A Fruitful Exchange/Conflict: Engineers and Mathematicians in Early Modern Italy

Summary

Exchanges of learning and controversies between engineers and mathematicians were important factors in the development of early modern science. This theme is discussed by focusing, first, on architectural and mathematical dynamism in mid 16th-century Milan. While some engineers-architects referred to Euclid and Vitruvius for improving their education and argued for an institutional reform of their profession, Girolamo Cardano and other mathematicians explained the De architectura and studied the inventions of the arts. Attention is drawn, then, to the entrance in the field of hydraulics of Benedetto Castelli, a Galilean mathematician who criticised some basic engineering methods but also relied on engineering experience. This mix of criticism and appropriation of engineering notions was particularly highlighted by Domenico Guglielmini, who saw in the study of the nature of rivers the key for their control and improvement. A further facet was the growing distinction among the practitioners. In 16th-century Milan professional distinctions appeared in a grand scale cadastral survey of the whole state and in the establishment of a professional body of engineers-architects and land surveyors. It was not a peculiarly Italian trend. In the postscript some remarks are added to show that a partly similar differentiation was also taking place among E.G.R. Taylor's mathematical practitioners of Elizabethan England.

Acknowledgements

I wrote a first version of this paper in 2011, while I was a visiting scholar at the Department of History and Philosophy of Science of the University of Cambridge. I thank Simon Schaffer, with whom I discussed the broad theme of my research, who kindly acted as my sponsor. I also thank Francesco Repishti and Mario Signori for archival and bibliographic information, David Cahan for his encouragement to write this paper and the two referees for their useful comments and suggestions. I finally acknowledge my debt to Allen Shotwell, who has generously accepted to revise the English text.

Notes

¹The Renaissance Vitruvian architect took e.g. part in engineering activities such as devising engines, measuring land, designing fortifications and waterworks (see below, § 2.1). And, conversely, ‘the word engineer (…), when it came to the design and construction of building’, was ‘virtually interchangeable with architect’ (J.R. Hale, Renaissance fortification: art or engineering?, London, 1977, 18). As for the diversity within the group of Renaissance mathematicians see, for the cases of Italy and England, Mario Biagioli, ‘The social status of Italian mathematicians, 1450-1600’, History of science, 27 (1989), 41–95 and Stephen Johnston, ‘The identity of the mathematical practitioner in 16th-century England’, in Der “mathematicus”. Zur entwicklung und bedeutung einer neue berufsgruppe in der zeit Gerhard Mercators, edited by Irmgarde Hantsche (Bochum, 1996), 93–120.

²In this paper I consider early-modern engineers and architects as belonging to a single professional group. An early institutional example of this kind of merging is given by the 16th-century University of Architects, Engineers and Land Surveyors of Milan (see below, § 2.3).

³Cesare S. Maffioli, Out of Galileo: The science of waters 1628–1718 (Rotterdam, 1994).

⁴E.G.R. Taylor, The mathematical practitioners of Tudor & Stuart England (Cambridge, 1954), 3–7.

⁵This kind of experimentation was performed in the field, mainly for practical purposes. Some of its results were subsequently appropriated by the mathematicians and transformed in basic notions of the new mathematical philosophy. For more on this theme see Cesare S. Maffioli, La via delle acque (1500–1700). Appropriazione delle arti e trasformazione delle matematiche (Firenze, 2010).

⁶Daniela Lamberini, Il Sanmarino. Giovan Battista Belluzzi architetto militare e trattatista del Cinquecento, 2 vols (Firenze, 2007), I, 317.

⁷Giorgio Vasari il Giovane, Raccolto di varii instrumenti per misurare con la vista , edited by Filippo Camerota (Firenze, 1996).

⁸Another Italian mathematician who, as early as 1539, published the description of some surveying instruments was Girolamo Cardano (see below, § 2.2).

⁹See e.g. Mordechai Feingold, The mathematicians’ apprenticeship: science, universities and society in England, 1560–1640 (Cambridge, 1984), 176–81; Stephen Johnston, ‘Mathematical practitioners and instruments in Elizabethan England’, Annals of science, 48 (1991), 319–44, on 326 and 343; Johnston (note 1), 93–5 and 111; Katherine Hill, ‘“Juglers or Schollers?”: negotiating the role of a mathematical practitioner’, The British journal for the history of science, 31 (1998), 253–74; Jim Bennett, ‘“Braggers that by showe of their instrument win credit”: the Errours of Edward Worsop’, in The Whipple Museum of the History of Science, edited by Liba Taub and Frances Willmoth (Cambridge, 2006), 79–94, on 80–4 and 93–4. Eric H. Ash has warned about ‘the danger of recklessly characterizing all mathematically minded Englishmen as members of a single community of “mathematical practitioners”’ and he has even suggested avoiding ‘using the term mathematical practitioners’: see his Power, knowledge and expertise in Elizabethan England (Baltimore, 2004), 183–4.

¹⁰Vitruvius, On architecture, edited from the Harleian Ms. 2767 and translated into English by Frank Granger (London, 1931–1934), I.Pref.2; Frontinus, The aqueducts of Rome, edited by Mary B. McElwain, with a revision by Charles E. Bennett of the English translation of Clemens Herschel (London, 1925), § 25.

¹¹Vitruvius, De l'architecture. Livre VIII , edited by Louis Callebat (Paris, 1973), X, XXXVIII and 167. On Vitruvius’ career see e.g. Vitruvio, De architectura, edited by Pierre Gros (Torino, 1997), X-XIX. Serious doubts about Vitruvius’ first-hand knowledge of aqueducts or surveying have however been cast by some scholars: see e.g. Michael Lewis, ‘Vitruvius and Greek aqueducts’, Papers of the British School at Rome, 67 (1999), 145–72.

¹²Vitruvius, On architecture (note 10), VIII.5.3. The reference is to prop. 2 of Book I of Archimedes’ On floating bodies, where the mathematical behaviour of water is put in the geophysical context of the shape of the world. For a different explanation of this reference to Archimedes’ theory see M.J.T. Lewis, Surveying instruments of Greece and Rome (Cambridge, 2001), 35. On Renaissance illustrations and descriptions of the chorobates, see below § 2.2.

¹³Leon Battista Alberti, On the art of building, translated by Joseph Rykwert, Neil Leach and Robert Tavernor (Cambridge MA, 1988), X.7, 336.

¹⁴This is not to say that there was no scepticism about the encyclopaedic education of the Vitruvian architect. A well known case is that of Alberti, who emphasised that the only arts ‘that are useful, even vital, to the architect are painting and mathematics’: ibid., IX.10, 317. Another example is Pietro Cataneo, I quattro primi libri di architettura (Venezia, 1554), I.1, 1v.

¹⁵ Il codice Ashburnham 361 della Biblioteca Medicea Laurenziana di Firenze: Trattato di architettura di Francesco di Giorgio Martini, 2 vols, edited by Pietro C. Marani (Firenze, 1979), II, § 39, 19–20; Francesco di Giorgio Martini, La traduzione del De Architectura di Vitruvio: dal ms. 2.1.141 della Biblioteca nazionale centrale di Firenze, edited by Marco Biffi (Pisa, 2002), XVII–XIX and CV. Another example is Raphael who in the early 1510s commissioned a translation of De architectura from the learned philologist Marco Fabio Calvo. The extant 16th-century Italian translations of De architectura are listed in Il “Vitruvio Magliabechiano” di Francesco di Giorgio Martini, edited by Gustina Scaglia (Firenze, 1985), 59–64. For a concise account of Italian editions and translations of Vitruvius from the 1480s to the 1520s, see Pamela O. Long, Artisans/practitioners and the rise of the new sciences, 1400–1600 (Corvallis, 2011), 80–93.

¹⁶Hale (note 1), 19ff. In comparison with Renaissance architectural treatises, the subject of fortifications had received little attention in De architectura where it is discussed in a short section of Book I: Vitruvius, On architecture (note 10), I.5.1–8.

¹⁷Jim Bennett and Stephen Johnston, The geometry of war 1500–1750 (Oxford, 1996), 12–3.

¹⁸ Il codice Ashburnham 361 (note 15), I, ff. 27v–33r and II, §§ 102–39, 56–67; on the sources of this treatise of practical geometry see Francesco di Giorgio Martini, La praticha di gieometria dal codice Ashburnham 361 della Biblioteca Medicea Laurenziana di Firenze, edited by Gino Arrighi (Firenze, 1970), 2–3 and Giorgio Vasari il Giovane, Raccolto (note 7), 83–5. Another example is given by the revised edition of the architecture of Pietro Cataneo, to which four new books were added. One of them dealt with geometry and another with perspective: see Pietro Cataneo, L'architettura alla quale (…) sono aggiunti di più il quinto, sesto, settimo e ottavo libro (Venezia, 1567). The architect's geometry was not, in any event, a simple case of reformulation in a less specialized language of some Euclid's propositions. By pointing out the case of Cataneo and other writers on architecture, it has been e.g. recently emphasised that they were able to use Euclid's Elements and, at the same time, to develop a kind of practical geometry which was based on mathematical instruments: see Samuel Gessner, ‘Savoir manier les instruments: la géométrie dans les écrits italiens d'architecture (1545–1570)’, Revue d'histoire des mathématiques, 16 (2010), 1–62.

¹⁹ Di Lucio Vitruvio Pollione de Architectura libri dece traducti de latino in vulgare affigurati commentati & con mirabile ordine insigniti (Como, 1521), 2v–3r (commentary by Cesariano on Vitruvius I.1.1–2). The polemic against the ‘falsi architecti’ is taken up again in the commentary of VI.Pref.6–7 (ibid., 92v).

²⁰‘Essendo così quisti pseudi predicti [i.e. these pseudo architects] si de(vo)no deponere de(l) titulo & cognominarli solum fabricanti murarii vel operarii, & non architecti, perché el è dato da le lege publice di non insignire & decorare indignamente, acioché li veri studiosi di tal scientia possano più magnanimamente pervenire con la summa doctrina al verace & excellente doctorato vel epinome architectonico’ (ibid., 3r).

²¹‘O quanti già sono inganati da se stessi a farse professori de questa arte, essendo anchora idiota, & da quisti [i.e. the pseudo architects] già quante fabrice de aedificii & altre fabrice de molte sorte sono sta(te) deturpate (…) & quante fossature & aquaeductione [ditches and canals] già sono sta(te) abandonate & lassate le possessione & li altrui praedii & prati [farms and meadows] senza consequire la affluente ubertate del aqua, e così stan(n)o a la descretione & voluntà solum de la celeste pluvia & altro non po(sso)no havere’ (ibid., 92v).

²²Cataneo, I quattro libri (note 14), dedicatory epistle to Enea Piccolomini.

²³Aurora Scotti, ‘Il collegio degli architetti, ingegneri ed agrimensori tra il XVI e il XVIII secolo’, in Costruire in Lombardia. Aspetti e problemi di storia edilizia, edited by Aldo Castellano and Ornella Selvafolta (Milano, 1983), 92–108, on 92–4. See also below, § 2.3.

²⁴On 17 January 1499 Tommaso Piatti left his patrimony to the Ospedale Maggiore of Milan for creating a school in his own house, to teach Greek, dialectics, arithmetic, geometry and astronomy. The lectures began in 1503: see Filippo Argelati, Bibliotheca scriptorum Mediolanensium, 4 vols (Milano, 1745), II.1, 1106–7; Attilio Simioni, ‘Un umanista milanese, Piattino Piatti’, Archivio storico lombardo, 31/3–4 (1904), 5–50 and 227–301, on 9-12; Arnaldo Masotti, ‘Matematica e matematici’, in Storia di Milano, 17 vols (Milano, 1953–1966), XVI, 713–814, on 733–4.

²⁵In a letter to Tartaglia of 5 January 1540 Cardano wrote that Zuane Colle (or da Coi) was in Milan because he had heard that Cardano would renounce in favour of him to ‘one of my lectureships, namely that of arithmetic’: see Nicolò Tartaglia, Quesiti et inventioni diverse (Venezia, 1554), 124v. Therefore, when he wrote this letter Cardano was still the incumbent of the lectureship of arithmetic and of at least another lectureship (he referred in fact to the lectureship of arithmetic as ‘una delle mie letture [plural]’).

²⁶Silvia Fazzo, ‘Girolamo Cardano e lo Studio di Pavia’, in Girolamo Cardano. Le opere, le fonti, la vita, edited by Marialuisa Baldi and Guido Canziani (Milano, 1999), 521–74, on 522–3.

²⁷This is the Encomium geometriae, recitatum anno 1535 in Academia Platina Mediolani: see Girolamo Cardano, Opera omnia [hereafter O.C.], 10 vols, edited by Charles Spon (Lyon, 1663), IV, 440–5.

²⁸See the chronology of the composition of Cardano's works in Girolamo Cardano, De libris propriis. The editions of 1544, 1550, 1557, 1562, with supplementary material, edited by Ian Maclean (Milano, 2004), 56–60, M 23–27, M 29–31.

²⁹‘Ut vero magis audientes allicerem, pro geometria geographiam, pro arithmetica architecturam docebam’, Girolamo Cardano, De libris propriis, 1557 edition, in O.C., I, 60–95, on 64a; Maclean's edition (note 28), 167–226, on 178.

³⁰Girolamo Cardano, De libris propriis, 1544 edition, in O.C., I, 55–59, on 57b; Maclean's edition (note 28), 121–136, on 130. See also Ingo Schütze, Die naturphilosophie in Girolamo Cardanos “De subtilitate” (München, 2000), 131–2.

³¹Maffioli (note 5), 51–63, 67, 76–7, 326–9.

³²Girolamo Cardano, Practica arithmetice et mensurandi singularis (Milano, 1539). The two instruments are described in ch. 67, problem 12 (the ‘quadratum geometricum’) and in ch. 63, § 45 (the chorobates). The description of the latter in the Opera omnia (O.C., IV, 121b–122b) lacks the original drawing and, therefore, it is difficult to follow.

³³ M. Vitruvius per Iocundum solito castigatior factus cum figuris et tabula ut iam legi et intelligi possit (Venezia, 1511), 80v–81r; Vitruvio de Architectura (note 19), 137r–138r.

³⁴ De iudiciis geniturarum, in O.C., V, 433–57, on 436b–437a. We know of two manuscript copies of Costanzo da Bologna's translation of the Elements of Euclid. One is limited to the first two books and the beginning of the third and was made in 1539 for ‘Vincentio Seregno Architetto nella fabrica del Domo di Milano’ (Trivulziana Library of Milan, codex 187). Another one is divided into fifteen books and was made in 1541 for ‘Francesco Resta … Ingegnero de la Cesarea Camera et del comune di Milano’ (Ambrosiana Library of Milan, codex O.222 sup). Costanzo da Bologna exchanged astrological materials with Cardano and should be identified with Nicolò Simo, an addressee of the first Cartello of Ferrari against Tartaglia (dated 10 February 1547): Ludovico Ferrari & Niccolò Tartaglia, Cartelli di sfida matematica, edited by Arnaldo Masotti (Brescia, 1974), LXXIX–LXXX, note 70; Anthony Grafton, Cardano's cosmos. The worlds and works of a Renaissance astrologer (Cambridge MA, 1999), 72–3.

³⁵Cardano (note 32), ch. 51, §§ 17 and 36 (or Practica arithmetice, in O.C., IV, 13–216, on 78b and 87b); Artis arithmeticae tractatus de integris, in O.C., X, 117–28, on 119a. On the relations between Cardano and Aratore see also Ferrari & Tartaglia, Cartelli (note 34), LXIII–LXIV, note 27 and Veronica Gavagna, ‘Medieval heritage and new perspectives in Cardano's “Practica arithmetice”’, Bollettino di storia delle scienze matematiche, 30 (2010), 61–80, on 67.

³⁶Maffioli (note 5), 77–8.

³⁷ De exemplis centum geniturarum, in O.C., V, 458–502, geniture 96, on 500b–501a; Vita Lodovici Ferrarii, in O.C., IX, 568-569. Some information on his teaching of mathematics in Milan is also given in the fifth Cartello (October 1547), where Ferrari stated that he had taught Vitruvius’ Architecture and Ptolemy's Geography many times ‘in publico, & in privato’ (as for Ptolemy's work, both the Greek and the Latin text): see Ferrari & Tartaglia, Cartelli (note 34), 133 and 157.

³⁸Ferrari & Tartaglia, Cartelli (note 34), 91–3 (Tartaglia) and 136–7 (Ferrari). It goes without saying that Ferrari described Tartaglia as the unlearned and himself, by implication, as the learned mathematician. According to Ferrari Tartaglia, like Cesariano before him, was e.g. unable to render Vitruvius’ Latin text in a polished Italian (here Ferrari was referring to a passage of Book IX). The only difference between them was that Cesariano ‘traduce un poco più alla Milanese di voi, e voi traducete molto più alla Bresciana di lui [Tartaglia was a native from Brescia]’ (ibid., 134).

³⁹ Ferrari & Tartaglia, 53–8 (for the questions of Tartaglia) and 66–8 (for those of Ferrari).

⁴⁰Francesco Repishti, ‘Architetti, ingegneri e agrimensori a Milano: i “Dies utiles annorum” 1505–1561’, Libri & documenti, 24/2–3 (1998), 27–33, on 27–8. In this work some lists of acknowledged professionals, which were printed in Milan every two years in the so-called Dies utiles, are given.

⁴¹Guaro Coppola, ‘L'agricoltura di alcune pievi della pianura milanese nei dati catastali della metà del secolo XVI’, in Contributi dell'Istituto di storia economica e sociale, I, edited by Mario Romani (Milano, 1973), 185–286, on 269–72.

⁴²The two lists are dated 10 August 1549 (the measurements began two days later) and are kept in Archivio Storico Civico di Milano (hereafter ASCM), Materie, 187: ‘Censo, 1549’. A few cases apart, the land surveyors of the surrounding area were less paid than the professionals of the city of Milan.

⁴³One of the reasons adduced for this measure was the number of engineers-architects and land surveyors that had been matriculated in recent years, taking advantage of the easiness of the procedure. Another reason was the need to constitute a cohesive body of professionals, whose judgements had legal value: see some printed Ordines Universitatis Architectorum, seu Ingenieriorum Inclytae Civitatis, & Ducatus Mediolani, in ASCM, Materie, 551: ‘Ingegneri e loro Collegio, 1563–1800’. In this context, the term Universitas meant something like to a professional guild or ‘magistrate’, which could enforce its rules to the practitioners of the town and the surrounding area. Something comparable happened, in this respect, to the Florentine Compagnia ed Accademia del Disegno, which was founded in 1563. In the 1570s it acquired the functions of a guild and its formal title became Università, Compagnia ed Accademia del Disegno: Karen-edis Barzman, The Florentine Academy and the early modern state: the discipline of disegno (Cambridge, 2000), 1 and 59. For the occurrence from the second half of the 15th century of the term Universitas in the denomination of other professional guilds of Milan, e.g. of those of the ‘magistri a lignamine’ (woodworkers) and of the painters, see Janice Shell, ‘The Scuola di San Luca, or Universitas Pictorum, in Renaissance Milan’, Arte lombarda, 104/1 (1993), 78–99.

⁴⁴The so-called Libro Z lists, for the years 1564 and 1564–1565, twenty-nine ‘Ingenieri’ and twenty-seven ‘Agrimensores non Ingenieri’ of Milan and the surrounding area (ASCM, Materie, 556: ‘Ingegneri e loro Collegio, 1563-1793’). The names of the newly-appointed engineers-architects from 1564 to 1734 have been published by Maria Luisa Gatti Perer (‘Fonti per la storia dell'architettura milanese dal XVI al XVIII secolo: Il Collegio degli Agrimensori Ingegneri e Architetti’, Arte Lombarda, 10/2, 1965, 115–30, on 123–30), but the corresponding list of land surveyors is still unpublished.

⁴⁵Repishti (note 40), 31–2.

⁴⁶Out of a list of hundred-fifty-nine measurers, which was made in 1549 by the Milan's Office of Provvisione for the cadastral measurements, forty-six had e.g. to be left off because they were unable to write (ASCM, Materie, 187: ‘Censo, 1549’).

⁴⁷ Raccolta degli ordini e statuti del Collegio de’ signori ingegneri ed architetti di Milano (Milano, 1767), 10 (an exemplar of this work is in ASCM, Materie, 552). For some early examples of questions for the ‘prova de lj agrimensori’ and the ‘prova de li inginieri’ see Francesco Repishti, ‘«Martinus de laqua ingenearius et arch[itectus] subscripsi». Due codici milanesi del Cinquecento sull’Ars Mensoria’, Quaderni dell'Ateneo di scienze, lettere e arti di Bergamo (1999), 11–31, on 23–4.

⁴⁸ Ordines Universitatis Architectorum (note 43). Another example of the importance which was attributed to mathematics is given by a manuscript treatise kept in ASCM (codex E2, ff. 13v–63v) concerning the mensuration of distances and heights with the ‘quadratum geometricum’ and other instruments. It was written in the early 1530s by Martino dell'Acqua, an engineer-architect of the town (Repishti, note 47, 21–4). It is doubtful, however, that the original text was by Dell'Acqua. Not only is the treatise a sort of translation – simplified and enlarged with the addition of many practical examples – of Georg von Peuerbach's Quadratum geometricum (Nürnberg, 1516), but at its very beginning (f. 13v) the possible name of its real author is also given (Giovanni Domenico Atti da Legnago [or Legnano]). This does not exclude the possibility that in copying it Dell'Acqua might have added something substantial of his own.

⁴⁹See Vita Lodovici Ferrarii (note 37), 568 and Silvestro Gherardi, ‘Cenni della vita e delle fatiche di Lodovico Ferrari’, Nuovi annali delle scienze naturali, s. 3, 1 (1850), 213–24, on 217–9. As imperial commissioner Ferrari superintended the cadastral operations in several areas: he e.g. served as the commissioner of the seventh ‘squadra’ of Milan (Alessandra Fiocca, ‘Alcune opere inedite di Ludovico Ferrari’, Bollettino di storia delle scienze matematiche, 8 (1988), 239–305, on 257), as the commissioner of the eighth and (partly) of the sixth ‘squadra’ of Pavia (Anita Zappa, ‘Il paesaggio pavese, campagne, Lomellina e Oltrepò, attraverso le fonti catastali della metà del ‘500’, Nuova Rivista Storica, 70 (1986), 33–106, on 40, note 35) and as the commissioner of the first ‘squadra' of Bobbio (Archivio di Stato di Milano, Atti di governo, Censo p.a., 14).

⁵⁰Coppola (note 41), 270–4.

⁵¹ Coppola (note 41), 201–8.

⁵²Ferrari presumably began his service with Ercole Gonzaga in early 1549 (as it is attested by the Cartelli, in 1547–1548 he was still publicly teaching mathematics in Milan) and started in the same year to superintend the cadastral measurements. As for the time span of this job, the Vita Lodovici Ferrarii (note 37) says that Ferrari earned in eight years ‘aureos coronatos pene quater mille’ but it does not specify whether Ferrari was employed in the cadastral survey all this time or for a shorter period.

⁵³The patrician from Milan who acted as intermediary was Marco Antonio Arese; the letters to Cardano and Ferrari were dated Milan, 17 November 1563 and their answers Bologna, 25 November 1563 (ASCM, Materie, 196: ‘Censo, 1562–1563’). The letters of Cardano and Ferrari are published in Fiocca (note 49), 259–64.

⁵⁴The pertica (perch) was both a linear and a surface unit of measurement. In Milan the latter was equivalent to 654.5 square metres.

⁵⁵In addition to the ‘varietà de tempi’, other factors that according to Cardano influenced the price of a piece of land were the following: 1) its ‘nature’ (meadows were e.g. of greater value than vineyards because they had similar revenues and fewer expenses); 2) its proximity to a town; 3) its fertility; 4) the quality of farming.

⁵⁶The variation of prices had actually been greater than 1 to 2. Cardano reported that in the decade 1526–1536 the prices of land had been four times lower than in 1546–1556.

⁵⁷The question concerned, we would say today, the significance of the statistical sample. Cardano limited himself saying that he did not go further, because a ‘quinterno’ of paper and a whole month would be necessary to give a thorough answer.

⁵⁸The modest yearly salary – no more than 50 ‘coronatos’, fringe benefits included – that Cardano had received in the Piattine Schools (De libris propriis, 1562 edition, in O.C., I, 96–150, on 100b; Maclean's edition (note 28), 227–378, on 240) was in any event probably comparable with if not higher than the average salary of the relatively few teachers of arithmetic, geometry and astronomy in early and mid 16th-century Italian universities (in the Studium of Padua, e.g., the lecturer of astrology and mathematics in 1521–1522 received for the two lectureships respectively 40 and 20 florins: Antonio Favaro, Galileo Galilei e lo Studio di Padova (Padova, 1966), 102). On the other hand, if we consider that Ferrari's yearly emolument in c. 1549–1556 was more than ten times higher than Cardano's salary in the Piattine Schools (see above, note 52), it is plausible that money was for Ferrari a powerful reason for accepting the offers of Ercole and Ferrante Gonzaga.

⁵⁹At least twice Ferrari, who died in 1565, had however envisaged becoming a university teacher. His name appears in a first draft of the rolls of the Pavia Studium for 1548–1549, as a teacher ‘ad mathematicam’, but not in the officials rolls approved by the Senate for that year: Fazzo (note 26), 554–5, note 91. Ferrari's name also appears in the rolls of the Bologna Studium for the academic year 1565–1566, as a teacher ‘ad mathematicam et dependentes’: Umberto Dallari, I rotuli dei lettori legisti e artisti dello Studio Bolognese dal 1384 al 1799, 4 vols (Bologna, 1888–1919).

⁶⁰In the first Cartello against Tartaglia, Ferrari wrote that his master could not directly answer to Tartaglia's challenge because of his ‘grado’ (namely because he was socially superior to Tartaglia, being at that time a lecturer of theoretical medicine at the Studium of Pavia). Although Cardano had practised the mathematical disciplines as amusement, added Ferrari, such was the felicity of his mind that he was considered one of the first mathematicians of the age. From these claims, Cardano does not appear as an ancient (and relatively poor) teacher of mathematics. He is depicted as intellectually superior to Tartaglia because he did his mathematical research ‘a guisa di giuoco’: Ferrari & Tartaglia, Cartelli (note 34), 5–6. It is however worth mentioning that Cardano, in publishing his and Ferrari's mathematical research which was made as a consequence of Tartaglia's challenge, called it a ‘rarum subtilitatis exemplum’: Gerolamo Cardano, De subtilitate (Lyon, 1550), 496; O.C., III, 589a.

⁶¹In February 1624 Castelli was asked by the Archduchess Marie Magdalene of Austria (who, together with the Grand Duchess Christina of Lorraine, acted as regent of the Grand Duchy of Tuscany) to give advice about the best way of freeing a plain area north of Pisa from the floods of the Fiume Morto (the main ditch collecting the waters of this area). Castelli emphasised that the main mistake was made around 1560–1570 when, on advice of the Office of Ditches of Pisa, the mouth in the sea of the Fiume Morto was closed and the ditch was diverted into the Serchio River. Although the move had been a disaster and the ‘sementatori’ had petitioned for sending again the Fiume Morto into the sea, the Office still believed that the Serchio was a better recipient because the strong winds stopped the discharge in the sea. Castelli thought instead, also on the basis of an experiment recently made, that sending the ditch into the sea caused fewer evils and advised doing that: Benedetto Castelli, Scrittura intorno l'aprire la bocca di Fiume Morto in mare e chiuderla in Serchio, in Raccolta d'autori italiani che trattano del moto dell'acque, 10 vols, edited by Francesco Cardinali (Bologna, 1821–1826), III, 257–66.

⁶²Alfeo Giacomelli, ‘Appunti per una rilettura storico-politica delle vicende idrauliche del Primaro e del Reno e delle bonifiche nell'età del governo pontificio’, in La pianura e le acque tra Bologna e Ferrara: un problema secolare (Cento, 1983) 101–254, on 103–22. On the conflict between the Republic of Venice and the Papal States about the Porto Viro cut see also Francesco Ceccarelli, La città di Alcina. Architettura e politica alle foci del Po nel tardo Cinquecento (Bologna, 1998), 199–241.

⁶³See Maffioli (note 3), 41–4 and (note 5), 225–6 and 235–6.

⁶⁶Benedetto Castelli, Della misura dell'acque correnti (Roma, 1628), 30. According to the traditional engineering method followed by Roscelli, the expected rise h ₂ of the Po was simply obtained by dividing the liquid cross-section area A ₁ of the Reno for the breadth L ₂ of the Po, namely h ₂ = A ₁ : L ₂ . According to Castelli, instead, also the velocities of the two rivers should be considered. If V ₁ is the velocity of the Reno before the confluence and V ₂ the velocity of the Reno waters in the Po, after the confluence, h ₂ = (A ₁ x V ₁ ): (L ₂ x V ₂ ). This formula expresses in algebraic terms the contents of a proposition (prop. 4) that is part of Castelli's Demostrazioni geometriche, which were appended to the 1628 book and were conceived and expressed in the geometric language of the theory of proportions (ibid., 54–6).

⁶⁴‘As Father Benedetto Castelli has acutely and keenly noticed – acknowledged Corsini in the official report of the visit –, pondering the measures of a river through its breadth and depth is not sufficient to elucidate the truth, but it is also necessary to observe the velocity of the waters and the time; these things have not yet been considered by the periti and, therefore, it is neither possible to know with certainty the quantity of water that is carried by those rivers nor their rising’ (Ottavio Corsini, Relazione dell'acque del Bolognese e Ferrarese, in Raccolta d'autori (note 61), III, 231–42, on 239).

⁶⁵This debate took place on April 2, 1625. The next day an experiment in the field was performed that gave evidence that the traditional method (namely the one adopted by the Ferrara perito) was wrong. See the proceedings of Corsini's visit in Biblioteca Universitaria di Bologna (hereafter BUB), Ms. 1102, year 1625, N° 10 as well as Corsini, Relazione (note 64), 239.

⁶⁷ Benedetto Castelli, Della misura dell'acque correnti (Roma, 1628), 30–1.

⁶⁸It is however worth mentioning that, after Corsini's visit, Roscelli and Sassi produced rather different calculations. According to the perito of Ferrara, if the Reno was diverted into the Po Grande the existing banks should be reinforced and raised of 3½ feet (about 1.4 m). In his calculation Sassi, following Castelli's suggestions, instead also took into account the velocity of the two rivers. He stated that, when both rivers were in flood, the Po Grande was four times swifter than the Reno. In this case, the expected rise of the Po was only 4½ ounces (about 0.15 m) or even less. For these calculations see Maffioli (note 5), 228–9 and 246.

⁶⁹See above (note 66), for the contents of this proposition and the meaning of these symbols.

⁷⁰This point was grasped by Giovan Battista Baliani, to whom Castelli had sent a preliminary manuscript version of his treatise. On 20 February 1627 Baliani wrote to Castelli that he had only one remark to make, concerning the second appendix (which was to become the third in the printed version). Given that the Reno has a certain slope while the Po channel near Ferrara is about horizontal, ‘I believe that from your own doctrine it may be easily inferred that the Po would raise more than two feet. And this because the water of the Reno, once in the Po, would slow down and therefore make a greater cross-section’: Benedetto Castelli, Carteggio, edited by Massimo Bucciantini (Firenze, 1988), 95; Maffioli (note 5), 202–3.

⁷¹‘Corollary 2. (…), as a river becomes more and more full it usually gathers more and more velocity; this is the reason why the same floods of a stream which enters in a river add less and less height as the river runs more and more full’: Castelli, Della misura (note 66), 8.

⁷²Benedetto Castelli, Discorso sopra la laguna di Venezia, in Raccolta d'autori (note 61), III, 199–204, on 199–200.

⁷³Cassini's first university salary (1651) in Bologna had been of 600 lire: Anna Cassini, Gio: Domenico Cassini. Uno scienziato del Seicento (Perinaldo, 1994), 50. On March 20, 1659 the Assunti di Studio decided to see if it was possible to raise Cassini's salary (which was then of 1200 lire) ‘in riguardo delle fatiche fatte per l'Acque’ and, on July 3, 1659 they proposed to reappoint him for five years with a yearly salary of 2500 lire: Archivio di Stato di Bologna (hereafter ASB), Assunteria di Studio, Atti, 13, ff. 30v and 33v–34r. From the payrolls of the Studium this salary appears to have been paid in the years 1660–1663. In the next two years Cassini was not paid because of his absence from the town. In 1666 and 1667 he was paid again, this time 3800 lire per year. This big salary was nominally paid to Cassini also in 1668 and 1669, even though some quarterly payments were stopped partly because of the financial crisis of the Studium and partly because in 1669 he left for Paris: ASB, Riformatori dello Studio, Quartironi degli Stipendi (1657–1684), 9.

⁷⁴The technicians referred to were those of the 1610 visit of Cardinal Bonifacio Caetani, whose calculations had been discussed in 1625 during the Corsini's visit. A copy of this reply by the Ferrara representatives is in BUB, Ms. 1102, year 1657, N° 4, 118–120.

⁷⁵Tommaso Spinola was a land surveyor from Ravenna, Bartolomeo Breccioli worked as architect in Rome.

⁷⁶BUB, Ms. 1102, year 1657, N° 17. For more on this manuscript work of Cassini see Maffioli (note 5), 246–50.

⁷⁷BUB, Ms. 1102, year 1651, N° 4. Similar ideas about the restoration of the ‘natural state’ may be found in other debates on land reclamation: see e.g. Eric H. Ash, ‘Amending nature: draining the English Fens’, in The mindful hand: inquiry and invention from the late Renaissance to early industrialisation, edited by Lissa Roberts, Simon Schaffer and Peter Dear (Amsterdam, 2007), 117–43, on 119 and 125–34.

⁷⁸We are informed of this discussion between Macrini and Guglielmini by the minutes of the meeting of 20.07.1693 of the Bologna Assunti to the waters (a copy is in BUB, Ms. 1102, year 1693, N° 6). We also know the texts of Macrini's report and of Guglielmini's reply (they are in BUB, Ms. 1102, year 1693, N° 11: book A, texts 1 and 2).

⁷⁹BUB, Ms. 1102, year 1693, N° 11: book A, text 3. This text of Guglielmini is published in Raccolta d'autori (note 61), II, 131–2.

⁸⁰For more on Guglielmini's method see Maffioli (note 5), 4–5 and 275–6.

⁸¹The term balance is here to be understood both as conservation of the river branches and as quantitative invariance of the ratio of their rates of flow. In Guglielmini's organicistic and mechanical conception, a river is a sort of living being that may be studied with the categories of the ancient medicine (as that of internal balance or krasis) and of the mechanical philosophy (as the mathematical notions of velocity and rate of flow).

⁸²Domenico Guglielmini, Della natura de’ fiumi (Bologna, 1697), 295–6.

⁸³ Domenico Guglielmini, Della natura de’ fiumi (Bologna, 1697), 298–302.

⁸⁵Guglielmini, Natura de’ fiumi (note 82), 302–3.

⁸⁴In figure 3 these lateral gates or ‘paraporti’, whose sills are lower than the bottom of the canal, are indicated with F and K. They should not be mistaken for the overflow structures H and I.

⁸⁶Giovan Battista Barattieri, Architettura d'acque (Piacenza, 1656), address to the readers (a second part of the treatise was published in 1663).

⁸⁷Maffioli (note 5), 6–7 and 259–70.

⁸⁸The problem of identifying the mathematician's ‘role’ and the ‘mathematical community’ is much debated among historians (see e.g. the useful comments of Alexander Marr, Between Raphael and Galileo: Mutio Oddi and the mathematical culture of late Renaissance Italy, Chicago, 2011, 10–11 and 15–9), but no convincing solution of it seems actually in sight.

⁸⁹ Trattato del Radio Latino, instrumento giustissimo per prendere qual si voglia misura inventato dal Signor Latino Orsini. Con li Commentarij del Reverendo Padre Maestro Egnatio Danti (Roma, 1586), 63–4 (the first edition of this work was published in Rome in 1583).

⁹⁰Giovan Battista Aleotti, Della scienza et dell'arte del ben regolare le acque, edited by Massimo Rossi (Ferrara, 2000), 683–4. Although Danti had given only an indication about the length of the ditches (‘molte miglia’), he had also specified that the levelling operations were made at night in order to make ‘lunghissime livellature’. It is therefore possible that they exceeded by far the range indicated by Aleotti.

⁹¹Gatti Perer (note 44), 120–1; Aurora Scotti, ‘Per un profilo dell'architettura milanese (1535–1565)’, in Omaggio a Tiziano. La cultura artistica milanese nell'età di Carlo V (Milano, 1977), 97–111, on 100–105. The figures of the artist-architect and of the professional architect are contrasted, without mentioning the professionals of Milan, in Catherine Wilkinson, ‘The new professionalism in the Renaissance’, in The architect: chapters in the history of the profession, edited by Spiro Kostoff (New York, 1977), 124–60, on 134–42.

⁹²J.V. Field, ‘Perspective and the mathematicians: Alberti to Desargues’, in Mathematics from manuscript to print, 1300–1600, edited by Cynthia Hay (Oxford, 1988), 236–63, on 248–50 and 256–60; Filippo Camerota, La prospettiva del Rinascimento: arte, architettura, scienza (Milano, 2006), 28–34 and 160–81. It is however worth mentioning that in the works on perspective of Commandino, Benedetti and Guidobaldo ‘the mathematics – as Field puts it (on 260) – becomes independent of the craft’, i.e. these works had been primarily written for mathematicians.

⁹³From the academy's records it appears that, in the late 16th century, mathematicians such as Pier Antonio Cataldi, Ostilio Ricci and Antonio Santucci taught for a while in the Accademia del Disegno: Barzman (note 43), 152–7.

⁹⁴A staunch defender of the predominance of disegno in the education of the architect was e.g. Benvenuto Cellini (1500–1571), who was twice elected ‘console’ of the academy (in 1565 and 1570) and once ‘arroto’ (in 1566): Gli accademici del disegno: elenco cronologico, edited by Luigi Zangheri (Firenze, 1999), 2–3 and 5. By referring to the Ferrara engineer-architect Terzo de Terzi and to Antonio da Sangallo the Younger, Cellini acknowledged that people from the low arts (‘di bassa arte’) may also produce something worthwhile in the field of architecture. However, if they do not belong to the ‘professione del disegno’ they cannot reach the ‘bellissima maniera’ of the new art of architecture (Benvenuto Cellini, Della architettura, in Id., Opere, edited by Giuseppe Guido Ferrero, Torino, 1971, 813–21, on 813–4 and 816). Against this kind of attitude, the Florentine military engineer Bernardo Puccini (1521–1575) voiced the grievance of the engineers-architects who were not admitted in the Accademia del Disegno unless they were also sculptors or painters. In an unfinished treatise Puccini contrasted the engineer's with the artist's kind of drawing and emphasised the superiority of the broad, liberal education of the Vitruvian architect: Daniela Lamberini, Il principe difeso. Vita e opere di Bernardo Puccini (Firenze, 1990), 15, 127–8, 335 and 347–8.

⁹⁵Barzman (note 43), 91–4. Giovanni Coccapani, e.g., from 1638 to about 1640 read not only Euclid and perspective but also surveying, mechanics, fortifications, civil architecture and the use of mathematical instruments. The new climate favourable to the professional architect is also evidenced by the presence among the 1640s lecturers of mathematics of Baccio del Bianco and Vincenzo Viviani, namely of engineers or would-be engineers of the Magistrato della Parte, the Tuscan department of public works (ibid., 94–7 and 159–61). It is also worth mentioning that Andrea Arrighetti, a disciple and a friend of Galileo, was instrumental in this mathematical turn of the academy.

⁹⁶Girolamo Cardano, Contradicentium medicorum liber (Venezia, 1545), lib. I, tract. iii, contr. 20: ‘Medicina an ars coniecturalis’, 89r. In the following, enlarged edition Cardano added that the propositions of medicine, except for the ambiguous use of names, ‘sunt firmae ut mathematicae’: (Lyon, 1548), 139. On the conception of the arts, and particularly of medicine, of the Contradicentium medicorum see Alfonso Ingegno, Saggio sulla filosofia di Cardano (Firenze, 1980), 232–4, note 7 and Nancy G. Siraisi, The clock and the mirror. Girolamo Cardano and Renaissance medicine (Princeton, 1997), 46–52.

⁹⁷Guglielmini developed these corpuscular-mathematical ideas within his researches on the geometric figure of salts and on the mechanics of fluids: see, respectively, Maffioli (note 5), 301–11 and (note 3), 256–65.

⁹⁸Guglielmini, Natura de’ fiumi (note 82), address to the readers. Guglielmini's ideas on the physician's education were published after his move to the Studium of Padua (1698). His inaugural lecture from the chair of theoretical medicine, which was printed in Venice in 1702 and significantly titled Pro theoria medica adversus empiricam sectam, is particularly relevant in this regard.

¹⁰³ Anthony Gerbino and Stephen Johnston, Compass and rule: architecture as mathematical practice in England, 1500–1750, with a contribution by Gordon Higgott (New Haven, 2009), 12.

⁹⁹Rudolf Wittkower, Palladio and English Palladianism (London, 1974), 73–4 and 100.

¹⁰⁰ Rudolf Wittkower, Palladio and English Palladianism (London, 1974), 98–9. Dee's Praeface covers twenty-five folio pages of Billingsley's edition; the section on architecture covers two folio pages (sig. d.iij r-d.iiij v). On the latter, see Frances A. Yates, Theatre of the world (London, 1969), 24–9 and Jean-Marc Mandosio, ‘Alberti dans le miroir magique de John Dee’, Albertiana, 2 (1999), 57–78, on 66–78.

¹⁰¹ Rudolf Wittkower, Palladio and English Palladianism (London, 1974), 100.

¹⁰²Anthony Gerbino and Stephen Johnston, Compass and rule: architecture as mathematical practice in England, 1500–1750, with a contribution by Gordon Higgott (New Haven, 2009), 65.

¹⁰⁴Wittkower (note 99), 64. For Alberti's conception of beauty see On the art of building (note 13), VI.2, 156 and Christine Smith, Architecture in the culture of early humanism: ethics, aesthetics, and eloquence, 1400–1470 (New York, 1992), 80–97. Anthony Grafton has observed that the picture that ‘has emerged in the last generation of scholars’ is much different from Wittkower's image of Alberti as the deviser of ‘a totalizing theory, based on a limited, coherent set’ of architectural principles (Anthony Grafton, Leon Battista Alberti master builder of Italian Renaissance, London, 2001, 263). On Alberti and Vitruvius see ibid., 273–6 and Franco Borsi, Leon Battista Alberti (Milano, 1975), 316–9.

¹⁰⁵Gerbino's and Johnston's narrative is, however, rather nuanced in this respect. They e.g. have not only reminded us that Vitruvius ‘had defined fortification as an integral part of architecture’ but have also observed that Inigo Jones, the herald of the supremacy of architectural theory, was much interested in the mathematical instruments and in the layout of bastions that were illustrated in Buonaiuto Lorini's book on fortifications: Gerbino and Johnston (note 102), 72.

¹⁰⁶Cesariano and Worsop respectively referred to the knowledge of the Vitruvian subjects and of the three branches of surveying: the mathematical, the legal and the judicial. What makes the two cases similar is that both asked for a regulation of their professions. They also used similar professional categorizations: in the field of architecture Cesariano contrasted the ‘docti architetti’ with the ‘pseudo architecti’ (see above, § 2.1), while in the field of surveying Worsop contrasted the ‘learned’ with the ‘unlearned practitioners’.

¹⁰⁷Edward Worsop, A discoverie of sundrie errours and faults daily committed by landmeaters, ignorant of arithmetike and geometrie, to the damage and preiudice of many her Maiesties subiects, with manifest proofe that none ought to be admitted to that function, but the learned practisioners of those sciences: written dialoguewise, according to a certaine communication had of that matter (London, 1582), dedicatory epistle to William Cecil, sig. A.2v. In the text of the dialogue there are scattered references to the (real) ‘mathematicians’ (that Worsop contrasted with the unlearned astrologers), to the ‘geometers’ or ‘professors and geometers’ (teaching in several cities ‘in France, Germanie, and Italie’), and finally to the ‘learned mathematicians’ (sig. F.3r, F.4r, G.1r-v, G.3r-v).

¹⁰⁸Jim Bennett has acknowledged that Worsop's Sundrie errours might shed light ‘on the hierachy of mathematical practitioners and the relationship between more practical or everyday geometry and the learned mathematicians’: Bennett (note 9), 80.

¹⁰⁹John Dee, Mathematicall praeface to The elements of geometrie of the most ancient philosopher Euclide of Megara, translated by H. Billingsley (London, 1570), sig. c.j r. Dee here translated a passage of the dedicatory epistle of Benedetti's Resolutio omnis Euclidis problematum una tantummodo circini data apertura (Venezia, 1553), a book which was owned by him: see John Dee's library catalogue, edited by Julian Roberts and Andrew G. Watson (London, 1990), N. 403 and B160.

¹¹⁰Worsop, Sundrie errours (note 107), sign. K.1r.

¹¹¹A useful comparison of Dee's and Digges’ approaches to mathematics and philosophy is given by Stephen Johnston, ‘Like father, like son? John Dee, Thomas Digges and the identity of the mathematician’, in John Dee: interdisciplinary studies in English Renaissance thought, edited by Stephen Clucas (Dordrecht, 2006), 65–84, on 73–8.

¹¹² A useful comparison of Dee's and Digges’ approaches to mathematics and philosophy is given by Stephen Johnston, ‘Like father, like son? John Dee, Thomas Digges and the identity of the mathematician’, in John Dee: interdisciplinary studies in English Renaissance thought, edited by Stephen Clucas (Dordrecht, 2006), 70.

¹¹³I follow a suggestion of Eric Ash (note 9), although his ‘expert mediator’ is a too general category to be fully convincing.

¹¹⁴ An arithmeticall militare treatise, named Stratioticos: (…) long since attempted by Leonard Digges (…) and lately finished by Thomas Digges, his sonne (London, 1579), 181 and 188.

¹¹⁵ An arithmeticall militare treatise, named Stratioticos: (…) long since attempted by Leonard Digges (…) and lately finished by Thomas Digges, his sonne (London, 1579), 189. Ten out of twenty questions on the subject of ‘randons’ concern the curved part of the trajectory: Q. 11: ‘Whether the upper part of the circuite made by the bullet be a portion of a circle as Tartalea supposeth’; Q. 12: ‘Whether it be not rather a conical section, and different at every several randon’; Q. 13: ‘Whether it be not at the utmost randon a section parabolical (…)’; Q. 14: ‘Whether at al inferiour randons that arke by Tartalea imagined circular be not an eleipsis (…); an so forth until Q. 20 (ibid., 187–8).

¹¹⁷Thomas Digges, A perfit description of the caelestiall orbes according to the most aunciente doctrine of the Pythagoreans, lately revived by Copernicus and by geometricall demonstrations approved, in A prognostication everlastinge (…). Published by Leonard Digges (…) and augmented by Thomas Digges his sonne (London, 1576), O.2r.

¹¹⁶Francis R. Johnson and Sanford V. Larkey, ‘Thomas Digges, the Copernican system, and the idea of the infinity of the universe in 1576’, The Huntington Library Bulletin, 5 (1934), 69–117, on 99.