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.

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.

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.

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.

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

If a visible triangle has angles of certain apparent magnitudes, its angles really are of those magnitudes.

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.

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:

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.

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.