Correspondence on various mathematical theories and calculations, mainly pure, but some applied, and miscellaneous papers on scientific matters. The correspondents include, among others, A. Cayley, J.G. Galle, A. de Morgan, I. Todhunter, W.H. Miller, J.B. Biot, W.R. Hamilton and A.R. Clarke. The papers include a 'Supplement to a Proof of the Theorem that Every Algebraic Equation has a Root by G B Airy'; a paper by B. Valz titled 'Effai de resolution des Equations par les series et le logarithmes' (The effect of resolving the equation by series and logaritHMS); two printed papers by H. Perigal on his 'Contributions to Kinematics' and his diagrams of 'Transformations of Kinematic Bicircloids'; a paper by A. de Morgan 'On an unfair suppression of due acknowledgement to the writings of B Gompertz'; an examination paper from University College, London, for the Andrews Scholarship of papers on 'Recent Terminology in Mathematics'; a paper in French on the theory and usage of logaritHMS by M.M. Lefort; and various papers and letters on quadrature of the circle.