What does the phrase 'but that' mean? e.g., in On Walden Pond. 'why has man rooted himself thus firmly in the earth, but that he may rise in the same proportion into the heavens above? '
This is a rhetorical flourish. You can substitute "if not" for "but", and read in a "so" in front of the "that":
Why has man rooted himself thus firmly in the earth, if not (so) that he may rise in the same proportion into the heavens above?
Because it is in the form of a rhetorical question, we can interpret this as something like
What explanation can there be [for man rooting himself etc.], if the explanation is not so that [he may rise etc.]? Well, obviously, there isn't any other explanation!
This use of "but" to mean "if not" may be familiar to you from the phrase there but for the grace of God (go I). The rhetorical structure may also be familiar if we strip it down:
Why X, if not because of Y?
Jane Austen's 1813 Pride and Prejudice (via Project Gutenberg) contains a famous example at the end of Chapter 57, also using "but" in place of "if not":
For what do we live, but to make sport for our neighbours, and laugh at them in our turn?
Another example, from a nineteenth century sermon on the Lord's prayer:
For what is it we pray for, but that we may commit no evil, for which we should be separated from that holy Bread.
—A Select Library of the Nicene and Post-Nicene Fathers of the Christian Church, 1888, Philip Schaff, Ed.
And some more recent examples of the structure, using more modern language:
Why do concepts, inferences, etc. of subjects age 2 to 6 remain preoperational, if not because of illegitimate intrusions of causality and an insufficient Causality?
—Jean Piaget, 1974, quoted in Webster Callaway's Jean Piaget: A Most Outrageous Deception, 2001
And why is asbestos treated less favorably, if not because it is foreign? Plainly, because it is dangerous.
—Trade and Human Health and Safety, 2006, George Bermann & Petros Mavroidis, Ed. (Note that in this case the authors have answered the rhetorical question with "because Z".)
This is a pretty obscure usage, but in in this context, everything after the "but" is an answer to the "why" in the first clause. "Why is man rooted? The answer is so he may rise."
In this context, it means "unless."
In this case, "but that" make a opposite connector with the first sentence and last one, like a possibility, even if firmly in the earth, he isn't prevented and might of rise in the same proportion into the heavens above.