The first European set foot on what would later become the United States in 1513. The last uncontacted indigenous person was captured in 1911. Many people are familiar with the story of Ishi; the picture above is of the monument commemorating the place in Oroville, California where he came under the power of European-descended people. (Image credit to Ray_Explores on Flickr.) This marker commemorates not just Ishi, but the date and place at which an entire nation's worth of indigenous people had finally been completely assimilated or exterminated - on August 29, 1911.
If taken in terms of territory, if it's just a linear expansion of a certain amount per unit time, then the annual expansion was 9,450 sq mi (24,729 km^2) per year; meaning a square of land about 98 miles (157 km) on a side. If we break it down to the day, then the U.S. was taking 26 sq mi per day (blocks 5.1 miles on a side.) Given the average American County size, that means every 46.5 days (about a month and a half) the U.S. was taking a county's worth of land in our march to manifest destiny. (I realize it didn't happen this smoothly but it gives us an idea.)
If we go by percent expansion (the more land you already got, the faster you get more), and we assume Ponce de Leon's camp when he landed was a square a hundredth of a mile on a side, then Europeans expanded at an average annual rate of 6.31%. Again, assuming this was a completely smooth process just for visualization purposes, that means the last year the US would have added territory equal to an area bigger than California but smaller than Texas. If going down to the day, the day Ishi was captured/gave himself up the US added half a county's worth of land.
But people are what we're most interested in here, and in human terms, the annualized assimilation rate for the United States of 4.1% over those 398 years. Obviously the size of the territory matters; to get a second data set to investigate territory size effects, I tried to find a similar event in the history of Canada but could not. With more than one country, we could get an idea of how the territory affects the rate.
Although it's not a conquest in the same way, it's been noted that the time for Germanic people to take over the Western Roman Empire was about six centuries, which corresponds to a 2.45% annualized rate of assimilation. In territory, in linear terms it's more than seven times slower, gaining 1,292 sq mi/year (an area 35.9 miles on a side.) In percentage terms, it's a 3.89% annual increase, using the same assumptions, i.e. same starting area, except this time it's a camp in the Black Forest instead of the Florida Coast. This is to be expected given that transportation was not as good and the technology gap between the Germanic tribes and the Romans was not nearly as great as between Europeans and North Americans; same people have likened colonization to being invaded by people from 4,000 years in the future.
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