An approximation technique, and its use by Wallis and Taylor

Abstract

My aim here is to describe and analyse two calculations, by John Wallis and Brook Taylor, which to modern eyes are clearly based on the approximation properties of continued fractions, though neither author appears aware of such a connection. The calculation by Wallis, which he illustrates by generating approximations to tt, is well known and is frequently described in terms of continued fractions; thus I wish to give a more accurate and detailed account of his procedure than seems to have been attempted hitherto. Taylor's brief calculation, an evaluation of log10 2, is less well known and the method has indeed subsequently been re-discovered and published at least three times, the last time as late as 1954.