The rest of the Constitution is pretty clear that it's purpose is to lay out rights and responsibilities of the Government, and that anything not named therein belonged to the people... one of the biggest objections to the Bill of Rights was that they were fundamentally redundant -- since nothing in the Constitution restricted freedom of speech, there was no reason to guarantee it -- and therefore caused a fear that anything NOT captured in the Bill of Rights would not be considered a right, even if specifically not enumerated in the text. By and large both sides of that argument have turned out to be true -- the existence of the Bill of Rights has been absolutely vital in firmly defining what rights are, but it is generally assumed that things not explicitly named in the Bill of Rights are implicitly not rights at all, which is precisely what opponents were worried would happen.

(the right to life, liberty, and the pursuit of happiness you refer to are in the Declaration of Independence... not the Constitution).