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Arithmetica universalis

‘Small in bulk, and yet ample in Matter, not too much crowded with Rules and Precepts, and yet well furnished with choice Examples’.

Preface to Arithmetica universalis [translated into English by Joseph Raphson in 1720].


Arithmetica 1707 titlepage.

Worth had two editions of Newton’s Arithmetica universalis: the first edition printed at Cambridge in 1707 and Willem ‘s Gravesande’s 1732 Leiden edition. This was perhaps Newton’s least favourite work yet despite that it was reprinted several times during his lifetime. Though many of Newton’s mathematical breakthroughs had taken place in the 1660s he had been loath to publish on the subject, preferring to ‘communicate’ them in scribal form. The fact that the 1707 edition was published anonymously indicates that Newton viewed it with reserve. His distaste for it may have been because he felt forced to write it in order to gain his colleagues’ support in the elections to the 1705 parliament. Equally, as the preface makes clear, the work had ‘not being immediately intended for the Press’ and Newton may well have felt that the manuscript on which it was based needed some revision, given that his lectures on which it was supposedly based had been delivered ‘almost thirty years before’.


Arithmetica 1707 diagram.

It seems clear that if there was any guiding force behind the 1707 Arithmetica universalis it was Newton’s devoted follower William Whiston (1667–1752), his successor to the Lucasian chair at Cambridge. In his preface Whiston declared that the work was based on Newton’s Lucasian lectures on algebra but Stedall (2006) suggests that the Arithmetica universalis was actually based on notes Newton had written about Gerard Kinkhuysen’s Cartesian Algebra ofte Stelkost (1661). Whiteside (1972) argues that far from being a clear record of actual lectures given over a set period of time, the manuscript source for the Arithmetica universalis represents instead Newton’s summary of them at a later time. He suggests that the source material dated to the period 1673-83 and was written up by Newton c 1683-4. The resulting work was a strange combination of basic material (which accounted for its popularity) and some of Newton’s mathematical insights (which drew the attention of scholars such as ‘s Gravesande). Initially it made little impact, not even being reviewed in the Philosophical Transactions. Though it later became a standard text in Britain (due not least to an English translation of 1728) it was not influential on the continent.

Newton’s ambivalent attitude to the Arithmetica universalis also reflects the change of direction his mathematical studies had taken since the 1670s. The Arithmetica universalis was based on his mathematical work in algebra of the 1660s but this had later been superseded but Newton’s championing of geometrical method over algebraic method. We can see this heightened respect for geometry over its poorer sister algebra in the final section of the book.


Arithmetica universalis (Leiden, 1732), titlepage.
Arithmetica universalis (Leiden, 1732),
note by ‘s Gravesande.

Worth’s second edition of the Arithmetica Universalis was printed at Leiden in 1732 by Willem ‘s Gravesande (1688-1742) who had been appointed professor of mathematics at Leiden on Newton’s recommendation. As a good Newtonian ‘s Gravesande had given his inaugural lecture at Leiden on the importance of mathematics to physics specifically and science generally but, as his teaching career at Leiden demonstrates, ‘s Gravesande’s reception of Newtonianism was quite unusual (at least in an English context) and certainly changed over time. One of his most famous books, his Matheseos universalis elementa of 1727 had been devoted to explaining Newton’s Arithmetica universalis and this is referred to in ‘s Gravesande’s note to his readers. Here he explains that, just as Whiston’s 1707 edition had included work by Edmond Halley (1656-1742) on equations which had been communicated to the Philosophical Transactions so too ‘s Gravesande was augmenting his own 1732 edition by likewise including not only Halley’s text, but also relevant texts by John Colson’s (1680-1759), Abraham de Moivre (1667-1754), Colin MacLaurin (1698–1746) and George Campbell. Colson’s ‘The universal resolution of cubic and biquadratic equations’ had originally been published in the Philosophical Transactions in 1670 and Abraham de Moivre’s ‘analytic solution of certain equations’ in 1707. MacLaurin’s ‘A letter … concerning equations with impossible roots’ and George Campbell’s extension of the argument were more recent, having been published in the Philosophical Transactions in 1726 and 1728 respectively. As Whiteside (1972) states, ‘s Gravesande’s 1732 edition of the Arithmetica universalis became the standard continental edition.

Arithmetica universalis (Leiden, 1732),detail of a diagram.


Grattan-Guinness, Ivor (1997), The Fontana History of the Mathematical Sciences (London).

Guicciardini, Niccolò (2002), ‘Analysis and Synthesis in Newton’s mathematical work’, in I. Bernard Cohen and George E. Smith (eds) The Cambridge Companion to Newton (Cambridge), pp.308-328.

Guicciardini, Niccolò (2004), ‘Isaac Newton and the publication of his mathematical manuscripts’, Studies in the History and philosophy of Science 35, pp. 455-470.

Stedall, J. (2006), ‘Newton’s Algebra’, Mini-Worshop: On the Reception of Isaac Newton in Europe, Mathematisches Forschungsinstitut Oberwolfach Report no. 10.

Tattersall, James J. (2004) ‘Colson, John (1680–1759)’, Oxford Dictionary of National Biography, Oxford University Press.

Whiteside, D. T. (1972), The mathematical papers of Isaac Newton (Cambridge).



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