Both Uses

republic
in
Annexation
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Already, before these words are written, her Convention has undoubtedly ratified the acceptance, by her Congress, of our proffered invitation into the Union; and made the requisite changes in her already republican form of constitution to adapt it to its future federal relations.
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Instead of our present relations with Mexico—instead of the serious risks which have been run, and those plausibilities of opprobrium which we have had to combat, not without great difficulty, nor with entire success—instead of the difficulties which now throng the path to a satisfactory settlement of all our unsettled questions with Mexico—Texas might, by a more judicious and conciliatory diplomacy, have been as securely in the Union as she is now—her boundaries defined—California probably ours—and Mexico and ourselves united by closer ties than ever; of mutual friendship and mutual support in resistance to the intrusion of European interference in the affairs of the American republics.†

republics = governmental systems in which a majority of citizens elect representatives to make laws

Definitions:

(1) (republic as in: the country is a republic)
of a system of government in which a majority of citizens elect representatives to make laws; or someone in favor of such a form of government
(2) (meaning too common or too rare to warrant focus)
As a proper noun, the word form Republican is commonly used to describe one of the major U.S. political parties. It is and has been used by many other organizations such as The Irish Republican Army.