Leonard Evens , Jun 09, 2007; 12:18 p.m.
If the prints are going to be viewed from normal viewing distance, usually defined as the diagonal of the print, it suffices to set the resolution to produce an acceptable 8 x 10 inch print. That is because, as viewers move progressively further way, their tolerance for loss of fine detail in the print reduces proportionally. If you think scanning at 300 ppi would produce an adequate 8 x 10 inch print to be viewed at about 12 inches, then in principle that would also be adequate for a proportionately larger print viewed proportionately further away.
But there are some problems with that assumption. First, ignoring the printing issue, when you scan a 300 ppi, digital image processing theory says you will get at best 150 line pairs per inch---or about 5.9 lp/mm---resolution. (You need two pixels for each line pair.) But no scanner is going to deliver the theoretical maximum. So in practice you would be better off scanning at a higher scanning rate to achieve what you want. The second issue is that the printing process also degrades resolution. 300 ppi for what you send to the printer assumes a perfect image to start. But when you combine two steps, you are going to get something less than the resolution obtainable at either step. If each step delivers about 6 lp/mm, the combination might deliver something between 3 and 4.2 lp/mm. In this case, if you increase the resolution obtained by scanning, reducing it further to 300 ppi for printing will get you closer to what you could get with a perfect image.
If instead, you assume people are going to get close to the print, you have to take that into account. A 4 x 5 foot print, requires an enlargement factor of about 48/8 = 60/10 = 6 times. If such a print is going to be viewed from about 12 inches, you have to multiply the acceptable scanning resolution by 6, so you need the full 1800 ppi that the scanner is capable of. But again, remember that this would only work for a perfect scanner. Even if the scanner theoretically produced enough pixels, it is unlikely to deliver the theoretical resolution so obtainable.
In terms of the printer, if you want to send the printer 300 pixels per inch, you also have to take into account the print size. If you start with 1800 pixels per inch after scanning, and enlarge 6 times, that reduces you to 300 pixels per inch. As above, taking into acccount the combination, it seems unlikely that you can produce a 4 x 5 foot print from an 1800 ppi scan that would show the same level of fine detail, viewed at 12 inches, that you would see in an 8 x 10 inch contact print.
In connection with this analysis, keep in mind that it assumed a negative with no loss of fine detail. In practice, any lens-film combination is unlikely to produce anything much better than 40 lp/mm. Enlarging that 6 times would produce a bit over 6 lp/mm, which would be barely acceptable if the other steps in producing the print didn't further detract. So in practice, no matter how you go about it, you are probably not going to be able to produce a 4 x 5 foot print from an 8 x 10 negative, in which critical viewers, looking close up, won't be able to discern some loss of quality. The upshot, I think, is that you should operate under the assumption that viewers will be satisfied looking at the print from normal viewing distances and choose your scanning resolution to be as high as the scanner can deliver to minimize its contribution to loss of quality.