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I'm sure you've all noticed documents in English from the 1700's often have 'F' where, if written now, there would be an 'S'. You can see what I'm talking about a few times in this example, like at the beginning where it says "Prayers faid" or in the date "Tuefday November 26. 1700."

What's going on with this? When did it start? When did it stop?

There's a typographical distinction between an actual f and the ſ you're referring to in the text. See for instance the difference between 'magiſtrats' and 'behalf' in the second paragraph.

The 'ſ' is a long 's'; the Wikipedia article has a very long section on its history and decline of use.

In general, the long s fell out of use in Roman and italic typefaces in professional printing well before the middle of the 19th century. It rarely appears in good quality London printing after 1800, though it lingers provincially until 1824, and is found in handwriting into the second half of the nineteenth century" being sometimes seen later on in archaic or traditionalist printing such as printed collections of sermons.

See this Old English Alphabet for a more complete list of changes to the alphabet. And this somewhat related Linguistics SE question, with a long answer that explains how 'ſ' was just another way of writing 's' in some circumstances, rather than a letter that corresponded to a different pronunciation.

Other interesting posts courtesy of sumelic:

• Use of “f” instead of “s” in historic, printed English documents


• How exactly was the long S used and why did people stop using it?