What does "a government of laws, not men mean"? and how does that go with a constitution government?

Answers:
It means the powers and duties of every government agency and official is defined by laws that were themselves enacted by a process that was itself established by law; and that the laws apply equally to everyone; and no official can arrogate more power to himself than he is granted by law.

The Constitution is precisely the type of law that grants such powers and defines their limits.

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Do your own home work and earn your place in life through hard work and not cheating.

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The Constitution must be followed by elected officials.
The country is run in accordance with The Constitution, not at the whim of those officials.

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It means that a government passes the laws and everyone is forced to follow them. Further, it means that men cannot take a matter of justice or ethics into their own hands, that justice is controled, dispenced and decided by the government.

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I interpret this to mean that the constitution and laws are set up so they cannot easily be changed based on the fact that the current leader likes or dislikes them.

It's a strange example, but when the Nazis were taking over Europe, they made edicts and laws based solely on the individual leader's own whims. In Holland, for example, streets became one-way, street signs changed, and marigolds were banned because orange was the color of the Dutch royalty. In a democratic society, there is a preset system of deciding how laws are made or ammended. A president who is strongly pro-life, for example, can't just get in office and declare abortion illegal. There is a system, so that you don't run into the problems of laws changing every four or eight years when new leaders come in.

Our nation's leaders definately have pull, and power, but it's not a dictatorship (which would be an example of a "government of men, not laws").

Our government remains stable, regardless of who is in office.

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A government of laws means that the government and the people are held to a known set of laws.
They do not need to fear the whim of political leaders.
They can live their lives and be certain that they are obeying the laws.
A government of men is either a monarchy or a dictatorship.
Under this type of government all you need to do to get arrested is to offend someone in power.
You may not have broken any of the laws that the country has you are the victim of the whims of men.