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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Tue, 14 May 2013 12:45:18 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd"> <archimedes> <info> <author>Newton, Isaac</author> <title>Philosophia naturalis principia mathematica</title> <date>1713</date> <place>Cambridge</place> <translator/> <lang>la</lang> <cvs_file>newto_philo_039_la_1713.xml</cvs_file> <cvs_version/> <locator>039.xml</locator> </info> <text> <front> </front> <body> <chap> <pb xlink:href="039/01/001.jpg"/> <p type="main"> <s><emph type="center"/>PHILOSOPHIÆ <lb/>NATURALIS <lb/>PRINCIPIA <lb/>MATHEMATICA.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>AUCTORE <lb/>ISAACO NEWTONO, <lb/>EQUITE A RATO.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>EDITIO SECUNDA AUCTIOR ET EMENDATIOR.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>CANTABRIGIÆ, MDCCXIII.<emph.end type="center"/></s></p><pb xlink:href="039/01/002.jpg"/><pb xlink:href="039/01/003.jpg"/> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>TRACTATUM HUNC D.D.D.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>JS. NEWTONUS.<emph.end type="italics"/></s></p><pb xlink:href="039/01/004.jpg"/></chap><chap><pb xlink:href="039/01/005.jpg"/> <p type="main"> <s><emph type="center"/>IN <lb/>VIRI PRÆSTANTISSIMI <lb/>ISAACI NEWTONI <lb/>OPUS HOCCE <lb/>MATHEMATICO PHYSICUM<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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> <s><lb/>Intima panduntur victi penetralia Cæli, <lb/>Nec latet, extremos quæ Vis circumrotet Orbes. </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><lb/>Hinc patet, horrificis qua &longs;it via flexa Cometis: <lb/>Di&longs;cimus hinc tandem, qua cau&longs;a argentea Phœbe <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><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 xlink:href="039/01/006.jpg"/>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;piciis, 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/>Qui &longs;criptis primus Tabulis compe&longs;cere Cædes, <lb/>Furta & Adulteria, & perjuræ crimina Fraudis; <lb/>Quive vagis populis circumdare mœnibus 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/>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/>Phœbus ade&longs;t, totoQ.E.I.ce&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></p></chap><chap><pb xlink:href="039/01/007.jpg"/> <p type="main"> <s><emph type="center"/>AUCTORIS <lb/>PRÆFATIO <lb/>AD <lb/>LECTOREM.<emph.end type="center"/></s></p> <p type="main"> <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;tantialibus & 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 Practicam &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 attingat<emph.end type="italics"/><lb/>Geometriæ; <emph type="italics"/>dein, 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 xlink:href="039/01/008.jpg"/><emph type="italics"/>&longs;ed non Geometrica. </s> <s>Ex<emph.end type="italics"/>Mechanica <emph type="italics"/>po&longs;tulatur 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 reliqua. </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 mundani. </s> <s>Ibi enim, ex phæ­<lb/>nomenis cæle&longs;tibus, per Propo&longs;itiones in Libris prioribus Mathe­<lb/>matice demon&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 ea om­<emph.end type="italics"/><pb xlink:href="039/01/009.jpg"/><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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="italics"/>IS. NEWTON.<emph.end type="italics"/></s></p><pb xlink:href="039/01/010.jpg"/> <p type="main"> <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></p> <p type="main"> <s>Dabam <emph type="italics"/>Londini,<emph.end type="italics"/><lb/>Mar. </s> <s>28. 1713. </s></p> <p type="main"> <s><emph type="italics"/>IS. NEWTON.<emph.end type="italics"/></s></p></chap><chap><pb xlink:href="039/01/011.jpg"/> <p type="main"> <s><emph type="center"/>EDITORIS <lb/>PRÆFATIO.<emph.end type="center"/></s></p> <p type="main"> <s>NEWTONIANÆ Philo&longs;ophiæ novam tibi, Lector benevole, <lb/>diuQ.E.D.&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></p> <p type="main"> <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 itaQ.E.D.cent. </s> <s>Cumque toti &longs;int in rerum no­<lb/>minibus, non in ip&longs;is rebus; Sermonem Q.E.D.m 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></p> <p type="main"> <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 xlink:href="039/01/012.jpg"/>&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></p> <p type="main"> <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 veritate 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/>Q.E.D.gnam 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 & &longs;olus ex Ap­<lb/>parentiis demon&longs;trare potuit, & &longs;peculationibus egregiis firmi&longs;&longs;i­<lb/>mum ponere fundamentum. </s></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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 xlink:href="039/01/013.jpg"/>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 infinitum, ver&longs;us plagam in quam tendit pondus <lb/>majus: omnino contra Experientiam. </s> <s>ItaQ.E.D.cendum 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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p><pb xlink:href="039/01/014.jpg"/> <p type="main"> <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 co­<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 quacunQ.E.D.mum cau&longs;a oriri fingatur. </s></p> <p type="main"> <s>Quin & hæc quoque concedenda &longs;unt, & Mathematice demon­<lb/>&longs;trantur: Si corpora plura motu æquabili revolvantur in Circulis <lb/>concentricis, & quadrata temporum periodieorum &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;tantiarum reciproce. </s></p><pb xlink:href="039/01/015.jpg"/> <p type="main"> <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;u 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></p> <p type="main"> <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/>ventis 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;&longs;e 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 di&longs;tantias illa &longs;e&longs;e Virtus extendat, <pb xlink:href="039/01/016.jpg"/>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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <s>Solis virtutem attractivam quoquover&longs;um propagari ad ingen­<lb/>tes u&longs;Q.E.D.&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 xlink:href="039/01/017.jpg"/>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></p> <p type="main"> <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></p> <p type="main"> <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 xlink:href="039/01/018.jpg"/>ni&longs;i per hanc Regulam de rerum univer&longs;arum natura judica­<lb/>mus. </s></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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 xlink:href="039/01/019.jpg"/>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;Q.E.D.m a cau&longs;is omnibus vacua fuerit & probe purgata Phi­<lb/>lo&longs;ophia. </s></p> <p type="main"> <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></p> <p type="main"> <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 xlink:href="039/01/020.jpg"/>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></p> <p type="main"> <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></p> <p type="main"> <s>Si plures Vortices in eodem &longs;patio contineri, & &longs;e&longs;e mutuo pe­<lb/>netrare, motibu&longs;Q.E.D.ver&longs;is revolvi ponantur; quoniam hi mo­<lb/>tus debent e&longs;&longs;e conformes delatorum corporum motibus, qui <pb xlink:href="039/01/021.jpg"/>&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 Effectu &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></p> <p type="main"> <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></p><pb xlink:href="039/01/022.jpg"/> <p type="main"> <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 undiQ.E.I. 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></p> <p type="main"> <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, & loeorum <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 xlink:href="039/01/023.jpg"/>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></p> <p type="main"> <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 xlink:href="039/01/024.jpg"/>&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;Q.E.D.­<lb/>diti Materiæ, ut Spatium a corporibus vacuum nullo pacto ad­<lb/>mittendum credere velint; videamus quo tandem oporteat illos <lb/>pervenire. </s></p> <p type="main"> <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 neQ.E.D.ci 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 fæces &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></p> <p type="main"> <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 xlink:href="039/01/025.jpg"/>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></p> <p type="main"> <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 referatis, 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></p> <p type="main"> <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 xlink:href="039/01/026.jpg"/>debeo: Huic & Tuas quæ debentur gratias, Lector benevole, non <lb/>denegabis. </s> <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> <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></p> <p type="main"> <s><emph type="italics"/>Cantabrigiæ,<emph.end type="italics"/><lb/>Maii 12. 1713. </s></p> <p type="main"> <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></p></chap><chap><pb xlink:href="039/01/027.jpg"/> <p type="main"> <s><emph type="center"/>INDEX CAPITUM <lb/>TOTIUS OPERIS.<emph.end type="center"/></s></p> <p type="main"> <s>PAG. </s></p> <p type="main"> <s>DEFINITIONES. 1 </s></p> <p type="main"> <s>AXIOMATA, SIVE LEGES MOTUS. 12 </s></p> <p type="main"> <s><emph type="center"/>DE MOTU CORPORUM LIBER PRIMUS.<emph.end type="center"/></s></p> <p type="main"> <s>SECT. I. <emph type="italics"/>DE Methodo rationum primarum & ultima­<lb/>rum.<emph.end type="italics"/>24 </s></p> <p type="main"> <s>SECT. II. <emph type="italics"/>De inventione Virium centripetarum.<emph.end type="italics"/>34 </s></p> <p type="main"> <s>SECT. III. <emph type="italics"/>De motu corporum in Conicis &longs;ectionibus eccentri­<lb/>cis.<emph.end type="italics"/>48 </s></p> <p type="main"> <s>SECT. IV. <emph type="italics"/>De inventione Orbium Elliptieorum, Parabolieorum <lb/>& Hyperbolieorum ex Umbilico dato.<emph.end type="italics"/>59 </s></p> <p type="main"> <s>SECT. V. <emph type="italics"/>De inventione Orbium ubi Umbilicus neuter datur.<emph.end type="italics"/>66 </s></p> <p type="main"> <s>SECT. VI. <emph type="italics"/>De inventione Motuum in Orbibus datis.<emph.end type="italics"/>97 </s></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <s>SECT. IX. <emph type="italics"/>De Motu corporum in Orbibus mobilibus, deque <lb/>Motu Ap&longs;idum.<emph.end type="italics"/>121 </s></p> <p type="main"> <s>SECT. X. <emph type="italics"/>De Motu corporum in Superficiebus datis, deque <lb/>Funependulorum Motu reciproco.<emph.end type="italics"/>132 </s></p> <p type="main"> <s>SECT. XI. <emph type="italics"/>De Motu corporum Viribus centripetis &longs;e mutuo pe­<lb/>tentium.<emph.end type="italics"/>147 </s></p> <p type="main"> <s>SECT. XII. <emph type="italics"/>De corporum Sphærieorum Viribus attractivis.<emph.end type="italics"/>173 </s></p><pb xlink:href="039/01/028.jpg"/> <p type="main"> <s>SECT. XIII. <emph type="italics"/>De corporum non Sphærieorum Viribus attracti­<lb/>vis.<emph.end type="italics"/>192 </s></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>DE MOTU CORPORUM LIBER SECUNDUS.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <s>SECT. IV. <emph type="italics"/>De corporum Circulari motu in Mediis re&longs;i&longs;tentibus.<emph.end type="italics"/><lb/>253 </s></p> <p type="main"> <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></p> <p type="main"> <s>SECT. VI. <emph type="italics"/>De Motu & Re&longs;i&longs;tentia corporum Funependulorum.<emph.end type="italics"/><lb/>272 </s></p> <p type="main"> <s>SECT. VII. <emph type="italics"/>De motu Fluidorum & re&longs;i&longs;tentia Projectilium.<emph.end type="italics"/>294 </s></p> <p type="main"> <s>SECT. VIII. <emph type="italics"/>De motu per Fluida propagato.<emph.end type="italics"/>329 </s></p> <p type="main"> <s>SECT. IX. <emph type="italics"/>De motu Circulari Fluidorum.<emph.end type="italics"/>345 </s></p> <p type="main"> <s><emph type="center"/>DE MUNDI SYSTEMATE LIBER TERTIUS.<emph.end type="center"/></s></p> <p type="main"> <s>REGULÆ PHILOSOPHANDI 357 </s></p> <p type="main"> <s>PHÆNOMENA 359 </s></p> <p type="main"> <s>PROPOSITIONES 362 </s></p> <p type="main"> <s>SCHOLIUM GENERALE. 481 </s></p></chap><chap><pb xlink:href="039/01/029.jpg"/> <p type="main"> <s><emph type="center"/>PHILOSOPHIÆ <lb/>NATURALIS <lb/>Principia <lb/>MATHEMATICA.<emph.end type="center"/><lb/></s></p> <p type="main"> <s><emph type="center"/>DEFINITIONES.<emph.end type="center"/><lb/></s></p> <p type="main"> <s><emph type="center"/>DEFINITIO I.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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>Et par e&longs;t ratio corporum omnium, quæ per cau&longs;as qua&longs;cun­<lb/>Q.E.D.ver&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><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>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></p> <p type="main"> <s><emph type="center"/>DEFINITIO II.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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></p><pb xlink:href="039/01/030.jpg" pagenum="2"/> <p type="main"> <s><emph type="center"/>DEFINITIO III.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s>Hæc &longs;emper proportionalis e&longs;t &longs;uo corpori, neQ.E.D.ffert 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>Unde etiam vis in&longs;ita nomine &longs;ignifican­<lb/>ti&longs;&longs;imo Vis Inertiæ dici po&longs;&longs;it. </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>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></p> <p type="main"> <s><emph type="center"/>DEFINITIO IV.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>DEFINITIO V.<emph.end type="center"/></s></p> <p type="main"> <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 utcunque tendunt.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <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>Lapis, in funda circum-<pb xlink:href="039/01/031.jpg" 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>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, quamQ.E.I.eò Centri­<lb/>petam appello, abibunt in rectis lineis uniformi cum motu. </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>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>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>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 etiam <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>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 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>Requiritur quippe, ut &longs;it ju&longs;tæ magnitudinis: <lb/>& Mathematieorum 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></p><pb xlink:href="039/01/032.jpg" pagenum="4"/> <p type="main"> <s><arrow.to.target n="note1"/></s></p> <p type="margin"> <s><margin.target id="note1"/>NI­<lb/>ES.</s></p> <p type="main"> <s><emph type="center"/>DEFINITIO VI.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s>Ut vis Magnetica pro mole magnetis vel inten&longs;ione virtutis major <lb/>in uno magnete, minor in alio. </s></p> <p type="main"> <s><emph type="center"/>DEFINITIO VII.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>DEFINITIO VIII.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s>Uti Pondus majus in majore corpore, minus in minore; inque <lb/>corpore eodem majus prope terram, minus in cælis. </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></p> <p type="main"> <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 xlink:href="039/01/033.jpg" pagenum="5"/>centro vis gravitantis) &longs;ive alia aliqua quæ non apparet. </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></p> <p type="main"> <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>Nam &longs;umma actionum vis acceleratricis in <lb/>&longs;ingulas corporis particulas e&longs;t vis motrix totius. </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>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></p> <p type="main"> <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;cunQ.E.I. 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></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s>I. </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. </s></p><pb xlink:href="039/01/034.jpg" pagenum="6"/><p><s><arrow.to.target n="note2"/></s></p> <p type="margin"> <s><margin.target id="note2"/></s></p> <p type="main"> <s>II. </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></p> <p type="main"> <s>III. </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/>loeorum. </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/>loeorum partium, & propterea internus & in corpore toto. </s></p> <p type="main"> <s>IV. </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 xlink:href="039/01/035.jpg" 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></p> <p type="main"> <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 </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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p><pb xlink:href="039/01/036.jpg" pagenum="8"/> <p type="main"> <s><arrow.to.target n="note3"/>quum illud datam po&longs;itionem &longs;ervet necne; quies vera ex horum <lb/>&longs;itu inter &longs;e definiri nequit. </s></p> <p type="margin"> <s><margin.target id="note3"/></s></p> <p type="main"> <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></p> <p type="main"> <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;Q.E.D.m <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></p> <p type="main"> <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 xlink:href="039/01/037.jpg" 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>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></p> <p type="main"> <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 xlink:href="039/01/038.jpg" pagenum="10"/><arrow.to.target n="note4"/>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></p> <p type="margin"> <s><margin.target id="note4"/>NI­<lb/>ES.</s></p> <p type="main"> <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></p> <p type="main"> <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 xlink:href="039/01/039.jpg" 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>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> <s>Hunc enim in finem Tractatum &longs;equentem <lb/>compo&longs;ui. <pb xlink:href="039/01/040.jpg" pagenum="12"/><arrow.to.target n="note5"/></s></p></chap><chap> <p type="margin"> <s><margin.target id="note5"/>TA,</s></p> <p type="main"> <s><emph type="center"/>AXIOMATA, <lb/>SIVE <lb/>LEGES MOTUS.<emph.end type="center"/><lb/></s></p> <p type="main"> <s><emph type="center"/>LEX I.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>LEX II.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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 generatrice 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;Q.E.D.termina­<lb/>tionem componitur. </s></p><pb xlink:href="039/01/041.jpg" pagenum="13"/> <p type="main"> <s><emph type="center"/>LEX III.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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/>utrinQ.E.D.&longs;tentus eodem relaxandi &longs;e conatu urgebit equum ver­<lb/>&longs;us lapidem, ac lapidem ver&longs;us equum; tantumQ.E.I.pediet 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></p> <p type="main"> <s><emph type="center"/>COROLLARIUM I.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s>Si corpus dato tempore, vi &longs;ola <lb/><figure id="id.039.01.041.1.jpg" xlink:href="039/01/041/1.jpg"/><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 xlink:href="039/01/042.jpg" pagenum="14"/><arrow.to.target n="note6"/></s></p> <p type="margin"> <s><margin.target id="note6"/>TA, <lb/>E</s></p> <p type="main"> <s><emph type="center"/>COROLLARIUM II.<emph.end type="center"/></s></p> <p type="main"> <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.</s><s> <emph type="italics"/>Quæ quidem compo&longs;itio <lb/>& re&longs;olutio abunde confirmatur ex Mechanica.<emph.end type="italics"/></s></p> <p type="main"> <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="id.039.01.042.1.jpg" xlink:href="039/01/042/1.jpg"/><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></p> <p type="main"> <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 xlink:href="039/01/043.jpg" 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><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 ex<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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>COROLLARIUM III.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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, adeoQ.E.D.ffe­<lb/>rentia motuum factorum in contrarias partes manebit eadem. </s></p> <p type="main"> <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/>beatQ.E.D.as velocitatis partes; & <emph type="italics"/>B<emph.end type="italics"/>&longs;equatur in eadem recta cum ve-<pb xlink:href="039/01/044.jpg" pagenum="16"/><arrow.to.target n="note7"/>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>AtQ.E.I.a &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></p> <p type="margin"> <s><margin.target id="note7"/>TA,</s></p> <p type="main"> <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.</s> <pb xlink:href="039/01/045.jpg" pagenum="17"/> <s>COROLLARIUM IV.</s></p> <p type="main"> <s><emph type="italics"/>Commune gravitas Centrum, corporum duorum vel plurimum, ab actio­<lb/>nibus corporum inter &longs;e non mutat &longs;tatum &longs;uum vel motus vel quie­<lb/>tis; & propterea corporum omnium in &longs; mutuo agentium (exclu&longs;is<lb/>actionibus & impedimentis externis) commune Centrum gravitatis<lb/>vel quie&longs;cit vel movetur uniformiter in directum.</s></p> <p type="main"> <s>Nam &longs;i puncta duo progrediantur uniformi cum motu in lineis<lb/>rectis, & di&longs;tantia eorum dividatur in ratione data, punctum divi­<lb/>dens vel quie&longs;cit vel progreditur uniformiter in linea recta. </s> <s>Hoc<lb/>po&longs;tea in Lemmate XXIII demon&longs;tratur, &longs;i corpora quotcunque moventur uNI­<lb/>formiter in lineis rectis, commune centrum gravitatis duorum quo­rumvis vel quie&longs;cit vel progreditur uniformiter in linea recta; propterea quod linea, horum corporum centra in recta uniformiter<lb/>progredientia jungens, dividitur ab hoc centro communis corporum duo­<lb/>rum & centri communis tertii in data ratione.</s> <s>Eodem modo &<lb/>commune centrum horum trium & quarti cuju&longs;vis vel quie&longs;cit vel<lb/>progreditur uniformiter in linea recta; propterea quod ab eo divi­<lb/>ditur di&longs;tantia inter centrum commune trium & centrum quarti in<lb/>data ratione, & &longs;ic in infinitum.</s> <s>Igitur in &longs;y&longs;temate corporum quæ<lb/>actionibus in &longs;e invicem alii&longs;que omnibus in &longs;e extrin&longs;ecus impre&longs;­<lb/>&longs;is omnino vacant, adeoque moventur &longs;ingula uniformiter in rectis<lb/>&longs;ingulis, commune omnium centrum gravitatis vel quie&longs;cit vel mo­<lb/>vetur uniformiter in directum.</s></p> <p type="main"> <s>Porro in &longs;y&longs;temate duorum corporum in &longs;e invicem agentium,<lb/>cum distantiæ centrorum utriusque a communi gravitatis centro &longs;int<lb/>reciproce ut corpora; erunt motus relativi corporum eorundem, vel<lb/>accedendi ad centrum illud vel ab eodem recedendi, æqualibus mutationibus in<lb/>partes contrarias factis, atque adeo ab actionibus horum corpo­<lb/>rum inter &longs;e, nec promovetur nec retardatur nec mutationem pa­<lb/>titur in &longs;tatu &longs;uo quoad motum vel quietem.</s> <s>In &longs;y&longs;temate autem<lb/>corporum plurimum, quoniam duorum quorumvis in &longs;e mutuo agen­<lb/>tium commune gravitatis centrum ob actionem illam nullatenus<pb xlink:href="039/01/046.jpg" pagenum="18"/><arrow.to.target n="note8"/>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></p> <p type="margin"> <s><margin.target id="note8"/>IATA, <lb/>VF.</s></p> <p type="main"> <s><emph type="center"/>COROLLARIUM V.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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>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></p> <p type="main"> <s><emph type="center"/>COROLLARIUM VI.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s>Nam vires illæ æqualiter (pro quantitatibus movendorum corpo-<pb xlink:href="039/01/047.jpg" 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></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Hactenus principia tradidi a Mathematicis recepta & experien­<lb/>tia multiplici confirmata. </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>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="id.039.01.047.1.jpg" xlink:href="039/01/047/1.jpg"/><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 xlink:href="039/01/048.jpg" 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"/>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 verus & 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/>&longs;imul 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, idQ.E.I. 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="id.039.01.048.1.jpg" xlink:href="039/01/048/1.jpg"/><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 xlink:href="039/01/049.jpg" 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></p> <p type="margin"> <s><margin.target id="note9"/>LEGES<lb/>MOTUS</s></p> <p type="main"> <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 xlink:href="039/01/050.jpg" pagenum="22"/><arrow.to.target n="note10"/></s></p> <p type="margin"> <s><margin.target id="note10"/>AXIOMATA <lb/>SIVE</s></p> <p type="main"> <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"/>motuQ.E.I. &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></p> <p type="main"> <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="id.039.01.050.1.jpg" xlink:href="039/01/050/1.jpg"/><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 utramQ.E.I. 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/>partem 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></p> <p type="main"> <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 xlink:href="039/01/051.jpg" 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 impedita 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 cuneum, 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 ligni cedunt cuneo, &longs;ecundum <lb/>lineas faciebus cunei perpendiculares. </s> <s>Et par e&longs;t ratio Machina­<lb/>rum omnium. </s></p> <p type="main"> <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></p><pb xlink:href="039/01/052.jpg" pagenum="24"/> <p type="main"> <s><arrow.to.target n="note11"/>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/></s></p> <p type="margin"> <s><margin.target id="note11"/>DE MOTU <lb/>CORPORUM</s></p></chap><chap><subchap1><subchap2> <p type="main"> <s><emph type="center"/>DE <lb/>MOTU CORPORUM <lb/>LIBER PRIMUS.<emph.end type="center"/><lb/></s></p> <p type="main"> <s><emph type="center"/>SECTIO I.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>LEMMA I.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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></p><pb xlink:href="039/01/053.jpg" pagenum="25"/> <p type="main"> <s><emph type="center"/>LEMMA II.<emph.end type="center"/></s></p> <p type="main"> <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/>prehen&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&longs;<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="id.039.01.053.1.jpg" xlink:href="039/01/053/1.jpg"/><lb/><emph type="italics"/>lelogramma<emph.end type="italics"/>aKbl, bLcm, cMdn, &c. <lb/><emph type="italics"/>Dein horum parallelogrammorum lati­<lb/>tudo minuatur, & numerus augeatur <lb/>in infinitum: dico quod ultimæ rationes, <lb/>quas habent 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></p> <p type="main"> <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>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></p> <p type="main"> <s><emph type="center"/>LEMMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Eædem rationes ultimæ &longs;unt etiam rationes æqualitatis, ubi paral­<lb/>lelogrammorum 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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et multo magis Figura rectilinea, quæ chordis evane&longs;-<pb xlink:href="039/01/054.jpg" pagenum="26"/><arrow.to.target n="note12"/>centium arcuum <emph type="italics"/>ab, bc, cd, &c.<emph.end type="italics"/>comprehenditur, coincidit ultimo <lb/>cum Figura curvilinea. </s></p> <p type="margin"> <s><margin.target id="note12"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Ut & Figura rectilinea circum&longs;cripta quæ tangentibus <lb/>eorundem arcuum comprehenditur. </s></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>LEMMA IV.<emph.end type="center"/></s></p> <p type="main"> <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;itQ.E.I.em 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></p><figure id="id.039.01.054.1.jpg" xlink:href="039/01/054/1.jpg"/> <p type="main"> <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></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc &longs;i duæ cuju&longs;cunque generis quantitates in eundem <lb/>partium numerum utcunQ.E.D.vidantur; & 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>Nam &longs;i in Lemmatis hujus Figuris &longs;umantur pa-<pb xlink:href="039/01/055.jpg" 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></p> <p type="main"> <s><emph type="center"/>LEMMA V.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>LEMMA VI.<emph.end type="center"/></s></p> <p type="main"> <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="id.039.01.055.1.jpg" xlink:href="039/01/055/1.jpg"/><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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>LEMMA VII.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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"/>produci, & &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"/><pb xlink:href="039/01/056.jpg" pagenum="28"/><arrow.to.target n="note13"/>evane&longs;cent, & rationem ultimam habebunt æqualitatis. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note13"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <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="id.039.01.056.1.jpg" xlink:href="039/01/056/1.jpg"/><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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>LEMMA VIII.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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="id.039.01.056.2.jpg" xlink:href="039/01/056/2.jpg"/><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></p> <p type="main"> <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></p><pb xlink:href="039/01/057.jpg" pagenum="29"/> <p type="main"> <s><emph type="center"/>LEMMA IX.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s>Etenim dum puncta <emph type="italics"/>B, C<emph.end type="italics"/>acce­<lb/><figure id="id.039.01.057.1.jpg" xlink:href="039/01/057/1.jpg"/><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, Ae:<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></p> <p type="main"> <s><emph type="center"/>LEMMA X.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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 temporum <emph type="italics"/>AD, AE. <lb/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/><pb xlink:href="039/01/058.jpg" pagenum="30"/><arrow.to.target n="note14"/></s></p> <p type="margin"> <s><margin.target id="note14"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <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>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>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></p> <p type="main"> <s><emph type="center"/>LEMMA XI.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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 xlink:href="039/01/059.jpg" 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="id.039.01.059.1.jpg" xlink:href="039/01/059/1.jpg"/><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><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <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><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <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. I & 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></p> <p type="main"> <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></p> <p type="main"> <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 convergunt. </s> <s><lb/>Nam &longs;agittæ illæ &longs;unt ut &longs;ubten&longs;æ <emph type="italics"/>BD, bd.<emph.end type="italics"/></s></p> <p type="main"> <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></p> <p type="main"> <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>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></p><pb xlink:href="039/01/060.jpg" pagenum="32"/> <p type="main"> <s><arrow.to.target n="note15"/></s></p> <p type="margin"> <s><margin.target id="note15"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <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>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></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <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>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>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>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 &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>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>Sed & inter duos quo&longs;vis ex his angulis pote&longs;t &longs;eries <lb/>utrinQ.E.I. 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"/>8/3, <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>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>Neque novit natura limitem. </s></p> <p type="main"> <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>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 xlink:href="039/01/061.jpg" 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></p> <p type="main"> <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, ubi attingit, nullam e&longs;&longs;e. </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>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>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></p> <p type="main"> <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><lb/>Verum hæc Objectio fal&longs;æ innititur hypothe&longs;i. </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></p><pb xlink:href="039/01/062.jpg" pagenum="34"/> <p type="main"> <s><arrow.to.target n="note16"/>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></p> <p type="margin"> <s><margin.target id="note16"/>DE MOTU <lb/>CORPORUM</s></p></subchap2><subchap2> <p type="main"> <s><emph type="center"/>SECTIO II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Inventione Virium Centripetarum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO I. THEOREMA I.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s>Dividatur tempus in partes æquales, & prima temporis parte de­<lb/>&longs;cribat 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="id.039.01.062.1.jpg" xlink:href="039/01/062/1.jpg"/><lb/>pergeret ad <emph type="italics"/>c,<emph.end type="italics"/>(per <lb/>Leg. </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/>venit ad <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>1.) reperietur in <emph type="italics"/>C,<emph.end type="italics"/>in <pb xlink:href="039/01/063.jpg" 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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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 xlink:href="039/01/064.jpg" pagenum="36"/><arrow.to.target n="note17"/>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></p> <p type="margin"> <s><margin.target id="note17"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <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 Parabolieorum <lb/>quos projectilia eodem tempore de&longs;cribunt. </s></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>PROPOSITIO II. THEOREMA II.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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>XL, <lb/>Lib. </s> <s>1 Elem. </s> <s>& Leg. </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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p><pb xlink:href="039/01/065.jpg" pagenum="37"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>PROPOSITIO III. THEOREMA III.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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; atQ.E.D. 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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Et vice ver&longs;a, &longs;i vis reliqua tendit quamproxime ad <pb xlink:href="039/01/066.jpg" pagenum="38"/><arrow.to.target n="note18"/>corpus alterum <emph type="italics"/>T,<emph.end type="italics"/>erunt areæ illæ temporibus quamproxime pro­<lb/>portionales. </s></p> <p type="margin"> <s><margin.target id="note18"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>PROPOSITIO IV. THEOREMA IV.<emph.end type="center"/></s></p> <p type="main"> <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></p> <p type="main"> <s>Tendunt hæ vires ad centra circulorum per Prop.II. & Corol. </s> <s>II. <lb/>Prop. </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>IV. Prop. </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></p> <p type="main"> <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></p><pb xlink:href="039/01/067.jpg" pagenum="39"/> <p type="main"> <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 periodieorum appli­<lb/>cata ad circulorum radios; hoc e&longs;t, in ratione compo&longs;ita ex ratione <lb/>radiorum directe & ratione duplicata temporum periodieorum in­<lb/>ver&longs;e. </s></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <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. <pb xlink:href="039/01/068.jpg" pagenum="40"/><arrow.to.target n="note19"/></s></p> <p type="margin"> <s><margin.target id="note19"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <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><lb/>IX. </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></p> <p type="main"> <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></p> <p type="main"> <s><emph type="center"/>PROPOSITIO. V. PROBLEMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Data quibu&longs;cunQ.E.I. 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></p> <p type="main"> <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 qæ&longs;ito <emph type="italics"/>S.<emph.end type="italics"/></s></p><pb xlink:href="039/01/069.jpg" pagenum="41"/><figure id="id.039.01.069.1.jpg" xlink:href="039/01/069/1.jpg"/> <p type="main"> <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> <s>Un­<lb/>de facile colligitur quod puncta <lb/><emph type="italics"/>S, D, T,<emph.end type="italics"/>&longs;unt in una recta. </s> <s>Et &longs;imili <lb/>argumento puncta <emph type="italics"/>S, E, V<emph.end type="italics"/>&longs;unt eti­<lb/>am in una recta; & propterea centrum <emph type="italics"/>S<emph.end type="italics"/>in concur&longs;u rectarum <emph type="italics"/>TD, VE<emph.end type="italics"/><lb/>ver&longs;atur. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO VI. THEOREMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si corpus in &longs;patio non re&longs;i&longs;tente circa centrum immobile in Orbe quocun­<lb/>que revolvatur, & arcum quemvis jamjam na&longs;centem tempore quàm <lb/>minimo de&longs;cribat, & &longs;agitta arcus duci intelligatur quæ chordam bi­<lb/>&longs;ecet, & producta tran&longs;eat per centrum virium: erit vis centripeta <lb/>in medio arcus, ut &longs;agitta directe & tempus bis inver&longs;e.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam &longs;agitta dato tempore e&longs;t ut vis (per Corol.4 Prop.I,) & augen­<lb/>do tempus in ratione quavis, ob auctum arcum in eadem ratione &longs;a­<lb/>gitta augetur in ratione illa duplicata (per Corol. </s> <s>2 & 3, Lem. </s> <s>XI,) ad­<lb/>eoque e&longs;t ut vis &longs;emel & tempus bis. </s> <s>Subducatur duplicata ratio tempo­<lb/>ris utrinque, & fiet vis ut &longs;agitta directe & tempus bis inver&longs;e. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s>Idem facile demon&longs;tratur etiam per Corol. </s> <s>4 Lem. </s> <s>X. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Si corpus <emph type="italics"/>P<emph.end type="italics"/>revolvendo <lb/><figure id="id.039.01.069.2.jpg" xlink:href="039/01/069/2.jpg"/><lb/>circa centrum <emph type="italics"/>S<emph.end type="italics"/>de&longs;cribat lineam <lb/>curvam <emph type="italics"/>APQ,<emph.end type="italics"/>tangat verò recta <lb/><emph type="italics"/>ZPR<emph.end type="italics"/>curvam illam in puncto <lb/>quovis <emph type="italics"/>P,<emph.end type="italics"/>& ad tangentem ab alio <lb/>quovis Curvæ puncto <emph type="italics"/>Q<emph.end type="italics"/>agatur <lb/><emph type="italics"/>QR<emph.end type="italics"/>di&longs;tantiæ <emph type="italics"/>SP<emph.end type="italics"/>parallela, ac <lb/>demittatur <emph type="italics"/>QT<emph.end type="italics"/>perpendicularis <lb/>ad di&longs;tantiam illam <emph type="italics"/>SP:<emph.end type="italics"/>vis cen­<lb/>tripeta erit reciproce ut &longs;olidum <lb/>(<emph type="italics"/>SP quad.XQT quad./QR<emph.end type="italics"/>) &longs;i modo &longs;olidi illius ea &longs;emper &longs;umatur quan­<lb/>titas, quæ ultimò fit ubi coeunt puncta <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/>Nam <emph type="italics"/>QR<emph.end type="italics"/>æqualis </s></p><pb xlink:href="039/01/070.jpg" pagenum="42"/> <p type="main"> <s><arrow.to.target n="note20"/>e&longs;t &longs;agittæ dupli arcus <emph type="italics"/>QP,<emph.end type="italics"/>in cujus medio e&longs;t <emph type="italics"/>P,<emph.end type="italics"/>& duplum trian­<lb/>guli <emph type="italics"/>SQP<emph.end type="italics"/>&longs;ive <emph type="italics"/>SPXQT,<emph.end type="italics"/>tempori quo arcus i&longs;te duplus de&longs;cribitur <lb/>proportionale e&longs;t, ideoque pro temporis exponente &longs;cribi pote&longs;t. </s></p> <p type="margin"> <s><margin.target id="note20"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Eodem argumento vis centripeta e&longs;t reciprocè ut &longs;olidum <lb/>(<emph type="italics"/>SYqXQPq/QR<emph.end type="italics"/>), &longs;i modo <emph type="italics"/>SY<emph.end type="italics"/>perpendiculum &longs;it a centro virium in Or­<lb/>bis tangentem <emph type="italics"/>PR<emph.end type="italics"/>demi&longs;&longs;um. </s> <s>Nam rectangula <emph type="italics"/>SYXQP<emph.end type="italics"/>& <emph type="italics"/>SPXQT<emph.end type="italics"/><lb/>æquantur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Si Orbis vel circulus e&longs;t, vel angulum contactus cum cir­<lb/>culo quam minimum continet, eandem habens curvaturam eundem­<lb/>que radium curvaturæ ad punctum contactus <emph type="italics"/>P<emph.end type="italics"/>; & &longs;i <emph type="italics"/>PV<emph.end type="italics"/>chorda <lb/>&longs;it circuli hujus a corpore per centrum virium acta: erit vis centri­<lb/>peta reciproce ut &longs;olidum <emph type="italics"/>SYqXPV.<emph.end type="italics"/>Nam <emph type="italics"/>PV<emph.end type="italics"/>e&longs;t (<emph type="italics"/>QPq/QR<emph.end type="italics"/>). </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Ii&longs;dem po&longs;itis, e&longs;t vis centripeta ut velocitas bis directe, <lb/>& chorda illa inver&longs;e. </s> <s>Nam velocitas e&longs;t reciproce ut perpendicu­<lb/>lum <emph type="italics"/>SY<emph.end type="italics"/>per Corol. </s> <s>I Prop. </s> <s>I. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Hinc &longs;i detur figura quævis curvilinea <emph type="italics"/>APQ,<emph.end type="italics"/>& in ea <lb/>detur etiam punctum <emph type="italics"/>S<emph.end type="italics"/>ad quod vis centripeta perpetuo dirigitur, <lb/>inveniri pote&longs;t lex vis centripetæ, qua corpus quodvis <emph type="italics"/>P<emph.end type="italics"/>a cur&longs;u <lb/>rectilineo perpetuò retractum in figuræ illius perimetro detinebitur <lb/>eamque revolvendo de&longs;cribet. </s> <s>Nimirum computandum e&longs;t vel &longs;o­<lb/>lidum (<emph type="italics"/>SPqXQTq/QR<emph.end type="italics"/>) vel &longs;olidum <emph type="italics"/>SYqXPV<emph.end type="italics"/>huic vi reciproce pro­<lb/>portionale. </s> <s>Ejus rei dabimus exempla in Problematis &longs;equentibus. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO VII. PROBLEMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Gyretur corpus in circumferentia Circuli, requiritur Lex vis centri­<lb/>petæ tendentis ad punctum quodcunQ.E.D.tum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>E&longs;to Circuli circumferentia <lb/><figure id="id.039.01.070.1.jpg" xlink:href="039/01/070/1.jpg"/><lb/><emph type="italics"/>VQPA,<emph.end type="italics"/>punctum datum ad <lb/>quod vis ceu ad <expan abbr="centrũ">centrum</expan> <expan abbr="&longs;uũ">&longs;uum</expan> ten­<lb/>dit <emph type="italics"/>S,<emph.end type="italics"/>corpus in circumferentia <lb/>latum <emph type="italics"/>P,<emph.end type="italics"/>locus proximus in quem <lb/>movebitur <emph type="italics"/>Q,<emph.end type="italics"/>& circuli tangens <lb/>ad locum priorem <emph type="italics"/>PRZ.<emph.end type="italics"/>Per <lb/>punctum <emph type="italics"/>S<emph.end type="italics"/>ducatur chorda <emph type="italics"/>PV,<emph.end type="italics"/><lb/>& acta circuli diametro <emph type="italics"/>VA<emph.end type="italics"/>jun­<lb/>gatur <emph type="italics"/>AP,<emph.end type="italics"/>& ad <emph type="italics"/>SP<emph.end type="italics"/>demittatur <lb/>perpendiculum <emph type="italics"/>QT,<emph.end type="italics"/>quod productum occurrat tangenti <emph type="italics"/>PR<emph.end type="italics"/>in <emph type="italics"/>Z,<emph.end type="italics"/><pb xlink:href="039/01/071.jpg" pagenum="43"/>ac denique per punctum <emph type="italics"/>Q<emph.end type="italics"/>agatur <emph type="italics"/>LR<emph.end type="italics"/>quæ ip&longs;i <emph type="italics"/>SP<emph.end type="italics"/>parallela <lb/>&longs;it & occurrat tum circulo in <emph type="italics"/>L<emph.end type="italics"/>tum tangenti <emph type="italics"/>PZ<emph.end type="italics"/>in <emph type="italics"/>R.<emph.end type="italics"/>Et <lb/>ob &longs;imilia triangula <emph type="italics"/>ZQR, ZTP, VPA<emph.end type="italics"/>; erit <emph type="italics"/>RP quad.<emph.end type="italics"/>hoc <lb/>e&longs;t <emph type="italics"/>QRL<emph.end type="italics"/>ad <emph type="italics"/>QT quad.<emph.end type="italics"/>ut <emph type="italics"/>AV quad.<emph.end type="italics"/>ad <emph type="italics"/>PV quad.<emph.end type="italics"/>Ideoque <lb/>(<emph type="italics"/>QRLXPV quad./AV quad.<emph.end type="italics"/>) æquatur <emph type="italics"/>QT quad.<emph.end type="italics"/>Ducantur hæc æqualia in <lb/>(<emph type="italics"/>SP quad./QR<emph.end type="italics"/>) &, punctis <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>Q<emph.end type="italics"/>coeuntibus, &longs;cribatur <emph type="italics"/>PV<emph.end type="italics"/>pro <emph type="italics"/>RL.<emph.end type="italics"/><lb/>Sic fiet (<emph type="italics"/>SP quad.XPV cub./AV quad.<emph.end type="italics"/>) æquale (<emph type="italics"/>SP quad.XQT quad./QR<emph.end type="italics"/>) Ergo (per <lb/>Corol.1 & 5 Prop.VI.) vis centripeta e&longs;t reciproce ut (<emph type="italics"/>SPqXPV cub./AV quad<emph.end type="italics"/>) <lb/>id e&longs;t, (ob datum <emph type="italics"/>AV quad.<emph.end type="italics"/>) reciproce ut quadratum di&longs;tantiæ &longs;eu <lb/>altitudinis <emph type="italics"/>SP<emph.end type="italics"/>& cubus chordæ <emph type="italics"/>PV<emph.end type="italics"/>conjunctim. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Ad tangentem <emph type="italics"/>PR<emph.end type="italics"/>productam demittatur perpendiculum <emph type="italics"/>SY,<emph.end type="italics"/><lb/>& ob &longs;imilia triangula <emph type="italics"/>SYP, VPA<emph.end type="italics"/>; erit <emph type="italics"/>AV<emph.end type="italics"/>ad <emph type="italics"/>PV<emph.end type="italics"/>ut <emph type="italics"/>SP<emph.end type="italics"/>ad <lb/><emph type="italics"/>SY,<emph.end type="italics"/>ideoque (<emph type="italics"/>SPXPV/AV<emph.end type="italics"/>) æquale <emph type="italics"/>SY,<emph.end type="italics"/>& (<emph type="italics"/>SP quad.XPV cub./AV quad.<emph.end type="italics"/>) æquale <lb/><emph type="italics"/>SY quad.XPV.<emph.end type="italics"/>Et propterea (per Corol.3 & 5 Prop.VI.) vis centri­<lb/>peta e&longs;t reciproce ut (<emph type="italics"/>SPqXPV cub./AVq<emph.end type="italics"/>) hoc e&longs;t, ob datam <emph type="italics"/>AV,<emph.end type="italics"/>reci­<lb/>proce ut <emph type="italics"/>SPqXPV cub. </s> <s><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i punctum datum <emph type="italics"/>S<emph.end type="italics"/>ad quod vis centripeta &longs;em­<lb/>per tendit, locetur in circumferentia hujus circuli, puta ad <emph type="italics"/>V<emph.end type="italics"/>; erit <lb/>vis centripeta reciproce ut quadrato cubus altitudinis <emph type="italics"/>SP.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Vis qua corpus <emph type="italics"/>P<emph.end type="italics"/>in cir­<lb/><figure id="id.039.01.071.1.jpg" xlink:href="039/01/071/1.jpg"/><lb/>culo <emph type="italics"/>APTV<emph.end type="italics"/>circum virium centrum <lb/><emph type="italics"/>S<emph.end type="italics"/>revolvitur, e&longs;t ad vim qua corpus <lb/>idem <emph type="italics"/>P<emph.end type="italics"/>in eodem circulo & eodem <lb/>tempore periodico circum aliud quod­<lb/>vis virium centrum <emph type="italics"/>R<emph.end type="italics"/>revolvi pote&longs;t, <lb/>ut <emph type="italics"/>RP quad.XSP<emph.end type="italics"/>ad cubum rectæ <emph type="italics"/>SG<emph.end type="italics"/><lb/>quæ a primo virium centro <emph type="italics"/>S<emph.end type="italics"/>ad or­<lb/>bis tangentem <emph type="italics"/>PG<emph.end type="italics"/>ducitur, & di&longs;tan­<lb/>tiæ corporis a &longs;ecundo virium centro <lb/>parallela e&longs;t. </s> <s>Nam, per con&longs;tructionem hujus Propo&longs;itionis, vis <lb/>prior e&longs;t ad vim po&longs;teriorem, ut <emph type="italics"/>RPqXPT cub.<emph.end type="italics"/>ad <emph type="italics"/>SPqXPV cub.<emph.end type="italics"/><pb xlink:href="039/01/072.jpg" pagenum="44"/><arrow.to.target n="note21"/>id e&longs;t, ut <emph type="italics"/>SPXRPq<emph.end type="italics"/>ad (<emph type="italics"/>SP cub.XPV cub/PT cub.<emph.end type="italics"/>) &longs;ive (ob &longs;imilia <lb/>triangula <emph type="italics"/>PSG, TPV<emph.end type="italics"/>) ad <emph type="italics"/>SG cub.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note21"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Vis, qua corpus <emph type="italics"/>P<emph.end type="italics"/>in Orbe quocunque circum virium <lb/>centrum <emph type="italics"/>S<emph.end type="italics"/>revolvitur, e&longs;t ad vim qua corpus idem <emph type="italics"/>P<emph.end type="italics"/>in eodem <lb/>orbe eodemque tempore periodico circum aliud quodvis virium <lb/>centrum <emph type="italics"/>R<emph.end type="italics"/>revolvi pote&longs;t, ut <emph type="italics"/>SPXRPq<emph.end type="italics"/>contentum utique &longs;ub di­<lb/>&longs;tantia corporis a primo virium centro <emph type="italics"/>S<emph.end type="italics"/>& quadrato di&longs;tantiæ ejus <lb/>a &longs;ecundo virium centro <emph type="italics"/>R<emph.end type="italics"/>ad cubum rectæ <emph type="italics"/>SG<emph.end type="italics"/>quæ a primo vi­<lb/>rium centro <emph type="italics"/>S<emph.end type="italics"/>ad orbis tangentem <emph type="italics"/>PG<emph.end type="italics"/>ducitur, & corporis a &longs;e­<lb/>cundo virium centro di&longs;tantiæ <emph type="italics"/>RP<emph.end type="italics"/>parallela e&longs;t. </s> <s>Nam vires in <lb/>hoc Orbe, ad ejus punctum quodvis <emph type="italics"/>P,<emph.end type="italics"/>eædem &longs;unt ac in Circulo <lb/>eju&longs;dem curvaturæ. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO. VIII. PROBLEMA. III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Moveatur corpus in Circulo<emph.end type="italics"/>PQA: <emph type="italics"/>ad hunc effectum requiritur Lex <lb/>vis centripetæ tendentis ad punctum adeo longinquum<emph.end type="italics"/>S, <emph type="italics"/>ut lineæ <lb/>omnes<emph.end type="italics"/>PS, RS <emph type="italics"/>ad id ductæ, pro parallelis haberi po&longs;&longs;int.<emph.end type="italics"/></s></p> <p type="main"> <s>A Circuli centro <emph type="italics"/>C<emph.end type="italics"/>agatur &longs;emidiameter <emph type="italics"/>CA<emph.end type="italics"/>parallelas i&longs;tas <lb/>perpendiculariter &longs;ecans in <emph type="italics"/>M<emph.end type="italics"/>& <lb/><figure id="id.039.01.072.1.jpg" xlink:href="039/01/072/1.jpg"/><lb/><emph type="italics"/>N,<emph.end type="italics"/>& jungatur <emph type="italics"/>CP.<emph.end type="italics"/>Ob &longs;imilia <lb/>triangula <emph type="italics"/>CPM, PZT<emph.end type="italics"/>& <emph type="italics"/>RZQ<emph.end type="italics"/><lb/>e&longs;t <emph type="italics"/>CPq<emph.end type="italics"/>ad <emph type="italics"/>PMq<emph.end type="italics"/>ut <emph type="italics"/>PRq<emph.end type="italics"/>ad <lb/><emph type="italics"/>QTq<emph.end type="italics"/>& ex natura Circuli <emph type="italics"/>PRq<emph.end type="italics"/><lb/>æquale e&longs;t rectangulo <emph type="italics"/>QRX√RN+QN<emph.end type="italics"/>&c. <lb/></s> <s>&longs;ive coeuntibus punctis <emph type="italics"/>P, Q<emph.end type="italics"/>rect­<lb/>angulo <emph type="italics"/>QRX2PM.<emph.end type="italics"/>Ergo e&longs;t <lb/><emph type="italics"/>CPq<emph.end type="italics"/>ad <emph type="italics"/>PM quad.<emph.end type="italics"/>ut <emph type="italics"/>QRX2PM<emph.end type="italics"/><lb/>ad <emph type="italics"/>QT quad.<emph.end type="italics"/>adeoque (<emph type="italics"/>QT quad./QR<emph.end type="italics"/>) <lb/>æquale (2<emph type="italics"/>PM cub./CP quad.<emph.end type="italics"/>), & (<emph type="italics"/>QT quad.XSP quad./QR<emph.end type="italics"/>) æquale (2<emph type="italics"/>PM cub.XSP qu./CP quad.<emph.end type="italics"/>) <lb/>E&longs;t ergo (per Corol. </s> <s>1 & 5 Prop. </s> <s>VI.) vis centripeta reciproce ut <lb/>(2<emph type="italics"/>PMcub.XSP quad./CP quad.<emph.end type="italics"/>) hoc e&longs;t (neglecta ratione determinata (2<emph type="italics"/>SP quad./CP quad.<emph.end type="italics"/>)) <lb/>reciproce ut <emph type="italics"/>PM cub. </s> <s><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s>Idem facile colligitur etiam ex Propo&longs;itione præcedente. </s></p><pb xlink:href="039/01/073.jpg" pagenum="45"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Et &longs;imili argumento corpus movebitur in Ellip&longs;i vel etiam in <lb/>Hyperbola vel Parabola, vi centripeta quæ &longs;it reciproce ut cu­<lb/>bus ordinatim applicatæ ad centrum virium maxime longinquum <lb/>tendentis. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO IX. PROBLEMA IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Gyretur corpus in Spirali<emph.end type="italics"/>PQS <emph type="italics"/>&longs;ecante radios omnes<emph.end type="italics"/>SP, SQ, <emph type="italics"/>&c.<emph.end type="italics"/><lb/><figure id="id.039.01.073.1.jpg" xlink:href="039/01/073/1.jpg"/><lb/><emph type="italics"/>in angulo dato: requiritur Lex <lb/>vis centripetæ tendentis ad <lb/>centrum Spiralis.<emph.end type="italics"/></s></p> <p type="main"> <s>Detur angulus indefinite par­<lb/>vus <emph type="italics"/>PSQ,<emph.end type="italics"/>& ob datos omnes <lb/>angulos dabitur &longs;pecie figura <emph type="italics"/>SPQRT.<emph.end type="italics"/>Ergo datur ratio (<emph type="italics"/>QT/QR<emph.end type="italics"/>), e&longs;tque <lb/>(<emph type="italics"/>QT quad./QR<emph.end type="italics"/>) ut <emph type="italics"/>QT,<emph.end type="italics"/>hoc e&longs;t ut <emph type="italics"/>SP.<emph.end type="italics"/>Mutetur jam uteunque angulus <emph type="italics"/>PSQ,<emph.end type="italics"/><lb/>& recta <emph type="italics"/>QR<emph.end type="italics"/>angulum contactus <emph type="italics"/>QPR<emph.end type="italics"/>&longs;ubtendens mutabitur (per <lb/>Lemma XI.) in duplicata ratione ip&longs;ius <emph type="italics"/>PR<emph.end type="italics"/>vel <emph type="italics"/>QT.<emph.end type="italics"/>Ergo manebit <lb/>(<emph type="italics"/>QT quad./QR<emph.end type="italics"/>) eadem quæ prius, hoc e&longs;t ut <emph type="italics"/>SP.<emph.end type="italics"/>Quare (<emph type="italics"/>QTq.XSPq/QR<emph.end type="italics"/>) <lb/>e&longs;t ut <emph type="italics"/>SP cub.<emph.end type="italics"/>adeoque (per Corol. </s> <s>1 & 5 Prop. </s> <s>VI.) vis centripeta e&longs;t <lb/>reciproce ut cubus di&longs;tantiæ <emph type="italics"/>SP. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Perpendiculum <emph type="italics"/>SY<emph.end type="italics"/>in tangentem demi&longs;&longs;um, & circuli Spiralem <lb/>tangentis chorda <emph type="italics"/>PV<emph.end type="italics"/>&longs;unt ad altitudinem <emph type="italics"/>SP<emph.end type="italics"/>in datis rationibus; <lb/>ideoque <emph type="italics"/>SP cub.<emph.end type="italics"/>e&longs;t ut <emph type="italics"/>SYqXPV,<emph.end type="italics"/>hoc e&longs;t (per Corol. </s> <s>3 & 5 Prop.VI.) <lb/>reciproce ut vis centripeta. </s></p> <p type="main"> <s><emph type="center"/>LEMMA XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Parallelogramma omnia, circa datæ Ellip&longs;eos vel Hyperbolæ diametros <lb/>qua&longs;vis conjugatas de&longs;cripta, e&longs;&longs;e inter &longs;e æqualia.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Con&longs;tat ex Conicis. <pb xlink:href="039/01/074.jpg" pagenum="46"/><arrow.to.target n="note22"/></s></p> <p type="margin"> <s><margin.target id="note22"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO X. PROBLEMA. V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Gyretur corpus in Ellip&longs;i: requiritur lex vis centripetæ tendentis ad <lb/>centrum Ellip&longs;eos.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sunto <emph type="italics"/>CA, CB<emph.end type="italics"/>&longs;emiaxes Ellip&longs;eos; <emph type="italics"/>GP, DK<emph.end type="italics"/>diametri conju­<lb/>gatæ; <emph type="italics"/>PF, Qt<emph.end type="italics"/>perpendicula ad diametros; <emph type="italics"/>Qv<emph.end type="italics"/>ordinatim appli­<lb/>cata ad diametrum <lb/><figure id="id.039.01.074.1.jpg" xlink:href="039/01/074/1.jpg"/><lb/><emph type="italics"/>GP<emph.end type="italics"/>; & &longs;i compleatur <lb/>parallelogrammum <lb/><emph type="italics"/>QvPR,<emph.end type="italics"/>erit (ex CoNI­<lb/>cis) <emph type="italics"/>PvG<emph.end type="italics"/>ad <emph type="italics"/>Qv quad.<emph.end type="italics"/><lb/>ut <emph type="italics"/>PC quad.<emph.end type="italics"/>ad <emph type="italics"/>CD <lb/>quad.<emph.end type="italics"/>& (ob &longs;imilia <lb/>triangula <emph type="italics"/>Qvt, PCF<emph.end type="italics"/>) <lb/><emph type="italics"/>Qv quad.<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>Qt <lb/>quad.<emph.end type="italics"/>ut <emph type="italics"/>PC quad.<emph.end type="italics"/>ad <lb/><emph type="italics"/>PF quad.<emph.end type="italics"/>& conjun­<lb/>ctis rationibus, <emph type="italics"/>PvG<emph.end type="italics"/><lb/>ad <emph type="italics"/>Qt quad.<emph.end type="italics"/>ut <emph type="italics"/>PC <lb/>quad.<emph.end type="italics"/>ad <emph type="italics"/>CD quad.<emph.end type="italics"/><lb/>& <emph type="italics"/>PC quad.<emph.end type="italics"/>ad <emph type="italics"/>PF <lb/>quad.<emph.end type="italics"/>id e&longs;t, <emph type="italics"/>vG<emph.end type="italics"/>ad <lb/>(<emph type="italics"/>Qt quad./Pv<emph.end type="italics"/>) ut <emph type="italics"/>PC quad.<emph.end type="italics"/><lb/>ad (<emph type="italics"/>CDqXPFq/PCq<emph.end type="italics"/>). Scribe <emph type="italics"/>QR<emph.end type="italics"/>pro <emph type="italics"/>Pv,<emph.end type="italics"/>& (per Lemma XII.) <emph type="italics"/>BCXCA<emph.end type="italics"/><lb/>pro <emph type="italics"/>CDXPF,<emph.end type="italics"/>nec non, punctis <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>Q<emph.end type="italics"/>coeuntibus, 2<emph type="italics"/>PC<emph.end type="italics"/>pro <lb/><emph type="italics"/>vG,<emph.end type="italics"/>& ductis extremis & mediis in &longs;e mutuo, fiet (<emph type="italics"/>Qt quad.XPCq/QR<emph.end type="italics"/>) <lb/>æquale (2<emph type="italics"/>BCqXCAq/PC<emph.end type="italics"/>). E&longs;t ergo (per Corol. </s> <s>5 Prop. </s> <s>VI.) vis centri­<lb/>peta reciproce ut (2<emph type="italics"/>BCqXGAq;/PC<emph.end type="italics"/>) id e&longs;t (ob datum 2<emph type="italics"/>BCqXCAq<emph.end type="italics"/>) <lb/>reciproce ut (1/<emph type="italics"/>PC<emph.end type="italics"/>); hoc e&longs;t, directe ut di&longs;tantia <emph type="italics"/>PC. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>In <emph type="italics"/>PG<emph.end type="italics"/>ab altera parte puncti <emph type="italics"/>t<emph.end type="italics"/>po&longs;ita intelligatur <emph type="italics"/>tu<emph.end type="italics"/>æqualis ip&longs;i <lb/><emph type="italics"/>tv<emph.end type="italics"/>; deinde cape <emph type="italics"/>uV<emph.end type="italics"/>quæ &longs;it ad <emph type="italics"/>vG<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>DC quad.<emph.end type="italics"/>ad <emph type="italics"/>PC quad.<emph.end type="italics"/><lb/>Et quoniam ex Conicis est <emph type="italics"/>Qv quad.<emph.end type="italics"/>ad <emph type="italics"/>PvG,<emph.end type="italics"/>ut <emph type="italics"/>DC quad.<emph.end type="italics"/>ad <lb/><emph type="italics"/>PC quad:<emph.end type="italics"/>erit <emph type="italics"/>Qv quad.<emph.end type="italics"/>æquale <emph type="italics"/>PvXuV.<emph.end type="italics"/>Unde quadratum chor-<pb xlink:href="039/01/075.jpg" pagenum="47"/>dæ arcus <emph type="italics"/>PQ<emph.end type="italics"/>erit æquale rectangulo <emph type="italics"/>VPv<emph.end type="italics"/>; adeoque Circulus qui <lb/><arrow.to.target n="note23"/>tangit Sectionem Conicam in <emph type="italics"/>P<emph.end type="italics"/>& tran&longs;it per punctum <emph type="italics"/>Q,<emph.end type="italics"/>tran&longs;ibit <lb/>etiam per punctum <emph type="italics"/>V.<emph.end type="italics"/>Coeant puncta <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>Q,<emph.end type="italics"/>& hic circulus <lb/>eju&longs;dem erit curvaturæ cum &longs;ectione conica in <emph type="italics"/>P,<emph.end type="italics"/>& <emph type="italics"/>PV<emph.end type="italics"/>æqualis erit <lb/>(2<emph type="italics"/>DCq/PC<emph.end type="italics"/>). Proinde vis qua corpus <emph type="italics"/>P<emph.end type="italics"/>in Ellip&longs;i revolvitur, erit reci­<lb/>proce ut (2<emph type="italics"/>DCq/PC<emph.end type="italics"/>) in <emph type="italics"/>PFq<emph.end type="italics"/>(per Corol. </s> <s>3 Prop. </s> <s>VI.) hoc e&longs;t (ob <lb/>datum 2<emph type="italics"/>DCq<emph.end type="italics"/>in <emph type="italics"/>PFq<emph.end type="italics"/>) directe ut <emph type="italics"/>PC. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note23"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. E&longs;t igitur vis ut di&longs;tantia corporis a centro Ellip&longs;eos: & <lb/>vici&longs;&longs;im, &longs;i vis &longs;it ut di&longs;tantia, movebitur corpus in Ellip&longs;i centrum <lb/>habente in centro virium, aut forte in Circulo, in quem utique <lb/>Ellip&longs;is migrare pote&longs;t. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et æqualia erunt revolutionum in Ellip&longs;ibus univer&longs;is cir­<lb/>cum centrum idem factarum periodica tempora. </s> <s>Nam tempora <lb/>illa in Ellip&longs;ibus &longs;imilibus æqualia &longs;unt per Corol. </s> <s>3 & 8, Prop. </s> <s>IV: <lb/>in Ellip&longs;ibus autem communem habentibus axem majorem, &longs;unt ad <lb/>invicem ut Ellip&longs;eon areæ totæ directe & arearum particulæ &longs;imul <lb/>de&longs;criptæ inver&longs;e; id e&longs;t, ut axes minores directe & corporum ve­<lb/>locitates in verticibus principalibus inver&longs;e; hoc e&longs;t, ut axes illi mi­<lb/>nores directe & ordinatim applicatæ ad axes alteros inver&longs;e; & prop­<lb/>terea (ob æqualitatem rationum directarum & inver&longs;arum) in ra­<lb/>tione æqualitatis. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Si Ellip&longs;is, centro in infinitum abeunte vertatur in Parabolam, <lb/>corpus movebitur in hac Parabola; & vis ad centrum infinite di­<lb/>&longs;tans jam tendens evadet æquabilis. </s> <s>Hoc e&longs;t Theorema <emph type="italics"/>Galilæi.<emph.end type="italics"/><lb/>Et &longs;i coni &longs;ectio Parabolica, inclinatione plani ad conum &longs;ectum <lb/>mutata, vertatur in Hyperbolam, movebitur corpus in hujus pe­<lb/>rimetro, vi centripeta in centrifugam ver&longs;a. </s> <s>Et quemadmo­<lb/>dum in Circulo vel Ellip&longs;i, &longs;i vires tendunt ad centrum figuræ <lb/>in Ab&longs;ci&longs;&longs;a po&longs;itum, hæ vires augendo vel diminuendo Ordinatas in <lb/>ratione quacunQ.E.D.ta, vel etiam mutando angulum inclinationis <lb/>Ordinatarum ad Ab&longs;ci&longs;&longs;am, &longs;emper augentur vel diminuuntur in <lb/>ratione di&longs;tantiarum a centro, &longs;i modo tempora periodica maneant <lb/>æqualia: &longs;ic etiam in figuris univer&longs;is, &longs;i Ordinatæ augeantur vel di­<lb/>minuantur in ratione quacunQ.E.D.ta, vel angulus ordinationis ut­<lb/>cunque mutetur, manente tempore periodico; vires ad centrum <lb/>quodcunQ.E.I. Ab&longs;ci&longs;&longs;a po&longs;itum tendentes a binis quibu&longs;vis figurarum locis, ad quæ termi­<lb/>nantur Ordinatæ corre&longs;pondentibus Ab&longs;ci&longs;&longs;arum punctis in&longs;i&longs;tentes, augentur vel &c. </s> <s>augentur vel diminuun­<lb/>tur in ratione di&longs;tantiarum a centro. <pb xlink:href="039/01/076.jpg" pagenum="48"/><arrow.to.target n="note24"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note24"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>SECTIO III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De motu Corporum in Conicis Sectionibus excentricis.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XI. PROBLEMA VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Revolvatur corpus in Ellip&longs;i: requiritur Lex vis centripetæ tenden­<lb/>tis ad umbilicum Ellip&longs;eos.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>E&longs;to Ellip&longs;eos umbilicus <emph type="italics"/>S.<emph.end type="italics"/>Agatur <emph type="italics"/>SP<emph.end type="italics"/>&longs;ecans Ellip&longs;eos <lb/>tum diametrum <emph type="italics"/>DK<emph.end type="italics"/>in <emph type="italics"/>E,<emph.end type="italics"/>tum ordinatim applicatam <emph type="italics"/>Qv<emph.end type="italics"/>in <lb/><emph type="italics"/>x,<emph.end type="italics"/>& compleatur parallelogrammum <emph type="italics"/>QxPR.<emph.end type="italics"/>Patet <emph type="italics"/>EP<emph.end type="italics"/>æqua­<lb/>lem e&longs;&longs;e &longs;emiaxi ma­<lb/><figure id="id.039.01.076.1.jpg" xlink:href="039/01/076/1.jpg"/><lb/>jori <emph type="italics"/>AC,<emph.end type="italics"/>eo quod <lb/>acta ab altero Ellip­<lb/>&longs;eos umbilico <emph type="italics"/>H<emph.end type="italics"/>li­<lb/>nea <emph type="italics"/>HI<emph.end type="italics"/>ip&longs;i <emph type="italics"/>EC<emph.end type="italics"/>pa­<lb/>rallela, (ob æquales <lb/><emph type="italics"/>CS, CH<emph.end type="italics"/>) æquentur <lb/><emph type="italics"/>ES, EI,<emph.end type="italics"/>adeo ut <emph type="italics"/>EP<emph.end type="italics"/><lb/>&longs;emi&longs;umma &longs;it ip&longs;a­<lb/>rum <emph type="italics"/>PS, PI,<emph.end type="italics"/>id e&longs;t <lb/>(ob parallelas <emph type="italics"/>HI, <lb/>PR<emph.end type="italics"/>& angulos æqua­<lb/>les <emph type="italics"/>IPR, HPZ<emph.end type="italics"/>) <lb/>ip&longs;arum <emph type="italics"/>PS, PH,<emph.end type="italics"/><lb/>quæ <expan abbr="cõjunctim">conjunctim</expan> axem <lb/>totum 2<emph type="italics"/>AC<emph.end type="italics"/>adæ­<lb/>quant. </s> <s>Ad <emph type="italics"/>SP<emph.end type="italics"/>de­<lb/>mittatur perpendicularis <emph type="italics"/>QT,<emph.end type="italics"/>& Ellip&longs;eos latere recto principali <lb/>(&longs;eu (2<emph type="italics"/>BC quad./AC<emph.end type="italics"/>)) dicto <emph type="italics"/>L,<emph.end type="italics"/>erit <emph type="italics"/>LXQR<emph.end type="italics"/>ad <emph type="italics"/>LXPv<emph.end type="italics"/>ut <emph type="italics"/>QR<emph.end type="italics"/>ad <lb/><emph type="italics"/>Pv,<emph.end type="italics"/>id e&longs;t ut <emph type="italics"/>PE<emph.end type="italics"/>&longs;eu <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>PC<emph.end type="italics"/>; & <emph type="italics"/>LXPv<emph.end type="italics"/>ad <emph type="italics"/>GvP<emph.end type="italics"/>ut <emph type="italics"/>L<emph.end type="italics"/>ad <lb/><emph type="italics"/>Gv<emph.end type="italics"/>; & <emph type="italics"/>GvP<emph.end type="italics"/>ad <emph type="italics"/>Qv quad.<emph.end type="italics"/>ut <emph type="italics"/>PC quad.<emph.end type="italics"/>ad <emph type="italics"/>CD quad<emph.end type="italics"/>; & (per Corol. </s> <s><lb/>2 Lem. </s> <s>VII.) <emph type="italics"/>Qv quad.<emph.end type="italics"/>ad <emph type="italics"/>Qx quad,<emph.end type="italics"/>punctis <emph type="italics"/>Q<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>coeuntibus, <lb/>e&longs;t ratio æqualitatis; & <emph type="italics"/>Qx quad.<emph.end type="italics"/>&longs;eu <emph type="italics"/>Qv quad.<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>QT quad.<emph.end type="italics"/><lb/>ut <emph type="italics"/>EP quad.<emph.end type="italics"/>ad <emph type="italics"/>PF quad,<emph.end type="italics"/>id e&longs;t ut <emph type="italics"/>CA quad.<emph.end type="italics"/>ad <emph type="italics"/>PF quad.<emph.end type="italics"/>&longs;ive (per <lb/>Lem XII.) ut <emph type="italics"/>CD quad.<emph.end type="italics"/>ad <emph type="italics"/>CB quad.<emph.end type="italics"/>Et conjunctis his omnibus ratio­<lb/>nibus, <emph type="italics"/>LXQR<emph.end type="italics"/>fit ad <emph type="italics"/>QT quad.<emph.end type="italics"/>ut <emph type="italics"/><expan abbr="ACXLXPCq.XCDq.">ACXLXPCq.XCDque</expan><emph.end type="italics"/>&longs;eu 2<emph type="italics"/><expan abbr="CBq.">CBque</expan> <lb/><expan abbr="XPCq.XCDq.">XPCq.XCDque</expan><emph.end type="italics"/>ad <emph type="italics"/><expan abbr="PCXGvXCDq.XCBq.">PCXGvXCDq.XCBque</expan><emph.end type="italics"/>&longs;ive ut 2<emph type="italics"/>PC<emph.end type="italics"/>ad <emph type="italics"/>Gv.<emph.end type="italics"/><pb xlink:href="039/01/077.jpg" pagenum="49"/>Sed, punctis <emph type="italics"/>Q<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>coeuntibus, <expan abbr="æquãtur">æquantur</expan> 2<emph type="italics"/>PC<emph.end type="italics"/>& <emph type="italics"/>Gv.<emph.end type="italics"/>Ergo & his pro­<lb/><arrow.to.target n="note25"/>portionalia <emph type="italics"/>LXQR<emph.end type="italics"/>& <emph type="italics"/>QT quad.<emph.end type="italics"/>æquantur. </s> <s>Ducantur hæc æqualia in <lb/>(<emph type="italics"/>SPq/QR<emph.end type="italics"/>) & fiet <emph type="italics"/><expan abbr="LXSPq.">LXSPque</expan><emph.end type="italics"/>æquale (<emph type="italics"/>SPq.XQTq/QR<emph.end type="italics"/>). Ergo (per Corol. </s> <s>1 <lb/>& 5 Prop. </s> <s>VI.) vis centripeta reciproce e&longs;t ut <emph type="italics"/><expan abbr="LXSPq.">LXSPque</expan><emph.end type="italics"/>id e&longs;t, reci­<lb/>proce in ratione duplicata di&longs;tantiæ <emph type="italics"/>SP. Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note25"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Cum vis ad centrum Ellip&longs;eos tendens, qua corpus <emph type="italics"/>P<emph.end type="italics"/>in Ellip&longs;i <lb/>illa revolvi pote&longs;t, &longs;it (per Corol. </s> <s>I Prop. </s> <s>X) ut <emph type="italics"/>CP<emph.end type="italics"/>di&longs;tantia cor­<lb/>poris ab Ellip&longs;eos centro <emph type="italics"/>C<emph.end type="italics"/>; ducatur <emph type="italics"/>CE<emph.end type="italics"/>parallela Ellip&longs;eos tan­<lb/>genti <emph type="italics"/>PR:<emph.end type="italics"/>& vis qua corpus idem <emph type="italics"/>P,<emph.end type="italics"/>circum aliud quodvis Ellip­<lb/>&longs;eos punctum <emph type="italics"/>S<emph.end type="italics"/>revolvi pote&longs;t, &longs;i <emph type="italics"/>CE<emph.end type="italics"/>& <emph type="italics"/>PS<emph.end type="italics"/>concurrant in <emph type="italics"/>E,<emph.end type="italics"/>erit ut <lb/>(<emph type="italics"/>PE cub./SPq<emph.end type="italics"/>) (per Corol. </s> <s>3 Prop. </s> <s>VII,) hoc e&longs;t, &longs;i punctum <emph type="italics"/>S<emph.end type="italics"/>&longs;it umbili­<lb/>cus Ellip&longs;eos, adeoque <emph type="italics"/>PE<emph.end type="italics"/>detur, ut <emph type="italics"/>SPq<emph.end type="italics"/>reciproce. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s>Eadem brevitate qua traduximus Problema quintum ad Parabo­<lb/>lam, & Hyperbolam, liceret idem hic facere: verum ob dignita­<lb/>tem Problematis & u&longs;um ejus in &longs;equentibus, non pigebit ca&longs;us ce­<lb/>teros demon&longs;tratione confirmare. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XII. PROBLEMA. VII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Moveatur corpus in Hyperbola: requiritur Lex vis centripetæ ten­<lb/>dentis ad umbilicum figuræ.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sunto <emph type="italics"/>CA, CB<emph.end type="italics"/>&longs;emi-axes Hyperbolæ; <emph type="italics"/>PG, KD<emph.end type="italics"/>diametri con­<lb/>jugatæ; <emph type="italics"/>PF, Qt<emph.end type="italics"/>perpendicula ad diametros; & <emph type="italics"/>Qv<emph.end type="italics"/>ordinatim <lb/>applicata ad diametrum <emph type="italics"/>GP.<emph.end type="italics"/>Agatur <emph type="italics"/>SP<emph.end type="italics"/>&longs;ecans cum diametrum <lb/><emph type="italics"/>DK<emph.end type="italics"/>in <emph type="italics"/>E,<emph.end type="italics"/>tum ordinatim applicatam <emph type="italics"/>Qv<emph.end type="italics"/>in <emph type="italics"/>x,<emph.end type="italics"/>& compleatur pa­<lb/>rallelogrammum <emph type="italics"/>QRPx.<emph.end type="italics"/>Patet <emph type="italics"/>EP<emph.end type="italics"/>æqualem e&longs;&longs;e &longs;emiaxi tran&longs;­<lb/>ver&longs;o <emph type="italics"/>AC,<emph.end type="italics"/>eo quod, acta ab altero Hyperbolæ umbilico <emph type="italics"/>H<emph.end type="italics"/>linea <lb/><emph type="italics"/>HI<emph.end type="italics"/>ip&longs;i <emph type="italics"/>EC<emph.end type="italics"/>parallela, ob æquales <emph type="italics"/>CS, CH,<emph.end type="italics"/>æquentur <emph type="italics"/>ES, EI<emph.end type="italics"/>; <lb/>adeo ut <emph type="italics"/>EP<emph.end type="italics"/>&longs;emidifferentia &longs;it ip&longs;arum <emph type="italics"/>PS, PI,<emph.end type="italics"/>id e&longs;t (ob pa­<lb/>rallelas <emph type="italics"/>IH, PR<emph.end type="italics"/>& angulos æquales <emph type="italics"/>IPR, HPZ<emph.end type="italics"/>) ip&longs;arum <emph type="italics"/>PS, <lb/>PH,<emph.end type="italics"/>quarum differentia axem totum 2<emph type="italics"/>AC<emph.end type="italics"/>adæquat. </s> <s>Ad <emph type="italics"/>SP<emph.end type="italics"/>de­<lb/>mittatur perpendicularis <emph type="italics"/>QT.<emph.end type="italics"/>Et Hyperbolæ latere recto princi­<lb/>pali (&longs;eu (2<emph type="italics"/>BCq/AC<emph.end type="italics"/>)) dicto <emph type="italics"/>L,<emph.end type="italics"/>erit <emph type="italics"/>LXQR<emph.end type="italics"/>ad <emph type="italics"/>LXPv<emph.end type="italics"/>ut <emph type="italics"/>QR<emph.end type="italics"/>ad <emph type="italics"/>Pv,<emph.end type="italics"/><lb/>id e&longs;t, ut <emph type="italics"/>PE<emph.end type="italics"/>&longs;eu <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>PC<emph.end type="italics"/>; Et <emph type="italics"/>LXPv<emph.end type="italics"/>ad <emph type="italics"/>GvP<emph.end type="italics"/>ut <emph type="italics"/>L<emph.end type="italics"/>ad <pb xlink:href="039/01/078.jpg" pagenum="50"/><arrow.to.target n="note26"/><emph type="italics"/>Gv<emph.end type="italics"/>; & <emph type="italics"/>GvP<emph.end type="italics"/>ad <emph type="italics"/>Qv quad.<emph.end type="italics"/>ut <emph type="italics"/><expan abbr="PCq.">PCque</expan><emph.end type="italics"/>ad <emph type="italics"/>CDq<emph.end type="italics"/>; & (per Corol. </s> <s>2. <lb/>Lem. </s> <s>VII.) <emph type="italics"/>Qv quad.<emph.end type="italics"/>ad <emph type="italics"/>Qx quad.<emph.end type="italics"/>punctis <emph type="italics"/>Q<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>coeuntibus fit <lb/>ratio æqualitatis; & <emph type="italics"/>Qx quad.<emph.end type="italics"/>&longs;eu <emph type="italics"/>Qv quad.<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>QTq,<emph.end type="italics"/>ut <emph type="italics"/>EPq,<emph.end type="italics"/><lb/>ad <emph type="italics"/>PFq,<emph.end type="italics"/>id e&longs;t ut <emph type="italics"/>CAq,<emph.end type="italics"/>ad <emph type="italics"/>PFq,<emph.end type="italics"/>&longs;ive (per Lem. </s> <s>XII.) ut <emph type="italics"/>CDq,<emph.end type="italics"/><lb/>ad <emph type="italics"/>CBq:<emph.end type="italics"/>& conjunctis his omnibus rationibus <emph type="italics"/>LXQR<emph.end type="italics"/>fit ad <lb/><emph type="italics"/><expan abbr="QTq.">QTque</expan><emph.end type="italics"/>ut <emph type="italics"/>ACXLXPCqXCDq<emph.end type="italics"/>&longs;eu 2<emph type="italics"/>CBqXPCqXCDq<emph.end type="italics"/>ad <lb/><emph type="italics"/>PCXGvXCDqXCB quad.<emph.end type="italics"/>&longs;ive ut 2<emph type="italics"/>PC<emph.end type="italics"/>ad <emph type="italics"/>Gv.<emph.end type="italics"/>Sed punctis <lb/><emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>Q<emph.end type="italics"/>coeuntibus æquantur 2<emph type="italics"/>PC<emph.end type="italics"/>& <emph type="italics"/>Gv.<emph.end type="italics"/>Ergo & his propor­<lb/>tionalia <emph type="italics"/>LXQR<emph.end type="italics"/>& <emph type="italics"/><expan abbr="QTq.">QTque</expan><emph.end type="italics"/>æquantur. </s> <s>Ducantur hæc æqualia in <lb/>(<emph type="italics"/>SPq/QR<emph.end type="italics"/>). & fiet <emph type="italics"/><expan abbr="LXSPq.">LXSPque</expan><emph.end type="italics"/>æquale (<emph type="italics"/>SPqXQTq/QR<emph.end type="italics"/>). Ergo (per Corol. </s> <s>I <lb/><figure id="id.039.01.078.1.jpg" xlink:href="039/01/078/1.jpg"/><lb/>& 5 Prop. </s> <s>VI.) vis centripeta reciproce e&longs;t ut <emph type="italics"/>LXSPq,<emph.end type="italics"/>id e&longs;t <lb/>reciproce in ratione duplicata di&longs;tantiæ <emph type="italics"/>SP. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/><pb xlink:href="039/01/079.jpg" pagenum="51"/><arrow.to.target n="note27"/></s></p> <p type="margin"> <s><margin.target id="note26"/>DE MOTU <lb/>CORPORUM</s></p> <p type="margin"> <s><margin.target id="note27"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Inveniatur vis quæ tendit ab Hyperbolæ centro <emph type="italics"/>C.<emph.end type="italics"/>Prodibit hæc <lb/>di&longs;tantiæ <emph type="italics"/>CP<emph.end type="italics"/>proportionalis. </s> <s>Inde vero (per Corol. </s> <s>3 Prop. </s> <s>VII.) <lb/>vis ad umbilicum <emph type="italics"/>S<emph.end type="italics"/>tendens erit ut (<emph type="italics"/>PEcub/SPq<emph.end type="italics"/>), hoc e&longs;t, ob datam <emph type="italics"/>PE,<emph.end type="italics"/><lb/>reciproce ut <emph type="italics"/><expan abbr="SPq.">SPque</expan> Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s>Eodem modo demon&longs;tratur quod corpus, hac vi centripeta in <lb/>centrifugam ver&longs;a, movebitur in Hyperbola conjugata. </s></p> <p type="main"> <s><emph type="center"/>LEMMA XIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Latus rectum Parabolæ ad verticem quemvis pertinens, e&longs;t quadru­<lb/>plum di&longs;tantiæ verticis illius ab umbilico figuræ.<emph.end type="italics"/>Patet ex Conicis. </s></p> <p type="main"> <s><emph type="center"/>LEMMA XIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Perpendiculum quod ab umbilico Parabolæ ad tangentem ejus demitti­<lb/>tur, medium e&longs;t proportionale inter di&longs;tantias umbilici a puncto con­<lb/>tactus & a vertice principali figuræ.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit enim <emph type="italics"/>AQP<emph.end type="italics"/>Parabola, <emph type="italics"/>S<emph.end type="italics"/>umbilicus ejus, <emph type="italics"/>A<emph.end type="italics"/>vertex principa­<lb/>lis <emph type="italics"/>P<emph.end type="italics"/>punctum <lb/><figure id="id.039.01.079.1.jpg" xlink:href="039/01/079/1.jpg"/><lb/>contactus, <emph type="italics"/>PO<emph.end type="italics"/><lb/>ordinatim ap­<lb/>plicata ad dia­<lb/>metrum prin­<lb/>cipalem, <emph type="italics"/>PM<emph.end type="italics"/><lb/>tangens dia­<lb/>metro princi­<lb/>pali occurrens <lb/>in <emph type="italics"/>M,<emph.end type="italics"/>& <emph type="italics"/>SN,<emph.end type="italics"/><lb/>linea perpen­<lb/>dicularis ab umbilico in tangentem. </s> <s>Jungatur <emph type="italics"/>AN,<emph.end type="italics"/>& ob æquales <lb/><emph type="italics"/>MS<emph.end type="italics"/>& <emph type="italics"/>SP, MN<emph.end type="italics"/>& <emph type="italics"/>NP, MA<emph.end type="italics"/>& <emph type="italics"/>AO,<emph.end type="italics"/>parallelæ erunt rectæ <lb/><emph type="italics"/>AN<emph.end type="italics"/>& <emph type="italics"/>OP,<emph.end type="italics"/>& inde triangulum <emph type="italics"/>SAN<emph.end type="italics"/>rectangulum erit ad <emph type="italics"/>A<emph.end type="italics"/>& <lb/>&longs;imile triangulis æqualibus <emph type="italics"/>SNM, SNP:<emph.end type="italics"/>Ergo <emph type="italics"/>PS<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>SN,<emph.end type="italics"/><lb/>ut <emph type="italics"/>SN<emph.end type="italics"/>ad <emph type="italics"/>SA. Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. <emph type="italics"/><expan abbr="PSq.">PSque</expan><emph.end type="italics"/>e&longs;t ad <emph type="italics"/><expan abbr="SNq.">SNque</expan><emph.end type="italics"/>ut <emph type="italics"/>PS<emph.end type="italics"/>ad <emph type="italics"/>SA.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et ob datam <emph type="italics"/>SA,<emph.end type="italics"/>e&longs;t <emph type="italics"/><expan abbr="SNq.">SNque</expan><emph.end type="italics"/>ut <emph type="italics"/>PS.<emph.end type="italics"/><pb xlink:href="039/01/080.jpg" pagenum="52"/><arrow.to.target n="note28"/></s></p> <p type="margin"> <s><margin.target id="note28"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Et concur&longs;us tangentis cuju&longs;vis <emph type="italics"/>PM<emph.end type="italics"/>cum recta <emph type="italics"/>SN,<emph.end type="italics"/><lb/>quæ ab umbilico in ip&longs;am perpendicularis e&longs;t, incidit in rectam <emph type="italics"/>AN,<emph.end type="italics"/><lb/>quæ Parabolam tangit in vertice principali. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO. XIII. PROBLEMA VIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Moveatur corpus in perimetro Parabolæ: requiritur Lex vis centri­<lb/>petæ tendentis ad umbilicum hujus figuræ.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Maneat con&longs;tructio Lemmatis, &longs;itque <emph type="italics"/>P<emph.end type="italics"/>corpus in perimetro Pa­<lb/>rabolæ, & a loco <emph type="italics"/>Q<emph.end type="italics"/>in quem corpus proxime movetur, age ip&longs;i <emph type="italics"/>SP<emph.end type="italics"/><lb/>parallelam <emph type="italics"/>QR<emph.end type="italics"/>& perpendicularem <emph type="italics"/>QT,<emph.end type="italics"/>necnon <emph type="italics"/>Qv<emph.end type="italics"/>tangenti pa­<lb/>rallelam & occurrentem tum diametro <emph type="italics"/>YPG<emph.end type="italics"/>in <emph type="italics"/>v,<emph.end type="italics"/>tum di&longs;tantiæ <lb/><emph type="italics"/>SP<emph.end type="italics"/>in <emph type="italics"/>x.<emph.end type="italics"/>Jam ob &longs;imilia triangula <emph type="italics"/>Pxv, SPM<emph.end type="italics"/>& æqualia unius <lb/>latera <emph type="italics"/>SM, SP,<emph.end type="italics"/>æqualia &longs;unt alterius latera <emph type="italics"/>Px<emph.end type="italics"/>&longs;eu <emph type="italics"/>QR<emph.end type="italics"/>& <emph type="italics"/>Pv.<emph.end type="italics"/><lb/>Sed, ex Conicis, quadratum ordinatæ <emph type="italics"/>Qv<emph.end type="italics"/>æquale e&longs;t rectangulo &longs;ub <lb/>latere recto & &longs;egmento diametri <emph type="italics"/>Pv,<emph.end type="italics"/>id e&longs;t (per Lem. </s> <s>XIII.) rectangu­<lb/>lo 4 <emph type="italics"/>PSXPv,<emph.end type="italics"/>&longs;eu 4 <emph type="italics"/>PSXQR<emph.end type="italics"/>; & punctis <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>Q<emph.end type="italics"/>coeuntibus, ra­<lb/>tio <emph type="italics"/>Qv<emph.end type="italics"/>ad <emph type="italics"/>Qx<emph.end type="italics"/>per (per Corol. </s> <s>2 Lem. </s> <s>VII.) fit ratio æqualitatis. </s> <s>Er­<lb/>go <emph type="italics"/>Qxquad.<emph.end type="italics"/>eo <lb/><figure id="id.039.01.080.1.jpg" xlink:href="039/01/080/1.jpg"/><lb/>in ca&longs;u, æquale <lb/>e&longs;t rectangu­<lb/>lo 4 <emph type="italics"/>PSXQR.<emph.end type="italics"/><lb/>E&longs;t autem (ob <lb/>&longs;imilia trian­<lb/>gula <emph type="italics"/>QxT, <lb/>SPN) <expan abbr="Qxq.">Qxque</expan><emph.end type="italics"/><lb/>ad <emph type="italics"/><expan abbr="QTq.">QTque</expan><emph.end type="italics"/>ut <lb/><emph type="italics"/><expan abbr="PSq.">PSque</expan><emph.end type="italics"/>ad <emph type="italics"/><expan abbr="SNq.">SNque</expan><emph.end type="italics"/><lb/>hoc e&longs;t (per <lb/>Corol. </s> <s>1. Lem. </s> <s>XIV.) ut <emph type="italics"/>PS<emph.end type="italics"/>ad <emph type="italics"/>SA,<emph.end type="italics"/>id e&longs;t ut 4 <emph type="italics"/>PSXQR<emph.end type="italics"/><lb/>ad 4<emph type="italics"/>SAXQR,<emph.end type="italics"/>& inde (per Prop. </s> <s>IX. Lib. </s> <s>v. </s> <s>Elem.) <emph type="italics"/><expan abbr="QTq.">QTque</expan><emph.end type="italics"/>& <lb/>4<emph type="italics"/>SAXQR<emph.end type="italics"/>æquantur. </s> <s>Ducantur hæc æqualia in (<emph type="italics"/>SPq./QR<emph.end type="italics"/>), & fiet <lb/>(<emph type="italics"/>SPq.XQTq./QR<emph.end type="italics"/>) æquale <emph type="italics"/>SPq.X4SA:<emph.end type="italics"/>& propterea (per Corol. </s> <s>1 & 5 <lb/>Prop. </s> <s>VI.) vis centripeta e&longs;t reciproce ut <emph type="italics"/>SPq.X4SA,<emph.end type="italics"/>id e&longs;t, ob da­<lb/>tam 4<emph type="italics"/>SA,<emph.end type="italics"/>reciproce in duplicata ratione di&longs;tantiæ <emph type="italics"/>SP. Q.E.I.<emph.end type="italics"/></s></p><pb xlink:href="039/01/081.jpg" pagenum="53"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Ex tribus novi&longs;&longs;imis Propo&longs;itionibus con&longs;equens e&longs;t, quod </s></p> <p type="main"> <s><arrow.to.target n="note29"/>&longs;i corpus quodvis <emph type="italics"/>P,<emph.end type="italics"/>&longs;ecundum lineam quamvis rectam <emph type="italics"/>PR,<emph.end type="italics"/>qua­<lb/>cunque cum velocitate exeat de loco <emph type="italics"/>P,<emph.end type="italics"/>& vi centripeta quæ &longs;it re­<lb/>ciproce proportionalis quadrato di&longs;tantiæ loeorum a centro, &longs;imul <lb/>agitetur; movebitur hoc corpus in aliqua &longs;ectionum Conicarum <lb/>umbilicum habente in centro virium; & contra. </s> <s>Nam datis umbi­<lb/>lico & puncto contactus & po&longs;itione tangentis, de&longs;cribi pote&longs;t &longs;ectio <lb/>Conica quæ curvaturam datam ad punctum illud habebit. </s> <s>Datur <lb/>autem curvatura ex data vi centripeta: & Orbes duo &longs;e mutuo tan­<lb/>gentes, eadem vi centripeta de&longs;cribi non po&longs;&longs;unt. </s></p> <p type="margin"> <s><margin.target id="note29"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si velocitas, quacum corpus exit de loco &longs;uo <emph type="italics"/>P,<emph.end type="italics"/>ea <lb/>&longs;it, qua lineola <emph type="italics"/>PR<emph.end type="italics"/>in minima aliqua temporis particula de&longs;cribi <lb/>po&longs;&longs;it, & vis centripeta potis &longs;it eodem tempore corpus idem mo­<lb/>vere per &longs;patium <emph type="italics"/>QR:<emph.end type="italics"/>movebitur hoc corpus in Conica aliqua &longs;e­<lb/>ctione, cujus latus rectum principale e&longs;t quantitas illa (<emph type="italics"/>QTq./QR<emph.end type="italics"/>) quæ <lb/>ultimo fit ubi lineolæ <emph type="italics"/>PR, QR<emph.end type="italics"/>in infinitum diminuuntur. </s> <s>Circu­<lb/>lum in his Corollariis refero ad Ellip&longs;in, & ca&longs;um excipio ubi cor­<lb/>pus recta de&longs;cendit ad centrum. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XIV. THEOREMA VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si corpora plura revolvantur circa centrum commune, & vis centri­<lb/>peta &longs;it reciproce in duplicata ratione di&longs;tantiæ loeorum a centro; <lb/>dico quod Orbium Latera recta principalia &longs;unt in duplicata ratio­<lb/>one arearum quas corpora, radiis ad centrum ductis, eodem tempore <lb/>de&longs;cribunt.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam, per Corol. </s> <s>2. Prop. </s> <s>XIII, Latus rectum <emph type="italics"/>L<emph.end type="italics"/>æquale e&longs;t quan­<lb/>titati (<emph type="italics"/>QTq./QR<emph.end type="italics"/>) quæ ultimo fit ubi coeunt puncta <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/>Sed linea <lb/>minima <emph type="italics"/>QR,<emph.end type="italics"/>dato tempore, e&longs;t ut vis centripeta generans, hoc <lb/>e&longs;t (per Hypothe&longs;in) reciproce ut <emph type="italics"/><expan abbr="SPq.">SPque</expan><emph.end type="italics"/>Ergo (<emph type="italics"/>QTq./QR<emph.end type="italics"/>) e&longs;t ut <lb/><emph type="italics"/><expan abbr="QTq.XSPq.">QTq.XSPque</expan><emph.end type="italics"/>hoc e&longs;t, latus rectum <emph type="italics"/>L<emph.end type="italics"/>in duplicata ratione areæ <lb/><emph type="italics"/>QTXSP. Q.E.D.<emph.end type="italics"/><pb xlink:href="039/01/082.jpg" pagenum="54"/><arrow.to.target n="note30"/></s></p> <p type="margin"> <s><margin.target id="note30"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc Ellip&longs;eos area tota, eique proportionale rectangu­<lb/>lum &longs;ub axibus, e&longs;t in ratione compo&longs;ita ex &longs;ubduplicata ratione <lb/>lateris recti & ratione temporis periodici. </s> <s>Namque area tota e&longs;t <lb/>ut area <emph type="italics"/>QTXSP,<emph.end type="italics"/>quæ dato tempore de&longs;cribitur, ducta in &c. </s> <s>ducta in tempus periodicum. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XV. THEOREMA VII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Ii&longs;dem po&longs;itis, dico quod Tempora periodica in Ellip&longs;ibus &longs;unt in ratione <lb/>&longs;e&longs;quiplicata majorum axium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Namque axis minor e&longs;t medius proportionalis inter axem majo­<lb/>rem & latus rectum, atque adeo rectangulum &longs;ub axibus e&longs;t in ra­<lb/>tione compo&longs;ita ex &longs;ubduplicata ratione lateris recti & &longs;e&longs;quiplicata <lb/>ratione axis majoris. </s> <s>Sed hoc rectangulum, per Corollarium Prop. </s> <s><lb/>XIV. e&longs;t in ratione compo&longs;ita ex &longs;ubduplicata ratione lateris recti <lb/>& ratione periodici temporis. </s> <s>Dematur utrobique &longs;ubduplicata <lb/>ratio lateris recti, & manebit &longs;e&longs;quiplicata ratio majoris axis æqua­<lb/>lis rationi periodici temporis. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Sunt igitur tempora periodica in Ellip&longs;ibus eadem ac in <lb/>Circulis, quorum diametri æquantur majoribus axibus Ellip&longs;eon. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XVI. THEOREMA VIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, & actis ad corpora lineis rectis, quæ ibidem tangant Or­<lb/>bitas, demi&longs;&longs;i&longs;que ab umbilico communi ad has tangentes perpendi­<lb/>cularibus: dico quod Velocitates corporum &longs;unt in ratione compo&longs;i­<lb/>ta ex ratione perpendiculorum inver&longs;e & &longs;ubduplicata ratione la­<lb/>terum rectorum principalium directe.<emph.end type="italics"/></s></p> <p type="main"> <s>Ab umbilico <emph type="italics"/>S<emph.end type="italics"/>ad tangentem <emph type="italics"/>PR<emph.end type="italics"/>demitte perpendiculum <emph type="italics"/>SY<emph.end type="italics"/><lb/>& velocitas corporis <emph type="italics"/>P<emph.end type="italics"/>erit reciproce in &longs;ubduplicata ratione quan­<lb/>titatis (<emph type="italics"/>SYq/L<emph.end type="italics"/>). Nam velocitas illa e&longs;t ut arcus quam minimus <emph type="italics"/>PQ<emph.end type="italics"/><lb/>in data temporis particula de&longs;criptus, hoc e&longs;t (per Lem. </s> <s>VII.) ut <lb/>tangens <emph type="italics"/>PR,<emph.end type="italics"/>id e&longs;t (ob proportionales <emph type="italics"/>PR<emph.end type="italics"/>ad <emph type="italics"/>QT<emph.end type="italics"/>& <emph type="italics"/>SP<emph.end type="italics"/>ad <emph type="italics"/>SY<emph.end type="italics"/>) ut <lb/>(<emph type="italics"/>SPXQT/SY<emph.end type="italics"/>), &longs;ive ut <emph type="italics"/>SY<emph.end type="italics"/>reciproce & <emph type="italics"/>SPXQT<emph.end type="italics"/>directe; e&longs;tque <pb xlink:href="039/01/083.jpg" pagenum="55"/><emph type="italics"/>SPXQT<emph.end type="italics"/>ut area dato tempore de&longs;cripta, id e&longs;t, per Prop. </s> <s>XIV. </s></p> <p type="main"> <s><arrow.to.target n="note31"/>in &longs;ubduplicata ratione lateris recti. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note31"/>LIBER <lb/>PRIMUS.</s></p><figure id="id.039.01.083.1.jpg" xlink:href="039/01/083/1.jpg"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Latera recta principalia &longs;unt in ratione compo&longs;ita ex <lb/>duplicata ratione perpendiculorum & duplicata ratione veloci­<lb/>tatum. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Velocitates corporum in maximis & minimis ab umbi­<lb/>lico communi di&longs;tantiis, &longs;unt in ratione compo&longs;ita ex ratione di­<lb/>&longs;tantiarum inver&longs;e & &longs;ubduplicata ratione laterum rectorum princi­<lb/>palium directe. </s> <s>Nam perpendicula jam &longs;unt ip&longs;æ di&longs;tantiæ. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Ideoque velocitas in Conica &longs;ectione, in maxima vel <lb/>minima ab umbilico di&longs;tantia, e&longs;t ad velocitatem in Circulo in ea­<lb/>dem à centro di&longs;tantia, in &longs;ubduplicata ratione lateris recti princi­<lb/>palis ad duplam illam di&longs;tantiam. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Corporum in Ellip&longs;ibus gyrantium velocitates in medi­<lb/>ocribus di&longs;tantiis ab umbilico communi &longs;unt eædem quæ corporum <lb/>gyrantium in Circulis ad ea&longs;dem di&longs;tantias; hoc e&longs;t (per Corol 6. <lb/>Prop. </s> <s>IV.) reciproce in &longs;ubduplicata ratione di&longs;tantiarum. </s> <s>Nam <lb/>perpendicula jam &longs;unt &longs;emi-axes minores; & hi &longs;unt ut mediæ <lb/>proportionales inter di&longs;tantias & latera recta. </s> <s>Componatur hæc <lb/>ratio inver&longs;e cum &longs;ubduplicata ratione laterum rectorum directe, & <lb/>fiet ratio &longs;ubduplicata di&longs;tantiarum inver&longs;e. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. In eadem figura, vel etiam in figuris diver&longs;is, quarum <pb xlink:href="039/01/084.jpg" pagenum="56"/><arrow.to.target n="note32"/>latera recta principalia &longs;unt æqualia, velocitas corporis e&longs;t reciproce <lb/>ut perpendiculum demi&longs;&longs;um ab umbilico ad tangentem. </s></p> <p type="margin"> <s><margin.target id="note32"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. In Parabola, velocitas e&longs;t reciproce in &longs;ubduplicata ra­<lb/>tione di&longs;tantiæ corporis ab umbilico figuræ; in Ellip&longs;i magis varia­<lb/>tur, in Hyperbola minus, quam in hac ratione. </s> <s>Nam (per Corol. </s> <s><lb/>2. Lem. </s> <s>XIV.) perpendiculum demi&longs;&longs;um ab umbilico ad tangentem <lb/>Parabolæ e&longs;t in &longs;ubduplicata ratione di&longs;tantiæ. </s> <s>In Hyperbola per­<lb/>pendiculum minus variatur, in Ellip&longs;i magis. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. In Parabola, velocitas corporis ad quamvis ab umbili­<lb/>co di&longs;tantiam, e&longs;t ad velocitatem corporis revolventis in Circulo <lb/>ad eandem a centro di&longs;tantiam, in &longs;ubduplicata ratione numeri bi­<lb/>narii ad unitatem; in Ellip&longs;i minor e&longs;t, in Hyperbola major quam <lb/>in hac ratione. </s> <s>Nam per hujus Corollarium &longs;ecundum, velocitas <lb/>in vertice Parabolæ e&longs;t in hac ratione, & per Corollaria &longs;exta hu­<lb/>jus & Propo&longs;itionis quartæ, &longs;ervatur eadem proportio in omnibus <lb/>di&longs;tantiis. </s> <s>Hinc etiam in Parabola velocitas ubique æqualis e&longs;t ve­<lb/>locitati corporis revolventis in Circulo ad dimidiam di&longs;tantiam, in <lb/>Ellip&longs;i minor e&longs;t, in Hyperbola major. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>8. Velocitas gyrantis in Sectione quavis Conica e&longs;t ad ve­<lb/>locitatem gyrantis in Circulo in di&longs;tantia dimidii lateris recti princi­<lb/>palis Sectionis, ut di&longs;tantia illa ad perpendiculum ab umbilico in <lb/>tangentem Sectionis demi&longs;&longs;um. </s> <s>Patet per Corollarium quintum. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>9. Unde cum (per Corol. </s> <s>6. Prop. </s> <s>IV.) velocitas gyrantis <lb/>in hoc Circulo &longs;it ad velocitatem gyrantis in Circulo quovis alio, <lb/>reciproce in &longs;ubduplicata ratione di&longs;tantiarum; fiet ex æquo velo­<lb/>citas gyrantis in Conica &longs;ectione ad velocitatem gyrantis in Circulo <lb/>in eadem di&longs;tantia, ut media proportionalis inter di&longs;tantiam illam <lb/>communem & &longs;emi&longs;&longs;em principalis lateris recti &longs;ectionis, ad per­<lb/>pendiculum ab umbilico communi in tangentem &longs;ectionis de­<lb/>mi&longs;&longs;um. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XVII. PROBLEMA. IX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod vis centripeta &longs;it reciproce proportionalis quadrato di&longs;tan­<lb/>&longs;tantiæ loeorum a centro, & quod vis illius quantitas ab&longs;oluta &longs;it <lb/>cognita; requiritur Linea quam corpus de&longs;cribit, de loco dato, cum <lb/>data velocitate, &longs;ecundum datam rectam egrediens.<emph.end type="italics"/></s></p> <p type="main"> <s>Vis centripeta tendens ad punctum <emph type="italics"/>S<emph.end type="italics"/>ea &longs;it qua corpus <emph type="italics"/>p<emph.end type="italics"/>in or­<lb/>bita quavis data <emph type="italics"/>pq<emph.end type="italics"/>gyretur, & cogno&longs;catur hujus velocitas in loco <emph type="italics"/>p.<emph.end type="italics"/><pb xlink:href="039/01/085.jpg" pagenum="57"/>De loco <emph type="italics"/>P,<emph.end type="italics"/>&longs;ecundum lineam <emph type="italics"/>PR,<emph.end type="italics"/>exeat corpus <emph type="italics"/>P,<emph.end type="italics"/>cum data velo­<lb/><arrow.to.target n="note33"/>citate, & mox inde, cogente vi centripeta, deflectat illud in CoNI­<lb/>&longs;ectionem <emph type="italics"/><expan abbr="Pq.">Pque</expan><emph.end type="italics"/>Hanc igitur recta <emph type="italics"/>PR<emph.end type="italics"/>tanget in <emph type="italics"/>P.<emph.end type="italics"/>Tangat itidem <lb/>recta aliqua <emph type="italics"/>pr<emph.end type="italics"/>Orbitam <emph type="italics"/>pq<emph.end type="italics"/>in <emph type="italics"/>p,<emph.end type="italics"/>& &longs;i ab <emph type="italics"/>S<emph.end type="italics"/>ad eas tangentes demitti <lb/>intelligantur perpendicula, erit (per Corol. </s> <s>1. Prop. </s> <s>XVI.) latus re­<lb/>ctum principale Coni&longs;ectionis ad latus rectum principale Orbitæ, in <lb/>ratione compo&longs;ita ex duplicata ratione perpendiculorum & dupli­<lb/>cata ratione velocitatum, atque adeo datur. </s> <s>Sit i&longs;tud <emph type="italics"/>L.<emph.end type="italics"/>Da­<lb/>tur præterea Coni&longs;e­<lb/><figure id="id.039.01.085.1.jpg" xlink:href="039/01/085/1.jpg"/><lb/>ctionis umbilicus <emph type="italics"/>S.<emph.end type="italics"/><lb/>Anguli <emph type="italics"/>RPS<emph.end type="italics"/>com­<lb/>plementum ad du­<lb/>os rectos fiat angu­<lb/>lus <emph type="italics"/>RPH,<emph.end type="italics"/>& dabi­<lb/>tur po&longs;itione linea <lb/><emph type="italics"/>PH,<emph.end type="italics"/>in qua umbilicus <lb/>alter <emph type="italics"/>H<emph.end type="italics"/>locatur. </s> <s>De­<lb/>mi&longs;&longs;o ad <emph type="italics"/>PH<emph.end type="italics"/>perpen­<lb/>diculo <emph type="italics"/>SK,<emph.end type="italics"/>erigi intelligatur &longs;emiaxis conjugatus <emph type="italics"/>BC,<emph.end type="italics"/>& erit <lb/><emph type="italics"/>SPq.-2KPH+PHq.=SHq.=4CHq.=4BHq-4BCq.= <lb/>―SP+PH: quad. -LX―SP+PH=SPq.+2SPH+PHq. <lb/>-LX―SP+PH.<emph.end type="italics"/>Addantur utrobique 2<emph type="italics"/>KPH-SPq-PHq <lb/>+LX―SP+PH,<emph.end type="italics"/>& fiet <emph type="italics"/>LX―SP+PH=2SPH+2KPH,<emph.end type="italics"/><lb/>&longs;eu <emph type="italics"/>SP+PH,<emph.end type="italics"/>ad <emph type="italics"/>PH,<emph.end type="italics"/>ut 2<emph type="italics"/>SP+2KP<emph.end type="italics"/>ad <emph type="italics"/>L.<emph.end type="italics"/>Unde datur <emph type="italics"/>PH<emph.end type="italics"/><lb/>tam longitudine quam po&longs;itione. </s> <s>Nimirum &longs;i ea fit corporis &c. </s> <s>in <emph type="italics"/>P<emph.end type="italics"/><lb/>velocitas, ut latus rectum <emph type="italics"/>L<emph.end type="italics"/>minus fuerit quam 2 <emph type="italics"/>SP+2KP,<emph.end type="italics"/><lb/>jacebit <emph type="italics"/>PH<emph.end type="italics"/>ad eandem partem tangentis <emph type="italics"/>PR<emph.end type="italics"/>cum linea <emph type="italics"/>PS,<emph.end type="italics"/><lb/>adeoque figura erit Ellip&longs;is, & ex datis umbilicis <emph type="italics"/>S, H,<emph.end type="italics"/>& axe <lb/>principali <emph type="italics"/>SP+PH,<emph.end type="italics"/>dabitur: Sin tanta &longs;it corporis velocitas ut <lb/>latus rectum <emph type="italics"/>L<emph.end type="italics"/>æquale fuerit 2 <emph type="italics"/>SP+2KP,<emph.end type="italics"/>longitudo <emph type="italics"/>PH<emph.end type="italics"/>infi­<lb/>nita erit, & propterea figura erit Parabola axem habens <emph type="italics"/>SH<emph.end type="italics"/>paral­<lb/>lelum lineæ <emph type="italics"/>PK,<emph.end type="italics"/>& inde dabitur. </s> <s>Quod &longs;i corpus majori adhuc <lb/>cum velocitate de loco &longs;uo <emph type="italics"/>P<emph.end type="italics"/>exeat, capienda erit longitudo <emph type="italics"/>PH<emph.end type="italics"/><lb/>ad alteram partem tangentis, adeoque tangente inter umbilicos per­<lb/>gente, figura erit Hyperbola axem habens principalem æqualem dif­<lb/>ferentiæ linearum <emph type="italics"/>SP<emph.end type="italics"/>& <emph type="italics"/>PH,<emph.end type="italics"/>& inde dabitur. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note33"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc in omni Coni&longs;ectione ex dato vertice principali <emph type="italics"/>D,<emph.end type="italics"/><lb/>latere recto <emph type="italics"/>L,<emph.end type="italics"/>& umbilico <emph type="italics"/>S,<emph.end type="italics"/>datur umbilicus alter <emph type="italics"/>H<emph.end type="italics"/>capiendo <emph type="italics"/>DH,<emph.end type="italics"/><lb/>ad <emph type="italics"/>DS<emph.end type="italics"/>ut e&longs;t latus rectum ad differentiam inter latus rectum & <lb/>4 <emph type="italics"/>DS.<emph.end type="italics"/>Nam proportio <emph type="italics"/>SP+PH<emph.end type="italics"/>ad <emph type="italics"/>PH<emph.end type="italics"/>ut 2 <emph type="italics"/>SP+2KP<emph.end type="italics"/>ad <emph type="italics"/>L,<emph.end type="italics"/><pb xlink:href="039/01/086.jpg" pagenum="58"/><arrow.to.target n="note34"/>in ca&longs;u hujus Corollarii, &longs;it <emph type="italics"/>DS+DH<emph.end type="italics"/>ad <emph type="italics"/>DH<emph.end type="italics"/>ut 4 <emph type="italics"/>DS<emph.end type="italics"/>ad <emph type="italics"/>L,<emph.end type="italics"/>& <lb/>divi&longs;im <emph type="italics"/>DS<emph.end type="italics"/>ad <emph type="italics"/>DH<emph.end type="italics"/>ut 4 <emph type="italics"/>DS-L<emph.end type="italics"/>ad <emph type="italics"/>L.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note34"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Unde &longs;i datur corporis velocitas in vertice principali <emph type="italics"/>D,<emph.end type="italics"/><lb/>invenietur Orbita expedite, capiendo &longs;cilicet latus rectum ejus, ad <lb/>duplam di&longs;tantiam <emph type="italics"/>DS,<emph.end type="italics"/>in duplicata ratione velocitatis hujus datæ <lb/>ad velocitatem corporis in Circulo, ad di&longs;tantiam <emph type="italics"/>DS,<emph.end type="italics"/>gyrantis (per <lb/>Corol. </s> <s>3. Prop. </s> <s>XVI.) dein <emph type="italics"/>DH<emph.end type="italics"/>ad <emph type="italics"/>DS<emph.end type="italics"/>ut latus rectum ad differen­<lb/>tiam inter latus rectum & 4 <emph type="italics"/>DS.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Hinc etiam &longs;i corpus moveatur in Sectione quacunque <lb/>Conica, & ex Orbe &longs;uo impul&longs;u quocunque exturbetur; cogno&longs;ci <lb/>pote&longs;t Orbis in quo po&longs;tea cur&longs;um &longs;uum peraget. </s> <s>Nam componen­<lb/>do proprium corporis motum cum motu illo quem impul&longs;us &longs;olus <lb/>generaret, habebitur motus quocum corpus de dato impul&longs;us loco, <lb/>&longs;ecundum rectam po&longs;itione datam, exibit. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Et &longs;i corpus illud vi aliqua extrin&longs;ecus impre&longs;&longs;a conti­<lb/>nuo perturbetur, innote&longs;cet cur&longs;us quam proxime, colligendo mu­<lb/>tationes quas vis illa in punctis quibu&longs;dam inducit, & ex &longs;eriei ana­<lb/>logia mutationes continuas in locis intermediis æ&longs;timando. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Si corpus <emph type="italics"/>P<emph.end type="italics"/>vi centripeta ad <lb/><figure id="id.039.01.086.1.jpg" xlink:href="039/01/086/1.jpg"/><lb/>punctum quodcunQ.E.D.tum <emph type="italics"/>R<emph.end type="italics"/><lb/>tendente moveatur in perimetro <lb/>datæ cuju&longs;cunque Sectionis co­<lb/>nicæ cujus centrum &longs;it <emph type="italics"/>C,<emph.end type="italics"/>& re­<lb/>quiratur Lex vis centripetæ: du­<lb/>catur <emph type="italics"/>CG<emph.end type="italics"/>radio <emph type="italics"/>RP<emph.end type="italics"/>paralle­<lb/>la, & Orbis tangenti <emph type="italics"/>PG<emph.end type="italics"/>oc­<lb/>currens in <emph type="italics"/>G<emph.end type="italics"/>; & vis illa (per <lb/>Corol. </s> <s>1 & Schol. </s> <s>Prop. </s> <s>X, & Corol. </s> <s>3 Prop. </s> <s>VII.) erit ut <lb/>(<emph type="italics"/>CG cub./RP quad.<emph.end type="italics"/>) <pb xlink:href="039/01/087.jpg" pagenum="59"/><arrow.to.target n="note35"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note35"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>SECTIO IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Inventione Orbium Elliptieorum, Parabolieorum & Hyperbolico­<lb/>rum ex umbilico dato.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>LEMMA XV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si ab Ellip&longs;eos vel Hyperbolæ cuju&longs;vis umbilicis duobus<emph.end type="italics"/>S, H, <emph type="italics"/>ad <lb/>punctum quodvis tertium<emph.end type="italics"/>V <emph type="italics"/>inflectantur rectæ duæ<emph.end type="italics"/>SV, HV, <lb/><emph type="italics"/>quarum una<emph.end type="italics"/>HV <emph type="italics"/>æqualis &longs;it axi principali figuræ, altera<emph.end type="italics"/>SV <emph type="italics"/>a <lb/>perpendiculo<emph.end type="italics"/>TR <emph type="italics"/>in &longs;e demi&longs;&longs;o bi-<emph.end type="italics"/><lb/><figure id="id.039.01.087.1.jpg" xlink:href="039/01/087/1.jpg"/><lb/><emph type="italics"/>&longs;ecetur in<emph.end type="italics"/>T; <emph type="italics"/>perpendiculum illud<emph.end type="italics"/><lb/>TR <emph type="italics"/>&longs;ectionem Conicam alicubi tan­<lb/>get: & contra, &longs;i tangit, erit<emph.end type="italics"/>HV <lb/><emph type="italics"/>æqualis axi principali figuræ.<emph.end type="italics"/></s></p> <p type="main"> <s>Secet enim perpendiculum <emph type="italics"/>TR<emph.end type="italics"/>re­<lb/>ctam <emph type="italics"/>HV<emph.end type="italics"/>productam, &longs;i opus fuerit, <lb/>in <emph type="italics"/>R<emph.end type="italics"/>; & jungatur <emph type="italics"/>SR.<emph.end type="italics"/>Ob æquales <lb/><emph type="italics"/>TS, TV,<emph.end type="italics"/>æquales erunt & rectæ <emph type="italics"/>SR, VR<emph.end type="italics"/>& anguli <emph type="italics"/>TRS, TRV.<emph.end type="italics"/><lb/>Unde punctum <emph type="italics"/>R<emph.end type="italics"/>erit ad Sectionem Conicam, & perpendiculum <lb/><emph type="italics"/>TR<emph.end type="italics"/>tanget eandem: & contra. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XVIII. PROBLEMA X.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Datis umbilico & axibus principalibus de&longs;cribere Trajectorias Ellipti­<lb/>cas & Hyperbolicas, quæ tran&longs;ibunt per puncta data, & rectas po­<lb/>&longs;itione datas contingent.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>S<emph.end type="italics"/>communis umbilicus figurarum; <emph type="italics"/>AB<emph.end type="italics"/>longitudo axis prin­<lb/>cipalis Trajectoriæ cuju&longs;vis; <emph type="italics"/>P<emph.end type="italics"/>punctum per quod Trajectoria de­<lb/>bet tran&longs;ire; & <emph type="italics"/>TR<emph.end type="italics"/>recta quam debet tangere. </s> <s>Centro <emph type="italics"/>P<emph.end type="italics"/>inter­<lb/>vallo <emph type="italics"/>AB-SP,<emph.end type="italics"/>&longs;i orbita &longs;it Ellip&longs;is, vel <emph type="italics"/>AB+SP,<emph.end type="italics"/>&longs;i ea &longs;it Hy­<lb/>perbola, de&longs;cribatur circulus <emph type="italics"/>HG.<emph.end type="italics"/>Ad tangentem <emph type="italics"/>TR<emph.end type="italics"/>demittatur <lb/>perpendiculum <emph type="italics"/>ST,<emph.end type="italics"/>& producatur idem ad <emph type="italics"/>V,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>TV<emph.end type="italics"/>æqualis <lb/><emph type="italics"/>ST<emph.end type="italics"/>; centroque <emph type="italics"/>V<emph.end type="italics"/>& intervallo <emph type="italics"/>AB<emph.end type="italics"/>de&longs;cribatur circulus <emph type="italics"/>FH.<emph.end type="italics"/>Hac <pb xlink:href="039/01/088.jpg" pagenum="60"/><arrow.to.target n="note36"/>methodo &longs;ive dentur duo puncta <emph type="italics"/>P, p,<emph.end type="italics"/>&longs;ive duæ tangentes <emph type="italics"/>TR, <lb/>tr,<emph.end type="italics"/>&longs;ive punctum <emph type="italics"/>P<emph.end type="italics"/>& tangens <lb/><figure id="id.039.01.088.1.jpg" xlink:href="039/01/088/1.jpg"/><lb/><emph type="italics"/>TR,<emph.end type="italics"/>de&longs;cribendi &longs;unt circuli duo. </s> <s><lb/>Sit <emph type="italics"/>H<emph.end type="italics"/>eorum inter&longs;ectio com­<lb/>munis, & umbilicis <emph type="italics"/>S, H,<emph.end type="italics"/>axe illo <lb/>dato de&longs;cribatur Trajectoria. </s> <s><lb/>Dico factum. </s> <s>Nam Trajecto­<lb/>ctoria de&longs;cripta (eo quod <emph type="italics"/>PH <lb/>+SP<emph.end type="italics"/>in Ellip&longs;i, & <emph type="italics"/>PH-SP<emph.end type="italics"/><lb/>in Hyperbola æquatur axi) <lb/>tran&longs;ibit per punctum <emph type="italics"/>P,<emph.end type="italics"/>& <lb/>(per Lemma &longs;uperius) tanget <lb/>rectam <emph type="italics"/>TR.<emph.end type="italics"/>Et eodem argu­<lb/>mento vel tran&longs;ibit eadem per <lb/>puncta duo <emph type="italics"/>P, p,<emph.end type="italics"/>vel tanget re­<lb/>ctas duas <emph type="italics"/>TR, tr. </s> <s>q.E.F.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note36"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XIX. PROBLEMA XI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Circa datum umbilicum Trajectoriam Parabolicam de&longs;cribere, quæ <lb/>tran&longs;ibit per puncta data, & rectas po&longs;itione datas continget.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>S<emph.end type="italics"/>umbilicus, <emph type="italics"/>P<emph.end type="italics"/>punctum & <emph type="italics"/>TR<emph.end type="italics"/>tangens Trajectoriæ de&longs;cri­<lb/>bendæ. </s> <s>Centro <emph type="italics"/>P,<emph.end type="italics"/>intervallo <emph type="italics"/>PS<emph.end type="italics"/>de&longs;cribe cir­<lb/><figure id="id.039.01.088.2.jpg" xlink:href="039/01/088/2.jpg"/><lb/>culum <emph type="italics"/>FG.<emph.end type="italics"/>Ab umbilico ad tangentem demit­<lb/>te perpendicularem <emph type="italics"/>ST,<emph.end type="italics"/>& produc eam ad <emph type="italics"/>V,<emph.end type="italics"/><lb/>ut &longs;it <emph type="italics"/>TV<emph.end type="italics"/>æqualis <emph type="italics"/>ST.<emph.end type="italics"/>Eodem modo de&longs;cri­<lb/>bendus e&longs;t alter circulus <emph type="italics"/>fg,<emph.end type="italics"/>&longs;i datur alterum <lb/>punctum <emph type="italics"/>p<emph.end type="italics"/>; vel inveniendum alterum punctum <lb/><emph type="italics"/>v,<emph.end type="italics"/>&longs;i datur altera tangens <emph type="italics"/>tr<emph.end type="italics"/>; dein ducenda re­<lb/>cta <emph type="italics"/>IF<emph.end type="italics"/>quæ tangat duos circulos <emph type="italics"/>FG, fg<emph.end type="italics"/>&longs;i <lb/>dantur duo puncta <emph type="italics"/>P, p,<emph.end type="italics"/>vel tran&longs;eat per duo <lb/>puncta <emph type="italics"/>V, v,<emph.end type="italics"/>&longs;i dantur duæ tangentes <emph type="italics"/>TR, tr,<emph.end type="italics"/>vel <lb/>tangat circulum <emph type="italics"/>FG<emph.end type="italics"/>& tran&longs;eat per punctum <emph type="italics"/>V,<emph.end type="italics"/><lb/>&longs;i datur punctum <emph type="italics"/>P<emph.end type="italics"/>& tangens <emph type="italics"/>TR.<emph.end type="italics"/>Ad <emph type="italics"/>FI<emph.end type="italics"/>demitte perpendicula­<lb/>rem <emph type="italics"/>SI,<emph.end type="italics"/>eamque bi&longs;eca in <emph type="italics"/>K<emph.end type="italics"/>; & axe <emph type="italics"/>SK,<emph.end type="italics"/>vertice principali <emph type="italics"/>K<emph.end type="italics"/>de­<lb/>&longs;cribatur Parabola. </s> <s>Dico factum. </s> <s>Nam Parabola, ob æquales <lb/><emph type="italics"/>SK<emph.end type="italics"/>& <emph type="italics"/>IK, SP<emph.end type="italics"/>& <emph type="italics"/>FP,<emph.end type="italics"/>tran&longs;ibit per punctum <emph type="italics"/>P<emph.end type="italics"/>; & (per Lem­<lb/>matis XIV. Corol. </s> <s>3.) ob æquales <emph type="italics"/>ST<emph.end type="italics"/>& <emph type="italics"/>TV<emph.end type="italics"/>& angulum rectum <lb/><emph type="italics"/>STR,<emph.end type="italics"/>tanget rectam <emph type="italics"/>TR. q.E.F.<emph.end type="italics"/><pb xlink:href="039/01/089.jpg" pagenum="61"/><arrow.to.target n="note37"/></s></p> <p type="margin"> <s><margin.target id="note37"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XX. PROBLEMA XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Circa datum umbilicum Trajectoriam quamvis &longs;pecie datam de&longs;cribe­<lb/>re, quæ per data puncta tran&longs;ibit & rectas tanget pofitione datas.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Dato umbilico <emph type="italics"/>S,<emph.end type="italics"/>de&longs;cribenda &longs;it Trajectoria <emph type="italics"/>ABC<emph.end type="italics"/>per <lb/>puncta duo <emph type="italics"/>B, C.<emph.end type="italics"/>Quoniam Trajectoria datur &longs;pecie, dabitur ra­<lb/>tio axis principalis ad di&longs;tantiam <lb/><figure id="id.039.01.089.1.jpg" xlink:href="039/01/089/1.jpg"/><lb/>umbilieorum. </s> <s>In ea ratione cape <lb/><emph type="italics"/>KB<emph.end type="italics"/>ad <emph type="italics"/>BS,<emph.end type="italics"/>& <emph type="italics"/>LC<emph.end type="italics"/>ad <emph type="italics"/>CS.<emph.end type="italics"/>Cen­<lb/>tris <emph type="italics"/>B, C,<emph.end type="italics"/>intervallis <emph type="italics"/>BK, CL,<emph.end type="italics"/>de­<lb/>&longs;cribe circulos duos, & ad rectam <lb/><emph type="italics"/>KL,<emph.end type="italics"/>quæ tangat eo&longs;dem in <emph type="italics"/>K<emph.end type="italics"/>& <lb/><emph type="italics"/>L,<emph.end type="italics"/>demitte perpendiculum <emph type="italics"/>SG,<emph.end type="italics"/>idemque &longs;eca in <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>a,<emph.end type="italics"/>ita ut &longs;it <lb/><emph type="italics"/>GA<emph.end type="italics"/>ad <emph type="italics"/>AS<emph.end type="italics"/>& <emph type="italics"/>Ga<emph.end type="italics"/>ad <emph type="italics"/>aS,<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>KB<emph.end type="italics"/>ad <emph type="italics"/>BS,<emph.end type="italics"/>& axe &c. <emph type="italics"/>Aa,<emph.end type="italics"/>verticibus <lb/><emph type="italics"/>A, a,<emph.end type="italics"/>de&longs;cribatur Trajectoria. </s> <s>Dico factum. </s> <s>Sit enim <emph type="italics"/>H<emph.end type="italics"/>umbilicus <lb/>alter Figuræ de&longs;criptæ, & cum &longs;it <emph type="italics"/>GA<emph.end type="italics"/>ad <emph type="italics"/>AS<emph.end type="italics"/>ut <emph type="italics"/>Ga<emph.end type="italics"/>ad <emph type="italics"/>aS,<emph.end type="italics"/>erit di­<lb/>vi&longs;im <emph type="italics"/>Ga-GA<emph.end type="italics"/>&longs;eu <emph type="italics"/>Aa<emph.end type="italics"/>ad <emph type="italics"/>aS-AS<emph.end type="italics"/>&longs;eu <emph type="italics"/>SH<emph.end type="italics"/>in eadem &c. </s> <s>ratione, <lb/>adeoQ.E.I. ratione quam habet axis principalis Figuræ de&longs;cribendæ <lb/>ad di&longs;tantiam umbilieorum ejus; & propterea Figura de&longs;cripta e&longs;t <lb/>eju&longs;dem &longs;peciei cum de&longs;cribenda. </s> <s>Cumque &longs;int <emph type="italics"/>KB<emph.end type="italics"/>ad <emph type="italics"/>BS<emph.end type="italics"/>& <emph type="italics"/>LC<emph.end type="italics"/><lb/>ad <emph type="italics"/>CS<emph.end type="italics"/>in eadem ratione, tran&longs;ibit hæc Figura per puncta <emph type="italics"/>B, C,<emph.end type="italics"/>ut <lb/>ex Conicis manife&longs;tum e&longs;t. </s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Dato umbilico <emph type="italics"/>S,<emph.end type="italics"/>de&longs;cribenda &longs;it Trajectoria quæ rectas <lb/>duas <emph type="italics"/>TR, tr<emph.end type="italics"/>alicubi contingat. </s> <s>Ab umbilico in tangentes demitte <lb/>perpendicula <emph type="italics"/>ST, St<emph.end type="italics"/>& produc ea­<lb/><figure id="id.039.01.089.2.jpg" xlink:href="039/01/089/2.jpg"/><lb/>dem ad <emph type="italics"/>V, v,<emph.end type="italics"/>ut &longs;int <emph type="italics"/>TV, tv<emph.end type="italics"/>æ­<lb/>quales <emph type="italics"/>TS, tS.<emph.end type="italics"/>Bi&longs;eca <emph type="italics"/>Vv<emph.end type="italics"/>in <emph type="italics"/>O,<emph.end type="italics"/><lb/>& erige perpendiculum infinitum <lb/><emph type="italics"/>OH,<emph.end type="italics"/>rectamque <emph type="italics"/>VS<emph.end type="italics"/>infinite pro­<lb/>ductam &longs;eca in <emph type="italics"/>K<emph.end type="italics"/>& <emph type="italics"/>k<emph.end type="italics"/>ita, ut &longs;it <lb/><emph type="italics"/>VK<emph.end type="italics"/>ad <emph type="italics"/>KS<emph.end type="italics"/>& <emph type="italics"/>Vk<emph.end type="italics"/>ad <emph type="italics"/>kS<emph.end type="italics"/>ut e&longs;t <lb/>Trajectoriæ de&longs;cribendæ axis prin­<lb/>cipalis ad umbilieorum di&longs;tantiam. </s> <s><lb/>Super diametro <emph type="italics"/>Kk<emph.end type="italics"/>de&longs;cribatur <lb/>circulus &longs;ecans <emph type="italics"/>OH<emph.end type="italics"/>in <emph type="italics"/>H<emph.end type="italics"/>; & umbilicis <emph type="italics"/>S, H,<emph.end type="italics"/>axe principali ip&longs;am <lb/><emph type="italics"/>VH<emph.end type="italics"/>æquante, de&longs;cribatur Trajectoria. </s> <s>Dico factum. </s> <s>Nam bi&longs;eca <lb/><emph type="italics"/>Kk<emph.end type="italics"/>in <emph type="italics"/>X,<emph.end type="italics"/>& junge <emph type="italics"/>HX, HS, HV, Hv.<emph.end type="italics"/>Quoniam e&longs;t <emph type="italics"/>VK<emph.end type="italics"/>ad <emph type="italics"/>KS<emph.end type="italics"/><lb/>ut <emph type="italics"/>Vk<emph.end type="italics"/>ad <emph type="italics"/>kS<emph.end type="italics"/>; & compofite ut <emph type="italics"/>VK+Vk<emph.end type="italics"/>ad <emph type="italics"/>KS+kS<emph.end type="italics"/>; divi&longs;imque <pb xlink:href="039/01/090.jpg" pagenum="62"/><arrow.to.target n="note38"/>ut <emph type="italics"/>Vk-VK<emph.end type="italics"/>ad <emph type="italics"/>kS-KS,<emph.end type="italics"/>id e&longs;t ut 2 <emph type="italics"/>VX<emph.end type="italics"/>ad 2 <emph type="italics"/>KX<emph.end type="italics"/>& 2 <emph type="italics"/>KX<emph.end type="italics"/>ad <lb/>2 <emph type="italics"/>SX,<emph.end type="italics"/>adeoque ut <emph type="italics"/>VX<emph.end type="italics"/>ad <emph type="italics"/>HX<emph.end type="italics"/>& <emph type="italics"/>HX<emph.end type="italics"/>ad <emph type="italics"/>SX,<emph.end type="italics"/>&longs;imilia erunt tri­<lb/>angula <emph type="italics"/>VXH, HXS,<emph.end type="italics"/>& propterea <emph type="italics"/>VH<emph.end type="italics"/>erit ad <emph type="italics"/>SH<emph.end type="italics"/>ut <emph type="italics"/>VX<emph.end type="italics"/>ad <emph type="italics"/>XH,<emph.end type="italics"/><lb/>adeoque ut <emph type="italics"/>VK<emph.end type="italics"/>ad <emph type="italics"/>KS.<emph.end type="italics"/>Habet igitur Trajectoriæ de&longs;criptæ axis <lb/>principalis <emph type="italics"/>VH<emph.end type="italics"/>eam rationem ad ip&longs;ius umbilieorum di&longs;tantiam <emph type="italics"/>SH,<emph.end type="italics"/><lb/>quam habet Trajectoriæ de&longs;cribendæ axis principalis ad ip&longs;ius um­<lb/>bilieorum di&longs;tantiam, & propterea eju&longs;dem e&longs;t &longs;peciei. </s> <s>In&longs;uper cum <lb/><emph type="italics"/>VH, vH<emph.end type="italics"/>æquentur axi principali, & <emph type="italics"/>VS, vS<emph.end type="italics"/>a rectis <emph type="italics"/>TR, tr<emph.end type="italics"/><lb/>perpendiculariter bi&longs;ecentur, liquet, ex Lemmate XV, rectas illas <lb/>Trajectoriam de&longs;criptam tangere. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note38"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>3. Dato umbilico <emph type="italics"/>S<emph.end type="italics"/>de&longs;cribenda &longs;it Trajectoria quæ rect­<lb/>am <emph type="italics"/>TR<emph.end type="italics"/>tanget in puncto dato <emph type="italics"/>R.<emph.end type="italics"/>In rectam <emph type="italics"/>TR<emph.end type="italics"/>demitte perpen­<lb/>dicularem <emph type="italics"/>ST,<emph.end type="italics"/>& produc eandem ad <emph type="italics"/>V,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>TV<emph.end type="italics"/>æqualis <emph type="italics"/>ST.<emph.end type="italics"/>Junge <lb/><emph type="italics"/>VR,<emph.end type="italics"/>& rectam <emph type="italics"/>VS<emph.end type="italics"/>infinite productam &longs;eca in <emph type="italics"/>K<emph.end type="italics"/>& <emph type="italics"/>k,<emph.end type="italics"/>ita ut &longs;it <lb/><emph type="italics"/>VK<emph.end type="italics"/>ad <emph type="italics"/>SK<emph.end type="italics"/>& <emph type="italics"/>Vk<emph.end type="italics"/>ad <emph type="italics"/>Sk<emph.end type="italics"/>ut Ellip&longs;eos de&longs;cribendæ axis principalis <lb/>ad di&longs;tantiam umbilieorum; circuloque &longs;uper diametro <emph type="italics"/>Kk<emph.end type="italics"/>de­<lb/>&longs;cripto, &longs;ecetur producta recta <emph type="italics"/>VR<emph.end type="italics"/>in <emph type="italics"/>H,<emph.end type="italics"/>& umbilicis <emph type="italics"/>S, H,<emph.end type="italics"/>axe <lb/>principali rectam <emph type="italics"/>VH<emph.end type="italics"/>æquante, de&longs;cribatur Trajectoria. </s> <s>Dico fa­<lb/>ctum. </s> <s>Namque <emph type="italics"/>VH<emph.end type="italics"/>e&longs;&longs;e ad <lb/><figure id="id.039.01.090.1.jpg" xlink:href="039/01/090/1.jpg"/><lb/><emph type="italics"/>SH<emph.end type="italics"/>ut <emph type="italics"/>VK<emph.end type="italics"/>ad <emph type="italics"/>SK,<emph.end type="italics"/>atque adeo <lb/>ut axis principalis Trajectoriæ <lb/>de&longs;cribendæ ad di&longs;tantiam um­<lb/>bilieorum ejus, patet ex demon­<lb/>&longs;tratis in Ca&longs;u &longs;ecundo, & prop­<lb/>terea Trajectoriam de&longs;criptam <lb/>eju&longs;dem e&longs;&longs;e &longs;peciei cum de&longs;cri­<lb/>benda; rectam vero <emph type="italics"/>TR<emph.end type="italics"/>qua an­<lb/>gulus <emph type="italics"/>VRS<emph.end type="italics"/>bi&longs;ecatur, tangere Trajectoriam in puncto <emph type="italics"/>R,<emph.end type="italics"/>patet ex <lb/>Conicis. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>4. Circa umbilicum <emph type="italics"/>S<emph.end type="italics"/>de&longs;cribenda jam &longs;it Trajectoria <emph type="italics"/>APB,<emph.end type="italics"/><lb/>quæ tangat rectam <emph type="italics"/>TR,<emph.end type="italics"/>tran&longs;eatque per punctum quodvis <emph type="italics"/>P<emph.end type="italics"/>extra <lb/>tangentem datum, quæque &longs;imilis &longs;it Figuræ <emph type="italics"/>apb,<emph.end type="italics"/>axe principali <lb/><emph type="italics"/>ab<emph.end type="italics"/>& umbilicis <emph type="italics"/>s, h<emph.end type="italics"/>de&longs;criptæ. </s> <s>In tangentem <emph type="italics"/>TR<emph.end type="italics"/>demitte per­<lb/>pendiculum <emph type="italics"/>ST,<emph.end type="italics"/>& produc idem ad <emph type="italics"/>V,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>TV<emph.end type="italics"/>æqualis <emph type="italics"/>ST.<emph.end type="italics"/>An­<lb/>gulis autem <emph type="italics"/>VSP, SVP<emph.end type="italics"/>fac angulos <emph type="italics"/>hsq, shq<emph.end type="italics"/>æquales; cen­<lb/>troque <emph type="italics"/>q<emph.end type="italics"/>& intervallo quod &longs;it ad <emph type="italics"/>ab<emph.end type="italics"/>ut <emph type="italics"/>SP<emph.end type="italics"/>ad <emph type="italics"/>VS<emph.end type="italics"/>de&longs;cribe circu­<lb/>lum &longs;ecantem Figuram <emph type="italics"/>apb<emph.end type="italics"/>in <emph type="italics"/>p.<emph.end type="italics"/>Junge <emph type="italics"/>sp<emph.end type="italics"/>& age <emph type="italics"/>SH<emph.end type="italics"/>quæ &longs;it ad <lb/><emph type="italics"/>sh<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>SP<emph.end type="italics"/>ad <emph type="italics"/>sp,<emph.end type="italics"/>quæque angulum <emph type="italics"/>PSH<emph.end type="italics"/>angulo <emph type="italics"/>psh<emph.end type="italics"/>& angulum <lb/><emph type="italics"/>VSH<emph.end type="italics"/>angulo <emph type="italics"/>psq<emph.end type="italics"/>æquales con&longs;tituat. </s> <s>Denique umbilicis <emph type="italics"/>S, H,<emph.end type="italics"/><lb/>& axe principali <emph type="italics"/>AB<emph.end type="italics"/>di&longs;tantiam <emph type="italics"/>VH<emph.end type="italics"/>æquante, de&longs;cribatur &longs;ectio <lb/>Conica. </s> <s>Dico factum. </s> <s>Nam &longs;i agatur <emph type="italics"/>sv<emph.end type="italics"/>quæ &longs;it ad <emph type="italics"/>sp<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>sh<emph.end type="italics"/><pb xlink:href="039/01/091.jpg" pagenum="63"/>ad <emph type="italics"/>sq,<emph.end type="italics"/>quæque con&longs;tituat angulum <emph type="italics"/>vsp<emph.end type="italics"/>angulo <emph type="italics"/>hsq<emph.end type="italics"/>& angulum <lb/><arrow.to.target n="note39"/><emph type="italics"/>vsh<emph.end type="italics"/>angulo <emph type="italics"/>psq<emph.end type="italics"/>æquales, triangula <emph type="italics"/>svh, spq<emph.end type="italics"/>erunt &longs;imilia, & prop­<lb/>terea <emph type="italics"/>vh<emph.end type="italics"/>erit ad <emph type="italics"/>pq<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>sh<emph.end type="italics"/>ad <emph type="italics"/>sq,<emph.end type="italics"/>id e&longs;t (ob &longs;imilia triangula <lb/><figure id="id.039.01.091.1.jpg" xlink:href="039/01/091/1.jpg"/><lb/><emph type="italics"/>VSP, hsq<emph.end type="italics"/>) ut e&longs;t <emph type="italics"/>VS<emph.end type="italics"/>ad <emph type="italics"/>SP<emph.end type="italics"/>&longs;eu <emph type="italics"/>ab<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="pq.">pque</expan><emph.end type="italics"/>Æquantur ergo <lb/><emph type="italics"/>vh<emph.end type="italics"/>& <emph type="italics"/>ab.<emph.end type="italics"/>Porro ob &longs;imilia triangula <emph type="italics"/>VSH. vsh,<emph.end type="italics"/>e&longs;t <emph type="italics"/>VH<emph.end type="italics"/>ad <lb/><emph type="italics"/>SH<emph.end type="italics"/>ut <emph type="italics"/>vh<emph.end type="italics"/>ad <emph type="italics"/>sh,<emph.end type="italics"/>id e&longs;t, axis Conicæ &longs;ectionis jam de&longs;criptæ ad <lb/>illius umbilieorum intervallum, ut axis <emph type="italics"/>ab<emph.end type="italics"/>ad umbilieorum inter­<lb/>vallum <emph type="italics"/>sh<emph.end type="italics"/>; & propterea Figura jam de&longs;eripta &longs;imilis e&longs;t Figuræ <lb/><emph type="italics"/>apb.<emph.end type="italics"/>Tran&longs;it autem hæc Figura per punctum <emph type="italics"/>P,<emph.end type="italics"/>eo quod trian­<lb/>gulum <emph type="italics"/>PSH<emph.end type="italics"/>&longs;imile &longs;it triangulo <emph type="italics"/>psh<emph.end type="italics"/>; & quia <emph type="italics"/>VH<emph.end type="italics"/>æquatur ip&longs;ius <lb/>axi & <emph type="italics"/>VS<emph.end type="italics"/>bi&longs;ecatur perpendiculariter a recta <emph type="italics"/>TR,<emph.end type="italics"/>tangit eadem <lb/>rectam <emph type="italics"/>TR. q.E.F.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note39"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>LEMMA XVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>A datis tribus punctis ad quartum non datum inflectere tres rectas <lb/>quarum differentiæ vel dantur vel nullæ &longs;unt.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Sunto puncta illa data <emph type="italics"/>A, B, C<emph.end type="italics"/>& punctum quartum <emph type="italics"/>Z,<emph.end type="italics"/><lb/>quod invenire oportet; Ob datam differentiam linearum <emph type="italics"/>AZ, BZ,<emph.end type="italics"/><lb/>locabitur punctum <emph type="italics"/>Z<emph.end type="italics"/>in Hyperbola cujus umbilici &longs;unt <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B,<emph.end type="italics"/>& <lb/>principalis axis differentia illa data. </s> <s>Sit axis ille <emph type="italics"/>MN.<emph.end type="italics"/>Cape <emph type="italics"/>PM.<emph.end type="italics"/><pb xlink:href="039/01/092.jpg" pagenum="64"/><arrow.to.target n="note40"/>ad <emph type="italics"/>MA<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>MN<emph.end type="italics"/>ad <emph type="italics"/>AB,<emph.end type="italics"/>& erecta <emph type="italics"/>PR<emph.end type="italics"/>perpendiculari ad <emph type="italics"/>AB,<emph.end type="italics"/><lb/>demi&longs;&longs;aque <emph type="italics"/>ZR<emph.end type="italics"/>perpendiculari ad <emph type="italics"/>PR<emph.end type="italics"/>; erit, ex natura hujus Hy­<lb/>perbolæ, <emph type="italics"/>ZR<emph.end type="italics"/>ad <emph type="italics"/>AZ<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>MN<emph.end type="italics"/>ad <emph type="italics"/>AB.<emph.end type="italics"/>Simili di&longs;cur&longs;u punctum <lb/><emph type="italics"/>Z<emph.end type="italics"/>locabitur in alia Hyperbola, cujus umbilici &longs;unt <emph type="italics"/>A, C<emph.end type="italics"/>& princi­<lb/>palis axis differentia inter <emph type="italics"/>AZ<emph.end type="italics"/>& <emph type="italics"/>CZ,<emph.end type="italics"/>ducique pote&longs;t <emph type="italics"/>QS<emph.end type="italics"/>ip&longs;i <emph type="italics"/>AC<emph.end type="italics"/><lb/>perpendicularis, ad quam &longs;i ab Hyperbolæ hujus puncto quovis <emph type="italics"/>Z<emph.end type="italics"/><lb/>demittatur normalis <emph type="italics"/>ZS,<emph.end type="italics"/>hæc fuerit ad <emph type="italics"/>AZ<emph.end type="italics"/>ut e&longs;t differentia inter <lb/><emph type="italics"/>AZ<emph.end type="italics"/>& <emph type="italics"/>CZ<emph.end type="italics"/>ad <emph type="italics"/>AC.<emph.end type="italics"/>Dantur ergo rationes ip&longs;arum <emph type="italics"/>ZR<emph.end type="italics"/>& <emph type="italics"/>ZS<emph.end type="italics"/><lb/>ad <emph type="italics"/>AZ,<emph.end type="italics"/>& idcirco datur earun­<lb/><figure id="id.039.01.092.1.jpg" xlink:href="039/01/092/1.jpg"/><lb/>dem <emph type="italics"/>ZR<emph.end type="italics"/>& <emph type="italics"/>ZS<emph.end type="italics"/>ratio ad invicem; <lb/>ideoque &longs;i rectæ <emph type="italics"/>RP, SQ<emph.end type="italics"/>concur­<lb/>rant in <emph type="italics"/>T,<emph.end type="italics"/>& agatur <emph type="italics"/>TZ,<emph.end type="italics"/>figura <lb/><emph type="italics"/>TRZS,<emph.end type="italics"/>dabitur &longs;pecie, & recta <lb/><emph type="italics"/>TZ<emph.end type="italics"/>in qua punctum <emph type="italics"/>Z<emph.end type="italics"/>alicubi lo­<lb/>catur, dabitur po&longs;itione. </s> <s>Eadem <lb/>methodo per Hyperbolam ter­<lb/>tiam, cujus umbilici &longs;unt <emph type="italics"/>B<emph.end type="italics"/>& <emph type="italics"/>C<emph.end type="italics"/><lb/>& axis principalis differentia re­<lb/>ctarum <emph type="italics"/>BZ, CZ,<emph.end type="italics"/>inveniri pote&longs;t <lb/>alia recta in qua <expan abbr="pũctum">punctum</expan> <emph type="italics"/>Z<emph.end type="italics"/>locatur. </s> <s><lb/>Habitis autem duobus Locis recti­<lb/>lineis, habetur punctum quæ&longs;itum <emph type="italics"/>Z<emph.end type="italics"/>in eorum inter&longs;ectione. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note40"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Si duæ ex tribus lineis, puta <emph type="italics"/>AZ<emph.end type="italics"/>& <emph type="italics"/>BZ<emph.end type="italics"/>æquantur, pun­<lb/>ctum <emph type="italics"/>Z<emph.end type="italics"/>locabitur in perpendiculo bi&longs;ecante di&longs;tantiam <emph type="italics"/>AB,<emph.end type="italics"/>& lo­<lb/>cus alius rectilineus invenietur ut &longs;upra. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>3. Si omnes tres æquantur, locabitur punctum <emph type="italics"/>Z<emph.end type="italics"/>in centro <lb/>Circuli per puncta <emph type="italics"/>A, B, C<emph.end type="italics"/>tran&longs;euntis. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s>Solvitur etiam hoc Lemma problematicum per Librum Tactio­<lb/>num <emph type="italics"/>Apollonii<emph.end type="italics"/>a <emph type="italics"/>Vieta<emph.end type="italics"/>re&longs;titutum. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXI. PROBLEMA XIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam circa datum umbilicum de&longs;cribere, quæ tran&longs;ibit per <lb/>puncta data & rectas po&longs;itione datas continget.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Detur umbilicus <emph type="italics"/>S,<emph.end type="italics"/>punctum <emph type="italics"/>P,<emph.end type="italics"/>& tangens <emph type="italics"/>TR,<emph.end type="italics"/>& invenien­<lb/>dus &longs;it umbilicus alter <emph type="italics"/>H.<emph.end type="italics"/>Ad tangentem demitte perpendiculum <lb/><emph type="italics"/>ST,<emph.end type="italics"/>& produc idem ad <emph type="italics"/>Y,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>TY<emph.end type="italics"/>æqualis <emph type="italics"/>ST,<emph.end type="italics"/>& erit <emph type="italics"/>YH<emph.end type="italics"/>æ­<lb/>qualis axi principali. </s> <s>Junge <emph type="italics"/>SP, HP,<emph.end type="italics"/>& erit <emph type="italics"/>SP<emph.end type="italics"/>differentia inter <lb/><emph type="italics"/>HP<emph.end type="italics"/>& axem principalem. </s> <s>Hoc modo &longs;i dentur plures tangen-<pb xlink:href="039/01/093.jpg" pagenum="65"/>tes <emph type="italics"/>TR,<emph.end type="italics"/>vel plura puncta <emph type="italics"/>P,<emph.end type="italics"/>devenietur &longs;emper ad lineas totidem <lb/><arrow.to.target n="note41"/><emph type="italics"/>YH,<emph.end type="italics"/>vel <emph type="italics"/>PH,<emph.end type="italics"/>a dictis punctis <emph type="italics"/>Y<emph.end type="italics"/>vel <lb/><figure id="id.039.01.093.1.jpg" xlink:href="039/01/093/1.jpg"/><lb/><emph type="italics"/>P<emph.end type="italics"/>ad umbilicum <emph type="italics"/>H<emph.end type="italics"/>ductas, quæ vel <lb/>æquantur axibus, vel datis longitu­<lb/>dinibus <emph type="italics"/>SP<emph.end type="italics"/>differunt ab ii&longs;dem, at­<lb/>que adeo quæ vel æquantur &longs;ibi invi­<lb/>cem, vel datas habent differentias; & <lb/>inde, per Lemma &longs;uperius, datur umbi­<lb/>licus ille alter <emph type="italics"/>H.<emph.end type="italics"/>Habitis autem um­<lb/>bilicis una cum axis longitudine (quæ <lb/>vel e&longs;t <emph type="italics"/>YH<emph.end type="italics"/>; vel, &longs;i Trajectoria Ellip&longs;is e&longs;t, <emph type="italics"/>PH+SP<emph.end type="italics"/>; &longs;in Hy­<lb/>perbola, <emph type="italics"/>PH-SP<emph.end type="italics"/>) habetur Trajectoria. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note41"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Ca&longs;us ubi dantur tria puncta &longs;ic &longs;olvitur expeditius. </s> <s>Dentur <lb/>puncta <emph type="italics"/>B, C, D.<emph.end type="italics"/>Junctas <emph type="italics"/>BC, CD<emph.end type="italics"/>produc ad <emph type="italics"/>E, F,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>EB<emph.end type="italics"/>ad <lb/><emph type="italics"/>EC<emph.end type="italics"/>ut <emph type="italics"/>SB<emph.end type="italics"/>ad <emph type="italics"/>SC,<emph.end type="italics"/>& <emph type="italics"/>FC<emph.end type="italics"/>ad <emph type="italics"/>FD<emph.end type="italics"/>ut <emph type="italics"/>SC<emph.end type="italics"/>ad <emph type="italics"/>SD.<emph.end type="italics"/>Ad <emph type="italics"/>EF<emph.end type="italics"/>ductam <lb/>& productam demitte normales <emph type="italics"/>SG, BH,<emph.end type="italics"/>inque <emph type="italics"/>GS<emph.end type="italics"/>infinite <lb/>producta cape <emph type="italics"/>GA<emph.end type="italics"/>ad <emph type="italics"/>AS<emph.end type="italics"/>& <emph type="italics"/>Ga<emph.end type="italics"/>ad <emph type="italics"/>aS<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>HB<emph.end type="italics"/>ad <emph type="italics"/>BS<emph.end type="italics"/>; & erit <lb/><emph type="italics"/>A<emph.end type="italics"/>vertex, & <emph type="italics"/>Aa<emph.end type="italics"/>axis principalis Trajectoriæ: quæ, perinde ut <emph type="italics"/>GA<emph.end type="italics"/><lb/>major, æqualis, vel minor fuerit quam <emph type="italics"/>AS,<emph.end type="italics"/>erit Ellip&longs;is, Parabola <lb/>vel Hyperbola; pun­<lb/><figure id="id.039.01.093.2.jpg" xlink:href="039/01/093/2.jpg"/><lb/>cto <emph type="italics"/>a<emph.end type="italics"/>in primo ca&longs;u <lb/>cadente ad eandem <lb/>partem lineæ <emph type="italics"/>GF<emph.end type="italics"/><lb/>cum puncto <emph type="italics"/>A<emph.end type="italics"/>; in <lb/>&longs;ecundo ca&longs;u abeunte <lb/>in infinitum; in tertio <lb/>cadente ad contrari­<lb/>am partem lineæ <emph type="italics"/>GF.<emph.end type="italics"/><lb/>Nam &longs;i demittantur <lb/>ad <emph type="italics"/>GF<emph.end type="italics"/>perpendicula <lb/><emph type="italics"/>CI, DK<emph.end type="italics"/>; erit <emph type="italics"/>IC<emph.end type="italics"/>ad <emph type="italics"/>HB<emph.end type="italics"/>ut <emph type="italics"/>EC<emph.end type="italics"/>ad <emph type="italics"/>EB,<emph.end type="italics"/>hoc e&longs;t, ut <emph type="italics"/>SC<emph.end type="italics"/>ad <emph type="italics"/>SB<emph.end type="italics"/>; & vi­<lb/>ci&longs;&longs;im <emph type="italics"/>IC<emph.end type="italics"/>ad <emph type="italics"/>SC<emph.end type="italics"/>ut <emph type="italics"/>HB<emph.end type="italics"/>ad <emph type="italics"/>SB<emph.end type="italics"/>&longs;ive ut <emph type="italics"/>GA<emph.end type="italics"/>ad <emph type="italics"/>SA.<emph.end type="italics"/>Et &longs;imili argumento <lb/>probabitur e&longs;&longs;e <emph type="italics"/>KD<emph.end type="italics"/>ad <emph type="italics"/>SD<emph.end type="italics"/>in eadem ratione. </s> <s>Jacent ergo puncta <emph type="italics"/>B, <lb/>C, D<emph.end type="italics"/>in Coni&longs;ectione circa umbilicum <emph type="italics"/>S<emph.end type="italics"/>ita de&longs;cripta, ut rectæ omnes <lb/>ab umbilico <emph type="italics"/>S<emph.end type="italics"/>ad &longs;ingula Sectionis puncta ductæ, &longs;int ad perpendicula <lb/>a punctis ii&longs;dem ad rectam <emph type="italics"/>GF<emph.end type="italics"/>demi&longs;&longs;a in data illa ratione. </s></p> <p type="main"> <s>Methodo haud multum di&longs;&longs;imili hujus problematis &longs;olutionem <lb/>tradit Clari&longs;&longs;imus Geometra <emph type="italics"/>de la Hire,<emph.end type="italics"/>Conieorum &longs;uorum Lib. </s> <s><lb/>VIII. Prop. XXV. <pb xlink:href="039/01/094.jpg" pagenum="66"/><arrow.to.target n="note42"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note42"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>SECTIO V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Inventio Orbium ubi umbilicus neuter datur.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>LEMMA XVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si a datæ Conicæ Sectionis puncto quovis<emph.end type="italics"/>P, <emph type="italics"/>ad Trapezii alicujus<emph.end type="italics"/><lb/>ABDC, <emph type="italics"/>in Conica illa &longs;ectione in&longs;cripti, latera quatuor infinite <lb/>producta<emph.end type="italics"/>AB, CD, AC, DB, <emph type="italics"/>totidem rectæ<emph.end type="italics"/>PQ, PR, PS, PT <lb/><emph type="italics"/>in datis angulis ducantur, &longs;ingulæ ad &longs;ingula: rectangulum duc­<lb/>tarum ad oppo&longs;ita duo latera<emph.end type="italics"/>PQXPR, <emph type="italics"/>erit ad rectangulum duc­<lb/>tarum ad alia duo latera oppo&longs;ita<emph.end type="italics"/>PSXPT <emph type="italics"/>in data ratione.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Ponamus primo lineas ad <lb/><figure id="id.039.01.094.1.jpg" xlink:href="039/01/094/1.jpg"/><lb/>oppo&longs;ita latera ductas parallelas e&longs;­<lb/>&longs;e alterutri reliquorum laterum, <lb/>puta <emph type="italics"/>PQ<emph.end type="italics"/>& <emph type="italics"/>PR<emph.end type="italics"/>lateri <emph type="italics"/>AC,<emph.end type="italics"/>& <emph type="italics"/>PS<emph.end type="italics"/><lb/>ac <emph type="italics"/>PT<emph.end type="italics"/>lateri <emph type="italics"/>AB.<emph.end type="italics"/>SintQ.E.I.&longs;uper <lb/>latera duo ex oppo&longs;itis, puta <emph type="italics"/>AC<emph.end type="italics"/><lb/>& <emph type="italics"/>BD,<emph.end type="italics"/>&longs;ibi invicem paralle­<lb/>la. </s> <s>Et recta quæ bi&longs;ecat paralle­<lb/>la illa latera erit una ex diametris <lb/>Conicæ &longs;ectionis, & bi&longs;ecabit eti­<lb/>am <emph type="italics"/><expan abbr="Rq.">Rque</expan><emph.end type="italics"/>Sit <emph type="italics"/>O<emph.end type="italics"/>punctum in quo <lb/><emph type="italics"/>RQ<emph.end type="italics"/>bi&longs;ecatur, & erit <emph type="italics"/>PO<emph.end type="italics"/>ordinatim applicata ad diametrum illam. </s> <s><lb/>Produc <emph type="italics"/>PO<emph.end type="italics"/>ad <emph type="italics"/>K<emph.end type="italics"/>ut &longs;it <emph type="italics"/>OK<emph.end type="italics"/>æqualis <emph type="italics"/>PO,<emph.end type="italics"/>& erit <emph type="italics"/>OK<emph.end type="italics"/>ordinatim <lb/>applicata ad contrarias partes diametri. </s> <s>Cum igitur puncta <emph type="italics"/>A, B, <lb/>P<emph.end type="italics"/>& <emph type="italics"/>K<emph.end type="italics"/>&longs;int ad Conicam &longs;ectionem, & <emph type="italics"/>PK<emph.end type="italics"/>&longs;ecet <emph type="italics"/>AB<emph.end type="italics"/>in dato an­<lb/>gulo, erit (per Prop.17 & 18 Lib. </s> <s>III Conieorum <emph type="italics"/>Apollonii<emph.end type="italics"/>) rectangu­<lb/>lum <emph type="italics"/>PQK<emph.end type="italics"/>ad rectangulum <emph type="italics"/>AQB<emph.end type="italics"/>in data ratione. </s> <s>Sed <emph type="italics"/>QK<emph.end type="italics"/>& <emph type="italics"/>PR<emph.end type="italics"/><lb/>æquales &longs;unt, utpote æqualium <emph type="italics"/>OK, OP,<emph.end type="italics"/>& <emph type="italics"/>OQ, OR<emph.end type="italics"/>differentiæ, <lb/>& inde etiam rectangula <emph type="italics"/>PQK<emph.end type="italics"/>& <emph type="italics"/>PQXPR<emph.end type="italics"/>æqualia &longs;unt; at­<lb/>que adeo rectangulum <emph type="italics"/>PQXPR<emph.end type="italics"/>e&longs;t ad rectangulum <emph type="italics"/>AQB,<emph.end type="italics"/>hoc <lb/>e&longs;t ad rectangulum <emph type="italics"/>PSXPT<emph.end type="italics"/>in data ratione. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p><pb xlink:href="039/01/095.jpg" pagenum="67"/> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Ponamus jam Trapezii latera oppo&longs;ita <emph type="italics"/>AC<emph.end type="italics"/>& <emph type="italics"/>BD<emph.end type="italics"/>non <lb/><arrow.to.target n="note43"/>e&longs;&longs;e parallela. </s> <s>Age <emph type="italics"/>Bd<emph.end type="italics"/>parallelam <emph type="italics"/>AC<emph.end type="italics"/>& occurrentem tum rectæ <lb/><emph type="italics"/>ST<emph.end type="italics"/>in <emph type="italics"/>t,<emph.end type="italics"/>tum Conicæ &longs;ectioni in <emph type="italics"/>d.<emph.end type="italics"/>Junge <emph type="italics"/>Cd<emph.end type="italics"/>&longs;ecantem <emph type="italics"/>PQ<emph.end type="italics"/>in <emph type="italics"/>r,<emph.end type="italics"/><lb/>& ip&longs;i <emph type="italics"/>PQ<emph.end type="italics"/>parallelam age <emph type="italics"/>DM<emph.end type="italics"/><lb/><figure id="id.039.01.095.1.jpg" xlink:href="039/01/095/1.jpg"/><lb/>&longs;ecantem <emph type="italics"/>Cd<emph.end type="italics"/>in <emph type="italics"/>M<emph.end type="italics"/>& <emph type="italics"/>AB<emph.end type="italics"/>in <emph type="italics"/>N.<emph.end type="italics"/><lb/>Jam ob &longs;imilia triangula <emph type="italics"/>BTt, <lb/>DBN<emph.end type="italics"/>; e&longs;t <emph type="italics"/>Bt<emph.end type="italics"/>&longs;eu <emph type="italics"/>PQ<emph.end type="italics"/>ad <emph type="italics"/>Tt<emph.end type="italics"/>ut <lb/><emph type="italics"/>DN<emph.end type="italics"/>ad <emph type="italics"/>NB.<emph.end type="italics"/>Sic & <emph type="italics"/>Rr<emph.end type="italics"/>e&longs;t ad <lb/><emph type="italics"/>AQ<emph.end type="italics"/>&longs;eu <emph type="italics"/>PS<emph.end type="italics"/>ut <emph type="italics"/>DM<emph.end type="italics"/>ad <emph type="italics"/>AN.<emph.end type="italics"/><lb/>Ergo, ducendo antecedentes in <lb/>antecedentes & con&longs;equentes in <lb/>con&longs;equentes, ut rectangulum <emph type="italics"/>PQ<emph.end type="italics"/><lb/>in <emph type="italics"/>Rr<emph.end type="italics"/>e&longs;t ad rectangulum <emph type="italics"/>PS<emph.end type="italics"/>in <lb/><emph type="italics"/>Tt,<emph.end type="italics"/>ita rectangulum <emph type="italics"/>NDM<emph.end type="italics"/>e&longs;t <lb/>ad rectangulum <emph type="italics"/>ANB,<emph.end type="italics"/>& (per Ca&longs;.1) ita rectangulum <emph type="italics"/>PQ<emph.end type="italics"/>in <emph type="italics"/>Pr<emph.end type="italics"/>e&longs;t <lb/>ad rectangulum <emph type="italics"/>PS<emph.end type="italics"/>in <emph type="italics"/>Pt,<emph.end type="italics"/>ac divi&longs;im ita rectangulum <emph type="italics"/>PQXPR<emph.end type="italics"/><lb/>e&longs;t ad rectangulum <emph type="italics"/>PSXPT. Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note43"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>3. Ponamus denique lineas <lb/><figure id="id.039.01.095.2.jpg" xlink:href="039/01/095/2.jpg"/><lb/>quatuor <emph type="italics"/>PQ, PR, PS, PT<emph.end type="italics"/>non <lb/>e&longs;&longs;e parallelas lateribus <emph type="italics"/>AC, AB,<emph.end type="italics"/><lb/>&longs;ed ad ea utcunQ.E.I.clinatas. </s> <s>Ea­<lb/>rum vice age <emph type="italics"/>Pq, Pr<emph.end type="italics"/>parallelas <lb/>ip&longs;i <emph type="italics"/>AC<emph.end type="italics"/>; & <emph type="italics"/>Ps, Pt<emph.end type="italics"/>parallelas <lb/>ip&longs;i <emph type="italics"/>AB<emph.end type="italics"/>; & propter datos angu­<lb/>los triangulorum <emph type="italics"/>PQq, PRr, <lb/>PSs, PTt,<emph.end type="italics"/>dabuntur rationes <lb/><emph type="italics"/>PQ<emph.end type="italics"/>ad <emph type="italics"/>Pq, PR<emph.end type="italics"/>ad <emph type="italics"/>Pr, PS<emph.end type="italics"/><lb/>ad <emph type="italics"/>Ps,<emph.end type="italics"/>& <emph type="italics"/>PT<emph.end type="italics"/>ad <emph type="italics"/>Pt<emph.end type="italics"/>; atque adeo rationes compo&longs;itæ <emph type="italics"/>PQXPR<emph.end type="italics"/><lb/>ad <emph type="italics"/>PqXPr,<emph.end type="italics"/>& <emph type="italics"/>PSXPT<emph.end type="italics"/>ad <emph type="italics"/>PsXPt.<emph.end type="italics"/>Sed, per &longs;uperius de­<lb/>mon&longs;trata, ratio <emph type="italics"/>PqXPr<emph.end type="italics"/>ad <emph type="italics"/>PsXPt<emph.end type="italics"/>data e&longs;t: Ergo & ratio <lb/><emph type="italics"/>PQXPR<emph.end type="italics"/>ad <emph type="italics"/>PSXPT. Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>LEMMA XVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, &longs;i rectangulum ductarum ad oppo&longs;ita duo latera Tra­<lb/>pezii<emph.end type="italics"/>PQXPR <emph type="italics"/>&longs;it ad rectangulum ductarum ad reliqua duo late­<lb/>ra<emph.end type="italics"/>PSXPT <emph type="italics"/>in data ratione; punctum<emph.end type="italics"/>P, <emph type="italics"/>a quo lineæ ducuntur, <lb/>tanget Conicam &longs;ectionem circa Trapezium de&longs;criptam.<emph.end type="italics"/></s></p><pb xlink:href="039/01/096.jpg" pagenum="68"/> <p type="main"> <s>Per puncta <emph type="italics"/>A, B, C, D<emph.end type="italics"/>& aliquod infinitorum punctorum <emph type="italics"/>P,<emph.end type="italics"/>pu­<lb/><arrow.to.target n="note44"/>ta <emph type="italics"/>p,<emph.end type="italics"/>concipe Conicam &longs;ectionem de&longs;cribi: dico punctum <emph type="italics"/>P<emph.end type="italics"/>hanc <lb/>&longs;emper tangere. </s> <s>Si negas, <lb/><figure id="id.039.01.096.1.jpg" xlink:href="039/01/096/1.jpg"/><lb/>junge <emph type="italics"/>AP<emph.end type="italics"/>&longs;ecantem hanc <lb/>Conicam &longs;ectionem alibi <lb/>quam in <emph type="italics"/>P,<emph.end type="italics"/>&longs;i fieri pote&longs;t, <lb/>puta in <emph type="italics"/>b.<emph.end type="italics"/>Ergo &longs;i ab his <lb/>punctis <emph type="italics"/>p<emph.end type="italics"/>& <emph type="italics"/>b<emph.end type="italics"/>ducantur in <lb/>datis angulis ad latera Tra­<lb/>pezii rectæ <emph type="italics"/>pq, pr, ps, pt<emph.end type="italics"/><lb/>& <emph type="italics"/>bk, br, b&longs;, bd<emph.end type="italics"/>; erit <lb/>ut <emph type="italics"/>bkXb<emph.end type="italics"/>r ad <emph type="italics"/>b&longs;Xbd<emph.end type="italics"/>ita <lb/>(per Lem. </s> <s>XVII) <emph type="italics"/>pqXpr<emph.end type="italics"/><lb/>ad <emph type="italics"/>psXpt,<emph.end type="italics"/>& ita (per <lb/>Hypoth.) <emph type="italics"/>PQXPR<emph.end type="italics"/>ad <lb/><emph type="italics"/>PSXPT.<emph.end type="italics"/>E&longs;t & prop­<lb/>ter &longs;imilitudinem Trapeziorum <emph type="italics"/>bkA&longs;, PQAS,<emph.end type="italics"/>ut <emph type="italics"/>bk<emph.end type="italics"/>ad <emph type="italics"/>b&longs;<emph.end type="italics"/>ita <lb/><emph type="italics"/>PQ<emph.end type="italics"/>ad <emph type="italics"/>PS.<emph.end type="italics"/>Quare, applicando terminos prioris proportionis ad <lb/>terminos corre&longs;pondentes hujus, erit <emph type="italics"/>b<emph.end type="italics"/>r ad <emph type="italics"/>bd<emph.end type="italics"/>ut <emph type="italics"/>PR<emph.end type="italics"/>ad <emph type="italics"/>PT.<emph.end type="italics"/>Er­<lb/>go Trapezia æquiangula <emph type="italics"/>Dr bd, DRPT<emph.end type="italics"/>&longs;imilia &longs;unt, & eorum <lb/>diagonales <emph type="italics"/>Db, DP<emph.end type="italics"/>propterea coincidunt. </s> <s>Incidit itaque <emph type="italics"/>b<emph.end type="italics"/>in <lb/>inter&longs;ectionem rectarum <emph type="italics"/>AP, DP<emph.end type="italics"/>adeoque coincidit cum puncto <lb/><emph type="italics"/>P.<emph.end type="italics"/>Quare punctum <emph type="italics"/>P,<emph.end type="italics"/>ubicunque &longs;umatur, incidit in a&longs;&longs;ignatam <lb/>Conicam &longs;ectionem. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note44"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc &longs;i rectæ tres <emph type="italics"/>PQ, PR, PS<emph.end type="italics"/>a puncto communi <emph type="italics"/>P<emph.end type="italics"/><lb/>ad alias totidem po&longs;itione datas rectas <emph type="italics"/>AB, CD, AC,<emph.end type="italics"/>&longs;ingulæ ad <lb/>&longs;ingulas, in datis angulis ducantur, &longs;itque rectangulum &longs;ub duabus <lb/>ductis <emph type="italics"/>PQXPR<emph.end type="italics"/>ad quadratum tertiæ <emph type="italics"/>PS quad.<emph.end type="italics"/>in data ratione: <lb/>punctum <emph type="italics"/>P,<emph.end type="italics"/>a quibus rectæ ducuntur, locabitur in &longs;ectione Conica <lb/>quæ tangit lineas <emph type="italics"/>AB, CD<emph.end type="italics"/>in <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>C<emph.end type="italics"/>; & contra. </s> <s>Nam coeat linea <lb/><emph type="italics"/>BD<emph.end type="italics"/>cum linea <emph type="italics"/>AC<emph.end type="italics"/>manente po&longs;itione trium <emph type="italics"/>AB, CD, AC<emph.end type="italics"/>; de­<lb/>in coeat etiam linea <emph type="italics"/>PT<emph.end type="italics"/>cum linea <emph type="italics"/>PS:<emph.end type="italics"/>& rectangulum <emph type="italics"/>PSXPT<emph.end type="italics"/><lb/>evadet <emph type="italics"/>PS quad.<emph.end type="italics"/>rectæque <emph type="italics"/>AB, CD<emph.end type="italics"/>quæ curvam in punctis <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B, <lb/>C<emph.end type="italics"/>& <emph type="italics"/>D<emph.end type="italics"/>&longs;ecabant, jam Curvam in punctis illis coeuntibus non am­<lb/>plius &longs;ecare po&longs;&longs;unt &longs;ed tantum tangent. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Nomen Conicæ &longs;ectionis in hoc Lemmate late &longs;umitur, ita ut <lb/>&longs;ectio tam Rectilinea per verticem Coni tran&longs;iens, quam Circularis <lb/>ba&longs;i parallela includatur. </s> <s>Nam &longs;i punctum <emph type="italics"/>p<emph.end type="italics"/>incidit in rectam, qua <lb/>quævis ex punctis quatuor <emph type="italics"/>A, B, C, D<emph.end type="italics"/>junguntur, Conica &longs;ectio <pb xlink:href="039/01/097.jpg" pagenum="69"/>vertetur in geminas Rectas, quarum una e&longs;t recta illa in quam pun­<lb/><arrow.to.target n="note45"/>ctum <emph type="italics"/>p<emph.end type="italics"/>incidit, & altera e&longs;t recta qua alia duo ex punctis quatuor jun­<lb/>guntur. </s> <s>Si Trapezii anguli duo oppo&longs;iti &longs;imul &longs;umpti æquentur <lb/>duobus rectis, & lineæ quatuor <emph type="italics"/>PQ, PR, PS, PT<emph.end type="italics"/>ducantur ad <lb/>latera ejus vel perpendiculariter vel in angulis quibu&longs;vis æqualibus, <lb/>&longs;itque rectangulum &longs;ub duabus ductis <emph type="italics"/>PQXPR<emph.end type="italics"/>æquale rectangu­<lb/>lo &longs;ub duabus aliis <emph type="italics"/>PSXPT,<emph.end type="italics"/>Sectio conica evadet Circulus. </s> <s>Idem <lb/>fiet &longs;i lineæ quatuor ducantur in angulis quibu&longs;vis & rectangulum <lb/>&longs;ub duabus ductis <emph type="italics"/>PQXPR<emph.end type="italics"/>&longs;it ad rectangulum &longs;ub aliis duabus <lb/><emph type="italics"/>PSXPT<emph.end type="italics"/>ut rectangulum &longs;ub &longs;inubus angulorum <emph type="italics"/>S, T,<emph.end type="italics"/>in quibus <lb/>duæ ultimæ <emph type="italics"/>PS, PT<emph.end type="italics"/>ducuntur, ad rectangulum &longs;ub &longs;inubus angu­<lb/>lorum <emph type="italics"/>Q, R,<emph.end type="italics"/>in quibus duæ primæ <emph type="italics"/>PQ, PR<emph.end type="italics"/>ducuntur. </s> <s>Cæteris <lb/>in ca&longs;ibus Locus puncti <emph type="italics"/>P<emph.end type="italics"/>erit aliqua trium figurarum quæ vulgo <lb/>nominantur Sectiones Conicæ. </s> <s>Vice autem Trapezii <emph type="italics"/>ABCD<emph.end type="italics"/>&longs;ub­<lb/>&longs;titui pote&longs;t Quadrilaterum cujus latera duo oppo&longs;ita &longs;e mutuo in­<lb/>&longs;tar diagonalium decu&longs;&longs;ant. </s> <s>Sed & e punctis quatuor <emph type="italics"/>A, B, C, D<emph.end type="italics"/><lb/>po&longs;&longs;unt unum vel duo abire ad infinitum, eoque pacto latera fi­<lb/>guræ quæ ad puncta illa convergunt, evadere parallela: quo in <lb/>ca&longs;u Sectio Conica tran&longs;ibit per cætera puncta, & in plagas paralle­<lb/>larum abibit in infinitum. </s></p> <p type="margin"> <s><margin.target id="note45"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>LEMMA XIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Invenire <expan abbr="punctũ">punctum</expan><emph.end type="italics"/>P, <emph type="italics"/>a quo &longs;i rectæ<emph.end type="italics"/><lb/><figure id="id.039.01.097.1.jpg" xlink:href="039/01/097/1.jpg"/><lb/><emph type="italics"/>quatuor<emph.end type="italics"/>PQ, PR, PS, PT, <lb/><emph type="italics"/>ad alias totidem po&longs;itione da<lb/>tas rectas<emph.end type="italics"/>AB, CD, AC, BD, <lb/><emph type="italics"/>&longs;ingulæ ad &longs;ingulas in datis <lb/>angulis ducantur, <expan abbr="rectangulũ">rectangulum</expan> <lb/>&longs;ub duabus ductis,<emph.end type="italics"/>PQXPR, <lb/><emph type="italics"/>&longs;it ad rectangulum &longs;ub aliis <lb/>duabus,<emph.end type="italics"/>PSXPT, <emph type="italics"/>in data ra­<lb/>tione.<emph.end type="italics"/></s></p> <p type="main"> <s>Lineæ <emph type="italics"/>AB, CD,<emph.end type="italics"/>ad quas rectæ duæ <emph type="italics"/>PQ, PR,<emph.end type="italics"/>unum rectan­<lb/>gulorum continentes ducuntur, conveniant cum aliis duabus po&longs;i­<lb/>tione datis lineis in punctis <emph type="italics"/>A, B, C, D.<emph.end type="italics"/>Ab eorum aliquo <emph type="italics"/>A<emph.end type="italics"/>age <lb/>rectam quamlibet <emph type="italics"/>AH,<emph.end type="italics"/>in qua velis punctum <emph type="italics"/>P<emph.end type="italics"/>reperiri. </s> <s>Secet ea <lb/>lineas oppo&longs;itas <emph type="italics"/>BD, CD,<emph.end type="italics"/>nimirum <emph type="italics"/>BD<emph.end type="italics"/>in <emph type="italics"/>H<emph.end type="italics"/>& <emph type="italics"/>CD<emph.end type="italics"/>in <emph type="italics"/>I,<emph.end type="italics"/>& ob <lb/>datos omnes angulos figuræ, dabuntur rationes <emph type="italics"/>PQ<emph.end type="italics"/>ad <emph type="italics"/>PA<emph.end type="italics"/>& <emph type="italics"/>PA<emph.end type="italics"/><pb xlink:href="039/01/098.jpg" pagenum="70"/><arrow.to.target n="note46"/>ad <emph type="italics"/>PS,<emph.end type="italics"/>adeoque ratio <emph type="italics"/>PQ<emph.end type="italics"/>ad <lb/><figure id="id.039.01.098.1.jpg" xlink:href="039/01/098/1.jpg"/><lb/><emph type="italics"/>PS.<emph.end type="italics"/>Auferendo hanca data ra­<lb/>tione <emph type="italics"/>PQXPR<emph.end type="italics"/>ad <emph type="italics"/>PSXPT,<emph.end type="italics"/><lb/>dabitur ratio <emph type="italics"/>PR<emph.end type="italics"/>ad <emph type="italics"/>PT,<emph.end type="italics"/>& <lb/>addendo datas rationes <emph type="italics"/>PI<emph.end type="italics"/>ad <lb/><emph type="italics"/>PR,<emph.end type="italics"/>& <emph type="italics"/>PT<emph.end type="italics"/>ad <emph type="italics"/>PH<emph.end type="italics"/>dabitur <lb/>ratio <emph type="italics"/>PI<emph.end type="italics"/>ad <emph type="italics"/>PH<emph.end type="italics"/>atque adeo <lb/>punctum <emph type="italics"/>P. Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note46"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc etiam ad Loci <lb/>punctorum infinitorum <emph type="italics"/>P<emph.end type="italics"/>pun­<lb/>ctum quodvis <emph type="italics"/>D<emph.end type="italics"/>tangens duci <lb/>pote&longs;t. </s> <s>Nam chorda <emph type="italics"/>PD<emph.end type="italics"/>ubi <lb/>puncta <emph type="italics"/>P<emph.end type="italics"/>ac <emph type="italics"/>D<emph.end type="italics"/>conveniunt, hoc <lb/>e&longs;t, ubi <emph type="italics"/>AH<emph.end type="italics"/>ducitur per punctum <emph type="italics"/>D,<emph.end type="italics"/>tangens evadit. </s> <s>Quo in ca&longs;u, <lb/>ultima ratio evane&longs;centium <emph type="italics"/>IP<emph.end type="italics"/>& <emph type="italics"/>PH<emph.end type="italics"/>invenietur ut &longs;upra. </s> <s>Ip&longs;i <lb/>igitur <emph type="italics"/>AD<emph.end type="italics"/>due parallelam <emph type="italics"/>CF,<emph.end type="italics"/>occurrentem <emph type="italics"/>BD<emph.end type="italics"/>in <emph type="italics"/>F,<emph.end type="italics"/>& in ea ul­<lb/>tima ratione &longs;ectam in <emph type="italics"/>E,<emph.end type="italics"/>& <emph type="italics"/>DE<emph.end type="italics"/>tangens erit, propterea quod <emph type="italics"/>CF<emph.end type="italics"/><lb/>& evane&longs;cens <emph type="italics"/>IH<emph.end type="italics"/>parallelæ &longs;unt, & in <emph type="italics"/>E<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>fimiliter &longs;ectæ. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Hinc etiam Locus punctorum omnium <emph type="italics"/>P<emph.end type="italics"/>definiri pote&longs;t. </s> <s><lb/>Per quodvis punctorum <emph type="italics"/>A, B, C, D,<emph.end type="italics"/>puta <emph type="italics"/>A,<emph.end type="italics"/>duc Loci tangentem <lb/><emph type="italics"/>AE<emph.end type="italics"/>& per aliud quodvis punctum <emph type="italics"/>B<emph.end type="italics"/>duc tangenti parallelam <emph type="italics"/>BF<emph.end type="italics"/><lb/>occurrentem Loco in <emph type="italics"/>F.<emph.end type="italics"/>Invenie­<lb/><figure id="id.039.01.098.2.jpg" xlink:href="039/01/098/2.jpg"/><lb/>tur autem punctum <emph type="italics"/>F<emph.end type="italics"/>per Lem. </s> <s>XIX. </s> <s><lb/>Bi&longs;eca <emph type="italics"/>BF<emph.end type="italics"/>in <emph type="italics"/>G,<emph.end type="italics"/>& acta indefinita <lb/><emph type="italics"/>AG<emph.end type="italics"/>erit po&longs;itio diametri ad quam <lb/><emph type="italics"/>BG<emph.end type="italics"/>& <emph type="italics"/>FG<emph.end type="italics"/>ordinatim applicantur. </s> <s><lb/>Hæc <emph type="italics"/>AG<emph.end type="italics"/>occurrat Loco in <emph type="italics"/>H,<emph.end type="italics"/>& <lb/>erit <emph type="italics"/>AH<emph.end type="italics"/>diameter &longs;ive latus tran&longs;­<lb/>ver&longs;um, ad quod latus rectum erit <lb/>ut <emph type="italics"/><expan abbr="BGq.">BGque</expan><emph.end type="italics"/>ad <emph type="italics"/>AGH.<emph.end type="italics"/>Si <emph type="italics"/>AG<emph.end type="italics"/>nullibi <lb/>occurrit Loco, linea <emph type="italics"/>AH<emph.end type="italics"/>exi&longs;tente <lb/>infinita, Locus erit Parabola & la­<lb/>rum rectum ejus ad diametrum <emph type="italics"/>AG<emph.end type="italics"/><lb/>pertinens erit (<emph type="italics"/>BGq./AG<emph.end type="italics"/>) Sin ea alicubi occurrit, Locus Hyperbola erit <lb/>ubi puncta <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>H<emph.end type="italics"/>&longs;ita &longs;unt ad ea&longs;dem partes ip&longs;ius <emph type="italics"/>G:<emph.end type="italics"/>& Ellip&longs;is, <lb/>ubi <emph type="italics"/>G<emph.end type="italics"/>intermedium e&longs;t, ni&longs;i forte angulus <emph type="italics"/>AGB<emph.end type="italics"/>rectus &longs;it & in&longs;uper <lb/><emph type="italics"/>BG quad.<emph.end type="italics"/>æquale rectangulo <emph type="italics"/>AGH,<emph.end type="italics"/>quo in ca&longs;u Circulus habebitur. </s></p> <p type="main"> <s>AtQ.E.I.a Problematis Veterum de quatuor lineis ab <emph type="italics"/>Euclide<emph.end type="italics"/>incæp­<lb/>ti & ab <emph type="italics"/>Apollonio<emph.end type="italics"/>continuati non calculus, &longs;ed compo&longs;itio Geometri­<lb/>ca, qualem Veteres quærebant, in hoc Corollario exhibetur. <pb xlink:href="039/01/099.jpg" pagenum="71"/><arrow.to.target n="note47"/></s></p> <p type="margin"> <s><margin.target id="note47"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>LEMMA XX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Parallelogrammum quodvis<emph.end type="italics"/>ASPQ <emph type="italics"/>angulis duobus oppo&longs;itis<emph.end type="italics"/>A <emph type="italics"/>&<emph.end type="italics"/><lb/>P <emph type="italics"/>tangit &longs;ectionem quamvis Conicam in punctis<emph.end type="italics"/>A <emph type="italics"/>&<emph.end type="italics"/>P; <emph type="italics"/>&, lateri­<lb/>bus unius angulorum illorum infinite productis<emph.end type="italics"/>AQ, AS, <emph type="italics"/>occurrit <lb/>eidem &longs;ectioni Conicæ in<emph.end type="italics"/>B <emph type="italics"/>&<emph.end type="italics"/>C; <emph type="italics"/>a punctis autem occur&longs;uum<emph.end type="italics"/>B <emph type="italics"/>&<emph.end type="italics"/><lb/>C <emph type="italics"/>ad quintum quodvis &longs;ectionis Conicæ punctum<emph.end type="italics"/>D <emph type="italics"/>agantur rec­<lb/>tæ duæ<emph.end type="italics"/>BD, CD <emph type="italics"/>occurrentes alteris duobus infinite productis pa­<lb/>rallelogrammi lateribus<emph.end type="italics"/>PS, PQ <emph type="italics"/>in<emph.end type="italics"/>T <emph type="italics"/>&<emph.end type="italics"/>R: <emph type="italics"/>erunt &longs;emper ab&longs;ci&longs;&longs;æ <lb/>laterum partes<emph.end type="italics"/>PR <emph type="italics"/>&<emph.end type="italics"/>PT <emph type="italics"/>adinvicem in data ratione. </s> <s>Et contra, &longs;i <lb/>partes illæ ab&longs;ci&longs;&longs;æ &longs;unt ad invicem in data ratione, punctum<emph.end type="italics"/>D <emph type="italics"/>tan­<lb/>get Sectionem Conicam per puncta quatuor<emph.end type="italics"/>A, B, C, P <emph type="italics"/>tran&longs;euntem.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Jungantur <emph type="italics"/>BP, CP<emph.end type="italics"/>& a puncto <emph type="italics"/>D<emph.end type="italics"/>agantur rectæ duæ <lb/><emph type="italics"/>DG, DE,<emph.end type="italics"/>quarum prior <lb/><figure id="id.039.01.099.1.jpg" xlink:href="039/01/099/1.jpg"/><lb/><emph type="italics"/>DG<emph.end type="italics"/>ip&longs;i <emph type="italics"/>AB<emph.end type="italics"/>parallela &longs;it & <lb/>occurrat <emph type="italics"/>PB, PQ, CA<emph.end type="italics"/>in <lb/><emph type="italics"/>H, I, G<emph.end type="italics"/>; altera <emph type="italics"/>DE<emph.end type="italics"/>paral­<lb/>lela &longs;it ipfi <emph type="italics"/>AC<emph.end type="italics"/>& occurrat <lb/><emph type="italics"/>PC, PS, AB<emph.end type="italics"/>in <emph type="italics"/>F, K, E:<emph.end type="italics"/><lb/>& erit (per Lemma XVII.) re­<lb/>ctangulum <emph type="italics"/>DEXDF<emph.end type="italics"/>ad re­<lb/>ctangulum <emph type="italics"/>DGXDH<emph.end type="italics"/>in ra­<lb/>tione data. </s> <s>Sed e&longs;t <emph type="italics"/>PQ<emph.end type="italics"/>ad <lb/><emph type="italics"/>DE<emph.end type="italics"/>(&longs;eu <emph type="italics"/>IQ<emph.end type="italics"/>) ut <emph type="italics"/>PB<emph.end type="italics"/>ad <emph type="italics"/>HB,<emph.end type="italics"/><lb/>adeoque ut <emph type="italics"/>PT<emph.end type="italics"/>ad <emph type="italics"/>DH<emph.end type="italics"/>; & <lb/>vici&longs;&longs;im <emph type="italics"/>PQ<emph.end type="italics"/>ad <emph type="italics"/>PT<emph.end type="italics"/>ut <emph type="italics"/>DE<emph.end type="italics"/>ad <emph type="italics"/>DH.<emph.end type="italics"/>E&longs;t & <emph type="italics"/>PR<emph.end type="italics"/>ad <emph type="italics"/>DF<emph.end type="italics"/>ut <emph type="italics"/>RC<emph.end type="italics"/><lb/>ad <emph type="italics"/>DC,<emph.end type="italics"/>adeoque ut (<emph type="italics"/>IG<emph.end type="italics"/>vel) <emph type="italics"/>PS<emph.end type="italics"/>ad <emph type="italics"/>DG,<emph.end type="italics"/>& vici&longs;&longs;im <emph type="italics"/>PR<emph.end type="italics"/>ad <emph type="italics"/>PS<emph.end type="italics"/><lb/>ut <emph type="italics"/>DF<emph.end type="italics"/>ad <emph type="italics"/>DG<emph.end type="italics"/>; & conjunctis rationibus fit rectangulum <emph type="italics"/>PQXPR<emph.end type="italics"/><lb/>ad rectangulum <emph type="italics"/>PSXPT<emph.end type="italics"/>ut rectangulum <emph type="italics"/>DEXDF<emph.end type="italics"/>ad rectan­<lb/>gulum <emph type="italics"/>DGXDH,<emph.end type="italics"/>atque adeo in data ratione. </s> <s>Sed dantur <emph type="italics"/>PQ<emph.end type="italics"/><lb/>& <emph type="italics"/>PS<emph.end type="italics"/>& propterea ratio <emph type="italics"/>PR<emph.end type="italics"/>ad <emph type="italics"/>PT<emph.end type="italics"/>datur. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Quod &longs;i <emph type="italics"/>PR<emph.end type="italics"/>& <emph type="italics"/>PT<emph.end type="italics"/>ponantur in data ratione ad invi­<lb/>cem, tum &longs;imili ratiocinio regrediendo, &longs;equetur e&longs;&longs;e rectangulum <lb/><emph type="italics"/>DEXDF<emph.end type="italics"/>ad rectangulum <emph type="italics"/>DGXDH<emph.end type="italics"/>in ratione data, adeoque <lb/>punctum <emph type="italics"/>D<emph.end type="italics"/>(per Lemma XVIII.) contingere Conicam &longs;ectionem <lb/>tran&longs;euntem per puncta <emph type="italics"/>A, B, C, P. Q.E.D.<emph.end type="italics"/><pb xlink:href="039/01/100.jpg" pagenum="72"/><arrow.to.target n="note48"/></s></p> <p type="margin"> <s><margin.target id="note48"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i agatur <emph type="italics"/>BC<emph.end type="italics"/>&longs;ecans <emph type="italics"/>PQ<emph.end type="italics"/>in <emph type="italics"/>r,<emph.end type="italics"/>& in <emph type="italics"/>PT<emph.end type="italics"/>capiatur <lb/><emph type="italics"/>Pt<emph.end type="italics"/>in ratione ad <emph type="italics"/>Pr<emph.end type="italics"/>quam habet <emph type="italics"/>PT<emph.end type="italics"/>ad <emph type="italics"/>PR:<emph.end type="italics"/>erit <emph type="italics"/>Bt<emph.end type="italics"/>tangens <lb/>Conicæ &longs;ectionis ad punctum <emph type="italics"/>B.<emph.end type="italics"/>Nam concipe punctum <emph type="italics"/>D<emph.end type="italics"/>coire <lb/>cum puncto <emph type="italics"/>B<emph.end type="italics"/>ita ut, chorda <emph type="italics"/>BD<emph.end type="italics"/>evane&longs;cente, <emph type="italics"/>BT<emph.end type="italics"/>tangens eva­<lb/>dat; & <emph type="italics"/>CD<emph.end type="italics"/>ac <emph type="italics"/>BT<emph.end type="italics"/>coincident cum <emph type="italics"/>CB<emph.end type="italics"/>& <emph type="italics"/>Bt.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et vice ver&longs;a &longs;i <lb/><figure id="id.039.01.100.1.jpg" xlink:href="039/01/100/1.jpg"/><lb/><emph type="italics"/>Bt<emph.end type="italics"/>fit tangens, & ad quod­<lb/>vis Conicæ &longs;ectionis punc­<lb/>tum <emph type="italics"/>D<emph.end type="italics"/>conveniant <emph type="italics"/>BD, <lb/>CD<emph.end type="italics"/>; erit <emph type="italics"/>PR<emph.end type="italics"/>ad <emph type="italics"/>PT<emph.end type="italics"/>ut <lb/>ut <emph type="italics"/>Pr<emph.end type="italics"/>ad <emph type="italics"/>Pt.<emph.end type="italics"/>Et contra, <lb/>&longs;i &longs;it <emph type="italics"/>PR<emph.end type="italics"/>ad <emph type="italics"/>PT<emph.end type="italics"/>ut <emph type="italics"/>Pr<emph.end type="italics"/>ad <lb/><emph type="italics"/>Pt:<emph.end type="italics"/>convenient <emph type="italics"/>BD, CD<emph.end type="italics"/><lb/>ad Conicæ Sectionis punc­<lb/>um aliquod <emph type="italics"/>D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Conica &longs;ectio <lb/>non &longs;ecat Conicam &longs;ectio­<lb/>nem in punctis pluribus quam quatuor. </s> <s>Nam, &longs;i fieri pote&longs;t, tran&longs;­<lb/>eant duæ Conicæ &longs;ectiones per quinque puncta <emph type="italics"/>A, B, C, P, O<emph.end type="italics"/>; ea&longs;­<lb/>que &longs;ecet recta <emph type="italics"/>BD<emph.end type="italics"/>in punctis <emph type="italics"/>D, d,<emph.end type="italics"/>& ip&longs;am <emph type="italics"/>PQ<emph.end type="italics"/>&longs;ecet recta <emph type="italics"/>Cd<emph.end type="italics"/><lb/>in r. </s> <s>Ergo <emph type="italics"/>PR<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>PT<emph.end type="italics"/>ut <emph type="italics"/>P<emph.end type="italics"/>r ad <emph type="italics"/>PT<emph.end type="italics"/>; unde <emph type="italics"/>PR<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>r &longs;ibi <lb/>invicem æquantur, contra Hypothe&longs;in. </s></p> <p type="main"> <s><emph type="center"/>LEMMA XXI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si rectæ duæ mobiles & infinitæ<emph.end type="italics"/>BM, CM <emph type="italics"/>per data puncta<emph.end type="italics"/>B, C, <emph type="italics"/>ceu <lb/>polos ductæ, concur&longs;u &longs;uo<emph.end type="italics"/>M <emph type="italics"/>de&longs;cribant tertiam po&longs;itione da­<lb/>tam rectam<emph.end type="italics"/>MN; <emph type="italics"/>& aliæ duæ infinitæ rectæ<emph.end type="italics"/>BD, CD <emph type="italics"/>cum <lb/>prioribus duabus ad puncta illa data<emph.end type="italics"/>B, C <emph type="italics"/>datos angulos<emph.end type="italics"/><lb/>MBD, MCD <emph type="italics"/>efficientes ducantur; dico quod hæ duæ<emph.end type="italics"/>BD, <lb/>CD <emph type="italics"/>concur&longs;u &longs;uo<emph.end type="italics"/>D <emph type="italics"/>de&longs;cribent &longs;ectionem Conicam per puncta<emph.end type="italics"/><lb/>B, C <emph type="italics"/>tran&longs;euntem. </s> <s>Et vice ver&longs;a, &longs;i rectæ<emph.end type="italics"/>BD, CD <emph type="italics"/>concur&longs;u <lb/>&longs;uo<emph.end type="italics"/>D <emph type="italics"/>de&longs;cribant Sectionem Conicam per data puncta<emph.end type="italics"/>B, C, A <lb/><emph type="italics"/>tran&longs;euntem, & &longs;it angulus<emph.end type="italics"/>DBM <emph type="italics"/>&longs;emper æqualis angulo dato<emph.end type="italics"/><lb/>ABC, <emph type="italics"/>angulu&longs;que<emph.end type="italics"/>DCM <emph type="italics"/>&longs;emper æqualis angulo dato<emph.end type="italics"/>ACB: <lb/><emph type="italics"/>punctum<emph.end type="italics"/>M <emph type="italics"/>continget rectam po&longs;itione datam.<emph.end type="italics"/></s></p><pb xlink:href="039/01/101.jpg" pagenum="73"/> <p type="main"> <s><arrow.to.target n="note49"/></s></p> <p type="margin"> <s><margin.target id="note49"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>Nam in recta <emph type="italics"/>MN<emph.end type="italics"/>detur punctum <emph type="italics"/>N,<emph.end type="italics"/>& ubi punctum mobile <lb/><emph type="italics"/>M<emph.end type="italics"/>incidit in immotum <emph type="italics"/>N,<emph.end type="italics"/>incidat punctum mobile <emph type="italics"/>D<emph.end type="italics"/>in immo­<lb/>tum <emph type="italics"/>P,<emph.end type="italics"/>Junge <emph type="italics"/>CN, BN,<emph.end type="italics"/><lb/><figure id="id.039.01.101.1.jpg" xlink:href="039/01/101/1.jpg"/><lb/><emph type="italics"/>CP, BP,<emph.end type="italics"/>& a puncto <lb/><emph type="italics"/>P<emph.end type="italics"/>age rectas <emph type="italics"/>PT, PR<emph.end type="italics"/><lb/>occurrentes ip&longs;is <emph type="italics"/>BD, <lb/>CD<emph.end type="italics"/>in <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>R,<emph.end type="italics"/>& fa­<lb/>cientes angulum <emph type="italics"/>BPT<emph.end type="italics"/><lb/>æqualem angulo dato <lb/><emph type="italics"/>BNM,<emph.end type="italics"/>& angulum <lb/><emph type="italics"/>CPR<emph.end type="italics"/>æqualem angu­<lb/>gulo dato <emph type="italics"/>CNM.<emph.end type="italics"/>Cum <lb/>ergo (ex Hypothe&longs;i) <lb/>æquales &longs;int anguli <lb/><emph type="italics"/>MBD, NBP,<emph.end type="italics"/>ut & <lb/>anguli <emph type="italics"/>MCD, NCP<emph.end type="italics"/>; <lb/>aufer communes <emph type="italics"/>NBD<emph.end type="italics"/><lb/>& <emph type="italics"/>NCD,<emph.end type="italics"/>& re&longs;tabunt <lb/>æquales <emph type="italics"/>NBM<emph.end type="italics"/>& <emph type="italics"/>PBT, <lb/>NCM<emph.end type="italics"/>& <emph type="italics"/>PCR:<emph.end type="italics"/>adeoque triangula <emph type="italics"/>NBM, PBT<emph.end type="italics"/>&longs;imilia &longs;unt, ut <lb/>& triangula <emph type="italics"/>NCM, PCR.<emph.end type="italics"/>Quare <emph type="italics"/>PT<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>NM<emph.end type="italics"/>ut <emph type="italics"/>PB<emph.end type="italics"/>ad <lb/><emph type="italics"/>NB,<emph.end type="italics"/>& <emph type="italics"/>PR<emph.end type="italics"/>ad <emph type="italics"/>NM<emph.end type="italics"/>ut <emph type="italics"/>PC<emph.end type="italics"/>ad <emph type="italics"/>NC.<emph.end type="italics"/>Sunt autem puncta <emph type="italics"/>B, C, N, P<emph.end type="italics"/><lb/>immobilia. </s> <s>Ergo <emph type="italics"/>PT<emph.end type="italics"/>& <emph type="italics"/>PR<emph.end type="italics"/>datam habent rationem ad <emph type="italics"/>NM,<emph.end type="italics"/>pro­<lb/>indeQ.E.D.tam rationem inter &longs;e; atque adeo, per Lemma xx, <lb/>punctum <emph type="italics"/>D<emph.end type="italics"/>(perpetuus rectarum mobilium <emph type="italics"/>BT<emph.end type="italics"/>& <emph type="italics"/>CR<emph.end type="italics"/>concur&longs;us) <lb/>contingit &longs;ectionem Conicam, per puncta <emph type="italics"/>B, C, P<emph.end type="italics"/>tran&longs;euntem. <lb/><emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s>Et contra, &longs;i punctum mobile <emph type="italics"/>D<emph.end type="italics"/>contingat &longs;ectionem Conicam <lb/>tran&longs;euntem per data puncta <emph type="italics"/>B, C, A,<emph.end type="italics"/>& &longs;it angulus <emph type="italics"/>DBM<emph.end type="italics"/>&longs;emper <lb/>æqualis angulo dato <emph type="italics"/>ABC,<emph.end type="italics"/>& angulus <emph type="italics"/>DCM<emph.end type="italics"/>&longs;emper æqualis angu­<lb/>lo dato <emph type="italics"/>ACB,<emph.end type="italics"/>& ubi punctum <emph type="italics"/>D<emph.end type="italics"/>incidit &longs;ucce&longs;&longs;ive in duo quævis &longs;e­<lb/>ctionis puncta immobilia <emph type="italics"/>p, P,<emph.end type="italics"/>punctum mobile <emph type="italics"/>M<emph.end type="italics"/>incidat &longs;ucce&longs;&longs;ive <lb/>in puncta duo immobilia <emph type="italics"/>n, N:<emph.end type="italics"/>per eadem <emph type="italics"/>n, N<emph.end type="italics"/>agatur Recta <emph type="italics"/>n N,<emph.end type="italics"/><lb/>& hæc erit Locus perpetuus puncti illius mobilis <emph type="italics"/>M.<emph.end type="italics"/>Nam, &longs;i fieri <lb/>pote&longs;t, ver&longs;etur punctum <emph type="italics"/>M<emph.end type="italics"/>in linea aliqua Curva. </s> <s>Tanget ergo <lb/>punctum <emph type="italics"/>D<emph.end type="italics"/>&longs;ectionem Conicam per puncta quinque <emph type="italics"/>B, CA, p, P,<emph.end type="italics"/><lb/>tran&longs;euntem, ubi punctum <emph type="italics"/>M<emph.end type="italics"/>perpetuo tangit lineam Curvam. </s> <s>Sed <lb/>& ex jam demon&longs;tratis tanget etiam punctum <emph type="italics"/>D<emph.end type="italics"/>&longs;ectionem CoNI­<lb/>cam per eadem quinque puncta <emph type="italics"/>B, C, A, p, P<emph.end type="italics"/>tran&longs;euntem, ubi pun-</s></p><pb xlink:href="039/01/102.jpg" pagenum="74"/> <p type="main"> <s><arrow.to.target n="note50"/>ctum <emph type="italics"/>M<emph.end type="italics"/>perpetuo tangit lineam Rectam. </s> <s>Ergo duæ &longs;ectiones Co­<lb/>nicæ tran&longs;ibunt per eadem quinque puncta, contra Corol. </s> <s>3. Lem. </s> <s><lb/>xx. </s> <s>Igitur punctum <emph type="italics"/>M<emph.end type="italics"/>ver&longs;ari in linea Curva ab&longs;urdum e&longs;t. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note50"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXII. PROBLEMA. XIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam per data quinque puncta de&longs;cribere.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Dentur puncta quinque <emph type="italics"/>A, B, C, P, D.<emph.end type="italics"/>Ab eorum aliquo <emph type="italics"/>A<emph.end type="italics"/>ad <lb/>alia duo quævis <emph type="italics"/>B, C,<emph.end type="italics"/>quæ poli nominentur, age rectas <emph type="italics"/>AB, AC,<emph.end type="italics"/><lb/><figure id="id.039.01.102.1.jpg" xlink:href="039/01/102/1.jpg"/><lb/>hi&longs;que parallelas <emph type="italics"/>TPS, PRQ<emph.end type="italics"/>per punctum quartum <emph type="italics"/>P.<emph.end type="italics"/>De­<lb/>inde a polis duobus <emph type="italics"/>B, C<emph.end type="italics"/>age per punctum quintum <emph type="italics"/>D<emph.end type="italics"/>infiNI­<lb/>tas duas <emph type="italics"/>BDT, CRD,<emph.end type="italics"/>novi&longs;&longs;ime ductis <emph type="italics"/>TPS, PRQ<emph.end type="italics"/>(prio­<lb/>rem priori & po&longs;teriorem po&longs;teriori) occurrentes in <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>R.<emph.end type="italics"/>De­<lb/>niQ.E.D. rectis <emph type="italics"/>PT, PR,<emph.end type="italics"/>acta recta <emph type="italics"/>tr<emph.end type="italics"/>ip&longs;i <emph type="italics"/>TR<emph.end type="italics"/>parallela, ab­<lb/>&longs;cinde qua&longs;vis <emph type="italics"/>Pt, Pr<emph.end type="italics"/>ip&longs;is <emph type="italics"/>PT, PR<emph.end type="italics"/>proportionales; & &longs;i per <lb/>earum terminos <emph type="italics"/>t, r<emph.end type="italics"/>& polos <emph type="italics"/>B, C<emph.end type="italics"/>actæ <emph type="italics"/>Bt, Cr<emph.end type="italics"/>concurrant in <lb/><emph type="italics"/>d,<emph.end type="italics"/>locabitur punctum illud <emph type="italics"/>d<emph.end type="italics"/>in Trajectoria quæ&longs;ita. </s> <s>Nam punc­<lb/>tum illud <emph type="italics"/>d<emph.end type="italics"/>(per Lemma xx) ver&longs;atur in Conica Sectione per <lb/>puncta quatuor <emph type="italics"/>A, B, C, P<emph.end type="italics"/>tran&longs;eunte; &, lineis <emph type="italics"/>Rr, Tt<emph.end type="italics"/>evane­<lb/>&longs;centibus, coit punctum <emph type="italics"/>d<emph.end type="italics"/>cum puncto <emph type="italics"/>D.<emph.end type="italics"/>Tran&longs;it ergo &longs;ectio Co­<lb/>nica per puncta quinque <emph type="italics"/>A, B, C, P, D. Q.E.D.<emph.end type="italics"/></s></p><pb xlink:href="039/01/103.jpg" pagenum="75"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/><lb/><arrow.to.target n="note51"/></s></p> <p type="margin"> <s><margin.target id="note51"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>E punctis datis junge tria quævis <emph type="italics"/>A, B, C<emph.end type="italics"/>; &, circum duo eorum <lb/><emph type="italics"/>B, C<emph.end type="italics"/>ceu polos, rotando angulos magnitudine datos <emph type="italics"/>ABC, <lb/>ACB,<emph.end type="italics"/>applicentur cru­<lb/><figure id="id.039.01.103.1.jpg" xlink:href="039/01/103/1.jpg"/><lb/>ra <emph type="italics"/>BA, CA<emph.end type="italics"/>primo ad <lb/>punctum <emph type="italics"/>D,<emph.end type="italics"/>deinde <lb/>ad punctum <emph type="italics"/>P,<emph.end type="italics"/>& no­<lb/>tentur puncta <emph type="italics"/>M, N<emph.end type="italics"/>in <lb/>quibus altera crura <lb/><emph type="italics"/>BL, CL<emph.end type="italics"/>ca&longs;u utroque <lb/>&longs;e decu&longs;&longs;ant. </s> <s>Agatur <lb/>recta infinita <emph type="italics"/>MN,<emph.end type="italics"/>& <lb/>rotentur anguli illi mo­<lb/>biles circum polos &longs;uos <lb/><emph type="italics"/>B, C,<emph.end type="italics"/>ea lege ut cru­<lb/>rum <emph type="italics"/>BL, CL<emph.end type="italics"/>vel <lb/><emph type="italics"/>BM, CM<emph.end type="italics"/>inter&longs;ectio <lb/>quæ jam &longs;it <emph type="italics"/>m<emph.end type="italics"/>incidat <lb/>&longs;emper in rectam illam <lb/>infinitam <emph type="italics"/>MN<emph.end type="italics"/>& cru­<lb/>rum <emph type="italics"/>BA, CA,<emph.end type="italics"/>vel <emph type="italics"/>BD, CD<emph.end type="italics"/>inter&longs;ectio, quæ jam &longs;it <emph type="italics"/>d,<emph.end type="italics"/>Trajecto­<lb/>riam quæ&longs;itam <emph type="italics"/>PAD dB<emph.end type="italics"/>delineabit. </s> <s>Nam punctum <emph type="italics"/>d,<emph.end type="italics"/>per Lem. </s> <s><lb/>XXI, continget &longs;ectionem Conicam per puncta <emph type="italics"/>B, C<emph.end type="italics"/>tran&longs;euntem; & <lb/>ubi punctum <emph type="italics"/>m<emph.end type="italics"/>accedit ad puncta <emph type="italics"/>L, M, N,<emph.end type="italics"/>punctum <emph type="italics"/>d<emph.end type="italics"/>(per con­<lb/>&longs;tructionem) accedet ad puncta <emph type="italics"/>A, D, P.<emph.end type="italics"/>De&longs;cribetur itaque &longs;ec­<lb/>tio Conica tran&longs;iens per puncta quinque <emph type="italics"/>A, B, C, P, D. q.E.F.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc recta expedite duci pote&longs;t quæ Trajectoriam quæ­<lb/>&longs;itam, in puncto quovis dato <emph type="italics"/>B,<emph.end type="italics"/>continget. </s> <s>Accedat punctum <emph type="italics"/>d<emph.end type="italics"/>ad <lb/>punctum <emph type="italics"/>B,<emph.end type="italics"/>& recta <emph type="italics"/>Bd<emph.end type="italics"/>evadet tangens quæ&longs;ita. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Unde etiam Trajectoriarum Centra, Diametri & Latera <lb/>recta inveniri po&longs;&longs;unt, ut in Corollario &longs;ecundo Lemmatis XIX. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Con&longs;tructio prior evadet paulo &longs;implicior jungendo <emph type="italics"/>BP,<emph.end type="italics"/>& in ea, <lb/>&longs;i opus e&longs;t, producta capiendo <emph type="italics"/>Bp<emph.end type="italics"/>ad <emph type="italics"/>BP<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>PR<emph.end type="italics"/>ad <emph type="italics"/>PT<emph.end type="italics"/>; & <lb/>per <emph type="italics"/>p<emph.end type="italics"/>agendo rectam infinitam <emph type="italics"/>p<emph.end type="italics"/>d ip&longs;i <emph type="italics"/>SPT<emph.end type="italics"/>parallelam, inque ea <lb/>capiendo &longs;emper <emph type="italics"/>p<emph.end type="italics"/>d æqualem <emph type="italics"/>Pr<emph.end type="italics"/>; & agendo rectas <emph type="italics"/>Bd, Cr<emph.end type="italics"/>con­<lb/>currentes in <emph type="italics"/>d.<emph.end type="italics"/>Nam cum &longs;int <emph type="italics"/>Pr<emph.end type="italics"/>ad <emph type="italics"/>Pt, PR<emph.end type="italics"/>ad <emph type="italics"/>PT, pB<emph.end type="italics"/>ad <emph type="italics"/>PB, <lb/>p<emph.end type="italics"/>d ad <emph type="italics"/>Pt<emph.end type="italics"/>in eadem ratione; erunt <emph type="italics"/>p<emph.end type="italics"/>d & <emph type="italics"/>Pr<emph.end type="italics"/>&longs;emper æqua-<pb xlink:href="039/01/104.jpg" pagenum="76"/>les. </s> <s>Hac methodo puncta Trajectoriæ inveniuntur expediti&longs;&longs;ime, </s></p> <p type="main"> <s><arrow.to.target n="note52"/>ni&longs;i mavis Curvam, ut in con&longs;tructione &longs;ecunda, de&longs;eribere Me­<lb/>chanice. </s></p> <p type="margin"> <s><margin.target id="note52"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIII. PROBLEMA XV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam de&longs;cribere quæ per data quatuor puncta tran&longs;ibit, & rec­<lb/>tam continget po&longs;itione datam.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Dentur tangens <emph type="italics"/>HB,<emph.end type="italics"/>punctum contactus <emph type="italics"/>B,<emph.end type="italics"/>& alia tria <lb/>puncta <emph type="italics"/>C, D, P.<emph.end type="italics"/>Junge <emph type="italics"/>BC,<emph.end type="italics"/>& agendo <emph type="italics"/>PS<emph.end type="italics"/>parallelam <emph type="italics"/>BH,<emph.end type="italics"/><lb/>& <emph type="italics"/>PQ<emph.end type="italics"/>parallelam <emph type="italics"/>BC,<emph.end type="italics"/>comple parallelogrammum <emph type="italics"/><expan abbr="BSPq.">BSPque</expan><emph.end type="italics"/><lb/><figure id="id.039.01.104.1.jpg" xlink:href="039/01/104/1.jpg"/><lb/>Age <emph type="italics"/>BD<emph.end type="italics"/>&longs;ecantem <emph type="italics"/>SP<emph.end type="italics"/>in <emph type="italics"/>T,<emph.end type="italics"/>& <emph type="italics"/>CD<emph.end type="italics"/>&longs;ecantem <emph type="italics"/>PQ<emph.end type="italics"/>in <emph type="italics"/>R.<emph.end type="italics"/>De­<lb/>nique, agendo quamvis <emph type="italics"/>tr<emph.end type="italics"/>ip&longs;i <emph type="italics"/>TR<emph.end type="italics"/>parallelam, de <emph type="italics"/>PQ, PS<emph.end type="italics"/><lb/>ab&longs;cinde <emph type="italics"/>Pr, Pt<emph.end type="italics"/>ip&longs;is <emph type="italics"/>PR, PT<emph.end type="italics"/>proportionales re&longs;pective; & <lb/>actarum <emph type="italics"/>Cr, Bt<emph.end type="italics"/>concur&longs;us <emph type="italics"/>d<emph.end type="italics"/>(per Lem. </s> <s>xx) incidet &longs;emper in <lb/>Trajectoriam de&longs;cribendam. <pb xlink:href="039/01/105.jpg" pagenum="77"/><arrow.to.target n="note53"/></s></p> <p type="margin"> <s><margin.target id="note53"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Idem aliter.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Revolvatur tum angulus magnitudine datus <emph type="italics"/>CBH<emph.end type="italics"/>circa polum <lb/><emph type="italics"/>B,<emph.end type="italics"/>tum radius quilibet rectilineus & utrinque productus <emph type="italics"/>DC<emph.end type="italics"/>cir­<lb/>ca polum <emph type="italics"/>C.<emph.end type="italics"/>Notentur puncta <emph type="italics"/>M, N<emph.end type="italics"/>in quibus anguli crus <emph type="italics"/>BC<emph.end type="italics"/><lb/>&longs;ecat radium illum ubi crus alterum <emph type="italics"/>BH<emph.end type="italics"/>concurrit cum eodem ra­<lb/>dio in punctis <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>D.<emph.end type="italics"/>Deinde ad actam infinitam <emph type="italics"/>MN<emph.end type="italics"/>con­<lb/><figure id="id.039.01.105.1.jpg" xlink:href="039/01/105/1.jpg"/><lb/>currant perpetuo radius ille <emph type="italics"/>CP<emph.end type="italics"/>vel <emph type="italics"/>CD<emph.end type="italics"/>& anguli crus <emph type="italics"/>BC,<emph.end type="italics"/>& <lb/>cruris alterius <emph type="italics"/>BH<emph.end type="italics"/>concur&longs;us cum radio delineabit Trajectoriam <lb/>quæ&longs;itam. </s></p> <p type="main"> <s>Nam &longs;i in con&longs;tructionibus Problematis &longs;uperioris accedat punc­<lb/>tum <emph type="italics"/>A<emph.end type="italics"/>ad punctum <emph type="italics"/>B,<emph.end type="italics"/>lineæ <emph type="italics"/>CA<emph.end type="italics"/>& <emph type="italics"/>CB<emph.end type="italics"/>coincident, & linea <emph type="italics"/>AB<emph.end type="italics"/>in <lb/>ultimo &longs;uo &longs;itu fiet tangens <emph type="italics"/>BH,<emph.end type="italics"/>atque adeo con&longs;tructiones ibi po­<lb/>&longs;itæ evadent eædem cum con&longs;tructionibus hic de&longs;criptis. </s> <s>Delinea­<lb/>bit igitur cruris <emph type="italics"/>BH<emph.end type="italics"/>concur&longs;us cum radio &longs;ectionem Conicam per <lb/>puncta <emph type="italics"/>C, D, P<emph.end type="italics"/>tran&longs;euntem, & rectam <emph type="italics"/>BH<emph.end type="italics"/>tangentem in puncto <lb/><emph type="italics"/>B. q.E.F.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Dentur puncta quatuor <emph type="italics"/>B, C, D, P<emph.end type="italics"/>extra tangentem <lb/><emph type="italics"/>HI<emph.end type="italics"/>&longs;ita. </s> <s>Junge bina lineis <emph type="italics"/>BD, CP<emph.end type="italics"/>concurrentibus in <emph type="italics"/>G,<emph.end type="italics"/>tangen-<pb xlink:href="039/01/106.jpg" pagenum="78"/><arrow.to.target n="note54"/>tique occurrentibus in <emph type="italics"/>H<emph.end type="italics"/>& <emph type="italics"/>I.<emph.end type="italics"/>Secetur tangens in <emph type="italics"/>A,<emph.end type="italics"/>ita ut &longs;it <lb/><emph type="italics"/>HA<emph.end type="italics"/>ad <emph type="italics"/>AI,<emph.end type="italics"/>ut e&longs;t rectan­<lb/><figure id="id.039.01.106.1.jpg" xlink:href="039/01/106/1.jpg"/><lb/>gulum &longs;ub media proportio­<lb/>nali inter <emph type="italics"/>CG<emph.end type="italics"/>& <emph type="italics"/>GP<emph.end type="italics"/>& me­<lb/>dia proportionali inter <emph type="italics"/>BH<emph.end type="italics"/>& <lb/><emph type="italics"/>HD,<emph.end type="italics"/>ad rectangulum &longs;ub me­<lb/>dia proportionali inter <emph type="italics"/>DG<emph.end type="italics"/>& <lb/><emph type="italics"/>GB<emph.end type="italics"/>& media proportionali in­<lb/>ter <emph type="italics"/>PI<emph.end type="italics"/>& <emph type="italics"/>IC<emph.end type="italics"/>; & erit <emph type="italics"/>A<emph.end type="italics"/>punc­<lb/>tum contactus. </s> <s>Nam &longs;i rectæ <lb/><emph type="italics"/>PI<emph.end type="italics"/>parallela <emph type="italics"/>HX<emph.end type="italics"/>Trajecto­<lb/>riam &longs;ecet in punctis quibu&longs;­<lb/>vis <emph type="italics"/>X<emph.end type="italics"/>& <emph type="italics"/>Y:<emph.end type="italics"/>erit (ex Conicis) <lb/>punctum <emph type="italics"/>A<emph.end type="italics"/>ita locandum, ut fuerit <emph type="italics"/>HA quad.<emph.end type="italics"/>ad <emph type="italics"/>AI quad.<emph.end type="italics"/>in ra­<lb/>tione compo&longs;ita ex ratione rectanguli <emph type="italics"/>XHY<emph.end type="italics"/>ad rectangulum <emph type="italics"/>BHD<emph.end type="italics"/><lb/>&longs;eu rectanguli <emph type="italics"/>CGP<emph.end type="italics"/>ad rectangulum <emph type="italics"/>DGB<emph.end type="italics"/>& ex ratione rectan­<lb/>guli <emph type="italics"/>BHD<emph.end type="italics"/>ad rectangulum <emph type="italics"/>PIC.<emph.end type="italics"/>Invento autem contactus <lb/>puncto <emph type="italics"/>A,<emph.end type="italics"/>de&longs;cribetur Trajectoria ut in ca&longs;u primo. <emph type="italics"/>q.E.F.<emph.end type="italics"/><lb/>Capi autem pote&longs;t punctum <emph type="italics"/>A<emph.end type="italics"/>vel inter puncta <emph type="italics"/>H<emph.end type="italics"/>& <emph type="italics"/>I,<emph.end type="italics"/>vel extra; <lb/>& perinde Trajectoria dupliciter de&longs;cribi. </s></p> <p type="margin"> <s><margin.target id="note54"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIV. PROBLEMA XVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam de&longs;cribere quæ tran&longs;ibit per data tria puncta & rectas <lb/>duas po&longs;itione datas continget.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Dentur tangentes <emph type="italics"/>HI, KL<emph.end type="italics"/>& <lb/><figure id="id.039.01.106.2.jpg" xlink:href="039/01/106/2.jpg"/><lb/>puncta <emph type="italics"/>B, C, D.<emph.end type="italics"/>Per punctorum <lb/>duo quævis <emph type="italics"/>B, D<emph.end type="italics"/>age rectam in­<lb/>finitam <emph type="italics"/>BD<emph.end type="italics"/>tangentibus occur­<lb/>rentem in punctis <emph type="italics"/>H, K.<emph.end type="italics"/>Deinde <lb/>etiam per alia duo quævis <emph type="italics"/>C, D<emph.end type="italics"/><lb/>age infinitam <emph type="italics"/>CD<emph.end type="italics"/>tangentibus oc­<lb/>currentem in punctis <emph type="italics"/>I, L.<emph.end type="italics"/>Actas <lb/>ita &longs;eca in <emph type="italics"/>R<emph.end type="italics"/>& <emph type="italics"/>S,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>HR<emph.end type="italics"/>ad <lb/><emph type="italics"/>KR<emph.end type="italics"/>ut e&longs;t media proportionalis <lb/>inter <emph type="italics"/>BH<emph.end type="italics"/>& <emph type="italics"/>HD<emph.end type="italics"/>ad mediam <lb/>proportionalem inter <emph type="italics"/>BK<emph.end type="italics"/>& <emph type="italics"/>KD<emph.end type="italics"/>; <lb/>& <emph type="italics"/>IS<emph.end type="italics"/>ad <emph type="italics"/>LS<emph.end type="italics"/>ut e&longs;t media pro­<lb/>portionalis inter <emph type="italics"/>CI<emph.end type="italics"/>& <emph type="italics"/>ID<emph.end type="italics"/>ad me­<lb/>diam proportionalem inter <emph type="italics"/>CL<emph.end type="italics"/><pb xlink:href="039/01/107.jpg" pagenum="79"/>& <emph type="italics"/>LD.<emph.end type="italics"/>Seca autem pro lubitu vel inter puncta <emph type="italics"/>K<emph.end type="italics"/>& <emph type="italics"/>H,<emph.end type="italics"/><lb/><arrow.to.target n="note55"/><emph type="italics"/>I<emph.end type="italics"/>& <emph type="italics"/>L,<emph.end type="italics"/>vel extra eadem: dein age <emph type="italics"/>RS<emph.end type="italics"/>&longs;ecantem tangentes in <emph type="italics"/>A<emph.end type="italics"/><lb/>& <emph type="italics"/>P,<emph.end type="italics"/>& erunt <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>puncta contactuum. </s> <s>Nam &longs;i <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/><lb/>&longs;upponantur e&longs;&longs;e puncta contactuum alicubi in tangentibus &longs;i­<lb/>ta; & per punctorum <emph type="italics"/>H, I, K, L<emph.end type="italics"/>quodvis <emph type="italics"/>I,<emph.end type="italics"/>in tangente al­<lb/>terutra <emph type="italics"/>HI<emph.end type="italics"/>&longs;itum, agatur recta <emph type="italics"/>IY<emph.end type="italics"/>tangenti alteri <emph type="italics"/>KL<emph.end type="italics"/>paral­<lb/>lela, quæ occurrat curvæ in <emph type="italics"/>X<emph.end type="italics"/>& <emph type="italics"/>Y,<emph.end type="italics"/>& in ea &longs;umatur <emph type="italics"/>IZ<emph.end type="italics"/>me­<lb/>dia proportionalis inter <emph type="italics"/>IX<emph.end type="italics"/>& <emph type="italics"/>IY:<emph.end type="italics"/>erit, ex Conicis, rectangulum <lb/><emph type="italics"/>XIY<emph.end type="italics"/>&longs;eu <emph type="italics"/>IZ quad.<emph.end type="italics"/>ad <emph type="italics"/>LP quad.<emph.end type="italics"/>ut rectangulum <emph type="italics"/>CID<emph.end type="italics"/>ad rectan­<lb/>gulum <emph type="italics"/>CLD,<emph.end type="italics"/>id e&longs;t (per con&longs;tructionem) ut <emph type="italics"/>SI quad.<emph.end type="italics"/>ad <lb/><emph type="italics"/>SL quad:<emph.end type="italics"/>atque adeo <emph type="italics"/>IZ<emph.end type="italics"/>ad <emph type="italics"/>LP<emph.end type="italics"/>ut <emph type="italics"/>SI<emph.end type="italics"/>ad <emph type="italics"/>SL.<emph.end type="italics"/>Jacent ergo punc­<lb/>ta <emph type="italics"/>S, P, Z<emph.end type="italics"/>in una recta. </s> <s>Porro tangentibus concurrentibus in <emph type="italics"/>G,<emph.end type="italics"/>e­<lb/>rit (ex Conicis) rectangulum <emph type="italics"/>XIY<emph.end type="italics"/>&longs;eu <emph type="italics"/>IZ quad.<emph.end type="italics"/>ad <emph type="italics"/>IA quad.<emph.end type="italics"/>ut <lb/><emph type="italics"/>GP quad<emph.end type="italics"/>ad <emph type="italics"/>GA quad:<emph.end type="italics"/>adeoque <emph type="italics"/>IZ<emph.end type="italics"/>& <emph type="italics"/>IA<emph.end type="italics"/>ut <emph type="italics"/>GP<emph.end type="italics"/>ad <emph type="italics"/>GA.<emph.end type="italics"/>Jacent <lb/>ergo puncta <emph type="italics"/>P, Z<emph.end type="italics"/>& <emph type="italics"/>A<emph.end type="italics"/>in una recta, adeoque puncta <emph type="italics"/>S, P<emph.end type="italics"/>& <emph type="italics"/>A<emph.end type="italics"/><lb/>&longs;unt in una recta. </s> <s>Et eodem argumento probabitur quod puncta <lb/><emph type="italics"/>R, P<emph.end type="italics"/>& <emph type="italics"/>A<emph.end type="italics"/>&longs;unt in una recta. </s> <s>Jacent igitur puncta contactuum <emph type="italics"/>A<emph.end type="italics"/><lb/>& <emph type="italics"/>P<emph.end type="italics"/>in recta <emph type="italics"/>RS.<emph.end type="italics"/>Hi&longs;ce autem inventis, Trajectoria de&longs;eribetur <lb/>ut in ca&longs;u primo Problematis &longs;uperioris. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note55"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>LEMMA XXII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Figuras in alias eju&longs;dem generis figur as mutare.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Tran&longs;mutanda &longs;it figura quævis <emph type="italics"/>HGI.<emph.end type="italics"/>Ducantur pro lubitu <lb/>rectæ duæ parallelæ <emph type="italics"/>AO, BL<emph.end type="italics"/>tertiam quamvis po&longs;itione datam <lb/><emph type="italics"/>AB<emph.end type="italics"/>&longs;ecantes in <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B,<emph.end type="italics"/><lb/><figure id="id.039.01.107.1.jpg" xlink:href="039/01/107/1.jpg"/><lb/>& a figuræ puncto quo­<lb/>vis <emph type="italics"/>G,<emph.end type="italics"/>ad rectam <emph type="italics"/>AB<emph.end type="italics"/><lb/>ducatur quævis <emph type="italics"/>GD,<emph.end type="italics"/><lb/>ip&longs;i <emph type="italics"/>OA<emph.end type="italics"/>parallela. </s> <s>De­<lb/>inde a puncto aliquo <emph type="italics"/>O,<emph.end type="italics"/><lb/>in linea <emph type="italics"/>OA<emph.end type="italics"/>dato, ad <lb/>punctum <emph type="italics"/>D<emph.end type="italics"/>ducatur <lb/>recta <emph type="italics"/>OD,<emph.end type="italics"/>ip&longs;i <emph type="italics"/>BL<emph.end type="italics"/>oc­<lb/>currens in <emph type="italics"/>d,<emph.end type="italics"/>& a puncto <lb/>occur&longs;us erigatur recta <lb/><emph type="italics"/>dg<emph.end type="italics"/>datum quemvis angulum cum recta <emph type="italics"/>BL<emph.end type="italics"/>continens, atque eam <lb/>habens rationem ad <emph type="italics"/>Od<emph.end type="italics"/>quam habet <emph type="italics"/>DG<emph.end type="italics"/>ad <emph type="italics"/>OD<emph.end type="italics"/>; & erit <emph type="italics"/>g<emph.end type="italics"/>punc­<lb/>tum in figura nova <emph type="italics"/>hgi<emph.end type="italics"/>puncto <emph type="italics"/>G<emph.end type="italics"/>re&longs;pondens. </s> <s>Eadem ratione <lb/>puncta &longs;ingula figuræ primæ dabunt puncta totidem figura novæ. <pb xlink:href="039/01/108.jpg" pagenum="80"/><arrow.to.target n="note56"/>Concipe igitur punctum <emph type="italics"/>G<emph.end type="italics"/>motu continuo percurrere puncta om­<lb/>nia figuræ primæ, & punctum <emph type="italics"/>g<emph.end type="italics"/>motu itidem continuo percurret <lb/>puncta omnia figuræ novæ & eandem de&longs;cribet. </s> <s>Di&longs;tinctionis gra­<lb/>tia nominemus <emph type="italics"/>DG<emph.end type="italics"/>ordinatam primam, <emph type="italics"/>dg<emph.end type="italics"/>ordinatam novam; <lb/><emph type="italics"/>AD<emph.end type="italics"/>ab&longs;ci&longs;&longs;am primam, <emph type="italics"/>ad<emph.end type="italics"/>ab&longs;ci&longs;&longs;am novam; <emph type="italics"/>O<emph.end type="italics"/>polum, <emph type="italics"/>OD<emph.end type="italics"/>ra­<lb/>dium ab&longs;cidentem, <emph type="italics"/>OA<emph.end type="italics"/>radium ordinatum primum, & <emph type="italics"/>Oa<emph.end type="italics"/>(qno <lb/>parallelogrammum <emph type="italics"/>OABa<emph.end type="italics"/>completur) radium ordinatum novum. </s></p> <p type="margin"> <s><margin.target id="note56"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Dico jam quod, &longs;i punctum <emph type="italics"/>G<emph.end type="italics"/>tangit rectam Lineam po&longs;itione da­<lb/>tam, punctum <emph type="italics"/>g<emph.end type="italics"/>tanget etiam Lineam rectam po&longs;itione datam. </s> <s>Si <lb/>punctum <emph type="italics"/>G<emph.end type="italics"/>tangit Conicam &longs;ectionem, punctum <emph type="italics"/>g<emph.end type="italics"/>tanget etiam <lb/>Conicam &longs;ectionem. </s> <s>Conicis &longs;ectionibus hic Circulum annumero. </s> <s><lb/>Porro &longs;i punctum <emph type="italics"/>G<emph.end type="italics"/>tan­<lb/><figure id="id.039.01.108.1.jpg" xlink:href="039/01/108/1.jpg"/><lb/>git Lineam tertii ordinis <lb/>Analytici, punctum <emph type="italics"/>g<emph.end type="italics"/><lb/>tanget Lineam tertii iti­<lb/>dem ordinis; & &longs;ic de <lb/>curvis lineis &longs;uperiorum <lb/>ordinum. </s> <s>Lineæ duæ e­<lb/>runt eju&longs;dem &longs;emper or­<lb/>dinis Analytici quas pun­<lb/>cta <emph type="italics"/>G, g<emph.end type="italics"/>tangunt. </s> <s>Et­<lb/>enim ut e&longs;t <emph type="italics"/>ad<emph.end type="italics"/>ad <emph type="italics"/>OA<emph.end type="italics"/><lb/>ita &longs;unt <emph type="italics"/>Od<emph.end type="italics"/>ad <emph type="italics"/>OD, dg<emph.end type="italics"/>ad <emph type="italics"/>DG,<emph.end type="italics"/>& <emph type="italics"/>AB<emph.end type="italics"/>ad <emph type="italics"/>AD<emph.end type="italics"/>; adeoque <emph type="italics"/>AD<emph.end type="italics"/><lb/>æqualis e&longs;t (<emph type="italics"/>OAXAB/ad<emph.end type="italics"/>), & <emph type="italics"/>DG<emph.end type="italics"/>æqualis e&longs;t (<emph type="italics"/>OAXdg/ad<emph.end type="italics"/>). Jam &longs;i punc­<lb/>tum <emph type="italics"/>G<emph.end type="italics"/>tangit rectam Lineam, atque adeo in æquatione quavis, <lb/>qua relatio inter ab&longs;ci&longs;&longs;am <emph type="italics"/>AD<emph.end type="italics"/>& ordinatam <emph type="italics"/>DG<emph.end type="italics"/>habetur, in­<lb/>determinatæ illæ <emph type="italics"/>AD<emph.end type="italics"/>& <emph type="italics"/>DG<emph.end type="italics"/>ad unicam tantum dimen&longs;ionem <lb/>a&longs;cendunt, &longs;cribendo in hac æquatione (<emph type="italics"/>OAXAB/ad<emph.end type="italics"/>) pro <emph type="italics"/>AD,<emph.end type="italics"/>& <lb/>(<emph type="italics"/>OAXdg/ad<emph.end type="italics"/>) pro <emph type="italics"/>DG,<emph.end type="italics"/>producetur æquatio nova, in qua ab&longs;ci&longs;&longs;a no­<lb/>va <emph type="italics"/>ad<emph.end type="italics"/>& ordinata nova <emph type="italics"/>dg<emph.end type="italics"/>ad unicam tantum dimen&longs;ionem a&longs;cen­<lb/>dent, atque adeo quæ de&longs;ignat Lineam rectam. </s> <s>Sin <emph type="italics"/>AD<emph.end type="italics"/>& <emph type="italics"/>DG<emph.end type="italics"/><lb/>(vel earum alterutra) a&longs;cendebant ad duas dimen&longs;iones in æquati­<lb/>one prima, a&longs;cendent itidem <emph type="italics"/>ad<emph.end type="italics"/>& <emph type="italics"/>dg<emph.end type="italics"/>ad duas in æquatione &longs;ecun­<lb/>da. </s> <s>Et &longs;ic de tribus vel pluribus dimen&longs;ionibus. </s> <s>Indeterminatæ <lb/><emph type="italics"/>ad, dg<emph.end type="italics"/>in æquatione &longs;ecunda & <emph type="italics"/>AD, DG<emph.end type="italics"/>in prima a&longs;cendent &longs;em­<lb/>per ad eundem dimen&longs;ionum numerum, & propterea Lineæ, quas <lb/>puncta <emph type="italics"/>G, g<emph.end type="italics"/>tangunt, &longs;unt eju&longs;dem ordinis Analytici. </s></p><pb xlink:href="039/01/109.jpg" pagenum="81"/> <p type="main"> <s>Dico præterea quod &longs;i recta aliqua tangat lineam curvam in fi­<lb/><arrow.to.target n="note57"/>gura prima; hæc recta eodem modo cum curva in figuram novam <lb/>tran&longs;lata tanget lineam illam curvam in figura nova: & contra. </s> <s>Nam <lb/>&longs;i Curvæ puncta quævis duo accedunt ad invicem & coeunt in fi­<lb/>gura prima, puncta eadem tran&longs;lata accedent ad invicem & coibunt <lb/>in figura nova, atque adeo rectæ, quibus hæc puncta junguntur, &longs;i­<lb/>mul evadent curvarum tangentes in figura utraque. </s> <s>Componi po&longs;­<lb/>&longs;ent harum a&longs;&longs;ertionum Demon&longs;trationes more magis Geometrico. </s> <s><lb/>Sed brevitati con&longs;ulo. </s></p> <p type="margin"> <s><margin.target id="note57"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>Igitur &longs;i figura rectilinea in aliam tran&longs;mutanda e&longs;t, &longs;ufficit rec­<lb/>tarum a quibus conflatur inter&longs;ectiones transferre, & per ea&longs;dem <lb/>in figura nova lineas rectas ducere. </s> <s>Sin curvilineam tran&longs;mutare <lb/>oportet, transferenda &longs;unt puncta, tangentes & aliæ rectæ quarum <lb/>ope curva linea definitur. </s> <s>In&longs;ervit autem hoc Lemma &longs;olutioni <lb/>difficiliorum Problematum, tran&longs;mutando figuras propo&longs;itas in &longs;im­<lb/>pliciores. </s> <s>Nam rectæ quævis convergentes tran&longs;mutantur in pa­<lb/>rallelas, adhibendo pro radio ordinato primo, lineam quam­<lb/>vis rectam quæ per concur&longs;um convergentium tran&longs;it: id adeo quia <lb/>concur&longs;us ille hoc pacto abit in infinitum, lineæ autem parallelæ <lb/>&longs;unt quæ ad punctum infinite di&longs;tans tendunt. </s> <s>Po&longs;tquam autem <lb/>Problema &longs;olvitur in figura nova, &longs;i per inver&longs;as operationes tran&longs;­<lb/>mutetur hæc figura in figuram primam, habebitur &longs;olutio quæ&longs;ita. </s></p> <p type="main"> <s>Utile e&longs;t etiam hoc Lemma in &longs;olutione Solidorum Problema­<lb/>tum. </s> <s>Nam quoties duæ &longs;ectiones Conicæ obvenerint, quarum in­<lb/>ter&longs;ectione Problema &longs;olvi pote&longs;t, tran&longs;mutare licet earum alter­<lb/>utram, &longs;i Hyperbola &longs;it vel Parabola, in Ellip&longs;in: deinde Ellip&longs;is <lb/>facile mutatur in Circulum. </s> <s>Recta item & &longs;ectio Conica, in con­<lb/>&longs;tructione Planorum Problematum, vertuntur in Rectam & Cir­<lb/>culum. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXV. PROBLEMA XVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam de&longs;cribere qua per data duo puncta tran&longs;ibit & rectas <lb/>tres continget po&longs;itione datas.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Per concur&longs;um tangentium quarumvis duarum cum &longs;e invicem, & <lb/>concur&longs;um tangentis tertiæ cum recta illa, quæ per puncta duo data <lb/>tran&longs;it, age rectam infinitam; eaque adhibita pro radio ordinato pri­<lb/>mo, tran&longs;mutetur figura, per Lemma &longs;uperius, in figuram novam. </s> <s>In <pb xlink:href="039/01/110.jpg" pagenum="82"/><arrow.to.target n="note58"/>hac figura tangentes illæ duæ evadent &longs;ibi invicem parallelæ, & tan­<lb/>gens tertia fiet parallela rectæ per <lb/><figure id="id.039.01.110.1.jpg" xlink:href="039/01/110/1.jpg"/><lb/>puncta duo data tran&longs;eunti. </s> <s>Sunto <lb/><emph type="italics"/>hi, kl<emph.end type="italics"/>tangentes illæ duæ parallelæ, <lb/><emph type="italics"/>ik<emph.end type="italics"/>tangens tertia, & <emph type="italics"/>hl<emph.end type="italics"/>recta huic <lb/>parallela tran&longs;iens per puncta illa <lb/><emph type="italics"/>a, b,<emph.end type="italics"/>per quæ Conica &longs;ectio in hac <lb/>figura nova tran&longs;ire debet, & pa­<lb/>rallelogrammum <emph type="italics"/>hikl<emph.end type="italics"/>complens. </s> <s><lb/>Secentur rectæ <emph type="italics"/>hi, ik, kl<emph.end type="italics"/>in <emph type="italics"/>c, d, e,<emph.end type="italics"/><lb/>ita ut &longs;it <emph type="italics"/>hc<emph.end type="italics"/>ad latus quadratum <lb/>rectanguli <emph type="italics"/>ahb, ic<emph.end type="italics"/>ad <emph type="italics"/>id,<emph.end type="italics"/>& <emph type="italics"/>ke<emph.end type="italics"/><lb/>ad <emph type="italics"/>kd<emph.end type="italics"/>ut e&longs;t &longs;umma rectarum <emph type="italics"/>hi<emph.end type="italics"/><lb/>& <emph type="italics"/>kl<emph.end type="italics"/>ad &longs;ummam trium linea­<lb/>rum quarum prima e&longs;t recta <emph type="italics"/>ik,<emph.end type="italics"/>& alteræ duæ &longs;unt latera quadrata <lb/>rectangulorum <emph type="italics"/>ahb<emph.end type="italics"/>& <emph type="italics"/>alb<emph.end type="italics"/>& erunt <emph type="italics"/>c, d, e<emph.end type="italics"/>puncta contactuum. </s> <s>Et­<lb/>enim, ex Conicis, &longs;unt <emph type="italics"/>hc<emph.end type="italics"/>quadratum ad rectangulum <emph type="italics"/>ahb,<emph.end type="italics"/>& <lb/><emph type="italics"/>ic<emph.end type="italics"/>quadratum ad <emph type="italics"/>id<emph.end type="italics"/>quadratum, & <emph type="italics"/>ke<emph.end type="italics"/>quadratum ad <emph type="italics"/>kd<emph.end type="italics"/>quadratum, <lb/>& <emph type="italics"/>el<emph.end type="italics"/>quadratum ad rectangulum <emph type="italics"/>alb<emph.end type="italics"/>in eadem ratione; & propter­<lb/>ea <emph type="italics"/>hc<emph.end type="italics"/>ad latus quadratum ip&longs;ius <emph type="italics"/>ahb, ic<emph.end type="italics"/>ad <emph type="italics"/>id, ke<emph.end type="italics"/>ad <emph type="italics"/>kd,<emph.end type="italics"/>& <emph type="italics"/>el<emph.end type="italics"/>ad <lb/>latus quadratum ip&longs;ius <emph type="italics"/>alb<emph.end type="italics"/>&longs;unt in &longs;ubduplicata illa ratione, & <lb/>compo&longs;ite, in data ratione omnium antecedentium <emph type="italics"/>hi<emph.end type="italics"/>& <emph type="italics"/>kl<emph.end type="italics"/>ad <lb/>omnes con&longs;equentes, quæ &longs;unt latus quadratum rectanguli <emph type="italics"/>ahb<emph.end type="italics"/>& <lb/>recta <emph type="italics"/>ik<emph.end type="italics"/>& latus quadratum rectanguli <emph type="italics"/>alb.<emph.end type="italics"/>Habentur igitur ex <lb/>data illa ratione puncta contactuum <emph type="italics"/>c, d, e,<emph.end type="italics"/>in figura nova. </s> <s>Per <lb/>inver&longs;as operationes Lemmatis novi&longs;&longs;imi transferantur hæc pun­<lb/>cta in figuram primam & ibi, per Probl. </s> <s>XIV, de&longs;cribetur <lb/>Trajectoria. <emph type="italics"/>q.E.F.<emph.end type="italics"/>Ceterum perinde ut puncta <emph type="italics"/>a, b<emph.end type="italics"/>ja­<lb/>cent vel inter puncta <emph type="italics"/>h, l,<emph.end type="italics"/>vel extra, debent puncta <emph type="italics"/>c, d, e<emph.end type="italics"/>vel <lb/>inter puncta <emph type="italics"/>h, i, k, l<emph.end type="italics"/>capi, vel extra. </s> <s>Si punctorum <emph type="italics"/>a, b<emph.end type="italics"/>al­<lb/>terutrum cadit inter puncta <emph type="italics"/>h, l,<emph.end type="italics"/>& alterum extra, Problema im­<lb/>po&longs;&longs;ibile e&longs;t. </s></p> <p type="margin"> <s><margin.target id="note58"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVI. PROBLEMA XVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam de&longs;cribere quæ tran&longs;ibit per punctum datum & rectas <lb/>quatuor po&longs;itione datas continget.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Ab inter&longs;ectione communi duarum quarumlibet tangentium ad <lb/>inter&longs;ectionem communem reliquarum duarum agatur recta infini-<pb xlink:href="039/01/111.jpg" pagenum="83"/>ta, & eadem pro radio ordinato primo adhibita, tran&longs;mutetur fi­<lb/><arrow.to.target n="note59"/>gura (per Lem. </s> <s>XXII) in figuram novam, & tangentes binæ, quæ ad <lb/>radium ordinatum primum concurrebant, jam evadent parallelæ. </s> <s>Sun­<lb/>to illæ <emph type="italics"/>hi<emph.end type="italics"/>& <emph type="italics"/>kl, ik<emph.end type="italics"/>& <emph type="italics"/>hl<emph.end type="italics"/>continentes parallelogrammum <emph type="italics"/>hikl.<emph.end type="italics"/>Sit­<lb/>que <emph type="italics"/>p<emph.end type="italics"/>punctum in hac nova figura, puncto in figura prima dato <lb/>re&longs;pondens. </s> <s>Per figuræ centrum <emph type="italics"/>O<emph.end type="italics"/>agatur <emph type="italics"/>pq,<emph.end type="italics"/>& exi&longs;tente <emph type="italics"/>Oq<emph.end type="italics"/>æ­<lb/>quali <emph type="italics"/>Op,<emph.end type="italics"/>erit <emph type="italics"/>q<emph.end type="italics"/>punctum alterum per quod &longs;ectio Conica in hac <lb/>figura nova tran&longs;ire debet. </s> <s>Per Lemmatis XXII operationem in­<lb/>ver&longs;am transferatur hoc punctum in figuram primam, & ibi habe­<lb/>buntur puncta duo per quæ Trajectoria de&longs;cribenda e&longs;t. </s> <s>Per ea­<lb/>dem vero de&longs;cribi pote&longs;t Trajectoria illa per Prob. </s> <s>XVII. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note59"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>LEMMA XXIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si rectæ duæ po&longs;itione datæ<emph.end type="italics"/>AC, BD <emph type="italics"/>ad data puncta<emph.end type="italics"/>A, B, <emph type="italics"/>ter­<lb/>minentur, datamque habeant rationem ad invicem, & recta<emph.end type="italics"/><lb/>CD, <emph type="italics"/>qua puncta indeterminata<emph.end type="italics"/>C, D <emph type="italics"/>junguntur, &longs;ecetur in ra­<lb/>tione data in<emph.end type="italics"/>K: <emph type="italics"/>dico quod punctum<emph.end type="italics"/>K <emph type="italics"/>locabitur in recta po&longs;i­<lb/>tione data.<emph.end type="italics"/></s></p> <p type="main"> <s>Concurrant enim rectæ <emph type="italics"/>AC,<emph.end type="italics"/><lb/><figure id="id.039.01.111.1.jpg" xlink:href="039/01/111/1.jpg"/><lb/><emph type="italics"/>BD<emph.end type="italics"/>in <emph type="italics"/>E,<emph.end type="italics"/>& in <emph type="italics"/>BE<emph.end type="italics"/>capiatur <emph type="italics"/>BG<emph.end type="italics"/><lb/>ad <emph type="italics"/>AE<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>BD<emph.end type="italics"/>ad <emph type="italics"/>AC,<emph.end type="italics"/>&longs;it­<lb/>que <emph type="italics"/>FD<emph.end type="italics"/>&longs;emper æqualis datæ <lb/><emph type="italics"/>EG<emph.end type="italics"/>; & erit ex con&longs;tructione <lb/><emph type="italics"/>EC<emph.end type="italics"/>ad <emph type="italics"/>GD,<emph.end type="italics"/>hoc e&longs;t, ad <emph type="italics"/>EF<emph.end type="italics"/>ut <lb/><emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>BD,<emph.end type="italics"/>adeoQ.E.I. ratione <lb/>data, & propterea dabitur &longs;pecie <lb/>triangulum <emph type="italics"/>EFC.<emph.end type="italics"/>Secetur <emph type="italics"/>CF<emph.end type="italics"/><lb/>in <emph type="italics"/>L<emph.end type="italics"/>ut &longs;it <emph type="italics"/>CL<emph.end type="italics"/>ad <emph type="italics"/>CF<emph.end type="italics"/>in ratio­<lb/>ne <emph type="italics"/>CK<emph.end type="italics"/>ad <emph type="italics"/>CD<emph.end type="italics"/>; &, ob datam il­<lb/>lam rationem, dabitur etiam &longs;pecie triangulum <emph type="italics"/>EFL<emph.end type="italics"/>; proindeque <lb/>punctum <emph type="italics"/>L<emph.end type="italics"/>locabitur in recta <emph type="italics"/>EL<emph.end type="italics"/>po&longs;itione data. </s> <s>Junge <emph type="italics"/>LK,<emph.end type="italics"/>& <lb/>&longs;imilia erunt triangula <emph type="italics"/>CLK, CFD<emph.end type="italics"/>; &, ob datam <emph type="italics"/>FD<emph.end type="italics"/>& datam <lb/>rationem <emph type="italics"/>LK<emph.end type="italics"/>ad <emph type="italics"/>FD,<emph.end type="italics"/>dabitur <emph type="italics"/>LK.<emph.end type="italics"/>Huic æqualis capiatur <emph type="italics"/>EH,<emph.end type="italics"/><lb/>& erit &longs;emper <emph type="italics"/>ELKH<emph.end type="italics"/>parallelogrammum. </s> <s>Locatur igitur punc­<lb/>tum <emph type="italics"/>K<emph.end type="italics"/>in parallelogrammi illius latere po&longs;itione dato <emph type="italics"/>HK. Q.E.D.<emph.end type="italics"/><pb xlink:href="039/01/112.jpg" pagenum="84"/><arrow.to.target n="note60"/></s></p> <p type="margin"> <s><margin.target id="note60"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>LEMMA XXIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si rectæ tres tangant quamcunque Coni&longs;ectionem, quarum duæ pa­<lb/>rallelæ &longs;int ac dentur po&longs;itione; dico quod Sectionis &longs;emidia­<lb/>meter hi&longs;ce duabus parallela, &longs;it media proportionalis inter ha­<lb/>rum &longs;egmenta, punctis contactuum & tangenti tertiæ inter­<lb/>jecta.<emph.end type="italics"/></s></p> <p type="main"> <s>Sunto <emph type="italics"/>AF, GB<emph.end type="italics"/>pa­<lb/><figure id="id.039.01.112.1.jpg" xlink:href="039/01/112/1.jpg"/><lb/>rallelæ duæ Coni&longs;ec­<lb/>tionem <emph type="italics"/>ADB<emph.end type="italics"/>tan­<lb/>gentes in <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B; EF<emph.end type="italics"/><lb/>recta tertia Coni&longs;ec­<lb/>tionem tangens in <emph type="italics"/>I,<emph.end type="italics"/><lb/>& occurrens prioribus <lb/>tangentibus in <emph type="italics"/>F<emph.end type="italics"/>& <emph type="italics"/>G<emph.end type="italics"/>; <lb/>&longs;itque <emph type="italics"/>CD<emph.end type="italics"/>&longs;emidiame­<lb/>ter Figuræ tangenti­<lb/>bus parallela: Dico <lb/>quod <emph type="italics"/>AF, CD, BG<emph.end type="italics"/><lb/>&longs;unt continue proportionales. </s></p> <p type="main"> <s>Nam &longs;i diametri conjugatæ <emph type="italics"/>AB, DM<emph.end type="italics"/>tangenti <emph type="italics"/>FG<emph.end type="italics"/>occurrant <lb/>in <emph type="italics"/>E<emph.end type="italics"/>& <emph type="italics"/>H,<emph.end type="italics"/>&longs;eque mutuo &longs;ecent in <emph type="italics"/>C,<emph.end type="italics"/>& compleatur parallelogram­<lb/>mum <emph type="italics"/>IKCL<emph.end type="italics"/>; erit, ex natura Sectionum Conicarum, ut <emph type="italics"/>EC<emph.end type="italics"/>ad <lb/><emph type="italics"/>CA<emph.end type="italics"/>ita <emph type="italics"/>CA<emph.end type="italics"/>ad <emph type="italics"/>CL,<emph.end type="italics"/>& ita divi&longs;im <emph type="italics"/>EC-CA<emph.end type="italics"/>ad <emph type="italics"/>CA-CL,<emph.end type="italics"/>&longs;eu <lb/><emph type="italics"/>EA<emph.end type="italics"/>ad <emph type="italics"/>AL,<emph.end type="italics"/>& compo&longs;ite <emph type="italics"/>EA<emph.end type="italics"/>ad <emph type="italics"/>EA+AL<emph.end type="italics"/>&longs;eu <emph type="italics"/>EL<emph.end type="italics"/>ut <emph type="italics"/>EC<emph.end type="italics"/>ad <lb/><emph type="italics"/>EC+CA<emph.end type="italics"/>&longs;eu <emph type="italics"/>EB<emph.end type="italics"/>; adeoque (ob &longs;imilitudinem triangulorum <emph type="italics"/>EAF, <lb/>ELI, ECH, EBG) AF<emph.end type="italics"/>ad <emph type="italics"/>LI<emph.end type="italics"/>ut <emph type="italics"/>CH<emph.end type="italics"/>ad <emph type="italics"/>BG.<emph.end type="italics"/>E&longs;t itidem, <lb/>ex natura Sectionum Conicarum, <emph type="italics"/>LI<emph.end type="italics"/>(&longs;eu <emph type="italics"/>CK<emph.end type="italics"/>) ad <emph type="italics"/>CD<emph.end type="italics"/>ut <emph type="italics"/>CD<emph.end type="italics"/>ad <lb/><emph type="italics"/>CH<emph.end type="italics"/>; atque, adeo ex æquo perturbate, <emph type="italics"/>AF<emph.end type="italics"/>ad <emph type="italics"/>CD<emph.end type="italics"/>ut <emph type="italics"/>CD<emph.end type="italics"/>ad <emph type="italics"/>BG. <lb/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i tangentes duæ <emph type="italics"/>FG, PQ<emph.end type="italics"/>tangentibus parallelis <lb/><emph type="italics"/>AF, BG<emph.end type="italics"/>occurrant in <emph type="italics"/>F<emph.end type="italics"/>& <emph type="italics"/>G, P<emph.end type="italics"/>& <emph type="italics"/>Q,<emph.end type="italics"/>&longs;eque mutuo &longs;ecent in <emph type="italics"/>O<emph.end type="italics"/>; <lb/>erit (ex æquo perturbate) <emph type="italics"/>AF<emph.end type="italics"/>ad <emph type="italics"/>BQ<emph.end type="italics"/>ut <emph type="italics"/>AP<emph.end type="italics"/>ad <emph type="italics"/>BG,<emph.end type="italics"/>& divi&longs;im <lb/>ut <emph type="italics"/>FP<emph.end type="italics"/>ad <emph type="italics"/>GQ,<emph.end type="italics"/>atque adeo ut <emph type="italics"/>FO<emph.end type="italics"/>ad <emph type="italics"/>OG.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Unde etiam rectæ duæ <emph type="italics"/>PG, FQ<emph.end type="italics"/>per puncta <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>G, <lb/>F<emph.end type="italics"/>& <emph type="italics"/>Q<emph.end type="italics"/>ductæ, concurrent ad rectam <emph type="italics"/>ACB<emph.end type="italics"/>per centrum Figuræ & <lb/>puncta contactuum <emph type="italics"/>A, B<emph.end type="italics"/>tran&longs;euntem. <pb xlink:href="039/01/113.jpg" pagenum="85"/><arrow.to.target n="note61"/></s></p> <p type="margin"> <s><margin.target id="note61"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>LEMMA XXV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si parallelogrammi latera quatuor infinite producta tangant Sectio­<lb/>nem quamcunque Conicam, & ab&longs;cindantur ad tangentem quamvis <lb/>quintam; &longs;umantur autem laterum quorumvis duorum contermi­<lb/>norum ab&longs;ci&longs;&longs;æ terminatæ ad angulos oppo&longs;itos parallelogrammi: <lb/>dico quod ab&longs;ci&longs;&longs;a alterutra &longs;it ad latus illud a quo est ab&longs;ci&longs;&longs;a, ut <lb/>pars lateris alterius contermini inter punctum contactus & latus <lb/>tertium, est ad ab&longs;ci&longs;&longs;arum alteram.<emph.end type="italics"/></s></p> <p type="main"> <s>Tangant parallelogrammi <emph type="italics"/>MLIK<emph.end type="italics"/>latera quatuor <emph type="italics"/>ML, IK, KL, <lb/>MI<emph.end type="italics"/>&longs;ectionem Conicam in <emph type="italics"/>A, B, C, D,<emph.end type="italics"/>& &longs;ecet tangens quinta <emph type="italics"/>FQ<emph.end type="italics"/><lb/>hæc latera in <emph type="italics"/>F, Q, H<emph.end type="italics"/><lb/><figure id="id.039.01.113.1.jpg" xlink:href="039/01/113/1.jpg"/><lb/>& <emph type="italics"/>E<emph.end type="italics"/>; &longs;umantur autem <lb/>laterum <emph type="italics"/>MI, KI<emph.end type="italics"/>ab­<lb/>&longs;ci&longs;&longs;æ <emph type="italics"/>ME, KQ,<emph.end type="italics"/>vel <lb/>laterum <emph type="italics"/>KL, ML<emph.end type="italics"/>ab­<lb/>&longs;ci&longs;&longs;æ <emph type="italics"/>KH, MF:<emph.end type="italics"/>di­<lb/>co quod &longs;it <emph type="italics"/>ME<emph.end type="italics"/>ad <lb/><emph type="italics"/>MI<emph.end type="italics"/>ut <emph type="italics"/>BK<emph.end type="italics"/>ad <emph type="italics"/>KQ<emph.end type="italics"/>; <lb/>& <emph type="italics"/>KH<emph.end type="italics"/>ad <emph type="italics"/>KL<emph.end type="italics"/>ut <lb/><emph type="italics"/>AM<emph.end type="italics"/>ad <emph type="italics"/>MF.<emph.end type="italics"/>Nam <lb/>per Corollarium &longs;e­<lb/>cundum Lemmatis &longs;uperioris, e&longs;t <emph type="italics"/>ME<emph.end type="italics"/>ad <emph type="italics"/>EI<emph.end type="italics"/>ut (<emph type="italics"/>AM<emph.end type="italics"/>&longs;eu) <emph type="italics"/>BK<emph.end type="italics"/>ad <lb/><emph type="italics"/>BQ,<emph.end type="italics"/>& componendo <emph type="italics"/>ME<emph.end type="italics"/>ad <emph type="italics"/>MI<emph.end type="italics"/>ut <emph type="italics"/>BK<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="Kq.">Kque</expan> Q.E.D.<emph.end type="italics"/><lb/>Item <emph type="italics"/>KH<emph.end type="italics"/>ad <emph type="italics"/>HL<emph.end type="italics"/>ut (<emph type="italics"/>BK<emph.end type="italics"/>&longs;eu) <emph type="italics"/>AM<emph.end type="italics"/>ad <emph type="italics"/>AF,<emph.end type="italics"/>& dividendo <emph type="italics"/>KH<emph.end type="italics"/>ad <lb/><emph type="italics"/>KL<emph.end type="italics"/>ut <emph type="italics"/>AM<emph.end type="italics"/>ad <emph type="italics"/>MF. Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i datur parallelogramum <emph type="italics"/>IKLM,<emph.end type="italics"/>circa datam Sec­<lb/>tionem Conicam de&longs;eriptum, dabitur rectangulum <emph type="italics"/>KQXME,<emph.end type="italics"/>ut <lb/>& huic æquale rectangulum <emph type="italics"/>KHXMF.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et &longs;i &longs;exta ducatur tangens <emph type="italics"/>eq<emph.end type="italics"/>tangentibus <emph type="italics"/>KI, MI<emph.end type="italics"/><lb/>occurrens in <emph type="italics"/>q<emph.end type="italics"/>& <emph type="italics"/>e<emph.end type="italics"/>; rectangulum <emph type="italics"/>KQXME<emph.end type="italics"/>æquabitur rectan­<lb/>gulo <emph type="italics"/>KqXMe<emph.end type="italics"/>; eritque <emph type="italics"/>KQ<emph.end type="italics"/>ad <emph type="italics"/>Me<emph.end type="italics"/>ut <emph type="italics"/>Kq<emph.end type="italics"/>ad <emph type="italics"/>ME,<emph.end type="italics"/>& divi&longs;im ut <lb/><emph type="italics"/>Qq<emph.end type="italics"/>ad <emph type="italics"/>Ee.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Unde etiam &longs;i <emph type="italics"/>Eq, eQ<emph.end type="italics"/>jungantur & bi&longs;ecentur, & recta <lb/>per puncta bi&longs;ectionum agatur, tran&longs;ibit hæc per centrum Sectio­<lb/>nis Conicæ. </s> <s>Nam cum &longs;it <emph type="italics"/>Qq<emph.end type="italics"/>ad <emph type="italics"/>Ee<emph.end type="italics"/>ut <emph type="italics"/>KQ<emph.end type="italics"/>ad <emph type="italics"/>Me,<emph.end type="italics"/>tran&longs;ibit ea-<pb xlink:href="039/01/114.jpg" pagenum="86"/><arrow.to.target n="note62"/>dem recta per medium omnium <emph type="italics"/>Eq, eQ, MK<emph.end type="italics"/>; (per Lem. </s> <s>XXIII) <lb/>& medium rectæ <emph type="italics"/>MK<emph.end type="italics"/>e&longs;t centrum Sectionis. </s></p> <p type="margin"> <s><margin.target id="note62"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVII. PROBLEMA XIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam de&longs;cribere quæ rectas quinque po&longs;itione datas continget.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Dentur pofitione tangentes <emph type="italics"/>ABG, BCF, GCD, FDE, EA.<emph.end type="italics"/><lb/>Figuræ quadrilateræ &longs;ub quatuor quibu&longs;vis contentæ <emph type="italics"/>ABFE<emph.end type="italics"/>dia­<lb/>gonales <emph type="italics"/>AF, BE<emph.end type="italics"/>bi&longs;eca, & (per Corol. </s> <s>3. Lem. </s> <s>XXV) recta <emph type="italics"/>MN<emph.end type="italics"/><lb/>per puncta bi&longs;ectionum acta tran&longs;ibit per centrum Trajectoriæ. </s> <s><lb/>Rur&longs;us Figuræ quadrilateræ <emph type="italics"/>BGDF,<emph.end type="italics"/>&longs;ub aliis quibu&longs;vis quatuor <lb/><figure id="id.039.01.114.1.jpg" xlink:href="039/01/114/1.jpg"/><lb/>tangentibus contentæ, diagonales (ut ita dicam) <emph type="italics"/>BD, GF<emph.end type="italics"/>bi­<lb/>&longs;eca in <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>Q:<emph.end type="italics"/>& recta <emph type="italics"/>PQ<emph.end type="italics"/>per puncta bi&longs;ectionum acta tran&longs;­<lb/>ibit per centrum Trajectoriæ. </s> <s>Dabitur ergo centrum in concur&longs;u bi­<lb/>&longs;ecantium. </s> <s>Sit illud <emph type="italics"/>O.<emph.end type="italics"/>Tangenti cuivis <emph type="italics"/>BC<emph.end type="italics"/>parallelam age <emph type="italics"/>KL,<emph.end type="italics"/><lb/>ad eam di&longs;tantiam ut centrum <emph type="italics"/>O<emph.end type="italics"/>in medio inter parallelas locetur, <lb/>& acta <emph type="italics"/>KL<emph.end type="italics"/>tanget Trajectoriam de&longs;cribendam. </s> <s>Secet hæc tan-<pb xlink:href="039/01/115.jpg" pagenum="87"/>gentes alias qua&longs;vis duas <emph type="italics"/>GCD, FDE<emph.end type="italics"/>in <emph type="italics"/>L<emph.end type="italics"/>& <emph type="italics"/>K.<emph.end type="italics"/>Per harum <lb/><arrow.to.target n="note63"/>tangentium non parallelarum <emph type="italics"/>CL, FK<emph.end type="italics"/>cum parallelis <emph type="italics"/>CF, KL<emph.end type="italics"/><lb/>concur&longs;us <emph type="italics"/>C<emph.end type="italics"/>& <emph type="italics"/>K, F<emph.end type="italics"/>& <emph type="italics"/>L<emph.end type="italics"/>age <emph type="italics"/>CK, FL<emph.end type="italics"/>concurrentes in <emph type="italics"/>R,<emph.end type="italics"/>& rec­<lb/>ta <emph type="italics"/>OR<emph.end type="italics"/>ducta & producta &longs;ecabit tangentes parallelas <emph type="italics"/>CF, KL<emph.end type="italics"/>in <lb/>punctis contactuum. </s> <s>Patet hoc per Corol. </s> <s>2. Lem. </s> <s>XXIV. </s> <s>Ea­<lb/>dem methodo invenire licet alia contactuum puncta, & tum de­<lb/>mum per Probl. </s> <s>XIV. &c. </s> <s>Trajectoriam de&longs;cribere. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note63"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Problemata, ubi dantur Trajectoriarum vel centra vel A&longs;ymp­<lb/>toti, includuntnr in præcedentibus. </s> <s>Nam datis punctis & tangen­<lb/>tibus una cum centro, dantur alia totidem puncta aliæque tangen­<lb/>tes a centro ex altera ejus parte æqualiter di&longs;tantes. </s> <s>A&longs;ymptotos <lb/>autem pro tangente habenda e&longs;t, & ejus terminus infinite di&longs;tans <lb/>(&longs;i ita loqui fas &longs;it) pro puncto contactus. </s> <s>Concipe tangentis cu­<lb/>ju&longs;vis punctum contactus abire in infinitum, & tangens vertetur in <lb/>A&longs;ymptoton, atque con&longs;tructiones Problematis XIV & Ca&longs;us pri­<lb/>mi Problematis XV vertentur in con&longs;tructiones Problematum ubi <lb/>A&longs;ymptoti dantur. </s></p> <p type="main"> <s>Po&longs;tquam Trajectoria de&longs;cripta e&longs;t, invenire licet axes & umbi­<lb/>licos ejus hac methodo. </s> <s>In con&longs;tructione & figura Lemmatis XXI, <lb/>fac ut angulorum mobi­<lb/><figure id="id.039.01.115.1.jpg" xlink:href="039/01/115/1.jpg"/><lb/>lium <emph type="italics"/>PBN, PCN<emph.end type="italics"/>cru­<lb/>ra <emph type="italics"/>BP, CP,<emph.end type="italics"/>quorum <lb/>concur&longs;u Trajectoria de­<lb/>&longs;cribebatur, &longs;int &longs;ibi invi­<lb/>cem parallela, eumque <lb/>&longs;ervantia &longs;itum revolvan­<lb/>tur circa polos &longs;uos <emph type="italics"/>B, C<emph.end type="italics"/><lb/>in figura illa. </s> <s>Interea ve­<lb/>ro de&longs;cribant altera an­<lb/>gulorum illorum crura <lb/><emph type="italics"/>CN, BN,<emph.end type="italics"/>concur&longs;u <lb/>&longs;uo <emph type="italics"/>K<emph.end type="italics"/>vel <emph type="italics"/>k,<emph.end type="italics"/>Circulum <lb/><emph type="italics"/>IBKGC.<emph.end type="italics"/>Sit Circuli <lb/>hujus centrum <emph type="italics"/>O.<emph.end type="italics"/>Ab <lb/>hoc centro ad Regulam <lb/><emph type="italics"/>MN,<emph.end type="italics"/>ad quam altera illa crura <emph type="italics"/>CN, BN<emph.end type="italics"/>interea concurrebant <pb xlink:href="039/01/116.jpg" pagenum="88"/><arrow.to.target n="note64"/>dum Trajectoria de&longs;cribebatur, demitte normalem <emph type="italics"/>OH<emph.end type="italics"/>Circulo oc­<lb/>currentem in <emph type="italics"/>K<emph.end type="italics"/>& <emph type="italics"/>L.<emph.end type="italics"/>Et ubi crura illa altera <emph type="italics"/>CK, BK<emph.end type="italics"/>concur­<lb/>runt ad punctum illud <emph type="italics"/>K<emph.end type="italics"/>quod Regulæ propius e&longs;t, crura prima <lb/><emph type="italics"/>CP, BP<emph.end type="italics"/>parallela erunt axi majori, & perpendicularia minori; <lb/>& contrarium eveniet &longs;i crura eadem concurrunt ad punctum remo­<lb/>tius <emph type="italics"/>L.<emph.end type="italics"/>Unde &longs;i detur Trajectoriæ centrum, dabuntur axes. </s> <s>Hi&longs;ce <lb/>autem datis, umbilici &longs;unt in promptu. </s></p> <p type="margin"> <s><margin.target id="note64"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Axium vero quadrata &longs;unt ad invicem ut <emph type="italics"/>KH<emph.end type="italics"/>ad <emph type="italics"/>LH,<emph.end type="italics"/>& inde <lb/>facile e&longs;t Trajectoriam <lb/><figure id="id.039.01.116.1.jpg" xlink:href="039/01/116/1.jpg"/><lb/>&longs;pecie datam per data <lb/>quatuor puncta de&longs;cri­<lb/>bere. </s> <s>Nam &longs;i duo ex <lb/>punctis datis con&longs;titu­<lb/>antur poli <emph type="italics"/>C, B,<emph.end type="italics"/>tertium <lb/>dabit angulos mobiles <lb/><emph type="italics"/>PCK, PBK<emph.end type="italics"/>; his au­<lb/>tem datis de&longs;cribi pote&longs;t <lb/>Circulus <emph type="italics"/>IBKGC.<emph.end type="italics"/><lb/>Tum ob datam &longs;pecie <lb/>Trajectoriam, dabitur <lb/>ratio <emph type="italics"/>OH<emph.end type="italics"/>ad <emph type="italics"/>OK,<emph.end type="italics"/>ad­<lb/>eoQ.E.I.&longs;a <emph type="italics"/>OH.<emph.end type="italics"/>Cen­<lb/>tro <emph type="italics"/>O<emph.end type="italics"/>& intervallo <emph type="italics"/>OH<emph.end type="italics"/><lb/>de&longs;cribe alium circulum, <lb/>& recta quæ tangit hunc circulum, & tran&longs;it per concur&longs;um crurum <lb/><emph type="italics"/>CK, BK,<emph.end type="italics"/>ubi crura prima <emph type="italics"/>CP, BP<emph.end type="italics"/>concurrunt ad quartum da­<lb/>tum punctum erit Regula illa <emph type="italics"/>MN<emph.end type="italics"/>cujus ope Trajectoria de&longs;cri­<lb/>betur. </s> <s>Unde etiam vici&longs;&longs;im Trapezium &longs;pecie datum (&longs;i ca&longs;us qui­<lb/>dam impo&longs;&longs;ibiles excipiantur) in data quavis Sectione Conica in­<lb/>&longs;cribi pote&longs;t. </s></p> <p type="main"> <s>Sunt & alia Lemmata quorum ope Trajectoriæ &longs;pecie datæ, <lb/>datis punctis & tangentibus, de&longs;cribi po&longs;&longs;unt. </s> <s>Ejus generis <lb/>e&longs;t quod, &longs;i recta linea per punctum quodvis po&longs;itione datum <lb/>ducatur, quæ datam Coni&longs;ectionem in punctis duobus inter&longs;e­<lb/>cet, & inter&longs;ectionum intervallum bi&longs;ecetur, punctum bi&longs;ectionis <lb/>tanget aliam Coni&longs;ectionem eju&longs;dem &longs;peciei cum priore, atque <lb/>axes habentem prioris axibus parallelos. </s> <s>Sed propero ad magis <lb/>utilia. <pb xlink:href="039/01/117.jpg" pagenum="89"/><arrow.to.target n="note65"/></s></p> <p type="margin"> <s><margin.target id="note65"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>LEMMA XXVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Trianguli &longs;pecie & magnitudine dati tres angulos ad rectas tot­<lb/>idem po&longs;itione datas, quæ non &longs;unt omnes parallelæ, &longs;ingulos ad <lb/>&longs;ingulas ponere.<emph.end type="italics"/></s></p> <p type="main"> <s>Dantur po&longs;itione tres rectæ infinitæ <emph type="italics"/>AB, AC, BC,<emph.end type="italics"/>& opor­<lb/>tet triangulum <emph type="italics"/>DEF<emph.end type="italics"/>ita locare, ut angulus ejus <emph type="italics"/>D<emph.end type="italics"/>lineam <emph type="italics"/>AB,<emph.end type="italics"/><lb/>angulus <emph type="italics"/>E<emph.end type="italics"/>lineam <emph type="italics"/>AC,<emph.end type="italics"/><lb/><figure id="id.039.01.117.1.jpg" xlink:href="039/01/117/1.jpg"/><figure id="id.039.01.117.2.jpg" xlink:href="039/01/117/2.jpg"/><lb/>& angulus <emph type="italics"/>F<emph.end type="italics"/>lineam <lb/><emph type="italics"/>BC<emph.end type="italics"/>tangat. </s> <s>Super <emph type="italics"/>DE, <lb/>DF<emph.end type="italics"/>& <emph type="italics"/>EF<emph.end type="italics"/>de&longs;cribe <lb/>tria circulorum &longs;eg­<lb/>menta <emph type="italics"/>DRE, DGF, <lb/>EMF,<emph.end type="italics"/>quæ capiant <lb/>angulos angulis <emph type="italics"/>BAC, <lb/>ABC, ACB<emph.end type="italics"/>æquales <lb/>re&longs;pective. </s> <s>De&longs;criban­<lb/>tur autem hæc &longs;egmen­<lb/>ta ad eas partes linea­<lb/>rum <emph type="italics"/>DE, DF, EF<emph.end type="italics"/>ut <lb/>literæ <emph type="italics"/>DRED<emph.end type="italics"/>eodem <lb/>ordine cum literis <lb/><emph type="italics"/>BACB,<emph.end type="italics"/>literæ <emph type="italics"/>DGFD<emph.end type="italics"/><lb/>eodem cum literis <lb/><emph type="italics"/>ABCA,<emph.end type="italics"/>& literæ <lb/><emph type="italics"/>EMFE<emph.end type="italics"/>eodem cum <lb/>literis <emph type="italics"/>ACBA<emph.end type="italics"/>in orbem <lb/>redeant; deinde com­<lb/>pleantur hæc &longs;egmenta <lb/>in circulos integros. </s> <s>Se­<lb/>cent circuli duo prio­<lb/>res &longs;e mutuo in <emph type="italics"/>G,<emph.end type="italics"/>&longs;int­<lb/>que centra eorum <emph type="italics"/>P<emph.end type="italics"/>& <lb/><emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/>Junctis <emph type="italics"/>GP, PQ,<emph.end type="italics"/><lb/>cape <emph type="italics"/>Ga<emph.end type="italics"/>ad <emph type="italics"/>AB<emph.end type="italics"/>ut e&longs;t <lb/><emph type="italics"/>GP<emph.end type="italics"/>ad <emph type="italics"/>PQ,<emph.end type="italics"/>& cen­<lb/>tro <emph type="italics"/>G,<emph.end type="italics"/>intervallo <emph type="italics"/>Ga<emph.end type="italics"/><lb/>de&longs;cribe circulum, qui &longs;ecet circulum primum <emph type="italics"/>DGE<emph.end type="italics"/>in <emph type="italics"/>a.<emph.end type="italics"/>Jungatur <lb/>tum <emph type="italics"/>aD<emph.end type="italics"/>&longs;ecans circulum &longs;ecundum <emph type="italics"/>DFG<emph.end type="italics"/>in <emph type="italics"/>b,<emph.end type="italics"/>tum <emph type="italics"/>aE<emph.end type="italics"/>&longs;ecans cir-<pb xlink:href="039/01/118.jpg" pagenum="90"/><arrow.to.target n="note66"/>culum tertium <emph type="italics"/>EMF<emph.end type="italics"/>in <emph type="italics"/>c.<emph.end type="italics"/>Et compleatur Figura <emph type="italics"/>ABC def<emph.end type="italics"/>&longs;imi­<lb/>lis & æqualis Figuræ <emph type="italics"/>abcDEF.<emph.end type="italics"/>Dico factum. </s></p> <p type="margin"> <s><margin.target id="note66"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Agatur enim <emph type="italics"/>Fc<emph.end type="italics"/>ip&longs;i <emph type="italics"/>aD<emph.end type="italics"/>occurrens in <emph type="italics"/>n,<emph.end type="italics"/>& jungantur <emph type="italics"/>aG, bG, <lb/>QG, QD, PD.<emph.end type="italics"/>Ex con&longs;tructione e&longs;t angulus <emph type="italics"/>EaD<emph.end type="italics"/>æqualis an­<lb/>gulo <emph type="italics"/>CAB,<emph.end type="italics"/>& angulus <lb/><figure id="id.039.01.118.1.jpg" xlink:href="039/01/118/1.jpg"/><figure id="id.039.01.118.2.jpg" xlink:href="039/01/118/2.jpg"/><lb/><emph type="italics"/>acF<emph.end type="italics"/>æqualis angulo <lb/><emph type="italics"/>ACB,<emph.end type="italics"/>adeoque trian­<lb/>gulum <emph type="italics"/>anc<emph.end type="italics"/>triangulo <lb/><emph type="italics"/>ABC<emph.end type="italics"/>æquiangulum. </s> <s><lb/>Ergo angulus <emph type="italics"/>anc<emph.end type="italics"/>&longs;eu <lb/><emph type="italics"/>FnD<emph.end type="italics"/>angulo <emph type="italics"/>ABC,<emph.end type="italics"/><lb/>adeoque angulo <emph type="italics"/>FbD<emph.end type="italics"/><lb/>æqualis e&longs;t; & propter­<lb/>ea punctum <emph type="italics"/>n<emph.end type="italics"/>incidit in <lb/>punctum <emph type="italics"/>b.<emph.end type="italics"/>Porro an­<lb/>gulus <emph type="italics"/>GPQ,<emph.end type="italics"/>qui di­<lb/>midius e&longs;t anguli ad <lb/>centrum <emph type="italics"/>GPD,<emph.end type="italics"/>æqua­<lb/>lis e&longs;t angulo ad cir­<lb/>cumferentiam <emph type="italics"/>GaD<emph.end type="italics"/>; <lb/>& angulus <emph type="italics"/>GQP,<emph.end type="italics"/>qui <lb/>dimidius e&longs;t anguli ad <lb/>centrum <emph type="italics"/>GQD,<emph.end type="italics"/>æ­<lb/>qualis e&longs;t complemen­<lb/>to ad duos rectos an­<lb/>guli ad circumferenti­<lb/>am <emph type="italics"/>GbD,<emph.end type="italics"/>adeoque æ­<lb/>qualis angulo <emph type="italics"/>Gba<emph.end type="italics"/>; <lb/>funtQ.E.I.eo triangu­<lb/>la <emph type="italics"/>GPQ, Gab<emph.end type="italics"/>&longs;imi­<lb/>lia; & <emph type="italics"/>Ga<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>ab<emph.end type="italics"/><lb/>ut <emph type="italics"/>GP<emph.end type="italics"/>ad <emph type="italics"/>PQ<emph.end type="italics"/>; id e&longs;t <lb/>(ex con&longs;tructione) ut <lb/><emph type="italics"/>Ga<emph.end type="italics"/>ad <emph type="italics"/>AB.<emph.end type="italics"/>Æquan­<lb/>tur itaque <emph type="italics"/>ab<emph.end type="italics"/>& <emph type="italics"/>AB<emph.end type="italics"/>; & propterea triangula <emph type="italics"/>abc, ABC,<emph.end type="italics"/>quæ mo­<lb/>do &longs;imilia e&longs;&longs;e probavimus, &longs;unt etiam æqualia. </s> <s>Unde, cum tan­<lb/>gant in&longs;uper trianguli <emph type="italics"/>DEF<emph.end type="italics"/>anguli <emph type="italics"/>D, E, F<emph.end type="italics"/>trianguli <emph type="italics"/>abc<emph.end type="italics"/>latera <lb/><emph type="italics"/>ab, ac, bc<emph.end type="italics"/>re&longs;pective, compleri pote&longs;t Figura <emph type="italics"/>ABCdef<emph.end type="italics"/>Figuræ <lb/><emph type="italics"/>abc DEF<emph.end type="italics"/>&longs;imilis & æqualis, atque eam complendo &longs;olvetur Pro­<lb/>blema. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s></p><pb xlink:href="039/01/119.jpg" pagenum="91"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc recta duci pote&longs;t cujus partes longitudine datæ rectis <lb/><arrow.to.target n="note67"/>tribus po&longs;itione datis interjacebunt. </s> <s>Concipe Triangulum <emph type="italics"/>DEF,<emph.end type="italics"/><lb/>puncto <emph type="italics"/>D<emph.end type="italics"/>ad latus <emph type="italics"/>EF<emph.end type="italics"/>accedente, & lateribus <emph type="italics"/>DE, DF<emph.end type="italics"/>in di­<lb/>rectum po&longs;itis, mutari in lineam rectam, cujus pars data <emph type="italics"/>DE<emph.end type="italics"/>rec­<lb/>tis po&longs;itione datis <emph type="italics"/>AB, AC,<emph.end type="italics"/>& pars data <emph type="italics"/>DF<emph.end type="italics"/>rectis po&longs;itione da­<lb/>tis <emph type="italics"/>AB, BC<emph.end type="italics"/>interponi debet; & applicando con&longs;tructionem præ­<lb/>cedentem ad hunc ca&longs;um &longs;olvetur Problema. </s></p> <p type="margin"> <s><margin.target id="note67"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVIII. PROBLEMA XX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam &longs;pecie & magnitudine datam de&longs;cribere, cujus partes da­<lb/>tæ rectis tribus po&longs;itione datis interjacebunt.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>De&longs;cribenda &longs;it Trajectoria quæ &longs;it &longs;imilis & æqualis Lineæ cur­<lb/>væ <emph type="italics"/>DEF,<emph.end type="italics"/>quæque a rectis tribus <emph type="italics"/>AB, AC, BC<emph.end type="italics"/>po&longs;itione datis, in <lb/><figure id="id.039.01.119.1.jpg" xlink:href="039/01/119/1.jpg"/><lb/>partes datis hujus partibus <emph type="italics"/>DE<emph.end type="italics"/>& <emph type="italics"/>EF<emph.end type="italics"/>&longs;imiles & æquales &longs;eca­<lb/>bitur. </s></p> <p type="main"> <s>Age rectas <emph type="italics"/>DE, EF, DF,<emph.end type="italics"/>& trianguli hujus <emph type="italics"/>DEF<emph.end type="italics"/>pone an­<lb/>los <emph type="italics"/>D, E, F<emph.end type="italics"/>ad rectas illas po&longs;itione datas (per Lem. </s> <s>XXVI) Dein <lb/>circa triangulum de&longs;cribe Trajectoriam Curvæ <emph type="italics"/>DEF<emph.end type="italics"/>&longs;imilem & <lb/>æqualem. <emph type="italics"/>q.E.F.<emph.end type="italics"/><pb xlink:href="039/01/120.jpg" pagenum="92"/><arrow.to.target n="note68"/></s></p> <p type="margin"> <s><margin.target id="note68"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>LEMMA XXVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Trapezium &longs;pecie datum de&longs;cribere cujus anguli ad rectas quatuor po­<lb/>&longs;itione datas, quæ neque omnes parallelæ &longs;unt, neque ad commune <lb/>punctum convergunt, &longs;inguli ad &longs;ingulas con&longs;i&longs;tent.<emph.end type="italics"/></s></p> <p type="main"> <s>Dentur po&longs;itione rectæ quatuor <emph type="italics"/>ABC, AD, BD, CE,<emph.end type="italics"/>qua­<lb/>rum prima &longs;ecet &longs;ecundam in <emph type="italics"/>A,<emph.end type="italics"/>tertiam in <emph type="italics"/>B,<emph.end type="italics"/>& quartam in <emph type="italics"/>C:<emph.end type="italics"/><lb/>& de&longs;cribendum &longs;it Trapezium <emph type="italics"/>fghi<emph.end type="italics"/>quod &longs;it Trapezio <emph type="italics"/>FGHI<emph.end type="italics"/><lb/><figure id="id.039.01.120.1.jpg" xlink:href="039/01/120/1.jpg"/><lb/>&longs;imile, & cujus angulus <emph type="italics"/>f,<emph.end type="italics"/>angulo dato <emph type="italics"/>F<emph.end type="italics"/>æqualis, tangat rectam <lb/><emph type="italics"/>ABC,<emph.end type="italics"/>cæterique anguli <emph type="italics"/>g, h, i,<emph.end type="italics"/>cæteris angulis datis <emph type="italics"/>G, H, I<emph.end type="italics"/>æqua­<lb/>les, tangant cæteras lineas <emph type="italics"/>AD, BD, CE<emph.end type="italics"/>re&longs;pective. </s> <s>Jungatur <lb/><emph type="italics"/>FH<emph.end type="italics"/>& &longs;uper <emph type="italics"/>FG, FH, FI<emph.end type="italics"/>de&longs;cribantur totidem circulorum &longs;eg­<lb/>menta <emph type="italics"/>FSG, FTH, FVI<emph.end type="italics"/>; quorum primum <emph type="italics"/>FSG<emph.end type="italics"/>capiat angu-<pb xlink:href="039/01/121.jpg" pagenum="93"/>lum æqualem angulo <emph type="italics"/>BAD,<emph.end type="italics"/>&longs;ecundum <emph type="italics"/>FTH<emph.end type="italics"/>capiat angulum æ­<lb/><arrow.to.target n="note69"/>qualem angulo <emph type="italics"/>CBD,<emph.end type="italics"/>ac tertium <emph type="italics"/>FVI<emph.end type="italics"/>capiat angulum æqualem <lb/>angulo <emph type="italics"/>ACE.<emph.end type="italics"/>De&longs;cribi autem debent &longs;egmenta ad eas partes li­<lb/>nearum <emph type="italics"/>FG, FH, FI,<emph.end type="italics"/>ut literarum <emph type="italics"/>FSGF<emph.end type="italics"/>idem &longs;it ordo circula­<lb/>ris qui literarum <emph type="italics"/>BADB,<emph.end type="italics"/>utque literæ <emph type="italics"/>FTHF<emph.end type="italics"/>eodem ordine cum <lb/>literis <emph type="italics"/>CBDC,<emph.end type="italics"/>& literæ <emph type="italics"/>FVIF<emph.end type="italics"/>eodem cum literis <emph type="italics"/>ACEA<emph.end type="italics"/>in or­<lb/>bem redeant. </s> <s>Compleantur &longs;egmenta in circulos integros, &longs;itque <emph type="italics"/>P<emph.end type="italics"/><lb/>centrum circuli primi <emph type="italics"/>FSG,<emph.end type="italics"/>& <emph type="italics"/>Q<emph.end type="italics"/>centrum &longs;ecundi <emph type="italics"/>FTH.<emph.end type="italics"/>Jungatur <lb/>& utrinque producatur <emph type="italics"/>PQ,<emph.end type="italics"/>& in ea capiatur <emph type="italics"/>QR<emph.end type="italics"/>in ea ratione ad <lb/><emph type="italics"/>PQ<emph.end type="italics"/>quam habet <emph type="italics"/>BC<emph.end type="italics"/>ad <emph type="italics"/>AB.<emph.end type="italics"/>Capiatur autem <emph type="italics"/>QR<emph.end type="italics"/>ad eas partes <lb/>puncti <emph type="italics"/>Q<emph.end type="italics"/>ut literarum <emph type="italics"/>P, Q, R<emph.end type="italics"/>idem &longs;it ordo atque literarum <lb/><emph type="italics"/>A, B, C:<emph.end type="italics"/>centroque <emph type="italics"/>R<emph.end type="italics"/>& intervallo <emph type="italics"/>RF<emph.end type="italics"/>de&longs;cribatur circulus quartus <lb/><emph type="italics"/>FNc<emph.end type="italics"/>&longs;ecans circulum tertium <emph type="italics"/>FVI<emph.end type="italics"/>in <emph type="italics"/>c.<emph.end type="italics"/>Jungatur <emph type="italics"/>Fc<emph.end type="italics"/>&longs;ecans <lb/>circulum primum in <emph type="italics"/>a<emph.end type="italics"/>& &longs;ecundum in <emph type="italics"/>b.<emph.end type="italics"/>Agantur <emph type="italics"/>a G, b H, c I,<emph.end type="italics"/>& <lb/>Figuræ <emph type="italics"/>abc FGHI<emph.end type="italics"/>&longs;imilis con&longs;tituatur Figura <emph type="italics"/>ABCfghi:<emph.end type="italics"/>Eritque <lb/>Trapezium <emph type="italics"/>fghi<emph.end type="italics"/>illud ip&longs;um quod con&longs;tituere oportebat. </s></p> <p type="margin"> <s><margin.target id="note69"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>Secent enim circuli duo primi <emph type="italics"/>FSG, FTH<emph.end type="italics"/>&longs;e mutuo in <emph type="italics"/>K.<emph.end type="italics"/><lb/>Jungantur <emph type="italics"/>PK, QK, RK, a K, b K, c K,<emph.end type="italics"/>& producatur <emph type="italics"/>QP<emph.end type="italics"/>ad <emph type="italics"/>L.<emph.end type="italics"/><lb/>Anguli ad circumferentias <emph type="italics"/>FaK, FbK, FcK<emph.end type="italics"/>&longs;unt &longs;emi&longs;&longs;es an­<lb/>gulorum <emph type="italics"/>FPK, FQK, FRK<emph.end type="italics"/>ad centra, adeoque angulorum <lb/>illorum dimidiis <emph type="italics"/>LPK, LQK, LRK<emph.end type="italics"/>æquales. </s> <s>E&longs;t ergo Figura <lb/><emph type="italics"/>PQRK<emph.end type="italics"/>Figuræ <emph type="italics"/>abcK<emph.end type="italics"/>æquiangula & &longs;imilis, & propterea <emph type="italics"/>ab<emph.end type="italics"/>e&longs;t <lb/>ad <emph type="italics"/>bc<emph.end type="italics"/>ut <emph type="italics"/>PQ<emph.end type="italics"/>ad <emph type="italics"/>QR,<emph.end type="italics"/>id e&longs;t, ut <emph type="italics"/>AB<emph.end type="italics"/>ad <emph type="italics"/>BC.<emph.end type="italics"/>Angulis in&longs;uper <emph type="italics"/>FaG, <lb/>FbH, FcI<emph.end type="italics"/>æquantur <emph type="italics"/>fAg, fBh, fCi<emph.end type="italics"/>per con&longs;tructionem. </s> <s>Er­<lb/>go Figuræ <emph type="italics"/>abcFGHI<emph.end type="italics"/>Figura &longs;imilis <emph type="italics"/>ABCfghi<emph.end type="italics"/>compleri pote&longs;t <lb/>Quo facto Trapezium <emph type="italics"/>fghi<emph.end type="italics"/>con&longs;tituetur &longs;imile Trapezio <emph type="italics"/>FGHI<emph.end type="italics"/><lb/>& angulis &longs;uis <emph type="italics"/>f, g, h, i<emph.end type="italics"/>tanget rectas <emph type="italics"/>ABC, AD, BD, CE <lb/>q.E.F.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc recta duci pote&longs;t cujus partes, rectis quatuor po&longs;i­<lb/>tione datis dato ordine interjectæ, datam habebunt proportionem <lb/>ad invicem. </s> <s>Augeantur anguli <emph type="italics"/>FGH, GHI<emph.end type="italics"/>u&longs;que eo, ut rectæ <emph type="italics"/>FG, <lb/>GH, HI<emph.end type="italics"/>in directum jaceant, & in hoc ca&longs;u con&longs;truendo Proble­<lb/>ma, ducetur recta <emph type="italics"/>fghi<emph.end type="italics"/>cujus partes <emph type="italics"/>fg, gh, hi,<emph.end type="italics"/>rectis quatuor po­<lb/>&longs;itione datis <emph type="italics"/>AB<emph.end type="italics"/>& <emph type="italics"/>AD, AD<emph.end type="italics"/>& <emph type="italics"/>BD, BD<emph.end type="italics"/>& <emph type="italics"/>CE<emph.end type="italics"/>interjectæ, e­<lb/>runt ad invicem ut lineæ <emph type="italics"/>FG, GH, HI,<emph.end type="italics"/>eundemque &longs;ervabunt or­<lb/>dinem inter &longs;e. </s> <s>Idem vero &longs;ic fit expeditius. <pb xlink:href="039/01/122.jpg" pagenum="94"/><arrow.to.target n="note70"/></s></p> <p type="margin"> <s><margin.target id="note70"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Producantur <emph type="italics"/>AB<emph.end type="italics"/>ad <emph type="italics"/>K,<emph.end type="italics"/>& <emph type="italics"/>BD<emph.end type="italics"/>ad <emph type="italics"/>L,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>BK<emph.end type="italics"/>ad <emph type="italics"/>AB<emph.end type="italics"/>ut <lb/><emph type="italics"/>HI<emph.end type="italics"/>ad <emph type="italics"/>GH<emph.end type="italics"/>; & <emph type="italics"/>DL<emph.end type="italics"/>ad <emph type="italics"/>BD<emph.end type="italics"/>ut <emph type="italics"/>GI<emph.end type="italics"/>ad <emph type="italics"/>FG<emph.end type="italics"/>; & jungatur <emph type="italics"/>KL<emph.end type="italics"/><lb/>occurrens rectæ <emph type="italics"/>CE<emph.end type="italics"/>in <emph type="italics"/>i.<emph.end type="italics"/>Producatur <emph type="italics"/>iL<emph.end type="italics"/>ad <emph type="italics"/>M,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>LM<emph.end type="italics"/>ad <emph type="italics"/>iL<emph.end type="italics"/><lb/>ut <emph type="italics"/>GH<emph.end type="italics"/>ad <emph type="italics"/>HI,<emph.end type="italics"/>& agatur tum <emph type="italics"/>MQ<emph.end type="italics"/>ip&longs;i <emph type="italics"/>LB<emph.end type="italics"/>parallela rectæque <lb/><emph type="italics"/>AD<emph.end type="italics"/>occurrens in <emph type="italics"/>g,<emph.end type="italics"/>tum <emph type="italics"/>gi<emph.end type="italics"/>&longs;ecans <emph type="italics"/>AB, BD<emph.end type="italics"/>in <emph type="italics"/>f, h.<emph.end type="italics"/>Dico <lb/>factum. </s></p> <p type="main"> <s>Secet enim <emph type="italics"/>Mg<emph.end type="italics"/>rectam <emph type="italics"/>AB<emph.end type="italics"/>in <emph type="italics"/>Q,<emph.end type="italics"/>& <emph type="italics"/>AD<emph.end type="italics"/>rectam <emph type="italics"/>KL<emph.end type="italics"/>in <emph type="italics"/>S,<emph.end type="italics"/>& <lb/>agatur <emph type="italics"/>AP<emph.end type="italics"/>quæ &longs;it ip&longs;i <emph type="italics"/>BD<emph.end type="italics"/>parallela & occurrat <emph type="italics"/>iL<emph.end type="italics"/>in <emph type="italics"/>P,<emph.end type="italics"/>& <lb/>erunt <emph type="italics"/>gM<emph.end type="italics"/>ad <emph type="italics"/>Lh (gi<emph.end type="italics"/>ad <emph type="italics"/>hi, Mi<emph.end type="italics"/>ad <emph type="italics"/>Li, GI<emph.end type="italics"/>ad <emph type="italics"/>HI, AK<emph.end type="italics"/>ad <lb/><emph type="italics"/>BK<emph.end type="italics"/>) & <emph type="italics"/>AP<emph.end type="italics"/>ad <emph type="italics"/>BL<emph.end type="italics"/>in eadem ratione. </s> <s>Secetur <emph type="italics"/>DL<emph.end type="italics"/>in <emph type="italics"/>R<emph.end type="italics"/>ut &longs;it <lb/><figure id="id.039.01.122.1.jpg" xlink:href="039/01/122/1.jpg"/><lb/><emph type="italics"/>DL<emph.end type="italics"/>ad <emph type="italics"/>RL<emph.end type="italics"/>in eadem illa ratione, & ob proportionales <emph type="italics"/>gS<emph.end type="italics"/>ad <lb/><emph type="italics"/>gM, AS<emph.end type="italics"/>ad <emph type="italics"/>AP,<emph.end type="italics"/>& <emph type="italics"/>DS<emph.end type="italics"/>ad <emph type="italics"/>DL<emph.end type="italics"/>; erit, ex æquo, ut <emph type="italics"/>gS<emph.end type="italics"/>ad <emph type="italics"/>Lh<emph.end type="italics"/>ita <lb/><emph type="italics"/>AS<emph.end type="italics"/>ad <emph type="italics"/>BL<emph.end type="italics"/>& <emph type="italics"/>DS<emph.end type="italics"/>ad <emph type="italics"/>RL<emph.end type="italics"/>; & mixtim, <emph type="italics"/>BL-RL<emph.end type="italics"/>ad <emph type="italics"/>Lh-BL<emph.end type="italics"/><lb/>ut <emph type="italics"/>AS-DS<emph.end type="italics"/>ad <emph type="italics"/>gS-AS.<emph.end type="italics"/>Id e&longs;t <emph type="italics"/>BR<emph.end type="italics"/>ad <emph type="italics"/>Bh<emph.end type="italics"/>ut <emph type="italics"/>AD<emph.end type="italics"/>ad <emph type="italics"/>Ag<emph.end type="italics"/>ad­<lb/>eoque ut <emph type="italics"/>BD<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="gq.">gque</expan><emph.end type="italics"/>Et vici&longs;&longs;im <emph type="italics"/>BR<emph.end type="italics"/>ad <emph type="italics"/>BD<emph.end type="italics"/>ut <emph type="italics"/>Bh<emph.end type="italics"/>ad <emph type="italics"/>gQ,<emph.end type="italics"/>&longs;eu <lb/><emph type="italics"/>fh<emph.end type="italics"/>ad <emph type="italics"/>fg.<emph.end type="italics"/>Sed ex con&longs;tructione linea <emph type="italics"/>RL<emph.end type="italics"/>eadem ratione &longs;ecta fuit <lb/>in <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>R<emph.end type="italics"/>atque linea <emph type="italics"/>FI<emph.end type="italics"/>in <emph type="italics"/>G<emph.end type="italics"/>& <emph type="italics"/>H:<emph.end type="italics"/>ideoque e&longs;t <emph type="italics"/>BR<emph.end type="italics"/>ad <emph type="italics"/>BD<emph.end type="italics"/><lb/>ut <emph type="italics"/>FH<emph.end type="italics"/>ad <emph type="italics"/>FG.<emph.end type="italics"/>Ergo <emph type="italics"/>fh<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>fg<emph.end type="italics"/>ut <emph type="italics"/>FH<emph.end type="italics"/>ad <emph type="italics"/>FG.<emph.end type="italics"/>Cum igitur <lb/>&longs;it etiam <emph type="italics"/>gi<emph.end type="italics"/>ad <emph type="italics"/>hi<emph.end type="italics"/>ut <emph type="italics"/>Mi<emph.end type="italics"/>ad <emph type="italics"/>Li,<emph.end type="italics"/>id e&longs;t, ut <emph type="italics"/>GI<emph.end type="italics"/>ad <emph type="italics"/>HI,<emph.end type="italics"/>patet li­<lb/>neas <emph type="italics"/>FI, fi<emph.end type="italics"/>in <emph type="italics"/>g<emph.end type="italics"/>& <emph type="italics"/>h, G<emph.end type="italics"/>& <emph type="italics"/>H<emph.end type="italics"/>&longs;imiliter &longs;ectas e&longs;&longs;e. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s></p><pb xlink:href="039/01/123.jpg" pagenum="95"/> <p type="main"> <s>In con&longs;tructione Corollarii hujus po&longs;tquam ducitur <emph type="italics"/>LK<emph.end type="italics"/>&longs;ecans </s></p> <p type="main"> <s><arrow.to.target n="note71"/><emph type="italics"/>CE<emph.end type="italics"/>in <emph type="italics"/>i,<emph.end type="italics"/>producere licet <emph type="italics"/>iE<emph.end type="italics"/>ad <emph type="italics"/>V,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>EV<emph.end type="italics"/>ad <emph type="italics"/>Ei<emph.end type="italics"/>ut <emph type="italics"/>FH<emph.end type="italics"/>ad <emph type="italics"/>HI,<emph.end type="italics"/><lb/>& agere <emph type="italics"/>Vf<emph.end type="italics"/>parallelam ip&longs;i <emph type="italics"/>BD.<emph.end type="italics"/>Eodem recidit &longs;i centro <emph type="italics"/>i,<emph.end type="italics"/>in­<lb/>tervallo <emph type="italics"/>IH,<emph.end type="italics"/>de&longs;cribatur circulus &longs;ecans <emph type="italics"/>BD<emph.end type="italics"/>in <emph type="italics"/>X,<emph.end type="italics"/>& producatur <lb/><emph type="italics"/>iX<emph.end type="italics"/>ad <emph type="italics"/>Y,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>iY<emph.end type="italics"/>æqualis <emph type="italics"/>IF,<emph.end type="italics"/>& agatur <emph type="italics"/>Yf<emph.end type="italics"/>ip&longs;i <emph type="italics"/>BD<emph.end type="italics"/>parallela. </s></p> <p type="margin"> <s><margin.target id="note71"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>Problematis hujus &longs;olutiones alias <emph type="italics"/>Wrennus<emph.end type="italics"/>& <emph type="italics"/>Walli&longs;ius<emph.end type="italics"/>olim ex­<lb/>cogitarunt. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIX. PROBLEMA XXI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam &longs;pecie datam de&longs;cribere, quæ a rectis quatuor po&longs;itione <lb/>datis in partes &longs;ecabitur, ordine, &longs;pecie & proportione datas.<emph.end type="italics"/><emph.end type="center"/></s></p><figure id="id.039.01.123.1.jpg" xlink:href="039/01/123/1.jpg"/> <p type="main"> <s>De&longs;cribenda &longs;it Trajectoria <lb/><figure id="id.039.01.123.2.jpg" xlink:href="039/01/123/2.jpg"/><lb/><emph type="italics"/>fghi,<emph.end type="italics"/>quæ &longs;imilis &longs;it Lincæ curvæ <lb/><emph type="italics"/>FGHI,<emph.end type="italics"/>& cujus partes <emph type="italics"/>fg, gh, hi<emph.end type="italics"/><lb/>illius partibus <emph type="italics"/>FG, GH, HI<emph.end type="italics"/>&longs;i­<lb/>miles & proportionales, rectis <lb/><emph type="italics"/>AB<emph.end type="italics"/>& <emph type="italics"/>AD, AD<emph.end type="italics"/>& <emph type="italics"/>BD, BD<emph.end type="italics"/><lb/>& <emph type="italics"/>CE<emph.end type="italics"/>po&longs;itione datis, prima pri­<lb/>mis, &longs;ecunda &longs;ecundis, tertia ter­<lb/>tiis interjaceant. </s> <s>Actis rectis <emph type="italics"/>FG, <lb/>GH, HI, FI,<emph.end type="italics"/>de&longs;cribatur (per <lb/>Lem. </s> <s>XXVII.) Trapezium <emph type="italics"/>fghi<emph.end type="italics"/><lb/>quod &longs;it Trapezio <emph type="italics"/>FGHI<emph.end type="italics"/>&longs;imile & cujus anguli <emph type="italics"/>f, g, h, i<emph.end type="italics"/>tangant <lb/>rectas illas po&longs;itione datas <emph type="italics"/>AB, AD, BD, CE,<emph.end type="italics"/>&longs;inguli &longs;ingulas <lb/>dicto ordine. </s> <s>Dein circa hoc Trapezium de&longs;cribatur Trajectoria <lb/>curvæ Lineæ <emph type="italics"/>FGHI<emph.end type="italics"/>con&longs;imilis. <pb xlink:href="039/01/124.jpg" pagenum="96"/><arrow.to.target n="note72"/></s></p> <p type="margin"> <s><margin.target id="note72"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Con&longs;trui etiam pote&longs;t hoc Problema ut &longs;equitur. </s> <s>Junctis <emph type="italics"/>FG, <lb/>GH, HI, FI<emph.end type="italics"/>produc <emph type="italics"/>GF<emph.end type="italics"/>ad <emph type="italics"/>V,<emph.end type="italics"/>jungeque <emph type="italics"/>FH, IG,<emph.end type="italics"/>& angulis <lb/><emph type="italics"/>FGH, VFH<emph.end type="italics"/>fac angulos <emph type="italics"/>CAK, DAL<emph.end type="italics"/>æquales. </s> <s>Concurrant <lb/><emph type="italics"/>AK, AL<emph.end type="italics"/>cum recta <emph type="italics"/>BD<emph.end type="italics"/>in <emph type="italics"/>K<emph.end type="italics"/>& <emph type="italics"/>L,<emph.end type="italics"/>& inde agantur <emph type="italics"/>KM, LN,<emph.end type="italics"/><lb/>quarum <emph type="italics"/>KM<emph.end type="italics"/>con&longs;tituat angulum <emph type="italics"/>AKM<emph.end type="italics"/>æqualem angulo <emph type="italics"/>GHI,<emph.end type="italics"/><lb/>&longs;itque ad <emph type="italics"/>AK<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>HI<emph.end type="italics"/>ad <emph type="italics"/>GH<emph.end type="italics"/>; & <emph type="italics"/>LN<emph.end type="italics"/>con&longs;tituat angulum <lb/><emph type="italics"/>ALN<emph.end type="italics"/>æqualem angulo <emph type="italics"/>FHI,<emph.end type="italics"/>&longs;itque ad <emph type="italics"/>AL<emph.end type="italics"/>ut <emph type="italics"/>HI<emph.end type="italics"/>ad <emph type="italics"/>FH.<emph.end type="italics"/>Du­<lb/>cantur autem <emph type="italics"/>AK, KM, AL, LN<emph.end type="italics"/>ad eas partes linearum <emph type="italics"/>AD, <lb/>AK, AL,<emph.end type="italics"/>ut literæ <emph type="italics"/>CAKMC, ALKA, DALND<emph.end type="italics"/>eodem <lb/>ordine cum literis <emph type="italics"/>FGHIF<emph.end type="italics"/>in orbem redeant; & act <emph type="italics"/>MN<emph.end type="italics"/>oc­<lb/>currat rectæ <emph type="italics"/>CE<emph.end type="italics"/>in <emph type="italics"/>i.<emph.end type="italics"/>Fac angulum <emph type="italics"/>iEP<emph.end type="italics"/>æqualem angulo <emph type="italics"/>IGF,<emph.end type="italics"/><lb/><figure id="id.039.01.124.1.jpg" xlink:href="039/01/124/1.jpg"/><lb/>&longs;itque <emph type="italics"/>PE<emph.end type="italics"/>ad <emph type="italics"/>Ei<emph.end type="italics"/>ut <emph type="italics"/>FG<emph.end type="italics"/>ad <emph type="italics"/>GI;<emph.end type="italics"/>& per <emph type="italics"/>P<emph.end type="italics"/>agatur <emph type="italics"/>PQf,<emph.end type="italics"/>quæ <lb/>cum recta <emph type="italics"/>ADE<emph.end type="italics"/>contineat angulum <emph type="italics"/>PQE<emph.end type="italics"/>æqualem angulo <lb/><emph type="italics"/>FIG,<emph.end type="italics"/>rectæque <emph type="italics"/>AB<emph.end type="italics"/>occurrat in <emph type="italics"/>f,<emph.end type="italics"/>& jungatur <emph type="italics"/>fi.<emph.end type="italics"/>Agantur au­<lb/>rem <emph type="italics"/>PE<emph.end type="italics"/>& <emph type="italics"/>PQ<emph.end type="italics"/>ad eas partes linearum <emph type="italics"/>CE, PE,<emph.end type="italics"/>ut literarum <lb/><emph type="italics"/>PEiP<emph.end type="italics"/>& <emph type="italics"/>PEQP<emph.end type="italics"/>idem &longs;it ordo circularis qui literarum <emph type="italics"/>FGHIF,<emph.end type="italics"/><lb/>& &longs;i &longs;uper linea <emph type="italics"/>fi<emph.end type="italics"/>eodem quoque literarum ordine con&longs;tituatur <lb/>Trapezium <emph type="italics"/>fghi<emph.end type="italics"/>Trapezio <emph type="italics"/>FGHI<emph.end type="italics"/>&longs;imile, & circum&longs;cribatur Tra­<lb/>jectoria &longs;pecie data, &longs;olvetur Problema. </s></p> <p type="main"> <s>Hactenus de Orbibus inveniendis. </s> <s>Supere&longs;t ut Motus corpo­<lb/>rum in Orbibus inventis determinemus. <pb xlink:href="039/01/125.jpg" pagenum="97"/><arrow.to.target n="note73"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note73"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>SECTIO VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Inventione Motuum in Orbibus datis.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXX. PROBLEMA XXII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corporis in data Trajectoria Parabolica moti invenire locum ad <lb/>tempus a&longs;&longs;ignatum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>S<emph.end type="italics"/>umbilicus & <emph type="italics"/>A<emph.end type="italics"/>vertex principa­<lb/><figure id="id.039.01.125.1.jpg" xlink:href="039/01/125/1.jpg"/><lb/>lis Parabolæ, &longs;itque 4 <emph type="italics"/>ASXM<emph.end type="italics"/>æquale <lb/>areæ Parabolicæ ab&longs;cindendæ <emph type="italics"/>APS,<emph.end type="italics"/><lb/>quæ radio <emph type="italics"/>SP,<emph.end type="italics"/>vel po&longs;t exce&longs;&longs;um cor­<lb/>poris de vertice de&longs;cripta fuit, vel an­<lb/>te appul&longs;um ejus ad verticem de&longs;cri­<lb/>benda e&longs;t. </s> <s>Innote&longs;cit quantitas areæ il­<lb/>lius ab&longs;cindendæ ex tempore ip&longs;i pro­<lb/>portionali. </s> <s>Bi&longs;eca <emph type="italics"/>AS<emph.end type="italics"/>in <emph type="italics"/>G,<emph.end type="italics"/>erigeque <lb/>perpendiculum <emph type="italics"/>GH<emph.end type="italics"/>æquale 3 M, & <lb/>Circulus centro <emph type="italics"/>H,<emph.end type="italics"/>intervallo <emph type="italics"/>HS<emph.end type="italics"/><lb/>de&longs;criptus &longs;ecabit Parabolam in loco <lb/>quæ&longs;ito <emph type="italics"/>P.<emph.end type="italics"/>Nam, demi&longs;&longs;a ad axem <lb/>perpendiculari <emph type="italics"/>PO<emph.end type="italics"/>& ducta <emph type="italics"/>PH,<emph.end type="italics"/>e&longs;t <lb/><emph type="italics"/>AGq+GHq (=HP q=―AO-AG: quad.+―PO-GH: quad.)= <lb/>AOq+POq-2 <expan abbr="GAO-2GHXPO+AGq+GHq.">GAO-2GHXPO+AGq+GHque</expan><emph.end type="italics"/>Unde <lb/>2 <emph type="italics"/>GHXPO (=AOq+POq-2GAO)=AOq+1/4 <expan abbr="POq.">POque</expan><emph.end type="italics"/><lb/>Pro <emph type="italics"/>AOq<emph.end type="italics"/>&longs;cribe (<emph type="italics"/>AOXPOq/4AS<emph.end type="italics"/>); &, applicatis terminis omnibus ad <lb/>3<emph type="italics"/>PO<emph.end type="italics"/>ducti&longs;Q.E.I. 2<emph type="italics"/>AS,<emph.end type="italics"/>fiet 4/3 <emph type="italics"/>GHXAS(=1/6AOXPO+1/2 ASXPO <lb/>=(AO+3AS/6)XPO=(4AO-3SO/6)XPO<emph.end type="italics"/>=areæ ―<emph type="italics"/>APO-SPO)<emph.end type="italics"/><lb/>=areæ <emph type="italics"/>APS.<emph.end type="italics"/>Sed <emph type="italics"/>GH<emph.end type="italics"/>erat 3 M, & inde 4/3 <emph type="italics"/>GHXAS<emph.end type="italics"/>e&longs;t 4 <emph type="italics"/>AS<emph.end type="italics"/>XM. </s> <s><lb/>Ergo area ab&longs;ci&longs;&longs;a <emph type="italics"/>APS<emph.end type="italics"/>æqualis e&longs;t ab&longs;cindendæ 4<emph type="italics"/>ASXM. Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc <emph type="italics"/>GH<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>AS,<emph.end type="italics"/>ut tempus quo corpùs de&longs;crip­<lb/>&longs;it arcum <emph type="italics"/>AP<emph.end type="italics"/>ad tempus quo corpus de&longs;crip&longs;it arcum inter verti­<lb/>cem <emph type="italics"/>A<emph.end type="italics"/>& perpendiculum ad axem ab umbilico <emph type="italics"/>S<emph.end type="italics"/>erectum. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et Circulo <emph type="italics"/>ASP<emph.end type="italics"/>per corpus motum <emph type="italics"/>P<emph.end type="italics"/>perpetuo tran&longs;­<lb/>eunte, velocitas puncti <emph type="italics"/>H<emph.end type="italics"/>e&longs;t ad velocitatem quam corpus habuit <pb xlink:href="039/01/126.jpg" pagenum="98"/><arrow.to.target n="note74"/>in vertice <emph type="italics"/>A,<emph.end type="italics"/>ut 3 ad 8; adeoQ.E.I. ea etiam ratione e&longs;t linea <emph type="italics"/>GH<emph.end type="italics"/><lb/>ad lineam rectam quam corpus tempore motus &longs;ui ab <emph type="italics"/>A<emph.end type="italics"/>ad <emph type="italics"/>P,<emph.end type="italics"/>ea <lb/>cum velocitate quam habuit in vertice <emph type="italics"/>A,<emph.end type="italics"/>de&longs;cribere po&longs;&longs;et. </s></p> <p type="margin"> <s><margin.target id="note74"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Hinc etiam vice ver&longs;a inveniri pote&longs;t tempus quo cor­<lb/>pus de&longs;crip&longs;it arcum quemvis a&longs;&longs;ignatum <emph type="italics"/>AP.<emph.end type="italics"/>Junge <emph type="italics"/>AP<emph.end type="italics"/>& ad <lb/>medium ejus punctum erige perpendiculum rectæ <emph type="italics"/>GH<emph.end type="italics"/>occur­<lb/>rens in <emph type="italics"/>H.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>LEMMA XXVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Nulla extat Figura Ovalis cujus area, rectis pro lubitu ab&longs;ci&longs;&longs;a, po&longs;&longs;it <lb/>per æquationes numero terminorum ac dimen&longs;ionum finitas genera­<lb/>liter inveniri.<emph.end type="italics"/></s></p> <p type="main"> <s>Intra Ovalem detur punctum quodvis, circa quod ceu polum re­<lb/>volvatur perpetuo linea recta, uniformi cum motu, & interea in rec­<lb/>ta illa exeat punctum mobile de polo, pergatque &longs;emper ea cum <lb/>velocitate, quæ &longs;it ut rectæ illius intra Ovalem quadratum. </s> <s>Hoc <lb/>motu punctum illud de&longs;cribet Spiralem gyris infinitis. </s> <s>Jam &longs;i areæ <lb/>Ovalis a recta illa ab&longs;ci&longs;&longs;æ incrementum per finitam æquationem <lb/>inveniri pote&longs;t, invenietur etiam per eandem æquationem di&longs;tantia <lb/>puncti a polo, quæ huic areæ proportionalis e&longs;t, adeoque om­<lb/>nia Spiralis puncta per æquationem finitam inveniri po&longs;&longs;unt: & <lb/>propterea rectæ cuju&longs;vis po&longs;itione datæ inter&longs;ectio cum Spirali in­<lb/>veniri etiam pote&longs;t per æquationem finitam. </s> <s>Atqui recta omnis <lb/>infinite producta Spiralem &longs;ecat in punctis numero infinitis, & æqua­<lb/>tio, qua inter&longs;ectio aliqua duarum linearum invenitur, exhibet ea­<lb/>rum inter&longs;ectiones omnes radicibus totidem, adeoque a&longs;cendit ad <lb/>rot dimen&longs;iones quot &longs;unt inter&longs;ectiones. </s> <s>Quoniam Circuli duo &longs;e <lb/>mutuo &longs;ecant in punctis duobus, inter&longs;ectio una non invenietur <lb/>ni&longs;i per æquationem duarum dimen&longs;ionum, qua inter&longs;ectio altera <lb/>etiam inveniatur. </s> <s>Quoniam duarum &longs;ectionum Conicarum quatuor <lb/>e&longs;&longs;e po&longs;&longs;unt inter&longs;ectiones, non pote&longs;t aliqua earum generaliter in­<lb/>veniri ni&longs;i per æquationem quatuor dimen&longs;ionum, qua omnes &longs;i­<lb/>mul inveniantur. </s> <s>Nam &longs;i inter&longs;ectiones illæ &longs;eor&longs;im quærantur, quo­<lb/>niam eadem e&longs;t omnium lex & conditio, idem erit calculus in ca&longs;u <lb/>unoquoque & propterea eadem &longs;emper conclu&longs;io, quæ igitur de­<lb/>bet omnes inter&longs;ectiones &longs;imul complecti & indifferenter exhibere. <pb xlink:href="039/01/127.jpg" pagenum="99"/>Unde etiam inter&longs;ectiones Sectionum Conicarum & Curvarum ter­<lb/><arrow.to.target n="note75"/>tiæ pote&longs;tatis, eo quod &longs;ex e&longs;&longs;e po&longs;&longs;unt, &longs;imul prodeunt per æqua­<lb/>tiones &longs;ex dimen&longs;ionum, & inter&longs;ectiones duarum Curvarum tertiæ <lb/>pote&longs;tatis, quia novem e&longs;&longs;e po&longs;&longs;unt, &longs;imul prodeunt per æqua­<lb/>tiones dimen&longs;ionum novem. </s> <s>Id ni&longs;i nece&longs;&longs;ario fieret, reducere licc­<lb/>ret Problemata omnia Solida ad Plana, & plu&longs;quam Solida ad Soli­<lb/>da. </s> <s>Loquor hic de Curvis pote&longs;tate irreducibilibus. </s> <s>Nam &longs;i æqua­<lb/>tio per quam Curva definitur, ad inferiorem pote&longs;tatem reduci <lb/>po&longs;&longs;it: Curva non erit unica, &longs;ed ex duabus vel pluribus compo&longs;i­<lb/>ta, quarum inter&longs;ectiones per calculos diver&longs;os &longs;eor&longs;im inveniri <lb/>po&longs;&longs;unt. </s> <s>Ad eundem modum inter&longs;ectiones binæ rectarum & &longs;ecti­<lb/>onum Conicarum prodeunt &longs;emper per æquationes duarum dimen­<lb/>&longs;ionum; ternæ rectarum & Curvarum irreducibilium tertiæ pote&longs;tatis <lb/>per æquationes trium, quaternæ rectarum & Curvarvm irreducibi­<lb/>lium quartæ pote&longs;tatis per æquationes dimen&longs;ionum quatuor, & &longs;ic <lb/>in infinitum. </s> <s>Ergo rectæ & Spiralis inter&longs;ectiones numero infinitæ, cum <lb/>Curva hæc &longs;it &longs;implex & in Curvas plures irreducibilis, requirunt æ­<lb/>quationes numero dimen&longs;ionum & radicum infinitas, quibus omnes <lb/>po&longs;&longs;unt &longs;imul exhiberi. </s> <s>E&longs;t enim eadem omnium lex & idem calculus. </s> <s><lb/>Nam &longs;i a polo in rectam illam &longs;ecantem demittatur perpendiculum, <lb/>& perpendiculum illud una cum &longs;ecante revolvatur circa polum, in­<lb/>ter&longs;ectiones Spiralis tran&longs;ibunt in &longs;e mutuo, quæque prima erat &longs;eu <lb/>proxima, po&longs;t unam revolutionem &longs;ecunda erit, po&longs;t duas tertia, <lb/>& &longs;ic deinceps: nec interea mutabitur æquatio ni&longs;i pro mutata mag­<lb/>nitudine quantitatum per quas po&longs;itio &longs;ecantis determinatur. </s> <s>Unde <lb/>cum quantitates illæ po&longs;t &longs;ingulas revolutiones redeunt ad magNI­<lb/>tudines primas, æquatio redibit ad formam primam, adeoque una <lb/>eademque exhibebit inter&longs;ectiones omnes, & propterea radices ha­<lb/>bebit numero infinitas, quibus omnes exhiberi po&longs;&longs;unt. </s> <s>Nequit <lb/>ergo inter&longs;ectio rectæ & Spiralis per æquationem finitam generali­<lb/>ter inveniri, & idcirco nulla extat Ovalis cujus area, rectis impe­<lb/>ratis ab&longs;ci&longs;&longs;a, po&longs;&longs;it per talem æquationem generaliter exhiberi. </s></p> <p type="margin"> <s><margin.target id="note75"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>Eodem argumento, &longs;i intervallum poli & puncti, quo Spiralis de­<lb/>&longs;cribitur, capiatur Ovalis perimetro ab&longs;ci&longs;&longs;æ proportionale, pro­<lb/>bari pote&longs;t quod longitudo perimetri nequit per finitam æquatio­<lb/>nem generaliter exhiberi. </s> <s>De Ovalibus autem hic loquor quæ non <lb/>tanguntur a figuris conjugatis in infinitum pergentibus. <pb xlink:href="039/01/128.jpg" pagenum="100"/><arrow.to.target n="note76"/></s></p> <p type="margin"> <s><margin.target id="note76"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Hinc area Ellip&longs;eos, quæ radio ab umbilico ad corpus mobile <lb/>ducto de&longs;cribitur, non prodit ex dato tempore, per æquationem <lb/>finitam; & propterea per de&longs;criptionem Curvarum Geometrice ra­<lb/>tionalium determinari nequit. </s> <s>Curvas Geometrice rationales ap­<lb/>pello quarum puncta omnia per longitudines æquationibus defiNI­<lb/>tas, id e&longs;t, per longitudinum rationes complicatas, determinari <lb/>po&longs;&longs;unt; cætera&longs;que (ut Spirales, Quadratrices, Trochoides) Geo­<lb/>metrice irrationales. </s> <s>Nam longitudines quæ &longs;unt vel non &longs;unt ut <lb/>numerus ad numerum (quemadmodum in decimo Elementorum) <lb/>&longs;unt Arithmetice rationales vel irrationales. </s> <s>Aream igitur Ellip&longs;eos <lb/>tempori proportionalem ab&longs;cindo per Curvam Geometrice irratio­<lb/>nalem ut &longs;equitur. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXI. PROBLEMA XXIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corporis in data Trajectoria Elliptica moti invenire locum ad <lb/>tempus a&longs;&longs;ignatum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Ellip&longs;eos <emph type="italics"/>APB<emph.end type="italics"/>&longs;it <emph type="italics"/>A<emph.end type="italics"/>vertex principalis, <emph type="italics"/>S<emph.end type="italics"/>umbilicus, & <emph type="italics"/>O<emph.end type="italics"/><lb/>centrum, &longs;itque <emph type="italics"/>P<emph.end type="italics"/>corporis locus inveniendus. </s> <s>Produc <emph type="italics"/>OA<emph.end type="italics"/>ad <emph type="italics"/>G,<emph.end type="italics"/><lb/>ut &longs;it <emph type="italics"/>OG<emph.end type="italics"/>ad <emph type="italics"/>OA<emph.end type="italics"/>ut <emph type="italics"/>OA<emph.end type="italics"/>ad <emph type="italics"/>OS.<emph.end type="italics"/>Erige perpendiculum <emph type="italics"/>GH,<emph.end type="italics"/>centroque <lb/><figure id="id.039.01.128.1.jpg" xlink:href="039/01/128/1.jpg"/><lb/><emph type="italics"/>O<emph.end type="italics"/>& intervallo <emph type="italics"/>OG<emph.end type="italics"/>de&longs;cribe circulum <emph type="italics"/>EFG,<emph.end type="italics"/>& &longs;uper regula <emph type="italics"/>GH,<emph.end type="italics"/><lb/>ceu fundo, progrediatur Rota <emph type="italics"/>GEF<emph.end type="italics"/>revolvendo circa axem <lb/>&longs;uum, & interea puncto &longs;uo <emph type="italics"/>A<emph.end type="italics"/>de&longs;cribendo Trochoidem <emph type="italics"/>ALI.<emph.end type="italics"/><pb xlink:href="039/01/129.jpg" pagenum="101"/>Quo facto, cape <emph type="italics"/>GK<emph.end type="italics"/>in ratione ad Rotæ perimetrum <emph type="italics"/>GEFG,<emph.end type="italics"/>ut <lb/><arrow.to.target n="note77"/>e&longs;t tempus quo corpus progrediendo ab <emph type="italics"/>A<emph.end type="italics"/>de&longs;crip&longs;it arcum <emph type="italics"/>AP,<emph.end type="italics"/>ad <lb/>tempus revolutionis unius in Ellip&longs;i. </s> <s>Erigatur perpendiculum <emph type="italics"/>KL<emph.end type="italics"/><lb/>occurrens Trochoidi in <emph type="italics"/>L,<emph.end type="italics"/>& acta <emph type="italics"/>LP<emph.end type="italics"/>ip&longs;i <emph type="italics"/>KG<emph.end type="italics"/>parallela occurret <lb/>Ellip&longs;i in corporis loco quæ&longs;ito <emph type="italics"/>P.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note77"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>Nam centro <emph type="italics"/>O,<emph.end type="italics"/>intervallo <emph type="italics"/>OA<emph.end type="italics"/>de&longs;cribatur &longs;emicirculus <emph type="italics"/>AQB,<emph.end type="italics"/><lb/>& arcui <emph type="italics"/>AQ<emph.end type="italics"/>occurrat <emph type="italics"/>LP<emph.end type="italics"/>producta in <emph type="italics"/>Q,<emph.end type="italics"/>junganturque <emph type="italics"/>SQ, <expan abbr="Oq.">Oque</expan><emph.end type="italics"/><lb/>Arcui <emph type="italics"/>EFG<emph.end type="italics"/>occurrat <emph type="italics"/>OQ<emph.end type="italics"/>in <emph type="italics"/>F,<emph.end type="italics"/>& in eandem <emph type="italics"/>OQ<emph.end type="italics"/>demittatur per­<lb/>pendiculum <emph type="italics"/>SR.<emph.end type="italics"/>Area <emph type="italics"/>APS<emph.end type="italics"/>e&longs;t ut area <emph type="italics"/>AQS,<emph.end type="italics"/>id e&longs;t, ut diffe­<lb/>rentia inter &longs;ectorem <emph type="italics"/>OQA<emph.end type="italics"/>& triangulum <emph type="italics"/>OQS,<emph.end type="italics"/>&longs;ive ut differen­<lb/>tia rectangulorum 1/2 <emph type="italics"/>OQXAQ<emph.end type="italics"/>& 1/2 <emph type="italics"/>OQXSR,<emph.end type="italics"/>hoc e&longs;t, ob datam <lb/>1/2 <emph type="italics"/>OQ,<emph.end type="italics"/>ut differentia inter arcum <emph type="italics"/>AQ<emph.end type="italics"/>& rectam <emph type="italics"/>SR,<emph.end type="italics"/>adeoque (ob <lb/>æqualitatem datarum rationum <emph type="italics"/>SR<emph.end type="italics"/>ad &longs;inum arcus <emph type="italics"/>AQ, OS<emph.end type="italics"/>ad <emph type="italics"/>OA, <lb/>OA<emph.end type="italics"/>ad <emph type="italics"/>OG, AQ<emph.end type="italics"/>ad <emph type="italics"/>GF,<emph.end type="italics"/>& divi&longs;im <emph type="italics"/>AQ-SR<emph.end type="italics"/>ad <emph type="italics"/>GF<emph.end type="italics"/>-&longs;in. </s> <s>arc. <emph type="italics"/>AQ<emph.end type="italics"/>) <lb/>ut <emph type="italics"/>GK<emph.end type="italics"/>differentia inter arcum <emph type="italics"/>GF<emph.end type="italics"/>& &longs;inum arcus <emph type="italics"/><expan abbr="Aq.">Aque</expan> <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Cæterum, cum difficilis &longs;it hujus Curvæ de&longs;criptio, præ&longs;tat &longs;olu­<lb/>tionem vero proximam adhibere. </s> <s>Inveniatur tum angulus quidam <lb/>B, qui &longs;it ad angulum graduum 57,29578, quem arcus radio æqualis <lb/>&longs;ubtendit, ut e&longs;t umbilieorum di&longs;tantia <emph type="italics"/>SH<emph.end type="italics"/>ad Ellip&longs;eos diame­<lb/>trum <emph type="italics"/>AB<emph.end type="italics"/>; tum etiam longitudo quædam L, quæ &longs;it ad radium in <lb/>eadem ratione inver&longs;e. </s> <s>Quibus &longs;emel inventis, Problema deinceps <lb/>confit per &longs;equentem Analy&longs;in. </s> <s>Per con&longs;tructionem quamvis (vel. </s> <s><lb/>utcunque conjec­<lb/><figure id="id.039.01.129.1.jpg" xlink:href="039/01/129/1.jpg"/><lb/>turam faciendo) <lb/>cogno&longs;catur cor­<lb/>poris locus <emph type="italics"/>P<emph.end type="italics"/>pro­<lb/>ximus vero ejus lo­<lb/>co <emph type="italics"/>p.<emph.end type="italics"/>Demi&longs;&longs;aque ad <lb/>axem Ellip&longs;eos or­<lb/>dinatim applicata <lb/><emph type="italics"/>PR,<emph.end type="italics"/>ex propor­<lb/>tione diametrorum <lb/>Ellip&longs;eos, dabitur <lb/>Circuli circum&longs;cri­<lb/>pti <emph type="italics"/>AQB<emph.end type="italics"/>ordinatim applicata <emph type="italics"/>RQ,<emph.end type="italics"/>quæ &longs;inus e&longs;t anguli <emph type="italics"/>AOQ<emph.end type="italics"/>exi­<lb/>&longs;tente <emph type="italics"/>AO<emph.end type="italics"/>radio. </s> <s>Sufficit angulum illum rudi calculo in numeris <lb/>proximis invenire. </s> <s>Cogno&longs;catur etiam angulus tempori propor-<pb xlink:href="039/01/130.jpg" pagenum="102"/><arrow.to.target n="note78"/>tionalis, id e&longs;t, qui &longs;it ad quatuor rectos, ut e&longs;t tempus quo corpus <lb/>de&longs;crip&longs;it arcum <emph type="italics"/>Ap,<emph.end type="italics"/>ad tempus revolutionis unius in Ellip&longs;i. </s> <s>Sit <lb/>angulus i&longs;te N. </s> <s>Tum capiatur & angulus D ad angulum B, ut <lb/>e&longs;t &longs;inus i&longs;te anguli <emph type="italics"/>AOQ<emph.end type="italics"/>ad radium, & angulus E ad angulum <lb/>N-<emph type="italics"/>AOQ<emph.end type="italics"/>+D, ut e&longs;t longitudo L ad longitudinem eandem L <lb/>co&longs;inu anguli <emph type="italics"/>AOQ<emph.end type="italics"/>diminutam, ubi angulus i&longs;te recto minor e&longs;t, <lb/>auctam ubi major. </s> <s>Po&longs;tea capiatur tum angulus F ad angulum B, <lb/>ut e&longs;t &longs;inus anguli <emph type="italics"/>AOQ<emph.end type="italics"/>+E ad radium, tum angulus G ad angu­<lb/>lum N-<emph type="italics"/>AOQ<emph.end type="italics"/>-E+F ut e&longs;t longitudo L ad longitudinem ean­<lb/>dem co&longs;inu anguli <emph type="italics"/>AOQ<emph.end type="italics"/>+E diminutam ubi angulus i&longs;te recto mi­<lb/>nor e&longs;t, auctam ubi major. </s> <s>Tertia vice capiatur angulus H ad an­<lb/>gulum B, ut e&longs;t &longs;inus anguli <emph type="italics"/>AOQ<emph.end type="italics"/>+E+G ad radium; & angu­<lb/>lus I ad angulum N-<emph type="italics"/>AOQ<emph.end type="italics"/>-E-G+H, ut e&longs;t longitudo L ad <lb/>eandem longitudinem co&longs;inu anguli <emph type="italics"/>AOQ<emph.end type="italics"/>+E+G diminutam, <lb/>ubi angulus i&longs;te re­<lb/><figure id="id.039.01.130.1.jpg" xlink:href="039/01/130/1.jpg"/><lb/>cto minor e&longs;t, auc­<lb/>tam ubi major. </s> <s>Et <lb/>&longs;ic pergere licet in <lb/>infinitum. </s> <s>DeNI­<lb/>que capiatur angu­<lb/>lus <emph type="italics"/>AOq<emph.end type="italics"/>æqualis <lb/>angulo <emph type="italics"/>AOQ<emph.end type="italics"/>+E <lb/>+G+I+&c. </s> <s>e t <lb/>ex co&longs;inu ejus <emph type="italics"/>Or<emph.end type="italics"/><lb/>& ordinata <emph type="italics"/>pr,<emph.end type="italics"/>quæ <lb/>e&longs;t ad &longs;inum ejus <lb/><emph type="italics"/>qr<emph.end type="italics"/>ut Ellip&longs;eos axis minor ad axem majorem, habebitur corporis <lb/>locus correctus <emph type="italics"/>p.<emph.end type="italics"/>Si quando angulus N-<emph type="italics"/>AOQ<emph.end type="italics"/>+D negativus <lb/>e&longs;t, debet &longs;ignum+ip&longs;ius E ubique mutari in-, & &longs;ignum-in+. <lb/>Idem intelligendum e&longs;t de &longs;ignis ip&longs;orum G & I, ubi anguli <lb/>N-<emph type="italics"/>AOQ<emph.end type="italics"/>-E+F, & N-<emph type="italics"/>AOQ<emph.end type="italics"/>-E-G+H negativi prodeunt. </s> <s><lb/>Convergit autem &longs;eries infinita <emph type="italics"/>AOQ<emph.end type="italics"/>+E+G+I+&c. </s> <s>quam <lb/>celerrime, adeo ut vix unquam opus fuerit ultra progredi quam <lb/>ad terminum &longs;ecundum E. </s> <s>Et fundatur calculus in hoc Theore­<lb/>mate, quod area <emph type="italics"/>APS<emph.end type="italics"/>&longs;it ut differentia inter arcum <emph type="italics"/>AQ<emph.end type="italics"/>& <lb/>rectam ab umbilico <emph type="italics"/>S<emph.end type="italics"/>in Radium <emph type="italics"/>OQ<emph.end type="italics"/>perpendiculariter de­<lb/>mi&longs;&longs;am. </s></p> <p type="margin"> <s><margin.target id="note78"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Non di&longs;&longs;imili calculo conficitur Problema in Hyperbola. </s> <s>Sit <lb/>ejus Centrum <emph type="italics"/>O,<emph.end type="italics"/>Vertex <emph type="italics"/>A,<emph.end type="italics"/>Umbilicus <emph type="italics"/>S<emph.end type="italics"/>& A&longs;ymptotos <emph type="italics"/>OK.<emph.end type="italics"/>Cog-<pb xlink:href="039/01/131.jpg" pagenum="103"/>no&longs;catur quantitas areæ ab&longs;cindendæ tempori proportionalis. </s> <s>Sit ea <lb/><arrow.to.target n="note79"/>A, & fiat conjectura de po&longs;itione rectæ <emph type="italics"/>SP,<emph.end type="italics"/>quæ aream <emph type="italics"/>APS<emph.end type="italics"/><lb/>ab&longs;cindat veræ proximam. </s> <s>Jun­<lb/><figure id="id.039.01.131.1.jpg" xlink:href="039/01/131/1.jpg"/><lb/>gatur <emph type="italics"/>OP,<emph.end type="italics"/>& ab <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>ad <lb/>A&longs;ymptoton agantur <emph type="italics"/>AI, PK<emph.end type="italics"/><lb/>A&longs;ymptoto alteri parallelæ, & per <lb/>Tabulam Logarithmorum dabi­<lb/>tur Area <emph type="italics"/>AIKP,<emph.end type="italics"/>eique æqualis <lb/>area <emph type="italics"/>OPA,<emph.end type="italics"/>quæ &longs;ubducta de tri­<lb/>angulo <emph type="italics"/>OPS<emph.end type="italics"/>relinquet aream ab­<lb/>&longs;ci&longs;&longs;am <emph type="italics"/>APS.<emph.end type="italics"/>Applicando areæ <lb/>ab&longs;cindendæ A & ab&longs;ci&longs;&longs;æ <emph type="italics"/>APS<emph.end type="italics"/><lb/>differentiam duplam 2 <emph type="italics"/>APS<emph.end type="italics"/>-2 A <lb/>vel 2 A-2 <emph type="italics"/>APS<emph.end type="italics"/>ad lineam <emph type="italics"/>SN,<emph.end type="italics"/>quæ ab umbilico <emph type="italics"/>S<emph.end type="italics"/>in tangentem <lb/><emph type="italics"/>PT<emph.end type="italics"/>perpendicularis e&longs;t, orietur longitudo chordæ <emph type="italics"/><expan abbr="Pq.">Pque</expan><emph.end type="italics"/>In&longs;cri­<lb/>batur autem chorda illa <emph type="italics"/>PQ<emph.end type="italics"/>inter <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>P,<emph.end type="italics"/>&longs;i area ab&longs;ci&longs;&longs;a <emph type="italics"/>APS<emph.end type="italics"/><lb/>major &longs;it area ab&longs;cindenda A, &longs;ecus ad puncti <emph type="italics"/>P<emph.end type="italics"/>contrarias partes: <lb/>& punctum <emph type="italics"/>Q<emph.end type="italics"/>erit locus corporis accuratior. </s> <s>Et computatione <lb/>repetita invenietur idem accuratior in perpetuum. </s></p> <p type="margin"> <s><margin.target id="note79"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>Atque his calculis Problema generaliter confit Analytice. </s> <s>Ve­<lb/>rum u&longs;ibus A&longs;tronomicis accommodatior e&longs;t calculus particularis <lb/>qui &longs;equitur. </s> <s>Exi&longs;tentibus <emph type="italics"/>AO, OB, OD<emph.end type="italics"/>&longs;emiaxibus Ellip&longs;eos, & <lb/>L ip&longs;ius latere recto, ac D differentia inter &longs;emiaxem minorem <emph type="italics"/>OD<emph.end type="italics"/><lb/>& lateris recti &longs;emi&longs;&longs;em 1/2 L; quære tum angulum Y, cujus &longs;inus <lb/>&longs;it ad Radium ut e&longs;t rectangu­<lb/><figure id="id.039.01.131.2.jpg" xlink:href="039/01/131/2.jpg"/><lb/>lum &longs;ub differentia illa D, & <lb/>&longs;emi&longs;umma axium <emph type="italics"/>AO+OD<emph.end type="italics"/><lb/>ad quadratum axis majoris <emph type="italics"/>AB<emph.end type="italics"/>; <lb/>tum angulum Z, cujus &longs;inus <lb/>&longs;it ad Radium ut e&longs;t duplum <lb/>rectangulum &longs;ub umbilieorum <lb/>di&longs;tantia <emph type="italics"/>SH<emph.end type="italics"/>& differentia <lb/>illa D ad triplum quadratum <lb/>&longs;emiaxis majoris <emph type="italics"/>AO.<emph.end type="italics"/>His <lb/>angulis &longs;emel inventis; locus corporis &longs;ic deinceps determinabitur. </s> <s><lb/>Sume angulum T proportionalem tempori quo arcus <emph type="italics"/>BP<emph.end type="italics"/>de&longs;crip­<lb/>tus e&longs;t, &longs;cu motui medio (ut loquuntur) æqualem; & angulum <lb/>V (primam medii motus æquationem) ad angulum Y (æquatio­<lb/>nem maximam primam) ut e&longs;t &longs;inus dupli anguli T ad Radium; <pb xlink:href="039/01/132.jpg" pagenum="104"/><arrow.to.target n="note80"/>atque angulum X (æquationem &longs;ecundam) ad angulum Z (æqua­<lb/>tionem maximam &longs;ecundam) ut e&longs;t cubus &longs;inus anguli T ad cubum <lb/>Radii. </s> <s>Angulorum T, V, X vel &longs;ummæ T+X+V, &longs;i angulus <lb/>T recto minor e&longs;t, vel differentiæ T+X-V, &longs;i is recto major e&longs;t <lb/>recti&longs;Q.E.D.obus minor, æqualem cape angulum <emph type="italics"/>BHP<emph.end type="italics"/>(motum <lb/>medium æquatum;) &, &longs;i <emph type="italics"/>HP<emph.end type="italics"/>occurrat Ellip&longs;i in <emph type="italics"/>P,<emph.end type="italics"/>acta <emph type="italics"/>SP<emph.end type="italics"/>ab­<lb/>&longs;cindet aream <emph type="italics"/>BSP<emph.end type="italics"/>tempori proportionalem quamproxime. </s> <s>Hæc <lb/>Praxis &longs;atis expedita videtur, <lb/><figure id="id.039.01.132.1.jpg" xlink:href="039/01/132/1.jpg"/><lb/>propterea quod angulorum per­<lb/>exiguorum V & X (in minutis <lb/>&longs;ecundis, &longs;i placet, po&longs;itorum) <lb/>figuras duas ter&longs;ve primas in­<lb/>venire &longs;ufficit. </s> <s>Sed & &longs;atis ac­<lb/>curata e&longs;t ad Theoriam Planeta­<lb/>rum. </s> <s>Nam in Orbe vel Martis <lb/>ip&longs;ius, cujus Æquatio centri ma­<lb/>xima e&longs;t graduum decem, error <lb/>vix &longs;uperabit minutum unum <lb/>&longs;ecundum. </s> <s>Invento autem angulo motus medii æquati <emph type="italics"/>BHP,<emph.end type="italics"/>an­<lb/>gulus veri motus <emph type="italics"/>BSP<emph.end type="italics"/>& di&longs;tantia <emph type="italics"/>SP<emph.end type="italics"/>in promptu &longs;unt per <lb/><emph type="italics"/>Wardi<emph.end type="italics"/>methodum noti&longs;&longs;imam. </s></p> <p type="margin"> <s><margin.target id="note80"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Hactenus de Motu corporum in lineis Curvis. </s> <s>Fieri autem po­<lb/>te&longs;t ut mobile recta de&longs;cendat vel recta a&longs;cendat, & quæ ad i&longs;tiu&longs;­<lb/>modi Motus &longs;pectant, pergo jam exponere. <pb xlink:href="039/01/133.jpg" pagenum="105"/><arrow.to.target n="note81"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note81"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>SECTIO VII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Corporum A&longs;cen&longs;u & De&longs;cen&longs;u Rectilineo.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXII. PROBLEMA XXIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod Vis centripeta &longs;it reciproce proportionalis quadrato di­<lb/>&longs;tantiæ loeorum a centro, Spatia definire quæ corpus recta cadendo <lb/>datis temporibus de&longs;cribit.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Si Corpus non cadit perpendicu­<lb/><figure id="id.039.01.133.1.jpg" xlink:href="039/01/133/1.jpg"/><lb/>lariter de&longs;cribet id, per Corol. </s> <s>1. Prop. </s> <s>XIII, <lb/>Sectionem aliquam Conicam cujus umbili­<lb/>cus congruit cum centro virium. </s> <s>Sit Sec­<lb/>tio illa Conica <emph type="italics"/>ARPB<emph.end type="italics"/>& umbilicus ejus <emph type="italics"/>S.<emph.end type="italics"/><lb/>Et primo &longs;i Figura Ellip&longs;is e&longs;t, &longs;uper hu­<lb/>jus axe majore <emph type="italics"/>AB<emph.end type="italics"/>de&longs;cribatur Semicirculus <lb/><emph type="italics"/>ADB,<emph.end type="italics"/>& per corpus decidens tran&longs;eat rec­<lb/>ta <emph type="italics"/>DPC<emph.end type="italics"/>perpendicularis ad axem; acti&longs;que <lb/><emph type="italics"/>DS, PS<emph.end type="italics"/>erit area <emph type="italics"/>ASD<emph.end type="italics"/>areæ <emph type="italics"/>ASP<emph.end type="italics"/>at­<lb/>que adeo etiam tempori proportionalis. </s> <s>Ma­<lb/>nente axe <emph type="italics"/>AB<emph.end type="italics"/>minuatur perpetuo latitudo <lb/>Ellip&longs;eos, & &longs;emper manebit area <emph type="italics"/>ASD<emph.end type="italics"/><lb/>tempori proportionalis. </s> <s>Minuatur latitudo <lb/>illa in infinitum: &, Orbe <emph type="italics"/>APB<emph.end type="italics"/>jam coin­<lb/>cidente cum axe <emph type="italics"/>AB<emph.end type="italics"/>& umbilico <emph type="italics"/>S<emph.end type="italics"/>cum <lb/>axis termino <emph type="italics"/>B,<emph.end type="italics"/>de&longs;cendet corpus in recta <lb/><emph type="italics"/>AC,<emph.end type="italics"/>& area <emph type="italics"/>ABD<emph.end type="italics"/>evadet tempori pro­<lb/>portionalis. </s> <s>Dabitur itaque Spatium <emph type="italics"/>AC,<emph.end type="italics"/><lb/>quod corpus de loco <emph type="italics"/>A<emph.end type="italics"/>perpendiculariter <lb/>cadendo tempore dato de&longs;cribit, &longs;i modo tempori proportiona­<lb/>lis capiatur area <emph type="italics"/>ABD,<emph.end type="italics"/>& a puncto <emph type="italics"/>D<emph.end type="italics"/>ad rectam <emph type="italics"/>AB<emph.end type="italics"/>demit­<lb/>tatur perpendicularis <emph type="italics"/>DC. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/><pb xlink:href="039/01/134.jpg" pagenum="106"/><arrow.to.target n="note82"/></s></p> <p type="margin"> <s><margin.target id="note82"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Si Figura illa <emph type="italics"/>RPB<emph.end type="italics"/>Hyperbola e&longs;t, de&longs;cribatur ad ean­<lb/>dem diametrum principalem <emph type="italics"/>AB<emph.end type="italics"/>Hyperbola rectangula <emph type="italics"/>BED:<emph.end type="italics"/><lb/>& quoniam areæ <emph type="italics"/>CSP, CBfP, SPfB<emph.end type="italics"/>&longs;unt ad areas <emph type="italics"/>CSD, <lb/>CBED, SDEB,<emph.end type="italics"/>&longs;ingulæ ad &longs;ingulas, in data ratione altitudi­<lb/>num <emph type="italics"/>CP, CD<emph.end type="italics"/>; & area <emph type="italics"/>SPfB<emph.end type="italics"/><lb/><figure id="id.039.01.134.1.jpg" xlink:href="039/01/134/1.jpg"/><lb/>proportionalis e&longs;t tempori quo <lb/>corpus <emph type="italics"/>P<emph.end type="italics"/>movebitur per arcum <lb/><emph type="italics"/>PfB<emph.end type="italics"/>; erit etiam area <emph type="italics"/>SDEB<emph.end type="italics"/>ei­<lb/>dem tempori proportionalis. </s> <s><lb/>Minuatur latus rectum Hyper­<lb/>bolæ <emph type="italics"/>RPB<emph.end type="italics"/>in infinitum ma­<lb/>nente latere tran&longs;ver&longs;o, & coibit <lb/>arcus <emph type="italics"/>PB<emph.end type="italics"/>cum recta <emph type="italics"/>CB<emph.end type="italics"/>& um­<lb/>bilicus <emph type="italics"/>S<emph.end type="italics"/>cum vertice <emph type="italics"/>B<emph.end type="italics"/>& recta <lb/><emph type="italics"/>SD<emph.end type="italics"/>cum recta <emph type="italics"/>BD.<emph.end type="italics"/>Proinde a­<lb/>rea <emph type="italics"/>BDEB<emph.end type="italics"/>proportionalis erit <lb/>tempori quo corpus <emph type="italics"/>C<emph.end type="italics"/>recto <lb/>de&longs;cen&longs;u de&longs;cribit lineam <emph type="italics"/>CB. <lb/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>3. Et &longs;imili argumento &longs;i <lb/>Figura <emph type="italics"/>RPB<emph.end type="italics"/>Parabola e&longs;t, & <lb/>eodem vertice principali <emph type="italics"/>B<emph.end type="italics"/>de­<lb/>&longs;cribatur alia Parabola <emph type="italics"/>BED,<emph.end type="italics"/><lb/>quæ &longs;emper maneat data interea <lb/>dum Parabola prior in cujus perimetro corpus <emph type="italics"/>P<emph.end type="italics"/>movetur, dimi­<lb/>nuto & in nihilum redacto ejus latere recto, conveniat cum linea <lb/><emph type="italics"/>CB<emph.end type="italics"/>; fiet &longs;egmentum Parabolicum <emph type="italics"/>BDEB<emph.end type="italics"/>proportionale tempori <lb/>quo corpus illud <emph type="italics"/>P<emph.end type="italics"/>vel <emph type="italics"/>C<emph.end type="italics"/>de&longs;cendet ad centrum <emph type="italics"/>S<emph.end type="italics"/>vel <emph type="italics"/>B. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXIII. THEOREMA IX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;itis jam inventis, dico quod corporis cadentis Velocitas in loco quo­<lb/>vis<emph.end type="italics"/>C <emph type="italics"/>est ad velocitatem corporis centro<emph.end type="italics"/>B <emph type="italics"/>intervallo<emph.end type="italics"/>BC <emph type="italics"/>Circu­<lb/>lum de&longs;cribentis, in &longs;ubduplicata ratione quam<emph.end type="italics"/>AC, <emph type="italics"/>di&longs;tantia cor­<lb/>poris a Circuli vel Hyperbolæ rect angulæ vertice ulteriore<emph.end type="italics"/>A, <emph type="italics"/>habet <lb/>ad Figuræ &longs;emidiametrum principalem<emph.end type="italics"/>1/2 AB. </s></p> <p type="main"> <s>Bi&longs;ecetur <emph type="italics"/>AB,<emph.end type="italics"/>communis utriu&longs;que Figuræ <emph type="italics"/>RPB, DEB<emph.end type="italics"/>dia­<lb/>meter, in <emph type="italics"/>O<emph.end type="italics"/>; & agatur recta <emph type="italics"/>PT<emph.end type="italics"/>quæ tangat Figuram <emph type="italics"/>RPB<emph.end type="italics"/>in <emph type="italics"/>P,<emph.end type="italics"/>atque <pb xlink:href="039/01/135.jpg" pagenum="107"/>etiam &longs;ecet communem illam diametrum <emph type="italics"/>AB<emph.end type="italics"/>(&longs;i opus e&longs;t productam) </s></p> <p type="main"> <s><arrow.to.target n="note83"/>in <emph type="italics"/>T<emph.end type="italics"/>; &longs;itque <emph type="italics"/>SY<emph.end type="italics"/>ad hanc rectam, & <emph type="italics"/>BQ<emph.end type="italics"/>ad <lb/><figure id="id.039.01.135.1.jpg" xlink:href="039/01/135/1.jpg"/><lb/>hanc diametrum perpendicularis, atque Figu­<lb/>ræ <emph type="italics"/>RPB<emph.end type="italics"/>latus rectum ponatur L. </s> <s>Con&longs;tat <lb/>per Cor. </s> <s>9. Prop. </s> <s>XVI, quod corporis in <lb/>linea <emph type="italics"/>RPB<emph.end type="italics"/>circa centrum <emph type="italics"/>S<emph.end type="italics"/>moventis velo­<lb/>citas in loco quovis <emph type="italics"/>P<emph.end type="italics"/>&longs;it ad velocitatem cor­<lb/>poris intervallo <emph type="italics"/>SP<emph.end type="italics"/>circa idem centrum Cir­<lb/>culum de&longs;cribentis in &longs;ubduplicata ratione rec­<lb/>tanguli 1/2 LX<emph type="italics"/>SP<emph.end type="italics"/>ad <emph type="italics"/>SY<emph.end type="italics"/>quadratum. </s> <s>E&longs;t au­<lb/>tem ex Conicis <emph type="italics"/>ACB<emph.end type="italics"/>ad <emph type="italics"/>CPq<emph.end type="italics"/>ut 2 <emph type="italics"/>AO<emph.end type="italics"/>ad L, <lb/>adeoque (2<emph type="italics"/>CPqXAO/ACB<emph.end type="italics"/>) æquale L. </s> <s>Ergo ve­<lb/>locitates illæ &longs;unt ad invicem in &longs;ubduplicata <lb/>ratione (<emph type="italics"/>CPqXAOXSP/ACB<emph.end type="italics"/>) ad <emph type="italics"/>SY quad.<emph.end type="italics"/>Por­<lb/>ro ex Conicis e&longs;t <emph type="italics"/>CO<emph.end type="italics"/>ad <emph type="italics"/>BO<emph.end type="italics"/>ut <emph type="italics"/>BO<emph.end type="italics"/>ad <emph type="italics"/>TO,<emph.end type="italics"/><lb/>& compo&longs;ite vel divi&longs;im ut <emph type="italics"/>CB<emph.end type="italics"/>ad <emph type="italics"/>BT.<emph.end type="italics"/><lb/>Unde vel dividendo vel componendo fit <lb/><emph type="italics"/>BO<emph.end type="italics"/>-vel+<emph type="italics"/>CO<emph.end type="italics"/>ad <emph type="italics"/>BO<emph.end type="italics"/>ut <emph type="italics"/>CT<emph.end type="italics"/>ad <emph type="italics"/>BT,<emph.end type="italics"/>id e&longs;t <lb/><emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>AO<emph.end type="italics"/>ut <emph type="italics"/>CP<emph.end type="italics"/>ad <emph type="italics"/>BQ<emph.end type="italics"/>; indeque (<emph type="italics"/>CPqXAOXSP/ACB<emph.end type="italics"/>) æquale e&longs;t <lb/>(<emph type="italics"/>BQqXACXSP/AOXBC.<emph.end type="italics"/>) Minuatur jam in infinitum Figuræ <emph type="italics"/>RPB<emph.end type="italics"/>latitu­<lb/>do <emph type="italics"/>CP,<emph.end type="italics"/>&longs;ic ut punctum <emph type="italics"/>P<emph.end type="italics"/>coeat cum puncto <emph type="italics"/>C,<emph.end type="italics"/>punctumque <emph type="italics"/>S<emph.end type="italics"/>cum <lb/>puncto <emph type="italics"/>B,<emph.end type="italics"/>& linea <emph type="italics"/>SP<emph.end type="italics"/>cum linea <emph type="italics"/>BC,<emph.end type="italics"/>lineaque <emph type="italics"/>SY<emph.end type="italics"/>cum linea <emph type="italics"/>BQ<emph.end type="italics"/>; <lb/>& corporis jam recta de&longs;cendentis in linea <emph type="italics"/>CB<emph.end type="italics"/>velocitas fiet ad <lb/>velocitatem corporis centro <emph type="italics"/>B<emph.end type="italics"/>intervallo <emph type="italics"/>BC<emph.end type="italics"/>Circulum de&longs;cribentis, <lb/>in &longs;ubduplicata ratione ip&longs;ius (<emph type="italics"/>BQqXACXSP/AOXBC<emph.end type="italics"/>) ad <emph type="italics"/>SYq,<emph.end type="italics"/>hoc e&longs;t (neg­<lb/>lectis æqualitatis rationibus <emph type="italics"/>SP<emph.end type="italics"/>ad <emph type="italics"/>BC<emph.end type="italics"/>& <emph type="italics"/>BQq<emph.end type="italics"/>ad <emph type="italics"/>SYq<emph.end type="italics"/>) in &longs;ub­<lb/>duplicata ratione <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>AO<emph.end type="italics"/>&longs;ive 1/2 <emph type="italics"/>AB. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note83"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Punctis <emph type="italics"/>B<emph.end type="italics"/>& <emph type="italics"/>S<emph.end type="italics"/>coeuntibus, fit <emph type="italics"/>TC<emph.end type="italics"/>ad <emph type="italics"/>TS<emph.end type="italics"/>ut <emph type="italics"/>AC<emph.end type="italics"/><lb/>ad <emph type="italics"/>AO.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Corpus ad datam a centro di&longs;tantiam in Circulo quo­<lb/>vis revolvens, motu &longs;uo &longs;ur&longs;um ver&longs;o a&longs;cendet ad duplam &longs;uam a <lb/>centro di&longs;tantiam. <pb xlink:href="039/01/136.jpg" pagenum="108"/><arrow.to.target n="note84"/></s></p> <p type="margin"> <s><margin.target id="note84"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXIV. THEOREMA X.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Figura<emph.end type="italics"/>BED <emph type="italics"/>Parabola e&longs;t, dico<emph.end type="italics"/><lb/><figure id="id.039.01.136.1.jpg" xlink:href="039/01/136/1.jpg"/><lb/><emph type="italics"/>quod corporis cadentis Veloci­<lb/>tas in loco quovis<emph.end type="italics"/>C <emph type="italics"/>æqualis e&longs;t <lb/>velocitati qua corpus centro<emph.end type="italics"/>B <lb/><emph type="italics"/>dimidio intervalli &longs;ui<emph.end type="italics"/>BC <emph type="italics"/>Cir­<lb/>culum uniformiter de&longs;cribere <lb/>potest.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam corporis Parabolam <lb/><emph type="italics"/>RPB<emph.end type="italics"/>circa centrum <emph type="italics"/>S<emph.end type="italics"/>de&longs;cri­<lb/>bentis velocitas in loco quovis <lb/><emph type="italics"/>P<emph.end type="italics"/>(per Corol. </s> <s>7. Prop. </s> <s>XVI) æ­<lb/>qualis e&longs;t velocitati corporis di­<lb/>midio intervalli <emph type="italics"/>SP<emph.end type="italics"/>Circulum cir­<lb/>ca idem centrum <emph type="italics"/>S<emph.end type="italics"/>uniformiter <lb/>de&longs;cribentis. </s> <s>Minuatur Parabolæ <lb/>latitudo <emph type="italics"/>CP<emph.end type="italics"/>in infinitum eo, ut <lb/>arcus Parabolicus <emph type="italics"/>PfB<emph.end type="italics"/>cum rec­<lb/>ta <emph type="italics"/>CB,<emph.end type="italics"/>centrum <emph type="italics"/>S<emph.end type="italics"/>cum vertice <emph type="italics"/>B,<emph.end type="italics"/><lb/>& intervallum <emph type="italics"/>SP<emph.end type="italics"/>cum intervallo <emph type="italics"/>BC<emph.end type="italics"/>coincidat, & con&longs;tabit Pro­<lb/>po&longs;itio. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXV. THEOREMA XI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, dico quod area Figuræ<emph.end type="italics"/>DES, <emph type="italics"/>radio indefinito<emph.end type="italics"/>SD <emph type="italics"/>de­<lb/>&longs;cripta, æqualis &longs;it areæ quam corpus, radio dimidium lateris recti <lb/>Figuræ<emph.end type="italics"/>DES <emph type="italics"/>æquante, circa centrum<emph.end type="italics"/>S <emph type="italics"/>uniformiter gyrando, eo­<lb/>dem tempore de&longs;cribere potest.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam concipe corpus <emph type="italics"/>C<emph.end type="italics"/>quam minima temporis particula lineo­<lb/>lam <emph type="italics"/>Cc<emph.end type="italics"/>cadendo de&longs;cribere, & interea corpus aliud <emph type="italics"/>K,<emph.end type="italics"/>uniformi­<lb/>ter in Circulo <emph type="italics"/>OKk<emph.end type="italics"/>circa centrum <emph type="italics"/>S<emph.end type="italics"/>gyrando, arcum <emph type="italics"/>Kk<emph.end type="italics"/>de&longs;cri­<lb/>bere. </s> <s>Erigantur perpendicula <emph type="italics"/>CD, cd<emph.end type="italics"/>occurrentia Figuræ <emph type="italics"/>DES<emph.end type="italics"/><lb/>in <emph type="italics"/>D, d.<emph.end type="italics"/>Jungantur <emph type="italics"/>SD, Sd, SK, Sk<emph.end type="italics"/>& ducatur <emph type="italics"/>Dd<emph.end type="italics"/>axi <emph type="italics"/>AS<emph.end type="italics"/>oc­<lb/>rens in <emph type="italics"/>T,<emph.end type="italics"/>& ad eam demittatur perpendiculum <emph type="italics"/>SY.<emph.end type="italics"/></s></p><pb xlink:href="039/01/137.jpg" pagenum="109"/> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>1. Jam &longs;i Figura <emph type="italics"/>DES<emph.end type="italics"/>Circulus e&longs;t vel Hyperbola, bi&longs;ece­<lb/><arrow.to.target n="note85"/>tur ejus tran&longs;ver&longs;a diameter <emph type="italics"/>AS<emph.end type="italics"/>in <emph type="italics"/>O,<emph.end type="italics"/>& erit <lb/><figure id="id.039.01.137.1.jpg" xlink:href="039/01/137/1.jpg"/><lb/><emph type="italics"/>SO<emph.end type="italics"/>dimidium lateris recti. </s> <s>Et quoniam e&longs;t <lb/><emph type="italics"/>TC<emph.end type="italics"/>ad <emph type="italics"/>TD<emph.end type="italics"/>ut <emph type="italics"/>Cc<emph.end type="italics"/>ad <emph type="italics"/>Dd,<emph.end type="italics"/>& <emph type="italics"/>TD<emph.end type="italics"/>ad <emph type="italics"/>TS<emph.end type="italics"/>ut <lb/><emph type="italics"/>CD<emph.end type="italics"/>ad <emph type="italics"/>SY,<emph.end type="italics"/>erit ex æquo <emph type="italics"/>TC<emph.end type="italics"/>ad <emph type="italics"/>TS<emph.end type="italics"/>ut <lb/><emph type="italics"/>CDXCc<emph.end type="italics"/>ad <emph type="italics"/>SYXDd.<emph.end type="italics"/>Sed per Corol. </s> <s>1. Prop. </s> <s><lb/>XXXIII, e&longs;t <emph type="italics"/>TC<emph.end type="italics"/>ad <emph type="italics"/>TS<emph.end type="italics"/>ut <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>AO,<emph.end type="italics"/>puta &longs;i <lb/>in coitu punctorum <emph type="italics"/>D, d<emph.end type="italics"/>capiantur linearum <lb/>rationes ultimæ. </s> <s>Ergo <emph type="italics"/>AC<emph.end type="italics"/>e&longs;t ad (<emph type="italics"/>AO<emph.end type="italics"/>&longs;eu) <emph type="italics"/>SK<emph.end type="italics"/><lb/>ut <emph type="italics"/>CDXCc<emph.end type="italics"/>ad <emph type="italics"/>SYXDd.<emph.end type="italics"/>Porro corporis <lb/>de&longs;cendentis velocitas in <emph type="italics"/>C<emph.end type="italics"/>e&longs;t ad velocitatem <lb/>corporis Circulum intervallo <emph type="italics"/>SC<emph.end type="italics"/>circa cen­<lb/>trum <emph type="italics"/>S<emph.end type="italics"/>de&longs;cribentis in &longs;ubduplicata ratione <lb/><emph type="italics"/>AC<emph.end type="italics"/>ad (<emph type="italics"/>AO<emph.end type="italics"/>vel) <emph type="italics"/>SK<emph.end type="italics"/>(per Prop. </s> <s>XXXIII.) Et <lb/>hæc velocitas ad velocitatem corporis de&longs;cri­<lb/>bentis Circulum <emph type="italics"/>OKk<emph.end type="italics"/>in &longs;ubduplicata ratione <lb/><emph type="italics"/>SK<emph.end type="italics"/>ad <emph type="italics"/>SC<emph.end type="italics"/>per Cor. </s> <s>6. Prop. </s> <s>IV, & ex æquo velo­<lb/>citas prima ad ultimam, hoc e&longs;t lineola <emph type="italics"/>Cc<emph.end type="italics"/>ad <lb/>arcum <emph type="italics"/>Kk<emph.end type="italics"/>in &longs;ubduplicata ratione <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>SC,<emph.end type="italics"/><lb/>id e&longs;t in ratione <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>CD.<emph.end type="italics"/>Quare e&longs;t <emph type="italics"/>CDXCc<emph.end type="italics"/><lb/>æquale <emph type="italics"/>ACXKk,<emph.end type="italics"/>& propterea <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>SK<emph.end type="italics"/>ut <lb/><emph type="italics"/>ACXKk<emph.end type="italics"/>ad <emph type="italics"/>SYXDd,<emph.end type="italics"/><expan abbr="indeq;">indeque</expan> <emph type="italics"/>SKXKk<emph.end type="italics"/>æqua­<lb/>le <emph type="italics"/>SYXDd,<emph.end type="italics"/>& 1/2 <emph type="italics"/>SKXKk<emph.end type="italics"/>æquale 1/2 <emph type="italics"/>SYXDd,<emph.end type="italics"/><lb/>id e&longs;t area <emph type="italics"/>KSk<emph.end type="italics"/>æqualis areæ <emph type="italics"/>SDd.<emph.end type="italics"/>Singulis <lb/>igitur temporis particulis generantur arearum <lb/>duarum particulæ <emph type="italics"/>KSk,<emph.end type="italics"/>& <emph type="italics"/>SDd,<emph.end type="italics"/>quæ, &longs;i mag­<lb/>nitudo earum minuatur & numerus augeatur in infinitum, ratio­<lb/>nem obtinent æqualitatis, & propterea (per Corollarium Lem­<lb/>matis IV) areæ totæ &longs;imul genitæ &longs;unt &longs;emper æquales, <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note85"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>2. Quod &longs;i Figura <emph type="italics"/>DES<emph.end type="italics"/>Parabola &longs;it, invenietur e&longs;&longs;e ut &longs;u­<lb/>pra <emph type="italics"/>CDXCc<emph.end type="italics"/>ad <emph type="italics"/>SYXDd<emph.end type="italics"/>ut <emph type="italics"/>TC<emph.end type="italics"/>ad <emph type="italics"/>TS,<emph.end type="italics"/>hoc e&longs;t ut 2 ad 1, ad­<lb/>eoque 1/4 <emph type="italics"/>CDXCc<emph.end type="italics"/>æquale e&longs;&longs;e 1/2 <emph type="italics"/>SYXDd.<emph.end type="italics"/>Sed corporis caden­<lb/>tis velocitas in <emph type="italics"/>C<emph.end type="italics"/>æqualis e&longs;t velocitati qua Circulus intervallo 1/2 <emph type="italics"/>SC<emph.end type="italics"/><lb/>uniformiter de&longs;cribi po&longs;&longs;it (per Prop. </s> <s>XXXIV) Et hæc velocitas ad ve­<lb/>locitatem qua Circulus radio <emph type="italics"/>SK<emph.end type="italics"/>de&longs;cribi po&longs;&longs;it, hoc e&longs;t, lineola <lb/><emph type="italics"/>Cc<emph.end type="italics"/>ad arcum <emph type="italics"/>Kk<emph.end type="italics"/>(per Corol. </s> <s>6. Prop. </s> <s>IV) e&longs;t in &longs;ubduplicata ratione <lb/><emph type="italics"/>SK<emph.end type="italics"/>ad 1/2 <emph type="italics"/>SC,<emph.end type="italics"/>id e&longs;t, in ratione <emph type="italics"/>SK<emph.end type="italics"/>ad 1/2 <emph type="italics"/>CD.<emph.end type="italics"/>Quare e&longs;t 1/2 <emph type="italics"/>SKXKk<emph.end type="italics"/><lb/>æquale 1/4 <emph type="italics"/>CDXCc,<emph.end type="italics"/>adeoque æquale 1/2 <emph type="italics"/>SYXDd,<emph.end type="italics"/>hoc e&longs;t, area <emph type="italics"/>KSk<emph.end type="italics"/><lb/>æqualis areæ <emph type="italics"/>SDd,<emph.end type="italics"/>ut &longs;upra. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/><pb xlink:href="039/01/138.jpg" pagenum="110"/><arrow.to.target n="note86"/></s></p> <p type="margin"> <s><margin.target id="note86"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXVI. PROBLEMA XXV.<emph.end type="center"/></s></p><figure id="id.039.01.138.1.jpg" xlink:href="039/01/138/1.jpg"/> <p type="main"> <s><emph type="italics"/>Corporis de loco dato<emph.end type="italics"/>A <emph type="italics"/>cadentis determinare Tem­<lb/>pora de&longs;cen&longs;us.<emph.end type="italics"/></s></p> <p type="main"> <s>Super diametro <emph type="italics"/>AS<emph.end type="italics"/>(di&longs;tantia corporis a cen­<lb/>tro &longs;ub initio) de&longs;cribe Semicirculum <emph type="italics"/>ADS,<emph.end type="italics"/>ut & <lb/>huic æqualem Semicirculum <emph type="italics"/>OKH<emph.end type="italics"/>circa centrum <lb/><emph type="italics"/>S.<emph.end type="italics"/>De corporis loco quovis <emph type="italics"/>C<emph.end type="italics"/>erige ordinatim ap­<lb/>plicatam <emph type="italics"/>CD.<emph.end type="italics"/>Junge <emph type="italics"/>SD,<emph.end type="italics"/>& areæ <emph type="italics"/>ASD<emph.end type="italics"/>æqua­<lb/>lem con&longs;titue &longs;ectorem <emph type="italics"/>OSK.<emph.end type="italics"/>Patet per Prop.<lb/>XXXV, quod corpus cadendo de&longs;cribet &longs;patium <emph type="italics"/>AC<emph.end type="italics"/><lb/>eodem Tempore quo corpus aliud uniformiter cir­<lb/>ca centrum <emph type="italics"/>S<emph.end type="italics"/>gyrando, de&longs;cribere pote&longs;t arcum <lb/><emph type="italics"/>OK. <expan abbr="q.">que</expan> E. F.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXVII. PROBLEMA XXVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corporis de loco dato &longs;ur&longs;um vel deor&longs;um projecti definire Tempora <lb/>a&longs;cen&longs;us vel de&longs;cen&longs;us.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Exeat corpus de loco dato <emph type="italics"/>G<emph.end type="italics"/>&longs;ecundum <lb/><figure id="id.039.01.138.2.jpg" xlink:href="039/01/138/2.jpg"/><lb/>lineam <emph type="italics"/>ASG<emph.end type="italics"/>cum velocitate quacunque. </s> <s><lb/>In duplicata ratione hujus velocitatis ad <lb/>uniformem in Circulo velocitatem, qua cor­<lb/>pus ad intervallum datum <emph type="italics"/>SG<emph.end type="italics"/>circa centrum <lb/><emph type="italics"/>S<emph.end type="italics"/>revolvi po&longs;&longs;et, cape <emph type="italics"/>GA<emph.end type="italics"/>ad 1/2 <emph type="italics"/>AS.<emph.end type="italics"/><lb/>Si ratio illa e&longs;t numeri binarii ad unita­<lb/>tem, punctum <emph type="italics"/>A<emph.end type="italics"/>infinite di&longs;tat, quo ca­<lb/>&longs;u Parabola vertice <emph type="italics"/>S,<emph.end type="italics"/>axe <emph type="italics"/>SC,<emph.end type="italics"/>latere quo­<lb/>vis recto de&longs;cribenda e&longs;t. </s> <s>Patet hoc per <lb/>Prop. </s> <s>XXXIV. </s> <s>Sin ratio illa minor vel ma­<lb/>jor e&longs;t quam 2 ad 1, priore ca&longs;u Circulus, <lb/>po&longs;teriore Hyperbola rectangula &longs;uper di­<lb/>ametro <emph type="italics"/>SA<emph.end type="italics"/>de&longs;cribi debet. </s> <s>Patet per <lb/>Prop. </s> <s>XXXIII. </s> <s>Tum centro <emph type="italics"/>S,<emph.end type="italics"/>intervallo <lb/>æquante dimidium lateris recti, de&longs;cribatur <lb/>Circulus <emph type="italics"/>HKk,<emph.end type="italics"/>& ad corporis a&longs;cenden­<lb/>tis vel de&longs;cendentis loca duo quævis <emph type="italics"/>G, C,<emph.end type="italics"/><lb/>erigantur perpendicula <emph type="italics"/>GI, CD<emph.end type="italics"/>occurren­<lb/>tia Conicæ Sectioni vel Circulo in <emph type="italics"/>I<emph.end type="italics"/>ac <emph type="italics"/>D.<emph.end type="italics"/><pb xlink:href="039/01/139.jpg" pagenum="111"/>Dein junctis <emph type="italics"/>SI, SD,<emph.end type="italics"/>fiant &longs;egmentis <emph type="italics"/>SEIS, SEDS,<emph.end type="italics"/>&longs;ec­<lb/><arrow.to.target n="note87"/>tores <emph type="italics"/>HSK, HSk<emph.end type="italics"/>æquales, & per Prop. </s> <s>XXXV, corpus <emph type="italics"/>G<emph.end type="italics"/>de&longs;cri­<lb/>bet &longs;patium <emph type="italics"/>GC<emph.end type="italics"/>eodem Tempore quo corpus <emph type="italics"/>K<emph.end type="italics"/>de&longs;cribere po­<lb/>te&longs;t arcum <emph type="italics"/>Kk. </s> <s><expan abbr="q.">que</expan> E. F.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note87"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXVIII. THEOREMA XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod Vis centripeta proportionalis &longs;it altitudini &longs;eu di&longs;tantiæ lo­<lb/>eorum a centro, dico quod cadentium Tempora, Velocitates & Spa­<lb/>tia de&longs;cripta &longs;unt arcubus, arcuumque finibus rectis & &longs;inibus <lb/>ver&longs;is re&longs;pective proportionalia.<emph.end type="italics"/></s></p> <p type="main"> <s>Cadat corpus de loco quovis <emph type="italics"/>A<emph.end type="italics"/>&longs;ecun­<lb/><figure id="id.039.01.139.1.jpg" xlink:href="039/01/139/1.jpg"/><lb/>dum rectam <emph type="italics"/>AS<emph.end type="italics"/>; & centro virium <emph type="italics"/>S,<emph.end type="italics"/>in­<lb/>tervallo <emph type="italics"/>AS,<emph.end type="italics"/>de&longs;cribatur Circuli quadrans <lb/><emph type="italics"/>AE,<emph.end type="italics"/>&longs;itque <emph type="italics"/>CD<emph.end type="italics"/>&longs;inus rectus arcus cuju&longs;­<lb/>vis <emph type="italics"/>AD<emph.end type="italics"/>; & corpus <emph type="italics"/>A,<emph.end type="italics"/>Tempore <emph type="italics"/>AD,<emph.end type="italics"/>ca­<lb/>dendo de&longs;cribet Spatium <emph type="italics"/>AC,<emph.end type="italics"/>inque loco <lb/><emph type="italics"/>C<emph.end type="italics"/>acquiret Velocitatem <emph type="italics"/>CD.<emph.end type="italics"/></s></p> <p type="main"> <s>Demon&longs;tratur eodem modo ex Propo&longs;i­<lb/>tione X, quo Propo&longs;itio XXXII, ex Propo­<lb/>&longs;itione XI demon&longs;trata fuit. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc æqualia &longs;unt Tempora quibus corpus unum de loco <lb/><emph type="italics"/>A<emph.end type="italics"/>cadendo pervenit ad centrum <emph type="italics"/>S,<emph.end type="italics"/>& corpus aliud revolvendo de­<lb/>&longs;cribit arcum quadrantalem <emph type="italics"/>ADE.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Proinde æqualia &longs;unt Tempora omnia quibus corpora de <lb/>locis quibu&longs;vis ad u&longs;que centrum cadunt. </s> <s>Nam revolventium tem­<lb/>pora omnia periodica (per Corol. </s> <s>3. Prop. </s> <s>IV.) æquantur. <pb xlink:href="039/01/140.jpg" pagenum="112"/><arrow.to.target n="note88"/></s></p> <p type="margin"> <s><margin.target id="note88"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXIX. PROBLEMA XXVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;ita cuju&longs;cunque generis Vi centripeta, & conce&longs;&longs;is figurarum <lb/>curvilinearum quadraturis, requiritu, corporis recta a&longs;cenden­<lb/>tis vel de&longs;cendentis tum Velocitas in locis &longs;ingulis, tum Tempus <lb/>quo corpus ad locum quemvis perveniet: Et contra.<emph.end type="italics"/></s></p> <p type="main"> <s>De loco quovis <emph type="italics"/>A<emph.end type="italics"/>in recta <emph type="italics"/>ADEC<emph.end type="italics"/>cadat corpus <emph type="italics"/>E,<emph.end type="italics"/>deque loco <lb/>ejus <emph type="italics"/>E<emph.end type="italics"/>erigatur &longs;emper perpendicularis <emph type="italics"/>EG,<emph.end type="italics"/>vi centripetæ in loco <lb/>illo ad centrum <emph type="italics"/>C<emph.end type="italics"/>tendenti proportio­<lb/><figure id="id.039.01.140.1.jpg" xlink:href="039/01/140/1.jpg"/><lb/>nalis: Sitque <emph type="italics"/>BFG<emph.end type="italics"/>linea curva quam <lb/>punctum <emph type="italics"/>G<emph.end type="italics"/>perpetuo tangit. </s> <s>Coinci­<lb/>dat autem <emph type="italics"/>EG<emph.end type="italics"/>ip&longs;o motus initio cum <lb/>perpendiculari <emph type="italics"/>AB,<emph.end type="italics"/>& erit corporis Ve­<lb/>locitas in loco quovis <emph type="italics"/>E<emph.end type="italics"/>ut areæ cur­<lb/>vilineæ <emph type="italics"/>ABGE<emph.end type="italics"/>latus quadratum. <lb/><emph type="italics"/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s>In <emph type="italics"/>EG<emph.end type="italics"/>capiatur <emph type="italics"/>EM<emph.end type="italics"/>lateri quadra­<lb/>to areæ <emph type="italics"/>ABGE<emph.end type="italics"/>reciproce proportio­<lb/>nalis, & &longs;it <emph type="italics"/>ALM<emph.end type="italics"/>linea curva quam <lb/>punctum <emph type="italics"/>M<emph.end type="italics"/>perpetuotangit, & erit Tem­<lb/>pus quo corpus cadendo de&longs;cribit li­<lb/>neam <emph type="italics"/>AE<emph.end type="italics"/>ut area curvilinea <emph type="italics"/>ALME. <lb/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s>Etenim in recta <emph type="italics"/>AE<emph.end type="italics"/>capiatur linea <lb/>quam minima <emph type="italics"/>DE<emph.end type="italics"/>datæ longitudinis, <lb/>&longs;itque <emph type="italics"/>DLF<emph.end type="italics"/>locus lineæ <emph type="italics"/>EMG<emph.end type="italics"/>ubi <lb/>corpus ver&longs;abatur in <emph type="italics"/>D<emph.end type="italics"/>; & &longs;i ea &longs;it vis centripeta, ut areæ <emph type="italics"/>ABGE<emph.end type="italics"/><lb/>latus quadratum &longs;it ut de&longs;cendentis velocitas, erit area ip&longs;a in du­<lb/>plicata ratione velocitatis, id e&longs;t, &longs;i pro velocitatibus in <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>E<emph.end type="italics"/><lb/>&longs;cribantur V & V+I, erit area <emph type="italics"/>ABFD<emph.end type="italics"/>ut VV, & area <emph type="italics"/>ABGE<emph.end type="italics"/>ut <lb/>VV+2 VI+II, & divi&longs;im area <emph type="italics"/>DFGE<emph.end type="italics"/>ut 2 VI+II, adeoque <lb/>(<emph type="italics"/>DFGE/DE<emph.end type="italics"/>) ut (2VI+II/<emph type="italics"/>DE<emph.end type="italics"/>), id e&longs;t, &longs;i primæ quantitatum na&longs;centium <lb/>rationes &longs;umantur, longitudo <emph type="italics"/>DF<emph.end type="italics"/>ut quantitas (2VI/<emph type="italics"/>DE<emph.end type="italics"/>), adeoque e­<lb/>tiam ut quantitatis hujus dimidium (IXV/<emph type="italics"/>DE<emph.end type="italics"/>). E&longs;t autem tempus quo <pb xlink:href="039/01/141.jpg" pagenum="113"/>corpus cadendo de&longs;cribit lineolam <emph type="italics"/>DE,<emph.end type="italics"/>ut lineola illa directe & <lb/><arrow.to.target n="note89"/>velocitas V inver&longs;e, e&longs;tque vis ut velocitatis incrementum I directe <lb/>& tempus inver&longs;e, adeoque &longs;i primæ na&longs;centium rationes &longs;uman­<lb/>tur, ut (IXV/<emph type="italics"/>DE<emph.end type="italics"/>), hoc e&longs;t, ut longitudo <emph type="italics"/>DF.<emph.end type="italics"/>Ergo Vis ip&longs;i <emph type="italics"/>DF<emph.end type="italics"/>vel <emph type="italics"/>EG<emph.end type="italics"/><lb/>proportionalis facit ut corpus ea cum Velocitate de&longs;cendat quæ &longs;it <lb/>ut areæ <emph type="italics"/>ABGE<emph.end type="italics"/>latus quadratum. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note89"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>Porro cum tempus, quo quælibet longitudinis datæ lineola <emph type="italics"/>DE<emph.end type="italics"/><lb/>de&longs;cribatur, &longs;it ut velocitas inver&longs;e adeoque ut areæ <emph type="italics"/>ABFD<emph.end type="italics"/>latus <lb/>quadratum inver&longs;e; &longs;itque <emph type="italics"/>DL,<emph.end type="italics"/>atque adeo area na&longs;cens <emph type="italics"/>DLME,<emph.end type="italics"/><lb/>ut idem latus quadratum inver&longs;e: erit tempus ut area <emph type="italics"/>DLME,<emph.end type="italics"/>& <lb/>&longs;umma omnium temporum ut &longs;umma omnium arearum, hoc e&longs;t <lb/>(per Corol. </s> <s>Lem. </s> <s>IV) Tempus totum quo linea <emph type="italics"/>AE<emph.end type="italics"/>de&longs;cribitur ut <lb/>area tota <emph type="italics"/>AME. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Si <emph type="italics"/>P<emph.end type="italics"/>&longs;it locus de quo corpus cadere debet, ut, urgen­<lb/>te aliqua uniformi vi centripeta nota (qualis vulgo &longs;upponitur <lb/>Gravitas) velocitatem acquirat in loco <emph type="italics"/>D<emph.end type="italics"/>æqualem velocitati <lb/>quam corpus aliud vi quacunque cadens acqui&longs;ivit eodem loco <emph type="italics"/>D,<emph.end type="italics"/><lb/>& in perpendiculari <emph type="italics"/>DF<emph.end type="italics"/>capiatur <emph type="italics"/>DR,<emph.end type="italics"/>quæ &longs;it ad <emph type="italics"/>DF<emph.end type="italics"/>ut vis illa <lb/>uniformis ad vim alteram in loco <emph type="italics"/>D,<emph.end type="italics"/>& compleatur rectangulum <lb/><emph type="italics"/>PDRQ,<emph.end type="italics"/>eique æqualis ab&longs;cindatur area <emph type="italics"/>ABFD;<emph.end type="italics"/>erit <emph type="italics"/>A<emph.end type="italics"/>locus <lb/>de quo corpus alterum cecidit. </s> <s>Namque completo rectangulo <lb/><emph type="italics"/>DRSE,<emph.end type="italics"/>cum &longs;it area <emph type="italics"/>ABFD<emph.end type="italics"/>ad aream <emph type="italics"/>DFGE<emph.end type="italics"/>ut VV ad <lb/>2VI, adeoque ut 1/2 V ad I, id e&longs;t, ut &longs;emi&longs;&longs;is velocitatis totius <lb/>ad incrementum velocitatis corporis vi inæquabili cadentis; & &longs;i­<lb/>militer area <emph type="italics"/>PQRD<emph.end type="italics"/>ad aream <emph type="italics"/>DRSE<emph.end type="italics"/>ut &longs;emi&longs;&longs;is velocitatis to­<lb/>tius ad incrementum velocitatis corporis uniformi vi cadentis; <lb/>&longs;intQ.E.I.crementa illa (ob æqualitatem temporum na&longs;centium) <lb/>ut vires generatrices, id e&longs;t, ut ordinatim applicatæ <emph type="italics"/>DF, DR,<emph.end type="italics"/><lb/>adeoque ut areæ na&longs;centes <emph type="italics"/>DFGE, DRSE<emph.end type="italics"/>; erunt (ex æquo) <lb/>areæ totæ <emph type="italics"/>ABFD, PQRD<emph.end type="italics"/>ad invicem ut &longs;emi&longs;&longs;es totarum ve­<lb/>locitatum, & propterea (ob æqualitatem velocitatum) æquantur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Unde &longs;i corpus quodlibet de loco quocunque <emph type="italics"/>D<emph.end type="italics"/>data <lb/>cum velocitate vel &longs;ur&longs;um vel deor&longs;um projiciatur, & detur lex vis <lb/>centripetæ, invenietur velocitas ejus in alio quovis loco <emph type="italics"/>e,<emph.end type="italics"/>erigen­<lb/>do ordinatam <emph type="italics"/>eg,<emph.end type="italics"/>& capiendo velocitatem illam ad velocitatem in <lb/>loco <emph type="italics"/>D<emph.end type="italics"/>ut e&longs;t latus quadratum rectanguli <emph type="italics"/>PQRD<emph.end type="italics"/>area curvili­<lb/>nea <emph type="italics"/>DFge<emph.end type="italics"/>vel aucti, &longs;i locus <emph type="italics"/>e<emph.end type="italics"/>e&longs;t loco <emph type="italics"/>D<emph.end type="italics"/>inferior, vel diminuti, <lb/>&longs;i is &longs;uperior e&longs;t, ad latus quadratum rectanguli &longs;olius <emph type="italics"/>PQRD,<emph.end type="italics"/>id <lb/>e&longs;t, ut √<emph type="italics"/>PQRD<emph.end type="italics"/>+vel-<emph type="italics"/>DFge<emph.end type="italics"/>ad √<emph type="italics"/>PQRD.<emph.end type="italics"/><pb xlink:href="039/01/142.jpg" pagenum="114"/><arrow.to.target n="note90"/></s></p> <p type="margin"> <s><margin.target id="note90"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Tempus quoQ.E.I.note&longs;cet erigendo ordinatam <emph type="italics"/>em<emph.end type="italics"/>re­<lb/>ciproce proportionalem lateri quadrato ex <emph type="italics"/>PQRD<emph.end type="italics"/>+vel-<emph type="italics"/>DFge,<emph.end type="italics"/><lb/>& capiendo tempus quo corpus de&longs;crip&longs;it lineam <emph type="italics"/>De<emph.end type="italics"/>ad tempus <lb/>quo corpus alterum vi uniformi cecidit a <emph type="italics"/>P<emph.end type="italics"/>& cadendo pervenit ad <lb/><emph type="italics"/>D,<emph.end type="italics"/>ut area curvilinea <emph type="italics"/>DLme<emph.end type="italics"/>ad rectangulum 2<emph type="italics"/>PDXDL.<emph.end type="italics"/>Nam­<lb/>que tempus quo corpus vi uniformi de&longs;cendens de&longs;crip&longs;it lineam <lb/><emph type="italics"/>PD<emph.end type="italics"/>e&longs;t ad tempus quo corpus idem de&longs;crip&longs;it lineam <emph type="italics"/>PE<emph.end type="italics"/>in &longs;ub­<lb/>duplicata ratione <emph type="italics"/>PD<emph.end type="italics"/>ad <emph type="italics"/>PE,<emph.end type="italics"/>id e&longs;t (lineola <emph type="italics"/>DE<emph.end type="italics"/>jamjam na&longs;cen­<lb/>te) in ratione <emph type="italics"/>PD<emph.end type="italics"/>ad <emph type="italics"/>PD<emph.end type="italics"/>+1/2 <emph type="italics"/>DE<emph.end type="italics"/>&longs;eu 2<emph type="italics"/>PD<emph.end type="italics"/>ad 2<emph type="italics"/>PD+DE,<emph.end type="italics"/><lb/>& divi&longs;im, ad tempus quo corpus idem de&longs;crip&longs;it lineolam <emph type="italics"/>DE<emph.end type="italics"/><lb/>ut 2<emph type="italics"/>PD<emph.end type="italics"/>ad <emph type="italics"/>DE,<emph.end type="italics"/>adeoque ut rectangulum 2<emph type="italics"/>PDXDL<emph.end type="italics"/>ad aream <lb/><emph type="italics"/>DLME<emph.end type="italics"/>; e&longs;tque tempus quo corpus utrumQ.E.D.&longs;crip&longs;it lineo­<lb/>lam <emph type="italics"/>DE<emph.end type="italics"/>ad tempus quo corpus alterum inæquabili motu de&longs;crip­<lb/>&longs;it lineam <emph type="italics"/>De<emph.end type="italics"/>ut area <emph type="italics"/>DLME<emph.end type="italics"/>ad aream <emph type="italics"/>DLme,<emph.end type="italics"/>& ex æquo <lb/>tempus primum ad tempus ultimum ut rectangulum 2<emph type="italics"/>PDXDL<emph.end type="italics"/><lb/>ad aream <emph type="italics"/>DLme.<emph.end type="italics"/></s></p></subchap2><subchap2> <p type="main"> <s><emph type="center"/>SECTIO VIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Inventione Orbium in quibus corpora Viribus quibu&longs;cunque cen­<lb/>tripetis agitata revolvuntur.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XL. THEOREMA XIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si corpus, cogente Vi quacunque centripeta, moveatur utcunque, & <lb/>corpus aliud recta a&longs;cendat vel de&longs;cendat, &longs;intque eorum Velocita­<lb/>tes in aliquo æqualium altitudinum ca&longs;u æquales, Velocitates eorum <lb/>in omnibus æqualibus altitudinibus erunt æquales.<emph.end type="italics"/></s></p> <p type="main"> <s>De&longs;cendat corpus aliquod ab <emph type="italics"/>A<emph.end type="italics"/>per <emph type="italics"/>D, E,<emph.end type="italics"/>ad centrum <emph type="italics"/>C,<emph.end type="italics"/>& <lb/>moveatur corpus aliud a <emph type="italics"/>V<emph.end type="italics"/>in linea curva <emph type="italics"/>VIKk,<emph.end type="italics"/>Centro <emph type="italics"/>C<emph.end type="italics"/>in­<lb/>tervallis quibu&longs;vis de&longs;cribantur circuli concentrici <emph type="italics"/>DI, EK<emph.end type="italics"/>rectæ <lb/><emph type="italics"/>AC<emph.end type="italics"/>in <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>E,<emph.end type="italics"/>curvæque <emph type="italics"/>VIK<emph.end type="italics"/>in <emph type="italics"/>I<emph.end type="italics"/>& <emph type="italics"/>K<emph.end type="italics"/>occurrentes. </s> <s>Junga­<lb/>tur <emph type="italics"/>IC<emph.end type="italics"/>occurrens ip&longs;i <emph type="italics"/>KE<emph.end type="italics"/>in <emph type="italics"/>N;<emph.end type="italics"/>& in <emph type="italics"/>IK<emph.end type="italics"/>demittatur perpendi­<lb/>culum <emph type="italics"/>NT<emph.end type="italics"/>; &longs;itque circumferentiarum circulorum intervallum <emph type="italics"/>DE<emph.end type="italics"/><lb/>vel <emph type="italics"/>IN<emph.end type="italics"/>quam minimum, & habeant corpora in <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>I<emph.end type="italics"/>velocita-<pb xlink:href="039/01/143.jpg" pagenum="115"/>tes æquales. </s> <s>Quoniam di&longs;tantiæ <emph type="italics"/>CD, CI<emph.end type="italics"/>æquantur, erunt vi­</s></p> <p type="main"> <s><arrow.to.target n="note91"/>res centripetæ in <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>I<emph.end type="italics"/>æquales. </s> <s>Exponantur hæ vires per æ­<lb/>quales lineolas <emph type="italics"/>DE, IN<emph.end type="italics"/>; & &longs;i vis una <emph type="italics"/>IN<emph.end type="italics"/>(per Legum Corol. </s> <s>2.) <lb/>re&longs;olvatur in duas <emph type="italics"/>NT<emph.end type="italics"/>& <emph type="italics"/>IT,<emph.end type="italics"/>vis <emph type="italics"/>NT,<emph.end type="italics"/>agendo &longs;ecundum lineam <lb/><emph type="italics"/>NT<emph.end type="italics"/>corporis cur&longs;ui <emph type="italics"/>ITK<emph.end type="italics"/>perpendicularem, nil mutabit velocita­<lb/>tem corporis in cur&longs;u illo, &longs;ed retrahet &longs;olummodo corpus a cur­<lb/>&longs;u rectilineo, facietQ.E.I.&longs;um de Orbis tangente perpetuo deflecte­<lb/>re, inque via curvilinea <emph type="italics"/>ITKk<emph.end type="italics"/>progredi. </s> <s>In hoc effectu produ­<lb/>cendo vis illa tota con&longs;umetur: vis autem altera <emph type="italics"/>IT,<emph.end type="italics"/>&longs;ecundum <lb/>corporis cur&longs;um agendo, tota accelerabit illud, ac dato tem­<lb/>pore quam minimo accelerationem generabit &longs;ibi ip&longs;i proportiona­<lb/>lem. </s> <s>Proinde corporum in <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>I<emph.end type="italics"/>accelerationes æqualibus tem­<lb/>poribus factæ (&longs;i &longs;umantur linearum na&longs;centium <emph type="italics"/>DE, IN, IK, <lb/>IT, NT<emph.end type="italics"/>rationes primæ) &longs;unt ut lineæ <emph type="italics"/>DE, IT:<emph.end type="italics"/>temporibus au­<lb/>tem inæqualibus ut lineæ illæ & tempora conjunctim. </s> <s>Tempora <lb/>autem quibus <emph type="italics"/>DE<emph.end type="italics"/>& <emph type="italics"/>IK<emph.end type="italics"/>de&longs;cribuntur, ob æqualitatem velocita­<lb/><figure id="id.039.01.143.1.jpg" xlink:href="039/01/143/1.jpg"/><lb/>tum &longs;unt ut viæ de&longs;criptæ <emph type="italics"/>DE<emph.end type="italics"/>& <emph type="italics"/>IK,<emph.end type="italics"/>adeoque accelerationes, in <lb/>cur&longs;u corporum per lineas <emph type="italics"/>DE<emph.end type="italics"/>& <emph type="italics"/>IK,<emph.end type="italics"/>funt ut <emph type="italics"/>DE<emph.end type="italics"/>& <emph type="italics"/>IT, DE<emph.end type="italics"/>& <lb/><emph type="italics"/>IK<emph.end type="italics"/>conjunctim, id e&longs;t ut <emph type="italics"/>DE quad<emph.end type="italics"/>& <emph type="italics"/>ITXIK rectangulum.<emph.end type="italics"/>Sed <lb/><emph type="italics"/>rectangulum ITXIK<emph.end type="italics"/>æquale e&longs;t <emph type="italics"/>IN quadrato,<emph.end type="italics"/>hoc e&longs;t, æquale <lb/><emph type="italics"/>DE quadrato;<emph.end type="italics"/>& propterea accelerationes in tran&longs;itu corporum a <lb/><emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>I<emph.end type="italics"/>ad <emph type="italics"/>E<emph.end type="italics"/>& <emph type="italics"/>K<emph.end type="italics"/>æquales generantur. </s> <s>Æquales igitur &longs;unt cor-<pb xlink:href="039/01/144.jpg" pagenum="116"/><arrow.to.target n="note92"/>porum velocitates in <emph type="italics"/>E<emph.end type="italics"/>& <emph type="italics"/>K<emph.end type="italics"/>& eodem argumento &longs;emper reperi­<lb/>entur æquales in &longs;ub&longs;equentibus æqualibus di&longs;tantiis. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note91"/>LIBER <lb/>PRIMUS.</s></p> <p type="margin"> <s><margin.target id="note92"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Sed & eodem argumento corpora æquivelocia & æqualiter a cen­<lb/>tro di&longs;tantia, in a&longs;cen&longs;u ad æquales di&longs;tantias æqualiter retarda­<lb/>buntur. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i corpus vel funipendulum o&longs;cilletur, vel im­<lb/>pedimento quovis politi&longs;&longs;imo & perfecte lubrico cogatur in li­<lb/>nea curva moveri, & corpus aliud recta a&longs;cendat vel de&longs;cendat, <lb/>&longs;intque velocitates eorum in eadem quacunque altitudine æquales: <lb/>erunt velocitates eorum in aliis quibu&longs;cunque æqualibus altitudi­<lb/>nibus æquales. </s> <s>NamQ.E.I.pedimento va&longs;is ab&longs;olute lubrici idem <lb/>præ&longs;tatur quod vi tran&longs;ver&longs;a <emph type="italics"/>NT.<emph.end type="italics"/>Corpus eo non retardatur, <lb/>non acceleratur, &longs;ed tantum cogitur de cur&longs;u rectilineo di&longs;cedere. </s></p><figure id="id.039.01.144.1.jpg" xlink:href="039/01/144/1.jpg"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Hinc etiam &longs;i quantitas P &longs;it maxima a centro di&longs;tan­<lb/>tia, ad quam corpus vel o&longs;cillans vel in Trajectoria quacunque re­<lb/>volvens, deque quovis Trajectoriæ puncto, ea quam ibi habet <lb/>velocitate &longs;ur&longs;um projectum a&longs;cendere po&longs;&longs;it; &longs;itque quantitas A <lb/>di&longs;tantia corporis a centro in alio quovis Orbitæ puncto, & vis <lb/>centripeta &longs;emper &longs;it ut ip&longs;ius A dignitas quælibet A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>, cujus <lb/>Index <emph type="italics"/>n<emph.end type="italics"/>-1 e&longs;t numerus quilibet <emph type="italics"/>n<emph.end type="italics"/>unitate diminutus; velocitas <lb/>corporis in omni altitudine A erit ut √P<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>-A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>, atque adeo da­<lb/>tur. </s> <s>Namque velocitas recta a&longs;cendentis ac de&longs;cendentis (per Prop. </s> <s><lb/>XXXIX) e&longs;t in hac ip&longs;a ratione. <pb xlink:href="039/01/145.jpg" pagenum="117"/><arrow.to.target n="note93"/></s></p> <p type="margin"> <s><margin.target id="note93"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLI. PROBLEMA XXVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;ita cuju&longs;cunque generis Vi centripeta & conce&longs;&longs;is Figurarum <lb/>curvilinearum quadraturis, requiruntur tum Trajectoriæ in qui­<lb/>bus corpora movebuntur, tum Tempora motuum in Trajectoriis <lb/>inventis.<emph.end type="italics"/></s></p> <p type="main"> <s>Tendat vis quælibet ad centrum <emph type="italics"/>C<emph.end type="italics"/>& invenienda &longs;it Trajectoria <lb/><emph type="italics"/>VITKk.<emph.end type="italics"/>Detur Circulus <emph type="italics"/>VXY<emph.end type="italics"/>centro <emph type="italics"/>C<emph.end type="italics"/>intervallo quovis <emph type="italics"/>CV<emph.end type="italics"/><lb/>de&longs;criptus, centroque eodem de&longs;cribantur alii quivis circuli <emph type="italics"/>ID, <lb/>KE<emph.end type="italics"/>Trajectoriam &longs;ecantes in <emph type="italics"/>I<emph.end type="italics"/>& <emph type="italics"/>K<emph.end type="italics"/>rectamque <emph type="italics"/>CV<emph.end type="italics"/>in <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>E.<emph.end type="italics"/><lb/>Age tum rectam <emph type="italics"/>CNIX<emph.end type="italics"/>&longs;ecantem circulos <emph type="italics"/>KE, VY<emph.end type="italics"/>in <emph type="italics"/>N<emph.end type="italics"/>& <emph type="italics"/>X,<emph.end type="italics"/><lb/>tum rectam <emph type="italics"/>CKY<emph.end type="italics"/>occurrentem circulo <emph type="italics"/>VXY<emph.end type="italics"/>in <emph type="italics"/>Y.<emph.end type="italics"/>Sint autem <lb/>puncta <emph type="italics"/>I<emph.end type="italics"/>& <emph type="italics"/>K<emph.end type="italics"/>&longs;ibi invicem vicini&longs;&longs;ima, & pergat corpus ab <emph type="italics"/>V<emph.end type="italics"/>per <lb/><emph type="italics"/>I, T<emph.end type="italics"/>& <emph type="italics"/>K<emph.end type="italics"/>ad <emph type="italics"/>k;<emph.end type="italics"/>&longs;itque punctum <emph type="italics"/>A<emph.end type="italics"/>locus ille de quo corpus aliud <lb/>cadere debet ut in loco <emph type="italics"/>D<emph.end type="italics"/>velocitatem acquirat æqualem veloci­<lb/>tati corporis prioris in <emph type="italics"/>I<emph.end type="italics"/>; & &longs;tantibus quæ in Propo&longs;itione XXXIX, <lb/>lineola <emph type="italics"/>IK,<emph.end type="italics"/>dato tempore quam minimo de&longs;cripta, erit ut ve­<lb/>locitas atque adeo ut latus quadratum areæ <emph type="italics"/>ABFD,<emph.end type="italics"/>& triangu­<lb/>lum <emph type="italics"/>ICK<emph.end type="italics"/>tempori proportionale dabitur, adeoque <emph type="italics"/>KN<emph.end type="italics"/>erit reci­<lb/>proce ut altitudo <emph type="italics"/>IC,<emph.end type="italics"/>id e&longs;t, &longs;i detur quantitas aliqua Q, & alti­<lb/>tudo <emph type="italics"/>IC<emph.end type="italics"/>nominetur A, ut Q/A. </s> <s>Hanc quantitatem Q/A nominemus Z, <lb/>& ponamus eam e&longs;&longs;e magnitudinem ip&longs;ius Q ut &longs;it in aliquo <lb/>ca&longs;u √ <emph type="italics"/>ABFD<emph.end type="italics"/>ad Z ut e&longs;t <emph type="italics"/>IK<emph.end type="italics"/>ad <emph type="italics"/>KN,<emph.end type="italics"/>& erit in omni ca&longs;u <lb/>√<emph type="italics"/>ABFD<emph.end type="italics"/>ad Z ut <emph type="italics"/>IK<emph.end type="italics"/>ad <emph type="italics"/>KN,<emph.end type="italics"/>& <emph type="italics"/>ABFD<emph.end type="italics"/>ad ZZ ut <emph type="italics"/><expan abbr="IKq.">IKque</expan><emph.end type="italics"/>ad <emph type="italics"/><expan abbr="KNq.">KNque</expan><emph.end type="italics"/><lb/>& divi&longs;im <emph type="italics"/>ABFD<emph.end type="italics"/>-ZZ ad ZZ ut <emph type="italics"/>IN quad<emph.end type="italics"/>ad <emph type="italics"/>KN quad,<emph.end type="italics"/>ad­<lb/>eoque √<emph type="italics"/>ABFD<emph.end type="italics"/>-ZZ ad (Z &longs;eu)Q/A ut <emph type="italics"/>IN<emph.end type="italics"/>ad <emph type="italics"/>KN,<emph.end type="italics"/>& propterea <lb/>AX<emph type="italics"/>KN<emph.end type="italics"/>æquale (QX<emph type="italics"/>IN/√ABFD<emph.end type="italics"/>-ZZ). Unde cum <emph type="italics"/>YXXXC<emph.end type="italics"/>&longs;it ad <lb/>AX<emph type="italics"/>KN<emph.end type="italics"/>ut <emph type="italics"/>CXq<emph.end type="italics"/>ad AA, erit rectangulum <emph type="italics"/>YXXXC<emph.end type="italics"/>æquale <lb/>(QX<emph type="italics"/>INXCX quad.<emph.end type="italics"/>/AA√<emph type="italics"/>ABFD<emph.end type="italics"/>-ZZ). Igitur &longs;i in perpendiculo <emph type="italics"/>DF<emph.end type="italics"/>capiantur <lb/>&longs;emper <emph type="italics"/>Db, Dc<emph.end type="italics"/>ip&longs;is (Q/2√<emph type="italics"/>ABFD<emph.end type="italics"/>-ZZ) & (QX<emph type="italics"/>CX quad.<emph.end type="italics"/>/2AA√<emph type="italics"/>ABFD<emph.end type="italics"/>-ZZ) <lb/>æquales re&longs;pective, & de&longs;cribantur curvæ lineæ <emph type="italics"/>ab, cd<emph.end type="italics"/>quas <pb xlink:href="039/01/146.jpg" pagenum="118"/><arrow.to.target n="note94"/>puncta <emph type="italics"/>b, c<emph.end type="italics"/>perpetuo tangunt; deque puncto <emph type="italics"/>V<emph.end type="italics"/>ad lineam <emph type="italics"/>AC<emph.end type="italics"/>eri­<lb/>gatur perpendiculum <emph type="italics"/>Vad<emph.end type="italics"/>ab&longs;cindens areas curvilineas <emph type="italics"/>VDba, <lb/>VDcd,<emph.end type="italics"/>& erigantur etiam ordinatæ <emph type="italics"/>Ez, Ex:<emph.end type="italics"/>quoniam rectan­<lb/>gulum <emph type="italics"/>DbXIN<emph.end type="italics"/>&longs;eu <emph type="italics"/>DbzE<emph.end type="italics"/>æquale e&longs;t dimidio rectanguli <lb/>AX<emph type="italics"/>KN,<emph.end type="italics"/>&longs;eu triangulo <emph type="italics"/>ICK<emph.end type="italics"/>; & rectangulum <emph type="italics"/>DcXIN<emph.end type="italics"/>&longs;eu <lb/><emph type="italics"/>DcxE<emph.end type="italics"/>æquale e&longs;t dimidio rectanguli <emph type="italics"/>YXXXC,<emph.end type="italics"/>&longs;eu triangulo <lb/><emph type="italics"/>XCY;<emph.end type="italics"/>hoc e&longs;t, quoniam arearum <emph type="italics"/>VDba, VIC<emph.end type="italics"/>æquales &longs;emper <lb/>&longs;unt na&longs;centes particulæ <emph type="italics"/>DbzE, ICK,<emph.end type="italics"/>& arearum <emph type="italics"/>VDcd, <lb/>VCX<emph.end type="italics"/>æquales &longs;emper &longs;unt na&longs;centes particulæ <emph type="italics"/>DcxE, XCY,<emph.end type="italics"/><lb/>erit area genita <emph type="italics"/>VDba<emph.end type="italics"/>æqualis areæ genitæ <emph type="italics"/>VIC,<emph.end type="italics"/>adeoque tem­<lb/>pori proportionalis, & area genita <emph type="italics"/>VDcd<emph.end type="italics"/>æqualis Sectori ge­<lb/>nito <emph type="italics"/>VCX.<emph.end type="italics"/>Dato igitur tempore quovis ex quo corpus di&longs;ce&longs;­<lb/>&longs;it de loco <emph type="italics"/>V,<emph.end type="italics"/>dabitur area ip&longs;i proportionalis <emph type="italics"/>VDba,<emph.end type="italics"/>& inde <lb/>dabitur corporis altitudo <emph type="italics"/>CD<emph.end type="italics"/>vel <emph type="italics"/>CI<emph.end type="italics"/>; & area <emph type="italics"/>VDcd,<emph.end type="italics"/>eique <lb/>æqualis Sector <emph type="italics"/>VCX<emph.end type="italics"/>una cum ejus angulo <emph type="italics"/>VCI.<emph.end type="italics"/>Datis autem <lb/>angulo <emph type="italics"/>VCI<emph.end type="italics"/>& altitudine <emph type="italics"/>CI<emph.end type="italics"/>datur locus <emph type="italics"/>I,<emph.end type="italics"/>in quo corpus com­<lb/>pleto illo tempore reperietur. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note94"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc maximæ minimæque corporum altitudines, id e&longs;t <lb/>Ap&longs;ides Trajectoriarum expedite inveniri po&longs;&longs;unt. </s> <s>Sunt enim <lb/>Ap&longs;ides puncta illa in quibus recta <emph type="italics"/>IC<emph.end type="italics"/>per centrum ducta incidit <lb/>perpendiculariter in Trajectoriam <emph type="italics"/>VIK:<emph.end type="italics"/>id quod &longs;it ubi rectæ <emph type="italics"/>IK<emph.end type="italics"/><lb/>& <emph type="italics"/>NK<emph.end type="italics"/>æquantur, adeoque ubi area <emph type="italics"/>ABFD<emph.end type="italics"/>æqualis e&longs;t ZZ. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Sed & angulus <emph type="italics"/>KIN,<emph.end type="italics"/>in quo Trajectoria alibi &longs;ecat <lb/>lineam illam <emph type="italics"/>IC,<emph.end type="italics"/>ex data corporis altitudine <emph type="italics"/>IC<emph.end type="italics"/>expedite inveNI­<lb/>tur; nimirum capiendo &longs;inum ejus ad radium ut <emph type="italics"/>KN<emph.end type="italics"/>ad <emph type="italics"/>IK,<emph.end type="italics"/>id <lb/>e&longs;t, ut Z ad latus quadratum areæ <emph type="italics"/>ABFD.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Si centro <emph type="italics"/>C<emph.end type="italics"/>& vertice principali <emph type="italics"/>V<emph.end type="italics"/>de&longs;cribatur Sectio quæ­<lb/>libet Conica <emph type="italics"/>VRS,<emph.end type="italics"/>& a quovis ejus puncto <emph type="italics"/>R<emph.end type="italics"/>agatur Tangens <emph type="italics"/>RT<emph.end type="italics"/><lb/>occurrens axi infinite producto <emph type="italics"/>CV<emph.end type="italics"/>in puncto <emph type="italics"/>T;<emph.end type="italics"/>dein juncta <emph type="italics"/>CR<emph.end type="italics"/><lb/>ducatur recta <emph type="italics"/>CP,<emph.end type="italics"/>quæ æqualis &longs;it ab&longs;ci&longs;&longs;æ <emph type="italics"/>CT,<emph.end type="italics"/>angulumque <emph type="italics"/>VCP<emph.end type="italics"/><lb/>Sectori <emph type="italics"/>VCR<emph.end type="italics"/>proportionalem con&longs;tituat; tendat autem ad centrum <emph type="italics"/>C<emph.end type="italics"/><lb/>Vis centripeta Cubo di&longs;tantiæ loeorum a centro reciproce propor­<lb/>tionalis, & exeat corpus de loco <emph type="italics"/>V<emph.end type="italics"/>ju&longs;ta cum Velocitate &longs;ecundum <lb/>lineam rectæ <emph type="italics"/>CV<emph.end type="italics"/>perpendicularem: progredietur corpus illud in <lb/>Trajectoria quam punctum <emph type="italics"/>P<emph.end type="italics"/>perpetuo tangit; adeoque &longs;i Conica <lb/>&longs;ectio <emph type="italics"/>CVRS<emph.end type="italics"/>Hyperbola &longs;it, de&longs;cendet idem ad centrum: Sin <lb/>ea Ellip&longs;is &longs;it, a&longs;cendet illud perpetuo & abibit in infinitum. </s> <s>Et con­<lb/>tra, &longs;i corpus quacunque cum Velocitate exeat de loco <emph type="italics"/>V,<emph.end type="italics"/>& perin­<lb/>de ut incæperit vel obliQ.E.D.&longs;cendere ad centrum, vel ab eo ob-<pb xlink:href="039/01/147.jpg" pagenum="119"/>lique a&longs;cendere, Figura <emph type="italics"/>CVRS<emph.end type="italics"/>vel Hyperbola &longs;it vel Ellip&longs;is, in­<lb/><arrow.to.target n="note95"/>veniri pote&longs;t Trajectoria augendo vel minuendo angulum <emph type="italics"/>VCP<emph.end type="italics"/><lb/>in data aliqua ratione. </s> <s>Sed &, Vi centripeta in centrifugam ver&longs;a, <lb/><figure id="id.039.01.147.1.jpg" xlink:href="039/01/147/1.jpg"/><lb/>a&longs;cendet corpus obliQ.E.I. Trajectoria <emph type="italics"/>VPQ<emph.end type="italics"/>quæ invenitur capi­<lb/>endo angulum <emph type="italics"/>VCP<emph.end type="italics"/>Sectori Elliptico <emph type="italics"/>CVRC<emph.end type="italics"/>proportionalem, & <lb/>longitudinem <emph type="italics"/>CP<emph.end type="italics"/>longitudini <emph type="italics"/>CT<emph.end type="italics"/>æqualem ut &longs;upra. </s> <s>Con&longs;equun­<lb/>tur hæc omnia ex Propo&longs;itione præcedente, per Curvæ cuju&longs;dam <lb/>quadraturam, cujus inventionem, ut &longs;atis facilem, brevitatis gratia <lb/>mi&longs;&longs;am facio. </s></p> <p type="margin"> <s><margin.target id="note95"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLII. PROBLEMA XXIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Data lege Vis centripetæ, requiritur motus corporis de loco dato <lb/>data cum Velocitate &longs;ecundum datam rectam egre&longs;&longs;i.<emph.end type="italics"/></s></p> <p type="main"> <s>Stantibus quæ in tribus Propo&longs;itionibus præcedentibus: exeat <lb/>corpus de loco <emph type="italics"/>I<emph.end type="italics"/>&longs;ecundum lineolam <emph type="italics"/>IT,<emph.end type="italics"/>ea cum Velocitate quam <lb/>corpus aliud, vi aliqua uniformi centripeta, de loco <emph type="italics"/>P<emph.end type="italics"/>cadendo ac­<lb/>quirere po&longs;&longs;et in <emph type="italics"/>D:<emph.end type="italics"/>&longs;itque hæc vis uniformis ad vim qua corpus <pb xlink:href="039/01/148.jpg" pagenum="120"/><arrow.to.target n="note96"/>primum urgetur in <emph type="italics"/>I,<emph.end type="italics"/>ut <emph type="italics"/>DR<emph.end type="italics"/>ad <emph type="italics"/>DF.<emph.end type="italics"/>Pergat autem corpus ver&longs;us <lb/><emph type="italics"/>k;<emph.end type="italics"/>centroque <emph type="italics"/>C<emph.end type="italics"/>& intervallo <emph type="italics"/>Ck<emph.end type="italics"/>de&longs;cribatur circulus <emph type="italics"/>ke<emph.end type="italics"/>occurrens <lb/>rectæ <emph type="italics"/>PD<emph.end type="italics"/>in <emph type="italics"/>e,<emph.end type="italics"/>& erigantur curvarum <emph type="italics"/>ALMm, BFGg, abzv, dcxw<emph.end type="italics"/><lb/><figure id="id.039.01.148.1.jpg" xlink:href="039/01/148/1.jpg"/><lb/>ordinatim applicatæ <emph type="italics"/>em, eg, ev, ew.<emph.end type="italics"/>Ex dato rectangulo <emph type="italics"/>PDRQ,<emph.end type="italics"/><lb/>dataque lege vis centripetæ qua corpus primum agitatur, dantur cur­<lb/>væ lineæ <emph type="italics"/>BFGg, ALMm,<emph.end type="italics"/>per con&longs;tructionem Problematis XXVII, <lb/>& ejus Corol. </s> <s>1. Deinde ex dato angulo <emph type="italics"/>CIT<emph.end type="italics"/>datur proportio na&longs;cen­<lb/>tium <emph type="italics"/>IK, KN,<emph.end type="italics"/>& inde, per con&longs;tructionem Prob. </s> <s>XXVIII, datur <lb/>quantitas Q, una cum curvis lineis <emph type="italics"/>abzv, dcxw:<emph.end type="italics"/>adeoque com­<lb/>pleto tempore quovis <emph type="italics"/>Dbve,<emph.end type="italics"/>datur tum corporis altitudo <emph type="italics"/>Ce<emph.end type="italics"/>vel <emph type="italics"/>Ck,<emph.end type="italics"/><lb/>tum area <emph type="italics"/>Dcwe,<emph.end type="italics"/>eique æqualis Sector <emph type="italics"/>XCy,<emph.end type="italics"/>angulu&longs;que <emph type="italics"/>ICk<emph.end type="italics"/>& <lb/>locus <emph type="italics"/>k<emph.end type="italics"/>in quo corpus tunc ver&longs;abitur. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note96"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Supponimus autem in his Propo&longs;itionibus Vim centripetam in <lb/>rece&longs;&longs;u quidem a centro variari &longs;ecundum legem quamcunque quam <lb/>quis imaginari pote&longs;t, in æqualibus autem a centro di&longs;tantiis e&longs;&longs;e <lb/>undeque eandem. </s> <s>Atque hactenus Motum corporum in Orbibus <lb/>immobilibus con&longs;ideravimus. </s> <s>Supere&longs;t ut de Motu eorum in Orbi­<lb/>bus qui circa centrum virium revolvuntur adjiciamus pauca. <pb xlink:href="039/01/149.jpg" pagenum="121"/><arrow.to.target n="note97"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note97"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>SECTIO IX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Motu Corporum in Orbibus mobilibus, deque motu Ap&longs;idum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLIII. PROBLEMA XXX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Efficiendum est ut corpus in Trajectoria quacunque circa centrum <lb/>Virium revolvente perinde moveri po&longs;&longs;it, atque corpus aliud in <lb/>eadem Trajectoria quie&longs;cente.<emph.end type="italics"/></s></p> <p type="main"> <s>In Orbe <emph type="italics"/>VPK<emph.end type="italics"/>po­<lb/><figure id="id.039.01.149.1.jpg" xlink:href="039/01/149/1.jpg"/><lb/>&longs;itione dato revolvatur <lb/>corpus <emph type="italics"/>P<emph.end type="italics"/>pergendo a <lb/><emph type="italics"/>V<emph.end type="italics"/>ver&longs;us <emph type="italics"/>K.<emph.end type="italics"/>A centro <lb/><emph type="italics"/>C<emph.end type="italics"/>agatur &longs;emper <emph type="italics"/>Cp,<emph.end type="italics"/><lb/>quæ &longs;it ip&longs;i <emph type="italics"/>CP<emph.end type="italics"/>æqualis, <lb/>angulumque <emph type="italics"/>VCp<emph.end type="italics"/>an­<lb/>gulo <emph type="italics"/>VCP<emph.end type="italics"/>proportio­<lb/>nalem con&longs;tituat; & a­<lb/>rea quam linea <emph type="italics"/>Cp<emph.end type="italics"/>de­<lb/>&longs;cribit erit ad aream <lb/><emph type="italics"/>VCP<emph.end type="italics"/>quam linea <emph type="italics"/>CP<emph.end type="italics"/><lb/>&longs;imul de&longs;cribit, ut velo­<lb/>citas lineæ de&longs;cribentis <lb/><emph type="italics"/>Cp<emph.end type="italics"/>ad velocitatem li­<lb/>neæ de&longs;cribentis <emph type="italics"/>CP<emph.end type="italics"/>; <lb/>hoc e&longs;t, ut angulus <emph type="italics"/>VCp<emph.end type="italics"/>ad angulum <emph type="italics"/>VCP,<emph.end type="italics"/>adeoQ.E.I. data ra­<lb/>tione, & propterea tempori proportionalis. </s> <s>Cum area tempori <lb/>proportionalis &longs;it quam linea <emph type="italics"/>Cp<emph.end type="italics"/>in plano immobili de&longs;cribit, ma­<lb/>nife&longs;tum e&longs;t quod corpus, cogente ju&longs;tæ quantitatis Vi centripeta, <lb/>revolvi po&longs;&longs;it una cum puncto <emph type="italics"/>p<emph.end type="italics"/>in Curva illa linea quam punctum <lb/>idem <emph type="italics"/>p<emph.end type="italics"/>ratione jam expo&longs;ita de&longs;cribit in plano immobili. </s> <s>Fiat angu­<lb/>lus <emph type="italics"/>VCu<emph.end type="italics"/>angulo <emph type="italics"/>PCp,<emph.end type="italics"/>& linea <emph type="italics"/>Cu<emph.end type="italics"/>lineæ <emph type="italics"/>CV,<emph.end type="italics"/>atque Figura <emph type="italics"/>uCp<emph.end type="italics"/>Fi­<lb/>guræ <emph type="italics"/>VCP<emph.end type="italics"/>æqualis, & corpus in <emph type="italics"/>p<emph.end type="italics"/>&longs;emper exi&longs;tens movebitur in <pb xlink:href="039/01/150.jpg" pagenum="122"/><arrow.to.target n="note98"/>perimetro Figuræ revolventis <emph type="italics"/>uCp,<emph.end type="italics"/>eodemque tempore de&longs;cribet <lb/>arcum ejus <emph type="italics"/>up<emph.end type="italics"/>quo corpus aliud <emph type="italics"/>P<emph.end type="italics"/>arcum ip&longs;i &longs;imilem & æqualem <lb/><emph type="italics"/>VP<emph.end type="italics"/>in Figura quie&longs;cente <emph type="italics"/>VPK<emph.end type="italics"/>de&longs;cribere pote&longs;t. </s> <s>Quæratur igi­<lb/>tur, per Corollarium quintum propo&longs;itionis VI, Vis centripeta qua <lb/>corpus revolvi po&longs;&longs;it in Curva illa linea quam punctum <emph type="italics"/>p<emph.end type="italics"/>de&longs;cribit <lb/>in plano immobili, & &longs;olvetur Problema. <emph type="italics"/>q.E.F.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note98"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLIV. THEOREMA XIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Differentia Virium, quibus corpus in Orbe quie&longs;cente, & corpus a­<lb/>liud in eodem Orbe revolvente æqualiter moveri po&longs;&longs;unt, est <lb/>in triplicata ratione communis altitudinis inver&longs;e.<emph.end type="italics"/></s></p> <p type="main"> <s>Partibus Orbis quie­<lb/><figure id="id.039.01.150.1.jpg" xlink:href="039/01/150/1.jpg"/><lb/>&longs;centis <emph type="italics"/>VP, PK<emph.end type="italics"/>&longs;unto <lb/>&longs;imiles & æquales Or­<lb/>bis revolventis partes <lb/><emph type="italics"/>up, pk<emph.end type="italics"/>; & punctorum <lb/><emph type="italics"/>P, K<emph.end type="italics"/>di&longs;tantia intelli­<lb/>gatur e&longs;&longs;e quam miNI­<lb/>ma. </s> <s>A puncto <emph type="italics"/>k<emph.end type="italics"/>in re­<lb/>ctam <emph type="italics"/>pC<emph.end type="italics"/>demitte per­<lb/>pendiculum <emph type="italics"/>kr,<emph.end type="italics"/>idem­<lb/>que produc ad <emph type="italics"/>m,<emph.end type="italics"/>ut &longs;it <lb/><emph type="italics"/>mr<emph.end type="italics"/>ad <emph type="italics"/>kr<emph.end type="italics"/>ut angulus <lb/><emph type="italics"/>VCp<emph.end type="italics"/>ad angulum <emph type="italics"/>VCP.<emph.end type="italics"/><lb/>Quoniam corporum al­<lb/>titudines <emph type="italics"/>PC<emph.end type="italics"/>& <emph type="italics"/>pC, KC<emph.end type="italics"/><lb/>& <emph type="italics"/>kC<emph.end type="italics"/>&longs;emper æquan­<lb/>tur, manife&longs;tum e&longs;t quod linearum <emph type="italics"/>PC<emph.end type="italics"/>& <emph type="italics"/>pC<emph.end type="italics"/>incrementa vel <lb/>decrementa &longs;emper &longs;int æqualia, ideoque &longs;i corporum in locis <lb/><emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>p<emph.end type="italics"/>exi&longs;tentium di&longs;tinguantur motus &longs;inguli (per Legum <lb/>Corol. </s> <s>2.) in binos, quorum hi ver&longs;us centrum, &longs;ive &longs;ecundum <lb/>lineas <emph type="italics"/>PC, pC<emph.end type="italics"/>determinentur, & alteri prioribus tran&longs;ver&longs;i &longs;int, <lb/>& &longs;ecundum lineas ip&longs;is <emph type="italics"/>PC, pC<emph.end type="italics"/>perpendiculares directionem <lb/>habeant; motus ver&longs;us centrum erunt æquales, & motus tran&longs;­<lb/>ver&longs;us corporis <emph type="italics"/>p<emph.end type="italics"/>erit ad motum tran&longs;ver&longs;um corporis <emph type="italics"/>P,<emph.end type="italics"/>ut mo­<lb/>tus angularis lineæ <emph type="italics"/>pC,<emph.end type="italics"/>ad motum angularem lineæ <emph type="italics"/>PC,<emph.end type="italics"/>id e&longs;t, <pb xlink:href="039/01/151.jpg" pagenum="123"/>ut angulus <emph type="italics"/>VCp<emph.end type="italics"/>ad angulum <emph type="italics"/>VCP.<emph.end type="italics"/>Igitur eodem tempore quo <lb/><arrow.to.target n="note99"/>corpus <emph type="italics"/>P<emph.end type="italics"/>motu &longs;uo utroque pervenit ad punctum <emph type="italics"/>K,<emph.end type="italics"/>corpus <emph type="italics"/>p<emph.end type="italics"/>æ­<lb/>quali in centrum motu æqualiter movebitur a <emph type="italics"/>p<emph.end type="italics"/>ver&longs;us <emph type="italics"/>C,<emph.end type="italics"/>adeoque <lb/>completo illo tempore reperietur alicubi in linea <emph type="italics"/>mkr,<emph.end type="italics"/>quæ per <lb/>punctum <emph type="italics"/>k<emph.end type="italics"/>in lineam <emph type="italics"/>pC<emph.end type="italics"/>perpendicularis e&longs;t; & motu tran&longs;ver&longs;o <lb/>acquiret di&longs;tantiam a linea <emph type="italics"/>pC,<emph.end type="italics"/>quæ &longs;it ad di&longs;tantiam quam cor­<lb/>pus alterum <emph type="italics"/>P<emph.end type="italics"/>acquirit a linea <emph type="italics"/>PC,<emph.end type="italics"/>ut e&longs;t motus tran&longs;ver&longs;us cor­<lb/>poris <emph type="italics"/>p<emph.end type="italics"/>ad motum tran&longs;ver&longs;um corporis alterius <emph type="italics"/>P.<emph.end type="italics"/>Quare cum <lb/><emph type="italics"/>kr<emph.end type="italics"/>æqualis &longs;it di&longs;tantiæ quam corpus <emph type="italics"/>P<emph.end type="italics"/>acquirit a linea <emph type="italics"/>PC,<emph.end type="italics"/>&longs;itque <lb/><emph type="italics"/>mr<emph.end type="italics"/>ad <emph type="italics"/>kr<emph.end type="italics"/>ut angulus <emph type="italics"/>VCp<emph.end type="italics"/>ad angulum <emph type="italics"/>VCP,<emph.end type="italics"/>hoc e&longs;t, ut motus <lb/>tran&longs;ver&longs;us corporis <emph type="italics"/>p<emph.end type="italics"/>ad motum tran&longs;ver&longs;um corporis <emph type="italics"/>P,<emph.end type="italics"/>manife­<lb/>&longs;tum e&longs;t quod corpus <emph type="italics"/>p<emph.end type="italics"/>completo illo tempore reperietur in loco <lb/><emph type="italics"/>m.<emph.end type="italics"/>Hæc ita &longs;e habebunt ubi corpora <emph type="italics"/>p<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>æqualiter &longs;ecundum <lb/>lineas <emph type="italics"/>pC<emph.end type="italics"/>& <emph type="italics"/>PC<emph.end type="italics"/>moventur, adeoque æqualibus Viribus &longs;ecundum <lb/>lineas illas urgentur. </s> <s>Capiatur autem angulum <emph type="italics"/>pCn<emph.end type="italics"/>ad angulum <lb/><emph type="italics"/>pCk<emph.end type="italics"/>ut e&longs;t angulus <emph type="italics"/>VCp<emph.end type="italics"/>ad angulus <emph type="italics"/>VCP,<emph.end type="italics"/>&longs;itque <emph type="italics"/>nC<emph.end type="italics"/>æqualis <lb/><emph type="italics"/>kC,<emph.end type="italics"/>& corpus <emph type="italics"/>p<emph.end type="italics"/>completo illo tempore revera reperietur in <emph type="italics"/>n<emph.end type="italics"/>; ad­<lb/>eoque Vi majore urgetur quam corpus <emph type="italics"/>P,<emph.end type="italics"/>&longs;i modo angulus <emph type="italics"/>mCp<emph.end type="italics"/><lb/>angulo <emph type="italics"/>kCp<emph.end type="italics"/>major e&longs;t, id e&longs;t &longs;i Orbis <emph type="italics"/>upk<emph.end type="italics"/>vel movetur in con­<lb/>&longs;equentia, vel movetur in antecedentia majore celeritate quam <lb/>&longs;it dupla ejus qua linea <emph type="italics"/>CP<emph.end type="italics"/>in con&longs;equentia fertur; & Vi mino­<lb/>re &longs;i Orbis tardius movetur in antecedentia. </s> <s>E&longs;tque Virium dif­<lb/>ferentia ut loeorum intervallum <emph type="italics"/>mn,<emph.end type="italics"/>per quod corpus illud <emph type="italics"/>p<emph.end type="italics"/><lb/>ip&longs;ius actione, dato illo temporis &longs;patio, transferri debet. </s> <s>Centro <lb/><emph type="italics"/>C<emph.end type="italics"/>intervallo <emph type="italics"/>Cn<emph.end type="italics"/>vel <emph type="italics"/>Ck<emph.end type="italics"/>de&longs;cribi intelligatur Circulus &longs;ecans <lb/>lineas <emph type="italics"/>mr, mn<emph.end type="italics"/>productas in <emph type="italics"/>s<emph.end type="italics"/>& <emph type="italics"/>t,<emph.end type="italics"/>& erit rectangulum <emph type="italics"/>mnXmt<emph.end type="italics"/>æ­<lb/>quale rectangulo <emph type="italics"/>mkXms,<emph.end type="italics"/>adeoque <emph type="italics"/>mn<emph.end type="italics"/>æquale (<emph type="italics"/>mkXms/mt<emph.end type="italics"/>). Cum <lb/>autem triangula <emph type="italics"/>pCk, pCn<emph.end type="italics"/>dentur magnitudine, &longs;unt <emph type="italics"/>kr<emph.end type="italics"/>& <emph type="italics"/>mr,<emph.end type="italics"/><lb/>earumQ.E.D.fferentia <emph type="italics"/>mk<emph.end type="italics"/>& &longs;umma <emph type="italics"/>ms<emph.end type="italics"/>reciproce ut altitudo <emph type="italics"/>pC,<emph.end type="italics"/><lb/>adeoque rectangulum <emph type="italics"/>mkXms<emph.end type="italics"/>e&longs;t reciproce ut quadratum altitudi­<lb/>nis <emph type="italics"/>pC.<emph.end type="italics"/>E&longs;t & <emph type="italics"/>mt<emph.end type="italics"/>directe ut 1/2 <emph type="italics"/>mt,<emph.end type="italics"/>id e&longs;t, ut altitudo <emph type="italics"/>pC.<emph.end type="italics"/>Hæ <lb/>&longs;unt primæ rationes linearum na&longs;centium; & hinc fit (<emph type="italics"/>mkXms/mt<emph.end type="italics"/>), id <lb/>e&longs;t lineola na&longs;cens <emph type="italics"/>mn,<emph.end type="italics"/>eique proportionalis Virium differentia reci­<lb/>proce ut cubus altitudinis <emph type="italics"/>pC. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note99"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc differentia virium in locis <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>p<emph.end type="italics"/>vel <emph type="italics"/>K<emph.end type="italics"/>& <emph type="italics"/>k,<emph.end type="italics"/>e&longs;t <lb/>ad vim qua corpus motu Circulari revolvi po&longs;&longs;it ab <emph type="italics"/>R<emph.end type="italics"/>ad <emph type="italics"/>K<emph.end type="italics"/>eodem <lb/>tempore quo corpus <emph type="italics"/>P<emph.end type="italics"/>in Orbe immobili de&longs;cribit arcum <emph type="italics"/>PK,<emph.end type="italics"/>ut <lb/>lineola na&longs;cens <emph type="italics"/>mn<emph.end type="italics"/>ad &longs;inum ver&longs;um arcus na&longs;centis <emph type="italics"/>RK,<emph.end type="italics"/>id e&longs;t <pb xlink:href="039/01/152.jpg" pagenum="124"/><arrow.to.target n="note100"/>ut (<emph type="italics"/>mkXms/mt<emph.end type="italics"/>) ad (<emph type="italics"/>rkq/2kC<emph.end type="italics"/>), vel ut <emph type="italics"/>mkXms<emph.end type="italics"/>ad <emph type="italics"/>rk<emph.end type="italics"/>quadratum; hoc e&longs;t, &longs;i <lb/>capiantur datæ quantitates F, G in ea ratione ad invicem quam <lb/>habet angulus <emph type="italics"/>VCP<emph.end type="italics"/>ad angulum <emph type="italics"/>VCp,<emph.end type="italics"/>ut GG-FF ad FF. </s> <s>Et <lb/>propterea, &longs;i centro <emph type="italics"/>C<emph.end type="italics"/>intervallo quovis <emph type="italics"/>CP<emph.end type="italics"/>vel <emph type="italics"/>Cp<emph.end type="italics"/>de&longs;cribatur <lb/>Sector circularis æqualis areæ toti <emph type="italics"/>VPC,<emph.end type="italics"/>quam corpus <emph type="italics"/>P<emph.end type="italics"/>tempore <lb/>quovis in Orbe immobili revolvens radio ad centrum ducto de­<lb/>&longs;crip &longs;it: differentia virium, quibus corpus <emph type="italics"/>P<emph.end type="italics"/>in Orbe immobili & <lb/>corpus <emph type="italics"/>p<emph.end type="italics"/>in Orbe mobili revolvuntur, erit ad vim centripetam qua <lb/>corpus aliquod radio ad centrum ducto Sectorem illum, eodem tem­<lb/>pore quo de&longs;cripta &longs;it area <emph type="italics"/>VPC<emph.end type="italics"/>uniformiter de&longs;eribere potui&longs;&longs;et, <lb/>ut GG-FF ad FF. </s> <s>Namque Sector ille & area <emph type="italics"/>pCk<emph.end type="italics"/>&longs;unt ad in­<lb/>vicem ut tempora quibus de&longs;cribuntur. </s></p> <p type="margin"> <s><margin.target id="note100"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si Orbis <emph type="italics"/>VPK<emph.end type="italics"/>Ellip&longs;is &longs;it umbilicum habens <emph type="italics"/>C<emph.end type="italics"/>& Ap­<lb/>&longs;idem &longs;ummam <emph type="italics"/>V;<emph.end type="italics"/>eique &longs;imilis & æqualis ponatur Ellip&longs;is <emph type="italics"/>upk,<emph.end type="italics"/><lb/>ita ut &longs;it &longs;emper <emph type="italics"/>pC<emph.end type="italics"/>æqualis <emph type="italics"/>PC,<emph.end type="italics"/>& angulus <emph type="italics"/>VCp<emph.end type="italics"/>&longs;it ad angulum <lb/><emph type="italics"/>VCP<emph.end type="italics"/>in data ratione G ad F; pro altitudine autem <emph type="italics"/>PC<emph.end type="italics"/>vel <emph type="italics"/>pC<emph.end type="italics"/><lb/>&longs;cribatur A, & pro Ellip&longs;eos latere recto ponatur 2 R: erit vis qua <lb/>corpus in Ellip&longs;i mobili revolvi pote&longs;t, ut (FF/AA)+(RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>) <lb/>& contra. </s> <s>Exponatur enim vis qua corpus revolvatur in immota <lb/>Ellip&longs;i per quantitatem (FF/AA), & vis in <emph type="italics"/>V<emph.end type="italics"/>erit (FF/<emph type="italics"/>CV quad.<emph.end type="italics"/>). Vis au­<lb/>tem qua corpus in Circulo ad di&longs;tantiam <emph type="italics"/>CV<emph.end type="italics"/>ea cum velocitate <lb/>revolvi po&longs;&longs;et quam corpus in Ellip&longs;i revolvens habet in <emph type="italics"/>V,<emph.end type="italics"/><lb/>e&longs;t ad vim qua corpus in Ellip&longs;i revolvens urgetur in Ap&longs;ide <emph type="italics"/>V,<emph.end type="italics"/><lb/>ut dimidium lateris recti Ellip&longs;eos. </s> <s>ad Circuli &longs;emidiametrum <emph type="italics"/>CV,<emph.end type="italics"/><lb/>adeoque valet (RFF/<emph type="italics"/>CV cub.<emph.end type="italics"/>): & vis quæ &longs;it ad hanc ut GG-FF ad <lb/>FF, valet (RGG-RFF/<emph type="italics"/>CV cub.<emph.end type="italics"/>): e&longs;tque hæc vis (per hujus Corol. </s> <s>1.) <lb/>differentia virium in <emph type="italics"/>V<emph.end type="italics"/>quibus corpus <emph type="italics"/>P<emph.end type="italics"/>in Ellip&longs;i immota <emph type="italics"/>VPK,<emph.end type="italics"/><lb/>& corpus <emph type="italics"/>p<emph.end type="italics"/>in Ellip&longs;i mobili <emph type="italics"/>upk<emph.end type="italics"/>revolvuntur. </s> <s>Unde cum (per <lb/>hanc Prop.) differentia illa in alia quavis altitudine A &longs;it ad &longs;e­<lb/>ip&longs;am in altitudine <emph type="italics"/>CV<emph.end type="italics"/>ut (1/A <emph type="italics"/>cub.<emph.end type="italics"/>) ad (1/<emph type="italics"/>CV cub.<emph.end type="italics"/>), eadem differentia <lb/>in omni altitudine. </s> <s>A valebit (RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>). Igitur ad vim (FF/AA) <lb/>qua corpus revolvi pote&longs;t in Ellip&longs;i immobili <emph type="italics"/>VPK,<emph.end type="italics"/>addatur ex­<lb/>ce&longs;&longs;us (RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>) & componetur vis tota (FF/AA)+(RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>) <pb xlink:href="039/01/153.jpg" pagenum="125"/>qua corpus in Ellip&longs;i mobili <emph type="italics"/>upk<emph.end type="italics"/>ii&longs;dem temporibus revolvi <lb/><arrow.to.target n="note101"/>po&longs;&longs;it. </s></p> <p type="margin"> <s><margin.target id="note101"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Ad eundem modum colligetur quod, &longs;i Orbis immo­<lb/>bilis <emph type="italics"/>VPK<emph.end type="italics"/>Ellip&longs;is &longs;it centrum habens in virium centro <emph type="italics"/>C<emph.end type="italics"/>; ei­<lb/>que &longs;imilis, æqualis & concentrica ponatur Ellip&longs;is mobilis <emph type="italics"/>upk;<emph.end type="italics"/><lb/>&longs;itque 2 R Ellip&longs;eos hujus latus rectum principale, & 2T latus <lb/>tran&longs;ver&longs;um &longs;ive axis major, atque angulus <emph type="italics"/>VCp<emph.end type="italics"/>&longs;emper &longs;it ad <lb/>angulum <emph type="italics"/>VCP<emph.end type="italics"/>ut G ad F; vires quibus corpora in Ellip&longs;i im­<lb/>mobili & mobili temporibus æqualibus revolvi po&longs;&longs;unt, erunt ut <lb/>(FFA/T <emph type="italics"/>cub.<emph.end type="italics"/>) & (FFA/T <emph type="italics"/>cub.<emph.end type="italics"/>)+(RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>) re&longs;pective. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Et univer&longs;aliter, &longs;i corporis altitudo maxima <emph type="italics"/>CV<emph.end type="italics"/>no­<lb/>minetur T, & radius curvaturæ quam Orbis <emph type="italics"/>VPK<emph.end type="italics"/>habet in <emph type="italics"/>V,<emph.end type="italics"/>id <lb/>e&longs;t radius Circuli æqualiter curvi, nominetur R, & vis centripeta <lb/>qua corpus in Trajectoria quacunQ.E.I.mobili <emph type="italics"/>VPK<emph.end type="italics"/>revolvi po­<lb/>te&longs;t, in loco <emph type="italics"/>V<emph.end type="italics"/>dicatur (VFF/TT), atque aliis in locis <emph type="italics"/>P<emph.end type="italics"/>indefinite dica­<lb/>tur X, altitudine <emph type="italics"/>CP<emph.end type="italics"/>nominata A, & capiatur G ad F in data <lb/>ratione anguli <emph type="italics"/>VCp<emph.end type="italics"/>ad angulum <emph type="italics"/>VCP:<emph.end type="italics"/>erit vis centripeta qua <lb/>corpus idem eo&longs;dem motus in eadem Trajectoria <emph type="italics"/>upk<emph.end type="italics"/>circula­<lb/>riter mota temporibus ii&longs;dem peragere pote&longs;t, ut &longs;umma virium <lb/>X+(VRGG-VRFF/A <emph type="italics"/>cub.<emph.end type="italics"/>). </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Dato igitur motu corporis in Orbe quocunQ.E.I.mo­<lb/>bili, augeri vel minui pote&longs;t ejus motus angularis circa centrum <lb/>virium in ratione data, & inde inveniri novi Orbes immobiles in <lb/>quibus corpora novis viribus centripetis gyrentur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Igitur &longs;i ad rectam <emph type="italics"/>CV<emph.end type="italics"/>po­<lb/><figure id="id.039.01.153.1.jpg" xlink:href="039/01/153/1.jpg"/><lb/>&longs;itione datam erigatur perpendiculum <lb/><emph type="italics"/>VP<emph.end type="italics"/>longitudinis indeterminatæ, jun­<lb/>gaturque <emph type="italics"/>CP,<emph.end type="italics"/>& ip&longs;i æqualis agatur <lb/><emph type="italics"/>Cp,<emph.end type="italics"/>con&longs;tituens angulum <emph type="italics"/>VCp,<emph.end type="italics"/>qui &longs;it <lb/>ad angulum <emph type="italics"/>VCP<emph.end type="italics"/>in data ratione; <lb/>vis qua corpus gyrari pote&longs;t in Curva <lb/>illa <emph type="italics"/>Vpk<emph.end type="italics"/>quam punctum <emph type="italics"/>p<emph.end type="italics"/>perpetuo <lb/>tangit, erit reciproce ut cubus altitu­<lb/>dinis <emph type="italics"/>Cp.<emph.end type="italics"/>Nam corpus <emph type="italics"/>P,<emph.end type="italics"/>per vim inertiæ, nulla alia vi urgente, <lb/>uniformiter progredi pote&longs;t in recta <emph type="italics"/>VP.<emph.end type="italics"/>Addatur vis in centrum <lb/><emph type="italics"/>C,<emph.end type="italics"/>cubo altitudinis <emph type="italics"/>CP<emph.end type="italics"/>vel <emph type="italics"/>Cp<emph.end type="italics"/>reciproce proportionalis, & (per <lb/>jam demon&longs;trata) detorQ.E.I.ur motus ille rectilineus in lineam <pb xlink:href="039/01/154.jpg" pagenum="126"/><arrow.to.target n="note102"/>curvam <emph type="italics"/>Vpk.<emph.end type="italics"/>E&longs;t autem hæc Curva <emph type="italics"/>Vpk<emph.end type="italics"/>eadem cum Curva illa <lb/><emph type="italics"/>VPQ<emph.end type="italics"/>in Corol. </s> <s>3. Prop. </s> <s>XLI inventa, in qua ibi diximus corpora <lb/>huju&longs;modi viribus attracta oblique a&longs;cendere. </s></p> <p type="margin"> <s><margin.target id="note102"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLV. PROBLEMA XXXI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Orbium qui &longs;unt Circulis maxime finitimi requiruntur motus Ap­<lb/>&longs;idum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Problema &longs;olvitur Arithmetice faciendo ut Orbis, quem corpus <lb/>in Ellip&longs;i mobili (ut in Propo&longs;itionis &longs;uperioris Corol. </s> <s>2, vel 3) <lb/>revolvens de&longs;cribit in plano immobili, accedat ad formam Orbis <lb/>cujus Ap&longs;ides requiruatur, & quærendo Ap&longs;ides Orbis quem cor­<lb/>pus illud in plano immobili de&longs;cribit. </s> <s>Orbes autem eandem ac­<lb/>quirent formam, &longs;i vires centripetæ quibus de&longs;cribuntur, inter &longs;e <lb/>collatæ, in æqualibus altitudinibus reddantur proportionales. </s> <s>Sit <lb/>punctum <emph type="italics"/>V<emph.end type="italics"/>Ap&longs;is &longs;umma, & &longs;cribantur T pro altitudine maxima <lb/><emph type="italics"/>CV,<emph.end type="italics"/>A pro altitudine quavis alia <emph type="italics"/>CP<emph.end type="italics"/>vel <emph type="italics"/>Cp,<emph.end type="italics"/>& X pro alti­<lb/>titudinum differentia <emph type="italics"/>CV-CP<emph.end type="italics"/>; & vis qua corpus in Ellip&longs;i <lb/>circa umbilicum &longs;uum <emph type="italics"/>C<emph.end type="italics"/>(ut in Corollario 2.) revolvente move­<lb/>tur, quæQ.E.I. Corollario 2. erat ut (FF/AA)+(RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>), id e&longs;t <lb/>ut (FFA+RGG-RFF/A <emph type="italics"/>cub.<emph.end type="italics"/>), &longs;ub&longs;tituendo T-X pro A, erit ut <lb/>(RGG-RFF+TFF-FFX/A <emph type="italics"/>cub.<emph.end type="italics"/>). Reducenda &longs;imiliter e&longs;t vis alia <lb/>quævis centripeta ad fractionem cujus denominator &longs;it A <emph type="italics"/>cub.,<emph.end type="italics"/>& <lb/>numeratores, facta homologorum terminorum collatione, &longs;tatuendi <lb/>&longs;unt analogi. </s> <s>Res Exemplis patebit. </s></p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/>1. Ponamus vim centripetam uniformem e&longs;&longs;e, adeoque <lb/>ut (A <emph type="italics"/>cub.<emph.end type="italics"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>), &longs;ive (&longs;cribendo T-X pro A in Numeratore) ut <lb/>(T <emph type="italics"/>cub.<emph.end type="italics"/>-3TTX+3TXX-X <emph type="italics"/>cub.<emph.end type="italics"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>); & collatis Numeratorum ter­<lb/>minis corre&longs;pondentibus, nimirum datis cum datis & non datis <lb/>cum non datis, fiet RGG-RFF+TFF ad T <emph type="italics"/>cub.<emph.end type="italics"/>ut-FFX ad <lb/>-3TTX+3TXX-X<emph type="italics"/>cub.<emph.end type="italics"/>&longs;ive ut-FF ad-3TT+3TX <lb/>-XX. </s> <s>Jam cum Orbis ponatur Circulo quam maxime finitimus, <lb/>coeat Orbis cum Circulo; & ob factas R, T æquales, atque X in infi-<pb xlink:href="039/01/155.jpg" pagenum="127"/>nitum diminutam, rationes ultimæ erunt RGG ad T <emph type="italics"/>cub.<emph.end type="italics"/>ut-FF <lb/><arrow.to.target n="note103"/>ad-3TT &longs;eu GG ad TT ut FF ad 3TT & vici&longs;&longs;im GG ad <lb/>FF ut TT ad 3 TT id e&longs;t, ut 1 ad 3; adeoque G ad F, <lb/>hoc e&longs;t angulus <emph type="italics"/>VCp<emph.end type="italics"/>ad angulum <emph type="italics"/>VCP,<emph.end type="italics"/>ut 1 ad √3. Er­<lb/>go cum corpus in Ellip&longs;i immobili, ab Ap&longs;ide &longs;umma ad Ap­<lb/>&longs;idem imam de&longs;cendendo conficiat angulum <emph type="italics"/>VCP<emph.end type="italics"/>(ut ita di­<lb/>cam) gradum 180; corpus aliud in Ellip&longs;i mobili, atque adeo in <lb/>Orbe immobili de quo agimus, ab Ap&longs;ide &longs;umma ad Ap&longs;idem <lb/>imam de&longs;cendendo conficiet angulum <emph type="italics"/>VCp<emph.end type="italics"/>gradum (180/√3): id <lb/>adeo ob &longs;imilitudinem Orbis hujus, quem corpus agente uniformi <lb/>vi centripeta de&longs;cribit, & Orbis illius quem corpus in Ellip&longs;i re­<lb/>volvente gyros peragens de&longs;cribit in plano quie&longs;cente. </s> <s>Per &longs;u­<lb/>periorem terminorum collationem &longs;imiles redduntur hi Orbes, non <lb/>univer&longs;aliter, &longs;ed tunc cum ad formam circularem quam maxime <lb/>appropinquant. </s> <s>Corpus igitur uniformi cum vi centripeta in <lb/>Orbe propemodum circulari revolvens, inter Ap&longs;idem &longs;ummam <lb/>& Ap&longs;idem imam conficiet &longs;emper angulum (180/√3) graduum, &longs;eu <lb/>103 <emph type="italics"/>gr.<emph.end type="italics"/>55 <emph type="italics"/>m.<emph.end type="italics"/>23 <emph type="italics"/>&longs;ec.<emph.end type="italics"/>ad centrum; perveniens ab Ap&longs;ide &longs;umma ad <lb/>Ap&longs;idem imam ubi &longs;emel confecit hunc angulum, & inde ad Ap&longs;i­<lb/>dem &longs;ummam rediens ubi iterum confecit eundem angulum; & <lb/>&longs;ic deinceps in infinitum. </s></p> <p type="margin"> <s><margin.target id="note103"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/>2. Ponamus vim centripetam e&longs;&longs;e ut altitudinis A dig­<lb/>nitas quælibet A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-3<emph.end type="sup"/> &longs;eu (A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>/A<emph type="sup"/>3<emph.end type="sup"/>): ubi <emph type="italics"/>n<emph.end type="italics"/>-3 & <emph type="italics"/>n<emph.end type="italics"/>&longs;ignificant digNI­<lb/>tatum indices quo&longs;cunQ.E.I.tegros vel fractos, rationales vel irratio­<lb/>nales, affirmativos vel negativos. </s> <s>Numerator ille A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> &longs;eu ―T-X<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/><lb/>in &longs;eriem indeterminatam per Methodum no&longs;tram Serierum conver­<lb/>gentium reducta, evadit T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>-<emph type="italics"/>n<emph.end type="italics"/>XT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>+(<emph type="italics"/>nn-n<emph.end type="italics"/>/2)XXT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-2<emph.end type="sup"/> &c. </s> <s><lb/>Et collatis hujus terminis cum terminis Numeratoris alterius <lb/>RGG-RFF+TFF-FFX, fit RGG-RFF+TFF ad T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/><lb/>ut-FF ad-<emph type="italics"/>n<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>+(<emph type="italics"/>nn-n<emph.end type="italics"/>/2)XT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-2<emph.end type="sup"/> &c. </s> <s>Et &longs;umendo ratio­<lb/>nes ultimas ubi Orbes ad formam circularem accedunt, fit RGG <lb/>ad T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> ut-FF ad-<emph type="italics"/>n<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>, &longs;eu GG ad T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/> ut FF ad <emph type="italics"/>n<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>, <lb/>& vici&longs;&longs;im GG ad FF ut T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/> ad <emph type="italics"/>n<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/> id e&longs;t ut 1 ad <emph type="italics"/>n<emph.end type="italics"/>; <lb/>adeoque G ad F, id e&longs;t angulus <emph type="italics"/>VCp<emph.end type="italics"/>ad angulum <emph type="italics"/>VCP,<emph.end type="italics"/><pb xlink:href="039/01/156.jpg" pagenum="128"/><arrow.to.target n="note104"/>ut 1 ad √<emph type="italics"/>n.<emph.end type="italics"/>Quare cum angulus <emph type="italics"/>VCP,<emph.end type="italics"/>in de&longs;cen&longs;u corporis <lb/>ab Ap&longs;ide &longs;umma ad Ap&longs;idem imam in Ellip&longs;i confectus, &longs;it <lb/>graduum 180; conficietur angulus <emph type="italics"/>VCp,<emph.end type="italics"/>in de&longs;cen&longs;u corporis <lb/>ab Ap&longs;ide &longs;umma ad Ap&longs;idem imam, in Orbe propemodum Cir­<lb/>culari quem corpus quodvis vi centripeta dignitati A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-3<emph.end type="sup"/> pro­<lb/>portionali de&longs;cribit, æqualis angulo graduum (180/√<emph type="italics"/>n<emph.end type="italics"/>); & hoc angulo <lb/>repetito corpus redibit ab Ap&longs;ide ima ad Ap&longs;idem &longs;ummam, & <lb/>&longs;ic deinceps in infinitum. </s> <s>Ut &longs;i vis centripeta &longs;it ut di&longs;tantia cor­<lb/>poris a centro, id e&longs;t, ut A &longs;eu (A<emph type="sup"/>4<emph.end type="sup"/>/A<emph type="sup"/>3<emph.end type="sup"/>), erit <emph type="italics"/>n<emph.end type="italics"/>æqualis 4 & √<emph type="italics"/>n<emph.end type="italics"/>æqualis 2; <lb/>adeoque angulus inter Ap&longs;idem &longs;ummam & Ap&longs;idem imam æ­<lb/>qualis (180/2) <emph type="italics"/>gr.<emph.end type="italics"/>&longs;eu 90 <emph type="italics"/>gr.<emph.end type="italics"/>Completa igitur quarta parte revolutio­<lb/>nis unius corpus perveniet ad Ap&longs;idem imam, & completa alia <lb/>quarta parte ad Ap&longs;idem &longs;ummam, & &longs;ic deinceps per vices in <lb/>infinitum. </s> <s>Id quod etiam ex Propo&longs;itione x. </s> <s>manife&longs;tum e&longs;t. </s> <s>Nam <lb/>corpus urgente hac vi centripeta revolvetur in Ellip&longs;i immobili, <lb/>cujus centrum e&longs;t in centro virium. </s> <s>Quod &longs;i vis centripeta &longs;it reci­<lb/>proce ut di&longs;tantia, id e&longs;t directe ut 1/A &longs;eu (A<emph type="sup"/>2<emph.end type="sup"/>/A<emph type="sup"/>3<emph.end type="sup"/>), erit <emph type="italics"/>n<emph.end type="italics"/>æqualis 2, ad­<lb/>eoQ.E.I.ter Ap&longs;idem &longs;ummam & imam angulus erit graduum (180/√2) <lb/>&longs;eu 127 <emph type="italics"/>gr.<emph.end type="italics"/>16 <emph type="italics"/>m.<emph.end type="italics"/>45 <emph type="italics"/>&longs;ec.<emph.end type="italics"/>& propterea corpus tali vi revolvens, perpe­<lb/>tua anguli hujus repetitione, vicibus alternis ab Ap&longs;ide &longs;umma ad <lb/>imam & ab ima ad &longs;ummam perveniet in æternum. </s> <s>Porro &longs;i vis <lb/>centripeta &longs;it reciproce ut latus quadrato-quadratum undecimæ <lb/>dignitatis altitudinis, id e&longs;t reciproce ut A (11/4), adeoQ.E.D.recte ut <lb/>(1/A<emph type="sup"/>11/4<emph.end type="sup"/>) &longs;eu ut (A<emph type="sup"/>1/4<emph.end type="sup"/>/A<emph type="sup"/>3<emph.end type="sup"/>) erit <emph type="italics"/>n<emph.end type="italics"/>æqualis 1/4, & (180/√<emph type="italics"/>n<emph.end type="italics"/>) <emph type="italics"/>gr.<emph.end type="italics"/>æqualis 360 <emph type="italics"/>gr.<emph.end type="italics"/>& prop­<lb/>terea corpus de Ap&longs;ide &longs;umma di&longs;cedens & &longs;ubinde perpetuo de­<lb/>&longs;cendens, perveniet ad Ap&longs;idem imam ubi complevit revolutionem <lb/>integram, dein perpetuo a&longs;cen&longs;u complendo aliam revolutionem in­<lb/>regram, redibit ad Ap&longs;idem &longs;ummam: & &longs;ic per vices in æternum. </s></p> <p type="margin"> <s><margin.target id="note104"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/>3. A&longs;&longs;umentes <emph type="italics"/>m<emph.end type="italics"/>& <emph type="italics"/>n<emph.end type="italics"/>pro quibu&longs;vis indicibus dignitatum <lb/>Altitudinis, & <emph type="italics"/>b, c<emph.end type="italics"/>pro numeris quibu&longs;vis datis, ponamus vim cen­<lb/>tripetam e&longs;&longs;e ut (<emph type="italics"/>b<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>), id e&longs;t, ut (<emph type="italics"/>b<emph.end type="italics"/>in ―T-X<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>in ―T-X<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>) <lb/>&longs;eu (per eandem Methodum no&longs;tram Serierum convergentium) ut <lb/>(<emph type="italics"/>b<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>-<emph type="italics"/>mb<emph.end type="italics"/>XT<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>-<emph type="italics"/>nc<emph.end type="italics"/>XT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>+(<emph type="italics"/>mm-mb<emph.end type="italics"/>/2)XXT<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-2<emph.end type="sup"/>+(<emph type="italics"/>nn-nc<emph.end type="italics"/>/2)XXT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-2<emph.end type="sup"/> <emph type="italics"/>&c.<emph.end type="italics"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>) <pb xlink:href="039/01/157.jpg" pagenum="129"/>& collatis numeratorum terminis, fiet RGG-RFF+TFF <lb/><arrow.to.target n="note105"/>ad <emph type="italics"/>b<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>, ut -FF ad -<emph type="italics"/>mb<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>-<emph type="italics"/>nc<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/><lb/>+(<emph type="italics"/>mm-m<emph.end type="italics"/>/2)<emph type="italics"/>b<emph.end type="italics"/>XT<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-2<emph.end type="sup"/>+(<emph type="italics"/>nn-n<emph.end type="italics"/>/2)<emph type="italics"/>c<emph.end type="italics"/>XT<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-2<emph.end type="sup"/> &c. </s> <s>Et &longs;umendo rationes ulti­<lb/>mas quæ prodeunt ubi Orbes ad formam circularem accedunt, fit <lb/>GG ad <emph type="italics"/>b<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>, ut FF ad <emph type="italics"/>mb<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>+<emph type="italics"/>nc<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>, & <lb/>vici&longs;&longs;im GG ad FF ut <emph type="italics"/>b<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/> ad <emph type="italics"/>mb<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>+<emph type="italics"/>nc<emph.end type="italics"/>T<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>. </s> <s><lb/>Quæ proportio, exponendo altitudinem maximam <emph type="italics"/>CV<emph.end type="italics"/>&longs;eu T Arith­<lb/>metice per Unitatem, fit GG ad FF ut <emph type="italics"/>b+c<emph.end type="italics"/>ad <emph type="italics"/>mb+nc,<emph.end type="italics"/>adeoque ut <lb/>1 ad (<emph type="italics"/>mb+nc/b+c<emph.end type="italics"/>). Unde e&longs;t G ad F, id e&longs;t angulus <emph type="italics"/>VCp<emph.end type="italics"/>ad angulum <lb/><emph type="italics"/>VCP,<emph.end type="italics"/>ut 1 ad √(<emph type="italics"/>mb+nc/b+c<emph.end type="italics"/>). Et propterea cum angulus <emph type="italics"/>VCP<emph.end type="italics"/>inter <lb/>Ap&longs;idem &longs;ummam & Ap&longs;idem imam in Ellip&longs;i immobili &longs;it 180 <emph type="italics"/>gr.<emph.end type="italics"/><lb/>erit angulus <emph type="italics"/>VCp<emph.end type="italics"/>inter ea&longs;dem Ap&longs;ides, in Orbe quem corpus vi <lb/>centripeta quantitati (<emph type="italics"/>b<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>c<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>) proportionali de&longs;cribit, æqua­<lb/>lis angulo graduum 180 √(<emph type="italics"/>b+c/mb+nc<emph.end type="italics"/>). Et eodem argumento &longs;i vis cen­<lb/>tripeta &longs;it ut (<emph type="italics"/>b<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>-<emph type="italics"/>c<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>), angulus inter Ap&longs;ides invenietur graduum <lb/>180 √(<emph type="italics"/>b-c/mb-nc<emph.end type="italics"/>). Nec &longs;ecus re&longs;olvetur Problema in ca&longs;ibus diffi­<lb/>cilioribus. </s> <s>Quantitas cui vis centripeta proportionalis e&longs;t, re­<lb/>&longs;olvi &longs;emper debet in Series convergentes denominatorem ha­<lb/>bentes A <emph type="italics"/>cub.<emph.end type="italics"/>Dein pars data numeratoris qui ex illa operatione <lb/>provenit ad ip&longs;ius partem alteram non datam, & pars data nu­<lb/>meratoris hujus RGG-RFF+TFF-FFX ad ip&longs;ius partem <lb/>alteram non datam in eadem ratione ponendæ &longs;unt: Et quantitates <lb/>&longs;uperfluas delendo, &longs;cribendoque Unitatem pro T, obtinebitur <lb/>proportio G ad F. </s></p> <p type="margin"> <s><margin.target id="note105"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i vis centripeta &longs;it ut aliqua altitudinis digNI­<lb/>tas, inveniri pote&longs;t dignitas illa ex motu Ap&longs;idum; & contra. </s> <s><lb/>Nimirum &longs;i motus totus angularis, quo corpus redit ad Ap&longs;idem <lb/>eandem, &longs;it ad motum angularem revolutionis unius, &longs;eu graduum <lb/>360, ut numerus aliquis <emph type="italics"/>m<emph.end type="italics"/>ad numerum alium <emph type="italics"/>n,<emph.end type="italics"/>& altitudo no­<lb/>minetur A: erit vis ut altitudinis dignitas illa A<emph type="sup"/>(<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3<emph.end type="sup"/>, cujus In-<pb xlink:href="039/01/158.jpg" pagenum="130"/><arrow.to.target n="note106"/>dex e&longs;t (<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3. Id quod per Exempla &longs;ecunda manife&longs;tum e&longs;t. </s> <s><lb/>Unde liquet vim illam in majore quam triplicata altitudinis ratione, <lb/>in rece&longs;&longs;u a centro, decre&longs;cere non po&longs;&longs;e: Corpus tali vi revolvens <lb/>deque Ap&longs;ide di&longs;cedens, &longs;i cæperit de&longs;cendere nunquam perveniet <lb/>ad Ap&longs;idem imam &longs;eu altitudinem minimam, &longs;ed de&longs;cendet u&longs;que ad <lb/>centrum, de&longs;cribens Curvam illam lineam de qua egimus in Cor. </s> <s>3. <lb/>Prop. </s> <s>XLI. </s> <s>Sin cæperit illud, de Ap&longs;ide di&longs;cedens, vel minimum <lb/>a&longs;cendere; a&longs;cendet in infinitum, neque unquam perveniet ad Ap­<lb/>&longs;idem &longs;ummam. </s> <s>De&longs;cribet enim Curvam illam lineam de qua ac­<lb/>tum e&longs;t in eodem Corol. </s> <s>& in Corol. </s> <s>6, Prop. </s> <s>XLIV. </s> <s>Sic & ubi <lb/>vis, in rece&longs;&longs;u a centro, decre&longs;cit in majore quam triplicata ratione <lb/>altitudinis, corpus de Ap&longs;ide di&longs;cedens, perinde ut cæperit de&longs;cen­<lb/>dere vel a&longs;cendere, vel de&longs;cendet ad centrum u&longs;que vel a&longs;cendet <lb/>in infinitum. </s> <s>At &longs;i vis, in rece&longs;&longs;u a centro, vel decre&longs;cat in minore <lb/>quam triplicata ratione altitudinis, vel cre&longs;cat in altitudinis ratione <lb/>quacunque; corpus nunquam de&longs;cendet ad centrum u&longs;que, &longs;ed ad <lb/>Ap&longs;idem imam aliquando perveniet: & contra, &longs;i corpus de Ap&longs;i­<lb/>de ad Ap&longs;idem alternis vicibus de&longs;cendens & a&longs;cendens nunquam <lb/>appellat ad centrum; vis in rece&longs;&longs;u a centro aut augebitur, aut in <lb/>minore quam triplicata altitudinis ratione decre&longs;cet: & quo ci­<lb/>tius corpus de Ap&longs;ide ad Ap&longs;idem redierit, eo longius ratio virium <lb/>recedet a ratione illa triplicata. </s> <s>Ut &longs;i corpus revolutionibus 8 vel <lb/>4 vel 2 vel 1 1/2 de Ap&longs;ide &longs;umma ad Ap&longs;idem &longs;ummam alterno de­<lb/>&longs;cen&longs;u & a&longs;cen&longs;u redierit; hoc e&longs;t, &longs;i fuerit <emph type="italics"/>m<emph.end type="italics"/>ad <emph type="italics"/>n<emph.end type="italics"/>ut 8 vel 4 vel <lb/>2 vel 1 1/2 ad 1, adeoque (<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3 valeat (1/64)-3 vel (1/16) -3 vel 1/4-3 <lb/>vel 4/9-3: erit vis ut A<emph type="sup"/>(1/64)-3<emph.end type="sup"/> vel A<emph type="sup"/>(1/16)-3<emph.end type="sup"/> vel A<emph type="sup"/>1/4-3<emph.end type="sup"/> vel A<emph type="sup"/>4/9-3<emph.end type="sup"/>, <lb/>id e&longs;t, reciproce ut A<emph type="sup"/>3-(1/64)<emph.end type="sup"/> vel A<emph type="sup"/>3-(1/16)<emph.end type="sup"/> vel A<emph type="sup"/>3-1/4<emph.end type="sup"/> vel A<emph type="sup"/>3-4/9<emph.end type="sup"/>. </s> <s><lb/>Si corpus &longs;ingulis revolutionibus redierit ad Ap&longs;idem eandem immo­<lb/>tam; erit <emph type="italics"/>m<emph.end type="italics"/>ad <emph type="italics"/>n<emph.end type="italics"/>ut 1 ad 1, adeoque A (<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3 æqualis A<emph type="sup"/>-2<emph.end type="sup"/> &longs;eu (1/AA<gap/>) <lb/>& propterea decrementum virium in ratione duplicata altitudinis, <lb/>ut in præcedentibus demon&longs;tratum e&longs;t. </s> <s>Si corpus partibus revo­<lb/>lutionis unius vel tribus quartis, vel duabus tertiis, vel una ter­<lb/>tia, vel una quarta, ad Ap&longs;idem eandem redierit; erit <emph type="italics"/>m<emph.end type="italics"/>ad <emph type="italics"/>n<emph.end type="italics"/>ut <lb/>1/4 vel 2/3 vel 1/3 vel 1/4 ad 1, adeoque A(<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3 æqualis A<emph type="sup"/>(16/9)-3<emph.end type="sup"/> vel <lb/>A<emph type="sup"/>9/4-3<emph.end type="sup"/> vel A<emph type="sup"/>9-3<emph.end type="sup"/> vel A<emph type="sup"/>16-3<emph.end type="sup"/>; & propterea vis aut reciproce ut <pb xlink:href="039/01/159.jpg" pagenum="131"/>A<emph type="sup"/>(11/9)<emph.end type="sup"/> vel A<emph type="sup"/>1/4<emph.end type="sup"/>, aut directe ut A<emph type="sup"/>6<emph.end type="sup"/> vel A <emph type="sup"/>13<emph.end type="sup"/>. </s> <s>Denique &longs;i corpus pergendo <lb/><arrow.to.target n="note107"/>ab Ap&longs;ide &longs;umma ad Ap&longs;idem &longs;ummam confecerit revolutionem in­<lb/>tegram, & præterea gradus tres, adeoque Ap&longs;is illa &longs;ingulis corporis <lb/>revolutionibus confecerit in con&longs;equentia gradus tres; erit <emph type="italics"/>m<emph.end type="italics"/>ad <emph type="italics"/>n<emph.end type="italics"/>ut <lb/>363 <emph type="italics"/>gr.<emph.end type="italics"/>ad 360<emph type="italics"/>gr.<emph.end type="italics"/>&longs;ive ut 121 ad 120, adeoque A<emph type="sup"/>(<emph type="italics"/>nn/mm<emph.end type="italics"/>)-3<emph.end type="sup"/> erit æquale <lb/>A<emph type="sup"/>-(29523/14641)<emph.end type="sup"/>; & propterea vis centripeta reciproce ut A <emph type="sup"/>(29523/14641)<emph.end type="sup"/> &longs;eu re­<lb/>ciproce ut A<emph type="sup"/>2 (4/2+3)<emph.end type="sup"/> proxime. </s> <s>Decre&longs;cit igitur vis centripeta in ratio­<lb/>ne paulo majore quam duplicata, &longs;ed quæ vicibus 59 3/4 propius ad <lb/>duplicatam quam ad triplicatam accedit. </s></p> <p type="margin"> <s><margin.target id="note106"/>DE MOTU <lb/>CORPORUM</s></p> <p type="margin"> <s><margin.target id="note107"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Hinc etiam &longs;i corpus, vi centripeta quæ &longs;it reciproce <lb/>ut quadratum altitudinis, revolvatur in Ellip&longs;i umbilicum haben­<lb/>te in centro virium, & huic vi centripetæ addatur vel auferatur <lb/>vis alia quævis extranea; cogno&longs;ci pote&longs;t (per Exempla tertia) <lb/>motus Ap&longs;idum qui ex vi illa extranea orietur: & contra. </s> <s>Ut &longs;i <lb/>vis qua corpus revolvitur in Ellip&longs;i &longs;it ut (1/AA), & vis extranea ab­<lb/>lata ut <emph type="italics"/>c<emph.end type="italics"/>A, adeoque vis reliqua ut (A-<emph type="italics"/>c<emph.end type="italics"/>A<emph type="sup"/>4<emph.end type="sup"/>/A <emph type="italics"/>cub.<emph.end type="italics"/>); erit (in Exemplis ter­<lb/>tiis) <emph type="italics"/>b<emph.end type="italics"/>æqualis 1, <emph type="italics"/>m<emph.end type="italics"/>æqualis 1, <emph type="italics"/>n<emph.end type="italics"/>æqualis 4, adeoque angulus revo­<lb/>lutionis inter Ap&longs;ides æqualis angulo graduum 180 √(1-<emph type="italics"/>c<emph.end type="italics"/>/1-4<emph type="sup"/><emph type="italics"/>c<emph.end type="italics"/><emph.end type="sup"/>). Po­<lb/>natur vim illam extraneam e&longs;&longs;e 357,<emph type="sup"/>45<emph.end type="sup"/> partibus minorem quam vis <lb/>altera qua corpus revolvitur in Ellip&longs;i, id e&longs;t <emph type="italics"/>c<emph.end type="italics"/>e&longs;&longs;e (100/35745), exi&longs;tente A <lb/>vel T æquali 1; & 180 √(1-<emph type="italics"/>c<emph.end type="italics"/>/1-4<emph type="sup"/><emph type="italics"/>c<emph.end type="italics"/><emph.end type="sup"/>) evadet 180 √(35645/35345), &longs;eu 180, 7623, <lb/>id e&longs;t, 180 <emph type="italics"/>gr.<emph.end type="italics"/>45 <emph type="italics"/>m.<emph.end type="italics"/>44 <emph type="italics"/>&longs;.<emph.end type="italics"/>Igitur corpus de Ap&longs;ide &longs;umma di&longs;ce­<lb/>dens, motu angulari 180 <emph type="italics"/>gr.<emph.end type="italics"/>45 <emph type="italics"/>m.<emph.end type="italics"/>44. <emph type="italics"/>&longs;.<emph.end type="italics"/>perveniet ad Ap&longs;idem <lb/>imam, & hoc motu duplicato ad Ap&longs;idem &longs;ummam redibit: adeo­<lb/>que Ap&longs;is &longs;umma &longs;ingulis revolutionibus progrediendo conficiet <lb/>1 <emph type="italics"/>gr.<emph.end type="italics"/>31 <emph type="italics"/>m.<emph.end type="italics"/>28 <emph type="italics"/>&longs;ec.<emph.end type="italics"/></s></p> <p type="main"> <s>Hactenus de Motu corporum in Orbibus quorum plana per <lb/>centrum Virium tran&longs;eunt. </s> <s>Supere&longs;t ut Motus etiam determine­<lb/>mus in planis excentricis. </s> <s>Nam Scriptores qui Motum gravium <lb/>tractant, con&longs;iderare &longs;olent a&longs;cen&longs;us & de&longs;cen&longs;us ponderum, <lb/>tam obliquos in planis quibu&longs;cunQ.E.D.tis, quam perpendicu­<lb/>lares: & pari jure Motus corporum Viribus quibu&longs;cunque cen-<pb xlink:href="039/01/160.jpg" pagenum="132"/><arrow.to.target n="note108"/>tra petentium, & planis excentricis innitentium hic con&longs;iderandus <lb/>venit. </s> <s>Plana autem &longs;upponimus e&longs;&longs;e politi&longs;&longs;ima & ab&longs;olute lubrica <lb/>ne corpora retardent. </s> <s>Quinimo, in his demon&longs;trationibus, vi­<lb/>ce planorum quibus corpora incumbunt quæque tangunt incum­<lb/>bendo, u&longs;urpamus plana his parallela, in quibus centra corpo­<lb/>rum moventur & Orbitas movendo de&longs;cribunt. </s> <s>Et eadem lege <lb/>Motus corporum in &longs;uperficiebus Curvis peractos &longs;ubinde de­<lb/>terminamus. </s></p> <p type="margin"> <s><margin.target id="note108"/>DE MOTU <lb/>CORPORUM</s></p></subchap2><subchap2> <p type="main"> <s><emph type="center"/>SECTIO X.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Motu Corporum in Superficiebus datis, deque Funipendulorum <lb/>Motu reciproco.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLVI. PROBLEMA XXXII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;ita cuju&longs;cunque generis Vi centripeta, datoque tum Virium cen­<lb/>tro tum Plano quocunQ.E.I. quo corpus revolvitur, & conce&longs;­<lb/>&longs;is Figurarum curvilinearum quadraturis: requiritur Motus cor­<lb/>poris de loco dato, data cum Velocitate, &longs;ecundum rectam in <lb/>Plano illo datam egre&longs;&longs;i.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>S<emph.end type="italics"/>centrum Virium, <emph type="italics"/>SC<emph.end type="italics"/>di&longs;tantia minima centri hujus a Plano <lb/>dato, <emph type="italics"/>P<emph.end type="italics"/>corpus de loco <emph type="italics"/>P<emph.end type="italics"/>&longs;ecundum rectam <emph type="italics"/>PZ<emph.end type="italics"/>egrediens, <emph type="italics"/>Q<emph.end type="italics"/><lb/>corpus idem in Trajectoria &longs;ua revolvens, & <emph type="italics"/>PQR<emph.end type="italics"/>Trajectoria <lb/>illa, in Plano dato de&longs;cripta, quam invenire oportet. </s> <s>Jungantur <emph type="italics"/>CQ <lb/>QS,<emph.end type="italics"/>& &longs;i in <emph type="italics"/>QS<emph.end type="italics"/>capiatur <emph type="italics"/>SV<emph.end type="italics"/>proportionalis vi centripetæ qua <lb/>corpus trahitur ver&longs;us centrum <emph type="italics"/>S,<emph.end type="italics"/>& agatur <emph type="italics"/>VT<emph.end type="italics"/>quæ fit parallela <lb/><emph type="italics"/>CQ<emph.end type="italics"/>& occurrat <emph type="italics"/>SC<emph.end type="italics"/>in <emph type="italics"/>T:<emph.end type="italics"/>Vis <emph type="italics"/>SV<emph.end type="italics"/>re&longs;olvetur (per Legum Corol. </s> <s>2.) <lb/>in vires <emph type="italics"/>ST, TV;<emph.end type="italics"/>quarum <emph type="italics"/>ST<emph.end type="italics"/>trahendo corpus &longs;ecundum lineam <lb/>plano perpendicularem, nil mutat motum ejus in hoc plano. </s> <s>Vis <lb/>autem altera <emph type="italics"/>TV,<emph.end type="italics"/>agendo &longs;ecundum po&longs;itionem plani, trahit cor­<lb/>pus directe ver&longs;us punctum <emph type="italics"/>C<emph.end type="italics"/>in plano datum, adeoque facit illud <lb/>in hoc plano perinde moveri ac &longs;i vis <emph type="italics"/>ST<emph.end type="italics"/>tolleretur, & corpus vi <lb/>&longs;ola <emph type="italics"/>TV<emph.end type="italics"/>revolveretur circa centrum <emph type="italics"/>C<emph.end type="italics"/>in &longs;patio libero. </s> <s>Data autem <pb xlink:href="039/01/161.jpg" pagenum="133"/>vi centripeta <emph type="italics"/>TV<emph.end type="italics"/>qua corpus <emph type="italics"/>Q<emph.end type="italics"/>in &longs;patio libero circa centrum <lb/><arrow.to.target n="note109"/>datum <emph type="italics"/>C<emph.end type="italics"/>revolvitur, datur per Prop. </s> <s>XLII, tum Trajectoria <emph type="italics"/>PQR<emph.end type="italics"/><lb/>quam corpus de&longs;cribit, tum locus <emph type="italics"/>Q<emph.end type="italics"/>in quo corpus ad datum quod­<lb/>vis tempus ver&longs;abitur, tum denique velocitas corporis in loco illo <lb/><emph type="italics"/>Q<emph.end type="italics"/>; & contra. <emph type="italics"/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note109"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLVII. THEOREMA XV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod Vis centripeta proportionalis &longs;it di&longs;tantiæ corporis a <lb/>centro; corpora omnia in planis quibu&longs;cunque revolventia de­<lb/>&longs;cribent Ellip&longs;es, & revolutiones Temporibus æqualibus peragent; <lb/>quæque moventur in lineis rectis, ultro citroQ.E.D.&longs;currendo, <lb/>&longs;ingulas eundi & redeundi periodos ii&longs;dem Temporibus ab&longs;ol­<lb/>vent.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam, &longs;tantibus quæ <lb/><figure id="id.039.01.161.1.jpg" xlink:href="039/01/161/1.jpg"/><lb/>in &longs;uperiore Propo&longs;itio­<lb/>ne, vis <emph type="italics"/>SV<emph.end type="italics"/>qua corpus <lb/><emph type="italics"/>Q<emph.end type="italics"/>in plano quovis <emph type="italics"/>PQR<emph.end type="italics"/><lb/>revolvens trahitur ver­<lb/>&longs;us centrum <emph type="italics"/>S<emph.end type="italics"/>e&longs;t ut di­<lb/>&longs;tantia <emph type="italics"/><expan abbr="Sq;">Sque</expan><emph.end type="italics"/>atque adeo <lb/>ob proportionales <emph type="italics"/>SV<emph.end type="italics"/><lb/>& <emph type="italics"/>SQ, TV<emph.end type="italics"/>& <emph type="italics"/>CQ,<emph.end type="italics"/>vis <lb/><emph type="italics"/>TV<emph.end type="italics"/>qua corpus trahi­<lb/>tur ver&longs;us punctum <emph type="italics"/>C<emph.end type="italics"/><lb/>in Orbis plano datum, <lb/>e&longs;t ut di&longs;tantia <emph type="italics"/>C Q.<emph.end type="italics"/>Vi­<lb/>res igitur, quibus cor­<lb/>pora in plano <emph type="italics"/>PQR<emph.end type="italics"/><lb/>ver&longs;antia trahuntur ver­<lb/>&longs;us punctum <emph type="italics"/>C,<emph.end type="italics"/>&longs;unt pro <lb/>ratione di&longs;tantiarum æquales viribus quibus corpora undiquaque <lb/>trahuntur ver&longs;us centrum <emph type="italics"/>S<emph.end type="italics"/>; & propterea corpora movebuntur ii&longs;­<lb/>dem Temporibus, in ii&longs;dem Figuris, in plano quovis <emph type="italics"/>PQR<emph.end type="italics"/>circa <lb/>punctum <emph type="italics"/>C,<emph.end type="italics"/>atQ.E.I. &longs;patiis liberis circa centrum <emph type="italics"/>S<emph.end type="italics"/>; adeoque (per <lb/>Corol. </s> <s>2. Prop. </s> <s>X, & Corol. </s> <s>2. Prop. </s> <s>XXXVIII) Temporibus &longs;emper <pb xlink:href="039/01/162.jpg" pagenum="134"/><arrow.to.target n="note110"/>æqualibus, vel de&longs;cribent Ellip&longs;es in plano illo circa centrum <emph type="italics"/>C,<emph.end type="italics"/><lb/>vel periodos movendi ultro citroQ.E.I. lineis rectis per centrum <emph type="italics"/>C<emph.end type="italics"/><lb/>in plano illo ductis, complebunt. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note110"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>His affines &longs;unt a&longs;cen&longs;us ac de&longs;cen&longs;us corporum in &longs;uperficiebus <lb/>curvis. </s> <s>Concipe lineas curvas in plano de&longs;cribi, dein circa axes <lb/>quo&longs;vis datos per centrum Virium tran&longs;euntes revolvi, & ea revo­<lb/>lutione &longs;uperficies curvas de&longs;cribere; tum corpora ita moveri ut <lb/>eorum centra in his &longs;uperficiebus perpetuo reperiantur. </s> <s>Si cor­<lb/>pora illa oblique a&longs;cendendo & de&longs;cendendo currant ultro citroque <lb/>peragentur eorum motus in planis per axem tran&longs;euntibus, atque <lb/>adeo in lineis curvis quarum revolutione curvæ illæ &longs;uperficies ge­<lb/>nitæ &longs;unt. </s> <s>I&longs;tis igitur in ca&longs;ibus &longs;ufficit motum in his lineis cur­<lb/>vis con&longs;iderare. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLVIII. THEOREMA XVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Rota Globo extrin&longs;ecus ad angulos rectos in&longs;i&longs;tat, & more ro­<lb/>tarum revolvendo progrediatur in circulo maximo; longitudo <lb/>Itineris curvilinei, quod punctum quodvis in Rotæ perimetro da­<lb/>tum, ex quo Globum tetigit, confecit, (quodque Cycloidem vel <lb/>Epicycloidem nominare licet) erit ad duplicatum &longs;inum ver&longs;um <lb/>arcus dimidii qui Globum ex eo tempore inter eundum tetigit, <lb/>ut &longs;umma diametrorum Globi & Rotæ ad &longs;emidiametrum Globi.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLIX. THEOREMA XVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Rota Globo concavo ad rectos angulos intrin&longs;ecus in&longs;i&longs;tat & re­<lb/>volvendo progrediatur in circulo maximo; longitudo Itineris <lb/>curvilinei quod punctum quodvis in Rotæ perimetro datum, ex <lb/>quo Globum tetigit, confecit, erit ad duplicatum &longs;inum ver&longs;um <lb/>arcus dimidii qui Globum toto hoc tempore inter eundum teti­<lb/>git, ut differentia diametrorum Globi & Rotæ ad &longs;emidiame­<lb/>trum Globi.<emph.end type="italics"/></s></p><pb xlink:href="039/01/163.jpg" pagenum="135"/> <p type="main"> <s>Sit <emph type="italics"/>ABL<emph.end type="italics"/>Globus, <emph type="italics"/>C<emph.end type="italics"/>centrum ejus, <emph type="italics"/>BPV<emph.end type="italics"/>Rota ei in&longs;i&longs;tens, <emph type="italics"/>E<emph.end type="italics"/><lb/><arrow.to.target n="note111"/>centrum Rotæ, <emph type="italics"/>B<emph.end type="italics"/>punctum contactus, & <emph type="italics"/>P<emph.end type="italics"/>punctum datum in pe­<lb/>rimetro Rotæ. </s> <s>Concipe hanc Rotam pergere in circulo maximo <lb/><emph type="italics"/>ABL<emph.end type="italics"/>ab <emph type="italics"/>A<emph.end type="italics"/>per <emph type="italics"/>B<emph.end type="italics"/>ver&longs;us <emph type="italics"/>L,<emph.end type="italics"/>& inter eundum ita revolvi ut ar­<lb/>cus <emph type="italics"/>AB, PB<emph.end type="italics"/>&longs;ibi invicem &longs;emper æquentur, atque punctum illud <lb/><emph type="italics"/>P<emph.end type="italics"/>in perimetro Rotæ datum interea de&longs;cribere Viam curvilineam <lb/><emph type="italics"/>AP.<emph.end type="italics"/>Sit autem <emph type="italics"/>AP<emph.end type="italics"/>Via tota curvilinea de&longs;cripta ex quo Rota <lb/>Globum tetigit in <emph type="italics"/>A,<emph.end type="italics"/>& erit Viæ hujus longitudo <emph type="italics"/>AP<emph.end type="italics"/>ad duplum <lb/><figure id="id.039.01.163.1.jpg" xlink:href="039/01/163/1.jpg"/><lb/>&longs;inum ver&longs;um arcus 1/2 <emph type="italics"/>PB,<emph.end type="italics"/>ut 2 <emph type="italics"/>CE<emph.end type="italics"/>ad <emph type="italics"/>CB.<emph.end type="italics"/>Nam recta <emph type="italics"/>CE<emph.end type="italics"/>(&longs;i <lb/>opus e&longs;t producta) occurrat Rotæ in <emph type="italics"/>V,<emph.end type="italics"/>junganturque <emph type="italics"/>CP, BP, <lb/>EP, VP,<emph.end type="italics"/>& in <emph type="italics"/>CP<emph.end type="italics"/>productam demittatur normalis <emph type="italics"/>VF.<emph.end type="italics"/>Tan­<lb/>gant <emph type="italics"/>PH, VH<emph.end type="italics"/>Circulum in <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>V<emph.end type="italics"/>concurrentes in <emph type="italics"/>H,<emph.end type="italics"/>&longs;ecetque <lb/><emph type="italics"/>PH<emph.end type="italics"/>ip&longs;am <emph type="italics"/>VF<emph.end type="italics"/>in <emph type="italics"/>G,<emph.end type="italics"/>& ad <emph type="italics"/>VP<emph.end type="italics"/>demittantur normales <emph type="italics"/>GI, HK.<emph.end type="italics"/><pb xlink:href="039/01/164.jpg" pagenum="136"/><arrow.to.target n="note112"/>Centro item <emph type="italics"/>C<emph.end type="italics"/>& intervallo quovis de&longs;cribatur circulus <emph type="italics"/>nom<emph.end type="italics"/>&longs;e­<lb/>cans rectam <emph type="italics"/>CP<emph.end type="italics"/>in <emph type="italics"/>n,<emph.end type="italics"/>Rotæ perimetrum <emph type="italics"/>BP<emph.end type="italics"/>&c. </s> <s>in <emph type="italics"/>o,<emph.end type="italics"/>& Viam curvi­<lb/>lineam <emph type="italics"/>AP<emph.end type="italics"/>in <emph type="italics"/>m;<emph.end type="italics"/>centroque <emph type="italics"/>V<emph.end type="italics"/>& intervallo <emph type="italics"/>Vo<emph.end type="italics"/>de&longs;cribatur circu­<lb/>lus &longs;ecans <emph type="italics"/>VP<emph.end type="italics"/>productam in <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note111"/>LIBER <lb/>PRIMUS.</s></p> <p type="margin"> <s><margin.target id="note112"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Quoniam Rota eundo &longs;emper revolvitur circa punctum con­<lb/>tactus <emph type="italics"/>B,<emph.end type="italics"/>manife&longs;tum e&longs;t quod recta <emph type="italics"/>BP<emph.end type="italics"/>perpendicularis e&longs;t ad <lb/><figure id="id.039.01.164.1.jpg" xlink:href="039/01/164/1.jpg"/><lb/>lineam illam curvam <emph type="italics"/>AP<emph.end type="italics"/>quam Rotæ punctum <emph type="italics"/>P<emph.end type="italics"/>de&longs;cribit, atque <lb/>adeo quod recta <emph type="italics"/>VP<emph.end type="italics"/>tanget hanc curvam in puncto <emph type="italics"/>P.<emph.end type="italics"/>Circuli <lb/><emph type="italics"/>nom<emph.end type="italics"/>radius &longs;en&longs;im auctus vel diminutus æquetur tandem di&longs;tantiæ <lb/><emph type="italics"/>CP<emph.end type="italics"/>; &, ob &longs;imilitudinem Figuræ evane&longs;centis <emph type="italics"/>Pnomq<emph.end type="italics"/>& Figuræ <lb/><emph type="italics"/>PFGVI,<emph.end type="italics"/>ratio ultima lineolarum evane&longs;centium <emph type="italics"/>Pm, Pn, Po, Pq,<emph.end type="italics"/><pb xlink:href="039/01/165.jpg" pagenum="137"/>id e&longs;t, ratio mutationum momentanearum curvæ <emph type="italics"/>AP,<emph.end type="italics"/>rectæ <lb/><arrow.to.target n="note113"/><emph type="italics"/>CP,<emph.end type="italics"/>arcus circularis <emph type="italics"/>BP,<emph.end type="italics"/>ac rectæ <emph type="italics"/>VP,<emph.end type="italics"/>eadem erit quæ linea­<lb/>rum <emph type="italics"/>PV, PF, PG, PI<emph.end type="italics"/>re&longs;pective. </s> <s>Cum autem <emph type="italics"/>VF<emph.end type="italics"/>ad <emph type="italics"/>CF<emph.end type="italics"/>& <lb/><emph type="italics"/>VH<emph.end type="italics"/>ad <emph type="italics"/>CV<emph.end type="italics"/>perpendiculares &longs;unt, angulique <emph type="italics"/>HVG, VCF<emph.end type="italics"/>prop­<lb/>terea æquales; & angulus <emph type="italics"/>VHG<emph.end type="italics"/>(ob angulos quadrilateri <emph type="italics"/>HVEP<emph.end type="italics"/><lb/>ad <emph type="italics"/>V<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>rectos) angulo <emph type="italics"/>CEP<emph.end type="italics"/>æqualis e&longs;t, &longs;imilia erunt tri­<lb/>angula <emph type="italics"/>VHG, CEP<emph.end type="italics"/>; & inde fiet ut <emph type="italics"/>EP<emph.end type="italics"/>ad <emph type="italics"/>CE<emph.end type="italics"/>ita <emph type="italics"/>HG<emph.end type="italics"/>ad <emph type="italics"/>HV<emph.end type="italics"/><lb/>&longs;eu <emph type="italics"/>HP<emph.end type="italics"/>& ita <emph type="italics"/>KI<emph.end type="italics"/>ad <emph type="italics"/>KP,<emph.end type="italics"/>& compo&longs;ite vel divi&longs;im ut <emph type="italics"/>CB<emph.end type="italics"/>ad <lb/><emph type="italics"/>CE<emph.end type="italics"/>ita <emph type="italics"/>PI<emph.end type="italics"/>ad <emph type="italics"/>PK,<emph.end type="italics"/>& duplicatis con&longs;equentibus ut <emph type="italics"/>CB<emph.end type="italics"/>ad 2 <emph type="italics"/>CE<emph.end type="italics"/><lb/>ita <emph type="italics"/>PI<emph.end type="italics"/>ad <emph type="italics"/>PV,<emph.end type="italics"/>atQ.E.I.a adeo <emph type="italics"/>Pq<emph.end type="italics"/>ad <emph type="italics"/>Pm.<emph.end type="italics"/>E&longs;t igitur decremen­<lb/>tum lineæ <emph type="italics"/>VP,<emph.end type="italics"/>id e&longs;t, incrementum lineæ <emph type="italics"/>BV-VP<emph.end type="italics"/>ad incremen­<lb/>tum lineæ curvæ <emph type="italics"/>AP<emph.end type="italics"/>in data ratione <emph type="italics"/>CB<emph.end type="italics"/>ad 2 <emph type="italics"/>CE,<emph.end type="italics"/>& prop­<lb/>terea (per Corol. </s> <s>Lem. </s> <s>IV.) longitudines <emph type="italics"/>BV-VP<emph.end type="italics"/>& <emph type="italics"/>AP,<emph.end type="italics"/>in­<lb/>crementis illis genitæ, &longs;unt in eadem ratione. </s> <s>Sed, exi&longs;tente <emph type="italics"/>BV<emph.end type="italics"/>ra­<lb/>dio, e&longs;t <emph type="italics"/>VP<emph.end type="italics"/>co-&longs;inus anguli <emph type="italics"/>BVP<emph.end type="italics"/>&longs;eu 1/2 <emph type="italics"/>BEP,<emph.end type="italics"/>adeoque <emph type="italics"/>BV-VP<emph.end type="italics"/><lb/>&longs;inus ver&longs;us eju&longs;dem anguli; & propterea in hac Rota, cujus radius <lb/>e&longs;t 1/2 <emph type="italics"/>BV,<emph.end type="italics"/>erit <emph type="italics"/>BV-VP<emph.end type="italics"/>duplus &longs;inus ver&longs;us arcus 1/2 <emph type="italics"/>BP.<emph.end type="italics"/>Ergo <lb/><emph type="italics"/>AP<emph.end type="italics"/>e&longs;t ad duplum &longs;inum ver&longs;um arcus 1/2 <emph type="italics"/>BP<emph.end type="italics"/>ut 2 <emph type="italics"/>CE<emph.end type="italics"/>ad <emph type="italics"/>CB. <lb/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note113"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>Lineam autem <emph type="italics"/>AP<emph.end type="italics"/>in Propo&longs;itione priore Cycloidem extra <lb/>Globum, alteram in po&longs;teriore Cycloidem intra Globum di&longs;tincti­<lb/>onis gratia nominabimus. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i de&longs;cribatur Cyclois integra <emph type="italics"/>ASL<emph.end type="italics"/>& bi&longs;ecetur <lb/>ea in <emph type="italics"/>S,<emph.end type="italics"/>erit longitudo partis <emph type="italics"/>PS<emph.end type="italics"/>ad longitudinem <emph type="italics"/>VP<emph.end type="italics"/>(quæ du­<lb/>plus e&longs;t &longs;inus anguli <emph type="italics"/>VBP,<emph.end type="italics"/>exi&longs;tente <emph type="italics"/>EB<emph.end type="italics"/>radio) ut 2 <emph type="italics"/>CE<emph.end type="italics"/>ad <emph type="italics"/>CB,<emph.end type="italics"/><lb/>atque adeo in ratione data. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et longitudo &longs;emiperimetri Cycloidis <emph type="italics"/>AS<emph.end type="italics"/>æquabitur <lb/>lineæ rectæ quæ e&longs;t ad Rotæ diametrum <emph type="italics"/>BV,<emph.end type="italics"/>ut 2 <emph type="italics"/>CE<emph.end type="italics"/>ad <emph type="italics"/>CB.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO L. PROBLEMA XXXIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Facere ut Corpus pendulum o&longs;cilletur in Cycloide data.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Intra Globum <emph type="italics"/>QVS,<emph.end type="italics"/>centro <emph type="italics"/>C<emph.end type="italics"/>de&longs;criptum, detur Cyclois <emph type="italics"/>QRS<emph.end type="italics"/><lb/>bi&longs;ecta in <emph type="italics"/>R<emph.end type="italics"/>& punctis &longs;uis extremis <emph type="italics"/>Q<emph.end type="italics"/>& <emph type="italics"/>S<emph.end type="italics"/>&longs;uperficiei Globi hinc <lb/>inde occurrens. </s> <s>Agatur <emph type="italics"/>CR<emph.end type="italics"/>bi&longs;ecans arcum <emph type="italics"/>QS<emph.end type="italics"/>in <emph type="italics"/>O,<emph.end type="italics"/>& produca­<lb/>tur ea ad <emph type="italics"/>A,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>CA<emph.end type="italics"/>ad <emph type="italics"/>CO<emph.end type="italics"/>ut <emph type="italics"/>CO<emph.end type="italics"/>ad <emph type="italics"/>CR.<emph.end type="italics"/>Centro <emph type="italics"/>C<emph.end type="italics"/>in-<pb xlink:href="039/01/166.jpg" pagenum="138"/><arrow.to.target n="note114"/>tervallo <emph type="italics"/>CA<emph.end type="italics"/>de&longs;eribatur Globus exterior <emph type="italics"/>ABD,<emph.end type="italics"/>& intra hunc Glo­<lb/>bum a Rota, cujus diameter &longs;it <emph type="italics"/>AO,<emph.end type="italics"/>de&longs;cribantur duæ Semicycloides <lb/><emph type="italics"/>AQ, AS,<emph.end type="italics"/>quæ Globum interiorem tangant in <emph type="italics"/>Q<emph.end type="italics"/>& <emph type="italics"/>S<emph.end type="italics"/>& Globo ex­<lb/>teriori occurrant in <emph type="italics"/>A.<emph.end type="italics"/>A puncto illo <emph type="italics"/>A,<emph.end type="italics"/>Filo <emph type="italics"/>APT<emph.end type="italics"/>longitudinem <lb/><emph type="italics"/>AR<emph.end type="italics"/>æquante, pendeat corpus <emph type="italics"/>T,<emph.end type="italics"/>& ita intra Semicycloides <emph type="italics"/>AQ, <lb/>AS<emph.end type="italics"/>o&longs;cilletur, ut quoties pendulum digreditur a perpendiculo <emph type="italics"/>AR,<emph.end type="italics"/><lb/><figure id="id.039.01.166.1.jpg" xlink:href="039/01/166/1.jpg"/><lb/>Filum parte &longs;ui &longs;uperiore <emph type="italics"/>AP<emph.end type="italics"/>applicetur ad Semicycloidem illam <lb/><emph type="italics"/>APS<emph.end type="italics"/>ver&longs;us quam peragitur motus, & circum eam ceu ob&longs;tacu­<lb/>lum flectatur, parteque reliqua <emph type="italics"/>PT<emph.end type="italics"/>cui Semicyclois nondum obji­<lb/>citur, protendatur in lineam rectam; & pondus <emph type="italics"/>T<emph.end type="italics"/>o&longs;cillabitur in <lb/>Cycloide data <emph type="italics"/>QRS. <expan abbr="q.">que</expan> E. F.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note114"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Occurrat enim Filum <emph type="italics"/>PT<emph.end type="italics"/>tum Cycloidi <emph type="italics"/>QRS<emph.end type="italics"/>in <emph type="italics"/>T,<emph.end type="italics"/>tum circulo <lb/><emph type="italics"/>QOS<emph.end type="italics"/>in <emph type="italics"/>V,<emph.end type="italics"/>agaturque <emph type="italics"/>CV;<emph.end type="italics"/>& ad Fili partem rectam <emph type="italics"/>PT,<emph.end type="italics"/>e punctis <lb/>extremis <emph type="italics"/>P<emph.end type="italics"/>ac <emph type="italics"/>T,<emph.end type="italics"/>erigantur perpendicula <emph type="italics"/>PB, TW,<emph.end type="italics"/>occurrentia re­<lb/>ctæ <emph type="italics"/>CV<emph.end type="italics"/>in <emph type="italics"/>B<emph.end type="italics"/>& <emph type="italics"/>W.<emph.end type="italics"/>Patet, ex con&longs;tructione & gene&longs;i &longs;imilium Fi­<lb/>gurarum <emph type="italics"/>AS, SR,<emph.end type="italics"/>perpendicula illa <emph type="italics"/>PB, TW<emph.end type="italics"/>ab&longs;cindere de <emph type="italics"/>CV<emph.end type="italics"/>lon­<lb/>gitudines <emph type="italics"/>VB, VW<emph.end type="italics"/>Rotarum diametris <emph type="italics"/>OA, OR<emph.end type="italics"/>æquales. </s> <s>E&longs;t igi­<lb/>tur <emph type="italics"/>TP<emph.end type="italics"/>ad <emph type="italics"/>VP<emph.end type="italics"/>(duplum &longs;inum anguli <emph type="italics"/>VBP<emph.end type="italics"/>exi&longs;tente 1/2 <emph type="italics"/>BV<emph.end type="italics"/>ra-<pb xlink:href="039/01/167.jpg" pagenum="139"/>dio) ut <emph type="italics"/>BW<emph.end type="italics"/>ad <emph type="italics"/>BV,<emph.end type="italics"/>&longs;eu <emph type="italics"/>AO+OR<emph.end type="italics"/>ad <emph type="italics"/>AO,<emph.end type="italics"/>id e&longs;t (cum &longs;int <emph type="italics"/>CA<emph.end type="italics"/><lb/><arrow.to.target n="note115"/>ad <emph type="italics"/>CO, CO<emph.end type="italics"/>ad <emph type="italics"/>CR<emph.end type="italics"/>& divi&longs;im <emph type="italics"/>AO<emph.end type="italics"/>ad <emph type="italics"/>OR<emph.end type="italics"/>proportionales,) ut <lb/><emph type="italics"/>CA+CO<emph.end type="italics"/>ad <emph type="italics"/>CA<emph.end type="italics"/>vel, &longs;i bi&longs;ecetur <emph type="italics"/>BV<emph.end type="italics"/>in <emph type="italics"/>E,<emph.end type="italics"/>ut 2 <emph type="italics"/>CE<emph.end type="italics"/>ad <emph type="italics"/>CB.<emph.end type="italics"/><lb/>Proinde, per Corol. </s> <s>1. Prop. </s> <s>XLIX, longitudo partis rectæ Fili <emph type="italics"/>PT<emph.end type="italics"/><lb/>æquatur &longs;emper Cycloidis arcui <emph type="italics"/>PS,<emph.end type="italics"/>& Filum totum <emph type="italics"/>APT<emph.end type="italics"/>æquatur <lb/>&longs;emper Cycloidis arcui dimidio <emph type="italics"/>APS,<emph.end type="italics"/>hoc e&longs;t (per Corol. </s> <s>2. Prop. </s> <s><lb/>XLIX) longitudini <emph type="italics"/>AR.<emph.end type="italics"/>Et propterea vici&longs;&longs;im &longs;i Filum manet &longs;em­<lb/>per æquale longitudini <emph type="italics"/>AR<emph.end type="italics"/>movebitur punctum <emph type="italics"/>T<emph.end type="italics"/>in Cycloide <lb/>data <emph type="italics"/>QRS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note115"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Filum <emph type="italics"/>AR<emph.end type="italics"/>æquatur Semicycloidi <emph type="italics"/>AS,<emph.end type="italics"/>adeoque ad &longs;emi­<lb/>diametrum <emph type="italics"/>AC<emph.end type="italics"/>eandem habet rationem quam &longs;imilis illi Semicy­<lb/>clois <emph type="italics"/>SR<emph.end type="italics"/>habet ad &longs;emidiametrum <emph type="italics"/>CO.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LI. THEOREMA XVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Vis centripeta tendens undique ad Globi centrum<emph.end type="italics"/>C <emph type="italics"/>&longs;it in locis <lb/>&longs;ingulis ut di&longs;tantia loci cuju&longs;que a centro, & hac &longs;ola Vi a­<lb/>gente corpus<emph.end type="italics"/>T <emph type="italics"/>o&longs;cilletur (modo jam de&longs;cripto) in perimetro Cy­<lb/>cloidis<emph.end type="italics"/>QRS: <emph type="italics"/>dico quod o&longs;cillationum utcunQ.E.I.æqualium <lb/>æqualia erunt Tempora.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam in Cycloidis tangentem <emph type="italics"/>TW<emph.end type="italics"/>infinite productam cadat per­<lb/>pendiculum <emph type="italics"/>CX<emph.end type="italics"/>& jungatur <emph type="italics"/>CT.<emph.end type="italics"/>Quoniam vis centripeta qua cor­<lb/>pus <emph type="italics"/>T<emph.end type="italics"/>impellitur ver&longs;us <emph type="italics"/>C<emph.end type="italics"/>e&longs;t ut di&longs;tantia <emph type="italics"/>CT,<emph.end type="italics"/>atque hæc (per Legum <lb/>Corol. </s> <s>2.) re&longs;olvitur in partes <emph type="italics"/>CX, TX,<emph.end type="italics"/>quarum <emph type="italics"/>CX<emph.end type="italics"/>impellen­<lb/>do corpus directe a <emph type="italics"/>P<emph.end type="italics"/>di&longs;tendit filum <emph type="italics"/>PT<emph.end type="italics"/>& per ejus re&longs;i&longs;tentiam <lb/>tota ce&longs;&longs;at, nullum alium edens effectum; pars autem altera <emph type="italics"/>TX,<emph.end type="italics"/><lb/>urgendo corpus tran&longs;ver&longs;im &longs;eu ver&longs;us <emph type="italics"/>X,<emph.end type="italics"/>directe accelerat motum <lb/>ejus in Cycloide; manife&longs;tum e&longs;t quod corporis acceleratio, huic <lb/>vi acceleratrici proportionalis, &longs;it &longs;ingulis momentis ut longitudo <lb/><emph type="italics"/>TX,<emph.end type="italics"/>id e&longs;t, (ob datas <emph type="italics"/>CV, WV<emph.end type="italics"/>ii&longs;que proportionales <emph type="italics"/>TX, TW,<emph.end type="italics"/>) <lb/>ut longitudo <emph type="italics"/>TW,<emph.end type="italics"/>hoc e&longs;t (per Corol. </s> <s>1. Prop. </s> <s>XLIX,) ut longitudo <lb/>arcus Cycloidis <emph type="italics"/>TR.<emph.end type="italics"/>Pendulis igitur duobus <emph type="italics"/>APT, Apt<emph.end type="italics"/>de per­<lb/>pendiculo <emph type="italics"/>AR<emph.end type="italics"/>inæqualiter deductis & &longs;imul dimi&longs;&longs;is, acceleratio­<lb/>nes eorum &longs;emper erunt ut arcus de&longs;cribendi <emph type="italics"/>TR, tR.<emph.end type="italics"/>Sunt au­<lb/>tem partes &longs;ub initio de&longs;criptæ ut accelerationes, hoc e&longs;t, ut totæ <lb/>&longs;ub initio de&longs;cribendæ, & propterea partes quæ manent de&longs;criben-<pb xlink:href="039/01/168.jpg" pagenum="140"/><arrow.to.target n="note116"/>dæ & accelerationes &longs;ub&longs;equentes, his partibus proportionales, &longs;unt <lb/>etiam ut totæ; & &longs;ic deinceps. </s> <s>Sunt igitur accelerationes atque <lb/>adeo velocitates genitæ & partes his velocitatibus de&longs;criptæ par­<lb/>te&longs;Q.E.D.&longs;cribendæ, &longs;emper ut totæ; & propterea partes de&longs;criben­<lb/>dæ datam &longs;ervantes rationem ad invicem &longs;imul evane&longs;cent, id e&longs;t, <lb/>corpora duo o&longs;cillantia &longs;imul pervenient ad perpendiculum <emph type="italics"/>AR.<emph.end type="italics"/><lb/>Cumque vici&longs;&longs;im a&longs;cen&longs;us perpendiculorum de loco in&longs;imo <emph type="italics"/>R,<emph.end type="italics"/>per <lb/>eo&longs;dem arcus Cycloidales motu retrogrado facti, retardentur in <lb/>locis &longs;ingulis a viribus ii&longs;dem a quibus de&longs;cen&longs;us accelerabantur, <lb/>patet velocitates a&longs;cen&longs;uum ac de&longs;cen&longs;uum per eo&longs;dem arcus fa­<lb/>ctorum æquales e&longs;&longs;e, atque adeo temporibus æqualibus fieri; & <lb/>propterea, cum Cycloidis partes duæ <emph type="italics"/>RS<emph.end type="italics"/>& <emph type="italics"/>RQ<emph.end type="italics"/>ad utrumque per­<lb/>pendiculi latus jacentes &longs;int &longs;imiles & æquales, pendula duo o&longs;cil­<lb/>lationes &longs;uas tam totas quam dimidias ii&longs;dem temporibus &longs;emper <lb/>peragent. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note116"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Vis qua corpus <emph type="italics"/>T<emph.end type="italics"/>in loco quovis <emph type="italics"/>T<emph.end type="italics"/>acceleratur vel retar­<lb/>tur in Cycloide, e&longs;t ad totum corporis eju&longs;dem Pondus in loco <lb/>alti&longs;&longs;imo <emph type="italics"/>S<emph.end type="italics"/>vel <emph type="italics"/>Q,<emph.end type="italics"/>ut Cycloidis arcus <emph type="italics"/>TR<emph.end type="italics"/>ad eju&longs;dem arcum <emph type="italics"/>SR<emph.end type="italics"/><lb/>vel <emph type="italics"/>QR.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LII. PROBLEMA XXXIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Definire & Velocitates Pendulorum in locis &longs;ingulis, & Tempora <lb/>quibus tum o&longs;cillationes totæ, tum &longs;ingulæ o&longs;cillationum partes <lb/>peraguntur.<emph.end type="italics"/></s></p> <p type="main"> <s>Centro quovis <emph type="italics"/>G,<emph.end type="italics"/>intervallo <emph type="italics"/>GH<emph.end type="italics"/>Cycloidis arcum <emph type="italics"/>RS<emph.end type="italics"/>æquante, <lb/>de&longs;cribe &longs;emicirculum <emph type="italics"/>HKMG<emph.end type="italics"/>&longs;emidiametro <emph type="italics"/>GK<emph.end type="italics"/>bi&longs;ectum. </s> <s>Et <lb/>&longs;i vis centripeta, di&longs;tantiis loeorum a centro proportionalis, tendat <lb/>ad centrum <emph type="italics"/>G,<emph.end type="italics"/>&longs;itque ea in perimetro <emph type="italics"/>HIK<emph.end type="italics"/>æqualis vi centripetæ <lb/>in perimetro Globi <emph type="italics"/>QOS (Vide Fig. </s> <s>Prop.<emph.end type="italics"/>L.) ad ip&longs;ius cen­<lb/>trum tendenti; & eodem tempore quo pendulum <emph type="italics"/>T<emph.end type="italics"/>dimittitur e <lb/>loco &longs;upremo <emph type="italics"/>S,<emph.end type="italics"/>cadat corpus aliquod <emph type="italics"/>L<emph.end type="italics"/>ab <emph type="italics"/>H<emph.end type="italics"/>ad <emph type="italics"/>G:<emph.end type="italics"/>quoniam <lb/>vires quibus corpora urgentur &longs;unt æquales &longs;ub initio & &longs;patiis <lb/>de&longs;cribendis <emph type="italics"/>TR, LG<emph.end type="italics"/>&longs;emper proportionales, atque adeo, &longs;i æ­<lb/>quantur <emph type="italics"/>TR<emph.end type="italics"/>& <emph type="italics"/>LG,<emph.end type="italics"/>æquales in locis <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>L<emph.end type="italics"/>; patet corpora illa <lb/>de&longs;cribere &longs;patia <emph type="italics"/>ST, HL<emph.end type="italics"/>æqualia &longs;ub initio, adeoque &longs;ubinde per­<lb/>gere æqualiter urgeri, & æqualia &longs;patia de&longs;cribere. </s> <s>Quare, per Prop. </s> <s><lb/>XXXVIII, tempus quo corpus de&longs;cribit arcum <emph type="italics"/>ST<emph.end type="italics"/>e&longs;t ad tempus <pb xlink:href="039/01/169.jpg" pagenum="141"/>o&longs;cillationis unius, ut arcus <emph type="italics"/>HI<emph.end type="italics"/>(tempus quo corpus <emph type="italics"/>H<emph.end type="italics"/>perveniet <lb/><arrow.to.target n="note117"/>ad <emph type="italics"/>L<emph.end type="italics"/>) ad &longs;emiperipheriam <emph type="italics"/>HKM<emph.end type="italics"/>(tempus quo corpus <emph type="italics"/>H<emph.end type="italics"/>per­<lb/>veniet ad <emph type="italics"/>M.<emph.end type="italics"/>) Et velocitas corporis penduli in loco <emph type="italics"/>T<emph.end type="italics"/>e&longs;t ad ve­<lb/>locitatem ip&longs;ius in loco infimo <emph type="italics"/>R,<emph.end type="italics"/>(hoc e&longs;t, velocitas corporis <emph type="italics"/>H<emph.end type="italics"/>in <lb/>loco <emph type="italics"/>L<emph.end type="italics"/>ad velocitatem ejus in loco <emph type="italics"/>G,<emph.end type="italics"/>&longs;eu incrementum momenta­<lb/>neum lineæ <emph type="italics"/>HL<emph.end type="italics"/>ad incrementum momentaneum lineæ <emph type="italics"/>HG,<emph.end type="italics"/>arcu­<lb/>bus <emph type="italics"/>HI, HK<emph.end type="italics"/>æquabili fluxu cre&longs;centibus) ut ordinatim applicata <lb/><emph type="italics"/>LI<emph.end type="italics"/>ad radium <emph type="italics"/>GK,<emph.end type="italics"/>&longs;ive ut √<emph type="italics"/><expan abbr="SRq.-TRq.">SRq.-TRque</expan><emph.end type="italics"/>ad <emph type="italics"/>SR.<emph.end type="italics"/>Unde cum, <lb/>in o&longs;cillationibus inæqualibus, de&longs;cribantur æqualibus temporibus <lb/>arcus totis o&longs;cillationum arcubus proportionales; habentur, ex da­<lb/>tis temporibus, & velocitates & arcus de&longs;cripti in o&longs;cillationibus <lb/>univer&longs;is. </s> <s>Quæ erant primo invenienda. </s></p> <p type="margin"> <s><margin.target id="note117"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>O&longs;cillentur jam Funipendula <lb/><figure id="id.039.01.169.1.jpg" xlink:href="039/01/169/1.jpg"/><lb/>corpora in Cycloidibus diver&longs;is <lb/>intra Globos diver&longs;os, quorum <lb/>diver&longs;æ &longs;unt etiam Vires ab&longs;olu­<lb/>tæ, de&longs;criptis: &, &longs;i Vis ab&longs;olu­<lb/>ta Globi cuju&longs;vis <emph type="italics"/>QOS<emph.end type="italics"/>dicatur V, <lb/>Vis acceleratrix qua <expan abbr="Pendulũ">Pendulum</expan> urge­<lb/>tur in circumferentia hujus Globi, <lb/>ubi incipit directe ver&longs;us centrum <lb/>ejus moveri, erit ut di&longs;tantia Cor­<lb/>poris penduli a centro illo & Vis ab&longs;oluta Globi conjunctim, hoc <lb/>e&longs;t, ut <emph type="italics"/>CO<emph.end type="italics"/>XV. </s> <s>Itaque lineola <emph type="italics"/>HY,<emph.end type="italics"/>quæ &longs;it ut hæc Vis accelera­<lb/>trix <emph type="italics"/>CO<emph.end type="italics"/>XV, de&longs;cribetur dato tempore; &, &longs;i erigatur normalis <emph type="italics"/>YZ<emph.end type="italics"/><lb/>circumferentiæ occurrens in <emph type="italics"/>Z,<emph.end type="italics"/>arcus na&longs;cens <emph type="italics"/>HZ<emph.end type="italics"/>denotabit datum <lb/>illud tempus. </s> <s>E&longs;t autem arcus hic na&longs;cens <emph type="italics"/>HZ<emph.end type="italics"/>in &longs;ubduplicata ra­<lb/>tione rectanguli <emph type="italics"/>GHY,<emph.end type="italics"/>adeoque ut √<emph type="italics"/>GHXCO<emph.end type="italics"/>XV. </s> <s>Unde Tem­<lb/>pus o&longs;cillationis integræ in Cycloide <emph type="italics"/>QRS<emph.end type="italics"/>(cum &longs;it ut &longs;emiperi­<lb/>pheria <emph type="italics"/>HKM,<emph.end type="italics"/>quæ o&longs;cillationem illam integram denotat, directe, <lb/>utque arcus <emph type="italics"/>HZ,<emph.end type="italics"/>qui datum tempus &longs;imiliter denotat, inver&longs;e) fiet <lb/>ut <emph type="italics"/>GH<emph.end type="italics"/>directe & √<emph type="italics"/>GHXCO<emph.end type="italics"/>XV inver&longs;e, hoc e&longs;t, ob æquales <emph type="italics"/>GH<emph.end type="italics"/><lb/>& <emph type="italics"/>SR,<emph.end type="italics"/>ut √(<emph type="italics"/>SR/CO<emph.end type="italics"/>XV), &longs;ive (per Corol. </s> <s>Prop. </s> <s>L) ut √(<emph type="italics"/>AR/AC<emph.end type="italics"/>XV). <lb/>Itaque O&longs;cillationes in Globis & Cycloidibus omnibus, quibu&longs;­<lb/>cunque cum Viribus ab&longs;olutis factæ, &longs;unt in ratione quæ compo­<lb/>nitur ex &longs;ubduplicata ratione longitudinis Fili directe, & &longs;ubdu­<lb/>plicata ratione di&longs;tantiæ inter punctum &longs;u&longs;pen&longs;ionis & centrum <pb xlink:href="039/01/170.jpg" pagenum="142"/><arrow.to.target n="note118"/>Globi inver&longs;e, & &longs;ubduplicata ratione Vis ab&longs;olutæ Globi etiam <lb/>inver&longs;e. <emph type="italics"/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note118"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc etiam O&longs;cillantium, Cadentium & Revolventium <lb/>corporum tempora po&longs;&longs;unt inter &longs;e conferri. </s> <s>Nam &longs;i Rotæ, qua Cy­<lb/>clois intra globum de&longs;cribitur, diameter con&longs;tituatur æqualis &longs;emi­<lb/>diametro globi, Cyclois evadet Linea recta per centrum globi tran­<lb/>&longs;iens, & O&longs;cillatio jam erit de&longs;cen&longs;us & &longs;ub&longs;equens a&longs;cen&longs;us in hac <lb/>recta. </s> <s>Unde datur tum tempus de&longs;cen&longs;us de loco quovis ad <lb/>centrum, tum tempus huic æquale quo corpus uniformiter cir­<lb/>ca centrum globi ad di&longs;tantiam quamvis revolvendo arcum qua­<lb/>drantalem de&longs;cribit. </s> <s>E&longs;t enim hoc tempus (per Ca&longs;um &longs;ecun­<lb/>dum) ad tempus &longs;emio&longs;cillationis in Cycloide quavis <emph type="italics"/>QRS<emph.end type="italics"/>ut <lb/>1 ad √(<emph type="italics"/>AR/AC<emph.end type="italics"/>). </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Hinc etiam con&longs;ectantur quæ <emph type="italics"/>Wrennus<emph.end type="italics"/>& <emph type="italics"/>Hugenius<emph.end type="italics"/>de <lb/>Cycloide vulgari adinvenerunt. </s> <s>Nam &longs;i Globi diameter augeatur <lb/>in infinitum: mutabitur ejus &longs;uperficies &longs;phærica in planum, Vi&longs;que <lb/>centripeta aget uniformiter &longs;ecundum lineas huic plano perpendi­<lb/>culares, & Cyclois no&longs;tra abibit in Cycloidem vulgi. </s> <s>I&longs;to autem <lb/>in ca&longs;u longitudo arcus Cycloidis, inter planum illud & punctum <lb/>de&longs;cribens, æqualis evadet quadruplicato &longs;inui ver&longs;o dimidii arcus <lb/>Rotæ inter idem planum & punctum de&longs;cribens; ut invenit <emph type="italics"/>Wren­<lb/>nus:<emph.end type="italics"/>Et Pendulum inter duas eju&longs;modi Cycloides in &longs;imili & æ­<lb/>quali Cycloide temporibus æqualibus O&longs;cillabitur, ut demon&longs;travit <lb/><emph type="italics"/>Hugenius.<emph.end type="italics"/>Sed & De&longs;cen&longs;us gravium, tempore O&longs;cillationis unius, <lb/>is erit quem <emph type="italics"/>Hugenius<emph.end type="italics"/>indicavit. </s></p> <p type="main"> <s>Aptantur autem Propo&longs;itiones a nobis demon&longs;tratæ ad veram <lb/>con&longs;titutionem Terræ, quatenus Rotæ eundo in ejus circulis maxi­<lb/>mis de&longs;cribunt motu Clavorum, perimetris &longs;uis infixorum, Cycloi­<lb/>des extra globum; & Pendula inferius in fodinis & cavernis Terra <lb/>&longs;u&longs;pen&longs;a, in Cycloidibus intra globos O&longs;cillari debent, ut O&longs;cilla­<lb/>tiones omnes evadant I&longs;ochronæ. </s> <s>Nam Gravitas (ut in Libro <lb/>tertio docebitur) decre&longs;cit in progre&longs;&longs;u a &longs;uperficie Terræ, &longs;ur­<lb/>&longs;um quidem in duplicata ratione di&longs;tantiarum a centro ejus, de <lb/>or&longs;um vero in ratione &longs;implici. <pb xlink:href="039/01/171.jpg" pagenum="143"/><arrow.to.target n="note119"/></s></p> <p type="margin"> <s><margin.target id="note119"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LIII. PROBLEMA XXXV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Conce&longs;&longs;is Figurarum curvilinearum quadraturis, invenire Vires qui­<lb/>bus corpora in datis curvis lineis O&longs;cillationes &longs;emper I&longs;ochro­<lb/>nas peragent.<emph.end type="italics"/></s></p> <p type="main"> <s>O&longs;cilletur corpus <emph type="italics"/>T<emph.end type="italics"/>in curva quavis linea <emph type="italics"/>STRQ,<emph.end type="italics"/>cujus axis &longs;it <lb/><emph type="italics"/>OR<emph.end type="italics"/>tran&longs;iens per virium centrum <emph type="italics"/>C.<emph.end type="italics"/>Agatur <emph type="italics"/>TX<emph.end type="italics"/>quæ curvam il­<lb/>lam in corporis loco quovis <emph type="italics"/>T<emph.end type="italics"/>contingat, inque hac tangente <emph type="italics"/>TX<emph.end type="italics"/><lb/><figure id="id.039.01.171.1.jpg" xlink:href="039/01/171/1.jpg"/><lb/>capiatur <emph type="italics"/>TY<emph.end type="italics"/>æqualis arcui <emph type="italics"/>TR.<emph.end type="italics"/>Nam longitudo arcus illius ex Fi­<lb/>gurarum quadraturis (per Methodos vulgares) innote&longs;cit. </s> <s>De pun­<lb/>cto <emph type="italics"/>Y<emph.end type="italics"/>educatur recta <emph type="italics"/>YZ<emph.end type="italics"/>tangenti perpendicularis. </s> <s>Agatur <emph type="italics"/>CT<emph.end type="italics"/>per­<lb/>pendiculari illi occurrens in <emph type="italics"/>Z,<emph.end type="italics"/>& erit Vis centripeta proportiona­<lb/>lis rectæ <emph type="italics"/>TZ. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/><pb xlink:href="039/01/172.jpg" pagenum="144"/><arrow.to.target n="note120"/></s></p> <p type="margin"> <s><margin.target id="note120"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Nam &longs;i vis, qua corpus trahitur de <emph type="italics"/>T<emph.end type="italics"/>ver&longs;us <emph type="italics"/>C,<emph.end type="italics"/>exponatur per <lb/>rectam <emph type="italics"/>TZ<emph.end type="italics"/>captam ip&longs;i proportionalem, re&longs;olvetur hæc in vires <lb/><emph type="italics"/>TY, YZ<emph.end type="italics"/>; quarum <emph type="italics"/>YZ<emph.end type="italics"/>trahendo corpus &longs;ecundum longitudinem <lb/>Fili <emph type="italics"/>PT,<emph.end type="italics"/>motum ejus nil mutat, vis autem altera <emph type="italics"/>TY<emph.end type="italics"/>motum ejus <lb/>in curva <emph type="italics"/>STRQ<emph.end type="italics"/>directe accelerat vel directe retardat. </s> <s>Proinde <lb/>cum hæc &longs;it ut via de&longs;cribenda <emph type="italics"/>TR,<emph.end type="italics"/>accelerationes corporis vel re­<lb/>tardationes in O&longs;cillationum duarum (majoris & minoris) parti­<lb/>bus proportionalibus de&longs;cribendis, erunt &longs;emper ut partes illæ, & <lb/>propterea facient ut partes illæ &longs;imul de&longs;cribantur. </s> <s>Corpora autem <lb/>quæ partes totis &longs;emper proportionales &longs;imul de&longs;cribunt, &longs;imul de­<lb/>&longs;cribent totas. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i corpus <emph type="italics"/>T<emph.end type="italics"/>Filo rectilineo <emph type="italics"/>AT<emph.end type="italics"/>a centro <emph type="italics"/>A<emph.end type="italics"/>pen­<lb/>dens, de&longs;cribat arcum circularem <emph type="italics"/>STRQ,<emph.end type="italics"/>& interea urgeatur &longs;e­<lb/>cundum lineas parallelas deor&longs;um a vi aliqua, quæ &longs;it ad vim uNI­<lb/>formem Gravitatis, ut arcus <emph type="italics"/>TR<emph.end type="italics"/>ad ejus &longs;inum <emph type="italics"/>TN:<emph.end type="italics"/>æqualia e­<lb/>runt O&longs;cillationum &longs;ingularum tempora. </s> <s>Etenim ob parallelas <lb/><emph type="italics"/>TZ, AR,<emph.end type="italics"/>&longs;imilia erunt triangula <emph type="italics"/>ATN, ZTY<emph.end type="italics"/>; & propterea <lb/><emph type="italics"/>TZ<emph.end type="italics"/>erit ad <emph type="italics"/>AT<emph.end type="italics"/>ut <emph type="italics"/>TY<emph.end type="italics"/>ad <emph type="italics"/>TN<emph.end type="italics"/>; hoc e&longs;t, (&longs;i Gravitatis vis unifor­<lb/>mis exponatur per longitudinem datam <emph type="italics"/>AT<emph.end type="italics"/>) vis <emph type="italics"/>TZ,<emph.end type="italics"/>qua O&longs;cil­<lb/>lationes evadent I&longs;ochronæ, erit ad vim Gravitatis <emph type="italics"/>AT,<emph.end type="italics"/>ut arcus <lb/><emph type="italics"/>TR<emph.end type="italics"/>ip&longs;i <emph type="italics"/>TY<emph.end type="italics"/>æqualis ad arcus illius &longs;inum <emph type="italics"/>TN.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Igitur in Horologiis, &longs;i vires a Machina in Pendulum <lb/>ad motum con&longs;ervandum impre&longs;&longs;æ ita cum vi Gravitatis componi <lb/>po&longs;&longs;int, ut vis tota deor&longs;um &longs;emper &longs;it ut linea quæ oritur appli­<lb/>cando rectangulum &longs;ub arcu <emph type="italics"/>TR<emph.end type="italics"/>& radio <emph type="italics"/>AR<emph.end type="italics"/>ad &longs;inum <emph type="italics"/>TN,<emph.end type="italics"/><lb/>O&longs;cillationes omnes erunt I&longs;ochronæ. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LIV. PROBLEMA XXXVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Conce&longs;&longs;is Figurarum curvilinearum quadraturis, invenire Tempora <lb/>quibus corpora Vi qualibet centripeta in lineis quibu&longs;cunque cur­<lb/>vis, in plano per centrum Virium tran&longs;eunte de&longs;criptis, de&longs;cen­<lb/>dent & a&longs;cendent.<emph.end type="italics"/></s></p> <p type="main"> <s>De&longs;cendat corpus de loco quovis <emph type="italics"/>S<emph.end type="italics"/>per lineam quamvis curvam <lb/><emph type="italics"/>STtR,<emph.end type="italics"/>in plano per virium centrum <emph type="italics"/>C<emph.end type="italics"/>tran&longs;eunte datam. </s> <s>Junga­<lb/>tur <emph type="italics"/>CS<emph.end type="italics"/>& dividatur eadem in partes innumeras æquales, &longs;itque <emph type="italics"/>Dd<emph.end type="italics"/><pb xlink:href="039/01/173.jpg" pagenum="145"/>partium illarum aliqua. </s> <s>Centro <emph type="italics"/>C,<emph.end type="italics"/>intervallis <emph type="italics"/>CD, Cd<emph.end type="italics"/>de&longs;criban­</s></p> <p type="main"> <s><arrow.to.target n="note121"/>tur circuli <emph type="italics"/>DT, dt,<emph.end type="italics"/>lineæ curvæ <emph type="italics"/>STtR<emph.end type="italics"/>occurrentes in <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>t.<emph.end type="italics"/>Et <lb/>ex data tum lege vis centripetæ, tum <lb/><figure id="id.039.01.173.1.jpg" xlink:href="039/01/173/1.jpg"/><lb/>altitudine <emph type="italics"/>CS<emph.end type="italics"/>de qua corpus cecidit; <lb/>dabitur velocitas corporis in alia qua­<lb/>vis altitudine <emph type="italics"/>CT,<emph.end type="italics"/>per Prop. </s> <s>XXXIX. </s> <s><lb/>Tempus autem, quo corpus de&longs;cribit <lb/>lineolam <emph type="italics"/>Tt,<emph.end type="italics"/>e&longs;t ut lineolæ hujus lon­<lb/>gitudo (id e&longs;t ut &longs;ecans anguli <emph type="italics"/>tTC<emph.end type="italics"/>) <lb/>directe, & velocitas inver&longs;e. </s> <s>Tempori <lb/>huic proportionalis &longs;it ordinatim appli­<lb/>cata <emph type="italics"/>DN<emph.end type="italics"/>ad rectam <emph type="italics"/>CS<emph.end type="italics"/>per punctum <lb/><emph type="italics"/>D<emph.end type="italics"/>perpendicularis, & ob datam <emph type="italics"/>Dd<emph.end type="italics"/><lb/>erit rectangulum <emph type="italics"/>DdXDN,<emph.end type="italics"/>hoc e&longs;t <lb/>area <emph type="italics"/>DNnd,<emph.end type="italics"/>eidem tempori propor­<lb/>tionale. </s> <s>Ergo &longs;i <emph type="italics"/>SNn<emph.end type="italics"/>&longs;it curva illa li­<lb/>nea quam punctum <emph type="italics"/>N<emph.end type="italics"/>perpetuo tangit, <lb/>erit area <emph type="italics"/>SNDS<emph.end type="italics"/>proportionalis tem­<lb/>pori quo corpus de&longs;cendendo de&longs;crip­<lb/>&longs;it lineam <emph type="italics"/>ST<emph.end type="italics"/>; proindeque ex inventa illa area dabitur Tempus. <lb/><emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note121"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LV. THEOREMA XIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si corpus movetur in &longs;uperficie quacunque curva, cujus axis per <lb/>centrum Virium tran&longs;it, & a corpore in axem demittatur per­<lb/>pendicularis, eique parallela & æqualis ab axis puncto quovis <lb/>dato ducatur: dico quod parallela illa aream tempori proportio­<lb/>nalem de&longs;cribet.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>BSKL<emph.end type="italics"/>&longs;uperficies curva, <emph type="italics"/>T<emph.end type="italics"/>corpus in ea revolvens, <emph type="italics"/>STtR<emph.end type="italics"/><lb/>Trajectoria quam corpus in eadem de&longs;cribit, <emph type="italics"/>S<emph.end type="italics"/>initium Trajecto­<lb/>riæ, <emph type="italics"/>OMNK<emph.end type="italics"/>axis &longs;uperficiei curvæ, <emph type="italics"/>TN<emph.end type="italics"/>recta a corpore in axem <lb/>perpendicularis, <emph type="italics"/>OP<emph.end type="italics"/>huic parallela & æqualis a puncto <emph type="italics"/>O<emph.end type="italics"/>quod in <lb/>axe datur educta, <emph type="italics"/>AP<emph.end type="italics"/>ve&longs;tigium Trajectoriæ a puncto <emph type="italics"/>P<emph.end type="italics"/>in lineæ <lb/>volubilis <emph type="italics"/>OP<emph.end type="italics"/>plano <emph type="italics"/>AOP<emph.end type="italics"/>de&longs;criptum, <emph type="italics"/>A<emph.end type="italics"/>ve&longs;tigii initium puncto <emph type="italics"/>S<emph.end type="italics"/><lb/>re&longs;pondens, <emph type="italics"/>TC<emph.end type="italics"/>recta a corpore ad centrum ducta; <emph type="italics"/>TG<emph.end type="italics"/>pars ejus <lb/>vi centripetæ qua corpus urgetur in centrum <emph type="italics"/>C<emph.end type="italics"/>proportionalis; <lb/><emph type="italics"/>TM<emph.end type="italics"/>recta ad &longs;uperficiem curvam perpendicularis, <emph type="italics"/>TI<emph.end type="italics"/>pars ejus vi <lb/>pre&longs;&longs;ionis, qua corpus urget &longs;uperficiem vici&longs;&longs;imque urgetur ver&longs;us <emph type="italics"/>M<emph.end type="italics"/><pb xlink:href="039/01/174.jpg" pagenum="146"/><arrow.to.target n="note122"/>a &longs;uperficie, proportiona­<lb/><figure id="id.039.01.174.1.jpg" xlink:href="039/01/174/1.jpg"/><lb/>lis; <emph type="italics"/>PHTF<emph.end type="italics"/>recta axi <lb/>parallela per corpus tran­<lb/>&longs;iens, & <emph type="italics"/>GF, IH<emph.end type="italics"/>rectæ <lb/>a punctis <emph type="italics"/>G<emph.end type="italics"/>& <emph type="italics"/>I<emph.end type="italics"/>in pa­<lb/>rallelam illam <emph type="italics"/>PHTF<emph.end type="italics"/><lb/>perpendiculariter demi&longs;­<lb/>&longs;æ. </s> <s>Dico jam quod area <lb/><emph type="italics"/>AOP,<emph.end type="italics"/>radio <emph type="italics"/>OP<emph.end type="italics"/>ab iNI­<lb/>tio motus de&longs;cripta, &longs;it <lb/>tempori proportionalis. </s> <s><lb/>Nam vis <emph type="italics"/>TG<emph.end type="italics"/>(per Le­<lb/>gum Corol. </s> <s>2.) re&longs;olvitur <lb/>in vires <emph type="italics"/>TF, FG<emph.end type="italics"/>; & vis <lb/><emph type="italics"/>TI<emph.end type="italics"/>in vires <emph type="italics"/>TH, HI:<emph.end type="italics"/><lb/>Vires autem <emph type="italics"/>TF, TH<emph.end type="italics"/><lb/>agendo &longs;ecundum lineam <lb/><emph type="italics"/>PF<emph.end type="italics"/>plano <emph type="italics"/>AOP<emph.end type="italics"/>per­<lb/>pendicularem mutant &longs;o­<lb/>lummodo motum cor­<lb/>poris quatenus huic plano perpendicularem. </s> <s>Ideoque motus ejus <lb/>quatenus &longs;ecundum po&longs;itionem plani factus, hoc e&longs;t, motus pun­<lb/>cti <emph type="italics"/>P<emph.end type="italics"/>quo Trajectoriæ ve&longs;tigium <emph type="italics"/>AP<emph.end type="italics"/>in hoc plano de&longs;cri­<lb/>bitur, idem e&longs;t ac &longs;i vires <emph type="italics"/>TF, TH<emph.end type="italics"/>tollerentur, & corpus &longs;olis vi­<lb/>ribus <emph type="italics"/>FG, HI<emph.end type="italics"/>agitaretur; hoc e&longs;t, idem ac &longs;i corpus in plano <lb/><emph type="italics"/>AOP,<emph.end type="italics"/>vi centripeta ad centrum <emph type="italics"/>O<emph.end type="italics"/>tendente & &longs;ummam virium <lb/><emph type="italics"/>FG<emph.end type="italics"/>& <emph type="italics"/>HI<emph.end type="italics"/>æquante, de&longs;criberet curvam <emph type="italics"/>AP.<emph.end type="italics"/>Sed vi tali de&longs;cribi­<lb/>tur area <emph type="italics"/>AOP<emph.end type="italics"/>(per Prop. </s> <s>1.) tempori proportionalis. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note122"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Eodem argumento &longs;i corpus a viribus agitatum ad centra <lb/>duo vel plura in eadem quavis recta <emph type="italics"/>CO<emph.end type="italics"/>data tendentibus, de&longs;cri­<lb/>beret in &longs;patio libero lineam quamcunque curvam <emph type="italics"/>ST<emph.end type="italics"/>; foret area <lb/><emph type="italics"/>AOP<emph.end type="italics"/>tempori &longs;emper proportionalis. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LVI. PROBLEMA XXXVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Conce&longs;&longs;is Figurarum curvilinearum quadraturis, dati&longs;que tum lege <lb/>Vis centripetæ ad centrum datum tendentis, tum &longs;uperficie cur­<lb/>va cujus axis per centrum illud træn&longs;it; invenieuda est Traje­<lb/>ctoria quam corpus in eadem &longs;uperficie de&longs;cribet, de loco dato, data <lb/>cum Velocitate, ver&longs;us plagam in &longs;uperficie illa datam egre&longs;&longs;um.<emph.end type="italics"/></s></p><pb xlink:href="039/01/175.jpg" pagenum="147"/> <p type="main"> <s>Stantibus quæ in &longs;uperiore Propo&longs;itione con&longs;tructa &longs;unt, exeat <lb/><arrow.to.target n="note123"/>corpus de loco <emph type="italics"/>S<emph.end type="italics"/>in Trajectoriam inveniendam <emph type="italics"/>STtR<emph.end type="italics"/>; &, ex da­<lb/>ta ejus velocitate in altitudine <emph type="italics"/>SC,<emph.end type="italics"/>dabitur ejus velocitas in alia <lb/>quavis altitudine <emph type="italics"/>TC.<emph.end type="italics"/>Ea cum velocitate, dato tempore quam <lb/>minimo, de&longs;cribat corpus Trajectoriæ &longs;uæ particulam <emph type="italics"/>Tt,<emph.end type="italics"/>&longs;itque <lb/><emph type="italics"/>Pp<emph.end type="italics"/>ve&longs;tigium ejus in plano <emph type="italics"/>AOP<emph.end type="italics"/>de&longs;criptum. </s> <s>Jungatur <emph type="italics"/>Op,<emph.end type="italics"/>& <lb/>Circelli centro <emph type="italics"/>T<emph.end type="italics"/>intervallo <emph type="italics"/>Tt<emph.end type="italics"/>in &longs;uperficie curva de&longs;cripti &longs;it <emph type="italics"/>PpQ<emph.end type="italics"/><lb/>ve&longs;tigium Ellipticum in eodem plano <emph type="italics"/>OAPp<emph.end type="italics"/>de&longs;criptum. </s> <s>Et ob <lb/>datum magnitudine & po&longs;itione Circellum, dabitur Ellip&longs;is illa <lb/><emph type="italics"/><expan abbr="Ppq.">Ppque</expan><emph.end type="italics"/>Cumque area <emph type="italics"/>POp<emph.end type="italics"/>&longs;it tempori proportionalis, atque ad­<lb/>eo ex dato tempore detur, dabitur <emph type="italics"/>Op<emph.end type="italics"/>po&longs;itione, & inde dabitur <lb/>communis ejus & Ellip&longs;eos inter&longs;ectio <emph type="italics"/>p,<emph.end type="italics"/>una cum angulo <emph type="italics"/>OPp,<emph.end type="italics"/><lb/>in quo Trajectoriæ ve&longs;tigium <emph type="italics"/>APp<emph.end type="italics"/>&longs;ecat lineam <emph type="italics"/>OP.<emph.end type="italics"/>Inde au­<lb/>tem invenietur Trajectoriæ ve&longs;tigium illud <emph type="italics"/>APp,<emph.end type="italics"/>eadem methodo <lb/>qua curva linea <emph type="italics"/>VIKk,<emph.end type="italics"/>in Propo&longs;itione XLI, ex &longs;imilibus datis <lb/>inventa fuit. </s> <s>Tum ex &longs;ingulis ve&longs;tigii punctis <emph type="italics"/>P<emph.end type="italics"/>erigendo ad pla­<lb/>num <emph type="italics"/>AOP<emph.end type="italics"/>perpendicula <emph type="italics"/>PT<emph.end type="italics"/>&longs;uperficiei curvæ occurrentia in <emph type="italics"/>T,<emph.end type="italics"/><lb/>dabuntur &longs;ingula Trajectoriæ puncta <emph type="italics"/>T. Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note123"/>LIBER <lb/>PRIMUS.</s></p></subchap2><subchap2> <p type="main"> <s><emph type="center"/>SECTIO XI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Motu Corporum Viribus centripetis &longs;e mutuo petentium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Hactenus expo&longs;ui Motus corporum attractorum ad centrum Im­<lb/>mobile, quale tamen vix extat in rerum natura. </s> <s>Attractiones enim <lb/>fieri &longs;olent ad corpora; & corporum trahentium & attractorum <lb/>actiones &longs;emper mutuæ &longs;unt & æquales, per Legem tertiam: ad­<lb/>eo ut neque attrahens po&longs;&longs;it quie&longs;cere neque attractum, &longs;i duo &longs;int <lb/>corpora, &longs;ed ambo (per Legum Corollarium quartum) qua&longs;i at­<lb/>tractione mutua, circum gravitatis centrum commune revolvantur: <lb/>& &longs;i plura &longs;int corpora (quæ vel ab unico attrahantur vel omnia <lb/>&longs;e mutuo attrahant) hæc ita inter &longs;e moveri debeant, ut gravitatis <lb/>centrum commune vel quie&longs;cat vel uniformiter moveatur in direc­<lb/>tum. </s> <s>Qua de cau&longs;a jam pergo Motum exponere corporum &longs;e mu­<lb/>tuo trahentium, con&longs;iderando Vires centripetas tanquam Attractio­<lb/>nes, quamvis forta&longs;&longs;e, &longs;i phy&longs;ice loquamur, verius dicantur Im­<lb/>pul&longs;us. </s> <s>In Mathematicis enim jam ver&longs;amur, & propterea mi&longs;&longs;is <lb/>di&longs;putationibus Phy&longs;icis, familiari utimur &longs;ermone, quo po&longs;&longs;imus <lb/>a Lectoribus Mathematicis facilius intelligi. <pb xlink:href="039/01/176.jpg" pagenum="148"/><arrow.to.target n="note124"/></s></p> <p type="margin"> <s><margin.target id="note124"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LVII. THEOREMA XX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corpora duo &longs;e invicem trahentia de&longs;cribunt, & circum commune <lb/>centrum gravitatis, & circum &longs;e mutuo, Figuras &longs;imiles.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sunt enim di&longs;tantiæ a communi gravitatis centro reciproce pro­<lb/>portionales corporibus, atque adeo in data ratione ad invicem, & <lb/>componendo, in data ratione ad di&longs;tantiam totam inter corpora. </s> <s><lb/>Feruntur autem hæ di&longs;tantiæ circum terminos &longs;uos communi motu <lb/>angulari, propterea quod in directum &longs;emper jacentes non mutant <lb/>inclinationem ad &longs;e mutuo. </s> <s>Lineæ autem rectæ, quæ &longs;unt in data <lb/>ratione ad invicem, & æquali motu angulari circum terminos &longs;uos <lb/>feruntur, Figuras circum eo&longs;dem terminos (in planis quæ una cum <lb/>his terminis vel quie&longs;cunt vel motu quovis non angulari moven­<lb/>tur) de&longs;cribunt omnino &longs;imiles. </s> <s>Proinde &longs;imiles &longs;unt Figuræ quæ <lb/>his di&longs;tantiis circumactis de&longs;cribuntur. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LVIII. THEOREMA XXI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si corpora duo Viribus quibu&longs;vis &longs;e mutuo trahunt, & interea re­<lb/>volvuntur circa gravitatis centrum commune: dico quod Fi­<lb/>guris, quas corpora &longs;ic mota de&longs;cribunt circum &longs;e mutuo, potest <lb/>Figura &longs;imilis & æqualis, circum corpus alterutrum immotum, <lb/>Viribus ii&longs;dem de&longs;cribi.<emph.end type="italics"/></s></p> <p type="main"> <s>Revolvantur corpora <emph type="italics"/>S, P<emph.end type="italics"/>circa commune gravitatis centrum <lb/><emph type="italics"/>C,<emph.end type="italics"/>pergendo de <emph type="italics"/>S<emph.end type="italics"/>ad <emph type="italics"/>T<emph.end type="italics"/>deque <emph type="italics"/>P<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/>A dato puncto <emph type="italics"/>s<emph.end type="italics"/>ip&longs;is <lb/><figure id="id.039.01.176.1.jpg" xlink:href="039/01/176/1.jpg"/><lb/><emph type="italics"/>SP, TQ<emph.end type="italics"/>æquales & parallelæ ducantur &longs;emper <emph type="italics"/>sp, sq<emph.end type="italics"/>; & Curva <lb/><emph type="italics"/>pqv<emph.end type="italics"/>quam punctum <emph type="italics"/>p,<emph.end type="italics"/>revolvendo circum punctum immotum <emph type="italics"/>s,<emph.end type="italics"/><pb xlink:href="039/01/177.jpg" pagenum="149"/>de&longs;cribit, erit &longs;imilis & æqualis Curvis quas corpora <emph type="italics"/>S, P<emph.end type="italics"/>de&longs;cri­<lb/><arrow.to.target n="note125"/>bunt circum &longs;e mutuo: proindeque (per Theor. </s> <s>XX) &longs;imilis Curvis <lb/><emph type="italics"/>ST<emph.end type="italics"/>& <emph type="italics"/>PQV,<emph.end type="italics"/>quas eadem corpora de&longs;cribunt circum commune <lb/>gravitatis centrum <emph type="italics"/>C:<emph.end type="italics"/>id adeo quia proportiones linearum <emph type="italics"/>SC, CP<emph.end type="italics"/><lb/>& <emph type="italics"/>SP<emph.end type="italics"/>vel. <emph type="italics"/>sp<emph.end type="italics"/>ad invicem dantur. </s></p> <p type="margin"> <s><margin.target id="note125"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Commune illud gravitatis centrum <emph type="italics"/>C,<emph.end type="italics"/>per Legum Co­<lb/>rollarium quartum, vel quie&longs;cit vel movetur uniformiter in direc­<lb/>tum. </s> <s>Ponamus primo quod id quie&longs;cit, inque <emph type="italics"/>s<emph.end type="italics"/>& <emph type="italics"/>p<emph.end type="italics"/>locentur cor­<lb/>pora duo, immobile in <emph type="italics"/>s,<emph.end type="italics"/>mobile in <emph type="italics"/>p,<emph.end type="italics"/>corporibus <emph type="italics"/>S<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>&longs;imilia <lb/>& æqualia. </s> <s>Dein tangant rectæ <emph type="italics"/>PR<emph.end type="italics"/>& <emph type="italics"/>pr<emph.end type="italics"/>Curvas <emph type="italics"/>PQ<emph.end type="italics"/>& <emph type="italics"/>pq<emph.end type="italics"/>in <lb/><emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>p,<emph.end type="italics"/>& producantur <emph type="italics"/>CQ<emph.end type="italics"/>& <emph type="italics"/>sq<emph.end type="italics"/>ad <emph type="italics"/>R<emph.end type="italics"/>& <emph type="italics"/>r.<emph.end type="italics"/>Et, ob &longs;imilitudi­<lb/>nem Figurarum <emph type="italics"/>CPRQ, sprq,<emph.end type="italics"/>erit <emph type="italics"/>RQ<emph.end type="italics"/>ad <emph type="italics"/>rq<emph.end type="italics"/>ut <emph type="italics"/>CP<emph.end type="italics"/>ad <emph type="italics"/>sp,<emph.end type="italics"/>ad­<lb/>eoQ.E.I. data ratione. </s> <s>Proinde &longs;i vis qua corpus <emph type="italics"/>P<emph.end type="italics"/>ver&longs;us cor­<lb/>pus <emph type="italics"/>S,<emph.end type="italics"/>atque adeo ver&longs;us centrum intermedium <emph type="italics"/>C<emph.end type="italics"/>attrahitur, e&longs;&longs;et <lb/>ad vim qua corpus <emph type="italics"/>p<emph.end type="italics"/>ver&longs;us centrum <emph type="italics"/>s<emph.end type="italics"/>attrahitur in eadem illa ra­<lb/>tione data; hæ vires æqualibus temporibus attraherent &longs;emper cor­<lb/>pora de tangentibus <emph type="italics"/>PR, pr<emph.end type="italics"/>ad arcus <emph type="italics"/>PQ, pq,<emph.end type="italics"/>per intervalla ip&longs;is <lb/>proportionalia <emph type="italics"/>RQ, rq;<emph.end type="italics"/>adeoque vis po&longs;terior efficeret ut corpus <lb/><emph type="italics"/>p<emph.end type="italics"/>gyraretur in Curva <emph type="italics"/>pqv,<emph.end type="italics"/>quæ &longs;imilis e&longs;&longs;et Curvæ <emph type="italics"/>PQV,<emph.end type="italics"/>in qua <lb/>vis prior efficit ut corpus <emph type="italics"/>P<emph.end type="italics"/>gyretur, & revolutiones ii&longs;dem tem­<lb/>poribus complerentur. </s> <s>At quoniam vires illæ non &longs;unt ad invi­<lb/>cem in ratione <emph type="italics"/>CP<emph.end type="italics"/>ad <emph type="italics"/>sp,<emph.end type="italics"/>&longs;ed (ob &longs;imilitudinem & æqualitatem <lb/>corporum <emph type="italics"/>S<emph.end type="italics"/>& <emph type="italics"/>s, P<emph.end type="italics"/>& <emph type="italics"/>p,<emph.end type="italics"/>æqualitatem di&longs;tantiarum <emph type="italics"/>SP, sp<emph.end type="italics"/>) <lb/>&longs;ibi mutuo æquales; corpora æqualibus temporibus æqualiter tra­<lb/>hentur de tangentibus: & propterea, ut corpus po&longs;terius <emph type="italics"/>p<emph.end type="italics"/>trahatur <lb/>per intervallum majus <emph type="italics"/>rq,<emph.end type="italics"/>requiritur tempus majus, idQ.E.I. &longs;ub­<lb/>duplicata ratione intervallorum; propterea quod (per Lemma de­<lb/>cimum) &longs;patia, ip&longs;o motus initio de&longs;cripta, &longs;unt in duplicata ratione <lb/>temporum. </s> <s>Ponatur igitur velocitas corporis <emph type="italics"/>p<emph.end type="italics"/>e&longs;&longs;e ad velocita­<lb/>tem corporis <emph type="italics"/>P<emph.end type="italics"/>in &longs;ubduplicata ratione di&longs;tantiæ <emph type="italics"/>sp<emph.end type="italics"/>ad di&longs;tantiam <lb/><emph type="italics"/>CP,<emph.end type="italics"/>eo ut temporibus quæ &longs;int in eadem &longs;ubduplicata ratione de­<lb/>&longs;cribantur arcus <emph type="italics"/>pq, PQ,<emph.end type="italics"/>qui &longs;unt in ratione integra: Et corpora <lb/><emph type="italics"/>P, p<emph.end type="italics"/>viribus æqualibus &longs;emper attracta de&longs;cribent circum centra <lb/>quie&longs;centia <emph type="italics"/>C<emph.end type="italics"/>& <emph type="italics"/>s<emph.end type="italics"/>Figuras &longs;imiles <emph type="italics"/>PQV, pqv,<emph.end type="italics"/>quarum po&longs;terior <lb/><emph type="italics"/>pqv<emph.end type="italics"/>&longs;imilis e&longs;t & æqualis Figuræ quam corpus <emph type="italics"/>P<emph.end type="italics"/>circum corpus <lb/>mobile <emph type="italics"/>S<emph.end type="italics"/>de&longs;cribit. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Ponamus jam quod commune gravitatis centrum, una <lb/>cum &longs;patio in quo corpora moventur inter &longs;e, progreditur unifor­<lb/>miter in directum; &, per Legum Corollarium &longs;extum, motus <lb/>omnes in hoc &longs;patio peragentur ut prius, adeoque corpora de&longs;cri-<pb xlink:href="039/01/178.jpg" pagenum="150"/><arrow.to.target n="note126"/>bent circum &longs;e mutuo Figuras ea&longs;dem ac prius, & propterea Figuræ <lb/><emph type="italics"/>pqv<emph.end type="italics"/>&longs;imiles & æquales. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note126"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc corpora duo Viribus di&longs;tantiæ &longs;uæ proportionali­<lb/>bus &longs;e mutuo trahentia, de&longs;cribunt (per Prop. </s> <s>X,) & circum com­<lb/>mune gravitatis centrum, & circum &longs;e mutuo, Ellip&longs;es concentri­<lb/>cas: & vice ver&longs;a, &longs;i tales Figuræ de&longs;cribuntur, &longs;unt Vires di&longs;tan­<lb/>tiæ proportionales. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et corpora duo Viribus quadrato di&longs;tantiæ &longs;uæ recipro­<lb/>ce proportionalibus de&longs;cribunt (per Prop. </s> <s>XI, XII, XIII) & circum <lb/>commune gravitatis centrum, & circum &longs;e mutuo, Sectiones conicas <lb/>umbilicum habentes in centro circum quod Figuræ de&longs;cribuntur. </s> <s>Et <lb/>vice ver&longs;a, &longs;i tales Figuræ de&longs;cribuntur, Vires centripetæ &longs;unt qua­<lb/>drato di&longs;tantiæ reciproce proportionales. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Corpora duo quævis cirum gravitatis centrum com­<lb/>mune gyrantia, radiis & ad centrum illud & ad &longs;e mutuo ductis, <lb/>de&longs;cribunt areas temporibus proportionales. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LIX. THEOREMA XXII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Corporum duorum<emph.end type="italics"/>S <emph type="italics"/>&<emph.end type="italics"/>P <emph type="italics"/>circa commune gravitatis centrum<emph.end type="italics"/>C <lb/><emph type="italics"/>revolventium Tempus periodicum e&longs;&longs;e ad Tempus periodicum cor­<lb/>poris alterutrius<emph.end type="italics"/>P, <emph type="italics"/>circa alterum immotum<emph.end type="italics"/>S <emph type="italics"/>gyrantis & Figu­<lb/>ris quæ corpora circum &longs;e mutuo de&longs;cribunt Figuram &longs;imilem & <lb/>æqualem de&longs;cribentis, in &longs;ubduplicata ratione corporis alterins<emph.end type="italics"/>S, <lb/><emph type="italics"/>ad &longs;ummam corporum<emph.end type="italics"/>S+P. </s></p> <p type="main"> <s>Namque, ex demon&longs;tratione &longs;uperioris Propo&longs;itionis, tempora <lb/>quibus arcus quivis &longs;imiles <emph type="italics"/>PQ<emph.end type="italics"/>& <emph type="italics"/>pq<emph.end type="italics"/>de&longs;cribuntur, &longs;unt in &longs;ub­<lb/>duplicata ratione di&longs;tantiarum <emph type="italics"/>CP<emph.end type="italics"/>& <emph type="italics"/>SP<emph.end type="italics"/>vel <emph type="italics"/>sp,<emph.end type="italics"/>hoc e&longs;t, in &longs;ub­<lb/>duplicata ratione corporis <emph type="italics"/>S<emph.end type="italics"/>ad &longs;ummam corporum <emph type="italics"/>S+P.<emph.end type="italics"/>Et com­<lb/>ponendo, &longs;ummæ temporum quibus arcus omnes &longs;imiles <emph type="italics"/>PQ<emph.end type="italics"/>& <emph type="italics"/>pq<emph.end type="italics"/><lb/>de&longs;cribuntur, hoc e&longs;t, tempora tota quibus Figuræ totæ &longs;imiles de­<lb/>&longs;cribuntur, &longs;unt in eadem &longs;ubduplicata ratione. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p><pb xlink:href="039/01/179.jpg" pagenum="151"/> <p type="main"> <s><emph type="center"/>PROPOSITIO LX. THEOREMA XXIII.<emph.end type="center"/><lb/><arrow.to.target n="note127"/></s></p> <p type="margin"> <s><margin.target id="note127"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>St corpora duo<emph.end type="italics"/>S <emph type="italics"/>&<emph.end type="italics"/>P, <emph type="italics"/>Viribus quadrato di&longs;tantiæ &longs;uæ reciproee <lb/>proportionalibus &longs;e mutuo trahentia, revalvuntur circa gravi­<lb/>tatis centrum commune: dico quod Ellip&longs;eos, quam corpus al­<lb/>terutrum<emph.end type="italics"/>P <emph type="italics"/>hoc motu circa alterum<emph.end type="italics"/>S <emph type="italics"/>de&longs;cribit, Axis principa­<lb/>lis erit ad Axem principalem Ellip&longs;eos, quam corpus idem<emph.end type="italics"/>P <lb/><emph type="italics"/>circa alterum quie&longs;cens<emph.end type="italics"/>S <emph type="italics"/>eodem tempore periodico de&longs;cribere <lb/>po&longs;&longs;et, ut &longs;umma corporum duorum<emph.end type="italics"/>S+P <emph type="italics"/>ad primam duarum <lb/>medie proportionalium inter hanc &longs;ummam & corpus illud al­<lb/>terum<emph.end type="italics"/>S. </s></p> <p type="main"> <s>Nam &longs;i de&longs;criptæ Ellip&longs;es e&longs;&longs;ent &longs;ibi invicem æquales, tempora <lb/>periodica (per Theorema &longs;uperius) forent in &longs;ubduplicata ratione <lb/>corporis <emph type="italics"/>S<emph.end type="italics"/>ad &longs;ummam corporum <emph type="italics"/>S+P.<emph.end type="italics"/>Minuatur in hac ratione <lb/>tempus periodicum in Ellip&longs;i po&longs;teriore, & tempora periodica eva­<lb/>dent æqualia; Ellip&longs;eos autem axis principalis (per Prop. </s> <s>XV.) minu­<lb/>etur in ratione cujus hæc e&longs;t &longs;e&longs;quiplicata, id e&longs;t in ratione, cujus <lb/>ratio <emph type="italics"/>S<emph.end type="italics"/>ad <emph type="italics"/>S+P<emph.end type="italics"/>e&longs;t triplicata; adeoque erit ad axem principalem <lb/>Ellip&longs;eos alterius, ut prima duarum medie proportionalium inter <lb/><emph type="italics"/>S+P<emph.end type="italics"/>& <emph type="italics"/>S<emph.end type="italics"/>ad <emph type="italics"/>S+P.<emph.end type="italics"/>Et inver&longs;e, axis principalis Ellip&longs;eos circa <lb/>corpus mobile de&longs;criptæ erit ad axem principalem de&longs;criptæ circa <lb/>immobile, ut <emph type="italics"/>S+P<emph.end type="italics"/>ad primam duarum medie proportionalium in­<lb/>ter <emph type="italics"/>S+P<emph.end type="italics"/>& <emph type="italics"/>S. Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXI. THEOREMA XXIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si corpora duo Viribus quibu&longs;vis &longs;e mutuo trahentia, neque alias <lb/>agitata vel impedita, quomodocunque moveantur; motus eo­<lb/>rum perinde &longs;e habebunt ac &longs;i non traherent &longs;e mutuo, &longs;ed u­<lb/>trumque a corpore tertio in communi gravitatis centro con&longs;tituto <lb/>Viribus ii&longs;dem traberetur: Et Virium trahentium eadem erit Lex <lb/>re&longs;pectu di&longs;tantiæ corporum a centro illo communi atque re&longs;pe­<lb/>ctu di&longs;tantiæ totius inter corpora.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam vires illæ, quibus corpora &longs;e mutuo trahunt, tendendo <lb/>ad corpora, tendunt ad commune gravitatis centrum interme-</s></p><pb xlink:href="039/01/180.jpg" pagenum="152"/> <p type="main"> <s><arrow.to.target n="note128"/>dium, adeoque eædem &longs;unt ac &longs;i a corpore intermedio mana­<lb/>rent. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note128"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Et quoniam data e&longs;t ratio di&longs;tantiæ corporis utriu&longs;vis a centro <lb/>illo communi ad di&longs;tantiam corporis eju&longs;dem a corpore altero, da­<lb/>bitur ratio cuju&longs;vis pote&longs;tatis di&longs;tantiæ unius ad eandem pote&longs;ta­<lb/>tem di&longs;tantiæ alterius; ut & ratio quantitatis cuju&longs;vis, quæ ex una <lb/>di&longs;tantia & quantitatibus datis utcunQ.E.D.rivatur, ad quantitatem <lb/>aliam, quæ ex altera di&longs;tantia & quantitatibus totidem datis da­<lb/>tamQ.E.I.lam di&longs;tantiarum rationem ad priores habentibus &longs;imiliter <lb/>derivatur. </s> <s>Proinde &longs;i vis, qua corpus unum ab altero trahitur, &longs;it <lb/>directe vel inver&longs;e ut di&longs;tantia corporum ab invicem; vel ut quæ­<lb/>libet hujus di&longs;tantiæ pote&longs;tas; vel denique ut quantitas quævis ex <lb/>hac di&longs;tantia & quantitatibus datis quomodocunQ.E.D.rivata: erit <lb/>eadem vis, qua corpus idem ad commune gravitatis centrum tra­<lb/>hitur, directe itidem vel inver&longs;e ut corporis attracti di&longs;tantia a cen­<lb/>tro illo communi, vel ut eadem di&longs;tantiæ hujus pote&longs;tas, vel de­<lb/>nique ut quantitas ex hac di&longs;tantia & analogis quantitatibus da­<lb/>tis &longs;imiliter derivata. </s> <s>Hoc e&longs;t, Vis trahentis eadem erit Lex re&longs;pe­<lb/>ctu di&longs;tantiæ utriu&longs;que. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXII. PROBLEMA XXXVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Corporum duorum quæ Viribus quadrato di&longs;tantiæ &longs;uæ reciproce <lb/>proportionalibus &longs;e mutuo trahunt, ac de locis datis demittun­<lb/>tur, determinare Motus.<emph.end type="italics"/></s></p> <p type="main"> <s>Corpora (per Theorema novi&longs;&longs;imum) perinde movebuntur ac <lb/>&longs;i a corpore tertio, in communi gravitatis centro con&longs;tituto, trahe­<lb/>rentur; & centrum illud ip&longs;o motus initio quie&longs;cet per Hypothe­<lb/>&longs;in; & propterea (per Legum Corol. </s> <s>4.) &longs;emper quie&longs;cet. </s> <s>Deter­<lb/>minandi &longs;unt igitur motus corporum (per Prob. </s> <s>XXV,) perinde <lb/>ac &longs;i a viribus ad centrum illud tendentibus urgerentur, & habe­<lb/>buntur motus corporum &longs;e mutuo trahentium. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXIII. PROBLEMA XXXIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Corporum duorum quæ Viribus quadrato di&longs;tantiæ &longs;uæ reciproce pro­<lb/>portionalibus &longs;e mutuo trahunt, deque locis datis, &longs;ecundum datas <lb/>rectas, datis cum Velocitatibus exeunt, determinare Motus.<emph.end type="italics"/></s></p><pb xlink:href="039/01/181.jpg" pagenum="153"/> <p type="main"> <s>Ex datis corporum motibus &longs;ub initio, datur uniformis motus <lb/><arrow.to.target n="note129"/>centri communis gravitatis, ut & motus &longs;patii quod una cum hoc <lb/>centro movetur uniformiter in directum, nec non corporum mo­<lb/>tus initiales re&longs;pectu hujus &longs;patii. </s> <s>Motus autem &longs;ub&longs;equentes <lb/>(per Legum Corollarium quintum, & Theorema novi&longs;&longs;imum) <lb/>perinde fiunt in hoc &longs;patio, ac &longs;i &longs;patium ip&longs;um una cum commu­<lb/>ni illo gravitatis centro quie&longs;ceret, & corpora non traherent &longs;e <lb/>mutuo, &longs;ed a corpore tertio &longs;ito in centro illo traherentur. </s> <s>Cor­<lb/>poris igitur alterutrius in hoc &longs;patio mobili, de loco dato, &longs;ecun­<lb/>dum datam rectam, data cum velocitate exeuntis, & vi centripeta <lb/>ad centrum illud tendente correpti, determinandus e&longs;t motus per <lb/>Problema nonum & vice&longs;imum &longs;extum: & habebitur &longs;imul mo­<lb/>tus corporis alterius e regione. </s> <s>Cum hoc motu componendus <lb/>e&longs;t uniformis ille Sy&longs;tematis &longs;patii & corporum in eo gyrantium <lb/>motus progre&longs;&longs;ivus &longs;upra inventus, & habebitur motus ab&longs;olutus <lb/>corporum in &longs;patio immobili. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note129"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXIV. PROBLEMA XL.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Viribus quibus Corpora &longs;e mutuo trahunt cre&longs;centibus in &longs;implici ra­<lb/>tione di&longs;tantiarum a centris: requiruntur Motus plurium Cor­<lb/>porum inter &longs;e.<emph.end type="italics"/></s></p> <p type="main"> <s>Ponantur primo corpora duo <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>L<emph.end type="italics"/>commune habentia gravi­<lb/>tatis centrum <emph type="italics"/>D.<emph.end type="italics"/>De&longs;cribent hæc (per Corollarium primum Theo­<lb/>rematis XXI) Ellip&longs;es centra habentes in <emph type="italics"/>D,<emph.end type="italics"/>quarum magnitudo ex <lb/>Problemate V, innote&longs;cit. </s></p> <p type="main"> <s>Trahat jam corpus tertium <lb/><figure id="id.039.01.181.1.jpg" xlink:href="039/01/181/1.jpg"/><lb/><emph type="italics"/>S<emph.end type="italics"/>priora duo <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>L<emph.end type="italics"/>viri­<lb/>bus acceleratricibus <emph type="italics"/>ST, SL,<emph.end type="italics"/><lb/>& ab ip&longs;is vici&longs;&longs;im trahatur. </s> <s><lb/>Vis <emph type="italics"/>ST<emph.end type="italics"/>(per Legum Cor. </s> <s>2.) <lb/>re&longs;olvitur in vires <emph type="italics"/>SD, DT<emph.end type="italics"/>; <lb/>& vis <emph type="italics"/>SL<emph.end type="italics"/>in vires <emph type="italics"/>SD, DL.<emph.end type="italics"/><lb/>Vires autem <emph type="italics"/>DT, DL,<emph.end type="italics"/>quæ <lb/>&longs;unt ut ip&longs;arum &longs;umma <emph type="italics"/>TL,<emph.end type="italics"/><lb/>atque adeo ut vires accelera­<lb/>trices quibus corpora <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>L<emph.end type="italics"/>&longs;e mutuo trahunt, additæ his viri­<lb/>bus corporum <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>L,<emph.end type="italics"/>prior priori & po&longs;terior po&longs;teriori, com­<lb/>ponunt vires di&longs;tantiis <emph type="italics"/>DT<emph.end type="italics"/>ac <emph type="italics"/>DL<emph.end type="italics"/>proportionales, ut prius, &longs;ed <pb xlink:href="039/01/182.jpg" pagenum="154"/><arrow.to.target n="note130"/>viribus prioribus majores; adeoque (per Corol. </s> <s>1. Prop. </s> <s>X. & Corol. </s> <s><lb/>1 & 8. Prop, IV) efficiunt ut corpora illa de&longs;cribant Ellip&longs;es ut prius, <lb/>&longs;ed motu celeriore. </s> <s>Vires reliquæ acceleratrices <emph type="italics"/>SD<emph.end type="italics"/>& <emph type="italics"/>SD,<emph.end type="italics"/>actio­<lb/>nibus motricibus <emph type="italics"/>SDXT<emph.end type="italics"/>& <emph type="italics"/>SDXL,<emph.end type="italics"/>quæ &longs;unt ut corpora, tra­<lb/>hendo corpora illa æqualiter & &longs;ecundum lineas <emph type="italics"/>TI, LK,<emph.end type="italics"/>ip&longs;i <emph type="italics"/>DS<emph.end type="italics"/><lb/>parallelas, nil mutant &longs;itus eorum ad invicem, &longs;ed faciunt ut ip&longs;a <lb/>æqualiter accedant ad lineam <emph type="italics"/>IK<emph.end type="italics"/>; quam ductam concipe per me­<lb/>dium corporis <emph type="italics"/>S,<emph.end type="italics"/>& lineæ <emph type="italics"/>DS<emph.end type="italics"/>perpendicularem. </s> <s>Impedietur au­<lb/>tem i&longs;te ad lineam <emph type="italics"/>IK<emph.end type="italics"/>acce&longs;&longs;us faciendo ut Sy&longs;tema corporum <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>L<emph.end type="italics"/><lb/>ex una parte, & corpus <emph type="italics"/>S<emph.end type="italics"/>ex altera, ju&longs;tis cum velocitatibus, gyren­<lb/>tur circa commune gravitatis centrum <emph type="italics"/>C.<emph.end type="italics"/>Tali motu corpus <emph type="italics"/>S<emph.end type="italics"/><lb/>(eo quod &longs;umma virium motricium <emph type="italics"/>SDXT<emph.end type="italics"/>& <emph type="italics"/>SDXL,<emph.end type="italics"/>di&longs;tan­<lb/>tiæ <emph type="italics"/>CS<emph.end type="italics"/>proportionalium, tendit ver&longs;us centrum <emph type="italics"/>C<emph.end type="italics"/>) de&longs;cribit El­<lb/>lip&longs;in circa idem <emph type="italics"/>C;<emph.end type="italics"/>& punctum <emph type="italics"/>D,<emph.end type="italics"/>ob proportionales <emph type="italics"/>CS, CD,<emph.end type="italics"/><lb/>de&longs;cribet Ellip&longs;in con&longs;imilem e regione. </s> <s>Corpora autem <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>L<emph.end type="italics"/><lb/>viribus motricibus <emph type="italics"/>SDXT<emph.end type="italics"/><lb/><figure id="id.039.01.182.1.jpg" xlink:href="039/01/182/1.jpg"/><lb/>& <emph type="italics"/>SDXL,<emph.end type="italics"/>(prius priore, <lb/>po&longs;terius po&longs;teriore) æqua­<lb/>liter & &longs;ecundum lineas pa­<lb/>rallelas <emph type="italics"/>TI<emph.end type="italics"/>& <emph type="italics"/>LK<emph.end type="italics"/>(ut dic­<lb/>tum e&longs;t) attracta, pergent <lb/>(per Legum Corollarium <lb/>quintum & &longs;extum) circa cen­<lb/>trum mobile <emph type="italics"/>D<emph.end type="italics"/>Ellip&longs;es &longs;uas <lb/>de&longs;cribere, ut prius. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note130"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Addatur jam corpus quartum <emph type="italics"/>V,<emph.end type="italics"/>& &longs;imili argumento conclude­<lb/>tur hoc & punctum <emph type="italics"/>C<emph.end type="italics"/>Ellip&longs;es circa omnium commune centrum <lb/>gravitatis <emph type="italics"/>B<emph.end type="italics"/>de&longs;cribere; manentibus motibus priorum corporum <lb/><emph type="italics"/>T, L<emph.end type="italics"/>& <emph type="italics"/>S<emph.end type="italics"/>circa centra <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>C,<emph.end type="italics"/>&longs;ed paulo acceleratis. </s> <s>Et eadem <lb/>methodo corpora plura adjungere licebit. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s>Hæc ita &longs;e habent ubi corpora <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>L<emph.end type="italics"/>trahunt &longs;e mutuo viribus <lb/>acceleratricibus majoribus vel minoribus quam quibus trahunt cor­<lb/>pora reliqua pro ratione di&longs;tantiarum. </s> <s>Sunto mutuæ omnium at­<lb/>tractiones acceleratrices ad invicem ut di&longs;tantiæ ductæ in corpo­<lb/>ra trahentia, & ex præcedentibus facile deducetur quod corpora <lb/>omnia æqualibus temporibus periodicis Ellip&longs;es varias, circa om­<lb/>nium commune gravitatis centrum <emph type="italics"/>B,<emph.end type="italics"/>in plano immobili de&longs;cri­<lb/>bunt. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p><pb xlink:href="039/01/183.jpg" pagenum="155"/> <p type="main"> <s><emph type="center"/>PROPOSITIO LXV. THEOREMA XXV.<emph.end type="center"/><lb/><arrow.to.target n="note131"/></s></p> <p type="margin"> <s><margin.target id="note131"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corpora plura, quorum Vires decre&longs;cunt in duplicata ratione di­<lb/>&longs;tantiarum ab eorundem centris, moveri po&longs;&longs;e inter &longs;e in El­<lb/>lip&longs;ibus; & radiis ad umbilicos ductis areas de&longs;cribere tempo­<lb/>ribus proportionales quam proxime.<emph.end type="italics"/></s></p> <p type="main"> <s>In Propo&longs;itione &longs;uperiore demon&longs;tratus e&longs;t ca&longs;us ubi motus plu­<lb/>res peraguntur in Ellip&longs;ibus accurate. </s> <s>Quo magis recedit Lex vi­<lb/>rium a Lege ibi po&longs;ita, eo magis corpora perturbabunt mutuos <lb/>motus; neque fieri pote&longs;t ut corpora, &longs;ecundum Legem hic po&longs;itam <lb/>&longs;e mutuo trahentia, moveantur in Ellip&longs;ibus accurate, ni&longs;i &longs;ervando <lb/>certam proportionem di&longs;tantiarum ab invicem. </s> <s>In &longs;equentibus au­<lb/>tem ca&longs;ibus non multum ab Ellip&longs;ibus errabitur. </s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Pone corpora plura minora circa maximum aliquod ad <lb/>varias ab eo di&longs;tantias revolvi, tendantque ad &longs;ingula vires ab&longs;olu­<lb/>tæ proportionales ii&longs;dem corporibus. </s> <s>Et quoniam omnium com­<lb/>mune gravitatis centrum (per Legum Corol. </s> <s>quartum) vel quie­<lb/>&longs;cit vel movetur uniformiter in directum, fingamus corpora mi­<lb/>nora tam parva e&longs;&longs;e, ut corpus maximum nunquam di&longs;tet &longs;en&longs;ibi­<lb/>liter ab hoc centro: & maximum illud vel quie&longs;cet vel movebitur <lb/>uniformiter in directum, ab&longs;que errore &longs;en&longs;ibili; minora autem re­<lb/>volventur circa hoc maximum in Ellip&longs;ibus, atque radiis ad idem <lb/>ductis de&longs;cribent areas temporibus proportionales; ni&longs;i quatenus <lb/>errores inducuntur, vel per errorem maximi a communi illo gravi­<lb/>tatis centro, vel per actiones minorum corporum in &longs;e mutuo. </s> <s>Di­<lb/>minui autem po&longs;&longs;unt corpora minora u&longs;Q.E.D.nec error i&longs;te & ac­<lb/>tiones mutuæ &longs;int datis quibu&longs;vis minores, atque adeo donec Orbes <lb/>cum Ellip&longs;ibus quadrent, & areæ re&longs;pondeant temporibus, ab&longs;que <lb/>errore qui non &longs;it minor quovis dato. <emph type="italics"/>q.E.O.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Fingamus jam Sy&longs;tema corporum minorum modo jam <lb/>de&longs;cripto circa maximum revolventium, aliudve quodvis duorum <lb/>circum &longs;e mutuo revolventium corporum Sy&longs;tema progredi unifor­<lb/>miter in directum, & interea vi corporis alterius longe maximi & <lb/>ad magnam di&longs;tantiam &longs;iti urgeri ad latus. </s> <s>Et quoniam æquales <lb/>vires acceleratrices, quibus corpora &longs;ecundum lineas parallelas ur­<lb/>gentur, non mutant &longs;itus corporum ad invicem, &longs;ed ut Sy&longs;tema <lb/>totum, &longs;ervatis partium motibus inter &longs;e, &longs;imul transferatur effici­<lb/>unt: manife&longs;tum e&longs;t quod, ex attractionibus in corpus maximum, </s></p><pb xlink:href="039/01/184.jpg" pagenum="156"/> <p type="main"> <s><arrow.to.target n="note132"/>nulla pror&longs;us orietur mutatio motus attractorum inter &longs;e, ni&longs;i vel <lb/>ex attractionum acceleratricum inæqualitate, vel ex inclinatione li­<lb/>nearum ad invicem, &longs;ecundum quas attractiones fiunt. </s> <s>Pone ergo <lb/>attractiones omnes acceleratrices in corpus maximum e&longs;&longs;e inter &longs;e <lb/>reciproce ut quadrata di&longs;tantiarum; &, augendo corporis maximi <lb/>di&longs;tantiam, donec rectarum ab hoc ad reliqua ductarum differen­<lb/>tiæ re&longs;pectu earum longitudinis & inclinationes ad invicem mino­<lb/>res &longs;int quam datæ quævis, per&longs;everabunt motus partium Sy&longs;tema­<lb/>tis inter &longs;e ab&longs;que erroribus qui non &longs;int quibu&longs;vis datis minores. </s> <s><lb/>Et quoniam, ob exiguam partium illarum ab invicem di&longs;tantiam, <lb/>Sy&longs;tema totum ad modum corporis unius attrahitur; movebitur <lb/>idem hac attractione ad modum corporis unius; hoc e&longs;t, centro <lb/>&longs;uo gravitatis de&longs;cribet circa corpus maximum Sectionem aliquam <lb/>Conicam (<emph type="italics"/>viz.<emph.end type="italics"/>Hyperbolam vel Parabolam attractione languida, <lb/>Ellip&longs;in fortiore,) & Radio ad maximum ducto de&longs;cribet areas <lb/>temporibus proportionales, ab&longs;que ullis erroribus, ni&longs;i quas par­<lb/>tium di&longs;tantiæ (perexiguæ &longs;ane & pro lubitu minuendæ) valeant <lb/>efficere. <emph type="italics"/>q.E.O.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note132"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Simili argumento pergere licet ad ca&longs;us magis compo&longs;itos in in­<lb/>finitum. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. In ca&longs;u &longs;ecundo; quo propius accedit corpus omnium <lb/>maximum ad Sy&longs;tema duorum vel plurium, eo magis turbabuntur <lb/>motus partium Sy&longs;tematis inter &longs;e; propterea quod linearum a cor­<lb/>pore maximo ad has ductarum jam major e&longs;t inclinatio ad invicem, <lb/>majorque proportionis inæqualitas. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Maxime autem turbabuntur, ponendo quod attractio­<lb/>nes acceleratrices partium Sy&longs;tematis ver&longs;us corpus omnium maxi­<lb/>mum, non &longs;int ad invicem reciproce ut quadrata di&longs;tantiarum a <lb/>corpore illo maximo; præ&longs;ertim &longs;i proportionis hujus inæqualitas <lb/>major &longs;it quam inæqualitas proportionis di&longs;tantiarum a corpore <lb/>maximo: Nam &longs;i vis acceleratrix, æqualiter & &longs;ecundum lineas pa­<lb/>rallelas agendo, nil perturbat motus inter &longs;e, nece&longs;&longs;e e&longs;t ut ex acti­<lb/>onis inæqualitate perturbatio oriatur, majorque &longs;it vel minor pro <lb/>majore vel minore inæqualitate. </s> <s>Exce&longs;&longs;us impul&longs;uum majorum, <lb/>agendo in aliqua corpora & non agendo in alia, nece&longs;&longs;ario muta­<lb/>bunt &longs;itum eorum inter &longs;e. </s> <s>Et hæc perturbatio, addita perturbatio­<lb/>ni quæ ex linearum inclinatione & inæqualitate oritur, majorem <lb/>reddet perturbationem totam. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Unde &longs;i Sy&longs;tematis hujus partes in Ellip&longs;ibus vel Cir­<lb/>culis &longs;ine perturbatione in&longs;igni moveantur; manife&longs;tum e&longs;t, quod <pb xlink:href="039/01/185.jpg" pagenum="157"/>eædem a viribus acceleratricibus ad alia corpora tendentibus, aut <lb/><arrow.to.target n="note133"/>non urgentur ni&longs;i levi&longs;&longs;ime, aut urgentur æqualiter & &longs;ecundum li­<lb/>neas parallelas quamproxime. </s></p> <p type="margin"> <s><margin.target id="note133"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXVI. THEOREMA XXVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Corpora tria, quorum Vires decre&longs;cunt in duplicata ratione di­<lb/>&longs;tantiarum, &longs;e mutuo trahant, & attractiones acceleratrices bi­<lb/>norum quorumcunQ.E.I. tertium &longs;int inter &longs;e reciproce ut qua­<lb/>drata di&longs;tantiarum; minora autem circa maximum revolvan­<lb/>tur: Dico quod interius circa intimum & maximum, radiis <lb/>ad ip&longs;um ductis, de&longs;cribet areas temporibus magis proportio­<lb/>nales, & Figuram ad formam Ellip&longs;eos umbilicum in concur­<lb/>&longs;u radiorum habentis magis accedentem, &longs;i corpus maximum <lb/>his attractionibus agitetur, quam &longs;i maximum illud vel a mi­<lb/>noribus non attractum quie&longs;cat, vel multo minus vel multo ma­<lb/>gis attractum aut multo minus aut multo magis agitetur.<emph.end type="italics"/></s></p> <p type="main"> <s>Liquet fere ex demon&longs;tratione Corollarii &longs;ecundi Propo&longs;itionis <lb/>præcedentis; &longs;ed argumento magis di&longs;tincto & latius cogente &longs;ic <lb/>evincitur. </s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Revolvantur <lb/><figure id="id.039.01.185.1.jpg" xlink:href="039/01/185/1.jpg"/><lb/>corpora minora <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>S<emph.end type="italics"/><lb/>in eodem plano circa <lb/>maximum <emph type="italics"/>T,<emph.end type="italics"/>quorum <lb/><emph type="italics"/>P<emph.end type="italics"/>de&longs;cribat Orbem in­<lb/>teriorem <emph type="italics"/>PAB,<emph.end type="italics"/>& <emph type="italics"/>S<emph.end type="italics"/><lb/>exteriorem <emph type="italics"/>SE.<emph.end type="italics"/>Sit <lb/><emph type="italics"/>SK<emph.end type="italics"/>mediocris di&longs;tan­<lb/>tia corporum <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>S<emph.end type="italics"/>; <lb/>& corporis <emph type="italics"/>P<emph.end type="italics"/>ver&longs;us <lb/><emph type="italics"/>S<emph.end type="italics"/>attractio acceleratrix in mediocri illa di&longs;tantia exponatur per e­<lb/>andem. </s> <s>In duplicata ratione <emph type="italics"/>SK<emph.end type="italics"/>ad <emph type="italics"/>SP<emph.end type="italics"/>capiatur <emph type="italics"/>SL<emph.end type="italics"/>ad <emph type="italics"/>SK,<emph.end type="italics"/>& e­<lb/>rit <emph type="italics"/>SL<emph.end type="italics"/>attractio acceleratrix corporis <emph type="italics"/>P<emph.end type="italics"/>ver&longs;us <emph type="italics"/>S<emph.end type="italics"/>in di&longs;tantia quavis <lb/><emph type="italics"/>SP.<emph.end type="italics"/>Junge <emph type="italics"/>PT,<emph.end type="italics"/>eique parallelam age <emph type="italics"/>LM<emph.end type="italics"/>occurrentem <emph type="italics"/>ST<emph.end type="italics"/>in <emph type="italics"/>M,<emph.end type="italics"/><lb/>& attractio <emph type="italics"/>SL<emph.end type="italics"/>re&longs;olvetur (per Legum Corol 2.) in attractiones <lb/><emph type="italics"/>SM, LM.<emph.end type="italics"/>Et &longs;ic urgebitur corpus <emph type="italics"/>P<emph.end type="italics"/>vi acceleratrice triplici: <pb xlink:href="039/01/186.jpg" pagenum="158"/><arrow.to.target n="note134"/>una tendente ad <emph type="italics"/>T<emph.end type="italics"/>& oriunda a mutua attractione corporum <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>P.<emph.end type="italics"/><lb/>Hac vi &longs;ola corpus <emph type="italics"/>P<emph.end type="italics"/>circum corpus <emph type="italics"/>T,<emph.end type="italics"/>&longs;ive immotum &longs;ive hac <lb/>attractione agitatum, de&longs;cribere deberet & areas, radio <emph type="italics"/>PT,<emph.end type="italics"/>tem­<lb/>poribus proportionales, & Ellip&longs;in cui umbilicus e&longs;t in centro cor­<lb/>poris <emph type="italics"/>T.<emph.end type="italics"/>Patet hoc per Prop. </s> <s>XI. & Corollaria 2 & 3 Theor. </s> <s>XXI. </s> <s>Vis <lb/>altera e&longs;t attractionis <emph type="italics"/>LM,<emph.end type="italics"/>quæ quoniam tendit a <emph type="italics"/>P<emph.end type="italics"/>ad <emph type="italics"/>T,<emph.end type="italics"/>&longs;uperad­<lb/>dita vi priori coincidet cum ip&longs;a, & &longs;ic faciet ut areæ etiamnum tem­<lb/>poribus proportionales de&longs;cribantur per Corol. </s> <s>3. Theor. </s> <s>XXI. </s> <s>At <lb/>quoniam non e&longs;t quadrato di&longs;tantiæ <emph type="italics"/>PT<emph.end type="italics"/>reciproce proportionalis, <lb/>componet ea cum vi priore vim ab hac proportione aberrantem, id­<lb/>que eo magis quo major e&longs;t proportio hujus vis ad vim priorem, <lb/>cæteris paribus. </s> <s>Proinde cum (per Prop. </s> <s>XI, & per Corol. </s> <s>2. <lb/>Theor. </s> <s>XXI) vis qua Ellip&longs;is circa umbilicum <emph type="italics"/>T<emph.end type="italics"/>de&longs;cribitur tendere <lb/>debeat ad umbilicum illum, & e&longs;&longs;e quadrato di&longs;tantiæ <emph type="italics"/>PT<emph.end type="italics"/>reciproce <lb/>proportionalis; vis illa <lb/><figure id="id.039.01.186.1.jpg" xlink:href="039/01/186/1.jpg"/><lb/>compo&longs;ita, aberrando <lb/>ab hac proportione, fa­<lb/>ciet ut Orbis <emph type="italics"/>PAB<emph.end type="italics"/><lb/>aberret a forma Ellip­<lb/>&longs;eos umbilicum haben­<lb/>tis in <emph type="italics"/>S;<emph.end type="italics"/>idque eo ma­<lb/>gis quo major e&longs;t ab­<lb/>erratio ab hac propor­<lb/>tione; atque adeo eti­<lb/>am quo major e&longs;t proportio vis &longs;ecundæ <emph type="italics"/>LM<emph.end type="italics"/>ad vim primam, cæ­<lb/>teris paribus. </s> <s>Jam vero vis tertia <emph type="italics"/>SM,<emph.end type="italics"/>trahendo corpus <emph type="italics"/>P<emph.end type="italics"/>&longs;ecun­<lb/>dum lineam ip&longs;i <emph type="italics"/>ST<emph.end type="italics"/>parallelam, componet cum viribus prioribus <lb/>vim quæ non amplius dirigitur a <emph type="italics"/>P<emph.end type="italics"/>in <emph type="italics"/>T,<emph.end type="italics"/>quæque ab hac determi­<lb/>natione tanto magis aberrat, quanto major e&longs;t proportio hujus ter­<lb/>tiæ vis ad vires priores, cæteris paribus; atque adeo quæ faciet ut <lb/>corpus <emph type="italics"/>P,<emph.end type="italics"/>radio <emph type="italics"/>TP,<emph.end type="italics"/>areas non amplius temporibus proportiona­<lb/>les de&longs;cribat, atque aberratio ab hac proportionalitate ut tanto ma­<lb/>jor &longs;it, quanto major e&longs;t proportio vis hujus tertiæ ad vires cæte­<lb/>ras. </s> <s>Orbis vero <emph type="italics"/>PAB<emph.end type="italics"/>aberrationem a forma Elliptica præfata hæc­<lb/>vis tertia duplici de cau&longs;a adaugebit, tum quod non dirigatur a <emph type="italics"/>P<emph.end type="italics"/><lb/>ad <emph type="italics"/>T,<emph.end type="italics"/>tum etiam quod non &longs;it proportionalis quadrato di&longs;tantiæ <emph type="italics"/>PT.<emph.end type="italics"/><lb/>Quibus intellectis, manife&longs;tum e&longs;t quod areæ temporibus tum max­<lb/>ime fiunt proportionales, ubi vis tertia, manentibus viribus cæte­<lb/>ris, fit minima; & quod Orbis <emph type="italics"/>PAB<emph.end type="italics"/>tum maxime accedit ad præ­<lb/>fatam formam Ellipticam, ubi vis tam &longs;ecunda quam tertia, &longs;ed præ­<lb/>cipue vis tertia, fit minima, vi prima manente. </s></p><pb xlink:href="039/01/187.jpg" pagenum="159"/> <p type="margin"> <s><margin.target id="note134"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Exponatur corporis <emph type="italics"/>T<emph.end type="italics"/>attractio acceleratrix ver&longs;us <emph type="italics"/>S<emph.end type="italics"/>per lineam <lb/><arrow.to.target n="note135"/><emph type="italics"/>SN;<emph.end type="italics"/>& &longs;i attractiones acceleratrices <emph type="italics"/>SM, SN<emph.end type="italics"/>æquales e&longs;&longs;ent; hæ, <lb/>trahendo corpora <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>æqualiter & &longs;ecundum lineas parallelas, <lb/>nil mutarent &longs;itum eorum ad invicem. </s> <s>Iidem jam forent corporum <lb/>illorum motus inter &longs;e (per Legum Corol. </s> <s>6.) ac &longs;i hæ attractio­<lb/>nes tollerentur. </s> <s>Et pari ratione &longs;i attractio <emph type="italics"/>SN<emph.end type="italics"/>minor e&longs;&longs;et at­<lb/>tractione <emph type="italics"/>SM,<emph.end type="italics"/>tolleret ip&longs;a attractionis <emph type="italics"/>SM<emph.end type="italics"/>partem <emph type="italics"/>SN,<emph.end type="italics"/>& ma­<lb/>neret pars &longs;ola <emph type="italics"/>MN,<emph.end type="italics"/>qua temporum & arearum proportionalitas <lb/>& Orbitæ forma illa Elliptica perturbaretur. </s> <s>Et &longs;imiliter &longs;i attra­<lb/>ctio <emph type="italics"/>SN<emph.end type="italics"/>major e&longs;&longs;et attractione <emph type="italics"/>SM,<emph.end type="italics"/>oriretur ex differentia &longs;ola <lb/><emph type="italics"/>MN<emph.end type="italics"/>perturbatio proportionalitatis & Orbitæ. </s> <s>Sic per attractio­<lb/>nem <emph type="italics"/>SN<emph.end type="italics"/>reducitur &longs;emper attractio tertia &longs;uperior <emph type="italics"/>SM<emph.end type="italics"/>ad attra­<lb/>ctionem <emph type="italics"/>MN,<emph.end type="italics"/>attractione prima & &longs;ecunda manentibus pror&longs;us im­<lb/>mutatis: & propterea areæ ac tempora ad proportionalitatem, & <lb/>Orbita <emph type="italics"/>PAB<emph.end type="italics"/>ad formam præfatam Ellipticam tum maxime acce­<lb/>dunt, ubi attractio <emph type="italics"/>MN<emph.end type="italics"/>vel nulla e&longs;t, vel quam fieri po&longs;&longs;it miNI­<lb/>ma; hoc e&longs;t, ubi corporum <emph type="italics"/>P & T<emph.end type="italics"/>attractiones acceleratrices, fa­<lb/>ctæ ver&longs;us corpus <emph type="italics"/>S,<emph.end type="italics"/>accedunt quantum fieri pote&longs;t ad æqualita­<lb/>tem; id e&longs;t, ubi attractio <emph type="italics"/>SN<emph.end type="italics"/>non e&longs;t nulla, neque minor minima <lb/>attractionum omnium <emph type="italics"/>SM,<emph.end type="italics"/>&longs;ed inter attractionum omnium <emph type="italics"/>SM<emph.end type="italics"/><lb/>maximam & minimam qua&longs;i mediocris, hoc e&longs;t, non multo major <lb/>neque multo minor attractione <emph type="italics"/>SK. Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note135"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Revolvantur jam corpora minora <emph type="italics"/>P, S<emph.end type="italics"/>circa maximum <emph type="italics"/>T<emph.end type="italics"/><lb/>in planis diver&longs;is; & vis <emph type="italics"/>LM,<emph.end type="italics"/>agendo &longs;ecundum lineam <emph type="italics"/>PT<emph.end type="italics"/>in pla­<lb/>no Orbitæ <emph type="italics"/>PAB<emph.end type="italics"/>&longs;itam, eundem habebit effectum ac prius, neque <lb/>corpus <emph type="italics"/>P<emph.end type="italics"/>de plano Orbitæ &longs;uæ deturbabit. </s> <s>At vis altera <emph type="italics"/>NM,<emph.end type="italics"/><lb/>agendo &longs;ecundum lineam quæ ip&longs;i <emph type="italics"/>ST<emph.end type="italics"/>parallela e&longs;t, (atque adco, <lb/>quando corpus <emph type="italics"/>S<emph.end type="italics"/>ver&longs;atur extra lineam Nodorum, inclinatur ad <lb/>planum Orbitæ <emph type="italics"/>PAB<emph.end type="italics"/>;) præter perturbationem motus in Longitu­<lb/>dinem jam ante expo&longs;itam, inducet perturbationem motus in Lati­<lb/>tudinem, trahendo corpus <emph type="italics"/>P<emph.end type="italics"/>de plano &longs;uæ Orbitæ. </s> <s>Et hæc per­<lb/>turbatio, in dato quovis corporum <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>T<emph.end type="italics"/>ad invicem &longs;itu, erit ut <lb/>vis illa generans <emph type="italics"/>MN,<emph.end type="italics"/>adeoque minima evadet ubi <emph type="italics"/>MN<emph.end type="italics"/>e&longs;t miNI­<lb/>ma, hoc e&longs;t (uti jam expo&longs;ui) ubi attractio <emph type="italics"/>SN<emph.end type="italics"/>non e&longs;t multo ma­<lb/>jor, neque multo minor attractione <emph type="italics"/>SK. Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Ex his facile colligitur quod, &longs;i corpora plura minora <lb/><emph type="italics"/>P, S, R,<emph.end type="italics"/>&c. </s> <s>revolvantur circa maximum <emph type="italics"/>T,<emph.end type="italics"/>motus corporis inti­<lb/>mi <emph type="italics"/>P<emph.end type="italics"/>minime perturbabitur attractionibus exteriorum, ubi corpus <lb/>maximum <emph type="italics"/>T<emph.end type="italics"/>pariter a cæteris, pro ratione virium acceleratricum, <lb/>attrahitur & agitatur atque cætera a &longs;e mutuo. <pb xlink:href="039/01/188.jpg" pagenum="160"/><arrow.to.target n="note136"/></s></p> <p type="margin"> <s><margin.target id="note136"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. In Sy&longs;temate vero trium corporum <emph type="italics"/>T, P, S,<emph.end type="italics"/>&longs;i attracti­<lb/>ones acceleratrices binorum quorumcunQ.E.I. tertium &longs;int ad invi­<lb/>cem reciproce ut quadrata di&longs;tantiarum; corpus <emph type="italics"/>P,<emph.end type="italics"/>radio <emph type="italics"/>PT,<emph.end type="italics"/>are­<lb/>am circa corpus <emph type="italics"/>T<emph.end type="italics"/>velocius de&longs;cribet prope Conjunctionem <emph type="italics"/>A<emph.end type="italics"/>& Op­<lb/>po&longs;itionem <emph type="italics"/>B,<emph.end type="italics"/>quam prope Quadraturas <emph type="italics"/>C, D.<emph.end type="italics"/>Namque vis omnis <lb/>qua corpus <emph type="italics"/>P<emph.end type="italics"/>urgetur & corpus <emph type="italics"/>T<emph.end type="italics"/>non urgetur, quæque non agit <lb/>&longs;ecundum lineam <emph type="italics"/>PT<emph.end type="italics"/>accelerat vel retardat de&longs;criptionem areæ, <lb/>perinde ut ip&longs;a in con&longs;equentia vel in antecedentia dirigitur. </s> <s>Talis <lb/>e&longs;t vis <emph type="italics"/>NM.<emph.end type="italics"/>Hæc in tran&longs;itu corporis <emph type="italics"/>P<emph.end type="italics"/>a <emph type="italics"/>C<emph.end type="italics"/>ad <emph type="italics"/>A<emph.end type="italics"/>tendit in con­<lb/>&longs;equentia, motumque accelerat; dein u&longs;que ad <emph type="italics"/>D<emph.end type="italics"/>in antecedentia, <lb/>& motum retardat; tum in con&longs;equentia u&longs;que ad <emph type="italics"/>B,<emph.end type="italics"/>& ultimo in <lb/>antecedentia tran&longs;eundo a <emph type="italics"/>B<emph.end type="italics"/>ad <emph type="italics"/>C.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Et eodem argumento patet quod corpus <emph type="italics"/>P,<emph.end type="italics"/>cæteris pa­<lb/>ribus, velocius movetur in Conjunctione & Oppo&longs;itione quam in <lb/>Quadraturis. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Orbita corporis <emph type="italics"/>P,<emph.end type="italics"/>cæteris paribus, curvior e&longs;t in Qua­<lb/>draturis quam in Conjunctione & Oppo&longs;itione. </s> <s>Nam corpora ve­<lb/>lociora minus deflec­<lb/><figure id="id.039.01.188.1.jpg" xlink:href="039/01/188/1.jpg"/><lb/>tunt a recto tramite. </s> <s>Et <lb/>præterea vis <emph type="italics"/>KL<emph.end type="italics"/>vel <lb/><emph type="italics"/>NM,<emph.end type="italics"/>in Conjunctione <lb/>& Oppo&longs;itione, con­<lb/>traria e&longs;t vi qua cor­<lb/>pus <emph type="italics"/>T<emph.end type="italics"/>trahit corpus <emph type="italics"/>P,<emph.end type="italics"/><lb/>adeoque vim illam mi­<lb/>nuit; corpus autem <emph type="italics"/>P<emph.end type="italics"/><lb/>minus deflectet a recto <lb/>tramite, ubi minus urgetur in corpus <emph type="italics"/>T.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Unde corpus <emph type="italics"/>P,<emph.end type="italics"/>cæteris paribus, longius recedet a cor­<lb/>pore <emph type="italics"/>T<emph.end type="italics"/>in Quadraturis, quam in Conjunctione & Oppo&longs;itione. </s> <s>Hæc <lb/>ita &longs;e habent exclu&longs;o motu Excentricitatis. </s> <s>Nam &longs;i Orbita corpo­<lb/>ris <emph type="italics"/>P<emph.end type="italics"/>excentrica &longs;it: Excentricitas ejus (ut mox in hujus Corol. </s> <s>9. <lb/>o&longs;tendetur) evadet maxima ubi Ap&longs;ides &longs;unt in Syzygiis; indeque <lb/>fieri pote&longs;t ut corpus <emph type="italics"/>P,<emph.end type="italics"/>ad Ap&longs;idem &longs;ummam appellans, ab&longs;it lon­<lb/>gius a corpore <emph type="italics"/>T<emph.end type="italics"/>in Syzygiis quam in Quadraturis. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Quoniam vis centripeta corporis centralis <emph type="italics"/>T,<emph.end type="italics"/>qua cor­<lb/>pus <emph type="italics"/>P<emph.end type="italics"/>retinetur in Orbe &longs;uo, augetur in Quadraturis per additio­<lb/>nem vis <emph type="italics"/>LM,<emph.end type="italics"/>ac diminuitur in Syzygiis per ablationem vis <emph type="italics"/>KL,<emph.end type="italics"/>& <lb/>ob magnitudinem vis <emph type="italics"/>KL,<emph.end type="italics"/>magis diminuitur quam augetur; e&longs;t au­<lb/>tem vis illa centripeta (per Corol. </s> <s>2, Prop. </s> <s>IV.) in ratione compo­<lb/>&longs;ita ex ratione &longs;implici radii <emph type="italics"/>TP<emph.end type="italics"/>directe & ratione duplicata tempo-<pb xlink:href="039/01/189.jpg" pagenum="161"/>ris periodici inver&longs;e: patet hanc rationem compo&longs;itam diminui per </s></p> <p type="main"> <s><arrow.to.target n="note137"/>actionem vis <emph type="italics"/>KL,<emph.end type="italics"/>adeoque tempus periodicum, &longs;i maneat Orbis <lb/>radius <emph type="italics"/>TP,<emph.end type="italics"/>augeri, idQ.E.I. &longs;ubduplicata ratione qua vis illa cen­<lb/>tripeta diminuitur: auctoque adeo vel diminuto hoc Radio, tem­<lb/>pus periodicum augeri magis, vel diminui minus quam in Radii hu­<lb/>jus ratione &longs;e&longs;quiplicata, per Corol. </s> <s>6. Prop. </s> <s>IV. </s> <s>Si vis illa corporis <lb/>centralis paulatim langue&longs;ceret, corpus <emph type="italics"/>P<emph.end type="italics"/>minus &longs;emper & minus <lb/>attractum perpetuo recederet longius a centro <emph type="italics"/>T<emph.end type="italics"/>; & contra, &longs;i vis <lb/>illa augeretur, accederet propius. </s> <s>Ergo &longs;i actio corporis longin­<lb/>qui <emph type="italics"/>S,<emph.end type="italics"/>qua vis illa diminuitur, augeatur ac diminuatur per vices; <lb/>augebitur &longs;imul ac diminuetur Radius <emph type="italics"/>TP<emph.end type="italics"/>per vices, & tempus pe­<lb/>riodicum augebitur ac diminuetur in ratione compo&longs;ita ex ratione <lb/>&longs;e&longs;quiplicata Radii & ratione &longs;ubduplicata qua vis illa centripeta <lb/>corporis centralis <emph type="italics"/>T,<emph.end type="italics"/>per incrementum vel decrementum actionis <lb/>corporis longinqui <emph type="italics"/>S,<emph.end type="italics"/>diminuitur vel augetur. </s></p> <p type="margin"> <s><margin.target id="note137"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. Ex præmi&longs;&longs;is con&longs;equitur etiam quod Ellip&longs;eos a cor­<lb/>pore <emph type="italics"/>P<emph.end type="italics"/>de&longs;criptæ Axis, &longs;eu Ap&longs;idum linea, quoad motum angula­<lb/>rem progreditur & regreditur per vices, &longs;ed magis tamen progre­<lb/>ditur, & in &longs;ingulis corporis revolutionibus per exce&longs;&longs;um progre&longs;­<lb/>&longs;ionis fertur in con&longs;equentia. </s> <s>Nam vis qua corpus <emph type="italics"/>P<emph.end type="italics"/>urgetur in <lb/>corpus <emph type="italics"/>T<emph.end type="italics"/>in Quadraturis, ubi vis <emph type="italics"/>MN<emph.end type="italics"/>evanuit, componitur ex vi <lb/><emph type="italics"/>LM<emph.end type="italics"/>& vi centripeta qua corpus <emph type="italics"/>T<emph.end type="italics"/>trahit corpus <emph type="italics"/>P.<emph.end type="italics"/>Vis prior <emph type="italics"/>LM,<emph.end type="italics"/><lb/>&longs;i augeatur di&longs;tantia <emph type="italics"/>PT,<emph.end type="italics"/>augetur in eadem fere ratione cum hac <lb/>di&longs;tantia, & vis po&longs;terior decre&longs;cit in duplicata illa ratione, adeo­<lb/>que &longs;umma harum virium decre&longs;cit in minore quam duplicata ra­<lb/>tione di&longs;tantiæ <emph type="italics"/>PT,<emph.end type="italics"/>& propterea (per Corol. </s> <s>1. Prop. </s> <s>XLV) efficit <lb/>ut Aux, &longs;eu Ap&longs;is &longs;umma, regrediatur. </s> <s>In Conjunctione vero & <lb/>Oppo&longs;itione, vis qua corpus <emph type="italics"/>P<emph.end type="italics"/>urgetur in corpus <emph type="italics"/>T<emph.end type="italics"/>differentia e&longs;t <lb/>inter vim qua corpus <emph type="italics"/>T<emph.end type="italics"/>trahit corpus <emph type="italics"/>P<emph.end type="italics"/>& vim <emph type="italics"/>KL<emph.end type="italics"/>; & differen­<lb/>tia illa, propterea quod vis <emph type="italics"/>KL<emph.end type="italics"/>augetur quamproxime in ratione <lb/>di&longs;tantiæ <emph type="italics"/>PT,<emph.end type="italics"/>decre&longs;cit in majore quam duplicata ratione di&longs;tan­<lb/>tiæ <emph type="italics"/>PT,<emph.end type="italics"/>adeoque (per Corol. </s> <s>1. Prop.XLV) efficit ut Aux progre­<lb/>diatur. </s> <s>In locis inter Syzygias & Quadraturas pendet motus Au­<lb/>gis ex cau&longs;a utraque conjunctim, adeo ut pro hujus vel alterius <lb/>exce&longs;&longs;u progrediatur ip&longs;a vel regrediatur. </s> <s>Unde cum vis <emph type="italics"/>KL<emph.end type="italics"/>in <lb/>Syzygiis &longs;it qua&longs;i duplo major quam vis <emph type="italics"/>LM<emph.end type="italics"/>in Quadraturis, ex­<lb/>ce&longs;&longs;us in tota revolutione erit penes vim <emph type="italics"/>KL,<emph.end type="italics"/>transferetque Au­<lb/>gem &longs;ingulis revolutionibus in con&longs;equentia. </s> <s>Veritas autem hujus <lb/>& præcedentis Corollarii facilius intelligetur concipiendo Sy&longs;tema <lb/>corporum duorum <emph type="italics"/>T, P<emph.end type="italics"/>corporibus pluribus <emph type="italics"/>S, S, S,<emph.end type="italics"/>&c, in Or­<lb/>be <emph type="italics"/>ESE<emph.end type="italics"/>con&longs;i&longs;tentibus, undique cingi. </s> <s>Namque horum actioni-<pb xlink:href="039/01/190.jpg" pagenum="162"/><arrow.to.target n="note138"/>bus actio ip&longs;ius <emph type="italics"/>T<emph.end type="italics"/>minuetur undique, decre&longs;cetQ.E.I. ratione plu&longs;­<lb/>quam duplicata di&longs;tantiæ. </s></p> <p type="margin"> <s><margin.target id="note138"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>8. Cum autem pendeat Ap&longs;idum progre&longs;&longs;us vel regre&longs;&longs;us <lb/>a decremento vis centripetæ facto in majori vel minori quam du­<lb/>plicata ratione di&longs;tantiæ <emph type="italics"/>TP,<emph.end type="italics"/>in tran&longs;itu corporis ab Ap&longs;ide ima <lb/>ad Ap&longs;idem &longs;ummam; ut & a &longs;imili incremento in reditu ad Ap­<lb/>&longs;idem imam; atque adeo maximus &longs;it ubi proportio vis in Ap&longs;ide <lb/>&longs;umma ad vim in Ap&longs;ide ima maxime recedit a duplicata ratione <lb/>di&longs;tantiarum inver&longs;a: manife&longs;tum e&longs;t quod Ap&longs;ides in Syzygiis <lb/>&longs;uis, per vim ablatitiam <emph type="italics"/>KL<emph.end type="italics"/>&longs;eu <emph type="italics"/>NM-LM,<emph.end type="italics"/>progredientur ve­<lb/>locius, inque Quadraturis &longs;uis tardius recedent per vim addititiam <lb/><emph type="italics"/>LM.<emph.end type="italics"/>Ob diuturnitatem vero temporis quo velocitas progre&longs;&longs;us vel <lb/>tarditas regre&longs;&longs;us continuatur, fit hæc inæqualitas longe maxima. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>9. Si corpus aliquod vi reciproce proportionali quadrato <lb/>di&longs;tantiæ &longs;uæ a centro, revolveretur circa hoc centrum in El­<lb/>lip&longs;i, & mox, in de&longs;cen&longs;u ab Ap&longs;ide &longs;umma &longs;eu Auge ad Ap&longs;idem <lb/>imam, vis illa per acce&longs;&longs;um perpetuum vis novæ augeretur in ra­<lb/>tione plu&longs;quam dupli­<lb/><figure id="id.039.01.190.1.jpg" xlink:href="039/01/190/1.jpg"/><lb/>cata di&longs;tantiæ diminu­<lb/>tæ: manife&longs;tum e&longs;t <lb/>quod corpus, perpe­<lb/>tuo acce&longs;&longs;u vis illius <lb/>novæ impul&longs;um &longs;em­<lb/>per in centrum, magis <lb/>vergeret in hoc cen­<lb/>trum, quam &longs;i urge­<lb/>retur vi &longs;ola cre&longs;cente <lb/>in duplicata ratione di&longs;tantiæ diminutæ, adeoque Orbem de&longs;cri­<lb/>beret Orbe Elliptico interiorem, & in Ap&longs;ide ima propius acce­<lb/>deret ad centrum quam prius. </s> <s>Orbis igitur, acce&longs;&longs;u hujus vis no­<lb/>væ, fiet magis excentricus. </s> <s>Si jam vis, in rece&longs;&longs;u corporis ab <lb/>Ap&longs;ide ima ad Ap&longs;idem &longs;ummam, decre&longs;ceret ii&longs;dem gradibus qui­<lb/>bus ante creverat, rediret corpus ad di&longs;tantiam priorem, adeoque <lb/>&longs;i vis decre&longs;cat in majori ratione, corpus jam minus attractum a&longs;­<lb/>cendet ad di&longs;tantiam majorem & &longs;ic Orbis Excentricitas adhuc ma­<lb/>gis augebitur. </s> <s>Igitur &longs;i ratio incrementi & decrementi vis centri­<lb/>petæ &longs;ingulis revolutionibus augeatur, augebitur &longs;emper Excentri­<lb/>citas; & e contra, diminuetur eadem &longs;i ratio illa decre&longs;cat. </s> <s>Jam <lb/>vero in Sy&longs;temate corporum <emph type="italics"/>T, P, S,<emph.end type="italics"/>ubi Ap&longs;ides Orbis <emph type="italics"/>PAB<emph.end type="italics"/><lb/>&longs;unt in Quadraturis, ratio illa incrementi ac decrementi minima e&longs;t, <pb xlink:href="039/01/191.jpg" pagenum="163"/>& maxima fit ubi Ap&longs;ides &longs;unt in Syzygiis. </s> <s>Si Ap&longs;ides con&longs;tituan­<lb/><arrow.to.target n="note139"/>tur in Quadraturis, ratio prope Ap&longs;ides minor e&longs;t & prope Syzy­<lb/>gias major quam duplicata di&longs;tantiarum, & ex ratione illa majori <lb/>oritur Augis motus veloci&longs;&longs;imus, uti jam dictum e&longs;t. </s> <s>At &longs;i con­<lb/>&longs;ideretur ratio incrementi vel decrementi totius in progre&longs;&longs;u inter <lb/>Ap&longs;ides, hæc minor e&longs;t quam duplicata di&longs;tantiarum. </s> <s>Vis in Ap­<lb/>&longs;ide ima e&longs;t ad vim in Ap&longs;ide &longs;umma in minore quam duplicata <lb/>ratione di&longs;tantiæ Ap&longs;idis &longs;ummæ ab umbilico Ellip&longs;eos ad di­<lb/>&longs;tantiam Ap&longs;idis imæ ab eodem umbilico: & e contra, ubi <lb/>Ap&longs;ides con&longs;tituuntur in Syzygiis, vis in Ap&longs;ide ima e&longs;t ad vim <lb/>in Ap&longs;ide &longs;umma in majore quam duplicata ratione di&longs;tantiarum. </s> <s><lb/>Nam vires <emph type="italics"/>LM<emph.end type="italics"/>in Quadraturis additæ viribus corporis <emph type="italics"/>T<emph.end type="italics"/>compo­<lb/>nunt vires in ratione minore, & vires <emph type="italics"/>KL<emph.end type="italics"/>in Syzygiis &longs;ubductæ <lb/>viribus corporis <emph type="italics"/>T<emph.end type="italics"/>relinquunt vires in ratione majore. </s> <s>E&longs;t igi­<lb/>tur ratio decrementi & incrementi totius, in tran&longs;itu inter Ap&longs;ides, <lb/>minima in Quadraturis, maxima in Syzygiis: & propterea in tran­<lb/>&longs;itu Ap&longs;idum a Quadraturis ad Syzygias perpetuo augetur, auget­<lb/>que Excentricitatem Ellip&longs;eos; inque tran&longs;itu a Syzygiis ad <lb/>Quadraturas perpetuo diminuitur, & Excentricitatem diminuit. </s></p> <p type="margin"> <s><margin.target id="note139"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>10. Ut rationem ineamus errorum in Latitudinem, finga­<lb/>mus planum Orbis <emph type="italics"/>EST<emph.end type="italics"/>immobile manere; & ex errorum expo­<lb/>&longs;ita cau&longs;a manife&longs;tum e&longs;t quod, ex viribus <emph type="italics"/>NM, ML,<emph.end type="italics"/>quæ &longs;unt <lb/>cau&longs;a illa tota, vis <emph type="italics"/>ML<emph.end type="italics"/>agendo &longs;emper &longs;ecundum planum Orbis <lb/><emph type="italics"/>PAB,<emph.end type="italics"/>nunquam perturbat motus in Latitudinem; quodque vis <emph type="italics"/>NM,<emph.end type="italics"/><lb/>ubi Nodi &longs;unt in Syzygiis, agendo etiam &longs;ecundum idem Orbis <lb/>planum, non perturbat hos motus; ubi vero &longs;unt in Quadraturis <lb/>eos maxime perturbat, corpu&longs;que <emph type="italics"/>P<emph.end type="italics"/>de plano Orbis &longs;ui perpetuo <lb/>trahendo, minuit inclinationem plani in tran&longs;itu corporis a Qua­<lb/>draturis ad Syzygias, augetque vici&longs;&longs;im eandem in tran&longs;itu a Syzy­<lb/>giis ad Quadraturas. </s> <s>Unde fit ut corpore in Syzygiis exi&longs;tente in­<lb/>clinatio evadat omnium minima, redeatque ad priorem magnitudi­<lb/>nem circiter, ubi corpus ad Nodum proximum accedit. </s> <s>At &longs;i Nodi <lb/>con&longs;tituantur in Octantibus po&longs;t Quadraturas, id e&longs;t, inter <emph type="italics"/>C<emph.end type="italics"/>& <emph type="italics"/>A, <lb/>D<emph.end type="italics"/>& <emph type="italics"/>B,<emph.end type="italics"/>intelligetur ex modo expo&longs;itis quod, in tran&longs;itu corporis <lb/><emph type="italics"/>P<emph.end type="italics"/>a Nodo alterutro ad gradum inde nonage&longs;imum, inclinatio pla­<lb/>ni perpetuo minuitur; deinde in tran&longs;itu per proximos 45 gradus, <lb/>u&longs;que ad Quadraturam proximam, inclinatio augetur, & po&longs;tea de­<lb/>nuo in tran&longs;itu per alios 45 gradus, u&longs;que ad Nodum proximum, <lb/>diminuitur. </s> <s>Magis itaQ.E.D.minuitur inclinatio quam augetur, & <lb/>propterea minor e&longs;t &longs;emper in Nodo &longs;ub&longs;equente quam in præce-<pb xlink:href="039/01/192.jpg" pagenum="164"/><arrow.to.target n="note140"/>dente. </s> <s>Et &longs;imili ratiocinio, inclinatio magis augetur quam diminui­<lb/>tur ubi Nodi &longs;unt in Octantibus alteris inter <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>D, B<emph.end type="italics"/>& <emph type="italics"/>C.<emph.end type="italics"/>In­<lb/>clinatio igitur ubi Nodi &longs;unt in Syzygiis e&longs;t omnium maxima. </s> <s>In <lb/>tran&longs;itu eorum a Syzygiis ad Quadraturas, in &longs;ingulis corporis ad <lb/>Nodos appul&longs;ibus, diminuitur, fitque omnium minima ubi Nodi <lb/>&longs;unt in Quadraturis & corpus in Syzygiis: dein cre&longs;cit ii&longs;dem gra­<lb/>dibus quibus antea decreverat, Nodi&longs;que ad Syzygias proximas ap­<lb/>pul&longs;is ad magnitudinem primam revertitur. </s></p> <p type="margin"> <s><margin.target id="note140"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>11. Quoniam corpus <emph type="italics"/>P<emph.end type="italics"/>ubi Nodi &longs;unt in Quadraturis per­<lb/>petuo trahitur de plano Orbis &longs;ui, idQ.E.I. partem ver&longs;us <emph type="italics"/>S,<emph.end type="italics"/>in <lb/>tran&longs;itu &longs;uo a Nodo <emph type="italics"/>C<emph.end type="italics"/>per Conjunctionem <emph type="italics"/>A<emph.end type="italics"/>ad Nodum <emph type="italics"/>D<emph.end type="italics"/>; & in <lb/>contrariam partem in tran&longs;itu a Nodo <emph type="italics"/>D<emph.end type="italics"/>per Oppo&longs;itionem <emph type="italics"/>B<emph.end type="italics"/>ad <lb/>Nodum <emph type="italics"/>C<emph.end type="italics"/>; manife&longs;tum e&longs;t quod in motu &longs;uo a Nodo <emph type="italics"/>C,<emph.end type="italics"/>corpus <lb/>perpetuo recedit ab Orbis &longs;ui plano primo <emph type="italics"/>CD,<emph.end type="italics"/>u&longs;Q.E.D.m per­<lb/>ventum e&longs;t ad Nodum proximum; adeoQ.E.I. hoc Nodo, longi&longs;&longs;i­<lb/>me di&longs;tans a plano illo primo <emph type="italics"/>CD,<emph.end type="italics"/>tran&longs;it per planum Orbis <emph type="italics"/>EST<emph.end type="italics"/><lb/>non in plani illius Nodo altero <emph type="italics"/>D,<emph.end type="italics"/>&longs;ed in puncto quod inde vergit <lb/>ad partes corporis <emph type="italics"/>S,<emph.end type="italics"/>quodque proinde novus e&longs;t Nodi locus in an­<lb/>teriora vergens. </s> <s>Et &longs;imili argumento pergent Nodi recedere in <lb/>tran&longs;itu corporis de hoc Nodo in Nodum proximum. </s> <s>Nodi igi­<lb/>tur in Quadraturis con&longs;tituti perpetuo recedunt; in Syzygiis (ubi <lb/>motus in Latitudinem nil perturbatur) quie&longs;cunt; in locis inter­<lb/>mediis, conditionis utriu&longs;que participes, recedunt tardius; adeoque, <lb/>&longs;emper vel retrogradi vel &longs;tationarii, &longs;ingulis revolutionibus ferun­<lb/>tur in antecedentia. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>12. Omnes illi in his Corollariis de&longs;cripti Errores &longs;unt pau­<lb/>lo majores in Conjunctione corporum <emph type="italics"/>P, S<emph.end type="italics"/>quam in eorum Op­<lb/>po&longs;itione, idque ob majores vires generantes <emph type="italics"/>NM<emph.end type="italics"/>& <emph type="italics"/>ML.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>13. Cumque rationes horum Corollariorum non pendeant <lb/>a magnitudine corporis <emph type="italics"/>S,<emph.end type="italics"/>obtinent præcedentia omnia, ubi corporis <lb/><emph type="italics"/>S<emph.end type="italics"/>tanta &longs;tatuitur magnitudo ut circa ip&longs;um revolvatur corporum duo­<lb/>rum <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>Sy&longs;tema. </s> <s>Et ex aucto corpore <emph type="italics"/>S<emph.end type="italics"/>auctaque adeo ip&longs;ius <lb/>vi centripeta, a qua errores corporis <emph type="italics"/>P<emph.end type="italics"/>oriuntur, evadent errores illi <lb/>omnes (paribus di&longs;tantiis) majores in hoc ca&longs;u quam in altero, ubi <lb/>corpus <emph type="italics"/>S<emph.end type="italics"/>circum Sy&longs;tema corporum <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>T<emph.end type="italics"/>revolvitur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>14. Cum autem vires <emph type="italics"/>NM, ML,<emph.end type="italics"/>ubi corpus <emph type="italics"/>S<emph.end type="italics"/>longin­<lb/>quum e&longs;t, &longs;int quamproxime ut vis <emph type="italics"/>SK<emph.end type="italics"/>& ratio <emph type="italics"/>PT<emph.end type="italics"/>ad <emph type="italics"/>ST<emph.end type="italics"/>con­<lb/>junctim, hoc e&longs;t, &longs;i detur tum di&longs;tantia <emph type="italics"/>PT,<emph.end type="italics"/>tum corporis <emph type="italics"/>S<emph.end type="italics"/>vis <lb/>ab&longs;oluta, ut <emph type="italics"/>ST cub.<emph.end type="italics"/>reciproce; &longs;int autem vires illæ <emph type="italics"/>NM, ML<emph.end type="italics"/><lb/>cau&longs;æ errorum & effectuum omnium de quibus actum e&longs;t in præce-<pb xlink:href="039/01/193.jpg" pagenum="165"/>dentibus Corollariis: manife&longs;tum e&longs;t quod effectus illi omnes, &longs;tan­<lb/><arrow.to.target n="note141"/>te corporum <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>Sy&longs;temate, & mutatis tantum di&longs;tantia <emph type="italics"/>ST<emph.end type="italics"/>& <lb/>vi ab&longs;oluta corporis <emph type="italics"/>S,<emph.end type="italics"/>&longs;int quamproxime in ratione compo&longs;ita ex <lb/>ratione directa vis ab&longs;olutæ corporis <emph type="italics"/>S<emph.end type="italics"/>& ratione triplicata inver&longs;a <lb/>di&longs;tantiæ <emph type="italics"/>ST.<emph.end type="italics"/>Unde &longs;i Sy&longs;tema corporum <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>revolvatur cir­<lb/>ca corpus longinquum <emph type="italics"/>S,<emph.end type="italics"/>vires illæ <emph type="italics"/>NM, ML<emph.end type="italics"/>& earum effectus <lb/>erunt (per Corol. </s> <s>2. & 6. Prop. </s> <s>IV.) reciproce in duplicata ratione <lb/>temporis periodici. </s> <s>Et inde etiam, &longs;i magnitudo corporis <emph type="italics"/>S<emph.end type="italics"/>propor­<lb/>tionalis &longs;it ip&longs;ius vi ab&longs;olutæ, erunt vires illæ <emph type="italics"/>NM, ML<emph.end type="italics"/>& earum <lb/>effectus directe ut cubus diametri apparentis longinqui corporis <emph type="italics"/>S<emph.end type="italics"/>e <lb/>corpore <emph type="italics"/>T<emph.end type="italics"/>&longs;pectati, & vice ver&longs;a. </s> <s>Namque hæ rationes eædem &longs;unt <lb/>atque ratio &longs;uperior compo&longs;ita. </s></p> <p type="margin"> <s><margin.target id="note141"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>15. Et quoniam &longs;i, manentibus Orbium <emph type="italics"/>ESE<emph.end type="italics"/>& <emph type="italics"/>PAB<emph.end type="italics"/><lb/>forma, proportionibus & inclinatione ad invicem, mutetur eorum <lb/>magnitudo, & &longs;i corporum <emph type="italics"/>S<emph.end type="italics"/>& <emph type="italics"/>T<emph.end type="italics"/>vel maneant vel mutentur vires <lb/>in data quavis ratio­<lb/><figure id="id.039.01.193.1.jpg" xlink:href="039/01/193/1.jpg"/><lb/>ne, hæ vires (hoc e&longs;t, <lb/>vis corporis <emph type="italics"/>T<emph.end type="italics"/>qua cor­<lb/>pus <emph type="italics"/>P<emph.end type="italics"/>de recto trami­<lb/>te in Orbitam <emph type="italics"/>PAB<emph.end type="italics"/><lb/>deflectere, & vis cor­<lb/>poris <emph type="italics"/>S<emph.end type="italics"/>qua corpus <lb/>idem <emph type="italics"/>P<emph.end type="italics"/>de Orbita illa <lb/>deviare cogitur) agunt <lb/>&longs;emper eodem mo­<lb/>do & eadem proportione: nece&longs;&longs;e e&longs;t ut &longs;imiles & proportiona­<lb/>les &longs;int effectus omnes & proportionalia effectuum tempora; hoc <lb/>e&longs;t, ut errores omnes lineares &longs;int ut Orbium diametri, angulares <lb/>vero iidem qui prius, & errorum linearium &longs;imilium vel angularium <lb/>æqualium tempora ut Orbium tempora periodica. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>16. Unde, &longs;i dentur Orbium formæ & inclinatio ad invi­<lb/>cem, & mutentur utcunque corporum magnitudines, vires & di­<lb/>&longs;tantiæ; ex datis erroribus & errorum temporibus in uno Ca&longs;u, col­<lb/>ligi po&longs;&longs;unt errores & errorum tempora in alio quovis, quam pro­<lb/>xime: Sed brevius hac Methodo. </s> <s>Vires <emph type="italics"/>NM, ML,<emph.end type="italics"/>cæteris &longs;tan­<lb/>tibus, &longs;unt ut Radius <emph type="italics"/>TP,<emph.end type="italics"/>& harum effectus periodici (per Corol.2, <lb/>Lem. </s> <s>X) ut vires & quadratum temporis periodici corporis <emph type="italics"/>P<emph.end type="italics"/>con­<lb/>junctim. </s> <s>Hi &longs;unt errores lineares corporis <emph type="italics"/>P<emph.end type="italics"/>; & hinc errores an­<lb/>gulares e centro <emph type="italics"/>T<emph.end type="italics"/>&longs;pectati (id e&longs;t, tam motus Augis & Nodorum, <lb/>quam omnes in Longitudinem & Latitudinem errores apparentes) <lb/>&longs;unt, in qualibet revolutione corporis <emph type="italics"/>P,<emph.end type="italics"/>ut quadratum temporis <pb xlink:href="039/01/194.jpg" pagenum="166"/><arrow.to.target n="note142"/>revolutionis quam proxime. </s> <s>Conjungantur hæ rationes cum ratio­<lb/>nibus Corollarii 14, & in quolibet corporum <emph type="italics"/>T, P, S<emph.end type="italics"/>Sy&longs;temate, <lb/>ubi <emph type="italics"/>P<emph.end type="italics"/>circum <emph type="italics"/>T<emph.end type="italics"/>&longs;ibi propinquum, & <emph type="italics"/>T<emph.end type="italics"/>circum <emph type="italics"/>S<emph.end type="italics"/>longinquum re­<lb/>volvitur, errores angulares corporis <emph type="italics"/>P,<emph.end type="italics"/>de centro <emph type="italics"/>T<emph.end type="italics"/>apparentes, <lb/>erunt, in &longs;ingulis revolutionibus corporis illius <emph type="italics"/>P,<emph.end type="italics"/>ut quadratum <lb/>temporis periodici corporis <emph type="italics"/>P<emph.end type="italics"/>directe & quadratum temporis pe­<lb/>riodici corporis <emph type="italics"/>T<emph.end type="italics"/>inver&longs;e. </s> <s>Et inde motus medius Augis erit in da­<lb/>ta ratione ad motum medium Nodorum; & motus uterque erit ut tempus periodicum corporis &c. </s> <s><lb/>quadratum temporis periodici corporis <emph type="italics"/>P<emph.end type="italics"/>directe & quadratum <lb/>temporis periodici corporis <emph type="italics"/>T<emph.end type="italics"/>inver&longs;e. </s> <s>Augendo vel minuendo <lb/>Excentricitatem & Inclinationem Orbis <emph type="italics"/>PAB<emph.end type="italics"/>non mutantur mo­<lb/>tus Augis & Nodorum &longs;en&longs;ibiliter, ni&longs;i ubi eædem &longs;unt nimis <lb/>magnæ. </s></p> <p type="margin"> <s><margin.target id="note142"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>17. Cum autem linea <emph type="italics"/>LM<emph.end type="italics"/>nunc major &longs;it nunc minor <lb/>quam radius <emph type="italics"/>PT,<emph.end type="italics"/>exponatur vis mediocris <emph type="italics"/>LM<emph.end type="italics"/>per radium il­<lb/>lum <emph type="italics"/>PT<emph.end type="italics"/>; & erit hæc ad <lb/><figure id="id.039.01.194.1.jpg" xlink:href="039/01/194/1.jpg"/><lb/>vim mediocrem <emph type="italics"/>SK<emph.end type="italics"/><lb/>vel <emph type="italics"/>SN<emph.end type="italics"/>(quam expo­<lb/>nere licet per <emph type="italics"/>ST<emph.end type="italics"/>) ut <lb/>longitudo <emph type="italics"/>PT<emph.end type="italics"/>ad lon­<lb/>gitudinem <emph type="italics"/>ST.<emph.end type="italics"/>E&longs;t au­<lb/>tem vis mediocris <emph type="italics"/>SN<emph.end type="italics"/><lb/>vel <emph type="italics"/>ST,<emph.end type="italics"/>qua corpus <emph type="italics"/>T<emph.end type="italics"/><lb/>retinetur in Orbe &longs;uo <lb/>circum <emph type="italics"/>S,<emph.end type="italics"/>ad vim qua <lb/>corpus <emph type="italics"/>P<emph.end type="italics"/>retinetur in Orbe &longs;uo circum <emph type="italics"/>T,<emph.end type="italics"/>in ratione compo&longs;ita ex <lb/>ratione radii <emph type="italics"/>ST<emph.end type="italics"/>ad radium <emph type="italics"/>PT,<emph.end type="italics"/>& ratione duplicata temporis pe­<lb/>riodici corporis <emph type="italics"/>P<emph.end type="italics"/>circum <emph type="italics"/>T<emph.end type="italics"/>ad tempus periodicum corporis <emph type="italics"/>T<emph.end type="italics"/><lb/>circum <emph type="italics"/>S.<emph.end type="italics"/>Et ex æquo, vis mediocris <emph type="italics"/>LM,<emph.end type="italics"/>ad vim qua corpus <lb/><emph type="italics"/>P<emph.end type="italics"/>retinetur in Orbe &longs;uo circum <emph type="italics"/>T<emph.end type="italics"/>(quave corpus idem <emph type="italics"/>P,<emph.end type="italics"/>eo­<lb/>dem tempore periodico, circum punctum quodvis immobile <emph type="italics"/>T<emph.end type="italics"/>ad <lb/>di&longs;tantiam <emph type="italics"/>PT<emph.end type="italics"/>revolvi po&longs;&longs;et) e&longs;t in ratione illa duplicata periodi­<lb/>eorum temporum. </s> <s>Datis igitur temporibus periodicis una cum di­<lb/>&longs;tantia <emph type="italics"/>PT,<emph.end type="italics"/>datur vis mediocris <emph type="italics"/>LM<emph.end type="italics"/>; & ea data, datur etiam vis <lb/><emph type="italics"/>MN<emph.end type="italics"/>quamproxime per analogiam linearum <emph type="italics"/>PT, MN.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>18. Ii&longs;dem legibus quibus corpus <emph type="italics"/>P<emph.end type="italics"/>circum corpus <emph type="italics"/>T<emph.end type="italics"/>re­<lb/>volvitur, fingamus corpora plura fluida circum idem <emph type="italics"/>T<emph.end type="italics"/>ad æqua­<lb/>les ab ip&longs;o di&longs;tantias moveri; deinde ex his contiguis factis confla­<lb/>ri Annulum fluidum, rotundum ac corpori <emph type="italics"/>T<emph.end type="italics"/>concentricum; & <lb/>&longs;ingulæ Annuli partes, motus &longs;uos omnes ad legem corporis <emph type="italics"/>P<emph.end type="italics"/>per-<pb xlink:href="039/01/195.jpg" pagenum="167"/>agendo, propius accedent ad corpus <emph type="italics"/>T,<emph.end type="italics"/>& celerius movebuntur <lb/><arrow.to.target n="note143"/>in Conjunctione & Oppo&longs;itione ip&longs;arum & corporis <emph type="italics"/>S,<emph.end type="italics"/>quam in <lb/>Quadraturis. </s> <s>Et Nodi Annuli hujus &longs;eu inter&longs;ectiones ejus cum <lb/>plano Orbitæ corporis <emph type="italics"/>S<emph.end type="italics"/>vel <emph type="italics"/>T,<emph.end type="italics"/>quie&longs;cent in Syzygiis; extra Syzy­<lb/>gias vero movebuntur in antecedentia, & veloci&longs;&longs;ime quidem in <lb/>Quadraturis, tardius aliis in locis. </s> <s>Annuli quoQ.E.I.clinatio varia­<lb/>bitur, & axis ejus &longs;ingulis revolutionibus o&longs;cillabitur, completaque <lb/>revolutione ad pri&longs;tinum &longs;itum redibit, ni&longs;i quatenus per præce&longs;&longs;i­<lb/>onem Nodorum circumfertur. </s></p> <p type="margin"> <s><margin.target id="note143"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>19. Fingas jam Globum corporis <emph type="italics"/>T,<emph.end type="italics"/>ex materia non fluida <lb/>con&longs;tantem, ampliari & extendi u&longs;que ad hunc Annulum, & alveo <lb/>per circuitum excavato continere Aquam, motuque eodem perio­<lb/>dico circa axem &longs;uum uniformiter revolvi. </s> <s>Hic liquor per vices <lb/>acceleratus & retardatus (ut in &longs;uperiore Corollario) in Syzygiis <lb/>velocior erit, in Quadraturis tardior quam &longs;uperficies Globi, & <lb/>&longs;ic fluet in alveo refluet que ad modum Maris. </s> <s>Aqua revolvendo cir­<lb/>ca Globi centrum quie&longs;cens, &longs;i tollatur attractio corporis <emph type="italics"/>S<emph.end type="italics"/>nullum <lb/>acquiret motum fluxus & refluxus. </s> <s>Par e&longs;t ratio Globi uniformiter <lb/>progredientis in directum & interea revolventis circa centrum <lb/>&longs;uum (per Legum Corol. </s> <s>5.) ut & Globi de cur&longs;u rectilineo uNI­<lb/>formiter tracti, per Legum Corol. </s> <s>6. Accedat autem corpus <emph type="italics"/>S,<emph.end type="italics"/><lb/>& ab ip&longs;ius inæquabili attractione mox turbabitur Aqua. </s> <s>Etenim <lb/>major erit attractio aquæ propioris, minor ea remotioris. </s> <s>Vis <lb/>autem <emph type="italics"/>LM<emph.end type="italics"/>trahet aquam deor&longs;um in Quadraturis, facietQ.E.I.­<lb/>&longs;am de&longs;cendere u&longs;que ad Syzygias; & vis <emph type="italics"/>KL<emph.end type="italics"/>trahet eandem &longs;ur­<lb/>&longs;um in Syzygiis, &longs;i&longs;tetQ.E.D.&longs;cen&longs;um ejus & faciet ip&longs;am a&longs;cendere <lb/>u&longs;que ad Quadraturas. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>20. Si Annulus jam rigeat & minuatur Globus, ce&longs;&longs;a­<lb/>bit motus fluendi & refluendi; &longs;ed O&longs;cillatorius ille inclinationis <lb/>motus & præce&longs;&longs;io Nodorum manebunt. </s> <s>Habeat Globus eundem <lb/>axem cum Annulo, gyro&longs;que compleat ii&longs;dem temporibus, & &longs;uper­<lb/>ficie &longs;ua contingat ip&longs;um interius, eiQ.E.I.hæreat; & participando <lb/>motum ejus, compages utriu&longs;que O&longs;cillabitur & Nodi regredien­<lb/>tur. </s> <s>Nam Globus, ut mox dicetur, ad &longs;u&longs;cipiendas impre&longs;&longs;iones <lb/>omnes indifferens e&longs;t. </s> <s>Annuli Globo orbati maximus inclinationis <lb/>angulus e&longs;t ubi Nodi &longs;unt in Syzygiis. </s> <s>Inde in progre&longs;&longs;u Nodo­<lb/>rum ad Quadraturas conatur is inclinationem &longs;uam minuere, & i&longs;to <lb/>conatu motum imprimit Globo toti. </s> <s>Retinet Globus motum im­<lb/>pre&longs;&longs;um u&longs;Q.E.D.m Annulus conatu contrario motum hunc tollat, <lb/>imprimatque motum novum in contrariam partem: Atque hac ra-<pb xlink:href="039/01/196.jpg" pagenum="168"/><arrow.to.target n="note144"/>tione maximus decre&longs;centis inclinationis motus fit in Quadraturis <lb/>Nodorum, & minimus inclinationis angulus in Octantibus po&longs;t <lb/>Quadraturas; dein maximus reclinationis motus in Syzygiis, & <lb/>maximus angulus in Octantibus proximis. </s> <s>Et eadem e&longs;t ratio Glo­<lb/>bi Annulo nudati, qui in regionibus æquatoris vel altior e&longs;t paulo <lb/>quam juxta polos, vel con&longs;tat ex nateria paulo den&longs;iore. </s> <s>Sup­<lb/>plet enim vicem Annuli i&longs;te materiæ in æquatoris regionibus exce&longs;­<lb/>&longs;us. </s> <s>Et quanquam, aucta utcunque Globi hujus vi centripeta, <lb/>tendere &longs;upponantur omnes ejus partes deor&longs;um, ad modum gra­<lb/>vitantium partium telluris, tamen Phænomena hujus & præceden­<lb/>tis Corollarii vix inde mutabuntur. </s></p> <p type="margin"> <s><margin.target id="note144"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>21. Eadem ratione qua materia Globi juxta æquatorem <lb/>redundans efficit ut Nodi regrediantur, atque adeo per hujus in­<lb/>crementum augetur i&longs;te regre&longs;&longs;us, per diminutionem vero diminui­<lb/>tur & per ablationem tollitur; &longs;i materia plu&longs;quam redundans tol­<lb/>latur, hoc e&longs;t, &longs;i Globus juxta æquatorem vel depre&longs;&longs;ior reddatur <lb/>vel rarior quam juxta polos, orietur motus Nodorum in con­<lb/>&longs;equentia. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>22. Et inde vici&longs;&longs;im, ex motu Nodorum innote&longs;cit con&longs;ti­<lb/>tutio Globi. </s> <s>Nimirum &longs;i Globus polos eo&longs;dem con&longs;tanter &longs;ervat, <lb/>& motus fit in antecedentia, materia juxta æquatorem redundat; <lb/>&longs;i in con&longs;equentia, deficit. </s> <s>Pone Globum uniformem & perfecte <lb/>circinatum in &longs;patiis liberis primo quie&longs;cere; dein impetu quocun­<lb/>que obliQ.E.I. &longs;uperficiem &longs;uam facto propelli, & motum inde <lb/>concipere partim circularem, partim in directum. </s> <s>Quoniam Glo­<lb/>bus i&longs;te ad axes omnes per centrum &longs;uum tran&longs;euntes indifferenter <lb/>&longs;e habet, neque propen&longs;ior e&longs;t in unum axem, unumve axis &longs;itum, <lb/>quam in alium quemvis; per&longs;picuum e&longs;t quod is axem &longs;uum axi&longs;­<lb/>Q.E.I.clinationem vi propria nunquam mutabit. </s> <s>Impellatur jam <lb/>Globus oblique, in eadem illa &longs;uperficiei parte qua prius, impul&longs;u <lb/>quocunque novo; & cum citior vel ferior impul&longs;us effectum nil <lb/>mutet, manife&longs;tum e&longs;t quod hi duo impul&longs;us &longs;ucce&longs;&longs;ive impre&longs;&longs;i <lb/>eundem producent motum ac &longs;i &longs;imul impre&longs;&longs;i fui&longs;&longs;ent, hoc e&longs;t, <lb/>eundem ac &longs;i Globus vi &longs;implici ex utroque (per Legum Corol. </s> <s>2.) <lb/>compo&longs;ita impul&longs;us fui&longs;&longs;et, atque adeo &longs;implicem, circa axem in­<lb/>clinatione datum. </s> <s>Et par e&longs;t ratio impul&longs;us &longs;ecundi facti in lo­<lb/>cum alium quemvis in æquatore motus primi; ut & impul&longs;us pri­<lb/>mi facti in locum quemvis in æquatore motus, quem impul&longs;us &longs;e­<lb/>cundus ab&longs;que primo generaret; atque adeo impul&longs;uum amborum <lb/>factorum in loca quæcunque: Generabunt hi eundem motum cir-<pb xlink:href="039/01/197.jpg" pagenum="169"/>cularem ac &longs;i &longs;imul & &longs;emel in locum inter&longs;ectionis æquatorum <lb/><arrow.to.target n="note145"/>motuum illorum, quos feor&longs;im generarent, fui&longs;&longs;ent impre&longs;&longs;i. </s> <s><lb/>Globus igitur homogeneus & perfectus non retinet motus plures <lb/>di&longs;tinctos, &longs;ed impre&longs;&longs;os omnes componit & ad unum reducit, & <lb/>quatenus in &longs;e e&longs;t, gyratur &longs;emper motu &longs;implici & uniformi circa <lb/>axem unicum, inclinatione &longs;emper invariabili datum. </s> <s>Sed nec vis <lb/>centripeta inclinationem axis, aut rotationis velocitatem mutare <lb/>pote&longs;t. </s> <s>Si Globus plano quocunque, per centrum &longs;uum & cen­<lb/>trum in quod vis dirigitur tran&longs;eunte, dividi intelligatur in duo he­<lb/>mi&longs;phæria; urgebit &longs;emper vis illa utrumque hemi&longs;phærium æqua­<lb/>liter, & propterea Globum, quoad motum rotationis, nullam in <lb/>partem inclinabit. </s> <s>Addatur vero alicubi inter polum & æquato­<lb/>rem materia nova in formam montis cumulata, & hæc, perpetuo <lb/>conatu recedendi a centro &longs;ui motus, turbabit motum Globi, fa­<lb/>cietque polos ejus errare per ip&longs;ius &longs;uperficiem, & circulos circum <lb/>&longs;e punctumque &longs;ibi oppo&longs;itum perpetuo de&longs;cribere. </s> <s>Neque corrige­<lb/>tur i&longs;ta vagationis enormitas, ni&longs;i locando montem illum vel in polo <lb/>alterutro, quo in Ca&longs;u (per Corol. </s> <s>21) Nodi æquatoris progredien­<lb/>tur; vel in æquatore, qua ratione (per Corol. </s> <s>20) Nodi regredi­<lb/>entur; vel denique ex altera axis parte addendo materiam novam, <lb/>qua mons inter movendum libretur, & hoc pacto Nodi vel pro­<lb/>gredientur, vel recedent, perinde ut mons & hæcce nova materia <lb/>&longs;unt vel polo vel æquatori propiores. </s></p> <p type="margin"> <s><margin.target id="note145"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXVII. THEOREMA XXVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;itis ii&longs;dem attractionum legibus, dico quod corpus exterius<emph.end type="italics"/>S, <lb/><emph type="italics"/>circa interiorum<emph.end type="italics"/>P, T <emph type="italics"/>commune gravitatis centrum<emph.end type="italics"/>C, <emph type="italics"/>radiis <lb/>ad centrum illud ductis, de&longs;cribit areas temporibus magis pro­<lb/>portionales & Orbem ad formam Ellip&longs;eos umbilicum in centro <lb/>eodem habentis magis accedentem, quam circa corpus intimum <lb/>& maximum<emph.end type="italics"/>T, <emph type="italics"/>radiis ad ip&longs;um ductis, de&longs;cribere potest.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam corporis <emph type="italics"/>S<emph.end type="italics"/>attractiones ver&longs;us <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>componunt ip&longs;ius at­<lb/>tractionem ab&longs;olutam, quæ magis dirigitur in corporum <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>com­<lb/>mune gravitatis centrum <emph type="italics"/>C,<emph.end type="italics"/>quam in corpus maximum <emph type="italics"/>T,<emph.end type="italics"/>quæque <lb/>quadrato di&longs;tantiæ <emph type="italics"/>SC<emph.end type="italics"/>magis e&longs;t proportionalis reciproce, quam <lb/>quadrato di&longs;tantiæ <emph type="italics"/>ST:<emph.end type="italics"/>ut rem perpendenti facile con&longs;tabit. <pb xlink:href="039/01/198.jpg" pagenum="170"/><arrow.to.target n="note146"/></s></p> <p type="margin"> <s><margin.target id="note146"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXVIII. THEOREMA XXVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;itis ii&longs;dem attractionum legibus, dico quod corpus exterius<emph.end type="italics"/>S, <lb/><emph type="italics"/>circa interiorum<emph.end type="italics"/>P & T <emph type="italics"/>commune gravitatis centrum<emph.end type="italics"/>C, <emph type="italics"/>ra­<lb/>diis ad centrum illud ductis, de&longs;cribit areas temporibus magis <lb/>proportionales, & Orbem ad formam Ellip&longs;eos umbilicum in <lb/>centro eodem habentis magis accedentem, &longs;i corpus intimum & <lb/>maximum his attractionibus perinde atque cætera agitetur, quam <lb/>&longs;i id vel non attractum quie&longs;cat, vel multo magis aut multo <lb/>minus attractum aut multo magis aut multo minus agitetur.<emph.end type="italics"/></s></p> <p type="main"> <s>Demon&longs;tratur eo­<lb/><figure id="id.039.01.198.1.jpg" xlink:href="039/01/198/1.jpg"/><lb/>dem fere modo cum <lb/>Prop. </s> <s>LXVI, &longs;ed ar­<lb/>gumento prolixiore, <lb/>quod ideo prætereo. </s> <s><lb/>Suffecerit rem &longs;ic æ&longs;ti­<lb/>mare. </s> <s>Ex demon&longs;tra­<lb/>tione Propo&longs;itionis <lb/>novi&longs;&longs;imæ liquet cen­<lb/>trum in quod corpus <lb/><emph type="italics"/>S<emph.end type="italics"/>conjunctis viribus urgetur, proximum e&longs;&longs;e communi centro gra­<lb/>vitatis duorum illorum. </s> <s>Si coincideret hoc centrum cum centro <lb/>illo communi, & quie&longs;ceret commune centrum gravitatis corporum <lb/>trium; de&longs;criberent corpus <emph type="italics"/>S<emph.end type="italics"/>ex una parte, & commune centrum <lb/>aliorum duorum ex altera parte, circa commune omnium centrum <lb/>quie&longs;cens, Ellip&longs;es accuratas. </s> <s>Liquet hoc per Corollarium &longs;ecun­<lb/>dum Propo&longs;itionis LVIII collatum cum demon&longs;tratis in Propo&longs;. </s> <s><lb/>LXIV & LXV. </s> <s>Perturbatur i&longs;te motus Ellipticus aliquantulum per <lb/>di&longs;tantiam centri duorum a centro in quod tertium <emph type="italics"/>S<emph.end type="italics"/>attrahitur. </s> <s><lb/>Detur præterea motus communi trium centro, & augebitur per­<lb/>turbatio. </s> <s>Proinde minima e&longs;t perturbatio ubi commune trium <lb/>centrum quie&longs;cit, hoc e&longs;t, ubi corpus intimum & maximum <emph type="italics"/>T<emph.end type="italics"/>lege <lb/>cæterorum attrahitur: fitque major &longs;emper ubi trium commune il­<lb/>lud centrum, minuendo motum corporis <emph type="italics"/>T,<emph.end type="italics"/>moveri incipit & ma­<lb/>gis deinceps magi&longs;que agitatur. </s></p><pb xlink:href="039/01/199.jpg" pagenum="171"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Et hinc, &longs;i corpora plura minora revolvantur circa maxi­<lb/><arrow.to.target n="note147"/>mum, colligere licet quod Orbitæ de&longs;criptæ propius accedent ad <lb/>Ellipticas, & arearum de&longs;criptiones fient magis æquabiles, &longs;i cor­<lb/>pora omnia viribus acceleratricibus, quæ &longs;unt ut eorum vires ab­<lb/>&longs;olutæ directe & quadrata di&longs;tantiarum inver&longs;e, &longs;e mutuo trahant <lb/>agitentque, & Orbitæ cuju&longs;que umbilicus collocetur in communi <lb/>centro gravitatis corporum omnium interiorum (nimirum umbi­<lb/>licus Orbitæ primæ & intimæ in centro gravitatis corporis maxi­<lb/>mi & intimi; ille Orbitæ &longs;ecundæ, in communi centro gravi­<lb/>tatis corporum duorum intimorum; i&longs;te tertiæ, in communi cen­<lb/>tro gravitatis trium interiorum; & &longs;ic deinceps) quam &longs;i corpus <lb/>intimum quie&longs;cat & &longs;tatuatur communis umbilicus Orbitarum <lb/>omnium. </s></p> <p type="margin"> <s><margin.target id="note147"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXIX. THEOREMA XXIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>In Sy&longs;temate corporum plurium<emph.end type="italics"/>A, B, C, D, <emph type="italics"/>&c. </s> <s>&longs;i corpus aliquod<emph.end type="italics"/><lb/>A <emph type="italics"/>trahit cætera omnia<emph.end type="italics"/>B, C, D, <emph type="italics"/>&c. </s> <s>viribus acceler atricibus <lb/>quæ &longs;unt reciproce ut quadrata di&longs;tantiarum a trahente; & <lb/>corpus aliud<emph.end type="italics"/>B <emph type="italics"/>trahit etiam cætera<emph.end type="italics"/>A, C, D, <emph type="italics"/>&c. </s> <s>viribus quæ <lb/>&longs;unt reciproce ut quadrata di&longs;tantiarum a trahente: erunt Ab­<lb/>&longs;olutæ corporum trahentium<emph.end type="italics"/>A, B <emph type="italics"/>vires ad invicem, ut &longs;unt <lb/>ip&longs;a corpora<emph.end type="italics"/>A, B, <emph type="italics"/>quorum &longs;unt vires.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam attractiones acceleratrices corporum omnium <emph type="italics"/>B, C, D<emph.end type="italics"/>ver­<lb/>&longs;us <emph type="italics"/>A,<emph.end type="italics"/>paribus di&longs;tantiis, &longs;ibi invicem æquantur ex Hypothe&longs;i; & <lb/>&longs;imiliter attractiones acceleratrices corporum omnium ver&longs;us <emph type="italics"/>B,<emph.end type="italics"/><lb/>paribus di&longs;tantiis, &longs;ibi invicem æquantur. </s> <s>E&longs;t autem ab&longs;oluta vis <lb/>attractiva corporis <emph type="italics"/>A<emph.end type="italics"/>ad vim ab&longs;olutam attractivam corporis <emph type="italics"/>B,<emph.end type="italics"/>ut <lb/>attractio acceleratrix corporum omnium ver&longs;us <emph type="italics"/>A<emph.end type="italics"/>ad attractionem <lb/>acceleratricem corporum omnium ver&longs;us <emph type="italics"/>B,<emph.end type="italics"/>paribus di&longs;tantiis; & <lb/>ita e&longs;t attractio acceleratrix corporis <emph type="italics"/>B<emph.end type="italics"/>ver&longs;us <emph type="italics"/>A,<emph.end type="italics"/>ad attractionem <lb/>acceleratricem corporis <emph type="italics"/>A<emph.end type="italics"/>ver&longs;us <emph type="italics"/>B.<emph.end type="italics"/>Sed attractio acceleratrix cor­<lb/>poris <emph type="italics"/>B<emph.end type="italics"/>ver&longs;us <emph type="italics"/>A<emph.end type="italics"/>e&longs;t ad attractionem acceleratricem corporis <emph type="italics"/>A<emph.end type="italics"/><lb/>ver&longs;us <emph type="italics"/>B,<emph.end type="italics"/>ut ma&longs;&longs;a corporis <emph type="italics"/>A<emph.end type="italics"/>ad ma&longs;&longs;am corporis <emph type="italics"/>B<emph.end type="italics"/>; propterea <lb/>quod vires motrices, quæ (per Definitionem &longs;ecundam, &longs;epti­<lb/>mam & octavam) ex viribus acceleratricibus in corpora attracta <lb/>ductis oriuntur, &longs;unt (per motus Legem tertiam) &longs;ibi invicem æqua-<pb xlink:href="039/01/200.jpg" pagenum="172"/><arrow.to.target n="note148"/>les. </s> <s>Ergo ab&longs;oluta vis attractiva corporis <emph type="italics"/>A<emph.end type="italics"/>e&longs;t ad ab&longs;olutam vim <lb/>attractivam corporis <emph type="italics"/>B,<emph.end type="italics"/>ut ma&longs;&longs;a corporis <emph type="italics"/>A<emph.end type="italics"/>ad ma&longs;&longs;am corpo­<lb/>ris <emph type="italics"/>B. Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note148"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i &longs;ingula Sy&longs;tematis corpora <emph type="italics"/>A, B, C, D,<emph.end type="italics"/>&c. <lb/></s> <s>&longs;eor&longs;im &longs;pectata trahant cætera omnia viribus acceleratricibus quæ <lb/>&longs;unt reciproce ut quadrata di&longs;tantiarum a trahente; erunt corpo­<lb/>rum illorum omnium vires ab&longs;olutæ ad invicem ut &longs;unt ip&longs;a cor­<lb/>pora. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Eodem argumento, &longs;i &longs;ingula Sy&longs;tematis corpora <lb/><emph type="italics"/>A, B, C, D,<emph.end type="italics"/>&c. </s> <s>&longs;eor&longs;im &longs;pectata trahant cætera omnia viribus <lb/>acceleratricibus quæ &longs;unt vel reciproce vel directe in ratione dig­<lb/>nitatis cuju&longs;cunQ.E.D.&longs;tantiarum a trahente, quæve &longs;ecundum Le­<lb/>gem quamcunque communem ex di&longs;tantiis ab unoquoque trahente <lb/>definiuntur; con&longs;tat quod corporum illorum vires ab&longs;olutæ &longs;unt <lb/>ut corpora. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. In Sy&longs;temate corporum, quorum vires decre&longs;cunt in <lb/>ratione duplicata di&longs;tantiarum, &longs;i minora circa maximum in Ellip&longs;i­<lb/>bus umbilicum communem in maximi illius centro habentibus quam <lb/>fieri pote&longs;t accurati&longs;&longs;imis revolvantur, & radiis ad maximum illud <lb/>ductis de&longs;cribant areas temporibus quam maxime proportionales: <lb/>erunt corporum illorum vires ab&longs;olutæ ad invicem, aut accurate aut <lb/>quamproxime in ratione corporum; & contra. </s> <s>Patet per Corol. </s> <s><lb/>Prop. </s> <s>LXVIII collatum cum hujus Corol. </s> <s>1. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>His Propo&longs;itionibus manuducimur ad analogiam inter vires cen­<lb/>tripetas & corpora centralia, ad quæ vires illæ dirigi &longs;olent. </s> <s>Ra­<lb/>tioni enim con&longs;entaneum e&longs;t, ut vires quæ ad corpora diriguntur <lb/>pendeant ab eorundem natura & quantitate, ut fit in Magneticis. </s> <s><lb/>Et quoties huju&longs;modi ca&longs;us incidunt, æ&longs;timandæ erunt corporum <lb/>attractiones, a&longs;&longs;ignando &longs;ingulis eorum particulis vires proprias, <lb/>& colligendo &longs;ummas virium. </s> <s>Vocem Attractionis hic generaliter <lb/>u&longs;urpo pro corporum conatu quocunque accedendi ad invicem; <lb/>&longs;ive conatus i&longs;te fiat ab actione corporum, vel &longs;e mutuo petentium, <lb/>vel per Spiritus emi&longs;&longs;os &longs;e invicem agitantium, &longs;ive is ab actione <lb/>Ætheris, aut Aeris, Mediive cuju&longs;cunque &longs;eu corporei &longs;eu incorpo­<lb/>rei oriatur corpora innatantia in &longs;e invicem utcunQ.E.I.pellentis. </s> <s><lb/>Eodem &longs;en&longs;u generali u&longs;urpo vocem Impul&longs;us, non &longs;pecies virium <pb xlink:href="039/01/201.jpg" pagenum="173"/>& qualitates Phy&longs;icas, &longs;ed quantitates & proportiones Mathema­<lb/><arrow.to.target n="note149"/>ticas in hoc Tractatu expendens, ut in Definitionibus explicui. </s> <s>In <lb/>Mathe&longs;i inve&longs;tigandæ &longs;unt virium quantitates & rationes illæ, quæ <lb/>ex conditionibus quibu&longs;cunque po&longs;itis con&longs;equentur: deinde, ubi <lb/>in Phy&longs;icam de&longs;cenditur, conferendæ &longs;unt hæ rationes cum Phæ­<lb/>nomenis, ut innote&longs;cat quænam virium conditiones &longs;ingulis cor­<lb/>porum attractivorum generibus competant. </s> <s>Et tum demum de vi­<lb/>rium &longs;peciebus, cau&longs;is & rationibus Phy&longs;icis tutius di&longs;putare lice­<lb/>bit. </s> <s>Videamus igitur quibus viribus corpora Sphærica, ex particu­<lb/>lis modo jam expo&longs;ito attractivis con&longs;tantia, debeant in &longs;e mutuo<lb/>agere, & quales motus inde con&longs;equantur. </s></p> <p type="margin"> <s><margin.target id="note149"/>LIBER <lb/>PRIMUS.</s></p></subchap2><subchap2> <p type="main"> <s><emph type="center"/>SECTIO XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Corporum Sphæriccrum Viribus attractivis.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXX. THEOREMA XXX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si ad Sphæricæ &longs;uperficiei puncta &longs;ingula tendant vires æquales cen­<lb/>tripetæ decre&longs;centes in duplicata ratione di&longs;tantiarum a punctis: <lb/>dico quod corpu&longs;culum intra &longs;uperficiem con&longs;titutum his viri­<lb/>bus nullam in partem attrahitur.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>HIKL<emph.end type="italics"/>&longs;uperficies illa Sphæri­<lb/><figure id="id.039.01.201.1.jpg" xlink:href="039/01/201/1.jpg"/><lb/>ca, & <emph type="italics"/>P<emph.end type="italics"/>corpu&longs;culum intus con&longs;titu­<lb/>tum. </s> <s>Per <emph type="italics"/>P<emph.end type="italics"/>agantur ad hanc &longs;uper­<lb/>ficiem lineæ duæ <emph type="italics"/>HK, IL,<emph.end type="italics"/>arcus <lb/>quam minimos <emph type="italics"/>HI, KL<emph.end type="italics"/>intercipi­<lb/>entes; &, ob triangula <emph type="italics"/>HPI, LPK<emph.end type="italics"/><lb/>(per Corol. </s> <s>3. Lem. </s> <s>VII) &longs;imilia, arcus <lb/>illi erunt di&longs;tantiis <emph type="italics"/>HP, LP<emph.end type="italics"/>pro­<lb/>portionales; & &longs;uperficiei Sphæricæ <lb/>particulæ quævis ad <emph type="italics"/>HI<emph.end type="italics"/>& <emph type="italics"/>KL,<emph.end type="italics"/>rec­<lb/>tis per punctum <emph type="italics"/>P<emph.end type="italics"/>tran&longs;euntibus un­<lb/>dique terminatæ, erunt in duplicata <lb/>illa ratione. </s> <s>Ergo vires harum particularum in corpus <emph type="italics"/>P<emph.end type="italics"/>exercitæ <lb/>&longs;unt inter &longs;e æquales. </s> <s>Sunt enim ut particulæ directe & quadrata <lb/>di&longs;tantiarum inver&longs;e. </s> <s>Et hæ duæ rationes componunt rationem <pb xlink:href="039/01/202.jpg" pagenum="174"/><arrow.to.target n="note150"/>æqualitatis. </s> <s>Attractiones igitur, in contrarias partes æqualiter fac­<lb/>tæ, &longs;e mutuo de&longs;truunt. </s> <s>Et &longs;imili argumento, attractiones omnes <lb/>per totam Sphæricam &longs;uperficiem a contrariis attractionibus de­<lb/>&longs;truuntur. </s> <s>Proinde corpus <emph type="italics"/>P<emph.end type="italics"/>nullam in partem his attractionibus <lb/>impellitur. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note150"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXI. THEOREMA XXXI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, dico quod corpu&longs;culum extra Sphæricam &longs;uperficiem <lb/>con&longs;titutum attrahitur ad centrum Sphæræ, vi reciproce propor­<lb/>tionali quadrato di&longs;tantiæ &longs;uæ ab eodem centro.<emph.end type="italics"/></s></p> <p type="main"> <s>Sint <emph type="italics"/>AHKB, ahkb<emph.end type="italics"/>æquales duæ &longs;uperficies Sphæricæ, centris <lb/><emph type="italics"/>S, s,<emph.end type="italics"/>diametris <emph type="italics"/>AB, ab<emph.end type="italics"/>de&longs;criptæ, & <emph type="italics"/>P, p<emph.end type="italics"/>corpu&longs;cula &longs;ita extrin­<lb/>&longs;ecus in diametris illis productis. </s> <s>Agantur a corpu&longs;culis lineæ <lb/><figure id="id.039.01.202.1.jpg" xlink:href="039/01/202/1.jpg"/><lb/><emph type="italics"/>PHK, PIL, phk, pil,<emph.end type="italics"/>auferentes a circulis maximis <emph type="italics"/>AHB, <lb/>ahb,<emph.end type="italics"/>æquales arcus <emph type="italics"/>HK, hk<emph.end type="italics"/>& <emph type="italics"/>IL, il:<emph.end type="italics"/>Et ad eas de­<lb/>mittantur perpendicula <emph type="italics"/>SD, sd; SE, se; IR, ir;<emph.end type="italics"/>quorum <lb/><emph type="italics"/>SD, sd<emph.end type="italics"/>&longs;ecent <emph type="italics"/>PL, pl<emph.end type="italics"/>in <emph type="italics"/>F<emph.end type="italics"/>& <emph type="italics"/>f:<emph.end type="italics"/>Demittantur etiam ad diame­<lb/>tros perpendicula <emph type="italics"/>IQ, <expan abbr="iq.">ique</expan><emph.end type="italics"/>Evane&longs;cant anguli <emph type="italics"/>DPE, dpe:<emph.end type="italics"/>& <lb/>(ob æquales <emph type="italics"/>DS<emph.end type="italics"/>& <emph type="italics"/>ds, ES<emph.end type="italics"/>& <emph type="italics"/>es,<emph.end type="italics"/>) lineæ <emph type="italics"/>PE, PF<emph.end type="italics"/>& <emph type="italics"/>pe, pf<emph.end type="italics"/><lb/>& lineolæ <emph type="italics"/>DF, df<emph.end type="italics"/>pro æqualibus habeantur; quippe quarum ra­<lb/>tio ultima, angulis illis <emph type="italics"/>DPE, dpe<emph.end type="italics"/>&longs;imul evane&longs;centibus, e&longs;t æ­<lb/>qualitatis. </s> <s>His itaque con&longs;titutis, erit <emph type="italics"/>PI<emph.end type="italics"/>ad <emph type="italics"/>PF<emph.end type="italics"/>ut <emph type="italics"/>RI<emph.end type="italics"/>ad <emph type="italics"/>DF,<emph.end type="italics"/><lb/>& <emph type="italics"/>pf<emph.end type="italics"/>ad <emph type="italics"/>pi<emph.end type="italics"/>ut <emph type="italics"/>df<emph.end type="italics"/>vel <emph type="italics"/>DF<emph.end type="italics"/>ad <emph type="italics"/>ri<emph.end type="italics"/>; & ex æquo <emph type="italics"/>PIXpf<emph.end type="italics"/>ad <emph type="italics"/>PFXpi<emph.end type="italics"/><lb/>ut <emph type="italics"/>RI<emph.end type="italics"/>ad <emph type="italics"/>ri,<emph.end type="italics"/>hoc e&longs;t (per Corol. </s> <s>3. Lem. </s> <s>VII,) ut arcus <emph type="italics"/>IH<emph.end type="italics"/>ad <lb/>arcum <emph type="italics"/>ih.<emph.end type="italics"/>Rur&longs;us <emph type="italics"/>PI<emph.end type="italics"/>ad <emph type="italics"/>PS<emph.end type="italics"/>ut <emph type="italics"/>IQ<emph.end type="italics"/>ad <emph type="italics"/>SE,<emph.end type="italics"/>& <emph type="italics"/>ps<emph.end type="italics"/>and <emph type="italics"/>pi<emph.end type="italics"/>ut <emph type="italics"/>se<emph.end type="italics"/><lb/>vel <emph type="italics"/>SE<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="iq;">ique</expan><emph.end type="italics"/>& ex æquo <emph type="italics"/>PIXps<emph.end type="italics"/>ad <emph type="italics"/>PSXpi<emph.end type="italics"/>ut <emph type="italics"/>IQ<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="iq.">ique</expan><emph.end type="italics"/>ET <lb/>conjunctis rationibus <emph type="italics"/>PI quad.XpfXps<emph.end type="italics"/>ad <emph type="italics"/>pi quad.XPFXPS,<emph.end type="italics"/><lb/>ut <emph type="italics"/>IHXIQ<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="ihXiq;">ihXique</expan><emph.end type="italics"/>hoc e&longs;t, ut &longs;uperficies circularis, quam <pb xlink:href="039/01/203.jpg" pagenum="175"/>arcus <emph type="italics"/>IH<emph.end type="italics"/>convolutione &longs;emicirculi <emph type="italics"/>AKB<emph.end type="italics"/>circa diametrum <emph type="italics"/>AB<emph.end type="italics"/><lb/><arrow.to.target n="note151"/>de&longs;cribet, ad &longs;uperficiem circularem, quam arcus <emph type="italics"/>ih<emph.end type="italics"/>convolutione <lb/>&longs;emicirculi <emph type="italics"/>akb<emph.end type="italics"/>circa diametrum <emph type="italics"/>ab<emph.end type="italics"/>de&longs;cribet. </s> <s>Et vires, quibus <lb/>hæ &longs;uperficies &longs;ecundum lineas ad &longs;e tendentes attrahunt corpu&longs;cu­<lb/>la <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>p,<emph.end type="italics"/>&longs;unt (per Hypothe&longs;in) ut ip&longs;æ &longs;uperficies applicatæ <lb/>ad quadrata di&longs;tantiarum &longs;uarum a corporibus, hoc e&longs;t, ut <emph type="italics"/>pfXps<emph.end type="italics"/><lb/>ad <emph type="italics"/>PFXPS.<emph.end type="italics"/>Suntque hæ vires ad ip&longs;arum partes obliquas <lb/>quæ (facta per Legum Corol. </s> <s>2. re&longs;olutione virium) &longs;ecundum <lb/>lineas <emph type="italics"/>PS, ps<emph.end type="italics"/>ad centra tendunt, ut <emph type="italics"/>PI<emph.end type="italics"/>ad <emph type="italics"/>PQ,<emph.end type="italics"/>& <emph type="italics"/>pi<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="pq;">pque</expan><emph.end type="italics"/>id <lb/>e&longs;t (ob &longs;imilia triangula <emph type="italics"/>PIQ<emph.end type="italics"/>& <emph type="italics"/>PSF, piq<emph.end type="italics"/>& <emph type="italics"/>psf<emph.end type="italics"/>) ut <emph type="italics"/>PS<emph.end type="italics"/>ad <lb/><emph type="italics"/>PF<emph.end type="italics"/>& <emph type="italics"/>ps<emph.end type="italics"/>ad <emph type="italics"/>pf.<emph.end type="italics"/>Unde, ex æquo, fit attractio corpu&longs;culi hujus <emph type="italics"/>P<emph.end type="italics"/><lb/>ver&longs;us <emph type="italics"/>S<emph.end type="italics"/>ad attractionem corpu&longs;culi <emph type="italics"/>p<emph.end type="italics"/>ver&longs;us <emph type="italics"/>s,<emph.end type="italics"/>ut (<emph type="italics"/>PFXpfXps/PS<emph.end type="italics"/>) ad <lb/>(<emph type="italics"/>pfXPFXPS/ps<emph.end type="italics"/>), hoc e&longs;t, ut <emph type="italics"/>ps quad.<emph.end type="italics"/>ad <emph type="italics"/>PS quad.<emph.end type="italics"/>Et &longs;imili argu­<lb/>mento vires, quibus &longs;uperficies convolutione arcuum <emph type="italics"/>KL, kl<emph.end type="italics"/>de­<lb/>&longs;criptæ trahunt corpu&longs;cula, erunt ut <emph type="italics"/>ps quad.<emph.end type="italics"/>ad <emph type="italics"/>PS quad.<emph.end type="italics"/>; inque <lb/>eadem ratione erunt vires &longs;uperficierum omnium circularium in quas <lb/>utraque &longs;uperficies Sphærica, capiendo &longs;emper <emph type="italics"/>sd<emph.end type="italics"/>æqualem <emph type="italics"/>SD<emph.end type="italics"/>& <lb/><emph type="italics"/>se<emph.end type="italics"/>æqualem <emph type="italics"/>SE,<emph.end type="italics"/>di&longs;tingui pote&longs;t. </s> <s>Et, per compo&longs;itionem, vires <lb/>totarum &longs;uperficierum Sphæricarum in corpu&longs;cula exercitæ erunt <lb/>in eadem ratione. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note151"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXII. THEOREMA XXXII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si ad Sphæræ cuju&longs;vis puncta &longs;ingula tendant vires æquales cen­<lb/>tripetæ decre&longs;centes in duplicata ratione di&longs;tantiarum a punctis, <lb/>ac detur tum Sphæræ den&longs;itas, tum ratio diametri Sphæræ ad <lb/>di&longs;tantiam corpu&longs;culi a centro ejus; dico quod vis qua corpu&longs;­<lb/>culum attrahitur proportionalis erit &longs;emidiametro Sphæræ.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam concipe corpu&longs;cula duo &longs;eor&longs;im a Sphæris duabus attrahi, <lb/>unum ab una & alterum ab altera, & di&longs;tantias eorum a Sphæra­<lb/>rum centris proportionales e&longs;&longs;e diametris Sphærarum re&longs;pective, <lb/>Sphæras autem re&longs;olvi in particulas &longs;imiles & &longs;imiliter po&longs;itas ad <lb/>corpu&longs;cula. </s> <s>Et attractiones corpu&longs;culi unius, factæ ver&longs;us &longs;ingulas <lb/>particulas Sphæræ unius, erunt ad attractiones alterius ver&longs;us ana­<lb/>logas totidem particulas Sphæræ alterius, in ratione compo&longs;ita ex <lb/>ratione particularum directe & ratione duplicata di&longs;tantiarum in-<pb xlink:href="039/01/204.jpg" pagenum="176"/><arrow.to.target n="note152"/>ver&longs;e. </s> <s>Sed particulæ &longs;unt ut Sphæræ, hoc e&longs;t, in ratione triplicata <lb/>diametrorum, & di&longs;tantiæ &longs;unt ut diametri, & ratio prior directe <lb/>una cum ratione po&longs;teriore bis inver&longs;e e&longs;t ratio diametri ad diame­<lb/>trum. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note152"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i corpu&longs;cula in Circulis, circa Sphæras ex materia <lb/>æqualiter attractiva con&longs;tantes, revolvantur; &longs;intQ.E.D.&longs;tantiæ a cen­<lb/>tris Sphærarum proportionales earundem diametris: Tempora peri­<lb/>odica erunt æqualia. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et vice ver&longs;a, &longs;i Tempora periodica &longs;unt æqualia; <lb/>di&longs;tantiæ erunt proportionales diametris. </s> <s>Con&longs;tant hæc duo per <lb/>Corol. </s> <s>3. Prop. </s> <s>IV. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Si ad Solidorum durorum quorumvis &longs;imilium & æquali­<lb/>ter den&longs;orum puncta &longs;ingula tendant vires æquales centripetæ de­<lb/>cre&longs;centes in duplicata ratione di&longs;tantiarum a punctis: vires qui­<lb/>bus corpu&longs;cula, ad Solida illa duo &longs;imiliter &longs;ita, attrahentur ab ii&longs;­<lb/>dem, erunt ad invicem ut diametri Solidorum. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXIII. THEOREMA XXXIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si ad Sphæræ alicujus datæ puncta &longs;ingula tendant æquales vires <lb/>centripetæ decre&longs;centes in duplicata ratione di&longs;tantiarum a pun­<lb/>ctis: dico quod corpu&longs;culum intra Sphæram con&longs;titutum attra­<lb/>bitur vi proportionali di&longs;tantiæ &longs;uæ ab ip&longs;ius centro.<emph.end type="italics"/></s></p> <p type="main"> <s>In Sphæra <emph type="italics"/>ABCD,<emph.end type="italics"/>centro <emph type="italics"/>S<emph.end type="italics"/>de&longs;cripta, <lb/><figure id="id.039.01.204.1.jpg" xlink:href="039/01/204/1.jpg"/><lb/>locetur corpu&longs;culum <emph type="italics"/>P<emph.end type="italics"/>; & centro eodem <emph type="italics"/>S,<emph.end type="italics"/><lb/>intervallo <emph type="italics"/>SP,<emph.end type="italics"/>concipe Sphæram interiorem <lb/><emph type="italics"/>PEQF<emph.end type="italics"/>de&longs;cribi. </s> <s>Manife&longs;tum e&longs;t, per Prop. </s> <s><lb/>LXX, quod Sphæricæ &longs;uperficies concentri­<lb/>cæ ex quibus Sphærarum differentia <emph type="italics"/>AEBF<emph.end type="italics"/><lb/>componitur, attractionibus per attractiones <lb/>contrarias de&longs;tructis, nil agunt in corpus <lb/><emph type="italics"/>P.<emph.end type="italics"/>Re&longs;tat &longs;ola attractio Sphæræ interioris <lb/><emph type="italics"/>PEQF.<emph.end type="italics"/>Et per Prop. </s> <s>LXXII, hæc e&longs;t ut <lb/>di&longs;tantia <emph type="italics"/>PS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Superficies ex quibus &longs;olida componuntur, hic non &longs;unt pure <lb/>Mathematicæ, &longs;ed Orbes adeo tenues ut eorum cra&longs;&longs;itudo in&longs;tar <pb xlink:href="039/01/205.jpg" pagenum="177"/>nihili &longs;it; nimirum Orbes evane&longs;centes ex quibus Sphæra ultimo <lb/><arrow.to.target n="note153"/>con&longs;tat, ubi Orbium illorum numerus augetur & cra&longs;&longs;itudo minui­<lb/>tur in infinitum. </s> <s>Similiter per Puncta, ex quibus lineæ, &longs;uperficies <lb/>& &longs;olida componi dicuntur, intelligendæ &longs;unt particulæ æquales <lb/>magnitudinis contemnendæ. </s></p> <p type="margin"> <s><margin.target id="note153"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXIV. THEOREMA XXXIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, dico quod corpu&longs;culum extra Sphæram con&longs;titutum <lb/>attrabitur vi reciproce proportionali quadrato di&longs;tantiæ &longs;uæ ab <lb/>ip&longs;ius centro.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam di&longs;tinguatur Sphæra in &longs;uperficies Sphæricas innumeras <lb/>concentricas, & attractiones corpu&longs;culi a &longs;ingulis &longs;uperficiebus <lb/>oriundæ erunt reciproce proportionales quadrato di&longs;tantiæ cor­<lb/>pu&longs;culi a centro, per Prop. </s> <s>LXXI. </s> <s>Et componendo, fiet &longs;um­<lb/>ma attractionum, hoc e&longs;t attractio corpu&longs;culi in Sphæram totam, in <lb/>eadem ratione. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc in æqualibus di&longs;tantiis a centris homogenearum <lb/>Sphærarum, attractiones &longs;unt ut Sphæræ. </s> <s>Nam per Prop. </s> <s>LXXII, <lb/>&longs;i di&longs;tantiæ &longs;unt proportionales diametris Sphærarum, vires erunt <lb/>ut diametri. </s> <s>Minuatur di&longs;tantia major in illa ratione; &, di&longs;tan­<lb/>tiis jam factis æqualibus, augebitur attractio in duplicata illa ratio­<lb/>ne, adeoque erit ad attractionem alteram in triplicata illa ratione, <lb/>hoc e&longs;t, in ratione Sphærarum. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. In di&longs;tantiis quibu&longs;vis attractiones &longs;unt ut Sphæræ ap­<lb/>plicatæ ad quadrata di&longs;tantiarum. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Si corpu&longs;culum, extra Sphæram homogeneam po&longs;itum, <lb/>trahitur vi reciproce proportionali quadrato di&longs;tantiæ &longs;uæ ab ip&longs;ius <lb/>centro, con&longs;tet autem Sphæra ex particulis attractivis; decre&longs;cet vis <lb/>particulæ cuju&longs;Q.E.I. duplicata ratione di&longs;tantiæ a particula. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXV. THEOREMA XXXV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si ad Sphæræ datæ puncta &longs;ingula tendant vires æquales centripe­<lb/>tæ, decre&longs;centes in duplicata ratione di&longs;tantiarum a punctis; dico <lb/>quod Sphæra quævis alia &longs;imilaris ab eadem attrahitur vi reci­<lb/>proce proportionali quadrato di&longs;tantiæ centrorum.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam particulæ cuju&longs;vis attractio e&longs;t reciproce ut quadratum di­<lb/>&longs;tantiæ &longs;uæ a centro Sphæræ trahentis, (per Prop. </s> <s>LXXIV) & prop-<pb xlink:href="039/01/206.jpg" pagenum="178"/><arrow.to.target n="note154"/>terea eadem e&longs;t ac &longs;i vis tota attrahens manaret de corpu&longs;culo uNI­<lb/>co &longs;ito in centro hujus Sphæræ. </s> <s>Hæc autem attractio tanta e&longs;t <lb/>quanta foret vici&longs;&longs;im attractio corpu&longs;culi eju&longs;dem, &longs;i modo illud a <lb/>&longs;ingulis Sphæræ attractæ particulis eadem vi traheretur qua ip&longs;as <lb/>attrahit. </s> <s>Foret autem illa corpu&longs;culi attractio (per Prop. </s> <s>LXXIV) <lb/>reciproce proportionalis quadrato di&longs;tantiæ &longs;uæ a centro Sphæ­<lb/>ræ; adeoque huic æqualis attractio Sphæræ e&longs;t in eadem ratio­<lb/>ne. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note154"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Attractiones Sphærarum, ver&longs;us alias Sphæras homoge­<lb/>neas, &longs;unt ut Sphæræ trahentes applicatæ ad quadrata di&longs;tantiarum <lb/>centrorum &longs;uorum a centris earum quas attrahunt. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Idem valet ubi Sphæra attracta etiam attrahit. </s> <s>Nam­<lb/>que hujus puncta &longs;ingula trahent &longs;ingula alterius, eadem vi qua ab <lb/>ip&longs;is vici&longs;&longs;im trahuntur, adeoque cum in omni attractione urgea­<lb/>tur (per Legem III) tam punctum attrahens, quam punctum at­<lb/>tractum, geminabitur vis attractionis mutuæ, con&longs;ervatis propor­<lb/>tionibus. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Eadem omnia, quæ &longs;uperius de motu corporum circa <lb/>umbilicum Conicarum Sectionum demon&longs;trata &longs;unt, obtinent ubi <lb/>Sphæra attrahens locatur in umbilico & corpora moventur extra <lb/>Sphæram. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Ea vero quæ de motu corporum circa centrum Co­<lb/>nicarum Sectionum demon&longs;trantur, obtinent ubi motus peraguntur <lb/>intra Sphæram. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXVI. THEOREMA XXXVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Sphæræ in progre&longs;&longs;u a centro ad circumferentiam (quoad mate­<lb/>riæ den&longs;itatem & vim attractivam) utcunQ.E.D.&longs;&longs;imilares, in <lb/>progre&longs;&longs;u vero per circuitum ad datam omnem a centro di&longs;tan­<lb/>tiam &longs;unt undique &longs;imilares, & vis attractiva puncti cuju&longs;que <lb/>decre&longs;cit in duplicata ratione di&longs;tantiæ corporis attracti: dico <lb/>quod vis tota qua huju&longs;modi Sphæra una attrahit aliam &longs;it reci­<lb/>proce proportionalis quadrato di&longs;tantiæ centrorum.<emph.end type="italics"/></s></p> <p type="main"> <s>Sunto Sphæræ quotcunque concentricæ &longs;imilares <emph type="italics"/>AB, CD, EF,<emph.end type="italics"/><lb/>&c. </s> <s>quarum interiores additæ exterioribus componant materiam <pb xlink:href="039/01/207.jpg" pagenum="179"/>den&longs;iorem ver&longs;us centrum, vel &longs;ubductæ relinquant tenuiorem; & <lb/><arrow.to.target n="note155"/>hæ (per Prop. </s> <s>LXXV) trahent Sphæras alias quotcunque concentri­<lb/>cas &longs;imilares <emph type="italics"/>GH, IK, LM,<emph.end type="italics"/>&c. </s> <s>&longs;ingulæ &longs;ingulas, viribus reci­<lb/>proce proportionalibus quadrato di&longs;tantiæ <emph type="italics"/>SP.<emph.end type="italics"/>Et componendo <lb/>vel dividendo, &longs;umma virium illarum omnium, vel exce&longs;&longs;us ali­<lb/>quarum &longs;upra alias, hoc e&longs;t, vis quas Sphæra tota ex concen­<lb/>tricis quibu&longs;cunque vel concentricarum differentiis compo&longs;ita <emph type="italics"/>AB,<emph.end type="italics"/><lb/>trahit totam ex concentricis quibu&longs;cunque vel concentricarum dif­<lb/>ferentiis compo&longs;itam <emph type="italics"/>GH,<emph.end type="italics"/>erit in eadem ratione. </s> <s>Augeatur nu­<lb/>merus Sphærarum concentricarum in infinitum &longs;ic, ut materiæ den­<lb/>&longs;itas una cum vi attractiva, in progre&longs;&longs;u a circumferentia ad cen­<lb/>trum, &longs;ecundum Legem quamcunque cre&longs;cat vel decre&longs;cat: &, ad­<lb/><figure id="id.039.01.207.1.jpg" xlink:href="039/01/207/1.jpg"/><lb/>dita materia non attractiva, compleatur ubivis den&longs;itas deficiens, eo <lb/>ut Sphæræ acquirant formam quamvis optatam; & vis qua harum <lb/>una attrahet alteram erit etiamnum (per argumentum &longs;uperius) in <lb/>eadem illa di&longs;tantiæ quadratæ ratione inver&longs;a. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note155"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i eju&longs;modi Sphæræ complures, &longs;ibi invicem per <lb/>omnia &longs;imiles, &longs;e mutuo trahant; attractiones acceleratrices &longs;ingula­<lb/>rum in &longs;ingulas erunt, in æqualibus quibu&longs;vis centrorum di&longs;tantiis, <lb/>ut Sphæræ attrahentes. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. InQ.E.D.&longs;tantiis quibu&longs;vis inæqualibus, ut Sphæræ attra­<lb/>hentes applicatæ ad quadrata di&longs;tantiarum inter centra. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Attractiones vero motrices, &longs;eu pondera Sphærarum in <lb/>Sphæras erunt, in æqualibus centrorum di&longs;tantiis, ut Sphæræ attra­<lb/>hentes & attractæ conjunctim, id e&longs;t, ut contenta &longs;ub Sphæris per <lb/>multiplicationem producta. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. InQ.E.D.&longs;tantiis inæqualibus, ut contenta illa applicata <lb/>ad quadrata di&longs;tantiarum inter centra. <pb xlink:href="039/01/208.jpg" pagenum="180"/><arrow.to.target n="note156"/></s></p> <p type="margin"> <s><margin.target id="note156"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Eadem valent ubi attractio oritur a Sphæræ utriu&longs;que <lb/>virtute attractiva, mutuo exercita in Sphæram alteram. </s> <s>Nam viri­<lb/>bus ambabus geminatur attractio, proportione &longs;ervata. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Si huju&longs;modi Sphæræ aliquæ circa alias quie&longs;centes re­<lb/>volvantur, &longs;ingulæ circa &longs;ingulas, &longs;intQ.E.D.&longs;tantiæ inter centra re­<lb/>volventium & quie&longs;centium proportionales quie&longs;centium diame­<lb/>tris; æqualia erunt Tempora periodica. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. Et vici&longs;&longs;im, &longs;i Tempora periodica &longs;unt æqualia; di&longs;tan­<lb/>tiæ erunt proportionales diametris. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>8. Eadem omnia, quæ &longs;uperius de motu corporum circa <lb/>umbilicos Conicarum Sectionum demon&longs;trata &longs;unt, obtinent ubi <lb/>Sphæra attrahens, formæ & conditionis cuju&longs;vis jam de&longs;criptæ, lo­<lb/>catur in umbilico. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>9. Ut & ubi gyrantia &longs;unt etiam Sphæræ attrahentes, con­<lb/>ditionis cuju&longs;vis jam de&longs;criptæ. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXVII. THEOREMA XXXVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si ad &longs;ingula Sphærarum puncta tendant vires centripetæ, proper­<lb/>tionales di&longs;tantiis punctorum a corporibus attractis: dico quod <lb/>vis compo&longs;ita, qua Sphæræ duæ &longs;e mutuo trahent, est ut di­<lb/>&longs;tantia inter centra Sphærarum.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Sit <emph type="italics"/>AEBF<emph.end type="italics"/>Sphæra, <emph type="italics"/>S<emph.end type="italics"/><lb/><figure id="id.039.01.208.1.jpg" xlink:href="039/01/208/1.jpg"/><lb/>centrum ejus, <emph type="italics"/>P<emph.end type="italics"/>corpu&longs;culum at­<lb/>tractum, <emph type="italics"/>PASB<emph.end type="italics"/>axis Sphæræ per <lb/>centrum corpu&longs;culi tran&longs;iens, <emph type="italics"/>EF, <lb/>ef<emph.end type="italics"/>plana duo quibus Sphæra &longs;e­<lb/>catur, huic axi perpendicularia & <lb/>hinc inde æqualiter di&longs;tantia a <lb/>centro Sphæræ; <emph type="italics"/>G, g<emph.end type="italics"/>inter&longs;ectio­<lb/>nes planorum & axis, & <emph type="italics"/>H<emph.end type="italics"/>pun­<lb/>ctum quodvis in plano <emph type="italics"/>EF.<emph.end type="italics"/>Pun­<lb/>cti <emph type="italics"/>H<emph.end type="italics"/>vis centripeta in corpu&longs;culum <emph type="italics"/>P,<emph.end type="italics"/>&longs;ecundum lineam <emph type="italics"/>PH<emph.end type="italics"/>exer­<lb/>cita, e&longs;t ut di&longs;tantia <emph type="italics"/>PH<emph.end type="italics"/>; & (per Legum Corol. </s> <s>2.) &longs;ecundum li­<lb/>neam <emph type="italics"/>PG,<emph.end type="italics"/>&longs;eu ver&longs;us centrum <emph type="italics"/>S,<emph.end type="italics"/>ut longitudo <emph type="italics"/>PG.<emph.end type="italics"/>Igitur pun­<lb/>ctorum omnium in plano <emph type="italics"/>EF,<emph.end type="italics"/>hoc e&longs;t plani totius vis, qua corpu&longs;­<lb/>culum <emph type="italics"/>P<emph.end type="italics"/>trahitur ver&longs;us centrum <emph type="italics"/>S,<emph.end type="italics"/>e&longs;t ut numerus punctorum <lb/>ductus in di&longs;tantiam <emph type="italics"/>PG:<emph.end type="italics"/>id e&longs;t, ut contentum &longs;ub plano ip&longs;o <emph type="italics"/>EF<emph.end type="italics"/><lb/>& di&longs;tantia illa <emph type="italics"/>PG.<emph.end type="italics"/>Et &longs;imiliter vis plani <emph type="italics"/>ef,<emph.end type="italics"/>qua corpu&longs;culum <emph type="italics"/>P<emph.end type="italics"/><pb xlink:href="039/01/209.jpg" pagenum="181"/>trahitur ver&longs;us centrum <emph type="italics"/>S,<emph.end type="italics"/>e&longs;t ut planum illud ductum in di&longs;tantiam </s></p> <p type="main"> <s><arrow.to.target n="note157"/>&longs;uam <emph type="italics"/>Pg,<emph.end type="italics"/>&longs;ive ut huic æquale planum <emph type="italics"/>EF<emph.end type="italics"/>ductum in di&longs;tantiam <lb/>illam <emph type="italics"/>Pg<emph.end type="italics"/>; & &longs;umma virium plani utriu&longs;que ut planum <emph type="italics"/>EF<emph.end type="italics"/>duc­<lb/>tum in &longs;ummam di&longs;tantiarum <emph type="italics"/>PG+Pg,<emph.end type="italics"/>id e&longs;t, ut planum illud <lb/>ductum in duplam centri & corpu&longs;culi di&longs;tantiam <emph type="italics"/>PS,<emph.end type="italics"/>hoc e&longs;t, ut <lb/>duplum planum <emph type="italics"/>EF<emph.end type="italics"/>ductum in di&longs;tantiam <emph type="italics"/>PS,<emph.end type="italics"/>vel ut &longs;umma æ­<lb/>qualium planorum <emph type="italics"/>EF+ef<emph.end type="italics"/>ducta in di&longs;tantiam eandem. </s> <s>Et &longs;i­<lb/>mili argumento, vires omnium planorum in Sphæra tota, hinc in­<lb/>de æqualiter a centro Sphæræ di&longs;tantium, &longs;unt ut &longs;umma planorum <lb/>ducta in di&longs;tantiam <emph type="italics"/>PS,<emph.end type="italics"/>hoc e&longs;t, ut Sphæra tota ducta in di&longs;tan­<lb/>tiam centri &longs;ui <emph type="italics"/>S<emph.end type="italics"/>a corpu&longs;culo <emph type="italics"/>P. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note157"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Trahat jam corpu&longs;culum <emph type="italics"/>P<emph.end type="italics"/>Sphæram <emph type="italics"/>AEBF.<emph.end type="italics"/>Et eo­<lb/>dem argumento probabitur quod vis, qua Sphæra illa trahitur, erit: <lb/>ut di&longs;tantia <emph type="italics"/>PS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>3. Componatur jam Sphæra altera ex corpu&longs;culis innume­<lb/>ris <emph type="italics"/>P<emph.end type="italics"/>; & quoniam vis; qua corpu&longs;culum unumquodque trahitur, <lb/>e&longs;t ut di&longs;tantia corpu&longs;culi a centro Sphæræ primæ ducta in Sphæ­<lb/>ram eandem, atque adeo eadem e&longs;t ac &longs;i prodiret tota de corpu&longs;­<lb/>culo unico in centro Sphæræ; vis tota qua corpu&longs;cula omnia in <lb/>Sphæra &longs;ecunda trahuntur, hoc e&longs;t, qua Sphæra illa tota trahitur, <lb/>eadem erit ac &longs;i Sphæra illa traheretur vi prodeunte de corpu&longs;culo <lb/>unico in centro Sphæræ primæ, & propterea proportionalis e&longs;t di­<lb/>&longs;tantiæ inter centra Sphærarum. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>4. Trahant Sphæræ &longs;e mutuo, & vis geminata proportio­<lb/>nem priorem &longs;ervabit. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>5. Locetur jam corpu&longs;culum <emph type="italics"/>p<emph.end type="italics"/>intra Sphæram <emph type="italics"/>AEBF<emph.end type="italics"/>; & <lb/>quoniam vis plani <emph type="italics"/>ef<emph.end type="italics"/>in corpu&longs;culum e&longs;t ut contentum &longs;ub plano <lb/>illo & di&longs;tantia <emph type="italics"/>pg<emph.end type="italics"/>; & vis contraria plani <emph type="italics"/>EF<emph.end type="italics"/>ut contentum &longs;ub <lb/>plano illo & di&longs;tantia <emph type="italics"/>pG<emph.end type="italics"/>; erit vis ex utraque compo&longs;ita ut diffe­<lb/>rentia contentorum, hoc e&longs;t, ut &longs;umma æqualium planorum ducta <lb/>in &longs;emi&longs;&longs;em differentiæ di&longs;tantiarum, id e&longs;t, ut &longs;umma illa ducta in <lb/><emph type="italics"/>pS<emph.end type="italics"/>di&longs;tantiam corpu&longs;culi a centro Sphæræ. </s> <s>Et &longs;imili argumento, <lb/>attractio planorum omnium <emph type="italics"/>EF, ef<emph.end type="italics"/>in Sphæra tota, hoc e&longs;t, at­<lb/>tractio Sphæræ totius, e&longs;t ut &longs;umma planorum omnium, &longs;eu Sphæra <lb/>tota, ducta in <emph type="italics"/>pS<emph.end type="italics"/>di&longs;tantiam corpu&longs;culi a centro Sphæræ. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>6. Et &longs;i ex corpu&longs;culis innumeris <emph type="italics"/>p<emph.end type="italics"/>componatur Sphæra <lb/>nova, intra Sphæram priorem <emph type="italics"/>AEBF<emph.end type="italics"/>&longs;ita; probabitur ut prius <lb/>quod attractio, &longs;ive &longs;implex Sphæræ unius in alteram, &longs;ive mutua <lb/>utriu&longs;Q.E.I. &longs;e invicem, erit ut di&longs;tantia centrorum <emph type="italics"/>pS. Q.E.D.<emph.end type="italics"/><pb xlink:href="039/01/210.jpg" pagenum="182"/><arrow.to.target n="note158"/></s></p> <p type="margin"> <s><margin.target id="note158"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXVIII. THEOREMA XXXVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Sphæræ in progre&longs;&longs;u a centro ad circumferentiam &longs;int utcunque <lb/>di&longs;&longs;imilares & inæquabiles, in progre&longs;&longs;u vero per circuitum ad <lb/>datam omnem a centro di&longs;tantiam &longs;int undique &longs;imilares; & <lb/>vis attractiva puncti cuju&longs;que &longs;it ut di&longs;tantia corporis attracti: <lb/>dico quod vis tota qua huju&longs;modi Sphæræ duæ &longs;e mutuo trahunt <lb/>&longs;it proportionalis di&longs;tantiæ inter centra Sphærarum.<emph.end type="italics"/></s></p> <p type="main"> <s>Demon&longs;tratur ex Propo&longs;itione præcedente, eodem modo quo <lb/>Propo&longs;itio LXXVI ex Propo&longs;itione LXXV demon&longs;trata fuit. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Quæ &longs;uperius in Propo&longs;itionibus X & LXIV de motu <lb/>corporum circa centra Conicarum Sectionum demon&longs;trata &longs;unt, <lb/>valent ubi attractiones omnes fiunt vi Corporum Sphærieorum <lb/>conditionis jam de&longs;criptæ, &longs;untque corpora attracta Sphæræ con­<lb/>ditionis eju&longs;dem. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Attractionum Ca&longs;us duos in&longs;igniores jam dedi expo&longs;itos; nimi­<lb/>rum ubi Vires centripetæ decre&longs;cunt in duplicata di&longs;tantiarum ra­<lb/>tione, vel cre&longs;cunt in di&longs;tantiarum ratione &longs;implici; efficientes <lb/>in utroque Ca&longs;u ut corpora gyrentur in Conicis Sectionibus, & <lb/>componentes corporum Sphærieorum Vires centripetas eadem Lege, <lb/>in rece&longs;&longs;u a centro, decre&longs;centes vel cre&longs;centes cum &longs;eip&longs;is: Quod <lb/>e&longs;t notatu dignum. </s> <s>Ca&longs;us cæteros, qui conclu&longs;iones minus ele­<lb/>gantes exhibent, &longs;igillatim percurrere longum e&longs;&longs;et. </s> <s>Malim <lb/>cunctos methodo generali &longs;imul comprehendere ac determinare, <lb/>ut &longs;equitur. </s></p> <p type="main"> <s><emph type="center"/>LEMMA XXIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si de&longs;cribantur centro<emph.end type="italics"/>S <emph type="italics"/>circulus quilibet<emph.end type="italics"/>AEB, <emph type="italics"/>& centro<emph.end type="italics"/>P <emph type="italics"/>cir­<lb/>culi duo<emph.end type="italics"/>EF, ef, <emph type="italics"/>&longs;ecantes priorem in<emph.end type="italics"/>E, e, <emph type="italics"/>lineamque<emph.end type="italics"/>PS <emph type="italics"/>in<emph.end type="italics"/><lb/>F, f; <emph type="italics"/>& ad<emph.end type="italics"/>PS <emph type="italics"/>demittantur perpendicula<emph.end type="italics"/>ED, ed: <emph type="italics"/>dico quod, <lb/>fi di&longs;tantia arcuum<emph.end type="italics"/>EF, ef <emph type="italics"/>in infinitum minui intelligatur, ra­<lb/>tio ultima lineæ evane&longs;centis<emph.end type="italics"/>Dd <emph type="italics"/>ad lineam evane&longs;centem<emph.end type="italics"/>Ff <lb/><emph type="italics"/>ea &longs;it, quæ lineæ<emph.end type="italics"/>PE <emph type="italics"/>ad lineam<emph.end type="italics"/>PS. </s></p><pb xlink:href="039/01/211.jpg" pagenum="183"/> <p type="main"> <s>Nam &longs;i linea <emph type="italics"/>Pe<emph.end type="italics"/>&longs;ecet arcum <emph type="italics"/>EF<emph.end type="italics"/>in <emph type="italics"/>q<emph.end type="italics"/>; & recta <emph type="italics"/>Ee,<emph.end type="italics"/>quæ cum <lb/><arrow.to.target n="note159"/>arcu evane&longs;cente <emph type="italics"/>Ee<emph.end type="italics"/>coincidit, producta occurrat rectæ <emph type="italics"/>PS<emph.end type="italics"/>in <emph type="italics"/>T<emph.end type="italics"/>; <lb/>& ab <emph type="italics"/>S<emph.end type="italics"/>demittatur in <emph type="italics"/>PE<emph.end type="italics"/>normalis <emph type="italics"/>SG:<emph.end type="italics"/>ob &longs;imilia triangula <lb/><emph type="italics"/>DTE, dTe, DES<emph.end type="italics"/>; erit <emph type="italics"/>Dd<emph.end type="italics"/>ad <emph type="italics"/>Ee,<emph.end type="italics"/>ut <emph type="italics"/>DT<emph.end type="italics"/>ad <emph type="italics"/>TE,<emph.end type="italics"/>&longs;eu <emph type="italics"/>DE<emph.end type="italics"/>ad <lb/><figure id="id.039.01.211.1.jpg" xlink:href="039/01/211/1.jpg"/><lb/><emph type="italics"/>ES<emph.end type="italics"/>; & ob triangula <emph type="italics"/>Eeq, ESG<emph.end type="italics"/>(per Lem. </s> <s>VIII, & Corol. </s> <s>3. <lb/>Lem. </s> <s>VII) &longs;imilia, erit <emph type="italics"/>Ee<emph.end type="italics"/>ad <emph type="italics"/>eq<emph.end type="italics"/>&longs;eu <emph type="italics"/>Ff,<emph.end type="italics"/>ut <emph type="italics"/>ES<emph.end type="italics"/>ad <emph type="italics"/>SG<emph.end type="italics"/>; & ex <lb/>æquo, <emph type="italics"/>Dd<emph.end type="italics"/>ad <emph type="italics"/>Ff<emph.end type="italics"/>ut <emph type="italics"/>DE<emph.end type="italics"/>ad <emph type="italics"/>SG<emph.end type="italics"/>; hoc e&longs;t (ob &longs;imilia triangula <lb/><emph type="italics"/>PDE, PGS<emph.end type="italics"/>) ut <emph type="italics"/>PE<emph.end type="italics"/>ad <emph type="italics"/>PS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note159"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXIX. THEOREMA XXXIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si &longs;uperficies ob latitudinem infinite diminutam jamjam evane&longs;cens<emph.end type="italics"/><lb/>EF fe, <emph type="italics"/>convolutione &longs;ui circa axem<emph.end type="italics"/>PS, <emph type="italics"/>de&longs;cribat &longs;olidum <lb/>Sphæricum concavo convexum, ad cujus particulas &longs;ingulas æqua­<lb/>les tendant æquales vires centripetæ: dico quod Vis, qua &longs;oli­<lb/>dum illud trahit corpu&longs;culum &longs;itum in<emph.end type="italics"/>P, <emph type="italics"/>est in ratione compo­<lb/>ta ex ratione &longs;olidi<emph.end type="italics"/>DE<emph type="italics"/>q<emph.end type="italics"/>XFf <emph type="italics"/>& ratione vis qua particula <lb/>data in loco<emph.end type="italics"/>Ff <emph type="italics"/>traheret idem corpu&longs;culum.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam &longs;i primo con&longs;ideremus vim &longs;uperficiei Sphæricæ <emph type="italics"/>FE,<emph.end type="italics"/>quæ <lb/>convolutione arcus <emph type="italics"/>FE<emph.end type="italics"/>generatur, & a linea <emph type="italics"/>de<emph.end type="italics"/>ubivis &longs;ecatur in <emph type="italics"/>r<emph.end type="italics"/>; <lb/>erit &longs;uperficiei pars annularis, convolutione arcus <emph type="italics"/>rE<emph.end type="italics"/>genita, ut <lb/>lineola <emph type="italics"/>Dd,<emph.end type="italics"/>manente Sphæræ radio <emph type="italics"/>PE,<emph.end type="italics"/>(uti demon&longs;travit <emph type="italics"/>Ar­<lb/>chimedes<emph.end type="italics"/>in Lib. </s> <s>de <emph type="italics"/>Sphæra<emph.end type="italics"/>& <emph type="italics"/>Cylindro.<emph.end type="italics"/>) Et hujus vis &longs;ecundum li­<lb/>neas <emph type="italics"/>PE<emph.end type="italics"/>vel <emph type="italics"/>Pr<emph.end type="italics"/>undiQ.E.I. &longs;uperficie conica &longs;itas exercita, ut <lb/>hæc ip&longs;a &longs;uperficiei pars annularis; hoc e&longs;t, ut lineola <emph type="italics"/>Dd<emph.end type="italics"/>vel, <lb/>quod perinde e&longs;t, ut rectangulum &longs;ub dato Sphæræ radio <emph type="italics"/>PE<emph.end type="italics"/>& <pb xlink:href="039/01/212.jpg" pagenum="184"/><arrow.to.target n="note160"/>lineola illa <emph type="italics"/>Dd:<emph.end type="italics"/>at &longs;ecundum lineam <emph type="italics"/>PS<emph.end type="italics"/>ad centrum <emph type="italics"/>S<emph.end type="italics"/>tendentem <lb/>minor, in ratione <emph type="italics"/>PD<emph.end type="italics"/>ad <emph type="italics"/>PE,<emph.end type="italics"/>adeoque ut <emph type="italics"/>PDXDd.<emph.end type="italics"/>Dividi <lb/>jam intelligatur linea <emph type="italics"/>DF<emph.end type="italics"/>in particulas innumeras æquales, quæ <lb/>&longs;ingulæ nominentur <emph type="italics"/>Dd<emph.end type="italics"/>; & &longs;uperficies <emph type="italics"/>FE<emph.end type="italics"/>dividetur in totidem <lb/>æquales annulos, quorum vires erunt ut &longs;umma omnium <emph type="italics"/>PDXDd,<emph.end type="italics"/><lb/>hoc e&longs;t, ut 1/2 <emph type="italics"/>PFq<emph.end type="italics"/>-1/2<emph type="italics"/>PDq,<emph.end type="italics"/>adeoque ut <emph type="italics"/>DE quad.<emph.end type="italics"/>Ducatur <lb/><figure id="id.039.01.212.1.jpg" xlink:href="039/01/212/1.jpg"/><lb/>jam &longs;uperficies <emph type="italics"/>FE<emph.end type="italics"/>in altitudinem <emph type="italics"/>Ef<emph.end type="italics"/>; & fiet &longs;olidi <emph type="italics"/>EFfe<emph.end type="italics"/>vis ex­<lb/>ercita in corpu&longs;culum <emph type="italics"/>P<emph.end type="italics"/>ut <emph type="italics"/>DEqXFf:<emph.end type="italics"/>puta &longs;i detur vis quam <lb/>particula aliqua data <emph type="italics"/>Ff<emph.end type="italics"/>in di&longs;tantia <emph type="italics"/>PF<emph.end type="italics"/>exercet in corpu&longs;culum <lb/><emph type="italics"/>P.<emph.end type="italics"/>At &longs;i vis illa non detur, fiet vis &longs;olidi <emph type="italics"/>EFfe<emph.end type="italics"/>ut &longs;olidum <lb/><emph type="italics"/>DEqXFf<emph.end type="italics"/>& vis illa non data conjunctim. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note160"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXX. THEOREMA XL.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si ad Sphæræ alicujus<emph.end type="italics"/>ABE, <emph type="italics"/>centro<emph.end type="italics"/>S <emph type="italics"/>de&longs;criptæ, particulas &longs;ingu­<lb/>las æquales tendant æquales vires centripetæ, & ad Sphæræ <lb/>axem<emph.end type="italics"/>AB, <emph type="italics"/>in quo corpu&longs;culum aliquod<emph.end type="italics"/>P <emph type="italics"/>locatur, erigantur de <lb/>punctis &longs;ingulis<emph.end type="italics"/>D <emph type="italics"/>perpendicula<emph.end type="italics"/>DE, <emph type="italics"/>Sphæræ occurrentia in<emph.end type="italics"/>E, <lb/><emph type="italics"/>& in ip&longs;is capiantur longitudines<emph.end type="italics"/>DN, <emph type="italics"/>quæ &longs;int ut quantitas<emph.end type="italics"/><lb/>(DE<emph type="italics"/>q<emph.end type="italics"/>XPS/PE) <emph type="italics"/>& vis quam Sphæræ particula &longs;ita in axe ad di­<lb/>&longs;tantiam<emph.end type="italics"/>PE <emph type="italics"/>exercet in corpu&longs;culum<emph.end type="italics"/>P <emph type="italics"/>conjunctim: dico quod <lb/>Vis tota, qua corpu&longs;culum<emph.end type="italics"/>P <emph type="italics"/>trahitur ver&longs;us Sphæram, est ut <lb/>area comprehen&longs;a &longs;ub axe Sphæræ<emph.end type="italics"/>AB <emph type="italics"/>& linea curva<emph.end type="italics"/>ANB, <lb/><emph type="italics"/>quam punctum<emph.end type="italics"/>N <emph type="italics"/>perpetuo tangit.<emph.end type="italics"/></s></p><pb xlink:href="039/01/213.jpg" pagenum="185"/> <p type="main"> <s>Etenim &longs;tantibus quæ in Lemmate & Theoremate novi&longs;&longs;imo <lb/><arrow.to.target n="note161"/>con&longs;tructa &longs;unt, concipe axem Sphæræ <emph type="italics"/>AB<emph.end type="italics"/>dividi in particulas <lb/>innumeras æquales <emph type="italics"/>Dd,<emph.end type="italics"/>& Sphæram totam dividi in totidem <lb/>laminas Sphæricas concavo-convexas <emph type="italics"/>EFfe<emph.end type="italics"/>; & erigatur perpen­<lb/>diculum <emph type="italics"/>dn.<emph.end type="italics"/>Per Theorema &longs;uperius, vis qua lamina <emph type="italics"/>EFfe<emph.end type="italics"/><lb/>trahit corpu&longs;culum <emph type="italics"/>P<emph.end type="italics"/>e&longs;t ut <emph type="italics"/>DEqXFf<emph.end type="italics"/>& vis particulæ unius ad <lb/>di&longs;tantiam <emph type="italics"/>PE<emph.end type="italics"/>vel <emph type="italics"/>PF<emph.end type="italics"/>exercita conjunctim. </s> <s>E&longs;t autem per Lem­<lb/>ma novi&longs;&longs;imum, <emph type="italics"/>Dd<emph.end type="italics"/>ad <emph type="italics"/>Ff<emph.end type="italics"/>ut <emph type="italics"/>PE<emph.end type="italics"/>ad <emph type="italics"/>PS,<emph.end type="italics"/>& inde <emph type="italics"/>Ff<emph.end type="italics"/>æqualis <lb/>(<emph type="italics"/>PSXDd/PE<emph.end type="italics"/>); & <emph type="italics"/>DEqXFf<emph.end type="italics"/>æquale <emph type="italics"/>Dd<emph.end type="italics"/>in (<emph type="italics"/>DEqXPS/PE<emph.end type="italics"/>), & propter­<lb/>ea vis laminæ <emph type="italics"/>EFfe<emph.end type="italics"/>e&longs;t ut <emph type="italics"/>Dd<emph.end type="italics"/>in (<emph type="italics"/>DEqXPS/PE<emph.end type="italics"/>) & vis particulæ ad <lb/>di&longs;tantiam <emph type="italics"/>PF<emph.end type="italics"/>exercita conjunctim, hoc e&longs;t (ex Hypothe&longs;i) ut <lb/><emph type="italics"/>DNXDd,<emph.end type="italics"/>&longs;eu area evane&longs;cens <emph type="italics"/>DNnd.<emph.end type="italics"/>Sunt igitur laminarum <lb/>omnium vires in corpus <emph type="italics"/>P<emph.end type="italics"/>exercitæ, ut areæ omnes <emph type="italics"/>DNnd,<emph.end type="italics"/>hoc <lb/>e&longs;t, Sphæræ vis tota ut area tota <emph type="italics"/>ABNA. Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note161"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i vis centripeta, ad particulas &longs;ingulas tendens, <lb/>eadem &longs;emper maneat in omnibus di&longs;tantiis, & fiat <emph type="italics"/>DN<emph.end type="italics"/>ut <lb/>(<emph type="italics"/>DEqXPS/PE<emph.end type="italics"/>): erit vis tota qua corpu&longs;culum a Sphæra attrahitur, <lb/>ut area <emph type="italics"/>ABNA.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si particularum vis centripeta &longs;it reciproce ut di&longs;tantia <lb/>corpu&longs;culi a &longs;e attracti, & fiat <emph type="italics"/>DN<emph.end type="italics"/>ut (<emph type="italics"/>DEqXPS/PEq<emph.end type="italics"/>): erit vis qua <lb/>corpu&longs;culum <emph type="italics"/>P<emph.end type="italics"/>a Sphæra tota attrahitur ut area <emph type="italics"/>ABNA.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Si particularum vis centripeta &longs;it reciproce ut cubus di­<lb/>&longs;tantiæ corpu&longs;culi a &longs;e attracti, & fiat <emph type="italics"/>DN<emph.end type="italics"/>ut (<emph type="italics"/>DEqXPS/PEqq<emph.end type="italics"/>): erit <lb/>vis qua corpu&longs;culum a tota Sphæra attrahitur ut area <emph type="italics"/>ABNA.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Et univer&longs;aliter &longs;i vis centripeta ad &longs;ingulas Sphæræ <lb/>particulas tendens ponatur e&longs;&longs;e reciproce ut quantitas V, fiat au­<lb/>tem <emph type="italics"/>DN<emph.end type="italics"/>ut (<emph type="italics"/>DEqXPS/PEXV<emph.end type="italics"/>); erit vis qua corpu&longs;culum a Sphæra tota <lb/>attrahitur ut area <emph type="italics"/>ABNA.<emph.end type="italics"/><pb xlink:href="039/01/214.jpg" pagenum="186"/><arrow.to.target n="note162"/></s></p> <p type="margin"> <s><margin.target id="note162"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXXI. PROBLEMA XLI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Stantibus jam po&longs;itis, men&longs;uranda est Area<emph.end type="italics"/>ABNA.<emph.end type="center"/></s></p> <p type="main"> <s>A puncto <emph type="italics"/>P<emph.end type="italics"/>ducatur recta <emph type="italics"/>PH<emph.end type="italics"/>Sphæram tangens in <emph type="italics"/>H,<emph.end type="italics"/>& ad <lb/>axem <emph type="italics"/>PAB<emph.end type="italics"/>demi&longs;&longs;a normali <emph type="italics"/>HI,<emph.end type="italics"/>bi&longs;ecetur <emph type="italics"/>PI<emph.end type="italics"/>in <emph type="italics"/>L;<emph.end type="italics"/>& erit <lb/>(per Prop. </s> <s>12, Lib. </s> <s>2. Elem.) <emph type="italics"/>PEq<emph.end type="italics"/>æquale <emph type="italics"/>PSq + SEq<emph.end type="italics"/>+ <lb/>2<emph type="italics"/>PSD.<emph.end type="italics"/>E&longs;t autem <emph type="italics"/>SEq<emph.end type="italics"/>&longs;eu <emph type="italics"/>SHq<emph.end type="italics"/>(ob &longs;imilitudinem triangu­<lb/>lorum <emph type="italics"/>SPH, SHI<emph.end type="italics"/>) æquale rectangulo <emph type="italics"/>PSI.<emph.end type="italics"/>Ergo <emph type="italics"/>PEq<emph.end type="italics"/>æquale <lb/>e&longs;t contento &longs;ub <emph type="italics"/>PS<emph.end type="italics"/>& <emph type="italics"/>PS+SI<emph.end type="italics"/>+2<emph type="italics"/>SD,<emph.end type="italics"/>hoc e&longs;t, &longs;ub <emph type="italics"/>PS<emph.end type="italics"/>& <lb/>2<emph type="italics"/>LS<emph.end type="italics"/>+2<emph type="italics"/>SD,<emph.end type="italics"/>id e&longs;t, &longs;ub <emph type="italics"/>PS<emph.end type="italics"/>& 2<emph type="italics"/>LD.<emph.end type="italics"/>Porro <emph type="italics"/>DE quad<emph.end type="italics"/>æquale <lb/>e&longs;t <emph type="italics"/>SEq-SDq,<emph.end type="italics"/>&longs;eu <emph type="italics"/>SEq -LSq<emph.end type="italics"/>+2<emph type="italics"/>SLD-LDq,<emph.end type="italics"/>id e&longs;t, <lb/>2<emph type="italics"/>SLD-LDq-ALB.<emph.end type="italics"/>Nam <emph type="italics"/>LSq-SEq<emph.end type="italics"/>&longs;eu <emph type="italics"/>LSq-SAq<emph.end type="italics"/><lb/><figure id="id.039.01.214.1.jpg" xlink:href="039/01/214/1.jpg"/><lb/>(per Prop. </s> <s>6, Lib. </s> <s>2. Elem.) æquatur rectangulo <emph type="italics"/>ALB.<emph.end type="italics"/>Scriba­<lb/>tur itaque 2<emph type="italics"/>SLD -LDq -ALB<emph.end type="italics"/>pro <emph type="italics"/>DEq<emph.end type="italics"/>; & quantitas <lb/>(<emph type="italics"/>DEqXPS/PEXV<emph.end type="italics"/>), quæ &longs;ecundum Corollarium quartum Propo&longs;itionis <lb/>præcedentis e&longs;t ut longitudo ordinatim applicatæ <emph type="italics"/>DN,<emph.end type="italics"/>re&longs;olvet <lb/>&longs;e&longs;e in tres partes (2<emph type="italics"/>SLDXPS/PE<emph.end type="italics"/>XV)-(<emph type="italics"/>LDqXPS/PE<emph.end type="italics"/>XV)-(<emph type="italics"/>ALBXPS/PE<emph.end type="italics"/>XV): <lb/>ubi &longs;i pro V &longs;cribatur ratio inver&longs;a vis centripetæ, & pro <emph type="italics"/>PE<emph.end type="italics"/>me­<lb/>dium proportionale inter <emph type="italics"/>PS<emph.end type="italics"/>& 2<emph type="italics"/>LD<emph.end type="italics"/>; tres illæ partes evadent <lb/>ordinatim applicatæ linearum totidem curvarum, quarum areæ per <lb/>Methodos vulgatas innote&longs;cunt. <emph type="italics"/><expan abbr="q.">que</expan> E. F.<emph.end type="italics"/><pb xlink:href="039/01/215.jpg" pagenum="187"/><arrow.to.target n="note163"/></s></p> <p type="margin"> <s><margin.target id="note163"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/>1. Si vis centripeta ad &longs;ingulas Sphæræ particulas ten­<lb/>dens &longs;it reciproce ut di&longs;tantia; pro V &longs;cribe di&longs;tantiam <emph type="italics"/>PE<emph.end type="italics"/>; dein <lb/>2<emph type="italics"/>PSXLD<emph.end type="italics"/>pro <emph type="italics"/>PEq,<emph.end type="italics"/>& fiet <emph type="italics"/>DN<emph.end type="italics"/>ut <emph type="italics"/>SL-1/2LD-(ALB/2LD).<emph.end type="italics"/><lb/>Pone <emph type="italics"/>DN<emph.end type="italics"/>æqualem duplo ejus 2<emph type="italics"/>SL-LD-(ALB/LD)<emph.end type="italics"/>: & ordinatæ <lb/>pars data 2<emph type="italics"/>SL<emph.end type="italics"/>ducta in longitudinem <emph type="italics"/>AB<emph.end type="italics"/>de&longs;cribet aream rectan­<lb/>gulam 2<emph type="italics"/>SLXAB<emph.end type="italics"/>; & pars indefinita <emph type="italics"/>LD<emph.end type="italics"/>ducta normaliter in <lb/>eandem longitudinem per motum continuum, ea lege ut inter mo­<lb/>vendum cre&longs;cendo vel decre&longs;cendo æquetur &longs;emper longitudini <lb/><emph type="italics"/>LD,<emph.end type="italics"/>de&longs;cribet aream (<emph type="italics"/>LBq-LAq<emph.end type="italics"/>/2), id e&longs;t, aream <emph type="italics"/>SLXAB<emph.end type="italics"/>; quæ <lb/>&longs;ubducta de area priore 2<emph type="italics"/>SLXAB<emph.end type="italics"/>relinquit aream <emph type="italics"/>SLXAB.<emph.end type="italics"/><lb/>Pars autem tertia (<emph type="italics"/>ALB/LD<emph.end type="italics"/>) ducta itidem per motum localem norma­<lb/>liter in eandem longitudinem, de&longs;cribet <lb/><figure id="id.039.01.215.1.jpg" xlink:href="039/01/215/1.jpg"/><lb/>aream Hyperbolicam; quæ &longs;ubducta de <lb/>area <emph type="italics"/>SLXAB<emph.end type="italics"/>relinquet aream quæ&longs;itam <lb/><emph type="italics"/>ABNA.<emph.end type="italics"/>Unde talis emergit Proble­<lb/>matis con&longs;tructio. </s> <s>Ad puncta <emph type="italics"/>L, A, B<emph.end type="italics"/><lb/>erige perpendicula <emph type="italics"/>Ll, Aa, Bb,<emph.end type="italics"/>quorum <lb/><emph type="italics"/>Aa<emph.end type="italics"/>ip&longs;i <emph type="italics"/>LB,<emph.end type="italics"/>& <emph type="italics"/>Bb<emph.end type="italics"/>ip&longs;i <emph type="italics"/>LA<emph.end type="italics"/>æquetur. </s> <s><lb/>A&longs;ymptotis <emph type="italics"/>Ll, LB,<emph.end type="italics"/>per puncta <emph type="italics"/>a, b<emph.end type="italics"/>de­<lb/>&longs;cribatur Hyperbola <emph type="italics"/>ab.<emph.end type="italics"/>Et acta chor­<lb/>da <emph type="italics"/>ba<emph.end type="italics"/>claudet aream <emph type="italics"/>aba<emph.end type="italics"/>areæ quæ&longs;itæ <lb/><emph type="italics"/>ABNA<emph.end type="italics"/>æqualem. </s></p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/>2. Si vis centripeta ad &longs;ingulas Sphæræ particulas ten­<lb/>dens &longs;it reciproce ut cubus di&longs;tantiæ, vel (quod perinde e&longs;t) ut cubus <lb/>ille applicatus ad planum quodvis datum; &longs;cribe (<emph type="italics"/>PEcub/2ASq<emph.end type="italics"/>) pro V, <lb/>dein 2<emph type="italics"/>PSXLD<emph.end type="italics"/>pro <emph type="italics"/>PEq<emph.end type="italics"/>; & fiet <emph type="italics"/>DN<emph.end type="italics"/>ut <emph type="italics"/>(SLXASq/PSXLD)-(ASq/2PS) <lb/>-(ALBXASq/2PSXLDq),<emph.end type="italics"/>id e&longs;t (ob continue proportionales <emph type="italics"/>PS, AS, SI<emph.end type="italics"/>) <lb/>ut <emph type="italics"/>(LSI/LD)-1/2SI-(ALBXSI/2LDq).<emph.end type="italics"/>Si ducantur hujus partes tres <lb/>in longitudinem <emph type="italics"/>AB,<emph.end type="italics"/>prima (<emph type="italics"/>LSI/LD<emph.end type="italics"/>) generabit aream Hyper-</s></p><pb xlink:href="039/01/216.jpg" pagenum="188"/> <p type="main"> <s><arrow.to.target n="note164"/>bolicam; &longs;ecunda 1/2<emph type="italics"/>SI<emph.end type="italics"/>aream 1/2<emph type="italics"/>ABXSI<emph.end type="italics"/>; tertia (<emph type="italics"/>ALBXSI/2LDq<emph.end type="italics"/>) are­<lb/>am <emph type="italics"/>(ALBXSI/2LA)-(ALBXSI/2LB),<emph.end type="italics"/>id e&longs;t 1/2<emph type="italics"/>ABXSI.<emph.end type="italics"/>De prima &longs;ub­<lb/>ducatur &longs;umma &longs;ecundæ & tertiæ, & <lb/><figure id="id.039.01.216.1.jpg" xlink:href="039/01/216/1.jpg"/><lb/>manebit area quæ&longs;ita <emph type="italics"/>ABNA.<emph.end type="italics"/>Un­<lb/>de talis emergit Problematis con&longs;tru­<lb/>ctio. </s> <s>Ad puncta <emph type="italics"/>L, A, S, B<emph.end type="italics"/>erige <lb/>perpendicula <emph type="italics"/>Ll, Aa, Ss, Bb,<emph.end type="italics"/>quo­<lb/>rum <emph type="italics"/>Ss<emph.end type="italics"/>ip&longs;i <emph type="italics"/>SI<emph.end type="italics"/>æquetur, perque pun­<lb/>ctum <emph type="italics"/>s<emph.end type="italics"/>A&longs;ymptotis <emph type="italics"/>Ll, LB<emph.end type="italics"/>de&longs;cri­<lb/>batur Hyperbola <emph type="italics"/>asb<emph.end type="italics"/>occurrens per­<lb/>pendiculis <emph type="italics"/>Aa, Bb<emph.end type="italics"/>in <emph type="italics"/>a<emph.end type="italics"/>& <emph type="italics"/>b<emph.end type="italics"/>; & rect­<lb/>angulum 2<emph type="italics"/>ASI<emph.end type="italics"/>&longs;ubductum de area <lb/>Hyperbolica <emph type="italics"/>AasbB<emph.end type="italics"/>reliquet aream <lb/>quæ&longs;itam <emph type="italics"/>ABNA.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note164"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/>3. Si Vis centripeta, ad &longs;ingulas Sphæræ particulas <lb/>tendens, decre&longs;cit in quadruplicata ratione di&longs;tantiæ a particulis; <lb/>&longs;cribe (<emph type="italics"/>PEqq/2AScub<emph.end type="italics"/>) pro V, dein √2<emph type="italics"/>PSXLD<emph.end type="italics"/>pro <emph type="italics"/>PE,<emph.end type="italics"/>& fiet <emph type="italics"/>DN<emph.end type="italics"/>ut <lb/><emph type="italics"/>(SIqXSL/√2SI)X(1/√LDc),-(SIq/2√2SI)X(1/√LD),-(SIqXALB/2√2SI)X(1/√LDqc).<emph.end type="italics"/><lb/>Cujus tres partes ductæ in longitudinem <emph type="italics"/>AB,<emph.end type="italics"/>producunt areas tot­<lb/>idem, <emph type="italics"/>viz. (2SIqXSL/√2SI<emph.end type="italics"/>) in <emph type="italics"/>(1/√LA)-(1/√LB); (SIq/√2SI)<emph.end type="italics"/>in <emph type="italics"/>√LB-√LA<emph.end type="italics"/>; <lb/>& (<emph type="italics"/>SIqXALB/3√2SI<emph.end type="italics"/>) in <emph type="italics"/>(1/√LAcub)-(1/√LBcub).<emph.end type="italics"/>Et hæ po&longs;t debitam redu­<lb/>ctionem fiunt <emph type="italics"/>(2SIqXSL/LI), SIq,<emph.end type="italics"/>& <emph type="italics"/>SIq+(2SIcub/3LI).<emph.end type="italics"/>Hæ vero, &longs;ub­<lb/>ctis po&longs;terioribus de priore, evadunt (<emph type="italics"/>4SIcub/3LI<emph.end type="italics"/>). Igitur vis tota, qua <lb/>corpu&longs;culum <emph type="italics"/>P<emph.end type="italics"/>in Sphæræ centrum trahitur, e&longs;t ut <emph type="italics"/>(SIcub/PI),<emph.end type="italics"/>id e&longs;t, <lb/>reciproce ut <emph type="italics"/>PS cubXPI. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s>Eadem Methodo determinari pote&longs;t Attractio corpu&longs;culi &longs;iti in­<lb/>tra Sphæram, &longs;ed expeditius per Theorema &longs;equens. <pb xlink:href="039/01/217.jpg" pagenum="189"/><arrow.to.target n="note165"/></s></p> <p type="margin"> <s><margin.target id="note165"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXXII. THEOREMA XLI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>In Sphæra centro<emph.end type="italics"/>S <emph type="italics"/>intervallo<emph.end type="italics"/>SA <emph type="italics"/>de&longs;cripta, &longs;i capiantur<emph.end type="italics"/>SI, SA, <lb/>SP <emph type="italics"/>continue proportionales: dico quod corpu&longs;culi intra Sphæ­<lb/>ram in loco quovis<emph.end type="italics"/>I <emph type="italics"/>attractio est ad attractionem ip&longs;ius extra <lb/>Sphæram in loco<emph.end type="italics"/>P, <emph type="italics"/>in ratione compo&longs;ita ex &longs;ubduplicata ratione <lb/>di&longs;tantiarum a centro<emph.end type="italics"/>IS, PS <emph type="italics"/>& &longs;ubduplicata ratione virium <lb/>centripetarum, in locis illis<emph.end type="italics"/>P <emph type="italics"/>&<emph.end type="italics"/>I, <emph type="italics"/>ad centrum tendentium.<emph.end type="italics"/></s></p> <p type="main"> <s>Ut &longs;i vires centripetæ particularum Sphæræ &longs;int reciproce ut di­<lb/>&longs;tantiæ corpu&longs;culi a &longs;e attracti; vis, qua corpu&longs;culum &longs;itum in <emph type="italics"/>I<emph.end type="italics"/><lb/>trahitur a Sphæra tota, erit ad vim qua trahitur in <emph type="italics"/>P,<emph.end type="italics"/>in ratione <lb/><figure id="id.039.01.217.1.jpg" xlink:href="039/01/217/1.jpg"/><lb/>compo&longs;ita ex &longs;ubduplicata ratione di&longs;tantiæ <emph type="italics"/>SI<emph.end type="italics"/>ad di&longs;tantiam <emph type="italics"/>SP<emph.end type="italics"/><lb/>& ratione &longs;ubduplicata vis centripetæ in loco <emph type="italics"/>I,<emph.end type="italics"/>a particula aliqua <lb/>in centro oriundæ, ad vim centripetam in loco <emph type="italics"/>P<emph.end type="italics"/>ab eadem in cen­<lb/>tro particula oriundam, id e&longs;t, ratione &longs;ubduplicata di&longs;tantiarum <lb/><emph type="italics"/>SI, SP<emph.end type="italics"/>ad invicem reciproce. </s> <s>Hæ duæ rationes &longs;ubduplicatæ <lb/>componunt rationem æqualitatis, & propterea attractiones in <emph type="italics"/>I<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/><lb/>a Sphæra tota factæ æquantur. </s> <s>Simili computo, &longs;i vires particu­<lb/>larum Sphæræ &longs;unt reciproce in duplicata ratione di&longs;tantiarum, col­<lb/>ligetur quod attractio in <emph type="italics"/>I<emph.end type="italics"/>&longs;it ad attractionem in <emph type="italics"/>P,<emph.end type="italics"/>ut di&longs;tantia <emph type="italics"/>SP<emph.end type="italics"/><lb/>ad Sphæræ &longs;emidiametrum <emph type="italics"/>SA:<emph.end type="italics"/>Si vires illæ &longs;unt reciproce in tr­<lb/>plicata ratione di&longs;tantiarum, attractiones in <emph type="italics"/>I<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>erunt ad invi-<pb xlink:href="039/01/218.jpg" pagenum="190"/><arrow.to.target n="note166"/>cem ut <emph type="italics"/>SP quad<emph.end type="italics"/>ad <emph type="italics"/>SA quad:<emph.end type="italics"/>Si in quadruplicata, ut <emph type="italics"/>SP cub<emph.end type="italics"/>ad <lb/><emph type="italics"/>SA cub.<emph.end type="italics"/>Unde cum attractio in <emph type="italics"/>P,<emph.end type="italics"/>in hoc ultimo ca&longs;u, inventa <lb/>fuit reciproce ut <emph type="italics"/>PS cubXPI,<emph.end type="italics"/>attractio in <emph type="italics"/>I<emph.end type="italics"/>erit reciproce ut <lb/><emph type="italics"/>SA cubXPI,<emph.end type="italics"/>id e&longs;t (ob datum <emph type="italics"/>SA cub<emph.end type="italics"/>) reciproce ut <emph type="italics"/>PI.<emph.end type="italics"/>Et <lb/>&longs;imilis e&longs;t progre&longs;&longs;us in infinitum. </s> <s>Theorema vero &longs;ic demon­<lb/>&longs;tratur. </s></p> <p type="margin"> <s><margin.target id="note166"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Stantibus jam ante con&longs;tructis, & exi&longs;tente corpore in loco <lb/>quovis <emph type="italics"/>P,<emph.end type="italics"/>ordinatim applicata <emph type="italics"/>DN<emph.end type="italics"/>inventa fuit ut (<emph type="italics"/>DEqXPS/PEXV<emph.end type="italics"/>). <lb/>Ergo &longs;i agatur <emph type="italics"/>IE,<emph.end type="italics"/>ordinata illa ad alium quemvis locum <emph type="italics"/>I,<emph.end type="italics"/>mu­<lb/>tatis mutandis, evadet ut (<emph type="italics"/>DEqXIS/IEXV<emph.end type="italics"/>). Pone vires centripetas, e <lb/>Sphæræ puncto quovis <emph type="italics"/>E<emph.end type="italics"/>manantes, e&longs;&longs;e ad invicem in di&longs;tantiis <lb/><emph type="italics"/>IE, PE,<emph.end type="italics"/>ut <emph type="italics"/>PE<emph type="sup"/>n<emph.end type="sup"/><emph.end type="italics"/>ad <emph type="italics"/>IE<emph type="sup"/>n<emph.end type="sup"/>,<emph.end type="italics"/>(ubi numerus <emph type="italics"/>n<emph.end type="italics"/>de&longs;ignet indicem <lb/>pote&longs;tatum <emph type="italics"/>PE<emph.end type="italics"/>& <emph type="italics"/>IE<emph.end type="italics"/>) & ordinatæ illæ fient ut (<emph type="italics"/>DEqXPS/PEXPE<emph type="sup"/>n<emph.end type="sup"/><emph.end type="italics"/>) & <lb/>(<emph type="italics"/>DEqXIS/IEXIE<emph type="sup"/>n<emph.end type="sup"/><emph.end type="italics"/>), quarum ratio ad invicem e&longs;t ut <emph type="italics"/>PSXIEXIE<emph type="sup"/>n<emph.end type="sup"/><emph.end type="italics"/>ad <lb/><emph type="italics"/>ISXPEXPE<emph type="sup"/>n<emph.end type="sup"/>.<emph.end type="italics"/>Quoniam ob &longs;imilia triangula <emph type="italics"/>SPE, SEI,<emph.end type="italics"/>fit <lb/><emph type="italics"/>IE<emph.end type="italics"/>ad <emph type="italics"/>PE<emph.end type="italics"/>ut <emph type="italics"/>IS<emph.end type="italics"/>ad <emph type="italics"/>SE<emph.end type="italics"/>vel <emph type="italics"/>SA<emph.end type="italics"/>; pro ratione <emph type="italics"/>IE<emph.end type="italics"/>ad <emph type="italics"/>PE<emph.end type="italics"/>&longs;cribe <lb/>rationem <emph type="italics"/>IS<emph.end type="italics"/>ad <emph type="italics"/>SA<emph.end type="italics"/>; & ordinatarum ratio evadet <emph type="italics"/>PSXIE<emph type="sup"/>n<emph.end type="sup"/><emph.end type="italics"/>ad <lb/><emph type="italics"/>SAXPE<emph type="sup"/>n<emph.end type="sup"/>.<emph.end type="italics"/>Sed <emph type="italics"/>PS<emph.end type="italics"/>ad <emph type="italics"/>SA<emph.end type="italics"/>&longs;ubduplicata e&longs;t ratio di&longs;tantiarum <lb/><emph type="italics"/>PS, SI<emph.end type="italics"/>; & <emph type="italics"/>IE<emph type="sup"/>n<emph.end type="sup"/><emph.end type="italics"/>ad <emph type="italics"/>PE<emph type="sup"/>n<emph.end type="sup"/><emph.end type="italics"/>&longs;ubduplicata e&longs;t ratio virium in di&longs;tan­<lb/>tiis <emph type="italics"/>PS, IS.<emph.end type="italics"/>Ergo ordinatæ, & propterea areæ quas ordinatæ <lb/>de&longs;cribunt, hi&longs;que proportionales attractiones, &longs;unt in ratione com­<lb/>po&longs;ita ex &longs;ubduplicatis illis rationibus. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXXIII. PROBLEMA XLII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire vim qua corpu&longs;culum in centro Sphæræ locatum ad ejus <lb/>Segmentum quodcunque attrahitur.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>P<emph.end type="italics"/>corpus in centro Sphæræ, & <emph type="italics"/>RBSD<emph.end type="italics"/>Segmentum ejus <lb/>plano <emph type="italics"/>RDS<emph.end type="italics"/>& &longs;uperficie Sphærica <emph type="italics"/>RBS<emph.end type="italics"/>contentum. </s> <s>Superfi­<lb/>cie Sphærica <emph type="italics"/>EFG<emph.end type="italics"/>centro <emph type="italics"/>P<emph.end type="italics"/>de&longs;cripta &longs;ecetur <emph type="italics"/>DB<emph.end type="italics"/>in <emph type="italics"/>F,<emph.end type="italics"/>ac di­<lb/>&longs;tinguatur Segmentum in partes <emph type="italics"/>BREFGS, FEDG.<emph.end type="italics"/>Sit <lb/>autem &longs;uperficies illa non pure Mathematica, &longs;ed Phy&longs;ica, pro­<lb/>funditatem habens quam minimam. </s> <s>Nominetur i&longs;ta profundi-<pb xlink:href="039/01/219.jpg" pagenum="191"/><arrow.to.target n="note167"/>tas O, & erit hæc &longs;uperficies (per de­<lb/><figure id="id.039.01.219.1.jpg" xlink:href="039/01/219/1.jpg"/><lb/>mon&longs;trata <emph type="italics"/>Archimedis<emph.end type="italics"/>) ut <emph type="italics"/>PFXDFXO.<emph.end type="italics"/><lb/>Ponamus præterea vires attractivas par­<lb/>ticularum Sphæræ e&longs;&longs;e reciproce ut <lb/>di&longs;tantiarum dignitas illa cujus Index <lb/>e&longs;t <emph type="italics"/>n<emph.end type="italics"/>; & vis qua &longs;uperficies <emph type="italics"/>FE<emph.end type="italics"/>trahit <lb/>corpus <emph type="italics"/>P<emph.end type="italics"/>erit ut (<emph type="italics"/>DFXO/PF<emph type="sup"/>n-1<emph.end type="sup"/><emph.end type="italics"/>). Huic pro­<lb/>portionale &longs;it perpendiculum <emph type="italics"/>FN<emph.end type="italics"/>duc­<lb/>tum in O; & area curvilinea <emph type="italics"/>BDLIB,<emph.end type="italics"/><lb/>quam ordinatim applicata <emph type="italics"/>FN<emph.end type="italics"/>in lon­<lb/>gitudinem <emph type="italics"/>DB<emph.end type="italics"/>per motum continuum <lb/>ducta de&longs;cribit, erit ut vis tota qua <lb/>Segmentum totum <emph type="italics"/>RBSD<emph.end type="italics"/>trahit corpus <emph type="italics"/>P. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note167"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXXIV. PROBLEMA XLIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire vim qua corpu&longs;culum, extra centrum Sphæræ in axe Seg­<lb/>menti cuju&longs;vis locatum, attrahitur ab eodem Segmento.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>A Segmento <emph type="italics"/>EBK<emph.end type="italics"/>trahatur corpus <emph type="italics"/>P<emph.end type="italics"/>(Vide Fig. </s> <s>Prop. </s> <s>LXXIX, <lb/>LXXX, LXXXI) in ejus axe <emph type="italics"/>ADB<emph.end type="italics"/>locatum. </s> <s>Centro <emph type="italics"/>P<emph.end type="italics"/>interval­<lb/>lo <emph type="italics"/>PE<emph.end type="italics"/>de&longs;cribatur &longs;uperficies Sphærica <emph type="italics"/>EFK,<emph.end type="italics"/>qua di&longs;tinguatur <lb/>Segmentum in partes duas <emph type="italics"/>EBKF<emph.end type="italics"/>& <emph type="italics"/>EFKD.<emph.end type="italics"/>Quæratur vis par­<lb/>tis prioris per Prop. </s> <s>LXXXI, & vis partis po&longs;terioris per Prop. </s> <s><lb/>LXXXIII; & &longs;umma virium erit vis Segmenti totius <emph type="italics"/>EBKD. <lb/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Explicatis attractionibus corporum Sphærieorum, jam pergere <lb/>liceret ad Leges attractionum aliorum quorundam ex particulis at­<lb/>tractivis &longs;imiliter con&longs;tantium corporum; &longs;ed i&longs;ta particulatim <lb/>tractare minus ad in&longs;titutum &longs;pectat. </s> <s>Suffecerit Propo&longs;itiones <lb/>qua&longs;dam generaliores de viribus huju&longs;modi corporum, deque mo­<lb/>tibus inde oriundis, ob earum in rebus Philo&longs;ophicis aliqualem <lb/>u&longs;um, &longs;ubjungere. <pb xlink:href="039/01/220.jpg" pagenum="192"/><arrow.to.target n="note168"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note168"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>SECTIO XIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Corporum non Sphærieorum viribus attactivis.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXXV. THEOREMA XLII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si corporis attracti, ubi attrahenti contiguum est, attractio longe <lb/>fortior &longs;it, quam cum vel minimo intervallo &longs;eparantur ab in­<lb/>vicem: vires particularum trahentis, in rece&longs;&longs;u corporis attrac­<lb/>ti, decre&longs;cunt in ratione plu&longs;quam duplicata di&longs;tantiarum a <lb/>particulis.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam &longs;i vires decre&longs;cunt in ratione duplicata di&longs;tantiarum a par­<lb/>ticulis; attractio ver&longs;us corpus Sphæricum, propterea quod (per <lb/>Prop. </s> <s>LXXIV) &longs;it reciproce ut quadratum di&longs;tantiæ attracti corpo­<lb/>ris a centro Sphæræ, haud &longs;en&longs;ibiliter augebitur ex contactu; atque <lb/>adhuc minus augebitur ex contactu, &longs;i attractio in rece&longs;&longs;u corporis <lb/>attracti decre&longs;cat in ratione minore. </s> <s>Patet igitur Propo&longs;itio de <lb/>Sphæris attractivis. </s> <s>Et par e&longs;t ratio Orbium Sphærieorum conca­<lb/>vorum corpora externa trahentium. </s> <s>Et multo magis res con&longs;tat in <lb/>Orbibus corpora interius con&longs;tituta trahentibus, cum attractiones <lb/>pa&longs;&longs;im per Orbium cavitates ab attractionibus contrariis (per Prop. </s> <s><lb/>LXX) tollantur, ideoque vel in ip&longs;o contactu nullæ &longs;unt. </s> <s>Quod <lb/>&longs;i Sphæris hi&longs;ce Orbibu&longs;que Sphæricis partes quælibet a loco con­<lb/>tactus remotæ auferantur, & partes novæ ubivis addantur: mu­<lb/>tari po&longs;&longs;unt figuræ horum corporum attractivorum pro lubitu, nec <lb/>tamen partes additæ vel &longs;ubductæ, cum &longs;int a loco contactus re­<lb/>motæ, augebunt notabiliter attractionis exce&longs;&longs;um qui ex contactu <lb/>oritur. </s> <s>Con&longs;tat igitur Propo&longs;itio de corporibus Figurarum om­<lb/>nium. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/><pb xlink:href="039/01/221.jpg" pagenum="193"/><arrow.to.target n="note169"/></s></p> <p type="margin"> <s><margin.target id="note169"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXXVI. THEOREMA XLIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si particularum, ex quibus corpus attractivum componitur, vires <lb/>in rece&longs;&longs;u corporis attracti decre&longs;cunt in triplicata vel plu&longs;quam <lb/>triplicata ratione di&longs;tantiarum a particulis: attractio longe for­<lb/>tior erit in contactu, quam cum attrahens & attractum inter­<lb/>vallo vel minimo &longs;eparantur ab invicem.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam attractionem in acce&longs;&longs;u attracti corpu&longs;culi ad huju&longs;modi <lb/>Sphæram trahentem augeri in infinitum, con&longs;tat per &longs;olutionem Pro­<lb/>blematis XLI, in Exemplo &longs;ecundo ac tertio exhibitam. </s> <s>Idem, per <lb/>Exempla illa & Theorema XLI inter &longs;e collata, facile colligitur <lb/>de attractionibus corporum ver&longs;us Orbes concavo-convexos, &longs;ive <lb/>corpora attracta collocentur extra Orbes, &longs;ive intra in eorum cavi­<lb/>tatibus. </s> <s>Sed & addendo vel auferendo his Sphæris & Orbibus ubi­<lb/>vis extra locum contactus materiam quamlibet attractivam, eo ut <lb/>corpora attractiva induant figuram quamvis a&longs;&longs;ignatam, con&longs;tabit <lb/>Propo&longs;itio de corporibus univer&longs;is. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXXVII. THEOREMA XLIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si corpora duo &longs;ibi invicem &longs;imilia, & ex materia æqualiter attra­<lb/>ctiva con&longs;tantia, &longs;eor&longs;im attrahant corpu&longs;cula &longs;ibi ip&longs;is proporti­<lb/>onalia & ad &longs;e &longs;imiliter po&longs;ita: attractiones acceleratrices cor­<lb/>pu&longs;culorum in corpora tota erunt ut attractiones acceleratrices <lb/>corpu&longs;culorum in eorum particulas totis proportionales & in to­<lb/>tis &longs;imiliter po&longs;itas.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam &longs;i corpora di&longs;tinguantur in particulas, quæ &longs;int totis pro­<lb/>portionales & in totis &longs;imiliter &longs;itæ; erit, ut attractio in particulam <lb/>quamlibet unius corporis ad attractionem in particulam corre&longs;pon­<lb/>dentem in corpore altero, ita attractiones in particulas &longs;ingulas <lb/>primi corporis ad attractiones in alterius particulas &longs;ingulas corre&longs;­<lb/>pondentes; & componendo, ita attractio in totum primum corpus <lb/>ad attractionem in totum &longs;ecundum. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Ergo &longs;i vires attractivæ particularum, augendo di&longs;tan­<lb/>tias corpu&longs;culorum attractorum, decre&longs;cant in ratione dignitatis <pb xlink:href="039/01/222.jpg" pagenum="194"/><arrow.to.target n="note170"/>cuju&longs;vis di&longs;tantiarum: attractiones acceleratrices in corpora tota <lb/>erunt ut corpora directe & di&longs;tantiarum dignitates illæ inver&longs;e. </s> <s>Ut <lb/>&longs;i vires particularum decre&longs;cant in ratione duplicata di&longs;tantiarum <lb/>a corpu&longs;culis attractis, corpora autem &longs;int ut <emph type="italics"/>A cub.<emph.end type="italics"/>& <emph type="italics"/>B cub.<emph.end type="italics"/>ad­<lb/>eoque tum corporum latera cubica, tum corpu&longs;culorum attracto­<lb/>rum di&longs;tantiæ a corporibus, ut <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B:<emph.end type="italics"/>attractiones acceleratri­<lb/>ces in corpora erunt ut (<emph type="italics"/>Acub./Aquad.<emph.end type="italics"/>) & (<emph type="italics"/>Bcub./Bquad.<emph.end type="italics"/>) id e&longs;t, ut corporum la­<lb/>tera illa cubica <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B.<emph.end type="italics"/>Si vires particularum decre&longs;cant in ra­<lb/>tione triplicata di&longs;tantiarum a corpu&longs;culis attractis; attractiones <lb/>acceleratrices in corpora tota erunt ut (<emph type="italics"/>Acub./Acub.<emph.end type="italics"/>) & (<emph type="italics"/>Bcub./Bcub.<emph.end type="italics"/>), id e&longs;t, æqua­<lb/>les. </s> <s>Si vires decre&longs;cant in ratione quadruplicata; attractiones in <lb/>corpora erunt ut (<emph type="italics"/>Acub./Aqq.<emph.end type="italics"/>) & (<emph type="italics"/>Bcub./Bqq.<emph.end type="italics"/>) id e&longs;t, reciproce ut latera cubi­<lb/>ca <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B.<emph.end type="italics"/>Et &longs;ic in cæteris. </s></p> <p type="margin"> <s><margin.target id="note170"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Unde vici&longs;&longs;im, ex viribus quibus corpora &longs;imilia tra­<lb/>hunt corpu&longs;cula ad &longs;e &longs;imiliter po&longs;ita, colligi pote&longs;t ratio decre­<lb/>menti virium particularum attractivarum in rece&longs;&longs;u corpu&longs;culi at­<lb/>tracti; &longs;i modo decrementum illud &longs;it directe vel inver&longs;e in ratione <lb/>aliqua di&longs;tantiarum. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXXVIII. THEOREMA XLV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si particularum æqualium Corporis cuju&longs;cunque vires attractivæ <lb/>&longs;int ut di&longs;tantiæ loeorum a particulis: vis corporis totius ten­<lb/>det ad ip&longs;ius centrum gravitatis; & eadem erit cum vi Globi <lb/>ex materia con&longs;imili & æquali con&longs;tantis & centrum habentis <lb/>in ejus centro gravitatis.<emph.end type="italics"/></s></p> <p type="main"> <s>Corporis <emph type="italics"/>RSTV<emph.end type="italics"/>particulæ <emph type="italics"/>A, <lb/>B<emph.end type="italics"/>trahant corpu&longs;culum aliquod <lb/><figure id="id.039.01.222.1.jpg" xlink:href="039/01/222/1.jpg"/><lb/><emph type="italics"/>Z<emph.end type="italics"/>viribus quæ, &longs;i particulæ æ­<lb/>quantur inter &longs;e, &longs;int ut di&longs;tan­<lb/>tiæ <emph type="italics"/>AZ, BZ<emph.end type="italics"/>; &longs;in particulæ &longs;ta­<lb/>tuantur inæquales, &longs;int ut hæ par­<lb/>ticulæ in di&longs;tantias &longs;uas <emph type="italics"/>AZ, BZ<emph.end type="italics"/><lb/>re&longs;pective ductæ. </s> <s>Et exponan­<lb/>tur hæ vires per contenta illa <lb/><emph type="italics"/>AXAZ<emph.end type="italics"/>& <emph type="italics"/>BXBZ.<emph.end type="italics"/>Jungatur <emph type="italics"/>AB,<emph.end type="italics"/><lb/>& &longs;ecetur ea in <emph type="italics"/>G<emph.end type="italics"/>ut &longs;it <emph type="italics"/>AG<emph.end type="italics"/>ad <emph type="italics"/>BG<emph.end type="italics"/>ut particula <emph type="italics"/>B<emph.end type="italics"/>ad particulam <emph type="italics"/>A<emph.end type="italics"/>; <pb xlink:href="039/01/223.jpg" pagenum="195"/>& erit <emph type="italics"/>G<emph.end type="italics"/>commune centrum gravitatis particularum <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B.<emph.end type="italics"/>Vis <lb/><arrow.to.target n="note171"/><emph type="italics"/>AXAZ<emph.end type="italics"/>(per Legum Corol.2.) re&longs;olvitur in vires <emph type="italics"/>AXGZ<emph.end type="italics"/>& <emph type="italics"/>AXAG<emph.end type="italics"/><lb/>& vis <emph type="italics"/>BXBZ<emph.end type="italics"/>in vires <emph type="italics"/>BXGZ<emph.end type="italics"/>& <emph type="italics"/>BXBG.<emph.end type="italics"/>Vires autem <emph type="italics"/>AXAG<emph.end type="italics"/><lb/>& <emph type="italics"/>BXBG,<emph.end type="italics"/>ob proportionales <emph type="italics"/>A<emph.end type="italics"/>ad <emph type="italics"/>B<emph.end type="italics"/>& <emph type="italics"/>BG<emph.end type="italics"/>ad <emph type="italics"/>AG,<emph.end type="italics"/>æquantur; <lb/>adeoque cum dirigantur in partes contrarias, &longs;e mutuo de&longs;truunt. </s> <s><lb/>Re&longs;tant vires <emph type="italics"/>AXGZ<emph.end type="italics"/>& <emph type="italics"/>BXGZ.<emph.end type="italics"/>Tendunt hæ ab Z ver&longs;us cen­<lb/>trum <emph type="italics"/>G,<emph.end type="italics"/>& vim ―<emph type="italics"/>A+BXGZ<emph.end type="italics"/>componunt; hoc e&longs;t, vim eandem ac <lb/>&longs;i particulæ attractivæ <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B<emph.end type="italics"/>con&longs;i&longs;terent in eorum communi gra­<lb/>vitatis centro <emph type="italics"/>G,<emph.end type="italics"/>Globum ibi componentes. </s></p> <p type="margin"> <s><margin.target id="note171"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s>Eodem argumento, &longs;i adjungatur particula tertia <emph type="italics"/>C,<emph.end type="italics"/>& compo­<lb/>natur hujus vis cum vi ―<emph type="italics"/>A+BXGZ<emph.end type="italics"/>tendente ad centrum <emph type="italics"/>G<emph.end type="italics"/>; vis <lb/>inde oriunda tendet ad commune centrum gravitatis Globi illius <emph type="italics"/>G<emph.end type="italics"/><lb/>& particulæ <emph type="italics"/>C<emph.end type="italics"/>; hoc e&longs;t, ad commune centrum gravitatis trium par­<lb/>ticularum <emph type="italics"/>A, B, C<emph.end type="italics"/>; & eadem erit ac &longs;i Globus & particula <emph type="italics"/>C<emph.end type="italics"/>con&longs;i­<lb/>&longs;terent in centro illo communi, Globum majorem ibi componentes. </s> <s><lb/>Et &longs;ic pergitur in infinitum. </s> <s>Eadem e&longs;t igitur vis tota particula­<lb/>rum omnium corporis cuju&longs;cunque <emph type="italics"/>RSTV<emph.end type="italics"/>ac &longs;i corpus illud, &longs;er­<lb/>vato gravitatis centro, figuram Globi indueret. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc motus corporis attracti <emph type="italics"/>Z<emph.end type="italics"/>idem erit ac &longs;i corpus <lb/>attrahens <emph type="italics"/>RSTV<emph.end type="italics"/>e&longs;&longs;et Sphæricum: & propterea &longs;i corpus illud <lb/>attrahens vel quie&longs;cat, vel progrediatur uniformiter in directum; <lb/>corpus attractum movebitur in Ellip&longs;i centrum habente in attra­<lb/>hentis centro gravitatis. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LXXXIX. THEOREMA XLVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Corpora &longs;int plura ex particulis æqualibus con&longs;tantia, quarum vi­<lb/>res &longs;unt ut di&longs;tantiæ loeorum a &longs;ingulis: vis ex omnium viri­<lb/>bus compo&longs;ita, qua corpu&longs;culum quodcunque trahitur, tendet ad <lb/>trahentium commune centrum gravitatis, & eadem erit ac &longs;i <lb/>trahentia illa, &longs;ervato gravitatis centro communi, coirent & in <lb/>Globum formarentur.<emph.end type="italics"/></s></p> <p type="main"> <s>Demon&longs;tratur eodem modo, atque Propo&longs;itio &longs;uperior. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Ergo motus corporis attracti idem erit ac &longs;i corpora tra­<lb/>hentia, &longs;ervato communi gravitatis centro, coirent & in Globum <lb/>formarentur. </s> <s>Ideoque &longs;i corporum trahentium commune gravita­<lb/>tis centrum vel quie&longs;cit, vel progreditur uniformiter in linea recta: <lb/>corpus attractum movebitur in Ellip&longs;i, centrum habente in com­<lb/>muni illo trahentium centro gravitatis. <pb xlink:href="039/01/224.jpg" pagenum="196"/><arrow.to.target n="note172"/></s></p> <p type="margin"> <s><margin.target id="note172"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XC. PROBLEMA XLIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si ad &longs;ingula Circuli cuju&longs;cunque puncta tendant vires æquales cen­<lb/>tripetæ, decre&longs;centes in quacunQ.E.D.&longs;tantiarum ratione: inve­<lb/>nire vim qua corpu&longs;culum attrahitur ubivis po&longs;itum in recta <lb/>quæ plano Circuli ad centrum ejus perpendiculariter in&longs;i&longs;tit.<emph.end type="italics"/></s></p> <p type="main"> <s>Centro <emph type="italics"/>A<emph.end type="italics"/>intervallo quovis <emph type="italics"/>AD,<emph.end type="italics"/>in plano cui recta <emph type="italics"/>AP<emph.end type="italics"/>per­<lb/>pendicularis e&longs;t, de&longs;cribi intelligatur Circulus; & invenienda &longs;it vis <lb/>qua corpu&longs;culum quodvis <emph type="italics"/>P<emph.end type="italics"/>in eundem attrahitur. </s> <s>A Circuli puncto <lb/>quovis <emph type="italics"/>E<emph.end type="italics"/>ad corpu&longs;culum attractum <emph type="italics"/>P<emph.end type="italics"/>agatur recta <emph type="italics"/>PE:<emph.end type="italics"/>In re­<lb/>cta <emph type="italics"/>PA<emph.end type="italics"/>capiatur <emph type="italics"/>PF<emph.end type="italics"/>ip&longs;i <emph type="italics"/>PE<emph.end type="italics"/>æ­<lb/><figure id="id.039.01.224.1.jpg" xlink:href="039/01/224/1.jpg"/><lb/>qualis, & erigatur normalis <emph type="italics"/>FK,<emph.end type="italics"/><lb/>quæ &longs;it ut vis qua punctum <emph type="italics"/>E<emph.end type="italics"/>tra­<lb/>hit corpu&longs;culum <emph type="italics"/>P.<emph.end type="italics"/>Sitque <emph type="italics"/>IKL<emph.end type="italics"/><lb/>curva linea quam punctum <emph type="italics"/>K<emph.end type="italics"/>per­<lb/>petuo tangit. </s> <s>Occurrat eadem Cir­<lb/>culi plano in <emph type="italics"/>L.<emph.end type="italics"/>In <emph type="italics"/>PA<emph.end type="italics"/>capiatur <lb/><emph type="italics"/>PH<emph.end type="italics"/>æqualis <emph type="italics"/>PD,<emph.end type="italics"/>& erigatur per­<lb/>pendiculum <emph type="italics"/>HI<emph.end type="italics"/>curvæ prædictæ <lb/>occurrens in <emph type="italics"/>I<emph.end type="italics"/>; & erit corpu&longs;­<lb/>culi <emph type="italics"/>P<emph.end type="italics"/>attractio in Circulum ut area <lb/><emph type="italics"/>AHIL<emph.end type="italics"/>ducta in altitudinem <emph type="italics"/>AP. <lb/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s>Etenim in <emph type="italics"/>AE<emph.end type="italics"/>capiatur linea quam minima <emph type="italics"/>Ee.<emph.end type="italics"/>Jungatur <emph type="italics"/>Pe,<emph.end type="italics"/><lb/>& in <emph type="italics"/>PE, PA<emph.end type="italics"/>capiantur <emph type="italics"/>PC, Pf<emph.end type="italics"/>ip&longs;i <emph type="italics"/>Pe<emph.end type="italics"/>æquales. </s> <s>Et quoniam vis, <lb/>qua annuli punctum quodvis <emph type="italics"/>E<emph.end type="italics"/>trahit ad &longs;e corpus <emph type="italics"/>P,<emph.end type="italics"/>ponitur e&longs;&longs;e <lb/>ut <emph type="italics"/>FK,<emph.end type="italics"/>& inde vis qua punctum illud trahit corpus <emph type="italics"/>P<emph.end type="italics"/>ver&longs;us <emph type="italics"/>A<emph.end type="italics"/>e&longs;t ut <lb/>(<emph type="italics"/>APXFK/PE<emph.end type="italics"/>), & vis qua annulus totus trahit corpus <emph type="italics"/>P<emph.end type="italics"/>ver&longs;us <emph type="italics"/>A,<emph.end type="italics"/>ut <lb/>annulus & (<emph type="italics"/>APXFK/PE<emph.end type="italics"/>) conjunctim; annulus autem i&longs;te e&longs;t ut rectan­<lb/>gulum &longs;ub radio <emph type="italics"/>AE<emph.end type="italics"/>& latitudine <emph type="italics"/>Ee,<emph.end type="italics"/>& hoc rectangulum (ob pro­<lb/>portionales <emph type="italics"/>PE<emph.end type="italics"/>& <emph type="italics"/>AE, Ee<emph.end type="italics"/>& <emph type="italics"/>CE<emph.end type="italics"/>) æquatur rectangulo <emph type="italics"/>PEXCE<emph.end type="italics"/><lb/>&longs;eu <emph type="italics"/>PEXFf<emph.end type="italics"/>; erit vis qua annulus i&longs;te trahit corpus <emph type="italics"/>P<emph.end type="italics"/>ver&longs;us <lb/><emph type="italics"/>A,<emph.end type="italics"/>ut <emph type="italics"/>PEXFf<emph.end type="italics"/>& (<emph type="italics"/>APXFK/PE<emph.end type="italics"/>) conjunctim, id e&longs;t, ut contentum <lb/><emph type="italics"/>FfXFKXAP,<emph.end type="italics"/>&longs;ive ut area <emph type="italics"/>FKkf<emph.end type="italics"/>ducta in <emph type="italics"/>AP.<emph.end type="italics"/>Et propterea <lb/>&longs;umma virium, quibus annuli omnes in Circulo, qui centro <emph type="italics"/>A<emph.end type="italics"/>& in-<pb xlink:href="039/01/225.jpg" pagenum="197"/>tervallo <emph type="italics"/>AD<emph.end type="italics"/>de&longs;cribitur, trahunt corpus <emph type="italics"/>P<emph.end type="italics"/>ver&longs;us <emph type="italics"/>A,<emph.end type="italics"/>e&longs;t ut area <lb/><arrow.to.target n="note173"/>tota <emph type="italics"/>AHIKL<emph.end type="italics"/>ducta in <emph type="italics"/>AP. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note173"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i vires punctorum decre&longs;cunt in duplicata di­<lb/>&longs;tantiarum ratione, hoc e&longs;t, &longs;i &longs;it <emph type="italics"/>FK<emph.end type="italics"/>ut (1/<emph type="italics"/>PFquad.<emph.end type="italics"/>), atque adeo a­<lb/>rea <emph type="italics"/>AHIKL<emph.end type="italics"/>ut (1/<emph type="italics"/>PA<emph.end type="italics"/>-1/<emph type="italics"/>PH<emph.end type="italics"/>); erit attractio corpu&longs;culi <emph type="italics"/>P<emph.end type="italics"/>in Circu­<lb/>lum ut (1-<emph type="italics"/>PA/PH<emph.end type="italics"/>), id e&longs;t, ut (<emph type="italics"/>AH/PH<emph.end type="italics"/>). </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et univer&longs;aliter, &longs;i vires punctorum ad di&longs;tantias D &longs;int <lb/>reciproce ut di&longs;tantiarum dignitas quælibet D<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>, hoc e&longs;t, &longs;i &longs;it <emph type="italics"/>FK<emph.end type="italics"/><lb/>ut (1/D<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>), adeoque area <emph type="italics"/>AHIKL<emph.end type="italics"/>ut (1/<emph type="italics"/>PA<emph type="sup"/>n-1<emph.end type="sup"/><emph.end type="italics"/>-1/<emph type="italics"/>PH<emph type="sup"/>n-1<emph.end type="sup"/><emph.end type="italics"/>); erit attra­<lb/>ctio corpu&longs;culi <emph type="italics"/>P<emph.end type="italics"/>in Circulum ut (1/<emph type="italics"/>PA<emph type="sup"/>n-2<emph.end type="sup"/><emph.end type="italics"/>-<emph type="italics"/>PA/PH<emph type="sup"/>n-1<emph.end type="sup"/><emph.end type="italics"/>). </s></p> <p type="main"> <s><emph type="italics"/>Corol<emph.end type="italics"/>3. Et &longs;i diameter Circuli augeatur in infinitum, & nume­<lb/>rus <emph type="italics"/>n<emph.end type="italics"/>&longs;it unitate major; attractio corpu&longs;culi <emph type="italics"/>P<emph.end type="italics"/>in planum totum <lb/>infinitum erit reciproce ut <emph type="italics"/>PA<emph type="sup"/>n-2<emph.end type="sup"/>,<emph.end type="italics"/>propterea quod terminus al­<lb/>ter (<emph type="italics"/>PA/PH<emph type="sup"/>n-1<emph.end type="sup"/><emph.end type="italics"/>) evane&longs;cet. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XCI. PROBLEMA XLV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Invenire attractionem corpu&longs;culi &longs;iti in axe Solidi rotundi, ad cujus <lb/>puncta &longs;ingula tendunt vires æquales centripetæ in quacunque <lb/>di&longs;tantiarum ratione decre&longs;centes.<emph.end type="italics"/></s></p> <p type="main"> <s>In Solidum <emph type="italics"/>ADEFG<emph.end type="italics"/>tra­<lb/><figure id="id.039.01.225.1.jpg" xlink:href="039/01/225/1.jpg"/><lb/>hatur corpu&longs;culum <emph type="italics"/>P,<emph.end type="italics"/>&longs;itum in <lb/>ejus axe <emph type="italics"/>AB.<emph.end type="italics"/>Circulo quoli­<lb/>bet <emph type="italics"/>RFS<emph.end type="italics"/>ad hunc axem per­<lb/>pendiculari &longs;ecetur hoc Solidum, <lb/>& in ejus diametro <emph type="italics"/>FS,<emph.end type="italics"/>in pla­<lb/>no aliquo <emph type="italics"/>PALKB<emph.end type="italics"/>per axem <lb/>tran&longs;eunte, capiatur (per Prop. </s> <s><lb/>XC) longitudo <emph type="italics"/>FK<emph.end type="italics"/>vi qua cor­<lb/>pu&longs;culum <emph type="italics"/>P<emph.end type="italics"/>in circulum illum <lb/>attrahitur proportionalis. </s> <s>Tangat autem punctum <emph type="italics"/>K<emph.end type="italics"/>curvam line­<lb/>am <emph type="italics"/>LKI,<emph.end type="italics"/>planis extimorum circulorum <emph type="italics"/>AL<emph.end type="italics"/>& <emph type="italics"/>BI<emph.end type="italics"/>occurrentem in <lb/><emph type="italics"/>L<emph.end type="italics"/>& <emph type="italics"/>I<emph.end type="italics"/>; & erit attractio corpu&longs;culi <emph type="italics"/>P<emph.end type="italics"/>in Solidum ut area <emph type="italics"/>LABI. <lb/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/><pb xlink:href="039/01/226.jpg" pagenum="198"/><arrow.to.target n="note174"/></s></p> <p type="margin"> <s><margin.target id="note174"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Unde &longs;i Solidum <lb/><figure id="id.039.01.226.1.jpg" xlink:href="039/01/226/1.jpg"/><lb/>Cylindrus &longs;it, parallelogrammo <lb/><emph type="italics"/>ADEB<emph.end type="italics"/>circa axem <emph type="italics"/>AB<emph.end type="italics"/>revo­<lb/>luto de&longs;criptus, & vires centri­<lb/>petæ in &longs;ingula ejus puncta ten­<lb/>dentes &longs;int reciproce ut quadra­<lb/>ta di&longs;tantiarum a punctis: erit <lb/>attractio corpu&longs;culi <emph type="italics"/>P<emph.end type="italics"/>in hunc <lb/>Cylindrum ut <emph type="italics"/>AB-PE+PD.<emph.end type="italics"/><lb/>Nam ordinatim applicata <emph type="italics"/>FK<emph.end type="italics"/><lb/>(per Corol. </s> <s>1. Prop. </s> <s>XC) erit ut 1-(<emph type="italics"/>PF/PR<emph.end type="italics"/>). Hujus pars 1 ducta in lon­<lb/>gitudinem <emph type="italics"/>AB,<emph.end type="italics"/>de&longs;cribit aream 1X<emph type="italics"/>AB<emph.end type="italics"/>; & pars altera (<emph type="italics"/>PF/PR<emph.end type="italics"/>) ducta <lb/>in longitudinem <emph type="italics"/>PB,<emph.end type="italics"/>de&longs;cribit aream 1 in ―(<emph type="italics"/>PE-AD<emph.end type="italics"/>) (id quod <lb/>ex curvæ <emph type="italics"/>LIK<emph.end type="italics"/>quadratura facile o&longs;tendi pote&longs;t:) & &longs;imiliter pars <lb/>eadem ducta in longitudinem <emph type="italics"/>PA<emph.end type="italics"/>de&longs;cribit aream 1 in ―(<emph type="italics"/>PD-AD<emph.end type="italics"/>), <lb/>ductaQ.E.I. ip&longs;arum <emph type="italics"/>PB, PA<emph.end type="italics"/>differentiam <emph type="italics"/>AB<emph.end type="italics"/>de&longs;cribit arearum <lb/>differentiam 1 in ―(<emph type="italics"/>PE-PD<emph.end type="italics"/>). De contento primo 1X<emph type="italics"/>AB<emph.end type="italics"/>aufe­<lb/>ratur contentum po&longs;tremum 1 in ―(<emph type="italics"/>PE-PD<emph.end type="italics"/>), & re&longs;tabit area <emph type="italics"/>LABI<emph.end type="italics"/><lb/>æqualis 1 in ―(<emph type="italics"/>AB-PE+PD<emph.end type="italics"/>). Ergo vis, huic areæ proportiona­<lb/>lis, e&longs;t ut <emph type="italics"/>AB-PE+PD.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Hinc etiam <lb/><figure id="id.039.01.226.2.jpg" xlink:href="039/01/226/2.jpg"/><lb/>vis innote&longs;cit qua Sphæ­<lb/>rois <emph type="italics"/>AGBCD<emph.end type="italics"/>attrahit <lb/>corpus quodvis <emph type="italics"/>P,<emph.end type="italics"/>exte­<lb/>rius in axe &longs;uo <emph type="italics"/>AB<emph.end type="italics"/>&longs;i­<lb/>tum. </s> <s>Sit <emph type="italics"/>NKRM<emph.end type="italics"/>Se­<lb/>ctio Conica cujus ordi­<lb/>natim applicata <emph type="italics"/>ER,<emph.end type="italics"/>ip&longs;i <lb/><emph type="italics"/>PE<emph.end type="italics"/>perpendicularis, æ­<lb/>quetur &longs;emper longitu­<lb/>dini <emph type="italics"/>PD,<emph.end type="italics"/>quæ ducitur <lb/>ad punctum illud <emph type="italics"/>D,<emph.end type="italics"/>in <lb/>quo applicata i&longs;ta Sphæroidem &longs;ecat. </s> <s>A Sphæroidis verticibus <emph type="italics"/>A, B<emph.end type="italics"/><lb/>ad ejus axem <emph type="italics"/>AB<emph.end type="italics"/>erigantur perpendicula <emph type="italics"/>AK, BM<emph.end type="italics"/>ip&longs;is <emph type="italics"/>AP, BP<emph.end type="italics"/><lb/>æqualia re&longs;pective, & propterea Sectioni Conicæ occurrentia in <emph type="italics"/>K<emph.end type="italics"/><lb/>& <emph type="italics"/>M<emph.end type="italics"/>; & jungatur <emph type="italics"/>KM<emph.end type="italics"/>auferens ab eadem &longs;egmentum <emph type="italics"/>KMRK.<emph.end type="italics"/><lb/>Sit autem Sphæroidis centrum <emph type="italics"/>S<emph.end type="italics"/>& &longs;emidiameter maxima <emph type="italics"/>SC:<emph.end type="italics"/>& vis <pb xlink:href="039/01/227.jpg" pagenum="199"/>qua Sphærois trahit corpus <emph type="italics"/>P<emph.end type="italics"/>erit ad vim qua Sphæra, diametro <emph type="italics"/>AB<emph.end type="italics"/></s></p> <p type="main"> <s><arrow.to.target n="note175"/>de&longs;cripta, trahit idem corpus, ut (<emph type="italics"/>ASXCSq-PSXKMRK/PSq+CSq-ASq<emph.end type="italics"/>) <lb/>ad (<emph type="italics"/>AS cub/3PS quad<emph.end type="italics"/>). Et eodem computandi fundamento invenire licet <lb/>vires &longs;egmentorum Sphæroidis. </s></p> <p type="margin"> <s><margin.target id="note175"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Quod &longs;i corpu&longs;culum intra Sphæroidem, in data qua­<lb/>vis eju&longs;dem diametro, collocetur; attractio erit ut ip&longs;ius di&longs;tantia a <lb/>centro. </s> <s>Id quod facilius colligetur hoc argumento. </s> <s>Sit <emph type="italics"/>AGOF<emph.end type="italics"/><lb/>Sphærois attrahens, <emph type="italics"/>S<emph.end type="italics"/>centrum ejus & <emph type="italics"/>P<emph.end type="italics"/>corpus attractum. </s> <s>Per <lb/>corpus illud <emph type="italics"/>P<emph.end type="italics"/>agantur tum &longs;emidiameter <emph type="italics"/>SPA,<emph.end type="italics"/>tum rectæ duæ <lb/>quævis <emph type="italics"/>DE, FG<emph.end type="italics"/>Sphæroidi hinc inde occurrentes in <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>E, F<emph.end type="italics"/><lb/>& <emph type="italics"/>G:<emph.end type="italics"/>Sintque <emph type="italics"/>PCM, HLN<emph.end type="italics"/>&longs;uperficies Sphæroidum duarum in­<lb/>teriorum, exteriori &longs;imilium & concentricarum, quarum prior tran&longs;­<lb/>eat per corpus <emph type="italics"/>P<emph.end type="italics"/>& &longs;ecet rectas <emph type="italics"/>DE<emph.end type="italics"/>& <emph type="italics"/>FG<emph.end type="italics"/>in <emph type="italics"/>B<emph.end type="italics"/>& <emph type="italics"/>C,<emph.end type="italics"/>po&longs;terior <lb/>&longs;ecet ea&longs;dem rectas in <emph type="italics"/>H, I<emph.end type="italics"/>& <emph type="italics"/>K, L.<emph.end type="italics"/>Habeant autem Sphæroides <lb/>omnes axem communem, & erunt rect­<lb/><figure id="id.039.01.227.1.jpg" xlink:href="039/01/227/1.jpg"/><lb/>arum partes hinc inde interceptæ <emph type="italics"/>DP<emph.end type="italics"/><lb/>& <emph type="italics"/>BE, FP<emph.end type="italics"/>& <emph type="italics"/>CG, DH<emph.end type="italics"/>& <emph type="italics"/>IE, FK<emph.end type="italics"/><lb/>& <emph type="italics"/>LG<emph.end type="italics"/>&longs;ibi mutuo æquales; propterea <lb/>quod rectæ <emph type="italics"/>DE, PB<emph.end type="italics"/>& <emph type="italics"/>HI<emph.end type="italics"/>bi&longs;ecan­<lb/>tur in eodem puncto, ut & rectæ <emph type="italics"/>FG, <lb/>PC<emph.end type="italics"/>& <emph type="italics"/>KL.<emph.end type="italics"/>Concipe jam <emph type="italics"/>DPF, <lb/>EPG<emph.end type="italics"/>de&longs;ignare Conos oppo&longs;itos, an­<lb/>gulis verticalibus <emph type="italics"/>DPF, EPG<emph.end type="italics"/>infi­<lb/>nite parvis de&longs;criptos, & lineas etiam <lb/><emph type="italics"/>DH, EI<emph.end type="italics"/>infinite parvas e&longs;&longs;e; & Conorum particulæ Sphæroidum <lb/>&longs;uperficiebus ab&longs;ci&longs;&longs;æ <emph type="italics"/>DHKF, GLIE,<emph.end type="italics"/>ob æqualitatem linearum <lb/><emph type="italics"/>DH, EI,<emph.end type="italics"/>erunt ad invicem ut quadrata di&longs;tantiarum &longs;uarum a <lb/>corpu&longs;culo <emph type="italics"/>P,<emph.end type="italics"/>& propterea corpu&longs;culum illud æqualiter trahent. </s> <s><lb/>Et pari ratione, &longs;i &longs;uperficiebus Sphæroidum innumerarum &longs;imilium <lb/>concentricarum & axem communem habentium dividantur &longs;patia <lb/><emph type="italics"/>DPF, EGCB<emph.end type="italics"/>in particulas, hæ omnes utrinque æqualiter tra­<lb/>hent corpus <emph type="italics"/>P<emph.end type="italics"/>in partes contrarias. </s> <s>Æquales igitur &longs;unt vires <lb/>Coni <emph type="italics"/>DPF<emph.end type="italics"/>& &longs;egmenti Conici <emph type="italics"/>EGCB,<emph.end type="italics"/>& per contrarietatem &longs;e <lb/>mutuo de&longs;truunt. </s> <s>Et par e&longs;t ratio virium materiæ omnis extra Sphæ­<lb/>roidem intimam <emph type="italics"/>PCBM.<emph.end type="italics"/>Trahitur igitur corpus <emph type="italics"/>P<emph.end type="italics"/>a &longs;ola Sphæ­<lb/>roide intima <emph type="italics"/>PCBM,<emph.end type="italics"/>& propterea (per Corol. </s> <s>3. Prop. </s> <s>LXXII) at­<lb/>tractio ejus e&longs;t ad vim, qua corpus <emph type="italics"/>A<emph.end type="italics"/>trahitur a Sphæroide tota <lb/><emph type="italics"/>AGOD,<emph.end type="italics"/>ut di&longs;tantia <emph type="italics"/>PS<emph.end type="italics"/>ad di&longs;tantiam <emph type="italics"/>AS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/><pb xlink:href="039/01/228.jpg" pagenum="200"/><arrow.to.target n="note176"/></s></p> <p type="margin"> <s><margin.target id="note176"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XCII. PROBLEMA XLVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Dato Corpore attractivo, invenire rationem decrementi virium cen­<lb/>tripetarum in ejus puncta &longs;ingula tendentium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>E Corpore dato formanda e&longs;t Sphæra vel Cylindrus aliave figu­<lb/>ra regularis, cujus lex attractionis, cuivis decrementi rationi con­<lb/>gruens (per Prop. </s> <s>LXXX, LXXXI, & XCI) inveniri pote&longs;t. </s> <s>Dein fa­<lb/>ctis experimentis invenienda e&longs;t vis attractionis in diver&longs;is di&longs;tan­<lb/>tiis, & lex attractionis in totum inde patefacta dabit rationem de­<lb/>crementi virium partium &longs;ingularum, quam invenire oportuit. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XCIII. THEOREMA XLVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Solidum ex una parte planum, ex reliquis autem partibus infiNI­<lb/>tum, con&longs;tet ex particulis æqualibus æqualiter attractivis, qua­<lb/>rum vires in rece&longs;&longs;u a Solido decre&longs;cunt in ratione pote&longs;tatis cu­<lb/>ju&longs;vis di&longs;tantiarum plu&longs;quam quadraticæ, & vi Solidi totius cor­<lb/>pu&longs;culum ad utramvis plani partem con&longs;titutum trahatur: dico <lb/>quod Solidi vis illa attractiva, in rece&longs;&longs;u ab ejus &longs;uperficie pla­<lb/>na, decre&longs;cet in ratione pote&longs;tatis, cujus latus est di&longs;tantia cor­<lb/>pu&longs;culi a plano, & Index ternario minor quam Index pote&longs;ta­<lb/>tis di&longs;tantiarum.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Sit <emph type="italics"/>LGl<emph.end type="italics"/>planum <lb/><figure id="id.039.01.228.1.jpg" xlink:href="039/01/228/1.jpg"/><lb/>quo Solidum terminatur. </s> <s><lb/>Jaceat Solidum autem ex <lb/>parte plani hujus ver&longs;us <lb/><emph type="italics"/>I,<emph.end type="italics"/>inque plana innumera <lb/><emph type="italics"/>mHM, nIN,<emph.end type="italics"/>&c. </s> <s>ip&longs;i <emph type="italics"/>GL<emph.end type="italics"/><lb/>parallela re&longs;olvatur. </s> <s>Et <lb/>primo collocetur corpus at­<lb/>tractum <emph type="italics"/>C<emph.end type="italics"/>extra Solidum. </s> <s><lb/>Agatur autem <emph type="italics"/>CGHI<emph.end type="italics"/>pla­<lb/>nis illis innumeris perpendicularis, & decre&longs;cant vires attractivæ <lb/>punctorum Solidi in ratione pote&longs;tatis di&longs;tantiarum, cujus index &longs;it <lb/>numerus <emph type="italics"/>n<emph.end type="italics"/>ternario non minor. </s> <s>Ergo (per Corol. </s> <s>3. Prop. </s> <s>XC) <pb xlink:href="039/01/229.jpg" pagenum="201"/>vis qua planum quodvis <emph type="italics"/>mHM<emph.end type="italics"/>trahit punctum <emph type="italics"/>C<emph.end type="italics"/>e&longs;t reciproce ut <lb/><arrow.to.target n="note177"/><emph type="italics"/>CH<emph type="sup"/>n-2<emph.end type="sup"/>.<emph.end type="italics"/>In plano <emph type="italics"/>mHM<emph.end type="italics"/>capiatur longitudo <emph type="italics"/>HM<emph.end type="italics"/>ip&longs;i <emph type="italics"/>CH<emph type="sup"/>n-2<emph.end type="sup"/><emph.end type="italics"/>re­<lb/>ciproce proportionalis, & erit vis illa ut <emph type="italics"/>HM.<emph.end type="italics"/>Similiter in planis &longs;in­<lb/>gulis <emph type="italics"/>lGL, nIN, oKO,<emph.end type="italics"/>&c. </s> <s>capiantur longitudines <emph type="italics"/>GL, IN, KO,<emph.end type="italics"/>&c. </s> <s><lb/>ip&longs;is <emph type="italics"/>CG<emph type="sup"/>n-2<emph.end type="sup"/>, CI<emph type="sup"/>n-2<emph.end type="sup"/>, CK<emph type="sup"/>n-2<emph.end type="sup"/>,<emph.end type="italics"/>&c. </s> <s>reciproce proportionales; & vi­<lb/>res planorum eorundem erunt ut longitudines captæ, adeoque <lb/>&longs;umma virium ut &longs;umma longitudinum, hoc e&longs;t, vis Solidi totius ut <lb/>area <emph type="italics"/>GLOK<emph.end type="italics"/>in infinitum ver&longs;us <emph type="italics"/>OK<emph.end type="italics"/>producta. </s> <s>Sed area illa (per <lb/>notas quadraturarum methodos) e&longs;t reciproce ut <emph type="italics"/>CG<emph type="sup"/>n-3<emph.end type="sup"/>,<emph.end type="italics"/>& prop­<lb/>terea vis Solidi totius e&longs;t reciproce ut <emph type="italics"/>CG<emph type="sup"/>n-3<emph.end type="sup"/>. </s> <s><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note177"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Collocetur jam corpu&longs;culum <emph type="italics"/>C<emph.end type="italics"/>ex parte plani <emph type="italics"/>lGL<emph.end type="italics"/>in­<lb/>tra Solidum, & capiatur di&longs;tantia <emph type="italics"/>CK<emph.end type="italics"/>æqualis di&longs;tantiæ <emph type="italics"/>CG.<emph.end type="italics"/>Et So­<lb/>lidi pars <emph type="italics"/>LGloKO,<emph.end type="italics"/>planis parallelis <emph type="italics"/>lGL, oKO<emph.end type="italics"/>terminata, cor­<lb/>pu&longs;culum <emph type="italics"/>C<emph.end type="italics"/>in medio &longs;itum nullam in partem trahet, contrariis op­<lb/>po&longs;itorum punctorum actionibus &longs;e mutuo per æqualitatem tollenti­<lb/>bus. </s> <s>Proinde corpu&longs;culum <emph type="italics"/>C<emph.end type="italics"/>&longs;ola vi Solidi ultra planum <emph type="italics"/>OK<emph.end type="italics"/>&longs;iti tra­<lb/>hitur. </s> <s>Hæc autem vis (per Ca&longs;um primum) e&longs;t reciproce ut <emph type="italics"/>CK<emph type="sup"/>n-3<emph.end type="sup"/>,<emph.end type="italics"/><lb/>hoc e&longs;t (ob æquales <emph type="italics"/>CG, CK<emph.end type="italics"/>) reciproce ut <emph type="italics"/>CG<emph type="sup"/>n-3<emph.end type="sup"/>. </s> <s><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i Solidum <emph type="italics"/>LGIN<emph.end type="italics"/>planis duobus infinitis pa­<lb/>rallelis <emph type="italics"/>LG, IN<emph.end type="italics"/>utrinque terminetur; innote&longs;cit ejus vis attra­<lb/>ctiva, &longs;ubducendo de vi attractiva Solidi totius infiniti <emph type="italics"/>LGKO<emph.end type="italics"/><lb/>vim attractivam partis ulterioris <emph type="italics"/>NICO,<emph.end type="italics"/>in infinitum ver&longs;us <emph type="italics"/>KO<emph.end type="italics"/><lb/>productæ. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si Solidi hujus infiniti pars ulterior, quando attractio e­<lb/>jus collata cum attractione partis citerioris nullius pene e&longs;t momen­<lb/>ti, rejiciatur: attractio partis illius citerioris augendo di&longs;tantiam de­<lb/>cre&longs;cet quam proxime in ratione pote&longs;tatis <emph type="italics"/>CG<emph type="sup"/>n-3<emph.end type="sup"/>.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Et hinc &longs;i corpus quodvis finitum & ex una parte pla­<lb/>num trahat corpu&longs;culum e regione medii illius plani, & di&longs;tantia <lb/>inter corpu&longs;culum & planum collata cum dimen&longs;ionibus corpo­<lb/>ris attrahentis perexigua &longs;it, con&longs;tet autem corpus attrahens ex <lb/>particulis homogeneis, quarum vires attractivæ decre&longs;cunt in <lb/>ratione pote&longs;tatis cuju&longs;vis plu&longs;quam quadruplicatæ di&longs;tantiarum; <lb/>vis attractiva corporis totius decre&longs;cet quamproxime in ratione <lb/>pote&longs;tatis, cujus latus &longs;it di&longs;tantia illa perexigua, & Index terna­<lb/>rio minor quam Index pote&longs;tatis prioris. </s> <s>De corpore ex particulis <lb/>con&longs;tante, quarum vires attractivæ decre&longs;cunt in ratione pote&longs;tatis <lb/>triplicatæ di&longs;tantiarum, a&longs;&longs;ertio non valet; propterea quod, in hoc <lb/>ca&longs;u, attractio partis illius ulterioris corporis infiniti in Corollario <lb/>&longs;ecundo, &longs;emper e&longs;t infinite major quam attractio partis citerioris. <pb xlink:href="039/01/230.jpg" pagenum="202"/><arrow.to.target n="note178"/></s></p> <p type="margin"> <s><margin.target id="note178"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Si corpus aliquod perpendiculariter ver&longs;us planum datum tra­<lb/>hatur, & ex data lege attractionis quæratur motus corporis: Sol­<lb/>vetur Problema quærendo (per Prop. </s> <s>XXXIX) motum corporis recta <lb/>de&longs;cendentis ad hoc planum, & (per Legum Corol. </s> <s>2.) componen­<lb/>do motum i&longs;tum cum uniformi motu, &longs;ecundum lineas eidem plano <lb/>parallelas facto. </s> <s>Et contra, &longs;i quæratur Lex attractionis in planum <lb/>&longs;ecundum lineas perpendiculares factæ, ea conditione ut corpus at­<lb/>tractum in data quacunque curva linea moveatur, &longs;olvetur Proble­<lb/>ma operando ad exemplum Problematis tertii. </s></p> <p type="main"> <s>Operationes autem contrahi &longs;olent re&longs;olvendo ordinatim appli­<lb/>catas in Series convergentes. </s> <s>Ut &longs;i ad ba&longs;em A in angulo quovis <lb/>dato ordinatim applicetur longitudo B, quæ &longs;it ut ba&longs;is dignitas <lb/>quælibet A<emph type="sup"/><emph type="italics"/>m/n<emph.end type="italics"/><emph.end type="sup"/>; & quæratur vis qua corpus, &longs;ecundum po&longs;itionem <lb/>ordinatim applicatæ, vel in ba&longs;em attractum vel a ba&longs;i fugatum, <lb/>moveri po&longs;&longs;it in curva linea quam ordinatim applicata termi­<lb/>no &longs;uo &longs;uperiore &longs;emper attingit: Suppono ba&longs;em augeri parte <lb/>quam minima O, & ordinatim applicatam ―(A+O)<emph type="italics"/>m/n<emph.end type="italics"/>re&longs;olvo in <lb/>Seriem infinitam A<emph type="sup"/><emph type="italics"/>m/n<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>m/n<emph.end type="italics"/>OA<emph type="sup"/>(<emph type="italics"/>m-n/n<emph.end type="italics"/>)<emph.end type="sup"/>+(<emph type="italics"/>mm-mn/2nn<emph.end type="italics"/>) OOA<emph type="sup"/>(<emph type="italics"/>m-2n/n<emph.end type="italics"/>)<emph.end type="sup"/> &c. </s> <s>at­<lb/>que hujus termino in quo O duarum e&longs;t dimen&longs;ionum, id e&longs;t, ter­<lb/>mino (<emph type="italics"/>mm-mn/2nn<emph.end type="italics"/>) OOA<emph type="sup"/>(<emph type="italics"/>m-2n/n<emph.end type="italics"/>)<emph.end type="sup"/> vim proportionalem e&longs;&longs;e &longs;uppono. </s> <s>E&longs;t <lb/>igitur vis quæ&longs;ita ut (<emph type="italics"/>mm-mn/nn<emph.end type="italics"/>)A<emph type="sup"/>(<emph type="italics"/>m-2n/n<emph.end type="italics"/>)<emph.end type="sup"/>, vel quod perinde e&longs;t, ut <lb/>(<emph type="italics"/>mm-mn/nn<emph.end type="italics"/>)B<emph type="sup"/>(<emph type="italics"/>m-2n/m<emph.end type="italics"/>)<emph.end type="sup"/>. </s> <s>Ut &longs;i ordinatim applicata Parabolam attingat, <lb/>exi&longs;tente <emph type="italics"/>m<emph.end type="italics"/>=2, & <emph type="italics"/>n<emph.end type="italics"/>=1: fiet vis ut data 2B°, adeoQ.E.D.bi­<lb/>tur. </s> <s>Data igitur vi corpus movebitur in Parabola, quemad­<lb/>modum <emph type="italics"/>Galilæus<emph.end type="italics"/>demon&longs;travit. </s> <s>Quod &longs;i ordinatim applicata <lb/>Hyperbolam attingat, exi&longs;tente <emph type="italics"/>m<emph.end type="italics"/>=o-1, & <emph type="italics"/>n<emph.end type="italics"/>=1; fiet vis ut <lb/>2A<emph type="sup"/>-3<emph.end type="sup"/> &longs;eu 2B<emph type="sup"/>3<emph.end type="sup"/>: adeoque vi, quæ &longs;it ut cubus ordinatim applicatæ, <lb/>corpus movebitur in Hyperbola. </s> <s>Sed mi&longs;&longs;is huju&longs;modi Propo&longs;iti­<lb/>onibus, pergo ad alias qua&longs;dam de Motu, quas nondum attigi. <pb xlink:href="039/01/231.jpg" pagenum="203"/><arrow.to.target n="note179"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note179"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>SECTIO XIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Motu corporum minimorum, quæ Viribus centripetis ad &longs;ingulas <lb/>magni alicujus corporis partes tendentibus agitantur.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XCIV. THEOREMA XLVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Media duo &longs;imilaria, &longs;patio planis parallelis utrinque terminato, <lb/>di&longs;tinguantur ab invicem, & corpus in tran&longs;itu per hoc &longs;patium <lb/>attrahatur vel impellatur perpendiculariter ver&longs;us Medium alter­<lb/>utrum, neque ulla alia vi agitetur vel impediatur: Sit autem <lb/>attractio, in æqualibus ab utroque plano di&longs;tantiis ad eandem <lb/>ip&longs;ius partem captis, ubique eadem: dico quod &longs;inus incidentiæ <lb/>in planum alterutrum erit ad &longs;inum emergentiæ ex plano altero <lb/>in ratione data.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Sunto <emph type="italics"/>Aa, Bb<emph.end type="italics"/><lb/><figure id="id.039.01.231.1.jpg" xlink:href="039/01/231/1.jpg"/><lb/>plana duo parallela. </s> <s>Inci­<lb/>dat corpus in planum pri­<lb/>us <emph type="italics"/>Aa<emph.end type="italics"/>&longs;ecundum lineam <lb/><emph type="italics"/>GH,<emph.end type="italics"/>ac toto &longs;uo per &longs;pati­<lb/>um intermedium tran&longs;itu <lb/>attrahatur vel impellatur <lb/>ver&longs;us Medium inciden­<lb/>tiæ, eaque actione de&longs;cri­<lb/>bat lineam curvam <emph type="italics"/>HI,<emph.end type="italics"/>& <lb/>emergat &longs;ecundum line­<lb/>am <emph type="italics"/>IK.<emph.end type="italics"/>Ad planum emer­<lb/>gentiæ <emph type="italics"/>Bb<emph.end type="italics"/>erigatur per­<lb/>pendiculum <emph type="italics"/>IM,<emph.end type="italics"/>occur­<lb/>rens tum lineæ inciden­<lb/>tiæ <emph type="italics"/>GH<emph.end type="italics"/>productæ in <emph type="italics"/>M,<emph.end type="italics"/><lb/>tum plano incidentiæ <emph type="italics"/>Aa<emph.end type="italics"/>in <emph type="italics"/>R<emph.end type="italics"/>; & linea emergentiæ <emph type="italics"/>KI<emph.end type="italics"/>producta <lb/>occurrat <emph type="italics"/>HM<emph.end type="italics"/>in <emph type="italics"/>L.<emph.end type="italics"/>Centro <emph type="italics"/>L<emph.end type="italics"/>intervallo <emph type="italics"/>LI<emph.end type="italics"/>de&longs;cribatur Circulus, <pb xlink:href="039/01/232.jpg" pagenum="204"/><arrow.to.target n="note180"/>&longs;ecans tam <emph type="italics"/>HM<emph.end type="italics"/>in <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>Q,<emph.end type="italics"/>quam <emph type="italics"/>MI<emph.end type="italics"/>productam in <emph type="italics"/>N,<emph.end type="italics"/>& primo <lb/>&longs;i attractio vel impul&longs;us ponatur uniformis, erit (ex demon&longs;tratis <lb/><emph type="italics"/>Galilæi<emph.end type="italics"/>) curva <emph type="italics"/>HI<emph.end type="italics"/>Parabola, cujus hæc e&longs;t proprietas, ut rectan­<lb/>gulum &longs;ub dato latere recto & linea <emph type="italics"/>IM<emph.end type="italics"/>æquale &longs;it <emph type="italics"/>HM<emph.end type="italics"/>quadrato; <lb/>&longs;ed & linea <emph type="italics"/>HM<emph.end type="italics"/>bi&longs;ecabitur in <emph type="italics"/>L.<emph.end type="italics"/>Unde &longs;i ad <emph type="italics"/>MI<emph.end type="italics"/>demittatur <lb/>perpendiculum <emph type="italics"/>LO,<emph.end type="italics"/>æ­<lb/><figure id="id.039.01.232.1.jpg" xlink:href="039/01/232/1.jpg"/><lb/>quales erunt <emph type="italics"/>MO, OR<emph.end type="italics"/>; <lb/>& additis æqualibus <emph type="italics"/>ON, <lb/>OI,<emph.end type="italics"/>fient totæ æquales <lb/><emph type="italics"/>MN, IR.<emph.end type="italics"/>Proinde cum <lb/><emph type="italics"/>IR<emph.end type="italics"/>detur, datur etiam <lb/><emph type="italics"/>MN<emph.end type="italics"/>; e&longs;tque rectangu­<lb/>lum <emph type="italics"/>NMI<emph.end type="italics"/>ad rectangu­<lb/>lum &longs;ub latere recto & <lb/><emph type="italics"/>IM,<emph.end type="italics"/>hoc e&longs;t, ad <emph type="italics"/>HMq,<emph.end type="italics"/><lb/>in data ratione. </s> <s>Sed rect­<lb/>angulum <emph type="italics"/>NMI<emph.end type="italics"/>æquale <lb/>e&longs;t rectangulo <emph type="italics"/>PMQ,<emph.end type="italics"/>id <lb/>e&longs;t, differentiæ quadrato­<lb/>rum <emph type="italics"/>MLq,<emph.end type="italics"/>& <emph type="italics"/>PLq<emph.end type="italics"/>&longs;eu <lb/><emph type="italics"/>LIq<emph.end type="italics"/>; & <emph type="italics"/>HMq<emph.end type="italics"/>datam <lb/>rationem habet ad &longs;ui ip&longs;ius quartam partem <emph type="italics"/>MLq:<emph.end type="italics"/>ergo datur <lb/>ratio <emph type="italics"/>MLq-LIq<emph.end type="italics"/>ad <emph type="italics"/>MLq,<emph.end type="italics"/>& divi&longs;im, ratio <emph type="italics"/>LIq<emph.end type="italics"/>ad <emph type="italics"/>MLq,<emph.end type="italics"/>& <lb/>ratio dimidiata <emph type="italics"/>LI<emph.end type="italics"/>ad <emph type="italics"/>ML.<emph.end type="italics"/>Sed in omni triangulo <emph type="italics"/>LMI,<emph.end type="italics"/>&longs;inus <lb/>angulorum &longs;unt proportionales lateribus oppo&longs;itis. </s> <s>Ergo datur <lb/>ratio &longs;inus anguli incidentiæ <emph type="italics"/>LMR<emph.end type="italics"/>ad &longs;inum anguli emergen­<lb/>tiæ <emph type="italics"/>LIR. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note180"/>DE MOTU <lb/>CORPORUM</s></p><figure id="id.039.01.232.2.jpg" xlink:href="039/01/232/2.jpg"/> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Tran&longs;eat jam corpus &longs;ucce&longs;&longs;ive per &longs;patia plura paralle­<lb/>lis planis terminata, <emph type="italics"/>AabB, BbcC,<emph.end type="italics"/>&c. </s> <s>& agitetur vi quæ &longs;it in <pb xlink:href="039/01/233.jpg" pagenum="205"/>&longs;ingulis &longs;eparatim uniformis, at in diver&longs;is diver&longs;a; & per jam de­<lb/><arrow.to.target n="note181"/>mon&longs;trata, &longs;inus incidentiæ in planum primum <emph type="italics"/>Aa<emph.end type="italics"/>erit ad &longs;inum <lb/>emergentiæ ex plano &longs;ecundo <emph type="italics"/>Bb,<emph.end type="italics"/>in data ratione; & hic &longs;inus, <lb/>qui e&longs;t &longs;inus incidentiæ in planum &longs;ecundum <emph type="italics"/>Bb,<emph.end type="italics"/>erit ad &longs;inum <lb/>emergentiæ ex plano tertio <emph type="italics"/>Cc,<emph.end type="italics"/>in data ratione; & hic &longs;inus ad <lb/>&longs;inum emergentiæ ex plano quarto <emph type="italics"/>Dd,<emph.end type="italics"/>in data ratione; & &longs;ic in <lb/>infinitum: & ex æquo, &longs;inus incidentiæ in planum primum ad &longs;i­<lb/>num emergentiæ ex plano ultimo in data ratione. </s> <s>Minuantur jam <lb/>planorum intervalla & augeatur numerus in infinitum, eo ut attra­<lb/>ctionis vel impul&longs;us actio, &longs;ecundum legem quamcunque a&longs;&longs;ignatam, <lb/>continua reddatur; & ratio &longs;inus incidentiæ in planum primum ad <lb/>&longs;inum emergentiæ ex plano ultimo, &longs;emper data exi&longs;tens, etiam­<lb/>num dabitur. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note181"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XCV. THEOREMA XLIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis; dico quod velocitas corporis ante incidentiam e&longs;t <lb/>ad ejus velocitatem po&longs;t emergentiam, ut &longs;inus emergentiæ ad <lb/>&longs;inum incidentiæ.<emph.end type="italics"/></s></p> <p type="main"> <s>Capiantur <emph type="italics"/>AH, Id<emph.end type="italics"/>æquales, & erigantur perpendicula <emph type="italics"/>AG, dK<emph.end type="italics"/><lb/>occurrentia lineis incidentiæ & emergentiæ <emph type="italics"/>GH, IK,<emph.end type="italics"/>in <emph type="italics"/>G<emph.end type="italics"/>& <emph type="italics"/>K.<emph.end type="italics"/><lb/>In <emph type="italics"/>GH<emph.end type="italics"/>capiatur <emph type="italics"/>TH<emph.end type="italics"/>æqualis <emph type="italics"/>IK,<emph.end type="italics"/>& ad planum <emph type="italics"/>Aa<emph.end type="italics"/>demittatur <lb/>normaliter <emph type="italics"/>Tv.<emph.end type="italics"/>Et (per Legum Corol. </s> <s>2) di&longs;tinguatur motus cor­<lb/>poris in duos, unum planis <emph type="italics"/>Aa, Bb, Cc,<emph.end type="italics"/>&c. </s> <s>perpendicularem, al­<lb/>terum ii&longs;dem parallelum. </s> <s>Vis attractionis vel impul&longs;us, agendo &longs;e­<lb/>cundum lineas perpendiculares, nil mutat motum &longs;ecundum paralle­<lb/>las, & propterea corpus hoc motu conficiet æqualibus temporibus <lb/>æqualia illa &longs;ecundum parallelas intervalla, quæ &longs;unt inter lineam <lb/><emph type="italics"/>AG<emph.end type="italics"/>& punctum <emph type="italics"/>H,<emph.end type="italics"/>interque punctum <emph type="italics"/>I<emph.end type="italics"/>& lineam <emph type="italics"/>dK<emph.end type="italics"/>; hoc e&longs;t, <lb/>æqualibus temporibus de&longs;cribet lineas <emph type="italics"/>GH, IK.<emph.end type="italics"/>Proinde velo­<lb/>citas ante incidentiam e&longs;t ad velocitatem po&longs;t emergentiam, ut <lb/><emph type="italics"/>GH<emph.end type="italics"/>ad <emph type="italics"/>IK<emph.end type="italics"/>vel <emph type="italics"/>TH,<emph.end type="italics"/>id e&longs;t, ut <emph type="italics"/>AH<emph.end type="italics"/>vel <emph type="italics"/>Id<emph.end type="italics"/>ad <emph type="italics"/>vH,<emph.end type="italics"/>hoc e&longs;t <lb/>(re&longs;pectu radii <emph type="italics"/>TH<emph.end type="italics"/>vel <emph type="italics"/>IK<emph.end type="italics"/>) ut &longs;inus emergentiæ ad &longs;inum inci­<lb/>dentiæ. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/><pb xlink:href="039/01/234.jpg" pagenum="206"/><arrow.to.target n="note182"/></s></p> <p type="margin"> <s><margin.target id="note182"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XCVI. THEOREMA L.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis & quod motus ante incidentiam velocior &longs;it quam <lb/>po&longs;tea: dico quod corpus, inclinando lineam incidentiæ, refle­<lb/>ctetur tandem, & angulus reflexionis fiet æqualis angulo inci­<lb/>dentiæ.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam concipe corpus inter parallela plana <emph type="italics"/>Aa, Bb, Cc,<emph.end type="italics"/>&c. </s> <s>de­<lb/>&longs;cribere arcus Parabolicos, ut &longs;upra; &longs;intque arcus illi <emph type="italics"/>HP, PQ, <lb/>QR,<emph.end type="italics"/>&c. </s> <s>Et &longs;it ea lineæ incidentiæ <emph type="italics"/>GH<emph.end type="italics"/>obliquitas ad planum pri­<lb/>mum <emph type="italics"/>Aa,<emph.end type="italics"/>ut &longs;inus incidentiæ &longs;it ad radium circuli, cujus e&longs;t &longs;inus, <lb/>in ea ratione quam habet idem &longs;inus incidentiæ ad &longs;inum emer­<lb/>gentiæ ex plano <emph type="italics"/>Dd,<emph.end type="italics"/>in &longs;patium <emph type="italics"/>DdeE:<emph.end type="italics"/>& ob &longs;inum emergen­<lb/>tiæ jam factum æqualem radio, angulus emergentiæ erit rectus, ad­<lb/>eoque linea emergentiæ coincidet cum plano <emph type="italics"/>Dd.<emph.end type="italics"/>Perveniat cor­<lb/>pus ad hoc planum in puncto <emph type="italics"/>R<emph.end type="italics"/>; & quoniam linea emergentiæ <lb/>coincidit cum eodem <lb/><figure id="id.039.01.234.1.jpg" xlink:href="039/01/234/1.jpg"/><lb/>plano, per&longs;picuum e&longs;t <lb/>quod corpus non po­<lb/>te&longs;t ultra pergere ver­<lb/>&longs;us planum <emph type="italics"/>Ee.<emph.end type="italics"/>Sed <lb/>nec pote&longs;t idem perge­<lb/>re in linea emergentiæ <lb/><emph type="italics"/>Rd,<emph.end type="italics"/>propterea quod <lb/>perpetuo attrahitur vel impellitur ver&longs;us Medium incidentiæ. </s> <s>Re­<lb/>vertetur itaQ.E.I.ter plana <emph type="italics"/>Cc, Dd,<emph.end type="italics"/>de&longs;cribendo arcum Parabolæ <lb/><emph type="italics"/>QRq,<emph.end type="italics"/>cujus vertex principalis (juxta demon&longs;trata <emph type="italics"/>Galilæi<emph.end type="italics"/>) e&longs;t in <lb/><emph type="italics"/>R<emph.end type="italics"/>; &longs;ecabit planum <emph type="italics"/>Cc<emph.end type="italics"/>in eodem angulo in <emph type="italics"/>q,<emph.end type="italics"/>ac prius in <emph type="italics"/>Q<emph.end type="italics"/>; dein <lb/>pergendo in arcubus parabolicis <emph type="italics"/>qp, ph,<emph.end type="italics"/>&c. </s> <s>arcubus prioribus <lb/><emph type="italics"/>QP, PH<emph.end type="italics"/>&longs;imilibus & æqualibus, &longs;ecabit reliqua plana in ii&longs;dem <lb/>angulis in <emph type="italics"/>p, h,<emph.end type="italics"/>&c. </s> <s>ac prius in <emph type="italics"/>P, H,<emph.end type="italics"/>&c. </s> <s>emergetque tandem ea­<lb/>dem obliquitate in <emph type="italics"/>h,<emph.end type="italics"/>qua incidit in <emph type="italics"/>H.<emph.end type="italics"/>Concipe jam planorum <lb/><emph type="italics"/>Aa, Bb, Cc, Dd, Ee,<emph.end type="italics"/>&c. </s> <s>intervalla in infinitum minui & nume­<lb/>rum augeri, eo ut actio attractionis vel impul&longs;us &longs;ecundum legem <lb/>quamcunque a&longs;&longs;ignatam continua reddatur; & angulus emergen­<lb/>tiæ &longs;emper angulo incidentiæ æqualis exi&longs;tens, eidem etiamnum <lb/>manebit æqualis. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/><pb xlink:href="039/01/235.jpg" pagenum="207"/><arrow.to.target n="note183"/></s></p> <p type="margin"> <s><margin.target id="note183"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Harum attractionum haud multum di&longs;&longs;imiles &longs;unt Lucis reflexi­<lb/>ones & refractiones, factæ &longs;ecundum datam Secantium rationem, ut <lb/>invenit <emph type="italics"/>Snellius,<emph.end type="italics"/>& per con&longs;equens &longs;ecundum datam Sinuum ratio­<lb/>nem, ut expo&longs;uit <emph type="italics"/>Carte&longs;ius.<emph.end type="italics"/>Namque Lucem &longs;ucce&longs;&longs;ive propagari <lb/>& &longs;patio qua&longs;i &longs;eptem vel octo minutorum primorum a Sole ad <lb/>Terram venire, jam con&longs;tat per Phænomena Satellitum <emph type="italics"/>Jovis,<emph.end type="italics"/>Ob­<lb/>&longs;ervationibus diver&longs;orum A&longs;tronomorum confirmata. </s> <s>Radii autem <lb/>in aere exi&longs;tentes (uti dudum <emph type="italics"/>Grimaldus,<emph.end type="italics"/>luce per foramen in te­<lb/>nebro&longs;um cubiculum admi&longs;&longs;a, invenit, & ip&longs;e quoque expertus <lb/>&longs;um) in tran&longs;itu &longs;uo prope corporum vel opaeorum vel per&longs;picuo­<lb/>rum angulos (quales &longs;unt nummorum ex auro, argento & ære cu­<lb/>&longs;orum termini rectanguli circulares, & cultrorum, lapidum aut fra­<lb/>ctorum vitrorum acies) incurvantur circum corpora, qua&longs;i attracti <lb/>in eadem; & ex his radiis, qui in tran&longs;itu illo propius accedunt <lb/>ad corpora incurvantur magis, qua­<lb/><figure id="id.039.01.235.1.jpg" xlink:href="039/01/235/1.jpg"/><lb/>&longs;i magis attracti, ut ip&longs;e etiam dili­<lb/>genter ob&longs;ervavi. </s> <s>In figura de&longs;ig­<lb/>nat <emph type="italics"/>s<emph.end type="italics"/>aciem cultri vel cunei cuju&longs;vis <lb/><emph type="italics"/>AsB<emph.end type="italics"/>; & <emph type="italics"/>gowog, fnunf, emtme, <lb/>dlsld,<emph.end type="italics"/>&longs;unt radii, arcubus <emph type="italics"/>owo, <lb/>nun, mtm, lsl<emph.end type="italics"/>ver&longs;us cultrum <lb/>incurvati; idque magis vel mi­<lb/>nus pro di&longs;tantia eorum a cultro. </s> <s><lb/>Cum autem talis incurvatio radio­<lb/>rum fiat in aere extra cultrum, de­<lb/>bebunt etiam radii, qui incidunt in cultrum, prius incurvari in aere <lb/>quam cultrum attingunt. </s> <s>Et par e&longs;t ratio incidentium in vitrum. </s> <s><lb/>Fit igitur refractio, non in puncto incidentiæ, &longs;ed paulatim per <lb/>continuam incurvationem radiorum, factam partim in aere ante­<lb/>quam attingunt vitrum, partim (ni fallor) in vitro, po&longs;tquam illud <lb/>ingre&longs;&longs;i &longs;unt: uti in radiis <emph type="italics"/>ckzkc, biyib, ahxha<emph.end type="italics"/>incidentibus ad <lb/><emph type="italics"/>r, q, p,<emph.end type="italics"/>& inter <emph type="italics"/>k<emph.end type="italics"/>& <emph type="italics"/>z, i<emph.end type="italics"/>& <emph type="italics"/>y, h<emph.end type="italics"/>& <emph type="italics"/>x<emph.end type="italics"/>incurvatis, delineatum e&longs;t. </s> <s><lb/>Igitur ob analogiam quæ e&longs;t inter propagationem radiorum lucis <lb/>& progre&longs;&longs;um corporum, vi&longs;um e&longs;t Propo&longs;itiones &longs;equentes in u&longs;us <lb/>Opticos &longs;ubjungere; interea de natura radiorum (utrum &longs;int cor­<lb/>pora necne) nihil omnino di&longs;putans, &longs;ed Trajectorias corporum <lb/>Trajectoriis radiorum per&longs;imiles &longs;olummodo determinans. <pb xlink:href="039/01/236.jpg" pagenum="208"/><arrow.to.target n="note184"/></s></p> <p type="margin"> <s><margin.target id="note184"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XCVII. PROBLEMA XLVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod &longs;inus incidentiæ in &longs;uperficiem aliquam &longs;it ad &longs;inum e­<lb/>mergentiæ in data ratione, quodQ.E.I.curvatio viæ corporum <lb/>juxta &longs;uperficiem illam fiat in &longs;patio brevi&longs;&longs;imo, quod ut pun­<lb/>ctum con&longs;iderari po&longs;&longs;it; determinare &longs;uperficiem quæ corpu&longs;cula <lb/>omnia de loco dato &longs;ucce&longs;&longs;ive manantia convergere faciat ad <lb/>alium locum datum.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>A<emph.end type="italics"/>locus a quo corpu&longs;cula divergunt; <emph type="italics"/>B<emph.end type="italics"/>locus in quem con­<lb/>vergere debent; <emph type="italics"/>CDE<emph.end type="italics"/>curva linea quæ circa axem <emph type="italics"/>AB<emph.end type="italics"/>revoluta <lb/>de&longs;cribat &longs;uperficiem quæ&longs;itam; <emph type="italics"/>D, E<emph.end type="italics"/>curvæ illius puncta duo quæ­<lb/>vis; & <emph type="italics"/>EF, EG<emph.end type="italics"/>perpendicula in corporis vias <emph type="italics"/>AD, DB<emph.end type="italics"/>demi&longs;&longs;a. </s> <s><lb/>Accedat punctum <emph type="italics"/>D<emph.end type="italics"/>ad punctum <emph type="italics"/>E<emph.end type="italics"/>; & lineæ <emph type="italics"/>DF<emph.end type="italics"/>qua <emph type="italics"/>AD<emph.end type="italics"/>au­<lb/>getur, ad lineam <emph type="italics"/>DG<emph.end type="italics"/>qua <emph type="italics"/>DB<emph.end type="italics"/>diminuitur, ratio ultima erit ea­<lb/>dem quæ &longs;inus incidentiæ ad &longs;inum emergentiæ. </s> <s>Datur ergo ratio <lb/><figure id="id.039.01.236.1.jpg" xlink:href="039/01/236/1.jpg"/><lb/>incrementi lineæ <emph type="italics"/>AD<emph.end type="italics"/>ad decrementum lineæ <emph type="italics"/>DB<emph.end type="italics"/>; & propterea <lb/>&longs;i in axe <emph type="italics"/>AB<emph.end type="italics"/>&longs;umatur ubivis punctum <emph type="italics"/>C,<emph.end type="italics"/>per quod curva <emph type="italics"/>CDE<emph.end type="italics"/><lb/>tran&longs;ire debet, & capiatur ip&longs;ius <emph type="italics"/>AC<emph.end type="italics"/>incrementum <emph type="italics"/>CM,<emph.end type="italics"/>ad ip&longs;ius <lb/><emph type="italics"/>BC<emph.end type="italics"/>decrementum <emph type="italics"/>CN<emph.end type="italics"/>in data illa ratione; centri&longs;que <emph type="italics"/>A, B,<emph.end type="italics"/>& in­<lb/>tervallis <emph type="italics"/>AM, BN<emph.end type="italics"/>de&longs;cribantur circuli duo &longs;e mutuo &longs;ecantes in <lb/><emph type="italics"/>D:<emph.end type="italics"/>punctum illud <emph type="italics"/>D<emph.end type="italics"/>tanget curvam quæ&longs;itam <emph type="italics"/>CDE,<emph.end type="italics"/>eandemque <lb/>ubivis tangendo determinabit. <emph type="italics"/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Faciendo autem ut punctum <emph type="italics"/>A<emph.end type="italics"/>vel <emph type="italics"/>B<emph.end type="italics"/>nunc abeat in in­<lb/>finitum, nunc migret ad alteras partes puncti <emph type="italics"/>C,<emph.end type="italics"/>habebuntur Fi­<lb/>guræ illæ omnes quas <emph type="italics"/>Carte&longs;ius<emph.end type="italics"/>in Optica & Geometria ad Refra­<lb/>ctiones expo&longs;uit. </s> <s>Quarum inventionem cum <emph type="italics"/>Carte&longs;ius<emph.end type="italics"/>maximi <lb/>fecerit & &longs;tudio&longs;e celaverit, vi&longs;um fuit hac propo&longs;itione expo­<lb/>nere. </s></p><pb xlink:href="039/01/237.jpg" pagenum="209"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si corpus in &longs;uperficiem quamvis <emph type="italics"/>CD,<emph.end type="italics"/>&longs;ecundum lineam <lb/><arrow.to.target n="note185"/>rectam <emph type="italics"/>AD<emph.end type="italics"/>lege quavis ductam incidens, emergat &longs;ecundum aliam <lb/>quamvis rectam <emph type="italics"/>DK,<emph.end type="italics"/><lb/><figure id="id.039.01.237.1.jpg" xlink:href="039/01/237/1.jpg"/><lb/>& a puncto <emph type="italics"/>C<emph.end type="italics"/>duci in­<lb/>telligantur Lineæ curvæ <lb/><emph type="italics"/>CP, CQ<emph.end type="italics"/>ip&longs;is <emph type="italics"/>AD, DK<emph.end type="italics"/><lb/>&longs;emper perpendiculares: <lb/>erunt incrementa linea­<lb/>rum <emph type="italics"/>PD, QD,<emph.end type="italics"/><expan abbr="atq;">atque</expan> ad­<lb/>eo lineæ ip&longs;æ <emph type="italics"/>PD, QD,<emph.end type="italics"/><lb/>incrementis i&longs;tis genitæ, <lb/>ut &longs;inus incidentiæ & e­<lb/>mergentiæ ad invicem: <lb/>& contra. </s></p> <p type="margin"> <s><margin.target id="note185"/>LIBER <lb/>PRIMUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XCVIII. PROBLEMA XLVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, & circa axem<emph.end type="italics"/>AB <emph type="italics"/>de&longs;cripta &longs;uperficie quacunque <lb/>attractiva<emph.end type="italics"/>CD, <emph type="italics"/>regulari vel irregulari, per quam corpora de <lb/>loco dato<emph.end type="italics"/>A <emph type="italics"/>exeuntia tran&longs;ire debent: invenire &longs;uperficiem &longs;e­<lb/>cundam attractivam<emph.end type="italics"/>EF, <emph type="italics"/>quæ corpora illa ad locum datum<emph.end type="italics"/>B <lb/><emph type="italics"/>convergere faciat.<emph.end type="italics"/></s></p> <p type="main"> <s>Juncta <emph type="italics"/>AB<emph.end type="italics"/>&longs;ecet &longs;uperficiem primam in <emph type="italics"/>C<emph.end type="italics"/>& &longs;ecundam in <emph type="italics"/>E,<emph.end type="italics"/><lb/>puncto <emph type="italics"/>D<emph.end type="italics"/>utcunque a&longs;&longs;umpto. </s> <s>Et po&longs;ito &longs;inu incidentiæ in &longs;uper­<lb/>ficiem primam ad &longs;inum emergentiæ ex eadem, & &longs;inu emergentiæ <lb/>e &longs;uperficie &longs;ecunda ad &longs;inum incidentiæ in eandem, ut quantitas <lb/>aliqua data M ad aliam datam N; produc tum <emph type="italics"/>AB<emph.end type="italics"/>ad <emph type="italics"/>G<emph.end type="italics"/>ut &longs;it <emph type="italics"/>BG<emph.end type="italics"/><lb/>ad <emph type="italics"/>CE<emph.end type="italics"/>ut M-N ad N, tum <emph type="italics"/>AD<emph.end type="italics"/>ad <emph type="italics"/>H<emph.end type="italics"/>ut &longs;it <emph type="italics"/>AH<emph.end type="italics"/>æqualis <emph type="italics"/>AG,<emph.end type="italics"/>tum <lb/>etiam <emph type="italics"/>DF<emph.end type="italics"/>ad <emph type="italics"/>K<emph.end type="italics"/>ut &longs;it <emph type="italics"/>DK<emph.end type="italics"/>ad <emph type="italics"/>DH<emph.end type="italics"/>ut N ad M. </s> <s>Junge <emph type="italics"/>KB,<emph.end type="italics"/>& <lb/>centro <emph type="italics"/>D<emph.end type="italics"/>intervallo <emph type="italics"/>DH<emph.end type="italics"/>de&longs;cribe circulum occurrentem <emph type="italics"/>KB<emph.end type="italics"/>pro­<lb/>ductæ in <emph type="italics"/>L,<emph.end type="italics"/>ip&longs;ique <emph type="italics"/>DL<emph.end type="italics"/>parallelam age <emph type="italics"/>BF:<emph.end type="italics"/>& punctum <emph type="italics"/>F<emph.end type="italics"/>tan­<lb/>get Lineam <emph type="italics"/>EF,<emph.end type="italics"/>quæ circa axem <emph type="italics"/>AB<emph.end type="italics"/>revoluta de&longs;cribet &longs;uperfi­<lb/>ciem quæ&longs;itam. <emph type="italics"/><expan abbr="q.">que</expan> E. F.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam concipe Lineas <emph type="italics"/>CP, CQ<emph.end type="italics"/>ip&longs;is <emph type="italics"/>AD, DF<emph.end type="italics"/>re&longs;pective, & Li­<lb/>neas <emph type="italics"/>ER, ES<emph.end type="italics"/>ip&longs;is <emph type="italics"/>FB, FD<emph.end type="italics"/>ubique perpendiculares e&longs;&longs;e, adeoque <lb/><emph type="italics"/>QS<emph.end type="italics"/>ip&longs;i <emph type="italics"/>CE<emph.end type="italics"/>&longs;emper æqualem; & erit (per Corol. </s> <s>2. Prop. </s> <s>XCVII) <lb/><emph type="italics"/>PD<emph.end type="italics"/>ad <emph type="italics"/>QD<emph.end type="italics"/>ut M ad N, adeoque ut <emph type="italics"/>DL<emph.end type="italics"/>ad <emph type="italics"/>DK<emph.end type="italics"/>vel <emph type="italics"/>FB<emph.end type="italics"/>ad <emph type="italics"/>FK<emph.end type="italics"/>; <pb xlink:href="039/01/238.jpg" pagenum="210"/><arrow.to.target n="note186"/>& divi&longs;im ut <emph type="italics"/>DL-FP<emph.end type="italics"/>&longs;eu <emph type="italics"/>PH-PD-FB<emph.end type="italics"/>ad <emph type="italics"/>FD<emph.end type="italics"/>&longs;eu <emph type="italics"/>FQ-QD<emph.end type="italics"/>; <lb/>& compo&longs;ite ut <emph type="italics"/>PH-FB<emph.end type="italics"/>ad <emph type="italics"/>FQ,<emph.end type="italics"/>id e&longs;t (ob æquales <emph type="italics"/>PH<emph.end type="italics"/><lb/>& <emph type="italics"/>CG, QS<emph.end type="italics"/>& <emph type="italics"/>CE) <lb/><figure id="id.039.01.238.1.jpg" xlink:href="039/01/238/1.jpg"/><lb/>CE+BG-FR<emph.end type="italics"/>ad <lb/><emph type="italics"/>CE-FS.<emph.end type="italics"/>Verum (ob <lb/>proportionales <emph type="italics"/>BG<emph.end type="italics"/>ad <lb/><emph type="italics"/>CE<emph.end type="italics"/>& M-N ad N) <lb/>e&longs;t etiam <emph type="italics"/>CE+BG<emph.end type="italics"/>ad <lb/><emph type="italics"/>CE<emph.end type="italics"/>ut M ad N: adeoque <lb/>divi&longs;im <emph type="italics"/>FR<emph.end type="italics"/>ad <emph type="italics"/>FS<emph.end type="italics"/>ut <lb/>M ad N, & propterea per <lb/>Corol. </s> <s>2. Prop. </s> <s>XCVII, <lb/>&longs;uperficies <emph type="italics"/>EF<emph.end type="italics"/>cogit cor­<lb/>pus, in ip&longs;am &longs;ecundum lineam <emph type="italics"/>DF<emph.end type="italics"/>incidens, pergere in linea <emph type="italics"/>FR<emph.end type="italics"/><lb/>ad locum <emph type="italics"/>B. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note186"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Eadem methodo pergere liceret ad &longs;uperficies tres vel plures. </s> <s><lb/>Ad u&longs;us autem Opticos maxime accommodatæ &longs;unt figuræ Sphæ­<lb/>ricæ. </s> <s>Si Per&longs;picillorum vitra Objectiva ex vitris duobus Sphæri­<lb/>ce figuratis & Aquam inter &longs;e claudentibus conflentur; fieri pote&longs;t <lb/>ut a refractionibus Aquæ errores refractionum, quæ fiunt in vitro­<lb/>rum &longs;uperficiebus extremis, &longs;atis accurate corrigantur. </s> <s>Talia au­<lb/>tem vitra Objectiva vitris Ellipticis & Hyperbolicis præferenda <lb/>&longs;unt, non &longs;olum quod facilius & accuratius formari po&longs;&longs;int, &longs;ed <lb/>etiam quod Penicillos radiorum extra axem vitri &longs;itos accurativs <lb/>refringant. </s> <s>Verum tamen diver&longs;a diver&longs;orum radiorum Refrangi­<lb/>bilitas impedimento e&longs;t, quo minus Optica per Figuras vel Sphæ­<lb/>ricas vel alias qua&longs;cunque perfici po&longs;&longs;it. </s> <s>Ni&longs;i corrigi po&longs;&longs;int er­<lb/>rores illinc oriundi, labor omnis in cæteris corrigendis imperite <lb/>collocabitur. <pb xlink:href="039/01/239.jpg" pagenum="211"/><arrow.to.target n="note187"/></s></p></subchap2></subchap1><subchap1><subchap2> <p type="margin"> <s><margin.target id="note187"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>DE <lb/>MOTU CORPORUM <lb/>LIBER SECUNDUS.<emph.end type="center"/><lb/></s></p> <p type="main"> <s><emph type="center"/>SECTIO I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Motu Corporum quibus re&longs;i&longs;titur in ratione <lb/>Velocitatis.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO I. THEOREMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Corporis, cui re&longs;i&longs;titur in ratione velocitatis, motus ex re&longs;i&longs;tentia <lb/>ami&longs;&longs;us e&longs;t ut &longs;patium movendo confectum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>NAm cum motus &longs;ingulis temporis particulis æqualibus ami&longs;&longs;us <lb/>&longs;it ut velocitas, hoc e&longs;t, ut itineris confecti particula: erit, <lb/>componendo, motus toto tempore ami&longs;&longs;us ut iter totum. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Igitur &longs;i corpus, gravitate omni de&longs;titutum, in &longs;patiis libe­<lb/>ris &longs;ola vi in&longs;ita moveatur; ac detur tum motus totus &longs;ub initio, tum <lb/>etiam motus reliquus po&longs;t &longs;patium aliquod confectum: dabitur &longs;pa­<lb/>tium totum quod corpus infinito tempore de&longs;cribere pote&longs;t. </s> <s>Erit <lb/>enim &longs;patium illud ad &longs;patium jam de&longs;criptum, ut motus totus &longs;ub <lb/>initio ad motus illius partem ami&longs;&longs;am. </s></p> <p type="main"> <s><emph type="center"/>LEMMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Quantitates differentiis &longs;uis proportionales, &longs;unt continue propor­<lb/>tionales.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sit A ad A-B ut B ad B-C & C ad C-D, &c. </s> <s>& dividendo <lb/>fiet A ad B ut B ad C & C ad D, &c. <emph type="italics"/>Q.E.D.<emph.end type="italics"/><pb xlink:href="039/01/240.jpg" pagenum="212"/><arrow.to.target n="note188"/></s></p> <p type="margin"> <s><margin.target id="note188"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO II. THEOREMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Corpori re&longs;i&longs;titur in ratione velocitatis, & idem &longs;ola vi in&longs;ita <lb/>per Medium &longs;imilare moveatur, &longs;umantur autem tempora æqua­<lb/>lia: velocitates in principiis &longs;ingulorum temporum &longs;unt in pro­<lb/>gre&longs;&longs;ione Geometrica, & &longs;patia &longs;ingulis temporibus de&longs;cripta <lb/>&longs;unt ut velocitates.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Dividatur tempus in particulas æquales; & &longs;i ip&longs;is parti­<lb/>cularum initiis agat vis re&longs;i&longs;tentiæ impul&longs;o unico, quæ &longs;it ut velo­<lb/>citas: erit decrementum velocitatis &longs;ingulis temporis particulis ut <lb/>eadem velocitas. </s> <s>Sunt ergo velocitates differentiis &longs;uis proportio­<lb/>nales, & propterea (per Lem. </s> <s>I. Lib. </s> <s>II.) continue proportionales. </s> <s><lb/>Proinde &longs;i ex æquali particularum numero componantur tempora <lb/>quælibet æqualia, erunt velocitates ip&longs;is temporum initiis, ut ter­<lb/>mini in progre&longs;&longs;ione continua, qui per &longs;altum capiuntur, omi&longs;&longs;o <lb/>pa&longs;&longs;im æquali terminorum intermediorum numero. </s> <s>Componuntur <lb/>autem horum terminorum rationes ex æqualibus rationibus termi­<lb/>norum intermediorum æqualiter repetitis, & propterea &longs;unt æqua­<lb/>les. </s> <s>Igitur velocitates, his terminis proportionales, &longs;unt in pro­<lb/>gre&longs;&longs;ione Geometrica. </s> <s>Minuantur jam æquales illæ temporum par­<lb/>ticulæ, & augeatur earum numerus in infinitum, eo ut re&longs;i&longs;tentiæ <lb/>impul&longs;us reddatur continuus; & velocitates in principiis æqualium <lb/>temporum, &longs;emper continue proportionales, erunt in hoc etiam <lb/>ca&longs;u continue proportionales. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Et divi&longs;im velocitatum differentiæ, hoc e&longs;t, earum partes <lb/>&longs;ingulis temporibus ami&longs;&longs;æ, &longs;unt ut totæ: Spatia autem &longs;ingulis <lb/>temporibus de&longs;cripta &longs;unt ut velocitatum partes ami&longs;&longs;æ, (per Prop. </s> <s><lb/>I. </s> <s>Lib II.) & propterea etiam ut totæ. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc &longs;i A&longs;ymptotis rectangulis <emph type="italics"/>ADC, CH<emph.end type="italics"/>de&longs;cribatur <lb/>Hyperbola <emph type="italics"/>BG,<emph.end type="italics"/>&longs;intque <emph type="italics"/>AB, DG<emph.end type="italics"/>ad A&longs;ymptoton <emph type="italics"/>AC<emph.end type="italics"/>perpen­<lb/>diculares, & exponatur tum corporis velocitas tum re&longs;i&longs;tentia Me­<lb/>dii, ip&longs;o motus initio, per lineam quam­<lb/><figure id="id.039.01.240.1.jpg" xlink:href="039/01/240/1.jpg"/><lb/>vis datam <emph type="italics"/>AC,<emph.end type="italics"/>elap&longs;o autem tempore ali­<lb/>quo per lineam indefinitam <emph type="italics"/>DC:<emph.end type="italics"/>exponi <lb/>pote&longs;t tempus per aream <emph type="italics"/>ABGD,<emph.end type="italics"/>& &longs;pa­<lb/>tium eo tempore de&longs;criptum per lineam <lb/><emph type="italics"/>AD.<emph.end type="italics"/>Nam &longs;i area illa per motum puncti <lb/><emph type="italics"/>D<emph.end type="italics"/>augeatur uniformiter ad modum tempo-<pb xlink:href="039/01/241.jpg" pagenum="213"/>ris, decre&longs;cet recta <emph type="italics"/>DC<emph.end type="italics"/>in ratione Geometrica ad modum veloci­<lb/><arrow.to.target n="note189"/>tatis, & partes rectæ <emph type="italics"/>AC<emph.end type="italics"/>æqualibus temporibus de&longs;criptæ decre­<lb/>&longs;cent in eadem ratione. </s></p> <p type="margin"> <s><margin.target id="note189"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO III. PROBLEMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Corporis, cui dum in Medio &longs;imilari recta a&longs;cendit vel de&longs;cendit, <lb/>re&longs;i&longs;titur in ratione velocitatis, quodque ab uniformi gravitate <lb/>urgetur, definire motum.<emph.end type="italics"/></s></p> <p type="main"> <s>Corpore a&longs;cendente, ex­<lb/><figure id="id.039.01.241.1.jpg" xlink:href="039/01/241/1.jpg"/><lb/>ponatur gravitas per datum <lb/>quodvis rectangulum <emph type="italics"/>BC,<emph.end type="italics"/>& <lb/>re&longs;i&longs;tentia Medii initio a&longs;­<lb/>cen&longs;us per rectangulum <emph type="italics"/>BD<emph.end type="italics"/><lb/>&longs;umptum ad contrarias par­<lb/>tes. </s> <s>A&longs;ymptotis rectangulis <lb/><emph type="italics"/>AC, CH,<emph.end type="italics"/>per punctum <emph type="italics"/>B<emph.end type="italics"/>de­<lb/>&longs;cribatur Hyperbola &longs;ecans per­<lb/>pendicula <emph type="italics"/>DE, de<emph.end type="italics"/>in <emph type="italics"/>G, g;<emph.end type="italics"/>& <lb/>corpus a&longs;cendendo, tempore <emph type="italics"/>DGgd,<emph.end type="italics"/>de&longs;cribet &longs;patium <emph type="italics"/>EGge,<emph.end type="italics"/>tem­<lb/>pore <emph type="italics"/>DGBA<emph.end type="italics"/>&longs;patium a&longs;cen&longs;us totius <emph type="italics"/>EGB<emph.end type="italics"/>; tempore <emph type="italics"/>AB<emph.end type="italics"/>2<emph type="italics"/>G<emph.end type="italics"/>2<emph type="italics"/>D<emph.end type="italics"/><lb/>&longs;patium de&longs;cen&longs;us <emph type="italics"/>BF<emph.end type="italics"/>2<emph type="italics"/>G,<emph.end type="italics"/>atque tempore 2<emph type="italics"/>D<emph.end type="italics"/>2<emph type="italics"/>G<emph.end type="italics"/>2<emph type="italics"/>g<emph.end type="italics"/>2<emph type="italics"/>d<emph.end type="italics"/>&longs;patium <lb/>de&longs;cen&longs;us 2<emph type="italics"/>GF<emph.end type="italics"/>2<emph type="italics"/>e<emph.end type="italics"/>2<emph type="italics"/>g<emph.end type="italics"/>: & velocitates corporis (re&longs;i&longs;tentiæ Medii <lb/>proportionales) in horum temporum periodis erunt <emph type="italics"/>ABED, <lb/>ABed,<emph.end type="italics"/>nulla, <emph type="italics"/>ABF<emph.end type="italics"/>2<emph type="italics"/>D, AB<emph.end type="italics"/>2<emph type="italics"/>e<emph.end type="italics"/>2<emph type="italics"/>d<emph.end type="italics"/>re&longs;pective; atque maxima <lb/>velocitas, quam corpus de&longs;cendendo pote&longs;t acquirere, erit <emph type="italics"/>BC.<emph.end type="italics"/></s></p> <p type="main"> <s>Re&longs;olvatur enim rectan­<lb/><figure id="id.039.01.241.2.jpg" xlink:href="039/01/241/2.jpg"/><lb/>gulum <emph type="italics"/>AH<emph.end type="italics"/>in rectangula <lb/>innumera <emph type="italics"/>Ak, Kl, Lm, Mn,<emph.end type="italics"/><lb/>&c. </s> <s>quæ &longs;int ut incrementa <lb/>velocitatum æqualibus tot­<lb/>idem temporibus facta; & e­<lb/>runt nihil, <emph type="italics"/>Ak, Al, Am, An,<emph.end type="italics"/><lb/>&c. </s> <s>ut velocitates totæ, at­<lb/>que adeo (per Hypothe&longs;in) <lb/>ut re&longs;i&longs;tentiæ Medii princi­<lb/>pio &longs;ingulorum temporum <lb/>æqualium. </s> <s>Fiat <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>AK<emph.end type="italics"/>vel <emph type="italics"/>ABHC<emph.end type="italics"/>ad <emph type="italics"/>ABkK,<emph.end type="italics"/>ut vis gra­<lb/>vitatis ad re&longs;i&longs;tentiam in principio temporis &longs;ecundi, deque vi gravi-<pb xlink:href="039/01/242.jpg" pagenum="214"/><arrow.to.target n="note190"/>tatis &longs;ubducantur re&longs;i&longs;tentiæ, & manebunt <emph type="italics"/>ABHC, KkHC, LlHC, <lb/>NnHC,<emph.end type="italics"/>&c. </s> <s>ut vires ab&longs;olutæ quibus corpus in principio &longs;ingu­<lb/>lorum temporum urgetur, atque adeo (per motus Legem 11) ut <lb/>incrementa velocitatum, id e&longs;t, ut rectangula <emph type="italics"/>Ak, Kl, Lm, Mn,<emph.end type="italics"/>&c; <lb/>& propterea (per Lem. </s> <s>I. Lib. </s> <s>II) in progre&longs;&longs;ione Geometrica. </s> <s>Qua­<lb/>re &longs;i rectæ <emph type="italics"/>Kk, Ll, Mm, Nn,<emph.end type="italics"/>&c. </s> <s>productæ occurrant Hyperbolæ <lb/>in <emph type="italics"/>q, r, s, t,<emph.end type="italics"/>&c. </s> <s>erunt areæ <emph type="italics"/>ABqK, KqrL, LrsM, MstN,<emph.end type="italics"/>&c. <lb/></s> <s>æquales, adeoque tum temporibus tum viribus gravitatis &longs;emper <lb/>æqualibus analogæ. </s> <s>E&longs;t autem area <emph type="italics"/>ABqK<emph.end type="italics"/>(per Corol. </s> <s>3. Lem. </s> <s>VII, <lb/>& Lem. </s> <s>VIII, Lib. </s> <s>I) ad aream <emph type="italics"/>Bkq<emph.end type="italics"/>ut <emph type="italics"/>Kq<emph.end type="italics"/>ad 1/2 <emph type="italics"/>kq<emph.end type="italics"/>&longs;eu <emph type="italics"/>AC<emph.end type="italics"/>ad 1/2 <emph type="italics"/>AK,<emph.end type="italics"/><lb/>hoc e&longs;t, ut vis gravitatis ad re&longs;i&longs;tentiam in medio temporis primi. </s> <s><lb/>Et &longs;imili argumento areæ <lb/><figure id="id.039.01.242.1.jpg" xlink:href="039/01/242/1.jpg"/><lb/><emph type="italics"/>qKLr, rLMs, sMNt,<emph.end type="italics"/>&c. <lb/></s> <s>&longs;unt ad areas <emph type="italics"/>qklr, rlms, <lb/>smnt,<emph.end type="italics"/>&c. </s> <s>ut vires gravi­<lb/>tatis ad re&longs;i&longs;tentias in me­<lb/>dio temporis &longs;ecundi, ter­<lb/>tii, quarti, &c. </s> <s>Proinde cum <lb/>areæ æquales <emph type="italics"/>BAKq, qKLr, <lb/>rLMs, sMNt,<emph.end type="italics"/>&c. </s> <s>&longs;int vi­<lb/>ribus gravitatis analogæ, e­<lb/>runt areæ <emph type="italics"/>Bkq, qklr, rlms, <lb/>smnt,<emph.end type="italics"/>&c. </s> <s>re&longs;i&longs;tentiis in mediis &longs;ingulorum temporum, hoc e&longs;t (per <lb/>Hypothe&longs;in) velocitatibus, atque adeo de&longs;criptis &longs;patiis analogæ. </s> <s><lb/>Sumantur analogarum &longs;ummæ, & erunt areæ <emph type="italics"/>Bkq, Blr, Bms, Bnt,<emph.end type="italics"/><lb/>&c. </s> <s>&longs;patiis totis de&longs;criptis analogæ; necnon areæ <emph type="italics"/>ABqK, ABrL, <lb/>ABsM, ABtN,<emph.end type="italics"/>&c. </s> <s>temporibus. </s> <s>Corpus igitur inter de&longs;cenden­<lb/>dum, tempore quovis <emph type="italics"/>ABrL,<emph.end type="italics"/>de&longs;cribit &longs;patium <emph type="italics"/>Blr,<emph.end type="italics"/>& tempore <lb/><emph type="italics"/>LrtN<emph.end type="italics"/>&longs;patium <emph type="italics"/>rlnt. </s> <s>Q.E.D.<emph.end type="italics"/>Et &longs;imilis e&longs;t demon&longs;tratio motus <lb/>expo&longs;iti in a&longs;cen&longs;u. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note190"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Igitur velocitas maxima, quam corpus cadendo pote&longs;t <lb/>acquirere, e&longs;t ad velocitatem dato quovis tempore acqui&longs;itam, ut<lb/>vis data gravitatis qua perpetuo urgetur, ad vim re&longs;i&longs;tentiæ qua in<lb/>fine temporis illius impeditur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Tempore autem aucto in progre&longs;&longs;ione Arithmetica, &longs;umma<lb/>velocitatis illius maximæ ac velocitatis in a&longs;cen&longs;u (atque etiam earun <lb/>dem differentia in de&longs;cen&longs;u) decre&longs;cit in progre&longs;&longs;ione Geometrica. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Sed & differentiæ &longs;patiorum, quæ in æqualibus tempo <lb/>rum differentiis de&longs;cribuntur, decre&longs;cunt in eadem progre&longs;&longs;ion <lb/>Geometrica. </s></p><pb xlink:href="039/01/243.jpg" pagenum="215"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Spatium vero a corpore de&longs;criptum differentia e&longs;t duo­<lb/><arrow.to.target n="note191"/>rum &longs;patiorum, quorum alterum e&longs;t ut tempus &longs;umptum ab initio <lb/>de&longs;cen&longs;us, & alterum ut velocitas, quæ etiam ip&longs;o de&longs;cen&longs;us initio <lb/>æquantur inter &longs;e. </s></p> <p type="margin"> <s><margin.target id="note191"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO IV. PROBLEMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod vis gravitatis in Medio aliquo &longs;imilari uniformis &longs;it, <lb/>ac tendat perpendiculariter ad planum Horizontis; definire mo­<lb/>tum Projectilis in eodem, re&longs;i&longs;tentiam velocitati proportionalem <lb/>patientis.<emph.end type="italics"/></s></p> <p type="main"> <s>Eloco quovis <emph type="italics"/>D<emph.end type="italics"/>egrediatur Pro­<lb/><figure id="id.039.01.243.1.jpg" xlink:href="039/01/243/1.jpg"/><lb/>jectile &longs;ecundum lineam quam­<lb/>vis rectam <emph type="italics"/>DP,<emph.end type="italics"/>& per longitu­<lb/>dinem <emph type="italics"/>DP<emph.end type="italics"/>exponatur eju&longs;dem <lb/>velocitas &longs;ub initio motus. </s> <s>A <lb/>puncto <emph type="italics"/>P<emph.end type="italics"/>ad lineam Horizonta­<lb/>lem <emph type="italics"/>DC<emph.end type="italics"/>demittatur perpendi­<lb/>culum <emph type="italics"/>PC,<emph.end type="italics"/>& &longs;ecetur <emph type="italics"/>DC<emph.end type="italics"/>in <emph type="italics"/>A<emph.end type="italics"/><lb/>ut &longs;it <emph type="italics"/>DA<emph.end type="italics"/>ad <emph type="italics"/>AC<emph.end type="italics"/>ut re&longs;i&longs;tentia <lb/>Medii, ex motu in altitudinem <lb/>&longs;ub initio orta, ad vim gravi­<lb/>tatis; vel (quod perinde e&longs;t) ut <lb/>&longs;it rectangulum &longs;ub <emph type="italics"/>DA<emph.end type="italics"/>& <emph type="italics"/>DP<emph.end type="italics"/><lb/>ad rectangulum &longs;ub <emph type="italics"/>AC<emph.end type="italics"/>& <emph type="italics"/>CP<emph.end type="italics"/><lb/>ut re&longs;i&longs;tentia tota &longs;ub initio mo­<lb/>tus ad vim gravitatis. </s> <s>A&longs;ymptotis <lb/><emph type="italics"/>DC, CP,<emph.end type="italics"/>de&longs;cribatur Hyperbo­<lb/>la quævis <emph type="italics"/>GTBS<emph.end type="italics"/>&longs;ecans perpen­<lb/>dicula <emph type="italics"/>DG, AB<emph.end type="italics"/>in <emph type="italics"/>G<emph.end type="italics"/>& <emph type="italics"/>B<emph.end type="italics"/>; & <lb/>compleatur parallelogrammum <lb/><emph type="italics"/>DGKC,<emph.end type="italics"/>cujus latus <emph type="italics"/>GK<emph.end type="italics"/>&longs;ecet <lb/><emph type="italics"/>AB<emph.end type="italics"/>in <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/>Capiatur linea N in <lb/>ratione ad <emph type="italics"/>QB<emph.end type="italics"/>qua <emph type="italics"/>DC<emph.end type="italics"/>&longs;it ad <lb/><emph type="italics"/>CP<emph.end type="italics"/>; & ad rectæ <emph type="italics"/>DC<emph.end type="italics"/>pun­<lb/>ctum quodvis <emph type="italics"/>R<emph.end type="italics"/>erecto perpen­<lb/>diculo <emph type="italics"/>RT,<emph.end type="italics"/>quod Hyperbolæ <lb/>in <emph type="italics"/>T,<emph.end type="italics"/>& rectis <emph type="italics"/>EH, GK, DP<emph.end type="italics"/><lb/>in <emph type="italics"/>I, t<emph.end type="italics"/>& <emph type="italics"/>V<emph.end type="italics"/>occurrat; in eo cape <emph type="italics"/>Vr<emph.end type="italics"/>æqualem (<emph type="italics"/>tGT<emph.end type="italics"/>/N), vel quod per-<pb xlink:href="039/01/244.jpg" pagenum="216"/><arrow.to.target n="note192"/>inde e&longs;t, cape <emph type="italics"/>Rr<emph.end type="italics"/>æqualem (<emph type="italics"/>GTIE<emph.end type="italics"/>/N); & Projectile tempore <emph type="italics"/>DRTG<emph.end type="italics"/><lb/>perveniet ad punctum <emph type="italics"/>r,<emph.end type="italics"/>de&longs;cribens curvam lineam <emph type="italics"/>DraF,<emph.end type="italics"/>quam <lb/>punctum <emph type="italics"/>r<emph.end type="italics"/>&longs;emper tangit, perveniens autem ad maximam altitudi­<lb/>nem <emph type="italics"/>a<emph.end type="italics"/>in perpendiculo <emph type="italics"/>AB,<emph.end type="italics"/>& po&longs;tea &longs;emper appropinquans ad A­<lb/>&longs;ymptoton <emph type="italics"/>PLC.<emph.end type="italics"/>E&longs;tque velocitas ejus in puncto quovis <emph type="italics"/>r<emph.end type="italics"/>ut Cur­<lb/>væ Tangens <emph type="italics"/>rL. <expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note192"/>DE MOTU <lb/>CORPORUN</s></p> <p type="main"> <s>E&longs;t enim N ad <emph type="italics"/>QB<emph.end type="italics"/>ut <emph type="italics"/>DC<emph.end type="italics"/>ad <emph type="italics"/>CP<emph.end type="italics"/>&longs;eu <emph type="italics"/>DR<emph.end type="italics"/>ad <emph type="italics"/>RV,<emph.end type="italics"/>adeoque <emph type="italics"/>RV<emph.end type="italics"/><lb/>æqualis (<emph type="italics"/>DRXQB<emph.end type="italics"/>/N), & <emph type="italics"/>Rr<emph.end type="italics"/>(id e&longs;t <emph type="italics"/>RV-Vr<emph.end type="italics"/>&longs;eu (<emph type="italics"/>DRXQB-tGT<emph.end type="italics"/>/N)) <lb/>æqualis (<emph type="italics"/>DRXAB-RDGT<emph.end type="italics"/>/N). Exponatur jam tempus per are­<lb/>am <emph type="italics"/>RDGT,<emph.end type="italics"/>& (per Legum <lb/><figure id="id.039.01.244.1.jpg" xlink:href="039/01/244/1.jpg"/><lb/>Corol. </s> <s>2.) di&longs;tinguatur motus <lb/>corporis in duos, unum a&longs;cen­<lb/>&longs;us, alterum ad latus. </s> <s>Et cum <lb/>re&longs;i&longs;tentia &longs;it ut motus, di&longs;tin­<lb/>guetur etiam hæc in partes duas <lb/>partibus motus proportionales <lb/>& contrarias: ideoque longitu­<lb/>do, a motu ad latus de&longs;cripta, e­<lb/>rit (per Prop. </s> <s>11. hujus) ut linea <lb/><emph type="italics"/>DR,<emph.end type="italics"/>altitudo vero (per Prop. </s> <s><lb/>111. hujus) ut area <emph type="italics"/>DRXAB <lb/>-RDGT,<emph.end type="italics"/>hoc e&longs;t, ut linea <emph type="italics"/>Rr.<emph.end type="italics"/><lb/>Ip&longs;o autem motus initio area <lb/><emph type="italics"/>RDGT<emph.end type="italics"/>æqualis e&longs;t rectangulo <lb/><emph type="italics"/>DRXAQ,<emph.end type="italics"/>ideoque linea illa <emph type="italics"/>Rr<emph.end type="italics"/><lb/>(&longs;eu (<emph type="italics"/>DRXAB-DRXAQ<emph.end type="italics"/>/N)) <lb/>tunc e&longs;t ad <emph type="italics"/>DR<emph.end type="italics"/>ut <emph type="italics"/>AB-AQ<emph.end type="italics"/><lb/>&longs;eu <emph type="italics"/>QB<emph.end type="italics"/>ad N, id e&longs;t, ut <emph type="italics"/>CP<emph.end type="italics"/><lb/>ad <emph type="italics"/>DC<emph.end type="italics"/>; atque adeo ut motus <lb/>in altitudinem ad motum in <lb/>longitudinem &longs;ub initio. </s> <s>Cum <lb/>igitur <emph type="italics"/>Rr<emph.end type="italics"/>&longs;emper &longs;it ut altitu­<lb/>do, ac <emph type="italics"/>DR<emph.end type="italics"/>&longs;emper ut longi­<lb/>tudo, atque <emph type="italics"/>Rr<emph.end type="italics"/>ad <emph type="italics"/>DR<emph.end type="italics"/>&longs;ub <lb/>initio ut altitudo ad longitudinem: nece&longs;&longs;e e&longs;t ut <emph type="italics"/>Rr<emph.end type="italics"/>&longs;emper &longs;it ad <lb/><emph type="italics"/>DR<emph.end type="italics"/>ut altitudo ad longitudinem, & propterea ut corpus movea­<lb/>tur in linea <emph type="italics"/>DraF,<emph.end type="italics"/>quam punctum <emph type="italics"/>r<emph.end type="italics"/>perpetuo tangit. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p><pb xlink:href="039/01/245.jpg" pagenum="217"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. E&longs;t igitur <emph type="italics"/>Rr<emph.end type="italics"/>æqualis (<emph type="italics"/>DRXAB<emph.end type="italics"/>/N)-(<emph type="italics"/>RDGT<emph.end type="italics"/>/N), ideoque <lb/><arrow.to.target n="note193"/>&longs;i producatur <emph type="italics"/>RT<emph.end type="italics"/>ad <emph type="italics"/>X<emph.end type="italics"/>ut &longs;it <emph type="italics"/>RX<emph.end type="italics"/>æqualis (<emph type="italics"/>DRXAB<emph.end type="italics"/>/N), (id e&longs;t, &longs;i <lb/>compleatur parallelogrammum <emph type="italics"/>ACPY,<emph.end type="italics"/>jungatur <emph type="italics"/>DY<emph.end type="italics"/>&longs;ecans <emph type="italics"/>CP<emph.end type="italics"/><lb/>in <emph type="italics"/>Z,<emph.end type="italics"/>& producatur <emph type="italics"/>RT<emph.end type="italics"/>donec occurrat <emph type="italics"/>DY<emph.end type="italics"/>in <emph type="italics"/>X<emph.end type="italics"/>;) erit <emph type="italics"/>Xr<emph.end type="italics"/>æqua­<lb/>lis (<emph type="italics"/>RDGT<emph.end type="italics"/>/N), & propterea tempori proportionalis. </s></p> <p type="margin"> <s><margin.target id="note193"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Unde &longs;i capiantur innumeræ <emph type="italics"/>CR<emph.end type="italics"/>vel, quod perinde e&longs;t, <lb/>innumeræ Z<emph type="italics"/>X,<emph.end type="italics"/>in progre&longs;&longs;ione Geometrica; erunt totidem <emph type="italics"/>Xr<emph.end type="italics"/>in <lb/>progre&longs;&longs;ione Arithmetica. </s> <s>Et hinc Curva <emph type="italics"/>DraF<emph.end type="italics"/>per tabulam Lo­<lb/>garithmorum facile delineatur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Si vertice <emph type="italics"/>D,<emph.end type="italics"/>diametro <emph type="italics"/>DE<emph.end type="italics"/>deor&longs;um producta, & La­<lb/>tere recto quod &longs;it ad 2<emph type="italics"/>DP<emph.end type="italics"/>ut re&longs;i&longs;tentia tota, ip&longs;o motus initio, <lb/>ad vim gravitatis, Parabola con&longs;truatur: velocitas quacum corpus <lb/>exire debet de loco <emph type="italics"/>D<emph.end type="italics"/>&longs;ecundum rectam <emph type="italics"/>DP,<emph.end type="italics"/>ut in Medio uNI­<lb/>formi re&longs;i&longs;tente de&longs;cribat Curvam <emph type="italics"/>DraF,<emph.end type="italics"/>ea ip&longs;a erit quacum ex­<lb/>ire debet de eodem loco <emph type="italics"/>D,<emph.end type="italics"/>&longs;ecundum eandem rectam <emph type="italics"/>DP,<emph.end type="italics"/>ut <lb/>in &longs;patio non re&longs;i&longs;tente de&longs;cribat Parabolam. </s> <s>Nam Latus re­<lb/>ctum Parabolæ hujus, ip&longs;o motus initio, e&longs;t (<emph type="italics"/>DVquad./Vr<emph.end type="italics"/>) & <emph type="italics"/>Vr<emph.end type="italics"/><lb/>e&longs;t (<emph type="italics"/>tGT<emph.end type="italics"/>/N) &longs;eu (<emph type="italics"/>DRXTt<emph.end type="italics"/>/2N). Recta autem quæ, &longs;i duceretur, Hy­<lb/>perbolam <emph type="italics"/>GTB<emph.end type="italics"/>tangeret in <emph type="italics"/>G,<emph.end type="italics"/>parallela e&longs;t ip&longs;i <emph type="italics"/>DK,<emph.end type="italics"/>ideoque <lb/><emph type="italics"/>Tt<emph.end type="italics"/>e&longs;t (<emph type="italics"/>CKXDR/DC<emph.end type="italics"/>) & N erat (<emph type="italics"/>QBXDC/CP<emph.end type="italics"/>). Et propterea <emph type="italics"/>Vr<emph.end type="italics"/>e&longs;t <lb/>(<emph type="italics"/>DRqXCKXCP/2DCqXQB<emph.end type="italics"/>), id e&longs;t, (ob proportionales <emph type="italics"/>DR<emph.end type="italics"/>& <emph type="italics"/>DC, DV<emph.end type="italics"/><lb/>& <emph type="italics"/>DP<emph.end type="italics"/>) (<emph type="italics"/>DVqXCKXCP/2DPqXQB<emph.end type="italics"/>), & Latus rectum (<emph type="italics"/>DVquad./Vr<emph.end type="italics"/>) prodit <lb/>(2<emph type="italics"/>DPqXQB/CKXCP<emph.end type="italics"/>), id e&longs;t (ob proportionales <emph type="italics"/>QB<emph.end type="italics"/>& <emph type="italics"/>CK, DA<emph.end type="italics"/>& <emph type="italics"/>AC<emph.end type="italics"/>) <lb/>(2<emph type="italics"/>DPqXDA/ACXCP<emph.end type="italics"/>), adeoque ad 2 <emph type="italics"/>DP,<emph.end type="italics"/>ut <emph type="italics"/>DPXDA<emph.end type="italics"/>ad <emph type="italics"/>CPXAC<emph.end type="italics"/>; hoc <lb/>e&longs;t, ut re&longs;i&longs;tentia ad gravitatem. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Unde &longs;i corpus de loco quovis <emph type="italics"/>D,<emph.end type="italics"/>data cum velocitate, <lb/>&longs;ecundum rectam quamvis po&longs;itione datam <emph type="italics"/>DP<emph.end type="italics"/>projiciatur; & re­<lb/>&longs;i&longs;tentia Medii ip&longs;o motus initio detur: inveniri pote&longs;t Curva <lb/><emph type="italics"/>DraF,<emph.end type="italics"/>quam corpus idem de&longs;cribet. </s> <s>Nam ex data velocitate <pb xlink:href="039/01/246.jpg" pagenum="218"/><arrow.to.target n="note194"/>datur latus rectum Parabolæ, ut <lb/>notum e&longs;t. </s> <s>Et &longs;umendo 2<emph type="italics"/>DP<emph.end type="italics"/><lb/>ad latus illud rectum, ut e&longs;t vis <lb/>gravitatis ad vim re&longs;i&longs;tentiæ, <lb/>datur <emph type="italics"/>DP.<emph.end type="italics"/>Dein &longs;ecando <emph type="italics"/>DC<emph.end type="italics"/><lb/>in <emph type="italics"/>A,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>CPXAC<emph.end type="italics"/>ad <lb/><emph type="italics"/>DPXDA<emph.end type="italics"/>in eadem illa rati­<lb/>one gravitatis ad re&longs;i&longs;tentiam, <lb/>dabitur punctum <emph type="italics"/>A.<emph.end type="italics"/>Et inde <lb/>datur Curva <emph type="italics"/>DraF.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note194"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Et contra, &longs;i datur <lb/><figure id="id.039.01.246.1.jpg" xlink:href="039/01/246/1.jpg"/><lb/>Curva <emph type="italics"/>DraF,<emph.end type="italics"/>dabitur & ve­<lb/>locitas corporis & re&longs;i&longs;tentia <lb/>Medii in locis &longs;ingulis <emph type="italics"/>r.<emph.end type="italics"/>Nam <lb/>ex data ratione <emph type="italics"/>CPXAC<emph.end type="italics"/>ad <lb/><emph type="italics"/>DPXDA,<emph.end type="italics"/>datur tum re&longs;i&longs;ten­<lb/>tia Medii &longs;ub initio motus, tum <lb/>latus rectum Parabolæ: & inde <lb/>datur etiam velocitas &longs;ub initio <lb/>motus. </s> <s>Deinde ex longitudine <lb/>tangentis <emph type="italics"/>rL,<emph.end type="italics"/>datur & huic <lb/>proportionalis velocitas, & ve­<lb/>locitati proportionalis re&longs;i&longs;ten­<lb/>tia in loco quovis <emph type="italics"/>r.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Cum autem longitu­<lb/>do 2<emph type="italics"/>DP<emph.end type="italics"/>&longs;it ad latus rectum <lb/>Parabolæ ut gravitas ad re&longs;i&longs;tentiam in <emph type="italics"/>D<emph.end type="italics"/>; & ex aucta velocitate <lb/>augeatur re&longs;i&longs;tentia in eadem ratione, at latus rectum Parabolæ au­<lb/>geatur in ratione illa duplicata: patet longitudinem 2<emph type="italics"/>DP<emph.end type="italics"/>augeri <lb/>in ratione illa &longs;implici, adeoque velocitati &longs;emper proportionalem <lb/>e&longs;&longs;e, neque ex angulo <emph type="italics"/>CDP<emph.end type="italics"/>mutato augeri vel minui, ni&longs;i mu­<lb/>tetur quoque velocitas. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. Unde liquet methodus determinandi Curvam <emph type="italics"/>DraF<emph.end type="italics"/><lb/>ex Phænomenis quamproxime, & inde colligendi re&longs;i&longs;tentiam & <lb/>velocitatem quacum corpus projicitur. </s> <s>Projiciantur corpora duo <lb/>&longs;imilia & æqualia eadem cum velocitate, de loco <emph type="italics"/>D,<emph.end type="italics"/>&longs;ecundum <lb/>angulos diver&longs;os <emph type="italics"/>GDP, cDp<emph.end type="italics"/>(minu&longs;cularum literarum locis &longs;ub­<lb/>intellectis) & cogno&longs;cantur loca <emph type="italics"/>F, f,<emph.end type="italics"/>abi incidunt in horizontale <lb/>planum <emph type="italics"/>DC.<emph.end type="italics"/>Tum, a&longs;&longs;umpta quacunque longitudine pro <emph type="italics"/>DP<emph.end type="italics"/><lb/>vel <emph type="italics"/>Dp,<emph.end type="italics"/>fingatur quod re&longs;i&longs;tentia in <emph type="italics"/>D<emph.end type="italics"/>&longs;it ad gravitatem in ra-<pb xlink:href="039/01/247.jpg" pagenum="219"/>tione qualibet, & exponatur ratio illa per longitudinem quamvis <lb/><arrow.to.target n="note195"/><emph type="italics"/>SM.<emph.end type="italics"/>Deinde per computationem, ex longitudine illa a&longs;&longs;umpta <lb/><emph type="italics"/>DP,<emph.end type="italics"/>inveniantur longitudines <emph type="italics"/>DF, Df,<emph.end type="italics"/>ac de ratione (<emph type="italics"/>Ef/DF<emph.end type="italics"/>) per <lb/>calculum inventa, auferatur ratio eadem <lb/><figure id="id.039.01.247.1.jpg" xlink:href="039/01/247/1.jpg"/><lb/>per experimentum inventa, & exponatur <lb/>differentia per perpendiculum <emph type="italics"/>MN.<emph.end type="italics"/>Idem <lb/>fac iterum ac tertio, a&longs;&longs;umendo &longs;emper <lb/>novam re&longs;i&longs;tentiæ ad gravitatem rationem <lb/><emph type="italics"/>SM,<emph.end type="italics"/>& colligendo novam differentiam <lb/><emph type="italics"/>MN.<emph.end type="italics"/>Ducantur autem differentiæ affirmativæ ad unam partem <lb/>rectæ <emph type="italics"/>SM,<emph.end type="italics"/>& negativæ ad alteram; & per puncta <emph type="italics"/>N, N, N<emph.end type="italics"/>agatur <lb/>ourva regularis <emph type="italics"/>NNN<emph.end type="italics"/>&longs;ecans rectam <emph type="italics"/>SMMM<emph.end type="italics"/>in <emph type="italics"/>X,<emph.end type="italics"/>& erit <emph type="italics"/>SX<emph.end type="italics"/><lb/>vera ratio re&longs;i&longs;tentiæ ad gravitatem, quam invenire oportuit. </s> <s>Ex <lb/>hac ratione colligenda e&longs;t longitudo <emph type="italics"/>DF<emph.end type="italics"/>per calculum; & longi­<lb/>tudo quæ &longs;it ad a&longs;&longs;umptam longitudinem <emph type="italics"/>DP,<emph.end type="italics"/>at longitudo <emph type="italics"/>DF<emph.end type="italics"/><lb/>per experimentum cognita ad longitudinem <emph type="italics"/>DF<emph.end type="italics"/>modo inventam, <lb/>erit vera longitudo <emph type="italics"/>DP.<emph.end type="italics"/>Qua inventa, habetur tum Curva linea <lb/><emph type="italics"/>DraF<emph.end type="italics"/>quam corpus de&longs;cribit, tum corporis velocitas & re&longs;i&longs;ten­<lb/>tia in locis &longs;ingulis. </s></p> <p type="margin"> <s><margin.target id="note195"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Cæterum, re&longs;i&longs;tentiam corporum e&longs;&longs;e in ratione velocitatis, Hy­<lb/>pothe&longs;is e&longs;t magis Mathematica quam Naturalis. </s> <s>Obtinet hæc ra­<lb/>tio quamproxime ubi corpora in Mediis rigore aliquo præditis tar­<lb/>di&longs;&longs;ime moventur. </s> <s>In Mediis antem quæ rigore omni vacant re­<lb/>&longs;i&longs;tentiæ corporum &longs;unt in duplicata ratione velocitatum. </s> <s>Etenim <lb/>actione corporis velocioris communicatur eidem Medii quantitati, <lb/>tempore minore, motus major in ratione majoris velocitatis; ad­<lb/>eoque tempore æquali (ob majorem Medii quantitatem perturba­<lb/>tam) communicatur motus in duplicata ratione major; e&longs;t que re­<lb/>&longs;i&longs;tentia (per motus Legem II & III) ut motus communicatus. </s> <s><lb/>Videamus igitur quades oriantur motus ex hac lege Re&longs;i&longs;tentiæ. <pb xlink:href="039/01/248.jpg" pagenum="220"/><arrow.to.target n="note196"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note196"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>SECTIO II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De motu Corporum quibus re&longs;i&longs;titur in duplicata ra­<lb/>tione Velocitatum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO V. THEOREMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Corpori re&longs;i&longs;iitur in velocitatis ratione duplicata, & idem &longs;ola <lb/>vi in&longs;ita per Medium &longs;imilare movetur; tempora vero &longs;uman­<lb/>tur in progre&longs;&longs;ione Geometrica a minoribus terminis ad majores <lb/>pergente: dico quod velocitates initio &longs;ingulorum temporum <lb/>&longs;unt in eadem progre&longs;&longs;ione Geometrica inver&longs;e, & quod &longs;patia <lb/>&longs;unt æqualia quæ &longs;ingulis temporibus de&longs;cribuntur.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam quoniam quadrato velocita­<lb/><figure id="id.039.01.248.1.jpg" xlink:href="039/01/248/1.jpg"/><lb/>tis proportionalis e&longs;t re&longs;i&longs;tentia Me­<lb/>dii, & re&longs;i&longs;tentiæ proportionale e&longs;t <lb/>decrementum velocitatis; &longs;i tempus <lb/>in particulas innumeras æquales divi­<lb/>datur, quadrata velocitatum &longs;ingulis <lb/>temporum initiis erunt velocitatum <lb/>earundem differentiis proportionalia. </s> <s><lb/>Sunto temporis particulæ illæ <emph type="italics"/>AK, <lb/>KL, LM,<emph.end type="italics"/>&c. </s> <s>in recta <emph type="italics"/>CD<emph.end type="italics"/>&longs;umptæ, <lb/>& erigantur perpendicula <emph type="italics"/>AB, Kk, <lb/>Ll, Mm,<emph.end type="italics"/>&c. </s> <s>Hyperbolæ <emph type="italics"/>BklmG,<emph.end type="italics"/><lb/>centro <emph type="italics"/>C<emph.end type="italics"/>A&longs;ymptotis rectangulis <emph type="italics"/>CD, CH<emph.end type="italics"/>de&longs;criptæ, occurrentia <lb/>in <emph type="italics"/>B, k, t, m,<emph.end type="italics"/>&c. </s> <s>& erit <emph type="italics"/>AB<emph.end type="italics"/>ad <emph type="italics"/>Kk<emph.end type="italics"/>ut <emph type="italics"/>CK<emph.end type="italics"/>ad <emph type="italics"/>CA,<emph.end type="italics"/>& divi&longs;im <lb/><emph type="italics"/>AB-Kk<emph.end type="italics"/>ad <emph type="italics"/>Kk<emph.end type="italics"/>ut <emph type="italics"/>AK<emph.end type="italics"/>ad <emph type="italics"/>CA,<emph.end type="italics"/>& vici&longs;&longs;im <emph type="italics"/>AB-Kk<emph.end type="italics"/>ad <emph type="italics"/>AK<emph.end type="italics"/><lb/>ut <emph type="italics"/>Kk<emph.end type="italics"/>ad <emph type="italics"/>CA,<emph.end type="italics"/>adeoque ut <emph type="italics"/>ABXKk<emph.end type="italics"/>ad <emph type="italics"/>ABXCA.<emph.end type="italics"/>Unde, cum <lb/><emph type="italics"/>AK<emph.end type="italics"/>& <emph type="italics"/>ABXCA<emph.end type="italics"/>dentur, erit <emph type="italics"/>AB-Kk<emph.end type="italics"/>ut <emph type="italics"/>ABXKk<emph.end type="italics"/>; & ultimo, <lb/>ubi coeunt <emph type="italics"/>AB<emph.end type="italics"/>& <emph type="italics"/>Kk,<emph.end type="italics"/>ut <emph type="italics"/><expan abbr="ABq.">ABque</expan><emph.end type="italics"/>Et &longs;imili argumento erunt <emph type="italics"/>Kk-Ll, <lb/>Ll-Mm,<emph.end type="italics"/>&c. </s> <s>ut <emph type="italics"/>Kkq, Llq,<emph.end type="italics"/>&c. </s> <s>Linearum igitur <emph type="italics"/>AB, Kk, Ll, Mm<emph.end type="italics"/><pb xlink:href="039/01/249.jpg" pagenum="221"/>quadrata &longs;unt ut earundem differentiæ; & idcirco cum quadrata ve­<lb/><arrow.to.target n="note197"/>locitatum fuerint etiam ut ip&longs;arum differentiæ, &longs;imilis erit amba­<lb/>rum progre&longs;&longs;io. </s> <s>Quo demon&longs;trato, con&longs;equens e&longs;t etiam ut areæ <lb/>his lineis de&longs;criptæ &longs;int in progre&longs;&longs;ione con&longs;imili cum &longs;patiis quæ <lb/>velocitatibus de&longs;cribuntur. </s> <s>Ergo &longs;i velocitas initio primi tempo­<lb/>ris <emph type="italics"/>AK<emph.end type="italics"/>exponatur per lineam <emph type="italics"/>AB,<emph.end type="italics"/>& velocitas initio &longs;ecundi <emph type="italics"/>KL<emph.end type="italics"/><lb/>per lineam <emph type="italics"/>Kk,<emph.end type="italics"/>& longitudo primo tempore de&longs;cripta per aream <lb/><emph type="italics"/>AKkB<emph.end type="italics"/>; velocitates omnes &longs;ub&longs;equentes exponentur per lineas <lb/>&longs;ub&longs;equentes <emph type="italics"/>Ll, Mm,<emph.end type="italics"/>&c. </s> <s>& longitudines de&longs;criptæ per areas <lb/><emph type="italics"/>Kl, Lm,<emph.end type="italics"/>&c. </s> <s>Et compo&longs;ite, &longs;i tempus totum exponatur per &longs;um­<lb/>mam partium &longs;uarum <emph type="italics"/>AM,<emph.end type="italics"/>longitudo tota de&longs;cripta exponetur per <lb/>&longs;ummam partium &longs;uarum <emph type="italics"/>AMmB.<emph.end type="italics"/>Concipe jam tempus <emph type="italics"/>AM<emph.end type="italics"/>ita <lb/>dividi in partes <emph type="italics"/>AK, KL, LM,<emph.end type="italics"/>&c. </s> <s>ut &longs;int <emph type="italics"/>CA, CK, CL, CM,<emph.end type="italics"/><lb/>&c. </s> <s>in progre&longs;&longs;ione Geometrica; & erunt partes illæ in eadem pro­<lb/>gre&longs;&longs;ione, & velocitates <emph type="italics"/>AB, Kk, Ll, Mm,<emph.end type="italics"/>&c. </s> <s>in progre&longs;&longs;ione ea­<lb/>dem inver&longs;a, atque &longs;patia de&longs;cripta <emph type="italics"/>Ak, Kl, Lm,<emph.end type="italics"/>&c. </s> <s>æqualia. <lb/><emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note197"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Pater ergo quod, &longs;i tempus exponatur per A&longs;ymptoti <lb/>partem quamvis <emph type="italics"/>AD,<emph.end type="italics"/>& velocitas in principio temporis per ordi­<lb/>natim applicatam <emph type="italics"/>AB<emph.end type="italics"/>; velocitas in fine temporis exponetur per <lb/>ordinatam <emph type="italics"/>DG,<emph.end type="italics"/>& &longs;patium totum de&longs;criptum per aream Hyper­<lb/>bolicam adjacentem <emph type="italics"/>ABGD<emph.end type="italics"/>; necnon &longs;patium quod corpus ali­<lb/>quod eodem tempore <emph type="italics"/>AD,<emph.end type="italics"/>velocitate prima <emph type="italics"/>AB,<emph.end type="italics"/>in Medio non <lb/>re&longs;i&longs;tente de&longs;cribere po&longs;&longs;et, per rectangulum <emph type="italics"/>ABXAD.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Unde datur &longs;patium in Medio re&longs;i&longs;tente de&longs;criptum, ca­<lb/>piendo illud ad &longs;patium quod velocitate uniformi <emph type="italics"/>AB<emph.end type="italics"/>in medio non <lb/>re&longs;i&longs;tente &longs;imul de&longs;cribi po&longs;&longs;et, ut e&longs;t area Hyperbolica <emph type="italics"/>ABGD<emph.end type="italics"/><lb/>ad rectangulum <emph type="italics"/>ABXAD.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Datur etiam re&longs;i&longs;tentia Medii, &longs;tatuendo eam ip&longs;o mo­<lb/>tus initio æqualem e&longs;&longs;e vi uniformi centripetæ, quæ in cadente cor­<lb/>pore, tempore <emph type="italics"/>AC,<emph.end type="italics"/>in Medio non re&longs;i&longs;tente, generare po&longs;&longs;et velo­<lb/>citatem <emph type="italics"/>AB.<emph.end type="italics"/>Nam &longs;i ducatur <emph type="italics"/>BT<emph.end type="italics"/>quæ tangat Hyperbolam in <emph type="italics"/>B,<emph.end type="italics"/><lb/>& occurrat A&longs;ymptoto in <emph type="italics"/>T<emph.end type="italics"/>; recta <emph type="italics"/>AT<emph.end type="italics"/>æqualis erit ip&longs;i <emph type="italics"/>AC,<emph.end type="italics"/>& <lb/>tempus exponet quo re&longs;i&longs;tentia prima uniformiter continuata tolle­<lb/>re po&longs;&longs;et velocitatem totam <emph type="italics"/>AB.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol<emph.end type="italics"/>4. Et inde datur etiam proportio hujus re&longs;i&longs;tentiæ ad vim <lb/>gravitatis, aliamve quamvis datam vim centripetam. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Et vicever&longs;a, &longs;i datur proportio re&longs;i&longs;tentiæ ad datam <lb/>quamvis vim centripetam; datur tempus <emph type="italics"/>AC,<emph.end type="italics"/>quo vis centripeta <lb/>re&longs;i&longs;tentiæ æqualis generare po&longs;&longs;it velocitatem quamvis <emph type="italics"/>AB<emph.end type="italics"/>; & in-<pb xlink:href="039/01/250.jpg" pagenum="222"/><arrow.to.target n="note198"/>de datur punctum <emph type="italics"/>B<emph.end type="italics"/>per quod Hyperbola, A&longs;ymptoris <emph type="italics"/>CH, CD,<emph.end type="italics"/><lb/>de&longs;cribi debet; ut & &longs;patium <emph type="italics"/>ABGD,<emph.end type="italics"/>quod corpus incipiendo <lb/>motum &longs;uum cum velocitate illa <emph type="italics"/>AB,<emph.end type="italics"/>tempore quovis <emph type="italics"/>AD,<emph.end type="italics"/>in Me­<lb/>dio &longs;imilari re&longs;i&longs;tente de&longs;cribere pote&longs;t. </s></p> <p type="margin"> <s><margin.target id="note198"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO VI. THEOREMA IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Corpora Spherica homogemea & æqualia, re&longs;i&longs;tentiis in duplicata <lb/>ratione velocitatum impedita, & &longs;olis viribus in&longs;itis incitata, <lb/>temporibus quæ &longs;unt reciproce ut velocitates &longs;ub initio, de&longs;cri­<lb/>bunt &longs;emper æqualia &longs;patia, & amittunt partes velocitatum pro­<lb/>portionales totis.<emph.end type="italics"/></s></p> <p type="main"> <s>A&longs;ymptotis rectangulis <emph type="italics"/>CD, <lb/><figure id="id.039.01.250.1.jpg" xlink:href="039/01/250/1.jpg"/><lb/>CH<emph.end type="italics"/>de&longs;cripta Hyperbola qua­<lb/>vis <emph type="italics"/>BbEe<emph.end type="italics"/>&longs;ecante perpendicula <lb/><emph type="italics"/>AB, ab, DE, de,<emph.end type="italics"/>in <emph type="italics"/>B, b, E, e,<emph.end type="italics"/><lb/>exponantur velocitates initi­<lb/>ales per perpendicula <emph type="italics"/>AB, <lb/>DE,<emph.end type="italics"/>& tempora per lineas <lb/><emph type="italics"/>Aa, Dd.<emph.end type="italics"/>E&longs;t ergo ut <emph type="italics"/>Aa<emph.end type="italics"/>ad <lb/><emph type="italics"/>Dd<emph.end type="italics"/>ita (per Hypothe&longs;in) <emph type="italics"/>DE<emph.end type="italics"/><lb/>ad <emph type="italics"/>AB,<emph.end type="italics"/>& ita (ex natura Hy­<lb/>perbolæ) <emph type="italics"/>CA<emph.end type="italics"/>ad <emph type="italics"/>CD<emph.end type="italics"/>; & com­<lb/>ponendo, ita <emph type="italics"/>Ca<emph.end type="italics"/>ad <emph type="italics"/>Cd.<emph.end type="italics"/>Ergo <lb/>areæ <emph type="italics"/>ABba, DEed,<emph.end type="italics"/>hoc e&longs;t, &longs;patia de&longs;cripta æquamtur inter &longs;e, <lb/>& velocitates primæ <emph type="italics"/>AB, DE<emph.end type="italics"/>&longs;unt ultimis <emph type="italics"/>ab, de,<emph.end type="italics"/>& propterea <lb/>(dividendo) partibus etiam &longs;uis ami&longs;&longs;is <emph type="italics"/>AB-ab, DE-de<emph.end type="italics"/>pro­<lb/>portionales. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO VII. THEOREMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Corpora Sphærica quibus re&longs;i&longs;titur in duplicata ratione velocitatum, <lb/>temporibus quæ &longs;unt ut motus primi directe & re&longs;i&longs;tentiæ pri­<lb/>mæ inver&longs;e, amittent partes motuum proportionales totis, & <lb/>&longs;patia de&longs;cribent temporibus i&longs;tis in velocitates primas ductis <lb/>proportionalia.<emph.end type="italics"/></s></p> <p type="main"> <s>Namque motuum partes ami&longs;&longs;æ &longs;unt ut re&longs;i&longs;tentiæ & tempora <pb xlink:href="039/01/251.jpg" pagenum="223"/>conjunctim. </s> <s>Igitur ut partes illæ &longs;int totis proportionales, debe­<lb/><arrow.to.target n="note199"/>bit re&longs;i&longs;tentia & tempus conjunctim e&longs;&longs;e ut motus. </s> <s>Proinde tem­<lb/>pus erit ut motus directe & re&longs;i&longs;tentia inver&longs;e. </s> <s>Quare temporam <lb/>particulis in ea ratione &longs;umptis, corpora amittent &longs;emper parti­<lb/>culas motuum proportionales totis, adeoque retinebunt velocita­<lb/>tes in ratione prima. </s> <s>Et ob datam velocitatum rationem, de&longs;cri­<lb/>bent &longs;emper &longs;patia quæ &longs;unt ut velocitates primæ & tempora con­<lb/>junctim. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note199"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Igitur &longs;i æquivelocibus corporibus re&longs;i&longs;titur in duplicata <lb/>ratione diametrorum: Globi homogenei quibu&longs;cunque cum velocita­<lb/>tibus moti, de&longs;cribendo &longs;patia diametris &longs;uis proportionalia, amit­<lb/>tent partes motuum proportionales totis. </s> <s>Motus enim Globi cu­<lb/>ju&longs;que erit ut ejus velocitas & Ma&longs;&longs;a conjunctim, id e&longs;t, ut veloci­<lb/>tas & cubus diametri; re&longs;i&longs;tentia (per Hypothe&longs;in) erit ut quadra­<lb/>tum diametri & quadratum velocitatis conjunctim; & tempus (per <lb/>hanc Propo&longs;itionem) e&longs;t in ratione priore directe & ratione po&longs;te­<lb/>riore inver&longs;e, id e&longs;t, ut diameter directe & velocitas inver&longs;e; ad­<lb/>eoque &longs;patium (tempori & velocitati proportionale) e&longs;t ut dia­<lb/>meter. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si æquivelocibus corporibus re&longs;i&longs;titur in ratione &longs;e&longs;quial­<lb/>tera diametrorum: Globi homogenei quibu&longs;cunque cum velocitati­<lb/>bus moti, de&longs;cribendo &longs;patia in &longs;e&longs;quialtera ratione diametrorum, <lb/>amittent partes motuum proportionales totis. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Et univer&longs;aliter, &longs;i æquivelocibus corporibus re&longs;i&longs;titur in <lb/>ratione dignitatis cuju&longs;cunQ.E.D.ametrorum: &longs;patia quibus Globi <lb/>homogenei, quibu&longs;cunque cum velocitatibus moti, amittent partes <lb/>motuum proportionales totis, erunt ut cubi diametrorum ad digNI­<lb/>tatem illam applicati. </s> <s>Sunto diametri D & E; & &longs;i re&longs;i&longs;tentiæ, <lb/>ubi velocitates æquales ponuntur, &longs;int ut D<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> & E<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>: &longs;patia quibus <lb/>Globi quibu&longs;cunque cum velocitatibus moti, amitteus partes mo­<lb/>tuum proportionales totis, erunt ut D<emph type="sup"/>3-<emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> & E<emph type="sup"/>3-<emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>. </s> <s>Igitur de&longs;cri­<lb/>bendo &longs;patia ip&longs;is D<emph type="sup"/>3-<emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> & E<emph type="sup"/>3-<emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> proportionalia, retinebunt veloci­<lb/>tates in eadem ratione ad invicem ac &longs;ub initio. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Quod &longs;i Globi non &longs;int homogenei, &longs;patium a Globo <lb/>den&longs;iore de&longs;criptum augeri debet in ratione den&longs;itatis. </s> <s>Motus <lb/>enim, &longs;ub pari velocitare, major e&longs;t in ratione den&longs;itatis, & tempus <lb/>(per hanc Propo&longs;itionem) augetur in ratione motus directe, ac <lb/>&longs;patium de&longs;criptum in ratione temporis. <pb xlink:href="039/01/252.jpg" pagenum="224"/><arrow.to.target n="note200"/></s></p> <p type="margin"> <s><margin.target id="note200"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Et &longs;i Globi moveantur in Mediis diver&longs;is; &longs;patium in <lb/>Medio, quod cæteris paribus magis re&longs;i&longs;tit, diminuendum erit in <lb/>ratione majoris re&longs;i&longs;tentiæ. </s> <s>Tempus enim (per hanc Propo&longs;itio­<lb/>nem) diminuetur in ratione re&longs;i&longs;tentiæ auctæ, & &longs;patium in ra­<lb/>tione temporis. </s></p> <p type="main"> <s><emph type="center"/>LEMMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Momentum Genitæ æquatur Momentis laterum &longs;ingulorum gene­<lb/>rantium in eorundem laterum indices dignitatum & coefficien­<lb/>tia continue ductis.<emph.end type="italics"/></s></p> <p type="main"> <s>Genitam voco quantitatem omnem quæ ex lateribus vel termi­<lb/>nis quibu&longs;cunque, in Arithmetica per multiplicationem, divi&longs;ionem, <lb/>& extractionem radicum; in Geometria per inventionem vel con­<lb/>tentorum & laterum, vel extremarum & mediarum proportionalium, <lb/>ab&longs;que additione & &longs;ubductione generatur. </s> <s>Eju&longs;modi quantita­<lb/>tes &longs;unt Facti, Quoti, Radices, Rectangula, Quadrata, Cubi, Latera <lb/>quadrata, Latera cubica, & &longs;imiles. </s> <s>Has quantitates ut indeterminatas <lb/>& in&longs;tabiles, & qua&longs;i motu fluxuve perpetuo cre&longs;centes vel decre­<lb/>&longs;centes, hic con&longs;idero; & earum incrementa vel decrementa momen­<lb/>tanea &longs;ub nomine Momentorum intelligo: ita ut incrementa pro <lb/>momentis addititiis &longs;eu affirmativis, ac decrementa pro &longs;ubductitiis <lb/>&longs;eu negativis habeantur. </s> <s>Cave tamen intellexeris particulas fiNI­<lb/>tas. </s> <s>Particulæ finitæ non &longs;unt momenta, &longs;ed quantitates ip&longs;æ ex <lb/>momentis genitæ. </s> <s>Intelligenda &longs;unt principia jamjam na&longs;centia fi­<lb/>nitarum magnitudinum. </s> <s>Neque enim &longs;pectatur in hoc Lemmate <lb/>magnitudo momentorum, &longs;ed prima na&longs;centium proportio. </s> <s>Eo­<lb/>dem recidit &longs;i loco momentorum u&longs;urpentur vel velocitates incre­<lb/>mentorum ac decrementorum, (quas etiam motus, mutationes <lb/>& fluxiones quantitatum nominare licet) vel finitæ quævis quanti­<lb/>tates velocitatibus hi&longs;ce proportionales. </s> <s>Lateris autem cuju&longs;que <lb/>generantis Coefficiens e&longs;t quantitas, quæ oritur applicando GeNI­<lb/>tam ad hoc latus. </s></p> <p type="main"> <s>Igitur &longs;en&longs;us Lemmatis e&longs;t, ut, &longs;i quantitatum quarumcunque <lb/>perpetuo motu cre&longs;centium vel decre&longs;centium A, B, C, &c. </s> <s>mo­<lb/>menta, vel mutationum velocitates dicantur <emph type="italics"/>a, b, c,<emph.end type="italics"/>&c. </s> <s>momentum <lb/>vel mutatio geniti rectanguli AB fuerit <emph type="italics"/>a<emph.end type="italics"/>B+<emph type="italics"/>b<emph.end type="italics"/>A, & geniti con­<lb/>tenti ABC momentum fuerit <emph type="italics"/>a<emph.end type="italics"/>BC+<emph type="italics"/>b<emph.end type="italics"/>AC+<emph type="italics"/>c<emph.end type="italics"/>AB: & genitarum <pb xlink:href="039/01/253.jpg" pagenum="225"/>dignitatum A<emph type="sup"/>3<emph.end type="sup"/>, A<emph type="sup"/>3<emph.end type="sup"/>, A<emph type="sup"/>4<emph.end type="sup"/>, A<emph type="sup"/>1/2<emph.end type="sup"/>, A<emph type="sup"/>1/3<emph.end type="sup"/>, A<emph type="sup"/>1/3<emph.end type="sup"/>, A<emph type="sup"/>2/3<emph.end type="sup"/>, A<emph type="sup"/>-1<emph.end type="sup"/>, A<emph type="sup"/>-2<emph.end type="sup"/>, & A<emph type="sup"/>-1/2<emph.end type="sup"/> momenta </s></p> <p type="main"> <s><arrow.to.target n="note201"/>2<emph type="italics"/>a<emph.end type="italics"/>A, 3<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>2<emph.end type="sup"/>, 4<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>3<emph.end type="sup"/>, 1/2<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>-1/2<emph.end type="sup"/>, 3/2<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>1/2<emph.end type="sup"/>, 1/3<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>-2/3<emph.end type="sup"/>, 2/3<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>-1/3<emph.end type="sup"/>, -<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>-2<emph.end type="sup"/>, <lb/>-2<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>-3<emph.end type="sup"/>, & -1/2<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>-1/2<emph.end type="sup"/> re&longs;pective. </s> <s>Et generaliter, ut dignitatis <lb/>cuju&longs;cunque A<emph type="sup"/><emph type="italics"/>n/m<emph.end type="italics"/><emph.end type="sup"/> momentum fuerit <emph type="italics"/>n/m a<emph.end type="italics"/>A<emph type="sup"/>(<emph type="italics"/>n-m/m<emph.end type="italics"/>)<emph.end type="sup"/>. </s> <s>Item ut Genitæ <lb/>A<emph type="sup"/>2<emph.end type="sup"/>B momentum fuerit 2<emph type="italics"/>a<emph.end type="italics"/>AB+<emph type="italics"/>b<emph.end type="italics"/>A<emph type="sup"/>2<emph.end type="sup"/>; & Genitæ A<emph type="sup"/>3<emph.end type="sup"/>B<emph type="sup"/>4<emph.end type="sup"/>C<emph type="sup"/>2<emph.end type="sup"/> momen­<lb/>tum 3<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>2<emph.end type="sup"/>B<emph type="sup"/>4<emph.end type="sup"/>C<emph type="sup"/>2<emph.end type="sup"/>+4<emph type="italics"/>b<emph.end type="italics"/>A<emph type="sup"/>3<emph.end type="sup"/>B<emph type="sup"/>3<emph.end type="sup"/>C<emph type="sup"/>2<emph.end type="sup"/>+2<emph type="italics"/>c<emph.end type="italics"/>A<emph type="sup"/>3<emph.end type="sup"/>B<emph type="sup"/>4<emph.end type="sup"/>C; & Genitæ (A<emph type="sup"/>3<emph.end type="sup"/>/B<emph type="sup"/>2<emph.end type="sup"/>) &longs;i­<lb/>ve A<emph type="sup"/>3<emph.end type="sup"/>B<emph type="sup"/>-2<emph.end type="sup"/> momentum 3<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>2<emph.end type="sup"/>B<emph type="sup"/>-2<emph.end type="sup"/>-2<emph type="italics"/>b<emph.end type="italics"/>A<emph type="sup"/>3<emph.end type="sup"/>B<emph type="sup"/>-3<emph.end type="sup"/>: & &longs;ic in cæteris. </s> <s><lb/>Demon&longs;tratur vero Lemma in hunc modum. </s></p> <p type="margin"> <s><margin.target id="note201"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Rectangulum quodvis motu perpetuo auctum AB, <lb/>ubi de lateribus A & B deerant momentorum dimidia 1/2<emph type="italics"/>a<emph.end type="italics"/>& 1/2<emph type="italics"/>b,<emph.end type="italics"/><lb/>fuit A-1/2<emph type="italics"/>a<emph.end type="italics"/>in B-1/2<emph type="italics"/>b,<emph.end type="italics"/>&longs;eu AB-1/2<emph type="italics"/>a<emph.end type="italics"/>B-1/2<emph type="italics"/>b<emph.end type="italics"/>A+1/4<emph type="italics"/>ab<emph.end type="italics"/>; & quam pri­<lb/>mum latera A & B alteris momentorum dimidiis aucta &longs;unt, eva­<lb/>dit A+1/2<emph type="italics"/>a<emph.end type="italics"/>in B+1/2<emph type="italics"/>b<emph.end type="italics"/>&longs;eu AB+1/2<emph type="italics"/>a<emph.end type="italics"/>B+1/2<emph type="italics"/>b<emph.end type="italics"/>A+1/4<emph type="italics"/>ab.<emph.end type="italics"/>De hoc rectan­<lb/>gulo &longs;ubducatur rectangulum prius, & manebit exce&longs;&longs;us <emph type="italics"/>a<emph.end type="italics"/>B+<emph type="italics"/>b<emph.end type="italics"/>A. </s> <s><lb/>Igitur laterum incrementis totis <emph type="italics"/>a<emph.end type="italics"/>& <emph type="italics"/>b<emph.end type="italics"/>generatur rectanguli incre­<lb/>mentum <emph type="italics"/>a<emph.end type="italics"/>B+<emph type="italics"/>b<emph.end type="italics"/>A. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Ponatur AB &longs;emper æquale G, & contenti ABC &longs;eu <lb/>GC momentum (per Cas. </s> <s>1.) erit <emph type="italics"/>g<emph.end type="italics"/>C+<emph type="italics"/>c<emph.end type="italics"/>G, id e&longs;t (&longs;i pro G & <emph type="italics"/>g<emph.end type="italics"/><lb/>&longs;cribantur AB & <emph type="italics"/>a<emph.end type="italics"/>B+<emph type="italics"/>b<emph.end type="italics"/>A) <emph type="italics"/>a<emph.end type="italics"/>BC+<emph type="italics"/>b<emph.end type="italics"/>AC+<emph type="italics"/>c<emph.end type="italics"/>AB. </s> <s>Et par e&longs;t ra­<lb/>tio contenti &longs;ub lateribus quotcunque. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>3. Ponantur latera A, B, C &longs;ibi mutuo &longs;emper æqualia; & <lb/>ip&longs;ius A<emph type="sup"/>2<emph.end type="sup"/>, id e&longs;t rectanguli AB, momentum <emph type="italics"/>a<emph.end type="italics"/>B+<emph type="italics"/>b<emph.end type="italics"/>A erit 2<emph type="italics"/>a<emph.end type="italics"/>A, ip­<lb/>&longs;ius autem A<emph type="sup"/>3<emph.end type="sup"/>, id e&longs;t contenti ABC, momentum <emph type="italics"/>a<emph.end type="italics"/>BC+<emph type="italics"/>b<emph.end type="italics"/>AC <lb/>+<emph type="italics"/>c<emph.end type="italics"/>AB erit 3<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>2<emph.end type="sup"/>. </s> <s>Et eodem argumento momentum dignitatis <lb/>cuju&longs;cunque A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> e&longs;t <emph type="italics"/>na<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1.<emph.end type="sup"/> <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>4. Unde cum 1/A in A &longs;it 1, momentum ip&longs;ius 1/A ductum <lb/>in A, una cum 1/A ducto in <emph type="italics"/>a<emph.end type="italics"/>erit momentum ip&longs;ius 1, id e&longs;t, NI­<lb/>hil. </s> <s>Proinde momentum ip&longs;ius 1/A &longs;eu ip&longs;ius A<emph type="sup"/>-1<emph.end type="sup"/> e&longs;t (-<emph type="italics"/>a<emph.end type="italics"/>/A<emph type="sup"/>2<emph.end type="sup"/>). Et ge­<lb/>neraliter cum (1/A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>) in A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> &longs;it 1, momentum ip&longs;ius (1/A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>) ductum in A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/><pb xlink:href="039/01/254.jpg" pagenum="226"/><arrow.to.target n="note202"/>una cum (1/A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>) in <emph type="italics"/>na<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/> erit nihil. </s> <s>Et propterea momentum ip­<lb/>&longs;ius (1/A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>) &longs;eu A<emph type="sup"/>-<emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> erit-(<emph type="italics"/>na<emph.end type="italics"/>/A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>+1). <emph type="italics"/>q.ED.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note202"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>5. Et cum A<emph type="sup"/>1/2<emph.end type="sup"/> in A<emph type="sup"/>1/2<emph.end type="sup"/> &longs;it A, momentum ip&longs;ius A<emph type="sup"/>1/2<emph.end type="sup"/> ductum in <lb/>2A<emph type="sup"/>1/2<emph.end type="sup"/> erit <emph type="italics"/>a,<emph.end type="italics"/>per Cas. </s> <s>3: ideoque momentum ip&longs;ius A<emph type="sup"/>1/2<emph.end type="sup"/> erit (<emph type="italics"/>a<emph.end type="italics"/>/2A 1/2) <lb/>&longs;ive 1/2<emph type="italics"/>a<emph.end type="italics"/>A<emph type="sup"/>-1/2<emph.end type="sup"/>. </s> <s>Et generaliter &longs;i ponatur A<emph type="sup"/><emph type="italics"/>m/n<emph.end type="italics"/><emph.end type="sup"/> æquale B, erit A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/> æ­<lb/>quale B<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>, ideoque <emph type="italics"/>ma<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/> æquale <emph type="italics"/>nb<emph.end type="italics"/>B<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1,<emph.end type="sup"/> & <emph type="italics"/>ma<emph.end type="italics"/>A<emph type="sup"/>-1<emph.end type="sup"/> æqua­<lb/>le <emph type="italics"/>nb<emph.end type="italics"/>B<emph type="sup"/>-1<emph.end type="sup"/> &longs;eu <emph type="italics"/>nb<emph.end type="italics"/>A<emph type="sup"/>-<emph type="italics"/>m/n<emph.end type="italics"/><emph.end type="sup"/>, adeoque <emph type="italics"/>m/n a<emph.end type="italics"/>A<emph type="sup"/>(<emph type="italics"/>m-n/n<emph.end type="italics"/>)<emph.end type="sup"/> æquale <emph type="italics"/>b,<emph.end type="italics"/>id e&longs;t, æquale <lb/>momento ip&longs;ius A<emph type="sup"/><emph type="italics"/>m/n<emph.end type="italics"/><emph.end type="sup"/>, <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>6. Igitur Genitæ cuju&longs;eunque A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>B<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> momentum e&longs;t mo­<lb/>mentum ip&longs;ius A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/> ductum in B<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>, una cum momento ip&longs;ius B<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> du­<lb/>cto in A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>, id e&longs;t <emph type="italics"/>ma<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/>-1<emph.end type="sup"/>B<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>+<emph type="italics"/>nb<emph.end type="italics"/>B<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>A<emph type="sup"/><emph type="italics"/>m<emph.end type="italics"/><emph.end type="sup"/>; idque &longs;ive dignita­<lb/>tum indices <emph type="italics"/>m<emph.end type="italics"/>& <emph type="italics"/>n<emph.end type="italics"/>&longs;int integri numeri vel fracti, &longs;ive affirmati­<lb/>vi vel negativi. </s> <s>Et par e&longs;t ratio contenti &longs;ub pluribus dignitati­<lb/>bus. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc in continue proportionalibus, &longs;i terminus unus <lb/>datur, momenta terminorum reliquorum erunt ut iidem termini <lb/>multiplicati per numerum intervallorum inter ip&longs;os & terminum <lb/>datum. </s> <s>Sunto A, B, C, D, E, F continue proportionales; & &longs;i <lb/>detur terminus C, momenta reliquorum terminorum erunt inter <lb/>&longs;e ut-2A, -B, D, 2E, 3F. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et &longs;i in quatuor proportionalibus duæ mediæ dentur, <lb/>momenta extremarum erunt ut eædem extremæ. </s> <s>Idem intelligen­<lb/>dum e&longs;t de lateribus rectanguli cuju&longs;cunQ.E.D.ti. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Et &longs;i &longs;umma vel differentia duorum quadratorum detur, <lb/>momenta laterum erunt reciproce ut latera. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>In literis quæ mihi cum Geometra periti&longs;&longs;imo <emph type="italics"/>G.G. Leibnitio<emph.end type="italics"/>an­<lb/>nis abhinc decem intercedebant, cum &longs;ignificarem me compotem <lb/>e&longs;&longs;e methodi determinandi Maximas & Minimas, ducendi Tangen­<lb/>tes, & &longs;imilia peragendi, quæ in terminis &longs;urdis æque ac in ratio­<lb/>nalibus procederet, & literis tran&longs;po&longs;itis hanc &longs;ententiam involven-<pb xlink:href="039/01/255.jpg" pagenum="227"/>tibus [<emph type="italics"/>Data Æquatione quotcunque Fluentes quantitates invelven-<emph.end type="italics"/><lb/><arrow.to.target n="note203"/><emph type="italics"/>te, Fluxiones invenire, & vice ver&longs;a<emph.end type="italics"/>] eandem celarem: re&longs;crip&longs;it <lb/>Vir Clari&longs;&longs;imus &longs;e quoQ.E.I. eju&longs;modi methodum incidi&longs;&longs;e, & me­<lb/>thodum &longs;uam communicavit a mea vix abludentem præterquam in <lb/>verborum & notarum formulis, & Idea generationis quantitatum. </s> <s><lb/>Utriu&longs;que fundamentum continetur in hoc Lemmate. </s></p> <p type="margin"> <s><margin.target id="note203"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO VIII. THEOREMA VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si corpus in Medio uniformi, Gravitate uniformiter agente, recta <lb/>a&longs;cendat vel de&longs;cendat, & &longs;patium totum de&longs;criptum di&longs;tingua­<lb/>tur in partes æquales, inque principiis &longs;ingularum partium <lb/>(addendo re&longs;i&longs;tentiam Medii ad vim gravitatis, quando cor­<lb/>pus a&longs;cendit, vel &longs;ubducendo ip&longs;am quando corpus de&longs;cendit) <lb/>colligantur vires ab&longs;olutæ; dico quod vires illæ ab&longs;olutæ &longs;unt <lb/>in progre&longs;&longs;ione Geometrica.<emph.end type="italics"/></s></p> <p type="main"> <s>Exponatur enim vis gravitatis per datam lineam <emph type="italics"/>AC<emph.end type="italics"/>; re&longs;i&longs;ten­<lb/>tia per lineam indefinitam <emph type="italics"/>AK<emph.end type="italics"/>; vis ab&longs;oluta in de&longs;cen&longs;u corporis <lb/>per differentiam <emph type="italics"/>KC<emph.end type="italics"/>; velocitas corporis per lineam <emph type="italics"/>AP<emph.end type="italics"/>(quæ &longs;it <lb/>media proportionalis inter <emph type="italics"/>AK<emph.end type="italics"/>& <emph type="italics"/>AC,<emph.end type="italics"/>ideoQ.E.I. &longs;ubduplicata <lb/>ratione re&longs;i&longs;tentiæ;) incrementum re&longs;i&longs;tentiæ data temporis particu­<lb/>la factum per lineolam <emph type="italics"/>KL,<emph.end type="italics"/>& contemporaneum velocitatis incre­<lb/>mentum per lineolam <emph type="italics"/>PQ<emph.end type="italics"/>; & centro <emph type="italics"/>C<emph.end type="italics"/>A&longs;ymptotis rectangulis <lb/><emph type="italics"/>CA, CH<emph.end type="italics"/>de&longs;cribatur Hyperbola quævis <emph type="italics"/>BNS,<emph.end type="italics"/>erectis perpendi­<lb/>culis <emph type="italics"/>AB, KN, LO, PR, QS<emph.end type="italics"/>occurrens in <emph type="italics"/>B, N, O, R, S.<emph.end type="italics"/>Quo­<lb/>niam <emph type="italics"/>AK<emph.end type="italics"/>e&longs;t ut <emph type="italics"/>APq,<emph.end type="italics"/>erit hujus momentum <emph type="italics"/>KL<emph.end type="italics"/>ut illius mo­<lb/>mentum 2<emph type="italics"/>APQ,<emph.end type="italics"/>id e&longs;t, ut <emph type="italics"/>AP<emph.end type="italics"/>in <emph type="italics"/>KC.<emph.end type="italics"/>Nam velocitatis incre­<lb/>mentum <emph type="italics"/>PQ,<emph.end type="italics"/>(per motus Leg.11.) proportionale e&longs;t vi generanti <emph type="italics"/>KC.<emph.end type="italics"/><lb/>Componatur ratio ip&longs;ius <emph type="italics"/>KL<emph.end type="italics"/>cum ratione ip&longs;ius <emph type="italics"/>KN,<emph.end type="italics"/>& fiet rect­<lb/>angulum <emph type="italics"/>KLXKN<emph.end type="italics"/>ut <emph type="italics"/>APXKCXKN<emph.end type="italics"/>; hoc e&longs;t, ob datum rect­<lb/>angulum <emph type="italics"/>KCXKN,<emph.end type="italics"/>ut <emph type="italics"/>AP.<emph.end type="italics"/>Atqui areæ Hyperbolicæ <emph type="italics"/>KNOL<emph.end type="italics"/><lb/>ad rectangulum <emph type="italics"/>KLXKN<emph.end type="italics"/>ratio ultima, ubi coeunt puncta <emph type="italics"/>K<emph.end type="italics"/>& <emph type="italics"/>L,<emph.end type="italics"/><lb/>e&longs;t æqualitatis. </s> <s>Ergo area illa Hyperbolica evane&longs;cens e&longs;t ut <emph type="italics"/>AP.<emph.end type="italics"/><lb/>Componitur igitur area tota Hyperbolica <emph type="italics"/>ABOL<emph.end type="italics"/>ex particulis <lb/><emph type="italics"/>KNOL<emph.end type="italics"/>velocitati <emph type="italics"/>AP<emph.end type="italics"/>&longs;emper proportionalibus, & propterea <lb/>&longs;patio velocitate i&longs;ta de&longs;cripto proportionalis e&longs;t. </s> <s>Dividatur jam <lb/>area illa in partes æquales <emph type="italics"/>ABMI, IMNK, KNOL,<emph.end type="italics"/>&c. </s> <s>& vi-<pb xlink:href="039/01/256.jpg" pagenum="228"/><arrow.to.target n="note204"/>res ab&longs;olutæ <emph type="italics"/>AC, IC, KC, LC,<emph.end type="italics"/>&c. </s> <s>erunt in progre&longs;&longs;ione Geo­<lb/>metrica. <emph type="italics"/>Q.E.D.<emph.end type="italics"/>Et &longs;imili argumento, in a&longs;cen&longs;u corporis, &longs;u­<lb/>mendo, ad contrariam partem puncti <emph type="italics"/>A,<emph.end type="italics"/>æquales areas <emph type="italics"/>ABmi, <lb/>imnk, knol,<emph.end type="italics"/>&c. </s> <s>con&longs;tabit quod vires ab&longs;olutæ <emph type="italics"/>AC, iC, kC, lC,<emph.end type="italics"/>&c. <lb/></s> <s>&longs;unt continue proportionales. </s> <s>Ideoque &longs;i &longs;patia omnia in a&longs;cen&longs;u & <lb/>de&longs;cen&longs;u capiantur æqualia; omnes vires ab&longs;olutæ <emph type="italics"/>lC, kC, iC, AC, <lb/>IC, KC, LC,<emph.end type="italics"/>&c. </s> <s>erunt continue proportionales. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note204"/>DE MOTU <lb/>CORPORUM</s></p><figure id="id.039.01.256.1.jpg" xlink:href="039/01/256/1.jpg"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i &longs;patium de&longs;criptum exponatur per aream Hy­<lb/>perbolicam <emph type="italics"/>ABNK<emph.end type="italics"/>; exponi po&longs;&longs;unt vis gravitatis, velocitas cor­<lb/>poris & re&longs;i&longs;tentia Medii per lineas <emph type="italics"/>AC, AP<emph.end type="italics"/>& <emph type="italics"/>AK<emph.end type="italics"/>re&longs;pective; <lb/>& vice ver&longs;a. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et velocitatis maximæ, quam corpus in infinitum de&longs;cen­<lb/>dendo pote&longs;t unquam acquirere, exponens e&longs;t linea <emph type="italics"/>AC.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Igitur &longs;i in data aliqua velocitate cogno&longs;catur re&longs;i&longs;ten­<lb/>tia Medii, invenietur velocitas maxima, &longs;umendo ip&longs;am ad veloci-<pb xlink:href="039/01/257.jpg" pagenum="229"/>tatem illam datam in &longs;ubduplicata ratione, quam habet vis Gravi­<lb/><arrow.to.target n="note205"/>tatis ad Medii re&longs;i&longs;tentiam illam cognitam. </s></p> <p type="margin"> <s><margin.target id="note205"/>LIBER <lb/>SECUMDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO IX. THEOREMA VII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;itis jam demon&longs;tratis, dico quod &longs;i Tangentes angulorum &longs;ecto <lb/>ris Circularis & &longs;ectoris Hyperbolici &longs;umantur velocitatibus <lb/>proportionales, exi&longs;tente radio ju&longs;tæ magnitudinis: erit tempus <lb/>omne a&longs;cen&longs;us futuri ut &longs;ector Circuli, & tempus omne de&longs;cen­<lb/>&longs;us præteriti ut &longs;ector Hyperbolæ.<emph.end type="italics"/></s></p> <p type="main"> <s>Rectæ <emph type="italics"/>AC,<emph.end type="italics"/>qua vis gravitatis exponitur, perpendicularis & æ­<lb/>qualis ducatur <emph type="italics"/>AD.<emph.end type="italics"/>Centro <emph type="italics"/>D<emph.end type="italics"/>&longs;emidiametro <emph type="italics"/>AD<emph.end type="italics"/>de&longs;cribatur tum <lb/>Circuli quadrans <emph type="italics"/>AtE,<emph.end type="italics"/>tum Hyperbola rectangula <emph type="italics"/>AVZ<emph.end type="italics"/>axem <lb/>habens <emph type="italics"/>AX,<emph.end type="italics"/>verticem principalem <emph type="italics"/>A<emph.end type="italics"/>& A&longs;ymptoton <emph type="italics"/>DC.<emph.end type="italics"/>Jun­<lb/>gantur <emph type="italics"/>Dp, DP,<emph.end type="italics"/>& erit &longs;ector Circularis <emph type="italics"/>AtD<emph.end type="italics"/>ut tempus a&longs;cen&longs;us <lb/>omnis futuri; & &longs;ector Hyperbolicus <emph type="italics"/>ATD<emph.end type="italics"/>ut tempus de&longs;cen&longs;us <lb/>omnis præteriti. </s> <s>Si modo &longs;ectorum Tangentes <emph type="italics"/>Ap, AP<emph.end type="italics"/>&longs;int ut <lb/>velocitates. </s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Agatur enim <emph type="italics"/>Dvq<emph.end type="italics"/>ab&longs;cindens &longs;ectoris <emph type="italics"/>ADt<emph.end type="italics"/>& trian­<lb/>guli <emph type="italics"/>ADp<emph.end type="italics"/>momenta, &longs;eu particulas quam minimas &longs;imul de&longs;crip­<lb/>tas <emph type="italics"/>tDv<emph.end type="italics"/>& <emph type="italics"/><expan abbr="pDq.">pDque</expan><emph.end type="italics"/>Cum particulæ illæ, ob angulum commu­<lb/>nem <emph type="italics"/>D,<emph.end type="italics"/>&longs;unt in duplicata ratione laterum, erit particula <emph type="italics"/>tDv<emph.end type="italics"/><lb/>ut (<emph type="italics"/>qDp/pDquad<emph.end type="italics"/>). Sed <emph type="italics"/>pDquad.<emph.end type="italics"/>e&longs;t <emph type="italics"/>ADquad+Apquad.<emph.end type="italics"/>id e&longs;t, <lb/><emph type="italics"/>ADquad+ADXAk<emph.end type="italics"/>&longs;eu <emph type="italics"/>ADXCk<emph.end type="italics"/>; & <emph type="italics"/>qDp<emph.end type="italics"/>e&longs;t 1/2 <emph type="italics"/><expan abbr="ADXpq.">ADXpque</expan><emph.end type="italics"/><lb/>Ergo &longs;ectoris particula <emph type="italics"/>tDv<emph.end type="italics"/>e&longs;t ut (<emph type="italics"/>pq/Ck<emph.end type="italics"/>), id e&longs;t, ut velocitatis de­<lb/>crementum quam minimum <emph type="italics"/>pq<emph.end type="italics"/>directe & vis illa <emph type="italics"/>Ck<emph.end type="italics"/>quæ velo­<lb/>citatem diminuit inver&longs;e, atque adeo ut particula temporis decre­<lb/>mento re&longs;pondens. </s> <s>Et componendo fit &longs;umma particularum om­<lb/>nium <emph type="italics"/>tDv<emph.end type="italics"/>in &longs;ectore <emph type="italics"/>ADt,<emph.end type="italics"/>ut &longs;umma particularum temporis <lb/>&longs;ingulis velocitatis decre&longs;centis <emph type="italics"/>Ap<emph.end type="italics"/>particulis ami&longs;&longs;is <emph type="italics"/>pq<emph.end type="italics"/>re&longs;pon­<lb/>dentium, u&longs;Q.E.D.m velocitas illa in nihilum diminuta eva­<lb/>nuerit; hoc e&longs;t, &longs;ector totus <emph type="italics"/>ADt<emph.end type="italics"/>e&longs;t ut a&longs;cen&longs;us totius futuri <lb/>tempus. <emph type="italics"/>Q.E.D.<emph.end type="italics"/><pb xlink:href="039/01/258.jpg" pagenum="230"/><arrow.to.target n="note206"/></s></p> <p type="margin"> <s><margin.target id="note206"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Agatur <emph type="italics"/>DQV<emph.end type="italics"/>ab&longs;cindens tum &longs;ectoris <emph type="italics"/>DAV,<emph.end type="italics"/>tum tri­<lb/>anguli <emph type="italics"/>DAQ<emph.end type="italics"/>particulas quam minimas <emph type="italics"/>TDV<emph.end type="italics"/>& <emph type="italics"/>PDQ<emph.end type="italics"/>; & e­<lb/>runt hæ particulæ ad invicem ut <emph type="italics"/>DTQ<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="DPq.">DPque</expan><emph.end type="italics"/>id e&longs;t (&longs;i <emph type="italics"/>TX<emph.end type="italics"/><lb/>& <emph type="italics"/>AP<emph.end type="italics"/>parallelæ &longs;int) ut <emph type="italics"/><expan abbr="DXq.">DXque</expan><emph.end type="italics"/>ad <emph type="italics"/><expan abbr="DAq.">DAque</expan><emph.end type="italics"/>vel <emph type="italics"/><expan abbr="TXq.">TXque</expan><emph.end type="italics"/>ad <emph type="italics"/><expan abbr="APq.">APque</expan><emph.end type="italics"/>& <lb/>divi&longs;im ut <emph type="italics"/>DXq-TXq<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="DAq-APq.">DAq-APque</expan><emph.end type="italics"/>Sed ex natura <lb/>Hyperbolæ <emph type="italics"/>DXq-TXq<emph.end type="italics"/>e&longs;t <emph type="italics"/>ADq,<emph.end type="italics"/>& per Hypothe&longs;in <emph type="italics"/>APq<emph.end type="italics"/><lb/>e&longs;t <emph type="italics"/>ADXAK.<emph.end type="italics"/>Ergo particulæ &longs;unt ad invicem ut <emph type="italics"/>ADq<emph.end type="italics"/>ad <lb/><figure id="id.039.01.258.1.jpg" xlink:href="039/01/258/1.jpg"/><lb/><emph type="italics"/>ADq-ADXAK<emph.end type="italics"/>; id e&longs;t, ut <emph type="italics"/>AD<emph.end type="italics"/>ad <emph type="italics"/>AD-AK<emph.end type="italics"/>&longs;eu <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>CK:<emph.end type="italics"/><lb/>ideoque &longs;ectoris particula <emph type="italics"/>TDV<emph.end type="italics"/>e&longs;t (<emph type="italics"/>PDQXAC/CK<emph.end type="italics"/>), atque adeo ob <lb/>datas <emph type="italics"/>AC<emph.end type="italics"/>& <emph type="italics"/>AD,<emph.end type="italics"/>ut (<emph type="italics"/>PQ/CK<emph.end type="italics"/>), id e&longs;t, ut incrementum velocitatis <lb/>directe utque vis generans incrementum inver&longs;e, atque adeo ut par­<lb/>ticula temporis incremento re&longs;pondens. </s> <s>Et componendo &longs;it &longs;um <lb/>ma particularum temporis, quibus omnes velocitatis <emph type="italics"/>AP<emph.end type="italics"/>particulæ <pb xlink:href="039/01/259.jpg" pagenum="231"/><emph type="italics"/>PQ<emph.end type="italics"/>generantur, ut &longs;umma particularum &longs;ectoris <emph type="italics"/>ATD,<emph.end type="italics"/>id e&longs;t, </s></p> <p type="main"> <s><arrow.to.target n="note207"/>tempus totum ut &longs;ector totus. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note207"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i <emph type="italics"/>AB<emph.end type="italics"/>æquetur quartæ parti ip&longs;ius <emph type="italics"/>AC,<emph.end type="italics"/>&longs;patium <lb/>quod corpus tempore quovis cadendo de&longs;cribit, erit ad &longs;patium <lb/>quod corpus velocitate maxima <emph type="italics"/>AC,<emph.end type="italics"/>eodem tempore uniformiter <lb/>progrediendo de&longs;cribere pote&longs;t, ut area <emph type="italics"/>ABNK,<emph.end type="italics"/>qua &longs;patium <lb/>cadendo de&longs;criptum exponitur, ad aream <emph type="italics"/>ATD<emph.end type="italics"/>qua tempus ex­<lb/>ponitur. </s> <s>Nam cum &longs;it <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>AP<emph.end type="italics"/>ut <emph type="italics"/>AP<emph.end type="italics"/>ad <emph type="italics"/>AK,<emph.end type="italics"/>erit (per <lb/>Corol. </s> <s>1, Lem. </s> <s>11 hujus) <emph type="italics"/>LK<emph.end type="italics"/>ad <emph type="italics"/>PQ<emph.end type="italics"/>ut 2<emph type="italics"/>AK<emph.end type="italics"/>ad <emph type="italics"/>AP,<emph.end type="italics"/>hoc e&longs;t, <lb/>ut 2<emph type="italics"/>AP<emph.end type="italics"/>ad <emph type="italics"/>AC,<emph.end type="italics"/>& inde <emph type="italics"/>LK<emph.end type="italics"/>ad 1/2<emph type="italics"/>PQ<emph.end type="italics"/>ut <emph type="italics"/>AP<emph.end type="italics"/>ad (1/4<emph type="italics"/>AC<emph.end type="italics"/>vel) <lb/><emph type="italics"/>AB<emph.end type="italics"/>; e&longs;t & <emph type="italics"/>KN<emph.end type="italics"/>ad (<emph type="italics"/>AC<emph.end type="italics"/>vel) <emph type="italics"/>AD<emph.end type="italics"/>ut <emph type="italics"/>AB<emph.end type="italics"/>ad <emph type="italics"/>CK<emph.end type="italics"/>; itaque ex <lb/>æquo <emph type="italics"/>LKN<emph.end type="italics"/>ad <emph type="italics"/>DPQ<emph.end type="italics"/>ut <emph type="italics"/>AP<emph.end type="italics"/>ad <emph type="italics"/>CK.<emph.end type="italics"/>Sed erat <emph type="italics"/>DPQ<emph.end type="italics"/>ad <lb/><emph type="italics"/>DTV<emph.end type="italics"/>ut <emph type="italics"/>CK<emph.end type="italics"/>ad <emph type="italics"/>AC.<emph.end type="italics"/>Ergo rur&longs;us ex æquo <emph type="italics"/>LKN<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>DTV<emph.end type="italics"/><lb/>ut <emph type="italics"/>AP<emph.end type="italics"/>ad <emph type="italics"/>AC<emph.end type="italics"/>; hoc e&longs;t, ut velocitas corporis cadentis ad veloci­<lb/>tatem maximam quam corpus cadendo pote&longs;t acquirere. </s> <s>Cum <lb/>igitur arearum <emph type="italics"/>ABNK<emph.end type="italics"/>& <emph type="italics"/>ATD<emph.end type="italics"/>momenta <emph type="italics"/>LKN<emph.end type="italics"/>& <emph type="italics"/>DTV<emph.end type="italics"/><lb/>&longs;unt ut velocitates, erunt arearum illarum partes omnes &longs;imul <lb/>genitæ ut &longs;patia &longs;imul de&longs;cripta, ideoque areæ totæ ab initio <lb/>genitæ <emph type="italics"/>ABNK<emph.end type="italics"/>& <emph type="italics"/>ATD<emph.end type="italics"/>ut &longs;patia tota ab initio de&longs;cen&longs;us de­<lb/>&longs;cripta. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Idem con&longs;equitur etiam de &longs;patio quod in a&longs;cen&longs;u de­<lb/>&longs;cribitur. </s> <s>Nimirum quod &longs;patium illud omne &longs;it ad &longs;patium, uNI­<lb/>formi cum velocitate <emph type="italics"/>AC<emph.end type="italics"/>eodem tempore de&longs;criptum, ut e&longs;t area <lb/><emph type="italics"/>ABnk<emph.end type="italics"/>ad &longs;ectorem <emph type="italics"/>ADt.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Velocitas corporis tempore <emph type="italics"/>ATD<emph.end type="italics"/>cadentis e&longs;t ad ve­<lb/>locitatem, quam eodem tempore in &longs;patio non re&longs;i&longs;tente acquire­<lb/>ret, ut triangulum <emph type="italics"/>APD<emph.end type="italics"/>ad &longs;ectorem Hyperbolicum <emph type="italics"/>ATD.<emph.end type="italics"/><lb/>Nam velocitas in Medio non re&longs;i&longs;tente foret ut tempus <emph type="italics"/>ATD,<emph.end type="italics"/>& <lb/>in Medio re&longs;i&longs;tente e&longs;t ut <emph type="italics"/>AP,<emph.end type="italics"/>id e&longs;t, ut triangulum <emph type="italics"/>APD.<emph.end type="italics"/>Et <lb/>velocitates illæ initio de&longs;cen&longs;us æquantur inter &longs;e, perinde ut areæ <lb/>illæ <emph type="italics"/>ATD, APD.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Eodem argumento velocitas in a&longs;cen&longs;u e&longs;t ad velocita­<lb/>tem, qua corpus eodem tempore in &longs;patio non re&longs;i&longs;tente omnem <lb/>&longs;uum a&longs;cendendi motum amittere po&longs;&longs;et, ut triangulum <emph type="italics"/>ApD<emph.end type="italics"/>ad <lb/>&longs;ectorem Circularem <emph type="italics"/>AtD<emph.end type="italics"/>; &longs;ive ut recta <emph type="italics"/>Ap<emph.end type="italics"/>ad arcum <emph type="italics"/>At.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. E&longs;t igitur tempus quo corpus in Medio re&longs;i&longs;tente caden­<lb/>do velocitatem <emph type="italics"/>AP<emph.end type="italics"/>acquirit, ad tempus quo velocitatem maximam <lb/><emph type="italics"/>AC<emph.end type="italics"/>in &longs;patio non re&longs;i&longs;tente cadendo acquirere po&longs;&longs;et, ut &longs;ector <lb/><emph type="italics"/>ADT<emph.end type="italics"/>ad triangulum <emph type="italics"/>ADC<emph.end type="italics"/>: & tempus, quo velocitatem <emph type="italics"/>Ap<emph.end type="italics"/>in <pb xlink:href="039/01/260.jpg" pagenum="232"/><arrow.to.target n="note208"/>Medio re&longs;i&longs;tente a&longs;cendendo po&longs;&longs;it amittere, ad tempus quo velo­<lb/>citatem eandem in &longs;patio non re&longs;i&longs;tente a&longs;cendendo po&longs;&longs;et amit­<lb/>tere, ut arcus <emph type="italics"/>At<emph.end type="italics"/>ad ejus tangentem <emph type="italics"/>Ap.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note208"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Hinc ex dato tempore datur &longs;patium a&longs;cen&longs;u vel de­<lb/>&longs;cen&longs;u de&longs;criptum. </s> <s>Nam corporis in infinitum de&longs;cendentis datur <lb/>velocitas maxima, per Corol. </s> <s>2, & 3, Theor. </s> <s>VI, Lib. </s> <s>11; indeque <lb/>datur tempus quo corpus velocitatem illam in &longs;patio non re&longs;i&longs;tente <lb/>cadendo po&longs;&longs;et acquirere. </s> <s>Et &longs;umendo Sectorem <emph type="italics"/>ADT<emph.end type="italics"/>vel <emph type="italics"/>ADt<emph.end type="italics"/><lb/>ad triangulum <emph type="italics"/>ADC<emph.end type="italics"/>in ratione temporis dati ad tempus modo <lb/>inventum; dabitur tum velocitas <emph type="italics"/>AP<emph.end type="italics"/>vel <emph type="italics"/>Ap,<emph.end type="italics"/>tum area <emph type="italics"/>ABNK<emph.end type="italics"/><lb/>vel <emph type="italics"/>ABnk,<emph.end type="italics"/>quæ e&longs;t ad &longs;ectorem <emph type="italics"/>ADT<emph.end type="italics"/>vel <emph type="italics"/>ADt<emph.end type="italics"/>ut &longs;patium quæ­<lb/>&longs;itum ad &longs;patium quod tempore dato, cum velocitate illa maxima <lb/>jam ante inventa, uniformiter de&longs;cribi pote&longs;t. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. Et regrediendo, ex dato a&longs;cen&longs;us vel de&longs;cen&longs;us &longs;patio <lb/><emph type="italics"/>ABnk<emph.end type="italics"/>vel <emph type="italics"/>ABNK,<emph.end type="italics"/>dabitur tempus <emph type="italics"/>ADt<emph.end type="italics"/>vel <emph type="italics"/>ADT.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO X. PROBLEMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Tendat uniformis vis gravitatis directe ad planum Horizontis, <lb/>&longs;itque re&longs;i&longs;tentia ut Medii den&longs;itas & quadratum velocitatis <lb/>conjunctim: requiritur tum Medii den&longs;itas in locis &longs;ingulis, <lb/>quæ faciat ut corpus in data quavis linea curva moveatur, <lb/>tum corporis velocitas & Medii re&longs;i&longs;tentia in locis &longs;ingulis.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>PQ<emph.end type="italics"/>planum illud pla­<lb/><figure id="id.039.01.260.1.jpg" xlink:href="039/01/260/1.jpg"/><lb/>no Schematis perpendicu­<lb/>lare; <emph type="italics"/>PFHQ<emph.end type="italics"/>linea curva <lb/>plano huic occurrens in <lb/>punctis <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/><expan abbr="q;">que</expan> G, H, I, K<emph.end type="italics"/><lb/>loca quatuor corporis in hac <lb/>curva ab <emph type="italics"/>F<emph.end type="italics"/>ad <emph type="italics"/>Q<emph.end type="italics"/>pergentis; <lb/>& <emph type="italics"/>GB, HC, ID, KE<emph.end type="italics"/>or­<lb/>dinatæ quatuor parallelæ ab <lb/>his punctis ad horizontem <lb/>demi&longs;&longs;æ & lineæ horizontali <emph type="italics"/>PQ<emph.end type="italics"/>ad puncta <emph type="italics"/>B, C, D, E<emph.end type="italics"/>in&longs;i&longs;ten­<lb/>tes; & &longs;int <emph type="italics"/>BC, CD, DE<emph.end type="italics"/>di&longs;tantiæ Ordinatarum inter &longs;e æqua­<lb/>les. </s> <s>A punctis <emph type="italics"/>G<emph.end type="italics"/>& <emph type="italics"/>H<emph.end type="italics"/>ducantur rectæ <emph type="italics"/>GL, HN<emph.end type="italics"/>curvam tan­<lb/>gentes in <emph type="italics"/>G<emph.end type="italics"/>& <emph type="italics"/>H,<emph.end type="italics"/>& Ordinatis <emph type="italics"/>CH, DI<emph.end type="italics"/>&longs;ur&longs;um productis occur­<lb/>rentes in <emph type="italics"/>L<emph.end type="italics"/>& <emph type="italics"/>N,<emph.end type="italics"/>& compleatur parallelogrammum <emph type="italics"/>HCDM.<emph.end type="italics"/><pb xlink:href="039/01/261.jpg" pagenum="233"/>Et tempora quibus corpus de&longs;cribit arcus <emph type="italics"/>GH, HI,<emph.end type="italics"/>erunt in <lb/><arrow.to.target n="note209"/>&longs;ubduplicata ratione altitudinum <emph type="italics"/>LH, NI<emph.end type="italics"/>quas corpus tempo­<lb/>ribus illis de&longs;cribere po&longs;&longs;et, a tangentibus cadendo: & velocitates <lb/>erunt ut longitudines de&longs;criptæ <emph type="italics"/>GH, HI<emph.end type="italics"/>directe & tempora in­<lb/>ver&longs;e. </s> <s>Exponantur tempora per T & <emph type="italics"/>t,<emph.end type="italics"/>& velocitates per <lb/>(<emph type="italics"/>GH<emph.end type="italics"/>/T) & (<emph type="italics"/>HI/t<emph.end type="italics"/>): & decrementum velocitatis tempore <emph type="italics"/>t<emph.end type="italics"/>factum ex­<lb/>ponetur per (<emph type="italics"/>GH<emph.end type="italics"/>/T)-(<emph type="italics"/>HI/t<emph.end type="italics"/>). Hoc decrementum oritur a re&longs;i&longs;tentia <lb/>corpus retardante & gravitate corpus accelerante. </s> <s>Gravitas in <lb/>corpore cadente & &longs;patium <emph type="italics"/>NI<emph.end type="italics"/>cadendo de&longs;cribente, generat ve­<lb/>locitatem qua duplum illud &longs;patium eodem tempore de&longs;cribi po­<lb/>tui&longs;&longs;et (ut <emph type="italics"/>Galilæus<emph.end type="italics"/>demon&longs;travit) id e&longs;t, velocitatem (2<emph type="italics"/>NI/t<emph.end type="italics"/>): at <lb/>in corpore arcum <emph type="italics"/>HI<emph.end type="italics"/>de&longs;cribente, auget arcum illum &longs;ola longi­<lb/>tudine <emph type="italics"/>HI-HN<emph.end type="italics"/>&longs;eu (<emph type="italics"/>MIXNI/HI<emph.end type="italics"/>), ideoque generat tantum velo­<lb/>citatem (2<emph type="italics"/>MIXNI/tXHI<emph.end type="italics"/>). Addatur hæc velocitas ad decrementum <lb/>prædictum, & habebitur decrementum velocitatis ex re&longs;i&longs;tentia <lb/>&longs;ola oriundum, nempe (<emph type="italics"/>GH<emph.end type="italics"/>/T)-<emph type="italics"/>(HI/t)+(2MIXNI/tXHI).<emph.end type="italics"/>Proindeque <lb/>cum gravitas eodem tempore in corpore cadente generet velocitatem <lb/>(2<emph type="italics"/>NI/t<emph.end type="italics"/>); Re&longs;i&longs;tentia erit ad Gravitatem ut (<emph type="italics"/>GH<emph.end type="italics"/>/T)-<emph type="italics"/>(HI/t)+(2MIXNI/tXHI)<emph.end type="italics"/><lb/>ad (<emph type="italics"/>2NI/t<emph.end type="italics"/>), &longs;ive ut (<emph type="italics"/>tXGH<emph.end type="italics"/>/T)-<emph type="italics"/>HI+(2MIXNI/HI)<emph.end type="italics"/>ad 2<emph type="italics"/>NI.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note209"/>LIBER <lb/>SECUNDUS</s></p> <p type="main"> <s>Jam pro ab&longs;ci&longs;&longs;is <emph type="italics"/>CB, CD, CE<emph.end type="italics"/>&longs;cribantur -<emph type="italics"/>o, o,<emph.end type="italics"/>20. Pro <lb/>Ordinata <emph type="italics"/>CH<emph.end type="italics"/>&longs;cribatur P, & pro <emph type="italics"/>MI<emph.end type="italics"/>&longs;cribatur &longs;eries quælibet <lb/>Q<emph type="italics"/>o<emph.end type="italics"/>+R<emph type="italics"/>oo<emph.end type="italics"/>+S<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/>+&c. </s> <s>Et &longs;eriei termini omnes po&longs;t primum, <lb/>nempe R<emph type="italics"/>oo<emph.end type="italics"/>+S<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/>+&c. </s> <s>erunt <emph type="italics"/>NI,<emph.end type="italics"/>& Ordinatæ <emph type="italics"/>DI, EK,<emph.end type="italics"/>& <emph type="italics"/>BG<emph.end type="italics"/><lb/>erunt P-Q<emph type="italics"/>o<emph.end type="italics"/>-R<emph type="italics"/>oo<emph.end type="italics"/>-S<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/>-&c, P-2Q<emph type="italics"/>o<emph.end type="italics"/>-4R<emph type="italics"/>oo<emph.end type="italics"/>-8S<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/>-&c, <lb/>& P+Q<emph type="italics"/>o<emph.end type="italics"/>-R<emph type="italics"/>oo<emph.end type="italics"/>+S<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/>-&c. </s> <s>re&longs;pective. </s> <s>Et quadrando diffe­<lb/>rentias Ordinatarum <emph type="italics"/>BG-CH<emph.end type="italics"/>& <emph type="italics"/>CH-DI,<emph.end type="italics"/>& ad quadrata pro­<lb/>deuntia addendo quadrata ip&longs;arum <emph type="italics"/>BC, CD,<emph.end type="italics"/>habebuntur arcuum <lb/><emph type="italics"/>GH, HI<emph.end type="italics"/>quadrata <emph type="italics"/>oo<emph.end type="italics"/>+QQ<emph type="italics"/>oo<emph.end type="italics"/>+2QR<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/>+&c, & <emph type="italics"/>oo<emph.end type="italics"/>+QQ<emph type="italics"/>oo<emph.end type="italics"/><lb/>+2QR<emph type="italics"/>o<emph.end type="italics"/>+&c. </s> <s>Quorum radices <emph type="italics"/>o<emph.end type="italics"/>√1+QQ-(QR<emph type="italics"/>oo<emph.end type="italics"/>/√1+QQ), & <pb xlink:href="039/01/262.jpg" pagenum="234"/><arrow.to.target n="note210"/><emph type="italics"/>o<emph.end type="italics"/>√1+QQ+(QR<emph type="italics"/>oo<emph.end type="italics"/>/√1+QQ) &longs;unt arcus <emph type="italics"/>GH<emph.end type="italics"/>& <emph type="italics"/>HI.<emph.end type="italics"/>Præterea &longs;i ab <lb/>Ordinata <emph type="italics"/>CH<emph.end type="italics"/>&longs;ubducatur &longs;emi&longs;umma Ordinatarum <emph type="italics"/>BG<emph.end type="italics"/>ac <emph type="italics"/>DI,<emph.end type="italics"/><lb/>& ab Ordinata <emph type="italics"/>DI<emph.end type="italics"/>&longs;ubducatur &longs;emi&longs;umma Ordinatarum <emph type="italics"/>CH<emph.end type="italics"/>& <lb/><emph type="italics"/>EK,<emph.end type="italics"/>manebunt arcuum <emph type="italics"/>GI<emph.end type="italics"/>& <emph type="italics"/>HK<emph.end type="italics"/>&longs;agittæ R<emph type="italics"/>oo<emph.end type="italics"/>& R<emph type="italics"/>oo<emph.end type="italics"/>+3S<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/>. </s> <s><lb/>Et hæ &longs;unt lineolis <emph type="italics"/>LH<emph.end type="italics"/>& <emph type="italics"/>NI<emph.end type="italics"/>proportionales, adeoQ.E.I. du­<lb/>plicata ratione temporum infinite parvorum T & <emph type="italics"/>t,<emph.end type="italics"/>& inde ratio <lb/><emph type="italics"/>t<emph.end type="italics"/>/T e&longs;t √(R+3S<emph type="italics"/>o<emph.end type="italics"/>/R) &longs;eu (R+3/2S<emph type="italics"/>o<emph.end type="italics"/>/R): & (<emph type="italics"/>tXGH<emph.end type="italics"/>/T)-<emph type="italics"/>HI+(2MIXNI/HI),<emph.end type="italics"/><lb/>&longs;ub&longs;tituendo ip&longs;orum <emph type="italics"/>t<emph.end type="italics"/>/T, <emph type="italics"/>GH, HI, MI<emph.end type="italics"/>& <emph type="italics"/>NI<emph.end type="italics"/>valores jam in­<lb/>ventos, evadit (3S<emph type="italics"/>oo<emph.end type="italics"/>/2R)√1+Qq. </s> <s>Et cum 2<emph type="italics"/>NI<emph.end type="italics"/>&longs;it 2R<emph type="italics"/>oo,<emph.end type="italics"/>Re­<lb/>&longs;i&longs;tentia jam erit ad Gravitatem ut (3S<emph type="italics"/>oo<emph.end type="italics"/>/2R)√1+QQ ad 2R<emph type="italics"/>oo,<emph.end type="italics"/><lb/>id e&longs;t, ut 3S√1+QQ ad 4RR. </s></p> <p type="margin"> <s><margin.target id="note210"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Velocitas autem ea e&longs;t quacum corpus de loco quovis <emph type="italics"/>H,<emph.end type="italics"/>&longs;e­<lb/>cundum tangentem <emph type="italics"/>HN<emph.end type="italics"/>egrediens, in Parabola diametrum <emph type="italics"/>HC<emph.end type="italics"/><lb/>& latus rectum (<emph type="italics"/>HNq/NI<emph.end type="italics"/>) &longs;eu (1+QQ/R) habente, deinceps in vacuo <lb/>moveri pote&longs;t. </s></p> <p type="main"> <s>Et re&longs;i&longs;tentia e&longs;t ut Medii den&longs;itas & quadratum velocitatis <lb/>conjunctim, & propterea Medii den&longs;itas e&longs;t ut re&longs;i&longs;tentia directe <lb/>& quadratum velocitatis inver&longs;e, id e&longs;t, ut (3S√1+QQ/4RR) directe <lb/>& (1+QQ/R) inver&longs;e, hoc e&longs;t, ut (S/R√1+QQ). <emph type="italics"/>q.EI.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Si tangens <emph type="italics"/>HN<emph.end type="italics"/>producatur utrinQ.E.D.nec occurrat <lb/>Ordinatæ cuilibet <emph type="italics"/>AF<emph.end type="italics"/>in <emph type="italics"/>T<emph.end type="italics"/>: erit (<emph type="italics"/>HT/AC<emph.end type="italics"/>) æqualis √1+QQ, adeo­<lb/>Q.E.I. &longs;uperioribus pro √1+QQ &longs;cribi pote&longs;t. </s> <s>Qua ratione <lb/>Re&longs;i&longs;tentia erit ad Gravitatem ut 3SX<emph type="italics"/>HT<emph.end type="italics"/>ad 4RRX<emph type="italics"/>AC,<emph.end type="italics"/>Velo­<lb/>citas erit ut (<emph type="italics"/>HT/AC<emph.end type="italics"/>√R), & Medii den&longs;itas erit ut (SX<emph type="italics"/>AC<emph.end type="italics"/>/RX<emph type="italics"/>HT<emph.end type="italics"/>). </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et hinc, &longs;i Curva linea <emph type="italics"/>PFHQ<emph.end type="italics"/>definiatur per rela­<lb/>tionem inter ba&longs;em &longs;eu ab&longs;ci&longs;&longs;am <emph type="italics"/>AC<emph.end type="italics"/>& ordinatim applicatam <pb xlink:href="039/01/263.jpg" pagenum="235"/><emph type="italics"/>CH,<emph.end type="italics"/>(ut moris e&longs;t) & valor ordinatim applicatæ re&longs;olvatur in &longs;e­<lb/><arrow.to.target n="note211"/>riem convergentem: Problema per primos &longs;eriei terminos expe­<lb/>dite &longs;olvetur, ut in exemplis &longs;equentibus. </s></p> <p type="margin"> <s><margin.target id="note211"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/>1. Sit Linea <emph type="italics"/>PFHQ<emph.end type="italics"/>Semicirculus &longs;uper diametro <emph type="italics"/>PQ<emph.end type="italics"/><lb/>de&longs;criptus, & requiratur Medii den&longs;itas quæ faciat ut Projectile <lb/>in hac linea moveatur. </s></p> <p type="main"> <s>Bi&longs;ecetur diameter <emph type="italics"/>PQ<emph.end type="italics"/>in <emph type="italics"/>A,<emph.end type="italics"/>dic <emph type="italics"/>AQ n, AC a, CH e,<emph.end type="italics"/>& <lb/><emph type="italics"/>CD o<emph.end type="italics"/>: & erit <emph type="italics"/>DIq<emph.end type="italics"/>&longs;eu <emph type="italics"/>AQq-ADq=nn-aa-2ao-oo,<emph.end type="italics"/>&longs;eu <lb/><emph type="italics"/>ee-2ao-oo,<emph.end type="italics"/>& radice per methodum no&longs;tram extracta, fiet <lb/><emph type="italics"/>DI=e-(ao/e)-(oo/2e)-(aaoo/2e<emph type="sup"/>3<emph.end type="sup"/>)-(ao<emph type="sup"/>3<emph.end type="sup"/>/2e<emph type="sup"/>3<emph.end type="sup"/>)-(a<emph type="sup"/>3<emph.end type="sup"/>o<emph type="sup"/>3<emph.end type="sup"/>/2e<emph type="sup"/>3<emph.end type="sup"/>)<emph.end type="italics"/>-&c. </s> <s>Hic &longs;cribatur <emph type="italics"/>nn<emph.end type="italics"/><lb/>pro <emph type="italics"/>ee+aa,<emph.end type="italics"/>& evadet <emph type="italics"/>DI=e-(ao/e)-(nnoo/2e<emph type="sup"/>3<emph.end type="sup"/>)-(anno<emph type="sup"/>3<emph.end type="sup"/>/2e<emph type="sup"/>3<emph.end type="sup"/>)<emph.end type="italics"/>-&c. </s></p> <p type="main"> <s>Huju&longs;modi &longs;eries di&longs;tinguo in terminos &longs;ucce&longs;&longs;ivos in hunc mo­<lb/>dum. </s> <s>Terminum primum appello in quo quantitas infinite par­<lb/>va <emph type="italics"/>o<emph.end type="italics"/>non extat; &longs;ecundum in quo quantitas illa e&longs;t unius dimen­<lb/>&longs;ionis, tertium in quo extat <lb/><figure id="id.039.01.263.1.jpg" xlink:href="039/01/263/1.jpg"/><lb/>duarum, quartum in quo <lb/>trium e&longs;t, & &longs;ic in infiNI­<lb/>tum. </s> <s>Et primus terminus <lb/>qui hic e&longs;t <emph type="italics"/>e,<emph.end type="italics"/>denotabit &longs;em­<lb/>per longitudinem Ordinatæ <lb/><emph type="italics"/>CH<emph.end type="italics"/>in&longs;i&longs;tentis ad initium <lb/>indefinitæ quantitatis <emph type="italics"/>o<emph.end type="italics"/>; &longs;e­<lb/>cundus terminus qui hic e&longs;t <lb/>(<emph type="italics"/>ao/e<emph.end type="italics"/>), denotabit differentiam <lb/>inter <emph type="italics"/>CH<emph.end type="italics"/>& <emph type="italics"/>DN,<emph.end type="italics"/>id e&longs;t, lineolam <emph type="italics"/>MN<emph.end type="italics"/>quæ ab&longs;cinditur com­<lb/>plendo parallelogrammum <emph type="italics"/>HCDM,<emph.end type="italics"/>atque adeo po&longs;itionem tan­<lb/>gentis <emph type="italics"/>HN<emph.end type="italics"/>&longs;emper determinat: ut in hoc ca&longs;u capiendo <emph type="italics"/>MN<emph.end type="italics"/>ad <lb/><emph type="italics"/>HM<emph.end type="italics"/>ut e&longs;t (<emph type="italics"/>ao/e<emph.end type="italics"/>) ad <emph type="italics"/>o,<emph.end type="italics"/>&longs;eu <emph type="italics"/>a<emph.end type="italics"/>ad <emph type="italics"/>e.<emph.end type="italics"/>Terminus tertius qui hic e&longs;t <lb/>(<emph type="italics"/>nnoo/2e<emph type="sup"/>3<emph.end type="sup"/><emph.end type="italics"/>) de&longs;ignabit lineolam <emph type="italics"/>IN<emph.end type="italics"/>quæ jacet inter tangentem & cur­<lb/>vam, adeoQ.E.D.terminat angulum contactus <emph type="italics"/>IHN<emph.end type="italics"/>&longs;eu curvatu­<lb/>ram quam curva linea habet in <emph type="italics"/>H.<emph.end type="italics"/>Si lineola illa <emph type="italics"/>IN<emph.end type="italics"/>finitæ e&longs;t <lb/>magnitudinis, de&longs;ignabitur per terminum tertium una cum &longs;e­<lb/>quentibus in infinitum. </s> <s>At &longs;i lineola illa minuatur in infinitum, <pb xlink:href="039/01/264.jpg" pagenum="236"/><arrow.to.target n="note212"/>termini &longs;ub&longs;equentes evadent infinite minores tertio, ideoque neg­<lb/>ligi po&longs;&longs;unt. </s> <s>Terminus quartus determinat variationem curva­<lb/>turæ, quintus variationem variationis, & &longs;ic deinceps. </s> <s>Unde obi­<lb/>ter patet u&longs;us non contemnendus harum Serierum in &longs;olutione <lb/>Problematum quæ pendent a tangentibus & curvatura curvarum. </s></p> <p type="margin"> <s><margin.target id="note212"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Conferatur jam &longs;eries <emph type="italics"/>e-(ao/e)-(nnoo/2e<emph type="sup"/>3<emph.end type="sup"/>)-(anno<emph type="sup"/>3<emph.end type="sup"/>/2e<emph type="sup"/>5<emph.end type="sup"/>)<emph.end type="italics"/>-&c, cum &longs;erie <lb/>P-Q<emph type="italics"/>o<emph.end type="italics"/>-R<emph type="italics"/>oo<emph.end type="italics"/>-S<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/>-&c. </s> <s>& perinde pro P, Q, R & S &longs;cribatur <lb/><emph type="italics"/>e, (a/e), (nn/2e<emph type="sup"/>3<emph.end type="sup"/>)<emph.end type="italics"/>& (<emph type="italics"/>ann/2e<emph type="sup"/>5<emph.end type="sup"/><emph.end type="italics"/>), & pro √1+QQ &longs;cribatur √1+(<emph type="italics"/>aa/ee<emph.end type="italics"/>) &longs;eu <emph type="italics"/>n/e,<emph.end type="italics"/>& <lb/>prodibit Medii den&longs;itas ut (<emph type="italics"/>a/ne<emph.end type="italics"/>), hoc e&longs;t, (ob datam <emph type="italics"/>n,<emph.end type="italics"/>) ut <emph type="italics"/>a/e,<emph.end type="italics"/>&longs;eu <lb/>(<emph type="italics"/>AC/CH<emph.end type="italics"/>), id e&longs;t, ut tangentis longitudo illa <emph type="italics"/>HT<emph.end type="italics"/>quæ ad &longs;emidiame­<lb/>trum <emph type="italics"/>AF<emph.end type="italics"/>ip&longs;i <emph type="italics"/>PQ<emph.end type="italics"/>normaliter in&longs;i&longs;tentem terminatur: & re&longs;i&longs;ten­<lb/>tia erit ad gravitatem ut 3<emph type="italics"/>a<emph.end type="italics"/>ad 2<emph type="italics"/>n,<emph.end type="italics"/>id e&longs;t, ut 3 <emph type="italics"/>AC<emph.end type="italics"/>ad Circuli <lb/>diametrum <emph type="italics"/>PQ<emph.end type="italics"/>: velocitas autem erit ut √<emph type="italics"/>CH.<emph.end type="italics"/>Quare &longs;i corpus <lb/>ju&longs;ta cum velocitate &longs;ecundum lineam ip&longs;i <emph type="italics"/>PQ<emph.end type="italics"/>parallelam exeat <lb/>de loco <emph type="italics"/>F,<emph.end type="italics"/>& Medii den&longs;itas in &longs;ingulis locis <emph type="italics"/>H<emph.end type="italics"/>&longs;it ut longi­<lb/>tudo tangentis <emph type="italics"/>HT,<emph.end type="italics"/>& re&longs;i&longs;tentia etiam in loco aliquo <emph type="italics"/>H<emph.end type="italics"/>&longs;it ad <lb/>vim gravitatis ut 3 <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>PQ,<emph.end type="italics"/>corpus illud de&longs;cribet Circuli <lb/>quadrantem <emph type="italics"/>FHQ. Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s>At &longs;i corpus idem de loco <emph type="italics"/>P,<emph.end type="italics"/>&longs;ecundum lineam ip&longs;i <emph type="italics"/>PQ<emph.end type="italics"/>per­<lb/>pendicularem egrederetur, & in arcu &longs;emicirculi <emph type="italics"/>PFQ<emph.end type="italics"/>moveri <lb/>inciperet, &longs;umenda e&longs;&longs;et <emph type="italics"/>AC<emph.end type="italics"/>&longs;eu <emph type="italics"/>a<emph.end type="italics"/>ad contrarias partes centri <emph type="italics"/>A,<emph.end type="italics"/><lb/>& propterea &longs;ignum ejus mutandum e&longs;&longs;et & &longs;cribendum -<emph type="italics"/>a<emph.end type="italics"/>pro <lb/>+<emph type="italics"/>a.<emph.end type="italics"/>Quo pacto prodiret Medii den&longs;itas ut -<emph type="italics"/>a/e<emph.end type="italics"/>. </s> <s>Negativam <lb/>autem den&longs;itatem, hoc e&longs;t, quæ motus corporum accelerat, Na­<lb/>tura non admittit: & propterea naturaliter fieri non pote&longs;t, ut <lb/>corpus a&longs;cendendo a <emph type="italics"/>P<emph.end type="italics"/>de&longs;cribat Circuli quadrantem <emph type="italics"/>PF.<emph.end type="italics"/>Ad <lb/>hunc effectum deberet corpus a Medio impellente accelerari, non <lb/>a re&longs;i&longs;tente impediri. </s></p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/>2. Sit linea <emph type="italics"/>PFHQ<emph.end type="italics"/>Parabola, axem habens <emph type="italics"/>AF<emph.end type="italics"/>ho­<lb/>rizonti <emph type="italics"/>PQ<emph.end type="italics"/>perpendicularem, & requiratur Medii den&longs;itas quæ <lb/>faciat ut Projectile in ip&longs;a moveatur. </s></p> <p type="main"> <s>Ex natura Parabolæ, rectangulum <emph type="italics"/>PDQ<emph.end type="italics"/>æquale e&longs;t rectan­<lb/>gulo &longs;ub ordinata <emph type="italics"/>DI<emph.end type="italics"/>& recta aliqua data: hoc e&longs;t, &longs;i dicantur <pb xlink:href="039/01/265.jpg" pagenum="237"/>recta illa <emph type="italics"/>b, PC a, PQ c, CH e<emph.end type="italics"/>& <emph type="italics"/>CD o<emph.end type="italics"/>; rectangulum <emph type="italics"/>a+o<emph.end type="italics"/><lb/><arrow.to.target n="note213"/>in <emph type="italics"/>c-a-o<emph.end type="italics"/>&longs;eu <emph type="italics"/>ac-aa-2ao+co-oo<emph.end type="italics"/>æquale e&longs;t rectangulo <lb/><emph type="italics"/>b<emph.end type="italics"/>in <emph type="italics"/>DI,<emph.end type="italics"/>adeoque <emph type="italics"/>DI<emph.end type="italics"/>æquale <emph type="italics"/>(ac-aa/b)+(c-2a/b)o-(oo/b).<emph.end type="italics"/>Jam &longs;cri­<lb/>bendus e&longs;&longs;et hujus &longs;eriei &longs;ecundus terminus <emph type="italics"/>(c-2a/b)o<emph.end type="italics"/>pro Q<emph type="italics"/>o,<emph.end type="italics"/>ter­<lb/>tius item terminus (<emph type="italics"/>oo/b<emph.end type="italics"/>) pro R<emph type="italics"/>oo.<emph.end type="italics"/>Cum vero plures non &longs;int ter­<lb/>mini, debebit quarti coefficiens S evane&longs;cere, & propterea quan­<lb/>titas (S/R√1+QQ) cui Medii den&longs;itas proportionalis e&longs;t, nihil <lb/>erit. </s> <s>Nulla igitur Medii den&longs;itate movebitur Projectile in Para­<lb/>bola, uti olim demon&longs;travit <emph type="italics"/>Galilæus, Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note213"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/>3. Sit linea <emph type="italics"/>AGK<emph.end type="italics"/>Hyperbola, A&longs;ymptoton habens <lb/><emph type="italics"/>NX<emph.end type="italics"/>plano horizontali <emph type="italics"/>AK<emph.end type="italics"/>perpendicularem; & quæratur Medii <lb/>den&longs;itas quæ faciat ut Projectile moveatur in hac linea. </s></p> <p type="main"> <s>Sit <emph type="italics"/>MX<emph.end type="italics"/>A&longs;ymptotos altera, ordinatim applicatæ <emph type="italics"/>DG<emph.end type="italics"/>productæ <lb/>occurrens in <emph type="italics"/>V,<emph.end type="italics"/>& ex natura Hyperbolæ, rectangulum <emph type="italics"/>XV<emph.end type="italics"/>in <emph type="italics"/>VG<emph.end type="italics"/><lb/>dabitur. </s> <s>Datur autem ratio <emph type="italics"/>DN<emph.end type="italics"/>ad <emph type="italics"/>VX,<emph.end type="italics"/>& propterea datur etiam <lb/>rectangulum <emph type="italics"/>DN<emph.end type="italics"/>in <emph type="italics"/>VG.<emph.end type="italics"/>Sit illud <emph type="italics"/>bb<emph.end type="italics"/>; & completo parallelogrammo <lb/><emph type="italics"/>DNXZ,<emph.end type="italics"/>dicatur <emph type="italics"/>BN a, BD o, NX c,<emph.end type="italics"/>& ratio data <emph type="italics"/>VZ<emph.end type="italics"/>ad <emph type="italics"/>ZX<emph.end type="italics"/><lb/>vel <emph type="italics"/>DN<emph.end type="italics"/>ponatur e&longs;&longs;e <emph type="italics"/>m/n<emph.end type="italics"/>. </s> <s>Et erit <emph type="italics"/>DN<emph.end type="italics"/>æqualis <emph type="italics"/>a-o, VG<emph.end type="italics"/>æqualis <lb/><emph type="italics"/>(bb/a-o), VZ<emph.end type="italics"/>æqualis <emph type="italics"/>m/n―a-o,<emph.end type="italics"/>& <emph type="italics"/>GD<emph.end type="italics"/>&longs;eu <emph type="italics"/>NX-VZ-VG<emph.end type="italics"/>æ­<lb/>qualis <emph type="italics"/>c-m/n a+m/n o-(bb/a-o).<emph.end type="italics"/>Re&longs;olvatur terminus (<emph type="italics"/>bb/a-o<emph.end type="italics"/>) in &longs;eriem <lb/>convergentem <emph type="italics"/>(bb/a)+(bb/aa)o+(bb/a<emph type="sup"/>3<emph.end type="sup"/>)oo+(bb/a<emph type="sup"/>4<emph.end type="sup"/>)o<emph type="sup"/>3<emph.end type="sup"/><emph.end type="italics"/>&c. </s> <s>& &longs;iet <emph type="italics"/>GD<emph.end type="italics"/>æqua­<lb/>lis <emph type="italics"/>c-m/n a-(bb/a)+m/n o-(bb/aa)o-(bb/a<emph type="sup"/>3<emph.end type="sup"/>)o<emph type="sup"/>2<emph.end type="sup"/>-(bb/a<emph type="sup"/>4<emph.end type="sup"/>)o<emph type="sup"/>3<emph.end type="sup"/><emph.end type="italics"/>&c. </s> <s>Hujus &longs;eriei termi­<lb/>nus &longs;ecundus <emph type="italics"/>m/no-(bb/aa)o<emph.end type="italics"/>u&longs;urpandus e&longs;t pro Q<emph type="italics"/>o,<emph.end type="italics"/>tertius cum &longs;igno <lb/>mutato <emph type="italics"/>(bb/a<emph type="sup"/>3<emph.end type="sup"/>)o<emph type="sup"/>2<emph.end type="sup"/><emph.end type="italics"/>pro R<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>2<emph.end type="sup"/>, & quartus cum &longs;igno etiam mutato <emph type="italics"/>(bb/a<emph type="sup"/>4<emph.end type="sup"/>)o<emph type="sup"/>1<emph.end type="sup"/><emph.end type="italics"/><lb/>pro S<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/>, eorumque coefficientes <emph type="italics"/>m/n-(bb/aa), (bb/a<emph type="sup"/>3<emph.end type="sup"/>)<emph.end type="italics"/>& (<emph type="italics"/>bb/a<emph type="sup"/>4<emph.end type="sup"/><emph.end type="italics"/>) &longs;cribendæ &longs;unt <lb/>in Regula &longs;uperiore, pro Q, R & S. </s> <s>Quo facto prodit medii den&longs;itas <pb xlink:href="039/01/266.jpg" pagenum="238"/><arrow.to.target n="note214"/>ut (<emph type="italics"/>(bb/a<emph type="sup"/>4<emph.end type="sup"/>)/(bb/a<emph type="sup"/>3<emph.end type="sup"/>)√1+(mm/nn)-(2mbb/naa)+(b<emph type="sup"/>4<emph.end type="sup"/>/a<emph type="sup"/>4<emph.end type="sup"/>)<emph.end type="italics"/>) &longs;eu (1/<emph type="italics"/>√aa+(mm/nn)aa-(2mbb/n)+(b<emph type="sup"/>4<emph.end type="sup"/>/aa)<emph.end type="italics"/>) id <lb/>e&longs;t, &longs;i in <emph type="italics"/>VZ<emph.end type="italics"/>&longs;umatur <emph type="italics"/>VY<emph.end type="italics"/>æqualis <emph type="italics"/>VG,<emph.end type="italics"/>ut (1/<emph type="italics"/>XY<emph.end type="italics"/>). Namque <emph type="italics"/>aa<emph.end type="italics"/>& <lb/><emph type="italics"/>(mm/nn)aa-(2mbb/n)+(b<emph type="sup"/>4<emph.end type="sup"/>/aa)<emph.end type="italics"/>&longs;unt ip&longs;arum <emph type="italics"/>XZ<emph.end type="italics"/>& <emph type="italics"/>ZY<emph.end type="italics"/>quadrata. </s> <s>Re&longs;i&longs;ten­<lb/>tia autem invenitur in ratione ad gravitatem quam habet 3 <emph type="italics"/>XY<emph.end type="italics"/>ad <lb/><figure id="id.039.01.266.1.jpg" xlink:href="039/01/266/1.jpg"/><lb/>2<emph type="italics"/>YG<emph.end type="italics"/>& velocitas ea e&longs;t quacum corpus in Parabola pergeret verti­<lb/>cem <emph type="italics"/>G,<emph.end type="italics"/>diametrum <emph type="italics"/>DG,<emph.end type="italics"/>& latus rectum (<emph type="italics"/>XYquad./VG<emph.end type="italics"/>) habente. </s> <s>Pona­<lb/>tur itaque quod Medii den&longs;itates in locis &longs;ingulis <emph type="italics"/>G<emph.end type="italics"/>&longs;int reciproce <lb/>ut di&longs;tantiæ <emph type="italics"/>XY,<emph.end type="italics"/>quodque re&longs;i&longs;tentia in loco aliquo <emph type="italics"/>G<emph.end type="italics"/>&longs;it ad gra­<lb/>vitatem ut 3<emph type="italics"/>XY<emph.end type="italics"/>ad 2<emph type="italics"/>YG<emph.end type="italics"/>; & corpus de loco <emph type="italics"/>A,<emph.end type="italics"/>ju&longs;ta cum veloci­<lb/>tate emi&longs;&longs;um, de&longs;cribet Hyperbolam illam <emph type="italics"/>AGK. Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note214"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Exempl.<emph.end type="italics"/>4. Ponatur indefinite, quod linea <emph type="italics"/>AGK<emph.end type="italics"/>Hyperbola &longs;it, <lb/>centro <emph type="italics"/>X<emph.end type="italics"/>A&longs;ymptotis <emph type="italics"/>MX, NX<emph.end type="italics"/>ea lege de&longs;cripta, ut con&longs;tructo <lb/>rectangulo <emph type="italics"/>XZDN<emph.end type="italics"/>cujus latus <emph type="italics"/>ZD<emph.end type="italics"/>&longs;ecet Hyperbolam in <emph type="italics"/>G<emph.end type="italics"/>& <pb xlink:href="039/01/267.jpg" pagenum="239"/>A&longs;ymptoton ejus in <emph type="italics"/>V,<emph.end type="italics"/>fuerit <emph type="italics"/>VG<emph.end type="italics"/>reciproce ut ip&longs;ius <emph type="italics"/>ZX<emph.end type="italics"/>vel <emph type="italics"/>DN<emph.end type="italics"/><lb/><arrow.to.target n="note215"/>dignitas aliqua <emph type="italics"/>DN<emph type="sup"/>n<emph.end type="sup"/>,<emph.end type="italics"/>cujus index e&longs;t numerus <emph type="italics"/>n<emph.end type="italics"/>: & quæratur <lb/>Medii den&longs;itas, qua Projectile progrediatur in hac curva. </s></p> <p type="margin"> <s><margin.target id="note215"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s>Pro <emph type="italics"/>BN, BD, NX<emph.end type="italics"/>&longs;cribantur A, O, C re&longs;pective, &longs;itque <emph type="italics"/>VZ<emph.end type="italics"/><lb/>ad <emph type="italics"/>XZ<emph.end type="italics"/>vel <emph type="italics"/>DN<emph.end type="italics"/>ut <emph type="italics"/>d<emph.end type="italics"/>ad <emph type="italics"/>e,<emph.end type="italics"/>& <emph type="italics"/>VG<emph.end type="italics"/>æqualis (<emph type="italics"/>bb/DN<emph type="sup"/>n<emph.end type="sup"/><emph.end type="italics"/>), & erit <emph type="italics"/>DN<emph.end type="italics"/>æqua­<lb/>lis A-O, <emph type="italics"/>VG<emph.end type="italics"/>=(<emph type="italics"/>bb<emph.end type="italics"/>/―<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>A-O), <emph type="italics"/>VZ<emph.end type="italics"/>=<emph type="italics"/>d/e<emph.end type="italics"/>―A-O, & <emph type="italics"/>GD<emph.end type="italics"/>&longs;eu <emph type="italics"/>NX-VZ <lb/>-VG<emph.end type="italics"/>æqualis C-<emph type="italics"/>d/e<emph.end type="italics"/>A+<emph type="italics"/>d/e<emph.end type="italics"/>O-(<emph type="italics"/>bb<emph.end type="italics"/>/―<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>A-O). Re&longs;olvatur terminus ille <lb/>(<emph type="italics"/>bb<emph.end type="italics"/>/―<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>A-O) in &longs;eriem infinitam (<emph type="italics"/>bb<emph.end type="italics"/>/A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>)+(<emph type="italics"/>nbb<emph.end type="italics"/>/A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>+1<emph.end type="sup"/>)O+(<emph type="italics"/>nn+n<emph.end type="italics"/>/2A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>+2<emph.end type="sup"/>)<emph type="italics"/>bb<emph.end type="italics"/>O<emph type="sup"/>2<emph.end type="sup"/>+ <lb/>(<emph type="italics"/>n<emph type="sup"/>3<emph.end type="sup"/>+3nn+2n<emph.end type="italics"/>/6A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>+3<emph.end type="sup"/>)<emph type="italics"/>bb<emph.end type="italics"/>O<emph type="sup"/>3<emph.end type="sup"/> &c. </s> <s>ac fiet <emph type="italics"/>GD<emph.end type="italics"/>æqualis C-<emph type="italics"/>d/e<emph.end type="italics"/>A-(<emph type="italics"/>bb<emph.end type="italics"/>/A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>)+ <lb/><emph type="italics"/>d/e<emph.end type="italics"/>O-(<emph type="italics"/>nbb<emph.end type="italics"/>/A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>+1<emph.end type="sup"/>)O-(<emph type="italics"/>+nn+n<emph.end type="italics"/>/2A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>+2<emph.end type="sup"/>)<emph type="italics"/>bb<emph.end type="italics"/>O<emph type="sup"/>2<emph.end type="sup"/>-(<emph type="italics"/>+n<emph type="sup"/>3<emph.end type="sup"/>+3nn+2n<emph.end type="italics"/>/6A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>+3<emph.end type="sup"/>)<emph type="italics"/>bb<emph.end type="italics"/>O<emph type="sup"/>3<emph.end type="sup"/> &c. </s> <s>Hu­<lb/>jus &longs;eriei terminus &longs;ecundus <emph type="italics"/>d/e<emph.end type="italics"/>O-(<emph type="italics"/>nbb<emph.end type="italics"/>/A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>+1<emph.end type="sup"/>)O u&longs;urpandus e&longs;t pro Q<emph type="italics"/>o,<emph.end type="italics"/><lb/>tertius (<emph type="italics"/>nn+n<emph.end type="italics"/>/2A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>+2<emph.end type="sup"/>)<emph type="italics"/>bb<emph.end type="italics"/>O<emph type="sup"/>2<emph.end type="sup"/> pro R<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>2<emph.end type="sup"/>, quartus (<emph type="italics"/>n<emph type="sup"/>3<emph.end type="sup"/>+3nn+2n<emph.end type="italics"/>/6A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>+3<emph.end type="sup"/>)<emph type="italics"/>bb<emph.end type="italics"/>O<emph type="sup"/>3<emph.end type="sup"/> pro <lb/>S<emph type="italics"/>o<emph.end type="italics"/><emph type="sup"/>3<emph.end type="sup"/>. </s> <s>Et inde Medii den&longs;itas (S/R√1+QQ), in loco quovis <emph type="italics"/>G,<emph.end type="italics"/>fit <lb/>(<emph type="italics"/>n<emph.end type="italics"/>+2/3√A<emph type="sup"/>2<emph.end type="sup"/>+(<emph type="italics"/>dd/ee<emph.end type="italics"/>)A<emph type="sup"/>2<emph.end type="sup"/>-(<emph type="italics"/>2dnbb/e<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>)A+(<emph type="italics"/>nnb<emph.end type="italics"/><emph type="sup"/>4<emph.end type="sup"/>/A<emph type="sup"/>2<emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>)), adeoque &longs;i in <emph type="italics"/>VZ<emph.end type="italics"/>capiatur <emph type="italics"/>VY<emph.end type="italics"/><lb/>æqualis <emph type="italics"/>nXVG,<emph.end type="italics"/>den&longs;itas illa e&longs;t reciproce ut <emph type="italics"/>XY.<emph.end type="italics"/>Sunt enim A<emph type="sup"/>2<emph.end type="sup"/><lb/>& (<emph type="italics"/>dd/ee<emph.end type="italics"/>)A<emph type="sup"/>3<emph.end type="sup"/>-(2<emph type="italics"/>dnbb/e<emph.end type="italics"/>A<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>)A+(<emph type="italics"/>nnb<emph.end type="italics"/><emph type="sup"/>4<emph.end type="sup"/>/A<emph type="sup"/>2<emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>) ip&longs;arum <emph type="italics"/>XZ<emph.end type="italics"/>& <emph type="italics"/>ZY<emph.end type="italics"/>quadrata. </s> <s>Re&longs;i&longs;ten­<lb/>tia autem in eodem loco <emph type="italics"/>G<emph.end type="italics"/>fit ad gravitatem ut 3S in (<emph type="italics"/>XY<emph.end type="italics"/>/A) ad 4RR, <lb/>id e&longs;t, <emph type="italics"/>XY<emph.end type="italics"/>ad (<emph type="italics"/>2nn+2n/n+2)VG.<emph.end type="italics"/>Et velocitas ibidem ea ip&longs;a e&longs;t qua­<lb/>cum corpus projectum in Parabola pergeret, verticem <emph type="italics"/>G,<emph.end type="italics"/>diametrum <lb/><emph type="italics"/>GD<emph.end type="italics"/>& latus rectum (1+QQ/R) &longs;eu (2<emph type="italics"/>XYquad./―nn+n<emph.end type="italics"/>in<emph type="italics"/>VG<emph.end type="italics"/>) habente. <emph type="italics"/>Q.E.I.<emph.end type="italics"/><pb xlink:href="039/01/268.jpg" pagenum="240"/><arrow.to.target n="note216"/></s></p> <p type="margin"> <s><margin.target id="note216"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Eadem ratione qua prodiit den&longs;itas Medii ut (SX<emph type="italics"/>AC<emph.end type="italics"/>/RX<emph type="italics"/>HT<emph.end type="italics"/>) in Co­<lb/>rollario primo, &longs;i re&longs;i&longs;tentia ponatur ut velocitatis V dignitas quæ­<lb/>libet V<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/> prodibit den&longs;itas Medii ut (S/R(4-<emph type="italics"/>n<emph.end type="italics"/>/2))X(―<emph type="italics"/>AC/HT<emph.end type="italics"/>|<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1.<emph.end type="sup"/>) </s></p> <p type="main"> <s>Et propterea &longs;i Curva inveniri pote&longs;t ea lege ut data fuerit ratio <lb/>(S/R(4-<emph type="italics"/>n<emph.end type="italics"/>/2)) ad (―<emph type="italics"/>HT/AC<emph.end type="italics"/>|<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>), vel (S<emph type="sup"/>2<emph.end type="sup"/>/R<emph type="sup"/>4-<emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>) ad (―1+QQ|<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>-1<emph.end type="sup"/>): corpus move­<lb/>bitur in hac Curva in uniformi Medio cum re&longs;i&longs;tentia quæ &longs;it ut <lb/>velocitatis dignitas V<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>. </s> <s>Sed redeamus ad Curvas &longs;impliciores. </s></p> <p type="main"> <s>Quoniam motus non fit in Parabola ni&longs;i in Medio non re&longs;i&longs;ten­<lb/>te, in Hyperbolis vero hic de&longs;criptis fit per re&longs;i&longs;tentiam perpetuam; <lb/>per&longs;picuum e&longs;t quod Linea, quam projectile in Medio uniformiter <lb/>re&longs;i&longs;tente de&longs;cribit, propius accedit ad Hyperbolas ha&longs;ce quam ad <lb/>Parabolam. </s> <s>E&longs;t utique linea illa Hyperbolici generis, &longs;ed quæ <lb/>circa verticem magis di&longs;tat ab A&longs;ymptotis; in partibus a vertice <lb/>remotioribus propius ad ip&longs;as accedit quam pro ratione Hyper­<lb/>bolarum quas hic de&longs;crip&longs;i. </s> <s>Tanta vero non e&longs;t inter has & illam <lb/>differentia, quin illius loco po&longs;&longs;int hæ in rebus practicis non in­<lb/>commode adhiberi. </s> <s>Et utiliores for&longs;an futuræ &longs;unt hæ, quam <lb/>Hyperbola magis accurata & &longs;imul magis compo&longs;ita. </s> <s>Ip&longs;æ vero <lb/>in u&longs;um &longs;ic deducentur. </s></p> <p type="main"> <s>Compleatur parallelogrammum <emph type="italics"/>XYGT,<emph.end type="italics"/>& recta <emph type="italics"/>GT<emph.end type="italics"/>tanget <lb/>Hyperbolam in <emph type="italics"/>G,<emph.end type="italics"/>ideoQ.E.D.n&longs;itas Medii in <emph type="italics"/>G<emph.end type="italics"/>e&longs;t reciproce ut <lb/>tangens <emph type="italics"/>GT,<emph.end type="italics"/>& velocitas ibidem ut √(<emph type="italics"/>GTq/GV<emph.end type="italics"/>), re&longs;i&longs;tentia autem ad <lb/>vim gravitatis ut <emph type="italics"/>GT<emph.end type="italics"/>ad <emph type="italics"/>(2nn+2n/n+2)GV.<emph.end type="italics"/></s></p> <p type="main"> <s>Proinde &longs;i corpus de loco <emph type="italics"/>A<emph.end type="italics"/>&longs;ecundum rectam <emph type="italics"/>AH<emph.end type="italics"/>projectum <lb/>de&longs;cribat Hyperbolam <emph type="italics"/>AGK,<emph.end type="italics"/>& <emph type="italics"/>AH<emph.end type="italics"/>producta occurrat A&longs;ymp­<lb/>toto <emph type="italics"/>MX<emph.end type="italics"/>in <emph type="italics"/>H,<emph.end type="italics"/>actaque <emph type="italics"/>AI<emph.end type="italics"/>eidem parallela occurrat alteri A&longs;ymp­<lb/>toto <emph type="italics"/>MX<emph.end type="italics"/>in <emph type="italics"/>I<emph.end type="italics"/>: erit Medii den&longs;itas in <emph type="italics"/>A<emph.end type="italics"/>reciproce ut <emph type="italics"/>AH,<emph.end type="italics"/>& cor­<lb/>poris velocitas ut √(<emph type="italics"/>AHq/AI<emph.end type="italics"/>), ac re&longs;i&longs;tentia ibidem ad gravitatem ut <lb/><emph type="italics"/>AH<emph.end type="italics"/>ad (<emph type="italics"/>2nn+2n/n+2<emph.end type="italics"/>) in <emph type="italics"/>AI.<emph.end type="italics"/>Unde prodeunt &longs;equentes Regulæ. </s></p><pb xlink:href="039/01/269.jpg" pagenum="241"/> <p type="main"> <s><emph type="italics"/>Reg.<emph.end type="italics"/>1. Si &longs;ervetur tum Medii den&longs;itas in <emph type="italics"/>A,<emph.end type="italics"/>tum velocitas qua­<lb/><arrow.to.target n="note217"/>cum corpus projicitur, & mutetur angulus <emph type="italics"/>NAH<emph.end type="italics"/>; manebunt lon­<lb/>gitudines <emph type="italics"/>AH, AI, HX.<emph.end type="italics"/>Ideoque &longs;i longitudines illæ in aliquo <lb/>ca&longs;u inveniantur, Hyperbola deinceps ex dato quovis angulo <emph type="italics"/>NAH<emph.end type="italics"/><lb/>expedite determinari pote&longs;t. </s></p> <p type="margin"> <s><margin.target id="note217"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Reg.<emph.end type="italics"/>2. Si &longs;ervetur tum angulus <emph type="italics"/>NAH,<emph.end type="italics"/>tum Medii den&longs;itas <lb/>in <emph type="italics"/>A,<emph.end type="italics"/>& mutetur velocitas quacum corpus projicitur; &longs;ervabitur <lb/>longitudo <emph type="italics"/>AH,<emph.end type="italics"/>& mutabitur <emph type="italics"/>AI<emph.end type="italics"/>in duplicata ratione velocitatis <lb/>reciproce. </s></p> <p type="main"> <s><emph type="italics"/>Reg.<emph.end type="italics"/>3. Si tam angulus <emph type="italics"/>NAH<emph.end type="italics"/>quam corporis velocitas in <emph type="italics"/>A,<emph.end type="italics"/><lb/>gravita&longs;que acceleratrix &longs;ervetur, & proportio re&longs;i&longs;tentiæ in <emph type="italics"/>A<emph.end type="italics"/>ad <lb/><figure id="id.039.01.269.1.jpg" xlink:href="039/01/269/1.jpg"/><lb/>gravitatem motricem augeatur in ratione quacunque: augebitur <lb/>proportio <emph type="italics"/>AH<emph.end type="italics"/>ad <emph type="italics"/>AI<emph.end type="italics"/>in eadem ratione, manente Parabolæ late­<lb/>re recto, eique proportionali longitudine (<emph type="italics"/>AHq/AI<emph.end type="italics"/>); & propterea mi­<lb/>nuetur <emph type="italics"/>AH<emph.end type="italics"/>in eadem ratione, & <emph type="italics"/>AI<emph.end type="italics"/>minuetur in ratione illa du­<lb/>plicata. </s> <s>Augetur vero proportio re&longs;i&longs;tentiæ ad pondus, ubi vel gra­<lb/>vitas &longs;pecifica &longs;ub æquali magnitudine fit minor, vel Medii den&longs;i­<lb/>tas major, vel re&longs;i&longs;tentia, ex magnitudine diminuta, diminuitur in <lb/>minore ratione quam pondus. <pb xlink:href="039/01/270.jpg" pagenum="242"/><arrow.to.target n="note218"/></s></p> <p type="margin"> <s><margin.target id="note218"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Reg.<emph.end type="italics"/>4. Quoniam den&longs;itas Medii prope verticem Hyperbolæ <lb/>major e&longs;t quam in loco <emph type="italics"/>A,<emph.end type="italics"/>ut habeatur den&longs;itas mediocris, debet <lb/>ratio minimæ tangentium <emph type="italics"/>GT<emph.end type="italics"/>ad tangentem <emph type="italics"/>AH<emph.end type="italics"/>inveniri, & <lb/>den&longs;itas in <emph type="italics"/>A<emph.end type="italics"/>angeri in ratione paudo majore quam &longs;emi&longs;ummæ <lb/>harum tangentium ad minimam tangentium <emph type="italics"/>GT.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Reg.<emph.end type="italics"/>5. Si dantur longitudines <emph type="italics"/>AH, AI,<emph.end type="italics"/>& de&longs;cribenda &longs;it Figu­<lb/>ra <emph type="italics"/>AGK:<emph.end type="italics"/>produc <emph type="italics"/>HN<emph.end type="italics"/>ad <emph type="italics"/>X,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>HX<emph.end type="italics"/>æqualis facto &longs;ub <emph type="italics"/>n<emph.end type="italics"/>+1 & <lb/><emph type="italics"/>AI<emph.end type="italics"/>; centroque <emph type="italics"/>X<emph.end type="italics"/>& A&longs;ymptotis <emph type="italics"/>MX, NX<emph.end type="italics"/>per punctum <emph type="italics"/>A<emph.end type="italics"/>de&longs;criba­<lb/>tur Hyperbola, ea lege, ut &longs;it <emph type="italics"/>AI<emph.end type="italics"/>ad quamvis <emph type="italics"/>VG<emph.end type="italics"/>ut <emph type="italics"/>XV<emph type="sup"/>n<emph.end type="sup"/><emph.end type="italics"/>ad <emph type="italics"/>XI<emph type="sup"/>n<emph.end type="sup"/>.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Reg.<emph.end type="italics"/>6. Quo major e&longs;t numerus <emph type="italics"/>n,<emph.end type="italics"/>eo magis accuratæ &longs;unt hæ <lb/>Hyperbolæ in a&longs;cen&longs;u corporis ab <emph type="italics"/>A,<emph.end type="italics"/>& minus accuratæ in ejus de­<lb/>&longs;cen&longs;u ad <emph type="italics"/>K<emph.end type="italics"/>; & contra. </s> <s>Hyperbola Conica mediocrem rationem <lb/>tenet, e&longs;t que cæteris &longs;implicior. </s> <s>Igitur &longs;i Hyperbola &longs;it hujus generis, <lb/>& punctum <emph type="italics"/>K,<emph.end type="italics"/>ubi corpus projectum incidet in rectam quamvis <emph type="italics"/>AN<emph.end type="italics"/><lb/>per punctum <emph type="italics"/>A<emph.end type="italics"/>tran&longs;euntem, quæratur: occurrat producta <emph type="italics"/>AN<emph.end type="italics"/><lb/>A&longs;ymptotis <emph type="italics"/>MX, NX<emph.end type="italics"/>in <emph type="italics"/>M<emph.end type="italics"/>& <emph type="italics"/>N,<emph.end type="italics"/>& &longs;umatur <emph type="italics"/>NK<emph.end type="italics"/>ip&longs;i <emph type="italics"/>AM<emph.end type="italics"/>æqualis. </s></p> <p type="main"> <s><emph type="italics"/>Reg.<emph.end type="italics"/>7. Et hinc liquet methodus expedita determinandi hanc <lb/>Hyperbolam ex Phænomenis. </s> <s>Projiciantur corpora duo &longs;imilia & <lb/>æqualia, eadem velocitate, in angulis diver&longs;is <emph type="italics"/>HAK, hAk,<emph.end type="italics"/>inci­<lb/>dantQ.E.I. planum Horizontis in <emph type="italics"/>K<emph.end type="italics"/>& <emph type="italics"/>k<emph.end type="italics"/>; & notetur proportio <emph type="italics"/>AK<emph.end type="italics"/><lb/>ad <emph type="italics"/>Ak.<emph.end type="italics"/>Sit ea <emph type="italics"/>d<emph.end type="italics"/>ad <emph type="italics"/>e.<emph.end type="italics"/>Tum erecto cuju&longs;vis longitudinis perpen­<lb/>diculo <emph type="italics"/>AI,<emph.end type="italics"/>a&longs;&longs;ume utcunque longitudinem <emph type="italics"/>AH<emph.end type="italics"/>vel <emph type="italics"/>Ah,<emph.end type="italics"/>& inde <lb/>collige graphice longitudines <emph type="italics"/>AK, Ak,<emph.end type="italics"/>per Reg. </s> <s>6. Si ratio <emph type="italics"/>AK<emph.end type="italics"/><lb/>ad <emph type="italics"/>Ak<emph.end type="italics"/>&longs;it eadem cum ratione <emph type="italics"/>d<emph.end type="italics"/>ad <emph type="italics"/>e,<emph.end type="italics"/>longitudo <emph type="italics"/>AH<emph.end type="italics"/>recte a&longs;&longs;ump­<lb/>ta fuit. </s> <s>Sin minus cape in recta infinita <emph type="italics"/>SM<emph.end type="italics"/>longitudinem <emph type="italics"/>SM<emph.end type="italics"/><lb/>æqualem a&longs;&longs;umptæ <emph type="italics"/>AH,<emph.end type="italics"/>& erige perpendiculum <emph type="italics"/>MN<emph.end type="italics"/>æquale ra­<lb/>tionum differentiæ <emph type="italics"/>(AK/Ak)-d/e<emph.end type="italics"/>ductæ in rectam quamvis datam. </s> <s>Si­<lb/>mili methodo ex a&longs;&longs;umptis pluribus longitudinibus <emph type="italics"/>AH<emph.end type="italics"/>invenien­<lb/>da &longs;unt plura puncta <emph type="italics"/>N,<emph.end type="italics"/>& per omnia a­<lb/><figure id="id.039.01.270.1.jpg" xlink:href="039/01/270/1.jpg"/><lb/>genda Curva linea regularis <emph type="italics"/>NNXN,<emph.end type="italics"/>&longs;e­<lb/>cans rectam <emph type="italics"/>SMMM<emph.end type="italics"/>in <emph type="italics"/>X.<emph.end type="italics"/>A&longs;&longs;umatur <lb/>demum <emph type="italics"/>AH<emph.end type="italics"/>æqualie ab&longs;ci&longs;&longs;æ <emph type="italics"/>SX<emph.end type="italics"/>& inde <lb/>denuo inveniatur longitudo <emph type="italics"/>AK<emph.end type="italics"/>; & lon­<lb/>gitudines, quæ &longs;int ad a&longs;&longs;umptam longitu­<lb/>dinem <emph type="italics"/>AI<emph.end type="italics"/>& hanc ultimam <emph type="italics"/>AH<emph.end type="italics"/>ut longitudo <emph type="italics"/>AK<emph.end type="italics"/>per experi­<lb/>mentum cognita ad ultimo inventam longitudinem <emph type="italics"/>AK,<emph.end type="italics"/>erunt veræ <lb/>illæ longitudines <emph type="italics"/>AI<emph.end type="italics"/>& <emph type="italics"/>AH,<emph.end type="italics"/>quas invenire oportuit. </s> <s>Hi&longs;ce vero <lb/>datis dabitur & re&longs;i&longs;tentia Medii in loco <emph type="italics"/>A,<emph.end type="italics"/>quippe quæ &longs;it ad vim <lb/>gravitatis ut <emph type="italics"/>AH<emph.end type="italics"/>ad 2<emph type="italics"/>AI.<emph.end type="italics"/>Augenda e&longs;t autem den&longs;itas. </s> <s>Medii per <lb/>Reg. </s> <s>4; & re&longs;i&longs;tentia modo inventa, &longs;i in eadem ratione augeatur, fiet <lb/>accuratior. </s></p><pb xlink:href="039/01/271.jpg" pagenum="243"/> <p type="main"> <s><emph type="italics"/>Reg.<emph.end type="italics"/>8. Inventis longitudinibus <emph type="italics"/>AH, HX<emph.end type="italics"/>; &longs;i jam de&longs;ideretur </s></p> <p type="main"> <s><arrow.to.target n="note219"/>po&longs;itio rectæ <emph type="italics"/>AH,<emph.end type="italics"/>&longs;ecundum quam Projectile, data illa cum veloci­<lb/>tate emi&longs;&longs;um, incidit in punctum quodvis <emph type="italics"/>K:<emph.end type="italics"/>ad puncta <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>K<emph.end type="italics"/><lb/>erigantur rectæ <emph type="italics"/>AC, KF<emph.end type="italics"/>horizonti perpendiculares, quarum <emph type="italics"/>AC<emph.end type="italics"/><lb/>deor&longs;um tendat, & æquetur ip&longs;i <emph type="italics"/>AI<emph.end type="italics"/>&longs;eu 1/2<emph type="italics"/>HX.<emph.end type="italics"/>A&longs;ymptotis <emph type="italics"/>AK, <lb/>KF<emph.end type="italics"/>de&longs;cribatur Hyperbola, cujus conjugata tran&longs;eat per punctum <lb/><emph type="italics"/>C,<emph.end type="italics"/>centroque <emph type="italics"/>A<emph.end type="italics"/>& intervallo <emph type="italics"/>AH<emph.end type="italics"/>de&longs;cribatur Circulus &longs;ecans Hy­<lb/>perbolam illam in puncto <emph type="italics"/>H;<emph.end type="italics"/>& Projectile &longs;ecundum rectam <emph type="italics"/>AH<emph.end type="italics"/><lb/>emi&longs;&longs;um incidet in punctum <emph type="italics"/>K. Q.E.I.<emph.end type="italics"/>Nam punctum <emph type="italics"/>H,<emph.end type="italics"/>ob <lb/>datam longitudinem <emph type="italics"/>AH,<emph.end type="italics"/>locatur alicubi in Circulo de&longs;cripto. </s> <s>A­<lb/>gatur <emph type="italics"/>CH<emph.end type="italics"/>occurrens ip&longs;is <emph type="italics"/>AK<emph.end type="italics"/>& <emph type="italics"/>KF,<emph.end type="italics"/>illi in <emph type="italics"/>E,<emph.end type="italics"/>huic in <emph type="italics"/>F;<emph.end type="italics"/>& ob <lb/><figure id="id.039.01.271.1.jpg" xlink:href="039/01/271/1.jpg"/><lb/>parallelas <emph type="italics"/>CH, MX<emph.end type="italics"/>& æquales <emph type="italics"/>AC, AI,<emph.end type="italics"/>erit <emph type="italics"/>AE<emph.end type="italics"/>æqualis <emph type="italics"/>AM,<emph.end type="italics"/><lb/>& propterea etiam æqualis <emph type="italics"/>KN.<emph.end type="italics"/>Sed <emph type="italics"/>CE<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>AE<emph.end type="italics"/>ut <emph type="italics"/>FH<emph.end type="italics"/>ad <lb/><emph type="italics"/>KN,<emph.end type="italics"/>& propterea <emph type="italics"/>CE<emph.end type="italics"/>& <emph type="italics"/>FH<emph.end type="italics"/>æquantur. </s> <s>Incidit ergo punctum <lb/><emph type="italics"/>H<emph.end type="italics"/>in Hyperbolam A&longs;ymptotis <emph type="italics"/>AK, KF<emph.end type="italics"/>de&longs;criptam, cujus conju­<lb/>gata tran&longs;it per punctum <emph type="italics"/>C,<emph.end type="italics"/>atque adeo reperitur in communi in­<lb/>ter&longs;ectione Hyperbolæ hujus & Circuli de&longs;cripti. <emph type="italics"/>Q.E.D.<emph.end type="italics"/>No­<lb/>tandum e&longs;t autem quod hæc operatio perinde &longs;e habet, &longs;ive recta <lb/><emph type="italics"/>AKN<emph.end type="italics"/>horizonti parallela &longs;it, &longs;ive ad horizontem in angulo quo­<lb/>vis inclinata: quodque ex duabus inter&longs;ectionibus <emph type="italics"/>H, H<emph.end type="italics"/>duo pro­<lb/>deunt anguli <emph type="italics"/>NAH, NAH<emph.end type="italics"/>; & quod in Praxi mechanica &longs;ufficit <pb xlink:href="039/01/272.jpg" pagenum="244"/><arrow.to.target n="note220"/>Circulum &longs;emel de&longs;cribere, deinde regulam interminatam <emph type="italics"/>CH<emph.end type="italics"/>ita ap­<lb/>plicare ad punctum <emph type="italics"/>C,<emph.end type="italics"/>ut ejus pars <emph type="italics"/>FH,<emph.end type="italics"/>Circulo & rectæ <emph type="italics"/>FK<emph.end type="italics"/>interje­<lb/>cta, æqualis &longs;it ejus parti <emph type="italics"/>CE<emph.end type="italics"/>inter punctum <emph type="italics"/>C<emph.end type="italics"/>& rectam <emph type="italics"/>AK<emph.end type="italics"/>&longs;itæ. </s></p> <p type="margin"> <s><margin.target id="note219"/>LIBER <lb/>SECUNDUS.</s></p> <p type="margin"> <s><margin.target id="note220"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Quæ de Hyperbolis dicta &longs;unt fa­<lb/><figure id="id.039.01.272.1.jpg" xlink:href="039/01/272/1.jpg"/><lb/>cile applicantur ad Parabolas. </s> <s>Nam <lb/>&longs;i <emph type="italics"/>XAGK<emph.end type="italics"/>Parabolam de&longs;ignet quam <lb/>recta <emph type="italics"/>XV<emph.end type="italics"/>tangat in vertice <emph type="italics"/>X,<emph.end type="italics"/>&longs;intque <lb/>ordinatim applicatæ <emph type="italics"/>IA, VG<emph.end type="italics"/>ut quæ­<lb/>libet ab&longs;ci&longs;&longs;arum <emph type="italics"/>XI, XV<emph.end type="italics"/>dignitates <lb/><emph type="italics"/>XI<emph type="sup"/>n<emph.end type="sup"/>, XV<emph type="sup"/>n<emph.end type="sup"/>;<emph.end type="italics"/>agantur <emph type="italics"/>XT, GT, AH,<emph.end type="italics"/><lb/>quarum <emph type="italics"/>XT<emph.end type="italics"/>parallela &longs;it <emph type="italics"/>VG,<emph.end type="italics"/>& <emph type="italics"/>GT, <lb/>AH<emph.end type="italics"/>Parabolam tangant in <emph type="italics"/>G<emph.end type="italics"/>& <emph type="italics"/>A:<emph.end type="italics"/>& <lb/>corpus de loco quovis <emph type="italics"/>A,<emph.end type="italics"/>&longs;ecundum <lb/>rectam <emph type="italics"/>AH<emph.end type="italics"/>productam, ju&longs;ta cum <lb/>velocitate projectum, de&longs;cribet hanc <lb/>Parabolam, &longs;i modo den&longs;itas Medii, <lb/>in locis &longs;ingulis <emph type="italics"/>G,<emph.end type="italics"/>&longs;it reciproce ut <lb/>tangens <emph type="italics"/>GT.<emph.end type="italics"/>Velocitas autem in <emph type="italics"/>G<emph.end type="italics"/>ea erit quacum Projectile per­<lb/>geret, in &longs;patio non re&longs;i&longs;tente, in Parabola Conica verticem <emph type="italics"/>G,<emph.end type="italics"/>dia­<lb/>metrum <emph type="italics"/>VG<emph.end type="italics"/>deor&longs;um productam, & latus rectum (<emph type="italics"/>2GTq./nn-nXVG<emph.end type="italics"/>) <lb/>habente. </s> <s>Et re&longs;i&longs;tentia in <emph type="italics"/>G<emph.end type="italics"/>erit ad vim gravitatis ut <emph type="italics"/>GT<emph.end type="italics"/>ad <lb/><emph type="italics"/>(2nn-2n/n-2) VG.<emph.end type="italics"/>Unde &longs;i <emph type="italics"/>NAK<emph.end type="italics"/>lineam horizontalem de&longs;ignet, & <lb/>manente tum den&longs;itate Medii in <emph type="italics"/>A,<emph.end type="italics"/>tum velocitate quacum corpus <lb/>projicitur, mutetur utcunque angulus <emph type="italics"/>NAH;<emph.end type="italics"/>manebunt longitu­<lb/>dines <emph type="italics"/>AH, AI, HX,<emph.end type="italics"/>& inde datur Parabolæ vertex <emph type="italics"/>X,<emph.end type="italics"/>& po&longs;itio <lb/>rectæ <emph type="italics"/>XI,<emph.end type="italics"/>& &longs;umendo <emph type="italics"/>VG<emph.end type="italics"/>ad <emph type="italics"/>IA<emph.end type="italics"/>ut <emph type="italics"/>XV<emph type="sup"/>n<emph.end type="sup"/><emph.end type="italics"/>ad <emph type="italics"/>XI<emph type="sup"/>n<emph.end type="sup"/>,<emph.end type="italics"/>dantur om­<lb/>nia Parabolæ puncta <emph type="italics"/>G,<emph.end type="italics"/>per quæ Projectile tran&longs;ibit. <pb xlink:href="039/01/273.jpg" pagenum="245"/><arrow.to.target n="note221"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note221"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>SECTIO III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><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"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XI. THEOREMA VIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Corpori re&longs;i&longs;titur partim in ratione velocitatis, partim in ve­<lb/>locitatis ratione duplicata, & idem &longs;ola vi in&longs;ita in Medio &longs;i­<lb/>milari movetur, &longs;umantur autem tempora in progre&longs;&longs;ione Arith­<lb/>metica: quantitates velocitatibus reciproce proportionales, datâ <lb/>quadam quantitate auctæ, erunt in progre&longs;&longs;ione Geometrica.<emph.end type="italics"/></s></p> <p type="main"> <s>Centro <emph type="italics"/>C,<emph.end type="italics"/>A&longs;ymptotis rectan­<lb/><figure id="id.039.01.273.1.jpg" xlink:href="039/01/273/1.jpg"/><lb/>gulis <emph type="italics"/>CADd<emph.end type="italics"/>& <emph type="italics"/>CH,<emph.end type="italics"/>de&longs;cribatur <lb/>Hyperbola <emph type="italics"/>BEeS,<emph.end type="italics"/>& A&longs;ympto­<lb/>to <emph type="italics"/>CH<emph.end type="italics"/>parallelæ &longs;int <emph type="italics"/>AB, DE, <lb/>de.<emph.end type="italics"/>In A&longs;ymptoto <emph type="italics"/>CD<emph.end type="italics"/>dentur <lb/>puncta <emph type="italics"/>A, G:<emph.end type="italics"/>Et &longs;i tempus ex­<lb/>ponatur per aream Hyperbolicam <lb/><emph type="italics"/>ABED<emph.end type="italics"/>uniformiter cre&longs;centem; <lb/>dico quod velocitas exponi pote&longs;t <lb/>per longitudinem <emph type="italics"/>DF,<emph.end type="italics"/>cujus reci­<lb/>proca <emph type="italics"/>GD<emph.end type="italics"/>una cum data <emph type="italics"/>CG<emph.end type="italics"/>com­<lb/>ponat longitudinem <emph type="italics"/>CD<emph.end type="italics"/>in progre&longs;&longs;ione Geometrica cre&longs;centem. </s></p> <p type="main"> <s>Sit enim areola <emph type="italics"/>DEed<emph.end type="italics"/>datum temporis incrementum quam <lb/>minimum, & erit <emph type="italics"/>Dd<emph.end type="italics"/>reciproce ut <emph type="italics"/>DE,<emph.end type="italics"/>adeoQ.E.D.recte ut <lb/><emph type="italics"/>CD.<emph.end type="italics"/>Ip&longs;ius autem (1/<emph type="italics"/>G-D<emph.end type="italics"/>) decrementum, quod (per hujus Lem. </s> <s>11) <lb/>e&longs;t (<emph type="italics"/>Dd/GDq<emph.end type="italics"/>), erit ut (<emph type="italics"/>CD/GDq<emph.end type="italics"/>) &longs;eu (<emph type="italics"/>CG+GD/GDq<emph.end type="italics"/>), id e&longs;t, ut (1/<emph type="italics"/>GD<emph.end type="italics"/>)+(<emph type="italics"/>CG/GDq<emph.end type="italics"/>). <lb/>Igitur tempore <emph type="italics"/>ABED<emph.end type="italics"/>peradditionem datarum particularum <emph type="italics"/>ED de<emph.end type="italics"/><lb/>uniformiter cre&longs;cente, decre&longs;cit (1/<emph type="italics"/>GD<emph.end type="italics"/>) in eadem ratione cum veloci­<lb/>tate. </s> <s>Nam decrementum velocitatis e&longs;t ut re&longs;i&longs;tentia, hoc e&longs;t (per <lb/>Hypothe&longs;in) ut &longs;umma duarum quantitatum, quarum una e&longs;t ut <pb xlink:href="039/01/274.jpg" pagenum="246"/><arrow.to.target n="note222"/>velocitas, altera ut quadratum velocitatis: & ip&longs;ius (1/<emph type="italics"/>GD<emph.end type="italics"/>) decremen­<lb/>tum e&longs;t ut &longs;umma quantitatum (1/<emph type="italics"/>GD<emph.end type="italics"/>) & (<emph type="italics"/>CG/GDq<emph.end type="italics"/>), quarum prior e&longs;t <lb/>ip&longs;a (1/<emph type="italics"/>GD<emph.end type="italics"/>), & po&longs;terior (<emph type="italics"/>CG/GDq<emph.end type="italics"/>) e&longs;t ut (1/<emph type="italics"/>GDq<emph.end type="italics"/>). Proinde (1/<emph type="italics"/>GD<emph.end type="italics"/>), ob an­<lb/>alogum decrementum, e&longs;t ut velocitas. </s> <s>Et &longs;i quantitas <emph type="italics"/>GD,<emph.end type="italics"/>ip&longs;i (1/<emph type="italics"/>GD<emph.end type="italics"/>) <lb/>reciproce proportionalis, quantitate data <emph type="italics"/>CG<emph.end type="italics"/>augeatur; &longs;umma <emph type="italics"/>CD,<emph.end type="italics"/><lb/>tempore <emph type="italics"/>ABED<emph.end type="italics"/>uniformiter cre&longs;cente, cre&longs;cet in progre&longs;&longs;ione <lb/>Geometrica. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note222"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Igitur. </s> <s>&longs;i, datis punctis <emph type="italics"/>A, G,<emph.end type="italics"/>exponatur tempus per <lb/>aream Hyperbolicam <emph type="italics"/>ABED,<emph.end type="italics"/>exponi pote&longs;t velocitas per ip&longs;ius <lb/><emph type="italics"/>GD<emph.end type="italics"/>reciprocam (1/<emph type="italics"/>GD<emph.end type="italics"/>). </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Sumendo autem <emph type="italics"/>GA<emph.end type="italics"/>ad <emph type="italics"/>GD<emph.end type="italics"/>ut velocitatis reciproca &longs;ub <lb/>initio, ad velocitatis reciprocam in fine temporis cuju&longs;vis <emph type="italics"/>ABED,<emph.end type="italics"/><lb/>invenietur punctum <emph type="italics"/>G.<emph.end type="italics"/>Eo autem invento, velocitas ex dato quo­<lb/>vis alio tempore inveniri pote&longs;t. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XII. THEOREMA IX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, dico quod &longs;i &longs;patia de&longs;cripta &longs;umantur in progre&longs;&longs;io­<lb/>ne Arithmetica, velocitates data quadam quantitate auctæ e­<lb/>runt in progre&longs;&longs;ione Geometrica.<emph.end type="italics"/></s></p> <p type="main"> <s>In A&longs;ymptoto <emph type="italics"/>CD<emph.end type="italics"/>detur pun­<lb/><figure id="id.039.01.274.1.jpg" xlink:href="039/01/274/1.jpg"/><lb/>ctum <emph type="italics"/>R,<emph.end type="italics"/>& erecto perpendiculo <emph type="italics"/>RS,<emph.end type="italics"/><lb/>quod occurrat Hyperbolæ in <emph type="italics"/>S,<emph.end type="italics"/>ex­<lb/>ponatur de&longs;criptum &longs;patium per a­<lb/>ream Hyperbolicam <emph type="italics"/>RSED<emph.end type="italics"/>; & <lb/>velocitas erit ut longitudo <emph type="italics"/>GD,<emph.end type="italics"/><lb/>quæ cum data <emph type="italics"/>CG<emph.end type="italics"/>componit longi­<lb/>tudinem <emph type="italics"/>CD,<emph.end type="italics"/>in progre&longs;&longs;ione Geo­<lb/>metrica decre&longs;centem, interea dum <lb/>&longs;patium <emph type="italics"/>RSED<emph.end type="italics"/>augetur in Arith­<lb/>metica. </s></p> <p type="main"> <s>Etenim ob datum &longs;patii incrementum <emph type="italics"/>EDde,<emph.end type="italics"/>lineola <emph type="italics"/>Dd,<emph.end type="italics"/>quæ <pb xlink:href="039/01/275.jpg" pagenum="247"/>decrementum e&longs;t ip&longs;ius <emph type="italics"/>GD,<emph.end type="italics"/>erit reciproce ut <emph type="italics"/>ED,<emph.end type="italics"/>adeoQ.E.D.­<lb/><arrow.to.target n="note223"/>recte ut <emph type="italics"/>CD,<emph.end type="italics"/>hoc e&longs;t, ut &longs;umma eju&longs;dom <emph type="italics"/>GD<emph.end type="italics"/>& longitudinis datæ <lb/><emph type="italics"/>CG.<emph.end type="italics"/>Sed velocitatis decrementum, tempore &longs;ibi reciproce pro­<lb/>portionali, quo data &longs;patii particula <emph type="italics"/>D de E<emph.end type="italics"/>de&longs;cribitur, e&longs;t ut re­<lb/>&longs;i&longs;tentia & tempus conjunctim, id e&longs;t, directe ut &longs;umma duarum <lb/>quantitatum, quarum una e&longs;t ut velocitas, altera ut velocitatis qua­<lb/>dratum, & inver&longs;e ut velocitas; adeoQ.E.D.recte ut &longs;umma duarum <lb/>quantitatum, quarum una datur, altera e&longs;t ut velocitas. </s> <s>Igitur de­<lb/>crementum tam velocitatis quam lineæ <emph type="italics"/>GD,<emph.end type="italics"/>e&longs;t ut quantitas data <lb/>& quantitas decre&longs;cens conjunctim, & propter analoga decremen­<lb/>ta, analogæ &longs;emper crunt quantitates decre&longs;centes: nimirum veloci­<lb/>tas & linea <emph type="italics"/>G.D. Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note223"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Igitur &longs;i velocitas exponatur per longitudinem <emph type="italics"/>GD,<emph.end type="italics"/>&longs;pa­<lb/>tium de&longs;criptum erit ut area Hyperbolica <emph type="italics"/>DESR.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et &longs;i utcunque a&longs;&longs;umatur punctum <emph type="italics"/>R,<emph.end type="italics"/>invenietur pun­<lb/>ctum <emph type="italics"/>G,<emph.end type="italics"/>capiendo <emph type="italics"/>GR<emph.end type="italics"/>ad <emph type="italics"/>GD,<emph.end type="italics"/>ut e&longs;t velocitas &longs;ub initio ad ve­<lb/>locitatem po&longs;t &longs;patium quodvis <emph type="italics"/>RSED<emph.end type="italics"/>de&longs;criptum. </s> <s>Invento au­<lb/>tem puncto <emph type="italics"/>G,<emph.end type="italics"/>datur &longs;patium ex data velocitate, & contra. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Unde cum, per Prop. </s> <s>XI. detur velocitas ex dato tem­<lb/>pore, & per hanc Propo&longs;itionem detur &longs;patium ex data velocitate; <lb/>dabitur &longs;patium ex dato tempore: & contra. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XIII. THEOREMA X.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod Corpus ab uniformi gravitate deor&longs;um attractum recta: <lb/>a&longs;cendit vel de&longs;cendit, & quod eidem re&longs;i&longs;titur partim in ra­<lb/>tione velocitatis, partim in eju&longs;dem ratione duplicata: dico quod <lb/>&longs;i Circuli & Hyperbolæ diametris parallelæ rectæ per conjuga­<lb/>tarum diametrorum terminos ducantur, & velocitates &longs;int ut <lb/>&longs;egmenta quædam parallelarum a dato puncto ducta, Tempora <lb/>erunt ut arearum Sectores, rectis a centro ad &longs;egmentorum ter­<lb/>minos ductis ab&longs;ci&longs;&longs;i: & contra.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>1. Ponamus primo quod corpus a&longs;cendit, centroque <emph type="italics"/>D<emph.end type="italics"/>& <lb/>&longs;emidiametro quovis <emph type="italics"/>DB<emph.end type="italics"/>de&longs;cribatur Circuli quadrans <emph type="italics"/>BETF,<emph.end type="italics"/>& <lb/>per &longs;emidiametri <emph type="italics"/>DB<emph.end type="italics"/>terminum <emph type="italics"/>B<emph.end type="italics"/>agatur infinita <emph type="italics"/>BAP,<emph.end type="italics"/>&longs;emidia­<lb/>metro <emph type="italics"/>DF<emph.end type="italics"/>parallela. </s> <s>In ea detur punctum <emph type="italics"/>A,<emph.end type="italics"/>& capiatur &longs;egmen­<lb/>tum <emph type="italics"/>AP<emph.end type="italics"/>velocitati proportionale. </s> <s>Et cum re&longs;i&longs;tentiæ pars aliqua &longs;it <pb xlink:href="039/01/276.jpg" pagenum="248"/><arrow.to.target n="note224"/>ut velocitas & pars altera ut <lb/><figure id="id.039.01.276.1.jpg" xlink:href="039/01/276/1.jpg"/><lb/>velocitatis quadratum, fit re­<lb/>&longs;i&longs;tentia tota in <emph type="italics"/>P<emph.end type="italics"/>ut <emph type="italics"/>AP quad<emph.end type="italics"/><lb/>+2 <emph type="italics"/>BAP.<emph.end type="italics"/>Jungantur <emph type="italics"/>DA, <lb/>DP<emph.end type="italics"/>Circulum &longs;ecantes in <emph type="italics"/>E<emph.end type="italics"/><lb/>ac <emph type="italics"/>T,<emph.end type="italics"/>& exponatur gravitas per <lb/><emph type="italics"/>DA quad,<emph.end type="italics"/>ita ut &longs;it gravitas ad <lb/>re&longs;i&longs;tentiam in <emph type="italics"/>P<emph.end type="italics"/>ut <emph type="italics"/>DAq<emph.end type="italics"/>ad <lb/><emph type="italics"/>APq<emph.end type="italics"/>+2<emph type="italics"/>BAP:<emph.end type="italics"/>& tempus <lb/>a&longs;cen&longs;us omnis &longs;uturi erit ut <lb/>Circuli &longs;ector <emph type="italics"/>EDTE.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note224"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Agatur enim <emph type="italics"/>DVQ,<emph.end type="italics"/>ab­<lb/>&longs;cindens & velocitatis <emph type="italics"/>AP<emph.end type="italics"/><lb/>momentum <emph type="italics"/>PQ,<emph.end type="italics"/>& Sectoris <lb/><emph type="italics"/>DET<emph.end type="italics"/>momentum <emph type="italics"/>DTV<emph.end type="italics"/>da­<lb/>to temporis momento re&longs;pondens: & velocitatis decrementum il­<lb/>lud <emph type="italics"/>PQ<emph.end type="italics"/>erit ut &longs;umma virium gravitatis <emph type="italics"/>DAq<emph.end type="italics"/>& re&longs;i&longs;tentiæ <lb/><emph type="italics"/>APq<emph.end type="italics"/>+2<emph type="italics"/>BAP,<emph.end type="italics"/>id e&longs;t (per Prop. </s> <s>12, Lib. </s> <s>2. Elem.) ut <emph type="italics"/>DPquad.<emph.end type="italics"/><lb/>Proinde area <emph type="italics"/>DPQ,<emph.end type="italics"/>ip&longs;i <emph type="italics"/>PQ<emph.end type="italics"/>proportionalis, e&longs;t ut <emph type="italics"/>DP quad<emph.end type="italics"/>; <lb/>& area <emph type="italics"/>DTV,<emph.end type="italics"/>(quæ e&longs;t ad aream <emph type="italics"/>DPQ<emph.end type="italics"/>ut <emph type="italics"/>DTq<emph.end type="italics"/>ad <emph type="italics"/>DPq<emph.end type="italics"/>) <lb/>e&longs;t ut datum <emph type="italics"/>DTQ<emph.end type="italics"/>Decre&longs;cit igitur area <emph type="italics"/>EDT<emph.end type="italics"/>uniformiter ad mo­<lb/>dum temporis futuri, per &longs;ubductionem datarum particularum <emph type="italics"/>DTV,<emph.end type="italics"/><lb/>& propterea tempori a&longs;cen&longs;us futuri proportionalis e&longs;t. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>2. Si veloci­<lb/><figure id="id.039.01.276.2.jpg" xlink:href="039/01/276/2.jpg"/><lb/>tas in a&longs;cen&longs;u cor­<lb/>poris exponatur per <lb/>longitudinem <emph type="italics"/>AP<emph.end type="italics"/><lb/>ut prius, & re&longs;i&longs;ten­<lb/>tia ponatur e&longs;&longs;e ut <lb/><emph type="italics"/>APq<emph.end type="italics"/>+2<emph type="italics"/>BAP,<emph.end type="italics"/>& <lb/>&longs;i vis gravitatis mi­<lb/>nor &longs;it quam quæ per <lb/><emph type="italics"/>DAq<emph.end type="italics"/>exponi po&longs;­<lb/>&longs;it; capiatur <emph type="italics"/>BD<emph.end type="italics"/>e­<lb/>jus longitudinis, ut <lb/>&longs;it <emph type="italics"/>ABq-BDq<emph.end type="italics"/><lb/>gravitati proportio­<lb/>nale, &longs;itque <emph type="italics"/>DF<emph.end type="italics"/>ip&longs;i <lb/><emph type="italics"/>DB<emph.end type="italics"/>perpendicularis & æqualis, & per verticem <emph type="italics"/>F<emph.end type="italics"/>de&longs;cribatur Hy­<lb/>perbola <emph type="italics"/>FTVE<emph.end type="italics"/>cujus &longs;emidiametri conjugatæ &longs;int <emph type="italics"/>DB<emph.end type="italics"/>& <emph type="italics"/>DF,<emph.end type="italics"/><lb/>quæque &longs;ecet <emph type="italics"/>DA<emph.end type="italics"/>in <emph type="italics"/>E,<emph.end type="italics"/>& <emph type="italics"/>DP, DQ<emph.end type="italics"/>in <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>V<emph.end type="italics"/>; & crit tempus <lb/>a&longs;cen&longs;us futuri ut Hyperbolæ &longs;ector <emph type="italics"/>TDE.<emph.end type="italics"/></s></p><pb xlink:href="039/01/277.jpg" pagenum="249"/> <p type="main"> <s>Nam velocitatis decrementum <emph type="italics"/>PQ,<emph.end type="italics"/>in data temporis particula <lb/><arrow.to.target n="note225"/>factum, e&longs;t ut &longs;umma re&longs;i&longs;tentiæ <emph type="italics"/>APq<emph.end type="italics"/>+2<emph type="italics"/>BAP<emph.end type="italics"/>& gravitatis <lb/><emph type="italics"/>ABq-BDq,<emph.end type="italics"/>id e&longs;t, ut <emph type="italics"/>BPq-BDq.<emph.end type="italics"/>E&longs;t autem area <emph type="italics"/>DTV<emph.end type="italics"/><lb/>ad aream <emph type="italics"/>DPQ<emph.end type="italics"/>ut <emph type="italics"/>DTq<emph.end type="italics"/>ad <emph type="italics"/>DPq<emph.end type="italics"/>adeoque, &longs;i ad <emph type="italics"/>DF<emph.end type="italics"/>demitta­<lb/>tur perpendiculum <emph type="italics"/>GT,<emph.end type="italics"/>ut <emph type="italics"/>GTq<emph.end type="italics"/>&longs;eu <emph type="italics"/>GDq-DFq<emph.end type="italics"/>ad <emph type="italics"/>BDq<emph.end type="italics"/><lb/>utque <emph type="italics"/>GDq<emph.end type="italics"/>ad <emph type="italics"/>BPq<emph.end type="italics"/>& divi&longs;im ut <emph type="italics"/>DFq<emph.end type="italics"/>ad <emph type="italics"/>BPq-BDq.<emph.end type="italics"/><lb/>Quare cum area <emph type="italics"/>DPQ<emph.end type="italics"/>&longs;it ut <emph type="italics"/>PQ,<emph.end type="italics"/>id e&longs;t, ut <emph type="italics"/>BPq-BDq<emph.end type="italics"/>; erit <lb/>area <emph type="italics"/>DTV<emph.end type="italics"/>ut datum <emph type="italics"/>DFq.<emph.end type="italics"/>Decre&longs;cit igitur area <emph type="italics"/>EDT<emph.end type="italics"/>unifor­<lb/>miter &longs;ingulis temporis particulis æqualibus, per &longs;ubductionem par­<lb/>ticularum totidem datarum <emph type="italics"/>DTV,<emph.end type="italics"/>& propterea tempori propor­<lb/>tionalis e&longs;t. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note225"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>3. Sit <emph type="italics"/>AP<emph.end type="italics"/>velocitas in de&longs;cen&longs;u corporis, & <emph type="italics"/>APq+2BAP<emph.end type="italics"/><lb/>re&longs;i&longs;tentia, & <emph type="italics"/>BDq-ABq<emph.end type="italics"/>vis gravitatis, exi&longs;tente angulo <emph type="italics"/>DBA<emph.end type="italics"/><lb/>recto. </s> <s>Et &longs;i centro <emph type="italics"/>D,<emph.end type="italics"/>vertice <lb/><figure id="id.039.01.277.1.jpg" xlink:href="039/01/277/1.jpg"/><lb/>principali <emph type="italics"/>B,<emph.end type="italics"/>de&longs;cribatur Hy­<lb/>perbola rectangula <emph type="italics"/>BETV<emph.end type="italics"/><lb/>&longs;ecans productas <emph type="italics"/>DA, DP<emph.end type="italics"/>& <lb/><emph type="italics"/>DQ<emph.end type="italics"/>in <emph type="italics"/>E, T<emph.end type="italics"/>& <emph type="italics"/>V<emph.end type="italics"/>; erit Hy­<lb/>perbolæ hujus &longs;ector <emph type="italics"/>DET<emph.end type="italics"/>ut <lb/>tempus de&longs;cen&longs;us. </s></p> <p type="main"> <s>Nam velocitatis <expan abbr="increm&etilde;tum">incrementum</expan> <lb/><emph type="italics"/>PQ,<emph.end type="italics"/>eique proportionalis area <lb/><emph type="italics"/>DPQ,<emph.end type="italics"/>e&longs;t ut exce&longs;&longs;us gravita­<lb/>tis &longs;upra re&longs;i&longs;tentiam, id e&longs;t, ut <lb/><emph type="italics"/>BDq-ABq-2BAP-APq<emph.end type="italics"/><lb/>&longs;eu <emph type="italics"/>BDq-BPq.<emph.end type="italics"/>Et area <lb/><emph type="italics"/>DTV<emph.end type="italics"/>e&longs;t ad aream <emph type="italics"/>DPQ<emph.end type="italics"/>ut <lb/><emph type="italics"/>DTq<emph.end type="italics"/>ad <emph type="italics"/>DPq,<emph.end type="italics"/>adeoque ut <lb/><emph type="italics"/>GTq<emph.end type="italics"/>&longs;eu <emph type="italics"/>GDq-BDq<emph.end type="italics"/>ad <lb/><emph type="italics"/>BPq<emph.end type="italics"/>utque <emph type="italics"/>GDq<emph.end type="italics"/>ad <emph type="italics"/>BDq<emph.end type="italics"/><lb/>& divi&longs;im ut <emph type="italics"/>BDq<emph.end type="italics"/>ad <emph type="italics"/>BDq-BPq.<emph.end type="italics"/>Quare cum area <emph type="italics"/>DPQ<emph.end type="italics"/><lb/>&longs;it ut <emph type="italics"/>BDq-BPq,<emph.end type="italics"/>erit area <emph type="italics"/>DTV<emph.end type="italics"/>ut datum <emph type="italics"/>BDq.<emph.end type="italics"/>Cre&longs;cit <lb/>igitur area <emph type="italics"/>EDT<emph.end type="italics"/>uniformiter &longs;ingulis temporis particulis æquali­<lb/>bus, per additionem totidem datarum particularum <emph type="italics"/>DTV,<emph.end type="italics"/>& prop­<lb/>terea tempori de&longs;cen&longs;us proportionalis e&longs;t. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Igitur velocitas <emph type="italics"/>AP<emph.end type="italics"/>e&longs;t ad velocitatem quam corpus tem­<lb/>pore <emph type="italics"/>EDT,<emph.end type="italics"/>in &longs;patio non re&longs;i&longs;tente, a&longs;cendendo amittere vel de­<lb/>&longs;cendendo acquirere po&longs;&longs;et, ut area trianguli <emph type="italics"/>DAP<emph.end type="italics"/>ad aream &longs;e­<lb/>ctoris centro <emph type="italics"/>D,<emph.end type="italics"/>radio <emph type="italics"/>DA,<emph.end type="italics"/>angulo <emph type="italics"/>ADT<emph.end type="italics"/>de&longs;cripti; ideoque ex <lb/>dato tempore datur. </s> <s>Nam velocitas, in Medio non re&longs;i&longs;tente, tem-<pb xlink:href="039/01/278.jpg" pagenum="250"/><arrow.to.target n="note226"/>pori atque adeo &longs;ectori huic proportionalis e&longs;t; in Medio re&longs;i&longs;ten­<lb/>te e&longs;t ut triangulum; & in Medio utroque, ubi quam minima e&longs;t, ac­<lb/>cedit ad rationem æqualitatis, pro more &longs;ectoris & trianguli. </s></p> <p type="margin"> <s><margin.target id="note226"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XIV. THEOREMA XI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, dico quod &longs;patium a&longs;cen&longs;u vel de&longs;cen&longs;u de&longs;criptum, <lb/>e&longs;t ut differentia areæ per quam tempus exponitur, & areæ cu­<lb/>ju&longs;dam alterius quæ augetur vel diminuitur in progre&longs;&longs;ione A­<lb/>rithmetica; &longs;i vires ex re&longs;i&longs;tentia & gravitate compo&longs;itæ &longs;u­<lb/>mantur in progre&longs;&longs;ione Geometrica.<emph.end type="italics"/></s></p> <p type="main"> <s>Capiatur <emph type="italics"/>AC<emph.end type="italics"/>(in Fig. </s> <s>tribus ultimis,) gravitati, & <emph type="italics"/>AK<emph.end type="italics"/>re&longs;i­<lb/>&longs;tentiæ proportionalis. </s> <s>Capiantur autem ad ea&longs;dem partes pun­<lb/>cti <emph type="italics"/>A<emph.end type="italics"/>&longs;i corpus de&longs;cendit, aliter ad contrarias. </s> <s>Erigatur <emph type="italics"/>Ab<emph.end type="italics"/>quæ <lb/>&longs;it ad <emph type="italics"/>DB<emph.end type="italics"/>ut <emph type="italics"/>DBq<emph.end type="italics"/>ad 4 <emph type="italics"/>BAC:<emph.end type="italics"/>& area <emph type="italics"/>AbNK<emph.end type="italics"/>augebitur vel <lb/>diminuetur in progre&longs;&longs;ione Arithmetica, dum vires <emph type="italics"/>CK<emph.end type="italics"/>in pro­<lb/>gre&longs;&longs;ione Geometrica &longs;umuntur. </s> <s>Dico igitur quod di&longs;tantia cor­<lb/>poris ab ejus altitudine maxima &longs;it ut exce&longs;&longs;us areæ <emph type="italics"/>AbNK<emph.end type="italics"/>&longs;upra <lb/>aream <emph type="italics"/>DET.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam cum <emph type="italics"/>AK<emph.end type="italics"/>&longs;it ut re&longs;i&longs;tentia, id e&longs;t, ut <emph type="italics"/>APq+2BAP<emph.end type="italics"/>: <lb/>a&longs;&longs;umatur data quævis quantitas Z, & ponatur <emph type="italics"/>AK<emph.end type="italics"/>æqualis <lb/>(<emph type="italics"/>APq+2BAP<emph.end type="italics"/>/Z); & (per hujus Lemma 11.) erit ip&longs;ius <emph type="italics"/>AK<emph.end type="italics"/>mo­<lb/>mentum <emph type="italics"/>KL<emph.end type="italics"/>æquale (2<emph type="italics"/>APQ+2BAXPQ<emph.end type="italics"/>/Z) &longs;eu (2<emph type="italics"/>BPQ<emph.end type="italics"/>/Z), & <lb/>areæ <emph type="italics"/>AbNK<emph.end type="italics"/>momentum <emph type="italics"/>KLON<emph.end type="italics"/>æquale (2<emph type="italics"/>BPQXLO<emph.end type="italics"/>/Z) &longs;eu <lb/>(<emph type="italics"/>BPQXBD cub.<emph.end type="italics"/>/2ZX<emph type="italics"/>CRXAB<emph.end type="italics"/>). </s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>1. Jam &longs;i corpus a&longs;cendit, &longs;itque gravitas ut <emph type="italics"/>ABq+BDq<emph.end type="italics"/><lb/>exi&longs;tente <emph type="italics"/>BET<emph.end type="italics"/>Circulo, (in Fig. </s> <s>Ca&longs;. </s> <s>1. Prop. </s> <s>XIII.) linea <emph type="italics"/>AC,<emph.end type="italics"/><lb/>quæ gravitati proportionalis e&longs;t, erit (<emph type="italics"/>ABq+BDq<emph.end type="italics"/>/Z), & <emph type="italics"/>DPq<emph.end type="italics"/>&longs;eu <lb/><emph type="italics"/>APq+2BAP+ABq+BDq<emph.end type="italics"/>erit <emph type="italics"/>AK<emph.end type="italics"/>XZ+<emph type="italics"/>AC<emph.end type="italics"/>XZ &longs;eu <lb/><emph type="italics"/>CK<emph.end type="italics"/>XZ; ideoque area <emph type="italics"/>DTV<emph.end type="italics"/>erit ad aream <emph type="italics"/>DPQ<emph.end type="italics"/>ut <emph type="italics"/>DTq<emph.end type="italics"/>vel <lb/><emph type="italics"/>DBq<emph.end type="italics"/>ad <emph type="italics"/>CK<emph.end type="italics"/>XZ. </s></p><pb xlink:href="039/01/279.jpg" pagenum="251"/> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>2. Sin corpus a&longs;cendit, & gravitas &longs;it ut <emph type="italics"/>ABq-BDq<emph.end type="italics"/><lb/><arrow.to.target n="note227"/>linea <emph type="italics"/>AC<emph.end type="italics"/>(Fig. </s> <s>Ca&longs;. </s> <s>2. Prop. </s> <s>XIII) erit (<emph type="italics"/>ABq-BDq<emph.end type="italics"/>/Z), & <emph type="italics"/>DTq<emph.end type="italics"/><lb/>erit ad <emph type="italics"/>DPq<emph.end type="italics"/>ut <emph type="italics"/>DFq<emph.end type="italics"/>&longs;eu <emph type="italics"/>DBq<emph.end type="italics"/>ad <emph type="italics"/>BPq-BDq<emph.end type="italics"/>&longs;eu <emph type="italics"/>APq+ <lb/>2BAP+ABq-BDq,<emph.end type="italics"/>id e&longs;t, ad <emph type="italics"/>AK<emph.end type="italics"/>XZ+<emph type="italics"/>AC<emph.end type="italics"/>XZ &longs;eu <emph type="italics"/>CK<emph.end type="italics"/>XZ. </s> <s><lb/>Ideoque area <emph type="italics"/>DTV<emph.end type="italics"/>erit ad aream <emph type="italics"/>DPQ<emph.end type="italics"/>ut <emph type="italics"/>DBq<emph.end type="italics"/>ad <emph type="italics"/>CK<emph.end type="italics"/>XZ. </s></p> <p type="margin"> <s><margin.target id="note227"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>3. Et eodem argumento, &longs;i corpus de&longs;cendit, & propterea <lb/>gravitas &longs;it ut <emph type="italics"/>BDq-ABq,<emph.end type="italics"/>& linea <emph type="italics"/>AC<emph.end type="italics"/>(Fig. </s> <s>Ca&longs;.3. Prop. </s> <s>præced.) <lb/>æquetur (<emph type="italics"/>BDq-ABq<emph.end type="italics"/>/Z) erit area <emph type="italics"/>DTV<emph.end type="italics"/>ad aream <emph type="italics"/>DPQ<emph.end type="italics"/>ut <emph type="italics"/>DBq<emph.end type="italics"/><lb/>ad <emph type="italics"/>CK<emph.end type="italics"/>XZ: ut &longs;upra. </s></p> <p type="main"> <s>Cum igitur areæ illæ &longs;emper &longs;int in hac ratione; &longs;i pro area <lb/><emph type="italics"/>DTV,<emph.end type="italics"/>qua momentum temporis &longs;ibimet ip&longs;i &longs;emper æquale ex­<lb/>ponitur, &longs;cribatur determinatum quodvis rectangulum, puta <lb/><emph type="italics"/>BDXm,<emph.end type="italics"/>erit area <emph type="italics"/>DPQ,<emph.end type="italics"/>id e&longs;t, 1/2<emph type="italics"/>BDXPQ<emph.end type="italics"/>; ad <emph type="italics"/>BDXm<emph.end type="italics"/>ut <lb/><emph type="italics"/>CK<emph.end type="italics"/>XZ ad <emph type="italics"/><expan abbr="BDq.">BDque</expan><emph.end type="italics"/>AtQ.E.I.de fit <emph type="italics"/>PQXBD cub.<emph.end type="italics"/>æquale <lb/>2<emph type="italics"/>BDXmXCK<emph.end type="italics"/>XZ, & areæ <emph type="italics"/>AbNK<emph.end type="italics"/>momentum <emph type="italics"/>KLON<emph.end type="italics"/>&longs;u­<lb/>perius inventum, fit (<emph type="italics"/>BPXBDXm/AB<emph.end type="italics"/>). Auferatur areæ <emph type="italics"/>DET<emph.end type="italics"/>mo­<lb/>mentum <emph type="italics"/>DTV<emph.end type="italics"/>&longs;eu <emph type="italics"/>BDXm,<emph.end type="italics"/>& re&longs;tabit (<emph type="italics"/>APXBDXm/AB<emph.end type="italics"/>). E&longs;t igi­<lb/>tur differentia momentorum, id e&longs;t, momentum differentiæ area­<lb/>rum, æqualis (<emph type="italics"/>APXBDXm/AB<emph.end type="italics"/>); & propterea (ob datum (<emph type="italics"/>BDXm/AB<emph.end type="italics"/>)) <lb/>ut velocitas <emph type="italics"/>AP,<emph.end type="italics"/>id e&longs;t, ut momentum &longs;patii quod corpus a&longs;cen­<lb/>dendo vel de&longs;cendendo de&longs;cribit. </s> <s>IdeoQ.E.D.fferentia arearum <lb/>& &longs;patium illud, proportionalibus momentis cre&longs;centia vel decre­<lb/>&longs;centia & &longs;imul incipientia vel &longs;imul evane&longs;centia, &longs;unt proportio­<lb/>nalia. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Igitur &longs;i longitudo aliqua V &longs;umatur in ea ratione ad du­<lb/>plum longitudinis M, quæ oritur applicando aream <emph type="italics"/>DET<emph.end type="italics"/>ad <emph type="italics"/>BD,<emph.end type="italics"/><lb/>quam habet linea <emph type="italics"/>DA<emph.end type="italics"/>ad lineam <emph type="italics"/>DE<emph.end type="italics"/>; &longs;patium quod corpus a&longs;cen­<lb/>&longs;u vel de&longs;cen&longs;u toto in Medio re&longs;i&longs;tente de&longs;cribit, erit ad &longs;patium <lb/>quod in Medio non re&longs;i&longs;tente eodem tempore de&longs;cribere po&longs;&longs;et, <lb/>ut arearum illarum differentia ad (<emph type="italics"/>BD<emph.end type="italics"/>XV<emph type="sup"/>2<emph.end type="sup"/>/4<emph type="italics"/>AB<emph.end type="italics"/>), ideoque ex dato tem­<lb/>pore datur. </s> <s>Nam &longs;patium in Medio non re&longs;i&longs;tente e&longs;t in dupli­<lb/>cata ratione temporis, &longs;ive ut V<emph type="sup"/>2<emph.end type="sup"/>, & ob datas <emph type="italics"/>BD<emph.end type="italics"/>& <emph type="italics"/>AB,<emph.end type="italics"/>ut <pb xlink:href="039/01/280.jpg" pagenum="252"/><arrow.to.target n="note228"/>(<emph type="italics"/>BD<emph.end type="italics"/>XV<emph type="sup"/>2<emph.end type="sup"/>/4<emph type="italics"/>AB<emph.end type="italics"/>). Momentum hujus areæ &longs;ive huic æqualis (<emph type="italics"/>DAqXBD<emph.end type="italics"/>XM<emph type="sup"/>2<emph.end type="sup"/>/<emph type="italics"/>DEqXAB<emph.end type="italics"/>) <lb/>e&longs;t ad momentum differentiæ arearum <emph type="italics"/>DET<emph.end type="italics"/>& <emph type="italics"/>AbNK,<emph.end type="italics"/>ut <lb/>(<emph type="italics"/>DAqXBD<emph.end type="italics"/>X2MX<emph type="italics"/>m<emph.end type="italics"/>/<emph type="italics"/>DEqXAB<emph.end type="italics"/>) ad (<emph type="italics"/>APXBDXm/AB<emph.end type="italics"/>), hoc e&longs;t, ut (<emph type="italics"/>DAqXBD<emph.end type="italics"/>XM/<emph type="italics"/>DEq<emph.end type="italics"/>) <lb/>ad 1/2<emph type="italics"/>BDXAP,<emph.end type="italics"/>&longs;ive ut (<emph type="italics"/>DAq/DEq<emph.end type="italics"/>) in <emph type="italics"/>DET<emph.end type="italics"/>ad <emph type="italics"/>DAP<emph.end type="italics"/>; adeoque ubi <lb/>areæ <emph type="italics"/>DET<emph.end type="italics"/>& <emph type="italics"/>DAP<emph.end type="italics"/>quam minimæ &longs;unt, in ratione æqualitatis. <lb/></s> <s>Æqualis igitur e&longs;t area quam minima (<emph type="italics"/>BD<emph.end type="italics"/>XV<emph type="sup"/>2<emph.end type="sup"/>/4<emph type="italics"/>AB<emph.end type="italics"/>) differentiæ quam <lb/>minimæ arearum <emph type="italics"/>DET<emph.end type="italics"/>& <emph type="italics"/>AbNK.<emph.end type="italics"/>Unde cum &longs;patia in Me­<lb/>dio utroque, in principio de&longs;cen&longs;us vel fine a&longs;cen&longs;us &longs;imul de&longs;crip­<lb/>ta accedunt ad æqualitatem, adeoque tunc &longs;unt ad invicem ut area <lb/>(<emph type="italics"/>BD<emph.end type="italics"/>XV<emph type="sup"/>2<emph.end type="sup"/>/4<emph type="italics"/>AB<emph.end type="italics"/>) & arearum <emph type="italics"/>DET<emph.end type="italics"/>& <emph type="italics"/>AbNK<emph.end type="italics"/>differentia; ob eorum ana­<lb/>loga incrementa nece&longs;&longs;e e&longs;t ut in æqualibus quibu&longs;cunque tempo­<lb/>ribus &longs;int ad invicem ut area illa (<emph type="italics"/>BD<emph.end type="italics"/>XV<emph type="sup"/>2<emph.end type="sup"/>/4<emph type="italics"/>AB<emph.end type="italics"/>) & arearum <emph type="italics"/>DET<emph.end type="italics"/>& <lb/><emph type="italics"/>AbNK<emph.end type="italics"/>differentia. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p><pb xlink:href="039/01/281.jpg" pagenum="253"/></subchap2><subchap2> <p type="margin"> <s><margin.target id="note228"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>SECTIO IV.<emph.end type="center"/><lb/><arrow.to.target n="note229"/></s></p> <p type="margin"> <s><margin.target id="note229"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Corporum Circulari Motu in Mediis re&longs;i&longs;tentibus.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>LEMMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Sit<emph.end type="italics"/>PQRr <emph type="italics"/>Spiralis quæ &longs;ecet radios omnes<emph.end type="italics"/>SP, SQ, SR, <emph type="italics"/>&c. </s> <s><lb/>in æqualibus angulis. </s> <s>Agatur recta<emph.end type="italics"/>PT <emph type="italics"/>quæ tangat eandem in <lb/>puncto quovis<emph.end type="italics"/>P, <emph type="italics"/>&longs;ecetque radium<emph.end type="italics"/>SQ <emph type="italics"/>in<emph.end type="italics"/>T; <emph type="italics"/>& ad Spiralem <lb/>erectis perpendiculis<emph.end type="italics"/>PO, QO <emph type="italics"/>concurrentibus in<emph.end type="italics"/>O, <emph type="italics"/>jungatur<emph.end type="italics"/><lb/>SO. <emph type="italics"/>Dico quod &longs;i puncta<emph.end type="italics"/>P <emph type="italics"/>&<emph.end type="italics"/>Q <emph type="italics"/>accedant ad invicem & co­<lb/>eant, angulus<emph.end type="italics"/>PSO <emph type="italics"/>evadet rectus, & ultima ratio rectanguli<emph.end type="italics"/><lb/>TQX2PS <emph type="italics"/>ad<emph.end type="italics"/>PQ<emph type="italics"/>quad. </s> <s>erit ratio æqualitatis.<emph.end type="italics"/></s></p> <p type="main"> <s>Etenim de angulis rectis <emph type="italics"/>OPQ, OQR<emph.end type="italics"/>&longs;ubducantur anguli <lb/>æquales <emph type="italics"/>SPQ, SQR,<emph.end type="italics"/>& manebunt anguli æquales <emph type="italics"/>OPS, OQS.<emph.end type="italics"/><lb/>Ergo Circulus qui tran&longs;it <lb/><figure id="id.039.01.281.1.jpg" xlink:href="039/01/281/1.jpg"/><lb/>per puncta <emph type="italics"/>O, S, P<emph.end type="italics"/>tran&longs;­<lb/>ibit etiam per punctum <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/><lb/>Coeant puncta <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>Q,<emph.end type="italics"/><lb/>& hic Circulus in loco co­<lb/>itus <emph type="italics"/>PQ<emph.end type="italics"/>tanget Spiralem, <lb/>adeoque perpendiculariter <lb/>&longs;ecabit rectam <emph type="italics"/>OP.<emph.end type="italics"/>Fiet <lb/>igitur <emph type="italics"/>OP<emph.end type="italics"/>diameter Cir­<lb/>culi hujus, & angulus <lb/><emph type="italics"/>OSP<emph.end type="italics"/>in &longs;emicirculo re­<lb/>ctus. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s>Ad <emph type="italics"/>OP<emph.end type="italics"/>demittantur perpendicula <emph type="italics"/>QD, SE,<emph.end type="italics"/>& linearum ratio­<lb/>nes ultimæ erunt huju&longs;modi: <emph type="italics"/>TQ<emph.end type="italics"/>ad <emph type="italics"/>PD<emph.end type="italics"/>ut <emph type="italics"/>TS<emph.end type="italics"/>vel <emph type="italics"/>PS<emph.end type="italics"/>ad <emph type="italics"/>PE,<emph.end type="italics"/><lb/>&longs;eu 2<emph type="italics"/>PO<emph.end type="italics"/>ad 2<emph type="italics"/>PS.<emph.end type="italics"/>Item <emph type="italics"/>PD<emph.end type="italics"/>ad <emph type="italics"/>PQ<emph.end type="italics"/>ut <emph type="italics"/>PQ<emph.end type="italics"/>ad 2<emph type="italics"/>PO.<emph.end type="italics"/>Et ex <lb/>æquo perturbate <emph type="italics"/>TQ<emph.end type="italics"/>ad <emph type="italics"/>PQ<emph.end type="italics"/>ut <emph type="italics"/>PQ<emph.end type="italics"/>ad 2<emph type="italics"/>PS.<emph.end type="italics"/>Unde fit <emph type="italics"/>PQq<emph.end type="italics"/><lb/>æquale <emph type="italics"/>TQX2PS. <expan abbr="q.">que</expan> E. D.<emph.end type="italics"/><pb xlink:href="039/01/282.jpg" pagenum="254"/><arrow.to.target n="note230"/></s></p> <p type="margin"> <s><margin.target id="note230"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XV. THEOREMA XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Medii den&longs;itas in locis &longs;ingulis &longs;it reciproce ut di&longs;tantia loeorum <lb/>a centro immobili, &longs;itque vis centripeta in duplicata ratione den­<lb/>&longs;itatis: dico quod corpus gyrari potest in Spirali, quæ radios <lb/>omnes a centro illo ductos inter&longs;ecat in angulo dato.<emph.end type="italics"/></s></p> <p type="main"> <s>Ponantur quæ in &longs;uperiore Lemmate, & producatur <emph type="italics"/>SQ<emph.end type="italics"/>ad <emph type="italics"/>V,<emph.end type="italics"/><lb/>ut &longs;it <emph type="italics"/>SV<emph.end type="italics"/>æqualis <emph type="italics"/>SP.<emph.end type="italics"/>Tempore quovis, in Medio re&longs;i&longs;tente, de­<lb/>&longs;cribat corpus arcum quam minimum <emph type="italics"/>PQ,<emph.end type="italics"/>& tempore duplo ar­<lb/>cum quam minimum <emph type="italics"/>PR<emph.end type="italics"/>; & decrementa horum arcuum ex re&longs;i­<lb/>&longs;tentia oriunda, &longs;ive defe­<lb/><figure id="id.039.01.282.1.jpg" xlink:href="039/01/282/1.jpg"/><lb/>ctus ab arcubus qui in Me­<lb/>dio non re&longs;i&longs;tente ii&longs;dem <lb/>temporibus de&longs;criberen­<lb/>tur, erunt ad invicem ut <lb/>quadrata temporum in <lb/>quibus generantur: E&longs;t <lb/>itaQ.E.D.crementum arcus <lb/><emph type="italics"/>PQ<emph.end type="italics"/>pars quarta decre­<lb/>menti arcus <emph type="italics"/>PR.<emph.end type="italics"/>Unde <lb/>etiam, &longs;i areæ <emph type="italics"/>PSQ<emph.end type="italics"/>æ­<lb/>qualis capiatur area <emph type="italics"/>QSr,<emph.end type="italics"/><lb/>erit decrementum arcus <lb/><emph type="italics"/>PQ<emph.end type="italics"/>æquale dimidio lineolæ <emph type="italics"/>Rr<emph.end type="italics"/>; adeoque vis re&longs;i&longs;tentiæ & vis cen­<lb/>tripeta &longs;unt ad invicem ut lineolæ 1/2<emph type="italics"/>Rr<emph.end type="italics"/>& <emph type="italics"/>TQ<emph.end type="italics"/>quas &longs;imul generant. </s> <s><lb/>Quoniam vis centripeta, qua corpus urgetur in <emph type="italics"/>P,<emph.end type="italics"/>e&longs;t reciproce ut <lb/><emph type="italics"/>SPq,<emph.end type="italics"/>& (per Lem. </s> <s>X. Lib. </s> <s>1,) lineola <emph type="italics"/>TQ,<emph.end type="italics"/>quæ vi illa generatur, e&longs;t <lb/>in ratione compo&longs;ita ex ratione hujus vis & ratione duplicata tem­<lb/>poris quo arcus <emph type="italics"/>PQ<emph.end type="italics"/>de&longs;cribitur, (Nam re&longs;i&longs;tentiam in hoc ca&longs;u, <lb/>ut infinite minorem quam vis centripeta, negligo) erit <emph type="italics"/>TQXSPq<emph.end type="italics"/><lb/>id e&longs;t (per Lemma novi&longs;&longs;imum) 1/2<emph type="italics"/>PQqXSP,<emph.end type="italics"/>in ratione duplicata <lb/>temporis, adeoque tempus e&longs;t ut <emph type="italics"/>PQX√SP<emph.end type="italics"/>; & corporis veloci­<lb/>tas, qua arcus <emph type="italics"/>PQ<emph.end type="italics"/>illo tempore de&longs;cribitur, ut (<emph type="italics"/>PQ/PQX√SP<emph.end type="italics"/>) &longs;eu <lb/>(1/√<emph type="italics"/>SP<emph.end type="italics"/>), hoc e&longs;t, in &longs;ubduplicata ratione ip&longs;ius <emph type="italics"/>SP<emph.end type="italics"/>reciproce. </s> <s>Et &longs;i­<lb/>mili argumento, velocitas qua arcus <emph type="italics"/>QR<emph.end type="italics"/>de&longs;cribitur, e&longs;t in &longs;ub-<pb xlink:href="039/01/283.jpg" pagenum="255"/>duplicata ratione ip&longs;ius <emph type="italics"/>SQ<emph.end type="italics"/>reciproce. </s> <s>Sunt autem arcus illi <emph type="italics"/>PQ<emph.end type="italics"/><lb/><arrow.to.target n="note231"/>& <emph type="italics"/>QR<emph.end type="italics"/>ut velocitates de&longs;criptrices ad invicem, id e&longs;t, in &longs;ubdupli­<lb/>cata ratione <emph type="italics"/>SQ<emph.end type="italics"/>ad <emph type="italics"/>SP,<emph.end type="italics"/>&longs;ive ut <emph type="italics"/>SQ<emph.end type="italics"/>ad √<emph type="italics"/>SPXSQ<emph.end type="italics"/>; & ob æqua­<lb/>les angulos <emph type="italics"/>SPQ, SQr<emph.end type="italics"/>& æquales areas <emph type="italics"/>PSQ, QSr,<emph.end type="italics"/>e&longs;t ar­<lb/>cus <emph type="italics"/>PQ<emph.end type="italics"/>ad arcum <emph type="italics"/>Qr<emph.end type="italics"/>ut <emph type="italics"/>SQ<emph.end type="italics"/>ad <emph type="italics"/>SP.<emph.end type="italics"/>Sumantur proportionalium <lb/>con&longs;equentium differentiæ, & fiet arcus <emph type="italics"/>PQ<emph.end type="italics"/>ad arcum <emph type="italics"/>Rr<emph.end type="italics"/>ut <emph type="italics"/>SQ<emph.end type="italics"/><lb/>ad <emph type="italics"/>SP-√SPXSQ,<emph.end type="italics"/>&longs;eu 1/2<emph type="italics"/>VQ<emph.end type="italics"/>; nam punctis <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>Q<emph.end type="italics"/>coeunti­<lb/>bus, ratio ultima <emph type="italics"/>SP-√SPXSQ<emph.end type="italics"/>ad 1/2<emph type="italics"/>VQ<emph.end type="italics"/>&longs;it æqualitatis. </s> <s><lb/>Quoniam decrementum arcus <emph type="italics"/>PQ,<emph.end type="italics"/>ex re&longs;i&longs;tentia oriundum, &longs;ive <lb/>hujus duplum <emph type="italics"/>Rr,<emph.end type="italics"/>e&longs;t ut re&longs;i&longs;tentia & quadratum temporis con­<lb/>junctim; erit re&longs;i&longs;tentia ut (<emph type="italics"/>Rr/PQqXSP<emph.end type="italics"/>). Erat autem <emph type="italics"/>PQ<emph.end type="italics"/>ad <emph type="italics"/>Rr,<emph.end type="italics"/><lb/>ut <emph type="italics"/>SQ<emph.end type="italics"/>ad 1/2<emph type="italics"/>VQ,<emph.end type="italics"/>& inde (<emph type="italics"/>Rr/PQqXSP<emph.end type="italics"/>) fit ut (1/2<emph type="italics"/>VQ/PQXSPXSQ<emph.end type="italics"/>) &longs;ive <lb/>ut (1/2<emph type="italics"/>OS/OPXSPq<emph.end type="italics"/>). Namque punctis <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>Q<emph.end type="italics"/>coeuntibus, <emph type="italics"/>SP<emph.end type="italics"/>& <emph type="italics"/>SQ<emph.end type="italics"/><lb/>coincidunt, & angulus <emph type="italics"/>PVQ<emph.end type="italics"/>fit rectus; & ob &longs;imilia triangula <lb/><emph type="italics"/>PVQ, PSO,<emph.end type="italics"/>fit <emph type="italics"/>PQ<emph.end type="italics"/>ad 1/2<emph type="italics"/>VQ<emph.end type="italics"/>ut <emph type="italics"/>OP<emph.end type="italics"/>ad 1/2<emph type="italics"/>OS.<emph.end type="italics"/>E&longs;t igitur <lb/>(<emph type="italics"/>OS/OPXSPq<emph.end type="italics"/>) ut re&longs;i&longs;tentia, id e&longs;t, in ratione den&longs;itatis Medii in <emph type="italics"/>P<emph.end type="italics"/><lb/>& ratione duplicata velocitatis conjunctim. </s> <s>Auferatur duplicata <lb/>ratio velocitatis, nempe ratio (1/<emph type="italics"/>SP<emph.end type="italics"/>), & manebit Medii den&longs;itas in <lb/><emph type="italics"/>P<emph.end type="italics"/>ut (<emph type="italics"/>OS/OPXSP<emph.end type="italics"/>). Detur Spiralis, & ob datam rationem <emph type="italics"/>OS<emph.end type="italics"/>ad <lb/><emph type="italics"/>OP,<emph.end type="italics"/>den&longs;itas Medii in <emph type="italics"/>P<emph.end type="italics"/>erit ut (1/<emph type="italics"/>SP<emph.end type="italics"/>). In Medio igitur cujus <lb/>den&longs;itas e&longs;t reciproce ut di&longs;tantia a centro <emph type="italics"/>SP,<emph.end type="italics"/>corpus gyrari po­<lb/>te&longs;t in hac Spirali. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note231"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Velocitas in loco quovis <emph type="italics"/>P<emph.end type="italics"/>ea &longs;emper e&longs;t quacum cor­<lb/>pus in Medio non re&longs;i&longs;tente gyrari pote&longs;t in Circulo, ad eandem a <lb/>centro di&longs;tantiam <emph type="italics"/>SP.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Medii den&longs;itas, &longs;i datur di&longs;tantia <emph type="italics"/>SP,<emph.end type="italics"/>e&longs;t ut (<emph type="italics"/>OS/OP<emph.end type="italics"/>), &longs;in <lb/>di&longs;tantia illa non datur, ut (<emph type="italics"/>OS/OPXSP<emph.end type="italics"/>). Et inde Spiralis ad quam­<lb/>libet Medii den&longs;itatem aptari pote&longs;t. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Vis re&longs;i&longs;tentiæ in loco quovis <emph type="italics"/>P,<emph.end type="italics"/>e&longs;t ad vim centripe-<pb xlink:href="039/01/284.jpg" pagenum="256"/><arrow.to.target n="note232"/>tam in eodem loco ut 1/2<emph type="italics"/>OS<emph.end type="italics"/>ad <emph type="italics"/>OP.<emph.end type="italics"/>Nam vires illæ &longs;unt ad invi­<lb/>vicem ut 1/4<emph type="italics"/>Rr<emph.end type="italics"/>& <emph type="italics"/>TQ<emph.end type="italics"/>&longs;ive ut (1/4<emph type="italics"/>VQXPQ/SQ<emph.end type="italics"/>) & (1/2<emph type="italics"/>PQq/SP<emph.end type="italics"/>), hoc e&longs;t, ut 1/2<emph type="italics"/>VQ<emph.end type="italics"/><lb/>& <emph type="italics"/>PQ,<emph.end type="italics"/>&longs;eu 1/2<emph type="italics"/>OS<emph.end type="italics"/>& <emph type="italics"/>OP.<emph.end type="italics"/>Data igitur Spirali datur proportio re­<lb/>&longs;i&longs;tentiæ ad vim centripetam, & vicever&longs;a ex data illa proportione <lb/>datur Spiralis. </s></p> <p type="margin"> <s><margin.target id="note232"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Corpus itaque gyrari nequit in hac Spirali, ni&longs;i ubi vis <lb/>re&longs;i&longs;tentiæ minor e&longs;t quam dimidium vis centripetæ. </s> <s>Fiat re&longs;i&longs;ten­<lb/>tia æqualis dimidio vis centripetæ & Spiralis conveniet cum linea <lb/>recta <emph type="italics"/>PS,<emph.end type="italics"/>inque hac recta corpus de&longs;cendet ad centrum, ea cum <lb/>velocitate quæ &longs;it ad velocitatem qua probavimus in &longs;uperioribus <lb/>in ca&longs;u Parabolæ (Theor. </s> <s>X, Lib. </s> <s>I,) de&longs;cen&longs;um in Medio non re&longs;i­<lb/>&longs;tente fieri, in &longs;ubduplicata ratione unitatis ad numerum binarium. </s> <s><lb/>Et tempora de&longs;cen&longs;us hic erunt reciproce ut velocitates, atque <lb/>adeo dantur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Et quoniam in æqualibus a centro di&longs;tantiis velocitas <lb/>eadem e&longs;t in Spirali <emph type="italics"/>PQR<emph.end type="italics"/>atQ.E.I. recta <emph type="italics"/>SP,<emph.end type="italics"/>& longitudo Spi­<lb/>ralis ad longitudinem rectæ <emph type="italics"/>PS<emph.end type="italics"/>e&longs;t in data ratione, nempe in <lb/>ratione <emph type="italics"/>OP<emph.end type="italics"/>ad <emph type="italics"/>OS<emph.end type="italics"/>; tempus de&longs;cen&longs;us in Spirali erit ad tem­<lb/>pus de&longs;cen&longs;us in recta <emph type="italics"/>SP<emph.end type="italics"/>in eadem illa data ratione, proinde­<lb/>Q.E.D.tur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Si centro <emph type="italics"/>S<emph.end type="italics"/>intervallis duobus quibu&longs;cunQ.E.D.tis de&longs;cri­<lb/>bantur duo Circuli; & manentibus hi&longs;ce Circulis, mutetur utcun­<lb/>que angulus quem Spiralis continet cum radio <emph type="italics"/>PS:<emph.end type="italics"/>numerus revo­<lb/>lutionum quas corpus intra Circulorum circumferentias, pergendo <lb/>in Spirali a circumferentia ad circumferentiam, complere pote&longs;t, e&longs;t <lb/>ut (<emph type="italics"/>PS/OS<emph.end type="italics"/>), &longs;ive ut Tangens anguli illius quem Spiralis continet cum <lb/>radio <emph type="italics"/>PS<emph.end type="italics"/>; tempus vero revolutionum earundem ut (<emph type="italics"/>OP/OS<emph.end type="italics"/>), id e&longs;t, ut <lb/>Secans anguli eju&longs;dem, vel etiam reciproce ut Medii den&longs;itas. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. Si corpus, in Medio cujus den&longs;itas e&longs;t reciproce ut di­<lb/>&longs;tantia loeorum a centro, revolutionem in Curva quacunque <emph type="italics"/>AEB<emph.end type="italics"/><lb/>circa centrum illud fecerit, & Radium primum <emph type="italics"/>AS<emph.end type="italics"/>in eodem an­<lb/>gulo &longs;ecuerit in <emph type="italics"/>B<emph.end type="italics"/>quo prius in <emph type="italics"/>A,<emph.end type="italics"/>idque cum velocitate quæ fue­<lb/>rit ad velocitatem &longs;uam primam in <emph type="italics"/>A<emph.end type="italics"/>reciproce in &longs;ubduplica­<lb/>ta ratione di&longs;tantiarum a centro (id e&longs;t, ut <emph type="italics"/>AS<emph.end type="italics"/>ad mediam pro­<lb/>portionalem inter <emph type="italics"/>AS<emph.end type="italics"/>& <emph type="italics"/>BS<emph.end type="italics"/>) corpus illud perget innume­<lb/>ras con&longs;imiles revolutiones <emph type="italics"/>BFC, CGD<emph.end type="italics"/>&c. </s> <s>facere, & inter&longs;e-<pb xlink:href="039/01/285.jpg" pagenum="257"/>ctionibus di&longs;tinguet Radium <emph type="italics"/>AS<emph.end type="italics"/>in partes <emph type="italics"/>AS, BS, CS, DS,<emph.end type="italics"/>&c. <lb/><arrow.to.target n="note233"/>continue proportionales. </s> <s>Revolutionum vero tempora erunt ut <lb/><figure id="id.039.01.285.1.jpg" xlink:href="039/01/285/1.jpg"/><lb/>perimetri Orbitarum <emph type="italics"/>AEB, BFC, CGD,<emph.end type="italics"/>&c. </s> <s>directe, & veloci­<lb/>tates in principiis <emph type="italics"/>A, B, C,<emph.end type="italics"/>inver&longs;e; id e&longs;t, ut <emph type="italics"/>AS<emph.end type="italics"/><emph type="sup"/>1/2<emph.end type="sup"/>, <emph type="italics"/>BS<emph.end type="italics"/><emph type="sup"/>1/2<emph.end type="sup"/>, <emph type="italics"/>CS<emph.end type="italics"/><emph type="sup"/>1/2<emph.end type="sup"/>. </s> <s>At­<lb/>que tempus totum, quo corpus perveniet ad centrum, erit ad tem­<lb/>pus revolutionis primæ, ut &longs;umma omnium continue proportiona­<lb/>lium <emph type="italics"/>AS<emph.end type="italics"/><emph type="sup"/>1/2<emph.end type="sup"/>, <emph type="italics"/>BS<emph.end type="italics"/><emph type="sup"/>1/2<emph.end type="sup"/>, <emph type="italics"/>CS<emph.end type="italics"/><emph type="sup"/>1/2<emph.end type="sup"/> pergentium in infinitum, ad terminum pri­<lb/>mum <emph type="italics"/>AS<emph.end type="italics"/><emph type="sup"/>1/2<emph.end type="sup"/>; id e&longs;t, ut terminus ille primus <emph type="italics"/>AS<emph.end type="italics"/><emph type="sup"/>1/2<emph.end type="sup"/> ad differentiam du­<lb/>orum primorum <emph type="italics"/>AS<emph.end type="italics"/><emph type="sup"/>1/2<emph.end type="sup"/>-<emph type="italics"/>BS<emph.end type="italics"/><emph type="sup"/>1/2<emph.end type="sup"/>, &longs;ive ut 2/3<emph type="italics"/>AS<emph.end type="italics"/>ad <emph type="italics"/>AB<emph.end type="italics"/>quam proxime. </s> <s><lb/>Unde tempus illud totum expedite invenitur. </s></p> <p type="margin"> <s><margin.target id="note233"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>8. Ex his etiam præter propter colligere licet motus cor­<lb/>porum in Mediis, quorum den&longs;itas aut uniformis e&longs;t, aut aliam <lb/>quamcunque legem a&longs;&longs;ignatam ob&longs;ervat. </s> <s>Centro <emph type="italics"/>S,<emph.end type="italics"/>intervallis con­<lb/>tinue proportionalibus <emph type="italics"/>SA, SB, SC,<emph.end type="italics"/>&c. </s> <s>de&longs;cribe Circulos quot­<lb/>cunque, & &longs;tatue tempus revolutionum inter perimetros duorum <lb/>quorumvis ex his Circulis, in Medio de quo egimus, e&longs;&longs;e ad tempus <lb/>revolutionum inter eo&longs;dem in Medio propo&longs;ito, ut Medii propo­<lb/>&longs;iti den&longs;itas mediocris inter hos Circulos ad Medii, de quo egimus, <lb/>den&longs;itatem mediocrem inter eo&longs;dem quam proxime: Sed & in ea­<lb/>dem quoque ratione e&longs;&longs;e Secantem anguli quo Spiralis præfinita, <lb/>in Medio de quo egimus, &longs;ecat radium <emph type="italics"/>AS,<emph.end type="italics"/>ad Secantem anguli <pb xlink:href="039/01/286.jpg" pagenum="258"/><arrow.to.target n="note234"/>quo Spiralis nova &longs;ecat radium eundem in Medio propo&longs;ito: At­<lb/>que etiam ut &longs;unt eorundem angulorum Tangentes ita e&longs;&longs;e numeros <lb/>revolutionum omnium inter Circulos eo&longs;dem duos quam proxime. </s> <s><lb/>Si hæc fiant pa&longs;&longs;im inter Circulos binos, continuabitur motus per <lb/>Circulos omnes. </s> <s>Atque hoc pacto haud difficulter imaginari po&longs;&longs;i­<lb/>mus quibus modis ac temporibus corpora in Medio quocunque re­<lb/>gulari gyrari debebunt. </s></p> <p type="margin"> <s><margin.target id="note234"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>9. Et quamvis motus excentrici in Spiralibus ad formam <lb/>Ovalium accedentibus peragantur; tamen concipiendo Spiralium <lb/>illarum &longs;ingulas revolutiones ii&longs;dem ab invicem intervallis di&longs;tare, <lb/>ii&longs;demque gradibus ad centrum accedere cum Spirali &longs;uperius de­<lb/>&longs;cripta, intelligemus etiam quomodo motus corporum in huju&longs;mo­<lb/>di Spiralibus peragantur. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XVI. THEOREMA XIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Medii den&longs;itas in locis &longs;ingulis &longs;it reciproce ut di&longs;tantia loco­<lb/>rum a centro immobili, &longs;itque vis centripeta reciproce ut dig­<lb/>nitas quælibet eju&longs;dem di&longs;tantiæ: dico quod corpus gyrari potest <lb/>in Spirali quæ radios omnes a centro illo ductos inter&longs;ecat in <lb/>angulo dato.<emph.end type="italics"/></s></p> <p type="main"> <s>Demon&longs;tratur eadem methodo cum Propo&longs;itione &longs;uperiore. </s> <s><lb/>Nam &longs;i vis centripeta in <emph type="italics"/>P<emph.end type="italics"/>&longs;it reciproce ut di&longs;tantiæ <emph type="italics"/>SP<emph.end type="italics"/>dignitas <lb/>quælibet <emph type="italics"/>SP<emph type="sup"/>n<emph.end type="italics"/>+1<emph.end type="sup"/> cujus index e&longs;t <emph type="italics"/>n<emph.end type="italics"/>+1; colligetur ut &longs;upra, <lb/>quod tempus quo corpus de&longs;cribit arcum quemvis <emph type="italics"/>PQ<emph.end type="italics"/>erit ut <lb/><emph type="italics"/>PQXSP<emph.end type="italics"/><emph type="sup"/>1/2<emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>, & re&longs;i&longs;tentia in <emph type="italics"/>P<emph.end type="italics"/>ut (<emph type="italics"/>Rr/PQqXSP<emph type="sup"/>n<emph.end type="sup"/><emph.end type="italics"/>), &longs;ive ut (―1-1/2<emph type="italics"/>nXVQ/PQXSP<emph type="sup"/>n<emph.end type="sup"/>XSQ<emph.end type="italics"/>), <lb/>adeoque ut (―1-1/2<emph type="italics"/>nXOS/OPXSP<emph type="sup"/>n+1<emph.end type="sup"/><emph.end type="italics"/>), hoc e&longs;t, ob datum (―1-1/2<emph type="italics"/>nXOS/OP<emph.end type="italics"/>), recipro­<lb/>ce ut <emph type="italics"/>SP<emph type="sup"/>n+1<emph.end type="sup"/>.<emph.end type="italics"/>Et propterea, cum velocitas &longs;it reciproce ut <emph type="italics"/>SP<emph.end type="italics"/><emph type="sup"/>1/2<emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>, <lb/>den&longs;itas in <emph type="italics"/>P<emph.end type="italics"/>erit reciproce ut <emph type="italics"/>SP.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Re&longs;i&longs;tentia e&longs;t ad vim centripetam, ut ―1-1/2<emph type="italics"/>nXOS<emph.end type="italics"/><lb/>ad <emph type="italics"/>OP.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si vis centripeta &longs;it reciproce ut <emph type="italics"/>SPcub,<emph.end type="italics"/>erit 1-1/2<emph type="italics"/>n=o<emph.end type="italics"/>; <lb/>adeoque re&longs;i&longs;tentia & den&longs;itas Medii nulla erit, ut in Propo&longs;itione <lb/>nona Libri primi. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Si vis centripeta &longs;it reciproce ut dignitas aliqua radii <lb/><emph type="italics"/>SP<emph.end type="italics"/>cujus index e&longs;t major numero 3, re&longs;i&longs;tentia affirmativa in nega­<lb/>tivam mutabitur. </s></p><pb xlink:href="039/01/287.jpg" pagenum="259"/> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><lb/><arrow.to.target n="note235"/></s></p> <p type="margin"> <s><margin.target id="note235"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s>Cæterum hæc Propo&longs;itio & &longs;uperiores, quæ ad Media inæquali­<lb/>ter den&longs;a &longs;pectant, intelligendæ &longs;unt de motu corporum adeo par­<lb/>vorum, ut Medii ex uno corporis latere major den&longs;itas quam ex al­<lb/>tero non con&longs;ideranda veniat. </s> <s>Re&longs;i&longs;tentiam quoque cæteris paribus <lb/>den&longs;itati proportionalem e&longs;&longs;e &longs;uppono. </s> <s>Unde in Mediis quorum <lb/>vis re&longs;i&longs;tendi non e&longs;t ut den&longs;itas, debet den&longs;itas eo u&longs;que augeri vel <lb/>diminui, ut re&longs;i&longs;tentiæ vel tollatur exce&longs;&longs;us vel defectus &longs;uppleatur. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XVII. PROBLEMA IV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Invenire & vim centripetam & Medii re&longs;i&longs;tentiam qua corpus <lb/>in data Spirali, data velocitatis Lege, revolvi potest.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit Spiralis illa <emph type="italics"/>PQR.<emph.end type="italics"/>Ex velocitate qua corpus percurrit ar­<lb/>cum quam minimum <emph type="italics"/>PQ<emph.end type="italics"/>dabitur tempus, & ex altitudine <emph type="italics"/>TQ,<emph.end type="italics"/><lb/>quæ e&longs;t ut vis centripeta & quadratum temporis, dabitur vis. </s> <s>De­<lb/>inde ex arearum, æqualibus temporum particulis confectarum <emph type="italics"/>PSQ<emph.end type="italics"/><lb/>& <emph type="italics"/>QSR,<emph.end type="italics"/>differentia <emph type="italics"/>RSr,<emph.end type="italics"/>dabitur corporis retardatio, & ex re­<lb/>tardatione invenietur re&longs;i&longs;tentia ac den&longs;itas Medii. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XVIII. PROBLEMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Data Lege vis centripetæ, invenire Medii den&longs;itatem in locis &longs;in­<lb/>gulis, qua corpus datam Spiralem de&longs;cribet.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Ex vi centripeta invenienda e&longs;t velocitas in locis &longs;ingulis, de­<lb/>inde ex velocitatis retardatione quærenda Medii den&longs;itas: ut in <lb/>Propo&longs;itione &longs;uperiore. </s></p> <p type="main"> <s>Methodum vero tractandi hæc Problemata aperui in hujus Pro­<lb/>po&longs;itione decima, & Lemmate &longs;ecundo; & Lectorem in huju&longs;modi <lb/>perplexis di&longs;qui&longs;itionibus diutius detinere nolo. </s> <s>Addenda jam <lb/>&longs;unt aliqua de viribus corporum ad progrediendum, deQ.E.D.n&longs;i­<lb/>tate & re&longs;i&longs;tentia Mediorum, in quibus motus hactenus expo&longs;iti & <lb/>his affines peraguntur. </s></p><pb xlink:href="039/01/288.jpg" pagenum="260"/></subchap2><subchap2> <p type="main"> <s><arrow.to.target n="note236"/></s></p> <p type="margin"> <s><margin.target id="note236"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>SECTIO V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Den&longs;itate & Compre&longs;&longs;ione Fluidorum, deque <lb/>Hydro&longs;tatica.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>Definitio Fluidi.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Fluidum e&longs;t corpus omne cujus partes cedunt vi cuicunQ.E.I.latæ, <lb/>& cedendo facile moventur inter &longs;e.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XIX. THEOREMA XIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Fluidi homogenei & immoti quod in va&longs;e quocunQ.E.I.moto clau­<lb/>ditur & undique comprimitur, partes omnes (&longs;epo&longs;ita conden­<lb/>&longs;ationis, gravitatis & virium omnium centripetarum con&longs;ide­<lb/>ratione) æqualiter premuntur undique, & ab&longs;que omni motu a <lb/>pre&longs;&longs;ione illa orto permanent in locis &longs;uis.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>1. In va&longs;e &longs;phærico <emph type="italics"/>ABC<emph.end type="italics"/>claudatur & uniformiter com­<lb/>primatur fluidum undique: dico quod eju&longs;dem pars nulla ex illa <lb/>pre&longs;&longs;ione movebitur. </s> <s>Nam &longs;i pars aliqua <emph type="italics"/>D<emph.end type="italics"/><lb/><figure id="id.039.01.288.1.jpg" xlink:href="039/01/288/1.jpg"/><lb/>moveatur, nece&longs;&longs;e e&longs;t ut omnes huju&longs;modi <lb/>partes, ad eandem a centro di&longs;tantiam un­<lb/>dique con&longs;i&longs;tentes, &longs;imili motu &longs;imul move­<lb/>antur; atque hoc adeo quia &longs;imilis & æ­<lb/>qualis e&longs;t omnium pre&longs;&longs;io, & motus omnis <lb/>exclu&longs;us &longs;upponitur, ni&longs;i qui a pre&longs;&longs;ione il­<lb/>la oriatur. </s> <s>Atqui non po&longs;&longs;unt omnes ad <lb/>centrum propius accedere, ni&longs;i fluidum ad <lb/>centrum conden&longs;etur; contra Hypothe&longs;in. </s> <s><lb/>Non po&longs;&longs;unt longius ab eo recedere, ni&longs;i <lb/>fluidum ad circumferentiam conden&longs;etur; <lb/>etiam contra Hypothe&longs;in. </s> <s>Non po&longs;&longs;unt &longs;ervata &longs;ua a centro di­<lb/>&longs;tantia moveri in plagam quamcunque, quia pari ratione movebun­<lb/>tur in plagam contrariam; in plagas autem contrarias non pote&longs;t <pb xlink:href="039/01/289.jpg" pagenum="261"/>pars eadem, eodem tempore, moveri. </s> <s>Ergo fluidi pars nulla de lo­<lb/><arrow.to.target n="note237"/>co &longs;uo movebitur. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note237"/>LIBER <lb/>SECUNDUS</s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>2. Dico jam quod fluidi hujus partes omnes &longs;phæricæ æqua­<lb/>liter premuntur undique: &longs;it enim <emph type="italics"/>EF<emph.end type="italics"/>pars &longs;phærica fluidi, & &longs;i <lb/>hæc undique non premitur æqualiter, augeatur pre&longs;&longs;io minor, u&longs;­<lb/>Q.E.D.m ip&longs;a undique prematur æqualiter; & partes ejus, per <lb/>Ca&longs;um primum, permanebunt in locis &longs;uis. </s> <s>Sed ante auctam pre&longs;­<lb/>&longs;ionem permanebunt in locis &longs;uis, per Ca&longs;um eundum primum, & <lb/>additione pre&longs;&longs;ionis novæ movebuntur de locis &longs;uis, per definitio­<lb/>nem Fluidi. </s> <s>Quæ duo repugnant. </s> <s>Ergo fal&longs;o dicebatur quod Sphæ­<lb/>ra <emph type="italics"/>EF<emph.end type="italics"/>non undique premebatur æqualiter. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>3. Dico præterea quod diver&longs;arum partium &longs;phæricarum æ­<lb/>qualis &longs;it pre&longs;&longs;io. </s> <s>Nam partes &longs;phæricæ contiguæ &longs;e mutuo pre­<lb/>munt æqualiter in puncto contactus, per motus Legem III. </s> <s>Sed &, <lb/>per Ca&longs;um &longs;ecundum, undique premuntur eadem vi. </s> <s>Partes igitur <lb/>duæ quævis &longs;phæricæ non contiguæ, quia pars &longs;phærica intermedia <lb/>tangere pote&longs;t utramque, prementur eadem vi. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>4. Dico jam quod fluidi partes omnes ubique premuntur <lb/>æqualiter. </s> <s>Nam partes duæ quævis tangi po&longs;&longs;unt a partibus Sphæ­<lb/>ricis in punctis quibu&longs;cunque, & ibi partes illas Sphæricas æquali­<lb/>ter premunt, per Ca&longs;um 3. & vici&longs;&longs;im ab illis æqualiter premuntur, <lb/>per Motus Legem tertiam. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>5. Cum igitur fluidi pars quælibet <emph type="italics"/>GHI<emph.end type="italics"/>in fluido reliquo <lb/>tanquam in va&longs;e claudatur, & undique prematur æqualiter, partes <lb/>autem ejus &longs;e mutuo æqualiter premant & quie&longs;cant inter &longs;e; ma­<lb/>nife&longs;tum e&longs;t quod Fluidi cuju&longs;cunque <emph type="italics"/>GHI,<emph.end type="italics"/>quod undique premi­<lb/>tur æqualiter, partes omnes &longs;e mutuo premunt æqualiter, & qui­<lb/>e&longs;cunt inter &longs;e. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>6. Igitur &longs;i Fluidum illud in va&longs;e non rigido claudatur, & <lb/>undique non prematur æqualiter, cedet idem pre&longs;&longs;ioni fortiori, per <lb/>Definitionem Fluiditatis. </s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>7. IdeoQ.E.I. va&longs;e rigido Fluidum non &longs;u&longs;tinebit pre&longs;&longs;io­<lb/>nem fortiorem ex uno latere quam ex alio, &longs;ed eidem cedet, idque <lb/>in momento temporis, quia latus va&longs;is rigidum non per&longs;equitur li­<lb/>quorem cedentem. </s> <s>Cedendo autem urgebit latus oppo&longs;itum, & <lb/>&longs;ic pre&longs;&longs;io undique ad æqualitatem verget. </s> <s>Et quoniam Fluidum, <lb/>quam primum a parte magis pre&longs;&longs;a recedere conatur, inhibetur per <lb/>re&longs;i&longs;tentiam va&longs;is ad latus oppo&longs;itum; reducetur pre&longs;&longs;io undique <lb/>ad æqualitatem, in momento temporis, ab&longs;que motu locali: & &longs;ub­<lb/>inde partes fluidi, per Ca&longs;um quintum, &longs;e mutuo prement æqua­<lb/>liter, & quie&longs;cent inter &longs;e. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/><pb xlink:href="039/01/290.jpg" pagenum="262"/><arrow.to.target n="note238"/></s></p> <p type="margin"> <s><margin.target id="note238"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Unde nec motus partium fluidi inter &longs;e, per pre&longs;&longs;ionem <lb/>fluido ubivis in externa &longs;uperficie illatam, mutari po&longs;&longs;unt, ni&longs;i qua­<lb/>tenus aut figura &longs;uperficiei alicubi mutatur, aut omnes fluidi partes <lb/>inten&longs;ius vel remi&longs;&longs;ius &longs;e&longs;e premendo difficilius vel facilius labun­<lb/>tur inter &longs;e. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XX. THEOREMA XV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Fluidi Sphærici, & in æqualibus a centro di&longs;tantiis homogenei, <lb/>fundo Sphærico concentrico incumbentis partes &longs;ingulæ ver&longs;us <lb/>centrum totius gravitent; &longs;u&longs;tinet fundum pondus Cylindri, cu­<lb/>jus bafis æqualis est &longs;uperficiei fundi, & altitudo eadem quæ <lb/>Fluidi incumbentis.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>DHM<emph.end type="italics"/>&longs;uperficies &longs;undi, & <emph type="italics"/>AEI<emph.end type="italics"/><lb/><figure id="id.039.01.290.1.jpg" xlink:href="039/01/290/1.jpg"/><lb/>&longs;uperficies &longs;uperior fluidi. </s> <s>Superficiebus <lb/>&longs;phæricis innumeris <emph type="italics"/>BFK, CGL<emph.end type="italics"/>di&longs;tin­<lb/>guatur fluidum in Orbes concentricos æ­<lb/>qualiter cra&longs;&longs;os; & concipe vim gravita­<lb/>tis agere &longs;olummodo in &longs;uperficiem &longs;upe­<lb/>riorem Orbis cuju&longs;que, & æquales e&longs;&longs;e a­<lb/>ctiones in æquales partes &longs;uperficierum om­<lb/>nium. </s> <s>Premitur ergo &longs;uperficies &longs;uprema <lb/><emph type="italics"/>AE<emph.end type="italics"/>vi &longs;implici gravitatis propriæ, qua & <lb/>omnes Orbis &longs;upremi partes & &longs;uperficies <lb/>&longs;ecunda <emph type="italics"/>BFK<emph.end type="italics"/>(per Prop. </s> <s>XIX.) pro men&longs;ura &longs;ua æqualiter pre­<lb/>muntur. </s> <s>Premitur præterea &longs;uperficies &longs;ecunda <emph type="italics"/>BFK<emph.end type="italics"/>vi propriæ <lb/>gravitatis, quæ addita vi priori facit pre&longs;&longs;ionem duplam. </s> <s>Hac <lb/>pre&longs;&longs;ione, pro men&longs;ura &longs;ua, & in&longs;uper vi propriæ gravitatis, id e&longs;t <lb/>pre&longs;&longs;ione tripla, urgetur &longs;uperficies tertia <emph type="italics"/>CGL.<emph.end type="italics"/>Et &longs;imiliter pre&longs;­<lb/>&longs;ione quadrupla urgetur &longs;uperficies quarta, quintupla quinta, & <lb/>&longs;ic deinceps. </s> <s>Pre&longs;&longs;io igitur qua &longs;uperficies unaquæque urgetur, <lb/>non e&longs;t ut quantitas &longs;olida fluidi incumbentis, &longs;ed ut numerus Or­<lb/>bium ad u&longs;que &longs;ummitatem fluidi; & æquatur gravitati Orbis infi­<lb/>mi multiplicatæ per numerum Orbium: hoc e&longs;t, gravitati &longs;olidi cu­<lb/>jus ultima ratio ad Cylindrum præfinitum, (&longs;i modo Orbium au­<lb/>geatur numerus & minuatur cra&longs;&longs;itudo in infinitum, &longs;ic ut actio <lb/>gravitatis a &longs;uperficie infima ad &longs;upremam continua reddatur) fiet <lb/>ratio æqualitatis. </s> <s>Su&longs;tinet ergo &longs;uperficies infima pondus Cylindri <pb xlink:href="039/01/291.jpg" pagenum="263"/>præfiniti. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/>Et &longs;imili argumentatione patet Propo&longs;itio, </s></p> <p type="main"> <s><arrow.to.target n="note239"/>ubi gravitas decre&longs;cit in ratione quavis a&longs;&longs;ignata di&longs;tantiæ a centro, <lb/>ut & ubi Fluidum &longs;ur&longs;um rarius e&longs;t, deor&longs;um den&longs;ius. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note239"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Igitur fundum non urgetur a toto fluidi incumbentis <lb/>pondere, &longs;ed eam &longs;olummodo ponderis partem &longs;u&longs;tinet quæ in <lb/>propo&longs;itione de&longs;cribitur; pondere reliquo a fluidi figura fornicata <lb/>&longs;u&longs;tentato. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. In æqualibus autem a centro di&longs;tantiis eadem &longs;emper e&longs;t <lb/>pre&longs;&longs;ionis quantitas, &longs;ive &longs;uperficies pre&longs;&longs;a &longs;it Horizonti parallela <lb/>vel perpendicularis vel obliqua; &longs;ive fluidum, a &longs;uperficie pre&longs;&longs;a &longs;ur­<lb/>&longs;um continuatum, &longs;urgat perpendiculariter &longs;ecundum lineam rectam, <lb/>vel &longs;erpit oblique per tortas cavitates & canales, ea&longs;que regulares <lb/>vel maxime irregulares, amplas vel angu&longs;ti&longs;&longs;imas. </s> <s>Hi&longs;ce circum­<lb/>&longs;tantiis pre&longs;&longs;ionem nil mutari colligitur, applicando demon&longs;tratio­<lb/>nem Theorematis hujus ad Ca&longs;us &longs;ingulos Fluidorum. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Eadem Demon&longs;tratione colligitur etiam (per Prop. </s> <s>XIX) <lb/>quod fluidi gravis partes nullum, ex pre&longs;&longs;ione ponderis incumben­<lb/>tis, acquirunt motum inter &longs;e, &longs;i modo excludatur motus qui ex <lb/>conden&longs;atione oriatur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Et propterea &longs;i aliud eju&longs;dem gravitatis &longs;pecificæ cor­<lb/>pus, quod &longs;it conden&longs;ationis expers, &longs;ubmergatur in hoc fluido, id <lb/>ex pre&longs;&longs;ione ponderis incumbentis nullum acquiret motum: non <lb/>de&longs;cendet, non a&longs;cendet, non cogetur figuram &longs;uam mutare. </s> <s>Si <lb/>&longs;phæricum e&longs;t manebit &longs;phæricum, non ob&longs;tante pre&longs;&longs;ione; &longs;i qua­<lb/>dratum e&longs;t manebit quadratum: idque &longs;ive molle &longs;it, &longs;ive fluidi&longs;&longs;i­<lb/>mum; &longs;ive fluido libere innatet, &longs;ive fundo incumbat. </s> <s>Habet e­<lb/>nim fluidi pars quælibet interna rationem corporis &longs;ubmer&longs;i, & par <lb/>e&longs;t ratio omnium eju&longs;dem magnitudinis, figuræ & gravitatis &longs;peci­<lb/>ficæ &longs;ubmer&longs;orum corporum. </s> <s>Si corpus &longs;ubmer&longs;um &longs;ervato pon­<lb/>dere lique&longs;ceret & indueret formam fluidi; hoc, &longs;i prius a&longs;cende­<lb/>ret vel de&longs;cenderet vel ex pre&longs;&longs;ione figuram novam indueret, etiam <lb/>nunc a&longs;cenderet vel de&longs;cenderet vel figuram novam induere coge­<lb/>retur: id adeo quia gravitas ejus cæteræque motuum cau&longs;æ per­<lb/>manent. </s> <s>Atqui, per Ca&longs;. </s> <s>5. Prop. </s> <s>XIX, jam quie&longs;ceret & figuram <lb/>retineret. </s> <s>Ergo & prius. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Proinde corpus quod &longs;pecifice gravius e&longs;t quam Flui­<lb/>dum &longs;ibi contiguum &longs;ub&longs;idebit, & quod &longs;pecifice levius e&longs;t a&longs;cen­<lb/>det, motumque & figuræ mutationem con&longs;equetur, quantum ex­<lb/>ce&longs;&longs;us ille vel defectus gravitatis efficere po&longs;&longs;it. </s> <s>Namque exce&longs;&longs;us <lb/>ille vel de&longs;ectus rationem habet impul&longs;us, quo corpus, alias in <pb xlink:href="039/01/292.jpg" pagenum="264"/><arrow.to.target n="note240"/>æquilibrio cum fluidi partibus con&longs;titutum, urgetur; & comparari <lb/>pote&longs;t cum exce&longs;&longs;u vel defectu ponderis in lance alterutra libræ. </s></p> <p type="margin"> <s><margin.target id="note240"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Corporum igitur in fluidis con&longs;titutorum duplex e&longs;t Gra­<lb/>vitas: altera vera & ab&longs;oluta, altera apparens, vulgaris & compa­<lb/>rativa. </s> <s>Gravitas ab&longs;oluta e&longs;t vis tota qua corpus deor&longs;um tendit: <lb/>relativa & vulgaris e&longs;t exce&longs;&longs;us gravitatis quo corpus magis tendit <lb/>deor&longs;um quam fluidum ambiens. </s> <s>Prioris generis Gravitate partes <lb/>fluidorum & corporum omnium gravitant in locis &longs;uis: ideoque <lb/>conjunctis ponderibus componunt pondus totius. </s> <s>Nam totum <lb/>omne grave e&longs;t, ut in va&longs;is liquorum plenis experiri licet; & pon­<lb/>dus totius æquale e&longs;t ponderibus omnium partium, ideoque ex ii&longs;­<lb/>dem componitur. </s> <s>Alterius generis Gravitate corpora non gravi­<lb/>tant in locis &longs;uis, id e&longs;t, inter &longs;e collata non prægravant, &longs;ed mu­<lb/>tuos ad de&longs;cendendum conatus impedientia permanent in locis <lb/>&longs;uis, perinde ac &longs;i gravia non e&longs;&longs;ent. </s> <s>Quæ in Aere &longs;unt & non <lb/>prægravant, vulgus gravia non judicat. </s> <s>Quæ prægravant vulgus <lb/>gravia judicat, quatenus ab Aeris pondere non &longs;u&longs;tinentur. </s> <s>Pon­<lb/>dera vulgi nihil aliud &longs;unt quam exce&longs;&longs;us verorum ponderum &longs;u­<lb/>pra pondus Aeris. </s> <s>Unde & vulgo dicuntur levia, quæ &longs;unt mi­<lb/>nus gravia, Aerique prægravanti cedendo &longs;uperiora petunt. </s> <s>Com­<lb/>parative levia &longs;unt, non vere, quia de&longs;cendunt in vacuo. </s> <s>Sic & <lb/>in Aqua, corpora, quæ ob majorem vel minorem gravitatem de­<lb/>&longs;cendunt vel a&longs;cendunt, &longs;unt comparative & apparenter gravia vel <lb/>levia, & eorum gravitas vel levitas comparativa & apparens e&longs;t ex­<lb/>ce&longs;&longs;us vel defectus quo vera eorum gravitas vel &longs;uperat gravita­<lb/>tem aque vel ab ea &longs;uperatur. </s> <s>Quæ vero nec prægravando de­<lb/>&longs;cendunt, nec prægravanti cedendo a&longs;cendunt, etiam&longs;i veris &longs;uis <lb/>ponderibus adaugeant pondus totius, comparative tamen & in &longs;en­<lb/>&longs;u vulgi non gravitant in aqua. </s> <s>Nam &longs;imilis e&longs;t horum Ca&longs;uum <lb/>Demon&longs;tratio. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. Quæ de gravitate demon&longs;trantur, obtinent in aliis qui­<lb/>bu&longs;cunque viribus centripetis. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>8. Proinde &longs;i Medium, in quo corpus aliquod movetur, <lb/>urgeatur vel a gravitate propria, vel ab alia quacunque vi centri­<lb/>peta, & corpus ab eadem vi urgeatur fortius: differentia virium <lb/>e&longs;t vis illa motrix, quam in præcedentibus Propo&longs;itionibus ut vim <lb/>centripetam con&longs;ideravimus. </s> <s>Sin corpus a vi illa urgeatur levius, <lb/>differentia virium pro vi centrifuga haberi debet. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>9. Cum autem fluida premendo corpora inclu&longs;a non <lb/>mutent eorum Figuras externas, patet in&longs;uper, per Corollarium <pb xlink:href="039/01/293.jpg" pagenum="265"/>Prop. </s> <s>XIX, quod non mutabunt &longs;itum partium internarum inter <lb/><arrow.to.target n="note241"/>&longs;e: proindeque, &longs;i Animalia immergantur, & &longs;en&longs;atio omnis a mo­<lb/>tu partium oriatur; nec lædent corpora immer&longs;a, nec &longs;en&longs;atio­<lb/>nem ullam excitabunt, ni&longs;i quatenus hæc corpora a compre&longs;&longs;ione <lb/>conden&longs;ari po&longs;&longs;unt. </s> <s>Et par e&longs;t ratio cuju&longs;cunque corporum Sy­<lb/>&longs;tematis fluido comprimente circundati. </s> <s>Sy&longs;tematis partes omnes <lb/>ii&longs;dem agitabuntur motibus, ac &longs;i in vacuo con&longs;tituerentur, ac &longs;o­<lb/>lam retinerent gravitatem &longs;uam comparativam, ni&longs;i quatenus flui­<lb/>dum vel motibus earum nonnihil re&longs;i&longs;tat, vel ad ea&longs;dem compre&longs;&longs;i­<lb/>one conglutinandas requiratur. </s></p> <p type="margin"> <s><margin.target id="note241"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXI. THEOREMA XVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Sit Fluidi cuju&longs;dam den&longs;itas compre&longs;&longs;ioni proportionalis, & partes <lb/>ejus a vi centripeta di&longs;tantiis &longs;uis a centro reciproce proportio­<lb/>nali deor&longs;um trabantur: dico quod, fi di&longs;tantiæ illæ &longs;umantur <lb/>continue proportionales, den&longs;itates Fluidi in ii&longs;dem di&longs;tantiis e­<lb/>runt etiam continue proportionales.<emph.end type="italics"/></s></p> <p type="main"> <s>De&longs;ignet <emph type="italics"/>ATV<emph.end type="italics"/>fundum Sphæricum cui fluidum incumbit, <emph type="italics"/>S<emph.end type="italics"/><lb/>centrum, <emph type="italics"/>SA, SB, SC, SD, SE,<emph.end type="italics"/>&c. </s> <s>di&longs;tantias continue propor­<lb/>tionales. </s> <s>Erigantur perpendicula <emph type="italics"/>AH, BI, CK, DL, EM, &c.<emph.end type="italics"/><lb/>quæ &longs;int ut den&longs;itates Medii in locis <emph type="italics"/>A, B, C, D, E<emph.end type="italics"/>; & &longs;pecificæ <lb/>gravitates in ii&longs;dem locis erunt ut <emph type="italics"/>(AH/AS), (BI/BS), (CK/CS),<emph.end type="italics"/>&c. </s> <s>vel, quod <lb/>perinde e&longs;t, ut <emph type="italics"/>(AH/AB), (BI/BC), (CK/CD),<emph.end type="italics"/>&c. </s> <s>Finge pri­<lb/><figure id="id.039.01.293.1.jpg" xlink:href="039/01/293/1.jpg"/><lb/>mum has gravitates uniformiter continuari ab <lb/><emph type="italics"/>A<emph.end type="italics"/>ad <emph type="italics"/>B,<emph.end type="italics"/>a <emph type="italics"/>B<emph.end type="italics"/>ad <emph type="italics"/>C,<emph.end type="italics"/>a <emph type="italics"/>C<emph.end type="italics"/>ad <emph type="italics"/>D,<emph.end type="italics"/>&c. </s> <s>factis per <lb/>gradus decrementis in punctis <emph type="italics"/>B, C, D,<emph.end type="italics"/>&c. </s> <s>Et <lb/>hæ gravitates ductæ in altitudines <emph type="italics"/>AB, BC, <lb/>CD,<emph.end type="italics"/>&c. </s> <s>conficient pre&longs;&longs;iones <emph type="italics"/>AH, BI, CK,<emph.end type="italics"/><lb/>quibus fundum <emph type="italics"/>ATV<emph.end type="italics"/>(juxta Theorema XV.) <lb/>urgetur. </s> <s>Su&longs;tinet ergo particula <emph type="italics"/>A<emph.end type="italics"/>pre&longs;&longs;iones <lb/>omnes <emph type="italics"/>AH, BI, CK, DL,<emph.end type="italics"/>pergendo in <lb/>infinitum; & particula <emph type="italics"/>B<emph.end type="italics"/>pre&longs;&longs;iones omnes <lb/>præter primam <emph type="italics"/>AH<emph.end type="italics"/>; & particula <emph type="italics"/>C<emph.end type="italics"/>omnes <lb/>præter duas primas <emph type="italics"/>AH, BI<emph.end type="italics"/>; & &longs;ic deinceps: adeoque parti­<lb/>culæ primæ <emph type="italics"/>A<emph.end type="italics"/>den&longs;itas <emph type="italics"/>AH<emph.end type="italics"/>e&longs;t ad particulæ &longs;ecundæ <emph type="italics"/>B<emph.end type="italics"/>den&longs;i-<pb xlink:href="039/01/294.jpg" pagenum="266"/><arrow.to.target n="note242"/>tatem <emph type="italics"/>BI<emph.end type="italics"/>ut &longs;umma omnium <emph type="italics"/>AH+BI+CK+DL,<emph.end type="italics"/>in infiNI­<lb/>tum, ad &longs;ummam omnium <emph type="italics"/>BI+CK+DL,<emph.end type="italics"/>&c. </s> <s>Et <emph type="italics"/>BI<emph.end type="italics"/>den­<lb/>&longs;itas &longs;ecundæ <emph type="italics"/>B,<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>CK<emph.end type="italics"/>den&longs;itatem tertiæ <emph type="italics"/>C,<emph.end type="italics"/>ut &longs;umma om­<lb/>nium <emph type="italics"/>BI+CK+DL,<emph.end type="italics"/>&c. </s> <s>ad &longs;ummam omnium <emph type="italics"/>CK+DL,<emph.end type="italics"/>&c. </s> <s><lb/>Sunt igitur &longs;ummæ illæ differentiis &longs;uis <emph type="italics"/>AH, BI, CK,<emph.end type="italics"/>&c. </s> <s>pro­<lb/>portionales, atque adeo continue proportionales, per hujus Lem. </s> <s>I. <lb/>proindeQ.E.D.fferentiæ <emph type="italics"/>AH, BI, CK,<emph.end type="italics"/>&c. </s> <s>&longs;ummis proportionales, <lb/>&longs;unt etiam continue proportionales. </s> <s>Quare cum den&longs;itates in locis <emph type="italics"/>A, <lb/>B, C,<emph.end type="italics"/>&c. </s> <s>&longs;int ut <emph type="italics"/>AH, BI, CK,<emph.end type="italics"/>&c. </s> <s>erunt etiam hæ continue propor­<lb/>tionales. </s> <s>Pergatur per &longs;altum, & (ex æquo) in di&longs;tantiis <emph type="italics"/>SA, SC, <lb/>SE<emph.end type="italics"/>continue proportionalibus, erunt den&longs;itates <emph type="italics"/>AH, CK, EM<emph.end type="italics"/><lb/>continue proportionales. </s> <s>Et eodem argumento, in di&longs;tantiis qui­<lb/>bu&longs;vis continue proportionalibus <emph type="italics"/>SA, SD, SG,<emph.end type="italics"/>den&longs;itates <emph type="italics"/>AH, DL, <lb/>GO<emph.end type="italics"/>erunt continue proportionales. </s> <s>Coeant jam puncta <emph type="italics"/>A, B, C, <lb/>D, E,<emph.end type="italics"/>&c. </s> <s>eo ut progre&longs;&longs;io gravitatum &longs;pecificarum a fundo <emph type="italics"/>A<emph.end type="italics"/>ad <lb/>&longs;ummitatem Fluidi continua reddatur, & in di&longs;tantiis quibu&longs;vis con­<lb/>tinue proportionalibus <emph type="italics"/>SA, SD, SG,<emph.end type="italics"/>den&longs;itates <emph type="italics"/>AH, DL, GO,<emph.end type="italics"/><lb/>&longs;emper exi&longs;tentes continue proportionales, manebunt etiamnum <lb/>continue proportionales. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note242"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc &longs;i detur den&longs;itas Fluidi in duobus locis, puta <emph type="italics"/>A<emph.end type="italics"/>& <lb/><emph type="italics"/>E,<emph.end type="italics"/>colligi pote&longs;t ejus den&longs;itas <lb/><figure id="id.039.01.294.1.jpg" xlink:href="039/01/294/1.jpg"/><lb/>in alio quovis loco <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/>Centro <lb/><emph type="italics"/>S,<emph.end type="italics"/>A&longs;ymptotis rectangulis <emph type="italics"/>SQ, <lb/>SX,<emph.end type="italics"/>de&longs;cribatur Hyperbola &longs;e­<lb/>cans perpendicula <emph type="italics"/>AH, EM, <lb/>QT<emph.end type="italics"/>in <emph type="italics"/>a, e, q,<emph.end type="italics"/>ut & perpendicu­<lb/>la <emph type="italics"/>HX, MY, TZ,<emph.end type="italics"/>ad A&longs;ymp­<lb/>toton <emph type="italics"/>SX<emph.end type="italics"/>demi&longs;&longs;a, in <emph type="italics"/>h, m<emph.end type="italics"/>& <emph type="italics"/>t.<emph.end type="italics"/><lb/>Fiat area <emph type="italics"/>ZYmtZ<emph.end type="italics"/>ad aream da­<lb/>tam <emph type="italics"/>YmhX<emph.end type="italics"/>ut area data <emph type="italics"/>EeqQ<emph.end type="italics"/><lb/>ad aream datam <emph type="italics"/>EeaA<emph.end type="italics"/>; & li­<lb/>nea <emph type="italics"/>Zt<emph.end type="italics"/>producta ab&longs;cindet li­<lb/>neam <emph type="italics"/>QT<emph.end type="italics"/>den&longs;itati proportio­<lb/>nalem. </s> <s>Namque &longs;i lineæ <emph type="italics"/>SA, SE, SQ<emph.end type="italics"/>&longs;unt continue proportiona­<lb/>les, erunt areæ <emph type="italics"/>EeqQ, EeaA<emph.end type="italics"/>æquales, & inde areæ his propor­<lb/>tionales <emph type="italics"/>YmtZ, XhmY<emph.end type="italics"/>etiam æquales, & lineæ <emph type="italics"/>SX, SY, SZ,<emph.end type="italics"/>id e&longs;t <lb/><emph type="italics"/>AH, EM, QT<emph.end type="italics"/>continue proportionales, ut oportet. </s> <s>Et &longs;i lineæ <lb/><emph type="italics"/>SA, SE, SQ<emph.end type="italics"/>obtinent alium quemvis ordinem in &longs;erie continue <lb/>proportionalium, lineæ <emph type="italics"/>AH, EM, QT,<emph.end type="italics"/>ob proportionales areas <lb/>Hyperbolicas, obtinebunt eundem ordinem in alia &longs;erie quantita­<lb/>tum continue proportionalium. <pb xlink:href="039/01/295.jpg" pagenum="267"/><arrow.to.target n="note243"/></s></p> <p type="margin"> <s><margin.target id="note243"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXII. THEOREMA XVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Sit Fluidi cuju&longs;dam den&longs;itas compre&longs;&longs;ioni proportionalis, & partes <lb/>ejus a gravitate quadratis di&longs;tantiarum &longs;uarum a centro reci­<lb/>proce proportionali deor&longs;um trabantur: dico quod, &longs;i di&longs;tantiæ <lb/>&longs;umantur in progre&longs;&longs;ione Mu&longs;ica, den&longs;itates Fluidi in bis di­<lb/>&longs;tantiis erunt in progre&longs;&longs;ione Geometrica.<emph.end type="italics"/></s></p> <p type="main"> <s>De&longs;ignet <emph type="italics"/>S<emph.end type="italics"/>centrum, & <emph type="italics"/>SA, SB, SC, SD, SE<emph.end type="italics"/>di&longs;tantias in pro­<lb/>gre&longs;&longs;ione Geometrica. </s> <s>Erigantur perpendicula <emph type="italics"/>AH, BI, CK,<emph.end type="italics"/>&c. </s> <s><lb/>quæ &longs;int ut Fluidi den&longs;itates in locis <emph type="italics"/>A, B, C, D, E,<emph.end type="italics"/>&c. </s> <s>& ip&longs;ius <lb/><figure id="id.039.01.295.1.jpg" xlink:href="039/01/295/1.jpg"/><lb/>gravitates &longs;pecificæ in ii&longs;dem locis erunt <emph type="italics"/>(AH/SAq), (BI/SBq), (CK/SCq),<emph.end type="italics"/>&c. </s> <s>Fin­<lb/>ge has gravitates uniformiter continuari, primam ab <emph type="italics"/>A<emph.end type="italics"/>ad <emph type="italics"/>B,<emph.end type="italics"/>&longs;e­<lb/>cundam a <emph type="italics"/>B<emph.end type="italics"/>ad <emph type="italics"/>C,<emph.end type="italics"/>tertiam a <emph type="italics"/>C<emph.end type="italics"/>ad <emph type="italics"/>D,<emph.end type="italics"/>&c. </s> <s>Et hæ ductæ in altitu­<lb/>dines <emph type="italics"/>AB, BC, CD, DE,<emph.end type="italics"/>&c. </s> <s>vel, quod perinde e&longs;t, in di&longs;tantias <lb/><emph type="italics"/>SA, SB, SC,<emph.end type="italics"/>&c. </s> <s>altitudinibus illis proportionales, conficient ex­<lb/>ponentes pre&longs;&longs;ionum <emph type="italics"/>(AH/SA), (BI/SB), (CK/SC),<emph.end type="italics"/>&c. </s> <s>Quare cum den&longs;itates <lb/>&longs;int ut harum pre&longs;&longs;ionum &longs;ummæ, differentiæ den&longs;itatum <emph type="italics"/>AH-BI, <lb/>BI-CK,<emph.end type="italics"/>&c. </s> <s>erunt ut &longs;ummarum differentiæ <emph type="italics"/>(AH/SA), (BI/SB), (CK/SC),<emph.end type="italics"/>&c. <pb xlink:href="039/01/296.jpg" pagenum="268"/><arrow.to.target n="note244"/>Centro <emph type="italics"/>S,<emph.end type="italics"/>A&longs;ymptotis <emph type="italics"/>SA, Sx,<emph.end type="italics"/>de&longs;cribatur Hyperbola quæ­<lb/>vis, quæ &longs;ecet perpendicula <emph type="italics"/>AH, BI, CK,<emph.end type="italics"/>&c. </s> <s>in <emph type="italics"/>a, b, c,<emph.end type="italics"/>&c. </s> <s>ut & <lb/>perpendicula ad A&longs;ymptoton <emph type="italics"/>Sx<emph.end type="italics"/>demi&longs;&longs;a <emph type="italics"/>Ht, Iu, Kw<emph.end type="italics"/>in <emph type="italics"/>h, i, k<emph.end type="italics"/>; <lb/>& den&longs;itatum differentiæ <emph type="italics"/>tu, uw,<emph.end type="italics"/>&c. </s> <s>erunt üt <emph type="italics"/>(AH/SA), (BI/SB),<emph.end type="italics"/>&c. </s> <s>Et <lb/>rectangula <emph type="italics"/>tuXth, uwXui,<emph.end type="italics"/>&c. </s> <s>&longs;eu <emph type="italics"/>tp, uq,<emph.end type="italics"/>&c. </s> <s>ut <emph type="italics"/>(AHXtb/SA), <lb/>(BIXui/SB),<emph.end type="italics"/>&c. </s> <s>id e&longs;t, ut <emph type="italics"/>Aa, Bb,<emph.end type="italics"/>&c. </s> <s>E&longs;t enim, ex natura Hyperbolæ, <lb/><emph type="italics"/>SA<emph.end type="italics"/>ad <emph type="italics"/>AH<emph.end type="italics"/>vel <emph type="italics"/>St,<emph.end type="italics"/>ut <emph type="italics"/>th<emph.end type="italics"/>ad <emph type="italics"/>Aa,<emph.end type="italics"/>adeoque (<emph type="italics"/>AHXth/SA<emph.end type="italics"/>) æquale <emph type="italics"/>Aa<emph.end type="italics"/><lb/><figure id="id.039.01.296.1.jpg" xlink:href="039/01/296/1.jpg"/><lb/>Et &longs;imili argumento e&longs;t (<emph type="italics"/>BIXui/SB<emph.end type="italics"/>) æquale <emph type="italics"/>Bb,<emph.end type="italics"/>&c. </s> <s>Sunt autem <emph type="italics"/>Aa, <lb/>Bb, Cc,<emph.end type="italics"/>&c. </s> <s>continue proportionales, & propterea differentiis &longs;u­<lb/>is <emph type="italics"/>Aa-Bb, Bb-Cc,<emph.end type="italics"/>&c. </s> <s>proportionales; ideoQ.E.D.fferentiis <lb/>hi&longs;ce proportionalia &longs;unt rectangula <emph type="italics"/>tp, uq,<emph.end type="italics"/>&c. </s> <s>ut & &longs;ummis diffe­<lb/>rentiarum <emph type="italics"/>Aa-Cc<emph.end type="italics"/>vel <emph type="italics"/>Aa-Dd<emph.end type="italics"/>&longs;ummæ rectangulorum <emph type="italics"/>tp+uq<emph.end type="italics"/><lb/>vel <emph type="italics"/>tp+uq+wr.<emph.end type="italics"/>Sunto eju&longs;modi termini quam plurimi, & &longs;um­<lb/>ma omnium differentiarum, puta <emph type="italics"/>Aa-Ff,<emph.end type="italics"/>erit &longs;ummæ omnium <lb/>rectangulorum, puta <emph type="italics"/>zthn,<emph.end type="italics"/>proportionalis. </s> <s>Augeatur numerus <lb/>terminorum & minuantur di&longs;tantiæ punctorum <emph type="italics"/>A, B, C,<emph.end type="italics"/>&c. </s> <s>in in­<lb/>nitum, & rectangula illa evadent æqualia areæ Hyperbolicæ <emph type="italics"/>zthn,<emph.end type="italics"/><lb/>adeoque huic areæ proportionalis e&longs;t differentia <emph type="italics"/>Aa-Ff.<emph.end type="italics"/>Suman-<pb xlink:href="039/01/297.jpg" pagenum="269"/>tur jam di&longs;tantiæ quælibet, puta <emph type="italics"/>SA, SD, SF<emph.end type="italics"/>in progre&longs;&longs;ione Mu­<lb/><arrow.to.target n="note245"/>&longs;ica, & differentiæ <emph type="italics"/>Aa-Dd, Dd-Ff<emph.end type="italics"/>erunt æquales; & propter­<lb/>ea differentiis hi&longs;ce proportionales areæ <emph type="italics"/>thlx, xlnz<emph.end type="italics"/>æquales erunt <lb/>inter &longs;e, & den&longs;itates <emph type="italics"/>St, Sx, Sz,<emph.end type="italics"/>id e&longs;t, <emph type="italics"/>AH, DL, FN,<emph.end type="italics"/>conti­<lb/>nue proportionales. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note244"/>DE MOTU <lb/>CORPORUM</s></p> <p type="margin"> <s><margin.target id="note245"/>LIBER <lb/>SECUNDUS</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc &longs;i dentur Fluidi den&longs;itates duæ quævis, puta <emph type="italics"/>AH<emph.end type="italics"/><lb/>& <emph type="italics"/>CK,<emph.end type="italics"/>dabitur area <emph type="italics"/>thkw<emph.end type="italics"/>harum differentiæ <emph type="italics"/>tw<emph.end type="italics"/>re&longs;pondens; & <lb/>inde invenietur den&longs;itas <emph type="italics"/>FN<emph.end type="italics"/>in altitudine quacunque <emph type="italics"/>SF,<emph.end type="italics"/>&longs;umen­<lb/>do aream <emph type="italics"/>thnz<emph.end type="italics"/>ad aream illam datam <emph type="italics"/>thkw<emph.end type="italics"/>ut e&longs;t differentia <lb/><emph type="italics"/>Aa-Ff<emph.end type="italics"/>ad differentiam <emph type="italics"/>Aa-Cc.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Simili argumentatione probari pote&longs;t, quod &longs;i gravitas particu­<lb/>larum Fluidi diminuatur in triplicata ratione di&longs;tantiarum a centro; <lb/>& quadratorum di&longs;tantiarum <emph type="italics"/>SA, SB, SC,<emph.end type="italics"/>&c. </s> <s>reciproca (nem­<lb/>pe <emph type="italics"/>(SAcub./SAq), (SAcub./SBq), (SAcub./SCq)<emph.end type="italics"/>) &longs;umantur in progre&longs;&longs;ione Arithme­<lb/>tica; den&longs;itates <emph type="italics"/>AH, BI, CK,<emph.end type="italics"/>&c. </s> <s>erunt in progre&longs;&longs;ione Geome­<lb/>trica. </s> <s>Et &longs;i gravitas diminuatur in quadruplicata ratione di&longs;tan­<lb/>tiarum, & cuborum di&longs;tantiarum reciproca (puta <emph type="italics"/>(SAqq/SAcub), (SAqq/SBcub), <lb/>(SAqq/SCcub.),<emph.end type="italics"/>&c.) &longs;umantur in progre&longs;&longs;ione Arithmetica; den&longs;itates <lb/><emph type="italics"/>AH, BI, CK,<emph.end type="italics"/>&c. </s> <s>erunt in progre&longs;&longs;ione Geometrica. </s> <s>Et &longs;ic in <lb/>infinitum. </s> <s>Rur&longs;us. </s> <s>&longs;i gravitas particularum Fluidi in omnibus di­<lb/>&longs;tantiis eadem &longs;it, & di&longs;tantiæ &longs;int in progre&longs;&longs;ione Arithmetica, <lb/>den&longs;itates erunt in progre&longs;&longs;ione Geometrica, uti Vir Cl. <emph type="italics"/>Edmundus <lb/>Hælleius<emph.end type="italics"/>invenit. </s> <s>Si gravitas &longs;it ut di&longs;tantia, & quadrata di&longs;tantia­<lb/>rum &longs;int in progre&longs;&longs;ione Arithmetica, den&longs;itates erunt in progre&longs;­<lb/>&longs;ione Geometrica. </s> <s>Et &longs;ic in infinitum. </s> <s>Hæc ita &longs;e habent ubi Fluidi <lb/>compre&longs;&longs;ione conden&longs;ati den&longs;itas e&longs;t ut vis compre&longs;&longs;ionis, vel, quod <lb/>perinde e&longs;t, &longs;patium a Fluido occupatum reciproce ut hæc vis. </s> <s><lb/>Fingi po&longs;&longs;unt aliæ conden&longs;ationis Leges, ut quod cubus vis com­<lb/>primentis &longs;it ut quadrato-quadratum den&longs;itatis, feu triplicata ra­<lb/>tio Vis æqualis quadruplicatæ rationi den&longs;itatis. </s> <s>Quo in ca&longs;u, &longs;i gra­<lb/>vitas e&longs;t reciproce ut quadratum di&longs;tantiæ a centro, den&longs;itas erit <lb/>reciproce ut cubus di&longs;tantiæ. </s> <s>Fingatur quod cubus vis compri­<lb/>mentis &longs;it ut quadrato-cubus den&longs;itatis, & &longs;i gravitas e&longs;t reciproce <lb/>ut quadratum di&longs;tantiæ, den&longs;itas erit reciproce in &longs;u&longs;quiplicata ra-<pb xlink:href="039/01/298.jpg" pagenum="270"/><arrow.to.target n="note246"/>tione di&longs;tantiæ. </s> <s>Fingatur quod vis comprimens &longs;it in duplicata <lb/>ratione den&longs;itatis, & gravitas reciproce in ratione duplicata di&longs;tan­<lb/>tiæ, & den&longs;itas erit reciproce ut di&longs;tantia. </s> <s>Ca&longs;us omnes percurre­<lb/>re longum e&longs;&longs;et. </s></p> <p type="margin"> <s><margin.target id="note246"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIII. THEOREMA XVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Fluidi ex particulis &longs;e mutuo fugientibus compo&longs;iti den&longs;itas &longs;it <lb/>ut compre&longs;&longs;io, vires centrifugæ particularum &longs;unt reciproce pro­<lb/>portionales di&longs;tantiis centrorum &longs;uorum. </s> <s>Et vice ver&longs;a, par­<lb/>ticulæ viribus quæ &longs;unt reciproce proportionales di&longs;tantiis cen­<lb/>trorum &longs;uorum &longs;e mutuo fugientes componunt Fluidum Ela&longs;ti­<lb/>cum, cujus den&longs;itas est compre&longs;&longs;ioni proportionalis.<emph.end type="italics"/></s></p> <p type="main"> <s>Includi intelligatur Fluidum in &longs;patio cubico <emph type="italics"/>ACE,<emph.end type="italics"/>dein com­<lb/>pre&longs;&longs;ione redigi in &longs;patium cubicum minus <emph type="italics"/>ace<emph.end type="italics"/>; & particularum, <lb/>&longs;imilem &longs;itum inter &longs;e in utro­<lb/><figure id="id.039.01.298.1.jpg" xlink:href="039/01/298/1.jpg"/><lb/>que &longs;patio obtinentium, di&longs;tan­<lb/>tiæ erunt ut cuborum latera <lb/><emph type="italics"/>AB, ab<emph.end type="italics"/>; & Medii den&longs;itates <lb/>reciproce ut &longs;patia continentia <lb/><emph type="italics"/>AB cub.<emph.end type="italics"/>& <emph type="italics"/>ab cub.<emph.end type="italics"/>In latere <lb/>cubi majoris <emph type="italics"/>ABCD<emph.end type="italics"/>capiatur <lb/>quadratum <emph type="italics"/>DP<emph.end type="italics"/>æquale lateri <lb/>cubi minoris <emph type="italics"/>db<emph.end type="italics"/>; & ex Hypo­<lb/>the&longs;i, pre&longs;&longs;io qua quadratum <emph type="italics"/>DP<emph.end type="italics"/>urget Fluidum inclu&longs;um, erit ad <lb/>pre&longs;&longs;ionem qua latus illud quadratum <emph type="italics"/>db<emph.end type="italics"/>urget Fluidum inclu&longs;um <lb/>ut Medii den&longs;itates ad invicem, hoc e&longs;t, ut <emph type="italics"/>ab cub.<emph.end type="italics"/>ad <emph type="italics"/>ABcub.<emph.end type="italics"/>Sed <lb/>pre&longs;&longs;io qua quadratum <emph type="italics"/>DB<emph.end type="italics"/>urget Fluidum inclu&longs;um, e&longs;t ad pre&longs;&longs;i­<lb/>onem qua quadratum <emph type="italics"/>DP<emph.end type="italics"/>urget idem Fluidum, ut quadratum <emph type="italics"/>DB<emph.end type="italics"/><lb/>ad quadratum <emph type="italics"/>DP,<emph.end type="italics"/>hoc e&longs;t, ut <emph type="italics"/>AB quad.<emph.end type="italics"/>ad <emph type="italics"/>ab quad.<emph.end type="italics"/>Ergo, ex <lb/>æquo, pre&longs;&longs;io qua latus <emph type="italics"/>DB<emph.end type="italics"/>urget Fluidum, e&longs;t ad pre&longs;&longs;ionem qua <lb/>latus <emph type="italics"/>db<emph.end type="italics"/>urget Fluidum, ut <emph type="italics"/>ab<emph.end type="italics"/>ad <emph type="italics"/>AB.<emph.end type="italics"/>Planis <emph type="italics"/>FGH, fgh,<emph.end type="italics"/>per <lb/>media cuborum ductis, di&longs;tinguatur Fluidum in duas partes, & hæ <lb/>&longs;e mutuo prement ii&longs;dem viribus, quibus premuntur a planis <emph type="italics"/>AC, ac,<emph.end type="italics"/><lb/>hoc e&longs;t, in proportione <emph type="italics"/>ab<emph.end type="italics"/>ad <emph type="italics"/>AB:<emph.end type="italics"/>adeoque vires centrifugæ, qui­<lb/>bus hæ pre&longs;&longs;iones &longs;u&longs;tinentur, &longs;unt in eadem ratione. </s> <s>Ob eundem <lb/>particularum numerum &longs;imilemque &longs;itum in utroque cubo, vires <lb/>quas particulæ omnes &longs;ecundum plana <emph type="italics"/>FGH, fgh<emph.end type="italics"/>exercent in om-<pb xlink:href="039/01/299.jpg" pagenum="271"/>nes, &longs;unt ut vires quas &longs;ingulæ exercent in &longs;ingulas. </s> <s>Ergo vires, <lb/><arrow.to.target n="note247"/>quas &longs;ingulæ exercent in &longs;ingulas &longs;ecundum planum <emph type="italics"/>FGH<emph.end type="italics"/>in cubo <lb/>majore, &longs;unt ad vires quas &longs;ingulæ exercent in &longs;ingulas &longs;ecundum <lb/>planum <emph type="italics"/>fgh<emph.end type="italics"/>in cubo minore ut <emph type="italics"/>ab<emph.end type="italics"/>ad <emph type="italics"/>AB,<emph.end type="italics"/>hoc e&longs;t, reciproce ut <lb/>di&longs;tantiæ particularum ad invicem. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note247"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s>Et vice ver&longs;a, &longs;i vires particularum &longs;ingularum &longs;unt reciproce <lb/>ut di&longs;tantiæ, id e&longs;t, reciproce ut cuborum latera <emph type="italics"/>AB, ab<emph.end type="italics"/>; &longs;ummæ <lb/>virium erunt in eadem ratione, & pre&longs;&longs;iones laterum <emph type="italics"/>DB, db<emph.end type="italics"/>ut <lb/>&longs;ummæ virium; & pre&longs;&longs;io quadrati <emph type="italics"/>DP<emph.end type="italics"/>ad pre&longs;&longs;ionem lateris <emph type="italics"/>DB<emph.end type="italics"/><lb/>ut <emph type="italics"/>ab quad.<emph.end type="italics"/>ad <emph type="italics"/>AB quad.<emph.end type="italics"/>Et, ex æquo, pre&longs;&longs;io quadrati <emph type="italics"/>DP<emph.end type="italics"/>ad pre&longs;­<lb/>&longs;ionem lateris <emph type="italics"/>db<emph.end type="italics"/>ut <emph type="italics"/>ab cub.<emph.end type="italics"/>ad <emph type="italics"/>AB cub.<emph.end type="italics"/>id e&longs;t, vis compre&longs;&longs;ionis ad <lb/>vim compre&longs;&longs;ionis ut den&longs;itas ad den&longs;itatem. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Simili argumento, &longs;i particularum vires centrifugæ &longs;int reciproce <lb/>in duplicata ratione di&longs;tantiarum inter centra, cubi virium compri­<lb/>mentium erunt ut quadrato-quadrata den&longs;itarum. </s> <s>Si vires centri­<lb/>fugæ &longs;int reciproce in triplicata vel quadruplicata ratione di&longs;tantia­<lb/>rum, cubi virium comprimentium erunt ut quadrato-cubi vel cubo­<lb/>cubi den&longs;itatum. </s> <s>Et univer&longs;aliter, &longs;i D ponatur pro di&longs;tantia, & <lb/>E pro den&longs;itate Fluidi compre&longs;&longs;i, & vires centrifugæ &longs;int reciproce <lb/>ut di&longs;tantiæ dignitas quælibet D<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/><emph.end type="sup"/>, cujus index e&longs;t numerus <emph type="italics"/>n<emph.end type="italics"/>; vi­<lb/>res comprimentes erunt ut latera cubica dignitatis E<emph type="sup"/><emph type="italics"/>n<emph.end type="italics"/>+2<emph.end type="sup"/>, cujus <lb/>index e&longs;t numerus <emph type="italics"/>n<emph.end type="italics"/>+2: & contra. </s> <s>Intelligenda vero &longs;unt hæc <lb/>omnia de particularum Viribus centrifugis quæ terminantur in par­<lb/>ticulis proximis, aut non longe ultra diffunduntur. </s> <s>Exemplum <lb/>habemus in corporibus Magneticis. </s> <s>Horum Virtus attractiva ter­<lb/>minatur fere in &longs;ui generis corporibus &longs;ibi proximis. </s> <s>Magnetis <lb/>virtus per interpo&longs;itam laminam ferri contrahitur, & in lamina fere <lb/>terminatur. </s> <s>Nam corpora ulteriora non tam a Magnete quam a <lb/>lamina trahuntur. </s> <s>Ad eundem modum &longs;i particulæ fugant alias &longs;ui <lb/>generis particulas &longs;ibi proximas, in particulas autem remotiores <lb/>virtutem nullam exerceant, ex huju&longs;modi particulis componentur <lb/>Fluida de quibus actum e&longs;t in hac Propo&longs;itione. </s> <s>Quod &longs;i particulæ <lb/>cuju&longs;que virtus in infinitum propagetur, opus erit vi majori ad æqua­<lb/>lem conden&longs;ationem majoris quantitatis Fluidi. </s> <s>An vero Fluida <lb/>Ela&longs;tica ex particulis &longs;e mutuo fugantibus con&longs;tent, Quæ&longs;tio Phy­<lb/>&longs;ica e&longs;t. </s> <s>Nos proprietatem Fluidorum ex eju&longs;modi particulis con­<lb/>&longs;tantium Mathematice demon&longs;travimus, ut Philo&longs;ophis an&longs;am præ­<lb/>beamus Quæ&longs;tionem illam tractandi. <pb xlink:href="039/01/300.jpg" pagenum="272"/><arrow.to.target n="note248"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note248"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>SECTIO VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Motu & Re&longs;i&longs;tentia Corporum Funependulorum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIV. THEOREMA XIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Quantitates materiæ in corporibus funependulis, quorum centra <lb/>o&longs;cillationum a centro &longs;u&longs;pen&longs;ionis æqualiter di&longs;tant, &longs;unt in ra­<lb/>tione compo&longs;ita ex ratione ponderum & ratione duplicata tem­<lb/>porum o&longs;cillationum in vacuo.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam velocitas, quam data vis in data materia dato tempore ge­<lb/>nerare pote&longs;t, e&longs;t ut vis & tempus directe, & materia inver&longs;e. </s> <s>Quo <lb/>major e&longs;t vis vel majus tempus vel minor materia, eo major gene­<lb/>rabitur velocitas. </s> <s>Id quod per motus Legem &longs;ecundam manife­<lb/>&longs;tum e&longs;t. </s> <s>Jam vero &longs;i Pendula eju&longs;dem &longs;int longitudinis, vires mo­<lb/>trices in locis a perpendiculo æqualiter di&longs;tantibus &longs;unt ut ponde­<lb/>ra: ideoque &longs;i corpora duo o&longs;cillando de&longs;cribant arcus æquales, & <lb/>arcus illi dividantur in partes æquales; cum tempora quibus cor­<lb/>pora de&longs;cribant &longs;ingulas arcuum partes corre&longs;pondentes &longs;int ut <lb/>tempora o&longs;cillationum totarum, erunt velocitates ad invicem in <lb/>corre&longs;pondentibus o&longs;cillationum partibus, ut vires motrices & tota <lb/>o&longs;cillationum tempora directe & quantitates materiæ reciproce: <lb/>adeoque quantitates materiæ ut vires & o&longs;cillationum tempora di­<lb/>recte & velocitates reciproce. </s> <s>Sed velocitates reciproce &longs;unt ut <lb/>tempora, atque adeo tempora directe & velocitates reciproce &longs;unt <lb/>ut quadrata temporum, & propterea quantitates materiæ &longs;unt ut <lb/>vires motrices & quadrata temporum, id e&longs;t, ut pondera & quadra­<lb/>ta temporum. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Ideoque &longs;i tempora &longs;unt æqualia, quantitates materiæ <lb/>in &longs;ingulis corporibus erunt ut pondera. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si pondera &longs;unt æqualia, quantitates materiæ erunt ut <lb/>quadrata temporum. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Si quantitates materiæ æquantur, pondera erunt reci­<lb/>proce ut quadrata temporum. </s></p><pb xlink:href="039/01/301.jpg" pagenum="273"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Unde cum quadrata temporum, cæteris paribus, &longs;int ut <lb/><arrow.to.target n="note249"/>longitudines pendulorum; &longs;i & tempora & quantitates materiæ æ­<lb/>qualia &longs;unt, pondera erunt ut longitudines pendulorum. </s></p> <p type="margin"> <s><margin.target id="note249"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Et univer&longs;aliter, quantitas materiæ pendulæ e&longs;t ut pon­<lb/>dus & quadratum temporis directe, & longitudo penduli inver&longs;e. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Sed & in Medio non re&longs;i&longs;tente quantitas materiæ pen­<lb/>dulæ e&longs;t ut pondus comparativum & quadratum temporis directe <lb/>& longitudo penduli inver&longs;e. </s> <s>Nam pondus comparativum e&longs;t vis <lb/>motrix corporis in Medio quovis gravi, ut &longs;upra explicui; adeoque <lb/>idem præ&longs;tat in tali Medio non re&longs;i&longs;tente atque pondus ab&longs;olutum <lb/>in vacuo. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. Et hinc liquet ratio tum comparandi corpora inter &longs;e, <lb/>quoad quantitatem materiæ in &longs;ingulis; tum comparandi pondera <lb/>eju&longs;dem corporis in diver&longs;is locis, ad cogno&longs;cendam variationem <lb/>gravitatis. </s> <s>Factis autem experimentis quam accurati&longs;&longs;imis inveni <lb/>&longs;emper quantitatem materiæ in corporibus &longs;ingulis eorum ponderi <lb/>proportionalem e&longs;&longs;e. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXV. THEOREMA XX:<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Corpora Funependula quibus, in Medio quovis, re&longs;i&longs;titur in ratione <lb/>momentorum temporis, & corpora Funependula quæ in eju&longs;dem <lb/>gravitatis &longs;pecificæ Medio non re&longs;i&longs;tente moventur, o&longs;cillatio­<lb/>nes in Cycloide eodem tempore peragunt, & arcuum partes pro­<lb/>portionales &longs;imul de&longs;cribunt.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>AB<emph.end type="italics"/>Cycloidis <lb/><figure id="id.039.01.301.1.jpg" xlink:href="039/01/301/1.jpg"/><lb/>arcus, quem corpus <lb/><emph type="italics"/>D<emph.end type="italics"/>tempore quovis in <lb/>Medio non re&longs;i&longs;tente <lb/>o&longs;cillando de&longs;cribit. </s> <s><lb/>Bi&longs;ecetur idem in <emph type="italics"/>C,<emph.end type="italics"/><lb/>ita ut <emph type="italics"/>C<emph.end type="italics"/>&longs;it infimum <lb/>ejus punctum; & erit <lb/>vis acceleratrix qua <lb/>corpus urgetur in lo­<lb/>co quovis <emph type="italics"/>D<emph.end type="italics"/>vel <emph type="italics"/>d<emph.end type="italics"/>vel <lb/><emph type="italics"/>E<emph.end type="italics"/>ut longitudo arcus <lb/><emph type="italics"/>CD<emph.end type="italics"/>vel <emph type="italics"/>Cd<emph.end type="italics"/>vel <emph type="italics"/>CE.<emph.end type="italics"/>Exponatur vis illa per eundem arcum; & <lb/>cum re&longs;i&longs;tentia &longs;it ut momentum temporis, adeoQ.E.D.tur, expona-<pb xlink:href="039/01/302.jpg" pagenum="274"/><arrow.to.target n="note250"/>tur eadem per datam arcus Cycloidis partem <emph type="italics"/>CO,<emph.end type="italics"/>& &longs;umatur ar­<lb/>cus <emph type="italics"/>Od<emph.end type="italics"/>in ratione ad arcum <emph type="italics"/>CD<emph.end type="italics"/>quam habet arcus <emph type="italics"/>OB<emph.end type="italics"/>ad arcum <lb/><emph type="italics"/>CB:<emph.end type="italics"/>& vis qua corpus in <emph type="italics"/>d<emph.end type="italics"/>urgetur in Medio re&longs;i&longs;tente, cum &longs;it ex­<lb/>ce&longs;&longs;us vis <emph type="italics"/>Cd<emph.end type="italics"/>&longs;upra re&longs;i&longs;tentiam <emph type="italics"/>CO,<emph.end type="italics"/>exponetur per arcum <emph type="italics"/>Od,<emph.end type="italics"/>ad­<lb/>eoque erit ad vim qua corpus <emph type="italics"/>D<emph.end type="italics"/>urgetur in Medio non re&longs;i&longs;tente, <lb/>in loco <emph type="italics"/>D,<emph.end type="italics"/>ut arcus <emph type="italics"/>Od<emph.end type="italics"/>ad arcum <emph type="italics"/>CD<emph.end type="italics"/>; & propterea etiam in lo­<lb/>co <emph type="italics"/>B<emph.end type="italics"/>ut arcus <emph type="italics"/>OB<emph.end type="italics"/>ad arcum <emph type="italics"/>CB.<emph.end type="italics"/>Proinde &longs;i corpora duo, <emph type="italics"/>D, d<emph.end type="italics"/><lb/>exeant de loco <emph type="italics"/>B,<emph.end type="italics"/>& his viribus urgeantur: cum vires &longs;ub initio <lb/>&longs;int ut arcus <emph type="italics"/>CB<emph.end type="italics"/>& <emph type="italics"/>OB,<emph.end type="italics"/>erunt velocitates primæ & arcus primo <lb/>de&longs;cripti in eadem ratione. </s> <s>Sunto arcus illi <emph type="italics"/>BD<emph.end type="italics"/>& <emph type="italics"/>Bd,<emph.end type="italics"/>& arcus <lb/>reliqui <emph type="italics"/>CD, Od<emph.end type="italics"/>erunt in eadem ratione. </s> <s>Proinde vires, ip&longs;is <lb/><emph type="italics"/>CD, Od<emph.end type="italics"/>proportionales, manebunt in eadem ratione ac &longs;ub initio, <lb/>& propterea corpora pergent arcus in eadem ratione &longs;imul de&longs;cri­<lb/>bere. </s> <s>Igitur vires & <lb/><figure id="id.039.01.302.1.jpg" xlink:href="039/01/302/1.jpg"/><lb/>velocitates & arcus re­<lb/>liqui <emph type="italics"/>CD, Od<emph.end type="italics"/>&longs;emper <lb/>erunt ut arcus toti <emph type="italics"/>CB, <lb/>OB,<emph.end type="italics"/>& propterea ar­<lb/>cus illi reliqui &longs;imul <lb/>de&longs;cribentur. </s> <s>Quare <lb/>corpora duo <emph type="italics"/>D, d<emph.end type="italics"/>&longs;i­<lb/>mul pervenient ad loca <lb/><emph type="italics"/>C<emph.end type="italics"/>& <emph type="italics"/>O,<emph.end type="italics"/>alterum qui­<lb/>dem in Medio non re­<lb/>&longs;i&longs;tente ad locum <emph type="italics"/>C,<emph.end type="italics"/>& <lb/>alterum in Medio re&longs;i&longs;tente ad locum <emph type="italics"/>O.<emph.end type="italics"/>Cum autem velocitates in <lb/><emph type="italics"/>C<emph.end type="italics"/>& <emph type="italics"/>O<emph.end type="italics"/>&longs;int ut arcus <emph type="italics"/>CB, OB<emph.end type="italics"/>; erunt arcus quos corpora ulterius <lb/>pergendo &longs;imul de&longs;cribunt, in eadem ratione. </s> <s>Sunto illi <emph type="italics"/>CE<emph.end type="italics"/>& <lb/><emph type="italics"/>Oe.<emph.end type="italics"/>Vis qua corpus <emph type="italics"/>D<emph.end type="italics"/>in Medio non re&longs;i&longs;tente retardatur in <emph type="italics"/>E<emph.end type="italics"/><lb/>e&longs;t ut <emph type="italics"/>CE,<emph.end type="italics"/>& vis qua corpus <emph type="italics"/>d<emph.end type="italics"/>in Medio re&longs;i&longs;tente retardatur in <emph type="italics"/>e<emph.end type="italics"/><lb/>e&longs;t ut &longs;umma vis <emph type="italics"/>Ce<emph.end type="italics"/>& re&longs;i&longs;tentiæ <emph type="italics"/>CO,<emph.end type="italics"/>id e&longs;t ut <emph type="italics"/>Oe<emph.end type="italics"/>; ideoque vi­<lb/>res, quibus corpora retardantur, &longs;unt ut arcubus <emph type="italics"/>CE, Oe<emph.end type="italics"/>propor­<lb/>tionales arcus <emph type="italics"/>CB, OB<emph.end type="italics"/>; proindeque velocitates, in data illa ratio­<lb/>ne retardatæ, manent in eadem illa data ratione. </s> <s>Velocitates igitur <lb/>& arcus ii&longs;dem de&longs;cripti &longs;emper &longs;unt ad invicem in data illa ratio­<lb/>ne arcuum <emph type="italics"/>CB<emph.end type="italics"/>& <emph type="italics"/>OB<emph.end type="italics"/>; & propterea &longs;i &longs;umantur arcus toti <emph type="italics"/>AB, <lb/>aB<emph.end type="italics"/>in eadem ratione, corpora <emph type="italics"/>D, d<emph.end type="italics"/>&longs;imul de&longs;cribent hos arcus, & <lb/>in locis <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>a<emph.end type="italics"/>motum omnem &longs;imul amittent. </s> <s>I&longs;ochronæ &longs;unt <lb/>igitur o&longs;cillationes totæ, & arcubus totis <emph type="italics"/>BA, Ba<emph.end type="italics"/>proportionales <lb/>&longs;unt arcuum partes quælibet <emph type="italics"/>BD, Bd<emph.end type="italics"/>vel <emph type="italics"/>BE, Be<emph.end type="italics"/>quæ &longs;imul de­<lb/>&longs;cribuntur. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p><pb xlink:href="039/01/303.jpg" pagenum="275"/> <p type="margin"> <s><margin.target id="note250"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Igitur motus veloci&longs;&longs;imus in Medio re&longs;i&longs;tente non incidit <lb/><arrow.to.target n="note251"/>in punctum infimum <emph type="italics"/>C,<emph.end type="italics"/>&longs;ed reperitur in puncto illo <emph type="italics"/>O,<emph.end type="italics"/>quo arcus <lb/>totus de&longs;criptus <emph type="italics"/>aB<emph.end type="italics"/>bi&longs;ecatur. </s> <s>Et corpus &longs;ubinde pergendo ad <emph type="italics"/>a,<emph.end type="italics"/><lb/>ii&longs;dem gradibus retardatur quibus antea accelerabatur in de&longs;cen&longs;u <lb/>&longs;uo a <emph type="italics"/>B<emph.end type="italics"/>ad <emph type="italics"/>O.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note251"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVI. THEOREMA XXI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Corporum Funependulorum, quibus re&longs;i&longs;titur in ratione velocitatum, <lb/>o&longs;cillationes in Cycloide &longs;unt I&longs;ochronæ.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam &longs;i corpora duo, a centris &longs;u&longs;pen&longs;ionum æqualiter di&longs;tantia, <lb/>o&longs;cillando de&longs;cribant arcus inæquales, & velocitates in arcuum par­<lb/>tibus corre&longs;pondentibus &longs;int ad invicem ut arcus toti: re&longs;i&longs;tentiæ <lb/>velocitatibus proportionales, erunt etiam ad invicem ut iidem ar­<lb/>cus. </s> <s>Proinde &longs;i viribus motricibus a gravitate oriundis, quæ &longs;int <lb/>ut iidem arcus, auferantur vel addantur hæ re&longs;i&longs;tentiæ, erunt dif­<lb/>ferentiæ vel &longs;ummæ ad invicem in eadem arcuum ratione: cumque <lb/>velocitatum incrementa vel decrementa &longs;int ut hæ differentiæ vel <lb/>&longs;ummæ, velocitates &longs;emper erunt ut arcus toti: Igitur velocitates, <lb/>&longs;i &longs;int in aliquo ca&longs;u ut arcus toti, manebunt &longs;emper in eadem ra­<lb/>tione. </s> <s>Sed in principio motus, ubi corpora incipiunt de&longs;cendere <lb/>& arcus illos de&longs;cribere, vires, cum &longs;int arcubus proportionales, ge­<lb/>nerabunt velocitates arcubus proportionales. </s> <s>Ergo velocitates &longs;em­<lb/>per erunt ut arcus toti de&longs;cribendi, & propterea arcus illi &longs;imul de­<lb/>&longs;cribentur. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVII. THEOREMA XXII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Corporibus Funependulis re&longs;i&longs;titur in duplicata ratione veloci­<lb/>tatum, differentiæ inter tempora o&longs;cillationum in Medio re&longs;i­<lb/>&longs;tente ac tempora o&longs;cillationum in eju&longs;dem gravitatis &longs;pecificæ <lb/>Medio non re&longs;i&longs;tente, erunt arcubus o&longs;cillando de&longs;criptis pro­<lb/>portionales, quam proxime.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam pendulis æqualibus in Medio re&longs;i&longs;tente de&longs;cribantur arcus <lb/>inæquales A, B; & re&longs;i&longs;tentia corporis in arcu A, erit ad re&longs;i&longs;ten­<lb/>tiam corporis in parte corre&longs;pondente arcus B, in duplicata ratio­<lb/>ne velocitatum, id e&longs;t, ut AA ad BB, quam proxime. </s> <s>Si re&longs;i-<pb xlink:href="039/01/304.jpg" pagenum="276"/><arrow.to.target n="note252"/>&longs;tentia in arcu B e&longs;&longs;et ad re&longs;i&longs;tentiam in arcu A ut AB ad AA; <lb/>tempora in arcubus A & B forent æqualia, per Propo&longs;itionem &longs;u­<lb/>periorem. </s> <s>Ideoque re&longs;i&longs;tentia AA in arcu A, vel AB in arcu B, <lb/>efficit exce&longs;&longs;um temporis in arcu A &longs;upra tempus in Medio non <lb/>re&longs;i&longs;tente; & re&longs;i&longs;tentia BB efficit exce&longs;&longs;um temporis in arcu B <lb/>&longs;upra tempus in Medio non re&longs;i&longs;tente. </s> <s>Sunt autem exce&longs;&longs;us illi <lb/>ut vires efficientes AB & BB quam proxime, id e&longs;t, ut arcus <lb/>A & B. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note252"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc ex o&longs;cillationum temporibus, in Medio re&longs;i&longs;tente, <lb/>in arcubus inæqualibus factarum, cogno&longs;ci po&longs;&longs;unt tempora o&longs;cilla­<lb/>tionum in eju&longs;dem gravitatis &longs;pecificæ Medio non re&longs;i&longs;tente. </s> <s>Nam <lb/>differentia temporum erit ad exce&longs;&longs;um temporis in arcu minore &longs;u­<lb/>pra tempus in Medio non re&longs;i&longs;tente, ut differentia arcuum ad ar­<lb/>cum minorem. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. O&longs;cillationes breviores &longs;unt magis I&longs;ochronæ, & bre­<lb/>vi&longs;&longs;imæ ii&longs;dem temporibus peraguntur ac in Medio non re&longs;i&longs;tente, <lb/>quam proxime. </s> <s>Earum vero quæ in majoribus arcubus fiunt, tem­<lb/>ra &longs;unt paulo majora, propterea quod re&longs;i&longs;tentia in de&longs;cen&longs;u cor­<lb/>poris qua tempus producitur, major &longs;it pro ratione longitudinis <lb/>in de&longs;cen&longs;u de&longs;criptæ, quam re&longs;i&longs;tentia in a&longs;cen&longs;u, &longs;ub&longs;equente qua <lb/>tempus contrahitur. </s> <s>Sed & tempus o&longs;cillationum tam brevium <lb/>quam longarum nonnihil produci videtur per motum Medii. </s> <s>Nam <lb/>corporibus tarde&longs;centibus paulo minus re&longs;i&longs;titur, pro ratione velo­<lb/>citatis, & corporibus acceleratis paulo magis quam iis quæ unifor­<lb/>miter progrediuntur: id adeo quia Medium, eo quem a corporibus <lb/>accepit motu, in eandem plagam pergendo, in priore ca&longs;u magis <lb/>agitatur, in po&longs;teriore minus; ac proinde magis vel minus cum <lb/>corporibus motis con&longs;pirat. </s> <s>Pendulis igitur in de&longs;cen&longs;u magis re­<lb/>&longs;i&longs;tit, in a&longs;cen&longs;u minus quam pro ratione velocitatis, & ex utraque <lb/>cau&longs;a tempus producitur. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVIII. THEOREMA XXIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Corpori Funependulo in Cycloide o&longs;cillanti re&longs;i&longs;titur in ratione <lb/>momentorum temporis, erit ejus re&longs;i&longs;tentia ad vim gravitatis <lb/>ut exce&longs;&longs;us arcus de&longs;cen&longs;u toto de&longs;cripti &longs;upra arcum a&longs;cen&longs;u <lb/>&longs;ub&longs;equente de&longs;criptum, ad penduli longitudinem duplicatam.<emph.end type="italics"/></s></p> <p type="main"> <s>De&longs;ignet <emph type="italics"/>BC<emph.end type="italics"/>arcum de&longs;cen&longs;u de&longs;criptum, <emph type="italics"/>Ca<emph.end type="italics"/>arcum a&longs;cen&longs;u de­<lb/>&longs;criptum, & <emph type="italics"/>Aa<emph.end type="italics"/>differentiam arcuum: & &longs;tantibus quæ in Propo-<pb xlink:href="039/01/305.jpg" pagenum="277"/>&longs;itione XXV con&longs;tructa & demon&longs;trata &longs;unt, erit vis qua corpus <lb/><arrow.to.target n="note253"/>olcnlans urgetur in loco quovis <emph type="italics"/>D,<emph.end type="italics"/>ad vim re&longs;i&longs;tentiæ ut arcus <lb/><emph type="italics"/>CD<emph.end type="italics"/>ad arcum <emph type="italics"/>CO,<emph.end type="italics"/>qui &longs;emi&longs;&longs;is e&longs;t differentiæ illius <emph type="italics"/>Aa.<emph.end type="italics"/>Ideoque <lb/>vis qua corpus o&longs;cillans urgetur in Cycloidis principio &longs;eu puncto <lb/>alti&longs;&longs;imo, id e&longs;t, vis gravitatis, erit ad re&longs;i&longs;tentiam ut arcus Cy­<lb/>cloidis inter punctum illud &longs;upremum & punctum infimum <emph type="italics"/>C<emph.end type="italics"/>ad <lb/>arcum <emph type="italics"/>CO<emph.end type="italics"/>; id e&longs;t (&longs;i arcus duplicentur) ut Cycloidis totius arcus, <lb/>&longs;eu dupla penduli longitudo, ad arcum <emph type="italics"/>Aa. </s> <s><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note253"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIX. PROBLEMA VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Po&longs;ito quod Corpori in Cycloide o&longs;cillanti re&longs;i&longs;titur in duplicata ra­<lb/>tione velocitatis: invenire re&longs;i&longs;tentiam in locis &longs;ingulis.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>Ba<emph.end type="italics"/>(Fig. </s> <s>Prop. </s> <s>XXV) arcus o&longs;cillatione integra de&longs;criptus, <lb/>&longs;itque <emph type="italics"/>C<emph.end type="italics"/>infimum Cycloidis punctum, & <emph type="italics"/>CZ<emph.end type="italics"/>&longs;emi&longs;&longs;is arcus Cycloi­<lb/>dis totius, longitudini Penduli æqualis; & quæratur re&longs;i&longs;tentia cor­<lb/><figure id="id.039.01.305.1.jpg" xlink:href="039/01/305/1.jpg"/><lb/>poris in loco quovis <emph type="italics"/>D.<emph.end type="italics"/>Secetur recta infinita <emph type="italics"/>OQ<emph.end type="italics"/>in punctis <emph type="italics"/>O, <lb/>C, P, Q,<emph.end type="italics"/>ea lege, ut (&longs;i erigantur perpendicula <emph type="italics"/>OK, CT, PI, QE,<emph.end type="italics"/><lb/>centroque <emph type="italics"/>O<emph.end type="italics"/>& A&longs;ymptotis <emph type="italics"/>OK, OQ<emph.end type="italics"/>de&longs;cribatur Hyperbola <emph type="italics"/>TIGE<emph.end type="italics"/><lb/>&longs;ecans perpendicula <emph type="italics"/>CT, PI, QE<emph.end type="italics"/>in <emph type="italics"/>T, I<emph.end type="italics"/>& <emph type="italics"/>E,<emph.end type="italics"/>& per punctum <emph type="italics"/>I<emph.end type="italics"/><lb/>agatur <emph type="italics"/>KF<emph.end type="italics"/>parallela A&longs;ymptoto <emph type="italics"/>OQ<emph.end type="italics"/>occurrens A&longs;ymptoto <emph type="italics"/>OK<emph.end type="italics"/>in <lb/><emph type="italics"/>K,<emph.end type="italics"/>& perpendiculis <emph type="italics"/>CT<emph.end type="italics"/>& <emph type="italics"/>QE<emph.end type="italics"/>in <emph type="italics"/>L<emph.end type="italics"/>& <emph type="italics"/>F<emph.end type="italics"/>) fuerit area Hyperboliea <lb/><emph type="italics"/>PIEQ<emph.end type="italics"/>ad aream Hyperbolicam <emph type="italics"/>PITC<emph.end type="italics"/>ut arcus <emph type="italics"/>BC<emph.end type="italics"/>de&longs;cen&longs;u cor­<lb/>poris de&longs;criptus ad arcum <emph type="italics"/>Ca<emph.end type="italics"/>a&longs;cen&longs;u de&longs;criptum, & area <emph type="italics"/>IEF<emph.end type="italics"/>ad <pb xlink:href="039/01/306.jpg" pagenum="278"/><arrow.to.target n="note254"/>aream <emph type="italics"/>ILT<emph.end type="italics"/>ut <emph type="italics"/>OQ<emph.end type="italics"/>ad <emph type="italics"/>OC.<emph.end type="italics"/>Dein perpendiculo <emph type="italics"/>MN<emph.end type="italics"/>ab&longs;cindatur <lb/>area Hyperbolica <emph type="italics"/>PINM<emph.end type="italics"/>quæ &longs;it ad aream Hyperbolicam <emph type="italics"/>PIEQ<emph.end type="italics"/><lb/>ut arcus <emph type="italics"/>CZ<emph.end type="italics"/>ad arcum <emph type="italics"/>BC<emph.end type="italics"/>de&longs;cen&longs;u de&longs;criptum. </s> <s>Et &longs;i perpendicu­<lb/>lo <emph type="italics"/>RG<emph.end type="italics"/>ab&longs;cindatur area Hyperbolica <emph type="italics"/>PIGR,<emph.end type="italics"/>quæ &longs;it ad aream <lb/><emph type="italics"/>PIEQ<emph.end type="italics"/>ut arcus quilibet <emph type="italics"/>CD<emph.end type="italics"/>ad arcum <emph type="italics"/>BC<emph.end type="italics"/>de&longs;cen&longs;u toto de­<lb/>&longs;criptum: erit re&longs;i&longs;tentia in loco <emph type="italics"/>D<emph.end type="italics"/>ad vim gravitatis, ut area <lb/><emph type="italics"/>(OR/OQ)IEF-IGH<emph.end type="italics"/>ad aream <emph type="italics"/>PIENM.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note254"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Nam cum vires a gravitate oriundæ quibus corpus in locis <emph type="italics"/>Z, B, D, <lb/>a<emph.end type="italics"/>urgetur, &longs;int ut arcus <emph type="italics"/>CZ, CB, CD, Ca,<emph.end type="italics"/>& arcus illi &longs;int ut areæ <lb/><emph type="italics"/>PINM, PIEQ, PIGR, PITC<emph.end type="italics"/>; exponantur tum arcus tum vi­<lb/>res per has areas re&longs;pective. </s> <s>Sit in&longs;uper <emph type="italics"/>Dd<emph.end type="italics"/>&longs;patium quam minimum <lb/>a corpore de&longs;cendente de&longs;criptum, & exponatur idem per aream <lb/>quam minimam <emph type="italics"/>RGgr<emph.end type="italics"/>parallelis <emph type="italics"/>RG, rg<emph.end type="italics"/>comprehen&longs;am; & pro­<lb/><figure id="id.039.01.306.1.jpg" xlink:href="039/01/306/1.jpg"/><lb/>ducatur <emph type="italics"/>rg<emph.end type="italics"/>ad <emph type="italics"/>h,<emph.end type="italics"/>ut &longs;int <emph type="italics"/>GHhg,<emph.end type="italics"/>& <emph type="italics"/>RGgr<emph.end type="italics"/>contemporanea arearum <lb/><emph type="italics"/>IGH, PIGR<emph.end type="italics"/>decrementa. </s> <s>Et areæ <emph type="italics"/>(OR/OQ)IEF-IGH<emph.end type="italics"/>incremen­<lb/>tum <emph type="italics"/>GHhg-(Rr/OQ)IEF,<emph.end type="italics"/>&longs;eu <emph type="italics"/>RrXHG-(Rr/OQ)IEF,<emph.end type="italics"/>erit ad areæ <lb/><emph type="italics"/>PIGR<emph.end type="italics"/>decrementum <emph type="italics"/>RGgr<emph.end type="italics"/>&longs;eu <emph type="italics"/>RrXRG,<emph.end type="italics"/>ut <emph type="italics"/>HG-(IEF/OQ)<emph.end type="italics"/><lb/>ad <emph type="italics"/>RG<emph.end type="italics"/>; adeoque ut <emph type="italics"/>ORXHG-(OR/OQ)IEF<emph.end type="italics"/>ad <emph type="italics"/>ORXGR<emph.end type="italics"/>&longs;eu <lb/><emph type="italics"/>OPXPI,<emph.end type="italics"/>hoc e&longs;t (ob æqualia <emph type="italics"/>ORXHG, ORXHR-ORXGR, <lb/>ORHK-OPIK, PIHR<emph.end type="italics"/>& <emph type="italics"/>FIGR+IGH<emph.end type="italics"/>) ut <emph type="italics"/>PIGR+ <lb/>IGH-(OR/OQ)IEF<emph.end type="italics"/>ad <emph type="italics"/>OPIK.<emph.end type="italics"/>Igitur &longs;i area <emph type="italics"/>(OR/OQ)IEF-IGH<emph.end type="italics"/><pb xlink:href="039/01/307.jpg" pagenum="279"/>dicatur Y, atque areæ <emph type="italics"/>PIGR<emph.end type="italics"/>decrementum <emph type="italics"/>RGgr<emph.end type="italics"/>detur, erit <lb/><arrow.to.target n="note255"/>incrementum areæ Y ut <emph type="italics"/>PIGR<emph.end type="italics"/>-Y. </s></p> <p type="margin"> <s><margin.target id="note255"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s>Quod &longs;i V de&longs;ignet vim a gravitate oriundam, arcui de&longs;cribendo <lb/><emph type="italics"/>CD<emph.end type="italics"/>proportionalem, qua corpus urgetur in <emph type="italics"/>D:<emph.end type="italics"/>& R pro re&longs;i&longs;ten­<lb/>tia ponatur: erit V-R vis tota qua corpus urgetur in <emph type="italics"/>D.<emph.end type="italics"/>E&longs;t <lb/>itaQ.E.I.crementum velocitatis ut V-R & particula illa temporis <lb/>in qua factum e&longs;t conjunctim: Sed & velocitas ip&longs;a e&longs;t ut incre­<lb/>mentum contemporaneum &longs;patii de&longs;cripti directe & particula ea­<lb/>dem temporis inver&longs;e. </s> <s>Unde, cum re&longs;i&longs;tentia (per Hypothe&longs;in) <lb/>&longs;it ut quadratum velocitatis, incrementum re&longs;i&longs;tentiæ (per Lem. </s> <s>II) <lb/>erit ut velocitas & incrementum velocitatis conjunctim, id e&longs;t, ut <lb/>momentum &longs;patii & V-R conjunctim; atque adeo, &longs;i momen­<lb/>tum &longs;patii detur, ut V-R; id e&longs;t, &longs;i pro vi V &longs;eribatur ejus ex­<lb/>ponens <emph type="italics"/>PIGR,<emph.end type="italics"/>& re&longs;i&longs;tentia R exponatur per aliam aliquam are­<lb/>am Z, ut <emph type="italics"/>PIGR<emph.end type="italics"/>-Z. </s></p> <p type="main"> <s>Igitur area <emph type="italics"/>PIGR<emph.end type="italics"/>per datorum momentorum &longs;ubductionem <lb/>uniformiter decre&longs;cente, cre&longs;cunt area Y in ratione <emph type="italics"/>PIGR<emph.end type="italics"/>-Y, <lb/>& area Z in ratione <emph type="italics"/>PIGR<emph.end type="italics"/>-Z. </s> <s>Et propterea &longs;i areæ Y & Z &longs;i­<lb/>mul incipiant & &longs;ub initio æquales &longs;int, hæ per additionem æqua­<lb/>lium momentorum pergent e&longs;&longs;e æquales, & æqualibus itidem mo­<lb/>mentis &longs;ubinde decre&longs;centes &longs;imul evane&longs;cent. </s> <s>Et vici&longs;&longs;im, &longs;i &longs;imul <lb/>incipiunt & &longs;imul evane&longs;cunt, æqualia habebunt momenta & &longs;em­<lb/>per erunt æquales: id adeo quia &longs;i re&longs;i&longs;tentia Z augeatur, veloci­<lb/>tas una cum arcu illo <emph type="italics"/>Ca,<emph.end type="italics"/>qui in a&longs;cen&longs;u corporis de&longs;cribitur, dimi­<lb/>nuetur; & puncto in quo motus omnis una cum re&longs;i&longs;tentia ce&longs;&longs;at <lb/>propius accedente ad punctum <emph type="italics"/>C,<emph.end type="italics"/>re&longs;i&longs;tentia citius evane&longs;cet quam <lb/>area Y. </s> <s>Et contrarium eveniet ubi re&longs;i&longs;tentia diminuitur. </s></p> <p type="main"> <s>Jam vero area Z incipit de&longs;initque ubi re&longs;i&longs;tentia nulla e&longs;t, hoc <lb/>e&longs;t, in principio & fine motus, ubi arcus <emph type="italics"/>CD, CD<emph.end type="italics"/>arcubus <emph type="italics"/>CB<emph.end type="italics"/>& <lb/><emph type="italics"/>Ca<emph.end type="italics"/>æquantur, adeoque ubi recta <emph type="italics"/>RG<emph.end type="italics"/>incidit in rectas <emph type="italics"/>QE<emph.end type="italics"/>& <emph type="italics"/>CT.<emph.end type="italics"/><lb/>Et area Y &longs;eu <emph type="italics"/>(OR/OQ)IEF-IGH<emph.end type="italics"/>incipit de&longs;initque ubi nulla e&longs;t, ad­<lb/>eoque ubi <emph type="italics"/>(OR/OQ)IEF<emph.end type="italics"/>& <emph type="italics"/>IGH<emph.end type="italics"/>æqualia &longs;unt: hoc e&longs;t (per con­<lb/>&longs;tructionem) ubi recta <emph type="italics"/>RG<emph.end type="italics"/>incidit in rectas <emph type="italics"/>QE<emph.end type="italics"/>& <emph type="italics"/>CT.<emph.end type="italics"/>Proin­<lb/>deque areæ illæ &longs;imul incipiunt & &longs;imul evane&longs;cunt, & propterea <lb/>&longs;emper &longs;unt æquales. </s> <s>Igitur area <emph type="italics"/>(OR/OQ)IEF-IGH<emph.end type="italics"/>æqualis e&longs;t <lb/>areæ Z, per quam re&longs;i&longs;tentia exponitur, & propterea e&longs;t ad aream <lb/><emph type="italics"/>PINM<emph.end type="italics"/>per quam gravitas exponitur, ut re&longs;i&longs;tentia ad gravita­<lb/>tem. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/><pb xlink:href="039/01/308.jpg" pagenum="280"/><arrow.to.target n="note256"/></s></p> <p type="margin"> <s><margin.target id="note256"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. E&longs;t igitur re&longs;i&longs;tentia in loco infimo <emph type="italics"/>C<emph.end type="italics"/>ad vim gravitatis, <lb/>ut area <emph type="italics"/>(OP/OQ) IEF<emph.end type="italics"/>ad aream <emph type="italics"/>PINM.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Fit autem maxima, ubi area <emph type="italics"/>PIHR<emph.end type="italics"/>e&longs;t ad aream <lb/><emph type="italics"/>IEF<emph.end type="italics"/>ut <emph type="italics"/>OR<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="Oq.">Oque</expan><emph.end type="italics"/>Eo enim in ca&longs;u momentum ejus (nimirum <lb/><emph type="italics"/>PIGR<emph.end type="italics"/>-Y) evadit nullum. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Hinc etiam innote&longs;cit velocitas in locis &longs;ingulis: quippe <lb/>quæ e&longs;t in &longs;ubduplicata ratione re&longs;i&longs;tentiæ, & ip&longs;o motus initio æ­<lb/>quatur velocitati corporis in eadem Cycloide ab&longs;que omni re&longs;i&longs;ten­<lb/>tia o&longs;cillantis. </s></p> <p type="main"> <s>Cæterum ob difficilem calculum quo re&longs;i&longs;tentia & velocitas per <lb/>hanc Propo&longs;itionem inveniendæ &longs;unt, vi&longs;um e&longs;t Propo&longs;itionem &longs;e­<lb/>quentem &longs;ubjungere, quæ & generalior &longs;it & ad u&longs;us Philo&longs;ophi­<lb/>cos abunde &longs;atis accurata. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXX. THEOREMA XXIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si recta<emph.end type="italics"/>aB <emph type="italics"/>æqualis &longs;it Cycloidis arcui quem corpus o&longs;cillando de­<lb/>&longs;cribit, & ad &longs;ingula ejus puncta<emph.end type="italics"/>D <emph type="italics"/>erigantur perpendicula<emph.end type="italics"/>DK, <lb/><emph type="italics"/>quæ &longs;int ad longitudinem Penduli ut re&longs;i&longs;tentia corporis in ar­<lb/>cus punctis corre&longs;pondentibus ad vim gravitatis: dico quod <lb/>differentia inter arcum de&longs;cen&longs;u toto de&longs;criptum, & arcum <lb/>a&longs;cen&longs;u toto &longs;ub&longs;equente de&longs;criptum, ducta in arcuum eorundem <lb/>&longs;emi&longs;ummam, æqualis erit areæ<emph.end type="italics"/>BKaB <emph type="italics"/>a perpendiculis omnibus<emph.end type="italics"/><lb/>DK <emph type="italics"/>occupatæ.<emph.end type="italics"/></s></p> <p type="main"> <s>Exponatur enim tum Cycloidis arcus, o&longs;cillatione integra de­<lb/>&longs;criptus, per rectam illam &longs;ibi æqualem <emph type="italics"/>aB,<emph.end type="italics"/>tum arcus qui de&longs;cribe­<lb/>retur in vacuo per longitudinem <emph type="italics"/>AB.<emph.end type="italics"/>Bi&longs;ecetur <emph type="italics"/>AB<emph.end type="italics"/>in <emph type="italics"/>C,<emph.end type="italics"/>& pun­<lb/>ctum <emph type="italics"/>C<emph.end type="italics"/>repræ&longs;entabit infimum Cycloidis punctum, & erit <emph type="italics"/>CD<emph.end type="italics"/>ut <lb/>vis a gravitate oriunda, qua corpus in <emph type="italics"/>D<emph.end type="italics"/>&longs;ecundum tangentem <lb/>Cycloidis urgetur, eamque habebit rationem ad longitudinem Pen­<lb/>duli quam habet vis in <emph type="italics"/>D<emph.end type="italics"/>ad vim gravitatis. </s> <s>Exponatur igitur vis <lb/>illa per longitudinem <emph type="italics"/>CD,<emph.end type="italics"/>& vis gravitatis per longitudinem pen­<lb/>duli, & &longs;i in <emph type="italics"/>DE<emph.end type="italics"/>capiatur <emph type="italics"/>DK<emph.end type="italics"/>in ea ratione ad longitudinem <pb xlink:href="039/01/309.jpg" pagenum="281"/>penduli quam habet re&longs;i&longs;tentia ad gravitatem, erit <emph type="italics"/>DK<emph.end type="italics"/>exponens </s></p> <p type="main"> <s><arrow.to.target n="note257"/>re&longs;i&longs;tentiæ. </s> <s>Centro <emph type="italics"/>C<emph.end type="italics"/>& intervallo <emph type="italics"/>CA<emph.end type="italics"/>vel <emph type="italics"/>CB<emph.end type="italics"/>con&longs;truatur Semi­<lb/>circulus <emph type="italics"/>BEeA.<emph.end type="italics"/>De&longs;cribat autem corpus tempore quam minimo <lb/>&longs;patium <emph type="italics"/>Dd,<emph.end type="italics"/>& erectis perpendiculis <emph type="italics"/>DE, de<emph.end type="italics"/>circumferentiæ oc­<lb/>currentibus in <emph type="italics"/>E<emph.end type="italics"/>& <emph type="italics"/>e,<emph.end type="italics"/>erunt hæc ut velocitates quas corpus in va­<lb/>cuo, de&longs;cendendo a puncto <emph type="italics"/>B,<emph.end type="italics"/>acquireret in locis <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>d.<emph.end type="italics"/>Patet <lb/>hoc per Prop. </s> <s>LII. Lib. </s> <s>1. Exponantur itaque hæ velocitates per <lb/>perpendicula illa <emph type="italics"/>DE, de<emph.end type="italics"/>; &longs;itque <emph type="italics"/>DF<emph.end type="italics"/>velocitas quam acquirit <lb/>in <emph type="italics"/>D<emph.end type="italics"/>cadendo de <emph type="italics"/>B<emph.end type="italics"/>in Medio re&longs;i&longs;tente. </s> <s>Et &longs;i centro <emph type="italics"/>C<emph.end type="italics"/>& inter­<lb/>vallo <emph type="italics"/>CF<emph.end type="italics"/>de&longs;cribatur Circulus <emph type="italics"/>FfM<emph.end type="italics"/>occurrens rectis <emph type="italics"/>de<emph.end type="italics"/>& <emph type="italics"/>AB<emph.end type="italics"/>in <lb/><emph type="italics"/>f<emph.end type="italics"/>& <emph type="italics"/>M,<emph.end type="italics"/>erit <emph type="italics"/>M<emph.end type="italics"/>locus ad quem deinceps ab&longs;que ulteriore re&longs;i&longs;ten­<lb/>tia a&longs;cenderet, & <emph type="italics"/>df<emph.end type="italics"/>velocitas quam acquireret in <emph type="italics"/>d.<emph.end type="italics"/>Unde etiam <lb/>&longs;i <emph type="italics"/>Fg<emph.end type="italics"/>de&longs;ignet velocitatis momentum quod corpus <emph type="italics"/>D,<emph.end type="italics"/>de&longs;cribendo <lb/>&longs;patium quam minimum <emph type="italics"/>Dd,<emph.end type="italics"/>ex re&longs;i&longs;tentia Medii amittit; & &longs;u­<lb/>matur <emph type="italics"/>CN<emph.end type="italics"/>æqualis <emph type="italics"/>Cg:<emph.end type="italics"/>erit <emph type="italics"/>N<emph.end type="italics"/>locus ad quem corpus deinceps <lb/>ab&longs;que ulteriore re&longs;i&longs;tentia a&longs;cenderet, & <emph type="italics"/>MN<emph.end type="italics"/>erit decrementum <lb/>a&longs;cen&longs;us ex velocitatis illius ami&longs;&longs;ione oriundum. </s> <s>Ad <emph type="italics"/>df<emph.end type="italics"/>demitta­<lb/>tur perpendiculum <emph type="italics"/>Fm,<emph.end type="italics"/>& velocitatis <emph type="italics"/>DF<emph.end type="italics"/>decrementum <emph type="italics"/>Fg<emph.end type="italics"/>a <lb/>re&longs;i&longs;tentia <emph type="italics"/>DK<emph.end type="italics"/>genitum, erit ad velocitatis eju&longs;dem incrementum <lb/><emph type="italics"/>fm<emph.end type="italics"/>a vi <emph type="italics"/>CD<emph.end type="italics"/>genitum, ut vis generans <emph type="italics"/>DK<emph.end type="italics"/>ad vim generantem <lb/><emph type="italics"/>CD.<emph.end type="italics"/>Sed & ob &longs;imilia <lb/><figure id="id.039.01.309.1.jpg" xlink:href="039/01/309/1.jpg"/><lb/>triangula <emph type="italics"/>Fmf, Fhg, <lb/>FDC,<emph.end type="italics"/>e&longs;t <emph type="italics"/>fm<emph.end type="italics"/>ad <emph type="italics"/>Fm<emph.end type="italics"/><lb/>&longs;eu <emph type="italics"/>Dd,<emph.end type="italics"/>ut <emph type="italics"/>CD<emph.end type="italics"/>ad <lb/><emph type="italics"/>DF<emph.end type="italics"/>; & ex æquo <emph type="italics"/>Fg<emph.end type="italics"/>ad <lb/><emph type="italics"/>Dd<emph.end type="italics"/>ut <emph type="italics"/>DK<emph.end type="italics"/>ad <emph type="italics"/>DF.<emph.end type="italics"/><lb/>Item <emph type="italics"/>Fh<emph.end type="italics"/>ad <emph type="italics"/>Fg<emph.end type="italics"/>ut <emph type="italics"/>DF<emph.end type="italics"/><lb/>ad <emph type="italics"/>CF<emph.end type="italics"/>; & ex æquo <lb/>perturbate, <emph type="italics"/>Fh<emph.end type="italics"/>&longs;eu <emph type="italics"/>MN<emph.end type="italics"/><lb/>ad <emph type="italics"/>Dd<emph.end type="italics"/>ut <emph type="italics"/>DK<emph.end type="italics"/>ad <emph type="italics"/>CF<emph.end type="italics"/><lb/>&longs;eu <emph type="italics"/>CM<emph.end type="italics"/>; ideoque &longs;umma omnium <emph type="italics"/>MNXCM<emph.end type="italics"/>æqualis erit &longs;ummæ <lb/>omnium <emph type="italics"/>DdXDK.<emph.end type="italics"/>Ad punctum mobile <emph type="italics"/>M<emph.end type="italics"/>erigi &longs;emper intelli­<lb/>gatur ordinata rectangula æqualis indeterminatæ <emph type="italics"/>CM,<emph.end type="italics"/>quæ motu <lb/>continuo ducatur in totam longitudinem <emph type="italics"/>Aa<emph.end type="italics"/>; & trapezium ex illo <lb/>motu de&longs;criptum &longs;ive huic æquale rectangulum <emph type="italics"/>Aa<emph.end type="italics"/>X1/2<emph type="italics"/>aB<emph.end type="italics"/>æquabitur <lb/>&longs;ummæ omnium <emph type="italics"/>MNXCM,<emph.end type="italics"/>adeoque &longs;ummæ omnium <emph type="italics"/>DdXDK,<emph.end type="italics"/><lb/>id e&longs;t, areæ <emph type="italics"/>BKkVTa. </s> <s>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note257"/>LIBER <lb/>SECUNDUS</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc ex lege re&longs;i&longs;tentiæ & arcuum <emph type="italics"/>Ca, CB<emph.end type="italics"/>differentia <emph type="italics"/>Aa,<emph.end type="italics"/><lb/>colligi pote&longs;t proportio re&longs;i&longs;tentiæ ad gravitatem quam proxime. <pb xlink:href="039/01/310.jpg" pagenum="282"/><arrow.to.target n="note258"/></s></p> <p type="margin"> <s><margin.target id="note258"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Nam &longs;i uniformis &longs;it re&longs;i&longs;tentia <emph type="italics"/>DK,<emph.end type="italics"/>Figura <emph type="italics"/>aBKkT<emph.end type="italics"/>rectangu­<lb/>lum erit &longs;ub <emph type="italics"/>Ba<emph.end type="italics"/>& <emph type="italics"/>DK<emph.end type="italics"/>; & inde rectangulum &longs;ub 1/2 <emph type="italics"/>Ba<emph.end type="italics"/>& <emph type="italics"/>Aa<emph.end type="italics"/><lb/>erit æquale rectangulo &longs;ub <emph type="italics"/>Ba<emph.end type="italics"/>& <emph type="italics"/>DK,<emph.end type="italics"/>& <emph type="italics"/>DK<emph.end type="italics"/>æqualis erit 1/2 <emph type="italics"/>Aa.<emph.end type="italics"/><lb/>Quare cum <emph type="italics"/>DK<emph.end type="italics"/>&longs;it exponens re&longs;i&longs;tentiæ, & longitudo penduli ex­<lb/>ponens gravitatis, erit re&longs;i&longs;tentia ad gravitatem ut 1/2 <emph type="italics"/>Aa<emph.end type="italics"/>ad longi­<lb/>tudinem Penduli; omnino ut in Prop. </s> <s>XXVIII demon&longs;tratum e&longs;t. </s></p> <p type="main"> <s>Si re&longs;i&longs;tentia &longs;it ut velocitas, Figura <emph type="italics"/>aBKkT<emph.end type="italics"/>Ellip&longs;is erit quam <lb/>proxime. </s> <s>Nam &longs;i corpus, in Medio non re&longs;i&longs;tente, o&longs;cillatione <lb/>integra de&longs;criberet longitudinem <emph type="italics"/>BA,<emph.end type="italics"/>velocitas in loco quovis <emph type="italics"/>D<emph.end type="italics"/><lb/>foret ut Circuli diametro <emph type="italics"/>AB<emph.end type="italics"/>de&longs;cripti ordinatim applicata <emph type="italics"/>DE.<emph.end type="italics"/><lb/>Proinde cum <emph type="italics"/>Ba<emph.end type="italics"/>in Medio re&longs;i&longs;tente, & <emph type="italics"/>BA<emph.end type="italics"/>in Medio non re&longs;i­<lb/>&longs;tente, æqualibus circiter temporibus de&longs;cribantur; adeoque velo­<lb/>citates in &longs;ingulis ip&longs;ius <lb/><figure id="id.039.01.310.1.jpg" xlink:href="039/01/310/1.jpg"/><lb/><emph type="italics"/>Ba<emph.end type="italics"/>punctis, &longs;int quam <lb/>proxime ad velocitates <lb/>in punctis corre&longs;pon­<lb/>dentibus longitudinis <lb/><emph type="italics"/>BA,<emph.end type="italics"/>ut e&longs;t <emph type="italics"/>Ba<emph.end type="italics"/>ad <emph type="italics"/>BA<emph.end type="italics"/>; <lb/>erit velocitas <emph type="italics"/>DK<emph.end type="italics"/>in <lb/>Medio re&longs;i&longs;tente ut Cir­<lb/>culi vel Ellip&longs;eos &longs;uper <lb/>diametro <emph type="italics"/>Ba<emph.end type="italics"/>de&longs;cripti <lb/>ordinatim applicata; adeoque Figura <emph type="italics"/>BKVTa<emph.end type="italics"/>Ellip&longs;is, quam pro­<lb/>xime. </s> <s>Cum re&longs;i&longs;tentia velocitati proportionalis &longs;upponatur, &longs;it <emph type="italics"/>OV<emph.end type="italics"/><lb/>exponens re&longs;i&longs;tentiæ in puncto Medio <emph type="italics"/>O<emph.end type="italics"/>; & Ellip&longs;is <emph type="italics"/>aBRVS,<emph.end type="italics"/><lb/>centro <emph type="italics"/>O,<emph.end type="italics"/>&longs;emiaxibus <emph type="italics"/>OB, OV<emph.end type="italics"/>de&longs;cripta, Figuram <emph type="italics"/>aBKVT,<emph.end type="italics"/><lb/>eique æquale rectangulum <emph type="italics"/>AaXBO,<emph.end type="italics"/>æquabit quamproxime. </s> <s>E&longs;t <lb/>igitur <emph type="italics"/>AaXBO<emph.end type="italics"/>ad <emph type="italics"/>OVXBO<emph.end type="italics"/>ut area Ellip&longs;eos hujus ad <emph type="italics"/>OVXBO<emph.end type="italics"/>: <lb/>id e&longs;t, <emph type="italics"/>Aa<emph.end type="italics"/>ad <emph type="italics"/>OV<emph.end type="italics"/>ut area &longs;emicirculi ad quadratum radii, &longs;ive ut <lb/>11 ad 7 circiter: Et propterea (1/11) <emph type="italics"/>Aa<emph.end type="italics"/>ad longitudinem penduli ut <lb/>corporis o&longs;cillantis re&longs;i&longs;tentia in <emph type="italics"/>O<emph.end type="italics"/>ad eju&longs;dem gravitatem. </s></p> <p type="main"> <s>Quod &longs;i re&longs;i&longs;tentia <emph type="italics"/>DK<emph.end type="italics"/>&longs;it in duplicata ratione velocitatis, Fi­<lb/>gura <emph type="italics"/>BKVTa<emph.end type="italics"/>Parabola erit verticem habens <emph type="italics"/>V<emph.end type="italics"/>& axem <emph type="italics"/>OV,<emph.end type="italics"/>id­<lb/>eoque æqualis erit rectangulo &longs;ub 2/3 <emph type="italics"/>Ba<emph.end type="italics"/>& <emph type="italics"/>OV<emph.end type="italics"/>quam proxime. </s> <s>E&longs;t <lb/>igitur rectangulum &longs;ub 1/2 <emph type="italics"/>Ba<emph.end type="italics"/>& <emph type="italics"/>Aa<emph.end type="italics"/>æquale rectangulo &longs;ub 2/3 <emph type="italics"/>Ba<emph.end type="italics"/><lb/>& <emph type="italics"/>OV,<emph.end type="italics"/>adeoque <emph type="italics"/>OV<emph.end type="italics"/>æqualis 1/4 <emph type="italics"/>Aa:<emph.end type="italics"/>& propterea corporis o&longs;cillan­<lb/>tis re&longs;i&longs;tentia in <emph type="italics"/>O<emph.end type="italics"/>ad ip&longs;ius gravitatem ut 1/4 <emph type="italics"/>Aa<emph.end type="italics"/>ad longitudi­<lb/>nem Penduli. </s></p> <p type="main"> <s>Atque has conclu&longs;iones in rebus practicis abunde &longs;atis accuratas <lb/>e&longs;&longs;e cen&longs;eo. </s> <s>Nam cum Ellip&longs;is vel Parabola <emph type="italics"/>BRVSa<emph.end type="italics"/>congruat <pb xlink:href="039/01/311.jpg" pagenum="283"/>cum Figura <emph type="italics"/>BKVTa<emph.end type="italics"/>in puncto medio <emph type="italics"/>V,<emph.end type="italics"/>hæc &longs;i ad partem al­</s></p> <p type="main"> <s><arrow.to.target n="note259"/>terutram <emph type="italics"/>BRV<emph.end type="italics"/>vel <emph type="italics"/>VSa<emph.end type="italics"/>excedit Figuram illam, deficiet ab eadem <lb/>ad partem alteram, & &longs;ic eidem æquabitur quam proxime. </s></p> <p type="margin"> <s><margin.target id="note259"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXI. THEOREMA XXV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Corporis o&longs;cillantis re&longs;i&longs;tentia in &longs;ingulis arcuum de&longs;criptorum <lb/>partibus proportionalibus augeatur vel minuatur in data ratio­<lb/>ne; differentia inter arcum de&longs;cen&longs;u de&longs;criptum & arcum &longs;ub­<lb/>&longs;equente a&longs;cen&longs;u de&longs;criptum, augebitur vel diminuetur in eadem <lb/>ratione.<emph.end type="italics"/></s></p> <p type="main"> <s>Oritur enim differentia illa ex retardatione Penduli per re&longs;i­<lb/>&longs;tentiam Medii, adeoque e&longs;t ut retardatio tota eique proportio­<lb/>nalis re&longs;i&longs;tentia retardans. </s> <s>In &longs;uperiore Propo&longs;itione rectangu­<lb/>lum &longs;ub recta 1/2 <emph type="italics"/>aB<emph.end type="italics"/>& arcuum illorum <emph type="italics"/>CB, Ca<emph.end type="italics"/>differentia <emph type="italics"/>Aa,<emph.end type="italics"/><lb/>æqualis erat areæ <emph type="italics"/>BKT.<emph.end type="italics"/>Et area illa, &longs;i maneat longitudo <emph type="italics"/>aB,<emph.end type="italics"/><lb/>augetur vel diminuitur in ratione ordinatim applicatarum <emph type="italics"/>DK<emph.end type="italics"/>; <lb/>hoc e&longs;t, in ratione re&longs;i&longs;tentiæ, adeoque e&longs;t ut longitudo <emph type="italics"/>aB<emph.end type="italics"/>& <lb/>re&longs;i&longs;tentia conjunctim. </s> <s>Proindeque rectangulum &longs;ub <emph type="italics"/>Aa<emph.end type="italics"/>& 1/2 <emph type="italics"/>aB<emph.end type="italics"/><lb/>e&longs;t ut <emph type="italics"/>aB<emph.end type="italics"/>& re&longs;i&longs;tentia conjunctim, & propterea <emph type="italics"/>Aa<emph.end type="italics"/>ut re&longs;i&longs;ten­<lb/>tia. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Unde &longs;i re&longs;i&longs;tentia &longs;it ut velocitas, differentia arcuum <lb/>in eodem Medio erit ut arcus totus de&longs;criptus: & contra. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si re&longs;i&longs;tentia &longs;it in duplicata ratione velocitatis, diffe­<lb/>rentia illa erit in duplicata ratione arcus totius: & contra. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Et univer&longs;aliter, &longs;i re&longs;i&longs;tentia &longs;it in triplicata vel alia <lb/>quavis ratione velocitatis, differentia erit in eadem ratione arcus <lb/>totius: & contra. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Et &longs;i re&longs;i&longs;tentia &longs;it partim in ratione &longs;implici velocita­<lb/>tis, partim in eju&longs;dem ratione duplicata, differentia erit partim in <lb/>ratione arcus totius & partim in ejus ratione duplicata: & contra. </s> <s><lb/>Eadem erit lex & ratio re&longs;i&longs;tentiæ pro velocitate, quæ e&longs;t differen­<lb/>tiæ illius pro longitudine arcus. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Ideoque &longs;i, pendulo inæquales arcus &longs;ucce&longs;&longs;ive de&longs;cri­<lb/>bente, inveniri pote&longs;t ratio incrementi ac decrementi differentiæ hu­<lb/>jus pro longitudine arcus de&longs;cripti; habebitur etiam ratio incrementi <lb/>ac decrementi re&longs;i&longs;tentiæ pro velocitate majore vel minore. <pb xlink:href="039/01/312.jpg" pagenum="284"/><arrow.to.target n="note260"/></s></p> <p type="margin"> <s><margin.target id="note260"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium Generale.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Ex his Propo&longs;itionibus, per o&longs;cillationes Pendulorum in Mediis <lb/>quibu&longs;cunque, invenire licet re&longs;i&longs;tentiam Mediorum. </s> <s>Aeris vero <lb/>re&longs;i&longs;tentiam inve&longs;tigavi per Experimenta &longs;equentia. </s> <s>Globum lig­<lb/>neum pondere unciarum <emph type="italics"/>Romanarum<emph.end type="italics"/>(57 7/22), diametro digitorum <lb/><emph type="italics"/>Londinen&longs;ium<emph.end type="italics"/>6 7/8 fabricatum, filo tenui ab unco &longs;atis firmo &longs;u&longs;pen­<lb/>di, ita ut inter uncum & centrum o&longs;cillationis Globi di&longs;tantia e&longs;&longs;et <lb/>pedum 10 1/2. In filo punctum notavi pedibus decem & uncia una <lb/>a centro &longs;u&longs;pen&longs;ionis di&longs;tans; & e regione puncti illius collocavi <lb/>Regulam in digitos di&longs;tinctam, quorum ope notarem longitudi­<lb/>nes arcuum a Pendulo de&longs;criptas. </s> <s>Deinde numeravi o&longs;cillationes <lb/>quibus Globus octavam motus &longs;ui partem amitteret. </s> <s>Si pendu­<lb/>lum deducebatur a perpendiculo ad di&longs;tantiam duorum digitorum, <lb/>& inde demittebatur; ita ut toto &longs;uo de&longs;cen&longs;u de&longs;criberet arcum <lb/>duorum digitorum, totaque o&longs;cillatione prima, ex de&longs;cen&longs;u & a&longs;cen­<lb/>&longs;u &longs;ub&longs;equente compo&longs;ita, arcum digitorum fere quatuor: idem <lb/>o&longs;cillationibus 164 ami&longs;it octavam motus &longs;ui partem, &longs;ic ut ultimo <lb/>&longs;uo a&longs;cen&longs;u de&longs;criberet arcum digiti unius cum tribus partibus <lb/>quartis digiti. </s> <s>Si primo de&longs;cen&longs;u de&longs;crip&longs;it arcum digitorum qua­<lb/>tuor; ami&longs;it octavam motus partem o&longs;cillationibus 121, ita ut a&longs;cen­<lb/>&longs;u ultimo de&longs;criberet arcum digitorum 3 1/2. Si primo de&longs;cen&longs;u de­<lb/>&longs;crip&longs;it arcum digitorum octo, &longs;exdecim, triginta duorum vel &longs;exa­<lb/>ginta quatuor; ami&longs;it octavam motus partem o&longs;cillationibus 69, 35 1/2, <lb/>18 1/2, 9 2/3, re&longs;pective. </s> <s>Igitur differentia inter arcus de&longs;cen&longs;u primo <lb/>& a&longs;cen&longs;u ultimo de&longs;criptos, erat in ca&longs;u primo, &longs;ecundo, tertio, <lb/>quarto, quinto, &longs;exto, digitorum 1/4, 1/2, 1, 2, 4, 8 re&longs;pective. </s> <s>Divi­<lb/>dantur eæ differentiæ per numerum o&longs;cillationum in ca&longs;u unoquo­<lb/>que, & in o&longs;cillatione una mediocri, qua arcus digitorum 3 1/4, 7 1/2, <lb/>15, 30, 60, 120 de&longs;criptus fuit, differentia arcuum de&longs;cen&longs;u & &longs;ub­<lb/>&longs;equente a&longs;cen&longs;u de&longs;criptorum, erit (1/656), (1/242), (1/69), (4/71), (8/37), (24/29) partes di­<lb/>giti re&longs;pective. </s> <s>Hæ autem in majoribus o&longs;cillationibus &longs;unt in du­<lb/>plicata ratione arcuum de&longs;criptorum quam proxime, in minoribus <lb/>vero paulo majores quam in ea ratione; & propterea (per Corol. </s> <s>2. <lb/>Prop. </s> <s>XXXI Libri hujus) re&longs;i&longs;tentia Globi, ubi celerius movetur, <lb/>e&longs;t in duplicata ratione velocitatis quam proxime; ubi tardius, pau­<lb/>lo major quam in ea ratione. </s></p><pb xlink:href="039/01/313.jpg" pagenum="285"/> <p type="main"> <s>De&longs;ignet jam V velocitatem maximam in o&longs;cillatione quavis, <lb/><arrow.to.target n="note261"/>&longs;intque A, B, C quantitates datæ, & fingamus quod differentia <lb/>arcuum &longs;it AV+BV 1/2+CV<emph type="sup"/>2<emph.end type="sup"/>. </s> <s>Cum velocitates maximæ &longs;int in <lb/>Cycloide ut &longs;emi&longs;&longs;es arcuum o&longs;cillando de&longs;criptorum, in Circu­<lb/>lo vero ut &longs;emi&longs;&longs;ium arcuum illorum chordæ; adeoque paribus <lb/>arcubus majores &longs;int in Cycloide quam in Circulo, in ratione <lb/>&longs;emi&longs;&longs;ium arcuum ad eorundem chordas; tempora autem in Cir­<lb/>culo &longs;int majora quam in Cycloide in velocitatis ratione reci­<lb/>proca; patet arcuum differentias (quæ &longs;unt ut re&longs;i&longs;tentia & qua­<lb/>dratum temporis conjunctim) ea&longs;dem fore, quamproxime, in utra­<lb/>que Curva: deberent enim differentiæ illæ in Cycloide augeri, una <lb/>cum re&longs;i&longs;tentia, in duplicata circiter ratione arcus ad chordam, ob <lb/>velocitatem in ratione illa &longs;implici auctam; & diminui, una cum <lb/>quadrato temporis, in eadem duplicata ratione. </s> <s>Itaque ut reductio <lb/>fiat ad Cycloidem, eædem &longs;umendæ &longs;unt arcuum differentiæ quæ <lb/>fuerunt in Circulo ob&longs;ervatæ, velocitates vero maximæ ponen­<lb/>dæ &longs;unt arcubus dimidiatis vel integris, hoc e&longs;t, numeris 1/2, 1, 2, <lb/>4, 8, 16 analogæ. </s> <s>Scribamus ergo in ca&longs;u &longs;ecundo, quarto & &longs;ex­<lb/>to numeros 1, 4 & 16 pro V; & prodibit arcuum differentia <lb/>(1/2/121)=A+B+C in ca&longs;u &longs;ecundo; (2/35 1/2)=4A+8B+16C in ca&longs;u <lb/>quarto; & (8/9 2/3)=16A+64B+256C in ca&longs;u &longs;exto. </s> <s>Et ex his æ­<lb/>quationibus, per debitam collationem & reductionem Analyticam, <lb/>fit A=0,0000916, B=0,0010847, & C=0,0029558. E&longs;t igitur <lb/>differentia arcuum ut 0,0000916V+0,0010847V1/2+0,0029558V<emph type="sup"/>2<emph.end type="sup"/>: <lb/>& propterea cum (per Corollarium Propo&longs;itionis XXX) re&longs;i&longs;tentia <lb/>Globi in medio arcus o&longs;cillando de&longs;cripti, ubi velocitas e&longs;t V, <lb/>&longs;it ad ip&longs;ius pondus ut (7/11)AV+(16/23)BV1/2+1/4CV<emph type="sup"/>2<emph.end type="sup"/> ad longitudinem <lb/>Penduli; &longs;i pro A, B & C &longs;cribantur numeri inventi, fiet re&longs;i&longs;tentia <lb/>Globi ad ejus pondus, ut 0,0000583V+0,0007546V1/2+0,0022169V<emph type="sup"/>2<emph.end type="sup"/><lb/>ad longitudinem Penduli inter centrum &longs;u&longs;pen&longs;ionis & Regulam, <lb/>id e&longs;t, ad 121 digitos. </s> <s>Unde cum V in ca&longs;u &longs;ecundo de&longs;ignet 1, <lb/>in quarto 4, in &longs;exto 16: erit re&longs;i&longs;tentia ad pondus Globi in ca&longs;u <lb/>&longs;ecundo ut 0,0030298 ad 121, in quarto ut 0,0417402 ad 121, in <lb/>&longs;exto ut 0,61675 ad 121. </s></p> <p type="margin"> <s><margin.target id="note261"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s>Arcus quem punctum in filo notatum in ca&longs;u &longs;exto de&longs;crip&longs;it, <lb/>erat 120-(8/9 2/3) &longs;eu (119 5/29) digitorum. </s> <s>Et propterea cum radius e&longs;&longs;et <lb/>121 digitorum, & longitudo Penduli inter punctum &longs;u&longs;pen&longs;ionis <pb xlink:href="039/01/314.jpg" pagenum="286"/><arrow.to.target n="note262"/>& centrum Globi e&longs;&longs;et 126 digitorum, arcus quem centrum Globi <lb/>de&longs;crip&longs;it erat (124 1/31) digitorum. </s> <s>Quoniam corporis o&longs;cillantis ve­<lb/>locitas maxima, ob re&longs;i&longs;tentiam Aeris, non incidit in punctum infi­<lb/>mum arcus de&longs;cripti, &longs;ed in medio fere loco arcus totius ver&longs;atur: <lb/>hæc eadem erit circiter ac &longs;i Globus de&longs;cen&longs;u &longs;uo toto in Medio <lb/>non re&longs;i&longs;tente de&longs;criberet arcus illius partem dimidiam digitorum <lb/>(62 1/62), idQ.E.I. Cycloide, ad quam motum Penduli &longs;upra reduxi­<lb/>mus: & propterea velocitas illa æqualis erit velocitati quam Glo­<lb/>bus, perpendiculariter cadendo & ca&longs;u &longs;uo de&longs;cribendo altitudinem <lb/>arcus illius &longs;inui ver&longs;o æqualem, acquirere po&longs;&longs;et. </s> <s>E&longs;t autem &longs;inus <lb/>ille ver&longs;us in Cycloide ad arcum i&longs;tum (62 1/62) ut arcus idem ad pen­<lb/>duli longitudinem duplam 252, & propterea æqualis digitis 15,278. <lb/>Quare velocitas ea ip&longs;a e&longs;t quam corpus cadendo & ca&longs;u &longs;uo &longs;pa­<lb/>tium 15,278 digitorum de&longs;cribendo acquirere po&longs;&longs;et. </s> <s>Tali igitur <lb/>cum velocitate Globus re&longs;i&longs;tentiam patitur, quæ &longs;it ad ejus pondus <lb/>ut 0,61675 ad 121, vel (&longs;i re&longs;i&longs;tentiæ pars illa &longs;ola &longs;pectetur quæ <lb/>e&longs;t in velocitatis ratione duplicata) ut 0,56752 ad 121. </s></p> <p type="margin"> <s><margin.target id="note262"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Experimento autem Hydro&longs;tatico inveni quod pondus Globi hu­<lb/>jus lignei e&longs;&longs;et ad pondus Globi aquei magnitudinis eju&longs;dem, ut 55 <lb/>ad 97: & propterea cum 121 &longs;it ad 213,4 in eadem ratione, erit <lb/>re&longs;i&longs;tentia Globi aquei præfata cum velocitate progredientis ad ip­<lb/>&longs;ius pondus, ut 0,56752 ad 213,4 id e&longs;t, ut 1 ad (376 1/50). Unde cum <lb/>pondus Globi aquei, quo tempore Globus cum velocitate unifor­<lb/>miter continuata de&longs;cribat longitudinem digitorum 30,556, veloci­<lb/>tatem illam omnem in Globo cadente generare po&longs;&longs;et; manife&longs;tum <lb/>e&longs;t quod vis re&longs;i&longs;tentiæ eodem tempore uniformiter continuata tol­<lb/>lere po&longs;&longs;et velocitatem minorem in ratione 1 ad (376 1/50), hoc e&longs;t, ve­<lb/>locitatis totius partem (1/(376 1/50)). Et propterea quo tempore Globus, <lb/>ea cum velocitate uniformiter continuata, longitudinem &longs;emidiame­<lb/>tri &longs;uæ, &longs;eu digitorum (3 7/16), de&longs;cribere po&longs;&longs;et, eodem amitteret mo­<lb/>tus &longs;ui partem (1/3342). </s></p> <p type="main"> <s>Numerabam etiam o&longs;cillationes quibus Pendulum quartam mo­<lb/>tus &longs;ui partem ami&longs;it. </s> <s>In &longs;equente Tabula numeri &longs;upremi deno­<lb/>tant longitudinem arcus de&longs;cen&longs;u primo de&longs;cripti, in digitis & par­<lb/>tibus digiti expre&longs;&longs;am: numeri medii &longs;ignificant longitudinem ar­<lb/>cus a&longs;cen&longs;u ultimo de&longs;cripti; & loco infimo &longs;tant numeri o&longs;cilla­<lb/>tionum. </s> <s>Experimentum de&longs;crip&longs;i tanquam magis accuratum quam <lb/>cum motus pars tantum octava amitteretur. </s> <s>Calculum tentet qui <lb/>volet. <pb xlink:href="039/01/315.jpg" pagenum="287"/><arrow.to.target n="note263"/><arrow.to.target n="table1"/><arrow.to.target n="table2"/></s></p> <p type="margin"> <s><margin.target id="note263"/>LIBER <lb/>SECUNDUS.</s></p><table><table.target id="table1"/><row><cell><emph type="italics"/>De&longs;cen&longs;us primus<emph.end type="italics"/></cell><cell>2</cell><cell>4</cell><cell>8</cell><cell>16</cell><cell>32</cell><cell>64</cell></row><row><cell><emph type="italics"/>A&longs;cen&longs;us ultimus<emph.end type="italics"/></cell><cell>1 1/2</cell><cell>3</cell><cell>6</cell><cell>12</cell><cell>34</cell><cell>48</cell></row><row><cell><emph type="italics"/>Numerus O&longs;cillat.<emph.end type="italics"/></cell><cell>374</cell><cell>272</cell><cell>162 1/2</cell><cell>83 1/3</cell><cell>41 2/3</cell><cell>22 2/3</cell></row></table> <p type="main"> <s>Po&longs;tea Globum plumbeum, diametro digitorum 2, & pondere <lb/> unciarum <emph type="italics"/>Romanarum<emph.end type="italics"/>26 1/4, &longs;u&longs;pendi filo eodem, &longs;ic ut inter cen­<lb/>trum Globi & punctum &longs;u&longs;pen&longs;ionis intervallum e&longs;&longs;et pedum 10 1/2, <lb/> & numerabam o&longs;cillationes quibus data motus pars amitteretur. <lb/> Tabularum &longs;ub&longs;equentium prior exhibet numerum o&longs;cillationum <lb/> quibus pars octava motus totius ce&longs;&longs;avit; &longs;ecunda numerum o&longs;cil­<lb/>lationum quibus eju&longs;dem pars quarta ami&longs;&longs;a fuit. <lb/></s></p> <table><row><cell><emph type="italics"/>De&longs;cen&longs;us primus<emph.end type="italics"/></cell><cell>1</cell><cell>2</cell><cell>4</cell><cell>8</cell><cell>16</cell><cell>32</cell><cell>64</cell></row><row><cell><emph type="italics"/>A&longs;cen&longs;us ultimus<emph.end type="italics"/></cell><cell>7/8</cell><cell>7/4</cell><cell>3 1/2</cell><cell>7</cell><cell>14</cell><cell>28</cell><cell>56</cell></row><row><cell><emph type="italics"/>Numerus O&longs;cillat.<emph.end type="italics"/></cell><cell>226</cell><cell>228</cell><cell>193</cell><cell>140</cell><cell>90 1/2</cell><cell>53</cell><cell>30</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell/><cell/><cell/></row><row><cell><emph type="italics"/>De&longs;cen&longs;us primus<emph.end type="italics"/></cell><cell>1</cell><cell>2</cell><cell>4</cell><cell>8</cell><cell>16</cell><cell>32</cell><cell>64</cell></row><row><cell><emph type="italics"/>A&longs;cen&longs;us ultimus<emph.end type="italics"/></cell><cell>3/4</cell><cell>1 1/2</cell><cell>3</cell><cell>6</cell><cell>12</cell><cell>24</cell><cell>48</cell></row><row><cell><emph type="italics"/>Numerus O&longs;cillat.<emph.end type="italics"/></cell><cell>510</cell><cell>518</cell><cell>420</cell><cell>318</cell><cell>204</cell><cell>121</cell><cell>70</cell></row></table> <p type="main"> <s>In Tabula priore &longs;eligendo ex ob&longs;ervationibus tertiam, quintam <lb/> & &longs;eptimam, & exponendo velocitates maximas in his ob&longs;erva­<lb/>tionibus particulatim per numeros 1, 4, 16 re&longs;pective, & genera­<lb/>liter per quantitatem V ut &longs;upra: emerget in ob&longs;ervatione tertia <lb/> (1/2/193)=A+B+C, in quinta (2/90 1/2)=4A+8B+16C, in &longs;eptima <lb/> (8/30)=16A+64B+256C. Hæ vero æquationes reductæ dant <lb/> A=0,001414, B=0,000297, C=0,000879. Et inde prodit re&longs;i­<lb/>&longs;tentia Globi cum velocitate V moti, in ea ratione ad pondus &longs;uum <lb/> unciarum 26 1/4, quam habet 0,0009V+0,000207V1/2+0,000659V<emph type="sup"/>2<emph.end type="sup"/><lb/>ad penduli longitudinem 121 digitorum. </s> <s>Et &longs;i &longs;pectemus eam &longs;o­<lb/>lummodo re&longs;i&longs;tentiæ partem quæ e&longs;t in duplicata ratione velocitatis, <lb/> hæc erit ad pondus Globi ut 0,000659V<emph type="sup"/>2<emph.end type="sup"/> ad 121 digitos. </s> <s>Erat au­<lb/>tem hæc pars re&longs;i&longs;tentiæ in experimento primo ad pondus Globi <lb/> lignei unciarum (57 7/22), ut 0,002217V<emph type="sup"/>2<emph.end type="sup"/> ad 121: & inde fit re&longs;i&longs;tentia <lb/> Globi lignei ad re&longs;i&longs;tentiam Globi plumbei (paribus eorum velocita­<lb/>tibus) ut (57 7/22) in 0,002217 ad 26 1/4 in 0,000659, id e&longs;t, ut 7 1/3 ad 1. <lb/> Diametri Globorum duorum erant 6 7/8 & 2 digitorum, & harum <lb/> quadrata &longs;unt ad invicem ut 47 1/4 & 4, &longs;eu (11 11/16) & 1 quamproxime. <lb/> Ergo re&longs;i&longs;tentiæ Globorum æquivelocium erant in minore ratione <lb/> quam duplicata diametrorum. </s> <s>At nondum con&longs;ideravimus re&longs;i-<pb xlink:href="039/01/316.jpg" pagenum="288"/><lb/><arrow.to.target n="note264"/>&longs;tentiam fili, quæ certe permagna erat, ac de pendulorum inventa <lb/> re&longs;i&longs;tentia &longs;ubduci debet. </s> <s>Hanc accurate definire non potui, &longs;ed <lb/> majorem tamen inveni quam partem tertiam re&longs;i&longs;tentiæ totius mi­<lb/>noris penduli; & inde didici quod re&longs;i&longs;tentiæ Globorum, dempta <lb/> fili re&longs;i&longs;tentia, &longs;unt quam proxime in duplicata ratione diametro­<lb/>rum. </s> <s>Nam ratio 7 1/3-1/3 ad 1-1/3, &longs;eu 10 1/2 ad 1, non longe abe&longs;t a <lb/> diametrorum ratione duplicata (11 11/16) ad 1. <lb/> </s></p> <p type="margin"> <s><margin.target id="note264"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s>Cum re&longs;i&longs;tentia fili in Globis majoribus minoris &longs;it momenti, <lb/> tentavi etiam experimentum in Globo cujus diameter erat 18 1/4 di­<lb/>gitorum. </s> <s>Longitudo penduli inter punctum &longs;u&longs;pen&longs;ionis & cen­<lb/>trum o&longs;cillationis erat digitorum 122 1/2, inter punctum &longs;u&longs;pen&longs;ionis <lb/> & nodum in filo 109 1/2 dig. Arcus primo penduli de&longs;cen&longs;u a no­<lb/>do de&longs;criptus, 32 dig. Arcus a&longs;cen&longs;u ultimo po&longs;t o&longs;cillationes <lb/> quinque ab eodem nodo de&longs;criptus, 28 dig. Summa arcuum &longs;eu <lb/> arcus totus o&longs;cillatione mediocri de&longs;criptus, 60 dig. Differentia <lb/> arcuum 4 dig. Ejus pars decima &longs;eu differentia inter de&longs;cen&longs;um & <lb/> a&longs;cen&longs;um in o&longs;cillatione mediocri 2/5 dig. Ut radius 109 1/2 ad radi­<lb/>um 122 1/2, ita arcus totus 60 dig. o&longs;cillatione mediocri a nodo de­<lb/>&longs;criptus, ad arcum totum 67 1/8 dig. o&longs;cillatione mediocri a centro <lb/> Globi de&longs;criptum: & ita differentia 2/5 ad differentiam novam 0,4475. <lb/> Si longitudo penduli, manente longitudine arcus de&longs;cripti, augere­<lb/>tur in ratione 126 ad 122 1/2; tempus o&longs;cillationis augeretur & velo­<lb/>citas penduli diminueretur in ratione illa &longs;ubduplicata, maneret <lb/> vero arcuum de&longs;cen&longs;u & &longs;ub&longs;equente a&longs;cen&longs;u de&longs;criptorum diffe­<lb/>rentia 0,4475. Deinde &longs;i arcus de&longs;criptus augeretur in ratione <lb/> (124 1/31) ad 67 1/8, differentia i&longs;ta 0,4475 augeretur in duplicata illa ra­<lb/>tione, adeoque evaderet 1,5295. Hæc ita &longs;e haberent, ex hy­<lb/>pothe&longs;i quod re&longs;i&longs;tentia Penduli e&longs;&longs;et in duplicata ratione velo­<lb/>citatis. </s> <s>Ergo &longs;i pendulum de&longs;criberet arcum totum (124 1/31) di­<lb/>gitorum, & longitudo ejus inter punctum &longs;u&longs;pen&longs;ionis & cen­<lb/>trum o&longs;cillationis e&longs;&longs;et 126 digitorum, differentia arcuum de­<lb/>&longs;cen&longs;u & &longs;ub&longs;equente a&longs;cen&longs;u de&longs;criptorum foret 1,5295 digito­<lb/>rum. </s> <s>Et hæc differentia ducta in pondus Globi penduli, quod erat <lb/> unciarum 208, producit 318,136. Rur&longs;us ubi pendulum &longs;uperius <lb/> ex Globo ligneo con&longs;tructum, centro o&longs;cillationis, quod a puncto <lb/> &longs;u&longs;pen&longs;ionis digitos 126 di&longs;tabat, de&longs;cribebat arcum totum (124 1/31) <lb/> digitorum, differentia arcuum de&longs;cen&longs;u & a&longs;cen&longs;u de&longs;criptum fuit <lb/> (126/121) in (8/9 2/3), quæ ducta in pondus Globi, quod erat unciarum (57 1/22), <lb/> producit 49,396. Duxi autem differentias ha&longs;ce in pondera Glo­<lb/>borum ut invenirem eorum re&longs;i&longs;tentias. </s> <s>Nam differentiæ ori-<pb xlink:href="039/01/317.jpg" pagenum="289"/><lb/>untur ex re&longs;i&longs;tentiis, &longs;untque ut re&longs;i&longs;tentiæ directe & pondera in­<lb/><arrow.to.target n="note265"/>ver&longs;e. </s> <s>Sunt igitur re&longs;i&longs;tentiæ ut numeri 318,136 & 49,396. Pars <lb/> autem re&longs;i&longs;tentiæ Globi minoris, quæ e&longs;t in duplicata ratione velo­<lb/>citatis, erat ad re&longs;i&longs;tentiam totam, ut 0,56752 ad 0,61675, id e&longs;t, ut <lb/> 45,453 ad 49,396; & pars re&longs;i&longs;tentiæ Globi majoris propemodum <lb/> æquatur ip&longs;ius re&longs;i&longs;tentiæ toti; adeoque partes illæ &longs;unt ut 318,136 <lb/> & 45,453 quamproxime, id e&longs;t, ut 7 & 1. Sunt autem Globorum <lb/> diametri 18 1/4 & 6 7/8; & harum quadrata (351 9/16) & (47 17/64) &longs;unt ut 7,438 <lb/> & 1, id e&longs;t, ut Globorum re&longs;i&longs;tentiæ 7 & 1 quamproxime. </s> <s>Diffe­<lb/>rentia rationum haud major e&longs;t quam quæ ex fili re&longs;i&longs;tentia oriri po­<lb/>tuit. </s> <s>Igitur re&longs;i&longs;tentiarum partes illæ quæ &longs;unt, paribus Globis, ut <lb/> quadrata velocitatum; &longs;unt etiam, paribus velocitatibus, ut qua­<lb/>drata diametrorum Globorum. <lb/> </s></p> <p type="margin"> <s><margin.target id="note265"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s>Cæterum Globorum, quibus u&longs;us &longs;um in his experimentis, max­<lb/>imus non erat perfecte Sphæricus, & propterea in calculo hic allato <lb/> minutias qua&longs;dam brevitatis gratia neglexi; de calculo accurato in <lb/> experimento non &longs;atis accurato minime &longs;ollicitus. </s> <s>Optarim itaque <lb/> (cum demon&longs;tratio Vacui ex his dependeat) ut experimenta cum <lb/> Globis & pluribus & majoribus & magis accuratis tentarentur. </s> <s>Si <lb/> Globi &longs;umantur in proportione Geometrica, puta quorum diametri <lb/> &longs;int digitorum 4, 8, 16, 32; ex progre&longs;&longs;ione experimentorum col­<lb/>ligetur quid in Globis adhuc majoribus evenire debeat. <lb/> </s></p> <p type="main"> <s>Jam vero conferendo re&longs;i&longs;tentias diver&longs;orum Fluidorum inter &longs;e <lb/> tentavi &longs;equentia. </s> <s>Arcam ligneam paravi longitudine pedum qua­<lb/>tuor, latitudine & altitudine pedis unius. </s> <s>Hanc operculo nuda­<lb/>tam implevi aqua fontana, fecique ut immer&longs;a pendula in medio <lb/> aquæ o&longs;cillando moverentur. </s> <s>Globus autem plumbeus pondere <lb/> 166 1/6 unciarum, diametro 3 5/8 digitorum, movebatur ut in Tabula <lb/> &longs;equente de&longs;crip&longs;imus, exi&longs;tente videlicet longitudine penduli a <lb/> puncto &longs;u&longs;pen&longs;ionis ad punctum quoddam in filo notatum 126 di­<lb/>gitorum, ad o&longs;cillationis autem centrum 134 1/8 digitorum.</s></p><table><row><cell><emph type="italics"/>Arcus de&longs;cen&longs;u primo a puncto in <lb/> filo notato de&longs;criptus, digitorum<emph.end type="italics"/></cell><cell>64</cell><cell>32</cell><cell>16</cell><cell>8</cell><cell>4</cell><cell>2</cell><cell>1</cell><cell>1/2</cell><cell>1/4</cell></row><row><cell><emph type="italics"/>Arcus a&longs;cen&longs;u ultimo de&longs;criptus, <lb/> digitorum<emph.end type="italics"/></cell><cell>48</cell><cell>24</cell><cell>12</cell><cell>6</cell><cell>3</cell><cell>1 1/4</cell><cell>1/4</cell><cell>1/8</cell><cell>(1/16)</cell></row><row><cell><emph type="italics"/>Arcuum differentia motui ami&longs;&longs;o <lb/> proportionalis, digitorum<emph.end type="italics"/></cell><cell>16</cell><cell>8</cell><cell>4</cell><cell>2</cell><cell>1</cell><cell>1/2</cell><cell>1/4</cell><cell>1/8</cell><cell>(1/16)</cell></row><row><cell><emph type="italics"/>Numerus O&longs;cillationum in aqua<emph.end type="italics"/></cell><cell/><cell/><cell>(29/60)</cell><cell>1 1/5</cell><cell>3</cell><cell>7</cell><cell>11 1/4</cell><cell>12 2/3</cell><cell>13 1/3</cell></row><row><cell><emph type="italics"/>Numerus O&longs;cillationum in aere<emph.end type="italics"/></cell><cell>85 1/2</cell><cell/><cell>287</cell><cell>535</cell><cell/><cell/><cell/><cell/><cell/></row></table><pb xlink:href="039/01/318.jpg" pagenum="290"/> <p type="margin"> <s><margin.target id="note266"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s>In experimento columnæ quartæ, motus æquales o&longs;cillationibus <lb/> 535 in aere, & 1 1/5 in aqua ami&longs;&longs;i &longs;unt. </s> <s>Erant quidem o&longs;cillationes <lb/> in aere paulo celeriores quam in aqua. </s> <s>At &longs;i o&longs;cillationes in aqua <lb/> in ea ratione accelerarentur ut motus pendulorum in Medio utro­<lb/>que fierent æquiveloces, maneret numerus idem o&longs;cillationum 1 1/5 <lb/> in aqua, quibus motus idem ac prius amitteretur; ob re&longs;i&longs;tentiam <lb/> auctam & &longs;imul quadratum temporis diminutum in eadem ratione <lb/> illa duplicata. </s> <s>Paribus igitur pendulorum velocitatibus motus æ­<lb/>quales in aere o&longs;cillationibus 535 & in aqua o&longs;cillationibus 1 1/5 ami&longs;&longs;i <lb/> &longs;unt; ideoque re&longs;i&longs;tentia penduli in aqua e&longs;t ad ejus re&longs;i&longs;tentiam in <lb/> aere ut 535 ad 1 1/5. Hæc e&longs;t proportio re&longs;i&longs;tentiarum totarum in <lb/> ca&longs;u columnæ quartæ. <lb/> </s></p> <p type="main"> <s>De&longs;ignet jam AV+CV differentiam arcuum in de&longs;cen&longs;u & &longs;ub­<lb/>&longs;equente a&longs;cen&longs;u de&longs;criptorum a Globo, in Aere cum velocitate maxi­<lb/>ma V moto; & cum velocitas maxima, in ca&longs;u columnæ quartæ, &longs;it <lb/> ad velocitatem maximam in ca&longs;u columnæ primæ, ut 1 ad 8; & diffe­<lb/>rentia illa arcuum, in ca&longs;u columnæ quartæ, ad differentiam in ca&longs;u <lb/> columnæ primæ ut (2/535) ad (16/85 1/2), &longs;eu ut 85 1/2 ad 4280: &longs;eribamus in <lb/> his ca&longs;ibus 1 & 8 pro velocitatibus, atque 85 1/2 & 4280 pro dif­<lb/>ferentiis arcuum, & fiet A+C=85 1/2 & 8A+64C=4280 &longs;eu <lb/> A+8C=535; indeque per reductionem æquationum proveniet <lb/> 7C=449 1/2 & C=(64 1/14) & A=21 1/7: atque adeo re&longs;i&longs;tentia, cum <lb/> &longs;it ut (7/11) AV+1/4 CV<emph type="sup"/>2<emph.end type="sup"/>, erit ut (13 6/11)V+(48 1/56)V<emph type="sup"/>2<emph.end type="sup"/>. Quare in ca&longs;u co­<lb/>lumnæ quartæ, ubi velocitas erat 1, re&longs;i&longs;tentia tota e&longs;t ad partem <lb/> &longs;uam quadrato velocitatis proportionalem, ut (13 6/11)+(48 2/56) &longs;eu <lb/> (61 12/17) ad (48 9/56); & idcirco re&longs;i&longs;tentia penduli in aqua e&longs;t ad re&longs;i&longs;ten­<lb/>tiæ partem illam in aere quæ quadrato velocitatis proportionalis <lb/> e&longs;t, quæque &longs;ola in motibus velocioribus con&longs;ideranda venit, ut (61 12/17) <lb/> ad (48 9/56) & 535 ad 1 1/5 conjunctim, id e&longs;t, ut 571 ad 1. Si penduli <lb/> in aqua o&longs;cillantis filum totum fui&longs;&longs;et immer&longs;um, re&longs;i&longs;tentia ejus <lb/> fui&longs;&longs;et adhuc major; adeo ut penduli in aere o&longs;cillantis re&longs;i&longs;tentia <lb/> illa quæ velocitatis quadrato proportionalis e&longs;t, quæque &longs;ola in <lb/> corporibus velocioribus con&longs;ideranda venit, &longs;it ad re&longs;i&longs;tentiam e­<lb/>ju&longs;dem penduli totius, eadem cum velocitate, in aqua o&longs;cillantis, <lb/> ut 800 vel 900 ad 1 circiter, hoc e&longs;t, ut den&longs;itas aquæ ad den&longs;ita­<lb/>tatem aeris quamproxime. <lb/> </s></p> <p type="main"> <s>In hoc calculo &longs;umi quoQ.E.D.beret pars illa re&longs;i&longs;tentiæ penduli <lb/> in aqua, quæ e&longs;&longs;et ut quadratum velocitatis, &longs;ed (quod mirum for­<lb/>te videatur) re&longs;i&longs;tentia in aqua augebatur in ratione velocitatis <pb xlink:href="039/01/319.jpg" pagenum="291"/><lb/>plu&longs;quam duplicata. </s> <s>Ejus rei cau&longs;am inve&longs;tigando, in hanc in­<lb/><arrow.to.target n="note267"/>cidi, quod Arca nimis angu&longs;ta e&longs;&longs;et pro magnitudine Globi pen­<lb/>duli, & motum aquæ cedentis præ angu&longs;tia &longs;ua nimis impedie­<lb/>bat. </s> <s>Nam &longs;i Globus pendulus, cujus diameter erat digiti u­<lb/>nius, immergeretur; re&longs;i&longs;tentia augebatur in duplicata ratione ve­<lb/>locitatis quam proxime. </s> <s>Id tentabam con&longs;truendo pendulum ex <lb/> Globis duobus, quorum inferior & minor o&longs;cillaretur in aqua, &longs;u­<lb/>perior & major proxime &longs;upra aquam filo affixus e&longs;&longs;et, & in Aere <lb/> o&longs;cillando, adjuvaret motum penduli eumQ.E.D.uturniorem redde­<lb/>ret. </s> <s>Experimenta autem hoc modo in&longs;tituta &longs;e habebant ut in Ta­<lb/>bula &longs;equente de&longs;cribitur. <lb/></s></p><table><row><cell><emph type="italics"/>Arcus de&longs;cen&longs;u primo de&longs;criptus<emph.end type="italics"/></cell><cell>16</cell><cell>8</cell><cell>4</cell><cell>2</cell><cell>1</cell><cell>1/2</cell><cell>1/4</cell></row><row><cell><emph type="italics"/>Arcus a&longs;cen&longs;u ultimo de&longs;criptus<emph.end type="italics"/></cell><cell>12</cell><cell>6</cell><cell>3</cell><cell>1 1/2</cell><cell>1/4</cell><cell>1/8</cell><cell>(1/16)</cell></row><row><cell><emph type="italics"/>Arcuum diff. motui ami&longs;&longs;o proport.<emph.end type="italics"/></cell><cell>4</cell><cell>2</cell><cell>1</cell><cell>1/2</cell><cell>1/4</cell><cell>1/8</cell><cell>(1/16)</cell></row><row><cell><emph type="italics"/>Numerus O&longs;cillationum<emph.end type="italics"/></cell><cell>3 1/8</cell><cell>6 1/2</cell><cell>(12 1/12)</cell><cell>21 1/5</cell><cell>34</cell><cell>53</cell><cell>62 1/5</cell></row></table> <p type="margin"> <s><margin.target id="note267"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s>Conferendo re&longs;i&longs;tentias Mediorum inter &longs;e, effeci etiam ut pen­<lb/>dula ferrea o&longs;cillarentur in argento vivo. </s> <s>Longitudo fili ferrei erat <lb/> pedum qua&longs;i trium, & diameter Globi penduli qua&longs;i tertia pars di­<lb/>giti. </s> <s>Ad filum autem proxime &longs;upra Mercurium affixus erat Glo­<lb/>bus alius plumbeus &longs;atis magnus ad motum penduli diutius conti­<lb/>nuandum. </s> <s>Tum va&longs;culum, quod capiebat qua&longs;i libras tres argenti <lb/> vivi, implebam vicibus alternis argento vivo & aqua communi, ut <lb/> pendulo in Fluido utroque &longs;ucce&longs;&longs;ive o&longs;cillante, invenirem propor­<lb/>tionem re&longs;i&longs;tentiarum: & prodiit re&longs;i&longs;tentia argenti vivi ad re&longs;i­<lb/>&longs;tentiam aquæ, ut 13 vel 14 ad 1 circiter: id e&longs;t, ut den&longs;itas argen­<lb/>ti vivi ad den&longs;itatem aquæ. Ubi Globum pendulum paulo majo­<lb/>rem adhibebam, puta cujus diameter e&longs;&longs;et qua&longs;i 1/3 vel 2/3 partes di­<lb/>giti, prodibat re&longs;i&longs;tentia argenti vivi in ea ratione ad re&longs;i&longs;tentiam <lb/> aquæ quam habet numerus 12 vel 10 ad 1 circiter. </s> <s>Sed experi­<lb/>mento priori magis fidendum e&longs;t, propterea quod in his ultimis <lb/> Vas nimis angu&longs;tum fuit pro magnitudine Globi immer&longs;i. </s> <s>Am­<lb/>pliato Globo, deberet etiam Vas ampliari. </s> <s>Con&longs;titueram quidem <lb/> huju&longs;modi experimenta in va&longs;is majoribus & in liquoribus tum <lb/> Metallorum fu&longs;orum, tum aliis quibu&longs;dam tam calidis quam fri­<lb/>gidis repetere: &longs;ed omnia experiri non vacat, & ex jam de&longs;criptis <lb/> &longs;atis liquet re&longs;i&longs;tentiam corporum celeriter motorum den&longs;itati Flu­<lb/>idorum in quibus moventur proportionalem e&longs;&longs;e quam proxime. <lb/> Non dico accurate. </s> <s>Nam Fluida tenaciora, pari den&longs;itate, procul-<pb xlink:href="039/01/320.jpg" pagenum="292"/><lb/><arrow.to.target n="note268"/>dubio magis re&longs;i&longs;tunt quam liquidiora, ut Oleum frigidum quam <lb/> calidum, calidum quam aqua pluvialis, aqua quam Spiritus Vini. <lb/> Verum in liquoribus qui ad &longs;en&longs;um &longs;atis fluidi &longs;unt, ut in Aere, in <lb/> Aqua &longs;eu dulci &longs;eu &longs;al&longs;a, in Spiritibus Vini, Terebinthi & Salium, <lb/> in Oleo a fæcibus per de&longs;tillationem liberato & calefacto, Oleoque <lb/> Vitrioli & Mercurio, ac Metallis liquefactis, & &longs;iqui &longs;int alii, qui <lb/> tam fluidi &longs;unt ut in va&longs;is agitati motum impre&longs;&longs;um diutius con­<lb/>&longs;ervent, effu&longs;ique liberrime in guttas decurrendo re&longs;olvantur, nul­<lb/>lus dubito quin regula allata &longs;atis accurate obtineat: præ&longs;ertim &longs;i <lb/> experimenta in corporibus pendulis & majoribus & velocius motis <lb/> in&longs;tituantur. <lb/> </s></p> <p type="margin"> <s><margin.target id="note268"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s>Denique cum recepti&longs;&longs;ima Philo&longs;ophorum ætatis hujus opinio <lb/> &longs;it, Medium quoddam æthereum & longe &longs;ubtili&longs;&longs;imum extare, <lb/> quod omnes omnium corporum poros & meatus liberrime per­<lb/>meet; a tali autem Medio per corporum poros fluente re&longs;i&longs;tentia <lb/> oriri debeat: ut tentarem an re&longs;i&longs;tentia, quam in motis corporibus <lb/> experimur, tota &longs;it in eorum externa &longs;uperficie, an vero partes eti­<lb/>am internæ in &longs;uperficiebus propriis re&longs;i&longs;tentiam notabilem &longs;enti­<lb/>ant, excogitavi experimentum tale. </s> <s>Filo pedum undecim longitu­<lb/>dinis, ab unco chalyoeo &longs;atis firmo, mediante annulo chalybeo, &longs;u­<lb/>&longs;pendebam pyxidem abiegnam rotundam, ad con&longs;tituendum pen­<lb/>dulum longitudinis prædictæ. Uncus &longs;ur&longs;um præacutus erat acie <lb/> concava, ut annulus arcu &longs;uo &longs;uperiore aciei innixus liberrime mo­<lb/>veretur. </s> <s>Arcui autem inferiori annectebatur filum. </s> <s>Pendulum ita <lb/> con&longs;titutum deducebam a perpendiculo ad di&longs;tantiam qua&longs;i pedum <lb/> &longs;ex, idque &longs;ecundum planum aciei unci perpendiculare, ne annu­<lb/>lus, o&longs;cillante pendulo, &longs;upra aciem unci ultro citroque laberetur. <lb/> Nam punctum &longs;u&longs;pen&longs;ionis, in quo annulus uncum tangit, immo­<lb/>tum manere debet. </s> <s>Locum igitur accurate notabam, ad quem de­<lb/>duxeram pendulum, dein pendulo demi&longs;&longs;o notabam alia tria loca ad <lb/> quæ redibat in fine o&longs;cillationis primæ, &longs;ecundæ ac tertiæ. Hoc re­<lb/>petebam &longs;æpius, ut loca illa quam potui accurati&longs;&longs;ime invenirem. <lb/> Tum pyxidem plumbo & gravioribus, quæ ad manus erant, me­<lb/>tallis implebam. </s> <s>Sed prius ponderabam pyxidem vacuam, una <lb/> cum parte fili quæ circum pyxidem volvebatur ac dimidio par­<lb/>tis reliquæ inter uncum & pyxidem pendulam tendebatur. <lb/> (Nam filum ten&longs;um dimidio ponderis &longs;ui pendulum a perpendiculo <lb/> digre&longs;&longs;um &longs;emper urget.) Huic ponderi addebam pondus Aeris <lb/> quem pyxis capiebat. </s> <s>Et pondus totum erat qua&longs;i pars &longs;eptuage­<lb/>&longs;ima octava pyxidis metallorum plenæ. Tum quoniam pyxis me-<pb xlink:href="039/01/321.jpg" pagenum="293"/><lb/>tallorum plena, pondere &longs;uo tendendo filum, augebat longitudi­<lb/><arrow.to.target n="note269"/>nem penduli, contrahebam filum ut penduli jam o&longs;cillantis eadem <lb/> e&longs;&longs;et longitudo ac prius. </s> <s>Dein pendulo ad locum primo notatum <lb/> retracto ac dimi&longs;&longs;o, numerabam o&longs;cillationes qua&longs;i &longs;eptuaginta & <lb/> &longs;eptem, donec pyxis ad locum &longs;ecundo notatum rediret, totidem­<lb/>que &longs;ubinde donec pyxis ad locum tertio notatum rediret, atque <lb/> rur&longs;us totidem donec pyxis reditu &longs;uo attingeret locum quartum. <lb/> Unde concludo quod re&longs;i&longs;tentia tota pyxidis plenæ non majorem <lb/> habebat proportionem ad re&longs;i&longs;tentiam pyxidis vacuæ quam 78 ad <lb/> 77. Nam &longs;i æquales e&longs;&longs;ent ambarum re&longs;i&longs;tentiæ, pyxis plena ob <lb/> vim &longs;uam in&longs;itam &longs;eptuagies & octies majorem vi in&longs;ita pyxidis <lb/> vacuæ, motum &longs;uum o&longs;cillatorium tanto diutius con&longs;ervare debe­<lb/>ret, atque adeo completis &longs;emper o&longs;cillationibus 78 ad loca illa <lb/> notata redire. </s> <s>Rediit autem ad eadem completis o&longs;cillationibus 77. <lb/> </s></p> <p type="margin"> <s><margin.target id="note269"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s>De&longs;ignet igitur A re&longs;i&longs;tentiam pyxidis in ip&longs;ius &longs;uperficie exter­<lb/>na, & B re&longs;i&longs;tentiam pyxidis vacuæ in partibus internis; & &longs;i re&longs;i­<lb/>&longs;tentiæ corporum æquivelocium in partibus internis &longs;int ut mate­<lb/>ria, &longs;eu numerus particularum quibus re&longs;i&longs;titur: erit 78 B re&longs;i&longs;ten­<lb/>tia pyxidis plenæ in ip&longs;ius partibus internis: adeoque pyxidis va­<lb/>cuæ re&longs;i&longs;tentia tota A+B erit ad pyxidis plenæ re&longs;i&longs;tentiam to­<lb/>tam A+78 B ut 77 ad 78, & divi&longs;im A+B ad 77 B, ut 77 ad 1, <lb/> indeque A+B ad B ut 77X77 ad 1, & divi&longs;im A ad B ut 5928 <lb/> ad 1. E&longs;t igitur re&longs;i&longs;tentia pyxidis vacuæ in partibus internis <lb/> quinquies millies minor quam eju&longs;dem re&longs;i&longs;tentia in externa &longs;uper­<lb/>ficie, & amplius. </s> <s>Sic vero di&longs;putamus ex Hypothe&longs;i quod ma­<lb/>jor illa re&longs;i&longs;tentia pyxidis plenæ, non ab alia aliqua cau&longs;a latente <lb/> oriatur, &longs;ed ab actione &longs;ola Fluidi alicujus &longs;ubtilis in metallum <lb/> inclu&longs;um. <lb/> </s></p> <p type="main"> <s>Hoc experimentum recitavi memoriter. </s> <s>Nam charta, in qua il­<lb/>lud aliquando de&longs;crip&longs;eram, intercidit. </s> <s>Unde fractas qua&longs;dam nu­<lb/>merorum partes, quæ memoria exciderunt, omittere compul&longs;us <lb/> &longs;um. </s> <s>Nam omnia denuo tentare non vacat. </s> <s>Prima vice, cum un­<lb/>co infirmo u&longs;us e&longs;&longs;em, pyxis plena citius retardabatur. </s> <s>Cau&longs;am <lb/> quærendo, reperi quod uncus infirmus cedebat ponderi pyxidis, & <lb/> ejus o&longs;cillationibus ob&longs;eQ.E.D. in partes omnes flectebatur. </s> <s>Para­<lb/>bam igitur uncum firmum, ut punctum &longs;u&longs;pen&longs;ionis immotum ma­<lb/>neret, & tunc omnia ita evenerunt uti &longs;upra de&longs;crip&longs;imus. <pb xlink:href="039/01/322.jpg" pagenum="294"/><lb/></s></p></subchap2><subchap2><p> <s><arrow.to.target n="note270"/><emph type="center"/>SECTIO VII.<emph.end type="center"/><lb/><emph type="center"/><emph type="italics"/>De Motu Fluidorum & Re&longs;i&longs;tentia Projectilium.<emph.end type="italics"/><emph.end type="center"/><lb/><emph type="center"/>PROPOSITIO XXXII. THEOREMA XXVI.<emph.end type="center"/><lb/></s></p> <p type="margin"> <s><margin.target id="note270"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Si Corporum Sy&longs;temata duo &longs;imilia ex æquali particularum numero <lb/> con&longs;tent, & particulæ corre&longs;pondentes &longs;imiles &longs;int & propor­<lb/>tionales, &longs;ingulæ in uno Sy&longs;temate &longs;ingulis in altero, & &longs;imiliter <lb/> &longs;itæ inter &longs;e, ac datam habeant rationem den&longs;itatis ad invicem, <lb/> & inter &longs;e temporibus proportionalibus &longs;imiliter moveri inci­<lb/>piant, (eæ inter &longs;e quæ in uno &longs;unt Sy&longs;temate & eæ inter &longs;e quæ <lb/> &longs;unt in altero) & &longs;i non tangant &longs;e mutuo quæ in eodem &longs;unt <lb/> Sy&longs;temate, ni&longs;i in momentis reflexionum, neque attrahant vel fu­<lb/>gent &longs;e mutuo, ni&longs;i viribus acceleratricibus quæ &longs;int ut particu­<lb/>larum corre&longs;pondentium diametri inver&longs;e & quadrata velocita. <lb/> tum directe: dico quod Sy&longs;tematum particulæ illæ pergent inter <lb/> &longs;e temporibus proportionalibus &longs;imiliter moveri.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Corpora &longs;imilia & &longs;imiliter &longs;ita temporibus proportionalibus in­<lb/>ter &longs;e &longs;imiliter moveri dico, quorum &longs;itus ad invicem in fine tem­<lb/>porum illorum &longs;emper &longs;unt &longs;imiles: puta &longs;i particulæ unius Sy&longs;te­<lb/>matis cum alterius particulis corre&longs;pondentibus conferantur. </s> <s>Un­<lb/>de tempora erunt proportionalia, in quibus &longs;imiles & proportiona­<lb/>les Figurarum &longs;imilium partes a particulis corre&longs;pondentibus de­<lb/>&longs;cribuntur. </s> <s>Igitur &longs;i duo &longs;int eju&longs;modi Sy&longs;temata, particulæ cor­<lb/>re&longs;pondentes, ob &longs;imilitudinem incæptorum motuum, pergent &longs;i­<lb/>militer moveri u&longs;Q.E.D.nec &longs;ibi mutuo occurrant. </s> <s>Nam &longs;i nullis <lb/> agitantur viribus, progredientur uniformiter in lineis rectis per mo­<lb/>tus Leg. 1. Si viribus aliquibus &longs;e mutuo agitant, & vires illæ &longs;int <lb/> ut particularum corre&longs;pondentium diametri inver&longs;e & quadrata ve­<lb/>locitatum directe; quoniam particularum &longs;itus &longs;unt &longs;imiles & vires <lb/> proportionales, vires totæ quibus particulæ corre&longs;pondentes agi­<lb/>tantur, ex viribus &longs;ingulis agitantibus (per Legum Corollarium <pb xlink:href="039/01/323.jpg" pagenum="295"/><lb/>fecundum) compo&longs;itæ, &longs;imiles habebunt determinationes, perin­<lb/><arrow.to.target n="note271"/>de ac &longs;i centra inter particulas &longs;imiliter &longs;ita re&longs;picerent; & erunt <lb/> vires illæ totæ ad invicem ut vires &longs;ingulæ componentes, hoc e&longs;t, <lb/> ut corre&longs;pondentium particularum diametri inver&longs;e, & quadrata <lb/> velocitatum directe: & propterea efficient ut corre&longs;pondentes par­<lb/>ticulæ figuras &longs;imiles de&longs;cribere pergant. </s> <s>Hæc ita &longs;e habebunt per <lb/> Corol. 1, & 8 Prop. IV, Lib. 1. &longs;i modo centra illa quie&longs;cant. <lb/> Sin moveantur, quoniam ob tran&longs;lationum &longs;imilitudinem, &longs;imiles <lb/> manent eorum &longs;itus inter Sy&longs;tematum particulas; &longs;imiles indu­<lb/>centur mutationes in figuris quas particulæ de&longs;cribunt. </s> <s>Similes igi­<lb/>tur erunt corre&longs;pondentium & &longs;imilium particularum motus u&longs;­<lb/>que ad occur&longs;us &longs;uos primos, & propterea &longs;imiles occur&longs;us, & &longs;i­<lb/>miles reflexiones, & &longs;ubinde (per jam o&longs;ten&longs;a) &longs;imiles motus in­<lb/>ter &longs;e donec iterum in &longs;e mutuo inciderint, & &longs;ic deinceps in in­<lb/>finitum. <emph type="italics"/>Q.E.D.<emph.end type="italics"/><lb/></s></p> <p type="margin"> <s><margin.target id="note271"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i corpora duo quævis, quæ &longs;imilia &longs;int & ad <lb/> Sy&longs;tematum particulas corre&longs;pondentes &longs;imiliter &longs;ita, inter ip&longs;as <lb/> temporibus proportionalibus &longs;imiliter moveri incipiant, &longs;intque <lb/> eorum magnitudines ac den&longs;itates ad invicem ut magnitudines ac <lb/> den&longs;itates corre&longs;pondentium particularum: hæc pergent tempori­<lb/>bus proportionalibus &longs;imiliter moveri. </s> <s>E&longs;t enim eadem ratio par­<lb/>tium majorum Sy&longs;tematis utriu&longs;que atque particularum. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et &longs;i &longs;imiles & &longs;imiliter po&longs;itæ Sy&longs;tematum partes om­<lb/>nes quie&longs;cant inter &longs;e: & earum duæ, quæ cæteris majores &longs;int, & <lb/> &longs;ibi mutuo in utroque Sy&longs;temate corre&longs;pondeant, &longs;ecundum lineas <lb/> &longs;imiliter &longs;itas &longs;imili cum motu utcunque moveri incipiant: hæ &longs;i­<lb/>miles in reliquis Sy&longs;tematum partibus excitabunt motus, & pergent <lb/> inter ip&longs;as temporibus proportionalibus &longs;imiliter moveri; atque <lb/> adeo &longs;patia diametris &longs;uis proportionalia de&longs;cribere. <lb/> <emph type="center"/>PROPOSITIO XXXIII. THEOREMA XXVII.<emph.end type="center"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, dico quod Sy&longs;tematum partes majores re&longs;i&longs;tituntur <lb/> in ratione compo&longs;ita ex duplicata ratione velocitatum &longs;uarum & <lb/> duplicata ratione diametrorum & ratione den&longs;itatis partium <lb/> Sy&longs;tematum.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Nam re&longs;i&longs;tentia oritur partim ex viribus centripetis vel centri­<lb/>fugis quibus particulæ Sy&longs;tematum &longs;e mutuo agitant, partim ex <lb/> occur&longs;ibus & reflexionibus particularum & partium majorum. <pb xlink:href="039/01/324.jpg" pagenum="296"/><lb/><arrow.to.target n="note272"/>Prioris autem generis re&longs;i&longs;tentiæ &longs;unt ad invicem ut vires totæ mo­<lb/>trices a quibus oriuntur, id e&longs;t, ut vires totæ acceleratrices & quan­<lb/>titates materiæ in partibus corre&longs;pondentibus; hoc e&longs;t (per Hy­<lb/>pothe&longs;in) ut quadrata velocitatum directe & di&longs;tantiæ particula­<lb/>rum corre&longs;pondentium inver&longs;e & quantitates materiæ in partibus <lb/> corre&longs;pondentibus directe: ideoque (cum di&longs;tantiæ particularum Sy­<lb/>&longs;tematis unius &longs;int ad di&longs;tantias corre&longs;pondentes particularum alte­<lb/>rius, ut diameter particulæ vel partis in Sy&longs;temate priore ad dia­<lb/>metrum particulæ vel partis corre&longs;pondentis in altero, & quantita­<lb/>tes materiæ &longs;int ut den&longs;itates partium & cubi diametrorum) re&longs;i­<lb/>&longs;tentiæ &longs;unt ad invicem ut quadrata velocitatum & quadrata dia­<lb/>metrorum & den&longs;itates partium Sy&longs;tematum. <emph type="italics"/>Q.E.D.<emph.end type="italics"/>Po&longs;te­<lb/>rioris generis re&longs;i&longs;tentiæ &longs;unt ut reflexionum corre&longs;pondentium nu­<lb/>meri & vires conjunctim. </s> <s>Numeri autem reflexionum &longs;unt ad in­<lb/>vicem ut velocitates partium corre&longs;pondentium directe, & &longs;patia <lb/> inter earum reflexiones inver&longs;e. </s> <s>Et vires reflexionum &longs;unt ut ve­<lb/>locitates & magnitudines & den&longs;itates partium corre&longs;pondentium <lb/> conjunctim; id e&longs;t, ut velocitates & diametrorum cubi & den&longs;ita­<lb/>tes partium. </s> <s>Et conjunctis his omnibus rationibus, re&longs;i&longs;tentiæ <lb/> partium corre&longs;pondentium &longs;unt ad invicem ut quadrata veloci­<lb/>tum & quadrata diametrorum & den&longs;itates partium conjunctim. <lb/> <emph type="italics"/>Q.E.D.<emph.end type="italics"/><lb/></s></p> <p type="margin"> <s><margin.target id="note272"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Igitur &longs;i Sy&longs;temata illa &longs;int Fluida duo Ela&longs;tica ad <lb/> modum Aeris, & partes eorum quie&longs;cant inter &longs;e: corpora autem <lb/> duo &longs;imilia & partibus fluidorum quoad magnitudinem & den&longs;ita­<lb/>tem proportionalia, & inter partes illas &longs;imiliter po&longs;ita, &longs;ecundum <lb/> lineas &longs;imiliter po&longs;itas utcunque projiciantur; vires autem acce­<lb/>leratrices, quibus particulæ Fluidorum &longs;e mutuo agitant, &longs;int ut <lb/> corporum projectorum diametri inver&longs;e, & quadrata velocitatum <lb/> directe: corpora illa temporibus proportionalibus &longs;imiles excita­<lb/>bunt motus in Fluidis, & &longs;patia &longs;imilia ac diametris &longs;uis propor­<lb/>tionalia de&longs;cribent. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Proinde in eodem Fluido projectile velox re&longs;i&longs;tentiam pa­<lb/>titur quæ e&longs;t in duplicata ratione velocitatis quam proxime. </s> <s>Nam <lb/> &longs;i vires, quibus particulæ di&longs;tantes &longs;e mutuo agitant, augerentur in <lb/> duplicata ratione velocitatis, re&longs;i&longs;tentia foret in eadem ratione du­<lb/>plicata accurate; ideoQ.E.I. Medio, cujus partes ab invicem di&longs;tan­<lb/>tes &longs;e&longs;e viribus nullis agitant, re&longs;i&longs;tentia e&longs;t in duplicata ratione ve­<lb/>locitatis accurate. </s> <s>Sunto igitur Media tria <emph type="italics"/>A, B, C<emph.end type="italics"/>ex partibus <lb/> &longs;imilibus & æqualibus & &longs;ecundum di&longs;tantias æquales regulariter <pb xlink:href="039/01/325.jpg" pagenum="297"/><lb/>di&longs;po&longs;itis con&longs;tantia. </s> <s>Partes Mediorum <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B<emph.end type="italics"/>fugiant &longs;e mutuo <lb/> <arrow.to.target n="note273"/>viribus quæ &longs;int ad invicem ut <emph type="italics"/>T<emph.end type="italics"/>& <emph type="italics"/>V,<emph.end type="italics"/>illæ Medii <emph type="italics"/>C<emph.end type="italics"/>eju&longs;mo­<lb/>di viribus omnino de&longs;tituantur. </s> <s>Et &longs;i corpora quatuor æqualia <lb/> <emph type="italics"/>D, E, F, G<emph.end type="italics"/>in his Mediis moveantur, priora duo <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>E<emph.end type="italics"/>in pri­<lb/>oribus duobus <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B,<emph.end type="italics"/>& altera duo <emph type="italics"/>F<emph.end type="italics"/>& <emph type="italics"/>G<emph.end type="italics"/>in tertio <emph type="italics"/>G<emph.end type="italics"/>; &longs;itque ve­<lb/>locitas corporis <emph type="italics"/>D<emph.end type="italics"/>ad velocitatem corporis <emph type="italics"/>E,<emph.end type="italics"/>& velocitas corpo­<lb/>ris <emph type="italics"/>F<emph.end type="italics"/>ad velocitatem corporis <emph type="italics"/>G,<emph.end type="italics"/>in &longs;ubduplicata ratione virium <emph type="italics"/>T<emph.end type="italics"/><lb/>ad vires <emph type="italics"/>V<emph.end type="italics"/>: re&longs;i&longs;tentia corporis <emph type="italics"/>D<emph.end type="italics"/>erit ad re&longs;i&longs;tentiam corporis <emph type="italics"/>E,<emph.end type="italics"/><lb/>& re&longs;i&longs;tentia corporis <emph type="italics"/>F<emph.end type="italics"/>ad re&longs;i&longs;tentiam corporis <emph type="italics"/>G,<emph.end type="italics"/>in velocitatum <lb/> ratione duplicata; & propterea re&longs;i&longs;tentia corporis <emph type="italics"/>D<emph.end type="italics"/>erit ad re&longs;i­<lb/>&longs;tentiam corporis <emph type="italics"/>F<emph.end type="italics"/>ut re&longs;i&longs;tentia corporis <emph type="italics"/>E<emph.end type="italics"/>ad re&longs;i&longs;tentiam corpo­<lb/>ris <emph type="italics"/>G.<emph.end type="italics"/>Sunto corpora <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>F<emph.end type="italics"/>æquivelocia ut & corpora <emph type="italics"/>E<emph.end type="italics"/>& <emph type="italics"/>G<emph.end type="italics"/>; <lb/> & augendo velocitates corporum <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>F<emph.end type="italics"/>in ratione quacunque, ac <lb/> diminuendo vires particularum Medii <emph type="italics"/>B<emph.end type="italics"/>in eadem ratione duplicata, <lb/> accedet Medium <emph type="italics"/>B<emph.end type="italics"/>ad formam & conditionem Medii <emph type="italics"/>C<emph.end type="italics"/>pro lubi­<lb/>tu, & idcirco re&longs;i&longs;tentiæ corporum æqualium & æquivelocium <emph type="italics"/>E<emph.end type="italics"/><lb/>& <emph type="italics"/>G<emph.end type="italics"/>in his Mediis, perpetuo accedent ad æqualitatem, ita ut ea­<lb/>rum differentia evadat tandem minor quam data quævis. </s> <s>Proinde <lb/> cum re&longs;i&longs;tentiæ corporum <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>F<emph.end type="italics"/>&longs;int ad invicem ut re&longs;i&longs;tentiæ cor­<lb/>porum <emph type="italics"/>E<emph.end type="italics"/>& <emph type="italics"/>G,<emph.end type="italics"/>accedent etiam hæ &longs;imiliter ad rationem æqualita­<lb/>tis. </s> <s>Corporum igitur <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>F,<emph.end type="italics"/>ubi veloci&longs;&longs;ime moventur, re&longs;i&longs;ten­<lb/>tiæ &longs;unt æquales quam proxime: & propterea cum re&longs;i&longs;tentia cor­<lb/>poris <emph type="italics"/>F<emph.end type="italics"/>&longs;it in duplicata ratione velocitatis, erit re&longs;i&longs;tentia corporis <lb/> <emph type="italics"/>D<emph.end type="italics"/>in eadem ratione quam proxime. <lb/> </s></p> <p type="margin"> <s><margin.target id="note273"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Igitur corporis in Fluido quovis Ela&longs;tico veloci&longs;&longs;ime <lb/> moti eadem fere e&longs;t re&longs;i&longs;tentia ac &longs;i partes Fluidi viribus &longs;uis <lb/> centrifugis de&longs;tituerentur, &longs;eque mutuo non fugerent: &longs;i modo <lb/> Fluidi vis Ela&longs;tica ex particularum viribus centrifugis oriatur, & <lb/> velocitas adeo magna &longs;it ut vires non habeant &longs;atis temporis ad <lb/> agendum. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Proinde cum re&longs;i&longs;tentiæ &longs;imilium & æquivelocium cor­<lb/>porum, in Medio cujus partes di&longs;tantes &longs;e mutuo non fugiunt, &longs;int <lb/> ut quadrata diametrorum; &longs;unt etiam æquivelocium & celerrime <lb/> motorum corporum re&longs;i&longs;tentiæ in Fluido Ela&longs;tico ut quadrata <lb/> diametrorum quam proxime. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Et cum corpora &longs;imilia, æqualia & æquivelocia, in <lb/> Mediis eju&longs;dem den&longs;itatis, quorum particulæ &longs;e mutuo non fu­<lb/>giunt, &longs;ive particulæ illæ &longs;int plures & minores, &longs;ive pauciores & <lb/> majores, in æqualem materiæ quantitatem temporibus æqualibus <lb/> inpingant, eique æqualem motus quantitatem imprimant, & vi-<pb xlink:href="039/01/326.jpg" pagenum="298"/><lb/><arrow.to.target n="note274"/>ci&longs;&longs;im (per motus Legem tertiam) æqualem ab eadem reactionem <lb/> patiantur, hoc e&longs;t, æqualiter re&longs;i&longs;tantur: manife&longs;tum e&longs;t etiam <lb/> quod in eju&longs;dem den&longs;itatis Fluidis Ela&longs;ticis, ubi veloci&longs;&longs;ime mo­<lb/>ventur, æquales &longs;int eorum re&longs;i&longs;tentiæ quam proxime; &longs;ive Fluida <lb/> illa ex particulis cra&longs;&longs;ioribus con&longs;tent, &longs;ive ex omnium &longs;ubtili&longs;&longs;i­<lb/>mis con&longs;tituantur. </s> <s>Ex Medii &longs;ubtilitate re&longs;i&longs;tentia projectilium ce­<lb/>lerrime motorum non multum diminuitur. <lb/> </s></p> <p type="margin"> <s><margin.target id="note274"/>DE MOTU <lb/> CORPORUM.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Hæc omnia ita &longs;e habent in Fluidis, quorum vis Ela­<lb/>&longs;tica ex particularum viribus centrifugis originem ducit. </s> <s>Quod &longs;i <lb/> vis illa aliunde oriatur, veluti ex particularum expan&longs;ione ad in&longs;tar <lb/> Lanæ vel ramorum Arborum, aut ex alia quavis cau&longs;a, qua motus <lb/> particularum inter &longs;e redduntur minus liberi: re&longs;i&longs;tentia, ob mi­<lb/>norem Medii fluiditatem, erit major quam in &longs;uperioribus Co­<lb/>rollariis. <lb/> <emph type="center"/>PROPOSITIO XXXIV. THEOREMA XXVIII.<emph.end type="center"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Si Globus & Cylindrus æqualibus diametris de&longs;cripti, in Medio <lb/> raro ex particulis æqualibus & ad æquales ab invicem di&longs;tan­<lb/>tias libere di&longs;po&longs;itis con&longs;tante, &longs;ecundum plagam axis Cylindri, <lb/> æquali cum velocitate moveantur: erit re&longs;i&longs;tentia Globi duplo <lb/> minor quam re&longs;i&longs;tentia Cylindri.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Nam quoniam actio Medii in corpus eadem e&longs;t (per Legum <lb/> Corol, 5.) &longs;ive corpus in Medio quie&longs;cente moveatur, &longs;ive Medii <lb/> particulæ eadem cum velocitate impingant in corpus quie&longs;cens: <lb/> con&longs;ideremus corpus tanquam quie&longs;cens, & videamus quo impetu <lb/> urgebitur a Medio movente. <lb/> <figure id="id.039.01.326.1.jpg" xlink:href="039/01/326/1.jpg"/><lb/>De&longs;ignet igitur <emph type="italics"/>ABKI<emph.end type="italics"/>cor­<lb/>pus Sphæricum centro <emph type="italics"/>C<emph.end type="italics"/>&longs;e­<lb/>midiametro <emph type="italics"/>CA<emph.end type="italics"/>de&longs;criptum, <lb/> & incidant particulæ Medii <lb/> data cum velocitate in cor­<lb/>pus illud Sphæricum, &longs;ecun­<lb/>dum rectas ip&longs;i <emph type="italics"/>AC<emph.end type="italics"/>paralle­<lb/>las: Sitque <emph type="italics"/>FB<emph.end type="italics"/>eju&longs;modi <lb/> recta. </s> <s>In ea capiatur <emph type="italics"/>LB<emph.end type="italics"/><lb/>&longs;emidiametro <emph type="italics"/>CB<emph.end type="italics"/>æqualis, <lb/> & ducatur <emph type="italics"/>BD<emph.end type="italics"/>quæ Sphæram tangat in <emph type="italics"/>B.<emph.end type="italics"/>In <emph type="italics"/>KC<emph.end type="italics"/>& <emph type="italics"/>BD<emph.end type="italics"/>de-<pb xlink:href="039/01/327.jpg" pagenum="299"/><lb/>mittantur perpendiculares <emph type="italics"/>BE, DL,<emph.end type="italics"/>& vis qua particula Medii, <lb/> <arrow.to.target n="note275"/>&longs;ecundum rectam <emph type="italics"/>FB<emph.end type="italics"/>obliQ.E.I.cidendo, Globum ferit in <emph type="italics"/>B,<emph.end type="italics"/>erit <lb/> ad vim qua particula eadem Cylindrum <emph type="italics"/>ONGQ<emph.end type="italics"/>axe <emph type="italics"/>ACI<emph.end type="italics"/>circa <lb/> Globum de&longs;criptum perpendiculariter feriret in <emph type="italics"/>b,<emph.end type="italics"/>ut <emph type="italics"/>LD<emph.end type="italics"/>ad <lb/> <emph type="italics"/>LB<emph.end type="italics"/>vel <emph type="italics"/>BE<emph.end type="italics"/>ad <emph type="italics"/>BC.<emph.end type="italics"/>Rur&longs;us efficacia hujus vis ad movendum <lb/> Globum &longs;ecundum incidentiæ &longs;uæ plagam <emph type="italics"/>FB<emph.end type="italics"/>vel <emph type="italics"/>AC,<emph.end type="italics"/>e&longs;t ad eju&longs;­<lb/>dem efficaciam ad movendum Globum &longs;ecundum plagam determi­<lb/>nationis &longs;uæ, id e&longs;t, &longs;ecundum plagam rectæ <emph type="italics"/>BC<emph.end type="italics"/>qua Globum di­<lb/>recte urget, ut <emph type="italics"/>BE<emph.end type="italics"/>ad <emph type="italics"/>BC.<emph.end type="italics"/>Et conjunctis rationibus, efficacia <lb/> particulæ, in Globum &longs;ecundum rectam <emph type="italics"/>FB<emph.end type="italics"/>obliQ.E.I.cidentis, ad <lb/> movendum eundem &longs;ecundum plagam incidentiæ &longs;uæ, e&longs;t ad effi­<lb/>caciam particulæ eju&longs;dem &longs;ecundum eandem rectam in Cylindrum <lb/> perpendiculariter incidentis, ad ip&longs;um movendum in plagam ean­<lb/>dem, ut <emph type="italics"/>BE<emph.end type="italics"/>quadratum ad <emph type="italics"/>BC<emph.end type="italics"/>quadratum. </s> <s>Quare &longs;i ad Cylin­<lb/>dri ba&longs;em circularem <emph type="italics"/>NAO<emph.end type="italics"/>erigatur perpendiculum <emph type="italics"/>bHE,<emph.end type="italics"/>& &longs;it <lb/> <emph type="italics"/>bE<emph.end type="italics"/>æqualis radio <emph type="italics"/>AC,<emph.end type="italics"/>& <emph type="italics"/>bH<emph.end type="italics"/>æqualis (<emph type="italics"/>BE quad/CB<emph.end type="italics"/>): erit <emph type="italics"/>bH<emph.end type="italics"/>ad <emph type="italics"/>bE<emph.end type="italics"/><lb/>ut effectus particulæ in Globum ad effectum particulæ in Cylin­<lb/>drum. </s> <s>Et propterea &longs;olidum quod à rectis omnibus <emph type="italics"/>bH<emph.end type="italics"/>occu­<lb/>patur erit ad &longs;olidum quod à rectis omnibus <emph type="italics"/>bE<emph.end type="italics"/>occupatur, ut <lb/> effectus particularum omnium in Globum ad effectum particu­<lb/>larum omnium in Cylindrum. </s> <s>Sed &longs;olidum prius e&longs;t Parabolois <lb/> vertice <emph type="italics"/>C,<emph.end type="italics"/>axe <emph type="italics"/>CA<emph.end type="italics"/>& latere recto <emph type="italics"/>CA<emph.end type="italics"/>de&longs;criptum, & &longs;olidum <lb/> po&longs;terius e&longs;t Cylindrus Paraboloidi circum&longs;criptus: & notum e&longs;t <lb/> quod Parabolois &longs;it &longs;emi&longs;&longs;is Cylindri circum&longs;cripti. </s> <s>Ergo vis <lb/> tota Medii in Globum e&longs;t duplo minor quam eju&longs;dem vis tota <lb/> in Cylindrum. </s> <s>Et propterea &longs;i particulæ Medii quie&longs;cerent, & <lb/> Cylindrus ac Globus æquali cum velocitate moverentur, foret re­<lb/>&longs;i&longs;tentia Globi duplo minor quam re&longs;i&longs;tentia Cylindri. <emph type="italics"/>Q.E.D.<emph.end type="italics"/><lb/><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><lb/></s></p> <p type="margin"> <s><margin.target id="note275"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s>Eadem methodo Figuræ aliæ inter &longs;e quo­<lb/><figure id="id.039.01.327.1.jpg" xlink:href="039/01/327/1.jpg"/><lb/>ad re&longs;i&longs;tentiam comparari po&longs;&longs;unt, eæQ.E.I.­<lb/>veniri quæ ad motus &longs;uos in Mediis re&longs;i&longs;ten­<lb/>tibus continuandos aptiores &longs;unt. </s> <s>Ut &longs;i ba&longs;e <lb/> circulari <emph type="italics"/>CEBH,<emph.end type="italics"/>quæ centro <emph type="italics"/>O,<emph.end type="italics"/>radio <emph type="italics"/>OC<emph.end type="italics"/><lb/>de&longs;cribitur, & altitudine <emph type="italics"/>OD,<emph.end type="italics"/>con&longs;truen­<lb/>dum &longs;it fru&longs;tum Coni <emph type="italics"/>CBGF,<emph.end type="italics"/>quod omni­<lb/>um eadem ba&longs;i & altitudine con&longs;tructorum & &longs;ecundum plagam <pb xlink:href="039/01/328.jpg" pagenum="300"/><lb/><arrow.to.target n="note276"/>axis &longs;ui ver&longs;us <emph type="italics"/>D<emph.end type="italics"/>progredientium fru&longs;torum minime re&longs;i&longs;tatur: bi­<lb/>&longs;eca altitudinem <emph type="italics"/>OD<emph.end type="italics"/>in <emph type="italics"/>Q<emph.end type="italics"/>& produc <emph type="italics"/>OQ<emph.end type="italics"/>ad <emph type="italics"/>S<emph.end type="italics"/>ut &longs;it <emph type="italics"/>QS<emph.end type="italics"/>æqua­<lb/>lis <emph type="italics"/>QC,<emph.end type="italics"/>& erit <emph type="italics"/>S<emph.end type="italics"/>vertex Coni cujus fru&longs;tum quæritur. <lb/> </s></p> <p type="margin"> <s><margin.target id="note276"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s>Unde obiter, cum angulus <emph type="italics"/>CSB<emph.end type="italics"/>&longs;emper &longs;it acutus, con&longs;equens <lb/> e&longs;t, quod &longs;i &longs;olidum <emph type="italics"/>ADBE<emph.end type="italics"/>convolutione figuræ Ellipticæ vel <lb/> Ovalis <emph type="italics"/>ADBE<emph.end type="italics"/>circa axem <emph type="italics"/>AB<emph.end type="italics"/>facta generetur, & tangatur figura <lb/> generans à rectis tribus <emph type="italics"/>FG, GH, HI<emph.end type="italics"/>in punctis <emph type="italics"/>F, B<emph.end type="italics"/>& <emph type="italics"/>I,<emph.end type="italics"/>ea <lb/> lege ut <emph type="italics"/>GH<emph.end type="italics"/>&longs;it perpendicularis ad axem in puncto contactus <emph type="italics"/>B,<emph.end type="italics"/><lb/>& <emph type="italics"/>FG, HI<emph.end type="italics"/>cum eadem <emph type="italics"/>GH<emph.end type="italics"/>contineant angulos <emph type="italics"/>FGB, BHI<emph.end type="italics"/><lb/>graduum 135: &longs;olidum, quod convolutione figuræ <emph type="italics"/>ADFGHIE<emph.end type="italics"/><lb/>circa axem eundem <emph type="italics"/>CB<emph.end type="italics"/>generatur, minus re&longs;i&longs;titur quam &longs;olidum <lb/> prius; &longs;i modo utrumque &longs;ecundum plagam axis &longs;ui <emph type="italics"/>AB<emph.end type="italics"/>progre­<lb/>diatur, & utriu&longs;que terminus <emph type="italics"/>B<emph.end type="italics"/>præcedat. </s> <s>Quam quidem propo&longs;i­<lb/>tionem in con&longs;truendis Navibus non inutilem futuram e&longs;&longs;e cen&longs;eo. <lb/> </s></p> <p type="main"> <s>Quod &longs;i Figura <emph type="italics"/>DNFG<emph.end type="italics"/><lb/>eju&longs;modi &longs;it curva ut, &longs;i ab <lb/> <figure id="id.039.01.328.1.jpg" xlink:href="039/01/328/1.jpg"/><lb/>ejus puncto quovis <emph type="italics"/>N<emph.end type="italics"/>ad <lb/> axem <emph type="italics"/>AB<emph.end type="italics"/>demittatur per­<lb/>pendiculum <emph type="italics"/>NM,<emph.end type="italics"/>& à pun­<lb/>cto dato <emph type="italics"/>G<emph.end type="italics"/>ducatur recta <lb/> <emph type="italics"/>GR<emph.end type="italics"/>quæ parallela &longs;it rectæ <lb/> figuram tangenti in <emph type="italics"/>N,<emph.end type="italics"/>& <lb/> axem productum &longs;ecet in <lb/> <emph type="italics"/>R,<emph.end type="italics"/>fuerit <emph type="italics"/>MN<emph.end type="italics"/>ad <emph type="italics"/>GR<emph.end type="italics"/>ut <lb/> <emph type="italics"/>GR cub<emph.end type="italics"/>ad 4 <emph type="italics"/>BRXGBq<emph.end type="italics"/>: <lb/> Solidum quod figuræ hujus revolutione circa axem <emph type="italics"/>AB<emph.end type="italics"/>facta de­<lb/>&longs;cribitur, in Medio raro prædicto ab <emph type="italics"/>A<emph.end type="italics"/>ver&longs;us <emph type="italics"/>B<emph.end type="italics"/>movendo, minus <lb/> re&longs;i&longs;tetur quam aliud quodvis eadem longitudine & latitudine de­<lb/>&longs;criptum Solidum circulare. <lb/> <emph type="center"/>PROPOSITIO XXXV. PROBLEMA VII.<emph.end type="center"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Si Medium rarum ex particulis quam minimis quie&longs;centibus æqua­<lb/>libus & ad æquales ab invicem di&longs;tantias libere di&longs;po&longs;itis con­<lb/>&longs;tet: invenire re&longs;i&longs;tentiam Globi in hoc Medio uniformitor pro­<lb/>gredientis.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Cylindrus eadem diametro & altitudine de&longs;criptus pro­<lb/>gredi intelligatur eadem velocitate &longs;ecundum longitudinem axis <lb/> &longs;ui in eodem Medio. </s> <s>Et ponamus quod particulæ Medii in quas <pb xlink:href="039/01/329.jpg" pagenum="301"/><lb/>Globus vel Cylindrus incidit, vi reflexionis quam maxima re&longs;iliant. <lb/> <arrow.to.target n="note277"/>Et cum re&longs;i&longs;tentia Globi (per Propo&longs;itionem novi&longs;&longs;imam) &longs;it duplo <lb/> minor quam re&longs;i&longs;tentia Cylindri, & Globus &longs;it ad Cylindrum ut <lb/> duo ad tria, & Cylindrus incidendo perpendiculariter in particulas <lb/> ip&longs;a&longs;que quam maxime reflectendo, duplam &longs;ui ip&longs;ius velocitatem <lb/> ip&longs;is communicet: Cylindrus quo tempore dimidiam longitudinem <lb/> axis &longs;ui de&longs;cribit communicabit motum particulis qui &longs;it ad totum <lb/> Cylindri motum ut den&longs;itas Medii ad den&longs;itatem Cylindri; & Glo­<lb/>bus quo tempore totam longitudinem diametri &longs;uæ de&longs;cribit, com­<lb/>municabit motum eundem particulis; & quo tempore duas tertias <lb/> partes diametri &longs;uæ de&longs;cribit communicabit motum particulis qui <lb/> &longs;it ad totum Globi motum ut den&longs;itas Medii ad den&longs;itatem Globi. <lb/> Et propterea Globus re&longs;i&longs;tentiam patitur quæ &longs;it ad vim qua totus <lb/> ejus motus vel auferri po&longs;&longs;it vel generari quo tempore duas tertias <lb/> partes diametri &longs;uæ de&longs;cribit, ut den&longs;itas Medii ad den&longs;itatem <lb/> Globi. <lb/> </s></p> <p type="margin"> <s><margin.target id="note277"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Ponamus quod particulæ Medii in Globum vel Cylin­<lb/>drum incidentes non reflectantur; & Cylindrus incidendo perpen­<lb/>diculariter in particulas &longs;implicem &longs;uam velocitatem ip&longs;is commu­<lb/>nicabit, ideoque re&longs;i&longs;tentiam patitur duplo minorem quam in pri­<lb/>ore ca&longs;u, & re&longs;i&longs;tentia Globi erit etiam duplo minor quam prius. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>3. Ponamus quod particulæ Medii vi reflexionis neque ma­<lb/>xima neque nulla, &longs;ed mediocri aliqua re&longs;iliant a Globo; & re&longs;i­<lb/>&longs;tentia Globi erit in eadem ratione mediocri inter re&longs;i&longs;tentiam in <lb/> primo ca&longs;u & re&longs;i&longs;tentiam in &longs;ecundo. <emph type="italics"/>Q.E.I.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i Globus & particulæ &longs;int infinite dura, & vi om­<lb/>ni ela&longs;tica & propterea etiam vi omni reflexionis de&longs;tituta: re­<lb/>&longs;i&longs;tentia Globi erit ad vim qua totus ejus motus vel auferri po&longs;&longs;it <lb/> vel generari, quo tempore Globus quatuor tertias partes diametri <lb/> &longs;uæ de&longs;cribit, ut den&longs;itas Medii ad den&longs;itatem Globi. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Re&longs;i&longs;tentia Globi, cæteris paribus, e&longs;t in duplicata ra­<lb/>tione velocitatis. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Re&longs;i&longs;tentia Globi, cæteris paribus, e&longs;t in duplicata ra­<lb/>tione diametri. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Re&longs;i&longs;tentia Globi, cæteris paribus, e&longs;t ut den&longs;itas Medii. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Re&longs;i&longs;tentia Globi e&longs;t in ratione quæ componitur ex du­<lb/>plicata ratione velocitatis & duplicata ratione diametri & ratione <lb/> den&longs;itatis Medii. <pb xlink:href="039/01/330.jpg" pagenum="302"/><lb/><arrow.to.target n="note278"/></s></p> <p type="margin"> <s><margin.target id="note278"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Et motus Globi cum ejus re&longs;i&longs;tentia &longs;ic exponi pote&longs;t. <lb/> Sit <emph type="italics"/>AB<emph.end type="italics"/>tempus quo Globus per re&longs;i&longs;tentiam &longs;uam uniformiter con­<lb/>tinuatam totum &longs;uum motum amit­<lb/><figure id="id.039.01.330.1.jpg" xlink:href="039/01/330/1.jpg"/><lb/>tere pote&longs;t. </s> <s>Ad <emph type="italics"/>AB<emph.end type="italics"/>erigantur per­<lb/>pendicula <emph type="italics"/>AD, BC.<emph.end type="italics"/>Sitque <emph type="italics"/>BC<emph.end type="italics"/><lb/>motus ille totus, & per punctum <emph type="italics"/>C<emph.end type="italics"/><lb/>A&longs;ymptotis <emph type="italics"/>AD, AB<emph.end type="italics"/>de&longs;cribatur <lb/> Hyperbola <emph type="italics"/>CF.<emph.end type="italics"/>Producatur <emph type="italics"/>AB<emph.end type="italics"/>ad <lb/> punctum quodvis <emph type="italics"/>E.<emph.end type="italics"/>Erigatur per­<lb/>pendiculum <emph type="italics"/>EF<emph.end type="italics"/>Hyperbolæ occur­<lb/>rens in <emph type="italics"/>F.<emph.end type="italics"/>Compleatur parallelo­<lb/>grammum <emph type="italics"/>CBEG,<emph.end type="italics"/>& agatur <emph type="italics"/>AF<emph.end type="italics"/><lb/>ip&longs;i <emph type="italics"/>BC<emph.end type="italics"/>occurrens in <emph type="italics"/>H.<emph.end type="italics"/>Et &longs;i Globus tempore quovis <emph type="italics"/>BE,<emph.end type="italics"/>motu <lb/> &longs;uo primo <emph type="italics"/>BC<emph.end type="italics"/>uniformiter continuato, in Medio non re&longs;i&longs;tente de­<lb/>&longs;cribat &longs;patium <emph type="italics"/>CBEG<emph.end type="italics"/>per aream parallelogrammi expo&longs;itum, idem <lb/> in Medio re&longs;i&longs;tente de&longs;cribet &longs;patium <emph type="italics"/>CBEF<emph.end type="italics"/>per aream Hyper­<lb/>bolæ expo&longs;itum, & motus ejus in fine temporis illius exponetur <lb/> per Hyperbolæ ordinatam <emph type="italics"/>EF,<emph.end type="italics"/>ami&longs;&longs;a motus ejus parte <emph type="italics"/>FG.<emph.end type="italics"/>Et <lb/> re&longs;i&longs;tentia ejus in fine temporis eju&longs;dem exponetur per longitudi­<lb/>nem <emph type="italics"/>BH,<emph.end type="italics"/>ami&longs;&longs;a re&longs;i&longs;tentiæ parte <emph type="italics"/>CH.<emph.end type="italics"/>Patent hæc omnia per <lb/> Corol. 1. Prop. v. Lib. II. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. Hinc &longs;i Globus tempore T per re&longs;i&longs;tentiam R unifor­<lb/>miter continuatam amittat motum &longs;uum totum M: idem Globus tem­<lb/>pore <emph type="italics"/>t<emph.end type="italics"/>in Medio re&longs;i&longs;tente, per re&longs;i&longs;tentiam R in duplicata velocitatis <lb/> ratione decre&longs;centem, amittet motus &longs;ui M partem (<emph type="italics"/>t<emph.end type="italics"/>M/T+<emph type="italics"/>t<emph.end type="italics"/>), manente <lb/> parte (TM/T+<emph type="italics"/>t<emph.end type="italics"/>), & de&longs;cribet &longs;patium quod &longs;it ad &longs;patium motu uni­<lb/>formi M eodem tempore <emph type="italics"/>t<emph.end type="italics"/>de&longs;criptum, ut Logarithmus numeri <lb/> (T+<emph type="italics"/>t<emph.end type="italics"/>/T) multiplicatus per numerum 2,302585092994 e&longs;t ad nume­<lb/>rum <emph type="italics"/>t<emph.end type="italics"/>/T. Nam area Hyperbolica <emph type="italics"/>BCFE<emph.end type="italics"/>e&longs;t ad rectangulum <lb/> <emph type="italics"/>BCGE<emph.end type="italics"/>in hac proportione. <lb/> <emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><lb/></s></p> <p type="main"> <s>In hac Propo&longs;itione expo&longs;ui re&longs;i&longs;tentiam & retardationem Pro­<lb/>jectilium Sphærieorum in Mediis non continuis, & o&longs;tendi quod <lb/> hæc re&longs;i&longs;tentia &longs;it ad vim qua totus Globi motus vel tolli po&longs;&longs;it vel <pb xlink:href="039/01/331.jpg" pagenum="303"/><lb/>generari quo tempore Globus duas tertias diametri &longs;uæ partes, ve­<lb/><arrow.to.target n="note279"/>locitate uniformiter continuata de&longs;cribat, ut den&longs;itas Medii ad <lb/> den&longs;itatem Globi, &longs;i modo Globus & particulæ Medii &longs;int &longs;umme <lb/> ela&longs;tica & vi maxima reflectendi polleant: quodque hæc vis &longs;it <lb/> duplo minor ubi Globus & particulæ Medii &longs;unt infinite dura & <lb/> vi reflectendi pror&longs;us de&longs;tituta. </s> <s>In Medus autem continuis qualia <lb/> &longs;unt Aqua, Oleum calidum, & Argentum vivum, in quibus Globus <lb/> non incidit immediate in omnes fluidi particulas re&longs;i&longs;tentiam gene­<lb/>rantes, &longs;ed premit tantum proximas particulas & hæ premunt alias <lb/> & hæ alias, re&longs;i&longs;tentia e&longs;t adhuc duplo minor. </s> <s>Globus utiQ.E.I. <lb/> huju&longs;modi Mediis fluidi&longs;&longs;imis re&longs;i&longs;tentiam patitur quæ e&longs;t ad vim <lb/> qua totus ejus motus vel tolli po&longs;&longs;it vel generari quo tempore, <lb/> motu illo uniformiter continuato, partes octo tertias diametri &longs;uæ <lb/> de&longs;cribat, ut den&longs;itas Medii ad den&longs;itatem Globi. </s> <s>Id quod in &longs;e­<lb/>quentibus conabimur o&longs;tendere. <lb/> <emph type="center"/>PROPOSITIO XXXVI. PROBLEMA VIII.<emph.end type="center"/><lb/></s></p> <p type="margin"> <s><margin.target id="note279"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Aquæ de va&longs;e Cylindrico per foramen in fundo factum effluentis <lb/> definire motum.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Sit <emph type="italics"/>ACDB<emph.end type="italics"/>vas cylindricum, <emph type="italics"/>AB<emph.end type="italics"/>ejus orificium &longs;uperius, <emph type="italics"/>CD<emph.end type="italics"/><lb/>fundum horizonti parallelum, <emph type="italics"/>EF<emph.end type="italics"/>foramen circulare in medio <lb/> fundi, <emph type="italics"/>G<emph.end type="italics"/>centrum foraminis, & <emph type="italics"/>GH<emph.end type="italics"/>axis cylindri horizonti per­<lb/>pendicularis. </s> <s>Et concipe cylindrum gla­<lb/><figure id="id.039.01.331.1.jpg" xlink:href="039/01/331/1.jpg"/><lb/>ciei <emph type="italics"/>APQB<emph.end type="italics"/>eju&longs;dem e&longs;&longs;e latitudinis <lb/> cum cavitate va&longs;is, & axem eundem ha­<lb/>bere, & uniformi cum motu perpetuo <lb/> de&longs;cendere, & partes ejus quam primum <lb/> attingunt &longs;uperficiem <emph type="italics"/>AB<emph.end type="italics"/>lique&longs;cere, & <lb/> in aquam conver&longs;as gravitate &longs;ua defluere <lb/> in vas, & cataractam vel columnam aquæ <lb/> <emph type="italics"/>ABNFEM<emph.end type="italics"/>cadendo formare, & per <lb/> foramen <emph type="italics"/>EF<emph.end type="italics"/>tran&longs;ire, idemque adæquate <lb/> implere. </s> <s>Ea vero &longs;it uniformis veloci­<lb/>tas glaciei de&longs;cendentis ut & aquæ con­<lb/>tiguæ in circulo <emph type="italics"/>AB,<emph.end type="italics"/>quam aqua caden­<lb/>do & ca&longs;u &longs;uo de&longs;cribendo altitudinem <lb/> <emph type="italics"/>IH<emph.end type="italics"/>acquirere pote&longs;t; & jaceant <emph type="italics"/>IH<emph.end type="italics"/>& <emph type="italics"/>HG<emph.end type="italics"/>in directum, & per <lb/> punctum <emph type="italics"/>I<emph.end type="italics"/>ducatur recta <emph type="italics"/>KL<emph.end type="italics"/>horizonti parallela & lateribus gla-<pb xlink:href="039/01/332.jpg" pagenum="304"/><lb/><arrow.to.target n="note280"/>ciei occurrens in <emph type="italics"/>K<emph.end type="italics"/>& <emph type="italics"/>L.<emph.end type="italics"/>Et velocitas aquæ effluentis per fora­<lb/>men <emph type="italics"/>EF<emph.end type="italics"/>ea erit quam aqua cadendo ab <emph type="italics"/>I<emph.end type="italics"/>& ca&longs;u &longs;uo de&longs;cribendo <lb/> altitudinem <emph type="italics"/>IG<emph.end type="italics"/>acquirere pote&longs;t. </s> <s>Ideoque per Theoremata <emph type="italics"/>Galilæi<emph.end type="italics"/><lb/>erit <emph type="italics"/>IG<emph.end type="italics"/>ad <emph type="italics"/>IH<emph.end type="italics"/>in duplicata ratione velocitatis aquæ per foramen <lb/> effluentis ad velocitatem aquæ in circulo <emph type="italics"/>AB,<emph.end type="italics"/>hoc e&longs;t, in dupli­<lb/>cata ratione circuli <emph type="italics"/>AB<emph.end type="italics"/>ad circulum <emph type="italics"/>EF<emph.end type="italics"/>; nam hi circuli &longs;unt re­<lb/>ciproce ut velocitates aquarum quæ per ip&longs;os, eodem tempore & <lb/> æquali quantitate, adæquate tran&longs;eunt. </s> <s>De velocitate aquæ hori­<lb/>zontem ver&longs;us hic agitur. </s> <s>Et motus horizonti parallelus quo par­<lb/>tes aquæ cadentis ad invicem accedunt, cum non oriatur a gravi­<lb/>tate, nec motum horizonti perpendicularem à gravitate oriundum <lb/> mutet, hic non con&longs;ideratur. </s> <s>Supponimus quidem quod partes <lb/> aquæ aliquantulum cohærent, & per cohæ&longs;ionem &longs;uam inter ca­<lb/>dendum accedant ad invicem per motus horizonti parallelos, ut <lb/> unicam tantum efforment cataractam & non in plures cataractas <lb/> dividantur: &longs;ed motum horizonti parallelum, a cohæ&longs;ione illa ori­<lb/>undum, hic non con&longs;ideramus. <lb/> </s></p> <p type="margin"> <s><margin.target id="note280"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Concipe jam cavitatem totam in va&longs;e, in circuitu aquæ <lb/> cadentis <emph type="italics"/>ABNFEM,<emph.end type="italics"/>glacie plenam e&longs;&longs;e, ut aqua per glaciem <lb/> tanquam per infundibulum tran&longs;eat. </s> <s>Et &longs;i aqua glaciem tantum <lb/> non tangat vel, quod perinde e&longs;t, &longs;i tangat & per glaciem propter <lb/> &longs;ummam ejus polituram quam liberrime & &longs;ine omni re&longs;i&longs;tentia la­<lb/>batur; hæc defluet per foramen <emph type="italics"/>EF<emph.end type="italics"/>eadem velocitate ac prius, & <lb/> pondus totum columnæ aquæ <emph type="italics"/>ABNFEM<emph.end type="italics"/>impendetur in deflu­<lb/>xum ejus generandum uti prius, & fundum va&longs;is &longs;u&longs;tinebit pon­<lb/>dus glaciei columnam ambientis. <lb/> </s></p> <p type="main"> <s>Lique&longs;cat jam glacies in va&longs;e; & effluxus aquæ quoad velocita­<lb/>tem, idem manebit ac prius. </s> <s>Non minor erit, quia glacies in aquam <lb/> re&longs;oluta conabitur de&longs;cendere: non major, quia glacies in aquam <lb/> re&longs;oluta non pote&longs;t de&longs;cendere ni&longs;i impediendo de&longs;cen&longs;um aquæ <lb/> alterius de&longs;cen&longs;ui &longs;uo æqualem. </s> <s>Eadem vis eandem aquæ effluen­<lb/>tis velocitatem generare debet. <lb/> </s></p> <p type="main"> <s>Sed foramen in fundo va&longs;is, propter obliquos motus particula­<lb/>rum aquæ effluentis, paulo majus e&longs;&longs;e debet quam prius. </s> <s>Nam par­<lb/>ticulæ aquæ jam non tran&longs;eunt omnes per foramen perpendicula­<lb/>riter; &longs;ed a lateribus va&longs;is undique confluentes & in foramen con­<lb/>vergentes, obliquis tran&longs;eunt motibus; & cur&longs;um &longs;uum deor&longs;um <lb/> flectentes in venam aquæ exilientis con&longs;pirant, quæ exilior e&longs;t pau­<lb/>lo infra foramen quam in ip&longs;o foramine, exi&longs;tente ejus diametro <lb/> ad diametrum foraminis ut 5 ad 6, vel 5 1/2 ad 6 1/2 quam proxime, &longs;i <pb xlink:href="039/01/333.jpg" pagenum="305"/><lb/>modo diametros recte dimen&longs;us &longs;um. </s> <s>Parabam utique laminam <lb/> <arrow.to.target n="note281"/>planam pertenuem in medio perforatam, exi&longs;tente circularis fora­<lb/>minis diametro partium quinque octavarum digiti. </s> <s>Et ne vena <lb/> aquæ exilientis cadendo acceleraretur & acceleratione redderetur <lb/> angu&longs;tior, hanc laminam non fundo &longs;ed lateri va&longs;is affixi &longs;ic, ut <lb/> vena illa egrederetur &longs;ecundum lineam horizonti parallelam. </s> <s>Dein <lb/> ubi vas aquæ plenum e&longs;&longs;et, aperui foramen ut aqua efflueret; & <lb/> venæ diameter, ad di&longs;tantiam qua&longs;i dimidii digiti â &longs;oramine quam <lb/> accurati&longs;&longs;ime men&longs;urata, prodiit partium viginti & unius quadrage&longs;i­<lb/>marum digiti. </s> <s>Erat igitur diameter foraminis hujus circularis ad <lb/> diametrum venæ ut 25 ad 21 quamproxime. </s> <s>Per experimenta vero <lb/> con&longs;tat quod quantitas aquæ quæ per foramen circulare in fundo <lb/> va&longs;is factum effluit, ea e&longs;t quæ, pro diametro venæ, cum velocitate <lb/> prædicta effluere debet. <lb/> </s></p> <p type="margin"> <s><margin.target id="note281"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s>In &longs;equentibus igitur, plano foraminis parallelum duci intelliga­<lb/>tur planum aliud &longs;uperius ad di&longs;tantiam diametro foraminis æqua­<lb/>lem vel paulo majorem & foramine majore pertu&longs;um, per quod <lb/> utique vena cadat quæ adæquate impleat <lb/> <figure id="id.039.01.333.1.jpg" xlink:href="039/01/333/1.jpg"/><lb/>foramen inferius <emph type="italics"/>EF,<emph.end type="italics"/>atque adeo cujus <lb/> diameter &longs;it ad diametrum foraminis in­<lb/>ferioris ut 25 ad 21 circiter. </s> <s>Sic enim <lb/> vena per foramen inferius perpendicu­<lb/>lariter tran&longs;ibit; & quantitas aquæ ef­<lb/>fluentis, pro magnitudine foraminis hu­<lb/>jus, ea erit quam &longs;olutio Problematis po­<lb/>&longs;tulat quamproxime. </s> <s>Spatium vero quod <lb/> planis duobus & vena cadente clauditur, <lb/> pro fundo va&longs;is haberi pote&longs;t. </s> <s>Sed ut <lb/> &longs;olutio Problematis &longs;implicior &longs;it & ma­<lb/>gis Mathematica, præ&longs;tat adhibere pla­<lb/>num &longs;olum inferius pro fundo va&longs;is, & <lb/> fingere quod aqua quæ per glaciem ceu per infundibulum deflue­<lb/>bat, & è va&longs;e per foramen <emph type="italics"/>EF<emph.end type="italics"/>egrediebatur, motum &longs;uum per­<lb/>petuo &longs;ervet & glacies quietem &longs;uam etiam&longs; in aquam fluidam <lb/> re&longs;olvatur. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Si foramen <emph type="italics"/>EF<emph.end type="italics"/>non &longs;it in medio fundi va&longs;is, &longs;ed fun­<lb/>dum alibi perforetur: aqua effluet eadem cum velocitate ac prius, <lb/> &longs;i modo eadem &longs;it foraminis magnitudo. </s> <s>Nam grave majori qui­<lb/>dem tempore de&longs;cendit ad eandem profunditatem per lineam ob­<lb/>liquam quam per lineam perpendicularem, &longs;ed de&longs;cendendo ean-<pb xlink:href="039/01/334.jpg" pagenum="306"/><lb/><arrow.to.target n="note282"/>dem velocitatem acquirit in utroque ca&longs;u, ut <emph type="italics"/>Galilæus<emph.end type="italics"/>demon­<lb/>&longs;travit. <lb/> </s></p> <p type="margin"> <s><margin.target id="note282"/>DE MOTU <lb/> CORPORUM.</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>3. Eadem e&longs;t aquæ velocitas effluentis per foramen in la­<lb/>tere va&longs;is. </s> <s>Nam &longs;i foramen parvum &longs;it, ut intervallum inter &longs;uper­<lb/>ficies <emph type="italics"/>AB<emph.end type="italics"/>& <emph type="italics"/>KL<emph.end type="italics"/>quoad &longs;en&longs;um evane&longs;cat, & vena aquæ hori­<lb/>zontaliter exilientis figuram Parabolicam efformet: ex latere recto <lb/> hujus Parabolæ colligetur, quod velocitas aquæ effluentis ea &longs;it <lb/> quam corpus ab aquæ in va&longs;e &longs;tagnantis altitudine <emph type="italics"/>HG<emph.end type="italics"/>vel <emph type="italics"/>IG<emph.end type="italics"/>ca­<lb/>dendo acquirere potui&longs;&longs;et. </s> <s>Facto utique experimento inveni quod, <lb/> &longs;i altitudo aquæ &longs;tagnantis &longs;upra foramen e&longs;&longs;et viginti digitorum <lb/> & altitudo foraminis &longs;upra planum horizonti parallelum e&longs;&longs;et quo­<lb/>que viginti digitorum, vena aquæ pro&longs;ilientis incideret in planum <lb/> illud ad di&longs;tantiam digitorum 37 circiter à perpendiculo quod in <lb/> planum illud à foramine demittebatur captam. </s> <s>Nam &longs;ine re&longs;i&longs;ten­<lb/>tia vena incidere debui&longs;&longs;et in planum illud ad di&longs;tantiam digitorum <lb/> 40, exi&longs;tente venæ Parabolicæ latere recto digitorum 80. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>4. Quinetiam aqua effluens, &longs;i &longs;ur&longs;um feratur, eadem egre­<lb/>ditur cum velocitate. </s> <s>A&longs;cendit enim aquæ exilientis vena parva <lb/> motu perpendiculari ad aquæ in va&longs;e &longs;tagnantis altitudinem <emph type="italics"/>GH<emph.end type="italics"/><lb/>vel <emph type="italics"/>GI,<emph.end type="italics"/>ni&longs;i quatenus a&longs;cen&longs;us ejus ab aeris re&longs;i&longs;tentia aliquantu­<lb/>lum impediatur; ac proinde ea effluit cum velocitate quam ab al­<lb/>titudine illa cadendo acquirere potui&longs;&longs;et. <lb/> <figure id="id.039.01.334.1.jpg" xlink:href="039/01/334/1.jpg"/><lb/>Aquæ &longs;tagnantis particula unaquæque <lb/> undique premitur æqualiter, per Prop. <lb/> XIX. Lib. II, & pre&longs;&longs;ioni cedendo æquali <lb/> impetu in omnes partes fertur, &longs;ive de­<lb/>&longs;cendat per foramen in fundo va&longs;is, &longs;ive <lb/> horizontaliter effluat per foramen in ejus <lb/> latere, &longs;ive egrediatur in canalem & inde <lb/> a&longs;cendat per foramen parvum in &longs;uperiore <lb/> canalis parte factum. </s> <s>Et velocitatem qua <lb/> aqua effluit, eam e&longs;&longs;e quam in hac Pro­<lb/>po&longs;itione a&longs;&longs;ignavimus, non &longs;olum rati­<lb/>one colligitus, &longs;ed etiam per experimenta <lb/> noti&longs;&longs;ima jam de&longs;cripta manife&longs;tum e&longs;t. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>5. Eadem e&longs;t aquæ effluentis velocitas &longs;ive figura foraminis <lb/> &longs;it circularis &longs;ive quadrata vel triangularis aut alia quæcunque cir­<lb/>culari æqualis. </s> <s>Nam velocitas aquæ effluentis non pendet à figura <lb/> foraminis &longs;ed ab ejus altitudine infra planum <emph type="italics"/>KL.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>6. Si va&longs;is <emph type="italics"/>ABDC<emph.end type="italics"/>pars inferior in aquam &longs;tagnantem im-<pb xlink:href="039/01/335.jpg" pagenum="307"/><lb/>mergatur, & altitudo aquæ &longs;tagnantis &longs;upra fundum va&longs;is &longs;it <emph type="italics"/>GR<emph.end type="italics"/>: <lb/> <arrow.to.target n="note283"/>velocitas quacum aqua quæ in va&longs;e e&longs;t, effluet per foramen <emph type="italics"/>EF<emph.end type="italics"/><lb/>in aquam &longs;tagnantem, ea erit quam aqua cadendo & ca&longs;u &longs;uo de­<lb/>&longs;cribendo altitudinem <emph type="italics"/>IR<emph.end type="italics"/>acquirere pote&longs;t. </s> <s>Nam pondus aquæ <lb/> omnis in va&longs;e quæ inferior e&longs;t &longs;uperficie aquæ &longs;tagnantis, &longs;u&longs;tine­<lb/>bitur in æquilibrio per pondus aquæ &longs;tagnantis, ideoque motum <lb/> aquæ de&longs;cendentis in va&longs;e minime accelerabit. </s> <s>Patebit etiam & <lb/> hic Ca&longs;us per Experimenta, men&longs;urando &longs;cilicet tempora qui­<lb/>bus aqua effluit. <lb/> </s></p> <p type="margin"> <s><margin.target id="note283"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i aquæ altitudo <emph type="italics"/>CA<emph.end type="italics"/>producatur ad <emph type="italics"/>K,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>AK<emph.end type="italics"/><lb/>ad <emph type="italics"/>CK<emph.end type="italics"/>in duplicata ratione areæ foraminis in quavis fundi parte <lb/> facti, ad aream circuli <emph type="italics"/>AB<emph.end type="italics"/>: velocitas aquæ effluentis æqualis erit <lb/> velocitati quam aqua cadendo & ca&longs;u &longs;uo de&longs;cribendo altitudinera <lb/> <emph type="italics"/>KC<emph.end type="italics"/>acquirere pote&longs;t. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et vis qua totus aquæ exilientis motus generari pote&longs;t, <lb/> æqualis e&longs;t ponderi Cylindricæ columnæ aquæ cujus ba&longs;is e&longs;t fora­<lb/>men <emph type="italics"/>EF,<emph.end type="italics"/>& altitudo 2<emph type="italics"/>GI<emph.end type="italics"/>vel 2<emph type="italics"/>CK.<emph.end type="italics"/>Nam aqua exiliens quo <lb/> tempore hanc columnam æquat, pondere &longs;uo ab altitudine <emph type="italics"/>GI<emph.end type="italics"/>ca­<lb/>dendo, velocitatem &longs;uam qua exilit, acquirere pote&longs;t. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Pondus aquæ totius in va&longs;e <emph type="italics"/>ABDC,<emph.end type="italics"/>e&longs;t ad ponderis <lb/> partem quæ in defluxum aquæ impenditur, ut &longs;umma circulorum <lb/> <emph type="italics"/>AB<emph.end type="italics"/>& <emph type="italics"/>EF,<emph.end type="italics"/>ad duplum circulum <emph type="italics"/>EF.<emph.end type="italics"/>Sit enim <emph type="italics"/>IO<emph.end type="italics"/>media pro­<lb/>portionalis inter <emph type="italics"/>IH<emph.end type="italics"/>& <emph type="italics"/>IG<emph.end type="italics"/>; & aqua per foramen <emph type="italics"/>EF<emph.end type="italics"/>egrediens, <lb/> quo tempore gutta cadendo ab <emph type="italics"/>I<emph.end type="italics"/>de&longs;cribere po&longs;&longs;et altitudinem <emph type="italics"/>IG,<emph.end type="italics"/><lb/>æqualis erit Cylindro cujus ba&longs;is e&longs;t circulus <emph type="italics"/>EF<emph.end type="italics"/>& altitudo e&longs;t 2<emph type="italics"/>IG,<emph.end type="italics"/><lb/>id e&longs;t, Cylindro cujus ba&longs;is e&longs;t circulus <emph type="italics"/>AB<emph.end type="italics"/>& altitudo e&longs;t 2<emph type="italics"/>IO,<emph.end type="italics"/><lb/>nam circulus <emph type="italics"/>EF<emph.end type="italics"/>e&longs;t ad circulum <emph type="italics"/>AB<emph.end type="italics"/>in &longs;ubduplicata ratione <lb/> altitudinis <emph type="italics"/>IH<emph.end type="italics"/>ad altitudinem <emph type="italics"/>IG,<emph.end type="italics"/>hoc e&longs;t, in &longs;implici ratione me­<lb/>diæ proportionalis <emph type="italics"/>IO<emph.end type="italics"/>ad altitudinem <emph type="italics"/>IG<emph.end type="italics"/>: & quo tempore gutta <lb/> cadendo ab <emph type="italics"/>I<emph.end type="italics"/>de&longs;cribere pote&longs;t altitudinem <emph type="italics"/>IH,<emph.end type="italics"/>aqua egrediens <lb/> æqualis erit Cylindro cujus ba&longs;is e&longs;t circulus <emph type="italics"/>AB<emph.end type="italics"/>& altitudo e&longs;t <lb/> 2<emph type="italics"/>IH<emph.end type="italics"/>: & quo tempore gutta cadendo ab <emph type="italics"/>I<emph.end type="italics"/>per <emph type="italics"/>H<emph.end type="italics"/>ad <emph type="italics"/>G<emph.end type="italics"/>de&longs;cribit <lb/> altitudinum differentiam <emph type="italics"/>HG,<emph.end type="italics"/>aqua egrediens, id e&longs;t, aqua tota in <lb/> &longs;olido <emph type="italics"/>ABNFEM<emph.end type="italics"/>æqualis erit differentiæ Cylindrorum, id e&longs;t, <lb/> Cylindro cujus ba&longs;is e&longs;t <emph type="italics"/>AB<emph.end type="italics"/>& altitudo 2<emph type="italics"/>HO.<emph.end type="italics"/>Et propterea <lb/> aqua tota in va&longs;e <emph type="italics"/>ABDC<emph.end type="italics"/>e&longs;t ad aquam totam cadentem in <lb/> &longs;olido <emph type="italics"/>ABNFEM<emph.end type="italics"/>ut <emph type="italics"/>HG<emph.end type="italics"/>ad 2<emph type="italics"/>HO,<emph.end type="italics"/>id e&longs;t, ut <emph type="italics"/>HO+OG<emph.end type="italics"/><lb/>ad 2<emph type="italics"/>HO,<emph.end type="italics"/>&longs;eu <emph type="italics"/>IH+IO<emph.end type="italics"/>ad 2<emph type="italics"/>IH.<emph.end type="italics"/>Sed pondus aquæ totius in <lb/> &longs;olido <emph type="italics"/>ABNFEM<emph.end type="italics"/>in aquæ defluxum impenditur: ac pro-<pb xlink:href="039/01/336.jpg" pagenum="308"/><lb/><arrow.to.target n="note284"/>inde pondus aquæ totius in va&longs;e e&longs;t ad ponderis partem quæ in <lb/> defluxum aquæ impenditur, ut <emph type="italics"/>IH+IO<emph.end type="italics"/>ad 2<emph type="italics"/>IH,<emph.end type="italics"/>atque adeo ut <lb/> &longs;umma circulorum <emph type="italics"/>EF<emph.end type="italics"/>& <emph type="italics"/>AB<emph.end type="italics"/>ad duplum circulum <emph type="italics"/>EF.<emph.end type="italics"/><lb/></s></p> <p type="margin"> <s><margin.target id="note284"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Et hinc pondus aquæ totius in va&longs;e <emph type="italics"/>ABDC,<emph.end type="italics"/>e&longs;t ad <lb/> ponderis partem alteram quam fundum va&longs;is &longs;u&longs;tinet, ut &longs;umma <lb/> circulorum <emph type="italics"/>AB<emph.end type="italics"/>& <emph type="italics"/>EF,<emph.end type="italics"/>ad differentiam eorundem circulorum. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Et ponderis pars quam fundum va&longs;is &longs;u&longs;tinet, e&longs;t ad <lb/> ponderis partem alteram quæ in defluxum aquæ impenditur, ut <lb/> differentia circulorum <emph type="italics"/>AB<emph.end type="italics"/>& <emph type="italics"/>EF,<emph.end type="italics"/>ad duplum circulum minorem <lb/> <emph type="italics"/>EF,<emph.end type="italics"/>&longs;ive ut area fundi ad duplum foramen. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Ponderis autem pars qua &longs;ola fundum urgetur, e&longs;t ad <lb/> pondus aquæ totius quæ fundo perpendiculariter incumbit, ut cir­<lb/>culus <emph type="italics"/>AB<emph.end type="italics"/>ad &longs;ummam circulorum <emph type="italics"/>AB<emph.end type="italics"/>& <emph type="italics"/>EF,<emph.end type="italics"/>&longs;ive ut circulus <lb/> <emph type="italics"/>AB<emph.end type="italics"/>ad exce&longs;&longs;um dupli circuli <emph type="italics"/>AB<emph.end type="italics"/>&longs;upra fundum. </s> <s>Nam ponderis <lb/> pars qua &longs;ola fundum urgetur, e&longs;t ad pondus aquæ totius in va&longs;e, <lb/> ut differentia circulorum <emph type="italics"/>AB<emph.end type="italics"/>& <emph type="italics"/>EF,<emph.end type="italics"/>ad &longs;ummam eorundem cir­<lb/>culorum, per Cor.4; & pondus aquæ totius in va&longs;e e&longs;t ad pondus <lb/> aquæ totius quæ fundo perpendiculariter incumbit, ut circulus <lb/> <emph type="italics"/>AB<emph.end type="italics"/>ad differentiam circulorum <emph type="italics"/>AB<emph.end type="italics"/>& <emph type="italics"/>EF.<emph.end type="italics"/>Itaque ex æquo <lb/> perturbate, ponderis pars qua &longs;ola fundum urgetur, e&longs;t ad pondus <lb/> aquæ totius quæ fundo perpendiculariter incumbit, ut circulus <lb/> <emph type="italics"/>AB<emph.end type="italics"/>ad &longs;ummam circulorum <emph type="italics"/>AB<emph.end type="italics"/>& <emph type="italics"/>EF<emph.end type="italics"/>vel exce&longs;&longs;um dupli cir­<lb/>culi <emph type="italics"/>AB<emph.end type="italics"/>&longs;upra fundum. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. Si in medio foraminis <emph type="italics"/>EF<emph.end type="italics"/><lb/><figure id="id.039.01.336.1.jpg" xlink:href="039/01/336/1.jpg"/><lb/>locetur Circellus <emph type="italics"/>PQ<emph.end type="italics"/>centro <emph type="italics"/>G<emph.end type="italics"/>de&longs;cri­<lb/>ptus & horizonti parallelus: pondus <lb/> aquæ quam circellus ille &longs;u&longs;tinet, majus <lb/> e&longs;t pondere tertiæ partis Cylindri a­<lb/>quæ cujus ba&longs;is e&longs;t circellus ille & al­<lb/>titudo e&longs;t <emph type="italics"/>GH.<emph.end type="italics"/>Sit enim <emph type="italics"/>ABNFEM<emph.end type="italics"/><lb/>cataracta vel columna aquæ cadentis <lb/> axem habens <emph type="italics"/>GH<emph.end type="italics"/>ut &longs;upra, & conge­<lb/>lari intelligatur aqua omnis in va&longs;e, tam <lb/> in circuitu cataractæ quam &longs;upra cir­<lb/>cellum, cujus fluiditas ad prompti&longs;&longs;imum <lb/> & celerrimum aquæ de&longs;cen&longs;um non requiritur. </s> <s>Et &longs;it <emph type="italics"/>PHQ<emph.end type="italics"/>co­<lb/>lumna aquæ &longs;upra circellum congelata, verticem habens <emph type="italics"/>H<emph.end type="italics"/>& alti­<lb/>tudinem <emph type="italics"/>GH.<emph.end type="italics"/>Et quemadmodum aqua in circuitu cataractæ con­<lb/>gelata <emph type="italics"/>AMEC, BNFD<emph.end type="italics"/>convexa e&longs;t in &longs;uperficie interna <emph type="italics"/>AME, <lb/> BNF<emph.end type="italics"/>ver&longs;us cataractam cadentem, &longs;ic etiam hæc columna <emph type="italics"/>PHQ<emph.end type="italics"/><pb xlink:href="039/01/337.jpg" pagenum="309"/><lb/>convexa erit ver&longs;us cataractam, & propterea major Cono cujus ba­<lb/><arrow.to.target n="note285"/>&longs;is e&longs;t circellus ille <emph type="italics"/>PQ<emph.end type="italics"/>& altitudo <emph type="italics"/>GH,<emph.end type="italics"/>id e&longs;t, major tertia parte <lb/> Cylindri eadem ba&longs;e & altitudine de&longs;cripti. </s> <s>Su&longs;tinet autem cir­<lb/>cellus ille pondus hujus columnæ, id e&longs;t, pondus quod pondere <lb/> Coni &longs;eu tertiæ partis Cylindri illius majus e&longs;t. <lb/> </s></p> <p type="margin"> <s><margin.target id="note285"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>8. Pondus aquæ quam circellus valde parvus <emph type="italics"/>PQ<emph.end type="italics"/>&longs;u&longs;tinet, <lb/> minor e&longs;t pondere duarum tertiarum partium Cylindri aquæ cujus <lb/> ba&longs;is e&longs;t circellus ille & altitudo e&longs;t <emph type="italics"/>HG.<emph.end type="italics"/>Nam &longs;tantibus jam po­<lb/>&longs;itis, de&longs;cribi intelligatur dimidium Sphæroidis cujus ba&longs;is e&longs;t cir­<lb/>cellus ille & &longs;emiaxis &longs;ive altitudo e&longs;t <emph type="italics"/>HG.<emph.end type="italics"/>Et hæc figura æqualis <lb/> erit duabus tertiis partibus Cylindri illius & comprehendet colum­<lb/>nam aquæ congelatæ <emph type="italics"/>PHQ<emph.end type="italics"/>cujus pondus circellus ille &longs;u&longs;tinet. <lb/> Nam ut motus aquæ &longs;it maxime directus, columnæ illius &longs;uper­<lb/>ficies externa concurret cum ba&longs;i <emph type="italics"/>PQ<emph.end type="italics"/>in angulo nonnihil acuto, <lb/> propterea quod aqua cadendo perpetuo acceleratur & propter ac­<lb/>celerationem fit tenuior; & cum angulus ille &longs;it recto minor, hæc <lb/> columna ad inferiores ejus partes jacebit intra dimidium Sphæroi­<lb/>dis. </s> <s>Eadem vero &longs;ur&longs;um acuta erit &longs;eu cu&longs;pidata, ne horizontalis <lb/> motus aquæ ad verticem Sphæroidis &longs;it infinite velocior quam ejus <lb/> motus horizontem ver&longs;us. </s> <s>Et quo minor e&longs;t circellus <emph type="italics"/>PQ<emph.end type="italics"/>eo <lb/> acutior erit vertex columnæ; & circello in infinitum diminuto, an­<lb/>gulus <emph type="italics"/>PHQ<emph.end type="italics"/>in infinitum diminuetur, & propterea columna ja­<lb/>cebit intra dimidium Sphæroidis. </s> <s>E&longs;t igitur columna illa minor <lb/> dimidio Sphæroidis, &longs;eu duabus tertiis partibus Cylindri cujus ba&longs;is <lb/> e&longs;t circellus ille & altitudo <emph type="italics"/>GH.<emph.end type="italics"/>Su&longs;tinet autem circellus vim aquæ <lb/> ponderi hujus columnæ æqualem, cum pondus aquæ ambientis in <lb/> defluxum ejus impendatur. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>9. Pondus aquæ quam circellus valde parvus <emph type="italics"/>PQ<emph.end type="italics"/>&longs;u&longs;ti­<lb/>net, æquale &longs;et ponderi Cylindri aquæ cujus ba&longs;is e&longs;t circellus ille <lb/> & altitudo e&longs;t 1/2<emph type="italics"/>GH<emph.end type="italics"/>quamproxime. </s> <s>Nam pondus hocce e&longs;t me­<lb/>dium Arithmeticum inter pondera Coni & Hemi&longs;phæroidis præ­<lb/>dictæ. At &longs;i circellus ille non &longs;it valde parvus, &longs;ed augeatur donec <lb/> æquet foramen <emph type="italics"/>EF<emph.end type="italics"/>; hic &longs;u&longs;tinebit pondus aquæ totius &longs;ibi per­<lb/>pendiculariter imminentis, id e&longs;t, pondus Cylindri aquæ cujus ba­<lb/>&longs;is e&longs;t circellus ille & altitudo e&longs;t <emph type="italics"/>GH.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>10. Et (quantum &longs;entio) pondus quod circellus &longs;u&longs;tinet, <lb/> e&longs;t &longs;emper ad pondus Cylindri aquæ cujus ba&longs;is e&longs;t circellus ille & <lb/> altitudo e&longs;t 1/2<emph type="italics"/>GH,<emph.end type="italics"/>ut <emph type="italics"/>EFq<emph.end type="italics"/>ad <emph type="italics"/>EFq<emph.end type="italics"/>-1/2<emph type="italics"/>PQq,<emph.end type="italics"/>&longs;ive ut circulus <lb/> <emph type="italics"/>EF<emph.end type="italics"/>ad exce&longs;&longs;um circuli hujus &longs;upra &longs;emi&longs;&longs;em circelli <emph type="italics"/>PQ<emph.end type="italics"/>quam­<lb/>proxime. <pb xlink:href="039/01/338.jpg" pagenum="310"/><lb/><arrow.to.target n="note286"/><emph type="center"/>LEMMA IV.<emph.end type="center"/><lb/></s></p> <p type="margin"> <s><margin.target id="note286"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Cylindri, qui &longs;ecundum longitudinem &longs;uam uniformiter progreditur, <lb/> re&longs;i&longs;tentia ex aucta vel diminuta ejus longitudine non mutatur; <lb/> ideoque eadem e&longs;t cum re&longs;i&longs;tentia Circuli eadem diametro de­<lb/>&longs;cripti & eadem velocitate &longs;ecundum lineam rectam plano ip­<lb/>&longs;ius perpendicularem progredientis.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Nam latera Cylindri motui ejus minime opponuntur: & Cy­<lb/>lindrus, longitudine ejus in infinitum diminuta, in Circulum <lb/> vertitur. <lb/> <emph type="center"/>PROPOSITIO XXXVII. THEOREMA XXIX.<emph.end type="center"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Cylindri, qui in fluide compre&longs;&longs;o infinito & non ela&longs;tico &longs;ecundum <lb/> longitudinem &longs;uam uniformiter progreditur, re&longs;i&longs;tentia quæ ori­<lb/>tur a magnitudine &longs;ectionis tran&longs;ver&longs;æ, e&longs;t ad vim qua totus <lb/> ejus motus interea dum quadruplum longitudinis &longs;uæ de&longs;cribit, <lb/> vel tolli po&longs;&longs;it vel generari, ut den&longs;itas Medii ad den&longs;itatem <lb/> Cylindri quamproxime.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Nam &longs;i vas <emph type="italics"/>ABDC<emph.end type="italics"/>fundo &longs;uo <emph type="italics"/>CD<emph.end type="italics"/>&longs;uperficiem aquæ &longs;tagnan­<lb/>tis tangat, & aqua ex hoc va&longs;e per ca­<lb/><figure id="id.039.01.338.1.jpg" xlink:href="039/01/338/1.jpg"/><lb/>nalem Cylindricum <emph type="italics"/>EFTS<emph.end type="italics"/>horizonti <lb/> perpendicularem in aquam &longs;tagnantem <lb/> effluat, locetur autem Circellus <emph type="italics"/>PQ<emph.end type="italics"/>ho­<lb/>rizonti parallelus ubivis in medio ca­<lb/>nalis, & producatur <emph type="italics"/>CA<emph.end type="italics"/>ad <emph type="italics"/>K,<emph.end type="italics"/>ut &longs;it <lb/> <emph type="italics"/>AK<emph.end type="italics"/>ad <emph type="italics"/>CK<emph.end type="italics"/>in duplicata ratione quam <lb/> habet exce&longs;&longs;us orificii canalis <emph type="italics"/>EF<emph.end type="italics"/>&longs;upra <lb/> circellum <emph type="italics"/>PQ<emph.end type="italics"/>ad circulum <emph type="italics"/>AB<emph.end type="italics"/>: mani­<lb/>fe&longs;tum e&longs;t (per Ca&longs;.5, Ca&longs;.6, & Cor. 1. <lb/> Prop.XXXVI.) quod velocitas aquæ tran­<lb/>&longs;euntis per &longs;patium annulare inter cir­<lb/>cellum & latera va&longs;is, ea erit quam aqua <lb/> cadendo & ca&longs;u &longs;uo de&longs;cribendo altitudinem <emph type="italics"/>KC<emph.end type="italics"/>vel <emph type="italics"/>IG<emph.end type="italics"/>acquirere <lb/> pote&longs;t. <pb xlink:href="039/01/339.jpg" pagenum="311"/><lb/></s></p> <p type="main"> <s>Et (per Cor. 10, Prop.XXXVI) &longs;i va&longs;is latitudo &longs;it infinita, ut li­<lb/><arrow.to.target n="note287"/>neola <emph type="italics"/>HI<emph.end type="italics"/>evane&longs;cat & altitudines <emph type="italics"/>IG, HG<emph.end type="italics"/>æquentur: vis aquæ <lb/> defluentis in circellum erit ad pondus Cylindri cujus ba&longs;is e&longs;t cir­<lb/>cellus ille & altitudo e&longs;t 1/2 <emph type="italics"/>IG,<emph.end type="italics"/>ut <emph type="italics"/>EFq<emph.end type="italics"/>ad <emph type="italics"/>EFq<emph.end type="italics"/>-1/2 <emph type="italics"/>PQq<emph.end type="italics"/>quam <lb/> proxime. </s> <s>Nam vis aquæ, uniformi motu defluentis per totum ca­<lb/>nalem, eadem erit in circellum <emph type="italics"/>PQ<emph.end type="italics"/>in quacunque canalis parte <lb/> locatum. <lb/> </s></p> <p type="margin"> <s><margin.target id="note287"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s>Claudantur jam canalis orificia <emph type="italics"/>EF, ST,<emph.end type="italics"/>& a&longs;cendat circellus in <lb/> fluido undique compre&longs;&longs;o & a&longs;cen&longs;u &longs;uo cogat aquam &longs;uperiorem <lb/> de&longs;cendere per &longs;patium annulare inter circellum & latera cana­<lb/>lis: & velocitas circelli a&longs;cendentis erit ad velocitatem aquæ <lb/> de&longs;cendentis ut differentia circulorum <emph type="italics"/>EF<emph.end type="italics"/>& <emph type="italics"/>PQ<emph.end type="italics"/>ad circulum <lb/> <emph type="italics"/>PQ,<emph.end type="italics"/>& velocitas circelli a&longs;cendentis ad &longs;ummam velocitatum, <lb/> hoc e&longs;t, ad velocitatem relativam aquæ de&longs;cendentis qua præ­<lb/>terfluit circellum a&longs;cendentem, ut differentia circulorum <emph type="italics"/>EF<emph.end type="italics"/>& <lb/> <emph type="italics"/>PQ<emph.end type="italics"/>ad circulum <emph type="italics"/>EF,<emph.end type="italics"/>&longs;ive ut <emph type="italics"/>EFq-PQq<emph.end type="italics"/>ad <emph type="italics"/>EFq.<emph.end type="italics"/>Sit illa <lb/> velocitas relativa æqualis velocitati qua &longs;upra o&longs;ten&longs;um e&longs;t <lb/> aquam tran&longs;ire per idem &longs;patium annulare dum circellus interea <lb/> immotus manet, id e&longs;t, velocitati quam aqua cadendo & ca&longs;u &longs;uo <lb/> de&longs;cribendo altitudinem <emph type="italics"/>IG<emph.end type="italics"/>acquirere pote&longs;t: & vis aquæ in cir­<lb/>cellum a&longs;cendentem eadem erit ac prius, per Legum Cor. 5, id e&longs;t, <lb/> Re&longs;i&longs;tentia circelli a&longs;cendentis erit ad pondus Cylindri aquæ cujus <lb/> ba&longs;is e&longs;t circellus ille & altitudo e&longs;t 1/2 <emph type="italics"/>IG,<emph.end type="italics"/>ut <emph type="italics"/>EFq<emph.end type="italics"/>ad <emph type="italics"/>EFq<emph.end type="italics"/>-1/2 <emph type="italics"/>PQq<emph.end type="italics"/><lb/>quamproxime. </s> <s>Velocitas autem circelli erit ad velocitatem quam <lb/> aqua cadendo & ca&longs;u &longs;uo de&longs;cribendo altitudinem <emph type="italics"/>IG<emph.end type="italics"/>acquirit, <lb/> ut <emph type="italics"/>EFq-PQq<emph.end type="italics"/>ad <emph type="italics"/>EFq.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Augeatur amplitudo canalis in infinitum: & rationes illæ inter <lb/> <emph type="italics"/>EFq-PQq<emph.end type="italics"/>& <emph type="italics"/>EFq,<emph.end type="italics"/>interque <emph type="italics"/>EFq<emph.end type="italics"/>& <emph type="italics"/>EFq<emph.end type="italics"/>-1/2 <emph type="italics"/>PQq<emph.end type="italics"/>acce­<lb/>dent ultimo ad rationes æqualitatis. </s> <s>Et propterea Velocitas cir­<lb/>celli ea nunc erit quam aqua cadendo & ca&longs;u &longs;uo de&longs;cribendo al­<lb/>titudinem <emph type="italics"/>IG<emph.end type="italics"/>acquirere pote&longs;t, Re&longs;i&longs;tentia vero ejus æqualis eva­<lb/>det ponderi Cylindri cujus ba&longs;is e&longs;t circellus ille & altitudo di­<lb/>midium e&longs;t altitudinis <emph type="italics"/>IG,<emph.end type="italics"/>a qua Cylindrus cadere debet ut velo­<lb/>citatem circelli a&longs;cendentis acquirat; & hac velocitate Cylindrus, <lb/> tempore cadendi, quadruplum longitudinis &longs;uæ de&longs;cribet. </s> <s>Re&longs;i­<lb/>&longs;tentia autem Cylindri, hac velocitate &longs;ecundum longitudinem &longs;uam <lb/> progredientis, eadem e&longs;t cum Re&longs;i&longs;tentia circelli per Lemma IV; <lb/> ideoque æqualis e&longs;t Vi qua motus ejus, interea dum quadruplum <lb/> longitudinis &longs;uæ de&longs;cribit, generari pote&longs;t quamproxime. <pb xlink:href="039/01/340.jpg" pagenum="312"/><lb/><arrow.to.target n="note288"/></s></p> <p type="margin"> <s><margin.target id="note288"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s>Si longitudo Cylindri augeatur vel minuatur: motus ejus ut & <lb/> tempus quo quadruplum longitudinis &longs;uæ de&longs;cribit, augebitur vel <lb/> minuetur in eadem ratione; adeoque Vis illa qua motus auctus vel <lb/> diminutus, tempore pariter aucto vel diminuto, generari vel tolli <lb/> po&longs;&longs;it, non mutabitur; ac proinde etiamnum æqualis e&longs;t re&longs;i­<lb/>&longs;tentiæ Cylindri, nam & hæc quoQ.E.I.mutata manet per Lem­<lb/>ma IV. <lb/> </s></p> <p type="main"> <s>Si den&longs;itas Cylindri augeatur vel minuatur: motus ejus ut & <lb/> Vis qua motus eodem tempore generari vel tolli pote&longs;t, in eadem <lb/> ratione augebitur vel minuetur. </s> <s>Re&longs;i&longs;tentia itaque Cylindri cu­<lb/>ju&longs;cunque erit ad Vim qua totus ejus motus, interea dum quadru­<lb/>plum longitudinis &longs;uæ de&longs;cribit, vel generari po&longs;&longs;it vel tolli, ut <lb/> den&longs;itas Medii ad den&longs;itatem Cylindri quamproxime. <emph type="italics"/>Q.E.D.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Fluidum autem comprimi debet ut &longs;it continuum, continuum <lb/> vero e&longs;&longs;e & non ela&longs;ticum ut pre&longs;&longs;io omnis quæ ab ejus compre&longs;&longs;i­<lb/>one oritur propagetur in in&longs;tanti &, in omnes moti corporis partes <lb/> æqualiter agendo, re&longs;i&longs;tentiam non mutet. </s> <s>Pre&longs;&longs;io utique quæ a <lb/> motu corporis oritur, impenditur in motum partium fluidi gene­<lb/>randum & Re&longs;i&longs;tentiam creat. </s> <s>Pre&longs;&longs;io autem quæ oritur a com­<lb/>pre&longs;&longs;ione fluidi, utcunque fortis &longs;it, &longs;i propagetur in in&longs;tanti, nul­<lb/>lum generat motum in partibus fluidi continui, nullam omnino in­<lb/>ducit motus mutationem; ideoque re&longs;i&longs;tentiam nec auget nec mi­<lb/>nuit. </s> <s>Certe Actio fluidi, quæ ab ejus compre&longs;&longs;ione oritur, fortior <lb/> e&longs;&longs;e non pote&longs;t in partes po&longs;ticas corporis moti quam in ejus par­<lb/>tes anticas, ideoque re&longs;i&longs;tentiam in hac Propo&longs;itione de&longs;criptam <lb/> minuere non pote&longs;t: & fortior non erit in partes anticas quam in <lb/> po&longs;ticas, &longs;i modo propagatio ejus infinite velocior &longs;it quam motus <lb/> corporis pre&longs;&longs;i. </s> <s>Infinite autem velocior erit & propagabitur in in­<lb/>&longs;tanti, &longs;i modo fluidum &longs;it continuum & non ela&longs;ticum. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Cylindrorum, qui &longs;ecundum longitudines &longs;uas in Mediis <lb/> continuis infinitis uniformiter progrediuntur, re&longs;i&longs;tentiæ &longs;unt in ra­<lb/>tione quæ componitur ex duplicata ratione velocitatum & dupli­<lb/>cata ratione diametrorum & ratione den&longs;itatis Mediorum. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si amplitudo canalis non augeatur in infinitum, &longs;ed Cy­<lb/>lindrus in Medio quie&longs;cente inclu&longs;o &longs;ecundum longitudinem &longs;uam <lb/> progrediatur, & interea axis ejus cum axe canalis coincidat: Re&longs;i­<lb/>&longs;tentia ejus erit ad vim qua totus ejus motus, quo tempore qua­<lb/>druplum longitudinis &longs;uæ de&longs;cribit, vel generari po&longs;&longs;it vel tolli, <lb/> in ratione quæ componitur ex ratione <emph type="italics"/>EFq<emph.end type="italics"/>ad <emph type="italics"/>EFq<emph.end type="italics"/>-1/2 <emph type="italics"/>PQq<emph.end type="italics"/><pb xlink:href="039/01/341.jpg" pagenum="313"/><lb/>&longs;emel, & ratione <emph type="italics"/>EFq<emph.end type="italics"/>ad <emph type="italics"/>EFq-PQq<emph.end type="italics"/>bis, & ratione den&longs;itatis <lb/> <arrow.to.target n="note289"/>Medii ad den&longs;itatem Cylindri. <lb/> </s></p> <p type="margin"> <s><margin.target id="note289"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Ii&longs;dem po&longs;itis, & quod longitudo L &longs;it ad quadru­<lb/>plum longitudinis Cylindri in ratione quæ componitur ex ratione <lb/> <emph type="italics"/>EFq<emph.end type="italics"/>-1/2 <emph type="italics"/>PQq<emph.end type="italics"/>ad <emph type="italics"/>EFq<emph.end type="italics"/>&longs;emel, & ratione <emph type="italics"/>EFq-PQq<emph.end type="italics"/>ad <emph type="italics"/>EFq<emph.end type="italics"/><lb/>bis: re&longs;i&longs;tentia Cylindri erit ad vim qua totus ejus motus, interea <lb/> dum longitudinem L de&longs;cribit, vel tolli po&longs;&longs;it vel generari, ut <lb/> den&longs;itas Medii ad den&longs;itatem Cylindri. <lb/> <emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><lb/></s></p> <p type="main"> <s>In hac Propo&longs;itione re&longs;i&longs;tentiam inve&longs;tigavimus quæ oritur a <lb/> &longs;ola magnitudine tran&longs;ver&longs;æ &longs;ectionis Cylindri, neglecta re&longs;i&longs;tentiæ <lb/> parte quæ ab obliquitate motuum oriri po&longs;&longs;it. </s> <s>Nam quemadmo­<lb/>dum in ca&longs;u primo Propo&longs;itionis XXXVI, obliquitas motuum qui­<lb/>bus partes aquæ in va&longs;e, undique convergebant in foramen <emph type="italics"/>EF,<emph.end type="italics"/><lb/>impedivit effluxum aquæ illius per foramen: &longs;ic in hac Propo&longs;iti­<lb/>one, obliquitas motuum quibus partes aquæ ab anteriore Cylindri <lb/> termino pre&longs;&longs;æ, cedunt pre&longs;&longs;ioni & undiQ.E.D.vergunt, retardat eo­<lb/>rum tran&longs;itum per loca in circuitu termini illius antecedentis ver­<lb/>&longs;us po&longs;teriores partes Cylindri, efficitque ut fluidum ad majorem <lb/> di&longs;tantiam commoveatur & re&longs;i&longs;tentiam auget, idQ.E.I. ea fere <lb/> ratione qua effluxum aquæ e va&longs;e diminuit, id e&longs;t, in ratione du­<lb/>plicata 25 ad 21 circiter. </s> <s>Et quemadmodum, in Propo&longs;itionis illius <lb/> ca&longs;u primo, effecimus ut partes aquæ perpendiculariter & maxima <lb/> copia tran&longs;irent per foramen <emph type="italics"/>EF,<emph.end type="italics"/>ponendo quod aqua omnis in <lb/> va&longs;e quæ in circuitu cataractæ congelata fuerat, & cujus motus <lb/> obliquus erat & inutilis, maneret &longs;ine motu: &longs;ic in hac Propo&longs;i­<lb/>tione, ut obliquitas motuum tollatur, & partes aquæ motu maxime <lb/> directo & brevi&longs;&longs;imo cedentes facillimum præbeant tran&longs;itum Cy­<lb/>lindro, & &longs;ola maneat re&longs;i&longs;tentia quæ oritur a magnitudine &longs;ecti­<lb/>onis tran&longs;ver&longs;æ, quæQ.E.D.minui non pote&longs;t ni&longs;i diminuendo dia­<lb/>metrum Cylindri, concipiendum e&longs;t quod partes fluidi quarum <lb/> motus &longs;unt obliqui & inutiles & re&longs;i&longs;tentiam creant, quie&longs;cant in­<lb/>ter &longs;e ad utrumque Cylindri ter­<lb/><figure id="id.039.01.341.1.jpg" xlink:href="039/01/341/1.jpg"/><lb/>minum, & cohæreant & Cylindro <lb/> jungantur. </s> <s>Sit <emph type="italics"/>ABCD<emph.end type="italics"/>rectan­<lb/>gulum, & &longs;int <emph type="italics"/>AE<emph.end type="italics"/>& <emph type="italics"/>BE<emph.end type="italics"/>arcus <lb/> duo Parabolici axe <emph type="italics"/>AB<emph.end type="italics"/>de&longs;cripti, <lb/> latere autem recto quod &longs;it ad &longs;pa-<pb xlink:href="039/01/342.jpg" pagenum="314"/><lb/><arrow.to.target n="note290"/>tium <emph type="italics"/>HG,<emph.end type="italics"/>de&longs;cribendum a Cylindro <lb/> <figure id="id.039.01.342.1.jpg" xlink:href="039/01/342/1.jpg"/><lb/>cadente dum velocitatem &longs;uam ac­<lb/>quirit, ut <emph type="italics"/>HG<emph.end type="italics"/>ad 1/2 <emph type="italics"/>AB.<emph.end type="italics"/>Sint etiam <lb/> <emph type="italics"/>CF<emph.end type="italics"/>& <emph type="italics"/>DF<emph.end type="italics"/>arcus alii duo Para­<lb/>bolici, axe <emph type="italics"/>CD<emph.end type="italics"/>& latere recto <lb/> quod &longs;it prioris lateris recti qua­<lb/>druplum de&longs;cripti; & convolutione figuræ circum axem <emph type="italics"/>EF<emph.end type="italics"/>ge­<lb/>neretur &longs;olidum cujus media pars <emph type="italics"/>ABDC<emph.end type="italics"/>&longs;it Cylindrus de quo <lb/> agimus, & partes extremæ <emph type="italics"/>ABE<emph.end type="italics"/>& <emph type="italics"/>CDF<emph.end type="italics"/>contineant partes fluidi <lb/> inter &longs;e quie&longs;centes & in corpora duo rigida concretas, quæ Cy­<lb/>lindro utrinque tanquam caput & cauda adhæreant. </s> <s>Et &longs;olidi <lb/> <emph type="italics"/>EACFDB,<emph.end type="italics"/>&longs;ecundum longitudinem axis &longs;ui <emph type="italics"/>FE<emph.end type="italics"/>in partes ver­<lb/>&longs;us <emph type="italics"/>E<emph.end type="italics"/>progredientis, re&longs;i&longs;tentia ea erit quamproxime quam in hac <lb/> Propo&longs;itione de&longs;crip&longs;imus, id e&longs;t, quæ rationem illam habet ad <lb/> vim qua totus Cylindri motus, interea dum longitudo 4 <emph type="italics"/>AC<emph.end type="italics"/>motu <lb/> illo uniformiter continuato de&longs;cribatur, vel tolli po&longs;&longs;it vel generari, <lb/> quam den&longs;itas Fluidi habet ad den&longs;itatem Cylindri quamproxime. <lb/> Et hac vi Re&longs;i&longs;tentia minor e&longs;&longs;e non pote&longs;t quam in ratione 2 ad 3, <lb/> per Corol. 7. Prop. XXXVI. <lb/> <emph type="center"/>LEMMA V.<emph.end type="center"/><lb/></s></p> <p type="margin"> <s><margin.target id="note290"/>DE MOTU <lb/> CORPORUM.</s></p> <p type="main"> <s><emph type="italics"/>Si Cylindrus, Sphæra & Sphærois, quorum latitudines &longs;unt æqua­<lb/>les, in medio canalis Cylindrici ita locentur &longs;ucce&longs;&longs;ive ut eo­<lb/>rum axes cum axe canalis coincidant: hæc corpora fluxum <lb/> aquæ per canalem æqualiter impedient.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Nam &longs;patia inter Canalem & Cylindrum, Sphæram, & Sphæroi­<lb/>dem per quæ aqua tran&longs;it, &longs;unt æqualia: & aqua per æqualia &longs;pa­<lb/>tia æqualiter tran&longs;it. <lb/> <emph type="center"/>LEMMA VI.<emph.end type="center"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis, corpora prædicta æqualiter urgentur ab aqua per <lb/> canalem fluente.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Patet per Lemma v & Motus Legem tertiam. </s> <s>Aqua utique & <lb/> corpora in &longs;e mutuo æqualiter agunt. <pb xlink:href="039/01/343.jpg" pagenum="315"/><lb/><arrow.to.target n="note291"/><emph type="center"/>LEMMA VII.<emph.end type="center"/><lb/></s></p> <p type="margin"> <s><margin.target id="note291"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Si aqua quie&longs;cat in canali, & hæc corpora in partes contrarias <lb/> æquali velocitate per canalem ferantur: æquales erunt eorum <lb/> re&longs;i&longs;tentiæ inter &longs;e.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Con&longs;tat ex Lemmate &longs;uperiore, nam motus relativi iidem inter <lb/> &longs;e manent. <lb/> <emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/><lb/></s></p> <p type="main"> <s>Eadem e&longs;t ratio corporum omnium convexorum & rotundo­<lb/>rum, quorum axes cum axe canalis coincidunt. </s> <s>Differentia aliqua <lb/> ex majore vel minore frictione oriri pote&longs;t; &longs;ed in his Lemmatis <lb/> corpora e&longs;&longs;e politi&longs;&longs;ima &longs;upponimus, & Medii tenacitatem & frictio­<lb/>nem e&longs;&longs;e nullam, & quod partes fluidi, quæ motibus &longs;uis obliquis <lb/> & &longs;uperfluis fluxum aquæ per canalem perturbare, impedire, & re­<lb/>tardare po&longs;&longs;unt, quie&longs;cant inter &longs;e tanquam gelu con&longs;trictæ, & cor­<lb/>poribus ad ip&longs;orum partes anticas & po&longs;ticas adhæreant, perinde <lb/> ut in Scholio Propo&longs;itionis præcedentis expo&longs;ui. </s> <s>Agitur enim in <lb/> &longs;equentibus de re&longs;i&longs;tentia omnium minima quam corpora rotunda, <lb/> datis maximis &longs;ectionibus tran&longs;ver&longs;is de&longs;cripta, habere po&longs;&longs;unt. <lb/> </s></p> <p type="main"> <s>Corpora fluidis innatantia, ubi moventur in directum, efficiunt <lb/> ut fluidum ad partem anticam a&longs;cendat, ad po&longs;ticam &longs;ub&longs;idat, præ­<lb/>&longs;ertim &longs;i figura &longs;int obtu&longs;a; & inde re&longs;i&longs;tentiam paulo majorem <lb/> &longs;entiunt quam &longs;i capite & cauda &longs;int acutis. </s> <s>Et corpora in fluidis <lb/> ela&longs;ticis mota, &longs;i ante & po&longs;t obtu&longs;a &longs;int, fluidum paulo magis <lb/> conden&longs;ant ad anticam partem & paulo magis relaxant ad po&longs;ticam; <lb/> & inde re&longs;i&longs;tentiam paulo majorem &longs;entiunt quam &longs;i capite & cau­<lb/>da &longs;int acutis. </s> <s>Sed nos in his Lemmatis & Propo&longs;itionibus non <lb/> agimus de fluidis ela&longs;ticis, &longs;ed de non ela&longs;ticis; non de in&longs;identibus <lb/> fluido, &longs;ed de alte immer&longs;is. </s> <s>Et ubi re&longs;i&longs;tentia corporum in fluidis <lb/> non ela&longs;ticis innote&longs;cit, augenda erit hæc re&longs;i&longs;tentia aliquantulum <lb/> tam in fluidis ela&longs;ticis, qualis e&longs;t Aer, quam in &longs;uperficiebus fluido­<lb/>rum &longs;tagnantium, qualia &longs;unt maria & paludes. <pb xlink:href="039/01/344.jpg" pagenum="316"/><lb/><arrow.to.target n="note292"/><emph type="center"/>PROPOSITIO XXXVIII. THEOREMA XXX.<emph.end type="center"/><lb/></s></p> <p type="margin"> <s><margin.target id="note292"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Globi, in Fluido compre&longs;&longs;o infinito & non ela&longs;tico uniformiter progre­<lb/>dientis, re&longs;i&longs;tentia e&longs;t ad vim qua totus ejus motus, quo tempore <lb/> octo tertias partes diametri &longs;uæ de&longs;cribit, vel tolli po&longs;&longs;it vel <lb/> generari, ut den&longs;itas Fluidi ad den&longs;itatem Globi quamproxime.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Nam Globus e&longs;t ad Cylindrum circum&longs;criptum ut duo ad tria; <lb/> & propterea Vis illa, quæ tollere po&longs;&longs;it motum omnem Cylindri <lb/> interea dum Cylindrus de&longs;cribat longitudinem quatuor diametro­<lb/>rum, Globi motum omnem tollet interea dum Globus de&longs;cribat <lb/> duas tertias partes hujus longitudinis, id e&longs;t, octo tertias partes <lb/> diametri propriæ. Re&longs;i&longs;tentia autem Cylindri e&longs;t ad hanc Vim <lb/> quamproxime ut den&longs;itas Fluidi ad den&longs;itatem Cylindri vel Globi, <lb/> per Prop.XXXVII; & Re&longs;i&longs;tentia Globi æqualis e&longs;t Re&longs;i&longs;tentiæ Cy­<lb/>lindri, per Lem. V, VI, VII. <emph type="italics"/>Q.E.D.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Globorum, in Mediis compre&longs;&longs;is infinitis, re&longs;i&longs;tentiæ &longs;unt <lb/> in ratione quæ componitur ex duplicata ratione velocitatis, & du­<lb/>plicata ratione diametri, & duplicata ratione den&longs;itatis Mediorum. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2 Velocitas maxima quacum Globus, vi ponderis &longs;ui com­<lb/>parativi, in fluido re&longs;i&longs;tente pote&longs;t de&longs;cendere, ea e&longs;t quam acqui­<lb/>rere pote&longs;t Globus idem, eodem pondere, ab&longs;que re&longs;i&longs;tentia caden­<lb/>do & ca&longs;u &longs;uo de&longs;cribendo &longs;patium quod &longs;it ad quatuor tertias <lb/> partes diametri &longs;uæ ut den&longs;itas Globi ad den&longs;itatem Fluidi. </s> <s>Nam <lb/> Globus tempore ca&longs;us &longs;ui, cum velocitate cadendo acqui&longs;ita, de­<lb/>&longs;cribet &longs;patium quod erit ad octo tertias diametri &longs;uæ, ut den&longs;itas <lb/> Globi ad den&longs;itatem Fluidi; & vis ponderis motum hunc generans, <lb/> erit ad vim quæ motum eundem generare po&longs;&longs;it quo tempore Glo­<lb/>bus octo tertias diametri &longs;uæ eadem velocitate de&longs;cribit, ut den&longs;itas <lb/> Fluidi ad den&longs;itatem Globi: ideoque per hanc Propo&longs;itionem, vis <lb/> ponderis æqualis erit vi Re&longs;i&longs;tentiæ, & propterea Globum accele­<lb/>rare non pote&longs;t. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Data & den&longs;itate Globi & velocitate ejus &longs;ub initio <lb/> motus, ut & den&longs;itate fluidi compre&longs;&longs;i quie&longs;centis in qua Globus <lb/> movetur; datur ad omne tempus & velocitas Globi & ejus re&longs;i­<lb/>ftentia & &longs;patium ab eo de&longs;criptum, per Corol. 7. Prop. XXXV. <pb xlink:href="039/01/345.jpg" pagenum="317"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Globus in fluido compre&longs;&longs;o quie&longs;cente eju&longs;dem &longs;ecum <lb/> <arrow.to.target n="note293"/>den&longs;itatis movendo, dimidiam motus &longs;ui partem prius amittet <lb/> quam longitudinem duarum ip&longs;ius diametrorum de&longs;crip&longs;erit, per <lb/> idem Corol. 7. <lb/> <emph type="center"/>PROPOSITIO XXXIX. THEOREMA XXXI.<emph.end type="center"/><lb/></s></p> <p type="margin"> <s><margin.target id="note293"/>LIBER <lb/> SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Globi, per Fluidum in canali Cylindrico clau&longs;um & compre&longs;&longs;um uni­<lb/>formiter progredientis, re&longs;i&longs;tentia e&longs;t ad vim qua totus ejus motus, <lb/> interea dum octo tertias partes diametri &longs;uæ de&longs;cribit, vel ge­<lb/>nerari po&longs;&longs;it vel tolli, in ratione quæ componitur ex ratione ori­<lb/>ficii canalis ad exce&longs;&longs;um hujus orificii &longs;upra dimidium circuli <lb/> maximi Globi, & ratione duplicata orificii canalis ad exce&longs;&longs;um <lb/> hujus orificii &longs;upra circulum maximum Globi, & ratione den­<lb/>&longs;itatis Fluidi ad den&longs;itatem Globi quamproxime.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Patet per Corol. 2. Prop. XXXVII; procedit vero demon&longs;tratio <lb/> quemadmodum in Propo&longs;itione præcedente. <lb/> <emph type="center"/>PROPOSITIO XL. PROBLEMA IX.<emph.end type="center"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Globi, in Medio fluidi&longs;&longs;imo compre&longs;&longs;o progredientis, invenire re&longs;i­<lb/>&longs;tentiam per Phænomena.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Sit A pondus Globi in vacuo, B pondus ejus in Medio re&longs;i­<lb/>&longs;tente, D diameter Globi, F &longs;patium quod &longs;it ad 4/3 D ut den&longs;itas <lb/> Globi ad den&longs;itatem Medii, id e&longs;t, ut A ad A-B, G tempus quo <lb/> Globus pondere B ab&longs;que re&longs;i&longs;tentia cadendo de&longs;cribit &longs;patium F, <lb/> & H velocitas quam Globus hocce ca&longs;u &longs;uo acquirit. </s> <s>Et erit H <lb/> velocitas maxima quacum Globus, pondere &longs;uo B, in Medio re&longs;i­<lb/>&longs;tente pote&longs;t de&longs;cendere, per Corol. 2, Prop. XXXVIII; & re&longs;i­<lb/>&longs;tentia quam Globus ea cum velocitate de&longs;cendens patitur, æqua­<lb/>lis erit ejus ponderi B: re&longs;i&longs;tentia vero quam patitur in alia qua­<lb/>cunque velocitate, erit ad pondus B in duplicata ratione velo­<lb/>citatis hujus ad velocitatem illam maximam <emph type="italics"/>H,<emph.end type="italics"/>&c. G, per Corol. 1, <lb/> Prop. XXXVIII. <pb xlink:href="039/01/346.jpg" pagenum="318"/><lb/><arrow.to.target n="note294"/></s></p> <p type="margin"> <s><margin.target id="note294"/>DE MOTU <lb/> CORPORUM</s></p> <p type="main"> <s>Hæc e&longs;t re&longs;i&longs;tentia quæ oritur ab inertia materiæ Fluidi. </s> <s>Ea <lb/> vero quæ oritur ab ela&longs;ticitate, tenacitate, & frictione partium <lb/> ejus, &longs;ic inve&longs;tigabitur. <lb/> </s></p> <p type="main"> <s>Demittatur Globus ut pondere &longs;uo B in Fluido de&longs;cendat; <lb/> & &longs;it P tempus cadendi, idQ.E.I. minutis &longs;ecundis &longs;i tempus <lb/> G in minutis &longs;ecundis habeatur. </s> <s>Inveniatur numerus ab&longs;o­<lb/>lutus N qui congruit Logarithmo 0,4342944819(2P/G), &longs;itque L <lb/> Logarithmus numer; (N+1/N): & velocitas cadendo acqui&longs;ita erit <lb/> (N-1/N+1)H, altitudo autem de&longs;cripta erit (2PF/G)-1,3862943611 F+ <lb/> 4,605170186LF. Si Fluidum &longs;atis profundum &longs;it, negligi pote&longs;t <lb/> terminus 4,605170186LF; & erit (2PF/G)-1,3862943611 F altitude <lb/> de&longs;cripta quamproxime. </s> <s>Patent hæc per Libri &longs;ecundi Propo­<lb/>&longs;itionem nonam & ejus Corollaria, ex Hypothe&longs;i quod Glo­<lb/>bus nullam aliam patiatur re&longs;i&longs;tentiam ni&longs;i quæ oritur ab inertia <lb/> materiæ. Si vero aliam in&longs;uper re&longs;i&longs;tentiam patiatur, de&longs;cen­<lb/>&longs;us erit tardior, & ex retardatione innote&longs;cet quantitas hujus re­<lb/>&longs;i&longs;tentiæ. <lb/> </s></p> <p type="main"> <s>Ut corporis in Fluido cadentis velocitas & de&longs;cen&longs;us facilius in­<lb/>note&longs;cant, compo&longs;ui Tabulam &longs;equentem, cujus columna prima <lb/> denotat tempora de&longs;cen&longs;us, &longs;ecunda exhibet velocitates cadendo <lb/> acqui&longs;itas exi&longs;tente velocitate maxima 100000000, tertia exhibet <lb/> &longs;patia temporibus illis cadendo de&longs;cripta, exi&longs;tente 2 F &longs;patio quod <lb/> corpus tempore G cum velocitate maxima de&longs;cribit, & quarta ex­<lb/>hibet &longs;patia ii&longs;dem temporibus cum velocitate maxima de&longs;cripta. <lb/> Numeri in quarta columna &longs;unt (2P/G), & &longs;ubducendo numerum <lb/> 1,3862944-4,6051702 L, inveniuntur numeri in tertia columna, & <lb/> multiplicandi &longs;unt hi numeri per &longs;patium F ut habeantur &longs;patia <lb/> cadendo de&longs;cripta. </s> <s>Quinta his in&longs;uper adjecta e&longs;t columna, quæ <lb/> continet &longs;patia de&longs;cripta ii&longs;dem temporibus a corpore, vi ponderis <lb/> &longs;ui comparativi B, in vacuo cadente. <pb xlink:href="039/01/347.jpg" pagenum="319"/><lb/><arrow.to.target n="note295"/></s></p><table><row><cell><emph type="italics"/>Tempora<emph.end type="italics"/><lb/>P</cell><cell><emph type="italics"/>Velocitates <lb/> cadentis in <lb/> fluido<emph.end type="italics"/></cell><cell><emph type="italics"/>Spatia caden­<lb/>do de&longs;cripta <lb/> in fluido<emph.end type="italics"/></cell><cell><emph type="italics"/>Spatia motu <lb/> maximo de­<lb/>&longs;cripta.<emph.end type="italics"/></cell><cell><emph type="italics"/>Spatia caden­<lb/>do de&longs;cripta <lb/> in vacuo.<emph.end type="italics"/></cell></row><row><cell>0,001G</cell><cell> (99999 29/30)</cell><cell>0,000001F</cell><cell>0,002F</cell><cell>0,000001F</cell></row><row><cell>0,01G</cell><cell> 999967</cell><cell>0,0001F</cell><cell>0,02F</cell><cell>0,0001F</cell></row><row><cell>0,1G</cell><cell> 9966799</cell><cell>0,0099834F</cell><cell>0,2F</cell><cell>0,01F</cell></row><row><cell>0,2G</cell><cell>19737532</cell><cell>0,0397361F</cell><cell>0,4F</cell><cell>0,04F</cell></row><row><cell>0,3G</cell><cell>29131261</cell><cell>0,0886815F</cell><cell>0,6F</cell><cell>0,09F</cell></row><row><cell>0,4G</cell><cell>37994896</cell><cell>0,1559070F</cell><cell>0,8F</cell><cell>0,16F</cell></row><row><cell>0,5G</cell><cell>46211716</cell><cell>0,2402290F</cell><cell>1,0F</cell><cell>0,25F</cell></row><row><cell>0,6G</cell><cell>53704957</cell><cell>0,3402706F</cell><cell>1,2F</cell><cell>0,36F</cell></row><row><cell>0,7G</cell><cell>60436778</cell><cell>0,4545405F</cell><cell>1,4F</cell><cell>0,49F</cell></row><row><cell>0,8G</cell><cell>66403677</cell><cell>0,5815071F</cell><cell>1,6F</cell><cell>0,64F</cell></row><row><cell>0,9G</cell><cell>71629787</cell><cell>0,7196609F</cell><cell>1,8F</cell><cell>0,81F</cell></row><row><cell>1G</cell><cell>76159416</cell><cell>0,8675617F</cell><cell>2F</cell><cell>1F</cell></row><row><cell>2G</cell><cell>96402758</cell><cell>2,6500055F</cell><cell>4F</cell><cell>4F</cell></row><row><cell>3G</cell><cell>99505475</cell><cell>4,6186570F</cell><cell>6F</cell><cell>9F</cell></row><row><cell>4G</cell><cell>99932930</cell><cell>6,6143765F</cell><cell>8F</cell><cell>16F</cell></row><row><cell>5G</cell><cell>99990920</cell><cell>8,6137964F</cell><cell>10F</cell><cell>25F</cell></row><row><cell>6G</cell><cell>99998771</cell><cell>10,6137179F</cell><cell>12F</cell><cell>36F</cell></row><row><cell>7G</cell><cell>99999834</cell><cell>12,6137073F</cell><cell>14F</cell><cell>49F</cell></row><row><cell>8G</cell><cell>99999980</cell><cell>14,6137059F</cell><cell>16F</cell><cell>64F</cell></row><row><cell>9G</cell><cell>99999997</cell><cell>16,6137057F</cell><cell>18F</cell><cell>81F</cell></row><row><cell>10G</cell><cell>99999999 1/5</cell><cell>18,6137056F</cell><cell>20F</cell><cell>100F</cell></row></table> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Ut re&longs;i&longs;tentias Fluidorum inve&longs;tigarem per Experimenta, paravi <lb/>vas ligneum quadratum, longitudine & latitudine interna digito­<lb/>rum novem pedis <emph type="italics"/>Londinen&longs;is,<emph.end type="italics"/>profunditate pedum novem cum <lb/>&longs;emi&longs;&longs;e, idemQ.E.I.plevi aqua pluviali; & globis ex cera & plum­<lb/>bo inclu&longs;o formatis, notavi tempora de&longs;cen&longs;us globorum, exi&longs;tente <lb/>de&longs;cen&longs;us altitudine 112 digitorum pedis. </s> <s>Pes &longs;olidus cubicus <lb/><emph type="italics"/>Londinen&longs;is<emph.end type="italics"/>continet 76 libras <emph type="italics"/>Romanas<emph.end type="italics"/>aquæ pluvialis, & pedis hu­<lb/>jus digitus &longs;olidus continet (19/36) uncias libræ hujus &longs;eu grana 253 1/3; <lb/>& globus aqueus diametro digiti unius de&longs;criptus continet grana <pb xlink:href="039/01/348.jpg" pagenum="320"/><arrow.to.target n="note328"/>132,645 in Medio aeris, vel grana 132,8 in vacuo; & globus qui­<lb/>libet alius e&longs;t ut exce&longs;&longs;us ponderis ejus in vacuo &longs;upra pondus ejus <lb/>in aqua. </s></p> <p type="margin"> <s><margin.target id="note328"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>1. Globus, cujus pondus erat 156 1/4 granorum in aere & <lb/>77 granorum in aqua, altitudinem totam digitorum 112 tempore <lb/>minutorum quatuor &longs;ecundorum de&longs;crip&longs;it. </s> <s>Et experimento repe­<lb/>tito, globus iterum cecidit eodem tempore minutorum quatuor &longs;e­<lb/>cundorum. </s></p> <p type="main"> <s>Pondus globi in vacuo e&longs;t (156 11/38) <emph type="italics"/>gran,<emph.end type="italics"/>& exce&longs;&longs;us hujus ponde­<lb/>ris &longs;upra pondus globi in aqua e&longs;t (79 11/38) <emph type="italics"/>gran.<emph.end type="italics"/>Unde prodit globi <lb/>diameter 0,84224 partium digiti. </s> <s>E&longs;t autem ut exce&longs;&longs;us ille ad <lb/>pondus globi in vacuo, ita den&longs;itas aquæ ad den&longs;itatem globi, <lb/>& ita partes octo tertiæ diametri globi (<emph type="italics"/>viz.<emph.end type="italics"/>2,24597 <emph type="italics"/>dig.<emph.end type="italics"/>) ad &longs;pa­<lb/>tium 2 F, quod proinde erit 4,4256 <emph type="italics"/>dig.<emph.end type="italics"/>Globus tempore minuti <lb/>unius &longs;ecundi, toto &longs;uo pondere granorum (156 11/38), cadendo in va­<lb/>cuo de&longs;cribet digitos 193 1/3; & pondere granorum 77, eodem tem­<lb/>pore, ab&longs;que re&longs;i&longs;tentia cadendo in aqua de&longs;cribet digitos 95,219; <lb/>& tempore G, quod &longs;it ad minutum unum &longs;ecundum in &longs;ubduplicata <lb/>ratione &longs;patii F &longs;eu 2,2128 <emph type="italics"/>dig.<emph.end type="italics"/>ad 95,219 <emph type="italics"/>dig,<emph.end type="italics"/>de&longs;cribet 2,2128 <emph type="italics"/>dig.<emph.end type="italics"/><lb/>& velocitatem maximam H acquiret quacum pote&longs;t in aqua de­<lb/>&longs;cendere. </s> <s>E&longs;t igitur tempus G 0″,15244. Et hoc tempore G, <lb/>cum velocitate illa maxima H, globus de&longs;cribet &longs;patium 2 F digi­<lb/>torum 4,4256; ideoque tempore minutorum quatuor &longs;ecundo­<lb/>rum de&longs;cribet &longs;patium digitorum 116,1245. Subducatur &longs;patium <lb/>1,3862944 F &longs;eu 3,0676 <emph type="italics"/>dig.<emph.end type="italics"/>& manebit &longs;patium 113,0569 digito­<lb/>rum quod globus cadendo in aqua, in va&longs;e ampli&longs;&longs;imo, tempore <lb/>minutorum quatuor &longs;ecundorum de&longs;cribet. </s> <s>Hoc &longs;patium, ob an­<lb/>gu&longs;tiam va&longs;is lignei prædicti, minui debet in ratione quæ compo­<lb/>nitur ex &longs;ubduplicata ratione orificii va&longs;is ad exce&longs;&longs;um orificii hu­<lb/>jus &longs;upra &longs;emicirculum maximum globi & ex &longs;implici ratione ori­<lb/>ficii eju&longs;dem ad exce&longs;&longs;um ejus &longs;upra circulum maximum globi, id <lb/>e&longs;t, in ratione 1 ad 0,9914. Quo facto, habebitur &longs;patium 112,08 <lb/>digitorum, quod Globus cadendo in aqua in hoc va&longs;e ligneo tem­<lb/>pore minutorum quatuor &longs;ecundorum per Theoriam de&longs;cribere <lb/>debuit quamproxime. </s> <s>De&longs;crip&longs;it vero digitos 112 per Experi­<lb/>mentum. </s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>2. Tres Globi æquales, quorum pondera &longs;eor&longs;im erant <lb/>76 1/3 granorum in aere & (5 1/16) granorum in aqua, &longs;ucce&longs;&longs;ive demitte­<lb/>bantur; & unu&longs;qui&longs;que cecidit in aqua tempore minutorum &longs;ecun­<lb/>dorum quindecim, ca&longs;u &longs;uo de&longs;cribens altitudinem digitorum 112. </s></p><pb xlink:href="039/01/349.jpg" pagenum="321"/> <p type="main"> <s>Computum ineundo prodcunt pondus globi in vacuo (76 1/12) <emph type="italics"/>gran,<emph.end type="italics"/><lb/><arrow.to.target n="note329"/>exce&longs;&longs;us hujus ponderis &longs;upra pondus in aqua (71 17/48) <emph type="italics"/>gran,<emph.end type="italics"/>diameter <lb/>globi 0,81296 <emph type="italics"/>dig,<emph.end type="italics"/>octo tertiæ partes hujus diametri 2,16789 <emph type="italics"/>dig,<emph.end type="italics"/><lb/>&longs;patium 2 F 2,3217 <emph type="italics"/>dig,<emph.end type="italics"/>&longs;patium quod globus pondere (5 1/16) <emph type="italics"/>gran,<emph.end type="italics"/><lb/>tempore 1″, ab&longs;que re&longs;i&longs;tentia cadendo de&longs;cribat 12,808 <emph type="italics"/>dig,<emph.end type="italics"/>& <lb/>tempus G 0′,301056. Globus igitur, velocitate maxima quacum <lb/>pote&longs;t in aqua vi ponderis (5 1/16) <emph type="italics"/>gran.<emph.end type="italics"/>de&longs;cendere, tempore 0′,301056 <lb/>de&longs;cribet &longs;patium 2,3217 <emph type="italics"/>dig.<emph.end type="italics"/>& tempore 15″ &longs;patium 115,678 <emph type="italics"/>dig.<emph.end type="italics"/><lb/>Subducatur &longs;patium 1,3862944 F &longs;eu 1,609 <emph type="italics"/>dig.<emph.end type="italics"/>& manebit &longs;patium <lb/>114,069 <emph type="italics"/>dig.<emph.end type="italics"/>quod proinde globus eodem tempore in va&longs;e lati&longs;li­<lb/>mo cadendo de&longs;cribere debet. </s> <s>Propter angu&longs;tiam va&longs;is no&longs;tri de­<lb/>trahi debet &longs;patium 0,895 <emph type="italics"/>dig.<emph.end type="italics"/>circiter. </s> <s>Et &longs;ic manebit &longs;patium <lb/>113,174 <emph type="italics"/>dig.<emph.end type="italics"/>quod globus cadendo in hoc va&longs;e, tempore 15″ de­<lb/>&longs;cribere debuit per Theoriam quamproxime. </s> <s>De&longs;crip&longs;it vero digi­<lb/>tos 112 per Experimentum. </s> <s>Differentia e&longs;t in&longs;en&longs;ibilis. </s></p> <p type="margin"> <s><margin.target id="note329"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>3. Globi tres æquales, quorum pondera &longs;eor&longs;im erant <lb/>121 <emph type="italics"/>gran.<emph.end type="italics"/>in aere & 1 <emph type="italics"/>gran.<emph.end type="italics"/>in aqua, &longs;ucce&longs;&longs;ive demittebantur; & <lb/>cadebant in aqua temporibus 46″, 47″, & 50″, de&longs;cribentes alti­<lb/>tudinem digitorum 112. </s></p> <p type="main"> <s>Per Theoriam hi globi cadere debuerunt tempore 40″ circiter. </s> <s><lb/>Quod tardius ceciderunt, vel bullulis nonnullis globo adhærenti­<lb/>bus, vel rarefactioni ceræ ad calorem vel tempe&longs;tatis vel manus <lb/>globum demittentis, vel erroribus in&longs;en&longs;ibilibus in ponderandis <lb/>globis in aqua, vel denique minori proportioni re&longs;i&longs;tentiæ quæ a <lb/>vi inertiæ in tardis motibus oritur ad re&longs;i&longs;tentiam quæ oritur ab <lb/>aliis cau&longs;is, tribuendum e&longs;&longs;e puto. </s> <s>Ideoque pondus globi in aqua <lb/>debet e&longs;&longs;e plurium granorum ut experimentum certum & fide dig­<lb/>num reddatur. </s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>4. Experimenta hactenus de&longs;cripta cæpi ut inve&longs;tigarem <lb/>re&longs;i&longs;tentias fluidorum antequam Theoria, in Propo&longs;itionibus pro­<lb/>xime præcedentibus expo&longs;ita, mihi innote&longs;ceret. </s> <s>Po&longs;tea, ut Theo­<lb/>riam inventam examinarem, paravi vas ligneum latitudine interna <lb/>digitorum 8 2/3, profunditate pedum quindecim cum triente. </s> <s>De­<lb/>inde ex cera & plumbo inclu&longs;o globos quatuor formavi, &longs;ingulos <lb/>pondere 139 1/4 granorum in aere & 7 1/8 granorum in aqua. </s> <s>Et hos <lb/>demi&longs;i ut tempora cadendi in aqua per pendulum, ad &longs;emi-minuta <lb/>&longs;ecunda o&longs;cillans, men&longs;urarem. </s> <s>Globi, ubi ponderabantur & po­<lb/>&longs;tea cadebant, frigidi erant & aliquamdiu frigidi man&longs;erant; quia <lb/>calor ceram rarefacit, & per rarefactionem diminuit pondus globi <lb/>in aqua, & cera rarefacta non &longs;tatim ad den&longs;itatem pri&longs;tinam per <pb xlink:href="039/01/350.jpg" pagenum="322"/><arrow.to.target n="note330"/>frigus reducitur. </s> <s>Antequam caderent, immergebantur penitus in <lb/>aquam; ne pondere partis alicujus ex aqua extantis de&longs;cen&longs;us eo­<lb/>rum &longs;ub initio acceleraretur. </s> <s>Et ubi penitus immer&longs;i quie&longs;cebant, <lb/>demittebantur quam cauti&longs;&longs;ime, ne impul&longs;um aliquem a manu de­<lb/>mittente acciperent. </s> <s>Ceciderunt autem &longs;ucce&longs;&longs;ive temporibus <lb/>o&longs;cillationum 47 1/2, 48 1/2, 50 & 51, de&longs;cribentes altitudinem pedum <lb/>quindecim & digitorum duorum. </s> <s>Sed tempe&longs;tas jam paulo frigi­<lb/>dior erat quam cum globi ponderabantur, ideoQ.E.I.eravi experi­<lb/>mentum alio die, & globi ceciderunt temporibus o&longs;cillationum <lb/>49, 49 1/2, 50 & 53, ac tertio temporibus o&longs;cillationum 49 1/2, 50, 51 <lb/>& 53. Et experimento &longs;æpius capto, Globi ceciderunt maxima <lb/>ex parte temporibus o&longs;cillationum 49 1/2 & 50. Ubi tardius ce­<lb/>cidere, &longs;u&longs;picor eo&longs;dem retardatos fui&longs;&longs;e impingendo in latera <lb/>va&longs;is. </s></p> <p type="margin"> <s><margin.target id="note330"/>DE MOTU <lb/>CORPORUM.</s></p> <p type="main"> <s>Jam computum per Theoriam ineundo, prodeunt pondus globi <lb/>in vacuo 139 2/5 granorum. </s> <s>Exce&longs;&longs;us hujus ponderis &longs;upra pondus <lb/>globi in aqua (132 11/40) <emph type="italics"/>gran.<emph.end type="italics"/>Diameter globi 0,99868 <emph type="italics"/>dig.<emph.end type="italics"/>Octo ter­<lb/>tiæ partes diametri 2,66315 <emph type="italics"/>dig.<emph.end type="italics"/>Spatium 2 F 2,8066 <emph type="italics"/>dig.<emph.end type="italics"/>Spatium <lb/>quod globus pondere 7 1/8 granorum, tempore minuti unius &longs;e­<lb/>cundi ab&longs;que re&longs;i&longs;tentia cadendo de&longs;cribit 9,88164 <emph type="italics"/>dig.<emph.end type="italics"/>Et tempus <lb/>G 0″,376843. Globus igitur, velocitate maxima quacum pote&longs;t in <lb/>aqua vi ponderis 7 1/8 granorum de&longs;cendere, tempore 0″,376843 de­<lb/>&longs;cribit &longs;patium 2,8066 digitorum, & tempore 1″ &longs;patium 7,44766 di­<lb/>gitorum, & tempore 25″ &longs;eu o&longs;cillationum 50 &longs;patium 186,1915 <emph type="italics"/>dig.<emph.end type="italics"/><lb/>Subducatur &longs;patium 1,386294 F, &longs;eu 1,9454 <emph type="italics"/>dig.<emph.end type="italics"/>& manebit &longs;pa­<lb/>tium 184,2461 <emph type="italics"/>dig.<emph.end type="italics"/>quod globus eodem tempore in va&longs;e lati&longs;&longs;imo <lb/>de&longs;cribet. </s> <s>Ob angu&longs;tiam va&longs;is no&longs;tri, minuatur hoc &longs;patium in ra­<lb/>tione quæ componitur ex &longs;ubduplicata ratione orificii va&longs;is ad <lb/>exce&longs;&longs;um hujus orificii &longs;upra &longs;emicirculum maximum globi, & &longs;im­<lb/>plici ratione eju&longs;dem orificii ad exce&longs;&longs;um ejus &longs;upra circulum ma­<lb/>ximum globi; & habebitur &longs;patium 181,86 digitorum, quod glo­<lb/>bus in hoc va&longs;e tempore o&longs;cillationum 50 de&longs;cribere debuit per <lb/>Theoriam quamproxime. </s> <s>De&longs;crip&longs;it vero &longs;patium 182 digitorum <lb/>tempore o&longs;cillationum 49 1/2 vel 50 per Experimentum. </s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>5. Globi quatuor pondere 154 1/8 <emph type="italics"/>gran.<emph.end type="italics"/>in aere & 21 1/2 <emph type="italics"/>gran.<emph.end type="italics"/><lb/>in aqua, &longs;æpe demi&longs;&longs;i, cadebant tempore o&longs;cillationum 28 1/2, 29, <lb/>29<gap/> & 30, & nonnunquam 31, 32 & 33, de&longs;cribentes altitudinem <lb/>pedum quindecim & digitorum duorum. </s></p> <p type="main"> <s>Per Theoriam cadere debuerunt tempore o&longs;cillationum 29 <lb/>quamproxime. </s></p><pb xlink:href="039/01/351.jpg" pagenum="323"/> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>6. Globi quinque pondere 212 1/8 <emph type="italics"/>gran.<emph.end type="italics"/>in aere & 79 1/2 in <lb/><arrow.to.target n="note331"/>aqua, &longs;æpe demi&longs;&longs;i, cadebant tempore o&longs;cillationum 15, 15 1/2, 16, <lb/>17 & 18, de&longs;cribentes altitudinem pedum quindecim & digitorum <lb/>duorum. </s></p> <p type="margin"> <s><margin.target id="note331"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s>Per Theoriam cadere debuerunt tempore o&longs;cillationum 15 <lb/>quamproxime. </s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>7. Globi quatuor pondere 293 1/8 <emph type="italics"/>gran.<emph.end type="italics"/>in aere & 35 1/8 <emph type="italics"/>gran.<emph.end type="italics"/><lb/>in aqua, &longs;æpe demi&longs;&longs;i, cadebant tempore o&longs;cillationum 29 1/2, 30, <lb/>30 1/2, 31, 32 & 33, de&longs;cribentes altitudinem pedum quindecim & <lb/>digiti unius cum &longs;emi&longs;&longs;e. </s></p> <p type="main"> <s>Per Theoriam cadere debuerunt tempore o&longs;cillationum 28 <lb/>quamproxime. </s></p> <p type="main"> <s>Cau&longs;am inve&longs;tigando cur globorum, eju&longs;dem ponderis & magNI­<lb/>tudinis, aliqui citius alii tardius caderent, in hanc incidi; quod glo­<lb/>bi, ubi primum demittebantur & cadere incipiebant, o&longs;cillarent cir­<lb/>cum centra, latere illo quod forte gravius e&longs;&longs;et, primum de&longs;cen­<lb/>dente, & motum o&longs;cillatorium generante. </s> <s>Nam per o&longs;cillationes <lb/>&longs;uas, globus majorem motum communicat aquæ, quam &longs;i &longs;ine o&longs;cil­<lb/>lationibus de&longs;cenderet; & communicando, amittit partem motus <lb/>proprii quo de&longs;cendere deberet: & pro majore vel minore o&longs;cil­<lb/>latione, magis vel minus retardatur. </s> <s>Quinetiam globus recedit <lb/>&longs;emper a latere &longs;uo quod per o&longs;cillationem de&longs;cendit, & receden­<lb/>do appropinquat lateribus va&longs;is & in latera nonnunquam impin­<lb/>gitur. </s> <s>Et hæc o&longs;cillatio in globis gravioribus fortior e&longs;t, & in <lb/>majoribus aquam magis agitat. </s> <s>Quapropter, ut o&longs;cillatio globo­<lb/>rum minor redderetur, globos novos ex cera & plumbo con&longs;truxi, <lb/>infigendo plumbum in latus aliquod globi prope &longs;uperficiem ejus; <lb/>& globum ita demi&longs;i, ut latus gravius, quoad fieri potuit, e&longs;&longs;et in­<lb/>fimum ab initio de&longs;cen&longs;us. </s> <s>Sic o&longs;cillationes factæ &longs;unt multo mi­<lb/>nores quam prius, & globi temporibus minus inæqualibus cecide­<lb/>runt, ut in experimentis &longs;equentibus. </s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>8. Globi quatuor pondere granorum 139 in aere & 6 1/2 in <lb/>aqua, &longs;æpe demi&longs;&longs;i, ceciderunt temporibus o&longs;cillationum non plu­<lb/>rium quam 52, non pauciorum quam 50, & maxima ex parte <lb/>tempore o&longs;cillationum 51 circiter, de&longs;cribentes altitudinem digi­<lb/>torum 182. </s></p> <p type="main"> <s>Per Theoriam cadere debuerunt tempore o&longs;cillationum 52 <lb/>circiter. </s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>9. Globi quatuor pondere granorum 273 1/4 in aere & <lb/>140 1/4 in aqua, &longs;æpius demi&longs;&longs;i, ceciderunt temporibus o&longs;cillationum <pb xlink:href="039/01/352.jpg" pagenum="324"/><arrow.to.target n="note332"/>non pauciorum quam 12, non plurium quam 13, de&longs;cribentes al­<lb/>titudinem digitorum 182. </s></p> <p type="margin"> <s><margin.target id="note332"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Per Theoriam vero hi globi cadere debuerunt tempore o&longs;cilla­<lb/>tionum 11 1/3 quamproxime. </s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>10. Globi quatuor pondere granorum 384 in aere & <lb/>119 1/2 in aqua, &longs;æpe demi&longs;&longs;i, cadebant temporibus o&longs;cillationum <lb/>17 1/4, 18, 18 1/2 & 19, de&longs;cribentes altitudinem digitorum 181 1/2. Et <lb/>ubi ceciderunt tempore o&longs;cillationum 19, nonnunquam audivi im­<lb/>pul&longs;um eorum in latera va&longs;is antequam ad fundum pervenerunt. </s></p> <p type="main"> <s>Per Theoriam vero cadere debuerunt tempore o&longs;cillationum <lb/>15 3/9 quamproxime. </s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>11. Globi tres æquales, pondere granorum 48 in aere <lb/>& (3 29/32) in aqua, &longs;æpe demi&longs;&longs;i, ceciderunt temporibus o&longs;cillationum <lb/>43 1/2, 44, 44 1/2, 45 & 46, & maxima ex parte 44 & 45, de&longs;cribentes <lb/>altitudinem digitorum 182 1/2 quamproxime. </s></p> <p type="main"> <s>Per Theoriam cadere debuerunt tempore o&longs;cillationum 46 5/9 <lb/>circiter. </s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>12. Globi tres æquales, pondere granorum 141 in aere <lb/>& 4 3/8 in aqua, aliquoties demi&longs;&longs;i, ceciderunt temporibus o&longs;cillatio­<lb/>num 61, 62, 63, 64 & 65, de&longs;cribentes altitudinem digitorum 182. </s></p> <p type="main"> <s>Et per Theoriam cadere debuerunt tempore o&longs;cillationum <lb/>64 1/2 quamproxime. </s></p> <p type="main"> <s>Per hæc Experimenta manife&longs;tum e&longs;t quod, ubi globi tarde ceci­<lb/>derunt, ut in experimentis &longs;ecundis, quartis, quintis, octavis, un­<lb/>decimis ac duodecimis, tempora cadendi recte exhibentur per <lb/>Theoriam: at ubi globi velocius ceciderunt, ut in experimentis <lb/>&longs;extis, nonis ac decimis, re&longs;i&longs;tentia paulo major extitit quam in <lb/>duplicata ratione velocitatis. </s> <s>Nam globi inter cadendum o&longs;cillant <lb/>aliquantulum; & hæc o&longs;cillatio in globis levioribus & tardius ca­<lb/>dentibus, ob motus languorem cito ce&longs;&longs;at; in gravioribus autem & <lb/>majoribus, ob motus fortitudinem diutius durat, & non ni&longs;i po&longs;t <lb/>plures o&longs;cillationes ab aqua ambiente cohiberi pote&longs;t. </s> <s>Quinetiam <lb/>globi, quo velociores &longs;unt, eo minus premuntur a fluido ad po­<lb/>&longs;ticas &longs;uas partes; & &longs;i velocitas perpetuo augeatur, &longs;patium va­<lb/>cuum tandem a tergo relinquent, ni&longs;i compre&longs;&longs;io fluidi &longs;imul au­<lb/>geatur. </s> <s>Debet autem compre&longs;&longs;io fluidi (per Prop. </s> <s>XXXII & XXXIII) <lb/>augeri in duplicata ratione velocitatis, ut re&longs;i&longs;tentia &longs;it in eadem <lb/>duplicata ratione. </s> <s>Quoniam hoc non fit, globi velociores paulo <lb/>minus premuntur a tergo, & defectu pre&longs;&longs;ionis hujus, re&longs;i&longs;tentia <lb/>eorum fit paulo major quam in duplicata ratione velocitatis. </s></p><pb xlink:href="039/01/353.jpg" pagenum="325"/> <p type="main"> <s>Congruit igitur Theoria cum phænomenis corporum caden­<lb/><arrow.to.target n="note333"/>tium in Aqua, reliquum e&longs;t ut examinemus phænomena caden­<lb/>tium in Aere. </s></p> <p type="margin"> <s><margin.target id="note333"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Exper.<emph.end type="italics"/>13. A culmine Eccle&longs;iæ <emph type="italics"/>S<emph type="sup"/>ti<emph.end type="sup"/> Pauli,<emph.end type="italics"/>in urbe <emph type="italics"/>Londini,<emph.end type="italics"/>globi <lb/>duo vitrei &longs;imul demittebantur, unus argenti vivi plenus, alter <lb/>aeris; & cadendo de&longs;cribebant altitudinem pedum <emph type="italics"/>Londinen&longs;ium<emph.end type="italics"/><lb/>220. Tabula lignea ad unum ejus terminum polis ferreis &longs;u&longs;pen­<lb/>debatur, ad alterum pe&longs;&longs;ulo ligneo incumbebat; & globi duo huic <lb/>Tabulæ impo&longs;iti &longs;imul demittebantur, &longs;ubtrahendo pe&longs;&longs;ulum, ut Ta­<lb/>bula polis ferreis &longs;olummodo innixa &longs;uper ii&longs;dem devolveretur, & <lb/>codem temporis momento pendulum ad minuta &longs;ecunda o&longs;cillans, <lb/>per filum ferreum a pe&longs;&longs;ulo ad imam Eccle&longs;iæ partem tendens, <lb/>dimitteretur & o&longs;cillare inciperet. </s> <s>Diametri & pondera globorum <lb/>ac tempora cadendi exhibentur in Tabula &longs;equente. <lb/><arrow.to.target n="table3"/> </s></p><table><table.target id="table3"/><row><cell><emph type="italics"/>Globorum mercurio plenorum.<emph.end type="italics"/></cell><cell><emph type="italics"/>Globorum aere plenorum.<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Pondera<emph.end type="italics"/></cell><cell><emph type="italics"/>Diametri<emph.end type="italics"/></cell><cell><emph type="italics"/>Tempora <lb/> cadendi.<emph.end type="italics"/></cell><cell><emph type="italics"/>Pondera<emph.end type="italics"/></cell><cell><emph type="italics"/>Diametri<emph.end type="italics"/></cell><cell><emph type="italics"/>Tempora <lb/> cadendi.<emph.end type="italics"/></cell></row><row><cell>908 <emph type="italics"/>gran.<emph.end type="italics"/></cell><cell>0,8 <emph type="italics"/>digit.<emph.end type="italics"/></cell><cell>4″</cell><cell>510 <emph type="italics"/>gran.<emph.end type="italics"/></cell><cell>5,1 <emph type="italics"/>digit.<emph.end type="italics"/></cell><cell>8″ 1/2</cell></row><row><cell>983</cell><cell>0,8</cell><cell>4-</cell><cell>642</cell><cell>5,2</cell><cell>8</cell></row><row><cell>866</cell><cell>0,8</cell><cell>4</cell><cell>599</cell><cell>5,1</cell><cell>8</cell></row><row><cell>747</cell><cell>0,75</cell><cell>4+</cell><cell>515</cell><cell>5,0</cell><cell>8 1/4</cell></row><row><cell>808</cell><cell>0,75</cell><cell>4</cell><cell>483</cell><cell>5,0</cell><cell>8 1/2</cell></row><row><cell>784</cell><cell>0,75</cell><cell>4+</cell><cell>641</cell><cell>5,2</cell><cell>8</cell></row></table> <p type="main"> <s>Cæterum tempora ob&longs;ervata corrigi debent. </s> <s>Nam globi mer­<lb/>curiales (per Theoriam <emph type="italics"/>Galilæi<emph.end type="italics"/>) minutis quatuor &longs;ecundis de&longs;cribent <lb/>pedes <emph type="italics"/>Londinen&longs;es<emph.end type="italics"/>257, & pedes 220 minutis tantum 3″ 42′. </s> <s>Ta­<lb/>bula lignea utique, detracto pe&longs;&longs;ulo, tardius devolvebatur quam par <lb/>erat, & tarda &longs;ua devolutione impediebat de&longs;cen&longs;um globorum <lb/>&longs;ub initio. </s> <s>Nam globi incumbebant Tabulæ prope medium ejus, <lb/>& paulo quidem propiores erant axi ejus quam pe&longs;&longs;ulo. </s> <s>Et hinc <lb/>tempora cadendi prorogata fuerunt minutis tertiis octodecim cir­<lb/>citer, & jam corrigi debent detrahendo illa minuta, præ&longs;ertim in <lb/>globis majoribus qui Tabulæ devolventi paulo diutius incumbe­<lb/>bant propter magnitudinem diametrorum. </s> <s>Quo facto, tempora <lb/>quibus globi &longs;ex majores cecidere, evadent, 8″, 12′, 7″ 42′, 7″ 42′, <lb/>7″ 57′, 8″ 12′, & 7″ 42′. <pb xlink:href="039/01/354.jpg" pagenum="326"/><arrow.to.target n="note334"/></s></p> <p type="margin"> <s><margin.target id="note334"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Globorum igitur aere plenorum quintus, diametro digitorum <lb/>quinque pondere granorum 483 con&longs;tructus, cecidit tempore <lb/>8″ 12′, de&longs;cribendo altitudinem pedum 220. Pondus aquæ huic <lb/>globo æqualis, e&longs;t 16600 granorum; & pondus aeris eidem æqualis <lb/>e&longs;t (16600/860) <emph type="italics"/>gran.<emph.end type="italics"/>&longs;eu (19 3/10) <emph type="italics"/>gran<emph.end type="italics"/>; ideoque pondus globi in vacuo e&longs;t <lb/>(502 3/10) <emph type="italics"/>gran<emph.end type="italics"/>; & hoc pondus e&longs;t ad pondus aeris globo æqualis, ut <lb/>(502 3/10) ad (19 3/10), & ita &longs;unt 2 F ad octo tertias partes diametri glo­<lb/>bi, id e&longs;t, ad (13 1/3) digitos. </s> <s>Unde 2 F prodeunt 28 <emph type="italics"/>ped.<emph.end type="italics"/>11 <emph type="italics"/>dig.<emph.end type="italics"/>Glo­<lb/>bus cadendo in vacuo, toto &longs;uo pondere (502 3/10) granorum, tempore <lb/>minuti unius &longs;ecundi de&longs;cribit digitos 193 1/3 ut &longs;upra, & pondere <lb/>483 <emph type="italics"/>gran.<emph.end type="italics"/>de&longs;cribit digitos 185,905, & eodem pondere 483 <emph type="italics"/>gran.<emph.end type="italics"/><lb/>etiam in vacuo de&longs;cribit &longs;patium F &longs;eu 14 <emph type="italics"/>ped.<emph.end type="italics"/>5 1/2 <emph type="italics"/>dig.<emph.end type="italics"/>tempore <lb/>57′ 58′, & velocitatem maximam acquirit quacum po&longs;&longs;it in aere <lb/>de&longs;cendere. </s> <s>Hac velocitate globus, tempore 8″ 12′, de&longs;cribet &longs;pa­<lb/>tium pedum 245 & digitorum 5 1/3. Aufer 1,3863 F &longs;eu 20 <emph type="italics"/>ped.<emph.end type="italics"/><lb/>0 1/2 <emph type="italics"/>dig.<emph.end type="italics"/>& manebunt 225 <emph type="italics"/>ped.<emph.end type="italics"/>5 <emph type="italics"/>dig.<emph.end type="italics"/>Hoc &longs;patium igitur globus, <lb/>tempore 8″ 12′, cadendo de&longs;cribere debuit per Theoriam. </s> <s>De­<lb/>&longs;crip&longs;it vero &longs;patium 220 pedum per Experimentum. </s> <s>Differentia <lb/>in&longs;en&longs;ibilis e&longs;t. </s></p> <p type="main"> <s>Similibus computis ad reliquos etiam globos aere plenos appli­<lb/>catis, confeci Tabulam &longs;equentem. <lb/><arrow.to.target n="table4"/> </s></p><table><table.target id="table4"/><row><cell><emph type="italics"/>Globorum <lb/> pondera<emph.end type="italics"/></cell><cell><emph type="italics"/>Dia­<lb/>metri<emph.end type="italics"/></cell><cell><emph type="italics"/>Tempora ca­<lb/>dendi ab al­<lb/>titudine pe­<lb/>dum<emph.end type="italics"/>220.</cell><cell><emph type="italics"/>Spatia de&longs;criben­<lb/>da per Theoriam.<emph.end type="italics"/></cell><cell><emph type="italics"/>Exce&longs;&longs;us<emph.end type="italics"/></cell></row><row><cell>510 <emph type="italics"/>gran.<emph.end type="italics"/></cell><cell>5,1 <emph type="italics"/>dig.<emph.end type="italics"/></cell><cell>8″</cell><cell>12′</cell><cell>226 <emph type="italics"/>ped.<emph.end type="italics"/></cell><cell>11 <emph type="italics"/>dig.<emph.end type="italics"/></cell><cell>6 <emph type="italics"/>ped.<emph.end type="italics"/></cell><cell>11 <emph type="italics"/>dig.<emph.end type="italics"/></cell></row><row><cell>642</cell><cell>5,2</cell><cell>7</cell><cell>42</cell><cell>230</cell><cell>9</cell><cell>10</cell><cell>9</cell></row><row><cell>599</cell><cell>5,1</cell><cell>7</cell><cell>42</cell><cell>227</cell><cell>10</cell><cell>7</cell><cell>10</cell></row><row><cell>515</cell><cell>5</cell><cell>7</cell><cell>57</cell><cell>224</cell><cell>5</cell><cell>4</cell><cell>5</cell></row><row><cell>483</cell><cell>5</cell><cell>8</cell><cell>12</cell><cell>225</cell><cell>5</cell><cell>5</cell><cell>5</cell></row><row><cell>641</cell><cell>5,2</cell><cell>7</cell><cell>42</cell><cell>230</cell><cell>7</cell><cell>10</cell><cell>7</cell></row></table> <p type="main"> <s>Globorum igitur tam in Aere quam in Aqua motorum re&longs;i­<lb/>&longs;tentia prope omnis per Theoriam no&longs;tram recte exhibetur, ac <lb/>den&longs;itati fluidorum, paribus globorum velocitatibus ac magnitudi­<lb/>nibus, proportionalis e&longs;t. </s></p><pb xlink:href="039/01/355.jpg" pagenum="327"/> <p type="main"> <s>In Scholio quod Sectioni &longs;extæ &longs;ubjunctum e&longs;t, o&longs;tendimus per </s></p> <p type="main"> <s><arrow.to.target n="note335"/>experimenta pendulorum quod globorum æqualium & æquivelo­<lb/>cium in Aere, Aqua, & Argento vivo motorum re&longs;i&longs;tentiæ &longs;unt ut <lb/>fluidorum den&longs;itates. </s> <s>Idem hic o&longs;tendimus magis accurate per <lb/>experimenta corporum cadentium in Aere & Aqua. </s> <s>Nam pendula <lb/>&longs;ingulis o&longs;cillationibus motum cient in fluido motui penduli re­<lb/>deuntis &longs;emper contrarium, & re&longs;i&longs;tentia ab hoc motu oriunda, ut <lb/>& re&longs;i&longs;tentia fili quo pendulum &longs;u&longs;pendebatur, totam Penduli re­<lb/>&longs;i&longs;tentiam majorem reddiderunt quam re&longs;i&longs;tentia quæ per experi­<lb/>menta corporum cadentium prodiit. </s> <s>Etenim per experimenta <lb/>pendulorum in Scholio illo expo&longs;ita, globus eju&longs;dem den&longs;itatis <lb/>cum Aqua, de&longs;cribendo longitudinem &longs;emidiametri &longs;uæ in Aere, <lb/>amittere deberet motus &longs;ui partem (1/3342). At per Theoriam in hac <lb/>&longs;eptima Sectione expo&longs;itam & experimentis cadentium confirma­<lb/>tam, globus idem de&longs;cribendo longitudinem eandem, amittere de­<lb/>beret motus &longs;ui partem tantum (1/4586), po&longs;ito quod den&longs;itas Aquæ &longs;it <lb/>ad den&longs;itatem Aeris ut 860 ad 1. Re&longs;i&longs;tentiæ igitur per experi­<lb/>menta pendulorum majores prodiere (ob cau&longs;as jam de&longs;criptas) <lb/>quam per experimenta globorum cadentium, idQ.E.I. ratione 4 ad <lb/>3 circiter. </s> <s>Attamen cum pendulorum in Aere, Aqua, & Argento <lb/>vivo o&longs;cillantium re&longs;i&longs;tentiæ a cau&longs;is &longs;imilibus &longs;imiliter augeantur, <lb/>proportio re&longs;i&longs;tentiarum in his Mediis, tam per experimenta pen­<lb/>dulorum, quam per experimenta corporum cadentium, &longs;atis recte <lb/>exhibebitur. </s> <s>Et inde concludi pote&longs;t quod corporum in fluidis <lb/>quibu&longs;cunque fluidi&longs;&longs;imis motorum re&longs;i&longs;tentiæ, cæteris paribus, <lb/>&longs;unt ut den&longs;itates fluidorum. </s></p> <p type="margin"> <s><margin.target id="note335"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s>His ita &longs;tabilitis, dicere jam licet quamnam motus &longs;ui partem <lb/>globus quilibet, in fluido quocunque projectus, dato tempore amit­<lb/>tet quamproxime. </s> <s>Sit D diameter globi, & V velocitas ejus &longs;ub <lb/>initio motus, & T tempus quo globus velocitate V in vacuo de­<lb/>&longs;cribet &longs;patium quod &longs;it ad &longs;patium 2/3D ut den&longs;itas globi ad den&longs;i­<lb/>tatem fluidi: & globus in fluido illo projectus, tempore quovis <lb/>alio <emph type="italics"/>t,<emph.end type="italics"/>amittet velocitatis &longs;uæ partem (<emph type="italics"/>t<emph.end type="italics"/>V/T+<emph type="italics"/>t<emph.end type="italics"/>), manente parte (TV/T+<emph type="italics"/>t<emph.end type="italics"/>), <lb/>& de&longs;cribet &longs;patium quod &longs;it ad &longs;patium uniformi velocitate V eo­<lb/>dem tempore de&longs;criptum in vacuo, ut logarithmus numeri (T+<emph type="italics"/>t<emph.end type="italics"/>/T) <lb/>multiplicatus per numerum 2,302585093 e&longs;t ad numerum <emph type="italics"/>t<emph.end type="italics"/>/T, per <pb xlink:href="039/01/356.jpg" pagenum="328"/><arrow.to.target n="note336"/>Corol. </s> <s>7, Prop.XXXV. </s> <s>In motibus tardis re&longs;i&longs;tentia pote&longs;t e&longs;&longs;e pau­<lb/>lo minor, propterea quod figura Globi paulo aptior &longs;it ad motum <lb/>quam figura Cylindri eadem diametro de&longs;cripti. </s> <s>In motibus ve­<lb/>locibus re&longs;i&longs;tentia pote&longs;t e&longs;&longs;e paulo major, propterea quod ela&longs;ti­<lb/>citas & compre&longs;&longs;io fluidi non augeantur in duplicata ratione ve­<lb/>locitatis. </s> <s>Sed huju&longs;modi minutias hic non expendo. </s></p> <p type="margin"> <s><margin.target id="note336"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Et quamvis Aer, Aqua, Argentum vivum & &longs;imilia fluida, per <lb/>divi&longs;ionem partium in infinitum, &longs;ubtiliarentur & fierent Media in­<lb/>finite fluida; tamen globis projectis haud minus re&longs;i&longs;terent. </s> <s>Nam <lb/>re&longs;i&longs;tentia, de qua agitur in Propo&longs;itionibus præcedentibus, oritur <lb/>ab inertia materiæ; & inertia materiæ corporibus e&longs;&longs;entialis e&longs;t & <lb/>quantitati materiæ &longs;emper proportionalis. </s> <s>Per divi&longs;ionem partium <lb/>fluidi, re&longs;i&longs;tentia quæ oritur a tenacitate & frictione partium, di­<lb/>minui quidem pote&longs;t: &longs;ed quantitas materiæ per divi&longs;ionem par­<lb/>tium ejus non diminuitur; & manente quantitate materiæ, manet <lb/>ejus vis inertiæ cui re&longs;i&longs;tentia, de qua hic agitur, &longs;emper proportio­<lb/>nalis e&longs;t. </s> <s>Ut hæc re&longs;i&longs;tentia diminuatur, diminui debet quantitas <lb/>materiæ in &longs;patiis per quæ corpora moventur. </s> <s>Et propterea &longs;pa­<lb/>tia Cœle&longs;tia, per quæ globi Planetarum & Cometarum in omnes <lb/>partes liberrime & ab&longs;que omni motus diminutione &longs;en&longs;ibili per­<lb/>petuo moventur, fluido omni corporeo de&longs;tituuntur, &longs;i forte vapo­<lb/>res longe tenui&longs;&longs;imos & trajectos lucis radios excipias. </s></p> <p type="main"> <s>Projectilia utique motum cient in fluidis progrediendo, & hic <lb/>motus oritur ab exce&longs;&longs;u pre&longs;&longs;ionis fluidi ad projectilis partes anti­<lb/>cas &longs;upra pre&longs;&longs;ionem ad ejus partes po&longs;ticas, & non minor e&longs;&longs;e po­<lb/>te&longs;t in Mediis infinite fluidis quam in Aere, Aqua, & Argento vivo <lb/>pro den&longs;itate materiæ in &longs;ingulis. </s> <s>Hic autem pre&longs;&longs;ionis exce&longs;&longs;us, <lb/>pro quantitate &longs;ua, non tantum motum ciet in fluido, &longs;ed etiam agit <lb/>in projectile ad motum ejus retardandum: & propterea re&longs;i­<lb/>&longs;tentia in omni fluido, e&longs;t ut motus in fluido a projectili excita­<lb/>tus, nec minor e&longs;&longs;e pote&longs;t in Æthere &longs;ubtili&longs;&longs;imo pro den&longs;itate <lb/>Ætheris, quam in Aere, Aqua, & Argento vivo pro den&longs;itatibus <lb/>horum fluidorum. <pb xlink:href="039/01/357.jpg" pagenum="329"/><arrow.to.target n="note337"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note337"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>SECTIO VIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De Motu per Fluida propagato.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLI. THEOREMA XXXII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Pre&longs;&longs;io non propagatur per Fluidum &longs;ecundum lineas rectas, ni&longs;i <lb/>ubi particulæ Fluidi in directum jacent.<emph.end type="italics"/></s></p> <p type="main"> <s>Si jaceant particulæ <emph type="italics"/>a, b, c, d, e<emph.end type="italics"/>in linea recta, pote&longs;t quidem <lb/>pre&longs;&longs;io directe propagari ab <emph type="italics"/>a<emph.end type="italics"/>ad <emph type="italics"/>e<emph.end type="italics"/>; at <lb/><figure id="id.039.01.357.1.jpg" xlink:href="039/01/357/1.jpg"/><lb/>particula <emph type="italics"/>e<emph.end type="italics"/>urgebit particulas oblique po­<lb/>&longs;itas <emph type="italics"/>f<emph.end type="italics"/>& <emph type="italics"/>g<emph.end type="italics"/>oblique, & particulæ illæ <emph type="italics"/>f<emph.end type="italics"/>& <emph type="italics"/>g<emph.end type="italics"/><lb/>non &longs;u&longs;tinebunt pre&longs;&longs;ionem illatam, ni&longs;i <lb/>fulciantur a particulis ulterioribus <emph type="italics"/>h<emph.end type="italics"/>& <emph type="italics"/>k<emph.end type="italics"/>; <lb/>quatenus autem fulciuntur, premunt par­<lb/>ticulas fulcientes; & hæ non &longs;u&longs;tinebunt <lb/>pre&longs;&longs;ionem ni&longs;i fulciantur ab ulterioribus <lb/><emph type="italics"/>l<emph.end type="italics"/>& <emph type="italics"/>m<emph.end type="italics"/>ea&longs;que premant, & &longs;ic deinceps in infinitum. </s> <s>Pre&longs;&longs;io igi­<lb/>tur, quam primum propagatur ad particulas quæ non in directum <lb/>jacent, divaricare incipiet & oblique propagabitur in infinitum; <lb/>& po&longs;tquam incipit oblique propagari, &longs;i inciderit in particulas <lb/>ulteriores, quæ non in directum jacent, iterum divaricabit; id­<lb/>que toties, quoties in particulas non accurate in directum ja­<lb/>centes inciderit. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Si pre&longs;&longs;ionis, a dato puncto per Fluidum propagatæ, pars <lb/>aliqua ob&longs;taculo intercipiatur; pars reliqua, quæ non intercipitur, <lb/>divaricabit in &longs;patia pone ob&longs;taculum. </s> <s>Id quod &longs;ic etiam de­<lb/>mon&longs;trari pote&longs;t. </s> <s>A puncto <emph type="italics"/>A<emph.end type="italics"/>propagetur pre&longs;&longs;io quaquaver­<lb/>&longs;um, idque &longs;i fieri pote&longs;t &longs;ecundum lineas rectas, & ob&longs;taculo <lb/><emph type="italics"/>NBCK<emph.end type="italics"/>perforato in <emph type="italics"/>BC,<emph.end type="italics"/>intercipiatur ea omnis, præter par­<lb/>tem Coniformem <emph type="italics"/>APQ,<emph.end type="italics"/>quæ per foramen circulare <emph type="italics"/>BC<emph.end type="italics"/>tran&longs;it. </s> <s><lb/>Planis tran&longs;ver&longs;is <emph type="italics"/>de, fg, hi<emph.end type="italics"/>di&longs;tinguatur conus <emph type="italics"/>APQ<emph.end type="italics"/>in fru&longs;ta; <lb/>& interea dum conus <emph type="italics"/>ABC,<emph.end type="italics"/>pre&longs;&longs;ionem propagando, urget fru-<pb xlink:href="039/01/358.jpg" pagenum="330"/><arrow.to.target n="note338"/>&longs;tum conicum ulterius <emph type="italics"/>degf<emph.end type="italics"/>in &longs;uperficie <emph type="italics"/>de,<emph.end type="italics"/>& hoc fru&longs;tum <lb/>urget fru&longs;tum proximum <emph type="italics"/>fgih<emph.end type="italics"/>in &longs;uperficie <emph type="italics"/>fg,<emph.end type="italics"/>& fru&longs;tum illud <lb/>urget fru&longs;tum tertium, & &longs;ic deinceps in infinitum; manife&longs;tum <lb/>e&longs;t (per motus Legem tertiam) quod fru&longs;tum primum <emph type="italics"/>defg,<emph.end type="italics"/>re­<lb/>actione fru&longs;ti &longs;ecundi <emph type="italics"/>fghi,<emph.end type="italics"/>tantum urgebitur & premetur in &longs;u­<lb/>perficie <emph type="italics"/>fg,<emph.end type="italics"/>quantum urget & premit fru&longs;tum illud &longs;ecundum. </s> <s><lb/>Fru&longs;tum igitur <emph type="italics"/>degf<emph.end type="italics"/>inter conum <emph type="italics"/>Ade<emph.end type="italics"/>& fru&longs;tum <emph type="italics"/>fhig<emph.end type="italics"/>com­<lb/>primitur utrinque, & propterea (per Corol. </s> <s>6. Prop. </s> <s>XIX.) figu­<lb/>ram &longs;uam &longs;ervare nequit, ni&longs;i vi eadem comprimatur undique. <lb/><figure id="id.039.01.358.1.jpg" xlink:href="039/01/358/1.jpg"/><lb/>Eodem igitur impetu quo premitur in &longs;uperficiebus <emph type="italics"/>de, fg,<emph.end type="italics"/>cona­<lb/>bitur cedere ad latera <emph type="italics"/>df, eg<emph.end type="italics"/>; ibique (cum rigidum non &longs;it, &longs;ed <lb/>omnimodo Fluidum) excurret ac dilatabitur, ni&longs;i Fluidum am­<lb/>biens ad&longs;it, quo conatus i&longs;te cohibeatur. </s> <s>Proinde conatu excur­<lb/>rendi, premet tam Fluidum ambiens ad latera <emph type="italics"/>df, eg<emph.end type="italics"/>quam fru&longs;tum <lb/><emph type="italics"/>fghi<emph.end type="italics"/>eodem impetu; & propterea pre&longs;&longs;io non minus propagabi­<lb/>tur a lateribus <emph type="italics"/>df, eg<emph.end type="italics"/>in &longs;patia <emph type="italics"/>NO, KL<emph.end type="italics"/>hinc inde, quam pro­<lb/>pagatur a &longs;uperficie <emph type="italics"/>fg<emph.end type="italics"/>ver&longs;us <emph type="italics"/>PQ. Q.E.D.<emph.end type="italics"/><pb xlink:href="039/01/359.jpg" pagenum="331"/><arrow.to.target n="note339"/></s></p> <p type="margin"> <s><margin.target id="note338"/>DE MOTU <lb/>CORPORUM.</s></p> <p type="margin"> <s><margin.target id="note339"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLII. THEOREMA XXXIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus omnis per Fluidum propagatus divergit a recto tramite <lb/>in &longs;patia immota.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Propagetur motus a puncto <emph type="italics"/>A<emph.end type="italics"/>per foramen <emph type="italics"/>BC,<emph.end type="italics"/>per­<lb/>gatque (&longs;i fieri pote&longs;t) in &longs;patio conico <emph type="italics"/>BCQP,<emph.end type="italics"/>&longs;ecundum li­<lb/>neas rectas divergentes a puncto <emph type="italics"/>C.<emph.end type="italics"/>Et ponamus primo quod <lb/>motus i&longs;te &longs;it undarum in &longs;uperficie &longs;tagnantis aquæ. </s> <s>Sintque <lb/><emph type="italics"/>de, fg, hi, kl,<emph.end type="italics"/>&c. </s> <s>undarum &longs;ingularum partes alti&longs;&longs;imæ, valli­<lb/>bus totidem intermediis ab invicem di&longs;tinctæ. </s> <s>Igitur quoniam <lb/>aqua in undarum jugis altior e&longs;t quam in Fluidi partibus immo­<lb/>tis <emph type="italics"/>LK, NO,<emph.end type="italics"/>defluet eadem de jugorum terminis <emph type="italics"/>e, g, i, l,<emph.end type="italics"/>&c. <lb/><emph type="italics"/>d, f, h, k,<emph.end type="italics"/>&c. </s> <s>hinc inde, ver&longs;us <emph type="italics"/>KL<emph.end type="italics"/>& <emph type="italics"/>NO<emph.end type="italics"/>: & quoniam in un­<lb/>darum vallibus depre&longs;&longs;ior e&longs;t quam in Fluidi partibus immotis <lb/><emph type="italics"/>KL, NO<emph.end type="italics"/>; defluet eadem de partibus illis immotis in undarum <lb/>valles. </s> <s>Defluxu priore undarum juga, po&longs;teriore valles hinc <lb/>inde dilatantur & propagantur ver&longs;us <emph type="italics"/>KL<emph.end type="italics"/>& <emph type="italics"/>NO.<emph.end type="italics"/>Et quo­<lb/>niam motus undarum ab <emph type="italics"/>A<emph.end type="italics"/>ver&longs;us <emph type="italics"/>PQ<emph.end type="italics"/>fit per continuum de­<lb/>fluxum jugorum in valles proximos, adeoque celerior non e&longs;t <lb/>quam pro celeritate de&longs;cen&longs;us; & de&longs;cen&longs;us aquæ, hinc inde, ver­<lb/>&longs;us <emph type="italics"/>KL<emph.end type="italics"/>& <emph type="italics"/>NO<emph.end type="italics"/>eadem velocitate peragi debet; propagabitur <lb/>dilatatio undarum, hinc inde, ver&longs;us <emph type="italics"/>KL<emph.end type="italics"/>& <emph type="italics"/>NO,<emph.end type="italics"/>eadem velo­<lb/>citate qua undæ ip&longs;æ ab <emph type="italics"/>A<emph.end type="italics"/>ver&longs;us <emph type="italics"/>PQ<emph.end type="italics"/>recta progrediuntur. </s> <s><lb/>Proindeque &longs;patium totum hinc inde, ver&longs;us <emph type="italics"/>KL<emph.end type="italics"/>& <emph type="italics"/>NO,<emph.end type="italics"/>ab <lb/>undis dilatatis <emph type="italics"/>rfgr, shis, tklt, vmnv,<emph.end type="italics"/>&c. </s> <s>occupabitur. <lb/><emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/>Hæc ita &longs;e habere quilibet in aqua &longs;tagnante expe­<lb/>riri pote&longs;t. </s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Ponamus jam quod <emph type="italics"/>de, fg, hi, kl, mn<emph.end type="italics"/>de&longs;ignent pul­<lb/>&longs;us a puncto <emph type="italics"/>A,<emph.end type="italics"/>per Medium Ela&longs;ticum, &longs;ucce&longs;&longs;ive propagatos. </s> <s><lb/>Pul&longs;us propagari concipe per &longs;ucce&longs;&longs;ivas conden&longs;ationes & rare­<lb/>factiones Medii, &longs;ic ut pul&longs;us cuju&longs;que pars den&longs;i&longs;&longs;ima &longs;phæricam <lb/>occupet &longs;uperficiem circa centrum <emph type="italics"/>A<emph.end type="italics"/>de&longs;criptam, & inter pul&longs;us <lb/>&longs;ucce&longs;&longs;ivos æqualia intercedant intervalla. </s> <s>De&longs;ignent autem lineæ <lb/><emph type="italics"/>de, fg, hi, kl,<emph.end type="italics"/>&c. </s> <s>den&longs;i&longs;&longs;imas pul&longs;uum partes, per foramen <emph type="italics"/>BC<emph.end type="italics"/><lb/>propagatas. </s> <s>Et quoniam Medium ibi den&longs;ius e&longs;t quam in &longs;patiis <lb/>hinc inde ver&longs;us <emph type="italics"/>KL<emph.end type="italics"/>& <emph type="italics"/>NO,<emph.end type="italics"/>dilatabit &longs;e&longs;e tam ver&longs;us &longs;patia illa <lb/><emph type="italics"/>KL, NO<emph.end type="italics"/>utrinque &longs;ita, quam ver&longs;us pul&longs;uum rariora intervalla; <pb xlink:href="039/01/360.jpg" pagenum="332"/><arrow.to.target n="note340"/>eoque pacto rarius &longs;emper evadens e regione intervallorum ac <lb/>den&longs;ius e regione pul&longs;uum, participabit eorundem motum. </s> <s>Et <lb/>quoniam pul&longs;uum progre&longs;&longs;ivus motus oritur a perpetua relaxa­<lb/>tione partium den&longs;iorum ver&longs;us antecedentia intervalla rariora; <lb/>& pul&longs;us eadem fere celeritate &longs;e&longs;e in Medii partes quie&longs;centes <lb/><emph type="italics"/>KL, NO<emph.end type="italics"/>hinc inde relaxare debent; pul&longs;us illi eadem fere cele­<lb/>ritate &longs;e&longs;e dilatabunt undiQ.E.I. &longs;patia immota <emph type="italics"/>KL, NO,<emph.end type="italics"/>qua <lb/>propagantur directe a centro <emph type="italics"/>A<emph.end type="italics"/>; adeoque &longs;patium totum <emph type="italics"/>KLON<emph.end type="italics"/><lb/>occupabunt. <emph type="italics"/>Q.E.D.<emph.end type="italics"/>Hoc experimur in Sonis, qui vel monte <lb/>interpo&longs;ito audiuntur, vel in cubiculum per fene&longs;tram admi&longs;&longs;i &longs;e&longs;e <lb/>in omnes cubiculi partes dilatant, inque angulis omnibus audiun­<lb/>tur, non tam reflexi a parietibus oppo&longs;itis, quam a fene&longs;tra directe <lb/>propagati, quantum ex &longs;en&longs;u judicare licet. </s></p> <p type="margin"> <s><margin.target id="note340"/>DE MOTU <lb/>CORPORUM</s><figure id="id.039.01.360.1.jpg" xlink:href="039/01/360/1.jpg"/></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>3. Ponamus denique quod motus cuju&longs;cunque generis <lb/>propagetur ab <emph type="italics"/>A<emph.end type="italics"/>per foramen <emph type="italics"/>BC<emph.end type="italics"/>: & quoniam propagatio i&longs;ta <lb/>non fit, ni&longs;i quatenus partes Medii centro <emph type="italics"/>A<emph.end type="italics"/>propiores urgent <lb/>commoventque partes ulteriores; & partes quæ urgentur fluidæ <lb/>&longs;unt, ideoque recedunt quaquaver&longs;um in regiones ubi minus pre-<pb xlink:href="039/01/361.jpg" pagenum="333"/>muntur: recedent eædem ver&longs;us Medii partes omnes quie&longs;centes, <lb/><arrow.to.target n="note341"/>tam laterales <emph type="italics"/>KL<emph.end type="italics"/>& <emph type="italics"/>NO,<emph.end type="italics"/>quam anteriores <emph type="italics"/>PQ,<emph.end type="italics"/>eoque pacto <lb/>motus omnis, quam primum per foramen <emph type="italics"/>BC<emph.end type="italics"/>tran&longs;iit, dilatari in­<lb/>cipiet & abinde, tanquam a principio & centro, in partes omnes <lb/>directe propagari. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note341"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLIII. THEOREMA XXXIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Corpus omne tremulum in Medio Ela&longs;tico propagabit motum pul­<lb/>&longs;uum undiQ.E.I. directum; in Medio vero non Ela&longs;tico motum <lb/>circularem excitabit.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Nam partes corporis tremuli vicibus alternis eundo & <lb/>redeundo, itu &longs;uo urgebunt & propellent partes Medii &longs;ibi proxi­<lb/>mas, & urgendo compriment ea&longs;dem & conden&longs;abunt; dein re­<lb/>ditu &longs;uo &longs;inent partes compre&longs;&longs;as recedere & &longs;e&longs;e expandere. </s> <s>Igi­<lb/>tur partes Medii corpori tremulo proximæ ibunt & redibunt per <lb/>vices, ad in&longs;tar partium corporis illius tremuli: & qua ratione <lb/>partes corporis hujus agitabant ha&longs;ce Medii partes, hæ &longs;imilibus <lb/>tremoribus agitatæ agitabunt partes &longs;ibi proximas, eæque &longs;imiliter <lb/>agitatæ agitabunt ulteriores, & &longs;ic deinceps in infinitum. </s> <s>Et <lb/>quemadmodum Medii partes primæ eundo conden&longs;antur & re­<lb/>deundo relaxantur, &longs;ic partes reliquæ quoties eunt conden&longs;abun­<lb/>tur, & quoties redeunt &longs;e&longs;e expandent. </s> <s>Et propterea non omnes <lb/>ibunt & &longs;imul redibunt (&longs;ic enim determinatas ab invicem di&longs;tan­<lb/>tias &longs;ervando, non rarefierent & conden&longs;arentur per vices) &longs;ed ac­<lb/>cedendo ad invicem ubi conden&longs;antur, & recedendo ubi rarefiunt, <lb/>aliquæ earum ibunt dum aliæ redeunt; idque vicibus alternis in <lb/>infinitum. </s> <s>Partes autem euntes & eundo conden&longs;atæ, ob motum <lb/>&longs;uum progre&longs;&longs;ivum quo feriunt ob&longs;tacula, &longs;unt pul&longs;us; & propte­<lb/>rea pul&longs;us &longs;ucce&longs;&longs;ivi a corpore omni tremulo in directum propaga­<lb/>buntur; idque æqualibus circiter ab invicem di&longs;tantiis, ob æqua­<lb/>lia temporis intervalla, quibus corpus tremoribus &longs;uis &longs;ingulis <lb/>&longs;ingulos pul&longs;us excitat. </s> <s>Et quanquam corporis tremuli par­<lb/>tes eant & redeant &longs;ecundum plagam aliquam certam & determi­<lb/>natam, tamen pul&longs;us inde per Medium propagati &longs;e&longs;e dilatabunt <lb/>ad latera, per Propo&longs;itionem præcedentem; & a corpore illo tre­<lb/>mulo tanquam centro communi, &longs;ecundum &longs;uperficies propemo­<lb/>dum &longs;phæricas & concentricas, undique propagabuntur. </s> <s>Cujus <pb xlink:href="039/01/362.jpg" pagenum="334"/><arrow.to.target n="note342"/>rei exemplum aliquod habemus in Undis, quæ &longs;i digito tremulo <lb/>excitentur, non &longs;olum pergent hinc inde &longs;ecundum plagam motus <lb/>digiti, &longs;ed, in modum circulorum concentrieorum, digitum &longs;tatim <lb/>cingent & undique propagabuntur. </s> <s>Nam gravitas Undarum &longs;up­<lb/>plet locum vis Ela&longs;ticæ. </s></p> <p type="margin"> <s><margin.target id="note342"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. Quod &longs;i Medium non &longs;it Ela&longs;ticum: quoniam ejus partes a <lb/>corporis tremuli partibus vibratis pre&longs;&longs;æ conden&longs;ari nequeunt, pro­<lb/>pagabitur motus in in&longs;tanti ad partes ubi Medium facillime ce­<lb/>dit, hoc e&longs;t, ad partes quas corpus tremulum alioqui vacuas a <lb/>tergo relinqueret. </s> <s>Idem e&longs;t ca&longs;us cum ca&longs;u corporis in Medio <lb/>quocunque projecti. </s> <s>Medium cedendo projectilibus, non rece­<lb/>dit in infinitum; &longs;ed in circulum eundo, pergit ad &longs;patia quæ <lb/>corpus relinquit a tergo. </s> <s>Igitur quoties corpus tremulum per­<lb/>git in partem quamcunque, Medium cedendo perget per circu­<lb/>lum ad partes quas corpus relinquit; & quoties corpus regredi­<lb/>tur ad locum priorem, Medium inde repelletur & ad locum &longs;uum <lb/>priorem redibit. </s> <s>Et quamvis corpus tremulum non &longs;it firmum, <lb/>&longs;ed modis omnibus flexile, &longs;i tamen magnitudine datum maneat, <lb/>quoniam tremoribus &longs;uis nequit Medium ubivis urgere, quin alibi <lb/>eidem &longs;imul cedat; efficiet ut Medium, recedendo a partibus <lb/>ubi premitur, pergat &longs;emper in orbem ad partes quæ eidem ce­<lb/>dunt <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hallucinantur igitur qui credunt agitationem partium <lb/>Flammæ ad pre&longs;&longs;ionem, per Medium ambiens, &longs;ecundum lineas <lb/>rectas propagandam conducere. </s> <s>Debebit eju&longs;modi pre&longs;&longs;io non <lb/>ab agitatione &longs;ola partium Flammæ, &longs;ed a totius dilatatione deri­<lb/>vari. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLIV. THEOREMA XXXV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si aqua in Canalis cruribus erectis<emph.end type="italics"/>KL, MN <emph type="italics"/>vicibus alternis <lb/>a&longs;cendat & de&longs;cendat; con&longs;truatur autem Pendulum cujus <lb/>longitudo inter punctum &longs;u&longs;pen&longs;ionis & centrum o&longs;cillationis <lb/>æquetur &longs;emi&longs;&longs;i longitudinis aquæ in Canali: dico quod aqua <lb/>a&longs;cendet & de&longs;cendet ii&longs;dem temporibus quibus Pendulum <lb/>o&longs;cillatur.<emph.end type="italics"/></s></p> <p type="main"> <s>Longitudinem aquæ men&longs;uro &longs;ecundum axes canalis & crurum, <lb/>eandem &longs;ummæ horum axium æquando; & re&longs;i&longs;tentiam aquæ quæ <pb xlink:href="039/01/363.jpg" pagenum="335"/>oritur ab attritu canalis, hic non con&longs;idero. </s> <s>De&longs;ignent igitur <emph type="italics"/>AB, <lb/><arrow.to.target n="note343"/>CD<emph.end type="italics"/>mediocrem altitudinem aquæ in crure utroque; & ubi aqua <lb/>in crure <emph type="italics"/>KL<emph.end type="italics"/>a&longs;cendit ad altitudinem <emph type="italics"/>EF,<emph.end type="italics"/>de&longs;cenderit aqua in <lb/>crure <emph type="italics"/>MN<emph.end type="italics"/>ad altitudinem <emph type="italics"/>GH.<emph.end type="italics"/>Sit autem <emph type="italics"/>P<emph.end type="italics"/>corpus pendulum, <lb/><emph type="italics"/>VP<emph.end type="italics"/>filum, <emph type="italics"/>V<emph.end type="italics"/>punctum &longs;u&longs;pen&longs;ionis, <emph type="italics"/>SPQR<emph.end type="italics"/>Cyclois quam Pen­<lb/>dulum de&longs;cribat, <emph type="italics"/>P<emph.end type="italics"/>ejus punctum infimum, <emph type="italics"/>PQ<emph.end type="italics"/>arcus altitudini <lb/><emph type="italics"/>AE<emph.end type="italics"/>æqualis. </s> <s>Vis, qua motus aquæ alternis vicibus acceleratur <lb/><figure id="id.039.01.363.1.jpg" xlink:href="039/01/363/1.jpg"/><lb/>& retardatur, e&longs;t exce&longs;&longs;us ponderis aquæ in alterutro crure &longs;upra <lb/>pondus in altero, ideoque, ubi aqua in crure <emph type="italics"/>KL<emph.end type="italics"/>a&longs;cendit ad <emph type="italics"/>EF,<emph.end type="italics"/><lb/>& in crure altero de&longs;cendit ad <emph type="italics"/>GH,<emph.end type="italics"/>vis illa e&longs;t pondus duplica­<lb/>tum aquæ <emph type="italics"/>EABF,<emph.end type="italics"/>& propterea e&longs;t ad pondus aquæ totius ut <lb/><emph type="italics"/>AE<emph.end type="italics"/>&longs;eu <emph type="italics"/>PQ<emph.end type="italics"/>ad <emph type="italics"/>VP<emph.end type="italics"/>&longs;eu <emph type="italics"/>PR.<emph.end type="italics"/>Vis etiam, qua pondus <emph type="italics"/>P<emph.end type="italics"/>in <lb/>loco quovis <emph type="italics"/>Q<emph.end type="italics"/>acceleratur & retardatur in Cycloide, (per Corol. </s> <s><lb/>Prop. </s> <s>LI.) e&longs;t ad ejus pondus totum, ut ejus di&longs;tantia <emph type="italics"/>YQ<emph.end type="italics"/>a loco <lb/>infimo <emph type="italics"/>P,<emph.end type="italics"/>ad Cycloidis longitudinem <emph type="italics"/>PR.<emph.end type="italics"/>Quare aquæ & pen­<lb/>duli, æqualia &longs;patia <emph type="italics"/>AE, PQ<emph.end type="italics"/>de&longs;cribentium, vires motrices &longs;unt <lb/>ut pondera movenda; ideoque, &longs;i aqua & pendulum in princi­<lb/>pio quie&longs;cunt, vires illæ movebunt eadem æqualiter tempori­<lb/>bus æqualibus, efficientque ut motu reciproco &longs;imul eant & re­<lb/>deant. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note343"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Igitur aquæ a&longs;cendentis & de&longs;cendentis, &longs;ive motus in­<lb/>ten&longs;ior &longs;it &longs;ive remi&longs;&longs;ior, vices omnes &longs;unt I&longs;ochronæ. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si longitudo aquæ totius in canali &longs;it pedum <emph type="italics"/>Pari&longs;ien­<lb/>&longs;ium<emph.end type="italics"/>6 1/9: aqua tempore minuti unius &longs;ecundi de&longs;cendet, & tem­<lb/>pore minuti alterius &longs;ecundi a&longs;cendet; & &longs;ic deinceps vicibus al­<lb/>ternis in infinitum. </s> <s>Nam pendulum pedum (3 1/18) longitudinis, <lb/>tempore minuti unius &longs;ecundi o&longs;cillatur. <pb xlink:href="039/01/364.jpg" pagenum="336"/><arrow.to.target n="note344"/></s></p> <p type="margin"> <s><margin.target id="note344"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Aucta autem vel diminuta longitudine aquæ, auge­<lb/>tur vel diminuitur tempus reciprocationis in longitudinis ratione <lb/>&longs;ubduplicata. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLV. THEOREMA XXXVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Undarum velocitas e&longs;t in &longs;ubduplicata ratione latitudinum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Con&longs;equitur ex con&longs;tructione Propo&longs;itionis &longs;equentis. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLVI. PROBLEMA X.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire velocitatem Undarum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Con&longs;tituatur Pendulum cujus longitudo, inter punctum &longs;u&longs;pen­<lb/>&longs;ionis & centrum o&longs;cillationis, æquetur latitudini Undarum: & quo <lb/>tempore pendulum illud o&longs;cillationes &longs;ingulas peragit, eodem Un­<lb/>dæ progrediendo latitudinem &longs;uam propemodum conficient. </s></p> <p type="main"> <s>Undarum latitudinem voco men&longs;uram tran&longs;ver&longs;am, quæ vel val­<lb/>libus imis, vel &longs;ummis culminibus interjacet. </s> <s>De&longs;ignet <emph type="italics"/>ABCDEF<emph.end type="italics"/><lb/>&longs;uperficiem aquæ &longs;tagnantis, undis &longs;ucce&longs;&longs;ivis a&longs;cendentem ac de&longs;­<lb/>cendentem; &longs;intque <emph type="italics"/>A, C, E,<emph.end type="italics"/>&c. </s> <s>undarum culmina, & <emph type="italics"/>B, D, F,<emph.end type="italics"/>&c. </s> <s><lb/>valles intermedii. </s> <s>Et quoniam motus undarum fit per aquæ &longs;uc­<lb/>ce&longs;&longs;ivum a&longs;cen&longs;um & de&longs;cen&longs;um, &longs;ic ut ejus partes <emph type="italics"/>A, C, E,<emph.end type="italics"/>&c. </s> <s><lb/>quæ nunc alti&longs;&longs;imæ &longs;unt, mox fiant infimæ; & vis motrix, qua <lb/>partes alti&longs;&longs;imæ de&longs;cendunt & infimæ a&longs;cendunt, e&longs;t pondus aquæ <lb/>elevatæ; alternus ille a&longs;cen&longs;us & de&longs;cen&longs;us analogus erit motui re­<lb/>ciproco aquæ in canali, ea&longs;demque temporis leges ob&longs;ervabit: & <lb/>propterea (per Prop. </s> <s>XLIV) &longs;i di&longs;tantiæ inter undarum loca alti&longs;­<lb/>&longs;ima <emph type="italics"/>A, C, E<emph.end type="italics"/>& infima <emph type="italics"/>B, D, F<emph.end type="italics"/>æquentur duplæ penduli longitu­<lb/>dini; partes alti&longs;&longs;imæ <emph type="italics"/>A, C, E,<emph.end type="italics"/>tempore o&longs;cillationis unius evadent <lb/>infimæ, & tempore o&longs;cillationis alterius denuo a&longs;cendent. </s> <s>Igitur <lb/>inter tran&longs;itum Undarum &longs;ingularum tempus erit o&longs;cillationum <lb/>duarum; hoc e&longs;t, Unda de&longs;cribet latitudinem &longs;uam, quo tempore <lb/>pendulum illud bis o&longs;cillatur; &longs;ed eodem tempore pendulum, cu­<lb/>jus longitudo quadrupla e&longs;t, adeoque æquat undarum latitudinem, <lb/>o&longs;cillabitur &longs;emel. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Igitur Undæ, quæ pedes <emph type="italics"/>Pari&longs;ien&longs;es<emph.end type="italics"/>(3 1/18) latæ &longs;unt, <lb/>tempore minuti unius &longs;ecundi progrediendo latitudinem &longs;uam con­<lb/>ficient; adeoque tempore minuti unius primi percurrent pedes <lb/>183 1/3, & horæ &longs;patio pedes 11000 quamproxime. </s><pb xlink:href="039/01/365.jpg" pagenum="337"/><figure id="id.039.01.365.1.jpg" xlink:href="039/01/365/1.jpg"/></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et undarum majorum vel minorum ve­</s></p> <p type="main"> <s><arrow.to.target n="note345"/>locitas augebitur vel diminuetur in &longs;ubduplicata <lb/>ratione latitudinis. </s></p> <p type="margin"> <s><margin.target id="note345"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s>Hæc ita &longs;e habent ex Hypothe&longs;i quod partes <lb/>aquæ recta a&longs;cendunt vel recta de&longs;cendunt; &longs;ed <lb/>a&longs;cen&longs;us & de&longs;cen&longs;us ille verius fit per circulum, <lb/>ideoque tempus hac Propo&longs;itione non ni&longs;i quam­<lb/>proxime definitum e&longs;&longs;e affirmo. </s></p> <p type="main"> <s><emph type="center"/>PROP. XLVII. THEOR. XXXVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Pul&longs;ibus per Fluidum propagatis, &longs;ingulæ Fluidi <lb/>particulæ, motu reciproco brevi&longs;&longs;imo euntes & <lb/>redeuntes, accelerantur &longs;emper & retardantur <lb/>pro lego o&longs;cillantis Penduli.<emph.end type="italics"/></s><figure id="id.039.01.365.2.jpg" xlink:href="039/01/365/2.jpg"/></p> <p type="main"> <s>De&longs;ignent <emph type="italics"/>AB, BC, CD,<emph.end type="italics"/><lb/>&c. </s> <s>pul&longs;uum &longs;ucce&longs;&longs;ivorum <lb/>æquales di&longs;tantias; <emph type="italics"/>ABC<emph.end type="italics"/><lb/>plagam motus pul&longs;uum ab <lb/><emph type="italics"/>A<emph.end type="italics"/>ver&longs;us <emph type="italics"/>B<emph.end type="italics"/>propagati; <emph type="italics"/>E, <lb/>F, G<emph.end type="italics"/>puncta tria Phy&longs;ica Me­<lb/>dii quie&longs;centis, in recta <emph type="italics"/>AC<emph.end type="italics"/><lb/>ad æquales ab invicem di­<lb/>&longs;tantias &longs;ita; <emph type="italics"/>Ee, Ff, Gg,<emph.end type="italics"/><lb/>&longs;patia æqualia perbrevia per <lb/>quæ puncta illa motu reciproco &longs;ingulis vibratio­<lb/>nibus eunt & redeunt; <foreign lang="greek">e, f, g</foreign> loca quævis inter­<lb/>media eorundem punctorum; & <emph type="italics"/>EF, FG<emph.end type="italics"/>lineolas <lb/>Phy&longs;icas &longs;eu Medii partes lineares punctis illis in­<lb/>terjectas, & &longs;ucce&longs;&longs;ive tran&longs;latas in loca <foreign lang="greek">ef, fg</foreign> & <lb/><emph type="italics"/>ef, fg.<emph.end type="italics"/>Rectæ <emph type="italics"/>Ee<emph.end type="italics"/>æqualis ducatur recta <emph type="italics"/>PS.<emph.end type="italics"/><lb/>Bi&longs;ecetur eadem in <emph type="italics"/>O,<emph.end type="italics"/>centroque <emph type="italics"/>O<emph.end type="italics"/>& intervallo <lb/><emph type="italics"/>OP<emph.end type="italics"/>de&longs;cribatur circulus <emph type="italics"/>SIPi.<emph.end type="italics"/>Per hujus cir­<lb/>cumferentiam totam cum partibus &longs;uis exponatur <lb/>tempus totum vibrationis unius cum ip&longs;ius parti­<lb/>bus proportionalibus; &longs;ic ut completo tempore <lb/>quovis <emph type="italics"/>PH<emph.end type="italics"/>vel <emph type="italics"/>PHSh,<emph.end type="italics"/>&longs;i demittatur ad <emph type="italics"/>PS<emph.end type="italics"/><lb/>perpendiculum <emph type="italics"/>HL<emph.end type="italics"/>vel <emph type="italics"/>hl,<emph.end type="italics"/>& capiatur <emph type="italics"/>E<emph.end type="italics"/><foreign lang="greek">e</foreign> æqua­<lb/>lis <emph type="italics"/>PL<emph.end type="italics"/>vel <emph type="italics"/>Pl,<emph.end type="italics"/>punctum Phy&longs;icum <emph type="italics"/>E<emph.end type="italics"/>reperiatur <pb xlink:href="039/01/366.jpg" pagenum="338"/><arrow.to.target n="note346"/>in <foreign lang="greek">e. </foreign></s> <s>Hac lege punctum quodvis <emph type="italics"/>E,<emph.end type="italics"/>eundo ab <emph type="italics"/>E<emph.end type="italics"/><lb/><figure id="id.039.01.366.1.jpg" xlink:href="039/01/366/1.jpg"/><lb/>per <foreign lang="greek">e</foreign> ad <emph type="italics"/>e,<emph.end type="italics"/>& inde redeundo per <foreign lang="greek">e</foreign> ad <emph type="italics"/>E,<emph.end type="italics"/>ii&longs;dem <lb/>accelerationis ac retardationis gradibus vibratio­<lb/>nes &longs;ingulas peraget cum o&longs;cillante Pendulo. </s> <s>Pro­<lb/>bandum e&longs;t quod &longs;ingula Medii puncta Phy&longs;ica <lb/>tali motu agitari debeant. </s> <s>Fingamus igitur Me­<lb/>dium tali motu a cau&longs;a quacunque cieri, & videa­<lb/>mus quid inde &longs;equatur. </s></p> <p type="margin"> <s><margin.target id="note346"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>In circumferentia <emph type="italics"/>PHSh<emph.end type="italics"/>capiantur æquales ar­<lb/>cus <emph type="italics"/>HI, IK<emph.end type="italics"/>vel <emph type="italics"/>hi, ik,<emph.end type="italics"/>eam habentes rationem <lb/>ad circumferentiam totam quam habent æquales <lb/>rectæ <emph type="italics"/>EF, FG<emph.end type="italics"/>ad pul&longs;uum intervallum totum <lb/><emph type="italics"/>BC.<emph.end type="italics"/>Et demi&longs;&longs;is perpendiculis <emph type="italics"/>IM, KN<emph.end type="italics"/>vel <lb/><emph type="italics"/>im, kn<emph.end type="italics"/>; quoniam puncta <emph type="italics"/>E, F, G<emph.end type="italics"/>motibus &longs;imili­<lb/>bus &longs;ucce&longs;&longs;ive agitantur, & vibrationes &longs;uas integras <lb/>ex itu & reditu compo&longs;itas interea peragunt dum <lb/><figure id="id.039.01.366.2.jpg" xlink:href="039/01/366/2.jpg"/><lb/>pul&longs;us transfertur a <emph type="italics"/>B<emph.end type="italics"/>ad <emph type="italics"/>C<emph.end type="italics"/>; <lb/>&longs;i <emph type="italics"/>PH<emph.end type="italics"/>vel <emph type="italics"/>PHSh<emph.end type="italics"/>&longs;it tem­<lb/>pus ab initio motus puncti <lb/><emph type="italics"/>E,<emph.end type="italics"/>erit <emph type="italics"/>PI<emph.end type="italics"/>vel <emph type="italics"/>PHSi<emph.end type="italics"/>tem­<lb/>pus ab initio motus puncti <lb/><emph type="italics"/>F,<emph.end type="italics"/>& <emph type="italics"/>PK<emph.end type="italics"/>vel <emph type="italics"/>PHSk<emph.end type="italics"/>tem­<lb/>pus ab initio motus puncti <lb/><emph type="italics"/>G<emph.end type="italics"/>; & propterea <emph type="italics"/>E<foreign lang="greek">e</foreign>, F<foreign lang="greek">f</foreign>, <lb/>G<emph.end type="italics"/><foreign lang="greek">g</foreign> erunt ip&longs;is <emph type="italics"/>PL, PM, <lb/>PN<emph.end type="italics"/>in itu punctorum, vel <lb/>ip&longs;is <emph type="italics"/>Pl, Pm, Pn<emph.end type="italics"/>in punctorum reditu, æqua­<lb/>les re&longs;pective. </s> <s>Unde <foreign lang="greek">eg</foreign> &longs;eu <emph type="italics"/>EG+G<foreign lang="greek">g</foreign>-E<emph.end type="italics"/><foreign lang="greek">e</foreign><lb/>in itu punctorum æqualis erit <emph type="italics"/>EG-LN,<emph.end type="italics"/>in re­<lb/>ditu autem æqualis <emph type="italics"/>EG+ln.<emph.end type="italics"/>Sed <foreign lang="greek">eg</foreign> latitudo e&longs;t <lb/>&longs;eu expan&longs;io partis Medii <emph type="italics"/>EG<emph.end type="italics"/>in loco <foreign lang="greek">eg</foreign>; & <lb/>propterea expan&longs;io partis illius in itu, e&longs;t ad ejus <lb/>expan&longs;ionem mediocrem, ut <emph type="italics"/>EG-LN<emph.end type="italics"/>ad <emph type="italics"/>EG<emph.end type="italics"/>; <lb/>in reditu autem ut <emph type="italics"/>EG+ln<emph.end type="italics"/>&longs;eu <emph type="italics"/>EG+LN<emph.end type="italics"/>ad <lb/><emph type="italics"/>EG.<emph.end type="italics"/>Quare cum &longs;it <emph type="italics"/>LN<emph.end type="italics"/>ad <emph type="italics"/>KH<emph.end type="italics"/>ut <emph type="italics"/>IM<emph.end type="italics"/>ad <lb/>radium <emph type="italics"/>OP,<emph.end type="italics"/>& <emph type="italics"/>KH<emph.end type="italics"/>ad <emph type="italics"/>EG<emph.end type="italics"/>ut circumferentia <lb/><emph type="italics"/>PHShP<emph.end type="italics"/>ad <emph type="italics"/>BC,<emph.end type="italics"/>id e&longs;t (&longs;i ponatur V pro ra­<lb/>dio circuli circumferentiam habentis æqualem in­<lb/>tervallo pul&longs;uum <emph type="italics"/>BC<emph.end type="italics"/>) ut <emph type="italics"/>OP<emph.end type="italics"/>ad V; & ex æ­<lb/>quo <emph type="italics"/>LN<emph.end type="italics"/>ad <emph type="italics"/>EG,<emph.end type="italics"/>ut <emph type="italics"/>IM<emph.end type="italics"/>ad V: erit expan&longs;io <lb/>partis <emph type="italics"/>EG<emph.end type="italics"/>punctive Phy&longs;ici <emph type="italics"/>F<emph.end type="italics"/>in loco <foreign lang="greek">eg</foreign>, ad ex-<pb xlink:href="039/01/367.jpg" pagenum="339"/>pan&longs;ionem mediocrem quam pars illa habet in loco &longs;uo primo <lb/><arrow.to.target n="note347"/><emph type="italics"/>EG,<emph.end type="italics"/>ut V-<emph type="italics"/>IM<emph.end type="italics"/>ad V in itu, utque V+<emph type="italics"/>im<emph.end type="italics"/>ad V in reditu. </s> <s>Unde <lb/>vis ela&longs;tica puncti <emph type="italics"/>F<emph.end type="italics"/>in loco <foreign lang="greek">eg</foreign>, e&longs;t ad vim ejus ela&longs;ticam medio­<lb/>crem in loco <emph type="italics"/>EG,<emph.end type="italics"/>ut (I/V-<emph type="italics"/>IM<emph.end type="italics"/>) ad I/V in itu, in reditu vero ut <lb/>(I/V+<emph type="italics"/>im<emph.end type="italics"/>) ad I/V. </s> <s>Et eodem argumento vires ela&longs;ticæ punctorum <lb/>Phy&longs;ieorum <emph type="italics"/>E<emph.end type="italics"/>& <emph type="italics"/>G<emph.end type="italics"/>in itu, &longs;unt ut (I/V-<emph type="italics"/>HL<emph.end type="italics"/>) & (I/V-<emph type="italics"/>KN<emph.end type="italics"/>) <lb/>ad I/V; & virium differentia ad Medii vim ela&longs;ticam mediocrem, <lb/>ut (<emph type="italics"/>HL-KN<emph.end type="italics"/>/VV-VX<emph type="italics"/>HL<emph.end type="italics"/>-VX<emph type="italics"/>KN+HLXKN<emph.end type="italics"/>) ad I/V. </s> <s>Hoc e&longs;t, ut <lb/>(<emph type="italics"/>HL-KN<emph.end type="italics"/>/VV) ad I/V, &longs;ive ut <emph type="italics"/>HL-KN<emph.end type="italics"/>ad V, &longs;i modo (ob angu­<lb/>&longs;tos limites vibrationum) &longs;upponamus <emph type="italics"/>HL<emph.end type="italics"/>& <emph type="italics"/>KN<emph.end type="italics"/>indefinite <lb/>minores e&longs;&longs;e quantitate V. </s> <s>Quare cum quantitas V detur, diffe­<lb/>rentia virium e&longs;t ut <emph type="italics"/>HL-KN,<emph.end type="italics"/>hoc e&longs;t (ob proportionales <lb/><emph type="italics"/>HL-KN<emph.end type="italics"/>ad <emph type="italics"/>HK,<emph.end type="italics"/>& <emph type="italics"/>OM<emph.end type="italics"/>ad <emph type="italics"/>OI<emph.end type="italics"/>vel <emph type="italics"/>OP,<emph.end type="italics"/>data&longs;que <emph type="italics"/>HK<emph.end type="italics"/>& <lb/><emph type="italics"/>OP<emph.end type="italics"/>) ut <emph type="italics"/>OM<emph.end type="italics"/>; id e&longs;t, &longs;i <emph type="italics"/>Ff<emph.end type="italics"/>bi&longs;ecetur in <foreign lang="greek">*w</foreign>, ut <foreign lang="greek">*wf. </foreign></s> <s>Et eodem <lb/>argumento differentia virium ela&longs;ticarum punctorum Phy&longs;ieorum <lb/><foreign lang="greek">e</foreign> & <foreign lang="greek">g</foreign>, in reditu lineolæ Phy&longs;icæ <foreign lang="greek">eg</foreign> e&longs;t ut <foreign lang="greek">*wf. </foreign></s> <s>Sed differentia <lb/>illa (id e&longs;t, exce&longs;&longs;us vis ela&longs;ticæ puncti <foreign lang="greek">e</foreign> &longs;upra vim ela&longs;ticam pun­<lb/>cti <foreign lang="greek">g</foreign>,) e&longs;t vis qua interjecta Medii lineola Phy&longs;ica <foreign lang="greek">eg</foreign> acceleratur; <lb/>& propterea vis acceleratrix lineolæ Phy&longs;icæ <foreign lang="greek">eg</foreign>, e&longs;t ut ip&longs;ius di­<lb/>&longs;tantia a medio vibrationis loco <foreign lang="greek">*w. </foreign></s> <s>Proinde tempus (per Prop. </s> <s><lb/>XXXVIII. Lib. </s> <s>1.) recte exponitur per arcum <emph type="italics"/>PI<emph.end type="italics"/>; & Medii pars <lb/>linearis <foreign lang="greek">eg</foreign> lege præ&longs;cripta movetur, id e&longs;t, lege o&longs;cillantis Pen­<lb/>duli: e&longs;tque par ratio partium omnium linearium ex quibus Me­<lb/>dium totum componitur. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note347"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc patet quod numerus pul&longs;uum propagatorum idem <lb/>&longs;it cum numero vibrationum corporis tremuli, neque multiplica­<lb/>tur in eorum progre&longs;&longs;u. </s> <s>Nam lineola Phy&longs;ica <foreign lang="greek">eg</foreign>, quamprimum <lb/>ad locum &longs;uum primum redierit, quie&longs;cet; neQ.E.D.inceps move­<lb/>bitur, ni&longs;i vel ab impetu corporis tremuli, vel ab impetu pul&longs;uum <lb/>qui a corpore tremulo propagantur, motu novo cieatur. </s> <s>Quie­<lb/>&longs;cet igitur quamprimum pul&longs;us a corpore tremulo propagari <lb/>de&longs;inunt. <pb xlink:href="039/01/368.jpg" pagenum="340"/><arrow.to.target n="note348"/></s></p> <p type="margin"> <s><margin.target id="note348"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLVIII. THEOREMA XXXVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Pul&longs;uum in Fluido Ela&longs;tico propagatorum velocitates, &longs;unt in ra­<lb/>tione compo&longs;ita ex &longs;ubduplicata ratione vis Ela&longs;ticæ directe & <lb/>&longs;ubduplicata ratione den&longs;itatis inver&longs;e; &longs;i modo Fluidi vis <lb/>Ela&longs;tica eju&longs;dem conden&longs;ationi proportionalis e&longs;&longs;e &longs;upponatur.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>1. Si Media &longs;int homogenea, & pul&longs;uum di&longs;tantiæ in his <lb/>Mediis æquentur inter &longs;e, &longs;ed motus in uno Medio inten&longs;ior &longs;it: <lb/>contractiones & dilatationes partium analogarum erunt ut iidem <lb/>motus. </s> <s>Accurata quidem non e&longs;t hæc proportio. </s> <s>Verum tamen <lb/>ni&longs;i contractiones & dilatationes &longs;int valde inten&longs;æ, non errabit <lb/>&longs;en&longs;ibiliter, ideoque pro Phy&longs;ice accurata haberi pote&longs;t. </s> <s>Sunt <lb/>autem vires Ela&longs;ticæ motrices ut contractiones & dilatationes; & <lb/>velocitates partium æqualium &longs;imul genitæ &longs;unt ut vires. </s> <s>Ideoque <lb/>æquales & corre&longs;pondentes pul&longs;uum corre&longs;pondentium partes, <lb/>itus & reditus &longs;uos per &longs;patia contractionibus & dilatationibus <lb/>proportionalia, cum velocitatibus quæ &longs;unt ut &longs;patia, &longs;imul pera­<lb/>gent: & propterea pul&longs;us, qui tempore itus & reditus unius lati­<lb/>tudinem &longs;uam progrediendo conficiunt, & in loca pul&longs;uum pro­<lb/>xime præcedentium &longs;emper &longs;uccedunt, ob æqualitatem di&longs;tantia­<lb/>rum, æquali cum velocitate in Medio utroque progredientur. </s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>2. Sin pul&longs;uum di&longs;tantiæ &longs;eu longitudines &longs;int majores in <lb/>uno Medio quam in altero; ponamus quod partes corre&longs;ponden­<lb/>tes &longs;patia latitudinibus pul&longs;uum proportionalia &longs;ingulis vicibus <lb/>eundo & redeundo de&longs;cribant: & æquales erunt earum contra­<lb/>ctiones & dilatationes. </s> <s>Ideoque &longs;i Media &longs;int homogenea, æqua­<lb/>les erunt etiam vires illæ Ela&longs;ticæ motrices quibus reciproco motu <lb/>agitantur. </s> <s>Materia autem his viribus movenda, e&longs;t ut pul&longs;uum <lb/>latitudo; & in eadem ratione e&longs;t &longs;patium per quod &longs;ingulis vici­<lb/>bus eundo & redeundo moveri debent. </s> <s>E&longs;tque tempus itus & <lb/>reditus unius in ratione compo&longs;ita ex ratione &longs;ubduplicata mate­<lb/>riæ & ratione &longs;ubduplicata &longs;patii, atque adeo ut &longs;patium. </s> <s>Pul&longs;us <lb/>autem temporibus itus & reditus unius eundo latitudines &longs;uas <lb/>conficiunt, hoc e&longs;t, &longs;patia temporibus proportionalia percurrunt; <lb/>& propterea &longs;unt æquiveloces. </s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;<emph.end type="italics"/>3 In Mediis igitur den&longs;itate & vi Ela&longs;tica paribus, pul&longs;us <lb/>omnes &longs;unt æquiveloces. </s> <s>Quod &longs;i Medii vel den&longs;itas vel vis Ela­<lb/>&longs;tica intendatur, quoniam vis motrix in ratione vis Ela&longs;ticæ, & <lb/>materia movenda in ratione den&longs;itatis augetur; tempus quo mo-<pb xlink:href="039/01/369.jpg" pagenum="341"/>tus iidem peragantur ac prius, augebitur in &longs;ubduplicata ratione <lb/><arrow.to.target n="note349"/>den&longs;itatis, ac diminuetur in &longs;ubduplicata ratione vis Ela&longs;ticæ. </s> <s>Et <lb/>propterea velocitas pul&longs;uum erit in ratione compo&longs;ita ex ratione <lb/>&longs;ubduplicata den&longs;itatis Medii inver&longs;e & ratione &longs;ubduplicata vis <lb/>Ela&longs;ticæ directe. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note349"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s>Hæc Propo&longs;itio ulterius patebit ex con&longs;tructione &longs;equentis. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLIX. PROBLEMA XI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Datis Medii den&longs;itate & vi Ela&longs;tica, invenire velocitatem pul­<lb/>&longs;uum.<emph.end type="italics"/></s></p> <p type="main"> <s>Fingamus Medium ab incumbente pondere, pro more Aeris <lb/>no&longs;tri comprimi; &longs;itque A altitudo Medii homogenei, cujus pon­<lb/>dus adæquet pondus incumbens, & cujus den&longs;itas eadem &longs;it cum <lb/>den&longs;itate Medii compre&longs;&longs;i, in quo pul&longs;us propagantur. </s> <s>Con&longs;ti­<lb/>tui autem intelligatur Pendulum, cujus longitudo inter punctum <lb/>&longs;u&longs;pen&longs;ionis & centrum o&longs;cillationis &longs;it A: & quo tempore Pen­<lb/>dulum illud o&longs;cillationem integram ex itu & reditu compo&longs;itam <lb/>peragit, eodem pul&longs;us eundo conficiet &longs;patium circumferentiæ <lb/>circuli radio A de&longs;cripti æquale. </s></p> <p type="main"> <s>Nam &longs;tantibus quæ in Propo&longs;itione XLVII con&longs;tructa &longs;unt, <lb/>&longs;i linea quævis Phy&longs;ica <emph type="italics"/>EF,<emph.end type="italics"/>&longs;ingulis vibrationibus de&longs;cribendo <lb/>&longs;patium <emph type="italics"/>PS,<emph.end type="italics"/>urgeatur in extremis itus & reditus cuju&longs;que locis <lb/><emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>S,<emph.end type="italics"/>a vi Ela&longs;tica quæ ip&longs;ius ponderi æquetur; peraget hæc <lb/>vibrationes &longs;ingulas quo tempore eadem in Cycloide, cujus peri­<lb/>meter tota longitudini <emph type="italics"/>PS<emph.end type="italics"/>æqualis e&longs;t, o&longs;cillari po&longs;&longs;et: id adeo <lb/>quia vires æquales æqualia corpu&longs;cula per æqualia &longs;patia &longs;imul im­<lb/>pellent. </s> <s>Quare cum o&longs;cillationum tempora &longs;int in &longs;ubduplicata <lb/>ratione longitudinis Pendulorum, & longitudo Penduli æquetur <lb/>dimidio arcui Cycloidis totius; foret tempus vibrationis unius ad <lb/>tempus o&longs;cillationis Penduli cujus longitudo e&longs;t A, in &longs;ubdupli­<lb/>cata ratione longitudinis 1/2 <emph type="italics"/>PS<emph.end type="italics"/>&longs;eu <emph type="italics"/>PO<emph.end type="italics"/>ad longitudinem A. </s> <s>Sed <lb/>vis Ela&longs;tica qua lineola Phy&longs;ica <emph type="italics"/>EG,<emph.end type="italics"/>in locis &longs;uis extremis <emph type="italics"/>P, S<emph.end type="italics"/><lb/>exi&longs;tens, urgetur, erat (in demon&longs;tratione Propo&longs;itionis XLVII) <lb/>ad ejus vim totam Ela&longs;ticam ut <emph type="italics"/>HL-KN<emph.end type="italics"/>ad V, hoc e&longs;t <lb/>(cum punctum <emph type="italics"/>K<emph.end type="italics"/>jam incidat in <emph type="italics"/>P<emph.end type="italics"/>) ut <emph type="italics"/>HK<emph.end type="italics"/>ad V: & vis illa <lb/>tota, hoc e&longs;t pondus incumbens, quo lineola <emph type="italics"/>EG<emph.end type="italics"/>comprimitur, <lb/>e&longs;t ad pondus lineolæ ut ponderis incumbentis altitudo A ad line­<lb/>olæ longitudinem <emph type="italics"/>EG<emph.end type="italics"/>; adeoque ex æquo, vis qua lineola <emph type="italics"/>EG<emph.end type="italics"/>in <lb/>locis &longs;uis <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>S<emph.end type="italics"/>urgetur, e&longs;t ad lineolæ illius pondus ut <emph type="italics"/>HK<emph.end type="italics"/>XA <lb/>ad VX<emph type="italics"/>EG,<emph.end type="italics"/>&longs;ive ut <emph type="italics"/>PO<emph.end type="italics"/>XA ad VV, nam <emph type="italics"/>HK<emph.end type="italics"/>erat ad <emph type="italics"/>EG<emph.end type="italics"/>ut <pb xlink:href="039/01/370.jpg" pagenum="342"/><arrow.to.target n="note350"/><emph type="italics"/>PO<emph.end type="italics"/>ad V. </s> <s>Quare cum tempora, quibus æqualia corpora per <lb/>æqualia &longs;patia impelluntur, &longs;int reciproce in &longs;ubduplicata ratione <lb/>virium, erit tempus vibrationis unius urgente vi illa Ela&longs;tica, ad <lb/>tempus vibrationis urgente vi ponderis, in &longs;ubduplicata ratione <lb/>VV ad <emph type="italics"/>PO<emph.end type="italics"/>XA, atque adeo ad tempus o&longs;cillationis Penduli cu­<lb/>jus longitudo e&longs;t A, in &longs;ubduplicata ratione VV ad <emph type="italics"/>PO<emph.end type="italics"/>XA, & <lb/>&longs;ubduplicata ratione <emph type="italics"/>PO<emph.end type="italics"/>ad A conjunctim; id e&longs;t, in ratione in­<lb/>tegra V ad A. </s> <s>Sed tempore vibrationis unius ex itu & reditu com­<lb/>po&longs;itæ, pul&longs;us progrediendo conficit latitudinem &longs;uam <emph type="italics"/>BC.<emph.end type="italics"/>Ergo <lb/>tempus quo pul&longs;us percurrit &longs;patium <emph type="italics"/>BC,<emph.end type="italics"/>e&longs;t ad tempus o&longs;cillati­<lb/>onis unius ex itu & reditu compo&longs;itæ, ut V ad A, id e&longs;t, ut <emph type="italics"/>BC<emph.end type="italics"/><lb/>ad circumferentiam circuli cujus radius e&longs;t A. </s> <s>Tempus autem, <lb/>quo pul&longs;us percurret &longs;patium <emph type="italics"/>BC,<emph.end type="italics"/>e&longs;t ad tempus quo percurret <lb/>longitudinem huic circumferentiæ æqualem, in eadem ratione; <lb/>ideoque tempore talis o&longs;cillationis pul&longs;us percurret longitudinem <lb/>huic circumferentiæ æqualem. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note350"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Velocitas pul&longs;uum ea e&longs;t quam acquirunt Gravia, æqua­<lb/>liter accelerato motu cadendo, & ca&longs;u &longs;uo de&longs;cribendo dimidium <lb/>altitudinis A. </s> <s>Nam tempore ca&longs;us hujus, cum velocitate cadendo <lb/>acqui&longs;ita, pul&longs;us percurret &longs;patium quod erit æquale toti altitu­<lb/>dini A, adeoque tempore o&longs;cillationis unius ex itu & reditu com­<lb/>po&longs;itæ, percurret &longs;patium æquale circumferentiæ circuli radio A <lb/>de&longs;cripti: e&longs;t enim tempus ca&longs;us ad tempus o&longs;cillationis ut radius <lb/>circuli ad eju&longs;dem circumferentiam. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Unde cum altitudo illa A &longs;it ut Fluidi vis Ela&longs;tica di­<lb/>recte & den&longs;itas eju&longs;dem inver&longs;e; velocitas pul&longs;uum erit in ratione <lb/>compo&longs;ita ex &longs;ubduplicata ratione den&longs;itatis inver&longs;e & &longs;ubdupli­<lb/>cata ratione vis Ela&longs;ticæ directe. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO L. PROBLEMA XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire pul&longs;uum di&longs;tantias.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Corporis, cujus tremore pul&longs;us excitantur, inveniatur numerus <lb/>Vibrationum dato tempore. </s> <s>Per numerum illum dividatur &longs;pa­<lb/>tium quod pul&longs;us eodem tempore percurrere po&longs;&longs;it, & pars in­<lb/>venta erit pul&longs;us unius latitudo. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Spectant Propo&longs;itiones novi&longs;&longs;imæ ad motum Lucis & Sonorum. </s> <s><lb/>Lux enim cum propagetur &longs;ecundum lineas rectas, in actione &longs;ola <pb xlink:href="039/01/371.jpg" pagenum="343"/>(per Prop. </s> <s>XLI. & XLII.) con&longs;i&longs;tere nequit. </s> <s>Soni vero propterea <lb/><arrow.to.target n="note351"/>quod a corporibus tremulis oriantur, nihil aliud &longs;unt quam aeris <lb/>pul&longs;us propagati, per Prop. </s> <s>XLIII. </s> <s>Confirmatur id ex tremoribus <lb/>quos excitant in corporibus objectis, &longs;i modo vehementes &longs;int & <lb/>graves, quales &longs;unt &longs;oni Tympanorum. </s> <s>Nam tremores celeriores <lb/>& breviores difficilius excitantur. </s> <s>Sed & &longs;onos quo&longs;vis, in chor­<lb/>das corporibus &longs;onoris uni&longs;onas impactos, exe&longs;tare tremores noti&longs;­<lb/>&longs;imum e&longs;t. </s> <s>Confirmatur etiam ex velocitate &longs;onorum. </s> <s>Nam cum <lb/>pondera &longs;pecifica Aquæ pluvialis & Argenti vivi &longs;int ad invicem <lb/>ut 1 ad 13 2/3 circiter, & ubi Mercurius in <emph type="italics"/>Barometro<emph.end type="italics"/>altitudinem <lb/>attingit digitorum <emph type="italics"/>Anglieorum<emph.end type="italics"/>30, pondus &longs;pecificum Aeris & <lb/>aquæ pluvialis &longs;int ad invicem ut 1 ad 870 circiter: erunt pon­<lb/>dera &longs;pecifica aeris & argenti vivi ut 1 ad 11890. Proinde cum <lb/>altitudo argenti vivi &longs;it 30 digitorum, altitudo aeris uniformis, <lb/>cujus pondus aerem no&longs;trum &longs;ubjectum comprimere po&longs;&longs;et, erit <lb/>356700 digitorum, &longs;eu pedum <emph type="italics"/>Anglieorum<emph.end type="italics"/>29725. E&longs;tque hæc <lb/>altitudo illa ip&longs;a quam in con&longs;tructione &longs;uperioris Problematis no­<lb/>minavimus A. </s> <s>Circuli radio 29725 pedum de&longs;cripti circumferen­<lb/>tia e&longs;t pedum 186768. Et cum Pendulum digitos 39 1/5 longum, <lb/>o&longs;cillationem ex itu & reditu compo&longs;itam, tempore minutorum <lb/>duorum &longs;ecundorum, uti notum e&longs;t, ab&longs;olvat; Pendulum pedes <lb/>29725, &longs;eu digitos 356700 longum, o&longs;cillationem con&longs;imilem tem­<lb/>pore minutorum &longs;ecundorum 190 3/4 ab&longs;olvere debebit. </s> <s>Eo igitur <lb/>tempore &longs;onus progrediendo con&longs;iciet pedes 186768, adeoque <lb/>tempore minuti unius &longs;ecundi pedes 979. </s></p> <p type="margin"> <s><margin.target id="note351"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s>Cæterum in hoc computo nulla habetur ratio cra&longs;&longs;itudinis &longs;oli­<lb/>darum particularum aeris, per quam &longs;onus utique propagatur in <lb/>in&longs;tanti. </s> <s>Cum pondus aeris &longs;it ad pondus aquæ ut 1 ad 870, & <lb/>&longs;ales &longs;int fere duplo den&longs;iores quam aqua; &longs;i particulæ aeris po­<lb/>nantur e&longs;&longs;e eju&longs;dem circiter den&longs;itatis cum particulis vel aquæ <lb/>vel &longs;alium, & raritas aeris oriatur ab intervallis particularum: <lb/>diameter particulæ aeris erit ad intervallum inter centra parti­<lb/>cularum, ut 1 ad 9 vel 10 circiter, & ad intervallum inter par­<lb/>ticulas ut 1 ad 8 vel 9. Proinde ad pedes 979 quos &longs;onus tem­<lb/>pore minuti unius &longs;ecundi juxta calculum &longs;uperiorem conficiet, <lb/>addere licet pedes (979/9) &longs;eu 109 circiter, ob cra&longs;&longs;itudinem particu­<lb/>larum aeris: & &longs;ie &longs;onus tempore minuti unius &longs;ecundi conficiet <lb/>pedes 1088 circiter. </s></p> <p type="main"> <s>His adde quod vapores in aere latentes, cum &longs;int alterius ela­<lb/>teris & alterius toni, vix aut ne vix quidem participant motum <lb/>aeris veri quo &longs;oni propagantur. </s> <s>His autem quie&longs;centibus, mo-<pb xlink:href="039/01/372.jpg" pagenum="344"/><arrow.to.target n="note352"/>tus ille celerius propagabitur per &longs;olum aerem verum, idQ.E.I. <lb/>&longs;ubduplicata ratione minoris materiæ. </s> <s>Ut &longs;i Atmo&longs;phæra con­<lb/>&longs;tet ex decem partibus aeris veri & una parte vaporum, motus <lb/>&longs;onorum celerior erit in &longs;ubduplicata ratione 11 ad 10, vel in in­<lb/>tegra circiter ratione 21 ad 20, quam &longs;i propagaretur per undecim <lb/>partes aeris veri: ideoque motus &longs;onorum &longs;upra inventus, augen­<lb/>dus erit in hac ratione. </s> <s>Quo pacto &longs;onus, tempore minuti unius <lb/>&longs;ecundi, conficiet pedes 1142. </s></p> <p type="margin"> <s><margin.target id="note352"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s>Hæc ita &longs;e habere debent tempore verno & autumnali, ubi aer <lb/>per calorem temperatum rare&longs;cit & ejus vis ela&longs;tica nonnihil in­<lb/>tenditur. </s> <s>At hyberno tempore, ubi aer per frigus conden&longs;atur, <lb/>& ejus vis ela&longs;tica remittitur, motus &longs;onorum tardior e&longs;&longs;e debet in <lb/>&longs;ubduplicata ratione den&longs;itatis; & vici&longs;&longs;im æ&longs;tivo tempore debet <lb/>e&longs;&longs;e velocior. </s></p> <p type="main"> <s>Con&longs;tat autem per experimenta quod &longs;oni tempore minuti uNI­<lb/>us &longs;ecundi eundo, conficiunt pedes <emph type="italics"/>Londinen&longs;es<emph.end type="italics"/>plus minus 1142, <lb/><emph type="italics"/>Pari&longs;ien&longs;es<emph.end type="italics"/>vero 1070. </s></p> <p type="main"> <s>Cognita &longs;onorum velocitate innote&longs;cunt etiam intervalla pul­<lb/>&longs;uum. </s> <s>Invenit utique <emph type="italics"/>D. Sauveur<emph.end type="italics"/>(factis a &longs;e experimentis) quod <lb/>fi&longs;tula aperta, cujus longitudo e&longs;t pedum <emph type="italics"/>Pari&longs;ien&longs;ium<emph.end type="italics"/>plus minus <lb/>quinque, &longs;onum edit eju&longs;dem toni cum &longs;ono chordæ quæ tempore <lb/>minuti unius &longs;ecundi centies recurrit. </s> <s>Sunt igitur pul&longs;us plus mi­<lb/>nus centum in &longs;patio pedum <emph type="italics"/>Pari&longs;ien&longs;ium<emph.end type="italics"/>1070, quos &longs;onus tem­<lb/>pore minuti unius &longs;ecundi percurrit; adeoque pul&longs;us unus occu­<lb/>pat &longs;patium pedum <emph type="italics"/>Pari&longs;ien&longs;ium<emph.end type="italics"/>qua&longs;i 10 (7/10), id e&longs;t, duplam circi­<lb/>ter longitudinem fi&longs;tulæ. </s> <s>Unde ver&longs;imile e&longs;t quod latitudines <lb/>pul&longs;uum, in omnium apertarum fi&longs;tularum &longs;onis, æquentur duplis <lb/>longitudinibus fi&longs;tularum. </s></p> <p type="main"> <s>Porro cur &longs;oni ce&longs;&longs;ante motu corporis &longs;onori &longs;tatim ce&longs;&longs;ant, ne­<lb/>Q.E.D.utius audiuntur ubi longi&longs;&longs;ime di&longs;tamus a corporibus &longs;ono­<lb/>ris, quam cum proxime ab&longs;umus, patet ex Corollario Propo&longs;itio­<lb/>nis XLVII Libri hujus. </s> <s>Sed & cur &longs;oni in Tubis &longs;tenterophoNI­<lb/>cis valde augentur, ex allatis principiis manife&longs;tum e&longs;t. </s> <s>Motus <lb/>enim omnis reciprocus &longs;ingulis recur&longs;ibus a cau&longs;a generante augeri <lb/>&longs;olet. </s> <s>Motus autem in Tubis dilatationem &longs;onorum impedienti­<lb/>bus, tardius amittitur & fortius recurrit, & propterea a motu <lb/>novo &longs;ingulis recur&longs;ibus impre&longs;&longs;o, magis augetur. </s> <s>Et hæc &longs;unt <lb/>præcipua Phænomena Sonorum. <pb xlink:href="039/01/373.jpg" pagenum="345"/><arrow.to.target n="note353"/></s></p></subchap2><subchap2> <p type="margin"> <s><margin.target id="note353"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>SECTIO IX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>De motu Circulari Fluidorum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/>HYPOTHESIS.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>RE&longs;i&longs;tentiam, quæ oritur ex defectu lubricitatis partium Fluidi, <lb/>cæteris paribus, proportionalem e&longs;&longs;e velocitati, qua partes <lb/>Fluidi &longs;eparantur ab invicem.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITION LI. THEOREMA XXXIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Cylindrus &longs;olidus infinite longus in Fluido uniformi & infinito <lb/>circa axem po&longs;itione datum uniformi cum motu revolvatur, & <lb/>ab hujus impul&longs;u &longs;olo agatur Fluidum in orbem, per&longs;everet <lb/>autera Fluidi pars unaquæque uniformiter in motu &longs;uo; dico <lb/>quod tempora periodica partium Fluidi &longs;unt ut ip&longs;arum di&longs;tantiæ <lb/>ab axe Cylindri.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>AFL<emph.end type="italics"/>Cylindrus uNI­<lb/><figure id="id.039.01.373.1.jpg" xlink:href="039/01/373/1.jpg"/><lb/>formiter circa axem <emph type="italics"/>S<emph.end type="italics"/>in or­<lb/>bem actus, & circulis con­<lb/>centricis <emph type="italics"/>BGM, CHN, <lb/>DIO, EKP,<emph.end type="italics"/>&c. </s> <s>di&longs;tin­<lb/>guatur Fluidum in Orbes cy­<lb/>lindricos innumeros concen­<lb/>tricos &longs;olidos eju&longs;dem cra&longs;&longs;i­<lb/>tudinis. </s> <s>Et quoniam homo­<lb/>geneum e&longs;t Fluidum, im­<lb/>pre&longs;&longs;iones contiguorum Or­<lb/>bium in &longs;e mutuo factæ, <lb/>erunt (per Hypothe&longs;in) ut <lb/>eorum tran&longs;lationes ab invicem & &longs;uperficies contiguæ in quibus <lb/>impre&longs;&longs;iones fiunt. </s> <s>Si impre&longs;&longs;io in Orbem aliquem major e&longs;t vel <pb xlink:href="039/01/374.jpg" pagenum="346"/><arrow.to.target n="note354"/>minor ex parte concava quam ex parte convexa; prævalebit im­<lb/>pre&longs;&longs;io fortior, & motum Orbis vel accelerabit vel retardabit, <lb/>prout in eandem regionem cum ip&longs;ius motu vel in contrariam di­<lb/>rigitur. </s> <s>Proinde ut Orbis unu&longs;qui&longs;Q.E.I. motu &longs;uo uniformiter <lb/>per&longs;everet, debent impre&longs;&longs;iones ex parte utraque &longs;ibi invicem æqua­<lb/>ri, & fieri in regiones contrarias. </s> <s>Unde cum impre&longs;&longs;iones &longs;unt ut <lb/>contiguæ &longs;uperficies & harum tran&longs;lationes ab invicem, erunt tran­<lb/>&longs;lationes inver&longs;e ut &longs;uperficies, hoc e&longs;t, inver&longs;e ut &longs;uperficierum di­<lb/>&longs;tantiæ ab axe. </s> <s>Sunt autem differentiæ motuum angularium circa <lb/>axem ut hæ tran&longs;lationes applicatæ ad di&longs;tantias, &longs;ive ut tran&longs;lati­<lb/>ones directe & di&longs;tantiæ inver&longs;e; hoc e&longs;t (conjunctis rationibus) <lb/>ut quadrata di&longs;tantiarum inver&longs;e. </s> <s>Quare &longs;i ad infinitæ rectæ <lb/><emph type="italics"/>SABCDEQ<emph.end type="italics"/>partes &longs;in­<lb/><figure id="id.039.01.374.1.jpg" xlink:href="039/01/374/1.jpg"/><lb/>gulas erigantur perpendicula <lb/><emph type="italics"/>Aa, Bb, Cc, Dd, Ee,<emph.end type="italics"/>&c. </s> <s><lb/>ip&longs;arum <emph type="italics"/>SA, SB, SC, SD, <lb/>SE,<emph.end type="italics"/>&c. </s> <s>quadratis reciproce <lb/>proportionalia, & per ter­<lb/>minos perpendicularium du­<lb/>ci intelligatur linea curva <lb/>Hyperbolica; erunt &longs;ummæ <lb/>differentiarum, hoc e&longs;t, mo­<lb/>tus toti angulares, ut re­<lb/>&longs;pondentes &longs;ummæ linearum <lb/><emph type="italics"/>Aa, Bb, Cc, Dd, Ee<emph.end type="italics"/>: id <lb/>e&longs;t, &longs;i ad con&longs;tituendum Me­<lb/>dium uniformiter fluidum, Orbium numerus augeatur & latitudo <lb/>minuatur in infinitum, ut areæ Hyperbolicæ his &longs;ummis analogæ <lb/><emph type="italics"/>AaQ, BbQ, CcQ, DdQ, EeQ,<emph.end type="italics"/>&c. </s> <s>Et tempora motibus an­<lb/>gularibus reciproce proportionalia, erunt etiam his areis reciproce <lb/>proportionalia. </s> <s>E&longs;t igitur tempus periodicum particulæ cuju&longs;vis <lb/><emph type="italics"/>D<emph.end type="italics"/>reciproce ut area <emph type="italics"/>DdQ,<emph.end type="italics"/>hoc e&longs;t, (per notas Curvarum qua­<lb/>draturas) directe ut di&longs;tantia <emph type="italics"/>SD. Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note354"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc motus angulares particularum fluidi &longs;unt reci­<lb/>proce ut ip&longs;arum di&longs;tantiæ ab axe cylindri, & velocitates ab&longs;o­<lb/>lutæ &longs;unt æquales. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si fluidum in va&longs;e cylindrico longitudinis infinitæ con­<lb/>tineatur, & cylindrum alium interiorem contineat, revolvatur <lb/>autem cylindrus uterque circa axem communem, &longs;intque revolu-<pb xlink:href="039/01/375.jpg" pagenum="347"/>tionum tempora ut ip&longs;orum &longs;emidiametri, & per&longs;everet fluidi pars <lb/><arrow.to.target n="note355"/>unaquæQ.E.I. motu &longs;uo: erunt partium &longs;ingularum tempora peri­<lb/>odica ut ip&longs;arum di&longs;tantiæ ab axe cylindrorum. </s></p> <p type="margin"> <s><margin.target id="note355"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Si cylindro & fluido ad hunc modum motis addatur <lb/>vel auferatur communis quilibet motus angularis; quoniam hoc <lb/>novo motu non mutatur attritus mutuus partium fluidi, non mu­<lb/>tabuntur motus partium inter &longs;e. </s> <s>Nam tran&longs;lationes partium ab <lb/>invicem pendent ab attritu. </s> <s>Pars quælibet in eo per&longs;everabit <lb/>motu, qui, attritu utrinQ.E.I. contrarias partes facto, non magis <lb/>acceleratur quam retardatur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Unde &longs;i toti cylindrorum & fluidi Sy&longs;temati auferatur <lb/>motus omnis angularis cylindri exterioris, habebitur motus fluidi <lb/>in cylindro quie&longs;cente. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Igitur &longs;i fluido & cylindro exteriore quie&longs;centibus, re­<lb/>volvatur cylindrus interior uniformiter; communicabitur motus <lb/>circularis fluido, & paulatim per totum fluidum propagabitur; <lb/>nec prius de&longs;inet augeri quam fluidi partes &longs;ingulæ motum Corol­<lb/>lario quarto definitum acquirant. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Et quoniam fluidum conatur motum &longs;uum adhuc latius <lb/>propagare, hujus impetu circumagetur etiam cylindrus exterior <lb/>ni&longs;i violenter detentus; & accelerabitur ejus motus quoad u&longs;que <lb/>tempora periodica cylindri utriu&longs;que æquentur inter &longs;e. </s> <s>Quod &longs;i <lb/>cylindrus exterior violenter detineatur, conabitur is motum fluidi <lb/>retardare; & ni&longs;i cylindrus interior vi aliqua extrin&longs;ecus impre&longs;&longs;a <lb/>motum illum con&longs;ervet, efficiet ut idem paulatim ce&longs;&longs;et. </s></p> <p type="main"> <s>Quæ omnia in Aqua profunda &longs;tagnante experiri licet. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LII. THEOREMA XL.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Sphæra &longs;olida, in Fluido uniformi & infinito, circa axem po&longs;i­<lb/>tione datum uniformi cum motu revolvatur, & ab hujus im­<lb/>pul&longs;u &longs;olo agatur Fluidum in orbem; per&longs;everet autem Fluidi <lb/>pars unaquæque uniformiter in motu &longs;uo: dico quod tem­<lb/>pora periodica partium Fluidi erunt ut quadrata di&longs;tantiarum <lb/>à centro Sphæræ.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>1. Sit <emph type="italics"/>AFL<emph.end type="italics"/>Sphæra uniformiter circa axem <emph type="italics"/>S<emph.end type="italics"/>in orbem <lb/>acta, & circulis concentricis <emph type="italics"/>BGM, CHN, DIO, EKP,<emph.end type="italics"/>&c. <pb xlink:href="039/01/376.jpg" pagenum="348"/><arrow.to.target n="note356"/>di&longs;tinguatur Fluidum in Orbes innumeros concentricos eju&longs;dem <lb/>cra&longs;&longs;itudinis. </s> <s>Finge autem Orbes illos e&longs;&longs;e &longs;olidos; & quoniam <lb/>homogeneum e&longs;t Fluidum, impre&longs;&longs;iones contiguorum Orbium in <lb/>&longs;e mutuo factæ, erunt (per Hypothe&longs;in) ut eorum tran&longs;lationes <lb/>ab invicem & &longs;uperficies contiguæ in quibus impre&longs;&longs;iones fiunt. </s> <s><lb/>Si impre&longs;&longs;io in Orbem aliquem major e&longs;t vel minor ex parte con­<lb/>cava quam ex parte convexa; prævalebit impe&longs;&longs;io fortior, & velo­<lb/>citatem Orbis vel accelerabit vel retardabit, prout in eandem regi­<lb/>onem cum ip&longs;ius motu vel in contrariam dirigitur. </s> <s>Proinde ut <lb/>Orbis unu&longs;qui&longs;Q.E.I. motu &longs;uo per&longs;everet uniformiter, debebunt <lb/>impre&longs;&longs;iones ex parte utraque &longs;ibi invicem æquari, & fieri in re­<lb/>giones contrarias. </s> <s>Unde cum impre&longs;&longs;iones &longs;int ut contiguæ &longs;u­<lb/>perficies & harum tran&longs;lationes ab invicem; erunt tran&longs;lationes <lb/>inver&longs;e ut &longs;uperficies, hoc e&longs;t, inver&longs;e ut quadrata di&longs;tantiarum &longs;u­<lb/>perficierum à centro. </s> <s>Sunt autem differentiæ motuum angularium <lb/>circa axem ut hæ tran&longs;lationes applicatæ ad di&longs;tantias, &longs;ive ut <lb/>tran&longs;lationes directe & di&longs;tantiæ inver&longs;e; hoc e&longs;t (conjunctis ra­<lb/>tionibus) ut cubi di&longs;tantiarum inver&longs;e. </s> <s>Quare &longs;i ad rectæ infi­<lb/>nitæ <emph type="italics"/>SABCDEQ<emph.end type="italics"/>partes &longs;ingulas erigantur perpendicula <emph type="italics"/>Aa, <lb/>Bb, Cc, Dd, Ee,<emph.end type="italics"/>&c. </s> <s>ip&longs;arum <emph type="italics"/>SA, SB, SC, SD, SE,<emph.end type="italics"/>&c. </s> <s><lb/>cubis reciproce proportionalia, erunt &longs;ummæ differentiarum, hoc <lb/>e&longs;t, motus toti angulares, ut re&longs;pondentes &longs;ummæ linearum <emph type="italics"/>Aa, <lb/>Bb, Cc, Dd, Ee<emph.end type="italics"/>: id e&longs;t (&longs;i ad con&longs;tituendum Medium uniformi­<lb/>ter fluidum, numerus Orbium augeatur & latitudo minuatur in in­<lb/>finitum) ut areæ Hyperbolicæ his &longs;ummis analogæ <emph type="italics"/>AaQ, BbQ, <lb/>CcQ, DdQ, EeQ,<emph.end type="italics"/>&c. </s> <s>Et tempora periodica motibus angu­<lb/>laribus reciproce proportionalia, erunt etiam his areis reciproce <lb/>proportionalia. </s> <s>E&longs;t igitur tempus periodicum Orbis cuju&longs;vis <lb/><emph type="italics"/>DIO<emph.end type="italics"/>reciproce ut area <emph type="italics"/>DdQ,<emph.end type="italics"/>hoc e&longs;t, (per notas Curvarum <lb/>quadraturas) directe ut quadratum di&longs;tantiæ <emph type="italics"/>SD.<emph.end type="italics"/>Id quod vo­<lb/>lui primo demon&longs;trare. </s></p> <p type="margin"> <s><margin.target id="note356"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>2. A centro Sphæræ ducantur infinitæ rectæ quam pluri­<lb/>mæ, quæ cum axe datos contineant angulos, æqualibus differen­<lb/>tiis &longs;e mutuo &longs;uperantes; & his rectis circa axem revolutis concipe <lb/>Orbes in annulos innumeros &longs;ecari; & annulus unu&longs;qui&longs;que habe­<lb/>bit annulos quatuor &longs;ibi contiguos, unum interiorem, alterum ex­<lb/>teriorem & duos laterales. </s> <s>Attritu interioris & exterioris non <lb/>pote&longs;t annulus unu&longs;qui&longs;que, ni&longs;i in motu juxta legem ca&longs;us primi <lb/>facto, æqualiter & in partes contrarias urgeri. </s> <s>Patet hoc ex de­<lb/>mon&longs;tratione ca&longs;us primi. </s> <s>Et propterea annulorum &longs;eries quælibet <pb xlink:href="039/01/377.jpg" pagenum="349"/>a Globo in infinitum recta pergens, movebitur pro lege ca&longs;us pri­<lb/><arrow.to.target n="note357"/>mi, ni&longs;i quatenus impeditur ab attritu annulorum ad latera. </s> <s>At <lb/>in motu hac lege facto, attritus annulorum ad latera nullus e&longs;t; <lb/>neque adeo motum, quo minus hac lege fiat, impediet. </s> <s>Si an­<lb/>nuli, qui a centro æqualiter di&longs;tant, vel citius revolverentur vel <lb/>tardius juxta polos quam juxta æquatorem; tardiores accelera­<lb/>rentur, & velociores retardarentur ab attritu mutuo, & &longs;ic verge­<lb/>rent &longs;emper tempora periodica ad æqualitatem, pro lege ca&longs;us <lb/>primi. </s> <s>Non impedit igitur hic attritus quo minus motus fiat &longs;e­<lb/>cundum legem ca&longs;us primi, & propterea lex illa obtinebit: hoc <lb/>e&longs;t, annulorum &longs;ingulorum tempora periodica erunt ut quadrata <lb/>di&longs;tantiarum ip&longs;orum à centro Globi. </s> <s>Quod volui &longs;ecundo de­<lb/>mon&longs;trare. </s></p> <p type="margin"> <s><margin.target id="note357"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Cas.<emph.end type="italics"/>3. Dividatur jam annulus unu&longs;qui&longs;que &longs;ectionibus tran&longs;­<lb/>ver&longs;is in particulas innumeras con&longs;tituentes &longs;ub&longs;tantiam ab&longs;olute <lb/>& uniformiter fluidam; & quoniam hæ &longs;ectiones non &longs;pectant ad <lb/>legem motus circularis, &longs;ed ad con&longs;titutionem Fluidi &longs;olummodo <lb/>conducunt, per&longs;everabit motus circularis ut prius. </s> <s>His &longs;ectionibus <lb/>annuli omnes quam minimi a&longs;peritatem & vim attritus mutui aut <lb/>non mutabunt aut mutabunt æqualiter. </s> <s>Et manente cau&longs;arum <lb/>proportione manebit effectuum proportio, hoc e&longs;t, proportio mo­<lb/>tuum & periodieorum temporum. <emph type="italics"/>Q.E.D.<emph.end type="italics"/>Cæterum cum motus <lb/>circularis, & abinde orta vis centrifuga, major &longs;it ad Eclipticam <lb/>quam ad Polos; debebit cau&longs;a aliqua ade&longs;&longs;e qua particulæ &longs;ingulæ <lb/>in circulis &longs;uis retineantur; ne materia quæ ad Eclipticam e&longs;t, rece­<lb/>dat &longs;emper à centro & per exteriora Vorticis migret ad Polos, in­<lb/>deque per axem ad Eclipticam circulatione perpetua revertatur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc motus angulares partium fluidi circa axem globi, <lb/>&longs;unt reciproce ut quadrata di&longs;tantiarum à centro globi, & veloci­<lb/>tates ab&longs;olutæ reciproce ut eadem quadrata applicata ad di&longs;tantias <lb/>ab axe. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Si globus in fluido quie&longs;cente &longs;imilari & infinito circa <lb/>axem po&longs;itione datum uniformi cum motu revolvatur, commuNI­<lb/>cabitur motus fluido in morem Vorticis, & motus i&longs;te paulatim <lb/>propagabitur in infin tum; neque prius ce&longs;&longs;abit in &longs;ingulis fluidi <lb/>partibus accelerari, quam tempora periodica &longs;ingularum partium <lb/>&longs;int ut quadrata di&longs;tantiarum à centro globi. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Quoniam Vorticis partes interiores ob majorem &longs;uam <lb/>velocitatem atterunt & urgent exteriores, motumQ.E.I.&longs;is ea acti-<pb xlink:href="039/01/378.jpg" pagenum="350"/><arrow.to.target n="note358"/>one perpetuo communicant, & exteriores illi eandem motus quan­<lb/>titatem in alios adhuc exteriores &longs;imul tran&longs;ferunt, eaque actione <lb/>&longs;ervant quantitatem motus &longs;ui plane invariatam; patet quod mo­<lb/>tus perpetuo transfertur à centro ad circumferentiam Vorticis, & <lb/>per infinitatem circumferentiæ ab&longs;orbetur. </s> <s>Materia inter &longs;phæri­<lb/>cas duas qua&longs;vis &longs;uperficies Vortici concentricas nunquam accele­<lb/>rabitur, eo quod motum omnem à materia interiore acceptum <lb/>transfert &longs;emper in exteriorem. </s></p> <p type="margin"> <s><margin.target id="note358"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Proinde ad con&longs;ervationem Vorticis con&longs;tanter in eo­<lb/>dem movendi &longs;tatu, requiritur principium aliquod activum, à quo <lb/>globus eandem &longs;emper quantitatem motus accipiat, quam imprimit <lb/>in materiam Vorticis. </s> <s>Ab&longs;que tali principio nece&longs;&longs;e e&longs;t ut globus <lb/>& Vorticis partes interiores, propagantes &longs;emper motum &longs;uum in <lb/>exteriores, neque novum aliquem motum recipientes, tarde&longs;cant <lb/>paulatim & in orbem agi definant. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Si globus alter huic Vortici ad certam ab ip&longs;ius centro <lb/>di&longs;tantiam innataret, & interea circa axem inclinatione datum vi <lb/>aliqua con&longs;tanter revolveretur; hujus motu raperetur fluidum in <lb/>Vorticem: & primo revolveretur hic Vortex novus & exiguus una <lb/>cum globo circa centrum alterius, & interea latius &longs;erperet ip&longs;ius <lb/>motus, & paulatim propagaretur in infinitum, ad modum Vorticis <lb/>primi. </s> <s>Et eadem ratione qua hujus globus raperetur motu Vorti­<lb/>cis alterius, raperetur etiam globus alterius motu hujus, &longs;ic ut <lb/>globi duo circa intermedium aliquod punctum revolverentur, &longs;e­<lb/>que mutuo ob motum illum circularem fugerent, ni&longs;i per vim <lb/>aliquam cohibiti. </s> <s>Po&longs;tea &longs;i vires con&longs;tanter impre&longs;&longs;æ, quibus <lb/>globi in motibus &longs;uis per&longs;everant, ce&longs;&longs;arent, & omnia legibus Me­<lb/>chanicis permitterentur, langue&longs;ceret paulatim motus globorum <lb/>(ob rationem in Corol. </s> <s>3. & 4. a&longs;&longs;ignatam) & Vortices tandem <lb/>conquie&longs;cerent. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Si globi plures datis in locis circum axes po&longs;itione da­<lb/>tos certis cum velocitatibus con&longs;tanter revolverentur, fierent Vor­<lb/>tices totidem in infinitum pergentes. </s> <s>Nam globi &longs;inguli, eadem <lb/>ratione qua unus aliquis motum &longs;uum propagat in infinitum, pro­<lb/>pagabunt etiam motus &longs;uos in infinitum, adeo ut fluidi infiniti <lb/>pars unaquæque eo agitetur motu qui ex omnium globorum acti­<lb/>onibus re&longs;ultat. </s> <s>Unde Vortices non definientur certis limitibus, <lb/>&longs;ed in &longs;e mutuo paulatim excurrent; globique per actiones Vorti­<lb/>cum in &longs;e mutuo, perpetuo movebuntur de locis &longs;uis, uti in <lb/>Corollario &longs;uperiore expo&longs;itum e&longs;t; neque certam quamvis inter &longs;e<pb xlink:href="039/01/379.jpg" pagenum="351"/>po&longs;itionem &longs;ervabunt, ni&longs;i per vim aliquam retenti. </s> <s>Ce&longs;&longs;antibus <lb/><arrow.to.target n="note359"/>autem viribus illis quæ in globos con&longs;tanter impre&longs;&longs;æ con&longs;ervant <lb/>ho&longs;ce motus, materia ob rationem in Corollario tertio & quarto <lb/>a&longs;&longs;ignatam, paulatim requie&longs;cet & in Vortices agi de&longs;inet. </s></p> <p type="margin"> <s><margin.target id="note359"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. Si fluidum &longs;imilare claudatur in va&longs;e &longs;phærico, ac <lb/>globi in centro con&longs;i&longs;tentis uniformi rotatione agatur in Vorticem, <lb/>globus autem & vas in eandem partem circa axem eundem revol­<lb/>vantur, &longs;intque eorum tempora periodica ut quadrata &longs;emidiame­<lb/>trorum: partes fluidi non prius per&longs;everabunt in motibus &longs;uis &longs;ine <lb/>acceleratione & retardatione, quam &longs;int eorum tempora periodica <lb/>ut quadrata di&longs;tantiarum à centro Vorticis. </s> <s>Alia nulla Vorticis <lb/>con&longs;titutio pote&longs;t e&longs;&longs;e permanens. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>8. Si vas, fluidum inclu&longs;um & globus &longs;ervent hunc mo­<lb/>tum, & motu præterea communi angulari circa axem quemvis da­<lb/>tum revolvantur; quoniam hoc motu novo non mutatur attritus <lb/>partium fluidi in &longs;e invicem, non mutabuntur motus partium in­<lb/>ter &longs;e. </s> <s>Nam tran&longs;lationes partium inter &longs;e pendent ab attritu. </s> <s><lb/>Pars quælibet in eo per&longs;everabit motu, quo fit ut attritu ex uno <lb/>latere non magis tardetur quam acceleretur attritu ex altero. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>9. Unde &longs;i vas quie&longs;cat ac detur motus globi, dabitur <lb/>motus fluidi. </s> <s>Nam concipe planum tran&longs;ire per axem globi & <lb/>motu contrario revolvi; & pone &longs;ummam temporis revolutionis <lb/>hujus & revolutionis globi e&longs;&longs;e ad tempus revolutionis globi, ut <lb/>quadratum &longs;emidiametri va&longs;is ad quadratum &longs;emidiametri globi: <lb/>& tempora periodica partium fluidi re&longs;pectu plani hujus, erunt ut <lb/>quadrata di&longs;tantiarum &longs;uarum à centro globi. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>10. Proinde &longs;i vas vel circa axem eundem cum globo, vel <lb/>circa diver&longs;um aliquem, data cum velocitate quacunque movea­<lb/>tur, dabitur motus fluidi. </s> <s>Nam &longs;i Sy&longs;temati toti auferatur v &longs;is <lb/>motus angularis, manebunt motus omnes iidem inter &longs;e qui prius, <lb/>per Corol. </s> <s>8. Et motus i&longs;ti per Corol. </s> <s>9. dabuntur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>11. Si vas & fluidum quie&longs;cant & globus uniformi cum <lb/>motu revolvatur, propagabitur motus paulatim per fluidum torum <lb/>in vas, & circumagetur vas ni&longs;i violenter detentum, neque prius <lb/>definent fluidum & vas accelerari, quam &longs;int eorum tempora peri­<lb/>odica æqualia temporibus periodicis globi. </s> <s>Quod &longs;i vas vi aliqua <lb/>detineatur vel revolvatur motu quovis con&longs;tanti & uniformi, de­<lb/>vemet Medium paulatim ad &longs;tatum motus in Corollariis 8. 9 & 10. <lb/>definiti, nes in alio unquam &longs;tatu quocunque per&longs;everabit. </s> <s>De­<lb/>inde vero &longs;i, viribus illis ce&longs;&longs;antibus quibus vas & globus certis <pb xlink:href="039/01/380.jpg" pagenum="352"/><arrow.to.target n="note360"/>motibus revolvebantur, permittatur Sy&longs;tema totum Legibus Me­<lb/>chanicis; vas & globus in &longs;e invicem agent mediante fluido, ne­<lb/>que motus &longs;uos in &longs;e mutuo per fluidum propagare prius ce&longs;&longs;a­<lb/>bunt, quam eorum tempora periodica æquentur inter &longs;e, & Sy&longs;te­<lb/>ma totum ad in&longs;tar corporis unius &longs;olidi &longs;imul revolvatur. </s></p> <p type="margin"> <s><margin.target id="note360"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>In his omnibus &longs;uppono fluidum ex materia quoad den&longs;itatem <lb/>& fluiditatem uniformi con&longs;tare. </s> <s>Tale e&longs;t in quo globus idem <lb/>codem cum motu, in eodem temporis intervallo, motus &longs;imiles & <lb/>æquales, ad æquales &longs;emper à &longs;e di&longs;tantias, ubivis in fluido con&longs;ti­<lb/>tutus, propagare po&longs;&longs;it. </s> <s>Conatur quidem materia per motum <lb/>&longs;uum circularem recedere ab axe Vorticis, & propterea premit <lb/>materiam omnem ulteriorem. </s> <s>Ex hac pre&longs;&longs;ione fit attritus par­<lb/>tium fortior & &longs;eparatio ab invicem difficilior; & per con&longs;equens <lb/>diminuitur materiæ fluiditas. </s> <s>Rur&longs;us &longs;i partes fluidi &longs;unt alicubi <lb/>cra&longs;&longs;iores &longs;eu majores, fluiditas ibi minor erit, ob pauciores &longs;uper­<lb/>ficies in quibus partes &longs;eparentur ab invicem. </s> <s>In huju&longs;modi ca&longs;i­<lb/>bus deficientem fluiditatem vel lubricitate partium vel lentore alia­<lb/>ve aliqua conditione re&longs;titui &longs;uppono. </s> <s>Hoc ni&longs;i fiat, materia ubi <lb/>minus fluida e&longs;t magis cohærebit & &longs;egnior erit, adeoque motum <lb/>tardius recipiet & longius propagabit quam pro ratione &longs;uperius <lb/>a&longs;&longs;ignata. </s> <s>Si figura va&longs;is non &longs;it Sphærica, movebuntur particulæ <lb/>in lineis non circularibus &longs;ed conformibus eidem va&longs;is figuræ, & <lb/>tempora periodica erunt ut quadrata mediocrium di&longs;tantiarum à <lb/>centro quamproxime. </s> <s>In partibus inter centrum & circumferen­<lb/>tiam, ubi latiora &longs;unt &longs;patia, tardiores erunt motus, ubi angu&longs;tiora <lb/>velociores, neque tamen particulæ velociores petent circumferen­<lb/>tiam. </s> <s>Arcus enim de&longs;cribent minus curvos, & conatus recedendi <lb/>à centro non minus diminuetur per decrementum hujus curva­<lb/>turæ, quam augebitur per incrementum velocitatis. </s> <s>Pergendo a <lb/>&longs;patiis angu&longs;tioribus in latiora recedent paulo longius a centro, <lb/>&longs;ed i&longs;to rece&longs;&longs;u tarde&longs;cent; & accedendo po&longs;tea de latioribus ad <lb/>angu&longs;tiora accelerabuntur, & &longs;ic per vices tarde&longs;cent & accelera­<lb/>buntur particulæ &longs;ingulæ in perpetuum. </s> <s>Hæc ita &longs;e habebunt in <lb/>va&longs;e rigido. </s> <s>Nam in fluido infinito con&longs;titutio Vorticum innote­<lb/>&longs;cit per Propo&longs;itionis hujus Corollarium &longs;extum. </s></p> <p type="main"> <s>Proprietates autem Vorticum hac Propo&longs;itione inve&longs;tigare co­<lb/>natus &longs;um, ut pertentarem &longs;iqua ratione Phænomena cœle&longs;tia per <pb xlink:href="039/01/381.jpg" pagenum="353"/>Vortices explicari po&longs;&longs;int. </s> <s>Nam Phænomenon e&longs;t, quod Planeta­<lb/><arrow.to.target n="note361"/>rum circa Jovem revolventium tempora periodica &longs;unt in ratione <lb/>&longs;e&longs;quiplicata di&longs;tantiarum a centro Jovis; & eadem Regula obti­<lb/>net in Planetis qui circa Solem revolvuntur. </s> <s>Obtinent autem hæ <lb/>Regulæ in Planetis utri&longs;que quam accurati&longs;&longs;ime, quatenus ob&longs;er­<lb/>vationes A&longs;tronomicæ hactenus prodidere. </s> <s>Ideoque &longs;i Planetæ <lb/>illi à Vorticibus circa Jovem & Solem revolventibus deferantur, <lb/>debebunt etiam hi Vortices eadem lege revolvi. </s> <s>Verum tempora <lb/>periodica partium Vorticis prodierunt in ratione duplicata di&longs;tan­<lb/>tiarum a centro motus: neque pote&longs;t ratio illa diminui & ad ra­<lb/>tionem &longs;e&longs;quiplicatam reduci, ni&longs;i vel materia Vorticis eo fluidior <lb/>&longs;it quo longius di&longs;tat a centro, vel re&longs;i&longs;tentia, quæ oritur ex de­<lb/>fectu lubricitatis partium fluidi, ex aucta velocitate qua partes <lb/>fluidi &longs;eparantur ab invicem, augeatur in majori ratione quam ea <lb/>e&longs;t in qua velocitas augetur. </s> <s>Quorum tamen neutrum rationi <lb/>con&longs;entaneum videtur. </s> <s>Partes cra&longs;&longs;iores & minus fluidæ (ni&longs;i gra­<lb/>ves &longs;int in centrum) circumferentiam petent; & veri&longs;imile e&longs;t <lb/>quod, etiam&longs;i Demon&longs;trationum gratia Hypothe&longs;in talem initio <lb/>Sectionis hujus propo&longs;uerim ut Re&longs;i&longs;tentia velocitati proportiona­<lb/>lis e&longs;&longs;et, tamen Re&longs;i&longs;tentia in minori &longs;it ratione quam ea velocita­<lb/>tis e&longs;t. </s> <s>Quo conce&longs;&longs;o, tempora periodica partium Vorticis erunt <lb/>in majori quam duplicata ratione di&longs;tantiarum ab ip&longs;ius centro. </s> <s><lb/>Quod &longs;i Vortices (uti aliquorum e&longs;t opinio) celerius moveantur <lb/>prope centrum, dein tardius u&longs;que ad certum limitem, tum denuo <lb/>celerius juxta circumferentiam; certe nec ratio &longs;e&longs;quiplicata neque <lb/>alia quævis certa ac determinata obtinere pote&longs;t. </s> <s>Viderint itaque <lb/>Philo&longs;ophi quo pacto Phænomenon illud rationis &longs;e&longs;quiplicatæ per <lb/>Vortices explicari po&longs;&longs;it. </s></p> <p type="margin"> <s><margin.target id="note361"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO LIII. THEOREMA XLI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Corpora quæ in Vortice delata in orbem redeunt, eju&longs;dem &longs;unt den­<lb/>&longs;itatis cum Vortice, & eadem lege cum ip&longs;ius partibus (quoad <lb/>velocitatem & cur&longs;us determinationem) moventur.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam &longs;i Vorticis pars aliqua exigua, cujus particulæ &longs;eu puncta <lb/>phy&longs;ica datum &longs;ervant &longs;itum inter &longs;e, congelari &longs;upponatur: hæc, <lb/>quoniam neque quoad den&longs;itatem &longs;uam, neque quoad vim in&longs;itam <lb/>aut figuram &longs;uam mutatur, movebitur eadem lege ac prius: & <pb xlink:href="039/01/382.jpg" pagenum="354"/><arrow.to.target n="note362"/>contra, &longs;i Vorticis pars congelata & &longs;olida eju&longs;dem &longs;it den&longs;itatis <lb/>cum reliquo Vortice, & re&longs;olvatur in fluidum; movebitur hæc ea­<lb/>dem lege ac prius, ni&longs;i quatenus ip&longs;ius particulæ jam fluidæ factæ <lb/>moveantur inter &longs;e. </s> <s>Negligatur igitur motus particularum inter <lb/>&longs;e, tanquam ad totius motum progre&longs;&longs;ivum nil &longs;pectans, & motus <lb/>totius idem erit ac prius. </s> <s>Motus autem idem erit cum motu alia­<lb/>rum Vorticis partium a centro æqualiter di&longs;tantium, propterea <lb/>quod &longs;olidum in Fluidum re&longs;olutum fit pars Vorticis cæteris parti­<lb/>bus con&longs;imilis. </s> <s>Ergo &longs;olidum, &longs;i &longs;it eju&longs;dem den&longs;itatis cum ma­<lb/>teria Vorticis, eodem motu cum ip&longs;ius partibus movebitur, in ma­<lb/>teria proxime ambiente relative quie&longs;cens. </s> <s>Sin den&longs;ius &longs;it, jam <lb/>magis conabitur recedere à centro Vorticis quam prius; adeoque <lb/>Vorticis vim illam, qua prius in Orbita &longs;ua tanquam in æquilibrio <lb/>con&longs;titutum retinebatur, jam &longs;uperans, recedet a centro & revol­<lb/>vendo de&longs;cribet Spiralem, non amplius in eundem Orbem rediens <lb/>Et eodem argumento &longs;i rarius &longs;it, accedet ad centrum. </s> <s>Igitur non <lb/>redibit in eundem Orbem ni&longs;i &longs;it eju&longs;dem den&longs;itatis cum fluido <lb/>Eo autem in ca&longs;u o&longs;ten&longs;um e&longs;t, quod revolveretur eadem lege cum <lb/>partibus fluidi à centro Vorticis æqualiter di&longs;tantibus. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note362"/>DE MOTU <lb/>CORPORUM</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Ergo &longs;olidum quod in Vortice revolvitur & in eundem <lb/>Orbem &longs;emper redit, relative quie&longs;cit in fluido cui innatat. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Et &longs;i Vortex &longs;it quoad den&longs;itatem uniformis, corpus <lb/>idem ad quamlibet a centro Vorticis di&longs;tantiam revolvi pote&longs;t. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Hinc liquet Planetas à Vorticibus corporeis non deferri. </s> <s>Nam<lb/>Planetæ &longs;ecundum Hypothe&longs;in <emph type="italics"/>Copernicæam<emph.end type="italics"/>circa Solem delati re­<lb/>volvuntur in Ellip&longs;ibus umbilicum habentibus in Sole, & radiis ad<lb/>Solem ductis areas de&longs;cribunt temporibus proportionales. </s> <s>At par­<lb/>tes Vorticis tali motu revolvi nequeunt. </s> <s>De&longs;ignent <emph type="italics"/>AD, BE, CF<emph.end type="italics"/>,<lb/>Orbes tres circa Solem <emph type="italics"/>S<emph.end type="italics"/>de&longs;criptos, quorum extimus <emph type="italics"/>CF<emph.end type="italics"/>circulus<lb/>&longs;it Soli concentricus, & interiorum duorum Aphelia &longs;int <emph type="italics"/>A, B<emph.end type="italics"/>&<lb/>Perihelia <emph type="italics"/>D, E.<emph.end type="italics"/>Ergo corpus quod revolvitur in Orbe <emph type="italics"/>CF,<emph.end type="italics"/>radio<lb/>ad Solem ducto areas temporibus proportionales de&longs;cribendo, mo­<lb/>vebitur uniformi cum motu. </s> <s>Corpus autem quod revolvitur in<lb/>Orbe <emph type="italics"/>BE,<emph.end type="italics"/>tardius movebitur in Aphelio <emph type="italics"/>B<emph.end type="italics"/>& velocius in Peri­<lb/>helio <emph type="italics"/>E,<emph.end type="italics"/>&longs;ecundum leges A&longs;tronomicas; cum tamen &longs;ecundum le­<lb/>ges Mechanicas materia Vorticis in &longs;patio angu&longs;tiore inter <emph type="italics"/>A<emph.end type="italics"/>& C<pb xlink:href="039/01/383.jpg" pagenum="355"/>velocius moveri debeat quam in &longs;patio latiore inter <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>F<emph.end type="italics"/>; id <lb/><arrow.to.target n="note363"/>e&longs;t, in Aphelio velocius quam in Perihelio. </s> <s>Quæ duo repugnant <lb/>inter &longs;e. </s> <s>Sic in principio Signi <lb/><figure id="id.039.01.383.1.jpg" xlink:href="039/01/383/1.jpg"/><lb/>Virginis, ubi Aphelium Martis <lb/>jam ver&longs;atur, di&longs;tantia inter or­<lb/>bes Martis & Veneris e&longs;t ad di­<lb/>&longs;tantiam eorundem orbium in <lb/>principio Signi Pi&longs;cium ut tria <lb/>ad duo circiter, & propterea <lb/>materia Vorticis inter Orbes il­<lb/>los in principio Pi&longs;cium debet <lb/>e&longs;&longs;e velocior quam in principio <lb/>Virginis in ratione trium ad duo. </s> <s><lb/>Nam quo angu&longs;tius e&longs;t &longs;patium <lb/>per quod eadem Materiæ quan­<lb/>titas eodem revolutionis unius <lb/>tempore tran&longs;it, eo majori cum <lb/>velocitate tran&longs;ire debet. </s> <s>Igitur &longs;i Terra in hac Materia cœ&longs;e­<lb/>&longs;ti relative quie&longs;cens ab ea deferretur, & una circa Solem re­<lb/>volveretur, foret hujus velocitas in principio Pi&longs;cium ad eju&longs;dem <lb/>velocitatem in principio Virginis in ratione &longs;e&longs;quialtera. </s> <s>Unde <lb/>Solis motus diurnus apparens in principio Virginis major e&longs;&longs;et <lb/>quam minutorum primorum &longs;eptuaginta, & in principio Pi&longs;cium <lb/>minor quam minutorum quadraginta & octo: cum tamen (expe­<lb/>rientia te&longs;te) apparens i&longs;te Solis motus major &longs;it in principio Pi­<lb/>&longs;cium quam in principio Virginis, & propterea Terra velocior in <lb/>principio Virginis quam in principio Pi&longs;cium. </s> <s>Itaque Hypothe&longs;is <lb/>Vorticum cum Phænomenis A&longs;tronomicis omnino pugnat, & non <lb/>tam ad explicandos quam ad perturbandos motus cœle&longs;tes, con­<lb/>ducit. </s> <s>Quomodo vero motus i&longs;ti in &longs;patiis liberis ab&longs;que Vorti­<lb/>cibus peraguntur intelligi pote&longs;t ex Libro primo, & in Mundi <lb/>Sy&longs;temate plenius docebitur. </s></p><pb xlink:href="039/01/384.jpg" pagenum="356"/></subchap2></subchap1><subchap1><subchap2> <p type="margin"> <s><margin.target id="note363"/>LIBER <lb/>SECUNDUS.</s></p> <p type="main"> <s><emph type="center"/>DE <lb/>MUNDI <lb/>SYSTEMATE <lb/>LIBER TERTIUS.<emph.end type="center"/></s></p> <p type="main"> <s>IN Libris præcedentibus principia Philo&longs;ophiæ tradidi, non ta­<lb/>men Philo&longs;ophica &longs;ed Mathematica tantum, ex quibus vide­<lb/>licet in rebus Philo&longs;ophicis di&longs;putari po&longs;&longs;it. </s> <s>Hæc &longs;unt mo­<lb/>tuum & virium leges & conditiones, quæ ad Philo&longs;ophiam ma­<lb/>xime &longs;pectant. </s> <s>Eadem tamen, ne &longs;terilia videantur, illu&longs;travi <lb/>Scholiis quibu&longs;dam Philo&longs;ophicis, ea tractans quæ generalia &longs;unt, <lb/>& in quibus Philo&longs;ophia maxime fundari videtur, uti corporum <lb/>den&longs;itatem & re&longs;i&longs;tentiam, &longs;patia corporibus vacua, motumque <lb/>Lucis & Sonorum. </s> <s>Supere&longs;t ut ex ii&longs;dem principiis doceamus con­<lb/>&longs;titutionem Sy&longs;tematis Mundani. </s> <s>De hoc argumento compo&longs;ue­<lb/>ram Librum tertium methodo populari, ut a pluribus legeretur. </s> <s><lb/>Sed quibus Principia po&longs;ita &longs;atis intellecta non fuerint, ii vim con­<lb/>&longs;equentiarum minime percipient, neque præjudicia deponent qui­<lb/>bus a multis retro annis in&longs;ueverunt: & propterea ne res in di&longs;pu­<lb/>tationes trahatur, &longs;ummam libri illius tran&longs;tuli in Propo&longs;itiones, <lb/>more Mathematico, ut ab iis &longs;olis legantur qui Principia prius <lb/>evolverint. </s> <s>Veruntamen quoniam Propo&longs;itiones ibi quam pluri­<lb/>mæ occurrant, quæ Lectoribus etiam Mathematice doctis moram <lb/>nimiam injicere po&longs;&longs;int, author e&longs;&longs;e nolo ut qui&longs;quam eas omnes <lb/>evolvat; &longs;uffecerit &longs;iquis Definitiones, Leges motuum & &longs;ectiones <lb/>tres priores Libri primi &longs;edulo legat, dein tran&longs;eat ad hunc Li­<lb/>brum de Mundi Sy&longs;temate, & reliquas Librorum priorum Propo­<lb/>&longs;itiones hic citatas pro lubitu con&longs;ulat. </s></p><pb xlink:href="039/01/385.jpg" pagenum="357"/> <p type="main"> <s><emph type="center"/>REGULÆ <lb/>PHILOSOPHANDI.<emph.end type="center"/><lb/><gap desc="hr tag"/></s></p> <p type="main"> <s><emph type="center"/>REGULA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Cau&longs;as rerum naturalium non plures admitti debere, quam quæ <lb/>& veræ &longs;int & earum Phænomenis explicandis &longs;ufficiant.<emph.end type="italics"/></s></p> <p type="main"> <s>DIcunt utique Philo&longs;ophi: Natura nihil agit fru&longs;tra, & fru&longs;tra <lb/>fit per plura quod fieri pote&longs;t per pauciora. </s> <s>Natura enim <lb/>&longs;implex e&longs;t & rerum cau&longs;is &longs;uperfluis non luxuriat. </s></p> <p type="main"> <s><emph type="center"/>REGULA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ideoque Effectuum naturalium eju&longs;dem generis eædem &longs;unt <lb/>Cau&longs;æ.<emph.end type="italics"/></s></p> <p type="main"> <s>Uti re&longs;pirationis in Homine & in Be&longs;tia; de&longs;cen&longs;us lapidum in <lb/><emph type="italics"/>Europa<emph.end type="italics"/>& in <emph type="italics"/>America<emph.end type="italics"/>; Lucis in Igne culinari & in Sole; reflexi­<lb/>onis Lucis in Terra & in Planetis. </s></p> <p type="main"> <s><emph type="center"/>REGULA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Qualitates corporum quæ intendi & remitti nequeunt, quæque <lb/>corporibus omnibus competunt in quibus experimenta in&longs;tituere <lb/>licet, pro qualitatibus corporum univer&longs;orum habendæ &longs;unt.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam qualitates corporum non ni&longs;i per experimenta innote&longs;cunt; <lb/>ideoque generales &longs;tatuendæ &longs;unt quotquot cum experimentis ge­<lb/>neraliter quadrant; & quæ minui non po&longs;&longs;unt, non po&longs;&longs;unt au­<lb/>ferri. </s> <s>Certe contra experimentorum tenorem &longs;omnia temere con­<lb/>fingenda non &longs;unt, nec a Naturæ ana logia recedendum e&longs;t, cum <pb xlink:href="039/01/386.jpg" pagenum="358"/><arrow.to.target n="note364"/>ea &longs;implex e&longs;&longs;e &longs;oleat & &longs;ibi &longs;emper con&longs;ona. </s> <s>Exten&longs;io corporum <lb/>non ni&longs;i per &longs;en&longs;us innote&longs;cit, nec in omnibus &longs;entitur: &longs;ed quia <lb/>&longs;en&longs;ibilibus omnibus competit, de univer&longs;is affirmatur, Corpora <lb/>plura dura e&longs;&longs;e experimur. </s> <s>Oritur autem durities totius a duritie <lb/>partium, & inde non horum tantum corporum quæ &longs;entiuntur, <lb/>&longs;ed aliorum etiam omnium particulas indivi&longs;as e&longs;&longs;e duras merito <lb/>concludimus. </s> <s>Corpora omnia impenetrabilia e&longs;&longs;e non ratione &longs;ed <lb/>&longs;en&longs;u colligimus. </s> <s>Quæ tractamus, impenetrabilia inveniuntur, & <lb/>inde concludimus impenetrabilitatem e&longs;&longs;e proprietatem corporum <lb/>univer&longs;orum. </s> <s>Corpora omnia mobilia o&longs;&longs;e, & viribus quibu&longs;dam <lb/>(quas vires inertiæ vocamus) per&longs;everare in motu vel quiete, ex <lb/>hi&longs;ce corporum vi&longs;orum proprietatibus colligimus. </s> <s>Exten&longs;io, du­<lb/>rities, impenetrabilitas, mobilitas & vis inertiæ totius, oritur ab <lb/>exten&longs;ione, duritie, impenetrabilitate, mobilitate & viribus iner­<lb/>tiæ partium: & inde concludimus omnes omnium corporum par­<lb/>tes minimas extendi & duras e&longs;&longs;e & impenetrabiles & mobiles &<lb/>viribus inertiæ præditas. </s> <s>Et hoc e&longs;t fundamentum Philo&longs;ophiæ <lb/>totius. </s> <s>Porro corporum partes divi&longs;as & &longs;ibi mutuo contiguas ab <lb/>invicem &longs;eparari po&longs;&longs;e, ex Phænomenis novimus, & partes indi­<lb/>vi&longs;as in partes minores ratione di&longs;tingui po&longs;&longs;e ex Mathematica <lb/>certum e&longs;t. </s> <s>Utrum vero partes illæ di&longs;tinctæ & nondum divi&longs;æ <lb/>per vires Naturæ dividi & ab invicem &longs;eparari po&longs;&longs;int, incertum <lb/>e&longs;t. </s> <s>At &longs;i vel unico con&longs;taret experimento quod particula aliqua <lb/>indivi&longs;a, frangendo corpus durum & &longs;olidum, divi&longs;ionem patere­<lb/>tur: concluderemus vi hujus Regulæ, quod non &longs;olum partes di­<lb/>vi&longs;æ &longs;eparabiles e&longs;&longs;ent, &longs;ed etiam quod indivi&longs;æ in infinitum dividi <lb/>po&longs;&longs;ent. </s></p> <p type="margin"> <s><margin.target id="note364"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Denique &longs;i corpora omnia in circuitu Terræ gravia e&longs;&longs;e in Ter­<lb/>ram, idque pro quantitate materiæ in &longs;ingulis, & Lunam gravem <lb/>e&longs;&longs;e in Terram pro quantitate materiæ &longs;uæ, & vici&longs;&longs;im mare no­<lb/>&longs;trum grave e&longs;&longs;e in Lunam, & Planetas omnes graves e&longs;&longs;e in &longs;e <lb/>mutuo, & Cometarum &longs;imilem e&longs;&longs;e gravitatem, per experimenta <lb/>& ob&longs;ervationes A&longs;tronomicas univer&longs;aliter con&longs;tet: dicendum erit <lb/>per hanc Regulam quod corpora omnia in &longs;e mutuo gravitant. </s> <s><lb/>Nam & fortius erit argumentum ex Phænomenis de gravitate uNI­<lb/>ver&longs;ali, quam de corporum impenetrabilitate: de qua utiQ.E.I. <lb/>corporibus Cœle&longs;tibus nullum experimentum, nullam pror&longs;us ob­<lb/>&longs;ervationem habemus. </s></p><pb xlink:href="039/01/387.jpg" pagenum="359"/></subchap2><subchap2> <p type="main"> <s><emph type="center"/>PHÆNOMENA.<emph.end type="center"/><lb/><arrow.to.target n="note365"/><gap desc="hr tag"/></s></p> <p type="margin"> <s><margin.target id="note365"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="center"/>PHÆNOMENON I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Planetas Circumjoviales, radiis ad centrum Jovis ductis, areas <lb/>de&longs;cribere temporibus proportionales, eorumque tempora periodica <lb/>e&longs;&longs;e in ratione &longs;e&longs;quiplicata di&longs;tantiarum ab ip&longs;ius centro.<emph.end type="italics"/></s></p> <p type="main"> <s>COn&longs;tat ex ob&longs;ervationibus A&longs;tronomicis. </s> <s>Orbes horum Pla­<lb/>netarum non differunt &longs;en&longs;ibiliter a circulis Jovi concentri­<lb/>cis, & motus eorum in his circulis uniformes deprehenduntur. </s> <s><lb/>Tempora vero periodica e&longs;&longs;e in &longs;e&longs;quiplicata ratione &longs;emidiame­<lb/>trorum Orbium con&longs;entiunt A&longs;tronomi; & idem ex Tabula &longs;e­<lb/>quente manife&longs;tum e&longs;t. <lb/><emph type="italics"/>Satellitum Jovialium tempora periodica.<emph.end type="italics"/><lb/><arrow.to.target n="table5"/><arrow.to.target n="table6"/></s></p><table><table.target id="table5"/><row><cell>1<emph type="sup"/>d<emph.end type="sup"/>.18<emph type="sup"/>h<emph.end type="sup"/>.27′.34″.</cell><cell>3<emph type="sup"/>d<emph.end type="sup"/>.13<emph type="sup"/>h<emph.end type="sup"/>.13′.42″.</cell><cell>7<emph type="sup"/>d<emph.end type="sup"/>.3<emph type="sup"/>h<emph.end type="sup"/>.42′.36″.</cell><cell>16<emph type="sup"/>d<emph.end type="sup"/>.16<emph type="sup"/>h<emph.end type="sup"/>.32′.9″.</cell></row></table><table><row><cell><emph type="italics"/>Di&longs;tantiæ Satellitum a centro Jovis.<emph.end type="italics"/><lb/></cell></row><row><cell><emph type="italics"/>Ex ob&longs;ervationibus<emph.end type="italics"/></cell><cell>1</cell><cell>2</cell><cell>3</cell><cell>4</cell><cell/></row><row><cell>Borelli</cell><cell>5 2/3</cell><cell>8 2/3</cell><cell>14</cell><cell>24 2/3</cell><cell>Semidiam. <lb/> Jovis</cell></row><row><cell>Townlei <emph type="italics"/>per Microm.<emph.end type="italics"/></cell><cell>5,52</cell><cell>8,78</cell><cell>13,47</cell><cell>24,72</cell></row><row><cell>Ca&longs;&longs;ini <emph type="italics"/>per Tele&longs;cop.<emph.end type="italics"/></cell><cell>5</cell><cell>8</cell><cell>13</cell><cell>23</cell></row><row><cell>Ca&longs;&longs;ini <emph type="italics"/>per Eclip&longs;. Satell.<emph.end type="italics"/></cell><cell>5 2/3</cell><cell>9</cell><cell>(14 23/60)</cell><cell>(25 1/10)</cell></row><row><cell><emph type="italics"/>Ex temporibus periodicis.<emph.end type="italics"/></cell><cell>5,667</cell><cell>9,017</cell><cell>14,384</cell><cell>25,299</cell></row></table><table><table.target id="table6"/><row><cell><emph type="italics"/>Satellitum Jovialium tempora periodica.<emph.end type="italics"/><lb/></cell></row></table> <p type="main"> <s><emph type="center"/>PHÆNOMENON II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Planetas Circum&longs;aturnios, radiis ad Saturnum ductis, areas de&longs;cri­<lb/>bere temporibus proportionales, & eorum tempora periodica <lb/>e&longs;&longs;e in ratione &longs;e&longs;quiplicata di&longs;tantiarum ab ip&longs;ius centro.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;&longs;inus<emph.end type="italics"/>utique ex ob&longs;ervationibus &longs;uis di&longs;tantias eorum a centro <lb/>Saturni & periodica tempora huju&longs;modi e&longs;&longs;e &longs;tatuit. <pb xlink:href="039/01/388.jpg" pagenum="360"/><arrow.to.target n="note366"/><arrow.to.target n="table7"/><arrow.to.target n="table8"/></s></p> <p type="margin"> <s><margin.target id="note366"/>DE MUNDI <lb/>SYSTEMATE</s></p><table><table.target id="table7"/><row><cell><emph type="italics"/>Satellitum Saturniorum tempora periodica.<emph.end type="italics"/><lb/></cell></row><row><cell>1<emph type="sup"/>d<emph.end type="sup"/>.21<emph type="sup"/><emph.end type="sup"/>.19′.</cell><cell>2<emph type="sup"/>d<emph.end type="sup"/>.17<emph type="sup"/>h<emph.end type="sup"/>.41′.</cell><cell>4<emph type="sup"/>d<emph.end type="sup"/>.13<emph type="sup"/>h<emph.end type="sup"/>.47′.</cell><cell>15<emph type="sup"/>d<emph.end type="sup"/>.22<emph type="sup"/>h<emph.end type="sup"/>.41′.</cell><cell>79<emph type="sup"/>d<emph.end type="sup"/>.22<emph type="sup"/>h<emph.end type="sup"/>.4′.</cell></row><row><cell><emph type="italics"/>Di&longs;tantiæ Satellitum a centro Saturni in &longs;emidiametris Annuli<emph.end type="italics"/></cell></row><row><cell><emph type="italics"/>Ex ob&longs;ervationibus<emph.end type="italics"/></cell><cell>(1 19/20).</cell><cell>2 1/2.</cell><cell>3 1/2.</cell><cell>8.</cell><cell>24.</cell></row><row><cell><emph type="italics"/>Ex temporibus periodicis<emph.end type="italics"/></cell><cell>1,95.</cell><cell>2,5.</cell><cell>3,52,</cell><cell>8,09.</cell><cell>23,71.</cell></row></table><p> <s>PHÆNOMENON III.<lb/></s></p> <p type="main"> <s><emph type="italics"/>Planetas quinque primarios Mercurium, Venerem, Martem, Jo­<lb/>vem & Saturnum Orbibus &longs;uis Solem cingere.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Mercurium & Venerem circa Solem revolvi ex eorum pha&longs;ibus <lb/> lunaribus demon&longs;tratur. </s> <s>Plena facie lucentes ultra Solem &longs;iti &longs;unt, <lb/> dimidiata è regione Solis, falcata cis Solem; per di&longs;cum ejus ad <lb/> modum macularum nonnunquam tran&longs;euntes. </s> <s>Ex Martis quoque <lb/> plena facie prope Solis conjunctionem, & gibbo&longs;a in quadraturis, <lb/> certum e&longs;t quod is Solem ambit. </s> <s>De Jove etiam & Saturno idem <lb/> ex eorum pha&longs;ibus &longs;emper plenis demon&longs;tratur. <lb/> PHÆNOMENON IV.<lb/></s></p> <p type="main"> <s><emph type="italics"/>Planetarum quinque primariorum, & (vel Solis circa Terram vel) <lb/> Terræ circa Solem tempora periodica e&longs;&longs;e in ratione &longs;e&longs;quipli­<lb/>cata mediocrium di&longs;tantiarum à Sole.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Hæc à <emph type="italics"/>Keplero<emph.end type="italics"/>inventa ratio in confe&longs;&longs;o e&longs;t apud omnes. </s> <s>Ea­<lb/>dem utique &longs;unt tempora periodica, eædemque orbium dimen­<lb/>&longs;iones, &longs;ive Sol circa Terram, &longs;ive Terra circa Solem revolvatur. <lb/> Ac de men&longs;ura quidem temporum periodieorum convenit inter <lb/> A&longs;tronomos univer&longs;os. </s> <s>Magnitudines autem Orbium <emph type="italics"/>Keplerus<emph.end type="italics"/>& <lb/> <emph type="italics"/>Bullialdus<emph.end type="italics"/>omnium diligenti&longs;&longs;ime ex Ob&longs;ervationibus determina­<lb/>verunt: & di&longs;tantiæ mediocres, quæ temporibus periodicis re&longs;pon­<lb/>dent, non differunt &longs;en&longs;ibiliter à di&longs;tantiis quas illi invenerunt, <lb/> &longs;untQ.E.I.ter ip&longs;as ut plurimum intermediæ; uti in Tabula &longs;e­<lb/>quente videre licet. <lb/> <pb xlink:href="039/01/389.jpg" pagenum="361"/><lb/><arrow.to.target n="note367"/></s></p><table><row><cell><emph type="italics"/>Planetarum ac Telluris di&longs;tantiæ mediocres à Sole.<emph.end type="italics"/></cell></row><row><cell/><cell><!--symbol10--></cell><cell><!--symbol17--></cell><cell><!--symbol8--></cell><cell><!--symbol18--></cell><cell><!--symbol9--></cell><cell><!--symbol19--></cell></row><row><cell>Secundum <emph type="italics"/>Keplerum<emph.end type="italics"/></cell><cell>951000.</cell><cell>519650.</cell><cell>152350.</cell><cell>100000.</cell><cell>72400.</cell><cell>38806.</cell></row><row><cell>Secundum <emph type="italics"/>Bullialdum<emph.end type="italics"/></cell><cell>954198.</cell><cell>522520.</cell><cell>152350.</cell><cell>100000.</cell><cell>72398.</cell><cell>38585.</cell></row><row><cell>Secundum tempora periodica</cell><cell>953806.</cell><cell>520116.</cell><cell>152399.</cell><cell>100000.</cell><cell>72333.</cell><cell>38710.</cell></row></table> <p type="margin"> <s><margin.target id="note367"/>LIBER <lb/> TERTIUS.</s></p> <p type="main"> <s>De di&longs;tantiis Mercurii & Veneris a Sole di&longs;putandi non e&longs;t locus, <lb/> cum hæ per eorum Elongationes à Sole determinentur.</s> <s> De di­<lb/>&longs;tantiis etiam &longs;uperiorum Planetarum à Sole tollitur omnis di&longs;pu­<lb/>tatio per Eclip&longs;es Satellitum Jovis.</s> <s> Etenim per Eclip&longs;es illas de­<lb/>terminatur po&longs;itio umbræ quam Jupiter projicit, & eo nomine <lb/> habetur Jovis longitudo Heliocentrica.</s> <s> Ex longitudinibus autem <lb/> Heliocentrica & Geocentrica inter &longs;e collatis determinatur di&longs;tan­<lb/>tia Jovis.</s> <s> <lb/> PHÆNOMENON V.</s></p> <p type="main"> <s><emph type="italics"/>Planetas primarios, radiis ad Terram ductis, areas de&longs;cribere tem­<lb/>poribus minime proportionales; at radiis ad Solem ductis, areas <lb/> temporibus proportionales percurrere.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Nam re&longs;pectu Terræ nunc progrediuntur, nunc &longs;tationarii &longs;unt, <lb/> nunc etiam regrediuntur: At Solis re&longs;pectu &longs;emper progrediuntur, <lb/> idque propemodum uniformi cum motu, &longs;ed paulo celerius tamen <lb/> in Periheliis ac tardius in Apheliis, &longs;ic ut arearum æquabilis &longs;it de­<lb/>&longs;criptio. </s> <s>Propo&longs;itio e&longs;t A&longs;tronomis noti&longs;&longs;ima, & in Jove apprime <lb/> demon&longs;tratur per Eclip&longs;es Satellitum, quibus Eclip&longs;ibus Helio­<lb/>centricas Planetæ hujus longitudines & di&longs;tantias à Sole determi­<lb/>nari diximus. <lb/> PHÆNOMENON VI.</s></p> <p type="main"> <s><emph type="italics"/>Lunam radio ad centrum Terræ ducto, aream tempori proporti­<lb/>onalem de&longs;cribere.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Patet ex Lunæ motu apparente cum ip&longs;ius diametro apparente <lb/> collato. </s> <s>Perturbatur autem motus Lunaris aliquantulum à vi So­<lb/>lis, &longs;ed errorum in&longs;en&longs;ibiles minutias in hi&longs;ce Phænomenis negligo. <lb/> <pb xlink:href="039/01/390.jpg" pagenum="362"/><lb/></s></p></subchap2><subchap2><p> <s><arrow.to.target n="note368"/>PROPOSITIONES.<lb/><gap desc="hr tag"/><lb/>PROPOSITIO I. THEOREMA I.<lb/></s></p> <p type="margin"> <s><margin.target id="note368"/>DE MUNDI <lb/> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Vires, quibus Planetæ Circumjoviales perpetuo retrahuntur à me­<lb/>tibus rectilineis & in Orbibus &longs;uis retinentur, re&longs;picere cen­<lb/>trum Jovis, & e&longs;&longs;e reciproce ut quadrata di&longs;tantiarum loco­<lb/>rum ab eodem centro.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>PAtet pars prior Propo&longs;itionis per Phænomenon primum, & <lb/> Propo&longs;itionem &longs;ecundam vel tertiam Libri primi: & pars <lb/> po&longs;terior per Phænomenon primum, & Corollarium &longs;extum Pro­<lb/>po&longs;itionis quartæ eju&longs;dem Libri. <lb/></s> </p> <p type="main"> <s>Idem intellige de Planetis qui Saturnum comitantur, per Phæ­<lb/>nomenon &longs;ecundum. <lb/> PROPOSITIO II. THEOREMA II.<lb/></s></p> <p type="main"> <s><emph type="italics"/>Vires, quibus Planetæ primarii perpetuo retrahuntur à motibus <lb/> rectilineis, & in Orbibus &longs;uis retinentur, re&longs;picere Solem, & <lb/> e&longs;&longs;e reciproce ut quadrata di&longs;tantiarum ab ip&longs;ius centro.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Patet pars prior Propo&longs;itionis per Phænomenon quintum, & <lb/> Propo&longs;itionem &longs;ecundam Libri primi: & pars po&longs;terior per Phæ­<lb/>nomenon quartum, & Propo&longs;itionem quartam eju&longs;dem Libri. <lb/> Accurati&longs;&longs;ime autem demon&longs;tratur hæc pars Propo&longs;itionis per <lb/> quietem Apheliorum. </s> <s>Nam aberratio quam minima à ratione <lb/> duplicata (per Corol. 1. Prop. XLV. Lib. I.) motum Ap&longs;idum in <lb/> &longs;ingulis revolutionibus notabilem, in plunibus enormem efficere <lb/> deberet. <lb/> <pb xlink:href="039/01/391.jpg" pagenum="363"/><lb/>PROPOSITIO III. THEOREMA III.<lb/><arrow.to.target n="note369"/></s></p> <p type="margin"> <s><margin.target id="note369"/>LIBER <lb/> TERTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Vim qua Luna retinetur in Orbe &longs;uo re&longs;picere Terram, & e&longs;&longs;e re­<lb/>citroce ut quadratum di&longs;tantiæ loeorum ab ip&longs;ius centro.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Patet a&longs;&longs;ertionis pars prior per Phænomenon &longs;extum, & Propo­<lb/>po&longs;itionem &longs;ecundam vel tertiam Libri primi: & pars po&longs;terior <lb/> per motum tardi&longs;&longs;imum Lunaris Apogæi. </s> <s>Nam motus ille, qui <lb/> &longs;ingulis revolutionibus e&longs;t graduum tantum trium & minutorum <lb/> trium in con&longs;equentia, contemni pote&longs;t. </s> <s>Patet enim (per Corol. 1. <lb/> Prop. XLV. Lib.I.) quod &longs;i di&longs;tantia Lunæ a centro Terræ &longs;it ad <lb/> &longs;emidiametrum Terræ ut D ad 1; vis a qua motus talis oriatur &longs;it <lb/> reciproce ut D (2 4/243), id e&longs;t, reciproce ut ea ip&longs;ius D dignitas cu­<lb/>jus index e&longs;t (2 4/243), hoc e&longs;t, in ratione di&longs;tantiæ paulo majore quam <lb/> duplicata inver&longs;e, &longs;ed quæ partibus 59 1/4 propius ad duplicatam <lb/> quam ad triplicatam accedit. </s> <s>Oritur vero ab actione Solis (uti <lb/> po&longs;thac dicetur) & propterea hic negligendus e&longs;t. </s> <s>Actio Solis <lb/> quatenus Lunam di&longs;trahit a Terra, e&longs;t ut di&longs;tantia Lunæ a Terra <lb/> quamproxime; ideoque (per ea quæ dicuntur in Corol. 2. Prop. <lb/> XLV. Lib. I.) e&longs;t ad Lunæ vim centripetam ut 2 ad 357,45 circi­<lb/>ter, &longs;eu 1 ad (178 29/40). Et neglecta Solis vi tantilla, vis reliqua qua <lb/> Luna retinetur in Orbe erit reciproce ut D<emph type="sup"/>2<emph.end type="sup"/>. Id quod etiam <lb/> plenius con&longs;tabit conferendo hanc vim cum vi gravitatis, ut fit <lb/> in Propo&longs;itione &longs;equente. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Si vis centripeta mediocris qua Luna retinetur in Orbe, <lb/> augeatur primo in ratione (177 29/40) ad (178 29/40), deinde etiam in rati­<lb/>one duplicata &longs;emidiametri Terræ ad mediocrem di&longs;tantiam centri <lb/> Lunæ a centro Terræ: habebitur vis centripeta Lunaris ad &longs;uper­<lb/>ficiem Terræ, po&longs;ito quod vis illa de&longs;cendendo ad &longs;uperficiem <lb/> Terræ, perpetuo augeatur in reciproca altitudinis ratione du­<lb/>plicata. <lb/> PROPOSITIO IV. THEOREMA IV.<lb/></s></p> <p type="main"> <s><emph type="italics"/>Lunam gravitare in Terram, & vi gravitatis retrahi &longs;emper a <lb/> motu rectilineo, & in Orbe &longs;uo retineri.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Lunæ di&longs;tantia mediocris a Terra in Syzygiis e&longs;t &longs;emidiametro­<lb/>rum terre&longs;trium, &longs;ecundum plero&longs;que A&longs;tronomorum 59, &longs;ecun­<lb/>dum <emph type="italics"/>Vendelinum<emph.end type="italics"/>60, &longs;ecundum <emph type="italics"/>Copernicum<emph.end type="italics"/>60 1/3, & &longs;ecundum <emph type="italics"/>Ty-<emph.end type="italics"/><lb/><pb xlink:href="039/01/392.jpg" pagenum="364"/><lb/><arrow.to.target n="note370"/><emph type="italics"/>chonem<emph.end type="italics"/>56 1/2. A&longs;t <emph type="italics"/>Tycho,<emph.end type="italics"/>& quotquot ejus Tabulas refractionum <lb/> &longs;equuntur, con&longs;tituendo refractiones Solis & Lunæ (omnino con­<lb/>tra naturam Lucis) majores quam Fixarum, idque &longs;crupulis qua&longs;i <lb/> quatuor vel quinque, auxerunt parallaxin Lunæ &longs;crupulis totidem, <lb/> hoc e&longs;t, qua&longs;i duodecima vel decima quinta parte totius paralla­<lb/>xeos. </s> <s>Corrigatur i&longs;te error, & di&longs;tantia evadet qua&longs;i 60 1/2 &longs;emi­<lb/>diametrorum terre&longs;trium, fere ut ab aliis a&longs;&longs;ignatum e&longs;t. </s> <s>A&longs;&longs;uma­<lb/>mus di&longs;tantiam mediocrem &longs;exaginta &longs;emidiametrorum; & Luna­<lb/>rem periodum re&longs;pectu Fixarum compleri diebus 27, horis 7, mi­<lb/>nutis primis 43, ut ab A&longs;tronomis &longs;tatuitur; atque ambitum Terræ <lb/> e&longs;&longs;e pedum Pari&longs;ien&longs;ium 123249600, uti a <emph type="italics"/>Gallis<emph.end type="italics"/>men&longs;urantibus de­<lb/>finitum e&longs;t: Et &longs;i Luna motu omni privari fingatur ac dimitti ut, <lb/> urgente vi illa omni qua in Orbe &longs;uo retinetur, de&longs;cendat in Ter­<lb/>ram; hæc &longs;patio minuti unius primi cadendo de&longs;cribet pedes Pari­<lb/>&longs;ien&longs;es (15 1/12). Colligitur hoc ex calculo vel per Propo&longs;itionem <lb/> XXXVI. Libri primi, vel (quod eodem recidit) per Corollarium <lb/> nonum Propo&longs;itionis quartæ eju&longs;dem Libri, confecto. </s> <s>Nam ar­<lb/>cus illius quem Luna tempore minuti unius primi, medio &longs;uo <lb/> motu, ad di&longs;tantiam &longs;exaginta &longs;emidiametrorum terre&longs;trium de­<lb/>&longs;cribat, &longs;inus ver&longs;us e&longs;t pedum Pari&longs;ien&longs;ium (15 1/12) circiter. </s> <s>Unde <lb/> cum vis illa accedendo ad Terram augeatur in duplicata di&longs;tantiæ <lb/> ratione inver&longs;a, adeoque ad &longs;uperficiem Terræ major &longs;it partibus <lb/> 60X60 quam ad Lunam; corpus vi illa in regionibus no&longs;tris ca­<lb/>dendo, de&longs;cribere deberet &longs;patio minuti unius primi pedes Pari­<lb/>&longs;ien&longs;es 60X60X(15 1/12), & &longs;patio minuti unius &longs;ecundi pedes (15 1/12). <lb/> Atqui corpora in regionibus no&longs;tris vi gravitatis cadendo, de&longs;cri­<lb/>bunt tempore minuti unius &longs;ecundi pedes Pari&longs;ien&longs;es (15 1/12), uti <lb/> <emph type="italics"/>Hugenius<emph.end type="italics"/>factis pendulorum experimentis & computo inde inito, <lb/> demon&longs;travit: & propterea (per Reg. 1. & 11.) vis qua Luna in <lb/> Orbe &longs;uo retinetur, illa ip&longs;a e&longs;t quam nos Gravitatem dicere &longs;ole­<lb/>mus. </s> <s>Nam &longs;i Gravitas ab ea diver&longs;a e&longs;t, corpora viribus utri&longs;que <lb/> conjunctis Terram petendo, duplo velocius de&longs;cendent, & &longs;patio <lb/> minuti unius &longs;ecundi cadendo de&longs;cribent pedes Pari&longs;ien&longs;es 30 1/6: <lb/> omnino contra Experientiam. <lb/> </s></p> <p type="margin"> <s><margin.target id="note370"/>DE MUNDI <lb/> SYSTEMATE</s></p> <p type="main"> <s>Calculus hic fundatur in hypothe&longs;i quod Terra quie&longs;cit. </s> <s>Nam <lb/> &longs;i Terra & Luna circum Solem moveantur, & interea quoque cir­<lb/>cum commune gravitatis centrum revolvantur: di&longs;tantia centro­<lb/>rum Lunæ ac Terræ ab invicem erit 60 1/2 &longs;emidiametrorum ter­<lb/>re&longs;trium; uti computationem (per Prop. LX. Lib. I.) ineunti <lb/> patebit. <lb/> <pb xlink:href="039/01/393.jpg" pagenum="365"/><lb/>PROPOSITIO V. THEOREMA V.<lb/><arrow.to.target n="note371"/></s></p> <p type="margin"> <s><margin.target id="note371"/>LIBER <lb/> TERTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Planetas Circumjoviales gravitare in Jovem, Circum&longs;aturnios in <lb/> Saturnum, & Circum&longs;olares in Solem, & vi gravitatis &longs;uæ <lb/> retrahi &longs;emper à motibus rectilineis, & in Orbibus curvili­<lb/>neis retineri.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Nam revolutiones Planetarum Circumjovialium circa Jovem, Cir­<lb/>cum&longs;aturniorum circa Saturnum, & Mercurii ac Veneris reliquo­<lb/>rumque Circum&longs;olarium circa Solem &longs;unt Phænomena eju&longs;dem ge­<lb/>neris cum revolutione Lunæ circa Terram; & propterea per <lb/> Reg. 11. à cau&longs;is eju&longs;dem generis dependent: præ&longs;ertim cum de­<lb/>mon&longs;tratum &longs;it quod vires, à quibus revolutiones illæ dependent, <lb/> re&longs;piciant centra Jovis, Saturni ac Solis, & recedendo à Jove, Sa­<lb/>turno & Sole decre&longs;cant eadem ratione ac lege, qua vis gravitatis <lb/> decre&longs;cit in rece&longs;&longs;u à Terra. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Gravitas igitur datur in Planetas univer&longs;os. </s> <s>Nam Ve­<lb/>nerem, Mercurium, cætero&longs;que e&longs;&longs;e corpora eju&longs;dem generis cum <lb/> Jove & Saturno, nemo dubitat. </s> <s>Et cum attractio omnis (per mo­<lb/>tus Legem tertiam) mutua &longs;it, Jupiter in Satellites &longs;uos omnes, <lb/> Saturnus in &longs;uos, TerraQ.E.I. Lunam, & Sol in Planetas omnes <lb/> primarios gravitabit. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Gravitatem, quæ Planetam unumquemque re&longs;picit, e&longs;&longs;e <lb/> reciproce ut quadratum di&longs;tantiæ loeorum ab ip&longs;ius centro. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Graves &longs;unt Planetæ omnes in &longs;e mutuo per Corol. 1. <lb/> & 2. Et hinc Jupiter & Saturnus prope conjunctionem &longs;e invicem <lb/> attrahendo, &longs;en&longs;ibiliter perturbant motus mutuos, Sol perturbat <lb/> motus Lunares, Sol & Luna perturbant Mare no&longs;trum, ut in <lb/> &longs;equentibus explicabitur. <lb/> PROPOSITIO VI. THEOREMA VI.<lb/></s></p> <p type="main"> <s><emph type="italics"/>Corpora omnia in Planetas &longs;ingulos gravitare, & pondera eorum <lb/> in eundem quemvis Planetam, paribus di&longs;tantiis à centro Pla­<lb/>netæ, proportionalia e&longs;&longs;e quantitati materiæ in &longs;ingulis.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>De&longs;cen&longs;us gravium omnium in Terram (dempta &longs;altem inæquali <lb/> retardatione quæ ex Aeris perexigua re&longs;i&longs;tentia oritur) æqualibus <lb/> <pb xlink:href="039/01/394.jpg" pagenum="366"/><lb/><arrow.to.target n="note372"/>temporibus fieri, jamdudum ob&longs;ervarunt alii; & accurati&longs;&longs;ime qui­<lb/>dem notare licet æqualitatem temporum in Pendulis. </s> <s>Rem tentavi <lb/> in Auro, Argento, Plumbo, Vitro, Arena, Sale communi, Ligno, <lb/> Aqua, Tritico. </s> <s>Comparabam pyxides duas ligneas rotundas & <lb/> æquales. </s> <s>Unam implebam Ligno, & idem Auri pondus &longs;u&longs;pende­<lb/>bam (quam potui exacte) in alterius centro o&longs;cillationis. </s> <s>Pyxides <lb/> ab æqualibus pedum undecim filis pendentes, con&longs;tituebant Pen­<lb/>dula, quoad pondus, figuram, & acris re&longs;i&longs;tentiam omnino paria: <lb/> Et paribus o&longs;cillationibus, juxta po&longs;itæ, ibant una & redibant di­<lb/>uti&longs;&longs;ime. </s> <s>Proinde copia materiæ in Auro (per Corol. 1. & 6. Prop. <lb/> XXIV. Lib. II.) erat ad copiam materiæ in Ligno, ut vis motricis <lb/> actio in totum Aurum ad eju&longs;dem actionem in totum Lignum; hoc <lb/> e&longs;t, ut pondus ad pondus. </s> <s>Et &longs;ic in cæteris. </s> <s>In corporibus eju&longs;­<lb/>dem ponderis differentia materiæ, quæ vel minor e&longs;&longs;et quam pars <lb/> mille&longs;ima materiæ totius, his experimentis manife&longs;to deprehendi <lb/> potuit. </s> <s>Jam vero naturam gravitatis in Planetas eandem e&longs;&longs;e atque <lb/> in Terram, non e&longs;t dubium. </s> <s>Elevari enim fingantur corpora hæc <lb/> Terre&longs;tria ad u&longs;que Orbem Lunæ, & una cum Luna motu omni <lb/> privata demitti, ut in Terram &longs;imul cadant; & per jam ante o&longs;ten&longs;a <lb/> certum e&longs;t quod temporibus æqualibus de&longs;cribent æqualia &longs;patia <lb/> cum Luna, adeoque quod &longs;unt ad quantitatem materiæ in Luna, ut <lb/> pondera &longs;ua ad ip&longs;ius pondus. </s> <s>Porro quoniam Satellites Jovis <lb/> temporibus revolvuntur quæ &longs;unt in ratione &longs;e&longs;quiplicata di&longs;tanti­<lb/>arum à centro Jovis, erunt eorum gravitates acceleratrices in Jo­<lb/>vem reciproce ut quadrata di&longs;tantiarum à centro Jovis; & prop­<lb/>terea in æqualibus a Jove di&longs;tantiis, eorum gravitates acceleratrices <lb/> evaderent æquales. </s> <s>Proinde temporibus æqualibus ab æqualibus <lb/> altitudinibus cadendo, de&longs;criberent æqualia &longs;patia; perinde ut fit <lb/> in gravibus, in hac Terra no&longs;tra. </s> <s>Et eodem argumento Planetæ <lb/> circum&longs;olares ab æqualibus à Sole di&longs;tantiis demi&longs;&longs;i, de&longs;cen&longs;u &longs;uo <lb/> in Solem æqualibus temporibus æqualia &longs;patia de&longs;criberent. </s> <s>Vires <lb/> autem, quibus corpora inæqualia æqualiter accelerantur, &longs;unt ut <lb/> corpora; hoc e&longs;t, pondera ut quantitates materiæ in Planetis. <lb/> Porro Jovis & ejus Satellitum pondera in Solem proportionalia <lb/> e&longs;&longs;e quantitatibus materiæ eorum, patet ex motu Satellitum quam <lb/> maxime regulari; per Corol. 3. Prop. LXV. Lib. I. Nam &longs;i ho­<lb/>rum aliqui magis traherentur in Solem, pro quantitate materiæ <lb/> &longs;uæ, quam cæteri: motus Satellitum (per Corol. 2. Prop. LXV. <lb/> Lib. I.) ex inæqualitate attractionis perturbarentur. </s> <s>Si (paribus <lb/> à Sole di&longs;tantiis) Satelles aliquis gravior e&longs;&longs;et in Solem pro quan­<lb/><pb xlink:href="039/01/395.jpg" pagenum="367"/><lb/>titate materiæ &longs;uæ, quam Jupiter pro quantitate materiæ &longs;uæ, in <lb/> <arrow.to.target n="note373"/>ratione quacunQ.E.D.ta, puta <emph type="italics"/>d<emph.end type="italics"/>ad <emph type="italics"/>e<emph.end type="italics"/>: di&longs;tantia inter centrum So­<lb/>lis & centrum Orbis Satellitis, major &longs;emper foret quam di&longs;tantia <lb/> inter centrum Solis & centrum Jovis in ratione &longs;ubduplicata quam <lb/> proxime; uti calculis quibu&longs;dam initis inveni. </s> <s>Et &longs;i Satelles mi­<lb/>nus gravis e&longs;&longs;et in Solem in ratione illa <emph type="italics"/>d<emph.end type="italics"/>ad <emph type="italics"/>e,<emph.end type="italics"/>di&longs;tantia centri <lb/> Orbis Satellitis à Sole minor foret quam di&longs;tantia centri Jovis à <lb/> Sole in ratione illa &longs;ubduplicata. </s> <s>Igitur &longs;i in æqualibus à Sole <lb/> di&longs;tantiis, gravitas acceleratrix Satellitis cuju&longs;vis in Solem major <lb/> e&longs;&longs;et vel minor quam gravitas acceleratrix Jovis in Solem, parte <lb/> tantum mille&longs;ima gravitatis totius, foret di&longs;tantia centri Orbis <lb/> Satellitis à Sole major vel minor quam di&longs;tantia Jovis à Sole <lb/> parte (7/2000) di&longs;tantiæ totius, id e&longs;t, parte quinta di&longs;tantiæ Satellitis <lb/> extimi à centro Jovis: Quæ quidem Orbis eccentricitas foret &c. valde <lb/> &longs;en&longs;ibilis. </s> <s>Sed Orbes Satellitum &longs;unt Jovi concentrici, & propte­<lb/>rea gravitates acceleratrices Jovis & Satellitum in Solem æquantur <lb/> inter &longs;e. </s> <s>Et eodem argumento pondera Saturni & Comitum ejus <lb/> in Solem, in æqualibus à Sole di&longs;tantiis, &longs;unt ut quantitates mate­<lb/>riæ in ip&longs;is: Et pondera Lunæ ac Terræ in Solem vel nulla &longs;unt, <lb/> vel earum ma&longs;&longs;is accurate proportionalia. </s> <s>Aliqua autem &longs;unt per <lb/> Corol. 1. & 3. Prop. V. <lb/> </s></p> <p type="margin"> <s><margin.target id="note372"/>DE MUNDI <lb/> SYSTEMATE</s></p> <p type="margin"> <s><margin.target id="note373"/>LIBER <lb/> TERTIUS.</s></p> <p type="main"> <s>Quinetiam pondera partium &longs;ingularum Planetæ cuju&longs;Q.E.I. <lb/> alium quemcunque, &longs;unt inter &longs;e ut materia in partibus &longs;ingulis. <lb/> Nam &longs;i partes aliquæ plus gravitarent, aliæ minus, quam pro quan­<lb/>titate materiæ: Planeta totus, pro genere partium quibus maxime <lb/> abundet, gravitaret magis vel minus quam pro quantitate materiæ <lb/> totius. </s> <s>Sed nec refert utrum partes illæ externæ &longs;int vel internæ. <lb/> Nam &longs;i verbi gratia corpora Terre&longs;tria, quæ apud nos &longs;unt, in <lb/> Orbem Lunæ elevari fingantur, & conferantur cum corporo Lunæ: <lb/> Si horum pondera e&longs;&longs;ent ad pondera partium externarum Lunæ <lb/> ut quantitates materiæ in ii&longs;dem, ad pondera vero partium in­<lb/>ternarum in majori vel minori ratione, forent eadem ad pondus <lb/> Lunæ totius in majori vel minori ratione: contra quam &longs;upra <lb/> o&longs;ten&longs;um e&longs;t. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc pondera corporum non pendent ab eorum for­<lb/>mis & texturis. </s> <s>Nam &longs;i cum formis variari po&longs;&longs;ent; forent ma­<lb/>jora vel minora, pro varietate formarum, in æquali materia: om­<lb/>nino contra Experientiam. <lb/> <pb xlink:href="039/01/396.jpg" pagenum="368"/><lb/><arrow.to.target n="note374"/></s></p> <p type="margin"> <s><margin.target id="note374"/>DE MUNDI <lb/> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Corpora univer&longs;a quæ circa Terram &longs;unt, gravia &longs;unt <lb/> in Terram; & pondera omnium, quæ æqualiter à centro Terræ <lb/> di&longs;tant, &longs;unt ut quantitates materiæ in ii&longs;dem. </s> <s>Hæc e&longs;t qualitas <lb/> omnium in quibus experimenta in&longs;tituere licet, & propterea per <lb/> Reg.111. de univer&longs;is affirmanda e&longs;t. </s> <s>Si Æther aut corpus aliud <lb/> quodcunque vel gravitate omnino de&longs;titueretur, vel pro quantitate <lb/> materiæ &longs;uæ minus gravitaret: quoniam id (ex mente <emph type="italics"/>Ari&longs;totelis, <lb/> Carte&longs;ii & aliorum<emph.end type="italics"/>non differet ab aliis corporibus ni&longs;i in forma<lb/>materiæ, po&longs;&longs;et idem per mutationem formæ gradatim tran&longs;mutari <lb/> in corpus eju&longs;dem conditionis cum iis quæ, pro quantitate materiæ, <lb/> quam maxime gravitant, & vici&longs;&longs;im corpora maxime gravia, fer­<lb/>mam illius gradatim induendo, po&longs;&longs;ent gravitatem &longs;uam gradatim <lb/> amittere. </s> <s>Ac proinde pondera penderent à formis corporum, <lb/> po&longs;&longs;entque cum formis variari, contra quam probatum e&longs;t in <lb/> Corollario &longs;uperiore. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Spatia omnia non &longs;unt æqualiter plena. </s> <s>Nam &longs;i &longs;patia <lb/> omnia æqualiter plena e&longs;&longs;ent, gravitas &longs;pecifica fluidi quo regio <lb/> aeris impleretur, ob &longs;ummam den&longs;itatem materiæ, nil cederet gra­<lb/>vitati &longs;pecificæ argenti vivi, vel auri, vel corporis alterius cuju&longs;­<lb/>cunQ.E.D.n&longs;i&longs;&longs;imi; & propterea nec aurum neque aliud quod­<lb/>cunque corpus in aere de&longs;cendere po&longs;&longs;et. </s> <s>Nam corpora in flui­<lb/>dis, ni&longs;i &longs;pecifice graviora &longs;int, minime de&longs;cendunt. </s> <s>Quod &longs;i <lb/> quantitas materiæ in &longs;patio dato per rarefactionem quamcunque <lb/> diminui po&longs;&longs;it, quidni diminui po&longs;&longs;it in infinitum? <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Si omnes omnium corporum particulæ &longs;olidæ &longs;int eju&longs;­<lb/>dem den&longs;itatis, neque ab&longs;que poris rarefieri po&longs;&longs;int, Vacuum da­<lb/>tur. </s> <s>Eju&longs;dem den&longs;itatis e&longs;&longs;e dico, quarum vires inertiæ &longs;unt ut <lb/> magnitudines. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Vis gravitatis diver&longs;i e&longs;t generis à vi magnetica. </s> <s>Nam <lb/> attractio magnetica non e&longs;t ut materia attracta. </s> <s>Corpora aliqua <lb/> magis trahuntur, alia minus, plurima non trahuntur. </s> <s>Et vis mag­<lb/>netica in uno & eodem corpore intendi pote&longs;t & remitti, e&longs;tque <lb/> nonnunquam longe major pro quantitate materiæ quam vis gra­<lb/>vitatis, & in rece&longs;&longs;u à Magnete decre&longs;cit in ratione di&longs;tantiæ non <lb/> duplicata, &longs;ed fere triplicata, quantum ex cra&longs;&longs;is quibu&longs;dam ob&longs;er­<lb/>vationibus animadvertere potui. <lb/> <pb xlink:href="039/01/397.jpg" pagenum="369"/><lb/>PROPOSITIO VII. THEOREMA VII.<lb/><arrow.to.target n="note375"/></s></p> <p type="margin"> <s><margin.target id="note375"/>LIBER <lb/> TERTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Gravitatem in corpora univer&longs;a fieri, eamque proportionalem e&longs;&longs;e <lb/> quantitati materiæ in &longs;ingulis.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Planetas omnes in &longs;e mutuo graves e&longs;&longs;e jam ante probavimus, <lb/> ut & gravitatem in unumquemque &longs;eor&longs;im &longs;pectatum e&longs;&longs;e reci­<lb/>proce ut quadratum di&longs;tantiæ loeorum à centro Planetæ. Et inde <lb/> con&longs;equens e&longs;t, (per Prop. LXIX. Lib. I. & ejus Corollaria) gra­<lb/>vitatem in omnes proportionalem e&longs;&longs;e materiæ in ii&longs;dem. <lb/> </s></p> <p type="main"> <s>Porro cum Pianetæ cuju&longs;vis <emph type="italics"/>A<emph.end type="italics"/>partes omnes graves &longs;int in Pla­<lb/>netam quemvis <emph type="italics"/>B,<emph.end type="italics"/>& gravitas partis cuju&longs;que &longs;it ad gravitatem <lb/> totius, ut materia partis ad materiam totius, & actioni omni re­<lb/>actio (per motus Legem tertiam) æqualis &longs;it; Planeta <emph type="italics"/>B<emph.end type="italics"/>in partes <lb/> omnes Planetæ <emph type="italics"/>A<emph.end type="italics"/>vici&longs;&longs;im gravitabit, & erit gravitas &longs;ua in par­<lb/>tem unamquamque ad gravitatem &longs;uam in totum, ut materia par­<lb/>tis ad materiam totius. <emph type="italics"/>Q.E.D.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Oritur igitur & componitur gravitas in Planetam to­<lb/>tum ex gravitate in partes &longs;ingulas. </s> <s>Cujus rei exempla habemus <lb/> in attractionibus Magneticis & Electricis. </s> <s>Oritur enim attractio <lb/> omnis in totum ex attractionibus in partes &longs;ingulas. </s> <s>Res intelli­<lb/>getur in gravitate, concipiendo Planetas plures minores in unum <lb/> Globum coire & Planetam majorem componere. </s> <s>Nam vis totius <lb/> ex viribus partium componentium oriri debebit. </s> <s>Siquis objiciat <lb/> quod corpora omnia, quæ apud nos &longs;unt, hac lege gravitare de­<lb/>berent in &longs;e mutuo, cum tamen cju&longs;modi gravitas neutiquam &longs;en­<lb/>tiatur: Re&longs;pondeo quod gravitas in hæc corpora, cum &longs;it ad gra­<lb/>vitatem in Terram totam ut &longs;unt hæc corpora ad Terram totam, <lb/> longe minor e&longs;t quam quæ &longs;entiri po&longs;&longs;it. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Gravitatio in &longs;ingulas corporis particulas æquales e&longs;t <lb/> reciproce ut quadratum di&longs;tantiæ loeorum à particulis. </s> <s>Patet per <lb/> Corol. 3. Prop. LXXIV. Lib. I. <lb/> <pb xlink:href="039/01/398.jpg" pagenum="370"/><lb/><arrow.to.target n="note376"/>PROPOSITIO VIII. THEOREMA VIII.<lb/></s></p> <p type="margin"> <s><margin.target id="note376"/>DE MUNDI <lb/> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Si Globorum duorum in &longs;e mutuo gravitantium materia undique, <lb/> in regionibus quæ à centris æqualiter di&longs;tant, homogenea &longs;it: <lb/> erit pondus Globi alterutrius in alterum reciproce ut quadra­<lb/>tum di&longs;tantiæ inter centra.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Po&longs;tquam inveni&longs;&longs;em gravitatem in Planetam totum oriri & <lb/> componi ex gravitatibus in partes; & e&longs;&longs;e in partes &longs;ingulas reci­<lb/>proce proportionalem quadratis di&longs;tantiarum a partibus: dubita­<lb/>bam an reciproca illa proportio duplicata obtineret accurate in vi <lb/> tota ex viribus pluribus compo&longs;ita, an vero quam proxime. </s> <s>Nam <lb/> fieri po&longs;&longs;et ut proportio, quæ in majoribus di&longs;tantiis &longs;atis accu­<lb/>rate obtineret, prope &longs;uperficiem Planetæ ob inæquales particu­<lb/>larum di&longs;tantias & &longs;itus di&longs;&longs;imiles, notabiliter erraret. </s> <s>Tandem <lb/> vero, per Prop. LXXV. & LXXVI. Libri primi & ip&longs;arum Corol­<lb/>laria, intellexi veritatem Propo&longs;itionis de qua hic agitur. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc inveniri & inter &longs;e comparari po&longs;&longs;unt pondera <lb/> corporum in diver&longs;os Planetas. </s> <s>Nam pondera corporum æqua­<lb/>lium circum Planetas in circulis revolventium &longs;unt (per Corol. 2. <lb/> Prop. IV. Lib.I.) ut diametri circulorum directe & quadrata tem­<lb/>porum periodieorum inver&longs;e; & pondera ad &longs;uperficies Planeta­<lb/>rum, alia&longs;ve qua&longs;vis a centro di&longs;tantias, majora &longs;unt vel minora <lb/> (per hanc Propo&longs;itionem) in duplicata ratione di&longs;tantiarum in­<lb/>ver&longs;a. Sic ex temporibus periodicis Veneris circum Solem die­<lb/>rum 224 & horarum 16 1/4, Satellitis extimi circumjovialis circum <lb/> Jovem dierum 16 & horarum (16 1/15), Satellitis Hugeniani circum <lb/> Saturnum dierum 15 & horarum 22 2/3, & Lunæ circum Terram <lb/> dierum 27, hor. 7. min. 43, collatis cum di&longs;tantia mediocri Vene­<lb/>ris a Sole & cum elongationibus maximis heliocentricis Satellitis <lb/> extimi circumjovialis a centro Jovis 8′. 21 1/2″, Satellitis Hugeniani <lb/> a centro Saturni 3′. 20″, & Lunæ a Terra 10′, computum ineundo <lb/> inveni quod corporum æqualium & a Sole, Jove, Saturno ac Terra <lb/> æqualiter di&longs;tantium pondera in Solem, Jovem, Saturnum ac Ter­<lb/>ram forent ad invicem ut 1, (1/1033), (1/2411), & (1/227512) re&longs;pective. </s> <s>E&longs;t enim <lb/> parallaxis Solis ex ob&longs;ervationibus novi&longs;&longs;imis qua&longs;i 10″, & <emph type="italics"/>Hal­<lb/>leius<emph.end type="italics"/>no&longs;ter per emer&longs;iones Jovis & Satellitum e parte ob&longs;cura <lb/> <pb xlink:href="039/01/399.jpg" pagenum="371"/><lb/>Lunæ, determinavit quod elongatio maxima heliocentrica Satelli­<lb/><arrow.to.target n="note377"/>tis extimi Jovialis a centro Jovis in mediocri Jovis a Sole di&longs;tan­<lb/>tia &longs;it 8′. 21 1/2″, & diameter Jovis 41″. Ex duratione Eclip&longs;eon <lb/> Satellitum in umbram Jovis incidentium prodit hæc diameter <lb/> qua&longs;i 40″, atque adeo &longs;emidiameter 20″. Men&longs;uravit autem <emph type="italics"/>Hu­<lb/>genius<emph.end type="italics"/>elongationem maximam heliocentricam Satellitis a &longs;e de­<lb/>tecti 3′. 20″ a centro Saturni, & hujus elongationis pars quarta, <lb/> nempe 50″, e&longs;t diameter annuli Saturni e Sole vi&longs;i, & diameter Sa­<lb/>turni e&longs;t ad diametrum annuli ut 4 ad 9, ideoque &longs;emidiameter <lb/> Saturni e Sole vi&longs;i e&longs;t 11″. Subducatur lux erratica quæ haud <lb/> minor e&longs;&longs;e &longs;olet quam 2″ vel 3″: Et manebit &longs;emidiameter Saturni <lb/> qua&longs;i 9″. Ex hi&longs;ce autem & Solis &longs;emidiametro mediocri 16′. 6″ <lb/> computum ineundo prodeunt veræ Solis, Jovis, Saturni ac Terræ <lb/> &longs;emidiametri ad invicem ut 10000, 1077, 889 & 104. Unde, <lb/> cum pondera æqualium corporum 2 centris Solis, Jovis, Saturni <lb/> ac Terræ æqualiter di&longs;tantium, &longs;int in Solem, Jovem, Saturnum <lb/> ac Terram, ut 1, (1/1033), (1/2411), & (1/227512) re&longs;pective, & auctis vel dimi­<lb/>nutis di&longs;tantiis pondera diminuantur vel augeantur in duplicata <lb/> ratione: pondera æqualium corporum in Solem, Jovem, Satur­<lb/>num ac Terram in di&longs;tantiis 10000, 1077, 889, & 104 ab eorum <lb/> centris, atque adeo in eorum &longs;uperficiebus, erunt ut 10000, 835, <lb/> 525, & 410 re&longs;pective. </s> <s>Quanta &longs;int pondera corporum in &longs;uper­<lb/>ficie Lunæ dicemus in &longs;equentibus. <lb/> </s></p> <p type="margin"> <s><margin.target id="note377"/>LIBER <lb/> TERTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Innote&longs;cit etiam quantitas materiæ in Planetis &longs;ingulis. <lb/> Nam quantitates materiæ in Planetis &longs;unt ut eorum vires in æqua­<lb/>libus di&longs;tantiis ab eorum centris, id e&longs;t, in Sole, Jove, Saturno ac <lb/> Terra &longs;unt ut 1, (1/1033), (1/2411), & (1/227512) re&longs;pective. </s> <s>Si parallaxis Solis <lb/> &longs;tatuatur major vel minor quam 10″, debebit quantitas materiæ in <lb/> Terra augeri vel diminui in triplicata ratione. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Innote&longs;cunt etiam den&longs;itates Planetarum. </s> <s>Nam pon­<lb/>dera corporum æqualium & homogeneorum in Sphæras homoge­<lb/>neas &longs;unt in &longs;uperficiebus Sphærarum ut Sphærarum diametri, per <lb/> Prop. LXXII. Lib. I. ideoque Sphærarum heterogenearum den&longs;i­<lb/>tates &longs;unt ut pondera illa applicata ad Sphærarum diametros. <lb/> Erant autem veræ Solis, Jovis, Saturni ac Terræ diametri ad invi­<lb/>cem ut 10000, 1077, 889, & 104, & pondera in eo&longs;dem ut 10000, <lb/> 835, 525, & 410, & propterea den&longs;itates &longs;unt ut 100, 78, 59, <lb/> & 396. Den&longs;itas Terræ quæ prodit ex hoc computo non pendet <lb/> a parallaxi Solis, &longs;ed determinatur per parallaxin Lunæ, & prop­<lb/><pb xlink:href="039/01/400.jpg" pagenum="372"/><lb/><arrow.to.target n="note378"/>terea hic recte definitur. </s> <s>E&longs;t igitur Sol paulo den&longs;ior quam Jupi­<lb/>ter, & Jupiter quam Saturnus, & Terra quadruplo den&longs;ior quam <lb/> Sol. </s> <s>Nam per ingentem &longs;uum calorem Sol rare&longs;cit. </s> <s>Luna vero <lb/> den&longs;ior e&longs;t quam Terra, ut in &longs;equentibus patebit. <lb/> </s></p> <p type="margin"> <s><margin.target id="note378"/>DE MUNDI <lb/> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Den&longs;iores igitur &longs;unt Planetæ qui &longs;unt minores, cæ­<lb/>teris paribus. </s> <s>Sic enim vis gravitatis in eorum &longs;uperficiebus ad <lb/> æqualitatem magis accedit. </s> <s>Sed & den&longs;iores &longs;unt Planetæ, cæte­<lb/>ris paribus, qui &longs;unt Soli propiores; ut Jupiter Saturno, & Terra <lb/> Jove. </s> <s>In diver&longs;is utiQ.E.D.&longs;tantiis a Sole collocandi erant Planetæ <lb/> ut quilibet pro gradu den&longs;itatis calore Solis majore vel minore <lb/> frueretur. </s> <s>Aqua no&longs;tra, &longs;i Terra locaretur in orbe Saturni, rige­<lb/>&longs;ceret, &longs;i in orbe Mercurii in vapores &longs;tatim abiret. </s> <s>Nam lux <lb/> Solis, cui calor proportionalis e&longs;t, &longs;eptuplo den&longs;ior e&longs;t in orbe <lb/> Mercurii quam apud nos: & Thermometro expertus &longs;um quod <lb/> &longs;eptuplo Solis æ&longs;tivi calore aqua ebullit. </s> <s>Dubium vero non e&longs;t <lb/> quin materia Mercurii ad calorem accommodetur, & propterea <lb/> den&longs;ior &longs;it hac no&longs;tra; cum materia omnis den&longs;ior ad operationes <lb/> Naturales obeundas majorem calorem requirat. <lb/> PROPOSITIO IX. THEOREMA IX.<lb/></s></p> <p type="main"> <s><emph type="italics"/>Gravitatem pergendo a &longs;uperficiebus Planetarum deor&longs;um de­<lb/>cre&longs;cere in ratione di&longs;tantiarum a centro quam proxime.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Si materia Planetæ quoad den&longs;itatem uniformis e&longs;&longs;et, obtineret <lb/> hæc Propo&longs;itio accurate: per Prop. LXXIII. Lib. I. Error igitur <lb/> tantus e&longs;t, quantus ab inæquabili den&longs;itate oriri po&longs;&longs;it. <lb/> PROPOSITIO X. THEOREMA X.<lb/><emph type="italics"/>Motus Planetarum in Cœlis diuti&longs;&longs;ime con&longs;ervari po&longs;&longs;e.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>In Scholio Propo&longs;itionis XL. Lib. II. o&longs;ten&longs;um e&longs;t quod globus <lb/> Aquæ congelatæ in Aere no&longs;tro, libere movendo & longitudinem <lb/> &longs;emidiametri &longs;uæ de&longs;cribendo, ex re&longs;i&longs;tentia Aeris amitteret motus <lb/> &longs;ui partem (1/4586). Obtinet autem eadem proportio quam proxime <lb/> in globis utcunque magnis & velocibus. </s> <s>Jam vero Globum Terræ <lb/> no&longs;træ den&longs;iorem e&longs;&longs;e quam &longs;i totus ex Aqua con&longs;taret, &longs;ic colligo. <lb/> Si Globue hicce totus e&longs;&longs;et aqueus, quæcunque rariora e&longs;&longs;ent quam <lb/> aqua, ob minorem &longs;pecificam gravitatem emergerent & &longs;upernata­<lb/><pb xlink:href="039/01/401.jpg" pagenum="373"/><lb/>rent. Ea Q.E.D. cau&longs;a Globus terreus aquis undique coopertus, <lb/> <arrow.to.target n="note379"/>&longs;i rarior e&longs;&longs;et quam aqua, emergeret alicubi, & aqua omnis inde <lb/> defluens congregaretur in regione oppo&longs;ita. </s> <s>Et par e&longs;t ratio <lb/> Terræ no&longs;træ maribus magna ex parte circumdatæ. Hæc &longs;i den­<lb/>&longs;ior non e&longs;&longs;et, emergeret ex maribus, & parte &longs;ui pro gradu levi­<lb/>tatis extaret ex Aqua, maribus omnibus in regionem oppo&longs;itam <lb/> confluentibus. </s> <s>Eodem argumento maculæ Solares leviores &longs;unt. <lb/> quam materia lucida Solaris cui &longs;upernatant. </s> <s>Et in formatione <lb/> qualicunque Planetarum, materia omnis gravior, quo tempore <lb/> ma&longs;&longs;a tota fluida erat, centrum petebat. </s> <s>Unde cum Terra com­<lb/>munis &longs;uprema qua&longs;i duplo gravior &longs;it quam aqua, & paulo infe­<lb/>rius in fodinis qua&longs;i triplo vel quadruplo aut etiam quintuplo gra­<lb/>vior reperiatur: veri&longs;imile e&longs;t quod copia materiæ totius in Terra <lb/> qua&longs;i quintuplo vel &longs;extuplo major &longs;it quam &longs;i tota ex aqua con­<lb/>&longs;taret; præ&longs;ertim cum Terram qua&longs;i quintuplo den&longs;iorem e&longs;&longs;e <lb/> quam Jovem jam ante o&longs;ten&longs;um &longs;it. </s> <s>Igitur &longs;i Jupiter paulo den­<lb/>&longs;ior &longs;it quam aqua, hic &longs;patio dierum triginta, quibus lon­<lb/> gitudinem 459 &longs;emidiametrorum &longs;uarum de&longs;cribit, amitteret in<lb/>Medio eju&longs;dem den&longs;itatis cum Aere no&longs;tro motus &longs;ui partem fere<lb/>decimam. </s> <s>Verum cum re&longs;i&longs;tentia Mediorum minuatur in ratione<lb/>ponderis ac den&longs;itatis, &longs;ic ut aqua, quæ partibus 13 2/3 levior e&longs;t <lb/> quam argentum vivum, minus re&longs;i&longs;tat in eadem ratione; & aer, <lb/> qui partibus 850 levior e&longs;t quam aqua, minus re&longs;i&longs;tat in eadem <lb/> ratione: &longs;i a&longs;cendatur in cœlos ubi pondus Medii, in quo Planetæ <lb/> moventur, diminuitur in immen&longs;um, re&longs;i&longs;tentia prope ce&longs;&longs;abit. <lb/> HYPOTHESIS I.<lb/><emph type="italics"/>Centrum Sy&longs;tematis Mundani quie&longs;cere.<emph.end type="italics"/><lb/></s></p> <p type="margin"> <s><margin.target id="note379"/>LIBER <lb/> TERTIUS.</s></p> <p type="main"> <s>Hoc ab omnibus conce&longs;&longs;um e&longs;t, dum aliqui Terram alii Solem <lb/> in centro Sy&longs;tematis quie&longs;cere contendant. </s> <s>Videamus quid inde <lb/> &longs;equatur. <lb/> PROPOSITIO XI. THEOREMA XI.<lb/></s></p> <p type="main"> <s><emph type="italics"/>Commune centrum gravitatis Terræ, Solis & Planetarum om­<lb/>nium quie&longs;cere.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Nam centrum illud (per Legum Corol. 4.) vel quie&longs;cet vel <lb/> progredietur uniformiter in directum. </s> <s>Sed centro illo &longs;emper <lb/> <pb xlink:href="039/01/402.jpg" pagenum="374"/><lb/><arrow.to.target n="note380"/>progrediente, centrum Mundi quoque movebitur contra Hy­<lb/>pothe&longs;in. <lb/> PROPOSITIO XII. THEOREMA XII.<lb/></s></p> <p type="margin"> <s><margin.target id="note380"/>DE MUNDI <lb/> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Solem motu perpetuo agitari, &longs;ed nunquam longe recedere a com­<lb/>muni gravitatis centro Planetarum omnium.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Nam cum (per Corol. 2. Prop. VIII.) materia in Sole &longs;it ad <lb/> materiam in Jove ut 1033 ad 1, & di&longs;tantia Jovis a Sole &longs;it ad <lb/> &longs;emidiametrum Solis in ratione paulo majore; incidet commune <lb/> centrum gravitatis Jovis & Solis in punctum paulo &longs;upra &longs;uper­<lb/>ficiem Solis. </s> <s>Eodem argumento cum materia in Sole &longs;it ad ma­<lb/>teriam in Saturno ut 2411 ad 1, & di&longs;tantia Saturni a Sole &longs;it ad <lb/> &longs;emidiametrum Solis in ratione paulo minore: incidet commune <lb/> centrum gravitatis Saturni & Solis in punctum paulo infra &longs;uper­<lb/>ficiem Solis. </s> <s>Et eju&longs;dem calculi ve&longs;tigiis in&longs;i&longs;tendo &longs;i Terra & <lb/> Planetæ omnes ex una Solis parte con&longs;i&longs;terent, commune omnium <lb/> centrum gravitatis vix integra Solis diametro a centro Solis di­<lb/>&longs;taret. </s> <s>Aliis in ca&longs;ibus di&longs;tantia centrorum &longs;emper minor e&longs;t. <lb/> Et propterea cum centrum illud gravitatis perpetuo quie&longs;cit, Sol <lb/> pro vario Planetarum &longs;itu in omnes partes movebitur, &longs;ed à cen­<lb/>tro illo nunquam longe recedet. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc commune gravitatis centrum Terræ, Solis & Pla­<lb/>netarum omnium pro centro Mundi habendum e&longs;t. </s> <s>Nam cum <lb/> Terra, Sol & Planetæ omnes gravitent in &longs;e mutuo, & propte­<lb/>rea, pro vi gravitatis &longs;uæ, &longs;ecundum leges motus perpetuo agi­<lb/>tentur: per&longs;picuum e&longs;t quod horum centra mobilia pro Mundi <lb/> centro quie&longs;cente haberi nequeunt. </s> <s>Si corpus illud in centro <lb/> locandum e&longs;&longs;et in quod corpora omnia maxime gravitant (uti <lb/> vulgi e&longs;t opinio) privilegium i&longs;tud concedendum e&longs;&longs;et Soli. <lb/> Cum autem Sol moveatur, eligendum erit punctum quie&longs;cens, <lb/> a quo centrum Solis quam minime di&longs;cedit, & a quo idem ad­<lb/>huc minus di&longs;cederet, &longs;i modo Sol den&longs;ior e&longs;&longs;et & major, ut <lb/> minus moveretur. <lb/> <pb xlink:href="039/01/403.jpg" pagenum="375"/><lb/><arrow.to.target n="note381"/>PROPOSITIO XIII. THEOREMA XIII.<lb/></s></p> <p type="margin"> <s><margin.target id="note381"/>LIBER <lb/> TERTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Planetæ moventur in Ellipfibus umbilicum habentibus in centro <lb/> Solis, & radiis ad centrum illud ductis areas de&longs;cribunt <lb/> temporibus proportionales.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Di&longs;putavimus &longs;upra de his motibus ex Phænomenis. </s> <s>Jam cog­<lb/>nitis motuum principiis, ex his colligimus motus cœle&longs;tes a pri­<lb/>ori. </s> <s>Quoniam pondera Planetarum in Solem &longs;unt reciproce ut <lb/> quadrata di&longs;tantiarum a centro Solis; &longs;i Sol quie&longs;ceret & Planetæ <lb/> reliqui non agerent in &longs;e mutuo, forent orbes eorum Elliptici, <lb/> Solem in umbilico communi habentes, & areæ de&longs;criberentur tem­<lb/>poribus proportionales (per Prop. I. & XI, & Corol. I. Prop. <lb/> XIII Lib. I.) Actiones autem Planetarum in &longs;e mutuo perexiguæ <lb/> &longs;unt (ut po&longs;&longs;int contemni) & motus Planetarum in Ellip&longs;ibus <lb/> circa Solem mobilem minus perturbant (per Prop. LXVI. Lib. I.) <lb/> quam &longs;i motus i&longs;ti circa Solem quie&longs;centem peragerentur. <lb/> </s></p> <p type="main"> <s>Actio quidem Jovis in Saturnum non e&longs;t omnino contemnenda. <lb/> Nam gravitas in Jovem e&longs;t ad gravitatem in Solem (paribus di­<lb/>&longs;tantiis) ut 1 ad 1033; adeoQ.E.I. conjunctione Jovis & Saturni, <lb/> quoniam di&longs;tantia Saturni a Jove e&longs;t ad di&longs;tantiam Saturni a Sole <lb/> fere ut 4 ad 9, erit gravitas Saturni in Jovem ad gravitatem Sa­<lb/>turni in Solem ut 81 ad 16X1033 &longs;eu 1 ad 204 circiter. </s> <s>Et <lb/> hinc oritur perturbatio orbis Saturni in &longs;ingulis Planetæ hujus <lb/> cum Jove conjunctionibus adeo &longs;en&longs;ibilis ut ad eandem A&longs;tronomi <lb/> hæreant. </s> <s>Pro vario &longs;itu Planetæ in his conjunctionibus, Eccen­<lb/>tricitas ejus nunc augetur nunc diminuitur, Aphelium nunc pro­<lb/>movetur nunc forte retrahitur, & medius motus per vices accele­<lb/>ratur & retardatur. </s> <s>Error tamen omnis in motu ejus circum So­<lb/>lem a tanta vi oriundus (præterquam in motu medio) evitari fere <lb/> pote&longs;t con&longs;tituendo umbilicum inferiorem Orbis ejus in communi <lb/> centro gravitatis Jovis & Solis (per Prop. LXVII. Lib. I.) & prop­<lb/>terea ubi maximus e&longs;t, vix &longs;uperat minuta duo prima. Et error <lb/> maximus in motu medio vix &longs;uperat minuta duo prima annuatim. <lb/> In conjunctione autem Jovis & Saturni gravitates acceleratrices <lb/> Solis in Saturnum, Jovis in Saturnum & Jovis in Solem &longs;unt fere <lb/> ut 16, 81 & (16X81X2411/25) &longs;eu 124986, adeoQ.E.D.fferentia gravi­<lb/>tatum Solis in Saturnum & Jovis in Saturnum e&longs;t ad gravitatem <lb/> <pb xlink:href="039/01/404.jpg" pagenum="376"/><lb/><arrow.to.target n="note382"/>Jovis in Solem ut 65 ad 124986 &longs;eu 1 ad 1923. Huic autem dif­<lb/>ferentiæ proportionalis e&longs;t maxima Saturni efficacia ad perturban­<lb/>dum motum Jovis, & propterea perturbatio orbis Jovialis longe <lb/> minor e&longs;t quam ea Saturnii. </s> <s>Reliquorum orbium perturbationes <lb/> &longs;unt adhuc longe minores, præterquam quod Orbis Terræ &longs;en&longs;i­<lb/>biliter perturbatur a Luna. </s> <s>Commune centrum gravitatis Terræ <lb/> & Lunæ, Ellip&longs;in circum Solem in umbilico po&longs;itum percurrit, & <lb/> radio ad Solem ducto areas in eadem temporibus proportionales <lb/> de&longs;cribit, Terra vero circum hoc centrum commune motu men­<lb/>&longs;truo revolvitur. <lb/> PROPOSITIO XIV. THEOREMA XIV.<lb/><emph type="italics"/>Orbium Aphelia & Nodi quie&longs;cunt.<emph.end type="italics"/><lb/></s></p> <p type="margin"> <s><margin.target id="note382"/>DE MUNDI <lb/> SYSTEMATE</s></p> <p type="main"> <s>Aphelia quie&longs;cunt, per Prop. XI. Lib. I. ut & Orbium plana, <lb/> per eju&longs;dem Libri Prop. 1. & quie&longs;centibus planis quie&longs;cunt Nodi. <lb/> Attamen a Planetarum revolventium & Cometarum actionibus in <lb/> &longs;e invicem orientur inæqualitates aliquæ, &longs;ed quæ ob parvitatem <lb/> hic contemni po&longs;&longs;unt. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Quie&longs;cunt etiam Stellæ fixæ, propterea quod datas ad <lb/> Aphelia Nodo&longs;que po&longs;itiones &longs;ervant. <lb/> </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Ideoque cum nulla &longs;it earum parallaxis &longs;en&longs;ibilis ex <lb/> Terræ motu annuo oriunda, vires earum ob immen&longs;am corporum <lb/> di&longs;tantiam nullos edent &longs;en&longs;ibiles effectus in regione Sy&longs;tematis <lb/> no&longs;tri. </s> <s>Quinimo Fixæ in omnes cæli partes æqualiter di&longs;per&longs;æ <lb/> contrariis attractionibus vires mutuas de&longs;truunt, per Prop. LXX. <lb/> Lib. I. <lb/> <emph type="italics"/>Scholium.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Cum Planetæ Soli propiores (nempe Mercurius, Venus, Terra, <lb/> & Mars) ob corporum parvitatem parum agant in &longs;e invicem: <lb/> horum Aphelia & Nodi quie&longs;cent, ni&longs;i quatenus a viribus Jovis, <lb/> Saturni, & corporum &longs;uperiorum turbentur. </s> <s>Et inde colligi po­<lb/>te&longs;t per theoriam gravitatis, quod horum Aphelia moventur ali­<lb/>quantulum in con&longs;equentia re&longs;pectu fixarum, idQ.E.I. proporti­<lb/>one &longs;e&longs;quiplicata di&longs;tantiarum horum Planetarum a Sole. </s> <s>Ut &longs;i <lb/> Aphelium Martis in annis centum conficiat 35′ in con&longs;equentia <lb/> re&longs;pectu fixarum; Aphelia Terræ, Veneris, & Mercurii in annis <lb/> centum conficient 18′. 36″, 11′. 27″, & 4′. 29″ re&longs;pective. </s> <s>Et hi <lb/> motus, ob parvitatem, negliguntur in hac Propo&longs;itione. <lb/> <pb xlink:href="039/01/405.jpg" pagenum="377"/><lb/><arrow.to.target n="note383"/>PROPOSITIO XV. PROBLEMA I.<lb/><emph type="italics"/>Invenire Orbium principales diametros.<emph.end type="italics"/><lb/></s> </p> <p type="margin"> <s><margin.target id="note383"/>LIBER <lb/> TERTIUS.</s></p> <p type="main"> <s>Capiendæ &longs;unt hæ in ratione &longs;ub&longs;e&longs;quiplicata temporum perio­<lb/>dieorum, per Prop. XV. Lib. I. deinde &longs;igillatim augendæ in rati­<lb/>one &longs;ummæ ma&longs;&longs;arum Solis & Planetæ cuju&longs;que revolventis ad <lb/> primam duarum medie proportionalium inter &longs;ummam illam & <lb/> Solem, per Prop. LX. Lib. I. <lb/> PROPOSITIO XVI. PROBLEMA II.<lb/><emph type="italics"/>Invenire Orbium Eccentricitates & Aphelia.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Problema confit per Prop. XVIII. Lib. I. <lb/> PROPOSITIO XVII. THEOREMA XV.<lb/></s></p> <p type="main"> <s><emph type="italics"/>Planetarum motus diurnos uniformes e&longs;&longs;e, & librationem Lunæ <lb/> ex ip&longs;ius motu diurno oriri.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Patet per motus Legem I, & Corol. 22. Prop. LXVI. Lib. I. <lb/> Quoniam vero Lunæ, circa axem &longs;uum uniformiter revolventis, <lb/> dies men&longs;truus e&longs;t; hujus facies eadem ulteriorem umbilicum or­<lb/>bis ip&longs;ius &longs;emper re&longs;piciet, & propterea pro &longs;itu umbilici illius <lb/> deviabit hinc inde a Terra. </s> <s>Hæc e&longs;t libratio in longitudinem. <lb/> Nam libratio in latitudinem orta e&longs;t ex inclinatione axis Lunaris <lb/> ad planum orbis. </s> <s>Porro hæc ita &longs;e habere, ex Phænomenis mani­<lb/>fe&longs;tum e&longs;t. <lb/> PROPOSITIO XVIII. THEOREMA XVI.<lb/></s></p> <p type="main"> <s><emph type="italics"/>Axes Planetarum diametris quæ ad eo&longs;dem axes normaliter du­<lb/>cuntur minores e&longs;&longs;e.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Planetæ &longs;ublato omni motu circulari diurno figuram Sphæricam, <lb/> ob æqualem undique partium gravitatem, affectare deberent. </s> <s>Per <lb/> motum illum circularem fit ut partes ab axe recedentes juxta <lb/> æquatorem a&longs;cendere conentur. </s> <s>Ideoque materia &longs;i fluida &longs;it <lb/> <pb xlink:href="039/01/406.jpg" pagenum="378"/><lb/><arrow.to.target n="note384"/>a&longs;cen&longs;u &longs;uo ad æquatorem diametros adaugebit, axem vero de­<lb/>&longs;cen&longs;u &longs;uo ad polos diminuet. </s> <s>Sic Jovis diameter (con&longs;entienti­<lb/>bus A&longs;tronomorum ob&longs;ervationibus) brevior deprehenditur inter <lb/> polos quam ab oriente in occidentem. </s> <s>Eodem argumento, ni&longs;i <lb/> Terra no&longs;tra paulo altior e&longs;&longs;et &longs;ub æquatore quam ad polos, Ma­<lb/>ria ad polos &longs;ub&longs;iderent, & juxta æquatorem a&longs;cendendo, ibi om­<lb/>nia inundarent. <lb/> PROPOSITIO XIX. PROBLEMA III.<lb/><emph type="italics"/>Invenire proportionem axis Planetæ ad diametros eidem <lb/> perpendiculares.<emph.end type="italics"/><lb/></s></p> <p type="margin"> <s><margin.target id="note384"/>DE MUNDI <lb/> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Picartus<emph.end type="italics"/>men&longs;urando arcum gradus unius & 22′. 55″ inter <lb/> <emph type="italics"/>Ambianum<emph.end type="italics"/>& <emph type="italics"/>Malvoi&longs;inam,<emph.end type="italics"/>invenit arcum gradus unius e&longs;&longs;e hexa­<lb/>pedarum Pari&longs;ien&longs;ium 57060. Unde ambitus Terræ e&longs;t pedum <lb/> Pari&longs;ien&longs;ium 123249600, ut &longs;upra. </s> <s>Sed cum error quadringente­<lb/>&longs;imæ partis digiti, tam in fabrica in&longs;trumentorum quam in ap­<lb/>plicatione eorum ad ob&longs;ervationes capiendas, &longs;it in&longs;en&longs;ibilis, & <lb/> in Sectore decempedali quo <emph type="italics"/>Galli<emph.end type="italics"/>ob&longs;ervarunt Latitudines loco­<lb/>rum re&longs;pondeat minutis quatuor &longs;ecundis, & in &longs;ingulis ob&longs;erva­<lb/>tionibus incidere po&longs;&longs;it tam ad centrum Sectoris quam ad ejus <lb/> circumferentiam, & errores in minoribus ar­<lb/>cubus &longs;int majoris momenti:<arrow.to.target n="note385"/> ideo <emph type="italics"/>Ca&longs;&longs;inus<emph.end type="italics"/><lb/>ju&longs;&longs;u Regio men&longs;uram Terræ per majora loco­<lb/>rum intervalla aggre&longs;&longs;us e&longs;t, & &longs;ubinde per <lb/> di&longs;tantiam inter Ob&longs;ervatorium Regium <emph type="italics"/>Pari&longs;ien&longs;e<emph.end type="italics"/>& villam <emph type="italics"/>Coli­<lb/>oure<emph.end type="italics"/>in <emph type="italics"/>Rou&longs;&longs;illon<emph.end type="italics"/>& Latitudinum differentiam 6<emph type="sup"/>gr.<emph.end type="sup"/> 18′, &longs;uppo­<lb/>nendo quod figura Terræ &longs;it Sphærica, invenit gradum unum e&longs;&longs;e <lb/> hexapedarum 57292, prope ut <emph type="italics"/>Norwoodus<emph.end type="italics"/>no&longs;ter antea invenerat. <lb/> Hic enim circa annum 1635, men&longs;urando di&longs;tantiam pedum Lon­<lb/>dinen&longs;ium 905751 inter <emph type="italics"/>Londinum<emph.end type="italics"/>& <emph type="italics"/>Eboracum,<emph.end type="italics"/>& ob&longs;ervando <lb/> differentiam Latitudinum 2<emph type="sup"/>gr.<emph.end type="sup"/> 28′, collegit men&longs;uram gradus unius <lb/> e&longs;&longs;e pedum Londinen&longs;ium 367196, id e&longs;t, hexapedarum Pari&longs;ien­<lb/>&longs;ium 57300. Ob magnitudinem intervalli a <emph type="italics"/>Ca&longs;&longs;ino<emph.end type="italics"/>mon&longs;urati, pro <lb/> men&longs;ura gradus unius in medio intervalli illius, id e&longs;t, inter La­<lb/>titudines 45<emph type="sup"/>gr.<emph.end type="sup"/> & 46<emph type="sup"/>gr.<emph.end type="sup"/> u&longs;urpabo hexapedas 57292. Unde, &longs;i <lb/> Terra &longs;it Sphærica, &longs;emidiameter ejus erit pedum Pari&longs;ien&longs;ium <lb/> 19695539. <lb/> <pb xlink:href="039/01/407.jpg" pagenum="379"/><lb/></s></p> <p type="foot"> <s><foot.target id="note385"/>Vide Hi&longs;toriam Aca­<lb/>demiæ Regiæ &longs;cientiarum <lb/> anno 1700.</s></p> <p type="main"> <s>Penduli in Latitudine <emph type="italics"/>Lutetiæ Pari&longs;iorum<emph.end type="italics"/>ad minuta &longs;ecunda <lb/> <arrow.to.target n="note386"/>o&longs;cillantis longitudo e&longs;t pedum trium Pari&longs;ien&longs;ium & linearum 8 5/9. <lb/> Et longitudo quod grave tempore minuti unius &longs;ecundi cadendo <lb/> de&longs;cribit, e&longs;t ad dimidiam longitudinem penduli hujus, in duplicata <lb/> ratione circumferentiæ circuli ad diametrum ejus (ut indicavit <lb/> <emph type="italics"/>Hugenius<emph.end type="italics"/>) ideoque e&longs;t pedum Pari&longs;ien&longs;ium 15, dig. 1, lin. (2 1/189), &longs;eu <lb/> linearum (2174 1/18). <lb/> </s></p> <p type="margin"> <s><margin.target id="note386"/>LIBER <lb/> TERTIUS.</s></p> <p type="main"> <s>Corpus in circulo, ad di&longs;tantiam pedum 19695539 a centro, <lb/> &longs;ingulis diebus &longs;idereis horarum 23. 56′. 4″ uniformiter revolvens, <lb/> tempore minuti unius &longs;ecundi de&longs;cribit arcum pedum 1436,223, <lb/> cujus &longs;inus ver&longs;us e&longs;t pedum 0,05236558, &longs;eu linearum 7,54064. <lb/> Ideoque vis qua gravia de&longs;cendunt in Latitudine <emph type="italics"/>Lutetiæ,<emph.end type="italics"/>e&longs;t ad <lb/> vim centrifugam corporum &c. in Æquatore, a Terræ motu diurno <lb/> oriundam, ut (2174 1/18) ad 7,54064. <lb/> </s></p> <p type="main"> <s>Vis centrifuga corporum in Æquatore, e&longs;t ad vim centrifugam <lb/> qua corpora directe tendunt a Terra in Latitudine <emph type="italics"/>Lutetiæ<emph.end type="italics"/>gra­<lb/>duum 48. 50′, in duplicata ratione Radii ad &longs;inum complementi <lb/> Latitudinis illius, id e&longs;t, ut 7,54064 ad 3,267. Addatur hæc vis <lb/> ad vim qua gravia de&longs;cendunt in Latitudine <emph type="italics"/>Lutetiæ,<emph.end type="italics"/>& corpus <lb/> in Latitudine <emph type="italics"/>Lutetiæ<emph.end type="italics"/>vi tota gravitatis cadendo, tempore minuti <lb/> unius &longs;ecundi de&longs;criberet lineas 2177,32, &longs;eu pedes Pari&longs;ien&longs;es 15, <lb/> dig. 1, & lin. 5,32. Et vis tota gravitatis in Latitudine illa, erit <lb/> ad vim centri&longs;ugam corporum &c. in Æquatore Terræ, ut 2177,32 <lb/> ad 7,54064, &longs;eu 289 ad 1. <lb/> </s></p> <p type="main"> <s>Unde &longs;i <emph type="italics"/>APBQ<emph.end type="italics"/>figuram Terræ de&longs;ignet jam non amplius <lb/> Sphæricam &longs;ed revolutione Ellip&longs;eos circum axem minorem <emph type="italics"/>PQ<emph.end type="italics"/><lb/>genitam, &longs;itque <emph type="italics"/>ACQqca<emph.end type="italics"/>canalis aquæ ple­<lb/><figure id="id.039.01.407.1.jpg" xlink:href="039/01/407/1.jpg"/><lb/>na, a polo <emph type="italics"/>Qq<emph.end type="italics"/>ad centrum <emph type="italics"/>Cc,<emph.end type="italics"/>& inde ad <lb/> Æquatorem <emph type="italics"/>Aa<emph.end type="italics"/>pergens: debebit pondus <lb/> aquæ in canalis crure <emph type="italics"/>ACca,<emph.end type="italics"/>e&longs;&longs;e ad pondus <lb/> aquæ in crure altero <emph type="italics"/>QCcq<emph.end type="italics"/>ut 289 ad 288, <lb/> eo quod vis centrifuga ex circulari motu <lb/> orta partem unam e ponderis partibus 289 <lb/> &longs;u&longs;tinebit ac detrahet, & pondus 288 in al­<lb/>tero crure &longs;u&longs;tinebit reliquas. </s> <s>Porro (ex <lb/> Propo&longs;itionis XCI. Corollario &longs;ecundo, Lib.I.) <lb/> computationem ineundo, invenio quod &longs;i Terra con&longs;taret ex uni­<lb/>formi materia, motuque omni privaretur, & e&longs;&longs;et ejus axis <emph type="italics"/>PQ<emph.end type="italics"/><lb/><pb xlink:href="039/01/408.jpg" pagenum="380"/><lb/><arrow.to.target n="note387"/>ad diametrum <emph type="italics"/>AB<emph.end type="italics"/>ut 100 ad 101: gravitas in loco <emph type="italics"/>Q<emph.end type="italics"/>in Terram, <lb/> foret ad gravitatem in eodem loco <emph type="italics"/>Q<emph.end type="italics"/>in Sphæram centro <emph type="italics"/>C<emph.end type="italics"/>radio <lb/> <emph type="italics"/>PC<emph.end type="italics"/>vel <emph type="italics"/>QC<emph.end type="italics"/>de&longs;criptam, ut 126 ad 125. Et eodem argumento <lb/> gravitas in loco <emph type="italics"/>A<emph.end type="italics"/>in Sphæroidem, convolutione Ellip&longs;eos <emph type="italics"/>APBQ<emph.end type="italics"/><lb/>circa axem <emph type="italics"/>AB<emph.end type="italics"/>de&longs;criptam, e&longs;t ad gravitatem in eodem loco <emph type="italics"/>A<emph.end type="italics"/>in <lb/> Sphæram centro <emph type="italics"/>C<emph.end type="italics"/>radio <emph type="italics"/>AC<emph.end type="italics"/>de&longs;criptam, ut 125 ad 126. E&longs;t au­<lb/>tem gravitas in loco <emph type="italics"/>A<emph.end type="italics"/>in Terram, media proportionalis inter <lb/> gravitates in dictam Sphæroidem & Sphæram: propterea quod <lb/> Sphæra, diminuendo diametrum <emph type="italics"/>PQ<emph.end type="italics"/>in ratione 101 ad 100, <lb/> vertitur in figuram Terræ; & hæc figura diminuendo in eadem <lb/> ratione diametrum tertiam, quæ diametris duabus <emph type="italics"/>AB, PQ<emph.end type="italics"/>per­<lb/>pendicularis e&longs;t, vertitur in dictam Sphæroidem; & gravitas in <lb/> <emph type="italics"/>A,<emph.end type="italics"/>in ca&longs;u utroque, diminuitur in eadem ratione quam proxime. <lb/> E&longs;t igitur gravitas in <emph type="italics"/>A<emph.end type="italics"/>in Sphæram centro <lb/> <figure id="id.039.01.408.1.jpg" xlink:href="039/01/408/1.jpg"/><lb/><emph type="italics"/>C<emph.end type="italics"/>radio <emph type="italics"/>AC<emph.end type="italics"/>de&longs;criptam, ad gravitatem in <lb/> <emph type="italics"/>A<emph.end type="italics"/>in Terram ut 126 ad 125 1/2, & gravitas <lb/> in loco <emph type="italics"/>Q<emph.end type="italics"/>in Sphæram centro <emph type="italics"/>C<emph.end type="italics"/>radio <emph type="italics"/>QC<emph.end type="italics"/><lb/>de&longs;criptam, e&longs;t ad gravitatem in loco <emph type="italics"/>A<emph.end type="italics"/>in <lb/> Sphæram centro <emph type="italics"/>C<emph.end type="italics"/>radio <emph type="italics"/>AC<emph.end type="italics"/>de&longs;criptam, <lb/> in ratione diametrorum (per Prop. LXXII. <lb/> Lib. I.) id e&longs;t, ut 100 ad 101. Conjungan­<lb/>tur jam hæ tres rationes, 126 ad 125, 126 <lb/> ad 125 1/2, & 100 ad 101: & fiet gravitas <lb/> in loco <emph type="italics"/>Q<emph.end type="italics"/>in Terram, ad gravitatem in loco <emph type="italics"/>A<emph.end type="italics"/>in Terram, ut <lb/> 126X126X100 ad 125X125 1/2X101, &longs;eu ut 501 ad 500. <lb/> </s></p> <p type="margin"> <s><margin.target id="note387"/>DE MUNDI <lb/> SYSTEMATE</s></p> <p type="main"> <s>Jam cum (per Corol. 3. Prop. XCI. Lib. I.) gravitas in canalis <lb/> crure utrovis <emph type="italics"/>ACca<emph.end type="italics"/>vel <emph type="italics"/>QCcq<emph.end type="italics"/>&longs;it ut di&longs;tantia loeorum a centro <lb/> Terræ; &longs;i crura illa &longs;uperficiebus tran&longs;ver&longs;is & æquidi&longs;tantibus di­<lb/>&longs;tinguantur in partes totis proportionales, erunt pondera partium <lb/> &longs;ingularum in crure <emph type="italics"/>ACca<emph.end type="italics"/>ad pondera partium totidem in crure <lb/> altero, ut magnitudines & gravitates acceleratrices conjunctim; id <lb/> e&longs;t, ut 101 ad 100 & 500 ad 501, hoc e&longs;t, ut 505 ad 501. Ac <lb/> proinde &longs;i vis centrifuga partis cuju&longs;Q.E.I. crure <emph type="italics"/>ACca<emph.end type="italics"/>ex motu <lb/> diurno oriunda, fui&longs;&longs;et ad pondus partis eju&longs;dem ut 4 ad 505, eo <lb/> ut de pondere partis cuju&longs;que, in partes 505 divi&longs;o, partes qua­<lb/>tuor detraheret; manerent pondera in utroque crure æqualia, & <lb/> propterea fluidum con&longs;i&longs;teret in æquilibrio. </s> <s>Verum vis centrifuga <lb/> partis cuju&longs;que e&longs;t ad pondus eju&longs;dem ut 1 ad 289, hoc e&longs;t, vis <lb/> centrifuga quæ deberet e&longs;&longs;e ponderis pars (4/505) e&longs;t tantum pars (1/289). <lb/> <pb xlink:href="039/01/409.jpg" pagenum="381"/><lb/>Et propterea dico, &longs;ecundum Regulam auream, quod &longs;i vis cen­<lb/><arrow.to.target n="note388"/>trifuga (4/505) faciat ut altitudo aquæ in crure <emph type="italics"/>ACca<emph.end type="italics"/>&longs;uperet altitu­<lb/>dinem aquæ in crure <emph type="italics"/>QCcq<emph.end type="italics"/>parte cente&longs;ima totius altitudinis: <lb/> vis centrifuga (1/289) faciet ut exce&longs;&longs;us altitudinis in crure <emph type="italics"/>ACca<emph.end type="italics"/>&longs;it <lb/> altitudinis in crure altero <emph type="italics"/>QCcq<emph.end type="italics"/>pars tantum (1/229). E&longs;t igitur dia­<lb/>meter Terræ &longs;ecundum æquatorem ad ip&longs;ius diametrum per polos <lb/> ut 230 ad 229. Ideoque cum Terræ &longs;emidiameter mediocris, juxta <lb/> men&longs;uram <emph type="italics"/>Ca&longs;&longs;ini,<emph.end type="italics"/>&longs;it. pedum Pari&longs;ien&longs;ium 19695539, &longs;eu milliarium <lb/> 3939 (po&longs;ito quod milliare &longs;it men&longs;ura pedum 5000) Terra altior <lb/> erit ad Æquatorem quam ad Polos exce&longs;&longs;u pedum 85820, &longs;eu <lb/> milliarum 17 1/6. <lb/> </s></p> <p type="margin"> <s><margin.target id="note388"/>LIBER <lb/> TERTIUS.</s></p> <p type="main"> <s>Si Planeta major &longs;it vel minor quam Terra manente ejus den­<lb/>&longs;itate ac tempore periodico revolutionis diurnæ, manebit pro­<lb/>portio vis centrifugæ ad gravitatem, & propterea manebit etiam <lb/> proportio diametri inter polos ad diametrum &longs;ecundum æquato­<lb/>rem. </s> <s>At &longs;i motus diurnus in ratione quacunque acceleretur vel <lb/> retardetur, augebitur vel minuetur vis centrifuga in duplicata illa <lb/> ratione, & propterea differentia diametrorum augebitur vel mi­<lb/>nuetur in eadem duplicata ratione quamproxime. </s> <s>Et &longs;i den&longs;itas <lb/> Planetæ augeatur vel minuatur in ratione quavis, gravitas etiam <lb/> in ip&longs;um tendens augebitur vel minuetur in eadem ratione, & <lb/> differentia diametrorum vici&longs;&longs;im minuetur in ratione gravitatis <lb/> auctæ vel augebitur in ratione gravitatis diminutæ. Unde cum <lb/> Terra re&longs;pectu fixarum revolvatur horis 23. 56′, Jupiter autem <lb/> horis 9. 56′, &longs;intque temporum quadrata ut 29 ad 5, & den&longs;itates <lb/> ut 5 ad 1: differentia diametrorum Jovis erit ad ip&longs;ius diame­<lb/>trum minorem ut (29/5)X(5/1)X(1/229) ad 1, &longs;eu 1 ad 8 quamproxime. </s> <s>E&longs;t <lb/> igitur diameter Jovis ab oriente in occidentem ducta, ad ejus dia­<lb/>metrum inter polos ut 9 ad 8 quamproxime, & propterea diame­<lb/>ter inter polos e&longs;t 35 1/2″. Hæc ita &longs;e habent ex hypothe&longs;i quod <lb/> uniformis &longs;it Planetarum materia. </s> <s>Nam &longs;i materia den&longs;ior &longs;it ad <lb/> centrum quam ad circumferentiam; diameter quæ ab oriente in <lb/> occidentem ducitur, erit adhuc major. <lb/> </s></p> <p type="main"> <s>Jovis vero diametrum quæ polis ejus interjacet minorem e&longs;&longs;e <lb/> diametro altera <emph type="italics"/>Ca&longs;&longs;inus<emph.end type="italics"/>dudum ob&longs;ervavit, & Terræ diametrum <lb/> inter polos minorem e&longs;&longs;e diametro altera patebit per ea quæ <lb/> dicentur in Propo&longs;itione &longs;equente. <lb/> <pb xlink:href="039/01/410.jpg" pagenum="382"/><lb/><arrow.to.target n="note389"/>PROPOSITIO XX. PROBLEMA IV.<lb/></s></p> <p type="margin"> <s><margin.target id="note389"/>DE MUNDI <lb/> SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Invenire & inter &longs;e comparare Pondera corporum in Terræ hujus <lb/> regionibus diver&longs;is.<emph.end type="italics"/><lb/></s></p> <p type="main"> <s>Quoniam pondera inæqualium crurum canalis aqueæ <emph type="italics"/>ACQqca<emph.end type="italics"/><lb/>æqualia &longs;unt; & pondera partium, cruribus totis proportionalium <lb/> & &longs;imiliter in totis &longs;itarum, &longs;unt ad invicem ut pondera totorum, <lb/> adeoque etiam æquantur inter &longs;e; erunt pondera æqualium & in <lb/> cruribus &longs;imiliter &longs;itarum partium reciproce ut crura, id e&longs;t, reci­<lb/>proce ut 230 ad 229. Et par e&longs;t ratio homogeneorum & æqua­<lb/>lium quorumvis & in canalis cruribus &longs;imiliter &longs;itorum corporum. <lb/> Horum pondera &longs;unt reciproce ut crura, id e&longs;t, reciproce ut di­<lb/>&longs;tantiæ corporum a centro Terræ. Proinde &longs;i corpora in &longs;upre­<lb/>mis canalium partibus, &longs;ive in &longs;uperficie Terræ con&longs;i&longs;tant; erunt <lb/> pondera eorum ad invicem reciproce ut di&longs;tantiæ eorum a centro. <lb/> Et eodem argumento pondera, in aliis quibu&longs;cunque per totam <lb/> Terræ &longs;uperficiem regionibus, &longs;unt reciproce ut di&longs;tantiæ loeorum <lb/> a centro; & propterea, ex Hypothe&longs;i quod Terra Sphærois &longs;it, <lb/> dantur proportione. <lb/> </s></p> <p type="main"> <s>Unde tale confit Theorema, quod incrementum ponderis per­<lb/>gendo ab Æquatore ad Polos, &longs;it quam proxime ut &longs;inus ver&longs;us <lb/> Latitudinis duplicatæ, vel, quod perinde e&longs;t, ut quadratum &longs;inus <lb/> recti Latitudinis. </s> <s>Et in eadem circiter ratione augentur arcus <lb/> graduum Latitudinis in Meridiano. </s> <s>Ideoque cum Latitudo <emph type="italics"/>Lu­<lb/>tetiæ Pari&longs;iorum<emph.end type="italics"/>&longs;it 48<emph type="sup"/>gr.<emph.end type="sup"/> 50′, ea loeorum &longs;ub Æquatore 00<emph type="sup"/>gr.<emph.end type="sup"/> 00′, <lb/> & ea loeorum ad Polos 90<emph type="sup"/>gr.<emph.end type="sup"/> & duplorum &longs;inus ver&longs;i &longs;int 11334, <lb/> 00000 & 20000, exi&longs;tente Radio 10000, & gravitas ad Polum &longs;it <lb/> ad gravitatem &longs;ub Æquatore ut 230 ad 229, & exce&longs;&longs;us gravi­<lb/>tatis ad Polum ad gravitatem &longs;ub Æquatore ut 1 ad 229: erit ex­<lb/>ce&longs;&longs;us gravitatis in Latitudine <emph type="italics"/>Lutetiæ<emph.end type="italics"/>ad gravitatem &longs;ub Æquatore, <lb/> ut 1X(11334/20000) ad 229, &longs;eu 5667 ad 2290000. Et propterea gravitates <lb/> totæ in his locis erunt ad invicem ut 2295667 ad 2290000. Quare <lb/> cum longitudines pendulorum æqualibus temporibus o&longs;cillantium <lb/> &longs;int ut gravitates, & in Latitudine <emph type="italics"/>Lutetiæ Pari&longs;iorum<emph.end type="italics"/>longitudo <lb/> penduli &longs;ingulis minutis &longs;ecundis o&longs;cillantis &longs;it pedum trium Pa­<lb/>ri&longs;ien&longs;ium & linearum 8 1/9: longitudo penduli &longs;ub Æquatore &longs;u­<lb/>perabitur a longitudine &longs;ynchroni penduli <emph type="italics"/>Pari&longs;ien&longs;is,<emph.end type="italics"/>exce&longs;&longs;u li­<lb/>neæ unius & 87 partium mille&longs;imarum lineæ. Et &longs;imili computo <lb/> confit Tabula &longs;equens. <lb/> <pb xlink:href="039/01/411.jpg" pagenum="383"/><lb/><arrow.to.target n="note390"/></s> </p><table><row><cell><emph type="italics"/>Latitudo <lb/> Loci<emph.end type="italics"/></cell><cell><emph type="italics"/>Longitudo <lb/> Penduli<emph.end type="italics"/></cell><cell><emph type="italics"/>Men&longs;ura <lb/> Gradus unius <lb/> in Meridiano<emph.end type="italics"/></cell></row><row><cell>Gr.</cell><cell>Ped.</cell><cell>Lin.</cell><cell>Hexaped.</cell></row><row><cell>0</cell><cell>3.</cell><cell>7,468</cell><cell>56909</cell></row><row><cell>5</cell><cell>3.</cell><cell>7,482</cell><cell>56914</cell></row><row><cell>10</cell><cell>3.</cell><cell>7,526</cell><cell>56931</cell></row><row><cell>15</cell><cell>3.</cell><cell>7,596</cell><cell>56959</cell></row><row><cell>20</cell><cell>3.</cell><cell>7,692</cell><cell>56996</cell></row><row><cell>25</cell><cell>3.</cell><cell>7,811</cell><cell>57042</cell></row><row><cell>30</cell><cell>3.</cell><cell>7,948</cell><cell>57096</cell></row><row><cell>35</cell><cell>3.</cell><cell>8,099</cell><cell>57155</cell></row><row><cell>40</cell><cell>3.</cell><cell>8,261</cell><cell>57218</cell></row><row><cell>1</cell><cell>3.</cell><cell>8,294</cell><cell>57231</cell></row><row><cell>2</cell><cell>3.</cell><cell>8,327</cell><cell>57244</cell></row><row><cell>3</cell><cell>3.</cell><cell>8,361</cell><cell>57257</cell></row><row><cell>4</cell><cell>3.</cell><cell>8,394</cell><cell>57270</cell></row><row><cell>45</cell><cell>3.</cell><cell>8,428</cell><cell>57283</cell></row><row><cell>6</cell><cell>3.</cell><cell>8,461</cell><cell>57296</cell></row><row><cell>7</cell><cell>3.</cell><cell>8,494</cell><cell>57309</cell></row><row><cell>8</cell><cell>3.</cell><cell>8,528</cell><cell>57322</cell></row><row><cell>9</cell><cell>3.</cell><cell>8,561</cell><cell>57335</cell></row><row><cell>50</cell><cell>3.</cell><cell>8,594</cell><cell>57348</cell></row><row><cell>55</cell><cell>3.</cell><cell>8,756</cell><cell>57411</cell></row><row><cell>60</cell><cell>3.</cell><cell>8,907</cell><cell>57470</cell></row><row><cell>65</cell><cell>3.</cell><cell>9,044</cell><cell>57524</cell></row><row><cell>70</cell><cell>3.</cell><cell>9,162</cell><cell>57570</cell></row><row><cell>75</cell><cell>3.</cell><cell>9,258</cell><cell>57607</cell></row><row><cell>80</cell><cell>3.</cell><cell>9,329</cell><cell>57635</cell></row><row><cell>85</cell><cell>3.</cell><cell>9,372</cell><cell>57652</cell></row><row><cell>90</cell><cell>3.</cell><cell>9,387</cell><cell>57657</cell></row></table> <p type="main"> <s>Con&longs;tat autem per hanc Tabulam, quod graduum inæqualitas <lb/>tam parva &longs;it, ut in rebus Geographicis figura Terræ pro Sphæ­<lb/>rica haberi po&longs;&longs;it, quodQ.E.I.æqualitas diametrorum Terræ faci­<lb/>lius & certius per experimenta pendulorum deprehendi po&longs;&longs;it vel <lb/>etiam per Eclip&longs;es Lunæ, quam per arcus Geographice men&longs;uratos <lb/>in Meridiano. <pb xlink:href="039/01/412.jpg" pagenum="384"/><arrow.to.target n="note415"/></s></p> <p type="margin"> <s><margin.target id="note415"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Hæc ita &longs;e habent ex hypothe&longs;i quod Terra ex uniformi ma­<lb/>teria con&longs;tat. </s> <s>Nam &longs;i materia ad centrum paulo den&longs;ior &longs;it quam <lb/>ad &longs;uperficiem, differentiæ pendulorum & graduum Meridiani <lb/>paulo majores erunt quam pro Tabula præcedente, propterea <lb/>quod &longs;i materia ad centrum redundans qua den&longs;itas ibi major <lb/>redditur, &longs;ubducatur & &longs;eor&longs;im &longs;pectetur, gravitas in Terram re­<lb/>liquam uniformiter den&longs;am, erit reciproce ut di&longs;tantia ponderis <lb/>a centro; in materiam vero redundantem reciproce ut quadratum <lb/>di&longs;tantiæ a materia illa quamproxime. </s> <s>Gravitas igitur &longs;ub æqua­<lb/>tore minor e&longs;t in materiam illam redundantem quam pro com­<lb/>puto &longs;uperiore: & propterea Terra ibi, propter defectum gravita­<lb/>tis, paulo altius a&longs;cendet, & exce&longs;&longs;us longitudinum Pendulorum & <lb/>graduum ad polos paulo majores erunt quam in præcedentibus <lb/>definitum e&longs;t. </s></p> <p type="main"> <s>Jam vero A&longs;tronomi aliqui in longinquas regiones ad ob&longs;erva­<lb/>tiones A&longs;tronomicas faciendas mi&longs;&longs;i, invenerunt quod horologia <lb/>o&longs;cillatoria tardius moverentur prope Æquatorem quam in regi­<lb/>onibus no&longs;tris. </s> <s>Et primo quidem <emph type="italics"/>D. Richer<emph.end type="italics"/>hoc ob&longs;ervavit anno <lb/>1672 in in&longs;ula <emph type="italics"/>Cayennæ.<emph.end type="italics"/>Nam dum ob&longs;ervaret tran&longs;itum Fixarum <lb/>per meridianum men&longs;e <emph type="italics"/>Augu&longs;to,<emph.end type="italics"/>reperit horologium &longs;uum tardius <lb/>moveri quam pro medio motu Solis, exi&longs;tente differentia 2′. </s> <s>28″ <lb/>&longs;ingulis diebus. </s> <s>Deinde faciendo ut Pendulum &longs;implex ad minuta <lb/>&longs;ingula &longs;ecunda per horologium optimum men&longs;urata o&longs;cillaret, <lb/>notavit longitudinem Penduli &longs;implicis, & hoc fecit &longs;æpius &longs;ingu­<lb/>lis &longs;eptimanis per men&longs;es decem. </s> <s>Tum in <emph type="italics"/>Galliam<emph.end type="italics"/>redux contulit <lb/>longitudinem hujus Penduli cum longitudine Penduli <emph type="italics"/>Pari&longs;ien&longs;is<emph.end type="italics"/><lb/>(quæ erat trium pedum Pari&longs;ien&longs;ium, & octo linearum cum tribus <lb/>quintis partibus lineæ) & reperit breviorem e&longs;&longs;e, exi&longs;tente diffe­<lb/>rentia lineæ unius cum quadrante. </s> <s>At ex tarditate horologii <lb/>o&longs;cillatorii in <emph type="italics"/>Cayenna,<emph.end type="italics"/>differentia Pendulorum colligitur e&longs;&longs;e lineæ <lb/>unius cum &longs;emi&longs;&longs;e. </s></p> <p type="main"> <s>Po&longs;tea <emph type="italics"/>Halleius<emph.end type="italics"/>no&longs;ter circa annum 1677 ad in&longs;ulam <emph type="italics"/>S<emph type="sup"/>sa<emph.end type="sup"/> Hel­<lb/>lenæ<emph.end type="italics"/>navigans, reperit horologium &longs;uum o&longs;cillatorium ibi tardius <lb/>moveri quam <emph type="italics"/>Londini,<emph.end type="italics"/>&longs;ed differentiam non notavit. </s> <s>Pendulum <lb/>vero brevius reddidit plu&longs;quam octava parte digiti, &longs;eu linea una <lb/>cum &longs;emi&longs;&longs;e. </s> <s>Et ad hoc efficiendum, cum longitudo cochleæ in <lb/>ima parte penduli non &longs;ufficeret, annulum ligneum thecæ cochleæ <lb/>& ponderi pendulo interpo&longs;uit. </s></p> <p type="main"> <s>Deinde anno 1682 <emph type="italics"/>D. Varin<emph.end type="italics"/>& <emph type="italics"/>D. </s> <s>Des Hayes<emph.end type="italics"/>invenerunt lon­<lb/>gitudinem Penduli &longs;ingulis minutis &longs;ecundis o&longs;cillantis in Ob&longs;er-<pb xlink:href="039/01/413.jpg" pagenum="385"/>vatorio Regio <emph type="italics"/>Pari&longs;ien&longs;i<emph.end type="italics"/>e&longs;&longs;e ped. </s> <s>3. lin. </s> <s>8 1/9. Et in in&longs;ula <emph type="italics"/>Gorea<emph.end type="italics"/></s></p> <p type="main"> <s><arrow.to.target n="note416"/>eadem methodo longitudinem Penduli &longs;ynchroni invenerunt e&longs;&longs;e <lb/>ped. </s> <s>3. lin. </s> <s>6 5/9, exi&longs;tente longitudinum differentia lin. </s> <s>2. Et eodem <lb/>anno ad in&longs;ulas <emph type="italics"/>Guadaloupam<emph.end type="italics"/>& <emph type="italics"/>Martinicam<emph.end type="italics"/>navigantes, invenerunt <lb/>longitudinem Penduli &longs;ynchroni in his in&longs;ulis e&longs;&longs;e ped. </s> <s>3. lin. </s> <s>6 1/3. </s></p> <p type="margin"> <s><margin.target id="note416"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Po&longs;thac <emph type="italics"/>D. Couplet<emph.end type="italics"/>filius anno 1697 men&longs;e <emph type="italics"/>Julio,<emph.end type="italics"/>horologium <lb/>&longs;uum o&longs;cillatorium ad motum Solis medium in Ob&longs;ervatorio Regio <lb/><emph type="italics"/>Pari&longs;ien&longs;i<emph.end type="italics"/>&longs;ic aptavit, ut tempore &longs;atis longo horologium cum motu <lb/>Solis congrueret. </s> <s>Deinde <emph type="italics"/>Uly&longs;&longs;ipponem<emph.end type="italics"/>navigans invenit quod <lb/>men&longs;e <emph type="italics"/>Novembri<emph.end type="italics"/>proximo horologium tardius iret quam prius, <lb/>exi&longs;tente differentia 2′. </s> <s>13″ in horis 24. Et men&longs;e <emph type="italics"/>Martio<emph.end type="italics"/>&longs;e­<lb/>quente <emph type="italics"/>Paraibam<emph.end type="italics"/>navigans invenit ibi horologium &longs;uum tardius <lb/>ire quam <emph type="italics"/>Pari&longs;iis,<emph.end type="italics"/>exi&longs;tente differentia 4′. </s> <s>12″ in horis 24. Et <lb/>affirmat Pendulum ad minuta &longs;ecunda o&longs;cillans brevius fui&longs;&longs;e <emph type="italics"/>Uly&longs;­<lb/>&longs;ipponi<emph.end type="italics"/>lineis 2 1/2 & <emph type="italics"/>Paraibæ<emph.end type="italics"/>lineis 3 2/3 quam <emph type="italics"/>Pari&longs;iis.<emph.end type="italics"/>Rectius po­<lb/>&longs;ui&longs;&longs;et differentias e&longs;&longs;e 1 1/3 & 2 5/9. Nam hæ differentiæ differen­<lb/>tiis temporum 2′. </s> <s>13″, & 4′. </s> <s>12″ re&longs;pondent. </s> <s>Cra&longs;&longs;ioribus hujus <lb/>Ob&longs;ervationibus minus fidendum e&longs;t. </s></p> <p type="main"> <s>Annis proximis (1699 & 1700) <emph type="italics"/>D. </s> <s>Des Hayes<emph.end type="italics"/>ad <emph type="italics"/>Americam<emph.end type="italics"/><lb/>denuo navigans, determinavit quod in in&longs;ulis <emph type="italics"/>Cayennæ<emph.end type="italics"/>& <emph type="italics"/>Granadæ<emph.end type="italics"/><lb/>longitudo Penduli ad minuta &longs;ecunda o&longs;cillantis, e&longs;&longs;et paulo minor <lb/>quam ped. </s> <s>3. lin. </s> <s>6 1/2, quodQ.E.I. in&longs;ula <emph type="italics"/>S. Chri&longs;tophori<emph.end type="italics"/>longitudo <lb/>illa e&longs;&longs;et ped. </s> <s>3. lin. </s> <s>6 1/4, & quod in in&longs;ula <emph type="italics"/>S. Dominici<emph.end type="italics"/>eadem e&longs;&longs;et <lb/>ped. </s> <s>3. lin. </s> <s>7. </s></p> <p type="main"> <s>Annoque 1704. <emph type="italics"/>P. Feuelleus<emph.end type="italics"/>invenit in <emph type="italics"/>Porto-belo<emph.end type="italics"/>in <emph type="italics"/>America<emph.end type="italics"/><lb/>longitudinem Penduli ad minuta &longs;ecunda o&longs;cillantis, e&longs;&longs;e pedum <lb/>trium Pari&longs;ien&longs;ium & linearum tantum (5 7/12), id e&longs;t, tribus fere li­<lb/>neis breviorem quam <emph type="italics"/>Lutetiæ Pari&longs;iorum,<emph.end type="italics"/>&longs;ed errante Ob&longs;erva­<lb/>tione. </s> <s>Nam deinde ad in&longs;ulam <emph type="italics"/>Martinicam<emph.end type="italics"/>navigans, invenit lon­<lb/>gitudinem Penduli i&longs;ochroni e&longs;&longs;e pedum tantum trium Pari&longs;ien­<lb/>&longs;ium & linearum (5 10/12). </s></p> <p type="main"> <s>Latitudo autem <emph type="italics"/>Paraibæ<emph.end type="italics"/>e&longs;t 6<emph type="sup"/>gr.<emph.end type="sup"/> 38′ ad au&longs;trum, & ea <emph type="italics"/>Porto­<lb/>beli<emph.end type="italics"/>9<emph type="sup"/>gr.<emph.end type="sup"/> 33′ ad boream, & Latitudines in&longs;ularum <emph type="italics"/>Cayennæ, Goreæ, <lb/>Guadaloupæ, Martinicæ, Granadæ, S<emph type="sup"/>ti.<emph.end type="sup"/> Chri&longs;tophori,<emph.end type="italics"/>& <emph type="italics"/>S<emph type="sup"/>ti.<emph.end type="sup"/> Domi­<lb/>nici<emph.end type="italics"/>&longs;unt re&longs;pective 4<emph type="sup"/>gr.<emph.end type="sup"/> 55′, 14<emph type="sup"/>gr.<emph.end type="sup"/> 40′, 14<emph type="sup"/>gr.<emph.end type="sup"/> 00′, 14<emph type="sup"/>gr.<emph.end type="sup"/> 44′, 12<emph type="sup"/>gr.<emph.end type="sup"/> 6′, <lb/>17<emph type="sup"/>gr.<emph.end type="sup"/> 19′, & 19<emph type="sup"/>gr.<emph.end type="sup"/> 48′ ad boream. </s> <s>Et exce&longs;&longs;us longitudinis Pen­<lb/>duli <emph type="italics"/>Pari&longs;ien&longs;is<emph.end type="italics"/>&longs;upra longitudines Pendulorum i&longs;ochronorum in <lb/>his latitudinibus ob&longs;ervatas, &longs;unt paulo majores quam pro Ta­<lb/>bula longitudinum Penduli &longs;uperius computata. </s> <s>Et propterea <lb/>Terra aliquanto altior e&longs;t &longs;ub Æquatore quam pro &longs;uperiore cal-<pb xlink:href="039/01/414.jpg" pagenum="386"/><arrow.to.target n="note417"/>culo, & den&longs;ior ad centrum quam in fodinis prope &longs;uperficiem, <lb/>ni&longs;i forte calores in Zona torrida longitudinem Pendulorum ali­<lb/>quantulum auxerint. </s></p> <p type="margin"> <s><margin.target id="note417"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Ob&longs;ervavit utique <emph type="italics"/>D. Picartus<emph.end type="italics"/>quod virga ferrea, quæ tempore <lb/>hyberno ubi gelabant frigora erat pedis unius longitudine, ad <lb/>ignem calefacta eva&longs;it pedis unius cum quarta parte lineæ. </s> <s>De­<lb/>inde <emph type="italics"/>D. de la Hire<emph.end type="italics"/>ob&longs;ervavit quod virga ferrea quæ tempore <lb/>con&longs;imili hyberno &longs;ex erat pedum longitudinis, ubi Soli æ&longs;tivo <lb/>exponebatur eva&longs;it &longs;ex pedum longitudinis cum duabus tertiis <lb/>partibus lineæ. </s> <s>In priore ca&longs;u calor major fuit quam in po&longs;te­<lb/>riore, in hoc vero major fuit quam calor externarum partium <lb/>corporis humani. </s> <s>Nam metalla ad Solem æ&longs;tivum valde incale­<lb/>&longs;cunt. </s> <s>At virga penduli in horologio o&longs;cillatorio nunquam ex­<lb/>poni &longs;olet calori Solis æ&longs;tivi, nunquam calorem concipit calori <lb/>externæ &longs;uperficiei corporis humani æqualem. </s> <s>Et propterea virga <lb/>Penduli in horologio tres pedes longa, paulo quidem longior <lb/>erit tempore æ&longs;tivo quam hyberno, &longs;ed exce&longs;&longs;u quartam partem <lb/>lineæ unius vix &longs;uperante. </s> <s>Proinde differentia tota longitudinis <lb/>pendulorum quæ in diver&longs;is regionibus i&longs;ochrona &longs;unt, diver&longs;o <lb/>calori attribui non pote&longs;t. </s> <s>Sed neque erroribus A&longs;tronomorum è <lb/><emph type="italics"/>Gallia<emph.end type="italics"/>mi&longs;&longs;orum tribuenda e&longs;t hæc differentia. </s> <s>Nam quamvis <lb/>eorum ob&longs;ervationes non perfecte congruant inter &longs;e, tamen erro­<lb/>res &longs;unt adeo parvi ut contemni po&longs;&longs;int. </s> <s>Et in hoc concordant <lb/>omnes, quod i&longs;ochrona pendula &longs;unt breviora &longs;ub Æquatore quam <lb/>in Ob&longs;ervatorio Regio <emph type="italics"/>Pari&longs;ien&longs;i,<emph.end type="italics"/>exi&longs;tente differentia duarum cir­<lb/>citer linearum &longs;eu &longs;extæ partis digiti. </s> <s>Per ob&longs;ervationes <emph type="italics"/>D. Ri­<lb/>cher<emph.end type="italics"/>in <emph type="italics"/>Cayenna<emph.end type="italics"/>factas, differentia fuit lineæ unius cum &longs;emi&longs;&longs;e. </s> <s><lb/>Error &longs;emi&longs;&longs;is lineæ facile committitur. </s> <s>Et <emph type="italics"/>D. des Hayes<emph.end type="italics"/>po&longs;tea <lb/>per ob&longs;ervationes &longs;uas in eadem in&longs;ula factas errorem correxit, <lb/>inventa differentia linearum (2 1/18). Sed & per ob&longs;ervationes in in­<lb/>&longs;ulis <emph type="italics"/>Gorea, Guadaloupa, Martinica, Granada, S. Chri&longs;tophori,<emph.end type="italics"/>& <lb/><emph type="italics"/>S. Dominici<emph.end type="italics"/>factas & ad Æquatorem reductas, differentia illa pro­<lb/>diit haud minor quam (1 19/20) lineæ, haud major quam 2 1/2 linearum. </s> <s><lb/>Et inter hos limites quantitas mediocris e&longs;t (2 9/40) linearum. </s> <s>Prop­<lb/>ter calores loeorum in Zona torrida negligamus (9/40) partes lineæ, <lb/>& manebit differentia duarum linearum. </s></p> <p type="main"> <s>Quare cum differentia illa per Tabulam præcedentem, ex hy­<lb/>pothe&longs;i quod Terra ex materia uniformiter den&longs;a con&longs;tat, &longs;it tan­<lb/>tum (1 87/1000) lineæ: exce&longs;&longs;us altitudinis Terræ ad æquatorem &longs;upra <lb/>altitudinem ejus ad polos, qui erat milliarium 17 1/6, jam auctus in <pb xlink:href="039/01/415.jpg" pagenum="387"/>ratione differentiarum, fiet milliarium (31 7/18). Nam tarditas Pen­<lb/><arrow.to.target n="note418"/>duli &longs;ub Æquatore defectum gravitatis arguit; & quo levior e&longs;t <lb/>materia eo major e&longs;&longs;e debet altitudo ejus, ut pondere &longs;uo mate­<lb/>riam &longs;ub Polis in æquilibrio &longs;u&longs;tineat. </s></p> <p type="margin"> <s><margin.target id="note418"/>LIBFR <lb/>TERTIUS.</s></p> <p type="main"> <s>Hinc figura umbræ Terræ per Eclip&longs;es Lunæ determinanda, non <lb/>erit omnino circularis, &longs;ed diameter ejus ab oriente in occidentem <lb/>ducta major erit quam diameter ejus ab au&longs;tro in boream ducta, <lb/>exce&longs;&longs;u 55″ circiter. </s> <s>Et parallaxis maxima Lunæ in Longitudi­<lb/>nem paulo major erit quam ejus parallaxis maxima in Latitudi­<lb/>nem. </s> <s>Ac Terræ &longs;emidiameter maxima erit podum Pari&longs;ien&longs;ium <lb/>19767630, minima pedum 19609820 & mediocris pedum 19688725<emph type="sup"/>1<emph.end type="sup"/><lb/>quamproxime. </s></p> <p type="main"> <s>Cum gradus unus men&longs;urante <emph type="italics"/>Picarto<emph.end type="italics"/>&longs;it hexapedarum 57060, <lb/>men&longs;urante vero <emph type="italics"/>Ca&longs;&longs;ino<emph.end type="italics"/>&longs;it hexapedarum 57292: &longs;u&longs;picantur ali­<lb/>qui gradum unumquemque, pergenda per <emph type="italics"/>Gallies<emph.end type="italics"/>au&longs;trum ver&longs;us <lb/>majorem e&longs;&longs;e gradu præcedente hexapedia plus minus: 72, &longs;eu <lb/>parte octingente&longs;ima gradus unius; exi&longs;tente Perra Sphæroide ob­<lb/>longa cujus partes ad polos &longs;unt alti&longs;&longs;imæ. </s> <s>Quo po&longs;ito, corpora <lb/>omnia ad polos Terræ leviora forent quam ad Æquatorem, & <lb/>altitudo Terræ ad polos &longs;uperaret altitudinem ejus ad æquatorem <lb/>milliaribus fere 95, & pendula i&longs;ochrona longiora forent ad Æ­<lb/>quatorem quem in Ob&longs;ervatorio Regio <emph type="italics"/>Pari&longs;ieu&longs;i<emph.end type="italics"/>exce&longs;&longs;u &longs;emi&longs;&longs;is <lb/>digiti circiter; ut con&longs;erenti proportiones hic po&longs;itas cum pro­<lb/>portionibus in Tabula præcedente po&longs;itis, facile con&longs;tabit. </s> <s>Sed <lb/>& diameter umbræ Terræ quæ ab au&longs;tro in boream ducitur, ma­<lb/>jor foret quam diameter ejus quæ ab oriente in occidentem duci­<lb/>tur, exce&longs;&longs;u 2′. </s> <s>46″, &longs;eu parte duodecima diametri Lunæ. </s> <s>Qui­<lb/>bus omnibus Experientia contrariatur. </s> <s>Certe <emph type="italics"/>Ca&longs;&longs;inus,<emph.end type="italics"/>definiendo <lb/>gradum unum e&longs;&longs;e hexapedarum 57292, medium inter men&longs;uras <lb/>&longs;uas omnes, ex hypothe&longs;i de æqualitate graduum a&longs;&longs;ump&longs;it. </s> <s>Et <lb/>quamvis <emph type="italics"/>Picartus<emph.end type="italics"/>in <emph type="italics"/>Galliæ<emph.end type="italics"/>limite boreali invenit gradum paulo <lb/>minorem e&longs;&longs;e, tamen <emph type="italics"/>Norwoodus<emph.end type="italics"/>no&longs;ter in regionibus magis bore­<lb/>alibus, men&longs;urando majus intervallum, invenit gradum paulo majo­<lb/>rem e&longs;&longs;e quam <emph type="italics"/>Ca&longs;&longs;inus<emph.end type="italics"/>invenerat. </s> <s>Et <emph type="italics"/>Ca&longs;&longs;inus<emph.end type="italics"/>ip&longs;e men&longs;uram <emph type="italics"/>Picarti,<emph.end type="italics"/><lb/>ob parvitatem intervalli men&longs;urati, non &longs;atis certam & exactam e&longs;&longs;e <lb/>judicavit ubi men&longs;uram gradus unius per intervallum longe majus <lb/>definire aggre&longs;&longs;us e&longs;t. </s> <s>Differentiæ vero inter men&longs;uras <emph type="italics"/>Ca&longs;&longs;ini, Pi­<lb/>carti,<emph.end type="italics"/>& <emph type="italics"/>Norwoodi<emph.end type="italics"/>&longs;unt prope in&longs;en&longs;ibiles, & ab in&longs;en&longs;ibilibus ob­<lb/>&longs;ervationum erroribus facilo oriri potuere, ut Nutationem axis <lb/>Terræ præteream. <pb xlink:href="039/01/416.jpg" pagenum="388"/><arrow.to.target n="note419"/></s></p> <p type="margin"> <s><margin.target id="note419"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXI. THEOREMA XVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Puncta Æquinoctialia regredi, & axem Terræ &longs;ingulis revoluti­<lb/>onibus annuis nutando bis inclinari in Eclipticam & bis re­<lb/>dire ad po&longs;itionem priorem.<emph.end type="italics"/></s></p> <p type="main"> <s>Patet per Corol. </s> <s>20. Prop. </s> <s>LXVI. Lib. </s> <s>I. </s> <s>Motus tamen i&longs;te <lb/>nutandi perexiguus e&longs;&longs;et debet, & vix aut ne vix quidem &longs;en­<lb/>&longs;ibilis. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXII. THEOREMA XVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus omnes Lunares, omne&longs;que motuum inæqualitates ex alla­<lb/>tis Principiis con&longs;equi.<emph.end type="italics"/></s></p> <p type="main"> <s>Planetas majores, interea dum circa Solem feruntur, po&longs;&longs;e alios <lb/>minores circum &longs;e revolventes Planetas deferre, & minores illos in <lb/>Ellip&longs;ibus, umbilicos in centris majorum habentibus, revolvi de­<lb/>bere patet per Prop. </s> <s>LXV. Lib. </s> <s>I. </s> <s>Actione autem Solis perturba­<lb/>buntur eorum motus multimode, ii&longs;que adficientur inæqualitati­<lb/>bus quæ in Luna no&longs;tra notantur. </s> <s>Hæc utique (per Corol. </s> <s>2, <lb/>3, 4, & 5. Prop. </s> <s>LXVI.) velocius movetur, ac radio ad Terram <lb/>ducto de&longs;cribit aream pro tempore majorem, Orbemque habet <lb/>minus curvum, atque adeo propius accedit ad Terram, in Syzygiis <lb/>quam in Quadraturis, ni&longs;i quatenus impedit motus Eccentricitatis. </s> <s><lb/>Eccentricitas enim maxima e&longs;t (per Corol. </s> <s>9. Prop. </s> <s>LXVI.) ubi <lb/>Apogæum Lunæ in Syzygiis ver&longs;atur, & minima ubi idem in Qua­<lb/>draturis con&longs;i&longs;tit; & inde Luna in Perigæo velocior e&longs;t & nobis <lb/>propior, in Apogæo autem tardior & remotior in Syzygiis quam <lb/>in Quadraturis. </s> <s>Progreditur in&longs;uper Apogæum, & regrediuntur <lb/>Nodi, &longs;ed motu inæquabili. </s> <s>Et Apogæum quidem (per Corol. </s> <s>7. <lb/>& 8. Prop. </s> <s>LXVI.) velocius progreditur in Syzygiis &longs;uis, tardius <lb/>regreditur in Quadraturis, & exce&longs;&longs;u progre&longs;&longs;us &longs;upra regre&longs;&longs;um <lb/>annuatim fertur in con&longs;equentia. </s> <s>Nodi autem (per Corol. </s> <s>11. <lb/>Prop. </s> <s>LXVI.) quie&longs;cunt in Syzygiis &longs;uis, & veloci&longs;&longs;ime regrediun­<lb/>tur in Quadraturis. </s> <s>Sed & major e&longs;t Lunæ latitudo maxima in <lb/>ip&longs;ius Quadraturis (per Corol. </s> <s>10. Prop. </s> <s>LXVI.) quam in Syzy­<lb/>giis: & motus medius tardior in Perihelio Terræ (per Corol. </s> <s>6. <pb xlink:href="039/01/417.jpg" pagenum="389"/>Prop. </s> <s>LXVI,) quam in ip&longs;ius Aphelio. </s> <s>Atque hæ &longs;unt inæquali­<lb/><arrow.to.target n="note420"/>tates in&longs;igniores ab A&longs;tronomis notatæ. </s></p> <p type="margin"> <s><margin.target id="note420"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Sunt etiam aliæ quædam nondum ob&longs;ervatæ inæqualitates, qui­<lb/>bus motus Lunares adeo perturbantur, ut nulla hactenus lege ad <lb/>Regulam aliquam certam reduci potuerint. </s> <s>Velocitates enim &longs;eu <lb/>motus horarii Apogæi & Nodorum Lunæ, & eorundem æquati­<lb/>ones, ut & differentia inter Eccentricitatem maximam in Syzygiis <lb/>& minimam in Quadraturis, & inæqualitas quæ Variatio dicitur, <lb/>augentur ac diminuuntur annuatim (per Corol. </s> <s>14. Prop. </s> <s>LXVI.) <lb/>in triplicata ratione diametri apparentis Solaris. </s> <s>Et Variatio præ­<lb/>terea augetur vel diminuitur in duplicata ratione temporis in­<lb/>ter quadraturas quam proxime (per Corol. </s> <s>1. & 2. Lem. </s> <s>X. & <lb/>Corol. </s> <s>16. Prop. </s> <s>LXVI. Lib. </s> <s>I.) Sed hæc inæqualitas in calculo <lb/>A&longs;tronomico, ad Pro&longs;thaphære&longs;in Lunæ referri &longs;olet, & cum ea <lb/>confundi. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIII. PROBLEMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Motus inæquales Satellitum Jovis & Saturni à motibus Luna­<lb/>ribus derivare.<emph.end type="italics"/></s></p> <p type="main"> <s>Ex motibus Lunæ no&longs;træ motus analogi Lunarum &longs;eu Satelli­<lb/>tum Jovis &longs;ic derivantur. </s> <s>Motus medius Nodorum Satellitis ex­<lb/>timi Jovialis, e&longs;t ad motum medium Nodorum Lunæ no&longs;træ, in ra­<lb/>tione compo&longs;ita ex ratione duplicata temporis periodici Terræ <lb/>circa Solem ad tempus periodicum Jovis circa Solem, & ratione <lb/>&longs;implici temporis periodici Satellitis circa Jovem ad tempus perio­<lb/>dicum Lunæ circa Terram: (per Corol. </s> <s>16. Prop. </s> <s>LXVI.) adeoque <lb/>annis centum conficit Nodus i&longs;te 8<emph type="sup"/>gr.<emph.end type="sup"/> 24′. </s> <s>in antecedentia. </s> <s>Motus <lb/>medii Nodorum Satellitum interiorum &longs;unt ad motum hujus, ut <lb/>illorum tempora periodica ad tempus periodicum hujus, per idem <lb/>Corollarium, & inde dantur. </s> <s>Motus autem Augis Satellitis cu­<lb/>ju&longs;Q.E.I. con&longs;equentia, e&longs;t ad motum Nodorum ip&longs;ius in antece­<lb/>dentia, ut motus Apogæi Lunæ no&longs;træ ad hujus motum Nodo­<lb/>rum, (per idem Corol.) & inde datur. </s> <s>Diminui tamen debet <lb/>motus Augis &longs;ic inventus in ratione 5 ad 9 vel 1 ad 2 circiter, ob <lb/>cau&longs;am quam hic exponere non vacat. </s> <s>Æquationes maximæ No­<lb/>dorum & Augis Satellitis cuju&longs;que fere &longs;unt ad æquationes maxi­<lb/>mas Nodorum & Augis Lunæ re&longs;pective, ut motus Nodorum & <lb/>Augis Satellitum tempore unius revolutionis æquationum prio-<pb xlink:href="039/01/418.jpg" pagenum="390"/><arrow.to.target n="note421"/>rum, ad motus Nodorum & Apogæi Lunæ tempore unius revo­<lb/>lutionis æquationum po&longs;teriorum. </s> <s>Variatio Satellitis è Jove &longs;pe­<lb/>ctati, e&longs;t ad Variationem Lunæ, ut &longs;unt ad invicem toti motus No­<lb/>dorum temporibus quibus Satelles & Luna ad Solem revolvuntur, <lb/>per idem Corollarium; adeoQ.E.I. Satellite extimo non &longs;uperat <lb/>5″. </s> <s>12′. </s></p> <p type="margin"> <s><margin.target id="note421"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIV. THEOREMA XIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Fluxum & refluxum Maris ab actionibus Solis ac <lb/>Lunæ oriri.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Mare &longs;ingulis diebus tam Lunaribus quam Solaribus bis intu­<lb/>me&longs;cere debere ac bis defluere, patet per Corol. </s> <s>19. Prop. </s> <s>LXVI. <lb/>Lib.I. ut & aquæ maximam altitudinem, in maribus profundis <lb/>& liberis, appul&longs;um Luminarium ad Meridianum loci, minori <lb/>quam &longs;ex horarum &longs;patio &longs;equi, uti fit in Maris <emph type="italics"/>Atlantici<emph.end type="italics"/>& <lb/><emph type="italics"/>Æthiopici<emph.end type="italics"/>tractu toto orientali inter <emph type="italics"/>Galliam<emph.end type="italics"/>& Promontorium <lb/><emph type="italics"/>Bonæ Spei,<emph.end type="italics"/>ut & in Maris <emph type="italics"/>Pacifici<emph.end type="italics"/>littore <emph type="italics"/>Chilen&longs;t<emph.end type="italics"/>& <emph type="italics"/>Peruviano<emph.end type="italics"/>: <lb/>in quibus omnibus littoribus æ&longs;tus in horam circiter tertiam in­<lb/>cidit, ni&longs;i ubi motus per loca vado&longs;a propagatus aliquantulum re­<lb/>tardatur. </s> <s>Horas numero ab appul&longs;u Luminaris utriu&longs;que ad Me­<lb/>ridianum loci, tam infra Horizontem quam &longs;upra, & per horas <lb/>diei Lunaris intelligo vige&longs;imas quartas partes temporis quo Luna <lb/>motu apparente diurno ad Meridianum loci revolvitur. </s></p> <p type="main"> <s>Motus autem bini, quos Luminaria duo excitant, non cernen­<lb/>tur di&longs;tincte, &longs;ed motum Q.E.D.m mixtum efficient. </s> <s>In Lumina­<lb/>rium Conjunctione vel Oppo&longs;itione conjungentur eorum effectus, <lb/>& componetur fluxus & refluxus maximus. </s> <s>In Quadraturis Sol <lb/>attollet aquam ubi Luna deprimit, deprimetque ubi Sol attollit; <lb/>& ex effectuum differentia æ&longs;tus omnium minimus orietur. </s> <s>Et <lb/>quoniam, experientia te&longs;te, major e&longs;t effectus Lunæ quam Solis, <lb/>incidet aquæ maxima altitudo in horam tertiam Lunarem. </s> <s>Ex­<lb/>tra Syzygias & Quadraturas, æ&longs;tus maximus qui &longs;ola vi Lunari <lb/>incidere &longs;emper deberet in horam tertiam Lunarem, & &longs;ola Solari <lb/>in tertiam Solarem, compo&longs;itis viribus incidet in tempus aliquod <lb/>intermedium quod tertiæ Lunari propinquius e&longs;t; adeoQ.E.I. <lb/>tran&longs;itu Lunæ a Syzygiis ad Quadraturas, ubi hora tertia Solaris <lb/>præcedit tertiam Lunarem, maxima aquæ altitudo præcedet etiam <pb xlink:href="039/01/419.jpg" pagenum="391"/>tertiam Lunarem, ideque maximo intervallo paulo po&longs;t Octantes <lb/><arrow.to.target n="note422"/>Lunæ; & paribus intervallis æ&longs;tus maximus &longs;equetur horam ter­<lb/>tiam Lunatem in tran&longs;itu Lunæ a Quadraturis ad Syzygias. </s> <s>Hæc <lb/>ita &longs;unt in Mari aperto. </s> <s>Nam in o&longs;tiis Fluviorum fluxus majo­<lb/>res cæteris paribus tardius ad <foreign lang="greek">a)kmlw\</foreign> venient. </s></p> <p type="margin"> <s><margin.target id="note422"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Pendent autem effectus Luminarium ex eorum di&longs;tantiis a Terra. </s> <s><lb/>In minoribus enim di&longs;tantiis majores &longs;unt eorum effectus, in ma­<lb/>joribus minores, idQ.E.I. triplicata ratione diametrorum appa­<lb/>rentium. </s> <s>Igitur Sol tempore hyberno, in Perigæo exi&longs;tens, ma­<lb/>jores edit effectus, efficitque ut æ&longs;tus in Syzygiis paulo majores <lb/>&longs;int, & in Quadraturis paulo minores (cæteris paribus) quam <lb/>tempore æ&longs;tivo; & Luna in Perigæo &longs;ingulis men&longs;ibus majores <lb/>ciet æ&longs;tus quam ante vel po&longs;t dies quindecim, ubi in Apogæo ver­<lb/>&longs;atur. </s> <s>Vnde fit ut æ&longs;tus duo omnino maximi in Syzygiis con­<lb/>tinuis &longs;e mutuo non &longs;equantur. </s></p> <p type="main"> <s>Pendet etiam effectus utriu&longs;que Luminaris ex ip&longs;ius Declina­<lb/>tione &longs;eu di&longs;tautia ab Æquatore. </s> <s>Nam &longs;i Luminare in polo con­<lb/>&longs;titueretur, traheret illud &longs;ingulas aquæ partes con&longs;tanter, ab&longs;que <lb/>actionis inten&longs;ione & remi&longs;&longs;ione, adeoque nullam motus recipro­<lb/>cationem cieret. </s> <s>Igitur Luminaria recedendo ab æquatore polum <lb/>ver&longs;us, effectus &longs;uos gradatim amittent, & propterea minores cie­<lb/>bunt æ&longs;tus in Syzygiis Sol&longs;titialibus quam in Æquinoctialibus. </s> <s><lb/>In Quadraturis autem Sol&longs;titialibus majores ciebunt æ&longs;tus quam <lb/>in Quadraturis Æquinoctialibus; eo quod Lunæ jam in æquatore <lb/>con&longs;titutæ effectus maxime &longs;uperat effectum Solis Incidunt igi­<lb/>tur æ&longs;tus maximi in Syzygias & minimi in Quadraturas Lumina­<lb/>rium, circa tempora Æquinoctii utriu&longs;que. </s> <s>Et æ&longs;tum maximum <lb/>in Syzygiis comitatur &longs;emper minimus in Quadraturis, ut experi­<lb/>entia compertum e&longs;t. </s> <s>Per minorem autem di&longs;tantiam Solis a <lb/>Terra, tempore hyberno quam tempore æ&longs;tivo, fit ut æ&longs;tus ma­<lb/>ximi & minimi &longs;æpius præcedant Æquinoctium vernum quam <lb/>&longs;equantur, & &longs;æpius &longs;equantur autumnale quam præcedant. </s></p> <p type="main"> <s>Pendent etiam effectus Luminarium ex loeorum latitudine. </s> <s>De­<lb/>&longs;ignet <emph type="italics"/>ApEP<emph.end type="italics"/>Tellurem aquis profundis undique coopertam; <emph type="italics"/>C<emph.end type="italics"/><lb/>centrum ejus; <emph type="italics"/>P, p<emph.end type="italics"/>polos, <emph type="italics"/>AE<emph.end type="italics"/>Æquatorem; <emph type="italics"/>F<emph.end type="italics"/>locum quemvis <lb/>extra Æquatorem; <emph type="italics"/>Ff<emph.end type="italics"/>parallelum loci; <emph type="italics"/>Dd<emph.end type="italics"/>parallelum ei re­<lb/>&longs;pondentem ex altera parte æquatoris; <emph type="italics"/>L<emph.end type="italics"/>locum quem Luna tri­<lb/>bus ante horis occupabat; <emph type="italics"/>H<emph.end type="italics"/>locum Telluris ei perpendiculariter <pb xlink:href="039/01/420.jpg" pagenum="392"/><arrow.to.target n="note423"/>&longs;ubjectum; <emph type="italics"/>h<emph.end type="italics"/>locum huic oppo&longs;itum; <emph type="italics"/>K, k<emph.end type="italics"/>loca inde gradibus 90 <lb/>di&longs;tantia, <emph type="italics"/>CH, Ch<emph.end type="italics"/>Maris altitudines maximas men&longs;uratas a cen­<lb/>tro Telluris; & <emph type="italics"/>CK, Ck<emph.end type="italics"/>altitudines minimas: & &longs;i axibus <emph type="italics"/>Hh, <lb/>Kk<emph.end type="italics"/>de&longs;cribatur Ellip&longs;is, deinde Ellip&longs;eos hujus revolutione circa <lb/>axem majorem <emph type="italics"/>Hh<emph.end type="italics"/>de&longs;cribatur Sphærois <emph type="italics"/>HPKhpk<emph.end type="italics"/>; de&longs;ignabit <lb/>hæc figuram Maris quam <lb/><figure id="id.039.01.420.1.jpg" xlink:href="039/01/420/1.jpg"/><lb/>proxime, & erunt <emph type="italics"/>CF, Cf, <lb/>CD, Cd<emph.end type="italics"/>altitudines Maris <lb/>in locis <emph type="italics"/>F, f, D, d.<emph.end type="italics"/>Quin­<lb/>etiam &longs;i in præfata Ellip&longs;eos <lb/>revolutione punctum quod­<lb/>vis <emph type="italics"/>N<emph.end type="italics"/>de&longs;cribat circulum <lb/><emph type="italics"/>NM,<emph.end type="italics"/>&longs;ecantem parallelos <lb/><emph type="italics"/>Ff, Dd<emph.end type="italics"/>in locis quibu&longs;vis <lb/><emph type="italics"/>R, T,<emph.end type="italics"/>& æquatorem <emph type="italics"/>AE<emph.end type="italics"/>in <lb/><emph type="italics"/>S<emph.end type="italics"/>; erit <emph type="italics"/>CN<emph.end type="italics"/>altitudo Maris <lb/>in locis omnibus <emph type="italics"/>R, S, T,<emph.end type="italics"/>&longs;itis in hoc circulo. </s> <s>Hinc in revolu­<lb/>tione diurna loci cuju&longs;vis <emph type="italics"/>F,<emph.end type="italics"/>affluxus erit maximus in <emph type="italics"/>F,<emph.end type="italics"/>hora <lb/>tertia po&longs;t appul&longs;um Lunæ ad Meridianum &longs;upra Horizontem; <lb/>po&longs;tea defluxus maximus in <emph type="italics"/>Q<emph.end type="italics"/>hora tertia po&longs;t occa&longs;um Lunæ; <lb/>dein affluxus maximus in <emph type="italics"/>f<emph.end type="italics"/>hora tertia po&longs;t appul&longs;um Lunæ ad <lb/>Meridianum infra Horizontem; ultimo defluxus maximus in <emph type="italics"/>Q<emph.end type="italics"/><lb/>hora tertia po&longs;t ortum Lunæ; & affluxus po&longs;terior in <emph type="italics"/>f<emph.end type="italics"/>erit mi­<lb/>nor quam affluxus prior in <emph type="italics"/>F.<emph.end type="italics"/>Di&longs;tinguitur enim Mare totum in <lb/>duos omnino fluctus Hemi&longs;phæricos, unum in Hemi&longs;phærio <lb/><emph type="italics"/>KHkC<emph.end type="italics"/>ad Boream vergentem, alterum in Hemi&longs;phærio oppo­<lb/>&longs;ito <emph type="italics"/>KhkC<emph.end type="italics"/>; quos igitur fluctum Borealem & fluctum Au&longs;tralem <lb/>nominare licet. </s> <s>Hi fluctus &longs;emper &longs;ibi mutuo oppo&longs;iti, veniunt <lb/>per vices ad Meridianos loeorum &longs;ingulorum, interpo&longs;ito inter­<lb/>vallo horarum Lunarium duodecim. </s> <s>Cumque regiones Boreales <lb/>magis participant fluctum Borealem, & Au&longs;trales magis Au&longs;tra­<lb/>lem, inde oriuntur æ&longs;tus alternis vicibus majores & minores, in <lb/>locis &longs;ingulis extra æquatorem, in quibus luminaria oriuntur & <lb/>occidunt. </s> <s>Æ&longs;tus autem major, Luna in verticem loci declinante, <lb/>incidet in horam circiter tertiam po&longs;t appul&longs;um Lunæ ad Meri­<lb/>dianum &longs;upra Horizontem, & Luna declinationem mutante verte­<lb/>tur in minorem. </s> <s>Et fluxuum differentia maxima incidet in tem­<lb/>pora Sol&longs;titiorum; præ&longs;ertim &longs;i Lunæ Nodus a&longs;cendens ver&longs;atur <lb/>in principio Arietis. </s> <s>Sic experientia compertum e&longs;t, quod æ&longs;tus <lb/>matutini tempore hyberno &longs;uperent ve&longs;pertinos & ve&longs;pertini tem-<pb xlink:href="039/01/421.jpg" pagenum="393"/>pore æ&longs;tivo matutinos, ad <emph type="italics"/>Plymuthum<emph.end type="italics"/>quidem altitudine qua&longs;i<lb/>pedis unius, ad <emph type="italics"/>Bri&longs;toliam<emph.end type="italics"/>vero altitudine quindecim digitorum:<lb/>ob&longs;ervantibus <emph type="italics"/>Colepre&longs;&longs;io<emph.end type="italics"/>& <emph type="italics"/>Sturmio<emph.end type="italics"/>.</s> <s>Motus autem hactenus de&longs;cripti mutantur aliquantulum per vim<lb/>illam reciprocationis aquarum, qua Maris a&longs;tus, etiam ce&longs;&longs;antibus Luminarium actionibus, po&longs;&longs;et aliquam diu per&longs;everare.</s> <s>Con&longs;er­<lb/>vatio hæcce motus impre&longs;&longs;i minuit differentiam æ&longs;tuum alterno­<lb/>rum; & a&longs;tus proxime po&longs;t Syzygias majores reddit, eo&longs;que pro­<lb/>xime po&longs;t Quadraturas minuit.</s> <s>Unde &longs;it ut æ&longs;tus alterni ad<emph type="italics"/>Ply­<lb/>muthum & Bri&longs;toliam<emph.end type="italics"/>non multo mafis differant ab invicem quam<lb/>altitudine pedis unius vel digitorum quindecim; utque æ&longs;tus om­<lb/>nium maximi in ii&longs;dem portubus, non &longs;int primi a Syzygiis, &longs;ed<lb/>tertii.</s> <s>Retardantur etiam motus omnes in tran&longs;itu per vada, adeo<lb/>ut æ&longs;tus omnium maximi, in fretis quibusdam & Fluviorum o&longs;tiis,<lb/>&longs;sint quarti vel etiam quinti a Syzygiis.</s> <s><lb/>Porro fieri pote&longs;t ut æ&longs;tus propagetur ab Oceano per freta di­<lb/>ver&longs;a ad eundem portum, & citius tran&longs;eat per aliqua freta quam<lb/>per alia: quo in ca&longs;u æ&longs;tus idem, in duos vel plures &longs;ucce&longs;&longs;ive ad­<lb/>venientis divi&longs;us, componere po&longs;&longs;it motus novos diver&longs;orum ge­<lb/>nerum.</s> <s>Fingamus æ&longs;tus duos æquales a diver&longs;is locis in eundem<lb/>portum venire, quorum prior præcedat alterum &longs;patio horarum<lb/>fex, incidatQ.E.I. horam tertiam ab appul&longs;u Lunæ ad Meridia­<lb/>num portus.</s> <s>Si Luna in hocce &longs;uo ad Meridianum appul&longs;u ver­<lb/>fabatur in æquatore, venient &longs;ingulis horis fenis æquales affluxus,<lb/>qui in motuos refluxus incidendo eo&longs;dem affluxibus æquabunt,<lb/> & &longs;ic &longs;patio diei illius efficient ut aqua tranquille &longs;tagnet.</s> <s>Si<lb/>Luna tunc declinabat ab Æquatore, fient æ&longs;tus in Oceano vici­<lb/>bus alternis majores & minores, uti dictum e&longs;t; &inde propaga­<lb/>buntur in hunc portum affluxus bini majores & bini minores, vi­<lb/>cibus alternis.</s> <s>Affluxus autem bini majores component aquam<lb/>alti&longs;&longs;imam in medio inter utrumque, affluxus major & minor fa­<lb/>ciet ut aqua a&longs;cendat ad mediocrem altitudinem in Medio ip&longs;o­<lb/>rum, & inter affluxus binos minores aqua a&longs;cendet ad altitudi­<lb/>dinem minimam.</s> <s>Sic &longs;patio viginti quatuor horarum, aqua non<lb/>bis ut fieri &longs;olet, sed &longs;emel tantum perveniet ad maximam altitu­<lb/>dinem & &longs;emel ad minimam; & altitudo maxima, &longs;i Luna decli­<lb/>nat in polum &longs;upra Horizontem loci, incidet in horam vel &longs;extam<lb/>vel trice&longs;imam ab appul&longs;u Lunæ ad Meridianum, atque Luna de­<lb/>clinationem mutante mutabitur in defluxum.</s> <s>Quorum omnium<lb/>exemplum, in portu regni <emph type="italics"/>Tunquini<emph.end type="italics"/>ad <emph type="italics"/>Bat&longs;ham<emph.end type="italics"/>, &longs;ub latitudine<pb xlink:href="039/01/422.jpg" pagenum="394"/>Boreali 20<emph type="sup"/>gr.<emph.end type="sup"/> 50′.</s> <s><emph type="italics"/>Halleius<emph.end type="italics"/>ex Nautarum Ob&longs;ervationibus pate­<lb/>fecit.</s> <s>Ibi aqua di transitum Lunæ per Æquatorem &longs;equente<lb/>&longs;tagnat, dein Luna ad Boream declinante incipit fluere & refluere,<lb/> non bis, ut in aliis portubus, &longs;ed &longs;emel &longs;ingulis diebus; & æ&longs;tus<lb/>incidit in occasum Lunæ, defluxus maximus in ortum.</s> <s>Cum<lb/> Lunæ declinatione augetur hic æ&longs;tus u&longs;que ad diem &longs;eptimum<lb/>vel octavum, dein per alios &longs;eptem dies iisdem gradibus decre&longs;cit,<lb/>quibus antea creverat; & Luna declinationem mutante ce&longs;&longs;at, ac<lb/>mox mutator in defluxum.</s> <s>Incidit enim &longs;ubinde defluxus in oc­<lb/>ca&longs;um Lunæ & affluxus in ortum, donec Luna iterum mutet de­<lb/>clinationem.</s> <s>Aditus ad hunc portum fretaque vicina duplex pa­<lb/>ter, alter ab Oceano <emph type="italics"/>Sinen&longs;i<emph.end type="italics"/>inter Continentem & In&longs;ulam <emph type="italics"/>Luco­<lb/>niam<emph.end type="italics"/>, alter a Mari <emph type="italics"/>Indico<emph.end type="italics"/>inter Continentem & In&longs;ulam <emph type="italics"/>Borneo<emph.end type="italics"/>.<lb/></s> <s>An æ&longs;tus &longs;patio horarum duodecim a Mari <emph type="italics"/>Indico<emph.end type="italics"/>& &longs;patio hora­<lb/>rum fex a Mari <emph type="italics"/>Sinen&longs;i<emph.end type="italics"/>per freta illa venientes, & &longs;ic in horam ter­<lb/>tiam & nonam Lunarem incidentes, componant huiu&longs;modi motus;<lb/>&longs;itne alia Marium illorum conditio, ob&longs;ervationibus vicinorum<lb/>littorum determinandum reliquo.</s></p> <p type="main"> <s>Hactenus cau&longs;as motuum Lunæ & Marium reddidi.</s> <s>De quan­<lb/>titate motuum jam convenit aliqua &longs;ubjungere.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXV. PROBLEMA VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Invenire vires Solis ad perturbandos motus Lunæ.<emph.end type="italics"/></s></p> <p type="main"> <s>De&longs;ignet <emph type="italics"/>S<emph.end type="italics"/>Solem, <emph type="italics"/>T<emph.end type="italics"/>Terram, <emph type="italics"/>P<emph.end type="italics"/>Lunam, <emph type="italics"/>P A D B<emph.end type="italics"/>orbem<lb/>Lunæ.</s> <s>In <emph type="italics"/>S P<emph.end type="italics"/>capiatur <emph type="italics"/>S K<emph.end type="italics"/>æqualis <emph type="italics"/>S T<emph.end type="italics"/>, &longs;itque <emph type="italics"/>S L<emph.end type="italics"/>ad <emph type="italics"/>S K<emph.end type="italics"/><figure id="id.039.01.422.1.jpg" xlink:href="039/01/422/1.jpg"/>in duplicata ratione <emph type="italics"/>S K<emph.end type="italics"/>ad <emph type="italics"/>S P<emph.end type="italics"/>, & ipsi <emph type="italics"/>P T<emph.end type="italics"/>agatur parallela<lb/><emph type="italics"/>L M<emph.end type="italics"/>; & &longs;i gravitas acceleratrix Terræ in Solem exponatur per<lb/>di&longs;tantiam <emph type="italics"/>S T<emph.end type="italics"/>vel <emph type="italics"/>S K<emph.end type="italics"/>, erit <emph type="italics"/>S L<emph.end type="italics"/>gravitas acceleratrix Lunæ in<pb xlink:href="039/01/423.jpg" pagenum="395"/>Solem. </s> <s>Ea componitur ex partibus <emph type="italics"/>SM, LM,<emph.end type="italics"/>quarum <emph type="italics"/>LM<emph.end type="italics"/>& <lb/><arrow.to.target n="note424"/>ip&longs;ius <emph type="italics"/>SM<emph.end type="italics"/>pars <emph type="italics"/>TM<emph.end type="italics"/>perturbat motum Lunæ, ut in Libri primi <lb/>Prop. </s> <s>LXVI. & ejus Corollariis expo&longs;itum e&longs;t. </s> <s>Quatenus Terra <lb/>& Luna circum commune gravitatis centrum revolvuntur, pertur­<lb/>babitur etiam motus Terræ circa centrum illud a viribus con&longs;imi­<lb/>libus; &longs;ed &longs;ummas tam virium quam motuum referre licet ad Lu­<lb/>nam, & &longs;ummas virium per lineas ip&longs;is analogas <emph type="italics"/>TM<emph.end type="italics"/>& <emph type="italics"/>ML<emph.end type="italics"/><lb/>de&longs;ignare. </s> <s>Vis <emph type="italics"/>ML<emph.end type="italics"/>(in mediocri &longs;ua quantitate) e&longs;t ad vim <lb/>centripetam, qua Luna in Orbe &longs;uo circa Terram quie&longs;centem ad <lb/>di&longs;tantiam <emph type="italics"/>PT<emph.end type="italics"/>revolvi po&longs;&longs;et, in duplicata ratione temporum <lb/>periodieorum Lunæ circa Terram & Terræ circa Solem, (per <lb/>Corol. </s> <s>17. Prop. </s> <s>LXVI. Lib.I.) hoc e&longs;t, in duplicata ratione die­<lb/>rum 27. <emph type="italics"/>hor.<emph.end type="italics"/>7. <emph type="italics"/>min.<emph.end type="italics"/>43. ad dies 365. <emph type="italics"/>hor.<emph.end type="italics"/>6. <emph type="italics"/>min.<emph.end type="italics"/>9. id e&longs;t, ut 1000 <lb/>ad 178725, &longs;eu 1 ad (178 39/40). Invenimus autem in Propo&longs;itione <lb/>quarta quod, &longs;i Terra & Luna circa commune gravitatis centrum <lb/>revolvantur, earum di&longs;tantia mediocris ab invicem erit 60 1/2 &longs;emi­<lb/>diametrorum mediocrium Terræ quamproxime. </s> <s>Et vis qua Luna <lb/>in Orbe circa Terram quie&longs;centem, ad di&longs;tantiam <emph type="italics"/>PT<emph.end type="italics"/>&longs;emidiame­<lb/>trorum terre&longs;trium 60 1/2 revolvi po&longs;&longs;et, e&longs;t ad vim, qua eodem <lb/>tempore ad di&longs;tantiam &longs;emidiametrorum 60 revolvi po&longs;&longs;et, ut <lb/>60 1/2 ad 60; & hæc vis ad vim gravitatis apud nos ut 1 ad <lb/>60X60 quamproxime. </s> <s>Ideoque vis mediocris <emph type="italics"/>ML<emph.end type="italics"/>e&longs;t ad vim <lb/>gravitatis in &longs;uperficie Terræ, ut 1X60 1/2 ad 60X60X60X(178 29/40), <lb/>&longs;eu 1 ad 638092, 6. Vnde ex proportione linearum <emph type="italics"/>TM, ML,<emph.end type="italics"/><lb/>datur etiam vis <emph type="italics"/>TM:<emph.end type="italics"/>& hæ &longs;unt vires Solis quibus Lunæ motus <lb/>perturbantur. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note423"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="margin"> <s><margin.target id="note424"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVI. PROBLEMA VII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Invenire incrementum borarium areæ quam Luna, radio ad Ter­<lb/>ram ducto, in Orbe circulari de&longs;cribit.<emph.end type="italics"/></s></p> <p type="main"> <s>Diximus aream, quam Luna radio ad Terram ducto de&longs;cribit, <lb/>e&longs;&longs;e tempori proportionalem, ni&longs;i quatenus motus Lunaris ab <lb/>actione Solis turbatur. </s> <s>Inæqualitatem momenti (vel incrementi <lb/>horarii) hic inve&longs;tigandam proponimus. </s> <s>Ut computatio facilior <lb/>reddatur, fingamus orbem Lunæ circularem e&longs;&longs;e, & inæqualitates <lb/>omnes negligamus, ea &longs;ola excepta, de qua hic agitur. </s> <s>Ob in­<lb/>gentem vero Solis di&longs;tantiam, ponamus etiam lineas <emph type="italics"/>SP, ST<emph.end type="italics"/>&longs;ibi <lb/>invicem parallelas e&longs;&longs;e. </s> <s>Hoc pacto vis <emph type="italics"/>LM<emph.end type="italics"/>reducetur &longs;emper <pb xlink:href="039/01/424.jpg" pagenum="396"/><arrow.to.target n="note425"/>ad mediocrem &longs;uam quantitatem <emph type="italics"/>TP,<emph.end type="italics"/>ut & vis <emph type="italics"/>TM<emph.end type="italics"/>ad medio­<lb/>crem &longs;uam quantitatem 3 <emph type="italics"/>PK.<emph.end type="italics"/>Hæ vires, per Legum Corol. </s> <s>2. <lb/>componunt vim <emph type="italics"/>TL<emph.end type="italics"/>; & hæc vis, &longs;i in radium <emph type="italics"/>TP<emph.end type="italics"/>demittatur <lb/>perpendiculum <emph type="italics"/>LE,<emph.end type="italics"/>re&longs;olvitur in vires <emph type="italics"/>TE, EL,<emph.end type="italics"/>quarum <emph type="italics"/>TE,<emph.end type="italics"/><lb/>agendo &longs;emper &longs;ecundum radium <emph type="italics"/>TP,<emph.end type="italics"/>nec accelerat nec retardat <lb/>de&longs;criptionem areæ <emph type="italics"/>TPC<emph.end type="italics"/>radio illo <emph type="italics"/>TP<emph.end type="italics"/>factam; & <emph type="italics"/>EL<emph.end type="italics"/>agendo <lb/>&longs;ecundum perpendiculum, accelerat vel retardat ip&longs;am, quan­<lb/>tum accelerat vel retardat Lunam. </s> <s>Acceleratio illa Lunæ, in <lb/>tran&longs;itu ip&longs;ius a Quadratura <emph type="italics"/>C<emph.end type="italics"/>ad Conjunctionem <emph type="italics"/>A,<emph.end type="italics"/>&longs;ingulis <lb/>temporis momentis facta, e&longs;t ut ip&longs;a vis accelerans <emph type="italics"/>EL,<emph.end type="italics"/>hoc e&longs;t, <lb/>ut (<emph type="italics"/>3PKXTK/TP<emph.end type="italics"/>). Exponatur tempus per motum medium Luna­<lb/>rem, vel (quod eodem fere recidit) per angulum <emph type="italics"/>CTP,<emph.end type="italics"/>vel <lb/><figure id="id.039.01.424.1.jpg" xlink:href="039/01/424/1.jpg"/>etiam per arcum <emph type="italics"/>CP.<emph.end type="italics"/>Ad <emph type="italics"/>CT<emph.end type="italics"/>erigatur normalis <emph type="italics"/>CG<emph.end type="italics"/>ip&longs;i <emph type="italics"/>CT<emph.end type="italics"/><lb/>æqualis. </s> <s>Et divi&longs;o arcu quadrantali <emph type="italics"/>AC<emph.end type="italics"/>in particulas innumeras <lb/>æquales <emph type="italics"/>Pp,<emph.end type="italics"/>&c. </s> <s>per quas æquales totidem particulæ temporis <lb/>exponi po&longs;&longs;int, ductaque <emph type="italics"/>pk<emph.end type="italics"/>perpendiculari ad <emph type="italics"/>CT,<emph.end type="italics"/>jungatur <lb/><emph type="italics"/>TG<emph.end type="italics"/>ip&longs;is <emph type="italics"/>KP, kp<emph.end type="italics"/>productis occurrens in <emph type="italics"/>F<emph.end type="italics"/>& <emph type="italics"/>f<emph.end type="italics"/>; & erit <emph type="italics"/>Kk<emph.end type="italics"/>ad <lb/><emph type="italics"/>PK<emph.end type="italics"/>ut <emph type="italics"/>Pp<emph.end type="italics"/>ad <emph type="italics"/>Tp,<emph.end type="italics"/>hoc e&longs;t in data ratione, adeoque <emph type="italics"/>FKXKk<emph.end type="italics"/><lb/>&longs;eu area <emph type="italics"/>FKkf,<emph.end type="italics"/>ut (<emph type="italics"/>3PKXTK/TP<emph.end type="italics"/>), id e&longs;t, ut <emph type="italics"/>EL<emph.end type="italics"/>; & compo&longs;ite, <lb/>area tota <emph type="italics"/>GCKF<emph.end type="italics"/>ut &longs;umma omnium virium <emph type="italics"/>EL<emph.end type="italics"/>tempore toto <lb/><emph type="italics"/>CP<emph.end type="italics"/>impre&longs;&longs;arum in Lunam, atque adeo etiam ut velocitas hac <pb xlink:href="039/01/425.jpg" pagenum="397"/>&longs;umma genita, id e&longs;t, ut acceleratio de&longs;criptionis areæ <emph type="italics"/>CTP,<emph.end type="italics"/>&longs;eu <lb/><arrow.to.target n="note426"/>incrementum momenti. </s> <s>Vis qua Luna circa Terram quie&longs;centem <lb/>ad di&longs;tantiam <emph type="italics"/>TP,<emph.end type="italics"/>tempore &longs;uo periodico <emph type="italics"/>CADBC<emph.end type="italics"/>dierum 27. <lb/><emph type="italics"/>hor.<emph.end type="italics"/>7. <emph type="italics"/>min.<emph.end type="italics"/>43. revolvi po&longs;&longs;et, efficeret ut corpus, tempore <emph type="italics"/>CT<emph.end type="italics"/><lb/>cadendo, de&longs;criberet longitudinem 1/2 <emph type="italics"/>CT,<emph.end type="italics"/>& velocitatem &longs;imul <lb/>acquireret æqualem velocitati, qua Luna in Orbe &longs;uo movetur. </s> <s><lb/>Patet hoc per Corol. </s> <s>9. Prop. </s> <s>IV. Lib. </s> <s>I. </s> <s>Cum autem perpen­<lb/>diculum <emph type="italics"/>Kd<emph.end type="italics"/>in <emph type="italics"/>TP<emph.end type="italics"/>demi&longs;&longs;um &longs;it ip&longs;ius <emph type="italics"/>EL<emph.end type="italics"/>pars tertia, & ip­<lb/>&longs;ius <emph type="italics"/>TP<emph.end type="italics"/>&longs;eu <emph type="italics"/>ML<emph.end type="italics"/>in Octantibus pars dimidia, vis <emph type="italics"/>EL<emph.end type="italics"/>in Octan­<lb/>tibus, ubi maxima e&longs;t, &longs;uperabit vim <emph type="italics"/>ML<emph.end type="italics"/>in ratione 3 ad 2, <lb/>adeoque erit ad vim illam, qua Luna tempore &longs;uo periodico circa <lb/>Terram quie&longs;centem revolvi po&longs;&longs;et, ut 100 ad 2/3X17872 1/2 &longs;eu <lb/>11915, & tempore <emph type="italics"/>CT<emph.end type="italics"/>velocitatem generare deberet quæ e&longs;&longs;et <lb/>pars (100/11915) velocitatis Lunaris, tempore autem <emph type="italics"/>CPA<emph.end type="italics"/>velocitatem <lb/>majorem generaret in ratione <emph type="italics"/>CA<emph.end type="italics"/>ad <emph type="italics"/>CT<emph.end type="italics"/>&longs;eu <emph type="italics"/>TP.<emph.end type="italics"/>Exponatur <lb/>vis maxima <emph type="italics"/>EL<emph.end type="italics"/>in Octantibus per aream <emph type="italics"/>FKXKk<emph.end type="italics"/>rectangulo <lb/>1/2 <emph type="italics"/>TPXPp<emph.end type="italics"/>æqualem. </s> <s>Et velocitas, quam vis maxima tempore <lb/>quovis <emph type="italics"/>CP<emph.end type="italics"/>generare po&longs;&longs;et, erit ad velocitatem quam vis omnis <lb/>minor <emph type="italics"/>EL<emph.end type="italics"/>eodem tempore generat, ut rectangulum 1/2 <emph type="italics"/>TPXCP<emph.end type="italics"/><lb/>ad aream <emph type="italics"/>KCGF<emph.end type="italics"/>: tempore autem toto <emph type="italics"/>CPA,<emph.end type="italics"/>velocitates ge­<lb/>nitæ erunt ad invicem ut rectangulum 1/2<emph type="italics"/>TPXCA<emph.end type="italics"/>& triangulum <lb/><emph type="italics"/>TCG,<emph.end type="italics"/>&longs;ive ut arcus quadrantalis <emph type="italics"/>CA<emph.end type="italics"/>& radius <emph type="italics"/>TP.<emph.end type="italics"/>Ideoque <lb/>(per Prop. </s> <s>IX. Lib. </s> <s>V. Elem.) velocitas po&longs;terior, toto tempore <lb/>genita, erit pars (100/11915) velocitatis Lunæ. </s> <s>Huic Lunæ velocitati, <lb/>quæ areæ momento mediocri analoga e&longs;t, addatur & auferatur <lb/>dimidium velocitatis alterius; & &longs;i momentum mediocre expona­<lb/>tur per numerum 11915, &longs;umma 11915+50 &longs;eu 11965 exhi­<lb/>bebit momentum maximum areæ in Syzygia <emph type="italics"/>A,<emph.end type="italics"/>ac differentia <lb/>11915-50 &longs;eu 11865 eju&longs;dem momentum minimum in Quadra­<lb/>turis. </s> <s>Igitur areæ temporibus æqualibus in Syzygiis & Quadra­<lb/>turis de&longs;criptæ, &longs;unt ad invicem ut 11965 ad 11865. Ad mo­<lb/>mentum minimum 11865 addatur momentum, quod &longs;it ad mo­<lb/>mentorum differentiam 100 ut trapezium <emph type="italics"/>FKCG<emph.end type="italics"/>ad triangu­<lb/>lum <emph type="italics"/>TCG<emph.end type="italics"/>(vel quod perinde e&longs;t, ut quadratum Sinus <emph type="italics"/>PK<emph.end type="italics"/>ad <lb/>quadratum Radii <emph type="italics"/>TP,<emph.end type="italics"/>id e&longs;t, ut <emph type="italics"/>Pd<emph.end type="italics"/>ad <emph type="italics"/>TP<emph.end type="italics"/>) & &longs;umma exhi­<lb/>bebit momentum areæ, ubi Luna e&longs;t in loco quovis interme­<lb/>dio <emph type="italics"/>P.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note425"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="margin"> <s><margin.target id="note426"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Hæc omnia ita &longs;e habent, ex Hypothe&longs;i quod Sol & Terra qui­<lb/>e&longs;cunt, & Luna tempore Synodico dierum 27. <emph type="italics"/>hor.<emph.end type="italics"/>7. <emph type="italics"/>min.<emph.end type="italics"/>43. re­<lb/>volvitur. </s> <s>Cum autem periodus Synodica Lunaris vere &longs;it die-<pb xlink:href="039/01/426.jpg" pagenum="398"/><arrow.to.target n="note427"/>rum 29. <emph type="italics"/>hor.<emph.end type="italics"/>12. & <emph type="italics"/>min.<emph.end type="italics"/>44. augeri debent momentorum incre­<lb/>menta in ratione temporis, id e&longs;t, in ratione 1080853 ad 1000000. <lb/>Hoc pacto incrementum totum, quod erat pars (100/11915) momenti <lb/>mediocris, jam fiet eju&longs;dem pars (100/11023). Ideoque momentum <lb/>areæ in Quadratura Lunæ erit ad ejus momentum in Syzygia <lb/>ut 11023-50 ad 11023+50, &longs;eu 10973 ad 11073, & ad ejus <lb/>momentum, ubi Luna in alio quovis loco intermedio <emph type="italics"/>P<emph.end type="italics"/>ver&longs;atur, <lb/>ut 10973 ad 10973+<emph type="italics"/>Pd,<emph.end type="italics"/>exi&longs;tente videlicet <emph type="italics"/>TP<emph.end type="italics"/>æquali 100. </s></p> <p type="margin"> <s><margin.target id="note427"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Area igitur, quam Luna radio ad Terram ducto &longs;ingulis tem­<lb/>poris particulis æqualibus de&longs;cribit, e&longs;t quam proxime ut &longs;umma <lb/>numeri 219,46 & Sinus ver&longs;i duplicatæ di&longs;tantiæ Lunæ a Quadra­<lb/>tura proxima, in circulo cujus radius e&longs;t unitas. </s> <s>Hæc ita &longs;e ha­<lb/>bent ubi Variatio in Octantibus e&longs;t magnitudinis mediocris. </s> <s>Sin <lb/>Variatio ibi major &longs;it vel minor, augeri debet vel minui Sinus ille <lb/>ver&longs;us in eadem ratione. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVII. PROBLEMA VIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Ex motu horario Lunæ invenire ip&longs;ius di&longs;tantiam a Terra.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Area, quam Luna radio ad Terram ducto, &longs;ingulis temporis <lb/>momentis, de&longs;cribit, e&longs;t ut motus horarius Lunæ & quadratum <lb/>di&longs;tantiæ Lunæ a Terra conjunctim; & propterea di&longs;tantia Lunæ <lb/>a Terra e&longs;t in ratione compo&longs;ita ex &longs;ubduplicata ratione Areæ di­<lb/>recte & &longs;ubduplicata ratione motus horarii inver&longs;e. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc datur Lunæ diameter apparens: quippe quæ &longs;it <lb/>reciproce ut ip&longs;ius di&longs;tantia a Terra. </s> <s>Tentent A&longs;tronomi quam <lb/>probe hæc Regula cum Phænomenis congruat. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Hinc etiam Orbis Lunaris accuratius ex Phænomenis <lb/>quam antehac definiri pote&longs;t. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXVIII. PROBLEMA IX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire diametros Orbis in quo Luna, ab&longs;que eccentricitate, <lb/>moveri deberet.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Curvatura Trajectoriæ, quam mobile, &longs;i &longs;ecundum Trajectoriæ <lb/>illius perpendiculum trahatur, de&longs;cribit, e&longs;t ut attractio directe & <lb/>quadratum velocitatis inver&longs;e, Curvaturas linearum pono e&longs;&longs;e in-<pb xlink:href="039/01/427.jpg" pagenum="399"/>ter &longs;e in ultima proportione Sinuum vel Tangentium angulorum <lb/><arrow.to.target n="note428"/>contactuum ad radios æquales pertinentium, ubi radii illi in infi­<lb/>nitum diminuuntur. </s> <s>Attractio autem Lunæ in Terram in Syzy­<lb/>giis e&longs;t exce&longs;&longs;us gravitatis ip&longs;ius in Terram &longs;upra vim Solarem <lb/>2 <emph type="italics"/>PK<emph.end type="italics"/>(Vide <emph type="italics"/>Figur. </s> <s>pag.<emph.end type="italics"/>394) qua gravitas acceleratrix Lunæ in <lb/>Solem &longs;uperat gravitatem acceleratricem Terræ in Solem. </s> <s>In Qua­<lb/>draturis autem attractio illa e&longs;t &longs;umma gravitatis Lunæ in Terram <lb/>& vis Solaris <emph type="italics"/>KT,<emph.end type="italics"/>qua Luna in Terram trahitur. </s> <s>Et hæ attra­<lb/>ctiones, &longs;i (<emph type="italics"/>AT+CT<emph.end type="italics"/>/2) dicatur N, &longs;unt ut (178725/<emph type="italics"/>ATq<emph.end type="italics"/>)-(2000/<emph type="italics"/>CTXN<emph.end type="italics"/>) & <lb/>(178725/<emph type="italics"/>CIq<emph.end type="italics"/>)+(1000/<emph type="italics"/>ATXN<emph.end type="italics"/>) quam proxime; &longs;eu ut 178725NX<emph type="italics"/>CTq<emph.end type="italics"/><lb/>-2000 <emph type="italics"/>ATqXCT<emph.end type="italics"/>& 178725 NX<emph type="italics"/>ATq<emph.end type="italics"/>+1000 <emph type="italics"/>CTqXAT.<emph.end type="italics"/>Nam <lb/>&longs;i gravitas acceleratrix Lunæ in Terram exponatur per numerum <lb/>178725, vis mediocris <emph type="italics"/>ML,<emph.end type="italics"/>quæ in Quadraturis e&longs;t <emph type="italics"/>PT<emph.end type="italics"/>vel <lb/><emph type="italics"/>TK<emph.end type="italics"/>& Lunam trahit in Ter­<lb/><figure id="id.039.01.427.1.jpg" xlink:href="039/01/427/1.jpg"/><lb/>ram, erit 1000, & vis me­<lb/>diocris <emph type="italics"/>TM<emph.end type="italics"/>in Syzygiis erit <lb/>3000; de qua, &longs;i vis medio­<lb/>cris <emph type="italics"/>ML<emph.end type="italics"/>&longs;ubducatur, mane­<lb/>bit vis 2000 qua Luna in <lb/>Syzygiis di&longs;trahitur a Terra, <lb/>quamque jam ante nominavi <lb/>2 <emph type="italics"/>PK.<emph.end type="italics"/>Velocitas autem Lu­<lb/>næ in Syzygiis <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B<emph.end type="italics"/>e&longs;t ad <lb/>ip&longs;ius velocitatem in Qua­<lb/>draturis <emph type="italics"/>C<emph.end type="italics"/>& <emph type="italics"/>D,<emph.end type="italics"/>ut <emph type="italics"/>CT<emph.end type="italics"/>ad <lb/><emph type="italics"/>AT<emph.end type="italics"/>& momentum areæ quam <lb/>Luna radio ad Terram du­<lb/>cto de&longs;cribit in Syzygiis ad <lb/>momentum eju&longs;dem areæ in <lb/>Quadraturis conjunctim; i.e. </s> <s><lb/>ut 11073 <emph type="italics"/>CT<emph.end type="italics"/>ad 10973 <emph type="italics"/>AT.<emph.end type="italics"/><lb/>Sumatur hæc ratio bis in­<lb/>ver&longs;e & ratio prior &longs;emel directe, & fiet curvatura Orbis Lu­<lb/>naris in Syzygiis ad eju&longs;dem curvaturam in Quadraturis ut <lb/>120406729X178725 <emph type="italics"/>ATqXCTq<emph.end type="italics"/>XN-120406729X2000 <emph type="italics"/>ATqq <lb/>XCT<emph.end type="italics"/>ad 122611329X178725 <emph type="italics"/>ATqXCTq<emph.end type="italics"/>XN+122611329X <lb/>1000 <emph type="italics"/>CTqqXAT, i.e.<emph.end type="italics"/>ut 2151969 <emph type="italics"/>ATXCT<emph.end type="italics"/>XN-24081 <emph type="italics"/>AT cub.<emph.end type="italics"/><lb/>ad 2191371 <emph type="italics"/>ATXCT<emph.end type="italics"/>XN+12261 <emph type="italics"/>CT cub.<emph.end type="italics"/><pb xlink:href="039/01/428.jpg" pagenum="400"/><arrow.to.target n="note429"/></s></p> <p type="margin"> <s><margin.target id="note428"/>LIBER <lb/>TERTIUS.</s></p> <p type="margin"> <s><margin.target id="note429"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Quoniam Figura orbis Lunaris ignoratur, hujus vice a&longs;&longs;uma­<lb/>mus Ellip&longs;in <emph type="italics"/>DBCA,<emph.end type="italics"/>in cujus centro <emph type="italics"/>T<emph.end type="italics"/>Terra collocetur, & cu­<lb/>jus axis major <emph type="italics"/>DC<emph.end type="italics"/>Quadraturis, minor <emph type="italics"/>AB<emph.end type="italics"/>Syzygiis interja­<lb/>ceat. </s> <s>Cum autem planum Ellip&longs;eos hujus motu angulari circa <lb/>Terram revolvatur, & Trajectoria cujus curvaturam con&longs;ideramus, <lb/>de&longs;cribi debet in plano quod omni motu angulari omnino de&longs;ti­<lb/>tuitur: con&longs;ideranda erit Figura, quam Luna in Ellip&longs;i illa revol­<lb/>vendo de&longs;cribit in hoc plano, hoc e&longs;t Figura <emph type="italics"/>Cpa,<emph.end type="italics"/>cujus puncta <lb/>&longs;ingula <emph type="italics"/>p<emph.end type="italics"/>inveniuntur capiendo punctum quodvis <emph type="italics"/>P<emph.end type="italics"/>in Ellip&longs;i, <lb/>quod locum Lunæ repre&longs;entet, & ducendo <emph type="italics"/>Tp<emph.end type="italics"/>æqualem <emph type="italics"/>TP,<emph.end type="italics"/>ea <lb/>lege ut angulus <emph type="italics"/>PTp<emph.end type="italics"/>æqualis &longs;it motui apparenti Solis a tem­<lb/>pore Quadraturæ <emph type="italics"/>C<emph.end type="italics"/>confecto; vel (quod eodem fere recidit) ut <lb/>angulus <emph type="italics"/>CTp<emph.end type="italics"/>&longs;it ad angulum <lb/><figure id="id.039.01.428.1.jpg" xlink:href="039/01/428/1.jpg"/><lb/><emph type="italics"/>CTP<emph.end type="italics"/>ut tempus revolutio­<lb/>nis Synodicæ Lunaris ad tem­<lb/>pus revolutionis Periodicæ <lb/>&longs;eu 29<emph type="sup"/>d.<emph.end type="sup"/> 12<emph type="sup"/>h.<emph.end type="sup"/> 44′, ad 27<emph type="sup"/>d.<emph.end type="sup"/> 7<emph type="sup"/>h.<emph.end type="sup"/> 43′. </s> <s><lb/>Capiatur igitur angulus <emph type="italics"/>CTa<emph.end type="italics"/><lb/>in eadem ratione ad angu­<lb/>lum rectum <emph type="italics"/>CTA,<emph.end type="italics"/>& &longs;it <lb/>longitudo <emph type="italics"/>Ta<emph.end type="italics"/>æqualis lon­<lb/>gitudini <emph type="italics"/>TA<emph.end type="italics"/>; & erit <emph type="italics"/>a<emph.end type="italics"/><lb/>Ap&longs;is ima & <emph type="italics"/>C<emph.end type="italics"/>Ap&longs;is &longs;um­<lb/>ma Orbis hujus <emph type="italics"/>Cpa.<emph.end type="italics"/>Ra­<lb/>tiones autem ineundo inve­<lb/>nio quod differentia inter <lb/>curvaturam Orbis <emph type="italics"/>Cpa<emph.end type="italics"/>in <lb/>vertice <emph type="italics"/>a,<emph.end type="italics"/>& curvaturam Cir­<lb/>culi centro <emph type="italics"/>T<emph.end type="italics"/>intervallo <emph type="italics"/>TA<emph.end type="italics"/><lb/>de&longs;cripti, &longs;it ad differentiam <lb/>inter curvaturam Ellip&longs;eos in <lb/>vertice <emph type="italics"/>A<emph.end type="italics"/>& curvaturam eju&longs;dem Circuli, in duplicata ratione an­<lb/>guli <emph type="italics"/>CTP<emph.end type="italics"/>ad angulum <emph type="italics"/>CTp<emph.end type="italics"/>; & quod curvatura Ellip&longs;eos in <emph type="italics"/>A<emph.end type="italics"/><lb/>&longs;it ad curvaturam Circuli illius, in duplicata ratione <emph type="italics"/>TA<emph.end type="italics"/>ad <emph type="italics"/>TC<emph.end type="italics"/>; <lb/>& curvatura Circuli illius ad curvaturam Circuli centro <emph type="italics"/>T<emph.end type="italics"/>in­<lb/>tervallo <emph type="italics"/>TC<emph.end type="italics"/>de&longs;cripti, ut <emph type="italics"/>TC<emph.end type="italics"/>ad <emph type="italics"/>TA<emph.end type="italics"/>; hujus autem curvatura ad <lb/>curvaturam Ellip&longs;eos in <emph type="italics"/>C,<emph.end type="italics"/>in duplicata ratione <emph type="italics"/>TA<emph.end type="italics"/>ad <emph type="italics"/>TC<emph.end type="italics"/>; & <lb/>differentia inter curvaturam Ellip&longs;eos in vertice <emph type="italics"/>C<emph.end type="italics"/>& curvaturam <lb/>Circuli novi&longs;&longs;imi, ad differentiam inter curvaturam Figuræ <emph type="italics"/>Tpa<emph.end type="italics"/><lb/>in vertice <emph type="italics"/>C<emph.end type="italics"/>& curvaturam eju&longs;dem Circuli, in duplicata ratione <pb xlink:href="039/01/429.jpg" pagenum="401"/>anguli <emph type="italics"/>CTp<emph.end type="italics"/>ad angulum <emph type="italics"/>CTP.<emph.end type="italics"/>Quæ quidem rationes ex &longs;inu­</s></p> <p type="main"> <s><arrow.to.target n="note430"/>bus angulorum contactus ac differentiarum angulorum facile colli­<lb/>guntur. </s> <s>His autem inter &longs;e collatis, prodit curvatura Figuræ <emph type="italics"/>Cpa<emph.end type="italics"/><lb/>in <emph type="italics"/>a<emph.end type="italics"/>ad ip&longs;ius curvaturam in <emph type="italics"/>C,<emph.end type="italics"/>ut <emph type="italics"/>AT cub<emph.end type="italics"/>+(16824/100000)<emph type="italics"/>CTqXAT<emph.end type="italics"/><lb/>ad <emph type="italics"/>CT cub<emph.end type="italics"/>+(16824/100000) <emph type="italics"/>ATqXCT.<emph.end type="italics"/>Ubi numerus (16824/100000) de&longs;ignat <lb/>differentiam quadratorum angulorum <emph type="italics"/>CTP<emph.end type="italics"/>& <emph type="italics"/>CTp<emph.end type="italics"/>appli­<lb/>catam ad quadratum anguli minoris <emph type="italics"/>CTP,<emph.end type="italics"/>&longs;eu (quod per­<lb/>inde e&longs;t) differentiam quadratorum temporum 27<emph type="sup"/>d.<emph.end type="sup"/> 7<emph type="sup"/>h.<emph.end type="sup"/> 43′, & <lb/>29<emph type="sup"/>d.<emph.end type="sup"/> 12<emph type="sup"/>h.<emph.end type="sup"/> 44′, applicatam ad quadratum temporis 27<emph type="sup"/>d.<emph.end type="sup"/> 7<emph type="sup"/>h.<emph.end type="sup"/> 43′, </s></p> <p type="margin"> <s><margin.target id="note430"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Igitur cum <emph type="italics"/>a<emph.end type="italics"/>de&longs;ignet Syzygiam Lunæ, & <emph type="italics"/>C<emph.end type="italics"/>ip&longs;ius Quadratu­<lb/>ram, proportio jam inventa eadem e&longs;&longs;e debet cum proportione <lb/>curvaturæ Orbis Lunæ in Syzygiis ad eju&longs;dem curvaturam in <lb/>Quadraturis, quam &longs;upra invenimus. </s> <s>Proinde ut inveniatur pro­<lb/>portio <emph type="italics"/>CT<emph.end type="italics"/>ad <emph type="italics"/>AT,<emph.end type="italics"/>duco extrema & media in &longs;e invicem. </s> <s>Et <lb/>termini prodeuntes ad <emph type="italics"/>ATXCT<emph.end type="italics"/>applicati, fiunt 2062, 79 <emph type="italics"/>CTqq<emph.end type="italics"/><lb/>-2151969 NX<emph type="italics"/>CTcub<emph.end type="italics"/>+368676 NX<emph type="italics"/>ATXCTq<emph.end type="italics"/>+36342 <emph type="italics"/>ATq <lb/>XCTq<emph.end type="italics"/>-362047 NX<emph type="italics"/>ATqXCT<emph.end type="italics"/>+2191371 NX<emph type="italics"/>AT cub<emph.end type="italics"/>+ <lb/>4051, 4 <emph type="italics"/>ATqq<emph.end type="italics"/>=0. Hic pro terminorum <emph type="italics"/>AT<emph.end type="italics"/>& <emph type="italics"/>CT<emph.end type="italics"/>&longs;emi&longs;um­<lb/>ma N &longs;cribo 1, & pro eorundem &longs;emidifferentia ponendo <emph type="italics"/>x,<emph.end type="italics"/>fit <lb/><emph type="italics"/>CT<emph.end type="italics"/>=1+<emph type="italics"/>x,<emph.end type="italics"/>& <emph type="italics"/>AT<emph.end type="italics"/>=1-<emph type="italics"/>x<emph.end type="italics"/>: quibus in æquatione &longs;criptis, & <lb/>æquatione prodeunte re&longs;oluta, obtinetur <emph type="italics"/>x<emph.end type="italics"/>æqualis 0,00719, & <lb/>inde &longs;emidiameter <emph type="italics"/>CT<emph.end type="italics"/>fit 1,00719, & &longs;emidiameter <emph type="italics"/>AT<emph.end type="italics"/>0,99281, <lb/>qui numeri &longs;unt ut (70 1/24) & (69 1/24) quam proxime. </s> <s>E&longs;t igitur di­<lb/>&longs;tantia Lunæ a Terra in Syzygiis ad ip&longs;ius di&longs;tantiam in Quadra­<lb/>turis (&longs;epo&longs;ita &longs;cilicet Eccentricitatis con&longs;ideratione) ut (69 1/24) ad <lb/>(70 1/24), vel numeris rotundis ut 69 ad 70. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXIX. PROBLEMA X.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire Variationem Lunæ.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Oritur hæc inæqualitas partim ex forma Elliptica orbis Luna­<lb/>ris, partim ex inæqualitate momentorum areæ, quam Luna radio <lb/>ad Terram ducto de&longs;cribit. </s> <s>Si Luna <emph type="italics"/>P<emph.end type="italics"/>in Ellip&longs;i <emph type="italics"/>DBCA<emph.end type="italics"/>circa <lb/>Terram in centro Ellip&longs;eos quie&longs;centem moveretur, & radio <emph type="italics"/>TP<emph.end type="italics"/><lb/>ad Terram ducto de&longs;criberet aream <emph type="italics"/>CTP<emph.end type="italics"/>tempori proportiona­<lb/>lem; e&longs;&longs;et autem Ellip&longs;eos &longs;emidiameter maxima <emph type="italics"/>CT<emph.end type="italics"/>ad &longs;emi­<lb/>diametrum minimam <emph type="italics"/>TA<emph.end type="italics"/>ut 70 ad 69: foret tangens anguli <lb/><emph type="italics"/>CTP<emph.end type="italics"/>ad tangentem anguli motus medii a Quadratura <emph type="italics"/>C<emph.end type="italics"/>compu­<lb/>tati, ut Ellip&longs;eos &longs;emidiameter <emph type="italics"/>TA<emph.end type="italics"/>ad eju&longs;dem &longs;emidiametrum <pb xlink:href="039/01/430.jpg" pagenum="402"/><arrow.to.target n="note431"/><emph type="italics"/>TC<emph.end type="italics"/>&longs;eu 69 ad 70. Debet autem de&longs;criptio areæ <emph type="italics"/>CTP,<emph.end type="italics"/>in pro­<lb/>gre&longs;&longs;u Lunæ a Quadratura ad Syzygiam, ea ratione accelerari, ut <lb/>ejus momentum in Syzygia Lunæ &longs;it ad ejus momentum in Qua­<lb/>dratura ut 11073 ad 10973, utque exce&longs;&longs;us momenti in loco <lb/>quovis intermedio <emph type="italics"/>P<emph.end type="italics"/>&longs;upra momentum in Quadratura &longs;it ut qua­<lb/>dratum &longs;inus anguli <emph type="italics"/>CTP.<emph.end type="italics"/>Id quod &longs;atis accurate fiet, &longs;i tan­<lb/>gens anguli <emph type="italics"/>CTP<emph.end type="italics"/>diminuatur in &longs;ubduplicata ratione numeri <lb/>10973 ad numerum 11073, id e&longs;t, in ratione numeri 68,6877 ad <lb/>numerum 69. Quo pacto <lb/><figure id="id.039.01.430.1.jpg" xlink:href="039/01/430/1.jpg"/><lb/>tangens anguli <emph type="italics"/>CTP<emph.end type="italics"/>jam e­<lb/>rit ad tangentem motus me­<lb/>dii ut 68,6877 ad 70, & an­<lb/>gulus <emph type="italics"/>CTP<emph.end type="italics"/>in Octantibus, <lb/>ubi motus medius e&longs;t 45<emph type="sup"/>gr.<emph.end type="sup"/><lb/>invenietur 44<emph type="sup"/>gr.<emph.end type="sup"/> 27′. </s> <s>28″. </s> <s>qui <lb/>&longs;ubductus de angulo motus <lb/>medii 45<emph type="sup"/>gr.<emph.end type="sup"/> relinquit Varia­<lb/>tionem maximam 32′. </s> <s>32″. </s> <s><lb/>Hæc ita &longs;e haberent &longs;i Luna, <lb/>pergendo a Quadratura ad <lb/>Syzygiam, de&longs;criberet angu­<lb/>lum <emph type="italics"/>CTA<emph.end type="italics"/>graduum tantum <lb/>nonaginta. </s> <s>Verum ob mo­<lb/>tum Terræ, quo Sol in con­<lb/>&longs;equentia motu apparente <lb/>transfertur, Luna, priu&longs;quam <lb/>Solem a&longs;&longs;equitur, de&longs;cribit <lb/>angulum <emph type="italics"/>CTa<emph.end type="italics"/>angulo recto majorem in ratione temporis revo­<lb/>lutionis Lunaris Synodicæ ad tempus revolutionis Periodicæ, id <lb/>e&longs;t, in ratione 29<emph type="sup"/>d.<emph.end type="sup"/> 12<emph type="sup"/>h.<emph.end type="sup"/> 44′. </s> <s>ad 27<emph type="sup"/>d.<emph.end type="sup"/> 7<emph type="sup"/>h.<emph.end type="sup"/> 43′. </s> <s>Et hoc pacto an­<lb/>guli omnes circa centrum <emph type="italics"/>T<emph.end type="italics"/>dilatantur in eadem ratione, & Va­<lb/>riatio maxima quæ &longs;ecus e&longs;&longs;et 32′. </s> <s>32″, jam aucta in eadem ratione <lb/>fit 35′. </s> <s>10″. </s></p> <p type="margin"> <s><margin.target id="note431"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Hæc e&longs;t ejus magnitudo in mediocri di&longs;tantia Solis a Terra, <lb/>neglectis differentiis quæ a curvatura Orbis magni majorique So­<lb/>lis actione in Lunam falcatam & novam quam in gibbo&longs;am & <lb/>plenam, oriri po&longs;&longs;int. </s> <s>In aliis di&longs;tantiis Solis a Terra, Variatio <lb/>maxima e&longs;t in ratione quæ componitur ex duplicata ratione tem­<lb/>poris revolutionis Synodicæ Lunaris (dato anni tempore) directe, <lb/>& triplicata ratione di&longs;tantiæ Solis a Terra inver&longs;e. </s> <s>IdeoQ.E.I. <pb xlink:href="039/01/431.jpg" pagenum="403"/>Apogæo Solis, Variatio maxima e&longs;t 33′. </s> <s>14″, & in ejus Perigæo <lb/><arrow.to.target n="note432"/>37′. </s> <s>11″, &longs;i modo Eccentricitas Solis &longs;it ad Orbis magni &longs;emidia­<lb/>metrum tran&longs;ver&longs;am ut (16 15/16) ad 1000. </s></p> <p type="margin"> <s><margin.target id="note432"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Hactenus Variationem inve&longs;tigavimus in Orbe non eccentrico, <lb/>in quo utique Luna in Octantibus &longs;uis &longs;emper e&longs;t in mediocri &longs;ua <lb/>di&longs;tantia a Terra. </s> <s>Si Luna propter eccentricitatem &longs;uam, magis <lb/>vel minus di&longs;tat a Terra quam &longs;i locaretur in hoc Orbe, Variatio <lb/>paulo major e&longs;&longs;e pote&longs;t vel paulo minor quam pro Regula hic <lb/>allata: &longs;ed exce&longs;&longs;um vel defectum ab A&longs;tronomis per Phænomena <lb/>determinandum relinquo. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXX. PROBLEMA XI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire motum borarium Nodorum Lunæ in Orbe circulari.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>De&longs;ignet <emph type="italics"/>S<emph.end type="italics"/>Solem, <emph type="italics"/>T<emph.end type="italics"/>Terram, <emph type="italics"/>P<emph.end type="italics"/>Lunam, <emph type="italics"/>NPn<emph.end type="italics"/>Orbem Lunæ, <lb/><emph type="italics"/>Npn<emph.end type="italics"/>ve&longs;tigium Orbis in plano Eclipticæ; <emph type="italics"/>N, n<emph.end type="italics"/>Nodos, <emph type="italics"/>nTNm<emph.end type="italics"/><lb/><figure id="id.039.01.431.1.jpg" xlink:href="039/01/431/1.jpg"/><lb/>lineam Nodorum infinite productam; <emph type="italics"/>PI, PK<emph.end type="italics"/>perpendicula de­<lb/>mi&longs;&longs;a in lineas <emph type="italics"/>ST, <expan abbr="Qq;">Qque</expan> Pp<emph.end type="italics"/>perpendiculum demi&longs;&longs;um in planum <pb xlink:href="039/01/432.jpg" pagenum="404"/>Eclipticæ; <emph type="italics"/>Q, q<emph.end type="italics"/>Quadraturas Lunæ in plano Eclipticæ, & <emph type="italics"/>p K<emph.end type="italics"/><lb/><arrow.to.target n="note433"/>perpendiculum in lineam <emph type="italics"/>Qq<emph.end type="italics"/>Quadraturis interjacentem. </s> <s>Vis <lb/>Solis ad perturbandum motum Lunæ (per Prop.xxv.) duplex e&longs;t, <lb/>altera lineæ <emph type="italics"/>LM,<emph.end type="italics"/>altera lineæ <emph type="italics"/>MT<emph.end type="italics"/>proportionalis. </s> <s>Et Luna vi <lb/>priore in Terram, po&longs;teriore in Solem &longs;ecundum lineam rectæ <emph type="italics"/>ST<emph.end type="italics"/><lb/>a Terra ad Solem ductæ parallelam trahitur. </s> <s>Vis prior <emph type="italics"/>LM<emph.end type="italics"/><lb/>agit &longs;ecundum planum orbis Lunaris, & propterea &longs;itum plani nil <lb/>mutat. </s> <s>Hæc igitur negligenda e&longs;t. </s> <s>Vis po&longs;terior <emph type="italics"/>MT<emph.end type="italics"/>qua planum <lb/>Orbis Lunaris perturbatur eadem e&longs;t cum vi 3<emph type="italics"/>PK<emph.end type="italics"/>vel 3<emph type="italics"/>IT.<emph.end type="italics"/><lb/>Et hæc vis (per Prop.xxv.) e&longs;t ad vim qua Luna in circulo circa <lb/><figure id="id.039.01.432.1.jpg" xlink:href="039/01/432/1.jpg"/><lb/>Terram quic&longs;centem tempore &longs;uo periodico uniformiter revolvi <lb/>po&longs;&longs;et, ut 3<emph type="italics"/>IT<emph.end type="italics"/>ad Radium circuli multiplicatum per numerum <lb/>178,725, &longs;ive ut <emph type="italics"/>IT<emph.end type="italics"/>ad Radium multiplicatum per 59,575. Cæte­<lb/>rum in hoc calculo & eo omni qui &longs;equitur, con&longs;idero lineas om­<lb/>nes a Luna ad Solem ductas tanquam parallelas lineæ quæ a Terra <lb/>ad Solem ducitur, propterea quod inclinatio tantum fere minuit <lb/>effectus omnes in aliquibus ca&longs;ibus, quantum auget in aliis; & <lb/>Nodorum motus mediocres quærimus, neglectis i&longs;tiu&longs;modi minu­<lb/>tiis, quæ calculum nimis impeditum redderent. </s></p><pb xlink:href="039/01/433.jpg" pagenum="405"/> <p type="margin"> <s><margin.target id="note433"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>De&longs;ignet jam <emph type="italics"/>PM<emph.end type="italics"/>arcum, quem Luna dato tempore quam <lb/><arrow.to.target n="note434"/>minimo de&longs;cribit, & <emph type="italics"/>ML<emph.end type="italics"/>lineolam quam Luna, impellente vi <lb/>præfata 3<emph type="italics"/>IT,<emph.end type="italics"/>eodem tempore de&longs;cribere po&longs;&longs;et. </s> <s>Jungantur <lb/><emph type="italics"/>PL, MP,<emph.end type="italics"/>& producantur eæ ad <emph type="italics"/>m<emph.end type="italics"/>& <emph type="italics"/>l,<emph.end type="italics"/>ubi &longs;ecent planum E­<lb/>clipticæ; inque <emph type="italics"/>Tm<emph.end type="italics"/>demittatur perpendiculum <emph type="italics"/>PH.<emph.end type="italics"/>Et quo­<lb/>niam recta <emph type="italics"/>ML<emph.end type="italics"/>parallela e&longs;t plano Eclipticæ, ideoque cum recta <lb/><emph type="italics"/>ml<emph.end type="italics"/>quæ in plano illo jacet concurrere non pote&longs;t, & tamen ja­<lb/>cent hæ rectæ in plano communi <emph type="italics"/>LMP ml<emph.end type="italics"/>; parallelæ erunt hæ­<lb/>rectæ, & propterea &longs;imilia erunt triangula <emph type="italics"/>LMP, Lmp.<emph.end type="italics"/>Jam <lb/>cum <emph type="italics"/>MPm<emph.end type="italics"/>&longs;it in plano Orbis, in quo Luna in loco <emph type="italics"/>P<emph.end type="italics"/>moveba­<lb/>tur, incidet punctum <emph type="italics"/>m<emph.end type="italics"/>in lineam <emph type="italics"/>Nn<emph.end type="italics"/>per Orbis illius Nodos. <lb/><emph type="italics"/>N, n<emph.end type="italics"/>dictam. </s> <s>Et quoniam vis qua lineola <emph type="italics"/>LM<emph.end type="italics"/>generatur, &longs;i <lb/>tota &longs;imul & &longs;emel in loco <emph type="italics"/>P<emph.end type="italics"/>impre&longs;&longs;a e&longs;&longs;et, efficeret ut Luna <lb/>moveretur in arcu, cujus chorda e&longs;&longs;et <emph type="italics"/>LP,<emph.end type="italics"/>atque adeo trans­<lb/>ferret Lunam de plano <emph type="italics"/>MPmT<emph.end type="italics"/>in planum <emph type="italics"/>LPIT<emph.end type="italics"/>; motus an­<lb/>gularis Nodorum a vi illa genitus, æqualis erit angulo <emph type="italics"/>mTl.<emph.end type="italics"/>E&longs;t <lb/>autem <emph type="italics"/>ml<emph.end type="italics"/>ad <emph type="italics"/>mP<emph.end type="italics"/>ut <emph type="italics"/>ML<emph.end type="italics"/>ad <emph type="italics"/>MP,<emph.end type="italics"/>adeoque cum <emph type="italics"/>MP<emph.end type="italics"/>ob da­<lb/>tum tempus data &longs;it, e&longs;t <emph type="italics"/>ml<emph.end type="italics"/>ut rectangulum <emph type="italics"/>MLXmP,<emph.end type="italics"/>id e&longs;t, <lb/>ut rectangulum <emph type="italics"/>ITXmP.<emph.end type="italics"/>Et angulus <emph type="italics"/>mTl,<emph.end type="italics"/>&longs;i modo angulus <lb/><emph type="italics"/>Tml<emph.end type="italics"/>rectus &longs;it, e&longs;t ut (<emph type="italics"/>ml/Tm<emph.end type="italics"/>), & propterea ut (<emph type="italics"/>ITXPm/Tm<emph.end type="italics"/>), id e&longs;t, <lb/>(ob proportionales <emph type="italics"/>Tm<emph.end type="italics"/>& <emph type="italics"/>mP, TP<emph.end type="italics"/>& <emph type="italics"/>PH<emph.end type="italics"/>) ut (<emph type="italics"/>ITXPH/TP<emph.end type="italics"/>), <lb/>adeoque ob datam <emph type="italics"/>TP,<emph.end type="italics"/>ut <emph type="italics"/>ITXPH.<emph.end type="italics"/>Quod &longs;i angulus <emph type="italics"/>Tml,<emph.end type="italics"/><lb/>&longs;eu <emph type="italics"/>STN<emph.end type="italics"/>obliquus fit, erit angulus <emph type="italics"/>mTl<emph.end type="italics"/>adhuc minor, in rati­<lb/>one &longs;inus anguli <emph type="italics"/>STN<emph.end type="italics"/>ad Radium. </s> <s>E&longs;t igitur velocitas No­<lb/>dorum ut <emph type="italics"/>ITXPHXAZ,<emph.end type="italics"/>&longs;ive ut contentum &longs;ub &longs;inubus trium <lb/>angulorum <emph type="italics"/>TPI, PTN<emph.end type="italics"/>& <emph type="italics"/>STN.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note434"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Si anguli illi, Nodis in Quadraturis & Luna in Syzygia exi&longs;ten­<lb/>tibus, recti &longs;int, lineola <emph type="italics"/>ml<emph.end type="italics"/>abibit in infinitum, & angulus <emph type="italics"/>mTl<emph.end type="italics"/><lb/>evadet angulo <emph type="italics"/>mPl<emph.end type="italics"/>æqualis. </s> <s>Hoc autem in ca&longs;u, angulus <emph type="italics"/>mPl<emph.end type="italics"/><lb/>e&longs;t ad angulum <emph type="italics"/>PTM,<emph.end type="italics"/>quem Luna eodem tempore motu &longs;uo <lb/>apparente circa Terram de&longs;cribit ut 1 ad 59,575. Nam angulus <lb/><emph type="italics"/>mPl<emph.end type="italics"/>æqualis e&longs;t angulo <emph type="italics"/>LPM,<emph.end type="italics"/>id e&longs;t, angulo deflexionis Lunæ <lb/>a recto tramite, quem &longs;ola vis præfata Solaris 3<emph type="italics"/>IT<emph.end type="italics"/>&longs;i tum ce&longs;&longs;a­<lb/>ret Lunæ gravitas dato illo tempore generare po&longs;&longs;et; & angulus <lb/><emph type="italics"/>PTM<emph.end type="italics"/>æqualis e&longs;t angulo deflexionis Lunæ a recto tramite, quem <lb/>vis illa, qua Luna in Orbe &longs;uo retinetur, &longs;i tum ce&longs;&longs;aret vis Sola­<lb/>ris 3<emph type="italics"/>IT<emph.end type="italics"/>eodem tempore generaret. </s> <s>Et hæ vires, ut &longs;upra dixi-<pb xlink:href="039/01/434.jpg" pagenum="406"/>mus, &longs;unt ad invicem ut 1 ad 59,575. Ergo cum motus medius <lb/><arrow.to.target n="note435"/>horarius Lunæ (re&longs;pectu fixarum) &longs;it 32′. </s> <s>56″. </s> <s>27′. </s> <s>12<emph type="sup"/>iv<emph.end type="sup"/>1/2, motus <lb/>horarius Nodi in hoc ca&longs;u erit 33″. </s> <s>10′. </s> <s>33<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>12<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>Aliis autem in <lb/>ca&longs;ibus motus i&longs;te horarius erit ad 33″. </s> <s>10′. </s> <s>33<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>12<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>ut conten­<lb/>tum &longs;ub &longs;inubus angulorum trium <emph type="italics"/>TPI, PTN,<emph.end type="italics"/>& <emph type="italics"/>STN<emph.end type="italics"/>(&longs;eu <lb/>di&longs;tantiarum Lunæ a Quadratura, Lunæ a Nodo, & Nodi a Sole) <lb/>ad cubum Radii. </s> <s>Et quoties &longs;ignum anguli alicujus de affirmativo <lb/>in negativum, deque negativo in affirmativum mutatur, debebit <lb/>motus regre&longs;&longs;ivus in progre&longs;&longs;ivum & progre&longs;&longs;ivus in regre&longs;&longs;ivum <lb/>mutari. </s> <s>Unde fit ut Nodi progrediantur quoties Luna inter Qua­<lb/>draturam alterutram & Nodum Quadraturæ proximum ver&longs;atur. </s> <s><lb/>Aliis in ca&longs;ibus regrediuntur, & per exce&longs;&longs;um regre&longs;&longs;us &longs;upra pro­<lb/>gre&longs;&longs;um, &longs;ingulis men&longs;ibus &longs;eruntur in antecedentia. </s></p> <p type="margin"> <s><margin.target id="note435"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i a dati arcus quam minimi <emph type="italics"/>PM<emph.end type="italics"/>terminis <emph type="italics"/>P<emph.end type="italics"/><lb/>& <emph type="italics"/>M<emph.end type="italics"/>ad lineam Quadraturas jungentem <emph type="italics"/>Qq<emph.end type="italics"/>demittantur perpen­<lb/>dicula <emph type="italics"/>PK, Mk,<emph.end type="italics"/>eademque producantur donec &longs;ecent lineam <lb/>Nodorum <emph type="italics"/>Nn<emph.end type="italics"/>in <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>d<emph.end type="italics"/>; erit motus horarius Nodorum ut area <lb/><emph type="italics"/>MPDd<emph.end type="italics"/>& quadratum lineæ <emph type="italics"/>AZ<emph.end type="italics"/>conjunctim. </s> <s>Sunto enim <lb/><figure id="id.039.01.434.1.jpg" xlink:href="039/01/434/1.jpg"/><lb/><emph type="italics"/>PK, PH<emph.end type="italics"/>& <emph type="italics"/>AZ<emph.end type="italics"/>prædicti tres &longs;inus. </s> <s>Nempe <emph type="italics"/>PK<emph.end type="italics"/>&longs;inus di­<lb/>&longs;tantiæ Lunæ a Quadratura, <emph type="italics"/>PH<emph.end type="italics"/>&longs;inus di&longs;tantiæ Lunæ a Nodo, & <lb/><emph type="italics"/>AZ<emph.end type="italics"/>&longs;inus di&longs;tantiæ Nodi a Sole: & erit velocitas Nodi ut conten­<lb/>tum <emph type="italics"/>PKXPHXAZ.<emph.end type="italics"/>E&longs;t autem <emph type="italics"/>PT<emph.end type="italics"/>ad <emph type="italics"/>PK<emph.end type="italics"/>ut <emph type="italics"/>PM<emph.end type="italics"/>ad <emph type="italics"/>Kk,<emph.end type="italics"/><lb/>adeoque ob datas <emph type="italics"/>PT<emph.end type="italics"/>& <emph type="italics"/>PM<emph.end type="italics"/>e&longs;t <emph type="italics"/>Kk<emph.end type="italics"/>ip&longs;i <emph type="italics"/>PK<emph.end type="italics"/>proportionalis. </s> <s><lb/>E&longs;t & <emph type="italics"/>AT<emph.end type="italics"/>ad <emph type="italics"/>PD<emph.end type="italics"/>ut <emph type="italics"/>AZ<emph.end type="italics"/>ad <emph type="italics"/>PH,<emph.end type="italics"/>& propterea <emph type="italics"/>PH<emph.end type="italics"/>rectangulo <pb xlink:href="039/01/435.jpg" pagenum="407"/><emph type="italics"/>PDXAZ<emph.end type="italics"/>proportionalis, & conjunctis rationibus, <emph type="italics"/>PKXPH<emph.end type="italics"/><lb/><arrow.to.target n="note436"/>e&longs;t ut contentum <emph type="italics"/>KkXPDXAZ,<emph.end type="italics"/>& <emph type="italics"/>PKXPHXAZ<emph.end type="italics"/>ut <lb/><emph type="italics"/>KkXPDXAZ qu.<emph.end type="italics"/>id e&longs;t, ut area <emph type="italics"/>PDdM<emph.end type="italics"/>& <emph type="italics"/>AZqu.<emph.end type="italics"/>con­<lb/>junctim. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note436"/>LIBER <lb/>TIRTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol<emph.end type="italics"/>2. In data quavis Nodorum po&longs;itione, motus horarius <lb/>mediocris e&longs;t &longs;emi&longs;&longs;is motus horarii in Syzygiis Lunæ, ideoque e&longs;t <lb/>ad 16″. </s> <s>35′. </s> <s>16<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>36<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>ut quadratum &longs;inus di&longs;tantiæ Nodorum a <lb/>Syzygiis ad quadratum Radii, five ut <emph type="italics"/>AZqu.<emph.end type="italics"/>AD <emph type="italics"/>AT.qu.<emph.end type="italics"/>Nam <lb/>&longs;i Luna uniformi cum motu perambulet &longs;emicirculum <emph type="italics"/>QAq,<emph.end type="italics"/>&longs;um­<lb/>ma omnium arearum <emph type="italics"/>PDdM,<emph.end type="italics"/>quo tempore Luna pergit a <emph type="italics"/>Q<emph.end type="italics"/>ad <lb/><emph type="italics"/>M,<emph.end type="italics"/>erit area <emph type="italics"/>QMdE<emph.end type="italics"/>quæ ad circuli tangentem <emph type="italics"/>QE<emph.end type="italics"/>termina­<lb/>tur; & quo tempore Luna attingit punctum <emph type="italics"/>n,<emph.end type="italics"/>&longs;umma illa erit <lb/>area tota <emph type="italics"/>EQAn<emph.end type="italics"/>quam linea <emph type="italics"/>PD<emph.end type="italics"/>de&longs;cribit, dein Luna pergente <lb/>ab <emph type="italics"/>n<emph.end type="italics"/>ad <emph type="italics"/>q,<emph.end type="italics"/>linea <emph type="italics"/>PD<emph.end type="italics"/>cadet extra circulum, & aream <emph type="italics"/>nqe<emph.end type="italics"/>ad <lb/>circuli tangentem <emph type="italics"/>qe<emph.end type="italics"/>terminatam de&longs;cribet; quæ, quoniam Nodi <lb/>prius regrediebantur, jam vero progrediuntur, &longs;ubduci debet de <lb/>area priore, & cum æqualis &longs;it areæ <emph type="italics"/>QEN,<emph.end type="italics"/>relinquet &longs;emicircu­<lb/>lum <emph type="italics"/>NQAn.<emph.end type="italics"/>Igitur &longs;umma omnium arearum <emph type="italics"/>PDdM,<emph.end type="italics"/>quo <lb/>tempore Luna &longs;emicirculum de&longs;cribit, e&longs;t area &longs;emicirculi; & <lb/>&longs;umma omnium quo tempore Luna circulum de&longs;cribit e&longs;t area cir­<lb/>culi totius. </s> <s>At area <emph type="italics"/>PDdM,<emph.end type="italics"/>ubi Luna ver&longs;atur in Syzygiis, e&longs;t <lb/>rectangulum &longs;ub arcu <emph type="italics"/>PM<emph.end type="italics"/>& radic <emph type="italics"/>MT<emph.end type="italics"/>; & &longs;umma omnium huic <lb/>æqualium arearum, quo tempore Luna circulum de&longs;cribit, e&longs;t <lb/>rectangulum &longs;ub circumferentia tota & radio circuli; & hoc <lb/>rectangulum, cum &longs;it æquale duobus circulis, duplo majus e&longs;t <lb/>quam rectangulum prius. </s> <s>Proinde Nodi, ea cum velocitate uNI­<lb/>formiter continuata quam habent in Syzygiis Lunaribus, &longs;patium <lb/>duplo majus de&longs;criberent quam revera de&longs;cribunt; & propterea <lb/>motus mediocris quocum, &longs;i uniformiter continuaretur, &longs;patium <lb/>a &longs;e inæquabili cum motu revera confectum de&longs;cribere po&longs;&longs;ent, e&longs;t <lb/>&longs;emi&longs;&longs;is motus quem habent in Syzygiis Lunæ. </s> <s>Unde cum mo­<lb/>tus horarius maximus, &longs;i Nodi in Quadraturis ver&longs;antur, &longs;it <lb/>33″. </s> <s>10′. </s> <s>33<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>12<emph type="sup"/>v<emph.end type="sup"/>, motus mediocris horarius in hoc ca&longs;u erit <lb/>16″. </s> <s>35′. </s> <s>16<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>36<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>Et cum motus horarius Nodorum &longs;emper &longs;it <lb/>ut <emph type="italics"/>AZqu.<emph.end type="italics"/>& area <emph type="italics"/>PDdM<emph.end type="italics"/>conjunctim, & propterea motus ho­<lb/>rarius Nodorum in Syzygiis Lunæ ut <emph type="italics"/>AZqu.<emph.end type="italics"/>& area <emph type="italics"/>PDdM<emph.end type="italics"/><lb/>conjunctim, id e&longs;t (ob datam aream <emph type="italics"/>PDdM<emph.end type="italics"/>in Syzygiis de­<lb/>&longs;criptam) ut <emph type="italics"/>AZqu.<emph.end type="italics"/>erit etiam motus mediocris ut <emph type="italics"/>AZqu.<emph.end type="italics"/>atque <lb/>adeo hic motus, ubi Nodi extra Quadraturas ver&longs;antur, erit ad <lb/>16″. </s> <s>35′. </s> <s>16<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>36<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>ut <emph type="italics"/>AZqu.<emph.end type="italics"/>ad <emph type="italics"/>ATqu. </s> <s>Q.E.D.<emph.end type="italics"/><pb xlink:href="039/01/436.jpg" pagenum="408"/><arrow.to.target n="note437"/></s></p> <p type="margin"> <s><margin.target id="note437"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXI. PROBLEMA XII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire motum horarium Nodorum Lunæ in Orbe Elliptico.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>De&longs;ignet <emph type="italics"/>Qpmaq<emph.end type="italics"/>Ellip&longs;in, axe majore <emph type="italics"/>Qq,<emph.end type="italics"/>minore <emph type="italics"/>ab<emph.end type="italics"/>de­<lb/>&longs;criptam, <emph type="italics"/>QAq<emph.end type="italics"/>Circulum circum&longs;criptum, <emph type="italics"/>T<emph.end type="italics"/>Terram in utriu&longs;que <lb/>centro communi, <emph type="italics"/>S<emph.end type="italics"/>Solem, <emph type="italics"/>p<emph.end type="italics"/>Lunam in Ellip&longs;i motam, & <emph type="italics"/>pm<emph.end type="italics"/>ar­<lb/>cum quem data temporis particula quam minima de&longs;cribit, <emph type="italics"/>N<emph.end type="italics"/>& <emph type="italics"/>n<emph.end type="italics"/><lb/>Nodos linea <emph type="italics"/>Nn<emph.end type="italics"/>junctos, <emph type="italics"/>pK<emph.end type="italics"/>& <emph type="italics"/>mk<emph.end type="italics"/>perpendicula in axem <emph type="italics"/>Qq<emph.end type="italics"/><lb/>demi&longs;&longs;a & hinc inde producta, donec occurrant Circulo in <emph type="italics"/>P<emph.end type="italics"/>& <emph type="italics"/>M,<emph.end type="italics"/><lb/><figure id="id.039.01.436.1.jpg" xlink:href="039/01/436/1.jpg"/><lb/>& lineæ Nodorum in <emph type="italics"/>D<emph.end type="italics"/>& <emph type="italics"/>d.<emph.end type="italics"/>Et &longs;i Luna, radio ad Terram du­<lb/>cto, aream de&longs;cribat tempori proportionalem, erit motus Nodi in <lb/>Ellip&longs;i ut area <emph type="italics"/>pDdm.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam &longs;i <emph type="italics"/>PF<emph.end type="italics"/>tangat Circulum in <emph type="italics"/>P,<emph.end type="italics"/>& producta occurrat <emph type="italics"/>TN<emph.end type="italics"/><lb/>in <emph type="italics"/>F,<emph.end type="italics"/>& <emph type="italics"/>pf<emph.end type="italics"/>tangat Ellip&longs;in in <emph type="italics"/>p<emph.end type="italics"/>& producta occurrat eidem <emph type="italics"/>TN<emph.end type="italics"/><pb xlink:href="039/01/437.jpg" pagenum="409"/>in <emph type="italics"/>f,<emph.end type="italics"/>conveniant autem hæ tangentes in axe <emph type="italics"/>TQ<emph.end type="italics"/>ad <emph type="italics"/>Y<emph.end type="italics"/>; & &longs;i <lb/><arrow.to.target n="note438"/><emph type="italics"/>ML<emph.end type="italics"/>de&longs;ignet &longs;patium quod Luna in Circulo revolvens, interea <lb/>dum de&longs;cribit arcum <emph type="italics"/>PM,<emph.end type="italics"/>urgente & impellente vi prædicta <lb/>3<emph type="italics"/>IT,<emph.end type="italics"/>motu tran&longs;ver&longs;o de&longs;cribere po&longs;&longs;et, & <emph type="italics"/>ml<emph.end type="italics"/>de&longs;ignet &longs;patium <lb/>quod Luna in Ellip&longs;i revolvens eodem tempore, urgente etiam vi <lb/>3<emph type="italics"/>IT,<emph.end type="italics"/>de&longs;cribere po&longs;&longs;et; & producantur <emph type="italics"/>LP<emph.end type="italics"/>& <emph type="italics"/>lp<emph.end type="italics"/>donec occurrant <lb/>plano Eclipticæ in <emph type="italics"/>G<emph.end type="italics"/>& <emph type="italics"/>g<emph.end type="italics"/>; & jungantur <emph type="italics"/>FG<emph.end type="italics"/>& <emph type="italics"/>fg,<emph.end type="italics"/>quarum <emph type="italics"/>FG<emph.end type="italics"/><lb/>producta &longs;ecet <emph type="italics"/>pf, pg<emph.end type="italics"/>& <emph type="italics"/>TQ<emph.end type="italics"/>in <emph type="italics"/>c, e<emph.end type="italics"/>& <emph type="italics"/>R<emph.end type="italics"/>re&longs;pective, & <emph type="italics"/>fg<emph.end type="italics"/>pro­<lb/>ducta &longs;ecet <emph type="italics"/>TQ<emph.end type="italics"/>in <emph type="italics"/>r<emph.end type="italics"/>: Quoniam vis 3<emph type="italics"/>IT<emph.end type="italics"/>&longs;eu 3<emph type="italics"/>PK<emph.end type="italics"/>in Circulo <lb/>e&longs;t ad vim 3<emph type="italics"/>IT<emph.end type="italics"/>&longs;eu 3<emph type="italics"/>pK<emph.end type="italics"/>in Ellip&longs;i, ut <emph type="italics"/>PK<emph.end type="italics"/>ad <emph type="italics"/>pK,<emph.end type="italics"/>&longs;eu <emph type="italics"/>AT<emph.end type="italics"/>ad <lb/><emph type="italics"/>aT<emph.end type="italics"/>; erit &longs;patium <emph type="italics"/>ML<emph.end type="italics"/>vi priore genitum, ad &longs;patium <emph type="italics"/>ml<emph.end type="italics"/>vi po­<lb/>&longs;teriore genitum, ut <emph type="italics"/>PK<emph.end type="italics"/>ad <emph type="italics"/>pK,<emph.end type="italics"/>id e&longs;t, ob &longs;imiles figuras <lb/><emph type="italics"/>PYKp<emph.end type="italics"/>& <emph type="italics"/>FYRc,<emph.end type="italics"/>ut <emph type="italics"/>FR<emph.end type="italics"/>ad <emph type="italics"/>cR.<emph.end type="italics"/>E&longs;t autem <emph type="italics"/>ML<emph.end type="italics"/>ad <emph type="italics"/>FG<emph.end type="italics"/>(ob <lb/>&longs;imilia triangula <emph type="italics"/>PLM, PGF<emph.end type="italics"/>) ut <emph type="italics"/>PL<emph.end type="italics"/>ad <emph type="italics"/>PG,<emph.end type="italics"/>hoc e&longs;t (ob <lb/>parallelas <emph type="italics"/>Lk, PK, GR<emph.end type="italics"/>) ut <emph type="italics"/>pl<emph.end type="italics"/>ad <emph type="italics"/>pe,<emph.end type="italics"/>id e&longs;t, (ob &longs;imilia trian­<lb/>gula <emph type="italics"/>plm, cpe<emph.end type="italics"/>) ut <emph type="italics"/>lm<emph.end type="italics"/>ad <emph type="italics"/>ce<emph.end type="italics"/>; & inver&longs;e ut <emph type="italics"/>LM<emph.end type="italics"/>e&longs;t ad <emph type="italics"/>lm,<emph.end type="italics"/>&longs;eu <lb/><emph type="italics"/>FR<emph.end type="italics"/>ad <emph type="italics"/>cR,<emph.end type="italics"/>ita e&longs;t <emph type="italics"/>FG<emph.end type="italics"/>ad <emph type="italics"/>ce.<emph.end type="italics"/>Et propterea &longs;i <emph type="italics"/>fg<emph.end type="italics"/>e&longs;&longs;et ad <emph type="italics"/>ce<emph.end type="italics"/>ut <lb/><emph type="italics"/>fY<emph.end type="italics"/>ad <emph type="italics"/>cY,<emph.end type="italics"/>id e&longs;t, ut <emph type="italics"/>fr<emph.end type="italics"/>ad <emph type="italics"/>cR<emph.end type="italics"/>(hoc e&longs;t, ut <emph type="italics"/>fr<emph.end type="italics"/>ad <emph type="italics"/>FR<emph.end type="italics"/>& <emph type="italics"/>FR<emph.end type="italics"/>ad <emph type="italics"/>cR<emph.end type="italics"/><lb/>conjunctim, id e&longs;t, ut <emph type="italics"/>fT<emph.end type="italics"/>ad <emph type="italics"/>FT<emph.end type="italics"/>& <emph type="italics"/>FG<emph.end type="italics"/>ad <emph type="italics"/>ce<emph.end type="italics"/>conjunctim,) quo­<lb/>niam ratio <emph type="italics"/>FG<emph.end type="italics"/>ad <emph type="italics"/>ce<emph.end type="italics"/>utrinque ablata relinquit rationes <emph type="italics"/>fg<emph.end type="italics"/>ad <emph type="italics"/>FG<emph.end type="italics"/><lb/>& <emph type="italics"/>fT<emph.end type="italics"/>ad <emph type="italics"/>FT,<emph.end type="italics"/>foret <emph type="italics"/>fg<emph.end type="italics"/>ad <emph type="italics"/>FG<emph.end type="italics"/>ut <emph type="italics"/>fT<emph.end type="italics"/>ad <emph type="italics"/>FT<emph.end type="italics"/>; atque adeo anguli, <lb/>quos <emph type="italics"/>FG<emph.end type="italics"/>& <emph type="italics"/>fg<emph.end type="italics"/>&longs;ubtenderent ad Terram <emph type="italics"/>T,<emph.end type="italics"/>æquarentur inter &longs;e. </s> <s><lb/>Sed anguli illi (per ca quæ in præcedente Propo&longs;itione expo&longs;ui­<lb/>mus) &longs;unt motus Nodorum, quo tempore Luna in Circulo ar­<lb/>cum <emph type="italics"/>PM,<emph.end type="italics"/>in Ellip&longs;i arcum <emph type="italics"/>pm<emph.end type="italics"/>percurrit: & propterea motus <lb/>Nodorum in Circulo & Ellip&longs;i æquarentur inter &longs;e. </s> <s>Hæc ita &longs;e <lb/>haberent, &longs;i modo <emph type="italics"/>fg<emph.end type="italics"/>e&longs;&longs;et ad <emph type="italics"/>ce<emph.end type="italics"/>ut <emph type="italics"/>fY<emph.end type="italics"/>ad <emph type="italics"/>cY,<emph.end type="italics"/>id e&longs;t, &longs;i <emph type="italics"/>fg<emph.end type="italics"/>æqua­<lb/>lis e&longs;&longs;et (<emph type="italics"/>ceXfY/cY<emph.end type="italics"/>). Verum ob &longs;imilia triangula <emph type="italics"/>fgp, cep,<emph.end type="italics"/>e&longs;t <emph type="italics"/>fg<emph.end type="italics"/><lb/>ad <emph type="italics"/>ce<emph.end type="italics"/>ut <emph type="italics"/>fp<emph.end type="italics"/>ad <emph type="italics"/>cp<emph.end type="italics"/>; ideoque <emph type="italics"/>fg<emph.end type="italics"/>æqualis e&longs;t (<emph type="italics"/>ceXfp/cp<emph.end type="italics"/>); & propterea <lb/>angulus, quem <emph type="italics"/>fg<emph.end type="italics"/>revera &longs;ubtendit, e&longs;t ad angulum priorem, quem <lb/><emph type="italics"/>FG<emph.end type="italics"/>&longs;ubtendit, hoc e&longs;t, motus Nodorum in Ellip&longs;i ad motum <lb/>Nodorum in Circulo, ut hæc <emph type="italics"/>fg<emph.end type="italics"/>&longs;eu (<emph type="italics"/>ceXfp/cp<emph.end type="italics"/>) ad priorem <emph type="italics"/>fg<emph.end type="italics"/>&longs;eu <lb/>(<emph type="italics"/>ceXfY/cY<emph.end type="italics"/>), id e&longs;t, ut <emph type="italics"/>fpXcY<emph.end type="italics"/>ad <emph type="italics"/>fYXcp,<emph.end type="italics"/>&longs;eu <emph type="italics"/>fp<emph.end type="italics"/>ad <emph type="italics"/>fY<emph.end type="italics"/>& <emph type="italics"/>cY<emph.end type="italics"/>ad <emph type="italics"/>cp,<emph.end type="italics"/><lb/>hoc e&longs;t, &longs;i <emph type="italics"/>ph<emph.end type="italics"/>ip&longs;i <emph type="italics"/>TN<emph.end type="italics"/>parallela occurrat <emph type="italics"/>FP<emph.end type="italics"/>in <emph type="italics"/>h,<emph.end type="italics"/>ut <emph type="italics"/>Fh<emph.end type="italics"/>ad <emph type="italics"/>FY<emph.end type="italics"/><lb/>& <emph type="italics"/>FY<emph.end type="italics"/>ad <emph type="italics"/>FP<emph.end type="italics"/>; hoc e&longs;t, ut <emph type="italics"/>Fh<emph.end type="italics"/>ad <emph type="italics"/>FP<emph.end type="italics"/>&longs;eu <emph type="italics"/>Dp<emph.end type="italics"/>ad <emph type="italics"/>DP,<emph.end type="italics"/>adeoque <lb/>ut area <emph type="italics"/>Dpmd<emph.end type="italics"/>ad aream <emph type="italics"/>DPMd.<emph.end type="italics"/>Et propterea, cum area po-<pb xlink:href="039/01/438.jpg" pagenum="410"/><arrow.to.target n="note439"/>&longs;terior proportionalis &longs;it motui Nodorum in Circulo, erit area <lb/>prior proportionalis motui Nodorum in Ellip&longs;i. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note438"/>LIBER <lb/>TERTIUS.</s></p> <p type="margin"> <s><margin.target id="note439"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Igitur cum, in data Nodorum po&longs;itione, &longs;umma omnium <lb/>arearum <emph type="italics"/>pDdm,<emph.end type="italics"/>quo tempore Luna pergit a Quadratura ad lo­<lb/>cum quemvis <emph type="italics"/>m,<emph.end type="italics"/>&longs;it area <emph type="italics"/>mpQEd,<emph.end type="italics"/>quæ ad Ellip&longs;eos tangentem <lb/><emph type="italics"/>QE<emph.end type="italics"/>terminatur; & &longs;umma omnium arearum illarum, in revolu­<lb/>tione integra, &longs;it area Ellip&longs;eos totius: motus mediocris Nodorum <lb/>in Ellip&longs;i erit ad motum mediocrem Nodorum in Circulo, ut El­<lb/>lip&longs;is ad Circulum; id e&longs;t, ut <emph type="italics"/>Ta<emph.end type="italics"/>ad <emph type="italics"/>TA,<emph.end type="italics"/>&longs;eu 69 ad 70. Et <lb/>propterea, cum motus mediocris horarius Nodorum in Circulo <lb/>&longs;it ad 16″. </s> <s>35′. </s> <s>16<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>36<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>ut <emph type="italics"/>AZqu.<emph.end type="italics"/>ad <emph type="italics"/>ATqu.<emph.end type="italics"/>&longs;i capiatur angu­<lb/>lus 16″. </s> <s>21′. </s> <s>3<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>30<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>ad angulum 16″. </s> <s>35′. </s> <s>16<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>36<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>ut 69 ad 70, <lb/>erit motus mediocris horarius Nodorum in Ellip&longs;i ad 16″. </s> <s>21′. </s> <s><lb/>3<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>30<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>ut <emph type="italics"/>AZq<emph.end type="italics"/>ad <emph type="italics"/>ATq<emph.end type="italics"/>; hoc e&longs;t, ut quadratum &longs;inus di&longs;tantiæ <lb/>Nodi a Sole ad quadratum Radii. </s></p> <p type="main"> <s>Cæterum Luna, radio ad Terram ducto, aream velocius de&longs;cri­<lb/>bit in Syzygiis quam in Quadraturis, & eo nomine tempus in Sy­<lb/>zygiis contrahitur, in Quadraturis producitur; & una cum tem­<lb/>pore motus Nodorum augetur ac diminuitur. </s> <s>Erat autem mo­<lb/>mentum areæ in Quadraturis Lunæ ad ejus momentum in Syzygiis <lb/>ut 10973 ad 11073, & propterea momentum mediocre in Octan­<lb/>tibus e&longs;t ad exce&longs;&longs;um in Syzygiis, defectumQ.E.I. Quadraturis, ut <lb/>numerorum &longs;emi&longs;umma 11023 ad eorundem &longs;emidifferentiam 50. <lb/>Unde cum tempus Lunæ in &longs;ingulis Orbis particulis æqualibus &longs;it <lb/>reciproce ut ip&longs;ius velocitas, erit tempus mediocre in Octantibus <lb/>ad exce&longs;&longs;um temporis in Quadraturis, ac defectum in Syzygiis, ab <lb/>hac cau&longs;a oriundum, ut 11023 ad 50 quam proxime. </s> <s>Pergendo <lb/>autem a Quadraturis ad Syzygias, invenio quod exce&longs;&longs;us momen­<lb/>torum areæ in locis &longs;ingulis, &longs;upra momentum minimum in Qua­<lb/>draturis, &longs;it ut quadratum &longs;inus di&longs;tantiæ Lunæ a Quadraturis <lb/>quam proxime; & propterea differentia inter momentum in loco <lb/>quocunque & momentum mediocre in Octantibus, e&longs;t ut diffe­<lb/>rentia inter quadratum &longs;inus di&longs;tantiæ Lunæ a Quadraturis & <lb/>quadratum &longs;inus graduum 45, &longs;eu &longs;emi&longs;&longs;em quadrati Radii; & <lb/>incrementum temporis in locis &longs;ingulis inter Octantes & Quadra­<lb/>turas, & decrementum ejus inter Octantes & Syzygias, e&longs;t in ea­<lb/>dem ratione. </s> <s>Motus autem Nodorum, quo tempore Luna per­<lb/>currit &longs;ingulas Orbis particulas æquales, acceleratur vel retardatur <lb/>in duplicata ratione temporis. </s> <s>E&longs;t enim motus i&longs;te, dum Luna <pb xlink:href="039/01/439.jpg" pagenum="411"/>percurrit <emph type="italics"/>PM,<emph.end type="italics"/>(cæteris paribus) ut <emph type="italics"/>ML,<emph.end type="italics"/>& <emph type="italics"/>ML<emph.end type="italics"/>e&longs;t in dupli­<lb/><arrow.to.target n="note440"/>cata ratione temporis. </s> <s>Quare motus Nodorum in Syzygiis, eo <lb/>tempore confectus quo Luna datas Orbis particulas percurrit, di­<lb/>minuitur in duplicata ratione numeri 11073 ad numerum 11023; <lb/>e&longs;tQ.E.D.crementum ad motum reliquum ut 100 ad 10973, ad <lb/>motum vero totum ut 100 ad 11073 quam proxime. </s> <s>Decre­<lb/>mentum autem in locis inter Octantes & Syzygias, & incremen­<lb/>tum in locis inter Octantes & Quadraturas, e&longs;t quam proxime ad <lb/>hoc decrementum, ut motus totus in locis illis ad motum totum <lb/>in Syzygiis & differentia inter quadratum &longs;inus di&longs;tantiæ Lunæ a <lb/>Quadratura & &longs;emi&longs;&longs;em quadrati Radii ad &longs;emi&longs;&longs;em quadrati Ra­<lb/>dii, conjunctim. </s> <s>Unde &longs;i Nodi in Quadraturis ver&longs;entur, & ca­<lb/>piantur loca duo æqualiter ab Octante hinc inde di&longs;tantia, & alia <lb/>duo a Syzygia & Quadratura ii&longs;dem intervallis di&longs;tantia, deque <lb/>decrementis motuum in locis duobus inter Syzygiam & Octantem, <lb/>&longs;ubducantur incrementa motuum in locis reliquis duobus, quæ <lb/>&longs;unt inter Octantem & Quadraturam; decrementum reliquum <lb/>æquale erit decremento in Syzygia: uti rationem ineunti facile <lb/>con&longs;tabit. </s> <s>ProindeQ.E.D.crementum mediocre, quod de Nodo­<lb/>rum motu mediocri &longs;ubduci debet, e&longs;t pars quarta decrementi in <lb/>Syzygia. </s> <s>Motus totus horarius Nodorum in Syzygiis (ubi Luna <lb/>radio ad Terram ducto aream tempori proportionalem de&longs;cribere <lb/>&longs;upponebatur) erat 32″. </s> <s>42′. </s> <s>7<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>Et decrementum motus Nodo­<lb/>rum, quo tempore Luna jam velocior de&longs;cribit idem &longs;patium, <lb/>diximus e&longs;&longs;e ad hunc motum ut 100 ad 11073; adeoQ.E.D.cre­<lb/>mentum illud e&longs;t 17′. </s> <s>43<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>11<emph type="sup"/>v<emph.end type="sup"/>, cujus pars quarta 4′. </s> <s>25<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>48<emph type="sup"/>v<emph.end type="sup"/>, <lb/>motui horario mediocri &longs;uperius invento 16″. </s> <s>21′. </s> <s>3<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>30<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>&longs;ub­<lb/>ducta, relinquit 16″. </s> <s>16′. </s> <s>37<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>42<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>motum mediocrem horarium <lb/>correctum. </s></p> <p type="margin"> <s><margin.target id="note440"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Si Nodi ver&longs;antur extra Quadraturas, & &longs;pectentur loca bina a <lb/>Syzygiis hinc inde æqualiter di&longs;tantia; &longs;umma motuum Nodo­<lb/>rum, ubi Luna ver&longs;atur in his locis, erit ad &longs;ummam motuum, <lb/>ubi Luna in ii&longs;dem locis & Nodi in Quadraturis ver&longs;antur, ut <lb/><emph type="italics"/>AZqu.<emph.end type="italics"/>ad <emph type="italics"/>ATqu.<emph.end type="italics"/>Et decrementa motuum, a cau&longs;is jam expo­<lb/>&longs;itis oriunda, erunt ad invicem ut ip&longs;i motus, adeoque motus reli­<lb/>qui erunt ad invicem ut <emph type="italics"/>AZqu.<emph.end type="italics"/>ad <emph type="italics"/>ATqu.<emph.end type="italics"/>& motus mediocres <lb/>ut motus reliqui. </s> <s>E&longs;t itaque motus mediocris horarius correctus, <lb/>in dato quocunque Nodorum &longs;itu, ad 16″. </s> <s>16′. </s> <s>37<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>42<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>ut. <emph type="italics"/>AZqu.<emph.end type="italics"/><lb/>ad <emph type="italics"/>ATqu.<emph.end type="italics"/>; id e&longs;t, ut quadratum &longs;inus di&longs;tantiæ Nodorum a Sy­<lb/>zygiis ad quadratum Radii. <pb xlink:href="039/01/440.jpg" pagenum="412"/><arrow.to.target n="note441"/></s></p> <p type="margin"> <s><margin.target id="note441"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXII. PROBLEMA XIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire motum medium Nodorum Lunæ.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Motus medius annuus e&longs;t &longs;umma motuum omnium horariorum <lb/>mediocrium in anno. </s> <s>Concipe Nodum ver&longs;ari in <emph type="italics"/>N,<emph.end type="italics"/>& &longs;ingulis <lb/>horis completis retrahi in locum &longs;uum priorem, ut non ob&longs;tante <lb/>motu &longs;uo proprio, datum &longs;emper &longs;ervet &longs;itum ad Stellas Fixas. </s> <s><lb/>Interea vero Solem <emph type="italics"/>S,<emph.end type="italics"/>per motum Terræ, progredi a Nodo, & <lb/>cur&longs;um annuum apparentem uniformiter complere. </s> <s>Sit autem <lb/><emph type="italics"/>Aa<emph.end type="italics"/>arcus datus quam minimus, quem recta <emph type="italics"/>TS<emph.end type="italics"/>ad Solem &longs;emper <lb/>ducta, inter&longs;ectione &longs;ui & circuli <emph type="italics"/>NAn,<emph.end type="italics"/>dato tempore quam mi­<lb/>nimo de&longs;cribit: & motus horarius mediocris (per jam o&longs;ten&longs;a) <lb/>erit ut <emph type="italics"/>AZq,<emph.end type="italics"/>id e&longs;t (ob proportionales <emph type="italics"/>AZ, ZY<emph.end type="italics"/>) ut rectan­<lb/>gulum &longs;ub <emph type="italics"/>AZ<emph.end type="italics"/>& <emph type="italics"/>ZY,<emph.end type="italics"/>hoc e&longs;t, ut area <emph type="italics"/>AZYa.<emph.end type="italics"/>Et &longs;umma om­<lb/>nium horariorum motuum mediocrium ab initio, ut &longs;umma om­<lb/>nium arearum <emph type="italics"/>aYZA,<emph.end type="italics"/>id e&longs;t, ut area <emph type="italics"/>NAZ.<emph.end type="italics"/>E&longs;t autem maxima <lb/><figure id="id.039.01.440.1.jpg" xlink:href="039/01/440/1.jpg"/><lb/><emph type="italics"/>AZYa<emph.end type="italics"/>æqualis rectangulo &longs;ub arcu <emph type="italics"/>Aa<emph.end type="italics"/>& radio circuli; & prop­<lb/>terea &longs;umma omnium rectangulorum in circulo toto ad &longs;ummam <lb/>totidem maximorum, ut area circuli totius ad rectangulum &longs;ub <lb/>circumferentia tota & radio; id e&longs;t, ut 1 ad 2. Motus autem ho­<lb/>rarius, rectangulo maximo re&longs;pondens, erat 16″. </s> <s>16′. </s> <s>37<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>42<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>Et <lb/>hic motus, anno toto &longs;idereo dierum 365. <emph type="italics"/>hor.<emph.end type="italics"/>6. <emph type="italics"/>min.<emph.end type="italics"/>9 fit <lb/>39<emph type="sup"/>gr.<emph.end type="sup"/> 38′. </s> <s>7″. </s> <s>50′. </s> <s>Ideoque hujus dimidium 19<emph type="sup"/>gr.<emph.end type="sup"/> 49′. </s> <s>3″. </s> <s>55′. </s> <s>e&longs;t mo-<pb xlink:href="039/01/441.jpg" pagenum="413"/>tus medius Nodorum circulo toti re&longs;pondens. </s> <s>Et motus Nodo­<lb/><arrow.to.target n="note442"/>rum, quo tempore Sol pergit ab <emph type="italics"/>N<emph.end type="italics"/>ad <emph type="italics"/>A,<emph.end type="italics"/>e&longs;t ad 19<emph type="sup"/>gr.<emph.end type="sup"/> 49′. </s> <s>3″. </s> <s>55′. </s> <s><lb/>ut area <emph type="italics"/>NAZ<emph.end type="italics"/>ad circulum totum. </s></p> <p type="margin"> <s><margin.target id="note442"/>LIBER. <lb/>TERTIUS.</s></p> <p type="main"> <s>Hæc ita &longs;e habent, ex Hypothe&longs;i quod Nodus horis &longs;ingulis in <lb/>locum priorem retrahitur, lic ut Sol anno toto completo ad No­<lb/>dum eundem redeat a quo &longs;ub initio digre&longs;&longs;us fuerat. </s> <s>Verum per <lb/>motum Nodi fit ut Sol citius ad Nodum revertatur, & compu­<lb/>tanda jam e&longs;t abbreviatio temporis. </s> <s>Cum Sol anno toto conficiat <lb/>360 gradus, & Nodus motu maximo eodem tempore conficeret <lb/>39<emph type="sup"/>gr.<emph.end type="sup"/> 38′. </s> <s>7″. </s> <s>50′, &longs;eu 39,6355 gradus; & motus mediocris. </s> <s>Nodi <lb/>in loco quovis <emph type="italics"/>N<emph.end type="italics"/>&longs;it ad ip&longs;ius motum mediocrem in Quadraturis <lb/>&longs;uis, ut <emph type="italics"/>AZq<emph.end type="italics"/>ad <emph type="italics"/>ATq<emph.end type="italics"/>: erit motus Solis ad motum Nodi in <emph type="italics"/>N,<emph.end type="italics"/>ut <lb/>360 <emph type="italics"/>ATq<emph.end type="italics"/>ad 39,6355 <emph type="italics"/>AZq<emph.end type="italics"/>; id e&longs;t, ut 9,0827646 <emph type="italics"/>ATq<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="AZq.">AZque</expan><emph.end type="italics"/><lb/>Unde &longs;i circuli totius circumferentia <emph type="italics"/>NAn<emph.end type="italics"/>dividatur in particu­<lb/>las æquales <emph type="italics"/>Aa,<emph.end type="italics"/>tempus quo Sol percurrat particulam <emph type="italics"/>Aa,<emph.end type="italics"/>&longs;i cir­<lb/>culus quie&longs;ceret, erit ad tempus quo percurrit eandem parti­<lb/>culam, &longs;i circulus una cum Nodis circa centrum <emph type="italics"/>T<emph.end type="italics"/>revolvatur, <lb/>reciproce ut 9,0827646 <emph type="italics"/><expan abbr="ATq.">ATque</expan><emph.end type="italics"/>ad 9,0827646 <emph type="italics"/><expan abbr="ATq+AZq.">ATq+AZque</expan><emph.end type="italics"/>Nam <lb/>tempus e&longs;t reciproce ut velocitas qua particula percurritur, & <lb/>hæc velocitas e&longs;t &longs;umma velocitatum Solis & Nodi. </s> <s>Igitur &longs;i tem­<lb/>pus, quo Sol ab&longs;que motu Nodi percurreret arcum <emph type="italics"/>NA,<emph.end type="italics"/>expo­<lb/>natur per Sectorem <emph type="italics"/>NTA,<emph.end type="italics"/>& particula temporis quo percurreret. </s> <s><lb/>arcum quam minimum <emph type="italics"/>Aa,<emph.end type="italics"/>exponatur per Sectoris particulam <lb/><emph type="italics"/>ATa<emph.end type="italics"/>; & (perpendiculo <emph type="italics"/>aY<emph.end type="italics"/>in <emph type="italics"/>Nn<emph.end type="italics"/>demi&longs;&longs;o) &longs;i in <emph type="italics"/>AZ<emph.end type="italics"/>capiatur <lb/><emph type="italics"/>dZ,<emph.end type="italics"/>ejus longitudinis ut &longs;it rectangulum <emph type="italics"/>dZ<emph.end type="italics"/>in <emph type="italics"/>ZY<emph.end type="italics"/>ad Sectoris <lb/>particulam <emph type="italics"/>ATa<emph.end type="italics"/>ut <emph type="italics"/>AZq<emph.end type="italics"/>ad 9,0827646 <emph type="italics"/>ATq+AZq,<emph.end type="italics"/>id e&longs;t, ut <lb/>&longs;it <emph type="italics"/>dZ<emph.end type="italics"/>ad 1/2 <emph type="italics"/>AZ<emph.end type="italics"/>ut <emph type="italics"/>ATq<emph.end type="italics"/>ad 9,0827646 <emph type="italics"/>ATq+AZq<emph.end type="italics"/>; rectangu­<lb/>lum <emph type="italics"/>dZ<emph.end type="italics"/>in <emph type="italics"/>ZY<emph.end type="italics"/>de&longs;ignabit decrementum temporis ex motu Nodi <lb/>oriundum, tempore toto quo arcus <emph type="italics"/>Aa<emph.end type="italics"/>percurritur. </s> <s>Et &longs;i pun­<lb/>ctum <emph type="italics"/>d<emph.end type="italics"/>tangit Curvam <emph type="italics"/>NdGn,<emph.end type="italics"/>area curvilinea <emph type="italics"/>NdZ<emph.end type="italics"/>erit decre­<lb/>mentum totum, quo tempore arcus totus <emph type="italics"/>NA<emph.end type="italics"/>percurritur; & <lb/>propterea exce&longs;&longs;us Sectoris <emph type="italics"/>NAT<emph.end type="italics"/>&longs;upra aream <emph type="italics"/>NdZ<emph.end type="italics"/>erit tempus <lb/>illud totum. </s> <s>Et quoniam motus Nodi tempore minore minor e&longs;t <lb/>in ratione temporis, debebit etiam area <emph type="italics"/>AaYZ<emph.end type="italics"/>diminui in eadem <lb/>ratione. </s> <s>Id quod fiet &longs;i capiatur in <emph type="italics"/>AZ<emph.end type="italics"/>longitudo <emph type="italics"/>eZ,<emph.end type="italics"/>quæ &longs;it <lb/>ad longitudinem <emph type="italics"/>AZ<emph.end type="italics"/>ut <emph type="italics"/>AZq<emph.end type="italics"/>ad 9,0827646 <emph type="italics"/><expan abbr="ATq+AZq.">ATq+AZque</expan><emph.end type="italics"/>Sic <lb/>enim rectangulum <emph type="italics"/>eZ<emph.end type="italics"/>in <emph type="italics"/>ZY<emph.end type="italics"/>erit ad aream <emph type="italics"/>AZYa<emph.end type="italics"/>ut decremen­<lb/>tum temporis quo arcus <emph type="italics"/>Aa<emph.end type="italics"/>percurritur, ad tempus totum quo <lb/>percurreretur &longs;i Nodus quie&longs;ceret: Et propterea rectangulum illud <lb/>re&longs;pondebit decremento motus Nodi. </s> <s>Et &longs;i punctum <emph type="italics"/>e<emph.end type="italics"/>tangat <pb xlink:href="039/01/442.jpg" pagenum="414"/><arrow.to.target n="note443"/>Curvam <emph type="italics"/>NeFn,<emph.end type="italics"/>area tota <emph type="italics"/>NeZ,<emph.end type="italics"/>quæ &longs;umma e&longs;t omnium decre­<lb/>mentorum, re&longs;pondebit decremento toti, quo tempore arcus <emph type="italics"/>AN<emph.end type="italics"/><lb/>percurritur; & area reliqua <emph type="italics"/>NAe<emph.end type="italics"/>re&longs;pondebit motui reliquo, qui <lb/>verus e&longs;t Nodi motus quo tempore arcus totus <emph type="italics"/>NA,<emph.end type="italics"/>per Solis & <lb/>Nodi conjunctos motus, percurritur. </s> <s>Jam vero area &longs;emicirculi <lb/>e&longs;t ad aream Figuræ <emph type="italics"/>NeFnT,<emph.end type="italics"/>per methodum Serierum infinita­<lb/>rum quæ&longs;itam, ut 793 ad 60 quamproxime. </s> <s>Motus autem qui <lb/>re&longs;pondet Circulo toti erat 19<emph type="sup"/>gr.<emph.end type="sup"/> 49′. </s> <s>3″. </s> <s>55′; & propterea motus, <lb/>qui Figuræ <emph type="italics"/>NeFnT<emph.end type="italics"/>duplicatæ re&longs;pondet, e&longs;t 1<emph type="sup"/>gr.<emph.end type="sup"/> 29′. </s> <s>58″. </s> <s>2′. </s> <s><lb/>Qui de motu priore &longs;ubductus relinquit 18<emph type="sup"/>gr.<emph.end type="sup"/> 19′. </s> <s>5″. </s> <s>53′. </s> <s>motum <lb/>totum Nodi inter &longs;ui ip&longs;ius Conjunctiones cum Sole; & hic mo­<lb/>tus de Solis motu annuo graduum 360 &longs;ubductus, relinquit 341<emph type="sup"/>gr.<emph.end type="sup"/><lb/>40′. </s> <s>54″. </s> <s>7′. </s> <s>motum Solis inter ea&longs;dem Conjunctiones. </s> <s>I&longs;te au­<lb/>tem motus e&longs;t ad motum annuum 360<emph type="sup"/>gr.<emph.end type="sup"/> ut Nodi motus jam in­<lb/>ventus 18<emph type="sup"/>gr.<emph.end type="sup"/> 19′. </s> <s>5″. </s> <s>53′. </s> <s>ad ip&longs;ius motum annuum, qui propterea <lb/>erit 19<emph type="sup"/>gr.<emph.end type="sup"/> 18′. </s> <s>1″. </s> <s>23′. </s> <s>Hic e&longs;t motus medius Nodorum in anno <lb/>Sidereo. </s> <s>Idem per Tabulas A&longs;tronomicas e&longs;t 19<emph type="sup"/>gr.<emph.end type="sup"/> 21′. </s> <s>21″. </s> <s>50′. </s> <s><lb/>Differentia minor e&longs;t parte trecente&longs;ima motus totius, & ab Or­<lb/>bis Lunaris Eccentricitate & Inclinatione ad planum Eclipticæ <lb/>oriri videtur. </s> <s>Per Eccentricitatem Orbis motus Nodorum nimis <lb/>acceleratur, & per ejus Inclinationem vici&longs;&longs;im retardatur aliquan­<lb/>tulum, & ad ju&longs;tam velocitatem reducitur. </s></p> <p type="margin"> <s><margin.target id="note443"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXIII. PROBLEMA XIV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire motum verum Nodorum Lunæ.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>In tempore quod e&longs;t ut area <emph type="italics"/>NTA-NdZ, (in Fig. </s> <s>præced.)<emph.end type="italics"/><lb/>motus i&longs;te e&longs;t ut area <emph type="italics"/>NAeN,<emph.end type="italics"/>& inde datur. </s> <s>Verum ob nimiam <lb/>calculi difficultatem, præ&longs;tat &longs;equentem Problematis con&longs;tructio­<lb/>nem adhibere. </s> <s>Centro <emph type="italics"/>C,<emph.end type="italics"/>intervallo quovis <emph type="italics"/>CD,<emph.end type="italics"/>de&longs;cribatur <lb/>circulus <emph type="italics"/>BEFD.<emph.end type="italics"/>Producatur <emph type="italics"/>DC<emph.end type="italics"/>ad <emph type="italics"/>A,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>AB<emph.end type="italics"/>ad <emph type="italics"/>AC<emph.end type="italics"/><lb/>ut motus medius ad &longs;emi&longs;&longs;em motus veri mediocris, ubi Nodi <lb/>&longs;unt in Quadraturis, (id e&longs;t, ut 19<emph type="sup"/>gr.<emph.end type="sup"/> 18′. </s> <s>1″. </s> <s>23′. </s> <s>ad 19<emph type="sup"/>gr.<emph.end type="sup"/> 49′. </s> <s><lb/>3″. </s> <s>55′, atque adeo <emph type="italics"/>BC<emph.end type="italics"/>ad <emph type="italics"/>AC<emph.end type="italics"/>ut motuum differentia 0<emph type="sup"/>gr.<emph.end type="sup"/> 31′. </s> <s><lb/>2″. </s> <s>32′, ad motum po&longs;teriorem 19′<emph type="sup"/>gr.<emph.end type="sup"/> 49. 3″. </s> <s>55′) hoc e&longs;t, ut <lb/>1 ad (38 1/10) dein per punctum <emph type="italics"/>D<emph.end type="italics"/>ducatur infinita <emph type="italics"/>Gg,<emph.end type="italics"/>quæ tangat <lb/>circulum in <emph type="italics"/>D<emph.end type="italics"/>; & &longs;i capiatur angulus <emph type="italics"/>BCE<emph.end type="italics"/>vel <emph type="italics"/>BCF<emph.end type="italics"/>æqualis <lb/>duplæ di&longs;tantiæ Solis a loco Nodi, per motum medium invento; <pb xlink:href="039/01/443.jpg" pagenum="415"/>& agatur <emph type="italics"/>AE<emph.end type="italics"/>vel <emph type="italics"/>AF<emph.end type="italics"/>&longs;ecans perpendiculum <emph type="italics"/>DG<emph.end type="italics"/>in <emph type="italics"/>G<emph.end type="italics"/>; & ca­<lb/><arrow.to.target n="note444"/>piatur angulus qui &longs;it ad motum totum Nodi inter ip&longs;ius Syzy­<lb/>gias (id e&longs;t, ad 9<emph type="sup"/>gr.<emph.end type="sup"/> 11′. </s> <s>3″.) ut tangens <emph type="italics"/>DG<emph.end type="italics"/>ad circuli <emph type="italics"/>BED<emph.end type="italics"/><lb/>circumferentiam totam; atque angulus i&longs;te (pro quo angulus <emph type="italics"/>DAG<emph.end type="italics"/><lb/>u&longs;urpari pote&longs;t) ad motum medium Nodorum addatur ubi Nodi <lb/><figure id="id.039.01.443.1.jpg" xlink:href="039/01/443/1.jpg"/><lb/>tran&longs;eunt a Quadraturis ad Syzygias, & ab eodem motu medio <lb/>&longs;ubducatur ubi tran&longs;eunt a Syzygiis ad Quadraturas; habebitur <lb/>eorum motus verus. </s> <s>Nam motus verus &longs;ic inventus congruet <lb/>quam proxime cum motu vero qui prodit exponendo tempus per <lb/>aream <emph type="italics"/>NTA-NdZ,<emph.end type="italics"/>& motum Nodi per aream <emph type="italics"/>NAeN<emph.end type="italics"/>; ut <lb/>rem perpendenti & computationes in&longs;tituenti con&longs;tabit. </s> <s>Hæc e&longs;t <lb/>æquatio &longs;eme&longs;tris motus Nodorum. </s> <s>E&longs;t & æquatio men&longs;trua, &longs;ed <lb/>quæ ad inventionem Latitudinis Lunæ minime nece&longs;&longs;aria e&longs;t. </s> <s>Nam <lb/>cum Variatio Inclinationis Orbis Lunaris ad planum Eclipticæ du­<lb/>plici inæqualitati obnoxia &longs;it, alteri &longs;eme&longs;tri, alteri autem men­<lb/>&longs;truæ; &c. </s> <s>hujus men&longs;trua inæqualitas & æquatio men&longs;trua Nodorum <lb/>ita &longs;e mutuo contemperant & corrigunt, ut ambæ in determinan­<lb/>da Latitudine Lunæ negligi po&longs;&longs;int. </s></p> <p type="margin"> <s><margin.target id="note444"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Ex hac & præcedente Propo&longs;itione liquet quod Nodi in <lb/>Syzygiis &longs;uis quie&longs;cunt, in Quadraturis autem regrediuntur motu <lb/>horario 16″. </s> <s>19′. </s> <s>26<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>Et quod æquatio motus Nodorum in <lb/>Octantibus &longs;it 1<emph type="sup"/>gr.<emph.end type="sup"/> 30′. </s> <s>Quæ omnia cum Phænomenis cœle&longs;tibus <lb/>probe quadrant. <pb xlink:href="039/01/444.jpg" pagenum="416"/><arrow.to.target n="note445"/></s></p> <p type="margin"> <s><margin.target id="note445"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXIV. PROBLEMA XV.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire Variationem horariam Inclinationis Orbis Lunaris ad <lb/>planum Eclipticæ.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>De&longs;ignent <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>a<emph.end type="italics"/>Syzygias; <emph type="italics"/>Q<emph.end type="italics"/>& <emph type="italics"/>q<emph.end type="italics"/>Quadraturas; <emph type="italics"/>N<emph.end type="italics"/>& <emph type="italics"/>n<emph.end type="italics"/>No­<lb/>dos; <emph type="italics"/>P<emph.end type="italics"/>locum Lunæ in Orbe &longs;uo; <emph type="italics"/>p<emph.end type="italics"/>ve&longs;tigium loci illius in plano <lb/>Eclipticæ, & <emph type="italics"/>mTl<emph.end type="italics"/>motum momentaneum Nodorum ut &longs;upra. </s> <s><lb/>Et &longs;i ad lineam <emph type="italics"/>Tm<emph.end type="italics"/>demittatur perpendiculum <emph type="italics"/>PG,<emph.end type="italics"/>jungatur <emph type="italics"/>pG,<emph.end type="italics"/><lb/>& producatur ea donec occurrat <emph type="italics"/>Tl<emph.end type="italics"/>in <emph type="italics"/>g,<emph.end type="italics"/>& jungatur etiam <emph type="italics"/>Pg<emph.end type="italics"/>: <lb/>erit angulus <emph type="italics"/>PGp<emph.end type="italics"/>Inclinatio orbis Lunaris ad planum Eclipticæ, <lb/><figure id="id.039.01.444.1.jpg" xlink:href="039/01/444/1.jpg"/><lb/>ubi Luna ver&longs;atur in <emph type="italics"/>P<emph.end type="italics"/>; & angulus <emph type="italics"/>Pgp<emph.end type="italics"/>Inclinatio eju&longs;dem po&longs;t <lb/>momentum temporis completum; adeoque angulus <emph type="italics"/>GPg<emph.end type="italics"/>Variatio <lb/>momentanea Inclinationis. </s> <s>E&longs;t autem hic angulus <emph type="italics"/>GPg<emph.end type="italics"/>ad an­<lb/>gulum <emph type="italics"/>GTg,<emph.end type="italics"/>ut <emph type="italics"/>TG<emph.end type="italics"/>ad <emph type="italics"/>PG<emph.end type="italics"/>& <emph type="italics"/>Pp<emph.end type="italics"/>ad <emph type="italics"/>PG<emph.end type="italics"/>conjunctim. </s> <s>Et prop­<lb/>terea &longs;i pro momento temporis &longs;ub&longs;tituatur hora; cum angulus <lb/><emph type="italics"/>GTg<emph.end type="italics"/>(per Propo&longs;it. </s> <s>xxx.) &longs;it ad angulum 33″. </s> <s>10′. </s> <s>33<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>ut <pb xlink:href="039/01/445.jpg" pagenum="417"/><emph type="italics"/>ITXPGXAZ<emph.end type="italics"/>ad <emph type="italics"/>ATcub,<emph.end type="italics"/>erit angulus <emph type="italics"/>GPg<emph.end type="italics"/>(&longs;eu Inclinationis <lb/><arrow.to.target n="note446"/>horaria Variatio) ad angulum 33″. </s> <s>10′. </s> <s>33<emph type="sup"/>iv<emph.end type="sup"/>, ut <emph type="italics"/>ITXAZXTG <lb/>X(Pp/PG)<emph.end type="italics"/>ad <emph type="italics"/>ATcub. </s> <s>q.EI.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note446"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Hæc ita &longs;e habent ex Hypothe&longs;i quod Luna in Orbe Circulari <lb/>uniformiter gyratur. </s> <s>Quod &longs;i Orbis ille Ellipticus &longs;it, motus me­<lb/>diocris Nodorum minuetur in ratione axis minoris ad axem majo­<lb/>rem; uti &longs;upra expo&longs;itum e&longs;t. </s> <s>Et in eadem ratione minuetur <lb/>etiam Inclinationis Variatio. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Si ad <emph type="italics"/>Nn<emph.end type="italics"/>erigatur perpendiculum <emph type="italics"/>TF,<emph.end type="italics"/>&longs;itque <emph type="italics"/>pM<emph.end type="italics"/><lb/>motus horarius Lunæ in plano Eclipticæ; & perpendicula <emph type="italics"/>pK, Mk<emph.end type="italics"/><lb/>in <emph type="italics"/>QT<emph.end type="italics"/>demi&longs;&longs;a & utrinque producta occurrant <emph type="italics"/>TF<emph.end type="italics"/>in <emph type="italics"/>H<emph.end type="italics"/>& <emph type="italics"/>h<emph.end type="italics"/>: <lb/>erit <emph type="italics"/>IT<emph.end type="italics"/>ad <emph type="italics"/>AT<emph.end type="italics"/>ut <emph type="italics"/>Kk<emph.end type="italics"/>ad <emph type="italics"/>Mp,<emph.end type="italics"/>& <emph type="italics"/>TG<emph.end type="italics"/>ad <emph type="italics"/>Hp<emph.end type="italics"/>ut <emph type="italics"/>TZ<emph.end type="italics"/>ad <emph type="italics"/>AT;<emph.end type="italics"/><lb/>ideoque <emph type="italics"/>ITXTG<emph.end type="italics"/>æquale (<emph type="italics"/>KkXHpXTZ/Mp<emph.end type="italics"/>), hoc e&longs;t, æquale areæ <lb/><emph type="italics"/>HpMh<emph.end type="italics"/>ductæ in rationem (<emph type="italics"/>TZ/Mp<emph.end type="italics"/>): & propterea Inclinationis Varia­<lb/>tio horaria ad 33″. </s> <s>10′. </s> <s>33<emph type="sup"/>iv<emph.end type="sup"/>, ut <emph type="italics"/>HpMh<emph.end type="italics"/>ducta in <emph type="italics"/>AZX(TZ/Mp)X(Pp/PG)<emph.end type="italics"/><lb/>ad <emph type="italics"/>AT cub.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Ideoque &longs;i Terra & Nodi &longs;ingulis horis completis re­<lb/>traherentur à locis &longs;uis novis, & in loca priora in in&longs;tanti &longs;emper <lb/>reducerentur, ut &longs;itus eorum, per men&longs;em integrum periodicum, <lb/>datus maneret; tota Inclinationis Variatio tempore men&longs;is illius <lb/>foret ad 33″. </s> <s>10′. </s> <s>33<emph type="sup"/>iv<emph.end type="sup"/>, ut aggregatum omnium arearum <emph type="italics"/>Hp Mh,<emph.end type="italics"/><lb/>in revolutione puncti <emph type="italics"/>p<emph.end type="italics"/>genitarum, & &longs;ub &longs;ignis propriis + & ­<lb/>conjunctarum, ductum in <emph type="italics"/>AZXTZX(Pp/PG)<emph.end type="italics"/>ad <emph type="italics"/>MpXAT cub.<emph.end type="italics"/>id <lb/>e&longs;t, ut circulus totus <emph type="italics"/>QAqa<emph.end type="italics"/>ductus in <emph type="italics"/>AZXTZX(Pp/PG)<emph.end type="italics"/>ad <emph type="italics"/>MpX <lb/>ATcub.<emph.end type="italics"/>hoc e&longs;t, ut circumferentia <emph type="italics"/>QAqa<emph.end type="italics"/>ducta in <emph type="italics"/>AZXTZX(Pp/PG)<emph.end type="italics"/><lb/>ad 2 <emph type="italics"/>MpXAT quad.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Proinde in dato Nodorum &longs;itu, Variatio mediocris <lb/>horaria, ex qua per men&longs;em uniformiter continuata Variatio illa <lb/>men&longs;trua generari po&longs;&longs;et, e&longs;t ad 33″. </s> <s>10′. </s> <s>33<emph type="sup"/>iv<emph.end type="sup"/>, ut <emph type="italics"/>AZXTZ <lb/>X(Pp/PG)<emph.end type="italics"/>ad 2 <emph type="italics"/>ATq,<emph.end type="italics"/>&longs;ive ut <emph type="italics"/>PpX(AZXTZ/1/2AT)<emph.end type="italics"/>ad <emph type="italics"/>PGX4AT,<emph.end type="italics"/>id <pb xlink:href="039/01/446.jpg" pagenum="418"/><arrow.to.target n="note447"/>e&longs;t (cum <emph type="italics"/>Pp<emph.end type="italics"/>&longs;it ad <emph type="italics"/>PG<emph.end type="italics"/>ut &longs;inus Inclinationis prædictæ ad ra­<lb/>dium, & (<emph type="italics"/>AZXTZ/1/2AT<emph.end type="italics"/>) &longs;it ad 4<emph type="italics"/>AT<emph.end type="italics"/>ut &longs;inus duplicati anguli <emph type="italics"/>ATn<emph.end type="italics"/><lb/>ad radium quadruplicatum) ut Inclinationis eju&longs;dem &longs;inus ductus <lb/>in &longs;inum duplicatæ di&longs;tantiæ Nodorum a Sole, ad quadruplum <lb/>quadratum radii. </s></p> <p type="margin"> <s><margin.target id="note447"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Quoniam Inclinationis horaria Variatio, ubi Nodi in <lb/>Quadraturis ver&longs;antur, e&longs;t (per hanc Propo&longs;itionem) ad angu­<lb/>lum 33″. </s> <s>10′. </s> <s>33<emph type="sup"/>iv<emph.end type="sup"/> ut <emph type="italics"/>ITXAZXTGX(Pp/PG)<emph.end type="italics"/>ad <emph type="italics"/>ATcub.<emph.end type="italics"/>id e&longs;t, <lb/>ut <emph type="italics"/>(ITXTG/1/2AT)X(Pp/PG)<emph.end type="italics"/>ad 2<emph type="italics"/>AT<emph.end type="italics"/>; hoc e&longs;t, ut &longs;inus duplicatæ di­<lb/>&longs;tantiæ Lunæ à Quadraturis ductus in (<emph type="italics"/>Pp/PG<emph.end type="italics"/>) ad radium duplica­<lb/>tum: &longs;umma omnium Variationum horariarum, quo tempore <lb/>Luna in hoc &longs;itu Nodorum tran&longs;it à Quadratura ad Syzygiam, <lb/>(id e&longs;t, &longs;patio horarum 177 1/6,) erit ad &longs;ummam totidem angulo­<lb/>rum 33″. </s> <s>10′. </s> <s>33<emph type="sup"/>iv<emph.end type="sup"/>, &longs;eu 5878″, ut &longs;umma omnium &longs;inuum dupli­<lb/>catæ di&longs;tantiæ Lunæ à Quadraturis ducta in (<emph type="italics"/>Pp/PG<emph.end type="italics"/>) ad &longs;ummam to­<lb/>tidem diametrorum; hoc e&longs;t, ut diameter ducta in (<emph type="italics"/>Pp/PG<emph.end type="italics"/>) ad cir­<lb/>cumferentiam; id e&longs;t, &longs;i Inclinatio &longs;it 5<emph type="sup"/>gr.<emph.end type="sup"/> 1′, ut 7X(874/10000) ad 22, <lb/>&longs;eu 278 ad 10000. Proindeque Variatio tota, ex &longs;umma om­<lb/>nium horariarum Variationum tempore prædicto conflata, e&longs;t <lb/>163″, &longs;eu 2′. </s> <s>43″. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXV. PROBLEMA XVI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Dato tempore invenire Inclinationem Orbis Lunaris ad planum <lb/>Eclipticæ.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sit <emph type="italics"/>AD<emph.end type="italics"/>&longs;inus Inclinationis maximæ, & <emph type="italics"/>AB<emph.end type="italics"/>&longs;inus Inclinatio­<lb/>nis minimæ. </s> <s>Bi&longs;ecetur <emph type="italics"/>BD<emph.end type="italics"/>in <emph type="italics"/>C,<emph.end type="italics"/>& centro <emph type="italics"/>C,<emph.end type="italics"/>intervallo <emph type="italics"/>BC,<emph.end type="italics"/><lb/>de&longs;cribatur Circulus <emph type="italics"/>BGD.<emph.end type="italics"/>In <emph type="italics"/>AC<emph.end type="italics"/>capiatur <emph type="italics"/>CE<emph.end type="italics"/>in ea ratione <lb/>ad <emph type="italics"/>EB<emph.end type="italics"/>quam <emph type="italics"/>EB<emph.end type="italics"/>habet ad 2<emph type="italics"/>BA:<emph.end type="italics"/>Et &longs;i dato tempore con&longs;ti­<lb/>tuatur angulus <emph type="italics"/>AEG<emph.end type="italics"/>æqualis duplicatæ di&longs;tantiæ Nodorum à <pb xlink:href="039/01/447.jpg" pagenum="419"/>Quadraturis, & ad <emph type="italics"/>AD<emph.end type="italics"/>demittatur perpendiculum <emph type="italics"/>GH<emph.end type="italics"/>: erit <lb/><arrow.to.target n="note448"/><emph type="italics"/>AH<emph.end type="italics"/>&longs;inus Inclinationis quæ&longs;itæ. </s></p> <p type="margin"> <s><margin.target id="note448"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Nam <emph type="italics"/>GEq<emph.end type="italics"/>æquale e&longs;t <emph type="italics"/>GHq+HEq=BHD+HEq= <lb/>HBD+HEq-BHq=HBD+BEq<emph.end type="italics"/>-2<emph type="italics"/>BHXBE= <lb/>BEq<emph.end type="italics"/>+2<emph type="italics"/>ECXBH<emph.end type="italics"/>=2<emph type="italics"/>ECXAB<emph.end type="italics"/>+2<emph type="italics"/>ECXBH<emph.end type="italics"/>=2<emph type="italics"/>ECXAH.<emph.end type="italics"/><lb/>Ideoque cum 2<emph type="italics"/>EC<emph.end type="italics"/>detur, e&longs;t <emph type="italics"/>GEq<emph.end type="italics"/>ut <emph type="italics"/>AH.<emph.end type="italics"/>De&longs;ignet jam <emph type="italics"/>AEg<emph.end type="italics"/><lb/>duplicatam di&longs;tantiam Nodorum à Quadraturis po&longs;t datum ali­<lb/>quod momentum temporis completum, & arcus <emph type="italics"/>Gg.,<emph.end type="italics"/>ob datum <lb/><figure id="id.039.01.447.1.jpg" xlink:href="039/01/447/1.jpg"/><lb/>angulum <emph type="italics"/>GEg,<emph.end type="italics"/>erit ut di&longs;tantia <emph type="italics"/>GE.<emph.end type="italics"/>E&longs;t autem <emph type="italics"/>Hh<emph.end type="italics"/>ad <emph type="italics"/>Gg<emph.end type="italics"/><lb/>ut <emph type="italics"/>GH<emph.end type="italics"/>ad <emph type="italics"/>GC,<emph.end type="italics"/>& propterea <emph type="italics"/>Hh<emph.end type="italics"/>e&longs;t ut contentum <emph type="italics"/>GHXGg,<emph.end type="italics"/><lb/>&longs;eu <emph type="italics"/>GHXGE<emph.end type="italics"/>; id e&longs;t, ut <emph type="italics"/>(GH/GE)XGEq<emph.end type="italics"/>&longs;eu <emph type="italics"/>(GH/GE)XAH,<emph.end type="italics"/>id e&longs;t, <lb/>ut <emph type="italics"/>AH<emph.end type="italics"/>& &longs;inus anguli <emph type="italics"/>AEG<emph.end type="italics"/>conjunctim. </s> <s>Igitur &longs;i <emph type="italics"/>AH<emph.end type="italics"/>in <lb/>ca&longs;u aliquo &longs;it &longs;inus Inclinationis, augebitur ea ii&longs;dem incremen­<lb/>tis cum &longs;inu Inclinationis, per Corol. </s> <s>3. Propo&longs;itionis &longs;uperioris, <lb/>& propterea &longs;inui illi æqualis &longs;emper manebit. </s> <s>Sed <emph type="italics"/>AH<emph.end type="italics"/>ubi <lb/>punctum <emph type="italics"/>G<emph.end type="italics"/>incidit in punctum alterutrum <emph type="italics"/>B<emph.end type="italics"/>vel <emph type="italics"/>D<emph.end type="italics"/>huic &longs;inui <lb/>æqualis e&longs;t, & propterea eidem &longs;emper æqualis manet. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s>In hac demon&longs;tratione &longs;uppo&longs;ui angulum <emph type="italics"/>BEG,<emph.end type="italics"/>qui e&longs;t du­<lb/>plicata di&longs;tantia Nodorum à Quadraturis, uniformiter augeri. </s> <s><lb/>Nam omnes inæqualitatum minutias expendeve non vacat. </s> <s>Con­<lb/>cipe jam angulum <emph type="italics"/>BEG<emph.end type="italics"/>rectum e&longs;&longs;e, & in hoc ea&longs;e <emph type="italics"/>Gg<emph.end type="italics"/>e&longs;&longs;e <lb/>augmentum horarium duplæ di&longs;tantiæ Nodorum & Solis ab invi­<lb/>cem; & Inclinationis Variatio horaria in eodem ca&longs;u (per Corol. </s> <s><lb/>3. Prop. </s> <s>novi&longs;&longs;imæ) erit ad 33′. </s> <s>10′. </s> <s>33<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>ut contentum &longs;ub In­<lb/>clinationis &longs;inu <emph type="italics"/>AH<emph.end type="italics"/>& &longs;inu anguli recti <emph type="italics"/>BEG,<emph.end type="italics"/>qui e&longs;t dupli­<lb/>cata di&longs;tantia Nodorum a Sole, ad quadruplum quadratum radii; <lb/>id. </s> <s>e&longs;t, ut mediocris Inclinationis &longs;inus <emph type="italics"/>AH<emph.end type="italics"/>ad radium quadru­<lb/>plicatum; hoc e&longs;t (cum Inclinatio illa mediocris &longs;it quafi 5<emph type="sup"/>gr.<emph.end type="sup"/> 8′1/2) <lb/>ut ejus &longs;inus 896 ad radium quadruplicatum 40000, &longs;ive ut 224 <lb/>ad 10000. E&longs;t autem Variatio tota, &longs;inuum differentiæ <emph type="italics"/>BD<emph.end type="italics"/><lb/>re&longs;pondens, ad Variationem illam horariam ut diameter <emph type="italics"/>BD<emph.end type="italics"/>ad <pb xlink:href="039/01/448.jpg" pagenum="420"/><arrow.to.target n="note449"/>arcum <emph type="italics"/>Gg<emph.end type="italics"/>; id e&longs;t, ut diameter <emph type="italics"/>BD<emph.end type="italics"/>ad &longs;emicircum ferentiam <lb/><emph type="italics"/>BGD<emph.end type="italics"/>& tempus horarum (2079 1/10), quo Nodus pergit à Quadra­<lb/>turis ad Syzygias, ad horam unam conjunctim; hoc e&longs;t, ut 7 ad <lb/>11 & (2079 7/10) ad 1. Quare &longs;i rationes omnes conjungantur, fiet <lb/>Variatio tota <emph type="italics"/>BD<emph.end type="italics"/>ad 33″. </s> <s>10′. </s> <s>33<emph type="sup"/>ix<emph.end type="sup"/> ut 224X7X2079 (7/10) ad <lb/>110000, id e&longs;t, ut 29645 ad 1000, & inde Variatio illa <emph type="italics"/>BD<emph.end type="italics"/><lb/>prodibit 16′. </s> <s>23″ 1/2. </s></p> <p type="margin"> <s><margin.target id="note449"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Hæc e&longs;t Inclinationis Variatio maxima quatenus locus Lunæ in <lb/>Orbe &longs;uo non con&longs;ideratur. </s> <s>Nam Inclinatio, &longs;i Nodi in Syzygiis <lb/>ver&longs;antur, nil mutatur ex vario &longs;itu Lunæ. </s> <s>At &longs;i Nodi in Qua­<lb/>draturis con&longs;i&longs;tunt, Inclinatio minor e&longs;t ubi Luna ver&longs;atur in Sy­<lb/>zygiis, quam ubi ea ver&longs;atur in Quadraturis, exce&longs;&longs;u 2′. </s> <s>43″; uti <lb/>in Propo&longs;itionis &longs;uperioris Corollario quarto indicavimus. </s> <s>Et <lb/>hujus exce&longs;&longs;us dimidio 1′. </s> <s>21″ 1/2. Variatio tota mediocris <emph type="italics"/>BD<emph.end type="italics"/>in <lb/>Quadraturis Lunaribus diminuta fit 15′, 2″, in ip&longs;ius autem Syzy­<lb/>giis aucta fit 17′. </s> <s>45″. </s> <s>Si Luna igitur in Syzygiis con&longs;tituatur, <lb/>Variatio tota, in tran&longs;itu Nodorum à Quadraturis ad Syzygias, <lb/>erit 17′. </s> <s>45″: adeoque &longs;i Inclinatio, ubi Nodi in Syzygiis ver&longs;an­<lb/>tur, &longs;it 5<emph type="sup"/>gr.<emph.end type="sup"/> 17′. </s> <s>20″; eadem, ubi Nodi &longs;unt in Quadraturis, & <lb/>Luna in Syzygiis, erit 4<emph type="sup"/>gr.<emph.end type="sup"/> 59′. </s> <s>35″. </s> <s>Atque hæc ita &longs;e habere <lb/>confirmatur ex Ob&longs;ervationibus. </s></p><figure id="id.039.01.448.1.jpg" xlink:href="039/01/448/1.jpg"/> <p type="main"> <s>Si jam de&longs;ideretur Orbis Inclinatio illa, ubi Luna in Syzygiis <lb/>& Nodi ubivis ver&longs;antur; fiat <emph type="italics"/>AB<emph.end type="italics"/>ad <emph type="italics"/>AD<emph.end type="italics"/>ut &longs;inus graduum 4. <lb/>59′. </s> <s>35″ ad &longs;inum graduum 5. </s> <s>17′, 20″, & capiatur angulus <emph type="italics"/>AEG<emph.end type="italics"/><lb/>æqualis duplicatæ di&longs;tantiæ Nodorum à Quadraturis; & erit <emph type="italics"/>AH<emph.end type="italics"/><lb/>&longs;inus Inclinationis quæ&longs;itæ. </s> <s>Huic Orbis Inclinationi æqualis e&longs;t <lb/>eju&longs;dem Inclinatio, ubi Luna di&longs;tat 90<emph type="sup"/>gr.<emph.end type="sup"/> à Nodis. </s> <s>In aliis Lunæ <lb/>locis inæqualitas men&longs;trua, quam Inclinationis variatio admittit, <lb/>in calculo Latitudinis Lunæ compen&longs;atur & quodammodo tolli­<lb/>tur per inæqualitatem men&longs;truam motus Nodorum, (ut &longs;upra dixi­<lb/>mus) adeoQ.E.I. calculo Latitudinis illius negligi pote&longs;t. <pb xlink:href="039/01/449.jpg" pagenum="421"/><arrow.to.target n="note450"/></s></p> <p type="margin"> <s><margin.target id="note450"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Hi&longs;ce motuum Lunarium computationibus o&longs;tendere volui, <lb/>quod motus Lunares, per Theoriam Gravitatis, a cau&longs;is &longs;uis com­<lb/>putari po&longs;&longs;int. </s> <s>Per eandem Theoriam inveni præterea quod Æ­<lb/>quatio Annua medii motus Lunæ oriatur a varia dilatatione Or­<lb/>bis Lunæ per vim Solis, juxta Corol. </s> <s>6. Prop. </s> <s>LXVI. Lib. </s> <s>I. </s> <s>Hæc <lb/>vis in Perigæo Solis major e&longs;t, & Orbem Lunæ dilatat; in Apo­<lb/>gæo ejus minor e&longs;t, & Orbem illum contrahi permittit. </s> <s>In Orbe <lb/>dilatato Luna tardius revolvitur, in contracto citius; & Æquatio <lb/>Annua per quam hæc inæqualitas compen&longs;atur, in Apogæo & <lb/>Perigæo Solis nulla e&longs;t, in mediocri Solis a Terra di&longs;tantia ad <lb/>11′. </s> <s>50″ circiter a&longs;cendit, in aliis locis Æquationi centri Solis <lb/>proportionalis e&longs;t; & additur medio motui Lunæ ubi Terra per­<lb/>git ab Aphelio &longs;uo ad Perihelium, & in oppo&longs;ita Orbis parte, &longs;ub­<lb/>ducitur. </s> <s>A&longs;&longs;umendo radium Orbis magni 1000 & Eccentricita­<lb/>tem Terræ 16 7/8, hæc Æquatio ubi maxima e&longs;t, per Theoriam Gra­<lb/>vitatis prodiit 11′. </s> <s>49″. </s> <s>Sed Eccentricitas Terræ paulo major e&longs;&longs;e <lb/>videtur, & aucta Eccentricitate hæc Æquatio augeri debet in ea­<lb/>dem ratione. </s> <s>Sit Eccentricitas (16 11/16), & Æquatio maxima erit <lb/>11′. </s> <s>52″. </s></p> <p type="main"> <s>Inveni etiam quod in Perihelio Terræ, propter majorem vim <lb/>Solis, Apogæum & Nodi Lunæ velocius moventur quam in Aphe­<lb/>lio ejus, idQ.E.I. triplicata ratione di&longs;tantiæ Terræ a Sole inver&longs;e, <lb/>Et inde oriuntur Æquationes Annuæ horum motuum Æquationi <lb/>centri Solis proportionales. </s> <s>Motus autem Solis e&longs;t in duplicata <lb/>ratione di&longs;tantiæ Terræ a Sole inver&longs;e, & maxima centri Æquatio <lb/>quam hæc inæqualitas generat, e&longs;t 1<emph type="sup"/>gr.<emph.end type="sup"/> 56′. </s> <s>26″ prædictæ Solis <lb/>Eccentricitati (16 15/16) congruens. </s> <s>Quod &longs;i motus Solis e&longs;&longs;et in tri­<lb/>plicata ratione di&longs;tantiæ inver&longs;e, hæc inæqualitas generaret Æqua­<lb/>tionem maximam 2<emph type="sup"/>gr.<emph.end type="sup"/> 56′. </s> <s>9″. </s> <s>Et propterea Æquationes maxi­<lb/>mæ quas inæqualitates motuum Apogæi & Nodorum Lunæ gene­<lb/>rant, &longs;unt ad 2<emph type="sup"/>gr.<emph.end type="sup"/> 56′. </s> <s>9″, ut motus medius diurnus Apogæi & <lb/>motus medius diurnus Nodorum Lunæ &longs;unt ad motum medium <lb/>diurnum Solis. </s> <s>Unde prodit Æquatio maxima medii motus <lb/>Apogæi 19′. </s> <s>52″: & Æquatio maxima medii motus Nodorum <lb/>9′. </s> <s>27″. </s> <s>Additur vero Æquatio prior & &longs;ubducitur po&longs;terior, ubi <lb/>Terra pergit a Perihelio &longs;uo ad Aphelium: & contrarium fit in <lb/>oppo&longs;ita Orbis parte. <pb xlink:href="039/01/450.jpg" pagenum="422"/><arrow.to.target n="note451"/></s></p> <p type="margin"> <s><margin.target id="note451"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Per Theoriam Gravitatis con&longs;titit etiam quod actio Solis in <lb/>Lunam paulo major &longs;it ubi tran&longs;ver&longs;a diameter Orbis Lunaris <lb/>tran&longs;it per Solem, quam ubi eadem ad rectos e&longs;t angulos cum <lb/>linea Terram & Solem jungente: & propterea Orbis Lunaris <lb/>paulo major e&longs;t in priore ca&longs;u quam in po&longs;teriore. </s> <s>Et hinc ori­<lb/>tur alia Æquatio motus medii Lunaris, pendens a &longs;itu Apogæi <lb/>Lunæ ad Solem, quæ quidem maxima e&longs;t cum Apogæum Lunæ <lb/>ver&longs;atur in Octante cum Sole; & nulla cum illud ad Quadraturas <lb/>vel Syzygias pervenit: & motui medio additur in tran&longs;itu Apo­<lb/>gæi Lunæ a Solis Quadratura ad Syzygiam, & &longs;ubducitur in tran­<lb/>&longs;itu Apogæi a Syzygia ad Quadraturam. </s> <s>Hæc Æquatio quam <lb/>Seme&longs;trem vocabo, in Octantibus Apogæi quando maxima e&longs;t, <lb/>a&longs;cendit ad 3′. </s> <s>45″ circiter, quantum ex Phænomenis colligere <lb/>potui. </s> <s>Hæc e&longs;t ejus quantitas in mediocri Solis di&longs;tantia a Terra. </s> <s><lb/>Augetur vero ac diminuitur in triplicata ratione di&longs;tantiæ Solis <lb/>inver&longs;e, adeoQ.E.I. maxima Solis di&longs;tantia e&longs;t 3′. </s> <s>34″, & in mi­<lb/>nima 3′. </s> <s>56″ quamproxime: ubi vero Apogæum Lunæ &longs;itum e&longs;t <lb/>extra Octantes, evadit minor; e&longs;tque ad Æquationem maximam, <lb/>ut &longs;inus duplæ di&longs;tantiæ Apogæi Lunæ a proxima Syzygia vel <lb/>Quadratura ad radium. </s></p> <p type="main"> <s>Per eandem Gravitatis Theoriam actio Solis in Lunam paulo <lb/>major e&longs;t ubi linea recta per Nodos Lunæ ducta tran&longs;it per So­<lb/>lem, quam ubi linea ad rectos e&longs;t angulos cum recta Solem ac <lb/>Terram jungente. </s> <s>Et inde oritur alia medii motus Lunaris Æqua­<lb/>tio, quam Seme&longs;trem &longs;ecundam vocabo, quæque maxima e&longs;t ubi <lb/>Nodi in Solis Octantibus ver&longs;antur, & evane&longs;cit ubi &longs;unt in Syzy­<lb/>giis vel Quadraturis, & in aliis Nodorum po&longs;itionibus proportio­<lb/>nalis e&longs;t &longs;inui duplæ di&longs;tantiæ Nodi alterutrius a proxima Syzygia <lb/>aut Quadratura: additur vero medio motui Lunæ dum Nodi <lb/>tran&longs;eunt a Solis Quadraturis ad proximas Syzygias, & &longs;ubduci­<lb/>tur in eorum tran&longs;itu a Syzygiis ad Quadraturas; & in Octanti­<lb/>bus ubi maxima e&longs;t, a&longs;cendit ad 47″ in mediocri Solis di&longs;tantia a <lb/>Terra, uti ex Theoria Gravitatis colligo. </s> <s>In aliis Solis di&longs;tantiis <lb/>hæe Æquatio, in Octantibus Nodorum, e&longs;t reciproce ut cubus di­<lb/>&longs;tantiæ Solis a Terra, ideoQ.E.I. Perigæo Solis ad 45″ in Apo­<lb/>gæo ejus ad 49″ circiter a&longs;cendit. </s></p> <p type="main"> <s>Per eandem Gravitatis Theoriam Apogæum Lunæ progreditur <lb/>quam maxime ubi vel cum Sole conjungitur vel eidem opponitur, <lb/>& regreditur ubi cum Sole Quadraturam facit. </s> <s>Et Eccentricitas <lb/>fit maxima in priore ca&longs;u & minima in po&longs;teriore, per Corol. <pb xlink:href="039/01/451.jpg" pagenum="423"/>7, 8 & 9. Prop. </s> <s>LXVI. Lib. </s> <s>I. </s> <s>Et hæ inæqualitates per eadem </s></p> <p type="main"> <s><arrow.to.target n="note452"/>Corollaria permagnæ &longs;unt, & Æquationem principalem Apogæi <lb/>generant, quam Seme&longs;trem vocabo. </s> <s>Et Æquatio maxima Seme­<lb/>&longs;tris e&longs;t 12<emph type="sup"/>gr.<emph.end type="sup"/> 18′ circiter, quantum ex Ob&longs;ervationibus colligere <lb/>potui. <emph type="italics"/>Horroxius<emph.end type="italics"/>no&longs;ter Lunam in Ellip&longs;i circum Terram, in ejus <lb/>umbilico inferiore con&longs;titutam, revolvi primus &longs;tatuit. <emph type="italics"/>Halleius<emph.end type="italics"/><lb/>centrum Ellip&longs;eos in Epicyclo locavit, cujus centrum uniformiter <lb/>revolvitur circum Terram. </s> <s>Et ex motu in Epicyclo oriuntur in­<lb/>æqualitates jam dictæ in progre&longs;&longs;u & regre&longs;&longs;u Apogæi & quanti­<lb/>tate Eccentricitatis. </s> <s>Dividi intelligatur di&longs;tantia mediocris Lunæ <lb/>a Terra in partes 100000, & referat <emph type="italics"/>T<emph.end type="italics"/>Terram & <emph type="italics"/>TC<emph.end type="italics"/>Eccentri­<lb/>citatem mediocrem Lunæ partium 5505. Producatur <emph type="italics"/>TC<emph.end type="italics"/>ad <emph type="italics"/>B,<emph.end type="italics"/><lb/>ut &longs;it <emph type="italics"/>CB<emph.end type="italics"/>&longs;inus Æquationis maximæ Seme&longs;tris 12<emph type="sup"/>gr.<emph.end type="sup"/> 18′ ad ra­<lb/>dium <emph type="italics"/>TC,<emph.end type="italics"/>& circulus <emph type="italics"/>BDA<emph.end type="italics"/>centro <emph type="italics"/>C<emph.end type="italics"/>intervallo <emph type="italics"/>CB<emph.end type="italics"/>de&longs;criptus, <lb/>erit Epicyclus ille in quo centrum Orbis Lunaris locatur & &longs;e­<lb/>cundum ordinem literarum <emph type="italics"/>BDA<emph.end type="italics"/>revolvitur. </s> <s>Capiatur angulus <lb/><emph type="italics"/>BCD<emph.end type="italics"/>æqualis duplo argumento annuo, &longs;eu duplæ di&longs;tantiæ veri <lb/>loci Solis ab Apogæo Lunæ &longs;emel æquato, & erit <emph type="italics"/>CTD<emph.end type="italics"/>Æquatio <lb/><figure id="id.039.01.451.1.jpg" xlink:href="039/01/451/1.jpg"/><lb/>Seme&longs;tris Apogæi Lunæ & <emph type="italics"/>TD<emph.end type="italics"/>Eccentricitas Orbis ejus in Apo­<lb/>gæum &longs;ecundo æquatum tendens. </s> <s>Habitis autem Lunæ motu <lb/>medio & Apogæo & Eccentricitate, ut & Orbis axe majore par­<lb/>tium 200000; ex his eruetur verus Lunæ locus in Orbe & di­<lb/>&longs;tantia ejus a Terra, idque per Methodos noti&longs;&longs;imas. </s></p> <p type="margin"> <s><margin.target id="note452"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>In Perihelio Terræ, propter majorem vim Solis, centrum Or­<lb/>bis Lunæ velocius movetur circum centrum <emph type="italics"/>C<emph.end type="italics"/>quam in Aphelio, <lb/>idQ.E.I. triplicata ratione di&longs;tantiæ Terræ a Sole inver&longs;e. </s> <s>Ob <lb/>Æquationem centri Solis in Argumento annuo comprehen&longs;am, cen­<lb/>trum Orbis Lunæ velocius movetur in Epicyclo <emph type="italics"/>BDA<emph.end type="italics"/>in du­<lb/>plicata ratione di&longs;tantiæ Terræ a Sole inver&longs;e. </s> <s>Ut idem adhuc <lb/>velocius moveatur in ratione &longs;implici di&longs;tantiæ inver&longs;e; ab Orbis <lb/>centro <emph type="italics"/>D<emph.end type="italics"/>agatur recta <emph type="italics"/>DE<emph.end type="italics"/>ver&longs;us Apogæum Lunæ, &longs;eu rectæ <lb/><emph type="italics"/>TC<emph.end type="italics"/>parallela, & capiatur angulus <emph type="italics"/>EDF<emph.end type="italics"/>æqualis exce&longs;&longs;ui Argu-<pb xlink:href="039/01/452.jpg" pagenum="424"/><arrow.to.target n="note453"/>menti annui prædicti &longs;upra di&longs;tantiam Apogæi Lunæ a Perigæo <lb/>Solis in con&longs;equentia; vel quod perinde e&longs;t, capiatur angulus <lb/><emph type="italics"/>CDF<emph.end type="italics"/>æqualis complemento Anomaliæ veræ Solis ad gradus 360. <lb/>Et &longs;it <emph type="italics"/>DF<emph.end type="italics"/>ad <emph type="italics"/>DC<emph.end type="italics"/>ut dupla Eccentricitas Orbis magni ad di&longs;tan­<lb/>tiam mediocrem Solis a Terra, & motus medius diurnus Solis ab <lb/>Apogæo Lunæ ad motum medium diurnum Solis ab Apogæo <lb/>proprio conjunctim, id e&longs;t, ut 33 7/8 ad 1000 & 52′. </s> <s>27″. </s> <s>16′ ad <lb/>59′. </s> <s>8″. </s> <s>10′ conjunctim, &longs;ive ut 3 ad 100. Et concipe centrum <lb/>Orbis Lunæ locari in puncto <emph type="italics"/>F,<emph.end type="italics"/>& in Epicyclo cujus centrum e&longs;t <lb/><emph type="italics"/>D<emph.end type="italics"/>& radius <emph type="italics"/>DF<emph.end type="italics"/>interea revolvi dum punctum <emph type="italics"/>D<emph.end type="italics"/>progreditur <lb/>in circumferentia circuli <emph type="italics"/>DABD.<emph.end type="italics"/>Hac enim ratione velocitas <lb/>qua centrum Orbis Lunæ in linea quadam curva circum centrum <lb/><emph type="italics"/>C<emph.end type="italics"/>de&longs;cripta movebitur, erit reciproce ut cubus di&longs;tantiæ Solis a <lb/>Terra quamproxime, ut oportet. </s></p> <p type="margin"> <s><margin.target id="note453"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Computatio motus hujus difficilis e&longs;t, &longs;ed facilior reddetur per <lb/>approximationem &longs;equentem. </s> <s>Si di&longs;tantia mediocris Lunæ a Terra <lb/>&longs;it partium 100000, & Eccentricitas <emph type="italics"/>TC<emph.end type="italics"/>&longs;it partium 5505 ut &longs;u­<lb/>pra: recta <emph type="italics"/>CB<emph.end type="italics"/>vel <emph type="italics"/>CD<emph.end type="italics"/>invenietur partium 1172 1/4, & recta <emph type="italics"/>DF<emph.end type="italics"/><lb/><figure id="id.039.01.452.1.jpg" xlink:href="039/01/452/1.jpg"/><lb/>partium 35 1/3. Et hæc recta ad di&longs;tantiam <emph type="italics"/>TC<emph.end type="italics"/>&longs;ubtendit angulum <lb/>ad Terram quem tran&longs;latio centri Orbis a loco <emph type="italics"/>D<emph.end type="italics"/>ad locum <emph type="italics"/>F<emph.end type="italics"/>ge­<lb/>nerat in motu centri hujus: & eadem recta duplicata in &longs;itu paral­<lb/>lelo ad di&longs;tantiam &longs;uperioris umbilici Orbis Lunæ a Terra, &longs;ubten­<lb/>dit eundem angulum, quem utique tran&longs;latio illa generat in motu <lb/>umbilici, & ad di&longs;tantiam Lunæ a Terra &longs;ubtendit angulum quem <lb/>eadem tran&longs;latio generat in motu Lunæ, quique propterea Æqua­<lb/>tio centri Secunda dici pote&longs;t. </s> <s>Et hæc Æquatio in mediocri Lunæ <lb/>di&longs;tantia a Terra, e&longs;t ut &longs;inus anguli quem recta illa <emph type="italics"/>DF<emph.end type="italics"/>cum recta <lb/>a puncto <emph type="italics"/>F<emph.end type="italics"/>ad Lunam ducta continet quamproxime, & ubi ma­<lb/>xima e&longs;t evadit 2′. </s> <s>25″. </s> <s>Angulus autem quem recta <emph type="italics"/>DF<emph.end type="italics"/>& recta <lb/>a puncto <emph type="italics"/>F<emph.end type="italics"/>ad Lunam ducta comprehendunt, invenitur vel &longs;ub­<lb/>ducendo angulum <emph type="italics"/>EDF<emph.end type="italics"/>ab Anomalia media Lunæ, vel addendo <lb/>di&longs;tantiam Lunæ a Sole ad di&longs;tantiam Apogæi Lunæ ab Apogæo <pb xlink:href="039/01/453.jpg" pagenum="425"/>Solis. </s> <s>Et ut radius e&longs;t ad &longs;inum anguli &longs;ic inventi, ita 2′. </s> <s>25″ <lb/><arrow.to.target n="note454"/>&longs;unt ad Æquationem centri Secundam, addendam &longs;i &longs;umma illa <lb/>&longs;it minor &longs;emicirculo, &longs;ubducendam &longs;i major. </s> <s>Sic habebitur ejus <lb/>Longitudo in ip&longs;is Luminarium Syzygiis. </s></p> <p type="margin"> <s><margin.target id="note454"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Si computatio accuratior de&longs;ideretur, corrigendus e&longs;t locus <lb/>Lunæ in Orbe ut &longs;upra inventus per Variationem duplicem. </s> <s>De <lb/>Variatione Prima & principali diximus &longs;upra, hæc maxima e&longs;t <lb/>in Octantibus Lunæ. </s> <s>Variatio altera maxima e&longs;t in Quadrantibus, <lb/>& oritur a varia Solis actione in Orbem Lunæ pro varia po&longs;itione <lb/>Apogæi Lunæ ad Solem, computatur vero in hunc modum. </s> <s><lb/>Ut radius ad &longs;inum ver&longs;um di&longs;tantiæ Apogæi Lunæ a Perigæo <lb/>Solis in con&longs;equentia, ita angulus quidam P ad quartum propor­<lb/>tionalem. </s> <s>Et ut radius ad &longs;inum di&longs;tantiæ Lunæ a Sole, ita &longs;um­<lb/>ma hujus quarti proportionalis & anguli cuju&longs;dam alterius Q ad <lb/>Variationem Secundam, &longs;ubducendam &longs;i Lunæ lumen augetur, ad­<lb/>dendam &longs;i diminuitur. </s> <s>Sic habebitur locus verus Lunæ in Orbe, <lb/>& per Reductionem loci hujus ad Eclipticam habebitur Longi­<lb/>tudo Lunæ. </s> <s>Anguli vero P & Q ex Ob&longs;ervationibus determi­<lb/>nandi &longs;unt. </s> <s>Et interea &longs;i pro angulo P u&longs;urpentur 2′, & pro <lb/>angulo Q 1′, non multum errabitur. </s></p> <p type="main"> <s>Cum Atmo&longs;phæra Terræ ad u&longs;que altitudinem milliarium 35 <lb/>vel 40 refringat lucem Solis, & refringendo &longs;pargat eandem in <lb/>Umbram Terræ, & &longs;pargendo lucem in confinio Umbræ dilatat <lb/>Umbram: ad diametrum Umbræ quæ per Parallaxim prodit, <lb/>addo minutum unum primum in Eclip&longs;ibus Lunæ, vel minutum <lb/>unum cum triente. </s></p> <p type="main"> <s>Theoria vero Lunæ primo in Syzygiis, deinde in Quadraturis, <lb/>& ultimo in Octantibus per Phænomena examinari & &longs;tabiliri de­<lb/>bet. </s> <s>Et opus hocce aggre&longs;&longs;urus motus medios Solis & Lunæ ad <lb/>tempus meridianum in Ob&longs;ervatorio Regio <emph type="italics"/>Grenovicen&longs;i,<emph.end type="italics"/>die ul­<lb/>timo men&longs;is <emph type="italics"/>Decembris<emph.end type="italics"/>anni 1700. &longs;t. </s> <s>vet. </s> <s>non incommode &longs;e­<lb/>quentes adhibebit: nempe motum medium Solis <gap/> 20<emph type="sup"/>gr.<emph.end type="sup"/> 43′. </s> <s>40″, & <lb/>Apogæi ejus <gap/> 7<emph type="sup"/>gr.<emph.end type="sup"/> 44′. </s> <s>30″, & motum medium Lunæ <gap/> 15<emph type="sup"/>gr.<emph.end type="sup"/><lb/>20′. </s> <s>00″, & Apogæi ejus <gap/> 8<emph type="sup"/>gr.<emph.end type="sup"/> 20′. </s> <s>00″, & Nodi a&longs;cendentis <lb/><gap/> 27<emph type="sup"/>gr.<emph.end type="sup"/> 24′. </s> <s>20″; & differentiam meridianorum Ob&longs;ervatorii hu­<lb/>jus & Ob&longs;ervatorii Regii <emph type="italics"/>Pari&longs;ien&longs;is<emph.end type="italics"/>0<emph type="sup"/>hor.<emph.end type="sup"/> 9<emph type="sup"/>min.<emph.end type="sup"/> 20<emph type="sup"/>&longs;ec.<emph.end type="sup"/>. <pb xlink:href="039/01/454.jpg" pagenum="426"/><arrow.to.target n="note455"/></s></p> <p type="margin"> <s><margin.target id="note455"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXVI. PROBLEMA XVII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire vim Solis ad Mare movendum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Solis vis <emph type="italics"/>ML<emph.end type="italics"/>&longs;eu <emph type="italics"/>PT,<emph.end type="italics"/>in Quadraturis Lunaribus, ad pertur­<lb/>bandos motus Lunares, erat (per Prop. </s> <s>XXV. hujus) ad vim <lb/>gravitatis apud nos, ut 1 ad 638092, 6. Et vis <emph type="italics"/>TM-LM<emph.end type="italics"/>&longs;eu <lb/>2<emph type="italics"/>PK<emph.end type="italics"/>in Syzygiis Lunaribus, e&longs;t duplo major. </s> <s>Hæ autem vires, <lb/>&longs;i de&longs;cendatur ad &longs;uperficiem Terræ, diminuuntur in ratione di­<lb/>&longs;tantiarum a centro Terræ, id e&longs;t, in ratione 60 1/2 ad 1; adeo­<lb/>que vis prior in &longs;uperficie Terræ, e&longs;t ad vim gravitatis, ut 1 ad <lb/>38604600. Hac vi Mare deprimitur in locis quæ 90 gradibus di&longs;tant <lb/><figure id="id.039.01.454.1.jpg" xlink:href="039/01/454/1.jpg"/><lb/>a Sole. </s> <s>Vi altera quæ duplo major e&longs;t, Mare elevatur & &longs;ub Sole <lb/>& in regione Soli oppo&longs;ita. </s> <s>Summa virium e&longs;t ad vim gravitatis <lb/>ut 1 ad 12868200. Et quoniam vis eadem eundem ciet motum, <lb/>&longs;ive ea deprimat Aquam in regionibus quæ 90 gradibus di&longs;tant à <lb/>Sole, &longs;ive elevet eandem in regionibus &longs;ub Sole & Soli oppo&longs;itis, <lb/>hæc &longs;umma erit tota Solis vis ad Mare agitandum; & eundem <lb/>habebit effectum ac &longs;i tota in regionibus &longs;ub Sole & Soli oppo­<lb/>&longs;itis Mare elevaret, in regionibus autem quæ 90 gradibus di&longs;tant <lb/>a Sole nil ageret. </s></p> <p type="main"> <s>Hæc e&longs;t vis Solis ad Mare ciendum in loco quo vis dato, ubi Sol <lb/>tam in vertice loci ver&longs;atur quam in mediocri &longs;ua di&longs;tantia a <lb/>Terra. </s> <s>In aliis Solis po&longs;itionibus vis ad Mare accollendum, e&longs;t <lb/>ut &longs;inus ver&longs;us duplæ altitudinis Solis &longs;upra horizontem loci di­<lb/>recte & cubus di&longs;tantiæ Solis a Terra inver&longs;e. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Cum vis centrifuga partium Terræ à diurno Terræ motu <lb/>oriunda, quæ e&longs;t ad vim gravitatis ut 1 ad 289, efficiat ut alti-<pb xlink:href="039/01/455.jpg" pagenum="427"/>tudo Aquæ &longs;ub Æquatore &longs;uperet ejus altitudinem &longs;ub Polis men­<lb/><arrow.to.target n="note456"/>&longs;ura pedum Pari&longs;ien&longs;ium 85820; vis Solaris de qua egimus, cum <lb/>&longs;it ad vim gravitatis ut 1 ad 12868200, atque adeo ad vim illam <lb/>centrifugam ut 289 ad 12868200 &longs;eu 1 ad 44527, efficiet ut al­<lb/>titudo Aquæ in regionibus &longs;ub Sole & Soli oppo&longs;itis, &longs;uperet alti­<lb/>tudinem ejus in locis quæ 90 gradibus di&longs;tant a Sole, men&longs;ura <lb/>tantum pedis unius Pari&longs;ien&longs;is & digitorum undecim cum octava <lb/>parte digiti. </s> <s>E&longs;t enim hæc men&longs;ura ad men&longs;uram pedum 85820 <lb/>ut 1 ad 44527. </s></p> <p type="margin"> <s><margin.target id="note456"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXVII. PROBLEMA XVIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire vim Lunæ ad Mare movendum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Vis Lunæ ad Mare movendum colligendà e&longs;t ex ejus propor­<lb/>tione ad vim Solis, & hæc proportio colligenda e&longs;t ex propor­<lb/>tione motuum Maris, qui ab his viribus oriuntur. </s> <s>Ante o&longs;tium <lb/>fluvii <emph type="italics"/>Avonæ<emph.end type="italics"/>ad lapidem tertium infra <emph type="italics"/>Bri&longs;toliam,<emph.end type="italics"/>tempore verno <lb/>& autumnali totus Aquæ a&longs;cen&longs;us in Conjunctione & Oppo&longs;itione <lb/>Luminarium (ob&longs;ervante <emph type="italics"/>Samuele Sturmio<emph.end type="italics"/>) e&longs;t pedum plus mi­<lb/>nus 45, in Quadraturis autem e&longs;t pedum tantum 25. Altitudo <lb/>prior ex &longs;umma virium, po&longs;terior ex earundem differentia oritur. </s> <s><lb/>Solis igitur & Lunæ in Æquatore ver&longs;antium & mediocriter a <lb/>Terra di&longs;tantium &longs;unto vires S & L, & erit L+S ad L-S ut <lb/>45 ad 25, &longs;eu 9 ad 5. </s></p> <p type="main"> <s>In portu <emph type="italics"/>Plymuthi<emph.end type="italics"/>Æ&longs;tus maris (ex ob&longs;ervatione <emph type="italics"/>Samuelis Cole­<lb/>pre&longs;&longs;i<emph.end type="italics"/>) ad pedes plus minus &longs;exdecim altitudine mediocri attolli­<lb/>tur, ac tempore verno & autumnali altitudo Æ&longs;tus in Syzygiis &longs;u­<lb/>perare pote&longs;t altitudinem ejus in Quadraturis, pedibus plus &longs;eptem <lb/>vel octo. </s> <s>Si maxima harum altitudinum differentia &longs;it pedum no­<lb/>vem, erit L+S ad L-S ut 20 1/2 ad 11 1/2 &longs;eu 41 ad 23. Quæ <lb/>proportio &longs;atis congruit cum priore. </s> <s>Ob magnitudinem Æ&longs;tus in <lb/>portu <emph type="italics"/>Bi&longs;toliæ,<emph.end type="italics"/>ob&longs;ervationibus <emph type="italics"/>Sturmii<emph.end type="italics"/>magis fidendum e&longs;&longs;e vi­<lb/>detur, ideoQ.E.D.nec aliquid certius con&longs;titerit, proportionem 9 <lb/>ad 5 u&longs;urpabimus. </s></p> <p type="main"> <s>Cæterum ob aquarum reciprocos motus, Æ&longs;tus maximi non in­<lb/>cidunt in ip&longs;as Luminarium Syzygias, &longs;ed &longs;unt tertii a Syzygiis <lb/>ut dictum fuit, &longs;eu proxime &longs;equuntur tertium Lunæ po&longs;t Syzy­<lb/>gias appul&longs;um ad meridianum loci, vel potius (ut a <emph type="italics"/>Sturmio<emph.end type="italics"/>no­<lb/>tatur) &longs;unt tertii po&longs;t diem novilunii vel plenilunii, &longs;eu po&longs;t ho-<pb xlink:href="039/01/456.jpg" pagenum="428"/><arrow.to.target n="note457"/>ram a novilunio vel plenilunio plus minus duodecimam, adeoque <lb/>incidunt in horam a novilunio vel plenilunio plus minus quadra­<lb/>ge&longs;imam tertiam. </s> <s>Incidunt vero in hoc portu in horam &longs;epti­<lb/>mam circiter ab appul&longs;u Lunæ ad meridianum loci; ideoque pro­<lb/>xime &longs;equuntur appul&longs;um Lunæ ad meridianum, ubi Luna di&longs;tat a <lb/>Sole vel ab oppo&longs;itione Solis gradibus plus minus octodecim vel <lb/>novendecim in con&longs;equentia. </s> <s>Æ&longs;tas & Hyems maxime vigent, <lb/>non in ip&longs;is Sol&longs;titiis, &longs;ed ubi Sol di&longs;tat a Sol&longs;titiis decima circi­<lb/>ter parte totius circuitus, &longs;eu gradibus plus minus 36 vel 37. Et <lb/>&longs;imiliter maximus Æ&longs;tus maris oritur ab appul&longs;u Lunæ ad meri­<lb/>dianum loci, ubi Luna di&longs;tat a Sole decima circiter parte motus <lb/>totius ab Æ&longs;tu ad Æ&longs;tum. </s> <s>Sit di&longs;tantia illa graduum plus mi­<lb/>nus 18 1/2. Et vis Solis in hac di&longs;tantia Lunæ a Syzygiis & Qua­<lb/>draturis, minor erit ad augendum & ad minuendum motum ma­<lb/>ris a vi Lunæ oriundum, quam in ip&longs;is Syzygiis & Quadraturis, in <lb/>ratione radii ad &longs;inum complementi di&longs;tantiæ hujus duplicatæ &longs;eu <lb/>anguli graduum 37, hoc e&longs;t, in ratione 10000000 ad 7986355. <lb/>IdeoQ.E.I. analogia &longs;uperiore pro S &longs;cribi debet 0, 7986355 S. </s></p> <p type="margin"> <s><margin.target id="note457"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Sed & vis Lunæ in Quadraturis, ob declinationem Lunæ ab <lb/>Æquatore, diminui debet. </s> <s>Nam Luna in Quadraturis, vel potius <lb/>in gradu 18 1/2 po&longs;t Quadraturas, in declinatione graduum plus <lb/>minus 22. 13′ ver&longs;atur. </s> <s>Et Luminaris ab Æquatore declinantis <lb/>vis ad Mare movendum diminuitur in duplicata ratione &longs;inus <lb/>complementi declinationis quamproxime. </s> <s>Et propterea vis <lb/>Lunæ in his Quadraturis e&longs;t tantum 0,8570327 L. </s> <s>E&longs;t igitur <lb/>L+0,7986355 S ad 0,8570327 L-0,7986355 S ut 9 ad 5. </s></p> <p type="main"> <s>Præterea diametri Orbis in quo Luna ab&longs;que Eccentricitate mo­<lb/>veri deberet, &longs;unt ad invicem ut 69 ad 70; ideoQ.E.D.&longs;tantia <lb/>Lunæ a Terra in Syzygiis e&longs;t ad di&longs;tantiam ejus in Quadraturis, <lb/>ut 69 ad 70, cæteris paribus. </s> <s>Et di&longs;tantiæ ejus in gradu 18 1/2 a <lb/>Syzygiis ubi Æ&longs;tus maximus generatur, & in gradu 18 1/2 a Qua­<lb/>draturis ubi Æ&longs;tus minimus generatur, &longs;unt ad mediocrem ejus <lb/>di&longs;tantiam, ut 69,098747 & 69,897345 ad 69 1/2. Vires autem Lu­<lb/>næ ad Mare movendum &longs;unt in triplicata ratione di&longs;tantiarum in­<lb/>ver&longs;e, ideoque vires in maxima & minima harum di&longs;tantiarum &longs;unt <lb/>ad vim in mediocri di&longs;tantia, ut 0,9830427 & 1,017522 ad 1. Unde fit <lb/>1,017522 L+0,7986355 S ad 0,9830427X0,8570327 L-0,7986355 S <lb/>ut 9 ad 5. Et S ad L ut 1 ad 4,4815. Itaque cum vis Solis fit <lb/>ad vim gravitatis ut 1 ad 12868200, vis Lunæ erit ad vim gravi­<lb/>tatis ut 1 ad 2871400. </s></p><pb xlink:href="039/01/457.jpg" pagenum="429"/> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Cum Aqua vi Solis agitata a&longs;cendat ad altitudinem <lb/><arrow.to.target n="note458"/>pedis unius & undecim digitorum cum octava parte digiti, eadem <lb/>vi Lunæ a&longs;cendet ad altitudinem octo pedum & digitorum octo, <lb/>& vi utraque ad altitudinem pedum decem cum &longs;emi&longs;&longs;e, & ubi <lb/>Luna e&longs;t in Perigæo ad altitudinem pedum duodecim cum &longs;emi&longs;&longs;e <lb/>& ultra, præ&longs;ertim ubi Æ&longs;tus ventis &longs;pirantibus adjuvatur. </s> <s>Tanta <lb/>autem vis ad omnes Maris motus excitandos abunde &longs;ufficit, & <lb/>quantitati motuum probe re&longs;pondet. </s> <s>Nam in maribus quæ ab <lb/>Oriente in Occidentem late patent, uti in Mari <emph type="italics"/>Pacifico,<emph.end type="italics"/>& Maris <lb/><emph type="italics"/>Atlantici<emph.end type="italics"/>& <emph type="italics"/>Æthiopici<emph.end type="italics"/>partibus extra Tropicos, aqua attolli &longs;o­<lb/>let ad altitudinem pedum &longs;ex, novem, duodecim vel quindecim. </s> <s><lb/>In Mari autem <emph type="italics"/>Pacifico,<emph.end type="italics"/>quod profundius e&longs;t & latius patet, Æ&longs;tus <lb/>dicuntur e&longs;&longs;e majores quam in <emph type="italics"/>Atlantico<emph.end type="italics"/>& <emph type="italics"/>Æthiopico.<emph.end type="italics"/>Etenim <lb/>ut plenus &longs;it Æ&longs;tus, latitudo Maris ab Oriente in Occidentem non <lb/>minor e&longs;&longs;e debet quàm graduum nonaginta. </s> <s>In Mari <emph type="italics"/>Æthiopico,<emph.end type="italics"/><lb/>a&longs;cen&longs;us aquæ intra Tropicos minor e&longs;t quam in Zonis tempera­<lb/>tis, propter angu&longs;tiam Maris inter <emph type="italics"/>Africam<emph.end type="italics"/>& Au&longs;tralem partem <lb/><emph type="italics"/>Americæ.<emph.end type="italics"/>In medio Mari aqua nequit a&longs;cendere, ni&longs;i ad littus <lb/>utrumque & orientale & occidentale &longs;imul de&longs;cendat: cum tamen <lb/>vicibus alternis ad littora illa in Maribus no&longs;tris angu&longs;tis de&longs;cen­<lb/>dere debeat. </s> <s>Ea de cau&longs;a fluxus & refluxus in In&longs;ulis, quæ à <lb/>littoribus longi&longs;&longs;ime ab&longs;unt, perexiguus e&longs;&longs;et &longs;olet. </s> <s>In Portubus <lb/>quibu&longs;dam, ubi aqua cum impetu magno per loca vado&longs;a, ad <lb/>Sinus alternis vicibus implendos & evacuandos, influere & effluere <lb/>cogitur, fluxus & refluxus debent e&longs;&longs;e &longs;olito majores, uti ad <lb/><emph type="italics"/>Plymuthum<emph.end type="italics"/>& pontem <emph type="italics"/>Chep&longs;towæ<emph.end type="italics"/>in <emph type="italics"/>Anglia<emph.end type="italics"/>; ad montes S. <emph type="italics"/>Mi­<lb/>chaelis<emph.end type="italics"/>& urbem <emph type="italics"/>Abrincatuorum<emph.end type="italics"/>(vulgo <emph type="italics"/>Auranches<emph.end type="italics"/>) in <emph type="italics"/>Normania<emph.end type="italics"/>; <lb/>ad <emph type="italics"/>Cambaiam<emph.end type="italics"/>& <emph type="italics"/>Pegu<emph.end type="italics"/>in <emph type="italics"/>India<emph.end type="italics"/>orientali. </s> <s>His in locis mare, <lb/>magna cum velocitate accedendo & recedendo, littora nunc in­<lb/>undat nunc arida relinquit ad multa milliaria. </s> <s>NeQ.E.I.petus <lb/>influendi & remeandi prius frangi pote&longs;t, quam aqua attollitur <lb/>vel deprimitur ad pedes 30, 40, vel 50 & amplius. </s> <s>Et par e&longs;t <lb/>ratio fretorum oblongorum & vado&longs;orum, uti <emph type="italics"/>Magellanici<emph.end type="italics"/>& ejus <lb/>quo <emph type="italics"/>Anglia<emph.end type="italics"/>circundatur. </s> <s>Æ&longs;tus in huju&longs;modi portubus & fretis, <lb/>per impetum cur&longs;us & recur&longs;us &longs;upra modum augetur. </s> <s>Ad littora <lb/>vero quæ de&longs;cen&longs;u præcipiti ad mare profundum & apertum <lb/>&longs;pectant, ubi aqua &longs;ine impetu effluendi & remeandi attolli & <lb/>&longs;ub&longs;idere pote&longs;t, magnitudo Æ&longs;tus re&longs;pondet viribus Solis & <lb/>Lunæ. <pb xlink:href="039/01/458.jpg" pagenum="430"/><arrow.to.target n="note459"/></s></p> <p type="margin"> <s><margin.target id="note458"/>LIBER <lb/>TERTIUS.</s></p> <p type="margin"> <s><margin.target id="note459"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Cum vis Lunæ ad Mare movendum, &longs;it ad vim gravi­<lb/>tatis ut 1 ad 2871400, per&longs;picuum e&longs;t quod vis illa &longs;it longe <lb/>minor quam quæ vel in experimentis Pendulorum, vel in Staticis <lb/>aut Hydro&longs;taticis quibu&longs;cunque &longs;entiri po&longs;&longs;it. </s> <s>In Æ&longs;tu &longs;olo ma­<lb/>rino hæc vis &longs;en&longs;ibilem edit effectum. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Quoniam vis Lunæ ad Mare movendum, e&longs;t ad Solis <lb/>vim con&longs;imilem ut 4,4815 ad 1, & vires illæ (per Corol. </s> <s>14. <lb/>Prop. </s> <s>LXVI. Lib. </s> <s>I.) &longs;unt ut den&longs;itates corporum Lunæ & Solis <lb/>& cubi diametrorum apparentium conjunctim; den&longs;itas Lunæ erit <lb/>ad den&longs;itatem Solis, ut 4,4815 ad 1 directe & cubus diametri <lb/>Lunæ ad cubum diametri Solis inver&longs;e: id e&longs;t (cum diametri me­<lb/>diocres apparentes Lunæ & Solis &longs;int 31′. </s> <s>16 1/2″ & 32′. </s> <s>12″) ut <lb/>4891 ad 1000. Den&longs;itas autem Solis erat ad den&longs;itatem Terræ, <lb/>ut 100 ad 396; & propterea den&longs;itas Lunæ e&longs;t ad den&longs;itatem <lb/>Terræ, ut 4891 ad 3960 &longs;eu 21 ad 17. E&longs;t igitur corpus Lunæ <lb/>den&longs;ius & magis terre&longs;tre quam Terra no&longs;tra. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Et cum vera diameter Lunæ (ex Ob&longs;ervationibus <lb/>A&longs;tronomicis) &longs;it ad veram diametrum Terræ, ut 100 ad 365; <lb/>erit ma&longs;la Lunæ ad ma&longs;&longs;am Terræ, ut 1 ad 39,371. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>5. Et gravitas acceleratrix in &longs;uperficie Lunæ, erit qua&longs;i <lb/>triplo minor quam gravitas acceleratrix in &longs;uperficie Terræ. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>6. Et di&longs;tantia centri Lunæ a centro Terræ, erit ad di­<lb/>&longs;tantiam centri Lunæ a communi gravitatis centro Terræ & Lunæ, <lb/>ut 40,371 ad 39,371. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>7. Et mediocris di&longs;tantia centri Lunæ a centro Terræ, erit <lb/>&longs;emidiametrorum maximarum Terræ 60 1/4 quamproxime. </s> <s>Nam <lb/>&longs;emidiameter maxima Terræ fuit pedum Pari&longs;ien&longs;ium 19767630, <lb/>& mediocris di&longs;tantia centrorum Terræ & Lunæ ex huju&longs;modi <lb/>&longs;emidiametris 60 1/4 con&longs;tans, æqualis e&longs;t pedibus 1190999707. Et <lb/>hæc di&longs;tantia (per Corollarium &longs;uperius) e&longs;t ad di&longs;tantiam centri <lb/>Lunæ a communi gravitatis centro Terræ & Lunæ, ut 40,371 ad <lb/>39,371, quæ proinde e&longs;t pedum 1161498340. Et cum Luna re­<lb/>volvatur re&longs;pectu Fixarum, diebus 27, horis 7 & minutis primis 43 1/5; <lb/>&longs;inus ver&longs;us anguli quem Luna, tempore minuti unius primi motu <lb/>&longs;uo medio, circa commune gravitatis centrum Terræ & Lunæ de­<lb/>&longs;cribit, e&longs;t 1275235, exi&longs;tente radio 100,000000,000000, Et ut <lb/>radius e&longs;t ad hunc &longs;inum ver&longs;um, ita &longs;unt pedes 1161498340 ad <lb/>pedes 14,811833. Luna igitur vi illa qua retinetur in Orbe, ca­<lb/>dendo in Terram, tempore minuti unius primi de&longs;cribet pedes <lb/>14,811833. Et &longs;i hæc vis augeatur in ratione (177 29/40) ad (178 29/40), ha-<pb xlink:href="039/01/459.jpg" pagenum="431"/>bebitur vis tota gravitatis in Orbe Lunæ, per Corol. </s> <s>Prop. </s> <s>III. </s></p> <p type="main"> <s><arrow.to.target n="note460"/>Et hac vi Luna cadendo, tempore minuti unius primi de&longs;cribere <lb/>deberet pedes 14,89517. Et ad &longs;exage&longs;imam partem hujus di­<lb/>&longs;tantiæ, id e&longs;t, ad di&longs;tantiam pedum 19849995 a centro Terræ, <lb/>corpus grave cadendo, tempore minuti unius &longs;ecundi de&longs;cribere <lb/>deberet etiam pedes 14,89517. Diminuatur hæc di&longs;tantia in &longs;ub­<lb/>duplicata ratione pedum 14,89517 ad pedes 15,12028, & habebitur <lb/>di&longs;tantia pedum 19701678 a qua grave cadendo, eodem tempore <lb/>minuti unius &longs;ecundi de&longs;cribet pedes 15,12028, id e&longs;t, pedes 15, <lb/>dig 1, lin. </s> <s>5,32. Et hac vi gravia cadunt in &longs;uperficie Terræ, in <lb/>Latitudine urbis <emph type="italics"/>Lutetiæ Pari&longs;iorum,<emph.end type="italics"/>ut &longs;upra o&longs;ten&longs;um e&longs;t. </s> <s>E&longs;t <lb/>autem di&longs;tantia pedum 19701678 paulo minor quam &longs;emidiame­<lb/>ter globi huic Terræ æqualis, & paulo major quam Terræ hujus <lb/>&longs;emidiameter mediocris, ut oportet. </s> <s>Sed differentiæ &longs;unt in&longs;en&longs;i­<lb/>biles. </s> <s>Et propterea vis qua Luna retinetur in Orbe &longs;uo, ad di­<lb/>&longs;tantiam maximarum Terræ &longs;emidiametrorum 60 1/4, ea e&longs;t quam <lb/>vis Gravitatis in &longs;uperficie Terræ requirit. </s></p> <p type="margin"> <s><margin.target id="note460"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>8. Di&longs;tantia mediocris centrorum Terræ & Lunæ, e&longs;t me­<lb/>diocrium Terræ &longs;emidiametrorum 60 1/2 quamproxime. </s> <s>Nam &longs;e­<lb/>midiameter mediocris, quæ erat pedum 19688725, e&longs;t ad &longs;emi­<lb/>diametrum maximam pedum 19767630, ut 60 1/4 ad 60 1/2 quam­<lb/>proxime. </s></p> <p type="main"> <s>In his computationibus Attractionem magneticam Terræ non <lb/>con&longs;ideravimus, cujus utique quantitas perparva e&longs;t & ignotatur. </s> <s><lb/>Siquando vero hæc Attractio inve&longs;tigari poterit, & men&longs;uræ gra­<lb/>duum in Meridiano, ac longitudines Pendulorum i&longs;ochronorum in <lb/>diver&longs;is parallelis, lege&longs;que motuum Maris, & parallaxis Lunæ <lb/>cum diametris apparentibus Solis & Lunæ ex Phænomenis accu­<lb/>ratius determinatæ fuerint: licebit calculum hunc omnem accura­<lb/>tius repetere. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXVIII. PROBLEMA XIX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire Figuram corporis Lunæ.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Si corpus Lunare fluidum e&longs;&longs;et ad in&longs;tar Maris no&longs;tri, vis Terræ <lb/>ad fluidum illud in partibus & citimis & ultimis elevandum, e&longs;&longs;et <lb/>ad vim Lunæ, qua Mare no&longs;trum in partibus & &longs;ub Luna & Lunæ <lb/>oppo&longs;itis attollitur, ut gravitas acceleratrix Lunæ in Terram ad <lb/>gravitatem acceleratricem Terræ in Lunam & diameter Lunæ ad <pb xlink:href="039/01/460.jpg" pagenum="432"/><arrow.to.target n="note461"/>diametrum Terræ conjunctim; id e&longs;t, ut 39,371 ad 1 & 100 ad <lb/>365 conjunctim, &longs;eu 1079 ad 100. Unde cum Mare no&longs;trum vi <lb/>Lunæ attollatur ad pedes 8 2/3, fluidum Lunare vi Terræ attolli de­<lb/>beret ad pedes 93 1/2. EaQ.E.D. cau&longs;a Figura Lunæ Sphærois e&longs;&longs;et, <lb/>cujus maxima diameter producta tran&longs;iret per centrum Terræ, & <lb/>&longs;uperaret diametros perpendiculares exce&longs;&longs;u pedum 187. Talem <lb/>igitur Figuram Luna affectat, eamque &longs;ub initio induere debuit. <lb/><emph type="italics"/><expan abbr="q.">que</expan> E. I.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note461"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Inde vero fit ut eadem &longs;emper Lunæ facies in Terram <lb/>obvertatur. </s> <s>In alio enim &longs;itu corpus Lunare quie&longs;cere non po­<lb/>te&longs;t, &longs;ed ad hunc &longs;itum o&longs;cillando &longs;emper redibit. </s> <s>Attamen o&longs;cil­<lb/>lationes, ob parvitatem virium agitantium, e&longs;&longs;ent longè tardi&longs;&longs;imæ: <lb/>adeo ut facies illa, quæ Terram &longs;emper re&longs;picere deberet, po&longs;&longs;it <lb/>alterum orbis Lunaris umbilicum, ob rationem in Prop. </s> <s>XVII. alla­<lb/>tam re&longs;picere, neque &longs;tatim abinde retrahi & in Terram converti. </s></p> <p type="main"> <s><emph type="center"/>LEMMA I.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si<emph.end type="italics"/>APEp <emph type="italics"/>Terram de&longs;ignet uniformiter den&longs;am, centroque <lb/>C & Polis<emph.end type="italics"/>P, p <emph type="italics"/>& Æquatore<emph.end type="italics"/>AE <emph type="italics"/>delineatam; & &longs;i centro<emph.end type="italics"/>C <lb/><emph type="italics"/>radio<emph.end type="italics"/>CP <emph type="italics"/>de&longs;cribi intelligatur Sphæra<emph.end type="italics"/>Pape; <emph type="italics"/>&longs;it autem<emph.end type="italics"/>QR <emph type="italics"/>pla­<lb/>num, cui recta a centro Solis ad centrum Terræ ducta normaliter <lb/>in&longs;i&longs;tit; & Terræ totius exterioris<emph.end type="italics"/>PapAPepE, <emph type="italics"/>quæ Sphæra <lb/>modo de&longs;cripta altior e&longs;t, particulæ &longs;ingulæ conentur recedere hinc <lb/>inde a plano<emph.end type="italics"/>QR, <emph type="italics"/>&longs;itque conatus particulæ cuju&longs;que ut eju&longs;dem <lb/>di&longs;tantia a plano: Dico primo, quod tota particularum omnium, in <lb/>Æquatoris circulo<emph.end type="italics"/>AE, <emph type="italics"/>extra globum uniformiter per totum cir­<lb/>cuitum in morem annuli di&longs;po&longs;itarum, vis & efficacia ad Terram <lb/>circum centrum ejus rotandam, &longs;it ad totam particularum totidem <lb/>in Æquatoris puncto<emph.end type="italics"/>A, <emph type="italics"/>quod a plano<emph.end type="italics"/>QR <emph type="italics"/>maxime di&longs;tat, con­<lb/>&longs;i&longs;tentium vim & efficaciam, ad Terram con&longs;imili motu circulari <lb/>circum centrum ejus movendam, ut unum ad duo. </s> <s>Et motus i&longs;te <lb/>circularis circum axem, in communi &longs;ectione Æquatoris & plani<emph.end type="italics"/><lb/>QR <emph type="italics"/>jacentem, peragetur.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam centro <emph type="italics"/>C<emph.end type="italics"/>diametro <emph type="italics"/>BD<emph.end type="italics"/>de&longs;cribatur &longs;emicirculus <lb/><emph type="italics"/>BAFDC.<emph.end type="italics"/>Dividi intelligatur &longs;emicircum ferentia <emph type="italics"/>BAD<emph.end type="italics"/>in <pb xlink:href="039/01/461.jpg" pagenum="433"/>partes innumeras æquales, & a partibus &longs;ingulis <emph type="italics"/>F<emph.end type="italics"/>ad diame­<lb/><arrow.to.target n="note462"/>trum <emph type="italics"/>BD<emph.end type="italics"/>demittantur &longs;inus <emph type="italics"/>FY.<emph.end type="italics"/>Et &longs;umma quadratorum ex <lb/>&longs;inibus omnibus <emph type="italics"/>FY<emph.end type="italics"/>æqualis erit &longs;ummæ quadratorum ex &longs;inibus <lb/>omnibus <emph type="italics"/>CY,<emph.end type="italics"/>& &longs;umma utraque æqualis erit &longs;ummæ quadrato­<lb/>rum ex totidem &longs;emidiametris <emph type="italics"/>CF<emph.end type="italics"/>; adeoque &longs;umma quadrato­<lb/>rum ex omnibus <emph type="italics"/>FY,<emph.end type="italics"/>erit duplo minor quam &longs;umma quadrato­<lb/>rum ex totidem &longs;emidiametris <emph type="italics"/>CF.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note462"/>LIBER <lb/>TERTIUS.</s></p><figure id="id.039.01.461.1.jpg" xlink:href="039/01/461/1.jpg"/> <p type="main"> <s>Jam dividatur perimeter circuli <emph type="italics"/>AE<emph.end type="italics"/>in particulas totidem æ­<lb/>quales, & ab earum unaquaque <emph type="italics"/>F<emph.end type="italics"/>ad planum <emph type="italics"/>QR<emph.end type="italics"/>demittatur <lb/>perpendiculum <emph type="italics"/>FG,<emph.end type="italics"/>ut & a puncto <emph type="italics"/>A<emph.end type="italics"/>perpendiculum <emph type="italics"/>AH.<emph.end type="italics"/>Et <lb/>vis qua particula <emph type="italics"/>F<emph.end type="italics"/>recedit a plano <emph type="italics"/>QR,<emph.end type="italics"/>erit ut perpendiculum <lb/>illud <emph type="italics"/>FG<emph.end type="italics"/>per hypothe&longs;in, & hæc vis ducta in di&longs;tantiam <emph type="italics"/>CG,<emph.end type="italics"/><lb/>erit efficacia particulæ <emph type="italics"/>F<emph.end type="italics"/>ad Terram circum centrum ejus con­<lb/>vertendam. </s> <s>Adeoque efficacia particulæ in loco <emph type="italics"/>F,<emph.end type="italics"/>erit ad effi­<lb/>caciam particulæ in loco <emph type="italics"/>A,<emph.end type="italics"/>ut <emph type="italics"/>FGXGC<emph.end type="italics"/>ad <emph type="italics"/>AHXHC,<emph.end type="italics"/>hoc <lb/>e&longs;t, ut <emph type="italics"/>FCq<emph.end type="italics"/>ad <emph type="italics"/>ACq<emph.end type="italics"/>; & propterea efficacia tota particularum <lb/>omnium in locis &longs;uis <emph type="italics"/>F,<emph.end type="italics"/>erit ad efficaciam particularum totidem in <lb/>loco <emph type="italics"/>A,<emph.end type="italics"/>ut &longs;umma omnium <emph type="italics"/>FCq<emph.end type="italics"/>ad &longs;ummam totidem <emph type="italics"/>ACq,<emph.end type="italics"/>hoc <lb/>e&longs;t, (per jam demon&longs;trata) ut unum ad duo. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s>Et quoniam particulæ agunt recedendo perpendiculariter a <lb/>plano <emph type="italics"/>QR,<emph.end type="italics"/>idque æqualiter ab utraque parte hujus plani: eædem <lb/>convertent circumferentiam circuli Æquatoris, eiQ.E.I.hærentem <lb/>Terram, circum axem tam in plano illo <emph type="italics"/>QR<emph.end type="italics"/>quam in plano Æqua­<lb/>toris jacentem. <pb xlink:href="039/01/462.jpg" pagenum="434"/><arrow.to.target n="note463"/></s></p> <p type="margin"> <s><margin.target id="note463"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>LEMMA II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis: Dico &longs;ecundo quod vis & efficacia tota parti­<lb/>cularum omnium extra globum undique &longs;itarum, ad Terram cir­<lb/>cum axem eundem rotandam, &longs;it ad vim totam particularum toti­<lb/>dem, in Æquatoris circulo<emph.end type="italics"/>AE, <emph type="italics"/>uniformiter per totum circuitum <lb/>in morem annuli di&longs;po&longs;itarum, ad Terram con&longs;imili motu circulari <lb/>movendam, ut duo ad quinque.<emph.end type="italics"/></s></p> <p type="main"> <s>Sit enim <emph type="italics"/>IK<emph.end type="italics"/>circulus quilibet minor Æquatori <emph type="italics"/>AE<emph.end type="italics"/>parallelus, <lb/>&longs;intque <emph type="italics"/>L, l<emph.end type="italics"/>particulæ duæ quævis æquales in hoc circulo extra <lb/>globum <emph type="italics"/>Pape<emph.end type="italics"/>&longs;itæ. </s> <s>Et &longs;i in planum <emph type="italics"/>QR,<emph.end type="italics"/>quod radio in Solem <lb/>ducto perpendiculare e&longs;t, demittantur perpendicula <emph type="italics"/>LM, lm:<emph.end type="italics"/><lb/>vires totæ quibus particulæ illæ fugiunt planum <emph type="italics"/>QR,<emph.end type="italics"/>proporti­<lb/>onales erunt perpendiculis illis <emph type="italics"/>LM, lm.<emph.end type="italics"/>Sit autem recta <emph type="italics"/>Ll<emph.end type="italics"/><lb/>plano <emph type="italics"/>Pape<emph.end type="italics"/>parallela & bi&longs;ecetur eadem in <emph type="italics"/>X,<emph.end type="italics"/>& per pun­<lb/>ctum <emph type="italics"/>X<emph.end type="italics"/>agatur <emph type="italics"/>Nn,<emph.end type="italics"/>quæ parallela &longs;it plano <emph type="italics"/>QR<emph.end type="italics"/>& perpendi­<lb/><figure id="id.039.01.462.1.jpg" xlink:href="039/01/462/1.jpg"/><lb/>culis <emph type="italics"/>LM, lm<emph.end type="italics"/>occurrat in <emph type="italics"/>N<emph.end type="italics"/>ac <emph type="italics"/>n,<emph.end type="italics"/>& in planum <emph type="italics"/>QR<emph.end type="italics"/>demit­<lb/>tatur perpendiculum <emph type="italics"/>XT.<emph.end type="italics"/>Et particularum <emph type="italics"/>L<emph.end type="italics"/>& <emph type="italics"/>l<emph.end type="italics"/>vires con­<lb/>trariæ, ad Terram in contrarias partes rotandam, &longs;unt ut <lb/><emph type="italics"/>LMXMC<emph.end type="italics"/>& <emph type="italics"/>lmXmC,<emph.end type="italics"/>hoc e&longs;t, ut <emph type="italics"/>LNXMC+NMXMC<emph.end type="italics"/>& <lb/><emph type="italics"/>lnXmC-nmXmC,<emph.end type="italics"/>&longs;eu <emph type="italics"/>LNXMC+NMXMC<emph.end type="italics"/>& <emph type="italics"/>LNXmC<emph.end type="italics"/><pb xlink:href="039/01/463.jpg" pagenum="435"/>-<emph type="italics"/>NMXmC<emph.end type="italics"/>: & harum differentia <emph type="italics"/>LNXMm-NMX―MC+mC,<emph.end type="italics"/><lb/><arrow.to.target n="note464"/>e&longs;t vis particularum ambarum &longs;imul &longs;umptarum ad Terram <lb/>rotandam. </s> <s>Hujus differentiæ pars affirmativa <emph type="italics"/>LNXMm<emph.end type="italics"/>&longs;eu <lb/>2<emph type="italics"/>LNXNX,<emph.end type="italics"/>e&longs;t ad particularum duarum eju&longs;dem magnitudi­<lb/>nis in <emph type="italics"/>A<emph.end type="italics"/>con&longs;i&longs;tentium vim 2<emph type="italics"/>AHXHC,<emph.end type="italics"/>ut <emph type="italics"/>LXq<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="ACq.">ACque</expan><emph.end type="italics"/><lb/>Et pars negativa <emph type="italics"/>NMX―MC+mC<emph.end type="italics"/>&longs;eu 2<emph type="italics"/>XYXCY,<emph.end type="italics"/>ad parti­<lb/>cularum earundem in <emph type="italics"/>A<emph.end type="italics"/>con&longs;i&longs;tentium vim 2<emph type="italics"/>AHXHC,<emph.end type="italics"/>ut <lb/><emph type="italics"/>CXq<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="ACq.">ACque</expan><emph.end type="italics"/>Ac proinde partium differentia, id e&longs;t, par­<lb/>ticularum duarum <emph type="italics"/>L<emph.end type="italics"/>& <emph type="italics"/>l<emph.end type="italics"/>&longs;imul &longs;umptarum vis ad Terram rotan­<lb/>dam, e&longs;t ad vim particularum duarum ii&longs;dem æqualium & in loco <lb/><emph type="italics"/>A<emph.end type="italics"/>con&longs;i&longs;tentium, ad Terram itidem rotandam, ut <emph type="italics"/>LXq-CXq<emph.end type="italics"/><lb/>ad <emph type="italics"/><expan abbr="ACq.">ACque</expan><emph.end type="italics"/>Sed &longs;i circuli <emph type="italics"/>IK<emph.end type="italics"/>circumferentia <emph type="italics"/>IK<emph.end type="italics"/>dividatur in par­<lb/>ticulas innumeras æquales <emph type="italics"/>L,<emph.end type="italics"/>erunt omnes <emph type="italics"/>LXq<emph.end type="italics"/>ad totidem <emph type="italics"/>IXq<emph.end type="italics"/><lb/>ut 1 ad 2, (per Lem. </s> <s>I.) atque ad totidem <emph type="italics"/>ACq,<emph.end type="italics"/>ut <emph type="italics"/>IXq<emph.end type="italics"/>ad <lb/>2<emph type="italics"/>ACq<emph.end type="italics"/>; & totidem <emph type="italics"/>CXq<emph.end type="italics"/>ad totidem <emph type="italics"/>ACq<emph.end type="italics"/>ut 2<emph type="italics"/>CXq<emph.end type="italics"/>ad 2<emph type="italics"/><expan abbr="ACq.">ACque</expan><emph.end type="italics"/><lb/>Quare vires conjunctæ particularum omnium in circuitu circuli <lb/><emph type="italics"/>IK,<emph.end type="italics"/>&longs;unt ad vires conjunctas particularum totidem in loco <emph type="italics"/>A,<emph.end type="italics"/>ut <lb/><emph type="italics"/>IXq<emph.end type="italics"/>-2<emph type="italics"/>CXq<emph.end type="italics"/>ad 2<emph type="italics"/>ACq<emph.end type="italics"/>: & propterea (per Lem. </s> <s>I.) ad vires <lb/>conjunctas particularum totidem in circuitu circuli <emph type="italics"/>AE,<emph.end type="italics"/>ut <lb/><emph type="italics"/>IXq<emph.end type="italics"/>-2<emph type="italics"/>CXq<emph.end type="italics"/>ad <emph type="italics"/><expan abbr="ACq.">ACque</expan><emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note464"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Jam vero &longs;i Sphæræ diameter <emph type="italics"/>Pp<emph.end type="italics"/>dividatur in partes innume­<lb/>ras æquales, quibus in&longs;i&longs;tant circuli totidem <emph type="italics"/>IK<emph.end type="italics"/>; materia in peri­<lb/>metro circuli cuju&longs;que <emph type="italics"/>IK<emph.end type="italics"/>erit ut <emph type="italics"/>IXq<emph.end type="italics"/>: ideoque vis materiæ <lb/>illius ad Terram rotandam, erit ut <emph type="italics"/>IXq<emph.end type="italics"/>in <emph type="italics"/>IXq<emph.end type="italics"/>-2<emph type="italics"/><expan abbr="CXq.">CXque</expan><emph.end type="italics"/>Et <lb/>vis materiæ eju&longs;dem, &longs;i in circuli <emph type="italics"/>AE<emph.end type="italics"/>perimetro con&longs;i&longs;teret, e&longs;&longs;et <lb/>ut <emph type="italics"/>IXq<emph.end type="italics"/>in <emph type="italics"/><expan abbr="ACq.">ACque</expan><emph.end type="italics"/>Et propterea vis particularum omnium ma­<lb/>teriæ totius, extra globum in perimetris circulorum omnium con­<lb/>&longs;i&longs;tentis, e&longs;t ad vim particularum totidem in perimetro circuli <lb/>maximi <emph type="italics"/>AE<emph.end type="italics"/>con&longs;i&longs;tentis, ut omnia <emph type="italics"/>IXq<emph.end type="italics"/>in <emph type="italics"/>IXq<emph.end type="italics"/>-2<emph type="italics"/>CXq<emph.end type="italics"/>ad <lb/>totidem <emph type="italics"/>IXq<emph.end type="italics"/>in <emph type="italics"/>ACq,<emph.end type="italics"/>hoc e&longs;t, ut omnia <emph type="italics"/>ACq-CXq<emph.end type="italics"/>in <lb/><emph type="italics"/>ACq<emph.end type="italics"/>-3<emph type="italics"/>CXq<emph.end type="italics"/>ad totidem <emph type="italics"/>ACq-CXq<emph.end type="italics"/>in <emph type="italics"/>ACq,<emph.end type="italics"/>id e&longs;t, ut <lb/>omnia <emph type="italics"/>ACqq<emph.end type="italics"/>-4<emph type="italics"/>ACqXCXq<emph.end type="italics"/>+3<emph type="italics"/>CXqq<emph.end type="italics"/>ad totidem <emph type="italics"/>ACqq <lb/>-ACqXCXq,<emph.end type="italics"/>hoc e&longs;t, ut tota quantitas fluens cujus fluxio <lb/>e&longs;t <emph type="italics"/>ACqq<emph.end type="italics"/>-4<emph type="italics"/>ACqXCXq<emph.end type="italics"/>+3<emph type="italics"/>CXqq,<emph.end type="italics"/>ad totam quantitatem flu­<lb/>entem cujus fluxio e&longs;t <emph type="italics"/>ACqq-ACqXCXq<emph.end type="italics"/>; ac proinde per Me­<lb/>thodum Fluxionum, ut <emph type="italics"/>ACqqXCX<emph.end type="italics"/>-4/3<emph type="italics"/>ACqxCXcub<emph.end type="italics"/>+3/5<emph type="italics"/>CXqc<emph.end type="italics"/><lb/>ad <emph type="italics"/>ACqqXCX<emph.end type="italics"/>-1/3<emph type="italics"/>ACqXCXcub,<emph.end type="italics"/>id e&longs;t, &longs;i pro <emph type="italics"/>CX<emph.end type="italics"/>&longs;cribatur <lb/>tota <emph type="italics"/>Cp<emph.end type="italics"/>vel <emph type="italics"/>AC,<emph.end type="italics"/>ut (4/15)<emph type="italics"/>ACqc<emph.end type="italics"/>ad 2/3<emph type="italics"/>ACqc,<emph.end type="italics"/>hoc e&longs;t, ut duo ad <lb/>quinque. <emph type="italics"/><expan abbr="q.">que</expan> E. D.<emph.end type="italics"/><pb xlink:href="039/01/464.jpg" pagenum="436"/><arrow.to.target n="note465"/></s></p> <p type="margin"> <s><margin.target id="note465"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>LEMMA III.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Ii&longs;dem po&longs;itis: Dico tertio quod motus Terræ totius circum axem <lb/>jam ante de&longs;criptum, ex motibus particularum omnium compo&longs;i­<lb/>tus, erit ad motum annuli prædicti circum axem eundem, in ra­<lb/>tione quæ componitur ex ratione materiæ in Terra ad materiam <lb/>in annulo, & ratione trium quadratorum ex arcu quadrantali <lb/>circuli cuju&longs;cunque ad duo quadrata ex diametro; id e&longs;t, in ra­<lb/>tione materiæ ad materiam & numeri<emph.end type="italics"/>925275 <emph type="italics"/>ad numerum<emph.end type="italics"/><lb/>1000000. </s></p> <p type="main"> <s>E&longs;t enim motus Cylindri circum axem &longs;uum immotum revol­<lb/>ventis, ad motum Sphæræ in&longs;criptæ & &longs;imul revolventis, ut quæ­<lb/>libet quatuor æqualia quadrata ad tres ex circulis &longs;ibi in&longs;criptis: <lb/>& motus Cylindri ad motum annuli tenui&longs;&longs;imi, Sphæram & Cy­<lb/>lindrum ad communem eorum contactum ambientis, ut duplum <lb/>materiæ in Cylindro ad triplum materiæ in annulo; & annuli <lb/>motus i&longs;te circum axem Cylindri uniformiter continuatus, ad <lb/>eju&longs;dem motum uniformem circum diametrum propriam, eodem <lb/>tempore periodico factum, ut circumferentia circuli ad duplum <lb/>diametri. </s></p> <p type="main"> <s><emph type="center"/>HYPOTHESIS II.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si annulus prædictus Terra omni reliqua &longs;ublata, &longs;olus in Orbe <lb/>Terræ, motu annuo circa Solem ferretur, & interea circa axem <lb/>&longs;uum, ad planum Eclipticæ in angulo graduum<emph.end type="italics"/>23 1/2 <emph type="italics"/>inclinatum, <lb/>motu diurno revolveretur: idem foret motus Punctorum Æqui­<lb/>noctialium &longs;ive annulus i&longs;te fluidus e&longs;&longs;et, &longs;ive is ex materia rigida <lb/>& firma con&longs;taret.<emph.end type="italics"/></s></p><pb xlink:href="039/01/465.jpg" pagenum="437"/> <p type="main"> <s><emph type="center"/>PROPOSITIO XXXIX. PROBLEMA XX.<emph.end type="center"/><lb/><arrow.to.target n="note466"/></s></p> <p type="margin"> <s><margin.target id="note466"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire Præce&longs;&longs;ionem Æquinoctiorum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Motus mediocris horarius Nodorum Lunæ in Orbe circulari, <lb/>ubi Nodi &longs;unt in Quadraturis, erat 16″. </s> <s>35′. </s> <s>16<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>36<emph type="sup"/>v<emph.end type="sup"/>. </s> <s>& hujus <lb/>dimidium 8′. </s> <s>17′. </s> <s>38<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>18<emph type="sup"/>v<emph.end type="sup"/>. (ob rationes &longs;upra explicatas) e&longs;t mo­<lb/>tus medius horarius Nodorum in tali Orbe; fitque anno toto <lb/>&longs;idereo 20<emph type="sup"/>gr.<emph.end type="sup"/> 11′. </s> <s>46″. </s> <s>Quoniam igitur Nodi Lunæ in tali Orbe <lb/>conficerent annuatim 20<emph type="sup"/>gr.<emph.end type="sup"/> 11′. </s> <s>46″. </s> <s>in antecedentia; & &longs;i plures <lb/>e&longs;&longs;ent Lunæ motus Nodorum cuju&longs;que, per Corol. </s> <s>16. Prop. </s> <s><lb/>LXVI. Lib. </s> <s>I. forent ut tempora periodica; &longs;i Luna &longs;patio <lb/>diei &longs;iderei juxta &longs;uperficiem Terræ revolveretur, motus annuus <lb/>Nodorum foret ad 20<emph type="sup"/>gr.<emph.end type="sup"/> 11′. </s> <s>46″. </s> <s>ut dies &longs;idereus horarum 23. 56′. </s> <s><lb/>ad tempus periodicum Lunæ dierum 27. 7 hor. </s> <s>43′; id e&longs;t, ut <lb/>1436 ad 39343. Et par e&longs;t ratio Nodorum annuli Lunarum <lb/>Terram ambientis; &longs;ive Lunæ illæ &longs;e mutuo non contingant, &longs;ive <lb/>lique&longs;cant & in annulum continuum formentur, &longs;ive denique an­<lb/>nulus ille rige&longs;cat & inflexibilis reddatur. </s></p> <p type="main"> <s>Fingamus igitur quod annulus i&longs;te, quoad quantitatem materiæ, <lb/>æqualis &longs;it Terræ omni <emph type="italics"/>PapAPepE<emph.end type="italics"/>quæ globo <emph type="italics"/>Pape<emph.end type="italics"/>&longs;uperior <lb/>e&longs;t; (<emph type="italics"/>Vid. </s> <s>Fig. </s> <s>pag.<emph.end type="italics"/>434.) & quoniam globus i&longs;te e&longs;t ad Terram illam <lb/>&longs;uperiorem ut <emph type="italics"/>aCqu.<emph.end type="italics"/>ad <emph type="italics"/>ACqu.-aCqu.<emph.end type="italics"/>id e&longs;t (cum Terræ diameter <lb/>minor <emph type="italics"/>PC<emph.end type="italics"/>vel <emph type="italics"/>aC<emph.end type="italics"/>&longs;it ad diametrum majorem <emph type="italics"/>AC<emph.end type="italics"/>ut 229 ad 230,) <lb/>ut 52441 ad 459; &longs;i annulus i&longs;te Terram &longs;ecundum Æquatorem <lb/>cingeret & uterque &longs;imul circa diametrum annuli revolveretur, <lb/>motus annuli e&longs;&longs;et ad motum globi interioris (per hujus Lem. </s> <s>III.) <lb/>ut 459 ad 52441 & 1000000 ad 925275 conjunctim, hoc e&longs;t, <lb/>ut 4590 ad 485223; ideoque motus annuli e&longs;&longs;et ad &longs;ummam mo­<lb/>tuum annuli ac globi, ut 4590 ad 489813. Unde &longs;i annulus glo­<lb/>bo adhæreat, & motum &longs;uum quo ip&longs;ius Nodi &longs;eu puncta Æqui­<lb/>noctialia regrediuntur, cum globo communicet: motus qui re&longs;ta­<lb/>bit in annulo erit ad ip&longs;ius motum priorem, ut 4590 ad 489813; <lb/>& propterea motus punctorum Æquinoctialium diminuetur in <lb/>eadem ratione. </s> <s>Erit igitur motus annuus punctorum Æqui­<lb/>noctialium corporis ex annulo & globo compo&longs;iti, ad motum <pb xlink:href="039/01/466.jpg" pagenum="438"/><arrow.to.target n="note467"/>20<emph type="sup"/>gr.<emph.end type="sup"/> 11′. </s> <s>46″, ut 1436 ad 39343 & 4590 ad 489813 conjun­<lb/>ctim, id e&longs;t, ut 100 ad 292369. Vires autem quibus Nodi Lu­<lb/>narum (ut &longs;upra explicui) atque adeo quibus puncta Æquinoctia­<lb/>lia annuli regrediuntur (id e&longs;t vires 3<emph type="italics"/>IT, in Fig. </s> <s>pag.<emph.end type="italics"/>403 & 404.) <lb/>&longs;unt in &longs;ingulis particulis ut di&longs;tantiæ particularum à plano <emph type="italics"/>QR,<emph.end type="italics"/><lb/>& his viribus particulæ illæ planum fugiunt; & propterea (per <lb/>Lem. </s> <s>II.) &longs;i materia annuli per totam globi &longs;uperficiem, in mo­<lb/>rem figuræ <emph type="italics"/>PapAPepE,<emph.end type="italics"/>ad &longs;uperiorem illam Terræ partem <lb/>con&longs;tituendam &longs;pargeretur, vis & efficacia tota particularum om­<lb/>nium ad Terram circa quamvis Æquatoris diametrum rotandam, <lb/>atque adeo ad movenda puncta Æquinoctialia, evaderet minor <lb/>quam prius in ratione 2 ad 5. Ideoque annuus Æquinoctiorum <lb/>regre&longs;&longs;us jam e&longs;&longs;et ad 20<emph type="sup"/>gr.<emph.end type="sup"/> 11′. </s> <s>46″, ut 10 ad 73092: ac proinde <lb/>fieret 9″. </s> <s>56′. </s> <s>50<emph type="sup"/>iv<emph.end type="sup"/>. </s></p> <p type="margin"> <s><margin.target id="note467"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Cæterum hic motus, ob inclinationem plani Æquatoris ad pla­<lb/>num Eclipticæ, minuendus e&longs;t, idQ.E.I. ratione &longs;inus 91706 (qui <lb/>&longs;inus e&longs;t complementi graduum 23 1/2) ad Radium 100000. Qua <lb/>ratione motus i&longs;te jam fiet 9″. </s> <s>7′. </s> <s>20<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>Hæc e&longs;t annua Præce&longs;&longs;io <lb/>Æquinoctiorum a vi Solis oriunda. </s></p> <p type="main"> <s>Vis autem Lunæ ad Mare movendum erat ad vim Solis, ut <lb/>4,4815 ad 1 circiter. </s> <s>Et vis Lunæ ad Æquinoctia movenda, e&longs;t <lb/>ad vim Soiis in eadem proportione. </s> <s>Indeque prodit annua Æ­<lb/>quinoctiorum Præce&longs;&longs;io a vi Lunæ oriunda 40″. </s> <s>52′. </s> <s>52<emph type="sup"/>iv<emph.end type="sup"/>; ac tota <lb/>Præce&longs;&longs;io annua a vi utraque oriunda 50″. </s> <s>00′. </s> <s>12<emph type="sup"/>iv<emph.end type="sup"/>. </s> <s>Et hic mo­<lb/>tus cum Phænomenis congruit. </s> <s>Nam Præce&longs;&longs;io Æquinoctiorum <lb/>ex Ob&longs;ervationibus A&longs;tronomicis e&longs;t minutorum &longs;ecundorum plus <lb/>minus quinquaginta. </s></p> <p type="main"> <s>Si altitudo Terræ ad Æquatorem &longs;uperet altitudinem ejus ad <lb/>Polos, milliaribus pluribus quam 17 1/6, materia ejus rarior erit ad <lb/>circumferentiam quam ad centrum: & Præce&longs;&longs;io Æquinoctiorum <lb/>ob altitudinem illam augeri, ob raritatem diminui debet. </s></p> <p type="main"> <s>De&longs;crip&longs;imus jam Sy&longs;tema Solis, Terræ, Lunæ, & Planetarum: <lb/>&longs;upere&longs;t ut de Cometis nonnulla adjiciantur. </s></p><pb xlink:href="039/01/467.jpg" pagenum="439"/> <p type="main"> <s><emph type="center"/>LEMMA IV.<emph.end type="center"/><lb/><arrow.to.target n="note468"/></s></p> <p type="margin"> <s><margin.target id="note468"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Cometas e&longs;&longs;e Luna &longs;uperiores & in regione Planetarum ver&longs;ari.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Ut defectus Parallaxeos diurnæ extulit Cometas &longs;upra regiones <lb/>&longs;ublunares, &longs;ic ex Parallaxi annua convincitur eorum de&longs;cen&longs;us in <lb/>regiones Planetarum. </s> <s>Nam Cometæ qui progrediuntur &longs;ecun­<lb/>dum ordinem &longs;ignorum &longs;unt omnes, &longs;ub exitu apparitionis, aut <lb/>&longs;olito tardiores aut retrogradi, &longs;i Terra e&longs;t inter ip&longs;os & Solem; <lb/>at ju&longs;to celeriores &longs;i Terra vergit ad oppo&longs;itionem. </s> <s>Et e contra, <lb/>qui pergunt contra ordinem &longs;ignorum &longs;unt ju&longs;to celeriores in fine <lb/>apparitionis, &longs;i Terra ver&longs;atur inter ip&longs;os & Solem; & ju&longs;to tar­<lb/>diores vel retrogradi &longs;i Terra &longs;ita e&longs;t ad contrarias partes. </s> <s>Con­<lb/>tingit hoc maxime ex motu Terræ in vario ip&longs;ius &longs;itu, perinde ut <lb/>fit in Planetis, qui, pro motu Terræ vel con&longs;pirante vel contra­<lb/>rio, nunc retrogradi &longs;unt, nunc tardius progredi videntur, nunc <lb/>vero celerius. </s> <s>Si Terra pergit ad eandem partem cum Cometa, <lb/>& motu angulari circa Solem tanto celerius fertur, ut recta per <lb/>Terram & Cometam perpetuo ducta convergat ad partes ultra <lb/>Cometam, Cometa e Terra &longs;pectatus, ob motum &longs;uum tardiorem, <lb/>apparet e&longs;&longs;e retrogradus; &longs;in Terra tardius fertur, motus Cometæ, <lb/><figure id="id.039.01.467.1.jpg" xlink:href="039/01/467/1.jpg"/><lb/>(detracto motu Terræ) fit &longs;altem tardior. </s> <s>At &longs;i Terra pergit in <lb/>contrarias partes, Cometa exinde velocior apparet. </s> <s>Ex accele­<lb/>ratione autem vel retardatione vel motu retrogrado di&longs;tantia Co­<lb/>metæ in hunc modum colligitur. </s> <s>Sunto <emph type="italics"/>r QA, r QB, r QC<emph.end type="italics"/><lb/>ob&longs;ervatæ tres longitudines Cometæ, &longs;ub initio motus, &longs;itque <lb/><emph type="italics"/>r QF<emph.end type="italics"/>longitudo ultimo ob&longs;ervata, ubi Cometa videri de&longs;init. <pb xlink:href="039/01/468.jpg" pagenum="440"/><arrow.to.target n="note469"/>Agatur recta <emph type="italics"/>ABC,<emph.end type="italics"/>cujus partes <emph type="italics"/>AB, BC<emph.end type="italics"/>rectis <emph type="italics"/>QA<emph.end type="italics"/>& <emph type="italics"/>QB, <lb/>QB<emph.end type="italics"/>& <emph type="italics"/>QC<emph.end type="italics"/>interjectæ, &longs;int ad invicem ut tempora inter ob&longs;er­<lb/>vationes tres primas. </s> <s>Producatur <emph type="italics"/>AC<emph.end type="italics"/>ad <emph type="italics"/>G,<emph.end type="italics"/>ut &longs;it <emph type="italics"/>AG<emph.end type="italics"/>ad <emph type="italics"/>AB<emph.end type="italics"/><lb/>ut tempus inter ob&longs;ervationem primam & ultimam, ad tempus <lb/>inter ob&longs;ervationem primam & &longs;ecundam, & jungatur <emph type="italics"/>QG.<emph.end type="italics"/>Et <lb/>&longs;i Cometa moveretur uniformiter in linea recta, atque Terra vel <lb/>quie&longs;ceret, vel etiam in linea recta, uniformi cum motu, progre­<lb/>deretur; foret angulus <emph type="italics"/>r QG<emph.end type="italics"/>longitudo Cometæ tempore Ob­<lb/>&longs;ervationis ultimæ. </s> <s>Angulus igitur <emph type="italics"/>FQG,<emph.end type="italics"/>qui longitudinum dif­<lb/>ferentia e&longs;t, oritur ab inæqualitate motuum Cometæ ac Terræ. </s> <s><lb/>Hic autem angulus, &longs;i Terra & Cometa in contrarias partes mo­<lb/>ventur, additur angulo <emph type="italics"/>rQG,<emph.end type="italics"/>& &longs;ic motum apparentem Co­<lb/>metæ velociorem reddit: Sin Cometa pergit in ea&longs;dem partes <lb/>cum Terra, eidem &longs;ubducitur, motumque Cometæ vel tardiorem <lb/>reddit, vel forte retrogradum; uti modo expo&longs;ui. </s> <s>Oritur igitur <lb/>hic angulus præcipue ex motu Terræ, & idcirco pro parallaxi Co­<lb/>metæ merito habendus e&longs;t, neglecto videlicet ejus incremento vel <lb/>decremento nonnullo, quod a Cometæ motu inæquabili in Orbe <lb/>proprio oriri po&longs;&longs;it. </s> <s>Di&longs;tantia vero Cometæ ex hac parallaxi &longs;ic <lb/>colligitur. </s> <s>De&longs;ignet <emph type="italics"/>S<emph.end type="italics"/>Solem, <emph type="italics"/>acT<emph.end type="italics"/>Orbem magnum, <emph type="italics"/>a<emph.end type="italics"/>locum <lb/>Terræ in ob&longs;ervatione prima, <emph type="italics"/>c<emph.end type="italics"/>locum <lb/><figure id="id.039.01.468.1.jpg" xlink:href="039/01/468/1.jpg"/><lb/>Terræ in ob&longs;ervatione tertia, <emph type="italics"/>T<emph.end type="italics"/>locum <lb/>Terræ in ob&longs;ervatione ultima, & <emph type="italics"/>Tr<emph.end type="italics"/>li­<lb/>neam rectam ver&longs;us principium Arietis <lb/>ductam. </s> <s>Sumatur angulus <emph type="italics"/>rTV<emph.end type="italics"/>æqua­<lb/>lis angulo <emph type="italics"/>rQF,<emph.end type="italics"/>hoc e&longs;t, æqualis lon­<lb/>gitudini Cometæ ubi Terra ver&longs;atur in <lb/><emph type="italics"/>T.<emph.end type="italics"/>Jungatur <emph type="italics"/>ac,<emph.end type="italics"/>& producatur ea ad <emph type="italics"/>g,<emph.end type="italics"/><lb/>ut &longs;it <emph type="italics"/>ag<emph.end type="italics"/>ad <emph type="italics"/>ac<emph.end type="italics"/>ut <emph type="italics"/>AG<emph.end type="italics"/>ad <emph type="italics"/>AC,<emph.end type="italics"/>& <lb/>erit <emph type="italics"/>g<emph.end type="italics"/>locus quem Terra tempore ob&longs;er­<lb/>vationis ultimæ, motu in recta <emph type="italics"/>ac<emph.end type="italics"/>uNI­<lb/>formiter continuato, attingeret. </s> <s>Ideo­<lb/>que &longs;i ducatur <emph type="italics"/>g r<emph.end type="italics"/>ip&longs;i <emph type="italics"/>Tr<emph.end type="italics"/>parallela, <lb/>& capiatur angulus <emph type="italics"/>rgV<emph.end type="italics"/>angulo <emph type="italics"/>rQG<emph.end type="italics"/><lb/>æqualis, erit hic angulus <emph type="italics"/>rgV<emph.end type="italics"/>æqualis <lb/>longitudini Cometæ e loco <emph type="italics"/>g<emph.end type="italics"/>&longs;pectati; <lb/>& angulus <emph type="italics"/>TVg<emph.end type="italics"/>parallaxis erit, quæ oritur a tran&longs;latione Terræ <lb/>de loco <emph type="italics"/>g<emph.end type="italics"/>in locum <emph type="italics"/>T<emph.end type="italics"/>: ac proinde <emph type="italics"/>V<emph.end type="italics"/>locus erit Cometæ in plano <lb/>Eclipticæ. </s> <s>Hic autem locus <emph type="italics"/>V<emph.end type="italics"/>Orbe Jovis inferior e&longs;&longs;e &longs;olet. </s></p><pb xlink:href="039/01/469.jpg" pagenum="441"/> <p type="margin"> <s><margin.target id="note469"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Idem colligitur ex curvatura viæ Cometarum. </s> <s>Pergunt hæc <lb/><arrow.to.target n="note470"/>corpora propemodum in circulis maximis quamdiu moventur cele­<lb/>rius; at in fine cur&longs;us, ubi motus apparentis pars illa quæ à pa­<lb/>rallaxi oritur, majorem habet proportionem ad motum totum ap­<lb/>parentem, deflectere &longs;olent ab his circulis, & quoties Terra mo­<lb/>vetur in unam partem, abire in partem contrariam. </s> <s>Oritur hæc <lb/>deflexio maxime ex Parallaxi, propterea quod re&longs;pondet motui <lb/>Terræ; & in&longs;ignis ejus quantitas, meo computo, collocavit di&longs;pa­<lb/>rentes Cometas &longs;atis longe infra Jovem. </s> <s>Unde con&longs;equens e&longs;t <lb/>quod in Perigæis & Periheliis, ubi propius ad&longs;unt, de&longs;cendunt <lb/>&longs;æpius infra orbes Martis & inferiorum Planetarum. </s></p> <p type="margin"> <s><margin.target id="note470"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Confirmatur etiam propinquitas Cometarum ex luce capitum. </s> <s><lb/>Nam corporis cœle&longs;tis a Sole illu&longs;trati & in regiones longinquas <lb/>abeuntis, diminuitur &longs;plendor in quadruplicata ratione di&longs;tantiæ: <lb/>in duplicata ratione videlicet ob auctam corporis di&longs;tantiam a <lb/>Sole, & in alia duplicata ratione ob diminutam diametrum appa­<lb/>rentem. </s> <s>Unde &longs;i detur & lucis quantitas & apparens diameter <lb/>Cometæ, dabitur di&longs;tantia, dicendo quod di&longs;tantia &longs;it ad di&longs;tan­<lb/>tiam Planetæ, in ratione diametri ad diametrum directe & ratione <lb/>&longs;ubduplicata lucis ad lucem inver&longs;e. </s> <s>Sic minima capillitii Co­<lb/>metæ anni 1682 diameter, per Tubum opticum &longs;exdecim pedum <lb/>a <emph type="italics"/>Flam&longs;tedio<emph.end type="italics"/>ob&longs;ervata & Micrometro men&longs;urata, æquabat 2′. </s> <s>0″. </s> <s><lb/>Nucleus autem &longs;eu &longs;tella in medio capitis vix decimam partem la­<lb/>titudinis hujus occupabat, adeoque lata erat tantum 11″ vel 12″. </s> <s><lb/>Luce vero & claritate capitis &longs;uperabat caput Cometæ anni 1680, <lb/>&longs;tella&longs;que primæ vel &longs;ecundæ magnitudinis æmulabatur. </s> <s>Ponamus <lb/>Saturnum cum annulo &longs;uo qua&longs;i quadruplo lucidiorem fui&longs;&longs;e: & <lb/>quoniam lux annuli propemodum æquabat lucem globi inter­<lb/>medii, & diameter apparens globi &longs;it qua&longs;i 21″, adeoque lux <lb/>globi & annuli conjunctim æquaret lucem globi, cujus diameter <lb/>e&longs;&longs;et 30″: erit di&longs;tantia Cometæ ad di&longs;tantiam Saturni ut 1 ad √ 4 <lb/>inver&longs;e, & 12″ ad 30″ directe, id e&longs;t, ut 24 ad 30 &longs;eu 4 ad 5. <lb/>Rur&longs;us Cometa anni 1665 men&longs;e <emph type="italics"/>Aprili,<emph.end type="italics"/>ut author e&longs;t <emph type="italics"/>Hevelius,<emph.end type="italics"/><lb/>claritate &longs;ua pene Fixas omnes &longs;uperabat, quinetiam ip&longs;um Satur­<lb/>num, ratione coloris videlicet longe vividioris. </s> <s>Quippe lucidior <lb/>erat hic Cometa altero illo, qui in fine anni præcedentis apparu­<lb/>erat & cum &longs;tellis primæ magnitudinis conferebatur. </s> <s>Latitudo <lb/>capillitii erat qua&longs;i 6′, at nucleus cum Planetis ope Tubi optici <lb/>collatus, plane minor erat Jove, & nunc minor corpore interme-<pb xlink:href="039/01/470.jpg" pagenum="442"/><arrow.to.target n="note471"/>dio Saturni, nunc ip&longs;i æqualis judicabatur. </s> <s>Porro cum diameter <lb/>capillitii Cometarum raro &longs;uperet 8′ vel 12′, diameter vero nu­<lb/>clei &longs;eu &longs;tellæ centralis &longs;it qua&longs;i decima vel forte decima quinta <lb/>pars diametri capillitii, patet Stellas ha&longs;ce ut plurimum eju&longs;dem <lb/>e&longs;&longs;e apparentis magnitudinis cum Planetis. </s> <s>Unde cum lux earum <lb/>cum luce Saturni non raro conferri po&longs;&longs;it, eamque aliquando &longs;u­<lb/>peret; manife&longs;tum e&longs;t quod Cometæ omnes in Periheliis vel in­<lb/>fra Saturnum collocandi &longs;int, vel non longe &longs;upra. </s> <s>Errant igitur <lb/>toto cœlo qui Cometas in regionem Fixarum prope ablegant: qua <lb/>certe ratione non magis illu&longs;trari deberent a Sole no&longs;tro, quam <lb/>Planetæ, qui hic &longs;unt, illu&longs;trantur a Stellis fixis. </s></p> <p type="margin"> <s><margin.target id="note471"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Hæc di&longs;putavimus non con&longs;iderando ob&longs;curationem Cometa­<lb/>rum per &longs;umum illum maxime copio&longs;um & cra&longs;&longs;um, quo caput <lb/>circundatur, qua&longs;i per nubem obtu&longs;e &longs;emper lucens. </s> <s>Nam quan­<lb/>to ob&longs;curius redditur corpus per hunc fumum, tanto propius ad <lb/>Solem accedat nece&longs;&longs;e e&longs;t, ut copia lucis a &longs;e reflexa Planetas æmu­<lb/>letur. </s> <s>Inde veri&longs;imile fit Cometas longe infra &longs;phæram Saturni <lb/>de&longs;cendere, uti ex Parallaxi probavimus. </s> <s>Idem vero quam ma­<lb/>xime confirmatur ex Caudis. </s> <s>Hæ vel ex reflexione fumi &longs;par&longs;i <lb/>per Æthera, vel ex luce capitis oriuntur. </s> <s>Priore ca&longs;u minuenda <lb/>e&longs;t di&longs;tantia Cometarum, ne fumus a capite &longs;emper ortus per <lb/>&longs;patia nimis ampla incredibili cum velocitate & expan&longs;ione pro­<lb/>pagetur. </s> <s>In po&longs;teriore referenda e&longs;t lux omnis tam caudæ quam <lb/>capillitii ad nucleum capitis. </s> <s>Igitur &longs;i concipiamus lucem hanc <lb/>omnem congregari & intra di&longs;cum nuclei coarctari, nucleus ille <lb/>jam certe, quoties caudam maximam & fulgenti&longs;&longs;imam emittit, <lb/>Jovem ip&longs;um &longs;plendore &longs;uo multum &longs;uperabit. </s> <s>Minore igitur <lb/>cum diametro apparente plus lucis emittens, multo magis illu&longs;tra­<lb/>bitur a Sole, adeoque erit Soli multo propior. </s> <s>Quinetiam capita <lb/>&longs;ub Sole delite&longs;centia, & caudas cum maximas tum fulgenti&longs;&longs;imas <lb/>in&longs;tar trabium ignitarum nonnunquam emittentia, eodem argu­<lb/>mento infra orbem Veneris collocari debent. </s> <s>Nam lux illa omnis <lb/>&longs;i in &longs;tellam congregari &longs;upponatur, ip&longs;am Venerem ne dicam Ve­<lb/>neres plures conjunctas quandoque &longs;uperaret. </s></p> <p type="main"> <s>Idem denique colligitur ex luce capitum cre&longs;cente in rece&longs;&longs;u <lb/>Cometarum a Terra Solem ver&longs;us, ac decre&longs;cente in eorum rece&longs;&longs;u <lb/>a Sole ver&longs;us Terram. </s> <s>Sic enim Cometa po&longs;terior Anni 1665 <lb/>(ob&longs;ervante <emph type="italics"/>Hevelio,<emph.end type="italics"/>) ex quo con&longs;pici cœpit, remittebat &longs;emper <pb xlink:href="039/01/471.jpg" pagenum="443"/>de motu &longs;uo apparente, adeoque præterierat Perigæum; Splen­<lb/><arrow.to.target n="note472"/>dor vero capitis nihilominus indies cre&longs;cebat, u&longs;Q.E.D.m Cometa <lb/>radiis Solaribus obtectus de&longs;iit apparere. </s> <s>Cometa Anni 1683, <lb/>ob&longs;ervante eodem <emph type="italics"/>Hevelio,<emph.end type="italics"/>in fine Men&longs;is <emph type="italics"/>Julii<emph.end type="italics"/>ubi primum con­<lb/>&longs;pectus e&longs;t, tardi&longs;&longs;ime movebatur, minuta prima 40 vel 45 circi­<lb/>ter &longs;ingulis diebus in Orbe &longs;uo conficiens. </s> <s>Ex eo tempore motus <lb/>ejus diurnus perpetuo augebatur u&longs;que ad <emph type="italics"/>Sept.<emph.end type="italics"/>4. quando eva&longs;it <lb/>graduum qua&longs;i quinque. </s> <s>Igitur toto hoc tempore Cometa ad <lb/>Terram appropinquabat. </s> <s>Id quod etiam ex diametro capitis <lb/>Micrometro men&longs;urata colligitur: quippe quam <emph type="italics"/>Hevelius<emph.end type="italics"/>reperit <lb/><emph type="italics"/>Aug.<emph.end type="italics"/>6. e&longs;&longs;e tantum 6′. </s> <s>5″ inclu&longs;a coma, at <emph type="italics"/>Sept.<emph.end type="italics"/>2. e&longs;&longs;e 9′. </s> <s>7″. </s> <s><lb/>Caput igitur initio longe minus apparuit quam in &longs;ine motus, at <lb/>initio tamen in vicinia Solis longe lucidius extitit quam circa <lb/>finem, ut refert idem <emph type="italics"/>Hevelius.<emph.end type="italics"/>Proinde toto hoc tempore, ob <lb/>rece&longs;&longs;um ip&longs;ius a Sole, quoad lumen decrevit, non ob&longs;tante ac­<lb/>ce&longs;&longs;u ad Terram. </s> <s>Cometa Anni 1618 circa medium Men&longs;is <emph type="italics"/>De­<lb/>cembris,<emph.end type="italics"/>& i&longs;te Anni 1680 circa finem eju&longs;dem Men&longs;is, celerrime <lb/>movebantur, adeoque tunc erant in Perigæis. </s> <s>Verum &longs;plendor <lb/>maximus capitum contigit ante duas fere &longs;eptimanas, ubi modo <lb/>exierant de radiis Solaribus; & &longs;plendor maximus caudarum <lb/>paulo ante, in majore vicinitate Solis. </s> <s>Caput Cometæ prioris, <lb/>juxta ob&longs;ervationes <emph type="italics"/>Cy&longs;ati, Decemb.<emph.end type="italics"/>1. majus videbatur &longs;tellis pri­<lb/>mæ magnitudinis, & <emph type="italics"/>Decemb.<emph.end type="italics"/>16. (jam in Perigæo exi&longs;tens) mag­<lb/>nitudine parum, &longs;plendore &longs;eu claritate luminis plurimum defe­<lb/>cerat. <emph type="italics"/>Jan.<emph.end type="italics"/>7. <emph type="italics"/>Keplerus<emph.end type="italics"/>de capite incertus finem fecit ob&longs;ervandi. </s> <s><lb/>Die 12 men&longs;is <emph type="italics"/>Decemb.<emph.end type="italics"/>con&longs;pectum & a <emph type="italics"/>Flam&longs;tedio<emph.end type="italics"/>ob&longs;ervatum <lb/>e&longs;t caput Cometæ po&longs;terioris, in di&longs;tantia novem graduum a Sole; <lb/>id quod &longs;tellæ tertiæ magnitudinis vix conce&longs;&longs;um fui&longs;&longs;et. <emph type="italics"/>Decemb.<emph.end type="italics"/><lb/>15. & 17 apparuit idem ut &longs;tella tertiæ magnitudinis, diminutum <lb/>utique &longs;plendore Nubium juxta Solem occidentem. <emph type="italics"/>Decemb.<emph.end type="italics"/>26. <lb/>veloci&longs;&longs;ime motus, inque Perigæo propemodum exi&longs;tens, cedebat <lb/>ori Pega&longs;i, Stellæ tertiæ magnitudinis. <emph type="italics"/>Jan.<emph.end type="italics"/>3. apparebat ut Stella <lb/>quartæ, <emph type="italics"/>Jan.<emph.end type="italics"/>9. ut Stella quintæ, <emph type="italics"/>Jan.<emph.end type="italics"/>13. ob &longs;plendorem Lunæ <lb/>cre&longs;centis di&longs;paruit. <emph type="italics"/>Jan.<emph.end type="italics"/>25. vix æquabat Stellas magnitudinis <lb/>&longs;eptimæ. </s> <s>Si &longs;umantur æqualia a Perigæo hinc inde tempora, ca­<lb/>pita quæ temporibus illis in longinquis regionibus po&longs;ita, ob <lb/>æquales a Terra di&longs;tantias, æqualiter lucere debui&longs;&longs;ent, in plaga <lb/>Solis maxime &longs;plenduere, ex altera Perigæi parte evanuere. </s> <s>Igi­<lb/>tur ex magna lucis in utroque &longs;itu differentia, concluditur magna <lb/>Solis & Cometæ vicinitas in &longs;itu priore. </s> <s>Nam lux Cometarum <pb xlink:href="039/01/472.jpg" pagenum="444"/><arrow.to.target n="note473"/>regularis e&longs;&longs;e &longs;olet, & maxima apparere ubi capita veloci&longs;&longs;ime <lb/>moventur, atque adeo &longs;unt in Perigæis; ni&longs;i quatenus ea major <lb/>e&longs;t in vicinia Solis. </s></p> <p type="margin"> <s><margin.target id="note472"/>LIBER <lb/>TERTIUS.</s></p> <p type="margin"> <s><margin.target id="note473"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Splendent igitur Cometæ luce Solis a &longs;e reflexa. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Ex dictis etiam intelligitur cur Cometæ tantopere fre­<lb/>quentant regionem Solis. </s> <s>Si cernerentur in regionibus longe <lb/>ultra Saturnum, deberent &longs;æpius apparere in partibus Soli oppo­<lb/>&longs;itis. </s> <s>Forent enim Terræ viciniores qui in his partibus ver&longs;a­<lb/>rentur, & Sol interpo&longs;itus ob&longs;curaret cæteros. </s> <s>Verum percur­<lb/>rendo hi&longs;torias Cometarum, reperi quod quadruplo vel quintuplo <lb/>plures detecti &longs;unt in Hemi&longs;phærio Solem ver&longs;us, quam in He­<lb/>mi&longs;phærio oppo&longs;ito, præter alios procul dubio non paucos quos <lb/>lux Solaris obtexit. </s> <s>Nimirum in de&longs;cen&longs;u ad regiones no&longs;tras <lb/>neque caudas emittunt, neque adeo illu&longs;trantur a Sole, ut nudis <lb/>oculis &longs;e prius detegendos exhibeant, quam &longs;int ip&longs;o Jove pro­<lb/>piores. </s> <s>Spatii autem tantillo intervallo circa Solem de&longs;cripti <lb/>pars longe major &longs;ita e&longs;t a latere Terræ quod Solem re&longs;picit; <lb/>inque parte illa majore Cometæ, Soli ut plurimum viciniores, <lb/>magis illuminari &longs;olent. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Hinc etiam manife&longs;tum e&longs;t, quod Cœli re&longs;i&longs;tentia de­<lb/>&longs;tituuntur. </s> <s>Nam Cometæ vias obliquas & nonnunquam cur&longs;ui <lb/>Planetarum contrarias &longs;ecuti, moventur omnifariam liberrime, & <lb/>motus &longs;uos etiam contra cur&longs;um Planetarum, diuti&longs;&longs;ime con&longs;er­<lb/>vant. </s> <s>Fallor ni genus Planetarum &longs;int, & motu perpetuo in or­<lb/>bem redeant. </s> <s>Nam quod Scriptores aliqui Meteora e&longs;&longs;e volunt, <lb/>argumentum a capitum perpetuis mutationibus ducentes, funda­<lb/>mento carere videtur. </s> <s>Capita Cometarum Atmo&longs;phæris ingen­<lb/>tibus cinguntur; & Atmo&longs;phæræ inferne den&longs;iores e&longs;&longs;e debent. </s> <s><lb/>Unde nubes &longs;unt, non ip&longs;a Cometarum corpora, in quibus muta­<lb/>tiones illæ vi&longs;untur. </s> <s>Sic Terra &longs;i e Planetis &longs;pectaretur, luce nu­<lb/>bium &longs;uarum proculdubio &longs;plenderet, & corpus firmum &longs;ub nu­<lb/>bibus prope delite&longs;ceret. </s> <s>Sic cingula Jovis in nubibus Planetæ <lb/>illius formata e&longs;t, quæ &longs;itum mutant inter &longs;e, & firmum Jovis <lb/>corpus per nubes illas difficilius cernitur. </s> <s>Et multo magis cor­<lb/>pora Cometarum &longs;ub Atmo&longs;phæris & profundioribus & cra&longs;&longs;iori­<lb/>bus ab&longs;condi debent. </s></p><pb xlink:href="039/01/473.jpg" pagenum="445"/> <p type="main"> <s><emph type="center"/>PROPOSITIO XL. THEOREMA XX.<emph.end type="center"/><lb/><arrow.to.target n="note474"/></s></p> <p type="margin"> <s><margin.target id="note474"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Cometas in Sectionibus Conicis umbilicos in centro Solis haben­<lb/>tibus moveri, & radiis ad Solem ductis areas temporibus pro­<lb/>portionales de&longs;cribere.<emph.end type="italics"/></s></p> <p type="main"> <s>Patet per Corol. </s> <s>1. Propo&longs;. </s> <s>XIII. </s> <s>Libri primi, collatum cum <lb/>Prop. </s> <s>VIII, XII & XIII. </s> <s>Libri tertii. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>1. Hinc &longs;i Cometæ in orbem redeunt: Orbes erunt Ellip­<lb/>&longs;es, & tempora periodica erunt ad tempora periodica Planetarum <lb/>in axium principalium ratione &longs;e&longs;quiplicata. </s> <s>Ideoque Cometæ <lb/>maxima ex parte &longs;upra Planetas ver&longs;antes, & eo nomine Orbes <lb/>axibus majoribus de&longs;cribentes, tardius revolventur. </s> <s>Ut &longs;i axis Or­<lb/>bis Cometæ &longs;it quadruplo major axe Orbis Saturni, tempus revo­<lb/>lutionis Cometæ erit ad tempus revolutionis Saturni, id e&longs;t, ad <lb/>annos 30, ut 4 √ 4 (&longs;eu 8) ad 1, ideoque erit annorum 240. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>2. Orbes autem erunt Parabolis adeo finitimi, ut eorum <lb/>vice Parabolæ, ab&longs;que erroribus &longs;en&longs;ibilibus, adhiberi po&longs;&longs;int. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>3. Et propterea, per Corol. </s> <s>7. Prop. </s> <s>XVI. Lib. </s> <s>I. velo­<lb/>citas Cometæ omnis, erit &longs;emper ad velocitatem Planetæ cuju&longs;vis <lb/>circa Solem in circulo revolventis, in &longs;ubduplicata ratione duplæ <lb/>di&longs;tantiæ Planetæ a centro Solis, ad di&longs;tantiam Cometæ a centro <lb/>Solis quamproxime. </s> <s>Ponamus radium Orbis magni, &longs;eu Ellip&longs;eos <lb/>in qua Terra revolvitur &longs;emidiametrum maximam, e&longs;&longs;e partium <lb/>100000000: & Terra motu &longs;uo diurno mediocri de&longs;cribet partes <lb/>1720212, & motu horario partes 71675 1/2. Ideoque Cometa in <lb/>eadem Telluris a Sole di&longs;tantia mediocri, ea cum velocitate quæ <lb/>&longs;it ad velocitatem Telluris ut √ 2 ad 1, de&longs;cribet motu &longs;uo diurno <lb/>partes 2432747, & motu horario partes 10136. In majoribus <lb/>autem vel minoribus di&longs;tantiis, motus tum diurnus tum horarius <lb/>erit ad hunc motum diurnum & horarium in &longs;ubduplicata ratione <lb/>di&longs;tantiarum reciproce, ideoQ.E.D.tur. </s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>4. Unde &longs;i Latus rectum Parabolæ quadruplo majus &longs;it <lb/>radio Orbis magni, & quadratum radii illius ponatur e&longs;&longs;e partium <lb/>100000000: area quam Cometa radio ad Solem ducto &longs;ingulis die­<lb/>bus de&longs;cribit, erit partium 1216373 1/4, & &longs;ingulis horis area illa <lb/>erit partium 50682 1/4. Sin latus rectum majus &longs;it vel minus in ra­<lb/>tione quavis, erit area diurna & horaria major vel minor in ea­<lb/>dem ratione &longs;ubduplicata. </s></p><pb xlink:href="039/01/474.jpg" pagenum="446"/> <p type="main"> <s><arrow.to.target n="note475"/></s></p> <p type="margin"> <s><margin.target id="note475"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>LEMMA V.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Invenire lineam curvam generis Parabolici, quæ per data <lb/>quotcunque puncta tran&longs;ibit.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Sunto puncta illa <emph type="italics"/>A, B, C, D, E, F,<emph.end type="italics"/>&c. </s> <s>& ab ii&longs;dem ad rectam <lb/>quamvis po&longs;itione datam <emph type="italics"/>HN<emph.end type="italics"/>demitte perpendicula quotcunque <lb/><emph type="italics"/>AH, BI, CK, DL, EM, FN.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>1. Si punctorum <emph type="italics"/>H, I, K, L, M, N<emph.end type="italics"/>æqualia &longs;unt inter­<lb/>valla <emph type="italics"/>HI, IK, KL,<emph.end type="italics"/>&c. </s> <s>collige perpendiculorum <emph type="italics"/>AH, BI, <lb/>CK,<emph.end type="italics"/>&c. </s> <s>differentias primas <emph type="italics"/>b,<emph.end type="italics"/>2<emph type="italics"/>b,<emph.end type="italics"/>3<emph type="italics"/>b,<emph.end type="italics"/>4<emph type="italics"/>b,<emph.end type="italics"/>5<emph type="italics"/>b,<emph.end type="italics"/>&c. </s> <s>&longs;ecundas <emph type="italics"/>c,<emph.end type="italics"/>2<emph type="italics"/>c,<emph.end type="italics"/><lb/>3<emph type="italics"/>c,<emph.end type="italics"/>4<emph type="italics"/>c,<emph.end type="italics"/>&c. </s> <s>tertias <emph type="italics"/>d,<emph.end type="italics"/>2<emph type="italics"/>d,<emph.end type="italics"/>3<emph type="italics"/>d,<emph.end type="italics"/>&c. </s> <s>id e&longs;t, ita ut &longs;it <emph type="italics"/>AH-BI=b, <lb/>BI-CK=2b, CK-DL=3b, DL+EM=4b,-EM+FN=5b,<emph.end type="italics"/><lb/><figure id="id.039.01.474.1.jpg" xlink:href="039/01/474/1.jpg"/><lb/>&c. </s> <s>dein <emph type="italics"/>b-2b=c,<emph.end type="italics"/>&c. <lb/></s> <s>& &longs;ic pergatur ad diffe­<lb/>rentiam ultimam quæ hic <lb/>e&longs;t <emph type="italics"/>f.<emph.end type="italics"/>Deinde erecta qua­<lb/>cunque perpendiculari <lb/><emph type="italics"/>RS,<emph.end type="italics"/>quæ fuerit ordina­<lb/>tim applicata ad curvam <lb/>quæ&longs;itam: ut inveniatur <lb/>hujus longitudo, pone <lb/>intervalla <emph type="italics"/>HI, IK, KL, <lb/>LM,<emph.end type="italics"/>&c. </s> <s>unitates e&longs;&longs;e, <lb/>& dic <emph type="italics"/>AH=a,-HS=p, <lb/>1/2p<emph.end type="italics"/>in -<emph type="italics"/>IS=q, 1/3q<emph.end type="italics"/>in <lb/>+<emph type="italics"/>SK=r, 1/4r<emph.end type="italics"/>in +<emph type="italics"/>SL=s, 1/5s<emph.end type="italics"/>in +<emph type="italics"/>SM=t<emph.end type="italics"/>; pergendo videlicet <lb/>ad u&longs;que penultimum perpendiculum <emph type="italics"/>ME,<emph.end type="italics"/>& præponendo &longs;igna <lb/>negativa terminis <emph type="italics"/>HS, IS,<emph.end type="italics"/>&c. </s> <s>qui jacent ad partes puncti <emph type="italics"/>S<emph.end type="italics"/>ver­<lb/>&longs;us <emph type="italics"/>A,<emph.end type="italics"/>& &longs;igna affirmativa terminis <emph type="italics"/>SK, SL,<emph.end type="italics"/>&c. </s> <s>qui jacent <lb/>ad alteras partes puncti <emph type="italics"/>S.<emph.end type="italics"/>Et &longs;ignis probe ob&longs;ervatis, erit <lb/><emph type="italics"/>RS=a+bp+cq+dr+es+ft,<emph.end type="italics"/>&c. </s></p> <p type="main"> <s><emph type="italics"/>Ca&longs;.<emph.end type="italics"/>2. Quod &longs;i punctorum <emph type="italics"/>H, I, K, L,<emph.end type="italics"/>&c. </s> <s>inæqualia &longs;int inter­<lb/>valla <emph type="italics"/>HI, IK,<emph.end type="italics"/>&c. </s> <s>collige perpendiculorum <emph type="italics"/>AH, BI, CK,<emph.end type="italics"/>&c. </s> <s><lb/>differentias primas per intervalla perpendiculorum divi&longs;as <emph type="italics"/>b,<emph.end type="italics"/>2<emph type="italics"/>b,<emph.end type="italics"/><lb/>3<emph type="italics"/>b,<emph.end type="italics"/>4<emph type="italics"/>b,<emph.end type="italics"/>5<emph type="italics"/>b<emph.end type="italics"/>; &longs;ecundas per intervalla bina divi&longs;as <emph type="italics"/>c,<emph.end type="italics"/>2<emph type="italics"/>c,<emph.end type="italics"/>3<emph type="italics"/>c,<emph.end type="italics"/>4<emph type="italics"/>c,<emph.end type="italics"/>&c. </s> <s><lb/>tertias per intervalla terna divi&longs;as <emph type="italics"/>d,<emph.end type="italics"/>2<emph type="italics"/>d,<emph.end type="italics"/>3<emph type="italics"/>d,<emph.end type="italics"/>&c. </s> <s>quartas per <pb xlink:href="039/01/475.jpg" pagenum="447"/>intervalla quaterna divi&longs;as <emph type="italics"/>e,<emph.end type="italics"/>2<emph type="italics"/>e,<emph.end type="italics"/>&c. </s> <s>& &longs;ic deinceps; id e&longs;t, ita <lb/><arrow.to.target n="note476"/>ut &longs;it <emph type="italics"/>b=(AH-BI/HI), 2b=(BI-CK/IK), 3b=(CK-DL/KL),<emph.end type="italics"/>&c. </s> <s>dein <lb/><emph type="italics"/>c=(b-2b/HK), 2c=(2b-3b/IL), 3c=(3b-4b/KM),<emph.end type="italics"/>&c. </s> <s>Po&longs;tea <emph type="italics"/>d=(c-2c/HL), <lb/>2d=(2c-3c/IM),<emph.end type="italics"/>&c. </s> <s>Inventis differentiis, dic <emph type="italics"/>AH=a, -HS=p, <lb/>p<emph.end type="italics"/>in -<emph type="italics"/>IS=q, q<emph.end type="italics"/>in +<emph type="italics"/>SK=r, r<emph.end type="italics"/>in +<emph type="italics"/>SL=s, s<emph.end type="italics"/>in +<emph type="italics"/>SM=t<emph.end type="italics"/>; <lb/>pergendo &longs;cilicet ad u&longs;que perpendiculum penultimum <emph type="italics"/>ME,<emph.end type="italics"/>& erit <lb/>ordinatim applicata <emph type="italics"/>RS=a+bp+cq+dr+es+ft,<emph.end type="italics"/>&c. </s></p> <p type="margin"> <s><margin.target id="note476"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Hinc areæ curvarum omnium inveniri po&longs;&longs;unt quampro­<lb/>xime. </s> <s>Nam &longs;i curvæ cuju&longs;vis quadrandæ inveniantur puncta ali­<lb/>quot, & Parabola per eadem duci intelligatur: erit area Parabolæ <lb/>hujus eadem quam proxime cum area curvæ illius quadrandæ. </s> <s><lb/>Pote&longs;t autem Parabola, per Methodos noti&longs;&longs;imas, &longs;emper quadrari <lb/>Geometrice. </s></p> <p type="main"> <s><emph type="center"/>LEMMA VI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Ex ob&longs;ervatis aliquot locis Cometæ invenive locum ejus ad <lb/>tempus quodvis intermedium datum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>De&longs;ignent <emph type="italics"/>HI, IK, KL, LM<emph.end type="italics"/>tempora inter ob&longs;ervationes, <lb/><emph type="italics"/>(in Fig. </s> <s>præced.) HA, IB, KC, LD, ME<emph.end type="italics"/>ob&longs;ervatas quinque <lb/>longitudines Cometæ, <emph type="italics"/>HS<emph.end type="italics"/>tempus datum inter ob&longs;ervationem pri­<lb/>mam & longitudinem quæ&longs;itam. </s> <s>Et &longs;i per puncta <emph type="italics"/>A, B, C, D, E<emph.end type="italics"/><lb/>duci intelligatur curva regularis <emph type="italics"/>ABCDE<emph.end type="italics"/>; & per Lemma &longs;upe­<lb/>rius inveniatur ejus ordinatim applicata <emph type="italics"/>RS,<emph.end type="italics"/>erit <emph type="italics"/>RS<emph.end type="italics"/>longitudo <lb/>quæ&longs;ita. </s></p> <p type="main"> <s>Eadem methodo ex ob&longs;ervatis quinque latitudinibus invenitur <lb/>latitudo ad tempus datum. </s></p> <p type="main"> <s>Si longitudinum ob&longs;ervatarum parvæ &longs;int differentiæ, puta gra­<lb/>duum tantum 4 vel 5; &longs;uffecerint ob&longs;ervationes tres vel quatuor <lb/>ad inveniendam longitudinem & latitudinem novam. </s> <s>Sin majores <lb/>&longs;int differentiæ, puta graduum 10 vel 20, debebunt ob&longs;ervationes <lb/>quinque adhiberi. <pb xlink:href="039/01/476.jpg" pagenum="448"/><arrow.to.target n="note477"/></s></p> <p type="margin"> <s><margin.target id="note477"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>LEMMA VII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Per datum punctum<emph.end type="italics"/>P <emph type="italics"/>ducere rectam lineam<emph.end type="italics"/>BC, <emph type="italics"/>cujus partes<emph.end type="italics"/><lb/>PB, PC, <emph type="italics"/>rectis duabus po&longs;itione datis<emph.end type="italics"/>AB, AC <emph type="italics"/>ab&longs;ci&longs;&longs;æ, da­<lb/>tam habeant rationem ad invicem.<emph.end type="italics"/></s></p><figure id="id.039.01.476.1.jpg" xlink:href="039/01/476/1.jpg"/> <p type="main"> <s>A puncto illo <emph type="italics"/>P<emph.end type="italics"/>ad rectarum al­<lb/>terutram <emph type="italics"/>AB<emph.end type="italics"/>ducatur recta quævis <lb/><emph type="italics"/>PD,<emph.end type="italics"/>& producatur eadem ver&longs;us <lb/>rectam alteram <emph type="italics"/>AC<emph.end type="italics"/>u&longs;que ad <emph type="italics"/>E,<emph.end type="italics"/>ut <lb/>&longs;it <emph type="italics"/>PE<emph.end type="italics"/>ad <emph type="italics"/>PD<emph.end type="italics"/>in data illa ratione. </s> <s><lb/>Ip&longs;i <emph type="italics"/>AD<emph.end type="italics"/>parallela &longs;it <emph type="italics"/>EC<emph.end type="italics"/>; & &longs;i <lb/>agatur <emph type="italics"/>CPB,<emph.end type="italics"/>erit <emph type="italics"/>PC<emph.end type="italics"/>ad <emph type="italics"/>PB<emph.end type="italics"/>ut <lb/><emph type="italics"/>PE<emph.end type="italics"/>ad <emph type="italics"/>PD. q.E.F.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>LEMMA VIII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Sit<emph.end type="italics"/>ABC <emph type="italics"/>Parabola umbilicum habens<emph.end type="italics"/>S. <emph type="italics"/>Chorda<emph.end type="italics"/>AC <emph type="italics"/>bi&longs;ecta <lb/>in<emph.end type="italics"/>I <emph type="italics"/>ab&longs;cindatur &longs;egmentum<emph.end type="italics"/>ABCI, <emph type="italics"/>cujus diameter &longs;it<emph.end type="italics"/>I <foreign lang="greek">m</foreign> <emph type="italics"/>& <lb/>vertex<emph.end type="italics"/><foreign lang="greek">m</foreign>. <emph type="italics"/>In<emph.end type="italics"/>I <foreign lang="greek">m</foreign> <emph type="italics"/>producta capiatur<emph.end type="italics"/><foreign lang="greek">m</foreign> O <emph type="italics"/>æqualis dimidio ip&longs;ius<emph.end type="italics"/><lb/><figure id="id.039.01.476.2.jpg" xlink:href="039/01/476/2.jpg"/><lb/>I <foreign lang="greek">m</foreign>. <emph type="italics"/>Jungatur<emph.end type="italics"/>OS, <emph type="italics"/>& producatur ea ad <foreign lang="greek">c</foreign>, ut &longs;it<emph.end type="italics"/>S <foreign lang="greek">c</foreign> <emph type="italics"/>æqualis<emph.end type="italics"/><lb/>2SO. <emph type="italics"/>Et &longs;i Cometa<emph.end type="italics"/>B <emph type="italics"/>moveatur in arcu<emph.end type="italics"/>CBA, <emph type="italics"/>& agatur<emph.end type="italics"/><lb/><foreign lang="greek">c</foreign> B <emph type="italics"/>&longs;ecans<emph.end type="italics"/>AC <emph type="italics"/>in<emph.end type="italics"/>E: <emph type="italics"/>dico quod punctum<emph.end type="italics"/>E <emph type="italics"/>ab&longs;cindet de chordo<emph.end type="italics"/><lb/>AC <emph type="italics"/>&longs;egmentum<emph.end type="italics"/>AE <emph type="italics"/>tempori proportionale quamproxime.<emph.end type="italics"/></s></p><pb xlink:href="039/01/477.jpg" pagenum="449"/> <p type="main"> <s>Jungatur enim <emph type="italics"/>EO<emph.end type="italics"/>&longs;ecans arcum Parabolicum <emph type="italics"/>ABC<emph.end type="italics"/>in <emph type="italics"/>Y,<emph.end type="italics"/>& aga­<lb/><arrow.to.target n="note478"/>tur <foreign lang="greek">m</foreign><emph type="italics"/>X<emph.end type="italics"/>quæ tangat eundem arcum in vertice <foreign lang="greek">m</foreign> & actæ <emph type="italics"/>EO<emph.end type="italics"/>occur­<lb/>rat in <emph type="italics"/>X<emph.end type="italics"/>; & erit area curvilinea <emph type="italics"/>AEX<foreign lang="greek">m</foreign>A<emph.end type="italics"/>ad aream curvilineam <lb/><emph type="italics"/>ACY<foreign lang="greek">m</foreign>A<emph.end type="italics"/>ut <emph type="italics"/>AE<emph.end type="italics"/>ad <emph type="italics"/>AC.<emph.end type="italics"/>Ideoque cum triangulum <emph type="italics"/>ASE<emph.end type="italics"/>&longs;it <lb/>ad triangulum <emph type="italics"/>ASC<emph.end type="italics"/>in eadem ratione, erit area tota <emph type="italics"/>ASEX<foreign lang="greek">m</foreign>A<emph.end type="italics"/><lb/>ad aream totam <emph type="italics"/>ASCY<foreign lang="greek">m</foreign>A<emph.end type="italics"/>ut <emph type="italics"/>AE<emph.end type="italics"/>ad <emph type="italics"/>AC.<emph.end type="italics"/>Cum autem <foreign lang="greek">c</foreign><emph type="italics"/>O<emph.end type="italics"/><lb/>&longs;it ad <emph type="italics"/>SO<emph.end type="italics"/>ut 3 ad 1, & <emph type="italics"/>EO<emph.end type="italics"/>ad <emph type="italics"/>XO<emph.end type="italics"/>in eadem ratione, erit <emph type="italics"/>SX<emph.end type="italics"/><lb/>ip&longs;i <emph type="italics"/>EB<emph.end type="italics"/>parallela: & propterea &longs;i jungatur <emph type="italics"/>BX,<emph.end type="italics"/>erit triangulum <lb/><emph type="italics"/>SEB<emph.end type="italics"/>triangulo <emph type="italics"/>XEB<emph.end type="italics"/>æquale. </s> <s>Unde &longs;i ad aream <emph type="italics"/>ASEX<foreign lang="greek">m</foreign>A<emph.end type="italics"/><lb/>addatur triangulum <emph type="italics"/>EXB,<emph.end type="italics"/>& de &longs;umma auferatur triangulum <lb/><emph type="italics"/>SEB,<emph.end type="italics"/>manebit area <emph type="italics"/>ASBX<foreign lang="greek">m</foreign>A<emph.end type="italics"/>areæ <emph type="italics"/>ASEX<foreign lang="greek">m</foreign>A<emph.end type="italics"/>æqualis, <lb/>atque adeo ad aream <emph type="italics"/>ASCY<foreign lang="greek">m</foreign>A<emph.end type="italics"/>ut <emph type="italics"/>AE<emph.end type="italics"/>ad <emph type="italics"/>AC.<emph.end type="italics"/>Sed areæ <lb/><emph type="italics"/>ASBX<foreign lang="greek">m</foreign>A<emph.end type="italics"/>æqualis e&longs;t area <emph type="italics"/>ASBY<foreign lang="greek">m</foreign>A<emph.end type="italics"/>quamproxime, & hæc <lb/>area <emph type="italics"/>ASBY<foreign lang="greek">m</foreign>A<emph.end type="italics"/>e&longs;t ad aream <emph type="italics"/>ASCY<foreign lang="greek">m</foreign>A,<emph.end type="italics"/>ut tempus de&longs;cripti <lb/>arcus <emph type="italics"/>AB<emph.end type="italics"/>ad tempus de&longs;cripti arcus totius <emph type="italics"/>AC.<emph.end type="italics"/>Ideoque <emph type="italics"/>AE<emph.end type="italics"/><lb/>e&longs;t ad <emph type="italics"/>AC<emph.end type="italics"/>in ratione temporum quamproxime. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note478"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Ubi punctum <emph type="italics"/>B<emph.end type="italics"/>incidit in Parabolæ verticem <foreign lang="greek">m</foreign>, e&longs;t <emph type="italics"/>AE<emph.end type="italics"/><lb/>ad <emph type="italics"/>AC<emph.end type="italics"/>in ratione temporum accurate. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Si jungatur <foreign lang="greek">mc</foreign> &longs;ecans <emph type="italics"/>AC<emph.end type="italics"/>in <foreign lang="greek">d</foreign> & in ea capiatur <foreign lang="greek">c</foreign><emph type="italics"/>n<emph.end type="italics"/>quæ &longs;it <lb/>ad <foreign lang="greek">m</foreign><emph type="italics"/>B<emph.end type="italics"/>ut 27 <emph type="italics"/>MI<emph.end type="italics"/>ad 16 <emph type="italics"/>M<emph.end type="italics"/><foreign lang="greek">m</foreign>: acta <emph type="italics"/>Bn<emph.end type="italics"/>&longs;ecabit chordam <emph type="italics"/>AC<emph.end type="italics"/>in <lb/>ratione temporum magis accurate quam prius. </s> <s>Jaceat autem <lb/>punctum <emph type="italics"/>n<emph.end type="italics"/>ultra punctum <foreign lang="greek">c</foreign>, &longs;i punctum <emph type="italics"/>B<emph.end type="italics"/>magis di&longs;tat a vertice <lb/>principali Parabolæ quam punctum <foreign lang="greek">m</foreign>; & citra, &longs;i minus di&longs;tat ab <lb/>eodem vertice. </s></p> <p type="main"> <s><emph type="center"/>LEMMA IX.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Rectæ<emph.end type="italics"/>I<foreign lang="greek">m</foreign> & <foreign lang="greek">m</foreign>M <emph type="italics"/>& longitudo (AIC/4S<foreign lang="greek">m</foreign>) æquantur inter &longs;e.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam 4<emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign> e&longs;t latus rectum Parabolæ pertinens ad verti­<lb/>cem <foreign lang="greek">m</foreign>. <pb xlink:href="039/01/478.jpg" pagenum="450"/><arrow.to.target n="note479"/></s></p> <p type="margin"> <s><margin.target id="note479"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="center"/>LEMMA X.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si producatur<emph.end type="italics"/>S<foreign lang="greek">m</foreign> <emph type="italics"/>ad<emph.end type="italics"/>N & P, <emph type="italics"/>ut<emph.end type="italics"/><foreign lang="greek">m</foreign>N <emph type="italics"/>&longs;it pars tertia ip&longs;ius<emph.end type="italics"/><foreign lang="greek">m</foreign>I, <lb/>& SP <emph type="italics"/>&longs;it ad<emph.end type="italics"/>SN <emph type="italics"/>ut<emph.end type="italics"/>SN <emph type="italics"/>ad<emph.end type="italics"/>S<foreign lang="greek">m</foreign>. <emph type="italics"/>Cometa, quo tempore de&longs;cri­<lb/>bit arcum<emph.end type="italics"/>A<foreign lang="greek">m</foreign>C, <emph type="italics"/>&longs;i progrederetur ea &longs;emper cum velocitate <lb/>quam habet in altitudine ip&longs;i<emph.end type="italics"/>SP <emph type="italics"/>æquali, de&longs;criberet longitudi­<lb/>nem æqualem chordæ<emph.end type="italics"/>AC. </s></p> <p type="main"> <s>Nam &longs;i Cometa velocitate quam habet in <foreign lang="greek">m</foreign>, eodem tempore <lb/>progrederetur uniformiter in recta quæ Parabolam tangit in <foreign lang="greek">m</foreign>; <lb/>area quam radio ad punctum <emph type="italics"/>S<emph.end type="italics"/>ducto de&longs;criberet, æqualis e&longs;&longs;et <lb/>areæ Parabolicæ <emph type="italics"/>ASC<emph.end type="italics"/><foreign lang="greek">m. </foreign></s> <s>Ideoque contentum &longs;ub longitudine in <lb/>tangente de&longs;cripta & longitudine <emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign>, e&longs;&longs;et ad contentum &longs;ub <lb/>longitudinibus <emph type="italics"/>AC<emph.end type="italics"/>& <emph type="italics"/>SM,<emph.end type="italics"/>ut area <emph type="italics"/>ASC<emph.end type="italics"/><foreign lang="greek">m</foreign> ad triangulum <lb/><emph type="italics"/>ASCM,<emph.end type="italics"/>id e&longs;t, ut <emph type="italics"/>SN<emph.end type="italics"/>ad <emph type="italics"/>SM.<emph.end type="italics"/>Quare <emph type="italics"/>AC<emph.end type="italics"/>e&longs;t ad longitudi­<lb/>nem in tangente de&longs;criptam, ut <emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign> ad <emph type="italics"/>SN.<emph.end type="italics"/>Cum autem velocitas <lb/><figure id="id.039.01.478.1.jpg" xlink:href="039/01/478/1.jpg"/><lb/>Cometæ in altitudine <emph type="italics"/>SP<emph.end type="italics"/>&longs;it (per Corol. </s> <s>6. Prop. </s> <s>XVI. Lib. </s> <s>I.) <lb/>ad velocitatem in altitudine <emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign>, in &longs;ubduplicata ratione <emph type="italics"/>SP<emph.end type="italics"/>ad <lb/><emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign> inver&longs;e, id e&longs;t, in ratione <emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign> ad <emph type="italics"/>SN<emph.end type="italics"/>; longitudo hac velo­<lb/>citate eodem tempore de&longs;cripta, erit ad longitudinem in tangente <lb/>de&longs;criptam, ut <emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign> ad <emph type="italics"/>SN,<emph.end type="italics"/>Igitur <emph type="italics"/>AC<emph.end type="italics"/>& longitudo hac nova ve­<lb/>locitate de&longs;cripta, cum &longs;int ad longitudinem in tangente de&longs;crip­<lb/>tam in eadem ratione, æquantur inter &longs;e. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="italics"/>Corol.<emph.end type="italics"/>Cometa igitur ea cum velocitate, quam habet in altitudine <lb/><emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign>+2/3<emph type="italics"/>I<emph.end type="italics"/><foreign lang="greek">m</foreign>, eodem tempore de&longs;criberet chordam <emph type="italics"/>AC<emph.end type="italics"/>quamproxime. <pb xlink:href="039/01/479.jpg" pagenum="451"/><arrow.to.target n="note480"/></s></p> <p type="margin"> <s><margin.target id="note480"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="center"/>LEMMA XI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Si Cometa motu omni privatus de altitudine<emph.end type="italics"/>SN <emph type="italics"/>&longs;eu<emph.end type="italics"/>S<foreign lang="greek">m</foreign>+1/3I<foreign lang="greek">m</foreign><lb/><emph type="italics"/>demitteretur, ut caderet in Solem, & ea &longs;emper vi uniformiter <lb/>continuata urgeretur in Solem, qua urgetur &longs;ub initio; idem &longs;e­<lb/>mi&longs;&longs;e temporis quo in Orbe &longs;uo de&longs;cribat arcum<emph.end type="italics"/>AC, <emph type="italics"/>de&longs;cen&longs;u <lb/>&longs;uo de&longs;criberet &longs;patium longitudini<emph.end type="italics"/>I<foreign lang="greek">m</foreign> <emph type="italics"/>æquale.<emph.end type="italics"/></s></p> <p type="main"> <s>Nam Cometa quo tempore de&longs;cribat arcum Parabolicum <emph type="italics"/>AC,<emph.end type="italics"/><lb/>eodem tempore ea cum velocitate quam habet in altitudine <emph type="italics"/>SP<emph.end type="italics"/><lb/>(per Lemma novi&longs;&longs;imum) de&longs;cribet chordam <emph type="italics"/>AC,<emph.end type="italics"/>adeoque (per <lb/>Corol. </s> <s>7. Prop. </s> <s>XVI. Lib. </s> <s>I.) eodem tempore in Circulo cujus &longs;emi­<lb/>diameter e&longs;&longs;et <emph type="italics"/>SP,<emph.end type="italics"/>vi gravitatis &longs;uæ revolvendo, de&longs;criberet arcum <lb/>cujus longitudo e&longs;&longs;et ad arcus Parabolici chordam <emph type="italics"/>AC,<emph.end type="italics"/>in &longs;ubdu­<lb/>plicata ratione unius ad duo. </s> <s>Et propterea eo cum pondere quod <lb/>habet in Solem in altitudine <emph type="italics"/>SP,<emph.end type="italics"/>cadendo de altitudine illa in <lb/>Solem, de&longs;criberet &longs;emi&longs;&longs;e temporis illius (per Corol.9. Prop. </s> <s>IV. <lb/>Lib. </s> <s>I.) &longs;patium æquale quadrato &longs;emi&longs;&longs;is chordæ illius applicato <lb/>ad quadruplum altitudinis <emph type="italics"/>SP,<emph.end type="italics"/>id e&longs;t, &longs;patium (<emph type="italics"/>AIq/4SP<emph.end type="italics"/>). Unde cum <lb/>pondus Cometæ in Solem in altitudine <emph type="italics"/>SN,<emph.end type="italics"/>&longs;it ad ip&longs;ius pondus <lb/>in Solem in altitudine <emph type="italics"/>SP,<emph.end type="italics"/>ut <emph type="italics"/>SP<emph.end type="italics"/>ad <emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign>: Cometa pondere <lb/>quod habet in altitudine <emph type="italics"/>SN<emph.end type="italics"/>eodem tempore, in Solem caden­<lb/>do, de&longs;cribet &longs;patium (<emph type="italics"/>AIq/4S<foreign lang="greek">m</foreign><emph.end type="italics"/>), id e&longs;t, &longs;patium longitudini <emph type="italics"/>I<emph.end type="italics"/><foreign lang="greek">m</foreign> vel <lb/><emph type="italics"/>M<emph.end type="italics"/><foreign lang="greek">m</foreign> æquale. <emph type="italics"/>Q.E.D.<emph.end type="italics"/></s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLI. PROBLEMA XXI.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Cometæ in Parabola moti Trajectoriam ex datis tribus <lb/>Ob&longs;ervationibus determinare.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Problema hocce longe difficillimum multimode aggre&longs;&longs;us, com­<lb/>po&longs;ui Problemata quædam in Libro primo quæ ad ejus &longs;olutio­<lb/>nem &longs;pectant. </s> <s>Po&longs;tea &longs;olutionem &longs;equentem paulo &longs;impliciorem <lb/>excogitavi. </s></p> <p type="main"> <s>Seligantur tres ob&longs;ervationes æqualibus temporum intervallis ab <lb/>invicem quamproxime di&longs;tantes. </s> <s>Sit autem temporis intervallum <lb/>illud ubi Cometa tardius movetur paulo majus altero, ita videlicet <pb xlink:href="039/01/480.jpg" pagenum="452"/><arrow.to.target n="note481"/>ut temporum differentia &longs;it ad &longs;ummam temporum, ut &longs;umma tem­<lb/>porum ad dies plus minus &longs;excentos; vel ut punctum <emph type="italics"/>E<emph.end type="italics"/>incidat in <lb/>punctum <emph type="italics"/>M<emph.end type="italics"/>quamproxime, & inde aberret ver&longs;us <emph type="italics"/>I<emph.end type="italics"/>potius quam <lb/>ver&longs;us <emph type="italics"/>A.<emph.end type="italics"/>Si tales ob&longs;ervationes non præ&longs;to &longs;int, inveniendus e&longs;t <lb/>novus Cometæ locus per Lemma &longs;extum. </s></p> <p type="margin"> <s><margin.target id="note481"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>De&longs;ignent <emph type="italics"/>S<emph.end type="italics"/>Solem, <emph type="italics"/>T, t,<emph.end type="italics"/><foreign lang="greek">t</foreign> tria loca Terræ in Orbe magno, <lb/><emph type="italics"/>TA, tB, <foreign lang="greek">t</foreign>C<emph.end type="italics"/>ob&longs;ervatas tres longitudines Cometæ, V tempus in­<lb/>ter ob&longs;ervationem primam & &longs;ecundam, W tempus inter &longs;ecun­<lb/>dam ac tertiam, X longitudinem quam Cometa toto illo tempore, <lb/>ea cum velocitate quam habet in mediocri Telluris à Sole di&longs;tan­<lb/>tia, de&longs;cribere po&longs;&longs;et, quæque per Corol. </s> <s>3. Prop. </s> <s>XL, Lib. </s> <s>III. <lb/>invenienda e&longs;t, & <emph type="italics"/>tV<emph.end type="italics"/>perpendiculum in chordam <emph type="italics"/>T<emph.end type="italics"/><foreign lang="greek">t. </foreign></s> <s>In longi­<lb/><figure id="id.039.01.480.1.jpg" xlink:href="039/01/480/1.jpg"/><lb/>tudine media <emph type="italics"/>tB<emph.end type="italics"/>&longs;umatur utcunque punctum <emph type="italics"/>B<emph.end type="italics"/>pro loco Co­<lb/>metæ in plano Eclipticæ, & inde ver&longs;us Solem <emph type="italics"/>S<emph.end type="italics"/>ducatur linea <lb/><emph type="italics"/>BE,<emph.end type="italics"/>quæ &longs;it ad &longs;agittam <emph type="italics"/>tV,<emph.end type="italics"/>ut contentum &longs;ub <emph type="italics"/>SB<emph.end type="italics"/>& <emph type="italics"/>St quad.<emph.end type="italics"/><lb/>ad cubum hypotenu&longs;æ trianguli rectanguli, cujus latera &longs;unt <emph type="italics"/>SB<emph.end type="italics"/>& <lb/>tangens latitudinis Cometæ in ob&longs;ervatione &longs;ecunda ad radium <emph type="italics"/>tB.<emph.end type="italics"/><pb xlink:href="039/01/481.jpg" pagenum="453"/>Et per punctum <emph type="italics"/>E<emph.end type="italics"/>agatur (per hujus Lem. </s> <s>VII.) recta <emph type="italics"/>AEC,<emph.end type="italics"/><lb/><arrow.to.target n="note482"/>cujus partes <emph type="italics"/>AE, EC<emph.end type="italics"/>ad rectas <emph type="italics"/>TA<emph.end type="italics"/>& <foreign lang="greek">t</foreign><emph type="italics"/>C<emph.end type="italics"/>terminatæ, &longs;int ad <lb/>invicem ut tempora V & W: & erunt <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>C<emph.end type="italics"/>loca Cometæ in <lb/>plano Eclipticæ in ob&longs;ervatione prima ac tertia quamproxime, &longs;i <lb/>modo <emph type="italics"/>B<emph.end type="italics"/>&longs;it locus ejus recte a&longs;&longs;umptus in ob&longs;ervatione &longs;ecunda. </s></p> <p type="margin"> <s><margin.target id="note482"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Ad <emph type="italics"/>AC<emph.end type="italics"/>bi&longs;ectam in <emph type="italics"/>I<emph.end type="italics"/>erige perpendiculum <emph type="italics"/>Ii.<emph.end type="italics"/>Per punctum <emph type="italics"/>B<emph.end type="italics"/><lb/>age occultam <emph type="italics"/>Bi<emph.end type="italics"/>ip&longs;i <emph type="italics"/>AC<emph.end type="italics"/>parallelam. </s> <s>Junge occultam <emph type="italics"/>Si<emph.end type="italics"/>&longs;ecan­<lb/>tem <emph type="italics"/>AC<emph.end type="italics"/>in <foreign lang="greek">l</foreign>, & comple parallelogrammum <emph type="italics"/>iI<emph.end type="italics"/><foreign lang="greek">lm. </foreign></s> <s>Cape <emph type="italics"/>I<emph.end type="italics"/><foreign lang="greek">s</foreign> æqua­<lb/>lem 3<emph type="italics"/>I<emph.end type="italics"/><foreign lang="greek">l</foreign>, & per Solem <emph type="italics"/>S<emph.end type="italics"/>age occultam <foreign lang="greek">sc</foreign> æqualem 3<emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">s</foreign>+3<emph type="italics"/>i<emph.end type="italics"/><foreign lang="greek">l</foreign>, <lb/>Et deletis jam literis <emph type="italics"/>A, E, C, I,<emph.end type="italics"/>a puncto <emph type="italics"/>B<emph.end type="italics"/>ver&longs;us punctum <foreign lang="greek">c</foreign><lb/>duc occultam novam <emph type="italics"/>BE,<emph.end type="italics"/>quæ &longs;it ad priorem <emph type="italics"/>BE<emph.end type="italics"/>in duplicata <lb/>ratione di&longs;tantiæ <emph type="italics"/>BS<emph.end type="italics"/>ad quantitatem <emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign>+1/3<emph type="italics"/>i<emph.end type="italics"/><foreign lang="greek">l. </foreign></s> <s>Et per punctum <lb/><emph type="italics"/>E<emph.end type="italics"/>iterum duc rectam <emph type="italics"/>AEC<emph.end type="italics"/>eadem lege ac prius, id e&longs;t, ita ut ejus <lb/>partes <emph type="italics"/>AE<emph.end type="italics"/>& <emph type="italics"/>EC<emph.end type="italics"/>&longs;int ad invicem, ut tempora inter ob&longs;ervationes <lb/>V & W. </s> <s>Et erunt <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>C<emph.end type="italics"/>loca Cometæ magis accurate. </s></p> <p type="main"> <s>Ad <emph type="italics"/>AC<emph.end type="italics"/>bi&longs;ectam in <emph type="italics"/>1<emph.end type="italics"/>erigantur perpendicula <emph type="italics"/>AM, CN, IO,<emph.end type="italics"/><lb/>quarum <emph type="italics"/>AM<emph.end type="italics"/>& <emph type="italics"/>CN<emph.end type="italics"/>&longs;int tangentes latitudinum in ob&longs;ervatione <lb/>prima ac tertia ad radios <emph type="italics"/>TA<emph.end type="italics"/>& <foreign lang="greek">t</foreign><emph type="italics"/>C.<emph.end type="italics"/>Jungatur <emph type="italics"/>MN<emph.end type="italics"/>&longs;ecans <emph type="italics"/>IO<emph.end type="italics"/><lb/>in <emph type="italics"/>O.<emph.end type="italics"/>Con&longs;tituatur rectangulum <emph type="italics"/>iI<emph.end type="italics"/><foreign lang="greek">lm</foreign> ut prius. </s> <s>In <emph type="italics"/>IA<emph.end type="italics"/>pro­<lb/>ducta capiatur <emph type="italics"/>ID<emph.end type="italics"/>æqualis <emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign>+2/3<emph type="italics"/>i<emph.end type="italics"/><foreign lang="greek">l</foreign>, & agatur occulta <emph type="italics"/>OD.<emph.end type="italics"/><lb/>Deinde in <emph type="italics"/>MN<emph.end type="italics"/>ver&longs;us <emph type="italics"/>N<emph.end type="italics"/>capiatur <emph type="italics"/>MP,<emph.end type="italics"/>quæ &longs;it ad longitudinem <lb/>&longs;upra inventam X, in &longs;ubduplicata ratione mediocris di&longs;tantiæ Tel­<lb/>luris a Sole (&longs;eu &longs;emidiametri Orbis magni) ad di&longs;tantiam <emph type="italics"/>OD.<emph.end type="italics"/><lb/>Si punctum <emph type="italics"/>P<emph.end type="italics"/>incidat in punctum <emph type="italics"/>N<emph.end type="italics"/>; erunt <emph type="italics"/>A, B, C<emph.end type="italics"/>tria loca Co­<lb/>metæ, per quæ Orbis ejus in plano Eclipticæ de&longs;cribi debet. </s> <s>Sin <lb/>punctum <emph type="italics"/>P<emph.end type="italics"/>non incidat in punctum <emph type="italics"/>N<emph.end type="italics"/>; in recta <emph type="italics"/>AC<emph.end type="italics"/>capiatur <lb/><emph type="italics"/>CG<emph.end type="italics"/>ip&longs;i <emph type="italics"/>NP<emph.end type="italics"/>æqualis, ita ut puncta <emph type="italics"/>G<emph.end type="italics"/>& <emph type="italics"/>P<emph.end type="italics"/>ad ea&longs;dem partes <lb/>rectæ <emph type="italics"/>NC<emph.end type="italics"/>jaceant. </s></p> <p type="main"> <s>Eadem methodo qua puncta <emph type="italics"/>E, A, C, G,<emph.end type="italics"/>ex a&longs;&longs;umpto puncto <lb/><emph type="italics"/>B<emph.end type="italics"/>inventa &longs;unt, inveniantur ex a&longs;&longs;umptis utcunque punctis aliis <lb/><emph type="italics"/>b<emph.end type="italics"/>& <foreign lang="greek">b</foreign> puncta nova <emph type="italics"/>e, a, c, g,<emph.end type="italics"/>& <foreign lang="greek">e, a, x, g. </foreign></s> <s>Deinde &longs;i per <emph type="italics"/>G, g,<emph.end type="italics"/><foreign lang="greek">g</foreign><lb/>ducatur circumferentia circuli <emph type="italics"/>Gg<emph.end type="italics"/><foreign lang="greek">g</foreign>, &longs;ecans rectam <foreign lang="greek">t</foreign><emph type="italics"/>C<emph.end type="italics"/>in <emph type="italics"/>Z<emph.end type="italics"/>: erit <lb/><emph type="italics"/>Z<emph.end type="italics"/>locus Cometæ in plano Eclipticæ. </s> <s>Et &longs;i in <emph type="italics"/>AC, ac,<emph.end type="italics"/><foreign lang="greek">ax</foreign> capi­<lb/>antur <emph type="italics"/>AF, af,<emph.end type="italics"/><foreign lang="greek">af</foreign> ip&longs;is <emph type="italics"/>CG, eg,<emph.end type="italics"/><foreign lang="greek">xg</foreign> re&longs;pective æquales, & per <lb/>puncta <emph type="italics"/>F, f,<emph.end type="italics"/><foreign lang="greek">f</foreign> ducatur circumferentia circuli <emph type="italics"/>Ff<emph.end type="italics"/><foreign lang="greek">f</foreign>, &longs;ecans rectam <lb/><emph type="italics"/>AT<emph.end type="italics"/>in <emph type="italics"/>X;<emph.end type="italics"/>erit punctum <emph type="italics"/>X<emph.end type="italics"/>alius Cometæ locus in plano Eclipticæ. </s> <s><lb/>Ad puncta <emph type="italics"/>X<emph.end type="italics"/>& <emph type="italics"/>Z<emph.end type="italics"/>erigantur tangentes latitudinum Cometæ ad ra­<lb/>dios <emph type="italics"/>TX<emph.end type="italics"/>& <foreign lang="greek">t</foreign><emph type="italics"/>Z<emph.end type="italics"/>; & habebuntur loca duo Cometæ in Orbe proprio. </s> <s><lb/>Denique (per Prop. </s> <s>XIX. Lib. </s> <s>I.) umbilico <emph type="italics"/>S,<emph.end type="italics"/>per loca illa duo de­<lb/>&longs;cribatur Parabola, & hæc erit Trajectoria Cometæ. <emph type="italics"/>Q.E.I.<emph.end type="italics"/><pb xlink:href="039/01/482.jpg" pagenum="454"/><arrow.to.target n="note483"/></s></p> <p type="margin"> <s><margin.target id="note483"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Con&longs;tructionis hujus demon&longs;tratio ex Lemmatibus con&longs;equitur: <lb/>quippe cum recta <emph type="italics"/>AC<emph.end type="italics"/>&longs;ecetur in <emph type="italics"/>E<emph.end type="italics"/>in ratione temporum, per <lb/>Lemma VII, ut oportet per Lem. </s> <s>VIII: & <emph type="italics"/>BE<emph.end type="italics"/>per Lem. </s> <s>XI. <lb/>&longs;it pars rectæ <emph type="italics"/>BS<emph.end type="italics"/>vel <emph type="italics"/>B<emph.end type="italics"/><foreign lang="greek">c</foreign> in plano Eclipticæ arcui <emph type="italics"/>ABC<emph.end type="italics"/>& <lb/>chordæ <emph type="italics"/>AEC<emph.end type="italics"/>interjecta; & <emph type="italics"/>MP<emph.end type="italics"/>(per Corol. </s> <s>Lem. </s> <s>X.) longi­<lb/>tudo &longs;it chordæ arcus, quem Cometa in Orbe proprio inter ob­<lb/>&longs;ervationem primam ac tertiam de&longs;cribere debet, ideoQ.E.I.&longs;i <lb/><emph type="italics"/>MN<emph.end type="italics"/>æqualis fuerit, &longs;i modo <emph type="italics"/>B<emph.end type="italics"/>&longs;it verus Cometæ locus in plano <lb/>Eclipticæ. </s></p><figure id="id.039.01.482.1.jpg" xlink:href="039/01/482/1.jpg"/> <p type="main"> <s>Cæterum puncta <emph type="italics"/>B, b,<emph.end type="italics"/><foreign lang="greek">b</foreign> non quælibet, &longs;ed vero proxima eli­<lb/>gere convenit. </s> <s>Si angulus <emph type="italics"/>AQt,<emph.end type="italics"/>in quo ve&longs;tigium Orbis in <lb/>plano Eclipticæ de&longs;criptum &longs;ecat rectam <emph type="italics"/>tB,<emph.end type="italics"/>præterpropter in­<lb/>note&longs;cat; in angulo illo ducenda erit recta occulta <emph type="italics"/>AC,<emph.end type="italics"/>quæ &longs;it <lb/>ad 4/3<emph type="italics"/>T<emph.end type="italics"/><foreign lang="greek">t</foreign> in &longs;ubduplicata ratione <emph type="italics"/>SQ<emph.end type="italics"/>ad <emph type="italics"/>St.<emph.end type="italics"/>Et agendo rectam <lb/><emph type="italics"/>SEB<emph.end type="italics"/>cujus pars <emph type="italics"/>EB<emph.end type="italics"/>æquetur longitudini <emph type="italics"/>Vt,<emph.end type="italics"/>determinabitur <lb/>punctum <emph type="italics"/>B<emph.end type="italics"/>quod prima vice u&longs;urpare licet. </s> <s>Tum recta <emph type="italics"/>AC<emph.end type="italics"/>de­<lb/>leta & &longs;ecundum præcedentem con&longs;tructionem iterum ducta, & <pb xlink:href="039/01/483.jpg" pagenum="455"/>inventa in&longs;uper longitudine <emph type="italics"/>MP<emph.end type="italics"/>; in <emph type="italics"/>tB<emph.end type="italics"/>capiatur punctum <emph type="italics"/>b,<emph.end type="italics"/></s></p> <p type="main"> <s><arrow.to.target n="note484"/>ea lege, ut &longs;i <emph type="italics"/>TA, <foreign lang="greek">t</foreign>C<emph.end type="italics"/>&longs;e mutuo &longs;ecuerint in <emph type="italics"/>Y,<emph.end type="italics"/>&longs;it di&longs;tantia <emph type="italics"/>Yb<emph.end type="italics"/><lb/>ad di&longs;tantiam <emph type="italics"/>YB,<emph.end type="italics"/>in ratione compo&longs;ita ex ratione <emph type="italics"/>MP<emph.end type="italics"/>ad <emph type="italics"/>MN<emph.end type="italics"/><lb/>& ratione &longs;ubduplicata <emph type="italics"/>SB<emph.end type="italics"/>ad <emph type="italics"/>Sb.<emph.end type="italics"/>Et eadem methodo inveNI­<lb/>endum erit punctum tertium <foreign lang="greek">b</foreign>, &longs;i modo operationem tertio repe­<lb/>tere lubet. </s> <s>Sed hac methodo operationes duæ ut plurimum &longs;uf­<lb/>fecerint. </s> <s>Nam &longs;i di&longs;tantia <emph type="italics"/>Bb<emph.end type="italics"/>perexigua obvenerit; po&longs;tquam <lb/>inventa &longs;unt puncta <emph type="italics"/>F, f<emph.end type="italics"/>& <emph type="italics"/>G, g,<emph.end type="italics"/>actæ rectæ <emph type="italics"/>Ff<emph.end type="italics"/>& <emph type="italics"/>Gg<emph.end type="italics"/>&longs;ecabunt <lb/><emph type="italics"/>TA<emph.end type="italics"/>& <foreign lang="greek">t</foreign><emph type="italics"/>C<emph.end type="italics"/>in punctis quæ&longs;itis <emph type="italics"/>X<emph.end type="italics"/>& <emph type="italics"/>Z.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note484"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Exemplum.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Proponatur Cometa anni 1680. Hujus motum a <emph type="italics"/>Flam&longs;tedio<emph.end type="italics"/><lb/>ob&longs;ervatum Tabula &longs;equens exhibet. <lb/><arrow.to.target n="table9"/></s></p><table><table.target id="table9"/><row><cell/><cell/><cell>Tem.appar.</cell><cell>Temp. verum</cell><cell>Long. Solis</cell><cell>Long. Cometæ</cell><cell>Lat. Cometæ</cell></row><row><cell/><cell/><cell>h.</cell><cell>′</cell><cell>h.</cell><cell>′</cell><cell>″</cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell>gr.</cell><cell>′</cell><cell>″</cell></row><row><cell>1680 <emph type="italics"/>Dec.<emph.end type="italics"/></cell><cell>12</cell><cell>4.</cell><cell>46</cell><cell>4.</cell><cell>46.</cell><cell>0</cell><cell><!--symbol1--> 1.</cell><cell>51.</cell><cell>23</cell><cell><!--symbol1--> 6.</cell><cell>31.</cell><cell>21</cell><cell>8.</cell><cell>26.</cell><cell>0</cell></row><row><cell/><cell>21</cell><cell>6.</cell><cell>32 1/2</cell><cell>6.</cell><cell>36.</cell><cell>59</cell><cell>11.</cell><cell>6.</cell><cell>44</cell><cell><!--symbol2--> 5.</cell><cell>7.</cell><cell>38</cell><cell>21.</cell><cell>45.</cell><cell>30</cell></row><row><cell/><cell>24</cell><cell>6.</cell><cell>12</cell><cell>6.</cell><cell>17.</cell><cell>52</cell><cell>14.</cell><cell>9.</cell><cell>26</cell><cell>18.</cell><cell>49.</cell><cell>10</cell><cell>25.</cell><cell>23.</cell><cell>24</cell></row><row><cell/><cell>26</cell><cell>5.</cell><cell>14</cell><cell>5.</cell><cell>20.</cell><cell>44</cell><cell>16.</cell><cell>9.</cell><cell>22</cell><cell>28.</cell><cell>24.</cell><cell>6</cell><cell>27.</cell><cell>0.</cell><cell>57</cell></row><row><cell/><cell>29</cell><cell>7.</cell><cell>55</cell><cell>8.</cell><cell>3.</cell><cell>2</cell><cell>19.</cell><cell>19.</cell><cell>43</cell><cell><!--symbol3--> 13.</cell><cell>11.</cell><cell>45</cell><cell>28.</cell><cell>10.</cell><cell>5</cell></row><row><cell/><cell>30</cell><cell>8.</cell><cell>2</cell><cell>8.</cell><cell>10.</cell><cell>26</cell><cell>20.</cell><cell>21.</cell><cell>9</cell><cell>17.</cell><cell>39.</cell><cell>5</cell><cell>28.</cell><cell>11.</cell><cell>12</cell></row><row><cell>1681 <emph type="italics"/>Jan.<emph.end type="italics"/></cell><cell>5</cell><cell>5.</cell><cell>51</cell><cell>6.</cell><cell>1.</cell><cell>38</cell><cell>26.</cell><cell>22.</cell><cell>18</cell><cell><!--symbol4--> 8.</cell><cell>49.</cell><cell>10</cell><cell>26.</cell><cell>15.</cell><cell>26</cell></row><row><cell/><cell>9</cell><cell>6.</cell><cell>49</cell><cell>7.</cell><cell>0.</cell><cell>53</cell><cell><!--symbol2--> 0.</cell><cell>29.</cell><cell>2</cell><cell>18.</cell><cell>43.</cell><cell>18</cell><cell>24.</cell><cell>12.</cell><cell>42</cell></row><row><cell/><cell>10</cell><cell>5.</cell><cell>54</cell><cell>6.</cell><cell>6.</cell><cell>10</cell><cell>1.</cell><cell>27.</cell><cell>43</cell><cell>20.</cell><cell>40.</cell><cell>57</cell><cell>23.</cell><cell>44.</cell><cell>0</cell></row><row><cell/><cell>13</cell><cell>6.</cell><cell>56</cell><cell>7.</cell><cell>8.</cell><cell>55</cell><cell>4.</cell><cell>33.</cell><cell>20</cell><cell>25.</cell><cell>59.</cell><cell>34</cell><cell>22.</cell><cell>17.</cell><cell>36</cell></row><row><cell/><cell>25</cell><cell>7.</cell><cell>44</cell><cell>7.</cell><cell>58.</cell><cell>42</cell><cell>16.</cell><cell>45.</cell><cell>36</cell><cell><!--symbol5--> 9.</cell><cell>35.</cell><cell>48</cell><cell>17.</cell><cell>56.</cell><cell>54</cell></row><row><cell/><cell>30</cell><cell>8.</cell><cell>7</cell><cell>8.</cell><cell>21.</cell><cell>53</cell><cell>21.</cell><cell>40.</cell><cell>58</cell><cell>13.</cell><cell>19.</cell><cell>36</cell><cell>16.</cell><cell>40.</cell><cell>57</cell></row><row><cell><emph type="italics"/>Feb.<emph.end type="italics"/></cell><cell>2</cell><cell>6.</cell><cell>20</cell><cell>6.</cell><cell>34.</cell><cell>51</cell><cell>24.</cell><cell>46.</cell><cell>59</cell><cell>15.</cell><cell>13.</cell><cell>48</cell><cell>16.</cell><cell>2.</cell><cell>2</cell></row><row><cell/><cell>5</cell><cell>6.</cell><cell>50</cell><cell>7.</cell><cell>4.</cell><cell>41</cell><cell>27.</cell><cell>49.</cell><cell>51</cell><cell>16.</cell><cell>59.</cell><cell>52</cell><cell>15.</cell><cell>27.</cell><cell>23</cell></row></table> <p type="main"> <s>His adde Ob&longs;ervationes qua&longs;dam e no&longs;tris. <lb/><arrow.to.target n="table10"/></s></p><table><table.target id="table10"/><row><cell/><cell/><cell>Temp. appar.</cell><cell>Cometæ Longit.</cell><cell>Com. Lat.</cell></row><row><cell><emph type="italics"/>Febr.<emph.end type="italics"/></cell><cell>25</cell><cell>8<emph type="sup"/>h<emph.end type="sup"/>.</cell><cell>30′</cell><cell><!--symbol5--> 26<emph type="sup"/>gr.<emph.end type="sup"/>.</cell><cell>18′.</cell><cell>17″</cell><cell>12<emph type="sup"/>gr.<emph.end type="sup"/>.</cell><cell>46′ 7/8</cell></row><row><cell/><cell>27</cell><cell>8.</cell><cell>15</cell><cell>27.</cell><cell>4.</cell><cell>24</cell><cell>12.</cell><cell>36 1/5</cell></row><row><cell><emph type="italics"/>Mart.<emph.end type="italics"/></cell><cell>1</cell><cell>11.</cell><cell>0</cell><cell>27.</cell><cell>53.</cell><cell>6</cell><cell>12.</cell><cell>24 6/7</cell></row><row><cell/><cell>2</cell><cell>8.</cell><cell>0</cell><cell>28.</cell><cell>12.</cell><cell>27</cell><cell>12.</cell><cell>20</cell></row><row><cell/><cell>5</cell><cell>11.</cell><cell>30</cell><cell>29.</cell><cell>20.</cell><cell>51</cell><cell>12.</cell><cell>3 1/2</cell></row><row><cell/><cell>9</cell><cell>8.</cell><cell>30</cell><cell><!--symbol6--> 0.</cell><cell>43.</cell><cell>4</cell><cell>11.</cell><cell>45 7/8</cell></row></table> <p type="main"> <s>Hæ Ob&longs;ervationes Tele&longs;copio &longs;eptupedali, & Micrometro fili&longs;­<lb/>Q.E.I. &longs;oco Tele&longs;copii locatis peractæ &longs;unt: quibus in&longs;trumentis <pb xlink:href="039/01/484.jpg" pagenum="456"/><arrow.to.target n="note485"/>& po&longs;itiones fixarum inter &longs;e & po&longs;itiones Cometæ ad fixas de­<lb/>terminavimus. </s> <s>De&longs;ignet <emph type="italics"/>A<emph.end type="italics"/>&longs;tellam in &longs;ini&longs;tro calcaneo Per&longs;ei <lb/><emph type="italics"/>(Bayero o) B<emph.end type="italics"/>&longs;tellam &longs;equentem in &longs;ini&longs;tro pede (<emph type="italics"/>Bayero<emph.end type="italics"/><foreign lang="greek">z</foreign>) & <lb/><emph type="italics"/>C, D, E, F, G, H, I, K, L, M, N, O<emph.end type="italics"/>&longs;tellas alias minores in eo­<lb/>dem pede. </s> <s>Sintque <emph type="italics"/>P, Q, R, S, T<emph.end type="italics"/>loca Cometæ in ob&longs;ervati­<lb/>onibus &longs;upra de&longs;criptis: & exi&longs;tente di&longs;tantia <emph type="italics"/>AB<emph.end type="italics"/>partium (80 7/12), <lb/>erat <emph type="italics"/>AC<emph.end type="italics"/>partium 52 1/4, <emph type="italics"/>BC<emph.end type="italics"/>58 5/6, <emph type="italics"/>AD<emph.end type="italics"/>(57 5/12), <emph type="italics"/>BD<emph.end type="italics"/>(82 6/11), <emph type="italics"/>CD<emph.end type="italics"/>23 2/3, <lb/><emph type="italics"/>AE<emph.end type="italics"/>29 4/7, <emph type="italics"/>CE<emph.end type="italics"/>57 1/2, <emph type="italics"/>DE<emph.end type="italics"/>(49 11/12), <emph type="italics"/>AI<emph.end type="italics"/>(27 7/12), <emph type="italics"/>BI<emph.end type="italics"/>52 1/6, <emph type="italics"/>CI<emph.end type="italics"/>(36 7/12), <lb/><figure id="id.039.01.484.1.jpg" xlink:href="039/01/484/1.jpg"/><lb/><emph type="italics"/>DI<emph.end type="italics"/>(53 5/11), <emph type="italics"/>AK<emph.end type="italics"/>38 2/3, <emph type="italics"/>BK<emph.end type="italics"/>43, <emph type="italics"/>CK<emph.end type="italics"/>31 5/9, <emph type="italics"/>FK<emph.end type="italics"/>29, <emph type="italics"/>FB<emph.end type="italics"/>23, <emph type="italics"/>FC<emph.end type="italics"/>36 1/4, <lb/><emph type="italics"/>AH<emph.end type="italics"/>18 6/7, <emph type="italics"/>DH<emph.end type="italics"/>50 7/8, <emph type="italics"/>BN<emph.end type="italics"/>(46 5/12), <emph type="italics"/>CN<emph.end type="italics"/>31 1/3, <emph type="italics"/>BL<emph.end type="italics"/>(45 5/12), <emph type="italics"/>NL<emph.end type="italics"/>31 5/7. <lb/><emph type="italics"/>HO<emph.end type="italics"/>erat ad <emph type="italics"/>HI<emph.end type="italics"/>ut 7 ad 6 & producta tran&longs;ibat inter &longs;tellas <lb/>D & <emph type="italics"/>E,<emph.end type="italics"/>&longs;ic ut di&longs;tantia &longs;tellæ <emph type="italics"/>D<emph.end type="italics"/>ab hac recta e&longs;&longs;et 1/6<emph type="italics"/>CD. LM<emph.end type="italics"/><lb/>erat ad <emph type="italics"/>LB<emph.end type="italics"/>ut 2 ad 9 & producta tran&longs;ibat per &longs;tellam <emph type="italics"/>H.<emph.end type="italics"/>His <lb/>interminabantur po&longs;itiones fixarum inter &longs;e. </s></p> <p type="margin"> <s><margin.target id="note485"/>E MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Die Veneris <emph type="italics"/>Feb.<emph.end type="italics"/>25. St. </s> <s>vet. </s> <s>Hor. </s> <s>8 1/2 P. M. </s> <s>Cometæ in <emph type="italics"/>p<emph.end type="italics"/>ex­<lb/>i&longs;tentis di&longs;tantia a &longs;tella <emph type="italics"/>E<emph.end type="italics"/>erat minor quam (3/13) <emph type="italics"/>AE,<emph.end type="italics"/>major quam <lb/>3/5 <emph type="italics"/>AE,<emph.end type="italics"/>adeoque æqualis (3/14)<emph type="italics"/>AE<emph.end type="italics"/>proxime; & angulus <emph type="italics"/>ApE<emph.end type="italics"/>non­<lb/>nihil obtu&longs;us erat, &longs;ed fere rectus. </s> <s>Nempe &longs;i demitteretur ad <lb/><emph type="italics"/>pE<emph.end type="italics"/>perpendiculum ab <emph type="italics"/>A,<emph.end type="italics"/>di&longs;tantiæ Cometæ a perpendiculo illo <lb/>erat 1/5<emph type="italics"/>pE.<emph.end type="italics"/></s></p> <p type="main"> <s>Eadem nocte, hora 9 1/2, Cometæ in <emph type="italics"/>P<emph.end type="italics"/>exi&longs;tentis di&longs;tantia a &longs;tella <lb/><emph type="italics"/>E<emph.end type="italics"/>erat major quam (1/(4 1/2))<emph type="italics"/>AE,<emph.end type="italics"/>minor quam (1/(5 1/4))<emph type="italics"/>AE,<emph.end type="italics"/>adeoque æqua-<pb xlink:href="039/01/485.jpg" pagenum="457"/>lis (1/(4 7/8))<emph type="italics"/>AE,<emph.end type="italics"/>&longs;eu (1/39)<emph type="italics"/>AE<emph.end type="italics"/>quamproxime. </s> <s>A perpendiculo autem a <lb/><arrow.to.target n="note486"/>&longs;tella <emph type="italics"/>A<emph.end type="italics"/>ad rectam <emph type="italics"/>PE<emph.end type="italics"/>demi&longs;&longs;o, di&longs;tantia Cometæ erat 4/5<emph type="italics"/>PE.<emph.end type="italics"/></s></p> <p type="margin"> <s><margin.target id="note486"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Die <!--symbol7--><emph type="sup"/>is<emph.end type="sup"/>, <emph type="italics"/>Feb.<emph.end type="italics"/>27. hor. </s> <s>8 1/4 P.M. </s> <s>Cometæ in <emph type="italics"/>Q<emph.end type="italics"/>exi&longs;tentis di­<lb/>&longs;tantia a &longs;tella <emph type="italics"/>O<emph.end type="italics"/>æquabat di&longs;tantiam &longs;tellarum <emph type="italics"/>O<emph.end type="italics"/>& <emph type="italics"/>H,<emph.end type="italics"/>& recta <lb/><emph type="italics"/>QO<emph.end type="italics"/>producta tran&longs;ibat inter &longs;tellas <emph type="italics"/>K<emph.end type="italics"/>& <emph type="italics"/>B.<emph.end type="italics"/>Po&longs;itionem hujus <lb/>rectæ ob nubes intervenientes, magis accurate definire non potui. </s></p> <p type="main"> <s>Die <!--symbol8--><emph type="sup"/>tis<emph.end type="sup"/>, <emph type="italics"/>Mart<emph.end type="italics"/>1, hor. </s> <s>11. P.M. </s> <s>Cometa in <emph type="italics"/>R<emph.end type="italics"/>exi&longs;tens, &longs;tellis <lb/><emph type="italics"/>K<emph.end type="italics"/>& <emph type="italics"/>C<emph.end type="italics"/>accurate interjacebat, & rectæ <emph type="italics"/>CRK<emph.end type="italics"/>pars <emph type="italics"/>CR<emph.end type="italics"/>paulo <lb/>major erat quam 1/3<emph type="italics"/>CK,<emph.end type="italics"/>& paulo minor quam 1/3<emph type="italics"/>CK<emph.end type="italics"/>+1/8<emph type="italics"/>CR,<emph.end type="italics"/><lb/>adeoque æqualis 1/3<emph type="italics"/>CK<emph.end type="italics"/>+(1/16)<emph type="italics"/>CR<emph.end type="italics"/>&longs;eu (16/45)<emph type="italics"/>CK.<emph.end type="italics"/></s></p> <p type="main"> <s>Die <!--symbol9--><emph type="sup"/>ii<emph.end type="sup"/>, <emph type="italics"/>Mart.<emph.end type="italics"/>2. hor. </s> <s>8. P.M. </s> <s>Cometæ exi&longs;tentis in <emph type="italics"/>S,<emph.end type="italics"/>di­<lb/>&longs;tantia a &longs;tella <emph type="italics"/>C<emph.end type="italics"/>erat 4/9<emph type="italics"/>FC<emph.end type="italics"/>quamproxime. </s> <s>Di&longs;tantia &longs;tellæ <emph type="italics"/>F<emph.end type="italics"/>a <lb/>recta <emph type="italics"/>CS<emph.end type="italics"/>producta erat (1/24)<emph type="italics"/>FC<emph.end type="italics"/>; & di&longs;tantia &longs;tellæ <emph type="italics"/>B<emph.end type="italics"/>ab eadem recta, <lb/>erat quintuplo major quam di&longs;tantia &longs;tellæ <emph type="italics"/>F.<emph.end type="italics"/>Item recta <emph type="italics"/>NS<emph.end type="italics"/><lb/>producta tran&longs;ibat inter &longs;tellas <emph type="italics"/>H<emph.end type="italics"/>& <emph type="italics"/>I,<emph.end type="italics"/>quintuplo vel &longs;extuplo pro­<lb/>pior exi&longs;tens &longs;tellæ <emph type="italics"/>H<emph.end type="italics"/>quam &longs;tellæ <emph type="italics"/>I.<emph.end type="italics"/></s></p> <p type="main"> <s>Die <!--symbol10--><emph type="sup"/>ni<emph.end type="sup"/>, <emph type="italics"/>Mart.<emph.end type="italics"/>5. hor. </s> <s>11 1/2. P. M. </s> <s>Cometa exi&longs;tente in <emph type="italics"/>T,<emph.end type="italics"/><lb/>recta <emph type="italics"/>MT<emph.end type="italics"/>æqualis erat 1/2<emph type="italics"/>ML,<emph.end type="italics"/>& recta <emph type="italics"/>LT<emph.end type="italics"/>producta tran&longs;ibat <lb/>inter <emph type="italics"/>B<emph.end type="italics"/>& <emph type="italics"/>F,<emph.end type="italics"/>quadruplo vel quintuplo propior <emph type="italics"/>F<emph.end type="italics"/>quam <emph type="italics"/>B,<emph.end type="italics"/>au­<lb/>ferens a <emph type="italics"/>BF<emph.end type="italics"/>quintam vel &longs;extam ejus partem ver&longs;us <emph type="italics"/>F.<emph.end type="italics"/>Et <emph type="italics"/>MT<emph.end type="italics"/><lb/>producta tran&longs;ibat extra &longs;patium <emph type="italics"/>BF<emph.end type="italics"/>ad partes &longs;tellæ <emph type="italics"/>B,<emph.end type="italics"/>quadru­<lb/>plo propior exi&longs;tens &longs;tellæ <emph type="italics"/>B<emph.end type="italics"/>quam &longs;tellæ <emph type="italics"/>F.<emph.end type="italics"/>Erat <emph type="italics"/>M<emph.end type="italics"/>&longs;tella pere­<lb/>xigua quæ per Tele&longs;copium videri vix potuit, & <emph type="italics"/>L<emph.end type="italics"/>&longs;tella major <lb/>qua&longs;i magnitudinis octavæ. </s></p> <p type="main"> <s>Ex huju&longs;modi ob&longs;ervationibus per con&longs;tructiones figurarum & <lb/>computationes (po&longs;ito quod &longs;tellarum <emph type="italics"/>A<emph.end type="italics"/>& <emph type="italics"/>B<emph.end type="italics"/>di&longs;tantia e&longs;&longs;et <lb/>2<emph type="sup"/>gr.<emph.end type="sup"/> 6′. </s> <s>46″, & &longs;tellæ <emph type="italics"/>A<emph.end type="italics"/>longitudo <!--symbol5--> 26<emph type="sup"/>gr.<emph.end type="sup"/> 41′. </s> <s>50″ & latitudo <lb/>borealis 12<emph type="sup"/>gr.<emph.end type="sup"/> 8′ 1/2, &longs;tellæque <emph type="italics"/>B<emph.end type="italics"/>longitudo <!--symbol5--> 28<emph type="sup"/>gr.<emph.end type="sup"/> 40′. </s> <s>24″ & lati­<lb/>tudo borealis 11<emph type="sup"/>gr.<emph.end type="sup"/> (17′ 9/10);) derivabam longitudines & latitudines <lb/>Cometæ. </s> <s>Micrometro parum affabre con&longs;tructo u&longs;us &longs;um, &longs;ed <lb/>longitudinum tamen & latitudinum errores (quatenus ab ob­<lb/>&longs;ervationibus no&longs;tris oriantur) dimidium minuti unius primi vix <lb/>&longs;uperant, præterquam in ob&longs;ervatione ultima <emph type="italics"/>Mart.<emph.end type="italics"/>9. ubi po&longs;i­<lb/>tiones &longs;tellarum minus accurate determinare potui. <emph type="italics"/>Ca&longs;&longs;inus<emph.end type="italics"/>qui <lb/>a&longs;cen&longs;ionem rectam Cometæ eodem tempore ob&longs;ervavit, decli­<lb/>nationem ejus tanquam invariatam manentem parum diligenter <lb/>definivit. </s> <s>Nam Cometa (juxta ob&longs;ervationes no&longs;tras) in fine <pb xlink:href="039/01/486.jpg" pagenum="458"/><arrow.to.target n="note487"/>motus &longs;ui notabiliter deflectere cœpit boream ver&longs;us, a paral­<lb/>lelo quem in fine Men&longs;is <emph type="italics"/>Februarii<emph.end type="italics"/>tenuerat. </s></p> <p type="margin"> <s><margin.target id="note487"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Jam ad Orbem Cometæ determinandum; &longs;elegi ex ob&longs;ervatio­<lb/>nibus hactenus de&longs;criptis tres, quas <emph type="italics"/>Flam&longs;tedius<emph.end type="italics"/>habuit <emph type="italics"/>Dec.<emph.end type="italics"/>21, <lb/><emph type="italics"/>Jan.<emph.end type="italics"/>5, & <emph type="italics"/>Jan.<emph.end type="italics"/>25. Ex his inveni <emph type="italics"/>St<emph.end type="italics"/>partium 9842,1 & <emph type="italics"/>Vt<emph.end type="italics"/>par­<lb/>tium 455, quales 10000 &longs;unt &longs;emidiameter Orbis magni. </s> <s>Tum <lb/>ad operationem primam a&longs;&longs;umendo <emph type="italics"/>tB<emph.end type="italics"/>partium 5657, inveni <lb/><emph type="italics"/>SB<emph.end type="italics"/>9747, <emph type="italics"/>BE<emph.end type="italics"/>prima vice 412, <emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">m</foreign> 9503, <emph type="italics"/>i<emph.end type="italics"/><foreign lang="greek">l</foreign> 413: <emph type="italics"/>BE<emph.end type="italics"/>&longs;ecun­<lb/>da vice 421, <emph type="italics"/>OD<emph.end type="italics"/>10186, X 8528,4, <emph type="italics"/>MP<emph.end type="italics"/>8450, <emph type="italics"/>MN<emph.end type="italics"/>8475, <lb/><emph type="italics"/>NP<emph.end type="italics"/>25. Unde ad operationem &longs;ecundam collegi di&longs;tantiam <lb/><emph type="italics"/>tb<emph.end type="italics"/>5640. Et per hanc operationem inveni tandem di&longs;tantias <lb/><emph type="italics"/>TX<emph.end type="italics"/>4775 & <foreign lang="greek">t</foreign><emph type="italics"/>Z<emph.end type="italics"/>11322. Ex quibus Orbem definiendo, inveni <lb/>Nodos ejus de&longs;cendentem in <!--symbol11--> & a&longs;cendentem in <!--symbol1--> 1<emph type="sup"/>gr.<emph.end type="sup"/> 53′; <lb/>Inclinationem plani ejus ad planum Eclipticæ 61<emph type="sup"/>gr.<emph.end type="sup"/> 20′ 2/3; verti­<lb/>cem ejus (&longs;eu Perihelium Cometæ) di&longs;tare a Nodo 8<emph type="sup"/>gr.<emph.end type="sup"/> 38′, & <lb/>e&longs;&longs;e in <!--symbol12--> 27<emph type="sup"/>gr.<emph.end type="sup"/> 43′ cum latitudine au&longs;trali 7<emph type="sup"/>gr.<emph.end type="sup"/> 34′; & ejus latus <lb/>rectum e&longs;&longs;e 236,8, areamque radio ad Solem ducto &longs;ingulis diebus <lb/>de&longs;criptam 93585, quadrato &longs;emidiametri Orbis magni po&longs;ito <lb/>100000000; Cometam vero in hoc Orbe &longs;ecundum &longs;eriem &longs;igno­<lb/>rum proce&longs;&longs;i&longs;&longs;e, & <emph type="italics"/>Decemb.<emph.end type="italics"/>8<emph type="sup"/>d<emph.end type="sup"/>. </s> <s>0<emph type="sup"/>h<emph.end type="sup"/>. </s> <s>4′. </s> <s>P. M. in vertice Orbis &longs;eu <lb/>Perihelio fui&longs;&longs;e. </s> <s>Hæc omnia per &longs;calam partium æqualium & <lb/>chordas angulorum ex Tabula &longs;inuum naturalium collectas, deter­<lb/>minavi Graphice; con&longs;truendo Schema &longs;atis amplum, in quo vide­<lb/>licet &longs;emidiameter Orbis magni (partium 10000) æqualis e&longs;&longs;et <lb/>digitis 16 2/3 pedis Anglicani. </s></p> <p type="main"> <s>Tandem ut con&longs;taret an Cometa in Orbe &longs;ic invento vere mo­<lb/>veretur, collegi per operationes partim Arithmeticas partim Gra­<lb/>phicas, loca Cometæ in hoc Orbe ad ob&longs;ervationum quarundam <lb/>tempora: uti in Tabula &longs;equente videre licet. <lb/><arrow.to.target n="table11"/> </s></p><table><table.target id="table11"/><row><cell/><cell/><cell>Di&longs;tant.Co­<lb/>metæ a Sole</cell><cell>Long.Collect.</cell><cell>Lat. Collect.</cell><cell>Long. Ob&longs;.</cell><cell>Lat. Ob&longs;.</cell><cell>Differ <lb/> Long.</cell><cell>Differ. <lb/> Lat.</cell></row><row><cell/><cell/><cell/><cell>gr.</cell><cell>′</cell><cell>gr.</cell><cell>′</cell><cell>gr.</cell><cell>′</cell><cell>gr.</cell><cell>′</cell><cell>′</cell><cell>′</cell></row><row><cell><emph type="italics"/>Dec.<emph.end type="italics"/></cell><cell>12</cell><cell>2792</cell><cell><!--symbol1--> 6.</cell><cell>32</cell><cell>8.</cell><cell>18 1/2</cell><cell><!--symbol1--> 6.</cell><cell>31 1/3</cell><cell>8.</cell><cell>26</cell><cell>+ 1</cell><cell>-7 1/2</cell></row><row><cell>29</cell><cell>8403</cell><cell><!--symbol3--> 13.</cell><cell>13 2/3</cell><cell>28.</cell><cell>0</cell><cell><!--symbol3--> 13.</cell><cell>11 3/4</cell><cell>28.</cell><cell>(10 1/12)</cell><cell>+ 2</cell><cell>-(10 1/12)</cell></row><row><cell><emph type="italics"/>Febr.<emph.end type="italics"/></cell><cell>5</cell><cell>16669</cell><cell><!--symbol5--> 17.</cell><cell>0</cell><cell>15.</cell><cell>29 2/3</cell><cell><!--symbol5--> 16.</cell><cell>59 7/8</cell><cell>15.</cell><cell>27 2/5</cell><cell>+ 0</cell><cell>+ 2 1/4</cell></row><row><cell><emph type="italics"/>Mar.<emph.end type="italics"/></cell><cell>5</cell><cell>21737</cell><cell>29.</cell><cell>19 1/4</cell><cell>12.</cell><cell>4</cell><cell>29.</cell><cell>20 6/7</cell><cell>12.</cell><cell>3 1/2</cell><cell>-1</cell><cell>+ 1/2</cell></row></table> <p type="main"> <s>Po&longs;tea vero <emph type="italics"/>Halleius<emph.end type="italics"/>no&longs;ter Orbitam, per calculum Arithmeti­<lb/>cum, accuratius determinavit quam per de&longs;eriptiones linearum <lb/>fieri licuit; & retinuit quidem locum Nodorum in <!--symbol11--> & <!--symbol1--> 1<emph type="sup"/>gr.<emph.end type="sup"/> 53′, <lb/>& Inclinationem plani Orbitæ ad Eclipticam 61<emph type="sup"/>gr.<emph.end type="sup"/> 20′ 1/3, ut & tem­<lb/>pus Perihelii Cometæ <emph type="italics"/>Decemb.<emph.end type="italics"/>8<emph type="sup"/>d<emph.end type="sup"/>. </s> <s>O<emph type="sup"/>h<emph.end type="sup"/>. </s> <s>4′: di&longs;tantiam vero Peri-<pb xlink:href="039/01/487.jpg" pagenum="459"/>helii a Nodo a&longs;cendente, in Orbita Cometæ men&longs;uratam, invenit <lb/><arrow.to.target n="note488"/>e&longs;&longs;e 9<emph type="sup"/>gr<emph.end type="sup"/> 20′, & Latus rectum Parabolæ e&longs;&longs;e 243 partium, ex­<lb/>i&longs;tente mediocri Solis a Terra di&longs;tantia partium 10000. Et ex his <lb/>datis, calculo itidem Arithmetico accurate in&longs;tituto, loca Cometæ <lb/>ad ob&longs;ervationum tempora computavit, ut &longs;equitur. <lb/><arrow.to.target n="table12"/> </s></p> <p type="margin"> <s><margin.target id="note488"/>LIBER <lb/>TERTIUS.</s></p><table><table.target id="table12"/><row><cell>Tempus verum</cell><cell>Di&longs;tantia</cell><cell>Long. comp.</cell><cell>Lat. comp.</cell><cell>Errores in</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell>Cometæ a <!--symbol7--></cell><cell/><cell/><cell/><cell/><cell/><cell/><cell/><cell>Long.</cell><cell>Lat.</cell></row><row><cell/><cell>d.</cell><cell>h.</cell><cell>′</cell><cell>″</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell/><cell>′</cell><cell>″</cell><cell>′</cell><cell>″</cell></row><row><cell><emph type="italics"/>Dec.<emph.end type="italics"/></cell><cell>12.</cell><cell>4.</cell><cell>46.</cell><cell>0</cell><cell>28028</cell><cell><!--symbol1--> 6.</cell><cell>29.</cell><cell>25</cell><cell>8.</cell><cell>26.</cell><cell>0</cell><cell>Bor.</cell><cell>-1.</cell><cell>56</cell><cell>+0.</cell><cell>0</cell></row><row><cell/><cell>21.</cell><cell>6.</cell><cell>36.</cell><cell>59</cell><cell>61076</cell><cell><!--symbol2--> 5.</cell><cell>6.</cell><cell>30</cell><cell>21.</cell><cell>43.</cell><cell>20</cell><cell>-1.</cell><cell>8</cell><cell>-2.</cell><cell>10</cell></row><row><cell/><cell>24.</cell><cell>6.</cell><cell>17.</cell><cell>52</cell><cell>70008</cell><cell>18.</cell><cell>48.</cell><cell>20</cell><cell>15.</cell><cell>22.</cell><cell>40</cell><cell>-0.</cell><cell>50</cell><cell>-0.</cell><cell>44</cell></row><row><cell/><cell>26.</cell><cell>5.</cell><cell>20.</cell><cell>44</cell><cell>75576</cell><cell>28.</cell><cell>22.</cell><cell>45</cell><cell>27.</cell><cell>1.</cell><cell>36</cell><cell>-1.</cell><cell>21</cell><cell>+0.</cell><cell>39</cell></row><row><cell/><cell>29.</cell><cell>8.</cell><cell>3.</cell><cell>2</cell><cell>84021</cell><cell><!--symbol3--> 13.</cell><cell>12.</cell><cell>40</cell><cell>28.</cell><cell>10.</cell><cell>10</cell><cell>+0.</cell><cell>55</cell><cell>+0.</cell><cell>5</cell></row><row><cell/><cell>30.</cell><cell>8.</cell><cell>10.</cell><cell>26</cell><cell>86661</cell><cell>17.</cell><cell>40.</cell><cell>5</cell><cell>28.</cell><cell>11.</cell><cell>20</cell><cell>+1.</cell><cell>0</cell><cell>+0.</cell><cell>8</cell></row><row><cell><emph type="italics"/>Jan.<emph.end type="italics"/></cell><cell>5.</cell><cell>6.</cell><cell>1.</cell><cell>38</cell><cell>101440</cell><cell><!--symbol4--> 8.</cell><cell>49.</cell><cell>49</cell><cell>26.</cell><cell>15.</cell><cell>15</cell><cell>+0.</cell><cell>39</cell><cell>-0.</cell><cell>11</cell></row><row><cell/><cell>9.</cell><cell>7.</cell><cell>0.</cell><cell>53</cell><cell>110959</cell><cell>18.</cell><cell>44.</cell><cell>36</cell><cell>24.</cell><cell>12.</cell><cell>54</cell><cell>+1.</cell><cell>18</cell><cell>+0.</cell><cell>12</cell></row><row><cell/><cell>10.</cell><cell>6.</cell><cell>6.</cell><cell>10</cell><cell>113162</cell><cell>20.</cell><cell>41.</cell><cell>0</cell><cell>23.</cell><cell>44.</cell><cell>10</cell><cell>+0.</cell><cell>3</cell><cell>+0.</cell><cell>10</cell></row><row><cell/><cell>13.</cell><cell>7.</cell><cell>8.</cell><cell>55</cell><cell>120000</cell><cell>26.</cell><cell>0.</cell><cell>21</cell><cell>22.</cell><cell>17.</cell><cell>30</cell><cell>+0.</cell><cell>47</cell><cell>-0.</cell><cell>6</cell></row><row><cell/><cell>25.</cell><cell>7.</cell><cell>58.</cell><cell>42</cell><cell>145370</cell><cell><!--symbol5--> 9.</cell><cell>33.</cell><cell>40</cell><cell>17.</cell><cell>57.</cell><cell>55</cell><cell>-2.</cell><cell>8</cell><cell>+1.</cell><cell>1</cell></row><row><cell/><cell>30.</cell><cell>8.</cell><cell>21.</cell><cell>53</cell><cell>155303</cell><cell>13.</cell><cell>17.</cell><cell>41</cell><cell>16.</cell><cell>42.</cell><cell>7</cell><cell>-1.</cell><cell>55</cell><cell>+1.</cell><cell>10</cell></row><row><cell><emph type="italics"/>Feb.<emph.end type="italics"/></cell><cell>2.</cell><cell>6.</cell><cell>34.</cell><cell>51</cell><cell>160951</cell><cell>15.</cell><cell>11.</cell><cell>11</cell><cell>16.</cell><cell>4.</cell><cell>15</cell><cell>-2.</cell><cell>37</cell><cell>+2.</cell><cell>13</cell></row><row><cell/><cell>5.</cell><cell>7.</cell><cell>4.</cell><cell>41</cell><cell>166686</cell><cell>16.</cell><cell>58.</cell><cell>25</cell><cell>15.</cell><cell>29.</cell><cell>13</cell><cell>-1.</cell><cell>27</cell><cell>+1.</cell><cell>50</cell></row><row><cell/><cell>25.</cell><cell>8.</cell><cell>19.</cell><cell>0</cell><cell>202570</cell><cell>26.</cell><cell>15.</cell><cell>46</cell><cell>12.</cell><cell>48.</cell><cell>0</cell><cell>-2.</cell><cell>31</cell><cell>+1.</cell><cell>8</cell></row><row><cell><emph type="italics"/>Mar.<emph.end type="italics"/></cell><cell>5.</cell><cell>11.</cell><cell>21.</cell><cell>0</cell><cell>216205</cell><cell>29.</cell><cell>18.</cell><cell>35</cell><cell>12.</cell><cell>5.</cell><cell>40</cell><cell>-2.</cell><cell>16</cell><cell>+2.</cell><cell>10</cell></row></table> <p type="main"> <s>Apparuit etiam hic Cometa men&longs;e <emph type="italics"/>Novembri<emph.end type="italics"/>præcedente, & <lb/>die undecimo hujus men&longs;is &longs;tylo veteri, ad horam quintam ma­<lb/>tutinam, <emph type="italics"/>Cantuariæ<emph.end type="italics"/>in <emph type="italics"/>Anglia,<emph.end type="italics"/>vi&longs;us fuit in <!--symbol13--> 12 1/2 cum latitudine <lb/>boreali 2<emph type="sup"/>gr.<emph.end type="sup"/> circiter. </s> <s>Cra&longs;&longs;i&longs;&longs;ima fuit hæc Ob&longs;ervatio: meliores &longs;unt <lb/>quæ &longs;equuntur. </s></p> <p type="main"> <s><emph type="italics"/>Nov.<emph.end type="italics"/>17, &longs;t. </s> <s>vet. <emph type="italics"/>Pontbæus<emph.end type="italics"/>& &longs;ocii hora &longs;exta matutina <emph type="italics"/>Romæ<emph.end type="italics"/><lb/>(id e&longs;t, hora 5, 10′ <emph type="italics"/>Londini<emph.end type="italics"/>) filis ad fixas applicatis Cometam <lb/>ob&longs;ervarunt in <!--symbol14--> 8. 30′, cum latitudine au&longs;trali 0<emph type="sup"/>gr.<emph.end type="sup"/> 40′. </s> <s>Extant <lb/>eorum Ob&longs;ervationes in tractatu quem <emph type="italics"/>Penthæus,<emph.end type="italics"/>de hoc Cometa, <lb/>in lucem edidit. <emph type="italics"/>Cellius<emph.end type="italics"/>qui aderat & ob&longs;ervationes &longs;uas in Epi­<lb/>&longs;tola ad <emph type="italics"/>D. Ca&longs;&longs;inum<emph.end type="italics"/>mi&longs;it, Cometam eadem hora vidit in <!--symbol14--> 8 <emph type="sup"/>gr.<emph.end type="sup"/><lb/>30′ cum latitudine au&longs;trali 0<emph type="sup"/>gr.<emph.end type="sup"/> 30′. </s> <s>Eadem hora <emph type="italics"/>Galletius<emph.end type="italics"/>etiam <lb/>Cometam vidit in <!--symbol14--> 8<emph type="sup"/>gr.<emph.end type="sup"/> &longs;ine latitudine. </s></p> <p type="main"> <s><emph type="italics"/>Nov.<emph.end type="italics"/>18. hora matutina 6. 30′ <emph type="italics"/>Romæ<emph.end type="italics"/>(id e&longs;t, hora 5, 40′ <emph type="italics"/>Lon­<lb/>dini) Ponthæus<emph.end type="italics"/>Cometam vidit in <!--symbol14--> 13<emph type="sup"/>gr.<emph.end type="sup"/> 30′ cum latitudine au­<lb/>&longs;trali 1<emph type="sup"/>gr.<emph.end type="sup"/> 20′. <emph type="italics"/>Cellius<emph.end type="italics"/>in <!--symbol14--> 13<emph type="sup"/>gr.<emph.end type="sup"/> 00′, cum latitudine au&longs;trali <lb/>1<emph type="sup"/>gr.<emph.end type="sup"/> 00′. <emph type="italics"/>Galletius<emph.end type="italics"/>autem hora matutina 5. 30′ <emph type="italics"/>Romæ,<emph.end type="italics"/>Cometam <lb/>vidit in <!--symbol14--> 13<emph type="sup"/>gr.<emph.end type="sup"/> 00′, cum latitudine au&longs;trali 1<emph type="sup"/>gr.<emph.end type="sup"/> 00′. </s> <s>Et <emph type="italics"/>R. P. <lb/>Ango<emph.end type="italics"/>in Academia <emph type="italics"/>Flexien&longs;i<emph.end type="italics"/>apud <emph type="italics"/>Galles,<emph.end type="italics"/>hora quinta matutina <lb/>(id e&longs;t, hora 5, 9′ <emph type="italics"/>Londini<emph.end type="italics"/>) Cometam vidit in medio inter &longs;tellas <pb xlink:href="039/01/488.jpg" pagenum="460"/><arrow.to.target n="note489"/>duas parvas, quarum una media e&longs;t trium in recta linea in Virgi­<lb/>nis au&longs;trali manu, & altera e&longs;t extrema alæ. </s> <s>Unde Cometa tunc <lb/>fuit in <!--symbol14--> 12. 46′, cum latitudine au&longs;trali 50′. </s> <s>Eodem die <emph type="italics"/>Bo­<lb/>&longs;toniæ<emph.end type="italics"/>in <emph type="italics"/>Nova-Anglia<emph.end type="italics"/>in Latitudine 42 1/2 graduum, hora quinta <lb/>matutina, (id e&longs;t <emph type="italics"/>Londini<emph.end type="italics"/>hora matutina 9. 44′) Cometa vi&longs;us <lb/>e&longs;t prope <!--symbol14--> 14, cum latitudine au&longs;trali 1<emph type="sup"/>gr.<emph.end type="sup"/> 30′, uti a <emph type="italics"/>Cl. </s> <s>Hal­<lb/>leio<emph.end type="italics"/>accepi. </s></p> <p type="margin"> <s><margin.target id="note489"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Nov.<emph.end type="italics"/>19. hora mat. </s> <s>4 1/2 <emph type="italics"/>Cantabrigiæ,<emph.end type="italics"/>Cometa (ob&longs;ervante ju­<lb/>vene quodam) di&longs;tabat a Spica <!--symbol13--> qua&longs;i 2<emph type="sup"/>gr.<emph.end type="sup"/> Boreazephyrum <lb/>ver&longs;us. </s> <s>Eodem die hor. </s> <s>5. mat. <emph type="italics"/>Bo&longs;toniæ<emph.end type="italics"/>in <emph type="italics"/>Nova-Anglia,<emph.end type="italics"/>Co­<lb/>meta di&longs;tabat a Spica <!--symbol13--> gradu uno, differentia latitudinum ex­<lb/>i&longs;tente 40′. </s> <s>Eodem die in In&longs;ula <emph type="italics"/>Jamaica,<emph.end type="italics"/>Cometa di&longs;tabat a Spica <lb/>intervallo qua&longs;i gradus unius. </s> <s>Et ex his ob&longs;ervationibus inter &longs;e <lb/>collatis colligo, quod hora 9. 44′. <emph type="italics"/>Londini,<emph.end type="italics"/>Cometa erat in <!--symbol14--> 18 <emph type="sup"/>gr.<emph.end type="sup"/><lb/>40′, cum latitudine au&longs;trali 1 <emph type="sup"/>gr.<emph.end type="sup"/> 18′ circiter. </s> <s>Eodem die D. <emph type="italics"/>Ar­<lb/>thurus Storer<emph.end type="italics"/>ad fluvium <emph type="italics"/>Patuxent,<emph.end type="italics"/>prope <emph type="italics"/>Hunting-Creek<emph.end type="italics"/>in <emph type="italics"/>Mary­<lb/>Land,<emph.end type="italics"/>in confinio <emph type="italics"/>Virginiæ<emph.end type="italics"/>in Lat. </s> <s>38 1/2<emph type="sup"/>gr.<emph.end type="sup"/> hora quinta matutina <lb/>(id e&longs;t, hora 10<emph type="sup"/>2<emph.end type="sup"/> <emph type="italics"/>Londini<emph.end type="italics"/>) Cometam vidit &longs;upra Spicam <!--symbol13-->, & <lb/>cum Spica propemodum conjunctum, exi&longs;tente di&longs;tantia inter eo&longs;­<lb/>dem qua&longs;i 3/4<emph type="sup"/>gr.<emph.end type="sup"/>. </s> <s>Ob&longs;ervator idem, eadem hora diei &longs;equentis, <lb/>Cometam vidit qua&longs;i 2<emph type="sup"/>gr.<emph.end type="sup"/> inferiorem Spica. </s> <s>Congruent hæ ob­<lb/>&longs;ervationes cum ob&longs;ervationibus in <emph type="italics"/>Nova-Anglia<emph.end type="italics"/>& <emph type="italics"/>Jamaica<emph.end type="italics"/>factis, <lb/>&longs;i modo di&longs;tantiæ (pro motu diurno Cometæ) nonnihil augean­<lb/>tur, ita ut Cometa die priore &longs;uperior e&longs;&longs;et Spica <!--symbol13-->, altitudine <lb/>1 <emph type="sup"/>gr.<emph.end type="sup"/> circiter, ac die po&longs;teriore inferior eadem &longs;tella, altitudine per­<lb/>pendiculari 3 <emph type="sup"/>gr.<emph.end type="sup"/> 40′. </s></p> <p type="main"> <s><emph type="italics"/>Nov.<emph.end type="italics"/>20. D. <emph type="italics"/>Montenarus<emph.end type="italics"/>A&longs;tronomiæ Profe&longs;&longs;or <emph type="italics"/>Paduen&longs;is,<emph.end type="italics"/>hora <lb/>&longs;exta matutina <emph type="italics"/>Venetiis<emph.end type="italics"/>(id e&longs;t, hora 5. 10′ <emph type="italics"/>Londini<emph.end type="italics"/>) Cometam <lb/>vidit in <!--symbol14--> 23 <emph type="sup"/>gr.<emph.end type="sup"/>, cum latitudine au&longs;trali 1 <emph type="sup"/>gr.<emph.end type="sup"/> 30′. </s> <s>Eodem die <lb/><emph type="italics"/>Bo&longs;toniæ,<emph.end type="italics"/>di&longs;tabat Cometa a Spica <!--symbol13-->, 4<emph type="sup"/>gr.<emph.end type="sup"/> longitudinis in orien­<lb/>tem, adeoque erat in <!--symbol14--> 23 <emph type="sup"/>gr.<emph.end type="sup"/> 24′ circiter. </s></p> <p type="main"> <s><emph type="italics"/>Nov.<emph.end type="italics"/>21. <emph type="italics"/>Ponthæus<emph.end type="italics"/>& &longs;ocii hor. </s> <s>mat. </s> <s>7 1/4 Cometam ob&longs;erva­<lb/>runt in <!--symbol14--> 27<emph type="sup"/>gr.<emph.end type="sup"/> 50′, cum latitudine au&longs;trali 1 <emph type="sup"/>gr.<emph.end type="sup"/> 16′; <emph type="italics"/>Ango<emph.end type="italics"/>hora <lb/>quinta matutina in <!--symbol14--> 27<emph type="sup"/>gr.<emph.end type="sup"/> 45′, <emph type="italics"/>Montenarus<emph.end type="italics"/>in <!--symbol14--> 27<emph type="sup"/>gr.<emph.end type="sup"/> 51′. </s> <s>Eo­<lb/>dem die in In&longs;ula <emph type="italics"/>Jamaica,<emph.end type="italics"/>Cometa vi&longs;us e&longs;t prope principium <lb/>Scorpii, eandemque circiter latitudinem habuit cum Spica Virgi­<lb/>nis, id e&longs;t, 2<emph type="sup"/>gr.<emph.end type="sup"/> 2′. </s></p> <p type="main"> <s><emph type="italics"/>Nov.<emph.end type="italics"/>22. Cometa vi&longs;us e&longs;t a <emph type="italics"/>Montenaro<emph.end type="italics"/>in <!--symbol15--> 2. 33′. <emph type="italics"/>Bo&longs;toniæ<emph.end type="italics"/><lb/>autem in <emph type="italics"/>Nova-Anglia<emph.end type="italics"/>apparuit in <!--symbol15--> 3<emph type="sup"/>gr.<emph.end type="sup"/> circiter, eadem fere <lb/>cum latitudine ac prius, id e&longs;t, 1 <emph type="sup"/>gr.<emph.end type="sup"/> 30′. </s> <s>Eodem die <emph type="italics"/>Londini,<emph.end type="italics"/><pb xlink:href="039/01/489.jpg" pagenum="461"/>hora mat. </s> <s>6 1/2 <emph type="italics"/>Hookius<emph.end type="italics"/>no&longs;ter Cometam vidit in <!--symbol15--> 3<emph type="sup"/>gr.<emph.end type="sup"/> 30′ cir­<lb/><arrow.to.target n="note490"/>citer, idQ.E.I. linea recta quæ tran&longs;it per Spicam Virginis & <lb/>Cor Leonis, non exacte quidem, &longs;ed a linea illa paululum defle­<lb/>ctentem ad boream. <emph type="italics"/>Montenarus<emph.end type="italics"/>itidem notavit quod linea a <lb/>Cometa per Spicam ducta, hoc die & &longs;equentibus tran&longs;ibat per <lb/>au&longs;trale latus Cordis Leonis, interpo&longs;ito perparvo intervallo inter <lb/>Cor Leonis & hanc lineam. </s> <s>Linea recta per Cor Leonis & <lb/>Spicam Virginis tran&longs;iens, Eclipticam &longs;ecuit in <!--symbol13--> 3<emph type="sup"/>gr.<emph.end type="sup"/> 46′, in an­<lb/>gulo 2<emph type="sup"/>gr.<emph.end type="sup"/> 51′. </s> <s>Et &longs;i Cometa locatus fui&longs;&longs;et in hac linea in <!--symbol15--> 3 <emph type="sup"/>gr.<emph.end type="sup"/>, <lb/>ejus latitudo fui&longs;&longs;et 2 <emph type="sup"/>gr<emph.end type="sup"/> 26′. </s> <s>Sed cum Cometa con&longs;entientibus <lb/><emph type="italics"/>Hookio<emph.end type="italics"/>& <emph type="italics"/>Montenaro,<emph.end type="italics"/>nonnihil di&longs;taret ab hac linea boream ver­<lb/>&longs;us, latitudo ejus fuit paulo minor. </s> <s>Die 20. ex ob&longs;ervatione <emph type="italics"/>Mon­<lb/>tenari,<emph.end type="italics"/>latitudo ejus propemodum æquabat latitudinem Spicæ <!--symbol13-->, <lb/>eratque 1<emph type="sup"/>gr.<emph.end type="sup"/> 30′ circiter, & con&longs;entientibus <emph type="italics"/>Hookio, Montenaro<emph.end type="italics"/>& <lb/><emph type="italics"/>Angone<emph.end type="italics"/>perpetuo augebatur, ideoque jam &longs;en&longs;ibiliter major erat <lb/>quam 1<emph type="sup"/>gr.<emph.end type="sup"/> 30′. </s> <s>Inter limites autem jam con&longs;titutos 2<emph type="sup"/>gr.<emph.end type="sup"/> 26′ & <lb/>1<emph type="sup"/>gr.<emph.end type="sup"/> 30′, magnitudine mediocri latitudo erit 1<emph type="sup"/>gr.<emph.end type="sup"/> 58′ circiter. </s> <s><lb/>Cauda Cometæ, con&longs;entientibus <emph type="italics"/>Hookio<emph.end type="italics"/>& <emph type="italics"/>Montenaro,<emph.end type="italics"/>dirigebatur <lb/>ad Spicam <!--symbol13-->, declinans aliquantulum a Stella i&longs;ta, juxta <emph type="italics"/>Hookium<emph.end type="italics"/><lb/>in au&longs;trum, juxta <emph type="italics"/>Montenarum<emph.end type="italics"/>in boream; ideoQ.E.D.clinatio illa <lb/>vix fuit &longs;en&longs;ibilis, & Cauda Æquatori fere parallela exi&longs;tens, ali­<lb/>quantulum deflectebatur ab oppo&longs;itione Solis boream ver&longs;us. </s></p> <p type="margin"> <s><margin.target id="note490"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="italics"/>Nov.<emph.end type="italics"/>24. Ante ortum Solis Cometa vi&longs;us e&longs;t a <emph type="italics"/>Montenaro<emph.end type="italics"/><lb/>in <!--symbol15--> 12<emph type="sup"/>gr.<emph.end type="sup"/> 52′, ad boreale latus rectæ quæ per Cor Leonis & Spicam <lb/>Virginis ducebatur, ideoque latitudinem habuit paulo minorem <lb/>quam 2<emph type="sup"/>gr.<emph.end type="sup"/> 38′. </s> <s>Hæc latitudo uti diximus, ex ob&longs;ervationibus <lb/><emph type="italics"/>Montenari, Angonis<emph.end type="italics"/>& <emph type="italics"/>Hookii,<emph.end type="italics"/>perpetuo augebatur; ideoque jam <lb/>paulo major erat quam 1<emph type="sup"/>gr.<emph.end type="sup"/> 58′; & magnitudine mediocri, ab&longs;que <lb/>notabili errore, &longs;tatui pote&longs;t 2<emph type="sup"/>gr.<emph.end type="sup"/> 18′. </s> <s>Latitudinem <emph type="italics"/>Ponthæus<emph.end type="italics"/>& <lb/><emph type="italics"/>Galletius<emph.end type="italics"/>jam decrevi&longs;&longs;e volunt, & <emph type="italics"/>Cellius<emph.end type="italics"/>& Ob&longs;ervator in <emph type="italics"/>Nova­<lb/>Anglia<emph.end type="italics"/>eandem fere magnitudinem retinui&longs;&longs;e, &longs;cilicet gradus unius <lb/>vel unius cum &longs;emi&longs;&longs;e. </s> <s>Cra&longs;&longs;iores &longs;unt ob&longs;ervationes <emph type="italics"/>Ponthæi<emph.end type="italics"/>& <lb/><emph type="italics"/>Cellii,<emph.end type="italics"/>eæ præ&longs;ertim quæ per Azimuthes & Altitudines capieban­<lb/>tur, ut & eæ <emph type="italics"/>Galletii<emph.end type="italics"/>: meliores &longs;unt eæ quæ per po&longs;itiones Co­<lb/>metæ ad fixas a <emph type="italics"/>Montenaro, Hookio, Angone<emph.end type="italics"/>& Ob&longs;ervatore in <lb/><emph type="italics"/>Nova-Anglia,<emph.end type="italics"/>& nonnunquam a <emph type="italics"/>Ponthæo<emph.end type="italics"/>& <emph type="italics"/>Cellio<emph.end type="italics"/>&longs;unt factæ. </s></p> <p type="main"> <s>Jam collatis Ob&longs;ervationibus inter &longs;e, colligere videor quod <lb/>Cometa hoc men&longs;e circulum fere maximum de&longs;crip&longs;it, &longs;ecantem <lb/>Eclipticam in <!--symbol13--> 25. 12′, idQ.E.I. angulo 3<emph type="sup"/>gr.<emph.end type="sup"/> 12′ quamproxime. </s> <s><lb/>Nam & <emph type="italics"/>Montenarus<emph.end type="italics"/>Orbitam ab Ecliptica in au&longs;trum, tribus &longs;al-<pb xlink:href="039/01/490.jpg" pagenum="462"/><arrow.to.target n="note491"/>tem gradibus declina&longs;&longs;e dicit. </s> <s>Et cognita cur&longs;us po&longs;itione, lon­<lb/>gitudines Cometæ ex ob&longs;ervationibus collectæ, ad incudem jam <lb/>revocari po&longs;&longs;unt & melius nonnunquam determinari, ut &longs;it in &longs;e­<lb/>quentibus. <emph type="italics"/>Cellius<emph.end type="italics"/>Novemb. </s> <s>17. ob&longs;ervavit di&longs;tantiam Cometæ a <lb/>Spica <!--symbol13-->, æqualem e&longs;&longs;e di&longs;tantiæ ejus a &longs;tella lucida in dextra ala <lb/>Corvi: & hinc locandus e&longs;t Cometa in inter&longs;ectione hujus circuli <lb/>quem Cometa motu apparente de&longs;crip&longs;it, cum circulo maximo <lb/>qui a fixis illis duabus æqualiter di&longs;tat, atque adeo in <!--symbol14--> 7<emph type="sup"/>gr.<emph.end type="sup"/> 54′, <lb/>cum latitudine au&longs;trali 43′. </s> <s>Præterea <emph type="italics"/>Montenarus, Novemb.<emph.end type="italics"/>20. <lb/>hora &longs;exta matutina <emph type="italics"/>Venetiis,<emph.end type="italics"/>Cometam vidit non totis quatuor <lb/>gradibus di&longs;tantiam a Spica; dicitque hanc di&longs;tantiam, vix æqua&longs;&longs;e <lb/>di&longs;tantiam &longs;tellarum duarum lucidarum in alis Corvi, vel duarum <lb/>in juba Leonis, hoc e&longs;t 3<emph type="sup"/>gr.<emph.end type="sup"/> & 30′ vel 32′. </s> <s>Sit igitur di&longs;tantia <lb/>Cometæ a Spica 3<emph type="sup"/>gr.<emph.end type="sup"/> 30′, & Cometa locabitur in <!--symbol14--> 22<emph type="sup"/>gr.<emph.end type="sup"/> 48′, cum <lb/>latitudine au&longs;trali 1<emph type="sup"/>gr.<emph.end type="sup"/> 30′. </s> <s>Adhæc <emph type="italics"/>Montenarus, Novemb.<emph.end type="italics"/>21, 22, <lb/>24 & 25 ante ortum Solis, Sextante æneo quintupedali ad mi­<lb/>nuta prima & &longs;emiminuta divi&longs;o & vitris Tele&longs;copicis armato, <lb/>di&longs;tantias men&longs;uravit Cometæ a Spica 8<emph type="sup"/>gr<emph.end type="sup"/> 28′, 13<emph type="sup"/>gr.<emph.end type="sup"/> 10′, 23<emph type="sup"/>gr.<emph.end type="sup"/><lb/>30′, & 28<emph type="sup"/>gr.<emph.end type="sup"/> 13′: & has di&longs;tantias, per refractionem nondum cor­<lb/>rectas, addendo longitudini Spicæ, collegit Cometam his tempo­<lb/>ribus fui&longs;&longs;e in <!--symbol14--> 27<emph type="sup"/>gr.<emph.end type="sup"/> 51′, <!--symbol15--> 2<emph type="sup"/>gr.<emph.end type="sup"/> 33′, <!--symbol15--> 12<emph type="sup"/>gr.<emph.end type="sup"/> 52′ & <!--symbol15--> 17<emph type="sup"/>gr.<emph.end type="sup"/> 45′. </s> <s><lb/>Si di&longs;tantiæ illæ per refractiones corrigantur, & ex di&longs;tantiis cor­<lb/>rectis differentiæ longitudinum inter Spicam & Cometam probe <lb/>deriventur, locabitur Cometa his temporibus in <!--symbol14--> 27<emph type="sup"/>gr.<emph.end type="sup"/> 52′, <lb/><!--symbol15--> 2<emph type="sup"/>gr.<emph.end type="sup"/> 36′, <!--symbol15--> 12<emph type="sup"/>gr.<emph.end type="sup"/> 58′ & <!--symbol15--> 17<emph type="sup"/>gr.<emph.end type="sup"/> 53′ circiter. </s> <s>Latitudines au­<lb/>tem ad has longitudines in via Cometæ captas, prodeunt 1 <emph type="sup"/>gr.<emph.end type="sup"/> 45′, <lb/>1<emph type="sup"/>gr.<emph.end type="sup"/> 58′, 2<emph type="sup"/>gr.<emph.end type="sup"/> 22′ & 2<emph type="sup"/>gr.<emph.end type="sup"/> 31′. </s> <s>Harum quatuor ob&longs;ervationum ho­<lb/>ras matutinas <emph type="italics"/>Montenarus<emph.end type="italics"/>non po&longs;uit. </s> <s>Priores duæ ante ho­<lb/>ram &longs;extam, po&longs;teriores (ob viciniam Solis) po&longs;t &longs;extam factæ <lb/>videntur. </s> <s>Die 22, ubi Cometa ex ob&longs;ervatione <emph type="italics"/>Montenari<emph.end type="italics"/>loca­<lb/>tur in <!--symbol15--> 2<emph type="sup"/>gr.<emph.end type="sup"/> 36′, <emph type="italics"/>Hookius<emph.end type="italics"/>no&longs;ter eundem locavit in <!--symbol15--> 3<emph type="sup"/>gr.<emph.end type="sup"/> 30′ <lb/>ut &longs;upra. <emph type="italics"/>Montenarus<emph.end type="italics"/>in defectu, <emph type="italics"/>Hookius<emph.end type="italics"/>in exce&longs;&longs;u erra&longs;&longs;e viden­<lb/>tur. </s> <s>Nam Cometa, ex &longs;erie ob&longs;ervationum, jam fuit in <!--symbol15--> 3<emph type="sup"/>gr.<emph.end type="sup"/> 56′ <lb/>vel <!--symbol15--> 3<emph type="sup"/>gr.<emph.end type="sup"/> circiter. </s></p> <p type="margin"> <s><margin.target id="note491"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Ob&longs;ervationum &longs;uarum ultimam inter vapores & diluculum <lb/>captam, <emph type="italics"/>Montenarus<emph.end type="italics"/>&longs;u&longs;pectam habebat. </s> <s>Et <emph type="italics"/>Cellius<emph.end type="italics"/>eodem tem­<lb/>pore (id e&longs;t, <emph type="italics"/>Novem.<emph.end type="italics"/>25) Cometam per ejus Altitudinem & Azi­<lb/>muthum locavit in <!--symbol15--> 15<emph type="sup"/>gr.<emph.end type="sup"/> 47′, cum latitudine au&longs;trali qua&longs;i gra­<lb/>dus unius Sed <emph type="italics"/>Cellius<emph.end type="italics"/>ob&longs;ervavit etiam eodem tempore, quod <lb/>Cometa erat in linea recta cum &longs;tella lucida in dextro &longs;emore<pb xlink:href="039/01/491.jpg" pagenum="463"/>Virginis & cum Lance au&longs;trali Libræ, & hæc linea &longs;ecat viam <lb/><arrow.to.target n="note492"/>Cometæ in <!--symbol15--> 18<emph type="sup"/>gr.<emph.end type="sup"/> 36′. <emph type="italics"/>Ponthæus<emph.end type="italics"/>etiam eodem tempore ob&longs;er­<lb/>vavit, quod Cometa erat in recta tran&longs;eunte per Chelam au&longs;tri <lb/>nam Scorpii & per &longs;tellam quæ Lancem borealem &longs;equitur: & <lb/>hæc recta &longs;ecat viam Cometæ in <!--symbol15--> 16<emph type="sup"/>gr.<emph.end type="sup"/> 34′. </s> <s>Ob&longs;ervavit etiam, <lb/>quod Cometa erat in recta tran&longs;eunte per &longs;tellam &longs;upra Lancem <lb/>au&longs;tralem Libræ & &longs;tellam in principio pedis &longs;ecundi Scorpii: & <lb/>hæc recta &longs;ecat viam Cometæ in <!--symbol15--> 17<emph type="sup"/>gr.<emph.end type="sup"/> 55′. </s> <s>Et inter longitu­<lb/>dines ex his tribus Ob&longs;ervationibus &longs;ic derivatas, longitudo me­<lb/>diocris e&longs;t <!--symbol15--> 17<emph type="sup"/>gr.<emph.end type="sup"/> 42′, quæ cum ob&longs;ervatione <emph type="italics"/>Montenari<emph.end type="italics"/>&longs;atis <lb/>congruit. </s></p> <p type="margin"> <s><margin.target id="note492"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Erravit igitur <emph type="italics"/>Cellius<emph.end type="italics"/>jam locando Cometam in <!--symbol15--> 15<emph type="sup"/>gr.<emph.end type="sup"/> 47′, <lb/>per ejus Azimuthum & Altitudinem. </s> <s>Et &longs;imilibus Azimuthorum <lb/>& Altitudinum ob&longs;ervationibus, <emph type="italics"/>Cellius<emph.end type="italics"/>& <emph type="italics"/>Ponthæus<emph.end type="italics"/>non minus <lb/>erraverunt locando Cometam in <!--symbol15--> 20 & <!--symbol15--> 24 diebus duobus <lb/>&longs;equentibus, ubi &longs;tellæ fixæ ob diluculum vix aut ne vix quidem <lb/>apparuere. </s> <s>Et corrigendæ &longs;unt hæ ob&longs;ervationes per additionem <lb/>duorum graduum, vel duorum cum &longs;emi&longs;&longs;e. </s></p> <p type="main"> <s>Ex omnibus autem Ob&longs;ervationibus inter &longs;e collatis & ad meri­<lb/>dianum <emph type="italics"/>Londini<emph.end type="italics"/>reductis, colligo Cometam huju&longs;modi cur&longs;um <lb/>quamproxime de&longs;crip&longs;i&longs;&longs;e. <lb/><arrow.to.target n="table13"/> </s></p><table><table.target id="table13"/><row><cell>Temp. med. &longs;t. vet.</cell><cell>Long. Cometæ</cell><cell>Lat. Cometæ</cell></row><row><cell/><cell>d.</cell><cell>h.</cell><cell>′</cell><cell>gr.</cell><cell>′</cell><cell>gr.</cell><cell>′</cell><cell/></row><row><cell><emph type="italics"/>Nov.<emph.end type="italics"/></cell><cell>15.</cell><cell>17.</cell><cell>10</cell><cell><!--symbol14--> 8.</cell><cell>0</cell><cell>0.</cell><cell>44</cell><cell>Au&longs;t.</cell></row><row><cell>17.</cell><cell>17.</cell><cell>10</cell><cell>12.</cell><cell>52</cell><cell>1.</cell><cell>0</cell></row><row><cell>18</cell><cell>21.</cell><cell>44</cell><cell>18.</cell><cell>40</cell><cell>1.</cell><cell>18</cell></row><row><cell>19</cell><cell>17.</cell><cell>10</cell><cell>22.</cell><cell>48</cell><cell>2.</cell><cell>30</cell></row><row><cell>20.</cell><cell>17</cell><cell>fere</cell><cell>27.</cell><cell>52</cell><cell>1.</cell><cell>48</cell></row><row><cell>22.</cell><cell>17</cell><cell>fere</cell><cell><!--symbol15--> 2.</cell><cell>56</cell><cell>1.</cell><cell>38</cell></row><row><cell>27.</cell><cell>17 1/4</cell><cell>&longs;ere</cell><cell>12.</cell><cell>58</cell><cell>2.</cell><cell>20</cell></row><row><cell>24.</cell><cell>17 1/2</cell><cell>&longs;ere</cell><cell>17.</cell><cell>53</cell><cell>2.</cell><cell>23</cell></row><row><cell>26.</cell><cell>18.</cell><cell>00</cell><cell>26 vel 27<emph type="sup"/>gr.<emph.end type="sup"/></cell><cell>2.</cell><cell>42</cell></row></table> <p type="main"> <s>Loca autem Cometæ in Orbe Parabolice computata, ita &longs;e habent. <lb/> <!-- tabelle fehlt--> <arrow.to.target n="table14"/> <pb xlink:href="039/01/492.jpg" pagenum="464"/><arrow.to.target n="note493"/></s></p> <p type="margin"> <s><margin.target id="note493"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Congruunt igitur Ob&longs;ervationes A&longs;tronomicæ, tam men&longs;e <emph type="italics"/>No­<lb/>vembri<emph.end type="italics"/>quam men&longs;ibus quatuor &longs;equentibus, cum motu Cometæ <lb/>circum Solem in Trajectoria hacce Parabolica, atque adeo unum <lb/>& cundem Cometam fui&longs;&longs;e, qui men&longs;e <emph type="italics"/>Novembri<emph.end type="italics"/>ad Solem de&longs;cen­<lb/>dir, & men&longs;ibus &longs;equentibus ab vodem a&longs;cendit, abunde confir­<lb/>mant, ut & hunc Cometam in Trajectoria hacce Parabolica dela­<lb/>tum fui&longs;&longs;e quamproxime. </s> <s>Men&longs;ibas <emph type="italics"/>Decembri, Januario, Fe­<lb/>bruario<emph.end type="italics"/>& <emph type="italics"/>Martio,<emph.end type="italics"/>ubi Ob&longs;ervationes hujus Cometæ &longs;unt &longs;atis ac­<lb/>curatæ, congruunt eædem cum motu ejus in hac Trajectoria, non <lb/>minus accurate quam ob&longs;ervationes Planetarum congruere &longs;olent <lb/>cum eorum Theoriis. </s> <s>Men&longs;e <emph type="italics"/>Novembri,<emph.end type="italics"/>ubi ob&longs;ervationes &longs;unt <lb/>cra&longs;&longs;æ, errores non &longs;unt majores quam qui cra&longs;&longs;itudini ob&longs;erva­<lb/>tionum tribuantur. </s> <s>Trajectoria Cometæ bis &longs;ecuit planum Eclip­<lb/>ticæ, & propterea non fuit rectilinea. </s> <s>Eclipticam &longs;ecuit non in <lb/>oppo&longs;itis cœli partibus, &longs;ed in fine Virginis & principio Capri­<lb/>corni, intervallo graduum 98 circiter; ideoque cur&longs;us Cometæ <lb/>plurimum deflectebatur a Circulo maximo. </s> <s>Nam & men&longs;e <emph type="italics"/>No­<lb/>vembri<emph.end type="italics"/>cur&longs;us ejus tribus &longs;altem gradibus ab Ecliptica in au&longs;trum <lb/>declinabat, & po&longs;tea men&longs;e <emph type="italics"/>Decembri<emph.end type="italics"/>gradibus 29 vergebat ab <lb/>Ecliptica in &longs;eptentrionem, partibus duabus Orbitæ in quibus <lb/>Cometa tendebat in Solem & redibat a Sole, angulo apparente <lb/>graduum plus triginta ab invicem declinantibus, ut ob&longs;ervavit <lb/><emph type="italics"/>Montenarus.<emph.end type="italics"/>Pergebat hic Cometa per &longs;igna fere novem, a Vir­<lb/>ginis &longs;cilicet duodecimo gradu ad principium Geminorum, præ­<lb/>ter &longs;ignum Leonis per quod pergebat antequam videri cœpit: & <lb/>nulla alia extat Theoria, qua Cometa tantam Cœli partem motu <lb/>regulari percurrat. </s> <s>Motus ejus fuit maxime inæquabilis. </s> <s>Nam <lb/>circa diem vige&longs;imum <emph type="italics"/>Novembris,<emph.end type="italics"/>de&longs;crip&longs;it gradus circiter quin­<lb/>que &longs;ingulis diebus; dein motu retardato inter <emph type="italics"/>Novemb.<emph.end type="italics"/>26 & <lb/><emph type="italics"/>Decemb.<emph.end type="italics"/>12, &longs;patio &longs;cilicet dierum quindecim cum &longs;emi&longs;&longs;e, de­<lb/>&longs;crip&longs;it gradus tantum 40; po&longs;tea vero motu iterum accelerato, <lb/>de&longs;crip&longs;it gradus fere quinque &longs;ingulis diebus, antequam motus <lb/>iterum retardari cœpir. </s> <s>Et Theoria quæ motui tam inæquabili <lb/>per maximam cœli partem probe re&longs;pondet, quæque ea&longs;dem ob­<lb/>&longs;ervat leges cum Theoria Planetarum, & cum accuratis ob&longs;erva­<lb/>tionibus A&longs;tronomicis accurate congruit, non pote&longs;t non e&longs;&longs;e vera. </s> <s><lb/>Cometa tamen &longs;ub finem motus deviabat aliquantulum ab hac <lb/>Trajectoria Parabolica ver&longs;us axem Parabolæ, ut ex erroribus mi­<lb/>nuti unius primi duorumve in latitudinem men&longs;e <emph type="italics"/>Februario<emph.end type="italics"/>& <lb/><emph type="italics"/>Martio<emph.end type="italics"/>con&longs;pirantibus, colligere videor; & propterea in Orbe El-<pb xlink:href="039/01/493.jpg"/><pb xlink:href="039/01/494.jpg"/><pb xlink:href="039/01/495.jpg"/><figure id="id.039.01.495.1.jpg" xlink:href="039/01/495/1.jpg"/><pb xlink:href="039/01/496.jpg" pagenum="465"/>liptico circum Solem movebatur, &longs;patio annorum plu&longs;quam quin­</s></p> <p type="main"> <s><arrow.to.target n="note494"/>gentorum, quantum ex erroribus illis judicare licuit, revolutio­<lb/>nem peragens. </s></p> <p type="margin"> <s><margin.target id="note494"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Cæterum Trajectoriam quam Cometa de&longs;crip&longs;it, & Caudam <lb/>veram quam &longs;ingulis in locis projecit, vi&longs;um e&longs;t annexo &longs;chemate <lb/>in plano Trajectoriæ optice delineatas exhibere: Ob&longs;ervationibus <lb/>&longs;equentibus in Cauda definienda adhibitis. </s></p> <p type="main"> <s><emph type="italics"/>Nov.<emph.end type="italics"/>17 Cauda gradus amplius quindecim longa <emph type="italics"/>Ponthæo<emph.end type="italics"/>ap­<lb/>paruit. <emph type="italics"/>Nov.<emph.end type="italics"/>18 Cauda 30<emph type="sup"/>gr.<emph.end type="sup"/> longa, SoliQ.E.D.recte oppo&longs;ita in <lb/><emph type="italics"/>Nova-Anglia<emph.end type="italics"/>cernebatur, & protendebatur u&longs;que ad &longs;tellam <!--symbol8-->, <lb/>quæ tunc erat in <!--symbol13--> 9<emph type="sup"/>gr.<emph.end type="sup"/> 54′. <emph type="italics"/>Nov.<emph.end type="italics"/>19 in <emph type="italics"/>Mary-Land<emph.end type="italics"/>cauda vi&longs;a <lb/>fuit gradus 15 vel 20 longa. <emph type="italics"/>Dec.<emph.end type="italics"/>10 Cauda (ob&longs;ervante <emph type="italics"/>Flam&longs;tedio<emph.end type="italics"/>) <lb/>tran&longs;ibat per medium di&longs;tantiæ inter caudam &longs;erpentis Ophiuchi & <lb/>&longs;tellam <foreign lang="greek">d</foreign> in Aquilæ au&longs;trali ala, & de&longs;inebat prope &longs;tellas <emph type="italics"/>A, <foreign lang="greek">w</foreign>, b<emph.end type="italics"/>in <lb/>Tabulis <emph type="italics"/>Bayeri.<emph.end type="italics"/>Terminus igitur erat in <emph type="sup"/>gr.<emph.end type="sup"/> 19 1/2<emph type="sup"/>gr.<emph.end type="sup"/> cum latitudine <lb/>boreali 34 1/4<emph type="sup"/>gr.<emph.end type="sup"/> circiter. <emph type="italics"/>Dec.<emph.end type="italics"/>11 &longs;urgebat ad u&longs;que caput Sagittæ <lb/>(<emph type="italics"/>Bayero,<emph.end type="italics"/><foreign lang="greek">a, b</foreign>,) de&longs;inens in <emph type="sup"/>gr.<emph.end type="sup"/> 26<emph type="sup"/>gr.<emph.end type="sup"/> 43′, cum latitudine boreali <lb/>38<emph type="sup"/>gr.<emph.end type="sup"/> 34′. <emph type="italics"/>Dec.<emph.end type="italics"/>13 tran&longs;ibat per medium Sagittæ, nec longe ultra <lb/>protendebatur, de&longs;inens in=4<emph type="sup"/>gr.<emph.end type="sup"/>, cum latitudine boreali 42 1/2<emph type="sup"/>gr.<emph.end type="sup"/> circi­<lb/>ter. </s> <s>Intelligenda &longs;unt hæc de longitudine caudæ clarioris. </s> <s>Nam luce <lb/>ob&longs;curiore, in cœlo for&longs;an magis &longs;ereno, cauda <emph type="italics"/>Dec.<emph.end type="italics"/>12, hora 5, 40′ <lb/><emph type="italics"/>Romæ<emph.end type="italics"/>(ob&longs;ervante <emph type="italics"/>Ponthæo<emph.end type="italics"/>) &longs;upra Cygni Uropygium ad gradus 10 <lb/>&longs;e&longs;e extulit; atque ab hac &longs;tella ejus latus ad occa&longs;um & boream <lb/>min. </s> <s>45 de&longs;titit. </s> <s>Lata autem erat cauda his diebus gradus 3, juxta <lb/>terminum &longs;uperiorem, ideoque medium ejus di&longs;tabat a Stella illa <lb/>2<emph type="sup"/>gr<emph.end type="sup"/> 15′ au&longs;trum ver&longs;us, & terminus &longs;uperior erat in <emph type="sup"/>gr.<emph.end type="sup"/> 22<emph type="sup"/>gr.<emph.end type="sup"/> cum <lb/>latitudine boreali 61<emph type="sup"/>gr.<emph.end type="sup"/>. <emph type="italics"/>Dec.<emph.end type="italics"/>21 &longs;urgebat fere ad cathedram <emph type="italics"/>Ca&longs;&longs;io­<lb/>peiæ,<emph.end type="italics"/>æqualiter di&longs;tans a <foreign lang="greek">b</foreign> & <emph type="italics"/>Schedir,<emph.end type="italics"/>& di&longs;tantiam ab utraque <lb/>di&longs;tantiæ earum ab invicem æqualem habens, adeoQ.E.D.&longs;inens <lb/>in <emph type="sup"/>gr.<emph.end type="sup"/> 24<emph type="sup"/>gr.<emph.end type="sup"/> cum latitudine 47 1/2<emph type="sup"/>gr.<emph.end type="sup"/>. <emph type="italics"/>Dec.<emph.end type="italics"/>29 tangebat <emph type="italics"/>Scheat<emph.end type="italics"/>&longs;itam ad <lb/>&longs;ini&longs;tram, & intervallum &longs;tellarum duarum in pede boreali <emph type="italics"/>Andro­<lb/>medæ<emph.end type="italics"/>accurate complebat, & longa erat 54<emph type="sup"/>gr.<emph.end type="sup"/> adeoQ.E.D.&longs;inebat <lb/>in 8 19<emph type="sup"/>gr.<emph.end type="sup"/> cum latitudine 35<emph type="sup"/>gr.<emph.end type="sup"/>. <emph type="italics"/>Jan.<emph.end type="italics"/>5 tetigit &longs;tellam <foreign lang="greek">p</foreign> in pectore <lb/><emph type="italics"/>Andromedæ,<emph.end type="italics"/>ad latus &longs;uum dextrum, & &longs;tellam <foreign lang="greek">m</foreign> in ejus cingulo <lb/>ad latus &longs;ini&longs;trum; & (juxta Ob&longs;ervationes no&longs;tras) longa erat <lb/>40<emph type="sup"/>gr.<emph.end type="sup"/>; curva autem erat & convexo latere &longs;pectabat ad au&longs;trum. </s> <s><lb/>Cum circulo per Solem & caput Cometæ tran&longs;eunte angulum <lb/>confecit graduum 4 juxta caput Cometæ; at juxta terminum al­<lb/>terum inclinabatur ad circulum illum in angulo 10 vel 11 graduum, <lb/>& chorda caudæ cum circulo illo continebat angulum graduum <pb xlink:href="039/01/497.jpg" pagenum="466"/><arrow.to.target n="note495"/>octo. <emph type="italics"/>Jan.<emph.end type="italics"/>13 Cauda luce &longs;atis &longs;en&longs;ibili terminabatur inter <emph type="italics"/>Ala­<lb/>mech<emph.end type="italics"/>& <emph type="italics"/>Algol,<emph.end type="italics"/>& luce tenui&longs;&longs;ima de&longs;inebat e regione &longs;tellæ <foreign lang="greek">x</foreign> in <lb/>latere <emph type="italics"/>Per&longs;ei.<emph.end type="italics"/>Di&longs;tantia termini caudæ a circulo Solem & Come­<lb/>tam ungente erat 3<emph type="sup"/>gr.<emph.end type="sup"/> 50′, & inclinatio chordæ caudæ ad circu­<lb/>lum illum 8 1/2<emph type="sup"/>gr<emph.end type="sup"/>. <emph type="italics"/>Jan.<emph.end type="italics"/>25 & 26 luce tenui micabat ad longitu­<lb/>dinem graduum 6 vel 7; & ubi cœlum valde &longs;erenum erat, luce <lb/>tenui&longs;&longs;ima & ægerrime &longs;en&longs;ibili attingebat longitudinem graduum <lb/>duodecim & paulo ultra. </s> <s>Dirigebatur autem ejus axis ad Luci­<lb/>dam in humero orientali Aurigæ accurate, adeoQ.E.D.clinabat ab <lb/>oppo&longs;itione Solis boream ver&longs;us in angulo graduum decem. </s> <s>De­<lb/>nique <emph type="italics"/>Feb.<emph.end type="italics"/>10 Caudam oculis armatis a&longs;pexi gradus duos lon­<lb/>gam. </s> <s>Nam lux prædicta tenuior per vitra non apparuit. <emph type="italics"/>Pon­<lb/>thæus<emph.end type="italics"/>autem <emph type="italics"/>Feb.<emph.end type="italics"/>7 &longs;e caudam ad longitudinem graduum 12 <lb/>vidi&longs;&longs;e &longs;cribit. </s></p> <p type="margin"> <s><margin.target id="note495"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Orbem jam de&longs;criptum &longs;pectanti & reliqua Cometæ hujus Phæ­<lb/>nomena in animo revolventi, haud difficulter con&longs;tabit quod cor­<lb/>pora Cometarum &longs;unt &longs;olida, compacta, fixa ac durabilia ad in­<lb/>&longs;tar corporum Planetarum. </s> <s>Nam &longs;i nihil aliud e&longs;&longs;ent quam vapo­<lb/>res vel exhalationes Terræ, Solis & Planetarum, Cometa hicce in <lb/>tran&longs;itu &longs;uo per viciniam Solis &longs;tatim di&longs;&longs;ipari debui&longs;&longs;et. </s> <s>E&longs;t enim <lb/>calor Solis ut radiorum den&longs;itas, hoc e&longs;t, reciproce ut quadratum <lb/>di&longs;tantiæ loeorum a Sole. </s> <s>Ideoque cum di&longs;tantia Cometæ a cen­<lb/>tro Solis <emph type="italics"/>Decemb.<emph.end type="italics"/>8 ubi in Perihelio ver&longs;abatur, e&longs;&longs;et ad di&longs;tan­<lb/>tiam Terræ a centro Solis ut 6 ad 1000 circiter, calor Solis apud <lb/>Cometam eo tempore erat ad calorem Solis æ&longs;tivi apud nos ut <lb/>1000000 ad 36, &longs;eu 28000 ad 1. Sed calor aquæ ebullientis e&longs;t <lb/>qua&longs;i triplo major quam calor quem terra arida concipit ad æ&longs;ti­<lb/>vum Solem, ut expertus &longs;um: & calor ferri candentis (&longs;i recte <lb/>conjector) qua&longs;i triplo vel quadruplo major quam calor aquæ ebul­<lb/>lientis; adeoque calor quem terra arida apud Cometam in Peri­<lb/>helio ver&longs;antem ex radiis Solaribus concipere po&longs;&longs;et, qua&longs;i 2000 <lb/>vicibus major quam calor ferri candentis. </s> <s>Tanto autem calore <lb/>vapores & exhalationes, omni&longs;que materia volatilis itatim con&longs;umi <lb/>ac di&longs;&longs;ipari debui&longs;&longs;ent. </s></p> <p type="main"> <s>Cometa igitur in Perihelio &longs;uo calorem immen&longs;um ad Solem <lb/>concepit, & calorem illum diuti&longs;&longs;ime con&longs;ervare pote&longs;t. </s> <s>Nam <lb/>globus ferri candentis digitum unum latus, calorem &longs;uum omnem <lb/>&longs;patio horæ unius in aere con&longs;i&longs;tens vix amitteret. </s> <s>Globus autem <lb/>major calorem diutius con&longs;ervaret in ratione diametri, propterea <lb/>quod &longs;uperficies (ad cujus men&longs;uram per contactum aeris ambi-<pb xlink:href="039/01/498.jpg" pagenum="467"/>entis refrigeratur) in illa ration minor e&longs;t pro quantitate mate­<lb/><arrow.to.target n="note496"/>riæ &longs;uæ calidæ inclu&longs;æ. </s> <s>Ideoque globus ferri candentis huic <lb/>Terræ æqualis, id e&longs;t, pedes plus minus 40000000 latus, diebus <lb/>totidem, & idcirco annis 50000, vix refrige&longs;ceret. </s> <s>Su&longs;picor ta­<lb/>men quod duratio Caloris, ob cau&longs;as latentes, augeatur in minore <lb/>ratione quam ea diametri: & optarim rationem veram per experi­<lb/>menta inve&longs;tigari. </s></p> <p type="margin"> <s><margin.target id="note496"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Porro notandum e&longs;t quod Cometa Men&longs;e <emph type="italics"/>Decembri,<emph.end type="italics"/>ubi ad <lb/>Solem modo incaluerat, caudam emittebat longe majorem & <lb/>&longs;plendidiorem quam antea Men&longs;e <emph type="italics"/>Novembri,<emph.end type="italics"/>ubi Periheliunt non­<lb/>dum attigerat. </s> <s>Et univer&longs;aliter caudæ omnes maximæ & fulgen­<lb/>ti&longs;&longs;imæ e Cometis oriuntur, &longs;tatim po&longs;t tran&longs;itum eorum per regi­<lb/>onem Solis. </s> <s>Conducit igitur calefactio Cometæ ad magnitudi­<lb/>nem caudæ. </s> <s>Et inde colligere videor quod cauda nihil aliud fit <lb/>quam vapor longe tenui&longs;&longs;imus, quem caput &longs;eu nucleus Cometæ <lb/>per calorem &longs;uum emittit. </s></p> <p type="main"> <s>Cæterum de Cometarum caudis triplex e&longs;t opinio; eas vel jubar <lb/>e&longs;&longs;e Solis per tran&longs;lucida Cometarum capita propagatum, vel oriri <lb/>ex refractione lucis in progre&longs;&longs;u ip&longs;ius a capite Comeræ in Ter­<lb/>ram, vel denique nubem e&longs;&longs;e &longs;eu vaporem a capite Comeræ jugi­<lb/>ter &longs;urgentem & abeuntem in partes a Sole aver&longs;as. </s> <s>Opinio pri­<lb/>ma eorum e&longs;t qui nondum imbuti &longs;unt &longs;cientia rerum Opticarum. </s> <s><lb/>Nam jubar Solis in cubiculo tenebro&longs;o non cernitur, ni&longs;i quatenus <lb/>lux reflectitur e pulverum & fumorum particulis per aerem &longs;em­<lb/>per volitantibus: adeoQ.E.I. aere fumis cra&longs;&longs;ioribus infecto &longs;plen­<lb/>didius e&longs;t, & &longs;en&longs;um fortius ferit; in aere clariore tenuius e&longs;t & <lb/>ægrius &longs;entitur: in cœlis autem ab&longs;que materia reflectente nullum <lb/>e&longs;&longs;e pote&longs;t. </s> <s>Lux non cernitur quatenus in jubare e&longs;t, &longs;ed quatenus <lb/>inde re&longs;tectitur ad oculos no&longs;tros. </s> <s>Nam vi&longs;io non &longs;it ni&longs;i per radios <lb/>qui in oculos impingunt. </s> <s>Requiritur igitur materia aliqua reflectens <lb/>in regione caudæ, ne cœlum totum luce Solis illu&longs;tratum unifor­<lb/>miter &longs;plendeat. </s> <s>Opinio &longs;ecunda multis premitur difficultatibus. </s> <s><lb/>Caudæ nunquam variegantur coloribus: qui tamen refractionum <lb/>&longs;olent e&longs;&longs;e comites in&longs;eparabiles. </s> <s>Lux Fixarum & Planetarum di­<lb/>&longs;tincte ad nos tran&longs;mi&longs;&longs;a, demon&longs;trat medium cœle&longs;te nulla vi re­<lb/>fractiva pollere. </s> <s>Nam quod dicitur Fixas ab <emph type="italics"/>Ægyptiis<emph.end type="italics"/>comatas <lb/>nonnunquam vi&longs;as fui&longs;&longs;e, id quoniam rari&longs;&longs;ime contingit, a&longs;cri­<lb/>bendum e&longs;t nubium refractioni fortuitæ. </s> <s>Fixarum quoque radia­<lb/>tio & &longs;cintillatio ad refractiones tum Oculorum tum Aeris tre­<lb/>muli referendæ &longs;unt: quippe quæ admotis oculo Tele&longs;copiis <pb xlink:href="039/01/499.jpg" pagenum="468"/><arrow.to.target n="note497"/>evane&longs;cunt, Aeris & a&longs;cendentium vaporum tremore fit ut radii <lb/>facile de angu&longs;to pupillæ &longs;patio per vices detorqueantur, de lati­<lb/>ore autem vitri objectivi apertura neutiquam. </s> <s>Inde e&longs;t quod <lb/>&longs;cintillatio in priori ca&longs;a generetur, in po&longs;teriore autem ce&longs;&longs;et: <lb/>& ce&longs;&longs;atio in po&longs;teriore ca&longs;u demon&longs;trat regularem tran&longs;mi&longs;&longs;ionem <lb/>lucis per cœlos ab&longs;que omni refractione &longs;en&longs;ibili. </s> <s>Nequis con­<lb/>tendat quod caudæ non &longs;oleant videri in Cometis cum eorum lux <lb/>non e&longs;t &longs;atis fortis, quia tunc radii &longs;ecundarii non habent &longs;itis vi­<lb/>rium ad oculos movendos, & propterea caudas Fixarum non cerni: <lb/>&longs;ciendum e&longs;t quod lux Fixarum plus centum vicibus augeri pote&longs;t <lb/>mediantibus Tele&longs;copiis, nec tamen caudæ cernuntur Planeta­<lb/>rum quoque lux copio&longs;ior e&longs;t, caudæ vero nunæ: Comeræ autem <lb/>&longs;æpe caudati&longs;&longs;imi &longs;unt, ubi capitum lux tenuis e&longs;t & valde obtu&longs;a: <lb/>&longs;ic enim Cometa Anni 1680, Men&longs;e <emph type="italics"/>Decembri,<emph.end type="italics"/>quo tempore ca­<lb/>put luce &longs;ua vix æquabat &longs;tellas &longs;ecundæ magnitudinis, caudam <lb/>emittebat &longs;plendore notabili u&longs;que ad gradus 40, 50, 60 longi­<lb/>tudinis & ultra: po&longs;tea <emph type="italics"/>Jan.<emph.end type="italics"/>27 & 28 caput apparebat ut &longs;tella <lb/>&longs;eptimæ tantum magnitudinis, cauda vero luce quidem pertenui <lb/>&longs;ed &longs;atis &longs;en&longs;ibili longa erat 6 vel 7 gradus, & luce ob&longs;curi&longs;&longs;ima, <lb/>quæ cerni vix po&longs;&longs;et, porrigebatur ad gradum u&longs;Q.E.D.odecimum <lb/>vel paulo ultra: ut &longs;upra dictum e&longs;t. </s> <s>Sed & <emph type="italics"/>F<foreign lang="greek">e</foreign>b.<emph.end type="italics"/>9 & 10 ubi <lb/>caput nudis oculis videri de&longs;ierat, caudam gradus duos longam <lb/>per Tele&longs;copium contemplatus &longs;um. </s> <s>Porro &longs;i cauda oriretur ex <lb/>refractione materiæ cœle&longs;tis, & pro figura cœlorum deflecteretur <lb/>de Solis oppo&longs;itione, deberet deflexio illa in ii&longs;dem cœli regioNI­<lb/>bus in eandem &longs;emper partem fieri. </s> <s>Atqui Cometa Anni 1680 <lb/><emph type="italics"/>Decemb.<emph.end type="italics"/>28. hora 8 1/2 P.M. <emph type="italics"/>Londini,<emph.end type="italics"/>ver&longs;abatur in <!--symbol3--> 8<emph type="sup"/>gr.<emph.end type="sup"/> 41′ cum <lb/>latitudine boreali 28<emph type="sup"/>gr.<emph.end type="sup"/> 6′, Sole exi&longs;tente in <!--symbol1--> 18<emph type="sup"/>gr.<emph.end type="sup"/> 26′. </s> <s>Et Co­<lb/>meta Anni 1577, <emph type="italics"/>Dec.<emph.end type="italics"/>29 ver&longs;abatur in <!--symbol3--> 8<emph type="sup"/>gr.<emph.end type="sup"/> 41′ cum latitu­<lb/>dine boreali 28<emph type="sup"/>gr.<emph.end type="sup"/> 40′, Sole etiam exi&longs;tente in <!--symbol1--> 18<emph type="sup"/>gr.<emph.end type="sup"/> 26′ circi­<lb/>ter. </s> <s>UtroQ.E.I. ca&longs;u Terra ver&longs;abatur in eodem loco, & Co­<lb/>meta apparebat in eadem cœli parte: in priori tamen ca&longs;u cauda <lb/>Cometæ (ex meis & aliorum Ob&longs;ervationibus) declinabat angulo <lb/>graduum 4 1/2 ab oppo&longs;itione Solis aquilonem ver&longs;us; in po&longs;te­<lb/>riore vero (ex Ob&longs;ervationibus <emph type="italics"/>Tychonis<emph.end type="italics"/>) declinatio erat gra­<lb/>duum 21 in au&longs;trum. </s> <s>Igitur repudiata cœlorum refractione, <lb/>&longs;upere&longs;t ut Phænomena Caudarum ex materia aliqua reflectente <lb/>deriventur. </s></p> <p type="margin"> <s><margin.target id="note497"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Caudas autem a capitibus oriri & in regiones a Sole aver&longs;as <lb/>a&longs;cendere confirmatur ex legibus quas ob&longs;ervant. </s> <s>Ut quod in <pb xlink:href="039/01/500.jpg" pagenum="469"/>planis Orbium Cometarum per Solem tran&longs;euntibus jacentes, de­<lb/><arrow.to.target n="note498"/>viant ab oppo&longs;itione Solis in eas &longs;emper partes, quas capita in <lb/>Orbibus iilis progredientia relinquunt. </s> <s>Quod &longs;pectatori in his <lb/>planis con&longs;tituto apparent in partibus a Sole directe aver&longs;is; di­<lb/>grediente autem &longs;pe&longs;tatore de his planis, deviatio paulatim &longs;en­<lb/>titur, & indies apparet major. </s> <s>Quod deviatio cæteris paribus <lb/>minor e&longs;t ubi cauda obliquior e&longs;t ad Orbem Cometæ, ut & ubi <lb/>caput Cometæ ad Solem propius accedit; præ&longs;ertim &longs;i &longs;pectetur <lb/>deviationis angulus juxta caput Cometæ. </s> <s>Præterea quod caudæ <lb/>non deviantes apparent rectæ, deviantes autem incurvantur. </s> <s>Quod <lb/>curvatura major e&longs;t ubi major e&longs;t deviatio, & magis &longs;en&longs;ibilis ubi <lb/>cauda cæteris paribus longior e&longs;t: nam in brevioribus curvatura <lb/>ægre animadvertitur. </s> <s>Quod deviationis angulus minor e&longs;t juxta <lb/>caput Cometæ, major juxta caudæ extremitatem alteram, atque <lb/>adeo quod cauda convexo &longs;ui latere partes re&longs;picit a quibus &longs;it <lb/>deviatio, quæQ.E.I. recta &longs;unt linea a Sole per caput Cometæ in <lb/>infinitum ducta. </s> <s>Et quod caudæ quæ prolixiores &longs;unt & latiores, <lb/>& luce vegetiore micant, &longs;int ad latera convexa paulo &longs;plendi­<lb/>diores & limite minus indi&longs;tincto terminatæ quam ad concava. </s> <s><lb/>Pendent igitur Phænomena caudæ a motu capitis, non autem a <lb/>regione cœli in qua caput con&longs;picitur; & propterea non fiunt per <lb/>refractionem cœlorum, &longs;ed a capite &longs;uppeditante materiam ori­<lb/>untur. </s> <s>Etenim ut in Aere no&longs;tro fumus corporis cuju&longs;vis igniti <lb/>petit &longs;uperiora, idque vel perpendiculariter &longs;i corpus quie&longs;cat, <lb/>vel oblique &longs;i corpus moveatur in latus: ita in Cœlis ubi corpora <lb/>gravitant in Solem, fumi & vapores a&longs;cendere debent à Sole (uti <lb/>jam dictum e&longs;t) & &longs;uperiora vel recta petere, &longs;i corpus fumans <lb/>quie&longs;cit; vel oblique, &longs;i corpus progrediendo loca &longs;emper de&longs;erit <lb/>a quibus &longs;uperiores vaporis partes a&longs;cenderant. </s> <s>Et obliquitas i&longs;ta <lb/>minor erit ubi a&longs;cen&longs;us vaporis velocior e&longs;t: nimirum in vicinia <lb/>Solis & juxta corpus fumans. </s> <s>Ex obliquitatis autem diver&longs;itate <lb/>incurvabitur vaporis columna: & quia vapor in columnæ latere <lb/>præcedente paulo recentior e&longs;t, ideo etiam is ibidem aliquanto <lb/>den&longs;ior erit, lucemque propterea copio&longs;ius reflectet, & limite mi­<lb/>nus indi&longs;tincto terminabitur. </s> <s>De Caudarum agitionibus &longs;ubita­<lb/>neis & incertis, deque earum figuris irregularibus, quas nonnulli <lb/>quandoQ.E.D.&longs;cribunt hic nihil adjicio; propterea quod vel a <lb/>mutationibus Aeris ne&longs;tri, & motibus nubium caudas aliqua ex <lb/>parte ob&longs;curantium oriantur; vel forte a partibus Viæ Lacteæ, <lb/>quæ cum caudis prætereuntibus confundi po&longs;&longs;int, ac tanquam ea­<lb/>rum partes &longs;pectari. <pb xlink:href="039/01/501.jpg" pagenum="470"/><arrow.to.target n="note499"/></s></p> <p type="margin"> <s><margin.target id="note498"/>LIBER <lb/>TERTIUS.</s></p> <p type="margin"> <s><margin.target id="note499"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Vapores autem, qui &longs;patiis tam immen&longs;is implendis &longs;ufficiant, <lb/>ex Cometarum Atmo&longs;phæris oriri po&longs;&longs;e, intelligetur ex ratitate <lb/>Aeris no&longs;tri. </s> <s>Nam Aer juxta &longs;uperficiem Terræ &longs;patium occupat <lb/>qua&longs;i 850 partibus majus quam Aqua eju&longs;dem ponderis, ideoque <lb/>Aeris columna cylindrica pedes 850 alta, eju&longs;dem e&longs;t ponderis <lb/>cum Aquæ columna pedali latitudinis eju&longs;dem. </s> <s>Columna autem <lb/>Aeris ad &longs;ummitatem Atmo&longs;phæræ a&longs;&longs;urgens æquat pondere &longs;uo <lb/>colurnnam Aquæ pedes 33 altam circiter; & propterea &longs;i colum­<lb/>næ totius Aereæ pars inferior pedum 850 altitudinis dematur, <lb/>pars reliqua &longs;uperior æquabit pondere &longs;uo columnam Aquæ altam <lb/>pedes 32. Inde vero (ex Hypothe&longs;i multis experimentis confir­<lb/>mata, quod compre&longs;&longs;io Aeris &longs;it ut pondus Atmo&longs;phæræ incum­<lb/>bentis, quodque gravitas &longs;it reciproce ut quadratum di&longs;tantiæ lo­<lb/>eorum a centro Terræ) computationem per Corol. </s> <s>Prop. </s> <s>XXII. <lb/>Lib. </s> <s>II. ineundo, inveni quod Aer, &longs;i a&longs;cendatur a &longs;uperficie <lb/>Terræ ad altitudinem &longs;emidiametri unius terre&longs;tris, rarior &longs;it quam <lb/>apud nos in ratione longe majori, quam &longs;patii omnis infra Or­<lb/>bem Saturni ad globum diametro digiti unius de&longs;criptum. </s> <s>Ideo­<lb/>que globus Aeris no&longs;tri digitum unum latus, ea cum raritate <lb/>quam haberet in altitudine &longs;emidiametri unius terre&longs;tris, impleret <lb/>omnes Planetarum regiones ad u&longs;que &longs;phæram Saturni & longe <lb/>ultra. </s> <s>Proinde cum Aer adhuc altior in immen&longs;um rare&longs;cat; & <lb/>coma &longs;eu Atmo&longs;phæra Cometæ, a&longs;cendendo ab illius centro, qua&longs;i <lb/>decuplo altior &longs;it quam &longs;uperficies nuclei, deinde cauda adhuc <lb/>altius a&longs;cendat, debebit cauda e&longs;&longs;e quam rari&longs;&longs;ima. </s> <s>Et quamvis, <lb/>ob longe cra&longs;&longs;iorem Cometarum Atmo&longs;phæram, magnamque cor­<lb/>porum gravitationem Solem ver&longs;us, & gravitationem particula­<lb/>rum Aeris & vaporum in &longs;e mutuo, fieri po&longs;&longs;it ut Aer in &longs;patiis <lb/>cœle&longs;tibus inque Cometarum caudis non adeo rare&longs;cat; perexi­<lb/>guam tamen quantitatem Aeris & vaporum, ad omnia illa cauda­<lb/>rum Phœnomena abunde &longs;ufficere, ex hac computatione per&longs;pi­<lb/>cuum e&longs;t. </s> <s>Nam & caudarum in&longs;ignis raritas colligitur ex a&longs;tris <lb/>pes eas tran&longs;lucentibus. </s> <s>Atmo&longs;phæra terre&longs;tris luce Solis &longs;plen­<lb/>dens, cra&longs;&longs;itudine &longs;ua paueorum milliarium, & a&longs;tra omnia & ip­<lb/>&longs;am Lunam ob&longs;curat & extinguit penitus: per immen&longs;am vero <lb/>caudarum cra&longs;&longs;itudinem, luce pariter Solari illu&longs;tratam, a&longs;tra mi­<lb/>nima ab&longs;que claritatis detrimento tran&longs;lucere no&longs;cuntur. </s> <s>Neque <lb/>major e&longs;&longs;e &longs;olet caudarum plurimarum &longs;plendor, quam Aeris no­<lb/>&longs;tri in tenebro&longs;o cubiculo latitudine digiti unius duorumve, lucem <lb/>Solis in jubare reflectentis. </s></p><pb xlink:href="039/01/502.jpg" pagenum="471"/> <p type="main"> <s>Quo temporis &longs;patio vapor a capite ad terminum caudæ a&longs;cen­</s></p> <p type="main"> <s><arrow.to.target n="note500"/>dit, cogno&longs;ci fere pote&longs;t ducendo rectam a termino caudæ ad So­<lb/>lem, & notando locum ubi recta illa Trajectoriam &longs;ecat. </s> <s>Nam <lb/>vapor in termino caudæ, &longs;i recta a&longs;cendat a Sole, a&longs;cendere cœpit <lb/>a capite quo tempore caput erat in loco inter&longs;ectionis. </s> <s>At vapor <lb/>non recta a&longs;cendit à Sole, &longs;ed motum Cometæ, quem aute a&longs;cen­<lb/>&longs;um &longs;uum habebat, retinendo, & cum motu a&longs;cen&longs;us &longs;ui eundem <lb/>componendo, a&longs;cendit oblique. </s> <s>Unde verior erit Problematis <lb/>&longs;olutio, ut recta illa quæ Orbem &longs;ecat, parallela &longs;it longitudini <lb/>caudæ, vel potius (ob motum curvilineum Cometæ) ut eadem a <lb/>linea caudæ divergat. </s> <s>Hoc pacto inveni quod vapor qui erat in <lb/>termino caudæ <emph type="italics"/>Jan.<emph.end type="italics"/>25, a&longs;cendere cœperat a capite ante <emph type="italics"/>Dec.<emph.end type="italics"/>11, <lb/>adeoque a&longs;cen&longs;u &longs;uo toto dies plus 45 con&longs;ump&longs;erat. </s> <s>At cauda <lb/>illa omnis quæ <emph type="italics"/>Dec.<emph.end type="italics"/>10 apparuit, a&longs;cenderat &longs;patio dierum illo­<lb/>rum duorum, qui a tempore Perihelii Cometæ elap&longs;i fuerant. </s> <s><lb/>Vapor igitur &longs;ub initio in vicinia Solis celerrime a&longs;cendebat, & <lb/>po&longs;tea cum motu per gravitatem &longs;uam &longs;emper retardato a&longs;cen­<lb/>dere pergebat; & a&longs;cendendo augebat longitudinem caudæ: cauda <lb/>autem quamdiu apparuit ex vapore fere omni con&longs;tabat qui a <lb/>tempore Perihelii a&longs;cenderat; & vapor, qui primus a&longs;cendit, & <lb/>terminum caudæ compo&longs;uit, non prius evanuit quam ob nimiam <lb/>&longs;uam tam a Sole illu&longs;trante quam ab oculis no&longs;tris di&longs;tantiam vi­<lb/>deri de&longs;iit. </s> <s>Unde etiam caudæ Cometarum aliorum quæ breves <lb/>&longs;unt, non a&longs;cendunt motu celeri & perpetuo a capitibus & mox <lb/>evane&longs;cunt, &longs;ed &longs;unt permanentes vaporum & exhalationum co­<lb/>lumnæ, a capitibus lenti&longs;&longs;imo multorum dierum motu propagatæ, <lb/>quæ, participando motum illum capitum quem habuere &longs;ub initio, <lb/>per cœlos una cum capitibus moveri pergunt. </s> <s>Et hinc rur&longs;us col­<lb/>ligitur &longs;patia cœle&longs;tia vi re&longs;i&longs;tendi de&longs;titui; utpote in quibus non <lb/>&longs;olum &longs;olida Planetarum & Cometarum corpora, &longs;ed etiam rari&longs;­<lb/>&longs;imi caudarum vapores motus &longs;uos veloci&longs;&longs;imos liberrime peragunt <lb/>ac diuti&longs;&longs;ime con&longs;ervant. </s></p> <p type="margin"> <s><margin.target id="note500"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>A&longs;cen&longs;um caudarum ex Atmo&longs;phæris capitum & progre&longs;&longs;um in <lb/>partes a Sole aver&longs;as <emph type="italics"/>Keplerus<emph.end type="italics"/>a&longs;cribit actioni radiorum lucis ma­<lb/>teriam caudæ &longs;ecum rapientium. </s> <s>Et auram longe tenui&longs;&longs;imam in <lb/>&longs;patiis liberrimis actioni radiorum cedere, non e&longs;t a ratione pror­<lb/>&longs;us alienum, non ob&longs;tante quod &longs;ub&longs;tantiæ cra&longs;&longs;æ, impediti&longs;&longs;imis <lb/>in regionibus no&longs;tris, a radiis Solis &longs;en&longs;ibiliter propelli nequeant. </s> <s><lb/>Alius particulas tam leves quam graves dari po&longs;&longs;e exi&longs;timat, & <lb/>materiam caudarum levitare, perque levitatem &longs;uam a Sole a&longs;cen-<pb xlink:href="039/01/503.jpg" pagenum="472"/><arrow.to.target n="note501"/>dere. </s> <s>Cum autem gravitas corporum terre&longs;trium &longs;it ut materia <lb/>in corporibus, ideoque &longs;ervata quantitate materiæ intendi & re­<lb/>mitti nequeat, &longs;u&longs;picor a&longs;cen&longs;um illum ex rarefactione materiæ <lb/>caudarum potius oriri. </s> <s>A&longs;cendit fumus in camino impul&longs;u Aeris <lb/>cui innatat. </s> <s>Aer ille per calorem rarefactus a&longs;cendit, ob diminu­<lb/>tam &longs;uam gravitatem &longs;pecificam, & fumum implicatum rapit &longs;e­<lb/>cum. </s> <s>Quidni cauda Cometæ ad eundem modum a&longs;cenderit a <lb/>Sole? </s> <s>Nam radii Solares non agitant Media quæ permeant, ni&longs;i <lb/>in reflexione & refractione. </s> <s>Particulæ reflectentes ea actione cale­<lb/>factæ calefacient auram ætheream cui implicantur. </s> <s>Illa calore &longs;ibi <lb/>communicato rarefiet, & ob diminutam ea raritate gravitatem <lb/>&longs;uam &longs;pecificam qua prius tendebat in Solem, a&longs;cendet & &longs;ecum <lb/>rapiet particulas reflectentes ex quibus cauda componitur: Ad <lb/>a&longs;cen&longs;um vaporum conducit etiam quod hi gyrantur circa Solem <lb/>& ea actione conantur a Sole recedere, at Solis Atmo&longs;phæra & <lb/>materia cœlorum vel plane quie&longs;cit, vel motu &longs;olo quem a Solis <lb/>rotatione acceperint, tardius gyratur. </s> <s>Hæ &longs;unt cau&longs;æ a&longs;cen&longs;us <lb/>caudarum in vicinia Solis, ubi Orbes curviores &longs;unt, & Cometæ <lb/>intra den&longs;iorem & ea ratione graviorem Solis Atmo&longs;phæram con­<lb/>&longs;i&longs;tunt, & caudas quam longi&longs;&longs;imas mox emittunt. </s> <s>Nam caudæ <lb/>quæ tunc na&longs;cuntur, con&longs;ervando motum &longs;uum & interea ver&longs;us <lb/>Solem gravitando, movebuntur circa Solem in Ellip&longs;ibus pro <lb/>more capitum, & per motum illum capita &longs;emper comitabuntur <lb/>& iis liberrime adhærebunt. </s> <s>Gravitas enim vaporum in Solem <lb/>non magis efficiet ut caudæ po&longs;tea decidant a capitibus Solem ver­<lb/>&longs;us, quam gravitas capitum efficere po&longs;&longs;it ut hæc decidant a cau­<lb/>dis. </s> <s>Communi gravitate vel &longs;imul in Solem cadunt, vel &longs;imul in <lb/>a&longs;cen&longs;u &longs;uo retardabuntur; adeoque gravitas illa non impedit, <lb/>quo minus caudæ & capita po&longs;itionem quamcunque ad invicem a <lb/>cau&longs;is jam de&longs;criptis, aut aliis quibu&longs;cunque, facillime accipiant & <lb/>po&longs;tea liberrime &longs;ervent. </s></p> <p type="margin"> <s><margin.target id="note501"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Caudæ igitur quæ in Cometarum Periheliis na&longs;cuntur, in regi­<lb/>ones longinquas cum eorum capitibus abibunt, & vel inde po&longs;t <lb/>longam annorum &longs;eriem cum ii&longs;dem ad nos redibunt, vel potius <lb/>ibi rarefactæ paulatim evane&longs;cent. </s> <s>Nam po&longs;tea in de&longs;cen&longs;u capi­<lb/>tum ad Solem caudæ novæ breviu&longs;culæ lento motu a capitibus <lb/>propagari debebunt, & &longs;ubinde, in Periheliis Cometarum illorum <lb/>qui adu&longs;que Atmo&longs;phæram Solis de&longs;cendunt, in immen&longs;um au­<lb/>geri. </s> <s>Vapor enim in &longs;patiis illis liberrimis perpetuo rare&longs;cit ac <lb/>dilatatur. </s> <s>Qua ratione fit ut cauda omnis ad extremitatem &longs;upe-<pb xlink:href="039/01/504.jpg" pagenum="473"/>riorem latior &longs;it quam juxta caput Cometæ. </s> <s>Ea autem rarefacti­<lb/><arrow.to.target n="note502"/>one vaporem perpetuo dilatatum diffundi tandem & &longs;pargi per <lb/>cœlos univer&longs;os, deinde paulatim in Planetas per gravitatem &longs;uam <lb/>attrahi & cum eorum Atmo&longs;phæris mi&longs;ceri, rationi con&longs;entaneum <lb/>videtur. </s> <s>Nam quemadmodum Maria ad con&longs;titutionem Terræ <lb/>hujus omnino requiruntur, idque ut ex iis per calorem Solis va­<lb/>pores copio&longs;e &longs;atis excitentur, qui vel in nubes coacti decidant <lb/>in pluviis, & terram omnem ad procreationem vegetabilium irri­<lb/>gent & nutriant; vel in frigidis montium verticibus conden&longs;ati <lb/>(ut aliqui cum ratione philo&longs;ophantur) decurrant in fontes & <lb/>flumina &longs;ic ad con&longs;ervationem marium & humorum in Planetis, <lb/>requiri videntur Cometæ, ex quorum exhalationibus & vapori­<lb/>bus conden&longs;atis, quicquid liquoris per vegetationem & putre­<lb/>factionem con&longs;umitur & in terram aridam convertitur, continuo <lb/>&longs;uppleri & refici po&longs;&longs;it. </s> <s>Nam vegetabilia omnia ex liquoribus <lb/>omnino cre&longs;cunt, dein magna ex parte in terram aridam per pu­<lb/>trefactionem abeunt, & limus ex liquoribus putrefactis perpetuo <lb/>decidit. </s> <s>Hinc moles Terræ aridæ indies augetur, & liquores, ni&longs;i <lb/>aliunde augmentum &longs;umerent, perpetuo decre&longs;cere deberent, ac <lb/>tandem deficere. </s> <s>Porro &longs;u&longs;picor Spiritum illum, qui Aeris no&longs;tri <lb/>pars minima e&longs;t &longs;ed &longs;ubtili&longs;&longs;ima & optima, & ad rerum omnium <lb/>vitam requiritur, ex Cometis præcipue venire. </s></p> <p type="margin"> <s><margin.target id="note502"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Atmo&longs;phæræ Cometarum in de&longs;cen&longs;u eorum in Solem, excur­<lb/>rendo in caudas, diminuuntur, & (ea certe in parte quæ Solem <lb/>re&longs;picit) angu&longs;tiores redduntur: & vici&longs;&longs;im in rece&longs;&longs;u eorum a <lb/>Sole, ubi jam minus excurrunt in caudas, ampliantur; &longs;i modo <lb/>Phænomena eorum <emph type="italics"/>Hevelius<emph.end type="italics"/>recte notavit. </s> <s>Minimæ autem ap­<lb/>parent ubi capita jam modo ad Solem calefacta in caudas maximas <lb/>& fulgenti&longs;&longs;imas abiere, & nuclei fumo for&longs;an cra&longs;&longs;iore & nigriore <lb/>in Atmo&longs;phærarum partibus infimis circundantur. </s> <s>Nam fumus <lb/>omnis ingenti calore excitatus, cra&longs;&longs;ior & nigrior e&longs;&longs;e &longs;olet. </s> <s>Sic <lb/>caput Cometæ de quo egimus, in æqualibus a Sole ac Terra di­<lb/>&longs;tantiis, ob&longs;curius apparuit po&longs;t Perihelium &longs;uum quam antea. </s> <s><lb/>Men&longs;e enim <emph type="italics"/>Decembri<emph.end type="italics"/>cum &longs;tellis tertiæ magnitudinis conferri &longs;ole­<lb/>bat, at Men&longs;e <emph type="italics"/>Novembri<emph.end type="italics"/>cum &longs;tellis primæ & &longs;ecundæ. </s> <s>Et qui <lb/>utrumque viderant, majorem de&longs;cribunt Cometam priorem. </s> <s>Nam <lb/>Juveni cuidam <emph type="italics"/>Cantabrigien&longs;i, Novemb.<emph.end type="italics"/>19, Cometa hicce luce &longs;ua <lb/>quantumvis plumbea & obtu&longs;a, æquabat Spicam Virginis, & cla­<lb/>rius micabat quam po&longs;tea. </s> <s>Et <emph type="italics"/>D. Storer<emph.end type="italics"/>literis quæ in manus no­<lb/>&longs;tras incidere, &longs;crip&longs;it caput ejus Men&longs;e <emph type="italics"/>Decembri,<emph.end type="italics"/>ubi caudam <pb xlink:href="039/01/505.jpg" pagenum="474"/><arrow.to.target n="note503"/>maximam & fulgenti&longs;&longs;imam emittebat, parvum e&longs;&longs;e & magnitu­<lb/>dine vi&longs;ibili longe cedere Cometæ, qui Men&longs;e <emph type="italics"/>Novembri<emph.end type="italics"/>ante <lb/>Solis ortum apparuerat. </s> <s>Cujus rei rationem e&longs;&longs;e conjectabatur, <lb/>quod materia capitis &longs;ub initio copio&longs;ior e&longs;&longs;et, & paulatim con­<lb/>&longs;umeretur. </s></p> <p type="margin"> <s><margin.target id="note503"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Eodem &longs;pectare videtur quod capita Cometarum aliorum, qui <lb/>caudas maximas & fulgenti&longs;&longs;imas emi&longs;erunt, apparuerint &longs;ubob­<lb/>&longs;cura & exigua. </s> <s>Nam Anno 1668 <emph type="italics"/>Mart.<emph.end type="italics"/>5. St. </s> <s>nov. </s> <s>hora &longs;eptima <lb/>ve&longs;pertina <emph type="italics"/>R. P. </s> <s>Vaientinus E&longs;tancius, Bra&longs;iliæ<emph.end type="italics"/>agens, Cometam <lb/>vidit Horizonti proximum ad occa&longs;um Solis brumalem, capite <lb/>minimo & vix con&longs;oicuo, cauda vero &longs;upra modum fulgente, ut <lb/>&longs;tantes in littore &longs;peciem ejus e mari reflexam facile cernerent. </s> <s><lb/>Speciem utique habebat trabis &longs;plendentis longitudine 23 gra­<lb/>duum, ab occidente in au&longs;trum vergens, & Horizonti fere para­<lb/>lela. </s> <s>Tantus autem &longs;plendor tres &longs;olum dies durabat, &longs;ubinde <lb/>notabiliter decre&longs;cens; & interea decre&longs;cente &longs;plendore aucta e&longs;t <lb/>magnitudine cauda. </s> <s>Unde etiam in <emph type="italics"/>Portugallia<emph.end type="italics"/>quartam fere <lb/>cœli partem (id e&longs;t, gradus 45) occupa&longs;&longs;e dicitur, ab occidente in <lb/>orientem &longs;plendore cum in&longs;igni proten&longs;a; nec tamen tota apparuit, <lb/>capite &longs;emper in his regionibus infra Horizontem delite&longs;cente. </s> <s><lb/>Ex incremento caudæ & decremento &longs;plendoris manife&longs;tum e&longs;t <lb/>quod caput a Sole rece&longs;&longs;it, eique proximum fuit &longs;ub initio, pro <lb/>more Cometæ anni 1680. Et &longs;imilis legitur Cometa anni 1101 <lb/>vel 1106, <emph type="italics"/>cujus Steila erat parva & ob&longs;cura<emph.end type="italics"/>(ut ille anni 1680) <lb/><emph type="italics"/>&longs;ed &longs;plendor qui ex ea exivit valde clarus & qua&longs;i ingens trabs ad <lb/>Orientem & Aquilonem tendebat,<emph.end type="italics"/>ut habet <emph type="italics"/>Hevelius<emph.end type="italics"/>ex <emph type="italics"/>Simeone <lb/>Dunelmen&longs;i<emph.end type="italics"/>Monacho. </s> <s>Apparuit initio Men&longs;is <emph type="italics"/>Februarii,<emph.end type="italics"/>circa ve­<lb/>&longs;peram, ad occa&longs;um Solis brumalem. </s> <s>Inde vero & ex &longs;itu caudæ col­<lb/>ligitur caput fui&longs;&longs;e Soli vicinum. <emph type="italics"/>A Sole,<emph.end type="italics"/>inquit Matthæus Pari­<lb/>&longs;ien&longs;is, <emph type="italics"/>di&longs;tabat qua&longs;i cubito uno, ab hora tertia<emph.end type="italics"/>[rectius &longs;exta] <emph type="italics"/>u&longs;­<lb/>que ad horam nonam radium ex &longs;e longum emittens.<emph.end type="italics"/>Talis etiam <lb/>erat ardenti&longs;&longs;imus ille Cometa ab <emph type="italics"/>Ari&longs;totele<emph.end type="italics"/>de&longs;criptus Lib. </s> <s>l. <lb/></s> <s>Meteor. </s> <s>6. <emph type="italics"/>cujus caput primo die non con&longs;pectum e&longs;t, eo quod ante <lb/>Solem vel &longs;altem &longs;ub radiis &longs;olaribus oceidi&longs;&longs;et, &longs;equente vero die <lb/>quantum potuit vi&longs;um e&longs;t. </s> <s>Nam quam minima fieri pote&longs;t di&longs;tantia <lb/>Solem reliquit, & mox occubuit. </s> <s>Ob nimium ardorem<emph.end type="italics"/>[caudæ &longs;cili­<lb/>cet] <emph type="italics"/>nondum apparebat capitis &longs;par&longs;us ignis, &longs;ed procedente tem­<lb/>pore<emph.end type="italics"/>(ait Ari&longs;toteles) <emph type="italics"/>cum<emph.end type="italics"/>[cauda] <emph type="italics"/>jam minus flagraret, reddita <lb/>e&longs;t<emph.end type="italics"/>[capiti] <emph type="italics"/>Cometæ &longs;ua facies. </s> <s>Et &longs;plendorem &longs;uum ad tertiam <lb/>u&longs;que cæli partem<emph.end type="italics"/>[id e&longs;t, ad 60<emph type="sup"/>gr.<emph.end type="sup"/>] <emph type="italics"/>extendit. </s> <s>Apparuit autem<emph.end type="italics"/><pb xlink:href="039/01/506.jpg" pagenum="475"/><emph type="italics"/>tempore hyberno, & a&longs;cendens u&longs;que ad cingulum Orionis ibi evanuit.<emph.end type="italics"/><lb/><arrow.to.target n="note504"/>Cometa ille anni 1618, qui c radiis Solaribus caudati&longs;&longs;imus emer&longs;it, <lb/>&longs;tellas primæ magnitudinis æquare vel paulo &longs;uperare videbatur, <lb/>&longs;ed majores apparuere Cometæ non pauci qui caudas breviores <lb/>habuere. </s> <s>Horum aliqui Jovem, alii Venerem vel etiam Lunam <lb/>æqua&longs;&longs;e traduntur. </s></p> <p type="margin"> <s><margin.target id="note504"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Diximus Cometas e&longs;&longs;e genus Planetarum in Orbibus valde ec­<lb/>centricis circa Solem revolventium. </s> <s>Et quemadmodum e Plane­<lb/>tis non caudatis, minores e&longs;&longs;e &longs;olent qui in Orbibus minoribus & <lb/>Soli propioribus gyrantur, &longs;ic etiam Cometas, qui in Perihcliis <lb/>&longs;uis ad Solem propius accedunt, ut plurimum minores e&longs;&longs;e, ne<lb/>Solem attractione &longs;ua nimis agitent, rationi con&longs;entaneum videtur. </s> <s><lb/>Orbium vero tran&longs;ver&longs;as diametros & revolutionum tempora <lb/>periodica, ex collatione Cometarum in ii&longs;dem Orbibus po&longs;t longa <lb/>temporum intervalla redeuntium, determinanda relinquo. </s> <s>Interea <lb/>huic negotio Propo&longs;itio &longs;equens lumen accendere pote&longs;t. </s></p> <p type="main"> <s><emph type="center"/>PROPOSITIO XLII. PROBLEMA XXII.<emph.end type="center"/></s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>Trajectoriam Cometæ Graphice inventam corrigere.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s><emph type="italics"/>Oper.<emph.end type="italics"/>1. A&longs;&longs;umatur po&longs;itio plani Trajectoriæ, per Propo&longs;itio­<lb/>nem &longs;uperiorem Graphice inventa; & &longs;eligantur tria loca Cometæ <lb/>ob&longs;ervationibus accurati&longs;&longs;imis de&longs;inita, & ab invicem quam ma­<lb/>xime di&longs;tantia; &longs;itque A tempus inter primam & &longs;ecundam, ac <lb/>B tempus inter &longs;ecundam ac tertiam. </s> <s>Cometam autem in eorum <lb/>aliquo in Perigæo ver&longs;ari convenit, vel &longs;altem non longe a Peri­<lb/>gæo abe&longs;&longs;e. </s> <s>Ex his locis apparentibus inveniantur, per opera­<lb/>tiones Trigonometricas, loca tria vera Cometæ in a&longs;&longs;umpto illo <lb/>plano Trajectoriæ. </s> <s>Deinde per loca illa inventa, circa centrum <lb/>Solis ceu umbilicum, per operationes Arithmeticas, ope Prop. </s> <s><lb/>XXI. Lib. </s> <s>I. in&longs;titutas, de&longs;cribatur Sectio Conica: & ejus areæ, <lb/>radiis a Sole ad loca inventa ductis terminatæ, &longs;unto D & E; <lb/>nempe D area inter ob&longs;ervationem primam & &longs;ecundam, & E <lb/>area inter &longs;ecundam ac tertiam. </s> <s>Sitque T tempus totum quo <lb/>area tota D+E, velocitate Cometæ per Prop. </s> <s>XVI. Lib. </s> <s>I. in­<lb/>venta, ce&longs;cribi debet. </s></p> <p type="main"> <s><emph type="italics"/>Oper.<emph.end type="italics"/>2. Augeatur longitudo Nodorum Plani Trajectoriæ, ad­<lb/>ditis ad longitudinem illam 20′ vel 30′, quæ dicantur P; & &longs;er­<lb/>vetur plani illius inclinatio ad planum Eclipticæ. </s> <s>Deinde ex <pb xlink:href="039/01/507.jpg" pagenum="476"/><arrow.to.target n="note505"/>prædictis tribus Cometæ locis ob&longs;ervatis, inveniantur in hoc novo <lb/>plano loca tria vera (at &longs;upra:) deinde etiam Orbis per loca <lb/>illa tran&longs;iens, & eju&longs;dem areæ duæ inter ob&longs;ervationes de&longs;criptæ, <lb/>quæ &longs;int <emph type="italics"/>d<emph.end type="italics"/>& <emph type="italics"/>e,<emph.end type="italics"/>nec non tempus totum <emph type="italics"/>t<emph.end type="italics"/>quo area tota <emph type="italics"/>d+e<emph.end type="italics"/>de­<lb/>&longs;cribi debeat. </s></p> <p type="margin"> <s><margin.target id="note505"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s><emph type="italics"/>Oper.<emph.end type="italics"/>3. Servetur Longitudo Nodorum in operatione prima, & <lb/>augeatur inclinatio Plani Trajectoriæ ad planum Eclipticæ, addi­<lb/>tis ad inclinationem illam 20′ vel 30′, quæ dicantur <expan abbr="q.">que</expan> Deinde <lb/>ex ob&longs;ervatis prædictis tribus Cometæ locis apparentibus, inve­<lb/>niantur in hoc novo Plano loca tria vera, Orbi&longs;que per loca <lb/>illa tran&longs;iens, ut & eju&longs;dem areæ duæ inter ob&longs;ervationes de­<lb/>&longs;criptæ, quæ &longs;int <foreign lang="greek">d</foreign> & <foreign lang="greek">e</foreign>, & tempus totum <foreign lang="greek">t</foreign> quo area tota <foreign lang="greek">d</foreign>+<foreign lang="greek">e</foreign><lb/>de&longs;cribi debeat. </s></p> <p type="main"> <s>Jam &longs;it C ad I ut A ad B, & G ad 1 ut D ad E, & <emph type="italics"/>g<emph.end type="italics"/>ad 1 ut <lb/><emph type="italics"/>d<emph.end type="italics"/>ad <emph type="italics"/>e,<emph.end type="italics"/>& <foreign lang="greek">g</foreign> ad 1 ut <foreign lang="greek">d</foreign> ad <foreign lang="greek">e</foreign>; &longs;itque S tempus verum inter ob&longs;erva­<lb/>tionem primam ac tertiam; & &longs;ignis + & -probe ob&longs;ervatis <lb/>quærantur numeri <emph type="italics"/>m<emph.end type="italics"/>& <emph type="italics"/>n,<emph.end type="italics"/>ea lege, ut &longs;it 2G-2C=<emph type="italics"/>m<emph.end type="italics"/>G-<emph type="italics"/>mg+ <lb/>n<emph.end type="italics"/>G-<emph type="italics"/>n<emph.end type="italics"/><foreign lang="greek">g</foreign>, & 2T-2S æquale <emph type="italics"/>m<emph.end type="italics"/>T-<emph type="italics"/>mt+n<emph.end type="italics"/>T-<emph type="italics"/>n<emph.end type="italics"/><foreign lang="greek">t. </foreign></s> <s>Et &longs;i, in <lb/>operatione prima, I de&longs;ignet inclinationem plani Trajectoriæ ad <lb/>planum Eclipticæ, & K longitudinem Nodi alterutrius, erit <lb/>I+<emph type="italics"/>n<emph.end type="italics"/>Q vera inclinatio Plani Trajectoriæ ad Planum Eclipticæ, & <lb/>K+<emph type="italics"/>m<emph.end type="italics"/>P vera longitudo Nodi. </s> <s>Ac denique &longs;i in operatione <lb/>prima, &longs;ecunda ac tertia, quantitates R, <emph type="italics"/>r<emph.end type="italics"/>& <foreign lang="greek">r</foreign> de&longs;ignent Latera <lb/>recta Trajectoriæ, & quantitates 1/L, 1/<emph type="italics"/>l,<emph.end type="italics"/>1/<foreign lang="greek">l</foreign> eju&longs;dem Latera tran&longs;­<lb/>ver&longs;a re&longs;pective: erit R+<emph type="italics"/>mr-m<emph.end type="italics"/>R+<emph type="italics"/>n<foreign lang="greek">r</foreign>-n<emph.end type="italics"/>R verum Latus re­<lb/>ctum, & (1/L+<emph type="italics"/>ml-m<emph.end type="italics"/>L+<emph type="italics"/>n<foreign lang="greek">l</foreign>-n<emph.end type="italics"/>L) verum Latus tran&longs;ver&longs;um Tra­<lb/>jectoriæ quam Cometa de&longs;cribit. </s> <s>Dato autem Latere tran&longs;ver&longs;o <lb/>datur etiam tempus periodicum Cometæ. <emph type="italics"/>Q.E.I.<emph.end type="italics"/></s></p> <p type="main"> <s>Cæterum Cometarum revolventium tempora periodica, & Or­<lb/>bium latera tran&longs;ver&longs;a, haud &longs;atis accurate determinabuntur, ni&longs;i <lb/>per collationem Cometarum inter &longs;e, qui diver&longs;is temporibus ap­<lb/>parent. </s> <s>Si plures Cometæ, po&longs;t æqualia temporum intervalla, <lb/>eundem Orbem de&longs;crip&longs;i&longs;&longs;e reperiantur, concludendum erit hos <lb/>omnes e&longs;&longs;e unum & eundem Cometam, in eodem Orbe revolven­<lb/>tem. </s> <s>Et tum demum ex revolutionum temporibus, dabuntur Or­<lb/>bium latera tran&longs;ver&longs;a, & ex his lateribus determinabuntur Or­<lb/>bes Elliptici. </s></p><pb xlink:href="039/01/508.jpg" pagenum="477"/> <p type="main"> <s>In hunc finem computandæ &longs;unt igitur Cometarum plurium <lb/><arrow.to.target n="note506"/>Traiectoriæ, ex hypothe&longs;i quod &longs;int Parabolicæ. </s> <s>Nam huju&longs;­<lb/>modi Trajectoriæ cum Phænomenis &longs;emper congruent quam­<lb/>proxime. </s> <s>Id liquet, non tantum ex Trajectoria Parabolica Co­<lb/>metæ anni 1680, quam cum ob&longs;ervationibus &longs;upra contuli, &longs;ed <lb/>etiam ex ea Cometæ illius in&longs;ignis, qui annis 1664 & 1665 appa­<lb/>ruit, & ab <emph type="italics"/>Hevelio<emph.end type="italics"/>ob&longs;ervatus fuit. </s> <s>Is ex ob&longs;ervationibus &longs;uis <lb/>longitudines & latitudines hujus Cometæ computavit, &longs;ed minus <lb/>accurate. </s> <s>Ex ii&longs;dem ob&longs;ervationibus, <emph type="italics"/>Halleius<emph.end type="italics"/>no&longs;ter loca Co­<lb/>metæ hujus denuo computavit, & tum demum ex locis &longs;ic inven­<lb/>tis Trajectoriam Cometæ determinavit. </s> <s>Invenit autem ejus No­<lb/>dum a&longs;cendentem in II 21<emph type="sup"/>gr.<emph.end type="sup"/> 13′. </s> <s>55″, Inclinationem Orbitæ ad <lb/>planum Eclipticæ 21<emph type="sup"/>gr.<emph.end type="sup"/> 18′. </s> <s>40″, di&longs;tantiam Perihelii a Nodo in<lb/>Orbita 49<emph type="sup"/>gr.<emph.end type="sup"/> 27′. </s> <s>30″. </s> <s>Perihelium in <!--symbol16--> 8<emph type="sup"/>gr.<emph.end type="sup"/> 40′. </s> <s>30′ cum Lati­<lb/>tudine au&longs;trina heliocentrica 16<emph type="sup"/>gr.<emph.end type="sup"/> 1′. </s> <s>45″. </s> <s>Cometam in Perihelio <lb/><emph type="italics"/>Novemb.<emph.end type="italics"/>24<emph type="sup"/>d<emph.end type="sup"/>. </s> <s>11<emph type="sup"/>h<emph.end type="sup"/>. </s> <s>52′. </s> <s>P. M. tempore æquato <emph type="italics"/>Londini,<emph.end type="italics"/>vel 13<emph type="sup"/>h<emph.end type="sup"/>. </s> <s>8′ <lb/><emph type="italics"/>Gedani,<emph.end type="italics"/>&longs;tylo veteri, & Latus rectum Parabolæ 410286, exi&longs;tente <lb/>mediocri Terræ a Sole di&longs;tantia 100000. Quam probe loca <lb/>Cometæ in hoc Orbe computata, congruunt cum ob&longs;ervationibus, <lb/>patebit ex Tabula &longs;equente ab <emph type="italics"/>Halleio<emph.end type="italics"/>&longs;upputata. <lb/><arrow.to.target n="table15"/> <pb xlink:href="039/01/509.jpg" pagenum="478"/><arrow.to.target n="note507"/><arrow.to.target n="table16"/> </s></p> <p type="margin"> <s><margin.target id="note507"/>DE MUNDI <lb/>SYSTEMATE</s></p><table><table.target id="table15"/><row><cell>Temp. Appar. <lb/> <emph type="italics"/>Gedani<emph.end type="italics"/></cell><cell>Ob&longs;ervata Cometæ di&longs;tantia</cell><cell>Loca ob&longs;ervata</cell><cell>Loca compu­<lb/>tata in Orbe</cell></row><row><cell/><cell/><cell/><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell></row><row><cell><emph type="italics"/>Decemb.<emph.end type="italics"/></cell><cell>a Corde Leonis</cell><cell>46.</cell><cell>24.</cell><cell>20</cell><cell>Long. <!--symbol14--></cell><cell>7.</cell><cell>1.</cell><cell>0</cell><cell><!--symbol14--></cell><cell>7.</cell><cell>1.</cell><cell>29</cell></row><row><cell>3d.</cell><cell>18<emph type="sup"/>h<emph.end type="sup"/>.</cell><cell>29 1/2</cell><cell>a Spica Virginis</cell><cell>22.</cell><cell>52.</cell><cell>10</cell><cell>L<gap/>au&longs;t.</cell><cell>21.</cell><cell>39.</cell><cell>0</cell><cell/><cell>21.</cell><cell>38.</cell><cell>50</cell></row><row><cell>4.</cell><cell>18.</cell><cell>1 1/2</cell><cell>a Corde Leonis</cell><cell>46.</cell><cell>2.</cell><cell>45</cell><cell>Long. <!--symbol14--></cell><cell>6.</cell><cell>15.</cell><cell>0</cell><cell><!--symbol14--></cell><cell>6.</cell><cell>16.</cell><cell>5</cell></row><row><cell>a Spica Virginis</cell><cell>23.</cell><cell>52.</cell><cell>40</cell><cell>Lat. a.</cell><cell>22.</cell><cell>24.</cell><cell>0</cell><cell/><cell>22.</cell><cell>24.</cell><cell>0</cell></row><row><cell>7.</cell><cell>17.</cell><cell>48</cell><cell>a Corde Leonis</cell><cell>44.</cell><cell>48.</cell><cell>0</cell><cell>Long. <!--symbol14--></cell><cell>3.</cell><cell>6.</cell><cell>0</cell><cell><!--symbol14--></cell><cell>3.</cell><cell>7.</cell><cell>33</cell></row><row><cell>a Spica Virginis</cell><cell>27.</cell><cell>56.</cell><cell>40</cell><cell>Lat. a.</cell><cell>25.</cell><cell>22.</cell><cell>0</cell><cell/><cell>25.</cell><cell>21.</cell><cell>40</cell></row><row><cell>17.</cell><cell>14.</cell><cell>43</cell><cell>a Corde Leonis</cell><cell>53.</cell><cell>15.</cell><cell>15</cell><cell>Long. <!--symbol16--></cell><cell>2.</cell><cell>56.</cell><cell>0</cell><cell><!--symbol16--></cell><cell>2.</cell><cell>56.</cell><cell>0</cell></row><row><cell>ab Humero Orionis dext.</cell><cell>45.</cell><cell>43.</cell><cell>30</cell><cell>Lat. a.</cell><cell>49.</cell><cell>25.</cell><cell>0</cell><cell/><cell>49.</cell><cell>25.</cell><cell>0</cell></row><row><cell>19.</cell><cell>9.</cell><cell>25</cell><cell>a Procyone</cell><cell>35.</cell><cell>13.</cell><cell>50</cell><cell>Long. II</cell><cell>28.</cell><cell>40.</cell><cell>30</cell><cell>II</cell><cell>28.</cell><cell>43.</cell><cell>0</cell></row><row><cell>a Lucid. Mandio. Geti</cell><cell>52.</cell><cell>56.</cell><cell>0</cell><cell>Lat. a.</cell><cell>45.</cell><cell>48.</cell><cell>0</cell><cell/><cell>45.</cell><cell>46.</cell><cell>0</cell></row><row><cell>20.</cell><cell>9.</cell><cell>53 1/2</cell><cell>a Procyone</cell><cell>40.</cell><cell>49.</cell><cell>0</cell><cell>Long. II</cell><cell>13.</cell><cell>3.</cell><cell>0</cell><cell>II</cell><cell>13.</cell><cell>5.</cell><cell>0</cell></row><row><cell>a Lucid. Mandib. Ceti</cell><cell>40.</cell><cell>4.</cell><cell>0</cell><cell>Lat. a.</cell><cell>39.</cell><cell>54.</cell><cell>0</cell><cell/><cell>39.</cell><cell>53.</cell><cell>0</cell></row><row><cell>21.</cell><cell>9.</cell><cell>9 1/2</cell><cell>ab Hum. dext. Orionis</cell><cell>26.</cell><cell>21.</cell><cell>25</cell><cell>Long. II</cell><cell>2.</cell><cell>16.</cell><cell>0</cell><cell>II</cell><cell>2.</cell><cell>18.</cell><cell>30</cell></row><row><cell>a Lucid. Mandib. Ceti</cell><cell>29.</cell><cell>28.</cell><cell>0</cell><cell>Lat. a.</cell><cell>33.</cell><cell>41.</cell><cell>0</cell><cell/><cell>33.</cell><cell>39.</cell><cell>40</cell></row><row><cell>22.</cell><cell>9.</cell><cell>0</cell><cell>ab Hum. dext. Orionis</cell><cell>29.</cell><cell>47.</cell><cell>0</cell><cell>Long. <!--symbol5--></cell><cell>24.</cell><cell>24.</cell><cell>0</cell><cell><!--symbol5--></cell><cell>24.</cell><cell>27.</cell><cell>0</cell></row><row><cell>a Lucid. Mandib. Ceti</cell><cell>20.</cell><cell>29.</cell><cell>30</cell><cell>Lat. a.</cell><cell>27.</cell><cell>45.</cell><cell>0</cell><cell/><cell>27.</cell><cell>46.</cell><cell>0</cell></row><row><cell>26.</cell><cell>7.</cell><cell>58</cell><cell>a Lucida Arietis</cell><cell>23.</cell><cell>20.</cell><cell>0</cell><cell>Long. <!--symbol5--></cell><cell>9.</cell><cell>0.</cell><cell>0</cell><cell><!--symbol5--></cell><cell>9.</cell><cell>2.</cell><cell>28</cell></row><row><cell>ab Aldebaran</cell><cell>26.</cell><cell>44.</cell><cell>0</cell><cell>Lat. a.</cell><cell>12.</cell><cell>36.</cell><cell>0</cell><cell/><cell>12.</cell><cell>34.</cell><cell>13</cell></row></table><table><table.target id="table16"/><row><cell>Temp. Appar. <lb/> <emph type="italics"/>Gedani<emph.end type="italics"/></cell><cell>Ob&longs;ervata Cometæ di&longs;tantia</cell><cell>Loca ob&longs;ervata</cell><cell>Loca compu­<lb/>tata in Orbe.</cell></row><row><cell>d.</cell><cell>h.</cell><cell>′</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell></row><row><cell>27.</cell><cell>6.</cell><cell>45</cell><cell>a Lucida Arictis</cell><cell>20.</cell><cell>45.</cell><cell>0</cell><cell>Long. <!--symbol5--></cell><cell>7.</cell><cell>5.</cell><cell>40</cell><cell><!--symbol5--></cell><cell>7.</cell><cell>8.</cell><cell>54</cell></row><row><cell>ab Aldebaran</cell><cell>28.</cell><cell>10.</cell><cell>0</cell><cell>Lat. a.</cell><cell>10.</cell><cell>23.</cell><cell>0</cell><cell/><cell>10.</cell><cell>23.</cell><cell>13</cell></row><row><cell>28.</cell><cell>7.</cell><cell>39</cell><cell>a Lucida Arictis</cell><cell>18.</cell><cell>29.</cell><cell>0</cell><cell>Long. <!--symbol5--></cell><cell>5.</cell><cell>24.</cell><cell>45</cell><cell><!--symbol5--></cell><cell>5.</cell><cell>27.</cell><cell>52</cell></row><row><cell>a Palilicio</cell><cell>29.</cell><cell>37.</cell><cell>0</cell><cell>Lat. a.</cell><cell>8.</cell><cell>22.</cell><cell>50</cell><cell/><cell>8.</cell><cell>23.</cell><cell>37</cell></row><row><cell>31.</cell><cell>6.</cell><cell>45</cell><cell>a Cing. Androm.</cell><cell>30.</cell><cell>48.</cell><cell>10</cell><cell>Long. <!--symbol5--></cell><cell>2.</cell><cell>7.</cell><cell>40</cell><cell><!--symbol5--></cell><cell>2.</cell><cell>8.</cell><cell>20</cell></row><row><cell>a Palilicio</cell><cell>32.</cell><cell>53.</cell><cell>30</cell><cell>Lat. a.</cell><cell>4.</cell><cell>13.</cell><cell>0</cell><cell/><cell>4.</cell><cell>16.</cell><cell>25</cell></row><row><cell><emph type="italics"/>Jan.<emph.end type="italics"/></cell><cell>a Cing. Androm.</cell><cell>25.</cell><cell>11.</cell><cell>0</cell><cell>Long. <!--symbol4--></cell><cell>28.</cell><cell>24.</cell><cell>47</cell><cell><!--symbol4--></cell><cell>28.</cell><cell>24.</cell><cell>0</cell></row><row><cell>7.</cell><cell>7.</cell><cell>37 1/2</cell><cell>a Palilicio</cell><cell>37.</cell><cell>12.</cell><cell>25</cell><cell>Lat. bor.</cell><cell>0.</cell><cell>54.</cell><cell>0</cell><cell/><cell>0.</cell><cell>53.</cell><cell>0</cell></row><row><cell>24.</cell><cell>7.</cell><cell>29</cell><cell>a Palilicio</cell><cell>40.</cell><cell>5.</cell><cell>0</cell><cell>Long. <!--symbol4--></cell><cell>26.</cell><cell>29.</cell><cell>15</cell><cell><!--symbol4--></cell><cell>26.</cell><cell>28.</cell><cell>50</cell></row><row><cell>a Cing. Androm.</cell><cell>20.</cell><cell>32.</cell><cell>15</cell><cell>Lat. bor.</cell><cell>5.</cell><cell>25.</cell><cell>50</cell><cell/><cell>5.</cell><cell>26.</cell><cell>0</cell></row><row><cell><emph type="italics"/>Mar.<emph.end type="italics"/></cell><cell>Cometa ab <emph type="italics"/>Hookio<emph.end type="italics"/>prope &longs;ecundam <lb/> Arictis ob&longs;ervabatur, <emph type="italics"/>Mar.<emph.end type="italics"/>1<emph type="sup"/>d.<emph.end type="sup"/> 7<emph type="sup"/>h.<emph.end type="sup"/> 0′ <lb/> <emph type="italics"/>Loudini,<emph.end type="italics"/>cum</cell><cell>Long. <!--symbol4--></cell><cell>29.</cell><cell>17.</cell><cell>20</cell><cell><!--symbol4--></cell><cell>29.</cell><cell>18.</cell><cell>20</cell></row><row><cell>1.</cell><cell>8</cell><cell>6</cell><cell>Lat. bor.</cell><cell>8.</cell><cell>37.</cell><cell>10</cell><cell/><cell>8.</cell><cell>36.</cell><cell>12</cell></row></table> <p type="main"> <s>Apparuit hic Cometa per men&longs;es tres, &longs;ignaque fere &longs;ex de­<lb/>&longs;crip&longs;it, & uno die gradus fere viginti confecit. </s> <s>Cur&longs;us ejus <lb/>a circulo maximo plurimum deflexit, in boream incurvatus; & <lb/>motus ejus &longs;ub finem ex retrogrado factus e&longs;t directus. </s> <s>Et non <lb/>ob&longs;tante cur&longs;u tam in&longs;olito, Theoria a principio ad finem cum <lb/>ob&longs;ervationibus non minus accurate congruit, quam Theoriæ <lb/>Planetarum cum eorum ob&longs;ervationibus congruere &longs;olent, ut in­<lb/>&longs;picienti Tabulam patebit. </s> <s>Subducenda tamen &longs;unt minuta duo <lb/>prima circiter, ubi Cometa veloci&longs;&longs;imus fuit; id quod fiet au­<lb/>ferendo duodecim minuta &longs;ecunda. </s> <s>prima ab angulo inter Nodum a&longs;cen­<lb/>dentem & Perihelium, &longs;eu con&longs;tituendo angulum illum 49<emph type="sup"/>gr.<emph.end type="sup"/><lb/>27′. </s> <s>18″. </s> <s>Cometæ utriu&longs;que (& hujus & &longs;uperioris) parallaxis <lb/>annua in&longs;ignis fuit, & inde demon&longs;tratur motus annuus Terræ in <lb/>Orbe magno. </s></p> <p type="main"> <s>Confirmatur etiam Theoria per motum Cometæ qui apparuit <lb/>anno 1683. Hic fuit retrogradus in Orbe cujus planum cum <lb/>plano Eclipticæ angulum fere rectum continebat. </s> <s>Hujus Nodus <lb/>a&longs;cendens (computante <emph type="italics"/>Halleio<emph.end type="italics"/>) erat in <!--symbol13--> 23<emph type="sup"/>gr<emph.end type="sup"/> 23′; Inclinatio <lb/>Orbitæ ad Eclipticam 83<emph type="sup"/>gr.<emph.end type="sup"/> 11′; Perihelium in II 25<emph type="sup"/>gr.<emph.end type="sup"/> 29′. </s> <s>30″; <lb/>Di&longs;tantia perihelia a Sole 56020, exi&longs;tente radio Orbis magni <lb/>100000, & tempore Perihelii <emph type="italics"/>Julii<emph.end type="italics"/>2<emph type="sup"/>d<emph.end type="sup"/>. </s> <s>3<emph type="sup"/>h<emph.end type="sup"/>. </s> <s>50′. </s> <s>Loca autem Co­<lb/>metæ in hoc Orbe ab <emph type="italics"/>Halleio<emph.end type="italics"/>computata, & cum locis a <emph type="italics"/>Flam­<lb/>&longs;tedio<emph.end type="italics"/>ob&longs;ervatis collata, exhibentur in Tabula &longs;equente. <pb xlink:href="039/01/510.jpg" pagenum="479"/><arrow.to.target n="table17"/> <lb/><arrow.to.target n="note508"/></s></p> <p type="margin"> <s><margin.target id="note508"/>LIBER <lb/>TERTIUS.</s></p><table><table.target id="table17"/><row><cell>1683</cell><cell>Locus Solis</cell><cell>Cometæ</cell><cell>Lat. Bor.</cell><cell>Cometæ</cell><cell>Lat. Bor.</cell><cell>Differ.</cell><cell>Differ.</cell></row><row><cell>Temp. Æquat.</cell><cell/><cell>Long. Comp.</cell><cell>Comp.</cell><cell>Long. Ob&longs;.</cell><cell>Ob&longs;er.</cell><cell>Long.</cell><cell>Lat.</cell></row><row><cell/><cell>d.</cell><cell>h.</cell><cell>′</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell>′</cell><cell>″</cell><cell>′</cell><cell>″</cell></row><row><cell><emph type="italics"/>Jul.<emph.end type="italics"/></cell><cell>13.</cell><cell>12.</cell><cell>55</cell><cell><!--symbol16--></cell><cell>1.</cell><cell>2.</cell><cell>30</cell><cell><!--symbol11--></cell><cell>13.</cell><cell>5.</cell><cell>42</cell><cell>29.</cell><cell>28.</cell><cell>13</cell><cell><!--symbol11--></cell><cell>13.</cell><cell>6.</cell><cell>42</cell><cell>29.</cell><cell>28.</cell><cell>20</cell><cell>+ 1.</cell><cell>0</cell><cell>+ 0.</cell><cell>7</cell></row><row><cell>15.</cell><cell>11.</cell><cell>15</cell><cell>2.</cell><cell>53.</cell><cell>12</cell><cell>11.</cell><cell>37</cell><cell>48</cell><cell>29.</cell><cell>34.</cell><cell>0</cell><cell>11.</cell><cell>39.</cell><cell>43</cell><cell>29.</cell><cell>34.</cell><cell>50</cell><cell>+ 1.</cell><cell>55</cell><cell>+ 0.</cell><cell>50</cell></row><row><cell>17.</cell><cell>10.</cell><cell>20</cell><cell>4.</cell><cell>45.</cell><cell>45</cell><cell>10.</cell><cell>7.</cell><cell>6</cell><cell>29.</cell><cell>33.</cell><cell>30</cell><cell>10.</cell><cell>8.</cell><cell>40</cell><cell>29.</cell><cell>34.</cell><cell>0</cell><cell>+ 1.</cell><cell>34</cell><cell>+ 0.</cell><cell>30</cell></row><row><cell>23.</cell><cell>13.</cell><cell>40</cell><cell>10.</cell><cell>38.</cell><cell>21</cell><cell>5.</cell><cell>10.</cell><cell>27</cell><cell>28.</cell><cell>51.</cell><cell>42</cell><cell>5.</cell><cell>11.</cell><cell>30</cell><cell>28.</cell><cell>50.</cell><cell>28</cell><cell>+ 1.</cell><cell>3</cell><cell>-1.</cell><cell>14</cell></row><row><cell>25.</cell><cell>14.</cell><cell>5</cell><cell>12.</cell><cell>35.</cell><cell>28</cell><cell>3.</cell><cell>27.</cell><cell>53</cell><cell>24.</cell><cell>24.</cell><cell>47</cell><cell>3.</cell><cell>27.</cell><cell>0</cell><cell>28.</cell><cell>23.</cell><cell>40</cell><cell>-0.</cell><cell>53</cell><cell>-1.</cell><cell>7</cell></row><row><cell>31.</cell><cell>9.</cell><cell>42</cell><cell>18.</cell><cell>9.</cell><cell>22</cell><cell>II</cell><cell>27.</cell><cell>55.</cell><cell>3</cell><cell>26.</cell><cell>22.</cell><cell>52</cell><cell>II</cell><cell>27.</cell><cell>54.</cell><cell>24</cell><cell>26.</cell><cell>22.</cell><cell>25</cell><cell>-0.</cell><cell>39</cell><cell>-0.</cell><cell>27</cell></row><row><cell>31.</cell><cell>14.</cell><cell>55</cell><cell>18.</cell><cell>21.</cell><cell>53</cell><cell>27.</cell><cell>41.</cell><cell>7</cell><cell>26.</cell><cell>16.</cell><cell>57</cell><cell>27.</cell><cell>41.</cell><cell>8</cell><cell>26.</cell><cell>14.</cell><cell>50</cell><cell>+ 0.</cell><cell>1</cell><cell>-2.</cell><cell>7</cell></row><row><cell><emph type="italics"/>Aug.<emph.end type="italics"/></cell><cell>2.</cell><cell>14.</cell><cell>56</cell><cell>20.</cell><cell>17.</cell><cell>16</cell><cell>25.</cell><cell>29.</cell><cell>32</cell><cell>25.</cell><cell>16.</cell><cell>19</cell><cell>25.</cell><cell>28.</cell><cell>46</cell><cell>25.</cell><cell>17.</cell><cell>28</cell><cell>-0.</cell><cell>46</cell><cell>+ 1.</cell><cell>9</cell></row><row><cell>4.</cell><cell>10.</cell><cell>49</cell><cell>22.</cell><cell>2.</cell><cell>50</cell><cell>23.</cell><cell>18.</cell><cell>20</cell><cell>24.</cell><cell>10.</cell><cell>49</cell><cell>23.</cell><cell>16.</cell><cell>55</cell><cell>24.</cell><cell>12.</cell><cell>19</cell><cell>-1.</cell><cell>25</cell><cell>+ 1.</cell><cell>30</cell></row><row><cell>6.</cell><cell>10.</cell><cell>9</cell><cell>23.</cell><cell>56.</cell><cell>45</cell><cell>20.</cell><cell>42.</cell><cell>23</cell><cell>22.</cell><cell>47.</cell><cell>5</cell><cell>20.</cell><cell>40.</cell><cell>32</cell><cell>22.</cell><cell>49.</cell><cell>5</cell><cell>-1.</cell><cell>51</cell><cell>+ 2.</cell><cell>0</cell></row><row><cell>9.</cell><cell>10.</cell><cell>26</cell><cell>26.</cell><cell>50.</cell><cell>52</cell><cell>16.</cell><cell>7.</cell><cell>57</cell><cell>20.</cell><cell>6.</cell><cell>37</cell><cell>16.</cell><cell>5.</cell><cell>55</cell><cell>20.</cell><cell>6.</cell><cell>10</cell><cell>-2.</cell><cell>2</cell><cell>-0.</cell><cell>27</cell></row><row><cell>15.</cell><cell>14.</cell><cell>1</cell><cell><!--symbol13--></cell><cell>2.</cell><cell>47.</cell><cell>13</cell><cell>3.</cell><cell>30.</cell><cell>48</cell><cell>11.</cell><cell>37.</cell><cell>33</cell><cell>3.</cell><cell>26.</cell><cell>18</cell><cell>11.</cell><cell>32.</cell><cell>1</cell><cell>-4.</cell><cell>30</cell><cell>-5.</cell><cell>32</cell></row><row><cell>16.</cell><cell>15.</cell><cell>10</cell><cell>3.</cell><cell>48.</cell><cell>2</cell><cell>0</cell><cell>43.</cell><cell>7</cell><cell>9.</cell><cell>34.</cell><cell>16</cell><cell>0.</cell><cell>41.</cell><cell>55</cell><cell>9.</cell><cell>34.</cell><cell>13</cell><cell>-1.</cell><cell>12</cell><cell>-0.</cell><cell>3</cell></row><row><cell>18.</cell><cell>15.</cell><cell>44</cell><cell>5.</cell><cell>45.</cell><cell>33</cell><cell><!--symbol5--></cell><cell>24.</cell><cell>52.</cell><cell>53</cell><cell>5.</cell><cell>11.</cell><cell>15</cell><cell><!--symbol5--></cell><cell>24.</cell><cell>49.</cell><cell>5</cell><cell>5.</cell><cell>9.</cell><cell>11</cell><cell>-3.</cell><cell>48</cell><cell>-2.</cell><cell>4</cell></row><row><cell/><cell/><cell/><cell/><cell/><cell/><cell/><cell/><cell/><cell>Au&longs;tr.</cell><cell/><cell/><cell/><cell>Au&longs;tr.</cell><cell/><cell/><cell/><cell/></row><row><cell>22.</cell><cell>14.</cell><cell>44</cell><cell>9.</cell><cell>35.</cell><cell>49</cell><cell>11.</cell><cell>7.</cell><cell>14</cell><cell>5.</cell><cell>16.</cell><cell>53</cell><cell>11.</cell><cell>7.</cell><cell>12</cell><cell>5.</cell><cell>16.</cell><cell>50</cell><cell>-0.</cell><cell>2</cell><cell>-0.</cell><cell>3</cell></row><row><cell>23.</cell><cell>15.</cell><cell>52</cell><cell>10.</cell><cell>36.</cell><cell>48</cell><cell>7.</cell><cell>2.</cell><cell>18</cell><cell>8.</cell><cell>17.</cell><cell>9</cell><cell>7.</cell><cell>1.</cell><cell>17</cell><cell>8.</cell><cell>16.</cell><cell>41</cell><cell>-1.</cell><cell>1</cell><cell>-0.</cell><cell>28</cell></row><row><cell>26.</cell><cell>16.</cell><cell>2</cell><cell>13.</cell><cell>31.</cell><cell>10</cell><cell><!--symbol4--></cell><cell>24.</cell><cell>45.</cell><cell>31</cell><cell>16.</cell><cell>38.</cell><cell>0</cell><cell><!--symbol4--></cell><cell>24.</cell><cell>44.</cell><cell>0</cell><cell>16.</cell><cell>38.</cell><cell>20</cell><cell>-1.</cell><cell>31</cell><cell>+ 0.</cell><cell>20</cell></row></table> <p type="main"> <s>Confirmatur etiam Theoria per motum Cometæ retrogradi qui <lb/>apparuit anno 1682. Hujus Nodus a&longs;cendens (computante <emph type="italics"/>Hal­<lb/>leio<emph.end type="italics"/>) erat in 8 21<emph type="sup"/>gr.<emph.end type="sup"/> 16′. </s> <s>30″. </s> <s>Inclinatio Orbitæ ad planum Eclip­<lb/>ticæ 17<emph type="sup"/>gr.<emph.end type="sup"/> 56′. </s> <s>0″. </s> <s>Perihelium in = 2<emph type="sup"/>gr.<emph.end type="sup"/> 52′. </s> <s>50″. </s> <s>Di&longs;tantia peri­<lb/>helia a Sole 58328. Et tempus æquatum Perihelii <emph type="italics"/>Sept.<emph.end type="italics"/>4<emph type="sup"/>d<emph.end type="sup"/>. </s> <s>7<emph type="sup"/>h<emph.end type="sup"/>. </s> <s>39′. </s> <s><lb/>Loca vero ex ob&longs;ervationibus <emph type="italics"/>Flam&longs;tedii<emph.end type="italics"/>computata, & cum locis <lb/>per Theoriam computatis collata, exhibentur in Tabula &longs;e­<lb/>quente. <lb/><arrow.to.target n="table18"/> </s></p><table><table.target id="table18"/><row><cell>1682</cell><cell>Locus Solis</cell><cell>Cometæ</cell><cell>Lat. Bor.</cell><cell>Cometæ</cell><cell>Lat. Bor.</cell><cell>Differ.</cell><cell>Differ.</cell></row><row><cell>Temp. Appar.</cell><cell/><cell>Long. Comp.</cell><cell>Comp.</cell><cell>Long. Ob&longs;.</cell><cell>Ob&longs;er.</cell><cell>Long.</cell><cell>Lat.</cell></row><row><cell/><cell>d.</cell><cell>h.</cell><cell>′</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell/><cell>gr.</cell><cell>′</cell><cell>″</cell><cell>gr.</cell><cell>′</cell><cell>″</cell><cell>′</cell><cell>″</cell><cell>′</cell><cell>″</cell></row><row><cell><emph type="italics"/>Aug.<emph.end type="italics"/></cell><cell>19.</cell><cell>16.</cell><cell>38</cell><cell><!--symbol13--></cell><cell>7.</cell><cell>0.</cell><cell>7</cell><cell><!--symbol16--></cell><cell>18.</cell><cell>14.</cell><cell>28</cell><cell>25.</cell><cell>50</cell><cell>7</cell><cell><!--symbol16--></cell><cell>18.</cell><cell>14.</cell><cell>40</cell><cell>25.</cell><cell>49.</cell><cell>55</cell><cell>-0.</cell><cell>12</cell><cell>+ 0.</cell><cell>12</cell></row><row><cell>20.</cell><cell>15.</cell><cell>38</cell><cell>7.</cell><cell>55.</cell><cell>52</cell><cell>24.</cell><cell>46.</cell><cell>23</cell><cell>26.</cell><cell>14</cell><cell>42</cell><cell>24.</cell><cell>46.</cell><cell>22</cell><cell>26.</cell><cell>12.</cell><cell>52</cell><cell>+ 0.</cell><cell>1</cell><cell>+ 1.</cell><cell>50</cell></row><row><cell>21.</cell><cell>8.</cell><cell>21</cell><cell>8.</cell><cell>36.</cell><cell>14</cell><cell>29.</cell><cell>37.</cell><cell>15</cell><cell>26.</cell><cell>20.</cell><cell>3</cell><cell>29.</cell><cell>38.</cell><cell>2</cell><cell>26.</cell><cell>17.</cell><cell>37</cell><cell>-0.</cell><cell>47</cell><cell>+ 2.</cell><cell>26</cell></row><row><cell>22.</cell><cell>8.</cell><cell>8</cell><cell>9.</cell><cell>33.</cell><cell>55</cell><cell><!--symbol13--></cell><cell>6.</cell><cell>29.</cell><cell>53</cell><cell>26.</cell><cell>8.</cell><cell>42</cell><cell><!--symbol13--></cell><cell>6.</cell><cell>30.</cell><cell>3</cell><cell>26.</cell><cell>7.</cell><cell>12</cell><cell>-0.</cell><cell>10</cell><cell>+ 1.</cell><cell>30</cell></row><row><cell>29.</cell><cell>8.</cell><cell>20</cell><cell>16.</cell><cell>22.</cell><cell>40</cell><cell><!--symbol14--></cell><cell>12.</cell><cell>37</cell><cell>54</cell><cell>18.</cell><cell>37.</cell><cell>47</cell><cell><!--symbol14--></cell><cell>12.</cell><cell>37.</cell><cell>49</cell><cell>18.</cell><cell>34.</cell><cell>5</cell><cell>+ 0.</cell><cell>5</cell><cell>+ 3.</cell><cell>42</cell></row><row><cell>30.</cell><cell>7.</cell><cell>45</cell><cell>17.</cell><cell>19.</cell><cell>41</cell><cell>15.</cell><cell>36.</cell><cell>1</cell><cell>17.</cell><cell>26.</cell><cell>43</cell><cell>15.</cell><cell>35.</cell><cell>18</cell><cell>17.</cell><cell>27.</cell><cell>17</cell><cell>+ 0.</cell><cell>43</cell><cell>-0.</cell><cell>34</cell></row><row><cell><emph type="italics"/>Sept.<emph.end type="italics"/></cell><cell>1.</cell><cell>7.</cell><cell>33</cell><cell>19.</cell><cell>16.</cell><cell>9</cell><cell>20.</cell><cell>30.</cell><cell>53</cell><cell>15.</cell><cell>13.</cell><cell>0</cell><cell>20.</cell><cell>27.</cell><cell>4</cell><cell>15.</cell><cell>9.</cell><cell>49</cell><cell>+ 3.</cell><cell>49</cell><cell>+ 3.</cell><cell>11</cell></row><row><cell>4.</cell><cell>7.</cell><cell>22</cell><cell>22.</cell><cell>11.</cell><cell>28</cell><cell>25.</cell><cell>42.</cell><cell>0</cell><cell>12.</cell><cell>23.</cell><cell>48</cell><cell>25.</cell><cell>40.</cell><cell>58</cell><cell>12.</cell><cell>22.</cell><cell>0</cell><cell>+ 1.</cell><cell>2</cell><cell>+ 1.</cell><cell>43</cell></row><row><cell>5.</cell><cell>7.</cell><cell>32</cell><cell>23.</cell><cell>10.</cell><cell>29</cell><cell>27.</cell><cell>0.</cell><cell>46</cell><cell>11.</cell><cell>33.</cell><cell>8</cell><cell>26.</cell><cell>59.</cell><cell>24</cell><cell>11.</cell><cell>33.</cell><cell>51</cell><cell>+ 1.</cell><cell>22</cell><cell>-0.</cell><cell>43</cell></row><row><cell>8.</cell><cell>7.</cell><cell>16</cell><cell>26.</cell><cell>5.</cell><cell>58</cell><cell>29.</cell><cell>58.</cell><cell>44</cell><cell>9.</cell><cell>26.</cell><cell>46</cell><cell>29.</cell><cell>58.</cell><cell>45</cell><cell>9.</cell><cell>26.</cell><cell>43</cell><cell>-0.</cell><cell>1</cell><cell>+ 0.</cell><cell>3</cell></row><row><cell>9.</cell><cell>7.</cell><cell>26</cell><cell>27.</cell><cell>5.</cell><cell>9</cell><cell><!--symbol15--></cell><cell>0.</cell><cell>44.</cell><cell>10</cell><cell>8.</cell><cell>49.</cell><cell>10</cell><cell><!--symbol15--></cell><cell>0.</cell><cell>44.</cell><cell>4</cell><cell>8.</cell><cell>48.</cell><cell>25</cell><cell>+ 0.</cell><cell>6</cell><cell>+ 0.</cell><cell>45</cell></row></table> <p type="main"> <s>His exemplis abunde &longs;atis manife&longs;tum e&longs;t, quod motus Come­<lb/>tarum per Theoriam a nobis expo&longs;itam non minus accurate ex-<pb xlink:href="039/01/511.jpg" pagenum="480"/><arrow.to.target n="note509"/>hibentur, quam &longs;olent motus Planetarum per eorum Theovias. </s> <s>Et <lb/>propterea Orbes Cometarum per hanc Theoriam enumerari po&longs;­<lb/>&longs;unt, & tempus periodicum Cometæ in quolibet Orbe revolventis <lb/>tandem &longs;ciri, & tum demum Orbium Elliptieorum latera tran&longs;­<lb/>ver&longs;a & Apheliorum altitudines innote&longs;cent. </s></p> <p type="margin"> <s><margin.target id="note509"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Cometa retrogradus qui apparuit anno 1607, de&longs;crip&longs;it Orbem <lb/>cujus Nodus a&longs;cendens (computante <emph type="italics"/>Halleio<emph.end type="italics"/>) erat in 8 20<emph type="sup"/>gr.<emph.end type="sup"/> 21′. </s> <s><lb/>Inclinatio plani Orbis ad planum Eclipticæ erat 17<emph type="sup"/>gr.<emph.end type="sup"/> 2′. </s> <s>Peri­<lb/>helium erat in = 2<emph type="sup"/>gr.<emph.end type="sup"/> 16′, & di&longs;tantia perihelia a Sole erat 58680, <lb/>exi&longs;tente radio Orbis magni 100000. Et Cometa erat in Peri­<lb/>helio <emph type="italics"/>Octob.<emph.end type="italics"/>16<emph type="sup"/>d<emph.end type="sup"/>. </s> <s>3<emph type="sup"/>h<emph.end type="sup"/>. </s> <s>50′. </s> <s>Congruit hic Orbis quamproxime cum <lb/>Orbe Cometæ qui apparuit anno 1682. Si Cometæ hi duo fue­<lb/>rint unus & idem, revolvetur hic Cometa &longs;patio annorum 75, & <lb/>axis major Orbis ejus erit ad axem majorem Orbis magni, ut <lb/>√<emph type="italics"/>c<emph.end type="italics"/>:75X75 ad 1, &longs;eu 1778 ad 100 circiter. </s> <s>Et di&longs;tantia aphe­<lb/>lia Cometæ hujus a Sole, erit ad di&longs;tantiam mediocrem Terræ a <lb/>Sole, ut 35 ad 1 circiter. </s> <s>Quibus cognitis, haud difficile fuerit <lb/>Orbem Ellipticum Cometæ hujus determinare. </s> <s>Atque hæc ita <lb/>&longs;e habebunt &longs;i Cometa, &longs;patio annorum &longs;eptuaginta quinque, in <lb/>hoc Orbe po&longs;thac redierit. </s> <s>Cometæ reliqui majori tempore re­<lb/>volvi videntur & altius a&longs;cendere. </s></p> <p type="main"> <s>Cæterum Cometæ, ob magnum eorum numerum, & magnam <lb/>Apheliorum a Sole di&longs;tantiam, & longam moram in Apheliis, per <lb/>gravitates in &longs;e mutuo nonnihil turbari debent, & eorum eccen­<lb/>tricitates & revolutionum tempora nunc augeri aliquantulum, <lb/>nunc diminui. </s> <s>Proinde non e&longs;t expectandum ut Cometa idem, <lb/>in eodem Orbe & ii&longs;dem temporibus periodicis, accurate redeat. </s> <s><lb/>Sufficit &longs;i mutationes non majores obvenerint, quam quæ a cau&longs;is <lb/>prædictis oriantur. </s></p> <p type="main"> <s>Et hinc ratio redditur cur Cometæ non comprehendantur Zo­<lb/>diaco (more Planetarum) &longs;ed inde migrent & motibus variis in <lb/>omnes cœlorum regiones ferantur. </s> <s>Scilicet eo fine, ut in Apheliis <lb/>&longs;uis ubi tardi&longs;&longs;ime moventur, quam longi&longs;&longs;ime di&longs;tent ab invicem <lb/>& &longs;e mutuo quam minime trahant. </s> <s>Qua de cau&longs;a Cometæ qui <lb/>altius de&longs;cendunt, adeoque tardi&longs;&longs;ime moventur in Apheliis, de­<lb/>bent altius a&longs;cendere. </s></p> <p type="main"> <s>Cometa qui anno 1680 apparuit, minus di&longs;tabat a Sole in Peri­<lb/>helio. </s> <s>&longs;uo quam parte &longs;exta diametri Solis; & propter &longs;ummam <lb/>velocitatem in vicinia illa, & den&longs;itatem aliquam Atmo&longs;phæræ So­<lb/>lis, re&longs;i&longs;tentiam nonnullam &longs;entire debuit, & aliquantulum retar-<pb xlink:href="039/01/512.jpg" pagenum="481"/>dari & propius ad Solem accedere: & &longs;ingulis revolutionibus ac­<lb/><arrow.to.target n="note510"/>cedendo ad Solem, incidet is tandem in corpus Solis. </s> <s>Sed & in <lb/>Aphelio ubi tardi&longs;&longs;ime movetur, aliquando per attractionem alio­<lb/>rum Cometarum retardari pote&longs;t & &longs;ubinde in Solem incidere. </s> <s><lb/>Sic etiam Stellæ fixæ quæ paulatim expirant in lucern & vapores, <lb/>Cometis in ip&longs;as incidentibus refici po&longs;&longs;unt, & novo alimento <lb/>accen&longs;æ pro Stellis Novis haberi. </s> <s>Vapores autem qui ex Sole & <lb/>Stellis fixis & caudis Cometarum oriuntur, incidere po&longs;&longs;unt per <lb/>gravitatem &longs;uam in Atmo&longs;phæras Planetarum, & ibi conden&longs;ari <lb/>& converti in aquam & &longs;piritus humidos, & &longs;ubinde per lentum <lb/>calorem in &longs;ales, & &longs;ulphura, & tincturas, & limum, & lutum, & <lb/>argillam, & arenam, & lapides, & coralla, & &longs;ub&longs;tantias alias <lb/>terre&longs;tres paulatim migrare. </s> <s>Decre&longs;cente autem corpore Solis <lb/>motus medii Planetarum circum Solem paulatim tarde&longs;cent, & <lb/>cre&longs;cente Terra motus medius Lunæ circum Terram paulatim au­<lb/>gebitur. </s> <s>Et collatis quidem ob&longs;ervationibus Eclip&longs;ium <emph type="italics"/>BabyloNI­<lb/>cis<emph.end type="italics"/>cum iis <emph type="italics"/>Albategnii<emph.end type="italics"/>& cum hodiernis, <emph type="italics"/>Halleius<emph.end type="italics"/>no&longs;ter motum <lb/>medium Lunæ cum motu diurno Terræ collatum, paulatim acce­<lb/>lerari, primus omnium quod &longs;ciam deprehendit. </s></p> <p type="margin"> <s><margin.target id="note510"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>SCHOLIUM GENERALE.<emph.end type="italics"/><emph.end type="center"/></s></p> <p type="main"> <s>Hypothe&longs;is Vorticum multis premitur difficultatibus. </s> <s>Ut Pla­<lb/>neta unu&longs;qui&longs;que radio ad Solem ducto areas de&longs;cribat tempori <lb/>proportionales, tempora periodica partium Vorticis deberent e&longs;&longs;e <lb/>in duplicata ratione di&longs;tantiarum a Sole. </s> <s>Ut periodica Plane­<lb/>tarum tempora &longs;int in proportione &longs;e&longs;quiplicata di&longs;tantiarum a <lb/>Sole, tempora periodica partium Vorticis deberent e&longs;&longs;e in eadem <lb/>di&longs;tantiarum proportione. </s> <s>Ut Vortices minores circum Satur­<lb/>num, Jovem & alios Planetas gyrati con&longs;erventur & tranquille <lb/>natent in Vortice Solis, tempora periodica partium Vorticis So­<lb/>laris deberent e&longs;&longs;e æqualia. </s> <s>Revolutiones Solis & Planetarum cir­<lb/>cum axes &longs;uos ab omnibus hi&longs;ce proportionibus di&longs;crepant. </s> <s>Mo­<lb/>tus Cometarum &longs;unt &longs;umme regulares, & ea&longs;dem leges cum Pla­<lb/>netarum motibus ob&longs;ervant, & per Vortices explicari nequeunt. </s> <s><lb/>Feruntur Cometæ motibus valde eccentricis in omnes cælorum <lb/>partes, quod fieri non pote&longs;t ni&longs;i Vortices tollantur. </s></p> <p type="main"> <s>Projectilia, in aere no&longs;tro, &longs;olam aeris re&longs;i&longs;tentiam &longs;entiunt. </s> <s><lb/>Sublato aere, ut fit in Vacuo <emph type="italics"/>Boyliano,<emph.end type="italics"/>re&longs;i&longs;tentia ce&longs;&longs;at, &longs;iqui­<lb/>dem pluma tenuis & aurum &longs;olidum æquali cum velocitate in hoc <pb xlink:href="039/01/513.jpg" pagenum="482"/><arrow.to.target n="note511"/>Vacuo cadunt. </s> <s>Et par e&longs;t ratio &longs;patiorum cæle&longs;tium quæ &longs;unt <lb/>&longs;upra atmo&longs;phæram Terræ. </s> <s>Corpora omnia in i&longs;tis &longs;patiis liber­<lb/>rime moveri debent; & propterea Planetæ & Cometæ in orbi­<lb/>bus &longs;pecie & po&longs;itione datis, &longs;ecundum leges &longs;upra expo&longs;itas, per­<lb/>petuo revolvi. </s> <s>Per&longs;everabunt quidem in orbibus &longs;uis per leges <lb/>gravitatis, &longs;ed regularem orbium &longs;itum primitus acquirere per <lb/>leges ha&longs;ce minime potuerunt. </s></p> <p type="margin"> <s><margin.target id="note511"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Planetæ &longs;ex principales revolvuntur circum Solem in circulis <lb/>Soli concentricis, eadem motus directione, in eodem plano quam­<lb/>proxime. </s> <s>Lunæ decem revolvuntur circum Terram, Jovem & Sa­<lb/>turnum in circulis concentricis, eadem motus directione, in planis <lb/>orbium Planetarum quamproxime. </s> <s>Et hi omnes motus regulares <lb/>originem non habent ex cau&longs;is Mechanicis; &longs;iquidem Cometæ in <lb/>Orbibus valde eccentricis, & in omnes cælorum partes libere <lb/>feruntur. </s> <s>Quo motus genere Cometæ per Orbes Planetarum ce­<lb/>lerrime & facillime tran&longs;eunt, & in Apheliis &longs;uis ubi tardiores <lb/>&longs;unt & diutius morantur, quam longi&longs;&longs;ime di&longs;tant ab invicem, <lb/>& &longs;e mutuo quam minime trahunt. </s> <s>Eleganti&longs;&longs;ima hæcce Solis, <lb/>Planetarum & Cometarum compages non ni&longs;i con&longs;ilio & dominio <lb/>Entis intelligentis & potentis oriri potuit. </s> <s>Et &longs;i Stellæ fixæ &longs;int <lb/>centra &longs;imilium &longs;y&longs;tematum; hæc omnia &longs;imili con&longs;ilio con&longs;tructa, <lb/>&longs;uberunt <emph type="italics"/>Unius<emph.end type="italics"/>dominio: præ&longs;ertim cum lux Fixarum &longs;it eiu&longs;dem <lb/>naturæ ac lux Solis, & &longs;y&longs;temata omnia lucem in omnia invicem <lb/>immittant. </s></p> <p type="main"> <s>Hic omnia regit, non ut Anima mundi, &longs;ed ut univer&longs;orum Do­<lb/>minus; & propter dominium &longs;uum Dominus Deus <lb/> <foreign lang="greek">pantik<gap/>t<gap/>r</foreign> dici &longs;olet. </s> <s>Nam <emph type="italics"/>Deus<emph.end type="italics"/>e&longs;t vox relativa <lb/>& ad &longs;ervos refertur: & <emph type="italics"/>Deitas<emph.end type="italics"/>e&longs;t dominatio Dei <lb/>non in corpus proprium, &longs;ed in &longs;ervos. <emph type="italics"/>Deus &longs;ummus<emph.end type="italics"/>e&longs;t Ens <lb/>æternum, infinitum, ab&longs;olute perfectum; &longs;ed Ens utcunque per­<lb/>fectum &longs;ine dominio, non e&longs;t <emph type="italics"/>Dominus Deus.<emph.end type="italics"/>Dicimus enim <emph type="italics"/>Deus <lb/>meus, Deus ve&longs;ter, Deus I&longs;raelis:<emph.end type="italics"/>&longs;ed non dicimus <emph type="italics"/>Æternus meus, <lb/>Æternus ve&longs;ter, Æternus I&longs;raelis<emph.end type="italics"/>; non dicimus <emph type="italics"/>Infinitus meus, <lb/>Infinitus ve&longs;ter, Infinitus I&longs;raelis<emph.end type="italics"/>; non dicimus <emph type="italics"/>Perfectus meus, Per­<lb/>fectus ve&longs;ter, Perfectus I&longs;raelis.<emph.end type="italics"/>Hæ appellationes relationem non <lb/>habent ad &longs;ervos. </s> <s>Vox <emph type="italics"/>Deus<emph.end type="italics"/>pa&longs;&longs;im &longs;ignificat <emph type="italics"/>Dominum,<emph.end type="italics"/>&longs;ed <lb/>omnis Dominus non e&longs;t Deus. </s> <s>Dominatio Entis &longs;piritualis <emph type="italics"/>Deum<emph.end type="italics"/><lb/>con&longs;tituit, vera verum, &longs;umma &longs;ummum, ficta fictum. </s> <s>Et ex do­<lb/>minatione vera &longs;equitur, Deum verum e&longs;&longs;e vivum, intelligentem & <lb/>potentem; ex reliquis perfectionibus &longs;ummum e&longs;&longs;e vel &longs;umme per-<pb xlink:href="039/01/514.jpg" pagenum="483"/>fectum. <emph type="italics"/>Æternus<emph.end type="italics"/>e&longs;t & <emph type="italics"/>Infinitus, Omnipotens<emph.end type="italics"/>& <emph type="italics"/>Omni&longs;ciens,<emph.end type="italics"/>id <lb/><arrow.to.target n="note512"/>e&longs;t, durat ab æterno in æternum & ade&longs;t ab infinito in infinitum, <lb/>omnia regit & omnia cogno&longs;cit quæ fiunt aut &longs;ciri po&longs;&longs;unt. </s> <s>Non <lb/>e&longs;t æternitas vel infinitas, &longs;ed æternus & infinitus; non e&longs;t duratio <lb/>vel &longs;patium, &longs;ed durat & ade&longs;t. </s> <s>Durat &longs;emper & ade&longs;t ubique, & <lb/>exi&longs;tendo &longs;emper & ubiQ.E.D.rationem & &longs;patium, æternitatem <lb/>& infinitatem con&longs;tituit. </s> <s>Cum unaquæque &longs;patii particula &longs;it <lb/><emph type="italics"/>&longs;emper,<emph.end type="italics"/>& unumquodQ.E.D.rationis indivi&longs;ibile momentum <emph type="italics"/>ubique<emph.end type="italics"/>; <lb/>certe rerum omnium Fabricator ac Dominus non erit <emph type="italics"/>nunquam <lb/>nu&longs;quam.<emph.end type="italics"/>Omnipræ&longs;ens e&longs;t nen per <emph type="italics"/>virtutem<emph.end type="italics"/>&longs;olam, &longs;ed etiam <lb/>per <emph type="italics"/>&longs;ub&longs;tantiam<emph.end type="italics"/>: nam virtus &longs;ine &longs;ub&longs;tantia <lb/>&longs;ub&longs;i&longs;tere non pote&longs;t. </s> <s>In ip&longs;o continentur <lb/>& moventur univer&longs;a, &longs;ed ab&longs;que mutua <emph type="italics"/>pa&longs;­<lb/>&longs;ione.<emph.end type="italics"/>Deus nihil patitur ex corporum moti­<lb/>bus: illa nullam &longs;entiunt re&longs;i&longs;tentiam ex om­<lb/>nipræ&longs;entia Dei. </s> <s>Deum &longs;ummum nece&longs;&longs;ario <lb/>exi&longs;tere in con&longs;e&longs;&longs;o e&longs;t: Et eadem nece&longs;&longs;itate <lb/><emph type="italics"/>&longs;emper<emph.end type="italics"/>e&longs;t & <emph type="italics"/>ubique.<emph.end type="italics"/>Unde etiam totus e&longs;t &longs;ui &longs;imilis, totus oculus, <lb/>totus auris, totus cerebrum, totus brachium, totus vis &longs;entiendi, <lb/>intelligendi & agendi; &longs;ed more minime humano, more minime <lb/>corporeo, more nobis pror&longs;us incognito. </s> <s>Ut cæcus ideam non <lb/>habet colorum, &longs;ic nos ideam non habemus modorum quibus <lb/>Deus &longs;apienti&longs;&longs;imus &longs;entit & intelligit omnia. </s> <s>Corpore omni & <lb/>figura corporea pror&longs;us de&longs;tituitur, ideoque videri non pote&longs;t, <lb/>nec audiri, nec tangi, nec &longs;ub &longs;pecie rei alicujus corporei coli de­<lb/>bet. </s> <s>Ideas habemus attributorum ejus, &longs;ed quid &longs;it rei alicujus <lb/>Sub&longs;tantia minime cogno&longs;cimus. </s> <s>Videmus tantum corporum figu­<lb/>ras & colores, audimus tantum &longs;onos, tangimus tantum &longs;uper­<lb/>ficies externas, olfacimus odores &longs;olos, & gu&longs;tamus &longs;apores; In­<lb/>timas &longs;ub&longs;tantias nullo &longs;en&longs;u, nulla actione reflexa cogno&longs;cimus, & <lb/>multo minus ideam habemus &longs;ub&longs;tantiæ Dei. </s> <s>Hunc cogno&longs;cimus <lb/>&longs;olummodo per proprietates &longs;uas & attributa, & per &longs;apienti&longs;&longs;i­<lb/>mas & optimas rerum &longs;tructuras, & cau&longs;as finales; veneramur au­<lb/>tem & colimus ob dominium. </s> <s>Deus enim &longs;ine dominio, provi­<lb/>dentia, & cau&longs;is finalibus, nihil aliud e&longs;t quam Fatum & Na­<lb/>tura. </s> <s>Et hæc de Deo; de quo utique ex Phænomenis di&longs;&longs;erere, <lb/>ad <emph type="italics"/>Philo&longs;ophiem Experimentalem<emph.end type="italics"/>pertinet. </s></p> <p type="margin"> <s><margin.target id="note512"/>LIBER <lb/>TERTIUS.</s></p> <p type="main"> <s>Hactenus Phænomena cælorum & maris no&longs;tri per Vim gravi­<lb/>tatis expo&longs;ui, &longs;ed cau&longs;am Gravitatis nondum a&longs;&longs;ignavi. </s> <s>Oritur <lb/>utique hæc Vis a cau&longs;a aliqua quæ penetrat ad u&longs;que centra Solis <pb xlink:href="039/01/515.jpg" pagenum="484"/><arrow.to.target n="note513"/>& Planetarum, &longs;ine virtutis diminutione; quæque agit non pro <lb/>quantitate <emph type="italics"/>&longs;uperficierum<emph.end type="italics"/>particularum in quas agit (ut &longs;olent cau&longs;æ <lb/>Mechanicæ,) &longs;ed pro quantitate materiæ <emph type="italics"/>&longs;olidæ<emph.end type="italics"/>; & cujus actio in <lb/>immen&longs;as di&longs;tantias undique extenditur, decre&longs;cendo &longs;emper in <lb/>duplicata ratione di&longs;tantiarum. </s> <s>Gravitas in Solem componitur <lb/>ex gravitatibus in &longs;ingulas Solis particulas, & recedendo a Sole <lb/>decre&longs;cit accurate in duplicata ratione di&longs;tantiarum ad u&longs;que or­<lb/>bem Saturni, ut ex quiete Apheliorum Planetarum manife&longs;tum e&longs;t, <lb/>& ad u&longs;que ultima Cometarum Aphelia, &longs;i modo Aphelia illa <lb/>quie&longs;cant. </s> <s>Rationem vero harum Gravitatis proprietatum ex <lb/>Phænomenis nondum potui deducere, & Hypothe&longs;es non &longs;ingo. </s> <s><lb/>Quicquid enim ex Phænomenis non deducitur, <emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/>vo­<lb/>canda e&longs;t; & Hypothe&longs;es &longs;eu Metaphy&longs;icæ, &longs;eu Phy&longs;icæ, &longs;eu Qua­<lb/>litatum occultarum, &longs;eu Mechanicæ, in <emph type="italics"/>Philo&longs;ophia Experimentali<emph.end type="italics"/><lb/>locum non habent. </s> <s>In hac Philo&longs;ophia Propo&longs;itiones deducun­<lb/>tur ex Phænomenis, & redduntur generales per Inductionem. </s> <s>Sie <lb/>impenetrabilitas, mobilitas, & impetus corporum & leges motuum <lb/>& gravitatis innotuerunt. </s> <s>Et &longs;atis e&longs;t quod Gravitas revera ex­<lb/>i&longs;tat, & agat &longs;ecundum leges a nobis expo&longs;itas, & ad corporum <lb/>cæle&longs;tium & maris no&longs;tri motus omnes &longs;ufficiat. </s></p> <p type="margin"> <s><margin.target id="note513"/>DE MUNDI <lb/>SYSTEMATE</s></p> <p type="main"> <s>Adjicere jam liceret nonnulla de Spiritu quodam &longs;ubtili&longs;&longs;imo cor­<lb/>pora cra&longs;&longs;a pervadente, & in ii&longs;dem latente; cujus vi & actionibus <lb/>particulæ corporum ad minimas di&longs;tantias &longs;e mutuo attrahunt, <lb/>& contiguæ factæ cohærent; & corpora Electrica agunt ad di­<lb/>&longs;tantias majores, tam repellendo quam attrahendo corpu&longs;cula vi­<lb/>cina; & Lux emittitur, reflectitur, refringitur, inflectitur, & cor­<lb/>pora calefacit; & Sen&longs;atio omnis excitatur, & membra Anima­<lb/>lium ad voluntatem moventur, vibrationibus &longs;cilicet hujus Spiri­<lb/>tus per &longs;olida nervorum capillamenta ab externis &longs;en&longs;uum orga­<lb/>nis ad cerebrum & a cerebro in mu&longs;culos propagatis. </s> <s>Sed hæc <lb/>paucis exponi non po&longs;&longs;unt; neque ade&longs;t &longs;ufficiens copia Experi­<lb/>mentorum, quibus leges actionum hujus Spiritus accurate deter­<lb/>minari & mon&longs;trari debent. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>FINIS.<emph.end type="italics"/><emph.end type="center"/></s></p><pb xlink:href="039/01/516.jpg"/></subchap2></subchap1></chap><chap> <p type="main"> <s><emph type="center"/>INDEX RERUM <lb/>ALPHABETICUS.<emph.end type="center"/></s></p> <p type="main"> <s>N.B. <emph type="italics"/>Citationes factæ &longs;unt ad normam &longs;equentis Exempli.<emph.end type="italics"/>III, 10: 444, 20: <lb/>471, 28 <emph type="italics"/>de&longs;ignant Libri tertii Propo&longs;itionem decimam: Paginæ<emph.end type="italics"/>444<emph type="italics"/><emph type="sup"/>ta<emph.end type="sup"/> <emph type="italics"/>lineam<emph.end type="italics"/><lb/>20<emph type="italics"/><emph type="sup"/>æm<emph.end type="sup"/>: Paginæ<emph.end type="italics"/>471<emph type="italics"/><emph type="sup"/>æm<emph.end type="sup"/> lineam<emph.end type="italics"/>28<emph type="italics"/><emph type="sup"/>æm<emph.end type="sup"/><emph.end type="italics"/>. <lb/></s></p> <p type="main"> <s><emph type="center"/>A.<emph.end type="center"/></s></p> <p type="main"> <s>ÆQuinoctiorum præce&longs;&longs;io </s></p> <p type="main"> <s>cau&longs;æ hujus motus indicantur III, <lb/>21 </s></p> <p type="main"> <s>quantitas motus ex cau&longs;is computatur III, 39 <lb/>Aeris </s></p> <p type="main"> <s>den&longs;itas ad quamlibet altitudinem colligitur <lb/>ex Prop. </s> <s>22. Lib. </s> <s>II. quanta &longs;it ad altitu­<lb/>dinem unius &longs;emidiametri Terre&longs;tris o&longs;ten­<lb/>ditur 470, 11 </s></p> <p type="main"> <s>ela&longs;tica vis quali cau&longs;æ tribui po&longs;&longs;it II, 23 </s></p> <p type="main"> <s>gravitas cum Aquæ gravitate collata 470, 3 </s></p> <p type="main"> <s>re&longs;i&longs;tentia quanta &longs;it, per Experimenta Pen­<lb/>dulorum colligitur 286, 28; per Experi­<lb/>menta corporum cadentium & Theoriam <lb/>accuratius invenitur 327, 13 </s></p> <p type="main"> <s>Anguli contactus non &longs;unt omne; eja&longs;dem gene­<lb/>ris, &longs;ed alii aliis in&longs;inite minores p. </s> <s>32 </s></p> <p type="main"> <s>Ap&longs;idum motus expendltur I, Sect. </s> <s>9 </s></p> <p type="main"> <s>Areæ quas corpora in gyros acta, radiis ad con­<lb/>trum virium ductis, de&longs;cribunt, conferuntur <lb/>cum temporibus de&longs;criptionum I, 1, 2, 3, <lb/>58, 65 </s></p> <p type="main"> <s>Attractio corporum univer&longs;orum demon&longs;tratur <lb/>III, 7; qualis &longs;it hujus demon&longs;trationis certi­<lb/>tudo o&longs;tenditur 358, 28: 484, 11 </s></p> <p type="main"> <s>Attractionis cau&longs;am vel modum nullibi definit <lb/>Author 5, 17: 147, 32: 172, 31: 483, 34. </s></p> <p type="main"> <s><emph type="center"/>C.<emph.end type="center"/></s></p> <p type="main"> <s>Cali </s></p> <p type="main"> <s>re&longs;i&longs;tentia de&longs;tituuntur III, 10: 444, 20: <lb/>471, 28; & propterea Fluido omni corpo­<lb/>rco 328, 18 </s></p> <p type="main"> <s>tran&longs;itum Luci præbent ab&longs;que ulla refracti­<lb/>one 467, 33 </s></p> <p type="main"> <s>Calore virga ferrea comperta e&longs;t augeri longi­<lb/>tudine 386, 4 </s></p> <p type="main"> <s>Calor Solis quantus &longs;it in diver&longs;is a Sole di&longs;tantiis <lb/>466, 20 </s></p> <p type="main"> <s>quantus apud Mercurium 372, 12 <lb/></s></p> <p type="main"> <s>quantus apud Cometam anni 1680 in Peri­<lb/>helio ver&longs;antem 466, 22 </s></p> <p type="main"> <s>Centrum commune gravitatis corporum plu­<lb/>rium, ab actionibus corporum inter &longs;e, non <lb/>mutat &longs;tatum &longs;uum vel motus vel quietis <lb/>p. </s> <s>17 </s></p> <p type="main"> <s>Centrum commune gravitatis Terræ, Solis & <lb/>Planctarum omnium quic&longs;cere III, 11; con­<lb/>fir matur ex Cor. </s> <s>2. Prop. </s> <s>14. Lib. </s> <s>III. </s></p> <p type="main"> <s>Centrum commune gravitatis Terræ & Lunæ <lb/>motu annuo percurrit Orbem magnum 376, 6 <lb/>quibur intervallis di&longs;tata Terra & Luna 430, 22 </s></p> <p type="main"> <s>Centrun Virium quibus corpora revolventia in <lb/>Orbibus retinentur </s></p> <p type="main"> <s>quali Arearum indicio invenitur 38, 14 </s></p> <p type="main"> <s>qua ratione ex datis revolventium velocitati­<lb/>bus invenitur I, 5 </s></p> <p type="main"> <s>Circuli circum&longs;erentia, qua lege vis centripetæ <lb/>tendentis ad punctum quodcunQ.E.D.tum de­<lb/>&longs;cribi pote&longs;t a corpore revolvente I, 4, 7, 8 </s></p> <p type="main"> <s>Cometæ </s></p> <p type="main"> <s>Genus &longs;unt Planetarum, non Meteororum <lb/>444, 24: 466, 15 </s></p> <p type="main"> <s>Luna &longs;uperiores &longs;unt, & in regione Planeta­<lb/>rum ver&longs;antur p. </s> <s>439 </s></p> <p type="main"> <s>Di&longs;tantia eorum qua ratione per Ob&longs;ervatio­<lb/>nes colligi pote&longs;t quamproxime 439, 21 </s></p> <p type="main"> <s>Plures ob&longs;ervati &longs;unt in hemi&longs;phærio Solem <lb/>ver&longs;us, quam in hemi&longs;phærio oppo&longs;ito; & <lb/>unde hoc fiat 444, 5 </s></p> <p type="main"> <s>Splendent luce Solis a &longs;e reflexa 444, 4; Lux <lb/>illa quanta e&longs;&longs;et &longs;olet 441, 12 </s></p> <p type="main"> <s>Cinguntur Atmo&longs;phæris ingentibus 442, 12: <lb/>444, 27 </s></p> <p type="main"> <s>Qui ad Solem propius accedunt ut plurimum <lb/>minores e&longs;&longs;e exi&longs;timantur 475, 7 </s></p> <p type="main"> <s>Quo fine non comprehenduntur Zodiaco <lb/>(more Planetarum) &longs;ed in omnes tælorum <lb/>regiones varie feruntur 480, 30 </s></p> <p type="main"> <s>Po&longs;&longs;unt aliquando in Solem incidere & no­<lb/>vum illi alimentum ignis præbere 480, 37 </s></p> <p type="main"> <s>U&longs;us eorum &longs;uggeritur 473, 1: 481, 7 </s></p><pb xlink:href="039/01/517.jpg"/> <p type="main"> <s>Cometaram caudr </s></p> <p type="main"> <s>avertuntur a Sole 408, 39 </s></p> <p type="main"> <s>maximæ &longs;unt & &longs;ulgenti&longs;&longs;imæ &longs;tatim po&longs;t <lb/>tran&longs;itum per vicinam Solis 467, 8 </s></p> <p type="main"> <s>in&longs;ignis earum raritas 470, 32 </s></p> <p type="main"> <s>origo & natura earundem 442. 19: 467, 13 </s></p> <p type="main"> <s>quo tempori; &longs;patio a capite a&longs;cendunt 471, 1 </s></p> <p type="main"> <s>Cometæ </s></p> <p type="main"> <s>Moventur in Sectionibus Conicis umbilicos <lb/>in centro Solis habentibus, & radiis ad So­<lb/>lem ductis de&longs;cribunt areas temporibus pro­<lb/>portionales. </s> <s>Et quidem in Ellip&longs;ibus mo­<lb/>ventur &longs;i in Orbem redeunt, hæ tamen <lb/>Parabolis erunt maximæ &longs;initimæ III, 40 </s></p> <p type="main"> <s>Trajectoria Paral olica ex datis tribus Ob&longs;er­<lb/>vationibus invenitur III, 41; Inventa cor­<lb/>rigitur III, 42 </s></p> <p type="main"> <s>Locus in Parabola invenitur ad tempus da­<lb/>tum 445, 30: I, 30 </s></p> <p type="main"> <s>Velocitas cum velocitate Planetarum con&longs;er­<lb/>tur 445, 17 </s></p> <p type="main"> <s>Cometa annorum 1664 & 1665 </s></p> <p type="main"> <s>Huius motus ob&longs;ervatus expenditur, & cum <lb/>Theoria accurate congruere deprehenditur <lb/>p. </s> <s>477 </s></p> <p type="main"> <s>Cometa annorum 1680 & 1681 </s></p> <p type="main"> <s>Hujus motus ob&longs;ervatus cum Theoria accu­<lb/>rate congruere invenitur p. </s> <s>455 & <expan abbr="&longs;eqq.">&longs;eqque</expan> </s></p> <p type="main"> <s>Videbatur in Ellip&longs;i revolvi &longs;patio annorum <lb/>plu&longs;quam quingentorum 464, 37 </s></p> <p type="main"> <s>Trajectoria illius & Cauda &longs;ingulis in locis <lb/>delineantur p. </s> <s>465 </s></p> <p type="main"> <s>Cometa anni 1682 </s></p> <p type="main"> <s>Hajus motus accurate te&longs;pondet Theoriæ <lb/>p. </s> <s>479 </s></p> <p type="main"> <s>Comparui&longs;&longs;e vi&longs;us e&longs;t anno 1607, iterumque re­<lb/>diturus videtur periodo 75 annorum 480, 6 </s></p> <p type="main"> <s>Cometa anni 1683 </s></p> <p type="main"> <s>Hujus motus accurate re&longs;pondet Theoriæ <lb/>p. </s> <s>478 </s></p> <p type="main"> <s>Curvæ di&longs;tinguuntur in Geometrice rationales & <lb/>Geometrice irrationales 100, 5 </s></p> <p type="main"> <s>Curvatura figurarum qua ratione æ&longs;timanda &longs;it <lb/>235, 28: 398, 33 </s></p> <p type="main"> <s>Cycloidis &longs;eu Epicycloidis <lb/>rectificatio I, 48, 49: 142, 18 </s></p> <p type="main"> <s>ëvoluta I, 50: 142, 22 </s></p> <p type="main"> <s>Cylindri attractio ex particulis trahentibus com­<lb/>po&longs;iti quarum vires &longs;unt reciproce ut qua­<lb/>drata di&longs;tantiarum 198, 1 </s></p> <p type="main"> <s><emph type="center"/>D.<emph.end type="center"/></s></p> <p type="main"> <s>Dei Natura p. </s> <s>482 & 483 </s></p> <p type="main"> <s>De&longs;cen&longs;us graviuni in vacuo quantus &longs;it, ex lon­<lb/>gitudine Penduii colligitur 379, 1 </s></p> <p type="main"> <s>De&longs;cen&longs;us vel A&longs;cen&longs;us rectilinci &longs;patia de&longs;cri­<lb/>pta, tempora de&longs;criptionum & velocitates ac­<lb/><lb/>qui&longs;itæ conferuntur, po&longs;ita cuju&longs;cunque ge­<lb/>neris vi centripeta I, Sect. </s> <s>7 </s></p> <p type="main"> <s>De&longs;cen&longs;us & A&longs;cen&longs;us corporum in Mediis re­<lb/>&longs;i&longs;tentibus II, 3, 8, 9, 40, 13, 14 </s></p> <p type="main"> <s><emph type="center"/>E.<emph.end type="center"/></s></p> <p type="main"> <s>Ellip&longs;is </s></p> <p type="main"> <s>qua lege vis contripetæ tendentis ad centrum <lb/>figuræ de&longs;cribitur a corpore revolvente <lb/>I, 10, 64 </s></p> <p type="main"> <s>qua lege vis centripetæ tendentis ad umbili­<lb/>cum figuræ de&longs;cribitur a corpore revol­<lb/>vente I, 11 </s></p> <p type="main"> <s><emph type="center"/>F.<emph.end type="center"/></s></p> <p type="main"> <s>Fleidi definitio p. </s> <s>260 </s></p> <p type="main"> <s>Flaidorum den&longs;itas & compre&longs;&longs;io quas leges ha­<lb/>bent, o&longs;tenditur II, Sect. </s> <s>5 </s></p> <p type="main"> <s>Fluidorum per foramen in va&longs;e factum effluen­<lb/>tium determinatur motus II, 36 </s></p> <p type="main"> <s>Fumi in camino a&longs;cen&longs;us obiter explicatur 472, 4 </s></p> <p type="main"> <s><emph type="center"/>G.<emph.end type="center"/></s></p> <p type="main"> <s>Graduum in Meridiano Terre&longs;tri men&longs;ura exhi­<lb/>betur, & quam &longs;it exigua inæqualitas o&longs;ten­<lb/>ditur ex Theoria III, 20 </s></p> <p type="main"> <s>Gravitas </s></p> <p type="main"> <s>diver&longs;i e&longs;t generis a vi Magnetica 368, 29 </s></p> <p type="main"> <s>mutua e&longs;t inter Terram & ejus partes 22, 18 </s></p> <p type="main"> <s>ejus cau&longs;a non a&longs;&longs;ignatur 483, 34 </s></p> <p type="main"> <s>datur in Planetas univer&longs;os 365, 15; & per­<lb/>gendo a &longs;uperficiebus Planetarum &longs;ur&longs;um <lb/>decre&longs;cit in duplicata ratione di&longs;tantiarum <lb/>a centro III, 8, deor&longs;um decre&longs;cit in &longs;im­<lb/>plici ratione quamproxime III, 9 </s></p> <p type="main"> <s>datur in corpora omnia, & proportionalis e&longs;t <lb/>quantitati materiæ in &longs;ingulis III, 7 </s></p> <p type="main"> <s>Gravitatem e&longs;&longs;e vim illam qua Luna retinetur <lb/>in Orbe III, 4, computo accuratiori com­<lb/>probatur 430, 25 </s></p> <p type="main"> <s>Gravitatem e&longs;&longs;e vim illam qua Planetæ primarii <lb/>& Satellites Jovis & Saturni retinentur in <lb/>Orbibus III, 5 </s></p> <p type="main"> <s><emph type="center"/>H.<emph.end type="center"/></s></p> <p type="main"> <s>Hydro&longs;taticæ principia traduntur II, Sect. </s> <s>5 </s></p> <p type="main"> <s>Hyperbola </s></p> <p type="main"> <s>qua lege vis centrifugæ tendentis a figuræ cen­<lb/>tro de&longs;cribitur a corpore revolvente 47, 26 </s></p> <p type="main"> <s>qua lege vis centrifugæ tendentis ab umbilico <lb/>figuræ de&longs;cribitur a corpore revolvente 51, 6 </s></p> <p type="main"> <s>qua lege vis centripetæ tendentis ad umbilicum <lb/>figuræde&longs;cribitur a corpore revolvente I, 12 </s></p> <p type="main"> <s>Hypothe&longs;es cuju&longs;cunque generis rejiciuntur ab <lb/>hac Philo&longs;ophia 484, 8. </s></p><pb xlink:href="039/01/518.jpg"/> <p type="main"> <s><emph type="center"/>I.<emph.end type="center"/></s></p> <p type="main"> <s>Inertiæ vis de&longs;initur p. </s> <s>2 </s></p> <p type="main"> <s>Jovis </s></p> <p type="main"> <s>di&longs;tantia a Sole 361, </s></p> <p type="main"> <s>&longs;emidiameter apparens 371, 3 </s></p> <p type="main"> <s>&longs;emidiameter vera 371, 14 </s></p> <p type="main"> <s>attractiva vis quanta &longs;it 370, 33 </s></p> <p type="main"> <s>pondus corporum in ejus &longs;uperficie 371, 19 </s></p> <p type="main"> <s>deniitas 371, 37 </s></p> <p type="main"> <s>quantitas materiæ 3: 1, 27 </s></p> <p type="main"> <s>perturbatio a Saturno quanta &longs;it 375, 33 </s></p> <p type="main"> <s>diametrorum proportio computo exhibetur <lb/>381, 27 </s></p> <p type="main"> <s>conver&longs;io citcum axem quo tempore ab&longs;olvi­<lb/>tur 381, 25 </s></p> <p type="main"> <s>cingulæ cau&longs;a &longs;ubindicatur 444 32. </s></p> <p type="main"> <s><emph type="center"/>L.<emph.end type="center"/></s></p> <p type="main"> <s>Locus definitur, & di&longs;tinguitur in ab&longs;olutum & <lb/>relativum 6, 12 </s></p> <p type="main"> <s>Loca corporum in Sectionibus conicis moto­<lb/>rum inveniuntur ad tempus a&longs;&longs;ignatum I, <lb/>Sect. </s> <s>6 </s></p> <p type="main"> <s>Lucis </s></p> <p type="main"> <s>propagatio non e&longs;t in&longs;tantanea 207, 5; non <lb/>fit per agitationem Medii alicujus Ætherci <lb/>342, 36 </s></p> <p type="main"> <s>velocitas in diver&longs;is Mediis diver&longs;a I, 95 </s></p> <p type="main"> <s>reflexio quædam explicatur I, 96 </s></p> <p type="main"> <s>refractio explicatur I, 94; non &longs;it in puncto <lb/>&longs;olum incidentiæ 207, 29 </s></p> <p type="main"> <s>incurvatio prope corporum terminos Expe­<lb/>rimentis ob&longs;ervata 207, 8 </s></p> <p type="main"> <s>Lunæ </s></p> <p type="main"> <s>corporis figura computo colligitur III, 38 </s></p> <p type="main"> <s>inde cau&longs;a patefacta, cur candem &longs;emper fa­<lb/>ciem in Terram obvertat 432, 9 </s></p> <p type="main"> <s>& libra ioncs explicantur III, 17 </s></p> <p type="main"> <s>diameter meliocris apparens 430, 12 </s></p> <p type="main"> <s>diameter mediocris 430, 17 </s></p> <p type="main"> <s>pondus corporum in ejus &longs;uperficie 430, 20 </s></p> <p type="main"> <s>den&longs;itas 430, 15 </s></p> <p type="main"> <s>quantitas materiæ 430, 19 </s></p> <p type="main"> <s>di&longs;tantia mediocris a Terra quot continet <lb/>maximas Terræ &longs;emidiametros 430, 25, <lb/>quot mediocres 431, 18 </s></p> <p type="main"> <s>parallaxis maxima in longitudinem paulo ma­<lb/>jor e&longs;t quam paraliaxis maxima in latitu­<lb/>dinem 387, 8 </s></p> <p type="main"> <s>vis ad Mare movendum quanta &longs;it III, 37; <lb/>non &longs;entiri pote&longs;t in Experimentis pendu­<lb/>lorum, vel in Staticis aut Hydro&longs;taticis <lb/>quibu&longs;cunque 430, 1 </s></p> <p type="main"> <s>tempus periodicum 430, 32 </s></p> <p type="main"> <s>tempus revolutionis &longs;ynodicæ 398, 1 </s></p> <p type="main"> <s>motus medius cum diurno motu Terræ col­<lb/><lb/>latus paulatim accelerari deprehenditur ab <lb/><emph type="italics"/>Helleio<emph.end type="italics"/>481, 16 </s></p> <p type="main"> <s>Lunæ motus & motuum inæqualitates a cau&longs;is <lb/>&longs;uis derivantur III, 22: p. </s> <s>421 & <expan abbr="&longs;eqq.">&longs;eqque</expan> </s></p> <p type="main"> <s>tardius revolvitur Luna dilatato Orbe, in pe­<lb/>rihelio Terræ, citius in ophelio, contracto <lb/>Orbe III, 22: 421, 6 </s></p> <p type="main"> <s>tardius revolvitur, dilatato Orbe, in Apogæi <lb/>Syzygiis cum Sole; citius in Quadraturis <lb/>Apogæi, contracto Orbe 422, 1 </s></p> <p type="main"> <s>tardius revolvitur, dilatato Orbe, in Syzygiis <lb/>Nodi cum Sole; citius in Quadraturis No­<lb/>di, contracto Orbe 422, 21 </s></p> <p type="main"> <s>tardius movetur in Quadraturis &longs;uis cum Sole, <lb/>citius in Syzygiis; & radio ad Terram <lb/>ducto de&longs;eribit aream pro tempere mino­<lb/>rem in priore ca&longs;u, majorem in po&longs;teriore <lb/>III, 22: Inæqualitas harum Arearum com­<lb/>putatur III, 26. Orbem in&longs;uper habet ma­<lb/>gis curvum & longius a Terra recedit in <lb/>priore ca&longs;u, minus curvum habet Orbem <lb/>& propius ad Terram accedit in po&longs;teriore <lb/>III, 22. Orbis hujus figura & proportio <lb/>diametrorum ejus computo colligitur III, <lb/>28. Et &longs;abinde proponitur methodus in­<lb/>veniendi di&longs;tantiam Lunæ a Terra ex motu <lb/>ejus horario III, 27 </s></p> <p type="main"> <s>Apogæum tardius movetur in Aphelio Terræ, <lb/>velocius in Perihclio III, 22: 421, 21 </s></p> <p type="main"> <s>Apogæum ubi e&longs;t in Solis Syzygiis, maxime <lb/>progreditur; in Quadraturis regreditur III, <lb/>22: 422, 37 </s></p> <p type="main"> <s>Eccentricitas maxima e&longs;t in Apogæi Syzygiis <lb/>cum Sole, minima in Quadraturis III, 22: <lb/>422, 39 </s></p> <p type="main"> <s>Nodi tardius moventur in Aphelio Terræ, ve­<lb/>locius in Perihelio III, 22: 421, 21 </s></p> <p type="main"> <s>Nodi quie&longs;cunt in Syzygiis &longs;uis cum Sole, & <lb/>veloci&longs;&longs;ime regrediuntur in Quadraturis <lb/>III, 22. Nodorum motus & inæqualitates <lb/>motuum computantur ex Theoria Gravi­<lb/>tatis III, 30, 31, 32, 33 </s></p> <p type="main"> <s>Inclinatio Oibis ad Ec&longs;ipticam maxima e&longs;t in <lb/>Syzygiis Nodorum cum Sole, minima in <lb/>Quadraturis I, 66 Cor. </s> <s>10. Inclinationis va­<lb/>riationes computantur ex Theoria Gravita­<lb/>tis III, 34, 35 </s></p> <p type="main"> <s>Lunarium motuum Æquationes ad u&longs;us A&longs;tro­<lb/>nomicos p. </s> <s>421 & <expan abbr="&longs;eqq.">&longs;eqque</expan> </s></p> <p type="main"> <s>Motus medii Lunæ </s></p> <p type="main"> <s>Æquatio annua 421, 4 </s></p> <p type="main"> <s>Æquatio &longs;eme&longs;tris prima 412, 1 </s></p> <p type="main"> <s>Æquatio &longs;eme&longs;tris &longs;ecunda 422, 21 </s></p> <p type="main"> <s>Æquatio centri prima 423, 20: p. </s> <s>101 & <lb/><expan abbr="&longs;eqq.">&longs;eqque</expan> </s></p> <p type="main"> <s>Æquatio centri &longs;ecunda 424, 15 </s></p> <p type="main"> <s>Variatio prima III, 29 </s></p> <p type="main"> <s>Variatio &longs;ecunda 425, 5 </s></p><pb xlink:href="039/01/519.jpg"/> <p type="main"> <s>Motus medii Apogæi </s></p> <p type="main"> <s>Æquatio annua 421, 21 </s></p> <p type="main"> <s>Æquatio &longs;eme&longs;tris 422, 37 </s></p> <p type="main"> <s>Eccentricitatis </s></p> <p type="main"> <s>Æquatio &longs;eme&longs;tris 422, 37 </s></p> <p type="main"> <s>Motus medii Nodorum </s></p> <p type="main"> <s>Æquatio annua 421, 21 </s></p> <p type="main"> <s>Æquatio &longs;eme&longs;tris III, 33 </s></p> <p type="main"> <s>Inclinationis Orbitæ ad Eclipticam </s></p> <p type="main"> <s>Æquatio &longs;eme&longs;tris 420, 22 </s></p> <p type="main"> <s>Lunarium motuum Theoria, qua Methodo &longs;ta­<lb/>bilienda &longs;it per Ob&longs;ervationes 425, 33. </s></p> <p type="main"> <s><emph type="center"/>M.<emph.end type="center"/></s></p> <p type="main"> <s>Magnetica vis 22, 13: 271, 25: 368, 29: <lb/>431, 23 </s></p> <p type="main"> <s>Maris æ&longs;tus a cau&longs;is &longs;uis derivatur III, 24, 36, 37 </s></p> <p type="main"> <s>Martis </s></p> <p type="main"> <s>di&longs;tantia a Sole 361, 1 </s></p> <p type="main"> <s>Aphelii motus 376, 33 </s></p> <p type="main"> <s>Materie</s></p> <p type="main"> <s>quantitas definitur p. </s> <s>1 </s></p> <p type="main"> <s>vis in&longs;ita &longs;eu vis inertiæ definitur p. </s> <s>2 </s></p> <p type="main"> <s>vis impre&longs;&longs;a definitur p. </s> <s>2 </s></p> <p type="main"> <s>exten&longs;io, durities, impenetrabilitas, mobilitas, <lb/>vis inertiæ, gravitas, qua ratione innote&longs;­<lb/>cunt 357, 16: 484, 10 </s></p> <p type="main"> <s>divi&longs;ibilitas nondum con&longs;tat 358, 18 </s></p> <p type="main"> <s>Materia &longs;ubtilis <emph type="italics"/>Carte&longs;ianorum<emph.end type="italics"/>ad examen quod­<lb/>dam revocatur 292, 12 </s></p> <p type="main"> <s>Materia vel &longs;ubtiii&longs;&longs;ima Gravitate non de&longs;titui­<lb/>tur 368, 1 </s></p> <p type="main"> <s>Mechanicæ, quæ dicuntur, Potentiæ explicantur <lb/>& demon&longs;trantur p. </s> <s>14 & 15: p. </s> <s>23 </s></p> <p type="main"> <s>Mercurii </s></p> <p type="main"> <s>di&longs;tantia a Sole 361, 1 </s></p> <p type="main"> <s>Aphelii motus 376, 33 </s></p> <p type="main"> <s>Methodus </s></p> <p type="main"> <s>Rationum primarum & ultimarum I, Sect. </s> <s>1 </s></p> <p type="main"> <s>Tran&longs;mutandi figuras in alias quæ &longs;unt eju&longs;­<lb/>dem Ordinis Analytici I, Lem. </s> <s>22. pag. </s> <s>79 </s></p> <p type="main"> <s>Fluxionum II, Lem. </s> <s>2. p. </s> <s>224 </s></p> <p type="main"> <s>Differentialis III, Lemm. </s> <s>5 & 6. pagg. </s> <s>446 <lb/>& 447 </s></p> <p type="main"> <s>Inveniendi Curvarum omnium quadraturas <lb/>proxime veras 447, 8 </s></p> <p type="main"> <s>Serierum convergentium adhibetur ad &longs;olu­<lb/>tionem Problematum difficiliorum p. </s> <s>127: <lb/>128: 202: 235: 414 </s></p> <p type="main"> <s>Motus quantitas definitur p. </s> <s>1 </s></p> <p type="main"> <s>Motus ab&longs;olutus & relativus p. </s> <s>6: 7: 8: 9 2b <lb/>invicem &longs;ecerni po&longs;&longs;unt, exemplo demon&longs;tra­<lb/>tur p. </s> <s>10 </s></p> <p type="main"> <s>Motus Leges p. </s> <s>12 & <expan abbr="&longs;eqq.">&longs;eqque</expan> </s></p> <p type="main"> <s>Motuum compo&longs;itio & re&longs;olutio p. </s> <s>14 </s></p> <p type="main"> <s>Motus corporum congredientium po&longs;t reflexio­<lb/>nem, quali Experimento recte colligi po&longs;&longs;unt, <lb/><lb/>o&longs;tenditur 19, 21 </s></p> <p type="main"> <s>Motus corporum </s></p> <p type="main"> <s>in Conicis &longs;ectionibus eccentricis I, Sect. </s> <s>3 </s></p> <p type="main"> <s>in Orbibus mobilibus I, Sect. </s> <s>9 </s></p> <p type="main"> <s>in Super&longs;iciebus datis & Funependulorum <lb/>motus reciprocus I, Sect. </s> <s>10 </s></p> <p type="main"> <s>Motus corporum viribus centripetis &longs;e mutuo <lb/>petentium I, Sect. </s> <s>11 </s></p> <p type="main"> <s>Motus corporum Minimorum, quæ viribus cen­<lb/>tripetis ad &longs;ingulas Magni alicujus corporis <lb/>partes tendentibus agitantur I, Sect. </s> <s>14 </s></p> <p type="main"> <s>Motus corporum quibus re&longs;i&longs;titur </s></p> <p type="main"> <s>in ratione velocitatis II, Sect. </s> <s>1 </s></p> <p type="main"> <s>in duplicata ratione velocitatis II, Sect. </s> <s>2 </s></p> <p type="main"> <s>partim in ratione velocitatis, partim in eju&longs;­<lb/>dem ratione duplicata II, Sect. </s> <s>3 </s></p> <p type="main"> <s>Motus </s></p> <p type="main"> <s>corporum &longs;ola vi in&longs;ita progredientium in <lb/>Mediis re&longs;i&longs;tentibus II, 1, 2, 5, 6, 7, 11, <lb/>12: 302, 1 </s></p> <p type="main"> <s>corporum recta a&longs;cendentium vel de&longs;cenden­<lb/>tium in Mediis re&longs;i&longs;tentibus, agente vi Gra­<lb/>vitatis uniformi II, 3, 8, 9, 40, 13, 14 </s></p> <p type="main"> <s>corporum projectorum in Mediis re&longs;iftenti­<lb/>bus, agente vi Gravitatis unifor mi II, 4, 10 </s></p> <p type="main"> <s>corporum circumgyrantium in Mediis re&longs;i­<lb/>&longs;tentibus II, Sect. </s> <s>4 </s></p> <p type="main"> <s>corporum Funependulorum in Mediis re&longs;i­<lb/>&longs;tentibus II, Sect. </s> <s>6 </s></p> <p type="main"> <s>Motus & re&longs;i&longs;tentia Fluidorum II, Sect. </s> <s>7 </s></p> <p type="main"> <s>Motus per Fluida propagatus II, Sect. </s> <s>8 </s></p> <p type="main"> <s>Motus circularis &longs;eu Vortico&longs;us Fluidorum II, <lb/>Sect. </s> <s>9 </s></p> <p type="main"> <s>Mundus originem non habet ex cau&longs;is Mecha­<lb/>nicis p. </s> <s>482, 12. </s></p> <p type="main"> <s><emph type="center"/>N.<emph.end type="center"/></s></p> <p type="main"> <s>Navium con&longs;tructioni Propo&longs;itio non inutilis <lb/>300, 4. </s></p> <p type="main"> <s><emph type="center"/>O.<emph.end type="center"/></s></p> <p type="main"> <s>Opticarum ovalium inventio quam <emph type="italics"/>Carte&longs;ius<emph.end type="italics"/>ce­<lb/>laverat I, 97. <emph type="italics"/>Carte&longs;iani<emph.end type="italics"/>Problematis genera­<lb/>lior &longs;olutio I, 98 </s></p> <p type="main"> <s>Orbitarum inventio </s></p> <p type="main"> <s>quas corpora de&longs;cribunt, de loco dato data <lb/>cum velocitate, &longs;ecundum datum rectam <lb/>egre&longs;&longs;a; ubi vis centripeta e&longs;t reciproce ut <lb/>quadratum di&longs;tantiæ & vis illius quantitas <lb/>ab&longs;oluta cogno&longs;citur I, 17 </s></p> <p type="main"> <s>quas corpora de&longs;cribunt ubi vires centripetæ <lb/>&longs;unt reciproce ut cubi di&longs;tantiarum 45, 18: <lb/>118, 27: 125, 25 </s></p> <p type="main"> <s>quas corpora viribus quibu&longs;cunque centripetis <lb/>agitata de&longs;cribunt I, Sect. </s> <s>8. </s></p><pb xlink:href="039/01/520.jpg"/> <p type="main"> <s><emph type="center"/>P.<emph.end type="center"/></s></p> <p type="main"> <s>Parabola, qua lege vis centripetæ tendentis ad <lb/>umbilicum figuræ, de&longs;cribitur a corpore revol­<lb/>vente I, 13 </s></p> <p type="main"> <s>Pendulorum affectiones explicantur I, 50, 51, <lb/>52, 53: II, Sect. </s> <s>6. </s></p> <p type="main"> <s>Pendulotum i&longs;ochronorum longitudines diver&longs;æ <lb/>in diver&longs;is loeorum Latitudinibus inter &longs;e <lb/>con&longs;eruntur, tum per Ob&longs;ervatienes, tum per <lb/>Theoriam Gravitatis III, 20 </s></p> <p type="main"> <s>Philo&longs;ophandi Regulæ p. </s> <s>357 </s></p> <p type="main"> <s>Planetæ </s></p> <p type="main"> <s>non deferuntur a Vorticibus corporeis 352, <lb/>37: 354, 25: 481, 21 </s></p> <p type="main"> <s>Primarii </s></p> <p type="main"> <s>Solem cingunt 360, 7 </s></p> <p type="main"> <s>moventur in Ellip&longs;ibus umbilicum habenti­<lb/>bus in centro Solis III, 13 </s></p> <p type="main"> <s>radiis ad Solem ductis de&longs;cribunt areas tem­<lb/>poribus proportionales 361, 15: III, 13 </s></p> <p type="main"> <s>temporibus periodicis revolvuntur quæ &longs;unt <lb/>in &longs;e&longs;quiplicata ratione di&longs;tantiarum a <lb/>Sole 360, 17: III, 13 & I, 15 </s></p> <p type="main"> <s>retinentur in Orbibus &longs;uis a vi Gravitatis <lb/>quæ re&longs;picit Solem, & e&longs;t reciproce ut <lb/>quadratum di&longs;tantiæ ab ip&longs;ius centro <lb/>III, 2, 5 </s></p> <p type="main"> <s>Secundarii </s></p> <p type="main"> <s>moventur in Ellip&longs;ibus umbilicum habenti­<lb/>bus in centro Primariorum III, 22 </s></p> <p type="main"> <s>radiis ad Primarios &longs;uos ductis de&longs;cribunt <lb/>areas temporibus proportionales 359, 3, <lb/>22: 361, 27: III, 22 </s></p> <p type="main"> <s>temporibus periodicis revolvuntur quæ &longs;unt <lb/>in &longs;e&longs;quiplicata ratione di&longs;tantiarum a <lb/>Primariis &longs;uis 359, 3, 22: III, 22 & I, 15 </s></p> <p type="main"> <s>retinentur in Orbibus &longs;uis a vi Gravitatis <lb/>quæ re&longs;picit Primarios, & e&longs;t reciproce <lb/>ut quadratum di&longs;tantiæ ab eorum centris <lb/>III, 1, 3, 4, 5 </s></p> <p type="main"> <s>Planetarum </s></p> <p type="main"> <s>di&longs;tantiæ a Sole 361, 1 </s></p> <p type="main"> <s>Orbium Aphelia & Nodi prope quie&longs;cunt <lb/>III, 14 </s></p> <p type="main"> <s>Orbes determinantur III, 15, 16 </s></p> <p type="main"> <s>loca in Orbibus inveniuntur I, 31 </s></p> <p type="main"> <s>den&longs;itas calori quem a Sole recipiunt, ac­<lb/>commodatur 372, 7 </s></p> <p type="main"> <s>conver&longs;iones diurnæ &longs;unt æquabiles III, 17</s></p> <p type="main"> <s>axes &longs;unt minores diametris quæ ad eo&longs;dem<lb/>axes normaliter ducuntur III, 18 </s></p> <p type="main"> <s>Pondera corporum </s></p> <p type="main"> <s>in Terram vel Solem vel Planetam quemvis, <lb/>paribus di&longs;tantiis ab eorum centris, &longs;unt ut <lb/>quantitates materiæ in corporibus III, 6 </s></p> <p type="main"> <s>non pendent ab eorum formis & texturis <lb/>367, 35 <lb/></s></p> <p type="main"> <s>in diver&longs;is Terræ regionibus inveniuntur & <lb/>inter &longs;e comparantur III, 20 </s></p> <p type="main"> <s>Problematis </s></p> <p type="main"> <s><emph type="italics"/>Kepleriani<emph.end type="italics"/>&longs;olutio per Trochoidem & per <lb/>Approximationes I, 31 </s></p> <p type="main"> <s><emph type="italics"/>Veterum<emph.end type="italics"/>de quatuor lineis, a <emph type="italics"/>Pappo<emph.end type="italics"/>memorati, <lb/>a <emph type="italics"/>Carte&longs;io<emph.end type="italics"/>par calculum Analyticum tentati, <lb/>compo&longs;itio Geometrica 70, 19 </s></p> <p type="main"> <s>Projectilia, &longs;epo&longs;ita Medii re&longs;i&longs;tentia, moveri in <lb/>Parabola colligitur 47, 23: 202, 23: 236, 29 </s></p> <p type="main"> <s>Projectilium motus in Mediis re&longs;i&longs;tentibus II, <lb/>4, 10 </s></p> <p type="main"> <s>Pul&longs;uam Aeris, quibes Soni propagantur, deter­<lb/>minantur intervalla &longs;eu latitudines II, 50: 344, <lb/>18. Hæc intervalla in apertarum Fi&longs;tularum <lb/>&longs;onis æquari duplis longitudinibus Fi&longs;tularum <lb/>vero&longs;imile e&longs;t 344, 26 </s></p> <p type="main"> <s><emph type="center"/><expan abbr="q.">que</expan><emph.end type="center"/></s></p> <p type="main"> <s>Quadratura generalis Ovalium dari non pote&longs;t <lb/>per finitos terminos I, Lem, 28. p. </s> <s>98 </s></p> <p type="main"> <s>Qualitates corporum qua ratione innote&longs;cunt & <lb/>admittuntur 357, 16 </s></p> <p type="main"> <s>Quies vera & relativa p. </s> <s>6, 7, 8, 9. </s></p> <p type="main"> <s><emph type="center"/>R.<emph.end type="center"/></s></p> <p type="main"> <s>Re&longs;i&longs;tentiæ quantitas </s></p> <p type="main"> <s>in Mediis non continuis II, 35 </s></p> <p type="main"> <s>in Mediis continuis II, 38 </s></p> <p type="main"> <s>in Mediis cuju&longs;cunque generis 302, 32 </s></p> <p type="main"> <s>Re&longs;i&longs;tentiarum Theoria confirmatur </s></p> <p type="main"> <s>per Experimenta Pendulorum II, 30, 31, Sch. </s> <s><lb/>Gen. </s> <s>p. </s> <s>284 </s></p> <p type="main"> <s>per Experimenta corporum cadentium II, 40, <lb/>Sch. </s> <s>p. </s> <s>319 </s></p> <p type="main"> <s>Re&longs;i&longs;tentia Mediorum </s></p> <p type="main"> <s>e&longs;t ut eorundem den&longs;itas, cæteris paribu, <lb/>290, 29: 291, 35: II, 33, 35, 38: 327, 14 </s></p> <p type="main"> <s>e&longs;t in duplicata ratione velocitatis corporum <lb/>quibus re&longs;i&longs;titur, cæteris paribus 219, 24: <lb/>284, 33; II, 33, 35, 38: 324, 23 </s></p> <p type="main"> <s>e&longs;t in duplicata ratione diametri corporum <lb/>Sphærieorum quibus re&longs;i&longs;titur, cæteris pa­<lb/>ribus 288, 4: 289, 11: II, 33, 35, 38: <lb/>Sch. </s> <s>p. </s> <s>319 </s></p> <p type="main"> <s>non minuitur ab actione Fluidi in partes po­<lb/>&longs;ticas corporis moti 312, 23 </s></p> <p type="main"> <s>Re&longs;i&longs;tentia Fluidorum duplex e&longs;t; oriturque vel <lb/>ab Inertia materiæ fluidæ, vel ab Ela&longs;ticitate, <lb/>Tenacitate & Frictione partium ejus 318, 1. <lb/>Re&longs;i&longs;tentia quæ &longs;entitur in Fluidis fere tota <lb/>e&longs;t prioris generis 326, 32, & minui non po­<lb/>te&longs;t per &longs;ubtilitatem partium Fluidi, manente <lb/>den&longs;itate 328, 7 </s></p> <p type="main"> <s>Re&longs;i&longs;tentiæ Globi ad re&longs;i&longs;tentiam Cylindri pro­<lb/>portio, in Mediis non continuis II, 34 </s></p><pb xlink:href="039/01/521.jpg"/> <p type="main"> <s>Re&longs;i&longs;tentia quam patitur a Fluido &longs;ru&longs;tum Co­<lb/>nicum, qua ratione fiat minima 299, 30 </s></p> <p type="main"> <s>Re&longs;i&longs;tentiæ minimæ Solidum 300, 15. </s></p> <p type="main"> <s><emph type="center"/>S.<emph.end type="center"/></s></p> <p type="main"> <s>Satellitis</s></p> <p type="main"> <s>Jovialis extimi elongatio maxima heliocentrica <lb/>a centro Jovis 370, 35 </s></p> <p type="main"> <s><emph type="italics"/>Hugeniani<emph.end type="italics"/>elongatio maxima heliocentrica a <lb/>centro Saturni 371, 5 </s></p> <p type="main"> <s>Satellitum </s></p> <p type="main"> <s>Jovialium tempora periodica & di&longs;tantiæ a <lb/>centro Jovis 359, 12 </s></p> <p type="main"> <s>Saturniorum tempora periodica & di&longs;tantiæ a <lb/>centro Saturni 360, 1 </s></p> <p type="main"> <s>Jorialium & Saturniorum inæquales motus <lb/>a motibus Lanæ derivari po&longs;&longs;e o&longs;&longs;enditur <lb/>III, 23 </s></p> <p type="main"> <s>Saturni </s></p> <p type="main"> <s>di&longs;tantia a Sole 361, 1 </s></p> <p type="main"> <s>&longs;emidiameter apparens 371, 9 </s></p> <p type="main"> <s>&longs;emidiameter vera 371, 14 </s></p> <p type="main"> <s>vis attractiva quanta &longs;it 370, 33 </s></p> <p type="main"> <s>pondus corporum in ejus &longs;uperficie 371, 19 </s></p> <p type="main"> <s>den&longs;itas 371, 37 </s></p> <p type="main"> <s>quantitas materiæ 371, 27 </s></p> <p type="main"> <s>perturbatio a Jove quanta &longs;it 375, 16 </s></p> <p type="main"> <s>diameter apparens Annuli quo cingitur 371, 8 </s></p> <p type="main"> <s>Sectiones Conicæ, qua lege vis centripetæ ten­<lb/>dentis ad punctum quodcunQ.E.D.tum, de&longs;cri­<lb/>buntur a corporibus revolventibus 58, 20 </s></p> <p type="main"> <s>Sectionum Conicarum de&longs;criptio Geometrica </s></p> <p type="main"> <s>ubi dantur Umbilici I, Sect. </s> <s>4 </s></p> <p type="main"> <s>ubi non dantur Umbilici I, Sect. </s> <s>5. ubi dan­<lb/>tur Centra vel A&longs;ymptoti 87, 9 </s></p> <p type="main"> <s>Se&longs;quiplicata ratio definitur 31, 40 </s></p> <p type="main"> <s>Sol </s></p> <p type="main"> <s>circum Planetarum omnium commune gravi­<lb/>tatis centrum movetur III, 12 </s></p> <p type="main"> <s>&longs;emidiameter ejus mediocris apparens 371, 12 </s></p> <p type="main"> <s>&longs;emidiameter vera 371, 14 </s></p> <p type="main"> <s>parallaxis ejus horizontalis 370, 33 </s></p> <p type="main"> <s>parallaxis men&longs;trua 376, 4 </s></p> <p type="main"> <s>vis ejus attractiva quanta &longs;it 370, 33 </s></p> <p type="main"> <s>pondus corporum in ejus &longs;uperficie 371, 19 </s></p> <p type="main"> <s>den&longs;itas ejus 371, 37 </s></p> <p type="main"> <s>quantitas mater æ 371, 27 </s></p> <p type="main"> <s>vis ejus ad perturbandos motus Lunæ 363, <lb/>15: III, 25 </s></p> <p type="main"> <s>vis ad Mare movendum III, 36 </s></p> <p type="main"> <s>Soaorum </s></p> <p type="main"> <s>natura explicatur II, 43, 47, 48, 49, 50 </s></p> <p type="main"> <s>propagatio divergit a recto tramite 332, 9, <lb/>fit per agitationem Aeris 343, 1 </s></p> <p type="main"> <s>velocitas computo colligitur 343. 8, paulu­<lb/>lum major e&longs;&longs;e debet Æ&longs;tivo quam Hyber­<lb/>no tempore, per Thecriam 344, 11 </s></p> <p type="main"> <s>ce&longs;&longs;atio fit &longs;tatim ubi ce&longs;&longs;at motus corporis <lb/>&longs;onori 344, 29 <lb/></s></p> <p type="main"> <s>augmentatio per tubos &longs;tenterophonicos <lb/>344, 32 </s></p> <p type="main"> <s>Spatium </s></p> <p type="main"> <s>ab&longs;olutum & relativum p. </s> <s>6, 7 </s></p> <p type="main"> <s>non e&longs;t æqualiter plenum 368, 16 </s></p> <p type="main"> <s>Sphæroidis attractio, cujus particularum vires <lb/>&longs;unt reciproce ut quadrata di&longs;tantiarum <lb/>198, 21 </s></p> <p type="main"> <s>Spiralis quæ &longs;ecat radios &longs;uos omnes in angulo <lb/>dato, qua lege vis centripetæ tendenti ad <lb/>centrum Spiralis de&longs;cribi pote&longs;t a corpore <lb/>revolvente, o&longs;tenditur I, 9: II, 15, 16 </s></p> <p type="main"> <s>Spiritum Q.E.D.m corpora pervadentem & in <lb/>corporibus latentem, ad plurima naturæ phæ­<lb/>nomena &longs;olvenda, requiri &longs;uggeritur 484, 17 </s></p> <p type="main"> <s>Stellarum fixarum </s></p> <p type="main"> <s>quies demon&longs;tratur 376, 18 </s></p> <p type="main"> <s>radiatio & &longs;cintillatio quibus cau&longs;is referendæ <lb/>&longs;int 467, 38 </s></p> <p type="main"> <s>Stellæ Novæ unde oriri po&longs;&longs;int 481, 5 </s></p> <p type="main"> <s>Sub&longs;tantiæ rerum omnium occultæ &longs;unt 483, 22 </s></p> <p type="main"> <s><emph type="center"/>T.<emph.end type="center"/></s></p> <p type="main"> <s>Tempus ab&longs;olutum & relativum p. </s> <s>5, 7 </s></p> <p type="main"> <s>Temporis Æquatio A&longs;tronomica per Horolo­<lb/>gium o&longs;cillatorium & Eclip&longs;es Satellitum Jo­<lb/>vis comprobatur 7, 15 </s></p> <p type="main"> <s>Tempora periodica corporum revolventium in <lb/>Ellip&longs;ibus, ubi vires centripetæ ad umbilicum <lb/>tendunt I, 15 </s></p> <p type="main"> <s>Terræ </s></p> <p type="main"> <s>dimen&longs;io per <emph type="italics"/>Picartum<emph.end type="italics"/>378, 11, per <emph type="italics"/>Ca&longs;&longs;inum<emph.end type="italics"/><lb/>378, 21, per <emph type="italics"/>Norwoodum<emph.end type="italics"/>378, 28 </s></p> <p type="main"> <s>figura invenitur, & proportio diametrorum, <lb/>& men&longs;ura graduum in Meridiano III, <lb/>19, 20 </s></p> <p type="main"> <s>altitudinis ad Æquatorem &longs;upra altitudinem ad <lb/>Polos quantus &longs;it exce&longs;&longs;us 381, 7: 387, 1 </s></p> <p type="main"> <s>&longs;emidiameter maxima, minima & mediocris <lb/>387, 10 </s></p> <p type="main"> <s>globus den&longs;ior e&longs;t quam &longs;i totus ex Aqua con­<lb/>&longs;taret 372, 31 </s></p> <p type="main"> <s>globus den&longs;ior e&longs;t ad centrum quam ad &longs;uper­<lb/>ficiem 386, 1 </s></p> <p type="main"> <s>molem indies augeri vero&longs;imile e&longs;t 473, 18 <lb/>481, 13 </s></p> <p type="main"> <s>axis nutatio III, 21 </s></p> <p type="main"> <s>motus annuus in Orbe magno demon&longs;tratur <lb/>III, 12, 13: 478, 26 </s></p> <p type="main"> <s>Eccentricitas quanta &longs;it 421, 15 </s></p> <p type="main"> <s>Aphelii motus quantus &longs;it 376, 33. </s></p> <p type="main"> <s><emph type="center"/>V.<emph.end type="center"/></s></p> <p type="main"> <s>Vacuum datur, vel &longs;patia omnia (&longs;i dicantur <lb/>e&longs;&longs;e plena) non &longs;unt æqualiter plena 328, 18: <lb/>368, 25 </s></p><pb xlink:href="039/01/522.jpg"/> <p type="main"> <s>Velocitas maxima quam Globus, in Medio re­<lb/>&longs;i&longs;tente cadendo, pote&longs;t acquirere II, 38, <lb/>Cor. </s> <s>2 </s></p> <p type="main"> <s>Velocitates corporum in Sectionibus conicis mo­<lb/>torum, ubi vires centripetæ ad umbilicum <lb/>tendunt I, 16 </s></p> <p type="main"> <s>Veneris </s></p> <p type="main"> <s>di&longs;tantia a Sole 361, 1 </s></p> <p type="main"> <s>tempus periodicum 370, 23 </s></p> <p type="main"> <s>Aphelii motus 376, 33 </s></p> <p type="main"> <s>Virium compo&longs;itio & re&longs;olutio p. </s> <s>14 </s></p> <p type="main"> <s>Vires attractivæ corporum </s></p> <p type="main"> <s>&longs;phærieorum ex particulis quacunque lege <lb/>trahentibus compo&longs;itorum, expenduntur <lb/>I, Sect. </s> <s>12 </s></p> <p type="main"> <s>non &longs;phærieorum ex particulis quacunque <lb/>lege trahentibus compo&longs;itorum, expendun­<lb/>tur I, Sect. </s> <s>13 </s></p> <p type="main"> <s>Vis centrifuga corporum in Æquatore Terræ <lb/>quanta &longs;it 379. 22 </s></p> <p type="main"> <s>Vis centripeta de&longs;initur p. </s> <s>2 </s></p> <p type="main"> <s>quantitas ejus ab&longs;oluta definitur p. </s> <s>4 </s></p> <p type="main"> <s>quantitas acceleratrix definitur, p. </s> <s>4 </s></p> <p type="main"> <s>quantitas motrix definitur p. </s> <s>4 </s></p> <p type="main"> <s>proportio ejus ad vim quamlibet notam, qua <lb/>ratione colligenda &longs;it, o&longs;tenditur 40, 1 </s></p> <p type="main"> <s>Virium centripetarum inventio, ubi corpus in <lb/>&longs;patio non re&longs;i&longs;tente, circa centrum immo­<lb/>bile, in Orbe quocunque revolvitur I, 6: I, <lb/>Sect. </s> <s>2 & 3 </s></p> <p type="main"> <s>Viribus centripetis datis ad quodcunque pun­<lb/>ctum tendentibus, quibus Figura quævis a <lb/><lb/>corpore revolvente de&longs;cribi pote&longs;t; dantur <lb/>vires centripetæ ad aliud quodvis punctum <lb/>tendentes, quibus eadem Figura eodem tem­<lb/>pore periodico de&longs;cribi pote&longs;t 44, 3 </s></p> <p type="main"> <s>Viribus centripetis datis quibus Figura qurvis <lb/>de&longs;cribitur a corpore revolvente; dantur vires <lb/>quibus Figura nova de&longs;cribi pote&longs;t, &longs;i Ordi­<lb/>natæ augeantur vel minuantur in ratione qua­<lb/>cunQ.E.D.ta, vel angulus Ordinationis utcun­<lb/>que mutetur, manente tempore periodico <lb/>47, 28 </s></p> <p type="main"> <s>Viribus centripetis in duplicata ratione di&longs;tantia­<lb/>rum decre&longs;centibus, quænam Figura de&longs;cribi <lb/>po&longs;&longs;unt, o&longs;tenditur 53, 1: 150, 8 </s></p> <p type="main"> <s>Vicentripeta </s></p> <p type="main"> <s>quæ &longs;it reciproce ut cubus ordinatim applica­<lb/>tæ tendentis ad centrum virium maxime <lb/>longinquum, corpus movebitur in data <lb/>quavis coni &longs;ectione 45, 1 </s></p> <p type="main"> <s>quæ &longs;it ut cubus ordinatim applicatæ tenden­<lb/>tis ad centrum virium maxime longinquum, <lb/>corpus movebitur in Hyperbola 202, 26 </s></p> <p type="main"> <s>Umbra Terre&longs;tris in Eclip&longs;ibus Lunæ augenda e&longs;t, <lb/>propter Atmo&longs;phæræ refractionem 425, 27 </s></p> <p type="main"> <s>Umbræ Terre&longs;tris dian etri non &longs;unt æquales; <lb/>quanta &longs;it differentia o&longs;tenditur 387, 8 </s></p> <p type="main"> <s>Undarum in aquæ &longs;tagtantis &longs;uperficie propa­<lb/>gatarum velocitas invenitur II, 46 </s></p> <p type="main"> <s>Vorticum natura & con&longs;titutio ad examen re­<lb/>vocatur II, Sect. </s> <s>9: 481, 21 </s></p> <p type="main"> <s><emph type="italics"/>Ut.<emph.end type="italics"/>Hujus voculæ fignificatio Mathematica de­<lb/>fiuitur 30, 19. </s></p> <p type="main"> <s><emph type="center"/><emph type="italics"/>FINIS.<emph.end type="italics"/><emph.end type="center"/></s></p> </chap> <pb xlink:href="039/01/523.jpg"/> </body> <back/> </text></archimedes>