Thomas Jefferson was a firm believer in a strict interpretation of the Constitution; whereas Hamilton favored a broader more lose interpretation, particularly with regard to the powers of Congress under Article I Section 8, specifically the last clause of that section which gave Congress the power:
...to make all laws which shall be necessary and proper for carrying into execution the forgoing powers.
The issue thus turned on whether Congress had the constitutional power to create a bank. Jefferson said no; while Hamilton said yes.
Jefferson, along with James Madison, argued that Congress had only those powers specifically enumerated in Article I Section 8. There was no provision therein for the creation or operation of a bank, therefore the power to create it did not exist. They based their argument on the 10th Amendment:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
They argued that the language of Article I Section 8 quoted above was merely a :convenience."
Hamilton argued that it was the end, not the means, which was the salient issue. Congress had power to collect taxes and regulate currency, it would need a bank for that purpose. Since the end purpose was legitimate, so also was the means. His argument turned on the theory that if the end is not specifically prohibited by the Constitution, then
it may safely be deemed to come within the compass of the national authority
Therein lay the argument until it was purportedly settled years later by Chief Justice John Marshall in McCullough vs. Maryland, in which Marshall appeared to accept Hamilton's argument.