It appears this sentence was written over one and a half centuries ago. Its structure and grammar are a bit outdated and considered unusually formal by today's standards. There is a couple of grammar points in this sentence worthy of analysis.
(just) as + clause, so + clause
This structure is very formal and dated-sounding, used to draw a comparison between two things. It is useful because it compares two clauses which can go very long. This function made it a handy tool for formal writing in the olden days.
used to compare two people or things, when they are similar
Just as the French love their wine, so the English love their beer. (Longman)
used for showing that there is a connection between two situations or actions
Just as the body needs regular exercise, so too an engine needs to be run at regular intervals.(Macmillan)
Another thing to note is that this sentence also employs a subject-auxiliary inversion in the second clause. The word "so" often leads an inverted elliptical clause to express agreement or sameness/similarity.
"I like chocolate ice cream." "So do I."
"She should sign up now. So should you."
The sentence in you example goes further in that it is not elliptical, which means it does not omit any structural elements in its comparison of the two clauses and has both clauses fully written out.
"Admit of" is a phase meaning to allow or to permit.
Having gone over all the relevant grammar points, now we can look at the sentence on the whole. All these put together, we get:
The origin and progress of the United States are unparalleled in history. In a similar way, her (referring to the United States as a nation) political career and prospects should not be compared to the rise and fall of ancient republics.