Yeah log(n) would be the space to hold the digits of the counter. Presumably (unless things got super-silly), this would be dominated by the buffer size for the line being read.

The point there is that you don’t need a buffer for whole line, you only have to find the line terminators. Something like:

    printf(“0: “);
    ln = 1;
    while ((ch = getch()) != EOF) {
      putch(ch);
      if (ch == EOL) {
        printf(“%d: “, ln);
        ln++;
      }
    }

And I believe that original unix implementation of nl is mostly this, although with the stdio functions replaced with raw read/write.

Yeah, I think the above poster said O(logn) space because that's the size of the line count in bits

The usual computation model includes "log n sized words", so it really is O (1). You usually are implicitly using that model, otherwise any algorithm that used pointers would have an extra log n tacked on, e.g. linked lists append would be log n not constant.