The meaning of this quote "A leader is best when people barely know he exists; when his work is done, his aim fulfilled, they will say; we did it ourselves" means if the leader is unknown to many people once he has finished or achieved a remarkable goal other people (such as Laozi) can steal the glory and be remembered for something they didn't do.
These two men seated at the table represent the Irish , with top hat and vest, and German (smoking pipe and drinking german beer) inmigrants in the 1800s in the US. The cartoon refers to the 1882's Chinese Exclusion Act, which was the first American law preventing the inmigration of an specific ethnic/national group. This can be apreciated in the "New Declaration of indepence" at the back of the cartoon.
The legend in the bottom ("If the Yankee Congress can keep the yellow man out, what is to hinder them from calling us green and keeping us out too?") refers to the raising fear in Irish and German groups of facing the same fate of Chinese inmigrants. This let us know American attitude towards inmigrants who weren't seen as equals and were discrimitated for their origins and skin color.
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Control of both elective and appointed posts also gives a machine control of government salaries and revenues, which can be used to enrich the party at the public's expense
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They were written in the midst of the debate over electing representatives to a ratifying convention in New York (state), and the immediate target audience was New York readers, particularly those who would be shaping the state’s approach to ratifying the Constitution.