An annotation on Passage 3373 of James Madison's Notes of Debates in the Federal Convention of 1787 3373

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John Adams, the first president of the Senate. Courtesy of the Library of Congress.

The original idea was that a senator would be president of the senate.  However, this led to a couple of problems.  First, if the president could vote only during a tie, the state would be under-represented.  Second, if the president could vote at all times, the state would be over-represented as the senator would have two voting seats. 

To solve these problems, the Brearly Committee decided that the vice president would be president of the Senate. 
Reference: Sidney M. Milkis and Michael Nelson, The American Presidency 55