Pittsburgh is a colonial-era city, but downtown has been rebuilt so many times that not much is left from before the Civil War. This building probably dates from the late 1840s, making it one of the oldest remaining downtown. It probably came after the Great Fire of 1845, but it appears in this engraving of the Diamond as it appeared before 1852, which was the year the old courthouse in the middle was torn down.
The building in the background, with smoke rising from its chimneys, is clearly meant to be this one. There are eleven columns of windows in the engraving instead of the nine columns of windows we see today, but old Pa Pitt suspects the engraver was working from a rough sketch and simply gave us his best guess.