Berkeley, in agreement with Locke (who was in agreement with Molyneux), says "no": it is not possible for someone born blind, who learned shape-names by touch, to then tell by vision alone, which of two shapes presented is a sphere, and which a cube.
Berkeley uses this opportunity to argue for the doctrine of proper sensibles—the view that there is no overlap among the ideas proper to different senses. In other words, Berkeley maintains that there are no ideas that originally enter the mind through more than one sense.
It is easy to see what Berkeley has in mind if we put the issue this way: Call the idea you get through touch of one side of a cube T-SQUARE (for tangible square). Call the idea you get through one vision of one side of a cube V-SQUARE (for visible square). Berkeley proposes that, if some ideas (such as the idea of a square) come in through both sight and touch, then T-SQUARE would be identical to V-SQUARE, and the only difference would be in the way you acquired them. But if T-SQUARE and V-SQUARE are identical, then, Berkeley argues, the formerly blind individual should be able to identify the cube, since they know that a cube is a body terminated by squares, and they can also see some squares.*
The most interesting part of Berkeley's discussion, though, comes in NTV 141 to 143(ish). And what makes this interesting is the startling similarity between what Berkeley says here, and the position Leibniz takes in New Essays On Human Understanding.
Right before NTV 141, Berkeley has just responded to the worry that V-SQUARE and T-SQUARE are called by a common name ('square') because they are of a common species, by appeal to the view that we often use the same name for the sign as well as for the thing signified. This, in combination with the view that V-SQUARE is a sign of T-SQUARE is intended to address that worry. The discussion moves on, then, to another potential worry:
But, say you, surely a tangible square is liker to a visible square than to a visible circle: it has four angles and as many sides: so also has the visible square: but the visible circle has no such thing, being bounded by one uniform curve without right lines or angles, which makes it unfit to represent the tangible square but very fit to represent the tangible circle. Whence it clearly follows that visible figures are patterns of, or of the same species with ,the respective tangible figures represented by them: that they are like unto them, and of their own nature fitted to represent them, as being of the same sort: and that they are in no respect arbitrary signs, as words. (NTV 141)
The worry of this passage rests on what I'll call the "greater fitness" claim: Some visible ideas have greater fitness than others to serve as signs of a given tangible idea. The worry attributes this fitness to a cross-modal commonality of species. In NTV 142, Berkeley responds to this worry by noting that the fitness of representation can be accounted for in terms of the complexity or simplicity of the ideas, without appeal to a common species. Importantly, Berkeley does not deny the greater fitness claim. Rather, he tries to show that a canonical instance of arbitrary representation exhibits a parallel case of differential fitness.
But it will not hence follow that any visible figure is like unto, or of the same species with, its corresponding tangible figure, unless it be also shewn that not only the number but also the kind of the parts be the same in both. To illustrate this, I observe that visible figures represent tangible figures much after the same manner that written words do sounds. Now, in this respect words are not arbitrary, it not being indifferent what written word stands for any sound: but it is requisite that each word contain in it so many distinct characters as there are variations in the sound it stands for. Thus, the single letter a is proper to mark one simple uniform sound; and the word adultery is accommodated to represent the sound annexed to it...It is indeed arbitrary that, in general, letters of any language represent sounds at all: but when that is once agreed, it is not arbitrary what combination of letters shall represent this or that particular sound. I leave this with the reader to pursue, and apply it in his own thoughts.
In the New Essays, Leibniz (through the mouth of Theophilus), answers Molyneux's question thus:
[Y]ou will see that I have included in [my reply] a condition which can be taken to be implicit in the question: namely that it is merely a problem of telling which is which, and that the blind man knows that the two shaped bodies which he has to discern are before him and thus that each of the appearances which he sees is either that of a cube or that of a sphere. Given this condition, it seems to me past question that the blind man whose sight is restored could discern them by applying rational principles to the sensory knowledge which he has already acquired by touch...My view rests on the fact that in the case of the sphere, there are no distinguished points on the surface of the sphere taken in itself, since everything there is uniform and without angles, whereas in the case of the cube there are eight points which are distinguished from all the others. (NEHU, book 2, chapter 9)
Leibniz claims that the formerly blind person could reason their way to the right answer, if they are told that the two visual appearances are of shapes with which they are already familiar (and further, told the specific pair of shapes that the two visual appearances are of). Berkeley concedes that, taking for granted that the visual is to be a sign of the tangible, it is not arbitrary which visible figures represent which tangible figures.
To give credit where credit is due; Leibniz himself indicated that he thinks he is on pretty much the same page with people who want to give a "no" answer; he just thinks they are giving a fine answer to the wrong question.
Anyway, it was interesting for me to find out that Berkeley pushes what is essentially the Leibnizian line on Molyneux's problem.
*Berkeley's argument is actual given in terms of numerical and specific difference, which is good, because it avoids an issue present in my quick reconstruction, having to do with token vs. type identity. To frame it so as to avoid this issue, we can take T-SQUARE to name a particular idea you got through touch. Then the question is whether T-SQUARE and V-SQUARE are of the same kind (i.e. intrinsically alike, for a certain sense of intrinsic), not whether they are identical. That way of putting it captures Berkeley's language more clearly: "upon the supposition that a visible and tangible square differ only in numero it follows that he might know, by the unerring mark of the square surfaces, which was the cube, and which not, while he only saw them" (NTV 133).