One of the earliest places where an infinite string of 9s is rounded off to a finite string is Euler's text on algebra around 1777. Here he gets the answer 9.999... and makes a comment that this is virtually indistinguishable from 10.

One of the earliest places where an infinite string of 9s is rounded off to a finite string is Euler's text on algebra around 1777. Here he gets the answer 9.999... and makes a comment that this is virtually indistinguishable from 10. That was one of the earliest places as I mentioned. Conifold found an earlier source (see comment above).