Manuscript volume containing writings on mathematics, [1668], and entitled 'The use of the rule and compass in drawings of lines, angles, parallels and geometrical superficial figures comprehended in eight practices, together with notes on decimal arithmetic, logarithms, mensuration, fortification, dialling and other branches of mathematics. The handwriting and watermarks are of the seventeenth century, and the date 1688 appears on the verso of leaf 68, where adjoining a 'Table for the declination of the sunne', is written 'Example for the 22th May 1668'. On leaf 112 appears 'A particular of all my mathematical instruments and the severall prices I paid for them'.

Manual of mathematics, including sections on calculating interest, 'brokage' and the purchase of stock, compiled by James Harrison of Leeds in 1775. At the end of the volume are added medical remedies and recipes in various hands, [1828].

Scientific and other papers sent to the Royal Society, presented at meetings of Fellows, or commissioned by the Society. They form a complementary series to the Early Letters, both of which were superseded by the Letters and Papers. Many of these items, referred to as the 'Guard Books', are duplicated in the Register Book of the Society. The classification is a simplified form of the 'Philosophical Transactions' abridgment by John Lowthorp. This arrangement was completed in 1741 by Thomas Birch. The majority of the papers in these volumes are manuscript, but a few printed documents occur throughout the series. Some of the papers are earlier in date than the grant on 15 July 1662 of the First Charter to the Society. The Committee of Trades seems to have been associated with the earlier meetings of those philosophers who subsequently became Fellows, and produced a number of practical papers, some of which were written in 1639 and which are mostly found in Volume 3(i). There are still earlier documents, mostly in Volume 25, which may have been included in the gift, in 1667, of the Arundel Library.

Papers of the Gregory family, volume one includes writings by Sir Isaac Newton, friend of David Gregorie, Savilian Professor of Astronomy at Oxford, the volume is entitled 'Notae in Newtonii Principia Mathematica Philosophiae Naturalis' and his 'Theory of the Moon' which was incorporated in the 'Astronomia Physica' published by the Savilian Professor of Astronomy at Oxford; volume two is a volume of letters and papers of the Gregory family, including some papers of Sir Isaac Newton; David Gregorie of Kinnairdie; James Gregorie author of 'Optica Promota'; David Gregorie, Savilian Professor of Geometry at Oxford; James Gregorie Professor of Mathematics at Edinburgh; and Charles Gregory, Professor of Mathematics at Edinburgh University.

Tables prepared by Sir George Everest for the use of the Great Trigonometrical Survey of India.

These are letters and papers of various members of the Gregory family including David Gregorie of Kinnairdie; James Gregory, author of Optical Promota; David Gregorie, Savillian Professor of Geometry at Oxford University; James Gregorie, Professor of Mathematics at Edinburgh University and Charles Gregory.

Mathematical works of Pappus Alexandrinus, stated by JO Halliwell that this was copied from Savilian MS number 9 for William Jones the mathematician.

A commentary on Newton's 'Principia' by Nicholas Fatio, with an introductory note in English by Dr Johnstone about Fatio, his life, religious beliefs, death and dispersal of his books.

A complete ternary square table up to 256,000 followed by a quaternary square table by Solomon Moses Drach, 1866.

Two manuscripts by Louis Joel Mordell: 1. 'On the geometry of numbers in some non-convex regions' 60 pages, some manuscript corrections and 2. ' Hardy's mathematician, apology' review article 10 pages, photocopy.

A collection of letters to Abraham Sharp between the years 1701-2 and 1719.

Papers of Thomas Wallace, 1719, comprising a volume written in Latin, containing case notes of patients; lists of symptoms, diseases, and treatments; rules and definitions of arithmetical progression; an article describing the volume, by E Muirhead Little, from the British Medical Journal, 8 Dec 1928; and a letter from the donor, W Reeve Wallace, 27 Mar 1934.

Papers of Professor Albrecht Fröhlich, 1916-2001, reflecting his mathematical research, publications and correspondence with colleagues. Papers are divided into six sections comprising biographical papers; research papers; publications; papers relating to visits, conferences and lectures; correspondence and theses and examinations.

Biographical papers include obituaries, memoirs and papers relating to memorials including a service in the Chapel of Robinson College Cambridge, 10 Feb 2002; autobiographical papers including copy of Fröhlich's entry for Personal Records of Fellows of the Royal Society; papers relating to Fröhlich's career, honours and awards including his Ph.D. thesis 'On some topics in the theory of representation of groups and in class field theory', and correspondence and papers relating to his university appointments and some of his principal honours, including election to the Fellowship of the Royal Society.

Research papers chiefly comprise mathematical workings, notes and drafts: notably relating to work on the decomposition of primes [1950] and work on Galois module theory and Gauss sums.

Publications comprise published material by Fröhlich, 1950-2000, drafts and unpublished drafts, including work with Charles Terence Clegg Wall on Brauer groups, 'Galois modules and the functional equation', and draft of a longer text entitled 'mathematical theory of ostensive and consequently of empirical predicates'.

Visits, conferences and lectures include papers relating to Fröhlich's conference attendance and overseas visits 1956-1998, documentation of his participation in mathematics meetings held at the Mathematisches Forschungsinstitut, Oberwolfach, Germany and papers relating to Visiting Professorships, including invitations, programmes, and texts or abstracts of lectures.

Correspondence,1950-2000, on subjects including progress of Fröhlich's research, discussions of problems with mathematical workings and drafts of papers. Notable correspondents include Fröhlich's Ph.D. supervisor Hans Arnold Heilbronn, colleagues Jean-Pierre Serre, Olga Taussky-Todd, Stephen Virgil Ullom and Charles Terence Clegg Wall, and former research students Colin J. Bushnell, Martin J. Taylor and Stephen Mark Johnson Wilson.

Theses and examinations comprise, theses of Ph.D. students supervised or examined by Fröhlich, including his research students at King's College London and examination papers, some with Fröhlich's manuscript calculations of answers, 1980-1990.

Papers of Hudson comprising 'Angle Generator for use in the study of trigonometry' by W B Bushell, 1873; a letter from H Orfleur concerning surfaces, with photograph of geometrical model attached, 1898; invitation to Hudson from Principal Henry Wace to the Medical Department Prize Distribution, with geometrical drawing on the reverse; mathematical model (quadratic surface: ellipsoid).

Collection comprising three manuscript notebooks of theoretical problems relating to mechanics compiled by James Clerk Maxwell mainly during his employment at King's College London, including exam questions on aspects of mechanics and geometry, 1860-1865; problems on the collision of elastic spheres and the dynamic relation between other solid bodies; the refraction of light; the relationship between heat and volume in materials; attraction between particles, 1860-1873; notes on the works of leading authorities on mathematics and physics including Arthur Cayley, Franz Neumann and Siméon Poisson, 1867-1873; bibliography of textbooks mainly on mechanics, [1860-1865]; list of scientific instrument suppliers, [1860-1865]; outline of the contents of Maxwell's Treatise on electricity and magnetism (1873).

Letter from Baldassarre Boncompagni-Ludovisi, 1856 to [To Augustus De Morgan], asking for a copy of 'The Elements of the Mathematical art commonly called Algebra, expounded in two books, by John Kersey. 1717'.