Should the people of America divide themselves into 3 or 4 nations, would not the same thing happen? Would not similar jealousies arise, and be in like manner cherished?
What does be in like manner mean?
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In like manner means in the same way, similarly. So your sentence means, "[if America split up,] wouldn't similar conflicts come up, and wouldn't they be cherished (fuelled, strengthened, supported) in the same way [as now]?".
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I appears you are citing The Federalist
The Federalist, Or the New Constitution; Written in 1788 by Mr. Hamilton, Mr Jay and Mr. Madison. A New Edition, with the Names and Portraits of the Several Writers
b. in like manner: in a similar way, similarly.**OED
Should the people of American divide themselves into 3 or 4 nations, would not the same thing happen? Would not similar jealousies arise, and be in like manner cherished?
If these United States of America divided into 4 nations, would not new jealousies arise within making further sub-divisions necessary?
"Be in like manner" means "likewise be" or "in a similar fashion be."
That said, the reason you're having difficulty understanding that second sentence is it doesn't make sense. What's being cherished, "jealousies"? That is the only possibility, but it is counter-intuitive. What's more, the sentence completely loses all meaning when you realize there is no manner given for the cherishing of jealousy to be akin to.
Basically, that second sentence is nonsense, just gibberish. It looks like it should make sense because all the words fit together, but don't be deceived by that. The phrase "and be likewise cherished" sends the sentence careening right into incoherency.