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The percentage of Americans traveling overseas doubled between 1860 and 1900, but overseas tourism was still very rare at the end of the century (only .16% of the population per annum). Americans in 2009 were around 10 times as likely to visit Europe as were Americans in 1900.
The Historical Statistics of the United States records how many Americans were "Ocean-Bound Tourists" each year from 1820 to the present. H.W. Brands asserts that for the late 19th century, most of these tourists were headed to Europe (American Colossus, 608). So the following shows what percentage of Americans traveled on the ocean in a given year, which is a decent estimate of what percentage of Americans visited Europe: