How do amendments ix and x apply to the constitutional principles of separation of powers and balance between government authority and individual rights?

Amendment 9 -- rights defined here don't mean other rights aren't for real. Amendment 10 -- Rights not delegated to the federal government by the constitution, and not denied to the states by the feds explicitly, are reserved to the states or to the people themselves. If the feds don't define it, go for it. The ninth and tenth amendments represent the end product of a feud between the Federalists and Anti-Federalist camps at the Constitutional Convention, which is still seeing heat today, usually with Roman candle terms such as "expansion of powers," "Big Government," "States' Rights" and so on.

South carolina was so angry about the election of lincoln that they seceded from the union on december 20, 1860. they attacked fort sumter later, on april 12, 1861.