"On a little promontory it stood isolated" (line 2)
This line puts the idea of isolation right out there. Thinking of the spider as not just alone, but "isolated," gives us an emotional connection to this little critter. It makes him a part of something bigger, rather than just a spider doing something. But, how exactly are we supposed to feel about this isolation? Is it a bad thing, a sort of loneliness, or is it more like solitude, a peaceful separation from the rest of the world?
"the vacant vast surrounding" (line 3)
Here, again, Whitman picks an intense word that he doesn’t necessarily need to use. The space around the spider is clearly big, but why does it have to be vacant, too? With that word dropped in there, we get feelings of isolation and hopelessness in the spider’s work. He tries to reach out with his web and hold onto something, but, what if there’s nothing around him? Differences in size also help create this isolating effect. Whitman takes something which everyone knows is tiny (a spider), and puts it in a "vast surrounding." This makes the emotional effect of isolation even more intense.
"Surrounded, detached in measureless oceans of space" (line 7)
It turns out that the speaker’s soul shares this isolation with the spider. This line almost exactly mirrors lines two and three above. Again, the choice of words is important. Not only is the soul separate, it is "detached" from anything else. Like the spider, it is an island in the middle of nothingness. Here, the feeling of separation is even more intense. The surrounding space is not just vast, but "measureless."