In 1620 he published Refutatio Libelli de Regimine Ecclesiae Scoticanae— an answer to a tract of Calderwood, who replied in the Vindiciae subjoined to his Altare Damascenum. The only other writing of Spotswood published during his lifetime was the sermon he preached at the Perth assembly. His most considerable work appeared posthumously— The History of the Church and State of Scotland, beginning the year of our Lord 203 and continued to the end of the reign of James VI. of ever blessed memory, London, 1655, fol. It displays considerable research and sagacity, and even when dealing with contemporary events gives a favourable impression, upon the whole, of the author’s candour and truth. An appendix was afterwards added by Thomas Middleton.

Spotswood left two sons,—Sir John Spotswoode of Dairsie in Fife, where the archbishop erected a church and bridge, which are still extant, and Sir Robert, a lawyer of great learning, who became president of the Court of Session, and was executed in 1646 for taking part in the expedition of Montrose.

As a mathematician he occupied himself with many branches of his favourite science, more especially with higher algebra, including the theory of determinants, with the general calculus of symbols, and with the application of analysis to geometry and mechanics. The following brief review of his mathematical work is quoted from the obituary notice which appeared in the Proceedings of the Royal Society (vol. xxxviii. p. 34):—“The interesting series of communications on the contact of curves and surfaces which are contained in the Philosophical Transactions of 1862 and subsequent years would alone account for the high rank he obtained as a mathematician. . . . The mastery which he had obtained over the mathematical symbols was so complete that he never shrank from the use of expressions, however complicated—nay, the more complicated they were the more he seemed to revel in them—provided they did not sin against the ruling spirit of all his work,— symmetry. To a mind imbued with the love of mathematical symmetry the study of determinants had naturally every attraction. In 1851 Mr Spottiswoode published in the form of a pamphlet an account of some elementary theorems on the subject. This having fallen out of print, permission was sought by the editor of Crelle to reproduce it in the pages of that journal. Mr Spottiswoode granted the request and undertook to revise his work. The subject had, however, been so extensively developed in the interim that it proved necessary not merely to revise it but entirely to rewrite the work, which became a memoir of 116 pages. To this, the first elementary treatise on determinants, much of the rapid development of the subject is due. The effect of the study on Mr Spottiswoode’s own methods was most pronounced; there is scarcely a page of his mathematical writings that does not bristle with determinants.” The Royal Society’s Catalogue of Scientific Papers (vols. i. - viii. ) shows a list of 49 papers by Spottiswoode, to which must be added about 66 more, the titles of which have not yet been printed in that catalogue. These were published principally in the Philosophical Transactions, Proceedings of the Royal Society, Quarterly Journal of Mathematics, Proceedings of the London Mathematical Society, and Crelle, aud one or two in the Comptes Rendus of the Paris Academy. Another list of his papers, arranged according to the several journals in which they originally appeared, with short notes upon the less familiar memoirs, is given in Nature, vol. xxvii. p. 599.