A Dispute Over Superposition: John Wallis, Honoré Fabri, and Giovanni Alfonso Borelli

Abstract

This paper aims first and foremost to unravel and clarify an interesting 17^th century controversy around superposition in projectiles, which allegedly existed between the French Jesuit Honoré Fabri and the Italian physicist and astronomer Giovanni Alfonso Borelli. This conflict – initially described by the English mathematician John Wallis in a letter from 1670 to the secretary of the Royal Society – has been erroneously identified with Fabri's Dialogi physici (1669), a work written in response to Borelli's De vi percussionis (1669). In fact, this “conflict” was nothing but Wallis's account of a contradiction between Borelli's above mentioned work and Fabri's Tractatus physicus de motu locali from 1646, while Fabri's 1669 work expressed views very different from those contained in his Tractatus physicus. I will try here to reconstruct Fabri's change of heart between 1646 and 1669 concerning projectiles and superposition, while tracing the real bone of contention between (the later) Fabri and Borelli – superimposing contrary motions – to its Aristotelian origins. My analysis will lead me to problematize the way modern historians usually interpret the relation between Aristotle's physical thinking and projectile theories of early modern theoreticians (e.g. Nicollò Tartaglia's).

Notes

¹John Wallis to Henry Oldenburg, 15 November 1670, in The Correspondence of Henry Oldenburg, edited and translated by Alfred Rupert Hall and Marie Boas Hall, 11 vols (Madison, Wis.: University of Wisconsin Press, 1965–1977), vol. 7, pp. 283–284.

²Galileo Galilei, Dialogue Concerning the Two Chief World Systems, Ptolemaic & Copernican (Berkeley: University of California Press, 2nd ed., 1967), translated by Stillman Drake, p. 155.

⁸Namely Fabri's Tractatus physicus de motu locali, in quo effectus omnes, qui ad impetum, motum naturalem, violentum, & mixtum pertinent, explicantur, & ex principiis physicis demonstrantur physicus de motu locali (Lyon, 1646).

⁹ The Correspondence of Henry Oldenburg (note 1), vol. 7, pp. 424–425.

³ The Correspondence of Henry Oldenburg (note 1), vol. 7, p. 285, n. 6.

⁴This definition, taken from G. Holton and D. Roller, Foundations of Modern Physical Science (Reading, Mass: Addison-Wesley, 1959), p. 47, appears in Gad Prudovsky, ‘The Confirmation of the Superposition Principle’, Studies in History and Philosophy of Science, 20:4 (1989), 453–468, p. 453.

⁵ The Correspondence of Henry Oldenburg (note 1), vol. 7, p. 285, n. 6.

⁶ The Correspondence of Henry Oldenburg (note 1), vol. 7, p. 285, n. 6. The full title of Fabri's book is Dialogi physici, quorum primus est de lumine, secundus et tertius de vi percussionis et motu, quartus de humoris elevatione per canaliculum, quintus et sextus de variis selectis (Lyon, 1669). It is not to be confused with an earlier “physical dialogues” published by Fabri: Dialogi physici in quibus de motu terrae disputatur, marini aestus nova causa proponitur necnon aquarum et mercurii supra libellum elevatio examinatur (Lyon, 1665); in this paper the title Dialogi physici refers to Fabri's 1669 work (unless otherwise stated).

⁷A. Rupert Hall, ‘Gunnery, Science, and the Royal Society’, in The Uses of Science in the Age of Newton, edited by John G. Burke (Berkeley : University of California Press, 1983), p. 126; Rob Iliffe, ‘Making Correspondents Network: Henry Oldenburg, Philosophical Commerce, and Italian Science 1660–72’, in The Accademia del Cimento and its European Context, edited by M. Beretta, A. Clericuzio, & L. M. Principe (Sagamore Beach, MA : Science History Publications, 2009), 211–228, p. 219. This false claim appears also in ‘An account of Some Books’, Philosophical Transactions (1665–1678), vol. 5 (1670), 2052–2062, p. 2058.

¹⁰John Wallis to Robert Moray, 14 July 1668, in The Correspondence of John Wallis, edited by Philip Beeley and Christoph J. Scriba, 2 vols (Oxford: Oxford University Press, 2003–2005), vol. 2, pp. 494–495.

¹¹Though in his letter to Moray Wallis had the page numbers wrong: the correct numbers are 169–173 and not 269–272.

¹²Contrary to Fabri and his Aristotelian colleagues, who typically regard impetus as a necessary cause of motion, in Borelli's vocabulary “impetus” and “velocity” are for all practical purpose synonyms.

¹³“Grave semper eodem impetu tendit deorsum sive a quiete descensus initium sumat sive a quocumque motu sursum deorsum aut transversali” (Giovanni Alfonso Borelli, De vi percussionis [Bologna, 1667], cap. XXIII, p. 169).

¹⁴“Porro modus facilis quo haec omnia experimento comprobavi talis fuit; in rota ABCD parieti affixa clavo E circa quem eadem convertebatur in plano perpendiculari ad horizontem, atque ex eadem proeminebat lignea regula C, & ductis diametris AC & DB adinvicem perpendicularibus connexae erant in terminis D, A, B tres fistulae quarum latera rotae peripheriam tangebant, earumque orificia ad easdem partes convertebantur, & in infimo loco aderat clavus F parieti fixus ut vertigo rotae sisti posset quando diameter DB horizonti aequidistabat. His praeparatis translato infimo rotae termino C ad situm B posui in cavitatibus fistularum tres pilas plumbeas R, R, R inter se aequales, aderat postea socius qui quartam pilam R sustinebat in situ F, postmodum revoluta vehementer rota eius terminus C a puncto B ad F impulsus fuit, & ibidem ob clavi F impedimentum tres proiectiones per rectas lineas circulum DAB tangentes eodem temporis instanti exactae fuerunt perpendicularis sursum DG horizontalis AH, & perpendicularis deorsum BK, atque simul socius pilam demittens a termino quietis F pertransijt ea ad terminum M” (ibid., pp. 171–172).

¹⁵“Si unum grave corpus impellatur horizontaliter, aliud vero simplici descensu feratur, excurrent eodem tempore spatia perpendicularia ad horizontem aequalia” (ibid., p. 169).

¹⁶“Si vero tertium corpus sursum oblique eodem impetu proijciatur eodemque tempore, percurret spatium directum aequale transverso horizontali pariterque spatium perpendiculare aequale prioribus” (ibid., p. 170).

¹⁷“Et si quartum corpus eodem impetu proijciator deorsum perpendiculariter ad horizontem eodem tempore excurret spatium aequale horizontali, & descensivo” (ibid.).

¹⁸“portio descensus KL pertinet ad virtutem externam impulsivam, reliqua vero eius pars LM efficietur a vi nativa gravitatis” (ibid., pp. 170–171).

²⁰“Tandem idem grave B ab eadem virtute proijciente impellatur sursum perpendiculariter ad horizontem a termino infimo N, & perducatur usque ad P, & quia virtus externa impulsiva semper eadem supponitur, pertransibit eodem tempore T id ipsum spatium NO aequale KL vel GH aut DE, reperiturque spatium transitus apparentis NP aequale differentiae eiusdem transitus impulsivi NO, & descensus OP aequalis eidem AC a quiete incepto, quare manifestum est motum descensus eiusdem gravis eodem tempore T nunquam alterari, scilicet non augeri neque minui, ex eo quod grave B quocunque motu afficiatur sive horizontali sive obliquo sive perpendiculari sursum aut deorsum, sed semper eiusdem mensurae esse & aequalem descensui libero a quiete incoato” (ibid.).

¹⁹“Si tandem quintum corpus eadem vi proijciatur sursum perpendiculariter ad horizontem, pertransibit eodem tempore spatium aequale differentiae horizontalis motionis, & descensivae” (ibid., p. 171).

²¹“Motum mixtum eum esse non dico, qui ex pluribus aliis motibus componatur; seu misceatur; nec enim plures motus simul esse possunt in eodem mobili; cum tantum esse possit uno dumtaxat instanti unica migratio ex loco in locum; nec plura loca naturali virtute simul acquiri possunt; igitur nec simul esse duo motus; Itaque motus mixtus simplex est, si consideretur ratio, & linea motus”; Honoré Fabri , Tractatus physicus de motu locali [Lyon, 1646], lib. IV, p. 153).

²²This is how Fabri defines local motion: “Motus localis est transitus mobilis e loco in locum continuo fluxu” (ibid., lib. I, def. 1, p. 1).

²³“Definitio 1: motus mixtus est, qui sequitur ex multiplici impetu ad eamdem, vel diversas lineas determinato, vel eodem ad diversas” (ibid., lib. IV, p. 153).

²⁴That is, gravity.

²⁵See Michael Elazar, Honoré Fabri and the Concept of Impetus: A Bridge between Conceptual Frameworks (Dordrecht: Springer, Boston Studies in the Philosophy of Science, 2011), p. 179.

²⁶Fabri refers to Mersenne's Ballistica, prop. 25, which states that in one of those “experiments” (which were nothing but observations held during battle, e.g. the siege on the French town of Thionville within the Thirty Years’ War), a horizontally projected missile remained in the air for 8 seconds, while a different missile – projected from the same height – travelled in the air for only 6 seconds (Fabri, Tractatus physicus, lib. IV, th. 46, p. 164; Marin Mersenne, De ballistica et acontismologia, seu de sagittarum, prop. 25, pp. 82–83, in F. Marini Mersenni minimi Cogitata physico mathematica. In quibus tam naturæ quam artis effectus admirandi certissimis demonstrationibus explicantur [Paris, 1644]); this statement contradicts of course Galileo's quoted paragraph from the Dialogo (see p. 3 above). It should be noted that Fabri accepted Galileo's opinion that in principal – i.e. ignoring air resistance – all bodies fall in the same rate.

²⁷Axiom vi from the first book of the Tractatus physicus claims “Quidquid est, frustra non est” (Fabri, Tractatus physicus, lib. I, p. 6).

²⁸As Edoardo Benvenuto explains, “thanks to the simple proportionality between force and velocity that characterizes the Peripatetic dynamics” the Aristotelian kinematic parallelogram of velocities (Mechanical Problems, 1, 848b10-b22) was readily interpreted in dynamic terms; Edoardo Benvenuto, “The Parallelogram of Forces”, Meccanica 20 (1985), 99–109, p. 101.

²⁹For more details see Michael Elazar (note 25), pp. 173, 183–185.

³⁰On this point also see Michael Elazar (note 25), pp. 185–187.

³¹It was from this very criticism that Newton learned of probably the most important Jesuit contribution to 17^th century physics: Grimaldi's discovery of diffraction; see A. R. Hall, “Beyond the Fringe: Diffraction as Seen by Grimaldi, Fabri, Hooke and Newton”, Notes and Records of the Royal Society of London, 44; 1 (Jan., 1990), 13–23.

³³“Transit deinde ad vim motricem gravium, ac praemittit quatuor experimenta, nimirum quod corpus grave libere demissum idem spatium decurrit in perpendiculari, quod decurrit in eadem, per horizontalem projectum, eodem scilicet tempore; item per inclinatam aequalem projectum; item si proijciatur deorsum, eadem vi, decurret spatium compositum ex utroque spatio, horizontalis scilicet, & liberi descensus; item si proijciatur sursum eadem velocitate, detrahendum est eidem spatio in horizontalem decurso, spatium decursum in perpendiculari” (Fabri, Dialogi physici, p. 221).

³²Fabri used this name to represent himself also in other dialogues, for example Opusculum Geometricum de linea sinuum et cycloide (Rome, 1659), Pithanophilus, seu Dialogus vel opusculum de opinione probabili (Rome, 1659), Dialogi physici in quibus de motu terrae disputatur (Lyon, 1665).

³⁴“Cuncta haec, opinor, admittis, cum mera puta sit Galilei doctrina; & ex tuis principiis facile demonstras” (Fabri, Dialogi physici, p. 222). In the beginning of the two dialogues attacking Borelli Fabri explicitly mentions the “excerpts on motion and impetus” published by Mousnier – necessarily referring to the Tractatus physicus and clarifying that this is where we should look for his “own principles” (Fabri, Dialogi physici, p. 99).

³⁵Fabri, Dialogi physici, p. 222.

³⁶That is, if it were thrown perpendicularly upwards with the same initial velocity. This is the second proposition in Torricelli's De motu poriectorum; see Robinson, ‘Evangelista Torricelli’, The Mathematical Gazette, Vol. 78, No. 481 (Mar., 1994), 37–44, p. 40.

³⁷Point H – the intersection of the “y axis” (i.e. the line continuing AF) with the line parallel to the “x axis” (the line containing AC) and going through G – does not appear in the figure.

³⁸“Quod autem medij resintentia hujusmodi progressus impediat & mutet, optime observat auctor, prop. 114, uti ante Galileus & multi alij observarant. Item quod projectum per inclinatam eat per semiparabolam, cuius axis sit subduplus altitudinis perpendicularis; ad quam eodem impetu attoleretur; v.g. sit inclinata AEG [pic. 95 in fig. 4], impetus impressus, quo attolli possit ad altitudinem perpendicularem AF, ibit per semiparabolam AD, cuius axis erit BD, subdupla BE, vel AF; si vero major sit impetus, quo scilicet attolli possit ad altitudinem AH, vel CG aequalem, ibit per semiparabolam AK, eritque CK subdupla CG, haec omnino trita sunt, & jam a multis demonstrata; item quod in projectis sursum, impetu inaequali, altitudines seu spatia, sint in duplicata impetuum impressorum, vel temporum; item in projectis per eamdem inclinatam” (Fabri, Dialogi physici, p. 260).

³⁹For example, Fabri, Dialogi physici, pp. 181–182, p. 258.

⁴⁰Giovanni Alfonso Borelli, De vi percussionis, prop. 66, p. 129.

⁴¹“… fallitur demum auctor in prop. 66, in qua dicit, nullam proportionem impetus destrui ratione motus mixti, quod falsissimum est, cum aliqua impetus portio sit frustra; sit enim globus projectus in plano horizontali per lineam AB, motu AB; ubi autem pervenit in B, impellatur per CBD, motu BD; ibit per BI, motu BI … aggregatum impetus in B, post impressionem per CB, est ut ABD, sed BI est minor ABD; igitur aliquid impetus est frustra; ergo aliquid destruitur; & illud, quod remanet, est ad aggregatum ut BI, ad ABD” (Fabri, Dialogi physici, p. 198).

⁴²On Fabri's view of impetus as a scalar magnitude, whose “vector counterpart” is determinatio, see Gideon Freudenthal, ‘A Rational Controversy over Compounding Forces’, in Scientific Controversies: Philosophical and Historical Perspectives, edited by Peter Machamer, Marcello Pera and Aristides Baltas (New York: OxfordUniversity Press, 2000), 125–142, pp. 130–135. On Fabri's notion of determinatio and its implementation within collisions see Michael Elazar (note 25), pp. 157–164 and David Lukens, An Aristotelian Response to Galileo: Honoré Fabri, S.J.(1608–1688) on the Causal Analysis of Motion (Ph.D. Thesis, University of Toronto, 1979), pp. 233–246. Since the concept of determinatio plays no part in Fabri's projectile theory I shall not discuss it in this paper.

⁴³Apparently Fabri supposes that in this case, the iterative way of obtaining the trajectory he suggested in the Tractatus physicus will result in Galileo's parabola. As already hinted, Fabri never says so in so many words, and never discusses (or even admits!) the huge difference in projectile theory between the two works.

⁴⁵Which claims that two contrary motions – constant and accelerated – may exist and be exercised in the same body during the same time (Borelli, De vi percussionis, prop. 112, p. 253).

⁴⁶“Repetit in prop. 112 ea, quae iam ante dixerat, nimirum motus oppositos simul esse v.g. sursum & deorsum; quod tamen repugnat; hinc reijcies áδυ´νατóν illud, quod iterum adstruit, motui scilicet sursum totum illud spatium detrahendum esse, quod grave interim deorsum conficeret; sit enim datum tempus, quo motu accelerato corpus grave decurrit spatium AC, & aequabili, AB, quod ut fiat, assumenda est velocitas subdupla acquisitae in descensu AC; iam vero eadem velocitate, qua fertur per AB, mittatur sursum recta per AE; nullus esset motus sursum quod repugnat; quia detrahendum esset totum spatium AC, vel aequale AE, ut vult auctor, nihil ergo restaret, si aequale auferatur aequali” (Fabri, Dialogi physici, p. 258).

⁴⁴“Corrige tamen id, quod ab auctore adstruitur, grave scilicet, quod sursum recta pellitur, eodem tempore deorsum moveri; quod falsum est, quia repugnat, idem corpus sursum simul ac deorsum per lineas oppositas simul moveri” (Fabri, Dialogi physici, p. 223).

⁴⁷Borelli's third phenomenon amounts of course to X(t) = v₀t + gt²/2, while the first phenomenon is equivalent simply to X(t) = – gt²/2.

⁴⁸Fabri, Dialogi physici, p. 258.

⁴⁹“Si rem ita accipias do manus; haec enim mera est & trita Galilei doctrina, inde tamen non sequitur, moveri grave per AE motibus contrariis; ut enim motus acceleratus est simplicissimus, ita & retardatus sursum perpendiculariter” (Fabri, Dialogi physici, p. 259).

⁵²“Acquired impetus” is the impetus accumulated by a freely falling body; see Michael Elazar (note 25), p. 84. As mentioned above, Fabri accepts the perfect Galilean symmetry between rise and fall. “Falsum est etiam, crescere impetum, & augeri gradus velocitatis ab ipsa virtute gravitatis descensiva; ita ut in fine motus eamdem velocitatem habeat corpus grave, quam in descensu aequali acquisivisset; falsum est etiam, manere semper eamdem virtutem proiectitiam intactam & aequalem in eodem gradu; alioquin maneret etiam in descensu, atque adeo in aeternum; quo quid absurdius dici possit, non video. Itaque dicendum est, non crescere impetum innatum, sed eumdem manere & decrescere semper singulis instantibus impetus proiecto impressum, in eadem proportione, in qua crescit naturalis” (Fabri, Dialogi physici, p. 259).

⁵⁰Borelli, De vi percussionis, prop. 113, pp. 254–255; see also note 20 above (“virtus externa impulsiva semper eadem supponitur”). This description corresponds to the modern equation (the derivative of the former) v(t) = v₀ – gt; in accordance with the Pre-classical conception of force, Borelli would have conceived both members of this equation as “forces”, while we of course recognize here only one force, i.e. one acceleration (one more differentiation will obtain merely a(t) = -g).

⁵¹See Michael Elazar (note 25), p. 83.

⁵³“Axioma I. Contradictoria simul esse non possunt, vel non esse … 1. Impossibile est idem simul esse, & non esse. 2. Quodlibet est, vel non est. 3. De eodem alterum contradictoriorum vere affirmatur, & alterum vere negatur, non simul utrumque” (Fabri, Tractatus physicus, lib. I, p. 5).

⁵⁴See Peter Damerow, Gideon Freudenthal, Peter McLlaughlin and Jürgen Renn, Exploring the Limits of Preclassical Mechanics (New York: Springer, 2^nd ed., 2004), pp. 82–84.

⁵⁵Aristotle, Metaphysics (tr. W. D. Ross), IV, 3, 1005b18-27.

⁵⁶Aristotle, Physics (tr. R. P. Hardie and R. K. Gaye), VIII, 8, 263b15-26.

⁵⁷ Physics, VIII, 8, 264a14-20. For Aristotle, of course, both changing colour and changing place belong to the general category of “kinesis”.

⁵⁸ Physics, VIII, 8, 262a6-12.

⁵⁹Thomas Aquinas, Commentary on Aristotle's Physics, edited and translated by Richard J Blackwell, Richard J Spath, and W Edmund Thirlkel (Notre Dame, Ind.: Dumb ox books, 1999), p. 589.

⁶⁰Aristotle, Meteorology (tr. E. W. Webster), 1, 4, 342a22-28.

⁶¹It was within this tradition that Archimedes – who so inspired Galileo – later generated “a spiral as the locus of a point that moves radially and tangentially at the same time”; John L. Heilbron, Galileo (Oxford: Oxford University Press, 2010), p. 140.

⁶²Edward Hussey, ‘Aristotle's Mathematical Physics: A Reconstruction’, in Aristotle's Physics: A Collection of Essays (Oxford: Clarendon Press, 1995), 213–242, p. 221. For example, in his Physics Aristotle characterizes the motion of a projectile as “when the movent causes a motion away from itself more violent than the natural locomotion of the thing moved, which continues its course so long as it is controlled by the motion imparted to it” (Physics, VII, 2, 243a20-b2).

⁶³Edward Hussey (note 62), p. 221, n. 24. But Hussey does not explain why Aristotle would consider meteors an exception; nor does he discuss here Aristotle's abovementioned aversion of compounding contrary velocities, and whether (or how) it relates to this “swamping” phenomenon.

⁶⁴Quoted in Marshall Clagett, Science of Mechanics in the Middle Ages (Madison: University of Wisconsin Press, 1959), p. 535.

⁶⁵Gideon Freudenthal (note 42), p. 131.

⁶⁶Buridan himself invokes the example of a fly climbing upwards on a falling lance; Edward Grant, God and Reason in the Middle Ages (Cambridge: Cambridge University Press, 2001), pp. 170–172.

⁶⁷Jochen Büttner, Peter Damerow, Jürgen Renn, and Matthias Schemmel, ‘The Challenging Images of Artillery: Practical Knowledge at the Roots of the Scientific Revolution’, in The Power of Images in Early Modern Science, edited by Wolfgang Lefèvre, Jürgen Renn, and Urs Schoepflin (Basel: Birkhäuser Verlag, 2003), 3–27, p. 13.

⁷¹Quoted in Otto Helbing, ‘Mechanics and Natural Philosophy in Late 16th-Century Pisa Mechanics and Natural Philosophy in Late 16th-Century Pisa', in Mechanics and Natural Philosophy before the Scientific Revolution, edited by Walter Roy Laird and Sophie Roux (Dordrecht: Springer, 2008), p. 190.

⁶⁸Stillman Drake & I. E. Drabkin (ed. and tr.), Mechanics in Sixteenth-Century Italy: Selections from Tartaglia, Benedetti, Guido Ubaldo, & Galileo (Madison: University of Wisconsin Press, 1969), p. 80.

⁶⁹Aristotle, On the Heavens (tr. J. L. Stocks), I, 2, 269a9-11.

⁷⁰Stillman Drake & I. E. Drabkin (note 68), p. 80, n. 20.

⁷²Quoted in Peter Damerow et al. (note 54), pp. 155–156; the quote is from Galileo's De motu in Drabkin's translation: Galileo Galilei, On Motion, and On Mechanics, ed. & tr. I. E. Drabkin and Stillman Drake (Madison: The University of Wisconsin Press, 1960), p. 114.

⁷³As Galileo's “mature” treatment of oblique projectiles shows (i.e. in the Discorsi), it might be doubted whether he ever assimilated superposition the way his successors (e.g. Torricelli and Borelli) did; see Peter Damerow et al. (note 54), pp. 251–254, 284–286, 351. On the question whether Galileo first discovered the correct law of fall and afterwards deduced the parabolic trajectory of projectiles or vice versa (and when did this happen) see Jürgen Renn, Peter Damerow, and Simone Rieger, ‘Hunting the White Elephant: When and How Did Galileo Discover the Law of Fall?’, in Galileo in Context, edited by Jürgen Renn (Cambridge: Cambridge University Press), 29–149.

⁷⁴The Saggi di naturali espeienze explains this discrepancy as the effect of air resistance. All of these results appear in W. E. Knowles Middleton, The Experimenters: A Study of the Accademia del Cimento (London: Johns Hopkins Press, 1971), pp. 240–241 (especially n. 301).

⁷⁵Thomas Birch, The History of the Royal Society of London: For Improving of Natural Knowledge, from its First Rise, 4 vols (New York: Johnson Reprint Corp., 1968), vol. 2, p. 454.

⁷⁶Thomas Birch, The History of the Royal Society of London: For Improving of Natural Knowledge, from its First Rise, 4 vols (New York: Johnson Reprint Corp., 1968), vol. 2, p. 461.

⁷⁷Thomas Birch, The History of the Royal Society of London: For Improving of Natural Knowledge, from its First Rise, 4 vols (New York: Johnson Reprint Corp., 1968), vol. 2, p. 464.

⁷⁸Thomas Birch, The History of the Royal Society of London: For Improving of Natural Knowledge, from its First Rise, 4 vols (New York: Johnson Reprint Corp., 1968), vol. 2, p. 465.

⁷⁹Thomas Birch, The History of the Royal Society of London: For Improving of Natural Knowledge, from its First Rise, 4 vols (New York: Johnson Reprint Corp., 1968), vol. 2, p. 467.

⁸⁰Thomas Birch, The History of the Royal Society of London: For Improving of Natural Knowledge, from its First Rise, 4 vols (New York: Johnson Reprint Corp., 1968), vol. 2, p. 469.

⁸¹Thomas Birch, The History of the Royal Society of London: For Improving of Natural Knowledge, from its First Rise, 4 vols (New York: Johnson Reprint Corp., 1968), vol. 2, p. 309.

⁸²Descartes, for example, was upset by the success of Fabri's Tractatus physicus and feared that this might hamper his own plans of “dethroning Aristotle and of taking his place in the Schools”; David Lukens (note 42), p. 14.

⁸³Franciscus Eschinardus, De impetu tractatus duplex: Primus de impetu in communi, de motu locali et de machinis. Secundus de fluidis in communi, de comparatione fluidorum cum solidis et de mensura aquarum currentium (Rome, 1684), pp. 49–50, 60–64; Eschinardus specifically agrees to the claim contained in Borelli's first phenomenon, i.e. the issue raised by Wallis's first letter to Oldenburg (ibid., p. 64).

⁸⁴Although as mentioned above, Hussey's explanation might have been fuller if he had also taken into account Aristotle's logic of contrariety, and had clarified why meteors and projectiles are treated so differently (see note 63 above).

⁸⁵Mary J. Henninger-Voss, ‘How the “New Science” of Cannons Shook up the Aristotelian Cosmos’, Journal of the History of Ideas, 63:3 (2002), 371–397, p. 382.

⁸⁶Karin J. Ekholm, ‘Tartaglia's ragioni: A maestro d'abaco's mixed approach to the bombardier's problem’, The British Journal for the History of Science, 43:2 (2010), 181–207, p. 188.

⁸⁷As mentioned above (see note 72), in his De motu Galileo had already applied Aristotle's explicit assertion that orthogonal motions do not interfere with each other.

⁸⁸Ernst Mach, The Science of Mechanics: A Critical and Historical Account of its Development, translated by Thomas J. McCormack (La Salle, Ill.: Open Court Publications, 1960), p. 526.

⁸⁹Max Jammer, Concepts of Force: A Study in the Foundation of Dynamics (Cambridge, Mass.: Harvard University Press, 1957), p. 132.

⁹⁰Quoted in Gideon Freudenthal (note 42), p. 141, note 15; see also ibid., p. 131. As Freudenthal shows in his paper, Fabri raises in his Dialogi physici an objection to claiming that the components of a parallelogram of forces (impetuses) “really exist” along with the resultant – an objection which from first sight seems ridiculous, but actually makes sense (ibid., pp. 133–134), and in a way foreshadows Russell's remark. It should be added that Fabri's primary objective in bringing up the objection described by Freudenthal (not discussed itself in this paper) is to refute Borelli's claim that within a body in rest two equal and contrary forces (or impetuses) are exercised simultaneously, contrary to Fabri's assertion that being totally “in vain” the contrary impetuses are both annihilated (Fabri, Dialogi physici, pp. 201–202).

⁹¹Gideon Freudenthal (note 42), p. 131.