Correspondence on various mathematical theories and calculations, mainly pure, but some applied, and miscellaneous papers on scientific matters. The correspondents include, among others, A. de Morgan, I. Todhunter, J. Challis, A.R. Clarke, W.H. Miller, J. Henry and Colonel Sir H. James. The papers include newspaper cuttings giving a brief history of the study of pi, with a table by W. Shanks showing pi to 607 decimal places; a paper by Airy titled 'On a Supposed Failure of the Calculus of Variations'; a paper on the Solution of a Problem in the Calculus of Variations by J. Challis; a printed paper on computing the time of descent in a circular arc; papers by Airy titled 'Section II History of Trigonometry' and dated 3 March 1825; printed papers related to the law of human mortality; manuscript papers 'On the Algebraical and Numerical Theory of Errors of Observations and the Combination of Observations' by Airy; and a paper by J. Davidson on 'Lunar Motion, Magnetism and Breakwaters'.