This paper examines, compares, and contrasts ideas about motion, especially the motion of a body in a resisting medium, proposed by Galileo, Newton, and Tartaglia, the author of the first text on exterior ballistics, within the context of the Aristotelian philosophy prevalent when these scholars developed their ideas. This historical perspective offers insights on the emergence of a scientific paradigm for motion, particularly with respect to the challenge of incorporating into this paradigm the role played by the medium.

## References

1.

Galileo

Galilei

, Two New Sciences

, translated with a new introduction and notes by Stillman Drake, 2nd ed. (Wall and Emerson

, Toronto

, 1989

), p. 147

.2.

Aristotle

, Physics, or Natural Hearing

, translated and introduced by Glen Coughlin (St. Augustine's Press

, South Bend, IN

, 2005

), p. ix

.3.

Aristotle

, On the Heavens

, translated by J. L. Stocks, in *The Complete Works of Aristotle*, edited by

J.

Barnes

(Princeton U.P.

, Princeton, NJ

, 1984

), Vol. 1, pp. 447

–511

. *On the Heavens*deals in more depth with some topics introduced in Aristotle's

*Physics*, and it contrasts motion in the celestial and terrestrial realms.

4.

Thomas S.

Kuhn

, The Essential Tension: Selected Studies in Scientific Tradition and Change

(University of Chicago Press

, Chicago

, 1977

), pp. x

–xiii

. Kuhn's encounter with Aristotle's *Physics*occurred in 1947 when, as a doctoral student in physics, he was asked to give a series of talks on the origins of mechanics.

5.

Reference 2, 201a10–201a14. Bekker numbers, the standard method for indexing and citing Aristotle's writings, are used in this paper. They refer to page, column, and line numbers in A. I. Bekker's 1831 edition of Aristotle's complete works.

6.

Reference 2, 225b5–225b9.

7.

Stillman

Drake

and I. E.

Drabkin

, Mechanics in Sixteenth-Century Italy: Selections from Tartaglia, Benedetti, Guido Ubaldo, and Galileo

(The University of Wisconsin Press

, Madison, WI

, 1969

), pp. 16

–26

.8.

Reference 7, pp. 61–97.

9.

A.

Rupert Hall

, Ballistics in the Seventeenth Century

(Cambridge U.P.

, Cambridge

, 1952

), p. 36

.10.

Reference 2, 254b7–256a3.

12.

Peter

Damerow

, Gideon

Freundenthal

, Peter

McLaughlin

, and Jürgen

Renn

, Exploring the Limits of Preclassical Mechanics

(Springer-Verlag

, New York

, 1992

), pp. 78

–91

.13.

Reference 7, pp. 70–72.

14.

Reference 2, 216a15–216a19: “it divides either by its shape or by the inclination which … the projectile has.”

15.

Reference 7, pp. 73–78. Tartaglia's recognition of acceleration in free fall is consistent with that of Aristotle, who states that things “borne on a straight line” are “borne faster” as they are “borne further away from … resting” (Ref. 2, 265b10–265b14). Tartaglia associates acceleration loosely with a body moving “more swiftly the more it shall depart from its beginning or the more it shall approach its end” (p. 74). This differs sharply from Galileo's precise notion of uniform acceleration associated with time (not distance) of fall.

16.

17.

Reference 7, pp. 89–91: “when the straight parts of violent trajectories … correspond, their distances also correspond, otherwise it would be very contradictory.”

18.

Reference 7, pp. 91–94: a cannonball shot horizontally “would complete its violent motion farther below the plane of the horizon than when elevated in any other direction” and, shot vertically, violent motion would “terminate higher … above the plane of the horizon than … at any other elevation;” thus, “there will be an elevation … to make this [violent motion] terminate precisely in the plane of the horizon,” and this is “that elevation … midway between those … most contrary in results” (p. 93); Tartaglia then invokes Supposition IV in Book II: “the most distant effect … that can be made … in violent motion upon any plane … is [in] that … which terminates precisely in this plane” (p. 85).

19.

Reference 7, pp. 94–97. In the proof, Tartaglia takes “as an assumption … that the distance of the violent trajectory … elevated at 45 degrees above the horizon is about ten times the straight trajectory made along the horizontal plane” (p. 94). He uses this and Proposition VIII to obtain a quadratic equation for the ratio of the initial straight part of the trajectory of a body projected at 45° to that of the same body shot horizontally using the same power, and he finds this ratio to be $200\u221210$, roughly $417$.

20.

Mary J.

Henninger-Voss

, “How the ‘new science’ of Cannons shook up the Aristotelian cosmos

,” J. Hist. Ideas

63

, 371

–397

(2002

).21.

Reference 7, pp. 84–85.

22.

Galileo

Galilei

, On Motion and On Mechanics

, comprising *De Motu*(ca. 1590), translated with introduction and notes by I. E. Drabkin, and

*Le Meccaniche*(ca. 1600), translated with introduction and notes by Stillman Drake (

The University of Wisconsin Press

, Madison, WI

, 1960

), pp. 3

–4

.23.

Reference 2, 215a25–216a20. Aristotle's explicit use of proportions (ratios) here shows that he is not averse to applying mathematics to some terrestrial phenomena.

24.

Reference 3, p 309b14.

25.

Reference 22, pp. 13–40.

26.

Reference 7, p. 149. Benedetti refers to

*On Floating Bodies*within the argument (pp. 148–152) for this doctrine in his*Resolutio*(1553).27.

Stillman

Drake

, *History of Free Fall: Aristotle to Galileo*, printed as an Appendix to Ref. 1, pp. 27–29.

28.

Reference 2, pp. 266b25–267b25. When “things thrown” are no longer “touched by the mover,” they continue to move because “the air or the water moves, being divisible,” as “things contiguous to each other” in a succession of motions, with successively decreasing “power for moving,” initiated by the thrower and extending to the projectile, which eventually stops.

29.

Reference 27, pp. 8–9 and 18–19.

30.

Reference 22, pp. 76–105.

31.

Reference 22, pp. 110–114.

32.

Reference 12, pp. 144–147.

33.

Reference 1, pp. xiii–xxxv.

34.

Reference 1, pp. 159–160.

35.

Reference 12, pp. 126–264, takes a very careful look at the transition from

*De Motu*to Two New Sciences, including the use of impressed force and the logic of contraries to explain projectile motion in the latter (pp. 251–262) and links from theory in*De Motu*to both the time-squared rule for distance fallen and the parabolic path of a projectile (pp. 149–158).36.

Reference 1, p. 156. If Salviati's claim is understood in terms of ratios, as seems likely, then it agrees and disagrees with classical mechanics, in that an effect defined this way is determined by the striking body's ability to do work, which is proportional to its initial height (via potential energy) and to the

*square*of its impact speed (via kinetic energy).37.

Reference 1, p. 256.

38.

Reference 27, p. 54.

40.

Reference 1, pp. 153–216.

41.

Reference 1, pp. 165–167. Proposition I follows from the

*Medieval Mean Speed Theorem*, originated at Merton College, Oxford, in the 14th century (Ref. 27, pp. 17–18) and developed further in the geometrical “configuration of qualities” method of Nicole Oresme. Some essentials of this method were not passed on to the 17th century, but Galileo used geometric aspects of it in the proof of this proposition (see Ref. 12, pp. 15–19 and 228–230). Galileo's statement of Proposition II and others adheres to Euclidean proportion theory (*Elements*, Book V), which Galileo uses to compare continuous magnitudes of the same kind (see Ref. 33). Euclidean proportions, though less flexible than modern equations, allow physical laws to be stated without constants like*g*.42.

Reference 1, p. 162. Salviati later (pp. 169–170) describes experiments involving a “wooden beam” and a “hard bronze ball.” In classical mechanics, a solid ball has a constant acceleration of $(5/7)g\u2009sin\u2009\theta $ on a plane inclined at angle

*θ*when subject to just enough friction to make it roll without slipping and assuming all its potential energy is converted to kinetic energy, as any physics undergrad can confirm using moment of inertia.43.

Note that

*De Motu*also deals with motion on inclined planes, claiming that the ratio of descent speeds on two such planes varies inversely as the ratio of their lengths (Ref. 22, pp. 63–69).44.

Reference 1, pp. 217–260.

47.

Reference 1, pp. 223–229.

48.

Reference 1, pp. 226–229. The first of these two experiments involves “two equal lead balls” hanging by “two equal threads,” both balls moved from the vertical, one “by 80 degrees or more,” the other “by no more than four or five degrees,” and released. Salviati claims the two pendulums remain in tandem for up to “hundreds” of oscillations, but Drake reports (footnote 12) that they vary by “one beat after about the first thirty.” If the pendulums stayed in tandem and a pendulum's period did not increase with amplitude, Sagredo's conclusion that “speed … is itself both the cause and the measure of … resistance” would seem more valid. The second (thought) experiment invokes a “bullet [fired] from an arquebus vertically downward on a stone pavement,” once from a great height and again from a few feet: Salviati believes the bullet fired from great height is “less flattened” because drag reduces its impact speed.

49.

Reference 1, pp. 249–255.

50.

Isaac

Newton

, The Principia: Mathematical Principles of Natural Philosophy

, a New Translation by I. Bernard Cohen and Anne Whitman assisted by Julia Budenz, preceded by *A Guide to Newton's Principia*(pp. 1–370) by

I.

Bernard Cohen

(University of California Press

, Berkeley

, 1999

), pp. 11

–13

. Editions of *Principia*were published in 1687, 1713, and 1726.

51.

John

Herivel

, The Background to Newton's Principia: A Study of Newton's Dynamical Researches in the Years 1664–84

(Oxford U.P.

, Oxford

, 1965

), pp. 35

–41

.52.

Reference 50, p. 424.

53.

George E.

Smith

, “The methodology of the Principia

,” in The Cambridge Companion to Newton

, edited by I.

Bernard Cohen

and George E.

Smith

(Cambridge U.P.

, Cambridge

, 2002

), pp. 138

–173

.54.

Reference 51, p. 1.

55.

Reference 50, p. 793.

56.

Reference 50, pp. 164–165, 779–790, and 939–944.

57.

Reference 50, pp. 633–641.

59.

Reference 50, pp. 655–669. On the generalized hyperbola, vertical distance from its oblique asymptote varies inversely as an arbitrary power—unity in the case of a conic hyperbola—of distance from its vertical asymptote.

60.

Michael

Nauenberg

, “Proposition 10, Book 2, in the Principia, revisited

,” Arch. Hist. Exact Sci.

65

, 567

–587

(2011

). This details Newton's analysis for Proposition 10 and the conceptual error in it (which persisted somewhat in his “corrected” version). See also Ref. 50, pp. 168–171.61.

Reference 9, pp. 154–156.

62.

Johann

Bernoulli

, “Responsio ad nonneminis provocationem, ejusque solutio quaestionis ipsi ab eodem propositae, de invenienda linea curva quam describit projectile in medio resistente

,” Acta Eruditorum

38

, 216

–226

(1719

).63.

William W.

Hackborn

, “Projectile motion: Resistance is fertile

,” Am. Math. Mon.

115

, 813

–819

(2008

); available at http://www.jstor.org/stable/27642609.64.

Edward

John Routh

, A Treatise on Dynamics of a Particle with Numerous Examples

(Cambridge U.P.

, Cambridge

, 1898

), p. 96

.65.

Reference 50, pp. 670–679.

66.

Reference 50, pp. 700–761. Newton reports on his pendulum (pp. 713–723) and vertical fall experiments (pp. 749–761) at great length and with careful attention to details.

67.

Lyle N.

Long

and Howard

Weiss

, “The velocity dependence of aerodynamic drag: A primer for mathematicians

,” Am. Math. Mon.

106

, 127

–135

(1999

). For a sphere of diameter d moving at speed s in fluid of density ρ and viscosity μ, the Reynolds number is $Re=s\rho d/\mu $. When $Re\u226a1$, the drag is dominated by viscous forces (yielding Stokes' Drag Law, $FD=6\pi \mu rs$ for a sphere of radius r, with drag proportional to speed; when $Re\u226b1$, inertial forces prevail). Fluid tenacity is expressed by the no-slip condition on the body's surface (see also Ref. 58).68.

Neville

de Mestre

, The Mathematics of Projectiles in Sport

(Cambridge U.P.

, Cambridge

, 1990

), pp. 49

–50

, 58–60, 123–126, and 135–137.69.

Reference 3, pp. 311b1–311b5.

70.

Reference 1, pp. xiii–xxxv. Drake firmly adopts this position, calling

*Two New Sciences*“a new kind of mathematical physics” (p. xvii) in his introduction to it.71.

Reference 12, p. 5. Damerow

*et al*. assert that the conceptual framework of classical mechanics “begins with such figures as Descartes and Galileo and takes shape with … their successors.”72.

A. P.

French

, Newtonian Mechanics

, The M.I.T. Introductory Physics Series (W.W. Norton & Co.

, New York, NY

, 1971

), p. 153

. See http://books.wwnorton.com/books/searchresults.aspx?searchtext=newtonian+mechanics.73.

Reference 4, pp. 31–65. In the physical sciences, Kuhn distinguishes the mathematical tradition from the newer “experimental tradition,” which includes sciences such as chemistry, and claims these traditions “remained distinct” into the 19th century (p. 48). Kuhn fails to consider the motion experiments conducted by Galileo and Newton, which make his claim questionable.

74.

Salomon

Bochner

, The Role of Mathematics in the Rise of Science

(Princeton U.P.

, Princeton, NJ

, 1966

), p. 7

.75.

Leonhard

Euler

, “Recherches sur la veritable courbe que décrivent les corps jettés dans l'air ou dans un autre fluide quelconque

,” Mem. de L'Acad. Sci. Berlin

9

, 321

–352

(1755

).76.

Edward J.

McShane

, John L.

Kelley

, and Franklin V.

Reno

, Exterior Ballistics

(The University of Denver Press

, Denver

, 1953

), p. 258

and pp. 305–310.77.

Gilbert

Ames Bliss

, Mathematics for Exterior Ballistics

(Wiley

, New York

, 1944

), pp. 17

–28

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