The Berkeley-Reid debate on vision concerns the question: does visual sensory data contain metrical information reflecting an objective reality or are such impressions merely "in the mind" (as colours are known to be)?
Berkeley argued that all our geometrical knowledge of the world is ultimately tactile in nature. Visual data contains no metrical information about the world, except insofar as visual impressions are reliably associated with tactile ones. To prove this Berkeley noted that "visible extensions 'have no settled determinate greatness'" since "a given visual extension, which we may take to be an arc containing a given number of degrees of the visual field, can be associated with tangible objects of different sizes" (p. 54; i.e., a man being as tall as a house because he is closer to us, etc.).
Reid tries to argue against this by claiming that the "geometry of visibles" does in fact have "determinate greatness." In fact, he maintains, this geometry is spherical geometry: to a single eye incapable of depth-perception, the visual impression of any object will be identical to the visual impression of that object's projection onto a sphere centred at the eye, which suggests the metric that "the common and natural measure of length, is an infinite right line [i.e., a great circle], which ... bears a finite ratio to every other line [segment]" (Reid, Inquiry, 6.IX).
Daniels is convinced by this. He summarises Berkeley's argument as saying that "we must be able to assign determinate magnitude to something if it is to be a possible object of geometry" (p. 55), claims that "Reid provides the basis for just such a metric for his geometry of visibles" (p. 55), and concludes that Reid has won: "So much for Berkeley's argument" (p. 56).
But this is a terrible argument. Read the summary of Berkeley again---"we must be able to assign determinate magnitude to *something* if it is to be a possible object of geometry"---and ask yourself what the "something" is that Reid is assigning magnitude to. Saying that the "somethings" are real objects external to the mind comes at the huge cost of ascribing to them wild fluctuations in size. As I go for a walk, trees grow and shrink all around me. Even worse, it also come at a price of all-out subjectivity: the sizes I observe have nothing to do with those observed by others. What kind of defence of "realism" is this? Since this approach is obviously untenable, we must conclude that Reid has only assigned a metric to the visual field itself, not to anything external to it, so his point fails to have any bearing on realism whatsoever.
However, some doubt still lingers. It seems strange to hinge our argument on issues of subjectivity---shouldn't the question be decidable even if we were alone in the universe? There remains the point that adopting the Reid metric forces us to hold that things fluctuate in size, but is this really such a terrible thing? This point may be tested by the following amusing thought experiment alleging a symmetry between the tactile and the visual:
"Berkeley's argument can ... be turned around to show that tangible extension has no 'determinate greatness'. That is, [if Reid's metric is taken as the starting point,] different visible extensions ... can be associated with the same tangible extension. According to Berkeley's original argument, this would show that the tangible extension is not of 'determinate greatness.'" (p. 55)
In other words:
Berkeley may say: "Here I have a pen. As I move it further away from me it looks smaller even though I can feel that it is the same size. Therefore visual impressions cannot be trusted."
But Reid may say: "Here I have a pen. As I shrink it [which is how he may interpret the same action] it still feels the same size. Therefore tactile impressions cannot be trusted."
Where does the symmetry break down? I think one way in which it breaks down is that Berkeley can explain why he can throw his pen out the window but not his desk, whereas Reid cannot explain the analogous state of affairs with respect to shrinking. Here is a follow-up question though: suppose I have both the tactile and visual metrics but that I cannot actually move any object---then will I still be able to decide in favour of Berkeley? But let's save that question for another day, since it would take us too far afield from the original issue of realism. As far as this issue is concerned I think we are justified in concluding that Reid's metric is utterly futile.