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Honoré Fabri and the Concept of Impetus: A Bridge between Paradigms pp 157–168Cite as

The Inherent Linearity of Impetus

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

Abstract

This chapter discusses the inherent linearity of Fabri’s impetus, which entails specifically conservation of rectilinear motion rather than of both linear and circular motion (as, for example, Isaac Beeckman and Pierre Gassendi maintained). Fabri, following Descartes, employs the old scholastic notion of determinatio to describe the necessary basic linearity of impetus (and consequently motion): “an impetus”, he declares in De impetu, “must be determined (determinatus) along a certain line of motion”. Fabri’s use of the concept of determinatio, within his analysis of reflection from totally elastic planes, is subsequently described. Finally, Fabri’s view concerning circular motion is outlined: as a direct consequence of his (relatively) modern conception of motion as inherently linear, Fabri regards circular motion as arising from an impeded straight motion, and accordingly observes that a stone tied to a sling will proceed along a straight line tangential to the circular original trajectory if the rope suddenly breaks.

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Notes

  1. 1.

    “Cum motus sequatur ex impetu, hic ad unam tantum lineam est determinatus, sive simplicem, sive mixtam, de hac determinatione fuse actum est tom. 2. l. 1” (Fabri 1648, lib. 12, prop. 20, p. 496).

  2. 2.

    As mentioned above, in 1585 Benedetti came quite close to CRM. See the beginning of Chapter 14 (especially Note 7), and also Clagett 1959, pp. 663–664; Koyré 1978, pp. 21–27.

  3. 3.

    The full meaning of this rather obscure remark – Non simul agit impetus in orbem – will be explained in Section 15.2 below.

  4. 4.

    The first role is to “exact” motion ad intra; the second role is to “create” a new impetus ad extra in another body following a collision.

  5. 5.

    “Non simul agit impetus in orbem sed tantum per lineam sui motus… Observabis tertio, impetum in utroque munere causae subesse tantum uni lineae; scilicet exigit motum per unam lineam; cum per plures simul motus esse non possit; ne idem mobile simul esset in pluribus locis” (Fabri 1646, lib. 1, th. 45, pp. 32–33).

  6. 6.

    “Impetus debet determinari ad aliquam lineam motus; probatur, quia non potest esse impetus, nisi exigat motum per Th. 14, nec exigere motum, nisi per aliquam lineam, ut patet; sed hoc est impetum esse determinatum ad aliquam lineam motus; praeterea si non est determinatus ad aliquam lineam; igitur indeterminatus, & indifferens per Ax. 5. Sed indifferens manere non potest; cur enim potius haberet motum per unam lineam, quam per aliam? igitur debet determinari” (Fabri 1646, lib. 1, th. 112, pp. 60–61). The text says “per Ax. 1”, but since axiom 1 discusses a completely different matter, and it is axiom 5 which asserts that what is “one” is also “determined” and “not indifferent”, it must be a typo.

  7. 7.

    “Observabis primo determinationem hanc nihil esse aliud, nisi ipsum impetum cum tali linea comparatum, seu coniunctum” (Fabri 1646, lib. 1, th. 114, scholium, p. 61). Gideon Freudenthal demonstrates Fabri’s preference (on account of ontological considerations) to regard impetus itself as a scalar magnitude, rather than a vector, while using nonetheless the parallelogram rule; Freudenthal 2000, pp. 130–135. In any case, we just saw that the impetus “cannot stay indifferent” and has to become “determined,” i.e. directed along a line.

  8. 8.

    Gabbey 1980, pp. 248–249 and p. 309 (n. 99).

  9. 9.

    “Quod unum est, determinatum est. Quia quod unum est, est hoc,& nihil aliud; nihil enim aliud est unum, nisi indivisum in se, & divisum a quolibet alio: quippe indifferentia, vel indeterminatio ibi tantum est, ubi sunt plura… si enim tantum unus est, certe indifferens non est” (Fabri 1646, lib. 1, ax. 5, p. 6).

  10. 10.

    Gabbey 1980, p. 249, and p. 309, n. 99.

  11. 11.

    “…etiam ex hoc phaenomeno duci potest vera mensura, seu regula refractionum, quod ingeniosissime excogitavit vir illustris Renatus Descartes in sua Dioptrica” (Fabri 1646, lib. 6, th. 85, p. 264).

  12. 12.

    Damerow et al. 2004, p. 131.

  13. 13.

    Damerow et al. 2004, p. 109.

  14. 14.

    Damerow et al. 2004, pp. 80, 109.

  15. 15.

    Lukens provides a fuller account of Fabri’s theory of collisions (Lukens 1979, pp. 233–246).

  16. 16.

    DO should be parallel and equal to GN, though unfortunately this is not so obvious in the figure Fabri supplies.

  17. 17.

    Fabri 1646, lib. 6, th. 40, pp. 245–249. See also Lukens 1979, pp. 235–236.

  18. 18.

    Damerow et al. 2004, p. 114. In the field of optics, this principle “was known in the fourth century B.C., and no doubt earlier”; Clagett 1955, p. 102.

  19. 19.

    “Hinc angulus reflexionis est aequalis angulo incidentiae” (Fabri 1646, lib. 6, th. 33, p. 243).

  20. 20.

    Not to be confused with the determinatio exerted by the wall on the plane, which was mentioned just above, and should be vectorially added to the original determinatio in order to obtain the final outcome.

  21. 21.

    “Quando linea incidentiae cadit perpendiculariter in planum reflectens est maximum impedimentum; quia scilicet est maximus ictus, ut probavimus lib. 1” (Fabri 1646, lib. 6, th. 22, p. 241); Fabri means theorems 50 and 55 in De impetu.

  22. 22.

    “Quo linea incidentiae cadit obliquius in planum, est minus impedimentum, quia est minor ictus… ictus per lineam GD est ad ictum per lineam AD, ut AD ad AB” (Fabri 1646, lib. 6, th. 23, p. 241).

  23. 23.

    Fabri 1646, lib. 6, th. 23, p. 241. Fabri refers to theorem 16 of book 5, De motu in diversis planis, p. 202. This claim is very similar to Jordanus of Nemora’s concept of “Positional Gravity”, discussed in Clagett 1959, pp. 74–75.

  24. 24.

    “Hinc decrescit determinatio, quam confert planum iuxta rationem sinuum versorum in GD. v. g. si sit linea incidentiae AD; ducatur APH parallela FB, determinatio quam confert planum, decrescit sinu verso PG” (Fabri 1646, lib. 6, th. 32, p. 243).

  25. 25.

    Assuming that θ represents the angle between the line of incidence and the perpendicular to the plane, e.g. the angle GDA in the case of the incident line AD.

  26. 26.

    “v. g. sit linea incidentiae AD, linea reflexionis DH; non tantum determinatur haec linea a plano FB, alioqui esset DG, nec est eadem cum prima; alioqui esset DE, sed partim determinatur a plano FB per DG partimque retinet aliquid primae determinationis, & ex utraque fit DH, ut constat, quia quo linea incidentiae est obliquior, planum minus determinat” (Fabri 1646, lib. 6, th. 29, p. 242).

  27. 27.

    “Hinc qua proportione planum minus confert ad novam determinationem, plus remanet prioris determinationis; quo vero plus illud confert, huius minus restat; hinc, cum planum totam confert novam determinationem ut in perpendiculari DG, nihil prioris remanet; hinc si linea incidentiae sit parallela plano BF nulla fiet nova determinatio, tota priore intacta; si vero sit perpendicularis GD, tota determinatio est nova, & nihil prioris remanet; si demum lineae incidentiae sint aliae, confert utrumque ad novam determinationem pro rata” (Fabri 1646, lib. 6, th. 30, p. 243).

  28. 28.

    “… at vero crescit portio prioris determinationis lineae incidentiae iuxta rationem sinuum rectorum in DB, v. g. si sit linea incidentiae AD, crescit sinu recto AP aequali BD, si sit ID crescit sinu recto IL vel RD” (Fabri 1646, lib. 6, th. 32, p. 243).

  29. 29.

    Fabri 1646, lib. 6, th. 33, p. 243.

  30. 30.

    Quoted in Gabbey 1980, p. 252.

  31. 31.

    “Motus circularis in sublunaribus oritur ex recto impedito; quia, scilicet, determinatur tantum impetus ad lineam rectam: hinc quidam motus circularis est mere per accidens, ut cum retinetur extremitas funependuli, seu fundae, quae si demittatur, sequitur motus rectus” (Fabri 1646, “Synopsis amplior”, “De motu circulari,” par. 1; this section of the Tractatus is unnumbered).

  32. 32.

    Clagett 1959, p. 534. See also beginning of Section 5.3 above.

  33. 33.

    “Theorema 1: Datur motus circularis. Probatur infinitis fere experimentis; primo in libra cuius brachia motu tantum circulari descendunt. Secundo in vecte, qui etiam movetur circulari motu; Tertio in turbine, rota molari, liquore contento intra vas sphaericum; Quarto in funependulo vibrato” (Fabri 1646, lib. 7, th. 1, p. 273).

  34. 34.

    “Probatur secundo; quia potest imprimi impetus utrique extremitati cilindri in partes oppositas, sit enim cilindrus, vel parallelipedum LC, cuius extremitati imprimatur impetus, per lineam CP, itemque extremitati L aequalis per lineam LM [page 273 says ‘LG’, but it is corrected in the errata into ‘LM’. M. E.] oppositam Cp. Dico, quod movebitur circulariter circa centrum K, ita vt extremitas L conficiat arcum LB & C arcum CE; nec enim C moveri potest per CP neque L per LM; quippe cum sit aequalis impetus, neutra extremitas praevalere potest: non utraque, quia MP est maior LC; nec dici potest neutram moveri, cum moveri possit L per arcum LT, & C per arcum CS” (Fabri 1646, lib. 7, th. 1, p. 273).

  35. 35.

    “Theorema 2: Nisi impediretur impetus determinatio per lineam rectam, non daretur motus circularis saltem in sublunaribus. v. g. nisi impediretur determinatio impetus, qui inest puncto L per lineam LM; haud dubie non moveretur per arcum LB, sed per rectam LM; igitur ille motus non esset circularis” (Fabri 1646, lib. 7, th. 2, p. 273).

  36. 36.

    “Theorema 3: Hinc motus circularis oritur ex recto impedito in singulis punctis… Theorema 4: Hinc singulis instantibus punctum dum movetur circa centrum K determinatur ad novam lineam… Theorema 5: Hinc tot sunt determinationes singulis instantibus respondentes, quot sunt Tangentes in circulo; quippe in singulis punctis determinatur ad Tangentem; sed impeditur denuo pro sequenti instanti; igitur ad novam Tangentem determinatur” (Fabri 1646, lib. 7, ths. 3–5, pp. 273–274).

  37. 37.

    Fabri describes curved motion as “mixtus ex recto & circulari” (Fabri 1646, lib. 9, th. 27, p. 364); the second postulate of the 4th book claims: “Illa linea vocetur curva quae constat infinitis prope lateribus polygoni” (Fabri 1646, lib. 4, post. 2, p. 154).

  38. 38.

    “Motum mixtum eum esse non dico, qui ex pluribus aliis motibus componatur; seu misceatur; nec enim plures motus simul esse possunt in eodem mobili… Motus mixtus est, qui sequitur ex multiplici impetu ad eamdem, vel diversas lineas determinato, vel eodem ad diversas” (Fabri 1646, lib. 4, p. 153; before the first definition).

  39. 39.

    “observabis dictum esse supra in sublunaribus quia corpora coelestia moventur motu circulari non habita ulla ratione motus recti” (Fabri 1646, lib. 7, th. 3, p. 273).

  40. 40.

    “Severior Geometria, ut omittam Astronomiam, motum supponit, cum ex fluxu seu motu puncti infinitas fere lineas describat” (see Section 3.1, Note 18 above).

  41. 41.

    The definition (in the beginning of Chapter 3 above, Note 1 above) “motus localis est transitus mobilis e loco in locum continuo fluxu”.

  42. 42.

    “Tertio: linea motus non differt ab ipso motu continuo tractu, seu fluxu quasi labenti” (Fabri 1646, lib. 1, th. 114, scholium, p. 61).

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    Michael Elazar

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Elazar, M. (2011). The Inherent Linearity of Impetus. 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_15

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