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This diagram appears on page 872 of “College Physics”, Zears and Zemansky, 3rd edition, Addison and Wesley, 1973. It mentions solid angle, saying that it is defined by the set of lines radiating from a point.

It does not say how.

Instead, in the left hand diagram, it gives a quite different definition, one which is expressed not as the ratio of two sets of lines, but of two areas—namely, A and R²—and, of course, this is the Inverse Square Law, as we have something proportional to the inverse (reciprocal) of a squared quantity The solid angle is “unitised” (calibrated) as indicated on the right-hand diagram.

There is no count of lines.

I surmise that the “solid angle” is to meant to be conceived of as a summed collection of little cones, each of which, “in the limit” of littleness, becomes a line. These lines then share the available flux equally among themselves, and it is thereafter supposed to “thin out” as the volume of the sphere increases, which of course implies and assumes a constant number of lines. I remember being told that if the flux radiates from an electric charge, e coulombs, say, there are taken to be (a constant) e lines of electric flux in all. There ought to be an infinite whole number of lines. This was for computational convenience only, when the talk turned to Faraday’s fields as manifolds of lines of force.

Now this summing of cones as “lines in the limit” is a seductive notion, but it is nonsense for all that, as a line is never a cone, not even approximately. In any case, if a light beam has thickness, it cannot be represented by a line, or lines, without even zero thickness.

What actually might be happening to produce fields is mysterious, and is much more intriguing than the stock, quasi-mathematical claptrap.

Incidentally, one often hears of “point-sources” of light, or of “point-masses”, or of a “point-charge”. Indeed, one needs such things to come to the notion of the inverse square law, above. But how do you conceive of them? What are they? What would you expect to observe? For if a point is just a place, and masses, charges and light sources are not places, how can they be points?