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In his Inquiry, Reid claims, against Berkeley, that there is a science of the perspectival shapes of objects (‘visible figures’): they are geometrically equivalent to shapes projected onto the surfaces of spheres. This claim should be understood as asserting that for every theorem regarding visible figures there is a corresponding theorem regarding spherical projections; the proof of the theorem regarding spherical projections can be used to construct a proof of the theorem regarding visible figures, and vice versa. I reconstruct Reid's argument for this claim, and expose its mathematical underpinnings: it is successful, and depends on no empirical assumptions to which he was not entitled about the workings of the human eye. I also argue that, although Reid may or may not have been aware of it, the geometry of spherical projections is not the only geometry of visible figure.

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