This animation shows the change in Manhattan's built-up area population density from 1800 - 2010. It is based on data collected by Shlomo Angel and Patrick Lamson-Hall of the NYU Stern Urbanization Project.
The lessons, in short? Densities in Manhattan as a whole rose in the 19th century, peaked in 1910, fell for 70 years, and have been rising slowly since 1980.
You can find their related working paper here: http://marroninstitute.nyu.edu/content/working-papers/the-rise-and-fall-of-manhattans-densities-1800-2010