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Projectile Motion and the Rejection of Superposition

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Honoré Fabri and the Concept of Impetus: A Bridge between Paradigms

Part of the book series: Boston Studies in the Philosophy of Science ((BSPS,volume 288))

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Abstract

The last chapter of Part III outlines Fabri’s theory of projectiles, the rather peculiar synthesis between some New Science principles and old notions, in which Fabri adheres to basic conservation of rectilinear motion but rejects Galileo’s principle of superposition, in favor of an Aristotelian-style “frustra” mechanism which is responsible for the destruction of violent impetus. This chapter shows that while adhering to his basic “inertial framework”, and devising a theory which purports to “save the phenomena”, Fabri failed to develop a useful theory of projectiles which could be regarded as an advance vis-à-vis the pioneering theory of Galileo and his disciples.

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Notes

  1. 1.

    Mechanical Problems, 1, 848b10–b22, in Aristotle 1936 (this text was probably written by one of Aristotle’s students). See also Clagett 1959, p. 94.

  2. 2.

    Physics [Aristotle 1930], 8, 8, 262a12.

  3. 3.

    Meteorology [Aristotle 1931], 1, 4, 342a22–28.

  4. 4.

    Oresme 1968, lib. 2, ch. 25, pp. 525, 527.

  5. 5.

    It was actually Cavalieri who first published, in his Lo Specchio Ustorio (1632), a demonstration of the parabolic shape of a projectile’s trajectory; however, “it is clear that Cavalieri was convinced he had merely repeated a result achieved by Galileo and known among Galileo’s disciples to have been achieved by him” (Damerow et al. 2004, p. 284; see also Koyré 1978, pp. 237–241). It is worth emphasizing that while the “traditional” application of superposition (inspired by the Pseudo-Aristotelian Mechanical Problems) involves uniform motions only, a projectile’s trajectory has of course a non-uniform component; Torricelli in fact composed uniform motion with motions of higher and higher degrees (\(v \propto {t^2}\), \(v \propto {t^3}\) etc.) to obtain parabolas of higher and higher degrees – cubic, quartic etc. (Boyer 1949, pp. 130–132).

  6. 6.

    “Et lors la qualité naturelle de la chose meue, si comme est pesanteur, fait appeticier ceste qualité ou redeur qui enclinoit contre le mouvement naturel de la chose, et va le mouvement en retardant et la violence en appetiçant et finablement cesse” (Oresme 1968, lib. 2, ch. 13, p. 417).

  7. 7.

    In fact, Oresme’s theory of projectiles contradicts classical mechanics even more than Buridan’s, because Oresme’s impetus is not permanent, and also since he opts for an “initial acceleration” of projectiles, thus claiming that after the projectile is thrown it accelerates for a short while and only then starts to decelerate (Clagett 1959, pp. 552–553, 681).

  8. 8.

    Galilei 1989, p. 222; text in brackets – Drake’s.

  9. 9.

    Damerow et al. 2004, pp. 251–254, 284–286, 351.

  10. 10.

    Damerow et al. 2004, pp. 261–262.

  11. 11.

    Renn et al. 2001, pp. 30, 51, 92–104, 113–126, 131–132. The curve of the catenary is of course not a parabola, but a hyperbolic function which resembles a parabola “if the distance between the two suspension points substantially exceeds the vertical distance between the suspension points and the lowest point of the chain” (ibid., p. 38).

  12. 12.

    “Motus mixtus est, qui sequitur ex multiplici impetu ad eandem, vel diversas lineas determinato, vel eodem ad diversas… observabis tantum ad motum mixtum sufficere duplicem impetum ad eandem lineam determinatam, deorsum, v.g. in mobili proiecto; nec enim est motus pure naturalis, nec etiam violentus, ut constat; igitur mixtus” (Fabri 1646, lib. 4, def. 1, pp. 153–154).

  13. 13.

    For Fabri’s notion of “hypothesis” see Chapter 2 above.

  14. 14.

    Fabri 1646, lib. 4, hyps. 1–2, p. 154.

  15. 15.

    “Proiectum per horizontalem sub finem motus minus ferit quam initio, imo & proiectum per inclinatam deorsum; haec hypothesis centies probata fuit; nec in dubium revocari potest” (Fabri 1646, lib. 4, hyp. 3, p. 154). According to Fabri, the only case in which the projectile is accelerated is a purely vertical downward throw (or fall).

  16. 16.

    When Fabri later discusses – and rejects – the possibility that it is air resistance which is responsible for the decrease in velocity, he claims that the effect of air resistance is so marginal that in vacuum the projectile would behave in the same way (see Section 16.1, Note 36 below).

  17. 17.

    Fabri 1646, lib. 1, th. 137, p. 66 and ths. 141–142, p. 67. When the impetuses coincide, i.e. point to the same direction, no impetus is destroyed.

  18. 18.

    “Hinc destruitur aliquid impetus per Th. 141. & 142 l.1, idque pro rata, ne aliquid sit frustra” (Fabri 1646, lib. 4, th. 6, p. 155).

  19. 19.

    “Hinc determinari potest portio utriusque impetus destructi, v.g. si sint aequales, portio detracta utrique aequalibus temporibus est differentia diagonalis & compositae ex DA, AB, quod clarum est; si vero impetus sint inaequales, portio destructa erit semper differentia diagonalis, v.g. AF & compositae ex AC.AD” (Fabri 1646, lib. 4, th. 7, p. 155).

  20. 20.

    Fabri 1646, lib. 4, th. 21, p. 159.

  21. 21.

    Fabri 1646, lib. 4, th. 30, p. 161.

  22. 22.

    Fabri 1646, lib. 4, th. 24, p. 159; see also Fabri 1646, lib. 2, th. 59, p. 95.

  23. 23.

    “… quia numquam destruitur impetus innatus” (Fabri 1646, lib. 4, th. 25, p. 160).

  24. 24.

    “… nulla est causa, a qua violentus possit accelerari” (Fabri 1646, lib. 4, th. 23, p. 159).

  25. 25.

    Fabri 1646, lib. 4, th. 17, p. 158.

  26. 26.

    “Haec linea est Parabola; quod ipse Galileus toties insinuavit, & quivis etiam rudior Geometra intelliget; in quo diutius non haereo, praesertim cum nullus sit motus, qui constet ex aequabili, & naturaliter accelerato, ut demonstrabimus infra” (Fabri 1646, lib. 4, th. 18, p. 158).

  27. 27.

    “Non est mixtus ex naturali accelerato & violento aequabili; demonstratur, primo, quia sub finem motus esset maior impetus; quippe nihil detraheretur violento, sed multum accederet naturali; igitur esset maior, igitur esset maior ictus contra hyp. 3” (Fabri 1646, lib. 4, th. 26, p. 160).

  28. 28.

    “… secundo, quotiescunque sunt duo impetus in eodem mobili ad diversas lineas determinati, aliquid illorum destruitur per Th. 141.l.1. tertio si esset uterque aequabilis, aliquid destrueretur per Theorema 6, igitur potiori iure, si impetus naturalis crescat” (Fabri 1646, lib. 4, th. 26, p. 160).

  29. 29.

    “Diceret forte aliquis impetum destrui ab aere, sed iam supra responsum est modicum inde imminui; nec enim unquam aer in corpore gravi destruit tantum impetus, quantum producitur naturalis si sit acceleratus; alioquin motus deorsum non cresceret contra experientiam” (Fabri 1646, lib. 4, th. 26, p. 160).

  30. 30.

    Fabri 1646, lib. 2, th. 61, p. 96.

  31. 31.

    “Hinc reiicies Galileum, qui in dialogis haec semper supposuit, sed nunquam probavit, nec probare unquam potuit; hoc etiam supponunt multi Galilei sectatores, qui censent impetum nunquam destrui nisi a resistentia medii; sed quaero ab illis quodnam medium destruat partem impetus in motu mixto; nec enim linea motus mixti adaequat duas alias ex quibus quasi resultat” (Fabri 1646, lib. 4, th. 28, p. 160).

  32. 32.

    “Hinc reiicies Galileum, & alios eius sectatores qui volunt impetum corpori impressum destrui tantum ab aere” (Fabri 1646, lib. 3, th. 45, p. 144).

  33. 33.

    Fabri 1646, lib. 3, th. 20, p. 138.

  34. 34.

    With the exception of a projectile thrown vertically downwards, where the natural and the violent impetuses are directed along the same line and therefore do not “fight” each other.

  35. 35.

    “… certe hoc non potest explicari cum infinitis fere aliis, nisi dicatur impetum destrui ab alio impetu, eo modo quo saepe diximus, hoc est ne sit frustra; igitur impetus violentus destruitur ab innato” (Fabri 1646, lib. 4, th. 28, p. 160).

  36. 36.

    “Hinc ratio clara cur sit minor ictus in fine huius motus; quia scilicet est minus impetus, quia plus detractum est quam additum; nec est quod tribuant hanc retardationem medio; quippe aer non plus resistit motui violento quam naturali; sed id quod detrahitur ab aere corpori gravi, v. g. pilae plumbeae est insensibile, ut fatentur omnes; igitur idem dicendum est de motu violento & mixto, hinc hoc ipsum etiam fieret in vacuo” (Fabri 1646, lib. 4, th. 36, p. 162).

  37. 37.

    Which perhaps could be seen (certainly in retrospect) as the weakest part of Fabri’s argumentation.

  38. 38.

    Fabri’s theorem 13, explained in Part II, Chapter 8 above.

  39. 39.

    “Non est mixtus ex naturali accelerato eo modo quo acceleratur deorsum per lineam perpendicularem & ex violento retardato: Probatur, si ita est, tantum additur naturali, quantum detrahitur violento, imo plus; igitur semper est in eo mobili aequalis vel maior impetus; igitur aequalis est semper, vel maior ictus contra hyp. 3” (Fabri 1646, lib. 4, th. 29, p. 160).

  40. 40.

    “… adde quod non minus impeditur ab impetu violento naturalis motus, quam ab inclinato plano”.

  41. 41.

    “Itaque motus praedictus mixtus est ex violento retardato & naturali accelerato, non eo quidem modo quo acceleratur in perpendiculari, sed eo quo acceleratur in plano inclinato, quod hic singulis instantibus mutatur” (Fabri 1646, lib. 4, th. 30, p. 161).

  42. 42.

    “Impetus naturalis concurrit ad hunc motum; probatur, quia alioquin esset rectus” (Fabri 1646, lib. 4, th. 37, p. 162).

  43. 43.

    “Si impetus naturalis non concurreret ad hunc motum, proiectum moveretur per lineam horizontalem rectam, ut constat, motu aequabili; posito quod non retardaretur in horizontali, eodem modo moveretur quo in verticali sursum” (Fabri 1646, lib. 4, th. 38, p. 162).

  44. 44.

    Clagett 1959, p. 535. The traditional view still appeared even in the writings of Nicolo Tartaglia (1558) and Diego Ufano (1628), though Tartaglia was actually aware of the continuous curvature of the trajectory. On this peculiar situation, see Büttner et al. 2003, pp. 13–16.

  45. 45.

    Fabri 1646, lib. 4, th. 59, cor. 9, p. 170.

  46. 46.

    Fabri 1646, lib. 4, th. 39, pp. 162–163.

  47. 47.

    Fabri 1646, lib. 4, th. 41, p. 163.

  48. 48.

    Fabri 1646, lib. 4, th. 52, p. 166.

  49. 49.

    Fabri 1646, lib. 4, th. 53, p. 166.

  50. 50.

    Fabri 1646, lib. 4, th. 55, pp. 166–167 (the result is Fig. 47 in Fig. 16.4 below).

  51. 51.

    Fabri 1646, lib. 4, th. 56, p. 166.

  52. 52.

    “Observabis nondum esse a nobis determinatam proportionem illam, in qua destruitur impetus violentus in motu mixto, quae tamen ex dictis supra potest colligi; quippe destruitur pro rata, idest qua proportione linea motus mixti est minor linea composita ex utroque” (Fabri 1646, lib. 4, th. 56, scholium, p. 167).

  53. 53.

    Judging from Fig. 49 (in Fig. 16.4), it seems that each of the “trajectoral” impetuses is not aligned along the tangents to the curve, despite his general reduction of motions to linear tangential motions described above (Section 15.2). In any case, Fabri – who does not decompose the trajectory into a vertical (accelerating/decelerating) component and a horizontal (uniform) one – is indeed still far from classical mechanics.

  54. 54.

    It is “in vain” because, owing to the fact that the diagonal is always smaller then the sum of the components (i.e. the two sides), it is – as it were – “not needed”. The only case in which no amount is frustra occurs when the projecting impetus and the natural impetus are aligned in the same direction, i.e. when an object is thrown perpendicularly downwards (see Note 17 above); in any other case the “destruction mechanism” operates to continuously diminish the “trajectoral” impetus (hence Fabri’s hypothesis 3).

  55. 55.

    Again, natural innate impetus can never diminish (see Chapter 8 above), so the frustra mechanism can never be applied to it; in the descending part, the natural impetus of course increases, by the accumulation of acquired impetus (OP < RT < XB in diagram 49).

  56. 56.

    Fabri means here a relation of arithmetical difference, not geometrical proportion.

  57. 57.

    “Destruitur impetus violentus pro rata, id est, qua proportione est frustra; v.g. sit impetus per AD inclinatam sursum, & alius per AB perpendicularem deorsum; haud dubie motus erit per AC; igitur concurrunt ad motum AC motus AB & AD, vel potius impetus; igitur debet destrui impetus in ea proportione, in qua AC est minor AN, id est composita ex AD, DC; quod impetus AB non possit destrui, totum id quod destruetur detrahetur impetui AD; igitur assumatur DF scilicet differentia AC, & AN; impetus destructus ita se habet ad impetum AD, ut DF ad AD, & ad residuum impetum ex AD, ut DF ad FA, quae omnia constant ex Th. 7” (Fabri 1646, lib. 4, th. 59, p. 168; I have corrected “G” in the original text, a letter which does not appear at all in diagram 48, to “N” – the obvious meaning. This error is probably a typo or a result of an incompatibility between the original text and diagram 48). It should be noted that David Lukens has already explained (in too little detail though) Fabri’s implementation of the frustra mechanism to projectiles (Lukens 1979, p. 227).

  58. 58.

    “Sit ergo AC Fig. 49 perpendicularis sursum, AD inclinata, AB horizontalis; sit impetus violentus respondens AD, & naturalis DG, ducatur AGK, ex AD detrahatur DF, id est differentia AG & compositae ex AD, DG, superest AF, cui assumitur aequalis GK, ex qua detrahitur KH, id est differentia GL, & compositae ex GK, KL, superest GH, cui LO accipitur aequalis, cui detrahitur OM, id est differentia LP & compositae ex LO, OP, superest ML, cui aequalis accipitur PR, atque ita deinceps” (Fabri 1646, lib. 4, th. 59, p. 168).

  59. 59.

    Of course, the ever-increasing natural impetus in the descending part renders the horizontal distance traversed shorter than in the ascending part (in which the natural impetus does not increase). Fabri’s frustra mechanism replaces the Galilean (and the classical) “destruction mechanism”, namely air resistance, as the explanation for the quick destruction of the horizontal component of an actual projectile. Fabri then is “wrong” from an anachronistic point of view but non inconsistent.

  60. 60.

    “Hinc reiicio Galileum qui nulla prorsus fultus ratione physica vult utrumque esse aequalem, quod tamen omnibus experimentis repugnat, & ipsi etiam pueri, qui disco ludunt observare possunt arcum descensus sui disci esse longe minorem, nec est quod ad suam Parabolam confugiat, quae duo falsa supponit principia, scilicet aequabilitatem motus violenti, & accelerationem naturalis eo scilicet modo quo fieret in perpendiculari” (Fabri 1646, lib. 4, th. 59, cor. 8, p. 169).

  61. 61.

    This is what constitutes, in the eyes of Fabri, an “experiment” (see e.g. Section 4.2 above).

  62. 62.

    Marin Mersenne also expresses belief in this Aristotelian principle, by claiming that “the movement of missiles which are moved violently go [sic] much more slowly as they are farther from their origin, that is, from the force by which they have been thrown”; Dear 1984, pp. 243–244.

  63. 63.

    Barbour 1989, p. 433; Barbour’s emphasis.

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Elazar, M. (2011). Projectile Motion and the Rejection of Superposition. In: Honoré Fabri and the Concept of Impetus: A Bridge between Paradigms. Boston Studies in the Philosophy of Science, vol 288. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1605-6_16

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