David Gregory, John Arbuthnot and their roles in the early development of probability in Great Britain

Abstract

The first English translation of Huygens' (1657) De Ratiociniis in Ludo Aleae was anonymously published in 1692; since the early eighteenth century it has been assigned to the Scottish physician John Arbuthnot (1667–1735). We have found evidence showing that Arbuthnot's work on probability was influenced by Archibald Pitcairne (1652–1713) and, most significantly, by David Gregory (1659–1708). Manuscripts found in Gregory's collection at the Edinburgh University Library suggest that Gregory had a key role in the translation. The circumstantial evidence presented in this article reveals that the circle of individuals led by Gregory, Arbuthnot and Pitcairne probably originated earlier than previously established and that the role of Arbuthnot was more important than originally thought. We have also shown that the first developments of probability theory in Great Britain took place in Scotland thanks to David Gregory.

Acknowledgments

The authors would like to thank two anonymous referees for their comments. Marcio Diniz would like to thank Mirian Ou, William Ou, Osvaldo Anacleto and the faculty and staff members of the Department of Statistical and Actuarial Sciences of Western University.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 See (Huygens 1920, 3–4). Even though the Pascal–Fermat correspondence preceded Huygens's book, it was only partially published in (Fermat 1679). (Pascal 1665) also has parts dedicated to probability. Cardano's work and Galileo's letter were also published after 1657.

2 See (Browne 1714) and (Todhunter 1865).

3 See (Guerrini 2008).

4 See (Anderson 1898).

5 According to Guicciardini (2004), without due acknowledgement to his uncle.

6 See (Du Rieu 1875, 631).

7 Between 1675 and 1683 mathematics was taught at Edinburgh by a student called James Young, acting only as tutor. See (Stewart 1901).

8 See (Eagles 1977).

9 See (Guerrini 2004) and (Eagles 1977).

10 See (Guerrini 2008).

11 Gregory was writing to Newton at least from 1684, with no response. See (Turnbull 1960a).

12 See (Beeley 2020) for more details on Gregory's election to Oxford.

13 See (Aitken 1892).

14 (Aitken 1892). Ross (2006) reports that Arbuthnot went to London in 1691, but from his correspondence we only know for certain that he was in the city by 1695.

15 See (Arbuthnot 1710), a paper on the sex ratio.

16 See (Anderson 1898), 251–253.

17 Pate, a rich and influential man, hosted Arbuthnot in London while he worked there as mathematics tutor.

18 See (Ross 2006, 47). Newton's recommendation letter written in 1691 in Gregory's favour was addressed to Charlett, then a Fellow of Trinity College.

19 See (Ross 2006, 49), (Shuttleton 1995) and (Johnston 1979).

20 See (Pitcairne 1715) for his complete works.

21 See (Hutton 1815, 605).

22 According to Bellhouse (2001) Arbuthnot had the first 1718 edition of De Moivre's Doctrine of Chances, De Moivre's Miscellanea Analytica published in 1730 and other mathematical books. In (Arbuthnot 1701), an anonymous pamphlet later attributed to him, he explains the usefulness of mathematics and recommends its teaching, among other reasons, for the influence it has ‘on philosophy and all useful learning, as well as the concerns of the public’. When discussing the ‘more immediate usefulness of mathematics in civil affairs’, he does not forget to write a paragraph underlining the importance of probability to gamblers or gamesters. According to Beattie (1935, 337), Arbuthnot himself was a good card player.

23 Edinburgh University Library (EUL) Ms Dc.1.61 (Coll-33, Folio C), 153. One-page manuscript dated ‘Leyden, 1679’. The last paragraph says: ‘Experimented and demonstrated 10 March 1680’.

24 EUL Ms Dc.1.61 (Coll-33, Folio C), 84. One-page manuscript dated on top ‘30 Septr. 1691’.

25 EUL Ms Dc.1.61 (Coll-33, Folio C), 49, one page.

26 EUL Ms Dc.1.61 (Coll-33, Folio C), 22.

27 ‘Item si a eveniat p fors; b, q fors; et c, r fors et quilibet casus sit eque facilis erit expectatio = $\frac{pa+qb+rc}{p+q+r}$. Et sic deinceps.’

28 Cambridge University Library, MS Add. 4000, 90r–92r.

29 See (Sheynin 1971) and (Bellhouse 2016) for more on Newton's notes on probability.

30 Simpson (1740) was the first to publish the solution.

31 See (Aitken 1892, 17–18).

32 See (Westfall 1980, 523) and (Turnbull 1960b, 336).

33 In 1684 Gregory sent Newton a copy of the Exercitatio, his first publication, based on his uncle's results. See also (Turnbull 1960a, 400) for an undated manuscript of Newton mentioning the works of James (uncle) and David Gregory.

34 John Freind (1675–1728) was English.