Beating untrodden paths: James Gregory and his Italian readers

Abstract

In this paper, I shall reconstruct the stay in Italy of James Gregory (1638–1675), Regius professor of mathematics at St Andrews. According to a standard account, Gregory spent four years (1664–1668) in Padua, as Stephano degli Angeli's student. However, this claim is problematic. First, Gregory's stay in Padua is confirmed only for the years 1667–1668. Second, the existence of a partial scribal copy of Vera quadratura circuli, ellipseos et hyperbolae in sua propria specie inventa et demonstrata, Gregory's debut work in the domain of quadrature problems, as well as a number of letters preserved at the National Library of Florence, suggest that relations between Gregory and Italian mathematicians were more complex and varied than have been suspected. On the basis of new, albeit scarce, textual evidence, I shall advance a few conjectures regarding scholars and philosophers that Gregory could have met in Padua, Rome and perhaps Florence.

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Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 For the importance of Gregory for Leibniz's mathematical thought, especially regarding the impossibility of squaring the circle and the emergence of the mathematical concept of transcendence, see (Lützen 2014; Serfati 2018; Crippa 2019). Gregory's mathematical contributions and his influence on Newton's mathematics have been studied in (Malet 1989).

2 See, for instance (Malet 1990; Simpson 1992).

3 Gregory's plan to meet Huygens is documented by letters exchanged between Robert Moray to Huygens; see (Huygens 1888–1950, vol 4, pp 330, 351). See also Gregory's letter to Huygens from 28 September 1667: (Huygens 1888–1950, vol 6, p 154).

4 The manuscript can be consulted online at the website of the National Library of Florence: https://opac.bncf.firenze.sbn.it/opac/controller.jsp;jsessionid=351F7C3881305F4F2BBAFB31BABCB8EA?action=notizia_view&notizia_idn=cf96006607&query_action=search_basesearch&query_filterterm=&query_position=6&query_maxposition=16&query_orderby=&query_filterterm=&query_querystring_1=james+gregory&query_fieldname_1=keywords. For an overview, see also (Truci and Zangheri 1994, 315–319).

5 Whiteside's account contains a few mistakes. Gabriele Manfredi, who was born in 1681, had actually met and studied with degli Angeli, but long after Gregory's stay in Padua. Degli Angeli was Cavalieri's, rather than Torricelli's student.

6 Apart from Whiteside, whose source was probably (Brown 1921, 165), see also (Baron 1969, 229), and (Bascelli et al. 2018, 135): ‘Gregory studied under Italian indivisibilists and specifically Stephano degli Angeli during his years 1664–1668 in Padua’. Among partial exceptions to this common opinion, we have (Simpson 1992, 83), where it is claimed, with no justification, that Gregory's stay in Padua ended in May 1667. Another exception is (Rawson 2015), quoted below in the main text.

7 A second chair of mathematics was created only in 1741 (Borgato 2006, 136).

8 We possess no further detailed information about the teaching of mathematics during Gregory's time, because the university notices (rotuli) from the years 1664 to 1667 are not extant.

9 See for instance, (Malet 1989, 27): ‘[Gregory] appears truly conversant with the content of the works of Angeli, whom he mentions appreciatively more than once’.

10 It is worth mentioning the case of Newton, whose copy of the Geometriae pars universalis is known to bear ‘dog-ears’ as a sign of an intense study (Malet 1989, 29).

11 According to (Baron 1969, 230–231), among Gregory's main sources there were Jesuit mathematicians such as André Tacquet (1612–1660), and Grégoire of St. Vincent (1584–1667).

12 ‘His examinavimus tria spiralium genera; quorum primum idem est cum illis duobus, quae mensuravit R.P. Stephanus de Angelis in libro suo de Spiralibus et in fine lib. 5 de parabolis; secundum quoque idem est cum illis duobus, quae dimensus est in libro de Spiralibus inversis; tertium etiam excogitavit idem Mathematicus sagacissimus, illudque mihi nuper communicavit’.

13

E notino lorsignori quello, che ho sentito dire dal Sig. Giacomo Gregorii, Scozzese, eccellentissimo Mattematico. Diceva egli essergli stato riferito da quelli, che in Inghilterra cavano il carbone di minera dentro cavità profonde, che ivi muovo[no] con gran agevolezza pezzi grandissimi, i quali poi condotti all'alto non li possono muovere se non con molta maggior forza.

For Gregory's interest in the effects of weight and magnetism on weight, see (Malet 1989, 32).

14 Padua, Biblioteca Del seminario arcivescovile, Codice 634, p 218. The list is printed in (Brown 1921); see also (Enriques 1939, 465).

15 Our findings agree with (Malet 1989, 27).

16 As we read in the review that appeared in 1668 in the Philosophical Transactions: ‘Only few of these Books were printed by the author for his own use, and that of his Friends’ (Account 1668a, 641).

17 Florence, National Library, MS Gal. 213.

18 ‘[F]igura aere incisa inserenda est inter pagina 12 et 13, si modo lector illam inserere velit; commodius tamen tenetur soluta, ut diversis propositionibus inserviat’.

19 The Huntington copy of the Vera quadratura is catalogued here: https://catalog.huntington.org/record=b1026300. Florence National Library also preserves a 1668 copy of the Vera quadratura bound together with the Geometria pars universalis which contains the sheet with illustrations.

20 Proposi.ni. Jacopo Gregorij Scozzese della Quadratura del Cerchio e dell'Iperbola, Mandatami dal Ser. Sig. Principe Leopoldo a dirne mio parere. Mand. poi [a] stampare in Padova nel 1667, 4 con altre proposi.ni aggiunte e tale titolo: Nova circuli et hyperbolae quadratura in propria sua proportionis specie inventa et demonstrata a Jacobo Gregorio Abredonensi Scoto. Patavii. Ex typis Jacobi de Cadorinis. 1667 in 4.

21 Gregory's definition of a convergent sequence differs from our own mainly in one respect: for him, a convergent sequence was actually a two-term recursion $(a_n,b_n)$, such that the sequence $\left(a_n\right)$ was monotonically increasing, while $\left(b_n\right)$ was monotonically decreasing (or vice versa), and the sequence of their differences tended to 0. This notion of convergent sequence was clearly abstracted from the well-known construction of a sequence of polygons approximating a sector of the circle (or the hyperbola). See (Crippa 2019, 52) for further details.

22 ‘All the propositions of my book De Circuli et Hyperbolae Quadratura, after the eleventh, are for facilitating the practic’ (Turnbull 1939, 50–51: Gregory to Collins, 26 March 1668).

23 ‘Advertendum est Scriptorem harum rerum imperitum hoc + signum additionis cum x littera quantitatem ignota significante saepius confundere, plerumque etiam notam hanc 2 significantem potestatem quadraticam saepe sumere in eadem linea cum quantitate’ (Florence, National Library, Gal. 213, fol 46r).

24 ‘In Italy the Royal Society has an excellent priviledge of receiving, and imparting Experiments, by the help of one of their own fellows, who has the opportunity of being Resident there for them, as well as for the King’ (Sprat 1667, 126). This fellow was without doubt John Finch.

25 For an overview of Finch's life and activities see (Villani 2005).

26 As we can read from Finch's accounts to Leopoldo, the state of the art in Padua university was judged very poor: ‘Qui in Padova trovo lo Studio tutto sconcertato, o per culpa di Lettori, o Riformatori non ne so dire, ma certo è che li Professori megliori sono a segno tale disgustati da questa Sig.ria che sono di partirsene volentieri trovando partito’ (Florence, National Library, Gal 277, fol 23v).

27 On the relations between these two scientific societies, see (Villani 2005).

28 It is also worth mentioning that, in a letter to Leopoldo de' Medici, Magliabechi listed both degli Angeli and Rinaldini among ‘Amici e servidori di V.A.R.' (Mirto 2012, 187).

29 More precisely, Rinaldini started his teaching duties in Padua only in October 1667, at least according to (Papadopoli 1726, vol 1, p 172).

30 On Rinaldini, see the classic (Patin 1982, 52). More recent presentations of Rinaldini's life and scientific activity can be found in (Pepe 1993; Petti 1996; Brigaglia 1999; Giannini 2016).

31 The letter is published in (Renaldini 1682, 73). In this letter, whose subject was a teardrop-shaped glass for experiments, Rinaldini also referred to previous exchanges between the two scholars, which seem to be lost.

32 According (Mirto 2012, 188), Renaldini left Florence for Padua in 1666. This would add plausibility to the hypothesis that Renaldini transmitted Gregory's manuscript to Florence. However, I cannot find any confirmation of Mirto's statement in the existing literature.

33 Florence, National Library, Gal. 278, fol 76r; Gal. 315, fol 407r.

34 ‘Credo che il Sig. Giacomo Gregorio Scozzese avrà inviato a V.A.R. il suo libro de Quadratura circuli et hyperbolae nuovamente stampato in Padova: ma quando ciò non sia seguito, ne ho due copie, una delle quali la manderò subbito a V.A.; essendo l'Autore di sottile ingegno, ed inventivo. Desidera ch'io dica il mio parere con libertà, ma richiede tanta fatica e tempo la multitud.ne dè numeri che bisogna fare per assicursarsi della verità, che non mi dà l'animo di stancarmici la testa, che del continuo devo tenere applicata alle faccende di queste con.ni, dove intervengo’ (Florence, National Library, Gal. 278, fol 76r).

35 ‘Mando a V.A.S il libro dello Scozzese de Quadratura hyperbolae’ (Florence, National Library, Gal. 315, fol 407r: Rome, 12 November 1667).

36 In a copy of Ricci's book preserved in the G W Leibniz Bibliothek in Hannover (see: https://opac.tib.eu/DB=3/SET=1/TTL=1/SHW?FRST=5) and in a copy preserved in Switzerland (https://www.e-rara.ch/doi/10.3931/e-rara-7457) Ricci's Exercitatio is bound together with Mercator's treatise and with Gregory's Exercitationes geometricae, published in late Summer 1668. It seems likely that Gregory's book was sold both bound together with Ricci's and Mercator's texts, and separately. Perhaps Gregory had brought or sent a copy of Ricci's book to London for the reprint and had chosen a similar title for his own small book. I thank Siegmund Probst for this insightful suggestion.

37 A noteworthy exception is (Rotta 1967).

38 For the history of this manuscript, see (Malet 1989, 99ff).

39 Ricci's work had certainly appeared in print in July 1666 or later, as mentioned in the preface to the first edition (Ricci 1666, preface, unnumbered).

40 ‘quisquiliis quibusdam opticis’.

41 ‘trita, vulgaria, ab Archimede & Lalouvera soluta, neque responsione digna’.

42 ‘In philosophiae, quam Matheseos, studiis potius versatus’.

43 Massa Esteve's article also contains a detailed analysis of Mengoli's technique of quadrature.

44 ‘il piú vivace geometra, che habbia io mai letto sino all'hora presente’.

45 As we read in the letter: ‘resi un'esatta censura e approvazione, secondo l'arte’ (Baroncini and Marta 1986, 47–48). Mengoli also added that Cassini possessed a copy of the book. We thus infer that Cassini must have been another recipient of one of the 150 copies that Gregory had originally ‘scattered through the world’.

46 The only review of the Vera Quadratura published in Italy appeared in the Giornale dei letterati (Vera Quadratura 1668b). It is an Italian translation of another, well-known review which had appeared in the July 1668 issue of the Journal des sçavans (Vera Quadratura 1668a), at the hands of Christiaan Huygens.

47 Apart from the already-mentioned review by Huygens, the Vera Quadratura and Geometriae Pars Universalis were also reviewed in appreciative terms in the Philosophical Transactions (Account 1668a, 1668b).

48 Bascelli et al. (2018) and, in particular, Collins' letter to Gregory, dated 25 November 1669:

One Mr. Norris a Master's Mate recently come from Venice, saith it was there reported that your bookes were suppressed, not a booke of them to be had anywhere, but from Dr. Caddenhead to whom application being made for one of them, he presently sent him one (though a stranger) refusing any thing for it' (Turnbull 1939, 74).

Afterwards, in a letter from 1670, Collins noted that: ‘Father Bertet sayth your Bookes are in great esteeme, but not to be procured in Italy’ (Turnbull 1939, 107). The Father Bertet mentioned here was Jean Bertet (1622–1692), a Jesuit, as well as one of Leibniz's acquaintances and correspondents. For a short biography, see (Agostini 2009, vol 1, p 11).

49 For instance, John Finch's papers, deposited at the Leicestershire Record Office, may contain further information regarding his alleged meeting with Gregory (Villani 2005, 164).

Additional information

Funding

This work was supported by Grantová Agentura České Republiky (CZ) [9-03125Y].