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According to this article, in 2009, around 13 million people from the United States travelled overseas, of which 35% visited Europe. Given the United States' population of 320 million, we can estimate that currently, around 1.4% of Americans visit Europe annually.

What would that percentage have been in the mid- to late 19th century? All transatlantic travel back then was by ship, and likely considerably more expensive, so presumably, the fraction of Americans able to afford a trip to the Old World was much smaller than it is today. However, out of those who could afford to travel overseas today, only a relatively small fraction actually do, so that does not necessarily mean much.

Note: In the period that I am asking about, millions of people immigrated to the U.S. from Europe, and had thus "visited" Europe before they every reached American soil. Those people should not count towards the total unless they returned to Europe afterwards, with the intent to return to America later. A visit is a trip with the intention of returning.

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    Also keep in mind that you should add to the monetary cost of the trip the cost of "not working for some weeks", which might easily be much greater. – o0'. Jan 12 '15 at 9:40
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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:

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