For reference, Peter Henry Emerson was a photographer working in the last decades of the 19th century, bringing the ideas of Impressionism into Pictorialism, roughly speaking.
I have run across two or three descriptions of Emerson's ideas of focusing, and how to produce an impressionistic effect in the camera. Writers all copy from one another, not as much as people on the Internet, but a lot. The result is that the references I have seen make more or less the same vague reference to the sketchy theories about how eyes work that Emerson was basing his thinking on, and then state briefly that he used "differential focus" to make his pictures soft. Sometimes they mention putting the subject a little out of focus.
There are a couple of points worth making here. Emerson's procedure is considerably more detailed and sensible than these writers make out, and Emerson was right even if his Eye Science was wrong.
The procedure is pretty simple. Focus on the subject, with the aperture wide open. Then, using a combination of narrower apertures (but use the widest one possible) and swings of the camera back, place everything except the subject out of focus to a greater or lesser degree. The use of the swing back may not be familiar to all photographers, although the use of wider apertures surely is. If you don't immediately know what rotating the back of a large camera relative to the lens does, read up on the Scheimpflug principle.
The degree to which Emerson suggests you should put things out of focus is very specific. It is to reproduce as far as possible the impression you have when looking at the scene. You place things which are not the subject out of focus not because of some theory of the eye, but because you are not looking at those things in the scene.
Finally, Emerson suggests putting the subject itself very very slightly out of focus, and only sometimes, to render it more natural looking. This is arguably an error, but even that is not quite clear to me.
Why is Emerson right?
Emerson's Eye Science may have been wrong, but it turns out that he had a pair of these things himself, could use them, and thought about how it felt to use them to look at things. This, not the flawed Eye Science, is essentially what his method is based on. I assume he put the Eye Science into his book to back up his theory, not as the actual source, but that is naturally just a guess.
When we look at an object we perceive that object as sharp, as clearly delineated and detailed. The rest of the scene isn't soft or out of focus, per se but it is out of our attention. The idea of sharp versus soft isn't really all that meaningful in speaking of what we perceive, in any case. What we "see" is a mental construct, bearing only a loose relationship to the actual image captured by the eye. We "see" it all as "sharp" but the stuff in the periphery is less within our attention, it is more extrapolated, interpolated, and just plain made up by the visual cortex, and we're not looking at it.
Imagine, if you will, an 8x10 print made with a normal lens. This print captures a fairly wide scene, one which if we had seen it ourselves, we would have been filling in a lot of periphery with the machinery of peripheral vision. Much of the scene, at any moment, would have been outside our attention. We take this whole thing, and cram it into a small object, a print, right in the center of our vision, where we can "see" the whole thing at once.
This is profoundly unnatural. Put in these terms, it's a little surprising to me that we can make sense of this at all, but we do. I think an argument can be made that the straight photography guys, and in particular the f/64 people, are characterized mainly by a willingness to embrace this unnaturalness. This is a sort of natural endpoint for photography, we trace nature with total accuracy, and then cram it, in a single chunk, into your visual cortex to see what happens. It's not natural, but it's definitely photography.
Emerson correctly perceived that this presents us with a problem, We're going to "see" the scene in the print all at once, we're going to "see" it much more as a single object, with all the elements of the scene as details of that singular object. Presented with the same scene in the real world, we would "see" it quite differently. Perhaps we would perceive the subject fully, in a moment, and only gradually over seconds or even minutes come to perceive the details of everything in the scene.
Emerson proposed quite an elegant solution with his differential focusing. It's not quite modern shallow depth of field, I think Emerson would have had no truck with bokeh. Things were to be soft, but the structure was always to be preserved. We might not "see" the surrounding scene in as much detail as we perceive the subject, but we are aware of it. We know there's a road to the left, a cottage behind, and a lake in the foreground. We just don't care, because we're looking at the girl. So, the photographer should not throw the rest of the scene into a mass of blur, but only soften the surrounding material, and render it less detailed and less important in the print.
In the modern era, I think we might more closely approximate the effect Emerson was looking for by taking a sharp picture, and then rendering a blurred layer over it varying the transparency of that layer from one area to another. Almost transparent over the subject and almost opaque over unimportant background, but never fully transparent or opaque. It is worth noting that extremely high-end modern portrait lenses do almost exactly this, at shocking expense.
In the modern era, especially the acolytes of f/64, we use other constructs to manage the viewer's attention. The use of contrast, saturation, leading lines, and so on, can also help the viewer sort out what to look at. There are a bunch of principles of composition and design which can help to arrange the masses within the frame in some sort of hierarchy of importance. Degree of softness is but one, but it's a neglected one.
Emerson's solution was pretty elegant, and really nothing like "just make it all soft, you know, impressionistic."