What was the population of the main area of settlement in New York City in 1890?
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It looks like the problem you are going to run into here is that the detailed records for the 1890 US Census were damaged in a fire, then (purposely!) destroyed by the US government.
However, you are in a bit of luck here, because NYC was suspicious it was getting cheated out of congressional representation by the 1890 census, so they commissioned their own. However, given the motivations (proving the official census was low) there are legitimate concerns that the extra 200,000 residents it discovered were made up.
If you are willing to live with that, it should be possible to take the data and exclude whatever areas you desire.
Unfortunately, even the NYC Police Census supposedly isn't easily available online, and some of its records were lost too. I can vouch that I'm having trouble finding online resources at the granularity you need. It seems like most folks interested in the data want to know about individual addresses for genealogical purposes.
My best guess is that if you were to go through all the trouble to look it up offline, and to weed out the more rural areas, you'd likely come out with roughly the official US figure for NYC's population (the rural population being roughly balanced by the extra count NYC's Census made). Either way, the actual size of the city in 1890 is in dispute by about 200,000, which is likely going to swamp any extra precision you'd get by excluding rural areas.
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