I have just thought it would be interesting to check what is the Carpenter's "sphere of unaided vision". The resolving capacity of a human eye is about 0.5 arc-minute. Say, you are standing on a reasonably flat surface (the maximal size of irregularities not exceeding 1 foot, or 0.3 m), the Sun is shining brightly and the air is absolutely clear. We can find the distance x, at which a 0.3 m object would be visible at an angle less than 0.5 arc-minute, or 1/120 of a degree:

tg(1/120) = 0.3/x

x=0.3/(tg(1/120)) = 0.3/0.000145 = 2063 m

Thus, a man (6 ft) would become "invisible" at a distance 6 times greater, about 12 km, or 7 miles. It is roughly twice more than Carpenter's figure, but his text apparently describes an observation in real conditions, so the opacity of air, amount of light and other factors might play their role.

Added: Sorry, I am not aware if someone has performed calculations like those on this forum. If yes, I would like to ask the moderator to remove this post and simply give the reference.