Abstract
In this section, biographical sketches and philological-historical-epistemological reflections are reported. In particular, we present Tartaglia’s study of mathematics, geometry, arithmetic, ballistics and fortifications.
Ma poi fra me pensando un giorno, mi parve cosa biasimevole, vituperosa e crudele & degna di non puoca punitione appresso Iddio & alli uomini a voler studiare di assottigliare tal essercitio dannoso al prossimo, anzi destruttore della specie umana & massime de Cristiani in lor continue guerre.
(Tartaglia 1537, Epistola, 5rv, line 25).
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Notes
- 1.
- 2.
- 3.
Tartaglia 1554, XXII, footnote 5. The translation is above in the running text.
- 4.
“Io Nicolo Tartaia Dottor di Mathematice […] ritrovandomi hora in letto aggravato da molto male, ho deliberato ordinar i fatti miei.” (The translation is ours; see also Filza 168.VII; N.119; Boncompagni 1881).
- 5.
Curzio Troiano Navò (or de Navò) was one of the most important editors and book sellers during the sixteenth century in Venice. His French origins are not clear. Some historians report about a family-publishing composed of him and his brothers. They and their heirs edited and published ca. 30 books between 1537 and 1599.
- 6.
Curzio Troiano Navò posthoumously published two other works by Tartaglia: Iordani Opusculum de ponderositate (de Nemore [Tartaglia’s editor] 1565) and Esperienze fatte da Nicolo Tartalea from 1541, 14 April to 1551, 7 April (Tartaglia 1541–1551). Philological notes regarding this point are provided in the following paragraphs concerning Book VI and Gionta in the Quesiti et inventioni diverse (Tartaglia [1554] 1959).
- 7.
“Io mi attrovo libri del mio general trattato de numeri et misure p.a (prima) 2.da (seconda) 3.a (terza) et 4.a (quarta) parte, et di miei Quesiti et invention diverse circa quatro cento […] Item mi attrovo circa .60. opere della travagliata inventione et ragionamenti […] Item libri de diverse sorte per lo mio studiare, per la valuta di cento ducati in circa […] Item mi attrovo circa quaranta libri di nuova scientia […] Io mi attrovo una balla de libri de Paris di diverse sorte, quali io sto per vendere”. (The translation is ours. In the Notary Archive of Venezia, a document (Filza 168.VII; N.119) which includes the testament exists; (see also Boncompagni 1881; Pizzamiglio 2007, 40).
- 8.
“Degna di nota ci sembra di conseguenza la circostanza per cui il Tartaglia, al momento della sua morte, non fosse in possesso di nessuna delle due edizioni latine dell’Euclide da lui utilizate, che erano in–f°, cioè nè quella di B. Zamberti [v. 1505] nè quella di G. Campano-L. Pacioli [see 1509b]”. Pizzamiglio in Tartaglia 2007, XXXIII (Author’s brackets and Italics). Recently on Euclid by Campano see Busard (2005) and on Aristotle-Archimedes and Euclid see Renn, Damerow and McLaughlin (2003). On early editions of Euclid’s elements see Stanford (1926).
- 9.
“1567. Nicolò Tartaglia Bresciano d’humile nascimento attese alle cose Matematiche e particolarmente alla Geometria & all’Aritmetica con tanto genio, che si lasciò molti adietro. Trasferì costui in lingua volgare gl’Elementi d’Euclide, ch’egli leggeva publicamente in Venetia. Scrisse molte opere appartenenti al moto de corpi gravi, a’ tiri dell’Artigliarie, a fortificationi de luoghi, a misurar con la vista, & altre cose tali, e finalmente scrisse due gran volumi, ne quali raccolse tutto quello che s’appartiene ad una compita specolatione e pratica delle cose dell’Aritmetica e della Geometria. Fu egli grand’avversario di Girolamo Cardano e scrisseli contro alcune opere. Attese nondimeno così poco alla bontà della lingua, che muove a riso talhora chi legge le cose sue.” (Baldi, 1707, 133).
- 10.
With regard to the second half of the past century, we should include works by Bortolotti and, of course, the crucial works by Masotti and recently by Pizzamiglio. Most important works cited in Table 1.1 are detailed in the References section below.
- 11.
Hereafter Quesiti.
- 12.
“Micheletto” (Little Michele) due his low stature. “Cavallo” in English is “horse”. “Cavallaro” is an ancient Italian word derived from “Cavallo” and means, more or less, a man busy with horses or using horses.
- 13.
The surname Fontana appears in his testament: Zuampiero Fontana.
- 14.
Tartaglia (1554, Book IX, Q. I.
- 15.
Masotti 1970–1980, 13, 258. (Author’s quotation marks).
- 16.
Tartaglia 1554, Book VI, Q 8.
- 17.
On Tartaglia’s Euclid, see Tartaglia 1543a, 2007). On Euclid see also Commandino edition (1575) and on Archimedes and Euclid see Knorr (1978–1979, 1985).
- 18.
She was Oscar Chisini’s (1889–1967) pupil and collaborated closely on historical studies with Masotti. She wrote two important memoirs on Luca Pacioli (1445–1517; Pisano 2013).
- 19.
On Masotti’s contributions about Tartaglia see: Masotti (1957, 1958a, b, 1960–1962a, 1960a, b, 1961–1962, 1962a, b, c,d, e, 1963a, b, c, 1964, 1971, 1972, 1973–1974, 1975, 1976a, b, 1979, 1980b).
- 20.
“È proprio quell’iniziativa che giunge ora a compimento, in occasione della celebrazione del 450° anniversario della morte del grande matematico bresciano, ed è quindi giusto e doveroso che questo volume delle “Opere di Niccolò Tartaglia” venga dedicato proprio al prof. Arnaldo Masotti. [Transl.: ours]. See also: Tartaglia 2007. “1990. Rendiconti dell’Istituto Lombardo, col. 124, pp 157–166 (L. Amerio) Nastasi, Lettera matematica, 23.
- 21.
We specify that Masotti reported the existence of some documents (Archivio di Stato di Verona) that declared his stay in Verona to be around 1529–1533 (Masotti 1970–1980, 13, 259). In this period 17 Quesiti concerning Book IX were proposed to him to solve.
- 22.
Until 1557 and except a short stay in Brescia (March 1548–October 1549).
- 23.
With the exception of his return to Brescia from 1548 to 1549 (ca. 18 months) he taught at Sant’Afra, San Barnaba, San Lorenzo and at the Academy near Rezzato, a small village.
- 24.
Heath 2002, On the Sphere and Cylinder, Book II, 62.
- 25.
The news spread and a mathematical contest made up of thirty problems was organized (12 febbraio 1535). Only Tartaglia succeeded in solving these problems in the allotted time.
- 26.
We remark that among the 31 inquiries which Ferrari sent to Tartaglia in Terzo Cartello di matematica disfida (1547–1548), there are two inherent to the inscription and reciprocal circonscription of regular polygons, which can also be found in Commentaria in Euclidis Elementa geometrica by Cardano (Cardano 1574; see also Masotti 1974b, 1974c, pp 66–68).
- 27.
Ferrari, Quinto cartello (Milan, October 1547), [25–39], 141–155.
- 28.
- 29.
Although he didn’t publish his discovery, before his death, Scipione dal Ferro revealed it to one of his students, the Venetian Anton Maria Florido (Floridus).
- 30.
It must be noted that a different historiography opinion exists according with Cardano who waited for six years so that Tartaglia could have the chance to publish it. About the role played by historiography of science in historical investigations see as very relevant Kragh 1987.
- 31.
- 32.
Cfr.: Pizzamiglio 2007.
- 33.
Directly on Greek codes, as yet unidentified, however of rather low quality.
- 34.
Cfr.: Pizzamiglio 2007.
- 35.
Gabrieli (1986, 29–67).
- 36.
Tartaglia (1554, Book IX, Q 22).
- 37.
The triangular method by means of a different configuration is possible to see in other early scholarly works, e.g., in Pascal’s Traité du triangle arithmétique (1653). Nevertheless, the earliest explicit depictions of a triangle of binomial coefficients occur in the 10th century in commentaries on the Chandas Shastra, an Ancient Indian book on Sanskrit prosody written (fl. 2nd century BC) by Pingala. (Edwards 2002, 30–31).
- 38.
Two years before his death (1556), Tartaglia worked on his larger compendium, which unfortunately, he was unable to finish and publish.
- 39.
Generally speaking, the Trattato was intended (at that time) as research work not necessarily large, and well structured mostly based on known principles. The Summa, typically within Meddle Ages, had the prerogative to be a largely and organically exhaustive for monastic schools and universities (Pisano 2013a, b, c, d).
- 40.
At the end of the book, this part includes the following quotation “in Vinagia per Comin da Tridino MDLVI” even though the title page reads “1560”. It circulated after Tartaglia’s death. An even more interesting fact is that in the inventory this book is cited “in folio”, that is, printed but not in hardcover.
- 41.
The correlation between Euclid’s propositions (IV: 1–16) and respectively Tartaglia’s propositions (Tartaglia 1556–1560, Part V, IX: 1–17, 13r–16r) is an interesting historical matter.
- 42.
Buridan, also in Latin Johannes Buridanus (ca. 1300 – ca. 1360). The historical genesis of the impetus theory – later applied to the motion of projectiles – is quite complex and varied. Aside from Aristotle’s initial theory (384–322 B.C.), among the scholars who dealt with the topic, we note: Johannes Philoponus (active in VI century), Pūr Sina’ (Persian) son of Sina called Avicenna (980–1037), Roger Bacon (1214–1292), Thomas Aquina (1225–1274), Pierre Jean Olivi (1248–1298), Francesco of Marchia or of Esculo, of Ascoli (fl. XIV century), William of Ockham (ca. 1280 – ca. 1349), and for some considerations, Jordanus de Nemore, too. Here, for the sake of brevity, and since there is already a vast literature on the topic, we refer only to that which historians consider a true cultural background of projectile theory until the Renaissance (Giannetto, Maccarone, Pappalardo and Tiné 1992).
- 43.
Buridan 1509. Nicole Oresme (ca. 1320/1325–1382) version should also be considered. An English study is in The Science of Mechanics in the Middle Ages (Clagett 1959) and in turn reproduced by Maier ([1509] 1968) which, in turn, includes – with some modifications – the Parisian edition from 1509. For the comments of Subtilissimae Quaestiones, at first glance, one can see Clagett (Clagett 1959 and secondary literatures cited). Clagett dated Buridan’s manuscript around 1357. It is archived at the Vatican Library in Roma (Vat. Lat. 2136, 1r.).
- 44.
Galilei Ms. 72, 116v; see also: Hill 1986, 283–291. On Galilei and mechanization of nature see recently: Bertoloni 2006; Garber and Roux 2013, Biagioli 2003.
- 45.
Tartaglia criticized Aristotle’s theory of the lever in regard to the sensitivity of a scale according to which (wrongly) the Stagirite supported that the greater the length of the arms, the greater the sensitivity of the instrument (Tartaglia 1554, Book VII, Qs IV–V–VI, 80v–82v). Still exciting about Aristotelian mechanics is Cartelon (1975).
- 46.
1504. Mortar’ model (Codex Atlanticus, 33r.). See also: Gille 1964 (and English version: 1966), 219; Pisano and Capecchi 2010a, Pisano 2009a, c; Vilain 2008). In 16th century an interesting study about ballistic arguments taking into account a straight-line path, the velocity lost and consequent downwards of the cannonball was done by Noviomago (1561).
- 47.
Here, Tartaglia also gave his contribution, which we will later discuss.
- 48.
Later, other scholars took up the questions of the range of projectile motion. Mainly (17th centuries): Galilei (The Dialogues Concerning the Two New Sciences), Torricelli (De Motu) on the geometrical way of calculating the range of a projectile, and Newton (Principia) on the proportion between air resistance and the square of the speed of the projectile. Recently on Newton, science-and-revolutions see Buchwald and Feingold (2011), Cohen (1985). On Newton and Geneva edition see: Pisano 2013b, 2014a, 2015a, b; Bussotti and Pisano 2014a, b; see also Newton (1687), (1713), ([1726; 1739–1742]; (1822), (1739–1742), (1972).
- 49.
See also Riccardi’s quotation in his Biblioteca Matematica Italiana dalla origine della stampa ai primi anni del secolo (Riccardi 1870–1880, II, 497; see also Riccardi (1870–1928, 1952, 1985) and Pizzamiglio 1989).
- 50.
Galileo’s notes were made more legible by transcribing the content of the folio (Drake 1985, 3–14; 1992, 113–116).
- 51.
Biblioteca istorico-critica di fortificazione permanente (Marini 1810, p XII).
- 52.
Nowadays we find an undue use of the term axiomatization concerning non-modern theories in the history of science. Usually, in mathematics and mathematical physics, the term axiomatization of a scientific theory represents a formulation of a scientific system of statements (e.g., axioms/primitive terms) in order to build a consistent-coherent corpus of statements (e.g., propositions) which may be logically and deductively derived from these statements; and the proof of any statement (i.e., theorems) should be taken into account and traceable back to these axioms. Of course, the latter is a difficult condition to be universally claimed: i.e., see the case-study of Archimedean’s On the equilibrium of planes (Capecchi and Pisano 2007, 2010b, Pisano 2009b, Pisano and Capecchi 2008, 2010b), and non-Euclidean geometry. Therefore, the use of axioms (in the history of science) as self-evident statements in a theory does not mean that this theory-system is axiomatically built (Pisano 2008). In fact, three fundamental properties should be formally respected: 1) an axiomatic system is said to be consistent if it lacks contradiction, i.e. the ability to derive both a statement and its denial from the system’s axioms; 2) in an axiomatic system, an axiom is called independent if it is not a theorem that can be derived from other axioms in the system; a system will be called independent if each of its underlying axioms is independent. Although independence is not a necessary requirement for a system, consistency is; 3) An axiomatic system will be called complete if for every statement, either itself or its negation is derivable. For example, Euclid of Alexandria authored the earliest extant axiomatic geometry and number theory presentation that can be formally considered: an axiomatic system, a model theory, and mathematical proofs within a formal system. All of that evidently is lacking in Tartaglia. Therefore a random use of axioms (i.e., in Tartaglia) only means a tentative step toward ordering a new theory – or simply to order a scientific reasoning extrapolated from a known theory – by means of primitive statements and eventually derived proportions. This aspect belongs to several periods of the history of science (see Pisano’s references). Recently on physico-mathematics as case study in Descartes-Agonistes see wonderful Schuster 2013, and on physico-mathematics in Descartes’ physical works see Bussotti and Pisano 2013.
- 53.
Stillman Drake translated it as “A body is called uniformly heavy […]” (Drake and Drabkin 1969, p 70). A remark is necessary. Now, following Tartaglia’s text (just after First definition) we note his recalls Avicenna’s work (see “Fen”, that is a section of the Liber canonis). Particularly Averroes’ fourth book of the De caelo et mundo, text 29 is cited by Tartaglia (Ibidem). In addition, the tentative correlation with geometric forms of bodies, the kind of the matter of bodies, the concept of shared gravity where “[…] each body, compounded of four elements, one of which is air, shares gravity […]” with bodies’ qualities (Ibidem), make evident his difficulties to distinguish equally bodies from – as Drake proposed – uniformly bodies. Of course the knowledge of a physical magnitude lacks: let us think to uniformly term which can be addressed (ambiguously) both constant velocity and no-acceleration. Moreover, one should also add equally bodies between them like i.e., Tartaglia correctly wrote “Equally heavy bodies are said to be similar and equal when they do not show [among each other] any substantial or accidental differences” (Tartaglia 1537, Book I, def. II, 9v). On that we would add that we prefer both equally and uniformly or more simply constant bodies since at that time the concept of constant gravity was already proposed in many works during the 1300s–1400s, i.e., one can see Subtilissimae Quaestiones super octo Physicorum libros Aristotelis (Buridanus 1509, 1513, 1942) by Johannes Buridanus and Tractatus de configurationibus qualitatum et motuum by Nicole Oresme also edited by Clagett as A treatise on the configuration of qualities and motions (Oresme 1968; of course see Clagett 1959; Brown 1967–1968; Moody and Clagett ([1952] 1960). Now, by avoiding Latinism-and-vulgare philological analysis since within a dictionary the term “egualmente” can be translated by “uguale a se”, “uniforme”, “costante” (equally itself, uniform, constant) we remark that an a posteriori reflection related to physical proprieties of a body during the motion, i.e. an ideal rigid geometric body and its tendency to fall down, may suggest, at that time, the idea of constant, that is a sort of invariant of the motion. In Tartaglia’s words: “[…] is not perceptibly influenced by air opposition during its motion” (Ibidem). On the contrary let us think about a paper or a leaf falling down. Finally in our opinion, since he refers to ancient conceptions of the fifth elements, Aristotelian and Medieval streams (i.e., gravitas ex figura), early attempts to formalize the friction as resistance by corpo offeso (offended bodies) concerning weapons etc., we prefer to literally translate it with equally heavy adding the term uniformly to both to give the idea that some physical substance (not clear at that time) does not change and for the modern-specialist-reader, avoiding attribution to Tartaglia – at this stage –of advanced mathematical abstract concepts within physics –mathematics relationships of subjects that are still hard to make historically and epistemologically clear and since the mathematization of the nature was still far from complete. (Pisano 2011; Pisano and Bussotti 2013b, c; on the relationship between physics and mathematics in the nineteenth century see: Pisano and Bussotti 2015c; Pisano 2013e, 2014a, d, e, 2015a, b; Pisano and Capecchi 2013; Barbin and Pisano 2013; see also Numbers 2006; Olschki 1919–1927; Pedersen 1992).
- 54.
“Diffinitione Prima. Corpo egualmente graue è detto quello, che ſecondo la grauita della materia, et la figura di quella è atto à non patire ſenſibilmente la oppoſition di l’aere in alcun ſuo moto.” (Tartaglia 1537, Book I, 9r).
- 55.
“Diffinitione. IIII. Il Tempo e una miſura del mouimento, et della quiete, li termini del quale ſon dui iſtanti.” (Tartaglia 1537, Book I, 9v).
- 56.
“Diffinitione. VI. Mouimento naturale di corpi egualmente graui e quello che naturalmente fanno da un luogo ſuperiore a un’altro inferiore perpendicularmente ſenza uiolenza alcuna.” (Tartaglia 1537, Book I, 10v).
- 57.
“Diffinitione. VII. Mouimento uiolente di corpi egualmente graui e quello che fanno sforzatamente di giuſo in ſuſo, di ſuſo in giuſo, di qua et di la, per cauſa di alcuna poſſanza mouente.” (Tartaglia 1537, Book I, 10v).
- 58.
Based on previous comments on axiomatization, we note that, in order to argue on statics in his “Scientia di Pesi” (Science of Weights) only in the Book VIII of the Quesiti et invention diverse (Tartaglia 1554, Book VIII, 83rv–97rv; see Chap. 3) Tartaglia proposes a sort of prologue to the statics writing his definitive conceptual ideas concerning the role played by Proper principles (also called Proper Principles by Aristotle as sentences strictly related to the subject of theory: Aristotle 1853, On the Definition and Division of Principles, Book I, Chap. X, p 266), Propositions (or also called by him conclusions which can confirm the science of weights), Suppositions (also called by him true principles) and Petitions (as sentences which can go against science of weights). We will return to that idea (see Chap. 3). For further readings see Pisano and Capecchi 2010a, b; Pisano 2009b.
- 59.
i.e., one can see Book I-The foundations: theories of triangles, parallels, and area of the Euclid’s Elements where after an initial 23 Definitions follow 5 Postulates, 5 Common Notions and 48 Propositions.
- 60.
A difference with regard to bodies in motion with respect to Archimedean statics studies.
- 61.
By physical reasonings.
- 62.
“[…] cioè a che segno si dovesse assettare un pezzo de arteglieria che facesse il maggior tiro che far possa sopra un piano”. (Tartaglia 1537, Book I, 3rv).
- 63.
“Serenissimo” is, e.g., a title for some Principe and Doge of the Republic of Venice. “Altezza” is also commonly used.
- 64.
Tartaglia (1537, Book I, 3rv, line 1).
- 65.
On invasion of Italy, particularly North-East (especially Venezia). Since Francesco Maria della Rovere, Duke of Urbino (interlocutor of Tartaglia’s letter) was employed by the Venetian Republic to organize a defense, Tartaglia’s words are particularly important at this stage of the Quesiti. We note that in Book I of the Quesiti, Tartaglia also describes technical results on 20–pound culverin as being 10 feet in length (ca. 3 mt.), and weighing 4300 pounds (ca. 1950 Kg).
- 66.
Tartaglia (1537, 4rv, line 37).
- 67.
The figure and other observations are below.
- 68.
From the beginning of the last century until today, we have not been exempt from seeing similar situations faced by Nobel Prize winners and involved scholars.
- 69.
We note that in the subsequent pages we can also see the explicit observation against the Aristotelean conception of violent and natural motion in effect at the time. (Tartaglia 1554, Book I, Q I, 5rv–7rv; Q. III, 11rv; Qs I–II–III–VI, 5rv–13rv, Q I, 6rv).
- 70.
Tartaglia (1537, Book I, 5rv, line 23).
- 71.
The following works during 1531–1532 which, in general, from a historical point of view, had a certain influence on society, should also be noted. Gerolamo Fracastoro observes the tails of comets and concludes that they are always facing opposite the Sun; 1535–38. Fracastoro publishes Homocentricorum sive de stellis, in which the system of the world starting with the geometric motion of the planets defined by the uniform rotations of homocentric spheres is discussed; 1536. Calvino publishes Istituzioni della religione cristiana.
- 72.
“[…] primo scritto di balistica […] basato saldamente sull’esperienza viva e concreta dei fatti e svolto con l’ausilio della geometria e del calcolo numerico […].” (Bolletti 1958, 14, line 8).
- 73.
- 74.
A clarification. Within 7rv folia (in-between Book I and Book II) of the Nova scientia, Tartaglia proposed his main arguments concerning the 3 parts-composition of the trajectory of a projectile: rectilinear segment, arc of circumference and a final rectilinear segment towards the centre of the Earth (Tartaglia 1537, Book I, 13rv–20rv; see also Book I, IV–V Props., 14r–15r; for the representation of various distances with respect to various inclinations see Ivi, 20v). These parts are described by some figures (Tartaglia 1537, Book I, 15r, 16r) which are divided into letters corresponding to natural motion, violent motion and mixed natural motion. Of course without a modern vectorial and mathematical interpretation of a composed motion (particularly along a curved path where the change of vectorial orientation produces an acceleration), then it is obvious that in Tartaglia’s context a body cannot assume (in a point long the path) negative and positive values at the same time.
- 75.
At that time many practical instruments were in use, so it is reasonable to think that the instruments often cited by Tartaglia were not originally invented by himself. For example, Tartaglia cites a frequent use of the quadrant at that time and without mentioning which version of quadrant he preferred. For sure we do not have historical proof if he really did or did not invent the quadrant that he often cited in his own manuscripts. Thus, even if similar instruments are reported in secondary literature (e.g., see: Alberti fl. 15th, 10rv–11rv (retrieved via web); Essenwein and Germanisches 1873), we cannot claim an historical hypothesis within history and historical epistemology of science studies concerning his eventual (or not) invention.
- 76.
On that Drake (Drake and Drabkin 1969, 66) pointed out that in the next editions Tartaglia avoided the word Archimedean (“Archimedeane”) and wrote “[…] con ragion natural […]” (by physical reasonings). In any case the relationship between Archmedean and natural reasoning is confirmed since the inductive method was adopted.
- 77.
“Da poi (Signor humanissimo) con ragion Archimedeane qualmente la distantia dil sopra ditto tiro elleuato alli 45 gradi sopra al orizonte, era circa decupla al tramito retto dun tiro fatto per il piano del orizonte: che da bombardiere è ditto tiro de ponto in bianco, con la qual evidentia, Magnanimo Duca, trovai con ragione geometrice e algebratice qualmente balla tirata vesro li detti 45 gradi sopra a l’orizzonte va circa a quattro volte tanto per l’aere di quello che va essendo tirato per il pian de l’orizzonte, che dà borbandieri è chiamato (come ho detto) tirar de punto in bianco [cioè tirare orizzontalmente].” (Tartaglia 1537, 5rv, line 28).
- 78.
Tartaglia 1537, 5rv–9rv, line 7. Incidentally, literature on military arguments was current at that time, (i.e., see Alberti). Thus, Tartaglia’s novelties might be merely part of these shared studies.
- 79.
- 80.
It should be noted that in the paradigm of Aristotelean science, it was necessary for the projectile trajectory to be composed of three parts: an inclined rectilinear branch (“violent motion”), a circular branch (“mixed motion”) and a vertical branch (“natural motion”). That is to say, that as gravity prevails it decreases speed and the “balla” falls vertically. Subsequent developments of this vision hypothesized the decrease of the speed of the “balla” was due to the impetus action. In Book I of Quesiti (Tartaglia 1554, Book I, Q III), Tartaglia denies this thesis, affirming that gravity, which is always present, acts on the “balla” from the beginning (of the shot) of its path until it touches the ground. According to Bolletti, Tartaglia’s explanation is essentially based on the fact that the “balla”, shot with whatever initial speed, would favor the composition –so to speak – of gravity and of the impetus of the “balla” itself (Bolletti 1958, 61–62). It must be noted, however, that if this was Tartaglia’s intention, in Q III of Quesiti, I don’t believe he was as explicit and precise as it seems in Bolletti’s analysis.
- 81.
We specify that in Book I Tartaglia suggests to the reader that, before proceeding in his ballistic theory, it is opportune to examine elements of the science of weights (Tartaglia 1554, Book I, Q II, 7rv–10rv). On Tartaglia’s dynamics see Koyré (1960); recently see Pisano and Bussotti 2015b in: Pisano, Agassi and Drozdova (eds). Hypotheses and Perspectives within History and Philosophy of Science - Hommage to Alexandre Koyré 1964–2014. Dordrecht Springer.
- 82.
Tartaglia (1554, Book VI, Q III, 65r).
- 83.
Tartaglia (1554, Book VI, Q IV, 66r, line 10).
- 84.
Galileo, as we will see in the following paragraph, considers this “quality” without referring to Tartaglia (Galilei 1888–1909c, II, pp 107–109, pp 118–120). In this sense, we will also see that the Galilean work feels the effects of the content from Tartaglia’s Book VI and Gionta; even given the different historical period and different aim (also didactic) of Galilei’s text compared to that of Tartaglia’s, in Trattato di Fortificazione, important theoretical advances can be noted (Pisano and Capecchi 2012).
- 85.
The subject of the small model in Renaissance architecture will be dealt with later in the analysis of Delle Fortificationi by Lorini who considers the matter (Pisano and Capecchi 2009, II, 797–808; see also Pisano and Capecchi 2014a, b). On mechanics and architecture an indispensable work is Entre Mécanique et Architecture by Patricia Radelet-de Grave and Edoardo Benvenuto (Radelet-de Grave and Benvenuto 1995).
- 86.
The original text, which is not necessary to comment upon, is presented in the Appendix to this chapter. (see also Vol. II).
- 87.
Tartaglia (1554, Book VI, Q VII, 67rv).
- 88.
The original text, which is not necessary to comment upon here, is presented in Chap. 4. It should be noted that the suggestive autobiographical information is found at the end of Book VI (see also Pizzamiglio 2005).
- 89.
The dialogue form (Puer’s questions and Magister’s answers) was perfectly integrated in the typically Renaissance scientific context (Altieri Biagi 1984, 891–847) both as advanced research, and teaching science.
- 90.
Tartaglia 1554, Gionta, Q VI, 76rv, line 2.
- 91.
Tartaglia (1554, Gionta, Q I, 70v, line 34). The translation is ours.
- 92.
The parianette, also called traverse, are structural elements placed along the walls of the curtains. They are usually arranged vertically. The aim was to limit the effects of enfilade fire. As is clear from the text, Tartaglia shows personal innovation for the construction of the traverse by assuming an inclination of and a height greater than that of a man.
- 93.
A small gun.
- 94.
Tartaglia 1554, Gionta, Q I, 71r, line 10. The image in the quoted text is suitably enlarged and rotated. Maybe due to an editorial pagination, the reader will find two similar images in the Quesiti (Tartaglia 1554, Gionta, Q I, 71r and 72v).
- 95.
See also Tartaglia’s figure in the text.
- 96.
This is an important fact for this work. It involves the ability to absorb kinetic energy from the “ball” in reference to the type of material used; in this case the dirt should absorb the shot better than the stone. (Tartaglia 1554, Gionta, Q VI, 76rv). Tartaglia also provides an entertaining geometric analogy with the moon (Ivi, Q VI, 76rv, line 19).
- 97.
In accordance with Masotti’s unverified hypothesis (Tartaglia 1554, Qs L–LI) it could be about Rusconi (di) Zanantonio, architect and painter, student of Tartaglia who, in Quesiti, Tartaglia explicitly names when introducing a problem on artillery and on ballistics in Vitruvio’s work (Tartaglia 1554, Book II, Q X, 34–35; here the entire “quesito” is dedicated to him: “done by […]”); and from a solution to a geometric problem (Ivi, Book IX, Q XXXVIII, 123; see also: Ivi, Q VII, 76rv, line 1).
- 98.
For Archimedes works see Heath 2002.
- 99.
- 100.
“Il primo libro di Archimede Siracusano, da me trovato & tradotto da uno latinamente scritto, qual era andato quassi in strazzaria & in mano di un salzizaro in Verona l’anno 1531. Del qual libro molte parti erano totalmente rotte & annullate, onde accioche una così degna sua opra non restasse del tutto morta, mi sono sforzato di redrizzarla & d’interpretar le parti che mancavano, talmente che ogni commune impegno potrà gustar dimostrativamente la sua gran dottrina in tal materia”. (Tartaglia 1560, Parte IV, Book III, 43v–44r).
- 101.
We note that Tartaglia did not mention the existence of the second book. Later (Tartaglia 1565) his editor, Curtio Troiano, published both the Archimedean books on the floating bodies as credited manuscripts from Tartaglia for his editorial job. (Heath 2001, XXVII–XVIII). Some historians have conjectured that Tartaglia had all Archmedean works and did not publish some of them freely. Nevertheless, this only means that Curtio Troiano produced an editorial job after Tartaglia’s death, and this it is not sufficient to claim (historically) that Tartaglia truly had the whole Archimedean corpus.
- 102.
“Si dichiara volgarmente quell libro di Archimede Siracusano, ditto, de insidentibus aquae, materia di non poca speculation, & intellettual dilettatione” (Tartaglia 1551a, [part of the subtitle of] Ragionamento Primo). Translation is ours.
- 103.
Heath 2002, p XXVII, line 10. (Author’s italics and quotations marks). The codices mentioned by Heath are: B = Codex Parisinus 2360, olim Mediceus; C = Codex Parisinus 2361, Fonteblandensis. Others codexes are mentioned, so we refer to Heath for a full reading. (Author’s symbol and quotations).
- 104.
In 1904, in the proceedings of the Congresso internazionale di scienze storiche held in Roma (1903) and edited by Sezione VIII di Storia delle Scienze Fisiche, Matematiche, Naturali e Mediche, a paper (Tonni-Bazza 1904b, 293–307), reported a discussion on the last results concerning Tartaglia’s death, the controversy on some content published in 1546 and/or 1554 of Quesiti et inventioni diverse and others things around the Brisciano. We note that the paper begins with the typical title page of Quesiti et inventioni diverse, but without including the date et al., so it is unclear which edition it is.
- 105.
Tonni-Bazza (1904b, 303, line 13). (Author’s italics and quotations marks).
- 106.
“Il Tartaglia, come si vede, rispondendo al Castrioti, si rallegra che i loro singoli studi sulle fortificazioni conducano a risultati conformi; e ciò, dice il Tartaglia, si vedrà nel Book dei quesiti fatto da me nuovamente nel sesto Book. I Quesiti et inuentioni diverse, già erano stati pubblicati la prima volta nel 1546; ma nel 1554 sopravvenne la ristampa […] con la appendice al sesto Book cui allude il Tartaglia […]. Ivi figurano alcuni problemi propostigli dal Magnifico e Clarissimo sig. Marc’ Antonio Morosini dottore e Philosopho Eccellentissimo. Non figura il Castrioti; sebbene vi si trattino argomenti contenuti nei « discorsi » di lui, e nella sua lettera, il Tartaglia, prometta una risposta partichulare et generale.” (Tonni-Bazza 1904b, 303, line 13 (Author’s italics and quotations marks)).
- 107.
“Tartaglia, Niccolo, Quesiti et inventioni diverse, 1554”
Permanent URL: http://echo.mpiwg-berlin.mpg.de/MPIWG:KQ9TP5T3
- 108.
Tartaglia (1554, Gionta, Q VII, 77v, line 2).
- 109.
Ibidem, line 16.
- 110.
“[…] ne far vendere in Venetia, ne in alcuno altro luoco, ò terra del Dominio Veneto, per anni diece, sotto pena de duc. 300 & perdere le opere in qual si voglia […]” (Tartaglia 1546, 77v).
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Tartaglia N (1541–1551) Esperienze fatte da Nicolo Tartalea. In: Tartaglia [de Nemore] 1565 folia 20rv–23v
Tartaglia N (1543a) Euclide Megarense acutissimo philosopho: solo introduttore delle scientie mathematice: diligentemente rassettato, et alla integrita ridotto, per il degno professore di tal Scientie Nicolo Tartalea, Brisciano, Secondo le due tradottioni: e per commune commodo & utilita di latino in volgar tradotto, con una ampla esposizione dello istesso tradottore di novo aggionta. Talmente chiara, che ogni mediocre ingegno, senza la notitia, over suffragio di alcun’altra scientia con facilità, sera capace a’ poterlo intendere. Stampato in Vinegia per Venturino Rossinelli ad instantia e requisitione de Guilielmo de Monferra, & de Pietro di Facolo da Vinegia libraro, & de Nicolo Tartalea Brisciano Tradottore: Nel Mese di Febraro. Anno no nostra salute MDXLIII
Tartaglia N (1543b) Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi per Nicolaum Tartaleam Brixianum (mathematicarum scientiarum cultorem) multis erroribus emendata, expurgata, ac in luce posita, multisque necessariis additis, quae plurimis locis intellectu difficillima erant, commentariolis sane luculentis & eruditissimis aperta, explicata atque illustrata existunt, appositisque manu propria figuris quae graeco exemplari deformatae ac depravatae erant, ad rectissimam symetriam omnia instaurata reducta & reformata elucent, apud Venturinum Ruffinellum, sumptu & requisizione Nicolai de Tartaleis Brixiani, mense Aprili
Tartaglia (1543c) Archimedis Siracusani Tetragonismus In: Tartaglia 1543b, 19v–29r
Tartaglia (1543d) Archimedis Syracusani Liber In: Tartaglia 1543b, 29v–31r
Tartaglia (1543e) Archimedis de Insidentibus Aquae In: Tartaglia 1543b, 31v–[36r]. See also: Tartaglia 1565–Insidentibus, 1rv–16rv
Tartaglia N (1546) Quesiti et inventioni diverse de Nicolo Tartalea Brisciano. Stampata in Venetia per Venturino Ruffinelli ad instantia et requisitione, & a proprie spese de Nicolo Tartalea Brisciano Autore. Nel mese di luio L’anno di nostra salute. M.D.XLVI
Tartaglia N (1551a) Ragionamenti de Nicolo Tartaglia sopra la sua Travagliata inventione. Nelli quali se dechiara volgarmente quel libro di Archimede Siracusano intitolato. De insidentibus aquae, con altre speculatiue pratiche da lui ritrouate sopra le materie, che stano, & chi non stano sopra lacqua ultimamente se assegna la ragione et causa naturale di tutte le sottile et oscure particularità dette et dechiarate nella detta sua Travagliata inue[n]tione co[n] molte altre da quelle dependent. Stampata in Venetia per Nicolo Bascarini à istantia & requisistione, & a proprie spese de Nicolo Tartaglia Autore. Nel mese di Maggio L’anno di nostra salute. 1551
Tartgalia N (1551b) [Ragionamenti I-III and Supplimento] Regola Generale da Sulevare con Ragione e Misura no[n] solame[n]te ogni affondata Nave: ma una Torre Solida di Mettallo Trovata da Nicolo Tartaglia, delle discipline Mathematice amatore intitolata la Travagliata Inventione. Insieme co un artificioso modo di poter andare, & stare plogo tepo sotto acqua, a ricercare le materie affondate, & in loco profundo. Giontovi anchor un tratttato, di segni della mutationi dell’Aria, over di te[m]pi, material no[n] men utile, che necessaria, a Nauiganti, & altri. Nicolo Bascarini, Venetia.
Tartaglia N (1554) Quesiti et inventioni diverse de Nicolo Tartaglia, di novo restampati con una gionta al sesto libro, nella quale si mostra duoi modi di redur una citta inespugnabile. In: Venetia per Nicolo de Bascarini, ad istantia & requisistione, & a proprie spese de Nicolo Tartaglia Autore. Nell’anno di nostra Salute. MDLIIII
Tartaglia N (1556–1560) La prima [–sesta] parte del general trattato di numeri, et misure di Nicolo Tartaglia, nella quale in diecisette libri si dichiara tutti gli atti operativi, pratiche, et regole necessarie non solamente in tutta l’arte negotiaria, & mercantile, ma anchorin ogni altra arte, scientia, over disciplina, dove interuenghi il calculo. In Vinegia per Curtio Troiano de i Navò. MDLVI [–MDLX]
Tartaglia N ([1558] 1562) La nova scientia de Nicolo Tartaglia con una giurita al terzo libro. Curtio Troiano dei Navo, Vinegia (see here: http://www.sudoc.fr/147532981)
Tartaglia N (1562) [Regola Generale with Supplimento and Ragionamenti I–II] Regola Generale di solevare ogni fondata Nave & navilli con Ragione. In: Vinegia, per Curtio Troiano de i Navò. MDLXII
Tartaglia N (1565a) Iordani Opusculvm de Ponderositate, Nicolai Tartaleae Studio Correctum Novisqve Figuris avctum. Cum Privilegio Traiano Curtio, Venetiis, Apud Curtium Troianum. MDLXV
Tartaglia N (1565b–Euclid) Euclide Megarense acutissimo philosopho, solo introduttore delle scientie mathematice. Diligentemente rassettato, et alla integrita ridotto, per il degno professore di tal Scientie Nicolo Tartalea, Brisciano. Secondo le due tradottioni. Con una ampla esposizione dello istesso tradottore di nuono aggiunta. Talmente chiara, che ogni mediocre ingegno, senza la notitia, over suffragio di alcun’altra scientia con facilità, sera capace a’ poterlo intendere. In Venetia, appresso Curtio Troiano, 1565
Tartaglia N (1565c–Insidentibus) Archimedis de Insidentibus Aquae Liber Primus […] Liber Secundus. Cum Privilegio Traiano Curtio, Venetiis, Apud Curtium Troianum. MDLXV, 1rv–16rv
Tartaglia N (1569) Euclide Megarense acutissimo philosopho, solo introduttore delle scientie mathematice. Diligentemente rassettato, et alla integrita ridotto, per il degno professore di tal Scientie Nicolo Tartalea, Brisciano. Secondo le due tradottioni. Con una ampla esposizione dello istesso tradottore di nuono aggiunta. Talmente chiara, che ogni mediocre ingegno, senza la notitia, over suffragio di alcun’altra scientia con facilità, sera capace a’ poterlo intendere. In Venetia, appresso Giovanni Bariletto, 1569
Tartaglia N (1583) [La nova scientia de Nicolo Tartaglia con una gionta al terzo libro] Disciplinae mathematicae loquuntur[.] Qui cupitis rerum varias cognoscere causas disate nos cunctis hac patet una uia. In Vinetia, Appresso Camillo Castelli, 1583
Tartaglia N (1876) I sei scritti di matematica disfida di Lodovico Ferrari coi sei contro–cartelli in risposta di Niccolò Tartaglia, comprendenti le soluzioni de’ quesiti dall’una e dall’altra parte proposti. Raccolti, Autografi e Pubblicati da Enrico Giordani, Bolognese. Premesse notizie bibliografiche ed illustrazioni sui Cartelli medesimi, estratte da documenti già a stampa ed altri manoscritti favoriti dal Comm. Prof. Silvestro Gherardi, Preside del’Istit. Tecn. Prov. di Firenze. R. Sabilimento litografico di Luigi Ronchi e tipografia degl’Ingegneri, Milano
Tartaglia N (1949) Tartaglia. Entry In: Enciclopedia Italiana Treccani, vols, XXII, XXXIII, VIII. Istituto Poligrafico di Stato, Roma
Tartaglia N ([1554] 1959) La nuova edizione dell’opera “Quesiti et inventioni diverse de Nicolo Tartaglia brisciano, Riproduzione in facsimile dell’edizione del 1554 [with Gionta al sesto Libro] Masotti A (ed) Commentari dell’Ateneo di Brescia, Tipografia La Nuova Cartografica, Brescia
Tartaglia N (2000) Tutte Le Opere. 3 Cd-Rom. Riproduzione dei testi originali. Pizzamiglio L (ed). Synergy Solutions, Brescia
Tartaglia N (2007) Euclide Megarense. Pizzamiglio P (ed). Ateneo di Brescia, Brescia
Tartaglia N (2010) Niccolo Tartaglia: Questions et inventions diverses, Livre IX ou L’invention de la résolution des équations du troisième degré. Hamon G, Degryse L (eds). Hermann, Paris
Tartaglia N (2013) The Nova Scientia of Nicolò Tartaglia. Valleriani M (ed). The Open Access Edition, Berlin
Tonni-Bazza V (1904b) Frammenti di nuove ricerche intorno a Nicolò Tartaglia. Comunicazione dell’ingegnere Vincenzo Tonni-Bazza In: Atti del Congresso internazionale di scienze storiche (Roma 1903) Tipografia della R. Accademia dei Lincei, Roma, s. XXXIII, pp 7–8, pp 293–307 (Tonni-Bazza also published a prologue of this communication in Tonni-Bazza 1904c. See Tonni-Bazza 1904b, p IX)
Villa M (1963–1964) Niccolo Tartaglia a quattro secoli dalla morte. Atti della Accademia delle Scienze dell’Istituto di Bologna, Classe di Scienze Fisiche. Rendiconti s XII(1):5–24
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Pisano, R., Capecchi, D. (2016). Niccolò Tartaglia and the Renaissance Society Between Science and Technique. In: Tartaglia’s Science of Weights and Mechanics in the Sixteenth Century. History of Mechanism and Machine Science, vol 28. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9710-8_1
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