http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf

Euler derives 1779 (age 72) this algorithm for Pi:

Which is suted for calculation by hand in the decimal system.

Let's see how (and if) it works:

piterms = 
  With[{n = 19}, 
   Table[(16/10 2^k + 12/10) (2 k)!!/(2 k + 1)!!, {k, 0, n}]];

Grid[Join[
  pi = MapIndexed[First[RealDigits[#1, 10, 19, #2[[1]] - 1]] &, 
    piterms], {First[
    RealDigits[Apply[Plus, Map[FromDigits, pi]]]]}, {First[
    RealDigits[N[Pi, 19]]]}], Frame -> All, 
 Background -> {None, {{LightBlue, LightYellow}, LightGreen, 
    LightRed}}]

Here 19 digits are computed and 20 terms are added (the 26th would be zero, so 25 terms would be the maximum)

The last line is Pi computed with Mathematica directly for comparison.