Abstract
The fourteenth-century concept of impetus denotes an impressed force and was used to explain the continuation of the motion of projectiles and the acceleration of falling bodies. This chapter deals with the use of this concept in the period between Galileo’s death (1642) and the publication of Newton’s Philosophiæ naturalis principia mathematica (1687). Focusing on three major figures among contemporary Jesuit thinkers, the French Honoré Fabri (1608–1688) and the Italians Paolo Casati (1617–1707) and Francesco Eschinardi (1623–1703), this chapter shows how these Jesuits employed the concept of impetus in their own versions of preclassical mechanics.
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Notes
- 1.
- 2.
See Chap. 1 in this volume.
- 3.
Galileo EN VIII, 201/157 f. Translation from Damerow et al. (2004).
- 4.
See Chap. 1 in this volume.
- 5.
See Galluzzi (2001).
- 6.
- 7.
The full title is Tractatus physicus de motu locali, in quo effectus omnes, qui ad impetum, motum naturalem, violentum, & mixtum pertinent, explicantur, & exprincipiis physicis demonstrantur; auctore Petro Mousnerio Doctore Medico; cuncta excerpta ex praelectionibus R. P. Honorati Fabry, Societatis Iesu. In any case, in later years Fabri treats this work as his own.
- 8.
See, e.g., Baldini (1999, 256–260).
- 9.
- 10.
“Cum enim impetus sit causa immediata motus, ut fuse demonstrabimus infra; & cum propter quid sit res cognosci non possit, nisi eius causa cognoscatur; dubium esse non potest, quin praemittenda sit tractatio illa, quae est. de impetu, ut deinde affectiones ipsius motus per causam eiusdem demonstrentur” Fabri (1646, 1). See also Elazar (2011, 3).
- 11.
“Immo ausim dicere ex unius impetus cognitione, non modo motum ipsum, verum etiam totam rem physicam pendere” Fabri (1646, 1).
- 12.
This claim has been demonstrated in Elazar (2011) and will be shown here only in brief for the sake of comparison with Casati and Eschinardi.
- 13.
This key concept has of course a much longer history, which will not be dealt with in this article. Concerning the possible ancient origins of impetus, ranging from Hipparchus of Nicaea (second century BC) to John Philoponus (ca. 490–570 AD), see Franco (2003, 525), as well as Clagett (1959, 508–510, 543–547).
- 14.
Franciscus Marchia, for example, adopted such “self-expending” vis derelicta, and Nicole Oresme also supported a similar kind of impetus, Clagett (1959, 529); so did Galileo in his early De motu antiquiora (ca. 1590).
- 15.
Fabri (1646, lib. 1, th. 12): “Impetus est. qualitas permanens.”
- 16.
“Definitio III: impetus est. qualitas exigens motum, seu fluxum localem sui subiecti, vel quae est. causa proxima motus illius mobilis, cui inest, eo scilicet modo, quo potest esse causa motus” (Fabri 1646, lib. 1, def. 3, p. 2). The verb “exacting” (exigere) means “requiring,” or “compelling.”
- 17.
Elazar (2008, 7–8). The ens and non ens to which Fabri refers signify a place just acquired and a place already left during motion.
- 18.
For example, in theorem 12, Fabri (1646, 18) refers to it as vis impressa.
- 19.
- 20.
“Cum motus sequatur ex impetu, hic ad unam tantum lineam est. determinatus, sive simplicem, sive mixtam” Fabri (1648, lib. 12, prop. 20, p. 496).
- 21.
“Non simul agit impetus in orbem sed tantum per lineam sui motus” Fabri (1646, lib. 1, th. 45, pp. 32–33).
- 22.
- 23.
- 24.
The full title of Casati’s work is Mechanicorum libri octo, in quibus uno eodemque principio vectis vires physice explicantur & geometrice demonstrantur, atque Machinarum omnis generis componendarum methodus proponitur.
- 25.
The title of the second chapter of De causis motus Machinalis promises to explain the nature of impetus as a cause immediately effecting motion (“Impetus motum proxime efficientis natura explicatur” Casati (1684, 136)). The translation to be found in the English translation of Duhem (1991, 570, n. 111), “The Nature of the impetus causing motion is explained in detail” is incorrect, for adverb proxime in the title therefore relates to efficientis and not to explicatur (furthermore, Casati states, in the next page that “impetus motum proxime effecit”; besides, proxime cannot mean “in detail” Casati (1684, 137).
- 26.
“Hinc nimirum in uberiorem ipsius resistantiae nototiam venientes ad explicandam motus machinalis causam propius accedemus” Casati (1684, 145).
- 27.
“Quicquid movetur, qualecumque est., causam habeat moventem necesse est., ut hoc quidem sponte sua, illud vero aliena vi ex alio in alium locum migret” Casati (1684, 136).
- 28.
“Etiamsi moventi naturæ, quæ ex Philosophi decretis substantia est., motus originem ultimam tribuamus, jure tamen optimo aliquid naturæ ipsi ac motui, interjectum agnoscimus (impetum nominamus) cujus intentionem ac remissionem velocitas ac tarditas consequatur” Casati (1684, 136).
- 29.
Which is indeed regarded as a substance by Aristotle (Metaphysics, 1071b13–22; Aristotle 1908).
- 30.
“... motui per se, quia ex alio in alium locum continuata migratio est., eficientiam aegre tribuere possumus; quippe qui, cum in fluxione consistat, ita ut locus loco, seu potius, ut scholae loquuntur, ubicatio ubicationi, priori scilicet pereunti succedat, posterior aeque fugax, inferioris notae censendus est. quam impetus natura sua aliquandiu permanens: labentia enim stantibus deteriora esse, caeteris paribus, quis neget? Casati (1684, 138).
- 31.
“Effectum autem causa praestabiliorem esse non posse ipsa originis notio suadet, ne quid effectus habeat, quod non acceperit, aut aliquid causa dederit, quo ipsa caereret. Non igitur impetum motus, sed motum impetus efficit” Casati (1684, 138).
- 32.
For 5.25 = 1.5 ⋅ 7 ⋅ 4 / 8 (“Quare ad movendas libras 5¼ velocitate ut 8, requiritur conatus sesquialter conatus necessarii ad movendas libras 4 velocitate ut 7”) Casati (1684, 142).
- 33.
- 34.
The full title is 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: additur in fine quamplurimum problematum seu quaesitorum solutio ex doctrinis praecedentibus.
- 35.
This of course must not be confused with the modern term of impulse (the integral of force over time)—an effect of motion rather than its cause.
- 36.
“Nomine igitur impulsus, sive impetus, intelligimus hic universaliter non solum illam qualitatem, quae ab impellente extrinseco proucitur in aliquo corpore, cum proiicitur; sed possumus etiam intelligere quamcomque; cuiusmodi puto esse, non solum qualitatem productam in saxo a proiiciente, sed etiam causam immediatam motus in Gravibus, in rarefactione & addensatione, in attractione magnetica, in motu animali etc” Eschinardi (1684, 2).
- 37.
“Impetus vero est. aliquid de sua natura permanens; licet per accidens possit fluere” Eschinardi (1684, 2).
- 38.
“Probatur autem existentia huius qualitatis, prout distinctae a substantia, & a quocumque alio accidente. Primo, quia lapis proiectus in altum, non moveretur; nisi in eo producta fuisset a manu proijcente qualitas permanens causativa dicti motus; lapis enim ille separatus a manu proijcente, de sua natura petit descendere; nec motus antecedens potest esse causa efficiens motus sequentis, ut mox ostendemus; ergo &c” Eschinardi (1684, 3). The second part of Eschinardi’s “proof” concerns naturally accelerated motion, and is described in Sect. 10.4.3.
- 39.
“Singuli enim impetus habent suam propriam directionem” Eschinardi (1684, 5).
- 40.
“Tamdiu conservatur impetus, quamdiu nihil exigit eius destructionem; quia destruitur tantum ad exigentiam alicuius, quidquid tandem illud sit, de quo infra, per Ax.14.num.2. certe tamdiu non destruitur, quamdiu nihil est., quod exigat eius destructionem; igitur tamdiu conservatur per Ax. 14.num. 3” Fabri (1646, lib. 1, th. 147, 68–69).
- 41.
It should be added though that Fabri objected to Torricelli’s claim to have produced vacuum de facto; for a detailed description of Fabri’s view, see Elazar (2011, 135–156).
- 42.
For Aristotle’s opinion, see Physics IV, 215a28–215b23; Physics, IV, 216a11–216a20 Physics, IV, 8, 215a19 (Aristotle 1930).
- 43.
“Concedo ultro fore, ut in totali vacuo lapis semel motus, semper moveatur, nisi ab extrinseco retineatur” Fabri (1648, 564).
- 44.
Neglecting, of course, air resistance. “Theorema 38: si impetus naturalis non concurreret ad hunc motum, proiectum moveretur per lineam horizontalem rectam, ut constat, motu æquabili” Fabri (1646, 74).
- 45.
“Theorema 71: in vacuo si corpus grave descenderet, prædictæ proportiones accuratissime servarentur; quia scilicet nullum esse impedimentum; at vero si aliquod intercedit impedimentum; haud dubie non servantur accurate” Fabri (1646, 115).
- 46.
- 47.
“Impetus productus secundo instanti in medio libero conservatur tertio, & productus tertio conservatur quarto, atque ita deinceps; quia scilicet nec conservantur a causa primo productiva per Th.144.libri: nec aliquid exigit destructionem; non contrarius impetus, quia nullus est. applicatus, ut constat; non resistentia medii, quae quidem alicuius momenti est.; sed non tanti, ut impedire possit motum omnino, ut constat; nam suppono liberum medium, igitur nec destruere impetum; cum tamdiu duret causa quamdiu durat effectus, ut patet; igitur nihil est. quod exigat impetus huius destructionem” Fabri (1646, lib. 2, th. 13, 82).
- 48.
“Cum itaque impetus acquisitus, aut aliunde impressus, sit qualitas propter motum instituta, quae non nisi in motu producitur, ita pariter nisi in motu, & cum motu non conservatur” Casati (1684, 155).
- 49.
Casati (1684, 155).
- 50.
Casati (1684, 152–153).
- 51.
“Contra vero inanimum non suo arbitrio motus intentionem moderatur, sed naturae juribus obsequens nihil prætermittit impetus, & quantum eniti potest, opportunum in locum, sibique a natura constitutum, contendit. Cave tamen existimes parem esse lapidis ejusdem, & in aere, & in aqua descendentis impetum: natura scilicet ex medio dividendo, in quo perficiendus est. motus, metitur impetus modum” Casati (1684, 138).
- 52.
See, e.g., Baldini (2004, 135).
- 53.
“Et sane conceptum impetum, natura sua, nec flabilem semper permanere, nec ad unicum temporis punctum durare, satis constat: sive enim sponte profluat ex natura debitum sibi locum quærente, sive aliena vi impressus suo loco corpus extrudat, perpetuus esse nequit; omnis scilicet motus terminum habeat necesse est.; nam si violentus quidem est., perennis utique non est.; sin autem naturalis, quem violentus præcesserit, certis definitur terminis; a loco enim, in quo quietem natura indixit, corpus infinito intervallo non abest, ac proinde ubi eum attigerit, demum conquiescet” Casati (1684, 146).
- 54.
See Damerow et al. (2004, ch. 1).
- 55.
Casati, Vacuum proscriptum: disputatio physica authore Paulo Casato Placentino Societatis Iesu, in qua nullum esse in rerum natura vacuum ostenditur; & potissimum examinatur, an ab argento vivo descendente in fistula superne clausa vacuum relinquatur, huiusque experimenti symptomata exlpicantur.
- 56.
Magni was to subsequently claim that when he published his book he had not even heard of Torricelli; be that as it may, Casati apparently—even though he lived in Rome—learned of this experiment only from Magni’s published work; see Middleton (1964, 31, n. 32; 42–43).
- 57.
- 58.
“Gravia & levia moventur habita tantum ratione contingui corporis” Casati (1649, cap. 2, n. 11, p. 9).
- 59.
“At si sensu iudice res decernatur, non relinquitur ambigendi locus, an quodcunque corpus naturae impetu translatum simplici, ac recto motu cietur, contingui tantummodo, non vero procul positi, corporis rationem habeat, ut a quiete dimoveatur, & ad motum extimuletur” Casati (1649, cap. 2, n. 11, p. 9).
- 60.
“Qui ergo fieri poterit, ut corpus in vacuo constitutum, a natura ad motum excitetur? Nam, cum a nullo contiguo corpore ambiatur, quorsum moveatur? Nihil enim plane est. seu gravius, seu levius, in quod levitatis suae, aut gravitatis vires exerceat” Casati (1649, cap. 2, n. 11, p. 10).
- 61.
“Quamvis autem quae simplici, & recto gravium seu levium motu cientur, in vacuo constituta, nullique seu graviori, seu leviori corpori adhaerentia, immota prorsus consistere debeant, si sibi ipsa relinquantur, non negarim tamen hominem in vacuo collocatum posse gladium versare, hastam tractare, pendentemque ex dextera fundam in gyrum contorquere, aut saxum ex uno in alio locum transferre” Casati (1649, cap. 2, n. 13, p. 11).
- 62.
“Quia nimirum singulis corporibus hoc a natura comparatum est. (id quod effecta produnt) ut se mutuo nexu arctissime complecttantur, non dimissura, nisi aliud in recedentis locum subrogetur” Casati (1649, cap. 4, n. 36, p. 32). This passage in entitled “quo nexu contigua invicem cohaereant.”
- 63.
“Ex hac mutua corporum adhaesione inferre potes, quos motus exercere non posset homo in vacuo constitutus: si enim cubiculi vacui pavimento pedes insisterent, progredi nullatenus posset, aut saltus edere quia (cum in vacuo collocatus ponatur) nullum adesset corpus, quod in abeuntium pedum locum sufficeretur” Casati (1649, n. 40, p. 35).
- 64.
“At quamvis pendulum libere brachium circumagens dexteram saxo gravem contorqueret; non tamen saxum posse excuti, sed manui adhaereret. Angelus vero saxum in vacuo proijcere posset impresso impetu, quia cum omni corporeae molis concretione careat, nullo adhaesionis vinculo saxum cum eo connecteretur” Casati (1649, cap. 4, n. 40, p. 35–36).
- 65.
Des Chene (1996, 175–176). The last two citations are phrases quoted by Des Chene (apparently in his translation) from the Coimbran’s commentary to Aristotle’s physics.
- 66.
As stated above (Sect. 10.2.3), Eschinardi emphasizes that every impetus has its “proper direction.” “Putant primo aliqui, non produci impetum de novo; cum non potest per illum obtineri motus exigitus ab illo impetu; hoc est. iuxta ipsius directionem (singuli enim impetus habent suam propriam directionem)” Eschinardi (1684, 5).
- 67.
“Et quidem aliquis auctor assert exemplum de rupe percussa, in quo videtur errare: hoc ipso enim quod rupes percussa sonat; signum est., quod tremit; adeoque movetur secundum partes; adeoque partes sunt impulsae” Eschinardi (1684, 5).
- 68.
“Impetus non producitur in eo mobili, quod moveri non potest a potentia motrice applicata, licet a fortiori moveri possit” Fabri (1646, th. 31, p. 24).
- 69.
We have seen above how impetus “rejoices” while fulfilling that goal, i.e. succeeding im moving the subject in which it inheres.
- 70.
“Probatur, quia impetus est. tantum propter motum, qui eius effectus est., & finis, per Th. 15. & 16. Igitur ubi non est. motus, frustra est. impetus; sed quod frustra est., non est.; id est. non est., quod frustra esset, si esset per Ax. 6” Fabri (1646, th. 31, p. 24). Fabri immediately adds that innate impetus is an exception to this rule—see Sect. 10.4.1 below.
- 71.
“& quidem pro rata plus, minus; prout plus, minus frustratur fine” Eschinardi (1684, 4).
- 72.
“...difficultas est. in assignando adaequate fines intentos a natura per impetum” Eschinardi (1684, 4).
- 73.
“... puto: Primo, quod impetus producatur de novo, quoties causa est. applicata; sed non conservetur, nisi cum potest haberi motus per directionem debitam tali impulsui; ita tamen, ut plus minus destruatur, prout plus minus habet de motu iuexta eam directionem” Eschinardi (1684, 6).
- 74.
“sic semper fit aequilibrium per duos impetus aequales contrarios fluentes, quorum uterque frustratur motu” Eschinardi (1684, 7).
- 75.
“Adverte hic, Grave in descensu semper esse in aliquo medio corporeo fluido, adeoque nunquam nos experiri eius gravitatem absolutam, sed solum comparativam” Eschinardi (1684, 23).
- 76.
“Moneo iam primo fieri non posse, ut sciamus mathematice gravitatem, sive pondus absolutum a medio; tum quia non sumus in vacuo, tem etiam quia semper datur impedimentum contactus hastae, sive brachii staterae” Eschinardi (1684, 333).
- 77.
“Secundo etiamsi essemus in vacuo, & non impediret contactus; adhuc nesciremus libram, unciam & c.; nam cum istae mensurae fuerint impositae in medio aere, nec sint nota corpora, quibus tunc usi sunt antiqui; hinc sequitur non posse mathematice cognosci, quid intellexerint per libram &c” Eschinardi (1684, 333).
- 78.
“Quoad levia vero, quia probabilius est. nihil dari leve simpliciter, & absolute, seu positive; sed solum comparative, hinc quae dicuntur levia in tantum ascendunt, in quantum sursum impellantur a gravioribus petentibus maiori nisu locum inferiorum” Eschinardi (1684, 24).
- 79.
“Innatus est. qui vel a generante simul cum corpore gravi productus est., porro cum in corpore gravi duplex quasi proprietas sensibilis esse videatur, scilicet gravitas, seu pondus & motus deorsum... Ex his dicendum est. hunc impetum nativum nunquam destrui, quia nunquam est. frustra, habet enim semper aliquem effectum” Fabri (1646, lib. 2, th. 14, scholium, p. 83).
- 80.
“Immo ostendam infra, nullam dari levitatem absolutam” Fabri (1646, prop. 28, 254).
- 81.
Fabri (1646, prop. 28, p. 254); Elazar (2008, 18–19). Galileo’s “formula” can be (anachronistically) reconstructed as (vo—v)/vo = ρm/ρb, where v = the speed of the falling body in medium m; vo = the speed of fall of all bodies in a void; ρm = the specific weight of medium m; ρb = the specific weight of the falling body, Elazar (2008, 17).
- 82.
See Elazar (2011, 75–76).
- 83.
“Igitur non movetur corpus grave a causa prima... nec ab aere, nec a virtute magnetica... nec a caelo pellente, nec a vi sympathica, nec a generante proxime & immediate... nec ab ullo alio extrinseco, ut constat inductione; igitur ab aliqua vi intrinseca, quidquid sit, de qua alibi” Fabri (1646, lib.2, th. 1, p. 79).
- 84.
“Hinc motus naturalis est. ab impetu. Probatur; est. ab aliqua causa per Ax.8. lib.1; ab aliqua intrinseca per Th. 1; non a substantia corporis gravis per Th. 3; non a gravitate per Th. 4; igitur ab impetu, quia nihil aliud esse potest intrinsecum, a quo sit motus” Fabri (1646, lib.2, th. 5, p. 80).
- 85.
“Aliquis forte diceret potentiam motricem exigere primam partem motus, quæ deinde secundam exigit, & secunda tertiam, tertia quartam, &c. Sed contra; quæro quid sit prima illa pars motus” Fabri (1646, lib 1, th. 5, p. 14.)
- 86.
“Cum enim eadem descendentis lapidis natura perseveret, nec illa in sua potestate sit, aut optione delata, ut eligat utrum velit, motum arbitrio suo incitare, aut remittere valeat; qui fieri possit, ut descendens velocitatem augeat, nisi ei, quem primum produxit, alium atque alium momentis singulis impetum adjiciat?” Casati (1684, 136–137).
- 87.
Casati (1684, 146). Casati’s acquired impetus will be soon described.
- 88.
Aristotle insists that earth, water and air each has weight in its proper place, while fire is the only element that has no weight anywhere, even in its natural place (On the Heavens, Book 4, Ch. 4, 311b8 & 312b3; Aristotle 1953). On the other hand Galileo in his De motu strictly denies that any element gravitates in its proper place, see Heilbron (2010, 150).
- 89.
“Hinc nullum aquae impetum tribuo intra aquam consistenti; sed tunc solim cum situla plena e lacu extrahitur, ea. aquae pars impetum habet, quae supra subjectam lacus superficiem aere circumfusa motum expetit, quo suum repetat locum repugnans sustinteni. Impetum hunc, qui naturali se movendi facultati respondet, & est. ipsa gravitatio, seu naturalis ad descensum propensio, Innatum voco, & is est., cui extrinseca causa repugnat motum impediens” Casati (1684, 146).
- 90.
“Periit ergo innatus impetus, quo aqua situlam replens descensum moliebatur” Casati (1684, 147).
- 91.
“Innatum videlicet gravitatis impetum vis ipsa suspendens aut sustentans elidit” Casati (1684, 154).
- 92.
“...ab iis impetum concipi statuo, quae aut loco naturali dimota, aut incongruam partium positionem nacta id repetunt, quod natura exigit” Casati (1684, 145). As far as we could see, Casati does not discuss this issue; perhaps he would have evoked here his “mutual adhesion of bodies” principle (see Sect. 10.3.2 above).
- 93.
Chapter 1 is entitled “Quid sit Centrum gravium, & levium” and Chap. 2 is called “An corpora praedita sint graitate, & levitate.”
- 94.
- 95.
See Chap. 9 in this volume.
- 96.
“Non ausim tamen in lapide, qui gravitatem in aqua & aere, levitatem in mercurio, aut plumbo liquente obtinet, duplicem statuere virtutem, quarum altera sursum, altera deorsum connitatur; cum enim impetus motus efficiens (ut infra constabit) ejudem naturae sit, in quamcunque demum orbis plagam dirigatur motus” Casati (1684, 11).
- 97.
- 98.
See Chap. 9 in this volume.
- 99.
“Sed quomodo deorsum ferretur in vacuo, in quo nulla est positionum differentia? An innato quodam impetu universae corporeae molis centrum peteret? Sed hoc multorum pace dictum sit, inter vulgi errores est. recensendum; ut quid enim singulis gravibus natura centri appetitum insereret, quem pene in omnibus frustrandum praecognosceret? Cum praesertim universi concinnitas diversas corporum dispari gravitate praeditorum sedes exigeret. Non igitur singulorum gravium motus eo tendit, ut corpus motum in centro quiescat, sed ut debitam sibi sedem in universo occupet, iuxta insitae gravitatis speciem, molis scilicet aequalitate spectata” Casati (1649, cap. 2, n. 10, 9).
- 100.
“Quare idem de impetu extrinsecus asumpto dicendum est., quod de acquisito” Casati (1649, 153).
- 101.
“Secundo. Consideranda est. gravitas elicita; sive actus gravitandi; qui dicitur etiam gravitatio; & est. illa, quam hic & nunc exercet grave in alia corpora sibi resistentia... quae gravitas elicita nihil aliud est., quam impetus productus a gravitate innata; & de isto impetu certumest, quod distinguitur a substantia corporis gravis” Eschinardi (1684, 2).
- 102.
“Probatur iterum in motu naturali gravium libere descendentium in aere; nam grave descendens, semper velocius & velocius movetur; sed causa immediata, et adaequata huius motus velocioris non est. substantia, caeteraque accidentia; nam haec plerumque remanent omnino eadem; invariata autem causa immediata necesaria, non variatur effectus; neque dici potest motus antecedens, impererfectior, quippe minus velox, causa adaequata motus sequentis perfectioris; praecipue cum sit mera translatio formalis de loco in locum” Eschinardi (1684, 3).
- 103.
- 104.
“Ut autem habeamus totam simul methodum doctrinae Galilei de gravibus cadentibus, concinnavi sequentem discursum, inhaerendo tamen iis, quae sparsim inveniuntur praecipue in ipsius dialoi, addita aliqua maiori explicatione ubi oportebat” Eschinardi (1684, 26).
- 105.
“In proprtione gravitatum [specificarum] detrahitur de velocitate, ut ipse [Galileo] hic supponit: de velocitate inquam, qum haberent [descendentes] in vacuo” Eschinardi (1684, 27). Thus Eschinardi paraphrases the “formula” from the Two New Sciences which Fabri also adopted, as shown above.
- 106.
“Patuit etiam, quod glans plumbea e sclopo emmisa deorsum in altitudine 100 ulnarum minus est. compressa quam in tribus ulnis, ut Galileus dixerat” Eschinardi (1684, 29).
- 107.
- 108.
- 109.
This is guaranteed by the important theorem 13 of his De motu naturali deorsum, discussed above in Sect. 10.2.3, p. 24.
- 110.
- 111.
“The time in which a certain space is traversed by a moveable in uniformly accelerated movement from rest is equal to the time in which the same space would be traversed by the same moveable carried in uniform motion whose degree of speed is one–half the maximum and final degree of speed of the previous, uniformly accelerated, motion” Galileo (1989, 165).
- 112.
If, following naturally accelerated motion (lasting for time AC), a body continues “to be moved with the same degree of speed BC, without accelerating further, then in the ensuing time CI it would pass a space double that which it passed in the equal time AC with degree of uniform speed EC, one–half the degree BC” Galileo (1989, 168).
- 113.
In figure 23: the quadrangles LMNR, IKRP, GHPO and ABOF (assuming an “instant” RN=PR = OP etc.); these areas behave of course like the natural numbers series.
- 114.
Fabri (1646, lib. 2, th. 61, p. 106). In triangle AFN, Consecutive areas over consecutive equal periods of time (e.g. GIPO, ILRP and LNR in figure 23) behave of course like the series 1, 3, 5 …
- 115.
This chapter, entitled “Quam dispares ex motus velocitate sint percussiones,” is the seventh chapter of the seventh book—De cuneo & percussionibus.
- 116.
“Hinc duae fere ex diametro oppositæ sententiæ cavendæ sunt, quarum altera gravium inæqualium motum statuit ipsorum gravitatibus analogum, ut decuplo velocius moveatur illud, quod est. decuplo gravius; altera æqualem omnibus velocitatem tribuit” Casati (1684, 682).
- 117.
Since, explains Casati, surfaces relate to each other as the squares of homologus sides (in duplicata ratione laterum homologorum), while weights—like volumes—as their cubes (in triplicata ratione laterum homologorum); he simply concludes that “Non igitur velocitates, quatenus ab impedimento temperantur, sunt directe gravitatibus analogae” Casati (1684, 684).
- 118.
“Ubi autem corpora ... si vero & dissimiles figuræ, & inæquales gravitates ponantur, ex omnibus simul compositis quodammodo conflari resistentiarum Rationem facile est. opinari; sed Rationis terminos temere definire non ausim” Casati (1684, 684).
- 119.
“... spatia vero cum temporibus... sine ullo fuco veritatis color.” The last sentence is copied from Cicero’s Brutus, cap. XLIV; the translation is taken from the Loeb edition.
- 120.
“Plura huiusmodi experimenta (si te a tentando absterreat labor) suppeditabit Ricciolius tom.1. Almag. lib.9. Sect. 4. cap.16. ex quibus demum infertur velocitatis incrementa fieri juxta incrementum progressionis Arithmeticæ numerorum imparium ab unitate incipientis 1.3.5.7.9.11.13, &c. adeo ut, si quod spatium primo momento percurritur, statuatur ut 1, triplo velocius moveatur corpus grave descendens in secundo momento, quintuplo velocius in tertio, septuplo velocius in quarto, atque ita deinceps” Casati (1684, 686–687).
- 121.
As Domenico Meli explains, Riccioli claimed that “if the earth rotated, falling bodies would move with a uniform speed along a circular arc and therefore they would hit the earth regardless of their height of fall” Meli (2006, 134).
- 122.
“Incrementa hæc velocitatis ex concepti impetus incremento desumenda esse nullus dubito; sed operosum videri posset augescentis impetus causam exponere. Cum junior Aristotelem interpretarer, & primas curas hujusmodi rerum contemplationi impenderem, hanc excogitavi hypothesim; videlicet impetus producti diuturnitatem maximam duobus tantum momentis circumscribebam, ita ut primo momento oriretur, secundo æqualem sibi impetum gigneret, in quo superstes esset, tertio periret” Casati (1684, 687–688).
- 123.
“Hinc corpus descendens primo momento primum habebat impetus gradum a gravitate productum; secundo momento primus ille gradus alium gradum gignebat præter eum, qui a gravitate tunc oriebatur; quare duo novi gradus cum uno antiquo tres gradus constituebant. Tertio momento primus gradus peribat, duo secundi gradus duos pariter producebant, & gravitas suum tertium gradum; quare quinque gradus erant. Quarto momento duobus secundis gradibus pereuntibus, tres gradus tertio momento producti reliqui erant, & sibi tres alios gradus addebant, quos producebant, atque gravitas suum quartum gradum efficiebat, ut in universum essent septem gradus. Ex his septem quinto momento peribant tres tertij gradus; quatuor reliqui item alios quatuor adjiciebant quinto gradui a gravitate proficiscenti, & erant novem” Casati (1684, 688).
- 124.
“Huic hypothesi cum Phenomeno optime conveniebat, & ea. statuebatur impetus intensio, quæ velocitati efficiendæ par esset, servata incrementorum Ratione, quæ ex iteratis experimentis innotuerat. Verum commentitia, & fabulæ proxima videbatur tam brevis impetus vita, quam non nisi duo momenta metirentur: in iis sane, quæ vi externa moventur, & longius projiciuntur, aut in gyrum aguntur, licet extincta effectrice causa impressus impetus diutius permanet; quidni & impetus sponte sua conceptus, suæque origini cohærens aliquandiu perseveret? quippe qui aut ejusdem, aut saltem non deterioris naturæ censendus est” Casati (1684, 688–689).
- 125.
- 126.
“Quare impetum a gravitate descendente productum, ex tam expedito interitu vendicandum, & virtute se novo impetu augendi spoliandum, longe probabiliore conjectura censui” Casati (1684, 689).
- 127.
“Singulis momentis, quibus potest descendere, singulos impetus gradus æquales prioribus adjicit, adeo ut, quot momenta motum metiuntur, tot gradus impetus postremo momento intensionem constituant, cui motus velocitas respondeat. Velocitatum igitur incrementa fiunt juxta naturalem numerorum progressionem 1.2.3.4.5, &c. nam juxta hanc eandem seriem impetus, velocitatis causa, augetur, singulis gradibus in singula momenta additis” Casati (1684, 690).
- 128.
“Habemus experimento globum argillaceum unciarum octo percurrere uno minuto Secundo horæ pedes 15, & duobus Secundis pedes 60, hoc est. spatium quadruplum, & quia tempora sunt ut 1 ad 2, spatia sunt ut quadrata, scilicet ut 1 ad 4” Casati (1684, 691).
- 129.
We can easily derive the “formula” which Casati uses from the arithmetical series sum ∑ = n(a1 + a2)/2. Casati calls this sum “summa mimimorum spatiorum.”
- 130.
“Ponamus in uno Secundo esse momenta 10,000; sunt igitur ultimo momento 10,000 gradus velocitatis similes & æquales primo gradui primi momenti, & spatium ultimo hoc momento decursum, ad spatium primi momenti est. ut 10,000 ad 1. Coge igitur in summam omnia spatia incipiendo ab unitate usque ad 10,000, videlicet ultimi termini dimidiato quadrato adde ejusdem ultimi termini semissem, & prodibit omnium spatiorum summa. Ultimi termini 10,000 quadratum est. 100,000,000, cui adde ipsum ultimum terminum; & hujus summæ medietas 50,005,000 est. summa minimorum spatiorum, quibus conflantur pedes 15. In duobus Secundis erunt momenta 20,000, & similiter invenitur summa 200,010,000 minimorum spatiorum, quibus constant pedes 60” Casati (1684, 691–692).
- 131.
“Non sunt quidem duæ hujusmodi summæ hic inventæ 50,005,000 & 200,010,000, omnino ut 1 ad 4, sed ut 1 ad \( 3\frac{49995}{50005} \): verùm tantula differentia \( \frac{10}{50005} \) quid officit allato experimento? an potuit observari?” Casati 1684, 692).
- 132.
“Cum itaque constituta hypothesis incrementi spatiorum, velocitatis, atque impetus juxta seriem naturalem numerorum sit naturæ consentanea, neque Physice repugnet experimentis, non debemus esse solliciti, ut aliam quæramus hypothesim ad statuenda incrementa exquisite juxta numeros impares” Casati (1684, 693).
- 133.
“... lapis, dum libere descendit, producit successive novum, & novum impetum; qui, dum obtinet intentum movendi lapidem deorsum, non destruitur, adeoque ex nova additione impetus, fit maior veluti cumulus, seu impetus intensior; adeoque causa velocioris motus: Sed hoc ipsum quomod fiat; deinde explicabimus” Eschinardi (1684, 3).
- 134.
“Quaeritur nunc ratio a priori huius incrementi; seu qua methodo praecisa utatur natura in hoc negotio. Putant aliqui singulis instantibus produci in singulis partibus materiae aequalem gradum, seu particulam impetus... Sed si hoc est., videtur res processura non per numeros solos impares; sed per omnnes sic 1,2,3,4,5 & c., ut bene ostendit Mousnerius ex P. Fabry pag. 105 de Motu Locali” Eschinardi (1684, 20).
- 135.
“Placet tamen sequi sententiam Galilei... intelligando tamen de partibus temporis physicis non mathematicis” Eschinardi (1684, 21).
- 136.
“Iuxta mensuram impetus est. velocitas, & iuxta velicitatem fiunt spatia (nam spatium pendet totaliter a velocitate); velocitas autem non habet aliam causam immediatam, quam impetum” Eschinardi (1684, 21).
- 137.
“Impetus producitur veluti per punctum fluens ita ut sicut fluit tempus; ita fluat impetus productio; ac proinde per dictam productionem sit linea, vel potius superficies, quod semper intelligas” Eschinardi (1684, 21).
- 138.
“Sed quia impetus productus conservatur, quando obtinet intentum motus, ut in praesenti casu; ideo in praesenti figura [Figure 2] sit AB tempus, & simul productio impetus aequaliter fluens, ac tempori correspondens; lineae vero transversae D1. E2 &c. sint impetus hic, & nunc existentes eo instanti, quo grave consideratur in ipso descensu, nempe totidem, quod fuerunt hactenus producti; ita ut D1 sit aequalis ad A1 &c.; quod si in singulis quodammodo punctis concipiantur dictae linae transversae: imo & fiat tota superficies, seu triangulum AD1, AE2 etc. impetuosae pro toto illo tempore impressae ab impetu non solum producto; sed conservato successive, figura correspondebit rei ipsi” Eschinardi (1684, 21).
- 139.
“Adverte ne concipias lineam AB supradictam significare spatium; sed solum tempus; alioquin pro spatio deberet esse longior &c” Eschinardi (1684, 21).
- 140.
“Posita dicta figura, habes tempus, & productionem impetus in linea AB; totum vero impetum, tam productum, quam conservatum ab initio usque adhuc.”
- 141.
That is, from rest.
- 142.
Each represents time in Figure 2.
- 143.
“Nempe primo spatia facta a gravi deorsum libere descendente sunt in ratione duplicata temporum: seu sicut quadrata temporum, in quibus fiunt, inter se, & incipiendo semper ab initio. Probatur, nam sequuntur proportionem superficiei triangulris descriptae a toto impetu, tum successive producto; tum successive conservato; sed istae partes communicantes trianguli, incipiendo semper ab initio A sunt ut quadrata, seu in ratione duplicata temporum, ergo &c. Probatur, nam sunt in ratione duplicata lineae A2 ad lineam A1 ob similitudinem triangulorum communicantium AD1 & AE2 &c” Eschinardi (1684, 22).
- 144.
“2. Singula spatia facta in descensu, crescunt per numeros impares; nam primo in triangulo AE2 cum hoc sit quadruplum trianguli AD1, trapezium DE21 est. 3 & reliquum AD1 est. unum; deinde triangulum AF3 est. novem; ergo trapezium EF23 additum est. 5 etc” Eschinardi (1684, 22).
- 145.
See Damerow et al. (2004, 240–242).
- 146.
In other words, Fabri can easily show (and has indeed shown) that consecutive rectangles are to each other as the corresponding spaces traversed.
- 147.
- 148.
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Elazar, M., Feldhay, R. (2018). Jesuit Conceptions of Impetus After Galileo: Honoré Fabri, Paolo Casati, and Francesco Eschinardi. In: Feldhay, R., Renn, J., Schemmel, M., Valleriani, M. (eds) Emergence and Expansion of Preclassical Mechanics. Boston Studies in the Philosophy and History of Science, vol 270. Springer, Cham. https://doi.org/10.1007/978-3-319-90345-3_10
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