In Walt Whitman's "A Noiseless Patient Spider," Whitman's love for nature--an ideal founded in Transcendentalism--is communicated as the spider becomes a metaphor for the individual soul. In its isolation, in its individuality--also a precept of Transcendentalism --the spider casts out its gossamer threads "to explore the vacant, vast surrounding." If this line is not reminiscient of Henry David Thoreau's quest into the woods to "live deliberately" and "to learn what it had to teach," then, certainly the last stanza is.
And, like Emerson's nonconformist who must be himself, the spider/soul of Whitman stands alone, "surrounded by measureless oceans of space...seeking the spheres to connect them"--that is, "living deliberately," as Thoreau writes.
This poem shows Transcendentalist ideas in a couple of ways.
First of all, the Transcendentalists felt that people were really a part of nature and that nature was a part of all people as well. We can see that in this poem as the speaker likens his own soul to a spider.
Second, the Transcendentalists believed that people should do their own thing. They should act as their conscience tells them. I see this in the first stanza as the spider figures out for itself how to fill the void. And then the spider fills it with things from within the spider itself. So this is talking about making your world with things from within you.
Whitman's version of transcendentalism was, in my view, a bit more passionate and personal than, say, Emerson's. They both believed that nature was a kind of outward manifestation of an inward spiritual state, but whereas Emerson sought to "disappear" or dissolve into nature (think of his metaphor of the "transparent eyeball" from his essay "Nature"), Whitman embraces it as one would a lover -- the personality of the poet, far from disappearing, is an essential element in creating the kind of unity of the physical and spiritual that defines the Transcendentalist project.
In "A Noiseless Patient Spider," we have an example of Whitman's intense involvement with nature. In the first stanza, Whitman describes seeing the spider, tiny in its vast surroundings, sending "filament, filament, filament, out of itself" as a way to explore its world. In the second stanza, Whitman characteristically shifts perspective, and turns the spider a metaphor for his own spiritual quest ("Ceaselessly musing, venturing, throwing, seeking the spheres to connect them"), in which he compares the spider's silk to his own need for connection in the last two lines: "Till the bridge you will need be form’d, till the ductile anchor hold, / Till the gossamer thread you fling catch somewhere, O my soul."
It's easy to see how the poet identifies with the spider -- simply put, they are both tiny beings in a vast and mysterious universe, casting about to find their way. I think, however, that Whitman's point is more complex than that. For one thing, while the spider spurs the poet to reflect on his own soul, Whitman does not make the spider subordinate to himself; the kinship the poet feels with the spider is one of equals. Second, the Whitman responds to the physicality of the spider, which uses its body (or the silk its body produces) to anchor itself in its surroundings; Whitman keenly feels this physicality, and imagines his soul also sending forth "gossamer threads," which he thinks of as "ductile anchors." The word "ductile" is a little unexpected here, but suggests how his soul might be physically deformed or stretched, in the way someone might stretch out a piece of metal, or pull it into a wire. These details serve to emphasize Whitman's unique fusion of the personal and the universal, or of the physical and the spiritual.