In 1715 he was expelled on account of his correspondence with his cousins, who were members of the Keir and Garden families, who were noted Jacobites, and had been accessory to the "Gathering of the Brig o' Turk" in 1708.
In London he remained for ten years, being most part of the time connected with an academy in Tower Street, and devoting his leisure to mathematics and correspondence with eminent mathematicians.
In 1730 his most important work was published, the Methodus differentialis, sive tractatus de summatione et interpolatione serierum infinitarum (4to, London), which is something more than an expansion of the paper of 1718.
[10] His next paper to the Royal Society was concerned, not with pure but with applied sciences; specifically, a trompe, i.e., a water-powered air compressor that was used by a Scottish lead mine.
A considerable collection of literary remains, consisting of papers, letters and two manuscript volumes of a treatise on weights and measures, are still preserved at Garden.