In December 1829, in his first State of the Union letter to Congress, Andrew Jackson noted that we had morphed into a broad-based democracy, and he asked Congress to propose amendments to the Constitution to reflect that. He asked for:
His second request was finally granted in 1913.
The Constitution is precisely what makes this a democratic republic. And we have moved this Republic in a more democratic direction through the Amendment process.
All kinds of governments refer to themselves as republics but what makes our republic different from China, or Florence under the Medici's, is that democracy is part and parcel of Constitution itself. It's a republic that can only function within a democratic framework.