And I never thought the change in focus as you move the easel up and down would be linear.
It should be linear, depending on how you judge "focus," I guess.
Here's a simple way to look at the situation, geometrically. Say that you have a "point" on the negative that you want to project onto the paper. Now imagine a "cone of light" originating from the entire lens aperture, and coming to a point, ideally at the paper surface. Of course we can't get to a perfect point, but this is a good approximation. So if the paper is at the wrong distance then it will intersect the cone at some point, and the diameter of the cone at that point represents the smallest detail that can be on the print. So it seems clear that the size of this "blur circle" is gonna linear with the distance error at the paper.
Here's a concrete example. Say that Bill is making 11x14" prints from a 35mm film negative, using a 50 mm fl enlarging lens set to f/4. I'm from the US, so I like to work in inch units (to convert, there are exactly 25.4 mm per inch.) OK, the diameter of the aperture is about 50mm/4 =~ 12.5 mm, or very close to 1/2 inch diameter. This is the base of the light cone.
For the height of the cone, one can just measure the approximate distance from lens to paper. This should be about 16.5 inches (I just calculated it).
So now we have a cone-of-light, from lens aperture to a point near center of the image. The cone has a base diameter of 0.5 inches, and a height of 16.5 inches. One can look at this a "rise over run" case where the diameter of the the cone increases by 0.5 inches for every 16.5 inches of "run." Or 0.5/16.5 = 0.030 inch diameter increase per inch of "run."
Now for the result of an error in paper position. If the paper is 1 inch out of position, either too close or too far, the smallest possible point of detail is about 0.030 inch (or about 3/4 mm) diameter. As a long-time car guy, I see this as about equal to an old-time spark plug gap. Or the thickness of three business cards. Or, consider that the thickness of a human hair is about 2 to 3 thousandths of an inch - so this 0.030 inch "blur circle" is about ten times larger than the width of a human hair. One might ask, can I see this amount of detail? Well, personally I don't have any trouble seeing a human hair on a piece of white paper from 5 feet away. So to me, this would be a pretty unhappy print if held in my hand.
If, instead of a 1 inch paper placement error, it was only 1/2 inch error, the smallest possible point shrinks to half of that, or 0.015 inch diameter - still considerably more than the thickness of a business card. Or 5 times the width of a human hair.
So what does it take, hypothetically, to appear "sharp?" Well, sharpness is about more than just detail, but if you wanted to be able to show hair-sized details on the print you'd want to be able to lay down "blur circle" diameters of about 3 thousandths of an inch, or smaller. With our 0.030 inch diameter change per inch of paper distance error, this means about one tenth of an inch paper error. Fwiw a typical color paper print is about the same thickness as a business card, about one hundredth of an inch. So in this specific example the paper would have to be positioned within about ten paper thicknesses of ideal.
If we start looking at much finer detail being delivered to the paper then we probably have to consider the effect of diffraction. In this case (lens, aperture, and lens to paper distance), I'd guestimate the smallest possible spot (the diameter of the Airy disc) as roughly 0.001 inch, one-thousandth of an inch. (This is a rough guess, so anyone please feel free to supply the correct number). The reason for such a large Airy disc is the amount of enlargement, leading to a long lens-to-paper distance. So the lens, although set to f/4, acts as though it is f/32 with respect to diffraction from the point of view of the paper.
Anyway, this is my view of how the effect of paper position errors come into play. I'd guess that a very finicky person, either nearsighted or with a loupe, would be able to see the difference between a perfectly focused print and one that is about five paper thicknesses off. In THIS EXAMPLE, and assuming that the negative has not moved at all. With less enlargement, say to a 5x7", same lens, aperture, they'd likely be able to see a difference with perhaps a 2 or 3 paper thickness error.
Ps, thanks to Bill Burk for doing the physical tests.