John Machin and Robert Simson on inverse-tangent series for \(\pi\) (Q1205956)

The author gives an English translation and analysis of a 1723 paper by the Scottish mathematician, Robert Simson (1687-1768). Using the formulae for the tangents of angle sums and differences, Simson derives a technique for decomposing inverse tangents as sums of others to obtain several efficient approximations for \(\pi\), such as \[ \pi=32\tan^{-1} {1\over 10}-16\tan^{-1}{1\over 515}-4\tan^{-1}{1\over 239} \] with power series expansions. These sorts of representations were discovered by several contemporaries, such as Hermann, Klingenstierna, Machin and Sharp.