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Papers of Harry Bateman including three notebooks, two (numbered six and seven) containing notes from the lectures of E T Whittaker at Trinity College Cambridge 1903-1904; one containing formulae connected with the Legendre and Bessel functions, with Trinity examination paper 1880 and printed testimonial by Rutherford.

Mathematical and scientific papers of George Boole.

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 They form a complementary series to the Early Letters, both of which were superseded by the Letters and Papers.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 Thomas George Cowling including correspondence with Neil Oscar Weiss, 1964-1989 and P A Gilman, 1980-1982.

A collection of tracts on fortification and geometry, 17th and 18th centuries: Folio 1 Della Fortificatione; Folio 53 Geometrica Prattica; Folio 83 Della sfera; Folio 104-115 Blank sheets and oddments; Folio 116 Del Misurare da Contano; Folio Il quaduarete Geometrico; Folio Aritmetica.

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

A system of mathematical tables inscribed on title page ' James Elgar May 8th 1756'. Originally entitled 'A system of mathematical tables by James Elgar and John Newton' and bound with 'Tabulae Mathematicae' by John Newton.

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.

Copy of 'The radix; a new way of looking at logarithms...in five problems by Robert Flower' made in 1877 by Alexander John Ellis with supplementary notes by Ellis including biographical information on Flower.

Papers of the Gregory family. Volume One includes writings by Sir Isaac Newton, 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 contains letters and papers of the Gregory family: David Gregory of Kinnairdie; James Gregorie; David Gregorie; and Charles Gregory (Professor of Mathematics at Edinburgh University); also including some papers of Sir Isaac Newton.

Papers on mathematics and physics collected by the Reverend Samuel Horsley DD, FRS.

Some correspondence, papers, notebooks and publications of Sir James Hopwood Jeans. Early manuscripts in the series relate to Jean's education at Trinity College, Cambridge, in the form of early lecture notebooks, largely on mathematical topics. A collection of letters, 1901-1907, documents his treatment for tuberculosis at Ringwood Sanatorium, where he completed work on the theory of gases; correspondents included G H Hardy and Adam Sedgewick among others. Jean's engagements in popularizing science are represented by proofs and typescript of lectures and essays, especially those written for the BBC, 1934-1935, together with associated letters and press cuttings. The series contains collections of offprints, reprints, and extracts of published works by Jeans and others, notably from the Philosophical Transactions and the Philosophical Magazine.

The correspondence of Sir John William Lubbock, providing information on a wide range of Lubbock's contemporaries, not exclusively in the field of science. The largest collections of scientific letters are from George Biddell Airy (113 letters), John George Children (128 letters), Joshua Alwyn Compton, 2nd Marquis of Northampton (98 letters), G P D de Pontecoulant (67 letters) and William Whewell (87 letters). There are smaller but no less important groups of letters from Charles Babbage, Francis Baily, Francis Beaufort, Charles Darwin, John Couch Adams, J F W Herschel, Baden Powell and W H F Talbot.

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.

Papers of Thomas Muir comprising notebooks: a series of six loosely-bound collections of notes and drafts; 'Articles for Pogendorff VI', notes and first draft completed 1933; 'Oblong Arrays' material for paper 304 1-30, 31-60, 61-93 and 'Dunkel Notes', material for paper 305 1-30, 31-51.

Manuscript volume of 'Philosophiae Naturalis Principia Mathematica', by Isaac Newton. Manuscript written out by Humphrey Newton, Isaac Newton's assistant and amanuensis, and annotated by Newton and Halley. Manuscript from which the First Edition was printed.

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

Methodo Inverso dos Limites ou Desenvoluimento geral das Funcoens algorithmicas em Series par Francisco de Borja Garcao Stockler.

Correspondence of Dr Brook Taylor, Secretary of the Royal Society on mathematical subjects with Professor Keil, Mr Machin, Bernouilli, the Abbot Conti, the Comte Raymond de Montmort, 1712-1717.

Written in England. Contains the curious characters known as the Boetian contractions, and is a valuable document for the history of arithmetic. The treatise commences on the Roman abacal system, and afterwards proceeds to give an explanation of the Eastern Boetian system with altered characters, and with a complete adoption of the advantages of local position. In Folio 2, recto, line 5, the author says that he will expound 'quaedam de numerorum scientia. regulis uidelicet algorismi' and on Folio 3, verso, he refers to Pithagoras, Nicomachus, Apuleius and Boethius. The text ends on the first line of Folio 34, recto, and is followed by eight lines in a semi-current 13th century hand.

Folio 1, recto ' Prefatio sequentis operis. Vt ait salomon.. punge oculum. et profert lacrimam. punge cor' et profert sensum.....'

Folio 4, verso ' De impari numero..'

Folio 6, verso, line 6, ' Divisio algorismi '. The divisions are given as folows; ' digitus, mediato, multiplicatio, divisio '

Folio 8, recto, title in red in margin, ' Qualiter digitus multiplicat digitum '

Folio 9, recto, line 15 ' Qualiter articulus multiplicatur per articulum '

Folio 9, verso, line 13, heading in red, ' Qualiter digitus multiplicat articulum '

Folio 10, recto, line 10, heading in red, ' Qualiter compositus per compositum multiplicatur '

Folio 10, verso, heading in margin, ' Qualiter compositus multiplicat compositum non eundem habens articulum '

Folio 11, recto, bottom line, heading in red ' Regulo ab uno inequalibus '

Folio 12, recto, line 6, heading in red, ' Regula ab uno in equalibus binario adjecto '

Folio 12, verso, line 8, heading in red ' De duplicatione '

Folio 13, recto, line 5, heading in red, ' De pari adjecto '

Folio 13, verso, line 6, heading in red, ' De impari adjecto '

Folio 14, verso, heading in margin, ' De proposito secundum novenarium dispositum '

Folio 15, recto, line 7-8, heading in red, ' De significationibus et nominibus figarum et locorum '
Transcript extract from line 14 ' Prima itaque omnium est igin. 1. id est unitas. Secunda andras. 2. id est binarius. Tercia ormis. 3. id est ternarius. Quarta arbas. 4. id est quaternarius. Quinta quimas. 5. id est quinarius. Sexta caltis. 6. id est senarius. Septima zemis. 7. id est septenarius. octava celentis. 8. id est octonarius. Nona cemenias. 9. id est novenarius. Singule ataque figure in quocumque loco sint sui ipsius exprimunt significationem. Nunquam enim figura unitatis officium binarii sibi usurpabit, nec binarius ternarii, nec ternarius quaternarii, nec quelibet ceterarum quod alter significare poterit. Ad majorem itaque evidentiam significationes locorum ponamus. primo ita loco posita igin seipsam, id es unitatem, significat. Secundo denarium. Tercio centenarium. Quarto millenarium. Quinto decem milia. Sexto centum milia. Septimo mille milia. Decimo mille mille milia. Undecimo decies milies mille milia. Duodecimo centies milies mille milia. Terciodecimo milies milies mille milia, et sic usque in infinitam singulorum decuplationem locorum extenditur. Similiter autem et andras rimo loco seipsam, id est binarium, significat. Secundo xx. Tercio cc. Quarto duo milia. Quinto xx. Sexto cc. Septima mm. et sic per cetera loca eodem cemate et cetere figure secundum propriam differentiam decuplabutn singulos articulos per loca singula. Est adhucet decima figura cujus nulla significatio est, sed tamen per loca disposita ceterarum significationes auget vel minuit. Dicitur autemzifera, et taliter depingitur 0. Sciendum autem quod omnes figure versus sinistram scribi debent, quia sic vim extendunt. Sed e converso literali ordine legi. '

Folio 17, verso, line 6, heading in red, ' De subtractione'

Folio 18, verso, line 8, heading in red, 'De dupli``catione'.