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Tartaglia's ragioni: A maestro d'abaco's mixed approach to the bombardier's problem

Published online by Cambridge University Press:  04 March 2010

Indiana University, Bloomington, USA. Email:


In La nova scientia (1537), Niccolò Tartaglia analyses trajectories of cannonballs by means of different forms of reasoning, including ‘physical and geometrical reasoning’, ‘demonstrative geometrical reasoning’, ‘Archimedean reasoning’, and ‘algebraic reasoning’. I consider what he understood by each of these methods and how he used them to render the quick succession of a projectile's positions into a single entity that he could explore and explain. I argue that our understanding of his methods and style is greatly enriched by considering the abacus tradition in which he worked. As a maestro d'abaco in sixteenth-century Venice he had access to a great variety of mathematical and natural-philosophical works. This paper traces how Tartaglia drew elements from a vast spectrum of sources and combined them in an innovative manner. I examine his use of algebra and geometry, consider what he knew about Archimedes and suggest a reading of his enigmatic phrase ‘Archimedean reasoning’, which has eluded satisfactory interpretation.

Research Article
Copyright © British Society for the History of Science 2010

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1 Niccolò Tartaglia, Nova scientia, Venice: Per Stephano da Sabio, ad instantia di Nicolo Tartalea brisciano il qual habita a san Saluador, 1537, 2r–3r; English translation in Stillman Drake and Israel E. Drabkin, Mechanics in Sixteenth-Century Italy, Madison: University of Wisconsin Press, 1969, pp. 63–64. The Duke of Urbino to whom the Nova scientia was dedicated was Francesco Maria della Rovere (1490–1538).

2 Tartaglia, op. cit. (1), (1537), 2r; (1969), p. 64, emphasis added. ‘Et dipoi che hebbi ben masticata & ruminata tal materia, gli conclusi, et dimostrai con ragioni naturale & geometrice’.

3 Tartaglia, Opera Archimedes, Venice: Per Venturinum Ruffinellum sumptu & requisitione Nicolai de Tartaleis Brixiani, which consisted of four Latin translations, including On the Measurement of the Circle, On the Quadrature of the Parabola, On the Equilibrium of Planes and On Floating Bodies, Book I. On the recovery of classical mathematical manuscripts and their influence see Paul Lawrence Rose, The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from Petrarch to Galileo, Geneva: Librairie Droz, 1975. Paolo Freguglia, ‘Nicolò Tartaglia e il Rinnovamento delle Matematiche nel Cinquecento’, in Giovanni Battista Benedetti (ed.), Cultura, Scienze e Tecniche nella Venezia del Cinquecento, Atti del Convegno Internazionale di Studio Giovan Battista Benedetti e il suo Tempo, Venice: Istituto veneto di scienze, lettere ed arti, 1987, pp. 203–215.

4 Cuomo, Serafina, ‘Shooting by the book: notes on Niccolò Tartaglia's Nova scientia’, History of Science (1997) 35, pp. 157188CrossRefGoogle Scholar, p. 184 n. 39.

5 Apart from Gerhard Arend's simplifying of Tartaglia's equation, which he presents in modern notation, and his brief discussion thereof, the use of algebra in the Nova scientia has remained virtually unobserved. See Gerhard Arend, Die Mechanik des Niccolo Tartaglia: Im Kontext der zeitgenössischen Erkenntnis- und Wissenschaftstheorie, Munich: Algorismus, Heft 24, 1998, pp. 191–192. Arend prefaces his simpler justification of the calculation, ‘Mit Hilfe von Figur 4.22 kann man die umständlichen Ausführungen erheblich kürzer fassen und dennoch den Kern der Argumentation veranschaulichen’. Arend's detailed study of the Nova scientia, of theories of motion before Tartaglia and of the influence of his work is exceptionally comprehensive. While his use of algebra in his first publication has been largely overlooked, Tartaglia is of course well known for his solution to cubic equations and the entailing dispute with Girolamo Cardano. See Moritz Cantor, Vorlesungen über Geschichte der Mathematik, vol. 2, Leipzig: B.G. Teubner, 1900, pp. 485–497; Giovanni Battista Gabrieli, Nicolò Tartaglia: Invenzioni, Disfide e Sfortune, Siena: Università degli studi di Siena, 1986; Leonardo Olschki, Galilei und Seine Zeit, Halle: Max Niemeyer, 1927, pp. 87–113. The most thorough account of the disfida is Renato Acampora's Die ‘Cartelli di matematica disfida’, Der Streit zwischen Nicolò Tartaglia und Ludovico Ferrari, Munich: Algorismus, Heft 35, 2000, which includes the enunciations of the problems posed to Tartaglia and Ferrari.

6 Tartaglia, Quesiti et Inventioni Diverse, Venice: per Venturino Ruffinelli, 1546, Book IX.

7 For biographical information and discussion of Tartaglia's influence on late sixteenth- and early seventeenth-century mathematics and mechanics in Italy see Antonio Favaro, ‘Per la Biografia di Niccolò Tartaglia’, Archivo Storico Italiano (1913) 71, pp. 335–372; Olschki, op. cit. (5), pp. 69–87; Arnaldo Masotti also supplies a biographical sketch in his introduction to Tartaglia's Quesiti et Inventioni Diverse, Brescia: Ateneo di Brescia, 1959, pp. xix–xxii. Prosper J. Charbonnier, Essais sur l'Historie de la balistique, Paris, Société d'éditions géographiques, maritimes et coloniales, 1928, pp. 1–40, provides a general introduction to theories of motion before Tartaglia and a French translation of parts of the Nova scientia and the Quesiti. Arend, op. cit. (5), gives the original Italian and a German translation of the definitions, suppositions and enunciations of propositions. Arend, op. cit. (5); Cuomo, op. cit. (4); and Henninger-Voss, Mary, ‘How the “New Science” of cannons shook up the Aristotelian cosmos’, Journal of the History of Ideas (2002) 63, pp. 371397CrossRefGoogle Scholar, provide careful analyses of the text and attention to its context that have greatly added to the biographical and bibliographical accounts of classic studies. Historical studies to date have focused on the following aspects of the Nova scientia: (a) its natural philosophy, in particular regarding Aristotelian views of local motion: Drake op. cit. (1), ‘Introduction’, pp. 16–26, and Charles B. Schmitt's review of Drake and Drabkin, ‘A fresh look at mechanics in 16th-century Italy’, History and Philosophy of Science (1970) 1, pp. 161–175; Arend, op. cit. (5), pp. 13–50, pp. 176–85, pp. 495–527; and Henninger-Voss (op. cit.), passim; (b) mathematics and social standing: Biagioli, Mario, ‘The social status of Italian mathematicians, 1450–1600’, History of Science (1989) 27, pp. 4194CrossRefGoogle Scholar; Cuomo op. cit. (4); and idem, ‘Niccolò Tartaglia, mathematics, ballistics and the power of possession of knowledge’, Endeavour (1998) 22, pp. 31–35; (c) changes in his views between the publications of the Nova scientia and the Quesiti (1546): Arend, op. cit. (5), pp. 312–330; Henninger-Voss (op. cit.), especially pp. 385–391; (d) the large number of editions and translations and their influence on ballistics treatises as well as more or less contemporary treatments of trajectories: A. Rupert Hall, Ballistics in the Seventeenth Century: A Study in the Relations of Science and War with Reference Principally to England, Cambridge: Cambridge University Press, 1969; Arend, op. cit. (5), pp. 237–311; Henninger-Voss (op. cit.), p. 392, n. 46; Jochen Büttner, Peter Damerow, Jürgen Renn and Matthias Schemmel, ‘The challenging images of artillery’, in Wolfgang Lefèvre, Jürgen Renn, and Urs Schoepflin (eds.), The Power of Images in Early Modern Science, Basel: Birkhauser, 2003, pp. 3–27, compare contemporary representations of trajectories, including those of Tartaglia; (e) the deductive structure of the treatise and his efforts to explain trajectories by means of geometry: Olschki, op. cit. (5), pp. 79–87; Arend, op. cit. (5), pp. 127–175.

8 Tartaglia, Quesiti, op. cit. (6), pp. 74–75v.

9 Garibotto, Eloisa, ‘Le Scuole d'abbaco a Verona’, Atti e memorie della Accademia di agricoltura, scienze e lettere di Verona (1923) 99, pp. 315328Google Scholar, references Verona's public records from the late thirteenth to the sixteenth century, which provide an account of the names and salaries of abacus teachers, the locations of classes, and appeals made by teachers to keep rent low. The records are incomplete, and Garibotto notes that while Tartaglia is not named in the public documents, there is a gap in the records during the time he would have been in Verona. Tartaglia claims, in General Trattato de numeri et misure, Part III, Venice: Per Curtio Troiano dei Navò, 1560, chapter 2, fol. 7r., that this is where he was first employed.

10 Warren Van Egmond, ‘The commercial revolution and the beginnings of Western mathematics in Renaissance Florence, 1300–1500’, Ph.D. thesis, DAI No.77–10 932, Indiana University, 1976, p. 74, and idem, Practical Mathematics in the Italian Renaissance: A Catalog of Italian Abbacus Manuscripts and Printed Books to 1600, Florence: Istituto e Museo di Storia della Scienza, 1980; Paul F. Grendler, Schooling in Renaissance Italy: Literacy and Learning 1300–1600, Baltimore: Johns Hopkins University Press, 1989, pp. 306–319, relies in large part on Van Egmond, ‘The commercial revolution’ (op. cit.). Also see Enrico Gamba and Vico Montebelli, ‘La Matematica Abachista tra Ricupero della Tradizione e Rinnovamento Scientificio’, in Benedetti, op. cit. (3), pp. 169–202, p. 178; and Goldthwaite, Richard, ‘Schools and teachers of commercial arithmetic in Renaissance Florence’, Journal of European Economic History (1972) 1, pp. 418433.Google Scholar

11 The reason that these lists are included in the city records is that the teachers in question were petitioning that the rent remain at a certain rate, and they noted the skills they could teach in efforts to convince the city of their worth. Recorded in Ant. Arch. Ver. Atti Cons. K K, p. 39r, quoted in Garibotto op. cit. (9), p. 325.

12 Biagioli, op. cit. (7), includes a very helpful bibliography in which he lists twelve articles on public teaching of the abacus. These include studies of abacus schools in Pisa, Lucca, Sicily, Bologna, Florence and Tuscany, Verona, Pistoia and Siena, and one general article on schools in fourteenth- and fifteenth-century Venice. The most comprehensive study of the abacus in sixteenth-century Venice is Gamba and Montebelli, op. cit. (10), which is not listed by Biagioli. Gamba and Montebelli draw upon Tartaglia's later writings, including his Euclide Megarense Philosopho solo introduttore delle scientie mathematicae, Venice: per Venturino Roffinelli, 1543; his Quesiti, op. cit. (6), and above all his six-part General trattato di numeri et misure, 1556–1560, op. cit. (9), but they do not consider the influence of abacus mathematics in his first work. Van Egmond's work has largely focused on Tuscan manuscripts and books.

13 For examples of the problems see Raffaella Franci and Laura Toti Rigatelli, Introduzione all'Arithmetica Mercantile, Siena: Quattro Venti, 1982. Van Egmond, op. cit. (10), surveys roughly 250 Italian abacus manuscripts and books produced from the early fourteenth century to the mid-sixteenth.

14 On Tartaglia's language see Mario Piotti, La lingua di Niccolò Tartaglia, Milan: Edizioni Universitarie di Lettere Economia Diritto, 1988.

15 See op. cit. (7). Tartaglia, General trattato, Part II, Venice: Per Curtio Troiano dei Navò, 1556, 41v, notes that his final dispute with Ferrari took place on church grounds, in the Giardino di Frati Zoccolanti.

16 Marino Sanuto, I diarii di Marino Sanuto, Bologna: Forni (ed. Rinaldo Fulin et al.), 1969–1970, a fifty-eight-volume account of life in Venice from 1496 to 1533.

17 Sanuto, op. cit. (16), vol. 3, p. 1146, 8 December 1500.

18 Sanuto, op. cit. (16), vol. 31, p. 60, 15 July 1521, vol. 40, p. 763, 2 February 1526, and February 1533.

19 Sanuto, op. cit. (16), vol. 54, pp. 96–97.

20 The manner in which Tartaglia follows the structure of the Elements is noted in most discussions of the Nova scientia; for thorough treatments thereof see Olschki, op. cit. (5), pp. 79–87; and Arend, op. cit. (5), pp. 127–175.

21 Tartaglia, Euclide, op. cit. (12).

22 Tartaglia, op. cit. (1), (1537), 2r–2v; (1969), pp. 63–64.

23 Henninger-Voss, op. cit. (7), p. 382.

24 Tartaglia, op. cit. (1), (1537), 2r–2v; (1969), p. 64.

25 Tartaglia, op. cit. (1) (1537), 3v–4r; (1969), p. 65. Tartaglia recounts the circumstances surrounding a wager between the prefect and the chief bombardier at Padua a year after he himself had solved the challenge: while the former used a squadra to conclude that firing at a 45-degree inclination to the horizon produces the furthest shot, the latter held that the longest range is achieved at a lesser elevation. When the bombardiers tested their propositions they, together with witnesses, ‘saw the truth of our determination, though before this experiment they had been in disagreement’.

26 Hall, op. cit. (7), p. 33. Max Jähns, Geschichte der Kriegswissenschaften, vol. 1, Munich: R. Oldenbourg, 1889, pp. 410, 598. On the uses of the quadrant see James A. Bennett, ‘The impact of geometry on surveying’, in idem, The Divided Circle: A History of Instruments for Astronomy, Navigation and Surveying, Oxford: Phaidon, 1987, pp. 38–50. Also see Annalisa Simi, ‘Celerimensura e strumenti in manoscritti dei secoli XII–XV’, in Rafaella Franci, Paolo Pagli and Laura Toti Rigatelli (eds.), Itinera Mathematica Studi in onore di Gino Arrighi per il suo 90° compleanno, Siena: Università di Siena, 1996, pp. 72–121.

27 ‘By physical reasoning’ is the standard translation for ‘on ragione naturale’, a concept predominantly associated with Thomas Aquinas to describe knowledge of God acquired by reasoning from experience of natural things rather than from authority. Arend, op. cit. (5), pp. 121–122, discusses why Tartaglia needed to supplement geometrical reasons with natural reasoning.

28 Tartaglia, op. cit. (1), Definition 3, (1537), 7r–7v; (1969), pp. 70–71, specifies that in order to compare the range of projectiles fired under a variety of conditions, the cannonballs must be of the same weight, uniformly heavy throughout, and spherical so that they suffer the least amount of resistance whatever way they are shot.

29 Aristotle, De Caelo, I.2; Physics, IV.8, VIII.4, 5 and 8. Tartaglia, op. cit. (1), (1537), 7r; (1969), p. 70, discusses Avicenna's and Averroës's different views on which elements have lightness and which have heaviness.

30 Tartaglia, op. cit. (1), Book I, Proposition 5, (1537), 14v–15r; (1969), pp. 80–81.

31 Libor Koudela, ‘Curves in the history of mathematics: the Late Renaissance’, WDS'05 Proceedings of Contributed Papers, Part I (2005), p. 198. Books V–VII of Apollonius' treatise on conic sections were discovered only in the seventeenth century, in Arabic translation. See Michael Fried and Sabetai Unguru, Apollonius of Perga's Conica: Text, Context, Subtext, Leiden: Mnemosyne, Bibliotheca Classica Batava Supplementum, 2001, p. 8.

32 Tartaglia, op. cit. (6), Book IX.

33 Galileo, Two New Sciences (tr. Henry Crew and Alfonso de Salvio with an introduction by Antonio Favaro), New York: Dover Publications, 1954. Also see Domenico Bertoloni Meli, Thinking with Objects, Baltimore: Johns Hopkins University Press, 2007, pp. 96–104. On Galileo's discovery of the parabolic trajectory see Jürgen Renn, Peter Damerow and Simone Rieger, ‘Hunting the white elephant: when and how did Galileo discover the law of fall’, in Jürgen Renn (ed.), Galileo in Context, Cambridge: Cambridge University Press, 2001, pp. 29–149; and Emil Wohlwill, ‘The discovery of the parabolic shape of the projectile trajectory’, originally published 1899, reprinted in Renn, op. cit., pp. 375–410.

34 Tartaglia, op. cit. (1), Supposition 2, (1537), 18v; (1969), pp. 84–85.

35 Tartaglia, op. cit. (1), Suppositions 2 and 3, (1537), 18v–19r; (1969), pp. 84–85.

36 Galileo, op. cit. (33), pp. 207–210, discusses the need to discount the material hindrances in order to treat physical processes geometrically. Similarly, Filippo Pigafetta points out the discrepancy between empirical knowledge of pulleys and theory of statics, ‘For in mathematical demonstrations, all lines are assumed to be without thickness, and all things are abstracted from actual matter, so that it is easy to persuade ourselves of the mathematical truth. But experience very often shows something different … for actual matter changes things quite a bit’. Quoted in John J. Roche, ‘The semantics of graphics in mathematical natural philosophy’, in Renato G. Mazzolini (ed.), Non-verbal Communication in Science Prior to 1900, Biblioteca di Nuncius Studi e testi XI, Florence: Leo S. Olschki, 1993, pp. 197–233, 200–201.

37 Tartaglia, op. cit. (1), Book II, Supposition 2, (1537), 18v; (1969), pp. 84–85, added emphases. Cf. idem, op. cit. (6), Book IX.

38 Tartaglia, op. cit. (1), Supposition 1, (1537), 18r and v; (1969), p. 84.

39 Tartaglia, op. cit. (1), Supposition 1, (1537), 18r and v; (1969), p. 84.

40 Tartaglia, op. cit. (1), Supposition 4, (1537), 19v–20v; (1969), pp. 85–86. Tartaglia bases his argument on the supposition that for projectiles fired at any elevation, the straight line drawn from the motive force to the end point of the violent motion is longer than the distance achieved along this very same line by a shot fired at any other elevation.

41 Tartaglia, op. cit. (1), Supposition 4, (1537), 19v–20v; (1969), pp. 85–86.

42 Tartaglia, op. cit. (1), Book II, Proposition 8, (1537), 28r and v; (1969), pp. 93–94.

43 Tartaglia, op. cit. (1), (1537), 4v; (1969), p. 66.

44 Drake and Drabkin, op. cit. (1), n. 6, notes this alteration was made in the posthumously published 1558 edition, but I found that the change is already present in the rare 1550 edition which Tartaglia oversaw. Marshall Clagett, Archimedes in the Middle Ages, 5 vols., Philadelphia: American Philosophical Society, 1978, vol. 3, p. 549, n. 8, notes that he had not seen the 1550 edition. While there are several very minor differences between the first and second editions of the Nova scientia (most notably differences in the salutations ascribed to the duke), the only other significant difference is an addition to the first proposition of Book I, (1550), 4; (1969), pp. 75–76. Here he considers what would happen to a falling heavy body if there were a tunnel at the center of the earth. This addition is discussed by Alexandre Koyré, ‘La Dynamique de Nicolo Tartaglia’, Études d'histoire de la pensée scientifique, Paris: Gallimard, 1973, pp. 117–139, 120–121; Pierre Costabel, ‘Vers une Mécanique nouvelle,’ in August Buck (ed.), Sciences de la renaissance, Paris: J. Vrin, 1973, pp. 127–142, p. 134; and Drake and Drabkin, op. cit. (1), p. 76, n. 18.

45 Although Tartaglia does not discuss what causes a projectile to continue moving once it is separated from the cannon, it is clear from his treatise that he subscribes to the impetus theory. See Edward Grant, ‘The physics of motion’, in idem, Physical Science in the Middle Ages, Cambridge: Cambridge University Press, 1977, pp. 36–59, for a survey of Aristotle's treatment of motion and the problems that it raised as well as medieval responses to and alterations of Aristotle's views. In fourteenth-century Oxford and Paris interest developed in quantifying natural processes, but the objects of the calculators' studies, as well as the manner in which they use images, differ significantly from Tartaglia's. While the former use lines to represent abstract magnitudes, such as degrees of velocity, Tartaglia's lines trace a physical path. See John Murdoch and Edith Sylla, ‘The science of motion’, in David C. Lindberg (ed.), Science in the Middle Ages, Chicago: University of Chicago Press, 1978, pp. 206–264. Also see Christopher Lewis, The Merton Tradition and Kinematics in Late Sixteenth and Early Seventeenth Century Italy, Padua: Antenore, 1980.

46 Roche, op. cit. (36), p. 198, points out the distinction between representing physical properties (such as time and weight) by abstract lines and representing physical bodies by abstract geometrical outlines. He notes that Archimedes may have introduced the latter, which is still ‘recognized as a diagram of theoretical mechanics’. Tartaglia's drawings present a third form of diagram insofar as the lines represent a physical process. On diagrams in Archimedes manuscripts see Reviel Netz, The Works of Archimedes, vol. 1, Cambridge: Cambridge University Press, 2004, pp. 8–10.

47 Archimedes, Opera Omnia, cum commentariis Eutocii, vol. 3 (ed. Johan Ludvig Heiberg), Prolegomena, Stuttgart: B.G. Teubner, 1972; and Clagett, op. cit. (44), have traced the Archimedes manuscripts during the middle ages and Renaissance. Clagett, p. 327, n.17, notes that he has checked Heiberg's references against the manuscripts. Also see Clagett, ‘The impact of Archimedes on medieval science’, Isis (1959) 50, pp. 420–425. Moerbecke first collected and translated Greek manuscripts of Archimedes into Latin. Rose, op. cit. (3), p. 80, notes that for two centuries Moerbecke's translation of Archimedes remained almost unknown and had no influence at all on scholastic mechanics or theories of motion.

48 Clagett, op. cit. (44), pp. 333 and 329. Translations of MS A are now in the Vatican, Paris, Madrid and Florence. Overview of the four pertinent manuscripts:

49 Clagett, op. cit. (44), pp. 551–552, on the basis of a comparison of the translations, argues that Tartaglia used MS O.

50 Arnaldo Masotti, ‘Tartaglia, Niccolo’ in DSB, 1970–1980, vol. 13, pp. 258–262. Tartaglia, op. cit. (6), p. 123. Rose, Paul Lawrence and Drake, Stillman, ‘The pseudo-Aristotelian questions of mechanic in Renaissance culture’, Studies in the Renaissance (1971) 18, pp. 65104CrossRefGoogle Scholar, p. 82; cf. Rose, op. cit. (3), pp. 154 and 285. Rose, p. 46, describes Mendoza's lending records from the Marciana, and he comments that in view of his friendship with Tartaglia, ‘these loans are of great significance for the renaissance of mathematics’. On Mendoza's manuscript collecting see Anthony Hobson, Renaissance Book Collecting: Jean Grolier and Diego Hurtado De Mendoza, Their Books and Bindings, Cambridge: Cambridge University Press, 1999, pp. 70–91; Jean Irigoin, ‘Les Ambassadeurs à Venise et le commerce des manuscrits grecs dans les années 1540–50’, in Hans-Georg Beck, Manoussos Manoussacas and Agostino Pertusi (eds.), Venezia centro di mediazione tra Oriente e Occidente (secoli XV–XVI), vol. 2, Florence: L.S. Olschi, 1977, pp. 399–415.

51 Masotti, op. cit. (50); Drake and Drabkin, op. cit. (1), ‘Introduction,’ p. 23; Clagett, op. cit. (44), p. 549.

52 In the Quesiti, op. cit. (6), Book VIII, (1546), 81r–97v. The English translation, (1969), pp. 111–143, includes only a few excerpts of this book. Tartaglia grapples with Jordanus and the science of weights and modifies several of his views on motion that he had articulated in the Nova scientia.

53 Drake and Drabkin, op. cit. (1), ‘Introduction’, p. 12; Laird, W. Roy, ‘Archimedes among the humanists’, Isis (1991) 82, pp. 628638.CrossRefGoogle Scholar

54 On Giorgio Valla's ownership of MS A see Rose, op. cit. (3), p. 35, p. 48.

55 Giorgio Valla's De expetendis et fugiendis rebus opus (Venice, 1501), XI, Chapter 8, treats quadrature, and XIII, Chapter 3 includes translations of On the Sphere and Cylinder.

56 Valla, op. cit. (55), Chapter 2 (no page numbers). Eutocius' commentary is printed in Archimedes, op. cit. (47), pp. 265–269. For the Archimedean postulate discussed by Eutocius see ‘On equilibrium of planes’, Book I, postulate 5, in The Works of Archimedes (ed. Sir Thomas Heath), Cambridge: Cambridge University Press, 1897, p. 189.

57 These were the first complete printed works of Archimedes.

58 Works of Archimedes, op. cit. (56), p. 234.

59 Works of Archimedes, op. cit. (56), ‘Quadrature of the parabola’, Propositions VI–XVX, pp. 238–243.

60 Laird, op. cit. (53), pp. 626–638, 631–632.

61 Rose, op. cit. (3), p. 266, discusses Bernardino Baldi's instrumental role in erecting the ‘new myth of Archimedes’ by means of his account of Archimedes in his Vite. Jens Høyrup, ‘Archimedism, not Platonism: on a malleable ideology of Renaissance mathematicians’, in Corrado Dollo, ed., Archimede: Mito Tradizione Scienza, Florence: Leo S. Olschki, 1992, pp. 81–110, p. 92.

62 Simms, Dennis L., ‘Archimedes and the invention of artillery and gunpowder’, Technology and Culture (1987) 28, pp. 6779CrossRefGoogle Scholar, p. 71. Simms, pp. 73–74, notes that Robertus Valturius, whom Tartaglia cites in the Quesiti, op. cit. (6), paraphrases Petrarch's crediting Archimedes with the invention of these implements of war in De re militari, Verona, 1472. While this suggests that Tartaglia was familiar with the ascription of the invention of cannons and gunpowder to Archimedes, we do not know when Tartaglia had read Valturius.

63 Van Egmond, op. cit. (10), Thesis, p. 218.

64 The Drake and Drabkin translation, op. cit. (1), p. 97, unfortunately introduced modern notation.

65 Tartaglia, op. cit. (1), (1537), 30v, ‘procederemo per algebra ponendo che il semidiametro del cerchio sia una cosa … multiplicando PI (che è posto esser una cosa) fia la mita di LO che è 10 fara, 10 cose per l'area del triangolo PLO laqual saluaremo da parte, da poi multiplicaremo la perpendicolare PH (che è pur una cosa) fia la mita de AL che sara Radice 50 ne venira Radice de 50 censi (per l'area del triangolo APL’ and triangle APO is the same as APL. That is to say that half of AL and of AO are both equal to half of the square root of 200 (i.e. of 14.142), which comes to the square root of 50 (i.e. 7.071).

66 Tartaglia, op. cit. (1), (1537), 30v and 31r, ‘faranno in suma Radice 200 censi piu 10 cose & questa suma sara eguale a l'area de tutto il triangolo ALO laqual è 100 onde levando quella Radice de 200 censi & restorando le parti et reccando a un censo haveremo uno censo piu 20 cose egual a 100 onde seguendo il capitolo trovamo la cosa valer Radice 200 men 10 & tanto fu lo semidiametro del cerchio cioe la linea PH over PI over PM & perche la linea AH è eguale alla linea HP (come di sopra fu dimostrato) seguita adonque che la detta linea AH sia etiam lei Radice 200 men 10 il qual residuo saria circa 4 1/7 onde la detta retta AH venneria a esser circa a quatro volte tanto e un settimo della retta AE che e il proposito’.

67 For examples see Yvonne Dold-Samplonius, ‘Problem of the two towers’, in Franci, Pagli and Toti Rigatelli, op. cit. (26), pp. 45–69.

68 In the Quesiti, op. cit. (6), Book I, (1546), 10r; (1969), 100–101, he explains that the violent motion of a projectile outside the perpendicular can never have any part that is straight, as he had claimed in Book II, Supposition 2 of the Nova scientia, but that he claimed it was in order to be understood by the common people, ‘Per esser inteso dal volgo’, who call the part that is insensibly curved straight.

69 Henninger-Voss, op. cit. (7), p. 386. In the Quesiti, op. cit. (6), Tartaglia reworks the theory of the relation between the speeds and effects of cannonballs in terms of weights.

70 Pseudo-Aristotle, ‘Mechanical problem’, in idem, Minor Works (tr. W.S. Hett), Cambridge: Loeb Clasical Library, 1936, Chapter 19. See Rose and Drake, op. cit. (50), pp. 65–104; and Laird, W. Roy, ‘The scope of Renaissance mechanics’, Osiris (1986) 2, pp. 4368.CrossRefGoogle Scholar

71 While Rose, op. cit. (3), p. 154, suggests that Tartaglia was only introduced to the Mechanica by Mendoza in 1539, i.e. two years after the publication of the Nova scientia, he does not put forward arguments to this end. Mendoza translated the Mechanica into Catalan; the translation was published by Raymond Foulché-Delbosc, ‘Mechánica de Aristóteles’, Revue Hispanique (1898) 5, pp. 365–405. See Erika Spivakovsky, Son of the Alhambra: Don Diego Hurtado de Mendoza, 1504–1575, Austin: University of Texas Press, 1970, p. 413, on Mendoza's manuscripts of his translation, complete with marginal and textual corrections in his hand, which are now in the Escorial. On the influence of Mendoza's library see Victor Navarro Brotons, ‘Mechanics in Spain at the end of the 16th Century and the Madrid Academy of Mathematics’, in W. Roy Laird and Sophie Roux (eds.), Mechanics and Natural Philosophy before the Scientific Revolution, Boston: Springer, 2008, pp. 239–258. Book VII of the Quesiti, op. cit. (6), 76r–80v; Drake's English translation, op. cit. (1), pp. 104–110, includes only a few excerpts of this book.

72 While discussions of effect being contingent on speed are also found in the medieval Scholastic tradition such as Jean Buridan, thus far I have not been able to establish that Tartaglia (who was not university-trained) would have had access to these texts while working on the Nova scientia.

73 N. Tartaglia, General trattato, Part I, Venice: per Curtio Troiano dei Navò, op. cit. (9), 1556, 171v.

74 Gamba and Montebelli, op. cit. (10), p. 179.

75 Michael E. Mallett and John R. Hale, The Military Organisation of a Renaissance State: Venice c. 1400 to 1617, Cambridge: Cambridge University Press, 1984, pp. 227–231. In the end, the Ottoman Empire did not actually attack Venice, but the latter became embroiled in hostilities to protect its interests. Tartaglia mentions the threat both in the dedication to the Nova scientia, (1546), f.5r of dedicatory letter, (1969), p. 69; and the Quesiti, op. cit. (6), f.1r of dedicatory letter, (1969), p. 98, Arend, op. cit. (5), pp. 312–315, provides a German translation of the dedicatory letter. See Cuomo, op. cit. (4), pp. 167–170; and Henninger-Voss, op. cit. (7), p. 376, for discussions of why Venetians are likely to have been especially receptive of Tartaglia's work.

76 Bert S. Hall, Weapons and Warfare in Renaissance Europe: Gunpowder, Technology, and Tactics, Baltimore: Johns Hopkins University Press, 1997, pp. 138–141, analyses reasons for the unpredictability of spherical cannonballs shot from smoothbore barrels, especially the incalculable spin placed on cannon balls by their final point of contact with the barrel.

77 ‘War, Technology of’, in Britannia Macropedia (2005), p. 543.

78 Hall, op. cit. (76), p. 155, on tactics, and pp. 161–162 on fortifications. Tartaglia, op. cit. (1), (1537), 5v, (1969), p. 68.

79 Only the first three of the five books that Tartaglia describes in the dedication were actually included in the Nova scientia. In the first two he presents a systematic analysis of trajectories, and the third book explains how to use sightings and make calculations in order to ascertain distances of shots.

80 Several studies have focused on Tartaglia's attempts at negotiating a higher social status and have provided a partial explanation of why he sought a mathematical solution to the bombardier's challenge. See Cuomo, op. cit. (4); Henninger-Voss, op. cit. (7); and Biagioli, op. cit. (7), pp. 41–95. Serafina Cuomo has pointed out instances of how Tartaglia used mathematics for political ends. A typical example occurs in the dedicatory letter where he mentions the bombardier's inability to understand ‘our reasons, not being well grounded in mathematics’ (emphasis added). Tartaglia thus sets himself apart from the uneducated bombardier, whose doubts were dispelled by his observations rather than by mathematical demonstrations, and he creates an exclusive niche for those who are competent in mathematics. Cuomo has shown that this appraisal of the bombardier's knowledge is an example of Tartaglia's ubiquitous attempts to establish complicity with the duke, with whom he shares knowledge of mathematics.

81 For a meticulous account of how mathematicians in late fifteenth- and sixteenth-century Italy benefited greatly from Renaissance humanism see Rose, op. cit. (3).

82 Rose, op. cit. (3), p. 14.

83 Hall, op. cit. (7), p. 29; Henninger-Voss, op. cit. (7), p. 375.

84 Aristotle himself does not make this claim.

85 In Two New Sciences, op. cit. (33), Galileo shows that a naturally accelerated body traverses a distance proportional to the square of its requisite time, and he then argues that projectile motion is essentially a variant of naturally accelerated motion. In the absence of a horizontal impression, a falling body traverses a single unit of distance in one unit of time, four units of distance in two units of time, nine units of distance in three units of time, etc. If a uniform horizontal motion is compounded with the uniform vertical motion of a body, the projectile describes a parabolic trajectory.

86 ‘The mathematical disciplines say: “You who desire to know the various causes of things, become acquainted with us”, by means of them [i.e. the mathematical disciplines] a single road leads to all [knowledge of causes].’ My translation. Drake and Drabkin, op. cit. (1), ‘Introduction’, p. 19, cite Edward W. Strong's description of the frontispiece from Procedures and Metaphysics: A Study in the Philosophy of Mathematical, Berkeley: University of California Press, 1936, pp. 57–59; Cuomo, op. cit. (4), pp. 157–161 also provides several interesting insights regarding the frontispiece.

87 ‘Let no one unskilled in geometry enter.’ The anecdote of the inscription was popular in medieval discussion of geometry and demonstration, but it dates no further back than late antique commentators on Aristotle. Eva T.H. Brann, Paradoxes of Education in a Republic, Chicago: University of Chicago Press, 1979, p. 162, n. 63, cites the entry ‘“ageometretos”, a person who is ignorant of geometry’, in Henry G. Lidell and Robert Scott, Greek–English Lexicon, Oxford: Oxford University Press, 1925, which in turn attributes the phrase ‘ageometretos medeis eisito’ to Elias Philosophus, Aristotelis categorias commentaria, and Joannes Philoponus, Aristotelis de anima libros commentaria. Neal W. Gilbert, Renaissance Concepts of Method, New York: Columbia University Press, 1960, p. 88, notes that the earliest account of this inscription seems to be in Philoponus's commentary on Aristotle's De anima. The third inscription on the frontispiece located above the illustration advertises that as gold is revealed by fire, ingenuity shows itself in mathematics (‘Aurum probator igni, et ingenium mathematicis’).

88 Neoplatonic works engaged in philosophical debates over the role of numbers and proportions in explaining the structure of the universe, the certainty of mathematical truths, and whether mathematics can serve as a means by which to ascend from the sensible world to realms of metaphysics and theology.

89 Gilbert op. cit. (87), pp. 88–90. The latter was initially publicized during the 1540s by Alessandro Piccolomini, an Aristotelian, a teacher of mathematics and an acquaintance of Tartaglia. Mendoza and Tartaglia met Piccolomini in 1539 and by the following year the Spanish ambassador had persuaded him to publish a paraphrase of the Mechanica. The work was bound with an essay entitled Quaestio de certitudine mathematicarum (Rome, 1547). See Rose, op. cit. (3), 12; Cuomo, op. cit. (4), p. 174, for late antique references to classical questions concerning the degree of certainty of mathematics kindled humanists' interest.

90 Mancosu, Paolo, ‘Aristotelian logic and Euclidean mathematics: seventeenth-century developments of the Questio de Certitudine Mathematicarum’, Studies in the History of Science (1992) 23, pp. 241265CrossRefGoogle Scholar, p. 243, emphasizes the importance of referring back to the Renaissance debates on this question in order to understand philosophical, especially epistemological, reflections on mathematics among seventeenth-century mathematicians and philosophers including Wallis, Hobbes, Barrow and Gassendi. Also see Nicholas Jardine, ‘The epistemology of the sciences’, in Charles B. Schmitt, Quentin Skinner and Eckhard Kessler (eds.), The Cambridge History of Renaissance Philosophy, Cambridge: Cambridge University Press, 1988, pp. 693–697. Rose, ‘“Certitudo Mathematicarum” from Leonardo to Galileo’, in Atti del Simposio Internazionale ‘Leonardo Disquisitio anatomica Vinci nella Scienza e nella Tecnica’, Florence: Giunti Barbèra, 1975, pp. 43–49; and William Wallace, ‘The certitude of science in late medieval and Renaissance thought’, History of Philosophy Quarterly (1986) 3, p. 286. Also see Hermann Schulling, Die Geschichte der Axiomatischen Methode im 16. und beginnenden 17. Jahrhundert, Hildesheim: Georg Olms Verlag, 1969.

91 Rose and Drake, op. cit. (50), pp. 68, 86, note that Picolomini's work marks this shift. Moreover, they note, at p. 88, that Tartaglia's study of Aristotle's treatment of the balance then gave rise to what was the first mathematical commentary on the text and ‘perhaps the first open attack on Aristotle's scientific accuracy to appear in print’.

92 Georg H.F. Nesselmann's classical study, Versuch einer kritischen Geschichte der Algebra: die Algebra der Griechen, Berlin: Reimer, 1842, pp. 302–306, on the basis of notation, divides algebra into three stages: (1) ‘rhetorische Algebra’ uses complete words; (2) ‘synkopierte Algebra’ uses abbreviated words; and finally (3) ‘symbolische Algebra’.

93 Jens Høyrup, ‘The tortuous ways toward a new understanding of algebra in the Italian abbacus school (14th–16th Centuries)’, in Proceedings of the Joint Meeting of PME 32 and PME-NA XXX, Morelia, 2008, pp. 1–15, argues that historians must look toward the Italian abacus schools, not the works of al-Khwarizmi and Fibonacci, for the roots of developments in algebra during the second half of the sixteenth century and the first half of the seventeenth.