Abstract
In this chapter I consider De Volder’s treatment of the idea of material substance and of the problem of its source of activity, i.e. of the cause and laws of motion. These issues dominated not only his correspondence with Leibniz, but also his natural-philosophical disputations and handwritten texts, in which he aimed at not reverting to the metaphysical idea of God in accounting for the physical problem of the movement of bodies. This notwithstanding, De Volder never arrived at a solution for the problem of activity, as he did not accept either Leibniz’s views on substance or Malebranche’s occasionalism, while at the same time he accepted the idea that the quantity of movement is not necessarily conserved in the world. His detachment of physics from metaphysics, moreover, is observed in his treatment of the problem of the cohesion of bodies – traced back by Descartes to the metaphysical idea of rest, and which De Volder solved by experimental means (viz. by considering the pressure of air). In turn, it was by experiments that De Volder came to accept the correct laws of impact for elastic and non-elastic bodies, already formulated by Huygens, Mariotte, Wallis and Wren.
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Notes
- 1.
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“In huius autem substantiae essentiam duplici methodo possumus inquirere. Una, […] supponendo nimirum corporis essentiam consistere in quocumque demum libuerit modo id ipsum in suo conceptu nullam involvit obscuritatem, et omnia rerum phoenomena ex hac posita et assumpta corporis natura deduci queant. Altera, inquirendo quidnam illud sit, quod commune sit omnibus iis substantiis quae vel mediate vel immediate sensus nostros afficiunt,” De Volder 1671a, thesis 5; as to Descartes’s Meditationes, cf. AT VII, 31. Please note that in commenting upon Descartes’s Meditationes, De Volder referred to Descartes’s Principia as to the explanation of the essence of corporeal and immaterial things: see Warsaw dictata, 36v.
- 4.
De Volder 1671a, thesis 19. Cf. Hamburg 273, on I.53, claiming that from extension as primary attribute (viz., the attribute which is primary according to our way of conceiving it, see Sect. 3.1.3, De Volder on substance monism) one can infer its substantiality, because it cannot be reduced to anything else, and – with an anti-Spinozist stance – that no substance can have more than a primary attribute: “[r]eferuntur. Hac in parte quia plurimi solent falli id accurate explicandum: id solet dici quo plura in re sunt attributa, eo rem esse clariorem, verum attributa cum re concipiuntur, aut non: hinc {q[uae]} attributa ex rei natura sequuntur non differunt ab ipsa rei natura, hinc[que] latet {…}, quod voluit proprietates diversas esse a natura rei, verum concipiamus corpus, sed per eius attributa, quod si extensum sit, quod divisibile, quod impenetrabile sit, quidnam attributum quaeritur ex his rei naturam constituat, recte sequis quaerat cur sit corpus? Ratio est quia extensum est, eadem ratio est, si q[uae]ras, cur divisibile, cur impenetrabile sit, si vero quaerat cur moveatur, recte causa externa q[uae]renda est, quia nunc respiciatur cum re, ut diximus, non possunt separare a et sunt eadem cum ipsa: sic idem est sive dicam corpus extensum aut divisibile esse, aut impenetrabile, quia haec omnia reciprocantur inter se, qui tamen inter res, inter quas nullum discrimen est, mens solet {…} distinctionem facere, hinc mens facit differentiam quod unum ex attributis assumat pro primario ex quo reliqua omnia deducit: sic extensionem assumimus pro primario attributo corporis, licet in corpore non differat ab impenetrabilitate aut divisibilitate, sed solum in nostro ratiocinio, similiter sive [triangulum] concipiam tanquam figuram 3 laterum, sive trianguli aequales sunt duobus rectis […]. Manifestum. Rem hanc explanat mentis et corporis exemplis, quicquid in iis animadverto est extensio aut cogitatio extensionem in corpore sumo pro primario attributo, ex quo alia deduco, dum itaq[ue] ex eo omnia deducimus et per se concipiamus, sequitur esse substantiam, eodem modo cogitatio concipitur se sola nihil in ea involvitur praeter cogitationem hinc eodem modo substantia est, non quod actionem continuam sed potentiam involvat, itaq[ue] cogitatio et res cogitans extensio et res extensa sunt eadem: id itaq[que] asserimus, non aliud quid esse in re cogitante rem non aliud cogitantem, si enim id quis dicat de re nihil intelligi, si enim duo velit concipere. R[espondeo] quid eis id, quod vocat rerum? Si idem intelligit, quod intelligit cum dicit esse rem cogitantem duo diversissima confundit,” Hamburg 273, 43–45. Notably, De Volder conceives mind or res cogitans not as a perpetual act of thought, but as a faculty: see Sect. 3.1.3.1, The relation between material and immaterial substances. In the Hamburg 274 series, De Volder restates his explanation of the supposed primacy of one attribute over the others (which depends only on our ways of considering them): Hamburg 274, 20–21. On the idea of matter and body in early modern philosophy, with particular attention to the Cartesian discussions, see Garber 1992, chapters 3–5; Woolhouse 1994; Slowik 1996, 2002, chapter 4; Des Chene 1996, chapters 4–5; Garber et al. 1998; Gaukroger 2002, chapter 4; Lennon 2007; Normore 2008; Smith 2010; Anstey and Jalobeanu 2011; Pasnau 2011, chapters 4 and 8; Zepeda 2014.
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“Unica enim efficiens omnium quae in corpore observantur mutationum causa motus est. Nam is nisi adsit, nullum inter haec vel illa corpora dari discrimen, ullam iis accidere vel levissimam mutationem ne cogitare quidem licet,” De Volder 1671b, thesis 2. See also theses 3 and 6–7; cf. Descartes’s Principia, II.23. On the Cartesian theories of motion, see Gabbey 1971, 1980; Garber 1992, chapters 6–8, 2001, 2002, 2006; Slowik 1996, 2002, chapters 3 and 5; Des Chene 1996, chapters 2 and 8; McLaughlin 2000; Lennon 2007; Anstey and Jalobeanu 2011; Thomas 2015.
- 6.
“[…] manifestum est, quod motus localis loci, h[oc] e[st] situs, in quo natura loci consistit, inferat mutationem. Nonne enim, cum corpora moveri concipio, concipio distantiam, quae inter haec et alia corpora intercedit continue mutari? Quid autem aliud est distantiam mutari, quam mutari situm? […] Situs […] cum relationem sive ordinem quendam inter diversa corpora, sive inter diversas unius corporis partes designet (quis enim situs unico corpori in partes non discreto nisi aliorum corporum respectu tribui potest?) […]” De Volder 1675, theses 1–2.
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De Volder 1671b, thesis 6.
- 8.
See Burgersdijk 1626, 46: “[s]itus est ordo partium inter se. […] Etsi autem situs mutari non possit sine motu locali, qui est motus ad ubi, non est tamen situs cum ubi confundendus. […] Situs alius naturalis est […] alius voluntarius: ut sedere, stare, iacere, pronum, supinum esse, et c.” For the Conimbricenses situs relates more strictly to locus: “[u]no modo situs dicitur ordo partium in loco, quo pacto, sedere, stare, […]. Altero modo definitur situs ordo partium in toto, nulla videlicet ratione loci habita,” Conimbricenses 1604, 134. On Scholastic and Cartesian ideas of place, see Des Chene 1996, chapter 8; Schmaltz 2002, 54–64; Ariew 2011a, 87–94.
- 9.
See Categoriae, 1b25–2a4, 6b11–14 and 11b10–14. On the idea of situs in Aristotle and the Aristotelian tradition, see Almeida 2014.
- 10.
See Burgersdijk 1626, 26.
- 11.
See AT VII, 43: “situm, quem diversa figurata inter se obtinent, et motum, sive mutationem istius situs.”
- 12.
Descartes 1982, 51.
- 13.
- 14.
“Situs autem cum relationem sive ordinem quendam inter diversa corpora […] designet, […] facile poterit contingere, ut unum idemque corpus respectu variorum, ad quae refertur, situm simul immutet et non immutet. Quibus consectarium est, nisi praecise magis hunc determinemus situm, unum idemque corpus posse simul moveri et quiescere. […] Quod ut effugiamus incommodum, quid superest remedii, quam ut ex innumeris locis, quae ratione situum diversorum uni eidemque corpori tribui queunt, unum eligamus, cuius mutationem motum vocemus, quique talis sit, ut non possit simul mutari et non mutari? Quis autem, quaeso, locus, situsque inter externa corpora erit huic rei magis conveniens quam is, quo refertur hoc corpus, de cuius motu agitur, ad ea corpora, quae ipsi proxime sunt contigua?” De Volder 1675, theses 2–3; cf. Descartes’s Principia, II.29–30 and II.62.
- 15.
In commenting on II.13, for instance, De Volder underlines that a fixed frame of reference is needed because one cannot determine the situs of bodies, if these are in continuous movement with respect to each other: Hamburg 273, 86. In the other series of dictata, he claims that if situs is changed, also place (locus) has to change, against the ‘vulgar’ way of speaking: Hamburg 274, 36. In commenting on the articles from 24 onwards in the Hamburg 273 series, De Volder criticizes the obscure definition of movement given by the Aristotelians (as actum entis in potentia, quatenus in potentia), as well as the ‘restriction’ of the idea of movement to that of locus, which makes both obscure (II.24; see Hamburg 273, 94); he approves Descartes’s definition of movement as transference, and remarks that the differentiation between parts and bodies is given by air (II.25: see Hamburg 273, 94); on II.26, see infra, n. 22; he remarks that movement and rest are modes (II.27; see Hamburg 273, 97). More noteworthy is that he claims that finding a frame of reference solves the problem of the linguistic ambiguity arising from attributing both movement and rest to the same body. The choice of this frame of reference is justified because, from it, two bodies cannot be both in movement and at rest with respect to each other, such as in the case of the internal movements of the mass of air, composed by moving particles (II.28–30; see Hamburg 273, 97–98). In commenting on II.31 (where Descartes explains how a single body participates in many different movements, like a clock on a boat), he notes that this “rule” had been more extensively explained by Gassendi and Galileo: Hamburg 273, 98.
- 16.
“[…] determinatam naturam non habeat, siquidem hoc pacto mutationem loci eodem tempore fieri, et non fieri, necesse etiam erit eodem tempore concipere eamdem rem moveri, et non moveri, quod si fit patet motum non haberem determinatam quandam naturam, quippe hoc si foret, non posset eodem tempore eadem de re et negari et affirmari, sed consistere eius naturam in quidam respecti, qui {exque} ac loci dependet a nostra cogitatione,” Hamburg 274, 39.
- 17.
Notably, De Volder remarks that Descartes used the ‘ambiguous’ idea of locus, because the idea of situs entails the relativity of motion: “vicinia corporum contiguorum. Author in hac definitione […] conatur, non relativam sed absolutam motus determinare naturam, utitur termino ambiguo loci, quia varie potest sumi, {mavult} locum circumscribere, eum {q[uod]} determinare, non per situm, ut antea, quia diverse poterat considerari pro diversis corporibus ad q[uos] refertur, sed per viciniam et {uidem} non quorum cum {q} corporum (haec enim modo, in idem quod evitare vult incideret incommodum) sed tantum contiguorum, {q[uae]} corpora contigua, cum {una} tantum sunt, nec possunt varie intelligi. Sequitur unius corporis unicum tantum fieri motum,” Hamburg 274, 40.
- 18.
“Quia tanquam quiescentia spectantur. Haec verba author addidisse videtur, non q[uo]d philosophicam involvant veritatem, sed ne nimium a loquendi consuetudine deviaret, si enim motus tantum esset translatio, et ea translatio, ut manifestum est, sit reciproca, patet eandem mutationem fieri in corpore CD cum ex eius vicinia transfertur corpus AB {q uitur} corpore AC ac proinde si tantum ad absolutam motus naturam attendamus, tantum erit motus in uno corpore quam in alio, et tantundem actionis in uno, q[ua]m in alio, quod quidem, ut verissimum est, ita minimum recederet a communi loquendi lege. {Cogeremu…} enim fateri minus corpis, ut demonstratum in sequenti, posse eodem tempore contrariis moveri motubus, quod prima fronte videtur {…}, quamquam si explicemus, non {…}; verumtamen cum rationi sit consentaneum, ut nos quantum veritas patiatur, accomoderemus etiam vulgari loquendi modo, addit author haec verba q[uae] tanquam quiescentia spectantur. Hoc ipso non obscure invicem posse corpora pro arbitrio spectari quiescientia, ita nempe ut si fiat {qui…dam} maior propinquitas, inter A et B, quam antea fuit, mei arbitri sit res, utriumq[ue] corpus dicere moveri, utriumq[ue] quiescere, quicquid enim reale est in A, quod ad motum spectat, esse in eodem modo in B, et quicquid est in B, etiam est in A. It ut pro arbitrio possum B spectare tanquam quiescens, et sic movebitur A, vel A tanquam quiescens spectare, et sic movebitur B, quo pacto tollitur illa in modo loquendi absurditas,” Hamburg 274, 40–41.
- 19.
See Burgersdijk 1626, 133; Burgersdijk 1652, 16–17.
- 20.
This idea was maintained first by Averroes: see his commentary to Physica, book 5: Aristotle and Averroes 1562–1574, commentary 3, 208D–F. Amongst De Volder’s sources, listed in the Bibliotheca Volderina, see Sennert 1651, 143: “[m]otui opponitur quies, quae est privatio motus eius, quod est aptum natum moveri; et quatenus propter quietem motus fit, potest etiam dici, quod sit finis et perfectio motus,” and Plempius 1654, 440: “[r]ogabis: quomodo statui possit, quietem a facultate fieri, cum et illa et omnis alia quies sit privatio motus: privatio autem non habet causam positivam. Respon. quies dupliciter potest sumi: uno modo prout est privatio motus, altero prout est detentio et permanentia, vel etiam praesentia in loco; et sic est quid positivum, habetque positivam causam: quo posteriori modo hic sumitur quies.” Johannes Clauberg, in turn, would combine the idea of rest as a positive permanence in a place with Descartes’s replacement of locus with situs, as he would define rest as permanence in a situs in his Disputationes physicae (held in 1656–1661): “[q]uies est permansio in eodem situ. […] Permansio enim positivum quid est, […] situs denique positivum,” Clauberg 1691, Disputationes physicae, 146.
- 21.
On the idea of rest in Descartes, see Schmaltz 2015.
- 22.
“[N]on arduum erit exterminare insitum illud plurimis philosophorum […] animis praeiudicium, multo plus actionis in motu esse quam in quiete, plusque requiri virium et efficaciae in causa, quam motum corpori, quam in ea quam eidem quietem conciliare oportet. […] Hosce autem interrogatos velim quidnam ipsos moveat, ut credant quietem tantum esse motus privationem? Nonne hoc ipsum quod plus actionis supponant in motu, quam in quiete? […] Sed considerent velim hoc non tam a rei natura, quam a nostro considerandi dependere modo. […] Demonstrandum itaque est, tantundem exigi efficaciae, in ea causa, quae corpus motum reducet ad quietem, quam in ea, quae corpus quiescens ad eundem gradum deducet motus. Corpus etenim movetur, causa igitur externa est, quae movet, cum ne dissidentibus quidem adversariis, nihil moveantur a se ipso. Quaecunque igitur causa, quae corpus motum ad quietem reducet, tollere necesse habet vim et efficaciam illius causae, quae confert motum,” De Volder 1675, theses 6–7 and 9. In the Hamburg 273 series of dictata, commenting upon II.26, several arguments are provided to support this theses, such as that the same action is needed for movement and rest, as one is the negation of the other; that the persistence in a state of rest is an aliquid, and it requires a cause like movement does; that both movement and rest do not follow from the essence of matter, so that they both need an external cause: see Hamburg 273, 94–95. The latter argument is provided also in the other series of dictata (on II.26): see Hamburg 274, 40. More arguments for this theses are provided in a 1684 disputation De motu, which I discuss in Sect. 4.2.3.2, A ‘purely relational’ standpoint on impact: the 1684 disputation De motu.
- 23.
The notions of hardness and solidity, as I am going to show in Sect. 4.2.1.1, Descartes’s laws of motion and idea of force, are not the same concept in Descartes’s theory.
- 24.
Descartes 1982, 23. Cf. the original versions: “[p]er substantiam nihil aliud intelligere possumus, quam rem quae ita existit, ut nulla alia re indigeat ad existendum,” AT VIII-A, 24; “[l]orsque nous concevons la substance, nous concevons seulement une chose qui existe en telle façon qu’elle n’a besoin que de soi-même pour exister,” AT IX-B, 47.
- 25.
Cf. the Latin version: “quod sint res, quae solo Dei concursu egent ad existendum,” AT VIII-A, 25.
- 26.
“[I]l faut seulement, pour entendre que ce sont des substances, que nous apercevions qu’elles peuvent exister sans l’aide d’aucune chose créée,” AT IX-B, 47.
- 27.
“Per se subsistere nihil aliud est, quam non esse in aliquo, ut in subiecto. Ut ergo substantia dicatur per se subsistere, non est necesse, ut ab alio non pendeat: sed hoc tantum requiritur, ut ab alio non pendeat, tanquam a subiecto,” Burgersdijk 1626, 12.
- 28.
“[…] si enim per substantiam velimus intelligere {…} nulla alia re indigeat ad existendum,” Hamburg 274, 19.
- 29.
“Per substantiam. Conceptus substantiae formatur tantum per nostro inadaequatos concipiendi modos […], videmus sane omnia Dei attributa plane diversa esse ab attributis rerum finitarum, nec posse communem formari conceptum substantiae quatenus competat Deo, et quatenus competit creaturis sive rebus finitis,” Hamburg 274, 19.
- 30.
“Substantia est, {q[uod]} nullius ope existit, {q[uod]} sua natura habet fundamentum existentiae, ergo unica dabitur, {q[uae]} est Deus, […] qui sua natura est quod est,” Hamburg 273, 41.
- 31.
“[…] non enim maior differentia est intra vitam nostram et aeternitatem, quam inter no[str]am et Dei voluntatem, ita ut certum sit, aut nulla attributa communicari, aut omnia, vero si verum sit omnia Dei attributa esse eius naturam, hinc sequitur si unum nobis communicetur attributum, communicari essentiam i.e. nos esse Deos,” Hamburg 273, 42. In commenting upon I.35, De Volder clarifies that Descartes’s idea that will is ‘infinite’ was subjected to many “misrepresentations,” and clarifies Descartes’s position by claiming that will is more extended as intellect because intellect works only on its own data, whereas will can combine them, by affirmations and negations: Hamburg 273, 29. In the other series of dictata, De Volder claims that anyway will cannot go beyond the data of the intellect: Hamburg 273, 15–16.
- 32.
See De Volder 1681, theses 46 and 48: “formis omnibus licet diversissimae naturae, id tamen competere, quod dent rebus hanc illamve essentiam, atque hac ratione omnem materiam ad unicum genus, omnesque formas ad unicum formarum conceptum reduci. Sed quis non videt id ipsum admodum metaphysicum sive logicum esse, nec ex physicis principiis aut phaenomenum natura, sed ex abstractis et si ad res referantur, phantasticis logicorum notionibus, desumptum esse? […] Nec dissimili ratione in rebus incorporeis progredi licebit. […] Habes […] formam. Qui igitur conceptus cum non sint corporibus peculiares, quis non videt eos non ex physicis sed ex generalibus quas de rebus formamus, hoc est logicis, metaphysicisve notionibus desumtos esse? […] Profecto non sine causa Verulamius frequentissime conqueritur, Aristotelem, quem hac in parte longe antecedunt Scholastici, logicis suis notionibus, et abstractis de rebus conceptibus totam penitus philosophiam pervertisse.” Cf. Bacon’s Novum Organum, part 1, aphorisms 14–15; cf. De Raey’s Specimen logicae interpretationis (a series of disputations held in 1669–1671), referring to aphorisms 2, 20–21 and 97: De Raey 1692, Specimen logicae interpretationis, 536. On De Raey’s criticism to Aristotelian logic and metaphysics, see Strazzoni 2012, 2015.
- 33.
“Sic autem videbimus illis utrisq[ue] commune esse, quod solo Deo ad existendum indigeant, in quo sane {differunt} a modis, qui praeter Dei concursum requirunt substantiam, in qua insunt, ita ut {…} substantiam definiri posse per id quod sit res solo Dei concursu egere ad existendum. Possunt autem. In gallico textu clarius. Verum cum inter res creatas quaedam sunt, {quae} non possunt existere sine aliis […],” Hamburg 274, 20. De Volder owned the 1651 edition of the French version of Descartes’s Principia: see Bibliotheca Volderina, 6.
- 34.
“Tentemus tamen, an ex hisce tenebris quidpiam lucis possimus haurire. Aiunt itaque ante cuiusque rei productionem formas non existere, quippe quae producuntur generanturque, quod certe, si iam adessent, fieri non posset. Neque tamen prorsus non existere, ne natura quae producit formam ex nihil quidpiam faceret, sed esse potestate in materia et naturalium causarum efficacia de illius potentia et quasi gremio exire, et ad actum perfectionemque deduci. Conimbr. in Phys. l. I, c. IX [quaestio XII] art. IV. Quis quaeso haec conciliet? Formae ante rei productionem non sunt, neque tamen prorsus non sunt. Quae profecto, si ulla quaepiam, manifesta est contradictio, nec tamen difficultatem solvit. Restat enim explicandum, quo pacto naturalium causarum efficacia ex gremio materiae educantur formae. Quas si accidentia esse demus, plana sunt omnia. Quis enim non facile percipiat, motum, unicam illam mutationum omnium in corporibus causam, capacem esse, ut producat varias in materia figuras, varium ordinem, varia deinque accidentia, quae huius illiusve materiae formas constituant?” De Volder 1681, thesis 85. Cf. Conimbricenses 1592, 205–206. Cf. Gorlaeus 1620, 267, and Basson 1621, 159, referred to in Voet 1643, thesis 4. On the Cartesian and pre-Cartesian rejection of substantial forms, see Van Ruler 1995; Lüthy 1997, 2012, 44–49; Hattab 2009, parts 1 and 3; Manning 2012.
- 35.
“[…] omnis quid est a se solo, aut se solo indiget ad existendum, estq[ue] Deus, aut ind[ig]et quidem alio, sed nulla praeter Deum quales sunt creatae, aut praeter Deum alio {ad…} indiget, quales sunt modi, et accidentia: res plana est in exemplo concipiamus corpus id nil indiget ut existat praeter causam efficientem quandam a se diversam, sed cum corpus se solo concipitur modo producatur, hinc existere potest sola ope Dei,” Hamburg 273, 42.
- 36.
“Sed si concipiam figuram corporis, haec concipi potest existere et non existere, hinc sequitur non existere per se, sed per aliud, {restum itaq[ue]} est non existere posse nisis restat corpus, cuius figura sit: eodem modo res se habet circa motum, et quietem, corpus potest existere sine motu, non vice versa motus sine corpore: corpus unicam tantum extensionem requirit, motus vero duas, et corpus et causam efficientem corporis,” Hamburg 273, 42.
- 37.
“Existendum. Ut enim corpus in suo conceptu involvit unicam, simplicem et individuam naturam […], idq[ue] est quod solo Deo eget, quid enim evidentius, quam id quod {possum} separatim concipere etiam existere posse seorsim, unde si possum concipere solam extensionem, solam cogitationem, cum illi conceptibus nil praeterea involvant, sane se solis possunt existere modo producantur,” Hamburg 273, 42.
- 38.
See Sect. 3.2.2.2, De Volder on the ‘truth’ of Cartesian principles.
- 39.
De Volder claims that Descartes was not entirely clear on this point: yet, his idea of extension entailed that real division is impossible: “[s]ed de eo non multum laborabit Cartesius, utrum corporis natura in extensione ponatur, an vero in eo, quod habeat partes a partibus distinctas. Existimabit enim, nec opinor iniuria, dicere corpus extensum esse, vel corpus habere partes a partibus distinctas, esse duobus diversis loquendi modis, rem unam dicere eandemque. Dico autem partes a partibus distinctas, non vero […] partes a partibus secretas et divisas, ne forte hic loquendi modus fucum alicui faciat, qui distinctas partes etiam in continuo et indiviso corpore facile intelligi putet, non ita vero partes a partibus secretas et divisas,” De Volder 1695, De corpore, thesis 21. See also infra, n. 47. On the uniqueness of substance according to Descartes, see Lennon 2007; Smith 2010; Thomas 2015; Slowik 2001.
- 40.
See Sect. 4.1.1, De Volder’s Cartesian standpoint on the ontology of physics.
- 41.
As seen in Sect. 3.1.3, De Volder on substance monism, indeed, Leibniz noted that by this criterion of substantiality only God is a substance: “[v]erum ut ad alterum veniam circa quod monere aliquid volebam, applicationem nempe ad substantiam, nonne etiam ad substantiae conceptum indigemus attributis? quodsi exigas, ut saltem non alia indigeretur substantia, circulum committeremus in definiendo. Deinde praeter primam substantiam nulla est (credo) quae per se possit concipi,” Leibniz to De Volder, 31 December 1700, in GP II, 223. In turn, De Volder maintained that we can conceive created substances by themselves, and without conceiving their cause, because this cause is God and falls beyond our comprehension: “[q]uod de causa substantiae dicis, existimem si nullius substantiae essentia concipiatur absque causa eius possibili, nullius essentiam concipi. Modorum siquidem causas possibiles concipere mihi videor, sed substantiarum nullas. Hinc vulgo eas dicimus creari, hoc est, habere causam a qua producantur, sed modo nescio quo,” De Volder to Leibniz, 7 October 1701, in GP II, 229. Leibniz did not reply further on this point. For a thorough discussion, see Lodge 2013, Introduction.
- 42.
See Sect. 3.1.3, De Volder on substance monism.
- 43.
See the letter of De Volder to Leibniz of 18 February 1699: “[e]xtensio[…], si quidpiam, per se concipiatur, hoc est, ita concipiatur, ut unum quid conceptui repraesentet […] maioris laboris est partes realiter distinctas concipere in extensione, quam unitatem. Nam, siquidem, ut concedis, nullum detur inane, non poterit una pars, quam quis sibi fingat, concipi absque altera,” GP II, 166. In his letter of 19 November 1703, Leibniz would criticize De Volder’s monism on material substance as a form of Spinozism: “aut admittenda Tibi mea sententia est, aut confugiendum ad ἀλλόγλωσσον illud, quod subjicis: forte totum universum unam tantum esse substantiam; quod qui dicit substantiae vocem a sensu aliorum detorquet. Nec video ullum argumentum verisimile tanti paradoxi, nam quae B. de S. eo attulit, ne umbram quidem (si quid iudico) demonstrationis habent,” GP II, 257–258. De Volder would then point out that Leibniz mistook his words, in his letter to Leibniz of 5 January 1704: “[u]bi dixi forte totum universum unam tantum esse substantiam[,] expresse locutus sum de universo corporeo, secutus hac in parte communem tantum non omnium opinionem statuentium corpora non esse substantias, quatenus sunt haec vel illa Corpora, sed quatenus sunt Corpora. Quare non capio, quid te moverit, ut opinionem B. D. S. quasi similem meae hic immiscueris. Quod enim illi peculiare est, non consistit in eo quod de universo corporeo dixi, sed quod cogitationem et corpus pro eadem substantia habeat, quod mihi semper visum est absurdissimum,” GP II, 260.
- 44.
“Fateor, et iam in prioribus fassus sum, in corpore mathematico non inveniri unitates indivisibiles; sed addam simul, me haerere tamen, an non in infinita extensionis massa unitas haec reperiatur, si quidem partes, quas in hac massa concipimus diversas, realiter distingui non videntur, quoniam nulla pars aut poni aut concipi potest, nisi positis, conceptisque omnibus,” De Volder to Leibniz, 14 November 1704, in GP II, 272.
- 45.
Descartes 1982, 29.
- 46.
On this topic, see especially Slowik 2001.
- 47.
“Longitudinem. Si concipiamus ceram {quae} potest omnes formas figurasq[ue] induere, eadem manet materia, licet moveatur aut sit sphaerior eodem modo minus licet nunc his nunc illis {vocet} cogitationibus, tamen est una eademq[ue], et nulla est diversitas. Sic cogitatio et extensio haberi possunt pro modis,” Hamburg 273, 65; the other series of dictata (see Hamburg 274, 24) does not contain references to different material substances either. In his letter to Leibniz of 31 May 1704, De Volder maintained that Descartes and all those following mechanical philosophy “meant” (voluerunt) that material substance is one. See the letter of De Volder: “[u]bi dixi, universum corporeum, hoc enim expresse addidi, esse forte unam substantiam, nihil aliud dixi, quam quod Cartesius, et omnes illi, qui mechanicam philosophandi rationem secuti sunt, voluerunt, rerum omnium corporalium unicam esse substantiam,” GP II, 266. See also the quotation from De Volder 1695, De corpore, thesis 21, given supra, n. 39.
- 48.
- 49.
De Volder 1681, theses 35–36; cf. Van Helmont’s Elementa, in Van Helmont 1707, 51; Astra necessitant; non inclinant, nec significant de vita, corpore vel fortunis nati, in Van Helmont 1707, 115; Progymnasma meteori, in Van Helmont 1707, 64. These texts were originally published in his Ortus medicinae (1648). On Van Helmont, see Redgrove and Redgrove 1922; Giglioni 2000; Clericuzio 2000, 54–60, 89–96, 108–114 and 152–163; Debus 2001, 2002; Pagel 2002; Newman and Principe 2002; Hedesan 2016. De Volder’s criticisms to the ‘monist sect’ were rebutted in a disputation taking place on 22 October 1681 in Copenhagen, presided over by Nicolaus Nicolai Seerup, professor of philosophy and medicine: namely the Disputationum philosophicarum sive Cogitationibus rationalium de Disputationibus philosophicis sive Cogitationibus, uti inscribuntur, rationalibus Viri Clarissimi et Doctissimi Dn. Burcheri de Volder de rerum naturalium principiis prima qua contra eundem solidis, ut spes est, argumentis ac rationibus, aqua solum materiale omnium corporum principium adstruitur, et simul eadem assertio ab animadversionibus clarissimorum aliquot virorum vindicatur.
- 50.
“Ad secundam sectam pertinet Democritus, licet infinita agnovisse principia dicatur {Aristoteli}, cum ex eodem constet infinitatem istam non in diversitate consistere substantiarum, sed in innumeris unius eiusdemque substantiae modis. Quae principia dudum reiecta nostro demum saeculo in lucem revocarunt Gassendus, Verulamius, Cartesius, Boylaeus, et quantum est ingeniosorum hominum, qui corpusculari, ut Angli vocant, addicti sunt philosophiae,” De Volder 1681, thesis 30. As to the uniqueness of the material substance according to a mechanical perspective, see supra, n. 47.
- 51.
- 52.
De Volder 1681, theses 33 and 149–156. See esp. thesis 155, referring to Voet 1661, 71–72. Daniel Voet also admitted blood circulation: see Voet 1661, 81–83. On Daniel Voet, see Dieckhöfer 1970, 1978. Voet’s Physiologia was present, in the 1661 edition, in De Volder’s library: see Bibliotheca Volderina, 5.
- 53.
“[N]on possum non laudare mechanicam explicandi rationem, qua per figuram, magnitudinem, et motum rerum explicantur phaenomena. Agnoscere enim ex infinitorum corpusculorum figura et magnitudine discrepantium coalitu dissolutioneque omnes mutationes fieri corporeas, est, ut in sequentibus demonstrabimus, ipsam veritatem profiteri. […] Verum tamen si dicendum est id quod res est, putem tamen in hac opinione quaedam reperiri, quae merito in reprehensionem incurrant. Primo enim id me male habet, quod haecce corpuscula individua faciant, omnisque mutationis expertia […],” De Volder 1681, theses 114–115.
- 54.
“[Q]uod in hac opinione mihi non arridet, est, quod gravitate hasce atomos donet Epicurus. […] Putem enim sufficere, si ostendam hanc gravitatem non percipi, ut eandem ex principiorum proprietatibus eliminem. Huius autem erroris Epicuri haec forte erit causa, quod non putaret id ipsum sufficere, si cum Democrito motum tribueret materiae, nisi eius motus causam assignaret. Qua in re cum ad Deum verum motus auctorem confugere nollet, utpote quem impie ab omni rerum mundanarum gubernatione secludit, maluit gravitatem quandam, licet incognitam, pro causa huius motus assignare. Qua in re et a se ipso et a mechanica philosophandi via descivit non parum,” De Volder 1681, theses 119 and 121.
- 55.
“Considerabo de nihilo nullas affirmari posse proprietates, atque spatium cum habeat aliquas, etiam rem imo substantiam quampiam esse asseverabo,” De Volder 1681, thesis 127.
- 56.
“Duplices quippe […] Phys. sect l. I, § 1, fingit dimensiones, alias corporeas corporis, […] alias spatiales […]. […] ut enim huic rationi obviam iret, quod pace tanti viri dictum sit, in longe difficiliorem se coniecti labyrinthum. Hoc enim spatium statuit rem quidem esse veram, sed tamen neque substantiam neque accidens, rem esse a Deo improductam, independentem, immutabilem. […] Imo, inquit, mirum videri non debet Deum non esse authorem spatii,” De Volder 1681, theses 130 and 133; cf. Gassendi 1649, 614–619, 1658, volume 1, Syntagma philosophicum, 181–182. On Gassendi’s theory of matter and space, see Joy 1987, chapter 8; Lennon 1993, chapter 2; LoLordo 2007, chapters 5–6.
- 57.
This passage follows the quotation given supra, n. 53: “[p]rimo enim id me male habet, quod {haecce} corpuscula individua faciant, omnisque mutationis expertia. Quippe nulli divisioni natura repugnat corporis, utpote quod cuiuscunque etiam sit parvitas, cum extensum sit, partes habeat, in quas dividi possit,” De Volder 1681, thesis 115.
- 58.
See supra, n. 39.
- 59.
“Naturam divisionem admitteret, non dari tamen causam satis efficacem, quae hanc divisione sive ob parvitatem sive ob soliditatem possit instituere. Sed frustra. Nihil quippe putem manifestius, quam corpusculorum exiguitatem divisionem reddere, non difficilem. Nonne enim evidens est, ad divisionem corporum id unice requiri, ut pars, quaedam moveatur reliqua immota, aut certe ut moveatur separatim a reliqua? Unde quid aliud sequitur, quam, cum facilius corpus dimidii pedis quam totius moveatur, multo etiam facilius caeteris paribus dividi bifariam id corpus, quod pedem, quam quod duos exaequat?” De Volder 1681, thesis 115.
- 60.
In his 1695 Exercitationes De Volder clarifies how, according to Descartes, hardness (durities) is properly speaking the sensation that results from the state of rest of parts of the body: De Volder 1695, De corpore, thesis 33. On the concept of solidity, which in fact means more than the mere resistance to division, and which has an important role in cosmology and theory of gravity, see Sect. 6.1.3, Descartes’s theory of planetary stability, weight and pressure. In the present section, I will take solidity and hardness as synonyms of resistance to division.
- 61.
In a Cartesian cosmos, such are the globules of second matter, which are perfectly continuous, internally. This idea can be found in the disputation presided over by De Volder, but authored by the respondens (Paulus von der Lahr), De absoluta quiete (1684): the author maintains that simple bodies are those which are not discrete, namely, they are pure, continuous volumes of extension i.e. they are internally motionless, even if with respect to other bodies they change their distances or situs. Absolute rest can be found in the particles or spheres constituting Descartes’s second matter, which are the most solid bodies, in which rest is their glue (thesis 12). This disputation may have been an exercise against Boyle’s Discourse about the Absolute Rest in Bodies (published in the 1669 edition of his Certain Physiological Essays and Other Tracts), and then translated into Latin and published in Amsterdam in 1671 as De absoluta quiete in corporibus, where Boyle maintains that according to the Cartesian cosmological hypothesis, based on the idea that subtle matter flows in all bodies, no absolute rest can be found in nature.
- 62.
“[…] consideremus lignum solidissimum firmissimumque, aut aliud quodcunque corpus ligno solidius. Illud certe vel ab hominibus dividetur, vel ab aëre longo temporis tractu tandem corrumpetur dissolveturque. Constituamus porro in hoc ligno dimidiam partem porosam, dimidiam solidam esse: nonne ut aliud nunc corpus dividam, quod totum sit solidum, quodque sit dimidium huius, eadem vis requiritur, quae ad dividendum hocce lignum; ut enim lignum dividatur, dimidia solum superficiei pars, ut separetur necesse est, quam eandem in altero corpore separare solummodo oportet,” De Volder 1681, thesis 117.
- 63.
“Deinde si putes centesimam aut millesimam solummodo partem plenam et cohaerentem, reliquas porosas et secretas esse, ea certe vis, quae lignum hocce dividit, dividit quoque corpus solidum aequale centesimae millesimaeve parti istius ligni. Unde manifeste et illud sequitur, quo corpus solidum minus est, eo facilius dividi, et ex inde licet atomi solidae sint, cum tamen sint parvulae, dari vim in natura sufficientem, quae ipsas dividat. Imo certe, si nulla solidi corporis poterit fieri divisio, nunquam fiet ulla. Sumamus enim corpus quam maxime porosum, profecto nulla hic divisio fiet pororum ratione, illic enim partes iam separatas obtinent, sed, ut manifestum est, fiet omnis ratione partium suis superficiebus cohaerentium. Ad quas superficies si respiciamus, illa corpora non porosa sed solida sunt, unde si vera est haec opinio, non licuit unamquamque superficieculam a reliqua separare, quia corpus hic solidum est nullis meatibus distinctum, Unde aut agnoscendum est, nullam unquam divisionem fieri, aut fieri solidi corporis,” De Volder 1681, thesis 117.
- 64.
“Mathematicorum effatum est qualibet potentia etiam minima quodlibet corpus etiam maximum moveri posse. Quid itaque? Si minima vis maximum movere possit corpus, nonne vis maior poterit movere minusculi corporis partem, et eo ipso motu id corpus dividere,” De Volder 1681, thesis 117.
- 65.
“[…] ne in mathematicis videar haerere, consideremus lignum […],” De Volder 1681, thesis 117. Cf. supra, n. 62.
- 66.
In his De rerum naturalium principiis De Volder refrains to deal with the nature of rest: “[n]eque vero arduum erit […] coniicere haec nobis principia praeter materiam et motum (nisi quis, de quo non laboramus, addere forte velit quietem) nulla alia esse,” De Volder 1681, thesis 161.
- 67.
In the De rerum naturalium principiis, the thesis in which the idea of rest and continuity are discussed is the 117th one, present in disputation 10 (now lost). Disputation 9 took place in 1675, while disputation 11 took place in 1676 (they are both preserved at Erlangen University Library). Disputation 10, therefore, was almost certainly coeval with experiments 1–2, taking place in early 1676 (12–13 March).
- 68.
The apparatus of Johannes Joosten van Musschenbroek was more advanced than the one used by De Volder in the 1670s (probably built for him by Samuel van Musschenbroek, who died in 1681). The use of Johannes Joosten’s model is described in Senguerd’s Rationis et experientiae connubium, chapter 19, in a long quotation from a pro gradu disputation of a student of his, Johannes Franciscus de Witte van Schooten, who “exactissime in Collegiis meis multoties observaverit, annotaverit, atque in elaboratissima disputatione inaugurali (quam pro gradu doctoratus consequendo in hac Academia publice defendit) methodice exposuerit, et ad mentem meam tradiderit.” The disputation was held in 1712, and published as Dissertatio philosophica inauguralis de solido, eiusque partium, nec non hemisphaeriorum concavorum, et cylindrorum solidorum cohaerentia, and appeared also in the Acta Eruditorum of February 1714 (as Wolferdi Senguerdi […] Annotationes circumstantiarum singularium circa cohaerentiam hemispheriorum concavorum et cylindrorum solidorum). Senguerd used different pairs of cylinders, with different diameters and materials, and disposing them either vertically or horizontally. E.g., cylinders of white marble with a diameter of 2 inches and 7 lines, 2 inches and 1 line, 0 inches and 10 lines, sustaining respectively 1,150, 900 and 200 pounds; cylinders of black marble of 2 inches and 2 lines diameter, sustaining 900 pounds, and two other pairs of ivory and brass. Senguerd noted that cylinders sustained the same weight even if disposed horizontally: see Senguerd 1715, 164, 166–167 and 171–172. The pair described by Morley (2 inches and circa
$$ \frac{1}{3} $$diameter), does not really fit with those ones described by Senguerd – who performed experimental lectures also at home, with his own instrument – which correspond to 2 inches and 4 lines, since, according to the English system, 12 lines = 1 inch.
- 69.
“Then he hang them up, as before, and into the scales he added more, et more weight, till they disunited and the one fell down with the weight which appeared to be very nigh 450 lb. […] He said Boyle could only hang on two such cylinders (being even something broader i.e. of 3 fingers breadth) 80 lb weight, which is not the 4th part of this experiment,” De Volder 1676–1677, 80v. As Boyle reports in his Defence against Francis Line (1662), in open air he could make one marble hang to the other up to a maximum of 1,300 (English) ounces only, which in De Volder’s statement are converted into 80 Dutch pounds (with a ratio of 1 pound to 16 ounces, corresponding to the Dutch (i.e. Amsterdam) commercial measurements, rather than to the English apothecary system), as due – according to Boyle – to the fact that their surfaces were not so polished: see Boyle 1662a, 84–86. In experiment 27 (as I am going to show in Sect. 5.1.2, The weighing of a separate volume of air) Morley states that De Volder followed the “Amsterdam weight” (De Volder 1676–1677, 138v), which is moreover followed in his calculations on the weight of a determined volume of air in disputation 5 of his De aëris gravitate. The same system of measurement was followed by Senguerd in his Connubium (see Senguerd 1715, Ad lectorem). Notably, De Volder’s hanging of weights of 560 and 600 pounds (probably, English ones) to cohering marbles of circa 3 inches diameter are reported also in Dalrymple 1686, 539: “[h]abeo tabulas marmoreas quarum diameter trium pollicum est, quae circiter 500 lib. sustinent. Sunt in Academia Leydensi tabulae marmoreae eiusdem fere diametri, quae uno tentamine a Doctiss. Professore Voldero demonstrato, 560 lib., in alio tentamine ab eodem ultra 600 lib. sustinuerunt.”
- 70.
Notably, one can also find the experiment with marbles in Boyle’s New Experiments (1660), as experiment 31, which Boyle notoriously performed in what was supposed to be an evacuated container: yet, even in the (supposed) absence of air, the plates of marble did not tear apart, even with an additional weight of 4 ounces appended to the lower one. Boyle justified the failure of the experiment by taking into account a leakage in the pump (Boyle 1660, 230–231). Later, he was attacked by Hobbes in his Dialogus physicus de natura aëris (1661): for a full account, see Shapin and Schaffer 1985.
- 71.
De Volder 1676–1677, 82r and 83r.
- 72.
“[A]t first glance there seems to be no reason why, for example, an iron nail (or any other body which is not large but extremely solid), cannot be divided into two parts solely by the force of our hands. For each half of this nail may be considered to be an individual body; and since one half is smaller than our hand, it seems that it ought to be possible to move it by the force of our hand and thus separate it from the other half. It must, however, be noted that our hands are extremely yielding, or closer to the nature of fluid bodies than to that of solid ones; for that reason, they are not accustomed to act as a whole against a body which they have to move; only that part of our hands which touches that body brings all its pressure to bear upon it at the same time. […] It is because this part can more easily be separated from the rest of the hand than the part of the nail from the rest of the nail, and because this separation cannot occur without the sensation of pain, that we cannot break the iron nail by means of our hand alone. If, however, in order to divide the body, we strengthen our hand by applying the force of a hammer, file, pair of cutters, or other tool to a part of the body to be divided which is smaller than the tool being used; it will be easy to overcome its hardness,” Descartes 1982, 76. Cf. De Volder 1676–1677, 82r: “[u]nto this Descartes answereth that the softness of our hands the pain we feel and our only touching said things s[ecun]dum extimam superficiem is the cause we cannot separate not so much as the least part of many solid things.”
- 73.
De Volder 1676–1677, 82r.
- 74.
De Volder 1676–1677, 82v–83r.
- 75.
“Dirò prima del vacuo, mostrando con chiare esperienze quale e quanta sia la sua virtù. E prima, il vedersi, quando ne piaccia, due piastre di marmo, di metallo o di vetro, esquisitamente spianate pulite e lustre, che, posata l’una su l’altra, senza veruna fatica se gli muove sopra strisciando (sicuro argumento che nissun glutine le congiugne), ma che volendo separarle, mantenendole equidistanti, tal repugnanza si trova, che la superiore solleva e si tira dietro l’altra e perpetuamente la ritiene sollevata, ancorché assai grossa e grave, evidentemente ci mostra l’orrore della natura nel dover ammettere, se ben per breve momento di tempo, lo spazio voto che tra di quelle rimarrebbe avanti che il concorso delle parti dell’aria circostante l’avesse occupato e ripieno. […] Tal resistenza, che così sensatamente si scorge tra le due lastre, non si può dubitare che parimente non risegga tra le parti di un solido, e che nel loro attaccamento non entri almanco a parte e come causa concomitante. […] Dal seguir dunque che fa l’inferior lastra la superiore, si raccoglie […] che pur tra le medesime piastre resti qualche vacuo, almeno per brevissimo tempo, cioè per tutto quello che passa nel movimento dell’ambiente, mentre concorre a riempiere il vacuo; ché se vacuo non vi restasse, né di concorso né di moto di ambiente vi sarebbe bisogno. Converrà dunque dire che, pur per violenza o contro a natura, il vacuo talor si conceda […]. Dirò il modo dell’appartar la virtù del vacuo dall’altre, e poi la maniera del misurarla. E per appartarla, piglieremo una materia continua, le cui parti manchino di ogni altra resistenza alla separazione fuor che di quella del vacuo, quale a lungo è stato dimostrato in certo trattato del nostro Accademico esser l’acqua: […] Ed io mercé di questi discorsi ritrovo la causa di un effetto che lungo tempo m’ha tenuto la mente ingombrata di maraviglia e vota d’intelligenza. Osservai già una citerna, nella quale, per trarne l’acqua, fu fatta fare una tromba, da chi forse credeva, ma vanamente, di poterne cavar con minor fatica l’istessa o maggior quantità che con le secchie ordinarie; ed ha questa tromba il suo stantuffo e animella su alta, sì che l’acqua si fa salire per attrazzione, e non per impulso, come fanno le trombe che hanno l’ordigno da basso. Questa, sin che nella citerna vi è acqua sino ad una determinata altezza, la tira abbondantemente; ma quando l’acqua abbassa oltre a un determinato segno, la tromba non lavora più,” Galileo 1890–1909, volume 8, 59–64. On Galileo’s discussion of vacuum, see Grant 1981, 60–65; Maffioli 2011; Zepeda 2013.
- 76.
“E chi sa che altri minutissimi vacui non lavorino per le minutissime particole, sì che per tutto sia dell’istessa moneta quello con che si tengono tutte le parti congiunte?” Galileo 1890–1909, volume 8, 66. On Galileo’s discussion of cohesion, see Baldini 1976; Le Grand 1978; Redondi 1983, 1985; Shea 1989, 2004; Palmerino 2000, 2001; Bertoloni Meli 2006, 91–95.
- 77.
“Il donne deux causes de ce que les parties d’un cors continu s’entretienent: l’une est la crainte du vuide, l’autre certaine cole ou liaison qui les tient, ce qu’il explique encore apres par le vuide; et ie les croy toutes deux tres fausses. Ce qu’il attribuë au vuide […] ne se doit attribuer qu’a la pesanteur de l’air; et il est certain que, si c’estoit la crainte du vuide qui empeschast que deux cors ne se separassent, il n’y auroit aucune force qui fust capable de les separer. […] Il examine la cole qu’il adiouste avec le vuide pour la liaison des parties des cors, et il l’attribuë a d’autres petits vuids qui ne sont nullement imaginables. Et ce qu’il dit […] pour prouver ces petits vuids, est un sophisme,” AT II, 382. De Volder owned the first edition of Descartes’s letters published by Clerselier, in three volumes (1657–1667): see Bibliotheca Volderina, 6.
- 78.
“La seconde partie contient vos remarques touchant Galilée, ou i’avoüe que ce qui empesche la separation des cors terrestres contigus, est la pesanteur du cylindre d’air qui est sur eux iusques à l’athmosphere, lequel cylindre peut bien peser moins de cent livres. Mais ie n’avouë pas que la force de la continuité des cors vienne de là; car elle ne consiste qu’en la liaison ou en l’union de leurs parties,” AT II, 439–440.
- 79.
AT III, 481–482.
- 80.
De rerum natura, book 1, verses 384–398; book 6, verses 1087–1089: “quasi ut anellis hamisque plicata / inter se quaedam possint coplata teneri / quod magis in lapide hoc fieri ferroque videtur.”
- 81.
“Hence also he inferred, as above that it was not the metus vacui which did hinder the separation of the cylinders with this dilemma: aut non datur talis metus vacui, et coë[u]nt illa, talis metu, cum non detur, nequit impedire avulsionem cylindrorum. Aut si datur, et ille sit qui impediat disiunctos, iam profecto disiunximus illos, et coë[u]ntes datur via, et modus, introducendi vacuii in na[tur]am: imo fecimus ipsi mo[do] vacuum, q[uo]d utrumq[ue] negare de[be]nt, qui vacuum tantopere abohrrere na[tur]am aiunt,” De Volder 1676–1677, 81v.
- 82.
“[…] these scrapings are no longer so firmly joined together in these droplets, and the droplets no longer adhere to one another by immediate contact, but are fastened together as if by certain hooks or barbs. And therefore the steel becomes soft and flexible rather than extremely hard, rigid, and fragile,” Descartes 1982, 248.
- 83.
“Si rursus ignis admovet globulos, partes, si non avelluntur, {concitiurtur} saltem, et lente si refrigeretur fiunt asperiores, nec cohaerent {superficibus} immediatis, sicque faciunt ferrum flexile {ea it erg} differentia est inter corpora mollia, flexilia, et dura fragilia, quod illa habeant cohaesione. {\.} uncis cohaerent, non a sola quiete, ut dura fragilia quia superficies est admodum parva,” Hamburg 273, 520; The same explanation is given also in the other dictata: “[u]ncis quibusdam. Ea enim cohaesio q[uae] fiet per uncos, multo minus firma est, quam ea, q[uae] fit immediato contactu, licet plurimis secus videatur,” Hamburg 274, 114.
- 84.
- 85.
“Ab hac authoris nostri opinione dissentit Gassendus, {dicendo} firmiorem connexionem fieri {po…}, si fingamus uncis et hamulis partes sibi invicem implicitas {ea}, sed fallitur profecto, concipiatur {e[nim]} id sic {ea}, ergo omnis cohaesio orietur a firmitate uncinorum et hamulorum, sed tunc {…} inde horum firmitas? Si {requirat} a prioribus, {…} quo unde earum firmitas, donec tandem {…} ad primos, qui nullis aliis connectuntur, cumque sic etiam adest connectio, qua a prioribus ea non posse supponimus, nullum aliud erit glutinum quam quies,” Hamburg 273, 114. Cf. the other series of dictata on the same article of Descartes’s Principia: “Ipsarum quies, sunt, qui corporis alicuius cohaerentiam ascribunt hamulis uncisque quibusdam, quibus {coagmentatam}, cohaereantque partes, non vero sola quiete hac {…} existimant, primo quidem ut fieri {…} non puto, sed ab illis petere velim, qua vi istarum hamularum {…} partes coahereant, non hic alias adferent opinor hamulos, quia de iisdem unde redibit quaestio, sed necesse habebunt confiteri, se nihil nimis, quo cohaereant, considerent, quam hanc ipsam iuxta se invicem positarum partium quietem, ni quia proinde sola {…} partium corporis glutinum consistat,” Hamburg 274, 47–48.
- 86.
See Gassendi 1658, volume 1, Syntagma philosophicum, 279 and 395. This theory is analyzed in Fisher 2005, 269–270.
- 87.
- 88.
Boyle 1669a, 210.
- 89.
Boyle 1669a, 213.
- 90.
Boyle 1660, 229–230.
- 91.
“XI. Nullum […] datur glutinum efficacius ipsa quiete. XII. A qua quiete omnis pendet vis connexionis, quae fit ab hamulis uncisve, et c.” De Volder 1673b, corollaries 11–12.
- 92.
“VIII. Durities corporum dependet a partium iuxta se invicem positarum quiete,” De Volder 1676–1678, disputation 4, corollary 8.
- 93.
“II. Nullas motus causa assignari potest praeter Deum, qui certam quandam quantitatem produxit, an vero semper eandem sive aequalem conservet, incertum est. III. Omne corpus cuius particulae iuxta se invicem quiescunt durum est. IV. Illa autem quies particularum non est unica causa duritiei,” De Volder 1690–1693, disputation 23, De corpore, annexa 2–4.
- 94.
See Sect. 2.2.3, The mid-1670s clash at Leiden and the foundation of the experimental theatre.
- 95.
“Possint. Haec altera difficultas est, ostendimus in regulis motus corpus minus {moveri} a maiori. {Quod} itaque unde cohaesio corporum, et quid requiritur ut corpus dividam? {Respondemus}, ut moveam, sed {cur cum} totius forcipem moveam, non etiam movere possum partem. Divelli. In {eo secando} difficultatis est, habeo corpus, {ad} ut dividatur, nihil est, ut unam partem moveam non motam altera, sed id non possumus. Ergo requiritur aliud glutinum praeter quietem,” Hamburg 273, 121. In the other series of dictata, De Volder is more constant to Descartes’s text, i.e. he does not challenge his explanation: Hamburg 274, 50.
- 96.
“Possumus. Primum inquit considerandum est, corpori manibus minori non {possunt totae manus} applicari, sed tantum partem {q[uae]} ei magnitudine est aequalis et contigua. {2. manus} pars illa non tota agit in ferrum, quia non omnes quiescunt iuxta se {invicem}, sed molles sunt, ergo {illae} tantum partes agere possunt, quae {et eis eas} accedunt, non quae recedunt. Sed ut {vera fatemur}, non videntur {haec} sufficere, quid ergo restat, si glutinum non consistat in quiete sola? {R[espondeo]} puto {sic} plurimum conferre {aerem}. Quid ergo fiet in corpore duro? Partes iuxta se invicem {quiescerunt}, ergo excluditur {aer}, qui proinde magna vi premit illud corpus, {q[uo]d} q[uam]mdiu non superatur, non potest {tolli} glutinum,” Hamburg 273, 121–122. In the other series of dictata, there is no recourse to the action of air as to this point: cf. Hamburg 274, 47–48.
- 97.
“In ipso {tamen} authoris {R} et hac {…} est, quod vires omnes sunt applicandae {quodque} eadem vis q[uae] potest attollere corpus 100 lb ad {…} unum pedem, possit 200 lb ad pedem dimidium. Et ut {…} regulae veritas pateat, concipio duas vires, quarum una potest corpus 10 lb elevare, ad unum pedem, alia 100 lb ad unum pedem, illa vis posterior est decupla prioris {…}, videamque porro, an non vis quae tollit 1000 lb ad pedis {altitudinem} est decupla {eius q[uod]} 100 lb ad pedem attollit? Unde sequitur illas vires, quarum una, 100 ad {decem} pedes, et altera 1000 ad unum potest tollere {eam aequale}, sed {…} potest tollere 1000 lb, unde {hoc}? Quia omnes vires non applicantur, id {…} est axioma mathematicu{m}, vim minima movere posse corpus maximum modo adhibetur, id itaque {…}, si quis utatur vecte, trochlea etc.: si solummodo vires {applicet} movebit profecto quodlibet corpus. Notissumum est, si quis vecte {velit} uti opus {ea} sustentaculo, {cui} quo corpus movendum propinquius, manus vero remotior est, eo facilius corpus tollitur: quia si tum adhibeantur vires {omnes, sic} ergo {…} frangere non possumus, quia non possumus adhibere vires,” Hamburg 273, 122.
- 98.
“Agnoscimus. Quies inquit unicum est glutinum per q[uo]d partes cohaereant, et res manifesta est, q[uo]d {e[nim] q[uae]so}, est cohaerere, nonne quiescere iuxta se invicem? Si {e[nim]} q[uas]dam partes moveantur, q[uas]dam quiescant, {nec} cohaerebunt, sed separabuntur, ergo si posita cohaesione ponatur q[ui]es, et vice versa, patet ambas unum idemq[ue] {…}. Autor {[n]a[m]} hic utitur argumento metaphysico magis quam physico,” Hamburg 273, 114.
- 99.
I discuss the point further in Sect. 6.2.2.3.2, De Volder’s Huygenian theory of weight (and its relation to the idea of cohesion).
- 100.
Descartes 1982, 59. Cf. the original versions: “unaquaeque res, quantum in se est, semper in eodem statu perseveret; sicque quod semel movetur, semper moveri pergat,” AT VIII-A, 62; “chaque chose demeure en l’etat qu’elle est pendant que rien ne le change,” AT IX-B, 89. On Descartes’s principle of conservation, see the entries given supra, n. 5.
- 101.
“God Himself […] in the beginning created matter with both movement and rest; and now maintains in the sum total of matter, by His normal participation, the same quantity of motion and rest as He placed in it at that time,” Descartes 1982, 58. On Descartes’s ‘metaphysical physics’, i.e. on the use of rational theology in the foundation of natural philosophy, see Hatfield 1979; Clarke 1982; Garber 1992, chapters 7–8, 2001, 2006; Voss 1993, section 1; Schuster 1993, 2013; Nadler 1993; Gaukroger 2002; Hattab 2007; Dobre 2017.
- 102.
“[…] from this same immutability of God, we can obtain knowledge of the rules or laws of nature, which are the secondary and particular causes of the diverse movements which we notice in individual bodies. The first of these laws is that each thing, provided that it is simple and undivided, always remains in the same state as far as is in its power, and never changes except by external causes. […] because experience seems to have proved it to us on many occasions, we are still inclined to believe that all movements cease by virtue of their own nature, or that bodies have a tendency toward rest. Yet this is assuredly in complete contradiction with the laws of nature; for rest is the opposite of movement, and nothing moves by virtue of its own nature toward its opposite or its own destruction,” Descartes 1982, 59.
- 103.
See the previous note, and the Latin and French versions of the article: AT VIII-A, 62; AT IX-B, 89.
- 104.
“[…] quod motus non sit ex natura materiae, et quod ab immediata causa non fluit, necessario a prima causa est sc. Deo, a corpore {a} non pendere ex eo liquet, quia quodlibet corpus possum concipere sine motu, si itaque non fluat ex natura extensionis, et sic non agatur de motu particulari, causa necessario nulla erit praeter Deum,” Hamburg 273, 101. The column b of The Hague dictata, as to this article, contains the opening phrase of the commentary on the same article of Hamburg 274, 42, and British Library dictata, 92v: cf. The Hague dictata, 19b.
- 105.
“Nihil hactenus percepi quam cogitationem, et extensionem. Quaeritur itaque an cogitatio possit augere vel diminuere motus? Resp: nequaquam, quia cogitatio a motu aliena est. Non etiam ex natura materiae fluit, sine iis ne res quidem concipi potest, atqui materia sine maiori minorive motu concipi potest, ergo maior minorve motus ex materiae natura non fluit, non etiam est ex natura motus, quia quemlibet gradum motus sine augmento aut decremento concipere, nec est quod {q[ui]s} dicat, motum tolli a motu {a[ut]} quiete, motus {e[nim]} motui non est contrarius, {q[uae] o[mn]ia} manifeste ostendunt nullis causis naturalibus fieri posse motus in universo augmentum aut decrementum,” Hamburg 273, 102.
- 106.
“Deum esse causam universalem omnium motuum. Ex eo manifestum est, q[uo]d sit causa materiae, omniumque materiae accidentium. Certam habet quantitatem. Quantitate motuum aeq[ue] possumus mensurare, ac omnia alia quanta, si tamen certam {quidam} mensuram motus ex arbitrio nostro sumamus, ad {quem} tamen omnem aliorum motuum quantitatem expendamus, motus autem quantitas dependet a duobus, vel a magnitudine corporis quod movetur, vel a celeritate motus, sic corpus A quod duplex sit illius B et eadem moveatur velocitate, habebit duplum plus motus quam B etenim si A dividatur in duas partes aequales ipsi B habebit unaquaeq[ue] pars cum {aecq[ue]} velocitate moveatur ac B aequalem cum B motus quantitatem, ergo in duabus partibus simul sumptis, hoc est, in corpore A erit duplex quantitas motus, qui est in corpore B nec minus {manifestam} est fere duplam quantitatem motus, in corpore A, {quam qui est} in corpore B si ambo quidem supponantur aequalia, sed tamen A duplo celerius movetur quam B. [Tantundem motus. Si vel augeretur vel minuetur motus quantitas, fieret illud augmentum vel decrementum, vel ab ipsa materia vel a Deo, non ab hoc, quia Deus semper eodem operat in modo, nec ab illa, quia eius essentia nec motum possit producere nec quietem],” Hamburg 274, 42–43. The part between brackets [ ] can be found in Pretoria dictata, 34, British Library dictata, 92v–93r, and The Hague dictata, 20b. Please note that the original Latin text of Descartes states that “Deum esse primariam motus causam.”
- 107.
The marginal note comes after “accidentium”: “[v]id. Rohault Part. 1 cap. 10 § 12–13.” In such paragraphs, Rohault relies on a standard Cartesian justification of the conservation of the quantity of motion, and points out – with an anti-metaphysical stance common to De Volder, that the natural philosopher should direct himself to natural causes: “[c]omme il n’y a que les proprietez essentielles d’un sujet qui se puissent deduire de son Essence, quand elle est connuë, ce seroit inutilement que nous tascherions de découvrir, comment le mouvement a pû estre produit la premiere fois dans le corps, puis que ce n’en est pas une Proprieté essentielle; nous ne nous arresterons donc pas à raisonner sur ce sujet; Et comme nous reconnoissons Dieu pour le createur de la matiere, de mesme le reconnoissons-nous pour son Premier Moteur. Mais parce que ce ne seroit pas philosopher que de luy faire faire à tous momens des miracles, & d’avoir perpétuellement recours à sa puissance, nous suposerons seulement qu’en creant la matiere de ce monde, il a imprimé une certaine quantité de mouvement dans ses parties, & qu’ensuitte il ne fait plus que prester son concours ordinaire, pour empescher que les choses ne retournent dans le neant d’où il les a tirées, & conserver ainsi incessamment en la matiere une égale quantité de mouvement; Si bien que ce que nous avons maintenant à faire, est de rechercher les autres circonstances du mouvement, & d’en estudier les causes Secondes, ou Naturelles,” Rohault 1671, volume 1, 62–63. See Des Chene 2002; Dobre 2013a; Milani 2015; Spink 2018.
- 108.
“Quatenus est simplex. Corporis compositi existentia potest constitui in continua mutatione, ut ignis ex. g. essentia consistit in continua agitatione, et continua variatione, sed haec fit, quia ignis non denotat simplex extensione, sed extensionem ita formatam, ita constitutam, ita motam, verum si tamen considerem extensionem, patet ex eius essentia nullam posse oriri mutationem, {q[uae]} sine ea mutatione supponitur existere posse extensio, ergo quaeq[ue] ei accidat mutatio, ea fit a causis externis,” Hamburg 274, 43.
- 109.
“Vid. Rohault part[em] 1. Cap. 11, § 1. 2. 3,” Pretoria dictata, 34. Cf. Rohault 1671, volume 1, 63–63: “I. Que ce qui est en repos ne peut jamais de soy commencer à mouvoir, et que ce qui a commencé à se muovoir ne peut jamais de soy cesser de se mouvoir. […] II. Que c’est une erreur de croire que les corps qui se meuvent tendent d’eux-mesme au repos, […] III. Que l’opinion d’Aristote n’est pas prouvée par l’experience.” The Pretoria copy also contains an additional commentary, in the same hand as the references to Rohault, in a unpaginated sheet between page 34 and 35, in which it is discussed the behaviour of a arrow in air, where it loses its movement by the impact with air itself.
- 110.
“Prima lex id vult, {q[od]d quaecumque} mutatio non ex natura corporis, sed externis causis fluit, quam rem, ut demonstremus, velim concipi in genere, quicquid mutationis est, esse vel ex natura natura rei, vel a re aliqua ab ea diversa, non potest esse a natura rei, quia […] certum est naturas esse aeternas, ergo si mutetur natura, etiam destruatur, unde (cum nihil tendit in destructionem sui ipsius) sequitur mutationem non esse ex natura rei. […] Motus inquit autor non inducet quietem, quia ipsi est contraria,” Hamburg 273, 102. Also in the other dictata there is no mention of the role of God, but only to the idea that essences do not change by themselves: Hamburg 274, 43 (the contents of this series, as to this article, are partially given in The Hague dictata, 20b).
- 111.
“Ut haec res clarius evadat, considerandum est a nostra contemplatione aliena esse miracula, i.e. illa phaenomena q[uae] non obtemperant legibus naturae, cum itaque dicimus eandem semper motus quantitatem manere, nihil aliud dicimus, quam nullas esse causas naturales, q[uae] motum augeant vel minuant, easq[ue] non esse sic demonstro,” Hamburg 273, 102.
- 112.
“De inertia materiae […] non percipio, quid illa notet ab extensione distinctum. Omnis enim res habet ex sua natura vim permanendi in suo statu, quae vis ab ipsa rei natura non differt, et in hoc exemplo extensionis ipsa inertia est,” De Volder to Leibniz, 18 February 1699, in GP II, 166. De Volder was commenting upon the letter of Leibniz to him of 27 December 1698, in which Leibniz stated that “[q]uod attinet ea quae attribuenda sunt materiae praeter extensionem, satis manifestum puto, inertiam (de qua supra) inter alia esse aliquid quod extensione nuda non continetur,” A II3B, 511. Leibniz was referring to the idea, expressed in the second letter of the correspondence, that mere extension cannot account for its activity; if one considers extension only, as well as its impenetrability, one would conclude that when two bodies collide, they lose their force: see Leibniz to Johann Bernoulli, 12/22 July 1698, in A III7B, 827. I deal with this problem in Sect. 4.2.3.3.1, The quantification of motive force: see esp. infra, n. 261.
- 113.
“Fateor unumquodque manere in statu suo, donec ratio sit mutationis, quod est metaphysicae necessitatis principium, sed aliud est statum retinere donec sit quod mutet, quod etiam facit per se indifferens ad utrumque, aliud est multoque plus continet rem non esse indifferentem sed vim habere et velut inclinationem ad statum retinendum atque adeo resistere mutanti. Itaque olim adolescens edito quodam libello, sumens materiam ut indifferentem per se ad motum et quietem, inde colligebam, maximum corpus quiescens moveri debere a minimo quocunque impellente, sine impellentis debilitatione, atque inde abstractas a systemate, motus regulas colligebam. Et fingi posset sane talis mundus, utique possibilis, in quo materia quiescens motori obediret sine ullo renisu; sed is profecto mundus merum chaos foret,” Leibniz to De Volder, 24 March/3 April 1699, in A II3B, 546–547. Leibniz refers to his Theoria motus abstracti (1670). I will comment upon this passage further in Sect. 4.2.3.4.2, De Volder on elastic impacts: see esp. infra, n. 319.
- 114.
“Deinde Inertiam materiae ab extensione differre existimas, quia per extensionem, indifferens est ad motum et quietem. Per inertiam vero etiam resistat mutanti. Me quod attinet, nullam aliam in extensione indifferentiam agnosco, quam ea est, qua suscipit motum et quietem, si causa fit, quae alterum efficiat. Neque putem ex illa sequi, ut quaelibet vis etiam minima quemlibet motum efficiat etiam maximum. Si nulla materiae Inertia ascribatur ab extensione distincta, putasne fore, ut minimum quoddam corpus, quocunque motu praeditum, corpus impelleret maximum sine ulla motus sui debilitatione? Huic rei repugnare videtur ipsa natura causae et effecti, quae certam quandam inter se proportionem servant. Et sane maioris efficaciae est, pari velocitate movere extensum maius, quam minus, vel hoc ipso, quod effectum maius est. Resistentia praeterea ipsa non videtur mera passio,” De Volder to Leibniz, 13 May 1699, in GP II, 179.
- 115.
See, for instance the letter of De Volder to Bernoulli – directed to Leibniz – of 21 November 1698: “[n]am si a priori demonstratum haberemus, omnem substantiam esse activam, facile mihi persuadeo ex hoc foecundissimo veritatum fonte non horum tantum scrupulorum secuturam enodationem, sed et earum difficultatum, quae hactenus omnes quotquot sunt Physicos presserunt. Huius etenim rei ignorantia, compulsi sunt causam motus coniicere in Deum, et nonnulli etiam ad quoslibet corporum occursus Deum e machina arcessere,” GP II, 151.
- 116.
- 117.
“Hoc tantum addam hac occasione, nunquam mihi placuisse illam Malebranchii, et quorundam Cartesianorum opinionem, motum fieri non vi concursus corporum, sed vi immediata Dei occasione concursus, corpus hoc illudve moventis,” De Volder to Leibniz, 12 November 1699, in GP II, 199. “Cuius ut exemplum adducam, ex hac assumtione, manifestum erit, illam vim translationem producturam, donec a vi alia, priori contraria inhibeatur, adeoque explicatu erit facile, cur corpora mota suo in motu perseverent, quod mihi, ut lenissime loquar, in Cartesiana hypothesi semper admodum arduum visum est. Nam cum motus non sit ipsis vis quaedam permanens, ex qua translatio sequatur, sed ipsa solummodo translatio, quae consistit in perpetua situs mutatione, debebit ex eadem regula quam ad nodum hunc solvendum adhibent, singulis momentis nova dari huius in situ mutationis causa; non secus ac recte volunt in motu per curvam requiri ut singulis momentis nova sit causa, quae determinationem mutet; qua sublata, ut hoc corpus non amplius determinationem suam, sic illud prius non amplius mutabit suum locum. Ex eadem illa vi, sequitur corporum inter se actio et passio, nullo interveniente ex Machina Deo, et quae plura sunt id genus,” De Volder to Leibniz, 25 July 1702, in GP II, 241–242. See also the letter of De Volder to Leibniz, 30 October 1703, in GP II, 254–256.
- 118.
“Tuam equidem hypothesin Malebranchianae longe praeferrem vel eo quod ille ad singulos actus Deum requirat, tu vero φυσικώτερον ad primum solummodo productionis rerum actum. Hoc tamen utrique videtur commune, quod illam aut primae ἐντελεχειῶν productionis, aut in singulis Divinae actionis rationem relinquat obscuram admodum; deinde et illud, quod per leges Mechanicas, utpote quae utraque assumit, everti neutra possit. […] Dein ne sic quidem satis percipio, quid id sit, quod substantiam voces. Si quid enim, nihil certe praeter vim activam, et resistentiam percipio, harum vero virium subiectum, quae tibi, ut opinor, ipsa demum substantia est, nequaquam percipio,” GP II, 254–255.
- 119.
See the letter of De Volder to Leibniz of 5 April 1700: “[…] de veritate conclusionis tuae a posteriori stabilitae dubitavi nunquam, verum ex ipsa substantiae notione demonstrationem quaesivi a priori, sed hactenus frustra,” GP II, 210; De Volder to Leibniz, 3 April 1702: “[t]ecum equidem sentio per experientiam constare, nequaquam omni vi destitui corpora; sed huius rei per experientiam notae demonstrationem quaero, ex ipsa natura substantiae petitam,” GP II, 238; Leibniz to De Volder, 19 November 1703: “[p]orro si mecum agnoscis systema causarum occasionalium, non esse dignum philosopho, si influxum substantiae in substantiam (de veris loquor) inexplicabilem arbitraris; non video quomodo dubitare possis de intrinseca rerum tendentia ad mutationem; cum mutationes adesse in rebus experientia phaenomenorum edoceamur et ab intrinseco mutationes exhibeant vel ipsae operationes mentis. Τὸ ὅτι ergo a posteriori demonstratum puto,” GP II, 256; De Volder to Leibniz, 30 November 1703: “[p]raeter quae in tua causa id me insuper male habet, quod videaris gratis assumere omnem substantiam esse activam, cum tuam vocis substantiae definitionem ingrediatur agendi principium. Quare, ne de voce lis sit, mihi {adhucdum} demonstrandum videtur primum illud, quod inter nos quaesitum fuit, nihil posse existere, quod agendi potentiam non habeat,” GP II, 254–255.
- 120.
See his letter to Leibniz of 12 November 1699 (GP II, 196–200).
- 121.
See the letters of Leibnz to De Volder of 6 July 1701 (GP II, 224–228), 20 June 1703 (GP II, 248–253), 11 October 1705 (GP II, 278–279). On Leibniz’s dynamics, see Gabbey 1971, 1980; Iltis 1971, 1974; Costabel 1973; Gale 1973, 1988; Bernstein 1981; Lariviere 1989; Bertoloni Meli 1993; Fichant 1998; Freudenthal 2002; Lodge 2003; Roberts 2003; Jauernig 2008; Rey 2009a, b.
- 122.
“[…] dicam interim, notionem virium seu virtutis, (quam Germani vocant Krafft, Galli la force) cui ego explicandae peculiarem Dynamices scientiam destinavi, plurimum lucis afferre ad veram notionem substantiae intelligendam. Differt enim vis activa a potentia nuda vulgo scholis cognita, quod potentia activa Scholasticorum, seu facultas, nihil aliud est quam propinqua agendi possibilitas, quae tamen aliena excitatione, et velut stimulo indiget, ut in actum transferatur. Sed vis activa actum quendam sive ἐντελέκειαν continet, atque inter facultatem agendi actionemque ipsam media est, et conatum involvit; atque ita per se ipsam in operationem fertur; nec auxiliis indiget, sed sola sublatione impedimenti. Quod exemplis gravis suspensi funem sustinentem intendentis, aut arcus tensi, illustrari potest. Etsi enim gravitas aut vis elastica mechanice explicari possint debeantque ex aetheris motu; ultima tamen ratio motus in materia, est vis in creatione impressa, quae in unoquoque corpore inest, sed ipso conflictu corporum varie in natura limitatur et coercetur. Et hanc agendi virtutem omni substantiae inesse aio, semperque aliquam ex ea actionem nasci; adeoque nec ipsam substantiam corpoream, (non magis quam spiritualem) ab agendo cessare unquam; quod illi non satis percepisse videntur, qui essentiam eius in sola extensione, vel etiam impenetrabilitate collocaverunt, et corpus omnimode quiescens concipere sibi sunt visi,” GP IV, 469–470.
- 123.
“Quare exultavi gaudio, ubi in Actis Lips. legi te, quem nihil absque summa ratione affirmare, et a quo, si ab ullo tale quid expectari posse, eram persuasissimus, asseverare, Omni substantiae vim agendi inesse,” GP II, 199.
- 124.
“In rebus corporeis esse aliquid praeter extensionem, imo extensione prius, alibi admonuimus, nempe ipsam vim naturae ubique ab Autore inditam […]. Quod si iam Deo per miraculum transcribi non debet, certe oportet, ut vis illa in ipsis corporibus ab ipso producatur, imo ut intimam corporum naturam constituat, quando agere est character substantiarum, extensioque nil aliud, quam iam […] substantiae continuationem sive diffusionem, dicit; […] motus (perinde ac tempus) nunquam existit, […], quia nunquam totus existit, quando partes coexistentes non habet. Nihilque adeo in ipso reale est, quam momentaneum illud, quod in vi ad mutationem nitente constitui debet. Huc igitur redit quicquid est in natura corporea praeter Geometriae obiectum seu extensionem,” GM IV, 235.
- 125.
“Duplex autem est vis activa […], nempe ut primitiva, quae in omni substantia corporea per se inest […], aut derivativa, quae primitivae velut limitatione, per corporum inter se conflictus resultans, varie exercetur. Et primitiva quidem (quae nihil aliud est, quam ἐντελέχεια ἡ πρώτη) animae vel formae substantiali respondet, sed vel ideo non nisi ad generales causas pertinet, quae phaenomenis explicandis sufficere non possunt. […] Et quidem vis primitiva patiendi seu resistendi id ipsum constituit, quod materia prima, si recte interpreteris, in Scholis appellatur, qua scilicet fit, ut corpus a corpore non penetretur; sed eidem obstaculum faciat, et simul ignavia quadam, ut sic dicam, id est ad motum repugnatione sit praeditum, neque adeo nisi fracta nonnihil vi agentis impelli se patiatur. Unde postea vis derivativa varie in materia secunda sese ostendit. Sed nostrum est, generalibus illis ac primitivis sepositis suppositisque quibus ob formam corpus omne semper agere et ob materiam corpus omne semper pati ac resistere docemur, nunc quidem pergere ulterius, et in hac doctrina de virtutibus et resistentiis derivativis tractare, quatenus variis nisibus pollent corpora aut rursus varie renituntur; his enim accommodantur leges actionum, quae non ratione tantum intelliguntur, sed et sensu ipso per phaenomena comprobantur,” GM IV, 236–237. As seen above, De Volder did not label resistance to motion as a passion: in any case, he agreed with Leibniz that resistance to motion is the result of a force.
- 126.
“Velocitas sumta cum directione conatus appellatur; impetus autem est factum ex mole corporis in velocitatem, eiusque adeo quantitas est, quod cartesiani appellare solent quantitatem motus,” GM IV, 237.
- 127.
“Hinc vis quoque duplex: alia elementaris, quam et mortuam appello, quia in ea nondum existit motus, sed tantum sollicitatio ad motum, qualis est globi in tubo, aut lapidis in funda, etiam dum adhuc vinculo tenetur; alia vero vis ordinaria est, cum motu actuali coniuncta, quam voco vivam. Et vis mortuae quiddem exemplum est ipsa vis […] qua elastrum tensum se restituere incipit. Sed in percussione, quae nascitur […] ab arcu se aliquamdiu restituente, aut a simili causa vis est viva, ex infinitis vis mortuae impressionibus continuatis nata,” GM IV, 238.
- 128.
Antognazza 2009, 423.
- 129.
“Caeterum in Monada seu substantiam simplicem completam cum Entelechia non coniungo nisi vim passivam primitivam relatam ad totam massam corporis organici, cuius quidem partem non faciunt reliquae monades subordinatae in organis positae, ad eam tamen requiruntur immediate, et cum primaria Monade concurrunt ad substantiam corpoream organicam, seu animal plantamve. Distinguo ergo (1) Entelechiam primitivam seu Animam, (2) Materiam nempe primam seu potentiam passivam primitivam, (3) Monada his duabas completam, (4) Massam seu materiam secundam, sive Machinam organicam, ad quam innumerae concurrunt Monades subordinatae, (5) Animal seu substantiam corpoream, quam Unam facit Monas dominans in Machinam,” GP II, 252. On the ideas of substance and monad in Leibniz, see Mugnai 1992; Adams 1994; Rutherford 1995; Fichant 1998; Cover et al. 1999; Look 1999; Phemister 2005; Hartz 2006; Smith and Nachtomy 2011.
- 130.
“Si nullae essent divisiones materiae in natura, nullae essent diversae res, imo nihil esset nisi mera rerum possibilitas: actualis vero divisio in massis facit res apparentes distinctas, et supponit substantias simplices,” Leibniz to De Volder, January 1705 (?), in GP II, 276.
- 131.
“[…] iudicari potest, debere in corporea substantia reperiri entelechiam primam, tandem πρώτον δεχτιχόν activitatis; vim scilicet motricem primitivam, quae praeter extensionem (seu id quod est mere geometricum) et praeter molem (seu id quod est mere materiale) superaddita, semper quidem agit, sed tamen varie ex corporum concursibus per conatus impetusve modificatur. Atque hoc ipsum substantiale principium est, quod in viventis anima, in aliis forma substantialis appellatur, et quatenus cum materia substantiam vere unam, sed unum per se constituit, id facit quod ego monadem appello; cum sublatis his vere, et realibus unitatibus, non nisi entia per aggregationem, imo quod hinc sequitur nulla vera entia in corporibus sint superfutura. Etsi enim dentur atomi substantiae, nostrae scilicet monades partibus carentes, nullae tamen dantur atomi molis, seu minimae extensionis, vel ultima elementa; cum ex punctis continuum non componatur,” GM IV, 511; recalled in Leibniz to De Volder, 6 July 1701, in GP II, 224–228.
- 132.
“Quod primum attinet, eo ipso quod corpus mathematicum non potest resolvi in prima constitutiva, id utique non esse reale colligitur; sed mentale quiddam, nec aliud designans quam possibilitatem partium non aliquid actuale,” Leibniz to De Volder, 30 June 1704, in GP II, 268. “Universum Corporeum compositum esse ex una substantia infinitis diversis modis affecta non dixerim, etsi materiam in se spectatam […] ubique esse sibi similem dici possit. […] [I]n extensione Mathematica, qua possibilia intelliguntur, nulla sit divisio actualis nec partes nisi quas cogitando facimus, nec prima Elementa; non magis quam inter numeros fractos minimus datur velut Elementum caeterorum,” Leibniz to De Volder, January 1705 (?), in GP II, 276. “[S]patium ut tempus non substantiale est quiddam, sed ideale […]. Itaque nullae ibi divisiones nisi quas mens facit et pars toto posterior est. Contra in realibus unitates multitudine sunt priores, nec existunt multitudines nisi per unitates,” Leibniz to De Volder, 11 October 1705, in GP II, 278–279.
- 133.
De Volder and Lufneu 1676, thesis 3; cf. Lucretius, De rerum natura, book 1, verses 615–622.
- 134.
“V. Divisibile est omne corpus in particulas numero infinitas. VI. Infinitum infinito potest esse maius. VII. Unde liquet omnes illas ratiocinationes, quae ex eo quod infinitum infinito maius esse non possit, deducuntur, mancas esse,” De Volder 1690–1693, disputation 23, annexa 5–7. On the idea of divisibility in De Volder and Leibniz, see MacDonald Ross 1987.
- 135.
“Extensio est abstractum Extensi nec magis est substantia quam numerus vel {multitudo substantia} censeri potest exprimitque nihil aliud quam quandam non successivam (ut duratio) sed simultaneam diffusionem vel repetitionem cuiusdam naturae seu quod eodem redit, multitudinem rerum eiusdem naturae, simul cum aliquo inter se ordine existentium,” Leibniz to De Volder, 30 November 1704, in GP II, 269. On Leibniz’s theory of extension and space, see Fox 1970; Sayre-McCord 1984; Hartz and Cover 1988; Khamara 1988, 1993; Arthur 1994, 2013; Garber et al. 1998; De Risi 2007; Futch 2008.
- 136.
“Diffusionem quam in extensione concipio et quae Tibi suspicionem nescio cuius paradoxi latentis iniecisse videtur nihil aliud esse volo quam continuationem qua pars est similis toti, ut albedinem concipimus in lacte diffusam et eandem ubique directionem in recta, et in circuli peripheria aequalem curvitatem. Neque vero […] substantiae simplices diffunduntur […], aut totum homogeneum constituunt homogeneitas enim materiae non nisi abstractione mentis habetur,” Leibniz to De Volder, January 1705 (?), in GP II, 277. This topic is discussed in Lodge 2004.
- 137.
“Diuturni mei silentii causa non tam est valetudo, quamquam et ea non satis commoda fortasse etiam nonnihil eo contulerit, quam quod ad tuas nihil novi quod reponerem, haberem, verererque ne earundem rerum repetitione tibi nauseam crearem. Nescio enim, qui fiat, ut etiamnum in iisdem, quas in ultimis meis posui, difficultatibus haeream. Neque enim vim illam primitivam mihi videor intelligere, nec capio, qui ex indivisibilibus extensum fieri, aut, si ita malis, apparere queat. Invasit me insuper desperatio quaedam, tuae opinionis rite percipiendae, cum post tot tentamina tam parum promoverim, ut ea requiri ad eius intelligentiam existimem, quae tu instar ultramundanorum ne quaerenda quidem censes,” GP II, 279.
- 138.
- 139.
See AT XI, 34–35.
- 140.
“From this it follows necessarily that from the time they begin to move, they also begin to change and diversify their motions by colliding with one another. Thus, while God subsequently preserves them in the same way He created them, He does not preserve them in the same state. That is to say, if God always acts in the same way and consequently always produces substantially the same effect, many differences in this effect occur, as if by accident. And it is easy to accept that God, who is, as everyone must know, immutable, always acts in the same way,” Descartes 2004, 25.
- 141.
Descartes 2004, 25–26.
- 142.
Descartes 2004, 27.
- 143.
- 144.
See supra, n. 140, and “these two rules [sc. the first and second one] follow manifestly from the sole fact that God is immutable and that, acting always in the same way, He always produces the same effect. For on the assumption that He placed a certain amount of motion in matter in general at the first instant He created it, we must admit either that He preserves the same amount of motion in it, or not believe that He always acts in the same way,” Descartes 2004, 28–29. On Descartes’s principle of conservation, see in particular Menn 1990; Garber 1992, chapter 7; Slowik 2002, chapter 5.
- 145.
Descartes 1982, 58.
- 146.
Descartes 1982, 59.
- 147.
Descartes 1982, 60.
- 148.
Descartes 1982, 61.
- 149.
Descartes 1982, 58.
- 150.
Cf. the original text: “virium, sive ad movendum, sive ad motui resistendum,” AT VIII-A, 67; “force en chacun de ces corps pour mouuoir ou pour resister au mouuement,” AT IX-B, 89. On Descartes’s theory of force, see in particular Hatfield 1979; Gabbey 1980; Guéroult 1980; Arthur 2007; Hattab 2007; Schmaltz 2008; Manchak 2009.
- 151.
“In what the force of each body to drive or to resist consists. We must however notice carefully at this time in what the force of each body to act against another or to resist the action of that other consists: namely, in the single fact that each thing strives, as far as is in its power, to remain in the same state, in accordance with the first law stated above. From this it follows that a body which is joined to another has some force to resist being separated from it, while a body which is separate has some force to remain separate. One which is at rest has some force to remain at rest, and consequently to resist everything which can change it; while a moving body has some force to continue its motion, i.e., to continue to move at the same speed and in the same direction. Furthermore, this force must be measured not only by the size of the body in which it is, and by the […] surface which separates this body from those around it; but also by the speed and nature of its movement, and by the different ways in which bodies come in contact with one another,” Descartes 1982, 63.
- 152.
See Sect. 4.2.1.3, Descartes’s idea of determination, and rules of reflection and refraction.
- 153.
This concept is discussed in Schuster 2013, 465–468 and 627–631.
- 154.
“Thus, it can happen that star N has less solidity, or less ability to continue its movement, than the globules of the second element which surround it; even though it may be very large and covered with fairly many layers of spots. For these globules, in proportion to their size, are as solid as any body can be, because we understand that they contain no pores filled with other […] matter; and because their figure is spherical; the sphere being the figure which has the least surface area in proportion to its volume, as Geometers know,” Descartes 1982, 153–154.
- 155.
“Finally, it can happen that the same star may be less solid than certain globules of the heaven, and more solid than some other rather smaller ones, both for the reason just stated, {namely, that the forces of several globules are less unified than those of one larger body equal to all of them}, and also because, although the quantity of the matter of the second element in all the globules which occupy a given [amount of] space may be the same whether they are {very} small or {quite} large, the smaller ones have {less force, because they have} more surface area {in proportion to the quantity of their matter}; and therefore they can be drawn off their course and turned aside in other directions more easily than the larger ones, either by the matter of the first element filling the spaces which they leave around themselves, or by any other bodies {which they encounter},” Descartes 1982, 155. The text between brackets thus { } is present in the French version of Descartes’s Principia.
- 156.
See Sect. 4.2.1.3, Descartes’s idea of determination, and rules of reflection and refraction.
- 157.
As argued by Alan Gabbey, the formulation and the addition of the seven rules of impact to the draft of the Principia traces back to the last years of the composition of this treatise: see Gabbey 1980. On Descartes’s rules of impact, see also Blackwell 1966; Aiton 1972, chapter 3; Garber 1992, chapter 8; McLaughlin 2000.
- 158.
“In order to determine, from the preceding laws, how individual bodies increase or decrease their movements or turn aside in different directions because of encounters with other bodies; it is only necessary to calculate [in Latin: calculo subducere; in French: calculer: see AT VIII-A, 67; AT IX-B, 89] how much force to move or to resist movement there is in each body; and to accept as a certainty that the one which is the stronger will always produce its effect. Moreover, this could easily be calculated if only two bodies were to come in contact, and if they were perfectly solid, and separated from all others {both solid and fluid} in such a way that their movements would be neither impeded nor aided by any other surrounding bodies, for then they would observe the following rules,” Descartes 1982, 64.
- 159.
“That movement is not contrary to movement, but to rest; and that determination in one direction is the opposite of determination in another. It must also be noticed that one movement is in no way contrary to another movement of equal speed; but that, strictly speaking, only a twofold opposition is found here. One is between movement and rest, or even between rapidity of movement and slowness of movement (i.e., to the extent that this slowness partakes of the nature of rest): the other is between the determination of a body to move in a given direction and the encounter, in its path, with a body which is either at rest or moving in a contrary manner; and this opposition is greater or smaller according to the direction in which the body which encounters the other is moving,” Descartes 1982, 63.
- 160.
As I am going to show in Sect. 4.2.1.3, Descartes’s idea of determination, and rules of reflection and refraction, the determination of motion is a concept allowing the decomposition of the quantity of motion in more directions: so, it applies to the case of reflection and refraction, where changes in the angles of impact are considered. In the seven rules of impact, on the other hand, the angle of impact and that of reflection or rebounding are always 90°.
- 161.
See McLaughlin 2000.
- 162.
See Regius’s Fundamenta physices (1646), chapter 1, and Rohault’s Traité de physique, part 1, chapter 10.
- 163.
The Cartesian quantification of force or quantity of motion is clearly presented in Rohault’s Traité de physique, part 1, chapter 10, §§ 7–10, where the model of the lever is explicitly taken into account.
- 164.
See Gabbey 1980, 137–139.
- 165.
“First, if these two bodies, for example Band C, were completely equal in size and were moving at equal speeds, B from right to left, and C toward B in a straight line from left to right; when they collided, they would spring back and subsequently continue to move, B toward the right and C toward the left, without having lost any of their speed. {For, in this case, there is no cause which could take their speed from them, but there is a very obvious one which must force them to spring back; and because it would be equal in each, they would both spring back in the same way},” Descartes 1982, 64–65.
- 166.
“[C]es regles […] ne dépendent que d’un seul principe, qui est, Que lors que deux cors se rencontrent qui ont en eux des Modes incompatibles, il se doit veritablement faire quelque changement en ces modes pour les rendre compatibles, mais que ce changement est tousiours le moindre qui puisse estre, c’est à dire, que si certaine quantité de ces modes estant changée ils peuvent devenir compatibles, il ne s’en changera point une plus grande quantité. Et il faut considerer dans le mouvement deux divers modes, l’un est la motion seule ou la vitesse, et l’autre est la determination de cette motion vers certain costé, lesquels deux modes se changent aussi difficilement l’un que l’autre,” AT IV, 185. For a thorough analysis of Descartes’s own discussion of his rules of impact, see Garber 1992, chapter 8.
- 167.
“Second, if B were slightly larger than C, and everything else were as previously described, then only C would spring back, and both would move toward the left at the same speed. {For B, having more force than C, could not be obliged by C to spring back},” Descartes 1982, 65.
- 168.
“Third, if the two bodies were equal in size, but if B were moving slightly more rapidly than C; after their collision not only would {C alone spring back and} both continue their movement toward the left, {that is, in the direction from which C came}, but also one half of B’s additional speed would be transferred from it to C, {since B could not move more rapidly than C which would be ahead of it}. For example, if B had initially been travelling at six degrees of speed, and C at a speed of only four, {B would transfer to C one of its two additional degrees of speed, and} both would subsequently move toward the left at five degrees of speed. {This would occur because it is much easier for B to transfer one of its additional degrees of speed to C than for C to change the course of all the movement which is in B},” Descartes 1982, 65. It is worth to note that also in this case – viz. in the French version of the article – Descartes overtly adopts the principle of least modal change.
- 169.
“Fourth, if the body C were entirely at rest, {that is, if it not only had no apparent motion but also were not surrounded by air or any other fluid (which makes the hard bodies immersed in such a fluid very easily movable, as I shall show)}, and if C were slightly larger than B; the latter could never {have the force to} move C, no matter how great the speed at which B might approach C. Rather, B would be driven back by C in the opposite direction: because {for B to move C, C would have to be driven as rapidly as B subsequently moves and} a body which is at rest puts up more resistance to high speed than to low speed; and this resistance increases in proportion to the difference in the speeds. Consequently, there would always be more force in C to resist than in B to drive, {because C is larger. For example, if B is one half as large as C and is travelling at three degrees of speed, then because B is only as large as each of C’s halves and because it cannot continue in the same direction more rapidly than it pushes C ahead of it; B cannot move C without transferring to it two thirds of its quantity of motion, one third for each of C’s halves, keeping for itself only one degree of speed. Similarly, if B has thirty degrees of speed, twenty will have to be communicated to C; if B has three hundred, two hundred will have to be transferred, and so on. But since C is at rest, its resistance to receiving twenty degrees of speed is ten times as great as its resistance to receiving two, and so on. Thus, the greater B’s speed, the proportionally greater C’s resistance will be. And because each half of C has as much force to remain at rest as B has to drive it, and because both halves resist at the same time, it is obvious that they must succeed in forcing B to spring back. So that, no matter how great the speed at which B approaches C, B can never have the force to move C},” Descartes 1982, 66.
- 170.
“Fifth, if the body C were at rest and {even very slightly} smaller than B; then, no matter how slowly B might advance toward C, it would move C with it by transferring to C as much of its motion as would permit the two to travel subsequently at the same speed. Thus if B were twice as large as C, it would transfer to C {only} one third of its quantity of motion; because that one third would move the body C at the same speed as the remaining two thirds would move the body B which {we are supposing} is twice as large as C. Therefore, after B had collided with C, its speed would be reduced by one third; that is to say, B would then need as much time to travel a distance of two feet as it previously did to travel a distance of three feet. Similarly, if B were three times as large as C, it would transfer to C one quarter of its motion; and so on. {And it is impossible for B to have so little force that it would ever be insufficient to move C; for it is certain that weaker motions must observe the same laws as stronger ones, and must produce, proportionally, the same result. Although we often think we see the opposite on this earth; this is because of the air and other fluids which always surround solid moving bodies and which can greatly increase or decrease their speed, as we shall see later},” Descartes 1982, 67.
- 171.
“Sixth, if the body C were at rest and exactly equal in size to body B, which was moving toward it; necessarily, C would be to some extent driven forward by B and would to some extent drive B back in the opposite direction. Thus, if B were k) approach C with four degrees of speed, it would {have to} communicate one degree to C, and be driven back in the opposite direction with the remaining three. {Because it must necessarily be the case that either B moves C without springing back, thus transferring two degrees of its speed to C; or that B springs back without moving C, retaining those two degrees of speed as well as the two which cannot be taken from it; or else that B springs back, retaining some portion of those two degrees of speed, and at the same time moves C with the remainder of those two degrees. Since Band C are equal and there is consequently no more reason for B to spring back than to move C; it is obvious that these two effects must be equally shared: that is, B must transfer one of these degrees of speed and spring back while retaining the other},” Descartes 1982, 67–68.
- 172.
Garber 1992, 242–251. This model is also presented in the letter of Descartes to Clerselier of 17 February 1645, in which Descartes discusses rule 4. In rule 4, B resists being moved by A as much as A strives to move it, regardless of the speed of A: so, the resistance it exerts is proportional to the effect that A would exert on B, if it had managed to move it: see AT IV, 184; cf. Garber’s presentation: “[n]ow, the faster B [sc. A] is going, the more motion it would have to transfer to C [sc. B] for the result to obtain. And so, if we think of the force of resisting in C as measured by the quantity of motion it would have if B were to succeed in pushing it after the collision, then the faster B goes, the more the resistance C offers,” Garber 1992, 244. Descartes’s explanation of rule 4 given in his letter would be then incorporated in the French version of the article: see supra, n. 169.
- 173.
“Finally, if B and C were travelling in the same direction, C more slowly than B, so that B (which would be following C) would eventually strike it; and if C were larger than B but B’s speed exceeded C’s by a greater extent than C’s size exceeded B’s: then B would transfer to C as much of its speed as would be required to permit them both to travel subsequently at the same speed and in the same direction,” Descartes 1982, 68.
- 174.
“However, if, on the contrary, B’s speed exceeded C’s by a smaller extent than C’s size exceeded B’s; B would be driven back in the opposite direction, and would retain all its movement,” Descartes 1982, 68.
- 175.
“{And, finally, when the ratio in which C’s size exceeds B’s is exactly equal to the ratio in which B’s speed exceeds that of C, B must transfer some of its motion to C and spring back with the rest},” Descartes 1982, 68.
- 176.
- 177.
McLaughlin 2000, 87.
- 178.
McLaughlin 2000, 92–93.
- 179.
As De Volder states in commenting upon Descartes’s Principia II.32 (where the decomposition of movement is theorized: “[h]ow, properly understood, the single movement peculiar to each body may also be regarded as multiple,” Descartes 1982, 55), he notes that this article “is the most worth of attention, because by this multiple phenomena, especially in catoptric and in dioptric, are explained” (“{[h]ic articulum} attentione {dignissimum} est, quia per hunc plurima ϕαινόμενα praesertim in catoptrica et dioptrica explicantur. Describat. Motus o[mn]is concipi potest instar plurimum etiam {simplicissimus}. Nec est q[uo]d dicunt motum rotae compositum e[ss]e, quia umunq[uo]dq[ue] punctum unam describat lineam sive mixtam sive simplicem, non e[nim] concipi potest linea ut circularis sola, quia recta adest, nec sola recta, quia adest circularis. Ergo motus o[mn]is est compositus, et licet nulla sit linea simplicior recta, composita tamen concipi potest. Circularis. Id vult autor sive concipiamus lineam simp[licem] sive compositam eandem fore conclusionem, o[mn]ia phaenomena eadem, atq[ue] {ad eo non} referre, sive simplicem, sive compositam concipiam vide dioptric. 2. §§ i” Hamburg 273, 98), and underlines that the decomposition of motion “often depends on our understanding, which however as in many things is grounded on the truth of thing [itself], it is indeed certain, that each point describe only one line” (“[i]nstar plurium. Hoc saepe ab intellectui nostro dependet, qui tamen ut plurimum fundatur in rei veritate, certum enim est, unumquod[ue] punctum tantum describere unam lineam. Et uno moti potest describi, {q[uae]} licet varie sit contorta, tamen una est linea, ita quidem, ut eadem effecta {fiat consectura}, sive concipiamus punctum {rotae}, ut revera sit, moveri per lineam illam contortam, sive concipiamus punctum id describere lineam curvam nimirum {-} circularem, dum ille circulus movetur per lineam rectam, utroq[ue] enim modo eadem describitur curva. Idem sequitur effectus, unde ex hac ipsa distinctionem motus {…} esset ex variis compositus, nullus error {suborri} potest, quod opprime notandum propter usum, quem habet in Dioptricis in {materiae} reflexionum et refractionum. Vide dioptricam cap. 2,” Hamburg 274, 41).
- 180.
Cf. Descartes’s Dioptrique, Discours second, AT VI, 93–96.
- 181.
Cf. Descartes’s Dioptrique, Discours second, AT VI, 96–100.
- 182.
- 183.
On the sine law, see in particular Schuster 2013, appendix 1.
- 184.
Schuster 2013, 188–189.
- 185.
Schuster 2013, 191.
- 186.
Schuster 2013, 208.
- 187.
See Sect. 4.1.1, De Volder’s Cartesian standpoint on the ontology of physics.
- 188.
Wallis communicated his results to the Royal Society by letter in November 1668; Wallis explained them at a session on 17 December 1668, Huygens communicated them by letter in January 1669 (see Huygens 1669a, 925–926). For a full account, see Murray et al. 2011. Please note that, according to the Bibliotheca Volderina, De Volder owned the issues of the Philosophical Transactions from 1665 to 1705 (13 volumes); the Latin edition published in Amsterdam in 1674 (as 5 volumes are listed, probably De Volder owned the series appearing in 5 volumes between 1671 and 1676, covering the years 1665–1669; in 1681 a Continuatio, covering the year 1670, followed); the 1705 abridgement of the Philosophical Transactions, by John Lowthorp: see Bibliotheca Volderina, 6 and 10. Also, De Volder owned the full series of the Journal des Sçavans (1665–1708): see Bibliotheca Volderina, 93.
- 189.
The text used by De Volder and Fullenius coincides with that of a manuscript beloning to Huygens, written by Denis Papin and presenting Huygens’s own corrections (ms. HUG 26 A, 91r–108v (Leiden University Library)), and based on a 1656 manuscript of Huygens (in Huygens 1888–1950, volume 16, 137–149; ms. HUG 26 A, 58r–61v): for a commentary, see Huygens 1888–1950, volume 16, 12 and 30; Yoder 2013, 117–119. In their preface to the reader, De Volder and Fullenius do not indicate any intervention by them in the final text: “[…] succedit tractatus de Motu, in quo regulae motus demonstrantur, quas iam ante 30, et quod excurrit, annos in Diario Gallico 18 Martii anni 1669 cum orbe litterario communicavit auctor,” Huygens 1703, Lectoribus, 11 (unnumbered). The manuscript contains some annotations by the editors: see Yoeder 2013, 118.
- 190.
Besides these two printed expositions of Huygens’s rules, there are extant working papers of Huygens on the topic, which have been analysed by Westfall (Westfall 1971, chapter 4), and dating back to 1652, 1654, 1656, 1659 and 1667: see Huygens 1888–1950, volume 16, 92–168. Moreover, Huygens presented to Oldenburg, together with his 1669 communication, an incomplete, larger treatise on collision (published in Murray et al. 2011). See also Andriesse 2005, chapter 6.
- 191.
See the letter of De Volder to Leibniz of 18/28 February 1699: “[…] post obitum Illustris Viri, libellum ejus de motu videre licuit,” A II3B, 529.
- 192.
“Si duo corpora aequalia perfecte dura aeque celeriter in contrarias partes mota, inter se colliderentur, unumquodque in eam partem unde venit reflecteretur nulla parte […] celeritatis amissa,” Huygens 1888–1950, volume 16, 92.
- 193.
See his 1656 manuscript on collision, where he considers the cases of rule 2 and 5: “[h]isce Cartesij regulis multi assensum praebuere haud scio verisimilitudine magis an autoritate subtilissimi Viri Philosophi permoti. Sed me primum dubitare coegit nimia illarum cum experimentis discrepantia nam quiescente sphaera ab aequali pulsam hanc ab ictu quiescere motumque omnem in illam transferri saepissime observaveram, atque alia praeterea notaveram percussionis accidentia quae superius recensui longe diversa regulisque istis adversantia. Deinde vero et ipsas inter se dissidere leges ipsius animadverti. quod hic obiter commonstrare expediet. Quinta nimirum docet quod si corpus majus B occurrat minori C quiescenti, aliquid de celeritate sua amittet. At ex lege secunda si occurrat B eidem minori corpori C, venienti ex adverso cum pari celeritate, nihil amittet B de celeritate sua quae quidem inter se pugnant, nisi dicamus corpori moto magis resistere corpus quiescens quam si ipsi ex adverso veniens impingatur. quod absonum est. Hisce itaque regulis minime fidens de alijs cogitare coepi, omnibusque tentatis. Itaque diligentius haec inspicere cepi tandemque veriora, uti existimo,” Huygens 1888–1950, volume 16, 139; Westfall 1971, 147.
- 194.
“I. Corpus quodlibet semel motum, si nihil obstet, pergere moveri eadem perpetuo celeritate, et secundum lineam rectam. II. Cum corpora duo dura, inter se aequalia, aequali celeritate ac directe sibi mutuo occurrunt, resilire utrumque eadem qua advenit celeritate; dicuntur autem directe occurrere, cum in eadem linea recta, utriusque centra gravitatis conjiungente, moventur, punctumque contactus accidit in eadem recta,” Huygens 1703, 369. As a third hypothesis, Huygens used a principle of relativity of motion. This principle, stating that the outcome of a collision is not influenced by the movement to which all the involved bodies participate, makes possible to foresee outcomes of collision which are consistent in all frames of reference (contra Descartes), for the reason that one can add or subtract to each body the same quantity of speed, without influencing the outcome of the collision: “III. Motum Corporum celeritatesque aequales aut inaequales respective intelligendas esse, facta relatione ad alia corpora, quae tanquam quiescentia considerantur; etsi fortasse et haec et illa alio communi motu involvantur; ac proinde, cùm corpora duo sibi mutuo occurrunt, etiamsi alteri praeterea motui utrumque simul obnoxium fuerit, haud aliter illa se invicem impellere respectu ejus qui eodem quoque motu defertur, ac si omnibus adventitius iste motus abesset,” Huygens 1703, 369–370. These three hypotheses served to establish the rules of impact for equal bodies.
- 195.
See Westfall 1971, 146–158; Slowik 2002, 177–186. Cf. Huygens’s 1652 manuscript on collision, stating as an axiom grounding his work on impact between unequal bodies, that “[s]i corpus A majus occurrat B minori, sed velocitas in B sit ad velocitatem in A reciprocè ut magnitudo A ad B, tum utrumque cum eadem qua venit celeritate resiliet. Hoc concesso omnia demonstrari possunt. Cartesius autem concedere cogitur,” Huygens 1888–1950, volume 16, 96. This axiom, for Huygens, had to be conceded by Descartes as it conforms to the conservation of m·s, even if it contradicts Descartes’s rule 7c. In any case, he added to this axiom the annotation “[s]ed videndum an demonstrari per notiora queat,” Huygens 1888–1950, volume 16, 96: see Westfall 1671, 149–150. Indeed, this axiom would become proposition 8 in his De motu, demonstrated on the basis of two additional hypotheses he set out in this work: see Huygens 1703, 381. On Huygens’s theory if impact, see also Blackwell 1977; Mormino 1990a, b.
- 196.
“[…] Leges motus Hugenianas, quas absque demonstratione edidit in Diariis,” De Volder to Leibniz, 18/28 February 1699, in A II3B, 528.
- 197.
In fact, in his De motu corporum ex percussione, as seen above, Huygens based the deduction of the outcomes of this kind of collision on three hypotheses.
- 198.
Murray et al. 2011, 156; cf. the original text: “I. Si corpori quiescenti duro aliud aequale corpus durum occurrat, post contactum hoc quidem quiescet, quiescenti vero acquiretur eadem quae fuit in impellente celeritas. II. At si alterum illud corpus aequale etiam moveatur, feraturque in eadem linea recta, post contactum permutatis invicem celeritatibus ferentur,” Huygens 1669a, 927; given also in Huygens 1703, as propositions 1 and 2.
- 199.
See Torricelli 1644, 99: “[p]raemittimus. Duo gravia simul coniuncta ex se moveri non posse, nisi centrum commune gravitatis ipsorum descendat. Quando enim duo gravia ita inter se coniuncta fuerint, ut ad motum unius motus etiam alterius consequatur, erunt duo illa gravia tamquam grave unum ex duobus compositum, sive id libra fiat, sive trochlea, sive qualibet alia Mechanica ratione, grave autem huiusmodi non movebitur unquam, nisi centrum gravitatis ipsius descendat.” From this, it follows the principle for which the motion acquired by two falling bodies cannot raise their common centre of gravity above their point of fall: Huygens overtly used this idea in the demonstration of proposition 8 of his De motu corporum: “certissimum enim in mechanicis est axioma, motu corporum qui a gravitate ipsorum proficiscitur, centrum commune gravitatis ipsorum non posse attolli,” Huygens 1703, 382. Leibniz would use this idea to criticize the quantification of force of motion by m·s, as I am going to show: see Sect. 4.2.3.3.1, The quantification of motive force. For a thorough discussion of Torricelli’s principle, see Pisano and Capecchi 2010; Capecchi 2012, chapter 6, 2017, 315–324.
- 200.
“Quoties duo corpora inter se colliduntur, eadem est, mutuo respectu, discendentibus celeritas, quae fuit appropinquantibus,” Huygens 1703, 375.
- 201.
“Hypothesis IV. Si corpus maius minori quiescenti occurrat, aliquem ei motum dare, ac proinde de suo aliquid amittere. […] V. Corporibus duobus duris sibi mutuo occurrentibus, si, post impulsum, contigat alteri eorum omnem quem habebat motum conservari, etiam alterius motui nihil decedere neque adiici,” Huygens 1703, 373 and 375. The demonstration of the proposition is as follows (see Fig. 4.25, Huygens 1703, De motu, plate 2, figure 5) – (please note that point E is the centre of gravity of the two bodies): “[d]e aequalibus corporibus manifestum est ex propositione II. Sint igitur nunc inaequalia, primumque is casus proponatur, quo, maiori A quiescenti, impingitur corpus minus B, celeritate BA dextram versus pergens. Dico ipsa post contactum eadem celeritate BA separatum iri, adeo ut, si parte una temporis corpus B confecerit spatium BA, post alteram similem temporis partem, rursus spatio, quod ipsi BA aequale sit, separata inveniantur. Constat enim A celeritatem aliquam accipere impulsu corporis B; sit ea AC, minorem autem esse oportet celeritate BA qua ipsum B movebatur: nam si ipsi B aequale esset A, tum demum celeritatem BA ex impulsu acciperet. Dividatur AC in duo aequalia puncto D, sitque AE aequalis AD. Si igitur in navigio hosce motus contingere existimemus, quod sinistram versus praetervehatur celeritate DA: necesse est ut ante impulsum, corpus A quod in navigio quiescebat, motum fuerit respectu ripae dicta celeritate DA, sinistram versus; post impulsum vero, cum in navigio motum dicatur celeritate AC dextrorsum, ipsum vero navigium celeritate DA in partem contrariam feratur, movebitur A, ripae respectu, celeritate DC seu AD in partem dextram. Itaque, respectu ripae, corpus A ante et post impulsum, eandem servat celeritatem. Quare etiam B, eiusdem respectu, nihil de sua celeritate perdidisse oportet. […] Hoc casu demonstrato reliqui facile consequuntur, supersunt autem quatuor diversi, nam vel minus corpus quiescit, vel utraque adversis motibus cientur, vel celeriore motu minus insequitur maius, vel contra; quos omnes simul proponere licebit,” Huygens 1703, 373–374.
- 202.
- 203.
Cf. Huygens 1669a, 928 (in a paragraph immediately below Huygens’s communication): “[…] subiungit, notasse se miram quandam naturae legem, quam demonstrare se posse affirmat in corporibus sphaericis, quaeque generalis ipsi videtur in reliquis omnibus sive duris sive mollibus, sive directe sive oblique sibi occurrentibus, videlicet centrum commune gravitatis duorum, trium, vel quotlibet corporum, aequaliter semper promoveri versus eandem partem in linea recta, ante et post occursum.”
- 204.
See Huygens 1888–1950, volume 16, 160–161. According to Huygens, if two soft i.e. non-elastic bodies with equal m·s impact, they come to rest: “A et B corpora mollia, A movetur celeritate AC B celeritate BC, quae sunt reciproce ut ipsa corpora. Dico post concursum in C, manere utrumque immotum.”
- 205.
See Huygens 1669a, rule 5: “[q]uantitas motus duorum corporum augeri minuive potest per eorum occursum; at semper ibi remanet eadem quantitas versus eandem partem, ablata inde quantitate motus contrarii.” Cf. Huygens 1703, proposition 6. This means that, in all collisions, the quantity of motion exerted in each direction (after having subtracted the quantity of motion exerted in the opposite direction) is equal in each body.
- 206.
- 207.
Cf. Westfall’s own words: “[t]he exit of the Cartesian ‘quantity of motion’, that is, was only the cue for the entrance of a new quantity (mv2) onto the stage […]. In some sense, the quantity had been implicit already in Galileo’s kinematics of free fall, in the demonstration that the velocity of a body falling from rest is proportional to the square root of the distance it has fallen […]. Perhaps Huygens’ derivation of the new equation of conservation was connected to his perception of the relation of free fall to the impact of bodies inversely proportional to their speeds. His demonstration of that case, it will be recalled, employed the kinematics of vertical motion. The source of his insight is purely speculative, however. What is certain is the fact that Huygens first made explicit a quantitative relation destined to play a central role in the future history of mechanics,” Westfall 1971, 157.
- 208.
Westfall 1971, 152.
- 209.
Mahoney 1995. Cf. the original text: “[q]uaecunque sit causa corporibus duris a mutuo contactu resiliendi cum in se invicem impinguntur; ponimus, cum corpora duo inter se aequalia, aequali celeritate, ex adverso ac directe sibi mutuo occurrunt, resilire utrumque eadem qua advenit celeritate,” Huygens 1703, 369.
- 210.
“Nunquam magnus ille Florentinus Galilaeus de Galilaeo admirabilem illam detexisset in motus acceleratione proportionem, nisi in geometricis demonstrationibus fuisset versatissimus. […] Quod inventum ut illustri Hugenio debet orbis litteratus, ita illi debuisset nunquam, nisi caeteris cum scientiis, in quibus excellit vir magnus, coniunxisse mathematicarum artium notitiam. Sed non est quod unicam hanc motus proprietatem urgeam, cum ipsae, quas vocant, motus leges, quibus corpora corporibus occurrentia vel subsistunt, vel suis in viis progrediuntur, vel versus contrarias partes repercutiuntur, mathematica agnoscant principia,” De Volder 1682, 17 (unnumbered). In his 1698 Oratio, De Volder refers to Huygens without overtly mentioning his discovery of the laws of impact: “[o]stenderunt magna huius seculi nostri lumina Hugenius, Newtonus, Leibnitzius, ne simplicium quidem corporum motus, viresque investigandas unquam, non dicam absque notitia matheseos, sed addam absque recondita harum artium scientia. Qua qui instructus non est, in physicis hospes ut sit, necessum est. Tanta igitur cum inter has disciplinas sit affinitas, eo meliori iure inquiremus, num eadem methodo tractari queant,” De Volder 1698, 26.
- 211.
The article is devoted to the demonstration of the second part of the law (i.e. “that, upon coming in contact with a weaker one, it [i.e. a stronger body] loses as much as it transfers to that weaker body”): “it is obvious that when God first created the world, He not only moved its parts in various ways, but also simultaneously caused some of the parts to push others and to transfer their motion to these others. So in now maintaining the world by the same action and with the same laws with which He created it,” Descartes 1982, 61–62.
- 212.
“Argumentum. Concipio corpus maius et efficacius impingere in minus et imbecillius, certum est c[or]pus q[uo]d efficacius est, progredi debere in suo motu, quia non est q[uo]d determinatio mutetur, nisi a corpore fortiori, sed {…} quod impingit, supponitur imbecillius, sed cum in motu progreditur, debet aliud corpus secum movere, ac per consequens aliq[uo]d sui motu in id transferre. Ex hac {[n]a[m]} lege patet, non po[ss]e dari motum, quis omnia in perpetuo sint constituta motu, ex quo seq[ui]tur muta[ti]onem {q[uam]} ex motu {seq[ue]tur} e[ss]e constantem et continuam,” Hamburg 273, 106–109. Hamburg 274, 44: “[i]n secunda parte regulae supponitur nullam dari causam efficientem ad mutandam determinationem, ac proinde ea non mutabitur, non potest autem corpus motu eandem servare motus determinationem, nisi id corpus {cui} occurrit, secum propellat, nec potest propellere nisi {quandam} sui {motus} partem ei conferat, quam proinde amittit, ne augetur motus in universo quantitas.” The commentary on articles 40–44 present some divergent numbering across the different copies of the Hamburg 274 series of his dictata. All the copies of the series lack of a separate commentary on II.40 and II.43, but all their commentaries on II.44 are in fact devoted also to II.43. Moreover, the British Library dictata has the commentary on II.41 under the heading of II.39, and the commentary of II.42 under the heading of II.40.
- 213.
“Perdit. Progreditur sic rursus author methodo geometrica, hoc articulo non demonstrat, sed praeponit, q[uod] in seqq. probat. Duo corpora quiescentia non agunt in se invicem, nec mota in easdem partes, sed unum motum in alterum {quiescentem}, sed diverso modo videamus itaq[ue] quid ex his contingat: si impingens sit imbecillius, resilit, et reflectitur, servatq[que] motum amittendo determinationem. 2. Si {c[or]pus} motum impingi in quiescens imbecillius, tantum amittit de suo motu, q[ua]ntum quiescens recipit. Si vero pila incidat in arenam {a[ut]} lutum, cum luti partes sunt disiunctae potest una moveri immota manente alia, nonnulla igitur luti partes simul cum pila movebuntur, perdetq[ue] tantundem motus pila, q[ua]ntum lutum recipit,” Hamburg 273, 106. The commentary on this article is absent in the Hamburg 274 series of dictata. The commentary numbered II.40 in the British Library dictata in fact comments upon II.42.
- 214.
“Remanente. Aliud motus, aliud eius determinatio est, si {ex} diversam habeant causam, et motus po[ss]it manere, mutata determinatione, et vice versa, sequitur etiam diversas a se invicem e[ss]e. Motum {[n]a[m] eundem} e[ss]e po[ss]e, mutata determinatione, ex eo sequitur, q[uo]d 2 c[or]pora possint eadem velocitate moveri versus occidentem et orientem. Diversus {[n]a[m]} motus est non mutata determinatione, cum 2 c[or]pora versus easdem plagas inaequaliter moventur. Debere. Pila incidat in parietem, is non opponitur motui pilae, sed determinationi: q[uo]d cum ita sit, quid quaeso manifestius quam parietem sollumodo agere in pilae determinationem. Non {[n]a[m]} opponitur motui quia est corpus fortius, nec a pila moveri potest, sed situ solo eius determinationi contrariatur, servat itaq[ue] pila motum, sed non progreditur versus lineam rectam,” Hamburg 273, 106; cf. “[s]aepissime diversas dari causas motus, et determinatio motus patet ex iis, {q[uas]} dicit autor dioptr. 2. Patet ibide enim situm et dispositionem reticuli esse causam determinationis motus, vis vero {q[uae]} reticulum agit in pilam esse causam motus, ita ut eadem vi possit pilae motus conferri, sed in alias partes, modo aliter disponatur reticulum, ut vel maior, vel minor possit imprimi motus pilae, manente eadem determinatione,” Hamburg 274, 44.
- 215.
“Tum a magnitudine. Id e[nim] constat reliquis paribus, quo c[or]pus maius est, eo maiorem superficie: suppono c[or]pus sphaericum et quadratum eiusdem molis, prius ambiatur 5, alterum 6 pedibus, quoniam igitur sphaericum 5 po[ss]it ambiri pedibus, seq[ue]tur {e[ti]am} e[ss]e imbecillius. Modi. Si e[nim] c[or]pus v.g. pila recta feri perpendiculariter incidat in parietem, paries {totus} aget in pilam, si vero oblique non {totus}, sed motu tantum laterali resistet,” Hamburg 273, 109; cf. the other series: “[c]um maior requiratur vis ad mutationem inducendam maiori corpori q[ua]m minori, patet etiam vim resistendi maiorem esse omni mutationem, vi corpore maiori qui an vi minori. Sit enim A corpus duplum ipsius B, supponitur enim eandem omnia: in A itaq[ue] erit vis resistendi cuiuscunq[ue] mutationi duplo maior q[ua]m ea est, q[uae] est in B, non tamen sic spectanda sola magnitudo, sed et externa superficies. Nam ea corpora, q[uae] eiusdem cum aliis sunt molis, plus tamen superficiei habent, non possunt attingi a tot corporibus, ac proinde affici quam illa. Varius etiam occursus {multim} etiam potest variare, quia corpus q[uo]d perpendiculariter in aliud incidit, tota sua mole et omni sua celeritate et determinatione motus in illud agit. Illud vero corpus q[uo]d ad angulos valde obliquos, aliud corpus pertingit, nec tota mole, nec tota motus quantitate in id cui occurrit corpus operetur,” Hamburg 274, 44 (in this series the commentary on II.43 is indicated as II.44).
- 216.
For instance, in commenting on II.44, where Descartes set the various forms of opposition between bodies (i.e. (1) in states of movement and rest, (2) determination or direction of motion, (3) different speeds) De Volder overtly declares that these oppositions guide Descartes’s treatment of impact, in three different macro-cases (i.e. rules 1–3, 4–6 and 7): “[m]ovetur. Duplex est contrarietas, si sc[ilicet] 2 corpora pari velocitate, vel in easdem partes moveantur, aut si ambo quiescunt, nulla est contrarietas. Si vero duorum corporum unum moveatur, alterum q[ui]escat, necesse est, ut motus quieti sit contrarius, et in se invicem agunt; aut si 2 corpora in pari celeritate, aut in diversas moveatur plagas, necessaria est contrarietas. In pari velocitate i.e. si id q[uo]d praecedit moveatur tardius, {q[ue]m} id q[uo]d {seq[ue]tur}, si e[nim] praecedens moveatur celerius, nulla erit contrarietas, si vero tardius, sequens necessario impinget in illud, et eadem est contrarietas, ac si unum c[or]pus nempe id q[uo]d {seq[ue]tur}, unicum, praecedens nullum motus gradum haberet. Determinationem. Ut cum 2 c[or]pora moventur in contrarias partes, tunc nulla ratione motus, sed determinationis est contrarietas. Pro ratione partis. Ut si corpus motum incurrat in aliud motum, contrarietas est maior, quia incurrunt directe in superficies mutuas, ut cum concipimus 2 motos circulares, sibi invicem contrarias,” Hamburg 273, 109. A commentary on Principia II.44 is absent from the Hamburg 274 series of dictata.
- 217.
“Regulas sequentes. Haec regulae motus ab authore allatae nec experientia confirmari, nec infirmari queunt, quoniam sic supponuntur corpora perfecte dura, et nullo fluido {cinta}, quod possit motum multis modis variare, qualia corpora in rerum natura nullibi reperiuntur. Nec mirum est, hasce regulas multum discrepare ab illis nob. Hugenii, aliorum anglorum, quia si ratiocinati sunt de corporibus fluido undiq[ue] cunctis, in quibus suas regulas nihil posse agnoscit author. Cum confitetur §§ 55 corpus B in fluido constitutum, quantumvis exigua vi moveri posse. Q[uo]d denuo repetiti parag. LXII. quod apprime convenit cum regula Hugenii, directe autem contrariatur {regulae 4tae} authoris,” Hamburg 274, 44–45. Both the British Library dictata and The Hague dictata report the correct indication to II.61. The commentary to this article, in the other series of dictata, serves only to introduce the next articles: “[v]idimus generalia, quibus autor nunc porro applicat particularia exempla,” Hamburg 273, 109.
- 218.
- 219.
“In Gallica editione multo clarius proponit autor. Revera saepe contingit experientiam prima fronte videri repugnare six explicatis regulis. Sed ratio eius rei perspicua est, hae enim regulae supponunt duo corpora B et C plane dura, et ita separata ab omnibus aliis, ut nullum circa se habeant, quod possit eorum motum aut sistere, aut promovere, talia autem corpora in rerum natura nulla observantur, ita ut antequam iudicium ferre possumus, {…} hae regulae in mundo locum habeant an minus, [non] sufficiat {…} quo pacto 2 corpora B et C sibi mutuo occurrentia, in se invicem possunt agere, sed amplius necesse sit advertere, quo pacto corpora, {q[uae]} ambiunt ea, possunt vel augere vel imminuere eorum actionem,” Hamburg 274, 46–47; the addition of “non,” in brackets, is from British Library dictata, 95r. Please note that the part quoted by De Volder includes, in the 1647 French version, some phrases which were translated into French from the 1644 Latin edition; however, De Volder does not use the original Latin, and re-translates into Latin also these sentences: cf. the Latin version: “[s]ed quia nulla in mundo corpora esse possunt a reliquis omnibus ita divisa, et nulla circa nos esse solent plane dura, ideo multo difficilius iniri potest calculus, ad determinandum quantum cuiusque corporis motus ob aliorum occursum mutetur. Simul enim habenda est ratio eorum omnium, quae illud circumquaque contingunt, eaque, quantum ad hoc, valde diversos habent effectus, prout sunt dura vel fluida: quorum ideo diversitas in quo consistat, hic est quaerendum,” AT VIII-A, 70, with the French version (the new parts included in the French version are in italics, following to the AT edition): “[e]n effet, il arrive souvent que l’expérience peut sembler d’abord répugner aux règles que je viens d’expliquer, mais la raison en est évidente. Car elles présupposent que les deux corps B et C sont parfaitement durs, et tellement séparés de tous les autres, qu’il n’y en a aucun autour d’eux qui puisse aider ou empêcher leur mouvement; et nous n’en voyons point de tels en ce monde. C’est pourquoi, avant qu’on puisse juger si elles s’y observent ou non, il ne suffit pas de savoir comment deux corps, tels que B et C, peuvent agir l’un contre l’autre, lorsqu’ils se rencontrent; mais il faut, outre cela, considérer comment tous les autres corps qui les environnent peuvent augmenter ou diminuer leur action. Et parce qu’il n’y a rien qui leur fasse avoir en ceci des effets différents, sinon la différence qui 94 est entre eux, en ce que les uns sont liquides ou mous, et les autres durs, il est besoin que nous examinions, en cet endroit, en quoi consistent ces deux qualités d’être dur et d’être liquide,” AT IX-B, 93–94. Notably, the Pretoria copy, in this and in other cases in which De Volder dictated his Latin version of the French Principia, includes as marginal annotations also the original French text. Moreover, in the second column of The Hague copy, the Latin translations from the French Principia (i.e. those given in the Hamburg 274 series) are often provided as an additional commentary.
- 220.
“Q[uae]rendum. Vidimus has regulas, q[uo]d si consulamus experientiam, o[mn]ia prater has {fient}, super his regulis multi viri docti scripsere, sunt regulae Wallisis, Hugenii, etc. Belgarum, Anglorum, Gallorumq[ue]: q[uae]ritur unde diversitas sententiarum? {R[es]p[ondeo]} considerandum est nonnullos inter quos autor considerare 2 corpora ab o[mn]ibus aliis, motum promoventibus vel impedientibus remota (q[ua]tenus in rerum natura non dantur) si quidem omne c[or]pus per fluidum q[uo]ddam moventur{)} alios contra considerare corpora circumquaq[ue] aliis corporibus cuncta. Ne {[n]a[m]} q[ui]s putet regulas has falsas e[ss]e, quia repugnant experientiae, pergit sic ostendere, quid possint corpora fluida ad movenda alia, {a.} dura ad sistenda, et {c[um]} hoc constet, cur hae regulae falsae appareant; notandumq[ue] sic est c[or]pus in fluido haerere in aequilibrio, adeoq[ue] minimum c[or]pus id movere po[ss]e,” Hamburg 273, 113.
- 221.
See Sect. 5.5.2, Descartes’s corpuscular standpoint on hydrostatics.
- 222.
The disputation was followed on 6 December 1684 by the disputation De absoluta quiete (which has been mentioned above: see Sect. 4.1.3, De Volder’s ideas on cohesion and divisibility) covertly criticizing Boyle’s Discourse about the Absolute Rest in Bodies (1669), as in the 1684 disputation the author defended the idea that, notwithstanding that all parts of matter participate in some motion, some of them do not change their situs with respect to each other (this being the case of the sub-parts or sub-volumes of the globules of second matter: De Volder and Von der Lahr 1684, thesis 12). Boyle, on the other hand, supported the idea that the Cartesian theory of matter entails the impossibility of absolute rest, as all parts of matter are moved by subtler particles: Boyle 1669a, Discourse about the Absolute Rest in Bodies, 7–9.
- 223.
See supra, n. 18.
- 224.
“IV. Ex his sic detectis quaedam deinceps de motu sequuntur proprietates, quae ex legitima eius definitione ordine ratiocinando sumus deducturi: quarum prima sic se habet: motum esse mutuum, hoc est mutationem situs utriusque corpori esse communem, tam illius quod moveri, quam illius quod vulgo quiescere concipitur: quod etsi prima fronte videatur quid esse paradoxum, ut tamen melius innotescat, consideremus ex. gr. duo corpora A et B sibi invicem contigua, B non potest dici ipsum A tangere nisi ipsum A tangat B, atque ut iste contactus sit mutuus sic et omnis translatio illorum corporum erit mutua vel reciproca: ut pateat quid velim, supponamus rursus illa duo corpora A et B sibi mutuo esse vicina, et illam viciniam in posterum mutari; nimirum quando A movetur a B, atque ideo situm suum (respectu habito ad corpus B, cui prius vicinum erat) immutat, nunc autem ab ipso removetur et distat, eamque distantiam recedendo continue mutat: verisimile erit maxime, illam situs variationem, aequali modo in corpore B obtinere quam in A, hoc et, corpus B, aeque situm suum mutat respectu ipsius A, ac A respectu ipsius B: quod idem est ac si dicas B aeque distat ab A ac A a B, ex quo tunc patet evidentissime, eam distantiae variationem aequali modo prorsus fieri inter B et A ac inter A et B. V. Cum igitur manifestissimum est motum esse reciprocum seu relativum, tamen ut illam relationem evidentius percipiamus supponenda sunt haec duo corpora A et B prius adhuc continua, nullis aliis annexa esse, sed quasi poni in vacuo: atqui tum evidentissime patebit si consideremus vim aliunde accedere A non posse divelli a B in medio ac alio transferri, nisi in B vis quaedam esset nitendi et tendendi in contrarium, nam sine vi illa contraria impetu translato in alterum vel in A vel in B, non a se mutuo divellerentur, sed tota moles simul verteretur versus unam aliquam partem, vel versus dextram vel versus sinistram. Quod ut rudi exemplo declarem, consideremus duo navigia sibi obviam ferri; periculum quod utriusque imminet aequale erit, et ut propulsetur indifferens erit in utro horum consistat aliquis homo, qui harpagine motum sui et alterius navigii sistat simul et eadem opera; eodem quoque prorsus modo si aliquis stans in terra fune navigium stipiti alicui attrahat, sive alius stans in navi mota funem fixum stipiti alicui attrahat, eaedem plane hoc in casu requiruntur vires. Sic in illis duobus, ille qui est in terra alter in mota navi, manibus simul eundem funem utrimque trahentes, simul et eadem vi navigium ripae, et ripam navi appropinquare faciunt. VI. Evidentissime igitur patet, motum inter corpora esse relativum, seu eius naturam esse reciprocam, quod ex eius definitione clarissime patet: motus enim essentia nihil considerandum praeter vicinitatem et distantiam nobis proponit, atque hae ambae non aliter quam mutuae possunt concipi, cum enim A concipitur distare a B simul etiam necessario B distat ab A et cum A vicinum sit ipsi B, sic eodem tempore debet esse vicium ipsi A […]. Primum itaque quod in motu concipimus est quod mutuus sit, quia quicquid ad naturam eius pertinet mutuum est,” De Volder and Vander Codde 1684, theses 4–6. It is noteworthy that, in the theses 1–3 the author stresses that movement induces the succession of our thoughts (thesis 1), that movement, understood as a mutation of situs, is a matter of absence or presence of one body with respect to another (thesis 2), and that movement, which does not take in place in an instant, is a succession of our thoughts (“motus igitur natura […], quod consistat nempe in tali relatione nostrarum cogitationum, qua consideramus aliqua prius sibi invicem vicina deinde a se mutuo distantia, evidentissime sequitur motum non posse concipi sine tempore; hoc est […] hoc ipso quaedam successio inde resultat, quae nobis tempus dicitur,” thesis 3). The author, however, does not avail a purely mental characterization of movement, which is a real change in distance.
- 225.
“VII. Motum itaque in se considerando, seu illud solummodo attendentes, quod revera inter corpora quae dicuntur moveri deprenditur, et quapropter proprie moveri considerantur, quoties duo corpora sibi mutuo vicina, unum in hanc alterum in illam transferuntur partem, atque a se invicem separantur, concipimus esse tantundem motus in uno quam in altero: considerare tamen assueti sumus aliqua corpora quiescere: ast cum illa situs mutatio seu relatio utrumque aeque spectat, penitus a nostro dependet arbitrio, cuinam corpori motus sit assignandus, cuinam vero quies, cum eadem fiat relatio respectu utriusque corporis, eadem distantiae mutatio. VIII. Quod ut evidentius innotescat, cogitandum est, etsi verum sit motu introducto nihil immotum posse reperiri, cunctaque vero moveri: possumus tamen corpus aliquod particulare abstrahere, illudque sic abstractum cogitare, quod comparatum cum uno pluribusve aliis, relationem sui situs non mutet, et eatenus illud corpus (comparatione sic instituta) in illa mea cogitatione quiescet. Cum interea dum, alia quaedam alterius cuiusdam hominis cogitatio illud conferat cum alius, quibuscum (alio respectu habito) moveri concipietur. Sic ex. gr. homo navigio vectus movebitur respectu littorum, si vero rationem habeamus respectu partium navigii, revera censebitur quiescere,” De Volder and Vander Codde 1684, theses 7–8.
- 226.
“IX. His sic detectis pergam ad alte nobis infixum praeiudicium, quo extimamus plus realitatis esse in motu quam in quiete, quod praeiudicium ut eximatur, considerandum velim omnia esse in motu, adeoque necessario causam aequipollentem requiri, quae aliquod corpus iam in motu positum ad statum quietis reducet, maneret utique alias illud corpus in motu, nec unquam quieturum esse conciperetur. Quod cumque igitur fuerit, quo corpus motum ad quietem sit reducendum, necesse habet vim et efficaciam illius causae tollere qua motus introductus est, ad hoc autem praestandum quid quaeso valeret nisi minimum tanta vi polleret, quanta pollet causa movens. Ex. gr. consideremus duo corpora A B, perfecte dura, nullis aliis corporibus annexa. Concipiamus iam ambo esse magnitudine et soliditate aequalia, dein consideremus A moveri, B quiescere. Aio ubi A attigerit ipsum B, moturum quidem B, sed rursus repelletur ab B, adeo ut ambi aeque celeriter pergant moveri in partes contrarias, quia sunt ambo magnitudine et soliditate aequalia. Nam B tantundem virium habet ad resistendum quam A ad impellendum. X. Dein et hoc meretur considerationem, quod putemus plus actionis requiri ad motum quam ad quietem: quod praeiudicium facile exuemus; si in memoria revocemus […] quietem requirere aeque realem causam, quam motum. Vel hoc evincet, quod extensio non maiorem habet conatum ad motum quam ad quietem. Corpus igitur aeque quiescit a causa externa quam a causa externa movetur. Sed haec est causa cur putemus quietem non requirere tantundem actionis quam motum, quod quies in corporibus mundi sublunaris producatur ob concursum plurium causarum quae saepe nos latent, e contra vis et actio manifestius deprehenditur ad motum inter eadem corpora producendum,” De Volder and Vander Codde 1684, theses 9–10.
- 227.
This condition is stated by Huygens only in his De motu corporum ex percussione, as hypothesis 3.
- 228.
Lodge 2013, Introduction, xxvii–xlv.
- 229.
The Papin-Leibniz controversy has been magisterially reconstructed in Freudenthal 2002.
- 230.
“Cum nuper Leyda transirem, Volderum conveni saepius […]. Vidi praeterea illum aliquoties apud alios Professores, sensi illum esse summum Tui admiratorem. […] Intellexi quoque quod a Papini parte stet circa aestimationem virium, dicens a Te gratis supponi corporum elasticitatem, hanc enim corporibus tantum esse accidentalem; si corpora perfecte dura supponantur Tua ratiocinia amplius locum non habere: duritie perfecta supposita sequi ex. gr. quod duo corpora aequalia et aequali celeritate sibi mutuo centraliter occurrentia, non debeant reflecti sed sisti in ipso momento occursus; et quod generaliter corpora duo perfecte dura cujuscunque molis et cujuscunque celeritatis post conflictum separari non debeant, quia separatio proveniat ab elasticitate, quam autem supponendam dicit,” Bernoulli to Leibniz, 5/15 July 1699, in A III7B, 205.
- 231.
“[C]um denuo inspiciens, quae his de rebus in Actis Lipsiensibus habentur perspexerim primariam meam difficultatem a Clar. Papino praeoccupatam. Quibus accedit quod perpendens argumenta hinc inde allata, ea talia inveniam, ut me utrinque omni scrupulo liberare nequeam, adeoque neutram partem pro certa amplecti audeam,” De Volder to Bernoulli for Leibniz, 21 November/December 1698, in A II3B, 477. De Volder was referring to the exchange between Papin and Leibniz in the Acta Eruditorum, i.e. to Papin’s De gravitatis causa et proprietatibus observationes (1689), Leibniz’s De causa gravitatis et defensio sententiae suae de veris naturae legibus contra Cartesianos (1690), Papin’s Mechanicorum de viribus motricibus sententia (1691), Leibniz’s De legibus naturae et vera aestimatione virium motricium contra Cartesianos (1691). Also Leibniz’s Brevis demonstratio erroris memorabilis Cartesii appeared in the Acta Eruditorum (March 1686).
- 232.
“Me quod attinet, de quantitate virium an eadem an diversa, quod certo asseverem, nihil habeo. Causam, quae vires augeat, nullam reperio; verum cum vires sibi invicem sint contrariae, quid impediat, quominus altera alteram tollat non video. Corpora mollia aequalia, et aequali velocitate sibi invicem occurrentia cessant a motu suo. Idem de corporibus duris non elasticis, quidquid etiam supponat Hugenius, ut credam admodum proclivis sum. Mihi quippe verisimile apparet, corpus quod aequali vi, sive eam nunc demum accipiat, sive jam ante aliunde acceperit, in contrarias pellitur partes, quieturum. Quod si verum est corpora aequalia sibi invicem aequali motu occurrentia, quiescent ambo. Nec liquet, cur non et quantitas motus, et vires hoc in casu dicantur deperditae,” De Volder to Bernoulli for Leibniz, 21 November/December 1698, in A II3B, 480–481.
- 233.
See the letter of De Volder to Leibniz of 12/22 November 1699: “[c]onstat autem jam inter nos, id quod in corporum elasticorum concursu conservatur, esse ut quadratum celeritatum ductum in molem,” A II3B, 611.
- 234.
In his letter to Leibniz of 21 November/1 December 1698, De Volder points out how Leibniz’s theory of the conservation of living force (m·s2) agrees with Huygens’s rules of collisions presented in the Philosophical Transactions, which, for De Volder, Huygens based on the hypothesis – which he could read, as a hypothesis, in the manuscript of De motu corporum ex percussione – that Descartes’s first rule of collision is true: “[q]uod autem monet Vir Ill. se vires vocare, quae semper maneant pari quantitate, cum motus quantitas augeatur minuaturve, id consentit cum Regulis motus Hugenii, quas in Transact. Anglicis videre est, in quibus, quantitas motus licet augeatur, minuaturve, summam tamen productorum factorum a mole cujuslibet corporis duri, ducta[m] in quadratum suae celeritatis, eandem semper esse ante et post eorum occursum, affirmat. Deducit has regulas Hugenius ex hypothesi, corpora dura aequalia, aequaliterque in se invicem mota, post occursum reflecti singula, eadem qua prius, velocitate, qua tamen cum Hypothesi, qui consentiat id, quod ait Reg. V. semper remanere eandem quantitatem motus versus eandem partem, ablata inde quantitate motus contrarii, perceptu mihi arduum videtur,” De Volder to Bernoulli for Leibniz, 21 November/1 December 1698, in A II3B, 480. Cf. Huygens 1669a, 928, 1703, 369. In claiming that Descartes’s first rule contradicts Huygens’s 1669 rule 5, De Volder (probably) pointed out that in Descartes’s first rule the difference between the two quantities of motion (which are equal, and exerted in opposite directions) is 0. In other cases, Huygens’s rule 5 is valid, i.e. the quantity of motion is always equal in both directions, once the contrary quantities are subtracted.
- 235.
“[…] si a priori demonstratum haberemus, omnem substantiam esse activam, facile mihi persuadeo ex hoc foecundissimo veritatum fonte non horum tantum scrupulorum secuturam enodationem, sed et earum difficultatum, quae hactenus omnes quotquot sunt, Physicos presserunt. Hujus etenim rei ignorantia, compulsi sunt causam motus conjicere in Deum, et nonnulli etiam ad quoslibet corporum occursus Deum e machina arcessere,” De Volder to Bernoulli for Leibniz, 21 November/December 1698, in A II3B, 481. See also the letter of De Volder to Leibniz of 12/22 November 1699: “[a]n mens mea a falsis notionibus vulgi, et Cartesianorum circa materiam et motum et substantiam corpoream satis libera sit, asserere non ausim. Id tamen me satis intelligere existimo, virium actionumque regulas ex eorum principiis deduci non posse, quascunque etiam leges sequamur, adeoque vel cum iis ad Deum confugiendum esse […], quod minime probo, vel vires corporum ex substantia corporea deducendas, quod utinam possem,” in A II3B, 612.
- 236.
“Accedo nunc ad id quod primarium est, quod sc. spectat substantiae activitatem; quo discusso, reliqua de quibus per occasionem mentio facta est, facile, ut opinor, enodari poterunt,” De Volder to Leibniz, 5 April 1700, in A II3B, 632.
- 237.
“Te scilicet praeter extensionem et impenetrabilitatem tertium quid requirere ad essentiam corporis quod consistat in vi ingenita ad conservationem virium, unde necessario vim elasticam omnibus corporibus ex natura sua competere. Ille autem regessit Te aliquid statuere quod concipere non possis; an illud tertium sit substantia an modus? si modus, nihil novi esse. Sin substantia an spiritus an corpus? aut forte tertium? hoc autem tertium explicari non posse, nisi cum veteribus ad formam substantialem diu explosam recurrere velis,” Bernoulli to Leibniz, 23 July/2 August 1698, in A III7B, 847. See also the letter of De Volder to Leibniz of 18/28 February 1699, commenting upon Huygens’s rules of impact: “[v]erum, licet hoc ita eveniat in corporibus perfecte elasticis, hanc tamen virium aestimationem omnibus corporibus qui applicare possis, non video; idque eo minus, quia hae ipsae vires in elasticis sequuntur ex quadam virium amissione in non elasticis, mea quidem in hypothesi. Ubi agitur de viribus corporum, non id spectandum est, quod ex peculiari corporum proprietate sequitur, sed id, quod ex generica omnium corporum fluens natura, omnibus, cujuscunque de caetero sint conditionis, corporibus ex aequo inest. Quare, Vir Amplissime, si velis ut absque omni scrupulo tibi assentiamur, descendendum erit, opinor, ad notionem substantiae, demonstrandumque eam ex natura sua necessario esse activam, aut certe eam esse naturam substantiae corporeae, ut necesse sit ejus vires conservari semper. Hoc enim nisi a priori demonstratum sit, non ita facile, mihi certe, erit concipere, corporis mathematici existentiam plane esse contradictoriam,” De Volder to Leibniz, 18/28 February 1699, in A II3B, 530.
- 238.
Lodge 2013, 143; cf. A II3B, 611: “[…] si demonstratum sit, omnem substantiam esse ex natura sua activam, adeoque eam, quae ex substantiae natura manat vim activam necessario conservari, patebit omnia quae contrariis motibus in se invicem impingunt corpora, esse elastica.”
- 239.
“Ac primum non magis mihi gaudeo, quam Tibi gratulor repertam veritatem, et comprobatas meas aestimandarum virium leges. […] Desideras postremo, Vir Nobilissime, Substantiae activitatem ex se, non tantum velut a posteriori (ut vocant) et ex eo probari quod ab alia creata substantia nihil accipere potest, et agit tamen, ut experientia constat; sed etiam a priori seu ex ipsa substantiae natura derivari. Verum ita vides constituendam prius esse Substantiae definitionem ac de ea nobis conveniendum, quod etsi non exiguae fortasse operae, non tamen fateor sine operae pretio foret,” Leibniz to De Volder, 6 September 1700, in A II3B, 654–656.
- 240.
“Sed haec ipsa virium aestimatio mentem praeparat, ut facilius sustineat majorem illam lucem, qua perstringuntur nimis velut […] caligine, qui vulgi et Cartesianorum etiam notionibus circa naturam materiae, motus, et virium sunt assueti. Itaque per hanc portam a Mathesi ad metaphysicam transeundum censeo,” Leibniz to Bernoulli, 1/11 September 1699, in A III8, 227.
- 241.
“D. Volderus postulat demonstrationem activitatis substantiarum; jam vides quam difficile sit demonstrare res Mathematico-physicas quales mechanicae sunt, quid de Metaphysicis futurum putas? Demonstratio est ratiocinatio, cujus vis sit evidens, et a qua indubitatam Tibi adversariorum convictionem promittas; talem etiam cum omnia animo jam comprehensa habemus exprimere verbis difficile est. Quanto magis cum nondum ipsi satis ordinavimus cogitationes nostras?” Leibniz to Bernoulli, 12/22 January 1700, in A III8, 301.
- 242.
For this reason, in his letter to Leibniz of 6/16 December 1698, Bernoulli encouraged Leibniz to compose a system of philosophy: “non dubito quin systema si quod componeres, felicissimum successum habiturum esset, sunt enim in Batavis nonnulli egregii Viri qui hunc Tuum ingenii foetum fortiter foverent et defenderent, interque illos ipse Dn. Volderus qui cum Cartesiana principia tanquam insufficientia et plurimum falsa ut ipse mihi fassus est jam a longo tempore deseruerit, si Tua semel probe percepisset et imbibisset, dubium non est quin ea gnaviter propagaturus suisque Discipulis quorum semper insignem numerum habet adeo esset inculcaturus,” A III7B, 960.
- 243.
See supra, n. 229.
- 244.
“Ego vero, ut ostendam quantum inter haec duo intersit, suppono, primo corpus cadens ex certa altitudine acquirere vim eousque rursus assurgendi, si directio ejus ita ferat, nec quicquam externorum impediat: exempli causa, pendulum ad altitudinem ex qua dimissum est praecise rediturum esse, nisi aeris resistentia similiaque impedimenta exigua alia nonnihil de vi ejus absorberent a quibus nos quidem nunc animum abstrahimus. Suppono item secundo, tanta vi opus esse ad elevandum corpus A unius librae usque ad altitudinem CD quatuor ulnarum, quanta opus est ad elevandum corpus B quatuor librarum, usque ad altitudinem EF unius ulna. Omnia haec a Cartesianis pariter ac caeteris Philosophis et Mathematicis nostri temporis conceduntur,” A VI4C, 2028. As he would clarify in his letter to De Volder of 27 December 1698, the second premise opens Descartes’s Traité de la mécanique (published in 1668): “[e]quidem Cartesius eam admiserat eaque erat usus in Mechanico suo Schediasmate, quod et fecerat Pascalius in tr. de aequilibrio liquorum,” Leibniz to De Volder, 27 December 1698, in A II3B, 495. Cf. AT II, 432: “[l]’invention de tous ces engins n’est fondée que sur un seul principe, qui est que la même force qui peut lever un poids, par exemple, de cent livres à la hauteur de deux pieds, en peut aussi lever un de deux cents livres à la hauteur d’un pied, ou un de quatre cents à la hauteur d’un demi-pied, et ainsi des autres, si tant est qu’elle lui soit appliquée.”
- 245.
- 246.
“Dissimulare tamen non possum, in illo exemplo, quo Vir Ill. utitur, in quo vires quae corpus 4 lb. attollere poterant ad pedem, postmodum tollunt ad quatuor, esse quidpiam, ex quo me satis commode extricare nequeo. Verum huic oppono illud, quod ex Ampl. Leibnitzii opinione sequitur, vires corporum admodum inaequales sibi tamen invicem in statera aequipollere. Paradoxum hoc vocat Vir Ill. sed nescio qui fiat, ut hoc paradoxum mihi aeque absurdum videatur, ac illud, quod modo attuli,” A II3B, 479–480.
- 247.
As Tzuchien Tho has put it, “[v]is viva describes the action of vis in extended motion while vis mortua only expresses momentaneous solicitation to move. In other words, the concept of vis mortua allows us to apply our understanding of gravitational solicitation as the “starting point” of an extended motion. […] [J]ust as a point constitutes the extremum of a line segment, vis mortua constitutes the “impetus” of a motion that has not yet begun to move. In turn, any motion can be analyzed into smaller line segments at every scale. In turn each of those line segments, smaller than can be assigned, will have (at least one) extremum. In this sense, the distinction between vis viva and mortua allows for the analysis of the continuity of motion without the reciprocal idea of the composition of motion from tendencies to move (tendencies which are themselves immobile),” Tho 2017, 82. As seen in Sect. 4.1.4, The search for physical unity and activity: De Volder vis-à-vis Leibniz, indeed, Leibniz claimed in his Specimen dynamicum that movement is not ‘real’ because it cannot be considered as taking place in a single instant; however, dead force is exerted in instants.
- 248.
“Transitio enim nunquam sit, nisi per inassignabilia incrementa seu infinite parva, adeoque per vires mortuas: et aequilibrii lex nunquam nisi quoad vires mortuas exercetur, idque sive corpora gravia impetu adhuc careant, ut fit in statera, ubi utrinque tantum descendere conantur, sive impetum jam conceperint, ut in concursu: nunquam enim aliud agunt invicem, quam quod mutuo sibi detrahunt aequalem motus (sed infinite parvi seu mortui) quantitatem. Si enim corpus A sit mole ut 4 celeritate ut 1, et corpus B mole ut 1 celeritate ut 4, tunc in conflictu sive staterae cum tantum conantur, sive ictus vel concursus, cum se impetu suo urgent, continue perit ipsi A gradus celeritatis infinite parvus ut 1, ipsi vero B gradus celeritatis infinite parvus ut 4, reciprocus scilicet corporibus. Nempe ipsi A quantitas motus mortui 4 in 1. seu 4, et ipsi B quantitas motus mortui 1 in 4 seu itidem 4. Idque ex lege aequilibrii Archimedea; neque enim demonstrationes Archimedis et aliorum nisi in conatibus istis primis per se mutuo tollendis locum habent. Quemadmodum fit cum pondera 4 et 1. ita in statera posita sunt ut nequeant descendere nisi illud incipiat descendere vel ascendere celeritate ut 1. hoc ascendere vel descendere celeritate ut 4,” Leibniz to De Volder, 27 December 1698, in A II3B, 497–498.
- 249.
Lodge 2013, 31. Cf. A II3B, 498–499: “[u]t autem cum corporum descensus impeditur aequilibrio, totum utriusque pondus sustinet fulcrum librae et pressionem aliquam flexionemque seu fibrarum tensionem accipit: ita quicquid corporibus perit virium transfertur in Elasma ipsorum corporum vel vinculi quo conjunguntur vel etiam si non satis Elastica sint (ut mollia), in motum partium toti (quod Elastrum faceret) non reddendum, sed plane ipsius respectu pereuntem. […] Hinc consequens est ut corpora A et B in casu praesenti se tandem mutuo sistant […]. Nam cum eandem habeant quantitatem (vivi) motus, et aequalia utrinque sint inter confligendum decrementa quantitatis motus ex lege aequilibrii per detractionem continuam virium mortuarum seu celeritatum molibus reciprocarum, hinc simul exhauriri utriusque motum necesse est.”
- 250.
“Interim uti reperitur in ipsis differentiis seu crementis semper servari legem aequilibrii, ita reperitur mira naturae arte, in ipsis terminis integris seu viribus vivis, computando tam id quod in corporibus residuum est, quam totum id quod est detractum et in Elasma translatum, semper eandem conservari vim vivam secundum legem aequipollentiae. Idque maxime apparet post restitutionem Elastri, cum Corpora a se invicem denuo recedentia, vim totam recepere. Tunc enim si fingas ea impetum suum sursum convertere ut si progrediendo se pendulo implicarent, reperitur semper eadem ascensus totalis quantitas, quae initio ante concursum obtineri potuisset; posito nihil accidentibus absorberi. In medio autem conflictu si corpora vim residuam sursum convertere ponas, fingendum praeterea erit Elasma corporum vel aliud elasma aequipollens vim suam in concursu acceptam totam exercere in aliquod corpus impositum, sursum ejaculandum. Ita rursus reperietur in summa eadem quae ante concursum ascensus quantitas,” Leibniz to De Volder, 27 December 1698, in A II3B, 499.
- 251.
“Hinc jam porro deprehenditur, naturam elegantissime conciliare legem aequilibrii confligentium, quae est respectiva, cum lege aequipollentiae causarum et effectuum, quae est absoluta. Idque mediante lege transitionis gradualis saltum omnem evitantis,” A II3B, 497–498.
- 252.
“Sed quantumcunque exiguum sit corpus tamen rationem habet ad magnum, et est aliqua vis conflictus, licet fateor exigua et corporis utriusque renitente Elastro eorum fit inflexio, per quam vis conflictus paulatim absumitur et in elastrum transfertur similiter paulatim ab eo restituenda. Caeterum Mollia corpora absorbent vim non destruendo, sed recipiendo in suis particulis exiguis, aut impendendo in abruptiones filorum ex quibus ipsa leviter connectantur; quemadmodum si globus per multa folia chartae trajiciatur. Vis autem filo rumpendo impensa iterum non perit, sed recipitur in materiam adhuc subtiliorem, prorsus ut vis quae [impenderetur] duabus Tabulis politis a se invicem divellendis, de quibus tam praeclara a Te experimenta facta accepi,” A II3B, 509.
- 253.
“Celeberrimus Hugenius, qui Te Hanovera transiisse credebat in Epistola quadam sua inter alia et de Te quaesivit, meminitque nescio cuius experimenti a Te insinuati, quod nondum sibi Volderoque successerit Mercurium per Siphonem attrahere volentibus, credo mente tua non satis percepta,” A V3, 16. Cf. the letter of Christiaan Huygens to Leibniz of 18 November 1690, claiming that the experiment, according to Spener and De Volder, could not be successful: “[j]e ne scay s’il vous a debitè une experience avec du mercure attirè par un siphon, que je ne pus croire et que j’ay aussi trouvée fausse et Mr. de Volder de mesme,” Huygens 1888–1950, volume 9, 540.
- 254.
De Volder’s experimental activities are mentioned in a letter of Bernoulli to Leibniz of 8/18 January 1698, but no detail is provided: “[o]rdines nostri novam mihi imposuerunt docendi provinciam, atque in eum finem certam decreverunt summam ad emenda instrumenta experimentalia, ut exemplo Volderi Lugdunensis Studiosos nostros etiam experimentis Mathematico-physicis exerceam et delectem,” A VIIB, 702. As to Dalrymple, see supra, n. 69.
- 255.
“Quod ad difficultatem meam, de staterae, aut etiam, ut recte addis, corporum reciproca ad magnitudinem celeritate in se invicem incurrentium aequilibrio attinet, eam (si quidem constiterit, aequi[li]brii legem nunquam nisi quoad vires mortuas exerceri, aut potius, si quidem demonstratum sit axioma tuum, nullam transitionem fieri per saltum, (ex hoc enim prius illud sequi video,) solutam esse agnosco,” De Volder to Leibniz, 18/28 February 1699, in A III8, 527–528.
- 256.
- 257.
“Idem praecise praesens systema per omnia et in omnibus non nisi uno modo produci posse arbitror, nec causas ejus diversas possibiles agnosco. Electio igitur mea sententia locum habet inter diversa systemata possibilia, non inter diversos ejusdem pleni systematis producendi modos. Bonum, perfectionem, ordinem putem non minus liquidis rationibus constare quam numeros et figuras. Finge triangulum produci debere, nullam autem esse rationem aliam qua species trianguli determinetur, haud dubie aequilaterum producetur,” A II3B, 570.
- 258.
“Quod autem corpora inflexibiliter dura attinet, quales Atomos fingimus, Tecum plane sentio, in illis nullam esse posse reflexionem, neque etiam virium conservationem, sed vicissim nulla talia in natura esse arbitror, quod non aegre credo admittes. Elasma ego corporibus essentiale puto ex rerum ordine et metaphysicis principiis,” Leibniz to De Volder, 27 December 1698, in A II3B, 509. Cf. also Leibniz to Bernoulli, 20/30 September 1698, in A III7B, 912–913: “[h]aec autem evitatio saltus in mutationibus corporum obtinetur per vim elasticam ipsis inexistentem. Ita enim fit ut corpora in concursu sese comprimentia et mox restituentia paulatim sibi cedant et graduali translatione directiones, viresque et ipsas ut demonstratum vidisti actionum motricium quantitates (longe a vulgo intellecta Quantitate motus diversas) conservent.”
- 259.
See the letter of De Volder to Leibniz of 18/28 February 1698, referring to Leibniz’s idea of a ‘principle of order’ (expressed in the letter of Leibniz to De Volder of 24 March/3 April 1699, as grounding the law of continuity: “[a]xioma quo utor: nullam transitionem fieri per saltum. Id fluere arbitror ex ordinis lege,” A II3B, 545): “[h]aec obstant, quo illa argumenta, quae ex ordine Malbranchius imprimis, desumit, minus me afficiant, ut et illa, quae ex beneplacito Dei deducuntur, hoc est, ut mihi videtur, ex principio obscurissimo, et quod saepenumero nullum habet fundamentum praeter ignorantiam nostram,” A II3B, 560.
- 260.
See Sect. 4.2.3.4, Collision in De Volder’s late thought.
- 261.
“Elasma ego corporibus essentiale puto ex rerum ordine et metaphysicis principiis: etsi in natura non aliter quam per fluidum intercurrens pergatur. In quo plane assentior Cartesio,” Leibniz to De Volder, 27 December 1698, in A II3B, 509. Cf. the quotation given supra, n. 258. The same position was also given in the letter of Leibniz to Bernoulli of 12/22 July 1698, commenting upon the first letter of the De Volder-Bernoulli-Leibniz correspondence (5/15 July 1698), where Bernoulli reported De Volder’s idea that when two hard bodies collide, they lose all their movement (see supra, n. 230). In his letter, Leibniz points out that De Volder was following the same kind of reasoning (neglecting elasticity) that he, Leibniz, adopted in his Theoria motus abstracti (1670–1671) – in which Leibniz came to the same conclusion as De Volder. This idea of collision in which forces are lost, for Leibniz, is derived from the consideration of the extension and impenetrability of matter alone, without considering elasticity, which he anyway conceives not as an internal quality of matter, but rather as a specific structure of it (à la Descartes): “[e]go ipse olim adolescens cum de legibus motus scriberem libellum, in ea eram sententia quae nunc est Domini Volderi, duo corpora aequalia et aequivelocia sua natura post concursum directum non debere reflecti, sed potius se sistere mutuo. Idque etiam sequitur ex vulgari notione materiae cum nempe nihil aliud in ea concipitur quam extensio et ἀντιτυπία seu impenetrabilitas. Sed ex his ipsis et similibus postea agnovi longe aliam esse naturam materiae in systema mundi redactae quam vulgo creditur; et vim Elasticam omni corpori esse essentialem, non ita quasi ea vis sit aliqua qualitas inexplicabilis, sed ex eo quod omne corpus utcunque parvum est machina ex cujus structura resilitionem, ubi opus ea est ad virium conservationem, oriri oportet,” Leibniz to Bernoulli, 12/22 July 1698, in A III7B, 827. Cf. Leibniz’s Theoria motus abstracti: “[t]heoremata. 1. Si corpus impingit in aliud quiescens, vel tardius directe occurrens vel tardius antecedens, secum abripit (id est movet in eandem plagam) differentia celeritatum. […] 12. Si non detur angulus, qui sit bisectilis (non datur autem angulus omnino in occursu recto, non datur angulus bisectilis in alio impactu, si impingitur linea motus recta et curva, vel curva et curva figurae dissimilis aut inequalis) et impactus aequivelox est, utrumque quiescet,” GM VI, 72–73.
- 262.
Lodge 2013, 73. Cf. the original text: “[c]aeterum quod ista virium aestimatio omnibus corporibus applicari possit, nunc quidem confirmo, satisfaciendo difficultatibus quae contra opponi possunt. Nempe hypothesi mea, quatenus corpora perfecte elastica non sunt, vis intestinis partibus quae et ipsae Elasticae sunt, recipitur, neque adeo perit, sed tantum sensibus subducitur, quae quidem consuetudini naturae et ordini id est experientiae et rationi consentanea esse non negabis. Nec sine Elasmate axiomata aut vitandorum saltuum, aut conservandarum virium tam absolutarum quam respectivarum vel conciliationes legum vis mortuae et vivae, compositionisque motuum cum quantitate virium obtineri possent. Et vero ista non nisi ex lege ordinis supremi demonstrari possunt, neque enim sunt absolutae necessitatis, ut contrarium implicet contradictionem. Innumerabilibus modis poterat constitui systema rerum, sed praevaluit, quod majore ratione nitebatur. Substantiae tamen activitas magis est metaphysicae necessitatis, et locum ni fallor habitura erat in systemate quocunque,” A II3B, 545–546. In his letter to Leibniz of 18/28 February 1699, De Volder replied that, if one supposes a law of order, one cannot drawn but absolutely necessary conclusions: “[p]rimum equidem non capio, qua ex lege ordinis determinare possimus ea, quae absolutae necessitatis non sunt,” A II3B, 560.
- 263.
- 264.
See the letter of Bernoulli to De Volder of 13 February 1700, published also in A III8, 413–414. Cf. Wolff 1713, volume 1, 594–595. See also the letters of Bernoulli to De Volder of 3 January 1699, 8 April 1699, and 31 May 1699. The argument was re-formulated and presented by De Volder to Leibniz in his letter of 5 April 1700 (A II3B, 630–633). Bernoulli was later to discuss Huygens’s laws of collision, as formulated in his posthumous treatise, in his letters to De Volder of 10 July 1703, 11 December 1703, 5 January 1704, 7 January 1704. The demonstration is also mentioned in the letters of Bernoulli to Jakob Hermann of 21 July 1718, to Wolff of 27 March 1722, to Willem Jacob ’s Gravesande of 31 October 1722 (see Sect. 2.4.2, De Volder’s retirement, death and legacy) – also printed in the article ’s Gravesande in the Dictionnaire historique of Prosper Marchand (see Marchand 1758–1759, volume 1, 234) –, to Georg Bernhard Bilfinger of 30 September 1722.
- 265.
See De Volder to Leibniz, Leiden, 5 April 1700, in A II3B, 630–633.
- 266.
Lodge 2013, 179–180; cf. Leibniz to De Volder, 6 September 1700: “[i]nterim haec quae veritatem optime comprobant, et contraria ad absurdum redigunt, cum 5 tamen specialibus assumendis indigeant velut gravitate, elastro, concursu obliquo, non satis aperiunt fontes causasque, quod sola facere potest mea actionis aestimatio; quam Tibi jam arridere non dubito, cum illud pulcherrimum appareat dum eadem servatur quantitas virium, etiam eandem aequalibus temporibus quantitatem Actionis motricis in universo conservari: quod Cartesius velut per nebulam cum transvidisset expetissetque, non bene est assecutus, celeritate pro viribus substituta,” A II3B, 655.
- 267.
- 268.
“Si quae sic impingunt corpora, intelligantur non absolute dura (prout hactenus supposuimus) sed ita ictui cedentia, ut elastica tamen vi se valeant restituere, hinc fieri poterit ut a se mutuo resiliant ea corpora […],” Wallis 1668, 866. I will deal with Wallis’s treatment of collision in his Mechanica (1670–1671) below.
- 269.
Mariotte 1673, 1–2. Wren also, according to his 1668 communication, carried out several experiments on collisions: Wren 1668, 857. Huygens and Wallis do not mention any experimental confirmation or development of their theories. De Volder probably met Mariotte during his visit to Paris in 1681; he owned Mariotte’s Traité: see Bibliotheca Volderina, 10. As to De Volder’s experiments with collisions, see Sect. 4.2.3.4.2, De Volder on elastic impacts.
- 270.
Descartes 1982, 242.
- 271.
“Ductum est. {Q[uaero]}: unde resiliendi illa vis, videtur {e[nim]} id repugnare regulae naturae, qua o[mn]ia c[or]pora conantur manere in statu in quo sunt? {R[espondeo]} causam q[uae]rendam esse in aere, {q[uae]} haec non solum in vitro, sed o[mn]ibus {q[uae]} se immediate conting[un]t, efficit. Poss[un]t. Id vult, utamur arcu laxato, dum eius partes cohaerent immediate, multi in iis sunt pori materiae {i. expervi}, ubi itaq[ue] inflectitur arcus, ita tamen ut non frangatur, {q[uo]d} fit? Pori mutantur, et una diameter fit altera maior, accedat itaq[ue] mat[eria] subtilis, haec per maiorem diametrum facile ferri potens impingit in minorem, donec cessante externa vi excluduntur latera, et arcus ad priorem statum reducitur. Aptantur. Si vero lignum sit nimis durum, {par[ticu]lae advellentur, q[uae]} transitum impediebant, sicq[ue] continuatur tensio,” Hamburg 273, 315. See also the text of the other series of dictata: “[e]x iis, q[uae] hoc articulo dicuntur, causae pendent, propter quas omnia corpora, q[uae] flecti nequeunt, resiliant, hoc enim ea fit de causa, quia corpora ea rigida non possunt omnibus ita disponi modis, ut liberum transitum praebeant materiae caelesti per ea fluenti, sed cum eorum {pori a} materia ea formati, eiusq[ue] magnitudini ac motui accomodati sint, si imminuentur, aliquod materiae motui impedimentum possunt praebere, cum autem constent {particulis} rigidis, non facile earum figura mutari potest, hinc in eas agere caelestis materia omnes suas intendit vires, ut eas in pristinum statum reducat, inde vis illa resiliendi dependet, sensim vis illa perit, quia ea materia, quum non possit totas in pristinum statum reducere, tracti temporis hoc agit, ut quantum singulis paulum immutandum tales poros efformet, per quos sine difficultate fluere queat,” Hamburg 274, 113–114.
- 272.
“Idemque est ratiocinium, in quacunque durorum inflexione, aut expansione, nam licet non semper pori angustentur, figura saltem ipsorum mutatur, ita ut v. gr. quae antea erat rotunda, expansione ista, fiat elliptica, quare, cum in generatione pori ipsorum, figurae materiae caelestis sint accommodati, hac mutatione in suo per eosdem transfluxu, impeditur,” Schuyl 1688a, chapter 3, 9 (unnumbered). As to the references to Huygens, see infra, n. 295.
- 273.
In fact, Huygens adopted Descartes’s first law of nature as part of his first hypothesis in his De motu corporum ex percussione: De Volder, however, had no access to this text until the death of Huygens.
- 274.
As I am going to show, in his letter to Leibniz of 18/28 February 1699 De Volder stated that Huygens’s rules are valid only for elastic bodies: see Sect. 4.2.3.4.1, De Volder on non-elastic impacts.
- 275.
De Volder could not find a solution to this problem, as far as collision is concerned, in the other treatise that he used in his lectures, namely Rohault’s Traité de physique. Most probably, he taught on this in the late 1690s, as his dictata on it (which have so far not been found) are dated 1698–1699. In chapter 11 of the first part of his treatise, Rohault avails himself of Descartes’s idea of conservation of m·s, and considers only Descartes’s rules 5 and 2 or 7a/2 (i.e. when one of the two bodies is bigger and faster than the other, regardless of the direction of movement), which are avowed: see Rohault 1671, volume 1, 65–66; (in Rohault and Le Grand 1691, 47–48, this section is not commented on by Antoine Le Grand. Samuel Clarke comments on it only in the 1710 edition of this work, proposing, in order to correct it, algebraic rules which perfectly fit with those of Wallis and De Volder: see Rohault and Clarke 1710, 45–48). Please note that the two cases considered by Rohault do not involve any rebounding, rather they are elastic as far as force (m·s) is conserved. In chapter 13, as he considers reflection, Rohault maintains that when one body cannot move another, it is necessarily rebounded by it, regardless of its features (i.e. softness or shape): see Rohault 1671, volume 1, 103; in his commentary, Clarke notes that this is not true, as the colliding body will stop: see Rohault and Clarke 1710, 74. Rohault does not reserve any separate treatment to elasticity understood as the capacity of bodies to restore their shape.
- 276.
“Memini me ante multos annos, cum inquirerem in Leges motus Hugenianas, quas absque demonstratione edidit in Diariis, easdem deduxisse in corporibus elasticis ex duplici hac hypothesi […],” De Volder to Leibniz, 18/28 February 1699, in A II3B, 528.
- 277.
See infra, n. 295.
- 278.
“Ex qua conclusione [i.e. De Volder’s own finding that m·s2 is conserved], cum sequi animadverterem regulas motus Hugenianas, neutiquam dubitabam, quin meae hae hypotheses cum Hugenianis consentirent, in quo tamen me falsum esse deprehendi, postquam post obitum Illustris Viri, libellum ejus de motu videre licuit. Illic enim nulla elasmatis facta mentione omnia sua deducit ex hac Hypothesi, duo corpora dura aequalia, et aequaliter mota, directe in se invicem impingentia repercuti eadem utrinque velocitatis aequalitate servata. Nec mirum ex ea hypothesi sequi leges elasticorum corporum, cum ea vera non foret, nisi ipsam elastri naturam tacite involveret,” De Volder to Leibniz, 18/28 February 1699, in A II3B, 529.
- 279.
See supra, n. 209. Ça va sans dire, this kind of criticism could also be brought against Descartes, as his rule 1 presupposes, for De Volder, the elasticity of the bodies involved.
- 280.
Please note that Huygens, as seen above (see Sect. 4.2.2, Huygens’s laws of collision) devoted to non-elastic impacts only a very short handwritten note (1667).
- 281.
See supra, n. 278.
- 282.
“[…] miratus saepe sum Hugenium nostrum aliunde ista quam ex Elastro deducere sperasse, quod tamen ut recte mones non potuit quin tacite supponeret. Libellum ejus de motu fortasse aliquando, si vobis ita videbitur, quibus scripta sua testamento credidit, possem addere dynamicis meis, et tam praeclaro emblemate exornare opus meum, favente connexione rerum, et similitudine argumenti,” Leibniz to De Volder, 24 March/3 April 1699, in A II3B, 545.
- 283.
“Quantitatem motus binorum corporum, (quam ego vires vocabam) quae esset versus contrarias partes, se invicem destruere (quod ex aequalitate virium contrariarum planum existimabam) reliquam vero quantitatem, ut et quae esset versus easdem partes, conservari, et in utrumque corpus, pro ratione magnitudinis corporum distribui. Ex qua hypothesi Leges motus in corporibus mollibus deducebam,” De Volder to Leibniz, 18/28 February 1699, in A II3B, 528.
- 284.
“Quibus positis ita ratiocinabar. Sit corpus a velocitate c, corpus b velocitate f, quae in se invicem impingant sive in contrarias partes, sive in easdem ferantur. Si corpora non forent elastica progrederentur ambo post occursum velocitate (ac ± bf)/(a + b) ab A versus B, siquidem pono in casu determinationis versus contrarias partes, ac majorem esse quam bf,” De Volder to Leibniz, 18/28 February 1699, in A II3B, 529.
- 285.
See Sect. 4.2.3.4, Collision in De Volder’s late thought.
- 286.
“Definitiones. […] II. Per conatum ad motum talem in corporibus intelligo dispositionem, qua, licet quiescant, actu tamen modo removeatur obstacula, moveri incipiunt. […] Axiomata. I. Nihil movetur a se ipso. […] IV. Corpus, quod ex Diametro aequaliter premitur vel aequalem, versus oppositas ex Diametro partes, ad motum conatum habet, necessario quiescit, minima tamen vi versus utramvis partem moveri potest,” Schuyl 1688a, 1–2 (unnumbered).
- 287.
“Ponamus enim duo corpora, A et B, perfecte dura, hoc est, nullis poris, ac proinde, nec vi elastica praedita, aequali vi ab opposita parte in se invicem impingere, dico, quod in puncto concursus ambo debeant quiescere. Utrumque enim hoc in puncto aequalem versus oppositam ex diametro partem, progrediendi conatum acquirit, in quantum enim A propria vi motus versus orientem v. gr. in tantum praecise a B determinatur versus occidentem. Idemque verum est de B, ergo […] ambo in puncto concursus quiescere debent. Q.E.D.,” Schuyl 1688a, chapter 3, 9 (unnumbered).
- 288.
In chapter 4, indeed, the author considers all three possible cases of impact (1. when they both move against each other; 2. when they move in the same direction; 3. when one is at rest). As to the first case, he states: “[s]it autem ac [namely, m1·s1] = 3 | 2 = bd [namely, m2·s2]: certe, cum B destruat in A vires, viribus suis aequales, […] non poterit A post concursum cum corpore B maiori vi progredi, et B secum abripere, quam quam praecise vim ipsius B superat, quapropter ambo simul spectata pergent moveri versus B, vi ac − bd [namely, m1·s1 − m2·s2].” As to the second: “[s]i vero ambo versus eundem terminum progrediantur, ut obtinet in caus secundo, quia hic utriusque vires combinantur, pergent moveri ambo simul spectata, vi ac + bd,” Schuyl 1688a, chapter 4, 10–11 (unnumbered). In the third case, of course, one just needs to consider m2·s2 = 0 (see page 11). “|” is a separation mark. I have partially modernized the mathematical notation.
- 289.
This can be confirmed by considering, on one hand, that the author claims that in the world perfect solidity cannot be found, so that all bodies are provided with an elastic force: “[c]um vero nulla dentur corpora perfecte dura, sed omnia vi elastica sint praedita, hinc fit, ut non diu illa in quiete haereant […],” Schuyl 1688a, chapter 3, 9 (unnumbered). This statement leaves unsolved the problem of the conservation of motion: as is evident from corollary 8: “[q]uaestioni, cur corpus in motu constitutum, aliquamdiu pergat moveri, a nemine hactenus est satisfactum,” Schuyl 1688a, Corollaria.
- 290.
“Altera erat, cum vis ictus inter eadem corpora eadem sit, quaecunque inter ea sit velocitatis ratio, modo intermedia distantia eodem superetur tempore, elastri vim ex hac celeritate aestimandam, eamque idcirco restitui reciproce pro corporum magnitudine,” De Volder to Leibniz, 18/28 February 1699, in A II3B, 528–529.
- 291.
“Quod si sint elastica, erit vis elastri ut c ± f, quae idcirco celeritas distribuenda erit in utrumque corpus reciproce pro ratione magnitudinis. Quare si fiat ut a + b ad c ± f, ita b ad celeritatem, quam elastrum reddit corpori a, et ita a ad celeritatem, quam elastrum reddit corpori b; habebit A ex occursu velocitatem ab A versus B, ut (ac ± bf)/(a + b) et ex elastro acquiret velocitatem a B versus A, ut (bc ± bf)/(a + b),” De Volder to Leibniz, 18/28 February 1699, in A II3B, 529. Cf. Schuyl 1688a, chapter 4, 11–12 (unnumbered): “[c]onsideremus iam corpora, pro ut vi elastica sunt praedita, ut, quanta in singulis ex mutuo ipsorum impulsu, vel actione, secundum tres illos iam nominatos casus, oriatur vis elastica, determinemus. Primum ergo ab opposita parte in se invicem impingant: quo in casu, cum ambo totis suis viribus in se invicem agant, erit ut (a + b)/(c + d) = 2 | 2 = a/((ac + ad)/(a + b)) vis elastica ipsius A, qua propellit B. b/((bc + bd)/(a + b)) vis elast. ipsius B, qua propellit A. Si vero ambo versus eandem partem moveantur, quia hoc in casu non agunt in se invicem, nisi quatenus velocitas ipsius A superet velocitatem ipsius B, hinc erit, ut (a + b)/(c − d) = 2 | 2 = a/((ac − ad)/(a + b)) vis elast. ipsius A, qua propellit B. b/((bc − bd)/(a + b)) vis elast. ipsius B, qua propellit A.” A description of an experimental proof of Descartes’s first rule is given in Mariotte 1673, 8–22. Mariotte certainly used an elastic and inelastic collision apparatus, of which however he provides no depiction in his Traité de la percussion ou choc des corps. I discuss below the fortunes and adoption of such apparatus by De Volder. Wren, Robert Hooke and Lawrence Rooke performed experiments on collision at the Royal Society: see Bertoloni Meli 2006, chapter 8. As to De Volder’s collision apparatus, see below.
- 292.
“Cum autem A propria vi motus, posita ac = 3 | 2 = bd secum abripiat ipsum B, velocitate (ac ± bd)/(a + b) in utroque sc. casu, a vi vero elastica ipsius B repercutiatur versus contrariam partem, velocitate (bc ± bd)/(a + b). Hinc detracta hac ab illa remanebit (ac − bc ± 2bd)/(a + b) quae proinde quantitas denotat velocitatem qua A, posita vi elastica, post concursum cum corpore B, moveri debeat versus e,” Schuyl 1688a, chapter 4, 12 (unnumbered).
- 293.
“B vero cum a vi elastica ipsius A, versus eandem impellatur partem, versus quam ab eodem A motu suo progressivo impellitur, hinc, combinatis illis quantitatibus, erit velocitas, qua B post concursum cum corpore A, moveri debet versus e. (2ac ± ad ± bd)/(a + b). Quod si B ponatur quiescens, deleatur duntaxat d, eritque (ac − bc)/(a + b) velocitas ipsius A. Velocitas vero ipsius B, erit 2ac/(a + b),” Schuyl 1688a, chapter 4, 12 (unnumbered). In the letter to Leibniz De Volder provides these quantifications in a more synthetic way (following the quotation given supra, n. 291): “[q]uarum ergo differentia [sc. between the ‘non elastic’ speed of A, and its ‘elastic’ speed after impact] erit velocitas ipsius A = (ac ± 2bf − bc)/(a + b) idque ab A versus B, si haec quantitas sit affirmata, a B vero versus A si sit negata. Itidem B ex occursu habebit velocitatem ab A versus B ut (ac ± bf)/(a + b), ex elastro vero acquiret velocitatem etiam ab A versus B = (ac ± ad)/(a + b) quarum ergo summa erit velocitas ipsius B = (2ac ± bf ± af)/(a + b), qua pergit ab A versus B,” De Volder to Leibniz, 18/28 February 1699, in A II3B, 529.
- 294.
“Ex his autem ipsis, quod ad rem nostram multum facit, sequitur in corporum perfecte elasticorum concursu, remanere semper idem productum corporum in celeritatum quadrata. Nam si quadr. (ac ± 2bf − bc)/(a + b) ducatur in a, et quadr. (2ac ± bf ± ab)/(a + b) ducatur in B, horum productorum summa aequabitur acc + bff. Juxta hanc autem mensuram vires corporum elasticorum aestimandas confirmatur ex eo, quod si illa corpora a et b celeritatibus post primum occursum acquisitis in se invicem iterum impingant, corpus A, post alterum hunc occursum, habiturum sit celeritatem ut c, B celeritatem ut f, adeoque hic causa effectui aequipolleat exactissime, cum alterum ex altero praecise sequatur, et vice versa,” De Volder to Leibniz, 18/28 February 1699, in A II3B, 529.
- 295.
In the disputation, the cases involved in Huygens’s first and second rule are explicitly considered, and the rules themselves are referred to. However, the cases in which the two bodies are not equal are considered: “[e]x quibus [i.e. from the algebraic formulas given in the disputation] innumera de motu corporum ex mutuo ipsorum impulsu orto determinari possunt, et variae construi regulae cum experientia convenientes. Patet enim evidenter, quod, si corpori quiescenti duro B v. gr. aliud aequale corpus durum A occurrat, post contactum hoc quidem quiescat, quiescens vero eandem, quae fuit in impellente, acquirat celeritatem a enim eum, hoc posito, sit aequalis b, erit et ac = bc, ac proinde A quiescere debet, cumque a + b sit = 2a, movebitur B celeritate c, quae est celeritas ipsius A. Quod si a ponatur maior quam b, progredietur quidem A versus eundem terminum cum B, sed B semper ipso celerius movebitur; et si a minor ponatur, quam b, reflectetur A versus contrariam partem. Patet etiam, quod, si alterum illud corpus aequale etiam moveatur, sive ab opposita parte in se invicem impingant […] sive versus eandem partem ambo progrediantur […] quod, inquam, post contactum permutatis invicem celeritatibus sequerentur. Ita ut A moveatur celeritate d; B vero celeritate c, sic tamen ut A in priori casu ea celeritate reflectatur, in posteriori vero secundum lineam rectam progrediatur. Vid. has duas regulas Societati Regiae Angl. inter alias communicatas ab Illustrissimo Viro Christiano Hugenio in Philos. transaction. pag. 927. Quia autem vel ei, qui prima duntaxat didict calculi algebraici elementa, reliqua omnia hinc deducere sit facillimum, ulteriori ipsorum deductioni supersedebo,” Schuyl 1688a, chapter 4, 12–13 (unnumbered). Cf. supra, n. 272.
- 296.
“Een hout instrument staande op syn kruck, onder met twee stucken van kopere afgedeelde circuls ad regulas motus. Hier toe 16 ivore en 3 houte bollen. Nogh een hout gootje tot het selfde gebruik,” Molhuysen 1913–1924, volume 4, 105∗.
- 297.
“Ein kleiner Sextans von Holz, den motum pendulorum damit zu machen,” Uffenbach 1753–1754, volume 3, 426.
- 298.
In an unpaginated sheet entitled Catalogus van alle d’instrumenten die by mij gemaakt werde, inserted into a letter of Van Musschenbroek dated 4 March 1694, Van Musschenbroek noted that “[m]ijn heer de Volder heeft een instrument om de wetten van de beweegingh te toonen dat heel fray is kost ontrent 60 guld[en],” Hessisches Staatsarchiv Marburg, file UniA 305a Nr 5947, 1–3 (transcribed also in De Clercq 1991, 119–120). According to a letter of 20 September 1694, the apparatus was not yet ready: file UniA 305a Nr 5947, 23. I thank Dr. Katharina Schaal, librarian at the Universitätsarchiv – Philipps-Universität Marburg, for her help in locating these manuscripts.
- 299.
File UniA 305a Nr 5947, 17. This price is included in a letter dated 11 May 1695. As to the ivory balls, see infra, n. 300.
- 300.
In the sheets extant at Marburg, ending with a letter dated 11 May 1695, one can find the description of the collision apparatus: “[l]’instrument pour voir les le loix de mouvement et de telle figure aa est un planche dans le quelle sont fait deux pieces d’un {mesh} circle de cuivre marque avec degrez en haut le mesme, en haut pendent <{…}> \en/ un petit {corde} deux boules d’yvoire le quelle on laisse egalement tombe{r} et se recontrent on void a quelle degrez se rejettent, on laisse un tomber est l’autre en repos on void combien de mouvement qu’il donne a l’autre et combie qu’il peut, {on} prend aussi des boules de diverses grandeurs et aussi de bois de metail, de terre, de tout sortes,” file UniA 305a Nr 5947, 14; see Fig. 4.29, Hessisches Staatsarchiv Marburg, file UniA 305a Nr 5947, 14. In the letter itself, it is implied that this instrument was based on Mariotte’s apparatus, to whose Traité, as well as to the Journal des voyages of De Monconys (viz. to the 1695 edition), Van Musschenbroek re-directs the reader: “[l]’instrument pour le choq du corps l’usage trouvera vous dans le livre Mariotte Le choq du corps. J’ay seulement pendu 2 boules du bois dur. L’yvoire est mieux parce qu’ils sont plus dur, mais ils sont fort cher, ils voudrent bien 20 florin pour les deux. Et vous pouvez tujours laissez faire. Et aussi de toutes sortes de bois, de cuivre \de fer/ pour voir la difference de reflexions. Pu pend cet instrument contre une muraille <{…}> en un clou a: le piece du plomb pour le pendre au niveau. Si Monsr desire les boules d’yvoire le lasseray bien faire. […] En le Voyage de Monconijs est l’instrument de choq du corps ecrit dans tom 5 fol 232,” file UniA 305a Nr 5947, 15 and 18. The place of De Monconys’s Journal mentioned by Van Musschnebroek is in fact a letter of a certain Monsieur Regnault (not to be mistaken with Valère Regnault (1545–1623) and with Noël Regnault (1683–1762)), dated 21 December 1655, in which Regnault describes his experiments on collision performed with pendulums (De Monconys 1695, volume 4, 231–244). Regnault was also the author of a De humidorum aequilibrio in siphonibus (De Monconys 1695, volume 4, 176–191), based on the principle of Torricelli. On De Monconys’s Journal (which included several observations on scientific instruments), see De Clercq 2016.
- 301.
In a letter dated 28 July 1696, Van Musschenbroek refers again to Mariotte, noting that in his treatise there are no illustrations of the apparatus. This illustration can however be found, as Van Musschenbroek notes, in the Cours entier de philosophie, ou Systeme general selon les principes de M. Descartes, contenant la logique, la metaphysique, la physique, et la morale (1691) of Pierre-Sylvain Régis, in which it is depicted exactly like Mariotte’s apparatus: “touchant une explication pour l’instrument le choq du corps je vous a envoye le livre de Monsr Mariotte il a \2/ figures qui manquent en ce livre nous avons achette ces deux livres \Du choq et mouvement des eaux/ en un auction car on ne puis en Hollande avoir ces livres, mais la description de l’instrument est en \Le/ Physique de Regis tom 1 chap 21 pag 366: j’espere que Monsr par ces deux livres aurez contentement, car ces choses sont trop long en une lettre {sil} tost que je puis trouver une livre de Monsr Mariotte je vous envoyer par le poste en une lettre les deux figures,” file UniA 305a Nr 5947, 20–21. See Régis 1691, volume 1, 366–367, containing Fig. 4.30, Régis 1691, volume 1, 366. On Régis’s physics and impact rules, see Clarke 1989; Des Chene 2002, 2005; Milani 2010, 2013, 2015. In a letter of 12 December 1696, Van Musschenbroek comments again upon the use of the instrument: “j’ai le mesure pour les boules de l’instrument de mouvement. Et Monsr le puis prendre aussi selon son desin, par exemple prenex une boule de 3 ponces de diametre et prennez un de 2 pounce et faite l’experience combien ils se reflechent, on prend deux egales boules aussi bien inegaux. Mais quand on veut faire exactement l’experience le boules doivent estre si grand que le <deux> distance des {elon} qui sont en {hout} de l’instrument car ils faut que le boules ne sont plus grand autrement on ne puis exactement mesurer, car ils douvent seulement se toucher quand ils sont en repos. Mais on le puis prendre bien petit et Monsr le faut faire deverses grandeurs de cette grandeur comme j’ay envoye jusque a une pouce de diametre, et aussi de diverses matiere, de terre, du bois, yvoire est le mieux pour le peu de ressort car ils reflechent le mieux j’ay fait ma delegence pour les figures de Mariotte du choq des corps, et je n’ay pas trouvex, la guerre mons empeche pour avoir ces livres mais si tost que je puis trouver je ne manqueray pas pour Monsr le faire avoir,” file UniA 305a Nr 5947, unnumbered page.
- 302.
- 303.
“Mathematica. Libri in folio. […] 26. Joh. Wallis Opera omnia Mathematica, Oxon. 1699. 3 voll. […] Mathematici in quarto. […] 99. [Joh. Wallisii] […] Mechanica sive de Motu, Lond. 1670. […] 101. […] de motu, Lond. 1670,” Bibliotheca Volderina, 13, 14–15. 1699 is the date of the last volume of Wallis’s Opera, published from 1693 (when also the second volume was published), to 1699 (third volume).
- 304.
“Perfecte durum, appello, quod ictui nequaquam cedit, adeoque nec molle, nec elasticum. Molle, appello, quod ictui ita cedit ut pristinam figuram amittat […]. Elasticum appello, quod, utut ictui aliquantisper cedat, se tamen in pristinam formam suopte marte restituit,” Wallis 1670–1671, 661. The numbering of the three volumes is continuous.
- 305.
Wallis 1670–1671, 16–17.
- 306.
“Prop. II. Si grave motum, gravi quiescenti directe impingat, sed ita constituto, ut aliunde ne moveatur non impediatur: utrumque iunctim movebitur, ea celeritate quam calculus (ponderum rationis et pristina celeritate rite computabis) indicabit. Nempe, si momentum (ex moti gravis pondere et celeritate compositum) per utriusque simul pondus dividatur, habebitur futura celeritas. […] Cum itaque vi mrPC, movendum sit pondus mP + nP, id fiet celeritate m/(m + n)rC (divisio scilicet momento mrPC, per pondus mP + nP),” Wallis 1670–1671, 662–663. In proposition 1 Wallis considers the case in which a body is thrown onto an unmoveable object, like a wall. In this case, he uses the idea of resistance, which is equal to the momentum of the body in movement: see Wallis 1670–1671, 660.
- 307.
Cf. propositions 3–4: Wallis 1670–1671, 664–665.
- 308.
This foreruns Newton’s third law of nature. Newton, indeed, cites the rules of impact of Huygens, Wren, Wallis and Mariotte to support his account: see Home 1968; Cook 1986.
- 309.
“Est autem ea vis comprimens, dupla momenti gravis impingentis. Nam vis ab impingente gravi illata, est ipsius momento aequalis (utpote cuius motus totus sistitur,) puta mrPC. Sed et, propter aequalem obicis resistentiam, (quae tantundem ad compressionem confert ac si aequalis vis contraria occurreret,) tantundem inde advenit, puta alterum mrPC, (sed cuius hic minor habenda ratio, utpote quod aliunde compensatur). Adeoque vis tota compressiva, 2mrPC. Dico porro, vim impellentem seu comprimentem, ubi ad aequipollentiam res redigitur cessare (utpote quae tota flectendo elatere impensa est). Vis enim extrinsecus impressa, qualem supponimus impellentem (secus quam vis insita, qualis est gravitas, et vis elastica compressa,) ubi semel ad quietem redigitur, perit, neque valet seipsam restituere,” Wallis 1670–1671, 688.
- 310.
“Dico itaque tertio, cessante […] motu impellente, seu comprimente, vim elasticam, iam liberam, et […] vi pridem comprimenti aequalem, aequaliter utrinque se explicare nitentem, vim dimidiam seu alterum mrPC in obicem impendere (utut irrito conatu) alterum vero mrPC in advectum grave. Quod itaque eadem vi, adeoque et eadem celeritate, corpus A repellit, qua advectum fuerat. (Nempe, vi mrPC, adeoque, propter pondus mP, celeritate rC),” Wallis 1670–1671, 688. Wallis does not provide a natural-philosophical explanation of how the elastic force is exerted: rather, in definitions 1–2 of chapter 13, he defines elasticity as the force by which a body re-aquires its previous shape, and provides a short etymological analysis of the term ‘elater’: Wallis 1670–1671, 686–687. In any case, an elastic force is required, for Wallis, to explain rebounding, as he clarifies in the scholium of the first proposition of chapter 13: “[n]on ignoro, plerosque mechanicorum vim nescio quam in motu incepto ponere, qua […] mutata directione (etiam absque nova causa) resiliat. Cum vero hoc gratis dictum videatur (ut novus motus absque nova causa incipiat,) et postulatum, atque multis porro incommodis urgeri possit,” Wallis 1670–1671, 690. See also supra, n. 304. Wallis might have appropriated the idea that elastic force doubles the initial momentum of the moving body from Giovanni Alfonso Borelli, who held similar positions in his De vi percussionis (1667). Borelli maintained that the force of collision of a body (calculated by considering size and speed) is first destroyed, and then ‘restituted’ as doubled in the case where one of the bodies is elastic. He provided a corpuscular explanation of collision not involving the idea of subtle matter: according to him, the particles of elastic bodies are flexible machinulae, whose interactions are however not explained by him: see Borelli 1667, 144 and 148–149. De Volder owned only the second edition of this treatise, published in Leiden in 1686: Bibliotheca Volderina, 3.
- 311.
“Quodque iam demonstratum est, posita in obice vi elastica: perinde obtinet si vis illa sit in corpore impingente, vel etiam in utroque. Quippe illud utcunque obtinet, non prius sisti vim impellentem, quam aequipolleat elateris (sive singularis, sive gemini,) adversa resistentia, sed neque ultra perseverare. Unde reliqua sequuntur,” Wallis 1670–1671, 688.
- 312.
See Wallis 1670–1671, 705–707.
- 313.
See Sect. 4.1.4, The search for physical unity and activity: De Volder vis-à-vis Leibniz.
- 314.
“Plures hic demonstrationes congessi […] non quod diffidam singulis […], inest enim singulis sua vis. Sed, cum pro vario lectorum gustu modo hic modo ille demonstrandi modus placeat magis, ut quam magis velit lector eligat. Prima quidem, aut etiam secunda […] physicam rei causam magis explicant, quam tamen in sequentibus parcius prosequor, ut quae ab his dependent,” Wallis 1670–1671, 699.
- 315.
See Sect. 4.2.3.4.2, De Volder on elastic impacts.
- 316.
This makes sense of De Volder’s remark, given in the dictata, that rules of impact alternative to Descartes’s were provided by Huygens and Wallis: Huygens, indeed, provided rules for elastic bodies, Wallis provided rules for non-elastic ones. No other author is specifically mentioned by De Volder.
- 317.
Murray et al. 2011, 179. Cf. the original text: “[s]i quae sic impingunt Corpora, intelligantur non absolute dura (prout hactenus supposuimus) sed ita ictui cedentia, ut Elastica tamen vi se valeant restituere, hinc fieri poterit ut a se mutuo resiliant ea corpora, quae secus essent simul processura,” Wallis 1668, 866.
- 318.
These passages have been considered in Sect. 4.1.4, The search for physical unity and activity: De Volder vis-à-vis Leibniz.
- 319.
See Leibniz to De Volder, 24 March/3 April 1699, quoted supra, n. 113.
- 320.
“Modum quo regulas motus binorum corporum concurrentium, ex destructione motus per conflictum, et restitutione per Elastrum deducis, puto esse rectissimum; si modo uberioris intelligentiae causa accedat, quemadmodum jam inter nos convenit, utrumque fieri successive ex lege aequilibrii interventu virium mortuarum,” Leibniz to De Volder, 24 March/3 April 1699, in A II3B, 545.
- 321.
“Quae Joachimus Jungius, Marcus Marci, Joh. Alph. Borellus, Hugenius, Wrennus, Wallisius, Mariotus de his recte (pro parte) meditati sunt, inaedificata ab ipsis partim experimentis, partim hypothesibus particularibus, ea non tantum ex veris fontibus explico, sed et longius profero ad casus quos vel nullo modo vel non bene attigere. […] Omnibus praejudicio fuit, lex aequilibrii ex qua sumsere, ut vires simpliciter aestimarent in ratione composita corporis et celeritatis neque adeo (explosa Cartesiana conservatione quantitatis motus) virium conservationem aliquam animadverterent; quae tamen superest mirifica et multimoda. Unde Wallisius in libro de motu, etiam agnoscens quantitatem motus non servari, nihilo minus tamen per eam vim aestimavit,” Leibniz to De Volder, 27 December 1698, in A II3B, 503–504. Moreover, according to Leibniz Huygens accepted the idea that motive force (m·s2) is conserved only after Leibniz himself had pointed this out to him, but meant by this only the “ascensional forces.” These are dealt with, but not overtly labelled as ‘vires’, in the second part of Huygens’s Horologium oscillatorium (1673), concerning the law of free fall. Cf. the same letter: “[i]pse Dn. Hugenius de virium conservatione fassus est se non cogitasse, postea a me admonitus vires quae conservantur appellavit Ascensionales, non male quidem, sed tamen non satis plene. Idem enim in omni viva vi locum habet, sive ad gravitatem, sive ad Elastrum, sive simpliciter ad motum, aliumve effectum qualemcunque applicetur,” A II3B, 503. The idea that motive force (m·s2) is ascensional force itself is given by Huygens in his marginalia to Leibniz’s Demonstratio brevis, as he comments upon Leibniz’s words “[h]inc sequitur, corpus A delapsum ex altitudine CD, praecise tantum acquisivisse virium, quantum corpus B lapsum ex altitudine EF,” with “virium ascensionalium, nam si acquisitis celeritatibus sibi mutuo occurrant, non aequaliter resistunt,” Huygens 1888–1950, volume 22, 790, and Leibniz’s article Schediasma de resistentia medii, et motu proiectorum gravium in medio resistente (Acta Eruditorum, January 1689), as he comments on Leibniz’s words “sunt enim vires ut quadrata velocitatum” with “vires ascensionales. Sed his hîc nihil opus,” Huygens 1888–1950, volume 22, 793. Leibniz could know the contents of these marginalia, as Bernoulli communicated them to him in a letter of 12/22 September 1696: see A III7A, 130–131. However, no direct communication between Leibniz and Huygens on ascensional forces is extant in the Akademie Ausgabe edition of Leibniz’s correspondence. See also the letter of Leibniz to Guillaume de L’Hôpital, 4/14 December 1696, in A III7A, 214: “[q]uant aux Dynamiques je croy que M. Hugens estoit de mon sentiment dans le fonds; et qu’il reconnoissoit qu’il se conserve tousjours la meme force, comme j’avois avancé. Apres avoir examiné mon sentiment, il trouva à propos d’appeller cette force Ascensionale, parce qu’il se conserve tousjours autant qu’il faut precisement pour faire montrer le même poids à la même hauteur. Mais comme cette même force a lieu, soit qu’on employe des corps pesans ou des ressorts ou autre chose, parce qu’il se conserve tousjours autant qu’il faut précisement pour faire monter le même poids à la même hauteur.” Pace Leibniz, Huygens understood the scope of the idea of the conservation of m·s2, well beyond the case of ascensional forces: see the letter of Huygens to Henri Basnage de Beauval of July 1690 (letter 2606), in Huygens 1888–1950, volume 9, 463: “[…] ma raison est, qu’il se perd souvent du mouvement, sans qu’on puisse dire qu’il s’est consumé à rien, comme dans plusieurs cas du choc de deux corps durs, suivant ce que j’ay remarqué en publiant les loix de ces sortes de mouvements dans le Journal des Sçavans en 1669. au mois de Fevrier): de sorte que ce n’est pas une necessité que la quantité de mouvement se conserve toûjours, si elle ne se consume à quelque chose; mais c’est une loy constante, que les corps doivent garder leur force ascensionelle, et que pour cela la somme des quarrez de leurs vitesses doit demeurer la même. Ce qui n’a pas seulement lieu dans les poids des pendules, et dans le choc des corps durs, comme je l’ay remarqué au même endroit, mais aussi en beaucoup d’autres recherches de Mechanique.”
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Strazzoni, A. (2019). The Principles of Natural Philosophy. In: Burchard de Volder and the Age of the Scientific Revolution. Studies in History and Philosophy of Science, vol 51. Springer, Cham. https://doi.org/10.1007/978-3-030-19878-7_4
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