Galileo's Construction of Idealized Fall in the Void

In effect, the document in which Galileo derived the law from the “erroneous” principle communicated to Sarpi, though traditionally associated with the 1604 letter, cannot be dated with certainty. It was first published in Le opere di Galileo Galilei, ed. by Favaro Antonio (Edizione Nazionale, 20 vols, Florence, 1890–1909; cited below as Galilei 1890–1909, followed by the Roman numeral of the volume and the page numbers in Arabic numerals). Cf. Galilei 1890–1909, viii, 373–4.

Koyré Alexandre , Études galiléennes (Paris, 1966), 86ff., 96.

Shea William , Galileo's intellectual revolution (New York, 1972), pp. vii–viii.

Shea , Galileo's intellectual revolution (ref. 3), 3.

Damerow P. , Freudenthal G. McLaughlin P. Renn J. , Exploring the limits of preclassical mechanics (New York and Berlin, 1992; but cf. also the second edition, New York and Berlin, 2004, with a few relevant revisions, especially in chap. 3, pp. 135–41); Drake Stillman , “Galileo's 1604 fragment on falling bodies”, The British journal for the history of science, iv (1969), 340–58, reprinted in Drake Stillman , Essays on Galileo and the history and philosophy of science (3 vols, Toronto, Buffalo and London, 1999; hereafter Essays), ii, 187–207; idem, “Uniform acceleration, space, and time”, The British journal for the history of science, xvii (1970), 21–43, reprinted in Drake , Essays, ii, 208–32; idem, Galileo studies: Personality, tradition and revolution (Ann Arbor, 1970); idem, “The uniform motion equivalent to a uniformly accelerated motion from rest”, Isis, lxiii (1972), 28–38, reprinted in Drake , Essays, ii, 233–47; idem, “Galileo's experimental confirmation of horizontal inertia: Unpublished manuscripts”, Isis, lxiv (1973), 291–305, reprinted in Drake , Essays, ii, 147–59; idem, “Galileo's discovery of the law of free fall”, Scientific American, ccxxviii (1973), 84–92, reprinted in Drake , Essays, ii, 248–64; idem, “Galileo's work on free fall in 1604”, Physis, xvi (1974), 309–22, reprinted in Drake , Essays, ii, 281–91; idem, “Free fall from Albert of Saxony to Honoré Fabri”, Studies in history and philosophy of science, v (1975), 347–66, reprinted in Drake , Essays, iii, 239–57; idem, “Galileo's accuracy in measuring horizontal projections”, Annali dell' Istituto e Museo di Storia della Scienza di Firenze, x (1985), 3–14, reprinted in Drake , Essays, ii, 321–31; idem, History of free fall: Aristotle to Galileo (Toronto, 1989); and idem, Galileo at work: His scientific biography (New York, 1995; 1st edn, Chicago, 1978).

Cf. Favaro Antonio , Galileo Galilei e lo studio di Padova (2 vols, Padua, 1966; 1st edn, Florence, 1883); idem, Galileo Galilei a Padova (Padua, 1968); Drake , Galileo at work (ref. 5); and the interesting essays in Santinello G. (ed.), Galileo e la cultura padovana (Padua, 1992), on various aspects of the Padua period. See also Hill D. , “Galileo's work on 116v: A new analysis”, Isis, lxxvii (1986), 283–91; idem, “Dissecting trajectories: Galileo's early experiments on projectile motion and the law of fall”, Isis, lxxix (1988), 646–68; idem, “Pendulums and planes: What Galileo didn't publish”, Nuncius, ix (1994), 499–515; Naylor Ronald , “The evolution of an experiment: Guidobaldo del Monte and Galileo's Discorsi demonstration of the parabolic trajectory”, Physis, xvi (1974), 323–46; idem, “Galileo and the problem of free fall”, The British journal for the history of science, vii (1974), 105–34; idem, “Galileo's simple pendulum”, Physis, xvi (1974), 23–46; idem, “Galileo: Real experiment and didactic demonstration”, Isis, lxvii (1976), 398–419; idem, “Galileo: The search for the parabolic trajectory”, Annals of science, xxxiii (1976), 153–72; idem, “Galileo's need for precision: The point of the fourth day pendulum experiment”, Isis, lxviii (1977), 97–103; idem, “Galileo's theory of motion: Processes of conceptual change in the period 1604–1610”, Annals of science, xxxiv (1977), 365–92; idem, “Mathematics and experiment in Galileo's new sciences”, Annali dell' Istituto e Museo di Storia della Scienza di Firenze, iv (1979), 55–63; idem, “Galileo's theory of projectile motion”, Isis, lxxi (1980), 550–70; idem, “Galileo's method of analysis and synthesis”, Isis, lxxxi (1990), 695–707; Renn J. , “Galileo's manuscripts on mechanics: The project of an edition with full critical apparatus of MSS. GAL. Codex 72”, Nuncius, iii (1988), 193–241; Renn J. Damerow P. Rieger S. , “Hunting the white elephant: When and how did Galileo discover the law of fall?”, Science in context, xiii (2000), 299–419: Settle Thomas , “Galilean science: Essays in the mechanics and dynamics of the ‘Discorsi’”, Ph.D. Thesis, Cornell University, 1966, copy consulted on line at: http://www.mpiwg-berlin.mpg.de/litserv/diss/settle/html/Page001.htm; Wisan W. , “A new science of motion: A study of Galileo's De motu locali”, Archive for history of exact sciences, xiii (1974), 103–306; and idem, “Galileo and the process of scientific creation”, Isis, lxxv (1984), 269–286.

For the sake of brevity, I have quoted Manuscript 72 with arbitrary dates as “Galilei 1600–38”, followed by the arabic numeral of the folio. The manuscript is available on-line, at the ECHO, European Cultural Heritage Online, website: http://xserve02.mpiwg-berlin.mpg.de:18880/echo_nav/echo_pages/content/scientific_revolution/galileo. Cf. Favaro Antonio , “Documenti inediti per la storia dei manoscritti Galileiani nella Biblioteca Nazionale di Firenze”, Bullettino di bibliografia e di storia della scienze matematiche e fisiche, xviii (1885), 1–112 and 151–230, for the intriguing story of the formation of the entire manuscript collection at Florence. Most of the content of Manuscript 72, however, has been known since it was first published in Galilei 1890–1909, viii, 363–448, cf. Galluzzi Paolo , Momento: Studi galileiani (Rome, 1979), 276–83. Ronald Naylor claims that in experiments aimed at studying the motion of projectiles Galileo found results conflicting with the sameness of ratios hypothesized in 1604; cf. his “Galileo's theory of projectile motion” (ref. 6), 562ff. Damerow et al. argue that this cannot be the case, and that the process of “discovery of the proportionality between the degree of velocity and time” was mostly a mathematical process; cf. Damerow , Freudenthal McLaughlin Renn , Exploring the limits of preclassical mechanics (ref. 5), 177ff. In W. Wisan's view, the discovery was based on a crucial experiment concerning projection, documented in Galilei 1600–38, folio 116 verso. See Wisan , “Galileo and the process of scientific creation” (ref. 6), 227–9. Hill , “Galileo's work on 116v” (ref. 6), rejects Wisan's reconstruction, and suggests that the experiment was tried to test a mathematical connection between degree of speed and the square root of the distance fallen through. All things considered I believe that these interpretations of single folios of Manuscript 72, though interesting, raise mostly unresolvable questions since they are based on reconstructions which are so underdetermined that a verdict cannot be reached. Finally, cf. Sylla E. , “Galileo and the Oxford calculatores: Analytical languages and the mean-speed theorem for accelerated motion”, in Wallace W. (ed.), Reinterpreting Galileo (Washington, DC, 1986), 53–110, on some technical aspects of the mathematics underlying the 1604 reasoning, and Palmieri P. , “Galileo's mathematical natural philosophy”, Ph.D. Thesis, London University, 2002, 43–45, for a criticism of Sylla's argument.

Cf. Galilei 1890–1909, ii, 261–6. The editor of the National Edition of Galileo's works, Antonio Favaro (Galilei 1890–1909, ii, 259–60), believed it to have been written about 1604, in connection with the 1604 letter to Sarpi. Koyré, Études galiléennes (ref. 2), 138, thought it to have been written in 1609, because in it Galileo claims that it took him a long time to come up with the “correct” definition. Wisan , “A new science of motion” (ref. 6), 277–8, argues that the De motu accelerato should be dated even later, to about 1630, on the basis of scant evidence concerning the quality of paper of the folios. Damerow , Freudenthal McLaughlin Renn , Exploring the limits of preclassical mechanics (ref. 5), 226, follow Wisan. As was already noted by Koyré, on the other hand, the nineteenth-century editor of a collection of Galileo's works, Eugenio Albèri, printed it among the Pisa period writings, De motu antiquiora, in Galilei G. , Le opere di Galileo Galilei, ed. by Albèri Eugenio (17 vols, Florence, 1842–56), xi. For the date and the internal chronology of De motu antiquiora, cf. Drabkin I. , “A note on Galileo's De motu”, Isis, li (1960), 271–7; Fredette Raymond , “Les De motu ‘plus anciens’ de Galileo Galilei: Prolegomenes”, Ph.D. Thesis, Montreal University, 1969, copy consulted on line at: http://www.mpiwg-berlin.mpg.de/litserv/diss/fred_l/HTML/Page001.htm; Drake Stillman , “The evolution of De motu”, Isis, lxvii (1976), 239–50, reprinted in Drake , Essays (ref. 5), i, 201–14; idem, “Galileo's pre-Paduan writings: Years, sources, motivations”, Studies in history and philsophy of science, xvii (1986), 429–48, reprinted in Drake , Essays (ref. 5), i, 215–35; Hooper W. , “Galileo and the problem of motion”, Ph.D. Thesis, Indiana University, 1992, 62–138; Camerota M. , Gli scritti ‘De motu antiquiora’ di Galileo Galilei: Il Ms Gal 71 (Cagliari, 1992); and Giusti E. , “Elements for the relative chronology of Galilei's De motu antiquiora”, Nuncius, xiii (1998), 427–60. The only exception to the early date is Carugo A. Crombie A. C. , “The Jesuits and Galileo's ideas of science and nature”, Annali dell' Istituto e Museo di Storia della Scienza di Firenze, viii (1983), 3–68, convincingly rejected, I believe, by Camerota in his Gli scritti ‘De motu antiquiora’ di Galileo Galilei cited above in this endnote.

Palmieri P. , “Mental models in Galileo's early mathematization of nature”, Studies in history and philosophy of science, xxxiv (2003), 229–64, and Mišcevic Nenad , “Mental models and thought experiments”, International studies in the philosophy of science, vi (1992), 215–16. Mental models are mental representations analogical with the states of affairs they are intended to represent.

Cf. Armijo Carmen , “Un nuevo rol para las definiciones”, in Montesinos J. Solís C. (ed.), Largo campo di filosofare: Eurosymposium Galileo 2001 (La Orotava, Tenerife, 2001), 85–99; Drake Stillman , “Velocity and Eudoxan proportion theory”, Physis, xv (1973), 49–64, reprinted in Drake , Essays (ref. 5), ii, 265–80; idem, “Mathematics and discovery in Galileo's physics”, Historia mathematica, i (1974), 129–50, reprinted in Drake , Essays (ref. 5), ii, 292–306; idem, “Euclid Book V from Eudoxus to Dedekind”, Cahiers d'histoire et de philosophie des sciences, n.s., xxi (1987), 52–64, reprinted in Drake , Essays (ref. 5), iii, 61–75; and Frajese A. , Galileo matematico (Rome, 1964). For a general treatment of the various aspects of the Euclidean theory of samenesses of ratios, I have relied on: Grattan-Guinness I. , “Numbers, magnitudes, ratios, and proportions in Euclid's Elements: How did he handle them?”, Historia mathematica, xxiii (1996), 355–75; Sasaki C. , “The acceptance of the theory of proportions in the sixteenth and seventeenth centuries”, Historia scientiarum, xxix (1985), 83–116; Saito K. , “Compounded ratio in Euclid and Apollonius”, Historia scientiarum, xxxi (1986), 25–59; and idem, “Duplicate ratio in Book VI of Euclid's Elements”, Historia scientiarum, 1 (1993), 115–35. Rose P. , The Italian renaissance of mathematics: Studies on humanists and mathematicians from Petrarch to Galileo (Geneva, 1975), is an extensive, immensely erudite survey of Renaissance mathematics in Italy from a non-technical point of view. Cf. also Sylla E. , “Compounding ratios: Bradwardine, Oresme, and the first edition of Newton's Principia”, in Mendelsohn E. (ed.), Transformation and tradition in the sciences: Essays in honor of I. Bernard Cohen (Cambridge, 1984), 11–43.

Giusti E. , “Aspetti matematici della cinematica Galileiana”, Bollettino di storia delle scienze matematiche, i (1981), 3–42; idem, “Ricerche galileiane: Il trattato ‘De motu equabili’ come modello della teoria delle proporzioni”, Bollettino di storia delle scienze matematiche, vi (1986), 89–108; idem, “Galilei e le leggi del moto”, in Galilei G. , Discorsi e dimostrazioni matematiche intorno a due nuove scienze attinenti alla meccanica ed i movimenti locali, ed. by Giusti Enrico (Turin, 1990), pp. ix–lx; idem, “La teoria galileiana delle proporzioni”, in Conti L. (ed.), La matematizzazione dell' universo: Momenti della cultura matematica tra ‘500e’600 (Perugia, 1992), 207–22; idem, Euclides reformatus: La teoria delle proporzioni nella scuola galileiana (Turin, 1993); idem, “Il filosofo geometra: Matematica e filosofia naturale in Galileo”, Nuncius, ix (1994), 485–98; idem, “Il ruolo dela matematica nella meccanica di Galileo”, in Tenenti A. , Galileo Galilei e la cultura veneziana (Venice, 1995), 321–38; Maracchia S. , “Galileo e Archimede”, in Montesinos Solís (ed.), Largo campo di filosofare (ref. 10), 119–30; Palladino F. , “La teoria delle proporzioni nel Seicento”, Nuncius, vi (1991), 33–81; Palmieri P. , “The obscurity of the equimultiples: Clavius' and Galileo's foundational studies of Euclid's theory of proportions”, Archive for history of exact sciences, lv (2001), 555–97; and idem, “Galileo's mathematical natural philosophy” (ref. 7).

The Postils to Rocco is a set of comments and notes that Galileo wrote in response to a book published in 1633 by an Aristotelian philosopher, Antonio Rocco, and attacking the Dialogue concerning the two chief world systems (published by Galileo the year before). The Postils to Rocco has been published in Galilei 1890–1909, vii, 569–750. On Rocco, cf. Favaro Antonio Rocco A. , Atti del Reale Istituto Veneto di Scienze Lettere e Arti, 7th series, iii (1892), 615–36.

Galilei 1890–1909, i, 307.

In general, on the possible influence on Galileo of Francesco Buonamici, a professor at Pisa in the 1590s, cf. Koyré , Études galiléennes (ref. 2), 24–47. M. O. Helbing has studied in detail Buonamici's natural philosophy and concluded that there seems to be some affinity between the two authors' philosophical attitudes towards mathematics; cf. his La filosofia di Francesco Buonamici (Pisa, 1989), 352–71. On Galileo and Girolamo Borri (1512–92), a professor at Pisa when Galileo was a student there, see de Pace Anna , “Galileo lettore di Girolamo Borri nel De motu”, in Rambaldi Enrico (ed.), De motu: Studi di storia del pensiero su Galileo, Hegel, Huygens e Gilbert (Milan, 1990), 3–70. Cf. also Camerota M. Helbing M. O. , “Galileo and Pisan Aristotelianism: Galileo's De motu antiquiora and the Quaestiones de motu elementorum of the Pisan professors”, Early science and medicine, v (2000), 319–65. This paper “tries to show that some of the main topics discussed in De motu antiquiora (the famous experiments on falling bodies and the questions of acceleration and Archimedean extrusion) are connected with the debate [emphasis added] on the ‘motion of the elements’ (motus elementorum) which took place in the second half of the sixteenth century at the ‘Studio Pisano’ among the local professors of philosophy” (ibid., 323). However, to my mind, aside from emphasizing vague similarities between scant passages in the De motu antiquiora and texts by two Aristotelian professors at Pisa, the authors fail to explain what the ‘connections’ between Galileo and the debate on the motion of the elements would really have consisted in. Camerota and Helbing also claim that Galileo's view on Archimedean extrusion (another vague expression that the paper's authors never clarify) was influenced by Girolamo Borri, and exactly the opposite of Buonamici's. For a radically different analysis of Galileo's views on Archimedean buoyancy in De motu antiquiora, which were firmly grounded in Archimedes's mathematical treatment of floating bodies, cf. Palmieri P. , “The cognitive development of Galileo's theory of buoyancy”, Archive for history of exact sciences, lix (2005), 189–222. A fundamental work is Lewis C. , The Merton tradition and kinematics in late sixteenth and early seventeenth century Italy (Padua, 1980), especially 127–70, with further bibliography. See also Schmitt C. , “The faculty of arts at Pisa at the time of Galileo”, Physis, xiv (1972), 243–72, “Towards a reassessment of Renaissance Aristotelianism”, History of science, xi (1973), 159–93, “The University of Pisa in the Renaissance”, History of education, iii (1974), 3–17, and Aristotle and the Renaissance (Cambridge, MA, 1983).

Galilei 1890–1909, i, 251.

Only Latin versions of On floating bodies had circulated in Western Europe until Heiberg's discovery of the Constantinople palimpsest, in the early twentieth century. See Archimedes , Archimedis opera omnia, ed. and transl. by Heiberg J. L. , ii (Leipzig, 1913).

Cf. Galilei 1890–1909, i, 215ff., and 379 (on the problem of Hiero's crown), and ibid., 233ff. (postils to On the sphere and the cylinder). Cf. also Shea , Galileo's intellectual revolution (ref. 3), 1–11, and Dollo C. , Galileo Galilei e la cultura della tradizione (Soveria Mannelli, Catanzaro, 2003), 63–86, in regard to the general influence of Archimedes on Galileo.

Cf. Galilei 1890–1909, i, 187ff., and Di Girolamo G. , “L' influenza archimedea nei ‘Theoremata’ di Galilei”, Physis, xxxvi (1999), 21–54.

Archimedes , Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi (Venice, 1543) and De iis quae vehuntur in aqua libri duo, a Federico Commandino urbinate in pristinum nitorem restituti et commentariis illustrati (Bologna, 1565).

“… sit aquae status, ante quam magnitudo in ipsam demittatur, abcd…. Necessarium itaque est, ut, dum magnitudo f demergitur, aqua attollatur.” Cf. Galilei 1890–1909, i, 255–6.

Cf. Galilei 1890–1909, i, 300. Galileo used to teach a course at Padua on traditional astronomy, preparing notes for a Trattato della sfera, which is published in Galilei 1890–1909, ii, 211–55. Unfortunately he does not comment on the Earth's size.

Galilei 1890–1909, i, 258. It is worth noting that the diagram of the watery spheres was repeated by Galileo in the Trattato della sfera (ref. 21), 219–20. Here Galileo wished to prove that the surface of water is spherical.

“… ita ut, nempe, mobile naturale unius ponderis in lancem vicem gerat; tanta autem moles medii, quanta est mobilis moles, alterum in lance pondus repraesentet” (Galilei 1890–1909, i, 259).

Galilei 1890–1909, i, 260–2. Galileo explicitly refers to Aristotle's passage in Physics, IV, at 215b. Cf. Aristotle , The complete works of Aristotle, ed. by Barnes J. (2 vols, Princeton, 1984), i, 366.

“De illis mobilibus quae sunt eiusdem speciei dixit Aristoteles, illud velocius moveri quod maius est”, Galilei 1890–1909, i, 262–3.

Galilei 1890–1909, i, 263. Cf. Aristotle's De caelo, at 301b. The passage is rather difficult. See, for example, a Renaissance edition in Latin, commented by Simplicius, in In quatuor libros De coelo Aristotelis (Venice, 1563), 206–7, and the modern English translations by Guthrie W. K. C. , in Aristotle, De caelo (Cambridge, MA, 1939), 277 (with the Greek original text on the facing page), and by Barnes, in Aristotle , The complete works of Aristotle (ref. 24), i, 494. Cf. also Palmieri , “Mental models in Galileo's early mathematization of nature” (ref. 9), 258ff., for a discussion of an interesting interpretation of this passage by a defender of Aristotle, a contemporary of Galileo and professor of Greek at Pisa University.

Galilei 1890–1909, i, 263.

It is worth quoting the original passage of this startling analogy. “Qua conclusione [i.e., that all mobiles of the same kind fall at the same speed] qui mirantur, mirabuntur etiam, tam maximam trabem quam parvum lignum aquae supernatare posse: Eadem enim est ratio.” Galilei 1890–1909, i, 263–4.

Galilei 1890–1909, i, 264. Galileo also specifies that it must be proven that volumes of the same matter have the same ratio as their weights. He proves this claim further on in De motu, cf. Galilei 1890–1909, i, 348–50.

Galilei 1890–1909, i, 264.

Galilei G. , De motu, trans. by Fredette R. (ECHO, European Cultural Heritage Online website, at http://xserve02.mpiwg-berlin.mpg.de:18880/echo_nav/echo_pages/content/scientific_revolution/galileo), 17. On this website, there are available both the translation and images of the manuscript folios, the so-called Manuscript 71. Cf. the original, in Galilei 1890–1909, i, 264.

Galilei 1890–1909, i, 264.

This can be gathered by looking at the Latin versions of commentaries on Aristotle circulating in the Renaissance. Cf, for example, Simplicius's commentary on De caelo, 301b, in Simplicius, In quatuor libros De coelo Aristotelis (ref. 26), 207, in which he asserts that Aristotle's demonstration [ostensio] is “by reductio ad absurdum [per deductionem ad impossibilem]”, and then concludes his paraphrase of Aristotle's text with the same formula used by Galileo, “quod est inconveniens”.

Aristotle , The complete works of Aristotle (ref. 24), i, 366.

Galilei , De motu (ref. 31), 28. Cf. the original in Galilei 1890–1909, i, 277.

Galilei , De motu (ref. 31), 28–29, with a few changes. Original in Galilei 1890–1909, i, 277–8.

Galilei , De motu (ref. 31), 33, with a few changes. Original in Galilei 1890–1909, i, 282.

Galilei 1890–1909, i, 273.

Galilei 1890–1909, ii, 262.

Galilei 1890–1909, ii, 263.

If you have difficulty, as I had at first, in focusing on the diagram because of the disturbing effects of optical illusionism, imagine drawing vertical lines passing through o and i.

Galilei 1890–1909, ii, 265.

In effect the origin of the tradition to date De motu accelerato to 1604 or later may have been Antonio Favaro's decision to publish the short writing in the second volume of Galilei 1890–1909 (the so-called National Edition), according to his conviction that the piece should have been written in connection with the letter to Sarpi (Galilei 1890–1909, ii, 259–60). No other reason, however, was brought by Favaro to support his decision, though he knew that Albèri had published it together with the De motu manuscript material, with which it is still bound today.

The De motu accelerato ends with the following definition of uniformly accelerated motion: “Motum uniformiter, seu aequabiliter, acceleratum dico ilium, cuius momenta, seu gradus, celeritatis a discessu ex quiete augentur iuxta ipsiusmet temporis incrementum a primo instanti lationis” (Galilei 1890–1909, ii, 266). In Two new sciences Galileo rephrased this definition, as follows: “Motum aequabiliter, seu uniformiter, acceleratum disco ilium, qui, a quiete recedens, temporibus aequalibus aequalia celeritatis momenta sibi superaddit” (Galilei 1890–1909, viii, 198).

Galilei 1890–1909, viii, 197–8.

Galilei 1890–1909, ii, 263 (De motu accelerato version), and ibid. , viii, 198 (Two new sciences version, with only a slight grammatical difference due to sentence construction).

Euclid , The thirteen books of the Elements, translated with commentary by Heath Thomas (2nd edn, 3 vols, New York, 1956), ii, 112–86. See Armijo , “Un nuevo rol para las definiciones” (ref. 10); Drake , “Velocity and Eudoxan proportion theory” (ref. 10); Drake , “Mathematics and discovery in Galileo's physics” (ref. 10); Drake , “Euclid Book V from Eudoxus to Dedekind” (ref. 10); Frajese , Galileo matematico (ref. 10); Giusti , “Aspetti matematici della cinematica Galileiana” (ref. 11); Giusti , “Ricerche galileiane: Il trattato ‘De motu equabili’ come modello della teoria delle proporzioni” (ref. 11); Giusti , “Galilei e le leggi del moto” (ref. 11); Giusti , “La teoria galileiana delle proporzioni” (ref. 11); Giusti , Euclides reformatus (ref. 11); Giusti , “Il filosofo geometra” (ref. 11); and Giusti , “Il ruolo dela matematica nella meccanica di Galileo” (ref. 11).

On the non-algebraic character of Galileo's approach to sameness of ratios, cf. Drake , “Velocity and Eudoxan proportion theory” (ref. 10); Drake , “Mathematics and discovery in Galileo's physics” (ref. 10); Drake , “Euclid Book V from Eudoxus to Dedekind” (ref. 10); Palmieri , “The obscurity of the equimultiples” (ref. 10); Palmieri , “Galileo's mathematical natural philosophy” (ref. 7); and Palmieri , “Mental models in Galileo's early mathematization of nature” (ref. 10).

Cf. the details of the definition of “sameness” of ratios, in Giusti , “Aspetti matematici della cinematica Galileiana” (ref. 11); Giusti , “Ricerche galileiane” (ref. 11); idem, “Galilei e le leggi del moto” (ref. 11); idem, “La teoria galileiana delle proporzioni” (ref. 11); idem, Euclides reformatus (ref. 11); and Palmieri , “The obscurity of the equimultiples” (ref. 10). Euclid's definition of “sameness” of ratios is complex. Galileo considered it to be obscure, and tried to replace it with a new one, in the very final months of his life. On Galileo and the equal multiples definition, cf. especially Giusti , Euclides reformatus (ref. 11), 57–82, and Palmieri , “The obscurity of the equimultiples” (ref. 11).

I have translated this definition from a common Renaissance edition of Euclid's Elements, that edited and commented upon by Christoph Clavius. The original is as follows: “In eadem ratione magnitudines dicuntur esse, prima ad secundam, et tertiam ad quartam, cum primae et tertiae aeque multiplicia, a secundae et quartae aeque multiplicibus, qualiscunque sit haec multiplicatio, utrumque ab utroque vel una deficiunt, vel una equalia sunt, vel una excedunt; si ea sumantur quae inter se respondent.” Clavius Christoph , Commentaria in Euclidis Elementa geometrica (Hildesheim, 1999; facsimile edn of the first volume of Christophori Clauii Bambergensis e Societate Iesu Opera mathematica V tomis distributa (Mainz, 1611–12)), 209.

“Io … stimo vera l' una e l' altra proposizione: Essendo certo che il continuo costa di parti sempre divisibili, dico che è verissimo e necessario che la linea sia composta di punti, e il continuo d' indivisibili; e cosa forse più inopinata vi aggiungo, cioè che, essendo il vero uno solo, conviene che il dire che il continuo costa di parti sempre divisibili, col dire che il continuo costa d' indivisibili, siano una medesima cosa…” ( Galilei 1890–1909, vii, 745). In addition, we know that Galileo had planned to write a tract on the composition of the continuum well before 1610 ( Galilei 1890–1909, x, 352). All studies of Galileo's views concerning the continuum that I am aware of, traditionally based on Two new sciences, miss the fundamental unorthodoxy of Galileo with respect to the then long-standing dichotomy between endless divisibility and indivisibility. In his last work, Galileo's discussion of atomism and the continuum meanders through paradoxes and questions that tend to hide his conviction of the equivalence between infinite indivisibles and endless divisibility, explicitly stated in his postils to Rocco. Cf. Lasswitz K. , Geschichte der Atomistik (2 vols, Hamburg and Leipzig, 1890), ii, 37–54; Dijksterhuis E. J. , The mechanization of the world picture (London, 1961), 419–24; Shea W. , “Galileo's atomic hypothesis”, Ambix, xviii (1970), 13–27; Smith A. Mark , “Galileo's theory of indivisibles: Revolution or compromise?”, Journal of the history of ideas, xxxvii (1976), 571–88; Quan S. , “Galileo and the problem of infinity: A refutation and a solution. Part 1: The geometrical demonstrations”, Annals of science, xxvi (1970), 115–51, and “Part 2: The dialectical arguments, and the solution”, Annals of science, xxviii (1972), 237–84; Redondi P. , Galileo eretico (Turin, 1988), 11–31; idem, “I problemi dell' atomismo”, in Montesinos Solís (ed.), Largo campo di filosofare (ref. 9), 661–76; Palmerino C. R. , “Una nuova scienza della materia per la scienza nova del moto: La discussione dei paradossi dell'infinito nella prima giornata dei Discorsi galileiani”, in Gatto R. Festa E. (ed.), Atomismo e continuo nel XVII secolo (Naples, 2000), 275–319; and idem, “Galileo's and Gassendi's solutions to the rota Aristotelis paradox: A bridge between matter and motion theories”, in Murdoch J. E. Newman W. R. Lüthy C. H. (ed.), Medieval and early modern corpuscular matter theories (Leiden, 2001), 381–422.

Unfortunately, this folio of Manuscript 72 ( Galilei 1600–38, viii, folio 91 verso), much as the folio containing the “erroneous” proof of the times-squared law, cannot be dated with certainty. Damerow , Freudenthal McLaughlin Renn , Exploring the limits of preclassical mechanics (ref. 5), 178ff., interpret this document as a proof of the correct sameness of ratios between speeds and times, and thus assume that it was written after the 1604 erroneous attempt (ibid., 181). Their argument, however, is based on the questionable assumption that truth must have followed error. Date is not a problem for my purposes here, on the other hand, since all I wish to show is that more or less at the same time as the 1604 “error”, Galileo was trying to express his early analysis of De motu accelerato in terms of samenesses of ratios.

“In motu ex quiete eadem ratione intenditur velocitatis momentum, et tempus ipsius motus” ( Galilei 1600–38, viii, folio 91 verso). On the meanings of “momento” in Galileo , cf. Galluzzi , Momento (ref. 7).

Damerow , Freudenthal McLaughlin Renn , Exploring the limits of preclassical mechanics (ref. 5), 181, claim that this result amounts to a contradiction because Galileo had previously used the sameness of ratios between speeds and spaces.

Galilei 1890–1909, viii, 203–4.

Galilei 1890–1909, xi, 85.

Galilei 1890–1909, viii, 203–4. The details of the counterargument are irrelevant for our purposes. Cf. Drake , History of free fall (ref. 5), 74–75.

Galilei 1890–1909, viii, 203.

Galilei 1890–1909, i, 296–302.

In Two new sciences, this is assumed as the fundamental postulate of the entire axiomatic structure of the treatise on motion. Galileo only furnished a corroborative argument from the motion of pendula. Cf. Galilei 1890–1909, viii, 205–8. Eventually Galileo found a “mechanical” proof of this postulate in the final years of his life, cf. Galilei 1890–1909, viii, 442–5.

Galilei 1890–1909, viii, 383–4.

Galilei 1890–1909, vii, 51–52.

Cf. Galilei 1890–1909, viii, 212–13, for Galileo's account, and Settle Thomas , “An experiment in the history of science: With a simple but ingenious device Galileo could obtain relatively precise time measurements”, Science, cxxxiii (1961), 19–23.

Galilei 1890–1909, i, 348–50. Cf. Euclid , The thirteen books of the Elements (ref. 50), ii, 114.

Galilei 1890–1909, vii, 731.

In De motu, Galileo had already given this definition of weight, but had not recognized its relevance for the case of free falling bodies. “We say that we feel burdened [gravari] when some weight is placed upon us which tends downwards because of its gravity, in which case we have to oppose a force so that the weight no longer descends; that opposing is what we call to feel burdened [… tunc dicimur gravari, quando super nos incumbit aliquod pondus quod sua gravitate deorsum tendit, nobis autem opus est nostra vi resistere ne amplius descendat; illud autem resistere est quod gravari appellamus]”. See Galilei 1890–1909, i, 288, 388. Since ‘gravari’ is a passive infinitive we might render it more literally as “be burdened”. However, let me explain why I prefer “feel burdened”. The question has to do with meaning reconstruction. We cannot simply attach the dictionary or grammatical value to a cognate. The context of the passage suggests that Galileo has in mind a human behaviour (that of opposing a force). It goes as follows: “… tunc dicimur gravari, quando super nos incumbit aliquod pondus quod sua gravitate deorsum tendit, nobis autem opus est nostra vi resistere ne aplius descendat; illud autem resistere est quod gravari appellamus.” So “We say that we feel burdened [gravari] when some weight is placed upon us which tends downwards because of its gravity, in which case we have to oppose a force so that the weight no longer descends; that opposing is what we call to feel burdened [gravari]”. It is significant that Galileo uses “gravari”, the passive infinitive, not “gravare”, the active infinitive. Why? Precisely because Galileo has in mind a behaviour typical of human beings. When burdened with a weight we need to exert an opposing force [“… quando super nos incumbit aliquod pondus … nobis autem opus est nostra vi resistere”]. So by translating “feel burdened” I wish to convey this idea that weight is for human beings to experience. Stones too support weights, but do not feel burdened (how awkward it would have been if Galileo had said “lapidibus autem opus est sua vi resistere” ?!). In humans action and feeling are interconnected.

“… l' unire e soprapporre l' uno all' altro de' sopranominati mattoni” ( Galilei 1890–1909, vii, 733).

Galilei 1890–1909, viii, 324–5.

Cf. Galileo's treatise, in Galilei 1890–1909, ii, 223–4, and the relevant passages by Ptolemy, Copernicus, and Clavius, in, respectively, Ptolemy , Almagestum Cl. Ptolemei (Venice, 1515), 4 (this edition of the Almagest was in Galileo's personal library), Copernicus , De revolutionibus orbium coelestium, Libri VI (Nuremberg, 1543), 5r–v, and Clavius Christoph , In sphaeram Ioannis de Sacro Bosco commentarius (Rome, 1585), 196.

Ong Walter , Orality and literacy: The technologizing of the word (London, 1988), 38.

Galilei 1890–1909, vii, 743.

There is, I think, a phrasing problem in this passage. Galileo has by now long realized that it is space that increases as the square of time. Here he claims that it is a body's “innate speed” that increases as the square of time. This may have been a slip of the pen, or he may have been thinking of “innate speed” as a global quantity, measurable by the space traversed, the latter being the quantity which in effect increases as the square of time.

Clearly Galileo believed that to make experiments in the void might turn out to be impossible. Cf. Galilei 1890–1909, vii, 743ff. Here I need to point out that in a short piece of writing, which cannot be dated, but presumably is part of a family of similar works concerning hydraulics problems written about the very early 1630s, before the Postils to Rocco, Galileo already asserted that he believed that bodies of all types of matter would move in the void with the same speed. He supported the claim with the argument that will be discussed in Section 6, based on the divergence of speeds of bodies in different media. At the beginning of this document, Galileo addresses one Bertizzolo, who apparently had made some objections to Galileo, as if his critic were still alive. Antonio Favaro identified the “Bertizzolo” of the writing with the Mantua engineer, Bertazzolo Emanuele (1570–1626), whom Galileo may have met in 1604 on a visit to that city. Thus this piece might have been written earlier than 1630. If this is case then by the mid-1630s Galileo had for some time been convinced that all bodies fall at the same speed in the void. What he still had not realized is the weightless condition of fall. Cf. Favaro A. , Scampoli galileiani, ed. by Rossetti L. Soppelsa M. L. (2 vols, Trieste, 1992), ii, 510–15.

Galilei 1890–1909, vii, 743.

Euclid , The thirteen books of the Elements (ref. 50), ii, 112–86.

Galilei 1890–1909, i, 276ff.

Thus, for example, the arithmetical ratio of 5 to 8 is the same as that of 20 to 23, since 8–5 = 3, which is equal to 23 - 20. Galilei 1890–1909, i, 278ff.

Galilei 1890–1909, vii, 743–4.

Galilei 1890–1909, vii, 734–42.

Galilei 1890–1909, viii, 128.

Galilei G. , Two new sciences. Including centres of gravity and force of percussion, ed. and transl. by Drake Stillman (Madison, 1974), 87, with small changes. Cf. the original, in Galilei 1890–1909, viii, 128–9.

Hill David , “Pendulums and planes: What Galileo didn't publish”, Nuncius, ix (1994), 499–515. Cf. also Drake S. , Galileo: Pioneer scientist (Toronto, 1990), 9–31, who proposes a different interpretation of the manuscripts studied by Hill.

Galilei 1890–1909, viii, 202–3.

Palmerino C. R. , “Galileo's and Gassendi's solutions to the Rota Aristotelis paradox: A bridge between matter and motion theories”, in Murdoch J. E. Newman W. R. Lüthy C. H. (ed.), Medieval and early modern corpuscular matter theories (Leiden, 2001), 381–422; idem, “Infinite degress of speed: Marin Mersenne and the debate over Galileo's law of free fall”, Early science and medicine, v (1999), 269–328; idem, “Two Jesuit responses to Galileo's science of motion: Honoré Fabri and Pierre La Cazre”, in Feingold M. (ed.), The new science and Jesuit science: Seventeenth century perspectives (Dordrecht, Boston and London, 2003), 187–227; and idem, “Galileo's theories of free fall and projectile motion as interpreted by Pierre Gassendi”, in Palmerino C. R. Thijssen J. M. M. H. (ed.), The reception of the Galilean science of motion in seventeenth-century Europe (Dordrecht, Boston and London, 2004), 137–64. Palmerino's articles have further interesting bibliography. Cf. also Giusti E. , “A master and his pupils: Theories of motion in the Galilean school”, in Palmerino C. R. Thijssen J. M. M. H. (ed.), The reception of the Galilean science of motion in seventeenth-century Europe (Dordrecht, Boston and London, 2004), 119–36; and Borgato M. T. , “Riccioli e la caduta dei gravi”, in Borgato M. T. (ed.), Giambattista Riccioli e il merito scientifico dei Gesuiti nell' età Barocca (Forence, 2002), 79–118, on Riccioli's experiments on falling bodies in Bologna about the middle of the seventeenth century.

Little is known of Coresio, see De Ceglia F. P. , “Giorgio Coresio: Note in merito a un difensore dell' opinione di Aristotele”, Physis, xxxvii (2000), 393–437. The passage by Aristotle discussed by Coresio was rendered into English by Guthrie W. K. C. as follows: “Suppose a body A to be weightless, and another body B to have weight, and let the weightless body move a distance CD and the body B move in an equal time CE. (The heavy body will move farther.) Now if the heavy body be divided in the proportion in which CE stands to CD (and it can quite well bear such a relationship to one of its parts), then if the whole traverses the whole distance CE, the part must traverse CD in an equal time. Thus that which has weight will traverse the same distance as that which has none, and this is impossible.” See Aristotle , On the heavens, transl. by Guthrie W. K. C. (Cambridge, MA, and London, 1986; 1st edn, 1939), 277–8.

Galilei 1890–1909, iv, 240–1. Coresio's argument is in his Operetta intorno al galleggiare de corpi solidi ( Galilei 1890–1909, iv, 197–244).

Galileo's pupil Benedetto Castelli compiled a list of ‘errors’ contained in Coresio's Operetta, but, significantly, he had nothing to say about the latter's counter-argument to Mazzoni-Galileo's critique of Aristotle ( Galilei 1890–1909, iv, 246–85).

As for Galileo's atomism in general, I have relied on: Lasswitz K. , Geschichte der Atomistik (2 vols, Hamburg and Leipzig, 1890), ii, 37–54; Dijksterhuis E. J. , The mechanization of the world picture (London, 1961), 419–24; Shea W. , “Galileo's atomic hypothesis”, Ambix, xvii (1970), 13–27; Smith A. Mark , “Galileo's theory of indivisibles: Revolution or compromise?”, Journal of the history of ideas, xxxvii (1976), 571–88; and Redondi P. , Galileo eretico (Turin, 1988; 1st edn, Turin, 1983), 11–31.

Aristotle , Minor works, transl. by Hett W. H. (Cambridge, MA, 1993; 1st edn, 1936), 387ff. Cf. also Drabkin J. E. , “Aristotle's wheel”, Osiris, ix (1950), 346–59, still the best discussion on this question.

Galilei , Two new sciences (ref. 84), 29ff.

Smith , “Galileo's theory of indivisibles” (ref. 91), 571–2.

Ibid., 583–4.

Palmerino , “Galileo's and Gassendi's solutions to the Rota Aristotelis paradox” (ref. 87), 384–5.

Ibid., 398.

Galilei , Two new sciences (ref. 84), 56, and Palmerino , “Galileo's and Gassendi's solutions to the Rota Aristotelis paradox” (ref. 87), 398–400.

Shea , “Galileo's atomic hypothesis” (ref. 91), 13.

Palmerino , “Galileo's and Gassendi's solutions to the Rota Aristotelis paradox” (ref. 87), 405ff.

Drake , History of free fall (ref. 5).

Fermat P. , Oeuvres, ed. by Tannery P. Henry C. (5 vols, Paris, 1894–1922), ii, 267–76, and iii, 302–9.

Torricelli E. , Opera geometrica (Florence, 1644). I have based my analysis on Torricelli E. , Opere scelte, ed. by Belloni L. (Turin, 1975), and Torricelli E. , Opere, ed. by Loria G. Vassura G. (5 vols, Faenza, 1919–44).

‘Geometrical foundation’ has to be taken in the sense that Torricelli furnishes a geometrical proof of Galileo's postulate, which in Two new sciences had been justified on the basis of an analogy with mechanical principles. Cf. Torricelli , Opere scelte (ref. 103), 156ff.

Torricelli's proposal of reform of Euclid's theory of samenesses of ratios was probably inspired by Galileo. Cf. Giusti , Euclides reformatus (ref. 9), 83–114. Torricelli attempted to reproduce all the basic results of the Fifth and Sixth Books of the Elements without having recourse to the notion of equimultiple. To do so he introduced nine definitions and six postulates.

It goes without saying that such a reconstruction of the genesis of Archimedes's tract on spiral lines is purely the product of Torricelli's fantasy.

Torricelli , Opere scelte (ref. 103), 173, and Torricelli , Opere (ref. 103), iii, 384.

Torricelli , Opere scelte (ref. 103), 173.