Manuscript volume, 1694: Rechenbuch. An arithmetic book containing problems, including calculations for finding the date of Judgement Day, the Golden Number, etc, together with astrological information. Some of the problems are set out in verse.

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 They form a complementary series to the Early Letters, both of which were superseded by the Letters and Papers.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.

Copy of the draft of the first work published by Augustus de Morgan, 'The Elements of Arithmetic', 1830.

In Arithmetica diversche questyen met hare solutien: door C.J. Broersz, liefhebber der Reeckenconst ... Tot Haarlem, Bij Gillis Rooman, woonende in de Jacobine-strate, in de vergulden Parsse, Anno 1590. The transcription of the book by Broersz occupies the first 33 leaves, and is signed F.[?] J. Riett, 1612. Ff. 35-88 contain further arithmetical problems, including some in verse form illustrated by pen-and-ink drawings.

Letter from Baldassarre Boncompagni-Ludovisi, 1858, [To Augustus De Morgan]. Thanking De Morgan for his letter of 14 November 1858, giving information about Francesco Galigai's 'Pratica d'Arithmetica' (1548), and asking for further details. Letter enquires about the library of Dr [Daniel Mitford] Peacock and asks De Morgan to get Count [Guglielmo] Libri to send catalogues of sales he was to hold [in March and August 1859] in England and in Paris. Enquires about further editions of De Morgan's 'Arithmetical Books' (1847) and offers to get one printed in Rome.

Of the twelve volumes in this class, two relate to the society of East India Commanders; one, 1780 to 1833, records wagers between members and promises of gifts to the society upon such eventualities as marriage or leaving the service; the other is an account book, 1825 to 1879. The remaining eight volumes are 'Elements of Navigation' by pupils of Christ's Hospital Mathematical School; the earliest is dated 1723 and the latest is c 1845. All are very carefully executed and the subjects include arithmetic, geometry, astronomy, gunnery and navigation; the majority are illustrated with diagrams, maps, pen and ink sketches or watercolours.

Manuscript volume, 17th century, containing: Mohammed Ibn Al-Hussain Karkhi, Kitab al-kafi fi 'ilm al-hisab wa-'ilm al-misahah wa-al-khawafi (Treatise on arithmetic, geometry, land surveying etc)

(1) Letter from William Frend De Morgan of The Vale, King's Road, Chelsea to H F Heath, Librarian of the University of London, 20 Oct 1900. (2) Letter from William Frend De Morgan of Via Lorenzo Magnifico 25, Florence, Italy to H F Heath, Librarian of the University of London, 4 Nov 1900. Both letters relate to De Morgan's gift of notes and additions made by his father, Augustus De Morgan, for his Arithmetical Books.

Both letters are autograph, with signatures. Filed with a typescript copy of Dr Heath's reply to the first letter.

Manuscript volume, 15th or 16th century, containing the 'Perspectiva Communis' of John Peckham, in three parts, beginning 'Inter philosophie consideracionis studia ... '. Bound (perhaps from the first) with two printed works, the Arithmetica of Jordanus Nemorarius, edited by Jacques le Fêvre (Johannes Higman and Wolfgang Hopyl, Paris, 1496), and the Geometria speculatiua of Bradwardine (Paris, 1495).

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'.