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version 1.8, 2002/07/09 23:38:57

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]><archimedes> <info> <author>Newton, Isaac</author> <title>Philosophiae Naturalis Principia Mathmatica</title> <date>1713</date>

<info>

<author>Isaac Newton</author>

<place>Cambridge</place> <editor></editor> <publisher></publisher> <translator></translator> <lang>la</lang> <chunk unit="page*">page</chunk> <locator>0000000039</locator> </info> <text> <front> </front> <body> <chap> <pb/>

<title>Philosophiae Naturalis Principia Mathmatica</title>

<place>Cambridge</place>

</info>

<text>

<front>

</front>

<body>

<chap>

<pb/>

<s><emph type="center"/>PHILOSOPHIÆ <lb/>NATURALIS <lb/>PRINCIPIA <lb/>MATHEMATICA.<emph.end type="center"/></s>

<s><emph type="center"/>AUCTORE <lb/>ISAACO NEWTONO, <lb/>EQUITE A RATO.<emph.end type="center"/></s>

<s><emph type="center"/>EDITIO SECUNDA AUCTIOR ET EMBNDATIOR.<emph.end type="center"/></s>

<s><emph type="center"/>CANTABRIGIÆ, MDCCXIII.<emph.end type="center"/></s>

<pb/>

<s><emph type="center"/>ILLUSTRISSIMÆ <lb/>SOCIETATI REGALI, <lb/>A <lb/>SERENISSIMO REGE <lb/>CAROLO II <lb/>AD PHILOSOPHIAM PROMOVENDAM <lb/>FUNDATÆ, <lb/>ET <lb/>AUSPICIIS <lb/>AUGUSTISSIMÆ REGINÆ <lb/>ANNÆ <lb/>FLORENTI,<emph.end type="center"/></s>

<s><emph type="center"/>TRACTATUM HUNC D.D.D.<emph.end type="center"/></s>

<s><emph type="italics"/>JS. NEWTONUS.<emph.end type="italics"/></s>

<pb/>

<s><emph type="center"/>IN <lb/>VIRI PRÆSTANTISSIMI <lb/>ISAACI NEWTONI <lb/>OPUS HOCCE <lb/>MATHEMATICO PHYSICUM<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Sæculi Genti&longs;que no&longs;træ Decus egregium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>EN tibi norma Poli, & divæ libramina Molis, <lb/>Computus en Jovis; & quas, dum primordia rerum. </s>

<s><emph type="center"/>PHILOSOPHIÆ <lb/>NATURALIS <lb/>PRINCIPIA <lb/>MATHEMATICA.<emph.end type="center"/></s>

<s><emph type="center"/>AUCTORE <lb/>ISAACO NEWTONO, <lb/>EQUITE A RATO.<emph.end type="center"/></s>

<s><lb/>Conderet, omnipotens &longs;ibi Leges ip&longs;e Creator <lb/>Dixerit, atque operum quæ fundamenta locarit. </s>

<s><emph type="center"/>EDITIO SECUNDA AUCTIOR ET EMBNDATIOR.<emph.end type="center"/></s><figure></figure>

<s><emph type="center"/>CANTABRIGIÆ, MDCCXIII.<emph.end type="center"/></s><pb/>

<s><lb/>Intima panduntur victi penetralia Cæli, <lb/>Nec latet, extremos quæ Vis circumrotet Orbes. </s>

<s><emph type="center"/>TRACTATUM HUNC D.D.D.<emph.end type="center"/></s>

<s><lb/>Sol &longs;olio re&longs;idens ad &longs;e jubet omnia prono <lb/>Tendere de&longs;cen&longs;u, nec recto tramite currus <lb/>Sidereos patitur va&longs;tum per inane moveri; <lb/>Sed rapit immotis, &longs;e centro, &longs;ingula gyris. </s>

<s><emph type="italics"/>JS. NEWTONUS.<emph.end type="italics"/></s><pb/>

<s><emph type="center"/>IN <lb/>VIRI PRÆSTANTISSIMI <lb/>ISAACI NEWTONI <lb/>OPUS HOCCE <lb/>MATHEMATICO PHYSICUM<emph.end type="center"/></s>

<s><lb/>Hinc patet, horrificis qua &longs;it via flexa Cometis: <lb/>Di&longs;cimus hinc tandem, qua cau&longs;a argentea Ph&oelig;be <lb/>Pa&longs;&longs;ibus haud æquis eat, & cur &longs;ubdita nulli <lb/>Hactenus A&longs;tronomo numerorum fræna recu&longs;et: <lb/>Cur remeent Nodi, curque Auges progrediantur. </s>

<s><emph type="center"/><emph type="italics"/>Sæculi Genti&longs;que no&longs;træ Decus egregium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>EN tibi norma Poli, & divæ libramina Molis, <lb/>Computus en Jovis; & quas, dum primordia rerum. </s>

<s><lb/>Conderet, omnipotens &longs;ibi Leges ip&longs;e Creator <lb/>Dixerit, atque operum quæ fundamenta locarit. </s>

<pb/>Quæ toties animos veterum tor&longs;ere Sophorum, <lb/>Quæque Scholas hodie rauco certamine vexant, <lb/>Obvia con&longs;picimus; nubem pellente Mathe&longs;i: <lb/>Quæ &longs;uperas penetrare domos, atque ardua Cæli, <lb/>NEWTONI au&longs;picils, jam dat contingere Templa. </s>

<s><lb/>Intima panduntur victi penetralia Cæli, <lb/>Nec latet, extremos quæ Vis circumrotet Orbes. </s>

<s><lb/>Surgite Mortales, terrenas mittite curas; <lb/>Atque hinc cæligenæ vites cogno&longs;cite Mentis, <lb/>A pecudum vita longe longeque remotæ. </s>

<s><lb/>Qui &longs;criptis primus Tabulis compe&longs;cere Cædes, <lb/>Furta & Adulteria, & perjuræ crimina Fraudis; <lb/>Quive vagis populis circumdare m&oelig;nibus Urbes <lb/>Auctor erat; Cereri&longs;ve beavit munere gentes; <lb/>Vel qui curarum lenimen pre&longs;&longs;it ab Uva; <lb/>Vel qui Niliaca mon&longs;travit arundine pictos <lb/>Con&longs;ociare &longs;onos, oculi&longs;que exponere Voces; <lb/>Humanam &longs;ortem minus extulit; utpote pauca <lb/>In commune ferens mi&longs;eræ &longs;olatia vitæ. </s>

<s><lb/>Di&longs;cimus, & quantis refluum vaga Cynthia Pontum <lb/>Viribus impellat; fe&longs;&longs;is dum fluctibus ulvam <lb/>De&longs;erit, ac nautis &longs;u&longs;pectas nudat arenas; <lb/>Alterni&longs;ve ruens &longs;pumantia littora pul&longs;at. <pb/>Quæ toties animos veterum tor&longs;ere Sophorum, <lb/>Quæque Scholas hodie rauco certamine vexant, <lb/>Obvia con&longs;picimus; nubem pellente Mathe&longs;i: <lb/>Quæ &longs;uperas penetrare domos, atque ardua Cæli, <lb/>NEWTONI au&longs;picils, jam dat contingere Templa. </s>

<s><lb/>Surgite Mortales, terrenas mittite curas; <lb/>Atque hinc cæligenæ vites cogno&longs;cite Mentis, <lb/>A pecudum vita longe longeque remotæ. </s>

<s><lb/>Jam vero Superis convivæ admittimur, alti <lb/>Jura poli tractare licet, jamque abdita diæ <lb/>Clau&longs;tra patent Naturæ, & rerum immobilis ordo; <lb/>Et quæ præteritis latuere incognita &longs;æclis. </s>

<s><lb/>Talia mon&longs;trantem ju&longs;tis celebrate Camænis, <lb/>Vos qui cæle&longs;ti gaudetis nectare ve&longs;ci, <lb/>NEWTONUM clau&longs;i re&longs;erantem &longs;crinia Veri, <lb/>NEWTONUM Mu&longs;is carum, cui pectore puro <lb/>Ph&oelig;bus ade&longs;t, totoque ince&longs;&longs;it Numine mentem: <lb/>Nec fas e&longs;t propius Mortali attingere Divos. <lb/><emph type="italics"/>EDM. HALLET.<emph.end type="italics"/></s><pb/>

<s><emph type="center"/>AUCTORIS <lb/>PRÆFATIO <lb/>AD <lb/>LECTOREM.<emph.end type="center"/></s>

<pb/>

<s><emph type="center"/>AUCTORIS <lb/>PRÆFATIO <lb/>AD <lb/>LECTOREM.<emph.end type="center"/></s>

<s><emph type="italics"/>CUM Veteres<emph.end type="italics"/> Mechanicam (<emph type="italics"/>uti Auctor e&longs;t<emph.end type="italics"/> Pappus) <emph type="italics"/>in rerum <lb/>Naturalium inve&longs;tigatione maximi fecerint; & Recentiores, <lb/>mi&longs;&longs;is formis &longs;ub&longs;tautialibus & qualitatibus occultis, Phænomena <lb/>Naturæ ad leges Mathematicas revocare aggre&longs;&longs;i fint: Vi&longs;um e&longs;t <lb/>in hoc Tractatu<emph.end type="italics"/> Mathe&longs;in <emph type="italics"/>excolere, quatenus ea ad<emph.end type="italics"/> Philo&longs;ophiam <lb/><emph type="italics"/>&longs;pectat.<emph.end type="italics"/> Mechanicam <emph type="italics"/>vero duplicem Veteres con&longs;tituerunt<emph.end type="italics"/>: Ra<lb/>tionalem <emph type="italics"/>quæ per Demon&longs;trationes accurate procedit, &<emph.end type="italics"/> Practi<lb/>cam. <emph type="italics"/>Ad <gap/>acticam &longs;pectant Artes omnes Manuales, a quibus <lb/>utique<emph.end type="italics"/> Mechanica <emph type="italics"/>nomen mutuata e&longs;t. </s>

<s>Cum autem Artifices pa<lb/>rum accurate operari &longs;oleant, fit ut<emph.end type="italics"/> Mechanica <emph type="italics"/>omnis a<emph.end type="italics"/> Geome<lb/>tria <emph type="italics"/>ita di&longs;tinguatur, ut quicquid accuratum &longs;it ad<emph.end type="italics"/> Geometriam <lb/><emph type="italics"/>referatur, quicquid minus accuratum ad<emph.end type="italics"/> Mechanicam. <emph type="italics"/>Attamen <lb/>errores non &longs;unt Artis &longs;ed Artificum. </s>

<s>Qui minus accurate ope<lb/>ratur, imperfectior e&longs;t Mechanicus, & &longs;i quis accurati&longs;&longs;ime ope<lb/>rari po&longs;&longs;et, hic foret Mechanicus omnium perfecti&longs;&longs;imus. </s>

<s>Nam & <lb/>Linearum rectarum & Circulorum de&longs;criptiones in quibus<emph.end type="italics"/> Geo<lb/>metria <emph type="italics"/>fundatur, ad<emph.end type="italics"/> Mechanicam <emph type="italics"/>pertinent. </s>

<s>Has lineas de&longs;cri<lb/>bere<emph.end type="italics"/> Geometria <emph type="italics"/>non docet &longs;ed po&longs;tulat. </s>

<s>Po&longs;tulat enim ut Tyro <lb/>ea&longs;dem accurate de&longs;cribere prius didicerit quam linen atting at<emph.end type="italics"/><lb/>Geometriæ; <emph type="italics"/>deia, quomodo per has operationes Problemata &longs;ol<lb/>uantur, docet. </s>

<s>Rectas & Circulos de&longs;cribere Problemata &longs;unt,<emph.end type="italics"/><pb/><emph type="italics"/>&longs;ed non Geometrica. </s>

<s>Rectas & Circulos de&longs;cribere Problemata &longs;unt,<emph.end type="italics"/>

<pb/><emph type="italics"/>&longs;ed non Geometrica. </s>

<s>Ex<emph.end type="italics"/> Mechanica <emph type="italics"/>po&longs;lulatur horum &longs;olutio, in<emph.end type="italics"/><lb/>Geometria <emph type="italics"/>docetur &longs;olutorum u&longs;us. </s>

<s>Ac gloriatur<emph.end type="italics"/> Geometria <lb/><emph type="italics"/>quod tam paucis principiis aliunde petitis tam multa præ&longs;tet. </s>

<s>Fun<lb/>datur igitur<emph.end type="italics"/> Geometria <emph type="italics"/>in praxi Mechanica, & nihil aliud e&longs;t <lb/>quam<emph.end type="italics"/> Mechanicæ univer&longs;alis <emph type="italics"/>pars illa quæ artem men&longs;urandi ac<lb/>curate proponit ac demon&longs;trat. </s>

<s>Cum autem artes Manuales in <lb/>corporibus movendis præcipue ver&longs;entur, fit ut<emph.end type="italics"/> Geometria <emph type="italics"/>ad mag<lb/>nitudinem,<emph.end type="italics"/> Mechanica <emph type="italics"/>ad motum vulgo referatur. </s>

<s>Quo &longs;en&longs;u<emph.end type="italics"/> Me<lb/>chanica rationalis <emph type="italics"/>erit Scientia Motuum qui ex viribus quibu&longs;<lb/>cunque re&longs;ultant, & Virium quæ ad motus quo&longs;cunque requirun<lb/>tur, accurate propo&longs;ita ac demon&longs;trata. </s>

<s>Pars hæc<emph.end type="italics"/> Mechanicæ <emph type="italics"/>a <lb/>Veteribus in<emph.end type="italics"/> Potentiis quinque <emph type="italics"/>ad artes manuales &longs;pectantibus <lb/>exculta fuit, qui Gravitatem (cum potentia manualis non &longs;it) vix <lb/>aliter quam in ponderibus per potentias illas movendis con&longs;iderarunt. </s>

<s><lb/>Nos autem non Artibus &longs;ed Philo&longs;ophiæ con&longs;ulentes, deque poten<lb/>tiis non manualibus &longs;ed naturalibus &longs;cribentes, ea maxime tracta<lb/>mus quæ ad Gravitatem, Levitatem, vim Ela&longs;ticam, re&longs;i&longs;tentiam <lb/>Fluidorum & eju&longs;modi vires &longs;eu attractivas &longs;eu impul&longs;ivas &longs;pe<lb/>ctant: Et ea propter, hæc no&longs;tra tanquam Philo&longs;ophiæ principia <lb/>Mathematica proponimus. </s>

<s>Omnis enim Philo&longs;ophiæ difficultas in <lb/>eo ver&longs;ari videtur, ut a Phænomenis motuum inve&longs;tigemus vires <lb/>Naturæ, deinde ab his viribus demon&longs;tremus phænomena <gap/>liquæ. </s>

<s><lb/>Et huc &longs;pectant Propo&longs;itiones generales quas Libro primo & &longs;ecundo <lb/>pertractavimus. </s>

<s>In Libro autem tertio Exemplum hujus rei propo<lb/>&longs;uimus per explicationem Sy&longs;tematis mundaui. </s>

<s>Ibi enim, ex phæ<lb/>nomenis cæleflibus, per Propo&longs;itiones in Libris prioribus Mathe<lb/>matice aemon&longs;tratas, derivantur vires Gravitatis quibus corpora <lb/>ad Solem & Planetas &longs;ingulos tendunt. </s>

<s>Deinde ex his viribus <lb/>per Propo&longs;itiones etiam Mathematicas, deducuntur motus Planeta<lb/>rum, Cometarum, Lunæ & Maris. </s>

<s>Utinam cætera Naturæ phæ<lb/>nomena ex principiis Mechanicis eodem argumentandi genere deri<lb/>vare liceret. </s>

<s>Nam multa me movent ut nonnihil &longs;u&longs;picer e<gap/> om-<emph.end type="italics"/><pb/><emph type="italics"/>nia ex viribus quibu&longs;dam pendere po&longs;&longs;e, quibus corporum particulæ <lb/>per cau&longs;as nondum cognitas vel in &longs;e mutuo impelluntur & &longs;e<lb/>cundum figuras regulares cohærent, vel ab invicem fugantur & <lb/>recedunt: quibus viribus ignotis, Philo&longs;ophi hactenus Naturam fru<lb/>&longs;tra tentarunt. </s>

<s>Nam multa me movent ut nonnihil &longs;u&longs;picer e<gap/> om-<emph.end type="italics"/>

<s>Spero autem quod vel huic Philo&longs;ophandi modo, <lb/>vel veriori alicui, Principia hic po&longs;ita lucem aliquam præbebunt.<emph.end type="italics"/></s>

<pb/><emph type="italics"/>nia ex viribus quibu&longs;dam pendere po&longs;&longs;e, quibus corporum particulæ <lb/>per cau&longs;as nondum cognitas vel in &longs;e mutuo impelluntur & &longs;e<lb/>cundum figuras regulares cohærent, vel ab invicem fugantur & <lb/>recedunt: quibus viribus ignotis, Philo&longs;ophi hactenus Naturam fru<lb/>&longs;tra tentarunt. </s>

<s>Spero autem quod vel huic Philo&longs;ophandi modo, <lb/>vel veriori alicui, Principia hic po&longs;ita lucem aliquam præbebunt.<emph.end type="italics"/></s>

<s><emph type="italics"/>In his edendis, Vir acuti&longs;&longs;imus & in omni literarum genere <lb/>eruditi&longs;&longs;imus<emph.end type="italics"/> Edmundus Halleius <emph type="italics"/>operam navavit, nec &longs;olum <lb/>Typothetarum Sphalmata correxit & Schemata incidi curavit, &longs;ed <lb/>etiam Auctor fuit ut horum editionem aggrederer. </s>

<s>Quippe cum <lb/>demon&longs;tratam a me Figuram Orbium cæle&longs;tium impetraverat, ro<lb/>gare non de&longs;titit ut eandem cum<emph.end type="italics"/> Societate Regali <emph type="italics"/>communicarem, <lb/>Quæ deinde hortatibus & benignis &longs;uis au&longs;piciis effecit ut de ea<lb/>dem in lucem emittenda cogitare inciperem. </s>

<s>At po&longs;tquam Mo<lb/>tuum Lunarium inæqualitates aggre&longs;&longs;us e&longs;&longs;em, deinde etiam ælia <lb/>tentare cæpi&longs;&longs;em quæ ad leges & men&longs;uras Gravitatis & aliarum <lb/>virium, & Figuras a corporibus &longs;ecundum datas qua&longs;cunque leges <lb/>attractis de&longs;cribendas, ad motus corporum plurium inter &longs;e, ad <lb/>motus corporum in Mediis re&longs;i&longs;tentibus, ad vires, den&longs;itates & <lb/>motus Mediorum, ad Orbes Cometarum & &longs;imilia &longs;pectant, edi<lb/>tionem in aliud tempus differendam e&longs;&longs;e putavi, ut cætera rima<lb/>rer & una in publicum darem. </s>

<s>Quæ ad motus Lunares &longs;pectant, <lb/>(imperfecta cum &longs;int,) in Corollariis Propo&longs;itionis<emph.end type="italics"/> LXVI. <emph type="italics"/>&longs;imul <lb/>complexus &longs;um, ne &longs;ingula methodo prolixiore quam pro rei digni<lb/>tate proponere, & &longs;igillatim demon&longs;trare tenerer, & &longs;eriem reli<lb/>quarum Propo&longs;itionum interrumpere. </s>

<s>Nonnulla &longs;ero inventa lo<lb/>cis minus idoneis in&longs;erere malui, quam numerum Propo&longs;itionum <lb/>& citationes mutare. </s>

<s>Ut omnia candide legantur, & defectus, <lb/>in materia tam difficili non tam reprehendantur, quam novis Le<lb/>ctorum conatibus inve&longs;tigentur, & benigne &longs;uppleantur, enixe rogo.<emph.end type="italics"/></s>

<s>Dabam <emph type="italics"/>Cantabrigiæ,<emph.end type="italics"/> e Collegio <lb/><emph type="italics"/>S. Trinitatis,<emph.end type="italics"/> Maii 8. 1686. </s>

<s>Dabam <emph type="italics"/>Cantabrigiæ,<emph.end type="italics"/> e Collegio <lb/><emph type="italics"/>S. Trinitatis,<emph.end type="italics"/> Maii 8. 1686. </s>

<s><emph type="italics"/>IS. NEWTON.<emph.end type="italics"/></s>

<pb/>

<s><emph type="italics"/>IN hac Secunda Principiorum Editione, multa &longs;par&longs;im emen<lb/>dantur & nonnulla adjiciuntur. </s>

<s><emph type="italics"/>IS. NEWTON.<emph.end type="italics"/></s><pb/>

<s><emph type="italics"/>IN hac Secunda Principiorum Editione, multa &longs;par&longs;im emen<lb/>dantur & nonnulla adjiciuntur. </s>

<s>In Libri primi Sectione<emph.end type="italics"/> II, <lb/><emph type="italics"/>Inventio virium quibus corpora in Orbibus datis revolvi po&longs;&longs;int, <lb/>facilior redditur & amplior. </s>

<s>In Libri &longs;ecundi Sectione<emph.end type="italics"/> VII, <lb/><emph type="italics"/>Theoria re&longs;i&longs;tentiæ Fluidorum accuratius inve&longs;tigatur & novis <lb/>Experimentis confirmatur. </s>

<s>In Libro tertio Theoria Lunæ & Præ<lb/>ce&longs;&longs;io Æquinoctiorum ex Principiis &longs;uis plenius deducuntur, & <lb/>Theoria Cometarum pluribus & accuratius computatis Orbium <lb/>exemplis confirmatur.<emph.end type="italics"/></s>

<s>Dabam <emph type="italics"/>Londini,<emph.end type="italics"/><lb/>Mar. </s>

<s><emph type="italics"/>IS. NEWTON.<emph.end type="italics"/></s><pb/>

<s><emph type="italics"/>IS. NEWTON.<emph.end type="italics"/></s>

<pb/>

<s><emph type="center"/>EDITORIS <lb/>PRÆFATIO.<emph.end type="center"/></s>

<s>NEWTONIANÆ Philo&longs;ophiæ novam tibi, Lector benevole, <lb/>diuque de&longs;ideratam Editionem, plurimum nunc emenda<lb/>tam atque auctiorem exhibemus. </s>

<s><emph type="center"/>EDITORIS <lb/>PRÆFATIO.<emph.end type="center"/></s>

<s>NEWTONIANÆ Philo&longs;ophiæ novam tibi, Lector benevole, <lb/>diuque de&longs;ideratam Editionem, plurimum nunc emenda<lb/>tam atque auctiorem exhibemus. </s>

<s>Quæ poti&longs;&longs;imum conti<lb/>neantur in hoc Opere celeberrimo, intelligere potes ex Indicibus <lb/>adjectis: quæ vel addantur vel immutentur, ip&longs;a te fere docebit <lb/>Auctoris Præfatio. </s>

<s>Reliquum e&longs;t, ut adjiciantur nonnulla de Me<lb/>thodo hujus Philo&longs;ophiæ. </s>

<s>Reliquum e&longs;t, ut adjiciantur nonnulla de Me<lb/>thodo hujus Philo&longs;ophiæ. </s>

<s>Qui Phy&longs;icam tractandam &longs;u&longs;ceperunt, ad tres fere cla&longs;&longs;es re<lb/>vocari po&longs;&longs;unt. </s>

<s>Extiterunt enim, qui &longs;ingulis rerum &longs;peciebus Quali<lb/>tates &longs;pecificas & occultas tribuerint; ex quibus deinde corporum <lb/>&longs;ingulorum operationes, ignota quadam ratione, pendere volue<lb/>runt. </s>

<s>In hoc po&longs;ita e&longs;t &longs;umma doctrinæ Schola&longs;ticæ, ab <emph type="italics"/>Ari&longs;totele<emph.end type="italics"/><lb/>& Peripateticis derivatæ: Affirmant utique &longs;ingulos Effectus ex <lb/>corporum &longs;ingularibus Naturis oriri; at unde &longs;int illæ Naturæ <lb/>non docent; nihil itaque docent. </s>

<s>Cumque toti &longs;int in rerum no<lb/>minibus, non in ip&longs;is rebus; Sermonem quendam Philo&longs;ophicum <lb/>cen&longs;endi &longs;unt adinveni&longs;&longs;e, Philo&longs;ophiam tradidi&longs;&longs;e non &longs;unt cen<lb/>&longs;endi. </s>

<s>Alii ergo melioris diligentiæ laudem con&longs;equi &longs;perarunt, rejecta <lb/>Vocabulorum inutili farragine. </s>

<s>Statuerunt itaque Materiam uni<lb/>ver&longs;am homogeneam e&longs;&longs;e, omnem vero Formarum varietatem, quæ <lb/>in corporibus cernitur, ex particularum componentium &longs;implici&longs;&longs;i<lb/>mis quibu&longs;dam & intellectu facillimis affectionibus oriri. </s>

<s>Et recte <lb/>quidem progre&longs;&longs;io in&longs;tituitur a &longs;implicioribus ad magis compo&longs;ita, <lb/>&longs;i particularum primariis illis affectionibus non alios tribuunt mo<lb/>dos, quam quos ip&longs;a tribuit Natura. </s>

<s>Verum ubi licentiam &longs;ibi <lb/>a&longs;&longs;umunt, ponendi qua&longs;cunque libet ignotas partium figuras & <lb/>magnitudines, incerto&longs;que &longs;itus & motus; quin & fingendi Fluida <lb/>quædam occulta, quæ corporum poros liberrime permeent, omni<lb/>potente prædita &longs;ubtilitate, motibu&longs;que occultis agitata; jam ad <lb/>&longs;omnia delabuntur, neglecta rerum con&longs;titutione vera: quæ fane <lb/>fru&longs;tra petenda e&longs;t ex fallacibus conjecturis, cum vix etiam per <lb/>certi&longs;&longs;imas Ob&longs;ervationes inve&longs;tigari po&longs;&longs;it. </s>

<s>Qui &longs;peculationum <pb/>&longs;uarum fundamentum de&longs;umunt ab Hypothe&longs;ibus, etiam&longs;i deinde <lb/>&longs;ecundum leges Mechanicas accurati&longs;&longs;ime procedant; Fabulam qui<lb/>dem elegantem forte & venu&longs;tam, Fabulam tamen concinnare di<lb/>cendi &longs;unt. </s>

<s>Qui &longs;peculationum

<pb/>&longs;uarum fundamentum de&longs;umunt ab Hypothe&longs;ibus, etiam&longs;i deinde <lb/>&longs;ecundum leges Mechanicas accurati&longs;&longs;ime procedant; Fabulam qui<lb/>dem elegantem forte & venu&longs;tam, Fabulam tamen concinnare di<lb/>cendi &longs;unt. </s>

<s>Relinquitur adeo tertium genus, qui Philo&longs;ophiam &longs;cilicet Ex<lb/>perimentalem profitentur. </s>

<s>Hi quidem ex &longs;implici&longs;&longs;imis quibus <lb/>po&longs;&longs;unt principiis rerum omnium cau&longs;as derivandas e&longs;&longs;e volunt: <lb/>nihil autem Principii loco a&longs;&longs;umunt, quod nondum ex Phænome<lb/>nis comprobatum fuerit. </s>

<s>Hypothe&longs;es non commini&longs;cuntur, neque <lb/>in Phy&longs;icam recipiunt, ni&longs;i ut Quæ&longs;tiones de quarum veritare di&longs;<lb/>putetur. </s>

<s>Duplici itaque Methodo incedunt, Analytica & Syn<lb/>thetica. </s>

<s>Naturæ vires lege&longs;que virium &longs;impliciores ex &longs;electis <lb/>quibu&longs;dam Phænomenis per Analy&longs;in deducunt, ex quibus deinde <lb/>per Synthe&longs;in reliquorum con&longs;titutionem tradunt. </s>

<s>Hæc illa e&longs;t <lb/>Philo&longs;ophandi ratio longe optima, quam præ ceteris merito am<lb/>plectendam cen&longs;uit Celeberrimus Auctor no&longs;ter. </s>

<s>Hanc &longs;olam uti<lb/>que dignam judicavit, in qua excolenda atque adornanda operam <lb/>&longs;uam collocaret. </s>

<s>Hujus igitur illu&longs;tri&longs;&longs;imum dedit Exemplum, <lb/>Mundani nempe Sy&longs;tematis explicationem e Theoria Gravitatis <lb/>felici&longs;&longs;ime deductam. </s>

<s>Gravitatis virtutem univer&longs;is corporibus in<lb/>e&longs;&longs;e, &longs;u&longs;picati &longs;unt vel finxerunt alii: primus Ille & folus ex Ap<lb/>parentiis demon&longs;trare potuit, & &longs;peculationibus egregiis firmi&longs;&longs;i<lb/>mum ponere fundamentum. </s>

<s>Scio equidem nonnullos magni etiam nominis Viros, præjudiciis <lb/>quibu&longs;dam plus æquo occupatos, huic novo Principio ægre a&longs;&longs;en<lb/>tiri potui&longs;&longs;e, & certis incerta identidem prætuli&longs;&longs;e. </s>

<s>Horum famam vel<lb/>licare non e&longs;t animus: Tibi potius, Benevole Lector, illa paucis ex<lb/>ponere lubet, ex quibus Tute ip&longs;e judicium non iniquum feras. </s>

<s>Horum famam vel<lb/>licare non e&longs;t animus: Tibi potius, Benevole Lector, illa paucis ex<lb/>ponere lubet, ex quibus Tute ip&longs;e judicium non iniquum feras. </s>

<s>Igitur ut Argumenti &longs;umatur exordium a &longs;implici&longs;&longs;imis & pro<lb/>ximis; de&longs;piciamus pauli&longs;per qualis &longs;it in Terre&longs;tribus natura Gra<lb/>vitatis, ut deinde tutius progrediamur ubi ad corpora Cæle&longs;tia, lon<lb/>gi&longs;&longs;ime a &longs;edibus no&longs;tris remota, perventum fuerit. </s>

<s>Convenit jam <lb/>inter omnes Philo&longs;ophos, corpora univer&longs;a circumterre&longs;tria gra<lb/>vitare in Terram. </s>

<s>Nulla dari corpora vere levia, jamdudum <lb/>confirmavit Experientia multiplex. </s>

<s>Quæ dicitur Levitas relativa, <lb/>non e&longs;t vera Levitas, &longs;ed apparens &longs;olummodo; & oritur a præ<lb/>pollente Gravitate corporum contiguorum. </s>

<s>Quæ dicitur Levitas relativa, <lb/>non e&longs;t vera Levitas, &longs;ed apparens &longs;olummodo; & oritur a præ<lb/>pollente Gravitate corporum contiguorum. </s>

<s>Porro, ut corpora univer&longs;a gravitant in Terram, ita Terra vici&longs;<lb/>&longs;im in corpora æqualiter gravitat; Gravitatis enim actionem e&longs;&longs;e <lb/>mutuam & utrinque æqualem, &longs;ic o&longs;tenditur. </s>

<s>Di&longs;tinguatur Terræ <pb/>totius moles in binas qua&longs;cunque partes, vel æquales vel utcunque <lb/>inæquales: jam &longs;i pondera partium non e&longs;&longs;ent in &longs;e mutuo æqua<lb/>lia; cederet pondus minus majori, & partes conjunctæ pergerent <lb/>recta moveri ad in&longs;initum, ver&longs;us plagam in quam tendit pondus <lb/>majus: omnino contra Experientiam. </s>

<s>Di&longs;tinguatur Terræ

<s>Itaque dicendum erit, pon<lb/>dera partium in æquilibrio e&longs;&longs;e con&longs;tituta: hoc e&longs;t, Gravitatis <lb/>actionem e&longs;&longs;e mutuam & utrinque æqualem. </s>

<pb/>totius moles in binas qua&longs;cunque partes, vel æquales vel utcunque <lb/>inæquales: jam &longs;i pondera partium non e&longs;&longs;ent in &longs;e mutuo æqua<lb/>lia; cederet pondus minus majori, & partes conjunctæ pergerent <lb/>recta moveri ad in&longs;initum, ver&longs;us plagam in quam tendit pondus <lb/>majus: omnino contra Experientiam. </s>

<s>Itaque dicendum erit, pon<lb/>dera partium in æquilibrio e&longs;&longs;e con&longs;tituta: hoc e&longs;t, Gravitatis <lb/>actionem e&longs;&longs;e mutuam & utrinque æqualem. </s>

<s>Pondera corporum, æqualiter a centro Terræ di&longs;tantium, &longs;unt ut <lb/>quantitates materiæ in corporibus. </s>

<s>Hoc utique colligitur ex <lb/>æquali acceleratione corporum omnium, e quiete per ponderum <lb/>vires cadentium: nam vires quibus inæqualia corpora æqualiter <lb/>accelerantur, debent e&longs;&longs;e proportionales quantitatibus materiæ <lb/>movendæ. </s>

<s>Jam vero corpora univer&longs;a cadentia æqualiter acce<lb/>lerari, ex eo patet, quod in Vacuo <emph type="italics"/>Boyliano<emph.end type="italics"/> temporibus æqualibus <lb/>æqualia &longs;patia cadendo de&longs;cribunt, &longs;ublata &longs;cilicet Aeris re&longs;i&longs;tentia: <lb/>accuratius autem comprobatur per Experimenta Pendulorum. </s>

<s>Vires attractivæ corporum, in æqualibus di&longs;tantiis, &longs;unt ut <lb/>quantitates materiæ in corporibus. </s>

<s>Nam cum corpora in Ter<lb/>ram & Terra vici&longs;&longs;im in corpora momentis æqualibus gravitent; <lb/>Terræ pondus in unumquodque corpus, &longs;eu vis qua corpus Ter<lb/>ram attrahit, æquabitur ponderi corporis eju&longs;dem in Terram. </s>

<s><lb/>Hoc autem pondus erat ut quantitas materiæ in corpore: itaque <lb/>vis qua corpus unumquodque Terram attrahit, &longs;ive corporis vis <lb/>ab&longs;oluta, erit ut eadem quantitas materiæ. </s>

<s>Oritur ergo & componitur vis attractiva corporum integrorum <lb/>ex viribus attractivis partium: &longs;iquidem aucta vel diminuta mole <lb/>materiæ, o&longs;ten&longs;um e&longs;t, proportionaliter augeri vel diminui ejus vir<lb/>tutem. </s>

<s>Actio itaque Telluris ex conjunctis partium Actionibus <lb/>conflari cen&longs;enda erit; atque adeo corpora omnia Terre&longs;tria &longs;e <lb/>mutuo trahere oportet viribus ab&longs;olutis, quæ &longs;int in ratione ma<lb/>teriæ trahentis. </s>

<s>Hæc e&longs;t natura Gravitatis apud Terram: videa<lb/>mus jam qualis &longs;it in Cælis. </s>

<s>Hæc e&longs;t natura Gravitatis apud Terram: videa<lb/>mus jam qualis &longs;it in Cælis. </s>

<s>Corpus omne per&longs;everare in &longs;tatu &longs;uo vel quie&longs;cendi vel movendi <lb/>uniformiter in directum, ni&longs;i quatenus a viribus impre&longs;&longs;is cogitur <lb/>&longs;tatum illum mutare; Naturæ lex e&longs;t ab omnibus recepta Philo&longs;o<lb/>phis. </s>

<s>Inde vero &longs;equitur, corpora quæ in Curvis moventur, atque <lb/>adeo de lineis rectis Orbitas &longs;uas tangentibus jugiter abeunt, Vi <lb/>aliqua perpetuo agente retineri in itinere curvilineo. </s>

<s>Planetis <lb/>igitur in Orbibus curvis revolventibus nece&longs;&longs;ario aderit Vis aliqua, <lb/>per cujus actiones repetitas inde&longs;inenter a Tangentibus deflectantur. </s><pb/>

<s>Planetis <lb/>igitur in Orbibus curvis revolventibus nece&longs;&longs;ario aderit Vis aliqua, <lb/>per cujus actiones repetitas inde&longs;inenter a Tangentibus deflectantur. </s>

<pb/>

<s>Jam illud concedi æquum e&longs;t, quod Mathematicis rationibus <lb/>colligitur & certi&longs;&longs;ime demon&longs;tratur; Corpora nempe omnia, quæ <lb/>moventur in linea aliqua curva in plano de&longs;cripta, quæque radio <lb/>ducto ad punctum vel quie&longs;cens vel utcunque motum de&longs;cribunt <lb/>areas circa punctum illud temporibus proportionales, urgeri a <lb/>Viribus quæ ad idem punctum tendunt. </s>

<s>Cum igitur in confe&longs;&longs;o <lb/>&longs;it apud A&longs;tronomos, Planetas primarios circum Solem, &longs;ecunda<lb/>rios vero circum &longs;uos primarios, areas de&longs;cribere temporibus pro<lb/>portionales; con&longs;equens e&longs;t ut Vis illa, qua perpetuo detorquen<lb/>tur a Tangentibus rectilineis, & in Orbitis curvilineis revolvi ce<lb/>guntur, ver&longs;us corpora dirigatur quæ &longs;ita &longs;unt in Orbitarum cen<lb/>tris. </s>

<s>Hæc itaque Vis non inepte vocari pote&longs;t, re&longs;pectu quidem <lb/>corporis revolventis, Centripeta; re&longs;pectu autem corporis cen<lb/>tralis, Attractiva; a quacunque demum cau&longs;a oriri fingatur. </s>

<s>Quin & hæc quoque concedenda &longs;unt, & Mathematice demon<lb/>&longs;trantur: Si corpora plura motu æquabili revolvantur in Circulis <lb/>concentricis, & quadrata temporum periodicorum &longs;int ut cubi di<lb/>&longs;tantiarum a centro communi; Vires centripetas revolventium <lb/>fore reciproce ut quadrata di&longs;tantiarum. </s>

<s>Vel, &longs;i corpora revol<lb/>vantur in Orbitis quæ &longs;unt Circulis finitimæ, & quie&longs;cant Orbita<lb/>rum Ap&longs;ides; Vires centripetas revolventium fore reciproce ut <lb/>quadrata di&longs;tantiarum. </s>

<s>Obtinere ca&longs;um alterutrum in Planetis <lb/>univer&longs;is con&longs;entiunt A&longs;tronomi. </s>

<s>Itaque Vires centripetæ Plane<lb/>tarum omnium &longs;unt reciproce ut quadrata di&longs;tantiarum ab Or<lb/>bium centris. </s>

<s>Si quis objiciat Planetarum, & Lunæ præ&longs;ertim, <lb/>Ap&longs;ides non penitus quie&longs;cere; &longs;ed motu quodam lento ferri in <lb/>con&longs;equentia: re&longs;ponderi pote&longs;t, etiam&longs;i concedamus hunc mo<lb/>tum tardi&longs;&longs;imum exinde profectum e&longs;&longs;e quod Vis centripetæ pro<lb/>portio aberret aliquantum a duplicata, aberrationem illam per <lb/>computum Mathematicum inveniri po&longs;&longs;e & plane in&longs;en&longs;ibilem <lb/>e&longs;&longs;e. </s>

<s>Ip&longs;a enim ratio Vis centripetæ Lunaris, quæ omnium ma<lb/>xime turbari debet, paululum quidem duplicatam &longs;uperabit; ad <lb/>hanc vero &longs;exaginta fere vicibus propius accedet quam ad tripli<lb/>catam. </s>

<s>Sed verior erit re&longs;pon&longs;io, &longs;i dicamus hanc Ap&longs;idum progre&longs;<lb/>&longs;ionem, non ex aberratione a duplicata proportione, &longs;ed ex alia <lb/>pror&longs;us diver&longs;a cau&longs;a oriri, quemadmodum egregie common&longs;tratur <lb/>in hac Philo&longs;ophia. </s>

<s>Re&longs;tat ergo ut Vires centripetæ, quibus Pla<lb/>netæ primarii tendunt ver&longs;us Solem & &longs;ecundarii ver&longs;us primarios <lb/>&longs;uos, &longs;int accurate ut quadrata di&longs;tantisrum reciproce. </s><pb/>

<pb/>

<s>Ex iis quæ hactenus dicta &longs;unt, con&longs;tat Planetas in Orbitis &longs;uis <lb/>retineri per Vim aliquam in ip&longs;os perpetuo agentem: con&longs;tat <lb/>Vim illam dirigi &longs;emper ver&longs;us Orbitarum centra: con&longs;tat hujus <lb/>efficaciam augeri in acce&longs;&longs;u ad centrum, diminui in rece&longs;&longs;n ab eo<lb/>dem: & augeri quidem in eadem proportione qua diminuitur qua<lb/>dratum di&longs;tantiæ, diminui in eadem proportione qua di&longs;tantiæ <lb/>quadratum augetur. </s>

<s>Videamus jam, comparatione in&longs;tituta inter <lb/>Planetarum Vires centripetas & Vim Gravitatis, annon eju&longs;dem <lb/>forte &longs;int generis. </s>

<s>Eju&longs;dem vero generis erunt, &longs;i deprehendan<lb/>tur hinc & inde leges eædem eædemque affectiones. </s>

<s>Primo ita<lb/>que Lunæ, quæ nobis proxima e&longs;t, Vim centripetam expendamus. </s>

<s>Primo ita<lb/>que Lunæ, quæ nobis proxima e&longs;t, Vim centripetam expendamus. </s>

<s>Spatia rectilinea, quæ a corporibus e quiete demi&longs;&longs;is dato tem<lb/>pore &longs;ub ip&longs;o motus initio de&longs;eribuntur, ubi a viribus quibu&longs;cun<lb/>que urgentur, proportionalia &longs;unt ip&longs;is viribus: Hoc utique con<lb/>&longs;equitur ex ratiociniis Mathematicis. </s>

<s>Erit igitur Vis centripeta <lb/>Lunæ in Orbita &longs;ua revolventis, ad Vim Gravitatis in &longs;uperficie <lb/>Terræ, ut &longs;patium quod tempore quam minimo de&longs;criberet Luna <lb/>de&longs;cendendo per Vim centripetam ver&longs;us Terram, &longs;i circulari om<lb/>ni motu privari fingeretur, ad &longs;patium quod eodem tempore quam <lb/>minimo de&longs;cribit grave corpus in vicinia Terræ, per Vim gravita<lb/>tis &longs;uæ cadendo. </s>

<s>Horum &longs;patiorum prius æquale e&longs;t arcus a Luna <lb/>per idem tempus de&longs;cripti &longs;inui ver&longs;o, quippe qui Lunæ tran&longs;la<lb/>tionem de Tangente, factam a Vi centripeta, metitur; atque adeo <lb/>computari pote&longs;t ex datis tum Lunæ tempore periodico tum di<lb/>&longs;tantia ejus a centro Terræ. </s>

<s>Spatium po&longs;terius invenitur per Ex<lb/>perimenta Pendulorum, quemadmodum docuit <emph type="italics"/>Hugenius.<emph.end type="italics"/> Inito <lb/>itaque calculo, &longs;patium prius ad &longs;patium pofterius, &longs;eu vis cen<lb/>tripeta Lunæ in Orbita &longs;ua revolventis ad vim Gravitatis in &longs;u<lb/>perficie Terræ, erit ut quadratum &longs;emidiametri Terræ ad Orbitæ <lb/>&longs;emidiametri quadratum. </s>

<s>Eandem habet rationem, per ea quæ <lb/>&longs;uperius o&longs;tenduntur, vis centripeta Lunæ in Orbita &longs;ua revol<lb/>venris ad vim Lunæ centripetam prope Terræ &longs;uperficiem. </s>

<s>Vis <lb/>itaque centripeta prope Terræ &longs;uperficiem æqualis e&longs;t vi Gravita<lb/>tis. </s>

<s>Non ergo diver&longs;æ &longs;unt vires, &longs;ed una atque eadem: &longs;i enim <lb/>diver&longs;æ e&longs;&longs;ent, corpora viribus conjunctis duplo celerius in Ter<lb/>ram caderent quam ex vi &longs;ola Gravitatis. </s>

<s>Con&longs;tat igitur Vim <lb/>illam centripetam, qua Luna perpetuo de Tangente vel trahitur <lb/>vel impellitur & in Orbita retinetur, ip&longs;am e&longs;fe vim Gravitatis <lb/>terre&longs;tris ad Lunam u&longs;que pertingentem. </s>

<s>Et rationi quidem con<lb/>&longs;entaneum e&longs;t ut ad ingentes diftantias illa &longs;e&longs;e Virtus extendat, <pb/>cum nullam ejus &longs;en&longs;ibilem imminutionem, vel in alti&longs;&longs;imis montium <lb/>cacuminibus, ob&longs;ervare licet. </s>

<s>Et rationi quidem con<lb/>&longs;entaneum e&longs;t ut ad ingentes diftantias illa &longs;e&longs;e Virtus extendat,

<pb/>cum nullam ejus &longs;en&longs;ibilem imminutionem, vel in alti&longs;&longs;imis montium <lb/>cacuminibus, ob&longs;ervare licet. </s>

<s>Gravitat itaque Luna in Terram: <lb/>quin & actione mutua, Terra vici&longs;&longs;im in Lunam æqualiter gravitat: <lb/>id quod abunde quidem confirmatur in hac Philo&longs;ophia, ubi agi<lb/>tur de Maris æ&longs;tu & Æquinoctiorum præce&longs;&longs;ione, ab actione tum <lb/>Lunæ tum Solis in Terram oriundis. </s>

<s>Hinc & illud tandem edo<lb/>cemur, qua nimirum lege vis Gravitatis decre&longs;cat in majoribus a <lb/>Tellure di&longs;tantiis. </s>

<s>Nam cum Gravitas non diver&longs;a &longs;it a Vi cen<lb/>tripeta Lunari, hæc vero &longs;it reciproce proportionalis quadrato <lb/>di&longs;tantiæ; diminuetur & Gravitas in eadem ratione. </s>

<s>Nam cum Gravitas non diver&longs;a &longs;it a Vi cen<lb/>tripeta Lunari, hæc vero &longs;it reciproce proportionalis quadrato <lb/>di&longs;tantiæ; diminuetur & Gravitas in eadem ratione. </s>

<s>Progrediamur jam ad Planetas reliquos. </s>

<s>Quoniam revolu<lb/>tiones primariorum circa Solem & &longs;ecundariorum circa Jovem & <lb/>Saturnum &longs;unt Phænomena generis eju&longs;dem ac revolutio Lunæ <lb/>circa Terram, quoniam porro demon&longs;tratum e&longs;t vires centripetas <lb/>primariorum dirigi ver&longs;us centrum Solis, &longs;ecundariorum ver&longs;us <lb/>centra Jovis & Saturni, quemadmodum Lunæ vis centripeta ver&longs;us <lb/>Terræ centrum dirigitur; adhæc, quoniam omnes illæ vires &longs;unt <lb/>reciproce ut quadrata di&longs;tantiarum a centris, quemadmodum vis <lb/>Lunæ e&longs;t ut quadratum di&longs;tantiæ a Terra: concludendum erit <lb/>eandem e&longs;&longs;e naturam univer&longs;is. </s>

<s>Itaque ut Luna gravitat in Ter<lb/>ram, & Terra vici&longs;&longs;im in Lunam; &longs;ic etiam gravitabunt omnes <lb/>&longs;ecundarii in primarios &longs;uos, & primarii vici&longs;&longs;im in &longs;ecundarios; <lb/>&longs;ic & omnes primarii in Solem, & Sol vici&longs;&longs;im in primarios. </s>

<s>Igitur Sol in Planetas univer&longs;os gravitat & univer&longs;i in Solem. </s>

<s><lb/>Nam &longs;ecundarii dum primarios &longs;uos comitantur, revolvuntur in<lb/>terea circum Solem una cum primariis. </s>

<s>Eodem itaque argumento, <lb/>utriu&longs;que generis Planetæ gravitant in Solem, & Sol in ip&longs;os. </s>

<s><lb/>Secundarios vero Planetas in Solem gravitare abunde in&longs;uper <lb/>con&longs;tat ex inæqualitatibus Lunaribus; quarum accurati&longs;&longs;imam <lb/>Theoriam, admiranda &longs;agacitate patefactam, in tertio hujus Operis <lb/>libro expo&longs;itam habemus. </s>

<s>Solis virtutem attractivam quoquover&longs;um propagari ad ingen<lb/>tes u&longs;que di&longs;tantias, & &longs;e&longs;e diffundere ad &longs;ingulas circumjecti &longs;pa<lb/>tii partes, aperti&longs;&longs;ime colligi pote&longs;t ex motu Cometarum; qui ab <lb/>immen&longs;is intervallis profecti feruntur in viciniam Solis, & non<lb/>nunquam adeo ad ip&longs;um proxime accedunt ut Globum ejus, in <lb/>Periheliis &longs;uis ver&longs;antes, tantum non contingere videantur. </s>

<s>Ho<lb/>rum Theoriam ab A&longs;tronomis antehac fru&longs;tra quæ&longs;itam, no&longs;tro <lb/>tandem &longs;æculo feliciter inventam & per Ob&longs;ervationes certi&longs;<lb/>&longs;ime demon&longs;tratam, Præ&longs;tanti&longs;&longs;imo no&longs;tro Auctori debemus. </s>

<s>Patet <pb/>igitur Cometas in Sectionibus Conicis umbilicos in centro Solis <lb/>habentibus moveri, & radiis ad Solem ductis areas temporibus <lb/>proportionales de&longs;cribere. </s>

<s>Patet

<pb/>igitur Cometas in Sectionibus Conicis umbilicos in centro Solis <lb/>habentibus moveri, & radiis ad Solem ductis areas temporibus <lb/>proportionales de&longs;cribere. </s>

<s>Ex hi&longs;ce vero Phænomenis manife<lb/>&longs;tum e&longs;t & Mathematice comprobatur, vires illas, quibus Cometæ <lb/>retinentur in orbitis &longs;uis, re&longs;picere Solem & e&longs;&longs;e reciproce ut qua<lb/>drata di&longs;tantiarum ab ip&longs;ius centro. </s>

<s>Gravitant itaque Cometæ <lb/>in Solem: atque adeo Solis vis attractiva non tantum ad corpora <lb/>Planetarum in datis di&longs;tantiis & in eodem fere plano collocata, <lb/>&longs;ed etiam ad Cometas in diver&longs;i&longs;&longs;imis Cælorum regionibus & in <lb/>diver&longs;i&longs;&longs;imis di&longs;tantiis po&longs;itos pertingit. </s>

<s>Hæc igitur e&longs;t natura <lb/>corporum gravitantium, ut vires &longs;uas edant ad omnes di&longs;tantias in <lb/>omnia corpora gravitantia. </s>

<s>Inde vero &longs;equitur, Planetas & Co<lb/>metas univer&longs;os &longs;e mutuo trahere, & in &longs;e mutuo graves e&longs;&longs;e: <lb/>quod etiam confirmatur ex perturbatione Jovis & Saturni, A&longs;tro<lb/>nomis non incognita, & ab actionibus horum Planetarum in &longs;e in<lb/>vicem oriunda; quin & ex motu illo lenti&longs;&longs;imo Ap&longs;idum, qui &longs;u<lb/>pra memoratus e&longs;t, quique a cau&longs;a con&longs;imili profici&longs;citur. </s>

<s>Eo demum pervenimus ut dicendum &longs;it, & Terram & Solem & <lb/>corpora omnia cæle&longs;tia, quæ Solem comitantur, &longs;e mutuo attrahere. </s>

<s><lb/>Singulorum ergo particulæ quæque minimæ vires &longs;uas attractivas <lb/>habebunt, pro quantitate materiæ pollentes; quemadmodum &longs;u<lb/>pra de Terre&longs;tribus o&longs;ten&longs;um e&longs;t. </s>

<s>In diver&longs;is autem di&longs;tantiis, <lb/>erunt & harum vires in duplicata ratione di&longs;tantiarum reciproce: <lb/>nam ex particulis hac lege trahentibus componi debere Globos <lb/>eadem lege trahentes, Mathematice demon&longs;tratur. </s>

<s>Conclu&longs;iones præcedentes huic innituntur Axiomati, quod a <lb/>nullis non recipitur Philo&longs;ophis; Effectuum &longs;cilicet eju&longs;dem ge<lb/>neris, quorum nempe quæ cogno&longs;cuntur proprietates eædem &longs;unt, <lb/>ea&longs;dem e&longs;&longs;e cau&longs;as & ea&longs;dem e&longs;&longs;e proprietates quæ nondum cog<lb/>no&longs;cuntur. </s>

<s>Quis enim dubitat, &longs;i Gravitas &longs;it cau&longs;a de&longs;cen&longs;us <lb/>Lapidis in <emph type="italics"/>Europa,<emph.end type="italics"/> quin eadem &longs;it cau&longs;a de&longs;cen&longs;us in <emph type="italics"/>America?<emph.end type="italics"/><lb/>Si Gravitas mutua fuerit inter Lapidem & Terram in <emph type="italics"/>Europa<emph.end type="italics"/>; <lb/>quis negabit mutuam e&longs;&longs;e in <emph type="italics"/>America?<emph.end type="italics"/> Si vis attractiva Lapidis <lb/>& Terræ componatur, in <emph type="italics"/>Europa,<emph.end type="italics"/> ex viribus attractivis partium; <lb/>quis negabit &longs;imilem e&longs;&longs;e compo&longs;itionem in <emph type="italics"/>America?<emph.end type="italics"/> Si attractio <lb/>Terræ ad omnia corporum genera & ad omnes di&longs;tantias propa<lb/>getur in <emph type="italics"/>Europa<emph.end type="italics"/>; quidni pariter propagari dicamus in <emph type="italics"/>America?<emph.end type="italics"/><lb/>In hac Regula fundatur omnis Philo&longs;ophia: quippe qua &longs;ublata <lb/>nihil affirmare po&longs;&longs;imus de Univer&longs;is. </s>

<s>Con&longs;titutio rerum &longs;ingula<lb/>rum innote&longs;cit per Ob&longs;ervationes & Experimenta: inde vero non <pb/>ni&longs;i per hanc Regulam de rerum univer&longs;arum natura judica<lb/>mus. </s>

<s>Con&longs;titutio rerum &longs;ingula<lb/>rum innote&longs;cit per Ob&longs;ervationes & Experimenta: inde vero non

<pb/>ni&longs;i per hanc Regulam de rerum univer&longs;arum natura judica<lb/>mus. </s>

<s>Jam cum Gravia &longs;int omnia corpora, quæ apud Terram vel in <lb/>Cælis reperiuntur, de quibus Experimenta vel Ob&longs;ervationes in<lb/>&longs;tituere licet; omnino dicendum erit, Gravitatem corporibus uni<lb/>ver&longs;is competere. </s>

<s>Et quemadmodum nulla concipi debent cor<lb/>pora, quæ non &longs;int Exten&longs;a, Mobilia, & Impenetrabilia; ita nulla <lb/>concipi debere, quæ non &longs;int Gravia. </s>

<s>Corporum Exten&longs;io, Mobi<lb/>litas, & Impenetrabilitas non ni&longs;i per Experimenta innote&longs;cunt: <lb/>eodem plane modo Gravitas innote&longs;cit. </s>

<s>Corpora omnia de qui<lb/>bus Ob&longs;ervationes habemus, Exten&longs;a &longs;unt & Mobilia & Impene<lb/>trabilia: & inde concludimus corpora univer&longs;a, etiam illa de qui<lb/>bus Ob&longs;ervationes non habemus, Exten&longs;a e&longs;&longs;e & Mobilia & Im<lb/>penetrabilia. </s>

<s>Ita corpora omnia &longs;unt Gravia, de quibus Ob&longs;er<lb/>vationes habemus: & inde concludimus corpora univer&longs;a, etiam <lb/>illa de quibus Ob&longs;ervationes non habemus, Gravia e&longs;&longs;e. </s>

<s>Si quis <lb/>dicat corpora Stellarum inerrantium non e&longs;&longs;e Gravia, quandoqui<lb/>dem eorum Gravitas nondum e&longs;t ob&longs;ervata; eodem argumento <lb/>dicere licebit neque Exten&longs;a e&longs;&longs;e, nec Mobilia, nec Impenetrabilia, <lb/>cum hæ Fixarum affectiones nondum &longs;int ob&longs;ervatæ. </s>

<s>Quid opus <lb/>e&longs;t verbis? </s>

<s>Inter primarias qualitates corporum univer&longs;orum vel <lb/>Gravitas habebit locum; vel Exten&longs;io, Mobilitas, & Impenetra<lb/>bilitas non habebunt. </s>

<s>Et natura rerum vel recte explicabitur <lb/>per corporum Gravitatem, vel non recte explicabitur per corpo<lb/>rum Exten&longs;ionem, Mobilitatem, & Impenetrabilitatem. </s>

<s>Et natura rerum vel recte explicabitur <lb/>per corporum Gravitatem, vel non recte explicabitur per corpo<lb/>rum Exten&longs;ionem, Mobilitatem, & Impenetrabilitatem. </s>

<s>Audio nonnullos hanc improbare conclu&longs;ionem, & de occultis <lb/>qualitatibus ne&longs;cio quid mu&longs;&longs;itare. </s>

<s>Gravitatem &longs;cilicet Occultum <lb/>e&longs;&longs;e quid, perpetuo argutari &longs;olent; occultas vero cau&longs;as pro<lb/>cul e&longs;&longs;e ablegandas a Philo&longs;ophia. </s>

<s>His autem facile re&longs;pon<lb/>detur; occultas e&longs;&longs;e cau&longs;as, non illas quidem quarum exi&longs;tentia <lb/>per Ob&longs;ervationes clari&longs;&longs;ime demon&longs;tratur, &longs;ed has &longs;olum quarum <lb/>occulta e&longs;t & ficta exi&longs;tentia nondum vero comprobata. </s>

<s>Gravitas <lb/>ergo non erit occulta cau&longs;a motuum cæle&longs;tium; &longs;iquidem ex Phæ<lb/>nomenis o&longs;ten&longs;um e&longs;t, hanc virtutem revera exi&longs;tere. </s>

<s>Hi potius <lb/>ad occultas confugiunt cau&longs;as; qui ne&longs;cio quos Vortices, materiæ <lb/>cuju&longs;dam pror&longs;us fictitiæ & &longs;en&longs;ibus omnino ignotæ, motibus <lb/>ii&longs;dem regendis præficiunt. </s>

<s>Ideone autem Gravitas occulta cau&longs;a dicetur, eoque nomine <lb/>rejicietur e Philo&longs;ophia, quod cau&longs;a ip&longs;ius Gravitatis occulta e&longs;t <lb/>& nondum inventa? </s>

<s>Qui &longs;ic &longs;tatuunt, videant nequid &longs;tatu-<pb/>ant ab&longs;urdi, unde totius tandem Philo&longs;ophiæ fundamenta convel<lb/>lantur. </s>

<s>Qui &longs;ic &longs;tatuunt, videant nequid &longs;tatu-

<pb/>ant ab&longs;urdi, unde totius tandem Philo&longs;ophiæ fundamenta convel<lb/>lantur. </s>

<s>Etenim cau&longs;æ continuo nexu procedere &longs;olent a compo<lb/>&longs;itis ad &longs;impliciora: ubi ad cau&longs;am &longs;implici&longs;&longs;imam perveneris, jam <lb/>non licebit ulterius progredi. </s>

<s>Cau&longs;æ igitur &longs;implici&longs;&longs;imæ nulla <lb/>dari pote&longs;t mechanica explicatio: &longs;i daretur enim, cau&longs;a non<lb/>dum e&longs;&longs;et &longs;implici&longs;&longs;ima. </s>

<s>Has tu proinde cau&longs;as &longs;implici&longs;&longs;imas <lb/>appellabis occultas, & exulare jubebis? </s>

<s>&longs;imul vero exulabunt <lb/>& ab his proxime pendentes & quæ ab illis porro pendent, <lb/>u&longs;que dum a cau&longs;is omnibus vacua fuerit & probe purgata Phi<lb/>lo&longs;ophia. </s>

<s>&longs;imul vero exulabunt <lb/>& ab his proxime pendentes & quæ ab illis porro pendent, <lb/>u&longs;que dum a cau&longs;is omnibus vacua fuerit & probe purgata Phi<lb/>lo&longs;ophia. </s>

<s>Sunt qui Gravitatem præter naturam e&longs;&longs;e dicunt, & Miraculum <lb/>perpetuum vocant. </s>

<s>Itaque rejiciendam e&longs;&longs;e volunt, cum in Phy<lb/>&longs;ica præternaturales cau&longs;æ locum non habeant. </s>

<s>Huic ineptæ <lb/>pror&longs;us objectioni diluendæ, quæ & ip&longs;a Philo&longs;ophiam &longs;ubruit <lb/>univer&longs;am, vix operæ pretium e&longs;t immorari. </s>

<s>Vel enim Gravita<lb/>tem corporibus omnibus inditam e&longs;&longs;e negabunt, quod tamen dici <lb/>non pote&longs;t: vel eo nomine præter naturam e&longs;&longs;e affirmabunt, quod <lb/>ex aliis corporum affectionibus atque adeo ex cau&longs;is Mechanicis <lb/>originem non habeat. </s>

<s>Dantur certe primariæ corporum affecti<lb/>ones; quæ, quoniam &longs;unt primariæ, non pendent ab aliis. </s>

<s>Vide<lb/>rint igitur annon & hæ omnes &longs;int pariter præter naturam, eo<lb/>que pariter rejiciendæ: viderint vero qualis &longs;it deinde futura <lb/>Philo&longs;ophia. </s>

<s>Vide<lb/>rint igitur annon & hæ omnes &longs;int pariter præter naturam, eo<lb/>que pariter rejiciendæ: viderint vero qualis &longs;it deinde futura <lb/>Philo&longs;ophia. </s>

<s>Nonnulli &longs;unt quibus hæc tota Phy&longs;ica cæle&longs;tis vel ideo minus <lb/>placet, quod cum <emph type="italics"/>Carte&longs;ii<emph.end type="italics"/> dogmatibus pugnare & vix conciliari <lb/>po&longs;&longs;e videatur. </s>

<s>His &longs;ua licebit opinione frui; ex æquo autem <lb/>agant oportet: non ergo denegabunt aliis eandem libertatem <lb/>quam &longs;ibi concedi po&longs;tulant. </s>

<s>NEWTONIANAM itaque Philo&longs;ophi<lb/>am, quæ nobis verior habetur, retinere & amplecti licebit, & cau&longs;as <lb/>&longs;equi per Phænomena comprobatas, potius quam fictas & nondum <lb/>comprobatas. </s>

<s>Ad veram Philo&longs;ophiam pertinet, rerum naturas <lb/>ex cau&longs;is vere exi&longs;tentibus derivare: eas vero leges quærere, qui<lb/>bus voluit Summus opifex hunc Mundi pulcherrimum ordinem <lb/>&longs;tabilire; non eas quibus potuit, &longs;i ita vi&longs;um fui&longs;&longs;et. </s>

<s>Rationi enim <lb/>con&longs;onum e&longs;t, ut a pluribus cau&longs;is, ab invicem nonnihil diver&longs;is, <lb/>idem po&longs;&longs;it Effectus profici&longs;ci: hæc autem vera erit cau&longs;a, ex qua <lb/>vere atque actu profici&longs;citur; reliquæ locum non habent in Philo<lb/>&longs;ophia vera. </s>

<s>In Horologiis automatis idem Indicis horarii mo<lb/>tus vel ab appen&longs;o Pondere vel ab intus conclu&longs;o Elatere oriri po<lb/>te&longs;t. </s>

<s>Quod &longs;i oblatum Horologium revera &longs;it in&longs;tructum Pondere; <pb/>ridebitur qui finget Elaterem, & ex Hypothe&longs;i &longs;ic præpropere con<lb/>ficta motum Indicis explicare &longs;u&longs;cipiet: oportuit enim internam <lb/>Machinæ fabricam penitius per&longs;crutari, ut ita motus propo&longs;iti prin<lb/>cipium verum exploratum habere po&longs;&longs;et. </s>

<s>Quod &longs;i oblatum Horologium revera &longs;it in&longs;tructum Pondere;

<pb/>ridebitur qui finget Elaterem, & ex Hypothe&longs;i &longs;ic præpropere con<lb/>ficta motum Indicis explicare &longs;u&longs;cipiet: oportuit enim internam <lb/>Machinæ fabricam penitius per&longs;crutari, ut ita motus propo&longs;iti prin<lb/>cipium verum exploratum habere po&longs;&longs;et. </s>

<s>Idem vel non ab&longs;imile <lb/>feretur judicium de Philo&longs;ophis illis, qui materia quadam &longs;ubti<lb/>li&longs;&longs;ima Cælos e&longs;&longs;e repletos, hanc autem in Vortices inde&longs;inenter <lb/>agi voluerunt. </s>

<s>Nam &longs;i Phænomenis vel accurati&longs;&longs;ime &longs;atisfacere <lb/>po&longs;&longs;ent ex Hypothe&longs;ibus &longs;uis; veram tamen Philo&longs;ophiam tradi<lb/>di&longs;&longs;e, & veras cau&longs;as motuum cæle&longs;tium inveni&longs;&longs;e nondum di<lb/>cendi &longs;unt; ni&longs;i vel has revera exi&longs;tere, vel &longs;altem alias non ex<lb/>i&longs;tere demon&longs;traverint. </s>

<s>Igitur &longs;i o&longs;ten&longs;um fuerit, univer&longs;orum <lb/>corporum Attractionem habere verum locum in rerum natura; <lb/>quinetiam o&longs;ten&longs;um fuerit, qua ratione motus omnes cæle&longs;tes ab<lb/>inde &longs;olutionem recipiant; vana fuerit & merito deridenda objectio, <lb/>&longs;i quis dixerit eo&longs;dem motus per Vortices explicari debere, etiam&longs;i <lb/>id fieri po&longs;&longs;e vel maxime conce&longs;&longs;erimus. </s>

<s>Non autem concedimus: <lb/>Nequeunt enim ullo pacto Phænomena per Vortices explicari; <lb/>quod ab Auctore no&longs;tro abunde quidem & clari&longs;&longs;imis rationibus <lb/>evincitur; ut &longs;omniis plus æquo indulgeant oporteat, qui inep<lb/>ti&longs;&longs;imo figmento re&longs;arciendo, novi&longs;que porro commentis ornando <lb/>infelicem operam addicunt. </s>

<s>Si corpora Planetarum & Cometarum circa Solem deferantur <lb/>a Vorticibus; oportet corpora delata & Vorticum partes proxime <lb/>ambientes eadem velocitate eademque cur&longs;us determinatione mo<lb/>veri, & eandem habere den&longs;itatem vel eandem Vim inertiæ pro <lb/>mole materiæ. </s>

<s>Con&longs;tat vero Planetas & Cometas, dum ver&longs;an<lb/>tur in ii&longs;dem regionibus Cælorum, velocitatibus variis variaque <lb/>cur&longs;us determinatione moveri. </s>

<s>Nece&longs;&longs;ario itaque &longs;equitur, ut <lb/>Fluidi cæle&longs;tis partes illæ, quæ &longs;unt ad ea&longs;dem di&longs;tantias a Sole, <lb/>revolvantur eodem tempore in plagas diver&longs;as cum diver&longs;is ve<lb/>locitatibus: etenim alia opus erit directione & velocitate, ut tran<lb/>&longs;ire po&longs;&longs;int Planetæ; alia, ut tran&longs;ire po&longs;&longs;int Cometæ. </s>

<s>Quod cum <lb/>explicari nequeat; vel fatendum erit, univer&longs;a corpora cæle&longs;tia <lb/>non deferri a materia Vorticis; vel dicendum erit, eorundem mo<lb/>tus repetendos e&longs;le non ab uno eodemque Vortice, &longs;ed a pluribus <lb/>qui ab invicem diver&longs;i &longs;int, idemque &longs;patium Soli circumjectum <lb/>pervadant. </s>

<s>Si plures Vortices in eodem &longs;patio contineri, & &longs;e&longs;e mutuo pe<lb/>netrare, motibu&longs;que diver&longs;is revolvi ponantur; quoniam hi mo<lb/>tus debent e&longs;&longs;e conformes delatorum corporum motibus, qui <pb/>&longs;unt &longs;umme regulares, & peraguntur in Sectionibus Conicis, nunc <lb/>valde eccentricis, nunc ad Circulorum proxime formam acceden<lb/>tibus; jure quærendum erit, qui fieri po&longs;&longs;it, ut iidem integri con<lb/>&longs;erventur, nec ab actionibus materiæ occur&longs;antis per tot &longs;æcula <lb/>quicquam perturbentur. </s>

<pb/>&longs;unt &longs;umme regulares, & peraguntur in Sectionibus Conicis, nunc <lb/>valde eccentricis, nunc ad Circulorum proxime formam acceden<lb/>tibus; jure quærendum erit, qui fieri po&longs;&longs;it, ut iidem integri con<lb/>&longs;erventur, nec ab actionibus materiæ occur&longs;antis per tot &longs;æcula <lb/>quicquam perturbentur. </s>

<s>Sane &longs;i motus hi fictitii &longs;unt magis com<lb/>po&longs;iti & difficilius explicantur, quam veri illi motus Planetarum <lb/>& Cometarum; fru&longs;tra mihi videntur in Philo&longs;ophiam recipi: <lb/>omnis enim Cau&longs;a debet e&longs;&longs;e E&longs;fectu &longs;uo &longs;implicior. </s>

<s>Conce&longs;&longs;a <lb/>Fabularum licentia, affirmaverit aliquis Planetas omnes & Cometas <lb/>circumcingi Atmo&longs;phæris, adin&longs;tar Telluris no&longs;træ; quæ quidem <lb/>Hypothe&longs;is rationi magis con&longs;entanea videbitur quam Hypothe<lb/>&longs;is Vorticum. </s>

<s>Affirmaverit deinde has Atmo&longs;phæras, ex natura <lb/>&longs;ua, circa Solem moveri & Sectiones Conicas de&longs;cribere; qui <lb/>&longs;ane motus multo facilius concipi pote&longs;t, quam con&longs;imilis motus <lb/>Vorticum &longs;e invicem permeantium. </s>

<s>Denique Planetas ip&longs;os & <lb/>Cometas circa Solem deferri ab Atmo&longs;phæris &longs;uis credendum e&longs;&longs;e <lb/>&longs;tatuat, & ob repertas motuum cæle&longs;tium cau&longs;as triumphum agat. </s>

<s><lb/>Qui&longs;quis autem hanc Fabulam rejiciendam e&longs;&longs;e putet, idem & alte<lb/>ram Fabulam rejiciet: nam ovum non e&longs;t ovo &longs;imilius, quam Hy<lb/>pothe&longs;is Atmo&longs;phærarum Hypothe&longs;i Vorticum. </s>

<s>Docuit <emph type="italics"/>Galilæus,<emph.end type="italics"/> lapidis projecti & in Parabola moti deflexio<lb/>nem a cur&longs;u rectilineo oriri a Gravitate lapidis in Terram, ab oc<lb/>culta &longs;cilicet qualitate. </s>

<s>Fieri tamen pote&longs;t ut alius aliquis, na&longs;i <lb/>acutioris, Philo&longs;ophus cau&longs;am aliam commini&longs;catur. </s>

<s>Finget igi<lb/>tur ille materiam quandam &longs;ubtilem, quæ nec vi&longs;u, nec tactu, <lb/>neque ullo &longs;en&longs;u percipitur, ver&longs;ari in regionibus quæ proxime <lb/>contingunt Telluris &longs;uperficiem. </s>

<s>Hanc autem materiam, in di<lb/>ver&longs;as plagas, variis & plerumque contrariis motibus ferri, & li<lb/>neas Parabolicas de&longs;cribere contendet. </s>

<s>Deinde vero lapidis de<lb/>flexionem pulchre &longs;ic expediet, & vulgi plau&longs;um merebitur. </s>

<s>La<lb/>pis, inquiet, in Fluido illo &longs;ubtili natat; & cur&longs;ui ejus ob&longs;equen<lb/>do, non pote&longs;t non eandem una &longs;emitam de&longs;cribere. </s>

<s>Fluidum <lb/>vero movetur in lineis Parabolicis; ergo lapidem in Parabola <lb/>moveri nece&longs;&longs;e e&longs;t. </s>

<s>Quis nunc non mirabitur acuti&longs;&longs;imum huju&longs;ce <lb/>Philo&longs;ophi ingenium, ex cau&longs;is Mechanicis, materia &longs;cilicet & <lb/>motu, phænomena Naturæ ad vulgi etiam captum præclare de<lb/>ducentis? </s>

<s>Quis vero non &longs;ub&longs;annabit bonum illum <emph type="italics"/>Galilæum,<emph.end type="italics"/> qui <lb/>magno molimine Mathematico qualitates occultas, e Philo&longs;ophia <lb/>feliciter exclu&longs;as, denuo revocare &longs;u&longs;tinuerit? </s>

<s>Sed pudet nugis <lb/>diutius immorari. </s><pb/>

<s>Sed pudet nugis <lb/>diutius immorari. </s>

<pb/>

<s>Summa rei huc tandem redìt: Cometarum ingens e&longs;t numerus; <lb/>motus eorum &longs;unt &longs;umme regulares, & ea&longs;dem leges cum Plane<lb/>tarum motibus ob&longs;ervant. </s>

<s>Moventur in Orbibus Conicis, hi or<lb/>bes &longs;unt valde admodum eccentrici. </s>

<s>Feruntur undique in omnes <lb/>Cælorum partes, & Planetarum regiones liberrime pertran&longs;eunt, <lb/>& &longs;æpe contra Signorum ordinem incedunt. </s>

<s>Hæc Phænomena <lb/>certi&longs;&longs;ime confirmantur ex Ob&longs;ervationibus A&longs;tronomicis: & per <lb/>Vortices nequeunt explicari: Imo, ne quidem cum Vorticibus <lb/>Planetarum con&longs;i&longs;tere po&longs;&longs;unt. </s>

<s>Cometarum motibus omnino lo<lb/>cus non erit; ni&longs;i materia illa fictitia penitus e Cælis amo<lb/>veatur. </s>

<s>Cometarum motibus omnino lo<lb/>cus non erit; ni&longs;i materia illa fictitia penitus e Cælis amo<lb/>veatur. </s>

<s>Si enim Planetæ circum Solem a Vorticibus devehuntur; Vor<lb/>ticum partes, quæ proxime ambiunt unumquemque Planetam, eju&longs;<lb/>dem den&longs;itatis erunt ac Planeta; uti &longs;upra dictum e&longs;t. </s>

<s>Itaque <lb/>materia illa omnis quæ contigua e&longs;t Orbis magni perimetro, pa<lb/>rem habebit ac Tellus den&longs;itatem: quæ vero jacet intra Orbem <lb/>magnum atque Orbem Saturni, vel parem vel majorem habebit. </s>

<s><lb/>Nam ut con&longs;titutio Vorticis permanere po&longs;&longs;it, debent partes mi<lb/>nus den&longs;æ centrum occupare, magis den&longs;æ longius a centro abire. </s>

<s><lb/>Cum enim Planetarum tempora periodica &longs;int in ratione &longs;e&longs;qui<lb/>plicata di&longs;tantiarum a Sole, oportet partium Vorticis periodos <lb/>eandem rationem &longs;ervare. </s>

<s>Inde vero &longs;equitur, vires centrifugas <lb/>harum partium fore reciproce ut quadrata di&longs;tantiarum. </s>

<s>Quæ <lb/>igitur majore intervallo di&longs;tant a centro, nituntur ab eodem re<lb/>cedere minore vi: unde &longs;i minus den&longs;æ fuerint, nece&longs;&longs;e e&longs;t ut ce<lb/>dant vi majori, qua partes centro propiores a&longs;cendere conantur. </s>

<s><lb/>A&longs;cendent ergo den&longs;iores, de&longs;cendent minus den&longs;æ, & locorum <lb/>fiet invicem permutatio; donec ita fuerit di&longs;po&longs;ita atque ordinata <lb/>materia fluida totius Vorticis, ut conquie&longs;cere jam po&longs;&longs;it in æqui<lb/>librio con&longs;tituta. </s>

<s>Si bina Fluida, quorum diver&longs;a e&longs;t den&longs;itas, <lb/>in eodem va&longs;e continentur; utique futurum e&longs;t ut Fluidum, cu<lb/>jus major e&longs;t den&longs;itas, majore vi Gravitatis infimum petat locum: <lb/>& ratione non ab&longs;imili omnino dicendum e&longs;t, den&longs;iores Vorticis <lb/>partes majore vi centrifuga petere &longs;upremum locum. </s>

<s>Tota igi<lb/>tur illa & multo maxima pars Vorticis, quæ jacet extra Telluris <lb/>orbem, den&longs;itatem habebit atque adeo vim inertiæ pro mole ma<lb/>teriæ, quæ non minor erit quam den&longs;itas & vis inertiæ Telluris: <lb/>inde vero Cometis trajectis orietur ingens re&longs;i&longs;tentia, & valde ad<lb/>modum &longs;en&longs;ibilis; ne dicam, quæ motum eorundem penitus &longs;i&longs;tere <lb/>atque ab&longs;orbere po&longs;&longs;e merito videatur. </s>

<s>Con&longs;tat autem ex motu Co-<pb/>metarum pror&longs;us regulari, nullam ip&longs;os re&longs;i&longs;tentiam pati quæ vel <lb/>minimum &longs;entiri pote&longs;t; atque adeo neutiquam in materiam ul<lb/>lam incur&longs;are, cujus aliqua &longs;it vis re&longs;i&longs;tendi, vel proinde cujus ali<lb/>qua &longs;it den&longs;itas &longs;eu vis Inertiæ. </s>

<s>Con&longs;tat autem ex motu Co-

<pb/>metarum pror&longs;us regulari, nullam ip&longs;os re&longs;i&longs;tentiam pati quæ vel <lb/>minimum &longs;entiri pote&longs;t; atque adeo neutiquam in materiam ul<lb/>lam incur&longs;are, cujus aliqua &longs;it vis re&longs;i&longs;tendi, vel proinde cujus ali<lb/>qua &longs;it den&longs;itas &longs;eu vis Inertiæ. </s>

<s>Nam re&longs;i&longs;tentia Mediorum ori<lb/>tur vel ab inertia materiæ fluidæ, vel a defectu lubricitatis. </s>

<s>Quæ <lb/>oritur a defectu lubricitatis, admodum exigua e&longs;t; & &longs;ane vix <lb/>ob&longs;ervari pote&longs;t in Fluidis vulgo notis, ni&longs;i valde tenacia fuerint <lb/>adin&longs;tar Olei & Mellis. </s>

<s>Re&longs;i&longs;tentia quæ &longs;entitur in Aere, Aqua, <lb/>Hydrargyro, & huju&longs;modi Fluidis non tenacibus fere tota e&longs;t <lb/>prioris generis; & minui non pote&longs;t per ulteriorem quemcunque <lb/>gradum &longs;ubtilitatis, manente Fluidi den&longs;itate vel vi inertiæ, cui <lb/>&longs;emper proportionalis e&longs;t hæc re&longs;i&longs;tentia; quemadmodum clari&longs;<lb/>&longs;ime demon&longs;tratum e&longs;t ab Auctore no&longs;tro in peregregia Re&longs;i&longs;ten<lb/>tiarum Theoria, quæ paulo nunc accuratius exponitur, hac &longs;e<lb/>cunda vice, & per Experimenta corporum cadentium plenius <lb/>confirmatur. </s>

<s>Corpora progrediendo motum &longs;uum Fluido ambienti paulatim <lb/>communicant, & communicando amittunt, amittendo autem re<lb/>tardantur. </s>

<s>E&longs;t itaque retardatio motui communicato proportio<lb/>nalis; motus vero communicatus, ubi datur corporis progredientis <lb/>velocitas, e&longs;t ut Fluidi den&longs;itas; ergo retardatio &longs;eu re&longs;i&longs;tentia <lb/>erit ut eadem Fluidi den&longs;itas; neque ullo pacto tolli pote&longs;t, ni&longs;i <lb/>a Fluido ad partes corporis po&longs;ticas recurrente re&longs;tituatur motus <lb/>ami&longs;&longs;us. </s>

<s>Hoc autem dici non poterit, ni&longs;i impre&longs;&longs;io Fluidi in cor<lb/>pus ad partes po&longs;ticas æqualis fuerit impre&longs;&longs;ioni corporis in Flui<lb/>dum ad partes anticas, hoc e&longs;t, ni&longs;i velocitas relativa qua Flui<lb/>dum irruit in corpus a tergo, æqualis fuerit velocitati qua cor<lb/>pus irruit in Fluidum, id e&longs;t, ni&longs;i velocitas ab&longs;oluta Fluidi re<lb/>currentis duplo major fuerit quam velocitas ab&longs;oluta Fluidi pro<lb/>pul&longs;i; quod fieri nequit. </s>

<s>Nullo igitur modo tolli pote&longs;t Flui<lb/>dorum re&longs;i&longs;tentia, quæ oritur ab corundem den&longs;itate & vi in<lb/>ertiæ. </s>

<s>Itaque concludendum erit; Fluidi cæle&longs;tis nullam e&longs;&longs;e <lb/>vim inertiæ, cum nulla &longs;it vis re&longs;i&longs;tendi: nullam e&longs;&longs;e vim qua <lb/>motus communicetur, cum nulla &longs;it vis inertiæ: nullam e&longs;&longs;e vim <lb/>qua mutatio quælibet vel corporibus &longs;ingulis vel pluribus indu<lb/>catur, cum nulla &longs;it vis qua motus communicetur: nullam e&longs;&longs;e <lb/>omnino efficaciam, cum nulla &longs;it facultas mutationem quamlibet <lb/>inducendi. </s>

<s>Quidni ergo hanc Hypothe&longs;in, quæ fundamento <lb/>plane de&longs;tituitur, quæque naturæ rerum explicandæ ne minimum <lb/>quidem in&longs;ervit, inepti&longs;&longs;imam vocare liceat & Philo&longs;opho pror-<pb/>&longs;us indignam. </s>

<pb/>&longs;us indignam. </s>

<s>Qui Cælos materia fluida repletos e&longs;&longs;e volunt, <lb/>hanc vero non inertem e&longs;&longs;e &longs;tatuunt; Hi verbis tollunt Vacuum, <lb/>re ponunt. </s>

<s>Nam cum huju&longs;modi materia fluida ratione nulla <lb/>&longs;ecerni po&longs;&longs;it ab inani Spatio; di&longs;putatio tota fit de rerum no<lb/>minibus, non de naturis. </s>

<s>Quod &longs;i aliqui &longs;int adeo u&longs;que de<lb/>diti Materiæ, ut Spatium a corporibus vacuui nullo pacto ad<lb/>mittendum credere velint; videamus quo tandem oporteat illos <lb/>pervenire. </s>

<s>Vel enim dicent hanc, quam confingunt, Mundi per omnia <lb/>pleni con&longs;titutionem ex voluntate Dei profectam e&longs;&longs;e, propter <lb/>eum finem, ut operationibus Naturæ &longs;ub&longs;idium præ&longs;ens haberi <lb/>po&longs;&longs;et ab Æthere &longs;ubtili&longs;&longs;imo cuncta permeante & implente; <lb/>quod tamen dici non pote&longs;t, &longs;iquidem jam o&longs;ten&longs;um e&longs;t ex Co<lb/>metarum phænomenis, nullam e&longs;&longs;e hujus Ætheris efficaciam: vel <lb/>dicent ex voluntate Dei profectam e&longs;&longs;e, propter finem aliquem <lb/>ignotum; quod neque dici debet, &longs;iquidem diver&longs;a Mundi con<lb/>&longs;titutio eodem argumento pariter &longs;tabiliri po&longs;&longs;et: vel denique <lb/>non dicent ex voluntate Dei profectam e&longs;&longs;e, &longs;ed ex nece&longs;&longs;itate <lb/>quadam Naturæ. </s>

<s>Tandem igitur delabi oportet in &longs;æces &longs;ordi<lb/>das Gregis impuri&longs;&longs;imi. </s>

<s>Hi &longs;unt qui &longs;omniant Fato univer&longs;a <lb/>regi, non Providentia; Materiam ex nece&longs;&longs;itate &longs;ua &longs;emper & ubi<lb/>que extiti&longs;&longs;e, infinitam e&longs;&longs;e & æternam. </s>

<s>Quibus po&longs;itis, erit <lb/>etiam undiquaque uniformis: nam varietas formarum cum nece&longs;<lb/>&longs;itate omnino pugnat. </s>

<s>Erit etiam immota: nam &longs;i nece&longs;&longs;ario <lb/>moveatur in plagam aliquam determinatam, cum determinata ali<lb/>qua velocitate; pari nece&longs;&longs;itate movebitur in plagam diver&longs;am <lb/>cum diver&longs;a velocitate; in plagas autem diver&longs;as, cum diver&longs;is <lb/>velocitatibus, moveri non pote&longs;t; oportet igitur immotam e&longs;&longs;e. </s>

<s><lb/>Neutiquam profecto potuit oriri Mundus, pulcherrima forma<lb/>rum & motuum varietate di&longs;tinctus, ni&longs;i ex liberrima voluntate <lb/>cuncta providentis & gubernantis Dei. </s>

<s><lb/>Neutiquam profecto potuit oriri Mundus, pulcherrima forma<lb/>rum & motuum varietate di&longs;tinctus, ni&longs;i ex liberrima voluntate <lb/>cuncta providentis & gubernantis Dei. </s>

<s>Ex hoc igitur fonte promanarunt illæ omnes quæ dicuntur <lb/>Naturæ leges: in quibus multa &longs;ane &longs;apienti&longs;&longs;imi con&longs;ilii, nulla <lb/>nece&longs;&longs;itatis apparent ve&longs;tigia. </s>

<s>Has proinde non ab incertis con<lb/>jecturis petere, &longs;ed Ob&longs;ervando atque Experiendo addi&longs;cere de<lb/>bemus. </s>

<s>Qui veræ Phy&longs;icæ principia Lege&longs;que rerum, &longs;ola men<lb/>tis vi & interno rationis lumine fretum, invenire &longs;e po&longs;&longs;e confi<lb/>dit; hunc oportet vel &longs;tatuere Mundum ex nece&longs;&longs;itate fui&longs;le, Le<lb/>ge&longs;que propo&longs;itas ex eadem nece&longs;&longs;itate &longs;equi; vel &longs;i per volun<lb/>tatem Dei con&longs;titutus &longs;it ordo Naturæ, &longs;e tamen, homuncionem <pb/>mi&longs;ellum, quid optimum factu &longs;it per&longs;pectum habere. </s>

<pb/>mi&longs;ellum, quid optimum factu &longs;it per&longs;pectum habere. </s>

<s>Sana om<lb/>nis & vera Philo&longs;ophia fundatur in Phænomenis rerum: quæ &longs;i <lb/>nos vel invitos & reluctantes ad huju&longs;modi principia deducunt, <lb/>in quibus clari&longs;&longs;ime cernuntur Con&longs;ilium optimum & Dominium <lb/>&longs;ummum &longs;apienti&longs;&longs;imi & potenti&longs;&longs;imi Entis; non erunt hæc ideo <lb/>non admittenda principia, quod quibu&longs;dam for&longs;an hominibus <lb/>minus grata &longs;int futura. </s>

<s>His vel Miracula vel Qualitates occultæ <lb/>dicantur, quæ di&longs;plicent: verum nomina malitio&longs;e indita non &longs;unt <lb/>ip&longs;is rebus vitio vertenda; ni&longs;i illud fateri tandem velint, utique <lb/>debere Philo&longs;ophiam in Athei&longs;mo fundari. </s>

<s>Horum hominum <lb/>gratia non erit labefactanda Philo&longs;ophia, &longs;iquidem rerum ordo <lb/>non vult immutari. </s>

<s>Horum hominum <lb/>gratia non erit labefactanda Philo&longs;ophia, &longs;iquidem rerum ordo <lb/>non vult immutari. </s>

<s>Obtinebit igitur apud probos & æquos Judices præ&longs;tanti&longs;&longs;ima <lb/>Philo&longs;ophandi ratio, quæ fundatur in Experimentis & Ob&longs;erva<lb/>tionibus. </s>

<s>Huic vero, dici vix poterit, quanta lux accedat, quanta <lb/>dignitas, ab hoc Opere præclaro Illu&longs;tri&longs;&longs;imi no&longs;tri Auctoris; cujus <lb/>eximiam ingenii felicitatem, difficillima quæque Problemata eno<lb/>dantis, & ad ea porro pertingentis ad quæ nec &longs;pes erat humanam <lb/>mentem a&longs;&longs;urgere potui&longs;&longs;e, merito admirantur & &longs;u&longs;piciunt qui<lb/>cunque paulo profundius in hi&longs;ce rebus ver&longs;ati &longs;unt. </s>

<s>Clau&longs;tris <lb/>ergo re&longs;eratis, aditum Nobis aperuit ad pulcherrima rerum my<lb/>&longs;teria. </s>

<s>Sy&longs;tematis Mundani compagem eleganti&longs;&longs;imam ita tan<lb/>dem patefecit & penitius per&longs;pectandam dedit; ut nec ip&longs;e, &longs;i <lb/>nunc revivi&longs;ceret, Rex <emph type="italics"/>Alphon&longs;us<emph.end type="italics"/> vel &longs;implicitatem vel harmoniæ <lb/>gratiam in ea de&longs;ideraret. </s>

<s>Itaque Naturæ maje&longs;tatem propius jam <lb/>licet intueri, & dulci&longs;&longs;ima contemplatione frui, Conditorem vero <lb/>ac Dominum Univer&longs;orum impen&longs;ius colere & venerari, qui fructus <lb/>e&longs;t Philo&longs;ophiæ multo uberrimus. </s>

<s>Cæcum e&longs;&longs;e oportet, qui ex <lb/>optimis & &longs;apienti&longs;&longs;imis rerum &longs;tructuris non &longs;tatim videat Fabri<lb/>catoris Omnipotentis infinitam &longs;apientiam & bonitatem: in&longs;anum, <lb/>qui profiteri nolit. </s>

<s>Extabit igitur Eximium NEWTONI Opus adver&longs;us Atheorum <lb/>impetus muniti&longs;&longs;imum præ&longs;idium: neque enim alicunde felicius, <lb/>quam ex hac pharetra, contra impiam Catervam tela depromp&longs;eris. </s>

<s><lb/>Hoc &longs;en&longs;it pridem, & in pereruditis Concionibus Anglice Latineque <lb/>editis, primus egregie demon&longs;travit Vir in omni Literarum genere <lb/>præclarus idemque bonarum Artium fautor eximius RICHARDUS <lb/>BENTLEIUS, Sæculi &longs;ui & Academiæ no&longs;træ magnum Orna<lb/>mentum, Collegii no&longs;tri <emph type="italics"/>S. Trinitatis<emph.end type="italics"/> Magi&longs;ter digni&longs;&longs;imus & in<lb/>tegerrimus. </s>

<s>Huic ego me pluribus nominibus ob&longs;trictum fateri <pb/>debeo: Huic & Tuas quæ debentur gratias, Lector benevole, non <lb/>denegabis. </s>

<s>Huic ego me pluribus nominibus ob&longs;trictum fateri

<s>Is enim, cum a longo tempore Celeberrimi Auctoris <lb/>amicitia intima frueretur, (qua etiam apud Po&longs;teros cen&longs;eri non <lb/>minoris æ&longs;timat, quam propriis Scriptis quæ literato orbi in de<lb/>liciis &longs;unt inclare&longs;cere) Amici &longs;imul famæ & &longs;cientiarum incre<lb/>mento con&longs;uluit. </s>

<pb/>debeo: Huic & Tuas quæ debentur gratias, Lector benevole, non <lb/>denegabis. </s>

<s>Itaque cum Exemplaria prioris Editionis rari&longs;<lb/>&longs;ima admodum & immani pretio coemenda &longs;upere&longs;&longs;ent; &longs;ua&longs;it Ille <lb/>crebris efflagitationibus & tantum non objurgando perpulit deni<lb/>que Virum Præ&longs;tanti&longs;&longs;imum, nec mode&longs;tia minus quam eruditi<lb/>one &longs;umma In&longs;ignem, ut novam hanc Operis Editionem, per om<lb/>nia elimatam denuo & egregiis in&longs;uper acce&longs;&longs;ionibus ditatam, &longs;uis <lb/>&longs;umptibus & au&longs;piciis prodire pateretur: Mihi vero, pro jure <lb/>&longs;uo, pen&longs;um non ingratum demandavit, ut quam po&longs;&longs;et emendate <lb/>id fieri curarem. </s>

<s><emph type="italics"/>Cantabrigiæ,<emph.end type="italics"/><lb/>Maii 12. 1713. </s>

<s>ROGERUS COTES Collegii <emph type="italics"/>S. Trinitatis<emph.end type="italics"/> Socius, <lb/>A&longs;tronomiæ & Philo&longs;ophiæ Experimentalis <lb/>Profe&longs;&longs;or <emph type="italics"/>Plumianus.<emph.end type="italics"/></s><pb/>

<s><emph type="center"/>INDEX CAPITUM <lb/>TOTIUS OPERIS.<emph.end type="center"/></s>

<s><emph type="italics"/>Cantabrigiæ,<emph.end type="italics"/><lb/>Maii 12. 1713. </s>

<pb/>

<s><emph type="center"/>INDEX CAPITUM <lb/>TOTIUS OPERIS.<emph.end type="center"/></s>

<s>DEFINITIONES. 1 </s>

<s>AXIOMATA, SIVE LEGES MOTUS. 12 </s>

<s><emph type="center"/>DE MOTU CORPORUM LIBER PRIMUS.<emph.end type="center"/></s>

<s>SECT. I. <emph type="italics"/>DE Methodo rationum primarum & ultima<lb/>rum.<emph.end type="italics"/> 24 </s>

<s>SECT. II. <emph type="italics"/>De inventione Virium centripetarum.<emph.end type="italics"/> 34 </s>

<s>SECT. III. <emph type="italics"/>De motu corporum in Conicis &longs;ectionibus eccentri<lb/>cis.<emph.end type="italics"/> 48 </s>

<s>SECT. IV. <emph type="italics"/>De inventione Orbium Ellipticorum, Parabolicorum <lb/>& Hyperbolicorum ex Umbilico dato.<emph.end type="italics"/> 59 </s>

<s>SECT. V. <emph type="italics"/>De inventione Orbium ubi Umbilicus neuter datur.<emph.end type="italics"/> 66 </s>

<s>SECT. VI. <emph type="italics"/>De inventione Motuum in Orbibus datis.<emph.end type="italics"/> 97 </s>

<s>SECT. VII. <emph type="italics"/>De corporum A&longs;cen&longs;u & De&longs;cen&longs;u rectilineo.<emph.end type="italics"/> 105 </s>

<s>SECT. VII. <emph type="italics"/>De inventione Orbium in quibus corpora Viribus <lb/>quibu&longs;cunque centripetis agitata revolvuntur.<emph.end type="italics"/> 114 </s>

<s>SECT. IX. <emph type="italics"/>De Motu corporum in Orbibus mobilibus, deque <lb/>Motu Ap&longs;idum.<emph.end type="italics"/> 121 </s>

<s>SECT. X. <emph type="italics"/>De Motu corporum in Superficiebus datis, deque <lb/>Funependulorum Motu reciproco.<emph.end type="italics"/> 132 </s>

<s>SECT. XI. <emph type="italics"/>De Motu corporum Viribus centripetis &longs;e mutuo pe<lb/>tentium.<emph.end type="italics"/> 147 </s>

<s>SECT. XII. <emph type="italics"/>De corporum Sphæricorum Viribus attractivis.<emph.end type="italics"/> 173 </s>

<pb/>

<s>SECT. XIII. <emph type="italics"/>De corporum non Sphæricorum Viribus attracti<lb/>vis.<emph.end type="italics"/> 192 </s>

<s>SECT. XIV. <emph type="italics"/>De Motu corporum Minimorum, quæ Veribus cen<lb/>tripetis ad &longs;ingulas Magni alicujus corporis partes ten<lb/>dentibus agitantur.<emph.end type="italics"/> 203 </s>

<s><emph type="center"/>DE MOTU CORPORUM LIBER SECUNDUS.<emph.end type="center"/></s>

<s>SECT. I. <emph type="italics"/>DE Motu corporum quibus re&longs;i&longs;titur in ratione <lb/>Velocitatis.<emph.end type="italics"/> 211 </s>

<s>SECT. II. <emph type="italics"/>De Motu corporum quibus re&longs;i&longs;titur in duplicata ra<lb/>tione Velocitatis.<emph.end type="italics"/> 220 </s>

<s>SECT. III. <emph type="italics"/>De Motu corporum quibus re&longs;i&longs;titur partim in ratione <lb/>Velocitatis, partim in eju&longs;dem ratione duplicata.<emph.end type="italics"/> 245 </s>

<s>SECT. IV. <emph type="italics"/>De corporum Circulari motu in Mediis re&longs;i&longs;tentibus.<emph.end type="italics"/><lb/>253 </s>

<s>SECT. V. <emph type="italics"/>De den&longs;itate & compre&longs;&longs;ione Fluidorum, deque Hy<lb/>dro&longs;tatica.<emph.end type="italics"/> 260 </s>

<s>SECT. VI. <emph type="italics"/>De Motu & Re&longs;i&longs;tentia corporum Funependulorum.<emph.end type="italics"/><lb/>272 </s>

<s>SECT. VII. <emph type="italics"/>De motu Fluidorum & re&longs;i&longs;tentia Projectilium.<emph.end type="italics"/> 294 </s>

<s>SECT. VIII. <emph type="italics"/>De motu per Fluida propagato.<emph.end type="italics"/> 329 </s>

<s>SECT. IX. <emph type="italics"/>De motu Circulari Fluidorum.<emph.end type="italics"/> 345 </s>

<s><emph type="center"/>DE MUNDI SYSTEMATE LIBER TERTIUS.<emph.end type="center"/></s>

<s>REGULÆ PHILOSOPHANDI 357 </s>

<s>PHÆNOMENA 359 </s>

<s>PROPOSITIONES 362 </s>

<s>SCHOLIUM GENERALE. 481 </s>

<pb/>

<s><emph type="center"/>PHILOSOPHIÆ <lb/>NATURALIS <lb/>Principia <lb/>MATHEMATICA.<emph.end type="center"/><lb/><gap desc="hr tag"/></s>

<s><emph type="center"/>DEFINITIONES.<emph.end type="center"/><lb/><gap desc="hr tag"/></s>

<s><emph type="center"/>DEFINITIO I.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Quantitas Materiæ e&longs;t men&longs;ura eju&longs;dem orta ex illius Den&longs;itate & <lb/>Magnitudine conjunctim.<emph.end type="italics"/><emph.end type="center"/></s>

<s>AER, den&longs;itate duplicata, in &longs;patio etiam duplicato fit qua<lb/>druplus; in triplicato &longs;extuplus. </s>

<s>DEFINITIONES. 1 </s>

<s>Idem intellige de Nive & <lb/>Pulveribus per compre&longs;&longs;ionem vel liquefactionem conden<lb/>&longs;atis. </s>

<s>AXIOMATA, SIVE LEGES MOTUS. 12 </s>

<s><emph type="center"/>DE MOTU CORPORUM LIBER PRIMUS.<emph.end type="center"/></s>

<s>Et par e&longs;t ratio corporum omnium, quæ per cau&longs;as qua&longs;cun<lb/>que diver&longs;imode conden&longs;antur. </s>

<s>SECT. I. <emph type="italics"/>DE Methodo rationum primarum & ultima<lb/>rum.<emph.end type="italics"/> 24 </s>

<s>SECT. II. <emph type="italics"/>De inventione Virium centripetarum.<emph.end type="italics"/> 34 </s>

<s>Medii interea, &longs;i quod fuerit, in<lb/>ter&longs;titia partium libere pervadentis, hic nullam rationem habeo. </s>

<s>SECT. III. <emph type="italics"/>De motu corporum in Conicis &longs;ectionibus eccentri<lb/>cis.<emph.end type="italics"/> 48 </s>

<s>SECT. IV. <emph type="italics"/>De inventione Orbium Ellipticorum, Parabolicorum <lb/>& Hyperbolicorum ex Umbilico dato.<emph.end type="italics"/> 59 </s>

<s><lb/>Hanc autem Quantitatem &longs;ub nomine Corporis vel Ma&longs;&longs;æ in &longs;e<lb/>quentibus pa&longs;&longs;im intelligo. </s>

<s>SECT. V. <emph type="italics"/>De inventione Orbium ubi Umbilicus neuter datur.<emph.end type="italics"/> 66 </s>

<s>SECT. VI. <emph type="italics"/>De inventione Motuum in Orbibus datis.<emph.end type="italics"/> 97 </s>

<s>Innote&longs;cit ea per corporis cuju&longs;que <lb/>Pondus. </s>

<s>SECT. VII. <emph type="italics"/>De corporum A&longs;cen&longs;u & De&longs;cen&longs;u rectilineo.<emph.end type="italics"/> 105 </s>

<s>SECT. VII. <emph type="italics"/>De inventione Orbium in quibus corpora Viribus <lb/>quibu&longs;cunque centripetis agitata revolvuntur.<emph.end type="italics"/> 114 </s>

<s>Nam Ponderi proportionalem e&longs;&longs;e reperi per experi<lb/>menta Pendulorum accurati&longs;&longs;ime in&longs;tituta, uti po&longs;thac docebitur. </s>

<s>SECT. IX. <emph type="italics"/>De Motu corporum in Orbibus mobilibus, deque <lb/>Motu Ap&longs;idum.<emph.end type="italics"/> 121 </s>

<s><emph type="center"/>DEFINITIO II.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Quantitas Motus e&longs;t men&longs;ura eju&longs;dem orta ex Velocitate & Quan<lb/>titate Materiæ conjunctim.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Motus totius e&longs;t &longs;umma motuum in partibus &longs;ingulis; adeoque <lb/>in corpore duplo majore æquali cum velocitate duplus e&longs;t, & du<lb/>pla cum velocitate quadruplus. </s>

<s><emph type="center"/>DEFINITIO III.<emph.end type="center"/></s>

<s><emph type="italics"/>Materiæ Vis In&longs;ita e&longs;t potentia re&longs;i&longs;tendi, qua corpus unumquodque, <lb/>quantum in &longs;e e&longs;t, per&longs;everat in &longs;tatu &longs;uo vel quie&longs;cendi vel <lb/>movendi uniformiter in directum.<emph.end type="italics"/></s>

<s>Hæc &longs;emper proportionalis e&longs;t &longs;uo corpori, neque differt quic<lb/>quam ab Inertia ma&longs;&longs;æ, ni&longs;i in modo concipiendi. </s>

<s>SECT. X. <emph type="italics"/>De Motu corporum in Superficiebus datis, deque <lb/>Funependulorum Motu reciproco.<emph.end type="italics"/> 132 </s>

<s>SECT. XI. <emph type="italics"/>De Motu corporum Viribus centripetis &longs;e mutuo pe<lb/>tentium.<emph.end type="italics"/> 147 </s>

<s>Per inertiam <lb/>materiæ, fit ut corpus omne de &longs;tatu &longs;uo vel quie&longs;cendi vel moven<lb/>di difficulter deturbetur. </s>

<s>SECT. XII. <emph type="italics"/>De corporum Sphæricorum Viribus attractivis.<emph.end type="italics"/> 173 </s><pb/>

<s>SECT. XIII. <emph type="italics"/>De corporum non Sphæricorum Viribus attracti<lb/>vis.<emph.end type="italics"/> 192 </s>

<s>Unde etiam vis in&longs;ita nomine &longs;ignifican<lb/>ti&longs;&longs;imo Vis Inertiæ dici po&longs;&longs;it. </s>

<s><emph type="center"/>DE MOTU CORPORUM LIBER SECUNDUS.<emph.end type="center"/></s>

<s>Exercet vero corpus hanc vim &longs;olum<lb/>modo in mutatione &longs;tatus &longs;ui per vim aliam in &longs;e impre&longs;&longs;am facta; <lb/><expan abbr="e&longs;tq;">e&longs;tque</expan> exercitium ejus &longs;ub diver&longs;o re&longs;pectu & Re&longs;i&longs;tentia & Impetus: <lb/>re&longs;i&longs;tentia, quatenus corpus ad con&longs;ervandum &longs;tatum &longs;uum relucta<lb/>tur vi impre&longs;&longs;æ; impetus, quatenus corpus idem, vi re&longs;i&longs;tentis ob<lb/>&longs;taculi difficulter cedendo, conatur &longs;tatum ejus mutare. </s>

<s>SECT. I. <emph type="italics"/>DE Motu corporum quibus re&longs;i&longs;titur in ratione <lb/>Velocitatis.<emph.end type="italics"/> 211 </s>

<s>SECT. II. <emph type="italics"/>De Motu corporum quibus re&longs;i&longs;titur in duplicata ra<lb/>tione Velocitatis.<emph.end type="italics"/> 220 </s>

<s>Vulgus <lb/>re&longs;i&longs;tentiam quie&longs;centibus & impetum moventibus tribuit: &longs;ed mo<lb/>tus & quies, uti vulgo concipiuntur, re&longs;pectu &longs;olo di&longs;tinguuntur <lb/>ab invicem; <expan abbr="neq;">neque</expan> &longs;emper vere quie&longs;cunt quæ vulgo tanquam quie<lb/>&longs;centia &longs;pectantur. </s>

<s>SECT. III. <emph type="italics"/>De Motu corporum quibus re&longs;i&longs;titur partim in ratione <lb/>Velocitatis, partim in eju&longs;dem ratione duplicata.<emph.end type="italics"/> 245 </s>

<s><emph type="center"/>DEFINITIO IV.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Vis Impre&longs;&longs;a e&longs;t actio in corpus exercita, ad mutandum ejus &longs;tatum <lb/>vel quie&longs;cendi vel movendi uniformiter in directum.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Con&longs;i&longs;tit hæc vis in actione &longs;ola, neque po&longs;t actionem permanet <lb/>in corpore. </s>

<s>SECT. IV. <emph type="italics"/>De corporum Circulari motu in Mediis re&longs;i&longs;tentibus.<emph.end type="italics"/><lb/>253 </s>

<s>SECT. V. <emph type="italics"/>De den&longs;itate & compre&longs;&longs;ione Fluidorum, deque Hy<lb/>dro&longs;tatica.<emph.end type="italics"/> 260 </s>

<s>Per&longs;everat enim corpus in &longs;tatu omni novo per &longs;olam <lb/>vim inertiæ. </s>

<s>SECT. VI. <emph type="italics"/>De Motu & Re&longs;i&longs;tentia corporum Funependulorum.<emph.end type="italics"/><lb/>272 </s>

<s>SECT. VII. <emph type="italics"/>De motu Fluidorum & re&longs;i&longs;tentia Projectilium.<emph.end type="italics"/> 294 </s>

<s>E&longs;t autem vis impre&longs;&longs;a diver&longs;arum originum, ut ex <lb/>Ictu, ex Pre&longs;&longs;ione, ex vi Centripeta. </s>

<s>SECT. VIII. <emph type="italics"/>De motu per Fluida propagato.<emph.end type="italics"/> 329 </s>

<s><emph type="center"/>DEFINITIO V.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Vis Centripeta e&longs;t, qua corpora ver&longs;us punctum aliquod tanquam ad <lb/>Centrum undique trahuntur, impelluntur, vel utcunq tendunt.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Hujus generis e&longs;t Gravitas, qua corpora tendunt ad centrum ter<lb/>ræ; Vis Magnetica, qua ferrum petit magnetem; & Vis illa, <lb/><expan abbr="quæcunq;">quæcunque</expan> &longs;it, qua Planetæ perpetuo retrahuntur a motibus rectili<lb/>neis, & in lineis curvis revolvi coguntur. </s>

<s>SECT. IX. <emph type="italics"/>De motu Circulari Fluidorum.<emph.end type="italics"/> 345 </s>

<s><emph type="center"/>DE MUNDI SYSTEMATE LIBER TERTIUS.<emph.end type="center"/></s>

<s>Lapis, in funda circum-

<s>REGULÆ PHILOSOPHANDI 357 </s>

<pb pagenum="3"/>actus, a circumagente manu abire conatur; & conatu &longs;uo fundam <lb/>di&longs;tendit, <expan abbr="eoq;">eoque</expan> fortius quo celerius revolvitur; &, quamprimum di<lb/>mittitur, avolat. </s>

<s>PHÆNOMENA 359 </s>

<s>Vim conatui illi contrariam, qua funda lapidem <lb/>in manum perpetuò retrahit & in orbe retinet, quoniam in manum <lb/>ceu orbis centrum dirigitur, Centripetam appello. </s>

<s>PROPOSITIONES 362 </s>

<s>SCHOLIUM GENERALE. 481 </s><pb/>

<s>Et par e&longs;t ratio <lb/>corporum omnium, quæ in gyrum aguntur. </s>

<s><emph type="center"/>PHILOSOPHIÆ <lb/>NATURALIS <lb/>Principia <lb/>MATHEMATICA.<emph.end type="center"/><lb/><gap desc="hr tag"/></s>

<s><emph type="center"/>DEFINITIONES.<emph.end type="center"/><lb/><gap desc="hr tag"/></s>

<s>Conantur ea omnia a <lb/>centris orbium recedere; & ni&longs;i ad&longs;it vis aliqua conatui i&longs;ti contra<lb/>ria, qua cohibeantur & in orbibus retineantur, quamque ideò Centri<lb/>petam appello, abibunt in rectis lineis uniformi cum motu. </s>

<s><emph type="center"/>DEFINITIO I.<emph.end type="center"/></s>

<s>Pro<lb/>jectile, &longs;i vi Gravitatis de&longs;titueretur, non deflecteretur in terram, &longs;ed <lb/>in linea recta abiret in cælos; idque uniformi cum motu, &longs;i modo <lb/>aeris re&longs;i&longs;tentia tolleretur. </s>

<s>AER, den&longs;itate duplicata, in &longs;patio etiam duplicato fit qua<lb/>druplus; in triplicato &longs;extuplus. </s>

<s>Idem intellige de Nive & <lb/>Pulveribus per compre&longs;&longs;ionem vel liquefactionem conden<lb/>&longs;atis. </s>

<s>Per gravitatem &longs;uam retrahitur a cur&longs;u <lb/>rectilineo & in terram perpetuo flectitur, idque magis vel minus <lb/>pro gravitate &longs;ua & velocitate motus. </s>

<s>Et par e&longs;t ratio corporum omnium, quæ per cau&longs;as qua&longs;cun<lb/>que diver&longs;imode conden&longs;antur. </s>

<s>Medii interea, &longs;i quod fuerit, in<lb/>ter&longs;titia partium libere pervadentis, hic nullam rationem habeo. </s>

<s>Quo minor erit ejus gravitas pro quantitate materiæ vel major &c. </s>

<s><lb/>Hanc autem Quantitatem &longs;ub nomine Corporis vel Ma&longs;&longs;æ in &longs;e<lb/>quentibus pa&longs;&longs;im intelligo. </s>

<s>Innote&longs;cit ea per corporis cuju&longs;que <lb/>Pondus. </s>

<s><lb/>vel major velocitas quacum projicitur, eo minus deviabit a cur&longs;u <lb/>rectilineo & longius perget. </s>

<s>Nam Ponderi proportionalem e&longs;&longs;e reperi per experi<lb/>menta Pendulorum accurati&longs;&longs;ime in&longs;tituta, uti po&longs;thac docebitur. </s>

<s><emph type="center"/>DEFINITIO II.<emph.end type="center"/></s>

<s>Si Globus plumbeus, data cum velo<lb/>citate &longs;ecundum lineam horizontalem a montis alicujus vertice vi <lb/>pulveris tormentarii projectus, pergeret in linea curva ad di&longs;tantiam <lb/>duorum milliarium, priu&longs;quam in terram decideret: hic dupla cum <lb/>velocitate qua&longs;i duplo longius pergeret, & decupla cum velocitate <lb/>qua&longs;i decuplo longius: &longs;i modo aeris re&longs;i&longs;tentia tolleretur. </s>

<s>Motus totius e&longs;t &longs;umma motuum in partibus &longs;ingulis; adeoque <lb/>in corpore duplo majore æquali cum velocitate duplus e&longs;t, & du<lb/>pla cum velocitate quadruplus. </s><pb pagenum="2"/>

<s>Et augendo <lb/>velocitatem augeri po&longs;&longs;et pro lubitu di&longs;tantia in quam projiceretur, <lb/>& minui curvatura lineæ quam de&longs;criberet, ita ut tandem caderet <lb/>ad di&longs;tantiam graduum decem vel triginta vel nonaginta; vel eriam <lb/>ut terram totam circuiret priu&longs;quam caderet; vel denique ut in <lb/>terram nunquam caderet, &longs;ed in cælos abiret & motu abeundi per<lb/>geret in infinitum. </s>

<s><emph type="center"/>DEFINITIO III.<emph.end type="center"/></s>

<s>Et eadem ratione, qua Projectile vi gravitatis <lb/>in orbem flecti po&longs;&longs;et & terram totam circuire, pote&longs;t & Luna vel <lb/>vi gravitatis, &longs;i modo gravis &longs;it, vel alia quacunque vi, qua in ter<lb/>ram urgeatur, retrahi &longs;emper a cur&longs;u rectilineo terram ver&longs;us, & <lb/>in orbem &longs;uum flecti: & ab&longs;que tali vi Luna in orbe &longs;uo retineri <lb/>non pote&longs;t. </s>

<s>Hæc &longs;emper proportionalis e&longs;t &longs;uo corpori, neque differt quic<lb/>quam ab Inertia ma&longs;&longs;æ, ni&longs;i in modo concipiendi. </s>

<s>Per inertiam <lb/>materiæ, fit ut corpus omne de &longs;tatu &longs;uo vel quie&longs;cendi vel moven<lb/>di difficulter deturbetur. </s>

<s>Hæc vis, &longs;i ju&longs;to minor e&longs;&longs;et, non &longs;atis flecteret Lunam <lb/>de cur&longs;u rectilineo: &longs;i ju&longs;to major, plus &longs;atis flecteret, ac de orbe <lb/>terram ver&longs;us deduceret. </s>

<s>Unde etiam vis in&longs;ita nomine &longs;ignifican<lb/>ti&longs;&longs;imo Vis Inertiæ dici po&longs;&longs;it. </s>

<s><emph type="center"/>DEFINITIO IV.<emph.end type="center"/></s>

<s>Con&longs;i&longs;tit hæc vis in actione &longs;ola, neque po&longs;t actionem permanet <lb/>in corpore. </s>

<s>Per&longs;everat enim corpus in &longs;tatu omni novo per &longs;olam <lb/>vim inertiæ. </s>

<s>E&longs;t autem vis impre&longs;&longs;a diver&longs;arum originum, ut ex <lb/>Ictu, ex Pre&longs;&longs;ione, ex vi Centripeta. </s>

<s><emph type="center"/>DEFINITIO V.<emph.end type="center"/></s>

<s>Lapis, in funda circum-<pb pagenum="3"/>actus, a circumagente manu abire conatur; & conatu &longs;uo fundam <lb/>di&longs;tendit, <expan abbr="eoq;">eoque</expan> fortius quo celerius revolvitur; &, quamprimum di<lb/>mittitur, avolat. </s>

<s>Vim conatui illi contrariam, qua funda lapidem <lb/>in manum perpetuò retrahit & in orbe retinet, quoniam in manum <lb/>ceu orbis centrum dirigitur, Centripetam appello. </s>

<s>Et par e&longs;t ratio <lb/>corporum omnium, quæ in gyrum aguntur. </s>

<s>Per gravitatem &longs;uam retrahitur a cur&longs;u <lb/>rectilineo & in terram perpetuo flectitur, idque magis vel minus <lb/>pro gravitate &longs;ua & velocitate motus. </s>

<s>Quo minor erit ejus gravitas pro quantitate materiæ vel major &c. </s>

<s><lb/>vel major velocitas quacum projicitur, eo minus deviabit a cur&longs;u <lb/>rectilineo & longius perget. </s>

<s>Requiritur quippe, ut &longs;it ju&longs;tæ magnitudinis: <lb/>& Mathematicorum e&longs;t invenire Vim, qua corpus in dato quovis <lb/>orbe data cum velocitate accurate retineri po&longs;&longs;it; & vici&longs;&longs;im inve<lb/>nire Viam curvilineam, in quam corpus e dato quovis loco data <lb/>cum velocitate egre&longs;&longs;um a data vi flectatur. </s>

<s>E&longs;t autem vis hujus cen<lb/>tripetæ Quantitas trium generum, Ab&longs;oluta, Acceleratrix, & Motrix. </s><pb pagenum="4"/>

<s>E&longs;t autem vis hujus cen<lb/>tripetæ Quantitas trium generum, Ab&longs;oluta, Acceleratrix, & Motrix. </s>

<s>

<arrow.to.target n="note1"></arrow.to.target></s>

<s><emph type="center"/>DEFINITIO VI.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Vis centripetæ Quantitas Ab&longs;oluta e&longs;t men&longs;ura eju&longs;dem major vel minor <lb/>pro Efficacia cau&longs;æ eam propagantis a centro per regiones in circuitu.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Ut vis Magnetica pro mole magnetis vel inten&longs;ione virtutis major <lb/>in uno magnete, minor in alio. </s>

<s><emph type="center"/>DEFINITIO VII.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Vis centripetæ Quantitas Acceleratrix e&longs;t ip&longs;ius men&longs;ura Velocitati <lb/>proportionalis, quam dato tempore generat.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Uti Virtus magnetis eju&longs;dem major in minori di&longs;tantia, minor <lb/>in majori: vel vis Gravitans major in vallibus, minor in cacumini<lb/>bus præaltorum montium, atque adhuc minor (ut po&longs;thac patebit) <lb/>in majoribus di&longs;tantiis a globo terræ; in æqualibus autem di&longs;tan<lb/>tiis eadem undique, propterea quod corpora omnia cadentia (gra<lb/>via an levia, magna an parva) &longs;ublata Aeris re&longs;i&longs;tentia, æqualiter <lb/>accelerat. </s>

<s><emph type="center"/>DEFINITIO VIII.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Vis centripetæ Quantitas Motrix e&longs;t ip&longs;ius men&longs;ura proportionalis. </s>

<s><emph type="center"/>DEFINITIO VI.<emph.end type="center"/></s>

<s><lb/>Motui, quem dato tempore generat.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Uti Pondus majus in majore corpore, minus in minore; inque <lb/>corpore eodem majus prope terram, minus in cælis. </s>

<s>Ut vis Magnetica pro mole magnetis vel inten&longs;ione virtutis major <lb/>in uno magnete, minor in alio. </s>

<s><emph type="center"/>DEFINITIO VII.<emph.end type="center"/></s>

<s>Hæc Quantitas <lb/>e&longs;t corporis totius centripetentia &longs;eu propen&longs;io in centrum, & (ut ita <lb/>dicam) Pondus; & innote&longs;cit &longs;emper per vim ip&longs;i contrariam & æ<lb/>qualem, qua de&longs;cen&longs;us corporis impediri pote&longs;t. </s>

<s>Ha&longs;ce virium quantitates brevitatis gratia nominare licet vires <lb/>motrices, acceleratrices, & ab&longs;olutas; & di&longs;tinctionis gratia referre ad <lb/>Corpora, centrum petentia, ad corporum Loca, & ad Centrum virium: <lb/>nimirum vim motricem ad Corpus, tanquam conatum & propen&longs;io<lb/>nem totius in centrum ex propen&longs;ionibus omnium partium compo&longs;i<lb/>tam; & vim acceleratricem ad Locum corporis, tanquam efficaciam <lb/>quandam, de centro per loca &longs;ingula in circuitu diffu&longs;am, ad movenda <lb/>corpora quæ in ip&longs;is &longs;unt; vim autem ab&longs;olutam ad Centrum, tan<lb/>quam cau&longs;a aliqua præditum, &longs;ine qua vires motrices non propa<lb/>gantur per regiones in circuitu; &longs;ive cau&longs;a illa &longs;it corpus aliquod <lb/>centrale (quale e&longs;t Magnes in centro vis magneticæ, vel Terra in

<pb pagenum="5"/>centro vis gravitantis) &longs;ive alia aliqua quæ non apparet. </s>

<s><emph type="center"/>DEFINITIO VIII.<emph.end type="center"/></s>

<s>Mathe<lb/>maticus duntaxat e&longs;t hic conceptus. </s>

<s><emph type="center"/><emph type="italics"/>Vis centripetæ Quantitas Motrix e&longs;t ip&longs;ius men&longs;ura proportionalis. </s>

<s><lb/>Motui, quem dato tempore generat.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Nam virium cau&longs;as & &longs;edes phy<lb/>&longs;icas jam non expendo. </s>

<s>Uti Pondus majus in majore corpore, minus in minore; inque <lb/>corpore eodem majus prope terram, minus in cælis. </s>

<s>E&longs;t igitur vis acceleratrix ad vim motricem ut celeritas ad mo<lb/>tum. </s>

<s>Oritur enim quantitas motus ex celeritate ducta in quanti<lb/>tatem materiæ, & vis motrix ex vi acceleratrice ducta in quantita<lb/>tem eju&longs;dem materiæ. </s>

<s>Mathe<lb/>maticus duntaxat e&longs;t hic conceptus. </s>

<s>Nam virium cau&longs;as & &longs;edes phy<lb/>&longs;icas jam non expendo. </s>

<s>Nam &longs;umma actionum vis acceleratricis in <lb/>&longs;ingulas corporis particulas e&longs;t vis motrix totius. </s>

<s>E&longs;t igitur vis acceleratrix ad vim motricem ut celeritas ad mo<lb/>tum. </s>

<s>Oritur enim quantitas motus ex celeritate ducta in quanti<lb/>tatem materiæ, & vis motrix ex vi acceleratrice ducta in quantita<lb/>tem eju&longs;dem materiæ. </s>

<s>Unde juxta <lb/>&longs;uperficiem Terræ, ubi gravitas acceleratrix &longs;eu vis gravitans in <lb/>corporibus univer&longs;is eadem e&longs;t, gravitas motrix &longs;eu pondus e&longs;t ut <lb/>corpus: at &longs;i in regiones a&longs;cendatur ubi gravitas acceleratrix fit mi<lb/>nor, pondus pariter minuetur, eritque &longs;emper ut corpus in <lb/>gravitatem acceleratricem ductum. </s>

<s>Nam &longs;umma actionum vis acceleratricis in <lb/>&longs;ingulas corporis particulas e&longs;t vis motrix totius. </s>

<s>Sic in regionibus ubi gravitas <lb/>acceleratrix duplo minor e&longs;t, pondus corporis duplo vel triplo <lb/>minoris erit quadruplo vel &longs;extuplo minus. </s>

<s>Sic in regionibus ubi gravitas <lb/>acceleratrix duplo minor e&longs;t, pondus corporis duplo vel triplo <lb/>minoris erit quadruplo vel &longs;extuplo minus. </s>

<s>Porro attractiones & impul&longs;us eodem &longs;en&longs;u acceleratrices & mo<lb/>trices nomino. </s>

<s>Voces autem Attractionis, Impul&longs;us, vel Propen<lb/>&longs;ionis cuju&longs;cunque in centrum, indifferenter & pro &longs;e mutuo pro<lb/>mi&longs;cue u&longs;urpo; has vires non Phy&longs;ice &longs;ed Mathematice tantum con<lb/>&longs;iderando. </s>

<s>Unde caveat lector, ne per huju&longs;modi voces cogitet me <lb/>&longs;peciem vel modum actionis cau&longs;amve aut rationem Phy&longs;icam ali<lb/>cubi definire, vel centris (quæ &longs;unt puncta Mathematica) vires <lb/>vere & Phy&longs;ice tribuere; &longs;i forte aut centra trahere, aut vires cen<lb/>trorum e&longs;&longs;e dixero. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Hactenus voces minus notas, quo &longs;en&longs;u in &longs;equentibus acci<lb/>piendæ &longs;int, explicare vi&longs;um e&longs;t. </s>

<s>Nam Tempus, Spatium, Locum <lb/>& Motum, ut omnibus noti&longs;&longs;ima, non definio. </s>

<s>Notandum tamen, quod <lb/>vulgus quantitates ha&longs;ce non aliter quam ex relatione ad &longs;en&longs;ibilia <lb/>concipiat. </s>

<s>Et inde oriuntur præjudicia quædam, quibus tollendis <lb/>convenit ea&longs;dem in ab&longs;olutas & relativas, veras & apparentes, ma<lb/>thematicas & vulgares di&longs;tingui. </s>

<s>Et inde oriuntur præjudicia quædam, quibus tollendis <lb/>convenit ea&longs;dem in ab&longs;olutas & relativas, veras & apparentes, ma<lb/>thematicas & vulgares di&longs;tingui. </s>

<s>Tempus Ab&longs;olutum, verum, & mathematicum, in &longs;e & natura <lb/>&longs;ua <expan abbr="ab&longs;q;">ab&longs;que</expan> relatione ad externum quodvis, æquabiliter fluit, <expan abbr="alioq;">alioque</expan> <lb/>nomine dicitur Duratio: Relativum, apparens, & vulgare e&longs;t &longs;en&longs;ibilis <lb/>& externa quævis Durationis per motum men&longs;ura (&longs;eu accurata <lb/>&longs;eu inæquabilis) qua vulgus vice veri temporis utitur; ut Hora, <lb/>Dies, Men&longs;is, Annus. <pb pagenum="6"/><arrow.to.target n="note2"></arrow.to.target></s>

<arrow.to.target n="note2"></arrow.to.target></s>

<s>Spatium Ab&longs;olutum, natura &longs;ua ab&longs;que relatione ad externum <lb/>quodvis, &longs;emper manet &longs;imilare & immobile: Relativum e&longs;t &longs;patii <lb/>hujus men&longs;ura &longs;eu dimen&longs;io quælibet mobilis, quæ a &longs;en&longs;ibus no&longs;tris <lb/>per &longs;itum &longs;uum ad corpora definitur, & a vulgo pro &longs;patio immo<lb/>bili u&longs;urpatur: uti dimen&longs;io &longs;patii &longs;ubterranei, aerei vel cæle&longs;tis <lb/>definita per &longs;itum &longs;uum ad Terram. </s>

<s>Idem &longs;unt &longs;patium ab&longs;olutum <lb/>& relativum, &longs;pecie & magnitudine; &longs;ed non permanent idem &longs;em<lb/>per numero. </s>

<s>Nam &longs;i Terra, verbi gratia, movetur; &longs;patium Aeris <lb/>no&longs;tri, quod relative & re&longs;pectu Terræ &longs;emper manet idem, nunc <lb/>erit una pars &longs;patii ab&longs;oluti in quam Aer tran&longs;it, nunc alia pars ejus; <lb/>& &longs;ic ab&longs;olute mutabitur perpetuo. </s>

<s>Locus e&longs;t pars &longs;patii quam corpus occupat, <expan abbr="e&longs;tq;">e&longs;tque</expan> pro ratione <lb/>&longs;patii vel Ab&longs;olutus vel Relativus. </s>

<s>Pars, inquam, &longs;patii; non Situs <lb/>corporis, vel Superficies ambiens. </s>

<s>Nam &longs;olidorum æqualium <lb/>æquales &longs;emper &longs;unt loci; Superficies autem ob di&longs;&longs;imilitudinem <lb/>figurarum ut plurimum inæquales &longs;unt; Situs vero proprie loquen<lb/>do quantitatem non habent, <expan abbr="neq;">neque</expan> tam &longs;unt loca quam affectiones <lb/>locorum. </s>

<s>Motus totius idem e&longs;t cum &longs;umma motuum partium, <lb/>hoc e&longs;t, tran&longs;latio totius de &longs;uo loco eadem e&longs;t cum &longs;umma tran&longs;la<lb/>tionum partium de locis &longs;uis; <expan abbr="adeoq;">adeoque</expan> locus totius idem cum &longs;umma <lb/>locorum partium, & propterea internus & in corpore toto. </s>

<s>Motus Ab&longs;olutus e&longs;t tran&longs;latio corporis de loco ab&longs;oluto in <lb/>locum ab&longs;olutum, Relativus de relativo in relativum. </s>

<s>Sic in navi <lb/>quæ velis pa&longs;&longs;is fertur, relativus corporis Locus e&longs;t navigii regio illa <lb/>in qua corpus ver&longs;atur, &longs;eu cavitatis totius pars illa quam corpus <lb/>implet, <expan abbr="quæq;">quæque</expan> adeo movetur una cum navi: & Quies relativa e&longs;t <lb/>perman&longs;io corporis in eadem illa navis regione vel parte cavita<lb/>tis. </s>

<s>At quies Vera e&longs;t perman&longs;io corporis in eadem parte &longs;patii <lb/>illius immoti in qua navis ip&longs;a una cum cavitate &longs;ua & contentis <lb/>univer&longs;is movetur. </s>

<s>Unde &longs;i Terra vere quie&longs;cit, corpus quod rela<lb/>tive quie&longs;cit in navi, movebitur vere & ab&longs;olute ea cum velocitate <lb/>qua navis movetur in Terra. </s>

<s>Sin Terra etiam movetur; orietur <lb/>verus & ab&longs;olutus corporis motus, partim ex Terræ motu vero in <lb/>&longs;patio immoto, partim ex navis motu relativo in Terra: & &longs;i cor<lb/>pus etiam movetur relative in navi; orietur verus ejus motus, par<lb/>tim ex vero motu Terræ in &longs;patio immoto, partim ex relativis mo<lb/>tibus tum navis in Terra, tum corporis in navi; & ex his motibus <lb/>relativis orietur corporis motus relativus in Terra. </s>

<s>Ut &longs;i Terræ pars <lb/>illa, ubi navis ver&longs;atur, moveatur vere in orientem cum velocitate <lb/>partium 10010; & velis <expan abbr="ventoq;">ventoque</expan> feratur navis in occidentem cum <lb/>velocitate partium decem; Nauta autem ambulet in navi ori-<pb pagenum="7"/>entem ver&longs;us cum velocitatis parte una: movebitur Nauta vere & <lb/>ab&longs;olute in &longs;patio immoto cum velocitatis partibus 10001 in o<lb/>rientem, & relative in terra occidentem ver&longs;us cum velocitatis <lb/>partibus novem. </s>

<pb pagenum="7"/>entem ver&longs;us cum velocitatis parte una: movebitur Nauta vere & <lb/>ab&longs;olute in &longs;patio immoto cum velocitatis partibus 10001 in o<lb/>rientem, & relative in terra occidentem ver&longs;us cum velocitatis <lb/>partibus novem. </s>

<s>Tempus Ab&longs;olutum a relativo di&longs;tinguitur in A&longs;tronomia per Æ<lb/>quationem temporis vulgi. </s>

<s>Inæquales enim &longs;unt dies Naturales, <lb/>qui vulgo tanquam æquales promen&longs;ura temporis habentur. </s>

<s>Hanc <lb/>inæqualitatem corrigunt A&longs;tronomi, ut ex veriore tempore men&longs;urent motus &c. </s>

<s><lb/>motus cæle&longs;tes. </s>

<s>Po&longs;&longs;ibile e&longs;t, ut nullus &longs;it motus æquabilis quo <lb/>Tempus accurate men&longs;uretur. </s>

<s>Accelerari & retardari po&longs;&longs;unt motus <lb/>omnes, &longs;ed fluxus temporis Ab&longs;oluti mutari nequit. </s>

<s>Eadem e&longs;t du<lb/>ratio &longs;eu per&longs;everantia exi&longs;tentiæ rerum; &longs;ive motus &longs;int celeres, &longs;ive <lb/>tardi, &longs;ive nulli: proinde hæc a men&longs;uris &longs;uis &longs;en&longs;ibilibus merito <lb/>di&longs;tinguitur, & ex ii&longs;dem colligitur per Æquationem A&longs;tronomi<lb/>cam. </s>

<s>Hujus autem æquationis in determinandis Phænomenis ne<lb/>ce&longs;&longs;itas, tum per experimentum Horologii O&longs;cillatorii, tum etiam <lb/>per eclip&longs;es Satellitum Jovis evincitur. </s>

<s>Ut partium Temporis ordo e&longs;t immutabilis, &longs;ic etiam ordo par<lb/>tium Spatii. </s>

<s>Moveantur hæ de locis &longs;uis, & movebuntur (ut ita <lb/>dicam) de &longs;eip&longs;is. </s>

<s>Nam tempora & &longs;patia &longs;unt &longs;ui ip&longs;orum & <lb/>rerum omnium qua&longs;i Loca. </s>

<s>In Tempore quoad ordinem &longs;ucce&longs;&longs;i<lb/>onis; in Spatio quoad ordinem &longs;itus locantur univer&longs;a. </s>

<s>De illo<lb/>rum e&longs;&longs;entia e&longs;t ut &longs;int Loca: & loca primaria moveri ab&longs;urdum <lb/>e&longs;t. </s>

<s>Hæc &longs;unt igitur ab&longs;oluta Loca; & &longs;olæ tran&longs;lationes de his lo<lb/>cis &longs;unt ab&longs;oluti Motus. </s>

<s>Hæc &longs;unt igitur ab&longs;oluta Loca; & &longs;olæ tran&longs;lationes de his lo<lb/>cis &longs;unt ab&longs;oluti Motus. </s>

<s>Verum quoniam hæ Spatii partes videri nequeunt, & ab invi<lb/>cem per &longs;en&longs;us no&longs;tros di&longs;tingui; earum vice adhibemus men&longs;uras <lb/>&longs;en&longs;ibiles. </s>

<s>Ex po&longs;itionibus enim & di&longs;tantiis rerum a corpore ali<lb/>quo, quod &longs;pectamus ut immobile, de&longs;inimus loca univer&longs;a: deinde <lb/>etiam & omnes motus æ&longs;timamus cum re&longs;pectu ad prædicta loca, <lb/>quatenus corpora ab ii&longs;dem transferri concipimus. </s>

<s>Sic vice loco<lb/>rum & motuum ab&longs;olutorum relativis utimur; nec incommode in <lb/>rebus humanis: in Philo&longs;ophicis autem ab&longs;trahendum e&longs;t a &longs;en&longs;ibus. </s>

<s><lb/>Fieri etenim pote&longs;t, ut nullum revera quie&longs;cat corpus, ad quod loca <lb/>motu&longs;que referantur. </s>

<s><lb/>Fieri etenim pote&longs;t, ut nullum revera quie&longs;cat corpus, ad quod loca <lb/>motu&longs;que referantur. </s>

<s>Di&longs;tinguuntur autem Quies & Motus ab&longs;oluti & relativi ab invi<lb/>cem per Proprietates &longs;uas & Cau&longs;as & Effectus. </s>

<s>Quietis proprietas <lb/>e&longs;t, quod corpora vere quie&longs;centia quie&longs;cunt inter &longs;e. </s>

<s>Ideoque <lb/>cum po&longs;&longs;ibile &longs;it, ut corpus aliquod in regionibus Fixarum, aut longe <lb/>ultra, quie&longs;cat ab&longs;olute; &longs;ciri autem non po&longs;&longs;it ex &longs;itu corporum <lb/>ad invicem in regionibus no&longs;tris, horumne aliquod ad longin-</s><pb pagenum="8"/>

<s><arrow.to.target n="note3"></arrow.to.target><lb/>quum illud datam po&longs;itionem &longs;ervet necne; quies vera ex horum <lb/>&longs;itu inter &longs;e definiri nequit. </s>

<s>

<arrow.to.target n="note3"></arrow.to.target><lb/>quum illud datam po&longs;itionem &longs;ervet necne; quies vera ex horum <lb/>&longs;itu inter &longs;e definiri nequit. </s>

<s>Motus proprietas e&longs;t, quod partes, quæ datas &longs;ervant po&longs;itiones <lb/>ad tota, participant motus eorundem totorum. </s>

<s>Motus proprietas e&longs;t, quod partes, quæ datas &longs;ervant po&longs;itiones <lb/>ad tota, participant motus eorundem totorum. </s>

<s>Nam Gyrantium <lb/>partes omnes conantur recedere ab axe motus, & Progredientium <lb/>impetus oritur ex conjuncto impetu partium &longs;ingularum. </s>

<s>Motis <lb/>igitur corporibus ambientibus, moventur quæ in ambientibus rela<lb/>tive quie&longs;cunt. </s>

<s>Et propterea motus verus & ab&longs;olutus definiri ne<lb/>quit per tran&longs;lationem e vicinia corporum, quæ tanquam quie&longs;cen<lb/>tia &longs;pectantur. </s>

<s>Debent enim corpora externa non &longs;olum tanquam qui<lb/>e&longs;centia &longs;pectari, &longs;ed etiam vere quie&longs;cere. </s>

<s>Alioquin inclu&longs;a om<lb/>nia, præter tran&longs;lationem e vicinia ambientium, participabunt <lb/>etiam ambientium motus veros; & &longs;ublata illa tran&longs;latione non <lb/>vere quie&longs;cent, &longs;ed tanquam quie&longs;centia &longs;olummodo &longs;pectabun<lb/>tur. </s>

<s>Sunt enim ambientia ad inclu&longs;a, ut totius pars exterior ad <lb/>partem interiorem, vel ut cortex ad nucleum. </s>

<s>Moto autem cor<lb/>tice, nucleus etiam, <expan abbr="ab&longs;q;">ab&longs;que</expan> tran&longs;latione de vicinia corticis, ceu pars <lb/>totius movetur. </s>

<s>Moto autem cor<lb/>tice, nucleus etiam, <expan abbr="ab&longs;q;">ab&longs;que</expan> tran&longs;latione de vicinia corticis, ceu pars <lb/>totius movetur. </s>

<s>Præcedenti proprietati affinis e&longs;t, quod moto Loco movetur una <lb/>Locatum: adeoque corpus, quod de loco moto movetur, participat <lb/>etiam loci &longs;ui motum. </s>

<s>Motus igitur omnes, qui de locis motis <lb/>fiunt, &longs;unt partes &longs;olummodo motuum integrorum & ab&longs;olutorum: <lb/>& motus omnis integer componitur ex motu corporis de loco &longs;uo <lb/>primo, & motu loci hujus de loco &longs;uo, & &longs;ic deinceps; u&longs;que dum <lb/>perveniatur ad locum immotum, ut in exemplo Nautæ &longs;upra me<lb/>morato. </s>

<s>Unde motus integri & ab&longs;oluti non ni&longs;i per loca immota <lb/>definiri po&longs;&longs;unt: & propterea hos ad loca immota, relativos ad mo<lb/>bilia &longs;upra retuli. </s>

<s>Loca autem immota non &longs;unt, ni&longs;i quæ omnia <lb/>ab infinito in infinitum datas &longs;ervant po&longs;itiones ad invicem; atque <lb/>adeo &longs;emper manent immota, &longs;patiumque con&longs;tituunt quod Immo<lb/>bile appello. </s>

<s>Cau&longs;æ, quibus motus veri & relativi di&longs;tinguuntur ab invicem, <lb/>&longs;unt Vires in corpora impre&longs;&longs;æ ad motum generandum. </s>

<s>Motus <lb/>verus nec generatur nec mutatur, ni&longs;i per vires in ip&longs;um corpus mo<lb/>tum impre&longs;&longs;as: at motus relativus generari & mutari pote&longs;t <expan abbr="ab&longs;q;">ab&longs;que</expan> <lb/>viribus impre&longs;&longs;is in hoc corpus. </s>

<s>Sufficit enim ut imprimantur in <lb/>alia &longs;olum corpora ad quæ fit relatio, ut iis cedentibus mutetur <lb/>relatio illa in qua hujus quies vel motus relativus con&longs;i&longs;tit. </s>

<s>Rur<lb/>&longs;um motus verus a viribus in corpus motum impre&longs;&longs;is &longs;emper muta<lb/>tur; at motus relativus ab his viribus non mutatur nece&longs;&longs;ario. </s>

<s>Nam <lb/>&longs;i eædem vires in alia etiam corpora, ad quæ &longs;it relatio, &longs;ic impri-<pb pagenum="9"/>mantur ut &longs;itus relativus con&longs;ervetur, con&longs;ervabitur relatio in qua <lb/>motus relativus con&longs;i&longs;tit. </s>

<s>Nam <lb/>&longs;i eædem vires in alia etiam corpora, ad quæ &longs;it relatio, &longs;ic impri-

<s>Mutari igitur pote&longs;t motus omnis relati<lb/>vus ubi verus con&longs;ervatur, & con&longs;ervari ubi verus mutatur; & prop<lb/>terea motus verus in eju&longs;modi relationibus minime con&longs;i&longs;tit. </s>

<pb pagenum="9"/>mantur ut &longs;itus relativus con&longs;ervetur, con&longs;ervabitur relatio in qua <lb/>motus relativus con&longs;i&longs;tit. </s>

<s>Effectus quibus motus ab&longs;oluti & relativi di&longs;tinguuntur ab invi<lb/>cem, &longs;unt vires recedendi ab axe motus circularis. </s>

<s>Nam in motu <lb/>circulari nude relativo hæ vires nullæ &longs;unt, in vero autem & ab&longs;o<lb/>luto majores vel minores pro quantitate motus. </s>

<s>Si pendeat &longs;itula <lb/>a filo prælongo, agaturque perpetuo in orbem, donec filum a con<lb/>tor&longs;ione admodum rige&longs;cat, dein impleatur aqua, & una cum aqua <lb/>quie&longs;cat; tum vi aliqua &longs;ubitanea agatur motu contrario in orbem, <lb/>& filo &longs;e relaxante, diutius per&longs;everet in hoc motu; &longs;uperficies a<lb/>quæ &longs;ub initio plana erit, quemadmodum ante motum va&longs;is: at <lb/>po&longs;tquam, vi in aquam paulatim impre&longs;&longs;a, effecit vas, ut hæc quoque <lb/>&longs;en&longs;ibiliter revolvi incipiat; recedet ip&longs;a paulatim a medio, a&longs;cen<lb/>detque ad latera va&longs;is, figuram concavam induens, (ut ip&longs;e exper<lb/>tus &longs;um) & incitatiore &longs;emper motu a&longs;cendet magis & magis, do<lb/>nec revolutiones in æqualibus cum va&longs;e temporibus peragendo, <lb/>quie&longs;cat in eodem relative. </s>

<s>Indicat hic a&longs;cen&longs;us conatum rece<lb/>dendi ab axe motus, & per talem conatum innote&longs;cit & men&longs;ura<lb/>tur motus aquæ circularis verus & ab&longs;olutus, motuique relativo <lb/>hic omnino contrarius. </s>

<s>Initio, ubi maximus erat aquæ motus rela<lb/>tivus in va&longs;e, motus ille nullum excitabat conatum recedendi ab <lb/>axe: aqua non petebat circumferentiam a&longs;cendendo ad latera va<lb/>&longs;is, &longs;ed plana manebat, & propterea motus illius circularis verus <lb/>nondum inceperat. </s>

<s>Po&longs;tea vero, ubi aquæ motus relativus decre<lb/>vit, a&longs;cen&longs;us ejus ad latera va&longs;is indicabat conatum recedendi ab <lb/>axe; atque hic conatus mon&longs;trabat motum illius circularem verum <lb/>perpetuo cre&longs;centem, ac tandem maximum factum ubi aqua quie<lb/>&longs;cebat in va&longs;e relative. </s>

<s>Igitur conatus i&longs;te non pendet a tran&longs;la<lb/>tione aquæ re&longs;pectu corporum ambientium, & propterea motus cir<lb/>cularis verus per tales tran&longs;lationes definiri nequit. </s>

<s>Unicus e&longs;t cor<lb/>poris cuju&longs;que revolventis motus vere circularis, conatui unico tan<lb/>quam proprio & adæquato effectui re&longs;pondens: motus autem rela<lb/>tivi pro variis relationibus ad externa innumeri &longs;unt; & relationum <lb/>in&longs;tar, effectibus veris omnino de&longs;tituuntur, ni&longs;i quatenus verum <lb/>illum & unicum motum participant. </s>

<s>Unde & in Sy&longs;temate eorum <lb/>qui Cælos no&longs;tros infra Cælos Fixarum in orbem revolvi volunt, <lb/>& Planetas &longs;ecum deferre; &longs;ingulæ Cælorum partes, & Planetæ <lb/>qui relative quidem in Cælis &longs;uis proximis quie&longs;cunt, moven-<pb pagenum="10"/><arrow.to.target n="note4"></arrow.to.target><lb/>tur vere. </s>

<s>Mutant enim po&longs;itiones &longs;uas ad invicem (&longs;ecus quam fit <lb/>in vere quie&longs;centibus) unaque cum cælis delati participant eorum <lb/>motus, & ut partes revolventium totorum, ab eorum axibus rece<lb/>dere conantur. </s>

<arrow.to.target n="note4"></arrow.to.target><lb/>tur vere. </s>

<s>Igitur quantitates relativæ non &longs;unt eæ ip&longs;æ quantitates, quarum <lb/>nomina præ &longs;e ferunt, &longs;ed earum men&longs;uræ illæ &longs;en&longs;ibiles (veræ an <lb/>errantes) quibus vulgus loco quantitatum men&longs;uratarum utitur. </s>

<s>At <lb/>&longs;i ex u&longs;u definiendæ &longs;unt verborum &longs;ignificationes; per nomina il<lb/>la Temporis, Spatii, Loci & Motus proprie intelligendæ erunt hæ <lb/>men&longs;uræ; & &longs;ermo erit in&longs;olens & pure Mathematicus, &longs;i quantita<lb/>tes men&longs;uratæ hic intelligantur. </s>

<s>Proinde vim inferunt Sacris <lb/>Literis, qui voces ha&longs;ce de quantitatibus men&longs;uratis ibi interpre<lb/>tantur. </s>

<s>Neque minus contaminant Mathe&longs;in & Philo&longs;ophiam, <lb/>qui quantitates veras cum ip&longs;arum relationibus & vulgaribus men<lb/>furis confundunt. </s>

<s>Neque minus contaminant Mathe&longs;in & Philo&longs;ophiam, <lb/>qui quantitates veras cum ip&longs;arum relationibus & vulgaribus men<lb/>furis confundunt. </s>

<s>Motus quidem veros corporum &longs;ingulorum cogno&longs;cere, & ab <lb/>apparentibus actu di&longs;criminare, difficillimum. </s>

<s>e&longs;t propterea quod <lb/>partes &longs;patii illius immobilis, in quo corpora vere moventur, non <lb/>incurrunt in &longs;en&longs;us. </s>

<s>Cau&longs;a tamen non e&longs;t pror&longs;us de&longs;perata. </s>

<s>Nam <lb/>&longs;uppetunt argumenta, partim ex motibus apparentibus qui &longs;unt <lb/>motuum verorum differentiæ, partim ex viribus quæ &longs;unt mo<lb/>tuum verorum cau&longs;æ & effectus. </s>

<s>Ut &longs;i globi duo, ad datam ab in<lb/>vicem di&longs;tantiam filo intercedente connexi, revolverentur circa <lb/>commune gravitatis centrum; innote&longs;ceret ex ten&longs;ione fili cona<lb/>tus globorum recedendi ab axe motus, & inde quantitas motus <lb/>circularis computari po&longs;&longs;et. </s>

<s>Deinde &longs;i vires quælibet æquales in <lb/>alternas globorum facies ad motum circularem augendum vel mi<lb/>nuendum &longs;imul imprimerentur, innote&longs;ceret ex aucta vel diminuta <lb/>fili ten&longs;ione augmentum vel decrementum motus; & inde tandem <lb/>inveniri po&longs;&longs;ent facies globorum in quas vires imprimi deberent, <lb/>ut motus maxime augeretur; id e&longs;t, facies po&longs;ticæ, &longs;ive quæ in mo<lb/>tu circulari &longs;equuntur. </s>

<s>Cognitis autem faciebus quæ &longs;equuntur, <lb/>& faciebus oppo&longs;itis quæ præcedunt, cogno&longs;ceretur determinatio <lb/>motus. </s>

<s>In hunc modum inveniri po&longs;&longs;et & quantitas & determi<lb/>natio motus hujus circularis in vacuo quovis immen&longs;o, ubi nihil <lb/>extaret externum & &longs;en&longs;ibile quocum globi conferri po&longs;&longs;ent. </s>

<s>Si <lb/>jam con&longs;tituerentur in &longs;patio illo corpora aliqua longinqua datam <lb/>inter &longs;e po&longs;itionem &longs;ervantia, qualia &longs;unt Stellæ Fixæ in regionibus <lb/>no&longs;tris: &longs;ciri quidem non po&longs;&longs;et ex relativa globorum tran&longs;latione <lb/>inter corpora, utrum his an illis tribuendus e&longs;&longs;et motus. </s>

<s>At &longs;i <pb pagenum="11"/>attenderetur ad filum, & deprenderetur ten&longs;ionem ejus illam ip&longs;am <lb/>e&longs;&longs;e quam motus globorum requireret; concludere liceret mo<lb/>tum e&longs;&longs;e globorum, & corpora quie&longs;cere; & tum demum ex <lb/>tran&longs;latione globorum inter corpora, determinationem hujus <lb/>motus colligere. </s>

<s>At &longs;i

<s>Motus autem veros ex eorum cau&longs;is, effecti<lb/>bus, & apparentibus differentiis colligere; & contra ex motibus <lb/>&longs;eu veris &longs;eu apparentibus eorum cau&longs;as & effectus, docebitur <lb/>fu&longs;ius in &longs;equentibus. </s>

<pb pagenum="11"/>attenderetur ad filum, & deprenderetur ten&longs;ionem ejus illam ip&longs;am <lb/>e&longs;&longs;e quam motus globorum requireret; concludere liceret mo<lb/>tum e&longs;&longs;e globorum, & corpora quie&longs;cere; & tum demum ex <lb/>tran&longs;latione globorum inter corpora, determinationem hujus <lb/>motus colligere. </s>

<s>Hunc enim in finem Tractatum &longs;equentem <lb/>compo&longs;ui. <pb pagenum="12"/><arrow.to.target n="note5"></arrow.to.target></s>

<s><emph type="center"/>AXIOMATA, <lb/>SIVE <lb/>LEGES MOTUS.<emph.end type="center"/><lb/><gap desc="hr tag"/></s>

<s>Hunc enim in finem Tractatum &longs;equentem <lb/>compo&longs;ui.

<arrow.to.target n="note5"></arrow.to.target></s>

<s><emph type="center"/>AXIOMATA, <lb/>SIVE <lb/>LEGES MOTUS.<emph.end type="center"/><lb/><gap desc="hr tag"/></s>

<s><emph type="italics"/>Corpus omne per&longs;everare in &longs;tatu &longs;uo quie&longs;cendi vel movendi uni<lb/>formiter in directum, ni&longs;i quatenus a viribus impre&longs;&longs;is cogitur <lb/>&longs;tatum illum mutare.<emph.end type="italics"/></s>

<s>PRojectilia per&longs;everant in motibus &longs;uis, ni&longs;i quatenus a re&longs;i<lb/>&longs;tentia aeris retardantur, & vi gravitatis impelluntur deor&longs;um. </s>

<s><lb/>Trochus, cujus partes cohærendo perpetuo retrahunt &longs;e&longs;e a mo<lb/>tibus rectilineis, non ce&longs;&longs;at rotari, ni&longs;i quatenus ab aere retardatur. </s>

<s><lb/>Majora autem Planetarum & Cometarum corpora motus &longs;uos & <lb/>progre&longs;&longs;ivos & circulares in &longs;patiis minus re&longs;i&longs;tentibus factos con<lb/>&longs;ervant diutius. </s>

<s><emph type="center"/><emph type="italics"/>Mutationem motus proportionalem e&longs;&longs;e vi motrici impre&longs;&longs;æ, & fieri <lb/>&longs;ecundum lineam rectam qua vis illa imprimitur.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Si vis aliqua motum quemvis generet; dupla duplum, tripla tri<lb/>plum generabit, &longs;ive &longs;imul & &longs;emel, &longs;ive gradatim & &longs;ucce&longs;&longs;ive im<lb/>pre&longs;&longs;a fuerit. </s>

<s>Et hic motus (quoniam in eandem &longs;emper plagam <lb/>cum vi generat<gap/>ice determinatur) &longs;i corpus antea movebatur, mo<lb/>tui ejus vel con&longs;piranti additur, vel contrario &longs;ubducitur, vel obli<lb/>quo oblique adjicitur, & cum eo &longs;ecundum utriu&longs;que determina<lb/>tionem componitur. </s><pb pagenum="13"/>

<s><emph type="italics"/>Actioni contrariam &longs;emper & æqualem e&longs;&longs;e reactionem: &longs;ive cor<lb/>porum duorum actiones in &longs;e mutuo &longs;emper e&longs;&longs;e æquales & in par<lb/>tes contrarias dirigi.<emph.end type="italics"/></s>

<s>Quicquid premit vel trahit alterum, tantundem ab eo premitur <lb/>vel trahitur. </s>

<s>Si quis lapidem digito premit, premitur & hujus <lb/>digitus a lapide. </s>

<s>Si equus lapidem funi alligatum trahit, retrahe<lb/>tur etiam & equus (ut ita dicam) æqualiter in lapidem: nam funis <lb/>utrinque di&longs;tentus eodem relaxandi &longs;e conatu urgebit equum ver<lb/>&longs;us lapidem, ac lapidem ver&longs;us equum; tantumque impediet pro<lb/>gre&longs;&longs;um unius quantum promovet progre&longs;&longs;um alterius. </s>

<s>Si corpus <lb/>aliquod in corpus aliud impingens, motum ejus vi &longs;ua quomodo<lb/>cunque mutaverit, idem quoque vici&longs;&longs;im in motu proprio eandem <lb/>mutationem in partem contrariam vi alterius ob æqualitatem pre&longs;<lb/>&longs;ionis mutuæ) &longs;ubibit. </s>

<s>His actionibus æquales fiunt mutationes, <lb/>non velocitatum, &longs;ed motuum; &longs;cilicet in corporibus non aliunde <lb/>impeditis. </s>

<s>Mutationes enim velocitatum, in contrarias itidem <lb/>partes factæ, quia motus æqualiter mutantur, &longs;unt corporibus re<lb/>ciproce proportionales. </s>

<s>Obtinet etiam hæc Lex in Attractionibus, <lb/>ut in Scholio proximo probabitur. </s>

<s>Obtinet etiam hæc Lex in Attractionibus, <lb/>ut in Scholio proximo probabitur. </s>

<s><emph type="center"/>COROLLARIUM I.<emph.end type="center"/></s>

<s><emph type="center"/>COROLLARIUM I.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Corpus viribus conjunctis diagonalem parallelogrammi eodem tem<lb/>pore de&longs;cribere, quo latera &longs;eparatis.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Si corpus dato tempore, vi &longs;ola <lb/>

<arrow.to.target n="fig1"></arrow.to.target><lb/><emph type="italics"/>M<emph.end type="italics"/> in loco <emph type="italics"/>A<emph.end type="italics"/> impre&longs;&longs;a, ferretur uni<lb/>formi cum motu ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>B<emph.end type="italics"/>; & vi <lb/>&longs;ola <emph type="italics"/>N<emph.end type="italics"/> in eodem loco impre&longs;&longs;a, fer<lb/>retur ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>C:<emph.end type="italics"/> compleatur pa<lb/>rallelogrammum <emph type="italics"/>ABDC,<emph.end type="italics"/> & vi utra<lb/>que feretur id eodem tempore in diagonali ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>D.<emph.end type="italics"/> Nam quo<lb/>niam vis <emph type="italics"/>N<emph.end type="italics"/> agit &longs;ecundum lineam <emph type="italics"/>AC<emph.end type="italics"/> ip&longs;i <emph type="italics"/>BD<emph.end type="italics"/> parallelam, hæc vis per <lb/>Legem 11 nihil mutabit velocitatem accedendi ad lineam illam <emph type="italics"/>BD<emph.end type="italics"/><lb/>a vi altera genitam. </s>

<s>Si corpus dato tempore, vi &longs;ola <lb/><figure id="fig1"></figure><lb/><emph type="italics"/>M<emph.end type="italics"/> in loco <emph type="italics"/>A<emph.end type="italics"/> impre&longs;&longs;a, ferretur uni<lb/>formi cum motu ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>B<emph.end type="italics"/>; & vi <lb/>&longs;ola <emph type="italics"/>N<emph.end type="italics"/> in eodem loco impre&longs;&longs;a, fer<lb/>retur ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>C:<emph.end type="italics"/> compleatur pa<lb/>rallelogrammum <emph type="italics"/>ABDC,<emph.end type="italics"/> & vi utra<lb/>que feretur id eodem tempore in diagonali ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>D.<emph.end type="italics"/> Nam quo<lb/>niam vis <emph type="italics"/>N<emph.end type="italics"/> agit &longs;ecundum lineam <emph type="italics"/>AC<emph.end type="italics"/> ip&longs;i <emph type="italics"/>BD<emph.end type="italics"/> parallelam, hæc vis per <lb/>Legem 11 nihil mutabit velocitatem accedendi ad lineam illam <emph type="italics"/>BD<emph.end type="italics"/><lb/>a vi altera genitam. </s>

<s>Accedet igitur corpus eodem tempore ad lineam <lb/><emph type="italics"/>BD,<emph.end type="italics"/> &longs;ive vis <emph type="italics"/>N<emph.end type="italics"/> imprimatur, &longs;ive non; atque adeo in fine illius tempo<lb/>ris reperietur alicubi in linea illa <emph type="italics"/>BD.<emph.end type="italics"/> Eodem argumento in fine tem<lb/>poris eju&longs;dem reperietur alicubi in linea <emph type="italics"/>CD,<emph.end type="italics"/> & idcirco in utriu&longs;que <lb/>lineæ concur&longs;u <emph type="italics"/>D<emph.end type="italics"/> reperiri nece&longs;&longs;e e&longs;t. </s>

<s>Perget autem motu rectili<lb/>neo ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>D<emph.end type="italics"/> per Legem 1. <pb pagenum="14"/><arrow.to.target n="note6"></arrow.to.target></s>

<s>Perget autem motu rectili<lb/>neo ab <emph type="italics"/>A<emph.end type="italics"/> ad <emph type="italics"/>D<emph.end type="italics"/> per Legem 1.

<arrow.to.target n="note6"></arrow.to.target></s>

<s><emph type="center"/>COROLLARIUM II.<emph.end type="center"/></s>

<s><emph type="italics"/>Et hinc patet compo&longs;itio vis directæ<emph.end type="italics"/> AD <emph type="italics"/>ex viribus quibu&longs;vis <lb/>obliquis<emph.end type="italics"/> AB <emph type="italics"/>&<emph.end type="italics"/> BD, <emph type="italics"/>& vici&longs;&longs;im re&longs;olutio vis cuju&longs;vis directæ<emph.end type="italics"/><lb/>AD <emph type="italics"/>in obliquas qua&longs;cunque<emph.end type="italics"/> AB <emph type="italics"/>&<emph.end type="italics"/> BD. <emph type="italics"/>Quæ quidem compo&longs;itio <lb/>& re&longs;olutio abunde confirmatur ex Mechanica.<emph.end type="italics"/></s>

<arrow.to.target n="fig2"></arrow.to.target><lb/>de&longs;cribatur circulus occurrens filo <lb/><emph type="italics"/>MA<emph.end type="italics"/> in <emph type="italics"/>D:<emph.end type="italics"/> & actæ rectæ <emph type="italics"/>OD<emph.end type="italics"/> pa<lb/>rallela &longs;it <emph type="italics"/>AC,<emph.end type="italics"/> & perpendicularis <lb/><emph type="italics"/>DC.<emph.end type="italics"/> Quoniam nihil refert, utrum <lb/>filorum puncta <emph type="italics"/>K, L, D<emph.end type="italics"/> affixa &longs;int <lb/>an non affixa ad planum rotæ; pon<lb/>dera idem valebunt, ac &longs;i &longs;u&longs;pende<lb/>rentur a punctis <emph type="italics"/>K<emph.end type="italics"/> & <emph type="italics"/>L<emph.end type="italics"/> vel <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>L.<emph.end type="italics"/><lb/>Ponderis autem <emph type="italics"/>A<emph.end type="italics"/> exponatur vis to<lb/>ta per lineam <emph type="italics"/>AD,<emph.end type="italics"/> & hæc re&longs;olvetur <lb/>in vires <emph type="italics"/>AC, CD,<emph.end type="italics"/> quarum <emph type="italics"/>AC<emph.end type="italics"/> trahendo radium <emph type="italics"/>OD<emph.end type="italics"/> directe a cen<lb/>tro nihil valet ad movendam rotam; vis autem altera <emph type="italics"/>DC,<emph.end type="italics"/> trahen<lb/>do radium <emph type="italics"/>DO<emph.end type="italics"/> perpendiculariter, idem valet ac &longs;i perpendiculari<lb/>ter traheret radium <emph type="italics"/>OL<emph.end type="italics"/> ip&longs;i <emph type="italics"/>OD<emph.end type="italics"/> æqualem; hoc e&longs;t, idem atque <lb/>pondus <emph type="italics"/>P,<emph.end type="italics"/> &longs;i modo pondus illud &longs;it ad pondus <emph type="italics"/>A<emph.end type="italics"/> ut vis <emph type="italics"/>DC<emph.end type="italics"/> ad <lb/>vim <emph type="italics"/>DA,<emph.end type="italics"/> id e&longs;t (ob &longs;imilia triangula <emph type="italics"/>ADC, DOK,<emph.end type="italics"/>) ut <emph type="italics"/>OK<emph.end type="italics"/><lb/>ad <emph type="italics"/>OD<emph.end type="italics"/> &longs;eu <emph type="italics"/>OL.<emph.end type="italics"/> Pondera igitur <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>P,<emph.end type="italics"/> quæ &longs;unt reciproce ut <lb/>radii in directum po&longs;iti <emph type="italics"/>OK<emph.end type="italics"/> & <emph type="italics"/>OL,<emph.end type="italics"/> idem pollebunt, & &longs;ic con&longs;i<lb/>&longs;tent in æquilibrio: quæ e&longs;t proprietas noti&longs;&longs;ima Libræ, Vectis, & <lb/>Axis in Peritrochio. </s>

<s>Sin pondus alterutrum &longs;it majus quam in hac <lb/>ratione, erit vis ejus ad movendam rotam tanto major. </s>

<s><emph type="center"/>COROLLARIUM II.<emph.end type="center"/></s>

<pb pagenum="15"/>geret illud hæc plana viribus <emph type="italics"/>pN, HN<emph.end type="italics"/> perpendiculariter, nimirun <lb/>planum <emph type="italics"/>pQ<emph.end type="italics"/> vi <emph type="italics"/>pN,<emph.end type="italics"/> & planum <emph type="italics"/>pG<emph.end type="italics"/> vi <emph type="italics"/>HN.<emph.end type="italics"/> Ideoque &longs;i tollatur pla<lb/>num <emph type="italics"/>pQ,<emph.end type="italics"/> ut pondus tendat filum; quoniam filum &longs;u&longs;tinendo pon <lb/>dus jam vicem præ&longs;tat plani &longs;ublati, tendetur illud eadem vi <emph type="italics"/>pN,<emph.end type="italics"/><lb/>qua planum antea urgebatur. </s>

<s>Unde ten&longs;io fili hujus obliqui erit <lb/>ad ten&longs;ionem &longs;ili alterius perpendicularis <emph type="italics"/>PN,<emph.end type="italics"/> ut <emph type="italics"/>pN<emph.end type="italics"/> ad <emph type="italics"/>pH.<emph.end type="italics"/> Id. </s>

<s>Ut &longs;i de rotæ alicujus centro <emph type="italics"/>O<emph.end type="italics"/> exeuntes radii inæquales <emph type="italics"/>OM, <lb/>ON<emph.end type="italics"/> filis <emph type="italics"/>MA, NP<emph.end type="italics"/> &longs;u&longs;tineant pondera <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>P,<emph.end type="italics"/> & quærantur vi<lb/>res ponderum ad movendam rotam: Per centrum <emph type="italics"/>O<emph.end type="italics"/> agatur recta <lb/><emph type="italics"/>KOL<emph.end type="italics"/> filis perpendiculariter occurrens in <emph type="italics"/>K<emph.end type="italics"/> & <emph type="italics"/>L,<emph.end type="italics"/> centroque <emph type="italics"/>O<emph.end type="italics"/> & <lb/>intervallorum <emph type="italics"/>OK, OL<emph.end type="italics"/> majore <emph type="italics"/>OL<emph.end type="italics"/><lb/><figure id="fig2"></figure><lb/>de&longs;cribatur circulus occurrens filo <lb/><emph type="italics"/>MA<emph.end type="italics"/> in <emph type="italics"/>D:<emph.end type="italics"/> & actæ rectæ <emph type="italics"/>OD<emph.end type="italics"/> pa<lb/>rallela &longs;it <emph type="italics"/>AC,<emph.end type="italics"/> & perpendicularis <lb/><emph type="italics"/>DC.<emph.end type="italics"/> Quoniam nihil refert, utrum <lb/>filorum puncta <emph type="italics"/>K, L, D<emph.end type="italics"/> affixa &longs;int <lb/>an non affixa ad planum rotæ; pon<lb/>dera idem valebunt, ac &longs;i &longs;u&longs;pende<lb/>rentur a punctis <emph type="italics"/>K<emph.end type="italics"/> & <emph type="italics"/>L<emph.end type="italics"/> vel <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>L.<emph.end type="italics"/><lb/>Ponderis autem <emph type="italics"/>A<emph.end type="italics"/> exponatur vis to<lb/>ta per lineam <emph type="italics"/>AD,<emph.end type="italics"/> & hæc re&longs;olvetur <lb/>in vires <emph type="italics"/>AC, CD,<emph.end type="italics"/> quarum <emph type="italics"/>AC<emph.end type="italics"/> trahendo radium <emph type="italics"/>OD<emph.end type="italics"/> directe a cen<lb/>tro nihil valet ad movendam rotam; vis autem altera <emph type="italics"/>DC,<emph.end type="italics"/> trahen<lb/>do radium <emph type="italics"/>DO<emph.end type="italics"/> perpendiculariter, idem valet ac &longs;i perpendiculari<lb/>ter traheret radium <emph type="italics"/>OL<emph.end type="italics"/> ip&longs;i <emph type="italics"/>OD<emph.end type="italics"/> æqualem; hoc e&longs;t, idem atque <lb/>pondus <emph type="italics"/>P,<emph.end type="italics"/> &longs;i modo pondus illud &longs;it ad pondus <emph type="italics"/>A<emph.end type="italics"/> ut vis <emph type="italics"/>DC<emph.end type="italics"/> ad <lb/>vim <emph type="italics"/>DA,<emph.end type="italics"/> id e&longs;t (ob &longs;imilia triangula <emph type="italics"/>ADC, DOK,<emph.end type="italics"/>) ut <emph type="italics"/>OK<emph.end type="italics"/><lb/>ad <emph type="italics"/>OD<emph.end type="italics"/> &longs;eu <emph type="italics"/>OL.<emph.end type="italics"/> Pondera igitur <emph type="italics"/>A<emph.end type="italics"/> & <emph type="italics"/>P,<emph.end type="italics"/> quæ &longs;unt reciproce ut <lb/>radii in directum po&longs;iti <emph type="italics"/>OK<emph.end type="italics"/> & <emph type="italics"/>OL,<emph.end type="italics"/> idem pollebunt, & &longs;ic con&longs;i<lb/>&longs;tent in æquilibrio: quæ e&longs;t proprietas noti&longs;&longs;ima Libræ, Vectis, & <lb/>Axis in Peritrochio. </s>

<s>Sin pondus alterutrum &longs;it majus quam in hac <lb/>ratione, erit vis ejus ad movendam rotam tanto major. </s>

<s><lb/>eoque &longs;i pondus <emph type="italics"/>p<emph.end type="italics"/> &longs;it ad pondus <emph type="italics"/>A<emph.end type="italics"/> in ratione quæ componitur <gap/><lb/>ratione reciproca minimarum di&longs;tantiarum &longs;uorum &longs;uorum <emph type="italics"/>pN, <lb/>AM<emph.end type="italics"/> a centro rotæ, & ratione directa <emph type="italics"/>pH<emph.end type="italics"/> ad <emph type="italics"/>pN<emph.end type="italics"/>; pondera idem <lb/>valebunt ad rotam movendam, atque adeo &longs;e mutuo &longs;u&longs;tinebunt, <lb/>ut quilibet experiri pote&longs;t. </s>

<s>Quod &longs;i pondus <emph type="italics"/>p<emph.end type="italics"/> ponderi <emph type="italics"/>P<emph.end type="italics"/> æquale partim &longs;u&longs;pendatur filo <emph type="italics"/>Np,<emph.end type="italics"/><lb/>partim incumbat plano obliquo <emph type="italics"/>pG:<emph.end type="italics"/> agantur <emph type="italics"/>pH, NH,<emph.end type="italics"/> prior ho<lb/>rizonti, po&longs;terior plano <emph type="italics"/>pG<emph.end type="italics"/> perpendicularis; & &longs;i vis ponderis <emph type="italics"/>p<emph.end type="italics"/><lb/>deor&longs;um tendens, exponatur per lineam <emph type="italics"/>pH,<emph.end type="italics"/> re&longs;olvi pote&longs;t hæc in <lb/>vires <emph type="italics"/>pN, HN.<emph.end type="italics"/> Si filo <emph type="italics"/>pN<emph.end type="italics"/> perpendiculare e&longs;&longs;et planum aliquod <lb/><emph type="italics"/>pQ,<emph.end type="italics"/> &longs;ecans planum alterum <emph type="italics"/>pG<emph.end type="italics"/> in linea ad horizontem paral<lb/>lela; & pondas <emph type="italics"/>p<emph.end type="italics"/> his planis <emph type="italics"/>pQ, pG<emph.end type="italics"/> &longs;olummodo incumberet; ur-<pb pagenum="15"/>geret illud hæc plana viribus <emph type="italics"/>pN, HN<emph.end type="italics"/> perpendiculariter, nimirun <lb/>planum <emph type="italics"/>pQ<emph.end type="italics"/> vi <emph type="italics"/>pN,<emph.end type="italics"/> & planum <emph type="italics"/>pG<emph.end type="italics"/> vi <emph type="italics"/>HN.<emph.end type="italics"/> Ideoque &longs;i tollatur pla<lb/>num <emph type="italics"/>pQ,<emph.end type="italics"/> ut pondus tendat filum; quoniam filum &longs;u&longs;tinendo pon <lb/>dus jam vicem præ&longs;tat plani &longs;ublati, tendetur illud eadem vi <emph type="italics"/>pN,<emph.end type="italics"/><lb/>qua planum antea urgebatur. </s>

<s>Pondus autem <emph type="italics"/>p,<emph.end type="italics"/> planis illis duobus obliquis incumbens, rationem <lb/>habet cunei inter corporis fi&longs;&longs;i facies internas: & inde vires cunei <lb/>& mallei innote&longs;cunt: utpote cum vis qua pondus <emph type="italics"/>p<emph.end type="italics"/> urget planum <lb/><emph type="italics"/>pQ<emph.end type="italics"/> &longs;it ad vim, qua idem vel gravitate &longs;ua vel ictu mallei impellitur <lb/>&longs;ecundum lineam <emph type="italics"/>pH<emph.end type="italics"/> in plano, &c. </s>

<s>ut <emph type="italics"/>pN<emph.end type="italics"/> and <emph type="italics"/>pH<emph.end type="italics"/>; atque ad vim, qua <lb/>urget planum alterum <emph type="italics"/>pG,<emph.end type="italics"/> ut <emph type="italics"/>pN<emph.end type="italics"/> ad <emph type="italics"/>NH.<emph.end type="italics"/> Sed & vis Cochleæ per <lb/>&longs;imilem virium divi&longs;ionem colligitur; quippe quæ cuneus e&longs;t a ve<lb/>cte impul&longs;us. </s>

<s>U&longs;us igitur Corollarii hujus lati&longs;&longs;ime patet, & late <lb/>patendo veritatem &longs;uam evincit; cum pendeat ex jam dictis Mecha<lb/>nica tota ab Auctoribus diver&longs;imode demon&longs;trata. </s>

<s>Ex hi&longs;ce enim <lb/>facile derivantur vires Machinarum, quæ ex Rotis, Tympanis, <lb/>Trochleis, Vectibus, nervis ten&longs;is & ponderibus directe vel obli<lb/>que a&longs;cendentibus, cæteri&longs;que potentiis Mechanicis componi &longs;o<lb/>lent, ut & vires Tendinum ad animalium o&longs;&longs;a movenda. </s>

<s><emph type="center"/>COROLLARIUM III.<emph.end type="center"/></s>

<s><emph type="center"/>COROLLARIUM III.<emph.end type="center"/></s>

<s><emph type="italics"/>Quantitas motus quæ colligitur capiendo &longs;ummam motuum factorum <lb/>ad eandem partem, & differentiam factorum ad contrarias, non <lb/>mutatur ab actione corporum inter &longs;e.<emph.end type="italics"/></s>

<s>Etenim actio eique contraria reactio æquales &longs;unt per Legem 111, <lb/>adeoque per Legem 11 æquales in motibus efficiunt mutationes ver<lb/>&longs;us contrarias partes. </s>

<s>Ergo &longs;i motus fiunt ad eandem partem; quic<lb/>quid additur motui corporis fugientis, &longs;ubducetur motui corporis <lb/>in&longs;equentis &longs;ic, ut &longs;umma maneat eadem quæ prius. </s>

<s>Sin corpora ob<lb/>viam eant; æqualis erit &longs;ubductio de motu utriu&longs;que, adeoque diffe<lb/>rentia motuum factorum in contrarias partes manebit eadem. </s>

<s>Sin corpora ob<lb/>viam eant; æqualis erit &longs;ubductio de motu utriu&longs;que, adeoque diffe<lb/>rentia motuum factorum in contrarias partes manebit eadem. </s>

<s>Ut &longs;i corpus &longs;phæricum <emph type="italics"/>A<emph.end type="italics"/> &longs;it triplo majus corpore &longs;phærico <emph type="italics"/>B,<emph.end type="italics"/> ha<lb/>beatque duas velocitatis partes; & <emph type="italics"/>B<emph.end type="italics"/> &longs;equatur in eadem recta cum ve-<pb pagenum="16"/><arrow.to.target n="note7"></arrow.to.target><lb/>locitatis partibus decem, adeoque motus ip&longs;ius <emph type="italics"/>A<emph.end type="italics"/> &longs;it ad motum ip&longs;ius <lb/><emph type="italics"/>B,<emph.end type="italics"/> ut &longs;ex ad decem: ponantur motus illis e&longs;&longs;e partium &longs;ex & par<lb/>tium decem, & &longs;umma erit partium &longs;exdecim. </s>

<arrow.to.target n="note7"></arrow.to.target><lb/>locitatis partibus decem, adeoque motus ip&longs;ius <emph type="italics"/>A<emph.end type="italics"/> &longs;it ad motum ip&longs;ius <lb/><emph type="italics"/>B,<emph.end type="italics"/> ut &longs;ex ad decem: ponantur motus illis e&longs;&longs;e partium &longs;ex & par<lb/>tium decem, & &longs;umma erit partium &longs;exdecim. </s>

<s>In corporum igitur <lb/>concur&longs;u, &longs;i corpus <emph type="italics"/>A<emph.end type="italics"/> lucretur motus partes tres vel quatuor vel <lb/>quinque, corpus <emph type="italics"/>B<emph.end type="italics"/> amittet partes totidem, adeoque perget corpus <lb/><emph type="italics"/>A<emph.end type="italics"/> po&longs;t reflexionem cum partibus novem vel decem vel undecim, <lb/>& <emph type="italics"/>B<emph.end type="italics"/> cum partibus &longs;eptem vel &longs;ex vel quinque, exi&longs;tente &longs;emper &longs;um<lb/>ma partium &longs;exdecim ut prius. </s>

<s>Si corpus <emph type="italics"/>A<emph.end type="italics"/> lucretur partes novem <lb/>vel decem vel undecim vel duodecim, adeoque progrediatur po&longs;t <lb/>concur&longs;um cum partibus quindecim vel &longs;exdecim vel &longs;eptendecim <lb/>vel octodecim; corpus <emph type="italics"/>B,<emph.end type="italics"/> amittendo tot partes quot <emph type="italics"/>A<emph.end type="italics"/> lucratur, <lb/>vel cum una parte progredietur ami&longs;&longs;is partibus novem, vel qui<lb/>e&longs;cet ami&longs;&longs;o motu &longs;uo progre&longs;&longs;ivo partium decem, vel cum una par<lb/>te regredietur ami&longs;&longs;o motu &longs;uo & (ut ita dicam) una parte amplius, <lb/>vel regredietur cum partibus duabus ob detractum motum progre&longs;<lb/>&longs;ivum partium duodecim. </s>

<s>Atque ita &longs;ummæ motuum con&longs;pirantium <lb/>15+1 vel 16+c, & differentiæ contrariorum 17-1 & 18-2 &longs;emper <lb/>erunt partium &longs;exdecim, ut ante concur&longs;um & reflexionem. </s>

<s>Cogni<lb/>tis autem motibus quibu&longs;cum corpora po&longs;t reflexionem pergent, in<lb/>venietur cuju&longs;que velocitas, ponendo eam e&longs;&longs;e ad velocitatem ante <lb/>reflexionem, ut motus po&longs;t e&longs;t ad motum ante. </s>

<s>Ut in ca&longs;u ultimo, ubi <lb/>corporis <emph type="italics"/>A<emph.end type="italics"/> motus erat partium &longs;ex ante reflexionem & partium octo<lb/>decim po&longs;tea, & velocitas partium duarum ante reflexionem; in<lb/>venietur ejus velocitas partium &longs;ex po&longs;t reflexionem, dicendo, ut <lb/>motus partes &longs;ex ante reflexionem ad motus partes octodecim po&longs;t<lb/>ea, ita velocitatis partes duæ ante reflexionem ad velocitatis partes <lb/>&longs;ex po&longs;tea. </s>

<s>Quod &longs;i corpora vel non Sphærica vel diver&longs;is in rectis moventia <lb/>incidant in &longs;e mutuo oblique, & requirantur eorum motus po&longs;t refle<lb/>xionem; cogno&longs;cendus e&longs;t &longs;itus plani a quo corpora concurrentia tan<lb/>guntur in puncto concur&longs;us: dein corporis utriu&longs;que motus (per <lb/>Corol.11.) di&longs;tinguendus e&longs;t in duos, unum huic plano perpendicu<lb/>larem, alterum eidem parallelum: motus autem paralleli, propter<lb/>ea quod corpora agant in &longs;e invicem &longs;ecundum lineam huic plano <lb/>perpendicularem, retinendi &longs;unt iidem po&longs;t reflexionem atque an<lb/>tea; & motibus perpendicularibus mutationes æquales in partes con<lb/>trarias tribuendæ &longs;unt &longs;ic, ut &longs;umma con&longs;pirantium & differentia <lb/>contrariorum maneat eadem quæ prius. </s>

<s>Ex huju&longs;modi reflexio<lb/>nibus oriri etiam &longs;olent motus circulares corporum circa centra pro<lb/>pria. </s>

<s>Sed hos ca&longs;us in &longs;equentibus non con&longs;idero, & nimis longum <lb/>e&longs;&longs;et omnia huc &longs;pectantia demon&longs;trare. <pb pagenum="18"/><arrow.to.target n="note8"></arrow.to.target><lb/>mutat &longs;tatum &longs;uum; & reliquorum, quibu&longs;cum actio illa non in<lb/>tercedit, commune gravitatis centrum nihil inde patitur; di&longs;tantia <lb/>autem horum duorum centrorum dividitur a communi corporum <lb/>omnium centro in partes &longs;ummis totalibus corporum quorum <lb/>&longs;unt centra reciproce proportionales; adeoque centris illis duobus <lb/>&longs;tatum &longs;uum movendi vel quie&longs;cendi &longs;ervantibus, commune omni<lb/>um centrum &longs;ervat etiam &longs;tatum &longs;uum: manife&longs;tum e&longs;t quod com<lb/>mune illud omnium centrum ob actiones binorum corporum inter <lb/>&longs;e nunquam mutat &longs;tatum &longs;uum quoad motum & quietem. </s>

<s>Sed hos ca&longs;us in &longs;equentibus non con&longs;idero, & nimis longum <lb/>e&longs;&longs;et omnia huc &longs;pectantia demon&longs;trare.

<arrow.to.target n="note8"></arrow.to.target><lb/>mutat &longs;tatum &longs;uum; & reliquorum, quibu&longs;cum actio illa non in<lb/>tercedit, commune gravitatis centrum nihil inde patitur; di&longs;tantia <lb/>autem horum duorum centrorum dividitur a communi corporum <lb/>omnium centro in partes &longs;ummis totalibus corporum quorum <lb/>&longs;unt centra reciproce proportionales; adeoque centris illis duobus <lb/>&longs;tatum &longs;uum movendi vel quie&longs;cendi &longs;ervantibus, commune omni<lb/>um centrum &longs;ervat etiam &longs;tatum &longs;uum: manife&longs;tum e&longs;t quod com<lb/>mune illud omnium centrum ob actiones binorum corporum inter <lb/>&longs;e nunquam mutat &longs;tatum &longs;uum quoad motum & quietem. </s>

<s>In tali <lb/>autem &longs;y&longs;temate actiones omnes corporum inter &longs;e, vel inter bina <lb/>&longs;unt corpora, vel ab actionibus inter bina compo&longs;itæ; & propterea <lb/>communi omnium centro mutationem in &longs;tatu motus ejus vel quie<lb/>tis nunquam inducunt. </s>

<s>Quare cum centrum illud ubi corpora non <lb/>agunt in &longs;e invicem, vel quie&longs;cit, vel in recta aliqua progreditur uni<lb/>formiter; perget idem, non ob&longs;tantibus corporum actionibus inter <lb/>&longs;e, vel &longs;emper quie&longs;cere, vel &longs;emper progredi uniformiter in dire<lb/>ctum; ni&longs;i a viribus in &longs;y&longs;tema extrin&longs;ecus impre&longs;&longs;is deturbetur de hoc <lb/>&longs;tatu. </s>

<s>E&longs;t igitur &longs;y&longs;tematis corporum plurium Lex eadem quæ cor<lb/>poris &longs;olitarii, quoad per&longs;everantiam in &longs;tatu motus vel quietis. </s>

<s>Mo<lb/>tus enim progre&longs;&longs;ivus &longs;eu corporis &longs;olitarii &longs;eu &longs;y&longs;tematis corporum <lb/>ex motu centri gravitatis æ&longs;timari &longs;emper debet. </s>

<s>Mo<lb/>tus enim progre&longs;&longs;ivus &longs;eu corporis &longs;olitarii &longs;eu &longs;y&longs;tematis corporum <lb/>ex motu centri gravitatis æ&longs;timari &longs;emper debet. </s>

<s><emph type="center"/>COROLLARIUM V.<emph.end type="center"/></s>

<s><emph type="italics"/>Corporum dato &longs;patio inclu&longs;orum iidem &longs;unt motus inter &longs;e, &longs;ive &longs;pa<lb/>tium illud quie&longs;cat, &longs;ive moveatur idem uniformiter in directum <lb/>ab&longs;que motu circulari.<emph.end type="italics"/></s>

<s>Nam differentiæ motuum tendentium ad eandem partem, & &longs;um<lb/>mæ tendentium ad contrarias, eædem &longs;unt &longs;ub initio in <expan abbr="utroq;">utroque</expan> ca&longs;u (ex <lb/>hypothe&longs;i) & ex his &longs;ummis vel differentiis oriuntur congre&longs;&longs;us & im<lb/>petus quibus corpora &longs;e mutuo feriunt. </s>

<s><emph type="center"/>COROLLARIUM V.<emph.end type="center"/></s>

<s>Ergo per Legem 11 æquales e<lb/>runt congre&longs;&longs;uum effectus in <expan abbr="utroq;">utroque</expan> ca&longs;u; & propterea manebunt mo<lb/>tus inter &longs;e in uno ca&longs;u æquales motibus inter &longs;e in altero. </s>

<s>Idem com<lb/>probatur experimento luculento. </s>

<s>Motus omnes eodem modo &longs;e ha<lb/>bent in Navi, &longs;ive ea quie&longs;cat, &longs;ive moveatur uniformiter in directum. </s>

<s>Motus omnes eodem modo &longs;e ha<lb/>bent in Navi, &longs;ive ea quie&longs;cat, &longs;ive moveatur uniformiter in directum. </s>

<s><emph type="center"/>COROLLARIUM VI.<emph.end type="center"/></s>

<s><emph type="center"/>COROLLARIUM VI.<emph.end type="center"/></s>

<s><emph type="italics"/>Si corpora <expan abbr="moveãtur">moveantur</expan> <expan abbr="quomodocunq;">quomodocunque</expan> inter&longs;e, & a viribus acceler atrici<lb/>bus æqualibus &longs;ecundum lineas parallelas urgeantur; pergent omnia <lb/>eodem modo moveri inter &longs;e, ac &longs;i viribus illis non e&longs;&longs;ent incitata.<emph.end type="italics"/></s>

<s>Nam vires illæ æqualiter (pro quantitatibus movendorum corpo-

<pb pagenum="19"/>rum) & &longs;ecundum lineas parallelas agendo, corpora omnia æquali<lb/>ter (quoad velocitatem) movebunt per Legem 11. adeoque nunquam <lb/>mutabunt po&longs;itiones & motus eorum inter &longs;e. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Hactenus principia tradidi a Mathematicis recepta & experien<lb/>tia multiplici confirmata. </s>

<s>Nam vires illæ æqualiter (pro quantitatibus movendorum corpo-<pb pagenum="19"/>rum) & &longs;ecundum lineas parallelas agendo, corpora omnia æquali<lb/>ter (quoad velocitatem) movebunt per Legem 11. adeoque nunquam <lb/>mutabunt po&longs;itiones & motus eorum inter &longs;e. </s>

<s>Per Leges duas primas & Corollaria duo <lb/>prima <emph type="italics"/>Galilæus<emph.end type="italics"/> invenit de&longs;cen&longs;um Gravium e&longs;&longs;e in duplicata ratione <lb/>temporis, & motum Projectilium fieri in Parabola; con&longs;pirante ex<lb/>perientia, ni&longs;i quatenus motus illi per aeris re&longs;i&longs;tentiam aliquantu<lb/>lum retardantur. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Hactenus principia tradidi a Mathematicis recepta & experien<lb/>tia multiplici confirmata. </s>

<s>Ab ii&longs;dem Legibus & Corollariis pendent de<lb/>mon&longs;trata de temporibus o&longs;cillantium Pendulorum, &longs;uffragante Ho<lb/>rologiorum experientia quotidiana. </s>

<s>Ex his ii&longs;dem & Lege tertia <lb/><emph type="italics"/>Chri&longs;tophorus Wrennus<emph.end type="italics"/> Eques Auratus, <emph type="italics"/>Jobannes Walli&longs;ius S.T.D.<emph.end type="italics"/><lb/>& <emph type="italics"/>Chri&longs;tianus Hugenius,<emph.end type="italics"/> hujus ætatis Geometrarum facile prin<lb/>cipes, regulas congre&longs;&longs;uum & reflexionum duorum corporum &longs;e<lb/>or&longs;im invenerunt, & eodem fere tempore cum <emph type="italics"/>Societate Regia<emph.end type="italics"/><lb/>communicarunt, inter &longs;e (quoad has leges) omnino con&longs;pirantes: <lb/>& primus quidem <emph type="italics"/>Walli&longs;ius,<emph.end type="italics"/> deinde <emph type="italics"/>Wrennus<emph.end type="italics"/> & <emph type="italics"/>Hugenius<emph.end type="italics"/> inven<lb/>tum prodiderunt. </s>

<s>Sed & veritas comprobata e&longs;t a <emph type="italics"/>Wrenno<emph.end type="italics"/> co<lb/>ram <emph type="italics"/>Regia Societate<emph.end type="italics"/> per experimentum Pendulorum: quod etiam <lb/><emph type="italics"/>Clari&longs;&longs;imus Mariottus<emph.end type="italics"/> libro integro exponere mox dignatus e&longs;t. </s>

<s>Ve<lb/>rum, ut hoc experimentum cum Theoriis ad amu&longs;&longs;im congruat, ha<lb/>benda e&longs;t ratio cum re&longs;i&longs;tentiæ aeris, tum etiam vis Ela&longs;ticæ con<lb/>currentium corporum. </s>

<s>Pendeant corpora <emph type="italics"/>A, B<emph.end type="italics"/> filis parallelis & <lb/>æqualibus <emph type="italics"/>AC, BD,<emph.end type="italics"/> a centris <emph type="italics"/>C, D.<emph.end type="italics"/> His centris & intervallis de<lb/>&longs;cribantur &longs;emicirculi <emph type="italics"/>EAF, GBH<emph.end type="italics"/> radiis <emph type="italics"/>CA, DB<emph.end type="italics"/> bi&longs;ecti. </s>

<s>Tra<lb/>hatur corpus <emph type="italics"/>A<emph.end type="italics"/> ad arcus <emph type="italics"/>EAF<emph.end type="italics"/> punctum quodvis <emph type="italics"/>R,<emph.end type="italics"/> & (&longs;ubducto <lb/>corpore <emph type="italics"/>B<emph.end type="italics"/>) demittatur inde, redeatque po&longs;t unam o&longs;cillationem <lb/>ad punctum <emph type="italics"/>V.<emph.end type="italics"/> E&longs;t <emph type="italics"/>RV<emph.end type="italics"/> re<lb/><figure id="fig3"></figure><lb/>tardatio ex re&longs;i&longs;tentia aeris. </s>

<arrow.to.target n="fig3"></arrow.to.target><lb/>tardatio ex re&longs;i&longs;tentia aeris. </s>

<s><lb/>Hujus <emph type="italics"/>RV<emph.end type="italics"/> fiat <emph type="italics"/>ST<emph.end type="italics"/> pars quar<lb/>ta &longs;ita in medio, ita &longs;cilicet <lb/>ut <emph type="italics"/>RS<emph.end type="italics"/> & <emph type="italics"/>TV<emph.end type="italics"/> æquentur, &longs;it<lb/>que <emph type="italics"/>RS<emph.end type="italics"/> ad <emph type="italics"/>ST<emph.end type="italics"/> ut 3 ad 2. <lb/>Et i&longs;ta <emph type="italics"/>ST<emph.end type="italics"/> exhibebit retarda<lb/>tionem in de&longs;cen&longs;u ab <emph type="italics"/>S<emph.end type="italics"/> ad <emph type="italics"/>A<emph.end type="italics"/><lb/>quam proxime. </s>

<s>Re&longs;tituatur <lb/>corpus <emph type="italics"/>B<emph.end type="italics"/> in locum &longs;uum. </s>

<s>Cadat corpus <emph type="italics"/>A<emph.end type="italics"/> de puncto <emph type="italics"/>S,<emph.end type="italics"/> & velo<lb/>citas ejus in loco reflexionis <emph type="italics"/>A,<emph.end type="italics"/> ab&longs;que errore &longs;en&longs;ibili, tanta erit ae <pb pagenum="20"/>&longs;i in vacuo cecidi&longs;&longs;et de loco <emph type="italics"/>T.<emph.end type="italics"/> Exponatur igitur hæc velocitas <lb/><arrow.to.target n="note9"></arrow.to.target><lb/>per chordam arcus <emph type="italics"/>TA.<emph.end type="italics"/> Nam velocitatem Penduli in puncto in<lb/>fimo e&longs;&longs;e ut chordam arcus quem cadendo de&longs;crip&longs;it, Propo&longs;itio e&longs;t <lb/>e&longs;t Geometris noti&longs;&longs;ima. </s>

<pb pagenum="20"/>&longs;i in vacuo cecidi&longs;&longs;et de loco <emph type="italics"/>T.<emph.end type="italics"/> Exponatur igitur hæc velocitas <lb/>

<arrow.to.target n="note9"></arrow.to.target><lb/>per chordam arcus <emph type="italics"/>TA.<emph.end type="italics"/> Nam velocitatem Penduli in puncto in<lb/>fimo e&longs;&longs;e ut chordam arcus quem cadendo de&longs;crip&longs;it, Propo&longs;itio e&longs;t <lb/>e&longs;t Geometris noti&longs;&longs;ima. </s>

<s>Po&longs;t reflexionem perveniat corpus <emph type="italics"/>A<emph.end type="italics"/> ad <lb/>locum <emph type="italics"/>s,<emph.end type="italics"/> & corpus <emph type="italics"/>B<emph.end type="italics"/> ad locum <emph type="italics"/>k.<emph.end type="italics"/> Tollatur corpus <emph type="italics"/>B<emph.end type="italics"/> & invenia<lb/>tur locus <emph type="italics"/>v<emph.end type="italics"/>; a quo &longs;i corpus <emph type="italics"/>A<emph.end type="italics"/> demittatur & po&longs;t unam o&longs;cillatio<lb/>nem redeat ad locum <emph type="italics"/>r,<emph.end type="italics"/> &longs;it <emph type="italics"/>st<emph.end type="italics"/> pars quarta ip&longs;ius <emph type="italics"/>rv<emph.end type="italics"/> &longs;ita in medio, <lb/>ita videlicet ut <emph type="italics"/>rs<emph.end type="italics"/> & <emph type="italics"/>tu<emph.end type="italics"/> æquentur; & per chordam arcus <emph type="italics"/>tA<emph.end type="italics"/> ex<lb/>ponatur velocitas quam corpus <emph type="italics"/>A<emph.end type="italics"/> proxime po&longs;t reflexionem habuit <lb/>in loco <emph type="italics"/>A.<emph.end type="italics"/> Nam <emph type="italics"/>t<emph.end type="italics"/> erit locus ille vcrus & correctus, ad quem cor<lb/>pus <emph type="italics"/>A,<emph.end type="italics"/> &longs;ublata aeris re&longs;i&longs;tentia, a&longs;cendere debui&longs;&longs;et: Simili me<lb/>thodo corrigendus erit locus <emph type="italics"/>k,<emph.end type="italics"/> ad quem corpus <emph type="italics"/>B<emph.end type="italics"/> a&longs;cendit, & in<lb/>veniendus locus <emph type="italics"/>l,<emph.end type="italics"/> ad quem corpus illud a&longs;cendere debui&longs;&longs;et in va<lb/>cuo. </s>

<s>Hoc pacto experiri licet omnia perinde ac &longs;i in vacuo con<lb/>&longs;tituti e&longs;&longs;emus. </s>

<s>Tandem ducendum erit corpus <emph type="italics"/>A<emph.end type="italics"/> in chordam ar<lb/>cus <emph type="italics"/>TA<emph.end type="italics"/> (quæ velocitatem ejus exhibet) ut habeatur motus ejus in <lb/>loco <emph type="italics"/>A<emph.end type="italics"/> proxime ante reflexionem; deinde in chordam arcus <emph type="italics"/>tA,<emph.end type="italics"/> ut <lb/>habeatur motus ejus in loco <emph type="italics"/>A<emph.end type="italics"/> proxime po&longs;t reflexionem. </s>

<s>Et &longs;ic <lb/>corpus <emph type="italics"/>B<emph.end type="italics"/> ducendum erit in chordam arcus <emph type="italics"/>Bb,<emph.end type="italics"/> ut habeatur motus <lb/>ejus proxime po&longs;t reflexionem. </s>

<s>Et &longs;imili methodo, ubi corpora duo <lb/>fimul demittuntur de locis diver&longs;is, inveniendi &longs;unt motus <expan abbr="utriu&longs;q;">utriu&longs;que</expan> <lb/>tam ante, quam po&longs;t reflexionem; & tum demum conferendi &longs;unt <lb/>motus inter &longs;e & colligendi effectus reflexionis. </s>

<s>Hoc modo in <lb/>Pendulis pedum decem rem tentando, idque in corporibus tam <lb/>inæqualibus quam æqualibus, & faciendo ut corpora de intervallis <lb/>ampli&longs;&longs;imis, puta pedum octo vel duodecim vel &longs;exdecim, concurre<lb/>rent; reperi &longs;emper &longs;ine errore trium digitorum in men&longs;uris, ubi <lb/>corpora &longs;ibi mutuo directe occurrebant, quod æquales erant muta<lb/>tiones motuum corporibus in partes contrarias illatæ, atque adeo <lb/>quod actio & reactio &longs;emper <lb/><figure id="fig4"></figure><lb/>erant æquales. </s>

<arrow.to.target n="fig4"></arrow.to.target><lb/>erant æquales. </s>

<s>Ut &longs;i corpus <lb/><emph type="italics"/>A<emph.end type="italics"/> incidebat in corpus <emph type="italics"/>B<emph.end type="italics"/> cum <lb/>novem partibus motus, & a<lb/>mi&longs;&longs;is &longs;eptem partibus perge<lb/>bat po&longs;t reflexionem cum du<lb/>abus; corpus <emph type="italics"/>B<emph.end type="italics"/> re&longs;iliebat cum <lb/>partibus i&longs;tis &longs;eptem. </s>

<s>Si cor<lb/>pora obviam ibant <emph type="italics"/>A<emph.end type="italics"/> cum <lb/>duodecim partibus & <emph type="italics"/>B<emph.end type="italics"/> cum &longs;ex, & redibat <emph type="italics"/>A<emph.end type="italics"/> cum duabus; redi<lb/>bat <emph type="italics"/>B<emph.end type="italics"/> cum octo, facta detractione partium quatuordecim utrin<lb/>que. </s>

<s>De motu ip&longs;ius <emph type="italics"/>A<emph.end type="italics"/> &longs;ubducantur partes duodecim, & re&longs;tabit <pb pagenum="21"/>nihil: &longs;ubducantur aliæ partes duæ, & fiet motus duarum partium <lb/>in plagam contrariam: & &longs;ic de motu corporis <emph type="italics"/>B<emph.end type="italics"/> partium &longs;ex &longs;ub<lb/>ducendo partes quatuordecim, fient partes octo in plagam contra<lb/>riam. </s>

<s>De motu ip&longs;ius <emph type="italics"/>A<emph.end type="italics"/> &longs;ubducantur partes duodecim, & re&longs;tabit

<pb pagenum="21"/>nihil: &longs;ubducantur aliæ partes duæ, & fiet motus duarum partium <lb/>in plagam contrariam: & &longs;ic de motu corporis <emph type="italics"/>B<emph.end type="italics"/> partium &longs;ex &longs;ub<lb/>ducendo partes quatuordecim, fient partes octo in plagam contra<lb/>riam. </s>

<s>Quod &longs;i corpora ibant ad eandam plagam, <emph type="italics"/>A<emph.end type="italics"/> velocius cum <lb/>partibus quatuordecim, & <emph type="italics"/>B<emph.end type="italics"/> tardius cum partibus quinque, & po&longs;t <lb/>reflexionem pergebat <emph type="italics"/>A<emph.end type="italics"/> cum quinque partibus; pergebat <emph type="italics"/>B<emph.end type="italics"/> cum qua<lb/>tuordecim, facta tran&longs;latione partium novem de <emph type="italics"/>A<emph.end type="italics"/> in <emph type="italics"/>B.<emph.end type="italics"/> Et &longs;ic <lb/>in reliquis. </s>

<s>A congre&longs;&longs;u & colli&longs;ione corporum nunquam muta<lb/>batur quantitas motus, quæ ex &longs;umma motuum con&longs;pirantium & <lb/>differentia contrariorum colligebatur. </s>

<s>Nam errorem digiti unius <lb/>& alterius in men&longs;uris tribuerim difficultati peragendi &longs;ingula <lb/>&longs;atis accurate. </s>

<s>Difficile erat, tum pendula &longs;imul demittere fic, ut <lb/>corpora in &longs;e mutuo impingerent in loco infimo <emph type="italics"/>AB<emph.end type="italics"/>; tum loca <emph type="italics"/>s, <lb/>k<emph.end type="italics"/> notare, ad quæ corpora a&longs;cendebant po&longs;t concur&longs;um. </s>

<s>Sed & in <lb/>ip&longs;is pilis inæqualis partium den&longs;itas, & textura aliis de cau&longs;is irre<lb/>gularis, errores inducebant. </s>

<s>Sed & in <lb/>ip&longs;is pilis inæqualis partium den&longs;itas, & textura aliis de cau&longs;is irre<lb/>gularis, errores inducebant. </s>

<s>Porro nequis objiciat Regulam, ad quam probandam inventum <lb/>e&longs;t hoc experimentum, præ&longs;upponere corpora vel ab&longs;olute dura <lb/>e&longs;&longs;e, vel &longs;altem perfecte ela&longs;tica, cuju&longs;modi nulla reperiuntur in <lb/>compo&longs;itionibus naturalibus; addo quod Experimenta jam de&longs;crip<lb/>ta &longs;uccedunt in corporibus mollibus æque ac in duris, nimirum a <lb/>conditione duritiei neutiquam pendentia. </s>

<s>Nam &longs;i Regula illa in <lb/>corporibus non perfecte duris tentanda e&longs;t, debebit &longs;olummodo <lb/>reflexio minui in certa proportione pro quantitate vis Ela&longs;ticæ. </s>

<s>In <lb/>Theoria <emph type="italics"/>Wrenni<emph.end type="italics"/> & <emph type="italics"/>Hugenii<emph.end type="italics"/> corpora ab&longs;olute dura redeunt ab invi<lb/>cem cum velocitate congre&longs;&longs;us. </s>

<s>Certius id affirmabitur de perfecte <lb/>Ela&longs;ticis. </s>

<s>In imperfecte Ela&longs;ticis velocitas reditus minuenda e&longs;t &longs;i<lb/>mul cum vi Ela&longs;tica; propterea quod vis illa; (ni&longs;i ubi partes cor<lb/>porum ex congre&longs;&longs;u læduntur, vel exten&longs;ionem aliqualem qua&longs;i &longs;ub <lb/>malleo patiuntur,) certa ac determinata &longs;it (quantum &longs;entio) faci<lb/>atque corpora redire ab invicem cum velocitate relativa, quæ &longs;it ad <lb/>relativam velocitatem concur&longs;us in data ratione. </s>

<s>Id in pilis ex lana <lb/>arcte conglomerata & fortiter con&longs;tricta &longs;ic tentavi. </s>

<s>Primum demit<lb/>tendo Pendula & men&longs;urando reflexionem, inveni quantitatem vis <lb/>Ela&longs;ticæ; deinde per hanc vim determinavi reflexiones in aliis ca<lb/>&longs;ibus concur&longs;uum, & re&longs;pondebant Experimenta. </s>

<s>Redibant &longs;emper <lb/>pilæ ab invicem cum velocitate relativa, quæ e&longs;&longs;et ad velocitatem <lb/>relativam concur&longs;us ut 5 ad 9 circiter. </s>

<s>Eadem fere cum velocitate <lb/>redibant pilæ ex chalybe: aliæ ex &longs;ubere cum paulo minore: in vi<lb/>treis autem proportio erat 15 ad 16 circiter. </s>

<s>Atque hoc pacto Lex <lb/>tertia quoad ictus & reflexiones per Theoriam comprobata e&longs;t, quæ <lb/>cum experientia plane congruit. <pb pagenum="22"/><arrow.to.target n="note10"></arrow.to.target></s>

<s>Atque hoc pacto Lex <lb/>tertia quoad ictus & reflexiones per Theoriam comprobata e&longs;t, quæ <lb/>cum experientia plane congruit.

<arrow.to.target n="note10"></arrow.to.target></s>

<s>In Attractionibus rem &longs;ic breviter o&longs;tendo. </s>

<s>Corporibus duobus <lb/>quibu&longs;vis <emph type="italics"/>A, B<emph.end type="italics"/> &longs;e mutuo trahentibus, concipe ob&longs;taculum quodvis <lb/>interponi quo congre&longs;&longs;us eorum impediatur. </s>

<s>Si corpus alterutrum <lb/><emph type="italics"/>A<emph.end type="italics"/> magis trahitur ver&longs;us corpus alterum <emph type="italics"/>B,<emph.end type="italics"/> quam illud alterum <emph type="italics"/>B<emph.end type="italics"/><lb/>in prius <emph type="italics"/>A,<emph.end type="italics"/> ob&longs;taculum magis urgebitur pre&longs;&longs;ione corporis <emph type="italics"/>A<emph.end type="italics"/> quam <lb/>pre&longs;&longs;ione corporis <emph type="italics"/>B<emph.end type="italics"/>; proindeque non manebit in æquilibrio. </s>

<s>Præ<lb/>valebit pre&longs;&longs;io fortior, facietque ut &longs;y&longs;tema corporum duorum & <lb/>ob&longs;taculi moveatur in directum in partes ver&longs;us <emph type="italics"/>B,<emph.end type="italics"/> motuque in &longs;patiis <lb/>liberis &longs;emper accelerato abeat in infinitum. </s>

<s>Quod e&longs;t ab&longs;urdum & <lb/>Legi primæ contrarium. </s>

<s>Nam per Legem primam debebit &longs;y&longs;tema <lb/>per&longs;everare in &longs;tatu &longs;uo quie&longs;cendi vel movendi uniformiter in di<lb/>rectum, proindeque corpora æqualiter urgebunt ob&longs;taculum, & id<lb/>circo æqualiter trahentur in invicem. </s>

<s>Tentavi hoc in Magnete & <lb/>Ferro. </s>

<s>Si hæc in va&longs;culis propriis &longs;e&longs;e contingentibus &longs;eor&longs;im po<lb/>&longs;ita, in aqua &longs;tagnante juxta fluitent; neutrum propellet alterum, <lb/>&longs;ed æqualitate attractionis utrinque &longs;u&longs;tinebunt conatus in &longs;e mu<lb/>tuos, ac tandem in æquilibrio con&longs;tituta quie&longs;cent. </s>

<s>Sic etiam gravitas inter Terram & ejus partes, mutua e&longs;t. </s>

<s>Se<lb/>cetur Terra <emph type="italics"/>FI<emph.end type="italics"/> plano quovis <emph type="italics"/>EG<emph.end type="italics"/> in partes duas <emph type="italics"/>EGF<emph.end type="italics"/> & <emph type="italics"/>EGI:<emph.end type="italics"/><lb/>& æqualia erunt harum pondera in &longs;e mu<lb/><figure id="fig5"></figure><lb/>tuo. </s>

<arrow.to.target n="fig5"></arrow.to.target><lb/>tuo. </s>

<s>Nam &longs;i plano alio <emph type="italics"/>HK<emph.end type="italics"/> quod priori <lb/><emph type="italics"/>EG<emph.end type="italics"/> parallelum &longs;it, pars major <emph type="italics"/>EGI<emph.end type="italics"/> &longs;e<lb/>cetur in partes duas <emph type="italics"/>EGKH<emph.end type="italics"/> & <emph type="italics"/>HKI,<emph.end type="italics"/><lb/>quarum <emph type="italics"/>HKI<emph.end type="italics"/> æqualis &longs;it parti prius ab<lb/>&longs;ci&longs;&longs;æ <emph type="italics"/>EFG:<emph.end type="italics"/> manife&longs;tum e&longs;t quod pars <lb/>media <emph type="italics"/>EGKH<emph.end type="italics"/> pondere proprio in neu<lb/>tram partium extremarum propendebit, <lb/>&longs;ed inter utramque in æquilibrio, ut ita <lb/>dicam, &longs;u&longs;pendetur, & quie&longs;cet. </s>

<s>Pars autem extrema <emph type="italics"/>HKI<emph.end type="italics"/> toto <lb/>&longs;uo pondere incumbet in partem mediam, & urgebit illam in <lb/>partom alteram extremam <emph type="italics"/>EGF<emph.end type="italics"/>; ideoque vis qua partium <lb/><emph type="italics"/>HKI<emph.end type="italics"/> & <emph type="italics"/>EGKH<emph.end type="italics"/> &longs;umma <emph type="italics"/>EGI<emph.end type="italics"/> tendit ver&longs;us partem tertiam <lb/><emph type="italics"/>EGF,<emph.end type="italics"/> æqualis e&longs;t ponderi partis <emph type="italics"/>HKI,<emph.end type="italics"/> id e&longs;t ponderi partis ter<lb/>tiæ <emph type="italics"/>EGF.<emph.end type="italics"/> Et propterea pondera partium duarum <emph type="italics"/>EGI, EGF<emph.end type="italics"/><lb/>in &longs;e mutuo &longs;unt æqualia, uti volui o&longs;tendere. </s>

<s>Et ni&longs;i pondera illa <lb/>æqualia e&longs;&longs;ent, Terra tota in libero æthere fluitans ponderi majori <lb/>cederet, & ab eo fugiendo abiret in infinitum. </s>

<s>Et ni&longs;i pondera illa <lb/>æqualia e&longs;&longs;ent, Terra tota in libero æthere fluitans ponderi majori <lb/>cederet, & ab eo fugiendo abiret in infinitum. </s>

<s>Ut corpora in concur&longs;u & reflexione idem pollent, quorum ve<lb/>locitates &longs;unt reciproce ut vires in&longs;itæ: &longs;ic in movendis In&longs;tru<lb/>mentis Mechanicis agentia idem pollent & conatibus contrariis &longs;e <lb/>mutuo &longs;u&longs;tinent, quorum velocitates &longs;ecundum determinationem <pb pagenum="23"/>virium æ&longs;timatæ, &longs;unt reciproce ut vires. </s>

<pb pagenum="23"/>virium æ&longs;timatæ, &longs;unt reciproce ut vires. </s>

<s>Sie pondera æquipollent <lb/>ad movenda brachia Libræ, quæ o&longs;cillante Libra &longs;unt reciproce ut <lb/>eorum velocitates &longs;ur&longs;um & deor&longs;um: hoc e&longs;t, pondera, &longs;i recta <lb/>a&longs;cendunt & de&longs;cendunt, æquipollent, quæ &longs;unt reciproce ut pun<lb/>ctorum a quibus &longs;u&longs;penduntur di&longs;tantiæ ab axe Libræ; &longs;in planis <lb/>obliquis alii&longs;ve admotis ob&longs;taculis impedi<gap/>a a&longs;cendunt vel de&longs;cen<lb/>dunt oblique, æquipollent quæ &longs;unt reciproce ut a&longs;cen&longs;us & de&longs;cen<lb/>&longs;us, quatenus facti &longs;ecundum perpendiculum: id adeo ob determi<lb/>nationem gravitatis deor&longs;um. </s>

<s>Similiter in Trochlea &longs;eu Poly&longs;pa&longs;to <lb/>vis manus funem directe trahentis, quæ &longs;it ad pondus vel directe <lb/>vel oblique a&longs;cendens ut velocitas a&longs;cen&longs;us perpendicularis ad ve<lb/>locitatem manus funem trahentis, &longs;u&longs;tinebit pondus. </s>

<s>In Horolo<lb/>giis & &longs;imilibus in&longs;trumentis, quæ ex rotulis commi&longs;&longs;is con&longs;tructa <lb/>&longs;unt, vires contrariæ ad motum rotularum promovendum & impe<lb/>diendum, &longs;i &longs;unt reciproce ut velocitates partium rotularum in quas <lb/>imprimuntur, &longs;u&longs;tinebunt &longs;e mutuo. </s>

<s>Vis Cochleæ ad premendum <lb/>corpus e&longs;t ad vim manus manubrium circumagentis, ut circularis <lb/>velocitas manubrii ea in parte ubi a manu urgetur, ad velocitatem <lb/>progre&longs;&longs;ivam cochleæ ver&longs;us corpus pre&longs;&longs;um. </s>

<s>Vires quibus Cu<lb/>neus urget partes duas ligni fi&longs;&longs;i &longs;unt ad vim mallei in cuueum, ut <lb/>progre&longs;&longs;us cunei &longs;ecundum determinationem vis a malleo in ip&longs;um <lb/>impre&longs;&longs;æ, ad velocitatem qua partes <gap/>gni cedunt cuneo, &longs;ecundum <lb/>lineas faciebus cunei perpendiculares. </s>

<s>Et par e&longs;t ratio Machina<lb/>rum omnium. </s>

<s>Et par e&longs;t ratio Machina<lb/>rum omnium. </s>

<s>Harum efficacia & u&longs;us in eo &longs;olo con&longs;i&longs;tit, ut diminuendo velo<lb/>citatem augeamus vim, & contra: Unde &longs;olvitur in omni aptorum <lb/>in&longs;trumentorum genere Problema, <emph type="italics"/>Datum pondus data vi moven<lb/>di,<emph.end type="italics"/> aliamve datam re&longs;i&longs;tentiam vi data &longs;uperandi. </s>

<s>Nam &longs;i Ma<lb/>chinæ ita formentur, ut velocitates Agentis & Re&longs;i&longs;tentis &longs;ine reci<lb/>proce ut vires; Agens re&longs;i&longs;tentiam &longs;u&longs;tinebit: & majori cum veloci<lb/>tatum di&longs;paritate eandem vincet. </s>

<s>Certe &longs;i tanta &longs;ic velocitatum <lb/>di&longs;paritas, ut vincatur etiam re&longs;i&longs;tentia omnis, quæ tam ex conti<lb/>guorum & inter &longs;e labentium corporum attritione, quam ex con<lb/>tinuorum & ab invicem &longs;eparandorum cohæ&longs;ione & elevandorum <lb/>ponderibus orirj &longs;olet; &longs;uperata omni ea re&longs;i&longs;tentia, vis redun<lb/>dans accelerationem motus &longs;ibi proportionalem, partim in parti<lb/>bus machinæ, partim in corpore re&longs;i&longs;tente producet. </s>

<s>Ceterum <lb/>Mechanicam tractare non e&longs;t hujus in&longs;tituti. </s>

<s>Hi&longs;ce volui tan<lb/>tum o&longs;tendere, quam late pateat quamque certa &longs;it Lex tertia <lb/>Motus. </s>

<s>Nam &longs;i æ&longs;timetur Agentis actio ex ejus vi & veloci-</s><pb pagenum="24"/>

<s>Nam &longs;i æ&longs;timetur Agentis actio ex ejus vi & veloci-</s>

<s><arrow.to.target n="note11"></arrow.to.target><lb/>tate conjunctim; & &longs;imiliter Re&longs;i&longs;tentis reactio æ&longs;timetur conjun<lb/>ctim ex ejus partium &longs;ingularum velocitatibus & viribus re&longs;i&longs;tendi <lb/>ab earum attritione, cohæ&longs;ione, pondere, & acceleratione ori<lb/>undis; erunt actio & reactio, in omni in&longs;trumentorum u&longs;u, <lb/>&longs;ibi invicem &longs;emper æquales. </s>

<s>

<arrow.to.target n="note11"></arrow.to.target><lb/>tate conjunctim; & &longs;imiliter Re&longs;i&longs;tentis reactio æ&longs;timetur conjun<lb/>ctim ex ejus partium &longs;ingularum velocitatibus & viribus re&longs;i&longs;tendi <lb/>ab earum attritione, cohæ&longs;ione, pondere, & acceleratione ori<lb/>undis; erunt actio & reactio, in omni in&longs;trumentorum u&longs;u, <lb/>&longs;ibi invicem &longs;emper æquales. </s>

<s>Et quatenus actio propagatur per <lb/>in&longs;trumentum & ultimo imprimitur in corpus omne re&longs;i&longs;tens, <lb/>ejus ultima determinatio determinationi reactionis &longs;emper erit <lb/>contraria. <lb/><gap desc="hr tag"/></s>

<s><margin.target id="note11"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s><emph type="center"/>DE <lb/>MOTU CORPORUM <lb/>LIBER PRIMUS.<emph.end type="center"/><lb/><gap desc="hr tag"/></s>

<s><emph type="center"/>SECTIO I.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>De Methodo Rationum primarum & ultimarum, cujus ope &longs;equentia <lb/>demon&longs;trantur.<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="center"/>LEMMA I.<emph.end type="center"/></s>

<s><emph type="italics"/>QUantitates, ut & quantitatum rationes, quæ ad æqualitatem <lb/>tempore quovis finito con&longs;tanter tendunt, & ante finem tempo<lb/>ris illius propius ad invicem accedunt quam pro data quavis diffe<lb/>tia, fiunt ultimo æquales.<emph.end type="italics"/></s>

<s>Si negas; fiant ultimò inequales, & &longs;it earum ultima differentia <lb/><emph type="italics"/>D.<emph.end type="italics"/> Ergo nequeunt propius ad æqualitatem accedere quam pro <lb/>data differentia <emph type="italics"/>D:<emph.end type="italics"/> contra hypothe&longs;in. </s>

<s><emph type="center"/>LEMMA II.<emph.end type="center"/></s>

<arrow.to.target n="fig6"></arrow.to.target><lb/><emph type="italics"/>lelogramma<emph.end type="italics"/> aKbl, bLcm, cMdn, &c. <lb/><emph type="italics"/>Dein boru<gap/> parallelogr ammorum lati<lb/>tudo minuatur, & numerus augeatur <lb/>in infinitum: dico quod ultimæ rationes, <lb/>quas babent ad &longs;e invicem Figura in<lb/>&longs;cripta<emph.end type="italics"/> AKbLcMdD, <emph type="italics"/>circum&longs;cripta<emph.end type="italics"/><lb/>AalbmcndoE, <emph type="italics"/>& curvilinea<emph.end type="italics"/> AbcdE, <lb/><emph type="italics"/>&longs;unt rationes æqualitatis.<emph.end type="italics"/></s>

<s>Nam Figuræ in&longs;criptæ & circum&longs;criptæ differentia e&longs;t &longs;umma pa<lb/>rallelogrammorum <emph type="italics"/>Kl, Lm, Mn, Do,<emph.end type="italics"/> hoc e&longs;t (ob æquales om<lb/>nium ba&longs;es) rectangulum &longs;ub unius ba&longs;i <emph type="italics"/>Kb<emph.end type="italics"/> & altitudinum &longs;umma <lb/><emph type="italics"/>Aa,<emph.end type="italics"/> id e&longs;t, rectangulum <emph type="italics"/>ABla.<emph.end type="italics"/> Sed hoc rectangulum, eo quod <lb/>latitudo ejus <emph type="italics"/>AB<emph.end type="italics"/> in infinitum minuitur, fit minus quovis dato. </s>

<s><margin.target id="note11"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s>Er<lb/>go (per Lemma 1) Figura in&longs;cripta & circum&longs;cripta & multo magis <lb/>Figura curvilinea intermedia fiunt ultimo æquales. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s>

<s><emph type="center"/>DE <lb/>MOTU CORPORUM <lb/>LIBER PRIMUS.<emph.end type="center"/><lb/><gap desc="hr tag"/></s>

<s><emph type="center"/>LEMMA III.<emph.end type="center"/></s>

<s><emph type="italics"/>Eædem rationes ultimæ &longs;unt etiam rationes æqualitatis, ubi par al<lb/>lelogr ammorum latitudines<emph.end type="italics"/> AB, BC, CD, &c. <emph type="italics"/>&longs;unt inæquales, <lb/>& omnes minuuntur in infinitum.<emph.end type="italics"/></s>

<s>Sit enim <emph type="italics"/>AF<emph.end type="italics"/> æqualis latitudini maximæ, & compleatur paralle<lb/>logrammum <emph type="italics"/>FAaf.<emph.end type="italics"/> Hoc erit majus quam differentia Figuræ in<lb/>&longs;criptæ & Figuræ circum&longs;criptæ; at latitudine &longs;ua <emph type="italics"/>AF<emph.end type="italics"/> in infinitum <lb/>diminuta, minus fiet quam datum quodvis rectangulum. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;umma ultima parallelogrammorum evane&longs;centium <lb/>coincidit omni ex parte cum Figura curvilinea. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et multo magis Figura rectilinea, quæ chordis evane&longs;-

<arrow.to.target n="note12"></arrow.to.target><lb/>centium arcuum <emph type="italics"/>ab, bc, cd, &c.<emph.end type="italics"/> comprehenditur, coincidit ultimo <lb/>cum Figura curvilinea. </s>

<s><margin.target id="note12"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Ut & Figura rectilinea circum&longs;cripta quæ tangentibus <lb/>eorundem arcuum comprehenditur. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Et propterea hæ Figuræ ultimæ (quoad perimetros <emph type="italics"/>acE,<emph.end type="italics"/>) <lb/>non &longs;unt rectilineæ, &longs;ed rectilinearum limites curvilinei. </s>

<s><emph type="center"/>LEMMA IV.<emph.end type="center"/></s>

<s><emph type="italics"/>Si in duabus Figuris<emph.end type="italics"/> AacE, PprT, <emph type="italics"/>in&longs;cribantur (ut &longs;upra) duæ <lb/>parallelogrammorum &longs;eries, &longs;itque idem amborum numerus, & ubi <lb/>latitudines in infinitum diminuuntur, rationes ultimæ parallelo<lb/>grammorum in una Figura ad parallelogramma in altera, &longs;ingulorum <lb/>ad fingula, &longs;int eædem; dico quod Figuræ duæ<emph.end type="italics"/> AacE, PprT, <lb/><emph type="italics"/>&longs;unt ad invicem in eadem illa ratione.<emph.end type="italics"/></s>

<s>Etenim ut &longs;unt parallelogramma &longs;ingula ad &longs;ingula, ita (compo<lb/>nendo) fit &longs;umma omnium ad &longs;ummam omnium, & ita Figura ad <lb/>Figuram; exi&longs;tente nimirum Figura priore (per Lemma 111) ad &longs;um<lb/>mam priorem, & Figura po&longs;teriore ad &longs;ummam po&longs;teriorem in ra<lb/>tione æqualitatis. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> Hinc &longs;i duæ cuju&longs;cunque generis quantitates in eundem <lb/>partium numerum utcunque dividantur; & partes illæ, ubi numerus <lb/>earum augetur & magnitudo diminuitur in infinitum, datam obti<lb/>neant rationem ad invicem, prima ad primam, &longs;ecunda ad &longs;ecundam, <lb/>cæteræque &longs;uo ordine ad cæteras: erunt tota ad invicem in eadem <lb/>illa data ratione. </s>

<s><emph type="center"/>SECTIO I.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>De Methodo Rationum primarum & ultimarum, cujus ope &longs;equentia <lb/>demon&longs;trantur.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Nam &longs;i in Lemmatis hujus Figuris &longs;umantur pa-

<s><emph type="center"/>LEMMA I.<emph.end type="center"/></s>

<pb pagenum="27"/>rallelogramma inter &longs;e ut partes, &longs;ummæ partium &longs;emper erunt ut <lb/>&longs;ummæ parallelogrammorum; atque adeo, ubi partium & paralle<lb/>logrammorum numerus augetur & magnitudo diminuitur in infini<lb/>tum, in ultima ratione parallelogrammi ad parallelogrammum, id <lb/>e&longs;t (per hypothe&longs;in) in ultima ratione partis ad partem. </s>

<s><emph type="center"/>LEMMA V.<emph.end type="center"/></s>

<s><emph type="italics"/>Similium Figurarum latera omnia, quæ &longs;ibi mutuo re&longs;pondent, &longs;unt <lb/>proportionalia, tam curvilinea quam rectilinea; & areæ &longs;unt in <lb/>duplicata ratione laterum.<emph.end type="italics"/></s>

<s><emph type="center"/>LEMMA VI.<emph.end type="center"/></s>

<s><emph type="italics"/>Si arcus quilibet po&longs;itione datus<emph.end type="italics"/> AB <emph type="italics"/>&longs;ub-<emph.end type="italics"/><lb/>

<arrow.to.target n="fig7"></arrow.to.target><lb/><emph type="italics"/>tendatur chorda<emph.end type="italics"/> AB, <emph type="italics"/>& in puncto <lb/>aliquo<emph.end type="italics"/> A, <emph type="italics"/>in medio curvaturæ continuæ, <lb/>tangatur a recta utrinque producta<emph.end type="italics"/><lb/>AD; <emph type="italics"/>dein puncta<emph.end type="italics"/> A, B <emph type="italics"/>ad invicem <lb/>accedant & coëant; dico quod angulus<emph.end type="italics"/><lb/>BAD, <emph type="italics"/>&longs;ub chorda & tangente conten<lb/>tus, minuetur in infinitum & ultimo e<lb/>vane&longs;cet.<emph.end type="italics"/></s>

<s>Nam &longs;i angulus ille non evane&longs;cit, continebit arcus <emph type="italics"/>AB<emph.end type="italics"/> cum tan<lb/>gente <emph type="italics"/>AD<emph.end type="italics"/> angulum rectilineo æqualem, & propterea curvatura ad <lb/>ad punctum <emph type="italics"/>A<emph.end type="italics"/> non erit continua, contra hypothe&longs;in. </s>

<s><emph type="center"/>LEMMA VII.<emph.end type="center"/></s>

<s><emph type="italics"/>Ii&longs;dem po&longs;itis; dico quod ultima ratio arcus, chordæ, & tangentis <lb/>ad invicem est ratio æqualitatis.<emph.end type="italics"/></s>

<s>Nam dum punctum <emph type="italics"/>B<emph.end type="italics"/> ad punctum <emph type="italics"/>A<emph.end type="italics"/> accedit, intelligantur &longs;emper <lb/><emph type="italics"/>AB<emph.end type="italics"/> & <emph type="italics"/>AD<emph.end type="italics"/> ad puncta longinqua <emph type="italics"/>b<emph.end type="italics"/> ac <emph type="italics"/>d<emph.end type="italics"/> product, & &longs;ecanti <emph type="italics"/>BD<emph.end type="italics"/><lb/>parallela agatur <emph type="italics"/>bd.<emph.end type="italics"/> Sitque arcus <emph type="italics"/>Ab<emph.end type="italics"/> &longs;emper &longs;imilis arcui <emph type="italics"/>AB.<emph.end type="italics"/><lb/>Et punctis <emph type="italics"/>A, B<emph.end type="italics"/> coeuntibus, angulus <emph type="italics"/>dAb,<emph.end type="italics"/> per Lemma &longs;uperius, <lb/>evane&longs;cet; adeoque rectæ &longs;emper &longs;initæ <emph type="italics"/>Ab, Ad<emph.end type="italics"/> & arcus interme<lb/>dius <emph type="italics"/>Ab<emph.end type="italics"/> coincident, & propterea æquales erunt. </s>

<s>Unde & hi&longs;ce <lb/>&longs;emper proportionales rectæ <emph type="italics"/>AB, AD,<emph.end type="italics"/> & arcus intermedius <emph type="italics"/>AB<emph.end type="italics"/>

<s><emph type="center"/>LEMMA II.<emph.end type="center"/></s>

<arrow.to.target n="note13"></arrow.to.target><lb/>evane&longs;cent, & rationem ultimam habebunt æqualitatis. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s>

<s><margin.target id="note13"></margin.target>DE MOTU <lb/>CORPORUM</s>

<arrow.to.target n="fig8"></arrow.to.target><lb/>untem perpetuo &longs;ecans in <emph type="italics"/>F,<emph.end type="italics"/><lb/>hæc <emph type="italics"/>BF<emph.end type="italics"/> ultimo ad arcum e<lb/>vane&longs;centem <emph type="italics"/>AB<emph.end type="italics"/> rationem <lb/>habebit æqualitatis, eo quod <lb/>completo parallelogrammo <emph type="italics"/>AFBD<emph.end type="italics"/> rationem &longs;emper habet æqua<lb/>litatis ad <emph type="italics"/>AD.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et &longs;i per <emph type="italics"/>B<emph.end type="italics"/> & <emph type="italics"/>A<emph.end type="italics"/> ducantur plures rectæ <emph type="italics"/>BE, BD, AF, <lb/>AG,<emph.end type="italics"/> &longs;ecantes tangentem <emph type="italics"/>AD<emph.end type="italics"/> & ip&longs;ius parallelam <emph type="italics"/>BF<emph.end type="italics"/>; ratio ulti<lb/>ma ab&longs;ci&longs;&longs;arum omnium <emph type="italics"/>AD, AE, BF, BG,<emph.end type="italics"/> chordæque & ar<lb/>cus <emph type="italics"/>AB<emph.end type="italics"/> ad invicem erit ratio æqualitatis. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Et propterea hæ omnes lineæ, in omni de rationibus ul<lb/>timis argumentatione, pro &longs;e invicem u&longs;urpari po&longs;&longs;unt. </s>

<s><emph type="center"/>LEMMA VIII.<emph.end type="center"/></s>

<s><emph type="italics"/>Si rectæ datæ<emph.end type="italics"/> AR, BR <emph type="italics"/>cum arcu<emph.end type="italics"/> AB, <emph type="italics"/>chorda<emph.end type="italics"/> AB <emph type="italics"/>& tangente<emph.end type="italics"/><lb/>AD, <emph type="italics"/>triangula tria<emph.end type="italics"/> ARB, ARB, ARD <emph type="italics"/>con&longs;tituunt, dein <lb/>puncta<emph.end type="italics"/> A, B <emph type="italics"/>accedunt ad invicem: dico quod ultima forma <lb/>triangulorum evane&longs;centium est &longs;imilitudinis, & ultima ratio <lb/>æqualitatis.<emph.end type="italics"/></s>

<s>Nam dum punctum <emph type="italics"/>B<emph.end type="italics"/> ad punctum <emph type="italics"/>A<emph.end type="italics"/><lb/>

<arrow.to.target n="fig9"></arrow.to.target><lb/>accedit, <expan abbr="intelligãtur">intelligantur</expan> &longs;emper <emph type="italics"/>AB, AD, AR<emph.end type="italics"/><lb/>ad puncta longinqua <emph type="italics"/>b, d<emph.end type="italics"/> & <emph type="italics"/>r<emph.end type="italics"/> produci, <lb/>ip&longs;ique <emph type="italics"/>RD<emph.end type="italics"/> parallela agi <emph type="italics"/>rbd,<emph.end type="italics"/> & arcui <lb/><emph type="italics"/>AB<emph.end type="italics"/> &longs;imilis &longs;emper &longs;it arcus <emph type="italics"/>Ab.<emph.end type="italics"/> Et coe<lb/>untibus punctis <emph type="italics"/>A, B,<emph.end type="italics"/> angulus <emph type="italics"/>bAd<emph.end type="italics"/> eva<lb/>ne&longs;cet, & propterea triangula tria &longs;emper <lb/>finita <emph type="italics"/>rAb, rAb, rAd<emph.end type="italics"/> coincident, &longs;unt<lb/>que eo nomine &longs;imilia & æqualia. </s>

<s>Unde <lb/>& hi&longs;ce &longs;emper &longs;imilia & proportionalia <lb/><emph type="italics"/>RAB, RAB, RAD<emph.end type="italics"/> &longs;ient ultimo &longs;ibi <lb/>invicem &longs;imilia & æqualia. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> Et hinc triangula illa, in omni de rationibus ultimis argu<lb/>mentatione, pro &longs;e invicem u&longs;urpari po&longs;&longs;unt. </s>

<s><emph type="center"/>LEMMA IX.<emph.end type="center"/></s>

<s><emph type="italics"/>Si recta<emph.end type="italics"/> AE <emph type="italics"/>& curva<emph.end type="italics"/> ABC <emph type="italics"/>po&longs;itione datæ &longs;e mutuo &longs;ecent in <lb/>angulo dato<emph.end type="italics"/> A, <emph type="italics"/>& ad rectam illam in alio dato angulo ordina<lb/>tim applicentur<emph.end type="italics"/> BD, CE, <emph type="italics"/>curvæ occurrentes in<emph.end type="italics"/> B, C; <emph type="italics"/>dein <lb/>puncta<emph.end type="italics"/> B, C <emph type="italics"/>&longs;imul accedant ad punctum<emph.end type="italics"/> A: <emph type="italics"/>dico quod areæ tri<lb/>angulorum<emph.end type="italics"/> ABD, ACE <emph type="italics"/>erunt ultimo ad invicem in duplicata <lb/>ratione laterum.<emph.end type="italics"/></s>

<s>Etenim dum puncta <emph type="italics"/>B, C<emph.end type="italics"/> acce<lb/>

<arrow.to.target n="fig10"></arrow.to.target><lb/>dunt ad punctum <emph type="italics"/>A,<emph.end type="italics"/> intelligatur <lb/>&longs;emper <emph type="italics"/>AD<emph.end type="italics"/> produci ad puncta lon<lb/>ginqua <emph type="italics"/>d<emph.end type="italics"/> & <emph type="italics"/>e,<emph.end type="italics"/> ut &longs;int <emph type="italics"/>Ad, Ae<emph.end type="italics"/> ip<lb/>&longs;is <emph type="italics"/>AD, AE<emph.end type="italics"/> proportionales, & e<lb/>rigantur ordinatæ <emph type="italics"/>db, ec<emph.end type="italics"/> ordina<lb/>tis <emph type="italics"/>DB, EC<emph.end type="italics"/> parallelæ quæ occur<lb/>rant ip&longs;is <emph type="italics"/>AB, AC<emph.end type="italics"/> productis in <lb/><emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>c.<emph.end type="italics"/> Duci intelligatur, tum curva <lb/><emph type="italics"/>Abc<emph.end type="italics"/> ip&longs;i <emph type="italics"/>ABC<emph.end type="italics"/> &longs;imilis, tum recta <lb/><emph type="italics"/>Ag,<emph.end type="italics"/> quæ tangat curvam utramque <lb/>in <emph type="italics"/>A,<emph.end type="italics"/> & &longs;ecet ordinatim applica<lb/>tas <emph type="italics"/>DB, EC, db, ec<emph.end type="italics"/> in <emph type="italics"/>F, G, f, g.<emph.end type="italics"/><lb/>Tum manente longitudine <emph type="italics"/>Ae<emph.end type="italics"/> coeant puncta <emph type="italics"/>B, C<emph.end type="italics"/> cum puncto <emph type="italics"/>A<emph.end type="italics"/>; <lb/>& angulo <emph type="italics"/>cAg<emph.end type="italics"/> evane&longs;cente, coincident areæ curvilineæ <emph type="italics"/>Abd, Ace<emph.end type="italics"/><lb/>cum rectilineis <emph type="italics"/>Afd, Age:<emph.end type="italics"/> adeoque (per Lemma v) erunt in dupli<lb/>cata ratione laterum <emph type="italics"/>Ad, A<gap/>:<emph.end type="italics"/> Sed his areis proportionales &longs;emper <lb/>&longs;unt areæ <emph type="italics"/>ABD, ACE,<emph.end type="italics"/> & his lateribus latera <emph type="italics"/>AD, AE.<emph.end type="italics"/> Ergo & <lb/>areæ <emph type="italics"/>ABD, ACE<emph.end type="italics"/> &longs;unt ultimo in duplicata ratione laterum <emph type="italics"/>AD, <lb/>AE. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s>

<s><emph type="center"/>LEMMA X.<emph.end type="center"/></s>

<s><emph type="italics"/>Spatia quæ corpus urgente quacunque Vi finita de&longs;cribit, five Vis <lb/>illa determinata & immutabilis &longs;it, five eadem continuo auge<lb/>atur vel continuo diminuatur, &longs;unt ip&longs;o motus initio in duplica<lb/>ta ratione Temporum.<emph.end type="italics"/></s>

<s>Exponantur tempora per lineas <emph type="italics"/>AD, AE,<emph.end type="italics"/> & velocitates genitæ <lb/>per ordinatas <emph type="italics"/>DB, EC<emph.end type="italics"/>; & &longs;patia his velocitatibus de&longs;cripta, erunt <lb/>ut areæ <emph type="italics"/>ABD, ACE<emph.end type="italics"/> his ordinatis de&longs;criptæ, hoc e&longs;t, ip&longs;o motus <lb/>initio (per Lemma IX) in duplicata ratione remporum <emph type="italics"/>AD, AE. <lb/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/>

<arrow.to.target n="note14"></arrow.to.target></s>

<s><margin.target id="note14"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Et hinc facile colligitur, quod corporum &longs;imiles &longs;imi<lb/>lium Figurarum partes temporibus proportionalibus de&longs;cribentium <lb/>Errores, qui viribus quibu&longs;vis æqualibus ad corpora &longs;imiliter ap<lb/>plicatis generantur, & men&longs;urantur per di&longs;tantias corporum a Fi<lb/>gurarum &longs;imilium locis illis ad quæ corpora eadem temporibus ii&longs;<lb/>dem proportionalibus ab&longs;que viribus i&longs;tis pervenirent, &longs;unt ut qua<lb/>drata temporum in quibus generantur quam proxime. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Errores autem qui viribus proportionalibus ad &longs;imiles <lb/>Figurarum &longs;imilium partes &longs;imiliter applicatis generantur, &longs;unt ut <lb/>vires & quadrata temporum conjunctim. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Idem intelligendum e&longs;t de &longs;patiis quibu&longs;vis quæ corpo<lb/>ra urgentibus diver&longs;is viribus de&longs;cribunt. </s>

<s><emph type="italics"/>Si in Figura quavis<emph.end type="italics"/> AacE, <emph type="italics"/>rectis<emph.end type="italics"/> Aa, AE <emph type="italics"/>& curva<emph.end type="italics"/> acE <emph type="italics"/>com <lb/>preben&longs;a, in&longs;cribantur parallelogramma quotcunque<emph.end type="italics"/> Ab, Bc, Cd <lb/>&c. <emph type="italics"/>&longs;ub ba&longs;ibus<emph.end type="italics"/> AB, BC, CD, &c. <emph type="italics"/>æqualibus, & lateribu<emph.end type="italics"/><lb/>Bb, Cc, Dd, &c. <emph type="italics"/>Figuræ lateri<emph.end type="italics"/> Aa <emph type="italics"/>pa<lb/>rallelis contenta; & compleantur paral-<emph.end type="italics"/><lb/><figure id="fig6"></figure><lb/><emph type="italics"/>lelogramma<emph.end type="italics"/> aKbl, bLcm, cMdn, &c. <lb/><emph type="italics"/>Dein boru<gap/> parallelogr ammorum lati<lb/>tudo minuatur, & numerus augeatur <lb/>in infinitum: dico quod ultimæ rationes, <lb/>quas babent ad &longs;e invicem Figura in<lb/>&longs;cripta<emph.end type="italics"/> AKbLcMdD, <emph type="italics"/>circum&longs;cripta<emph.end type="italics"/><lb/>AalbmcndoE, <emph type="italics"/>& curvilinea<emph.end type="italics"/> AbcdE, <lb/><emph type="italics"/>&longs;unt rationes æqualitatis.<emph.end type="italics"/></s>

<s>Hæc &longs;unt, ip&longs;o motus ini<lb/>tio, ut vires & quadrata temporum conjunctim. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Ideoque vires &longs;unt ut &longs;patia, ip&longs;o motus initio, de&longs;cripta <lb/>directe & quadrata temporum inver&longs;e. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Et quadrata temporum &longs;unt ut de&longs;cripta &longs;patia directe <lb/>& vires inver&longs;e. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Si quantitates indeterminatæ diver&longs;orum generum conferantur <lb/>inter &longs;e, & earum aliqua dicatur e&longs;&longs;e ut e&longs;t alia quævis directe vel <lb/>inver&longs;e: &longs;en&longs;us e&longs;t, quod prior augetur vel diminuitur in eadem <lb/>ratione cum po&longs;teriore, vel cum ejus reciproca. </s>

<s><emph type="center"/>LEMMA III.<emph.end type="center"/></s>

<s>Et &longs;i earum aliqua <lb/>dicatur e&longs;&longs;e ut &longs;unt aliæ duæ vel plures directe vel inver&longs;e: &longs;en&longs;us <lb/>e&longs;t, quod prima augetur vel diminuitur in ratione quæ componitur <lb/>ex rationibus in quibus aliæ vel aliarum reciprocæ augentur vel di<lb/>minuuntur. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;umma ultima parallelogrammorum evane&longs;centium <lb/>coincidit omni ex parte cum Figura curvilinea. </s>

<s>Ut &longs;i A dicatur e&longs;&longs;e ut B directe & C directe & D in<lb/>ver&longs;e: &longs;en&longs;us e&longs;t, quod A augetur vel diminuitur in eadem ratione <lb/>cum BXCX1/D, hoc e&longs;t, quod A & (BC/D) &longs;unt ad invicem in ratio<lb/>ne data. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et multo magis Figura rectilinea, quæ chordis evane&longs;-<pb pagenum="26"/><arrow.to.target n="note12"></arrow.to.target><lb/>centium arcuum <emph type="italics"/>ab, bc, cd, &c.<emph.end type="italics"/> comprehenditur, coincidit ultimo <lb/>cum Figura curvilinea. </s>

<s><emph type="center"/>LEMMA XI.<emph.end type="center"/></s>

<s><margin.target id="note12"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s><emph type="italics"/>Subten&longs;a evane&longs;cens anguli contactus, in curvis omnibus curvatu<lb/>ram finitam ad punctum contactus habentibus, est ultimo in ra<lb/>tione duplicata &longs;ubten&longs;æ arcus contermini.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Ut & Figura rectilinea circum&longs;cripta quæ tangentibus <lb/>eorundem arcuum comprehenditur. </s>

<pb pagenum="31"/>concurrentes in <emph type="italics"/>G<emph.end type="italics"/>; dein accedant puncta <emph type="italics"/>D, B, G,<emph.end type="italics"/> ad puncta <emph type="italics"/>d, b, g,<emph.end type="italics"/><lb/>&longs;itque <emph type="italics"/>J<emph.end type="italics"/> inter&longs;ectio linearum <emph type="italics"/>BG, AG<emph.end type="italics"/> ultimo facta ubi puncta <emph type="italics"/>D, B<emph.end type="italics"/><lb/>accedunt u&longs;que ad <emph type="italics"/>A.<emph.end type="italics"/> Manife&longs;tum e&longs;t quod di&longs;tantia <emph type="italics"/>GJ<emph.end type="italics"/> minor <lb/>e&longs;&longs;e pote&longs;t quam a&longs;&longs;ignata quævis. </s>

<s><emph type="center"/>LEMMA IV.<emph.end type="center"/></s>

<s>E&longs;t autem (ex natura circulorum <lb/>per puncta <emph type="italics"/>ABG, Abg<emph.end type="italics"/> tran&longs;euntium) <emph type="italics"/>ABquad.<emph.end type="italics"/><lb/>

<arrow.to.target n="fig11"></arrow.to.target><lb/>æquale <emph type="italics"/>AGXBD,<emph.end type="italics"/> & <emph type="italics"/>Ab quad.<emph.end type="italics"/> æquale <emph type="italics"/>AgXbd,<emph.end type="italics"/><lb/>adeoque ratio <emph type="italics"/>AB quad.<emph.end type="italics"/> ad <emph type="italics"/>Ab quad.<emph.end type="italics"/> compo<lb/>nitur ex rationibus <emph type="italics"/>AG<emph.end type="italics"/> ad <emph type="italics"/>Ag<emph.end type="italics"/> & <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>bd.<emph.end type="italics"/><lb/>Sed quoniam <emph type="italics"/>GJ<emph.end type="italics"/> a&longs;&longs;umi pote&longs;t minor longitu<lb/>dine quavis a&longs;&longs;ignata, fieri pote&longs;t ut ratio <emph type="italics"/>AG<emph.end type="italics"/><lb/>ad <emph type="italics"/>Ag<emph.end type="italics"/> minus differat a ratione æqualitatis quam <lb/>pro differentia quavis a&longs;&longs;ignata, adeoque ut ra<lb/>tio <emph type="italics"/>AB quad.<emph.end type="italics"/> ad <emph type="italics"/>Ab quad.<emph.end type="italics"/> minus differat a ra<lb/>tione <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>bd<emph.end type="italics"/> quam pro differentia quavis <lb/>a&longs;&longs;ignata. </s>

<s>E&longs;t ergo, per Lemma 1, ratio ultima <lb/><emph type="italics"/>AB quad.<emph.end type="italics"/> ad <emph type="italics"/>Ab quad.<emph.end type="italics"/> æqualis rationi ultimæ <lb/><emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>bd. </s>

<s>Nam &longs;i in Lemmatis hujus Figuris &longs;umantur pa-<pb pagenum="27"/>rallelogramma inter &longs;e ut partes, &longs;ummæ partium &longs;emper erunt ut <lb/>&longs;ummæ parallelogrammorum; atque adeo, ubi partium & paralle<lb/>logrammorum numerus augetur & magnitudo diminuitur in infini<lb/>tum, in ultima ratione parallelogrammi ad parallelogrammum, id <lb/>e&longs;t (per hypothe&longs;in) in ultima ratione partis ad partem. </s>

<s><emph type="center"/>LEMMA V.<emph.end type="center"/></s>

<s><emph type="italics"/>Cas.<emph.end type="italics"/> 2. Inclinetur jam <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>AD<emph.end type="italics"/> in angulo <lb/>quovis dato, & eadem &longs;emper erit ratio ultima <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>bd<emph.end type="italics"/> quæ <lb/>prius, adeoque eadem ae <emph type="italics"/>AB quad.<emph.end type="italics"/> ad <emph type="italics"/>Ab quad. </s>

<s><emph type="center"/>LEMMA VI.<emph.end type="center"/></s>

<s><emph type="italics"/>Si arcus quilibet po&longs;itione datus<emph.end type="italics"/> AB <emph type="italics"/>&longs;ub-<emph.end type="italics"/><lb/><figure id="fig7"></figure><lb/><emph type="italics"/>tendatur chorda<emph.end type="italics"/> AB, <emph type="italics"/>& in puncto <lb/>aliquo<emph.end type="italics"/> A, <emph type="italics"/>in medio curvaturæ continuæ, <lb/>tangatur a recta utrinque producta<emph.end type="italics"/><lb/>AD; <emph type="italics"/>dein puncta<emph.end type="italics"/> A, B <emph type="italics"/>ad invicem <lb/>accedant & coëant; dico quod angulus<emph.end type="italics"/><lb/>BAD, <emph type="italics"/>&longs;ub chorda & tangente conten<lb/>tus, minuetur in infinitum & ultimo e<lb/>vane&longs;cet.<emph.end type="italics"/></s>

<s><emph type="italics"/>Cas.<emph.end type="italics"/> 3. Et quamvis angulus <emph type="italics"/>D<emph.end type="italics"/> non detur, &longs;ed recta <emph type="italics"/>BD<emph.end type="italics"/> ad da<lb/>tum punctum convergente, vel alia quacunque lege con&longs;tituatur; <lb/>tamen anguli <emph type="italics"/>D, d<emph.end type="italics"/> communi lege con&longs;tituti ad æqualitatem &longs;emper <lb/>vergent & propius accedent ad invicem quam pro differentia qua<lb/>vis a&longs;&longs;ignata, adeoque ultimo æquales erunt, per Lem. <gap/> & prop<lb/>terea lineæ <emph type="italics"/>BD, bd<emph.end type="italics"/> &longs;unt in eadem ratione ad invicem ac prius. <lb/><emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Unde eum tangentes <emph type="italics"/>AD, Ad,<emph.end type="italics"/> arcus <emph type="italics"/>AB, Ab,<emph.end type="italics"/> & eo<lb/>rum &longs;inus <emph type="italics"/>BC, bc<emph.end type="italics"/> fiant ultimo chordis <emph type="italics"/>AB, Ab<emph.end type="italics"/> æquales; erunt <lb/>etiam illorum quadrata ultimo ut &longs;ubten&longs;æ <emph type="italics"/>BD, bd.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Eorundem quadrata &longs;unt etiam ultimo ut &longs;unt arcuum <lb/>&longs;agittæ quæ chordas bi&longs;ecant & ad datum punctum conver gunt. </s>

<s><emph type="center"/>LEMMA VII.<emph.end type="center"/></s>

<s><lb/>Nam &longs;agittæ illæ &longs;unt ut &longs;ubten&longs;æ <emph type="italics"/>BD, bd.<emph.end type="italics"/></s>

<s><emph type="italics"/>Ii&longs;dem po&longs;itis; dico quod ultima ratio arcus, chordæ, & tangentis <lb/>ad invicem est ratio æqualitatis.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Ideoque &longs;agitta e&longs;t in duplicata ratione temporis quo <lb/>corpus data velocitate de&longs;cribit arcum. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Triangula rectilinea <emph type="italics"/>ADB, Adb<emph.end type="italics"/> &longs;unt ultimo in tripli<lb/>cata ratione laterum <emph type="italics"/>AD, Ad,<emph.end type="italics"/> inque &longs;e&longs;quiplicata laterum <emph type="italics"/>DB, <lb/>db<emph.end type="italics"/>; utpote in compo&longs;ita ratione laterum <emph type="italics"/>AD,<emph.end type="italics"/> & <emph type="italics"/>DB, Ad<emph.end type="italics"/> & <emph type="italics"/>db<emph.end type="italics"/><lb/>exi&longs;tentia. </s>

<s>Unde & hi&longs;ce <lb/>&longs;emper proportionales rectæ <emph type="italics"/>AB, AD,<emph.end type="italics"/> & arcus intermedius <emph type="italics"/>AB<emph.end type="italics"/><pb pagenum="28"/><arrow.to.target n="note13"></arrow.to.target><lb/>evane&longs;cent, & rationem ultimam habebunt æqualitatis. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s>

<s>Sic & triangula <emph type="italics"/>ABC, Abc<emph.end type="italics"/> &longs;unt ultimo in triplicata <lb/>ratione laterum <emph type="italics"/>BC, bc.<emph.end type="italics"/> Rationem vero Se&longs;quiplicatam voco tri<lb/>plicatæ &longs;ubduplicatam, quæ nempe ex &longs;implici & &longs;ubduplicata com<lb/>ponitur, quamque alias Se&longs;quialteram dicunt. </s>

<s><margin.target id="note13"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s>

<arrow.to.target n="note15"></arrow.to.target></s>

<s><margin.target id="note15"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Et quoniam <emph type="italics"/>DB, db<emph.end type="italics"/> &longs;unt ultimo parallelæ & in dupli<lb/>cata ratione ip&longs;arum <emph type="italics"/>AD, Ad:<emph.end type="italics"/> erunt areæ ultimæ curvilineæ <emph type="italics"/>ADB, <lb/>Adb<emph.end type="italics"/> (ex natura Parabolæ) duæ tertiæ partes triangulorum rectili<lb/>neorum <emph type="italics"/>ADB, Adb<emph.end type="italics"/>; & &longs;egmenta <emph type="italics"/>AB, Ab<emph.end type="italics"/> partes tertiæ eo<lb/>rundem triangulorum. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Unde &longs;i per <emph type="italics"/>B<emph.end type="italics"/> ducatur tangenti parallela <emph type="italics"/>BF,<emph.end type="italics"/> rectam <lb/>quamvis <emph type="italics"/>AF<emph.end type="italics"/> per <emph type="italics"/>A<emph.end type="italics"/> tran&longs;e<lb/><figure id="fig8"></figure><lb/>untem perpetuo &longs;ecans in <emph type="italics"/>F,<emph.end type="italics"/><lb/>hæc <emph type="italics"/>BF<emph.end type="italics"/> ultimo ad arcum e<lb/>vane&longs;centem <emph type="italics"/>AB<emph.end type="italics"/> rationem <lb/>habebit æqualitatis, eo quod <lb/>completo parallelogrammo <emph type="italics"/>AFBD<emph.end type="italics"/> rationem &longs;emper habet æqua<lb/>litatis ad <emph type="italics"/>AD.<emph.end type="italics"/></s>

<s>Et inde hæ areæ & hæc &longs;egmenta erunt in <lb/>triplicata ratione tum tangentium <emph type="italics"/>AD, Ad<emph.end type="italics"/>; tum chordarum & <lb/>arcuum <emph type="italics"/>AB, Ab.<emph.end type="italics"/></s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Cæterum in his omnibus &longs;upponimus angulum contactus nec in<lb/>finite majorem e&longs;&longs;e angulis contactuum, quos Circuli continent cum <lb/>tangentibus &longs;uis, nec ii&longs;dem infinite minorem; hoc e&longs;t, curvaturam <lb/>ad punctum <emph type="italics"/>A,<emph.end type="italics"/> nec infinite parvam e&longs;&longs;e nec infinite magnam, &longs;eu <lb/>intervallum <emph type="italics"/>AJ<emph.end type="italics"/> finitæ e&longs;&longs;e magnitudinis. </s>

<s><emph type="center"/>LEMMA VIII.<emph.end type="center"/></s>

<s>Capi enim pote&longs;t <emph type="italics"/>DB<emph.end type="italics"/><lb/>ut <emph type="italics"/>AD<emph type="sup"/>3<emph.end type="sup"/>:<emph.end type="italics"/> quo in ca&longs;u Circulus nullus per punctum <emph type="italics"/>A<emph.end type="italics"/> inter tangen<lb/>tem <emph type="italics"/>AD<emph.end type="italics"/> & curvam <emph type="italics"/>AB<emph.end type="italics"/> duci pote&longs;t, proindeque angulus contactus <lb/>erit infinite minor Circularibus. </s>

<s>Nam dum punctum <emph type="italics"/>B<emph.end type="italics"/> ad punctum <emph type="italics"/>A<emph.end type="italics"/><lb/><figure id="fig9"></figure><lb/>accedit, <expan abbr="intelligãtur">intelligantur</expan> &longs;emper <emph type="italics"/>AB, AD, AR<emph.end type="italics"/><lb/>ad puncta longinqua <emph type="italics"/>b, d<emph.end type="italics"/> & <emph type="italics"/>r<emph.end type="italics"/> produci, <lb/>ip&longs;ique <emph type="italics"/>RD<emph.end type="italics"/> parallela agi <emph type="italics"/>rbd,<emph.end type="italics"/> & arcui <lb/><emph type="italics"/>AB<emph.end type="italics"/> &longs;imilis &longs;emper &longs;it arcus <emph type="italics"/>Ab.<emph.end type="italics"/> Et coe<lb/>untibus punctis <emph type="italics"/>A, B,<emph.end type="italics"/> angulus <emph type="italics"/>bAd<emph.end type="italics"/> eva<lb/>ne&longs;cet, & propterea triangula tria &longs;emper <lb/>finita <emph type="italics"/>rAb, rAb, rAd<emph.end type="italics"/> coincident, &longs;unt<lb/>que eo nomine &longs;imilia & æqualia. </s>

<s>Et &longs;imili argumento &longs;i fiat <emph type="italics"/>DB<emph.end type="italics"/><lb/>&longs;ucce&longs;&longs;ive ut <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>4<emph.end type="sup"/>, <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>5<emph.end type="sup"/>, <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>6<emph.end type="sup"/>, <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>7<emph.end type="sup"/>, &c. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> Et hinc triangula illa, in omni de rationibus ultimis argu<lb/>mentatione, pro &longs;e invicem u&longs;urpari po&longs;&longs;unt. </s><pb pagenum="29"/>

<s><emph type="center"/>LEMMA IX.<emph.end type="center"/></s>

<s>habebitur &longs;eries an<lb/>gulorum contactus pergens in infinitum, quorum quilibet po&longs;te<lb/>rior e&longs;t infinite minor priore. </s>

<s>Etenim dum puncta <emph type="italics"/>B, C<emph.end type="italics"/> acce<lb/><figure id="fig10"></figure><lb/>dunt ad punctum <emph type="italics"/>A,<emph.end type="italics"/> intelligatur <lb/>&longs;emper <emph type="italics"/>AD<emph.end type="italics"/> produci ad puncta lon<lb/>ginqua <emph type="italics"/>d<emph.end type="italics"/> & <emph type="italics"/>e,<emph.end type="italics"/> ut &longs;int <emph type="italics"/>Ad, Ae<emph.end type="italics"/> ip<lb/>&longs;is <emph type="italics"/>AD, AE<emph.end type="italics"/> proportionales, & e<lb/>rigantur ordinatæ <emph type="italics"/>db, ec<emph.end type="italics"/> ordina<lb/>tis <emph type="italics"/>DB, EC<emph.end type="italics"/> parallelæ quæ occur<lb/>rant ip&longs;is <emph type="italics"/>AB, AC<emph.end type="italics"/> productis in <lb/><emph type="italics"/>b<emph.end type="italics"/> & <emph type="italics"/>c.<emph.end type="italics"/> Duci intelligatur, tum curva <lb/><emph type="italics"/>Abc<emph.end type="italics"/> ip&longs;i <emph type="italics"/>ABC<emph.end type="italics"/> &longs;imilis, tum recta <lb/><emph type="italics"/>Ag,<emph.end type="italics"/> quæ tangat curvam utramque <lb/>in <emph type="italics"/>A,<emph.end type="italics"/> & &longs;ecet ordinatim applica<lb/>tas <emph type="italics"/>DB, EC, db, ec<emph.end type="italics"/> in <emph type="italics"/>F, G, f, g.<emph.end type="italics"/><lb/>Tum manente longitudine <emph type="italics"/>Ae<emph.end type="italics"/> coeant puncta <emph type="italics"/>B, C<emph.end type="italics"/> cum puncto <emph type="italics"/>A<emph.end type="italics"/>; <lb/>& angulo <emph type="italics"/>cAg<emph.end type="italics"/> evane&longs;cente, coincident areæ curvilineæ <emph type="italics"/>Abd, Ace<emph.end type="italics"/><lb/>cum rectilineis <emph type="italics"/>Afd, Age:<emph.end type="italics"/> adeoque (per Lemma v) erunt in dupli<lb/>cata ratione laterum <emph type="italics"/>Ad, A<gap/>:<emph.end type="italics"/> Sed his areis proportionales &longs;emper <lb/>&longs;unt areæ <emph type="italics"/>ABD, ACE,<emph.end type="italics"/> & his lateribus latera <emph type="italics"/>AD, AE.<emph.end type="italics"/> Ergo & <lb/>areæ <emph type="italics"/>ABD, ACE<emph.end type="italics"/> &longs;unt ultimo in duplicata ratione laterum <emph type="italics"/>AD, <lb/>AE. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s>

<s>Et &longs;i fiat <emph type="italics"/>DB<emph.end type="italics"/> &longs;ucce&longs;&longs;ive ut <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>2<emph.end type="sup"/>, <lb/><emph type="italics"/>AD<emph.end type="italics"/>3/2, <emph type="italics"/>AD<emph.end type="italics"/>4/3, <emph type="italics"/>AD<emph.end type="italics"/>5/4, <emph type="italics"/>AD<emph.end type="italics"/>6/5, <emph type="italics"/>AD<emph.end type="italics"/>7/6, &c. </s>

<s><emph type="center"/>LEMMA X.<emph.end type="center"/></s>

<s>habebitur alia &longs;eries infinita <lb/>angulorum contactus, quorum primus e&longs;t eju&longs;dem generis cum Cir<lb/>cularibus, &longs;ecundus infinite major, & quilibet po&longs;terior infinite ma<lb/>jor priore. </s>

<s><margin.target id="note14"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s>Sed & inter duos quo&longs;vis ex his angulis pote&longs;t &longs;eries <lb/>utrinque in infinitum pergens angulorum intermediorum in&longs;eri, <lb/>quorum quilibet po&longs;terior erit infinite major minorve priore. </s>

<s>Ut <lb/>&longs;i inter terminos <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>2<emph.end type="sup"/> & <emph type="italics"/>AD<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/> in&longs;eratur &longs;eries <emph type="italics"/>AD<emph.end type="italics"/>(13/6), <emph type="italics"/>AD<emph.end type="italics"/>(1<gap/>/5), <lb/><emph type="italics"/>AD<emph.end type="italics"/>9/4, <emph type="italics"/>AD<emph.end type="italics"/>7/3, <emph type="italics"/>AD<emph.end type="italics"/>5/2, <emph type="italics"/>AD<emph.end type="italics"/><gap/>/3, <emph type="italics"/>AD<emph.end type="italics"/>(11/4), <emph type="italics"/>AD<emph.end type="italics"/>(14/5), <emph type="italics"/>AD<emph.end type="italics"/>(17/6), &c. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Idem intelligendum e&longs;t de &longs;patiis quibu&longs;vis quæ corpo<lb/>ra urgentibus diver&longs;is viribus de&longs;cribunt. </s>

<s>Hæc &longs;unt, ip&longs;o motus ini<lb/>tio, ut vires & quadrata temporum conjunctim. </s>

<s>Et rur<lb/>&longs;us inter binos quo&longs;vis angulos hujus &longs;eriei in&longs;eri pote&longs;t &longs;eries no<lb/>va angulorum intermediorum ab invicem infinitis intervallis diffe<lb/>rentium. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Ideoque vires &longs;unt ut &longs;patia, ip&longs;o motus initio, de&longs;cripta <lb/>directe & quadrata temporum inver&longs;e. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Et quadrata temporum &longs;unt ut de&longs;cripta &longs;patia directe <lb/>& vires inver&longs;e. </s>

<s>Neque novit natura limitem. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Quæ de curvis lineis deque &longs;uperficiebus comprehen&longs;is demon<lb/>&longs;trata &longs;unt, facile applicantur ad &longs;olidorum &longs;uperficies curvas & <lb/>contenta. </s>

<s>Præmi&longs;i vero hæc Lemmata, ut effugerem tædium dedu<lb/>cendi perplexas demon&longs;trationes, more veterum Geometrarum, ad <lb/>ab&longs;urdum. </s>

<s><emph type="center"/>LEMMA XI.<emph.end type="center"/></s>

<s>Contractiores enim redduntur demon&longs;trationes per me<lb/>thodum Indivi&longs;ibilium. </s>

<s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/> 1. Sit arcus ille <emph type="italics"/>AB,<emph.end type="italics"/> tangens ejus <emph type="italics"/>AD,<emph.end type="italics"/> &longs;ubten&longs;a anguli con<lb/>tactus ad tangentem perpendicularis <emph type="italics"/>BD,<emph.end type="italics"/> &longs;ubten&longs;a arcus <emph type="italics"/>AB.<emph.end type="italics"/> Huic <lb/>&longs;ubten&longs;æ <emph type="italics"/>AB<emph.end type="italics"/> & tangenti <emph type="italics"/>AD<emph.end type="italics"/> perpendiculares erigantur <emph type="italics"/>AG, BG,<emph.end type="italics"/><pb pagenum="31"/>concurrentes in <emph type="italics"/>G<emph.end type="italics"/>; dein accedant puncta <emph type="italics"/>D, B, G,<emph.end type="italics"/> ad puncta <emph type="italics"/>d, b, g,<emph.end type="italics"/><lb/>&longs;itque <emph type="italics"/>J<emph.end type="italics"/> inter&longs;ectio linearum <emph type="italics"/>BG, AG<emph.end type="italics"/> ultimo facta ubi puncta <emph type="italics"/>D, B<emph.end type="italics"/><lb/>accedunt u&longs;que ad <emph type="italics"/>A.<emph.end type="italics"/> Manife&longs;tum e&longs;t quod di&longs;tantia <emph type="italics"/>GJ<emph.end type="italics"/> minor <lb/>e&longs;&longs;e pote&longs;t quam a&longs;&longs;ignata quævis. </s>

<s>E&longs;t autem (ex natura circulorum <lb/>per puncta <emph type="italics"/>ABG, Abg<emph.end type="italics"/> tran&longs;euntium) <emph type="italics"/>ABquad.<emph.end type="italics"/><lb/><figure id="fig11"></figure><lb/>æquale <emph type="italics"/>AGXBD,<emph.end type="italics"/> & <emph type="italics"/>Ab quad.<emph.end type="italics"/> æquale <emph type="italics"/>AgXbd,<emph.end type="italics"/><lb/>adeoque ratio <emph type="italics"/>AB quad.<emph.end type="italics"/> ad <emph type="italics"/>Ab quad.<emph.end type="italics"/> compo<lb/>nitur ex rationibus <emph type="italics"/>AG<emph.end type="italics"/> ad <emph type="italics"/>Ag<emph.end type="italics"/> & <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>bd.<emph.end type="italics"/><lb/>Sed quoniam <emph type="italics"/>GJ<emph.end type="italics"/> a&longs;&longs;umi pote&longs;t minor longitu<lb/>dine quavis a&longs;&longs;ignata, fieri pote&longs;t ut ratio <emph type="italics"/>AG<emph.end type="italics"/><lb/>ad <emph type="italics"/>Ag<emph.end type="italics"/> minus differat a ratione æqualitatis quam <lb/>pro differentia quavis a&longs;&longs;ignata, adeoque ut ra<lb/>tio <emph type="italics"/>AB quad.<emph.end type="italics"/> ad <emph type="italics"/>Ab quad.<emph.end type="italics"/> minus differat a ra<lb/>tione <emph type="italics"/>BD<emph.end type="italics"/> ad <emph type="italics"/>bd<emph.end type="italics"/> quam pro differentia quavis <lb/>a&longs;&longs;ignata. </s>

<pb pagenum="33"/>evane&longs;centium &longs;ummas & rationes, prima&longs;que na&longs;centium, id e&longs;t, <lb/>ad limites &longs;ummarum & rationum deducere; & propterea limitum <lb/>illorum demon&longs;trationes qua potui brevitate præmittere. </s>

<s>His enim <lb/>idem præ&longs;tatur quod per methodum Indivi&longs;ibilium; & principiis de<lb/>mon&longs;tratis jam tutius utemur. </s>

<s>Proinde in &longs;equentibus, &longs;iquando <lb/>quantitates tanquam ex particulis con&longs;tantes con&longs;ideravero, vel &longs;i <lb/>pro rectis u&longs;urpavero lineolas curvas; nolim indivi&longs;ibilia, &longs;ed eva<lb/>ne&longs;centia divi&longs;ibilia, non &longs;ummas & rationes partium determinata<lb/>rum, &longs;ed &longs;ummarum & rationum limites &longs;emper intelligi; vimque <lb/>talium demon&longs;trationum ad methodum præcedentium Lemmatum <lb/>&longs;emper revocari. </s>

<s>Objectio e&longs;t, quod quantitatum evane&longs;centium nulla &longs;it ultima <lb/>proportio; quippe quæ, antequam evanuerunt, non e&longs;t ultima, ubi <lb/>evanuerunt, nulla e&longs;t. </s>

<s>Sed & eodem argumento æque contendi po&longs;&longs;et <lb/>nullam e&longs;&longs;e corporis ad certum locum pervenientis velocitatem ul<lb/>timam: hanc enim, antequam corpus attingit locum, non e&longs;&longs;e ulti<lb/>mam, ubiattingit, nullam e&longs;&longs;e. </s>

<s><lb/>Nam &longs;agittæ illæ &longs;unt ut &longs;ubten&longs;æ <emph type="italics"/>BD, bd.<emph.end type="italics"/></s>

<s>Et re&longs;pon&longs;io facilis e&longs;t: Per velocita<lb/>tem ultimam intelligi eam, qua corpus movetur neque antequam <lb/>attingit locum ultimum & motus ce&longs;&longs;at, neque po&longs;tea, &longs;ed tunc <lb/>cum attingit; id e&longs;t, illam ip&longs;am velocitatem quacum corpus attin<lb/>git locum ultimum & quacum motus ce&longs;&longs;at. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Ideoque &longs;agitta e&longs;t in duplicata ratione temporis quo <lb/>corpus data velocitate de&longs;cribit arcum. </s>

<s>Et &longs;imiliter per ulti<lb/>mam rationem quantitatum evane&longs;centium, intelligendam e&longs;&longs;e ratio<lb/>nem quantitatum non antequam evane&longs;cunt, non po&longs;tea, &longs;ed qua<lb/>cum evane&longs;cunt. </s>

<s>Pariter & ratio prima na&longs;centium e&longs;t ratio qua<lb/>cum na&longs;cuntur. </s>

<s><margin.target id="note15"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s>Et &longs;umma prima & ultima e&longs;t quacum e&longs;&longs;e (vel <lb/>augeri & minui) incipiunt & ce&longs;&longs;ant. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Extat limes quem velocitas <lb/>in fine motus attingere pote&longs;t, non autem tran&longs;gredi. </s>

<s>Hæc e&longs;t <lb/>velocitas ultima. </s>

<s>habebitur &longs;eries an<lb/>gulorum contactus pergens in infinitum, quorum quilibet po&longs;te<lb/>rior e&longs;t infinite minor priore. </s>

<s>Et par e&longs;t ratio limitis quantitatum & propor<lb/>tionum omnium incipientium & ce&longs;&longs;antium. </s>

<s>Cumque hic limes <lb/>&longs;it certus & definitus, Problema e&longs;t vere Geometricum eundem de<lb/>terminare. </s>

<s>Geometrica vero omnia in aliis Geometricis determi<lb/>nandis ac demon&longs;trandis legitime u&longs;urpantur. </s>

<s>Neque novit natura limitem. </s>

<s>Contendi etiam pote&longs;t, quod &longs;i dentur ultimæ quantitatum eva<lb/>ne&longs;centium rationes, dabuntur & ultimæ magnitudines: & &longs;ic quan<lb/>titas omnis con&longs;tabit ex Indivi&longs;ibilibus, contra quam <emph type="italics"/>Euclides<emph.end type="italics"/> de <lb/>Incommen&longs;urabilibus, in libro decimo Elementorum, demon&longs;travit. </s>

<s>Quæ de curvis lineis deque &longs;uperficiebus comprehen&longs;is demon<lb/>&longs;trata &longs;unt, facile applicantur ad &longs;olidorum &longs;uperficies curvas & <lb/>contenta. </s>

<s>Præmi&longs;i vero hæc Lemmata, ut effugerem tædium dedu<lb/>cendi perplexas demon&longs;trationes, more veterum Geometrarum, ad <lb/>ab&longs;urdum. </s>

<s><lb/>Verum hæc Objectio fal&longs;æ innititur hypothe&longs;i. </s>

<s>Contractiores enim redduntur demon&longs;trationes per me<lb/>thodum Indivi&longs;ibilium. </s>

<s>Sed quoniam durior e&longs;t Indivi&longs;ibilium hy<lb/>pothe&longs;is, & propterea methodus illa minus Geometrica cen&longs;etur; <lb/>malui demon&longs;trationes rerum &longs;equentium ad ultimas quantitatum <pb pagenum="33"/>evane&longs;centium &longs;ummas & rationes, prima&longs;que na&longs;centium, id e&longs;t, <lb/>ad limites &longs;ummarum & rationum deducere; & propterea limitum <lb/>illorum demon&longs;trationes qua potui brevitate præmittere. </s>

<s>Ultimæ rationes <lb/>illæ quibu&longs;cum quantitates evane&longs;cunt, revera non &longs;unt rationes <lb/>quantitatum ultimarum, &longs;ed limites ad quos quantitatum &longs;ine limi<lb/>te decre&longs;centium rationes &longs;emper appropinquant; & quas propius <lb/>a&longs;&longs;equi po&longs;&longs;unt quam pro data quavis differentia, nunquam vero </s>

<s>His enim <lb/>idem præ&longs;tatur quod per methodum Indivi&longs;ibilium; & principiis de<lb/>mon&longs;tratis jam tutius utemur. </s>

<s>

<arrow.to.target n="note16"></arrow.to.target><lb/>tran&longs;gredi, neque prius attingere quam quantitates diminuuntur in <lb/>infinitum. </s>

<s>Objectio e&longs;t, quod quantitatum evane&longs;centium nulla &longs;it ultima <lb/>proportio; quippe quæ, antequam evanuerunt, non e&longs;t ultima, ubi <lb/>evanuerunt, nulla e&longs;t. </s>

<s>Res clarius intelligetur in infinite magnis. </s>

<s>Pariter & ratio prima na&longs;centium e&longs;t ratio qua<lb/>cum na&longs;cuntur. </s>

<s>Et &longs;umma prima & ultima e&longs;t quacum e&longs;&longs;e (vel <lb/>augeri & minui) incipiunt & ce&longs;&longs;ant. </s>

<s>Extat limes quem velocitas <lb/>in fine motus attingere pote&longs;t, non autem tran&longs;gredi. </s>

<s>Hæc e&longs;t <lb/>velocitas ultima. </s>

<s>Et par e&longs;t ratio limitis quantitatum & propor<lb/>tionum omnium incipientium & ce&longs;&longs;antium. </s>

<s>Cumque hic limes <lb/>&longs;it certus & definitus, Problema e&longs;t vere Geometricum eundem de<lb/>terminare. </s>

<s>Geometrica vero omnia in aliis Geometricis determi<lb/>nandis ac demon&longs;trandis legitime u&longs;urpantur. </s>

<s><lb/>Verum hæc Objectio fal&longs;æ innititur hypothe&longs;i. </s>

<s><arrow.to.target n="note16"></arrow.to.target><lb/>tran&longs;gredi, neque prius attingere quam quantitates diminuuntur in <lb/>infinitum. </s>

<s>Res clarius intelligetur in infinite magnis. </s>

<s>Si quantitates <lb/>duæ quarum data e&longs;t differentia auges ntur in infinitum, dabitur <lb/>harum ultima ratio, nimirum ratio æqualitatis, nec tamen ideo da<lb/>buntur quantitates ultimæ &longs;eu maximæ quarum i&longs;ta e&longs;t ratio. </s>

<s>Igitur <lb/>in &longs;equentibus, &longs;iquando facili rerum conceptui con&longs;ulens dixero <lb/>quantitates quam minimas, vel evane&longs;centes, vel ultimas; cave in<lb/>telligas quantitates magnitudine determinatas, &longs;ed cogita &longs;emper <lb/>diminuendas &longs;ine limite. </s>

<s><margin.target id="note16"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s><margin.target id="note16"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s><emph type="center"/>SECTIO II.<emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>De Inventione Virium Centripetarum.<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="center"/>PROPOSITIO I. THE OREMA I.<emph.end type="center"/></s>

<s><emph type="italics"/>Areas, quas corpora in gyros acta radiis ad immobile centrum virium <lb/>ductis de&longs;cribunt, & in planis immobilibus con&longs;i&longs;tere, & e&longs;&longs;e tem<lb/>poribus proportionales.<emph.end type="italics"/></s>

<s><emph type="center"/>SECTIO II.<emph.end type="center"/></s>

<arrow.to.target n="fig12"></arrow.to.target><lb/>pergeret ad <emph type="italics"/>c,<emph.end type="italics"/> (per <lb/>Leg. </s>

<s><emph type="center"/><emph type="italics"/>De Inventione Virium Centripetarum.<emph.end type="italics"/><emph.end type="center"/></s>

<s>1.) de&longs;cribens <lb/>lineam <emph type="italics"/>Bc<emph.end type="italics"/> æqualem <lb/>ip&longs;i <emph type="italics"/>AB<emph.end type="italics"/>; adeo ut ra<lb/>diis <emph type="italics"/>AS, BS, cS<emph.end type="italics"/> ad <lb/>centrum actis, con<lb/>fectæ forent æqua<lb/>les areæ <emph type="italics"/>ASB, BSc.<emph.end type="italics"/><lb/>Verum ubi corpus <lb/>venitad <emph type="italics"/>B,<emph.end type="italics"/> agat vis <lb/>centripeta impul<lb/>&longs;u unico &longs;ed mag<lb/>no, efficiatque ut <lb/>corpus de recta <emph type="italics"/>Bc<emph.end type="italics"/><lb/>declinet & pergat <lb/>in recta <emph type="italics"/>BC.<emph.end type="italics"/> Ip&longs;i <lb/><emph type="italics"/>BS<emph.end type="italics"/> parallela agatur <emph type="italics"/>cC,<emph.end type="italics"/> occurens <emph type="italics"/>BC<emph.end type="italics"/> in <emph type="italics"/>C<emph.end type="italics"/>; & completa &longs;ecunda <lb/>temporis parte, corpus (per Legum Corol. </s>

<s><emph type="center"/>PROPOSITIO I. THE OREMA I.<emph.end type="center"/></s>

<s>1.) reperietur in <emph type="italics"/>C,<emph.end type="italics"/> in

<s>Dividatur tempus in partes æquales, & prima temporis parte de<lb/>&longs;eribat corpus vi in&longs;ita rectam <emph type="italics"/>AB.<emph.end type="italics"/> Idem &longs;ecunda temporis parte, &longs;i <lb/>nil impediret, recta <lb/><figure id="fig12"></figure><lb/>pergeret ad <emph type="italics"/>c,<emph.end type="italics"/> (per <lb/>Leg. </s>

<pb pagenum="35"/>eodem plano cum triangulo <emph type="italics"/>ASB.<emph.end type="italics"/> Junge <emph type="italics"/>SC<emph.end type="italics"/>; & triangulum <emph type="italics"/>SBC,<emph.end type="italics"/><lb/>ob parallelas <emph type="italics"/>SB, Cc,<emph.end type="italics"/> æquale erit triangulo <emph type="italics"/>SBc,<emph.end type="italics"/> atque adeo etiam <lb/>triangulo <emph type="italics"/>SAB.<emph.end type="italics"/> Simili argumento &longs;i vis centripeta &longs;ucce&longs;&longs;ive agat <lb/>in <emph type="italics"/>C, D, E,<emph.end type="italics"/> &c. </s>

<s>faciens ut corpus &longs;ingulis temporis particulis &longs;in<lb/>gulas de&longs;eribat rectas <emph type="italics"/>CD, DE, EF,<emph.end type="italics"/> &c. </s>

<s>1.) reperietur in <emph type="italics"/>C,<emph.end type="italics"/> in <pb pagenum="35"/>eodem plano cum triangulo <emph type="italics"/>ASB.<emph.end type="italics"/> Junge <emph type="italics"/>SC<emph.end type="italics"/>; & triangulum <emph type="italics"/>SBC,<emph.end type="italics"/><lb/>ob parallelas <emph type="italics"/>SB, Cc,<emph.end type="italics"/> æquale erit triangulo <emph type="italics"/>SBc,<emph.end type="italics"/> atque adeo etiam <lb/>triangulo <emph type="italics"/>SAB.<emph.end type="italics"/> Simili argumento &longs;i vis centripeta &longs;ucce&longs;&longs;ive agat <lb/>in <emph type="italics"/>C, D, E,<emph.end type="italics"/> &c. </s>

<s>faciens ut corpus &longs;ingulis temporis particulis &longs;in<lb/>gulas de&longs;eribat rectas <emph type="italics"/>CD, DE, EF,<emph.end type="italics"/> &c. </s>

<s>jacebunt hæ omnes in <lb/>eodem plano; & triangulum <emph type="italics"/>SCD<emph.end type="italics"/> triangulo <emph type="italics"/>SBC,<emph.end type="italics"/> & <emph type="italics"/>SDE<emph.end type="italics"/> ip&longs;i <lb/><emph type="italics"/>SCD,<emph.end type="italics"/> & <emph type="italics"/>SEF<emph.end type="italics"/> ip&longs;i <emph type="italics"/>SDE<emph.end type="italics"/> æquale erit. </s>

<s>Æqualibus igitur tempori<lb/>bus æquales areæ in plano immoto de&longs;cribuntur: & componendo, <lb/>&longs;unt arearum &longs;ummæ quævis <emph type="italics"/>SADS, SAFS<emph.end type="italics"/> inter &longs;e, ut &longs;unt tem<lb/>pora de&longs;criptionum. </s>

<s>Augeatur jam numerus & minuatur latitudo <lb/>triangulorum in infinitum; & eorum ultima perimeter <emph type="italics"/>ADF,<emph.end type="italics"/> (per <lb/>Corollarium quartum Lemmatis tertii) erit linea curva: adeoque vis <lb/>centripeta, qua corpus a tangente hujus curvæ perpetuo retrahitur, <lb/>aget inde&longs;inenter; areæ vero quævis de&longs;criptæ <emph type="italics"/>SADS, SAFS<emph.end type="italics"/><lb/>temporibus de&longs;criptionum &longs;emper proportionales, erunt ii&longs;dem tem<lb/>poribus in hoc ca&longs;u proportionales. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Velocitas corporis in centrum immobile attracti e&longs;t in <lb/>&longs;patiis non re&longs;i&longs;tentibus reciproce ut perpendiculum a centro illo in <lb/>Orbis tangentem rectilineam demi&longs;&longs;um. </s>

<s>E&longs;t enim velocitas in locis <lb/>illis <emph type="italics"/>A, B, C, D, E,<emph.end type="italics"/> ut &longs;unt ba&longs;es æqualium triangulorum <emph type="italics"/>AB, BC, <lb/>CD, DE, EF<emph.end type="italics"/>; & hæ ba&longs;es &longs;unt reciproce ut perpendicula in ip&longs;as <lb/>demi&longs;&longs;a. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Si arcuum duorum æqualibus temporibus in &longs;patiis non <lb/>re&longs;i&longs;tentibus ab eodem corpore &longs;ucce&longs;&longs;ive de&longs;criptorum chordæ <emph type="italics"/>AB, <lb/>BC<emph.end type="italics"/> compleantur in parallelogrammum <emph type="italics"/>ABCU,<emph.end type="italics"/> & hujus diagona<lb/>lis <emph type="italics"/>BU<emph.end type="italics"/> in ea po&longs;itione quam ultimo habet ubi arcus illi in infini<lb/>tum diminuuntur, producatur utrinque; tran&longs;ibit eadem per cen<lb/>trum virium. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Si arcuum æqualibus temporibus in &longs;patiis non re&longs;i&longs;ten<lb/>tibus de&longs;criptorum chordæ <emph type="italics"/>AB, BC<emph.end type="italics"/> ac <emph type="italics"/>DE, EF<emph.end type="italics"/> compleantur in <lb/>parallelogramma <emph type="italics"/>ABCU, DEFZ<emph.end type="italics"/>; vires in <emph type="italics"/>B<emph.end type="italics"/> & <emph type="italics"/>E<emph.end type="italics"/> &longs;unt ad invi<lb/>cem in ultima ratione diagonalium <emph type="italics"/>BU, EZ,<emph.end type="italics"/> ubi arcus i&longs;ti in infi<lb/>nitum diminuuntur. </s>

<s>Nam corporis motus <emph type="italics"/>BC<emph.end type="italics"/> & <emph type="italics"/>EF<emph.end type="italics"/> componun<lb/>tur (per Legum Corol. </s>

<s>1.) ex motibus <emph type="italics"/>Bc, BU<emph.end type="italics"/> & <emph type="italics"/>Ef, EZ:<emph.end type="italics"/> at<lb/>qui <emph type="italics"/>BU<emph.end type="italics"/> & <emph type="italics"/>EZ,<emph.end type="italics"/> ip&longs;is <emph type="italics"/>Cc<emph.end type="italics"/> & <emph type="italics"/>Ff<emph.end type="italics"/> æquales, in Demon&longs;tratione Pro<lb/>po&longs;itionis hujus generabantur ab impul&longs;ibus vis centripetæ in B & <lb/><emph type="italics"/>E,<emph.end type="italics"/> ideoque &longs;unt his impul&longs;ibus proportionales. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Vires quibus corpora quælibet in &longs;patiis non re&longs;i&longs;tenti<lb/>bus a motibus rectilineis retrahuntur ac detorquentur in orbes cur<lb/>vos &longs;unt inter &longs;e ut arcuum æqualibus temporibus de&longs;criptorum &longs;a<lb/>gittæ illæ quæ convergunt ad centrum virium, & chordas bi&longs;ecant <pb pagenum="36"/><arrow.to.target n="note17"></arrow.to.target><lb/>ubi arcus illi in infinitum diminuuntur. </s>

<s>Nam hæ &longs;agittæ &longs;unt &longs;e<lb/>mi&longs;&longs;es diagonalium de quibus egimus in Corollario tertio. </s>

<s><margin.target id="note17"></margin.target>DE MOTU <lb/>CORPORUM</s>

<arrow.to.target n="note17"></arrow.to.target><lb/>ubi arcus illi in infinitum diminuuntur. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Ideoque vires eædem &longs;unt ad vim gravitatis, ut hæ &longs;a<lb/>gittæ ad &longs;agittas horizonti perpendiculares arcuum Parabolicorum <lb/>quos projectilia eodem tempore de&longs;cribunt. </s>

<s>Nam hæ &longs;agittæ &longs;unt &longs;e<lb/>mi&longs;&longs;es diagonalium de quibus egimus in Corollario tertio. </s>

<s><margin.target id="note17"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 6. Eadem omnia obtinent per Legum Corol. </s>

<s>IV, ubi plana <lb/>in quibus corpora moventur, una cum centris virium quæ in ip&longs;is <lb/>fita &longs;unt, non quie&longs;cunt, &longs;ed moventur uniformiter in directum. </s>

<s>IV, ubi plana <lb/>in quibus corpora moventur, una cum centris virium quæ in ip&longs;is <lb/>fita &longs;unt, non quie&longs;cunt, &longs;ed moventur uniformiter in directum. </s>

<s><emph type="center"/>PROPOSITIO II. THEOREMA II.<emph.end type="center"/></s>

<s><emph type="center"/>PROPOSITIO II. THEOREMA II.<emph.end type="center"/></s>

<s><emph type="italics"/>Corpus omne, quod movetur in linea aliqua curva in plano de<lb/>&longs;cripta, & radio ducto ad punctum vel immobile, vel motu rectili<lb/>neo uniformiter progrediens, de&longs;cribit areas circa punctum illud <lb/>temporibus proportionales, urgetur a vi centripeta tendente ad idem <lb/>punctum.<emph.end type="italics"/></s>

<s><emph type="italics"/>Cas.<emph.end type="italics"/> 1. Nam corpus omne quod movetur in linea curva, detor<lb/>quetur de cur&longs;u rectilineo per vim aliquam in ip&longs;um agentem (per <lb/>Leg. </s>

<s>1.) Et vis illa qua corpus de cur&longs;u rectilineo detorquetur, & <lb/>cogitur triangula quam minima <emph type="italics"/>SAB, SBC, SCD,<emph.end type="italics"/> &c. </s>

<s>circa <lb/>punctum immobile <emph type="italics"/>S<emph.end type="italics"/> temporibus æqualibus æqualia de&longs;cribere, a<lb/>git in loco <emph type="italics"/>B<emph.end type="italics"/> &longs;ecundum lineam parallelam ip&longs;i <emph type="italics"/>cC<emph.end type="italics"/> (per Prop. </s>

<s>11.) hoc e&longs;t, &longs;ecundum lineam <emph type="italics"/>BS<emph.end type="italics"/>; & in loco <lb/><emph type="italics"/>C<emph.end type="italics"/> &longs;ecundum lineam ip&longs;i <emph type="italics"/>dD<emph.end type="italics"/> parallelam, hoc e&longs;t, &longs;ecundum lineam <lb/><emph type="italics"/>SC,<emph.end type="italics"/> &c. </s>

<s>Agit ergo &longs;emper &longs;ecundum lineas tendentes ad punctum <lb/>illud immobile <emph type="italics"/>S. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s>

<s><emph type="italics"/>Cas.<emph.end type="italics"/> 2. Et, per Legum Corollarium quintum, perinde e&longs;t &longs;ive <lb/>quie&longs;cat &longs;uperficies in qua corpus de&longs;cribit figuram curvilineam, <lb/>&longs;ive moveatur eadem una cum corpore, figura de&longs;cripta, & puncto <lb/>&longs;uo <emph type="italics"/>S<emph.end type="italics"/> uniformiter in directum. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. In Spatiis vel Mediis non re&longs;i&longs;tentibus, &longs;i areæ non &longs;unt <lb/>temporibus proportionales, vires non tendunt ad concur&longs;um radio<lb/>rum; &longs;ed inde declinant in con&longs;equentia &longs;eu ver&longs;us plagam in quam <lb/>fit motus, &longs;i modo arearum de&longs;criptio acceleratur: &longs;in retardatur, de<lb/>clinant in antecedentia. </s>

<s>Agit ergo &longs;emper &longs;ecundum lineas tendentes ad punctum <lb/>illud immobile <emph type="italics"/>S. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. In Mediis etiam re&longs;i&longs;tentibus, &longs;i arearum de&longs;criptio accele<lb/>ratur, virium directiones declinant a concur&longs;u radiorum ver&longs;us plagam <lb/>in quam &longs;it motus. </s><pb pagenum="37"/>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Urgeri pote&longs;t corpus a vi centripeta compo&longs;ita ex pluribus viri<lb/>bus. </s>

<s>In hoc ca&longs;u &longs;en&longs;us Propo&longs;itionis e&longs;t, quod vis illa quæ ex om<lb/>nibus componitur, tendit ad punctum <emph type="italics"/>S.<emph.end type="italics"/> Porro &longs;i vis aliqua agat <lb/>perpetuo &longs;ecundum lineam &longs;uperficiei de&longs;criptæ perpendicularem; <lb/>hæc faciet ut corpus deflectatur a plano &longs;ui motus: &longs;ed quantita<lb/>tem &longs;uperficiei de&longs;criptæ nec augebit nec minuet, & propterea in <lb/>compo&longs;itione virium negligenda e&longs;t. </s>

<s><emph type="center"/>PROPOSITIO III. THEOREMA III.<emph.end type="center"/></s>

<s><emph type="center"/>PROPOSITIO III. THEOREMA III.<emph.end type="center"/></s>

<s><emph type="italics"/>Corpus omne, quod radio ad centrum corporis alterius utcunque moti <lb/>ducto de&longs;cribit areas circa centrum illud temporibus proportiona<lb/>les, urgetur vi compo&longs;ita ex vi centripeta tendente ad corpus il<lb/>lud alterum, & ex vi omni acceleratrice qua corpus illud alterum <lb/>urgetur.<emph.end type="italics"/></s>

<s>Sit corpus primum <emph type="italics"/>L<emph.end type="italics"/> & corpus alterum <emph type="italics"/>T:<emph.end type="italics"/> & (per Legum Corol. </s>

<s><lb/>VI.) &longs;i vi nova, quæ æqualis & contraria &longs;it illi qua corpus alterum <lb/><emph type="italics"/>T<emph.end type="italics"/> urgetur, urgeatur corpus utrumque &longs;ecundum lineas parallelas; <lb/>perget corpus primum <emph type="italics"/>L<emph.end type="italics"/> de&longs;cribere circa corpus alterum <emph type="italics"/>T<emph.end type="italics"/> areas <lb/>ea&longs;dem ac prius: vis autem, qua corpus alterum <emph type="italics"/>T<emph.end type="italics"/> urgebatur, jam <lb/>de&longs;truetur per vim &longs;ibi æqualem & contrariam; & propterea (per <lb/>Leg. </s>

<s>1.) corpus illud alterum <emph type="italics"/>T<emph.end type="italics"/> &longs;ibimet ip&longs;i jam relictum vel qui<lb/>e&longs;cet vel movebitur uniformiter in directum: & corpus primum <emph type="italics"/>L<emph.end type="italics"/><lb/>urgente differentia virium, id e&longs;t, urgente vi reliqua perget areas <lb/>temporibus proportionales circa corpus alterum <emph type="italics"/>T<emph.end type="italics"/> de&longs;cribere. </s>

<s>Ten<lb/>dit igitur (per Theor. </s>

<s>11.) differentia virium ad corpus illud alte<lb/>rum <emph type="italics"/>T<emph.end type="italics"/> ut centrum. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Hinc &longs;i corpus unum <emph type="italics"/>L<emph.end type="italics"/> radio ad alterum <emph type="italics"/>T<emph.end type="italics"/> ducto de<lb/>&longs;cribit areas temporibus proportionales; atque de vi tota (&longs;ive &longs;im<lb/>plici, &longs;ive ex viribus pluribus, juxta Legum Corollarium &longs;ecundum, <lb/>compo&longs;ita,) qua corpus prius <emph type="italics"/>L<emph.end type="italics"/> urgetur, &longs;ubducatur (per idem Le<lb/>gum Corollarium) vis tota acceleratrix qua corpus alterum urgetur: <lb/>vis omnis reliqua qua corpus prius urgetur tendet ad corpus alte<lb/>rum <emph type="italics"/>T<emph.end type="italics"/> ut centrum. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et, &longs;i areæ illæ &longs;unt temporibus quamproxime propor<lb/>tionales, vis reliqua tendet ad corpus alterum <emph type="italics"/>T<emph.end type="italics"/> quamproxime. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Et vice ver&longs;a, &longs;i vis reliqua tendit quamproxime ad <pb pagenum="38"/><arrow.to.target n="note18"></arrow.to.target><lb/>corpus alterum <emph type="italics"/>T,<emph.end type="italics"/> erunt areæ illæ temporibus quamproxime pro<lb/>portionales. </s>

<s><margin.target id="note18"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Et vice ver&longs;a, &longs;i vis reliqua tendit quamproxime ad

<arrow.to.target n="note18"></arrow.to.target><lb/>corpus alterum <emph type="italics"/>T,<emph.end type="italics"/> erunt areæ illæ temporibus quamproxime pro<lb/>portionales. </s>

<s><margin.target id="note18"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 4. Si corpus <emph type="italics"/>L<emph.end type="italics"/> radio ad alterum corpus <emph type="italics"/>T<emph.end type="italics"/> ducto de&longs;cri<lb/>bit areas quæ, cum temporibus collatæ, &longs;unt valde inæquales; & <lb/>corpus illud alterum <emph type="italics"/>T<emph.end type="italics"/> vel quie&longs;cit vel movetur uniformiter in di<lb/>rectum: actio vis centripetæ ad corpus illud alterum <emph type="italics"/>T<emph.end type="italics"/> tendentis, <lb/>vel nulla e&longs;t, vel mi&longs;cetur & componitur cum actionibus admodum <lb/>potentibus aliarum virium: Vi&longs;que tota ex omnibus, &longs;i plures &longs;unt <lb/>vires, compo&longs;ita, ad aliud (&longs;ive immobile &longs;ive mobile) centrum <lb/>dirigitur. </s>

<s>Idem obtinet, ubi corpus alterum motu quocunque mo<lb/>vetur; &longs;i modo vis centripeta &longs;umatur, quæ re&longs;tat po&longs;t &longs;ubductio<lb/>nem vis totius in corpus illud alterum <emph type="italics"/>T<emph.end type="italics"/> agentis. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Quoniam æquabilis arearum de&longs;criptio Index e&longs;t Centri, quod <lb/>vis illa re&longs;picit qua corpus maxime afficitur, quaque retrahitur a mo<lb/>tu rectilineo & in orbita &longs;ua retinetur: quidni u&longs;urpemus in &longs;equen<lb/>tibus æquabilem arearum de&longs;criptionem, ut Indicem Centri circum <lb/>quod motus omnis circularis in &longs;patiis liberis peragitur? </s>

<s><emph type="center"/>PROPOSITIO IV. THEOREMA IV.<emph.end type="center"/></s>

<s><emph type="italics"/>Corporum, quæ diver&longs;os circulos æquabili motu de&longs;cribunt, vires cen<lb/>tripetas ad centra eorundem circulorum tendere; & e&longs;&longs;e inter &longs;e, <lb/>ut &longs;unt arcuum &longs;imul de&longs;criptorum quadrata applicata ad circulo<lb/>rum radios.<emph.end type="italics"/></s>

<s>Tendunt hæ vires ad centra circulorum per Prop.II. & Corol. </s>

<s><emph type="center"/>PROPOSITIO IV. THEOREMA IV.<emph.end type="center"/></s>

<s>Tendunt hæ vires ad centra circulorum per Prop.II. & Corol. </s>

<s>1; & &longs;unt inter &longs;e ut arcuum æqualibus temporibus quam mini<lb/>mis de&longs;criptorum &longs;inus ver&longs;i per Corol. </s>

<s>1; & &longs;unt inter &longs;e ut arcuum æqualibus temporibus quam mini<lb/>mis de&longs;criptorum &longs;inus ver&longs;i per Corol. </s>

<s>I; hoc e&longs;t, ut qua<lb/>drata arcuum eorundem ad diametros circulorum applicata per <lb/>Lem. </s>

<s>VII: & propterea, cum hi arcus &longs;int ut arcus temporibus <lb/>quibu&longs;vis æqualibus de&longs;cripti, & diametri &longs;int ut eorum radii; vi<lb/>res erunt ut arcuum quorumvis &longs;imul de&longs;criptorum quadrata ap<lb/>plicata ad radios circulorum. <emph type="italics"/>q.E.D.<emph.end type="italics"/></s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 1. Igitur, cum arcus illi &longs;int ut velocitates corporum, vi<lb/>res centripetæ &longs;unt ut velocitatum quadrata applicata ad radios <lb/>circulorum: hoc e&longs;t, ut cum Geometris loquar, vires &longs;unt in ra<lb/>tione compo&longs;ita ex duplicata ratione velocitatum directe & ratione <lb/>&longs;implici radiorum inver&longs;e. </s><pb pagenum="39"/>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 2. Et, cum tempora periodica &longs;int in ratione compo&longs;ita ex <lb/>ratione radiorum directe & ratione velocitatum inver&longs;e, vires cen<lb/>tripetæ &longs;unt reciproce ut quadrata temporum periodicorum appli<lb/>cata ad circulorum radios; hoc e&longs;t, in ratione compo&longs;ita ex ratione <lb/>radiorum directe & ratione duplicata temporum periodicorum in<lb/>ver&longs;e. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 3. Unde, &longs;i tempora periodica æquentur & propterea ve<lb/>locitates &longs;int ut radii; erunt etiam vires centripetæ ut radii: & <lb/>contra. </s>

<s><emph type="italics"/>Cor.<emph.end type="italics"/> 4. Si & tempora periodica & velocitates &longs;int in ratione &longs;ub<lb/>duplicata radiorum; æquales erunt vires centripetæ inter &longs;e: & <lb/>contra. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Si tempora periodica &longs;int ut radii & propterea veloci<lb/>tates æquales; vires centriperæ erunt reciproce ut radii: & contra. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 6. Si tempora periodica &longs;int in ratione &longs;e&longs;quiplicata radio<lb/>rum & propterea velocitates reciproce in radiorum ratione &longs;ubdu<lb/>plicata; vires centripetæ erunt reciproce ut quadrata radiorum: <lb/>& contra. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 7. Et univer&longs;aliter, &longs;i tempus periodicum &longs;it ut Radii <emph type="italics"/>R<emph.end type="italics"/><lb/>pote&longs;tas quælibet <emph type="italics"/>R<emph type="sup"/>n<emph.end type="sup"/>,<emph.end type="italics"/> & propterea velocitas reciproce ut Radii <lb/>pote&longs;tas <emph type="italics"/>R<emph type="sup"/>n-1<emph.end type="sup"/><emph.end type="italics"/>; erit vis centripeta reciproce ut Radii pote&longs;tas <emph type="italics"/>R<emph type="sup"/>2n-1<emph.end type="sup"/>:<emph.end type="italics"/><lb/>& contra. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 5. Si tempora periodica &longs;int ut radii & propterea veloci<lb/>tates æquales; vires centriperæ erunt reciproce ut radii: & contra. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 8. Eadem omnia de temporibus, velocitatibus, & viribus, qui<lb/>bus corpora &longs;imiles figurarum quarumcunque &longs;imilium, centraque <lb/>in figuris illis &longs;imiliter po&longs;ita habentium, partes de&longs;cribunt, con&longs;e<lb/>quuntur ex Demon&longs;tratione præcedentium ad ho&longs;ce ca&longs;us applicata. </s>

<s><lb/>Applicatur autem &longs;ub&longs;tituendo æquabilem arearum de&longs;criptionem <lb/>pro æquabili motu, & di&longs;tantias corporum a centris pro radiis u&longs;ur<lb/>pando. </s>

<s><lb/>Applicatur autem &longs;ub&longs;tituendo æquabilem arearum de&longs;criptionem <lb/>pro æquabili motu, & di&longs;tantias corporum a centris pro radiis u&longs;ur<lb/>pando. </s>

<s><emph type="italics"/>Corol.<emph.end type="italics"/> 9. Ex eadem demon&longs;tratione con&longs;equitur etiam; quod ar<lb/>cus, quem corpus in circulo data vi centripeta uniformiter revolven<lb/>do tempore quovis de&longs;cribit, medius e&longs;t proportionalis inter dia<lb/>metrum circuli, & de&longs;cen&longs;um corporis eadem data vi eodem que tem<lb/>pore cadendo confectum. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Ca&longs;us Corollarii &longs;exti obtinet in corporibus cæle&longs;tibus, (ut &longs;eor<lb/>&longs;um collegerunt etiam no&longs;trates <emph type="italics"/>Wrennus, Hookius<emph.end type="italics"/> & <emph type="italics"/>Hallæus<emph.end type="italics"/>) & <lb/>propterea quæ &longs;pectant ad vim centripetam decre&longs;centem in dupli<lb/>cata ratione di&longs;tantiarum a centris, decrevi fu&longs;ius in &longs;equentibus <lb/>exponere.

<arrow.to.target n="note19"></arrow.to.target></s>

<s><margin.target id="note19"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s>Porro præcedentis propo&longs;itionis & corollariorum ejus beneficio, <lb/>colligitur etiam proportio vis centripetæ ad vim quamlibet notam, <lb/>qualis e&longs;t ea Gravitatis. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Nam &longs;i corpus in circulo Terræ concen<lb/>trico vi gravitatis &longs;uæ revolvatur, hæc gravitas e&longs;t ip&longs;ius vis centri<lb/>peta. </s>

<s><margin.target id="note19"></margin.target>DE MOTU <lb/>CORPORUM</s>

<s>Porro præcedentis propo&longs;itionis & corollariorum ejus beneficio, <lb/>colligitur etiam proportio vis centripetæ ad vim quamlibet notam, <lb/>qualis e&longs;t ea Gravitatis. </s>

<s>Nam &longs;i corpus in circulo Terræ concen<lb/>trico vi gravitatis &longs;uæ revolvatur, hæc gravitas e&longs;t ip&longs;ius vis centri<lb/>peta. </s>

<s>Datur autem, ex de&longs;cen&longs;u gravium, & tempus revolutionis <lb/>unius, & arcus dato quovis tempore de&longs;criptus, per hujus Corol. </s>

<s>Et huju&longs;modi propo&longs;itionibus <emph type="italics"/>Hugenius,<emph.end type="italics"/> in eximio &longs;uo Tracta<lb/>tu <emph type="italics"/>de Horologio O&longs;cillatorio,<emph.end type="italics"/> vim gravitatis cum revolventium vi<lb/>ribus centrifugis contulit. </s>

<s>Demon&longs;trari etiam po&longs;&longs;unt præcedentia in hunc modum. </s>

<s>In cir<lb/>culo quovis de&longs;cribi intelligatur Polygonum laterum quotcunque. </s>

<s><lb/>Et &longs;i corpus, in polygoni lateribus data cum velocitate movendo, <lb/>ad ejus angulos &longs;ingulos a circulo reflectatur; vis qua &longs;ingulis re<lb/>flexionibus impingit in circulum erit ut ejus velocitas: adeoque <lb/>&longs;umma virium in dato tempore erit ut velocitas illa & numerus re<lb/>flexionum conjunctim: hoc e&longs;t (&longs;i polygonum detur &longs;pecie) ut longi<lb/>tudo dato illo tempore de&longs;cripta & longitudo eadem applicata ad <lb/>Radium circuli; id e&longs;t, ut quadratum longitudinis illius applicatum <lb/>ad Radium: adeoque, &longs;i polygonum lateribus infinite diminutis co<lb/>incidat cum circulo, ut quadratum arcus dato tempore de&longs;cripti ap<lb/>plicatum ad radium. </s>

<s>Hæc e&longs;t vis centrifuga, qua corpus urget cir<lb/>culum: & huic æqualis e&longs;t vis contraria, qua circulus continuo re<lb/>pellit corpus centrum ver&longs;us. </s>

<s>Hæc e&longs;t vis centrifuga, qua corpus urget cir<lb/>culum: & huic æqualis e&longs;t vis contraria, qua circulus continuo re<lb/>pellit corpus centrum ver&longs;us. </s>

<s><emph type="center"/>PROPOSITIO. V. PROBLEMA I.<emph.end type="center"/></s>

<s><emph type="center"/>PROPOSITIO. V. PROBLEMA I.<emph.end type="center"/></s>

<s><emph type="italics"/>Data quibu&longs;cunque in locis velocitate, qua corpus figuram datam vi<lb/>ribus ad commune aliquod centrum tendentibus de&longs;cribit, centrum <lb/>illud invenire.<emph.end type="italics"/></s>

<s>Figuram de&longs;criptam tangant rectæ tres <emph type="italics"/>PT, TQV, VR<emph.end type="italics"/> in <lb/>punctis totidem <emph type="italics"/>P, Q, R,<emph.end type="italics"/> concurrentes in <emph type="italics"/>T<emph.end type="italics"/> & <emph type="italics"/>V.<emph.end type="italics"/> Ad tangentes <lb/>erigantur perpendicula <emph type="italics"/>PA, QB, RC,<emph.end type="italics"/> velocitatibus corporis in <lb/>punctis illis <emph type="italics"/>P, Q, R<emph.end type="italics"/> a quibus eriguntur reciproce proportionalia; <lb/>id e&longs;t, ita ut &longs;it <emph type="italics"/>PA<emph.end type="italics"/> ad <emph type="italics"/>QB<emph.end type="italics"/> ut velocitas in <emph type="italics"/>Q<emph.end type="italics"/> ad velocitatem in <lb/><emph type="italics"/>P,<emph.end type="italics"/> & <emph type="italics"/>QB<emph.end type="italics"/> ad <emph type="italics"/>RC<emph.end type="italics"/> ut velocitas in <emph type="italics"/>R<emph.end type="italics"/> ad velocitatem in <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/> Per <lb/>perpendiculorum terminos <emph type="italics"/>A, B, C<emph.end type="italics"/> ad angulos rectos ducantur <emph type="italics"/>AD, <lb/>DBE, EC<emph.end type="italics"/> concurrentes in <emph type="italics"/>D<emph.end type="italics"/> & <emph type="italics"/>E:<emph.end type="italics"/> Et actæ <emph type="italics"/>TD, VE<emph.end type="italics"/> concur<lb/>rent in centro q<gap/>&longs;ito <emph type="italics"/>S.<emph.end type="italics"/></s>

<s>Nam perpendicula a centro <emph type="italics"/>S<emph.end type="italics"/><lb/>in tangentes <emph type="italics"/>PT, QT<emph.end type="italics"/> demi&longs;&longs;a (per <lb/>Corol. </s>

<s>1. Prop.I.) &longs;unt reciproce <lb/>ut velocitates corporis in punctis <lb/><emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>V<emph.end type="italics"/>; &c. </s>

<s>Nam perpendicula a centro <emph type="italics"/>S<emph.end type="italics"/><lb/>in tangentes <emph type="italics"/>PT, QT<emph.end type="italics"/> demi&longs;&longs;a (per <lb/>Corol. </s>

<s>1. Prop.I.) &longs;unt reciproce <lb/>ut velocitates corporis in punctis <lb/><emph type="italics"/>P<emph.end type="italics"/> & <emph type="italics"/>V<emph.end type="italics"/>; &c. </s>

<s>adeoque per con&longs;tructio<lb/>nem ut perpendicula <emph type="italics"/>AP, BQ<emph.end type="italics"/> di<lb/>recte, id e&longs;t ut perpendicula a pun<lb/>cto <emph type="italics"/>D<emph.end type="italics"/> in tangentes demi&longs;&longs;a. </s>