicians, as to require a demonstration before it can be believed" (ibid., 103).

What is the demonstration? Reid's case for the non-Euclidean geometry of visibles may be divided into two parts. First, he argues that because depth cannot be perceived, every visible figure has the same geometrical properties as some spherical figure--that is, some figure drawn on the surface of a sphere. Second, he points out that the geometrical properties of spherical figures are not the properties familiar to us from Euclidean plane geometry. For example, spherical triangles always have an angle sum greater than 180 degrees. The properties of figures on the sphere are in fact precisely those of the corresponding figures in Riemannian or doubly elliptical geometry. Putting the two parts together, we get the result that the geometry of visibles is not Euclidean, but Riemannian.

Let us now examine the two parts of Reid's case, beginning with the geometry of the sphere.

3. The Properties of Spherical Figures

Reid offers as a "small specimen of the geometry of visibles" a list of twelve propositions, each "not less true nor less evident [with regard to visible figure and space] than the propositions of Euclid, with regard to tangible figures" (Inq. 6.9, 105). I comment on several of these propositions below, noting how they come out true if the terms 'right line' and 'straight line' (which Reid uses interchangeably) are taken to denote great circles of the sphere (or segments of such circles).

1. Every right line being produced, will at last return into itself. Trace the equator (or any other great circle on the globe) and you will eventually return to the point from which you started.

5. Any two right lines being produced, will meet in two points, and mutually bisect each other Any two lines of longitude intersect each other at the poles; similarly, any other great circles on the globe intersect each other at two opposite points. Thus, no two straight lines are parallel to each other if parallels are defined in Euclid's fashion as lines that do not intersect. This proposition denies Euclid's parallel postulate in the Playfair version of it, which says that through any point outside a given line there is exactly one line parallel to the original line.

6. If two lines be parallel, that is, every where equally distant from each other, they cannot both be straight. If parallels are defined in terms of equidistance, there are still no parallel straight lines on the sphere. Consider, for example, the equator and any other line of latitude. They are everywhere equidistant, but the line of latitude (being a lesser circle) does not count as straight.

10. Of every right-lined triangle, the three angles taken together, are greater than two right angles. Consider a triangle composed of a stretch of the equator as base and segments of two lines of longitude meeting at the north pole as its legs. There are two right angles at the base, to which we must add the angle at the pole, which may be anything from just over zero degrees to just under 360 degrees. Triangles so composed may have angle sums ranging from just over 180 to just under 540 degrees. Triangles containing no right angles must also have angle sums exceeding 180 degrees (for example, there can be equilateral triangles with each angle equal to 61 degrees).

11. The angles of a right-lined triangle, may all be right angles, or all obtuse angles. The "all right angles" case has been covered above: take two lines of longitude that make right angles with the equator and with each other at the north pole. For the "all obtuse angles" case, consider a short segment of the equator as base and two legs each making obtuse angles with it. In a flat plane legs making obtuse angles with the base would never meet, but on the sphere they may wrap around and meet in an obtuse angle on the other side.

If the foregoing propositions are merely taken to hold of "right lines" interpreted as great circles on the sphere and of figures composed of segments of such lines, there is nothing inherently counterEuclidean about them. In fact, they all belong to the Euclidean geometry of the sphere--a body of truths that was known and used by the Greek astronomers who believed that the stars were embedded in a great sphere rotating around the earth. But if the propositions are taken to hold of genuinely straight lines and figures composed of them, we do get a non-Euclidean geometry; it is precisely the geometry that now bears the name of Riemann.

4. Depth Is Not Perceived

I turn now to the other half of Reid's case--that every visible figure has the same relevant geometrical properties as some spherical figure. Reid's reason for thinking so is bound up with a claim he takes over from Berkeley--that depth is not perceived by the eye:
   Supposing the eye placed in the centre of a sphere, every great
   circle of the sphere will have the same appearance to the eye as if
   it was a straight line. For the curvature of the eye being turned
   directly toward the eye, is not perceived by it. And for the same
   reason, any line which is drawn in the plane of a great circle of
   the sphere, whether it be in reality straight or curve, will appear
   straight to the eye. (Inq. 6.9,103)


The middle sentence from this passage recalls what Berkeley says in the second paragraph of the New Theory of Vision--the only argument he gives in that work for its central presupposition that distance is not perceived by sight:
   It is, I think, agreed by all that distance, of itself and
   immediately, cannot be seen. For, distance being a line directed
   endwise to the eye, it projects only one point in the fund of the
   eye, which point remains invariably the same, whether the distance
   be longer or shorter.


Lateral distance (distance in two dimensions) is perceivable by the eye; the distance between the endpoints of a line viewed broadside is proportional to the angle subtended by that line at the eye. But outward distance (distance out from the eye in three dimensional space, or what Berkeley calls "outness") is not. Whether a point is near or far along a line extending outward from the eye, it will produce the same impression, from which Berkeley and Reid both conclude that the eye is incapable of any depth discrimination. That is why a straight line and a line whose curvature is purely outward from the eye will appear exactly the same. All this is reflected in Reid's definition of "position of objects with regard to the eye":
   Objects that lie in the same right line drawn from the centre of the
   eye, have the same position, however different their distances from
   the eye may be: but objects which lie in different right lines drawn
   from the eye's centre, have a different position; and this
   difference of position is greater or less in proportion to the angle
   made at the eye by the right lines mentioned. (Inq. 6.7, 96)


There is room for debate over just how Berkeley and Reid's argument that distance is not perceived is to be understood and assessed, but I shall not go into that here. Instead, I shall simply assume that Berkeley and Reid are right, for it is of great interest to see if the rest of what Reid says about the geometry of visibles is sound if we concede this fundamental presupposition.

The consequence of the unperceivability of depth that will be important for us is this: every visible triangle is indistinguishable from some spherical triangle. That is, for any triangle that I see, there is a corresponding triangle composed of segments of great circles centered on my eye; the sides of this spherical triangle share endpoints with the sides of the original triangle, but curve out away from my eye. Since this outward curvature is undetectable by the eye, the spherical triangle presents exactly the same appearance to the eye as the original triangle. In this sense, the two are indistinguishable.

To forestall confusion, let me note three points about the relation I am calling "indistinguishability." First, when I call two figures "indistinguishable," I mean that one is capable of exactly occluding the other from view, or that one could be substituted for the other without occasioning any perceptible difference in their spatial properties. A sufficient condition for the indistinguishability of two lines or figures in this sense is that corresponding points of them occupy the same visual position in Reid's sense (that is, they lie on the same outward line from the center of the eye), as when one figure does exactly occlude another. (10) Second, there is a sense in which a perfectly straight line and a line containing minute side-to-side deviations too small for an eye of limited acuity to discern are sometimes called indistinguishable. As is well known, indistinguishability in this sense is not a transitive relation. But this is not the sense of indistinguishability that figures in Reid's argument. The unseen deviations from straightness Reid is talking about lie in the outward dimension that is unseen even by an eye of unlimited acuity. Indistinguishability as it figures in Reid's argument is a transitive relation. Third, Reid tell us that "as the real figure of a body consists in the situation of its several parts with regard to one another, so its visible figure consists in the position of its several parts with regard to the eye" (Inq. 6.7, 96). In other words, the visible figure of any object is determined by the totality of directions in which its various points lie from the eye. Since the same lines of direction may pass from my eye to every point on a tilted circle and to every point on an ellipse placed orthogonally to my line of sight, the tilted circle and the ellipse (when appropriately placed) have or present the same visible figure; they are visually indistinguishable. More generally, objects with very different real figures may be visually indistinguishable (as we shall see later in the case of objects that differ more dramatically than a circle and an ellipse).

5. The Argument of Paragraph 5

Two fundamental facts from which Reid derives his geometry--the properties of spherical figures and the eye's inability to distinguish visible figures from spherical figures--are now before us. They come together in the following paragraph, to which Reid attaches the numeral '5', and which is set forth as though it constitutes the core of his argument for the geometry of visibles:
   5. Hence it is evident, that every visible right-lined triangle,
   will coincide in all its parts with some spherical triangle. The
   sides of the one will appear equal to the sides of the other, and
   the angles of the one to the angles of the other, each to each; and
   therefore the whole of the one triangle will appear equal to the
   whole of the other. In a word, to the eye they will be one and the
   same, and have the same mathematical properties. The properties
   therefore of visible right-lined triangles, are not the same with
   the properties of plain [Reid's variant spelling of 'plane')
   triangles, but are the same with those of spherical triangles. (Inq.
   6.9, 104)


The context makes clear that when Reid says that every visible triangle "coincides in all its parts" with some spherical triangle, the relation he has in mind is not real coincidence, but the relation he elsewhere calls coincidence to the eye. In other words, it is the relation of indistinguishability, or so I shall assume. I shall also select having an angle sum greater than two right angles as a specimen property possessed by spherical but not by plane triangles. It then appears that the argument of paragraph 5--which I shall henceforth call the argument from indistinguishability-may be set out as follows:

1. Every visible triangle is visibly indistinguishable from some spherical triangle.

2. Spherical triangles have angle sums greater than two right angles.

3. Therefore, every visible triangle has an angle sum greater than two right angles.

If that is Reid's argument, however, it commits an alarming fallacy. It has the same form as the following argument:

1. Every visible straight line is visibly indistinguishable from some squiggly line. (Reid implies as much when he says "[A]ny line which is drawn in the plane of a great circle of the sphere, whether it be in reality straight or curve, will appear straight to the eye [placed at the centre of the sphere].")

2. Squiggly lines contain squiggles.

3. Therefore, every visible straight line contains squiggles.

That should make the fallacy obvious enough. To dramatize the difficulty with the argument, however, I now present two more counterexamples to it. (These examples may be overkill for present purposes, but they will be useful for other purposes as we continue.)

The first counterexample is an argument that stands on all fours with the original, but shows that every visible triangle has an angle sum of less than 180 degrees (as happens in Lobachevskian or hyperbolic geometry). Let lines of direction be drawn from my eye to every point of a plane triangle that I see, and let there be interposed between this triangle and my eye a saddle-shaped surface of the kind used to model Lobachevskian geometry. The lines of direction will intercept this surface in a triangle one side of which curves outward and two sides of which curve inward in relation to the eye. This inward or outward curvature being invisible to the eye, the saddle triangle will be visibly as indistinguishable from the plane triangle as any of the spherical triangles that are projections of the plane triangle in Reid's scheme. Yet the saddle triangle has an angle sum of less than 180 degrees. By the argument of paragraph 5 as we are presently understanding it, it should therefore follow that the visible triangle in this situation has an angle sum of less than 180 degrees. (11)

I have just shown that if the original argument from indistinguishability proves visible triangles to be Riemannian, another application of the argument proves them to be Lobachevskian. I show next that a further application of the argument proves the visual field as a whole to be neither Riemannian nor Lobachevskian. Euclidean, Riemannian, and Lobachevskian spaces are all members of the family of homoloidal spaces--that is to say, they are spaces of constant curvature, in which figures can be moved around without distortion. What I am going to show, then, is that the argument from indistinguishability proves space to be nonhomoloidal.

Take a sphere and at one or more points on its surface "pinch up" a bit of it to form a peak or a spike. On each of the peaks, let two straight lines cross at the apex to form four equal angles. By Euclid's definition of a right angle, the angles formed at such intersections will all be right angles, since all are equal to one another. (12) But these angles will clearly be narrower than right angles formed by lines intersecting at smooth places on the sphere. Hence we get a violation of Euclid's Postulate 4, which says that all right angles are equal. Since this is precisely the postulate that affirms that space is homoloidal, the spiky sphere is a nonhomoloidal space.

The next thing to notice is that the substitution of a spiky sphere centered on the eye for the normal sphere that Reid employs in his exposition of the geometry of visibles would occasion no difference in the visible figures presented to the eye. For example, two straight lines that bisect each other will present the same appearance to the eye whether they meet at a smooth place or at a peak. If they meet at a peak they will be receding from the eye as they approach the peak, but this recession will be as undetectable as the curvature possessed by lines crossing at a smooth place on the sphere. More generally, projections on a sphere and projections on a spiky sphere will present exactly similar visible figures to the eye. To see this, consider lines of direction from the eye to every point in some object, and consider surfaces of varying orientation or curvature intercepted by these lines. The real figures of the intercepts may vary considerably, but the visible figures of all will be the same--because in varying the intercepting surface, we do not alter the directions of the lines from the eye.

Reid tells us that "the whole surface of the sphere will represent the whole of visible space ... since every visible point coincides with some point of the surface of the sphere" (Inq. 6.9, 104). But in just the same way, every visible point coincides to the eye with some point on the surface of a spiky sphere. If it were legitimate, then, to conclude that visible space has the geometrical properties of the sphere representing it, it would be equally legitimate to conclude that visible space has the geometrical properties of the spiky sphere representing it. But the sphere and the spiky sphere have incompatible geometries, one being homoloidal and the other not.

The argument of paragraph 5, as we are now understanding it, is seriously deficient, and it is hard to believe that Reid would have offered so patent a fallacy. Our task in the next four sections is to find a better argument for the geometry of visibles.

6. Visibles as Sense Data?

The apparent fallacy in paragraph 5 is the assumption that anything visually indistinguishable from an F thing is itself an F thing. Without this assumption, the argument is invalid; yet the assumption seems plainly false. The first proposal I wish to explore is that the assumption, though not correct in general, works for the special entities Reid calls visibles. Well, then, what sorts of entities must visibles be, if they are to play the necessary role in Reid's reasoning? One natural suggestion is that they are phenomenal entities akin to classical sense data.

To introduce this possibility, I quote from Strawson's discussion of Kant on phenomenal geometry:
   The straight lines which are the objects of pure intuition ... are
   not physical objects, or physical edges, which, when we see them,
   look straight. They are rather just the looks themselves which
   physical things have when, and in so far as, they look straight.
   (13)


Strawson's "looks" are a special kind of phenomenal object, presumably satisfying the central axiom in the classical theory of sense data: if a physical object looks F to a perceiver, it does so by presenting the perceiver with a sense datum that really is F. Sense data themselves are the immediate objects of perception, and they have all (and perhaps only) the properties they appear to have. (14)

Alas, there are two things wrong with trying to salvage the geometry of visibles with "looks," construed as a special kind of phenomenal object. The first, as the reader will immediately surmise, is that the introduction of such objects threatens to jeopardize Reid's direct realism. We would perceive external things (if we perceive them at all) only by perceiving sense data. (15) The second is that the introduction of such objects, contrary to what one may at first suspect, does not suffice to make the argument from indistinguishability valid after all.

On the first point, note how out of character the introduction of special phenomenal objects would be with Reid's response to Hume's argument about the table. In assessing that argument, Reid said that if an object looks so big to an eye placed here, that is a fundamental dyadic relation between the object and the eye. It does not unfold into a triadic relation of Object's presenting Item to Eye. But under the current proposal, Reid would be saying that if a line looks straight to the eye, it does so by presenting to the eye an item that really is straight. That, of course, is the classic sense datum move. It is at odds with a theory of direct perception, because it implies that when a curved straw looks straight, I see the straw only by seeing something that really is straight, and that something must be other than the straw.

The second point is that even if we do construe Reid's visibles as sense data, that is not enough to save him from the fallacy above. The fallacy was embodied in the following principle, which is evidently what must be added to the argument from indistinguishability to make it valid: If x is visibly indistinguishable from y and y is F, then x is F, too. The supposedly saving sense datum suggestion would be that the following qualified version of the principle is true: If x is a visual sense datum visibly indistinguishable from y and y is F, then x is F, too. But the qualified principle turns out to be as false as the original. That may be shown by adapting an example already used above: a straight sense datum may be visibly indistinguishable from a physical line that contains squiggles without containing squiggles itself. So not even our modified principle is true, (16) and it remains fallacious to assume that visible triangles indistinguishable from physical triangles drawn on a sphere would have the same geometrical properties as the spherical triangles.

7. Coincidence as Identity?

Suppose we take literally (as I initially refused to do) Reid's claim in paragraph 5 that every visible triangle coincides in all its parts with some spherical triangle. (17) He would then be implying that every visible triangle is identical with some spherical triangle. The resulting argument would avoid the fallacy we have been discussing, for properties not transferred by indistinguishability are indisputably transferred by identity:

1'. Every visible triangle coincides in all its parts with, and is thus identical with, some spherical triangle.

2. Spherical triangles have angle sums greater than two right angles.

3. Therefore, every visible triangle has an angle sum greater than two right angles.

If real coincidence falls short of identity, as with the statue and the clay, it is nonetheless a strong enough relation to transfer all geometrical properties, so the argument would still reach the desired conclusion.

There are three reasons, however, for not construing Reid's argument as an argument from identity or real coincidence.

First, I think it is clear that the argument just presented is not the argument Reid intended to give. As I mentioned above, it is clear from the context that the "coincidence" he speaks of in the first sentence of paragraph 5 is not real coincidence, but the relation he elsewhere calls coincidence to the eye or apparent coincidence. A plane figure and a spherical figure may be coincident in this sense without being identical or even congruent.

Second, the identity interpretation of Reid's argument raises the question why visible figures should be identified with projections on the sphere, rather than with equivalent (that is, indistinguishable to the eye) projections on some other surface. I pointed out above that there are projections on surfaces other than the sphere (for example, the saddle and the spiky sphere) that yield visible figures indistinguishable from those yielded by projections on the sphere. So what justifies the choice of the sphere, and why identify visible figures with spherical figures in particular?

Daniels mentions two considerations in favor of the sphere. The first is that "the eye sees all points in its visual field (as if they were) equidistant." (18) As he explains further:
   [S]eeing no depth differences between visible points is just like
   seeing all visible points equidistant from the eye. But this is
   equivalent to projecting visible points onto an arbitrary sphere.
   (19)


It is true, of course, that the locus of all points equidistant from a given point is a sphere. But that the eye sees all points "as if" equidistant confers no privilege on the sphere as the surface of projection. The sense in which the eye sees all points as equidistant is simply this: if all points seen by the eye were replaced by points equidistant from the eye but retaining the same visual position as before, everything would look the same. But it is equally true that if all points seen by the eye were replaced by points on the surface of a spiky sphere but retaining the same visual position as before, everything would look the same. So the eye sees all points "as if" placed on a spiky sphere, which could with equal right have been chosen as the surface of projection.

Daniels also notes that the anatomy of the human eye--in particular, its being roughly spherical--seems to have been a motivating consideration in Reid's singling out of the sphere. (20) He says that this consideration makes the choice of a sphere natural and even inevitable, given that any other surface "would violate the symmetry considerations based on the anatomy of the eye" (21) But I doubt that such anatomical considerations can have had anything much to do with Reid's choice. For one thing, he thinks the geometry of visibles is exactly the geometry of the sphere, whereas the shape of the eye is only approximately round. For another, he would have realized that a camera obscura "sees" the same visibles as the eye despite having nothing spherical in its anatomy. I return to this point in section 10.

I come now to the third difficulty with the identity interpretation of Reid's argument. Let us grant that there is a good reason for choosing the sphere as the surface of projection (as I shall in fact propose before we are done). It would still not serve Reid's cause to identify visible figures with spherical figures--at least not if we wish to credit him with discovering a non-Euclidean geometry. With the argument from identity, rather than getting a non-Euclidean geometry of straight lines, we would get instead a Euclidean geometry of curved lines. (22) No one credits the ancient Greek astronomers who worked out the geometry of figures on the celestial sphere with being the first discoverers of nonEuclidean geometry.

8. Angell's Approach

In 1974 R. B. Angell published an essay entitled "The Geometry of Visibles" (see citation in note 3), worked out independently of Reid, but given its title in honor of geid's priority. Like Reid, Angell believes that the visual field has a Riemannian (or doubly elliptical) geometry, and he thinks so for reasons having in good part to do with the fact that the eye does not perceive depth. But Angell makes his case using a strategy different from the strategy I have so far imputed to Reid. Whereas Reid may seem to assign properties to visible figures by letting them "inherit" properties from spherical figures from which they are indistinguishable, Angell assigns properties to visible figures directly, by measurement or in some cases by simple inspection. He thereby avoids the non sequitur that is involved in the argument from indistinguishability. I shall nonetheless raise two difficulties for Angell's approach: that it can succeed only at the cost of an objectionable reification of visibles, and that even with the reification granted, there is reason to question his claims about what can be verified about visible figures by inspection or measurement.

Here is how Angell identifies the domain of visibles about which his geometry is supposed to be true:
   Visibles or visual objects are not the same as what would ordinarily
   be denoted by 'an object which is visible'. Thus, I might say that a
   certain tree was an object which was visible to me at a certain
   time; the "object which is visible" in this case is a physical,
   three-dimensional object, a tree. I might say of the tree that I
   judged it to be about seventy feet tall.... But I might also say
   that the tree appears to be no larger than my thumb appears when
   held at arm's length. The appearances thus compared and found equal
   are the visual objects, or visibles, in our present sense. (23)


Angell thus believes that when the tree and my thumb appear to be of the same length, there exist a tree-appearance and a thumb-appearance that are of the same length. That is the classical sense datum move. It is all of a piece with saying that as I retreat from the table, I see appearances that are successively smaller--even though the table itself "suffers no alteration." If so, I do not see the table itself, and direct realism is lost.

But let us grant for now the reification of visibles and see why Angell thinks their geometry is non-Euclidean. He lists a number of nonEuclidean theorems that he claims may be verified by careful measurement of visibles or in some cases by simple inspection of them. Here are two of the propositions supposedly verifiable by inspection, along with Angell's commentary on them:
   Every pair of straight lines intersects at two points. Imagine
   standing in the middle of a straight railroad track on a vast plane.
   The visual lines associated with the two rails are demonstrably
   visually straight in every segment--they appear perfectly straight,
   not curved, visually. Yet these visually straight lines meet at two
   points which are opposite each other on the horizon, and they
   enclose a substantial region on the visual field. (24)

   Two straight lines, cut by a third straight line perpendicular to
   both, always intersect. The two rails, both appearing visually
   straight, are cut by the straight edge of the railroad tie at our
   feet, and this tie is perpendicular visually, to both of them; yet
   the two visual rails intersect twice. (25)


I note that Angell's second example gives us in the bargain triangles with angle sums greater than 180 degrees: two base angles with 90 degrees each at our feet (where the rails cross the tie), and an apex angle with some positive magnitude at the horizon (where the rails appear to converge).

Now the striking thing about the example of the railroad track is that the purportedly non-Euclidean configuration it involves is never given in a single view. There is an appearance composed of two straight railappearances meeting at the horizon; there is also an appearance of two straight rail-appearances making right angles with a tie-appearance; but these appearances are never combined in a single view. As developed so far, Angell's geometry is a geometry of appearances that never appear.

It would be all right if the appearances of which Angell's geometry holds do not appear, just so long as they nonetheless exist. So he needs a principle assuring us that certain total appearances exist even though nothing more than various alleged parts of them are ever given to us at once. He recognizes the need for some such principle in the following passing remark: "We will therefore speak of a person's total visual field as that expanse which includes all possible continuous extensions of lines or regions in his momentary visual field." (26) But how exactly would we formulate a principle that guarantees the existence of a more-than-180-degree triangle composed of the appearance of right angles I get if I look at my feet and the appearance of an apex angle I get if I look toward the horizon? It would have to be a principle allowing us to identify extensions of the lines that meet at the horizon with extensions of the lines that cross the tie at our feet. I am skeptical whether there is any acceptable principle that will fill the bill.

Another purported example of an appearance that is inspectably non-Euclidean is described by J. R. Lucas, whom Angell quotes as follows:
   Let the reader look up at the four corners of the ceiling of his
   room, and judge what the apparent angle at each corner is; that is,
   at what angle the two lines where the walls meet the ceiling appear
   to him to intersect each other, If the reader imagines sketching
   each corner in turn, he will soon convince himself that all the
   angles are more than right angles, some considerably so. And yet the
   ceiling appears to be a quadrilateral. From which it would seem that
   the geometry of appearance is non-Euclidean. (27)


It is, of course, a theorem in Riemannian geometry that quadrilaterals contain more than four right angles, since every quadrilateral is composed of two triangles, and the triangles contain more than 180 degrees each. The question is whether the visual field really contains such quadrilaterals. Lucas's ceiling, like Angell's fie-and-track triangle, is evidently one of those appearances that never appears (in toto).

Angell is on safer ground, it seems to me, insofar as he rests his case on the measurement of visible figures that we can take in at a single view--for example, triangles and quadrilaterals that take up only a small portion of the visual field. Such figures are not noticeably nonEuclidean. As Reid points out, any figures small enough to be seen "distinctly and at one view" are approximately ("very nearly, although not strictly and mathematically") Euclidean (Inq. 6.9, 106). That is part of his explanation of why the alleged non-Euclidean character of visibles is so easy to overlook. But if inspection will not tell us that small figures are non-Euclidean, Angell tells us, careful measurement will. To measure a visible angle, fix a protractor in front of your eyes with its angles aligned with the angle in the visible; do that for each of the angles in a visible triangle and you will obtain a sum greater than 180 degrees.

Is he right about this? I find myself that I cannot measure small visible figures with any precision. Too much depends on how I hold the protractor or how I cock my head--problems that do not affect the measurement of stable lines on paper--and I seldom get the same result twice. So far as I can tell by measurement, visible triangles might contain 180 degrees, or 175 or 185.

To summarize the results of sections 5 through 8, I have made three main points. First, Reid's argument (or the only one we have so far identified, the argument of paragraph 5) for the non-Euclidean character of visibles rests on a fallacy--the fallacy of transferring properties from spherical figures to any figures indistinguishable from them. Second, Angell's argument does not rest on the same fallacy, but is still problematic. It must rest either on dubious claims about the existence of figures that are never given in one view or on dubious claims about the results of measuring the properties of figures that are given in one view. Third, even if these claims were correct, Angell's approach yields a non-Euclidean geometry only if we reify visibles in a way that jeopardizes direct realism.

9. The Argument of Paragraph 4

Can we find a better argument for Reid's geometry of visibles than any we have considered so far? I believe we can. Reid's paragraph 5, on which I have so far concentrated the search, is meant to follow in part from paragraph 4. Let us back up one paragraph and see whether we can find anything there that bolsters Reid's case. (28)

Here is paragraph 4, with labels inserted to mark its conclusion (C), its two main premises (Pl and P2), and one auxiliary premise (A1):
   (C) That the visible angle comprehended under two visible right
   lines, is equal to the spherical angle comprehended under the two
   great circles which are the representatives of these visible lines.
   For since (A1) the visible lines appear to coincide with the great
   circles, (Pl) the visible angle comprehended under the former, must
   be equal to the visible angle comprehended under the latter. But
   (P2) the visible angle comprehended under the two great circles,
   when seen from the centre, is of the same magnitude with the
   spherical angle which they really comprehend, as mathematicians
   know; therefore (C) the visible angle made by any two visible lines,
   is equal to the spherical angle made by the two great circles of the
   sphere which are their representatives. (Inq. 6.9, 104)


The main argument of this paragraph may be set out as follows:

P1. The visible angle made by any two visible straight lines = the visible angle made by the two great circles representing these lines. (When visible angles are spoken of, they are always angles as seen from a certain point of view. I often leave it implicit, as Reid does, that the point of view is the center of the sphere containing the great circle representatives.)

P2. The visible angle made by two great circles (when seen from the center of the sphere containing them) = the real angle made by these great circles.

C: The visible angle made by any two visible straight lines = the real angle made by the two great circles representing them. (29)

The conclusion follows from the premises by the transitivity of equality. The questions we need to address are these: Why are the premises true? And how does the conclusion contribute to Reid's overall argument?

Why is P1 true? Reid gives us half of the answer, namely A1: any two visible straight lines appear to coincide with two great circles. The other half of the answer, which Reid thought obvious enough to go unstated, is A2: if the angle-making lines 1 and 2 appear to coincide respectively with the angle-making lines 3 and 4, then the visible angle made by 1 and 2 is equal to the visible angle made by 3 and 4. A1 and A2 seem entirely unobjectionable, and the two together yield Reid's P1.

Why is P2 true? Reid gives no argument for P2; he simply says that it is something that "mathematicians know." What exactly did he have in mind? I am not sure, but I conjecture that he would have approved of the following argument, which in any case satisfies me that he is entirely correct in affirming P2:

A3. The real angle made by two great circles (X) = the plane angle made by lines tangent to them at their point of intersection (Y). (This premise is true by standard mathematical convention. In general, one measures the angle between curves by measuring the angle between lines tangent to the curves at their point of intersection and lying in the same plane with them.)

A4. The visible angle made by two great circles (Z) = the visible angle made by their tangents (W). (This follows from A2 and something very like Reid's A1: the great circles appear to coincide with their tangents.)

A5. The visible angle made by two such tangents (W) = the plane angle made by ("really comprehended by") the tangents (Y). (This is true given our assumption that the tangents are viewed from the center of the sphere containing the great circles, since in that case one's line of sight will be orthogonal to the plane of the tangents.)

P2. The visible angle made by two great circles (Z) = the real angle made by these great circles ("the spherical angle which they really comprehend") (X).

As before, the conclusion follows from the premises by the transitivity of equality. I have inserted the letters X, Y, Z, and W to make the transitivity easier to track.

For further light on why P2 is true, I now present an alternative argument for it. In this argument, as in the one just given, we need to consider two kinds of angle--visible angle, or the angle between two lines as seen from a certain point of view, and real angle, or what Reid calls "the angle really comprehended" by two lines. (Perhaps we should speak instead of the real magnitude of an angle and its visible magnitude from a certain point of view, but for brevity I shall often speak simply of real and visible angle.) The visible angle made by two lines AB and AC as seen from e may be equated with (or measured by) a certain dihedral angle--namely, the angle made by the two planes eAB and eAC. (30) The visible angle made by two curved lines may be equated with the dihedral angle of the planes containing them. The real angle made by two lines, if they are straight, is simply the plane angle made by them; if they are curved, it may be equated with the plane angle made by the two lines tangent to the curves at their point of intersection and lying in the same planes as the curves. (This is a generalization of the mathematical convention referred to above in justification of A3.) If we adopt the further convention that straight lines are identical with their own tangents, we may save words by saying that the real angle of two lines is their tangent angle. Thus, visible angle equals dihedral angle and real angle equals tangent angle. Now in the case of angles between segments of great circles seen from the center of the sphere, the dihedral angle and the tangent angle are the same. Hence the visible angle of two such segments and their real angle are the same--which is just what Reid's P2 says. (31)

How does conclusion C help ? How does conclusion C contribute to the overall case for the geometry of visibles? I believe it does so by making possible the following argument. First, for every visible triangle, there is a spherical triangle that is indistinguishable from it--a triangle whose sides and angles coincide visually with the sides and angles of the visible triangle. This is what Reid recapitulates in the opening sentences of paragraph 5. Second, each visible angle made by lines in the visible triangle is equal to the real angle made by the great circle representatives of these lines in the spherical triangle. This is conclusion C from paragraph 4. Third, as we know from the geometry of the sphere, the angles in a spherical triangle always add up to more than 180 degrees. Put these three premises together and it follows that the visible angles in any visible triangle add up to more than 180 degrees. Q.E.D.

Alternatively, as I find more perspicuous, we could make the overall argument using P1 and P2 from paragraph 4 rather than C:

1. Every visible triangle is indistinguishable from some spherical triangle, and therefore (by P1) has its visible angles equal to the visible angles in the spherical triangle.

2. The visible angles in a spherical triangle equal its real angles (from P2).

3. The real angles in a spherical triangle add up to more than 180 degrees.

4. Therefore, the visible angles in a visible triangle add up to more than 180 degrees.

With that argument, we have apparently arrived at last at a valid argument for a non-Euclidean geometry of visibles.

10. The Real Basis of the Geometry of Visibles

We are now in a position to understand the true basis of the geometry of visibles, disentangling it from several fallacies and irrelevancies.

Not an argument from indistinguishability. Contrary to the impression given by paragraph 5, Reid is not giving the argument from indistinguishability. He is not saying that visible triangles are indistinguishable from spherical triangles and for that reason alone share their geometric properties. That would be fallacious for all the reasons belabored above--that visible triangles are also indistinguishable from plane triangles, saddle triangles, and triangles drawn on all manner of exotic surfaces.

In what sense spherical figures represent visible figures. What may lead the reader astray on the previous point is that Reid says that visible figures have the geometrical properties of the spherical figures that represent them. (32) If the reader supposes that a visible figure may be represented by any figure indistinguishable from it, it may then appear that Reid is giving the argument from indistinguishability. But in fact, for a figure x to represent a visible figure y, it is not sufficient that x be indistinguishable from y. (Spherical triangles are indistinguishable from visible triangles, but so are appropriately chosen plane triangles, saddle triangles, and so on.) There is a further necessary condition on representation: figure x represents visible figure y only if the apparent magnitudes of angles in y are equal to the real magnitudes of angles in x. (33) This further condition is satisfied by spherical figures (as Reid's argument in paragraph 4 shows), but not by figures on the other surfaces of projection I have discussed. For example, it is not satisfied by figures on the spiky sphere. Two lines that bisect each other at a peak on the spiky sphere will make apparent angles (for an eye looking up from the center) equal to right angles, but they will make real angles very much narrower than right angles.

Perhaps surprisingly, not even plane figures meet the further necessary condition on representation. Consider a 90-90-90 triangle on the sphere, consisting of one quarter of the equator and legs running from its endpoints up to the north pole. Connect the vertices of this triangle by three really straight lines. To the eye looking out from the center of the sphere, the resulting plane triangle will be indistinguishable from the original spherical triangle (in the sense that one perfectly occludes the other), but it will be a 60-60-60 triangle rather than a 90-90-90 triangle. Reid's claim is that the visible triangle presented to the eye by either of these indistinguishable real triangles will have apparent angle magnitudes equal to the real angle magnitudes in the spherical triangle and not to those in the plane triangle. That is what we learn from paragraph 4.

How can the visible angles in the visible figure presented by the 6060-60 plane triangle not have 60 degrees? The answer is that when one views any of the 60-degree angles from the position of the central eye, one will be viewing it obliquely rather than head on, and in that case it will appear to have more than 60 degrees. This of course is why the angles in Lucas's ceiling appear from the center of the room to be obtuse, even though a carpenter's square would fit snugly in each corner.

What is special about the sphere? Now we can say what is special about the sphere as a surface of projection. That spherical figures are indistinguishable from visible figures confers no special privilege on them as representatives, for the same is true of appropriately chosen plane figures, saddle figures, and so on. But spherical figures are different from the other figures in this respect: among figures indistinguishable from a given visible figure, spherical figures alone have their real angle magnitudes equal to the apparent magnitudes of the angles in the visible figure.

It might be thought that there are plane angles that will represent the angles in any visible figure just as well as spherical triangles. If I trace the angle made by two edges of my ceiling on a flat sheet of plastic held normal to my line of sight, I will obtain a plane angle that meets both conditions of representation--it is visually coincident with the visible angle and its real magnitude equals the apparent magnitude of the visible angle. True enough. But if you compute the angle sum of the visible ceiling by adding up the angles in their planar representatives, you will get more than 360 degrees, just as Reid says. So this exception, if it is one, does not lead to unReidian results. Moreover, we do not really have an exception here to the italicized claim in the preceding paragraph. There is no one figure that can contain all four representative planar angles. If we want a single figure (as opposed to a system of four uncombinable plane angles) to represent the properties of a visible quadrilateral, the most obvious candidate is a spherical figure. (Are spherical figures the only candidates? I discuss that question in the Appendix.)

Angell and Lucas revisited. I complained above against Lucas that no ceiling ever appears in a single view as a quadrilateral having four obtuse angles, and I questioned whether we are entitled to assume the existence of non-Euclidean appearances if they never appear. (34) I also questioned whether figures small enough to be seen in one view are measurably non-Euclidean, as Angell contends. Reid can sidestep both objections, for he need not claim that visible figures are either inspectably or measurably non-Euclidean.

Regarding the first objection, Reid would concede that no figures that appear to the eye are ever noticeably non-Euclidean. He notes that when a triangle "is so small as to be seen distinctly at one view, and is placed directly before the eye, ... its three angles will be so nearly equal to two right angles, that the sense cannot discern the difference" (Inq. 6.11, 118-19, eliding several lines; see also Inq. 6.9, 106). Regarding the second objection, Reid need not claim that we could discover visible figures to be non-Euclidean by measurement. His case for the non-Euclidean character of visibles is the argument we have set forth in the previous section, drawing on the combined resources of paragraphs 4 and 5. The argument is a theoretical argument showing that the angles in any visible triangle are not "strictly and mathematically" equal to the angles in any Euclidean plane triangle, but are equal to the angles in a spherical triangle instead. It is an argument that applies to any visible triangle, even if the triangle is too big to be seen at once, too small to be noticeably non-Euclidean, or too evanescent to be accurately measured.

The relevance of the spherical eye. Contrary to the impression formed by a number of his commentators (for example, Daniels and Pastore), (35) Reid's case for the geometry of visibles has nothing to do with the roundness of the human eye. I supported this contention above by noting that a camera obscura "sees" the same things we do without having any spherical component. We are now in a position to go into this point more deeply.

What is it that the camera "sees"? Nothing, of course, if we take 'seeing' literally. But let us assume for expository purposes that the camera obeys a variant of the law of visible direction, which Reid takes to govern human vision. Reid says that any point in an object is seen in the direction of a straight line running from the point of retinal stimulation back through the center of the eye and into the environment. (36) The variant for a camera obscura would be that it "sees" any point in the direction of a straight line drawn from the image of the point on its rear wall back through the pinhole. Under this assumption, the visible figure of what the camera sees will be unaffected by the curvature of its rear wall--it will be the same regardless of whether the surface on which the image is cast is flat, hemispherical, or curved like a funhouse mirror. That is because the directions of all lines from image points through the pinhole will be unaffected by these variations--and Reid tells us that visible figure is uniquely determined by the totality of all such directions (which in the case of the eye pass through its center rather than through the camera's pinhole).

Of course, the real figure of the image will vary with the curvature of the surface on which it is cast: a quadrilateral projected on a flat retina or camera wall will have an angle sum of 360 degrees while one projected on a hemispherical surface will have an angle sum greater than 360 degrees. But the real figure in the rear of the camera is not what the camera sees, and the image painted on the retina is not what we see. Reid is at pains more than once to point out that we do not see retinal impressions. (37)

The relevance of retinal contour to visible figure seems to be just this: because of the happy accident that the retina is roughly hemispherical, it is correct for Reid to observe that the projection on a sphere that determines visible figure "is the same [species of] figure with that which is projected upon the tunica retina in vision" (Inq. 6.7, 95). If the retina were flat, Reid could not make this claim, for the real angles in the retinal image would not equal the apparent angles in the visible figure or the real angles in its spherical projection. But he could still make the same case for the geometry of visibles, for whether the retinal image is flat or curved, it will determine (in accordance with the law of visible direction) the same visible figure. (38)

The geometry of visibles is the geometry of the single point of view. This idea is implicit in the explanation I give above of why the sphere is special, but let me now be more explicit about it. If you view each of your ceiling corners from directly beneath it, you will see each corner as a right angle. The sum of the visible angles in your ceiling, as seen successively from these four points of view, will be 360 degrees. But if you insist on viewing all four corners from the same point of view, you will you come up with a sum of visible angles greater than 360 degrees. If the single viewpoint is near the center of the room, all four angles will appear obtuse. If the single viewpoint is directly beneath one of the corners, that corner and the two adjacent corners will appear as right angles, but the diagonally opposite corner will appear obtuse. Wherever you position yourself, so long as you confine yourself to a single viewpoint, the sum of the visible angles is bound to exceed four right angles. (Note that I am saying "single viewpoint" rather than "single view"--it may be necessary for the eye to rotate in order to take in all four angles of the ceiling.) (39)

What happens if we add a second eye? What happens to the geometry of visibles if we add a second eye? This is a question to which I cannot do justice here, but I wish to mark out two issues for further investigation: whether a second eye alters the geometry of visibles by enabling us to perceive depth, and how the concepts of visual geometry are to be defined if there are two eyes or viewpoints rather than one.

The most powerful mechanism of depth perception is retinal disparity, which of course requires two eyes and which was unknown to Reid. (40) If the eye (or rather, the eyes) can perceive depth after all, is Reid's case for the geometry of visibles undermined?

Reid himself certainly connects the geometry of visibles with the unperceivability of depth. In his numbered list of "evident principles" underlying the geometry of visibles, the unperceivability of depth is at the top: "For the curvature of the circle being turned directly toward the eye, is not perceived by it" (Inq. 6.9, 103). Moreover, in a manuscript version of the geometry of visibles he tells us this: "Now if it is allowed that an Idomenian [one of Reid's fictional creatures endowed with sight but not touch] can have no notion of distance or proximity betwixt himself & what he sees, I think after careful examination, their Geometry must be such as Apodemus hath described it." (41) In other words, it must be the non-Euclidean geometry that Reid himself has described; he repeats this claim when he retells the story of the Idomenians at the end of Inquiry 6.9. Finally, it is a key premise in the argument for the geometry of visibles as I have reconstructed it in the preceding section that any visible straight line appears to coincide with (or is visibly indistinguishable from) a great circle centered on the eye. This is the assumption A1 underlying both of the main premises P1 and P2 in the argument of paragraph 4, and Reid rests it squarely on the unperceivability of depth.

And yet it is not obvious to me exactly how Reid's argument would he undermined if we grant that depth can be perceived. There are two main ways of taking the question whether depth is perceived by sight, one epistemological (do we know through vision how far out things are?) and the other phenomenological (do things look to be more or less distant?). Suppose first that depth is perceived in an epistemological sense. Suppose, for example, that as in Abbott's Flatland, an ever-present fog makes more distant objects appear dimmer, in proportion to their distance. We would then be in a position to know that a line curving away from us really was curved (or to interpret it as being a curved line), even though it had the same appearance as a straight line. Would that affect Reid's case for the geometry of visibles? I think not. Reid would still be able to maintain that the visible triangle presented to us by three such lines of varying brightness or dimness has the angle sum of a spherical triangle.

Suppose next that depth is perceived in a phenomenological sense, as H.H. Price famously claimed when he said that tomatoes look bulgy. (42) Would that undermine Reid's argument? It is not obvious that it would. For is it not still true that a straight line can perfectly occlude a curved line, as assumed in the crucial assumption A1? It is also worthy of note that our visual sense of depth does not undo various key facts implied by Reid's monocular assumptions--for example, that right angles seen obliquely look acute or obtuse, and that two straight lines, if sufficiently prolonged, appear to converge.

However we resolve the issue whether Reid's argument depends on the unperceivability of depth, there is another way in which adding a second eye threatens to undermine his case. Visible figure, visible magnitude (of lines), and visible angle are all defined by reference to a point of view. The visible angle made by lines AB and AC is relative to a point e; we have equated it with the dihedral angle made by the planes eAB and eAC. How are we even to define such notions if there are two viewpoints, e and e'? Although I do not believe this difficulty is insuperable, I lack the space to address it here. (43)

11. Does the Geometry of Visibles Jeopardize Direct Realism?

Angell's way of developing the geometry of visibles explicitly involves positing visibles as entities akin to sense data. Does Reid's argument for the geometry of visibles involve a similar commitment? I shall now offer a prima facie case for answering yes.

Here, slightly reworded, is the reconstruction of Reid's argument we reached at the end of section 9:

1. Visible triangles are indistinguishable from spherical triangles. In particular, for every visible triangle v seen from e, there is a spherical triangle s centered on e such that the apparent magnitudes of angles in v are equal to the apparent magnitudes of angles in s.

2. In a spherical triangle, the apparent magnitudes of the angles (as seen from the center of the sphere) are equal to their real magnitudes.

3. The real magnitudes of the angles of a spherical triangle sum to more than 180 degrees.

4. Therefore, every visible triangle has apparent angle magnitudes that sum to more than 180 degrees. (44)

The first of these premises is what Reid takes to follow from the unperceivability of depth, the second is what he establishes in paragraph 4, and the third is geometrical fact already known to the ancient Greeks. The conclusion implies that visible figures are governed by a non-Euclidean geometry.

Or does it? It is arguable that to obtain a genuinely non-Euclidean geometry of visibles, we must add one more premise:

5. Visible triangles are what they appear to be in relevant geometrical respects. If a visible triangle has angles of certain apparent magnitudes, its angles really are of those magnitudes.

With that premise aboard, we may take one more step:

6. Therefore, every visible triangle has angles whose real magnitudes sum to more than 180 degrees.

The rub, of course, is that premise 5 is none other than the sense datum move. It implies, for example, that if a visible triangle appears to have one of its angles obtuse, that angle really is obtuse.

Why think the sense-datum move essential to Reid's case? For an answer, listen to a direct realist who resists the move: "That triangular piece of wood over there, which I am now directly seeing and which I call a visible, does indeed appear, when seen from a single vantage point, to have angle values that exceed 180 degrees. But it doesn't really have such values; nor is there anything else, a "visible" in your sense, intervening between it and me, that really has such values." In short, without the sense-datum move, the argument stops at step 4, and in that case we do not get any entities that have non-Euclidean properties. We only get entities that appear to have non-Euclidean properties. (45)

To obtain entities that really do have non-Euclidean properties, we must reify visible angles. But if we do that, a version of Hume's table argument against direct realism may be reinstated. When I view a rectangular tabletop from an oblique perspective at one end, its two nearer angles appear acute and its two farther angles appear obtuse. If I reify visible angles, I will say that what I see contains two angles that are acute and two that are obtuse. But then what I see cannot be the tabletop, for all of its angles are right angles.

12. What Are Visibles?

'Figure' can be either an object word, as in 'he drew a figure on the blackboard', or a property word, as in 'these two objects have the same figure'. Reid uses it both ways, but more often in the former way, and visible figures in the object sense are what he generally means by 'visibles'.

But what exactly are Reid's visibles? As he frames the question himself, "To what category of beings does visible figure then belong?" (Inq. 6.8, 98). The question proves to be a perplexing one, capable of eliciting at least four possible answers: visible figures might be taken to be mental entities, physical entities, abstract entities, or nonexistent Meinongian entities.

Some readers have entertained the suspicion that visible figures are mental entities. Cummins, for instance, says that by introducing visible figure as something distinct from tangible figure, Reid "came perilously close to reintroducing ideas of sense." (46) But this is a suggestion Reid explicitly repudiates. Visible figure cannot be an impression or an idea, he tells us, because "it may be long or short, broad or narrow, triangular, quadrangular, or circular: and therefore unless ideas and impressions are extended and figured, it cannot belong to that category" (Inq. 6.8, 98). Instead, "[T]he visible figure of bodies is a real and external object to the eye" (Inq. 6.8, 101). (47)

When I myself suggested in the preceding section that visibles are like sense data insofar as they satisfy the 'x looks F-x is F' formula that governs sense data, I was not thereby implying that they are mental. Sense data are not necessarily mental entities; they were not construed as such by Moore, Russell, and other classical sense datum theorists. And Frank Jackson, the most notable exponent of sense data in recent decades, explicitly holds that they exist at various distances from the perceiver, rather than being located in a special private space. (48)

If visible figures are external objects, what sort of external objects are they? In one place, Reid seems to identify visible figures with the projections on a sphere we have discussed above:
   Now I require no more knowledge in a blind man, in order to his
   being able to determine the visible figure of bodies, than that he
   can project the outline of a given body, upon the surface of a
   hollow sphere, whose centre is in the eye. This projection is the
   visible figure he wants ... (Inq. 6.7, 95) (49)


In most places, however, Reid refrains from identifying visible figures with projections on a sphere, using the language of representation rather than the language of identity. For example, he says "every visible figure is represented by that part of the surface of the sphere, on which it might be projected, the eye being at the centre" (Inq. 6.9, 104). And in the section that explicitly raises the question where visible figure belongs in the system of categories, he gives this coy answer: "A projection of the sphere ... is a representative in the very same sense as visible figure is, and wherever they have their lodging in the categories, they will be found to dwell next door to them" (Inq. 6.8, 99).

Reid does not explain his refusal to identify visible figures with projections on a sphere, but there are at least two good reasons for it (already familiar to us from section 7). First, though we have now seen why Reid chooses the sphere as his surface of projection, how could he nonarbitrarily identify visible figures with projections on a sphere of one size rather than another? Projections on spheres with the same center but different radii will all represent the same visible figure, but will be numerically different arrays of points. Second, if visible figures are identical with projections on a sphere, they are really curved. The geometry of visibles would simply be the Euclidean geometry of the sphere, and there would be nothing non-Euclidean about Reid's geometry.

These reasons for not identifying visible figures with projections on a sphere are in fact reasons for not identifying visible figures with any particular physical arrays of points whatsoever. If they were arrays of points external to the mind, just where would they be? Not, it seems, at one distance rather than another. Indeed, Reid goes so far as to say, "visible figure hath no distance from the eye" (Inq. 6.23, 188). Moreover, if visible figures were points in physical space, would their geometry not by Reid's own account be Euclidean?

This brings us to the third suggestion, which is that visible figures are not concrete arrays of points but abstract equivalence classes. Daniels suggests that Reid's visible points are really classes of points having the same visible position; (50) by extension, visible figures would be classes of arrays of points all possessing the same visible figure in the property sense. But this suggestion has its difficulties, too. Reid insists that visible figure is something extended--that is why he takes it to be nonmental. But classes, as abstract objects, are no more extended than mental entities. Moreover, Reid emphasizes that visible figures are "the immediate objects of sight"--they are things seen (Inq. 6.8, 102, and Inq. 6.9, 105). But I doubt that we ever see classes. In short, equivalence classes are neither figures nor things visible, and that makes them poor candidates to be visible figures.

The last possibility I wish to mention is that visible figures are nonexistent objects, perhaps of a Meinongian sort. (51) It is an important part of Reid's theory of conception that we may conceive of objects that do not exist at all, and that provides a possible fourth status for visibles within his system--they are nonexistent intentional objects. But this proposal runs against the whole tenor of Reid's discourse about visibles, which strongly suggests that they are existing things.

13. Direct Realism and Seeing What We Touch

Berkeley notoriously held that the realms of vision and touch are totally disparate. In explicit opposition to Berkeley, Reid affirms that we see and touch the same things: "When I hold my walking-cane upright in my hand, and look at it, I take it for granted, that I see and handle the same individual object.... I conceive the horizon as a fixed object both of sight and touch" (Inq. 6.11, 119). It is not clear that Reid is entitled to this affirmation, however, for it appears to be at odds with what he says about the geometry of visibles. Discussion of this inconsistency will once again highlight the worry that the geometry of visibles is inconsistent with direct realism.

Before I get to the main inconsistency, I want to dismiss a superficial one. Reid tells us in several places that whereas tangible objects are three-dimensional, visible objects are merely two-dimensional (Inq. 6.9, 106-8, and 6.23, 188) So how can we see and touch the same things? This difficulty is easily resolved. Reid could still hold that visible objects are parts of tangible objects, in the way that a two-dimensional square face is part of a three-dimensional cube, and that we see parts of the very objects we touch. This would fit well with what he says about visible and tangible space:
   [W]hen I use the names of tangible and visible space, I do not mean
   to adopt Bishop Berkeley's opinion, so far as to think that they are
   really different things, and altogether unlike. I take them to be
   different conceptions of the same thing; the one very partial, and
   the other more complete; but both distinct and just, as far as they
   reach. (EIP 2.19, 283)


Now for the deeper inconsistency, which is not so easily removed. Reid apparently affirms each of the propositions in the following inconsistent triad:

1. I sometimes see and touch the same things.

2. The geometry of what I touch is Euclidean.

3. The geometry of what I see is non-Euclidean.

If the objects of sight and touch are governed by different geometries, we are evidently precluded even from saying that I see parts of what I touch. The surface of a triangular block of wood has an angle sum of 180 degrees; any visible triangle has an angle sum of more than 180 degrees; therefore, nothing I see (no visible, at any rate) can ever be the facing surface of a block of wood or of any other tangible object.

We could get around this inconsistency by distinguishing a broader from a narrower sense of 'seeing'. What I see in the narrow sense are just the visibles; what I see in the broader sense includes as well things that are suggested to the mind by visibles, things to a conception of which the mind automatically passes upon being presented with the visible signs of them (compare Inq. 6.8, 101-2). Our triad would then look like this:

1. I sometimes see (in our new broader sense) things that I touch.

2. The geometry of what I touch is Euclidean.

3. The geometry of visibles (of things I see in the narrower sense) is non-Euclidean.

The problem with this suggestion is that it removes the inconsistency only at the cost of abandoning a direct realism of vision. Seeing in the broader sense--the seeing of tables and trees that is mediated by visibles--is not direct seeing if the visibles themselves are what we immediately see. And that is what Reid tells us:
   [In the geometry that mathematicians have been developing for two
   thousand years] not a single proposition do we find with regard to
   the figure and extension which are the immediate objects of sight.
   (Inq. 6.8, 102)

   [T]hose figures and that extension which are the immediate objects
   of sight, are not the figures and the extension about which common
   geometry is employed. (Inq. 6.9, 105)


Reid evidently closes off the option of saying that visibles function as signs in vision without being seen. (52) It would seem to follow that tables and trees are at best mediate objects of sight--that we do not see them directly.

I am not suggesting that Reid lapses back into the way of ideas. His visibles are clearly not ideas or mental entities of any sort--whatever they are, they are supposed to be external to the mind. Nonetheless, he fails to be a direct realist in the Pappas and Jackson sense we have employed above. He is committed to saying that my seeing the table in front of me depends on my seeing something else that is not even part of the table--an object whose geometrical properties are incompatible with those of the table and any of its parts.

14. Direct Realism Restored?

In the hope of reconciling Reid's geometry of visibles with direct realism after all, I now present one last conception of what visibles might be.

Recall that Angell distinguished between visibles, or visual objects in his sense, and objects that are visible. That may have struck the reader as a surprising distinction. The suggestion I wish to make now is that visibles simply are objects that are visible--they include tables, trees, and all the furniture of the earth. Visible figure is a property of such objects--not an intrinsic property, like real figure, but a relational property (or perhaps better, a relativized property), possessed only in relation to a point of view. (53) "As the real figure of a body consists in the position of its several parts with regard to one another," Reid tells us, "so its visible figure consists in the position of its several parts with regard to the eye" (Inq. 6.7, 96). Since all that is relevant about the eye is its location, we may as well say "with regard to a point of view." Elaborating Reid's distinction further, we may say that objects have certain shape properties in themselves (that is, absolutely or nonperspectivally) and other shape properties relative to various points of view. For example, the mouth of a bucket may be round in itself or absolutely, but elliptical to varying degrees from various oblique points of view. A line may be curved absolutely (because there is a dimension of space through which it curves) and yet straight to the eye (if the absolute curvature is turned away from the eye). An angle may be a right angle in itself (or have 90 degrees as its real magnitude), while being acute from some points of view and obtuse from others. Reid thinks that it is ultimately through touch that real or absolute geometrical values are ascertained, but I need not take a stand on that. The proposal I am making is that visibles are none other than the familiar objects around us, and that the geometry of visibles is the geometry that investigates the geometrical properties possessed by these objects from a point of view. Any triangle has an angle sum of 180 degrees absolutely, but a stun greater than that relative to any chosen point of view. The sum will vary depending on the point of view, but will always be more than 180 degrees.

This proposal may sound like the suggestion of our direct realist in section 11, who resisted the introduction of visibles as a special class of entities and insisted that visible objects merely appear to have non-Euclidean properties (or properties that would yield a non-Euclidean object if all combined in the same whole). The four angles in Lucas's ceiling merely look obtuse to the viewer in the center of the room, but are really right angles. In fact, however, I am going just part of the way with that direct realist. I am identifying visibles with ordinary objects, as he does, but in the relativized approach to visible figure I am suggesting, I am not saying that quadrilaterals merely appear to have angles that sum to more than 360 degrees. (That would sound too much like an error theory.) The corner of the ceiling does not merely look obtuse from here--it is obtuse, from here. Compare: the mouth of a soccer goal does not merely look narrow from the sidelines; it is narrow from there, as shown by the greater difficulty of putting the ball in the net from there. (54)

The present way of construing visibles leaves the way clear for a direct realism of vision. What we see are not special visible intermediaries that inherently possess non-Euclidean properties, but ordinary objects that possess non-Euclidean properties relative to our point of view. In the same way, the present approach gives us a way of avoiding the inconsistent triad about the objects of sight and touch that was brought to light in section 13. What we see are indeed the very things we touch. Things we touch (and so of course things we see) have Euclidean properties absolutely. Things we see (and so of course things we touch) have non-Euclidean properties relative to our point of view. There is no inconsistency in that. In fact, this way out of the triad is very like Reid's own way with Hume's table argument: what I see is nothing other than the table, which varies in its apparent or perspectival magnitude while remaining constant in its real or intrinsic magnitude.

Two questions remain to be asked about the conception of visibles as ordinary objects and visible figure as a relativized property of them. Did Reid think of things in this way? And does doing so give rise to a non-Euclidean geometry?

Did Reid think of things in this way? Probably not. He says that visible objects are two-dimensional (Inq. 6.23, 188, among many other places), whereas visibles on the conception I am now proposing are three-dimensional. Throughout Inq. 6.9 he talks of "visible right lines," implying that visible right lines are a species of right (that is, straight) lines, whereas the more apt phrase on the present conception would be "visibly right line," a phrase that may apply to a line that is absolutely curved. (55) Finally, his language tends to suggest two domains of objects--visible objects seen immediately and the tangible objects they suggest to the mind. But no deep doctrine of his philosophy would have prevented Reid from adopting the present conception of visibles, and he would perhaps have welcomed alternative suggestions about where visibles might have their lodging in the categories.

Does the present conception give us a non-Euclidean geometry? If to propound a non-Euclidean geometry it suffices to point out that there are objects that have non-Euclidean properties from any point of view--even though these same objects are Euclidean in themselves--the answer would be yes. But I shall now give reason to question the antecedent of that conditional. What does it mean to say that objects have non-Euclidean properties from a point of view? Here we may note that there is generally an equivalence between facts about the properties an object has in relation to a point p and facts about the nonrelational properties of larger configurations including the object and p. Here is one example:
   Triangle ABC has an angle sum of more than 180 degrees from p
   iff the pyramid with ABC as base and p as apex has more than 180
   degrees as the sum of the three dihedral angles made by its three
   walls. (56)


The right-hand side of that biconditional is a purely Euclidean fact. So how can the left-hand side report a non-Euclidean fact?

If we are to obtain genuinely non-Euclidean results, I think we must make an assumption like the following--as I suspect Reid implicitly does:
   If object O has property F in relation to point p, then if your eye
   were placed at p and directed at O, an object with F would be
   presented to you. (57)


This principle implies that if O is F in relation to p, then there is something that is F simpliciter. For example, if a circle is elliptical from p, an observer with eye at p would be presented with an object that is elliptical, period, If an ordinary triangle has an angle sum of more than 180 degrees from p, then there is a special visible triangle--whatever exactly its ontological status may be--that has more than 180 degrees, period.

We have now arrived at genuinely non-Euclidean facts, but only by introducing special visible objects for them to be facts about. That reinforces the worry I have had from the beginning: that Reid can secure a non-Euclidean geometry only at the cost of abandoning a direct realism of vision.

Brown University

Appendix: What Is Special about the Sphere?

In section 10, I advance the claim that what is special about the sphere as a surface of projection is this: among figures indistinguishable from a given visible figure, spherical figures alone have their real angle magnitudes equal to the apparent magnitudes of the angles in the visible figure. This turns out to be false, but for reasons that leave Reid's geometry of visibles unaffected. In this appendix, I explain both why it is false and why it does not matter.

Given that indistinguishable figures are alike in the apparent magnitudes of their corresponding angles, the claim I have italicized above would follow from the following proposition, once plausibly conjectured as a theorem by Gideon Yaffe:

Yaffe's conjecture: If the visible magnitudes of angles in a figure f (as seen from e) are equal to the real magnitudes of angles in f, then f is a spherical figure. In particular, f is composed of segments of great circles of a sphere centered on e.

Yaffe's conjecture could be proved if we were entitled to use each of the following two lemmas:

Lemma 1: The real angles in a figure equal its apparent (or visible) angles (as seen from e) only when one is viewing each angle head on--that is (roughly), only when the line of sight from e is orthogonal to the plane in which the angle is measured. (58)

Lemma 2: One can view each of the angles in a figure head on from a point of view e only if the figure is a spherical figure, composed of segments of great circles whose center is e. (59)

Together, these lemmas imply that if there is a point of view e such that the real angles in f equal the apparent angles in f as seen from e, then f is a spherical figure.

Unfortunately, however, both lemmas turn out to be false. Subsequent investigation has turned up counterexamples to each of them. (60)

As for Lemma 1, Yaffe has shown how to construct angles in which (surprisingly) the real magnitude and the visible magnitude as seen from e are the same even though the line from e to the plane of the angle is not orthogonal. The construction is too complex to present here, so I simply refer the reader to his article. (61)

As for Lemma 2, here is one counterexample: (62)

In this diagram I have highlighted six edges of a cube to form a six-edged figure and then rounded off three of the corners to obtain a figure with three curved edges. (63) Imagine what this figure would look like to an eye placed at e. (To view the figure properly, assume that the face of the cube containing vertices A, B, and e is frontmost.) The eye would see visibly right angles to the left at A, to the right at C, and overhead at B. It would see the curved lines connecting A with B, B with C, and C with A as straight lines, since their curvature lies in the unseen outward dimension. Thus the Reidian eye would see the three-line system as a triangle. (It would also see the six-line system without rounded corners as a triangle.) Now the eye at e views each of the angles at A, B, and C head on; yet the figure containing them is clearly not a spherical figure. So we have here a counterexample to Lemma 2.

The same figure serves as a counterexample to Yaffe's conjecture. The visible magnitudes of the angles at A, B, and C as seen from e are equal to the real magnitudes of these angles, which are right angles all around; yet the figure containing them is not spherical. The figure also serves as a counterexample to the claim in the first paragraph of this appendix, since it is a nonspherical figure having the same real angle magnitudes as the apparent magnitudes in a visible triangle indistinguishable from it.

Where does all this leave Reid? First of all, even if the claim in the first paragraph is false, that does not harm Reid's case for the geometry of visibles. His case rests on the fact that spherical figures (seen from the center) enjoy equality of real and visible angle magnitudes, but it does not require that spherical figures be unique in this regard, If there are other figures that also enjoy this property (as in the counterexample), that is all right, provided the other figures have the same real angle magnitudes as the relevant spherical figure. And they do. The figure in the diagram is not a spherical figure, but it has the same angle sum--namely, 270 degrees--as the spherical figure Reid would use in assigning an angle sum to the visible figure presented to the eye. (64) So the counterexample leaves intact all of Reid's claims about the angle sums of visible figures. Nor does any other counterexample I know of jeopardize any of Reid's claims about the geometry of visibles. In short, even if some nonspherical figures represent visible figures as well as spherical figures do, the geometry of visibles is still the same as the geometry of sphericals. (65)

Second, although Reid does not claim (or need to claim) that the sphere is unique as a surface of projection for modeling the properties of visible figures, there may yet be a sense in which this is true. What the counterexamples in this appendix show is that for a given visible figure, we can find nonspherical figures that represent it. (Recall how we defined 'figure x represents visible figure y' in section 10: (i) x is indistinguishable from y, and (ii) the apparent magnitudes of angles in y are equal to the real magnitudes of corresponding angles in x.) Yet it may still be true that there is something special about the sphere as a total surface of representation. It may be, or so I conjecture, that the sphere is the only surface with the following property: any figure seen from a given point can be represented on it. In other words:

Further conjecture: If S is a surface such that any figure seen from e can be represented by a figure on S, then S is a sphere centered on e.

If this conjecture is true, then although any visible figure may be represented by some nonspherical figure, that nonspherical figure will not lie on a surface on which all visible figures seen from e may be represented. The sphere alone will contain representatives of all figures seen from a given point of view. Reid says, "The whole surface of the sphere will represent the whole of visible space" (Inq. 6.9, 104); the present conjecture is that no other surface does that.

Notes

I presented earlier versions of this paper at the Second International Reid Symposium (University of Aberdeen, 2000), the NEH Summer Seminar in the philosophy of Thomas Reid (Brown University, 2000), and at philosophy department colloquia at the University of Arkansas, SUNY College at Brockport, Cornell University, the University of Iowa, and New York University. I am grateful to participants in all those occasions. I am also grateful to Panayot Butchvarov, Tim Chambers, Ryan Nichols, Ernest Sosa, Gideon Yaffe, and referees of this journal for their comments on earlier drafts.

(1) Thomas Reid, An Inquiry into the Human Mind on the Principles of Common Sense, ed. Derek R. Brookes (Edinburgh: Edinburgh University Press, 1997). The Inquiry was first published in 1764. The Brookes edition is the latest and best, and my page references are to it. For readers who do not have this edition at hand, however, I shall also give chapter and section numbers. The section entitled "Of the Geometry of Visibles" is chapter 6, section 9, which I shall denote as 'Inq. 6.9'.

(2) Norman Daniels, Thomas Reid's 'Inquiry': The Geometry of Visibles and the Case for Realism (Stanford: Stanford University Press, 1974). For Daniels, there is no doubt that Reid did establish that the visual field is non-Euclidean; the question is only how Reid could have done what he did working (as Daniels believes) in isolation from the mathematical practice of the day. Using Reid's unpublished manuscripts, Paul Wood has shown that Reid was not isolated from relevant mathematical practice--he taught Euclid's Elements as part of his duties at Aberdeen, was in contact with the leading Scottish geometers of the day, and was preoccupied for several decades of his life with proving Euclid's parallel postulate from some other proposition with a greater degree of self-evidence. See Paul Wood, "Reid, Parallel Lines, and the Geometry of Visibles," Reid Studies 2 (1998): 27-41. Wood does not address my main question here: what is the real basis of the geometry of visibles?

(3) R. B. Angell, "The Geometry of Visibles," Nous 8 (1974): 87-117. I discuss Angell's article at length below.

(4) In a Presidential Address delivered to the London Mathematical Society on November 8, 1888 ("On the Confluences and Bifurcations of Certain Theories"), Sir James Cockle said that the name of Reid "ought to head the roll on which will be inscribed the names of Lobatschewsky, Riemann, and other investigators."

(5) David Hume, An Enquiry Concerning Human Understanding, section 12, part 1.

(6) Thomas Reid, Essays on the Intellectual Powers of Man, essay 2, chapter 14, abbreviated hereafter in the style 'EIP 2.14'. My page references are to Baruch Brody's edition of this work (Cambridge: MIT Press, 1969). The Essays were first published in 1785.

(7) George S. Pappas, "Sensation and Perception in Reid," Nous 23 (1989): 155-67, at 156. To simplify exposition, I have omitted one further condition that Pappas includes in clause (2): "and where R is not a constituent or group of constituents of O, nor is O of R." I surmise that Pappas includes this so that a philosopher like Berkeley may qualify as believing in direct perception of everyday objects: I perceive the cherry only by perceiving something distinct from it (for example, a certain tartness or redness), but since the redness is one constituent in the congeries that constitutes the cherry, I still qualify as perceiving the cherry directly.

(8) Frank Jackson, Perception (Cambridge: Cambridge University Press, 1977), chap. 1, especially 19-20.

(9) I discuss strategies for defining direct perception at greater length in "Reid's Theory of Perception," forthcoming in The Cambridge Companion to Reid, ed. Terence Cuneo and Rene van Woudenberg (Cambridge: Cambridge University Press).

(10) Note that my definition says x and y are indistinguishable if one of them could occlude the other--not that either could occlude the other. That is, the definiens is the disjunction 'x could occlude y or y could occlude x', not the conjunction 'x could occlude y and y could occlude x'. This allows that figures of different sizes can be indistinguishable--the smaller could (exactly) occlude the larger from a given point of view though not vice versa. Figures of different colors may also be indistinguishable in the purely spatial sense that is at issue here.

(11) Tom Banchoff has pointed out to me that at least one side of the projection of a triangle on a saddle surface will fail to be a geodesic, in which case the projected figure is not really a triangle. It is nonetheless a figure indistinguishable from the original and having an angle sum of less than 180 degrees, so it stands as a counterexample to the argument from indistinguishability.

(12) "When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right" This is definition 10 in book 1 of Euclid's Elements, translated by Sir Thomas L. Heath (New York: Dover, 1956), vol. 1, 153.

(13) P. F. Strawson, The Bounds of Sense (London: Methuen, 1966), 282. Ironically, Strawson invokes phenomenal objects to secure a domain in which Euclidean geometry holds even if the geometry of the physical world is non-Euclidean. For Reid it is the other way around--physical geometry is Euclidean and visual geometry is non-Euclidean.

(14) Classical sense datum theory always assumed that sense data have all the properties they appear to have, but did not necessarily assume that sense data have only the properties they appear to have. See C. D. Broad, Scientific Thought (Paterson, N.J.: Littlefield, Adams and Co., 1959; reprint of 1923 edition), 244-45.

(15) Does my perceiving the physical thing really depend on my perceiving the phenomenal thing? I discuss this question in "Reid's Theory of Perception."

(16) What would be true is this: If x and y are both sense data and x is visibly indistinguishable from y, then if y is F, so is x. But the spherical figures that play the role of y in Reid's argument are not sense data.

(17) I thank Ed Minar for asking me to consider this possibility further.

(18) Daniels, Thomas Reid's 'Inquiry', 8.

(19) Ibid., 11.

(20) Ibid., 10.

(21) Ibid., 11.

(22) This danger is pointed out by Daniels, ibid., 11-12.

(23) Angell, "Geometry of Visibles," 89-90.

(24) Ibid., 95.

(25) Ibid.

(26) Ibid., 91.

(27) J. R. Lucas, British Journal for the Philosophy of Science 20 (1969); quoted in Angell, at 115. Lucas's passage adds one more sentence omitted by Angell: 'And so it is.'

(28) I am indebted to Tim Chambers for helping me get clear on the crucial role of paragraph 4 in Reid's overall argument.

(29) A visible straight line does not have a unique great circle representative, since a circle of any radius would do. The word 'the' in the formulation of P1 and C should therefore really be replaced by 'any', which could be done without harm to the argument. so For this suggestion I am indebted to Gideon Yaffe. It implies that the visible angles in a triangle as seen from point e are equal to the angles between the walls of the pyramid having that triangle as base and point e as its apex.

(31) In a manuscript at the Aberdeen University Library ("Of the Doctrine of the Sphere," 2131/5/II/36), I find that Reid defines the angle made by great circles as the dihedral angle made by the planes containing them: "The angles [in a spherical triangle] are measured by the inclination of the planes of the great circles under which the angles are comprehended." This definition would yield by a more direct route yet the equality of visible angle with real angle for spherical figures. The manuscript is undated, but it appears to consist of notes for Reid's lectures at Aberdeen, which would have been delivered prior to the publication of the Inquiry in 1764. Thanks to Sabina Tropea for directing my attention to this and several other relevant manuscripts.

(32) Reid uses 'represent' in two disparate senses: spherical figures represent (that is, model the properties of) visible figures, and visible figures represent (that is, signify or suggest to the mind) tangible figures. For instances of the first usage, see Inq. 6.9, 104, throughout. For instances of the second usage, see Inq. 6.8, 103, lines 1-3, and 6.9, 105, lines 36-40. I am currently making a comment on how to understand 'represent' in the first sense.

(33) Actually, it may be that Reid himself operates with a weaker notion of representation that does not incorporate this condition. He concludes that a visible straight line may be represented by a great circle merely from the fact that the two coincide to the eye, or have corresponding points in the same visual position (Inq. 6.9, 103, line 27, through 104, line 6). He similarly concludes that "the whole surface of the sphere will represent the whole of visible space" from the fact that "every visible point coincides with some point on the surface of the sphere, and has the same visible place" (Inq. 6.9, 104, lines 31-33). In these inferences, he is apparently taking indistinguishability as a sufficient condition of representation. In that case, my proposed additional condition should come in not as a component condition of the premise 'x represents y', but as an auxiliary condition for inferring from this premise to 'y has the same relevant geometrical properties as x'. The upshot would be this: we should not say that visible figures have the geometrical properties of any figures representing them, hut only that they have the properties of spherical figures representing them.

(34) Note, by the way, that my complaint against Lucas can be made against an inference of Reid's. In paragraph 5 he says, "The sides of the one will appear equal to the sides of the other, and the angles of the one to the angles of the other, each to each; and therefore the whole of the one triangle will appear equal to the whole of the other." If the whole of the one triangle never appeared at all, the premises of this inference would he true and the conclusion false. Fortunately, however, Reid does not need to affirm the conclusion of this inference, as I am about to explain in the text.

(35) Daniels, Thomas Reid's 'Inquiry', 10-11; Nicholas Pastore, Selective History of Theories of Visual Perception: 1650-1950 (New York: Oxford University Press, 1971), 114.

(36) Reid puts it thus: "[E]very point of the object is seen in the direction of a right line passing from the picture of that point on the retina through the centre of the eye" (Inq. 6.12, 122-23). Reid uses this law to solve the puzzle of how we see objects erect by means of inverted retinal images.

(37) "We have reason to believe, that the rays of light make some impression upon the retina; hut we are not conscious of this impression" (Inq. 6.8, 100). "Nor is there any probability that the mind perceives the pictures upon the retina. These pictures are no more objects of our perception, than the brain is, or the optic nerve" (Inq. 6.12, 121).

(38) I do have to convict Reid of one mistake, however. He says that what a blind man needs in order to determine the visible figure of a body is its projection on a sphere, "for it is the same figure with that which is projected upon the tunica retina in vision" (Inq. 6.7, 95). If I am right, Reid's 'for' should merely be an 'and'.

(39) Question: If the eye may be allowed to rotate, why did I complain above that Angell's tie-and-track triangle is never given in a single view? Answer: Insofar as Angell's argument is a phenomenological one, inviting us simply to "read off" the non-Euclidean character of what is before our eyes, it is important that we have some assurance that there actually are such figures as the tie-and-track triangle. We cannot have such assurance if the triangle is not presented in a single view. Reid is not open to this challenge, since he has a theoretical argument applying to triangles of any size. We can take a small triangle, of whose existence we are assured because we see it, and apply Reid's argument to show that it has an angle sum exceeding 180 degrees.

(40) In Inq. 6.22 Reid identifies five types of depth cue, which according to him enable an acquired perception of depth even though we have no original perception of depth. Of his five, the first four are monocular and the fifth--the angle of convergence between the eyes--is binocular. Reid evidently did not know about retinal disparity, the discovery of which is generally credited to Wheatstone, inventor of the stereoscope, in 1837. However, Reid's theory of corresponding retinal points, which he uses in Inq. 6.13 to explain why we see single with two eyes, is an important ingredient in the retinal disparity explanation of depth perception. On the connection between corresponding points and retinal disparity, see Julian Hochberg, Perception, 2d ed.(Englewood Cliffs, NJ.: Prentice-Hall, 1978), 55-60.

(41) Brookes edition of the Inquiry, 275. This is from one of Reid's Discourses delivered before the Aberdeen Philosophical Society, dated June 14, 1758. As far as I know, this Discourse is the first extant version of Reid's geometry of visibles.

(42) H. H. Price, Perception (London: Methuen, 1932), 3, with further details on 218-21.

(43) Essentially the same problem arises for the classical definition of the visible magnitude of a line as the angle it subtends at the eye: how are we to adapt this definition given that there are two eyes? Psychologists resort to the artifice of the "Cyclopean eye," the midpoint between the two eyes, defining visible magnitude as the angle subtended at this midpoint.

(44) One must take some care in stating 4, in which there is some danger of scope ambiguity. The point is not that visible triangles appear to have angle sums of more than 180 degrees, in which case they would "look non-Euclidean." Rather, they have sums of apparent angles that exceed 180 degrees. Formally, the difference is that between 'v appears to have angle magnitudes x, y, and z such that x + y + z > 180 degrees' and 'there are magnitudes x, y, and z such that (i) x + y + z > 180 degrees and (ii) v appears to have angles of those magnitudes x, y, and z'. 4 does not assert the former, but at most the latter. Even the latter may be in a certain way too strong, suggesting as it does that visible triangles appear to have angles of certain numerical magnitudes. It is rather that visible triangles have angles that appear equal to the angles in a spherical triangle with certain numerical magnitudes. Compare: a line may appear to be the same in length as a line that is two inches long without appearing to be two inches long.

(45) Remember the caveat in the preceding note about how this claim is to be understood.

(46) Phillip Cummins, "Reid's Realism," Journal of the History of Philosophy 12 (1974): 317-40, in n. 56.

(47) In light of Reid's insistence that visible figure is real and external, what he says about the visible figures of the Idomenians is quite mystifying. Lacking any notion of the third dimension because they lack the sense of touch, the Idomenians can conceive of visible figures neither as plane (that is, uncurved through the third dimension) nor as curved. Therefore, Reid says, their visible figures are neither plane nor curved (Inq. 6.9, 108). But what could authorize that inference besides the assumption that the esse of visible figures is concipi? And how could that assumption be true of visible figures if they are real figures external to the mind?

(48) Jackson, Perception, 102-3. Despite this, however, he holds that sense data are mental entities.

(49) The sentence continues as follows: "for it is the same figure with that which is projected upon the tunica retina in vision." I find that some students take this sentence to imply that visible figures are figures on the retina in particular; such may also be the assumption of Pastore (Selective History, 115). But when Reid says that visible figure is the "same figure" as that projected on the retina, I think the sameness he is talking about is just sameness of type, not sameness of token. He tells us unequivocally that visible figure is an "external object to the eye" (Inq. 6.8, 101) ; he also tells us that we do not see retinal images.

(50) Daniels, Thomas Reid's 'Inquiry', chap. 1.

(51) There is a hint of this suggestion in a passing remark by Lorne Falkenstein: "Though Reid does not come out and say it in so many words, he would appear to be forced to admit that our beliefs in visible figures are beliefs in something that does not actually exist in the external world, though they serve as signs for the things that do so exist" ("Reid's Theory of Localization," Philosophy and Phenomenological Research 61 (2000): 305-28, in n. 38).

(52) Other language Reid uses for our cognitive relation to visible figures includes the following: they are "presented to the mind by vision" (Inq. 6.9, 105, line 33); they are "presented to [the mathematician's] eye" (ibid., lines 37-38); they are among "the perceptions which we have purely by sight" (Inq. 6.9,106, lines 26-27). Of course, he tells us repeatedly that we seldom, and only with difficulty, attend to visible figures, but they are nonetheless there before the mind.

(53) For discussion of visible figure as a relational property, see Ryan Nichols, "Visible Figure and Reid's Theory of Visual Perception," Hume Studies 28 (2002): 49-82. For discussion of the distinction between relational and relativized properties (though I was there somewhat skeptical about it), see my Problems from Kant (New York: Oxford, 1999), 248.

(54) The relativized approach to visible figure is similar to the Multiple Location theory discussed by Price in Perception (55-58). How far should we go with such relativized predication? If one post obscures another from a certain point of view, should we say there is only one post from there?

(55) Note also that the present construal of visibles has trouble accommodating Reid's Proposition 1: "Every right line being produced, will at last return into itself." On the present construal, that proposition should be understood as follows: every line that is straight to the eye, when continued so as to make a longer line that is also straight to the eye, at last returns into itself. (A line is straight to an eye at p iff it lies in a plane containing p but does not pass through p.) That is true of some lines and methods of continuation: a segment of the equator is straight to the eye at the center and may be continued round the globe as a line that is straight to the eye and rejoins itself. But it is false of others: a really straight line may be continued infinitely in a really straight line that is straight to the eye all the way and never rejoins itself. So visibles under the present construal of them do not satisfy Reid's Proposition 1. But that may well be a problem with Proposition 1 itself rather than the construal: although Proposition 1 is true if 'right line' is interpreted as 'great circle or segment thereof', I have been unable to find any good construal of 'visible right line' that makes Proposition 1 true, as it is supposed to be, of all such lines.

(56) Recall that a dihedral angle is the angle between two planes, such as the wails pBA and pBC. Except in special cases, this will be broader or narrower than the angle ABC.

(57) Two questions may occur to the reader about this principle. (i) Does the F object exist regardless of whether there is an eye at p? I suspect Reid would say yes. To make this explicit, we could change the right-hand side to 'there is an F object that would be presented to any observer with eye at p and directed at O'. (ii) Would the F object be presented even in a fog or to an astigmatic eye? Here Reid is obviously making the idealizing assumption that what figures are presented to the eye is determined solely by geometrical considerations--the real figure of O and the location of p.

(58) Here, more precisely, is what I mean by viewing an angle head on: If the angle is composed of two straight lines AB and AC meeting at vertex A, then one's line of sight (that is, the straight line connecting e with A) must be orthogonal to the plane containing AB and AC. If the angle is composed of two curved lines 1 and 2 meeting at vertex A, then one's line of sight must be orthogonal to the plane in which the angle is measured--that is, the plane containing the straight lines tangent to 1 and 2 at A. (Recall what was said in the text about measuring angles between curved lines by the angles between their tangents.)

There is an exception to Lemma 1 when we are dealing with right angles. Suppose I am viewing a planar angle whose real magnitude is 90 degrees--for example, the angle between rail and tie. The apparent magnitude of this angle will equal 90 degrees just so long as my eye is either directly above the rail or directly above the tie, even if it is not directly above the point where they meet. So here the condition under which the real magnitude of the angle between AB and AC is equal to its apparent magnitude as seen from viewpoint e may be more relaxed than the head-on condition: e must lie outside the plane of AB and AC and in one of the planes erected upon either AB or AC and perpendicular to the plane containing them. This exception would not affect the use to be made of Lemma 1 provided the following were true: one can satisfy the more relaxed condition with respect to all the angles in a figure only if the figure is spherical.

(59) This lemma is clearly related to the fact that the following property is possessed by a surface S if and only if S is spherical: there is a point e (the center) such that the straight line from e to any point p on S is perpendicular to the tangent plane to S at p.

(60) I proposed Lemma 1 at the NEH Seminar on the philosophy of Thomas Reid in August of 2000. On that occasion, Yaffe advanced his conjecture and suggested combining Lemma 1 with Lemma 2 to prove it. His attempts at proof led instead to counterexamples such as those reported in the text.

(61) Gideon Yaffe, "Reconsidering Reid's Geometry of Visibles," Philosophical Quarterly 52 (2002): 602-20. Yaffe shows at 617-18 how to find single angles in which real and visible magnitude are the same despite an oblique line of sight. He has suggested to me in correspondence that the same method will yield complete figures in which each angle as seen from e has the same visible magnitude as its real magnitude, despite not being viewed head on.

(62) This is a variation on an example of Yaffe's in which the angles are all spherical angles (rather than flat angles as here) hut there is deviation from a spherical triangle somewhere between the vertices.

(63) Actually, neither the six straight lines nor the three rounded lines uniquely determine a figure; instead each set of lines forms the boundary of indefinitely many figures. But any of the figures bounded by these lines will do for my purposes.

(64) What about the six-edged figure that was there before we rounded three corners to get the figure in the diagram? That figure has an angle sum of 540 degrees rather than 270. This is not a problem. To reckon the angle sum of a visible, we do not sum all the angles in a representative of it; we sum only those of its angles that correspond to angles in the visible.

(65) See Yaffe for an argument that the geometry of visibles is in fact completely isomorphic to the geometry of the sphere.

What I am saying in this paragraph is that Reid's case for the geometry of visibles is not threatened by the difficulty I pointed out in section 5 for the argument from indistinguishability. I criticized that argument on the ground that it would lead to the assignment of incompatible geometries to visible figures. In section 10 I said that Reid gets around that problem by holding that we assign to visibles not the geometrical properties of any figures indistinguishable from them, but just those of figures that represent them (that is, have real angle magnitudes equal to the apparent magnitudes of angles in the visible). Now it turns out that there are nonspherieal figures that can represent visible But that is all right, because using those representatives will not lead us to assign to visibles any properties incompatible with those of spherical geometry.
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